Advances in Imaging and Electron Physics, Volume 98

ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 98

EDITOR-IN-CHIEF

PETER W. HAWKES CEMES/ Laboratoire d 'OptiqueElectronique du Centre National de la Recherche Scientifque Toulouse, France

ASSOCIATE EDITORS

BENJAMIN W A N Xerox Corporation Palo Alto Research Center Palo Alto, California

TOM MULVEY Department of Electronic Engineering and Applied Physics Aston University Birmingham, United Kingdom

Advances in

Imaging and Electron Physics EDITED BY PETER W. HAWKES CEMES / Laboratoire d’Optique Electronique du Centre National de la Recherche Scientifique Toulouse, France

VOLUME 98

ACADEMIC PRESS San Diego London Boston New York Sydney Tokyo Toronto

This book is printed on acid-free paper. @ Copyright 0 1996 by ACADEMIC PRESS All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

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96

4

3 2

I

CONTENTS CONTRIBUTORS ...................................... PREFACE ..........................................

Quantitative Particle Modeling DONALD GREENSPAN I. General Introduction ........................ 11. Melting Points ............................. 111. Colliding Microdrops of Water . . . . . . . . . . . . . . . . . IV. Crack Development in a Stressed Copper Plate ..... V. Liquid Drop Formation on a Solid Surface . . . . . . . . . VI. Fluid Bubbles ............................. VII. Rapid Kinetics ............................ VIII. Speculative Model of the Diatomic Molecular Bond . . References ...............................

Theory of the Recursive Dyadic Green’s Function for Inhomogeneous Ferrite Canonically Shaped Microstrip Circulators CLIFFORD M. KROWNE Introduction .............................. I. Introduction to the Two-Dimensional Treatment .... 11. Green’s Function Formalism . . . . . . . . . . . . . . . . . . 111. Two-Dimensional Field Relationships in Cylindrical Coordinates .............................. IV. Two-Dimensional Governing Helmholtz Wave Equation ................................ V. Two-DimensionalFields in the Inner Disk . . . . . . . . . VI. Two-DimensionalFields in the Annuli . . . . . . . . . . . VII. Two-Dimensional Boundary Conditions and the Disk-First Annulus Interface . . . . . . . . . . . . . . . . . . VIII. Two-DimensionalIntra-annuli Boundary Conditions . V

ix xi

2 6 13 21 30 44

61 67 74

78 79 81 83 86 87 88 90 92

vi

CONTENTS

IX. Two-Dimensional Nth-Annulus-Outer Region Boundary Conditions . . , . . , . . . . . . . . . . . . . . . . . . X. Two-Dimensional Dyadic Green’s Function within the Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI. Two-Dimensional Dyadic Green’s Function in the Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X I . Two-Dimensional Dyadic Green’s Function on the Outer Annulus-Port Interface . . . . . . . . . . . . . . . . . XIII. Circuit Parameters in the Two-Dimensional Model . . XIV. Scattering Parameters for a Three-Port Circulator in the Two-Dimensional Model . . . , . . . . . . . . . . . . . XV. Limiting Aspects of the Two-Dimensional Model . . . . XVI. Summary of the Two-DimensionalModel . . . . . . . . . XVII. Introduction to the Three-DimensionalTheory . . . . . XVIII. Three-Dimensional Field Equations . . . . . . . . . . . . . XIX. Diagonalization of Three-Dimensional Governing Equations . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . XX. Three-Dimensional Characteristic Equation through Rectangular Coordinate Formulation . . . . . . . . . . . . XXI. Transverse Fields in the Three-Dimensional Model . . XXII. Nonexistence of TE,TM, and TEM Modes in the Three-Dimensional Model . . . , . . . . . , . . . . . . . . . . XXIII. Three-Dimensional Fields in the Inner Cylinder Disk . XXIV. Three-Dimensional Fields in the Cylindrical Annuli . . XXV. z-Field Dependence . . . . . . . . . . . . . . . . . . . . . . . . . XXVI. Metallic Losses in the Three-Dimensional Circulator . XXVII. Three-Dimensional Boundary Conditions for the Cylinder Disk-First-Annulus Interface . . . . . . . . . . . XXVIII. Three-Dimensional Boundary Conditions for the Intra-annuli Interfaces . . . . . . . . . . . . . . . . . . . . . . . XXIX. Three-Dimensional Conditions for the Nth-AnnulusOuter Region Interface . . . . . . . . . . . . . . . . . . . . . . XXX. Three-Dimensional Dyadic Green’s Function within the Cylinder Disk . . . . . . . . . . . . . . . . . . . . . . . . . . XXXI. Three-Dimensional Dyadic Green’s Function within theAnnuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXXII. Three-Dimensional Dyadic Green’s Function on the Nth-Annulus-Outer Region Interface . . . . . . . . . . . . XXXIII. Scattering Parameters for Three-Dimensional Port Circulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXXIV. Limiting Aspects of the Three-Dimensional Model . . . XXXV. z-Ordered Layers in the Radially Ordered Circulator .

97 98 104 107 108 117 121 127 127 129 139 151 170 174 176 181 188 195 198 205 212 219 225 234 238 246 260

CONTENTS

XXXVI. Doubly Ordered Cavity: Radial Rings and Horizontal Layers .................................. XXXVII. Three-Dimensionai Impedance Wall Condition Effect on Modes and Fields ........................ XXXVIII. Summary of the Three-Dimensional Theory . . . . . . . . XXXIX. Numerical Results for the Two-Dimensional Circulator Model ........................... XXXX. Overall Conclusions ......................... References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Electron Holography and Lorentz Microscopy of Magnetic Materials MARIANMANKOS.M. R . SCHEINFEIN. AND J . M . COWLEY I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Lorentz Microscopy ......................... I11. Electron Holography ........................ IV. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Conclusions .............................. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii 283 288 301 303 316 317

323 333 362 387 422 424 427

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CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors’ contributions begin.

J. M. COWLEY (323), Department of Physics and Astronomy, Arizona State University, Tempe, Arizona 85287

DONALD GREENSPAN (l), Department of Mathematics, University of Texas at Arlington, Arlington, Texas 76019 CLIFFORDM. KROWNE(771, Microwave Technology Branch, Electronics Science & Technology Division, Naval Research Laboratory, Washington, DC 20375

MARIANMANKOS(323), IBM, T. J. Watson Research Center, Yorktown Heights, New York 10598 M. R. SCHEINFEIN (3231, Department of Physics and Astronomy, Arizona State University, Tempe, Arizona 85287

ix

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PREFACE

The three contributions to this volume examine three very different themes. We begin with a chapter by D. Greenspan, who has already written for these Advances some ten years ago, on quantitative particle modeling. In this new area of modeling, which has come into its own in the past decade, the response of a system to external forces is studied by accumulating the results of the responses of the individual constituents of the system. Greenspan concentrates on numerical methods that require only a (reasonably powerful) personal computer. This extended account of the subject, which includes numerous examples, should enable interested readers to try out these techniques for themselves. We continue with a chapter that has in fact the scale of a monograph on a theme that has been treated here several times over the years, namely, the behavior of microstrip circulators. C. K. Krowne sets out in great detail the theory needed to understand the performance of the electromagnetic circulators that are being used in modern integrated circuit technology. The approach adopted is designed not only to obtain accurate solutions efficiently and elegantly but also to relate the numerical findings to the physics of the problem. The recursive dyadic Green’s function approach is well suited to these requirements. The chapter begins with a full examination of the two-dimensional approximation, after which the full three-dimensional problem is studied. This extremely detailed treatment of an important problem will surely become the standard text on this topic. The volume concludes with an account, again a monograph rather than a review, of a subject that is rapidly acquiring importance in the world of electron microscopy, namely, electron holography. Invented nearly 50 years ago by Dennis Gabor in an attempt to improve the resolution of the microscopes of the time, the subject lay dormant for many years owing to the poor coherence of the electron sources then available and to the fact that the laser had not yet been invented. The first successful attempts to implement holography with electrons were made a quarter of a century ago, in Japan and Germany, but it is only in the past few years that holography has really entered the microscopist’s toolbox. Articles on various aspects of the subject have already appeared in this series and others are planned. The chapter included here, by M. Mankos, M. R. Scheinfein, and J. M. Cowley, is concerned with one of the most important xi

xii

PREFACE

applications of the technique, namely, the study of magnetic materials. In addition, the scanning transmission electron microscope (STEM) is the instrument of choice, a feature which distinguishes the present work from many of the earlier endeavors. As always, I thank all the authors most sincerely for all the trouble they have taken, not only in preparing their chapters but also in ensuring that their work is accessible to nonspecialists and to readers who are entering a new field of research. I conclude with a list of articles planned for future volumes. I also draw attention to the fact that volume 100 will contain a cumulative index of the entire series, from 1948 to the present day. Peter W. Hawkes

FORTHCOMING CONTRIBUTIONS Nanofabrication Finite-element methods for eddy-current problems Use of the hypermatrix Image processing with signal dependent noise The Wigner distribution Hexagon-based image processing Microscopic imaging with mass-selected secondary ions Modem map methods for particle optics Cadmium selenide field-effect transistors and display ODE methods Electron microscopy in mineralogy and geology Electron-beam deflection in color cathode-ray tubes Fuzzy morphology

H. Ahmed and W. Chen (vol. 101) R. Albanese and G. Rubinacci D. Antzoulatos H. H. Arsenault M. J. Bastiaans S. B. M. Bell M. T. Bernius M. Berz and colleagues T. P. Brody, A. van Calster, and J. F. Farrell J. C. Butcher P. E. Champness (vol. 99) B. Dasgupta E. R. Dougherty and D. Sinha

xiii

PREFACE

The study of dynamic phenomena in solids using field emission Gabor filters and texture analysis Miniaturization in electron optics Liquid metal ion sources The critical-voltage effect Stack filtering Median filters Structural analysis of quasicrystals Formal polynomials for image processing Contrast transfer and crystal images Morphological scale-spaces Optical interconnects Surface relief Spin-polarized SEM Sideband imaging Near-field optical imaging Vector transformation SAGCM InP/InGaAs avalanche photodiodes for optical fiber communications SEM image processing Electron holography of electrostatic fields The dual de Broglie wave Electronic tools in parapsychology Phase-space treatment of photon beams Aspects of mirror electron microscopy The imaging plate and its applications Representation of image operators Z-contrast in materials science HDTV The wave-particle dualism Electron holography Space-variant image restoration X-ray microscopy Accelerator mass spectroscopy

M. Drechsler J. M. H. Du Buf A. Feinerman (vol. 99) R. G. Forbes A. Fox M. Gabbouj N. C. Gallagher and E. Coyle K. Hiraga (vol. 99) A. Imiya K. Ishizuka P. Jackway (vol. 99) M. A. Karim and K. M. Iftekharuddin J. J. Koenderink and A. J. van Doorn K. Koike W. Krakow A. Lewis W. Li C. L. F. Ma, M. J. Deen, and L. E. Tarof (vol. 99) N. C. MacDonald G. Matteucci, G. F. Missiroli, and G. Pozzi M. Molski (vol. 99) R. L. Morris G. Nemes S. Nepijko (vol. 101) T. Oikawa and N. Mori (vol. 99) B. Olstad S. J. Pennycook E. Petajan H. Rauch D. Saldin A. de Santis (vol. 99) G. Schmahl J. P. F. Sellschop

XiV

PREFACE

Applications of mathematical morphology Set-theoretic methods in image processing Focus-deflection systems and their applications Mosaic color filters for imaging devices

Electron gun system for color cathode-ray tubes New developments in ferroelectrics Electron gun optics Very high resolution electron microscopy Morphology on graphs Analytical perturbation methods in charged-particle optics

J. Serra M. I. Sezan T. Soma T. Sugiura, K. Masui, K. Yamamoto, and M. Tni H. Suzuki J. Toulouse Y. Uchikawa D. van Dyck L. Vincent M. I. Yavor

Quantitative Particle Modeling DONALD GREENSPAN Mathematics Department. Unwersityof Teras at Arlington Arlington, Texas 76019

.................................. .................................. ............................. ............................... ...................................... ..................................... ................................ ............................... .................................. .................... ............................ ..................................... ................... ...................................... ................... ..................................... ................................. ...................................... ..................... ..................................... ............................... ................................ ............................ .............................. ...................................... ..................................... .................................... ................................. .............................. ..................................... ..................................... .................... ...................... ................................ ................ ...................................... ................ ....................... ...................... .......................................

I. General Introduction A . Particle Modeling B Classical Molecular Forces C. Numerical Methodology I1. Meltingpoints A . Introduction B. Formula Development C. Noble Gas Calculations D Helium (26 atm) E . Homogeneous. Diatomic Molecular Solids I11. Colliding Microdrops of Water A . Introduction B. Mathematical and Physical Considerations C. Examples IV. Crack Development in a Stressed Copper Plate A . Introduction B. Formula Derivation C. Examples V. Liquid Drop Formation on a Solid Surface A . Introduction B. Local Force Formulas C. Dynamical Equations D . Drop and Slab Stabilization E. Sessile Drop Formation VI . FluidBubbles A . Introduction B. FluidModels C. Basin Stabilization D. Motion of CO, Bubbles VII. RapidKinetics A . Introduction B. Mathematical and Physical Preliminaries C. Conservative Numerical Methodology D . Computer Examples VIII . Speculative Model of the Diatomic Molecular Bond A . Introduction B. Classical Simulations of the Hydrogen Molecule C. Modification of the Classical Model D . Extension to Liz. B,. C,. N,. and 0, References

.

.

1

2 2 3 4 6 6 6 9 11 13 13 13 14 16 21 21 23 26 30 30 32 39 40 42 44 44 45 50 50 61 61 62 63 64 67 67 68 71 72 74

Copyright Q 1996 by Academic Press. Inc. All rights of reproduction in any form reserved .

2

DONALD GREENSPAN

I. GENERALINTRODUCTION

A. Particle Modeling

Particle modeling is the study of the dynamical reaction of a material body to external forces. The term particle will be used in a comprehensive fashion to include atom, molecule, or collections of atoms or molecules. The mathematical equations of particle modeling are large systems of nonlinear, second-order ordinary differential equations, rather than small systems of partial differential equations. The feasibility of particle modeling is the result of the availability of modem, digital computer technology. Our dynamical equations will be Newtonian, for if one is interested in dynamics, that is, in how things change with time, then classical mechanics is an indispensable tool. The reason is that for N-body problems the time-dependent Schrodinger equation requires (3N + 1)-dimensional space. Thus simulation of the solar system by means of quantum mechanics requires 31 dimensions. On the other hand, relativity denies actionreaction, thus limiting N to be 1, so that solar system simulation is not possible at all. The general idea of particle modeling is as follows. A material body is given which has N molecules, total mass M, and total energy E. One wishes to approximate the response of the system to an applied force. If N is small, that is, if one has a micro system, an N-body problem with classical molecular potentials is approximated numerically using Newtonian mechanics. If N is not small, that is, if one has a macro system, the molecules are aggregated into n units, each containing many molecules, called particles, over which mass is distributed. A classical molecular-type formula is determined for particle motion in such a fashion that energy is conserved. The molecular system’s response is then approximated by that of the particle system using Newtonian mechanics. Unlike our previous paper (Greenspan, 1983, we will concentrate here only on quuntitutwe models which have been developed within the last 10 years. However, the discussion will be self-contained. Also, in order to invite the reader to venture into this new area of modeling, our choice of applications will require only the availability of a modem, scientific personal computer. Thus we will not discuss current applications which require massive vector or parallel supercomputers [see, e.g., Rapaport (1991)l.

QUANTITATIVE PARTICLE MODELING

0

3

r

pz FIGURE1.

B. Classical Molecular Forces From the classical, Newtonian point of view, both atoms and molecules exhibit the following behavior. Two molecules, for example, interact only locally, that is, when they are in close proximity to each other. Qualitatively, this interaction is of the following character (Feynman et al., 1963). If pushed together, the molecules repel; if pulled apart they attract; and the repulsive force is of a greater order of magnitude than is the attractive one. A mathematical formulation of this behavior can be given as follows (Hirschfelder et al., 1965). Consider two molecules P, and P2 on an X-axis, as shown in Fig. 1. Let P, be at the origin and let P2 be at a positive distance r from P,. Let the force F which P, exerts on P2 have magnitude F given by

where G, H , p, q are positive constants with q > p. Consider, for example, G = H = 1, p = 7, q = 13, which are good approximations for a variety of experimental results (Hirschfelder et al., 1965). Then 1 F = - - r7+ -

1 r13 ’

If, in (2), r = 1, then F = 0, so that P, exerts no force on P2. In this case, one can say that the molecules are in equilibrium. If r > 1, say, r = 2, then

which is negative, so that P, exerts an attractive force on P 2 . If, on the other hand, 0 < r < 1, say, r = 0.1, then

which is positive, so that P, exerts a repulsive force on P2. As r ap-

4

DONALD GREENSPAN

proaches zero, the force F in (2) becomes unbounded in magnitude. Mathematically, r is not allowed to be zero because, if it were, F in (2) would be undefined. Physically, r is not allowed to be zero because one assumes conservation of mass, so that the same position cannot be occupied simultaneously by different physical entities. If one sets F = 0 in (l), then, using the same reasoning as before for (2), one finds that equilibrium results if

with an attractive force resulting for larger values of r and a repulsive force for smaller values of r. It is important to observe that even though the gross motion of, for example, a fluid may be physically stable, the motion between two neighboring molecules of the fluid, in accordance with ( 0 , may be highly volatile. This volatility, however, is strictly local. In general, and for consistency, we will employ cgs units throughout. Thus, let P,, P2 be two particles, r cm apart, in three-dimensional xyz-space. To P, and P2 let there be associated a potential + ( r ) , which depends only on r. Let the units of be ergs (= g * cm2/s2). Then the force F between P, and P2 will be given in dynes ( = g cm/s2) and the magnitude F of F satisfies

+

d+ F = -dr '

C. Numerical Methodology It will be necessary in particle modeling to solve a system of nonlinear, second-order ordinary differential equations from given initial data. The only two numerical methods we will require are the leap-frog method, which is basically a central difference, low-order method which is efficient and easy to program, and a completely conservative method, which conserves differential system invariants. These are described as follows. Let h = A t be a positive time step. Let tk = k At, k = 0,1,2,. . . . For i = 1,2,. . .,N , let Pi have mass mi and at tk let Pi be located at ri,k,have at 1, is velocity vi, k , and have acceleration a i ,k. If the vector from Pi to denoted by rij,k,we define its magnitude by rij,k = llri~,kll. The leap-frog formulas, which relate position, velocity, and acceleration

5

QUANTITATIVE PARTICLE MODELING

for i

=

1,2,. , .,N , are (Greenspan, 1980) At

y l I 2= v i , o vi,k+l/2 ri,k+l

+ 2a i , o

-

- vi,k-1/2 = rj,k

(6)

(starter formula),

k

f (At)ai,k,

+ (At)vi,k+1/2,

=

1,2,3,...,

(7)

=

O, 1 , 2 , * * *

(8)

*

The name “leap frog” is derived from the way position and velocity are defined at alternate, sequential time points. Completely conservative numerical methodology can be described as follows. For clarity, we proceed in three dimensions with the basic N-body problem, that is, with N = 3. Extension to arbitrary N follows using entirely similar ideas and proofs as for N = 3. For i = 1,2,3, let Pi of mass mi be at ri = ( x i ,y i , z i ) at time t . Let the positive distance between Pi and q, i # j, be rij, with rij = rji. Let 4 ( r i j )= c$ij, given in ergs, be a potential for the pair Pi, Then the Newtonian dynamical equations for the three-body interactions are

q.

d2rj mix =

84 ri - rk -- - -, drij rij arik rik 8 4 ri - rj

--

i

=

1,2,3,

(9)

where j = 2 and k = 3 when i = 1, j = 1 and k = 3 when i = 2, and j = 1 and k = 2 when i = 3. This system conserves energy, linear momentum, and angular momentum. In addition, it is covariant; that is, it has the same functional form under translation, rotation, and uniform relative motion of coordinate frames. A numerical scheme for solving the system from given initial data so that the numerical scheme preserves the very same system invariants is given as follows. For h > 0, let tn = nh, n = 0 , 1 , 2 , . . . . At time t,, let P,, i = 1,2,3, be at rz,n = ( x i , n , Y L , ~ , z c , n ) and have velocity v i , n = (ui,.x,n , ~ , , y , n ,u i , z , n ) * Let the positive distances IP1P21,IP1P31,and IP2P31be represented by r 1 2 , n , r13,nrand rZ3,n , respectively. We then approximate the second-order differential system (9) by the first-order difference system rr,n+l

-

rr,n -

+ Vi,n 2

Vi,n+1

At V,,fl+l

-

Y , n -

_ -

4(rL/,n+l)

At

rij, n

+I

- 4 ( r i j , n ) ri,n+1 -

- 4(rLk,n+l) - 4(rrk,n) rik,n

+1

‘ik, n

‘I/,

+ ri,n

+ ri,n

-

+ rij,n

- ‘k,n+l

- rk,n

rrk,n+l

+ rik.n

-

r/,n+l

rij,n+l

n

ri,n+l

( 10)

9

3

r1.n

(11)

6

DONALD GREENSPAN

where j = 2 and k = 3 when i = 1, j = 1 and k = 3 when i = 2, and j = 1 and k = 2 when i = 3. This difference system constitutes 18 implicit recursion equations for the unknowns x i , n + yi,n+ zi, n+ vi,x , n + V i , y , n + l , V i , r , n + l in the 18 knowns X i , n , Y i , n , Z i , n , V i , x , n , V i , y , n , ~ i , r , n , i = 1,2,3. These equations can be solved readily by Newton’s method (Greenspan, 1980) to yield the numerical solution and the required invariance.

,,

11. MELTINGPOINTS A. Introduction The melting point of a solid characterizes, in a fundamental way, the transition between solid and fluid states. It is usually defined in terms of the average kinetic energy of a large ensemble of atoms or molecules (Hirschfelder et al., 1965; Cotterill et al., 1974). We will now develop a new approach for determining the melting point of a homogeneous atomic or molecular solid via the four-body problem. As a consequence of our approach, a new formula in terms of Planck‘s constant rather than Boltzmann’s constant, results.

B. Formula Development Consider first four identical atoms P I , P 2 , P3, P,, each of mass m . Let 4 ( r ) be a related classical interatomic potential and let F be the interatomic force defined by 4. Let F be zero when r equals r*, the equilibrium distance. Although r and r* will be given in angstroms, since this is customary, all other quantities will be given in cgs units. Next set Pi,i = 1,2,3,4, to be the vertices of a regular tetrahedron of edge length r*, at the respective points ( x i ,y i , zi), as shown in Fig. 2, in which, for convenience, (xl, y , , z,) = (O,O, [(r*I2 - ($r* sin 600)2 11 / 2 ), ( x 2 , y , , z2) = (0, $r* sin 60°, 01, ( x 3 , y , , z,) = ( i r * , -$-* sin 60°, O), ( x , , y , , 2,) = ( - $-*, - ir* sin 60°, 0). For this arrangement, P 2 , P3, P4 are in the XY-plane and are equidistant from the origin, while P, lies on the Z-axis. To derive a formula for the melting point of a solid, we begin by studying, in particular, copper. We will then show that the formula derived thereby will apply to other solids. For this purpose, note first that a

7

QUANTTTATIVE PARTICLE MODELING

PI

t

+x

7

I

Y

FIGURE2.

Lennard-Jones 6- 12 potential for copper is (Greenspan, 1989)

4(r)

=

-

1.398068 x 10-’O r6

1.55104

+x r12

erg.

(12)

In dynes, the force F then has magnitude F given by

F= -

8.388408 x r’

from which it follows readily that r* tions of motion for Pi are then

.!.

+ =

18.61248 r13

’

(13)

2.460486 A. The Newtonian equa-

8.388408

- dyn,

(14)

j= 1 izj

where rji is the vector from pi to pi and rij = Ilrijll. Since the mass of a g, Eq. (14) is equivalent to copper atom is 1.0542 X (1.0542 X 10-22)ai =

j= 1

j#i

(

18.61248 rji lo8 A

-8 . 3 y IJ

-k

‘ij13

)<(7)

8

DONALD GREENSPAN

or d2ri

7.957132

x lo2* +

r?. IJ

17.65555 13 ‘ij

j#i

i = 1,2,3,4. (15) To solve the system (15) efficiently, we make the time transformation T

=

1015t s,

from which it follows that dri dri dT dri v.= = - - = - )( 10 l5 =

‘

dt

dT

dT dt

d2ri dt2

dvi dt

x 1015,

( 16)

d2ri x 1030, dT2

-=-=-

and (15) reduces to d2ri

0.07957132 r?.

17.65555 rji +

‘ i 13 j

) - *‘ i j

( 18)

11

j+i

For initial data we have (xl,Yl,q)

=

(0,0,2.008978),

( x2 Y 2 , z , ) = (0,1.420562,0) , ( ~ 3 , ~ 3 ,= ~ 3(1.230243, ) -0.7102811,0), 9

(xq,y4,zq)

=

(-1.230243, -0.7102811,O)

and choose

v, = ( O , O , -V,),

v, = v3 = v4 = 0.

We will determine the minimum value V, for which P, passes through the plane of P,, P3, P4.Intuitively, such behavior is not that of a solid, and hence is fluidlike and should enable one to characterize transition. Beginning with V , = 0.1,0.2,0.3,0.4,. ..,1.0 and running each such choice for 750,000 time steps using the leap-frog formulas with AT = 0.0001, it is found that 0.2 < V , < 0.3. Then refining V , to V , = 0.20,0.21,0.22,. .. ,0.30 and running each case again, it is found that 0.20 < V, < 0.21. Continuing in the indicated fashion, it is found that, to five decimal places, V , = 0.20380. Thus, from (161, U, =

0.20380 X 10’’

A/s

=

0.20380 X lo7 m / s .

QUANTITATIVE PARTICLE MODELING

9

If b is the initial speed of PI relative to the mass center of the system, then b = i u z = 0.15285 X lo7 cm/s. We now define the melting point To of copper, in degrees Kelvin, by To

=

c(+ma2),

( 19)

where C is a constant which is determined as follows. Since the mass of a g and its melting point is 1357 K, Eq. (19) copper atom is 1.0542 X implies 1357 = C

(3 -

(1.0542 X 10-22)[(0.15285)2X loi4],

so that C

=

1.101935 X

but, to 0.2%, one finds

where h is Planck’s constant, 6.6251 X From the preceding results, we propose that the general formula for the melting point To of any system is

and proceed to examine the applicability of this formula to other atomic species. C. Noble Gas Calculations

For atomic interaction, potential formulas are available primarily for helium and the noble gases (Hirschfelder et al., 1965). We will direct our attention in this section to the noble gases. It may be noted immediately that the formula (12) for copper was derived by a least squares fit of an available data set and only a limited number of such data sets seems to be available. As we turn our attention to the noble gases, however, difficult problems of choice result immediately. Not only are there structurally different potential formulas available, such as those of Buckingham, Corner, and Lennard-Jones, but each particular form may have a variety of parameter values available. For example, for argon, there are at least five different parameter values available for the general Lennard-Jones

10

DONALD GREENSPAN

potential:

(

cp(r) = k [ zI);

-

(3'1.

In this section we will limit our attention to Lennard-Jones potentials and will consider exactly three different parameter sets for each noble gas. Using the Boltzmann constant, k = 1.38055 X lo-'', we have recorded in Table I the resulting parameters for (22) for each noble gas. Also listed are the respective masses, equilibrium distances r*, and minimum calculated values V,. The experimental melting point (Dean, 1985) and theoretical melting point To, from (20, are recorded in the final columns of Table I. These are given in degrees Celsius because the experimental results have been determined in degrees Celsius. The calculated values of To in Table I indicate quite clearly that they are sensitive to the choices of E and u.Thus the method requires highly accurate values of E and u for applicability. Nevertheless, for each noble gas, at least one value of To is very accurate.

TABLE I NOBLEGAS CALCULATIONS

Noble gas ( ~ 1 0 - erg) 1~ Ne

49.14758' 48.18120a 49.2856b

U

r*

(A)

(A)

Experimental melting Mass Computed point ( ~ 1 0 - 2 4g) V, ("C)

TO ("C)

2.749 3.085648 2.78 3.120444 2.789 3.130547

33.47954

0.04804 0.04764 0.04815

-249

-249 - 249 - 249

66.27881

0.06240 0.06292 0.06342

-189

-193 - 191 - 190

Ar

165.389ga 168.4271a 171.18gb

3.405 3.821983 3.40 3.816371 3.418 3.836575

Kr

236.07405' 218.1269' 227.1005'

3.60 4.040863 3.597 4.037496 3.60 4.040863

139.0348

0.05251 0.05066 0.05150

-157

-154 - 162 - 158

Xe

305.1016a 299.5794' 316.1460b

4.100 4.602094 3.963 4.448317 4.055 4.551584

217.8434

0.04881 0.04822 0.04952

-112

-112 - 116 - 107

' Parameters determined from second virial coefficients. Parameters determined from viscosity.

11

QUANTITATIVE PARTICLE MODELING

D. Helium (26 atm) Calculation of the melting point of helium requires special considerations. There are two reasons for this. First, although potential functions are available, the differentiation of these functions for dynamical calculations magnifies the errors. Differentiation simply does not have the stable character or integration. Second, in degrees Kelvin, the experimental melting point is 0.8 K, which is so close to zero that numerical accuracy, after differentiation is introduced, is not easily achievable. In this section, then, we will consider three structurally different potentials (Hirschfelder et al., 1965) and compare the results for each. These potentials are 2.556

4 ( r ) = 4(14.10922 X

(Lennard-Jones) ,

(Rosen-Marginau-Page) ,

(Slater-Kirkwood)

.

(25)

Recall also that the mass of helium is 6.64082 X g. The computation using the Lennard-Jones potential follows in the fashion described in Section 1I.C and, as recorded in Table 11, yields r* = 2.869013, V , = 0.05683, To = -266.34"C. The other two potentials require more extensive considerations, so let us consider next the Rosen-Marginau-Page potential in detail. From the Rosen-Marginau-Page potential (24), it follows that

From the dynamical equation (6.64082

X

10-24)a = F ,

12

DONALD GREENSPAN TABLE I1 HE CALCULATIONS ~

r*

Potential

V.

("C)

Computed

(A)

~~

Experimental melting point ("C)

TO

hnnard-Jones

2.869013

0.05683

- 272.2

- 266.34

Rosen-Marginau-Page

0.9795108 3.1835732

0.002379 0.008665

- 272.2

- 272.99 - 272.84

Slater-Kirkwood

2.4257225 2.94305228

0.006700 0.008010

- 272.2

- 272.91

- 272.87

one finds d2r

612.87612e-4.40r z=(

-

-

449.46257e~~.~~'

1.255869 - 3.6140115) x lom A r7 r9 s2 .

(26)

To determine r* from (26), one must solve the transcendental equation 6 12.87612e -4.40r - 449.46257e- 5.33r

-

1.255869 r7

-

3.6140115 r9

Interestingly enough, (27) has two solutions, namely, r* r* = 3.1835732. Transforming (27) by T = 1015t reduces (26) to d2r

a =612.87612e-4 ( -

.40r

=

=

0.

0.9795108 and

- 449.46257e- 5.33r

)

1.255869 3.6140115 x 10-lo. r7 r9

(28)

Applying the leap-frog formulas to (28) with AT = 0.0004 using r* = 0.9795108 yields a minimum V , = 0.002379 and, from (20, To = -272.99"C, as recorded in Table 11. For r* = 3.1835732, one finds a minimum V , = 0.008665 and To = 27234°C. Consider now the Slater-Kirkwood potential. It can be handled in a fashion entirely analogous to the Rosen-Marginau-Page potential as

13

QUANTITATIVE PARTICLE MODELING

follows. From (251, the dynamical equations are

(

d2r = 533.36787e-4.60'- 1'3462193) x lO-'O, dTZ r7 d2r = (676~5377e-~.~O' - 1.3552543 dT2 r7

-

r I 2.61, (29)

)

3.023723 x 10-11, r9 r > 2.61. (30)

Now the right-hand side of (29) has two values of r* which reduce it to zero. However, only one of the two is in the range r I; 2.61, and it is r* = 2.4257225. Similarly, for (301, there are two values of r*, but only one is in the range r 2 2.61, namely, r* = 2.94305228. For r* = 2.4257225, the minimum V , is 0.006700 and To = -272.91°C, while, for r* = 2.9430523, the minimum V , is 0.008010 and To = - 272.87"C, as recorded in Table 11.

E. Homogeneous, Diatomic Molecular Solids We now turn our attention to solids which are composed of homogeneous, diatomic molecules. For these, we will consider all available cases for which there exist at least two Lennard-Jones formulas of type (121, and we will apply the methodology of Section 1I.C. The molecules considered are H,, D,, N,, 0,, and C1, (Hirschfelder et al., 1965). Using the same tabular notation as in Table I, we have listed the parameter values and computational results in Table 111. Again, except for the C1, case, each molecule has at least one related parameter set for which the calculation yields excellent results. For Cl,, note that only two parameter sets were available, neither of which was derived from quantum mechanical considerations. Moreover, even in the C1, case, one of the results is in error by only 4%. 111. COLLIDING MICRODROPS OF WATER A. Introduction

Collisions of microdrops are important in microwave, chemical nucleation, and raindrop studies (Adam et al., 1968; Peterson, 1985; Simpson and Haller, 1988). In this section we will show how to simulate collisions of microdrops of water. Since the interaction during collisions will be independent of gravity, we will neglect long-range forces.

14

DONALD GREENSPAN TABLE I11

DIATOMIC MOLECULE CALCULATIONS

U

r*

(A)

(A)

2.928 2.968 2.915 2.87

3.28656888 3.33146736 3.27197687 3.22146608

3.3448

51.0803Sa 2.928 3.28656888 42.93510Sa 2.87 3.22146608 54.25561Sb 2.948 3.30901812

6.6896

Molecule (X10-l6 erg)

H,

D,

a

Experimental melting Mass Computed point g) V, (“C)

51.0803Sa 45.97232b 52.46Wb 40.31206a

0.14737 0.14004 0.14928 0.13 125

- 259.19

0.10529 0.09676 0.10842

- 252.89

TO (“C)

- 250.4 - 252.6 - 249.9 - 255.1

- 249.97 - 253.55 - 248.58

N,

132.3948’ 131.2213a 126.3203b 110.1679b

3.71 3.698 3.681 3.749

4.1643320 46.5028 4.15086465 4.13178280 4.20811021

0.06676 0.06650 0.06645 0.06681

- 209.86

- 208.6 - 209.1 - 209.2 - 208.5

0,

162.9049a 162.2146’ 156.0021Sb 121.4884b

3.46 3.58 3.433 3.541

3.88371869 53.1186 4.01841413 3.85341221 3.97463811

0.06855 0.06857 0.06715 0.05992

- 218.4

- 200.2 - 205.6 -213.6 - 188.1

C1,

492.8564b 354.80135b

4.115 4.61893133 117.7037 4.4 4.93883301

0.08047 0.06946

- 100.98

- 36.3 - 96.64

Parameters determined from second virial coefficients. Parameters determined from viscosity.

B. Mathematical and Physical Considerations An elementary water molecule potential is (Hirschfelder et al., 1965)

+( r )

=

(1.9646833 X

[( 2 . y - ( 2 . 7 6 1 erg,

(31)

where r is measured in oangstroms. From (31) the force F, in dynes, between two molecules r A apart has magnitude F given by F(r)

=

(4.325809 x lo-’)

In considering how to proceed, let us now recall that all mathematical models are only approximations of the real thing. For this reason, we need not use (32), but will develop a modified, simpler approach. Direct use of a

15

QUANTITATIVE PARTICLE MODELING

molecular formula has already been explored in Section 11, and can, if desired, be applied in this section also. Consider then a least squares fit of (32) by the function F*( r )

=

(4.325809 X lo-’)

( ? + ?). --

-

(33)

For the fit one can determine as many data points (r, F ( r ) ) as one desires from (32). For simplicity, let us consider only the five values r = 2.5, 2.75, 3.0, 3.25, 3.5, which straddle the equilibrium point r = 3.06. Then, from (32) with C = [(4.325809)-’ x lo’], CF(3.5)

=

-0.09616,

CF(3.25)

CF(2.75)

=

=

-0.08888,

0.83084, CF(2.5)

=

CF(3.0) 4.30357,

=

0.06291, (34)

from which it follows that the least squares fit is F * ( r ) = (4.325809 X lo-’)

(35)

Since F is in dynes and since the mass of a water molecule is 30.103 X g, a dynamical equation which describes the moticn of one water molecule which interacts with only one water molecule r A away is (4.325809 X lo-’)

=

(30.103

X

10-24)a, (36)

where a is measured in centimeters per second squared. Changing to angstroms per second squared and also introducing the computationally convenient time transformation T = 1013.’t yields the dynamical equation d2r

16.5

- - --r 3 dT2

158.6

+T.

(37)

For a system of N water molecules PI,P 2 , ...,P,,,,it follows from Eq. (37) that, from given initial data, the motion of each Pican be determined by solving the system of second-order, nonlinear, ordinary differential equations d2ri

16.5

158.6 rji

+ -)-, ri. rij

jZi

where ri is the position vector of Pi,rji is the vector from is the magnitude of rji.

to Pi,and rij

16

DONALD GREENSPAN

C. Examples Before studying the interaction of two water drops, it is necessary to generate a single drop, which is done as follows, and, for physical reasons, we will do this first in two dimensions. Since $~(2.725)= 0, let us consider, for variety, a regular triangular mosaic of points (xi,yi), i = 1,2,. . .,9000, given by X, = ~ 5 = 2

xi+

=

2.725

x i + l = 2.725 x I. = x ., - l o l ,

yi

-68.125,

y,

-66.7625,

y52

+ xi,

+xi, =

=

-58.997975,

=

-56.638056,

yi+, = y l , yi+,=y52,

4.719384

+yi-lol,

i

i

=

=

i

1 , 2 , . . .,50,

5 2 3 3 , . ..,100,

=

102,103,. . . ,9000.

From these we choose only those which satisfy x: + y ; I 2320, thus yielding 1128 points which lie in a relatively circular pattern. At each such point (xi,y i ) we place a water molecule Pi. Each of the 1128 water molecules Piis now allowed to interact with all other molecules in accordance with Eq. (38). For simplicity, we assume first that all initial velocities are zero. The leap-frog formulas are then applied numerically with AT = 0.0002 until T = 11.2. At this time the system has contracted maximally, so that its energy should be almost all potential. Thus, at T = 11.2, all velocities are reset to zero and the system is allowed to interact until T = 14.0, at which time all velocities are again reset to zero. Thereafter, the molecules are allowed to interact without further damping. The resulting system configurations are shown at T = 14.0, 16.8, 19.6, 22.4, 25.2, 28.0, 30.8, and 33.6 in Fig. 3a-h, respectively. Figure 3 shows the presence of surface waves, which, in fact, are due to the system’s contractions and expansions with time. Note also that the density at any time is always greater in the interior of the system than at the boundary, which is consistent with the surface tension theory which holds that the surface molecules are in an attraction mode. Our real interest, however, is in three dimensions, not two dimensions. The discussion has been limited thus far to two dimensions because it is easier in this case to demonstrate pictorially that molecular fluid models contain surface tension inherently. There is no need, as when one considers the Navier-Stokes equations, to impose surface tension on the model. In three dimensions (Greenspan and Heath, 1990, then, consider N = 4102 water molecules P,, Pz,...,PN which interact in accordance with the

17

QUANTITATIVE PARTICLE MODELING

a

- y ...4 <<.-'....

--.-CI-.-

b

C

d

FIGURE3. T = 14.0 (a), 16.8 (b), 19.6 (c), 22.4 (d), 25.2 (el, 28.0 (f), 30.8 (g), 33.6 (h). (Source: Greenspan, 1990; courtesy of Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, UK.)

dynamical system

Our objective will be to study the collision modes of two water drops, so, for convenience, a single water drop will be generated first. Consider that portion of three-dimensional space for which -31 Ix I 31, -31 IY I31, -31 Iz I31, and let water molecules be placed at the grid points which result for Ax = A y = Az = 3.1. This time, again for variety, we have chosen the value r = 3.1 which makes F equal to zero, rather than the value which makes 4 equal to zero. Again, however, we will have to impose a damping procedure, which indicates that neither initial choice of r seems to be the superior one. Next, molecules outside the sphere whose equation is x 2 + y 2 + z 2 = 26' are deleted, and each of the remaining molecules is assigned a random velocity in the range IuI I0.02. At initial time, then, there are 2517 molecules which are thereafter allowed to interact in accordance with (39) for 31,000 time steps. At T31,000, all molecules whose position coordinates satisfy r > 26 are deleted, reducing the number to 2051. The simulation is

18

DONALD GREENSPAN

FIGURE4. A microdrop of water at 45°C. (Source: Greenspan and Heath, 1991; courtesy of the Institute of Physics.)

then continued to T88000, but with all velocities reset to zero at T40,500, are T48,000, T53,000, T58,000, T63,000, T68,000, and T73,000* At T78,000, the damped by a factor of 0.5. The damping process so imposed cools the molecular configuration, so that at T88,000 the temperature (Hirschfelder et al., 1965) of the resulting drop is 45°C.This drop is shown in Fig. 4. In order to study collision modes, the single drop generated previously is duplicatFd by mirror imaging. The resulting two drops are set symmetrically 3 A apart about the YZ-plane, as shown in Fig. 5. To elucidate the motions of individual molecules during collision, the drops are displayed in different shades. To avoid complete symmetry, the velocity of any molecule and that of its mirror image molecule are taken to be the same. In addition, the time counter is reset to zero. To simulate collision, we will assume that each molecule of the light drop, on the left in Fig. 5, has its velocity increased initially by v*, while each molecule of the dark drop has its velocity decreased by v*. As a first case, let v* = (0,0,0), so that the two drops are allowed to interact with no changes in velocity. Then Fig. 6 shows, at the indicated times, an oblate spheroid oscillation mode. After an extended period of time, the large boundary gradients due to surface tension transform this mode into a relatively spherical drop which exhibits small oscillations throughout its surface. Indeed, in this example and in all cohesive interactions to be described, the so-called oscillation modes are, in reality,

FIGURE5. Two microdrops of water 3 courtesy of the Institute of Physics.)

A

*

apart. (Source: Greenspan and Heath, 1991;

QUANTITATIVE PARTICLE MODELING

19

b

d

C

FIGURE 6. Oscillating oblateness mode: (a) Tsso0,(b) T,,,,,,, (c) T2s,soo,(d) TJb,soo. (Source: Greenspan and Heath, 1991; courtesy of the Institute of Physics.)

dynamical configurations which, in time, transform into a spherical configuration. Next, set v* = (2.2,0.2,0). Figure 7 then shows, at the indicated times, the development of a raindrop mode (Peterson, 1985). Setting v* = (-2.0,4.5,0) yields, as shown in Fig. 8, a dumbbell mode (Adam et al., 1968; Simpson and Haller, 1988).

a

b

C

FIGURE 7. Raindrop mode: (a) T,,,,, (b) T18,500, (c) T,,,,,,. (Source: Greenspan and Heath, 1991; courtesy of the Institute of Physics.)

20

DONALD GREENSPAN

FIGURE 8. Dumbbell mode. (Source: Greenspan and Heath, 1991; courtesy of the Institute of Physics.)

Next, selecting the largest speed of any case yet considered by the choice v* = (0.2,8.0,0) yields, as shown in Fig. 9, a noncohesive, brush-type collision in which each of the drops forms a teardrop mode (Simpson and Haller, 1988). Finally, increasing the speed still further, but in a fashion which results in more direct collision, the choice v* = (5.0,10.0,0) yields, as shown in Fig. 10, a noncohesive collision which exhibits an initial clean slicing effect and the molecular transfer during and after separation. Of the additional cases considered, we found that choices of v* which yielded high speeds resulted in explosive-type reaction. Thus, for v* =

FIGURE 9. Brush-type collision with teardrop modes. (Source: Greenspan and Heath, 1991; courtesy of the Institute of Physics.)

QUANTITATIVE PARTICLE MODELING

21

b

C

d

FIGURE10. Soft collision: (a) Tdoo0, (b) TsOo0, (c) T13,500, (d) T18,000. (Source: Greenspan and Heath, 1991; courtesy of the Institute of Physics.)

(12.0,0.2,0), this type of reaction is shown in Fig. 11. Large momentum effects, as in the case v* = (0.2,5.0,0), would often have dumbbell modes develop into a peanut shape in their transition to sphericity, as shown in Fig. 12. Small perturbations of each collision mode described previously yielded entirely similar results confirming the physical stability of the modes.

Iv. CRACK DEVELOPMENT IN A

STRESSED COPPER PLATE

A. Introduction The problem we will consider in this section is that of crack development in a stressed plate. The ability to predict where a crack will develop in a

22

DONALD GREENSPAN

c

*

.i-

FIGUREl l . Direct, hard collision. (Source: Greenspan and Heath, 1991; courtesy of the Institute of Physics.)

FIGURE12. Peanut mode. (Source: Greenspan and Heath, 1991; courtesy of the Institute of Physics.)

23

QUANTITATIVE PARTICLE MODELING

stressed plate is of fundamental importance in the design of buildings, aircraft, and nuclear reactors. Specifically, we will simulate crack and fracture development in a stressed, slotted copper plate. B. Formula Derivation

As in Section 11, an fpproximate potential function for the interaction of two copper atoms r A apart is

4(r)

=

-

1.398068 X lo-'' r6

+

1.55104 X r''

erg.

(40)

From (401, it follows that !he magnitude F of the force F, in dynes, between two copper atoms r A apart is

F(r)

=

-

8.388408 X lo-' r7

+

1.861248 X 10 r13

The minimum 4 occurs when F ( r ) = 0, that is, at r

=

(41)

2.46 A, and yields

4(2.46) = -3.15045 X erg. (42) With these observations made, let us then consider a rectangular copper plate which is approximately 8 cm X 11.4 cm. To simulate the plate, let the points Piwith respective coordinates ( x i , yi), i = 1,2,. . ,2713, be defined bY ~ ( 1= ) -5.71576764, ~(1= ) -3.9,

.

~(41= ) -4.0,

+ 1) = x ( i ) + 0.2, x ( i + 1) = x(i) + 0.2, x(i

x( i)

=x(

i

-

81),

~ ( 4 1 )= -5.54256256,

+ 1) = y ( l ) , i = 1 , 2,..., 39, i = 41,42,. ..,80, y(i + 1) = y(41), y ( i ) = y ( i - 81) + 2(0.17320508), y(i

i = 82,83,. ..,2713. The resulting arrangement is shown in Fig. 13. The (xi,yi) are vertices of a regular triangular mosaic in which the distance from any Pi to an immediate neighbor is 0.2 cm. The 6 are assumed to represent particles, or aggregates of molecules, of an 8-cm X 11.43-cm rectangular copper plate. The neighbors of any Pi are those particles which are 0.2 cm from Pi.Since we are modeling a solid, we assume that the neighbors of any Pi are defined to be the neighbors of Pi for all time. In order to determine a mass rn for each Pi, we use total mass conservation. Suppose the rectangular plate were to be filled with copper

24

DONALD GREENSPAN

FIGURE13. The initial configuration. (Source: Greenspan, 1989; courtesy of Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, UK.)

atoms using, again, a regular triangular mosz$c, but in which the distance between two immediate neighbors is 2.46 A. Then the number N * of atoms in the plate is approximately 11.43 X lo8 lo8 = 1.745 X (43) 2.46 2.13 Since the mass of a copper atom is 1.0542 X g, the total mass M of these copper atoms is then M = 1.840 X lo-’ g. Distributing the mass over the 2713 particles yields a particle mass m given by

N*

=

8

X

-X

m = 6.782 x g. ( 44) To determine computationally convenient force and potential formulas, we utilize energy conservation. Since the minimum potential between two copper atoms is given by Eq. (421, it follows, under the assumption of zero

25

QUANTITATIVE PARTICLE MODELING

kinetic energy, that the total energy E* of the system of atoms is approximately

E*

3(1.745 x 1017)( -3.15045 x

-1.6493 x lo5 erg. (45) Let us assume now that the force F, in dynes, between two particles has magnitude F given by G H 5, F ( R ) = -R3 R where R is measured in centimeters. Hence =

=

+

0.5G

4(R)

0.25H

-R2 R4 erg.

=

(47)

+

Assuming 4 ( R ) is minimal for R

=

0.2, so that F(0.2)

=

0, implies

H

G

Approximating the total energy E of the particle system in the fashion used to obtain E* yields G

Equating E and E* implies G --

2(0.2)2

+-=

H

4(0.2)4

H

- 20.264.

The solution of Eqs. (48) and (50) is G = 3.24224, H = 0.12969, so that Eq. (46) takes the specific form 3.24224 0.12969 7 + 7 . (51) F(R) = R R Implicit in the preceding argument is the assumption that the molecular and particle systems will both have zero system kinetic energy at some given time or temperature. A less restrictive assumption which yields the same formulas is that at absolute zero temperature both systems have the same zero-point kinetic energy. Now, the force between two particles must be local in the presence of gravity. Thus we will introduce a normalizing constant a such that at a distance of 0.4 cm the force between two particles is small relative to gravity. This is essential because we have assumed that forces act only between neighbors, and initially the distance from any Pi to a neighbor is

26

DONALD GREENSPAN

0.2 cm. If we can define “small relative to gravity” to mean 0.1% of the effect of gravity and if we assume that

I

a -3.24224/(0.34)3

-

+ 0.12969/(0.34)’1

< (0.001)980m,

(52)

then a 1.25 x lo-’’. Since the effect of gravity will be relatively insignificant in the problems to be considered in this section, because the plate will be supported, the dynamical equation for the motion of each particle is then dZRi

mdt2 = (1.25 X lO-’O)C where Rji is the vector 5 to Pi and summation is taken over the neighbors of Pi. From Eq. (44) and introducing the computationally convenient transformations R* = 4R, T 2 = lot2, Eq. (53) reduces to d2R: dT2

-=

(54)

For the distance D* at which a particle bond breaks, we choose, as recommended by Ashurst and Hoover (1976), the value R* at which dF/dR* first becomes negative. From Eq. (54), then, D* = 1.033. C. Examples As a first example, consider a slotted plate, that is, as shown in Fig. 14, a plate in which the 15 particles Pi, i = 1070 + 41, k = 0,1,2,. .., 14, have been removed. At each time step, the particles in the bottom row are relocated so that their Y-coordinates are decreased by 0.00002 units. However, the particles in the top row also have their Y-coordinates increased by 0.00002 units per time step. Thereby, the plate is stretched. Solving system (54) numerically with the leap-frog formulas with time step AT = 0.0001 yields the following results. Figures 15-18 show the developing force field throughout the bottom half of the plate at the times T = 2.0, 6.0, 10.0, and 12.0, respectively. In these figures the force field is represented by vectors emanating from the centers of the particles. Figures 15-18 reveal the stress effect being transmitted to the interior of the plate. The fact that this transmission is not instantaneous serves to differentiate a nonimpulsive force from an impulsive one, which would simply tear the top and bottom of the plate. Figures 15 and 18 reveal large forces on the left and right sides of the slot, which yield a widening of the slot. Figure 19, at T = 12.8, shows clearly from the force field that the first crack occurs at

QUANTITATIVE PARTICLE MODELING

27

FIGURE14. The slotted plate. (Source: Greenspan, 1989; courtesy of Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, UK.)

........................................ ....................................... ......................................... ....................................... ........................................ ....................................... ......................................... ....................................... ......................................... ........................................ ......................................... ........................................ ......................................... ........................................ ......................................... ........................................ ......................................... ........................................ ......................................... ........................................ ......................................... ........................................ ......................................... ........................................ ......................................... 8

I

8

I

1

1

8

l

I

l

I

I

I

I

1

l

l

I

I

8

I

I

1

l

~

I

l

l

l

8

l

~

l

8

~

~

0

0

'........................................ -* 6-

N

I U 8 I I I I 8 I 8 8 I I I I I I 0

I I I I 8 8 I V I I I#

FIGURE15. T = 2.0. (Source: Greenspan, 1989; courtesy of Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, UK.)

0

~

28

DONALD GREENSPAN

........................................ ....................................... ........................................ ....................................... ........................................

....................................... ........................................ ....................................... ......................................... ... ......................................... ........................................ D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . * . . ~ D

o

o e e D D # D l D ~ # ~ l # l ~ l l l l o o l ~ l # l e o o ~ D D D # D @ I I I I I I I I I I I I I I I # I # I D I I I I I I I l D l l l l l

l

l

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e

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*

~

\ # a 1 l 1 1 1 l l 1 1 1 l l l 1 l l 1 l l l l l l l l l l l l l l l l l l l ~ * 1 1 ~ 1 1 1 1 l 1 l 1 1 1 l 1 1 l 1 1 1 l l l l l l l l ( l ) o ~ ~ I ~ l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l ~ l l ~ / 1 1 ~ I 1 1 1 1 l l 1 1 1 1 1 1 1 1 1 1 l l l l l l l l l l l l l l l l ~ l l l ~ 1 l O l O 1 O 1 1 1 1 1 1 1 1 1 l l ( ) l ( ) l o ( ) ( ( r l l l l l l l l l l l l 8 l l l l ~

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l t O . . . . . . * . . . . . . . . . . . . . . ~ . . . ~ ~ ~ . . .

........................................ ,. ........................................ -...............................,......\.......................................*...-# ......................................... ~

o

~

#

#

l

l

(

l

~

8

*

~

~

#

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#

l

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l

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~

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l

#

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l

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l

*

g

e . D . . o o . . . . . O O . . . . . . . . r . r . . r . . . . . . . . . . . . ~ ~ ~ ~ # . . ~

~

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#

l

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l

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~

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l

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~

#

r.......r...............*..........*.*u

).(e.#e . . . . . . . r . . . . . . . . . . . . . r . . . . . . ( . r( .~

FIGURE16. T = 6.0. (Source: Greenspan, 1989; courtesy of Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, UK.)

.................... ................... 8

O

~

O

l

#

#

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#

D

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#

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# D # D D D D D D D # # # # # O * *

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8

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@

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~

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D

@

) # ) # o i . . . . . . , , . r . . , o

*l D* D ~ l

*

l

l

D

D l l l @ @ @ @ @ @ @ @ @ # * * *

D

#

#

~

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l

l

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$

o

~

i

l (l ( l ~ l # l# ~l D l D 1 b 0 1 1 1 8 1 1 1 8 8 l

# I * @ @ @ @ @ * @ # ~ @ # D # ) ~(

a a a a ~ ~ e ~ o o e e e ~ e ~*

~

l

D

~

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l

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D

c

l

l

e

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l

l

t

r

l

l

c

l

#

i

D

~

l

~

l

t

l

r

c

c

~

l

l t * @ @ # D ~ @ D @ # l D~ ~ ( ) ~ l l # D l o l l * ~ l ~ l l 1 ~ l l # ~ ~ ~ ~ D D @ ~ D ~ D # ~ D ~ ~ ~ # # ~ ~ D D l l l l l 8 l O ~ O * ~ * ~ # D * * D D ~ (D I D ~. # D D ~ D D D i~ l l ~ I s * i * i ~ t e t D D ~ @ ~ 8 * ~ * ~ ~ * D D D D ~ D D 1 D l ~ O ~ l ~ l l ~ ~ ~ # ( ( # 8 * * * . . * .

1

............................. ......................................... .. ......................................... ......................................... ........................................ ......................................... ......................................... ........................................ ....... ........................................ ......................................... l

~

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l

l

111.*...*1*.*0**1,......0,*......-

...(...........*........................ ......................................... ........................................ ......................................... ........................................ ......................................... ........................................ ......................................... ........................................

FIGURE17. T = 10.0. (Source: Greenspan, 1989; courtesy of Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, UK)

.

.

r

l

e

~ .

o

~ .

l

~

29

QUANTITATIVE PARTICLE MODELING

.................... ................... ....................................... ....................................... ....................................... ....................................... ...................... ........................................ ....................................... ......................................... ........................................ ..... ......................................... ........................................ ......................................... ......................................... ........................................ ......................................... ........................................ ......................................... ........................................ ......................................... ........................................ ......................................... ........................................ ......................................... ........................................ ......................................... ........................................ 1 1 1 ~ * * " * ' * * " ' * * ' * *

e(o1)#na.......,....)))1)()).....

*

*

'

~

~

*

~

~

~

~

~

*

~

~

@

~

~

~

D

~

~

#

*

#

*

I

#

#

~

......................................... \........................................ -~~~**~..*.*............................. ........................................

?

FIGURE 18. T = 12.0. (Source: Greenspan, 1989; courtesy of Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 IGB, UK.)

.................... ................... ..................... .................... ................... .................. ...................... -.,,,................ ................. ........ ...................... i ,(.,... .......... v l I " * * * ' * . * * * * - - . * *

. * * * r . . l I # ( # * # . . . . . . . . . . m r # l l

I I O I O ~ ~ . . . . . . . .

........................................ t ........................................ .......................................... ........ ......................................... ........................................ ..........*....,,,(,,***......,................................................. ......................................... ........................................ ......................................... ........................................ ......................................... I I , . ' . 8 . 1 ) l ( * . . . . . . . . . .

.

r . . . . . * l c t l r r , r . . . . . ( ~ # ~ O l ~ S 8 1 * * ~ * * . g ~ * *

*.............

I , , , . . , * * * , # ~ ~ * * * m . * # , * s * ~ * ~ * * ~ .

........................................ ......................................... ........................................ ......................................... ........................................ ......................................... ........................................ ......................................... ........................................ ......................................... ........................................

FIGURE19. T = 12.8. (Source: Greenspan, 1989; courtesy of Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 IGB, UK.)

30

DONALD GREENSPAN

FIGURE20. T = 13.8. (Source: Greenspan, 1989; courtesy of Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, UK.)

the lower left of the slot and hence, by symmetry, simultaneously at the upper right. Figures 20-22 show the gross effect on the plate at the respective times T = 13.8, 15.8, 18.8. Figure 23 shows the associated row dislocation at T = 16.8. Figures 24 and 25 show the fracture of a full plate under shear at the times T = 20.0 and T = 30.0, respectively. In this case, the top and bottom rows were again moved 0.00002 units per time step in the Y-direction, while they were moved simultaneously, and again symmetrically with respect to the origin, 0.000005 units in the X-direction per time step.

V. LIQUIDDROPFORMATION ON A SOLIDSURFACE A. Zntroduction

In this section we will show how to simulate the formation of a liquid drop, taken to be water, on a horizontal solid surface, taken to be graphite. At present, for computational simplicity, attention will be restricted to twodimensional simulations. Nevertheless, all the ideas and methods do extend to three dimensions.

QUANTITATIVE PARTICLE MODELING

31

FIGURE 21. T = 15.8. (Source: Greenspan, 1989; courtesy of Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, UK.)

FIGURE 22. T = 18.8. (Source: Greenspan, 1989; courtesy of Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, UK.)

32

-

DONALD GREENSPAN

-

............ ..

%.

< w e . . . . . . . .

*................

FIGURE 23. Row dislocation at T = 16.8. (Source: Greenspan, 1989; courtesy of Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, UK.)

FIGURE 24. T = 20.0. (Source: Greenspan, 1989; courtesy of Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, UK.)

B. Local Force Fomulas

Consider the system of 823 water particles P I ,P 2 , .. . ,PSz3 shown in Fig. 26. These particles are arranged on a regular triangular mosaic but in a relatively circular pattern. The edge length of each triangle in the mosaic is 0.0305871 cm, the rationale of which will be explained shortly. The algorithm for generating the positions di)= ( x ( i ) , y(i>> and velocities

33

QUANTITATIVE PARTICLE MODELING

FIGURE25. T = 30.0. (Source: Greenspan, 1989; courtesy of Elsevier Science Boulevard, Langford Lane, Kidlington OX5 lGB, UK.)

v(i)

=

( u x ( i ) , u y ( i ) ) for each

id., The

4. is given as follows. From the 9500 points

with ~(1= ) -0.7946762,

~(1= ) -0.6117420, u,(i)

=

0.0,

u,(i)

=

10-9,

~(42= ) -0.7681870,

~ ( 4 2= ) -0.5964485, ~ ~ ( 4= 2 )lop9,

~ ~ ( 4= 2 )0.0,

+ 1) = x ( i ) + 0.0305871, y ( i + 1) = y ( l ) , i = 1 , 2 , 3,..., 40, u x ( i + 1) = 0.0, u,(i + 1) = uy(l), ~ ( +i 1) =x(i) + 0.0305871, y ( i + 1) =y(42), u y ( i + 1) = 0.0, i = 42,43,. . . ,80, u x ( i + 1) = u,(42), X ( i) = X ( i - 81), y( i) = y( i - 81) + 0.0529784, x(i

u,(i)

=

- u x ( i - 81),

u,(i)

=

- u y ( i - 81),

i

=

82,83,...,9500,

34

DONALD GREENSPAN Y

... ..... . . . . . . . . . . . . . ......... ....... . . . . . . . . ........... . . . . . . . . . .......... .......... ............ ............ .............. . . . . . . . . . . . ............ ............. ............... . . . . . . . . . . . . ............. .............. .............. .............. .............. .. ......................................... .............. .............. 0.47 .............. ............... ............... ............... 0.47.

.

I

............... .............. .............. ............... ............... ............... .............. .............. .............. .............. .............. .............. .............. .............. ............. ............. ............. ............ ..................................... ............ .......... .......... .......... .......... ........ ......... . . . . . . . ........ ..... . . . . . ... .

.

.

X

a

FIGURE26. (Source: Greenspan, 1991; courtesy of John Wiley & Sons, Ltd.)

choose only those which satisfy [ x ( i ) 1 2 + [ y ( i ) I 2 5 0.2123.

(55)

These are the 823 points shown in Fig. 26. Note that the horizontal radius of this relatively circular set contains 15 particles, so that the radius r is approximately r

=

15(0.0305871)

=

0.4588065 cm.

(56)

For the interaction of the particles, we assume a force F,, in dynes, between two particles R cm apart, with magnitude F , given by

From (571, then, in ergs,

35

QUANTITATIVE PARTICLE MODELING

For the system shown in Fig. 26, we assume that F , neighbors, so that -(0.0305871)’G

=

0 between two

+ H = 0.

(59) Assuming zero kinetic energy, the total energy El of the particle system is approximately El

=

3(823)

(

G -

2(0.0305871)’

+

Now, for actual water molecules we use the approximation (311, that is, # ~ ( r= ) (1.9646383 X 10-13)

where the distance r between two molecules is measured in angstroms. From (611, the magnitude of the force F, in dynes, is

F(r)

=

(2.725)

(1.9646383 X

- 6 (2*T:)6).

(62)

Note from (62) that F ( r ) = 0 implies r = 3.05871 A. (Our choice of 0.0305871 cm for the construction of Fig. 26 was based on the angstrom measurement for the purpose of simplifying later calculations.) We now fill the circle shown in Fig. 26 with moleculesowhich are vertices of a regular triangular grid with edge length 3.05871 A. The number N of molecules which fill the region is approximately 15(0.0305871) 3.05871 X

)’= 706.858 X 10”.

Assuming zero kinetic energy, the total energy E of the molecular system is approximately E

=

3(707 X 10”)(1.9646383

2.725

2.725

X

(64) or E = -104.1745. Note that the expression in the square brackets of (64) is simply [($>’ - ($)I. Equating E , and E implies -534.43326

+ 285,618.8H = -0.04219299.

(65)

36

DONALD GREENSPAN

The solution of (59) and (65) is H = 1.47725 X G = 1.57898 x l o v 4 , so that (57) and (58) can be given explicitly as 1.57898 1.47725 x 10-4 -x 10-7, F , ( R ) = -RS R3 7 3949 3.693125 4dR) = x 10-5 x R4 Note that (67) can be rewritten as

+

(66)

+

7

(67)

where el = 0.0421929 and u1= 0.0216284. Note also that since the mass of a single water molecule is approxig, distributing the total water molecule mass mately m = 30.103 X over the 823 particles yields an individual water particle mass Ml given by M,

=

2.586 x lo-" g.

(69) Using the very same line of reasoning as for water, let us proceed to develop approximate formulas for graphite. A five-row, 1003-particle slab of graphite particles is generated using a regular triangular mosaic with edge length 0.03834 cm by the formulas ~ ( 1=) -3.834, ~ ( 1= ) 0.0, ~ ( 2 0 2 )= 0.03317, ~ ( 2 0 2 )= -3.81085, i = 1 ,2,..., 200, x ( i + 1) =x(i) + 0.03834, y(i + 1) = y ( l ) , x(i

+ 1) =x(i) + 0.03834,

y(i

+ 1) =y(202),

202,203,. . . ,400, y( i) = 0.06634 y( i - 401), X ( i) = X ( i - 401), i = 402,403,. ..,1003. The resulting slab is 7688 cm wide and 0.13268 cm high and is shown in Fig. 27. One should note immediately the difference in units between Figs. 26 and 27.

+

i

=

f' -4.0

I

4.0

FIGURE 27. Initial slab. (Source: Greenspan, 1991; courtesy of John Wiley & Sons, Ltd.)

37

QUANTITATIVE PARTICLE MODELING

Let the force F,, in dynes, between two graphite particles R cm apart have magnitude F2 given by G H F 2 ( R ) = -3+ 7 , R R so that

Assuming F,

=

0 for two neighbors, (70) implies, in analogy with (591,

-(0.03834)'G

+ H = 0.

(72) Assuming zero kinetic energy, the total energy E, of the system is approximately -

E2 = 3(1003)( - 2(0.03834)2

+

4(0.03834)4

)*

(73)

However, for actual atoms of graphite (Girifalco and Lad, 1956; Kelly, 19811, with r in angstroms,

x lo-'' erg, F(r)

=

(74) (75)

where F(r) is measured in dynes. In (74) and (75), 4(3.41570) = 0 and F(3.83400) = 0. Then the number N of atoms whjch fill the slab on a regular triangular mosaic with edge length 3.83400 A is 7.668 X lo8 0.13268 X lo8 = 7.992 X 1014 3.83400 3.3203483 Assuming zero kinetic energy, the total energy E of the system is approximately

N =

E

=

3(7.992 X 1014)

38591.3 (3.83400) l2

-

(3 .834000)6

so that E = -9.17149. Note that the ratio of the numbers in the large parentheses of (76) is two. Equating E and E, implies G H + = -3.04802 X (77) 2( 0.03834)2 4( 0.03834)4

38

DONALD GREENSPAN

The solution of system (72) and (77) is approximately

G

=

1.792181 X l o p 5 ,

H

=

2.634426 X

Thus

2.634426 R5 x 10-8,

(78)

6.586065 x 10-9. R4 Note that 4,(0.0271105) = 0, so that 4, can be rewritten as

(79)

F2 = 4 2 =

-

-

1.792181 x 10-5 R3 8.960905 R2

x

+ +

where u, = 0.271105 and E , = 3.048013 X lop3. Note that since the mass of a carbon atom is approximately 1.9938 x g, the total atomic mass of the slab, when distributed over the 1003 graphite particles, yields a particle mass M2 given by

M2 = 1.588679 X lo-" g.

(81) Finally, to determine the force F3 between graphite and water particles, we use the empirical bonding law (Hirschfelder et al., 1965):

43= 4€,[ where e3 = Thus

( 34( -

3

2

]

7

d m = 0.0113404 and cr3 = ;(a, + u2)= 0.02436945. 43 -- - -2.6938 R2 x 10-5

1.5998

+x R4

Hence

5.3877 6.3992 x 10-5 + -x R3 R5 In summary, the water-water interparticle force F,, graphite-graphite interparticle force F,, and water-graphite interparticle force F3 yield, to four significant figures, 1.579 1.477 Fl( R) = - - x 10-4 + - x 10-7, R5 R3 1.792 2.634 F2( R ) = - - x 10-5 + - x 10-8, R3 R5 5.388 6.399 F3( R ) = - - x 10-5 + - x 10-8, R5 R3 F3=

--

39

QUANTITATIVE PARTICLE MODELING

while F,(0.03059)

The distances R , rium radii.

=

= F2(0.03834) = F3(0.03446) =

0.03059, R ,

=

0.03834, and R ,

=

0.

(86)

0.03446 are equilib-

C. Dynamical Equations In order to derive dynamical equations for particle motions, let us begin with the motion of a water particle Pi as it interacts with other water particles. The motion of Pi, in general, is given by d2Ri M , - - -980M1 dt2

+

1.579 ( ~ 1

1.477 x 10-4 + ( ~ i j ) ’

where the summation is taken over particles q which are within a prescribed distance D,from Pi, R i j is the distance between Pi and q, a, is a scaling factor which assures that the particle interaction is local relative to gravity, and M, is the mass of a water particle given by (69). Division by M , yields d2Ri -dt2

0.61060

0.57115 x 107 + -X l o 4 ) % . ( ~ i j ) ’ Rij

(88)

For variety, we now assume, as is common in molecular mechanics, that Pi is acted upon only by particles 5 which are within five equilibrium radii of Pi, so that D,= 5(0.03059) = 0.15295. By “local relative to gravity,” we will now assume the usual 5% experimental error allowance, so that ff1

I-

which yields d2Ri

-=

0.61060 (0.15295)

x 107 +

aj = 2.99095 X

-980+

0.57115 (0.15295)

’ x lo4

=

5%(980),

lo-’. Thus (88) reduces to

1.82627

x lo-’

dt2

1

1.70828 Rji +x 10-4 -. ( ~ i j ) ’

Rij

(89)

Finally, making the changes of variables

R = 10R,

T = lot,

(90)

40

DONALD GREENSPAN

(89) reduces to

d2Ri dT2

-=

(91)

Using the same line of reasoning, the dynamical equation for the interaction of a graphite particle with other graphite particles is d2Ri

0.359575

-=

dt2

+ 5.285275 x 10-4

i

( ~ i j ) ’

1

Rj i -, Rij

(92)

and the distance of local interaction is D, = 0.1917. Under transformations (901, the equation becomes d2Ri

35.9575

+

--

dT2

)-

Rji ( R ~ ~K )i j ~’

5.285275

The interaction of a water particle Pi with graphite particles erned by the equation d2Ri dt2

0.261085

-=

(93)

is gov-

3.10075

+-

( ~ i j ) ’

with the local interaction distance D , (90), the equation becomes

=

0.1723. Under transformations

d2Ri dT2

- = -98.0+

(95)

Note, incidentally, that the first transformation in (90) has the effect of multiplying all coordinates in Figs. 26 and 27 by a factor of 10. Hereafter, all discussion is in terms of the variables R and T . However, the random initial velocities prescribed for particles P1-PSz3will be retained.

D. Drop and Slab Stabilization If one uses the leap-frog formulas with AT = 0.00005 and if one allows the water particles to interact in accordance with (911, the system exhibits large expansion and contraction modes. The reason is that the initial potential energy is large. To overcome this situation, ?1-P823 were allowed to interact in accordance with (911, but every 1000 time steps all velocities were damped by a factor of 0.9. At the end of 37,000 time steps, the

41

QUANTITATIVE PARTICLE MODELING

Y

..

.

. X

. '*.. FIGURE28. (Source: Greenspan, 1991; courtesy of John Wiley & Sons, Ltd.)

damping was removed and the particles were allowed to interact for 9000 more time steps to t,,,,,,. The large oscillating modes were no longer present. The resulting stable configuration is shown in Fig. 28, where, most importantly, the outermost particles show a lower density than the inner particles, which is characteristic of liquid surface tension. The average diameter in Fig. 28 is approximately two-thirds of that shown in Fig. 26. The slab was stabilized in accordance with (93), but in the following fashion in order to maintain its solid state. Again, we chose AT = 0.00005. Whenever the total system kinetic energy exceeded 100, all velocities were damped by a factor of 0.25. The system was allowed to run to t25,000,at which time the slab had contracted vertically primarily to the relatively stable configuration shown in Fig. 29. More extensive time calculations led to crumbling, which was considered to be physically unreasonable. The result in Fig. 29 has a height which is approximately two-thirds of that shown in Fig. 27.

42

DONALD GREENSPAN

ty I

4-

*X

I -4.0 4.0 FIGURE 29. (Source: Greenspan, 1991; courtesy of John Wiley & Sons, Ltd.)

..................................... .................................... ..................................... .................................... ..................................... FIGURE 30. T

=

0.0. (Source: Greenspan, 1991; courtesy of John Wiley & Sons,Ltd.)

E. Sessile Drop Formation

The drop shown in Fig. 28 is now translated vertically upwards 4.5 units, so that it sits immediately above the slab shown in Fig. 29. This arrangement is shown in Fig. 30 with only the central slab particles being plotted. The mode of presentation in this and the next figures distinguishes between the water and carbon particles, so that the interaction can be discerned easily. The slab particles were allowed no further motion, but PI-P,,, were allowed to interact with themselves and with the graphite particles in accordance with (91) and (93). For the first 200,000 steps of the computer Thereafter, we used AT = 2 X simulation, we used AT = 1 X To account for the energy increase due to the effect of gravity, all velocities were damped by a factor of 0.9 every 2000 time steps throughout the calculations. This was not considered to be significant since we were only interested in a relatively steady-state configuration. The results are summarized in Figs. 31-34 at the respective times T = 0.28, 0.60, 0.92, and 1.40. The kinetic energy (KE) is also recorded in each of the captions. The figures show an interlocking of particles below the central fluid mass and a relative steady state at T = 1.40. Using linear least squares approximations with the four lowest boundary particles on the left and right sides of the system, as shown in Fig. 35, we found a left

QUANTITATIVE PARTICLE MODELING

43

..................................... .................................... ..................................... FIGURE31. T = 0.28,KE = 4500.(Source: Greenspan, 1991; courtesy of John Wiley & Sons, Ltd.)

..................................... .................................... ..................................... FIGURE32. T Sons, Ltd.)

=

0.60,KE = 1900. (Source: Greenspan, 1991; courtesy of John Wiley &

.................................... ..................................... FIGURE33. T = 0.92,KE = 840. (Source: Greenspan, 1991; courtesy of John Wiley &

Sons,Ltd.)

44

DONALD GREENSPAN

..................................... .................................... ..................................... FIGURE34. T = 1.40, KE Sons, Ltd.)

= 85.

(Source: Greenspan, 1991; courtesy of John Wiley &

.................................... ..................................... .................................... ..................................... FIGURE35. Contact angle fit. (Source: Greenspan, 1991; courtesy of John Wiley & Sons, Ltd.)

contact angle of 64" and a right contact angle of 68", the average being 66". An experimentally determined contact angle measurement reported

(Adamson, 1976) is 60".

VI. FLUIDBUBBLES A. htroduction

In this section we will develop a particle approach to a two-dimensional study of the motion of carbon dioxide bubbles in water. The discussion for this prototype problem will contain all the concepts and methods which are essential for three-dimensional studies of the motions of fluid drops within fluids.

45

QUANTITATIVE PARTICLE MODELING

B, Fluid Models Let us begin by considering first water. Unless otherwise specified, the term water molecule will now represent either an H 2 0 molecule or a D,O molecule. Differentiation between the two will be essential only when the discussion will require the concept of mass. Given two water molecules Pi and 5, a second classical molecular potential + ( r ) for the pair is the Rowlinson potential (Hirschfelder et al., 1965):

[( y)12(31

+( r i j ) = (2.098 X

-

erg,

(96)

where the distance rij between P, and is measured in angstroms. From (961, it follows that the force Fi on Pidue to 5 is

Fi = (2.098 X lo-’)

12(2.65)12 -

From (97), it follows that c ( r i j )= llFill is given by F , ( r j j )= (2.098 X lo-’)

-

6(2.65)6 r!. 11

from which it follows that Fi(7) = 0 implies 7 = 2.975 A,

(99)

which is the equilibrium distance for the force Fi. Consider now a two-dimensional, rectangular basin whose base is 23.8 cm and whose height is 23.713 cm. An XY-coordinate system is superimposed on the basin as shown in Fig. 36. The basin is symmetrical about the Y-axis and lies in the upper half-plane. On the basin we now construct a regular, triangular grid, as shown in Fig. 37. The triangular building block for the grid has edge 1.19 cm and altitude 1.031 cm, as shown in Fig. 38. The grid has 24 rows and 492 grid points, which are numbered 1-492 in the usual fashion, left to right on each row and bottom to top. At each of the constructed grid points, we wish to place a water particle, that is, an aggregate of water molecules. If a point has been numbered i , then the associated particle is denoted by Pi. The mass of each Piwill be determined by distributing equally over the particles the total mass of all the molecules in the basin. Use of (99) as the edge of a regular, triangular grid of water molecules implies that the number N of water molecules in

46

DONALD GREENSPAN

Y A

(-11.9,23.713)

D

0

B (-11.9,O.O)

FIGURE36.

C

(11.9,23.713)

-x

(11.9,O.O)

(Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

the basin is approximately

N=

(23.8)(23.713) (2.975 X 1OV8)(2.576X

=

7.364 x 1017.

(loo)

Now, the mass of a single H 2 0 molecule is 30.103 X g, so that the total H 2 0 molecular mass is 2.217 X lo-’ g. Distributing this over the 492 particles implies that the mass M , of an H 2 0 particle is Ml

=

4.506 x

g.

(101a)

In a similar fashion, since the mass of a D 2 0 molecule is twice that of an H 2 0 molecule, the mass M2 of a D 2 0 particle is M2 = 9.011 x

lo-* g.

(101b)

QUANTITATIVE PARTICLE MODELING

47

p47

p, p2

p.3

p21

FIGURE37. Triangular basin grid. (Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

Next, we wish to develop a formula for the force between the two particles Pi and 4. For variety, we assume here that the force has a magnitude G H F = - + - dyn, Rij R:j where the distance R i j between Pi and p/ is measured in centimeters. From (1021, then,

48

DONALD GREENSPAN

1.19 cm

FIGURE38. Regular triangular building block. (Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

Assuming that the basic edge length 1.19 cm of the triangular grid in Fig. 37 is the equilibrium distance for the force magnitude F in (102)implies G H

-+-

1.19

=

(1.19)3

0.

A second equation for G and H is determined by computing potential energies of the particle and molecular systems as follows. Assuming that all velocities are zero, the potential energy E of the particle system is approximately

2( 1.19)’

i= 1

or

3(492)( -0.173956 + 0.353088), while that for the molecular system is approximately E

=

7.364X 10”

(2.098 X lo-”)[

( 106)

(E 2.65 ) ” - (2.65 E ) ‘(lo7) ]],

i= 1

so that E

=

-1.159 x lo5 erg.

( 108)

Thus (106)and (108)imply

-0.49276 + H = -222.4. Finally, the solution of (104)and (109)is

G

=

116.51,

H

=

-164.99,

(109) (110)

49

QUANTITATIVE PARTICLE MODELING

so that (102)becomes

116.51 F = - - Rij

164.99 Rt

Next, let us develop dynamical equations for the H 2 0 and D 2 0 particles, Consider first a D20particle. From (101b) and (110, let the motion of Pi be determined by the dynamical equation

(9.011X lo-')-

=

d2Ri

dt2

-980(9.011

X

+a

10-')6

9

(112)

j= 1 j#i

where 6 = (0,l)and a is a normalization constant which is determined as follows. From (1121,

d2Ri - - - -9806+a dt2

492

C

[(

]

12.930 18.310) Rji -- - X 10' -. (113) Rij

j= 1 j+i

R$

Rij

The normalization constant a is now chosen so that each particle Pi in the very top row of the configuration shown in Fig. 37 is supported completely by the local interaction with any particle which lies 1.0305 units directly below it (see Fig. 38). Thus

.[(

12.930 1.0305

18.310) x 10' 1.03053

]

=

980,

from which one finds a

=

-234.19 X lo-'.

Hence (113) reduces to d 2-R i

dt2

- -9806

(--

+ j= 1 j#i

3028 Rij

+

(115)

-)-

4288 R j i ' R?. R 1.1. 11

For actual computation, we will make the convenient change of variables T

=

lot,

(117)

50

DONALD GREENSPAN

so that (115) becomes finally d2R, - - - -9.86 dT2

+ j= 1

(--

30.28

Rij

+

-)-

42.88 R,, R3. R 11. . ' 'I

(118a)

j#i

Observe immediately that since the mass of H,O is half that of D,O, the dynamical equation for an H,O particle is, from (118a), -=

-9.86+

60.56 85.76 ?(--+-)R?. Rij

j= 1 j#i

Rj, R . .'

'I

( 118b)

11

C. Basin Stabilization We next let a basin of H,O particles find its own equilibrium configuration dynamically as follows. Consider a basin of H,O particles at the grid points of the triangular grid shown in Fig 37. To avoid symmetry, a velocity of either fO.OOOOOO1 is assigned in the X-direction, at random, to each particle. From these initial data, the motion of the system is generated as follows. The dynamical equation of each Pi is taken to be d2R,

--

dT2

-

-9.86+

492

C

j= 1 j+i

(

A

--

R,,

B +-

Rji

-,

R:)R,,

( 119)

where R,, > 1.2 implies A = B = 0, while R,, I 1.2 implies A = 60.56, B = 85.76. In this fashion, the interparticle force is kept strictly local. The resulting 492-body problem is solved numerically by the leap-frog formulas with AT = 0.0002 on a Silicon Graphics workstation. Every 500 time steps, each velocity is damped by a factor of 0.1. Particle reflection due to wall collision is done symmetrically with velocity damped by a factor of 0.1. In the usual notation Tk = k AT, k = 0,1,2,. . ., the evolution of the basin through T140,oOo is shown in Figs. 39-41. With the same considerations, the stabilized D,O basin at T140,000 is shown in Fig. 42. At T140,000, the maximum value of y for the H,O basin is approximately 14.4, while that for the D,O basin is approximately 12.2. because we desired to have a fluid Calculations were halted at T140,000 with nontrivial internal and surface motions. D. Motionof CO, Bubbles

To simulate the motion of CO, bubbles in H,O and in D,O, we must first repeat for a CO, gas the considerations in Section V1.B. Since CO, gas in three dimensions at 0°C has a density approximately (Sears and Zemansky,

QUANTITATIVE PARTICLE MODELING

51

..................... ......................

0 0 0 0 o o 0 0 0 0 o 0 e 0 0 0 0 0 0 0

0 ~ 0 ~ e 0 0 0 ~ ~ 0 0 ~ ~ 0 0 0 0 0 0 0

eoeoeoooooomooooooee

oooeeeeooeeeoooooooo

oooeeoeeeoe0oooooooo0 eoeeeoeoooooeooooeoe eoeoeoeeoooeoooooeoeo eeoooeoooeoeoooooooo eooeooeeeoooooooooooo

~ o o o o o ~ ~ o e o e o o o o o o o o

eeeeeeeooooooooeoooe eooeo~e~oo0ooooooooo

ooooe~eooeooooooooooo

0 e 0 ~ 0 ~ e 0 0 ~ 0 ~ o o 0 0 0 0 0 0 0

e ~ o e e ~ ~ e m ~ ~ o o m m o o o o o ~ o o e ~ ~ o o e o o o o o o o o o ~ o ~ 00 0 0 0 e o e e 0 0 0 0 0 0 0 0 0 00 eeoooeeee~ooooomooo0 0 e o o ~ ~ ~ e ~ o o o o o o o o o ~ o ~ o 0 0 0 e ~ ~ e m m 0 e ~ 0 0 0 0 o ~ o o e o e ~ e e e o o o o o o o ~ oooo m e 0 e o e 0 0 0 e 0 0 0 oooo FIGURE39. H,O basin at T,,,,,,. (Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

FIGURE40. H,O basin at T8,,,,,. (Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

FIGURE41. H,O basin at T140,000.(Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

52

DONALD GREENSPAN

42. D,O basin at T,40,000.(Source: Greenspan, 1995; courtesy of Elsevier FIGURE Science Inc.)

1

(11.9,10.305)

(-11.9, 10.305)

(-5.95,O) (5.95,O) 43. CO, particles. (Source:Greenspan, 1995; courtesy of Elsevier Science Inc.) FIGURE

1957) 1/500 that of H,O, in two dimensions the density will be approximately 500-2/3, or approximately 1/63, that of H,O. Thus into the basin shown in Fig. 36 we place 1.169 X 1OI6 molecules and 492/63 or, for convenience, seven, CO, particles. The particles are arranged as shown in Fig. 43. For the molecular arrangement, note that a potential for CO, is (Hirschfelder et al., 1965)

(

4 ( r i j ) = (1.132051 X lowi3)[

?)12

-

(120)

so that F( r i j ) = (1.132051 x

-

6( 4.07)6

r!. 'I

53

QUANTITATIVE PARTICLE MODELING

which yields an equilibrium distance of 4.57 A. One then has an approximate total molecular potential energy of

(s)"(Zi"]i

1.17X 10l6

E

=

3

((1.132051 X lo-")[

1

(122)

-

or E

=

-993.4 erg.

For CO, particles, we assume A F = - + Rij

B

Rt'

with Rij measured in centimeters. Then B 2R,"i'

4 = -A log Rij + Assuming that R implies

=

11.9 cm is the CO, particle equilibrium distance

A B - + -= o . 11.9

(11.9)3

The total potential energy of the particle system is approximately

so that 2( 11.9)'

=

-993.4.

The solution of (126) and (127) is A = 27.800, B = -3936.9. Finally, since the mass of a CO, molecule is 7.3585 X g, the total molecular mass M is

M

=

(7.3585 X 10-23)(1.169X 10l6) = 8.602 X lo-' g

(128)

and the mass M3 of a CO, particle is M3 = M/7 = 1.2289 X Thus the equation of a CO, particle is M3

dZRi 7 = -9806M3 4- (Y dt

27.800

g.

-Rij

(129)

-)-

3936.9 R . . Rt

Rij

.

(130)

54

DONALD GREENSPAN

From (115) and (1291, d2Ri

-=

dt2

-9806

+

-234.2

X

lo-'

1.2289 X lo-'

27.800

x(Rij

3936.9 Rji Rt Rij (131)

-.-)-

or d2Ri dt2

-=

-9806

+ E(--

529.78 Rij

75,025 Rji R,"j R i j

+ -)-.

(132)

Hence, by (1171, we find d2Rj = -9.86 dT2

--

+

(133)

We next need the equations of motion for the C0,-H,O interaction. For H,O-H,O and C0,-CO, particle interactions, the equations are (118a) and (1331, respectively. For the C0,-H,O particle interaction, we use a simple law of empirical bonding (Hirschfelder et al., 1965) in which the local interaction constants are averaged. However, we will also impose a local interaction distance D to force local interaction only. Our choice is D = 1.2. Thus the following dynamical approach will be used. Let Pi and 5 be any two particles in the basin shown in Fig. 41. The motion of Pi is determined by the dynamical equation d2Ri -=

dT2

492

-9.86

+C

j= 1

j#i

(

--A Rij

B Rji +Rij '

(134)

If R i j > 1.2, then A = B = 0. If Rij 1.2, then A and B are determined as follows. If Pi, pi are both H,O particles, then A = 60.56, B = 85.76. If Pi, pi are both CO, particles, then A = 5.30, B = 750.25. In all other cases, A = 32.93 = 4(60.56 + 5.301, B = 418.01 = i(85.76 + 750.25). As a first example, consider the H,O basin shown in Fig. 41. The particles PlO8, P213, P 2 6 3 , PZg4, PZg8, and P 3 6 2 are now assumed to be CO, particles. No changes in positions or velocities are made. The initial configuration is shown in Fig. 44. The system (134) was solved numerically with AT = 0.0002 by the leap-frog formulas through T64,oOo. The natural, rapid bubble emergence from the basin is shown in Figs. 45-50 at the indicated times. As a second example, the previous example was repeated in each detail with the single exception that the basin used was the D,O basin in Fig. 42. The initial configuration is shown in Fig. 51. The emergence of the bubbles from the basin is shown typically in Figs. 52-54 at the indicated times. The

QUANTITATIVE PARTICLE MODELING

FIGURE 44. Initial C0,-H,O vier Science Inc.)

55

configuration.(Source: Greenspan, 1995; courtesy of Else-

FIGURE 45. T

=

T,,,,,,. (Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

46. T FIGURE

=

T,,,,,.

(Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

56

DONALD GREENSPAN

FIGURE 47. T = T,,,,,,. (Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

FIGURE48. T = T,zo,ooo.(Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

0

FIGURE49. T

=

T,,,,,,,. (Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

QUANTITATIVE PARTICLE MODELING

FIGURE 50. T

=

57

T240,000. (Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

FIGURE 51. Initial C0,-D20 configuration. (Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

FIGURE 52. T

=

T20,000. (Source:Greenspan, 1995; courtesy of Elsevier Science Inc.)

58

DONALD GREENSPAN

e

FIGURE53. T

=

n0

0 0

T40,000. (Source:Greenspan, 1995; courtesy of Elsevier Science Inc.)

0

FIGURE 54. T

=

Tso,ooo. (Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

FIGURE55. Initial C0,-H,O vier Science Inc.)

configuration.(Source: Greenspan, 1995; courtesy of Else-

QUANTITATIVE PARTICLE MODELING

59

emergence was approximately 0.7 times faster from the D,O basin than from the H 2 0 basin. Consider finally setting the seven CO, particles in the H,O basin in the positions P216, P2,1, P236, P237, P238,PZ5,,and PZs8,as shown in Fig. 55. The effect is to create a large compressed gas bubble. One must now expect the generation of a compression wave. With AT = 0.00002, the resulting motion is shown in Figs. 56-60 at the indicated times. Figure 56 shows the immediate compression wave effect directly above the bubble at the basin surface. The figures also show the disintegration of the bubble as it rises. Figure 61 shows at T160,oOo only those H 2 0 particles which were originally below the bubble and their formation into a wake below the CO, as it rises. Figure 62 shows, at this same time, how the particles originally at the top of the basin have moved downward toward the area vacated by

FIGURE56. T = T20,000. (Source: Greenspan, 1995; courtesy of Elsevier Science Inc.) 0

FIGURE57. T

=

T60,000. (Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

60

DONALD GREENSPAN 0

0

FIGURE58. T

=

T ~ 6 0 ~ o(Source: oo. Greenspan, 1995; courtesy of Elsevier Science Inc.)

FIGURE59. T

=

T240,000. (Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

FIGURE60. T

=

T,,,,,,.

(Source: Greenspan, 1995; courtesy of Elsevier Science Inc.)

QUANTITATIVE PARTICLE MODELING

61

FIGURE61. Wake flow at T,,,,,,,. (Source: Greenspan, 1995; courtesy of Elsevier Science Inc.) 0

o~oooe~~e~~e~oooee~oeoooeo~oo FIGURE62. Vertical flow of uppermost particles at T 1995; courtesy of Elsevier Science Inc.)

=

T,,o,,oo. (Source: Greenspan,

particles in the wake. A large rotational H,O motion is evident at this time.

A. Introduction Many chemical reactions occur within several picoseconds. Such reactions are called rapid kinetic reactions. In this section simulations are made of prototype, ground-state, rapid kinetic reactions for A + BC, where A , B, and C are hydrogen atoms and BC is a hydrogen molecule. We study cases in which B and C first unbind and then A , B , and C undergo

62

DONALD GREENSPAN

complex, three-body oscillatory behavior in accordance with the Morse potential (Hirschfelder et al., 1965). It is shown that, in every case, one of A, B, or C is ejected and the remaining two atoms form an H, bond with precisely correct ground-state energy, frequency, and bond length.

B. Mathematical and Physical Preliminaries We consider any H atom as a point source entity. The ground state of H is -2.17856 X lo-" erg and its mass is 1.6733 X g. The ground-state energy of an H, molecule is -5.11 X lo-', erg, its average diameter is 0.74 A, and its frequency of oscillation is 1.3 X 1014 Hz (Herzberg, 1965; Hirschfelder et al., 1965). For clarity of presentation, let us proceed first in one space dimension. The extensions of two and three space dimensions will be given later. Hence let PI, Pz, P, be three H atoms in motion on an X-axis. Of course, we use P,, P,, P, in place of A, B, C in order to take advantage of computer subscripting capability. Let the positive distance between Pi and pi, i # j, be rij,measured in angstroms. We will consider in this section only the relatively popular Morse potential 4(rij)for the pair Pi, q, that is (Hirschfelder et al., 19651,

4( r i j ) = (7.60429 X lo-'') x ( - 8.4646357e-1.9459562ri, + 17.912514e-3.8919123r1, 1 erg. (135) From (1351, the force Fij, in dynes, on Pi due to bY

has magnitude tigiven

C j = (125.25642 x l o p 4 ) (-e-1.9459562ru + 4.2323178e-3.8919123r1, )* ( 136) At time t, measured in seconds, let P1,,P2, P, be, respectively, at xl, x , , x,, measured in angstroms. Since xi A = x i x cm =Xi cm, the classical equations of motion for P,, P2, P,, that is, xi - x i xi - xk (1.6733 X 10-24)Xi= &j+ & , k -,

i

=

1,2,3, (137)

i' k

'ij

where j = 2 and k = 3 when i = 1, j = 1 and k = 1 and k = 2 when i = 3, are equivalent to

=

3 when i

=

2, and

=

1,2,3.

j

ui = [(1.6733)-'

x

(xi

10321

-xj)ej 'ij

+

(xi - X k ) & k i' k

,

i

63

QUANTITATIVE PARTICLE MODELING

Under the transformation T = 10-I6t, system (138) yields finally the dynamical system

-e-l

.9459562rI3 +

4.2323178e-3.8919123r,3

1,

(139)

d2x2 dT2

- = 74.855952( +e-1.9459562r,2 - 4.2323178e-3.8919123'12 - e - 1 .9459562ru

+

4.2323178e-3.8919123ru

d2x3 - 74.855952( +e- 1.9459562r~3- 4.2323178e-3.891923'1, dT2 +,-1.9459562r, - 4.2323178e-3.8919123ru

)7 (140)

--

1

( 141) For two-dimensional motion, (138) need only be expanded to include three additional equations for the y,, y,, and y3 coordinates, while for three-dimensional motion an additional three equations would be required for the zl, z 2 , and z3 coordinates. *

C. Conservative Numerical Methodology

Let us now show in detail how the conservative methodology of Section 1.C will be applied to solve (139)-(141) from given initial data. A completely analogous discussion is valid for the two- and three-dimensional cases. For AT > 0, let T,, = n AT. At time T,, let Pi be at xi,,, and have velocity ui,,,.At T,,, let the distance IP1P21,IP1P31,IP2P31be r12,n,rI3,,,, r23,n,respectively, so that rI2,,,= Iq,,- x2,,,l, r13,,,= Iq,, - x3,,,l, and rZ3,,,= Ix2,,, - x3,,,I. Relative to system (139)-(141), we now define the Morse-related potential Q(rij>by e - 1.9459562rij 4.2323178e-3.8919123rij + Q(rij) = 74.855952 1.9459562 3.8919123

(

In terms of Q, (139)-(141) can be rewritten as

64

DONALD GREENSPAN

The difference equation approximations of (143)-(145) which we use are, for i = 1,2,3,

rij,n+l

-

Q(rik,n+l)

'ik, n

+1

- Q(rik,n) rik, n

(Xi,n+l +Xi,n)

+ 'ij,n

-

(Xk,n+l +Xk,n) 9

rik,n+l

+ rik,n

( 147) where j = 2 and k = 3 when i = 1, j = 1 and k = 3 when i = 2, and = 1 and k = 2 when i = 3. System (146)-(147) constitutes six implicit recursion equations for the q n + i = 1,2,3, in terms of the six knowns x i , ,,, ui, n , unknowns i = 1,2,3, and these equations can be solved readily by Newton's method (Greenspan, 1980).

j

,,

,,

D. Computer Examples In all the examples to be discussed, motion is determined in the xy-plane. Throughout, P , and P, are set initially on the x-axis with x , = - x 2 = 0.37, y , = y , = 0.0, ~ 1 = , - u~ ~ = , ~-0.2338298, u , , = ~ ~ 2 = , 0.0, ~ which is consistent with the assumption that P , and P , form a ground-state H, molecule. Various initial data for P, will be studied and the consequences analyzed. Throughout, the numerical time step is AT = lop5. The Newtonian iteration tolerances are lo-'' for position and lo-' for velocity. All calculations were performed in double precision. However, most results are reported to only six decimal places, while vibrational constants are reported only to the same accuracy reported by experiment. Consider first P, initially at x, = 0.1, y, = 20.0, with u , , ~= 0.0, u , , ~= -0.1. Here P, is positioned so far from P , and P, that the potential energies for P , P , and P, P3 are negligible to 16 decimal places. The total energy of the resulting three-body system is -7.286118 X lo-" erg, both initially and at each time step. P I , P2, and P3 come into close proximity after t = 0.0162 ps. Extensive, unbonded, nonlinear, local, three-body interaction results through t = 0.0495 ps. Typical, simultaneous atomic

QUANTITATIVE PARTICLE MODELING

65

FIGURE63. Typical local interaction trajectories. (Source: Greenspan, 1992b; courtesy of Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, UK.)

trajectories during this period are shown in Fig. 63, where the time between successive particle positions is 0.00001 ps. Figure 64 shows the motion of P , , Pz, and P3 relative to the mass center in the time period t = 0.0533-0.0535 ps, during which time Pz is ejected from the system. The particles are simultaneously at the positions marked J and K in the figure. At these times, the relatively large kinetic energy of P , is transferred to Pz and results in the ejection of Pz from the system. Simultaneously, P, and P3 bond. By the time t = 0.06 ps, the particle locations are X , = 4.870412, y1 = - 15.408138, X Z = - 8.939519, yz = - 8.902167, ~3 = 4.169107, y 3 = -15.689693, with respective speeds u1 = 0.169495, u2 = 0.100506, u3 = 0.271556. For the bonded subsystem P, P3, the energy is -5.11 X lo-” erg, the average bond length is 0.74 A, and the frequency

66

DONALD GREENSPAN

-1.2

A

I

FIGURE64. Ejection of P2 and bonding of P, and P3 during the period 0.0533-0.0535 ps. Motion is relative to the mass center. (Source: Greenspan, 1992b; courtesy of Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 lGB, UK.)

of oscillation is 1.3 X 1014 Hz, in complete agreement with the experimental results. For the other cases considered, extensive descriptions, entirely analogous to the one given previously, can be presented. However, let us only summarize the major results in tabular form. This is done in Table IV, where it is seen that all resulting vibrational constants are in excellent agreement with experiment. Note that the two final entries in the table are examples of one-dimensional simulations. From Table IV, we conclude that any one of the reactions

can and does occur.

67

QUANTITATIVE PARTICLE MODELING TABLE IV RAPIDKINETICRESULTS

Initial data for P3 x3

0.1 0.1 0.1 0.1 0.1 -20 -20

Y3 u3,x

20 20 20 20 20 0 0

0 0 0 0 0 0.1 0.5

03,)'

-0.04 -0.06 -0.10 -0.30 -0.50 0 0

Vibrational constants of the Threeresulting bonding system body system E Particle Bonded erg) ejected subsystem E (lo-" erg) d (A) f -7.286820 -7.286142 -7.286118 -7.279424 -7.266038 -7.286118 -7.266038

P3 P3 P2

PI p2 PI p2

PI P3

p2 p3 PI p2 PIP2 p2 p3

P3

PI

PIP,

-5.11 -5.11 -5.11 -5.10 -5.10 -5.11 -5.11

0.74 0.75 0.74 0.74 0.75 0.75 0.74

Hz) 1.3 1.3 1.3 1.3 1.3 1.3 1.3

The times required for the ejection of the third particle varied extensively. A most surprising element of the calculations was that a particle was always ejected. It may be that the minimum escape velocity of the ejected particle is exactly what is required for the remaining two particles to form an H, bond, but the nonlinearity of the interactions does not seem to allow for a simple mathematical analysis.

VIII.

MODELOF THE DIATOMIC MOLECULAR BOND

SPECULATIVE

A. Introduction

The intimate relationship between wavelength and energy level enables one to use the steady-state Schrodinger wave equation to deduce many molecular vibrational constants without actually simulating the vibrational motions themselves. However, simulation of vibrational motions by means of the nonsteady Schrodinger equation presents difficulties at the present time (Borman, 1990; Polanyi, 1987). For this reason, a classical molecular approach was implemented in Section VII for fast reaction dynamics. In this section we will explore the possibility of simulating the vibrations of diatomic molecular bonds by a classical approach.

68

DONALD GREENSPAN

B. Classical Simulations of the Hydrogen Molecule The inadequacy of Newtonian mechanics on the atomic and molecular levels is readily apparent if one attempts to simulate a vibrating, groundstate, hydrogen molecule using only Coulombic forces. It will be instructive, however, to show this in detail in this section. Recall first that the ground-state energy of H, is -5.1104 X lo-" erg, the vibrationalofrequency of the protons is 1.3 X 1014 Hz, and the bond length is 0.74 A (Herzberg, 1965). In a ground-state H, molecule, denote the electrons by P,, P3 and the protons by P,,P4.Classically, assume PI, P,, P3, P4 are point sources and that the only forces of interaction are Coulombic. In cgs units, for i = 1,2,3,4, and at any time t, let Pi be located at ri = (xi,yi, zi), have velocity vi = ( i iyi, , ii),and have acceleration ai = (ii, yi, zi).Then the classical equations of motion for the Piare

jZi

where rjj is the vector from el

= e3 =

-e2

to Pi, rij = llrijll,and =

-e4

=

-4.8028

X

lo-'' esu,

lo-,* g,

m,

= m3 =

9.1085

m,

= m4 =

16,724 X lo-,* g.

X

( 149)

( 150)

(151) For computational convenience, we now set R j= ( X i , y ,Zi) and make the transformations

( 152)

R i = 1Ol2ri,

T = 102,t. (153) Then the system (148) of 12 equations in the 12 unknowns x i , y i , z i , i = 1,2,3,4, transforms readily into the following equivalent system: d2X,

- = 2.5324576

x, -x, x, - x 3 -

+

dT2

R:3

x'R;4x4)

3

(154)

'

(156)

d2Y,

= 2.5324576 dT2 d2Z,

= 2.5324576 dT2

Z, - Z ,

+

Z, -2, R:3

"R:," )

69

QUANTITATIVE PARTICLE MODELING

x, -x,

d'X, - - - (1.379269 X

-

x, -x3

dT2

d2Y2

--

dT2 d2Z, -=

dT2

R:3 Y , - Yl

- (1.379269 X lop3)

'

Y2 - Y3

z, - z, - z, - z3 +

(1.379269 X

R:3

d 'X3

x3-x,

dT2

R:3

d2Y3 dT2

- = 2.5324576( d '2,

- = 2.5324576

z3 -

x 3

-x,

-

R:3

Y3 - Y , R:3

~

-

-

)

x3Ri4x4 '

Y3 - Y, R:3

z, - z3 - z ,

-

R%

x4-xl

(1.379269 X

)

z2Ri4z4 '

-- -

dT2

d 'X4

)

24

-= 2.5324576(

-=

+ x2Ri4x4

dT2

)

z3Ri4z4 '

+ x4-x,

-

R/4

)

x4Ri4x3 '

d2& dT2

--

d2Z, -=

dT2

(1.3792969

z 4

X

- z,

+ z4 - z, R:4

-

z4Ri4z3

)

*

It is system (154)-(165) which will be solved numerically by implicit, conservative methodology from given initial data. For convenience, we set

, v.= and observe that vi

=

lO'OV,.

(- dq dZi) dXi

d T ' d T ' dT

70

DONALD GREENSPAN

Note finally that the total energy E of the system is given by 1 1 E = -(9.1085 X 1 0 - 2 s ) ( ~ + f u : ) + -(16,724 X 1 0 - 2 8 ) ( ~ ;+ u:) 2 2 1 1 1 +- -- -+ -- +(23.06689 X r12

r13

r23

r14

r24

r34

( 166) or, equivalently, by 1 E = -(9.1085 X lO-')>(V: 2

+ (23.06689 X lo-')

(

1

+ V l ) + -(16,724 2 -- + 'lI2

1

X

lO-')(V;

--

-- -+

1

1

R13

R14

R23

+ b2)

( 167)

We now consider several examples. Assume that R,

=

(0,6000,0), R 2 = (3742,0,0), R 3 = - R l ,

Vl

=

(O,O,VZ),

V2 = ( K X , O , O ) ,

V3 = -Vl,

R4 = - R 2 , (168) V4 = -V2.

(169)

First, set KX = 0.00025. Since the system energy is -5.1104 X lo-" erg, substitution into (167) yields VZ = 0.0143997. Thus all initial data are known. The system (154)-(165) was then solved numerically with AT = 2.0, 1.0, and 0.5. We report only on the 0.5 case, which was the most accurate. The numerical solution was generated for 10' time steps. At each time step, the resulting nonlinear algebraic system was solved by Newtonian iteration with tolerances lo-' for position and 10-01' for velocity. The average molecular diameter which resulted was 0.76 A and the frequency of oscillation was 2.1 X 1014 Hi.Recall that the average diameter is 0.74 A and the frequency is 1.3 X lo4 Hz. Although changes in EX and VZ did not alter the frequency by more than 0.1 x 1014 Hz, they did alter the molecular diameter more extensively. Thus the choice EX = 0.0003, VZ = 0.0125244, which increased the initial speed of the protons, yielded a frequency of 2.1 X 1014 Hz and a molecular diameter of 0.82 A. On the other hand, the choice KY = 0.00015, VZ = 0.0167569 yielded a frequency of 2.2 X 1014 Hz and a molecular diameter of 0.66 A. A variety of other examples were run in which P,, P2, P3, P4 were repositioned. In all three-dimensional calculations which incorporated the symmetry intrinsic in (168) and (1691, the results were entirely similar to those described previously. Nonsymmetric examples required time steps

71

QUANTITATIVE PARTICLE MODELING

AT smaller than 0.01 and invariably resulted in one electron in motion near the two protons and one electron relatively distant from the protons. Finally, note that for the choice of initial data R, = (0,3742,0), R, = (3742,0,0), R3 = - R l , R4 = - R 2 , ( 170) Vl

=

( - 0.0275604,0,0) , V,

=

0,

v3 = -v1,

v 4 =

v,, (171)

the molecule disintegrates into two slowly separating H atoms. Indeed, substitution of (170) and (171) into (167) yields a system energy of -4.35912 x lo-" erg, which is greater than that of H, and, indeed, is twice the energy of ground-state H. C. Modijication of the Classical Model

Since quantum mechanics implies that two electrons in the same orbital repel with an effective force which is less than that of full Coulombic repulsion, we repeated the classical calculation, but decreased the repulsive electron force by a factor of 0.9. Assuming conservation of energy, we adjusted the initial velocities of the electrons accordingly. The vibrational frequency then decreased to 2.13 X 1014 Hz. Encouraged by this reduction, we proceeded in the same spirit as before to decrease the electron repulsion until the factor of 0.9 was reduced to 0.0001, but the vibrational frequency decreased only to 1.78 x 1014 Hz. We then proceeded through zero to choose negative factors until the Coulombic force between the electrons was multiplied by -1.0, that is, until the force between the electrons was assumed to be fully attractive rather than fully repulsive. To us, the final results were astonishing. We then proceeded to modify the discussion in Section VII1.B so that the electrons attract rather than repel. It should be pointed out immediately that electron attraction is not unknown. For example, a quantum theory of superconductivity requires electron attraction (Bardeen et al., 1957). The basic changes to be made, then, are as follows. In system (1481, the term e1e3has to be replaced by -e1e3. The formulas (166) and (167) are then replaced by 1 1 E A -(9.1085 X 1 0 - 2 8 ) ( ~ + f u:) + -(16,724 X 1 0 - 2 8 ) ( ~+ i u:) 2 2 1 1 1 1 1 - -- -- -+ -- +(23.06689 X lo-,') r12

r13

r14

r23

r24

r34

72

DONALD GREENSPAN

TABLE V FREQUENCY AND DIAMETER CALCULA'ITONS

X

Y

vz

f (1014 HZ)

4,000 4,000 4,000 3,800 4,435 3,000 4,000

4,500 4,200 4,000

0.033020594 0.034271092 0.035125168 0.031242354 0.039437537 0.035006180 0.013916043

1.366 1.375 1.383 1.377 1.363 1.339 1.409

Case

5,OOO 5,oOO

4,360 10,Ooo

0.776 0.770 0.764 0.774 0.790 0.808 0.762

and E

=

1 -(9.1085 2

X

lO-')(V:

+(23.06689 X lo-')

1

+ V t ) + -(16,724 2 1

X

lO-')(V;

+ &')

1 1 - -- -+ R14

R23

R,4

R34

respectively. Table V then records the resulting average vibrational frequencies f and diameters for the indicated parameters X,Y,VZ,with KY = 0. The conservative numerical methodology is, of course, essential since the ground-state energy is time invariant. The results are all entirely within physically acceptable scientific limits (Greenspan, 1992).

D. Extension to Liz, B,, C,, N,, and 0, Classical calculation of the correct frequencies and bond lengths for the diatomic molecules Li,, B,, C,, N, and 0, can be accomplished by the method of Section VII1.B if one proceeds as follows. For Li,, B,, C,, and N,, consider the nuclei and electrons arranged as shown in Fig. 65.The nuclei are denoted by PI and P,. If in each case one allows attraction between pairs of electrons which are separated maximally, where one has X < 0 while the other has X > 0, then correct results follow (Greenspan, 1993). From Fig. 65a-d one would guess that the use of hexagons would yield correct results for 0,. However, this is not the case. A more complex division of the electrons is required (Greenspan, 1993).

QUANTITATIVE PARTICLE MODELING

a

73

I’ 7

/

/

b

t

B;

FIGURE65. Electron and nuclei configurations for Li;, B:, Ci2, and Ni4. (Source: Greenspan, 1992a; courtesy of Physics Essays.)

74

DONALD GREENSPAN C

7

13

9

11

d

N

i4

FIGURE65. (continued)

REFERENCES Adam, J. R., Lindblad, N. R., and Hendricks, C. D. (1968). The collision, coalescence, and disruption of water droplets. J . Appl. Phys. 39,5173. Adamson, A. W. (1976). “Physical Chemistry of Surfaces.” Interscience, New York. Ashurst, W. T., and Hoover, W. G. (1976). Microscopic fracture studies in the two-dimensional triangular lattice. Phys. Reu. B 14, 1465.

QUANTITATIVE PARTICLE MODELING

75

Bardeen, J, Cooper, L. N., and Schrieffer, J. R. (1957). Theory of superconductivity. Phys. Reu. 108, 1175. Borman, S. (1990). Theory, experiment team up to probe “simplest” reaction. Chem. Engrg. News 4,32. Cotterill, R. M. J., Kristensen, W. D., and Jensen, E. J. (1974). Molecular dynamics studies of melting. 111. Spontaneous dislocation generation and the dynamics of melting. Philos. Mag. 30, 245. Dean, J. A. (Ed.) (1985). “Lange’s Handbook of Chemistry,” 13th ed. McGraw-Hill, New York. Feynman, R. P., Leighton, R. B., and Sands, M. (1963). “The Feynman Lectures on Physics.” Addison-Wesley, Reading, Mass. Girifalco, L. A., and Lad, R. A. (1956). Energy of cohesion, compressibility, and the potential energy functions of the graphite system. J. Chem. Phys. 25, 693. Greenspan, D. (1980). “Arithmetic Applied Mathematics.” Pergamon, Oxford. Greenspan, D. (1985). Discrete mathematical physics and particle modelling. In “Advances in Electronics and Electron Physics,” p. 189. Academic Press, New York. Greenspan, D. (1989). Supercomputer simulation of cracks and fractures by quasimolecular dynamics. J. Phys. Chem. Solid 50, 1245. Greenspan, D. (1990). Supercomputer simulation of colliding microdrops of water. Comput. Math. Appl. 19, 91. Greenspan, D. (1991). Supercomputer simulation of liquid drop formation on a solid surface. Int. J. Num. Methods in Fluids 13, 895. Greenspan, D. (1992). Electron attraction as a mechanism for the molecular bond. Phys. Essays 5, 250. Greenspan, D. (1992a). On electron attraction in the diatomic bond. Physics Essays 5, 554. Greenspan, D. (1992b). Studies in rapid kinetic reactions by quasi-quantum mechanical, conservative methodology. Comput Math. Appl. 24, 11. Greenspan, D. (1993). Electron attraction and Newtonian methodology for approximating quantum mechanical phenomena. Comput. Math. Appl. 25, 75. Greenspan, D. (1995). Particle simulation of large carbon dioxide bubbles in water. Appl. Math. Modelling 19, 738. Greenspan, D., and Heath, L. (1991). Supercomputer simulation of the modes of colliding microdrops of water. J . Phys. D: Applied Physics 24, 2121. Herzberg, G. (1965). “Molecular Spectra and Molecular Structure,” 2nd ed. Van Nostrand, New York. Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B. (1965). “Molecular Theory of Gases and Liquids.” Wiley, New York. Kelly, B. T. (1981). “Physics of Graphite.” Applied Science, London. Peterson, I. (1985). Raindrop oscillation. Sci. News 2, 136. Polanyi, J. C. (1987). Some concepts in reaction dynamics. Science 236,680. Rapaport, D. C. (1991). Multi-million particle molecular dynamics. I. Comput. Phys. Comm. 62, 198. Sears, F. W., and Zemansky, M. W. (1957). “University Physics,” 2nd ed. Addison-Wesley, Reading, Mass. Simpson, S. F., and Haller, F. J. (1988). Effects of experimental variables on mixing of solutions by collisions of microdroplets. Anal. Chem. 60,2483.

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Theory of the Recursive Dyadic Green’s Function for Inhomogeneous Ferrite Canonically Shaped Microstrip Circulators CLIFFORD M . KROWNE Microwave TechnologyBranch, Electronics Science and Technology Division Naval Research Laboratory. Washington.DC 20375

.................................... ............... ..........................

Introduction I. Introduction to the Two-DimensionalTreatment I1. Green’s Function Formalism 111 Two-DimensionalField Relationships in Cylindrical Coordinates IV. Two-DimensionalGoverning Helmholtz Wave Equation V. Two-DimensionalFields in the Inner Disk VI Two-DimensionalFields in the Annuli VII. Two-Dimensional Boundary Conditions and the Disk-First-Annulus Interface VIII . Two-Dimensional Intra-annuli Boundary Conditions IX. Two-Dimensional Nth-Annulus-Outer Region Boundary Conditions X. Two-DimensionalDyadic Green’s Function within the Disk XI . Two-Dimensional Dyadic Green’s Function in the Annulus XI1. Two-Dimensional Dyadic Green’s Function on the Outer Annulus-Port Interface XI11. Circuit Parameters in the Two-Dimensional Model XIV. Scattering Parameters for a Three-Port Circulator in the ’ho-Dimensional Model XV . Limiting Aspects of the Two-Dimensional Model XVI. Summary of the Two-Dimensional Model XVII. Introduction to the Three-Dimensional Theory XVIII Three-Dimensional Field Equations XIX Diagonalization of Three-Dimensional Governing Equations XX. Three-Dimensional Characteristic Equation through Rectangular Coordinate Formulation XXI. Transverse Fields in the Three-Dimensional Model XXII Nonexistence of TE.TM. and TEM Modes in the Three-Dimensional Model XXIII. Three-Dimensional Fields in the Inner Cylinder Disk XXIV. Three-Dimensional Fields in the Cylindrical Annuli XXV r-Field Dependence XXVI Metallic Losses in the Three-Dimensional Circulator XXVII . Three-Dimensional Boundary Conditions for the Cylinder Disk-First-Annulus Interface

.

.

...... ..........

.................. ..................... ........................................ ............. ... ......... ........ .....................................

............. ............................

.............. .................. ............... ..................... .......

. . .

. .

............................ ............. ......................................

............ ............

............................... ............ ......................... 71

78 79 81 83 86 87 88 90 92 97 98 104 107 108 117 121 127 127 129 139 151 170 174 176 181 188 195 198

Copyright Q 1996 by Academic Press. Inc. All rights of reproduction in any form resewed .

78

CLIFFORD M. KROWNE

XXVIII. Three-Dimensional Boundary Conditions for the Intra-annuli Interfaces XXIX. Three-Dimensional Boundary Conditions for the Nth-Annulus-Outer Region Interface XXX. Three-Dimensional Dyadic Green’s Function within the Cylinder Disk. XXXI. Three-Dimensional Dyadic Green’s Function within the Annuli. XXXII. Three-Dimensional Dyadic Green’s Function on the Nth-Annulus-Outer Region Interface XXXIII. Scattering Parameters for a Three-Dimensional Three-Port Circulator XXXIV. Limiting Aspects of the Three-Dimensional Model XXXV. z-Ordered Layers in the Radially Ordered Circulator . . . . . . . . . . . . XXXVI. Doubly Ordered Cavity: Radial Rings and Horizontal Layers XXXVII. Three-Dimensional Impedance Wall Condition Effect on Modes andFields XXXVIII. Summary of the Three-Dimensional Theory XXXIX. Numerical Results for the Two-Dimensional Circulator Model XXXX. Overall Conclusions. References

..................................... ................................ . ..... .................... .. ............. ....... .................................... ................. ...... .............................. ....................................

205 212 219 225 234 238 246 260 283 288 301 303 316 317

INTRODUCTION This chapter is concerned with the upgrading of the theory for electromagnetic circulators which are realized today in the planar configuration compatible with integrated circuit technologies. The theoretical analysis is developed with the intent of engendering mathematical beauty and retaining clarity of presentation in regard to the mathematical physics involved in the problem. Therefore, some care is exercised when presenting derivations, proving important theoretical aspects of the mathematics or physics, or stating fully the mathematical results as they unfold. The electromagnetic theory is done by retaining the complete set of equations describing the electromagnetic and physical phenomena, so that one obtains, what are commonly referred to in the literature as, full-wave electromagnetic solutions. This is extremely desirable in microwave and millimeter wave applications where the propagation characteristics and field behavior require a realistic modeling of frequency dispersion. Quasi-static approximations to the complete set of field equations is not acceptable, and we have spared no trouble to retain the most complete formulations here. This fact will become very apparent to the reader once he or she makes some progress in reading this contribution. However, full-wave does not mean approximations are not made in regard to the geometry and coupling to the outside circuit environment. The circulator configuration examined here is purposely limited to the circular cross-section type in order to make use of canonical solution properties of partial differential equations. Use of magnetic and electric

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

79

walls, where they are reasonable to constrain the electromagnetic field, is done to make the problem more tractable and elegant. It is possible to develop for either the two-dimensional or the three-dimensional theoretical models a way to let the fields extend beyond the ferrite region in a canonical fashion, excluding the port windows which connect the device to the microstrip lines. Furthermore, it is possible to develop two- or threedimensional approaches (for the one- or two-dimensional contour surfaces) for describing the field distributions over each port aperture, thereby obtaining a more realistic match between the internal circulator fields and the external microstrip fields at the port interfaces. These extensions to the theory were not done here, but it has been pointed out how this would be done in the chapter, and the mathematics makes it very apparent how these objectives could be accomplished. Judicious choices for the two- and three-dimensional circulator models presented here have allowed theories to be realized which are not only mathematically satisfying but also capable of being turned into computer programs which may yield numerical results very efficiently. This has been the case for the two-dimensional theory, and there is little doubt that the three-dimensional theory provided here will have similar success in implementation because it follows much of the same philosophy employed in the simpler two-dimensional theory. As a further justification for developing more analytically based, closed-form solutions to the particular partial differential equations found in these electromagnetic problems, one has only to look at the computation times needed to acquire numerical data. Numerical results found with the recursive Green’s function approach used here, employing a two-dimensional model, typically take a few seconds per frequency point, whereas numerically intensive methods run on the order of several hours, making a 1 : 1000 to 1: 10,000 ratio of computation times comparing the two approaches!

I.

INTRODUCTION TO THE TWO-DIMENSIONAL

TREATMENT

Previous work in the area of multiport circulators has focused on the treatment of high-symmetry geometric configurations, a limited number of symmetrically disposed ports, and a homogeneous nonreciprocating medium (Okoshi and Miyoshi, 1972; Miyoshi et al., 1977; Ayasli, 1978; Helszajn and James, 1978; Miyoshi and Miyauchi, 1980; Lyon and Helszajn, 1982; Kishi and Okoshi, 1987; Helszajn and Lynch, 1992; Neidert, 1992; Davis and Sloan, 1993; Neidert and Phillips, 1993; Gaukel and El-Sharawy, 1994; Gentili and Macchiarella, 1994; How et al., 1994;

80

CLIFFORD M. KROWNE

Krowne, 1994a). The theoretical techniques for modeling the circulator have ranged from Green’s functions, boundary element methods, boundary contour integral methods, to finite-element methods. Each method has special advantages and disadvantages in relation to the other methods depending on what the researcher is interested in emphasizing in the problem. Discussion of these numerical techniques as well as other information on circulators and anisotropic media may be found in recent surveys (Krowne, 1994b, 1995). Because our interest is in obtaining a formulation which allows us to inspect the physics and electromagnetics of the solution, may be related to earlier simple results on homogeneous problems, and is numerically efficient to evaluate, an analytical approach was taken to deriving a Green’s function which would allow the circulator region to be divided into an arbitrary number of rings of definite radial thickness. The idea was to make the rings or annuli thin enough to accurately describe the actual arbitrary radial variation of the various inhomogeneities contributing to the permeability tensor. The recursion process to be employed in this paper is like that utilized for planar structures on highly anisotropic layered media (Krowne, 1984a). Here we develop a two-dimensional dyadic recursive Green’s function (Section 11) with elements Gf,! (Sections X, XI, and X I ) suitable for determining the electric field component E, and the magnetic field components H, and H+ anywhere within the circulator. The recursive nature of G,”i’is a reflection of the inhomogeneous region being broken up into one inner disk containing a singularity and N annuli. Gy(r,4) is found for any arbitrary point ( r , 4) within the disk region (Section V) and within any ith annulus (Section VI). Appropriate boundary conditions are applied at the disk-first-annulus (Section VII), intra-annuli (Section VIII), and lastannulus-outer region (Section IX) interfaces. Specification of G,”i’,i = E, j = H , s = z , r = 4 at the circulator diameter r = R leads to the determination of the circulator impedance matrix Z , (Section XIII). The ports have been separated into discretized ports with elements and continuous ports. An admittance matrix Y is found which relates the internal circulator field behavior to the finite number of external ports characterized by voltages and currents (Section XIII). It is also shown how Gi$(R, 4) enables s-parameters to be found for the simple case of a three-port ferrite circulator (Section XIV). Because of the general nature of the problem construction, the ports may be located at arbitrary azimuthal 4iand possess arbitrary line widths wi for the ith port. The line widths may also be measured in terms of the angular spread A 4 i on the outer edge of the circular disk of radius R. Inhomogeneities occur in the applied magnetic field Ha, magnetization 47-rMS, and de-

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

81

magnetization factor Nd. All inhomogeneity effects can be put into the frequency-dependent tensor elements of the anisotropic permeability tensor (Sections I11 and IV). The z 4 Green’s functions element has been numerically evaluated for the simpler but immensely practical case of symmetrically disposed ports of equal widths, taking into account these radial inhomogeneities, and the results are presented in Krowne and Neidert (1995) and are also discussed here in Section XXXIX. The computer code that evaluates the single 24 recursive Green’s function element is very efficient and its convergence properties are covered in that section.

11. GREEN’S FUNCTION FORMALISM

The Green’s function to be developed here, although of a recursive nature, may in the limit be shown to reduce to either the single circular disk case (Bosma, 1962, 1964) or a circular ring (Davis and Dmitriyev, 1992; Helszajn and Nisbet, 1992; Borjak and Davis, 1994). We develop the Green’s function as a response to a distribution function which represents a driving forcing function of magnetic field type H+s located on the azimuthal boundary of a circular contour of radius R. The distribution function has the property of limiting the field to finite values only at radius r = R and where i are specific points along the azimuthal angle locations 9 = 4i, enclosing circulator contour. The linear system of partial differential equations (PDEs) though which H+&r = R, 4 = c#+) imposes its forcing behavior may be written formally in terms of one governing PDE with the operator L acting on our prime field quantity of interest here, E,: LEz( r , 4)

=

H+s(R , 4i)

(1)

*

From E, the other field components, H+ and H,,can be determined in this two-dimensional problem. The distribution function solves the problem LGEH:D(r,

4; R , 4i) = D ( r , 4; R , 4i),

(2)

where D(r, 4; R, 4i) is the distribution function acting on the system producing an E-to-H coupling response G E H : D. It is G E H : which we will D the limit is exactly G E H , where first find. We can prove that G E H :in LGEH ( r 2 4 ; R

4i) = 8 ( r - R ) 8 ( 4 - 4i)

*

(3)

82

CLIFFORD M. KROWNE

The right-hand side of (3) consists of the product of two Dirac delta functions (a particular type of distribution function). The distribution function has the following behavior: S ( r - R ) 6 ( 4 - 4i) =

lim

AQ,+O,r+R

D(r,4;R,4i).

(4)

After the distribution function GEH: is found, the limiting process will be applied in a straightfornard way, giving

Let us identify the magnetic field at location r = R to be the contour field associated with the surface in the two-dimensional problem we are treating: H , , ( R , 4)

=H,(R

41,

(6)

where an explicit subscript is added to denote this association. H+ may be related to the physical forcing magnetic field H,, by the relationship q r - R)H,,(R,

4)

=H,,(K

4).

(7)

Replace the left-hand side of (7) with an integration so that

Referring to (3), we can equate the Dirac function double product in the integrand with the operator L acting on the cross-EH Green’s function, thereby eliminating it: 7r

j- * LG,H(r, 4; R , 4’)H+c(R,4’) d 4 ’ = H+,(R, 4)-

(9)

Since L operates on ( r , 4) and the integral is definite, we can invert the order of integration and partial differentiation implied by the partial differential (PD) operator, giving

Comparing (10) with (11, the electric field can be written as

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

83

111. TWO-DIMENSIONAL FIELDRELATIONSHIPS IN CYLINDRICAL COORDINATES

Maxwell’s sourceless curl equations are, for harmonic conditions with phasor time dependence exp(iwt) assumed

V X E = -iwB, ( 12) V X H = iwD. (13) These two equations are valid within the ferrite disk region (see Fig. 1 in Section XXXIX) which is considered to be inhomogeneously loaded with material (it may be a semiconductor region if we were to use a semiconductor relying on the magnetoplasma effect). The constitutive relationships are generally given by B

=

jlH,

(14)

D = 2.E. (15) In the ferrite disk region we will assume that the dielectric tensor reduces to a scalar & = &. (16) Of course, this would not be the case for a semiconductor employing magnetoplasma effects where we would retain the tensor permittivity and drop the tensor permeability (Krowne et al., 1988; Krowne, 1993). The general expression in matrix notation for the curl of an arbitrary vector field is A

where it is noted that the expansion of (17) is accomplished by keeping the unit vector terms outside of the partial operators a,, i = r, 4, z . It is also noted that we use r instead of the usual p for the cylindrical radius. For the two-dimensional problem we are constructing, it is sufficient to drop a dimension by setting d

_ - 0. dz

Curl A then expands as

84

CLIFFORD M. KROWNE

or

To be somewhat consistent with the notation in the circulator literature (Krowne and Neidert, 19951, we set the permeability tensor

$= By (14),

iK

p

0

- i0K

Po

I-

(21)

pHr - iKH+

which can be written in terms of the three component equations:

Following similar steps, the curl H equation becomes

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

85

and again listing the component equations,

If we consider the cases where there is at least one ground plane in the real device, then (18) implies E,(z)

=

+ E,(zo)

/‘O&

(27)

and the ground plane forms the Dirichlet tangential boundary condition in the third coordinate direction Er(Z0) = 0 if the conductor is assumed perfect. This leads to

(28)

E,(z) = 0 (29) and the dropping of this field component in the analysis. A similar null condition holds for E,(z)

=

0.

(30)

Examination of (2 4 ~ in ) light of (29) and (30) gives H J Z ) = 0. (31) Magnetic horizontal fields can be found from (24) by multiplying the first equation by i~ and the second by p and subtracting:

Solving for H+ in terms of partial derivatives of E,,

H+ = and the partial derivative of rH+ is

iK

dE,

(33)

86

CLIFFORD M. KROWNE

Multiplying the first equation by ip and the second equation by subtracting, ip dE, dE, - - + K - = w ( p’ r d4 dr

H,

-

K’)H,,

K

and

(35)

=

dr

[-

1

dHr

-=

w(p’-

d+

ip d’E,

+

r a+’

K’)

d’E,

K-I.

(37)

d+ d r

IV. TWO-DIMENSIONAL GOVERNING HELMHOLTZ WAVE EQUATION Inserting (34) and (37) into (26), we obtain

Since the Laplacian operator in cylindrical coordinates is 1 d’E,

V2E, - -

d’E,

where the second equality comes from (18), (38) can be expressed in a slightly reduced and more familiar form

V’E, + k&E,

=

0

(40)

with the definitions

kL

= 02&peff,

(41)

Form (40) for the governing Helmholtz equation in rectangular coordinates agrees with an earlier result provided in cylindrical coordinates (Bosma, 1964). Using definition (29) and (33) and (36) provides substan-

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

87

tially more compact expressions for H+, and H,: iK

H+=-

H,

['

=

WPeff

1 dEz

-1, -1.

+

t3Ez

'4 ar -i dEz + K dEz P r

r d4

P

v. TWO-DIMENSIONAL FIELDS IN

( 44)

dr

THE

(43)

INNER DISK

The inhomogeneous circular surface is broken up into one inner disk centered at r = 0 and N annuli, each annulus labeled by the index i. To be consistent in labeling notation, the inner disk is labeled with i = 0. The disk, as well as each annulus region, is sourceless, so that the homogeneous Helmholtz equation (40) holds. The solution to (40) in cylindrical coordinates is well known to be Bessel functions multiplied by azimuthal circular harmonics. For the problem at hand, azimuthal symmetry exists requiring that the separable circular harmonics be of type {exp(in4)), for any integer n. Helmholtz equation (40) will therefore yield Bessel functions of integer order. Because the inner disk contains the point r = 0, the only Bessel function to be well behaved, not possessing a singularity, will be the Bessel function of the first kind, J,. Therefore, the total electric field Ezo in the disk must be a superposition of

giving

88

CLIFFORD M. KROWNE

In order to standardize the notation and make transparent what is actually transpiring, a few definitions are made: Cneui(r)

Jn(ke,ir>,

(49)

1

c . e -@Peff, i

In these four definition equations, the general disk or annuli location index i has been used as the last index on the Cnhni,on the material tensor , on the effective propagation constant element parameters p i and K ~ and k e f f ,and i permeability p e f f , For i . the disk the index in (49)-(52) is merely i = 0, allowing us to rewrite (46)-(48) as -m

Ez,

=

C

n=

anoCneuo(r)ein',

(53)

anoC:hnO(r)einQ,

(54)

--m

m

~

4 =0

C

fl= - w

VI. TWO-DIMENSIONAL FIELDS IN

THE ANNULI

Because an annulus does not include the origin, a superposition of any two linearly independent Bessel functions will be required to construct the

89

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

radial part of the separable solution to (40). The electric field is therefore m

E , ~=

C

n=

[ a n i ~keff, J ;r)

--m

+ b , , ~ ,keff,ir)] ( ein+,

i

=

1,2,. ..,N . ( 56)

As in (491, let us define

Cnebi(r) Nn(ke,ir), so that (56) can be rewritten in the more transparent form m

Ezi =

c

n=

[aniCneai(r)+ bniCnebi(r)]ein4, i

=

(57)

1 , 2 , . . . ,N . (58)

-m

For the H-field components, referring to (43) for H+i again,

Using the coefficient definition in (51) for the ani factor and the additional definition

H+; can be expressed in the much more compact form

Similarly, H,; is, employing (44) again,

Using the coefficient definition in (52) for the ani factor and the additional definition

90

CLIFFORD M. KROWNE

Hri can be expressed in the much more compact form m

H , ~=

C [ u n i ~ ; h ari)( + b , , i ~ ~ hr )b]iein+. ( n= -m

VII. TWO-DIMENSIONAL BOUNDARY CONDITIONS AND THE DISK-FIRST-ANNULUS INTERFACE There are three distinct types of boundary condition interfaces. The first boundary condition type is at the disk-first-annulus interface. This interface must match the inner disk, which contains a potential singularity at r = 0 which has been specially excluded, to the first annulus, which contains two linearly independent Bessel functions out of which the &-field is constructed. Once the matching has been completed at this first interface the field information can be pulled through to the next interface, and the matching procedure repeated. Thus each internal interface due to two adjacent annulli involves the same matching process. These internal interfaces constitute the second type of boundary condition. If there are N annuli, then there will be exactly Ni = N - 1 interfaces of the second type. The third type of boundary condition occurs at the interface between the last annulus, the i = N annulus, and the external part of the circulator geometry. This is where the last annulus or ring abuts up against either an ideally imposed magnetic wall which approximately expresses the transition between the ferrite material and the outside dielectric (be it air or a surrounding dielectric) or the transition ports taking energy into or out of the circulator. For a three-port circulator, these ports are referred to as the input port, the output port, and the isolated port. Normal practical design strategy attempts to reduce the exiting signal from the isolated port to be a small value compared to either of the other two ports. There will be a total of Ni + 2 interfacial boundary conditions, all of the internal ones plus one disk-annulus interface and one Nth-annulusoutside interface. The inner disk has radius ro. Each annulus has radius ri measured from its center. The width of each annulus is Ari = rio - r i I , where the subscript “0”or “ I ” indicates the outer or inner radius of the ith annulus. It is sufficient to apply boundary constraints on either the ( B n ,0,) normal pair or the (Ef, H , ) tangential pair. We choose the second pair as it

91

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

is easily applied. For the first type of interfacial boundary condition, E*o(r = 1 0 )

(65)

= E Z l ( t= r l l ) ,

H40(t = ro) =

r

(66)

= rl,).

Using (53) and (58) for the E, constraint, (65) becomes m

c

n=

00

c

anOcneaO(rO)ein4=

n=

--m

[anlCneal(rll)

+bnlCnebl(rlI)]ein4*

(67)

--m

Utilizing (54) and (61) for the H4 constraint, (66) becomes m

c

m

anOC,?hoO(

=

c

[ anlCnhal(

ril)

+ bnlc,?hbl(

ril)]

ein4.

(68)

By the orthogonality of the azimuthal harmonics on ( - T , T ) , these equations may be written for each individual nth harmonic as follows: anOCneaOD

= anlCnealD

+ bnlCneblD,

(69a)

+ bnlct?hblD* (69b) Here the argument information of the C coefficients has been compressed into a single added subscript index D which denotes radial evaluation at the disk radius D = ro = r l l . Solution of (69) yields for the first-annulus field coefficients a,, and bnl: anOCf?haOD

= anlCf?hhalD

CneaOD

CneblD

CnhaOD

CnhblD

These expression may be considerably abbreviated by defining the disk-toannulus coupling numerator factors

92

CLIFFORD M. KROWNE

=I

and the determinant D iproviding the information in the ith annulus

Di

CneaiA

CnebiA

cn hai A

cn h bi A

In (72) the subscript combination L4 denotes a radial evaluation at the ith-annulus inner radius ri,, that is, r i A = ri,

= ri -

Ari/2.

(73)

Thus we may now write a,, and b,,, as

a,,

=

MDAa

-all0

7

D l

VIII. TWO-DIMENSIONAL INTRA-ANNULI BOUNDARY CONDITIONS The (Ef, Hf) tangential pair is used to match between two adjacent annuli. Following forms (65) and (661, 'zi(r

= ria) = E z ( i + l ) ( r = r(i+l),),

(75)

ria) = H , ( i + l ) ( r = r ( i + l ) , ) (76) Invoking the annuli &-field expression in (58) and inserting it into (751, H,i(r

aniCneai(riO)

-k

=

bniCnebi(riO)

= an(i+ l)cnea(i+ l)('(i+ 1 ) 1 )

+ bn(i+

l)cneb(i+

l)('(i+

1)1) *

(77)

1)1) *

(78)

Similarly for H,, recalling (61) and inserting it into (761, anicr$hai(riO)

+ bnicr$hbi(riO)

-

- ' n ( i + l)Cr$ha(i+ l)('(i+ 1 ) 1 ) + bn(i+ l)c$hb(i+

l)('(i+

These two equations may be compressed by defining the fifth index on the C coefficients to be the outer radius rio of the ith annulus or the inner ) , the (i + 11th annulus. This so defined radius is precisely radius T ( ~ + ~ of the value used to evaluate the radial arguments of the C coefficients: aniCneaii

+ bniCnebii

= a n ( i + l)cnea(i+ 1)i

+ bn(i+

l)cneb(i+

l)i,

(79a)

+ bniCr$hbii = an(i+ l)c$ha(i+ I)i + b n ( i + l ) C r $ h b ( i + I ) i * (79b) this set of equations can be solved for the (i + 11th-annulus field coeffianiCr$haii

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

cients

+

93

and bn,i+ 1):

an(i+l)

-

Lneii

Cneb(i+ 1)i

Lnhii

Cnhb(i+ 1)i

Cnea(i+ l ) i

Cneb(i+ 1)i

Cnha(i+l)

' n h b ( i + 1)i

Cnea(i+ I ) i

Lneii

Cnha(i+ 1)i

Lnhii

I'

bn(i+l) = Cnea(i+ 1)i

Cneb(i+ 1)i

Cnha(i+ 1)i

Cnhb(i+ l)i

Here the left-hand-equation information about the previous inner ith annulus is stored in Lneii = uniCneaii

+ bniCnebii,

(81a)

+ bniCr?hbii* (81b) The fifth index on the C coefficients represents the outer radius rio of the inner annulus i or the inner radius r(i+l)Iof the outer annulus (i + 1). Formulas (80a) and (8Ob) can be somewhat simplified by recognizing that the denominators have already been defined in (72). The fifth index A has now been replaced by the subscript i denoting the inner radius r(i+l)l of the outer annulus (i + 1) or the outer radius rio of the inner annulus i. Thus the fifth index represents the interfacial radius of the last two indices in the new notation and so is a unique specification. Using the more generally constructed determinant Lnhii = uniCr?haii

Oi+' =

Cnea(i+ l ) i

Cneb(i+ 1)i

Cnha(i+ I)i

Cnhb(i+ 1)i

1

u n ( i + 1) =

Lneii

Dj+ 1 " n h i i

I.

Cneb(i+ l ) i 'nhb(i+ l)i

These expressions implicitly contain forward-propagating recursion information from the previous annulus in the Lneiiand L n h i i terms. This information will now be explicitly inserted from (81) into (831, factoring out

94

CLIFFORD M. KROWNE

the previous annulus field coefficients, so that explicit forward-propagating recursion formulas result: 1 an(i+1 ) =

{ [ Cnhb(i+ t ) i C n e a i i - c n e b ( i + I ) i ~ n h a i i ] a n i Di+ 1 + [ C n h b ( i + l)iCnebii - ' n e b ( i +

t)i'nhbii]

bni},

(84a)

1

b n ( i + 1) =

-{ [ C n e a ( i + 1)iCnhaii - c n h a ( i + I ) i C n e a i i ] a n i Di+ 1

+ ['nea(i+

l)iCnhbii

- ' n h a ( i + I)iCnebii] b n i ] *

(84b)

Each term within the square brackets in (84a) and (84b) is a connection term linking the (i + 1) and i annuli. Therefore, we define them as = Cnhb(i+l)iCneaii

- Cneb(i+l)iCnhaii,

(85a)

= Cnhb(i+l)iCnebii

- Cneb(i+I)iCnhbii,

(85b)

ab(i + 1 , i)

= Cnea(i+ 1)iCnhnii

- C n h a ( i + 1)iCneaiir

(85c)

+ 1, i)

= Cnea(i+l)iCnhbii

- Cnha(i+t)iCnebii,

(854

a a ( i + 1,i) pa(i

Pb(i

+ 1, i,

With these assignments, the recursion expressions (84a) and (84b) are 1 l,i)uni pa(i + l,i)bni}, a n ( i + 1) - -{au(i (86a) Di+ 1

1 bn(i+l) =

-{ab(i Di+

+

+

+ l,i)uni+ pb(i + l,i)bni}.

(86b)

Since the coupling terms a,(i + 1, i) and pp(i + 1,i), p = a, b, can be determined once the material parameters of the different rings are specified and the ring geometries set, the field coefficients of any succeeding ring can be found by (86). Starting from the first annulus i = 1, (86) may be successively applied (recursively) until the outermost last i = N annulus is reached. The iterative process must be repeated N - 1 times for N annuli, taking us from the field coefficient information in the innermost first annulus a,, and bnl to the field coefficient information in the last annulus anN and bnN. Backward propagation formulas can be developed just as was done for the forward propagation formulas, leading to the compact expressions seen in (86). Return to (79). Let i + i - 1 and obtain a n ( i - l)Cnea(i-l)i

+ bn(i-l)cneb(i-

1)i = a n i c n e a i i

+ 'nicnebii,

(87')

an(i-l)c?ha(i-

+ bn(i-l)Ct$b(i-

1)i = aniC,?hhaii

+ bniC?hbii*

(8%)

1)i

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

95

This set of equations can be solved for the (i - 1)th-annulus field coefficients un(i-. and bn(iRneii

Cneb(i- I)i

Rnhii

Cnhb(i- 1)i

un(i-1) =

bn(i-1)

-

I

1

Cnea(i-l)i

Cneb(i-I)i

Cnha(i-l)i

Cnhb(i-l)i

Cnea(i

- l)i

‘neb(i

C n h a ( i - 1)i

- I)i

C n h b ( i - I)i

I

’

I’ 1

Here the right-hand-equation information about the outer ith annulus is stored in Rneii

= uniCnenii

-k bniCnebii,

(894

Rnhii

= uniCt!haii

-k bnict!hbii.

(89b)

The fifth index on the C coefficients represents the outer radius rCi- of the inner (i - 1) annulus or the inner radius ri, of the outer annulus i. It is important to realize that although (81) and (89) look the same, they are not based upon the correct identification of the fifth index meaning, and the argument evaluations made upon the assignment of radial distance made according to this meaning. Formulas (88a) and (88b) can be simplified by recognizing that the denominators have a form similar to (82). The fifth index still represents the interfacial radius of the last two indices and so is a unique specification. For the determinant we set Di-l

=I

Cnea(i- I)i

Cneb(i- 1)i

Cnha(i- I)i

C n h b ( i - I)i

The annulus field coefficients ufl+

and bn(i- in (89) reduce to

96

CLIFFORD M. KROWNE

These expressions implicitly contain backward-propagating recursion information from the outer annulus in the Rneiiand Rnhiiterms. This information will now be explicitly inserted from (89) into (911, factoring out the previous annulus field coefficients, so that explicit backward-propagating recursion formulas result: 1 a n ( i - 1) =

{[

- C n h b ( i - I)iCneaii - C n e b ( i - l ) i C n h a i i ] ani Di- 1

+ [ Cnhb(i-l)iCnebii

- Cneb(i- l)icnhbii]bni}y

(92a)

1

bn(i- 1 ) =

-{ [ C n e a ( i -

1)iCnhaii

- Cnha(i- I)iCneaii] a n i

Di- 1

+ [ Cnen(i-I)iCnhbii

- Cnha(i- l)iCnebii] ' n i } *

(92b)

Each term within the square brackets in (92a) and (92b) is a connection term linking the (i - 1) and i annuli. Therefore, we define them as aa(i

-

'3

i>

= 'nhb(i-1)iCneaii

- Cneb(i-l)iCnhaii,

(93a)

pa(i

-

' 9

i,

= 'nhb(i-l)iCnebii

- Cneb(i- l)iCnhbii,

(93b)

ab(i

-

'7

i,

= 'neo(i-l)iCnhaii

- Cnha(i- l)iCneaii,

(9 3 4

Pb(i

-

'2

i,

= Cneo(i-l)iCnhbii

- Cnha(i-

(9 3 4

l)iCnebii*

With these assignments, the recursion expressions (92a) and (92b) are 1

Since the coupling terms a,,(i - 1,i) and &(i - 1,i), p = a, by can be determined once the material parameters of the different rings are specified and the ring geometries set, the field coefficients of any succeeding inner ring can be found by (94). Starting from the Nth annulus i = N, (94) may be successively applied (recursively) until the innermost i = 1 annulus is reached. The iterative process must be repeated N - 1 times for N annuli, taking us from the field coefficient information in the outermost annulus an,,, and bnNto the field coefficient information in the innermost annulus a,, and bnl.

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

97

IX. TWO-DIMENSIONAL NTH-ANNULUS-OUTER REGIONBOUNDARY CONDITIONS The progression of annuli may be effectively truncated at the r = R boundary of the device where the last i = N annulus ends and the outer region of the device begins. It is here that ports exit from the device. It is also here that the device transitions from a ferrite medium to a dielectric medium. If one wishes to stop the two-dimensional field analysis at r = R , then approximating boundary conditions must be applied here to model the effect of the ports and the change at the other contour regions where the device becomes dielectric. The first requirement is met by imposing constraints typical of that describing a circulator-microstrip line interface. The second requirement is met by assuming magnetic wall conditions where the device transitions from ferrite to dielectric. At the perimeter r = R , the boundary condition on H+ consistent with both requirements is a Dirichlet boundary condition (BC): Ha, H?'(

R,4)

=

H6,

Hc, 0,

4 a - A 4 a , / 2 < 4 < 4 a + A4a,/2, 46 - A 4 b / 2 < 4 < 4 6 + 4 c - A4c,/2 < 4 < 4 c + A4c,/2,

(95)

nonport contour regions.

An arbitrary function like that specified in (95) can be represented by a one-dimensional Fourier series over the appropriate domain ( - T , T ) : W

H?'(R,~)=

C m=

A m ,im d .

(96)

--m

Multiplying both sides of (96) by exp( - in+), integrating over the domain, and using the orthogonality property

yields the nth coefficient of the expansion

These coefficients must be precisely the same as those found in the Bessel-Fourier expansion provided for the H+-field solution for the last annulus in (61). Setting i = N and r = R, W

98

CLIFFORD M. KROWNE

Equating H,p”‘(R, 4) and H4N and using the orthogonality property of the Fourier harmonic functions, we find that A n = anNcr?haN(R)

+ bnNCr?hbN(R)

-

- anNCr?hhoNO + bnNC?hbNO ( 100) + bnNCr?hbNR, where the second equality is consistent with our earlier convention of attributing the fifth index “0”to the fourth index i = N, thereby assigning the radius for argument evaluation of the C coefficient as rNo and where the third equality simply registers explicitly the radius for argument evaluation as r = R. Examination of (74) and the linear mapping process implied by (86) indicates that a n Nand bnN can be written as a n N = a,,(recur)a,,, (101a) = anNCr?haNR

b,

= b,

(recur) a,, ,

(101b)

Here aflN(recur)and b,,(recur) denote the quantities obtained by applying forward recursion formulas (86a) and (86b) N - 1 times starting with (74) and at the end factoring out the single factors a,, from the final a n N and bnNresults. The recipe for getting aflN(recur)and b,,(recur) requires a,, to be formally set to unity in (74) and the recursion process executed as described. Equations (101a) and (101b) are extremely important relationships. Inserting them into (100) and solving for a,, gives an0 =

an N

cr?ha

An NR + bn N ( recur) cr?h

bNR

.

(104

Because all the quantities are known on the right-hand side of (102), a,, is determined. Once a,, is determined, all the fields in all the annuli are known by the very nature of the recursion process. Thus the driving or forcing function contained in (95) and implicitly stored in A, leads to the fields to be specified. This relationship means that we can now find the various Green’s functions relating forcing contour field H4(R, 4 ) to H J r , +), H,.(r, 41, and E J r , 4). This we will be finding the various components of the dyadic Green’s functions.

X. TWO-DIMENSIONAL DYADICGREEN’SFUNCTION WITHIN THE

DISK

The direct-coupling dyadic elements relating forcing contour field H4(r, 4 ) to H4(r, 4 ) and Hr(r, 41, and the cross-coupling (or indirect-coupling)

99

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

dyadic elements relating forcing contour field H+(R,4 ) to EZ(r,4 ) will be found here. First the fields will be examined within the disk, then the fields within the annuli. Invoking (102) and putting a,, into (53)-(55) gives the three field components at an ( T , 4 ) location within the disk: m EZo(r,

4) =

An

.Zma,N(recur)C~h,NR+ b,N(recur)C!hbNR x

CneaO(

'

r)eino,

m

H+o(r,

(103) An

+

4) = n=-m

anN(recur)Cth,NR bnN(recur)c$hbNR

x C,$haO(r)ein+,

(104)

m

H,o(r, 4 )

=

c

-n= -

-m

anN(

C,$haNR

An + bn N (

cr?h b N R

x C,$hao(r)ein+. (105) In order to find the dyadic Green's function form of solution, the implicit forcing function information in A, must be made explicit by replacing A, with (98), properly extracting out of the integral operator the forcing field. Recalling that integral form,

it is expeditious to break up the contour regions ( N T r pof them) where H,P"'(R, 4 ) is nonzero into a total of NTp zones [equal to C(q)N,41, each of which is of infinitesimal size [see (93, for example, where there are three regions]. Each individual q region where H,p"'(R, 4 ) is nonzero is composed of N,4 segments or one-dimensional elements of angular extent A 4 2 = A4Tq/N;, where A$Tq represents the total angular spread on the circumferential perimeter for the q th region [A +Tq has been simplified to A & q = a, b, c in (95)]: NTrp

HP'(R, 4 )

=

N;

c c H;Pker(R, 42) a( 4

q=l k=l

-

42) A 4 2 9

(106)

100

CLIFFORD M. KROWNE

Inserting (106) into (98) and reversing the order of summations and integrations gives

Performing the integration,

Returning to (103) and substituting for A,, Ezo(r7 4)

(111) Reversing the order of Fourier azimuthal harmonic summation and the double port and element discretization summations produces

x e-in4zein+H$( R , 4,")A4,".

( 112) This can be considerably streamlined by defining the constant denominator term to be ( recur) C$haNR + bn N (recur) c,?hbNR and placing it into (112): 3/n N = a n N

1

Ezo(r,4)

=

Np"

NTrp

cc c 2?r

-

q=l k=l

CneaO(r) 3/nN

n=-m

x e-in+fein+HPer R +k(

+q) 7

k

From the discussion in Section 11, we can recognize H & , ( R ,4')

=

Hi;'(R , 42)

and perform the limiting process lim A+f - 0 . Np"+

m

A+q

k*

(113)

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

101

When these two activities are completed, the z+ cross-coupling dyadic Green's function element arises from (114) as

The electric field Ezo(r,4) is obtained from (117) by multiplying the Green's function by H$'(R, 42) and then applying to this product the discretization operator obtained from the integral operator by the assignment NTrp

N$

q=l k=l

--a

That is, in discretized form, (11) now reads NTrp

& ( r , 4)

=

Nj

c c Gi$o(r' 4; R , 4f)H$c(R , 42) A4R.

(119)

q=l k=l

The three dyadic Green's function elements based on the preceding discussion, when placed in the field expressions, make (103)-(105) become

E Z o ( r 4) ,

=

/* Gi$o(r, 4; R , 4')H+c(R, 4') d4', --a

(120a)

102

CLIFFORD M. KROWNE

It may be desirable to consider the case where the forcing contour field H,,(R, 4) is treated as constant over some regions. Therefore, we will consider N&, port regions where H,,(R, 4) can be removed from the integrations in (120). This will require a generalization of the integral-todiscretization operator mapping provided in (118):

There are now a total of NTrpport regions, some of which are discretized into elements and some of which are continuously treated: NTrp

= N$rp

+ N;rp

(124)

*

Equations (119) and (121) become N&p

&(r,

4)

=

N;

cc

9=l k=l N&p

+

GfiO(h

4 ; R , 42)H,,(R, 42) A 4 2

c H?5c(R74u)/

u=l

4"+ A 4,/2

4"- A 4"/2

Gifio(r, 4; R , 4') d4'9

(125a) NAP N j

H,o(r, 4)

=

c c G&o(r, 4; R,

42W,c(r9

42) A 4 2

q=l k=l

(125b)

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

103

where the definite integral evaluation has been used:

The second term can be made to look like the first term by defining a modified definite integral which is normalized to the finite angular width of the port region

104

CLIFFORD M. KROWNE

and defining a modified Green's function

XI. WO-DIMENSIONAL DYADICGREEN'SFUNCTION IN THE ANNULUS

Getting the dyadic Green's function elements within the annuli is a somewhat more difficult task than that for the disk because here the recursive process will have to be used and correctly truncated at the ith

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

105

annulus where we desire the field information. The effort starts with the field formulas (58), (611, and (641, and uses the same reasoning used to obtain (101) for an intermediate annulus i: ani = ani(recur) a,, ,

(133a)

bni= bni(recur)anO.

(133b)

Substituting these relationships into the annuli field formulas gives m

E , ~=

C n=

[u,,(recur)Cneai(r)+ b , i ( r e c u r ) ~ , e b i ( r ) ] u , ~ e i , + ,

--m

i

=

1,2,..., N, (134a)

m

H + ~=

C

n=

-m

[ani(recur>c,$,,(r)

+ b,i(recur)~~hbi(r)]a,,ei"+, i = 1 , 2,..., N, (134b)

m

H , ~=

C

[ uni(recur)CAhai(r ) + bni(recur)Cihbi(r)]u,,e'"+,

n=-m

i

=

1 , 2,..., N. (134c)

Again a,, must be replaced with (102), which may be written as

using (113). From this point on, we can follow the same reasoning process as was done for the disk. The results will be merely stated here. Equations (131a)-(131c) for the ith annulus become

106

CLIFFORD M. KROWNE

The dyadic Green's function elements are now given by the new expressions

(137a)

1

c

--

2~

uni(recur)Cihui(r)+ bni(recur)Cihbi( r)

-in+fein+

YnN

n=-m

(137c) and the modified expressions

1

--

2.rr

C

n=-m

uni(recur)Cneui(r) + bni(recur)Cnpbi(r)-

.

Z:ern + ,

Yn N

(138a)

1

--

2~

c m

n=-m

uni(recur)CthUi( r)

+ bni(recur)C,$bi(r )

-

.

I i e r n+ ,

Yn N

(138b)

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

107

(138c)

FUNCTION ON THE XII. TWO-DIMENSIONAL DYADICGREEN'S OUTERANNULUS-PORT INTERFACE Due to the separable nature of the governing equation (40) and the resulting sourceless solution being the product of radial and azimuthal functions, the dyadic Green's function elements evaluated on the contour r = R simplify significantly. These Green's functions and the fields found as a result from them are of importance in relating the solution found inside the ferrite circulator domain on 0 Ir IR and - .rr I4 < .rr to the outside structure, namely the interfacing ports. If we assign a notation similar to that found in (113) to the radial numerator factors in (137) and (1381, we find (139a)

?n%

= anN(recur)CneaN(R)

+ bnN(recur)CnebN(R),

y$

= anN(recur)C:h,N(R)

+ bnN(recur)C,$hbN(R), (139b)

Y:

= an.(recur)c~h,N(R) -

bnN(reCUr)CLhbN(R),

(139~)

Notice that (139b) is identical to (1131, but with the upgraded notation being employed here. Furthermore, let us define normalized quantities

With the definitions (139) and (1401, the fields and dyadic Green's function elements can be given by the expressions N$.p

&(R,

4)

=

N;

c c G Z N ( R , 4; R , 4,")HJR,

4,")A 4 2

q=l k=l N&p

+

c %v(R,

u=l

4; R , 4 " ) q d K4 u )

w,>,(14W

108

CLIFFORD M. KROWNE

XIII. CIRCUIT PARAMETERS IN

THE TWO-DIMENSIONAL

MODEL

Relating the field quantities determined from previous sections to circuit quantities is both a useful and necessary step if the field results are eventually to produce a circulator device coupling to the outside world. Thus we will find an equivalent form for the field results which makes the device appear as a multiport device with NT,p port terminals corresponding exactly to the number of port regions given in (124). This is done by associating field quantities at the device-external world interface located

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

109

at r = R with voltages and currents. Line integration of the field component normal to the top microstrip and bottom ground plane metal surfaces, which is E,, yields a uniquely defined voltage where the integration path is in the perpendicular direction z . Closed contour line integration around a path enclosing the microstrip region under consideration using Ampere’s law gives the current. Short discussions relating to port segmentation, covered in more detail here and involved in the process of coupling the distributed circulator fields to an external circuit model, may be found in Gaukel and El-Sharawy (1994, 1995). The basic relationships to be used are therefore

V=/E-dl

( 144)

I

(145)

and Ampere’s law =#H

dl.

For the discretized part of the r = R contour, the line integral for voltage gives, when the evaluation is done at the midpoint of the element,

hE,( R , 4f). ( 146) Here h is the separation between the microstrip and ground plane. For the continuous part of the contour, the line integral for voltage gives, again using the midpoint, =

v, = hE*(R,4”).

(147) The line integration around a closed path for the current on the discretized part of the contour is

110

CLIFFORD M. KROWNE

The first approximation reduces the line integration to an azimuthal integration. The second approximation reduces that integration to the current flow through a chord of a circle subtending the element. Factor F accounts for the relative contribution of the outer part of the contour further removed from the ground plane and above the microstrip compared to the contribution between the microstrip and the ground plane. A symmetrically disposed stripline, for example, would give F = 2. An extremely wide microstrip (almost like a parallel plate) would give F = 1. For the continuous part of the contour, the closed line integral for current gives, using the midpoint,

Expression (14614149) relate voltages and currents on the perimeter of the circulator to the field quantities E, and H,. Thus the appropriate dyadic equation to use for relating the circuit quantities to these field variables is (141a). First substitute into that equation current formulas (148) and (149):

This equation was obtained by replacing the contour field H,, imposed on the circulator with the current information. Equation (150) needs to be evaluated for the discretized and continuous ports, obtaining N&* EzN(

R , 42)

=

NpI

cc

r=l

s=l

GifiN(

R , 48; R , 4:)

4s

'4;

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

111

Now we must associate the electric fields in the Nth annulus (the last annulus) on the left-hand side of (151) with the voltages given in (146) and (147). Using (146) and (147),

(152a)

v, = h

N&

N p r

C

G;$N(R,

Zrs

4 u ; R ?4:)-

r=l s=l

'4;

FWrs

N&p

+

ciflN(R,

u=l

4u; R , 4 u ) -

IU

'4u'

(152b)

FWU

It is important to recognize that rs discretized or u continuous points act as sources, which is why they are distinguished from the response points qk and u for, respectively, discretized and continuous locations. Equation (152a) and (152b) may be put into a much more compacted form, which allows us to recognize the circuit nature of the problem, by defining the following terms in the summations: h Zqk,rs = G;$N(

z q k , u = G;$N(

R , 42; R , 4;)-

'4:3

R , 42; R , 4 u ) -

h

(153b)

FWU

h zu,rs

= Gi$N(

(153a)

FWrs

R , 4 u ; R , 4:)-

z u , u = ciffN(R,

4 ~R;, 4u)-

(153c)

FWU

h

A4u*

(153d)

FWU

The units of these matrix elements are impedance. Place these impedance

112

CLIFFORD M. KROWNE

elements back into the (152) expressions: N A P Npr vqk

=

Vu =

C C Zqk,rszrs

N;.V

-t

C

r=l

s=l

N&p

Npr

N+,p

r=l

s=l

u=l

Zqk,uzu,

(154a)

u=l

C C Zu,rsZrs + C

zu,uIu*

(154b)

Next associate the impedance matrix elements with a global impedance matrix for the entire circulator structure. Define it as

where the square brackets around each element indicate the matrix associated with that type of indexing. Inserting (153) into (1541, (154) may be written in matrix form as

The total number of discretized region elements for the entire circulator perimeter is

For consistency of notation we abbreviate the total number of continuous regions as N&,

=

N".

(158)

Then the sizes of the various submatrices in (156) are size{[zqk,rs]}= N~ x N ~ ,

(159a)

size{[zqk,u]}= N~ x N',

(159b)

size{[Zu,rs]}= N' x N d ,

(159c)

s i ~ e { [ Z ~ ,= ~ ]N'} x N',

(159d)

Let us now define overall voltage and current matrices for the entire

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

113

circulator, just as was done in (156) for the impedance: (160a)

( 160b) Placing (155) and (160) into (156) produces the most compact form of circuit expression for the circulator: v = ZI. (161) V and Z are column matrices whereas Z is a square matrix. The sizes of the global matrices in (161) are size{[V])

=

( N d + N') X 1,

(162a)

size{[Z])

=

( N d + N')

X

1,

(162b)

size{[ Z])

=

(N d

+ N')

X

(N d

+ N').

(162c)

Indexing in the global matrix system found in (161) produces N

6=

c

'cabrb*

b= I

Here N

=

Nd

+ N'.

The row and column global indices are related to the submatrix indices by N&,

a =k

j

j-1

i=l

NfrP

b

=s

NAP

+ C C NpiS(q-I),j + j

C NiScr,, + ~ a c r , 1 ,

NAP

+ C C N i S ( r - l ) , j+ C NiS,,,, + j=1

(165a)

i= 1

(165b)

i= 1

j=l

Here the S are Kronecker deltas and the cr and cc indices locate movement, in the global indexing scheme, from the discretized port counting sequence to the continuous port counting sequence: cr

=

1 . H ( v - l),

cc

=

1 * H ( u- 1).

(166a) (166b)

H(u - 1) is the Heaviside step function which is taken to be unity for its argument greater than or equal to zero. One may show that the index scheme in (165) reduces to the simple case in Gaukel and El-Sharawy

114

CLIFFORD M. K R O W

(1994) if only discretized ports are considered and each is broken into the same number of elements. In order to connect to the outside circuitry, each discretized port will need to have its current summed to yield the total current exiting (or entering) the port. This is Kirchoffs current law and assumes that the current leaving the circulator at that port is entering what is essentially a wire. Of course, that is not the case in that the current is really entering a microstrip line which may be characterized by a single total current. Characterizing the microstrip line here by a single current is the same as saying that the distributed two-dimensional circulator problem, solved with the ferrite material, has been reduced to a one-dimensional problem outside of the radius r = R. In order to completely reduce the problem to one dimension outside of the ferrite region, we must also characterize the element voltages along a particular port by a single voltage. This is essentially Kirchoff s voltage law stating that the net voltage drop around a closed loop is zero, the loop beginning on an element and stopping at the ground plane a distance h below the microstrip metal, then traveling along the ground plane just enough to be under the adjacent element, and then going distance h in the z-direction, with a final path along the contour between the two adjacent elements. Here we are taking the voltage drops along the contour or its projection on the ground plane to be zero. Since we will be adding currents, invert the form of (161) to obtain

Z=W,

Y = 2-'. In expanded form (167) reads

Here the summation occurs over discretized ports a = d and continuous ports a = c. We need the expressions for the currents on the discretized ports and on the continuous ports. The expressions will be different, and care must be exercised in deriving them. For a discretized port r, source contributions occur at both the discretized and the continuous ports and this must be reflected in the construction. Kirchhoffs current law at port r is N p r

=

c

Ib(rs),

(170)

s= 1

where we have uncompacted the indexing according to (165) and summed over the appropriate segment index s. Each individual current at the field point location s has contributions from discretized and continuous source

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

115

points, expressed as N$rp N," Irs

NC

cc

=

c ys,uK,

+

ys,kl'kl

k = l I=1

(171)

u=l

which comes from (169) by inserting the appropriate summation operators. Notice that we have uncompacted the indices completely in this formula for ease of further analysis. Insert (171) into the (170) sum: NpI

Ir

=

N;

N$rp N,"

ccc

+

%,klVkl

s = l k=l 1=1

NC

c c %,uK*

( 172)

s = l u=I

Now use the fact that

*

vk, = Vk,

Vkl

*vk,

11,112

(173)

SNpk,

which means that the segment voltages are assumed the same across the kth port, to reorder the summation indices and change the position of voltage in (172): N&p 'r =

c

k = l ',I(

scllz

NC

%,k l )

+l

:

K(

szl

q s , U)

*

(174)

In this formula v k , has been factored out of an 1 sum, and it is understood that any 1 within the summation range will work. Since (173) has been invoked, the collapsed form of voltage index should more properly be used to appear like that in the second sum. Define NpI

c k =

N,"

cc

(175a)

ys,kl,

s = l I=1

( 175b) s= 1

With these definitions (174) considerably simplifies: N.Aa

I,.=

c

k= 1

c k v k

+

c f,K. NC

u=l

For a continuous port u , applying (169) for b

= u,

116

CLIFFORD M. KROWNE

Changing the position of the voltage,

Identify

Substitution of (179) into (178) yields

Equations (176) and (180) can be combined to give the following expression which relates the internal field behavior inside the circulator to the external ports, where each port is identified by a single index and all explicit discretization has been removed. This is the final desired form we have been seeking:

In compact form, this becomes Z=W, where

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

117

XIV. SCATTERING PARAMETERS FOR

A THREE-PORT CIRCULATOR IN THE TWO-DIMENSIONAL MODEL

Here we will consider a particularly simple case where the circulator has either discretized ports or continuous ports but not both. Furthermore, if discretized ports are treated, then only one element per port is allowed. In effect, what that means is that the angular extent of the ports is considered so small that a single element is sufficient to approximate the contour port segments. This case is then so simple that we may refer to an earlier section and not use the most general circuit analysis of the preceding section. Thus (141) becomes N&p

c Gi$N(R,4; R , 49)H&3,

EzN(r, 4) =

49)

q= 1

N&p

+

c G R N ( R 9 4; R , 4u)K$c(R

A4u.

(186)

u=l

Limiting the device to three ports makes N&

=N&,=

3 and

3

EZN( R , 4) =

c di$,(R, 4; R , 4 4 ) q & ? ,

49)

(187)

q= 1

where

If we absorb the azimuthal spread into the Green's function by defining a modified form

G(4; 4q)= diRN(R,4; R , 49)

(189)

where the understood indices and arguments have been dropped, (187) can be expanded E z ~ ( R , ' $ )= 6 ( 4 , 4 0 ) ~+aG ( 4 , 4 6 ) H 6+ ' ( 4 * 4 ~ ) ~(190) ~' Now evaluate (190) at each of the ports, q = a, b, c , labeled counterclockwise, and simplify the notation for E z N ( R ,4) to E,4 by setting 4 = 49:

E," = E( 4 0 3 4 a ) H a

+ E(

E," = 6(4,,4a)Ha + E ( 4 6 , Ef

=

& 4 c , 4 a l H a + G(4c,

4 6 ) H 6 -I-

E( 4 a ,

4 6 l H 6 -k

E(46, ~

46IH6

+ G(4c,

~ c ) ~ c (191a) , c

)

4cIHc3

~

c

(191b) * (191c)

118

CLIFFORD M. KROWNE

Let us make a number of practical assumptions which will further simplify the forthcoming analysis. Assume that the input port a is subject to reflections from the microstrip-circulator interface. Therefore, s1 is nonzero and the match is imperfect for port u. However, assume that the other two ports, the output port b and the isolated port c, are loaded in a perfectly reflectionless manner to the microstrip lines. These assumptions translate into the relationships E,4in) # E:,

( 192a)

H&in)+ H:,

( 192b)

E,40Ut)= E,b,

(193a)

H&OUt)= H&

(193b)

E&It)= Ef,

(194a)

= H&

(194b)

%(out)

where the subscript indicates an inward or outward propagating wave along the microstrip in relation to the circulator. Each microstrip line is characterized by a wave impedance. Consequently, (195a) b

EZ(0Ut) - -

-lb,

(195b)

-5c.

(19%)

Hb(out)

Ef(0ut) - HC(0Ut)

Next we define the s-parameters which are to be determined by this process of analysis: (196a) E,"= ( 1 + sll)E:(in), H$

=

( 1 - sll)H&in),

(196b)

Formulas (192)-(196) must be combined to utilize only the total fields in the microstrip lines because at the circulator-microstrip interfaces

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

119

we relate the z- and +components by interfacial tangential boundary conditions ( 199a) E:( cir) = E!( mic) ,

HJ(cir)

=

HJ (mic) ,

(199b)

where formulas (199a) and (199b) relate total fields. When this is done, (200a) (200b) (200c) (201a) (201b) = 1, as it is free to set, and put it into (196a) Make the input field Elcin)

so that the H-field is determined in terms of the input s-parameter in

(200a). We obtain E," = (1 + s11),

(202)

l aH a -=

1. 1 - s11 Combining these two equation eliminates sI1:

E," = 2 - LaHa.

(204) Now using (201) and (204), remove the E-field unknowns from (191), obtaining a simulations set of three equations in three unknown H-fields: 2 - laHa

- 6 bHb - lcHc

=

GaaHa

+ GabHb + GacHc,

= Gba Ha

+ Gbb

= GcaHa

+ GcbHb + GccHc*

Hb

+ Gbc

Hc

(205a) 9

(205b)

(20%)

Rewriting (209,

+ &IH, + GabHb + GacHc GbaHa GcaHa

+ ( G b b + 5 b I H b + GbcHc GcbHb (Gee + 6 c ) H c

=

2,

(206a) (206b)

0.

(206c)

= =

120

CLIFFORD M. KROWNE

The solution for the H-fields is

where the H-field system determinant is (Gaa

Dp

=

+

Gba Gca

la)

Gab (Gbb

+

Gc b

Gac

lb)

-

Gbc (Gcc

+

(208)

lc)

The H-fields have been found and from them the s-parameters can also be obtained. Equation (203) gives (209a) (209b) (209c) where the latter two formulas came from using (201) and (202) in (200b) and (200~).Obviously, the E-fields have been obtained by this process, too.

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

xv. LIMITINGASPECTS

OF THE

121

TWO-DIMENSIONAL MODEL

It is instructive to show that the Green’s function dyadic elements for the two-dimensional recursive model involving annuli reduce to those of a single-region homogeneous circulator device. To do this, focus is directed at the EH-coupling element involving the z+components, Gig, which may be compared almost directly with the homogeneous result obtained by Bosma (1964) for this particular dyadic element. Since Bosma only provides a Green’s function result on the perimeter of the device, take his electric field result

and specialize it to r = R, nothing that his study looked at the special case of vanishingly small port angle, namely that A+‘ + 0. Furthermore, using the general Green’s function form in (111, write the equivalent Green’s function element to his cosinusoidal result, in an exponential azimuthal expansion:

Here x = kR and k = k e f f ,consistent with Bosma’s notation. Finally, there is one last issue to be addressed to get this Green’s function formula into the correct form to directly compare with our study. It is the issue of time dependence. Bosma’s two studies (1962, 1964) both assumed an exp( -i w t ) time variation. It is very easy to avoid redoing the analysis with this dependence, since we assumed the inverse, exp(i w t ) . Merely take the odd powers of w in the expression under consideration and make w + - w. That is, place a factor of ( - 1) outside of all such terms in the expression. The question is how to correctly identify these radian frequency terms in the preceding G ( R , 4; R , 4’) formula. The three contributing constants, Seff, K , and p, affecting the G form will be addressed in turn. The effective wave impedance of the ferrite Serf contains, as will become apparent when we examine the comparable recursive Green’s function, an explicit *dependence. To see this, we jump ahead and retrieve our

122

CLIFFORD M. KROWNE

effective recursive wave impedance

leff, r:

- WPeff 5eff.r

-9

k e ff

where keff is taken to be the positive root of (41): kzff = w2Epeff

9

(41)

so that the arguments of the Bessel function solutions are uniquely determined and that purely real permittivity and effective permeability generate a positive-valued effective propagation constant in the plane of the circulator. Therefore, placing (41) into (212) gives the fully explicit form of 5 e f f , r :

For

K

and p, they can be rewritten (Soohoo, 1960) as

+ p( w ) = p’( w ) + ip”(w ) . K( 0 ) = K’( W )

iK”( W),

(214a) (214b)

Although Soohoo derives these relationships assuming an exp(i w t ) dependence for the small perturbational radio frequency time variation of the fields, obviously, by an w + - w change, they can put in the inverse exponential form, too. The parts K ” ( w ) and p ‘ ( w ) are even functions of w , whereas the parts K Y O ) and p ’ ’ ( ~are ) odd functions of w . Therefore, putting these three equations, (2131, (214a), and (214b), which are descriptive of the constant variation with radian frequency into the Green’s function expression (210, a formula with all the radial frequency behavior is found under the inverse Bosma exp(iwt) assumption:

and applying w + - w to (2151, we obtain the homogeneous Green’s function result for the time dependence exp(i w t ) in the two-dimensional

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

123

recursive Green’s function formulation: G ( R , 4; R , +’)Im-

--o

It is to this result that we will compare the limiting case of a two-region inhomogeneous circulator device, with an inner disk region and a single additional annulus. The limiting inhomogeneous case will, in general, have different materials in the annuli regions, but here the material parameters will be selected for only two regions with the same constants in the i = 0 disk region as in the i = 1 annulus region. The inner disk radius is r = ro, and the outer radius of the annulus is r = R. The Green’s function corresponding to (217) is (142): i

;7

=

m

“nl(recur)Cneal(R) + bnl(recur)Cnebl(R)

yAh = “,l(recur)c?h,l(R)

9

(220a)

+ bnl(recur)C?h’,,,(R), (22Ob)

where N = 1, the k summation index has been dropped since the port zones have not been segmented, and only the q index has been retained to indicate different port locations. Here the recursion constants “,,(recur) and bnl(recur) in (74a) and (74b), although recursion constants representing propagation through all the annuli, are the same as the formulas relating the i = 0 disk to the first annulus. Therefore, formulas (74a) and (74b) apply, and, using (71) and (72), MDA a

anl = -“no Dl

9

MDA b

bnl = -‘ n o Dl MDAa

=

9

CneaOO

Cnebll

CnhaOO

cnhbl,

P

(221a)

124

CLIFFORD M. KROWNE

MD.4b

=

Cneoll

CneaOO

cn h n l l

CnhaOO

(221b)

The last index on the function elements has used the explicit instructional scheme to indicate the inner “I”or outer “0”radius of the region being treated. Before trying to evaluate 72,first examine D, to verify that it is well behaved for the limiting case under study. We will need the general formulas given in (49)-(51) and (57) and (601, which are restated here for convenience:

These function elements become, setting i

=

1,

Cneoll = Cneal(rO)

= Jn(keff,IrO),

(223a)

CneblI = cnebl(rO)

= Nn(keff,lrO)y

(223b)

(224b)

when properly evaluated at r = ro. Inserting (233)-(225) for the function elements at the interface into the first recursion determinant D, delin-

125

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

eated by (2221, Dl = CneallCnhbll

- Cne bllCnha ll

[

= Cl kef f, 1 Jn( keff, lro)

X (ke ff, 1'0) - Nn( keff, 1'0) JA(

keff, lro

>I

-

(226) Here W(x,) is the Wronskian of the argument [discussed in Krowne (1983)l: = Cl kef f, l W ( x 1 )

(227) Thus D, is well behaved, and we may proceed to the determination of. ; : 7 Placing yz and ydh from (220a) and (220b) into the definition (219) of 72 and invoking the a,, and bn, relationships (74a) and (74b), XI = keff, 1 r 0 *

-z e = MDAaCnea10 Yn1 MDAaCnho10

+ MDAbCneb10 -k MDAbCnhb10

Let us find the property of the second factor tional function elements needed are

MDAb.

(228) *

The relevant addi-

CnhaOO = Ct?haO(rO>

Putting these function elements, along with those previously found at the interface, into the M D A b expression (221b), MDAb

= CneallCnhaOO

- CneaOOCnhall

where the dimensionless radius (233) result as the first-annulus parameters

xo = k e f f , o r o

is employed. The limit of this

MDAb

126

CLIFFORD M. KROWNE

approach those of the inner disk region is lim

hfDAb(xC)’xI) =

0.

(234)

Xl+XO

In order to find the Green’s function factor yt:, it will no longer be necessay to find h f D A b . Instead, the formula reduces to the transparently simple result

These two first-annulus function elements are determined from (49) and (51) as (236) C n e a l O = C n e a l ( R ) = Jn( k e f f , 1’) lZK1 1 Cnha10 = c,?hal( R , = c1 keff, lJA(keff, 1 R, - - - Jn( ’eff, 1 R, * (237) Pl R Defining XI = k e f f , IR’ the Green’s function factor becomes in the limit 3

[

1

=-

kff

J;(x)

12K

- -ln(X)

PX

’

(239)

where the final argument of the Bessel functions is = keffR.

(240) The prefactor in (239) is an impedance which came from the recursion formulation limiting process, so it is defined as x

When this impedance definition is utilized along with 72 in the recursion Green’s function (2181, the final desired expression to compare with the homogeneous case [see (21711 results:

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

127

This is identical to the homogeneous case formula (217) if one notes that (6' and (6q have the same meaning. XVI.

SUMMARY OF THE TWO-DIMENSIONAL MODEL

Here we have developed a two-dimensional dyadic recursive Green's function with elements G,"i' suitable for determining the electric field component E, and the magnetic field components H, and H4 anywhere within the circulator (Sections V and VI). The problem was inhomogeneous because of variations in the applied magnetic field H a , magnetization 4.rrMS,and demagnetization factor Nd. All inhomogeneity effects can be put into the frequency-dependent tensor elements of the anisotropic permeability tensor jl. The recursive nature (Sections VII-IX) of G,"i' is a reflection of the inhomogeneous region being broken up into one inner disk containing a singularity and N annuli. G z ( r ,(6) was found for any arbitrary point ( r , (6) within the disk region and within any ith annulus (Sections X-XI). Specification of G;, i = E , j = H , s = z, r = (6, at the circulator diameter r = R led to the determination of the circulator impedance matrix 2, (Section XIII). Ports were separated into discretized ports with elements and continuous ports located at arbitrary azimuthal (6 and arbitrary line widths. An admittance matrix Y was found which relates the internal circulator field behavior to the finite number of external ports characterized by voltages and currents. It was also shown how Gi$(R, (6) enables s-parameters to be found for the simple case of a three-port ferrite circulator (Section XIV). Limiting aspects of the two-dimensional model were also covered to show how it reduces to the uniform case where inhomogeneities disappear (Section XV). XVII. INTRODUCTION

TO THE

THREE-DIMENSIONAL THEORY

As discussed in a previous paper (Krowne and Neidert; 1995) the ferrite research and development community, which has focused on producing ferrite-based circulators, has been in need of simple but accurate ways of calculating performance when the device is subject to radial variation of the bias field Happ, ferrite material magnetization 47rM,, and demagnetization factor Nd. The two-dimensional recursive Green's function employed in Krowne and Neidert (1995) allowed the inhomogeneous boundary value problem, subject to inhomogeneities in the parameters, to be solved in an orderly and systematic fashion. It utilized an integral-discretization map-

128

CLIFFORD M. KROWNE

ping operator and finally resulted in scattering parameters being expressed for a three-port circulator with unsymmetrically disposed ports. The theory requires the circulator region to be broken up into two different zones. The inner zone is made up of a disk containing the origin point at (O,O), and the outer zone is segmented or divided up into annuli, each one of unequal radial extent, layered as in an onion. Numerical calculations, based on a FORTRAN computer code developed from the theory, show that a few seconds are required per frequency point to obtain results including s-parameters (see Section XXXIX). So, not only is the theory elegant, but it is also readily coded into FORTRAN, making available quickly obtained numerical results. In contrast, two-and three-dimensional finite-element (FE)and finite-difference (FD) analyses are hundreds to thousands of times slower. For three-dimensional analysis using tetrahedral elements in an FE approach, several hours per point are the expected scenario with between 10,000 and 25,000 tetrahedra needed! Each approach has its advantages and disadvantages. The Green’s function (GF) method is best for geometries with some symmetry, even if it has to be imposed in a consistent manner. But, the GF method can lose its attractiveness when the geometry of the object under study becomes very irregular and complex. This is especially true for arbitrarily located inhomogeneities and jagged boundaries. Then the FE or FD methods become much more feasible and even necessary. However, for canonical or quasi-canonical structures like a radially inhomogeneous circular puck (two-dimensional) or a radially inhomogeneous circular pill box (three-dimensional), a GF method is a sound approach. This is especially true in that the results from such an analysis can be used as a check of noncanonically based approaches like FE or FD. A three-dimensional GF approach will be developed in the ensuing sections, following much of the reasoning used in the two-dimensional method. Integral-discretization operators will be employed, and spectral summation over the doubly infinite domain of azimuthal integers n will be maintained. However, because of the three-dimensional nature of the construction, neglect of some of the field components will not be necessary any more. Although most circulators are built to be thin in terms of electrical wavelengths compared to their planar extent, assumptions requiring the thickness h to approach zero to apply a two-dimensional model will no longer be required. Thus the actual effects of a finite-thickness substrate on the circulator behavior will now be possible. The one characteristic radial propagation constant found in the two-dimensional model will now break up into two radial propagation constants, both affected by the allowed normal z-directed propagation constant k,. The problem is no

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

129

longer reducible to or described by a single governing equation, but rather by two coupled governing equations which always stitch the field components together. Thus TEM,TM, and TE modes are not allowed in relation to any coordinates. That is, no coordinate transformation will ever allow such modes to be found. It is impossible to find such simple modes and therefore a much more careful approach to solving the three-dimensional problem must be enlisted.

XVIII. THREE-DIMENSIONAL FIELDEQUATIONS In three dimensions, Maxwell’s sourceless curl equations are, for harmonic conditions with phasor time dependence exp(i w t ) assumed, VXE

=

V

= ioD.

H

X

-ioB,

( 12) (13)

These two equations are valid within the ferrite circulator region which is considered to be inhomogeneously loaded with material (it may be a semiconductor region if we were to use a semiconductor relying on the magnetoplasma effect). The constitutive relationships (14) and (15) are still given by B

=

fiH,

( 14)

D

=

BE.

(15)

In the ferrite region, we will again assume that the dielectric tensor reduces to a scalar

8=

E.

( 16)

Of course, this would not be the case for a semiconductor employing magnetoplasma effects where we would retain the tensor permittivity and drop the tensor permeability. The general expression in matrix notation for the curl of an arbitrary vector field is 1 r

VXA=-

130

CLIFFORD M. KROWNE

where it is noted that the expansion of (17) is accomplished by keeping the unit vector terms outside of the partial operators 8, = d / d x i , x i = r , 4, z . It is also noted that we use r instead of the usual p for the cylindrical radius. Curl A then expands along the second row as

or

=

[----IF+ 1 dAz r

34

dA, dz

[z dAr ---Id dAz

The third term in (243) is new in the three-dimensional treatment of the problem compared to the two-dimensional approach which allowed us to drop it because we had set d / d z = 0. That is no longer the situation here and care must be exercised to include these new partial derivative terms. The permeability tensor is, by (21),

I;. p

G=

L

By (141,

0

; lo].

-iK

131

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

Using the expression fof curl E found in (19), A result, the curl E equation becomes 1 dE, r d4

dE+

dEr

=

E, and the preceding B

dE,

dz

2

which can be written in terms of the three component equations: (246a) (246b) (246c) Following similar steps, the curl H equation becomes

-I?+

1 dH, dH+ r d 4 - dz

[-

dHr

[dl-

- Ic$+ dH,

= iw&[

(247)

and, again listing the component equations, (248a) (248b) 1

r

(248c)

132

CLIFFORD M. KROWNE

We notice that the first two equations of each curl set of equations are only in terms of the partial derivatives of the z-components E, and H, if an exponential form exp(ik,z) is assumed for the z-directed propagation normal to the circulator surface. The other components which appear in these equations are the transverse components to the z-direction, ( E l ,E,) and (H,, H,): 1 dE, - - - ik,E, = -io( pHr - iKH,), (249a) r 84 JEZ ik,Er - - = -iw(iKH, + pH,), (249b) dr

1 dH, - - ik, H, r d4

= i WEE,,

(249c)

ik,Hr - - = ioEE,.

(249d)

dr

These are four equations in four unknown transverse field components, which may be rewritten in a more transparent form as 1 dH, WEE,+ 0 . E 4 + O.H, ik,H 4 - ; z , (250a)

+

O.E,

+ i m E , - ik,H, + O.H,

=

aHz -dr

1 dE,

0 * E,

+ ik, E,

ik,Er

+ 0 . E, + i20KHr + iwpH, = -.JEz dr

- iwpH,

’

+ i20KH, = -r d4 ’

(250b) (250c) (250d)

Solution of this 4 X 4 system of equations in terms of the partial derivatives acting on the z-component fields is readily delineated: 1 dH,

0

0

ik,

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

1

E4

=

0,

H = -

1

dH, dr

0

--

0

-r d4

1 dE,

iW&

0

0

iw&

OS 0

- ik,

0

-iwp

-WK

-WK

iwp

133

(253) ik, 0

0

1

H4

where the 4

X

=

0

WE

0,

(254) 0

ik,

ik,

0

4 system determinant 0,is iWE

0,=

0 0 ik,

0 iws ik, 0

ik, 0

0 -ik, -imp

-OK

-WK

iwp

After expanding the determinants for each field component solution,

134

CLIFFORD M. KROWNE

The coefficient notation FLL for the transverse field component solutions is that i = solution field component direction, j = direction of partial derivative operation, k = solution field type ( E or H ) and redundant with the basic coefficient itself, and m = the field type the partial derivative acts on. They are given by the following 16 expressions:

1 -: I

EL$

=

- i w p k: -

K 2 D;',

ELL

=

wKk:Ds-',

(25%)

EL:

=

w2EKkZDs-',

(257c)

EiL

=

ik,(k;

(257d)

- k2)Ds-',

(257a)

EA4 = wKk:Di',

(257e)

E$

(257f)

=

- i w ( p , k 2 - pk,')Ds-',

E24 = ik,(k: - k 2 ) D ; ' ,

(2576)

Ez

(257h)

K

=

--k

z k2D-1 s ,

CL

Hhr;t = ik,(k: - k2)Ds-', Hi$

'

K

=

- k , k 0,- , CL

(257i) (2571) (257k)

Hi:

=

H$

= - - k,k2Ds-',

i ~ ~ ( k-:k2)Ds-', K

(2571) (257m)

CL

H$'

=

-ik,(k2

- k;)D;l,

(257n)

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

HZ

=

i W s ( k 2 - k:)D;’,

135

(2570)

Here the system determinant 0, is evaluated to be 0, = - ( k 2 - k:)’

For the case where k,

=

(258)

0, it takes a particularly compact form

0, = ( W ’ E K )

2

- k4

= ~ 4 & 2 ( k 2-

=

+ ( W ’ O E K )2 ,

-k2kz.

P2>

(259)

This is a very special case, and it corresponds to the situation where the only contributing mode has no z-directed propagation constant. When will this hold? Only in those cases when a single normal mode is required and it is zero. Zero perpendicular mode number means that effectively we are dropping the t-dependence in the problem, exactly what we are trying to avoid in the three-dimensional problem. However, use of the k, = 0 limiting case will prove useful in checking, later, some of the properties of the three-dimensional formulation. Generally, k, f 0, and the only question remaining is whether the problem will require a superposition or k, eigenvalues (to be discussed in Section XX), or allow us to select one dominant perpendicular mode. For the ferrite material sandwiched between two highly conductive plates, one of which is the ground plane, we expect the perpendicular modes to act as if they are constrained to exist in a cavity (a one-dimensional cavity). This is made all the more plausible and acceptable as a picture since the entire circulator structure is sort of a magnetic bottle (cavity) in regard to the vertical azimuthal side wall, even if it is somewhat leaky. Of course, the leaky side wall leads to the desired circulator action. One can show that the general form of 0, agrees with Van Trier (1952-1954) because the limiting case of that work for only gyromagnetic anisotropy is A = -D,, his notation. In Van Trier (1952-19541, both the permeability and the permittivity tensors are assumed to be the result of a z-directed dc magnetic field, as would occur for a ferrite material and a plasma medium.

136

CLIFFORD M. KROWNE

Examination of the F#,, coefficients shows that many of them are positively or negatively equated to each other:

(260e)

-H E

= Hi: = iwg(k: - k 2 ) Ds- ' = u .

(260f)

The last equalities in each of these expressions enable a compact notation to be used later in the analysis, and also allow a correlation between the work here and that in Van Trier (1952-1954). There is an exact correspondence between the p, F, q, s, t, u associations here and those found in Van Trier (1952-1954). The bar notation over r , i., is done here to avoid confusion with the retention of the radial notation originally employed for the two-dimensional problem. Returning to the third equations of the curlE and curlH formulas, (246c) and ( 2 4 8 ~which )~ allow the determination of the perpendicular field components H, and E, in terms of the transverse (in plane cylindrical) field components, and inserting the transverse field components from (256a)-(256d),

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

137

Notice that the second equation may be obtained from the first by the procedure ELL

+ HLi : i , j = r , 4 ,

k,m

= e,h,

k: e

+

h

(263)

and - p o H z + EE, for changing the right-hand side of (261). Using the simplifying properties of the coefficients in (260a)-(260f), the mixed partial derivative terms become zero in both curl equations, and can be expressed in the streamlined forms

dE, 1 d2E, z(r7) 73-77]

1 d Hi?:[;

+

We readily recognize that the two-dimensional Laplacian operator (transverse field operator) acts on the perpendicular field components in (264) and (265). Defining it as

the previous two curl equations can be stated in the most simple forms

+ E$ V:H, + i w p o H , = 0, V:E, + H,$ V:H, - iwEE, = 0.

E$ V:E, H$

(267a) (26%)

It would be most advantageous to reduce the number of transverse Laplacian operators in each equation to one, in effect then mimicking an ordinary Helmholtz equation in cylindrical coordinates. This can easily be accomplished by selecting each V:Fz, where F = E or H , as the unknowns in a 2 X 2 system of equations. The solution to this system is (268a)

138

CLIFFORD M. KROWNE

-i upoH, i WEE,

(I

(268b)

Expanding the determinants in (268) and (269) gives us the desired two coupled equations, each with a single transverse Laplacian operator:

V;E,

iWE i +E g E , + -H UP0

DV

Q;H, -

DV

iWE E

DV

~ H =, 0,

i @Po

~ E -, -H,$;H= 0"

- H&rE&r = 42 DV - E&rH'$r ee hh he eh

=

- us.

0,

(270a)

(270b) (271)

The last equality in the transverse Laplacian determinant came about by invoking (260a)-(260f). The coupled radial Helmholtz equations can be streamlined further by defining (272a) (272b) (272c)

d

=

iWE --E&r=

DV

ee

i osq

42-us'

(272d)

The final coupled form of the Helmholtz equations is

V;E,

+ aE, + bH, = 0,

(273a)

V;H,

+ cH, + dE, = 0.

(273b)

Enlisting (260a)-(260f), D v can be evaluated in terms of the basic propagation constants of the device, allowing a, b, c, d in (272aH272d) to also

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

139

be simplified and similarly evaluated: D,

=

k2Ds,

a

= k: -

b

=

c

=

(274) (275a)

k:, K

- i o p o k , -,

(275b)

P

PO

-(k2

- kz),

(275c)

P K

d =imk, -

(275d)

P

XIX. DIAGONALIZATION OF THREE-DIMENSIONAL GOVERNING EQUATIONS The two governing equations may be written as

+ aE, + bH, = 0, V?H, + cH, + dE, = 0. V?E,

(273a) (273b)

These two equations link the E,- and H,-field components together and disallow the construction of TM, and TE, modes. In fact, the coupled nature of (273a) and (273b) disallow any TM, or TE, modes where i = x, y , z (to be covered in Section XXII). Furthermore, because of this coupled nature, no coordinate transformation will yield a new ith coordinate where TM, or TE, properties occur. Thus a completely new field behavior is admitted when the theory is upgraded from two or three dimensions. In the two-dimensional theory, E, existed, but H, did not. Now H, is present, as are all the other field components. That is, the theory has gone from a three-component theory utilizing E,, H,, and H, to a six-component theory utilizing (E,, E,, E,) and ( H , , H,, H,). Sometimes a great advantage in theoretical analysis or numerical evaluation results from diagonalization procedures. That is the case here as will become evident when the procedure allows much simpler separated equations to be obtained. The new governing equations will have familiar properties which can be exploited to find their solutions and, eventually, the general solution of the entire field problem. We start by rearranging the second equation (273b). Now the two can be written as

+ aE, + bH, = 0, V?H, + dE, + cH, = 0. V?E,

(276a) (276b)

140

CLIFFORD M. KROWNE

With this rearrangement, we recognize that (276a) and (276b) can be recast in matrix form =

0.

(277)

Coupling between field equations has been examined before in the context of different materials including uniaxial and biaxial dieledrics (Krowne, 1984b). The transverse Laplacian operator V: acting in the ( x , y ) - or ( r , 4)-plane may be removed from the leftmost vector in (2771, giving

v:[ H, E , ] + ["d

"I[

H,

c

=

0.

The vector with the components (E,, H,) as well as the matrix multiplying the vector can be defined as

allowing the single governing equation to be stated in the compact form V , ~ F+ M F

=

0.

(281) The way to determine if this equation may benefit from diagonalization, if it is possible, is to transform the entire equation into another transformed coordinate system. This is done by multiplying (281) from the left by the inverse of the transforming matrix H - ' , where H is the transforming matrix: H-' V:F + H-'MF = 0, (282) H-' V:F

+ H-'MHH-'F

=

0.

(283) Here the identity matrix I = H * H-' has been inserted between M and F in the second form of the governing equation. Noting that H-' can be pulled through the V; operator and grouping terms together, we find that V?(H-'F)

+ (H-'MH)(H-'F) V?F' + M'F'

= 0,

(284)

= 0,

(285)

where the new transformed vector field and matrix are given by

F'

=W

'F,

M' = H - ~ M H .

(286) (287)

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

141

For the matrix M,there exists an eigenequation Me

(288) For an N, x N, square matrix M,there will be N, eigenvalues A = hi, i = 1,2,. ..,N,, with N, associated eigenvectors e = ei, i = 1,2,. .. ,N,. There is no problem with diagonalization as long as all the eigenvalues are distinct (Protter and Morrey, 1964). That will turn out to be the case here as long as the substrate thickness is finite. Therefore, a more proper statement of (288) is =

he.

Mei = hiei, i = 1,2,..., N,. (289) Examination of (289) for M shows that it is a 2 X 2 matrix describing a 2 X 2 system having N, = 2. With distinct eigenvalues, it is known that the transformed matrix M’ will possess diagonal elements equal to the M eigenvalues of (289). The eigenvectors of the transformed system will only have one nonzero value in their components, the location of that nonzero entry being precisely equal to the ordering of the eigenvalues. That same ordering is also seen in the entries of the transformed matrix M’.Finally, it is possible to write down the transformed matrix once the eigenvectors have been found since

H

= [ei

e2

eN,

1

*

Because each eigenvector e, has N, elements, H must be of size N, x N,, as we expect it must be if it is to transform the original matrix of the system M by the similarity transformation given in (287). So, the job at hand is to find the eigenvalues of M,then obtain its eigenvectors, and lastly find the transforming or mapping matrix H. First start with the eigenvalue determination: Mei

- hiei =

0,

[M-hi]ei=O,

i = 1 , 2 ,..., N,.

Since (291) is a linear homogeneous equation, it can only have a solution if the characteristic equation i = 1 , 2 ,..., N,, (292) det[M-Ai]=O, holds for each given eigenvalue A = hi. Substituting (280) for M into (2921, we get

(a - A j ) ( c - A i ) - db = 0,

i

=

1,2.

(293)

142

CLIFFORD M. KROWNE

Note that this is of the same general form as that found for chiral bi-isotropic media with unequal cross-coupling coefficients (Lakhtakia, 1994). Of course, that is where the similarity ends, because here the actual elements of M are vastly different for a ferrite medium versus a chiral medium! Equation (293) is a quadratic equation describing the medium’s properties, and so it will have precisely the required two eigenvalues expected for the system. They are found by solving the equation A? - ( a

+ c ) A i + (ac - d b ) = 0 ,

i

=

1,2.

(294)

The solution of (294) is 1 I/’ - [ ( a + c)’ - 4(ac - d b ) ] . (295) 2 2 In order to identify and order the two eigenvalues, we first need to define A=

(a+c)

( a + c)

A =

R

2

= :[(a

’

+ c)’

- ~ ( U-Cd b ) ]

I/’

.

(297) The information needed to find A and the radical R, is found by referring to the previous section. By (273,

t)1, 2

P

( k z - k:)(k’ - k : ) - k’k:(

(299)

where we state for convenient reference again K

b

=

-iWpokz-,

(275b)

P

c

=

PO

-(k2

- k:),

(275c)

P

d

=

iwk,

K

-. P

In these equations recall that k 2 = o‘ep, ki

=

k:

= w’ep,,

w2ep0,

(275d)

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

143

where /A2

Pe =

-

K2

P

Using A and R in (299, h=A&R, and we may order the eigenvalue solutions according to

(302)

Al=A+R,

(303a)

h2=A-R.

(303b)

Eigenvectors are found using (292) for an N,

=

2 system:

[ M - A,]e, = 0,

i = 1,2. (304) Writing this equation out explicitly, we obtain the general eigenvector equations

+ m12ei2= 0,

(305a)

m21eil+ ( m 2 2- hi)ei2= 0 .

(305b)

(mil

-

hi)eil

Substituting into (305) the matrix elements of M found by using (280) gives (306a) ( a - h i ) e j 1+ b e i 2 = 0, de,,

+ ( c - hi)ei2= 0.

(306b)

In these types of homogeneous equations, the elements of the eigenvalue vectors can only be determined to within an arbitrary scaling constant or, stated somewhat differently, in reference to one component selected out of the N,,, available. Thus the first equation yields

and the second equation yields

It is easy to see that if the two eil’sare equated, the original determinantal characteristic equation is obtained. We can also easily show that if the first equation is used to describe the eigenvalue solutions, it indeed contains the second equation form implicitly since the second form can be obtained from the first one by a simple procedure. Substituting for the eigenvalue A

144

CLIFFORD M. KROWNE

and u and b in (3071,

Multiplying the denominator and numerator of this expression by [ ] f R, where the quantity within square brackets is just that found in the

preceding equations, el

=

-

iWPOkz(K/P) [ 1 fR e2 [k,Z-k:-A] T R [ ] f R - i ( 4 % M U P ) ( - 1)"

[kf

1 k R)

-A]' - R2

- kt

e2

and inserting the expressions for A and R into the denominator gives, after some tedious algebra, the second eigenvector equation (308). This is the expected result and constitutes a very important check on our analysis. The first and second eigenvectors can now be put down using the first eigenvector relationship (307) and the eigenvalues in (303a) and (303b):

1 Al;u]ell=[

el=[:::]=[

1 A+;-a]a,

(311a)

e 2 = [ ~ ~ ~ ] = [1A 2 i u ] e 2 1 =1[ A - R - a

p.

(311b)

b

Here (Y and p are arbitrary values, independent of one another. It is also possible to factor out the second eigenvector components in these expressions, obtaining

e2 =

[

=

[

b A, - a 1

'

D22

=

b A-R-a 1

p.

(312b)

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

145

However, this is less ideal than the first set of expressions since a radical occurs in the denominators. It is also possible to use the second eigenvector relationship instead of the first. Factoring out the first eigenvector components, 1

el

=

[z:i]=[ 1 ]el,=[ A, - c

1 1 ]a, A+R-c

1

(313a)

1

A-R-c

or, factoring out the second vector components,

A, - c (314a)

e2 =

[:;I

=

[

A, - c =

[

]

A-R-c

P.

(314)

In this case, because of the radical position, these are the preferable expressions. Mapping matrix H for out 2 X 2 system is given from (290) by H = [el e,]. Retrieving the eigenvector expressions in (311a) and (311b), a

A+R-a a b

P A-R-a b

P

1

.

(315)

(316)

Although the H transforming matrix itself is used to reverse-map from the transformed space (or primed space) to the original space, it is the inverse which is required to map out of the old space. H's inverse is found by constructing a cofactor matrix H c out of the original matrix, taking its transpose, and dividing by its determinant:

Sometimes other terminologies are used for the numerator in this con-

146

CLIFFORD M. KROWNE

struction, like assigning the name adjoint after the transpose operation. Here we work with H' directly and determine it on an element-byelement basis as HG = ( - l)i+jM.$, (318) where the minor M;, the same as used in the determinantal evaluation of H, is the determinant of the ( N , - 1 ) x ( N , - 1) submatrix of H, obtained by deleting its ith row and jth column. For a general 2 X 2 matrix H, its cofactor matrix is Hc =

[

h22

-h21

(319)

h,,]

- 4 2

and its transpose is

[

-h12

h22

( H c ) T= -h2,

h,,].

Lastly, we note that the denominator of (317), det H, is det H = hl,h22- h I 2 h 2 , . (321) Enlisting (320), the inverse of the transforming matrix is set down as A-R-a P 1 B H-l= - A +G ; - a f[ f

-P (322)

and its determinant as

ffP

detH= -2-R.

(323)

b

This inverse can be used to find the transformed eigenvectors spanning the new space, as well as finding the transformed matrix M'. Consider first the transformed eigenvectors e:. They can be found by employing the general formula (286) for mapping of a vector F from the old to the new space. Here we set F = e:, where i = 1,2, e; = H - l e ,

-

1

ffP

-2-R b

[

A-R-a A : r - a f fP

.j"l..j

-P

ff

-

(324)

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

r

-4

A-R-a fl

=

147

1

*I![

(325)

Both of these eigenvector results are expected since we have purposely constructed a transforming matrix to put us into a principal axis system. Notice that the results are completely independent of the coefficient choices a and P. This property is not true for an arbitrary vector undergoing the H mapping, and we will have some comments about this fact later. Go back to the equation giving the transformed matrix M': M'

= H-%H

A-R-a

P

1

-P

-

P

ff

-

A+R-a b

[

ff

A-R-a b

P

O]. 0

A2

The choice of a similarity transformation for mapping M to M' assured us of this result for distinct eigenvalues. Transformed field vector F' in the new space, which gives us the mapped z-component fields, is found using (286):

-

1

ffP

-2-R b

I

A-R-a P b A+R-a ff b

148

CLIFFORD M. KROWNE

Labeling the components of the new vector field as

we find that they can be expressed individually as (329a) F2=

--

--

(A

+ R - a ) E , + H,

(329b)

To find the reverse mapping, that is, the original field in terms of the transformed field, we multiply (286) by H, obtaining F = HF'. (330) This formula will give the original field components in terms of the transformed field components by merely inserting the mapping matrix H:

Thus the original field components are

E,

=

aF;

+ PF;,

A, - a

H,=cu-

b

A, - a F ; + P b F; .

(332a) (332b)

In order to reduce the number of unknown quantities in the problem solution and because they are free to choose depending upon the circumstances, a and p are selected to be ff=p=1.

(333)

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

149

This choice makes the two t-component fields

E,

= Fi

+ F;,

(334a)

A, - a A, - a -Fi + -F; . (334b) b b It is to be noted that the preceding rigorous method of determining the correct transformed space in which to work with the transformed governing equations does have some reference point in earlier literature. Although the previous work in the early 1950s used an ad hoc approach for getting at the results in some incomplete form, it was enough at that time to allow studies of guided wave propagation in gyromagnetic media (Kales, 1952, 1953) or simultaneous gyromagnetic and gyroelectric media (Van Trier, 1952-1954). The fundamental work of Kales (Kales, 1952; Kales et al., 1953) and Van Trier (1952-1954) was also quoted in texts dealing with the basic nature of ferrites in that decade, too (Soohoo, 1960; Lax and Button, 1962). Van Trier’s work, which cited Kales (19521, set H,

=

E, = (PI + (P2, H* =g,(P, + g z ( P 2 .

(335a) (335b)

Comparing this last set of equations and those given in (334a) and (334b) permits us to make the connection between Van Trier’s special case and notation and the general theory:

F;,

(336a)

9 2 = F;,

(336b)

( ~ = 1

(337a)

Kales (1952) uses a set of four constants to scale the transformed fields: Ez = P l U l

+P2U2,

(338a)

Hz

+ 42u2.

(338b)

= 414

Comparing this pair of equation to (334a) and (334b) allows us to relate Kales’s specialized case to the general theory: u, = F ; ,

(339a)

u2 = F ; ,

(339b)

150

CLIFFORD M. KROWNE p1=

ff7

(340a)

P2 =

P7

(340b)

A, - a

41 =

7

q2

7 P.

(341a)

ff,

A, - a

=

(341b)

When we examine Kales' work further and note that he selects p i and qi to be P1 = ' 1 7 (342a) (342b)

P2 = s 2 ,

(343a) (343b) it becomes clear that his particular selection corresponds to storing the eigenvalues of the characteristic determinantal equation (of M ) as the coefficients of the transformed variables making up the electric field component E,. Therefore, we see that (344a)

s1 =

s,

=

(344b)

A,.

We also finally note that the extensive paper of Suhl and Walker (1954a) was aware of earlier work of Kales (1953) and Gamo (1953) as well as their own work (Suhl and Walker, 19521, all of which looked at various ferrite and plasma (the gyroelectric case) effects. These researchers came out at the same time with two other related papers on transverse magnetization (Suhl and Walker, 1954b) and perturbation approaches (Suhl and Walker, 1954~). Now let us write the final transformed governing equation (285) explicitly, with the specific form of M' from (326) inserted into it:

V;F'

+

["

0

'IF' A2

=

0.

(345)

Since we were able to fully diagonalize M, it is useful to go back into component form and put (328) back into two governing equations, which are now de-coupled by the very nature of the theoretical process

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

151

undertaken: V,‘F; V,’F;

+ h,F; = 0, + A,F; = 0 .

(346a) (346b)

These equations can be solved independently, just as one would solve a single Helmholtz equation in cylindrical coordinates. Thus the approach utilized for the two-dimensional circulator problem can be used to good advantage here in this regard. Of course, the actual field component solutions in the z-direction must be found by a proper linear superposition given by the reverse transformations in (332a) and (332b) for E, and H,. From these two properly determined field component solutions, all of the other transverse field components in the plane of the circulator can also be acquired, as shown in Section XVIII. The radial solution parts of F;, i = 1,2, of the decoupled Helmholtz equations use the square root of the factor on the transformed field components as arguments of the Bessel function solutions. Namely, they use the M eigenvalues, the diagonal elements of the M’ matrix, and the square root of the V,‘ operator eigenvalues. Thus we can write the equations as

+ u:F; V,‘F; + u:F; V,’F;

=

0,

(347a)

=

0.

(34%)

The 0;’ values store the outward and inward propagating radial waves for the ith type of radial wave. That is, ui= k f i

i

=

1,2,

( 348)

where the actual linearly independent Bessel function solutions properly represent both signs in (3471, so that only the unique assignments ui=

6,i

=

1,2,

(349)

for the two radial modes need be taken.

xx.THREE-DIMENSIONAL CHARACTERISTIC EQUATION THROUGH REC~ANGULAR COORDINATE FORMULATION It is possible to use a spectral domain approach for obtaining the relationship between the radial propagation constants a,,i = 1,2, and the perpendicular propagation constant k,. Start with Maxwell’s equations in time-

152

CLIFFORD M. KROWNE

harmonic form (Krowne, 1984a) assuming an exp(i ot): V X E = -iwB, V

X

H

=

( 12)

iwD.

(13)

In order to expand the curl operator generating rectangular coordinates, we use for an arbitrary vector A the expansion

V x A =

-

JAY dAx dAz ( dAz a y- a ,) (x -ax) +

+

dA, dAx (ax - a,) . (350)

The result of using (350) in (12) and (13) is dE,

dE,

(dy- x) = -iwBx,

(351a) (351b)

(

dE,

-

z) dEx

-iwB,

(351c)

= iwDx,

(352a)

= iwD,,,

(352b)

=

for the curl E equation, and

dH,

dH,

(352c)

for the curlH equation. These six component equations can be put

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

153

together to form on equation:

0

0 d

-

0

dz d

--

0 d

--

dX

0

dz

0 d

JY

ax

0

0

0

0

0

0

dY d

d

--

Now we note that any arbitrary function f ( x , y ) in two-dimensional space can be represented by a double-Fourier series:

f(k,,k,)

=

jm lm f(x,y)e-ikxxe-ikyydudy. -m

(354b)

--m

Furthermore, for a three-dimensional problem, this transform pair still applies to the two-dimensional surface ( x , y ) : 1 f(k , ,k,, z)eikxxeikyYdk, dk, , (355a) z) = 2 (2r)

z ) = Jm -m

jm f( x , y , z)e-ikx*e-ikyydudy. --m

(355b)

154

CLIFFORD M. KROWNE

With these transform pairs, the 3 X 3 partial differential operator matrix found in (353) can be replaced as follows: 0

L,

d

=

dz d --

dy

a

d

--

-

0

d --

az

d

dx

dy

0

--

d dz

ik,

0

-ik,

ik,

0

d dz -ik,

dX

0

.

(356)

The 6 x 6 constitutive tensor M describing a general medium is

3.

M=

(357)

For a pure ferrite medium which we are considering here, the off-diagonal optical activity subtensor elements are

where I is the 3 X 3 identity tensor

[b

I= 0

:I

1 0 .

Thus we may now write the constitutive relationship between the fields in the Fourier transform (spectral) domain

qR =aL.

(361)

Here the field vectors in the spectral domain are defined as

VL = Ex E, Ez

[

D, D,

H,

H,

Dz B, By

'I

,

(362a)

I' .

(362b)

Hz B,

With the foregoing information, the constitutive relationship can be ex-

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

155

pressed with all the field components directly shown:

(We assume that the same relationship form holds in the spectral domain.) Fourier transforming the :ystem equation (353) and employing the general macroscopic tensor M,we obtain 0

1

d

--

0

0

dz

d

0

0

-

0 d

0

- ik,

ikx

dz

- ik,

0

0

0

ikx

0

0

-ikx

0

0

0

0

dz

=i

vL3 vL6,

o

m12

m13

m14

m15

m16

m21

m22

m23

m24

m25

m26

m31

m32

m33

m34

m35

m36

m41

m42

m43

m44

m45

m46

m51

m52

m53

m54

m55

m56

- m61

m62

m63

m64

m65

m66

-

Ex

EY 2'

.

(364)

fix

-

fi, fiz-

and vector components of f L (i.e.ygz and HzIyare algebraically expressible in terms of the other components using rows 3 and 6 of the previous system equation:

156

CLIFFORD M. KROWNE 6

i k y i x- i k x i y = i o

m6ifLi.

(365b)

i= 1

Written out explicitly. These two equations appear as -

ikyfix+ ikxHy

m31ix + m 3 2 i y+ m 3 3 i z+ m34fix+ m35fiy+ m , , f i Z ] , (366a) ikY& - i k x i y =

1

i o m 6 1 i x+ m 6 2 ~+ym 6 3 i z+ m6,fi,

+ m6,fiy+ m66fiZ]. (366b)

z and f i z components and placing them on the Extracting out the i left-hand sides of the two equations: m33Ez

+ m36Hz =

- [ m 3 , i x + m 3 2 i y+ (m34+ %o) f i x

+ ( m 3 5-

2)fiy], (367a)

= -

[(

m61-

:)i, +

(m62+ :)iY + m64fix+ m 6 5 f i y ] .

Let us define

The z-component fields

Ezand f i z are solvable from the 2 X 2 system

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

as

(D36= m33m66- m36m63in the preceding) or, in expanded form, E

=-([

1

D36

-m32m66

+

[

-m66(

m34

+ m36

):

+

- m36m64]Hx

These two z-component equations may be compacted by the formulas

157

158

CLIFFORD M. KROWNE

= a61VL1 -k a62fL2

+ a64fL.4 + a65VL5

6

=

c

a6j(1

- s3,j)(1

-

a6,j)VLj*

(374)

j= 1

Here the aij coefficients are defined by =

a; j -,

(375)

36

=

al,j

-,

(377)

36

abl = m31m63

- m33

(378a)

ak2 = m32m63

- m33

(378b) (378c)

a’ss

= m63( m35 -

2)

- m33m65’

(378d)

Returning to the system equation (3641, rows 1, 2, 4, and 5 are seen to be first-order linear differential equations. Let them be listed here

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

159

explicitly: dHy --

dz dHx

dfL5

+ ikYH2= - -+ ikyfL6= i w dz dfL4

-

- - ik,H,

=

d& --

+ ikx& =

c m2ifLi,

(379b)

i=l

dfL2

6

(379c) m4,fLi, dz i= 1 dfL1 6 - - + ikxfL3= i w m5ifLi. (3794

= --

dz

(379a)

6

dz

diy -- i k E

c mlifLi,

i= 1

-- ikxfL6= i w

dz

6

ikyfL3= i w

dz dz i= 1 Equations entirely in terms of the transverse field components Ey, I?x, and gycan be found by invoking (373) and (3741, giving the perpendicular components and f i z in terms of the transverse field components. When the perpendicular field components are eliminated from the system row equations 1, 2, 4,and 5, we find iw

dz

-=[

m13a31

- '61

+ m16a61 +

m13'32

+ m16'62

+

m13u34

+ m16'64

+

( (

kY

m12

- '62

m14

- '64

1 dHx

i w dz

m23'31

+ m26'61

+

w

w

1

(

+ m23a35+ m26u65+ m25+ '65

5)]fiY, (380b) w

160

CLIFFORD M. KROWNE

1dgy

= i w dz

[

m43a31

+ m46461 +

m43'32

+

1

m43u34

(

+ '31

m41

+ m46u62 +

(

+ m46u64 +

(

m42

+ '32 "w) ] ' y

m44

+ '34

")I.

w

m43u35+ m46u65+ m45 + a35")]fiyy (3 8 0 ~) w

--=[

1d i x m53'31

i w dz

+

[

+ m56'61

+

m53u32

+ m56'62(

m52

m53u34

+ m56'64

+

(

- '32 ' w) ] ' y m54

- '34

?)]fix

m53u35+ m56u65+ mS5- u35 ?)]I?..

(380d)

These four transverse component equations can be put into a much more streamlined form by defining (381a) (381b)

6 3 = m14

+ lt113'34

r&

+ m13u35+ '65

=mi5

+ '64(

m16

- :),

(381c) (381d)

for the first system equation row, or, put into a single equation, rAi = mi,

+ mI3u3,+

'60

(m16- ?).

(382)

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

161

For the second system equation row, rgl = m21

+ m23a31 + a61

(383a)

r!42 = m22

+ m23a32 + a62

(383b)

r;3 = m24

+ m23a34 + a64

(383c)

rj4 = m25

+ m23a35+ a65

(383d)

r;i = m 2 8

+ m23a38 + a68

or

For the third system equation row, (385a)

ri2 = m42 + m46a62+ a32

(385b)

ri3 = m44+ m46a64+ a34

(38%)

ri4 = m45 + m46a65+ a35

(385d)

or

rii = m4e + m46a68+ a3e For the fourth, or last system equation row, (387a)

(38%)

162

CLIFFORD M. KROWNE

(387c) (387d) or

In all of the preceding condensed equations, the function 8 of index i is given as 3i - 2 , i 2

=

even,

, i

=

odd,

2

i

=

1,2,3,4.

(389)

With all of these rji definitions, the system equation in rectangular coordinates can be restated as

-;I.

It may be slightly simplified after examining the prefactor matrix -1

s*=[

0

0

0

0

:;; 0

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

163

Consider the product of this matrix with itself

1

0

1

0

0

1

0

0

0

0

0

0

0

0

0

0

(392) which shows us that the product is merely the 4 X 4 identity matrix I. Thus we are led to multiply both sides of the revised system equation (390) by Sp to obtain the much simplified final descriptive equation

EX 1 d -i w -[H dzy

41 41

fix

-ril

.

4 2 4 2

6 3 4 3

4 4

-ri2

-ri3

-ri4

If we denote the old right-hand matrix by R' in (390), then R = SpR, and -ril -ri2 -ri3 -ri4

-ril

42

r;3

-ri2

-ri3

(393)

(394)

(395)

-ri4

We see that the relationship between the old and new R matrix elements is specified by r'I. . =

r;j, i -rij, i

2,3, = 1,4,

=

j

=

1,2,3,4.

(396)

Now define the transverse field vector

(397)

164

CLIFFORD M. KROWNE

With this definition, the final form of the descriptive system equation is

Next we will try to reduce the number of elements in R by using the subtensor characteristics peculiar to a material which only has permeability anisotropy. First, no special assumptions will be made regarding the type of magnetic anisotropy, thereby keeping the most generality, until we actually arrive at our special circulator case with the situation of z-directed dc bias field. Therefore, we have

2. = EZ

-

6=0 fit = 0

m12= mZ1= m13= m31= mZ3= m32 m,4 = m,,

-

=

0 , (399a)

= mI6 = mZ4= m25 = m26 = m34

(399b)

= m35= m36= 0 ,

m41= m42= m43= mS1= mS2= mS3= m61 (399c)

= m62= m63= 0,

ji = filled tensor.

(399d)

With this selection, D36

= m33m66

- m36m63

= m33m66,

(400)

a;, = 0,

(401a)

a;2 = 0,

(401b) (401c) (401d) (402a)

kx

ai2 = -m33 -,

(402b)

a& = -m33m64,

(402c)

-m33m65.

(402d)

w

a& =

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

165

We may define the values of a& and a& as elements in two constant vectors

When the matrix elements of R are determined, it is found that

m64

kx

m65

m66

IN

m66

m64

ky

m65

ky

m66

IN

m66

IN

kx

"

(405) Now let us treat the case where the dc bias magnetic field B, is parallel to the z-axis. In this case, the permeability tensor tremendously simplifies to -iK

0

m44

llZ4S

m46

(406) implying that m4b= mS6= mb4 = m65 = 0.

(407)

166

CLIFFORD M. KROWNE

Thus the R matrix reduces appreciably to

R=

0 0

0 0

.]

+;(y + kxkyl -

0

0

-iu

-7 3

0

0 0

where the second equality comes from making the correct tensor element substitutions. Also, 0 0

-po-

ky w

kxlr

pow

,

(409)

T

w

Solution of the compact system equation (398) has the form

+y( 2);

= e'k:z6+T(o)

(411)

in the mth layer, ordering layers from the bottom, for the ith normal z-directed eigenmode propagation constant k z . Here zk is the local

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

167

coordinate in the mth layer, and it is related to the z-coordinate by m-1

Z;

= Z-

C

hi,

j= 1

(412)

where hi is the thickness of the jth layer. Putting ~#(Y.zh)into the single governing differential equation (398) describing the biased ferrite circulator device, we find that [ ~ Z - R ] # J ~ ( O= O)

(413)

specifies the normal mode eigenvector solutions to the problem. This equation is a very general result, and is not limited by our special ferrite case under consideration. Equation (4131, a homogeneous equation, can only have a solution if its determinant is zero, that is, if - R] = 0.

det[

(414)

Let us reverse the order of the terms in the determinantal equation (4141, preparatory to inserting R from (408). This reversal will insure the least amount of manipulation of the large number of matrix elements involved in the algebra to ensue:

];

det[R -

=

0.

(415)

Putting R into this expression yields the determinant formula to be reduced:

168

CLIFFORD M. KROWNE

Expanding the determinant produces the final form

This can be made much more transparent by defining (300)

k2 = w 2 ~ p , k; = kz + ky”.

(418)

Therefore,

(

:[ Lo] I

[kzI4-2 k 2 - -

1+-

[

1

k: [ k z ] ’ + k 2 - - k ? Lo

[k2-k?]

This is a quartic equation in k;, and so has four eigenvalues ordered as 1, 2, 3, 4. But because it can be written as a quadratic equation in [k,:!]’,we expect two sets of eigenvalues, each set possessing propagation constants which are the negatives of each other, corresponding to forward and backward propagating waves. Thus the solution to (419) is 1 [ k z I 2 =k2 - 2

i

=

The two sets of solutions are expressed as [kz]=

.[x’-

k[l

+i l k :

i

=

1,2, (421a)

169

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

[ k a ] = f [ k 2- f[l

+ $]k: 1/2 9

i

=

3,4. (421b)

There are some special cases which reduce the [k,:!]’ equation solution (420) still further. For example, if the first two diagonal permeability tensor elements p are equal to the free-space, third diagonal permeability tensor element po,then

, .

[k,:! ]’= k 2 - k: f k

It is unlikely that the ratio P EL=-

(423)

PO

is exactlv iinitv. esneciallv nver a freoiiencnr rnnue. Hnwever. this is o n e of

examining waveguiding propagation. If ji bias is turned off, making K = 0, ..

If it is further stipulated that ji the familiar result

=

[ k z ]2

#

1 and the magnetic dc field ..

1

1

1, then this reduces even more to give = k2 -

k:,

(425)

with the provision that this equation really contains implicitly double degeneracy. The double degeneracy explicitly appears in the original quartic equation (419) as is easily verified by substituting the special permeability tensor values. In order to demonstrate that the general characteristic equation (419) for k z is exactly equivalent to that derived earlier giving the dependence of u 2 on k: [see (295) and (34911, replace

kf = k:

+ ky’ = u 2

everywhere in the characteristic equation, and solve it for ing solution is identical to that found previously.

(426) u 2 .The

result-

170

CLIFFORD M. KROWNE

m. TRANSVERSE FIELDSIN

THE

THREE-DIMENSIONAL MODEL

It is extremely useful to streamline the transverse field formulas given previously in (256) [See Section XVIII]. This can be done by enlisting (260a)-(260f) for p , T, q, s, t , u assignments to F& coefficients:

dHz 1 dH, 1 dE, Er=s-+F+q--+p-, r d4 r d4 dr

dEz ar

(427a)

dHz 1 dE, dEz E = T - - 1 dH, - s + p -4-, + r 84 r 84 dr dr

( 42%)

1 dHz dHz 1 dEz +u--+t-, r 34 + P a r r d4

(427c)

H,=q-

1 dHz H4=p--r 84

dHz

1 dEz

+t---u-. ‘ 7 r d 4

dEz dr

dE, dr

(427d)

Compact operator forms exist for the transverse E,-and H,-fields if the total fields are reconstructed from (427aH427d):

E, = ;Er

+ c$E+

= TV,H, =

H,

+ p V I E z - s.2

Vl(FHz + pE,)

= ?H,

(428)

-2X

X

V,Hz - 92 X V‘E,

V,( sH,

+ qEz),

+ c$H4

+ t VIEz- 92 X V,H, - u2. X VIEz = V,( pHz + tE,) - 2 X V,(qHz + uE,).

(429) (430)

=pV,H,

(431) Here the two-dimensional gradient operator in cylindrical coordinates is

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

171

given by

and its cross-product with the perpendicular z-direction is given by

The compact forms for E, and H,may be useful for deriving certain field properties of the anisotropic medium. In order to express the transverse fields in terms of the fundamental transformed field F’, the formulas for E, and H, in (334a) and (334b) of Section XIX must be inserted into (429) and (431): 1 b

E, = - V , [ ( p b- r a ) { F ; + F ; } -

1 b

+ r { A I F ; + A2F;}]

- 2 X V,[( q b - S U ) { F; + F;}

1 b

H,= - V,[(tb- P U ) { F ; 1

(434)

+ F;} + p { A , F ; + A,F;}]

- - 2 X V, [ ( ub - qa) { F; b

+ S{ A1F; + A , F ; } ] ,

+ Fi} + q{ AIF; + A2 F;} ] .

(435)

The separation into 4’ and A&’ parts may have beneficial results if some groupings go to zero, but alongside the need to separate mode-type terms, especially when we construct actual field components inside the circulator disk and annulus regions in future sections, what we do here is done more for completeness than absolute necessity. It will also allow the reader to make comparisons to Van Trier (1952-1954) if desired. Anyway, the coefficient groupings are found to be p b - r u =0 s~ [ k 2 ( ~ ) (l+;)(k2-kz)], ’ -

(436a)

(436c)

172

CLIFFORD M. KROWNE

ub - qa

=

(ti[

*(kZ - k : )

kz k2 --

1

+ ( k z - k’) .

P

0 s

(436d)

None of these constants is identically zero. However, if the fact that the sum of the M’ diagonal elements eliminates the radical term, A,

+ A,

=a

+ C,

(437)

as seen by examining (294) and (295) from Section XIX, is used to replace a in H, in (3341, giving

H,

c - A,

c - A,

=

F; + -F; , b b

(438)

then El and HI can be reevaluated. Some constant coefficient groupings will be seen to then vastly simplify. Notice that H, given here now has the second eigenvalue associated with the first-mode field vector component and the first eigenvalue associated with the second-mode field vector component, exactly the reverse of what occurred for the old formula: E,

=

1 - V,[(p b b 1

- -2 X

b

H,

=

- rc){F;

+ F i ) - r{ A,F; + A,F;)]

V, [ ( q b - SC) { Fi

+ F;)

- S{ A,F;

1 - V l [ ( t b+ p ~ ) { F + i Fi} -p{A,F; b 1 --2X b

+ A, F ; ) ] ,

(439)

+ AIF;}]

V , [ ( u b - q c ) ( F ; + F i ) -q{A,F;+A,Fi)],

(440)

where the groupings take on the values

+ rc = 0 , qb + sc = - i m p o ,

pb

tb

PO + pc = ik, , P

ub + qc

=

0.

(441a) (441b) (441c) (441d)

Placing the new constant groupings into the El and H,equations (439)

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

173

and (4401,

E, =

r b

- - V,[ A,F;

+ A,F;]

+ F i } - p ( A,F; + A,F;}

+ -4b2

X

V,[ A,F;

+ AIF;].

(443)

Returning to the individual transverse field component formulas (427a)-(427d) and using the second grouping constants simplification, and (438) for H,,

E

F d ( A2Fi

=

--

=

F --

+ AIF;) +-qb + sc

d(

F;

+ F;)

b dr br J4 +-p b +b r c d ( F ;dr+ F ; ) - -brs d ( A , F { + A , F i ) a4 b

d(

A,F;

+ h,F;)

dr

--i w p o d ( F ; br

+ F;)

d4

--s d ( A,Fi br

+ AIF;) Y

d4

(444a)

174 H4

CLIFFORD M. KROWNE

=--

ub

+ qc

d(F;

b

+ F ; ) +-tb + p c

dr q d ( h2Fi + h,F;)

+-

b

ikzpo d(Ff

=-

brCL

dr

a+

+ F;)

br

a+

p a( A,Ff

+ h,F;)

-br

+ F ; ) + -q

d(F;

d+

d(h,F;

b

+ h,F;) dr

p d(h,F;

-br

+ A,F;)

a+ (445b)

Finally, let us regroup the previous expressions according to their modal character, since when we develop the field relationships, it will be the modal coefficients which will become important, not the eigenvalue associations, and these are stored in Fi: FA, dF; impo + sh, dF; FA, dF; impo + sh, dF; Er = - - - b dr br d+ b dr br d+ ’ (446a) FA2 dFf

E ----+ 4br d+

impo + s h , d F ;

b

dr

FA, dF; br a+

+ impob+ s h ,

dF;

dr ’ (446b)

(447a)

XXII. NONEXISTENCEOF TE, TM,

AND TEM MODES IN THE THREE-DIMENSIONAL MODEL

Whether or not TE, TM, or TEM modes can exist is such an important subject, that a little space is devoted to it in this section. A relatively straightforward way to treat the question of simplified mode existence is to return to the undiagonalized governing equations, first presented in Sec-

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

tion XVIII: V:E,

V:H,

+ aE, + bH, = 0 , + cH, + dE, = 0.

175

(273a) (273b)

A quick reference to (273a) and (273b) shows that neither one of the cross-coupling constants between the two equations, b and d , is zero if k , # 0. Since this is generally the case in our three-dimensional model, these two wave equations must be coupled. It is from the strict coupling of these two equations that the impossibility of simplified modes arises. Let us consider the TE, mode situation first. This requires E, = 0, making bH, = 0, (448a)

V:H,

+ cH, = 0.

(448b)

By virtue of b f 0, H, = 0 from the first of this pair of formulas, with the second becoming identically satisfied because of the first condition. Thus we see that in order to have a null component of the electric field in the z-direction, a null component of the magnetic field must also hold. What about the possibility of a TM, mode existing? The argument for it goes the same way as for the TE, case, merely replacing the constant b with the constant d . That is, H, = 0 implies dE, = 0, (449) and again it is observed that both z-component fields must be zero. So, the final basic question remaining is this, can a TEM mode exist? That is, with E, = 0, H, = 0 (450) can a nontrivial solution be found with transverse field components present? These components are E, H,

V,(FH, + p E , ) - 2 X V,(sH, + qE,), = V,( pH, + tE,) - 2 X V,( qH, + u E Z ) . =

(429)

(431) The only remote possibility that E, and H, exist under rigid conditions (450) is for some of the constants r, p, s, q, t , u to approach the limit w. A careful examination of the expressions in Section XVIII show that the only hope of attaining such a limit resides in the common denominator 0,:

0,= - (k’ If the static limit w

+0

- k$

+

(wz&Ky.

(258)

is excluded, setting

- ( k 2 - k$

+

(02EK)

2

=

0,

(451)

176

CLIFFORD M. KROWNE

0,= 0 gives the double-valued equation, in terms of the off-diagonal permeability tensor element K ,

k 2 - k:

(452)

= ~ W ~ E K ,

where

k 2 = w2ep. Thus the system null determinant constraint (451) becomes k:

= w2e( p

T-

(453) Because K and p may take arbitrary values, it is extremely unlikely that k, will take the values in this equation. This is especially true for single k, mode of operation. However, even for the situation where many k, modes are superimposed, as will be addressed in Section XXIX, it is still very unlikely that (453) holds. This concludes our proof that TEM modes do not exist. K).

XXIII. THREE-DIMENSIONAL FIELDSIN INNER CYLINDER DISK

THE

To arrive at a complete solution for all the field components in the three-dimensional solid disk cylinder, of height h, we start with the basic z-component field solutions E, and H, found in Section XIX: (334a) E, = F; + F;, A, - a A, - a (334b) H,= -F; + -F; b b We recall in Section XXI that (334b) was changed into a form reversing the eigenvalue placement, which was a matter of convenience and of some value in creating some constant groupings which became zero because of the removal of the constant a from (334b) and its replacement by the constant c. Here the explicit appearance of constant c will be retained and the appearance of a suppressed. Thus, once again, the new form of (438) is invoked: c - A, c - A, H, = -F; + -F; . (438) b b On the issue of z-field dependence, that will be postponed until a later section, when definitive and essential knowledge about the transverse field components will have been acquired. However, it is mentioned here that &’ comes from the analysis of the decoupled, separable, Helmholtz govern-

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

177

ing equation in cylindrical coordinates ( r , +), and is therefore wholly independent of any z-directed dependence normal to the circulator surface. This dependence was chosen to vary as exp(ik,z), making the actual solution a superposition of the eigenmodes (discrete or continuous, depending on the nature of the boundary conditions) because of the linear property of the system equations (250). The equations controlling the dependence of FL are found in Section

XIX: V:F; V:F;

+ afF; = 0, + u:Fi = 0.

(347a) (347b)

Their solutions are F;

= uLoJn(u l , o r ) e i n 4 ,

(454a)

F;

= aioJn(c ~ ~ , ~ r ) e ' " @ ,

(454b)

Ci%O(r> = J n ( a l , o r ) ,

(455a)

C,Z,2,o(r)= Jn(

(455b)

~2,0r)*

The second set of equations is necessary to keep track of the many functions which will arise as a result of the specific field component constructions. The superscripts denote the field component direction and radial mode type. The subscripts denote the azimuthal mode order (as well as the integral Bessel function order), the field type ( E or H ) , the type of Bessel function (first [ a ] or second kind [ bl), and the inner cylindrical disk (0) or cylindrical annulus location (i). Due to azimuthal mode superposition, using the E, formula (365a) and the Fi expressions, m

EZO =

c

n=

[atoC,.T,o(r> + ~ ~ o c ~ e 2 a o ( r > 1 e ' n 4 .

(456)

-m

Similarly, using the H, formula, azimuthal mode superposition, and the Fi expressions, m

K a =

C n=

2 c22 nhua(r)]ein4,

[ a L a c L o ( r > + an0

(457)

--m

c - A, C,Zi,o(r) = b Jn(

U I , O ~ ) ~

(458a)

~ 2 , o r* )

(458b)

c - A1

C,'Lo(r>

=

b Jn(

178

CLIFFORD M. KROWNE

In order to carefully study the transverse field component formulas and to manipulate them into appropriate forms, we start out by restating them here for convenience: Er

FA, dF;

= - _ _ _ _

b

dr

FA, dF;

E -----+ + br 84

H,

=

iwpo

br iwpo

dF;

FA, dF;

b

d+

+ sh, dF; -.- FAl

b

dr

i k z ( P o / P ) -PA, JF; b dr

+ ikz( P O /bI I ) - p h i H+ =

+ sh,

dF;

-+ d+

+ sh,

br

dFi d4 ' (459a)

+

i w p o sh, dF; b dr ' (459b)

9 A 2 dF;

br

dF;

dr

i k z ( P o / P ) -PA, dF; br d+

br

dr

impo

dr#~

qh1 JF; br d+ '

+ -9A2 -b

(460a)

dF;

dr (460b)

Let us insert F; and F; in turn into each of these transverse field component expressions and study each resulting formula, streamlining it in the process and going on to the next transverse component expression until we have studied them all. For E,, for the nth azimuthal mode, using (459a),

The total transverse radial electric field solution can be tremendously abbreviated by adopting the notational definitions

179

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

r cLaao(r)

= -b

in '1,og2,0JA(~2,0~)- ~ ( i w +~"1,o)Jn(g2,or)* o (462b)

Eliminating these functions from the Ern, formula and performing azimuthal superposition, m

E,,

Ernoein4

=

n=

-m m

=

C

n= -m

[ a t o ~ ; L a o ( r+) a : , ~ ~ ; ~ , ( r ) ] e ~ ~ +(463) .

For E,, for the nth azimuthal mode, using (459b),

1

-

1b g2,,(iwp0 + ~A1,,)JA(u2,,r)]ein,.

(464)

The total transverse azimuthal electric field solution can be tremendously abbreviated by adopting the notational definitions 1 in? C,"b,,(r> = - AZ,OJn(~l,O~) + g1,oGwPo + sA*,o)JA(gl,o~), br (465a)

;

1 in? C,dkz,0(4 = - br Al,OJn(~Z,O~) +b %,o(iWPo + ~ A I , O ) J A ( % , O ~ ) * (465b) Eliminating these functions from the E,no formula and performing azimuthal superposition, m

180

CLIFFORD M. KROWNE

For H,, for the nth azimuthal mode, using (460a),

The total transverse radial magnetic field solution can be tremendously abbreviated by adopting the notational definitions

(468a)

(468b) Eliminating these functions from the Hrno formula and performing azimuthal superposition, m

For H+, for the nth azimuthal mode, using (460b),

(470) The total transverse azimuthal magnetic field solution can be tremen-

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

181

dously abbreviated by adopting the notational definitions

(471a)

(471b) formula and performing azi-

Eliminating these functions from the muthal superposition, m

XXIV. THREE-DIMENSIONAL FIELDS IN CYLINDRICAL ANNULI

THE

As in the cylinder disk, a combination of transformed field components must be used to construct the E, and H, z-directed field components. Restated from Section XXIII, (334a) and (4381,

E,

= Fi

+ F;,

(334a)

c - A2

-F; +

c - A,

F; . (438) b b are chosen to include both linearly independent Bessel Now, however, functions of the first and second kinds, because no singularity occurs for the Bessel function of the second kind N, due to an exclusion of the r = 0 point: F; = [ aXiJn(al, b,liNn(sly)] e i n 4 , (473a)

H,

=

+

F;

= [aXiJn(az,ir) +

bni N, ( a z , i r ) ] e i n + .

(473b)

For reasons similar to what was done for the inner cylinder disk, define

182

CLIFFORD M. KROWNE = Nn('l,ir),

(474c)

ci,26i(r)= N n ( ' 2 , i r ) *

(474d)

cibbi(r)

Due to azimuthal mode superposition, using the E, formula (334a) and the 6'expressions, m

E , ~=

C

n=

-m

[aAiciiai(r>+ b$i,!b;(r)

+ aiici,',;(r) + bni 2 C'2 nebi(r)]ein9. (475)

Similarly, using the H, formula, azimuthal mode superposition, and the F; expressions, m

Hzi

=

c

n=

[akiciiai(r>

1 cz' nhbi(r)

+ bni

-m

+

+ u i i C ~ , ' , i ( r ) b , ' i ~ i i ~ ~ ( r ) ] e ~ ~( 476) +,

(477a)

(477c) (477d) Also note for future reference that the indexing scheme employed here is meant to be consistent, whether one is inside the cylinder disk or any one of the annuli. Only for brevity of expressions, did we neglect to put the i = 0 index on the parameters depending upon the various material physical constants, for the cylinder disk region. In reality, we have u = u0,

b

=

b,,

c

=

c0,

d

(478)

= do,

T=T.,,

s =so, p =Po, q =qo. (479) Consider now the transverse field components. First treat E, by retrieving its formula ?A2 dF; i m p o + sh2 dF; FA, dF; i w p o sh, dF; Er = _ - - b dr br d+ b dr br 84 (466a)

+

183

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

and inserting the appropriate F; and F; equations from (473a) and (473b) for the nth azimuthal mode into it: in Erni= u!,~- - Az,iul,iJA( u l , i r )- - ( i w p o bir

[$

1

+ s i A 2 , i ) J n ( u1,jr) eind

in A2,iul,iZVi( ~ , , ~-r -)( i w p o bir

1

+ S ~ A ~ , ~ u) N, y, () eind

The total transverse radial electric field solution can be greatly abbreviated by adopting the following notational definitions, just as we did before for the cylinder disk, but the necessity being much more critical here:

Eliminating these functions from the Elni formula and performing azi-

184

CLIFFORD M. KROWNE

muthal superposition, m

Eri =

c

Erniein4

n= -m m

=

c

n=

-m

[ ukiCiLai(r ) + b,!,,C,libi
Next obtain E4, for the nth azimuthal mode, by getting its formula E

FA, dF; br d 4

=---

Ib

+ iopob+ sA,

dFI - - -FA, dF;

br

dr

d4

dF; + i w p ob+ sh, dr

(446b) and inserting the appropriate F; and Fi equations from (473a) and (473b) into it: A2,iJn(a , y )

1

+ -1 ~ , , ~ ( i w+ ps ~~A ~ , ~ ) aJ ,;,(i r ) ein4 bi

1 A2,iNn(u1,ir) - a,,i(iwl.co+siA2,i)N,'(a,,i~ bi

+

1

1 A,,iJn( a 2 2 , i r+ ) - ~ , , ~ ( i w l+. cs ~ A , , ~ ) J ;u( 2 y ) ein4 bi

(483) The total transverse azimuthal electric field solution can be tremendously streamlined by adopting the notational definitions

inFi

1

bir

bi

Col r ~) , , ~ ( i + w Sp~ ~ A , , ~ ) J a; (l , i r ) , neat. ( r ) = - - A2,iJn(( ~ ~ ,+~-

(484a)

inFi ct?elbi(r)

= -

1 A2,iNn(al,ir)

+ - al,i(iwpO + s i A 2 , i ) N , ' ( a l , i r > ,

bi

(484b)

185

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

Cr?Ai(r)

=

1 inFi - F A l , i J n ( u z . i r ) + -uz,i(iuPo + si'l,i)JA(uZ,ir),

bi

(484c) inFi 1 C$Ai(r) = -- ~1,iNn(u2,ir) + - uz,i(iuPo + siAl,i>N,'(uz.ir)* bir bi (484d) Eliminating these functions from the E+,,, formula and performing azimuthal superposition, m

E4i

E+nieind

= n=

--m

m

=

C [

n=

r)

--m

+ b,!iC,?2bi(r ) + ufiC,?Ai(r ) + biiiC,$$(

r ) ]einQ.

(485) To find H,, for the nth azimuthal mode, get its formula H,

=

ik,(Po/P) - P A , b

dFi

-PA1 + ik,( Po/P) b

dr

9 4 aF; br d+ 9 4 dF; br d+

dF; dr

(447a)

and insert the appropriate F; and F; equations from (473a) and (473b) into it: PO

inq, - p i A z , i ) J i ( u l , i r) b,r A z , i J n ( u l , i r ) inq,

PO

1 1

+ II,!~[Abi u l , iik, ( --P~A~,~)N u ,1' (y )- -A2,iNn(u l , i r ) einQ Pi bir + afi[

+z,i(

inq, ik,* Pi - P ~ A ~ , ~ ) Ju ; (2 y )- bir A l , i J n ( u z , i r ) ein4 -piAl,i)Ni(uz,ir)

1

inq, -- Al,iNn(u Z , , r )ein4. bir

(486)

186

CLIFFORD M. KROWNE

The total transverse radial magnetic field solution can be tremendously abbreviated by adopting the notational definitions

(487a)

(487c)

(487d) Eliminating these functions from the Hrni formula and performing azimuthal superposition, m

Hri =

=

Prniein4

[ a;,CLL,,(

r)

2 r ) ]e i n 4 . + bAjCLAbi(r ) + a ~ , C ~ ~r,), +( bniCnhbi( r2

Finally, to find H4, for the nth azimuthal mode, one obtains its formula

H+, =

ik,(Po/P) -PA, br 34

+--9b4

aF; dr

and inserts the appropriate F; and F; equations from (473a) and (473b)

187

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

into it:

+

P~A~,~)~ J , ,( , ~ r4i) Az,iul,iJA( u,y)

PO

bi

+ 4i

1

N,( ~ , , ~ r-) Az,iul,iNL( u l , i r ) ei"+ bi

(489) The total transverse azimuthal magnetic field solution can be tremendously abbreviated by adopting the notational definitions

(490a)

(490b)

(490c)

(490d) Eliminating these functions from the H,+,niformula and performing azimuthal superposition, W

Hsi

f14,iein4

= n=

-m m

=

n=

-m

[

r)

+ b,,1 c+l n h b i ( r ) + uiiC,$;ui(r ) + LfiC$$,,( r ) ]ein4. (491)

188

CLIFFORD M. KROWNE

XXV. Z-FIELDDEPENDENCE Total field dependence on the coordinates, including the z-field dependence, is stored in the curlE and curlH Maxwell equations in partial derivative form, given in (12) and (13) of Section 111. When we selected the exp(ik,z) z-field dependence in that section, we had, in effect, chosen one z-mode type of behavior. This selection can lead to transverse plane (xy-plane) dependent field solutions, as evidenced in Sections XIX, XXI, XXIII, and XXIV. But, the z-dependent field behavior is still missing. Thus the z-dependent field information contained in the original curl E and curl H equations, written in component form and partly repeated here for examination of the partial z-derivative operators: 1 -dE, - - - dE+ - -iw( pHr - i K H + ) , r 84 dz dEr ---=

dz

1 dH, r dcb

dE, dr

+ pH+),

dH+ = iwEE,, dz

dHr - - dH, dz

-iw(iKH,

dr

=

iwEE+,

(246a)

(246b) (248a)

(248b)

must be put back into the total field solutions. The best place to start is to return to the transformed field components, F; and F i , which, if we recall, only had the transverse coordinate dependent information ( r , 4 ) in them. These transformed fields can be upgraded to represent the actual total transformed field solution if they are multiplied by a z-dependent function Z b ) . A single function is chosen since radial mode differences do not appear in the direction perpendicular to the circulator surface. These upgraded transformed fields F/ can be denoted by a bar over them:

Fin(r , 4, z ) = Z ( z)F;,,(r , 4) = Z ( z ) R , , ( r ) e i n d ,

(492a)

F i n ( r , 4 , z ) = Z ( z ) F i n ( r , 4 )= Z ( z ) R , , ( r ) e ' " + .

(492b)

Indexing with an n azimuthal mode number is essential to retain the

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

189

proper summation procedure for total field construction. Clearly, m

n=

m

-m m

(493a)

= Z(Z)

C

n=

R2,,(r)e'"+.

(493b)

-m

These formulas contain the u:; or b,'; weights for superposition. Using the pre-superpositioned form of E, in (334a) and its superpositioned form in (456), and the preceding E ( r , 4, z ) , E, = Z ( z )E,( r , 4; k:,) = z(z)(

i

=

q(r , 4 , ~ +) F;( r , 4 , ~ ) m

4) +

~ ; n ( r ,

c

F;n(r, 4)).

(494)

It is critical at this juncture to recognize that all of the cylindrical coordinate dependence has been lumped into one term given in the second equality here and that the function is parameterized in terms of the square of the z-propagation constant. That this is so may be readily seen by going back to the formulas in (454) [used in the last equality of (494)l. The F; depend upon Bessel functions Jn(a,, ; r ) for the cylinder disk, and these in turn depend on a,,; which we know has a unique association with the M' eigenvalues A,. The eigenvalues given by (295) are built from the pieces A and R found in (296) and (297). They in turn are constructed from the governing equation constants a, b, c, d , which, when combined according to (298) and (299) yielding the combinations ( a + c) and (ac - db), only depend on k:. This quadratic dependence on k, is immensely important, and affects the way the k,,. propagation constant eigenvalues control the superposition of the fields, including the E, being presently studied. Index j denotes the eigenmodes for the z-direction. Because the z-directed propagation will have forward and backward waves, with eigenvalues k,,+ and k,,-, where the J' index itself indicates the magnitude as well as the sign of one of the directions (take it to be

190

CLIFFORD M. KROWNE

forward going here), the Ezj total solution can be set down as

Ezj(r , 4 3 2 )

=

Z ( z ; k z j + ) E z j ( r ,4 ; k:j+) + Z ( z ; kzj-)Ezj(r, 4; k:j-) (495)

=

[ Z ( z ;k z j + ) + Z ( ~ ; k z j - ) ] E z j ( 4r ,; k : j ) ,

(496)

k ZJ + = - k z j - . (497) The weighting coefficient for the k z j + and k z j - modes enable us to specify the two z-contributions in (496):

Z ( z ;k Z l.+ )

= K Z l.+ e i k v + ,

(498a)

Z ( z ; k21.- )

= K ZJ. - eikz,-.

(498b)

If we use the constraint K Z]+. = K 21-.

(499)

on weighting coefficients, as will be shown next to be the case after applying boundary conditions on the tangential electric field components, then

(500) Ezj(r, 4 , ~=) 2Kzj+ cos(kzj+z)Ezj(r,4 ; k:j+). Consider the E, transverse field formula from Section XXI, (446a). Then Er

?A, dF;

=----

b -

iwpo

dr

+ sh,

br

dF; d+

?A, dF;

b

dr

i m p o + s h , dF;

br 34 dF; + i w p o + sh, aF; dFi i w p o + sh, dF; - -;[FA,- + FA, + dr r d4 dr r d4 1 = - E r ( r ,4 ; k : ) . (501) kZ It is apparent from (501)that the factor of the leading inverse propagation constant has been identified as dF; i m p o + sh, dF; E r ( r ,4 ; k l ) = FA, - + dr r d4 dF; i m p o + s h , dF; + FA, - + (502) dr r d4 * This has been accomplished by examining the detailed expressions for the

-1

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

191

constants F and s and noting that it is already known that the explicitly appearing eigenvalues have

A,(k:), A2 = & ( k : ) , (503) and that the implicitly appearing arguments of the Bessel functions have A,

=

a,(k:), u2= u 2 ( k : ) . (504) Looking at the i and s formulas in (260b) and (260d) shows that the constants also yield u1=

i(k:), s = s(k:). (505) Consequently, it is seen that the elegantly simple form adopted in (500) for Ez,(r, 4, z) can be used here, too, in a somewhat carefully altered way: i

=

or

(507)

This result occurs if we upgrade the transformed fields as in (492a) and (492b), place them into the complicated Er expression (5021, and extract out the z-dependence, or go directly to Er(r, 4; k : ) and multiply it by the modifying z-dependence. Either procedure is equivalent. Inserting the proper Z ( z ) variations from (498a) and (498b) into (5071,

192

CLIFFORD M. KROWNE

Boundary conditions on E,, assuming perfectly conducting metal for the ground plane (at z = 0) and microstrip (at z = h) surface, are Erj(r , 4 , =~0)

=

0,

(509a)

Eli( r , 4 , z

=

0.

(509b)

=

h)

Applying these conditions to the last Erj(r,4, z ) form,

eikzj+hK

,

21 +

K z j + - K,j-= 0, - e-ikzj+hK = 0. ,

21

(510) (511)

-

For this 2 X 2 system to have a nontrivial solution for the kzj, constants, considered as unknowns here, the determinant must be zero:

+

e-ik,j+h

eik,j+h

-e-ik,j+h

eikzi+h

=

2isin(kzj+h)= 0. (512)

The propagation constant eigenvalues are constrained to be kzj+h = j r ,

j

=

0, f 1, f 2,...

.

Because kzj+ represents forward propagation in an exp(iwt the negative signs must only be chosen and

(513)

+ ik,z) form,

This means that the other semi-infinite set of eigenvalues must be associated with kzj-:

The equality given in (499) between K z j + and K z j - allows the relationship (508) for Erj(r,4, z ) to be reduced to its final form E r j ( r ,4 , z )

=

Kzj+

2i -sin(kzj+z)Erj(r,4 ; k:j). kzj+

(516)

Even if the first allowed eigenvalue for kzj+ is selected, setting it to zero, the entire field solution does not go to zero, as is easily seen by merely examining Ezj(r,4, z ) in (5001, whose z-dependence limits to a constant value of 1. In fact, this particular eigenvalue of kZj+ corresponds to the limiting two-dimensional case. Now consider the other electric field component, E,, on which a boundary condition will be imposed. Using the transverse field formula

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

from Section XXI, (446b), FA, dF; i w p o + sA, dF; E =---+ br a4 b dr

FA, dF;

br

@

34

+

iwpo

+ sA,

b

193

dFi dr

dF;

+ ( i m p o + sA,) dr FA, dFi

br 1 = - E+(r,

kz

84

+ ( i w p o + sA,)-

dr

4 ; k:).

(517)

From (517) the factor of the leading inverse propagation constant was identified as E+(r,+;k;)

FA, = --

r

dF;

dF;

34

dr

- + ( i m p o +sA,)-

FA, dF;

br

84

dFi

+ ( i w p o + sA,)-. dr

(518)

This identification with the functional behavior of the kz-dependence in the argument of E+(r, 4; k:) follows for exactly the same reasons as employed for the E,-field component. By superposition,

which becomes, by (497),

(520) This is precisely the same form as the relationship for Erj(r, 4, z) in (5161, and since we already know Z(z; kzj*) from (497), (498a1, and (498b), E+j(r, 4, z )

=

Kzj+ 2i -sin(kzj+z)E+j(r,4; k:j). kzj+

(521)

194

CLIFFORD M. KROWNE

Let us now determine the z-dependences in the magnetic field components. For the z-component, c - A, c - A, H, = -F; + -F; . (438) b b We already know that the transformed field components only contribute quadratic dependences on k,, which leaves only the prefactors to study. The eigenvalues only contribute quadratic dependences, too, and so we expect this to be case for the constants c, which is indeed the situation on checking its detailed expression in (27%): c

=

c(ki).

Of course, the remaining constant b has had its k,-dependence

b

=

b(k,)

used already to analyze the E-field components: 1 H, = - H , ( r , k i ) . kz Once we have the odd dependence on k,, superposition in

+;

leads to the final form

Moving on to the r-component of the magnetic field,

On examining the detailed forms for the constants p and q: 4 = q(k,), P = P(k,), it is noted that they are odd in k,, and so each of the coefficients of the partial derivatives must be even in k,. Therefore, the construction in (495)

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

195

for E, applies exactly here, starting with

by superposition and ending with the final result

Hrj(r, 43.)

=

2K,j+ cos(k,j+z)Hrj(r, 4 ; kZj).

(529)

Lastly, the H+ expression H+

=

ik,(P"/P) -PA, dFF; br 34

+ ik,( P d brP ) - P A ,

+ -9-b4

dF; dd,

dF; dr

9 4 dF; +b dr

(44%)

has the same coefficients as the H,-field component, with only sign differences in the radial mode superposition. So, although the partial derivatives are combined differently from the r-component formula, the same conclusions are reached regarding z-dependences. Thus superposition in

THE

mvI. METALLICLOSSESIN THREE-DIMENSIONAL CIRCULATOR

Total losses in the circulator may be determined by breaking up the loss contributions into a metallic conductor part and a dielectric part. In this section we will assume that the losses, whether metallic or dielectric, are small enough to be treated perturbationally. Consider the metallic losses first. There is a ground plane at z = 0 and a top microstrip circulator plate at z = h. Both metallic surfaces are lossy, and the loss may be character-

196

CLIFFORD M. KROWNE

ized by ground plane surface resistance R,, and top surface resistance R,, . The amount of average power dissipated in watts/m2 at the top and bottom conducting surfaces of the circulator (Harrington, 1961) are, respectively, per unit area, 15er =

IHtan,tlZ st,

(532)

2

E g =

IHtan ,g I Rsg

(533)

9

where H tan, and H tan, are the complex tangential vector magnetic fields at the conducting surfaces:

[

+~

Htan,, = H ~ , ~ = H ,4( ,~z, = h ) = H,? Htan,g=HI,,

=

H,(r, 4, z

=

0)

4

+ 4H,Z]

(534a)

[ Hr? + H,c$ + H,Z]I r = O .

=

(534b)

Here the first equalities come from the recognition that the tangential fields are in the transverse-to-2-direction orientation:

-

IHll2 = HI HT

=

lH,I2 + lH,I2

+ lH,I2 = lHrI2 + lH,I2.

(535)

The last equality arises from the restriction that vertical conducting walls are not being considered. However, when such walls are utilized for an integrated circuit circulator, this term will have to be retained. Let us examine the radial and azimuthal field components for the magnetic field in the ith annulus: m

Hri(

r , 4)

=

c

n=

-w

[ akicALui( r , + b i i c ; L b i ( r)

+U:ic$,i(

r,

+ biicA;bi( r ) ]e i n d ,

(488)

m

H+i(

r , 4)

=

c

n=

--m

[ akiC$i,i(

r,

+ brfic$ibi( r )

+a:iC$&(r)

+ b$$&(r)]ein9.

(491)

These components may be put into a much more powerful notation in order to deal with the manipulations required to determine power loss. Define the mode constant symbol as i=a,b (536) and the mode index s = 1,2 running over the two allowed radial modes. The formulas have been generalized to include the disk region by defining b,lo = 0. Then the field components become, including only one

J

(537) mode for k Z j +

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

197

for the z-dependence from (529) and (531), 2

c c c iiiC;iii(r)ein+, (538a) W

Hri = K z j +C O S ( ~ , ~ ~ + Z )

n=-m m

H,i

= K z j +cos(kzij+z)

H=a,b

s=l 2

c c c ZiiC?&(r)einb.

n=-w

(538b)

i=a,b j=s

We leave the Kzi+ weight in the formulas to remind us that some type of weighting of allowed k Z j +modes may become necessary. The total average power lost due to conductor losses, for the ith annulus, is (539) Inserting the surface resistance formulas (532) and (533) into this relationship and using the tangential field information from (538a) and (538b),

PTcti=

lriO j 2 " F C t i r d +dr ',I

0

PTcgi= jri0j 2 = F C g i r d dr 4 riI

*

4,O)l'

=Rsg/rio/2n[lHri(r, rir

0

+ IH+i(r,+ , 0 ) 1 2 ] r d dr. ~

(541b)

The squared magnitude quantities in these integrals are given by the following two expressions, each consisting of a six-level summation and evaluated at either z = 0 or h:

198

CLIFFORD M. KROWNE

x

c?iii(r ) [ c::~.~]* ( r ) ei("

(542b)

-m)d.

Including different k,,: modes adds two more summations to each of these expressions, making a grand total of eight summations per expression. It is apparent that the azimuthal part of the integrations in (541a) and (541b) may be factored out and analytically evaluated as

which effectively reduces the number of summations to five per term which must be retained for the power calculations. Therefore, the total conductor loss for all the annuli, including the inner disk, is

i=O

XXvII. THREE-DIMENSIONAL BOUNDARY CONDITIONS THE CYLINDER DISK-FIRST-ANNULUS INTERFACE

FOR

The finite thickness of the solid geometry of the cylindrical disk does not change the requirement for tangential boundary conditions at the interface between the inner cylinder disk and the first annulus or ring compared to the two-dimensional case. However, here there are two new added fields, which must now be recognized as being both nonzero, namely, H, and E+. Using the same nomenclature as employed for the two-dimensional case, the boundary conditions are stated as

r l l , 4, z ) ,

(545a)

= T I / , 4,217

(545b)

H z o ( r = ro, 4 , ~=) H z l ( r = r l l , 4 , z ) ,

(54%)

E+"(r = ro, 4 , ~=) & ( r

(545d)

&(r

=

ro, 4 , ~=)

H+o(r = ro, 4 9 2 )

=

=

Hddr

= r l l ,4

,~).

There are now twice as many boundary conditions as were present for the two-dimensional case. For all these equations, the z-dependence will divide out on both sides of each equation (refer to Section XXV), so only

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

199

the ( r , 4) behavior must be addressed. Let us consider Ez as typical. Referring back to (456) for Ezo and (475) for Ezl for, respectively, the cylinder disk and the first annulus, the first boundary condition becomes

2 cz2 neal(rll)

2

+ 'nl

C'2

nebl(rll)]ein'-

(546)

Applying the orthogonality of azimuthal harmonics, each side of this equation may be equated, drastically simplifying the analysis. Treating the electric field boundary conditions on E, and E+ first, and then the magnetic field boundary conditions on H+ and H,, we obtain 1 c z l an0 n e a O ( r 0 )

+ aioCi,2,o(ro)

1 c z l = 'nl neal(rll)

1 czl nhaO(r0)

an0

1 c z l nebl(rll)

+ 'nl

+

2 cz2 neal(rll)

+ bilcr$bl(rll), (547a)

2 cz2 nhaO(r0)

+ an0

1 c z l nhbl(rll)

= u ~ l c ~ ~ a l ( r i l )+ bnl

+ u ~ l c ~ ~ a l ( r l l )+ 'ilci:bl(rlI), (547b)

These may be considerably compressed by dropping the radial arguments by adding the subscript D to denote the interface at the disk diameter:

200

CLIFFORD M. KROWNE 1

'no

c41

nhaOD

2 c42 nhaOD

+ 'no

= u:lCr?ialD

1

'no

c41

neaOD

-k

1 bnl

c41

nhblD

2 c42 nhnlD

+ 'nl

+ b:lcr?:blD,

(548c)

+ 'iOCr?:aOD

1 c41 - ' n1l c4n1 e a l D + bnl n e b l D + ' i l C r ? k l D + b:lCr?AID,

(548d)

This 4 x 4 system of equations can be solved for the first annulus constants uA1,bAl, uil, and b:, in terms of the cylinder disk constants uko and uio:

I

Ll

'irblD

ci,",lD

';?blD

CiLblD

Cih2aID

c,'iblD

ct?iblD

cr?:alD

cr?:blD

I

1 = -ILlM1l

Dl

Here Li, i equations:

=

+ L3M31 - L4M41}*

- L2M21

(549)

1, 2, 3, 4, are the left-hand knowns of the 4 X 4 system of L l = u ! z O c ~ ~ a O D+ ' n2o c znea0DY 2

(550a)

L 2 = u:OC;laOD

+

2 c znhaOD, 2

(550b)

L 3 = u~OCr?iaOD

+

2 c4 nhaOD, 2

(550c)

L4 = u:Ocr!elaOD

-k ' n2o c4 neaOD' 2

(550d)

Mij are the minors obtained by eliminating the ith row (corresponding to the i in L j ) and the jth column (corresponding to the unknown being

ciiblD

cih2a1D

c,'h2blD

cr?iblD

cr?:alD

cr?iblD

cr?;blD

cr?AID

cr?AID

CirblD

ci?alD

ci,",lD

= cr?iblD

cr?/?alD

cr?:blD

cr?AID

cr?:blD

=

M21

c~.blD

(551a)

(551b)

201

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

(551c)

(551d)

Putting the Li from (550a)-(550d), noting the definitions of Mij from (551a)-(551d), into the ukl formula allows the compact determination

1

uA1 =

-[ MllCd:aOD

- M21cika0D

Dl

- M41cr?elaOD]un0

+ M31cr?;a0D

1

1

-k

-[ MllC,":aOD Dl

- M21c,',2a0D

- M41cr?:aOD]

$- M31Cr?;a0D

.',O*

(552)

For the second unknown constant bil, the determinantal expansion procedure gives

I b'n l

=

ciialD

Ll

cf,',lD

c,'zblD

-

cikalD

L2

czh2a1D

cih2b1D

Dl

cr?ialD

L3

Cr?ialD

cr?;blD

I

(553)

Here the minors are C k l D

MI2 = cr?,al D

c,'h2alD

ci,'blD

cr?;a

cr?ib 1D

1D

cr?ela 1D

cr?AID

ciralD

ci,',l D

cr?A 1D c,'e26 1D

1D

cr?ib 1D

M22 = cr?ialD

cr?ela

1D

cr?;a

42

CnealD

c?r;

1D

(554a)

(554b)

202

CLIFFORD M. KROWNE

(554c)

(554d)

Putting the Lifrom (550a)-(550d) into the bil expression (553) produces

1 + -[ -M12c,‘:a0D

+ M22c,‘h2a0D

Dl

+ M42c!AOD]

- M32c!:a0D

2

‘no’

(555) , determinantal expansion proceFor the third unknown constant u : ~the dure gives

1

Ufl =

Dl

‘,‘LalD

c,’ablD

Ll

cxzblD

‘,‘kalD

‘,‘kbID

L2

c,’h2b1D

‘!;a c!JalD

1D

‘!;b

1D

3

‘!>bID

L4

‘!/?bI

D

‘!AID

Here the minors are ‘,‘;blD

=

‘!;b

1D

‘!;b

1D

(557a)

‘,‘dbID ‘!;b

1D

‘!;b

1D

(55%)

203

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

(557c)

(557d)

Putting the Lifrom (550a)-(550d) into the u:l expression (556) produces 1

u:l

=

-[ M13C,‘faOD

- M23C,’;aOD

- M43cr?ehD]

+ M33cr?hhD

1

D l

1

+ -[ M13C,‘&OD

- M23c,’h2aOD

4

2

+ M33ct?fa0D

- M43Cr?LOD]an0.

(558)

For the fourth unknown constant b:l, the determinantal expansion procedure gives

b;,

=

1 Dl

Here the minors are ‘,‘;alD M14

92

=

‘nhalD

3

(560a)

>

(560b)

92

‘nealD

‘,‘falD M24

= ‘$alD 91

‘nealD

‘;:a

1D

92

‘nhalD

92

‘nealD

204

CLIFFORD M. KROWNE

(560c)

(560d) Putting the Li from (550a)-(550d) into the bi, expression (559) produces 1 1 bil = -[ -M14c,'ia0D + M24CiiuOD - M34cr!iu0D -k M44cr!2aOD] ' n o D l

-k

1 -[ -'14C,'zu0D Dl

+ M24c,'h2a0D

- M34cr!2a0D

+ M44cr!AOD]unO'

2

(561) The determinant D, for the 4

X 4

system treated here is

(562)

I

cr!:ulD

cr!;blD

c!AID

cr!ilD

It is instructive to realize that the indexing scheme on L ; was done as somewhat of a crutch. That is, the correct physical associations should have had the definitions + a:OC,'zaOD

L',eOO

=

L',hOO

= anO 1 c znhu0D l -k a:Oc,'h2aOD,

1 cz' neaOD

9

L%oo = a!zocr!iaoD + an0 2 c 4n h2a O D , L$eOO - 1 c 4 neuOD 1 -k 2 c 4neaOD' 2

(563a) (563b) (563c) (563d)

The last subscript index on, say Lteo0 (they all are similar), is a place location for noting that the correct interfacial radial value has been chosen, and is the same as the second-to-last index which is equal to the annulus number i, the last annulus under consideration. Something valuable can be learned here by comparing (563aH563d) with the short-form Li definitions. This is because the Li have direct equation system number ordering, a very valuable insight. This insight may be applied to the C,rsfpmk coefficients ( r = z or 4, s = 1 or 2, f = e or h,

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

205

p = a or b, m = 0 or i = 1,2,. .. ,k = particular notation or i index) by labeling each D , entry according to its row position, with each new row adding to the previous value. Thus in

our 4 x 4 system (548) or, in abbreviated form,

i = a, c.. =c 11 ( i - l)N,+j'

the element is

(565)

(566) This gives a unique single index number to each matrix 6 entry, allowing a very easy way to keep track of all the CGpmkcoefficients. Such a process can be so helpful in hand expansion or numerical work. It also reduces a two-subscript indexing scheme (for a two-dimensional matrix) to a single indexing scheme (for a one-dimensional matrix = vector). Here N, is the equation system size, known to be N, = 4.

m I I . THREE-DIMENSIONAL BOUNDARY CONDITIONS FOR THE INTRA-ANNULI INTERFACES Tangential boundary conditions at the interface between two annuli can be set down by extending the earlier form used for the cylinder-first-annulus interface in Section XXVII: Ezi(r=rio,+,z)

='z(i+l)(r

= r(i+l)I,+,z),

+,z),

(56%)

= r(i+l)I, 492)'

(567c)

H 4 i ( r = r i o , + , z ) =H+(j+l)(r=r(i+l)I,

Ki(r

= ria, 492) = H.(i+l)(r

r = ria, 4

, ~ =) E4(i+l)(r

(567a)

= r(j+l)I,

4,~).

(567d)

Making the correct substitutions from Section XXIV for the field components Eli, Hbi, Hzi, and E4i in (475), (491), (476), and (4851, one obtains

206

CLIFFORD M. KROWNE

Equations (568a)-(568d) are four equations in four unknowns a:(,+ and b&+ ,) in terms of the old previous known constants b,$+ 1 ) , a,2 u i i , bAi, a,,, 4' and b;, at the ith annulus or ring location. Define

To see what this set of equations would look like collapsing all the C&i+ I)i coefficients into ,a single indexing scheme as in Section XXVII for the coefficient matrix C, it is rewritten below in that form, including L ,

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

207

in the same process: ‘1

= a:(i+

+ bi(i+ 1)c2 + u i ( i + 1 ) c 3 + bi(i+

(571a)

‘2

= a:(i+l)Cs

+ b i ( i + I ) C , + ai(i+I)C, + ’i(i+l)cgy

(571b)

‘3

=

+ b&i+1)C10 + ai(i+1 ) c l I + b:(;+1 ) ~ 1 32

(571~)

‘4

=

a:(;+ l)clS

+ bi(i+ 1)c14 +

l)clS

+ br?(;+

(571d)

l)c16*

All of the mechanics of the unknown solution determination can be done in the single indexing method, and at the end reconverted to the uncollapsed form with the explicit physical information. The system determinant is C i t a ( i + 1);

c f i b ( i + 1);

C i k a ( i + 1)i

ciib(i+ 1);

Di+l =

61

Cnha(i+ l ) i

61

Cnea(i+ I)i

‘ih2b(i+ I ) i

61

42 Cnhb(i+ l)i

Cnhb(i+ l)i

.

(572)

61

Cneb(i+ l)i

For the first unknown constant a:(;+

I

‘;eii

Ciib(i+ 1);

22 C n e b ( i + 1);

1

L;hii

‘i;b(i+l)i

‘ih2b(i+ 1)i

Di+ 1

‘$hii

cr?ib(i+

a:(;+,) = -

62 Cnhb(i+ I);

I)i

62 Cneb(i+ 1);

where the Mij are 22 ‘ n h a ( i + 1);

cih2b(i+ 1)i

41

= Cnhb(i+ I)i

62 Cnha(i+ I);

62 Cnhb(i+ 1);

61

Cneb(i+ 1 ) ;

62 Cnea(i+ l ) i

62 Cneb(i+I)i

C i f b ( i + 1)i

22 Cnea(i+ 1);

cr$b(i+

‘!;a(;+

62 Cnhb(i+ 1)i

‘ik MI1

M ~= I

+

b(i 1)i

‘,Q;fb(i+l)i

61

Cneb(i+ 1)i

1);

62 ‘nea(i+ 1);

(574a)

1)i

62 Cneb(i+ I);

(574b)

208

CLIFFORD M. KROWNE c i : b ( i + 1)i

22 C n e a ( i + 1)i

c;:b(i+I)i

M 3 1 = c i i b ( i +l ) i

cih2a(i+ I)i

Cnhb(i+l)i 22

91

C n e b ( i +l ) i

C n e a ( i + 1)i

92

92 Cneb(i+ 1)i

Citb(i+l)i

22 C n e a ( i + 1)i

ci?b(i+

M41 = c i l b ( i + I ) i

Cih2a(i+l)i

cih2b(i+l)i

91

92

C n h b ( i + 1)i

Cnha(i+ l ) i

7

(574c)

.

(574d)

1)i

9.2

Cnhb(i+l)i

Putting the Li from (569a)-(569d) into the uk(i+l) formula, noting the definitions of M i from (574a)-(574d), allows the compact recursive formula to be constructed:

+a:&

+ l , i ) U i i + P,",(i + l,i)b,2,],

where

procedure gives 21 C n e a ( i + 1)i

'ieii

c::a(i+

Ci;n(i+1)i

Lihii

Cih2a(i+l)i

cih2b(i+ 1)i

L%ii

c!la(i+l)i

Cnhb(i+l)i

91

Cnha(i+l)i

91

Cnea(i+ 1)i

'%ii

+ i(:!c r

l)i

'i:b(i+l)i

92

l)i

92 C n e b ( i + 1)i

(575)

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

209

where the Mij are C i i a ( i + I)i M I 2 = ‘?hla(i+

I)i

61

cih2a(i+ 1)i

cih2b(i+ I ) i

Cnha(i+ 92 1)i

Cnhb(i+ 62 I)i

62

Cnea(i+ 1)i

Cnea(i+ I)i

Cneb(i+ l ) i

C&(i+

1)i

22 Cnea(i+ I)i

ci;b(i+

M22 = cr!hla(i+

l)i

Cnha(i+ 92 I)i

Cnhb(i+ 62 I)i

91

62

‘neb(i+ 1)i

Ci:a(i+ I)i

cih2a(i+ 1)i

‘i;b(i+l)i

Cih2a(i+ 1)i

cih2b(i+ I ) i

l)i

Cnea(i+ 1)i

62

9

(578b)

Y

(578c)

-

(578d)

92

Cnea(i+ 1)i

61

(578a)

1)i

Cneo(i+ 1)i

M32 = Cr&i+

9

92

92

Cnea(i+ I)i

Cneb(i+ 1)i

ci;a(i+

‘i;b(i+

21

Cnea(i+ I ) i M42 = C i i a ( i + l ) i

91

‘ n h a ( i + I)i

1)i

‘:h2a(i+I)i 92 Cnha(i+ 1)i

I)i

cih2b(i+l)i

62

Cnhb(i+ I)i

Putting the Li from (569a)-(569d) into the b,$+l) formula, noting the definitions of Mij from (578a)-(578d), allows the compact recursive formula to be constructed:

where

For the third unknown constant

1),

the determinantal expansion

CLIFFORD M. KROWNE

210 procedure gives

‘:;a(i+

l)i

‘;:b(i+

1);

‘:;a(;+

l)i

‘;;b(i+

I)i

41

61

‘nha(i+ l)i

41

‘neu(i+

‘nhb(i+ l)i

41

I);

Cneb(i+ l ) i

‘;:b(i+

1)i

‘‘,hii

‘;h2b(i+

I)i

L$hii

‘!;b(i+l)i

L$eii

‘!e%(i+

1);

Here the minors are ‘:;a(;+

I);

a ( i + l)i = ‘ n h41

‘:;b(i+

I);

‘;h2b(i+

‘!;b(i+

I);

‘ n h42 b ( i + I)i

41

61

I)i

‘ n e a ( i + l)i

‘neb(i+ l ) i

42 Cneb(i+ I ) i

C;ia(i+ I)i

‘;;b(i+

I);

22 ‘neb(i+ I)i

‘ n h41 a ( i + I)i

‘ n h91 b(i+ l)i

41

41

‘ n h42 b(i+ I)i

‘ n e b ( i + I);

‘;da(i+

I);

‘;fb(i+

I)i

22 ‘neb(i+l)i

I);

‘:;b(i+

I);

22 ‘nhb(i+ l)i

‘ n e a ( i + I);

‘ n e b ( ; + I);

42 ‘neb(;+ I)i

‘:ta(i+

‘:ib(i+

I)i

‘::b(i+

I)i

22 ‘nhb(i+l)i

61

61

I);

M 4 3 = ‘;/lza(i+

61

‘:/lzb(i+

1);

‘nha(i+ I)i

41

‘nhb(i+ l)i

(582a)

7

(582b)

9

(582c)

9

(582d)

62 ‘ n e b ( i + 1);

‘ n e a ( i + 1);

M 3 3 = ‘;;a(i+

9

I);

42 ‘nhb(i+ I)i

Putting the Li from (569a)-(569d) into the a&+,) formula, noting the definitions of Mij from (582a)-(582d), allows the compact recursive formula to be constructed: 1 a ; ( i + l= ) -[&(i Di+ 1

+ l,i)UAi + +&(i

+ l,i)bii

+ 1 , i ) ~ +; ~p,”,(i + l , i ) b i i ] ,

(583)

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

211

Here the minors are cikb(i+ 1)i

22 Cnha(i+ 1)i

41

41 Cnhb(i+l)i

42 Cnha(i+l)i

41

Cnen(i+ 1)i

41 Cneb(i+ 1)i

Cnea(i+l)i

C i i a ( i + 1)i

cidb(i+ 1)i

Ciza(i+ l)i

Cika(i+ 1)i M14 = Cnha(i+l)i

41

Cnha(i+ 1)i

c$&i+

41

Cnhb(i+ l ) i

42 Cnha(i+l)i Cnea(i+ 1)i

Cnea(i+ 1)i

ciib(i+ 1)i

ci&i+1)i

M 3 4 = Cika(i+ I)i 41 Cnea(i+ l ) i

cr$!b(i+ l ) i

Ciia(i+l)i

21

21

Cnea(i+ l ) i

M 4 4 = Ciia(i+l)i C n41 ha(i+I)i

41

(586b)

42

c$elb(i+ 1)i

I)i

(586a)

42

(586~)

42

Cneb(i+ 1)i

Cnea(i+ 1);

cidb(i+ I)i

' l z a ( i + 1)i

cilb(i+I)i

cr%a(i+l)i

Cnhb(i+ 41 I)i

Cnha(i+ 42 1)i

(586d)

212

CLIFFORD M. KROWNE

definitions of Mij from (586a)-(586d), allows the compact recursive formula to be constructed:

mix. THREE-DIMENSIONAL BOUNDARY CONDITIONS FOR NTH-ANNULUS-OUTER REGIONINTERFACE

THE

We realize, as in the two-dimensional case, that W

Hr‘( R,+,z)

ArneIrn4

= m=

(589)

--m

can be applied as a boundary condition on the azimuthal magnetic field component H4. This may be a reasonable condition as long as the substrate thickness h is not too large causing rapid field variations along the perimeter. As in the two-dimensional case, it will be assumed that fields do not penetrate the r = R wall, except at the port locations. In other words, once again we are constructing a leaky cavity, where the leaks are only allowed to be at specified locations, namely at the ports. The two-dimensional solution for ax1 and bil, using its compact notation, can be applied to the three-dimensional case here by including the other two mode constants and b;,: (S90a)

(590b)

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

213

(590c) (590d) where ML,,(a,b), the shorthand notation, k defined by referring to Section XXVII:

=

1 or 2 and m

MLla = M 11Cz' neaOD - M21CikaOD + M31c$hlaOD M;2a

cz2

= M 11 neaOD - M21ciiaOD

ML!)lb = -M12c,':aOD Mi26 =

-M

12 CneaOD z2

+ M31c$2aOD

(591a)

- M41c$t0D

9

(591b) (591c)

+ M22c,'h2a0D

- M32cr!?aOD

+ M42c%30D7

(591d)

- M43c$@0D ,

(591e)

- M43c?2a0D9

(591f)

+ M33c?hlaOD

Mi2a

= M13C,':aOD

- M23Cih2aOD + M33c?2a0D

c"neaOD + M24C,'iaOD + M24C,'h2aOD

M i 2 b = -M14c,':aOD

9

+ M42c$Ja0D

- M23CikaOD

14

- M41c?JaOD

- M32c?iaOD

= M13c~:aOD

-M

1 or 2, is

+ M22cikaOD

Mila

M i l b =

=

- M34cr?iaOD

+ M44cr!2aOD

- M34c$:aOD

+ M44cr!20D*

7

7

(591g) (591h)

It is clear that if ukO and afo are known, they map into uki and b,ki, k = 1 or 2, for the first i = 1 ring, and then, by recursive formulas, into all the other rings i = 2,3,. ..,N . Thus, for the Nth ring

u t N = u!&(recur)uAo + uA%(recur)ufo,

(592a)

+ uf" (recur) ufO,

(592b)

+ bi2,(recur)ufo, (recur) uAO+ b:; (recur) ufo.

uf

b:,,,

=

uffv(recur) uto

=

b;',(recur)a;,

(592c)

=

E b:y

(592d)

What is new here in these formulas, compared to the two-dimensional case, are the second radial mode contributions. This substantially complicates the analysis, although conceptually nothing else is damaged. From Section XXIV, the azimuthal magnetic field component H4 in the ith ring is m

H4i =

c

n=

1 c4l nhai(r)

['ni

1

+ bni

c41

nhbi(r)

--m

+ufiC,$tai(r )

+ b:iC$2bi( r ) ] e i n 4 .

(491)

214

CLIFFORD M. KROWNE

Therefore, for the last i

=N

annulus or ring, r

m

H+N

c

=

n=

-m

[ u!,Nc!iaNR

+ b:NC!ibNR

2

+ anN

= R,

c+2 nhaNR

+ b:NC~t?bNR]

""*

(593) Now equate the perimeter azimuthal magnetic field with the azimuthal field found in the Nth ring:

HF'(R,9 , z , =

H+N(R,

$3

(594)

2).

Making the proper substitutions found in (589) and (593), (594) becomes Arneirn+ m = -a m

=

c

n=

+ b:NC!ibNR

['!zNc!iaNR

+ aiNC!iaNR

+ btNC!ibNR]ein'*

-m

(595) Orthogonality leads to 2 + a n2N c4 n h a N R + bt?NC!t?bNR. (596) Inserting the recursion relations (592a)-(592d) into (596) gives, for the azimuthal magnetic field perimeter field coefficients, An

1

= anN

c+1 nhaNR

+ b:NC!J!bNR

An =Aka!,, + A i a i , , A', = a!,& (recur) c $ i a N ,

+ biT,(recur) C,Q;f,NR + (recur) c,$;aNR + bih (recur) C t i b N R , A; = a!,; (recur) C,$iaNR + b;; (recur)C!J!bNR + (recur) c,ffa,R + b,"N(recur) c $ i b N R

(597) (598a) (598b)

as a function of the modal constants in the cylinder disk. The A, can be considered to be known. Consequently, one other equation is needed to find the superposition constants a!,, and a:,. Consider using the azimuthal electric field component H, on the perimeter. Just as we are employing the H+-field component on the perimeter, similarly constrain H,:

\o,

nonport contour regions.

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

215

This perimeter Hz can be expanded in a manner similar to that already done for H+: m

H F r ( R , 4, z ) =

c

m=

B,eim9,

(600)

-m

where B, are determined from

Equating H?' and H z N will enable the imposition of the second major constraint at the Nth-annulus-external region interface:

Hpr(R , 4, z , = HzN(

R , 4, z ) . (602) From Section XXIV, the azimuthal field component Ezi in the ith ring is given as m

H.zi =

c

n=

-m

[u k i c i k a i ( r )

+ b:iCiibi(r)

+UtiCiiai( r ) -k biic&( r ) ] e i n 4 .

For the i

=N

(476)

annulus,

m

n=

-m

(603) Inserting the proper expansions for each of the field quantities into (6031, + btNc,"kbNR Following the A, analysis, Bn = u!tNCiiaNR

B, B:

2 cz2 nhbNR'

+ briN

+ B:u:~,

= u!,$(recur)CiiaNR

=

(604) (605)

= B,'u!to

+ a:" Bi

+ aiNC,'laNR

+ bAh(reCUr)c,$bNR

+ b$, (recur) CifbNR,

(606a)

+ bA~(recur)C~~b,, (recur) C,'iaN, + b i i (recur) CiibNR.

(606b)

(recur) CliaNR

ak$(recur)Ci,!,,,,

+

As a result of applying two constraints at the circulator wall,

216

CLIFFORD M. KROWNE

a 2 x 2 simple linear system of equations is acquired. Its solution is

So, a solution for and u:o has finally been built, based upon the Nth annulus-outer circulator region interface and the field propagation through all the internal parts of the circulator, namely, the center cylinder disk and the N annuli or rings. The recursive nature of the problem solution has enabled the internal circulator properties to be added into the problem solution 'in a systematic way. A, contains the driving or forcing azimuthal field component H4 information, just as in the two-dimensional case. In addition, now B, is needed, which contains the driving or forcing z-component H,-field information. Note that because the perimeter Hrr- and H,P"'-fields were expanded, as in the two-dimensional case, in only azimuthal harmonics, in the simplest of approximations to what actually occurs at the circulator-outside interface, including at the microstrip port parts of the perimeter, we used only the cylindrical (I, +)-field solution parts for and Hzi. The z-dependence, discussed in Section XXV, was neglected, as far as the last interfacial boundary conditions were concerned. This does not mean it cannot be taken into account in a more realistic fashion in the future. The work here provides that basis. Let us address the z-dependence briefly here. From Section XXV, we restate the field component results in order to assess the perpendicular coordinate effect:

E z j ( r ,4 , ~=) 2 K z j + COS(kzj+Z)Ezj(~, +; k:j), Erj(I ,

+, z )

=

Kzj+ 2i -sin(kzj+z)Erj(r , 4 , k:j), kzj+

(500)

(516)

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

217

H r j ( r , 492) =

2K,j+ COS(kzj+Z)Hrj(r,4;k;j)s

(529)

H.+j(r,4, z )

2KZj+cos(kzj+z)H,+j(r,4; k$).

(531)

=

Because the ( r , +)-field solution constants uki and bji can all be uniformly scaled by the leading prefactor in the Ezj given previously, which amounts to setting K2 ].+ = I2 , (610) all of the field expressions can be economized. Also the explicit k;j+dependence in the arguments of the ( r , 4) parts of the field dependences will be eliminated to compress the notation. Thus we recognize that ~ z i j r( ,

4; k:ij+)

(611)

= Ezij,

where the right-hand members of these equations are found in Sections XXIII and XXIV. Notice that the annulus index i has been added to specify the particular ring under consideration. We obtain Ezij(r, &j(

$ 9 2 )

r , 472)

=

cos(kzij+Z)Ezij(r, 4)>

= i sin( kzij+z)Erij(r ,

4)

(613a) (613b)

E.+ij(r , 6 2 ) = i sin( kzij+z)E4ij(r , 4),

(613c)

H z j j (r , 4, z )

(613d)

= i sin( kzij+Z)Hzij( r,

4),

Hrij(r, 492) =

COS(kzij+Z)Hrij(r,4),

(613e)

H.+ij(r,492)

cos(kzjj+Z)H+ij(r,4).

(613f)

=

This procedure is valid when only one jth perpendicular eigenmode of the propagation constant k, = kZj is selected. Such an assumption may be reasonable if one eigenmode is dominant over all the others. For a very thin substrate where h is approaching zero, k, + 0, too, and there is only one dominant eigenvalue. But, when h is considered large, and this may be thought of as allowing or requiring many perpendicular modes to properly describe the matching between the circulator and the external ports (a mismatch problem often characterized as a mode-matching problem), or even the leakage out of the supposedly perfect magnetic walls between the ports into the outlying dielectric (or whatever material is contained in the space outside of the circulator), a rigorous superposition of perpendicular modes must be done.

218

CLIFFORD M. KROWNE

In the general situation requiring perpendicular mode superposition, the weighting coefficient Kzi+ must be retained and only the factor of 2 may

(614a)

(614b)

(614c)

(614d)

(614e)

m

(614f) These summations over the z-directed propagation constant will be retained in deriving the complete Green's functions for the circulator.

219

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

XXX. THREE-DIMENSIONAL DYADICGREEN’SFUNCTION WITHIN THE m I N D E R DISK Here the dyadic Green’s function within the cylinder disk region, the i = 0 index case, w ill be derived based on H4 or H, (or both) sources. Cylindrical ( r , 4) information comes from (456) and (457) for E, and H,, from (463) and (466) for E, and E4,and from (469) and (472) for H, and H4. The z-dependence comes from Section XXV with the modal summation amendment from the end of Section XXIX

m

m

m

m

m

a

(615d)

m

m

220

CLIFFORD M. KROWNE

m

m

Placing the a!,jo and aijo constants into (615a)-(615f) and separating according to A , or B, gives the field component solutions

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

m

m

+C C j=O

221

Bn j iKzj+ sin(kzoj+z)DABj

fl=-m

x [ BijC,Q;faj0( r)

- BjjCt:ajo(

r ) ] ein4

Next insert the Fourier integral relationships for Anj and Bnj, found in (98) and (601), into the field component expressions (616)-(621). The field components can now be constructed in terms of the Green's functions: Ezo(r, 49 z ) N$rp N;

N,

=

C

c

C

s=l q = l k=l N,

+

G&,( r , 4 7 2 ; R'

42, zs)H'c( R , 429 zs)A 4 2

Ngrp

C C

s=l u=l

Gtfo(r9 49 z ; R , 4",zs)H'c(&

z,) A4u

222

CLIFFORD M. KROWNE

s=l u=l

s=l u = l

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

x [ B:jCf:ajo(r )

- Bnj 1 c zn e2a j O ( r ) ]e - ' " + f e i n 4 ,

223

(628a)

224

CLIFFORD M. KROWNE

(628b)

(628c)

(628d)

(628e)

(628f)

(6288)

(628h)

(6283)

(628j1

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

=

2.rr

c c

j=o

225

Kzj+ c o S ( k z o j + z ) A DABj

n=-m

x [ A',jC?kjo(r) -Anj C4' nhajO( r ) ] e - ' " $ f e i n 4 . (6281) The extra discretization provided in the z-direction was done to allow segmentation on the perimeter in the port windows, an apparent upgrade to the one-dimensional Fourier decomposition in 4 used to construct Anj and Bnj. Aij, A:j, Bij, Bij, and DABjwould then depend on z,. It is also possible to further upgrade the dyadic Green's function by allowing the entire substrate to be composed of many layers, each one with possibly different ferrite material constants. Then the layering discretization could be chosen the same as the perimeter perpendicular discretization, or, if found necessary, allowed to vary in a different way. Aspects of these points will be addressed in much later sections. the modified dyadic Green's function element is found by the prescription

G:f,= G,ff,(e-'"@f +ji).

(629)

XXXI. THREE-DIMENSIONAL DYADICGREEN'S FUNCTION WITHIN THE ANNULI Here the dyadic Green's function within the annuli regions, for i = 1,2,. .. ,N index cases, will be derived based upon H, or Hz (or both) sources. Cylindrical ( r , 4) information comes from (475) and (476) for Ez and H,, from (482) and (485) for E, and E,, and from (488) and (491) for H, and H+. The z-dependence comes from Section XXV with the modal summation amendment from the end of Section XXIX: m

Ezi(r, 472)

=

m

C C

j=o

Kzj+ COS(kzij+z)

n=-m

1 c z l

x [a!zjiCiLji(r) + bnji nebji(r) +anji 2 c zn 2e a j i ( r ) + bnji 2 c znebji(r)]ein'3 2

(630a)

226

CLIFFORD M. KROWNE m

Eri(r94,Z)

=

m

C C

j=O

iKzj+ sin(kzij+z)

n=-m

+ b,'fi( r)C,'tbji +a:jiC,':nji(r)+ b:jiCL:bji(r)]ein4, (630b)

x [ a;jiC;taji(r ) m

E& i (r, + , z ) =

m

C C j=O

iKzj+ sin(kzij+z)

n=-m

x [a!zjiC!elaji(r) + bnji 1 c4 n e1b j i ( r ) +a:jiC$Aji(r) + b:jiC,?2ji(r)]ein4, (630c) m

Hzi(r7

4 , ~ =)

m

C C j-0 n=-m

iKzj+ sin(kzij+z)

x [ a;jiCiioji(r) + b,'jiCiibji(r) +a:jiC,l&ji(r ) m

Hri(r,

492)

=

C C

j = o n=--m

m

1 crl

+ bnji

+u:jjC,'ioji(r ) =

(630d)

Kzj+ ~ ~ ~ ( k z i j + z )

x [aAjiC;:oji(r)

H.+i(r,+,z)

+ brTiiCLibji(r ) ]e i n 4 ,

m

nhbji(r)

+ b:jiC$bji( r ) ]ein9,

(630e)

m

C C j=O

Kzj+ COS(kzij+z)

n=-m

x [ aAjiC,?,aji(r) + b:jiC,?ibji(r)

+a:jiC!toji(r)

+ bnji 2 c&2 n h b j i ( r ) ] ein4. (630f)

One can find the recursion relationships giving the aAji, bjji, a:ji, biji constants for the ith ring or annulus in terms of the cylinder disk constants akjo and ai j o ,just as was done in Section XXIX, in (592a)-(592d), for the i = N last annulus. The results for the ith ring, in terms of akjo and a i j o , are (631a) af,j i = a$ (recur) af,j o + (recur) a,' j o ,

+ + b,'$(recur) a: j o , + b:;i (recur) a: j o .

a:ji = a~~.i(recur)aAjOa:;i(recur)a:jo,

b,'ji

= bjfi(recur) af,j o

b:ji = b$ (recur) a; j o

(631b) (631c) (631d)

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

227

Because of these recursion relationships, a major intermediate step must be added here in order to eventually arrive at the dyadic Green's function, compared to the disk case. Using the recursion relationships (63la)-(631d), placing them into the preceding six component field expressions, (630a)-(630f), an intermediate form of field component expressions is

(632a)

(63213)

(632c)

(632d)

(632e)

(632f) The T;ii(r)radial functions, u = r, 4, z and s given by the following recursion formulas: T;& ( r )

= akfi(recur) C& j i ( I)

+

(recur) C&

ji

=

1 or 2, f

=e

or h, are

+ b$ (recur) C,Z,lbji( r ) ( r ) + bi/i(recur) C,Z:bji( r ) ,

(633a)

228

CLIFFORD M. KROWNE

+ b;;;(recur)C,'abjj( r ) + ai;i(recur)C,':,ji( r ) + b;;j(recur)C,':bji( r ) , T;:~;(r ) = a!$(recur)CLtaji( r ) + b,';i(recur)C,'tbjj( r ) + a;);(recur)C~~,,,(r ) + b$( recur)CiIbji(r ) , T;::.;( r ) = a~;j(recur)C~f,jj( r ) + b$(recur)C~~bjj( r) + a~~j(recur)CL~,ji( r ) + b$(recur)C,'Ibji( r ) , T$;(r) = a~)i(recur)C$~ajj( r ) + b,$(recur)C$2bji( r ) + ai:j(recur)C,$2,jj(r) + bi;j(recur)C,$2bji(r ) , ~ $ ; ; ( r )= a$;(recur)C$Jaji( r ) + bi;;(recur)C$Jbji(r ) + ai;j(recur)C$2,ji( r ) + b;;i(recur)C,$2bji( r ) , ~ ; l l f ~ ~= ( r a~~.;(recur)CiAUji( ) r ) + b,$(recur)C,$bji( r ) + aijj(recur)C,'~,ji(r ) + b;ji(recur)C,'ibji( r ) , T;;~~(r ) = a$j(recur)C,$aji( r ) + b$(recur)C,'lbjj( r ) + a~;j(recur)C,'~,ji( r ) + b$(recur)C,Libji( r ) , T';~;( r ) = a$(recur)cLf,ajj( r ) + bAfi(recur)C,'ibjj(r ) + ai)i(recur)C,'iaji( r ) + b;;;(recur)C,'ibjj(r), T'iji( r ) = a$j(recur)CL~aji( r ) + b:?;(recur)CLibjj(r ) + a$(recur)C,!&ji( r ) + b$(recur)C,'ibjj( r ) , T,$$( r ) = at~i(recur)C$~,ji( r ) + b,';;(recur)C,$ibji(r ) + ai;i(recur)C,$&ji( r ) + b;;i(recur)C,$ibjj( r ) , T;$( r )

T,$$( r )

= a$(recur)C~~,ji( r)

= a:;;( recur)C,$llfaji( r)

(633b)

(633c)

(633d)

(633e)

(633f)

(6338)

(633h)

(633i)

(633j)

(633k)

+ b,$(recur)C,$llfbj;(r )

+ aE;;(recur)C,$faj;(r) + b,2j2i(recur)C,$;bj;( r ) . (6331) Just as we saw in going from the ith to the (i + 0 t h annulus in Section XXVIII, with the mode 1 coefficient being equal to a mixing of the first and second radial modes, there is modal mixing here too, although it looks very complicated. Now the first mode function T';.;(r) in the ith ring is composed of information from the first and second modes in all the annuli. The same argument is true for the second mode function T:ii(r). Now insert the akjo and a i j o formulas, (607) and (6081, into the reconstructed field relationships (632aH632f). This will result in the final field forms

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

preparatory to obtaining a dyadic Green's function form

m

A

m

m

m

Bn j iKZj+sin(kZij+z)-

C

+

j=o n = - m

m

DA B j

x

c

C k=O

An j

iKzj+ sin(kzij+z)-

DABj

n=-x

x [ BijTA;i(r ) m

f

f

i

-

BijTA$i(r ) ] e i n 4

+C C

Bnj iKZj+sin(kzij+z)-

j=O n=-x

DABj

x

m

c c i=o

Anj

iKZj+s i n ( k r i j + z ) -

DABj

n=-m

x [ BijT:lji( r ) m

+

m

C C j=o

n=-m

- BijTkiji(r ) ]ein4

Bnj iKZj+sin(kzij+z)DABj

229

230

CLIFFORD M. KROWNE

Next insert the Fourier integral relationship for A n j and Bnj,found in (98) and (6011, into the field component expressions (634)-(639). The field components can now be constructed in terms of the Green’s functions:

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

231

232

CLIFFORD M. KROWNE

1

m

m

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

233

(646c)

(646d)

(646e)

(646f)

(646g)

(646h)

(646i)

(6463')

( 646k)

234

CLIFFORD M. KROWNE

The modified dyadic Green's function element is found by the prescription GL/= G:f(e-'"+f+ (647)

mu).

=I.

THREE-DIMENSIONAL THE NTH-ANNULUS-OUTER

DYADIC GREEN'SFLJNCI'ION REGIONINTERFACE

ON

The field values and associated dyadic Green's function elements can be evaluated on the interface between the Nth annulus and the outer region by employing the field solutions and dyadic Green's function elements within the Nth annulus. This is done by taking the limit of the expressions provided in Section XXXI as r + R , with i = N. These resulting formulas will be very important for finding the s-parameters of the circulator, which will be discussed in the next section: EzN(

R , 49 z, N,

=

N&

N:

C C C

GiRN(R , 4, z; R , 4f9z,)H+,(R, 4f72,) A 4 2

s=l q=l k=l N,

+

c

W r p

N,

N&p

C G;$d R , 4 9 2 ; R , 4u'z,)H+,(R, 4 u 9 2,) A4u

s=l u=l

+

N:

C C C

G G N ( R ,4, z ; R , 42, z s ) H Z c ( R4f, 9 zs>A4f

s=l q=l k = l

N,

+

W r p

C C s=l v = l

G ~ N ( R , ~ , Z ; R , ~ ~ , Z s ) H ~ c ( R , '4U7 ~ ~ , Z (648) s )

ErN( R %49 z, N,

=

N$rP N;

C C C

G&N( R , 4 , ~R;, 4f, z,)H+,(R, 42, zs)A 4 8

s=l q = l k=l N,

+

c

NTrp

C

s=l u=l

GgIN(R' 492; R , 4%' Z,>H+,(R' 4U'

2,)

A4"

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

235

236

CLIFFORD M. KROWNE

i

m

m

x [A',jT;$,,(R)

1

- AZnj~i~jN(R)]e-inc$feinc$, (654b)

x [ B:jTi:jN( R ) - Bij7"&( R ) ] e-'"@feinc$, (654c)

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

237

(654d)

(654e)

(654f)

(6548)

(654h)

(654i)

(6541)

(654k)

(6541)

238

CLIFFORD M. KROWNE

The modified dyadic Green's function element is found by the prescription =

GAh(e-i"+f+ j i ) .

(655)

m I I . SCA'ITERING PARAMETERS FOR A THREE-DIMENSIONAL THREE-PORT CIRCULATOR We will follow as many of the simplifying assumptions for the twodimensional case to arrive at a form for discussion and examination of the three-dimensional circulator model. In that regard, we had already considered H+ a prime field in relation to the port boundary conditions like in the two-dimensional case, but also added H, for the three-dimensional case. It might be noted here that, in principle, another pair of prime fields could have been chosen. This is somewhat similar to what is done when solving the planar propagation problem where conducting strips exist at some of the interfaces. For the microstrip guiding problem, in which the propagation constant and fields are often determined, the dyadic Green's function elements can be found when treating surface current components on the strips as sources or by treating slot field components between the strips or side walls as sources. Likewise, for the three-dimensional circulator problem, another prime set of fields could be chosen instead of ( H z , H J , given generally by (F:, F:) where i, j = 1 or 2 selects an E- or H-field and t, s = r, 4, z. Furthermore, note that the mode-matching technique, which can be employed to exactly connect the interfacial circulator fields with those of the microstrip ports in a self-consistent manner, has not even been addressed here. To make the whole problem consistent, such an approach seeking even greater accuracy could let the nonport regions experience leaky walls, too, considerably altering the mechanics of the solution approach already developed, by requiring the boundary condition constraints at r = R to be modified to take this new information into account. Copying the two-dimensional approach in Section XIV and setting N; = 1, NirP = N&p = 3, N, = 1, and z = zs= a single chosen value, write the electric and magnetic field components from Section XXXII as N&l EzN(R,4,zs)

=

c

A4q

~ ~ ~ N ( R , ~ ~ z s ~ R , ~ q ~ Z s ) H + ~ ( R ~ ~ q ~ Z

q= 1

NAP

+

~ ~ H N ( R , ~ , Z , ; R , ~ ' , Z S ) ~ 4'92s) , , ( ~ ,

A4q,

q= 1

(656)

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

239

(658a)

(658b)

(659a)

GgHN( R ,6, 2,;R , 4q,2,)

=

GgHN( R , 4 , z , ;R , + q , z s ) ,

discretized,

c#J~,z , ) ,

continuous.

cgHN( R,4, z , ; R,

(659b) Dropping some of the superscripts and subscripts, when they are apparent, absorbing the azimuthal spread into the Green's function, and dropping interfacial arguments when known, (660a) (660b)

(661b)

240

CLIFFORD M. KROWNE

where the coordinate indices must be retained on the right-hand sides of the equations to distinguish the magnetic fields. Now evaluate the E,,(R, (b, zJ and H J R , (b,z,>fields in (662) and (663) at each of the ports, located at q = a, b, c, by assigning (b = (bq:

If a direct association between the port electric and magnetic field components is chosen in the same simple fashion as for the two-dimensional case,

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

241

for the sake of discussion, then the second set of triple equations can be used to find the perpendicular component H, of the magnetic field, which can be substituted into the first triple set, and then the s-parameters determined through the H+ and E, relationships. This will be our approach here, at first, since it is so straightforward, extendable from the two-dimensional method, and in view of a more realistic but complex discretized two-dimensional port interface surface in the three-dimensional model, quite tractable. Therefore,

E, Eb

=

2 - laHa,

=

-lbHb,

(204) (201a)

(201b) E, = - 5 c H C ' Returning to the second set of triple equations, this 3 X 3 inhomogeneous system of equations can be solved for H,,, Hzb,and Hzc in terms of the relatively large azimuthal magnetic field components Hba, H4b, and H+C :

H

='a

H

=zb

1

0,

1

0,

242

CLIFFORD M. KROWNE

Performing the implied expansions in (666)-(668), retaining the summation driving terms - Cf=, G,?h(ji)H+i,j = a, b, c, yields for the perpendicular magnetic field component HZa:

where Maa =

Mab =

Mac =

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

243

(674) (675a) (675b) (67%) (676a) (676b) (676c)

Placing the H,,, HZb, and H,, expressions (670, (674), and (6771, employing the Green's function scaling coefficients T:, into the first triple set of equations (664a)-(664c), the compact and reduced forms for the perpendicular electric field components E,, Eb, and E, are found Ea

= T:aH4a

Ti, T:b

=

+ T:bH+b + T L H + c ?

(680)

+ TAG:,( aa) + TiaGt,( ab) + TAG;,( a c ) , = G$( ab) + TAG:,( U U ) + TbhbG;,( ab) + TAG,',( G$( aa)

UC),

(681a) (681b)

244

CLIFFORD M. KROWNE

T,",

=

+ T,h,G,',(aa) + Tbh,G,h(ab) + T,h,G&(ac),

G$(uc)

+ TbbH+b+ TbcH+c,

Eb = Tbe,H+a

(682)

+ T,h,G,',( ba) + Tbh,G&(66) + T,h,G,',( b c ) , Tbb = G$( bb) + TahbG&(ba) + T,h,G,;,( bb) + ctG&( b c ) , T& = G$( bc) + T,h,G,',( ba) + T,hG,',( bb) + TAG,',( b c ) ,

Tia = G$( ba)

Ec

= caH+a

(681c)

+ G H + b + ycH+c,

(683a) (683b) (6 8 3 ~) (684)

+ T,h,G,",(ca)+ TLaG:h(~b)+ T,h,Gth(cc), TA = G$(cb) + T,h,G$,(ca) + T/bG:h(cb) + T,hbG;h(CC), T,", = G$(cc) + T,h,G,',(ca) + Tbh,Gfh(cb)+ TAG,',(cc). T,", = G$(ca)

(685a) (685b) (685c)

Examining the electric field component equations themselves Ea = T:aH+a + Eb = Tba H+a

Ec =

T,",H+a

T:bH+b

+ T:cff+c,

+ Tbb H+b + Tbc H+c

+ T&H+b + T,",H+c

(680) (682)

9

(684)

9

we see that the form is the same as for the two-dimensional circulator case if the Green's function coefficients Gij of H+i are replaced by qe, i,j = a, b, c (see Section XIV). Now using (201a), (201b), and (2041, remove the E-field unknowns from (6801, (682), and (6841, obtaining a simultaneous set of three equations in three unknown H-fields;

2-

5aH+a =

TiaH+a

TibH+b

T,",H+c,

(686a)

= TiaH+a

+ TbbH+b + TbeH+c,

(686b)

- 5cH+c = T A H + a

+ cbH+b + rcH+c*

(686c)

- lbH+b

Rewriting (686),

( Tia + l a ) H+a + T i b H+b + Tic TbaH+a

H+c

=

2,

(687a)

+ ( T b b + 6 b ) H + b + TbcH+c

=

O,

(68%)

=

0.

(687c)

T:aH+a + TcebH+b + (TA + l c ) H+c

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

245

The solutions for the H-field components are

3

where the H-field system determinant is

Once the fields have been determined, it is a simple matter to apply the prescription employed in the two-dimensional model to the s-parameter calculation being done here for the three-dimensional case. The three s-parameters are ~ 1 = 1

1 - LaHba,

(692a)

246

CLIFFORD M. KROWNE

m v . LIMITING ASPECTSOF THE THREE-DIMENSIONAL MODEL To gain some further understanding, including the numerical behavior, of the three-dimensional circulator model, we will look at the special case of just one annulus or ring beyond the center cylinder disk. The radial modal constants in the i = 1 annulus are given by (590) in Section XXIX, with the j index added to account for z-spatial variation (693a)

(693b)

(693c)

(693d) The coefficients of the cylinder disk constants a t j o and aijo may also be identified with recursive formulas (63la)-(631d) in Section XXXI:

aLjl = a~),(recur)aLj0+ a ~ l ( r e c u r ) a ~ j , ,

(631a)

+ a~~l(recur)a~jo, bijl = b ~ ~ l ( r e c u r ) a+~ jb$(recur)aijo, o b:jl = b,2ill(recur)aLjo+ b:;l(recur)a:jo.

(631b)

a i j l = a:fl(recur)aLjo

(631c) (631d)

Furthermore, in order to compare the three-dimensional case directly with the two-dimensional model, the limit k, + 0 will be imposed. Because kZj+=j.lr/h, the k, + 0 situation must correspond to j = 0 for finite thickness h, or no spatial variation in the z-direction. Additionally, for very thin h values, only the j = 0 selection will give reasonably small propagation constant numbers. Since the z-field dependence is a cosinusoidal function of kZj+z,driving the k, value to zero seems equivalent to allowing only extremely small values of the z-coordinate, the same as making the substrate very thin. For the two-dimensional model, d / d z + 0 was applied. But, for exponential eigenmodes exp(ik, z ) , the application of the z-partial derivative operator onto such a modal function yields

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

247

ik, exp(ik,z). Thus driving k , to zero is equivalent to the earlier application in (18) of d / d z -+ 0. When the circulator cylinder collapses to a very thin disk, the two separate radial modes resolve themselves into two limiting values, with the full weight of the field energy going into the first modal constants atjo,b i o ,a!,jl, b i l and nothing going into the second modal constants a f j o ,bnjo,a i j l ,bnjl.Thus the last two mode constant equations may be dropped, and the first two written as (694a)

(694b) It is instructive to study the first of these two equations for a t j l under the limiting behavior for the z-directed propagation constant. The numerator is given by = MjllCjdajOD

Mhjla

- M j 2 1 C f i a j O D + Mj31Cr!iajOD - Mj41cr!JajOD

(695)

from Section XXVII, (552). The determinant is given by (562):

1 Dlj=

CitajlD

CitbjlD

cizajlD

ci,26ilD

Ct$tajlD

ciibjlD

Cih2ajlD

c%jlD

ct!hlajlD

ct!ibjlD

cr?kjlD

cr!;bjlD

ct!elajlD

ct?elbjlD

ct!2ajlD

ct!AjlD

I '

I

(696)

I

Let us study the effect k , 0 has on the constants and parameters contributing to MAila and D l j ,which are composed of C ~ f a j l tD=, r , 4, z and s = 1,2,f = e, h. From section XVIII: -+

p

=

lim ik,(k; - k2)D;' = 0 ,

(697a)

k,+O

i.= lim wKkZD;l

=

(697b)

0,

k,+O

K

q = lim - k,k2DS-' = 0 , k*+O

s = lim -io( p k ; - p,k2)D,-' kz+ 0

(697c)

P

= iwpk;

lim Ds-',

kz+ 0

(697d)

248

CLIFFORD M. KROWNE K

t

=

lirn -w.s-k2Ds-l P

kz+ 0

u

=

K

= --WE-

k 2 lim Ds-',

P

lim iw.s(k: - k2)oS-' = -iw.sk2 lim D~-', k,-. 0

k,-0

lim 0, = lim

kz+ 0

k,+O

( - ( k 2 - k:)' + (

2

(697e)

kz-10

) = ( w2,m)'

W ~ E K )

- k4 =

(697f) -k2k,", (698)

a

=

lim [ k , " - k:] = k , " ,

(699a)

kz+ 0

b

K ._

=

lim -iwpu,kz - = 0,

c

=

PO

lim - ( k 2 - k:) = k ; , k-10

d

=

(699b)

P

k,+O

(699c)

P

lim i w k , kz-. 0

K

(699d)

- = 0.

P

Now focus on the radial eigenvalues Ai = q2,i = 1,2, whose solutions are provided in Section XIX. Their behavior is most interesting in the limiting k , + 0 situation. Examine A and R from (296) and (297):

= k:

+ k;, + F ) k : +k:+k:

4R2 = k,+O lim ([-(1

I'

P =

(k,"- k:)2.

(701)

Therefore, by (305) and the preceding results for A and R, A - lirn [ A + R ] = k , " , -,k,+O

A,

=

lirn [ A - R ]

kz+ 0

=

k:.

(702a) (702b)

What is essentially happening here, is that when k , # 0 and the substrate thickness is finite, the z-modal behavior and the ( r , 4) or ( x , y ) in the planar surface modal behavior are mixing. This is because, in the plane, the radial wave propagation can be characterized by k,, whereas out of the

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

249

plane or in a perpendicular direction to it along the z-axis, there is no nonreciprocating anisotropic effect and the propagation can be described by merely the free-space k , value. The mixing occurs in the threedimensional model in contrast to the two-dimensional model, because all field components are coupled. This is a fundamental property of the ungraded problem-it is simply more realistic! Now it is apparent what these last two equations are telling us. The amount of weight assigned to the second mode is zero in the limit as k , + 0 because the problem decouples critical field components and the second eigenvalue mode drops out all together. This behavior will be basic to finding the correct dependence of uAj? in the limit, as well as also finding the field component dependences in the limit. Now return to finding some of the limits of the C,$ajlD,using the basic limits discovered previously. In the cylinder disk, referring to Section XXIII, (458a),

and the z-directed function at the interface becomes

In any ith annulus, invoking (477a) and (477~1, (705a) (705b)

so that the results at the interface for i

=

1 will be

C;ialD= k,-r lim0 C;ial(ro)= 0,

(706a)

C;iblD = k,+O lim C i i b l ( r o )= 0.

(706b)

For the radial function (462a) in the cylinder disk,

-

=

r - A 2 , 0 a 1 , 0 J ~ ( ~ 1lim ,0r) k,+O b

in r

- -Jn( a l , , r ) lim k,+O

i w o + SA2,O b

250

CLIFFORD M. KROWNE

Let us evaluate these two limits in (707) carefully, for any region, whether in the disk with i = 0 or in an annulus with i 2 1,

r lim k,-0 b

lim

k,+O

impo + sA, b

=

=

P 1 = lim -ik, 0, k,+O Po k2k,2

lim

k,+O

(708)

impo - io( p k z - p,k2)D;'ki b

Thus

(710)

c;:,o(r) = 0 ,

and so the limit at the interface must be (711)

C,'~,,, = k,+O lim C ~ ~ , o ( r=o0.)

Turn your attention to radial functions within the rings, given by (481a) and (482b),

in

- -(impo bir

1

+

By virtue of (708) and (7091,

(712b)

c;t,i(r) = 0,

(713a)

r,

(713b)

C;ibi(

=

O,

and so their limits at the interface for the i C;t,,,

.

= =

=

1 annulus will be

lim C ~ ~ 4 1 (=r o 0 ,)

(714a)

l h

(714b)

kz+ 0 kz+ 0

c;:b1(r0) =

0.

251

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

Finally, we focus our attention on the azimuthal functions. Examine first the cylinder disk, enlisting (465a),

i

+

; ~ l . O ( ~ W+0~ A 2 , o ) J X q o r )

0, yielding at the interface =

(715)

C,?:,oD = lim C,?2ao(ro)= o kz+ 0

(716)

by recalling (708) and (709). For the annuli, recall (484a) and (484b):

i +-u,,i(iw~+ o siAz,i)K(c1,ir) bi

0, producing at the interface, for the first i =

(71%) =

1 annulus,

c,dtlD= k,-r lim0 C,?:,l(ro) = 0, C&lblD =

lim

k,+O

c,?:bj(

ro)

=

0,

(718a) (718b)

again recalling (708) and (709). Preparations for determining the limiting behavior of the field components with k, 3 0 have now been made. First study H, from (457): m

252

CLIFFORD M. KROWNE.

utilizing the function limit in (703) and the fact that all second-mode constants, like u:,,, must go to zero. In the annuli, by (476), m

H21. =

lim

kz-0

n=-m

+b

['xicrki +U:~C,&(T)

1c z l

~ i nhbi(r)

+ b 2n i c22n h b i ( r ) ] e i n +

0, (720) utilizing the function limits in (705a) and (705b) and the null properties of the second-mode constants. Seeing that the z-directed magnetic field component limits to the correct zero value, as expected and seen in the two-dimensional model, direct attention to E,. From (482), =

m

C

E , ~= lim kz-0

n=--m

[ u t , ~ ; t , ~ (+r )b:i~;tbi(r)

+u:jc;:,i( r ) + b;ic;;bi(r ) ]ein+ -

m

-

0, (721) utilizing the function limits in (713a) and (713b) and the null second-mode properties. Finally, look at the azimuthal electric field component E+ in (485): =

m

E+~ = lim kz-0

C [ u!&Jai(

r)

5 ix.[

n = --m

=

0,

r)

+ bii~!Ai(r)]ein+ ~ ! 2 ~ ~+( bii r ) lim c ! ~ ~ ~ ( r ) k,+O

+@,?Ai(r) =

+ b:ic,?jbi(

n=-m

lim

k,+O

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

253

utilizing the function limits in (717a) and (717b) and the null second-mode properties. At this point in the analysis, we can return to studying a:jl by examining the detailed behavior of the numerator factor M i j l a and the denominator factor Dlj. MAjla in (695) has its second and fourth terms drop out by (704) and (716), giving Ml!)jla

= MjllcijajOD

Getting the necessary two minors for the i

Mj31,i=l

Mjll,i=l

=

=

1 annulus:

CidblD

ci,',lD

cie261D

CiiblD

c,'h2alD

'ih2blD

cr?.blD

'r?AlD

cr?AID

c:h2a1D

cih2b1D

'!%alD

'r?;blD

'r?AlD

'r?AlD

= cr?iblD

(723)

+ Mj31Cr?iajOD*

(725)

f

Placing these reduced minors into the MAjla expansion (723) yields c,'h2alD

c,'h2blD

cr?AID

cr?AID

. (728)

254

CLIFFORD M. KROWNE

The determinantal factor in (7281, consisting only of the second-mode functions, has a particularly elegant reduction: u2.1 -(c - ~ l , l ) ( i W C L O+ S 1 4 , l )

b:

x [ Jn(

u 2 , 1 r0

N ( ~ 2 , l r o - J;( ~

2lro ,

Nn( uz,lro

I

2

u2.1 -(c

(729) b: r’a2.1ro employing the properties of the Bessel function Wronskians (Krowne, 1983) for the last factor in the right-hand side. Therefore, the final numerator result is =

MAjla

- hl,l)(iWPO

+S14,J-

9

. ‘41 + ‘‘1n e b l.l D ‘41nhalOD . ] [ - ‘1 nealOD nhbjlD ,

x

* a2 1

(c -

4 , W ~ C L + O ~

b l

2 1 ~ 1 , d =u2,1ro

.

(730)

Returning to the denominator D , . provided in (696) and substituting in the function elements from (706a), (706b), (718a), and (718b),

Dlj =

‘iiajlD

CitbjlD

‘izajlD

cr%jlD

0

0

‘ih2ajlD

‘ih2bjlD

cr?:ajlD

cr?;bjlD

‘r?;jlD

cr!e?kjlD

‘ k j l D

‘itbjlD

‘ic?bjlD

‘r?;ajlD

‘r?t!bjlD

cr?:bjlD

‘%jlD

0

-

-‘jh2ajlD

‘r?t!bjlD

0

0

‘r?;jlD

CidajlD

‘iibjlD

‘izajlD

cr?t!ajlD

‘%bjlD

c!;ajlD

0

=

0

[ -‘~h2ajlDC?~bjlD

0

+ ‘ih2bjlD

‘r?;ajlD

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

255

It is observed that DIi consists of two factors, the first being a determinantal form made up of only second-mode function elements, and the second being a determinantal form made up of only first-mode function elements. Let us tackle the first factor, obtaining the functional element expressions from (477b), (477d), (484a), and (484b) for the i = 1 annulus: (732a) (732b)

(734) where we have chosen to emphasize the Wronskian by writing for it

(735) This is the Wronskian for the second radial eigenmode. Now attack the second factor in (731), retrieving the appropriate functional element expressions from (474a), (474c), (490a), and (490b) for the

256

CLIFFORD M. KROWNE

(736a) (736b)

Although, in the last two formulas, p1 and q1 are known to be zero in the k , + 0 limit from (697a) and (697~1,no terms drop out because of the exact counterbalancing effect of the b, divisor. Only by working with all parts of the function elements does a relatively simple result come about. Placing these four formulas, (736a)-(73%), into the second factor,

where we have chosen to emphasize the Wronskian by writing for it

This is the Wronskian for the second radial eigenmode. Thus Dlj can be written as the negative product of (734) and (738):

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

257

Next find M k j l a ,but using the Wronskian in it, and write it as Mkjla =

[ -CirajODC$hlbjlD x

+(+2 1

(c

+ c~~bjlDc$~ajOD]

- Al,l)(iWO +SlAl,l)W(2).

(741)

bl

Then find the expression (694a) for a:jl, and replace the numerator and denominator by the last two formulas:

The q l / b , ratio is finite although the individual limits are zero as k , 1 41 - = bl

iwok,2,1 *

4

0:

(743)

The proper eigenvalue limits for A, and A, are found in (702a) and (702b). Thus the final akjl expression is

The natural question to ask now is how this result relates to the two-dimensional model formulation. To answer that question, it is best to retrieve the a,, formulas (74a), (71a), and (72), giving

(745) Expanding MDAaand reinserting it into a,, creates a form very similar to the three-dimensional form in (744): iCneaODCnhblD

- CneblDCnhaOD

an1 = D l

1

ano.

(746)

First let us evaluate the denominator here so that the two- and threedimensional forms may be compared. All of its function elements pertain to the first annulus, i = 1. The function elements are, from (49), (571, (501,

258

CLIFFORD M. KROWNE

(747a) (747b)

(748)

Evaluating these formulas at r = yo, dropping the superscript to put us into the implicit index notation (not much of an economization for the two-dimensional problem where the number of components is severely reduced compared to the three-dimensional model), and placing into (745) for D,, Dl

= CnealACnhblA

='1

- CneblACnhalA

ke, 1 [ Jn ( ke, 1'0

N ( k e , ro 1

- Nn( ke, 1'0

) JA( k e , 1'0

1

= clke, 1W( 1)

We see that this is exactly the same as the denominator in the threedimensional formula (742). This result makes the two- and threedimensional modal weighting coefficients uAjl. and a,, the same in form, recognizing that the function elements have different meanings in the two different formulations found in the numerators. It is to those numerators that we turn our final attention. Consider first the two-dimensional form. Since the first-annulus information required is known from (747b), (748), and (749b), write down the inner cylinder disk function elements: Cneao(r> = Jn(ke,or)7

(751)

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

259

When these expressions are placed into MDAa,the two-dimensional numerator becomes

Next consider the three-dimensional form. Examining (744), it is necessary to acquire four function elements:

c;:ao(r)

(755a)

=Jnhor)7

(755b) (75%)

41 + -~2,1(+1,1Nnl(~l,lr)*

(755d)

bl

To find the limiting form of the numerator, we need the limiting expressions of the constants multiplying the Bessel functions in the function element formulas: -PA,

)

=

Po in lim - ik, - - ik,[k: k,-+o b r ( p

-

using (697a), (698), (699b), and (702b). For the other constant multiplier, 4

lim - A l a l k,+O b

=

cike

(757)

260

CLIFFORD M. KROWNE

by (702a), and (702b), and (741). Enlisting these limiting properties in (756) and (7571, the function element limits are determined to be (758a)

(758b) (758c)

(758d) When these are inserted into the three-dimensional numerator Cr&jODCr!ibjlD

- ci~bjlDc?iajOD

9

(759)

an identical result to the two-dimensional form MDAaoccurs.

m v . Z-ORDERED

LAYERS IN THE RADIALLY

ORDERED CIRCULATOR It would be much easier to treat the circulator if it ideally looked like a cavity with magnetic side walls located at r = R for all 4 with electric walls located at z = 0 and z = h,. These are hard wall conditions in that they force fields to be zero, and furthermore insure that no energy flow exits the device. Such a supposition would only be approximately true for the case of relatively narrow slots of height h, and width R A 4 cut out for the ports at r = R . The interest in such ideal conditions is they they allow us to recognize that the modes in each cylindrical layer of thickness h , for the mth layer should possess orthogonality properties besides forming a complete set of basis functions capable of representing an arbitrary function. As the cylinders of heights h , are stacked vertically in the z-direction, mode-matching interfacial conditions are imposed on the tangential components of the fields. It is for these interfacial conditions that orthogonality (Kurokawa, 1958; Van Bladel, 1962; Marcuvitz, 1964) can be exploited on one side of each equation to simplify the extraction of unknowns. The other side of each equation will have nonorthogonality hold since the modes of the lower layer rn are projected onto those of the one above it m + 1. Furthermore, because the layers may be stacked with

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

261

different materials, there is no reason to suppose mode similarity (Reiter and Arndt, 1995). This is exactly the same type of mode behavior which occurs for the junction between two dissimilar geometrical cross sections of waveguides. Such characteristics have been observed for two cylindrical waveguides of different radius (Belhadj-Tahar and Fourrier-Lamer, 1986; Zaki et al., 1988, Li, 1993; Li and Chen, 1994) as well as for two rectangular waveguides of different height and width dimensions (Chu et al., 1986; Alessandri et al., 1988; Biswas and Bhat, 1988). Related cavity and waveguide papers and mode-matching papers are found in, respectively, Bolle (1962), Hord and Rosenbaum (1968), Zaki, and Atia (1968), Gerdine (19691, Kobayashi and Tanaka (1980), Fiedzuiszko (1982), Maystre et al. (19831, Zaki and Atia (1983), Maj and Pospieszalski (1984), Zaki and Chen (1985a, b), Zaki and Atia (19861, Hernandez-Gil et al. (19871, Kajfez (19871, Fiedziuszko (1988), and Mautz (1995) and Glisson and Wiltron (1980), Vahldieck (1984), Vahldieck and Bornemann (19851, Wade and MacPhie (19861, Rautio (1987), Zhang and Joines (19871, Rautio (1990), Wang (1991), and Reiter and Arndt (1992). Related anisotropic junction studies are in Tsai and Omar (1992, 1993), and material related to Li (1993) and Li and Chen (1994) on stratified cavity problems in Galejs (1969). Some recent spherical and planar stratified problems can be found in Huang and Tzuang (19941, Li et al. (1994), and Pan and Wolff (1994). Because of only partial orthogonality (when considering the whole collection of modes in all layers) and because the structure in fact has holes possibly breaking all orthogonality properties, we will not utilize or assume any orthogonality. However, when the problem is simplified for special limiting cases and some degree of orthogonality does hold, naturally the resulting mode-matching matrices method will also simplify in that more of the elements will go to zero. Thus for the simpler cases the already sparse matrix will become more sparse. The resulting matrix in the mode-matching method for many layers is sparse because the derivation procedure is to apply interfacial boundary conditions in a step-by-step process, examining all of the layers in turn. This results in nonzero elements clustered about the matrix diagonal. Slots in the entire circulator structure may make the cylindrical structure look electromagnetically like a solid cylinder with as many spokes as ports, with each spoke of height h , having finite width. Such a structure is noncanonical in that a well-definable orthogonal basis set cannot be found to represent an arbitrary field within [Reiter and Arndt (199511. Such noncanonical structures can still be studied by employing complete sets of basis functions found from conditions sufficient to assure such completeness. Completeness could be assured by instituting hard wall conditions over the entire structure or, in cases where the structure has leaks, truncating it.

262

CLIFFORD M. KROWNE

Consider breaking up in the circulator cylindrical height into just two layers first. The bottom layer, m = 1, lies on the hard electric wall which imposes r , 4, z )

=

0,

E i j ( r , 4 , Z ) = 0,

0,

(760a)

z = 0.

(760b)

z

=

These are just the same boundary conditions applied to the fields for the single-region, unlayered case, previously treated in Section XXV. There, however, the boundary condition on the upper electric wall discretized the k,. Here this discretization is no longer valid, and is indeed the subject under study. The form of the fields are the same, however, as determined earlier for m = 1. Our goal will be to determine the infinitely denumerable set of eigenvalues k z , for the m = 1 or 2 layer, j = j m , the index in the mth layer. For the m = 2 layer, E;(r, 4, z ) and E i j ( r , 4 , z ) are expressible from (498), (507) and (520) as

The null boundary conditions on these two fields at the top of the structure z = h, = h , + h, are

=o, Eij(r,4,Z) =o, E;.(r,f$,z)

z=h, +h,=h,,

(762a)

z=h, +h,=h,.

(762b)

Application of these boundary conditions to (761a) and (761b) determines the backward-wave z-coefficient:

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

263

where we have again absorbed the explicit sign-dependent information of the perpendicular propagation constant into the radial-azimuthal field function in the last factor of these formulas. For the two-layer case, there is just one interface at z = h,, and the fields must obey continuity of their tangential components:

E;(r,4,z)= E ; ( r , 4 , z ) ,

z=h,,

(765a)

Ef( r , 492)

z = hi

(765b)

( r , 4,z)

= E:( =

r , 492)

H; ( r , 4 , 2) ,

z

=

h,

7

(765c)

(765d) z = h, . H,'( r , 4 , z ) = H,'( r ,4 , z ) We note in these equations that an infinite sum over the k i j modes in the first layer must be taken to obtain the correct total field components on the left-hand sides. A similar statement holds for k:j on the right-hand sides. Regarding the construction of the z-dependence, H, and H+ have the same form as E, (Section XXV). Therefore, 7

H;j( r , 4 , ~ = ) [ eik:j+' + e-ik:j+2e2ik:j+h,] KZj+Hij(r , 4; k:,+), (766a) ~ ; ( r 4, , z)

[

= ei+

+ e - i k ~ j + z e 2 i k : + h , ] K Z j + H4;~k( :rj, + ) .

(766b)

Putting the field expressions for the two layers m = 1 and m = 2 into (765a) and (765b) yields the mode-matching equations for the two-layer problem:

c 2iKij+ sin(k:j+hl)E$j(r; k i j + ) Jl

j= 1

Jl

C

2iKLj+ sin(kij+h1)Efj(r;k i j + )

j= 1

Jl

2iKij+ sin(k:j+hl)Hij(r; k i j + ) j= 1

[ eik:j+hl

= j= 1

+ e - i k : j + h l e 2 i k 2Z.J + h

~ ] K i z , + H ; ~k( r: j; + ) , ( 7 6 7 ~ )

264

CLIFFORD M. KROWNE Jl

C 2 i ~ f sin( ~ + k i j + h l ) ~ $r;( k i j + ) j=1

-

J2

C [ e i k : j + h , + e - i k i j + h ~ e 2 i k : +Kzj+H;(r; ht] k:j+).

(767d)

j= 1

Here we have been able to remove the azimuthal functions due to identical orthogonality on both sides of each of the equations. The radial-azimuthal functions consequently reduce to one-argument dependence on r parameterized in terms of the z-dependent propagation constant. Next perform an integration procedure by projecting test functions onto these equations, producing inner products of the form ( t A , j 9 , e A e ,= j)

,,”wA,,.(r)F~(r)F~’(r)dr.

(768)

Here t A , j , is the test function, chosen to be a field component of type f = 4, r, in the layer rn of the z-index j’. Similarly, ek,, is the expansion modal field in the layer rn‘ of the z-index j. The field F = E or H. Note that the correct field expression must be inserted into (768) based upon the annulus being integrated through. The integration variable r will sweep through the disk and all annuli from the first i = 1 to the last i = N. w ~ , , , ( r is ) a weighting function. For the first mode-matching equation, we project tA,j , = Ekjr(r),f = 4, and m = 1, j‘ = 1,2,. ..,J,, onto it. For the second mode-matching equation, we project tA, = E$(r), f = r and rn = 2, j ’ = 1,2,. .. ,J,, onto it

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

j’

=

265

1 , 2,..., J 2 . (770)

A total of ( J , + J 2 ) equations in J , z-coefficient unknowns K i j + and J2 z-coefficient unknowns Ki.+ has been obtained, allowing a solution for the z-coefficients to be found from the resulting homogeneous set of equations. Let the following integral matrix element terms be defined:

j’ = 1,2,.. .,J1,

j

=

1,2,. ..,.I1, (771a)

j’

=

1,2,... ,J1,

j

=

1,2,...,J 2 , (771b)

j’

=

1,2,.. ., J 2 ,

j

=

1 , 2,..., J , , (771~)

j’

=

1,2,. .., J 2 ,

j

=

1 , 2,...,J 2 . (771d)

=

0,

(772a)

=

0.

(772b)

Inserting these definitions into (769) and (770), Jl

Z)iK:j+

+

Jl

j= 1

Z)j”K:j+ j= 1

j= 1

c Z;;Kfj+

J2

+

J2

c Z;fKL:.+

j= 1

266

CLIFFORD M. KROWNE

The actual double set of equations, when the summations are written out explicitly, has the appearance

+ + I ~ ~ l K f j +, +I;fK:l+ + I;;KZ2+ + .’. +Ijj2K:j2+ I$iKf,+ + I i l K f 2 + + ... + l ~ ~ l K i j + , +Ii;K:,+ + I$;K:2+ + ... +Ii:2K:J2+ I f i K f , + i- I;iKi2+

=

0,

=

0,

(773)

(774)

(775) The elements of are termed global elements because they relate to the entire system of equations involving all the layers. Here the number of matrix element defined in (771a)-(771d), for layers is M = 2. Each m = 1,2 and m ’ = 1,2, specifies the matrix elements in the local frame of reference formed by considering the interfacial constraints at the m(m + 1) interface. Here there is only one interface, so m(m 1) = 12. This terminology is not unlike that sometimes used in finite-element work. Indeed, later when we increase the number of layers to an arbitrarily large number, the utility of these concepts will become both apparent and necessary. The summation limits seen in (772a) and (772b) and implied in (775) are finite. One must recognize that only an infinite number of basis functions can result in faithfully reproducing the actual field behavior, but

+

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

267

a close approximation should be possible by choosing a suitable finite number of them. How to properly choose modal expansion limits has been a subject of considerable research that is understood well today (Wang, 1991). The ( J , + J 2 ) z-coefficient unknowns in (774) will only have a solution if det[

q ktl +

7

kfz +

9

. ..

kz,, + ;k,, + ,kr2+ ,* * - kZJ2+)] 2

1

9

2

9

=

0,

(776)

1 , 2 , . . ., M . (778) Each of the previous three equations has the same meaning, except that as we progress downward, the generality increases, and this generality will be needed to treat adequately the arbitrary number of layers, with an arbitrarily large number of z-indexed modes in each layer. Consider the first equation (776) for only one layer, the situation when there is no interface at all and it is known that the determinant, a simple quantity to evaluate, separates into jl = 1 , 2,..., J , ,

j,

=

rn

1 , 2 , . .. , J 2 ,

det[ j(kfjl+)] = 0,

j,

=

=

1 , 2 , ... , J , .

(779)

However, because ] sin(k:j,+h,) det[ j ( k i j 1 + ) =

and because the indexing scheme only allows one solution to be identified out of the infinitely denumerable set known to exist, we immediately come to the conclusion that

(781) This result agrees with our earlier finding for the three-dimensional model developed with no layering. Next consider two layers with J , = J2 = 1. The determinantal equation can be written as det[ f(ktl+ ;k : , + ) ]

( 782) We see that we have a single transcendental equation in two unknown z-propagation constants. In order to give us two nonlinear equations in two unknowns, providing us with a system of nonlinear equations to solve, we =

0.

268

CLIFFORD M. KROWNE

choose two sets of weighting functions

{ ' ~ 2m(, r ) ) = 'w f 1( r ) ;' w f 2 ( r> { w2,rn (r )} = w f 1( r 1; W f 2 ( r ) .

(783a)

9

(783b)

When the weighting functions are generalized in this way, the matrix elements must be similarly generalized. The result is

r

det['l(k;,+; k:,,)]

0,

(784a)

d e t [ 2 ~ ( k ~;l/c:~+)] + = 0.

(784b)

=

We must only look for one set of physically reasonable z-propagation constants for this system of two equations. For an arbitrary number of layers and an arbitrary number of z-indexed modes in each layer, the most general solution is

M

J = 1 , 2,...,

Ji,

j,

=

1 , 2,..., J,,

m

=

1,2,..., M , (785)

i= 1

where the weights are chosen as a set M

J

(Jk;,,.(r)],

=

1,2,...,

c Ji.

i=l

Now we turn our attention to the three-layer case, M = 3, There will be two interfaces with interfacial conditions of tangential electric field continuity. The tangential electric field components are

Erlj(r,z)

=

2iKfi+ sin(k:j+z)E,!j(r; k t + ) ,

(787a)

E;.( r , z )

=

[

(78%)

E3,(r , z )

=

[eik:,+z

E i j ( r ,z)

=

2iKij+ sin(k:j+z)E,$j(r; k : j + ) ,

r1

K;j+eikt+'

- K2. 21 - e - i k : J + z ] ~ ; . ( rk:j+), ;

),

- e - i k $ + z e 2 i k : , + h , ] K3. zI+ E3 r j ( r ;k:]+ '

(787d)

E4j 2( r , z ) = [ K ZJ2 .+ eik:J+z- K Z2l .- e - i k : ~ + Z ] E i jk(zrj;+ ) , E:j(r, z )

= [eik:J+z

(787')

(787e)

- e - i k : ~ + Z e 2 i k : ~ + h , ] K : +kE: j (+r);. (787f)

The two added continuity conditions easily follow from those between the first and second layers:

q r , 4,z)

=Ei(r,

4, z ) ,

E,?(r,+,z) = E , ? (r, + , z ),

z

= h,

+ h,,

(788a)

z

= h,

+ h,.

(788b)

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

269

Placing the field expressions (787a)-(787f) into the continuity conditions at the first interface (765a) and (765b) and at the second interface (788a) and (788b) gives Jl

C

2iKjj+ sin(k:j+ hl)E ij(r ;k i j + )

j= 1

(789a) Jl

C

2iKij+ sin(k:j+hl)E:j(r;k i j + )

j= 1

(789b)

Again performing test function projections onto these equations and carrying out the inner products as was done for the single-interface case,

270

CLIFFORD M. KROWNE

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

271

272

CLIFFORD M. KROWNE

With these matrix element definitions, the mode-matching equations (790)-(793) become

c I;;Kjj+ + c I;7'K:j+ Jl

J2

j= 1

j= 1

+

c I;7-KzjJ2

=

j' Jl

J2

j= 1

j= 1

c I>:Kjj+ + c I;;"K:j+

=

J2

j= 1

c I$4-K:jJ2

1,2,. ..,J , ,

(795a)

J2

+ j = 1 I;;'-K:ji-

=

j'

C Ii?j'++K:j++

0,

j= 1

0,

=

1,2,. . . ,J 2 , (795b)

J3

+ C I;;KZ:.+

=

0,

j= 1

j= 1

j'

c I;;+K,"~++ C I;;-K:~- + c I;;K;~+ J2

J2

J3

j= 1

j= 1

j= 1

=

=

j'

1,2,. ..,J 2 , (79%)

0,

=

1,2,. . .,J 3 . (795d)

In matrix form, they become the single equation

or

q({k;m+}jm>,)K=O, M

J = 1 , 2 ,...,

C J ~ , j,=1,2

m = 1 , 2 ,...,M , (797)

,..., J,,

i= 1

where M = 3 in the general expression (797). Equation (797) indicates that we must form ( J , + J,) matrix equations for the two-layered problem. We note that the use of weight sets based o n the index J attached left superscripts to the element notation, that is, The subscript indices of the matrix elements I j j , forming a submatrix with j' and j running over all appropriate values for a particular f index, are in the local representation system of the particular interface under consideration. Putting these local

'if.

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

273

representations into (796) implies an assembly process, not unlike that found in finite-element procedures. Also similar to finite-element construction, the matrix is seen to display banded behavior, a direct result of the interface conditions only involving a limited number of unknowns. In fact, these unknowns are limited to those associated with a particular interface. Because there will be for our field problem at most four unknowns per interface, we expect the local representation about each interface to consist of six or eight clustered submatrices. Six clustered submatrices are found for the first and last interfaces, and eight for all interfaces in between. For the two-layered problem at hand, there is no interface sandwiched between two others, and only the six cluster groups are found, each of size 2 X 3. The bandedness is just becoming apparent. The single interface case for the two-layered problem can have its matrix formula (772) written in submatrix form

It is too small to develop any banded behavior. As we add more and more layers, the banded behavior becomes much more apparent and the global matrix develops huge regions, above and below a single main band, with zero submatrices. This makes the global matrix especially sparse, and encourages sparse matrix methods to be used to solve it. Following the same reasoning process as employed for the two- and three-layered problems, the matrix equation for four layers, M = 4, looks like

=

0. (799)

For the first time, we see in the center of the global matrix a single cluster of size 2 x 4 with eight submatrices, associated with the single interface sandwiched between the top and bottom interfaces. The bottom interface cluster occurs in the top leftmost corner of the global matrix and is of size 2 X 3, as expected. The top interface cluster occurs in the bottom right-

274

CLIFFORD M. KROWNE

most corner of the global matrix and is of size 2 X 3, too, as expected. Notice that the notation has been slightly upgraded in the corner clusters, but otherwise the M = 4 case follows a notation growing out of the M = 3 case. Larger global matrices may be constructed, and they only add clusters of size 2 X 4. It is instructive to examine a significantly larger global matrix, so consider one for the M = 7 case: 0 0

0 0 0 0 1;4+ 1S5r+

I'i 1554+ I 'i

I65r+ 1'1

0

0

0

0

0

0

We will write out all the local matrix elements, and then ask how they can be put into a much more compact form. It is this ability to find local elements, and then move them into the global matrix, using a streamlined

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

275

notation, that allows this method to become so amenable to numerical computation by systematic programming techniques. Now let us determine how many elements we need. First, examine the size of the matrix for a circulator with M layers. Looking at the M = 3 and M = 4 cases in (796) and (799), the number of global submatrix columns in compacted form is Ncg=1+2(M-2)+1 =

2(M - 1).

(801)

Here we have done the bookkeeping in this way: one column for the first layer where only one unknown is required, two columns for the two unknowns associated with the forward and backward traveling waves in the second layer sandwiched between the top and bottom, and one column for the third layer where only one unknown is needed. The amount of sandwiched layers is clearly M - 2, just the quantity to appear in the Ncg global matrix size formula. The global matrix is of size Ncg X Ncg. Now, returning to the number of elements to be specified for M = 7, we can use the M = 3 results in (794)-(796), appropriately generalized in notation, up to the last column, but not including the last column. Noting that the weighting functions must be the same for each submatrix row in the global matrix, recognizing the number of 4 x 2 blocks within the entire global matrix as B = 5, where B=M-2,

(802)

and identifying these blocks starting from the second submatrix column of the global matrix and ending at the Ncg - 1 submatrix column, the submatrix elements are

j ’ = 1 , 2,..., J , ,

j = 1 , 2 ,..., J , , (803)

j ’ = 1 , 2 ,..., J , ,

j = 1 , 2 ,..., J 2 , (804a)

j ’ = 1 , 2,..., J , ,

j = 1 , 2 ,..., J 2 , (804b)

276

CLIFFORD M. KROWNE

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

277

278

CLIFFORD M. KROWNE = -,-2ikIj+(hl+h2+h,+h4)j~45$+

j145$-

i’i

I ‘i

j ’ = 1 , 2 ,...,J4,

j = 1 , 2 ,..., J,,

j1?4r+ i ‘i = ,ik;,+(hl + h 2 + h 3 + h 4 )d R j w 4 , 3 ( r ) ~ ; , k ( r: j; r + ) ~ : . ( k:j+) r;

j ’ = 1 , 2 ,..., J,, j1154r- =

i ‘i

-,-

i’i

R;

+h2+h3+h

’W2,4(

j155r- = - , - 2 i k : , + ( h l J ’i

j ’ = 1 , 2 ,..., J,,

-,-

2ikIj+(hl + hZ+ h , + h4 + hs)j155$+ JPJ

+ h2 + h3 + h4 + hs)

,-

i‘i

(811~)

j = 1 , 2 ,..., J,,

(811d)

k:j+) dr,

j = 1 , 2 ,..., J 6 ,

(812a)

i’i

-,-

j = 1 , 2 ,..., J6, (812b)

r)E;.,(r; k ZJ 6 . +, ) E l5] . ( r ;k5. Z j + ) dr,

eikI,+(hl+h2+h3+h4+h~)dRjW4,4(

j ’ = l , 2 ,..., J 6 , j165r- =

j = 1 , 2 ,..., J,,

2ik:,+(hl + h 2 + h 3 + h 4 + h , ) l z ~ 6 $ +

J ’ = l , 2 ,..., J,, ;165r+= i’i

(811b)

1’ ~ 1 , 5 ( r ) ~ ; j ,k:j,+)Eij(r; (r;

j ’ = 1 , 2 ,..., Js, = -

j = 1 , 2 ,..., J,,

’

j ’ = 1 , 2 ,..., J,,

i ‘i

(811a)

r ) q j < ( rk:jp+)E;i(r; ; k : j + )dr,

jz155$+ i ‘i = , i k : , + ( h , + h 2 + h , + h 4 + h ~ ) joR’WP,4(

i ‘i

j = 1 , 2 ,..., J,,

+ h , + h3+ h 4 ) j 1 5 5 r + i’i

j ’ = 1 , 2 , ...,J,,

j 1 5 6 $ + = -,ikEj+(hl

j = 1 , 2 ,...,J4, (810d)

.)E;,( r ; k:j’+)E;’(r ; k : j + ) dr,

j ’ = 1 , 2 ,..., J,,

J156rP-

j = 1 , 2 ,..., J4, ( 8 1 0 ~ )

’

‘)i

j155r+ i’i = - , i k t + ( h l

I ‘i

dr,

2ik;,+(hl + h 2 + h 3 +h 4 ) j ~ 5 4 r +

j ’ = 1 , 2 ,..., J,,

jI?5$- =

(810b)

j = 1 , 2 ,..., J,,

(812c)

.. J5,

(812d)

2 i k j j + ( h l+ h 2 + h 3 + h 4 + h , ) j ~ 6 5 r +

J’i

’

j ’ = 132,. . .

9

36,

j=1,29.

9

279

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

j ' = 1 , 2 ,..., J 6 , j166+

i'i

j167++ =

i 'i

-

[ e i k l j + ( h l + h 2 + h 3 + h , +hs + h 6 ) -

j'=1,2,.. ., J 6 ,

j = 1 , 2 ,... ,J 7 , (814a)

j f = 1 , 2,..., J 7 ,

j = 1 , 2 ,...,J 6 , (814b)

= - -2ik:j+(hl +h2+h3+h4+h, + h , ) j I 7 6 r + 1'1

j177r+ =

- [e i k l j + ( h l +h,+h,+h,+h,

j = 1 , 2 ,..., J 6 , (813d)

- i k z , + ( h l + h 2 + h 3 + h 4 + h5 + h 6 ) e 2 i k : j + h , ]

'

j ' = 1 , 2 ,..., J , , i'i

(8 1 3 ~)

- = - - Zik;, + ( h , + hz + h3 + h4 + h , + h,)j166+ + i'i 9

j f = 1 , 2,..., J 6 ,

jI;;r-

j = l , 2 , . .. , J 6 ,

+h,)

j = 1 , 2 ,..., J 6 , (8 1 4 ~)

- e - i k l , + ( h l + h2+h3+

j ' = 1 , 2 ,..., J , ,

h,+h,

+ h6)e2ik:,+h,

j = 1 , 2 ,..., J , .

I

(815)

Here the weighting elements must obey w3,1 = w1,2y

w4,1 = w 2 , 2 ,

W 3 , 2 = W1,3,

W4,2 = w2,3,

w3,3 = W1,4,

(816a) w4,3 = W 2 , 4 ,

W3,4 = W1,5,

w4,4 = w 2 , 5 .

(816b)

There is really no need to fill in the other cases of M below M = 7 or to obtain the higher-layer cases because a general set of submatrix relationships exist to describe the general case for M 2 3. With the index p denoting the particular 4 X 2 submatrix cluster (or block), from upper left

280

CLIFFORD M. KROWNE

in the global matrix to lower right in ascending order,

j ' = 1 , 2 ,..., J p ,

j = 1 , 2 ,..., JP+',

p = 1 , 2 ,..., M

-

2 , (81%)

~ i ~ ~ w ~ , ~ ( r k$??)Ez+'(r; ) E $ ( r ; k$:') dr,

j ' = l , 2 ,..., JP+',

j = 1 , 2 ,... ,J p + l ,

p = 1 , 2 ,..., M - 2 , (818a)

j ' = 1 , 2 ,..., Jpfl,

j = 1 , 2 ,..., J p f l ,

p = 1 , 2 ,..., M - 2 ,

X

(818b)

iRjw,,,( r ) E$ '(r ; k$?,!) E$: '(r ;k z ; ' ) dr ,

j ' = 1 , 2 ,...,JP+',

j = 1 , 2 ,..., JP+',

p = 1 , 2 ,..., M - 2 ,

j ' = 1 , 2 ,..., J p + l ,

j = 1 , 2 ,..., J p + l ,

p=1,2,.

.., M

(819a)

- 2 , (819b)

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

281

X i R j w 4 , p ( r ) E $ + 2 (k$?:)E$+'(r; r; k$:') dr, j ' = 1 , 2 ,..., J p + 2 ,

j = 1 , 2 ,..., .Ip+,,

p = l , 2 , . . ., M

j ' = 1 , 2 ,..., J p + , ,

j=1,2,...,JP+,,

p = 1 , 2 ,..., M - 2 .

-

2, (820a)

(820b)

For the first and last 2 X 1 clusters (most extreme upper left and lower right locations in the global matrix), j4?j1 ++= 2i sin(kfi+h,)/Riw,,,(r)E~j~(r; k t t + ) E i j ( r k; i j + )dr, 0

j ' = 1 , 2 ,..., J , ,

jI$j

r+ =

j = 1 , 2 ,..., J , , (821)

2i sin( k t j +h, ) l R j w z ,,( r ) E;., ( r ;ktjc+ ) Ejj( r ;k f i +) dr , 0

j r = 1 , 2,..., J , ,

j = 1 , 2 , ..., J1, (822)

I 'i

jIM,Mr+=

I 'i

X L ~ ' W ~ , M - ~ ( ~ )kE$ ;+() ~E ;t ( r ; k:+) dr,

j ' = 1 , 2 ,..., JIM,

j = 1 , 2 ,..., J M ,

(824)

where the weighting function relationships become p = 1 , 2 ,..., M - 3 . w4. P - w 4 , p + l , w3,p = Wl,p+l,

(825)

These submatrix expressions need to go through a conversion which places them correctly within the global matrix. The global matrix is denoted by j , that is, with a bar over the submatrix element symbol. In the

282

CLIFFORD M. KROWNE

following mappings, the arrow indicates translation into the global system from the local submatrix systems:

ipsp + 1 d + + i j d + I ’i

j’+Zr=l(J,- 1 + J i X 1

j ‘ = 1 , 2 ,..., J p ,

- 8 i l ) ,j+.Z:P-1(2- S i 1 ) J i

7

p = 1 , 2 ,..., M - 2 ,

j = 1 , 2 ,..., J P + ] ,

(826a)

$lp,p+ld-JjdI ‘i

j’+E:P-l(Ji-lJiX1 - S i l ) , j + E : P = I ( J i + J i + l )

J ‘ = 1 , 2 , . ..,J p , ip+l.p+lr++ijc+ I I 1

j=1,2,.

+ E:P=

(2 - S i 1 ) J i . i+E:P=1(2

j ’ = 1 , 2 ,..., J p + l , izr+l,p+lrII 1 +Iz;;

Er=, ( 2 -

- 6,l)Ji

8 i l ) J , ,j +

Er=, ( J i + J , +

.., M - 2,

(826b)

7

p = 1 , 2 ,..., M - 2 ,

(827a)

p = 1 , 2 ,..., M - 2 ,

(82%)

p = 1 , 2 ,..., M - 2 ,

(828a)

f

j = 1 , 2 ,...,J p f l ,

d++ij?+ I ’ + E P - , ( ~ , + J ~ + I ) , ~ + E .-P S- i~, )(J ~, 3

j ’ = 1 , 2 ,..., J p f l ,

i1t+2,p+lr+ I 1

ize + 2, p + 1 r -

j = 1 , 2 ,..., J P + ] ,

+’v:J, ’-

+2Z:P_, J i + , , j+.Z.P=1(2- S i 1 ) J i

j ‘ = 1 , 2 ,..., J p f 2 , I 1

p=1,2,.

j = 1 , 2 ,..., J p + l ,

j ’ = 1 , 2 ,..., J p f l , $f+l,P+l I 1

. . ,J p + l ,

9

3

j = 1 , 2 , . . ,,J p + l ,

p=1,2,.

. ., M - 2,

(829a)

.-

+”.;

J,

j ’ = 1 , 2 ,..., J p + 2 ,

+2E,=

J,+l,i+EP-l(J,+J,+l)

j = 1 , 2 ,..., J P + ] ,

p = 1 , 2 ,..., M - 2 .

(829b)

Besides the 4 X 2 submatrix clusters, there will be one 2 X 1 submatrix cluster at the top left-hand side of the global matrix and one 2 X 1 submatrix cluster at the bottom right-hand side of the global matrix. The submatrices in these clusters need to go through a conversion which places them correctly within the global matrix, that is, an assembly process: $ 1 1 ++ j ‘i

+ip+

i z 2 1 r+

Jj:+ lt+Jl,j,

i’i

j ’ = 1 , 2 ,..., J , , j ’ = 1 , 2 ,..., J 2 ,

j = 1 , 2 ,..., J , ,

(830)

j = 1 , 2 ,..., J , , (831)

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

j ' = 1 , 2 ,... , J M ,

j=l,2,...,JM.

283

(833)

is a diagonally centered sparse matrix. For M layers, its size (unclustered now) is Ng: M-2

J1

+2 C

M-2 Ji+l + J M

i= 1

What about the M = 2 case? This really means that p = 0 (no 4 x 2 clusters). We drop all larger clusters and only retain the smaller first and last 2 x 1 clusters. The submatrix representation of the global matrix for this simple system looks like

The procedure to acquire the k;, eigenvalues has been covered in this section. Once these eigenvalues are known in each layer, the fields in each layer may be determined. Solution was found under the condition of finite port widths, which changes a canonically shaped circular cylindrical structure into a noncanonically shaped cylinder with sections cut off forming facets at each port location. It was also indicated that for thin-width ports, a good approximation might be to find the k$+ eigenvalues by solving the canonical cavity problem for an M-layered structure. This is relatively easy to do once the procedure for handling a multilayered structure has been developed, as in this section. The next section will provide the cavity solution. ORDERED CAVITY: m v I . DOUBLY RINGSAND HORIZONTAL LAYERS

RADIAL

Estimates of the eigenvalues K z + can perhaps be found for the case where the ports of the circulator are so narrow in azimuthal width that they can be considered as small perturbations to the central cavity fields. In this situation, the field energy entering and existing the circulator at w

284

CLIFFORD M. KROWNE

can be treated as small. Thus looking for the resonant frequencies w may help us find K z + . Magnetic wall boundary conditions exist at r = R :

=

wI

H2(r,4,z) =0,

r=R,

(836a)

H,,,(r,4,z) = 0 ,

r=R.

(836b)

Dropping the z-dependence and building it in back later, as was done in Section XXXV, a rigorous derivation of the cavity fields and resonant frequencies is possible. We already have some of the framework necessary to treat the condition on the tangential azimuthal magnetic field from the Fourier expansion (589) (Section XXIX) applied to the perimeter field: m

H F r ( R , ~=)

n= -m

Using the field expression for the i perimeter condition given previously H?'(R, 4)

A$einc.

(837)

ring and equating it to the

=N

4),

= H,,,N(R,

(838)

and employing recursion relationships, the old constraint on u i o and can be used again if the superscript indices are generalized: A$ =A@'a' n no + A Y a L .

.ao

(839)

Now we have, by (836b) and (8371, A$

=

0

and A$'ako + A$2aao= 0. For the tangential perpendicular field H,, the expansion m

Hzi

=

c

1

in'[

C"'

nhoi(r)

n = -m

1

+ bni

cz'n h b i ( r )

applies, which may also be written as m

-

2 (72 2 C'2 + 'ni n h o i ( r ) + bni n h b i ( ' ) * (843) On the perimeter, a Fourier expansion can be performed as for the H6-

Hzni

= akicfiai(r)

1

C'l

-k bni n h b i ( r )

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

285

field [see (60011: m

Ane z in$ .

Hpr(R,4)=

n=

(844)

--m

As done for the azimuthal field, H, can be equated to the i at r = R:

ring field

=N

H?r( R , 4) = H z N ( R , 4). (845) Combining the Fourier expansion (844) and the perimeter condition (843, using the field expression (4761, gives m

2

n=

=

c

n = --m

--m

[utNciiaNR

+ biNcjibNR

+a:NCii,NR and, by orthogonality, A', = C" nN

nhaNR

1 c z l nhbNR

+ bnN

+ biNC;ibNR]eing

+ a:NC,"i,NR

(846)

(847)

-k bjNCih2bNR'

Now using the recursion relationships (592a)-(592d), restated here,

+ a!&(recur)a;,,, &,= u;L(recur)u:, + &(recur)a:,, biN = bjk(recur)a:, + bj2,(recur)af0, biN = b,2; (recur) .to + b,2k(recur) a:,, u t N = a$,(recur)a:,

(592a) (592b) (592c) (592d)

in (847), one obtains the perimeter azimuthal magnetic field coefficients as a function of the modal constants in the cylinder disk: A',

= A','u:,

+ Ai2a;,,

= u~,,,(recur)Cii,,,

+

(848)

+ biL(recur)CikbNR

(recur) C,'iaNR+ b,"k(recur) CiibNR,

(849a)

+ bj&(recur) C,Z;bNR + &,(recur) Cii, NR + b,"N(recur) C,'ibNR.

(849b)

A',2 = u:; (recur) Cii, N R

Applying the magnetic wall boundary condition (836a) to the Fourier expansion (844) makes A',

=

(850)

0,

which leads to A','a:,

+ A',*u;,

=

0.

(851)

286

CLIFFORD M. KROWNE

Thus we obtain a 2

X

2 system of equations

+ A$'u;, = 0, A','U;, + AZ,2~io = 0.

A$'u;,

(841) (851)

Whereas before, for the leaky cavity making up the central part of the circulator, the modal coefficients were determined by the port constraints which ultimately depend upon the external circuit conditions, here u;, and a;, are determined solely by the cavity properties. A solution to the homogeneous simultaneous system of two equations is only possible if

Each coefficient in the determinant formula depends on the perpendicular propagation constant k:+ and the radian frequency w when considering the mth layer (the radial propagation constants can be expressed in terms of k,",). That is, A$'

= A$'( k,",

,w ) ,

(853a)

A$2

= A$2( k,",

,w ) ,

(853b)

/l;1

= A i l ( k,", , w ) ,

(854a)

Ai2

= A",(

k,m, , w ) .

(854b)

Placing (853) and (854) into (852), the equation F&y(kz",,w)

=A$'(k,",,o)Ai2(kZm+,w)- A $ 2 ( k ~ + , w ) A ~ 1 ( k ~ + 90)

=o

(855)

is found which can be solved for k,", for each specified w. Since (855) is a transcendental equation, it has an infinitely denumerable set of eigenvalues, indicated as follows:

k,",

= k,mi+(w),

j

=

1,2, ....

(856)

Each j index represents a curve on a k,",-versus-w plot. The kT+ solution is different for each type of layer m because the materials in the rings can differ from layer to layer, although we assume division of the rings to be the same. For a chosen value of w = wu, this line will intersect the individual j = 1,j = 2,. .. curves at different points. Once the k,", are

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

287

known, (841) or (851) can be used to find the modal coefficients for the mth layer: 1

an, _ a n20

A:2

--

A:' '

(857a)

We note here that the form of (855) is similar to that found in Van Trier (1952- 1954) for the simple single-region cavity with an inner anisotropic disk and one outer annulus of air. The cavity was entirely electric walled, and the inner disk filled with dc magnetically biased plasma-ferrite material providing permittivity and permeability tensors. We also note here that it is possible to solve for the cavity problem when the walls are not necessarily hard electric or magnetic walls by requiring the fields to decay to zero at infinity (Chew and Moghaddam; 1987), as is the case for isolated dielectric resonators (Kajfez and Guillon, 1986)' and applying a constructive interference superposition within the cavity as a resonance condition. This might be a better way to represent the fields for the narrow-port circulator case where most of the field sees an interface between the ferrite region and the outer dielectric at r = R. Once the perpendicular propagation constant functions (856) are determined, they can be inserted into the mth-layer field expressions and employed in satisfying the electric wall boundary conditions at the bottom and top of the circulator, as well as the continuity conditions at all the interfaces between the horizontal layers. This procedure has already been developed in Section XXXV. By selecting out one weighting function in (778), say J = 1, the assembled layered system matrix must obey

J=1,

j m = j m O : j m O ~ ( Z } ,m = 1 , 2 ,..., M . (858)

This single secular equation for the resonant frequence is solved by selecting out a particular set of perpendicular propagation constant functions for every layer in the cavity. Once this is done, the single nonlinear transcendental equation in one unknown w can be solved. j m ois set to one index choice in the set of integers for each horizontal layer m. The weighting function set now is utilized only once

288

CLIFFORD M. KROWNE

XXXVII. THREE-DIMENSIONAL IMPEDANCE WALL CONDITION EFFECTON MODESAND FIELDS It may be desired to take into account the imperfect conducting walls presented to the circulator device by the bottom ground plane and the top microstrip metal (Fiedziuszko and Jelenski, 1994a, b). This would change the boundary conditions examined here on the top and bottom of the circulator from perfect electric walls to imperfect conducting walls. The net effect would be to perturb the fields within the circulator as well as provide a new field region within the imperfect metal, thereby extending the effective volume of the device structure. A study of such imperfect wall effects on dyadic Green’s functions was previously done with regard to layered structures and anisotropic impedance boundary conditions (Krowne, 1989). For ease of discussion and simplicity, and because most ordinary circulators use isotropic conductors, only a scalar impedance condition will be presented and developed into a modified theory which will properly treat the losses and penetration into the conductor experienced by the electromagnetic fields. The scalar impedance boundary condition, appearing the same in form on the top and bottom walls of the circulator, is Z,”J,,

=

E,”,

(860)

where the index m denotes the layer, starting from the bottom at m = 1. Thus this formula only makes sense for m = 1 (the bottom layer surface) and m = M (the top layer surface). The problem at hand is to relate the surface fields within the imperfect conductors to the circulator fields immediately adjacent to the surfaces. This is easy to do, noting that surface electric current and surface magnetic current lead to discontinuities in, respectively, magnetic and electric fields (Harrington, 1961; Jackson, 1975) across an interface. For a normal vector pointing from region 2 into region 1, with the surface separating the two regions, the magnetic field discontinuity must obey fi x [H(l) - H(2)] = JS .

(861)

On the bottom conductor,

where f i b points out of the circulator at the bottom. Therefore, (861) simplifies to

2 X Hi

=

JS1

(863a)

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

289

or, in component form,

Jsrl = - H '+by

(863b)

Js+l = H,b.

Substituting the vector current relationship into the impedance boundary condition (8601, the desired vector field relationship is acquired at the bottom surface: Zji

HL

X

=

Ei.

(864)

This derivation has used the fact that

[E(') - E(2) x 2 1

=

M s= 0

(865)

at the interface between the conductor and the first region. Since

E(1) = Elby

E(2) =

El

(866)

S )

the surface electric field is found to be related to the volume electric field in layer rn = 1 by

[Ei

-

Ef]

2 = 0 * E:

X

=

EL

(867)

because we do not expect currents perpendicular to the surface. Next consider the top conductor: A

n

A

A

~ ( 2= ) HM

H(') = 0,

= 11, = Z ,

f

5

J,

+

JsM,

(868)

where 2, points out of the circulator at the top. Thus (861) simplifies to HY

X

2 = JsM

(869)

or, in component form, JsiM = H$,

Js+M =

-Hf.

(870)

Again substituting the vector current relationship into the impedance boundary condition (860), the vector field relationship at the top surface is ZYHY X 2 = EY.

(871)

As for the bottom surface, there are no currents perpendicular to the surface and so

EY

=

EY

(872)

was used to obtain (871). Return to the bottom surface ( z = 0) so that a study can be conducted of the impedance boundary condition effect on the z-directed field coefficients K$+ and K $ - , as well as finding the z-directed propagation

290

CLIFFORD M. KROWNE

constants. First write out the transverse fields for rn azimuthal sum is not needed:

=

1, noting that the

c [ Kij+eik:j+'- K'. e-ik:~+z]13:j(r; kij+), Jl

E : ( r , 2)

=

ZJ

-

(873a)

Putting these field components into the vector surface relationship (8641, two equations from the r- and +component parts are found

(874a)

c [ K ; ~ ++ K ; ~ - ] H $ (k~i ;j + )

JI

J1

z;

j=1

=

C [ K:~+- K ; ~ - ] E ; ~k (i j~+;) . j=l

(874b) Form an inner product of the first equation with H i j Fand the weight w t l ( r ) and the second equation with Eij. and the same weight:

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

291

where the superscript B has been added to emphasize that we are working at the bottom conductor. Define the local submatrix elements for these two equations: J li'j llBr+=

-z,'"~ R j ~ f l ( r ) H j j jk. i(j rl +; ) H i j ( rk; i i + )dr

-kRiwfl(r)Hij,(r; kij,+) E: j ( r ;k i i + ) dr, j ' = l , 2 ,..., J , , j111Br-= i'i

j = 1 , 2 ,..., J , , (877a)

-z,lBkR'wf,(r)Hjj.(r;k i j r + ) H i j ( rk; : i + )dr

+ jdljw f 1( r ) Hij,(r ;kir ) (r ; kii ) dr , +

+

j ' = 1 , 2 ,..., J , , i'i

j = 1 , 2 ,..., J , ,

(877b)

Zf" kRjwtl(r)Eij,(r; k i j c + ) H A ( r/;t i j + )dr - k R j w f l ( r ) E i j , ( r ;k i i , + ) E i j ( r k; i j + )dr,

j ' = 1 , 2 ,..., J , , j~llB+-= ZlB

i'i

kRj

S

B

W,

+kRiwf

1(

j = 1 , 2 ,..., J , ,

(877~)

l ( r ) E i j s ( rk; i j t + ) H A ( rk; i i + ) dr r ) E i j l( r ;kfjt+) Eii( r ;kij+ ) dr, j ' = 1 , 2 ,..., J , ,

j = 1 , 2 ,..., J , .

(877d)

Placing these elements into (875) and (8761, the descriptive equations for the bottom conductor surface impedance condition become

c j~;j

Jl

J1

~j~ +

~ r +

+

C jl;jBr-Kjj-

Jl

J1

c ~ I ; ; B + + K ~ + C '111B4-Kjjzj+

j= 1

=

0,

j'

=

1 , 2,..., J , , (878a)

=

0,

j'

=

1 , 2,..., J1. (878b)

j= 1

j= 1

j= 1

I

I 'i

Now remember that layer 1 is connected to layer 2 and so on, so that we expect that in an assembly process these equations could be added as appropriate to the top of the global matrix.

292

CLIFFORD M. KROWNE

A very special simplifying case may be entertained here. It may, at times, be a rather severe approximation, so caution should be used in employing it. Consider the case when there is no mode conversion at the bottom surface. This means that the system matrix diagonalizes, and is a reflection of the satisfaction on a mode-by-mode basis of the impedance boundary condition

Z,"J,,,

=

E3,

j

=

1 , 2 , . ..,.TI.

(879)

The system equations (878a) and (878b) become two single equations ( j ' =j ) :

jIl1Br+K:,+ + l j y ? r - K z'l,ii

=

0,

j

=

1 , 2,..., J,,

(880a)

.iI1I B + + K z1 j + +i]j 1l 1 B + - K1ziii

=

0,

j

=

1 , 2,...,J,,

(880b)

each solvable for the backward z-coefficient in terms of the forward coefficient: j I 11B r +

K 211.

= - &I I 1 l B r -

Kfi+

7

(881a)

i'i

71; 184 +

K 2' 1. -

= -lIllB+-

Kh+ *

(881b)

i' j

If this approach is accurate, then both expressions for K f j - would give the same result when evaluated. Both expressions give the correct behavior as we limit the resistive and reactive parts of the surface impedance to 0. That is,

This is the result seen previously for a perfect conducting ground plane. Let us write the relationship between the z-coefficients in the compact form K Z'J.- = a ' + i K i j + , (883)

so that the field solutions in the region m

=

1 can be made definite:

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

c K'.

293

J,

Eld r , ~ =)

21

+

c K'. Jl

H:(r,z)

[eik;J+'-

"-+j

e-iki~+z]Eij(r;kij+ ( 8) 8, 4 ~ )

j= 1

=

Zl +

[eik:J+'+ ( y l + j e - i k i , + 2 ] H ~ ( r ; k : +(885b) ),

j= 1

Now consider the top surface at z = h,. Using the field expressions (873a)-(873d), altered to be valid for rn = M , JM

E,"

[ K$+eik!+'

- KM 21 - e - i k ! + 2 ] E f ( rk:+), ;

(886a)

c [Kz+eik:+' + KM e - i k ! + z ] H f ( rk:+), ;

(887a)

=

j= 1

JM

H,"

=

21 -

j= 1

c [ K$+eik!+' + KM e - i k z + z ] H { ( rk!+). ; JM

Hr

=

21 -

(887b)

j= 1

Putting these field components into the vector surface relationship (871), two equations from the r and &component parts are found

294

CLIFFORD M. KROWNE

Form an inner product of the first equation with H;, and the weight and the same weight:

wL,M ( r )and the second equation with E$

295

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

[

j I ?I 'i MT+-=

WT M, M ( r ) E $ ( r ;

-zyT/o.i

k $ + ) H t ( r ; k $ + ) dr

1

( r ) E & ( r ; k $ + ) E g ( r ; k$+) dr e-ikz+hi.(891d)

+L R 4 4 ; ,

Placing these elements into (889) and (8901, the descriptive equations for the top conductor surface impedance condition become JM

C

.

C I.Ii'iM M T r - K U 'M

iIMMTr+KM

i'j

ZJ

+

+

zJ-

=

0,

jr

=

1 , 2,..., JM, (892a)

=

0,

j'

=

1 , 2,..., JM. (892b)

j= 1

j= 1

-

'M

J II M 'i M T + - K Z j= 1

j= 1

Now remember that layer M is connected to layer M - 1 and so on in descending order, so that we expect that in an assembly process these equations could be added as appropriate to the bottom of the global matrix. Again consider the special case when there is no mode conversion at the top surface. Referring to (8711, j = 1,2,. .. ,J M . (893) EY, The system equations (892a) and (892b) become two single equations

ZyJ,,,

=

( j r =j ) : i Z M M T r t K M + j ] M M T r - K Mtl- = I1 zj+ IJ M M T+ t K M

ii

zj+

+jrIJMM T 4 - K zIM =

0,

j

=

1,2,..., J M ,

(894a)

0,

j

=

1,2,...,J M ,

(894b)

each solvable for the backward z-coefficient in terms of the forward coefficient: (895a)

(895b) Both expressions give the correct behavior as we limit the resistive and reactive parts of the surface impedance to 0. That is, lim

z,+o

iIMMTi+ J 'i IIMMTiI 'i

= -e2ik$+ht

i

=

r, 4

=$

K M ZJ

-

= ezik$+h,K;+.

(896)

296

CLIFFORD M. KROWNE

This is the result seen previously for a perfectly conducting ground plane at a nonzero location. Again write K$- = ‘Y!+~K$+. (897) It is possible to obtain a single transparent nonlinear transcendental equation in the t-propagation constant for a single layer by using the uncoupled mode expressions at the impedance boundary wall surfaces. Since A4 = 1, using (883), .=

f f+ Iy =. f :B+j.

(898) The last two statements in this equation connect the top and bottom surfaces, and allow the straightforward derivation of the secular equation for k t j + .Invoking (881a) and (895a), fff +I

or, in a slightly different form, F s ( k : j + ) = jzj l’ lj T r + ( k ; j + )izj 1’ 1j B r -

pij+)- -(r

j 11 ’ j Br+

(k;j+)jz;;Tr-(k;j+)

= 0. (900) The procedure to solve a single transcendental equation of this format is well known, and leads to the eigenvalue spectrum

F S ( k t j + )= 0, j = 1 ,2,..., J , . (901) This demonstrates what happens when mode conversion is not a serious factor at the top or bottom conducting surfaces. When that is no longer true and mode coupling is important, then take the r-component equations (880a) and (892a) as a system to solve:

c jZ;;Br+Kjj++ c iZ./jBr-K1 c jZ;jTr+Kij++ c jZ;iTr-K1 J1

Jl

j= 1

j= 1

Jl

J1

j= 1

j= 1

,..., J , , (902a)

rj- =

0,

j’

= ,1,2

=

0,

j’

=

zj-

1 , 2,..., J , . (902b)

In matrix form,

Again, we note here that the subscripts j ‘ j on the submatrix elements have been left on the global matrix entries to explicitly show the entire indexing scheme. Strictly speaking, these submatrices are matrices in their own

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

297

right, and should only have the subscripts broken out in the element definitions. [A similar comment applies for the z-coefficients regarding the j subscripts in (903).] To have a solution, the determinant of the system must obey

This determinant is of size 2J1 X 2J,, and provides the following constraint on the z-propagation constant eigenvalues k i j +:

loss( (kij+}) = 0.

(905)

There are J, elements in the set { k i j + } ,leading us to use the class of weights {'w} to require

iDl,,ss({k~j+}) = detp({kij+})] = 0,

i= 1,2, ..., .Il,

(906)

giving a simultaneous set of nonlinear transcendental equations to be solved. Here denotes the global matrix. For many horizontal layers, it must be recognized that no longer can the backward-wave z-coefficient be expressed in terms of the forward-move coefficient, thereby eliminating a variable in both the top and bottom layers, when hard electric walls exist as discussed in Section XXXV. Now the surface impedance conditions at the top and bottom surfaces, as we have seen for a single layer, require the use of both K j j + and K i j - . In effect, one more variable has been added to the global column containing submatrices for the bottom layer, rn = 1, consequently making the addition of another global row containing submatrices necessary also. These column and row additions for the bottom surface occur at the top left-hand side of the global matrix. The following global matrix shows the first few rows to demonstrate what has been altered beyond the hard wall case seen in (800) for the M = 7 example:

r

jzllBr+

i'i

jl11Br-

i 'i

0

0

0

;p+ .izll0i'i

i'i

$124+

+2+-

0

jz21r+

jz21r-

1122r+

j122r-

0

i'i

i'i

i'i

i'i

i'i

i'i

0

0

jz22$+

$210-

0

jz32r+

i132r-

0

0 0

0

...

i'i

i'i

0

i'i

i'i

0

... ...

... ... ...

298

CLIFFORD M. KROWNE

It is apparent that the first two column submatrix elements, for the second and third rows, must be modified from the hard electric bottom wall case. Column 2, rows 2 and 3, are completely new, but extended from the hard wall global matrix case. Of course, the first row is new, too, but it has already been studied at length previously in this section. So, four modified or new submatrix elements are needed:

= --e

jzZ1r-

i‘i

j r = l , 2,..., J1,

j = 1 , 2 ,..., J , , (908a)

j ’ = 1 , 2 ,..., J , ,

j = 1 , 2 ,..., J,, (908b)

j ’ = 1 , 2,..., J , ,

j = 1 , 2 ,..., J , , (909a)

- Zik:i+hl i I : l r +

’ j r = l , 2,...,J , ,

I 1

j = 1 , 2 ,..., J,. (909b)

For the top surface, the column and row additions occur at the bottom right-hand side of the global matrix. The global matrix is shown below for the last few rows to demonstrate what has been altered beyond the hard wall case:

...

... ...

I= 0 0

0

...

... ...

0 0 0 ~ I M - Z , M - I + +~ I M - Z , M - I + - 0 i’i I’i iIM-l.M-lr+ I ‘1

j~F-l,M-lr-

0

~IM-I,M-I++

~IF-I,M-I+-

0

j~M.M-lr+

0

0

i’i

I ’i

1’1

0

0

-

0 0

0

. (910)

I ‘i

jIM,M-lr-

i’i

0

It is apparent that the last two column submatrix elements, for the secondand third-to-last rows, must be modified from the hard electric top wall case. The second-to-last column, second- and third-to-last rows, are completely new, but extended from the hard wall global matrix case. Of course, the last row is new, too, but it has already been studied at length previously in this section. Therefore, four modified or new submatrix elements are

DYADIC GREEN'S FUNCTION FOR MICROSTRIP CIRCULATORS

299

needed:

x E g ( r; k;+) dr, j ' = 1 , 2 ,...,J M P I ,

X

j = 1 , 2 ,..., JM, (911a)

E;( r ; k$+) dr, j ' = 1 , 2 ,..., J M ,

j = 1 , 2 ,..., J M , (912a)

j ' = 1 , 2 ,..., J M ,

j = 1 , 2 ,..., J M . (912b)

Next the local coordinate indexed submatrices in the 4 X 2 and 3 x 2 clusters must be assembled into the global matrix. The following mappings show the translation from the local to the global system. First are listed the results for the 4 x 2 clusters. They have been found by adding J1 to both the j ' and the j indexes of the old hard electric wall global matrix submatrices: j z e ,P + 1 d + + j p + I 1

l'+Ji

j ' = 1 , 2 ,...,J p ,

j = 1 , 2 ,..., J p + l ,

ize,P + 1 4 - + if+ I 1

I'+JI

j ' = 1 , 2 ,..., J p ,

+

j l p 1,p I 'I

+ EP- i ( J i -

1

j = 1 , 2 ,..., J P + ' ,

I + J ~ + E P = I ( ~ - ~ ,9 ~ ) J ,

p = 1 , 2 ,..., M - 2 , 6,1),1+Ji + ZP-i ( J , + J , +

1)

p = 1 , 2 ,..., M - 2 ,

(913a) 9

(913b)

--

+ 1 r+

j ' = 1 , 2 ,..., J P + ' ,

+ E P - i ( J , - i + J , X l -6,ih

-+

'~~~J,+EP~,(2-6,,~J,,I+Jl+zp~l(2-s,,)~,

j = 1 , 2 ,..., J p + l ,

p = 1 , 2 ,..., M - 2 ,

(914a)

300

CLIFFORD M. KROWNE Iz" I 1

'"+

r-

+

j ' = 1 , 2 ,..., Jp+l,

l , p + 1 ++ +

...,Jp+,, 'ZY+ I1

j'

= 1,2,

ly+

]'+J,

I1

+

. . .,Jp+

j

1,

ize + 2, p + 1 r +

= 1,2,

+

= 1,2,.

j ' = l , 2 , ..., JP+,,

+Ef=, ( J i + J i + , )

p=1,2,

I , + , ,j + J ,

+ Ef- ,(2-

&)J,

(915a)

9

..., M -

2, (915b)

9

p = 1 , 2 ,... , M - 2 ,

j = 1 , 2 ,..., JP+,,

j i c + 2, P + 1 r I1

+ 2J1+ 2Ep=

j'

. .,JP+,,

- ..,Jp+ 1,

j j r+

Si,)J,

p = 1 , 2 ,..., M - 2 ,

j&?;J1+ Ep=,(Ji+Ji+,),j + J ,

I 1

j'

+ Ef= , ( J i + J , + , ) , j + J , + Efp l(2-

j = 1 , 2 ,..., JP+,,

9-

7

p = 1 , 2 , ..., M - 2, (914b)

j = 1 , 2 ,..., Jp + , ,

++ j'=1,2,

+ Ef- ,(2 - S , , ) J , , j + J , + EfF l ( J , + J i + ,)

"jYJl

(916a)

.-

"i'.:

,,j + Jl + Ef= ,(Ji+ J i + j = 1 , 2 ,... , J p + , , p = 1 , 2 , ..., M - 2. (916b) 2 J 1 + 2Ef= J , ,

1)

9

Now treat the 3 x 2 submatrix clusters at the top left-hand side of the global matrix and at the bottom right-hand side of the global matrix. The submatrices in these clusters need to go through a conversion which places them correctly within the global matrix, that is, an assembly process: j z l l E r + --t j f E r +

i'i

j'j

Y

j ' = 1 , 2 , . . . , J1 , i~llEri' 1 ~

JjBr; I'.I+JI

jj$+

I 'i

i'+JI,i

=

1 ,2,..., J,,

j

=

1,2,. . .,J,, (91%)

j

=

1 ,2,..., J1, (918a)

j

=

1 ,2 ,...,J 1 , (918b)

j

=

1,2,..., J,,

j

=

1,2,...,.11, (919b)

(917a)

9

j ' = 1 , 2,..., J , , ljlld+

j

'

j ' = 1 , 2 , . . . , J1 ,

ip+- $ ~

j

'i

]'+J,,j+J,

j' jz21r+

i 'i

=

3

1 , 2 , . . . , J1,

jj!+ ]'+2Jl.j

7

j' = 1 , 2,..., J , , jz21ri'i

+

jfr]'+ZJ,,j+J,

j'

(919a)

=

9

1 , 2,..., J2,

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

j'

=

1,2,.. ., J M ,

j

=

301

1 , 2,..., JM, (921a)

._ ' I ~ ' M r - ~ ' I ~ ~ 2 E Ji,j+2EE;' ~ ; '

j' j]M. M T+

= ~

1,2,...,J,,,,

j

Ji

9

=

1,2,..., J M , (921b)

jjT+ l't2E:fl;' J , t J M , j + 2 E E ; ' J ,

I 'i

j r = 1,2,..., J M , 'IM, M T I 'i

-,j j T 1' + 21:;

OF THE

=

Ji + I,, j t 2EE;'

j r = 1,2,. . .,J M ,

~ V I I I SUMMARY .

j

j

=

9

1 , 2,..., JIM, (922a) Ji

+JM

3

1,2,. ..,JIM, (922b)

THREE-DIMENSIONAL THEORY

Three-dimensional theory was developed in a more systematic way from the outset, compared to the two-dimensional model. This was desirable in order to determine the correct modal set in each annulus and the inner disk. Thus the governing equations (Section XVIII) were diagonalized (Section XIX), leading to a transparent single-matrix governing equation. The characteristic equation, relating the radial modes and the perpendicular propagation constants, was also found using a rectangular spectral domain formulation, to provide a check and show the equivalence of the two approaches (Section XX). Next the general forms of the transverse field components (Section XXI) in the plane of the circulator (xy-plane) were found, in preparation for obtaining the fields in the inner cylinder disk (Section XXIII) and cylindrical annuli (Section XXIV). With the addition of the z-field dependence (Section XXV), the boundary conditions at the disk-first-annulus interface (Section XXVII), between annuli (Section XXVIII), and at the Nth-annulus-outer region interface (Section XXIX) were imposed. This allowed the recursive dyadic Green's functions to be found in the inner disk and annuli (Sections XXX and

302

CLIFFORD M. KROWNE

XXXI), and specifically for the Nth annulus which is of particular interest in matching to the external circuit (Section XXXII). The scattering parameters were then determined for a three-port circulator (Section XXXIII), with a substantial upgrade in the theory resulting from the increase in the number of interfacial components available. An example of how this three-dimensional theory reduces to the twodimensional theoretical model was executed to confirm the relationship of the two theories (Section XXXIV). With the further addition of horizontal layering, a proper way to model the z-field dependence was developed (Section XXXV). A related structure to the doubly ordered circulator, the doubly ordered cavity, was considered because it requires no approximations along the walls, and allows the determination of the perpendicular propagation constant eigenvalues, which can be close to those of the circulator (Section XXXVI). The issue of losses was treated perturbationally, in a power sense, for imperfect conductors (Section XXVI), and similar constructions may also be done for the volume dielectric and permeability contributions. But the effect of losses in the permittivity and permeability tensors (volume effects) is automatically taken into account by the formulation here, and so the resultant fields contain this information. That information is obviously contained in the s-parameters, too. So, it is possible to extract rigorously from the fields (Sections XXIII-XXV) and s-parameters (Section XXXIII) the volume power loss information, obtained earlier in an approximate fashion (Section XXVI). To get the surface loss information rigorously, impedance wall conditions must be applied (Section XXXVII), and this leads to rigorously determined propagation constants, fields, and s-parameters. And one should be able to extract rigorously, if desired, from the surface fields, the surface power loss at individual surfaces. Here that loss would occur at the top microstrip metal conductor and the bottom ground plane. There is little doubt that the modeling and numerical simulation of circulator device structures with many radial layered rings, with possibly added horizontal layering, is a very sophisticated problem. Although the dyadic Green’s function appears to be stated in a closed form, this occurs because of the compact formalism introduced to deal with the recursive nature of the problem. Therefore, in reality, the Green’s function is a hybrid closed-form-algorithm type of solution to the partial differential equation system. For only a very few radial rings, the Green’s function can be written down in closed analytical form. Beyond that number, the algebra becomes prohibitive. However, even for this situation, the Green’s function can be extremely complex, as has been noted for the case of planar multilayers in propagating transmission structure problems (Das

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

303

and Pozar, 1987). When the horizontal layering is added to the circulator problem, a mode-matching approach was needed which did not even allow us the luxury of developing an additional streamlined recursive formula(s). The result is that the doubly layered problem is not doubly recursive, at least with the approach taken here.

xxxrx. NUMERICAL RESULTSFOR

THE TWO-DIMENSIONAL CIRCULATOR MODEL

The geometric shape of the circulator is shown in Fig. 1, which essentially shows the device structure in top view. Generally, the device has a finite thickness, and this is denoted in the bottom part of the figure by the dark region which shows a cross-sectional view of only the cylindrical part of the device, not the exiting and entry microstrip lines seen in the top part of the figure. Although the theoretical device can possess any number of arbitrarily located ports, here they are shown as three symmetrically disposed

4 = +n/3 OUTPUT

FIGURE1. Circulator computational regions.

304

CLIFFORD M. KROWNE

ports. The radial sectioning is drawn for a device with one inner disk and four annuli (any number can be chosen, depending upon the application). The labeling scheme here would be i = 0 (inner disk), and i = 1 (first annulus), 2, 3,4, (last annulus). Each annulus may be a different thickness, and made up of differing material characteristics. This includes the possibility that some annuli may be ferrite, whereas some other annuli may be dielectric (easily modeled by turning off the biasing dc magnetic field and adjusting the other physical parameters accordingly). An arbitrary point within the circulator is located at ( r , 4). The width of the microstrip lines is w (of course, all the lines can be selected to be different, if it is desired), and the angular extent of a port is A&. Here the ports are located at + = - ~ / (input), 3 ~ / (output), 3 and T (isolated). When the disk and annuli are selected to model rapid radial variations, the disk region radius may be made small and the annuli thicknesses made vanishingly tiny to allow an arbitrarily large number of them. When no radial variations in the parameters occur, the problem reduces to a uniform circulator which requires only the disk region. Most likely, an actual problem will be somewhere in between these two extremes. Parameter dependence comes about through the permeability tensor elements

where the real and imaginary parts of the diagonal and off-diagonal tensor elements are given by Soohoo (1960): p‘‘=l+

w m w o [wo” - w y 1 - a;)]

[ wo” - w 2 ( 1 + a ; ) ] 2+ 4w2wo”a; ’

(925a)

(925b)

K’

=

-

+ a;)] [ wo” - w 2 ( 1 + a ; ) ] 2+ 402wo”a; , w,w[ wo” - w y 1

The frequency quantities in these expressions are

(926a)

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

305

and w which are, respectively, the magnetization frequency, the Larmor precessional frequency of the electron in the interhal magnetic field Hi,, and the microwave rf frequency. y is the gyromagnetic ratio. In (925) and (926), a, appears. It is the dampening factor allowing the magnetization to line up with the prevailing field, and may be given by AH am = - y - . (929) 2w AH is the magnetic field linewidth. For the cases to be considered here, the actual magnetization is very nearly the saturation value

M, = M,. (930) This choice is consistent with the requirement to have a net internal effective magnetic field Hi, acting on the magnetic material. It is also consistent with (but not identical to) the decomposition (Soohoo, 1960): M,

=

Md, -k m,,eiof,

(931) where the net total magnetization M, is assumed to be essentially in the static magnetization direction Md, which has been taken in this study to lie in the z-direction. The internal magnetic field used in (928) is directly related to the applied dc biasing magnetic field and the static demagnetization factor rj,, which is, in general, a tensor:

Hi = Happ- lGdM.

(932) f i d takes into account the demagnetization effects due to the nonellipsoidal shape of the ferrite material, assuming the ferrite is uniform. Since our model assumes that the primary biasing field effect is in the z-direction, we need to obtain the z-component from (932). Equation (932) written out explicitly is

Thus the z-component of the internal dc biasing field must be from (932): = Happ,z

- r j , x x M x - rjdzyMy - fidzzMz*

(934)

If we recognize that the transverse components of the magnetization may be much smaller than the perpendicular components, namely, Mx, y << Mz

3

(935)

306

CLIFFORD M. KROWNE

then it may be appropriate to simplify (933) to = Happ,z

- NdrrMz'

(936)

In the calculation to follow, we have, in fact, equated M, to M,. Dielectric loss can be included by using a complex dielectric constant &

= E'

+ i&"

,

(937)

tan 6 . (938) Figure 2 shows a top view of the microstrip metallization used to match the circulator ports to the outside circuit, generally in the 50-R system. Each port in this diagram has a single quarter-wave transformer matching the circulator to the 50-fl system. The nominal geometric and material parameters used in the simulations were R = 0.279 cm, w 1 = 0.096 cm, w 2 = 0.030 cm, L = 0.241 cm, H = 0.051 cm, conductor thickness = 0.0005 cm, 47rM, = 1780 G, AH = 45 Oe, = 15.0, and tan S = 0.0002. Figure 3 gives the s-parameter results for s21 (through port), s31 (isolated port), and sI1 (input port), over the frequency range 6-11 GHz (Krowne and Neidert, 1995)'. The calculations were done for a uniform circulator with the number of azimuthal terms truncated at +n,,, = + 9 for the E" = E '

'Figures in this section appeared in Krowne and Neidert, 1995 which was first presented at 25th EuMC in Bologna.

R10.279cm W1=0.096cm L10.24lcm W2=0.030cm (10 Ohma) Three Way Symmetry

FERRITE: 4znMe=1780 Q, AH-41 00, Er=lS.O, Tanar0.0002 Subatrate Thlckneea=O.OSlcm Outalde Clrcle: Er-lS.0, Tan6=0.0002 Conductor Thlckneaa=O.OOOScrn

FIGURE 2. Circulator pattern and material parameters.

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

307

0 THROUQH

-5

-15

-20 6

9

8

7

FREQUENCY

10

11

(GHz)

FIGURE 3. s-parameters for a uniform circulator. -8

-9

-10

c e

-11

2

e -13

-14

-15

6

6.5

7

7.5

8

8.5

9

9.5

FREQUENCY (GHZ)

FIGURE 4. Uniform circulator isolation convergence versus nmax.

10

308

CLIFFORD M. KROWNE

.

t

L

.

I

I

I\

c e P

.

-10

Y

3

P

-15

0

5

10

15

20

n(max) FIGURE 5. Convergence of isolation at 7 and 9 GHz with a matching circuit microstrip line transformer. 0

-5

L

P

-10

-15

-20 0

5

10

15

20

n(max) FIGURE6. Convergence of isolation at 7 and 9 GHz with only a disk (reference impedance Z , = 50n).

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

309

0

-5

6

e 2

0

-10

-20

L 0

5

10

15

20

n(max)

Z,,

FIGURE7. Convergence of isolation at 7 and 9 GHz-disk = 280,).

only (reference impedance

upper and lower bounds. In this figure, we notice that the main operating bandwidth is 6-9 GHz. Figure 4 gives 8 3 1 versus frequency f over the 6-10-GHz band. The curves are parameterized in terms of n,,, = 3,6,9,18,36,72. At n,,, = 9 or beyond, the numerical result has converged to within about f0.4 dB of the final value. Figure 5 selects two frequencies, 7 and 9 GHz, in order to plot 8 3 1 versus nmax= 1,2,. . .,20. It is seen for these two frequencies that is extremely well converged beyond nmax= 15 and that the numerical oscillations seem to die down at and beyond n,,, = 12. When the quarter-wave matching transformers are removed from the device structure, the numerical oscillations drastically reduce, being essentially gone by nmax= 10 and small beyond nmax= 5 as seen in Fig. 6. Figure 7 shows similar results for the removed transformers when the reference system is changed to 28 a, a system more compatible with the small impedances characteristic of a thin-substrate circulator thickness. A possible variation of the applied magnetic field H,,,Cr/R) versus r / R is shown in Fig. 8, where there is a 4.5% climb in magnitude over the r = 0 value at r / R = 0.62 and an 8% decline over H(O) at r / R = 1. We would

310

CLIFFORD M. KROWNE ,

r -

,

,

,

,

t

0'95

0.9

0

0.2

0.4

0.6

0.8

r/R FIGURE8. Happverus r / R . 0

-5

-10

3

-I58 -20

7

Q

8

10

11

FREQ GHZ

FIGURE9. Circulator s-parametersversus frequency with Happnonuniform from Fig. 8.

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

311

1

0.9

0.8

0.7

0.8

0.5

0.4

0

0.1

0.05

0.2

0.15

Radius , r

FIGURE10. Nzzversus radius r for R

=

0.25

0.3

(cm)

0.2794 cm and H = 0.0508 cm.

0

-5

-10

-15

I

-20 6

7

,

.

8

FREQUENCY

,

,

B

10

11

(QHZ)

FIGURE11. Circulator s-parameters versus frequency with N,, nonuniform from Fig. 10.

312

CLIFFORD M. KROWNE

hope such a modest field nonuniformity would not significantly change or damage the circulator performance, and this is seen to be the case in Fig. 9, which plots S,, versus frequency. Five regions were chosen to model the nonuniform applied magnetic field, with the radius of the disk and outer radii of the annuli being 0.052,0.122,0.227,0.262,and 0.279 (the value of R , the radius of the whole circular circulator structure). The applied magnetic field was set in the five regions as Happ= 1783 (disk), 1809 (first annulus), 1844 (second annulus), 1798 (third annulus), and 1623 Oe (fourth annulus). Demagnetization factor Ndrt, unlike the applied magnetic field, may undergo a large variation over the radius of the entire circulator device. Such a variation is shown in Fig. 10. Although NdLzremains between 0.91 and 0.85 over two-thirds of the device radius, it drops to about 0.46 at r = R , a 50% reduction in size. Assessing the effect such a variation has on the circulator performance is of great interest, and can be observed in Fig. 11. Compared to a uniform circulator, the frequency of maximum isolation is shifted upward by about 0.5 GHz, and the resonant spike moves above 11 GHz. Five regions were chosen to model the nonuniform

-9

-1 0

-1 1

-12

-13

L

-1 -I4 5 6

6.5

7

7.5

8

8.5

9

9.5

10

FREQUENCY (GHZ)

FIGURE12. Isolation versus frequency with curves parameterized in terms of nmax.A single ferrite material is used for the circulator, with a five-region model employed to represent the demagnetizing factor.

DYADIC GREENS FUNCTION FOR MICROSTRIP CIRCULATORS

313

FREQUENCY (QHZ) FIGURE13. Isolation versus frequency with curves parameterized in terms of the number of regions NR used to model the demagnetizating factor.

demagnetization factor, with the radius of the disk and outer radii of the annuli being 0.150, 0.225, 0.250, 0.265, and 0.279 (the value of R, the radius of the whole circular circulator structure). The demagnetization factor was set in the five regions as Ndrr= 0.9 (disk), 0.84 (first annulus), 0.75 (second annulus), 0.64 (third annulus), and 0.50 (fourth annulus). Figure 12 gives sgl versus frequency f over the 6-10-GHz band with the curves parameterized in terms of nmax= 3,6,9,18,36,72. At n,,, = 18 or beyond, the numerical result has converged to within about kO.25 dB of the final value. We may wonder just how many annuli are required to obtain an acceptable approximation to the circulator performance, when the actual Ndrz variation is that provided in Fig. 10. Figure 13 gives us an idea what to expect in s-parameter behavior. Isolation versus frequency f over the 6-10-GHz band is given in the figure, with the curves parameterized in terms of the number of regions NR = 1 + N , where the number of annuli is N = 0,5,15,49. Each annulus, including the disk (i = O), has equal radial thickness. For NR = 6 or greater, the numerical result has converged to within about ItO.1 dB of the final value.

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CLIFFORD M. KROWNE

6

7

8

9

10

11

FREQUENCY (GHZ) FIGURE14. Circulator s-parameters versus frequency. A two-region ferrite material is used for the circulator: 47rMs = 1780 G (inner region); 4 r M S = 550 G (outer region).

As mentioned before, intentional use of different ferrite materials for the disk and annuli regions may be desirable for some circulator designs. Figure 14 gives the numerical results for a two-region device with the inner region (disk, I I R / 2 ) 4mMs = 1780 G and the outer region (first annulus, r > R / 2 ) 4mMs = 550 G. Uniform Happ= 1780 Oe and Ndrz = 1.0 were assumed. The shapes of the s-parameter curves are noticeably different than the other ones found earlier, but the bandwidth and band center are almost the same as before. Figure 15 demonstrates a tremendous advantage of using a recursive Green’s function approach for solving the circulatdr problem. That advantage is in the extremely quick computation time required for determining a set of s-parameter curves over the 6-11-GHz range involving about 25 sampling points. A comparable computation done using a general intensive numerical method like finite elements could take anywhere from a few hours to a day. This is based on the fact that a single frequency point takes a few seconds on a MacIntosh Quadra 650 using the Green’s function approach, but several hours on a workstation, Vax, or Cray

DYADIC GREEN’S FUNCTION FOR MICROSTRIP CIRCULATORS

315

Fa w

8

a a w a

y.

MACINTOSH QUADRA 660

4 Y

0

2

4

6

8

10

12

NUMBER OF REQIONS NR

FIGURE15. Calculation time versus the number of regions NR using one microsctrip matching transformer section.

supercomputer when acquiring both the s-parameters and the field distributions using a finite element code. It is the combination of the Green’s function approach on a canonical type of structure and mathematical format which allows this superb numerical efficiency. The computer code is in FORTRAN, but any other language appropriate for scientific computation would be just as acceptable. The code runs on MacIntosh or IBM-compatible 80 X 86 desktop computers. The results shown here were done on a MacIntosh Quadra computer. The algorithm for the NR = 1 number of regions case (the uniform case, no annuli used) is somewhat different than that for many regions, which is why the curve first goes down, then up, For NR 2 2, the calculation time per frequency point goes up linearly with the number of regions NR,and is roughly given, as seen by Fig. 15, by the formula t = N R / 2 (seconds) for one matching section and nmax= 9. The results in this figure represent average times for calculations done in the frequency band 6-11 GHz. Specific calculations can be twice as fast or twice as slow as the average time.

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XXXX. OVERALL CONCLUSIONS The first part of this chapter treated the two-dimensional modeling of electromagnetic behavior in inhomogeneous ferrite circulator devices (Sections I-XVI). The two-dimensional recursive Green’s function theory results in elegant formulas, which can be conveniently coded to make a computer program in FORTRAN. The computer program runs very quickly (Section XXXIX), far surpassing anything possible employing intensive numerical techniques such as the finite-element and the finite-difference methods. Naturally, there has been a trade-off taken in the approach here, namely that we are limited to canonical structures of cylindrical type. However, many circulators produced in the past and today possess much circular symmetry, and so there is a lot of justification for developing an analytically sophisticated theory and the consequent computer code. Furthermore, we expect many of the basic physical features of any shaped circulator to be approximately described by the study we have undertaken. Although the two-dimensional theory is very compact and elegant, the theory cannot account for thickness effects. Additionally, it is impossible for the two-dimensional theory to rigorously describe any horizontal layering, which has actually been examined by some industrial firms. Thus it was the intent of the second major thrust in this circulator research to obtain a more comprehensive theory which could undertake to model these new facets of the problem, while still retaining as much mathematical-physical elegance as possible. This has been accomplished by developing a three-dimensional recursive Green’s function theory capable of handling any thickness ferrite (Sections XVII-XXXIX). The horizontal layering has been accomplished by an addition to the basic three-dimensional theory involving mode matching. The three-dimensional theory is beautiful in its form, but there is no question that the case of a doubly ordered circulator with many layers in each space (radial and horizontal) will require considerable care in numerical evaluation. The electromagnetic circulator device is an active control component device in that it may allow interaction of an outside user to modify its response by a control variable, namely the dc biasing magnetic field. It is not, however, active in the sense of allowing energy exchange from one form of wave motion or flow to another, such as that found in heterojunction bipolar transistor devices (Krowne et al., 1995). Of course, the manufacturers of circulators may choose to fix the dc bias field once the optimum value is found for the intended application. Then the circulator becomes a standard control component enabling such devices as isolators

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317

and transmit/receive switches to be constructed. Use of circulators in either the active or the passive mode is possible. Although we have used a Green’s function approach involving recursion and mode matching, for the previously stated reasons, the reader may wish to examine the work done here in the context of other techniques. There are a number of reviews and books covering analytical and numerical methods in electromagnetics and physics, which we will not cover now. Besides an earlier work in this series by the author (Krowne, 1995), a very short but nice summary of numerical methods for passive electromagnetic components can be found in Sorrentino (1988).

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Fiedziuszko, S., and Jelenski, A. (1994a). The influence of conducting walls on resonant frequencies of the dielectric microwave resonator. IEEE Trans. Microwave Theory Tech. MTT-19, 778. Fiedziuszko,S., and Jelenski, A. (1994b). Double dielectric resonator. IEEE Trans. Microwave Theory Tech. MTT-19, 779. Galejs, J. (1969). “Antennas in Inhomogeneous Media,” International Set of Monographs in Electromagnetic Waves (A. L. Cullen, V. A. Foick, and J. R. Wait, Eds.). Pergamon Press, Oxford, England. Gamo, H. (1953). The Faraday rotation of waves in a circular waveguide. J . Phys. SOC.Japan 8, 176. Gaukel, K. M., and El-Sharawy, E.-B. (1994). Three-port disk circulator analysis using only port segmentation. IEEE Microwave Theory Tech. Symp. Dig., 925. Gaukel, K. M., and El-Sharawy, E.-B. (1995). Eigenvalue matrix analysis of segmented stripline junction disk circulator. IEEE Trans. Microwave Theory Tech. MTT-43, 1484. Gentili, G. G., and Macchiarella, G. (1994). Efficient analysis of planar circulators by a new boundary-integral technique. IEEE Trans. Microwave Theory Tech. MTT-42,489. Gerdine, M. A. (1969). A frequency-stabilized microwave band-rejection filter using high dielectric constant resonators. IEEE Tmns. Microwave Theory Tech. MTT-17, 354. Glisson, A. W., and Wiltron, D. R. (1980). Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces. IEEE Trans. Antennas and Propagation AP-28, 593. Harrington, R. F. (1961). “Time-Harmonic Electromagnetic Fields.” McGraw-Hill, New York. Helszajn, J., and James, D. S. (1978). Planar triangular resonators with magnetic walls. IEEE Trans. Microwave Theory Tech. MTT-26, 95. Helszajn, J., and Lynch, D. J. (1992). Cut-off space of cloverleaf resonators with electric and magnetic walls. IEEE Trans. Microwave Theory Tech. MTT-40, 1620. Helszajn, J., and Nisbet, W. T. (1992). Complex gyrator circuit of junction circulators using weakly magnetized ring resonators. IEE Proc., Part H, 139, 319. Hernandez-GI, F., Perze-Leal, R., and Gebauer, A. (1987). Resonant frequency stability analysis of dielectric resonators with tuning mechanisms. IEEE Microwave Theory Tech. Symp. Dig., 345. Hord, W. E., and Rosenbaum, F. J. (1968). Approximation technique for dielectric loaded waveguide. IEEE Trans. Microwave Theory Tech. MTT-16, 228. How, H., Fang, T.-M., Vittoria, C., and Schmidt, R. (1994). Design of six-port stripline ferrite junction circulators. IEEE Trans. Microwave Theory Tech. MTT-42, 1272. Huang, J.-W., and Tzuang, C.-K. C. (1994). Green’s impedance function approach for propagation characteristics of generalized stripline and slotlines on nonlayered substrates. IEEE Trans. Microwave Theory Tech. MTT-42, 2317. Jackson, J. D. (1975). “Classical Electrodynamics.” Wiley, New York. Kajfez, D. (1987). Indefinite integrals useful in the analysis of cylindrical dielectric resonators. IEEE Trans. Micrvwaue Theory Tech. MIT-35, 873. Kajfez, D., and Guillon, P. (Eds.) (1986). “Dielectric Resonators,” The Artech House Microwave Library. Artech House, Dedham, Mass. Kales, M. L. (1952). “Modes in Waveguides That Contain Ferrites,” NRL Report 4027, Naval Research Laboratory, Washington, D.C. Kales, M. L. (1953). Modes in wave guides containing ferrites. J . Appl. Phys. 24, 604. Kales, M. L., Cham, H. N., and Sakiotis, N. G. (1953). A nonreciprocal component. J . Appl. Phys. 24, 816.

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Kishi, N., and Okoshi, T. (1987). Proposal for a boundary-integral method without using Green’s function. IEEE Trans. Microwave Theory Tech. MlT-35, 887. Kobayashi, Y., and Tanaka, S. (1980). Resonant modes of a dielectric rod resonator shortcircuited at both ends by parallel conducting plates. IEEE Trans. Microwave Theory Tech. MlT-28, 1077. Krowne, C. M. (1983). Cylindrical-microstrip antenna. IEEE Trans. Antennas and Propagation AP-31, 194. Krowne, C. M. (1984a). Fourier transformed matrix method of finding propagation characteristics of complex anisotropic layered media. IEEE Trans. Microwave Theory Tech. MlT-32, 1617. Krowne, C. M. (1984b). Green’s function in the spectral domain for biaxial and uniaxial anisotropic planar dielectric structures. IEEE Trans. Antennas and Propagation AP-32, 1273. Krowne, C. M. (1989). Relationships for Green’s function spectral dyadics involving anisotropic imperfect conductors imbedded in layered anisotropic media. IEEE Trans. Antennas and Propagation AP37,1207. Krowne, C. M. (1993). Waveguiding structures employing the solid state magnetoplasma effect for microwave and millimeter wave propagation. IEE Proc., Part H, 140, 147. Krowne, C. M. (1994a). Vector variational and weighted residual finite element procedures for highly anisotropic media. IEEE Trans. Antennas and Propagation AP-42, 642. Krowne, C. M. (1994b). “Theoretical Techniques for Analyzing Nonreciprocal Microwave Planar Devices: Modeling, Simulation and Relationship to Experiment,” Workshop on CAE, Modeling and Measurement Verification, London, pp. 110-1 15. Krowne, C. M. (1995). “Electromagnetic Propagation and Field Behavior in Highly Anisotropic Media,” In Advances in Imaging and Electron Physics (P. W. Hawkes, Ed.), pp. 79-214. Academic Press, San Diego. Krowne, C. M., Ikossi-Anastasiou, K., and Kougianos, E. (1995). Early voltage in heterojunction bipolar transistors: quantum tunneling and base recombination effects. Solid State Electronics 38, 1979. Krowne, C. M., Mostafa, A. A., and Zaki, K. A. (1988). Slot and microstrip guiding structures using magnetoplasmons for nonreciprocal millimeter wave propagation. IEEE Trans. Microwave Theory Tech. MlT-36, 1850. Krowne, C. M., and Neidert, R. E. (1995). Inhomogeneous ferrite microstrip circulator: theory and numerical calculations using a recursive Green’s function. European Microwave Conf.Dig., 414. Kurokawa, K. (1958). The expansion of electromagnetic fields in cavities. IEEE Tmns. Microwave Theory Tech. MlT-6, 178. Lakhtakia, A. (1994). “Beltrami Fields in Chiral Media.” World Scientific [Series in Contemporary Chemical Physics, vol. 21, Singapore. Lax, B., and Button, K. J. (1962). “Microwave Ferrites and Ferrimagnetics.” McGraw-Hill, Lexington, Mass. Li, C.-L. (1993). “Measurement of Electromagnetic Properties of Materials Via Coaxial Probe Systems,” Ph.D. Thesis, Department of Electrical Engineering, Michigan State University. [MSU Report No. 7, Contract No. B-225383 to Boeing Defense and Space Group.] Li, C.-L., and Chen, K.-M. (1994). Determination of dielectric properties of materials using a coaxial cavity system driven by a coaxial line. IEEE Trans. Microwave Theory Tech. MlT-42, 2195. Li, L.-W., Kooi, P.-S., Leong, M.-S., and Yeo, T.4. (1994). Electromagnetic dyadic Green’s

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Electron Holography and Lorentz Microscopy of Magnetic Materials MARIAN MANKOS T.J. WatsonResearch Center, IBM Corporation YorktownHeights, New York 10598

M. R. SCHEINFEIN and J. M. COWLEY Department of Physics and Astronomy, Arizona State University Tempe,Arizona 85287-1504

. .. ........ . . . .. . ..... . . ... . ... .. . . . . . . . . ...... . . . . . . . .. . . ..... . . . .. . . ..... . . ..... . . . .... . . .... ... . .... .. . . .... .... . . .... ... . .. .. . . .... . .... ... . . ....... ..... . . . ... . . ....... . .. . . . . . . . ..... . . ..... . . .. . ...... ... . .. . . . . . .. .......... . . . . . . ... . . . ... ... .... . . . .. . . ... . . . . . . ...... . ... .. . ..... .. ... . . ... . ..... ...... ... ... . .... . . . ..... . . . . ..... ...... ... . ...... . . .....

I. Introduction .. . . . . .... . . . . . 11. Lorentz Microscopy. .... . . . ..... . . . ... . . . , ... A. Particle-Wave Duality of Electrons . . . . . .. .. B. Fresnel Mode . . . . . .. . . . ... . C. Differential Phase Contrast Mode. ... D. Small-Angle Electron Diffraction . E. Foucault Mode. . .. ..,. . . ... .. . 111. Electron Holography . ..... . . ... .. . . . . A. Historical Foundations , ... B. Basic Principles . . .. . . . . . C. STEM Holography Modes . . . . . . ....... . .... IV. Applications . . . . .... . A. Magnetic Thin Films .. .. . ....... B. Multilayer Structures . . . . ... ... . . C. Fine Magnetic Particles. . . . . ..... .. . . .. . V. Conclusions , . . .. References .... .. . . ...... . ..

323 333 333 335 350 358 360 362 362 363 373 387 387 400 408 422 424

I. INTRODUCTION The goal of this work is to give a comprehensive account of the principles, practice, implementation, and achievements of electron holography and Lorentz microscopy techniques applied to investigations of magnetic materials. The focus is on off-axis electron holography, a new absolute magnetornetric measurement method for thin magnetic specimens. When implemented in a scanning transmission electron microscope, it provides a powerful micromagnetic analysis technique with high sensitivity ( 10- l 6

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Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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emu) and high spatial resolution (a few nanometers). We begin with a brief historical review. The first observations of magnetism in materials were made at the beginning of modern civilization. It is claimed that the Chinese used the compass two and a half millennia BC; the power of lodestone (magnetite) to attract iron has been known since 600 BC. Socrates observed that magnetite can induce iron to acquire attractive powers. Many years passed before Gilbert (1540-1603) correctly assumed that the compass needle orients itself in the earth’s magnetic field. The first quantitative investigations of magnetism were performed by Coulomb who used torsion measurements to determine the forces between magnetic poles. A comprehensive description of electromagnetism was then published by Faraday and Maxwell in the second third of the 19th century. In 1907 Weiss introduced a molecular field in order to model the magnetic interaction, and Heisenberg later described in the framework of quantum mechanics this interaction as an exchange effect. Weiss also predicted that the state of magnetic saturation is the thermodynamic equilibrium state at temperatures well below the Curie point. The fact that magnetic materials are divided into regions of uniform magnetization (domain structure) was first discussed by Bloch in 1932, and subsequently Landau and Lifshitz predicted that domains are formed through the minimization of the total energy of the system. The domain concept was further refined by Nkel and Kittel, leading to the micromagnetic theory formulated by Brown. Although magnetic materials have been known for millennia, they continue to be the focus of current scientific investigations. Developments in quantum mechanics, statistical mechanics, and electromagnetism have accelerated the understanding of the physical underpinnings of magnetism, enabling the widespread applications of magnetic materials. In the last 25 years significant progress in the development of magnetic materials has been made. Permanent magnets evolved from AlNiCo alloys to strong rare earth magnets of SmCo, and Sm(Co,Fe,Cu,Zr),, and Nd-Fe-B magnets. The magnetic storage industry introduced completely new technologies using TbFe and Pt/Co superlattices to replace the Fe,O, and CrO, conventional recording media. The increasing demand for information storage has encouraged developments in hard disk media, recording heads, and magnetic sensors. Entire new technologies have developed around integrated optical and electronic devices. New material growth techniques such as molecular beam epitaxy and metal-organic chemical vapor deposition combined with high-resolution microfabrication techniques such as electron beam lithography have been used to create novel magnetic materials. The structure of these materials is controlled at the atomic level, thereby governing many important magnetic properties

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such as magnetic anisotropy and magnetostriction, surface and interface magnetism, coercivity and hysteresis, magnetoresistance, and multilayer coupling. As the crucial dimensions of these structures decrease into the subnanometer range, the demand for techniques capable of delivering qualitative and, more importantly, quantitative information at very high spatial resolution has become immanent. The spatial distribution of the magnitude and direction of the magnetization inside the specimen and the distribution of magnetic (leakage) fields near the specimen constitute the magnetic microstructure. Below the Curie temperature, with no applied external fields, the magnetization in a specimen is locally saturated. The magnitude of the magnetization is fixed, but the orientation is not. Regions of approximately uniform magnetization, or magnetic domains, exist within an otherwise unstructured sample. The transition region between magnetic domains with different magnetization orientation is the domain wall. The size, shape, and distribution of domains and the domain wall structure is governed by the factors contributing to the total energy. An almost infinite number of states (combinations of orientations of elementary magnets) is possible; however, only a few are realized. The magnetic microstructure is influenced, for example, by crystal anisotropy, magnetostriction, internal stresses, and specimen geometry. These factors, together with the exchange energy, self-consistent magnetostatic fields, and external magnetic fields, contribute to the total energy. Under the variational Landau-Lifshitz principle (Landau and Lifshitz, 1936), the magnetic microstructure with the smallest total energy is most likely to be realized. This leads to a set of differential equations (micromagnetic equations; Brown, 1963), which are nonlinear and nonlocal. They are difficult to solve and only a few domain structures have been derived directly. The strong exchange interaction, a fundamental property of a ferromagnet, tends to align all neighboring magnetic dipoles and favors a constant magnetization (in magnitude and orientation) throughout the sample. Any change in orientation of the magnetization away from uniform alignment increases the exchange energy and must therefore be compensated by the reduction of another energy contribution, for example, that due to stray magnetostatic fields. Consider a slab of magnetic material as shown in Fig. 1. The stray field, which is enormous for a single domain structure (Fig. la), is reduced by the formation of antiparallel domains (Fig. lb) and becomes negligible when flux closure domains are formed (Fig. lc). Crystal anisotropy and magnetostriction, two independent material properties, are related to spin-orbit coupling and cannot be derived from the Weiss theory of ferromagnetism. Crystalline magnetic materials possess crystal anisotropy, if the magnetization vector preferably assumes a

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b

C

FIGURE1. Closure domain formation in magnetic materials: (a) single domain, large magnetic stray field; (b) formation of two antiparallel domains results in a decrease of the stray field; (c) formation of closure domains with zero stray field.

direction along certain "easy" crystal axes. Uniaxial materials have a single easy axis and their micromagnetic structure consists of domains separated by 180" domain walls. An example of a typical domain pattern in a cobalt foil, the slab structure, is shown in Fig. 2a. Materials with cubic anisotropy, for example, iron, have six easy directions ({loo)) and their domain structure has characteristic 90" and 180" domain walls (Fig. 2b). Surfaces, interfaces, and imperfections in crystal structure also introduce certain preferred magnetization directions, giving rise to (local) induced anisotropy associated with the bulk termination or defects. Magnetostriction is usually a small effect when compared to anisotropy. A magnetic specimen deforms spontaneously under the influence of a magnetic interaction, depending on

t

a

b FIGURE2. Domain structure in crystalline materials (easy axes shown in upper left corner): (a) materials with uniaxial anisotropy; (b) materials with cubic anisotropy.

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the orientation of magnetization. The deformation, accounting for a change in total energy, is described by an elastic distortion tensor consisting of a symmetric (elastic strain) and antisymmetric (lattice rotation) part. Strains or rotations effectively change the directions of anisotropy minima, therefore influencing the domain structure. The resulting domain structure is determined by the interplay of all energy contributions subject to the minimum total energy constraint. The change in magnetization direction between two neighboring domains does not occur abruptly within one lattice spacing, since this would significantly increase the total energy due to high values of the exchange energy. Instead, neighboring dipoles (spins) are slightly misaligned over a certain distance until the magnetization changes from one orientation to another, thereby forming a domain wall. The Bloch 180" domain wall, where the magnetization direction stays in a plane parallel to the wall plane, is the simplest example (Fig. 3a). In thin films of thickness comparable to the domain wall width, the NCel wall (Fig. 3b) is energetically more favorable. The magnetization rotates in the film plane, perpendicular to the wall plane, thereby minimizing the stray fields at the film surface. These simplified domain structures become complicated as the parameters determining the magnetic microstructure (i.e., dimensionality, shape,

FIGURE3. Simple magnetic domain wall structures: (a) 180" Bloch domain wall, magnetization direction stays in a plane (dotted) parallel to the wall plane; (b) 180" NCel domain wall, magnetization direction stays in a plane (dotted) perpendicular to the wall plane; (c) 180" Bloch domain wall with Bloch line substructure.

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M. COWLEY

anisotropy, etc.) are relaxed and become more realistic. At a smaller length scale domain substructures exist. Domain walls can have two energetically equivalent forms, differing in sense of rotation, which coexist and are separated by dividing lines called Bloch lines (Fig. 3c). Bloch points are micromagnetic singularities, which have been predicted to exist in films with perpendicular anisotropy (i.e., the magnetization points out of the film plane). As overall size decreases boundary contributions become more dominant. In most magnetic thin films with thicknesses of a few hundred nanometers or less, the in-plane orientation of magnetization becomes energetically more favorable, even if the easy axis is perpendicular to the film plane. Small magnetic particles can exist with a specific domain structure or, when domain wall formation is energetically not favorable, as nearly single-domain particles. The length scale of micromagnetic structure spans many orders of magnitude. Domain dimensions range from several hundred micrometers to several hundred nanometers and domain walls are typically several tens to hundreds of nanometers wide; however, for hard magnetic materials (SmCo,) widths of only several nanometers are predicted. Bloch lines have a characteristic length of several nanometers and Bloch points are probably pinned to atomic sites. This wide length scale imposes a challenge for imaging techniques, which must be capable of high spatial resolution ( - 1 nm) and a wide magnification range (102-106 times). All techniques for magnetic microstructure imaging are based on the interaction between a probe (photon, neutron, electron, etc.) and either the magnetic microstructure itself or a physical quantity caused by the magnetization distribution (magnetostriction, magnetic induction, etc.). Techniques using a transmission probe are sensitive to the bulk magnetic microstructure and integrate along the direction of the incident beam such that information about surface magnetic structure is lost. On the other hand, reflection techniques interact only with a few atomic layers near the surface and are sensitive to the surface magnetic microstructure, therefore losing information about the bulk structure. A sample investigated by different techniques may give different results depending on the sensitivity of the imaging techniques. Thorough knowledge of the micromagnetic interactions and its underlying physical principles are required for a correct interpretation of the observed image contrast. A large variety of techniques have been developed and the work presented in the following sections is witness to our scientific pursuit and effort to contribute to this rich mosaic. The Bitter pattern technique (Bitter, 1931, 1932) is probably the oldest and most popular technique for the observation of magnetic domains and magnetic microstructure. A colloidal suspension of small magnetic parti-

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cles is dispersed and settles on the sample’s surface. The small magnetic particles accumulate in a field gradient emanating from the sample, which stabilizes the particles against thermal motion. The resulting pattern is then observed in a light microscope. The spatial resolution can be improved further for particles which are considerably smaller than the light wavelength when an electron microscope (Schwartze, 1957; Goto and Sakurai, 1977) is used for the observation of the dried surface. Ultimately, the spatial resolution is limited by the particle size to several tens of nanometers. However, this method is not suitable for specimens with no stray field leaking from the surface (e.g., common to high-permeability materials), and dynamic observations are limited due to the slow response of the magnetic particles. The interpretation of the patterns is not straightforward, since the correlation between the leakage field and the underlying magnetic domain structure is indirect. In magneto-optical techniques a linearly polarized light beam incident upon a magnetic sample is observed in reflection or transmission. Since optical constants such as the polarizability x,in general a tensor quantity, depend on the local magnetization of the specimen, the measurement of the change of the polarization plane with respect to the incident beam yields domain contrast utilizing the Faraday, magneto-optical Kerr, Cotton-Mouton, or Voigt effect (Freiser, 1968). In a typical scheme for the observation of magnetic microstructure, a linearly polarized beam is reflected from or transmitted through the magnetic specimen and then passed through an analyzer (polarizer), while the beam or the sample is raster-scanned. The Kerr effect has been used extensively for the characterization of surface magnetization in metallic films (Bader, 1991). The spatial resolution of magneto-optical methods is limited by diffraction. However, recently this resolution limit has been significantly improved using the near-field scanning scheme (Betzig et al., 1992) and spatial resolution of approximately 30 nm has been achieved. Magneto-optical techniques allow direct observation of the sample’s magnetization, with no or little effect (heating by incident beam) on the magnetization. High-speed dynamic observations are possible and the experimental setup enables an easy application of external fields and stresses. However, the surface of the sample has to be optically flat and the spatial resolution is limited due to the relatively large wavelengths. Two distinct phenomena account for the magnetic contrast observed when X-rays probe a magnetic specimen: In crystalline samples, magnetostriction near domain walls changes the lattice spacing, resulting in a charge in diffraction conditions. This phenomena is utilized for contrast in X-ray topography. This technique is based on Lang’s method (Lang, 19591, where a narrow beam from a cathode or synchrotron is diffracted at a

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Bragg condition and the transmitted or reflected beam is recorded on a photographic plate simultaneously raster-scanned with the sample. X-ray topography is well suited for thick samples and is therefore one of the few methods revealing internal bulk domain structure. The sensitivity of X-rays to crystal imperfections enables the investigation of interactions of defects and magnetic domains (Miltat and Kleman, 1979). The technique is limited to crystals of good quality with nonzero magnetostrictive constant. Since imperfections (dislocations, etc.) produce contrast similar to that produced by magnetic domains, careful observations are required in order to separate magnetic and structural contrast. The spatial resolution is relatively low ( - 1 pm) and long exposure times are required. Alternatively, X-ray dichroism gives contrast caused by a difference in the absorption of right and left circularly polarized X-rays in magnetic materials composed of magnetic domains. In this technique circularly polarized X-rays illuminate the magnetic sample. An ultrahigh vacuum (UHV) compatible photoelectron microscope is used to distinguish the difference in absorption between right and left circularly polarized X-rays by measuring the photoelectron yield at an inner-shell absorption edge (Stohr et al., 1993). The magnetic contrast is based on the fact that X-rays have spin-split densities of states when the sample is magnetized. Several unique features make this technique very promising for the future. First, the elemental and “chemical state” specificity is obtained by tuning to a characteristic atomic absorption edge. Investigations of the near-edge X-ray absorption fine structure, which is sensitive to the local charge state and local bonding, give additional information about the local structure. Further, a variable probing depth can be achieved by using different detection modes to measure electron (secondary or Auger) or fluorescence yields. The resolution is limited by the aberrations of the electrostatic lenses of the photoelectron microscope to approximately 20 nm. The resolution can be improved to the 10-nm range by using focused X-rays and scanning the sample, but more photon flux density is needed to obtain sufficient signal in a reasonable time. Neutron topography uses an approach similar to Lang’s X-ray method (Schlenker and Baruchel, 1978). Neutrons have the advantage of a direct magnetic interaction with matter. Neutrons carry spin which can interact with unpaired electron spins as well as with nuclear spins. When electron spins are ordered in parallel (ferromagnetic materials) or antiparallel (antiferromagnetic materials) arrays, the spin-dependent scattering gives rise to diffraction maxima. An antiferromagnetic arrangement of spins usually lowers the symmetry, resulting in larger unit cells which are responsible for “superlattice” reflections of purely magnetic origin. Neutron topography is a direct magnetic imaging technique well suited to the

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observation of bulk magnetic structure (where X-rays are strongly absorbed) and to the observation of antiferromagnetic domain structure. Wider applications are hampered by the requirement of a bright neutron source, rather low spatial resolution of tens of micrometers, and the limitation to good single crystals. Magnetic force microscopy (Saenz et al., 1987) is a special implementation of atomic force microscopy (Binnig et al., 1986) which is closely related to scanning tunneling microscopy (Binnig et al., 1982). The magnetostatic dipole-dipole interaction of a sharp magnetic tip with the stray field of a magnetic sample yields magnetic contrast. The magnetic tip is attached to a flexible cantilever and the interaction is measured as a function of position while the sample is piezoelectrically scanned (Griitter ef al., 1992). Magnetic force microscopy is a relatively inexpensive technique, requiring little or no sample preparation. At present the spatial resolution is limited by the tip size to approximately 50 nm and further improvements are possible with finer tips. The mutual tip-sample interaction and the nonuniqueness of the magnetization determination remain as major obstacles in the investigation of magnetic materials. Electron-optical methods represent the widest family of techniques for magnetic domain imaging. The magnetic microstructure observed in the various modes can be explained classically through the Lorentz force acting on an electron in a magnetic field. When the electron velocity is not parallel to the magnetic flux density, a nonzero net Lorentz force deflects the electrons. Such trajectory displacements result in magnetic contrast. The most advanced techniques provide the highest resolution so far and display high contrast and sensitivity to small magnetization changes. Usually structural information is also available at the same or better spatial resolution. Electron holography offers quantitative micromagnetic information at high spatial resolution, a feature missing in most of the electron-optical imaging techniques. Unfortunately, the electron microscope equipment is rather expensive and relatively strong criteria are imposed on sample preparation such as vacuum compatibility, sample thickness ( < 100 nm) for transmission modes, and surface quality in reflection modes. The field of view is limited and external fields are rather difficult to apply. Also, stray fields above and below the sample can influence the contrast. The most widely used electron-optical techniques are the various modes of Lorentz transmission microscopy and electron holography, which are described in detail in the following sections. Two different types of magnetic contrast in scanning electron microscopy (Newbury et al., 1986) are due to the Lorentz force acting on either secondary electrons (type I contrast) or backscattered electrons (type I1 contrast). In a scanning

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electron microscope a finely focused electron beam is scanned across the sample and secondary or backscattered primary electrons are collected by a detector. When a magnetic stray field exists above the surface of a sample, the low-energy (0-50 eV) secondary electrons (SE) ejected from the sample are deflected by this field (Wells, 1985), yielding type I contrast. Depending on the magnetization at the probe position on the specimen surface, electrons are deflected toward or away from the asymmetric SE detector, disturbing the symmetry of the secondary-electron angular distribution. The secondary-electron contrast is usually diffuse, showing only main magnetic patterns at relatively low spatial resolution of approximately 1 pm. The high-energy (typically > 30 keV) primary electrons, deflected by the magnetic flux density toward or away from the surface as they travel through the sample, have different backscattering yields depending on local magnetization, giving rise to type I1 contrast (Wells and Savoy, 1979). The domain contrast is very small, as low as 0.1% at 30 keV primary energy and 1% at 100 keV, and depends strongly on the angle of incidence. The spatial resolution, determined by the scattering process inside the sample, is typically approximately 0.5 pm, that is, worse than magneto-optical techniques. In mirror electron microscopy (Mayer, 19571, the electrons illuminating the sample do not penetrate or even impinge on the sample, but are reflected by stray fields above the sample and imaged on a screen. The magnetic contrast-forming deflection is caused by the component of the Lorentz force which stems from the radial components of the electron velocity, that is, the component of the magnetic field normal to the plane of the mirror specimen. This requires a nonzero beam tilt and the contrast increases with increasing tilt. An improvement in resolution of about an order of magnitude over magneto-optical techniques has been demonstrated and the parallel detection scheme allows observation of dynamics. The limited field of view obstructed by the electron gun, required stray fields, and complex image contrast are the main disadvantages of this technique. Some of the disadvantages are overcome in the low-energy electron microscope (Telieps and Bauer, 1985), which utilizes a magnetic prism to split the illuminating and imaging beams (Kolarik et al., 1991). A variety of electron polarization effects have been investigated (Kirschner, 1985), some of which have been further developed for imaging of magnetic microstructure. In spin-polarized photoemission the spin polarization of photoelectrons ejected from crystals is analyzed. Low-energy electrons diffracted from crystals yield polarized Bragg reflections due to spin-orbit coupling or/and exchange interaction. Auger electrons are polarized, irrespective of the way in which the core hole required for Auger electron emission is generated. Secondary electrons emitted from a

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ferromagnetic surface have their spins antiparallel (and their magnetic moments parallel) to the magnetization vector of the originating region. Two techniques based on polarization analysis have been developed into microscopies applicable to magnetic microstructure investigations: scanning electron microscopy with polarization analysis (SEMPA) and spin-polarized low-energy electron microscopy (SPLEEM). In SEMPA (Scheinfein et al., 1990), a finely focused electron beam is scanned across the sample, exciting spin-polarized secondary electrons at the surface. The secondary electrons are transported from the sample surface to a spin polarization analyzer and a magnetization map of the specimen surface is obtained point by point. High spatial resolution and direct determination of the magnetization are the main advantages of this technique. The main drawbacks are long acquisition times, extensive specimen preparation, and expensive instrumentation. SPLEEM (Altman et al., 1991) is based on the existing low-energy electron microscope, which is capable of imaging transverse surface details at 10 nm spatial resolution and atomic steps in the (longitudinal) direction perpendicular to the surface. For investigations of magnetic microstructure, a spin-polarized GaAs electron gun and electrostatic illumination lenses are incorporated. The magnetic contrast arises from the exchange scattering depending on the dot product of the spins of the incident and target electrons. This surface magnetic imaging technique has been established experimentally with a spatial resolution of approximately 50 nm, allowing simultaneous observation of magnetic structure and surface morphology. Very expensive instrumentation and the requirement of flat, conductive, or semiconductive samples compatible with UHV limit applications.

11. LQRENTZMICROSCOPY

A. Particle-Wave Duality of Electrons In 1923 Louis de Broglie formulated the hypothesis that particles possess wavelike properties and exhibit a wave-particle duality in analogy with electromagnetic radiation. Free particles (such as electrons) can be described by associated matter waves of frequency Y, related to the energy E and magnitude of momentum p through the Planck-de Broglie relationships

E y = -

h

and

h

h = -

P9

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

where A is the de Broglie wavelength of a particle. In order to observe wavelike properties such as interference or diffraction, the geometric parameters of apertures and gratings must have dimensions comparable to the wavelength, which for electrons accelerated to voltages common in electron microscopy is on the order of a few picometers. Therefore, the motion of electrons in classical electromagnetic fields shows no wavelike properties. The first diffraction and interference effects were observed by Davisson and Germer (1927) while studying the reflection of electrons from a crystal surface, that is, objects with lattice parameters of a few angstroms. They observed peaks in the scattered intensity which can only be explained by constructive interference of electrons scattered by the crystal lattice. The wave nature of matter is directly related to the finiteness of Planck’s constant h: if h were zero, the wavelength would be zero as well and the particles would behave classically. Since h is “small” on a macroscopic scale, the wave properties of matter are not typically observed. In accordance with the correspondence principle, geometrical optics is the short-wavelength limit of wave optics and classical mechanics is the short-wavelength limit of wave (or quantum) mechanics. In classical mechanics the trajectories of charged particles of charge q moving in an electromagnetic field (characterized by the electric field E and magnetic flux density B) are completely determined from Newton’s law, where the Lorentz force F is given as F = q(E v X B). For typical accelerating voltages used in electron microscopes, relativistic effects have to be taken into account. The wavelength A can be written as a product of two terms: h 1

+

where the first term represents the nonrelativistic wavelength, and the second the relativistic correction. For accelerating potentials U = 100 kV, the relativistic correlation is already significant (= 0.954) and A = 3.7 pm. Quantum-mechanically, electrons are described in the nonrelativistic case by a wave function U(r, t ) , which is a solution of the time-dependent Schrodinger equation

where 2‘ is the Hamiltonian describing the interaction of electrons with the specimen. In most of the practical cases, the Hamiltonian is timeindependent, and Schrodinger’s equation can be solved using the separa-

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tion of variables. In field-free space the Hamiltonian has only a kinetic mergy term, and Schrodinger’s equation for a free electron has solutions in the form of plane waves or spherical waves propagating from or to a [point) source. Electrons propagating in a microscope can therefore be treated as spherical waves, and most of the wave mechanical formalism developed in light optics can be utilized. The plane and spherical wave solutions are frequently used for the description of electrons propagating in the column of an electron microscope. For example, the electron source can be modeled as a small virtual point source emitting spherical waves and the action of the whole illuminating system of a conventional transmission electron microscope results in the plane wave illumination of the specimen. Position-dependent (r) electromagnetic fields present in a magnetic specimen are described by a scalar potential Wr) and vector potential A(r). For a correct description of the propagation of electrons through such a sample, the potential energy operator hcs to be added to the Hamiltonian, and the canonical momentum fi eA must be used in place of the classical momentum operator. Here the intrinsic magnetic moment of the electron, the spin, is neglected. A relativistically correct theory including spin, following from Dirac’s relativistic wave equation, is given by White (1983). The common procedure in (classical) electron optics is to solve the nonrelativistic Schrodinger equation and apply relativistic corrections to the mass and wavelength. It can be shown (Hawkes and Kasper, 1994) that Dirac’s equation containing spinors is reducible to a scalar wave equation (with a relativistically corrected mass) when the spin is neglected, the possibility of performing Lorentz transformations is sacrificed, and a fixed gauge is chosen for the electrostatic potential.

+

B. Fresnel Mode In the following discussion we examine the interaction of electrons, propagating in a scanning transmission electron microscope (STEM), with the electromagnetic fields present in the specimen. In a STEM the electrons emitted from a field emission gun (FEG) are collimated by the condenser and objective lenses, and form a finely focused probe at the specimen plane (Fig. 4). The field emission gun is crucial for the observation of interference patterns such as Fresnel fringes, lattice fringes, and holograms. Thermionic and Schottky electron guns do not provide sufficient electron current into this illumination angle and the high brightness, small virtual source size, and small energy spread FEGs are necessary in order to obtain a reasonable coherence width.

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

a

b

FIGURE4. STEM operation modes: (a) scanning; (b) shadow imaging.

In the conventional STEM mode the probe is raster-scanned across the specimen and a selected part of the transmitted and/or diffracted beam is detected yielding a signal for a television screen which is scanned at the same rate as the beam in the column (Fig. 4a). A STEM equipped with an appropriate detector system (scintillating screen and optics) provides an additional imaging mode. When the scanning is switched off and the objective lens is defocused, a real and magnified image of the specimen,

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the shadow image, is formed in the detector plane (Fig. 4b). In highresolution TEMs and STEMS the specimen is immersed into the strong magnetic field of the objective lens in order to decrease its focal length and therefore minimize lens aberrations. When a magnetic specimen is placed inside the objective lens, magnetic microstructure is strongly disturbed (usually saturated), resulting in a loss of magnetic contrast. Therefore, magnetic samples must be placed far enough outside the objective lens so that the effect of the lens field is negligible. For example, in the HB-5 STEM equipped with high-resolution pole pieces, the specimen is placed 12-13 mm above the center of the objective lens. A finite-element calculation of the lens stray field shows that the field decays rapidly outside the pole pieces and the residual flux density at the specimen is 5 mT (50 Oe) and can be further reduced by a specimen lift. In the following sections the observed magnetic contrast is explained first in terms of geometrical optics and then refined using the wave-optical approach.

1. Geometrical Optics In the Fresnel mode the magnetic microstructure becomes visible (Hale et al., 1959) when the objective lens is defocused, that is, in the shadow image. The electrons are focused in a plane above or beneath the specjmen (Fig. 5 ) and an under- or overfocus shadow image is formed in the detector plane. The magnification M depends on the defocus Af and is given in terms of geometrical optics theory as M = L/Af 1, where L is the camera length. In a typical setup a region of several micrometers is magnified on a YAG screen (25 mm diameter). The camera length can be varied by postspecimen lenses and typically L = 10 m, Af = 1 mm, and M = 10,000. The illumination half-angle ayiis small, ai= lop3rad. In Fig. 6 a series of shadow images, acquired at constant camera length L and varying defocus A f (and therefore varying magnification), demonstrates the limit of the geometrical optics theory. At small magnifications of several hundred to several thousand times (i.e., large defoci of several millimeters, Fig. 6a and b), the shadow image faithfully replicates the structure of the specimen. However, at a magnification of approximately M = 10,000 X (Fig. 6c), Fresnel fringes, which cannot be explained within the framework of geometrical optics, begin to appear near the particle edges and at higher magnifications (Fig. 6d, M = 29, 400 X ) these fringes significantly obscure the image of the particle. The image of a specimen containing magnetic domains, observed under defocus imaging conditions, shows a net of bright and dark lines in addition to the common diffraction contrast (Fig. 7). When the defocus of

+

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

detector plane

FIGURE 5. Geometry of shadow imaging.

the objective lens is changed from underfocus (Fig. 7a) to overfocus (Fig.

7b) or vice versa, the bright and dark lines switch contrast. A line scan across a domain in the under- and overfocus mode is shown in Fig. 7c. This line contrast, attributed to the magnetic microstructure of the specimen, is classically due to the action of a Lorentz force acting on the electrons passing through the specimen with a given magnetization. For simplicity, a one-dimensional derivation of the magnetic contrast is given and due to the small illumination half-angle cq a nearly parallel illumination is assumed. Further, consider a magnetic specimen of uniform thickness t with uniform magnetization (and magnetic flux density B,) within the domains separated by 180" domain walls. The angle of deflection E due to the magnetic flux density in the specimen is given by tan

E =

~

eB,th h *

(4)

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FIGURE6. Variation of magnificationwith objective lens defocus: (a) 180X ; (b) 2 3 0 0 ;~ (c) 9800X ; (d) 29,400X. Specimen: CrO, particles on holey amorphous carbon film.

For E = 100 keV, t = 10 nm, and a typical ferromagnet with B, = 1 T, the deflection angle assumes rather small values, E = tan E = 0.98 X rad. The electrons are deflected in opposite directions in adjacent domains (with oppositely oriented B,,), yielding a deficiency or excess of electrons in the position of the domain wall image. This argument, extended to two dimensions, explains the presence of bright and dark lines in the shadow image as a map of domain walls in the specimen. The contrast reversal of domain wall images, which occurs while switching from underfocus to

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

1.3 1.2 1.1

t

i

-c

1 0.9

0.8 -1

FIGURE 7. Underfocus (a) and overfocus (h) images of a [Co (1 nm)/Cu (3 nm& multilayer structure grown on a holey amorphous carbon film; (c) line scans taken along marked line in (a) and (h).

overfocus, is related to the fact that the sign of the magnification changes, and converging rays become diverging rays. Geometrical optics theory breaks down not only for very high magnification (i.e., very small defocus at constant camera length), but for very large defoci as well. In Fig. 8a and b images of the same area are acquired at the same magnification; however, the amount of underfocus is different (underfocus in Fig. 8b is much larger). An interference pattern consisting of approximately equidistantly spaced fringes appears at the position of the converging domain walls, replacing the simple line contrast described before. At larger defocus the overlap of the beams near a convergent domain wall image increases, resulting in interference effects (with sufficient coherence) which can be explained in the wave theory framework only.

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

34 1

FIGURE8. Defocus images of a Co (6 nm)[Cu (2 nm)/Co (1.5 nm)], multilayer structure grown on an amorphous carbon film: (a) typical underfocus; (b) large underfocus.

2. Wave Optics

The wave-optical approach is based on Huygens’ treatment of waves propagating through space and generating secondary waves at each point of a wavefront, where the envelope of the secondary waves forms a new wavefront. This intuitive description is expressed mathematically in the form of the Kirchhoff formula (Born and Wolf, 1985). In what follows some simplifying assumptions are made. First, when the spin of the electrons is not considered, only a scalar theory is needed and all dimen-

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

sions of the objects of interest are considered to be much larger than the electrons' wavelength ( A = 3.7 pm). The interaction of the specimen with the illuminating electron beam, resulting in a change of amplitude and phase of the transmitted electron wave, is described in terms of an object transfer function q(r): q(r)

(5)

= u(r)e-iq(r).

When q(r) is multiplied by the incident wave function, the effect of the specimen on the amplitude and phase of the incident wave is recovered. The amplitude a(r) and phase q(r) cannot be resolved in the direction of the beam (unless tomographic techniques are used); therefore, in transmission electron microscopy the projected amplitude a ( x , y) and phase q(x,y ) , averaged over the specimen thickness t, are considered only:

1

1

a(xyY)

=

fl Jspecimen 4 r ) d.2,

cp(X,Y> =

fi [pecime"(P(

r) dz, (6)

where f is a unit length vector. The change in amplitude, due to absorption and inelastic scattering, is usually extremely small for the ultrathin specimens used in (S)TEM and the approximation of a (pure) phase object with a h , y) = 1 is commonly used. A detailed and elegant formulation of the wave theory of imaging in an electron microscope has been presented by Cowley (1986). Here the relevant applications of the wave theory to STEM imaging are reviewed. The wave-optical treatment of imaging, based on Fourier transforms, was introduced in the 1940s by Duffieux (1946) and a comprehensive treatment is given by Born and Wolf (1985). Abbe's theory, in combination with the small-angle approximation, conveniently describes the essential properties of imaging in a STEM. For example, propagation through free space of length R is given by a convolution of the incident wave with the Fresnel propagator exp( - i k ( x 2 y2)/2R) and an ideal lens of focal length f can be represented as a planar object with a transmission function exp(ik(x2 + y2)/2f). A plane wave passing through an ideal lens is focused in the focal plane of the lens, giving a Sfunction spatial distribution. The effect of lens aberrations is accounted for by introducing higher-order terms in the exponent of the lens transfer function in the focal plane. A n imaging system consisting of any combination of sources, lenses, field-free space, and the object of interest can be modeled by an appropriate series of Fourier transforms, convolutions, propagators, and transmission functions. In a STEM the wave coherently illuminating the specimen can be written as t(r) = c(r) is(r), which is a Fourier transform of the objective

+

+

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

343

lens transfer function T(u): ~ ( u= ) A(u)eZTix("),

(7)

where u is the reciprocal space vector (spatial frequency), the magnitude of which is related to the scattering angle 6 through lul = [2 sin( 6/2)]/h. The lens transfer function T(u) describes the phase changes introduced by the objective lens aberrations [ x(u)l and apertures [Ah)] in the back focal plane of the objective lens. The aperture function A h ) describes the limitation of the spatial frequency range due to the physical aperture, A(u) = 1 for lul Iuo and A(u) = 0 for IuI > u,,. Here x(u) describes how the transmitted wavefront deviates from an ideal one due to defocus Af and spherical aberration coefficient C,, ~ ( u = ) A f hu2 + $Csh2u4.The influence of other aberrations is neglected; for example, the chromatic aberration contribution is small due to the small energy spread of a field emission source and astigmatism is assumed to be corrected by stigmators. The wave function at the exit plane r = ( x , y , O ) of the specimen is qe= q(r)t(r - R), where R = ( x , , y , , O ) represents the position of the scanning probe. The wave function distribution in the detection plane q(u) is then given by a Fourier transform 53of qe;using the multiplication theorem (Cowley, 1986) and the fact that a shift in coordinates Fourier transforms as a phase factor exp(2.rriu.R) yields T(u) = Q(u)* T(u)e2TiU'R, where * denotes a convolution. The actual observable quantity in the detection plane is the intensity Z(u): I(u)

=

l€?(u)* T(u)e

2aiu.RI2

(8)

A shadow image is formed with no aperture [A(u) = 1 for every u] and the scanning switched off (choose R = 0) and, for simplicity, without loss of generality, we consider a one-dimensional derivation only, that is, q ( u ) = Q(u)*T(u).

(9)

From the definition of convolution, q ( u ) = Q(u)T(u - U )dU. In practice, the postspecimen lenses strongly magnify only a small central portion of the transmitted beam which is subsequently detected (i.e., U is small) and therefore higher-order terms in U are neglected in the integration. Rewriting the argument of the exponential function yields (Mankos, 1994) *(u)

=

T(u){q(mu)*t(mu)),

( 10)

where rn is a function of u and the intensity Z(u) = I&>* T(u)12= Iq(rnu)* t(rnu)12.For the small scattering angles 6 encountered in electron microscopy ( 6 10 mrad), the spatial frequency u can be related to the

-

344

M. MANKOS, M. R. SCHEINFEIN AND J. M.COWLEY

detection plane coordinate x: u=

2sin(8/2) h

x

8

2 -

h

= -

AL

and the function m assumes the meaning of a coordinate-dependent magnification factor: X

The observed intensity represents a real image smeared out by the spread function t and magnified by the factor M a , which is not uniform in the image plane and depends on the microscope parameters Af and C,. A plot of the magnification ratio M a / M , as a function of the spatial frequency u for a typical value of C , = 153 mm (objective lens focal length is 13 mm) and three different under- and overfoci, shown in Fig. 9, demonstrates that under out-of-lens optical conditions (large A f ), the acquired shadow image represents a magnified and undistorted image. For the largest magnifications used ( M 100,000 times, Af = 100 km), the distortion is less than 0.2% up to the resolution limit (1 nm). The distortion may become significant for the in-lens position of the specimen (very small C, and A f 1. For simplicity, we assume that the magnification is one. Any specimen, whether magnetic or not, can be characterized by a spatially varying distribution of electromagnetic fields or potentials. For further development of the wave-optical theory, the relationship between the electromagnetic potentials and the object transfer function has to be

-

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

345

determined. In the following discussion the focus is on the phase changes imposed by the specimen and the amplitude changes are neglected (for a further discussion see Section 111) due to the fact that in STEM only very thin specimens are examined. The phase shift introduced by the specimen is a purely quantummechanical effect and was first discussed in detail by Ehrenberg and Siday (1949) and Aharonov and Bohm (1959). With no specimen present the transmitted electron wave is described by a free-particle wave function satisfying Schrodinger's equation (neglecting wave aberrations). The intensity which is proportional to lT(r,t)12 in the detection plane remains unchanged unless the illuminating electron beam is split into (at least) two parts which undergo different phase shifts. The time-averaged intensity Z(x, y ) , observed in the detection plane, depends on the phase difference cp,

- cp2:

assuming that Woj = ITojle-i'+'Oi, cpo = cpol - qO2, and, for simplicity, cpo is set to zero. Note that the phase difference is inversely proportional to h and therefore in the classical case ( h + 0) the argument of the cosine function changes infinitely rapidly, that is, the oscillations are smeared out and no effect is observed. A relativistic generalization of the phase and difference cpl - cp2 due to covariance involving the scalar potential the vector potential A yields (Aharonov and Bohm, 1959)

where the integral path is taken along a closed loop in space-time (dt and dl, Fig. 10). The phase difference cp, - rp2 can be rewritten applying

S

D

path 1 FIGURE10. Geometry and beam paths in the Aharonov-Bohm experiment.

346

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

Stokes theorem: 1 h

q1 - q 2 = - + p . d l -

e

-arn, h

where cPrn is the magnetic flux enclosed in the area S defined by the closed path. From an electron-optical point of view, a magnetic specimen represents a phase object with a transmission function q(x, y ) = e - i A q p ( x 3 Y ) , where the phase difference A q = q1 - q2 between two paths originating in the same point S (source) and ending in the same point D (detector) in the detector plane is given by (e/h)cP, (neglecting the nonmagnetic contributions, Fig. 10). For an estimate of the magnitude of the phase difference, consider again a magnetic film of thickness t = 10 nm and B = 1 T. If the split beams travel through two points separated by a distance x = 100 nm, their phase difference is A q = 1.519 rad; that is, magnetic specimens are strong phase objects and the widely used weak phase object approximation (WPOA) cannot be used. It can be seen immediately that, since the phase shift due to C , is negligible, a certain amount of defocus is required in order to obtain magnetic contrast. With the microscope in focus, A f = 0 and t(rnu) is a &function; therefore, Z(u) = Iq(rnu)* t(rnu)12= le-'A'P12= 1 and no magnetic contrast is observed. Let us consider a magnetic specimen represented by a phase object transfer function and, for simplicity, assume Ma = 1 and neglect the C , term. Within the validity of the Fresnel diffraction approximation, the wave function in the detection plane is then given by the convolution integral

W X , Y )

q(x,y)*t(x,y) = exp [ - i A q( x, y ) ] * exp [ - ik( x 2 =

+ y ')/2

Af ] .

For a one-dimensional derivation, the inverse Fourier transform of the object transfer function Q h ) , defined by q ( x ) = /:- Q(u)e-2aiuxdu, is introduced, yielding W x ) = JymQ(u)e-2niuxe ? r i A f h u 2 du. When 7~ Af Au2 is sufficiently small, that is, for Af < 0.1 mm and u < 0.01 nm-' (resolution limit 100 nm, corresponds to magnifications of a few thousand times as shown in Fig. 71, the second exponential can be approximated by the first two terms of its Taylor series, e"' 'f = 1 + Ti A f Au2 - -.. and the intensity Z(x) at a given defocus Af is then

Z(x)

=

AfA 1 - -A$' 2a

AfA AfA + (x) A q " ' + (7)

(15)

For defoci smaller than 0.1 mm only the term linear in Af makes a significant contribution. Now assume a commonly used model (Fig. 11) for

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY 1.2, 1.15

1

.

-1000

.

,

, , . . , . ,

347

. . . , , . . . . ,

A

-500

0

500

1000

x [nml FIGURE 11. Calculated intensity profile in the under- and overfocus condition for the shown wall model.

the distribution of the magnetic flux density B ( x ) in a domain wall, uniform along the beam direction and characterized by a wall parameter w , B ( x ) = B, arctan(x/w>, that is B ( k m ) = k(7r/2)B0. A plot of the intensity distribution (Fig. 11) for a 10-nm-thick film with B, = 1 T and w = 50 nm in the under- and overfocus condition (Af = f O . l mm) demonstrates the contrast reversal at a domain wall when switching the defocus (compare with Fig. 7c). Unfortunately, the same approximation used previously fails to predict the contrast at very large defoci (the Taylor expansion cannot be terminated and is slowly converging). Therefore, it is necessary to return to the full formula or preferably to the original Kirchhoff integral. It has been shown (Winthrop and Worthington, 1966) that the convolution integral for the wave function can be inverted if the amplitude and phase are known. In the Fresnel mode only intensities are recorded and the phase information is lost. Therefore, the determination of A q ( x , y ) is rather difficult and a numerical approach is required. Cohen (1967) used a parametric form of the phase distribution A q ( x , y ) for the evaluation of domain wall profiles and magnetization ripple images and in successive steps refined the parameters determining the phase difference (and therefore the magnetic structure). However, uniqueness of the solution and an accurate match with experimental data still remain a concern. Profiles of convergent and divergent domain walls have been calculated and compared from presumed magnetization distributions in the geometric and wave-optical frame theory in Reimer (1984) and the comparison of convergent wall images can be used to distinguish between different types o\E domain walls (Schwellinger, 1976). Hotherstall (1972) compared calculated and experimental profiles of domain walls and reached good qualitative (asymmetry) but only moderate quantitative agreement. Wade (1962)

348

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

obtained domain wall thickness with i-30% accuracy from measurements of convergent and divergent domain wall images. The accurate knowledge of imaging parameters (defocus, illumination aperture size), which is essential for the extraction of quantitative information, together with the inversion problem precludes the Fresnel mode from being used as a tool for accurate quantitative determination of magnetic microstructure. Limited progress can be made in the case of converging symmetric domain walls. An underfocus series of Fresnel images of an approximately 25-nm-thick Co film is shown in Fig. 12. Fringes of constant period appearing on both sides of the domain wall image can be recognized as biprism fringes [e.g., cosine fringes in Eq. (12)] formed by the overlap of two parts of the electron wave passing through neighboring domains (Boersch et al., 1960). Two neighboring intensity maxima must differ in phase by 27r, corresponding to an enclosed magnetic flux of Brn= h/e. However, as made clear in Fig. 13, the actual object position coinciding with the fringe in the image lies at the center (Ax/2) of the area enclosed by the two interfering waves (within a negligible error of E Ax). Therefore, fringe maxima create a map of the enclosed magnetic flux in units of h/2e. If the thickness is constant and known, this map corresponds to the distribution of the in-plane component of the magnetization (but at relatively low spatial resolution). A comparison of line scans taken from images acquired at different defoci, but at the same specimen position (Fig. 12e) and averaged across 60 nm, shows that the maxima appear at the same specimen position, independent of the defocus. The variation of the profiles with defocus (different peak-to-peak ratios) may be caused by the variation of the scattering contrast with defocus (Fresnel fringes), which overlaps with the cosine fringe pattern. The average spatial period x p of the fringes in Fig. 12b-d equals 80 f 10 nm, which yields for the magnitude of the field component (parallel to the fringes) Bp = h / 2 a p t 1.03 T. This value compares favorably with the component Bp(co)parallel to a 180" domain wall ( B , = 1.8096 T for Co), Bp(co)= B, cos54O = 1.06 T. The overlap region 2 p (in the specimen plane) is proportional to the defocus; that is, with increasing focus a broader flux map becomes visible. However, at the same time the magnification (and the resolution) decreases. The overlap measured from Fig. 12b-d as the distance of the two outermost maxima on each side of the domain wall equals 407, 825, and 1539 nm at 23,45, and 86 mm defocus and confirms the linear dependence. The Fresnel mode of Lorentz microscopy is often used for quick determination of the position of in-plane magnetization changes (domain walls, ripple, etc.) in magnetic specimens. It does not require any special

-

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

349

e .-3 v)

C

I C

.-

400

-200

0

200

400

x [nml FIGURE12. Underfocus series of a 25-nm-thick Co film grown on amorphous carbon at 8 (a), 23 (b), 45 (c), and 86 mm (d) defocus. Intensity profiles from (b)-(d) at the marked positions and averaged over 60 nm across are shown in (e).

350

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

Af

I

L

-

FIGURE 13. Geometry of electron wave overlap near a convergent domain wall.

detector or image processing and can be observed in any TEM as well. Due to its parallel acquisition scheme (whole image acquired in one frame), it is well suited for observation of domain wall dynamics on a time scale determined only by the frame acquisition speed. It is rather difficult to extract any quantitative information from Fresnel images and other techniques (described in the following sections) must be employed. C. Differential Phase Contrast Mode

The differential phase contrast (DPC) mode of Lorentz microscopy is a scanned technique (Fig. 4a). Phase contrast is obtained by splitting the

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

351

detector into two or more segments, using an annular detector, or a combination of both. Dekkers and de Lang (1974) demonstrated that such a detector system is capable of giving contrast proportional to the gradient of the phase of the object transmission function. The split-detector scheme has been applied to magnetic specimens by Chapman et al. (1978) and an extension to a symmetric quadrant detector (Chapman and Morrison, 1983) allows the determination of both components of the phase gradient proportional to the thickness-averaged in-plane component of B. 1. Geometrical Optics In the DPC mode a finely focused beam is scanned (Fig. 14) across a magnetic specimen and deflected by the local Lorentz force. After passing through the specimen region, the beam is descanned which assures that the transmitted beam remains symmetrically positioned around the center of the split detector when no (or a constant phase) specimen is present. The difference signal A - B is equal to zero for every beam position in this case. When a magnetic specimen is present, the transmitted electron beam is deflected by an angle &,y) which is proportional to the local thickness-averaged in-plane component B, = (l/t)ji B(x,y) dz and given by Eq. (4). When the beam is deflected in a direction perpendicular to the slit, different intensities will be registered by the two detectors A and B and the difference signal A - B is proportional to the deflection angle E ( X , y). This argument is easily extended for a two-dimensional quadranted detector. The x(y)-axis is now defined by the slits between detectors A and B (C and 0 ) and the difference signals A - B, C - D are proportional to the corresponding projected in-plane components of the flux density, that is, ZAP&, y) a B,(x, y), ZC-Jx, y) a B J x , y ) . For each probe position detector intensities A through D are collected and differences between oppositely oriented quandrant pairs are displayed, each yielding a two-dimensional distribution of one component of the projected flux density In the DPC mode, images of domains of uniform magnetization (and therefore flux density) can be identified as areas of bright and dark contrast. When two images sensitive to (preferably) perpendicular components of are acquired, the vector nature of the local in-plane magnetization can be revealed. A summation of all the signals A through D can be used to form a high-resolution (conventional) bright field image, allowing the correlation of the observed magnetic microstructure with the underlying microscopic structure. Within the geometric approximation the difference signals are directly related to the spatial distribution of Bin-p,ane-components (compare to the Fresnel mode) and quantitative information may be extracted. For domain wall thick-

352

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

B

A

A

I

II

a

I

b

FIGURE14. Electron-optical scheme of the DPC mode of Lorentz microscopy: (a) nonmagnetic specimen; (b) magnetic specimen.

nesses which are much larger than the probe size (one to a few nanometers), domain wall profiles can be determined at high spatial resolution. The quadranted detector cannot distinguish between magnetic contrast and contrast due to nonisotropic scattering, the contribution of which is comparable to or even larger than the magnetic component. Diffraction contrast from small crystallites and scattering from edges, interfaces, and defects can be interpreted as magnetic contrast by the quadrant detector. This spurious contrast can be partially eliminated when the magnetic

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

353

microstructure varies on a scale significantly larger than the size of individual crystallites (1-10 nm). An annular detector split into eight segments can be employed to control the efficiency of detection of high and low spatial frequencies (Chapman et al., 1990) and experimental images with improved signal-to-noise ratio and separation of microstructural and magnetic contrast have been obtained by a proper choice of difference signals. A simplified mode of DPC Lorentz microscopy, based on a single annular detector (Kraut and Cowley, 1993), has been implemented in the HB-5 STEM using the high-angle annular dark field (HAADF) detector. A postspecimen lens doublet magnifies the transmitted beam, defined by the (virtual) objective aperture, until the disk is comparable to the inner diameter of the annular detector (Fig. 15). The transmitted beam is, after descanning, positioned in the detector plane so that it overlaps slightly with the inner edge of the annular detector. The local beam deflection ex due to the perpendicular Lorentz force changes the amount (area) of overlap and therefore the detected intensity (the influence of E~ is neglected since E is small). When the beam is positioned at the opposite inner edge of the annular detector, the contrast is reversed. In principle, it is sufficient to measure at two locations of the disk rotated by 90" with respect to each other. In practice, four images are acquired at four

C

B

A

D FIGURE15. Detector configuration in the DPC mode with annular detector.

354

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

perpendicular positions (labeled A-D in Fig. 15) and enhanced magnetic contrast is obtained by subtracting images from opposite beam positions ( B - A , D - C ) . Similarly as for a quadranted detector, the sum of the signals gives a conventional bright field image. The transmitted beam must overlap the detector by a distance greater than &/aYi ( a i is the illumination half-angle) such that for any direction of deflection the beam may still be detected. It has been shown (Kraut and Cowley, 1993) that, for ai= 1 mrad, a maximum deflection angle E = 0.05 mrad, and a beam diameter equal to one-half of the inner diameter of the detector, the measured intensity is approximately linear with beam deflection and for an overlap of 10% the error is smaller than 1%. As the beam profile is expanded by the postspecimen lenses, the range of linearity increases. However, the problem of spurious signals from deflections in the direction perpendicular to the overlap arises. The effect of first-order diffracted beams is negligible for thin specimens since they appear at approximately 10 mrad, which is 10 times the beam size and out of reach of the annular detector. Relative to the quadranted detector, the annular detector has the disadvantage that separate scans must be carried out for each signal A through D. Also, problems arise with the accuracy of the positioning and amount of overlap of scans acquired at opposite edge positions, which ultimately limits the accuracy of quantitative information. The effect of overlap inaccuracy can be partially eliminated by subsequent image processing if knowledge of the domain structure exists prior to the experiment. First assume that the beams are perfectly aligned with respect to the detector (position A and B ) . The expected line scans from signals A , B across a magnetic domain wall are drawn in Fig. 16a. The difference signal B - A shows a perfect domain wall profile and A + B reflects thickness variations of the thin film. Note that the average intensity (or gray level, due to given by total overlap) and relative change (or gain, given by &/ai) magnetic contrast are identical for both signals A and B. In practice, the registered intensities A and B have different gray levels and gains and the magnetic contrast can be obscured by spurious signals (Fig. 16b). The difference in gray levels and gains can be removed by a proper scaling of the image data, but spurious signals may remain in the difference signal B - A . An example of the image processing is shown in Fig. 17. The first four images (Fig. 17a-d) are the raw images acquired at four perpendicular positions. The difference signals B - A and D - C show the projected component of in the direction of the arrow indicated in Fig. 17e and f, where white (black) corresponds to a component parallel (antiparallel) to the arrow. The summed signal C + D is a bright field image, which reveals the thickness variation across the scan field. A Fresnel mode image

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

I

355

I

a

b gray level

gain

spurious signals

B-A~!E~-l&---;lFIGURE16. Image processing in the DPC mode: (a) ideal conditions; (b) experimental conditions with varying gains, gray levels, and spurious signals from nonmagnetic scattering.

of the same region (marked in Fig. 17h) reveals the domain wall position only. A comparison of the difference signals (Fig. 17e and f) with the bright field image (Fig. 17g) reveals that a substantial contribution from misalignment and nonisotropic scattering is present in the magnetic structure images. The DPC images are extremely useful for the determination of the orientation of magnetization in domains. However, the accuracy of

356

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

FIGURE 17. DPC mode images acquired at different positions of the beam with respect to the annular detector. Images (a)-(d) correspond to positions A through D as marked in Fig. 15, (e) corresponds to B - A, (f) to D - C, (g) to C + D, and (h) is a Fresnel mode image from the same region of the specimen. Note the arrows in (e) and (f) which show the mapped component of in-plane magnetization.

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

357

the absolute magnitude of magnetization (or flux density) is not very good (even in the case of sophisticated quadrant detectors). A flux density rad, change of 10 mT (100 G) corresponds to a deflection smaller than which with a (optimistic) camera length L = 100 m results in a displacement of 10pm, which is difficult to quantify. The validity of the geometrical optics theory is limited by the relationship between the probe (size and illumination angle) and the scale of magnetization changes. In a STEM the probe can be focused to a diameter as small as 1 nm and the angle a; subtended by the probe is typically 1 mrad. As long as the magnetization changes occur over regions much larger than 1 nm and for deflection angles E << 1 mrad (which is true in almost all cases), geometrical optics and wave optics will yield essentially the same results. Wave mechanics must be applied to deconvolute the probe distribution from the recorded intensity only in cases where the magnetization changes on a very small length scale (e.g., hard magnets). 2. Wave Optics The wave function at the specimen exit plane is given by the product of the object transfer function q(r) and the lens transfer function t(r - R). The actual observable quantity in the detection plane is the intensity I(R), which now depends on the position of the scanning probe R and on the geometry of the detector characterized by a detector function D(u): Z(R)

=

1

detector

D(u)lQ(u) * T(u)e2rriu’RIZ du.

( 16)

For example, if the detector function N u ) equals a &function (idealization of the point detector), the intensity is proportional to IQ(u)* T(u)I2, that is, exactly the same as in a conventional TEM equipped with the same objective lens and a point electron source as expected from the reciprocity between TEM and STEM (Cowley, 1969). Restricting the analysis to a one-dimensional derivation and assuming q ( X ) = e P iA q p ( X ) allows us to simplify the expression for the intensity (Mankos, 1994):

Z ( X ) = -(Acp)’* I t ( X ) I * ; (17) that is, in the first-order approximation, the intensity is proportional to the gradient of the phase difference convoluted by lt(X)I2.Now let us return to the domain wall model (Section II.B.2) and assume that the probe size is much smaller than the scale of magnetization changes, that is, t ( X ) = 6 ( X ) . Within this approximation the recorded difference intensity is then directly proportional to the domain wall profile and the left and right domains separated by this domain wall will appear as dark and bright areas, in agreement with the geometrical optics theory. The convolution by

358

M. MANKOS. M. R. SCHEINFEIN AND J. M. COWLEY

the probe becomes important when the magnetization changes occur over a distance comparable to the probe size. Since Af = 0 and for the optimum aperture size a;= 1 mrad, u, = 0.27 nm-' and x ( u ) is negligible. The probe is then diffraction-limited by the Fourier transform of the (real) aperture function A(u). For a circular aperture (ayi= 1 mrad), the Fourier transform is a Bessel function of the first kind of order zero (Cowley, 1986) with the first zero at x p = 1.22/(2uC) I2.25 nm and the probe size is then approximately 4.5 nm, which confirms the assumption that the wave-optical treatment is not required for typical magnetic specimen, as conjectured in the previous section. In spite of the requirement for special detectors and image processing, the DPC mode has become a useful tool for obtaining quantitative information in the form of vector maps of magnetization and magnetic domain wall profiles. The ability to simultaneously observe magnetic and microscopic structure in the DPC mode allows for the correlation of micromagnetic features with the underlying structure, geometry, and composition.

D. Small-Angle Electron DifSraction This technique is normally practiced in a conventional TEM where the diffraction pattern may be substantially magnified by projector lenses and a much smaller aperture size (typically < lop2 mrad) is available. Since it is not used very frequently in a STEM, only a brief description is given here in order to point out some advantages and limitations in relation to the DPC mode. Classically, rays passing through domains with the same orientation of magnetization are deflected in the same direction by an angle E [Eq. (4)l and converge to a single point in the back focal plane of the objective lens. When the specimen consists of domains oriented only parallel or antiparallel to one direction, two sets of distinct spots separated by 28 are observed in the diffraction pattern. If the domains are oriented rather randomly in all directions, an annular ring of angular diameter 28 is formed instead. Since E is typically lo-* mrad or smaller, considerable magnification is required. In a TEM a relatively large area (1000 nm X 1000 nm) is illuminated and the diffraction pattern reflects a distribution of deflection angles from the whole illuminated region. This is advantageous for samples with periodic magnetic structure. In a STEM with the specimen in the field-free region the beam can be focused to a spot as small as 1 nm and a convergent electron beam diffraction (CBED) pattern can be observed. In principle, a map of beam shifts, which contain information about the magnitude of magnetization

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

359

(angular magnitude of E ) and its orientation (direction of shift), can be created for every point of interest. With the scanning switched off, the probe is focused to the desired spot size and positioned in the domain of interest, and an image of the highly magnified transmitted disk is acquired (Fig. 18). This image is then compared to the image when no specimen is present or the beam is located in a neighboring domain. The latter is demonstrated in Fig. 18, which shows a double exposure from two domains separated by a 180"domain wall. The objective lens is slightly defocused so that an area of approximately 100 nm in diameter is illuminated and the domain wall image is visible when the beam is moved across the wall. In Fig. 18 the domain wall is running approximately in the horizontal direction, that is, perpendicular to the measured beam shift. The beam shift due to the 180" domain wall amounts to 0.06 k 0.005 mrad, and corresponds to E where B, = Eh/eht = 1.68 T, which is within the error margin of the expected saturated value for cobalt (1.81 T). The accuracy of the measurement is limited ( - 10%) since the beam shift is barely visible even at the largest camera lengths. A finely focused beam requires a strong lateral demagnification of the source and therefore a large angular magnification, which results in poorer resolution in the detection of beam shifts. Since the DPC mode is, in principle, a scanning version of this technique, the

FIGURE18. Small-angle (convergent beam) electron diffraction from neighboring magnetic domains.

360

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

same error margin applies here to the accuracy of quantitative information derived from the angular deflection. The beam shift is readily derived in wave-optical terms, too. Assume, for simplicity, a one-dimensional derivation with B, perpendicular to x and the case of an ideal lens, that is, t ( x ) = S(x) and q ( x ) = e-iAvp(x). The phase shift can be written as

Acp(x)

e

=

i / / B * d S = -Boxt ti

and the intensity is proportional to I2

I

I ( u ) =I/exp[-iAcp(x)]exp(2aiur) drl

that is, the beam is shifted in reciprocal space by a spatial frequency uB = ( e / h ) B , t , which corresponds to a deflection angle E = u B h = eB,th/h as derived earlier in Eq. (4).

E. Foucault Mode The Foucault mode is typically implemented in a conventional TEM and cannot be implemented in our STEM (the objective aperture is not available and the angular magnification is too small). Magnetic contrast is obtained by inserting an aperture in the back focal plane of the objective lens and selecting one of the split beams for imaging (one step further than in small-angle diffraction). Classically, only electrons deflected in the direction selected by the aperture (Fig. 19) are transmitted and contribute to the image contrast. Domains deflecting the beam into the aperture appear as bright areas, while those deflected in any other direction appear dark. The technique is simple to implement, shows high contrast, and is sensitive to the direction of magnetization. However, the small value of the deflection angle E leads to the same difficulty described in the previous section. The aperture has to be positioned in the back focal plane with submicron accuracy and any contamination or charging of the aperture precludes the technique from accurate quantitative measurements of magnetization (McFadyen and Chapman, 1992). Even with an ideal aperture the technique is not reliable for regions with rapidly varying magnetization and the intensity variations near magnetic domain walls is not directly

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

361

I I I I I I I I

IE cl I

P

specimen

objective lens

back focal Dlane

FIGURE19. Geometry of the magnetic contrast in the Foucault mode.

related to the magnetization distribution (Chapman, 1984). The Foucault mode is typically used for a quick determination of the magnetization orientation in large regions of constant magnetization (large magnetic domains) and thereby complements the Fresnel mode, which yields information about regions with rapidly changing magnetization (Doole et al., 1993). Recently, a modification of this magnetic contrast mode (Chapman et al., 1994) reveals Foucault images in the form of interferograms, which are related to the magnetization distribution in magnetic domains. Here a

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

phase shifting rather than an opaque aperture is inserted in the back focal plane of the objective lens and the images can be interpreted as in-line holograms (see Section 111). Coherent Foucault images showing high contrast fringes have been obtained (Chapman et al., 19941, from which the magnetic flux distribution can be deduced since the spacing of the fringes is determined by the flux enclosed by the corresponding beam paths. An important advantage of this new Foucault technique is the immediate visibility of the interference fringes (and thereby flux distribution) on the screen without any image processing which is usually required in electron holography techniques (Section 111). However, the accuracy of quantitative information is limited by the variation of the fringe spacing with position, shape, and exact transfer function of the phase aperture.

111. ELECTRON HOLOGRAPHY

A. Historical Foundations The original concept of holography was developed by Gabor (1948, 1949) as a way of eliminating the effect of lens aberrations in electron microscopy. Holography is a two-step process: first, two (or more) waves interfere, forming a hologram, and then the hologram is reconstructed, where information about the interference process can be extracted. Traditionally, the hologram has been reconstructed (light-) optically. However, many of the practical difficulties associated with light reconstruction can be avoided when the reconstruction is performed on a computer (Hawkes and Kasper, 1994). Although conceived for electrons, the first successful applications of holography were carried out with light due to the availability of a bright coherent light source, the laser. k i t h and Upatnieks (1962) introduced the off-line scheme of holography into light optics as a means of eliminating the twin image. Since Gabor’s proposal, a variety of holography schemes have been developed in electron microscopy (Haine and Mulvey, 1952; Marton, 1952; Mollenstedt and Diiker, 1956) and currently more than 20 forms have been described (Cowley, 1992). Some suggested modes are equivalent in principle, but differ in the microscope implementation and are related through the reciprocity principle; other modes may be considered as in-line and off-line equivalents of one mode. Recent applications include the improvement of the spatial resolution of electron microscopes (Kirkland, 1984; Van Dyck and Opdebeeck, 1990; Lichte, 1993) through aberration correction and studies of electromagnetic potentials in and around specimens (Frabboni et al., 1987; Tonomura, 1987).

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Electron holography, in conjunction with the quantifications of electron phase shift as outlined by Ehrenberg, Siday, Aharonov, and Bohm, has become a quantitative research tool for the investigation of magnetic materials (Mankos et al., 1994a). Extensive applications of electron holographic methods to the investigation of magnetic materials have been demonstrated by Tonomura (1993) in a conventional TEM. In what follows the principles of electron holography are derived for applications in STEM.

B. Basic Principles The principles of in-line and off-axis holography are illustrated in Fig. 20. In in-line holography (Fig. 20a), a coherent electron beam illuminating the sample is partially diffracted while passing through the sample. The interference between the unscattered wave and the diffracted wave, which both propagate along the same axis, is the hologram. The diffracted wave, modulated by electromagnetic potentials inside the specimen, contains information about the specimen, which can be extracted from the hologram in the reconstruction process. The advantage of this arrangement lies in the beam path symmetry which reduces the effect of lens aberrations. However, this advantage is overshadowed by severe problems arising in the reconstruction process. As will be made clear, the reconstruction process yields two conjugate images, which in the in-line scheme overlap and are difficult if not impossible to separate. The twin-image problem is circumvented in the off-axis scheme (Fig. 20b), where the illuminating beam is split by a biprism (independently from the specimen). When the specimen is in a field-free region, the influence of increased aberrations due to the asymmetry of the beam paths in the off-axis mode is negligible. The basic principles of interference and reconstruction are common for both schemes and, without loss of generality, the off-axis scheme will be considered only. It may be noted for completeness that the shadow image discussed in Section I1 is in Gabor’s interpretation an in-line hologram and the difficulty in extracting quantitative information is related to the problem of twin-image separation. In the off-axis beam technique, first carried out with electrons by Mollenstedt and Wahl (19681, a separate wave is introduced using an electrostatic biprism which was previously developed by Mollenstedt and Diiker (1956). This separate (second) wave is incident on the detector at an appreciable angle and in most of the applications (but not necessarily all) passes through vacuum. This wave is often termed the reference wave. The

364

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

a

diffracted wave incident wave

detector specimen

b

-

object wave

, reference wave

FIGURE 20. The in-line (a) and off-axis (b) schemes of holography.

implementation of off-axis holography in an electron microscope is illustrated in Fig. 21. The biprism acts as a beam splitter electron-optically and images an axial point source as a pair of coherent virtual point sources symmetrically placed about the z-axis (direction of beam propagation). For simplicity, we assume that the two split waves are coherent and monochro-

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365

electron source

biprism(s) lenses specimen

c

detector

FIGURE21. Schematic diagram of off-axis holography implementation.

matic. The two coherent wave packets propagate through the microscope, interact with the specimen, and form an interference pattern which is then observed in the detector plane. It can be immediately seen that this holography scheme is identical to the arrangement of the Aharonov-Bohm experiment. The intensity distribution of the interference pattern between the two coherent waves, appearing as a fringe-modulated image in the detector plane (Fig. 22a and b), is described by Eq. (12) and can be rewritten as ~ ( x , y =) W(r)12 = I

~ I ~ +, ~I qI0~2 1 2 + I*olI

W O A ( eiAqp, + e - i A p t ) .

(19)

Here the wave propagation, beam splitting, interaction with the specimen, and image formation can be described in terms of a total phase shift A q , due to the electromagnetic potentials enclosed by the two paths originating at the source and ending at the same point D in the detector plane. The microscope aberrations are neglected and the magnification is set to one for simplicity. The effect of lens aberrations will be considered later. Figure 22a is a low-magnification hologram of MgO crystals on a Cu grid showing the overlap region (bright band) while a detail of the interference fringe field is shown in Fig. 22b. Before attempting the reconstruction process, it is advantageous to separate the phase shift associated with the specimen by removing the specimen from the area enclosed by the beam paths and recording a reference vacuum hologram (Fig. 22c). This hologram is a two-beam interferogram from the electrostatic biprism and the

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

FIGURE22. Holography of MgO smoke crystals: (a) low-magnification hologram; (b) detail of fringe field near a crystal; (c) reference hologram with specimen removed from field of view; (d) and (el Fourier transforms of (b) and (c).

imaging action of the microscope lenses. Due to the small angles involved, the interfering waves can be approximated (Hawkes and Kasper, 1994) as plane waves. The resulting intensity modulation in the detector plane is then given by c o s ( 2 ~ u ~ xwhere ), u, is the carrier frequency determined by the electrostatic potential applied to the biprism and the lens excitations, and the x-axis is perpendicular to the biprism wire. This formula is valid

367

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

only in the central part of the interference fringe field. Near the edges Fresnel fringes from the biprism shadow are superimposed on the interference pattern. The effect of these diffraction effects is neglected here, although, in practice, the out-of-focus image of the biprism could be removed. A small loss of contrast and attenuation of the fringes with increasing distance from the axis due to the finite size of the electron source and its energy spread are neglected as well. With the specimen absent the phase shift Aq” can be written as A q u = 27rucx and the total phase shift A p t can be separated into two parts, A q , = 27rucx A q , where A q is now due to the specimen only. The reconstruction of the recorded hologram is a two-step process. First, a Fourier transform of the hologram intensity distribution yields a diffractogram (Fig. 22d and e) with three distinct peaks, a central peak and two sidebands:

+

F(z(x,Y))

=F(I*~,I~) +F(IW,,I’) +F(I*,,~I

+

Iq,,21e2niuc+iAv)

[ q o 2 1 e ~ 2 n i u c x ->, iAv

(20)

where Fdenotes a Fourier transformation. Using the multiplication theo=F(*oi’P$i) 2) = rem, the first two terms can be written as F(lYroi1 F(TOi(r)) * F.(*,J- r)), that is, an autocorrelation function peaked around the origin. The last two terms are rewritten in the following way:

~(lq I q o~ 21e~ + ( 2 vli u c x + i A v ) ) -

1

/,I,

1qoll 1qo2le fiAvefvi((u+ space

U,)X+UY)

Y.

(21)

These two distributions are centered at u = f u , in reciprocal space and form the two sidebands separated by 2u, (Fig. 22d). It is important to realize that each of the (complex conjugate) sidebands carries complete information about the specimen, that is, both the amplitude and the phase. In the next step in the reconstruction process, one of the sidebands is isolated (as marked in Fig. 22d) and the center of the sideband is set to be the new origin, that is, u’ = u f u c , u’ = u. In the new coordinates the sideband distributions can be written as

and Fourier transformation yields a reconstructed (complex) image

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

Let us assume that one of the split waves, the reference wave, traveled through a vacuum only and did not interact with the specimen. The amplitude of the reference wave 1q02 = 1a, can be normalized and set to one. Then the reconstructed images directly yield the phase difference A&, y ) and amplitude lqo,l= a ( x , y ) of the object wave (possibly perturbed by the lens aberrations):

Y)

=

a( x , Y)e

?C

i A vP(x,Y ).

(22)

The amplitude a(x, y) is dominated by inelastic scattering processes and the phase A p(x, y ) carries information about the projected electromagnetic potentials in the specimen:

=

1 -$p h

.dl

e

- -Q*.

h

With the reconstruction process completed, let us discuss the physical quantities which can be extracted from the reconstructed image 1u,(x, y). First consider nonmagnetic specimens. Electrons passing through a specimen are influenced by an effective electrostatic potential N r ) , created by the superposition of single-atom potentials. In a crystalline material the potential energy V(r) = eWr) is periodic and can be expressed in , where the Fourier coefficients V, a Fourier series, V(r) = Call Vge2Tig’r are known as structure potentials and g are the reciprocal lattice vectors. The lowest-order Fourier coefficient Vo = eQi represents the spatial average of V(r) and @; is called the mean inner potential. This mean inner potential causes a phase shift in the wave penetrating the sample (the sample has a different refractive index than the vacuum) relative to a wave traveling in a vacuum. Here, in the low-spatial-resolution limit in the electron-holographic mode ( 1 nm), we neglect the strong local variations of V(r), which produce elastic scattering at higher spatial frequencies. The phase difference due to the nonmagnetic specimen (A = 0) is then A d x , y ) = ( l / h ) $ p * dl. Consider the geometry shown in Fig. 23. The only phase difference between the two paths is due to the difference of potentials (Q outside the specimen, Q + Qi inside) when traveling the distance equal to the specimen thickness and Acp(x,y) = ( l / h ) j i ( p i - p ) d l , where pi is the magnitude of momentum inside the specimen. Using the relativistic expressions

,

-

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

I I

X

369

*

FIGURE23. Geometry for mean inner potential or thickness measurements.

for momenta, the phase difference can be rewritten as (Mankos, 1994)

assuming that aidoes not change with thickness and is much smaller than the beam energy (e@ = 100 keV) and electron rest energy ( E , = 511 keV). In the experiment @ and A are constant and the phase difference becomes a local measure of the projected mean inner potential: A v ( x , y ) = C,@,(x,y)t, (25) where C , is a constant depending on the beam voltage @. This allows for the determination of either the @; (for known thickness dependence) or the thickness (for known composition). Typical values of the mean inner potential vary from approximately 7-25 V. For a specimen with ai= 20 V and 10 nm thick, the phase difference for a 100-keV electron due to the

370

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

mean inner potential (when compared to a vacuum) is approximately 0 . 5 9 ~ An . example of a measurement of the mean inner potential in magnesium oxide is shown in Fig. 24. MgO smoke crystals form nearly perfect cubes making it a suitable material for the determination of the mean inner potential. When misaligned from the [ 1001 orientation (misalignment is necessary to reduce dynamical scattering effects), the electron wave travels through a 90” wedge, where the projected thickness f changes linearly with increasing distance from the edge. A phase image is shown in Fig. 24a. Note that the phase retrieval process yields only the principal values in the interval [ - T , + T I . For strong phase objects, such as thin magnetic films or thicker crystals, phase differences of tens of radians are common. The phase images reveal this periodicity in the form of characteristic “wrapped” images. The wrapped images must be further processed in

2

-15.0

2 -20.0 a -25.0 -30.0

-35.0 0 50 100 150 200 250 300 350 400 thickness [nm]

FIGURE24. Mean inner potential measurement in MgO smoke crystals: (a) wrapped phase image; (b) unwrapped phase of section marked in (a); (c) line scan perpendicular to crystal edge, averaged over 15 nm across (note that the x-axis is distance calibrated directly in corresponding thickness; (d) three-dimensional plot of the unwrapped phase (b).

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

371

order to unwrap the actual phase by successive additions or subtractions of 2.rr rad. A section of the phase image is unwrapped in Fig. 24b, where the slope of the line scan perpendicular to the edge (Fig. 24c) gives a phase gradient of 0.124 f 0.003 rad/nm (thickness). The mean inner potential extracted from this line scan is aj = 13.4 k 0.4 V. This value agrees with the accepted value of Qi= 13.01 V 0.6% (Gajdardziska-Josifovska et al., 1993). A three-dimensional plot of the unwrapped phase is shown in Fig. 24d. The experiment is carried out with the specimen in the out-of-lens position, where the error limits result from the inaccuracy in magnification calibration and variations in the vacuum phase. For a more accurate determination of either the mean inner potential or the thickness, the effects of dynamical scattering have to be considered (Gajdardziska-Josifovska et al., 1993). The dependence of the phase difference on the projected local mean inner potential can be used to investigate interfaces between materials (aj varies across the interface; Weiss et al., 1993) or study built-in potentials (e.g., due to doping in semiconductor devices; McCartney, 1994) and intrinsic electric fields in ferroelectric materials (Zhang et al., 1993). For a magnetic specimen of constant thickness and composition, the electrostatic potentials contribute a constant phase shift (this will be considered later), but the phase difference is predominantly due to the magnetic flux enclosed between the two beam paths. Unlike in a nonmagnetic specimen, the phase difference at a given position depends not only on the local value of B but on all its values enclosed between the object and reference beam paths. A detailed discussion of the applications of phase difference measurements to magnetic materials is given in the following sections. The amplitude of the reconstructed image provides information about the normalized thickness t / h i , where t is the local thickness of the specimen and hi is the inelastic mean free path (McCartney and Gajdardziska-Josifovska, 1994). Here the angular dependence of the inelastic mean free path hi, the partial coherence of the source, and Fresnel diffraction at the biprism are neglected for the sake of simplicity. Highenergy electrons suffer loss when passing through the sample and the relationship between the inelastic intensity Zi and the incident intensity I, is given as Zi = 1,(1 - e-f’Ai) (Egerton, 1986) and t =

hi

-In(

7). z, - zi

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

The amplitude I,,- Ii includes the unscattered and elastically scattered electrons. Cowley (1995) has shown that images reconstructed from the off-axis sideband are formed predominantly by unscattered and elastically scattered electrons; the small contribution from electrons with energy losses smaller or comparable to the energy spread AE is neglected here (AE is smaller for a field emission source). The incident intensity Z,, is proportional to the square of the amplitude of the reference wave and I,, - Zi is proportional to the square of the amplitude of the object wave, that is,

The ratio a,/a, can be obtained simply be acquiring an image with the specimen present (amplitude image equals a,a,) and absent (amplitude image equals a,a,) and dividing these two amplitude images. The natural logarithm of this ratio then yields an image which is proportional to the local normalized thickness t / h i . For a specimen of constant composition this image presents a thickness map at the same spatial resolution as the hologram and may be used to eliminate the thickness dependence of the corresponding phase image. A specimen of known thickness may be used to determine the value of the inelastic mean free path Ai (McCartney and Gajdardziska-Josifovska, 1994). A measurement of Ai in MgO is demonstrated on the same hologram which has been used for the mean inner potential measurement (Fig. 24). The reconstructed amplitude image is shown in Fig. 25a. Since in

FIGURE25. Inelastic mean free path measurement in MgO smoke crystals: (a) reconstructed amplitude image; (b) image of normalized thickness t / h i .

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373

this case the vacuum reference hologram is not available, a constant, determined from the amplitude in vacuum near the crystal edge, is subtracted from the natural logarithm image (Fig. 25b). A line scan (marked in Fig. 25b) of the t / A i = - 2/ln(a,/a,) image yields Ai = 78 6 nm (thickness dependence known from phase), which is in good agreement with the accepted value of 71 f 6 nm (McCartney and GajdardziskaJosifovska, 1994).

C. STEM Holography Modes For completeness, the familiar forms of off-axis STEM holography are described briefly. The in-line modes are not considered due to the inherent twin-image problem. The basic scheme which has been used for off-axis STEM holography employing an electrostatic biprism is shown in Fig. 26a. The biprism is placed in the illumination system so that a point source of electrons gives two electron probes in the specimen region. It can be seen from Fig. 26b that this scheme corresponds to the conventional off-axis TEM scheme with electrons traveling in the opposite direction (reciprocity theorem) and the biprism in the projector system of the TEM. Using a point detector at the optical axis, the intensity in the detector plane is recorded as the probes are scanned across the specimen. The dimensions of the detector must be much smaller than the periodicity of the fringes and the scheme becomes too inefficient since only a small portion of the transmitted electrons is detected. Progress has been made by Leuthner et al. (1989) who proposed and experimentally demonstrated the scanning mode of STEM holography with the detector in a form of a grating having the periodicity of the interface fringes. However, the efficiency of the method is limited by the fact that the area of the detector grating must be small and an increase in detector size results in loss of resolution and decrease in useful imaging area on the specimen. Several STEM-based electron holography modes have been proposed and further details considering the feasibility of off-axis STEM holography modes have been discussed by Cowley (1992). Our experimental practice has shown that the far-out-of-focus mode of STEM holography (Cowley, 1992), discussed in detail in the next section, is a useful and easy to implement mode of STEM holography. 1. Far-Out-of-Focus Mode of STEM Holography An electron-optical schematic of the HB-5 STEM adapted for the

off-axis holography mode is shown in Fig. 27. The Mollenstedt biprism

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

detector

source 0

wave

objective lens

biprism

I

electron source

a

detector

b

FIGURE 26. Ray diagram in STEM (a) and TEM (b) off-axis holography.

(Mollenstedt and Duker, 1956) is placed in the STEM illuminating system between the gun and condenser lens. The electron wave emitted from the source is split by the biprism into two wave packets, which are transferred by the condenser lens and focused by the objective lens into two fine electron probes coherently illuminating the specimen. When the objective lens is operated at a relatively large defocus, a large specimen region may be observed for each incident beam position. In most applications one hologram is sufficient and the scanning is switched off. The two wave packets interfere after passing through the specimen and form an interference pattern, a hologram, which is subsequently magnified by two postspecimen lenses and appears as a fringe-modulated twin image in the detector plane. The fringe spacing and the separation of the two (twin) images in the hologram are determined by the electrostatic voltage applied

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

field emission gun

375

L U !

specimen

post-specimen lenses and detector system FIGURE27. Implementation of off-axis holography in an HB-5 STEM.

to the biprism and by the excitations of the condenser and objective lenses. The combination permits expanded flexibility of our system, comparable to the classical TEM holography scheme. The hologram is recorded on a slow-scan CCD camera and reconstructed on a computer. The recorded hologram is acquired under the same imaging conditions as in the Fresnel mode described in Section 11, and consists of two partially overlapping Fresnel images (Fig. 28). The magnetic contrast observed in

376

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

FIGURE 28. Far-out-of-focushologram of a 20-nm-thickfilm on amorphous carbon. Note the twin-image overlap and domain walls (bright lines).

the recorded hologram is the same as in a Fresnel image; that is, domain walls appear as bright and dark lines (this is quite helpful when orienting on the sample). Again, the variation of magnification in the shadow image is negligible due to the typical imaging conditions (100 pm < A f < 1 mm). We have isolated two distinct holography modes: the absolute and the differential (Mankos et al., 1994a). In the absolute mode (Fig. 29a), one of the two electron probes travels through vacuum, while the other passes through the specimen. Assuming zero phase in vacuum and prior knowledge of the specimen thickness, we can absolutely determine the phase shift caused by the electromagnetic fields present in the specimen. For a magnetic specimen the phase shift A rp is proportional to the magnetic flux enclosed by the two beam paths. In a uniformly magnetized domain located near the edge of a specimen of constant thickness, the phase difference A rp changes linearly with increasing distance from the edge and Arp a / / B dS = B,xt, where B, is the component of the magnetic field normal to the plane determined by the wave vectors of the two split electron waves, x is the distance from the edge, and t is the (constant) thickness, The gradient of the phase determines the magnitude of B, (averaged over the film thickness) in the domain. For a film of nearly

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

377

electron probes

FIGURE29. Geometry of the absolute (a) and differential (b) modes of STEM holography.

constant thickness, we can neglect the contribution of the constant phase of the electrostatic field present in the specimen, since quantitative information is derived from the gradient of the phase difference and the phase variation is small (-0.1 rad/nm of film thickness). The straightforward interpretation of the phase image cannot be done for magnetic fields which are not confined to the magnetic specimen and extend spatially to relatively large distances (strong leakage fields, small particles with magnetic dipole structure). In this case a three-dimensional computer simulation of the phase integrated along the beam path is necessary for correct quantitative interpretation and the flatness of the reference wave phase has to be reconfirmed. In the differential mode (Fig. 29b), both of the split electron waves pass through the specimen. The separation of the beam paths, which is adjustable by the prism voltage as well as the excitation of the condenser and/or objective lenses, can be made as small as 10 nm. The area defining the enclosed magnetic flux is approximately constant (illumination is almost parallel) for every point in the hologram. In this mode the phase of a uniformly magnetized domain in a specimen of constant thickness is constant, in contrast to the absolute mode, where the same domain has a linearly varying phase difference. The phase difference A 4 a // B dS E &st, where B,, is the component of the magnetic field normal to the plane determined by the wave vectors of the two split electron waves, s is the separation of the beam paths projected into the specimen plane, and r is the (constant) thickness. The differential mode is advantageous for the investigation of magnetic domain wall profiles and allows straightforward interpretation of the magnetic microstructure (Mankos et a/., 1994a). Since this mode does not require a vacuum reference phase, no assumption has to be made about the flatness of the phase in vacuum. This becomes

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

advantageous in the case of strong leakage fields and for the observation of features far away from the edge or hole in the specimen. The sensitivity to local changes in phase difference in the direction perpendicular to the fringes is limited in this mode by the separation of the two sources. Since the resolution limit in reconstructed holograms is about 2-3 times the cosine fringe spacing, a sufficient condition for maintaining resolution is that the separation of the two dual images in the recorded hologram should be less than approximately three fringes. If this condition is met, the reconstructed phase difference becomes a direct measure of the local electromagnetic fields. We restrict the wave-optical description to one dimension, since the extension to two dimensions is straightforward. The imaging properties of the microscope are given by the transfer function of the objective lens which is dominated by the defocus and spherical aberration:

T ( u ) = A(u)exp

Ti Af

Ti

Au2 + - C,A3u4 2

The aperture is usually not inserted, that is, A(u) = 1 for all spatial frequencies and IT(u)I2 = 1. The spread function t ( x ) is given by a Fourier transform of T(u), t ( x ) = Y ( T ( u ) }= C ( X ) is(x), where 4 x 1 = .F{cos(~Tx(u))} and s ( x ) = Y{sin(2q(u))). Let us first consider the absolute mode and assume that the two probes are separated by a distance a. The exit wave function of the wave passing through the specimen [with a transfer function q h ) ] equals q ( x ) t ( x ) and the exit wave function for the reference wave is given by t ( x - a), where the Fresnel propagator exp(-ikx2/2t) has been neglected due to the small thickness t. The total exit wave function is given by the sum of the object and reference waves:

+

*(X)

=q(x)t(x)

+t(X

-a).

(27) The wave function W u ) in the detector plane is the Fourier transform of Eq. (271, W u ) = Q(u)* T ( u ) T(u)e21riau, and the image intensity Z(u) = lU(u)12 can be rewritten as

+

Z(u)

=

1 + l Q ( u ) * T ( u ) l 2 + 24Re2{5} + Im2( t } c o s ( 2 ~ a u+ a); (28)

that is, the image in the detector plane is modulated by (shifted) cosine fringes and a is the phase of T(u)(Q(u)*T(u))* = 5, defined by cos a

=

Re{ (1 dRe2{51 + Im2{51

,

sin a

=

1 4 51 4Re2{5) + Im2{61

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

379

An example of a far-out-of-focus hologram of an MgO crystal is shown in

Fig. 30a. For the reconstruction process a Fourier transform of the intensity I(u) is performed and a carrier frequency u, = a/m is introduced (Mankos, 1994). The Fourier transform consists of a central distribution and two sidebands (Fig. 30b) centered at u = fu, and includes the

FIGURE30. Reconstruction process in far-out-of-focus holography: (a) acquired hologram; (b) Fourier transform of (a); (c) and (d) real and imaginary parts of T * ( u ) = e - 2 r r i y ( u ) , the complex conjugate of the transfer function T(u); (e) and (f) amplitude and phase images not corrected for defocus; (g) and (h) amplitude and phase images corrected for shown defoci.

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

h

FIGURE30. Continued.

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

381

effect of aberrations. Centering one of the sidebands yields

) 30c and a multiplication of this diffractogram by T*(u) = e - z v i x y ( u(Fig. and d) allows us to compensate for the aberrations introduced by the objective lens. An inverse Fourier transform of the aberration-corrected sideband yields the object transfer function q ( x ) . The aberration correction requires knowledge of the accurate values of the lens aberration coefficients, in our case the defocus Af and the spherical aberration coefficient C,. As a consequence of the nonzero defocus Af, Fresnel fringes appear near edges or holes and may severely obscure the magnetic contrast of reconstructed holograms, as demonstrated by the uncorrected amplitude and phase images (Fig. 30e and f). In far-out-of-focus holography, the spherical aberration contribution to the phase shift can be neglected for the electron-optical conditions employed here (long focal length and resolution limit approximately a few nanometers). A through focal series of amplitude and phase images, shown in Fig. 30g and h, has been generated from the hologram by applying successive defocus phase corrections. Note the false phase line contrast present in both under- and overfocus images. The minimization of the standard deviation of the amplitude in vacuum has been found to be the most reliable criterion for the determination of optimum defocus, which can be loosely described as the condition for the vanishing of Fresnel fringes. This is demonstrated in Fig. 31 where line scans from amplitude images, corrected by defocus values & 1.5% from the optimum defocus (as determined in Fig. 301, show the characteristic first Fresnel fringe near the edge of the MgO cube (arrow). A series of three line scans of the corresponding phase images (Fig. 31b) has been used for a quantitative evaluation of the phase error as a function of defocus. A maximum phase variation of f0.3 rad within a *1.5% interval of the apparent correct defocus is of the same order of magnitude as that due to noise in the phase measurement in vacuum and limits the accuracy of quantitative data extracted from phase images. It is obvious that without the aberration correction it is not possible to extract in this case the quantitative information as shown in Section 1II.B. For a magnetic specimen the reconstructed and aberration-corrected phase difference image has an electrostatic and magnetic contribution:

Let us assume that the magnetic flux does not leak out of the specimen and that the sample is of uniform composition and thickness such that the

382

MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

25.0

,

'

'

'

,

'

"

I

'

.

'

- - .I37 pn ~

'

'

'

-139 prn _.-_. 141 p.rn

20.0 15.0

10.0 5.00

a 0.00 0

"

" '

20

'

"

40

' .

'

I

"

60

"

'

.

'

80

100

80

100

distance [nm] 3.00

2.00 1.oo

0.00

-1 .oo -2.00 -3.00 0

20

40 60 distance [nm]

FIGURE31. Line scans from amplitude (in relative units) and phase images, corrected with defoci near the optimum value (139 wm). The arrows mark the edge of the MgO smoke crystal.

magnetization is uniform in magnitude along the beam direction. The enclosed area determining the magnetic flux lies in the plane of the two beam paths (xz-plane, Fig. 32) and the phase difference simplifies to e

A C P ( X , Y= ) C,@i(x,y)t(x,y)

-

//&(x,Y) hdz,

where B, is the component of the magnetic field normal to the plane determined by the wave vectors of the two split electron waves (here

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

a

383

b

FIGURE32. Magnetic structure (a) and corresponding phase difference image (b) in the absolute mode of STEM holography.

B,

= By). A

gradient of the phase difference with respect to x yields

For a specimen of uniform composition and with negligible thickness variations, the first two derivatives can be neglected and

dA4x9 y, =

-

dX

e B,(x,y)t(x,y);

n

that is, the phase gradient represents a direct measure of the local variations of the normal in-plane component of the magnetic flux density B,(x, y ) , averaged along the specimen thickness. The relationship between the measured B,(x, y ) and the magnetization distribution in the specimen is discussed in the next section. Equation (31) forms the basis for STEM holography as a quantitatwe magnetometric method (absolute magnetometry) at high spatial resolution. In cases where the magnetization is uniform in magnitude and only changing in direction and has in-plane components only (as in some hltrathin films), the phase difference image completely determines B [not just B , ( x , y ) ] . This can be demonstrated in the following way. Let us assume that the phase difference A&, y ) and the magnetic flux density

384

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

B(B,, B y ,0) are continuous functions of x and y and

and hence

Since the divergence of B equals zero,

and B, can be rewritten as B

=

- j dB L&= fi - dj2 A 4 D ( x , y ) & = -fi et

dY

dx

dy

et

d A P ( X , Y ) , (33) dy

and the gradients of the phase difference in two orthogonal directions yield the two orthogonal components of the in-plane flux density B. Note that this is true even when B, is nonvanishing but independent of z , that is, dB,/dz = 0. An example of a specimen with predominantly in-plane magnetization, shown in Fig. 33, illustrates the information which can be extracted from a phase image. The maximum gradient of the phase difference is 70.3 f 1.0 mrad/nm in domain I and 69.6 zk 1.0 mrad/nm in domain 11, which differs by less than 1%from the average (70.0 mrad/nm). A detailed analysis of the specimen is given in Section IV. In the case where flux leaks from the sample, the whole integral J B, dz has to be considered, and a straightforward interpretation cannot be done due to the three-dimensional character of the leakage fields; the phase difference images have to be accompanied by computer simulations. In the differential mode the two probes are separated by a distance a which is small when compared to magnetic features of interest. The exit wave function of the wave passing through the specimen is now given as

+q(x)t(x -a).

(34) The expressions for the recorded intensity are manipulated in a similar way as demonstrated for the absolute mode (Mankos, 1994). Again, as in the absolute mode, the image in the detector plane is modulated by cosine fringes. An example of a far-out-of-focus hologram of a thin Co film is shown in Fig. 34a. Both split waves are treated on equal footing and cannot be distinguished as either an object or a reference wave. As a consequence the twin images are separated by a small distance. The *(x)

=

q(x)t(x)

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

385

FIGURE33. Extraction of information from absolute mode phase images: (a) wrapped phase image; (b) unwrapped phase of region marked in (a); (c) contour image of (b), representing equimagnetization lines; (d) three-dimensional image of (b).

reconstruction process yields a more complicated image:

In order to gain insight some assumptions have to be made about the object transfer function q ( x ) and the spread function dx). Let us approximate to first order the spread function by a 8-function distribution. This is justified assuming that the lens transfer function T ( u )is dominated by the defocus contribution and therefore a Gaussian distribution in reciprocal space. The Fourier transform of T ( u ) is then a Gaussian distribution characterized by a width w = which approximately equals 20 nm for a typical defocus of 100 pm. This approximation is then appropriate for objects whose transfer function does not change appreciably on a length scale comparable to w.The expression for the reconstructed image

386

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

...,...,...,...I,,.,...

4.00

0

200

400

600

800

1000

1200

distance [nm] FIGURE34. Differential mode of STEM holography: (a) acquired hologram; (b) unwrapped phase image reconstructed from (a); (c) line scans across domain wall, averaged over 100 nm.

can be rewritten with t ( x ) = 6 ( x ) as

+ a)q*(x)(36) Let us further assume that the phase difference A p(x) varies slowly over a length scale comparable to the beam separation and express Acp(x + a) in a Taylor series: *r(x)

=q

(X

Keeping only the first two terms simplifies Eq. (36) to

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

387

that is, the phase of the reconstructed image is proportional to the gradient of the phase difference and therefore is a direct measure of the normal component of the projected in-plane flux density (and magnetization). A phase image, reconstructed from the hologram (Fig. 34a), illustrates the predicted flat phase distribution inside domains and a phase jump across the domain wall (Fig. 34b). Although the beam separation is rather large in this case, quantitative information in the form of a domain wall profile (Fig. 34c) can be retrieved under the assumptions that the magnetization distribution is approximately constant along the domain wall and that the beam separation projected onto the direction of magnetization change is small (perpendicular to the domain wall, as in Fig. 34). For small-scale magnetization changes and high-resolution studies, a full analysis of Eq. (35) is required. This rapidly becomes complicated and has to be considered case by case. The relationship among the spatial resolution, field of view, and detector parameters has been discussed in detail by Cowley et al. (1995). The differential mode of STEM holography yields a direct distribution of the in-plane component of magnetic flux density B, (or the in-plane component of the electric field E, in a ferroelectric specimen). Another important advantage of this mode is the fact that no hole or edge is required. However, the quantitative interpretation is straightforward only for simple magnetization distributions and at medium resolution ( 20 nm). The requirement of a small beam separation requires low biprism voltages, which in turn limits the field of view (small fringe field).

-

IV. APPLICATIONS A. Magnetic Thin Films

Magnetic thin films, due to their two-dimensional character, have certain common properties which justify their separate consideration. From energy minimization calculations of various magnetization configurations (Kittel, 19461, it can be seen for film thickness less than some critical thickness (-300 nm for Fe and 100 nm for Co) that the films are magnetized predominantly in plane regardless of the orientation of the crystal structure. Films of thickness below this critical thickness (but still thick enough so that the surface anisotropy cmtribution is negligible) behave like films with an in-plane anisotropy even if the easy axis is perpendicular to the surface. Thin-film preparation of magnetic materials was first realized by Blois (Blois, 1955) who prepared thin films by vapor

388

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

deposition. Structurally, these films are mostly polycrystalline, made of nanocrystallites with inherent crystal anisotropy and diameters ranging from approximately 2-10 nm. The anisotropy is generally not uniform and depends on the microscopic structure, which is largely influenced by the growth conditions. The magnetization direction may diverge from the overall easy direction in a domain, resulting in “magnetization ripple.” Magnetic thin films are used for applications in magnetic storage technology (recording media and heads) and magnetic sensors because of their nearly square hysteresis loops and fast magnetization reversal processes. Magnetic and structural properties of magnetic thin films are well understood (Craik and Tebble, 1965; Carey and Isaac, 1966). The predominant in-plane magnetization and constant (small) thickness make thin films an ideal specimen for testing newly developed holographic techniques (Section 111) and for making comparisons with the established conventional techniques (Section 11). With no magnetic field applied, the total micromagnetic energy is reduced by domain formation. The magnetic microstructure of domains and domain walls in thin films may have its origin in stray field reduction, local anisotropy, or the geometry of the specimen (or a combination). Below the critical thickness the domains are separated by NCel or asymmetric Bloch (LaBonte) walls rather than Bloch domain walls. Inside NCel walls the magnetization may also reverse its direction periodically (to reduce total energy), forming a more complicated wall structure. When Bloch line segments exist, a further energy reduction can occur in such a way that flux closure occurs between every second segment, resulting in the formation of cross-tie walls. Also, the simplest view of domain structure, with absolutely uniform magnetization inside domains and relatively abrupt changes at domain walls, cannot be applied strictly to thin films and rather continuous changes of magnetization extending into the domain can be seen (Craik and Tebble, 1965). For example, near a hole in the specimen the magnetization tends to run tangentially around the edge so as to minimize stray field energy. Further complications of the micromagnetic structure can be expected from long-range stray field interactions and interactions of NCel walls and Bloch lines with structural imperfections (defects, grain boundaries, etc.). Ripple patterns reflect the response of magnetization to the local distribution of individual nanocrystallites and the magnetic microstructure depends on the previous magnetization states, too. Experimentally, the fine structure of domains and domain walls is difficult to extract due to their small size. Domain walls appear in thin films in complicated networks and it is difficult to study an “ideal” isolated wall. The goal of this section is to elucidate quunfifutiuely the magnetic

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

389

microstructure in films made of different materials and thicknesses using the inherent high spatial resolution of the electron-holographic techniques. The holographic technique allows the extraction of quantitative information about the magnetic structure at high spatial resolution and therefore significantly expands the abilities of STEM as a tool for investigating magnetic materials. Since the Fresnel mode is still the simplest and fastest method for magnetic micrstructure observation and the DPC mode provides additional information about the magnetization orientation in the particular domains, the three techniques used in conjunction in the same instrument provide all the necessary information for complete, calibrated micromagnetic structure determination. However, we have to keep in mind that in the transmission modes the extracted information relates to the projected magnetic structure only and the bulk structure dominates over possible surface effects, which are washed out. A three-dimensional determination of the magnetic microstructure requires tomographic techniques, the description of which goes beyond the scope of this work. 1. Cobalt Thin Films

Cobalt is an element with hexagonal closed-packed structure at room temperature in bulk, and a face-centered cubic phase at temperatures above 400°C. Below 1120°C it is ferromagnetic and has a room temperature spontaneous magnetization of 1440 emu/cm3 ( = 1.8096 T). In the following analysis we neglect the variation of the magnetic film thickness over the analysis region (typical thickness variations are less than 1-pm field of view) as well as the phase shift a few nanometers over the caused by the mean inner potential of the sample. For a film of nearly constant thickness, we can neglect the contribution of the constant phase of the electrostatic field present in the specimen, since quantitative information is derived from the gradient of the phase difference and the phase variation due to the electrostatic potential is small ( - 0.1 rad/nm of film thickness). Any phase changes caused by thickness variations are small ( 0.2 rad/nm for cobalt) when compared to the absolute phase changes ~ caused by the magnetic field ( 1 0 rad). In order to examine the unique contrast revealed by STEM holography, we compare the micromagnetic structure of an approximately 25-nm-thick Co film, analyzed in Fig. 33, with the accepted Fresnel and DPC contrast modes of Lorentz microscopy. The wrapped phase image with the proposed magnetic structure is shown again in Fig. 35b. A bright field image of the same area, obtained as a summation DPC signal, reveals the microscopic structure in Fig. 35a. Figure 35c and d displays Fresnel mode images and Fig. 35e and f displays DPC images of the same region of the

-

-

-

390

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

FIGURE 35. Lorentz microscopy of a 25-nm-thick Co film: (a) bright field image; (b) absolute phase image with proposed magnetic microstructure; (c) and (d) underand overfocus Frensel images; (e) and (f) DPC images.

cobalt film. We can clearly correlate the domain wall structure emanating from the kink in Fig. 35c and d (black in Fig. 35c, white in Fig. 35d) as the same wall which divides the two regions of uniform magnetization in Fig. 35b. The orientation of the domains can be extracted straightforwardly from the DPC images and agrees with the proposed structure in Fig. 35b. What we are not able to extract from Fig. 35c-f is the absolute value of

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

-

391

--domain It

6 -6.00 Y

v)

c -8.00 n

-10.0

-12.0 0

"

'

I

'

"

' 40

"

'

I

'

I ' ' ' d

"

60 60 80 100 distance [nm] FIGURE 36. Line scans of phase along maximum gradients in domains I and I1 (in 20

Fig. 35b).

magnetization. Line scans of the phase image acquired in the absolute mode and taken along the maximum phase change (Fig. 36) reveal a nearly linear dependence of the phase with distance with a phase gradient of 70.3 f 1.0 mrad/nm (domain I) and 69.6 k 1.0 mrad/nm (domain 11). The average phase gradient of 70.0 mrad/nm is compared with the theoretically predicted value: assuming uniform bulk spontaneous magnetization of 1440 emu/cm3, the phase gradient/nanometer for 1 nm thickness equals dAcp

d

-= -

dx

dx

(-ne //

€3 d S )

e =

Bt

=

2.75 mrad/nm.

(38)

Assuming uniform in-plane magnetization inside domains I and 11, the average phase gradient of 70.0 mrad/nm corresponds to a projected flux density of 46.1 T-nm. This agrees favorably with the accepted value of the saturation magnetization in Co (1.81 T) for a 25.5-nm-thick film. A direct determination of the thickness-averaged flux density is not attempted here since the thickness is not known precisely enough ( f 15%). Also evident in Fig. 35b is the rotation angle of magnetization across the wall (108.3 + 2.0"). This structure is consistent with the presence of 71", 109", and 180" domain boundaries on (110) surfaces in fcc lattices (Craik and Tebble, 1965). A detailed analysis of the domain wall separating domains I and I1 is carried out in Fig. 37. Line scans acquired in a direction perpendicular (Fig. 37a) and parallel (Fig. 3%) to the domain wall reveal the domain wall profile. In the perpendicular direction, sensitive to the projected magnetization component parallel to the domain wall, the phase is symmetric

392

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY 60.0

40.0 20.0

0.00 -20.0

-40.0 -60.0

-80.0t , 0

'

" '

'

'

100

'

'

' ' ' '

200

'

"

"

300

I

'

'

'

400

500

distance [nm]

-8.00-

-12.01. 0

"

' 20

'

'

'

' 40

'

'

"

'

60

'

" '

80

'

'

100

distance [nm] FIGURE37. Quantitative analysis of domain wall structure in a 25-nm-thick Co film: (a) phase profile in the direction perpendicular to the wall and its gradient (smoothened in bold); (b) phase profile in the direction parallel to the wall at three different positions.

about the wall core and the phase gradient equals -33.2 mrad/nm (domain I) and 37.9 mrad/nm (domain 111, yielding an absolute average gradient of 35.6 mrad/nm in the domains and approximately zero at the wall core. Line scans in the parallel direction, sensitive to the projected magnitization component perpendicular to the domain wall, show a nearly linear phase change, characteristic for constant magnetization. The phase gradient in this case is 55.9 mrad/nm (domain I> and 63.8 mrad/nm (domain 11), yielding an average gradient of 59.9 mrad/nm in the domains. The orthogonal components, added in quadrature, yield a magnitude of

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

393

J(35.6)' + (59.9)' = 69.7 mrad/nm as expected. However, the parallel phase gradient at the wall core is 51.6 mrad/nm, about 30% less than the saturation value of 70 mrad/nm. This indicates that the domain wall cannot be a pure NCel wall and must have segments with out-of-plane magnetization, consistent with an asymmetric Bloch wall (LaBonte wall) structure, as expected for films in this thickness range. The wall profile along the core seems to have 10-15 nm regions of periodic high and low gradient. However, the signal-to-noise ratio of the phase is too small to allow a definite quantitative determination. The width of the domain wall (10-90% of maximum value), as determined from the phase gradient in Fig. 37a, is 128 L- 5 nm. A phase image of the same region, reconstructed from a hologram acquired in the differential mode, is shown in Fig. 38a. The phase is approximately constant in regions with constant magnetization, and changes across the domain wall. The domain wall magnetization profile, extracted

8.00

6.00 Y

4.00

2.00

0.00 -2.00t. 0

"

"

'

100

'

' ' " 200 300 distance [nm]

"

"

"

"

'

"

400

'

500

FIGURE38. Differential mode of holography in a 25-nm-thick Co fdm: (a) phase image and magnetization structure; (b) phase line scan in direction perpendicular to the wall from unwrapped phase.

394

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY 380 r . , .. . . , . . . . ,

.-

;I, ~,: ,,

320

0

50

,u

,.

,...., ..

,

..

,

60.0 , . . . . , . . . . , . . . , .

, ..

,.

,

,

,..

I\

J,

-40.0 :

-60.0-

100 150 200 250 300 350 400

-80.0LL-Ld-&' ' " ' ' ' ' ' ' ' ' " ' " 1 0 50 100 150 200 250 300 350 400

distance [nm]

distance [nm]

100,. . . . ,

8.00

.. ,

.,

,

,

,

,

.

,

6.00

6

-et S

n

4.00 2.00

0.00

-

-2.00

0

50

100 150 200 250 300 350 400

distance [rim]

0

50

100 150 200 250 300 350 400

distance [nm]

FIGURE39. Comparison of domain wall profiles extracted from Fresnel (a), absolute phase (b), differential phase (c), and DPC (d) images.

in a direction perpendicular to the domain wall and shown in Fig. 38b, yields a domain wall width of 145 k 15 nm. Figure 39 compares domain wall profiles, extracted from images acquired in the absolute and differential holography modes, DPC mode, and Fresnel mode in a direction perpendicular to the domain wall. The Fresnel mode profile (Fig. 39a) does not allow any direct interpretation of the domain wall thickness or the magnetization distribution across the domain wall. The sharp spikes in the absolute mode profile (Fig. 39b) are caused by the noise present in the phase image, which becomes more pronounced when the numerical derivative is performed. Comparing the wall profiles from the differential holography (Fig. 39c) and DPC (Fig. 39d) modes, we readily see that the latter displays strong ripple due to scattering contrast from small particles, hence significantly complicating the determination of the wall profile. This ripple is suppressed in the holography mode, because the phase changes caused by these effects are small when compared to the absolute change of phase from the magnetic flux. The width of the domain

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

395

wall determined from the DPC mode profile (116 k 15 nm) agrees (within the fluctuations due’to experimental error) with both holography modes. A hologram of a flux vortex in a 20-nm-thick Co film, acquired in the absolute mode, is shown in Fig. 40a. The wrapped phase image (Fig. 40b) reveals a nearly linear phase change in the domain to the right of the domain wall perpendicular to the film edge. The maximum phase gradient extracted from the unwrapped phase image (Fig. 40c and d) is 48.2 mrad/nm, in good agreement with the value for a saturated (approximately) 20-nm-thick Co film (55.0 mrad/nm). The magnetization rotates

FIGURE40. Holography of a 20-nm-thick Co film: (a) absolute hologram; (b) wrapped phase image at low magnification; (c) unwrapped phase image; (d) three-dimensional plot of (c); (el three-dimensional plot ofunwrapped phase, 450 nm X 450 nm; (f) phase profile along directions marked in (e).

396

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

by approximately 110" when crossing the domain wall. However, the maximum gradient decreases and is minimal along the long domain wall parallel to the edge. A high-resolution hologram (Fig. 40e) of the vortex does not include the specimen edge, and so does not permit the use of the Fresnel fringe criterion for an accurate defocus correction. Under these conditions the criterion of vanishing magnetic contrast in the amplitude can be applied. The amplitude image is, in principle, a Fresnel mode image, displaying a domain wall contrast switch when changing from an

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

397

undercorrected to an overcorrected condition. Two line scans (Fig. 40f) acquired perpendicular to each other (as marked in Fig. 40e) show that within an area 15 f 5 nm in diameter around the vortex center the phase is constant; that is, the magnitization is oriented completely out-of-plane. The whole vortex structure is symmetric about the long domain wall with an abrupt phase change at the center and a gradual change along the wall (Fig. 40f). The investigation of dynamic phenomena in Co thin films has been inspired by recent interest in slow relaxation in magnetic materials (Chamberlin and Scheinfein, 1992). Length scales of 1-100 nm and time scales of 1 ns-100 s are of interest for mesoscopic excitations such as the relaxation of magnons in ferromagnetic materials. Currently no technique exists for their direct measurement. A systematic study requires a variable external magnetic field in the specimen area. In order to demonstrate the spatial and temporal resolution of the STEM, experiments have been carried out with the specimen in the in-lens position and the objective lens switched off. When the lens is switched off, the specimen is in a field-free region; when the lens is switched on, the specimen is subjected to a strong magnetic flux density of approximately 1 T depending on the lens excitation. The inhomogeneity of the applied field and reduced spatial resolution are the obvious complications of this modification. First, the microscope is focused and aligned in the Fresnel mode, then the objective lens is switched on for several seconds and then off again. Time evolution images are recorded on a video recorder and the images are subsequently analyzed frame by frame. Usually the first five frames ( 150 ms) after the switch-off are washed out due to the relatively large time constant caused by the inductance of the objective lens. A series of consecutive frames (30 ms each) displaying relaxation in a 25-nm-thick Co film is shown in Fig. 41a. In each consecutive series, the same domain wall disappears (marked by arrows). However, there is a different delay time for each series. A plot of the distribution of delay times is shown in Fig. 41b. No direct evidence of the slow relaxation as described by Chamberlin and Scheinfein (1992) has been observed so far. This is also further complicated by the fact that static domains do not have to coincide with dynamically correlated domains (Chamberlin and Scheinfein, 1992) and distinguishing criteria for their imaging have yet to be found. N

2. Nickel Thin Films Nickel is an element with face-centered cubic structure. Below 360°C it is ferromagnetic and has at room temperature a value of spontaneous magnetization of 484 emu/cm3 ( = 0.6082 TI.

398

M. W K O S , M. R. SCHEINFEIN AND J. M. COWLEY

I=I

300 nm 8 7

6 5

4 3

2 1 0

270 300 330 360 390 420 450 480

Delay after switch-off [ms]

FIGURE41. Relaxation phenomena in a 25-nm-thick Co film: (a) a series of consecutive frames shows the gradual decay of a domain wall (marked by arrows); (b) the distribution of delay times after the external magnetic field is switched off.

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

399

As in the previous section, we neglect the variation of the magnetic film thickness over the analysis region as well as the phase shift caused by the mean inner potential of the sample. In the hologram of a 10-nm-thick Ni film, acquired in the absolute mode (Fig. 42a), two converging and one diverging wall oriented perpendicular to the edge can be recognized. The unwrapped phase image (Fig. 42b) and its three-dimensional plot (Fig. 42c) reveal a nearly linear phase change inside the domains and a small magnetic flux leak, recognizable as a slightly undulated phase near the specimen edge. Inside the domains, the magnetization is oriented parallel to the domain walls and rotates by 180" when crossing a domain wall. At the wall the magnetization is approximately parallel to the edge, resulting in almost no magnetic flux leak. The maximum flux leak can be observed near the center of each domain. The maximum phase gradients in the domains, extracted from the line scans of the phase acquired in a direction perpendicular to the domain walls (Fig. 42d), are 11.4, 9.4, and 12.1 mrad/nm, in good agreement with the value for a saturated 10-nm-thick Ni film (9.4 mrad/nm). A gradient of the phase (Fig. 42f, near the left domain wall in Fig. 42d), taken in a direction perpendicular to the domain wall, is proportional to the in-plane component of magnetization parallel to the domain wall. The domain wall is symmetric, with approximately equal phase gradients in the vicinity of the domain wall (k10 mrad/nm) and a domain wall width of 80 i-5 nm. Near the wall core the perpendicular gradient is zero. However, this phase gradient drop is exactly compensated in the direction parallel to the domain wall, where the phase changes linearly (Fig. 42g) with a slope of 10.7 mrad/nm. This allows us to conclude that the magnetization remains in plane everywhere and the wall is a pure NCel wall. The phase distribution acquired approximately 150 nm from the specimen edge (in vacuum) is shown in Fig. 42e (note that the scale is five times larger). The same analysis can be carried out assuming that the flux density remains in plane outside the specimen, too. The maximum phase gradients, as measured from Fig. 42e, are 2.1, 0.6, and 1.1 mrad/nm, indicating that the flux density drops rapidly with increasing distance from the edge. Rigorously, one cannot assume that the flux remains oriented in plane. The correct approach is to take a line scan in the direction perpendicular to the probe separation (Section 111). In this case the gradient drops to l/e ( 1 mrad/nm) of its maximum value (near edge) at a distance of approximately 400 nm from the edge. This is important to know, since long-range leakage fields could influence the measurement accuracy, due to a nonzero and varying phase of the reference beam.

-

400

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

B. Multilayer Structures Multilayer structures can be defined in general as thin films composed of two (or more) different materials, with a composition modulated with a wavelength (bilayer thickness in the case of two different materials) ranging from a fraction of a nanometer up to approximately 100 nm. The ideal multilayer structure should consist of bilayers of equal thickness and be free of defects, with sharply defined interfaces. This requires a near lattice match, low solubility of the constituents, and similar free energies. The highest-quality multilayers are grown by molecular beam epitaxy. This is impractical for the production of large quantities of superlattices and lattice-mismatched systems. More practical fabrication methods include vapor deposition, electron beam evaporation, and sputtering. Multilayer structures made of semiconductors with closely matched lattice constants have been applied in semiconductor laser technology. Another area of extensive multilayer research is X-ray optics. Here the effort is directed toward applications for optical elements due to the high normal-incidence reflectivity in the soft X-ray range. The application area we will be concerned with most is the field of magnetic multilayers. Magnetic multilayers are multilayer structures with at least one magnetic component (element or alloy). Investigations were accelerated when Baibich et al. (1988) discovered that magnetic multilayers exhibit a magnetoresistance (MR) effect, which for some compositions and geometries reaches extremely large values (giant magnetoresistance or GMR). The GMR effect is defined as the change in resistance from when the sample is placed in zero magnetic field to that in the saturation field divided by the saturation field resistance. The mechanisms underlying the (G)MR effect are of fundamental scientific interest. It should be noted here that strong magnetoresistance effects (although based on different mechanisms) were obtained in the 1960s in iron whiskers (Reed and Fawcett, 1964) and rediscovered in multilayers. In the first experiments an oscillation of the magnetoresistance with interlayer thickness was observed (Parkin et al., 1990), the period of which corresponds to the period of oscillations of the

FIGURE42. Holography of a 10-nm-thickNi film: (a) hologram acquired in absolute mode; (b) unwrapped phase image of section including the domain walls marked in (a), with proposed magnetic microstructure; (c) three-dimensional plot of unwrapped phase image (b); (d) line scan of phase (b), parallel to the edge and inside the film; (e) line scan of phase (b), parallel to the edge and outside the film (in vacuum; (f) phase gradient in direction perpendicular to the left domain wall in (d); (g) line scan of phase in direction parallel to the left domain wall in (d).

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ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

2.00

e6 3 5

-

-2.00 n

-4.00

-3.50

-6.00 0

domain wall 500

lo00

near domain wall

1500

-4.00 0

distance [nm] 15.0

,.. . . , . . . . , .. . . , .. . . , . . . . , -,

500

1000

1500

distance [nm]

.. .. ,... . #

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0

1

,

.

,

,

50

.. .

. ,

... .

100

distance [nm]

,

150

,

,

, ,

,

200

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

exchange coupling across the nonmagnetic layer. In a simplified picture high magnetoresistance ratios occur in antiferromagnetically coupled films. When a sufficiently strong field is applied, the internal coupling is overcome and the magnetization in all layers aligns parallel to the applied field, resulting in a decrease of resistance (less electron scattering). The giant magnetoresistance effect has been observed in several multilayer systems made of different magnetic and nonmagnetic layers, for example, Co/Cu and Fe/Cr (Baibich et al., 1988; Parkin et al., 1991). The exact nature of coupling and the value of the giant magnetoresistance ratio depend strongly on anisotropy and local microscopic structure; for example, the presence of a buffer layer gives much higher quality multilayer structures with flat layers and sharp, well-defined interfaces. Surface and interface effects complicate the interlayer coupling. In addition to the predicted antiferromagnetic 180" coupling, 90" coupling may coexist in multilayers and is attributed to thickness variations and interface imperfections (Slonczewski, 1991). Another attractive feature of magnetic multilayer structures is the presence of perpendicular (out-of-plane) anisotropy, reducing the domain size and therefore increasing the recording density. This makes, for example, Co/Pd multilayers a promising candidate for magneto-optical recording materials in the short-wavelength optical region (den Broeder et al., 1992). The favorable magnetic properties are mostly attributed to surface and interface effects. 1. Cobalt / Palladium Multilayers

In Co/Pd multilayers the magnetic properties are strongly influenced by the number and thickness of the magnetic and nonmagnetic layers, the sharpness of the interfaces, and the local microscopic structure, which in turn depend on the growth parameters. From hysteresis loop measurements of rf-sputtered Co/Pd multilayer structures (deHaan, 19921, it was found that the easy axis switches from in plane to a direction perpendicular to the surface when the Co layer is decreased below a critical thickness ( - 1 nm), which depends on the sputter rate, sputter gas (Ar,Kr), and its partial pressure. It is the increasing surface contribution that is responsible for the favorable perpendicular anisotropy. A Co/Pd multilayer structure, Pd (20 nm)/[Co (1 nm) Pd (1.1 nm)],,, grown on an amorphous carbon film, displayed dominant in-plane magnetization with saturation magnetization Ms = 1600 emu/cm3 and coercive field Hc,, = 88 Oe. Multilayers of this type are useful here as illustrative examples of the absolute magnetometric capabilities of electron holography since the thickness of the multilayer must be controlled at the percent

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

403

level. The Fresnel mode images (Fig. 42a and b) display a typical distribution of magnetic domain walls appearing as white and dark lines (note in Fig. 43a the broad dark biprism shadow and the edge which is parallel to the biprism at the bottom of Fig. 43a). Near an edge or hole, the domain walls become nearly parallel to each other, running approximately perpendicular to the specimen’s edge. Further away from the edge, the magnetization begins to curl forming typical “w”-shaped domain walls (Fig. 43b). A phase image (Fig. 44a), reconstructed and unwrapped from a hologram acquired in the absolute mode of STEM holography, and a threedimensional map of the marked area (Fig. 44b) show that the magnetization is oriented perpendicular to the edge of the sample and rotates by 180” when crossing the domain wall, a result consistently observed in different specimen regions. A line scan of the phase, taken along the edge and averaged over a region 80 nm across (Fig. 4 4 4 shows the linear dependence of the phase inside domains I and I1 and the location of the domain wall. The slope of the phase absolutely determines the magnitude of magnetization inside the domains for uniform thickness films. In this case the phase gradient is 28.0 mrad/nm in domain I and 10.9 mrad/nm in domain 11. The value in domain I agrees well with the theoretically predicted value for all Co layers ferromagnetically aligned throughout the superlattice stack. Assuming a total Co thickness of 10 X 1.0 nm = 10 nm with uniform bulk spontaneous magnetization of 1440 emu/cm3 (1.8096 T), the phase gradient equals 27.49 mrad/nm, that is, within 2% of the measured value. The magnetization in domain I1 is approximately 39.6% of the expected ferromagnetically aligned bulk value. This suggests that not all magnetic layers in domain I1 are magnetized in the same direction (assuming only in-plane magnetization). The measured value indicates that

FIGURE43. Fresnel mode images of Co/Pd multilayer structures.

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

distance [nm] FIGURE 44. Holography of Co/Pd multilayer structures: (a) partially unwrapped phase image of domain structure near the edge of a film; (b) three-dimensional plot of the phase in the region marked in (a); (c) line scan of phase along specimen edge, averaged across 80 nm.

the magnetization vectors in the layers must be oriented with seven layers in one direction and three layers in the opposite direction, producing a net projected (integrated) magnetization of 40% of the saturated value. The phase gradient at the wall core (in a direction parallel to the wall) is 11.5 mrad/nm. This suggests that not all layers rotate in the wall in the same direction, since this would correspond to a larger phase gradient. While observing the magnetic structure near the specimen edge, a strong magnetic flux leak into the surrounding vacuum was found (Fig. 45). The reconstructed, unwrapped phase (Fig. 45a) and contour image (Fig. 45b) of the same area display the periodically changing phase; the contours are equimagnetic-induction lines and make the flux flow more visible. The line scan in Fig. 45c, taken in a direction perpendicular to the film edge, shows the decay of the leakage field. The gradient of the phase, which is proportional to the projected component of the magnetic induction approximately parallel to the edge, reveals that this field falls to l / e of its maximum value at a distance approximately 250 nm from the edge. The

405

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

2.00

000

-2 00

-4.00

0

200

400

600

800

distance [nml

1000

1200

0

500

1000 1500

2000

2500

3000

distance [nm]

FIGURE 45. Leakage fields in Co/Pd multilayer films: (a) partially unwrapped phase image; (b) contour image of the same area as in (a) where one contour corresponds to r / l O rad; ( c ) line scan of the phase perpendicular to the edge and its gradient; (d) line scan of the phase parallel to the edge; (el three-dimensional plot of the phase outside the specimen.

ripple in the right part of the profile is due to the Fresnel fringes of the biprism. A comparison of the line scan parallel to the fiIm's edge ( 150 nm off edge, Fig. 45d) and a Fresnel image of the same area shows that the domain walls terminate at inflection points of the phase curve and near the center of a domain the phase is at maximum or minimum. The corresponding maximum phase gradients are 7.1, 5.3, and 9.8 mrad/nm. A threeN

406

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

-10 -

a

--12 v

2

L

-

,

<\

.-.'.&*

:

-14 l n ' Q)

'.

r n '

r Q-16

.

-18 -

-20 0 100 200 300 400 500 distance [nm]

-2 -

6

2

I

t

2a

-4

..-.*..-.

'-

c1

-

-6 -

-8 -10

distance [nm]

FIGURE46. Domain walls in Co/Pd multilayer films: (a) hologram with marked walls and line profile positions; (b) profiles of broader wall A; (c) profiles of wall B.

dimensional plot of the phase in the space near the edge is shown in Fig. 45e. Investigations of the magnetic microstructure in regions far away from a hole are carried out in the differential mode of STEM holography (Fig. 46). In this mode the phase represents a direct measure of the magnetic field in the specimen and displays a constant phase value in regions of constant magnetization. This is advantageous for the investigation of domain wall profiles. A series of line scans (positions 1-3 in Fig. 46a) yields a set of three domain wall profiles for each of the two marked domain walls, A , B. The average domain wall width, as determined from Fig. 46b and c, is 245 nm (wall A ) and 200 nm (wall B). The difference in mean domain wall width is likely related to the presence of partial antiferromagnetic coupling within the superlattice stack. 2. Cobalt/ Copper Multilayers Magnetic coupling between adjacent ferromagnetic layers in a superlattice composed of alternating ferromagnetic and nonmagnetic layers is present

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

407

when giant magnetoresistance is observed. A series of multilayer structures with varying seed layer thickness, number of bilayers, and bilayer geometry have been grown under ultrahigh vacuum conditions. Samples grown on thin amorphous holey carbon films are observed in the Fresnel mode with the beam perpendicular to the layers of the superlattice (not crosssectional view) and holograms of identical regions yield quantitative information about the magnetic microstructure. The variation of the magnitude of magnetization can be used to determine the interlayer coupling assuming in-plane magnetization (Mankos et al., 1994b). For example, the Fresnel image of a Co (6 nm)/[Cu (3 nm) Co (1.5 nm)],/Cu (3 nm) superlattice shows five domains aligned in a flux vortex (Fig. 47a). From the hologram, acquired in the absolute mode (Fig. 47b), the maximum

FIGURE47. Holography of Co/Cu multilayer structures: (a) Fresnel mode image (underfocus); (b) partially unwrapped phase image (absolute mode); (c) proposed magnetic structure in the rnultilayer stack.

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

phase gradients (as determined in each of the five domains) are 41.2, 41.8, 39.8, 36.7, and 40.7 mrad/nm. The average maximum phase gradient in domains 1,2, 3, and 5 is 41.0 k 0.8 mrad, which differs from the predicted bulk value for a cobalt film of 15 nm total thickness (41.4 mrad/nm) by less than 1%.This indicates that the domains penetrate the sandwich and are uniformly (ferromagneticallly) aligned. The phase gradient in domain 4 is 36.7 mrad/nm, which is approximately 90% of the expected uniformly magnetized value (37.3 mrad/nm). A proposed explanation of this magnetization amplitude loss is outlined in Fig. 47c. The magnetization in one of the layers (10% of the active thickness) is rotated by 90" with respect to the magnetization in the other layers. The amplitude must then be calculated as a vector sum; that is, magnetization amplitude equals 0 1) + (0 9) = 0.906 and is therefore approximately 10% lower than the aligned value. If a single layer were antiferromagnetically aligned in the superlattice stack within this domain, the phase gradient would have to be 80% of the maximum value. The existence of 90" coupling between layers has been confirmed by hysteresis loop measurements performed on the same sample (Yang and Scheinfein, 1995). This confirms that we are able to determine the magnetization orientation of domains in a superlattice and thereby are capable of correlating macroscopic giant magnetoresistance measurements with micromagnetic structure.

d

r

C. Fine Magnetic Particles

Fine magnetic particles are of great practical importance in many technology areas, for example, in recording media, ferrofluids, catalysts, and pigments. Ultrafine particles are commonly found in soils, rocks, and living organisms and can be used for medical diagnostics and drug delivery. Small (fine, ultrafine) particles scan be defined as structures with dimensions limited in more than one direction, that is, needles (limited in two dimensions) and genuine small particles (platelets, cubes, etc.). Properties of ultrafine magnetic particles differ considerably from those of the corresponding bulk materials (Dormann and Fiorani, 1992). Fine magnetic particles are difficult to fabricate in a controlled manner and their magnetic microstructure is difficult to investigate. Small particles can be produced by vapor deposition (Fe), chemical reduction (FeB), sputtering, or milling (Hadjipanayis et al., 1992). For example, the coercivity of Fe particles of approximately 20 nm in diameter is approximately two orders of magnitude higher then for bulk Fe and the saturation magnetization varies from 20-90% of the bulk value (Tasaki et al., 1974). Magnetic properties are strongly dependent on the surface layers which constitute a

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

409

relatively large fraction of ultrafine particles. It has been found from Mossbauer studies of small ferrimagnetic iron oxide particles that the atomic magnetic moments are not completely aligned parallel or antiparallel to the external field, which was explained by a noncollinear spin structure in the 1-nm-thick surface layer (Coey, 1971). Surface properties are sensitive to the environment and a rigorous investigation requires the application of ultrahigh vacuum techniques and in-situ growth and characterization. Electron holography, carried out in a STEM, provides quantitative information about the magnetization in the specimen at nanometer resolution and therefore represents a valuable tool for the determination of the magnetic microstructure in small particles. Microstructure calculations of the three-dimensional magnetic fields and phase differences must be carried out for comparison with experimental phase images. The intuitive and straightforward interpretation of phase images can be done only for in-plane distribution of the magnetization and flux density. 1. Phase Image Simulations

In what follows, examples of phase images corresponding to simple threedimensional distributions, a magnetic (fictitious) point charge and point dipole, are calculated first analytically and later using numerical calculations. The mean inner potential contribution is neglected. When calculating the electron-holographic phase shifts, we have to keep in mind that the contributions of the electric and magnetic fields are evaluated in a principally different way: the electrostatic fields are evaluated as a line integral, while the magnetic contribution is based on an area integral. Therefore, the same spatial field distribution yields different phase images for an electric and magnetic monopole or dipole. The magnetic contribution to the phase difference depends on the orientation, determined by the separation of the two probes. This is not important in the case of spherically symmetric fields (monopole), but will play a role in lower-symmetry cases (dipole). The phase difference is proportional to the projected component (i.e., integrated along z ) perpendicular to the probe separation. With the probe separation along the x-axis, the phase difference for a magnetic monopole located at the origin is proportional to the projected y-component of the magnetic flux density, that is,

e

x

=--1 n --m

2y ak. x2 +y2

~

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

Before continuing the integration it should be noted that the integrand is the projected component of the magnetic flux density and therefore displays the expected phase image distribution (Fig. 48a) in the differential mode (for a magnetic specimen) and no additional calculations are needed. The second integration step (along x ) then yields the phase difference distribution in the absolute mode (Fig. 48c and el, and

FIGURE48. Phase image simulations for a magnetic monopole: (a) and (b) projected y and x-components of magnetic flux density; (c) and (d) projected y - ( x - ) component integrates along x ( y ) ; (el three-dimensional view of (c).

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

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Due to the symmetry of the monopole, the same phase difference distribution is found when the probes are separated along y (Fig. 48b and d), with the phase discontinuity along the y-direction. Analogously, the phase difference is calculated for a point magnetic dipole, located at the origin and oriented along the positive x-axis (Fig 49). When the beam separation is along the x-axis, the phase difference depends on the y-component of the flux density:

=

e --I fi

4xy

x

2 h(x2+y2) The integrand, the projected y-component of the magnetic flux, is shown in Fig. 50a and coincides with the image observable in the differential mode. The second integration yields e -2y ACp(X7Y) = --m

-n

and the phase image is shown in Fig. 50c. With the probes separated in the y-direction, the phase difference is integrated in the same way and the phase images are shown in Fig. 50b, d, and e. Note the typical bright and dark contrast regions along the sides of the dipole. As it will be shown, this is a characteristic feature of a magnetic dipole. It is important to realize that, due to the three-dimensional character of the fields, we can quantitatively measure only one component, and the relevant line scan, which can be used for quantitative evaluation and comparison with the experimental image, is along the axis which is determined by the probe separation (the evaluation component is perpendicular to this direction).

"t

4

FIGURE49. Geometry of the calculated magnetic dipole fields.

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

FIGURE50. Phase image simulations for a magnetic dipole: (a) and (b) projected y- and x-components of magnetic flux density; (c) and (d) projected y - ( x - ) component integrated along n (y); (e) three-dime.isional view of (d).

For comparison, the phase images of an electrostatic dipole are calculated. Note that, due to the fact that the phase difference is evaluated from a line integral, only the z-integral is needed and no orientation preference (given by the probe separation) exists for electrostatic fields. The absolute phase image of an electrostatic dipole is shown in Fig. 51a. The regions of bright and dark contrast are rotated by 90" when compared to the magnetic dipole. A derivative of the phase difference with respect to x and y yields the projected x- and y-components of the electrostatic field, which are shown in Fig. 51b and c. These images coincide with the

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

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FIGURE51. Phase image simuIations for an electrostatic dipole: (a) absolute phase image; (b) and (c) projected x- and y-components of electric field; (d) three-dimensional view of (a).

differential mode images and are precisely the same as the differential images of a magnetic dipole, as conjectured earlier. For completeness let us consider the case when the magnetic dipole is perpendicular to the xy-plane. When the beam separation is along the x-axis, the phase difference depends on the y-component of the flux density and since the integrand ( B y ) is an odd function of z , the zIntegration must equal zero. Similar considerations apply for the case with

414

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

the probes separated along the y-direction, and no magnetic contrast is observed when the beam is parallel to the dipole axis. In the previous calculations we assumed a point dipole and therefore no information can be extracted about the inner structure of the dipole. For simulations accounting for the inner magnetic structure of the dipole (which differs for electrostatic and magnetic dipoles), numerical calculations must be carried out. An example of such a calculation for a 640 nm X 80 nm X40 nm rectangular particle with magnetization moment of 400 emu/cm3 is shown in Fig. 52. The integrated y- and x-components of

FIGURE 52. Phase image simulations for a physical magnetic dipole: (a) and (b) projected y - and x-components of the magnetic flux density; (c) and (d) projected y - ( x - ) component integrated along x ( y ) ; (e) three-dimensional view of (d).

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

415

FIGURE53. Phase image simulations near one end of a physical magnetic dipole: (a) unwrapped phase image; (b) contour image; (c) wrapped phase image; (d) threedimensional view of phase image.

the flux density, coinciding with the differential mode image, are shown in Fig. 52a and b. The corresponding phase images obtained by integrating By along x and B, along y (Fig. 52c and d) display the characteristic areas of bright and dark contrast outside the particle, the large gradient of the phase inside the particle, and the flux leak from the particle’s ends. A detailed simulation near one end of a dipole and its possible imaging representations are shown in Fig. 53. 2. Chromium Dioxide (CrO, ) Particles

-

CrO, particles are elongated “stretch shaped” particles ( 50-100 nm wide and 500-800 nm long) with a magnetic moment of 93 emu/g ( - 400 emu/cm2> and a coercive field of 405 Oe. Holograms of single particles acquired in the absolute mode confirm the prediction that the particles are typically uniformly magnetized. CrO, particles are flat needles of nearly constant cross section; therefore, we neglect the phase changes due to thickness variations. A typical example of a CrO, particle (640 nm long and 80 nm wide) with magnetic dipole structure is shown in Fig. 54a (see

416

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

f

5:

n

-30 1 0

'

'

'

'

100

,

,

,

,

200

,

,

' . . .. ' , 300 400

,

distance [nm]

,

,

, J

500

distance [nm]

FIGURE 54. Quantitative measurements of a CrO, particle: (a) wrapped absolute phase image, where the arrow marks the measured component of B; (b) line scan across particle in direction perpendicular to the arrow in (a); (c) line scan from calculated dipole (Fig. 52d).

simulations in Figs. 52 and 53). Typical bright and dark contrast observed outside the particle is in good qualitative agreement with the calculated phase image (Fig. 54d). Absolute phase shifts caused by the magnetic dipole can be determined. The experimental line scan across the particle (Fig. 54b) shows that the phase gradient inside the particle is approximately 30 mrad/nm, which compares favorably with the calculated image (28.1 mrad/nm, Fig. 54c). It must be noted here that the phase gradient for an in-plane magnetization distribution (no boundaries) of the same thickness corresponds to 30.5 mrad/nm; hence the effects of the demagne-

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

417

tization field are apparent. An accurate value of the thickness is required and is obtained by imaging the particle in a bright field using a tilt stage. The volume of the observed particle is approximately 2 X pm3, which corresponds to a magnetic moment of approximately lo-', emu. This means that we are capable of safely detecting magnetic moments of approximately four orders in magnitude smaller than those detectable with SQUID magnetometry. Under (near) ideal imaging conditions we can detect phase gradients of several milliradians per nanometer, corresponding to a minimum particle volume of approximately 1 nm x 1 nm x 5 nm. This corresponds to a magnetic moment of approximately 10-'6-10-'7 emu, our sensitivity limit for magnetization measurements. A cluster of strongly interacting CrO, particles is shown in Fig. 55. The magnetizations of the two CrO, particles near the edge of the supporting carbon film (Fig. 55a-c-phase and contour images) are aligned parallel,

FIGURE55. A cluster of CrO, particles with strong magnetic flux leak (a) wrapped absolute phase image; (b) unwrapped absolute phase image, where the arrow marks the measured component of B; (c) contour image of phase, where one contour corresponds to approximately ~ / l l (d) ; STEM bright field image of the cluster.

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

FIGURE56. Tip of CrO, particle with strong magnetic flux leak (a) unwrapped absolute phase image, where the arrow marks the measured component of B; (b) contour image of phase, where one contour corresponds to approximately ~ / l l (c) ; three-dimensional plot of a part of (a), compare to Fig. 53.

which manifests itself as a strong leakage field. A high-resolution investigation of the tip of the leaking particle (Fig. 56) shows striking resemblance to the simulated images of a dipole end in Fig. 53. By comparison, the two CrO, particles in Fig. 57a and b (phase and contour image) are aligned antiparallel; therefore, the magnetic flux is closed and no flux leak is observed.

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

419

6.0

-E

4.0

0

2.0

I

f

0.0

S

CL

-2.0 -4.0 ,

-6.0

0

50

100

,

,,

150 200 distance [nm]

, , ,

,

,

250

c.1 300

FIGURE57. Two CrO, particles with magnetizations aligned antiparallel: (a) unwrapped absolute phase image, where the thin arrow marks the measured component of B (b) contour image of phase, where one contour corresponds to approximately ~ / l l (c) ; three line scans of phase in direction perpendicular to the (thin) arrow in (a) at positions marked by lines.

Two CrO, particles, lying on top of each other at (nearly) right angles and forming a quadrupole-like distribution, are shown in Fig. 58. The magnetic flux leaks from the particle ends are connected to other nearby particles. A line scan in the direction of the probe separation yields a gradient of 51.0 mrad/nm in particle I (with magnetization nearly along the sensitive direction) and 4.2 mrad/nm in particle I1 (magnetization nearly perpendicular). The ratio of the phase gradients in particles I and I1

420

M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

a,

c m Q

1,El, ---.particle I ---particle Ii

-5.0

-6.0

0

50

,, ,

,

,

,

,

,

dP'L , ,

i

-

,,, ,, ,,

100 150 200 250 300

,;;,:E

350 400

distance [nm]

FIGURE 58. TWOCrO, particles forming a magnetic quadrupole field: (a) wrapped absolute phase image, where the arrow marks the measured component of B (b) contour image of phase, where one contour corresponds to approximately r / l l ; (c) line scans of phase in direction perpendicular to the arrow marked in (a).

is approximately 12.1. This agrees favorably with the ratios of the projected measured components, which are proportional to the cosines of the angle that the magnetization makes with the measured directions. The angles are 3.5" for particle I and 84" for particle 11, yielding a ratio of approximately 9.5. The difference may be given by the nonequal volumes of the particles. An example of a CrO, particle with internal domain structure is given in Fig. 59. At one end the particle magnetization is split into two domains separated by a 180" domain wall, demonstrated by the linearly changing phase with approximately equal phase gradients ( 30 mrad/nm) of oppo-

-

421

ELECTRON HOLOGRAPHY AND LORENTZ MICROSCOPY

0

20

40

60

80

100

distance [nml

20

40

60

80

distance [nm]

FIGURE59. CrO, particle with internal magnetic domain structure; (a) unwrapped absolute phase image; (b) image of the logarithm of the amplitude, proportional to thickness; (c) image of phase divided by the logarithm of the amplitude; (d) line scans from (a), (b), and (c) along the line marked in (a); (e) three-dimensional view of (a); (f) DPC image; (8) bright field image; (h) line scan of DPC image at the same position as marked in (a).

site sign in the phase line scan (Fig. 59d). A line scan taken at the same position in the image of the natural logarithm of the amplitude (proportional to the projected thickness) is nearly constant. The phase gradient is still present in the thickness-corrected phase image (phase image divided by image of the logarithm of the amplitude), therefore ruling out a

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M. MANKOS, M. R. SCHEINFEIN AND J. M. COWLEY

thickness-dependent effect. The maximum phase gradient corresponds to a thickness of approximately 40 nm. The domain wall is visible in the DPC image as well (Fig. 59f). A line scan of the difference signal from the DPC image reveals the two domains as regions of dark and bright contrast, corresponding to nearly uniform magnetization in opposite directions. The domain wall width equals approximately 10 nm in this particular case.

V. CONCLUSIONS We have developed a new method for the absolute measurement of magnetization in thin magnetic films using far-out-of-focus, off-axis STEM holography. A geometrical- and wave-optics description of the new holography modes and the conventional Fresnel and differential phase contrast (DPC) modes of Lorentz microscopy was given and the obtained contrast was compared. The Fresnel mode of Lorentz microscopy is often used for quick determination of the position of in-plane magnetization changes (domain wall, ripple, etc.) in magnetic specimens. It does not require any special detector or image processing and can be observed in a TEM as well. Due to its parallel acquisition scheme (whole image acquired in one frame), it is well suited for observation of domain wall dynamics on a time scale determined only by the frame acquisition speed. It is rather difficult to extract any quantitative information from Fresnel images and other techniques must be employed. The DPC mode is best used for the determination of magnetization orientation in domains. In spite of the requirement of special detectors and image processing, the DPC mode has become a useful tool for obtaining quantitative information in the form of vector maps of magnetization and magnetic domain wall profiles. The ability to simultaneously observe magnetic and microscopic structure in the DPC mode allows fcjr the correlation of micromagnetic features with the underlying structure, geometry, and composition. A large error margin applies to the accuracy of quantitative information derived from the angular deflection, since the finely focused beam requires a strong lateral demagnification of the source and therefore a large angular magnification, which results in poorer resolution in the detection of beam shifts. The holography modes permit quantitative measurement of magnetic induction and magnetization, the determination of equimagnetization lines in domains, and straightforward determination of domain wall and flux vortex profiles. The absolute mode of STEM holography displays a linear change in phase difference for regions with constant magnetization and

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423

the slope determines the absolute value and direction of magnetization. The differential mode of STEM holography displays a constant value of phase difference for regions with constant magnetization, which simplifies the identification of magnetic structures in the specimen. In addition, no edge or hole is necessary since neither wave packet need pass through a vacuum. Both modes of STEM holography accompanied by the conventional techniques have been applied to the characterization of thin magnetic films and magnetic multilayer structures. The magnetic microstructure was investigated quuntitutwely in thin films of different materials and thicknesses using the inherent high spatial resolution of the electronholographic techniques. In multilayer structures we were able to determine the orientation of domains in a superlattice stack. Magnetic fine particles have been characterized experimentally and compared with numerical simulations. Interactions of magnetic particles and their internal micromagnetic structure have been observed at high spatial resolution. Particles with magnetic moments of approximately lo-'' emu can be measured, which means that we are capable of detecting magnetic moments of approximately four orders in magnitude smaller than those detectable with SQUID magnetometry. Under (near) ideal imaging conditions we can detect phase gradients of several milliradians per nanometer, corresponding to a minimum particle volume of approximately 1 nm x 1 nm X 5 nm with a magnetic moment of approximately 10-'6-10-'7 emu. The holographic technique allows the extraction of quantitative information about the magnetic structure at high spatial resolution and therefore significantly expands the abilities of STEM as a tool for investigating different types of novel magnetic materials. Combining a variety of micromagnetic analysis techniques in one instrument provides a valuable tool for quantitative investigations of magnetic structures at the nanometer level. The work presented here opens a wide range of possibilities for future research. For example, reflection electron holography should become a sensitive technique for surface magnetism investigations. The implementation of a specimen stage with temperature control would permit investigations of magnetic phase transition phenomena near the Curie temperature and a variable external magnetic field would allow for the investigations of domain switching, hysteresis, and saturation phenomena. ACKNOWLEDGMENTS

The work that is part of Marian Mankos's Ph.D. dissertation was conducted over a four-year period at Arizona State University and would not

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have been possible without the help of numerous people concentrated in the Physics Department and at the Center for High Resolution Electron Microscopy. We would like to thank A. A. Higgs, C. Weiss, and J. Wheatley for keeping the HB-5 electron microscope in a flawless state and for helping with specimen preparation; R. Thornton for work on the HB-5 detector system; M. R. McCartney and J. K. Weiss for electron holographic image processing software; and P. Perkes for help with image processing. We would also like to thank Dr. G. Matteucci for providing samples of small magnetic particles; Dr. J. Simsova, V. Kambersky, and P. deHaan for the Co/Pd multilayer samples; and Z.-J. Yang for the Co/Cu multilayers, thin film samples, and hysteresis loop measurements.

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Index

A

Anisotropy, crystalline magnetic materials, 325-326 Argon, melting point, 9, 10 B

Basin stabilization, quantitative particle motion, 50-51 Bitter pattern technique, magnetic microstructure, 328-329 Bloch domain wall, 327, 388 Boron, diatomic molecular bond, 72-74 Boundary conditions three-dimensional circulator model, 198-212 impedance boundary condition, 288-301 two-dimensional circulator model, 90-98 Bubbles, quantitative particle motion, 44-61 C

Carbon, diatomic molecular bond, 72-74 Carbon dioxide, fluid bubbles in water, 44-61 Chromium dioxide, fine magnetic particles, 415-422 Circuit parameters, two-dimensional microstrip circulator model, 108-116 Circulators three-dimensional theory, 127-129, 301-303,316-317 427

boundary conditions, 198-212, 288-301 characteristic equation through rectangular coordinate formulation, 151-170 diagonalization of governing equation, 139-151 doubly ordered cavity, 283-287 dyadic recursive Green’s function, 128, 219-238,316 field equations, 129-139 impedance boundary condition, 288-301 limiting aspects, 246-260 metallic losses, 195-198 nonexistence of TE, TM,and TEM modes, 128, 174-176 Nth annulus-outer region interface, 212-218,234-238 radially ordered circulator, 260-283 scattering parameters, 238-245, 302 three-dimensional fields, 176-187 three-port circulator, 238-245 transverse fields, 170-174 z-field dependence, 188-195 z-ordered layers, 260-283 two-dimensional model, 80-81, 127, 316 boundary conditions, 90-98 circuit parameters, 108-116 cylindrical coordinates, 83-86 dyadic recursive Green’s functions, 79-81, 81-82, 98-108, 121-127, 316 governing Helmholtz wave equation, 86-87

INDEX Circulators (Continued) two-dimensional model (Continued) limiting aspects, 121-127 numerical results, 303-315 scattering parameters, 117-120 three-port circulator, 117-120 two-dimensional fields, 87-90 Cobalt, magnetic thin films, 389-397 Cobalt/copper, magnetic multilayers, 406-408 Cobalt/palladium, magnetic multilayers, 402-406 Collisions, microdrops of water, 13-21, 22 Copper crack development in stressed copper plate, 21, 22-30, 31-33 melting point, 6-9 Copper/cobalt, magnetic multilayers, 406-408 Cracks, quantitative particle modeling, 21, 22-30,31-33

D Diatomic molecules melting points, 13, 14 molecular bonds, 67-74 Differential phase contrast mode geometrical optics, 351-357 wave optics, 357-358 Domain structure, magnetic materials, 324, 325-327,388-389 Domain walls, 327-328 Double-ordered cavity, three-dimensional microstrip circulators, 283-287 DPC mode, see Differential phase contrast mode Drops collisions of microdrops, 13-21, 22 formation on solid surfaces, 30, 32-44 motion within fluids, 44-61 Dyadic recursive Green’s function, microstrip circulators three-dimensional model, 219-238 two-dimensional model, 79-81, 98-108, 121-127, 316

E Electromagnetic circulators, see Circulators

Electron holography, 331,362-363,422-423 off-axis electron holography, 323-324 principles, 363-373 STEM holography, 373-387,422-423 Electron microscopy, magnetic microstructure, 330-333 Electron-optical techniques, magnetic microstructure, 331-332 Electrons, particle-wave duality, 333-335

F Far-out-of-focus holography, 373-387, 422-423 Ferromagnets, strong exchange interactions, 325 Fine magnetic particles, 408-409 chromium dioxide, 415-422 phase image simulations, 409-415 Fluid bubbles, carbon dioxide in water, 44-61 Foucault mode, Lorentz microscopy, 360-362 Four-body problem, melting points, 4-13 Fractures, quantitative particle modeling, 21, 22-30,31-33 Fresnel mode, 335-337 geometric optics, 337-341 wave optics, 341-350 G

Geometrical optics differential phase contrast mode, 351-357 Fresnel mode, 337-341 Giant magnetoresistance, magnetic multilayers, 400, 402 Graphite, liquid drop formation on, 30,32-44 Green’s function, 81-82 microstrip circulators, 128 three-dimensional model, 219-238, 316 two-dimensional model, 79-81, 98-103, 121-127.316

H Helium, melting point, 11-13 Helmholtz wave equation three-dimensional circulator model, 137-138 two-dimensional circulator model, 86-87

INDEX

Holography, 362 electron holography, 331,362-363, 422-423 far-out-of-focusholography, 373-387 off-axis electron holography, 323-324 principles, 363-373 STEM holography, 373-387,422-423 Hydrogen molecule, simulations, 68-72 I

Imaging, magnetic microstructure, 328-333 Impedance boundary condition, three-dimensional microstrip circulators, 288-301 Information storage, magnetic materials, 324 Iron, fine magnetic particles, 408-409

K Kinetics, rapid kinetics, 61-67 Krypton, melting point, 10

L Landau-Lifshitz principle, 325 Leap-frog method, particle modeling, 4-6, 40 Lennard-Jones potential, 9-10, 11 Lithium, diatomic molecular bond, model, 72-74 Lorentz microscopy, 422-423 differential phase contrast mode, 350-351 geometrical optics, 351-357 wave optics, 357-358 Foucault mode, 360-362 Fresnel mode, 335-337 geometrical optics, 337-341 wave optics, 341-350 particle-wave duality of electrons, 333-335 small-angle electron diffraction, 358-360

M Magnetic electron microscopy, magnetic microstructure, 331-332 Magnetic force microscopy, magnetic microstructure, 331 Magnetic materials domain structure, 324, 325-327, 388-389 electron holography, 331,362-363, 422-423

429

principles, 363-373 STEM holography, 373-387,422-423 fine magnetic particles, 408-409 chromium dioxide particles, 415-422 phase image simulations, 409-415 history, 324 Lorentz microscopy, 422-423 differential phase contrast mode, 350-358,422 Foucault mode, 360-362 Fresnel mode, 335-350, 422 particle-wave duality, 333-335 small-angle electron diffraction, 358-360 microstructure imaging techniques, 328-333 multilayer structures, 400, 402 cobalt/copper, 406-408 cobalt/palladium, 402-406 production, 324-325 thin films, 387-389 cobalt, 389-397 nickel, 397-399,400-401 Magnetic thin films, 387-389 cobalt, 389-397 nickel, 397-399,400-401 Magneto-optical techniques, magnetic microstructure, 329 Magnetoresistance, magnetic multilayers, 400,402 Magnetostriction, 326 Melting points diatomic molecules, 13, 14 four-body problem, 4-13 Microdrops, collisions, 13-21, 22 Micromagnetic theory, 324,325, 327-328 Microscopy magnetic microstructure, 330-333 Lorentz microscopy, 333-362, 422-423 magnetic force microscopy, 331 mirror electron microscopy, 332 photoelectron microscopy, 330 scanning electron microscopy, 33 1-332 scanning electron microscopy with polarization analysis, 333 scanning transmission electron microscopy, 335-337,342,353,357,358, 360 spin-polarized low-energy electron microscopy, 333

430

INDEX

Molecular bonds, diatomic molecules, 67-74 Multilayer structures magnetic materials, 400, 402 cobalt/copper, 406-408 cobalt/palladium, 402-406

N NCel domain wall, 327, 388 Neon, melting point, 10 Neutron topography,magnetic microstructure, 330-331 Nickel, magnetic thin films, 397-399, 400-401 Nitrogen, diatomic molecular bond, 72-74 Noble gases, melting point, 9-10 0

Off-axis electron holography, 323-324 Oxygen, diatomic molecular bond, 72

P Palladium/copper, magnetic multilayers, 402-406 Particle modeling, 2 classical molecular forces, 3-4, 67-71 numerical methodology, 4-6, 63-64 quantitative, see Quantitative particle modeling Particle-wave duality, electrons, 333-335 Permanent magnets, 324 Photoelectron microscopy, magnetic microstructure, 330

Q Quantitative particle modeling, 2 classical molecular forces, 3-4 crack development, in stressed copper plate, 21, 23-30, 31-33 diatomic molecules melting points, 13, 14 molecular bonds, 67-74 drops colliding microdrops of water, 13-21,22 liquid drop formation on solid surfaces, 30,3244 fluid bubbles, 44-61

melting points, 6-13 numerical methodology, 4-6 rapid kinetics, 61-67

R Radially ordered circulator, z-ordered layers, 260-283 Rapid kinetics, quantitative particle motion, 61-67 Recursive dyadic Green’s function, microstrip circulators three-dimensional model, 219-238 two-dimensional model, 79-81, 98-108, 121-127,316 Rosen-Margenau-Page potential, 11 Rowlinson potential, 45 S

Scanning electron microscopy, magnetic microstructure, 331-332 Scanning electron microscopy with polarization analysis, magnetic microstructure, 333 Scanning transmission electron microscopy, magnetic microstructure, 335-337, 342,353, 357,358,360 Scattering parameters, three-port circulator, 117-120,238-245,302 SEM, see Scanning electron microscopy SEMPA, see Scanning electron microscopy with polarization analysis Sessile drop, liquid drop formation on a solid surface, 42-44 Slab stabilization, drop formation, 40-42 Slater-Kirkwood potential, 12 Small-angle electron diffraction, h r e n t z microscopy, 358-360 Spin-polarized low-energy electron microscopy, magnetic microstructure, 333 SPLEEM, see Spin-polarized low-energy electron microscopy STEM, see Scanning transmission electron microscopy STEM holography, 373-387,422-423 Strong exchange interactions, ferromagnets, 325

43 1

INDEX

T

W

Thin films, magnetic, 387-399,400-401, 402-408 Three-dimensional circulators, see Circulators Three-port circulator, scattering parameters, 117-120,238-245,302 Transverse field formulas, three-dimensional microstrip circulators, 133-134, 170-174 Two-dimensional circulators, see Circulators

Water carbon dioxide bubbles in, 44-61 collision of microdrops, 13-21, 22 fluid models, 45-52 liquid drop formation on a solid surface, 30, 32-44 Wave optics differential phase contrast mode, 357-358 Fresnel mode, 341-350

U Ultrafine magnetic particles, see Fine magnetic particles

X Xenon, melting point, 10 X-ray dichroism, magnetic microstructure, 330 X-ray topography, magnetic microstructure, 329

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