P1: OTA/XYZ P2: ABC fm JWBS036-Numai August 11, 2010
9:45
LASER DIODES AND THEIR APPLICATIONS TO COMMUNICATIONS AND IN...
34 downloads
985 Views
17MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
P1: OTA/XYZ P2: ABC fm JWBS036-Numai August 11, 2010
9:45
LASER DIODES AND THEIR APPLICATIONS TO COMMUNICATIONS AND INFORMATION PROCESSING
Printer Name: Yet to Come
P1: OTA/XYZ P2: ABC fm JWBS036-Numai August 11, 2010
9:45
Printer Name: Yet to Come
WILEY SERIES IN MICROWAVE AND OPTICAL ENGINEERING KAI CHANG, Editor Texas A&M University
A complete list of the titles in this series appears at the end of this volume.
P1: OTA/XYZ P2: ABC fm JWBS036-Numai August 11, 2010
9:45
Printer Name: Yet to Come
LASER DIODES AND THEIR APPLICATIONS TO COMMUNICATIONS AND INFORMATION PROCESSING
TAKAHIRO NUMAI Ritsumeikan University
A JOHN WILEY & SONS, INC., PUBLICATION
P1: OTA/XYZ P2: ABC fm JWBS036-Numai August 11, 2010
9:45
Printer Name: Yet to Come
C 2010 by John Wiley & Sons, Inc. All rights reserved. Copyright
Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Numai, Takahiro Laser diodes and their applications to communications and information processing / Takahiro Numai. ISBN 978-0-470-53668-1 (cloth) Printed in the Singapore 10 9 8 7 6 5 4 3 2 1
P1: OTA/XYZ P2: ABC fm JWBS036-Numai August 11, 2010
9:45
Printer Name: Yet to Come
To my grandparents in the United States Kenichiro and Asano Kanzaki
P1: OTA/XYZ P2: ABC fm JWBS036-Numai August 11, 2010
9:45
Printer Name: Yet to Come
CONTENTS
PREFACE
PART I
ix
PHYSICS REQUIRED TO DESIGN LASER DIODES
1 Energy Bands in Bulk and Quantum Structures 1.1 1.2 1.3 1.4
3
Introduction, 3 Bulk Structure, 4 Quantum Structures, 14 Superlattices, 19 References, 21
2 Optical Transitions 2.1 2.2 2.3 2.4 2.5
1
23
Introduction, 23 Direct and Indirect Transitions, 24 Light-Emitting Processes, 25 Spontaneous Emission, Stimulated Emission, and Absorption, 26 Optical Gains, 27 References, 37
3 Optical Waveguides
39
3.1 Introduction, 39 3.2 Two-Dimensional Optical Waveguides, 41 3.3 Three-Dimensional Optical Waveguides, 52 References, 54 vii
P1: OTA/XYZ P2: ABC fm JWBS036-Numai August 11, 2010
viii
9:45
Printer Name: Yet to Come
CONTENTS
4 Optical Resonators 4.1 4.2 4.3 4.4
Introduction, 57 Fabry–Perot Cavity, 59 Waveguide Grating, 65 Vertical Cavity, 82 References, 91
5 pn- and pnpn-Junctions 5.1 5.2 5.3 5.4
57
93
Intrinsic Semiconductor, 93 Extrinsic Semiconductor, 97 pn-Junction, 103 pnpn-Junction, 117 References, 121
PART II CONVENTIONAL LASER DIODES
123
6 Fabry–Perot Laser Diodes
125
6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9
Introduction, 125 Rate Equations, 128 Current versus Voltage Characteristics, 130 Current versus Light Output Characteristics, 131 Polarization of Light, 146 Transverse Modes, 148 Longitudinal Modes, 157 Modulation Characteristics, 160 Noises, 172 References, 189
7 Quantum Well Laser Diodes
191
7.1 Introduction, 191 7.2 Features of Quantum Well LDs, 191 7.3 Strained Quantum Well LDs, 201 References, 211 8 Single-Mode Laser Diodes 8.1 8.2 8.3 8.4
Introduction, 213 DFB LDs, 213 DBR LDs, 222 Vertical Cavity Surface-Emitting LDs, 224 References, 229
213
P1: OTA/XYZ P2: ABC fm JWBS036-Numai August 11, 2010
9:45
Printer Name: Yet to Come
CONTENTS
9 Semiconductor Optical Amplifiers 9.1 9.2 9.3 9.4
ix
233
Introduction, 233 Signal Gain, 233 Polarization, 238 Noises, 239 References, 241
PART III ADVANCED LASER DIODES AND RELATED DEVICES
247
10 Phase-Controlled DFB Laser Diodes
249
10.1 10.2 10.3 10.4 10.5
Introduction, 249 Theoretical Analysis, 249 Device Structure, 251 Device Characteristics and Discussion, 252 Summary, 254 References, 255
11 Phase-Shift-Controlled DFB Laser Diodes 11.1 11.2 11.3 11.4 11.5
Introduction, 257 Theoretical Analysis, 258 Device Structure, 262 Device Characteristics and Discussion, 264 Summary, 269 References, 269
12 Phase-Controlled DFB Laser Filter 12.1 12.2 12.3 12.4
271
Introduction, 271 Device Structure, 272 Device Characteristics and Discussion, 272 Summary, 276 References, 277
13 Phase-Shift-Controlled DFB Filter 13.1 13.2 13.3 13.4 13.5
257
Introduction, 279 Theoretical Analysis, 280 Device Structure, 282 Device Characteristics and Discussion, 283 Summary, 287 References, 287
279
P1: OTA/XYZ P2: ABC fm JWBS036-Numai August 11, 2010
x
9:45
Printer Name: Yet to Come
CONTENTS
14 Passive Phase-Shifted DFB Filter 14.1 14.2 14.3 14.4 14.5
Introduction, 289 Theoretical Analysis, 290 Device Structure, 294 Device Characteristics and Discussion, 298 Summary, 301 References, 301
15 Two-Section Fabry–Perot Filter 15.1 15.2 15.3 15.4 15.5
289
305
Introduction, 305 Theoretical Analysis, 306 Device Structure, 309 Device Characteristics and Discussion, 311 Summary, 314 References, 315
16 Optical Functional Devices with pnpn-Junctions
317
16.1 Introduction, 317 16.2 Edge-Emitting Optical Functional Device, 318 16.3 Surface-Emitting Optical Functional Device, 321 References, 336
PART IV SYSTEM DEMONSTRATIONS USING ADVANCED LASER DIODES AND RELATED DEVICES
339
17 Photonic Switching Systems
341
17.1 17.2 17.3 17.4
Introduction, 341 Wavelength Division Switching, 344 Wavelength- and Time-Division Hybrid Switching, 345 Summary, 350 References, 350
18 Optical Information Processing 18.1 18.2 18.3 18.4 18.5 18.6 18.7
Introduction, 353 Serial-to-Parallel Data Conversion, 354 Optical Self-Routing Switch, 355 Optical ATM Switch, 356 Optical Interconnection, 359 Optical Memory, 362 Optical Bus, 365 References, 366
353
P1: OTA/XYZ P2: ABC fm JWBS036-Numai August 11, 2010
9:45
Printer Name: Yet to Come
CONTENTS
xi
Appendix A: Density of States
367
Appendix B: Density of States Effective Mass
381
Appendix C: Conductivity Effective Mass
383
INDEX
385
P1: OTA/XYZ P2: ABC fm JWBS036-Numai August 11, 2010
9:45
Printer Name: Yet to Come
PREFACE
With the rapid increase in Internet users, photonics will be more important in the future, because photonics contributes to building an infrastructure for the Internet. Laser diodes, which are used commercially as light sources for optical fiber communications and the read/write ability of compact disks (CDs) and digital video disks (DVDs), have the potential to expand photonics technology if their operating principles are applied to photonic devices such as optical filters and optical functional devices. In this book, the potentials of laser diodes and their applications to optical functional devices and photonic systems are explained. To develop excellent photonic devices we have to fully understand the physics behind the operation of photonic devices. Therefore, the physics behind energy bands of semiconductors, optical transitions, optical waveguides, and semiconductor junctions is explained in detail. In addition, the physical characteristics of laser diodes, optical filters, and optical functional devices are reviewed. Using these photonic devices, photonic systems are demonstrated and some experimental results are described. The book consists of four parts: Part I, Physics Required to Design Laser Diodes; Part II, Conventional Laser Diodes; Part III, Advanced Laser Diodes and Related Devices; and Part IV, System Demonstrations Using Advanced Laser Diodes and Related Devices. First, the physics behind the operating principles of laser diodes is explained in detail. Second, concepts in the design of laser diodes, optical filters, and optical photonic devices are presented, and their characteristics and experimental results in system applications are reviewed.
xiii
P1: OTA/XYZ P2: ABC fm JWBS036-Numai August 11, 2010
xiv
9:45
Printer Name: Yet to Come
PREFACE
Features highlighted in the book are as follows: 1. We remove a large gap between journal articles and textbooks for universities and graduate schools; the reader will find that journal articles are readable after he or she has finished the book. However, only knowledge of electromagnetism and quantum mechanics obtained in an undergraduate course is required to read the book. 2. We give the reader analytical tools for Fabry–Perot laser diodes (LDs), distributed feedback (DFB) LDs, and vertical cavity surface-emitting LDs (VCSELs). If the reader follows the equations in this book, he or she will be enabled to analyze the characteristics of Fabry–Perot LDs, DFB LDs, and VCSELs. 3. We describe the types of potentials in laser diodes used for photonic devices, such as optical filters and optical functional devices. In addition, differences in the specifications required for light sources and those required for optical filters and optical functional devices are discussed. 4. We describe experimental results for system applications of laser diodes, optical filters, and optical functional devices. The reader is introduced to subjects that must be dealt with in the future, with the goal of stimulating research and development in photonics technology. Finally, I would like to thank Professor Emeritus of the University of Tokyo, Koichi Shimoda (former professor of Keio University); Professor Emeritus of Keio University, Kiyoji Uehara; Professor Tomoo Fujioka of Tokai University (former professor of Keio University); and Professor Minoru Obara of Keio University for their warm encouragement and helpful advice when I was a student. I am also indebted to NEC Corporation, where I began research on laser diodes after graduation from the Graduate School of Keio University. Thanks are extended to Mr. George Telecki of Wiley for his kind help and to Professor Kai Chang of Texas A&M University, Editor of Wiley’s Microwave and Optical Engineering Series, for giving me the opportunity to write the book. I am especially grateful to Springer Science+Business Media for allowing me to adapt material used in Chapters 1 and 8, published originally in my book Fundamentals of Semiconductor Lasers (Springer-Verlag Series in Optical Sciences, Vol. 93, 2004). Takahiro Numai
P1: OSO c01 JWBS036-Numai
August 9, 2010
10:8
Printer Name: Yet to Come
PART I PHYSICS REQUIRED TO DESIGN LASER DIODES
1
P1: OSO c01 JWBS036-Numai
August 9, 2010
10:8
Printer Name: Yet to Come
1 ENERGY BANDS IN BULK AND QUANTUM STRUCTURES
1.1
INTRODUCTION
Transitions of Electrons The emission and absorption of light are generated by the transitions of electrons. Light is emitted because electrons transit from high-energy states to lower-energy states, and light is absorbed in the reverse process. When electrons transit from highenergy states to lower-energy states, nonradiative transitions, which do not emit light, may exist as well as radiative transitions, which accompany light emissions.
Energy Bands When the atomic spacing is so large that mutual interactions of atoms may be neglected, the electron energies are discrete and energy levels are formed. With a decrease in the atomic spacing, the positions of the electrons of neighboring atoms start to overlap. Therefore, the energy levels begin to split to satisfy the Pauli exclusion principle. With a further decrease in atomic spacing, the number of electrons whose positions overlap with each other increases. As a result, the number of split energy levels goes up, and the energy differences in the adjacent energy levels are reduced. In semiconductor crystals, the number of atoms per cubic centimeter is on the order of 1022 , where the atomic spacing is about 0.2 nm. As a result, the spacing of energy levels is much narrower than the bandgap energy, on the order of Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright
3
P1: OSO c01 JWBS036-Numai
4
August 9, 2010
10:8
Printer Name: Yet to Come
ENERGY BANDS IN BULK AND QUANTUM STRUCTURES
electron volts. Therefore, the constituent energy levels are considered to be almost continuous, and energy bands are formed.
1.2 BULK STRUCTURE Bulk Semiconductors in which constituent atoms are placed periodically at a sufficiently long range compared with lattice spacing are called bulk semiconductors. In this section, the energy bands in bulk semiconductors are calculated.
k· p Perturbation Semiconductors have free electrons and holes only in the vicinity of band edges. As a result, the band shapes and effective masses of carriers near band edges often give us sufficient information about optical transitions. To analyze the energy bands in the neighbor of band edges, k · p perturbation theory [1–4] is often employed. The wave functions and energies of the bands are calculated with k = k − k0 as a perturbation parameter, where k is a wave vector near a band edge and k0 is a wave vector at a band edge. For simplicity, k0 = 0 is selected in the following.
Schr¨odinger Equation The Schr¨odinger equation in the steady state is given by [5, 6] 2 2 − ∇ + V (r) ψn (k, r) = E n (k)ψn (k, r), 2m
(1.1)
where = h/2π = 1.0546 × 10−34 J · s is Dirac’s constant, h = 6.6261 × 10−34 J · s is Planck’s constant, m = 9.1094 × 10−31 kg is the electron mass in vacuum, V (r) is a potential, ψn (k, r) is a wave function, E n (k) is an energy eigenvalue, n is a quantum number, and k is a wave vector. In single crystals where the atoms are placed periodically, the potential V (r) is also spatially periodic. Therefore, as a solution of (1.1), we can consider a Bloch function, such as ψn (k, r) = exp(i k · r)u n (k, r), u n (k, r) = u n (k, r + R),
(1.2) (1.3)
where R is a translational vector which represents the periodicity of the crystal. Equations (1.2) and (1.3) constitute the Bloch theorem. Substituting (1.2) into (1.1)
P1: OSO c01 JWBS036-Numai
August 9, 2010
10:8
Printer Name: Yet to Come
BULK STRUCTURE
5
leads to 2 2 ∇ + V (r) + H u n (k, r) = E n (k)u n (k, r), − 2m
(1.4)
where 2 k 2 + k· p, 2m m p = − i ∇.
H =
(1.5) (1.6)
Note that the k· p perturbation theory, whose name is derived from the second term on the right-hand side of (1.5), is valid only for small k, and we solve (1.4) by regarding (1.5) as the perturbation. First-Order Perturbation Theory For an energy band with n = 0, the wave equation for an unperturbed state with k = 0 is expressed as 2 2 − ∇ + V (r) u 0 (0, r) = E 0 (0)u 0 (0, r). 2m
(1.7)
In the following, for simplicity, the energy E 0 (0) is represented as E 0 . In first-order perturbation theory, the wave function u 0 (k, r) for a nondegenerate case is given by u 0 (k, r) = u 0 (0, r) +
− i (2 /m)k · α|∇|0 α=0
α|∇|0 =
E0 − Eα
u α (0, r),
u α ∗ (0, r)∇u 0 (0, r) d3 r.
(1.8) (1.9)
Here u 0 (k, r) and u α (k, r) are assumed to be orthonormal functions and α| and |0 are the bra and ket vectors, respectively, which were introduced by Dirac. Second-Order Perturbation Theory In second-order perturbation theory, an energy eigenvalue is obtained as E(k) = E 0 +
0| pi |αα| p j |0 2 k 2 2 ki k j . + 2 2m m i, j E0 − Eα α=0
(1.10)
P1: OSO c01 JWBS036-Numai
6
August 9, 2010
10:8
Printer Name: Yet to Come
ENERGY BANDS IN BULK AND QUANTUM STRUCTURES
The reciprocal effective mass tensor is defined as
1 m
ij
⎞ 0| pi |αα| p j |0 ⎠ 1 ∂ E 1 2 ≡ 2 = ⎝δi j + . ∂ki ∂k j m m α=0 E0 − Eα ⎛
2
(1.11)
Using (1.11), (1.10) is reduced to E(k) = E 0 +
2 1 ki k j . 2 i, j m i j
(1.12)
Equations (1.11) and (1.12) indicate that the effect of the periodic potential of the crystal is included in the effective mass of the electron, which makes analysis easier. In a cyclotron resonance experiment, the rest mass in vacuum in not measured, but the effective mass is measured. sp3 Hybrid Orbitals Next, we consider the energy bands of semiconductor crystals with zinc blende structures, which are used widely as material for light sources. In zinc blende structures, the atomic bonds are formed via sp3 hybrid orbitals. Therefore, the wave functions for electrons in zinc blende or diamond structures are expressed as superpositions of s- and p-orbital functions. We assume that the bottom of a conduction band and the tops of valence bands are placed at k = 0, as in direct transition semiconductors. When spin-orbit interaction is neglected, the tops of the valence bands are threefold degenerate, corresponding to the three p-orbitals ( px , p y , pz ). Here the s-orbital wave function for the bottom of the conduction band is u s (r), and the p-orbital wave functions for the tops of the valence bands are u x = x f (r), u y = y f (r), and u z = z f (r),where f (r) is a spherical function. Since the energy bands are degenerate, a perturbed wave equation is given by a linear superposition of u s (r) and u j (r) ( j = x, y, z), such as u n (k, r) = Au s (r) + Bu x (r) + Cu y (r) + Du z (r),
(1.13)
where A, B, C, and D are coefficients. To obtain the energy eigenvalues, (1.4) is rewritten −
2 k 2 2 2 ∇ + V (r) + Hd u n (k, r) = E n (k) − u n (k, r), 2m 2m i 2 Hd = k· p = − k·∇. m m
(1.14) (1.15)
P1: OSO c01 JWBS036-Numai
August 9, 2010
10:8
Printer Name: Yet to Come
BULK STRUCTURE
7
By setting k = 0 in (1.14), an unperturbed equation is obtained. For a conduction band, we set E n (0) = E c , which is the energy of the bottom of a conduction band, and u 0 (0, r) = u s (r); for valence bands, we set E n (0) = E v , which is the energy of the top of each valence band, and u 0 (0, r) = u j (r) ( j = x, y, z). Substituting (1.13) into (1.14); multiplying u s ∗ (r), u x ∗ (r), u y ∗ (r), and u z ∗ (r) from the left-hand side; and then integrating with respect to a volume over the space leads to + E c − λ)A + Hsx B + Hsy C + Hsz D = 0, (Hss Hxs A + (Hx x + E v − λ)B + Hx y C + Hx z D = 0, Hys A + Hyx B + (Hyy + E v − λ)C + Hyz D = 0, Hzs A + Hzx B + Hzy C + (Hzz + E v − λ)D = 0,
(1.16)
where Hi j = u i |Hd |u j =
u i ∗ (r)Hd u j (r) d3 r λ = E n (k) −
(i, j = s, x, y, z),
2 k 2 . 2m
(1.17) (1.18)
Note that the orthonormality of u s (r) and u j (r) ( j = x, y, z) was used to derive (1.16). The condition used to obtain solutions A, B, C, and D other than A = B = C = D = 0 is E c − λ Pk x Pk y Pk z P ∗kx Ev − λ 0 0 = 0, (1.19) P ∗k y 0 Ev − λ 0 P ∗ kz 0 0 Ev − λ where P = −i
2 m
us ∗
∂u j 3 d r, ∂r j
P∗ = − i
2 m
u j∗
∂u s 3 d r ∂r j
( j = x, y, z, r x = x, r y = y, r z = z).
(1.20)
From (1.19) we obtain 2 k 2 Ec + Ev + ± E 1,2 (k) = 2 2m
Ec − Ev 2
E 3,4 (k) = E v + where (1.18) was used.
1/2
2
2 k 2 , 2m
+ k |P| 2
2
,
(1.21) (1.22)
P1: OSO c01 JWBS036-Numai
8
August 9, 2010
10:8
Printer Name: Yet to Come
ENERGY BANDS IN BULK AND QUANTUM STRUCTURES
Electron
Electron
Nucleus
Nucleus
(a)
(b) FIGURE 1.1 Motions of an electron.
Spin-Orbit Interaction In addition to k· p perturbation, we consider spin-orbit interaction and second-order perturbation. First, let us consider spin-orbit interaction semiclassically. As shown in Fig. 1(a), an electron with electric charge −e = −1.6022 × 10−19 C rotates about the nucleus with electric charge +Z e. The velocity of the electron is v and the position vector of the electron is r, with the position of the nucleus as the initial point. If we see the nucleus from the electron as shown in Fig. 1.1(b), the nucleus seems to rotate about the electron with a velocity −v. As a result, a magnetic flux density B is produced at the position of the electron, which is written B=
r ×v µ0 Z e 1 µ0 Ze 3 = l. 4π r 4π m r 3
(1.23)
This equation is known as Biot–Savart’s law. In (1.23), µ0 is magnetic permeability of vacuum, and l is the orbital angular momentum, which is given by l = r × p = r × mv.
(1.24)
The spin magnetic moment µs is expressed as µs = −
2µB e s=− s, m
(1.25)
where s is the spin angular momentum and µB is the Bohr magneton, which is defined as µB ≡
e = 9.2732 × 10−24 A · m2 . 2m
(1.26)
As a result, the magnetic field, which is generated at the position of the electron due to the orbital motions of the nucleus, interacts with the electron’s spin magnetic moment. The interaction energy HSO between the magnetic flux density B and the
P1: OSO c01 JWBS036-Numai
August 9, 2010
10:8
Printer Name: Yet to Come
BULK STRUCTURE
9
spin magnetic moment µs is obtained as HSO = −µs · B =
µ0 Z e2 1 l · s. 4π m 2 r 3
(1.27)
Note that (1.27) is obtained using classical electromagnetism. From Dirac’s relativistic quantum mechanics, the interaction energy HSO is given by HSO =
µ0 Z e2 1 l · s, 4π 2m 2 r 3
(1.28)
which is half of (1.27). Pauli’s Spin Matrices Pauli’s spin matrices σ are defined as s = σ, 2 01 0 −i 1 0 σx = , σy = , σz = . 10 i 0 0 −1
(1.29) (1.30)
Using Pauli’s spin matrices, the spin-orbit interaction Hamiltonian HSO can be rewritten HSO =
µ0 Z e2 1 l · σ. 4π 2m 2 r 3 2
(1.31)
If the up-spin ↑ (sz = /2) and down-spin ↓ (sz = −/2) are expressed as α and β, respectively, they are written in matrix form: 1 0 α= , β= . 0 1
(1.32)
As a result, operations of σz on α and β are written σz α = α, σz β = −β.
(1.33)
When a spherical polar coordinate system is used, the spin-orbit interaction Hamiltonian HSO is expressed as HSO
l+ σ− + l− σ+ , = ξ (r) l · σ = ξ (r) l z σz + 2 2 2
(1.34)
P1: OSO c01 JWBS036-Numai
10
August 9, 2010
10:8
Printer Name: Yet to Come
ENERGY BANDS IN BULK AND QUANTUM STRUCTURES
where µ0 Z e2 1 , 4π 2m 2 r 3 l+ = l x + i l y , l− = l x − i l y , σ+ = σx + i σ y , σ− = σx − i σ y . ξ (r) =
(1.35)
When the spin-orbit interaction Hamiltonian HSO is added to (1.14) as a perturbation term, the Schr¨odinger equation is written
2 2 2 k 2 − ∇ + V (r) + Hd + HSO u n (k, r) = E n (k) − u n (k, r). 2m 2m
(1.36)
It should be noted that l operates on exp(i k · r) in the Bloch function, but this operation is neglected because the result is much smaller than the other terms. To solve (1.36), the wave functions are represented in the spherical polar coordinate system as us = us , u+ = −
ux + u y ux − u y x−y x+y ∼ √ , u z ∼ z. (1.37) ∼ − √ , u− = √ √ 2 2 2 2
In (1.37), √ the spherical function f (r) is omitted after ∼ to simplify expressions. Note that 2 is introduced in the denominators to normalize the wave functions. Using the spherical harmonic function Ylm , the wave functions u + , u − , and u z are expressed as 1 =− 2
u+ =
Y11
u− =
Y1−1
1 = 2
uz =
Y10
x +iy 3 1 =− 2 2 2 2π x + y + z 2 x −iy 3 1 = 2 2 2 2π x + y + z 2
1 = 2
3 exp(i φ) sin θ, 2π
3 exp(−i φ) sin θ, 2π
z 3 1 = 2 2 2 π x +y +z 2
(1.38)
3 cos θ, π
where x = r sin θ cos φ, y = r sin θ sin φ, and z = r cos θ . Including the up- and down-spins α and β, the following eight wave functions are present: u s α, u s β, u + α, u + β, u z α, u z β, u − α, u − β. Therefore, we have to calculate the elements of the 8 × 8 matrix to obtain energy eigenvalues from (1.36).
P1: OSO c01 JWBS036-Numai
August 9, 2010
10:8
Printer Name: Yet to Come
11
BULK STRUCTURE
For brevity, we assume that k = k x , k y , k z is a vector in the positive direction of the z-axis and express the elements of k as k z = k, k x = k y = 0.
(1.39)
In this case we only have to solve the determinant for the 4 × 4 matrix on four elements of u s α, u + β, u z α, u − β or those of u s β, u − α, u z β, u + α because of the symmetry in the 8 × 8 matrix. This determinant for the 4 × 4 matrix is written Ec − λ 0 0 0 Ev − λ − 3 √ 2 P ∗k 0 3 0 0
0 = 0, Ev − λ 0 0 0 Ev − λ + 3 √Pk 2 0 3
0
(1.40)
where the terms including 0 are the matrix elements of HSO , and the other terms are those of Hd . Here, using ξ (r) in (1.35), 0 is expressed as 0 2 2 = u + ∗ u + ξ (r) d3 r = u − ∗ u − ξ (r) d3 r 3 2 2 2 2 = (u x 2 + u y 2 )ξ (r) d3 r = u z 2 ξ (r) d3 r. 4 2
(1.41)
From (1.40), the energy of valence band 1 is obtained as
E v1 (k) = E v +
2 k 2 0 + . 3 2m
(1.42)
When |P|2 k 2 is small enough, the energy of the conduction band E c is reduced to 2 k 2 |P|2 k 2 E c (k) = E c + + 2m 3
2 1 + Eg E g + 0
,
(1.43)
where Eg = Ec − Ev −
0 . 3
(1.44)
P1: OSO c01 JWBS036-Numai
12
August 9, 2010
10:8
Printer Name: Yet to Come
ENERGY BANDS IN BULK AND QUANTUM STRUCTURES
Similarly, the energies of valence bands 2 and 3 are given by E v2 (k) = E v +
0 2 k 2 2|P|2 k 2 , + − 3 2m 3E g
|P|2 k 2 2 2 k 2 − . E v3 (k) = E v − 0 + 3 2m 3(E g + 0 )
(1.45) (1.46)
Note that these results were obtained under first-order k· p perturbation. Valence Bands Under second-order perturbation, the energies of the valence bands are given by 0 E v1,2 (k) = E v + + A2 k 2 3 1/2 ± B2 2 k 4 + C2 2 k x 2 k y 2 + k y 2 k z 2 + k z 2 k x 2 (1 → +, 2 → −), (1.47) 2 (1.48) E v3 (k) = E v − 0 + A2 k 2 . 3 Equations (1.43), (1.47), and (1.48) are shown in Fig. 1.2. From the definition of effective mass in (1.11), the band with energy E v1 (k) is referred to as a heavy hole band and that with E v2 (k) is called a light hole band. It should be noted that the heavy and light hole bands are degenerate at k = 0. The band with energy E v3 (k) is called the split-off band, and 0 is called the split-off energy. The coefficients A2 , B2 , and C2 in (1.47) and (1.48) are determined experimentally by cyclotron
Conduction Band
Heavy Hole Band Light Hole Band Split-off Band FIGURE 1.2 Energy bands of a bulk structure when the spin-orbit interaction is considered under a second-order perturbation.
P1: OSO c01 JWBS036-Numai
August 9, 2010
10:8
Printer Name: Yet to Come
BULK STRUCTURE
13
TABLE 1.1 Relations Between Operators and Eigenvalues Operator 2
l lz s2 sz j2 jz
Eigenvalue l(l + 1) (l = 0:s-orbital, l = 1: p-orbitals) m l , m l = 1, 0, −1 s(s + 1)2 , s = 1/2 m s , m s = 1/2, −1/2 j( j + 1)2 , j = 3/2, 1/2 m j , m j=3/2 = 3/2, 1/2, −1/2, −3/2, m j=1/2 = 1/2, −1/2 2
resonance. In general, the effective masses depend on the direction of k, and the energy bands are more complicated. Note that in the preceding analysis, the energy bands of direct transition semiconductors, in which the bottom of the conduction band and the tops of the valence bands are placed at k = 0, are calculated. In indirect transition, the k’s of the bottom of the conduction band and the k’s of the tops of the valence bands are different. Due to the spin-orbit interaction, the quantum states are indicated by j = l + s, where l is the angular momentum operator and s is the spin operator. Therefore, as indexes of the wave functions, we can use the quantum numbers j and m j , which represent the eigenvalues of operators j and jz , respectively. The relations between the operators and the eigenvalues are summarized in Table 1.1. When we express the wave functions as | j, m j , the wave functions of the valence bands under the second-order perturbation are expressed as follows: For a heavy hole band, 3 3 1 , 2 2 = √2 |(x + i y)α, 3 , − 3 = √1 |(x − i y)β, 2 2 2
(1.49)
for a light hole band, 3 1 1 , 2 2 = √6 |2zα + (x + i y)β, 3 , − 1 = √1 |2zβ − (x − i y)α, 2 2 6
(1.50)
P1: OSO c01 JWBS036-Numai
14
August 9, 2010
10:8
Printer Name: Yet to Come
ENERGY BANDS IN BULK AND QUANTUM STRUCTURES
and for a split-off band, 1 1 1 , 2 2 = √3 |zα − (x + i y)β, 1 , − 1 = √1 |zβ + (x − i y)α. 2 2 3
(1.51)
1.3 QUANTUM STRUCTURES Quantum Effects Semiconductor structures whose sizes are small enough that their quantum effects, such as splitting of energy bands and the tunneling effect, may be significant are called quantum structures. Square Well Electrons in quantum structures feel both the periodic potential of crystals and the quantum well potential. First, the energy eigenvalues and wave functions of a particle in a square well potential are reviewed briefly. As shown in Fig. 1.3, we assume that a carrier is present in a square potential well V (r) as V (r) =
0 ∞
inside the well, at the boundaries
(1.52)
When the potential well is a cube with side L, the boundary conditions for a wave function ϕ(x, y, z) are given by ϕ(0, y, z) = ϕ(L , y, z) = 0, ϕ(x, 0, z) = ϕ(x, L , z) = 0, ϕ(x, y, 0) = ϕ(x, y, L) = 0.
ϕ
ϕ
FIGURE 1.3 Square well potential.
(1.53)
P1: OSO c01 JWBS036-Numai
August 9, 2010
10:8
Printer Name: Yet to Come
QUANTUM STRUCTURES
15
FIGURE 1.4 Wave function ϕ and energy eigenvalues E in a one-dimensional square well potential.
Under these boundary conditions, the wave function ϕ(x, y, z) and energy eigenvalue E are obtained as
8 sin k x x · sin k y y · sin k z z, L3 2 (k x 2 + k y 2 + k z 2 ), E= 2m nyπ nx π nz π , ky = , kz = (n x , n y , n z = 1, 2, 3, . . .). kx = L L L ϕ(x, y, z) =
(1.54)
Figure 1.4 shows the wave function ϕ and energy eigenvalues E for a one-dimensional square well potential. The energy eigenvalues E are discrete and their values are proportional to a square of the quantum number n x . In addition, with a decrease in L, the energy separation between the energy levels increases.
Potential Well and Energy Barrier Figure 1.5 shows the energies of the conduction band and valence bands at k = 0 for GaAs, which is sandwiched by AlGaAs layers. The low-energy regions for electrons in the conduction band and holes in the valence bands are called potential wells. Note that in Fig. 1.5, the vertical line shows the energy of the electrons, and the energy of the holes decreases with an increase in the height of the vertical line. When the width of the potential well L z is on the order of less than several tens of nanometers, the potential well is called the quantum well. The bandgaps of AlGaAs layers are higher than those of GaAs. As a result, these AlGaAs layers become the energy barriers for GaAs and are called energy barrier layers. At the interfaces of the quantum well and the barriers, energy differences exist in the conduction band E c , and in the valence bands, E v , and are called band offsets.
P1: OSO c01 JWBS036-Numai
16
August 9, 2010
10:8
Printer Name: Yet to Come
ENERGY BANDS IN BULK AND QUANTUM STRUCTURES
Band Offset
Band Offset
Energy Barrier Potential Energy Barrier Layer Well Layer FIGURE 1.5 Quantum well structure.
Effective Mass Approximation The periods of the potential for semiconductor crystals are represented by lattice constants which are on the order of 0.5 nm. In contrast, the thickness of potential wells or barriers in quantum structures is between an order of nanometers and that of several tens of nanometers. Hence, in quantum structures, electrons and holes feel both periodic and quantum potentials. If we use effective mass, the effect of the periodic potential is included in the effective mass, as shown in (1.12), and we only have to consider the quantum potential, referred to as the effective mass approximation. Under effective mass approximation, a wave function in the quantum structure is obtained as a product of base function ψ and envelope function ϕ. As base function ψ we use a wave function for the periodic potential: ψn (k, r) = exp(i k · r)u nk (r),
u n (k, r) = u nk (r + R).
(1.55)
As the envelope function ϕ, we use a wave function for the quantum potential. For example, for a cube with a side length of L and infinite potential at the boundaries, the envelope function ϕ is given by ϕ(x, y, z) =
8 sin k x x · sin k y y · sin k z z. L3
(1.56)
Figure 1.6 shows one-, two-, and three-dimensional quantum wells. A sheet in which only L z is of quantum size, as shown in Fig. 1.6(a), is called a one-dimensional quantum well or simply, a quantum well. A stripe in which only L y and L z are quantum sizes, as shown in Fig. 1.6(b), is called a two-dimensional quantum well
P1: OSO c01 JWBS036-Numai
August 9, 2010
10:8
Printer Name: Yet to Come
QUANTUM STRUCTURES
(a)
(b)
17
(c)
FIGURE 1.6 (a) One-, (b) two-, and (c) three-dimensional quantum wells.
or a quantum wire. A box whose L x , L y , and L z are all quantum sizes, as shown in Fig. 1.6(c), is called a three-dimensional quantum well or a quantum box. The energies of the carriers, which are confined completely in the sheet shown in Fig. 1.6(a), are written
Ex y
E = Ex y + Ez , 2 π 2 2 2 π 2 2 = (n x + n y 2 ), E z = nz , ∗ 2 2m L 2m ∗ L z 2
(1.57)
where is Dirac’s constant; m ∗ is the effective mass of the carrier; and n x , n y , and n z are quantum numbers. If n x , n y , and n z are of the same order, we have E x y E z . Density of States As an example, let us calculate the density of states in a one-dimensional quantum well for n z = 1. The density of states is determined by the number of combinations of n x and n y . When n x and n y are large enough, the combinations (n x , n y ) for constant energy E x y are represented by the points on the circumference of a circle with radius r , which is given by r 2 = nx 2 + n y 2 =
2m ∗ L 2 Ex y . 2 π 2
(1.58)
Because both n x and n y are positive numbers, the number S of combinations (n x , n y ) is given by the area of a quarter circle with radius r . As a result, S is expressed as S=
1 2 π π 2m ∗ L 2 m∗ L 2 πr = (n x 2 + n y 2 ) = Ex y . E = x y 4 4 4 2 π 2 22 π
(1.59)
Considering the up- and down-spins, the number of states N is twice as large as S, which is written N = 2S =
m∗ L 2 Ex y . 2 π
(1.60)
P1: OSO c01 JWBS036-Numai
18
August 9, 2010
10:8
Printer Name: Yet to Come
ENERGY BANDS IN BULK AND QUANTUM STRUCTURES
Substituting E x y = E − E z=1 into (1.60), the electron concentration n for the energy between zero and E is obtained as n=
N L2 L
= z
m∗ 2 π L
(E − E z=1 ).
(1.61)
z
When we define the density of states per volume for the energy between E and E + dE as ρ1 (E), we have ρ1 (E) dE ≡ n.
(1.62)
From (1.61) and (1.62), we obtain ρ1 (E) ≡
m∗ dn = 2 . dE π Lz
(1.63)
The densities of states for n z = 2, 3, . . . are calculated similarly, and the results are shown in Fig. 1.7(a). Here L z is 3 nm; m ∗ is 0.08m, where m is the electron mass in vacuum; and the ρ1 (E) for n z = 1, 2, and 3 are indicated as ρ11 , ρ12 , and ρ13 , respectively. It should be noted that the density of states for a one-dimensional quantum well is a step function. In contrast, the bulk structures have a density of states such that ρ0 (E) =
(2m ∗ )3/2 1/2 E , 2π 2 3
(1.64)
which is proportional to E 1/2 as shown by a dashed line, because the number of states is represented by the number of the points existing in one-eighth of a sphere with radius r .
Bulk
3
5
5
4
4
/
4
/
/
5
3
3
2
2
2
1
1
1
0
0
0
2 4 6 Energy (eV)
(a)
8
0
2 4 6 Energy (eV)
(b)
8
0
0
2 4 6 Energy (eV)
8
(c)
FIGURE 1.7 Density of states for (a) one-, (b) two-, and (c) three-dimensional quantum wells.
P1: OSO c01 JWBS036-Numai
August 9, 2010
10:8
Printer Name: Yet to Come
19
SUPERLATTICES
If we set L y = L z = L, the energies of the carriers, which are confined completely in the wire shown in Fig. 1.6(b), are written E = E x + E yz , 2 π 2 2 2 π 2 2 n x , E yz = (n y + n z 2 ). Ex = 2 ∗ 2m L x 2m ∗ L 2
(1.65)
For a pair of quantum numbers (n y , n z ), the density of states ρ2 (E) is obtained as √ √ 2m ∗ −1/2 2m ∗ ρ2 (E) = E = (E − E yz )−1/2 . x π L 2 π L 2
(1.66)
The result calculated for (1.66) is shown in Fig. 1.7(b). When the energy E is equal to E yz , the density of states ρ2 (E) is infinity. When E exceeds E yz , ρ2 (E) decreases in proportion to (E − E yz )−1/2 , which leads to a density of states ρ2 (E) with a sawtoothed shape. If we set L x = L y = L z = L, the energies of the carriers, which are confined completely in the box shown in Fig. 1.6 (c), are written E = Ex + E y + Ez , 2 π 2 2 2 π 2 2 2 π 2 2 nx , E y = n y , Ez = nz . Ex = ∗ 2 ∗ 2 2m L 2m L 2m ∗ L 2
(1.67)
It should be noted that the energy eigenvalues are completely discrete. The density of states ρ3 (E) is a delta function, which is written ρ3 (E) = 2
δ(E − E x − E y − E z ).
(1.68)
n x ,n y ,n z
Figure 1.7(c) shows the number of states per volume and the density of states in a three-dimensional quantum well. With an increase in the dimension of the quantum wells, the energy bandwidths of the densities of states decrease. Therefore, the energy distribution of the electron concentrations narrows with an increase in the dimension of the quantum wells, as shown in Fig. 1.8. Therefore, the optical gain concentrates on a certain energy (wavelength). As a result, in quantum well lasers, a low threshold current, a high speed modulation, low chirping, and a narrow spectral linewidth are expected.
1.4
SUPERLATTICES
Array quantum structures and solitary structures are called superlattices. From the viewpoint of the potential, superlattices are classified as follows. Figure 1.9 shows three kinds of potentials of superlattices. The horizontal direction indicates the position of the layers, and the vertical direction represents the energy of the electrons.
P1: OSO c01 JWBS036-Numai
20
August 9, 2010
10:8
Printer Name: Yet to Come
ENERGY BANDS IN BULK AND QUANTUM STRUCTURES
10
10
8
8
8 /
/
/
10
6
6
6
4
4
4
2
2
2
0
0 20 40 60 80 100
0
0 20 40 60 80 100
0
0 20 40 60 80 100
(b)
(a)
(c)
FIGURE 1.8 Energy distribution of electron concentrations in quantum wells: (a) bulk structure; (b) one-dimensional quantum; (c) two-dimensional quantum structure.
With an increase in height, the energy of electrons increases and that of holes decreases. As shown in Fig. 1.9(a), in a type I superlattice , the position of the potential well for electrons in the conduction band is the same as that for holes in the valence band. Therefore, both electrons and holes are confined in semiconductor layer B, which has a narrower bandgap than that of semiconductor layer A. In the type II superlattice in Fig. 1.9(b), the electrons in the conduction band are confined in semiconductor layer B, and the holes in the valence band are confined in semiconductor layer A. In the type III superlattice in Fig. 1.9(c), the energy of the conduction band of semiconductor layer B overlaps that of the valence band of semiconductor layer A, which results in a semimetal. Note that in the literature, types II and III are sometimes called types I and II, respectively. From the perspective of the period, superlattices are classified as follows. Figure 1.10 shows the relationships between the characteristics of superlattices and the thickness of barriers and wells. When each layer thickness is larger than several tens of nanometers, only the bulk characteristics are observed. If the barrier thickness is less than several tens of nanometers, the quantum mechanical tunneling effect appears. When the barriers are thick and only the wells are thin, quantum energy levels are formed in the wells. If such wells are used as the active layers in light-
BABAB
BABAB
BABAB
(a)
(b)
(c)
FIGURE 1.9 Classification of super lattices by potential: (a) type I; (b) type II; and (c) type III.
P1: OSO c01 JWBS036-Numai
August 9, 2010
10:8
Printer Name: Yet to Come
REFERENCES
21
Electron Well Thickness (nm)
Bulk Region Tunneling Effect
Minizone
Quantum Level
Bending of Brillouin Zone
Quantum Level
Barrier Thickness (nm) FIGURE 1.10 Classification of superlattices by period.
emitting devices, narrow light emission spectra are obtained. When both barriers and wells are thinner than about 10 nm, the wave functions of a well start to penetrate adjacent wells. As a result, the wave functions of each well overlap each other, which produces minizones and induces Bloch oscillations or negative resistances. When the thickness of both barriers and wells decreases further, down to the order of atomic layers, bending of Brillouin zones appears, which will transform indirect transition materials into direct transition materials.
REFERENCES 1. E. O. Kane, “Band structure of indium antimonide,” J. Phys. Chem. Solids 1, 249 (1957). 2. C. Kittel, Quantum Theory of Solids, 2nd ed., Wiley, New York, 1987. 3. G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, Halsted Press, New York, 1988. 4. S. Datta, Quantum Phenomena, Addison-Wesley, Reading, MA, 1989. 5. L. I. Schiff, Quantum Mechanics, 3rd ed., McGraw-Hill, New York, 1968. 6. A. Messiah, Quantum Mechanics, Vols.1 and 2, North-Holland, Amsterdam, 1961. 7. T. Numai, Fundamentals of Semiconductor Lasers, Springer-Verlag, New York, 2004.
P1: OSO c02 JWBS036-Numai
August 4, 2010
15:51
Printer Name: Yet to Come
2 OPTICAL TRANSITIONS
2.1
INTRODUCTION
Excitation and Relaxation The electrons in semiconductors tend to stay in the lowest-energy state, because the lowest-energy state is most stable. Electrons that are excited by thermal energy, light, or electron beams absorb these energies and transit to high-energy states. Transitions of the electrons from low-energy states to high-energy states are called excitations. Electrons in high-energy states, however, transit to low-energy states in certain lifetimes, because high-energy states are unstable. Transitions of excited electrons from high-energy states to low-energy states are called relaxations. Excitation and relaxation processes between a valence band and a conduction band are shown in Fig. 2.1. Recombination In semiconductors, transitions of electrons from high-energy states to low-energy states are interpreted as recombinations of electrons and holes. Such recombinations are both radiative and nonradiative. Radiative recombinations emit photons, and the energies of photons are given by differences in the energies between the initial and final energy states related to the transitions. Nonradiative recombinations do not emit photons. When phonons are emitted to crystal lattices or electrons are trapped in defects, the transition energy is Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright
23
P1: OSO c02 JWBS036-Numai
24
August 4, 2010
15:51
Printer Name: Yet to Come
OPTICAL TRANSITIONS
Conduction Band
Electron
Electron
Electron
Hole
Hole
Electron Valence Band
Stable State
Excitation
Relaxation
Stable State
FIGURE 2.1 Excitation and relaxation.
transformed into heat. When three-body collisions happen among two electrons and one hole, or one electron and two holes, the transition energy is used to excite an electron, and these three-body collisions do not emit light. These processes, known as Auger processes, are also categorized as nonradiative recombinations. To obtain high light-emitting efficiency, nonradiative recombinations have to be minimized. However, to enhance modulation characteristics, nonradiative recombination centers may be induced intentionally in the active layers, because the carrier lifetimes are reduced.
2.2 DIRECT AND INDIRECT TRANSITIONS Wave Vector In a conduction band, electrons tend to reside at the bottom, because the energy of the electron is lowest at the bottom. In a valence band, holes tend to reside at the top, because the energy of the hole is lowest at the top. Therefore, recombinations of electrons and holes occur most frequently between the bottom of the conduction band and the top of the valence band. So, let us consider transitions of the electrons from the bottom of a conduction band to the top of a valence band. A semiconductor in which the bottom of the conduction band and the top of the valence band are placed at a common wave vector k is called a direct transition semiconductor. A semiconductor in which the bottom of the conduction band and the top of the valence band have different k-values is called an indirect transition semiconductor. Direct and indirect transitions are shown schematically in Fig. 2.2. Phonon In electron transitions, the energy and momentum are conserved. Therefore, phonons with momentum k do not take part in direct transitions, because a difference in the wave vectors k for the bottom of the conduction band and the top of the valence band is 0. In contrast, transitions of phonons with momentum k = 0 occur in indirect transitions to satisfy the momentum conservation law. Hence, in direct transitions,
P1: OSO c02 JWBS036-Numai
August 4, 2010
15:51
Printer Name: Yet to Come
LIGHT-EMITTING PROCESSES
(a)
25
(b)
FIGURE 2.2 (a) Direct and (b) indirect transition semiconductors.
the transition probabilities are determined by electron transition probabilities alone; in indirect transitions, the transition probabilities are given by the product of electron and phonon transition probabilities. As a result, the transition probabilities of direct transitions are much higher than those of indirect transitions. Consequently, direct transition semiconductors such as InGaAs, InGaAsP, and AlGaAs are superior to indirect transition semiconductors such as Si and Ge for light-emitting devices.
2.3
LIGHT-EMITTING PROCESSES
Luminescence Light emission due to radiative recombinations is called luminescence. Lightemitting processes are classified according to the lifetime, excitation methods, and energy states related to the transitions. Lifetime With regard to lifetime, there are two light emissions: fluorescence, with a short lifetime of 10−9 to 10−3 s, and phosphorescence, with a long lifetime of 10−3 s to 1 day. Excitation Luminescence that is accompanied by optical excitation or optical pumping, called photoluminescence, is used widely to characterize materials. Optical excitation is also used to pump dye lasers and solid-state lasers. When the photon energy of pumping light is ω1 and that of the luminescence is ω2 , luminescence with ω2 < ω1 is called Stokes luminescence and that with ω2 > ω1 is called anti-Stokes
P1: OSO c02 JWBS036-Numai
26
August 4, 2010
15:51
Printer Name: Yet to Come
OPTICAL TRANSITIONS
luminescence. Luminescence that is generated by electrical excitation, called electroluminescence, has been used for panel displays. In particular, luminescence by injection of electric current, called injection-type electroluminescence, has been used for light-emitting diodes (LEDs) and semiconductor lasers or laser diodes (LDs). In LEDs and LDs, the carriers are injected into the active layers by forward bias across the pn-junctions. Luminescence due to electron beam irradiation, called cathodoluminescence, has been adopted to characterize materials. Luminescence induced by mechanical excitation using stress is called triboluminescence, and that induced by thermal excitation is called thermoluminescence. Luminescence during a chemical reaction, referred to as chemiluminescence, has not been reported in semiconductors.
2.4 SPONTANEOUS EMISSION, STIMULATED EMISSION, AND ABSORPTION Emissions and Absorption Figure 2.3 shows spontaneous emission, stimulated emission (or induced emission), and absorption. In the spontaneous emission shown in Fig. 2.3(a), an excited electron decays in a certain lifetime and a photon is emitted irrespective of incident light. In contrast, in the stimulated emission shown in Fig. 2.3(b), incident light induces a radiative transition of an excited electron. Light emitted due to stimulated emission has the same wavelength, phase, and direction as those of incident light. Therefore, the light generated by stimulated emission is highly monochromatic, coherent, and directional. In stimulated emission, one incident photon generates two photons: the incident photon itself and a photon emitted due to stimulated emission. As a result, the incident light is amplified by stimulated emission. In the absorption shown in Fig. 2.3(c), an electron transits from a lower-energy state to a higher-energy state by absorbing energy from the incident light. Because this transition is induced
Electron
Electron
Electron Conduction Band
Valence Band Hole (a)
Hole (b)
Hole (c)
FIGURE 2.3 (a) Spontaneous emission; (b) stimulated emission; (c) absorption.
P1: OSO c02 JWBS036-Numai
August 4, 2010
15:51
Printer Name: Yet to Come
OPTICAL GAINS
27
by the incident light, it is sometimes called induced absorption. It should be noted that spontaneous absorption does not exist. Thermal Equilibrium When light is incident on a material, stimulated emission and absorption take place simultaneously. In thermal equilibrium, there are more electrons in a lower-energy state than in a higher-energy state. Therefore, in thermal equilibrium, the absorption exceeds the stimulated emission, and only the absorption is observed when light is incident on a material. Population Inversion To obtain a net optical gain, we have to make the number of electrons in a higherenergy state greater than the number in lower-energy state. This condition is called inverted population, or population inversion, because the electron population is inverted compared with that in thermal equilibrium. In semiconductors, the population inversion due to optical pumping or electric current injection generates many electrons at the bottom of the conduction band and many holes at the top of the valence bands. Laser Laser oscillators use fractions of spontaneous emission as optical input and amplify the fractions by stimulated emission under population inversion. Once the optical gains in laser oscillators exceed the optical losses, laser oscillations take place. The term laser is an acronym for “light amplification by stimulated emission of radiation” and is used alone as a noun with a meaning laser oscillator. Semiconductor light-emitting devices employ spontaneous emission and are used in remote-control transmitters, switch lights, brake lights, displays, and traffic signals. In contrast, laser diodes are oscillators of light using stimulated emission and are used as light sources in lightwave communications, compact disks (CDs), magnetooptical disks (MOs), digital videodisks (DVDs), laser beam printers, laser pointers, and bar-code readers.
2.5
OPTICAL GAINS
Laser Oscillator Figure 2.4 shows a laser oscillator where a fraction of the spontaneous emission is used as input and the optical gain is produced by the stimulated emission. To feed back light, optical resonators or optical cavities, which are composed of reflectors or mirrors, are adopted. Due to this configuration, characteristics of lasers are affected by the optical gains and optical resonators. It should be noted that all LDs use optical resonators.
P1: OSO c02 JWBS036-Numai
28
August 4, 2010
15:51
Printer Name: Yet to Come
OPTICAL TRANSITIONS
Optical Feedback by Reflectors Fraction of Spontaneous Emission
Optical Gain
Laser Light
FIGURE 2.4 Laser oscillator.
Spontaneous Emission As explained earlier, a fraction of the spontaneous emission is used as the input of a laser. However, note that all the light emitted spontaneously cannot be used as input, because it has different wavelengths, phases, and propagation directions. Among these, only light that has wavelengths within the optical gain spectrum and that satisfies the resonance conditions of optical resonators can be a source of laser oscillations. Other spontaneous emission, which does not satisfy the resonance conditions of optical resonators, is readily emitted outward without obtaining sufficient optical gain for laser oscillation. The light amplified by stimulated emission has the same wavelength, phase, and propagation direction as those of the input light. Therefore, the laser light is highly monochromatic, bright, coherent, and directional. Quasi-Fermi Levels When a lot of electrons are excited in the conduction band and a lot of holes are generated in the valence band by optical pumping or electrical current injection, the carrier distribution is in nonthermal equilibrium. As a result, one Fermi level E F cannot describe the distribution functions of the electrons and holes. In this case it is useful to determine the distribution functions by assuming that the electrons in the conduction band and the holes in the valence band are governed separately by Fermi–Dirac distribution. For this purpose we introduce quasi-Fermi levels E Fc and E Fv , which are defined as E c − E Fc , n = Nc exp − kB T E Fv − E v p = Nv exp − , kB T 2π m e kB T 3/2 Mc , Nc = 2 h2 2π m h kB T 3/2 . Nv = 2 h2
(2.1)
P1: OSO c02 JWBS036-Numai
August 4, 2010
15:51
Printer Name: Yet to Come
OPTICAL GAINS
29
Here n and p are the electron and hole concentration, respectively; E c and E v are the energy at the bottom of the conduction band and that at the top of the valence band, respectively; kB = 1.3807 × 10−23 J/K is the Boltzmann constant; T is the absolute temperature; Nc and Nv are the effective density of states for the electrons and holes, respectively; m e and m h are the effective mass of the electrons and holes, respectively; Mc is the number of the band edges of the conduction band; and h is Planck’s constant. From (2.1), these quasi-Fermi levels E Fc and E Fv are written as E Fc = E c + kB T ln E Fv
n , Nc
p = E v − kB T ln . Nv
(2.2)
Using E Fc and E Fv , we can express a distribution function for the electrons in the valence band f 1 with the energy E 1 and that for the electrons in the conduction band f 2 with the energy E 2 as f1 =
1 , exp [(E 1 − E Fv ) /kB T ] + 1 (2.3)
1 . f2 = exp [(E 2 − E Fc ) /kB T ] + 1 It should be noted that the distribution function for the holes in the valence band is given by [1 − f 1 ]. Interaction Between Light and a Direct Transition Semiconductor We assume that light, which has a photon energy of E 21 = E 2 − E 1 and a photon concentration of n ph (E 21 ), interacts with a direct transition semiconductor as shown in Fig. 2.5, where E c is the energy at the bottom of the conduction band and E v is the energy at the top of the valence band. Stimulated Emission Rate The concentration n 2 of an electron, which occupies a state with energy E 2 in a conduction band, is expressed as n 2 = ρc (E 2 − E c ) f 2 ,
(2.4)
where ρc (E 2 − E c ) is the density of states, which is a function of E 2 − E c , and f 2 is the distribution function in (2.3).
P1: OSO c02 JWBS036-Numai
30
August 4, 2010
15:51
Printer Name: Yet to Come
OPTICAL TRANSITIONS
Electron Conduction Band
Valence Band Hole FIGURE 2.5 Schematic model for radiation and absorption.
The concentration p1 of a hole, which occupies a state with energy E 1 in the valence band, is written p1 = ρv (E v − E 1 )[1 − f 1 ],
(2.5)
where ρv (E v − E 1 ) is the density of states, which is a function of E v − E 1 , and f 1 is the distribution function in (2.3). When we express the transition probability per unit time for the transition from E 2 to E 1 as B21 , the stimulated emission rate per unit volume r21 (stim) is given by r21 (stim) = B21 n 2 p1 n ph (E 21 ) = B21 n ph (E 21 )ρc (E 2 − E c )ρv (E v − E 1 ) f 2 [1 − f 1 ].
(2.6)
Absorption Rate The concentration p2 of an empty state, which is not occupied by electrons with energy E 2 in the conduction band, is expressed as p2 = ρc (E 2 − E c )[1 − f 2 ].
(2.7)
The concentration n 1 of the electron, which occupies a state with energy E 1 in the valence band, is written n 1 = ρv (E v − E 1 ) f 1 .
(2.8)
When we express the transition probability per unit time for the transition from E 1 to E 2 as B12 , the absorption rate per unit volume r12 (abs) is given by r12 (abs) = B12 p2 n 1 n ph (E 21 ) = B12 n ph (E 21 )ρc (E 2 − E c )ρv (E v − E 1 ) f 1 [1 − f 2 ].
(2.9)
P1: OSO c02 JWBS036-Numai
August 4, 2010
15:51
Printer Name: Yet to Come
OPTICAL GAINS
31
Spontaneous Emission Rate When we express the transition probability per unit time for the transition from E 2 to E 1 as A21 and use (2.4) and (2.5), the spontaneous emission rate per unit volume r21 (spon) is given by r21 (spon) = A21 n 2 p1 = A21 ρc (E 2 − E c )ρv (E v − E 1 ) f 2 [1 − f 1 ].
(2.10)
Blackbody Radiation First, we consider the emissions and absorption in thermal equilibrium. In thermal equilibrium, the carrier distributions are described by one Fermi level E F , and the emissions balance the absorption. Therefore, we obtain r21 (stim) + r21 (spon) = r12 (abs), E F1 = E F2 = E F .
(2.11)
Substituting (2.6), (2.9), and (2.10) into (2.11) results in n ph (E 21 ) =
A21 , B12 exp E 21 /(kB T ) − B21
(2.12)
and the blackbody radiation theory gives n ph (E 21 ) =
8π n r 3 E 21 2 . h 3 c3 exp(E 21 /kB T ) − h 3 c3
(2.13)
Here n r is the effective refractive index of a material, h is Planck’s constant, and c = 2.99792458 × 108 m/s is the speed of light in vacuum. Comparing (2.12) with (2.13), we have A21 = m(E 21 )B, B21 = B12 = B, 8π n r 3 E 21 2 m(E 21 ) = , h 3 c3
(2.14)
where A21 is called Einstein’s A coefficient, B is called Einstein’s B coefficient, and m(E 21 ) is the mode density. Net Stimulated Emission The third equation in (2.14) suggests that spontaneous emission takes place for all the modes with an energy of E 21 , whereas stimulated emission and absorption occur only for the mode corresponding to the incident light.
P1: OSO c02 JWBS036-Numai
32
August 4, 2010
15:51
Printer Name: Yet to Come
OPTICAL TRANSITIONS
In nonthermal equilibrium conditions, in which many free carriers exist, radiations do not balance with absorption. When light exists in a material, stimulated emission and absorption take place simultaneously. Therefore, the net stimulated emission rate r 0 (stim) is given by r 0 (stim) = r21 (stim) − r12 (abs) = Bn ph (E 21 )ρc (E 2 − E c )ρv (E v − E 1 )[ f 2 − f 1 ] A21 = n ph (E 21 )ρc (E 2 − E c )ρv (E v − E 1 )[ f 2 − f 1 ]. m(E 21 )
(2.15)
From (2.15), to obtain the net stimulated emission r 0 (stim) > 0, it is found that we need f2 > f1,
(2.16)
which indicates that population inversion is required in the semiconductors. With the help of (2.3), (2.16) is reduced to E Fc − E Fv > E 2 − E 1 = E 21 ,
(2.17)
which is known as the Bernard–Duraffourg relation. In LDs, a typical carrier concentration for r 0 (stim) > 0 is on the order of 1018 cm−3 . Optical Power Gain Coefficient per Unit Length The power optical gain coefficient per unit length g is defined as dI = g I, dz
(2.18)
where I is the light power per unit area and z is a coordinate for a propagation direction of the light. Note that the field optical gain coefficient gE , which is the gain coefficient for the electric field, is given by gE = g/2. The light power per unit area I is expressed as I = vg E 21 n ph (E 21 ) = vg ωn ph (E 21 ), dω c vg = = , dk nr dn ph (E 21 ) dI = vg E 21 = vg E 21 r 0 (stim), dt dt
(2.19)
where vg is a group velocity of the light in the material, E 21 = ω is the photon energy, and n ph (E 21 ) is the photon concentration.
P1: OSO c02 JWBS036-Numai
August 4, 2010
15:51
Printer Name: Yet to Come
OPTICAL GAINS
33
Because the position z is a function of the time t, we have dI dt dI = = dz dz dt
dz dt
−1
1 dI dI = , dt vg dt
(2.20)
dI = g I = g vg E 21 n ph (E 21 ). dz From (2.19) and (2.20), we have r 0 (stim) = g vg n ph (E 21 ).
(2.21)
As a result, a large r 0 (stim) leads to a large g. Using (2.15), (2.19), and (2.21), we can also express g as g=
nr r 0 (stim) n r = Bρc (E 2 − E c )ρv (E v − E 1 )[ f 2 − f 1 ]. n ph (E 21 ) c c
(2.22)
Einstein’s A and B Coefficients From time-dependent quantum mechanical perturbation theory, Einstein’s B coefficient in (2.14) is given by B=
e2 h 1| p|22 . 2m 2 ε0 n r 2 E 21
(2.23)
Here e is the elementary electric charge; h is Planck’s constant; m is the electron mass; ε0 is permittivity in vacuum; n r is the refractive index of the material; E 21 is the photon energy; p is the momentum operator; and 1| and |2 are the bra and ket vectors, corresponding to the wave functions of the valence band and the conduction band in a steady state, respectively. Substituting (2.23) into (2.14), we can write Einstein’s A coefficient as A21 =
4π e2 n r E 21 1| p|22 . m 2 ε0 h 2 c3
(2.24)
Einstein’s A and B coefficients can be written B=
π e2 ω πω 2 1|r|22 = µ, 2 ε0 n r ε0 n r 2 2n r ω3 2 µ, A21 = ε0 hc3 µ2 = 1|er|22 .
(2.25)
P1: OSO c02 JWBS036-Numai
34
August 4, 2010
15:51
Printer Name: Yet to Come
OPTICAL TRANSITIONS
Here µ is the dipole moment, and the following relations on the momentum operator p are used: p=m
dr , dt
(2.26)
1 d 1| p|2 = 1|r|2 = i ω1|r|2, m dt
(2.27)
where r ∝ e i ωt is assumed. In the explanations above, Einstein’s B coefficient is defined to be in proportion to the photon concentration. Therefore, the quantum mechanical transition rate for the stimulated emission is equal to Einstein’s B coefficient. Transitions Between Various Energy States In semiconductors, transitions take place between various energy states in the conduction and valence bands, as shown in Fig. 2.6. If we set E 21 = E and E 2 − E c = E , the electron energy in the valence band E for the transition allowed is given by E = E − E. Therefore, integrating (2.15) with respect to E gives the net stimulated emission rate r 0 (stim) as
e2 h n ph (E) r (stim) = 2m 2 ε0 n r 2 E 0
∞
1| p|22 ρc (E )ρv (E )[ f 2 (E ) − f 1 (E )] dE , (2.28)
0
where (2.23) is used. From(2.22) and (2.28), the power optical gain coefficient g(E) is given by g(E) =
e2 h 2 2m ε0 n r cE
∞
1| p|22 ρc (E )ρv (E )[ f 2 (E ) − f 1 (E )] dE .
0
Electron Conduction Band
Valence Band Hole FIGURE 2.6 Transition with a constant photon energy.
(2.29)
P1: OSO c02 JWBS036-Numai
August 4, 2010
15:51
Printer Name: Yet to Come
OPTICAL GAINS
35
k-Selection Rule In (2.29), ρc (E )ρv (E ) is considered to be the density- of-state pairs related to optical transitions. The density-of-state pairs are also expressed using the law of momentum conservation (k-selection rule) and the law of energy conservation (E 21 = E 2 − E 1 ). Under the k-selection rule, optical transitions take place for k = k1 = k2 only in direct transition semiconductors. Therefore, we can define the density-of-state pairs or the reduced density-of-states ρred (E 21 ) as ρred (E 21 ) dE 21 ≡ ρv (E 1 ) dE 1 = ρc (E 2 ) dE 2 ,
(2.30)
dE 21 = dE 1 + dE 2 .
(2.31)
where
From (2.30) and (2.31), we have
1 1 ρred (E 21 ) = + ρv (E 1 ) ρc (E 2 )
−1
.
(2.32)
When the conduction and valence bands are assumed to be parabolic, the energies E 1 and E 2 are written 2 k 2 , 2m h 2 k 2 E2 = Ec + , 2m e
E1 = Ev −
(2.33) (2.34)
where m h and m e are the effective masses of the holes and electrons, respectively. As a result, E 1 and E 2 are rewritten me me + mh mh E2 = Ec + me + mh
E1 = Ev −
E 21 − E g ,
(2.35)
E 21 − E g .
(2.36)
The number of state pairs per volume associated with the optical transitions for photon energies between E 21 and E 21 + dE 21 is given by ρred (E 21 ) [ f 2 (E 21 ) − f 1 (E 21 )] dE 21 .
(2.37)
P1: OSO c02 JWBS036-Numai
36
August 4, 2010
15:51
Printer Name: Yet to Come
OPTICAL TRANSITIONS
FIGURE 2.7 Power optical gain coefficient under the k-selection rule.
We assume that the energy states related to optical transitions have a Lorentzian spectrum with a width of /τin : L(E 21 ) =
/τin 1 , π (E 21 − E)2 + (/τin )2
(2.38)
where τin is the relaxation time due to electron scatterings and transitions. In this case, the power optical gain coefficient g(E) is expressed as [1] e2 h g(E) = 2m 2 ε0 n r cE
∞
1| p|22 ρred (E 21 ) [ f 2 (E 21 ) − f 1 (E 21 )] L(E 21 ) dE 21 ,
0
(2.39) which is plotted in Fig. 2.7 for bulk structures with τin = 10−13 s. It is found that relaxation is equivalent to band tailing, which contributes to optical transitions in E < E g . With an increase in the carrier concentration n, the gain peak shifts toward a higher energy. This shift in the gain peak is caused by the band filling effect [2], in which the electrons in the conduction band and the holes in the valence band occupy each band from the band edges. Because higher-energy states are more dense than lower-energy states, the optical gain in the higher-energy states is larger than that in the lower-energy states. From (2.10) and (2.24), the spontaneous emission rate r21 (spon) is expressed as 4π e2 n r E r21 (spon) = 2 2 3 m ε0 h c
∞
1| p|22 ρc (E )ρv (E ) f 2 (E )[1 − f 1 (E )] dE . (2.40)
0
Because f 2 and [1 − f 1 ] are always positive, spontaneously emitted light distributes at a higher energy level than that of the gain peak. Figure 2.8 shows optical power for stimulated emission (laser light) and spontaneous emission as a function of the photon energy.
P1: OSO c02 JWBS036-Numai
August 4, 2010
15:51
Printer Name: Yet to Come
Optical Power
REFERENCES
37
Stimulated Emission Spontaneous Emission
Photon Energy FIGURE 2.8 Stimulated and spontaneous emission.
REFERENCES 1. T. Numai, Fundamentals of Semiconductor Lasers, Springer-Verlag, New York, 2004. 2. J. I. Pankove, Optical Processes in Semiconductors, Dover, New York, 1975. 3. H. C. Casey, Jr., and M. B. Panish, Heterostructure Lasers A, Academic Press, San Diego, CA, 1978. 4. L. I. Schiff, Quantum Mechanics, 3rd ed., McGraw-Hill, New York, 1968.
P1: OSO c03 JWBS036-Numai
August 13, 2010
14:35
Printer Name: Yet to Come
3 OPTICAL WAVEGUIDES
3.1
INTRODUCTION
Free Space and Optical Waveguides An optical beam that propagates in free space expands its beam width due to diffraction. For example, if a light beam with a wavelength of 1 µm, which is emitted from a slit with a diameter of 1 mm, propagates by 100 m, its beam width widens to about 10 cm. In contrast, optical waveguides confine light to themselves during the propagation of light [1–6] and contribute to efficient amplification of light for LDs. In typical double heterostructure LDs with bulk active layers, the thickness of the active layers is on the order of 0.1 µm, the beam width is 0.3 µm, and the cavity length is 300 µm. If there are no optical waveguides, the beam width expands up to 500 µm by a single-way propagation, where a wavelength of light is assumed to be 0.5 µm. In LDs, only the active layer has an optical gain, and it is extremely inefficient to amplify a 500 µm-wide laser beam by a 0.1 µm-thick active layer. Therefore, to amplify light efficiently, optical waveguides are indispensable for LDs. Index and Gain Guiding Figure 3.1 shows schematic cross-sectional views of optical waveguides. The operating principles of the optical waveguides are divided into index guiding and gain guiding. In index guiding, light is confined in a shaded region with a high refractive index n f , surrounded by regions with low refractive indexes n c and n s , as shown in Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright
39
P1: OSO c03 JWBS036-Numai
40
August 13, 2010
14:35
Printer Name: Yet to Come
OPTICAL WAVEGUIDES
g
g
g
g g g
(a)
(b)
FIGURE 3.1 Cross sections of optical waveguides: (a) index guiding; (b) gain guiding.
Fig. 3.1(a). In gain guiding, light is confined in a shaded region with a high optical gain coefficient gf , surrounded by regions with low optical gain coefficients gc and gs , as shown in Fig. 3.1(b). Complex Refractive Index Using the complex refractive index n˜ = n r − i κ,
(3.1)
where n r is the refractive index and κ is the extinction coefficient, the field optical gain coefficient gE is expressed as ω gE = − κ. c
(3.2)
Note that the power optical gain coefficient g is equal to 2gE , because the optical power is proportional to the square of the electric field. From (3.1) and (3.2) it can be said that index guiding and gain guiding use the distribution of the refractive index n r and the extinction coefficient κ, respectively. Two- and Three-Dimensional Optical Waveguides From the viewpoint of shape, optical waveguides are classified into a twodimensional optical waveguide (planar optical waveguide) and a threedimensional optical waveguide (strip optical waveguide). A two-dimensional waveguide has a plane much larger than that of a wavelength of light and confines light one-dimensionally, as shown in Fig. 3.2(a). The size of its confinement direction is on the order of only a light wavelength or less. A three-dimensional waveguide confines light two-dimensionally, and the sizes along these two confinement directions are on the order of a light wavelength or less, as shown in Fig. 3.2(b). The remaining direction is the propagation direction of light.
P1: OSO c03 JWBS036-Numai
August 13, 2010
14:35
Printer Name: Yet to Come
TWO-DIMENSIONAL OPTICAL WAVEGUIDES
Propagation Direction
41
Propagation Direction
(a)
(b)
FIGURE 3.2 (a) Two- and (b) three-dimensional optical waveguides.
3.2
TWO-DIMENSIONAL OPTICAL WAVEGUIDES
Three-Layer Waveguides Figure 3.3 shows a two-dimensional waveguide where film is sandwiched between the cladding layer and the substrate. The refractive indexes of the film, the cladding layer, and the substrate are n f , n c , and n s , respectively. To confine light to the film, we need n f > n s ≥ n c , and n f − n s is usually on the order of 10−3 to 10−1 to confine only a fundamental mode. Snell’s Law Figure 3.4 shows propagation directions of light when light enters from the substrate to the cladding layer through the film. From Snell’s law, the angles θs , θf , and θc , which are formed by the interface normals and the directions of the light, and the refractive indexes n s , n f , and n c have the relation n s sin θs = n f sin θf = n c sin θc .
(3.3)
Critical Angle When θs is equal to π/2, light cannot propagate in the substrate, and the total reflection, with a power reflectivity of 100%, takes place at the interface of the
Cladding Layer Film Substrate FIGURE 3.3 Cross section of a two-dimensional optical waveguide.
P1: OSO c03 JWBS036-Numai
42
August 13, 2010
14:35
Printer Name: Yet to Come
OPTICAL WAVEGUIDES
FIGURE 3.4 Snell’s law.
substrate and the film. When θc is equal to π/2, the total reflection takes place at the interface of the cladding layer and the film. The minimum value of θf to obtain the total reflection is called a critical angle. According to θs = π/2 and θc = π/2 we have two critical angles, θfs and θfc , which are expressed as θfs = sin−1
ns nc , θfc = sin−1 . nf nf
(3.4)
Here θfs ≥ θfc is satisfied under the assumption of n f > n s ≥ n c . Propagation Modes Corresponding to a value of θf , there are three propagation modes, as shown in Fig. 3.5. (a) Radiation mode (θf < θfc ≤ θfs ). The light is not confined in the optical waveguide at all. (b) Substrate radiation mode (θfc < θf < θfs ). Total reflection occurs at the interface of the cladding layer and the film, while refraction takes place at the interface of the film and the substrate. As a result, a fraction of light is radiated to the substrate. (c) Guided mode (θfs < θf < π/2). At both interfaces, total reflections occur. Light is completely confined in the film during the propagation of light.
(a)
(b)
(c)
FIGURE 3.5 Propagation modes: (a) radiation mode; (b) substrate radiation mode; (c) guided mode.
P1: OSO c03 JWBS036-Numai
August 13, 2010
14:35
Printer Name: Yet to Come
TWO-DIMENSIONAL OPTICAL WAVEGUIDES
43
The guided mode shown in Fig. 3.5(c) is indispensable for LDs and photonic integrated circuits. Note that the guided mode does not exist in Fig. 3.5(a) and (b). We now examine the guided mode in detail.
TE and TM Modes In the following we consider a plane wave and define the plane of incidence as a plane on which all the directions of incident light, reflected light, and refracted light coexist. The transverse electric (TE) mode represents linearly polarized light whose electric field E is normal to the plane of incidence. The transverse magnetic (TM) mode represents linearly polarized light whose magnetic field H is normal to the plane of incidence. Figure 3.6 shows the TE and TM modes where the plane of incidence is parallel to the surface of the page. Note that only incident and reflected light are illustrated. From the Fresnel formulas, the amplitude reflectivities are given by
rTE,c =
rTE,s =
rTM,c
n f cos θf − n f cos θf + n f cos θf − n f cos θf +
n c 2 − n f 2 sin2 θf n c 2 − n f 2 sin2 θf
, (3.5)
n s 2 − n f 2 sin2 θf n s 2 − n f 2 sin2 θf
,
n c 2 cos θf − n f n c 2 − n f 2 sin2 θf = , n c 2 cos θf + n f n c 2 − n f 2 sin2 θf
rTM,s =
(a)
(3.6)
θf n s cos θf − n f n s − n f . 2 2 2 2 n s cos θf + n f n s − n f sin θf 2
2
2
sin2
(b)
FIGURE 3.6 (a) TE mode; (b) TM mode.
P1: OSO c03 JWBS036-Numai
44
August 13, 2010
14:35
Printer Name: Yet to Come
OPTICAL WAVEGUIDES
Here the subscripts TE and TM represent the TE and TM modes, respectively; and the subscripts c and s represent the reflection at the cladding layer–film interface and that at the substrate–film interface, respectively. Total Reflection When total reflections take place, |r |2 = 1 is satisfied, where r represents the amplitude reflectivities in (3.5) and (3.6), which are expressed by complex numbers. As a result, the amplitude reflectivity r can be written r = exp( i 2φ),
(3.7)
where φ is the phase shift, which is added to the reflected wave at the interface. Note that in the total reflection, the phase of the reflected wave is shifted at the interface of the reflection. When we rewrite (3.5) and (3.6) in the form of (3.7), the phase shifts φs are expressed as tan φTE,c
n f 2 sin2 θf − n c 2 = , n f cos θf
tan φTE,s
n f 2 sin2 θf − n s 2 = , n f cos θf
tan φTM,c
(3.8)
n f n f 2 sin2 θf − n c 2 = , n c 2 cos θf
tan φTM,s =
(3.9)
θf − n s nf nf , 2 n s cos θf 2
sin2
2
where the subscripts are the same as in (3.5) and (3.6). Monochromatic Coherent Plane Wave Let us consider a monochromatic coherent plane wave that propagates in an optical waveguide with a film thickness of h, as shown in Fig. 3.7. When a time-dependent factor is neglected, the propagating electric field E is expressed as E = E 0 exp[−i k0 n f (±x cos θf + z sin θf )],
(3.10)
where k0 = ω/c is a wave number in vacuum, and the ± sign preceding x represents the positive and negative propagation directions of light along the x-axis, respectively.
P1: OSO c03 JWBS036-Numai
August 13, 2010
14:35
Printer Name: Yet to Come
TWO-DIMENSIONAL OPTICAL WAVEGUIDES
45
FIGURE 3.7 Coordinate system for a guided mode.
FIGURE 3.8 Propagation constant.
As shown in Fig. 3.8, the propagation constant β and phase velocity vp along the z-axis are written β = k0 n f sin θf =
ω . vp
(3.11)
Phase Shifts If we see light from a coordinate system moving toward the positive z-axis direction with phase velocity vp , we observe that light is multireflected along the x-axis. The phase shifts during a round trip of the light along the x-axis are given by ⎧ 0 → h (the first half): ⎪ ⎪ ⎨ h (at the interface of the reflection): x= h → 0 (the second half): ⎪ ⎪ ⎩ 0 (at the interface of the reflection):
k0 n f h cos θf , −2φc , k0 n f h cos θf , −2φs .
Transverse Resonance Condition To obtain a lightwave propagating throughout the optical waveguide without decay, a total phase shift in the round trip must satisfy 2k0 n f h cos θf − 2φc − 2φs = 2 mπ
(m = 0, 1, 2, . . .),
(3.12)
where m is the mode number. Equation (3.12) shows a resonance condition along the x-axis which is normal to the z-axis and is called the transverse resonance condition.
P1: OSO c03 JWBS036-Numai
46
August 13, 2010
14:35
Printer Name: Yet to Come
OPTICAL WAVEGUIDES
In summary, the guiding condition is given by both the transverse resonance condition (3.12) and the total reflection condition θfs < θf < π/2.
Goos–H¨anchen Shift When a total reflection takes place, a phase of a lightwave shifts by −2φ at the interface of the reflection. In geometrical optics, this phase shift is considered equivalent to a shift in the optical path at the reflection interfaces, as shown in Fig. 3.9. This shift in the optical path Z , called the Goos–H¨anchen shift, is given by
Z=
dφ , dβ
(3.13)
where β is the propagation constant along the z-axis.
Evanescent Wave At each reflection interface, a lightwave propagates along the interface and has a component with a decay constant 1/ X along the x-axis, where X is called the penetration depth. This exponentially decaying wave represents the evanescent wave, and energy does not flow along the x-axis. Using (3.8) and (3.9), the Goos–H¨anchen shifts and penetration depths are expressed as k0 Z TE,c = (n f 2 sin2 θf − n c 2 )−1/2 · tan θf , k0 Z TE,s = (n f 2 sin2 θf − n s 2 )−1/2 · tan θf , Z TE,c Z TE,s , X TE,s = , X TE,c = tan θf tan θf
FIGURE 3.9 Goos–H¨anchen shift.
(3.14)
P1: OSO c03 JWBS036-Numai
August 13, 2010
14:35
Printer Name: Yet to Come
TWO-DIMENSIONAL OPTICAL WAVEGUIDES
k0 Z TM,s
−1 N2 N2 + − 1 , nc2 nf 2 2 −1 N N2 2 2 2 −1/2 = (n f sin θf − n s ) · tan θf + 2 −1 , ns2 nf Z TM,c Z TM,s , X TM,s = . X TM,c = tan θf tan θf
k0 Z TM,c = (n f 2 sin2 θf − n c 2 )−1/2 · tan θf
47
(3.15)
Effective Refractive Index Using a wave number k0 = ω/c in vacuum and the propagation constant β, we define the effective refractive index N as N≡
β = n f sin θf . k0
(3.16)
In LDs, the effective refractive index N is used to analyze the resonance conditions.
Optical Confinement Factor The optical confinement factor is defined as a ratio of the optical power of the light existing in the relevant layer to the total optical power. Because the light distributes as shown in Fig. 3.10, the optical confinement factor f for the film, which is shown as a hatched area, is given by
h
f = 0∞ −∞
|E(x)|2 dx . |E(x)|2 dx
Light Intensity
FIGURE 3.10 Distribution of optical power.
(3.17)
P1: OSO c03 JWBS036-Numai
48
August 13, 2010
14:35
Printer Name: Yet to Come
OPTICAL WAVEGUIDES
Similarly, the optical confinement factors c for the cladding layer and s for the substrate are written
∞
c = h ∞ −∞
|E(x)|2 dx |E(x)| dx 2
0
, s = −∞ ∞ −∞
|E(x)|2 dx .
(3.18)
|E(x)|2 dx
The optical confinement factor is important in the design of the optical losses or optical gains in optical waveguides.
Normalized Expressions for an Eigenvalue Equation The equation for the transverse resonance condition (3.12) is called an eigenvalue equation. Normalizing (3.12) gives us dispersion curves, which are common to all steplike two-dimensional optical waveguides. To obtain a normalized expression of an eigenvalue equation, the asymmetry measure a, normalized wave-guide refractive index b, and normalized frequency, or normalized waveguide thickness V , are introduced as follows: ns2 − nc2 , nf 2 − ns2 N 2 − ns2 , b= 2 nf − ns2 V = k0 h n f 2 − n s 2 . a=
(3.19) (3.20) (3.21)
Substituting (3.8), (3.9), and (3.19)–(3.21) into (3.12), the normalized eigenvalue equation is obtained as √ V 1 − b = mπ + tan−1 χs
b a+b + tan−1 χc , 1−b 1−b
(3.22)
where
χi =
1: (n f /n i )2 :
TE mode TM mode (i = s, c).
(3.23)
The guided modes for the gain guiding waveguides can also be calculated using (3.12) or (3.22) if we replace the refractive index n i (i = s, f, c) with the complex refractive index n˜ i .
P1: OSO c03 JWBS036-Numai
August 13, 2010
14:35
Printer Name: Yet to Come
Normalized Waveguide Index
TWO-DIMENSIONAL OPTICAL WAVEGUIDES
49
From the top
Normalized Frequency FIGURE 3.11 Normalized dispersion curves for guided TE modes.
Cutoff Figure 3.11 shows normalized dispersion curves for guided TE modes. If the refractive index and thickness of each layer are given, we can design optical waveguides using Fig. 3.11. To design optical waveguides, the cutoff condition, in which guided modes do not exist, is important. When the angle of incidence θf is equal to the critical angle θfs , the lightwave is no longer confined in the film, and a fraction of the optical power is radiated to the substrate. In this case, we have N = n s , which results in b = 0 from (3.20). Hence, from (3.22), a normalized frequency for the cutoff condition Vm is given by √ Vm = mπ + tan−1 χc a.
(3.24)
For Vm < V < Vm+1 , guided modes from zeroth to mth order exist as shown in Fig. 3.11, and a lower-order mode has a larger b for a common V value. Therefore, from (3.20), a lower-order mode has a larger effective refractive index, and the fundamental mode (m = 0) has the largest N among the guided modes. Figure 3.12 shows electric fields for fundamental (m = 0), first-order (m = 1), and second-order (m = 2) TE modes, in which the mth-order TE mode is indicated by TEm .
P1: OSO c03 JWBS036-Numai
50
August 13, 2010
14:35
Printer Name: Yet to Come
OPTICAL WAVEGUIDES
FIGURE 3.12 Distributions of electric fields.
Multiple-Layer Waveguide To analyze multiple-layer optical waveguides with more than three layers, as shown in Fig. 3.13, it is convenient to use matrices. When light propagates along the z-axis and the layers are stacked along the x-axis, the relationship between the light at x and xi is expressed by a matrix G, defined as
ψ y (x) ψ y (xi ) = G(x − xi ) . ψz (x) ψz (xi )
(3.25)
...
Here an electric field E and a magnetic field H of the light are assumed to be expressed by a separation-of-variables procedure. As a result, we have ψ y (x) = E y (x), ψz (x) = iZ 0 Hz (x) for the TE mode and ψ y (x) = Z 0 Hy (x), ψz (x) = −iE z (x) for the TM√mode, where a subscript indicates a component along each coordinate and Z 0 = µ0 /ε0 is the impedance of vacuum.
FIGURE 3.13 Optical waveguide with multilayers.
P1: OSO c03 JWBS036-Numai
August 13, 2010
14:35
Printer Name: Yet to Come
TWO-DIMENSIONAL OPTICAL WAVEGUIDES
51
We introduce parameters such as
κi = k0 2 n i 2 − β 2 ,
γi = β 2 − k0 2 n i 2 , ζi =
1: ni 2:
TE mode TM mode
(3.26) (3.27)
(3.28)
where k0 is the wave number of the light in vacuum, n i is the (complex) refractive index of the ith layer, and β is the propagation constant, which is calculated by solving the eigenvalue equation described below. For the ith layer with β < k0 n i , matrix G is written ⎡
⎤ k0 ζi sin κi x ⎥ ⎢ cos κi x κi G(x) = ⎣ κi ⎦. − sin κi x cos κi x k0 ζi
(3.29)
For the ith layer with β > k0 n i , matrix G is written ⎡
⎤ k0 ζi cos γ x sin γ x i i ⎥ ⎢ γi G(x) = ⎣ γi ⎦. sin γi x cos γi x k0 ζi
(3.30)
Because both ψ y (x) and ψz (x) are continuous at the boundaries, ψ y (x0 ), ψz (x0 ), ψ y (x N ), and ψz (x N ) have the relation
N −1 ψ y (x0 ) ψ y (x N ) A B ψ y (x N ) G(−di ) = = ψz (x0 ) ψz (x N ) ψz (x N ) C D
(3.31)
i
where di is the thickness of the ith layer. Because the propagation modes should have a field distribution of exp(γ0 x) in the region 0 and exp (−γ N x) in the region N , we have the eigenvalue equation k 0 2 ζ0 ζ N k 0 ζ0 k0 ζ N A−B− C+ D = 0. γN γ0 γ N γ0
(3.32)
By solving (3.32), we can obtain the propagation constant of the multilayer optical waveguide β and the field distribution.
P1: OSO c03 JWBS036-Numai
52
August 13, 2010
14:35
Printer Name: Yet to Come
OPTICAL WAVEGUIDES
Refractive Index
Horizontal Position
(a)
(b)
FIGURE 3.14 Antiguiding optical waveguides.
Antiguiding Figure 3.14 shows two examples of multilayer optical waveguides. The optical waveguide shown in Fig. 3.14(a) was proposed by Kawakami, and it was found that negative dispersion is obtained in an optical fiber [7]. In addition, the optical waveguide shown in Fig. 3.14(a) confines a fundamental mode and radiates higherorder transverse modes (antiguiding effect for higher-order transverse modes). As a result, it is expected that the kink level increases in LDs [8–10, 12]. The optical waveguide shown in Fig. 3.14(b) also confines a fundamental mode and has an antiguiding effect for higher-order transverse modes, which leads to a high kink level in LDs [11].
3.3 THREE-DIMENSIONAL OPTICAL WAVEGUIDES Confinement of Light Two-dimensional optical waveguides can confine light to an area sandwiched between two parallel planes. In contrast, three-dimensional optical waveguides confine light to an area surrounded by two axes other than the propagation direction. When the three-dimensional optical waveguides are adopted in LDs, the guided modes are efficiently amplified, which leads to low-threshold, high-efficiency laser operations. In an analysis of three-dimensional optical waveguides, however, we cannot obtain exact analytical solutions. As a result, approximate analytical methods or numerical analyses are used to design three-dimensional optical waveguides.
Approximate Analytical Methods As approximate analytical methods, the effective refractive index method and Marcatili’s method are well known. They can be applied to guided modes only for optical waveguides with an aspect ratio of a/ h > 1, where h is the film thickness and a is the width of the optical waveguide, under guiding conditions, which is far from cutoff conditions.
P1: OSO c03 JWBS036-Numai
August 13, 2010
14:35
Printer Name: Yet to Come
THREE-DIMENSIONAL OPTICAL WAVEGUIDES
53
Top View
Cross-Sectional View
FIGURE 3.15 Effective refractive index method.
Effective Refractive Index Method Figure 3.15 shows a ridge optical waveguide where the upper figure is a top view and the lower figure is a cross-sectional view. Here a is the width of the optical waveguide, h is the film thickness of the core region, and f is the film thickness of the surrounding regions. First, we separate the core and surrounding regions, and we regard each region as a two-dimensional optical waveguide. Therefore, the normalized frequency V and the normalized waveguide refractive index b for each region are written Vh = k0 h n f 2 − n s 2 , Vf = k0 f n f 2 − n s 2 , bh =
Nh − n s 2 Nf − n s 2 , bf = 2 , 2 2 nf − ns nf − ns2 2
(3.33)
2
(3.34)
where the subscripts h and f correspond to the core and surrounding regions, respectively; and Nh and Nf are the effective refractive indexes for each region. Second, we assume that film with a refractive index Nh and thickness a is sandwiched between layers with refractive index Nf , as shown in the top view of Fig. 3.15. As a result, the normalized frequency Vy of a three-dimensional optical waveguide is obtained as
Vy = k0 a Nh 2 − Nf 2 . (3.35) If we express the effective refractive index of a three-dimensional optical waveguide as Ns , the normalized waveguide refractive index bs of a three-dimensional optical waveguide is given by bs =
Ns 2 − Nf 2 . Nh 2 − Nf 2
(3.36)
Substituting Vy and bs into (3.22) leads to dispersion curves for a three-dimensional optical waveguide.
P1: OSO c03 JWBS036-Numai
54
August 13, 2010
14:35
Printer Name: Yet to Come
OPTICAL WAVEGUIDES
III, V,
I,
IV,
II,
FIGURE 3.16 Marcatili’s method.
Marcatili’s Method Figure 3.16 shows a cross-sectional view of a three-dimensional optical waveguide, where light propagates along the z-axis. If most guided modes are confined in region I, light field amplitudes decay drastically with increased distance from the interfaces. Therefore, the light distributed in the shaded areas in Fig. 3.16 can be neglected, which is referred to as Marcatili’s method [13].
REFERENCES 1. T. Tamir, ed., Integrated Optics, 2nd ed., Springer-Verlag, Berlin, 1979. 2. T. Tamir, ed., Guided-Wave Optoelectronics, 2nd ed., Springer-Verlag, Berlin, 1990. 3. R. G. Hunsperger, Integrated Optics: Theory and Technology, 3rd ed., Springer-Verlag, Berlin, 1991. 4. D. Marcuse, Theory of Dielectric Waveguides, 2nd ed., Academic Press, San Diego, CA, 1991. 5. K. J. Ebeling, Integrated Optoelectronics, Springer-Verlag, Berlin, 1992. 6. T. Numai, Fundamentals of Semiconductor Lasers, Springer-Verlag, New York, 2004. 7. S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a lowindex inner cladding,” IEEE J. Quantum Electron. 10, 879 (1974). 8. N. Shomura, M. Fujimoto, and T. Numai, “Fiber pump semiconductor lasers with optical antiguiding layers for horizontal transverse modes,” IEEE J. Quantum Electron. 44, 819 (2008). 9. N. Shomura, M. Fujimoto, and T. Numai, “Fiber-pump semiconductor lasers with optical antiguiding layers for horizontal transverse modes: dependence on mesa width,” Jpn. J. Appl. Phys. 48, 042103 (2009). 10. N. Shomura and T. Numai, “Ridge-type semiconductor lasers with optical antiguiding layers for horizontal transverse modes: dependence on step positions,” Jpn. J. Appl. Phys. 48, 042104 (2009).
P1: OSO c03 JWBS036-Numai
August 13, 2010
14:35
Printer Name: Yet to Come
REFERENCES
55
11. H. Takada and T. Numai, “Ridge-type semiconductor lasers with antiguiding cladding layers for horizontal transverse modes,” IEEE J. Quantum Electron. 45, 917 (2009). 12. H. Yoshida and T. Numai, “Ridge-type semiconductor lasers with antiguiding layers for horizontal transverse modes: dependence on space in the antiguiding layers,” Jpn. J. Appl. Phys. 48, 082105 (2009). 13. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071 (1969).
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
4 OPTICAL RESONATORS
4.1
INTRODUCTION
As shown in Fig. 4.1, lasers use a fraction of spontaneously emitted light as input and amplify the fraction by stimulated emission. To feed back the light, optical resonators, or optical cavities, which consist of reflectors, are adopted. Threshold gain and oscillation wavelength are determined by the resonance conditions of the optical resonators. Optical resonators are divided into three groups, whose constituent components are mirrors, diffraction gratings, and periodic multilayers. Figure 4.2 shows a Fabry–Perot cavity, a ring cavity with mirrors, and a ring cavity with a circular waveguide. In the Fabry–Perot cavity, which consists of two parallel mirrors, light propagates backward and forward between the two mirrors. In the ring cavity, which has more than two mirrors or a circular waveguide, light propagates clockwise or counterclockwise. In the circular waveguide shown in Fig. 4.2(c), the waveguide–air interface functions as a mirror, as explained in Chapter 3. Figure 4.3 shows distributed feedback (DFB) and a distributed Bragg reflector (DBR). The DFB has an active layer, which generates light, and an optical gain in its corrugated region, which functions by itself as an optical resonator. In contrast, the DBR does not have an active layer in its corrugated region and functions as a reflector, not as a resonator. Therefore, the DBR is combined with other DBRs or cleaved facets to form optical resonators. Note that both DFBs and DBRs use periodic modulation of the refractive index to reflect light.
Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright
57
P1: OSO c04 JWBS036-Numai
58
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
Optical Feedback by Reflectors Fraction of Spontaneous Emission
Laser Light
Optical Gain FIGURE 4.1 Laser.
Mirror Laser Light
Mirror
Mirror
Mirror
Laser Light
Laser Light
Circular Waveguide
Laser Light Mirror
(a)
(b)
(c)
FIGURE 4.2 Optical resonators with mirrors: (a) Fabry–Perot cavity with parallel mirrors; (b) ring cavity with mirrors; (c) ring cavity with a circular waveguide.
Active Layer
Laser Beam
Active Layer
Laser Beam
Laser Beam
Laser Beam
(a)
(b) FIGURE 4.3 (a) DFB; (b) DBR.
DBR Active Layer DBR Laser Beam FIGURE 4.4 Vertical cavity.
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
FABRY–PEROT CAVITY
Cleaved Facet
Laser Beam
59
Active Layer
Laser Beam
FIGURE 4.5 Fabry–Perot LD.
FIGURE 4.6 Analytical model for a Fabry–Perot cavity.
Figure 4.4 shows a vertical cavity, which consists of periodic multilayers and is used as a surface-emitting laser. Each set of multilayers is called DBR, because the operating principle of the periodic multilayers is common to that of diffraction grating without optical gain.
4.2
FABRY–PEROT CAVITY
Cleaved Facets In Fabry–Perot LDs, cleaved facets are used as mirrors, as illustrated in Fig. 4.5, because light is reflected at the semiconductor–air interface due to a difference in the refractive indexes. Cleaved facets are very flat, on the order of atomic layers, and they are much smoother than light wavelengths. Cleaved facets are often coated with dielectric films to control the reflectivities or to prevent the facets from being oxidized. Transmission Characteristics We assume that the amplitude reflectivities of the two mirrors are r1 and r2 and the amplitude transmissivities of the two mirrors are t1 and t2 , as shown in Fig. 4.6. The reflectivities and transmissivities depend on the angle of incidence and the polarization of light. Here the angle of incidence θ0 is supposed to be small enough, and r1 , r2 , t1 , and t2 are regarded as constant.
P1: OSO c04 JWBS036-Numai
60
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
If a time-dependent factor is neglected, the incident electric field E i through the surface at z = 0 is written in ω rt (x sin θ + z cos θ ) × exp [gE (x sin θ + z cos θ )] . E i = t1 E 0 exp − c
(4.1)
Here E 0 is an amplitude of the electric field of incident light, n rt is a real part of the complex refractive index of material placed between the two mirrors, ω is an angular frequency of the light, c is the speed of light in vacuum, and gE is a field optical gain coefficient. The electric field E t transmitted through the Fabry–Perot cavity is expressed as gE L i n rt ω (x sin θ + L cos θ ) + E t = t1 t2 E 0 exp − c cos θ 2gE L 2gE L + (r1r2 )2 exp 2 −i δ + × 1 + r1r2 exp −i δ + cos θ cos θ + ··· gE L i n rt ω (x sin θ + L cos θ ) + t1 t2 E 0 exp − c cos θ = , (4.2) 2gE L 1 − r1r2 exp −i δ + cos θ where δ=
2n rt ωL cos θ . c
(4.3)
The incident light intensity I0 and the transmitted light intensity It are related to the electric fields E 0 and E t as I0 ∝ E 0∗ E 0 , It ∝ E t∗ E t .
(4.4)
Therefore, from (4.2), we obtain It =
(t1 t2 )2 G s I0 , 1 + (r1r2 )2 G s 2 − 2r1r2 G s cos δ
(4.5)
where
2gE L G s = exp cos θ
.
(4.6)
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
FABRY–PEROT CAVITY
61
Because the power reflectivities R1 and R2 and the power transmissivities T1 and T2 are given by R1 = r1∗r1 , R2 = r2∗r2 , T1 = t1∗ t1 , T2 = t2∗ t2 ,
(4.7)
we can rewrite (4.5) as It =
T1 T2 G s I0 . √ 1 + G s R1 R2 − 2G s R1 R2 cos δ 2
(4.8)
Using the angle θf in the optical waveguide as shown in Fig. 3.7, the angle θ is written as θ = π/2 − θf . As a result, (4.3) becomes δ=
2n r ωL ω = 2n r k0 L , k0 = , c c
(4.9)
where cos θ = sin θf and the effective refractive index N = n rt sin θf ≡ n r are used. When the angle of incidence θ0 is small, as in a fundamental mode, (4.8) leads to It =
T1 T2 G s0 I0 , √ √ (1 − G s0 R1 R2 )2 + 4G s0 R1 R2 sin2 (n r k0 L)
(4.10)
where G s0 = exp (2gE L) = exp(gL).
(4.11)
Here (4.9) is used and g is the power optical gain coefficient. Figure 4.7 shows It /I0 as a function of n r k0 L, where G s0 = 1, and R1 = R2 = R, T1 = T2 = T = 0.98 − R with an optical power loss of 2% at each mirror. The transmissivities have maximum values at n r k0 L = nπ and minimum values at (n + 1/2)π , where n is a positive integer. With an increase in R, the transmission spectra narrow and the transmissivities decrease. If the optical power loss at the mirror is null, we have T = 1 − R, which results in a maximum power transmissivity of 1 (100%), irrespective of R. Resonance Condition The resonance condition is the condition at which the power transmissivity has a peak. From (4.10), the resonance condition is given by n r k0 L =
n r ωL n rt ωL = cos θ = nπ, c c
(4.12)
where n is a positive integer. From (4.12) it is found that the effective refractive index n r is useful to express the resonance condition. At normal incidence with
P1: OSO c04 JWBS036-Numai
62
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
Free Spectral Range
FIGURE 4.7 Transmission characteristics for a Fabry–Perot cavity.
θ0 = θ = 0, the resonance condition is written n rt ωL = n rt k0 L = nπ. c
(4.13)
Using a wavelength in vacuum λ0 , (4.13) results in L=n
λ0 . 2n rt
(4.14)
Because λ0 /n rt is a wavelength in a material, a product of a positive integer and a half-wavelength in a material is equal to the cavity length L at the resonance condition. Free Spectral Range A Fabry–Perot cavity is used as an optical filter because its transmissivity depends on a light wavelength. However, two light sources whose n r k0 L value is different by nπ , in which n is a positive integer, cannot be resolved by a Fabry–Perot cavity because there is a common transmissivity for the two lights. When (n − 1/2)π < n r k0 L < (n + 1/2)π is satisfied, the light is resolved, and the frequency or wavelength of the light becomes single-valued. This region of n r k0 L is called the free spectral range, because the light is resolved free from other light. The free spectral range in an angular frequency ωFSR is given by ωFSR =
c π. nr L
(4.15)
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
FABRY–PEROT CAVITY
63
Using the free spectral range λFSR in a wavelength, ωFSR is written as
ωFSR
1 1 = 2π c − λ0 λ0 + λFSR
.
(4.16)
As a result, we have λFSR
λ0 2 λ0 2 = ωFSR . 2n r L 2π c
(4.17)
Spectral Linewidth Half width at half maximum (HWHM) and full width at half maximum (FWHM) are often used as spectral linewidths. HWHM is the difference between the wavelength or frequency for maximum transmissivity and that for half of the maximum transmissivity. FWHM is the difference between two wavelengths or frequencies for half of the maximum transmissivity and is twice as large as HWHM. When the resonance condition (4.13) is satisfied, the transmissivity takes a maximum value. As a result, in the resonance condition, the denominator in (4.10) results in (1 − G s0 R1 R2 )2 . When the transmissivity is half-maximum, the denominator in (4.10) is twice as large as that in the resonance condition, and we obtain (1 − G s0 R1 R2 )2 + 4G s0 R1 R2 sin2 (n r k0 L) = 2(1 − G s0 R1 R2 )2 ,
(4.18)
which leads to 1 sin(n r k0 L) = ± (1 − G s0 R1 R2 ). 4 2 2 G s0 R1 R2
(4.19)
Expressing k0 = km on resonance and k0 = km ± k0 at half-maximum, then substituting k0 = km ± k0 into (4.19), we have 1 (1 − G s0 R1 R2 ). sin(n r k0 L) = 4 2 2 G s0 R1 R2
(4.20)
When n r k0 L 1 is satisfied, (4.20) results in k0 =
√ 1 − G s0 R1 R2 . 4 2n r L G s0 2 R1 R2
(4.21)
P1: OSO c04 JWBS036-Numai
64
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
Using k0 = ω/c, HWHM in an angular frequency ωH is expressed as √ c(1 − G s0 R1 R2 ) . ωH = ck0 = 4 2n r L G s0 2 R1 R2
(4.22)
Accordingly, FWHM ωF , which is twice HWHM, is given by ωF = 2ωH =
√ c(1 − G s0 R1 R2 ) . 4 n r L G s0 2 R1 R2
(4.23)
Using a wavelength in vacuum, FWHM in a wavelength λF is written √ λ0 2 λ0 2 (1 − G s0 R1 R2 ) ωF = . λF 4 2π c 2π n r L G s0 2 R1 R2
(4.24)
Note that the spectral linewidth narrows with an increase in the optical gain below the threshold of laser oscillation. Electric Field Inside a Fabry–Perot Cavity When the amplitude reflectivities are r1 = r2 = r and the amplitude transmissivities are t1 = t2 = t, the electric field E inside a Fabry–Perot cavity is written E = t E 0 exp(keff x sin θ ) {exp(keff z cos θ ) + r exp[keff (2L − z) cos θ ]}
× 1 + r 2 exp(2keff L cos θ ) + r 4 exp(4keff L cos θ ) + · · · =
t E 0 exp(keff x sin θ ) {exp(keff z cos θ ) + r exp[keff (2L − z) cos θ ]} , 1 − r 2 exp(2keff L cos θ )
(4.25)
where keff = −
i n rt ω + gE . c
(4.26)
In (4.25), exp(keff z cos θ ) represents a forward running wave along the z-axis, and exp[keff (2L − z) cos θ ] expresses a backward running wave along the z-axis after reflection at a plane with z = L. If we introduce a = r exp(2keff L cos θ ), (4.25) is reduced to E = t E 0 exp(keff x sin θ )
(1 − a) exp(keff z cos θ ) + 2a cosh(keff z cos θ ) . 1 − r 2 exp(2keff L cos θ )
(4.27)
In (4.27), the first term in the numerator represents a forward running wave, and the second term in the numerator shows a standing wave. From (4.25)–(4.27) it is found
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
WAVEGUIDE GRATING
65
that the light intensity inside a Fabry–Perot cavity takes a maximum value only when n rt ωL n r ωL cos θ = = n r k0 L = nπ. c c
(4.28)
Here n is a positive integer which shows the number of nodes existing between z = 0 and z = L for a standing wave. From (4.12) and (4.28) it is revealed that both the transmissivity and the internal light intensity of a Fabry–Perot cavity have the largest values at the resonance condition.
4.3
WAVEGUIDE GRATING
The DFB and DBR are optical waveguides which have diffraction gratings. They feed light back by spatially modulating the complex refractive indexes of optical waveguides. The difference between a DFB and a DBR is that the former has optical gain in the corrugated region and the latter does not. As described earlier, a DFB functions as an optical resonator, and a DBR forms an optical resonator with other DBRs or cleaved facets. Coupled Wave Theory When a lightwave is assumed to propagate along the z-axis, a propagation constant k in the optical waveguide is given by k = ω µ ε˜ = ω µ (εr − i εi ) k0 n r (z) 2
2
2
2
2
2α(z) 1+i , k0 n r (z)
(4.29)
where α(z) 2 ε˜ = εr − i εi = ε0 n r (z) + i . k0
(4.30)
Here ω is an angular frequency of the light, µ is the permeability of a material, ε˜ is a complex dielectric constant, k0 = ω/c = 2π/λ0 is a wave number in vacuum, λ0 is a wavelength of the light in vacuum, ε0 = 8.854 × 10−12 F/m is the permittivity of vacuum, n r (z) is a real part of a complex refractive index, and α(z) = gE is a field optical gain coefficient. For the usual optical materials, µ is almost equal to the permeability of vacuum, µ0 = 4π × 10−7 H/m. Because of |α(z)| k0 , a second-order term of α(z) is neglected in (4.29). Figure 4.8 shows a schematic cross-sectional view of diffraction grating. The effective refractive index of an optical waveguide is periodically modulated with a grating pitch of by the corrugations formed at the interface between the two layers with refractive indexes n A and n B (n A = n B ).
P1: OSO c04 JWBS036-Numai
66
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
FIGURE 4.8 Diffraction grating.
We assume that n r (z) and α(z) are sinusoidal functions of z: n r (z) = n r0 + n r1 cos(2β0 z + ), α(z) = α0 + α1 cos(2β0 z + ).
(4.31)
Here is a phase at z = 0 and β0 is given by β0 =
π ,
(4.32)
where is the grating pitch. Under the assumption that n r1 n r0 and α1 α0 , substituting (4.31) into (4.29) leads to k(z) = k0 n r0 + i 2k0 n r0 α0 + 4k0 n r0 2
2
2
π n r1 α1 +i λ0 2
cos(2β0 z + ).
(4.33)
When the refractive index is uniform (n r1 = 0) and the material is transparent (α0 = α1 = 0), the propagation constant k(z) in (4.33) is reduced to k(z) = β = k0 n r0 .
(4.34)
In optical waveguides with corrugations, a forward running wave and a backward running wave are coupled, due to reflections. To express this coupling, the coupling coefficient κ of the diffraction gratings is defined as κ=
π n r1 α1 +i , λ0 2
(4.35)
which is important in describing the resonance characteristics of DFBs and DBRs. Using (4.34) and (4.35), (4.33) is reduced to k(z)2 = β 2 + i 2βα0 + 4βκ cos(2β0 z + ).
(4.36)
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
WAVEGUIDE GRATING
67
A wave equation for the electric field E is given by d2 E + k(z)2 E = 0. dz 2
(4.37)
Substituting (4.36) into (4.37) gives us d2 E 2 + β + i 2βα0 + 4βκ cos(2β0 z + ) E = 0. dz 2
(4.38)
Coupled Wave Equation The electric field E(z), which is a solution of (4.38), is represented by superposition of a forward running field Er (z) and a backward running field E s (z), such as E(z) = Er (z) + E s (z), Er (z) = R(z) exp(−i β0 z),
(4.39)
E s (z) = S(z) exp( i β0 z ), where R(z) and S(z) are the field amplitudes of the forward and backward running waves, respectively [1]. Note that both R(z) and S(z) are functions of z. Inserting (4.39) into (4.38) gives the wave equations for R and S: −
dR + (α0 − i δ)R = i κ S exp(−i ), dz
(4.40)
dS + (α0 − i δ)S = i κ R exp( i ), dz where δ is defined as δ≡
β 2 − β0 2 β − β0 . 2β0
(4.41)
Here R and S were assumed to be slowly varying functions of z, and the second derivatives with respect to z are neglected. Because the forward running wave R and the backward running wave S are coupled by the coupling coefficient κ, (4.40) is called a coupled wave equation, and a theory based on (4.40) is referred to as coupled wave theory [2]. Bragg Wavelength The Bragg wavelength in vacuum λB , which satisfies δ = 0, is defined as λB =
2n r0 , m
(4.42)
P1: OSO c04 JWBS036-Numai
68
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
where (4.32), (4.34), and (4.41) are used and m is a positive integer called the order of diffraction. Transfer Matrix Using constants a and b, which are determined by the boundary conditions, general solutions of (4.38) are given by Er (z) = [a exp(γ z) + ρ exp(−i ) · b exp(−γ z)] exp(−i β0 z), E s (z) = [ρ exp( i ) · a exp( γ z) + b exp(−γ z)] exp( i β0 z),
(4.43)
where γ 2 = (α0 − i δ)2 + κ 2 , −γ + (α0 − i δ) ρ= . iκ
(4.44)
It is useful to introduce a transfer matrix F i [3], which is defined as
Er (0) Er (L i ) = Fi . E s (0) E s (L i )
(4.45)
Here L i is the length of a corrugated region and F i is written Fi =
F11 F21
F12 , F22
(4.46)
where α0 − i δ sinh (γ L i ) exp ( i β0 L i ), = cosh (γ L i ) − γ iκ sinh (γ L i ) exp [−i (β0 L i + )], = γ iκ sinh (γ L i ) exp [ i (β0 L i + )], =− γ α0 − i δ sinh (γ L i ) exp (−i β0 L i ). = cosh (γ L i ) + γ
F11 F12 F21 F22
(4.47)
When multiple regions are connected in series, as shown in Fig. 4.9, the total transfer matrix F is given by a product of transfer matrices in all regions, such as F=
Fi .
(4.48)
i
If the power reflectivities of both facets are R1 and R2 without optical loss, the electric fields at the facets in the air, Er (0), E s (0), Er (L), and E s (L), and the electric
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
69
WAVEGUIDE GRATING
Air
Semiconductor
Air
FIGURE 4.9 Analytical model for a diffraction grating.
fields at the facets inside the semiconductor, Er i (0), E si (0), Er i (L), and E si (L), are related as 1 − R1 Er (0) + R1 E si (0), E s (0) = R1 exp(iπ )Er (0) + 1 − R1 E si (0) = − R1 Er (0) + 1 − R1 E si (0), Er (L) = 1 − R2 Er i (L) + R2 exp(iπ )E s (L) = 1 − R2 Er i (L) − R2 E s (L), E si (L) = 1 − R2 E s (L) + R2 Er i (L). Er i (0) =
(4.49)
(4.50)
(4.51) (4.52)
Here the phase shift of π at the reflection of light is considered when the light is incident on a semiconductor from the air, because the refractive index of the semiconductor is larger than that of the air. From(4.49) and (4.50) we have
√ 1 Er (0) 1 − R1 Er i (0) √ =√ . E s (0) E si (0) 1 − R1 − R1 1
(4.53)
From(4.51) and (4.52) we have
√ 1 R2 Er i (L) Er (L) √1 =√ . E si (L) E s (L) R2 1 1 − R2
(4.54)
Combining (4.53) and (4.54) gives us the total transfer matrix F R as follows: √ √ 1 R2 1 − R1 1 √ √ FR = √ ×F× . R2 1 (1 − R1 )(1 − R2 ) − R1 1
(4.55)
When we assume that the corrugated region length is L and the input is Er (0) with E s (L) = 0, the output for the transmission is Er (L) and that for the reflection is E s (0). From the definition of a transfer matrix, the power transmissivity T and power
P1: OSO c04 JWBS036-Numai
70
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
FIGURE 4.10 Calculated transmission and reflection spectra of a diffraction grating.
reflectivity R are given by T =
1
, ∗ F11 F11
R=
∗ F21 F21 . ∗ F11 F11
(4.56)
Figure 4.10 shows calculated transmission and reflection spectra of a diffraction grating. The horizontal line is δ × L = δL and the vertical line is the power transmissivity T and power reflectivity R. Here it is assumed that the optical waveguide is transparent (α0 = α1 = 0), κ L = 2, and R1 = R2 = 0. A low-transmissivity (high-reflectivity) region that is symmetrical about δL = 0 is called a stopband. Application of a Transfer Matrix to a Fabry–Perot Cavity A Fabry–Perot cavity can also be analyzed by the total transfer matrix F R . Because a Fabry–Perot cavity does not have corrugations, κ = 0 is satisfied and (4.55) is simplified as 1 FR = √ (1 − R1 )(1 − R2 )
FR11 FR21
FR12 , FR22
(4.57)
where FR11 = e(−α+iβ)L − R1 R2 e(α−iβ)L , FR12 = R2 e(−α+iβ)L − R1 e(α−iβ)L , FR21 = − R1 e(−α+iβ)L + R2 e(α−iβ)L , FR22 = − R1 R2 e(−α+iβ)L + e(α−iβ)L .
(4.58)
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
WAVEGUIDE GRATING
71
The power transmissivity T is given by T =
1 ∗ FR11 FR11
=
(1 − R1 )(1 − R2 ) exp(2αL) . √ √ [1 − exp(2αL) R1 R2 ]2 + 4 exp(2αL) R1 R2 sin2 (β L) (4.59)
When exp(2αL) = G s0 , 1 − R1 = T1 , 1 − R2 = T2 , and β = n r k0 is satisfied, (4.59) agrees completely with It /I0 given by (4.10), which explains the power transmissivity of a Fabry–Perot cavity. Category of Diffraction Gratings From the viewpoint of the pitch and depth, the diffraction gratings are divided into four groups. A diffraction grating with uniform pitch and depth, which is shown in Fig. 4.11(a), is a uniform grating. Figure 4.11(b) shows a phase-shifted grating [4, 5], whose corrugations shift in its optical waveguide; this grating is especially important for longitudinal single-mode operations in DFB LDs. Figure 4.11(c) shows a tapered grating whose corrugation depth is spatially modulated along the propagation direction of light. Figure 4.11(d) shows a chirped grating whose pitch varies along the propagation direction. Phase-Shifted Grating Optical fibers have dispersion. As a result, if LDs operate with multiple longitudinal modes, optical pulses broaden in the time domain, and finally, adjacent optical pulses overlap with each other, which limits the transmission of signals. This overlap of adjacent pulses becomes serious with increases in the transmission distance and signal speed. Therefore, longitudinal single-mode LDs are required for long-haul, large-capacity optical fiber communication systems. To achieve stable single-mode operations, DFB LDs with phase-shifted gratings or gain-coupled gratings have been
(a)
(b)
(c)
(d)
FIGURE 4.11 Diffraction gratings used for DFBs and DBRs: (a) uniform; (b) phase-shifted; (c) tapered; (d) chirped.
P1: OSO c04 JWBS036-Numai
72
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
developed. In the following, phase-shifted gratings, which have been used commercially, are explained.
Transmission and Reflection Characteristics Figure 4.12 shows the structure of phase-shifted grating, in which the corrugation phase is shifted by − along the z-axis and the corrugations in uniform gratings are represented by a dashed line. It should be noted that the negative sign of the phase shift is based on a definition of the spatial distribution of the complex refractive index in (4.31). Figure 4.13 shows an analytical model in which the diffraction gratings consist of two regions, and the phase shift is introduced as a phase jump at the interface of the two regions. It is also assumed that both the pitch and depth are uniform in the two regions. When the phase at the left edge of region 1 is θ1 , the phase θ2 at the right edge of region 1 is given by θ2 = θ1 + 2β0 L 1 .
Phase Shift
Pitch
FIGURE 4.12 Phase-shifted grating.
Region 1
Region 2
FIGURE 4.13 Analytical model for phase-shifted grating.
(4.60)
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
73
Power Reflectivity
WAVEGUIDE GRATING
FIGURE 4.14 Reflection spectrum.
Due to the phase shift , the phase θ3 at the left edge of region 2 is obtained as θ3 = θ2 + = θ1 + 2β0 L 1 + .
(4.61)
By using (4.56), the transmission and reflection characteristics of the phase-shifted gratings are given by the transfer matrix F = F 1 × F 2 , where F 1 and F 2 are the transfer matrices of regions 1 and 2, respectively. Figure 4.14 shows calculated power reflectivity as a function of δ × L = δL for κ L = 2. Here it is assumed that a material is transparent and that the facet reflectivity is null. The solid and dashed lines represent phase-shifted grating with − = π and uniform grating with − = 0, respectively. Note that the phase-shifted grating has a passband within the stopband, and the phase-shifted DFB LDs oscillate at this transmission wavelength. The transmission wavelength depends on the value of the phase shift − . When the phase shift − is π , the transmission wavelength agrees with the Bragg wavelength, which is represented by δ = 0.
Comparison of Phase-Shifted Grating and Fabry–Perot Cavities Comparing phase-shifted grating with Fabry–Perot cavities, we can understand the physical meaning of the relationship between the transmission wavelength and the phase shift [6]. In addition, it is found that the operating principle of a vertical cavity is the same as that of phase-shifted grating. Figure 4.15 shows a Fabry–Perot cavity where the power reflectivity of the mirror is R0 and the cavity length is L. From (4.14), the resonance condition of the Fabry–Perot cavity is given by L=n
λ0 . 2n r
(4.62)
P1: OSO c04 JWBS036-Numai
74
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
Mirror
Mirror
FIGURE 4.15 Fabry-Perot cavity.
(b)
(a)
FIGURE 4.16 Resonance characteristics of a Fabry–Perot cavity: (a) power reflectivity R0 of a mirror; (b) power transmissivity T of a cavity.
Here n is a positive integer, n r is the effective refractive index of a material, and λ0 is a wavelength of light in vacuum. Figure 4.16 shows the power reflectivity R0 of a mirror and the power transmissivity T of a Fabry–Perot cavity as a function of wavelength λ. It should be noted that R0 is independent of λ if the dispersion is neglected, and sharp peaks are present for all wavelengths that satisfy the resonance condition (4.62). Figure 4.17 shows a phase-shifted grating with sawtoothed corrugations and one with rectangular corrugations. Note that the grating shapes affect the value of κ, but the concepts regarding the grating pitch are common to both structures. In the following, to focus solely on the grating pitch, we consider the rectangular shape. The phase shift − is defined as shown in Fig. 4.17. Such a phase-shifted grating can be regarded as a Fabry–Perot cavity with length L and two mirrors whose reflectivities depend on a wavelength. In Fig. 4.17(b), L and are related as | | | | 1+ , = + L= 2 2β0 2 π
(4.63)
where (4.32) is used. From (4.63) it is found that the cavity length L changes with the phase shift − , which varies a resonance (transmission) wavelength according to (4.14). When the phase-shift is π , (4.42) and (4.63) give
L= =m
λB , 2n r0
(4.64)
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
WAVEGUIDE GRATING
75
Phase Shift
(a) Phase Shift
Mirror
Mirror (b)
FIGURE 4.17 Phase-shifted gratings with (a) sawtoothed corrugations and (b) rectangular corrugations.
Stopband
Stopband
(a)
(b)
FIGURE 4.18 Resonance characteristics of phase-shifted grating: (a) power reflectivity R1 of a mirror; (b) power transmissivity T of a cavity.
where m is a positive integer called the order of diffraction, and n r0 , the average refractive index, is shown in (4.31). For first-order grating (m = 1), the resonant wavelength is the Bragg wavelength. Figure 4.18 shows the power reflectivity R1 of the mirror shown in Fig. 4.17(b) and the power transmissivity T of the phase-shifted grating with − = π as a function of a wavelength. The power reflectivity R1 of the mirror depends on a wavelength λ, and only the wavelength region within the stopband has high reflectivity. Therefore, only a resonant wavelength, which is located in the stopband, is selectively multireflected in the cavity. Also, because the cavity length L is on the order of a wavelength of light, the free spectral range is so large that only one transmission peak is present within the stopband. As described earlier, phase-shifted grating can be explained qualitatively as a Fabry–Perot cavity. The only difference between phase-shifted grating and a Fabry– Perot cavity is the dependence of the mirror reflectivities on a wavelength of light. In addition, from the viewpoint of distribution of the refractive index in the phase-shifted grating shown in Fig. 4.19, the operating principle of the phase-shifted grating is the same as that of the vertical cavity.
P1: OSO c04 JWBS036-Numai
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
Refractive Index
76
August 9, 2010
Position FIGURE 4.19 Distribution of a refractive index in phase-shifted grating.
Definition of the Phase Shift There are two definitions of the phase shift in a phase-shifted grating, as shown in Fig. 4.20 [7]. In Fig. 4.20(a), the phase shifts in region 1 in z < 0 and region 2 in z > 0 are symmetrical with respect to z = 0. In Fig. 4.20(b), the phase shift in region 1 is not present; only the phase shift in region 2 in z > 0 is present. Using the definition in Fig. 4.20(a), the refractive index n r (z) in region 1 is expressed as n r (z) = n r0 + n r1 cos(2β0 z + θ ),
(4.65)
whereas the refractive index in region 2 is written as n r (z) = n r0 + n r1 cos(2β0 z − θ ).
(4.66)
Using the definition in Fig. 4.20(b), the refractive index in region 2 is written n r (z) = n r0 + n r1 cos(2β0 z + θ + ),
Region 1
Region 2
(a)
(4.67)
Region 2
Region 1
(b) FIGURE 4.20 Definition of phase shift.
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
WAVEGUIDE GRATING
77
where (4.65) is used. Because (4.66) and (4.67) represent the same refractive index, the two phase shifts and θ are related as θ + = 2mπ − θ,
(4.68)
2θ + = 2mπ,
(4.69)
which is reduced to
where m is an integer. These two definitions of the phase shift are used in various journal articles and books, and we should be careful in discussing a value of the phase shift. Fabrication of Diffraction Gratings In DFB LDs and DBR LDs, oscillation wavelengths are at or in the vicinity of Bragg wavelengths. The oscillation wavelength for the light sources of optical fiber transmission systems is 1.3 µm, in which dispersion of non-dispersion-shifted optical fiber is the lowest, or 1.55 µm, in which the absorption loss of an optical fiber is the lowest. Because the effective refractive index n r0 of LDs is nearly 3.2, pitches of the first-order grating are approximately 0.2 µm for an oscillation wavelength of 1.3 µm and 0.24 µm for an oscillation wavelength of 1.55 µm from (4.42). The corrugation depth is about 0.1 µm = 100 nm just after grating fabrication and is reduced to several tens of nanometers after the epitaxial growth of semiconductor layers on diffraction grating, due to thermal decomposition of the grating surface during heating prior to epitaxial growth. To fabricate such fine diffraction gratings with high accuracy, holographic exposure [8–16], electron beam exposure [17], and x-ray exposure [18] systems have been developed. Holographic Exposure In holographic exposure, interference of two coherent laser beams is used to generate interference fringe patterns. The interference fringe patterns obtained are transferred to the photoresist coated on the substrate via developing the photoresists. Finally, the substrate is etched with a patterned photoresist as the etching mask. Figure 4.21 shows the principle of holographic exposure, where solid lines show wave fronts of two lightwaves and filled circles represent points at which the light intensity is enhanced due to interference. When wave fronts propagate toward the substrate, filled circles connected by dashed lines end in arrows. Consequently, the photoresist regions which are pointed to by the arrows are exposed to light, and the grating pitch is determined by the spacing of the dashed lines. Figure 4.22 shows the angles of incidence θ1 and θ2 , which are formed by the normals of the substrate plane and the propagation directions of the two laser beams.
P1: OSO c04 JWBS036-Numai
78
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
Pitch
Wave Front
Photoresist Substrate FIGURE 4.21 Principle of holographic exposure.
Propagation Direction of the Laser Beam Wavelength
Photoresist Substrate FIGURE 4.22 Angle of incidence in holographic exposure.
The grating pitch is given by =
λe , sin θ1 + sin θ2
(4.70)
where λe is a wavelength of the incident laser beams. Figure 4.23 shows a holographic exposure system. He-Cd lasers with λe = 441.6 nm, or 325 nm, or Ar ion lasers with λe = 488 nm, are widely used as light sources. A single laser beam emitted from the light source is divided into two laser beams by a beamsplitter, and these two laser beams are expanded by beam expanders. These expanded beams are collimated by collimating lenses and are finally incident on the photoresist coated on the substrate, which is represented as a sample in Fig. 4.23. Electron Beam Exposure In electron beam exposure, electron beams are scanned on the photoresist in vacuum, as shown in Fig. 4.24. Scanning the electron beams and transfer of samples are controlled by computers, and flexible patterns are easily fabricated. The problem in electron beam exposure is that the exposure time is very long.
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
WAVEGUIDE GRATING
Mirror
Shutter
He-Cd laser Ar ion laser
79
Mirror
Collimating Lens
Beamsplitter
Sample
Spatial Filter
Mirror Beam Expander
FIGURE 4.23 Holographic exposure system.
Electron Beam
Photoresist Substrate FIGURE 4.24 Electron beam exposure.
X-Ray Exposure In x-ray exposure, photoresist is irradiated by x-rays through a photomask, as shown in Fig. 4.25. Because wavelengths of the x-rays are short, diffraction angles are small. As a result, x-rays are suitable to transfer fine patterns of the photomask to the photoresist. To obtain sufficient x-ray intensity for exposure, synchrotron radiation, which needs a huge plant, is often used. In addition, it is difficult to obtain highly reliable photomasks for x-ray exposure. Fabrication of Phase-Shifted Grating Electron beam and x-ray exposure systems are suitable for fabricating various patterns. These systems have problems, however, such as high costs and low productivity. As a result, holographic exposure, which has high productivity with low costs, has attracted a lot of interest. The most stable single-longitudinal-mode laser operations are obtained in DFB LDs with a phase shift of | | = π , which corresponds to a shift in length /2 in first-order grating. Therefore, in first-order phase-shifted grating, the top and bottom of corrugations are reversed in an optical waveguide. From the viewpoint of the reverse of the corrugations, positive and negative photoresists are exposed simultaneously. By development, exposed areas are removed in a positive photoresist; unexposed areas are removed in a negative photoresist. Therefore, by selectively forming the positive and negative photoresists on the substrate,
P1: OSO c04 JWBS036-Numai
80
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
X-ray Photomask Photoresist Substrate FIGURE 4.25 X-ray exposure. Laser Beam
Laser Beam
Cyclized-Rubber Negative Photoresist
Novolak Negative Photoresist SiN
Novolak Positive Photoresist
(a)
Novolak Positive Photoresist
(b)
FIGURE 4.26 Holographic exposure using positive and negative photoresists.
a pattern and its reverse can be obtained on the same plane. Figure 4.26 shows examples of this method. In Fig. 4.26(a), novolak-type positive photoresist is formed selectively, and cyclized-rubber negative photoresist, which does not react chemically with novolak positive photoresist, is coated on the entire surface [8]. Novolak positive photoresist has high resolution, but cyclized-rubber negative photoresist has low resolution, owing to imbibition during developing. As a result, it is difficult to fabricate phase-shifted grating with fine patterns. In Fig. 4.26(b), both positive and negative novolak photoresists with high resolution are used, and SiN is inserted between novolak positive and negative photoresists to prevent chemical reactions between positive and negative photoresists [9]. In this method it is difficult to control etching time for SiN, and the production yield is low. From the viewpoint of the shift in the pitch, wave fronts of laser beams are shifted, as shown in Figs 4.27 and 4.28. In Fig. 4.27, a material with a larger refractive index than the refractive index of the air is placed selectively on a surface of a photoresist in order to shift both wave fronts of the two incident laser beams. The light incident on this material is refracted by Snell’s law, and the exposed positions are shifted. In Fig. 4.27(a) [11], the phase-shift plate is placed on the photoresist. If there is a tiny air gap, which is on the order of 1 µm between the phase-shift plate and the photoresist, exposed patterns are heavily degraded due to multireflections of the laser beams. As a result, precise position control of the phase-shift plate and the photoresist is required. In contrast, in Fig. 4.27(b) [12], the phase-shift layer is coated on a buffer layer, which results automatically in no air gap between the phase-shift and buffer layers. In Fig. 4.28, an optical element such as a phase-shift plate or hologram is inserted in the optical path for one laser beam in order to shift only one of the wave fronts of the two incident laser beams. In Fig. 4.28(a) [13], the wave fronts are disturbed due
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
WAVEGUIDE GRATING
Laser Beam
81
Laser Beam Phase-Shift Layer
Phase-Shift Plate
Buffer Layer
Photoresist
Photoresist
(a)
(b)
FIGURE 4.27 Holographic exposure shifting both wave fronts of the two incident laser beams.
Laser Beam
Laser Beam
Laser Beam
Laser Beam Phase-Shift Plate
Hologram
(a)
(b)
FIGURE 4.28 Holographic exposure shifting one of the wave fronts of the two incident laser beams.
to diffraction at steps in the phase-shift plate, and the grating formed area is limited to only a small region. Figure 4.28(b) [14] uses a volume hologram to generate a required phase shift on the photoresist, and distortions are not present in the wave fronts. However, a highly reliable volume hologram has not yet been obtained. Figure 4.29 shows a replica method [15] where one laser beam is incident on a replica of a master phase-shifted grating, which is placed above a photoresist. The fringe patterns, which are formed by interference of the light transmitted and the light diffracted from the replica, are transferred to the photoresist. Because of reproducibility, productivity, a large tolerance in lithography conditions, and costs, the phase-shift method shown in Fig. 4.27(b) [12] was first applied to manufacture phase-shifted gratings. Later, the replica method shown in Fig. 4.29 [15] was also used commercially. Recently, electron beam exposure systems have also been used in some factories.
Laser Beam Replica Photoresist FIGURE 4.29 Replica method.
P1: OSO c04 JWBS036-Numai
82
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
FIGURE 4.30 Analytical model for the discrete approach.
4.4 VERTICAL CAVITY Discrete Approach As shown in Fig. 4.4, the vertical cavity consists of periodic multilayers and is used for a surface-emitting laser. Each set of the periodic multilayers is also called a DBR, because the operating principle of periodic multilayers is common to that of diffraction grating without optical gain. Periodic multilayers can be analyzed using the discrete approach [19]. Figure 4.30 shows a model for analysis where a region with a complex refractive index n 2 and length h 2 and a region with n 3 and h 3 are placed alternately. The angles formed by the interface normal and the light propagation directions are supposed to be θ2 and θ3 in the former and latter regions, respectively. Characteristic Matrix The relationship between input and output light is expressed by a characteristic matrix M 2 , which is defined as
U (0) U (z) = M2 . V (0) V (z)
(4.71)
Here it is assumed that an electric field E and a magnetic field H are expressed by a separation-of-variables procedure. Dependence of E and Z 0 H on z are written √ as U (z) and V (z), where Z 0 = µ0 /ε0 is the impedance of vacuum, µ0 is the permeability of vacuum, and ε0 is the permittivity of vacuum. As shown in Fig. 4.30, the light propagates along the z-axis and the plane of incidence is the x z-plane. As a result we have U (z) = E y (z) and V (z) = Z 0 Hx (z) for the TE mode (E x = E z = 0) and U (z) = Z 0 Hy (z) and V (z) = −E x (z) for the TM mode (Hx = Hz = 0), where a subscript represents a component along each axis.
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
VERTICAL CAVITY
83
Derivation of the Characteristic Matrix A characteristic matrix M 2 is derived in the following. When the electric current does not flow, Maxwell’s equations are written rotE = ∇ × E = −µ0
∂H ∂E , rotH = ∇ × H = ε0 n r 2 , ∂t ∂t
(4.72)
where n r is the refractive index of a material. Each component of (4.72) is expressed as ∂ Ey ∂ Hy ∂ Ez ∂ Hz ∂ Hx ∂ Ex − = −µ0 , − = ε0 n r 2 , ∂y ∂z ∂t ∂y ∂z ∂t ∂ Hy ∂ Ey ∂ Ex ∂ Ez ∂ Hx ∂ Hz − = −µ0 , − = ε0 n r 2 , ∂z ∂x ∂t ∂z ∂x ∂t ∂ Ey ∂ Hy ∂ Hz ∂ Ez ∂ Ex ∂ Hx − = −µ0 , − = ε0 n r 2 . ∂x ∂y ∂t ∂x ∂y ∂t
(4.73)
It is assumed that light propagates along the z-axis, and the dependence of the electric field E and magnetic field H on time is expressed as exp(iωt). In this case, we obtain ∂/∂t = i ω. Therefore, (4.73) is reduced to ∂ Ey ∂ Hy ∂ Hz ∂ Ez k0 − = −ik0 Z 0 Hx , − = i nr2 E x , ∂y ∂z ∂y ∂z Z0 ∂ Ez ∂ Hz k0 ∂ Ex ∂ Hx − = −ik0 Z 0 Hy , − = i nr2 E y , ∂z ∂x ∂z ∂x Z0 ∂ Ey ∂ Hy ∂ Ex ∂ Hx k0 − = −ik0 Z 0 Hz , − = i nr2 Ez , ∂x ∂y ∂x ∂y Z0
(4.74)
√ where k0 = ω/c is a wave number in vacuum and c = 1/ ε0 µ0 is the speed of light in vacuum.
TE Mode We assume that the lightwave is uniform along the y-axis, which results in ∂/∂ y = 0. For the TE mode with E x = E z = 0 and Hy = 0, (4.74) is reduced to ∂ Ey = ik0 Z 0 Hx , ∂z
∂ Ey = −ik0 Z 0 Hz , ∂x
∂ Hx ∂ Hz k0 − = i nr2 E y . ∂z ∂x Z0
(4.75)
P1: OSO c04 JWBS036-Numai
84
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
From (4.75), we obtain a wave equation as ∂2 Ey ∂2 Ey + = ik0 Z 0 ∂x2 ∂z 2
∂ Hx ∂ Hz − ∂z ∂x
= −k0 2 n r 2 E y .
(4.76)
Here we assume that E and H are expressed by a separation-of-variables procedure. Substituting E y = X (x)U (z) into (4.76) and then dividing both sides by E y leads to 1 ∂ 2U 1 ∂2 X + = −k0 2 n r 2 . X ∂x2 U ∂z 2
(4.77)
From (4.77), if we set 1 ∂2 X = −k0 2 n r 2 sin2 θ, X ∂x2
1 ∂ 2U = −k0 2 n r 2 cos2 θ, U ∂z 2
(4.78)
where θ is the angle formed by the direction of of light and the z-axis, we can express E y as E y = U (z) exp [ i(ωt − k0 n r x sin θ ) ] .
(4.79)
Z 0 Hx = V (z) exp [ i(ωt − k0 n r x sin θ ) ] , Z 0 Hz = W (z) exp [ i(ωt − k0 n r x sin θ ) ] .
(4.80)
Similarly, we have
Substituting (4.79) and (4.80) into (4.75) results in dU = i k0 V, dz n r sin θ · U = W, dV + i k0 n r sin θ · W = i k0 n r 2 U. dz
(4.81) (4.82) (4.83)
From (4.82) and (4.83) we obtain dV = i k0 n r 2 cos2 θ · U. dz
(4.84)
Differentiating both sides of (4.84) with respect to z with the help of (4.81), we have d2 V + k0 2 n r 2 cos2 θ · V = 0. dz 2
(4.85)
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
VERTICAL CAVITY
85
In summary, U and V are related as d2 U d2 V 2 2 2 + k n cos θ · U = 0, + k0 2 n r 2 cos2 θ · V = 0, 0 r dz 2 dz 2 dU dV = i k0 V, = i k0 n r 2 cos2 θ · U. dz dz
(4.86) (4.87)
From (4.86) and (4.87), solutions for U and V are expressed as U = A cos(k0 n r z cos θ ) + B sin(k0 n r z cos θ ),
(4.88)
V = i n r cos θ [A sin(k0 n r z cos θ ) − B cos(k0 n r z cos θ )],
(4.89)
where A and B are constants, which are determined by boundary conditions. Because (4.86) is a linear differential equation of second order, we can express U and V as
U (z) F(z) = V (z) G(z)
f (z) g(z)
U (0) , V (0)
(4.90)
where U1 = f (z), U2 = F(z), V1 = g(z), V2 = G(z).
(4.91)
Also, from (4.87) we have the relations dU1 dU2 ≡ U1 = i k0 V1 , ≡ U2 = i k0 V2 , dz dz dV1 dV2 ≡ V1 = i k0 n r 2 cos2 θ · U1 , ≡ V2 = i k0 n r 2 cos2 θ · U2 , dz dz
(4.92) (4.93)
from which we obtain U1 V2 − V1 U2 = 0, V1 U2 − U1 V2 = 0.
(4.94)
d (U1 V2 − V1 U2 ) = 0, dz
(4.95)
As a result, we have
which leads to
U1 U2
V1 V2 = constant.
(4.96)
P1: OSO c04 JWBS036-Numai
86
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
As the matrix elements satisfying (4.96), we use f (0) = G(0) = 0, F(0) = g(0) = 1, (4.97) F(z)g(z) − f (z)G(z) = 1. From (4.90) and (4.97), the transfer matrix M is defined as
U (0) U (z) g(z) =M , M= V (0) V (z) −G(z)
− f (z) . F(z)
(4.98)
In addition, from (4.88), (4.89), and (4.97), we set f (z) =
i sin(k0 n r z cos θ ), n r cos θ
F(z) = cos(k0 n r z cos θ ),
(4.99)
g(z) = cos(k0 n r z cos θ ), G(z) = i n r cos θ · sin(k0 n r z cos θ ). If we place βi = k0 n r z cos θ,
pi = n r cos θ,
(4.100)
and substitute (4.99) and (4.100) into (4.98), the transfer matrix M can be expressed as M=
cos βi −i pi sin βi
−
i sin βi m 11 pi = m 21 cos βi
m 12 . m 22
(4.101)
The tangent of the electric field E y and that of the magnetic field Hx are continuous at each interface. Therefore, if we write the tangent components of electric fields for the incident, the reflected, and transmitted waves as A, R, and T , respectively, the boundary condition is expressed as A + R = U (0), T = U (z 1 ),
(4.102)
p0 (A − R) = V (0), p1 T = V (z 1 ),
(4.103)
p0 = n 0 cos θ, p1 = n 1 cos θ.
(4.104)
where
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
VERTICAL CAVITY
87
Here n 0 and n 1 are the refractive indexes of the first and last media, respectively. From (4.98) and (4.101)–(4.104), we obtain A + R = (m 11 + m 12 p1 )T, p0 (A − R) = (m 21 + m 22 p1 )T.
(4.105) (4.106)
From (4.105) and (4.106), the amplitude reflectivity r and the amplitude transmissivity t are given by R (m 11 + m 12 p1 ) p0 − (m 21 + m 22 p1 ) = , A (m 11 + m 12 p1 ) p0 + (m 21 + m 22 p1 ) T 2 p0 t= = . A (m 11 + m 12 p1 ) p0 + (m 21 + m 22 p1 )
r=
(4.107) (4.108)
Using the amplitude reflectivity r and amplitude transmissivity t, the power reflectivity R and power transmissivity T are expressed R = r ∗r, T =
p1 ∗ t t. p0
(4.109)
TM Mode For the TM mode with Hx = Hz = 0 and E y = 0, Hy , E x , and E z are written Z 0 Hy = U (z) exp [ i (ωt − k0 n r x sin θ ) ] , E x = −V (z) exp [ i (ωt − k0 n r x sin θ ) ] , E z = −W (z) exp [ i (ωt − k0 n r x sin θ ) ] .
(4.110)
∂ Hy k0 n r 2 = −i Ex , ∂z Z0 ∂ Hy k0 n r 2 =i Ez , ∂x Z0 ∂ Ex ∂ Ez − = −i k0 Z 0 Hy . ∂z ∂x
(4.111)
From (4.74) we have
Therefore, we obtain the wave equation ∂ 2 Hy ∂ 2 Hy k0 n r 2 + = −i ∂x2 ∂z 2 Z0
∂ Ex ∂ Ez − ∂z ∂x
= −k0 2 n r 2 Hy .
(4.112)
P1: OSO c04 JWBS036-Numai
88
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
Substituting (4.110) into (4.112) gives d2 U + (k0 n r cos θ )2 U = 0. dz 2
(4.113)
Inserting (4.110) into (4.111) results in dU = i k0 n r 2 V, dz sin θ · U = n r W, dV + i k0 n r sin θ · W = i k0 U. dz
(4.114) (4.115) (4.116)
From (4.115) and (4.116) we obtain dV = i k0 cos2 θ · U. dz
(4.117)
Differentiating (4.117) with respect to z with the help of (4.114) results in d2 V + (k0 n r cos θ )2 V = 0. dz 2
(4.118)
In summary, a relationship between U and V for the TM mode is written d2 U d2 V 2 2 2 + k n cos θ · U = 0, + k0 2 n r 2 cos2 θ · V = 0, 0 r dz 2 dz 2 dU dV = i k0 n r 2 V, = i k0 cos2 θ · U. dz dz
(4.119) (4.120)
For the TM mode, the transfer matrix M, amplitude reflectivity r , amplitude transmissivity t, power reflectivity R, and power transmissivity T are obtained by replacing pi for the TE mode with qi =
cos θ . nr
(4.121)
Elements of the Characteristic Matrix When we introduce parameters such as β2 =
2π 2π n 2 h 2 cos θ2 , β3 = n 3 h 3 cos θ3 , λ0 λ0 p2 = n 2 cos θ2 ,
p3 = n 3 cos θ3 ,
(4.122)
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
VERTICAL CAVITY
89
the characteristic matrix M 2 for the TE mode is written M2 =
cos β2
−
−i p2 sin β2
i cos β3 sin β2 p2 cos β2 −i p3 sin β3
−
i sin β3 p3 . cos β3
(4.123)
Using the number of periods N , the total characteristic matrix of the optical waveguide M is given by M = M2 N =
m 11 m 21
m 12 . m 22
(4.124)
If θ2 = θ3 = 0, we have common characteristic matrices for the TE and TM modes. From (4.124), when the outside of the vertical cavity is the air, the power transmissivity T and power reflectivity R are obtained as
2
2
, T =
(m 11 + m 12 ) + (m 21 + m 22 )
(m 11 + m 12 ) − (m 21 + m 22 ) 2
.
R= (m 11 + m 12 ) + (m 21 + m 22 )
(4.125)
Comparison of Coupled Wave Theory and the Discrete Approach The results of the coupled wave theory and the discrete approach are compared in the following. For simplicity, it is assumed that a material is transparent (α0 = α1 = 0), and reflectivity at the Bragg wavelength (δ = 0) in first-order diffraction gratings (m = 1) is considered. Coupled Wave Theory From (4.44), the assumption of α0 − i δ = 0 leads to γ = ±κ. Substituting this result into (4.46), (4.47), and (4.56), the power reflectivity R is obtained as R = tanh2 (κ L),
(4.126)
where L is the corrugated region length. Discrete Approach useful to set
To analyze DFBs and DBRs using the discrete approach, it is n 2 = n r0 + n, n 3 = n r0 − n, h2 = h3 =
, θ2 = θ3 = 0, 2
(4.127)
P1: OSO c04 JWBS036-Numai
90
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
where n r0 is an average refractive index of a material, n is a shift of refractive index from n r0 ; is the grating pitch; and L = N is the corrugated region length. When n n r0 is satisfied, the power reflectivity R is obtained as R = 1−
n r0 − n n r0 + n
2N 2
1+
n r0 − n n r0 + n
2N −2 ,
(4.128)
where (4.123)–(4.125) are used. The second terms in both sets of brackets are approximated as
n r0 − n n r0 + n
2N
4N n/n r0 n n r0 /n n 4N 1− 1− = n r0 n r0 n . (4.129) = exp −4N n r0
By using N = L and (4.42), the exponent of the right-hand side in (4.129) is written 4N
n 8n 8n L= = L. n r0 2n r0 λB
(4.130)
If we set 8n = 2π n r1
(4.131)
and substitute (4.131) into (4.130), we have 4N
n π n r1 =2 L = 2κ L , n r0 λB
(4.132)
where (4.35) and the assumption of α1 = 0 are used. Substituting (4.132) into (4.129) results in
n r0 − n n r0 + n
2N
e−2κ L .
(4.133)
= tanh2 (κ L).
(4.134)
Inserting (4.133) into (4.128) leads to
1 − e−2κ L R 1 + e−2κ L
2
From (4.126) and (4.134), it is found that the result of coupled wave theory agrees with that of the discrete approach when n n r0 is satisfied.
P1: OSO c04 JWBS036-Numai
August 9, 2010
9:8
Printer Name: Yet to Come
REFERENCES
91
It should be noted that coupled wave theory assumes that complex refractive indexes vary sinusoidally, while the discrete approach presumes that complex refractive indexes change abruptly. Also, in the example of the discrete approach, two layers were alternated. Therefore, with an increase in n, the results of coupled wave theory and those of the discrete approach differ. The diffraction grating has n, which is on the order of 10−3 ; the vertical cavity has n, which is on the order of 10−2 . Therefore, for analysis of a vertical cavity, the discrete approach has to be used.
REFERENCES 1. T. Numai, “1.5 µm phase-shift-controlled distributed feedback wavelength tunable optical filter,” IEEE J. Quantum Electron. 28, 1513 (1992). 2. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327 (1972). 3. M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental matrix approach,” Appl. Opt. 26, 3474 (1987). 4. H. A. Haus and C. V. Shank, “Antisymmetric taper of distributed feedback lasers,” IEEE J. Quantum Electron. 12, 532 (1976). 5. H. A. Haus, Waves and Fields in Optoelectronics, Prentice-Hall, Englewood Cliffs, NJ, 1984. 6. T. Numai, A study on semiconductor wavelength tunable optical filters and lasers. Ph.D. dissertation, Keio University, Yokohama, Japan, 1992. 7. T. Numai, “1.5-µm wavelength tunable phase-shift-controlled distributed feedback laser,” IEEE/OSA J. Lightwave Technol. 10 199 (1992). 8. K. Utaka, S. Akiba, K. Sakai, and Y. Matsushima, “λ/4-shifted InGaAsP/InP DFB lasers by simultaneous holographic exposure of positive and negative photoresists,” Electron. Lett. 20, 1008 (1984). 9. K. Utaka, S. Akiba, K. Sakai, and Y. Matsushima, “λ/4-shifted InGaAsP/InP DFB lasers,” IEEE J. Quantum Electron. 22, 1042 (1986). 10. M. Okai, S. Tsuji, M. Hirao, and H. Matsumura, “New high resolution positive and negative photoresist method for λ/4-shifted DFB lasers,” Electron. Lett. 23, 370 (1987). 11. M. Shirasaki, H. Soda, S. Yamakoshi, and H. Nakajima, “λ/4-shifted DFB-LD corrugation formed by a novel spatial phase modulating mask,” European Conference on Optical Communications/Integrated Optics and Optical Communications, 25 (1985). 12. T. Numai, M. Yamaguchi, I. Mito, and K. Kobayashi, “A new grating fabrication method for phase-shifted DFB LDs,” Jpn. J. Appl. Phys. Pt. 2 26, L1910 (1987). 13. S. Tsuji, A. Ohishi, M. Okai, M. Hirao, and H. Matsumura, “Quarter lambda shift DFB lasers by phase image projection method,” 10th Int. Semiconductor Laser Conf. 58 (1986). 14. Y. Ono, S. Takano, I. Mito, and N. Nishida, “Phase-shifted diffraction-grating fabrication using holographic wavefront reconstruction,” Electron. Lett. 23, 57 (1987). 15. M. Okai, S. Tsuji, N. Chinone, and T. Harada, “Novel method to fabricate corrugation for a λ/4-shifted distributed feedback laser using a grating photomask,” Appl. Phys. Lett. 55, 415 (1989).
P1: OSO c04 JWBS036-Numai
92
August 9, 2010
9:8
Printer Name: Yet to Come
OPTICAL RESONATORS
16. H. Sugimoto, Y. Abe, T. Matsui, and H. Ogata, “Novel fabrication method of quarterwave-shifted gratings using ECR-CVD SiNx films,” Electron. Lett. 23, 1260 (1987). 17. K. Sekartedjo, N. Eda, K. Furuya, Y. Suematsu, F. Koyama, and T. Tanbun-ek, “1.5-µm phase-shifted DFB lasers for single-mode operation,” Electron. Lett. 20, 80 (1984). 18. T. Nishida, M. Nakao, T. Tamamura, A. Ozawa, Y. Saito, K. Nishimura, and H. Yoshihara, “Synchrotron radiation lithography for DFB laser gratings,” Jpn. J. Appl. Phys. Pt. 1 28, 2333 (1989). 19. M. Born and E. Wolf, Principles of Optics, 7th ed., Cambridge University Press, Cambridge, UK, 1999. 20. T. Numai, Fundamentals of Semiconductor Lasers, Springer-verlag, New York, 2004.
P1: OSO c05 JWBS036-Numai
August 9, 2010
9:14
Printer Name: Yet to Come
5 PN- AND PNPN-JUNCTIONS
5.1
INTRINSIC SEMICONDUCTOR
Resistivity Semiconductors are solids whose resistivity at room temperature is in the range 10−2 to 109 · cm. Element semiconductors such as Si and Ge and compound semiconductors such as GaAs, InP, and GaN are widely used. Si and Ge have diamond structures and GaAs and InP have zinc blende structures, which are shown in Fig. 5.1. Thermal Excitation Intrinsic semiconductors are pure semiconductors without containing impurities. As shown in Fig. 5.2, the electrons existing in a valence band are excited to a conduction band by receiving thermal energy. As a result, holes are generated in the valence band. Both the holes in the valence band and the electrons in the conduction band contribute to electric conduction. Therefore, the electrons in the conduction band are called conduction electrons. With an increase in temperature, thermal excitation becomes active, and the conduction-electron and hole concentrations increase, which leads to a decrease in the electrical resistance of the intrinsic semiconductors.
Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright
93
P1: OSO c05 JWBS036-Numai
94
August 9, 2010
9:14
Printer Name: Yet to Come
PN- AND PNPN-JUNCTIONS
(a)
(b)
FIGURE 5.1 Crystal structure of semiconductors: (a) diamond structure; (b) zinc blende structure.
Conduction Band Conduction Electron Hole Valence Band FIGURE 5.2 Thermal excitation of electrons from a valence band to a conduction band.
Carrier Concentration In the intrinsic semiconductors, the conduction electrons and holes are generated by thermal excitation. As a result, the conduction-electron concentration n and the hole concentration p are equal, which is written n = p.
(5.1)
Both the conduction electrons and the holes are called carriers, because they carry electric charges. Conduction-Electron Concentration The electrons are Fermi particles with spin 1/2 and occupy the energy states according to Fermi statistics. As a result, the conduction-electron concentration n is given by n=
E0
g(E − E c ) f (E) dE Ec Ec − EF . = Nc exp − kB T
∞
g(E − E c ) f (E) dE
Ec
(5.2)
P1: OSO c05 JWBS036-Numai
August 9, 2010
9:14
Printer Name: Yet to Come
INTRINSIC SEMICONDUCTOR
95
Here E c is the energy of the bottom of the conduction band, g(E − E c ) = De (E − E c )/L 3 is the density of states per unit volume of the conduction band, f (E) is the Fermi–Dirac distribution function, E 0 is the vacuum level of a semiconductor, kB is Boltzmann’s constant, and T is the absolute temperature. Nc is the effective density of states for the conduction band, which is defined as Nc ≡ 2
2π m e kB T h2
32
Mc ,
(5.3)
where m e is the effective mass of the conduction electron, h is Planck’s constant, and Mc is the number of band edges of the conduction band. In general, the effective mass of the conduction electron has different values according to the direction of the wave vectors. When the effective masses of a conduction electron along the principal axes of wave vectors are expressed as m 1 ∗ , m 2 ∗ , and m 3 ∗ , the density-of-states effective mass of the conduction electron m e is given by 1 m e = m de ≡ m 1 ∗ m 2 ∗ m 3 ∗ 3 .
(5.4)
In Si and Ge, by using the transverse effective mass m t and the longitudinal effective mass m l , the density-of-states effective mass m e is written 1 (5.5) m e = m de = m t 2 m l 3 , where
m1∗ = m2∗ = mt, m3∗ = ml.
(5.6)
Hole Concentration The hole concentration p is given by p=
EF − Ev . g(E v − E)[1 − f (E)] dE Nv exp − kB T −∞ Ev
(5.7)
Here E v is the energy of the top of a valence band and g(E v − E) is the density of states per unit volume for the valence band. Nv is the effective density of states for the valence band, which is defined as Nv ≡ 2
2π m h kB T h2
32
.
(5.8)
Here, by using the effective mass of the heavy hole m hh and the effective mass of the light hole m lh , the effective mass of the hole m h is expressed as 23 3 3 m h = m dh ≡ m hh 2 + m lh 2 , where m dh is called the density-of-states effective mass for the valence band.
(5.9)
P1: OSO c05 JWBS036-Numai
96
August 9, 2010
9:14
Printer Name: Yet to Come
PN- AND PNPN-JUNCTIONS
Intrinsic Carrier Concentration The carrier concentration in the intrinsic semiconductor, that is, the intrinsic carrier concentration n i , has a relation such as n i = n = p.
(5.10)
Using the energy gap E g = E c − E v , we have Eg Ec − Ev = Nc Nv exp − . n i 2 = np = Nc Nv exp − kB T kB T
(5.11)
As a result, the intrinsic carrier concentration n i is written ni =
Eg Ec − Ev Nc Nv exp − = Nc Nv exp − . 2kB T 2kB T
(5.12)
In thermal equilibrium, both the intrinsic and extrinsic semiconductors satisfy (5.11) and (5.12). Intrinsic Fermi Level From (5.2), (5.7), and (5.10), the intrinsic carrier concentration n i is given by Ec − Ei Ei − Ev = Nv exp − , n i = Nc exp − kB T kB T
(5.13)
where E i is the Fermi level in the intrinsic semiconductor, that is, the intrinsic Fermi level. Therefore, by using the intrinsic carrier concentration n i and the intrinsic Fermi level E i , the effective density of states Nc and Nv can be rewritten as Ec − Ei , Nc = n i exp kB T Ei − Ev . Nv = n i exp kB T
(5.14) (5.15)
If we put E F = E i in (5.2) and (5.7), the intrinsic Fermi level E i is expressed as 1 1 Nv (E c + E v ) + kB T ln 2 2 Nc 1 1 Nv = E v + E g + kB T ln . 2 2 Nc
Ei =
(5.16)
From (5.16) it is found that the intrinsic Fermi level E i shifts slightly from the center of the energy gap E g /2.
P1: OSO c05 JWBS036-Numai
August 9, 2010
9:14
Printer Name: Yet to Come
EXTRINSIC SEMICONDUCTOR
5.2
97
EXTRINSIC SEMICONDUCTOR
Impurities Extrinsic semiconductors are semiconductors that contain impurities. To control the electric conductivity, impurities are added intentionally to the semiconductors, and the added impurities are called dopants. Dopants that give conduction electrons to the conduction band are called donors; dopants that accept electrons from the valence band are called acceptors. A semiconductor in which the conduction-electron concentration n is larger than the hole concentration p (n > p) is called an n-type semiconductor; a semiconductor in which n is smaller than p (n < p) is called a p-type semiconductor. Equations of Motion for the Carriers Using the elementary electric charge e, the electric charge of a carrier q is −e for the conduction electron, e for the hole. When the magnetic flux density B(= 0) is a vector in the positive direction of the z-axis, the x-, y-, and z-components of the equations of motion for the carriers in the semiconductor are given by
d 1 + vx = q E x + Bv y , dt τ d 1 + v y = q E y − Bvx , m∗ dt τ d 1 + vz = q E z . m∗ dt τ
m∗
(5.17)
In a steady state (d/dt = 0), (5.17) is reduced to q τ E x + ωc τ v y , m∗ q v y = ∗ τ E y − ωc τ vx , m q vz = ∗ τ E z . m vx =
(5.18) (5.19) (5.20)
Here ωc is the cyclotron angular frequency, which is defined as ωc ≡
qB . m∗
(5.21)
Hall Effect Let us consider a case in which the electric field E = (E x , 0, 0) and the magnetic flux density B = (0, 0, B) are applied to a semiconductor sample. When the electric current flows only along the x-axis under this condition, we have v y = 0 because of
P1: OSO c05 JWBS036-Numai
98
August 9, 2010
9:14
Printer Name: Yet to Come
PN- AND PNPN-JUNCTIONS
the y-component of the current density j y = 0. As a result, from (5.18), (5.19), and (5.21), we obtain E y = ωc τ E x =
q Bτ Ex . m∗
(5.22)
The x-component of the current density jx is given by jx = n c qvx =
ncq 2τ Ex , m∗
(5.23)
where n c is the carrier concentration. Using the carrier concentration n c and the x-component of the current density jx , the Hall coefficient RH is defined as RH ≡
Ey 1 = , jx B ncq
(5.24)
where n c is the carrier concentration. We have RH = −1/n c e < 0 for the conduction electron and RH = 1/n c e > 0 for the hole. Donor When a group V element such as P is added to a group IV element such as Si, the group V element combines with the group IV element. In this case, an outermost electron of the group V element is left, because the group V element has five outermost electrons while the group IV element has four outermost electrons. If the group V element receives energy such as heat, the group V element donates the excess outermost electron to the conduction band and the group V element becomes a positive ion. The electron donated to the conduction band from the group V element functions as a conduction electron. Therefore, the group V element can play a role as a conductionelectron donor and contributes to generating carriers. The energy level of the group V element is called the donor level. As shown in Fig. 5.3, the energy at the donor level E d is lower than the energy at the bottom of the conduction band E c . To obtain Conduction Electron
Conduction Band
Ionized Donor
Neutral Donor
Valence Band FIGURE 5.3 Donor level.
P1: OSO c05 JWBS036-Numai
August 9, 2010
9:14
Printer Name: Yet to Come
EXTRINSIC SEMICONDUCTOR
99
as many conduction electrons as possible from donors, we usually select the group V element whose donor ionization energy, E d = E c − E d , is at most several tens of millielectron volts, which is on the same order as the thermal energy at room temperature. Acceptor When a group III element such as B is added to a group IV element such as Si, the group III element combines with the group IV element. In this case, an outermost electron of the group III element is lacking, because the group III element has three outermost electrons whereas the group IV element has four outermost electrons. If the electron in the valence band receives energy such as heat, the group III element accepts the electron in the valence band and the group III element becomes a negative ion. The hole generated in the valence band can carry the electric charge. Therefore, the group III element functions as an electron acceptor and contributes to generating carriers. The energy level of the group III element is called the acceptor level. As shown in Fig. 5.4, the energy at the acceptor level E a is higher than the energy at the top of the valence band E v . To obtain as many holes as possible from acceptors, we usually select the group III element whose acceptor ionization energy, E a = E a − E v , is at most several tens of millielectron volts, which is on the same order as the thermal energy at room temperature. Figure 5.5 shows the Fermi level E F of n-type Si and that of p-type Si as a function of carrier concentration. Figure 5.6 shows the Fermi level E F of n-type Si and that of p-type Si as a function of absolute temperature with a difference in the carrier concentrations n = n − p as a parameter. Drift Current Density The carriers in semiconductors move through the semiconductors by colliding with the constituent atoms or impurities. Using the average collision time τ , the equation of motion for the carrier is expressed as m∗
m∗v dv = qE − . dt τ Conduction Band
Ionized Acceptor
Electron
Neutral Acceptor
Valence Band
Hole FIGURE 5.4 Acceptor level.
(5.25)
P1: OSO c05 JWBS036-Numai
100
August 9, 2010
9:14
Printer Name: Yet to Come
PN- AND PNPN-JUNCTIONS
n
p
FIGURE 5.5 Fermi levels of Si as a function of carrier concentration.
FIGURE 5.6 Fermi levels of Si as a function of absolute temperature with a difference in the carrier concentrations n = n − p as a parameter.
Here m ∗ is the effective mass of the carrier, v is the velocity of the carrier, t is a time, q is the electric charge of the carrier, and E is the electric field. Using the effective masses along the principal axes of the wave vector, m 1 ∗ , m 2 ∗ , and m 3 ∗ , the conductivity effective mass of the carrier m c is defined as 1 1 ≡ mc 3
1 1 1 + + m1∗ m2∗ m3∗
.
(5.26)
For example, in Ge and Si, from (5.6), the conductivity effective mass of the conduction-electron m c is given by mc =
3m t m l . m t + 2m l
(5.27)
In a steady state (d/dt = 0), the velocity of the carrier v is written v=
qτ E ≡ µE. m∗
(5.28)
P1: OSO c05 JWBS036-Numai
August 9, 2010
9:14
Printer Name: Yet to Come
EXTRINSIC SEMICONDUCTOR
101
Here µ is the mobility of the carrier, which is defined as µ≡
qτ . m∗
(5.29)
Using the carrier concentration n c , the drift current density j d , which is the electric current per unit area generated by the electric field E, is expressed as j d = n c qv = n c qµE =
ncq 2τ E ≡ σ E, m∗
(5.30)
where σ is the electric conductivity of the semiconductor, which is written σ =
ncq 2τ . m∗
(5.31)
When both conduction electrons and holes are present in a semiconductor, the drift current density jd is given by jd = e(nµn + pµ p )E.
(5.32)
Here n and p are the conduction-electron and hole concentrations, respectively, and µn and µ p are the mobility of the conduction electron and that of the hole, respectively. Diffusion Current Density When there is a gradient in the carrier concentration, electric current flows due to the diffusion. For example, if a gradient in the carrier concentration exists along the x-axis, the diffusion current density due to the conduction electron jn and the diffusion current density due to the j p are written dn dn = eDn , jn = −e −Dn dx dx dp dp = −eD p . j p = e −D p dx dx
(5.33) (5.34)
Here n is the conduction-electron concentration, p is the hole concentration, and Dn and D p are the diffusion coefficients of the conduction electron and the hole, respectively. The diffusion coefficients and mobilities are related as Dp Dn kB T , = = µn µp e which is known as Einstein’s relation.
(5.35)
P1: OSO c05 JWBS036-Numai
102
August 9, 2010
9:14
Printer Name: Yet to Come
PN- AND PNPN-JUNCTIONS
Quasi-Fermi Level If the conduction-electron concentration n and hole concentration p are much larger than the values in equilibrium due to optical excitation or injection of the electric current, we cannot describe the Fermi–Dirac distribution function with one Fermi level. As a result, we assume that the conduction and valence bands are expressed separately by Fermi–Dirac distribution functions with different Fermi levels, such as 1 , E − E Fc 1 + exp kB T 1 , f v (E) = E − E Fv 1 + exp kB T f c (E) =
(5.36)
(5.37)
where E Fc and E Fv are quasi-Fermi levels. When the quasi-Fermi level of the conduction band E Fc is uniform in a semiconductor, the conduction electrons are in a diffusive equilibrium state. Therefore, electric current due to the conduction electron does not flow. If the quasi-Fermi level has a gradient, electric current flows with electric current density j n such as j n = µn n grad E Fc .
(5.38)
In an extrinsic semiconductor with n i n Nc ,
(5.39)
replacing Fermi level E F with quasi-Fermi level E Fc in (5.2) results in E Fc = E c + kB T ln
n . Nc
(5.40)
Inserting (5.40) into (5.38) leads to j n = µn n grad E c + µn kB T grad n.
(5.41)
Using the electric potential ϕ, the gradient of the energy of the bottom of the conduction band E c is expressed as grad E c = − e grad ϕ = e E,
(5.42)
where E is the electric field. From (5.35), the diffusion coefficient Dn is given by Dn =
µn kB T . e
(5.43)
P1: OSO c05 JWBS036-Numai
August 9, 2010
9:14
Printer Name: Yet to Come
103
PN-JUNCTION
From (5.41)–(5.43), the electric current density j n is written j n = eµn n E + eDn grad n,
(5.44)
where the first term on the right-hand side, eµn n E, represents the drift current density and the second term on the right-hand side, eDn grad n, represents the diffusion current density. Using a procedure similar to the calculations above, the hole current density j p is obtained as j p = µ p p grad E Fv = eµ p p E − eD p grad p.
5.3
(5.45)
PN-JUNCTION
Depletion and Space-Charge Layers The pn-junction is a junction of p- and n-type semiconductors. At the interface of a pn-junction, the conduction electrons diffuse from an n-type semiconductor to a p-type semiconductor; the holes diffuse from a p-type semiconductor to a n-type semiconductor, so that Fermi levels in p- and n-type semiconductors may have a common value. Figure 5.7(a) shows the diffusion of conduction electrons and holes. By diffusion of the carriers, the carriers are depleted at the interface of the pn-junction, and this region is called the depletion layer. Space charges such as ionized acceptors and ionized donors are left in the depletion layer, as shown in Fig. 5.7(b), and the depletion layer is also called the space-charge layer. Diffusion and Built-in Potential Using the intrinsic Fermi level of an n-type semiconductor, E in , and that of a p-type semiconductor, E ip , the electric potential of an n-type semiconductor, φn , and that of
Conduction Electron
Hole
n
p Depletion Layer (a)
Ionized Donor
Ionized Acceptor
n
p
Depletion Layer (Space-Charge Layer) (b)
FIGURE 5.7 pn-junction diode: (a) carrier diffusion at the interface; (b)depletion layer (space-charge layer).
P1: OSO c05 JWBS036-Numai
104
August 9, 2010
9:14
Printer Name: Yet to Come
PN- AND PNPN-JUNCTIONS
a p-type semiconductor, φp , are defined as kB T Nd E F − E in = ln , e e ni E F − E ip kB T Na . φp ≡ =− ln e e ni φn ≡
(5.46) (5.47)
Here E F is the common Fermi level of the pn-junction, e is the elementary electric charge, kB is Boltzmann’s constant, T is the absolute temperature, Nd is the donor concentration in an n-type semiconductor, Na is the acceptor concentration in a ptype semiconductor, and n i is the intrinsic carrier concentration. For simplicity it is assumed that only donors are doped in an n-type semiconductor and only acceptors are doped in a p-type semiconductor. In addition, all the donors and acceptors are assumed to be ionized completely. At the interface of the pn-junction, the potential difference φD is generated and is written φD = φ n − φ p =
Eg Na Nd kB T Na Nd kB T ln + ln = , 2 e ni e e Nc Nv
(5.48)
where (5.12) is used. This potential difference φD is called the diffusion potential or built-in-potential. Owing to the diffusion potential φD , an electric field is generated at the interface of the pn-junction. In thermal equilibrium, the drift current due to the electric field balances the diffusion current, and net electric current does not flow in the pn-junction. Abrupt pn-Junction A pn-junction whose impurity concentration (Na − Nd ) changes abruptly at the interface, as shown in Fig. 5.8, is called an abrupt pn-junction. In Fig. 5.8 the interface of a pn-junction is placed at x = 0, an n-type semiconductor with donor concentration Nd is located at x ≤ 0, and a p-type semiconductor with acceptor concentration Na is located at 0 ≤ x. For simplicity it is assumed that only donors are doped in an n-type semiconductor and only acceptors are doped in a p-type semiconductor. Note that n and p in Fig. 5.8 represent the regions where n- and p-type semiconductors are located, respectively. The spatial distribution of the carrier concentrations n and p is shown in Fig. 5.9, where −ln and lp are the x-coordinates of depletion layer interfaces in n- and p-type semiconductors, respectively. Here, it is assumed that all donors and acceptors are ionized completely and the interface region of a pn-junction with −ln ≤ x ≤ lp is depleted completely. The spatial distribution of the electric charge density at the interface of a pnjunction ρ is shown in Fig. 5.10. Here it is assumed that the concentration of the carriers and that of the ionized impurities have a common value, and that an electrically neutral condition is satisfied in both x ≤ −ln and lp ≤ x.
P1: OSO c05 JWBS036-Numai
August 9, 2010
9:14
Printer Name: Yet to Come
PN-JUNCTION
n
p
0
FIGURE 5.8 Impurity concentration distribution in an abrupt pn-junction.
n
p
Depletion Layer FIGURE 5.9 Carrier concentration in an abrupt pn-junction.
n
p
Depletion Layer FIGURE 5.10 Electric charge density in an abrupt pn-junction.
105
P1: OSO c05 JWBS036-Numai
106
August 9, 2010
9:14
Printer Name: Yet to Come
PN- AND PNPN-JUNCTIONS
Poisson Equation are written
Poisson equations in the depletion layer of an abrupt pn-junction d2 φ eNd =− (−ln ≤ x ≤ 0), 2 dx εn ε0 eNa d2 φ = (0 ≤ x ≤ lp ). 2 dx εp ε0
(5.49) (5.50)
Here φ is the electric potential, εn is the relative dielectric constant of an n-type semiconductor, εp is the relative dielectric constant of a p-type semiconductor, and ε0 = 8.854 × 10−12 F/m is the permittivity of vacuum. Electric Field In electrically neutral regions with x ≤ −ln and lp ≤ x, the electric field is E x = −dφ/dx = 0. As a result, as the boundary condition for the electric field, E x = −dφ/dx = 0 is satisfied at both x = −ln and x = lp . Under this boundary condition, integrating (5.49) and (5.50) with respect to x results in ⎧ ⎪ eNd ⎪ ⎪ (x + ln ) ⎨− dφ εn ε0 = ⎪ eNa dx ⎪ ⎪ x − lp ⎩ εp ε0
(−ln ≤ x ≤ 0), (0 ≤ x ≤ lp ).
(5.51) (5.52)
From (5.51) and (5.52), the electric field E x = −dφ/dx is shown in Fig. 5.11. The electric field E x = −dφ/dx should have a common value at the interface of the pn-junction x = 0. As a result, from (5.51) and (5.52) we obtain Nd Na ln = lp . εn εp
n
(5.53)
p
Depletion Layer FIGURE 5.11 Electric field in an abrupt pn-junction.
P1: OSO c05 JWBS036-Numai
August 9, 2010
9:14
Printer Name: Yet to Come
PN-JUNCTION
107
If εn = εp is satisfied, we have Ndln = Nalp . Accordingly, when Nd > Na , we have ln < lp , which indicates that the depletion layer expands to a region with lower impurity concentration. Electric Potential Under the assumption of the electric potential φ = 0 at x = 0 as the boundary condition, integrating (5.51) and (5.52) with respect to x leads to ⎧ ⎪ eNd 2 ⎪ ⎪ ⎨− x + 2ln x (−ln ≤ x ≤ 0), 2εn ε0 φ= ⎪ eNa 2 ⎪ ⎪ ⎩ x − 2lp x (0 ≤ x ≤ lp ). 2εp ε0
(5.54) (5.55)
The electric potential φ and diffusion potential φD are shown in Fig. 5.12. The energy of the electron is represented by −eφ with the elementary electric charge e. Therefore, according to the electric potential shown in Fig. 5.12, the conduction and valence bands bend in the interface of the pn-junction, as shown in Fig. 5.13. Thickness of the Depletion Layer If the relative dielectric constants of n- and ptype semiconductors are common, such as εn = εp = εs , the electric field at x = 0, which is written as E m , is given by Em =
eNd eNa σ ln = lp ≡ . εs ε0 εs ε0 εs ε0
(5.56)
Here σ is the electric charge, which is stored in the depletion layer per unit area, and is expressed as σ = eNd ln = eNalp .
n
(5.57)
p
Depletion Layer FIGURE 5.12 Electric potential in an abrupt pn-junction.
P1: OSO c05 JWBS036-Numai
108
August 9, 2010
9:14
Printer Name: Yet to Come
PN- AND PNPN-JUNCTIONS
n
p
Depletion Layer FIGURE 5.13 Energy in an abrupt pn-junction.
When a forward bias voltage V > 0 is applied to a pn-junction, the electric potential φ(x), which is a function of a position x, is related to the diffusion potential φD as φD − V = φ (−ln ) − φ lp = φn − φp 1 σ2 e 1 . Nd l n 2 + Na l p 2 = = + 2εs ε0 2eεs ε0 Nd Na
(5.58)
From (5.57) and (5.58), ln and lp are expressed as σ ln = = eNd σ = lp = eNa
2εs ε0 Na (φD − V ) , eNd Na + Nd
(5.59)
2εs ε0 Nd (φD − V ) . eNa Na + Nd
(5.60)
The thickness of the depletion layer lD is given by lD = ln + lp .
(5.61)
Junction and Depletion Layer Capacitance In an abrupt pn-junction, the positive and negative electric charges are localized, as shown in Fig. 5.10, and it can be interpreted that a capacitor has been formed. The capacitance per unit area is called the junction capacitance or depletion layer capacitance, and its value CJ is given by dσ eεs ε0 εs ε0 Na Nd = CJ = = , · dV 2 (φD − V ) Na + Nd lD where (5.58) is used.
(5.62)
P1: OSO c05 JWBS036-Numai
August 9, 2010
9:14
Printer Name: Yet to Come
PN-JUNCTION
109
FIGURE 5.14 CJ −2 as a function of bias voltage V .
From (5.62) we have CJ −2 =
2 (φD − V ) Na + Nd · . eεs ε0 Na Nd
(5.63)
Figure 5.14 shows CJ −2 as a function of the reverse bias voltage V . From Fig. 5.14 it is found that CJ −2 = 0 is obtained at V = φD , which makes it possible to determine the diffusion potential φD experimentally. Graded pn-Junction A pn-junction whose impurity concentration (Na − Nd ) changes gradually at the interface, as shown in Fig. 5.15, is called a graded pn-junction. Here x = 0 is located at the interface of the pn-junction and the (Na − Nd ) is written Na − Nd = ax (−l0 ≤ x ≤ l0 ),
(5.64)
where a is positive and −l0 and l0 are the x-coordinates of the depletion layer interfaces in n- and p-type semiconductors, respectively. The pn-junction shown in Fig. 5.15 is called a linearly graded pn-junction. An n-type semiconductor with donor concentration Nd is located in x ≤ 0; a p-type semiconductor with acceptor concentration Na is located in 0 ≤ x. For simplicity it is assumed that only donors are doped in an n-type semiconductor and only acceptors are doped in a p-type semiconductor. Note that n and p in Fig. 5.15 represent the regions where the n- and p-type semiconductors are located, respectively. The spatial distribution of carrier concentrations n and p is shown in Fig. 5.16. Here it is assumed that all donors and acceptors are ionized completely and that the interface region of a pn-junction with −l0 ≤ x ≤ l0 is depleted completely. Poisson Equation The spatial distribution of the electric charge density at the interface of a pn-junction ρ is shown in Fig. 5.17. Here it is assumed that the
P1: OSO c05 JWBS036-Numai
110
August 9, 2010
9:14
Printer Name: Yet to Come
PN- AND PNPN-JUNCTIONS
n
p
Depletion Layer FIGURE 5.15 Impurity concentration in a graded pn-junction.
n
p
Depletion Layer FIGURE 5.16 Carrier concentration in a graded pn-junction.
n
p
Depletion Layer FIGURE 5.17 Electric charge density in a graded pn-junction.
P1: OSO c05 JWBS036-Numai
August 9, 2010
9:14
Printer Name: Yet to Come
PN-JUNCTION
111
concentration of the carriers and that of the ionized impurities have a common value, and an electrically neutral condition is satisfied in both x ≤ −l0 and l0 ≤ x. The Poisson equation in the depletion layer of a linearly graded pn-junction is written d2 φ eax = (−l0 ≤ x ≤ l0 ). dx 2 εs ε0
(5.65)
Here φ is the electric potential, εs is the relative dielectric constant of the semiconductor, and ε0 = 8.854 × 10−12 F/m is the permittivity of vacuum. For simplicity, the relative dielectric constants of n- and p-type semiconductors are assumed to have a common value. Electric Field In an electrically neutral region in x ≤ −l0 and l0 ≤ x, the electric field is E x = −dφ/dx = 0. As a result, E x = −dφ/dx = 0 is satisfied at both x = −l0 and x = l0 as the boundary condition for the electric field. Under this boundary condition, integrating (5.65) with respect to x results in ea x 2 − l0 2 dφ = (−l0 ≤ x ≤ l0 ). dx 2εs ε0
(5.66)
Figure 5.18 shows an electric field E x = −dφ/dx as a function of x. The electric field at x = 0 is given by Em =
eal0 2 . 2εs ε0
n
(5.67)
p
Depletion Layer FIGURE 5.18 Electric field distribution in a graded pn-junction.
P1: OSO c05 JWBS036-Numai
112
August 9, 2010
9:14
Printer Name: Yet to Come
PN- AND PNPN-JUNCTIONS
Electric Potential Under the assumption of an electric potential φ = 0 at x = 0 as the boundary condition, integrating (5.66) with respect to x leads to ea x 3 − 3l0 2 x φ= (−l0 ≤ x ≤ l0 ). 6εs ε0
(5.68)
The electric potential φ and the diffusion potential φD are shown in Fig. 5.19. The energy of the electron is represented by −eφ with elementary electric charge e. Therefore, according to the electric potential shown in Fig. 5.19, the conduction and valence bands bend in the interface of the pn-junction, as shown in Fig. 5.20.
n
p
Depletion Layer FIGURE 5.19 Electric potential in a graded pn-junction.
n
p
Depletion Layer FIGURE 5.20 Energy in a graded pn-junction.
P1: OSO c05 JWBS036-Numai
August 9, 2010
9:14
Printer Name: Yet to Come
PN-JUNCTION
113
Thickness of the Depletion Layer The electric charge σ , which is stored in the depletion layer per unit area, is written σ =
0
−l0
−eax dx =
0
l0
1 −eax dx = eal0 2 . 2
(5.69)
When the forward bias voltage V > 0 is applied to the pn-junction, the electric potential φ(x), which is a function of a position x, is related to the diffusion potential φD as √ 2eal0 3 4 2 σ3 . φD − V = φ (−l0 ) − φ (l0 ) = φn − φp = = 3εs ε0 3εs ε0 ea
(5.70)
As a result, l0 is expressed as l0 =
13 3εs ε0 (φD − V ) . 2ea
(5.71)
The thickness of the depletion layer lD is given by lD = 2l0 .
(5.72)
Junction and Depletion Layer Capacitance In a graded pn-junction, the positive and negative electric charges are localized, as shown in Fig. 5.17, and it is considered that a capacitor has been formed. From (5.70), the junction capacitance CJ is given by 1 dσ ea (εs ε0 )2 3 εs ε0 εs ε0 = CJ = = = . dV 12 (φD − V ) 2l0 lD
(5.73)
Diffusion Equations for the Carriers In thermal equilibrium, a pn-junction has a space-charge layer, which is sandwiched between the electrically neutral p- and n-layers. If these layers are placed along the x-axis as shown in Fig. 5.9, the diffusion equations for the carriers are expressed by ∂ pn pn − pn0 ∂ 2 pn =− + D pn ∂t τ pn ∂x2
(in an electrically neutral n-region),
(5.74)
n p − n p0 ∂ 2np ∂n p =− + Dn p ∂t τn p ∂x2
(in an electrically neutral p-region).
(5.75)
Here t is the time, pn is the hole concentration in the n-region, pn0 is the steady-state value of pn , τ pn is the lifetime of the hole in the n-region, and D pn is the diffusion
P1: OSO c05 JWBS036-Numai
114
August 9, 2010
9:14
Printer Name: Yet to Come
PN- AND PNPN-JUNCTIONS
coefficient of the hole in the n-region. In addition, n p is the conduction-electron concentration in the p-region, n p0 is the steady-state value of n p , τn p is the lifetime of a conduction electron in the p-region, and Dn p is the diffusion coefficient of a conduction electron in the p-region. Diffusion Length In a steady state (∂/∂t = 0), the excess carrier concentrations pn
and n p are written pn = pn − pn0 , n p = n p − n p0 .
(5.76)
We assume that all donors and acceptors are ionized completely. In this case, the hole concentration in the p-region in a steady state pp0 and the conduction-electron concentration in the n-region in a steady state n n0 are written pp0 = Na , n n0 = Nd .
(5.77)
Substituting (5.77) into (5.48) results in φD =
pp0 n n0 kB T ln , e ni2
(5.78)
where n i 2 = pp0 n p0 = pn0 n n0 .
(5.79)
From (5.78) and (5.79) we have pn0 n p0
eφD , = pp0 exp − kB T eφD . = n n0 exp − kB T
(5.80) (5.81)
Under the forward bias (V > 0), by replacing φD with (φD − V ) in (5.80) and (5.81), the hole concentration in the n-region pn and the conduction-electron concentration in the p-region n p are obtained as e (φD − V ) eV = pn0 exp , pn = pp0 exp − kB T kB T e (φD − V ) eV = n p0 exp , n p = n n0 exp − kB T kB T
(5.82) (5.83)
P1: OSO c05 JWBS036-Numai
August 9, 2010
9:14
Printer Name: Yet to Come
PN-JUNCTION
115
where (5.80) and (5.81) are used. Substituting (5.82) and (5.83) into (5.76) leads to eV −1 , pn = pn0 exp kB T eV −1 . n p = n p0 exp kB T
(5.84) (5.85)
The diffusion equations for these excess carrier concentrations in a steady state are expressed as ∂ 2 pn
pn
pn
= ≡ (in an electrically neutral n-region), ∂x2 D pn τ pn L pn 2 ∂ 2 n p ∂x2
=
n p Dn p τn p
≡
n p L np 2
(in an electrically neutral p-region).
(5.86) (5.87)
Here L pn is the diffusion length for a hole in the n-region and L n p is the diffusion length for a conduction electron in the p-region, which are defined as L pn ≡ L np ≡
D pn τ pn ,
(5.88)
Dn p τn p .
(5.89)
Excess Carrier Concentration From (5.86) and (5.87), the excess conductionelectron concentration in the electrically neutral p-region n p and the excess concentration of the hole in the electrically neutral n-region pn are obtained as x + ln eV − 1 exp , pn (x) = pn0 exp kB T L pn x − lp eV . − 1 exp − n p (x) = n p0 exp kB T L np
(5.90) (5.91)
Here, −ln and lp are the x-coordinates of the depletion layer interfaces in n- and p-type semiconductors, respectively. From (5.91) and (5.90), the conduction electron current density in an electrically neutral p-region due to diffusion Jn p (x) and the hole current density in an electrically neutral n-region due to diffusion J pn (x) are given by Dp eV x + ln d pn
, = e n pn0 exp − 1 exp dx L pn kB T L pn dn p Dn p x − lp eV Jn p (x) = −eDn p n p0 exp =e − 1 exp − . dx L np kB T L np
J pn (x) = eD pn
(5.92) (5.93)
P1: OSO c05 JWBS036-Numai
116
August 9, 2010
9:14
Printer Name: Yet to Come
PN- AND PNPN-JUNCTIONS
Saturation Current Density The total electric current density J is given by the sum of Jn p (lp ) and J pn (−ln ) and is expressed as eV J = Jn p (lp ) + J pn (−ln ) = Js exp −1 , kB T
(5.94)
where the saturation current density Js is given by Js = e
Dn p D pn n p0 + pn0 . L np L pn
(5.95)
Diffusion Capacitance When the forward bias is applied to the pn-junction, conduction electrons are injected from the n-region to the p-region and holes are injected from the p-region to the n-region. Both conduction electrons, which are injected to the p-region, and holes, which are injected to the n-region, are minority carriers. Due to the spatial distribution of these minority carriers, the forward electric current flows in the pn-junction and the carriers are accumulated at the interfaces of the depletion layer between the electrically neutral p- and n-regions. For example, from (5.90), the electric charge per unit area in the n-region σd p due to excess concentration of the hole in the n-region pn is given by σd p = e
ln −∞
eV −1 . pn (x) dx = eL pn pn0 exp kB T
(5.96)
From (5.96), the diffusion capacitance per unit area due to the holes Cd p is defined as dσd p e2 L pn pn0 eV = exp − 1 . Cd p ≡ dV kB T kB T
(5.97)
Similarly, the diffusion capacitance per unit area due to the conduction electrons Cdn is defined as Cdn ≡
e2 L n p n p0 eV exp −1 . kB T kB T
(5.98)
Breakdown When the absolute value of the reverse bias voltage exceeds a certain value, electric current flows abruptly from the n-region to the p-region in a pn-junction, as shown in Fig. 5.21. The direction of this electric current at reverse bias is opposite that at forward bias, a phenomenon called breakdown. In avalanche breakdown, collisions between the carriers and atoms occur in succession, and in Zener breakdown,
P1: OSO c05 JWBS036-Numai
August 9, 2010
9:14
Printer Name: Yet to Come
117
PNPN-JUNCTION
Reverse Bias
Forward Bias
FIGURE 5.21 Electric current I as a function of bias voltage V in a pn-junction.
electrons in the valence band penetrate the energy barrier due to the quantum mechanical tunneling effect, and electrons in the valence band transit to the conduction band. Ionization Coefficient The ionization coefficient α is defined as the number of electron–hole pairs that are generated when a carrier travels a unit length. Assuming that collisional ionizations occur in a region with −ln < −lc ≤ x ≤ la < lp , the current multiplication factor M is given by
M = 1−
la −lc
−1 α dx
.
(5.99)
Avalanche breakdown occurs at M = ∞, and this condition is expressed as
la
−lc
5.4
α dx = 1.
(5.100)
PNPN-JUNCTION
Shockley Diode A Shockley diode is a pnpn diode, shown in Fig. 5.22. A Shockley diode is a thyristor in a broad sense and has three pn-junctions: J1 , J2 , and J3 . When the anode voltage V is positive (V > 0), junctions J1 and J3 are forward biased and junction J2 is reverse biased. As a result, when the anode voltage V is lower than the switching voltage Vs , most of the voltage V is applied to junction J2 , and the depletion layer of junction J2 broadens. The energy band under this condition is shown in Fig. 5.23. In junction J2 , there is an energy barrier on the order of eV ,
P1: OSO c05 JWBS036-Numai
118
August 9, 2010
9:14
Printer Name: Yet to Come
PN- AND PNPN-JUNCTIONS
Anode
Cathode
FIGURE 5.22 Shockley diode.
Conduction Electron Reverse Biased
Forward Biased Hole Forward Biased FIGURE 5.23 Energy band of a Shockley diode when anode voltage V is in the range 0 < V < Vs .
where e is the elementary electric charge and V is the anode voltage. In junctions J1 and J3 there are energy barriers on the order of diffusion potentials. Therefore, in the condition band shown in Fig. 5.23, electric current scarcely flows, and this condition is called the OFF-state. When the anode voltage V becomes higher, the electric field that is applied to the depletion layer of junction J2 is higher. When the anode voltage V reaches the switching voltage Vs , avalanche breakover occurs in the depletion layer of junction J2 , leading to switching on, and electric current starts to flow. In this condition, the carriers are injected to junction J2 and the space-charge density in junction J2 goes low, due to the screening effect. As a result, the voltage between the anode and cathode decreases. In junctions J1 and J3 , electric current flows due to diffusion; in junction J2 , electric current flows due to drift. This condition is called the ON-state, and the energy band in this condition is shown in Fig. 5.24. When the anode voltage V is negative (V < 0), junction J2 is forward biased; junctions J1 and J3 are reverse biased. As a result, depletion layers are formed in junctions J1 and J3 , and electric current scarcely flows. When the absolute value of the bias voltage |V | increases further and avalanche breakdown occurs in the depletion layers of junctions J1 and J3 , the carriers just cross junctions J1 and J3 . As a result, space-charge densities of junctions J1 and J3 hardly change. Therefore, the
P1: OSO c05 JWBS036-Numai
August 9, 2010
9:14
Printer Name: Yet to Come
PNPN-JUNCTION
Diffusion
Drift
119
Diffusion Conduction Electron
Hole FIGURE 5.24 Energy band of a Shockley diode after switching.
Avalanche Diffusion Avalanche
Hole
Conduction Electron Reverse Biased Forward Biased Reverse Biased FIGURE 5.25 Energy band of a Shockley diode at breakdown.
voltage between the anode and the cathode is almost equal to the breakdown voltage. The energy band under this condition is shown in Fig. 5.25. Figure 5.26 shows the electric current versus voltage characteristics of a Shockley diode, where Vs is the switching voltage. Punch-Through When a positive voltage is applied to the anode with the cathode grounded in the Shockley diode in Fig. 5.27, let us calculate avalanche breakover voltage VBO and punch-through voltage VPT . The avalanche breakover voltage VBO is the voltage required to start avalanche breakover. The punch-through voltage VPT is the voltage at which the depletion layer of junction J2 contacts the depletion layer of junctions J1 . If VPT is smaller than VBO , the Shockley diode is never switched on. Therefore, we have to design the layer parameters of a Shockley diode so that VBO may be smaller than VPT . Here it is assumed that the avalanche breakover electric field is
P1: OSO c05 JWBS036-Numai
120
August 9, 2010
9:14
Printer Name: Yet to Come
PN- AND PNPN-JUNCTIONS
FIGURE 5.26 Electric current versus voltage characteristics of a Shockley diode.
Anode
Cathode
FIGURE 5.27 Depletion layer in the n1 region of a Shockley diode.
E BO and the thickness of the n1 region is W1 . In addition, it is supposed that the impurity concentration in p2 is much higher than that in n1 , and the depletion layer at the interface between n1 and p2 is formed in the n1 region. Here −l0 and 0 are the x-coordinates of the edges of the depletion layer in the interface between n1 and p2 . In addition, we assume that the impurity in the n1 region consists of donors only, that all donors are ionized completely, and that the depletion layer is depleted completely. The Poisson equation for the depletion layer is written d2 φ(x) eNd =− , 2 dx εn1 ε0
(5.101)
where φ(x) is the electric potential at x, e is the elementary electric charge, Nd is the donor concentration in the n1 region, εn1 is the relative dielectric constant in the n1 region, and ε0 is the permittivity of vacuum. Assuming E x (−l0 ) = 0 as the boundary condition, the x-component of the electric field E x (x) is expressed as E x (x) = −
dφ(x) eNd (x + l0 ) . = dx εn1 ε0
(5.102)
P1: OSO c05 JWBS036-Numai
August 9, 2010
9:14
Printer Name: Yet to Come
121
REFERENCES
Assuming φ (−l0 ) = 0 as the boundary condition, the electric potential φ(x) is obtained as φ(x) = −
eNd 2 eNd (x + l0 )2 . x + 2l0 x + l0 2 = − 2εn1 ε0 2εn1 ε0
(5.103)
If the avalanche breakover electric field E BO is set equal to E x (0), we have l0 =
εn1 ε0 εn1 ε0 E x (0) = E BO , eNd eNd
(5.104)
where (5.102) is used. When the avalanche breakover voltage VBO is applied only to the depletion layer at the interface between n1 and p2 , the avalanche breakover voltage VBO is given by VBO = φ(−l0 ) − φ(0) =
eNd 2 εn1 ε0 l0 = E BO 2 , 2εn1 ε0 2eNd
(5.105)
where (5.103) and (5.104) are used. Note that the switching voltage Vs is almost equal to the avalanche breakover voltage VBO , because most of the voltage applied between the anode and cathode is applied to junction J2 in the OFF-state. The punch-through voltage VPT , which is the voltage at l0 = W1 , is expressed as VPT = φ(−W1 ) − φ(0) =
eNd W1 2 , 2εn1 ε0
(5.106)
where (5.103) is used and the thickness of the depletion layer in junction J1 is neglected. It should be noted again that VBO has to be smaller than VPT to switch on a Shockley diode.
REFERENCES 1. W. Shockley, Electrons and Holes in Semiconductors, D. Van Nostrand, New York, 1950. 2. S. M. Sze, Physics of Semiconductor Devices, 2nd ed., Wiley, New York, 1981. 3. S. M. Sze, Semiconductor Devices: Physics and Technology, 2nd ed., Wiley, New York, 2002. 4. S. M. Sze, Semiconductor Devices: Pioneering Papers, World Scientific, Singapore, 1991. 5. A. S. Grove, Physics and Technology of Semiconductor Devices, Wiley, New York, 1967. 6. J. I. Pankove, Optical Processes in Semiconductors, Dover, New York, 1975. 7. T. Numai, Fundamentals of Semiconductor Lasers, Springer-Verlag, New York, 2004.
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
PART II CONVENTIONAL LASER DIODES
123
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
6 FABRY–PEROT LASER DIODES
6.1
INTRODUCTION
Fabry–Perot Cavity ¯ surfaces are used as two In Fabry–Perot LDs, cleaved facets such as {011} or {011} parallel mirrors, as shown in Fig. 6.1. When light is normally incident on a facet, the power reflectivity R0 is given by R0 =
n rt − 1 n rt + 1
2 ,
(6.1)
where n rt and 1 are the refractive indexes of the semiconductor and the air, respectively. When n rt is 3.5, R0 is about 31%. It should be noted that cleaved facets are flat on the order of atomic layers, much smaller than the wavelength of light. Therefore, cleaved facets can be used as mirrors with very high quality. To control the reflectivities or to protect the facets, dielectric films are often coated on cleaved facets. pn-Junction When carriers are injected into active layers, spontaneous emission is generated, and a fraction of the spontaneous emission, which satisfies the resonance condition Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright
125
P1: OSO c06 JWBS036-Numai
126
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
Cleaved Facet
Active Layer
Laser Beam
Laser Beam
FIGURE 6.1 Fabry–Perot LD.
p-Cladding Layer Active Layer n-Cladding Layer FIGURE 6.2 Cross section of a pn-junction in an LD.
with the wavelength within the optical gain spectrum, is amplified by the stimulated emission. To inject carriers into the active layer, the active layer is placed inside the pnjunction. Therefore, the active layer is sandwiched by the p-cladding layer and the n-cladding layer, as shown in Fig. 6.2. Applying a forward bias voltage which is positive on the p-side and negative on the n-side across this pn-junction, conduction electrons are injected from the n-cladding layer to the active layer and holes are injected from the p-cladding layer to the active layer. When population inversion is generated by carrier injection, net stimulated emission is obtained. Note that the impurities are often undoped in the active layer to achieve high radiation efficiency. However, background carriers whose concentration depends on epitaxial growth methods are present in the active layer. Therefore, the active layer is not an ideal intrinsic semiconductor layer. If impurities are doped in the active layer, the injected carriers combine with the impurities. Therefore, the carrier lifetime is reduced, and the modulation speed is enhanced. However, recombinations of injected carriers and impurities do not contribute to laser transitions, which decreases the radiation efficiency. As a result, the active layer is sometimes doped intentionally to achieve high-speed modulations as long as the radiation efficiency is not highly degraded.
Double Heterostructure A heterojunction is a junction consisting of different semiconductors; a homojunction is a junction composed of common semiconductors. Note that the different semiconductors include semiconductors with different compositions.
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
INTRODUCTION
p-Cladding Layer
Active Layer
127
n-Cladding Layer
(a)
(b) FIGURE 6.3 Double heterostructure: (a) energy of electrons; (b) distribution of refractive index.
The bandgap energies in semiconductors depend on the constituent elements and compositions. As a result, the heterostructures have energy barriers at the junction interfaces, and the carriers are confined in the well layers, due to the energy barriers. To achieve efficient recombinations of conduction electrons and holes, the conduction electrons and holes have to be confined in the active layer. Therefore, heterostructures are formed at both interfaces of an active layer, so the active layer may be the well layer. This structure is called a double heterostructure because double heterojunctions are present. Figure 6.3 shows the distributions of energy and the refractive index of a double heterostructure. At the junction interfaces, band offsets E c for the conduction band and E v for the valence band are present, as shown in Fig. 6.3(a). Under a forward bias, the holes are injected from the p-cladding layer to the active layer, and the energy barrier for the holes is E v at the interface of the n-cladding layer and the active layer; the conduction electrons are injected from the n-cladding layer to the active layer, and the energy barrier for the conduction electrons is E c at the interface of the p-cladding layer and the active layer. The refractive indexes of many semiconductors increase with a decrease in the bandgap energies. Hence, the refractive index of the active layer n a is usually greater than that of the p-cladding layer n p and that of the n-cladding layer n n . As a result, light is efficiently confined in the active layer, which results in a high light amplification rate. A double heterostructure confines both the carriers and light to the active layer. Therefore, a double heterostructure is indispensable to achieving continuous-wave (CW) laser oscillation at room temperature in LDs [1, 2].
P1: OSO c06 JWBS036-Numai
128
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
6.2 RATE EQUATIONS In the following it is assumed that the conduction-electron concentration in an active layer n is equal to the hole concentration in an active layer p. The rate equations for the carrier concentration n and the average photon concentration S = Nph /Vm of the laser light can be written dn J n = − G(n)S − , dt ed τn n S dS + βsp . = a G(n)S − dt τph τr
(6.2) (6.3)
Here Nph is the total number of photons; Vm is the mode volume; J is the injected current density, which is an electric current flowing through a unit area; e is the elementary electric charge; d is the active layer thickness; G(n) is the amplification rate due to the stimulated emission; τn is the carrier lifetime; a is the optical confinement factor of the active layer; τph is the photon lifetime; βsp is the spontaneous emission coupling factor; and τr is the radiative recombination lifetime for a spontaneous emission. Note that the optical confinement factor of the active layer a is given approximately by Va /Vm , where Va is the volume of the active layer. In (6.2), J/ed is an increasing rate of carrier concentration in the active layer due to injection of the carriers, −G(n)S shows a decay rate of the carrier concentration due to the stimulated emission, and −n/τn expresses a decay rate of the carrier concentration in the carrier lifetime τn . In (6.3), a G(n)S shows an increasing rate of photon concentration S due to the stimulated emission, −S/τph is a reduction rate of the photon concentration inside the optical cavity due to the absorption and light emission toward the outside of the optical cavity, and βsp n/τr represents a coupling rate of the spontaneous emission to the lasing mode, which is a resonant mode of the cavity. In the following, G(n), τn , and βsp are explained in detail. When the carrier concentration n is low, the active layer absorbs light. With an increase in n, the active layer has an optical gain, and at n = n 0 the active layer becomes transparent to light for the lasing mode. Using the transparent carrier concentration n 0 , in which a semiconductor is transparent, the amplification rate due to the stimulated emission G(n) can be written approximately as G(n) = g0 (n − n 0 ),
(6.4)
where g0 is the differential gain coefficient. Using the radiative recombination lifetime τr and the nonradiative recombination lifetime τnr , the carrier lifetime τn can be expressed as 1 1 1 = + . τn τr τnr
(6.5)
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
RATE EQUATIONS
129
Note that the radiative recombination lifetime τr is determined only by spontaneous emission and is not affected by stimulated emission. The nonradiative recombination lifetime τnr depends on collisions between the injected carriers and the defects or impurities, and these collisions do not contribute to emit light. The spontaneous emission coupling factor βsp is defined as βsp =
spontaneous emission coupling rate to the lasing mode . total spontaneous emission rate
(6.6)
When the spontaneous emission spectrum is assumed to be Lorentzian with center angular frequency ω0 and the FWHM ω, the spontaneous emission coupling rate to the lasing mode per unit time and unit volume rsp is given by rsp = rsp0
(ω/2)2 , (ω − ω0 )2 + (ω/2)2
(6.7)
where rsp0 is a coefficient. To calculate the total spontaneous emission rate, we consider the number of modes dN with two polarizations that exist in a volume V : a solid angle for propagation direction d and an angular frequency range dω. When the distribution of the modes is continuous, as in a free space, dN is given by n r 3 ω2 d d = V 2 3 dω , 4π π c 4π n r 3 ω2 m(ω) = 2 3 , π c
dN = V m(ω) dω
(6.8) (6.9)
where m(ω) is the mode density. From (6.7) and (6.8), the total spontaneous emission rate Rsp is obtained as Rsp =
rsp dN = rsp0
V n r 3 2 ω0 ω. 2π c
(6.10)
Using (6.6), (6.7), and (6.10), the spontaneous emission coupling factor βsp for the angular frequency ω0 (a wavelength in vacuum λ0 ) can be expressed as βsp = a
rsp 2π = a Rsp V
c nr
3
λ0 4 1 a = , ω0 2 ω 4π 2 n r 3 V λ
(6.11)
where a is the optical confinement factor of the active layer and λ is the FWHM in units of wavelength. From (6.11) it is found that the spontaneous emission coupling factor βsp increases with a decrease in the volume V and the spectral linewidth λ.
P1: OSO c06 JWBS036-Numai
130
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
6.3 CURRENT VERSUS VOLTAGE CHARACTERISTICS In a steady state, (6.2) and (6.3) are reduced to n 1 , S = −βsp τr a G(n) − 1/τph n . I = eVa G(n)S + τn
(6.12) (6.13)
Here I is the injected current and Va = Sa d is the volume of the active layer, where Sa is the area of the active layer and d is the thickness of the active layer. A flowing current in an LD consists of the diffusion current, the drift current, and the recombination current. Here it is assumed that the radiative recombination is dominant and the diffusion and drift current are neglected, as shown in (6.13). Under the forward bias (V > 0), the hole concentration in the n-region, pn , exceeds the steady-state value, pp0 . Using pn and the conduction-electron concentration in the n-region n n n n0 , where n n0 is the steady-state value, we have np = n n pn = n i 2 exp
eV kB T
,
(6.14)
where (5.79) and (5.82) are used. As a result, the carrier concentration n is given approximately by n = n i exp
eV 2kB T
.
(6.15)
From (6.12)–(6.15), the injected current I is related to the voltage V as n i eeV /2kB T G(n i eeV /2kB T ) n i eeV /2kB T I = eVa −βs , + τr a G(n i eeV /2kB T ) − 1/τph τn
(6.16)
which is shown in Fig. 6.4, where it is assumed that the photon lifetimes τph is 1 ps; the sum of a contact resistance and a bulk resistance is 4 , which is typical in conventional LDs. Other physical parameters are βs = 10−5 , n i = 2.7 × 1011 cm−3 , τr = τn = 1 ns, T = 293.15 K (20◦ C), a = 0.1, (∂G/∂n)n=n th = 2.5 × 10−6 cm3 /s, and Va = 40 µm3 = 4 × 10−11 cm3 . It is also supposed that the transparent carrier concentration is n 0 = 0.6 n th , where n th is the threshold carrier concentration, the effective refractive index is 3.5, and the group velocity vg is 8.57 × 109 cm/s.
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
131
Injected Current
CURRENT VERSUS LIGHT OUTPUT CHARACTERISTICS
Applied Voltage FIGURE 6.4 Current versus voltage (I –V ) characteristics.
6.4
CURRENT VERSUS LIGHT OUTPUT CHARACTERISTICS
With an increase in the current injected into an LD, the carrier concentration n in the active layer is enhanced. When the carrier concentration n exceeds the threshold carrier concentration n th , laser oscillation starts and the light output increases drastically over that below the threshold. This change in light output is considered to be an abrupt increase in the photon concentration. The current versus light output (I –L) characteristics can be analyzed by the rate equations on the carrier concentration n and the photon concentration S in the active layer.
Threshold Current Density Using the rate equations, the threshold current density Jth is calculated in the following. At first, the rate equations below the threshold are considered. For simplicity, it is assumed that net stimulated emission is negligible and S = 0 below the threshold. Under this assumption, (6.2) is reduced to J n dn = − . dt ed τn
(6.17)
From (6.17), the carrier concentration n in a steady state (d/dt = 0) is given by n=
J τn . ed
(6.18)
P1: OSO c06 JWBS036-Numai
132
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
When the carrier concentration n increases from 0 to the threshold carrier concentration n th , it is expected that (6.18) is still valid at the threshold. As a result, the threshold current density Jth is expressed as Jth =
ed n th . τn
(6.19)
Because the optical confinement factor a of the active layer depends on the active layer thickness d, the threshold current density n th is a function of d and an optimum d value is present to achieve the lowest Jth . Second, the threshold carrier concentration n th is calculated using rate equations above the threshold. In most LDs the spontaneous emission coupling factor βsp is on the order of 10−5 . Therefore, as a first approximation, the term βsp n/τr in (6.3) is neglected. Because (6.3) is valid for any S value in a steady state, we have a G(n) = a g0 (n − n 0 ) =
1 , τph
(6.20)
where (6.4) is used.
Threshold Gain The amplification rate due to the stimulated emission G(n) at the threshold is expressed by the threshold gain, which is the optical gain required for laser oscillation. The threshold gain is obtained by using the resonance condition of the Fabry–Perot cavity. From (4.10), the ratio of the transmitted light intensity It to the incident light intensity I0 is written It T1 T2 G s0 , = √ √ I0 (1 − G s0 R1 R2 )2 + 4G s0 R1 R2 sin2 (n r k0 L)
(6.21)
G s0 = exp(2gE L) = exp(gL).
(6.22)
where
Here T1 and T2 are the power transmissivities of the facets, R1 and R2 are the power reflectivities of the facets, n r is the effective refractive index, L is the cavity length, gE is the field optical gain coefficient for the electric field, and g is the power optical gain coefficient. Oscillation represents a state in which an output is present without an input from outside. As a result, the oscillation condition is given by I0 = 0 and It > 0 in (6.21). Therefore, at the oscillation condition, the denominator in (6.21) is 0. Hence, the
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
CURRENT VERSUS LIGHT OUTPUT CHARACTERISTICS
133
oscillation condition for Fabry–Perot LDs is expressed as Resonance condition: Gain condition:
sin(n r k0 L) = 0, √ 1 − G s0 R1 R2 = 0.
(6.23)
Resonance Condition From (6.23), the resonance condition is written n r ωL = n r k0 L = nπ, c
(6.24)
where n is a positive integer. Using a wavelength in vacuum λ0 , (6.24) is reduced to L=n
λ0 , 2n r
(6.25)
which is the same as the resonance condition of a Fabry–Perot cavity. Note that laser oscillation starts at the resonant wavelength nearest the gain peak. Gain Condition From (6.23), the gain condition is obtained as 1 − G s0 R1 R2 = 1 − R1 R2 exp(gL) = 0.
(6.26)
As a result, the power optical gain coefficient g is written g=
1 1 , ln √ L R1 R2
(6.27)
where the right-hand side is called the mirror loss. Guided modes propagate in optical waveguides while they are confined in the film (active layer), and the fields of guided modes penetrate the p- and n-cladding layers, as shown in Fig. 6.5. As a result, the light sees optical losses in the p- and n-cladding layers. Therefore, the power optical gain coefficient g for the entire region consists of optical gain in the active layer and optical losses in the active and cladding layers. Using optical confinement factors, the power optical gain coefficient, and optical power optical loss coefficients, the power optical gain coefficient g of the optical waveguide is written approximately as g = a ga − a αa − p αp − n αn .
(6.28)
Here a , p , and n are the optical confinement factors of the active, p-cladding, and n-cladding layers, respectively; ga is the power optical gain coefficient of the
P1: OSO c06 JWBS036-Numai
134
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
Light Intensity p-Cladding Layer Optical Waveguide
Active Layer n-Cladding Layer
FIGURE 6.5 Distribution of light intensity in an optical waveguide.
active layer; and αa , αp , and αn are the power optical loss coefficients of the active, p-cladding, and n-cladding layers, respectively. Note that the exact g value is obtained by solving eigenvalue equations, which include complex refractive indexes. The internal loss αi is defined as αi = a αa + p αp + n αn .
(6.29)
Substituting (6.29) into (6.28) leads to g = a ga − αi ,
(6.30)
where a ga is the modal gain. Inserting (6.30) into (6.27) results in a ga = αi +
1 1 1 1 ln √ ln = αi + , L 2L R1 R2 R1 R2
(6.31)
which is the threshold gain of Fabry–Perot LDs. If the light propagates along the z-axis, a derivative of the photon concentration S with respect to a time t is given by dz dS c dS dS = = . dt dt dz n r dz
(6.32)
From (6.31) and (6.32), G(n) is expressed as G(n) =
c ga , nr
(6.33)
where n r is the effective refractive index and c is the speed of light in vacuum. From (6.20) and (6.33), the photon lifetime τph is expressed as 1 c = τph nr
1 1 αi + ln 2L R1 R2
.
(6.34)
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
CURRENT VERSUS LIGHT OUTPUT CHARACTERISTICS
135
From (6.20), the carrier concentration n in a steady state is given by n=
1 + n0. a g0 τph
(6.35)
Because (6.35) holds even at the threshold, the threshold carrier concentration n th is written n th =
1 + n0. a g0 τph
(6.36)
In LDs, changes in the cavity length, facet reflectivities, and refractive indexes during laser operation are small. As a result, the right-hand sides of (6.35) and (6.36) are considered to be constant. Therefore, above the threshold, the carrier concentration n is clamped on the threshold carrier concentration n th when coupling of the spontaneous emission to the lasing mode is neglected. Accordingly, G(n) is constant above the threshold as long as the gain saturation and coupling of the spontaneous emission to the lasing mode are neglected. Substituting (6.36) into (6.19) results in Jth =
ed n th = J A + J B , τn
(6.37)
where JA =
ed n0, τn
JB =
1 ed . τn a g0 τph
(6.38)
From (6.37) and (6.38) it is found that the threshold current density Jth depends explicitly on the optical confinement factor a . Figure 6.6 shows the results calculated for the optical confinement factor a for Alx Ga1−x As/GaAs double heterostructures. With an increase in the active layer thickness d, a is enhanced. Note that a is proportional to d 2 when d is small. Figure 6.7 shows the threshold current density Jth as a function of the active layer thickness d. The threshold current density Jth takes a minimum value when d is approximately 0.1 µm. In Fig. 6.7, J A is a current density which is required to obtain the population inversion and J B is a current density in which the optical gain balances the loss in the optical cavity. The current density J A is proportional to the active layer thickness d as in (6.38); for a thin active layer, J B is inversely proportional to d, because a is proportional to d 2 . Because Jth is given by J A + J B , an optimum d value is present for a minimum Jth .
P1: OSO c06 JWBS036-Numai
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
Optical Confinement Factor
136
August 9, 2010
Active Layer Thickness
Threshold Current Density
FIGURE 6.6 Optical confinement factor a as a function of active layer thickness d.
Active Layer Thickness FIGURE 6.7 Threshold current density Jth as a function of active layer thickness d.
I–L Characteristics in CW Operation Without Coupling of Spontaneous Emission to the Lasing Mode When the spontaneous emission coupling factor βsp is small, coupling of the spontaneous emission to the lasing mode can be neglected. Under this condition, I –L characteristics in CW operation are analyzed using the rate equations (6.2) and (6.3). Below the threshold, the carrier concentration n increases with an increase in J , according to (6.18); the photon concentration S is 0. Above the threshold, n no longer increases and remains at the threshold carrier concentration n th ; S increases with J , because the excess carriers are converted to photons.
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
CURRENT VERSUS LIGHT OUTPUT CHARACTERISTICS
137
The steady-state photon concentration S above the threshold is obtained by setting dn/dt = 0 in (6.2) and is written 1 S= G(n)
J n th − ed τn
.
(6.39)
Substituting (6.19) and (6.20) into (6.39) results in S=
a τph (J − Jth ). ed
(6.40)
From the results above, the dependence of the carrier concentration n and photon concentration S on the injected current density J is summarized as follows: For J < Jth , n=
J τn , ed
Jth τn , ed
S=
S = 0;
(6.41)
and for J ≥ Jth , n=
a τph (J − Jth ). ed
(6.42)
The results calculated for (6.41) and (6.42) are shown in Fig. 6.8, where it is clearly indicated that the carrier concentration n is clamped on n th above the threshold current density Jth . With Coupling of Spontaneous Emission to the Lasing Mode For simplicity it is assumed that the nonradiative recombination is negligible, which leads to τr ≈ τn . In a steady state (d/dt = 0), when the coupling of spontaneous emission to the lasing
FIGURE 6.8 Carrier concentration n and photon concentration S as a function of injected current density J when coupling of the spontaneous emission to the lasing mode is neglected.
P1: OSO c06 JWBS036-Numai
138
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
FIGURE 6.9 Carrier concentration n and photon concentration S as a function of the injected current density J when coupling of the spontaneous emission to the lasing mode is included.
mode is included, (6.2) and (6.3) are reduced to J n = g0 (n − n 0 )S + , ed τn n S = a g0 (n − n 0 )S + βsp , τph τn
(6.43) (6.44)
where (6.4) is used. The results calculated for (6.43) and (6.44) are shown in Fig. 6.9, where the solid and dashed lines correspond to βsp > 0 and βsp = 0, respectively. Coupling the spontaneous emission to the lasing mode lowers n and enhances S, which results in a vague threshold. Note that the light emitted below the threshold is incoherent or amplified spontaneous emission. Radiation Efficiency According to Figs. 6.8 and 6.9, the current versus light output (I –L) characteristics of LDs look like Fig. 6.10, where Ith is the threshold current. To evaluate the radiation
FIGURE 6.10
I –L characteristics.
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
CURRENT VERSUS LIGHT OUTPUT CHARACTERISTICS
139
efficiency of LDs, slope efficiency and external differential quantum efficiency are often used. Slope Efficiency The slope efficiency per facet Sd j is defined as the ratio of the increase in light output power P j ( j = 1, 2) to the increase in injected current I , which is given by Sd j =
P j . I
(6.45)
From (6.45), Sd j is expressed in units of mW/mA or W/A. The slope efficiency for the total light output Sd,tot is obtained as Sd,tot =
P P1 + P2 = , I I
(6.46)
where P is an increase in the total light output power. External Differential Quantum Efficiency The external differential quantum efficiency ηd is defined as the number of photons emitted outward per carrier injected. The number of photons emitted outward per second is P/ω, where ω is an angular frequency of the light and is Dirac’s constant. The carrier injected per second is I /e, where e is the elementary electric charge. Therefore, the external differential quantum efficiency ηd for the total light output is given by ηd =
I P e e P ÷ = = Sd,tot . ω e I ω ω
(6.47)
From(6.47), ηd is expressed in no units. As shown in (6.31), the total loss is a sum of the internal loss and the mirror loss. When the light output is measured outside the optical cavity, the mirror loss indicates the light emission rate from the optical cavity. Using the internal quantum efficiency ηi , which is defined as the number of photons emitted inside the optical cavity per carrier injected, the external differential quantum efficiency ηd is expressed as
mirror loss = ηi ηd = ηi total loss
1 1 1 ln ln 2L R1 R2 R1 R2 = ηi , 1 1 1 ln αi + 2αi L + ln 2L R1 R2 R1 R2
(6.48)
where the optical losses at the facets due to absorption or scattering are assumed to be negligible.
P1: OSO c06 JWBS036-Numai
140
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
FIGURE 6.11 Optical powers inside and outside a Fabry–Perot cavity.
Light Output Ratio from Facets Optical powers, which are emitted from the two facets, are P1 and P2 . As shown in Fig. 6.11, it is assumed that the optical powers in the vicinity of the facets inside a Fabry–Perot cavity are Pa , Pb , Pc , and Pd ; the power reflectivities of the facets are R1 and R2 ; and the power transmissivities of the facets are T1 and T2 . Here, the arrows indicate the propagation directions of the light. In a steady state, the optical powers are related as Pa = R1 Pd , Pb = egL Pa , Pc = R2 Pb , Pd = egL Pc , P1 = T1 Pd , P2 = T2 Pb ,
(6.49)
where g is the power optical gain coefficient and L is the cavity length. Deleting Pk (k = a, b, c, d) from (6.49) results in √ P1 T1 R2 = √ . P2 T2 R1
(6.50)
From (6.50), to extract a large light output P j from a facet j, the power transmissivity of the facet T j should be high; the power reflectivity of the facet R j should be low. Substituting P = P1 + P2 into (6.49) leads to √ T1 R2 P1 = √ √ P, T1 R2 + T2 R1
√ T2 R1 P2 = √ √ P. T1 R2 + T2 R1
(6.51)
As a result, the external differential quantum efficiencies for each light output ηd1 and ηd2 are written ηd1
√ T1 R2 = √ √ ηd , T1 R2 + T2 R1
ηd2
√ T2 R1 = √ √ ηd , T1 R2 + T2 R1
(6.52)
where ηd is the external differential quantum efficiency for the total light output. When the optical losses at the facets are negligibly small, we can use T1 = 1 − R1 and T2 = 1 − R2 .
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
CURRENT VERSUS LIGHT OUTPUT CHARACTERISTICS
141
Light Output
AlGaAs/GaAs LD InGaAsP/InP LD
Pulsed
Injected Current FIGURE 6.12
I –L characteristics with temperature as a parameter.
Dependence of I–L on Temperature With a rise in temperature, the threshold current density Jth usually increases and the external differential quantum efficiency ηd decreases, as shown in Fig. 6.12. The reason for a decrease in the external quantum efficiency ηd with an increase in Jth is that the threshold carrier concentration n th increases with Jth , which enhances the free carrier absorption, which is derived below. An equation of motion for the electron in a crystal without a magnetic field is given by m∗
d2 x 1 dx = −eE, + m∗ dt 2 τ dt
(6.53)
where m ∗ is the effective mass of the electron; τ is the relaxation time, such as the mean free time of collision; e is the elementary electric charge; and E is the electric field. If we assume x, E ∝ e i ωt , where ω is the angular frequency of light, a position x of the electron is obtained as x=
m∗
eE
. − i ω/τ
ω2
(6.54)
The polarization P of a semiconductor is written P = P0 + Pi ; Pi = −nex,
(6.55)
P1: OSO c06 JWBS036-Numai
142
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
where P0 is the polarization caused by ionization of atoms constituting the crystal and Pi is the polarization induced by the motion of the electrons. Here we express the electric flux density D as D = ε0 E + P = (ε0 E + P0 ) + Pi = εε0 E + Pi = ε ε0 E,
(6.56)
where ε0 is the permittivity of vacuum and ε is the dielectric constant of the crystal based on ionization of the atoms. The dielectric constant ε , which is modified by the motion of the electrons, is written ε = εr − i εi = ε −
nex , ε0 E
(6.57)
where εr and εi are real and imaginary parts of ε , respectively. Substituting (6.54) into (6.57) leads to ne2 , m ∗ ω2 ε0 ne2 , εi ∗ 3 m ω ε0 τ
εr − ε −
(6.58) (6.59)
where ω 1/τ was used. Using the complex refractive index n r − i κ, we can express εr and εi as εr = n r 2 − κ 2 , εi = 2n r κ.
(6.60) (6.61)
Therefore, the optical power absorption coefficient α due to free carrier absorption is obtained as α=
ω εi ne2 2ω κ= , = ∗ 2 c c nr m ω ε0 n r cτ
(6.62)
where c is the speed of light in vacuum. Characteristic Temperature Dependence of the threshold current density Jth on temperature is expressed empirically as
Tj Jth = Jth0 exp T0
,
(6.63)
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
CURRENT VERSUS LIGHT OUTPUT CHARACTERISTICS
143
InGaAsP/InP LD AlGaAs/GaAs LD Pulsed
Temperature FIGURE 6.13 Dependence of threshold current density on temperature.
where Jth0 is a coefficient; Tj is the temperature in the active layer or junction temperature; and T0 , called the characteristic temperature, represents the dependence of the threshold current density on the temperature. A large characteristic temperature T0 seems to result in a small dJth /dTj value, which indicates a high-quality LD. However, T0 cannot always be used as an appropriate figure of merit of LDs, because a larger Jth leads to a greater T0 when dJth /dTj is constant. Therefore, only LDs with a common Jth at the same temperature can be compared with each other by using T0 . Figure 6.13 shows the dependence of the threshold current density on the temperature for an AlGaAs/GaAs LD and an InGaAsP/InP LD in pulsed operations. In an AlGaAs/GaAs LD with an oscillation wavelength of 0.85 µm, T0 is approximately 160 K from 25 to 80◦ C and 120 K above 80◦ C. It is considered that an increase in Jth with Tj is caused by broadening of the gain spectrum and the overflow of the carriers over the heterobarriers. To reduce the overflow of the carriers over the heterobarriers, the band offset E g between the active layer and the cladding layers has to be increased. In general, the band offset E g should be larger than 0.3 eV to suppress an increase in Jth with Tj . In an InGaAsP/InP LD with an oscillation wavelength of 1.3 µm, T0 is approximately 70 K from 25 to 65◦ C and 50 K above 65◦ C. These T0 values are lower than those of the AlGaAs/GaAs LD because of efficient overflow of the carriers due to the light effective mass of the conduction electrons and nonradiative recombinations due to Auger processes and valence band absorptions. The effective masses of the conduction electrons are 0.070m 0 for AlGaAs with a bandgap wavelength of 0.85 µm, and 0.059m 0 for InGaAsP with a bandgap wavelength of 1.3 µm, where m 0 is the electron mass in vacuum. Auger processes are shown in Fig. 6.14, where C, H, L, and S represent the conduction, heavy hole, light hole, and split-off bands, respectively. In Auger processes, two processes, such as CHSH and CHCC, are present. In the
P1: OSO c06 JWBS036-Numai
144
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
Conduction Band (C)
Heavy Hole Band (H) Light Hole Band (L) Split-off Band (S) (a)
(b)
FIGURE 6.14 Auger processes: (a) CHSH; (b) CHCC.
CHSH process, the energy emitted due to a recombination of conduction electron 1 in the conduction band (C) and hole 2 in the heavy hole band (H) excites valence electron 3 in the split-off band (S) to energy state 4 in the heavy hole band (H). In the CHCC process, the energy emitted due to a recombination of conduction electron 1 in the conduction band (C) and hole 2 in the heavy hole band (H) excites conduction electron 3 in the conduction band (C) to higher-energy state 4 in the conduction band (C). These processes are three-body collision processes, and a recombination rate RA for Auger processes is given by RA = Cp np 2 + Cn n 2 p,
(6.64)
where Cp and Cn are Auger coefficients for the CHSH and CHCC processes, respectively. The valence band absorptions are shown in Fig. 6.15, where an electron in the split-off band absorbs light which is generated by a recombination of a conduction electron in the conduction band and a hole in the heavy hole band. As a result, the electron in the split-off band is excited to the heavy hole band or the acceptor level. With an increase in temperature, the internal quantum efficiency ηi is decreased due to Auger processes and the external quantum efficiency ηd is reduced due to the valence band absorptions and free carrier absorption. Therefore, light emission efficiency is lowered with an increase in temperature. Derivative Light Output To detect the threshold current precisely, derivative measurements can be used. Using the photon concentration S in the active layer, the internal optical power P can be written P = ωvg SB S,
(6.65)
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
CURRENT VERSUS LIGHT OUTPUT CHARACTERISTICS
145
Conduction Band (C)
Acceptor Level Heavy Hole Band (H) Light Hole Band (L) Split-off Band (S) (a)
(b)
FIGURE 6.15 Valence band absorption: an electron is excited to (a) the heavy hole band and (b) the acceptor level.
where = h/2π is Dirac’s constant, h is Planck’s constant, ω is the angular frequency of light, vg is a group velocity of the light, and SB is a beam area. From (6.65), the derivative optical power with respect to the current injected, dP/dI , is written dS ∂ S ∂n dP = ωvg SB = ωvg SB , dI dI ∂n ∂ I
(6.66)
Light Output
where I is the current injected and n is the carrier concentration. Figure 6.16 shows calculated light output and derivative light output as a function of the current injected where (6.12), (6.13), (6.65), and (6.66) are used. The threshold current is indicated clearly by a sharp rise in dP/dI .
Injected Current FIGURE 6.16 Current versus light output and derivative light output.
P1: OSO c06 JWBS036-Numai
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
Light Output
146
August 9, 2010
Injected Current (a)
Injected Current (b)
FIGURE 6.17 (a) I –L and (b) I –dV /dI characteristics.
Derivative Electrical Resistance The derivative resistance dV /dI is expressed as ∂ V ∂n dV = , dI ∂n ∂ I
(6.67)
where V is the voltage that is applied to the LD. Figure 6.17(a) shows calculated I –L characteristics and Fig. 6.17(b) shows calculated I –dV /dI characteristics where (6.15), (6.16), and (6.67) are used. As shown in Fig. 6.17, each I –dV /dI curve has a kink at the threshold current. Therefore, the threshold current Ith can be determined by the derivative electrical resistance dV /dI , and this method will be especially useful for ring or disk LDs with extremely low light output. 6.5 POLARIZATION OF LIGHT Fabry–Perot LDs with bulk active layers oscillate in the TE mode. Because the bulk active layers do not have particular quantum mechanical axes, the optical gains for
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
POLARIZATION OF LIGHT
147
Active Layer Effective Refractive Index
Refractive Index
Refractive Index
(a)
Refractive Index
(b) FIGURE 6.18 Reflection at a facet.
bulk active layers are independent of the polarization of light. However, the facet reflectivities depend on the polarization of light. Figure 6.18(a) shows the reflection at a facet of an LD using a zigzag model in geometrical optics. We suppose that the effective refractive index of the optical waveguide is n A and that the refractive index of the outside of the optical waveguide is n B . This reflection is also considered as the reflection at the interface of a semiconductor with refractive index n A = n A / cos θ and that with the refractive index n B , as shown in Fig. 6.18(b), where θ is the angle of incidence. Fresnel formulas give us the power reflectivities RTE for the TE mode and RTM for the TM mode:
RTE
RTM
n cos θ − n 2 − n 2 sin2 θ 2 B A A = , n A cos θ + n B 2 − n A 2 sin2 θ n 2 cos θ − n n 2 − n 2 sin2 θ 2 A B A B = , n B 2 cos θ + n A n B 2 − n A 2 sin2 θ
(6.68)
(6.69)
Figure 6.19 shows the power reflectivity R when laser beams are emitted from GaAs with the refractive index n A = 3.6 to the air with n B = 1. As shown in Fig. 6.19, it is found that RTE ≥ RTM is kept for all values of θ . Therefore, from (6.31), the threshold gains for TE modes are lower than those of TM modes, which leads to lasing in the TE modes with lower threshold current than for the TM modes. With an increase in θ (i.e., with an increase in the order of vertical transverse modes), RTE is enhanced. Therefore, the threshold gains of higher-order vertical transverse modes are smaller than those of lower-order vertical transverse modes. However, from Fig. 6.7, to minimize the threshold current density, the active layer thickness d should be approximately 0.1 µm or less, in which the higher-order vertical transverse modes are cut off and only the fundamental vertical transverse mode oscillates. Note that the optical power ratio of the TE and TM modes is approximately 1 : 1 below the threshold, because most of the light emitted below the threshold is not guided light but spontaneously emitted light due to amplified spontaneous emission.
P1: OSO c06 JWBS036-Numai
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
Power Reflectivity
148
August 9, 2010
GaAs Air
TE
TM
Angle of Incidence FIGURE 6.19 Reflectivities for TE and TM modes.
6.6 TRANSVERSE MODES Transverse modes, also called lateral modes, show the light intensity distributions along the axes perpendicular to the cavity axis, and represent the shapes of the laser beams. Transverse modes are highly dependent on the structures of optical waveguides. Vertical transverse modes display light intensity distributions along the axes perpendicular to the active layer plane; horizontal transverse modes exhibit light intensity distributions along the axes parallel to the active layer plane. To examine transverse modes, near-field patterns (NFPs) and far-field patterns (FFPs), which are shown in Fig. 6.20, are generally used. An NFP is a light intensity Vertical NFP
Near-Field Pattern (NFP)
Beam Size
Horizontal NFP Beam Size
Vertical FFP
Facet of a LD
Radiation Angle
Far-Field Pattern (FFP) Horizontal FFP Radiation Angle
FIGURE 6.20 Near- and far-field patterns.
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
TRANSVERSE MODES
149
distribution on a facet of an LD. The sizes of the NFP are measured in units of the length of the emission regions W and W⊥ , illustrated in Fig. 6.20. Usually, the active layer thickness d is approximately 0.1 µm or less to achieve low threshold current; the active layer width is on the order of 2 µm or more to obtain high light output. As a result, the near-field pattern is asymmetric, which is long along the parallel axis and short along the perpendicular axis to the active layer plane. An FFP is a light intensity distribution at a position far enough from a facet of an LD. As shown in Fig. 6.20, the sizes of the FFP are measured in units of radiation angles θ and θ⊥ , which are independent of the distance between a facet of an LD and a photodetector. An FFP is considered a diffracted pattern of an NFP if the NFP is regarded as light emitted from a slit. With decreases in the sizes of the slit, the sizes of the diffracted patterns increase. Therefore, an FFP is large for a small NFP and small for a large NFP. Because of asymmetry in an NFP, an FFP is also asymmetric, with a small horizontal transverse mode and a large vertical transverse mode. To achieve a large coupling efficiency of a laser beam to a lens or an optical fiber, a symmetric laser beam with a narrow radiation angle is required. These requirements can be accomplished using vertical cavity surface-emitting lasers. Vertical Transverse Modes Guided Modes Because double heterostructures are adopted in LDs, light is confined in an area sandwiched by the planes, which are parallel to the active layer plane, due to index guiding. The guiding condition of vertical transverse modes is represented by the eigenvalue equation for the transverse resonance condition. Laser Oscillation in Higher-Order Modes Active Layer Thickness d hc The optical confinement factors m for the mthorder mode and m−1 for the (m − 1)th-order mode are related as m−1 > m .
(6.70)
When the active layer thickness d is slightly larger than the cutoff guiding layer thickness h c , the difference in the reflectivities between the adjacent higher-order modes is small. Therefore, from (6.37) and (6.38), the threshold current densities Jth,m for the mth-order mode and Jth,m−1 for the (m − 1)th-order mode are related as Jth,m−1 < Jth,m ,
(6.71)
which leads to a laser oscillation in the (m − 1)th-order mode, not the mth-order mode. Active Layer Thickness d hc As shown in Fig. 6.19, higher-order TE modes have larger reflectivities than those of lower-order TE modes. Therefore, when the active layer thickness d is much larger than the cutoff guiding layer thickness h c , a laser
P1: OSO c06 JWBS036-Numai
150
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
oscillation does not take place in an (m − 1)th-order mode but does in an mth-order mode. For example, in an AlGaAs/GaAs LD with n f = 3.6, (n f − n s )/n f = 5%, and λ0 = 0.85 µm, the cutoff guiding layer thickness h cm for the mth-order mode is obtained as h c1 = 0.38 µm, h c2 = 0.76 µm, h c3 = 1.13 µm. In this case, according to the active layer thickness d, the following laser oscillations in higher-order modes take place: ⎧ ⎨ 0.66 µm: 0.98 µm: ⎩ 1.30 µm:
first-order mode second-order mode third-order mode
In conventional LDs, the active layer thickness d is 0.1 µm or less, which leads to laser oscillations in the fundamental vertical transverse mode (m = 0). NFP and FFP When the active layer thickness d is larger than the wavelength of light in a semiconductor, the full width at half maximum (FWHM) of the beam size of a vertical NFP W⊥ decreases with a reduction in d because the light emission region narrows. However, when d is less than the wavelength of light in a semiconductor, W⊥ increases with a reduction in d because the guided light penetrates deeply into the cladding layer and the substrate. Figure 6.21 shows the results calculated for the FWHM of the beam size of a vertical NFP W⊥ and the FWHM of the radiation angle of a vertical FFP θ⊥ for the fundamental TE mode (m = 0) as a function of the active layer thickness d with = (n f − n s )/n f as a parameter. Here n f and n s are the refractive indexes of the active layer and substrate, respectively. Because the FFP is considered to be
Active Layer Thickness FIGURE 6.21 Sizes of NFP and FFP.
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
TRANSVERSE MODES
151
a diffracted pattern of the NFP, the FFP is large for a small NFP and small for a large NFP.
Horizontal Transverse Modes Horizontal transverse modes show light intensity distributions along the axes parallel to the active layer plane. To control horizontal transverse modes, gain guiding and index guiding have been used. In gain guiding, light propagates only in the optical gain region. In index guiding, light propagates in a high-refractive-index region which is surrounded by low-index regions. Gain guiding structures can be fabricated by a simpler method compared with index guiding structures, but the horizontal mode is unstable and the threshold current is higher than in index guiding structures. Gain Guiding In gain guiding structures, optical gain regions are formed by restricting the current flowing area. For example, electrodes are formed selectively, as shown in Fig. 6.22(a), which is a cross-sectional view of a gain guiding LD seen from a facet. The current injected flows from the selectively formed electrode along the arrows by diffusion. As a result, the carrier concentration is largest at the center of the stripe, and it decreases with an increase in distance from the center, as shown in Fig. 6.22(b). Therefore, the center region of the stripe has optical gain and the stripe edges have optical losses, as shown in Fig. 6.22(c). With an increase in the current injected, the refractive index changes due to the free carrier plasma effect, Joule heating in the active layer, and spatial hole burning.
Anode Active Layer Cathode (a) Carrier Concentration
Position (b)
Optical Gain Position Optical Loss (c) FIGURE 6.22 Gain guiding LD: (a) cross-sectional view; (b) distribution of carrier concentration; (c) distribution of optical gain.
P1: OSO c06 JWBS036-Numai
152
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
Free Carrier Plasma Effect When light is present in a semiconductor, free carriers in the semiconductor vibrate with the frequency of the light; the phases of the free carrier vibrations cancel out the electrical polarizations of the lattice atoms. This phenomenon is called the free carrier plasma effect. A change in the refractive index n rf due to the free carrier plasma effect is given in the following. When the carrier concentration increases by n, it is assumed that the real part of the dielectric constant εr and the real part of the complex refractive index n r change to εr + εr and n r + n r , respectively. In this case we have 2n r n r = εr .
(6.72)
Hence, we obtain n r =
εr εr − ε e2 = =− ∗ 2 n, 2n r 2n r 2m ω ε0 n r
(6.73)
where (6.58) is used. Equation (6.73) is proportional to the injected carrier concentration n. When the carrier concentration n is on the order of 1018 cm−3 , a decrease in the refractive index is on the order of 10−3 . Because the carrier concentration is greatest at the center of the stripe, the refractive index in the center is lower than that of the surrounding regions, according to (6.73). As a result, the light is not confined completely to the active layer but is radiated to the surrounding regions, which is called the antiguiding effect. Joule Heating With an increase in current injected, the active layer is heated by Joule heating. Therefore, the refractive index increases; from experimental results, this change n rT is given by n rT = (2 ∼ 5) × 10−4 T,
(6.74)
where T is an increase in the temperature of the active layer, which is expressed in kelvin units. As opposed to the free carrier plasma effect, an increase in the refractive index due to Joule heating of the active layer contributes to confining the light to the active layer, which is called the guiding effect. Spatial Hole Burning When the injected current increases further and a large light output is obtained, a lot of carriers recombine, due to the stimulated emission. Because the stimulated emission takes places efficiently in a large optical gain region, which is the center of the stripe, the carrier concentration at the center of the stripe is lower than that at its surrounding regions. This phenomenon, called spatial hole burning, increases the refractive index at the center region and enhances the guiding effect, resulting in a large confinement of the light to the active layer. The horizontal distribution of the refractive index is determined by both the antiguiding effect, which is due to the free carrier plasma effect, and the guiding effect,
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
TRANSVERSE MODES
153
Kink Kink
FIGURE 6.23 Kink in an I –L curve.
which is due to Joule heating of the active layer and spatial hole burning. Hence, according to the current injected, the horizontal transverse modes show complicated behaviors, such as changes in positions or multiple peaks. When the horizontal transverse modes change, kinks are observed in I –L curves, as shown in Fig. 6.23. In gain guiding structures, the wave fronts of horizontal transverse modes bend convexly in the propagation direction, while those of vertical transverse modes are close to plane waves. As a result, a minimum beam diameter, which is called the beam waist of a horizontal transverse mode, is placed inside the optical cavity; the beam waist of vertical transverse modes is placed on a facet of an LD. Such a difference in the positions of the beam waists for vertical and horizontal transverse modes is called astigmatism. Because of astigmatism, the vertical and horizontal transverse modes cannot be focused on a common plane by an axially symmetric convex lens, so only defocused images are obtained. Index Guiding Index guiding structures have an intentionally formed refractive index distribution, as shown in Fig. 6.24. In index guiding structures, horizontal and vertical transverse modes are both close to plane waves, which does not result in astigmatism. To obtain stable horizontal transverse modes, we need the following: (1) n r > |n rf |, where n r is the difference in the refractive indexes between the active layer and its surrounding layers and n rf is a change in the refractive index due to the free carrier plasma effect; (2) the cutoff condition for the higher-order modes is satisfied; and (3) the active layer width is shorter than the carrier diffusion length, which is 2 to 3 µm. Condition (1) is required to achieve index guiding even in a high injected current. Condition (2) is introduced to obtain a single horizontal transverse mode by confining the fundamental transverse mode to the optical waveguide. If multiple higher-order horizontal transverse modes are present in the optical waveguide, mode hopping or mode competition takes place according to operating conditions, which leads to unstable horizontal transverse modes. To satisfy conditions (1) and (2), n r /n r is approximately 10−2 . Finally, condition (3) is needed to achieve uniform
P1: OSO c06 JWBS036-Numai
154
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
Anode Active Layer Cathode (a) Refractive Index Position (b) FIGURE 6.24 Index guiding LD: (a) cross-sectional view; (b) distribution of refractive index.
distributions of the carrier concentration in the active layer, which reduces spatial hole burning. To date, many index guiding structures have been developed to confine the carriers and the light efficiently to the active layer, and the structures are classified into three categories: rib waveguides, ridge waveguides, and buried heterostructures (BHs). Rib Waveguide Rib waveguides are optical waveguides with convex or concave regions, which are suitable for LDs whose active layers are easily oxidized in the air during etching, and their emission efficiencies decrease drastically, due to oxidization. These active layers (e.g., AlGaAs layers) cannot be etched in the air, and the rib waveguides can be formed without etching the active layers in the air. As an example of a rib waveguide, Fig. 6.25 shows a planoconvex waveguide (PCW) structure in which the semiconductor layers are grown on a preetched substrate. Here arrow a, with a solid line, and arrow b, with a dashed line, represent the flowing paths of the electric current. Path a shows the direction of a forward current
p-diffused Region Anode
n p Active Layer n n-substrate
Cathode FIGURE 6.25 Rib waveguide.
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
TRANSVERSE MODES
(a) FIGURE 6.26
155
(b)
I –V characteristics in (a) pn- and (b) pnpn-junctions.
across the pn-junction, and the electrical resistance along this path is low; path b shows the direction of a forward current across the pnpn-junction. Figure 6.26 shows I –V characteristics in the pn- and pnpn-junctions. When the voltage V applied is below the switching voltage Vs , a pn-junction inside the pnpnjunction is reverse biased and electric current barely flows. Once the applied voltage V exceeds Vs , the pnpn-junction switches on and shows an I –V characteristic similar to that of a pn-junction in forward bias. Therefore, by designing a pnpn-junction so that Vs may be larger than the voltage applied across the pn-junction, little electric current flows through path b; the current flows mainly through path a. Note that the electric current flowing regions are broad, because a current constriction structure is not formed below the p-diffused region. As a result, it can be said that the rib waveguides are not optimized for confinement of the carriers to the active layers. Therefore, the horizontal transverse mode is not as stable with respect to the injected current, and the threshold current is relatively high. Ridge Waveguide Ridge waveguides are optical waveguides with a convex region. Because they are easily fabricated by etching after epitaxial growth, low-cost LDs are expected. Etching is stopped above the active layer, as shown in Fig. 6.27, and the easily oxidized layers are not exposed to the air.
Anode
p Active Layer n-substrate
Cathode FIGURE 6.27 Ridge waveguide.
P1: OSO c06 JWBS036-Numai
156
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
Anode p-Cladding Guiding Active Guiding n-Cladding n-Substrate Cathode (a)
(b)
FIGURE 6.28 Ridge LDs with antiguiding waveguides: (a) antiguiding layers; (b) antiguiding cladding layers.
Note that the electric current–flowing regions are broad, because a current constriction structure is not formed below the p-diffused region. As a result, it can be said that the ridge waveguides are not optimized for confinement of the carriers to the active layers. Therefore, the horizontal transverse mode is not as stable with respect to the injected current, and the threshold current is relatively high. As shown in Fig. 6.28, ridge LDs with antiguiding layers [3–6] and antiguiding cladding layers [7] have been proposed, and it is shown theoretically that the horizontal transverse mode becomes stable. Buried Heterostructure In buried heterostructures (BHs) the active layer is surrounded by regrown regions. At first, epitaxial layers are grown on a semiconductor substrate, which is followed by etching to form a stripe. This stripe is buried by the second epitaxial growth, and the buried regions prevent the injected current from flowing. Although fabrication processes are complicated, as described above, low-threshold, high-efficiency laser operations are obtained because of efficient confinement to the active layer of both the carriers and the light. However, because the active layers are exposed to the air during etching, only active layers insensitive to oxidization can be used for BHs. For example, in InGaAsP/InP LDs, which are the light sources of longhaul optical fiber communication systems, BHs are frequently adopted. Figure 6.29 shows a BH [8] and a double-channel planar buried heterostructure (DC-PBH) [9].
p-Diffused Region Anode p
n p
Insulator
p p
Active Layer n-Substrate
(a)
Cathode
(b)
FIGURE 6.29 Buried heterostructures: (a) BH; (b) DC-PBH.
n p p
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
LONGITUDINAL MODES
157
To constrict the current-flowing region, the surrounding regions of the stripe are pnpn-junctions. Therefore, electric current flows efficiently through path a but flows hardly at all through path b. In the DC-PBH shown in Fig. 6.29(b), path c is a pnpin-junction because the active layer is undoped and regarded as nearly intrinsic. Therefore, electric current through path c is smaller than that through path b. As a result, the DC-PBH has a lower threshold current and higher efficiency than those of the BH.
6.7
LONGITUDINAL MODES
Longitudinal modes, also called axial modes, determine the resonant wavelengths of the cavity and show the light intensity distributions along the cavity axes. Figure 6.30 shows examples of oscillation spectra for multimode and a single-mode operation. LDs use interband transitions to obtain the optical gain, and the optical gain spectrum has a width of about 10 nm. In addition, Fabry–Perot cavities have a lot of resonant modes, which leads to low mode selectivity. For these two reasons, Fabry–Perot LDs tend to oscillate in multimodes, as shown in Fig. 6.30(a). However, single-mode LDs are not always needed for applications such as compact disks, laser printers, bar-code readers, laser pointers, or short-haul optical fiber communication systems in which Fabry–Perot LDs are used as light sources. In long-haul, large-capacity optical fiber communication systems, single-mode LDs are needed because optical fibers demonstrate dispersion in that their refractive indexes depend on the wavelengths and modes of light. Due to the dispersion, the propagation speed of light changes according to the wavelengths and modes of the light. If LDs show multimode operations, the optical pulses broaden in the time domain during propagating through the optical fibers. With an increase in the transmission distance and a decrease in the pulse spacing, adjacent optical pulses tend to overlap each other. Finally, the photodetectors cannot resolve sequentially transmitted optical pulses, as shown in Fig. 6.31.
Wavelength (a)
Wavelength (b)
FIGURE 6.30 Oscillation spectra for (a) multimode and (b) single-mode operation.
P1: OSO c06 JWBS036-Numai
158
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
Optical Power
Optical Power
Time
Time (b)
(a)
FIGURE 6.31 Light pulses in (a) a transmitter and (b) a photodetector.
Note that the longitudinal modes change with the transverse modes, because the effective refractive indexes of the optical waveguides depend on the transverse modes. Single-transverse-mode operation is indispensable to achieving single-longitudinalmode operation. Static Characteristics of Fabry–Perot LDs Let us calculate the wavelength spacing of the longitudinal modes λ = λm − λm+1 in the Fabry–Perot LD shown in Fig. 6.32. Using a cavity length L, a positive integer m ( 1), and a refractive index n r (λ) that is a function of a light wavelength λ, resonant wavelengths λm and λm+1 are written λm =
2L 2L n r (λm ) , λm+1 = n r (λm+1 ) . m m+1
(6.75)
As a result, we have λ = λm − λm+1
1 λm dn r λ + n r (λm ) dλ 2L
λm n r (λm )
2 n r (λm ) ,
(6.76)
where m 1 is used. Therefore, we obtain λ =
λm 2 , 2n rλ L
(6.77)
Wavelength FIGURE 6.32 Wavelength spacing of the longitudinal modes in a Fabry–Perot LD.
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
LONGITUDINAL MODES
Oscillation Spectra
159
A
B
Optical Gain A B Wavelength FIGURE 6.33 Relationship between longitudinal modes and an optical gain spectrum.
where n rλ is the equivalent refractive index, which is an effective refractive index with dispersion and is given by dn r . n rλ = n r (λm ) 1 − λm dλ
(6.78)
Generally, we have dn r /dλ < 0, and the equivalent refractive index n rλ is larger than n r . Figure 6.33 shows a relationship between the longitudinal modes and the optical gain spectrum. Fabry–Perot LDs oscillate at the resonant wavelength that is closest to the gain peak. As a result, the oscillation wavelength for optical gain A is λ0A and that for optical gain B is λ0B . If the nonlinear effect and the coupling of the spontaneous emission to the lasing mode are negligibly small, the carrier concentration in the active layer is constant above the threshold, the free carrier plasma effect does not change the refractive index, and Joule heating of the active layer enhances the refractive index with an increase in the injected current I . Hence, the resonant wavelengths become longer with I (> Ith ) according to (6.25). Above the threshold, the band-filling effect is not dominant, and the peak wavelength of the optical gain spectrum becomes longer due to Joule heating of the optical waveguide with an increase in I . With an increase in I above Ith , both the resonant wavelengths and the peak wavelength of the optical gain spectrum become longer, due to Joule heating of the optical waveguide. Therefore, the oscillation wavelength increases with I . Here it should be noted that the change rate of the resonant wavelength and that of the peak
P1: OSO c06 JWBS036-Numai
9:17
Printer Name: Yet to Come
Peak Wavelength
-th -th
Peak Wavelength
Mode Intensity
FABRY–PEROT LASER DIODES
(nm)
160
August 9, 2010
(d)
(d)
(c)
Temperature
(c)
Injected Current
(a)
(b) (b)
d m dT j
Main Mode (Peak Wavelength d dT j
)
Optical Gain
Submode
(a) Wavelength
Injected Current (mA) FIGURE 6.34 Dependence of longitudinal modes on injected current.
wavelength of the optical gain spectrum with I differ from each other. As a result, with an increase in I , the lasing mode sometimes jumps, because the resonant wavelength closest to the peak wavelength of the optical gain spectrum at I=Ith departs from the peak wavelength of the optical gain spectrum at I>Ith , and other resonant wavelengths approach the peak wavelength of the optical gain spectrum at I>Ith . Figure 6.34 shows the dependence of the longitudinal modes on the injected current I . It is found that the wavelengths of longitudinal modes become longer, and the mode jumps to the other mode at several values of I . In addition, according to an increase or decrease in I , hysteresis loops are present. Similar phenomena are observed when the junction temperature Tj is changed while I is kept constant. The reason for these hysteresis loops is that the optical gain concentrates on the oscillating longitudinal mode, and the optical gains for the other modes are suppressed due to coupling of modes and intraband relaxation of the carriers. Carriers with the transition energy of the lasing mode are highly consumed by stimulated emission. These consumed carriers, however, are compensated by the intraband relaxation of the carriers, and the number of carriers with the transition energy of the nonlasing mode decreases. Therefore, the optical gain is concentrated on the lasing mode. 6.8 MODULATION CHARACTERISTICS Lightwave Transmission Systems and Modulation In lightwave transmission systems, an intensity modulation/direct detection system and a coherent system are present. With regard to the modulations of laser beams, direct modulation of LDs and external modulation using optical modulators exist.
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
MODULATION CHARACTERISTICS
161
Optical Signals Optical Fiber LD
Photodetector
FIGURE 6.35 Intensity modulation/direct detection system.
Intensity Modulation/Direct Detection System Figure 6.35 shows an intensity modulation/direct detection system. In this system, a transmitter sends optical signals by modulating the light intensity; a photodetector directly detects changes in the light intensity and transforms the optical signals into electric signals. This system is simpler and more cost-effective than a coherent system. Therefore, the intensity modulation/direct detection system has been used in conventional optical fiber communication systems. A problem with transmission distance, which was inferior to that of a coherent system, was solved by the advent of optical fiber amplifiers. Coherent System In a coherent system, modulation schemes such as amplitude shift keying (ASK), frequency shift keying (FSK), and phase shift keying (PSK) are present according to modulations of the laser beam. As shown in Fig. 6.36, modulated light from a master light source LD1 and light from a slave (local) light source LD2 are simultaneously incident on a photodetector. In the photodetector, interference of these two laser beams generates an optical beat signal, which is converted to an electric signal. With an increase in the light output of LD2 , the signal-to-noise (S/N) ratio is improved, which leads to long-haul optical fiber communication systems. However, to obtain an optical beat, we have to prepare two LDs whose laser light has narrow spectral linewidths, almost common wavelengths, and the same polarizations. In addition, we need polarization controllers to achieve the same polarizations, because polarization of the lightwave changes due to contortion of the optical fibers. Also,
Optical Fiber
Coupler LD1
Photodetector LD2 FIGURE 6.36 Coherent system.
P1: OSO c06 JWBS036-Numai
162
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
the electronic circuits used to process electric signals from a photodetector are complicated. As a result, the cost of a coherent system is much higher than that of an intensity modulation/direct detection system. Direct Modulation In direct modulation of LDs, the current injected to LDs is modulated. During direct modulations, multimode operations, chirping, and changes in the turn-on delay time take place. Fabry–Perot LDs can be used as light sources for short-haul optical fiber communication systems with a transmission distance of 1 km or so, as in a building, because these problems are not serious. In contrast, Fabry– Perot LDs with a transmission distance of more than several tens of kilometers cannot satisfy the system specifications for long-haul optical fiber communication systems. Therefore, DFB LDs with stable single-longitudinal-mode operations are used as light sources for long-haul optical fiber communication systems. External Modulation In external modulation of the laser beams, the current injected to LDs is kept constant, and the laser beams emitted from LDs are modulated by the optical modulators. Because the current injected into LDs is constant, relaxation oscillations do not take place, in contrast to direct modulation. Therefore, multimode operations can be avoided. In addition, chirping is low, because the change in the refractive index in optical modulators is small. To reduce costs and to achieve high optical coupling efficiencies between the LDs and optical modulators, integrated light sources of DFB LDs and optical modulators have been developed and used in conventional long-haul, large-capacity optical fiber communication systems. In the following we focus on the direct modulation of LDs because it is helpful to understand the peculiar characteristics of LDs. To examine the direct modulation of Fabry–Perot LDs, a turn-on delay time and a relaxation frequency are derived.
Direct Modulation Turn-on Delay Time It is assumed that a step pulsed current is injected into an LD. As shown in Fig. 6.37(a), a bias current density Jb is below the threshold current density Jth (Jb < Jth ), and a pulsed current density Jp is injected to the LD at time t = ton = 0. Here it is assumed that the pulse width is much larger than the carrier lifetime τn . The carrier concentration n increases from a biased value n b with a time constant τn . As shown in Fig. 6.37(b), when the carrier concentration n reaches the threshold carrier concentration n th at the turn-on delay time t = td , laser oscillation starts. Once oscillation begins, the carrier concentration n and photon concentration S show relaxation oscillations, as shown in Fig. 6.37(b) and (c). Note that Fig. 6.37 was drawn by analyzing (6.2) and (6.3) numerically, because exact analytical solutions are not present for the rate equations (6.2) and (6.3). The turn-on delay time td can be calculated using rate equations. For simplicity, the coupling of spontaneous emission to the lasing mode is neglected, and the photon
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
MODULATION CHARACTERISTICS
163
(a)
(b)
(c) FIGURE 6.37 Turn-on delay time and relaxation oscillations: (a) current density; (b) carrier concentration; (c) photon concentration.
concentration is S = 0 for n < n th . As a result, (6.2) is reduced to dn J n = − . dt ed τn
(6.79)
The current density J is assumed to be J = Jp · u(t) + Jb ,
(6.80)
where u(t) =
0 1
(t < 0), (t ≥ 0).
(6.81)
We substitute (6.80) into (6.79), then take the Laplace transform of both sides. If we express a Laplace transform of n(t) as N (s) and let n(0) = n b = τn Jb /ed, we
P1: OSO c06 JWBS036-Numai
164
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
obtain s N (s) − n(0) = s N (s) −
Jp + Jb 1 1 τn Jb = − N (s). ed ed s τn
(6.82)
τn (Jp + Jb ) τn Jb 1 + . ed ed s + τn −1
(6.83)
As a result, N (s) is written N (s) =
1 1 − s s + τn −1
From (6.83), the inverse Laplace transform of N (s) is written τn (Jp + Jb ) −t/τn τn Jb −t/τn τn (Jp + Jb ) u(t) − e e + ed ed ed τn (Jp + Jb ) τn Jp −t/τn = . u(t) − e ed ed
n(t) =
(6.84)
Because of u(t) = 1 in t ≥ 0, (6.84) is expressed as n(t) =
τn Jp −t/τn τn J − e , ed ed
(6.85)
Jp + Jb = J.
(6.86)
where
At t = td , the carrier concentration n reaches the threshold carrier concentration n th , and we have n(td ) = n th =
τn Jth , ed
(6.87)
where (6.19) is used. Using (6.85)–(6.87), the turn-on delay time td is obtained as td = τn ln
J − Jb . J − Jth
(6.88)
To generate high-speed optical signals by modulating the current injected to LDs, the turn-on delay time td should be as short as possible. From (6.88) it is found that a large bias current density Jb , a low threshold current density Jth , and a short carrier lifetime τn are suitable for direct high-speed modulations. Relaxation Oscillation If we use small-signal analysis, approximate analytical solutions can be obtained from (6.2) and (6.3), and the solutions clearly give us their
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
165
MODULATION CHARACTERISTICS
physical meanings. To proceed in small-signal analysis, the carrier concentration n, the photon concentration S, and the current density J are expressed as n = n c0 + δn, S = S0 + δS, J = J0 + δ J > Jth , n c0 δn, S0 δS, J0 δ J,
(6.89)
where n c0 , S0 , and J0 are steady-state values of the carrier concentration, photon concentration, and current density, respectively; δn, δS, and δ J are deviations from each steady-state value. If Jb Jp is assumed and several initial sharp peaks in the relaxation oscillation are excluded, the conditions for the small-signal analysis are satisfied. We set J0 = Jb and δ J = Jp in accord with Fig. 6.37. Neglecting coupling of the spontaneous emission to the lasing mode, (6.2) and (6.3) are reduced to J n dn = − G(n)S − , dt ed τn S dS = a G(n)S − . dt τph
(6.90) (6.91)
In a steady state (d/dt = 0), substituting (6.89) into (6.90) and (6.91) leads to J0 n c0 − G(n c0 )S0 − = 0, ed τn 1 . a G(n c0 ) = τph
(6.92) (6.93)
Substituting (6.89) into (6.4) results in G(n) = G(n c0 + δn) = g0 (n c0 + δn − n 0 ) = g0 (n c0 − n 0 ) + g0 δn = G(n c0 ) +
∂G δn, ∂n
(6.94)
where the differential gain, which is defined as g0 ≡
∂G , ∂n
(6.95)
is introduced. Inserting (6.89) into (6.90) and(6.91) with the help of (6.92)–(6.94) and then neglecting the second-order small term δn · S, the rate equations on the deviations δn and δS are written δJ δS d ∂G δn δn = − S0 δn − , − dt ed τph ∂n τn ∂G d δS = a S0 δn. dt ∂n
(6.96) (6.97)
P1: OSO c06 JWBS036-Numai
166
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
By removing δn from (6.96) and (6.97), we have d2 δS + dt 2
∂G 1 S0 + ∂n τn
d ∂G S0 ∂G S0 δS = a δS + a δ J, dt ∂n τph ∂n ed
(6.98)
which indicates that the deviation of the photon concentration δS shows a relaxation oscillation. From (6.98), the decay coefficient γ0 , decay time τre , and oscillation angular frequency ωr are written ∂G 1 1 S0 + = , ∂n τn τre ∂G S0 . ωr 2 = a ∂n τph
γ0 =
(6.99) (6.100)
From (6.100), the relaxation oscillation frequency f r is given by 1 fr = 2π
a
∂G S0 . ∂n τph
(6.101)
To generate high-speed optical signals by modulating the current injected to LDs, the decay coefficient γ0 and the relaxation oscillation frequency f r have to be large. From (6.99) and (6.101) it is found that a large differential gain a ∂G/∂n, a large photon concentration in a steady state S0 , a short carrier lifetime τn , and a short photon lifetime τph are required for high-speed modulations. High-speed modulations and other characteristics often have trade-offs. As shown in (6.37), (6.38), and (6.95), a large a ∂G/∂n leads to a low Jth ; short τn and τph increase Jth . As a result, to achieve simultaneous low-threshold and high-speed operations, we need to obtain a large a ∂G/∂n, which is accomplished in the quantum well LDs. With regard to system applications, if S0 is large, optical power is detected by a photodetector even when optical pulses are not transmitted. As a result, the extinction ratio decreases and the S/N ratio degrades with an increase in S0 . Therefore, S0 should be limited to satisfying the extinction ratio and the S/N ratio, which are specified in applications. The decay coefficient γ0 and the relaxation oscillation frequency f r can also be expressed using the current density J . From (6.18), the threshold carrier concentration n th and the transparent carrier concentration n 0 are written n th =
τn τn Jth , n 0 = J, ed ed 0
(6.102)
where J0 is the current density, which is required for a semiconductor to be transparent. From (6.36) we have n th − n 0 =
1 . a g0 τph
(6.103)
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
MODULATION CHARACTERISTICS
167
Substituting (6.95) and (6.102) into (6.103), we obtain a
ed ∂G = . ∂n τn τph (Jth − J0 )
(6.104)
Inserting (6.40) and (6.104) into (6.99) and (6.101), the decay coefficient γ0 and relaxation oscillation frequency f r are expressed as 1 J − J0 , γ0 = τn Jth − J0 a J − Jth 1 fr = . 2π τn τph Jth − J0
(6.105) (6.106)
Relationship Between Relaxation Oscillation and Longitudinal Modes As shown in Fig. 6.37(b), during relaxation oscillation, the carrier concentration n is modulated around n th . As a result, the optical gain and refractive index change simultaneously and the longitudinal modes are altered. The optical gain g exceeds the threshold optical gain gth when the carrier concentration n is larger than n th . In this case, longitudinal modes with optical gain g ≥ gth show laser operations. Therefore, during relaxation oscillation, multimode laser oscillations are often observed. When the relaxation oscillation is decayed, the number of lasing longitudinal modes decreases. The refractive index is modulated by the free carrier plasma effect. When a deviation in the carrier concentration δn is 2 × 1017 cm−3 , the wavelengths of the longitudinal modes become shorter by about 0.4 nm from their steady-state values. These dynamic changes in longitudinal modes, which are called chirping, broaden the timeaveraged light output spectra, as shown in Fig. 6.38. Multimode laser operations and chirping should be suppressed for long-haul, large-capacity optical fiber communication systems, because the optical fibers have dispersions, as described earlier. Note that the effect of Joule heating with a time scale of microseconds or more is negligible, because the decay time of the relaxation oscillation is on the order of nanoseconds. Dependence on Modulation Frequency When an absolute value of the deviation in the current density δ J is much smaller than the steady-state current density J0 (|δ J | J0 ), small-signal analysis can be used. It is assumed that the deviation δ J is expressed as δ J (ω) e i ωt . Accordingly, the deviation in the carrier concentration δn and that in the photon concentration δS are expressed as δn = δn(ω) e i ωt and δS = δS(ω) e i ωt , respectively. Substituting δ J , δn, and δS into (6.96) and (6.97), we have i ω δ J (ω) , D(ω) ed τph ωr 2 δ J (ω) , δS(ω) = − D(ω) ed δn(ω) = −
(6.107) (6.108)
P1: OSO c06 JWBS036-Numai
168
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
Average Longitudinal Mode During (1)−(3): Chirping
Longitudinal Mode at (1)
(1) (2)
Optical Power
Optical Gain
Longitudinal Mode at (3)
(3)
(1) (2) (3)
Wavelength FIGURE 6.38 Multimode operation during relaxation oscillation. The inset shows an optical pulse.
where D(ω) = ω2 − ωr 2 − i ωγ0 .
(6.109)
Modulation efficiencies δ(ω) and δ( f ), which are defined as the number of photons generated per injected electron, are written δS(ω) τph ωr 2 τph ωr 2 = = , δ(ω) = δ J (ω)/ed |D(ω)| (ω2 − ωr 2 )2 + ω2 γ0 2
(6.110)
2π τph f r 2 , δ( f ) = 4π 2 ( f 2 − f r 2 )2 + f 2 γ0 2
(6.111)
where (6.108) and (6.109) are used, and the modulation frequency f is ω/2π . From (6.110) we obtain δ(ω) ωr 2 , = δ(0) (ω2 − ωr 2 )2 + ω2 γ0 2
(6.112)
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
MODULATION CHARACTERISTICS
Modulation Frequency
169
(Hz)
FIGURE 6.39 Resonance phenomena.
which indicates that the modulation efficiency δ(ω) shows resonance characteristics, and the resonant angular frequency equals the relaxation oscillation angular frequency ωr . Figure 6.39 shows the modulation efficiency δ( f ) as a function of the modulation frequency f = ω/2π with the injected current density J as a parameter. The resonant frequency f r represents the highest limit in the modulation frequency. As shown in (6.106), with an increase in J , f r is enhanced, which results in a large modulation bandwidth. Note that the electrical resistance and capacitance of LDs also affect the modulation bandwidth. Analog and Digital Modulation In direct modulations of LDs, analog modulation and digital modulation are present, as shown in Fig. 6.40. In analog modulation, a change in the injected current I is transformed into a change in optical power P, and high linearity in the I –L curve is required. The upper limit in the modulation frequency for analog modulation is the resonant frequency f r , and nonlinearity in the I –L curve and Joule heating during large amplitude modulations cause highfrequency distortions. Typical noises are optical feedback noises, which are induced by light reflected from edges of optical fibers, and modal noises, which are generated in the optical fibers. Compared with digital modulation, analog modulation can transmit more information with lower modulation frequencies, but in contrast to digital modulation, it is easily affected by distortions of optical signals during transmissions. As a result, analog modulation is used for short-haul, large-capacity optical fiber communication systems such as cable television (CATV). In digital modulation, the signals 1 and 0 are assigned to the ON and OFF states of laser light. To obtain optical pulses, which immediately follow the pulsed injected current, dc bias current is injected into LDs to shorten the turn-on delay time. Compared with analog modulation, digital modulation is less affected by distortions of the optical signals during transmission. As a result, digital modulation is used
P1: OSO c06 JWBS036-Numai
170
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
Digital Modulation
Analog Modulation
Digital Modulation
Analog Modulation
FIGURE 6.40 Analog modulation and digital modulation.
for long-haul, large-capacity optical fiber communication systems. However, higher modulation frequency than that of analog modulation is required. According to the original signals 1 and 0, return-to-zero (RZ) and nonreturnto-zero (NRZ) signals, which are shown in Fig. 6.41, are often used. In an RZ signal, a signal level is returned to zero after the original signal 1 is transmitted, which leads to a large S/N ratio. However, a short pulse width is needed to generate an RZ signal, and a high modulation frequency is required. In an NRZ signal, a signal level is not
Original Signal
RZ Signal
NRZ Signal FIGURE 6.41 RZ and NRZ signals.
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
MODULATION CHARACTERISTICS
171
returned to zero after the original signal 1 is transmitted. The S/N ratio of an NRZ signal is inferior to that of an RZ signal, but the modulation speed is not as high as that of an RZ signal. When the modulation speed is higher, the pulse width is shorter, and a steady state, which is indicated as (3) in Fig. 6.38, does not exist within each optical pulse. In this case, the relaxation oscillation is sharp, the optical pulses have deep valleys, and the average optical power decreases. The carrier concentration n in the active layer of an LD changes according to preceding optical pulses, a phenomenon called the pattern effect. With a change in carrier concentration n in the active layer of an LD, the bias level of the LD is altered and the turn-on delay time td is modified, as shown in Fig. 6.42. To examine changes in the optical pulses, the pulses are intentionally overlapped with electrical pulses as the reference, as shown in Fig. 6.42(d), which is called the eye pattern. When large deviations in the turn-on delay time are present, resolutions in the photodetector degrade. In this case the eye patterns collapse and a lot of jitters appear. If deviations in the turn-on delay time are small, the eye patterns are wide open. Therefore, the quality of the signals transmitted can be evaluated using eye patterns.
Signal Reference Time (a)
(b)
a
b
c (c)
Reference Time
a c b Time Scale (d)
FIGURE 6.42 Pattern effect: (a) pulsed injected current; (b) carrier concentration n; (c) photon concentration S; (d) eye pattern.
P1: OSO c06 JWBS036-Numai
172
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
TABLE 6.1 Material Dispersion and Optical Loss in Quartz Optical Fibers Wavelength (µm)
˚ · km) Material Dispersion (ps/A
Optical Loss (dB/km)
0.85 1.30 1.55
9 ∼0 2
2–3 0.5 < 0.2
When the relaxation oscillation takes place, Fabry–Perot LDs oscillate with multiple longitudinal modes, which restrict the transmission distance due to dispersions in the optical fibers. The optical fibers have mode dispersion, material dispersion, and structural dispersion. In mode dispersion the effective refractive index of the optical fiber changes according to a mode which corresponds to a distribution of light fields. In material dispersion, or chromatic dispersion, the refractive index of the optical fiber depends on a wavelength of light. In structural dispersion the effective refractive index of the optical fiber for a common mode changes with a wavelength of light because the dependence of the refractive index of the core and that of the cladding of the optical fiber on a wavelength of light are different. In long-haul, large-capacity optical fiber communication systems, a fundamental transverse mode is used, and material dispersion is dominant. Table 6.1 shows the material dispersion and optical loss of quartz optical fibers for typical wavelengths of LDs. If five longitudinal modes are present during relaxation oscillation, the difference ˚ for LDs with a wavein the longest and shortest wavelengths is approximately 50 A length of 1.55 µm. When this optical signal is transmitted through the optical fibers by 10 km, the maximum delay in the optical signal is ˚ · km × 50 A ˚ × 10 km = 1 ns, 2 ps/A
(6.113)
and the optical pulse broadens.
6.9 NOISES Noises of LDs fall into two categories: noises with origins existing in and outside LDs. The former noises are quantum noises and noises on longitudinal modes. The latter noises are optical feedback noise and noises from the environment and from the driving circuits. Quantum Noises Fundamental Equations Quantum noises such as amplitude-modulating (AM) and frequency-modulating (FM) noise are caused by spontaneous emission with random amplitudes, frequencies, and phases of the light. FM noise is caused directly by
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
NOISES
173
spontaneous emission and is also affected by AM noise. Due to AM noise, the amplitude of a light field fluctuates and the carrier concentration in the active layer is modulated, leading to carrier noise. As a result, through the free carrier plasma effect, the refractive indexes of semiconductors are fluctuated, and FM noise is generated. In addition, the carrier noise induced by AM noise generates electric current noise, which changes Joule heating in the active layer. Therefore, the refractive indexes of semiconductors fluctuate and FM noise is generated. Note that AM noise is also altered by the carrier noise induced by AM noise itself. To analyze quantum noises, semiclassical theory, in which electromagnetic fields are treated classically and atomic systems in the fields are considered quantum mechanically, is often used [10]. From Maxwell’s equations, an equation for the electric field E inside an LD is written ∇ 2 E − µσ
∂2 E ∂E ∂2 − µε 2 = µ 2 ( P + p). ∂t ∂t ∂t
(6.114)
Here the assumption ∇ · E = 0 is used; µ is permeability; σ is conductivity, which represents optical loss; ε0 is the permittivity of vacuum; n r is a refractive index of the semiconductor (ε ≡ ε0 n r 2 ); P is polarization of a medium, contributing to the laser transition; and p is a polarization source for spontaneous emission, which is regarded as a Langevin source. It is also supposed that the electric field E and polarizations P and p are expressed as E = Re P = Re p = Re
m
m
E m (t)em (r) , Pm (t)em (r) ,
(6.115)
pm (t)em (r) .
m
Here a spatial distribution function em (r) satisfies both the boundary conditions of the LD and the relation ∇ 2 em (r) + ωm 2 µε em (r) = 0.
(6.116)
In addition, em (r) is orthonormalized as
em ∗ · en dV = δmn Vm , all volume
(6.117)
P1: OSO c06 JWBS036-Numai
174
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
where Vm is a mode volume. Substituting (6.115) into (6.114) and then taking the inner product with en ∗ (r) results in 1 ˙ 1 E¨ n + E n + ωn 2 E n = − ( P¨ n + p¨ n ), τph ε
(6.118)
where (6.116) and (6.117) are used. In (6.118), a dot and a double dot above E n , Pn , and pn represent the first and second derivatives with respect to a time t, respectively; ωn is the resonant angular frequency of the nth mode; τph = ε/σ is the photon lifetime of the optical cavity; and µ = µ0 , which is usually satisfied in the optical materials. Using the electric field E n (t), the polarization of a medium Pn (t) is written Pn (t) = ε0 X (1) + X (3) |E n (t)|2 E n (t).
(6.119)
Here X (1) and X (3) are expressed as X (1) = X (3) =
χ (1) Vm (3)
χ Vm
l , L medium 3l (en ∗ · en )2 dV ≈ χ (3) , 2L medium en ∗ · en dV ≈ χ (1)
(6.120) (6.121)
where χ (1) and χ (3) are the first and third optical susceptibilities, respectively; l is the crystal length; and L is the cavity length. Under single-longitudinal-mode laser operation, the electric field E n (t) and the polarization source for spontaneous emission, which is a Langevin source, pn (t), are expressed as E n (t) = [A0 + δ(t)] e i [ωm t+φ(t)] , −
1 ∂ pn = (t) e i [ωm t+φ(t)] , ε ∂t 2
(6.122)
2
(6.123)
where A0 is the average amplitude of the electric field, δ(t) is a deviation in the amplitude of the electric field from A0 , φ(t) is an instantaneous phase, (t) is a random function representing spontaneous emission, and ωm is the average angular frequency of the laser light. It is assumed that δ(t), φ(t), and (t) are slowly varying functions compared with ωm . For simplicity, the carrier fluctuations are neglected, and it is supposed that δ(t) A0 , δ(t) = φ(t) = (t) = 0, where · · · shows a time average or ensemble average.
(6.124)
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
NOISES
175
Substituting (6.119), (6.122), and (6.123) into (6.118) leads to
∂δ 3A0 2 X (3) ∂φ ∂δ 2 2 i ωm + i A0 + − ωm δ 2 i ωm ∂t ∂t nr2 ∂t 2 ωm (1) 2 2 2 (3) ωm − (X + A0 X ) 2 A0 = (t), + ωn − ωm + i τph nr
(6.125)
where ε = ε0 n r 2 is used. A real part and an imaginary part of the steady-state solution of (6.125) are given by −1 X (1) + A0 2 X r(3) ωm 2 = ωn 2 1 + r : real, nr2 1 nr2 : imaginary, A0 2 = − (3) X i(1) + ωm τph Xi
(6.126) (6.127)
where X (1,3) = X r(1,3) − i X i(1,3) . Therefore, (6.125) is reduced to 3A0 2 X i(3) ∂δ 3A0 2 ωm X r(3) ∂φ r (t) − + A0 δ=− : real, 2 2 ∂t nr ∂t 2n r 2ωm 3A0 2 ωm X i(3) 3A0 2 X r(3) ∂δ i (t) + 1+ δ= : imaginary, 2 2 nr ∂t 2n r 2ωm
(6.128) (6.129)
where (t) = r (t) + i i (t). In (6.128), the second term on the left-hand side is usually neglected, because it is much smaller than the other terms. In this case, the third term on the left-hand side relates the amplitude fluctuation δ and the phase fluctuation φ. Spectra of Laser Light To analyze the spectra of laser light, a power fluctuation spectrum expressing AM noise, a frequency fluctuation spectrum indicating FM noise, and a field spectrum are often used. Figure 6.43 shows a measurement system for these spectra. Note that a linewidth of the field spectrum is used as a spectral linewidth of laser light emitted from an LD. To obtain the spectra of laser light, the autocorrelation functions of the fluctuations are first calculated using (6.128) and (6.129). Then, using the Wiener– Khintchine theorem, spectral density functions are obtained. Autocorrelation Function of the Amplitude Fluctuation δ(t) For the usual laser field, A0 2 X r(3) /n r 2 1 is satisfied. As a result, (6.129) is reduced to ∂δ i (t) , + ω1 δ = ∂t 2ωm
(6.130)
P1: OSO c06 JWBS036-Numai
176
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
Test Laser
Spectrum Analyzer
Detector
Power Fluctuation Sectrum
Reference Laser Optical Mixer
Spectrum Analyzer
Limiter Discriminator Spectrum Analyzer
Frequency Fluctuation Spectrum
Field Spectrum FIGURE 6.43 Measurement system for spectra of laser light.
where ω1 =
3A0 2 ωm X i(3) > 0. 2n r 2
(6.131)
A Laplace transform of (6.130) is written ˜ ˜ + ω1 δ(s) ˜ = i (s) , −δ(0) + s δ(s) 2ωm
(6.132)
where
∞
˜ ≡ δ(s) 0
˜ i (s) ≡
∞
δ(t) e−st dt,
(6.133)
i (t) e−st dt.
(6.134)
0
Supposing that δ(0) = 0, (6.132) results in ˜ = δ(s)
˜ i (s) . 2ωm (s + ω1 )
(6.135)
Taking an inverse Laplace transform of (6.135) leads to δ(t) =
1 2ωm
t 0
i (λ)e−ω1 (t−λ) dλ.
(6.136)
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
NOISES
177
From (6.136), the autocorrelation function δ(t + τ )δ(t) of the amplitude fluctuation δ(t) is given by 1 δ(t + τ )δ(t) = 4ωm 2
t+τ
t
dλ2 i (λ1 )i (λ2 ) 0 −ω1 (t+τ −λ1 ) −ω1 (t−λ2 )
dλ1 0
×e
e
.
(6.137)
Here it is assumed that the correlation functions of the Langevin sources are written i (λ1 )i (λ2 ) = r (λ1 )r (λ2 ) = W · D(λ1 − λ2 ), i (λ1 )r (λ2 ) = r (λ1 )i (λ2 ) = 0,
(6.138)
where D(x) is a δ function and W is a coefficient representing spontaneous emission. Substituting (6.138) into (6.137) leads to δ(t + τ )δ(t) =
W e−ω1 |τ | (1 − e−2ω1 t ). 8ωm 2 ω1
(6.139)
From (6.139), in a steady state where t is long enough, we have δ(t + τ )δ(t) =
W e−ω1 |τ | . 8ωm 2 ω1
(6.140)
Autocorrelation Function of the Phase Fluctuation φ(t) Neglecting the second term on the left-hand side of (6.128), which is small compared with the other terms, results in A0
∂φ r (t) 3A0 2 ωm X r(3) δ=− . + ∂t 2n r 2 2ωm
(6.141)
Under the assumption that φ(0) = 0, integrating (6.141) with respect to t gives φ(t) = −
3 X r(3) A0 4 n r 2 ω1
t
t
i (λ) dλ −
0
i (λ)e−ω1 (t−λ) dλ
0
−
1 2A0 ωm
t
r (λ) dλ,
(6.142)
0
where (6.136) is used. From (6.130), (6.138), and (6.142), the autocorrelation function φ(t1 )φ(t2 ) of the phase fluctuation φ(t) is given by φ(t1 )φ(t2 ) =
W 4ωm 2 A0 2
(1 + α ) × 2
t1 (t1 < t2 ), t2 (t1 > t2 ),
(6.143)
P1: OSO c06 JWBS036-Numai
178
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
where α≡
X r(3)
(6.144)
X i(3)
is the α-parameter or the spectral linewidth enhancement factor. In X = X r − i X i , the real part X r and imaginary part X i are written X r = X r(1) + X r(3) |E n |2 ,
(6.145)
X i(3) |E n |2 .
(6.146)
Xi =
X i(1)
+
As a result, X r and X i are given by ∂ Xr , ∂|E n |2 ∂ Xi = . ∂|E n |2
X r(3) =
(6.147)
X i(3)
(6.148)
From (6.144), (6.147), and (6.148), the α-parameter is expressed as α=
∂ Xr ∂|E n |2
∂ Xi ∂|E n |2
−1
=
∂ Xr ∂n
∂ Xi ∂n
−1
,
(6.149)
where |E n |2 is assumed to be linearly proportional to the carrier concentration n. Using the complex refractive index n˜ = n r − i κ, the complex dielectric constant ε˜ is written ε˜ = ε0 n˜ 2 = ε0 [(n r 2 − κ 2 ) − i 2n r κ] = ε0 (X r − i X i ),
(6.150)
X r = n r 2 − κ 2 , X i = 2n r κ.
(6.151)
where
In the vicinity of the bandgap in semiconductors, n r κ is satisfied, and we obtain ∂κ ∂n r ∂n r ∂ Xr = 2n r − 2κ ≈ 2n r , ∂n ∂n ∂n ∂n ∂ Xi ∂κ ∂κ ∂n r = 2n r + 2κ ≈ 2n r . ∂n ∂n ∂n ∂n
(6.152) (6.153)
Substituting (6.152) and (6.153) into (6.149) leads to ∂n r α= ∂n
∂κ ∂n
−1
.
(6.154)
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
NOISES
179
Using the extinction coefficient κ, the power optical gain coefficient g is expressed as g=−
2ωm κ. c
(6.155)
Using G(n) in the rate equations (6.2) and (6.3), g is written a G(n) =
c g. nr
(6.156)
From (6.155) and (6.156), we have nr ∂κ ∂G =− . a ∂n 2ωm ∂n
(6.157)
Substituting (6.157) into (6.154) results in 2ωm ∂n r α=− n r ∂n
∂G −1 a . ∂n
(6.158)
When the optical gain increases with carrier injection (∂G/∂n > 0) and Joule heating of the active layer is negligibly small, the refractive index decreases with carrier injection (∂n r /∂n < 0), due to the free carrier plasma effect. As a result, the α-parameter shown in (6.158) is positive, and the measured values are between 1 and 7. From (6.158) it is found that we should increase a ∂G/∂n to reduce the value of α, which is achieved in quantum well LDs. Note that the α-parameter is an important parameter to characterize the spectral linewidth and the optical feedback noise in LDs. Autocorrelation Function of the Angular Frequency Fluctuation ω(t) Using the instantaneous angular frequency of the laser light ω(t), the average angular frequency of the laser light ωm , and the phase fluctuation φ(t), the fluctuation in the angular frequency ω(t) is written ω(t) ≡ ω(t) − ωm =
∂φ . ∂t
(6.159)
Therefore, the autocorrelation function ω(t1 )ω(t2 ) of the angular frequency fluctuation ω(t) is obtained as ˙ 1 )φ(t ˙ 2 ) ω(t1 )ω(t2 ) = φ(t α2 W −ω1 |t1 −t2 | D(t1 − t2 ) + , ω1 e = 2 4ωm 2 A0 2
(6.160)
P1: OSO c06 JWBS036-Numai
180
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
where (6.140), (6.141), and (6.149) are used, and the dot above φ represents the first derivative with respect to time. Spectral Density Function Applying the Wiener–Khintchine theorem to (6.140), the spectral density function Wδ () of the amplitude fluctuation δ(t) is obtained as Wδ () = =
1 π
∞
δ(t + τ )δ(t) e−iτ dτ
−∞
W 4π ωm
2 (2
+ ω1 2 )
.
(6.161)
Applying the Wiener–Khintchine theorem to (6.160), the spectral density function Wω () of the angular frequency fluctuation ω(t) is given by 1 π
∞
ω(t + τ )ω(t) e−iτ dτ α 2 ω1 2 W 1 + . = 2 + ω1 2 4π ωm 2 A0 2
Wω () =
−∞
(6.162)
Coefficient W The generation rate of the spontaneous emission is expressed as ωm E cv , where E cv is the number of photons, that are spontaneously emitted to the lasing mode per time. The dissipation rate of the spontaneous emission is given by Φs /τph , where Φs is the energy of spontaneous emission coupled to the lasing mode and τph is the photon lifetime. In a steady state, the generation rate is balanced with the dissipation rate, and we have Φs = ωm E cv τph .
(6.163)
The electric field of the spontaneous emission is given by solving 1 ˙ E n + ωn 2 E n = [r (t) + i i (t)] e i ωm t , E¨ n + τph
(6.164)
where ωn ≈ ωm is assumed, and the dielectric polarization due to the stimulated emission in (6.118) is neglected because only the spontaneous emission is considered. ˙ Assuming that E(0) = E(0), we have E(t) =
1 ωm
0
t
dτ [r (τ ) + i i (τ )] e i ωm τ e−(t−τ )/(2τph ) sin[ωm (t − τ )]
(6.165)
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
181
NOISES
as a solution of (6.164). Hence, E ∗ (t)E(t) is given by t t 1 E (t)E(t) = dλ1 dλ2 2W D(λ1 − λ2 ) e−t/τph e(λ1 +λ2 )/(2τph ) ωm 2 0 0 × sin[ωm (t − λ1 )] sin[ωm (t − λ2 )] W τph = , (6.166) ωm 2 ∗
where (6.138) is used. Using (6.166), a steady-state spontaneous emission energy Φs is expressed as Φs = εVm E ∗ (t)E(t) =
εVm W τph , ωm 2
(6.167)
where the mode volume Vm is given by Vm ≡
|en (r)|2 dV.
(6.168)
Here ε is the dielectric constant of a semiconductor and Vm is the mode volume. From (6.163) and (6.167), W is obtained as W =
ωm 3 E cv . εVm
(6.169)
Intensity Fluctuation Spectrum (AM Noise) The light emission rate γ from the optical cavity through the mirrors is given by γ =
1 c 1 ln , nr L R
(6.170)
where c is the speed of light in vacuum, n r is the refractive index of a semiconductor, L is the cavity length, and R is the power reflectivity of a facet where both facets are assumed to have a common reflectivity. Using the electric field of the laser light E n , the light output power P is expressed as P = εE n 2 Vm γ .
(6.171)
From (6.122), (6.123), and (6.171), the average light output power P0 and the fluctuation in light output power P = P − P0 are written P0 = ε A0 2 Vm γ , P = 2ε A0 δVm γ .
(6.172)
P1: OSO c06 JWBS036-Numai
182
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
From (6.161), (6.169), (6.170), and (6.172), the spectral density function WP () of the light output power fluctuation P is obtained as WP () =
1 π
∞
P(t + τ )P(t) e−iτ dτ
−∞
ωm ε A0 2 Vm E cv γ 2 π (2 + ω1 2 ) ωm E cv P0 γ ωm cE cv P0 ln(1/R) = = . 2 2 π ( + ω1 ) π (2 + ω1 2 )n r L = 4ε2 A0 2 Vm 2 γ 2 Wδ () =
(6.173)
Frequency Fluctuation Spectrum (FM Noise) From (6.162) and (6.169)–(6.172), the spectral density function Wω () of the angular frequency fluctuation ω(t) is given by α 2 ω1 2 ωm E cv 1 + 2 + ω1 2 4π εVm A0 2 α 2 ω1 2 ωm cE cv ln(1/R) . 1+ 2 = 4π P0 n r L + ω1 2
Wω () =
(6.174)
Field Spectrum The field spectrum is most frequently used as the laser light spectrum. Because the contribution of the amplitude fluctuation to the correlation function of the electric field is negligibly small, the correlation function E(t + τ )E(t) is given by 1 [E(t + τ ) + E ∗ (t + τ )][E(t) + E ∗ (t)] 4 A0 2 −i ωm τ iφ [e = e + c.c.], 4
E(t + τ )E(t) =
(6.175)
where the phase fluctuation φ is defined as φ ≡ φ(t + τ ) − φ(t).
(6.176)
The phase fluctuation φ is caused by a lot of independent spontaneous emission processes. As a result, the distribution of φ is given by the Gaussian distribution function g(φ) as 1 2 2 g(φ) = e−(φ) /2(φ) . 2 2π (φ)
(6.177)
Therefore, we obtain e
iφ
=
∞ −∞
g(φ) e iφ d(φ) = e− 2 (φ) . 1
2
(6.178)
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
NOISES
183
By assuming that t1 = t2 = τ in (6.143), (φ)2 on the right-hand side of (6.178) is written (φ)2 =
W 4 ωm 2 A0 2
(1 + α 2 )|τ |.
(6.179)
Substituting (6.178) and (6.179) into (6.175) results in W A0 2 2 exp − E(t + τ )E(t) = (1 + α )|τ | cos(ωm τ ). 2 8ωm 2 A0 2
(6.180)
Using the Wiener–Khintchine theorem, the spectral density function W E (ω) of the electric field is given by 1 W E (ω) = π ≈
∞ −∞
E(t + τ )E(t) e−i ωτ dτ
A0 2 ω0 , 4π (ω − ωm )2 + (ω0 /2)2
(6.181)
ωm cE cv ln(1/R) (1 + α 2 ). 4P0 n r L
(6.182)
where ω0 ≡
From (6.181), the field spectrum is Lorentzian with a FWHM of ω0 in (6.182). Equation (6.182), called the modified Schawlow–Townes linewidth formula, gives the spectral linewidth for LDs. In solid-state or gas lasers, the term α 2 can be neglected because α 2 1. In contrast, in LDs, the term α 2 is required because α 2 > 1. Noises on Longitudinal Modes Mode Partition Noise Mode partition noise is observed when a longitudinal mode is selected during laser operations with multiple longitudinal modes. This noise is large at low frequencies. For example, in Fabry–Perot LDs under pulsed operations or gain guiding LDs, noise for the total light output power is comparable to noise in single-longitudinal-mode LDs. However, noise for each longitudinal mode in multiple longitudinal modes is much larger than the noise in single-longitudinal-mode LDs, because the optical gain is randomly delivered to each longitudinal mode among multiple longitudinal modes. The mode partition noise causes a serious problem in mode selective systems such as optical fiber communication systems. To prevent mode partition noise, we need single-longitudinal-mode LDs. Mode Hopping Noise Mode hopping noise is generated when a longitudinal mode in single-mode LDs jumps to other modes. Mode hopping depends on driving conditions such as temperature and injected current. When mode hopping occurs, random
P1: OSO c06 JWBS036-Numai
184
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
oscillations between multiple longitudinal modes are repeated, and the noise increases due to a difference in light output powers between the relevant modes. When two competing modes are present, the noise is large in a low frequency range below 50 MHz; when three or more competing modes are present, the noise is large up to higher frequencies. Note that the mode partition noise is also large during mode hopping. For analog systems such as videodisks, the relative intensity noise (RIN) has to be lower than −140 dB/Hz, and for digital systems such as compact disks, the RIN should be lower than −120 dB/Hz. Mode hopping noise is caused by fluctuations of spontaneous emission and optical gain’s inclination to concentrate on the lasing mode. To avoid mode hopping noise, two opposite processes, such as laser operation with a single longitudinal mode and laser operation with multiple longitudinal modes are used. For systems with mode selectivity, such as optical fiber communication systems, single-longitudinal-mode LDs such as the DFB LDs are adopted. Bistable LDs, which contain saturable absorbers and have hysteresis in I –L curves, can also suppress mode competition. However, it is difficult for bistable LDs to keep stable single-longitudinal-mode operations with a large extinction ratio during modulation. As a result, bistable LDs are not used in conventional optical fiber communication systems. For systems without mode selectivity, such as video and compact disks, multiplelongitudinal-mode operations are widely used to reduce mode hopping noise. In these applications, optical systems have to be set up in a small space, and the optical feedback noise has to be reduced without using optical isolators. Therefore, multiple-longitudinal-mode operations are suitable for video and compact disks. Multiple-longitudinal-mode operations have a higher noise level than that of singlelongitudinal-mode operations, but the noise level is stable with changes in the temperature or injected current. Therefore, the maximum noise level is lower than the mode hopping noise. To obtain multiple-longitudinal-mode operations, high-frequency modulations and self-pulsations are adopted. High-frequency modulations are obtained when electric current pulses with frequencies above 600 MHz are injected into Fabry–Perot LDs and the minimum current is set below the threshold current. Selfpulsations, which are pulsed operations under dc bias, are obtained in Fabry–Perot LDs with saturable absorbers or combined structures of index and gain guidings. Due to the high-frequency modulations or self-pulsations, multiple-longitudinal-mode operations with low coherence take place. As a result, stable laser operations against optical feedback noise are obtained.
Optical Feedback Noise Optical feedback noise [11] is generated when laser light emitted from an LD is fed back to the LD itself. An external cavity is formed by a facet of the LD and reflective external objects, such as optical components, optical fibers, and optical disks. The external cavity and internal cavity of the LD produce a coupled cavity, which induces optical feedback noise.
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
NOISES
185
Optical feedback noise is noticeable even when the relative feedback optical power is on the order of 10−6 . Due to the optical feedback noise, the light output characteristics of an LD change intricately, according to the distance between reflective external optical objects and the LD, the feedback optical power, and driving conditions. In static or time-averaged characteristics, the light output power, the number of lasing modes, and light output spectra are modified. In dynamic characteristics, the noise level and shape of the light pulse are altered. Fundamental Equations Figure 6.44 shows a coupled cavity and its equivalent model in which the effect of the external cavity is expressed by the equivalent amplitude reflectivity reff . It is assumed that the power reflectivities for the facets of the LD are R1 and R2 , and the power reflectivity for the external reflector is R3 . Using an angular frequency of laser light and the round-trip time in the external cavity τ , the electric field of the reflected light E r e it is expressed as E r e it = E i e it
R2 + (1 − R2 ) R3 e−iτ + (1 − R2 ) R2 R3 e−i2τ + · · · ,
(6.183)
where E i e it is the electric field of the incident light. The third- and higher-order terms in the [· · · ] of (6.183) correspond to multireflections of light in the external cavity. Usually, the power reflectivity of the external reflector R3 is on the order of 10−2 or less. Therefore, multireflections of light in the external cavity are neglected in the following. As a result, reff is obtained as reff =
Er = R2 1 + a e−iτ , Ei
(6.184)
where a = (1 − R2 )
R3 1. R2
Reflective End Internal Cavity External (LD) Cavity (a)
(6.185)
LD
(b)
FIGURE 6.44 (a) Coupled cavity and (b) its equivalent model.
P1: OSO c06 JWBS036-Numai
186
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
Using the power decay rate in an LD itself, γ0 , the decay rate for the electric field γ0 /2 is given by 1 1 c γ0 = 2 2 n r0
1 1 αi + , ln 2L R1 R2
(6.186)
where c is the speed of light in vacuum, n r0 is the effective refractive index of the LD in a steady state, αi is the internal loss in the LD, and L is the length of the internal cavity. Using (6.186), the decay rate for the coupled cavity γ can be written 1 c 1 γ = 2 2 n r0
αi +
1 1 ln 2L R1reff 2
=
1 γ0 − κ e−iτ , 2
(6.187)
where c c a= (1 − R2 ) κ= 2n r0 L 2n r0 L
R3 . R2
(6.188)
Here κ is the coupling rate of feedback light to the LD. For R2 = 32%, R3 = 1%, n r0 = 3.5, and L = 300 µm, we have κ = 1.7 × 1010 s−1 . This value is between the decay rate for the carrier concentration, 1/τn ∼ 109 s−1 , and that for the photon concentration, 1/τph ∼ 1012 s−1 . Using the decay rate γ , an equation for the electric field E can be expressed as d E e it = dt
1 i ω N (n) + [a G(n) − γ ] 2
E e it ,
(6.189)
where ω N (n) is an angular frequency for the N th-order resonant mode, which is a function of the carrier concentration n. Substituting (6.187) into (6.189) results in d E(t) = dt
1 i [ω N (n) − ] + [a G(n) − γ0 ] 2
E(t) + κ E(t − τ ) e−iτ , (6.190)
where the final term on the right-hand side, κ E(t − τ ) e−iτ , represents a contribution of the feedback light. Note that (6.190) includes the phase of the laser light in contrast to (6.3), because the optical feedback noise is highly dependent on the phase of the feedback light. Effect of the Feedback Light on Static Characteristics It is assumed that the electric field E takes a steady-state value. From a real part in (6.190), the amplification rate at the threshold G th is given by a G th ≡ a G(n th ) = γ0 − 2κ cos(τ ).
(6.191)
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
NOISES
187
From an imaginary part in (6.190), the oscillation angular frequency at the threshold is obtained as ω N (n th ) = + κ sin(τ ).
(6.192)
Because the steady-state values ω N (n c0 ), 0 , and κ0 satisfy (6.191) and (6.192), we have a G(n c0 ) = γ0 − 2κ,
(6.193)
where it is supposed that n th = n c0 and ω N (n c0 ) = 0 . Because of the carrier lifetime τn ∼ 10−9 s and the photon lifetime τph ∼ 10−12 s, the carrier concentration n does not always take a steady-state value, even though the electric field E is in a steady state. Therefore, the threshold carrier concentration n th can be written n th = n c0 + n,
(6.194)
and the amplification rate at the threshold G th is expressed as G th = G(n c0 ) +
∂G n. ∂n
(6.195)
Substituting (6.195) into (6.191) results in ∂G −1 n = 2κ a [1 − cos(τ )], ∂n
(6.196)
where (6.193) is used, and it is assumed that κ = κ0 because 1/τn ∼ 109 s−1 and κ ∼ 1010 s−1 . The resonant angular frequency at the threshold ω N (n th ) is expressed as ω N (n th ) = ω N (n c0 ) −
ω N (n c0 ) ∂n r n. n r0 ∂n
(6.197)
Inserting (6.158) and (6.196) into (6.197) leads to ω N (n th ) = ω N (n c0 ) + ακ[1 − cos(τ )].
(6.198)
From the right-hand side of (6.198), it is found that the spectral linewidth enhancement factor α plays an important role in the optical feedback noise as well as the spectral linewidth of the LDs. Substituting (6.198) into (6.192) gives ω N (n c0 ) = + κ sin(τ ) − ακ[1 − cos(τ )].
(6.199)
P1: OSO c06 JWBS036-Numai
188
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
These results are summarized as follows: When the electric field E takes a steadystate value, the amplification rate at the threshold G th is given by a G th ≡ a G(n th ) = γ0 − 2κ cos(τ ),
(6.200)
and the oscillation angular frequency is expressed as ω N (n c0 ) = + κ sin(τ ) − ακ[1 − cos(τ )].
(6.201)
The optical feedback noise in LDs is larger than that in other lasers because LDs have a larger α, a lower R2 , and a shorter L than other lasers, which leads to a larger κ. For example, a gas laser with α 1, R2 = 98%, and L ∼ 1 m has κ ∼ 105 s−1 ; an LD with α = 1 to 7, R2 = 32%, and L ∼ 300 µm has κ ∼ 1010 s−1 . The terms including trigonometric functions in (6.200) and (6.201) indicate interference between the light in an LD and the feedback light. Due to this interference, hysteresis is present in both the oscillation angular frequency and the light output power. Figure 6.45 shows the resonant angular frequency ω N (n c0 ) as a function of the oscillation angular frequency for κτ = 1 and α = 3. Here the arrows represent the points where jumps. Figure 6.46 shows I –L curves calculated in continuous-wave (CW) operations with the spectral linewidth enhancement factor α as a parameter. Here Joule heating in the active layer is also considered. Hysteresis in the light output, which is observed experimentally, is reproduced theoretically as shown in Fig. 6.46.
FIGURE 6.45 Resonant angular frequency ω N (n c0 ) as a function of oscillation angular frequency .
P1: OSO c06 JWBS036-Numai
August 9, 2010
9:17
Printer Name: Yet to Come
189
Light Output (arb. units)
REFERENCES
Injected Current (mA) FIGURE 6.46
I –L characteristics in CW operation.
According to the phase of the feedback light, the dynamic characteristics are complicated; chaos and enhancement/suppression of the relaxation oscillation are observed. Enhancement of Noise Due to Feedback Light Due to feedback light, quantum noises increase in a certain frequency region. In addition, laser oscillations become unstable, and the noise increases in a low-frequency range which is less than several hundred megahertz by the feedback light. The reason for this is random mode hopping between the longitudinal modes in the internal cavity and those in the external cavity. Reducing Optical Feedback Noise To stabilize a longitudinal mode in the internal cavity, single-longitudinal-mode LDs such as DFB LDs or bistable LDs are needed. To suppress interference between the feedback light and the internal light in an LD, coherence of the laser light should be reduced by high-frequency modulation or selfpulsation. To decrease the optical power of feedback light, a low-coupling-rate κ is required, which is achieved by a large facet reflectivity and a long cavity, as shown in (6.187). However, a large reflectivity leads to a low light output, and a long cavity results in a large threshold current. Therefore, optical isolators are generally used to decrease the optical power of the feedback light in optical fiber communication systems, but the cost and size of the optical systems increase. REFERENCES 1. Z. I. Alferov, V. M. Andreev, E. L. Portnoy, and M. K. Trukan, “AlAs–GaAs heterojunction injection lasers with a low room-temperature threshold,” Sov. Phys. Semicond. 3, 1107 (1970).
P1: OSO c06 JWBS036-Numai
190
August 9, 2010
9:17
Printer Name: Yet to Come
FABRY–PEROT LASER DIODES
2. I. Hayashi, M. B. Panish, P. W. Foy, and S. Sumski, “Junction lasers which operate continuously at room temperature,” Appl. Phys. Lett. 17, 109 (1970). 3. N. Shomura, M. Fujimoto, and T. Numai, “Fiber pump semiconductor lasers with optical antiguiding layers for horizontal transverse modes,” IEEE J. Quantum Electron. 44, 819 (2008). 4. N. Shomura, M. Fujimoto, and T. Numai, “Fiber-pump semiconductor lasers with optical antiguiding layers for horizontal transverse modes: dependence on mesa width,” Jpn. J. Appl. Phys. 48, 042103 (2009). 5. N. Shomura and T. Numai, “Ridge-type semiconductor lasers with optical antiguiding layers for horizontal transverse modes: dependence on step positions,” Jpn. J. Appl. Phys. 48, 042104 (2009). 6. H. Yoshida and T. Numai, “Ridge-type semiconductor lasers with antiguiding layers for horizontal transverse modes: dependence on space in the antiguiding layers,” Jpn. J. Appl. Phys. 48, 082105 (2009). 7. H. Takada and T. Numai, “Ridge-type semiconductor lasers with antiguiding cladding layers for horizontal transverse modes,” IEEE J. Quantum Electron. 45, 917 (2009). 8. M. Hirao, A. Doi, S. Tsuji, M. Nakamura, and K. Aiki, “Fabrication and characterization of narrow stripe InGaAsP/InP buried heterostructure lasers,” J. Appl. Phys. 51, 4539 (1980). 9. I. Mito, M. Kitamura, K. Kobayashi, S. Murata, M. Seki, Y. Odagiri, H. Nishimoto, M. Yamaguchi, and K. Kobayashi, “InGaAsP double-channel-planar-buried-heterostructure laser diode (DC-PBH LD) with effective current confinement,” IEEE J. Lightwave Technol. 1, 195 (1983). 10. A. Yariv, Quantum Electronics, 3rd ed., Wiley, New York, 1989. 11. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347 (1980). 12. T. Numai, Fundamentals of Semiconductor Lasers, Springer-Verlag, New York, 2004.
P1: OSO c07 JWBS036-Numai
August 4, 2010
16:7
Printer Name: Yet to Come
7 QUANTUM WELL LASER DIODES
7.1
INTRODUCTION
Quantum well (QW) LDs [1, 2] are LDs that have quantum well active layers. A QW LD with one QW active layer is called a single quantum well (SQW) LD, and one with multiple QW active layers is called a multiple quantum well (MQW) LD. The density of states of a one-dimensional QW is a step function of the energy. Therefore, a low threshold current, high differential quantum efficiency, high-speed modulation, low chirping, and narrow spectral linewidth are obtained simultaneously in QW LDs.
7.2
FEATURES OF QUANTUM WELL LDs
Configurations of Quantum Wells Figure 7.1 shows the configurations of various QWs at a band edge. The horizontal and vertical directions represent a position and the energy of an electron, respectively. Here E c and E v are the band-edge energies of the conduction and valence bands, respectively. Figure 7.1(a) illustrates an SQW active layer in which the optical confinement factor a is small because the active layer thickness L z is as thin as 10 nm or less. This small a leads to a large threshold current density Jth . Therefore, to obtain a larger a than that of the SQW, structures (b)–(e) were developed. Figure 7.1(b) shows a MQW which has multiple QW active layers in the optical confinement region. However, due to the energy barriers between the adjacent QW Laser Diodes and Their Applications to Communications and Information Processing, By Takahiro Numai C 2010 John Wiley & Sons, Inc. Copyright
191
P1: OSO c07 JWBS036-Numai
192
August 4, 2010
16:7
Printer Name: Yet to Come
QUANTUM WELL LASER DIODES
(a)
(b)
(c)
(d)
(e)
FIGURE 7.1 Configurations of quantum wells: (a) SQW; (b) MQW; (c) modified MQW; (d) SCH; (e) GRIN-SCH.
active layers, the carrier injection efficiency decreases with propagation of the carriers. Hence, it is difficult to achieve uniform carrier distribution all over the multiple QW active layers. The modified MQW shown in Fig. 7.1(c) can improve the carrier injection efficiency and the uniformity of the carrier distribution in an MQW. The energy barriers between the adjacent QW active layers are lower than those in the cladding layers, which results in a high carrier injection efficiency and uniform carrier distribution all over the multiple QW active layers. A separate confinement heterostructure (SCH), which has two energy steps, as shown in in Fig. 7.1(d), is able to obtain a large optical confinement factor a in the SQW active layer. In the materials used conventionally for LDs, the refractive index increases with a decrease in the bandgap energy. The outer potential confines light to the QW active layer by the refractive index distribution; the inner potential confines the carriers by the energy barriers. Because the potentials to confine the light and the carriers are separate, this structure is called an SCH. Figure 7.1(e) illustrates a graded index SCH (GRIN-SCH) whose potential and refractive index distributions in the outside of the active layer are parabolic. When the active layer thickness L z is small, the optical confinement factor a of a GRIN-SCH is proportional to L z and a of an SQW is proportional to L z 2 . Therefore, when the active layer is thin, the optical confinement factor a of the GRIN-SCH is larger than that of the SQW. Characteristics of QW LDs Low Threshold Current The density of states per unit area ρ(E) in an SQW is written ρ(E) =
∞ m∗ H (E − n ), π 2 n=1
(7.1)
where E is the energy of the carrier, m ∗ is the effective mass of the carrier, is Dirac’s constant, and H (x) is the Heaviside function or step function, which is defined as H (x) =
0 1
(x < 0), (x ≥ 0).
(7.2)
P1: OSO c07 JWBS036-Numai
August 4, 2010
16:7
Printer Name: Yet to Come
FEATURES OF QUANTUM WELL LDs
193
When the barrier height or the barrier thickness is large so that each QW can be independent, the energy level of the nth subband n is expressed as n =
(nπ )2 . 2m ∗ L z 2
(7.3)
In this case, the density of states per unit area ρ(E) in an MQW is obtained as
ρ(E) = N
∞ m∗ H (E − n ), π 2 n=1
(7.4)
where N is the number of QWs. When the barrier height or barrier thickness is small, the wave functions for the carriers penetrate adjacent QWs. In this case the QWs are coupled to each other and degeneracy is removed. As a result, N quantum levels are generated per quantum level degenerated, and ρ(E) is given by
ρ(E) =
∞ N m∗ H (E − nk ), π 2 n=1 k=1
(7.5)
where nk (k = 1, 2, . . . , N ) is the energy of each quantum level, which is produced by the removal of degeneracy. Because the densities of states in QW LDs are step functions, QW LDs have narrow optical gain spectra. Hence, the optical gain concentrates on a certain energy, and the peak optical gain is enhanced. As a result, a threshold current density Jth is lower than that of a bulk double heterostructure (DH) LD. Under the k-selection rule and the assumption of a negligibly low nonradiative recombination rate, the linear optical gain g of the QW LD is written c ω g(E, n) = 2 χI (E, n). nr nr
(7.6)
Here c is the speed of light in vacuum, n r is the effective refractive index of the QW LD, E is the photon energy, ω is the angular frequency of light, and χI is an imaginary part of the relative electric susceptibility. The imaginary part of the relative electric susceptibility χI is expressed as ω χI (E, n) = nr2
∞ n=0 j=l,h
j
n, j
j ρred,n [ f c (E c,n ) − f v (E v,n )]χˆ I (E, ) d.
(7.7)
P1: OSO c07 JWBS036-Numai
194
August 4, 2010
16:7
Printer Name: Yet to Come
QUANTUM WELL LASER DIODES n, j
The imaginary part of the relative electric susceptibility χˆ I (E, ) for a photon with energy E and an electron–hole pair with energy is given by n, j
χˆ I (E, ) =
e2 h 2m 0 2 ε0 n r 2 E g
|Mn, j ()|2ave
/τin , (E − )2 + (/τin )2
(7.8)
where j = h and l represents the heavy hole (h) and the light hole (l), respectively; e is the elementary electric charge; h is Planck’s constant; m 0 is the electron mass in vacuum; ε0 is the permittivity of vacuum; E g is the bandgap energy; |Mn, j ()|2ave is a square of the averaged momentum matrix element; and τin is the intraband j relaxation time. The reduced density of states for the nth subband ρred,n is defined as j ρred,n
≡
1 1 + j ρc,n ρv,n
−1 ,
(7.9)
j
where ρc,n and ρv,n are the densities of states for the nth subband in the conduction j and valence bands, respectively; f c (E c,n ) and f v (E v,n ) are Fermi–Dirac distribution j functions of the conduction and valence bands, respectively. Here E c,n and E v,n , which are the energies of the conduction and valence bands, respectively, are expressed as j
E c,n =
j
j
m c v,n + m v E + m v c,n j
mc + mv j
j = E v,n
,
(7.10)
,
(7.11)
j
m c v,n − m c E + m v c,n j
mc + mv
j
where m c and m v are the effective masses of the conduction electron and hole, j respectively; c,n and v,n are the energies of the nth subband in the conduction and valence bands, respectively. Figure 7.2(a) shows the modal gain gmod = a g as a function of the injected current density J with the number of QWs N as a parameter. Here a is the optical confinement factor of the active layer. According to the optical loss in the optical cavity, which is equal to the threshold gain, the number of QWs N to minimize Jth are present. Also, in the SQW (N = 1), gain flattening is observed with an increase in the injected current, as shown in Fig. 7.2(b). The cause of the gain flattening is that the density of states is a step function H (E − E L ) of the photon energy E with a lasing photon energy E L . Characteristic Temperature Because the densities of states in QWs are step functions, changes in characteristics of QW LDs with temperature are expected to be small. However, even in a bulk DH LD, a characteristic temperature T0 as high as 200 K is obtained, and an advantage of QW LDs over bulk DH LDs has not yet been proved.
P1: OSO c07 JWBS036-Numai
August 4, 2010
16:7
Printer Name: Yet to Come
FEATURES OF QUANTUM WELL LDs
195
g
Modal Gain g
g
g
Current Density (a)
/ (b)
FIGURE 7.2 (a) Modal gain; (b) gain flattening.
Anisotropic Optical Gain A difference in the optical gains for TE and TM modes is about 20 cm−1 in bulk DH LDs and about 140 cm−1 in QW LDs. In bulk DH LDs, an anisotropic optical gain is generated because the optical confinement factor a of the TE mode is larger than that of the TM mode, due to the configurations of the optical waveguides. In QW LDs, anisotropic optical gain is also produced by the selection rule of optical transitions, which is explained in the following. In bulk structures, the heavy hole (hh) band with m j = 3/2 and the light hole (lh) band with m j = 1/2 are degenerate at k = 0; in QWs this degeneracy is removed. Figure 7.3 shows the energies of the valence band. Here solid lines represent the energies of the heavy hole bands E hh1 and E hh2 and dashed lines represent the
FIGURE 7.3 Valence band in a one-dimensional QW.
P1: OSO c07 JWBS036-Numai
196
August 4, 2010
16:7
Printer Name: Yet to Come
QUANTUM WELL LASER DIODES
FIGURE 7.4 Quantum well.
energies of the light hole bands E lh1 and E lh2 , where the subscripts 1 and 2 are the values of the principal quantum number n. The effective masses of the hole in QWs depend on the directions of the axes. As shown in Fig. 7.4, if the quantization axis is the z-axis, the effective mass along the z-axis of the heavy hole m z,hh is larger than that of the light hole m z,lh and the effective mass on the x y-plane of the heavy hole m x y,hh is smaller than that of the light hole m x y,lh . This result is summarized as m z,hh > m z,lh: along the z-axis, m x y,hh < m x y,lh: on the x y-plane. According to the px - or p y -like orbital, a wave function of the heavy hole distributes on the x y-plane. Therefore, the heavy hole moves on the x y-plane more easily than along the z-axis, which leads to m x y,hh < m z,hh . According to the pz -like orbital, a wave function of the light hole distributes along the z-axis. Therefore, the light hole moves along the z-axis more easily than on the x y-plane, which results in m z,lh < m x y,lh . For example, in GaAs we have m z,hh = 0.377m 0 , m z,lh = 0.09m 0 , m x y,hh = 0.11m 0 , and m x y,lh = 0.21m 0 , where m 0 is the electron mass in vacuum. Wave Function It is assumed that a wave vector k is along the z-axis and that a contribution of the split-off band is negligibly small. If the up-spin and down-spin are expressed as α and β, respectively, the wave functions of the conduction band are written |sα, |sβ.
(7.12)
Quantum states of the valence bands are indicated by j = l + s where l is the angular momentum operator and s is the spin operator. The wave function | j, m j can be expressed as follows: For the heavy hole, 3 3 1 , 2 2 = √2 |(x + i y)α, 3 , − 3 = √1 |(x − i y)β, 2 2 2
(7.13)
P1: OSO c07 JWBS036-Numai
August 4, 2010
16:7
Printer Name: Yet to Come
FEATURES OF QUANTUM WELL LDs
197
and for the light hole, 3 1 1 , 2 2 = √6 |2zα + (x + i y)β, 3 , − 1 = √1 |2zβ − (x − i y)α, 2 2 6
(7.14)
where j is the eigenvalue of j and m j is the eigenvalue of jz . Momentum Matrix Elements with k Along the z-Axis As shown in (2.29), the power optical gain coefficient g is in proportion to a square of the momentum matrix element 1| p|22 . Therefore, to examine the anisotropic optical gain, the momentum matrix elements for the optical transitions are compared. The x-, y-, and z-components of the momentum matrix elements between the conduction band and the heavy hole band Mx , M y , and Mz are given by 1 √ Mx = √ 3M, 2 1 √ M y = ±i √ 3M, 2 Mz = 0,
(7.15) (7.16) (7.17)
where M is defined as √ 3M ≡ s| px |x = s| p y |y = s| pz |z 1/2 1 E g (E g + 0 ) = m0 . 2m e E g + 23 0
(7.18)
Here (1.6) and (1.43) are used, m 0 is the electron mass in vacuum, m e is the effective mass of the conduction √ electron, E g is the bandgap energy, and 0 is the split-off energy. A coefficient 3 on the left-hand side in (7.18) is introduced so that a matrix element averaged over all directions of the wave vector k may be M. Momentum Matrix Elements with k Along an Arbitrary Axis As shown in Fig. 7.5, it is assumed that a quantization axis is the z-axis and a QW layer is placed on the x y-plane. If light propagates along the x-axis, an electric field E along the y-axis is the TE mode and an electric field E along the z-axis is the TM mode. For the wave vector k with an arbitrary direction, a direction of k can be expressed in a polar coordinate system, as shown in Fig. 7.5.
P1: OSO c07 JWBS036-Numai
198
August 4, 2010
16:7
Printer Name: Yet to Come
QUANTUM WELL LASER DIODES
(TM)
(TE) FIGURE 7.5 Direction of a wave vector k.
According to the directions of k, the x-, y-, and z-components of the momentum matrix elements Mx , M y , and Mz are given by 1 √ Mx = √ 3M(cos θ cos φ ∓ i sin φ), 2 1 √ My = √ 3M(cos θ sin φ ± i cos φ), 2 1 √ Mz = − √ 3M sin θ. 2
(7.19) (7.20) (7.21)
Note that the square of the optical transition matrix elements is proportional to 1|E · p|22 , where E is the electric field. Momentum Matrix Elements Between the Conduction Band and the Heavy Hole Band in QWs An average of the squared momentum matrix element M 2 hh,TE for the TE mode (E//y) is expressed as 2π 3M 2 1 · (cos2 θ sin2 φ + cos2 φ) dφ 2 2π 0
3M 2 kz 2 3M 2 = (1 + cos2 θ ) = 1+ 2 4 4 k 2
E z,n 3M , 1+ = 4 En
M 2 hh,TE =
(7.22)
where E z,n is the quantized energy of the nth subband and E n is the total energy of the nth subband. At the subband edge with E z,n = E n , we have M 2 hh,TE =
3M 2 . 2
(7.23)
P1: OSO c07 JWBS036-Numai
August 4, 2010
16:7
Printer Name: Yet to Come
FEATURES OF QUANTUM WELL LDs
199
An average of the squared momentum matrix element M 2 hh,TM for the TM mode (E//z) is given by 3M 2 3M 2 sin2 θ = (1 − cos2 θ ) 2 2
kz 2 E z,n 3M 2 3M 2 1− 2 = 1− . = 2 k 2 En
M 2 hh,TM =
(7.24)
At the subband edge with E z,n = E n , we obtain M 2 hh,TM = 0.
(7.25)
As shown in (7.23) and (7.25), a selection rule is present between the conduction band and the heavy hole band at the subband edge. Only the optical transition for the TE mode is allowed, and that for the TM mode is inhibited. Momentum Matrix Elements Between the Conduction Band and the Light Hole Band in QWs An average of the squared momentum matrix element M 2 lh,TE for the TE mode (E//y) is obtained as M2 (1 + cos2 θ ) + M 2 sin2 θ 4
E z,n E z,n M2 + M2 1 − . 1+ = 4 En En
M 2 lh,TE =
(7.26)
As a result, at the subband edge with E z,n = E n , we have M 2 lh,TE =
M2 . 2
(7.27)
An average of the squared momentum matrix element M 2 lh,TM for the TM mode (E//z) is written M2 sin2 θ + 2M 2 cos2 θ 2
E z,n E z,n M2 + 2M 2 1− . = 2 En En
M 2 lh,TM =
(7.28)
Therefore, at the subband edge with E z,n = E n , we obtain M 2 lh,TM = 2M 2 .
(7.29)
Optical Gains in QW LDs The concentration of a heavy hole is larger than that of a light hole. Therefore, optical gains for the TE mode, ge-hh,TE and ge-lh,TE , and that
P1: OSO c07 JWBS036-Numai
200
August 4, 2010
16:7
Printer Name: Yet to Come
QUANTUM WELL LASER DIODES
for the TM mode, ge-lh,TM , are related as ge-hh,TE > ge-lh,TM > ge-lh,TE , where the subscript e-hh represents recombination of the conduction electron and the heavy hole and the subscript e-lh represents recombination of the conduction electron and the light hole. Low-Loss Optical Waveguides In a QW, the density of states is a step function, and the absorption coefficient at the band edge changes sharply with a wavelength of light. Therefore, the absorption loss in optical waveguides with a QW is lower than that with a bulk DH. As a result, the threshold gain of a QW LD is lower than that of a bulk DH LD. High-Speed Modulation given by
From (6.101), the relaxation oscillation frequency f r is
1 fr = 2π
a
∂G S0 . ∂n τph
(7.30)
In a QW LD, based on the steplike density of states, the differential optical gain a ∂G/∂n is larger than that in a bulk DH LD. Hence, f r in a QW LD is larger than that in a bulk DH LD. As a result the modulation speed of a QW LD is higher than that of a bulk DH LD. Figure 7.6 shows a calculated differential optical gain (n r /c)a ∂G/∂n
QW- LD
DH-LD
Quasi-Fermi Level
(meV)
FIGURE 7.6 Differential optical gain.
P1: OSO c07 JWBS036-Numai
August 4, 2010
16:7
Printer Name: Yet to Come
STRAINED QUANTUM WELL LDs
201
as a function of the quasi-Fermi level of the conduction band E Fc . The differential optical gain of a QW LD is higher than that of a bulk DH LD. From (7.6)-(7.8), optical gain g can be reduced to g = |M|2ave ρ [ f c − f v ] .
(7.31)
A QW modifies the density of states ρ in (7.31); the intentionally doped active layers can change [ f c − f v ]. For example, a p-doped active layer reduces f v ; a relaxation oscillation frequency of about 30 GHz was reported. Narrow Spectral Linewidth is expressed as
From (6.182), the spectral linewidth ω0 of laser light
ω0 =
ωm cE cv ln(1/R) (1 + α 2 ), 4P0 n r L
(7.32)
where α is given by α=−
2ωm ∂n r n r ∂n
∂G −1 a
