Diffraction Coherence in Optics

THE COMMONWEALTH A N D INTERNATIONAL LIBRARY Joint Chairmen of t h Honorary ~ Editorial Advisory Board SIR ROBERT ROBINSON, O.M.. F.R.S., LONDON, AND DEAN ATHELSTAN SPILHAUS, MINNESOTA Publisher :ROBERT MAXWELL. M.c., M.P.

PHYSICS DIVISION General Editors:

W. ASHHURST AND

L. L. GREEN

DIFFRACTION Coherence in Optics

DIFFRACTION Coherence in Optics

TRANSLATED FROM THE FRENCH BY

BARBARA JEFFREY TRANSLATION EDITED BY

J . H. SANDERS Fellow of Oriel Collegu, Oxford

PERGAMON PRESS OXFORD * TORONTO

+

LONDON PARIS

NEW Y O R K FRANKFURT

Pergamon Press Ltd., Headington Hill Hall, Oxford 4 & 5 Fitzroy Square, London W. 1 Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh I Pergamon Press Inc., 4 4 4 1 21st Street, Long Island City, New York f I 101 Pcrgamon of Canada Ltd., 6 Adelaide Street East, Toronto, Ontario Pergamon Press s.A.J.L., 24 rue des Bco~es,Paris 5C Pergamon Press GmbH,Kaiserstrasse 75, Frankfurt-am-Main Copyright Q 1966 Pergarnon Press Ltd. First edition 1966 Library of Congress Catalog Card No. 65-28098 .-

. .

Printed in Great Britain by J . W. Arrowsmith Lrd.. Winrerstoke R w d , Brisrol3

This book is sold subject to the condition that it shall not, by way of trade, be lent, resold, hired out, or otherwise disposed of without the publisher's consent, in any form of binding or cover other than that in which it is published.

Contents PAGE

ix I. HUYGENS' PRINCIPLE AND. DIFFRACTION PHENOMENA FOR A MONOCHROMATIC POINTSOURCE 1. I

1.2 1.3 1.4 1.5 1.6

2.1 2.2 2.3 2.4 2.5

Diffraction at infinity and at a finite distance Light vibrations The Huygens-Fresnel principle Simplified expressions for the field Calculation of the path difference The general expression for the luminous intensity Diffraction by a rectangular aperture Diffraction by a narrow slit Diffraction by a circular aperture The asymptotic expansion for large values of Z The distribution of luminous flux in the Airy spot

III . THE FOURIERTRANSFORMATION 3.1 The representation of a periodic function by a Fourier series 3.2 The Fourier integral 3.3 Some Fourier transforms 3.4 General properties connecting the wave surface with the diffraction pattern 4.1 Diffraction by several apertures identical in shape and orientation 4.2 Diffraction by a large number of apertures, identical in shape and orientation, irregularly distributed over the diaphragm

27 28 30

36

41

42

Contents

4.3 4.4 4.5 4.6

Complementary screens. Babinet's theorem Diffraction by two identical slits Diffraction by three identical slits Diffraction by many slits

Spatial coherence and time-coherence The relation between the length of wave trains and the spectral width of the emitted radiation. Coherence length and coherence time The vibration emitted by an atom. The variation in the complex amplitude of the vibration during a wave train The successive wave trains emitted by an atom Vibrations from two different atoms which radiate the same mean frequency v, (quasimonochromatic light) Vibrations with different frequencies from a single atom Vibrations with the same frequency from a single atom The representation of the emission of an incoherent light source The influence of time-coherence on diffraction phenomena. The diffraction spot in quasimonochromatic light Spatially coherent and incoherent objects The image of an extended object illuminated with spatially incoherent light The image of an extended object illuminated with spatially wherent light

Dmcno~ PHENOMENA M PERFECT OPTICAL INSTRUMENTS

6.1 Resolving power and the limit of perception in optical instruments

Contents

vii PAGE

6.2

6.3 6.4 6.5

6.6

6.1.1 Resolving power and the limit of perception for an astronomical telescope 98 6.1.2 Resolving power and the limit of perception for a microscope 101 6.1.3 Resolving power of a prism spectroScope 103 Diffraction gratings 105 6.2.1 Description 105 6.2.2 Infinite grating 106 6.2.3 The dispersion of a diffraction grating 108 6.2.4 Superposition of the spectra of a grating 109 6.2.5 Finite grating 111 6.2.6 Real gratings 113 6.2.7 The mounting of a grating 114 6.2.8 Ghosts 114 Phase contrast 115 Dark ground method 119 Apodization 120 Filtering spatial frequencies (incoherent object) 12 1

7.1 Diffraction in the case of a spherical wave 7.2 Diffraction pattern when there is a focusing error 7.3 Precision o f focusing with an optical instrument 7.4 Diffraction spot in the presence of aberrations

126 129 131

132

Preface THIS book presents a detailed account of the course on Fraunhofer diffraction phenomena, i.e. diffraction at infinity, which is studied at the Faculty of Science in Paris. In the first few chapters these phenomena are investigated by the plane wave method. The effects due to focusing defects and aberrations are treated more logically by the spherical wave method in Chapter VII (paragraphs 2.4, 2.6, 6.6 and Chapter VII do not form part of the course). Chapter V, which deals with ekmentary concepts of coherence, is developed further than it is in the course given to the students (paragraphs 5.4, 5.5, 5.7, 5.8 and 5.9 are not covered). Current developments in optics show that it is essential for future engineers and research workers to be familiar with the basic concepts of coherence of vibrations from the second year of their degree courses. The account given in Chapter V is, of course, an elementary one but it will help the student to go on to tackle more complete treatises.

CHAPTER I

Huygens' Principle and Dtffraction Phenomena for a Monochromatic Point Source M&rction at hlhity and at a M t e distance Let 0 be a thin converging lens forming an image at S' of the point source S (Fig. 1.1). The effective surface of the lens is limited by a diaphragm D with an aperture T. The image S' is a small spot with a structure which depends on the form of the aperture T. The image S' cannot be investigated simply on the basis of geometrical optics. The concept of diffraction has to be introduced, as

1.1

it was by the astronomer Airy, who was the first to calculate the distribution of light within the image S, the so-called "diffraction spot". In the case shown in Fig. 1.1, investigating the structure of the image S involves the investigation of diffraction phenomena at infinity, or Fraunhofer phenomena The diffraction can be considered to be at infinity because the lens 0 could be replaced by two lenses 0,and 0, such that S and S remain conjugate (Fig. 1.2) : 0, has a focal length equal to the distance SO, and 0,a f o d length equal to the distance 0,S'. The effect is the same as if 4

Dzfraction : Coherence in Optics

lens O2 were illuminated by a source at infinity, that is a plane wave. The plane wave is limited by the aperture Tin the diaphragm D. The structure of the image can also be investigated in a plane very close to Sf. Provided that the error in focusing is small the diffraction phenomena still belong to the class of Fraunhofer phenomena,

but if they are observed in a plane x (Fig. 1.1) at some distance from S' they are no longer the same. In practice diffraction phenomena occur only towards the edge MM' of the beam, that is to say near the limit of the geometrical shadow of the stop D. Since in this case the diffraction is observed far away from the image S', it is equivalent to the case where the image is virtual (Fig. 1.3) or the lens is absent (Fig. 1.4): these are diffraction phenomena at a finite distance, or Fresael phenomena. Phenomena of this class are observed as shown in the diagram in Fig. 1.4: the diffracting aperture T is placed between the light source S and the observing screen n.

Huygens' Principle and Diffraction Phenomena

3

In general the effects produced by diffraction are limited to the edges MM' of the geometrical beam, but if the aperture T is sufficiently small, C is very close to the edge MM' of the geometrical beam and there is a small "diffraction spot" which covers the region M CM'. To summarize, two experimental arrangements can be considered : (a) a point source S, a tens 0 covered by a diaphragm D with an aperture T (T can be the surface of the lens itself limited by its mount) and an image S' of S. (b) a point source S, a diaphragm D with an aperture T and an observing screen n.

In case (a), the structure of the image S' is due to diffraction phenomena at infinity (Fraunhofer) and in case (b), the distribution of light on the screen x is due to diffraction phenomena at a finite distance (Fresnel). From now on we shall confine our investigations to diffraction at infinity or Fraunhofer diffraction. 1.2 Light vibrations Fresnel was the first to explain the phenomena of physical optics by assuming that light is made up of transverse vibrations. In his theory, Fresnel compares light vibrations with transverse elastic vibrations in solids. This hypothesis gave rise to a number of contradictions which led to the replacement of Fresnel's mechanical theory by Maxwell's electromagnetic theory. According to the electromagnetic theory light appears to be due to the simultaneous propagation of an electric field and a magnetic field, the

4

DlfJracrion : Coherence in Optics

vibrations of the electric field representing the light vibrations in the space in which the light is propagated. Let us consider an isotropic dielectric medium in which the velocity of propagation of the electric field E is equal to u. Maxwell's equations lead to the wave equation :

where V2E is the Laplacian of E and t is the time. Let us take a plane wave Z normal to the axis ox (Fig. 1.5) and let x be its abscissa at time t. The electric field E lies in the wave front and the Laplacian v2Ereduces to the derivative d213/dX2. Equation (1.1) becomes

Each rectangular component E, or E, (Ex = 0) of E satisfies equation (1.2). We shall see later that when we study diffraction phenomena in optical instruments with small apertures we can consider the light vibration as a scalar quantity. We shall represent this by a function U and (1.2) will be written in the form:

Huygens' Principle and Dlffmction Phenomena

5

If we put

(1.3) becomes :

which has the general solution :

G and H being two arbitrary functions. We shall retain only the function G(u) which represents a progressive wave propagated in the direction ox. Thus Maxwell's theory shows that a varying electromagnetic field does not -remain localized in one region in space: the electric field and the magnetic field are propagated. Huygens' principle allows us to specify the mechanism of this propagation (5 1.3). Since the function G(u) is arbitrary, equation (1.3) does not define the form of the variation of the vibration as a function of time. Let us take G to be a function which is sinusoidal with time. On the basis of (1.4) the vibration can be represented by the expression :

where a is the amplitude, 8 is a constant and o the angular frequency of the light vibration. a, 8 and o are constants. Representing U by a sinusoidal function means considering a monochromatic plane wave with vibrations of period T and frequency v Bven by:

Let A,, = UTbe the wavelength of the vibrations. Put:

The wavelength

A,, is characteristic of a given radiation in a

Dzfraction : Coherence in Optics

6

particular medium. If a given radiation passes from one medium to another its frequency remains the same but its wavelength changes. Radiation can be characterized by its wavelength in a vacuum

I

= c/v

with c = 3 x 108m/s.

If radiation of frequency v is propagated in a medium with a refractive index n, its wavelength will be

(1.8) can therefore be written in the form:

the product 6 = nx is the optical path between 0 and M, or the path difference between these two points. cp is the phase difference between 0 and M. Neglecting the constant 0 in expression (1.7), the light vibration in a plane wave can be written:

It is more advantageous to use complex notation by putting:

with

The physical vibration (3.1 1) is obtained by taking the real part of (1.12). The intensity of the vibration at a point is equal to the square of the modulus of the complex amplitude. It is assumed that this represents the luminous intensity at this point. The use of the complex notation is convenient since if the operations on U are linear the calculations can be carried out using a complex function. The physical magnitude is then the real part of the final expression obtained from calculation. Since @Or appears as a factor in all calculations it is omitted. The vibration is therefore simply represented by the expression :

U

= ae - j P

(1.13)

Huygens' Principle and Diffraction Phenomena

7

that is the complex amplitude of the vibration in the case of a plane wave. It is to be noted that expressions (1.12) and (1.13) represent a monochromatic vibration when the wave is not limited by any screen (unrestricted wave).

1.3 The HuygellgFreswl prindple Let us consider a source S (Fig. 1.6) and let Z be the wave surface at time t. In Huygens' hypothesis each point M of T is considered as a secondary source which emits a spherical wavelet (in a homogeneous medium). At time t+B, the radius of the wavelet is v0, where v is the velocity of propagation. At the time t+B the wave Z'is the envelope of the wavelets with radius ve. In a homogeneous and isotropic medium this is a sphere with radius v(t 9). Huygens thus showed the mechanism of propagation to be a step by step process between successive points in space. Huygens' construction was completed by Fresnel's hypothesis according to which interference can occur between the different wavelets. It therefore seems natural to assume that the secondary sources distributed on Z have exactly the same phase as that of the vibrations on Z. A rigorous investigation of the phenomena shows, however, that a phase lead of n/2 has to be introduced.

+

8

Dzflraction :Coherence in Optics

The Huygens-Fresnel principle not only gives an amplitude in the direction of propagation, but also an amplitude in the reverse direction, the wave Z" (Fig. 1.7),which is the other surface envelope of the wavelets. A mathematical investigation of the phenomena justified the Huygens-Fresnel principle and eliminated the possibility of a wave Z", which is not found in practice.

Diffraction phenomena can be calculated in a simple manner on the basis of the Huygens-Fresnel principle. Let us consider an objective 0 illuminated by a beam of parallel rays (Fig. 1.8). The incident plane wave is limited by a diaphragm D and the geometric image of the point source S is formed at S'. According to the

Huygens' Principle and D~flructionPhenomena

9

Huygens-Fresnel principle we can assume that all the elements of Z behave like secondary sources in phase. Figure 1.8 shows the rays which are diffracted in the direction a. These rays meet at P f in the focal plane n' of the objective 0.The vibration at P' is therefore the resultant of an infinite number of vibrations sent out by the secondary sources distributed over Z. Calculation of the vibratory state at P' thus becomes an interference calculation. We assume that the amplitude of the radiation emitted by the secondary sources is independent of the direction, that is of the angle a. This approximation is valid provided a remains small. 1.4 Simplieed expressions for the field Let us .consider Fig. 1.8. Expression (1.12)can be written in the vector form :

This represents a solution to (1.1) which does not introduce a diaphragm of any sort. Now the electric field E at some point on I: is not the same with and without a diaphragm D. A rigorous solution would consist of rewriting the equation of propagation taking into account the presence of the diaphragm D. This very complicated problem has not yet been solved. In all cases it is assumed that the diaphragm D does not perturb the wave apart from suppressing the parts which are masked, that is the electric field at some point on Z is the same with and without the diaphragm D. This approximation is valid provided the dimensions of the aperture are large in comparison with the wavelength. In other respects we shall confine ourselves in the following to the application of Huygens' principle in an intuitive form: if the field at (Fig. 1.91, the field produced by some point M on Z is E = this point at P' will be represented by (1.14)with

where ( M P f )is the optical path from M to P'. Neglecting the factor dwr,the vibrations at P due to M and C are represented by e - j k ( M P ' ) and e-jk(CP') where k = 21r/A. Let us take as the origin of the phases at Pf the phase of the vibration emitted from the point C. The vibration at P' due to M

D~+@ruction: Coherenee in Optics

is then written ejk(CP'-MP'). Let us now put

the field emitted by a small element dX of Z is proportional to dZ and the total field at P' will be given by :

where A is a constant. The integral (1.15) is taken over the unmasked part of Z. The electric field is in the same direction at different points in the wavefront. The fields at P' (which is always very close to S') will be in practically the same direction if all the rays converging at P' are at a small angle to the axis. This is the case if the aperture a' of the objective 0,stopped by the diaphragm D, is sufficiently small that cosa' can be taken as unity. It is therefore unnecessary to introduce the direction of the electric field. The light vibration can be considered as a scalar quantity, the amplitude at P' being written:

assuming'that the electric field is the same at all points of the wavefront (a = 1 for example).

I S Calculation of the path Werence It is assumed that the objective 0 is perfect. Let us calculate the path difference 6 = (CP')-(MP' at P It is represented by CH

Huygenr' Principle and Dzflruction Phenomena

11

in Figs. 1.10 and 1.1 1 since according to Malus's theorem if I;' is a plane normal to the rays, all optical paths from Z' to P' are equal. Figure 1.1 1 is a perspective view of Fig. 1.10. As before it is assumed that the incident rays form a beam of parallel rays normal to the diffracting diaphragm D. This hypothesis does not at all reduce the generality of the following calculations. The coordinates 11 and of the point M refer to two coordinate axes Cq and Cc in the plane of the diaphragm D. The coordinates y' and z' of the point P' refer to two axes S'y' and S'z' in the plane n'.

12

Diffraction : Coherence in Optics

Since P' is close to S' the direction cosines of CI (or OP') are u = y'/R and v = zr/R where R = 0s'. The path difference 6 = (CPf)-(MPr) is considered positive when the wave arrives at H later than at C. The difference

6 = CH

= (CPr)-(MP')

is equal in magnitude and sign to the projection on the direction CI of the vector CM which has components q and C. If q is a unit vector on CI : and the phase difference :

1.6 The general expression for the luminous intensity at P' According to (1.16), the amplitude at P' is

whence the intensity is

I=ff* The integral (1.19) assumes that the amplitude is the same at all points on the wave Z. The expression (1.19) can be generalized by assuming that there are both amplitude variations and phase variations in the wavefront. These variations can be produced by placing a plate of glass with variable absorption and thickness in the plane of the diaphragm D. Variations in absorption produce variations in the amplitude of the wave Z, and variations in thickness produce variations of phase. These variations may have a different origin: for example, if the transparency of the objective is not uniform and if it has aberrations. To calculate the amplitude at P' these defects can be transferred to the wavefront Z and the objective can thereafter be assumed perfect. The variations in transparency produce variations in amplitude on Z and the aberrations produce phase variations.

Huygens' Principle and Diffraction Phenomena

13

These two effects, variations in phase and variations in amplitude on Z,can be represented by a function F(q, 0 of the form:

A(q, () denotes the distribution of the amplitude on the wavefront and @(q,0 the phase, that is the deviation of this wavefront from a plane. If @(q,c) = 0, the wave X is perfectly plane and A(q, 5) gives the amplitude distribution on the wavefront. If A(q, 0 = const. and if @(q,5) # 0, the wavefront is deformed but the amplitude is the same at all points. Thus if F(rl,n is real, there are only variations in absorption and if F(q, 0 is complex there are also phase variations. In the general case the amplitude at P' is a function of u and v (or of y' and 2') which is written in the form :

f (u, v ) =

If

F(q, ( ) # ( Y 9 + v 4 )

dq dC

This expression shows that the development of diffraction phenomena is expressed mathematically by a Fourier transformation. The amplitude f(u, v ) at P' is a Fourier transform of the function F(q, 5). Since F(q, c) is zero outside the uncovered part Z of the wavefront, the limits of the integral (1.22) can be chosen in an infinite number of ways. The same is true in the case where the amplitude A(q, 0 is constant and @(q, c) = 0. All that is required is to put F(q, C) = 1 for all the points on the uncovered part X of the wavefront and F(q, 5) = 0 outside this.

CHAPTER I1

Dzfraction by an Aperture of Simple Form Difiaction by a redanguk aperture The diaphragm D is placed in front of the objective lens 0 (Fig. 2.1) to limit its aperture. It is assumed that D is sufficiently close to 0 for all the rays diffracted by the aperture to be incident on 0.The centre C of the diaphragm is on the principal axis COY of the lens. The incident rays are in the form of a bundle of rays parallel to COS' and the plane of the diaphragm D is normal to the incident rays. The sides of the rectangular diaphragm are of lengths 2t7, and 25,. The intensity is calculated for a point P' in the plane r' passing through the geometric image S of the source. From (1.19) the amplitude at P' is given by 2.1

The position of the point P' in the plane n' is determined by the direction cosines of the diffracted rays. We have:

sin kuq, sin kvt;, f (u, v) = 4tl0C0 kuqo kof;,

(2.2)

whence the intensity is:

Expression (2.3) gives the variations of intensity in the image of a monochromatic point source produced by an objective which is both perfect and accurately focused. It shows that the intensity at a point on the diffraction pattern is proportional to the square of the free surface area of the diaphragm. 14

Diffraction by an Aperture of Simple Form

The intensity is equal to the product of two factors : the first depends solely on the coordinate u and the second on the coordinate v. Two series of dark fringes in the form of a chequered pattern can be observed in the plane n'. The dark fringes parallel to S'z' are given by

p being an integer different from zero. The dark fringes parallel to Sy' are given by:

DifJraction : Coherence in Optics

16

The distribution of intensity along S'z' is calculated using the factor : *

=

( ku(, )1 sin kvCo

since the factor sin kuqo/ktqo tends to 1 as u tends to zero. The curve representing the intensity is shown in Fig. 2.2. All the minima are equidistant. The other factor in (2.3) gives the distribution of intensities along S'y'. Outside the two axes S'y'and Sz' the intensity is given by (2.3), that is by the product of the intensities along S'y' and S'z'. Thus away from the axes S'y' and S'z' the intensity is very low.

I

to-

0*0002

0.0007

0.0002

FIGURE 2.3

Figure 2.3 gives some numerical values for the maxima. Since it the diffracted light is spread out has been assumed that qo > more along S'z' than along S'y' (equations 2.4 and 2.5). Figure 2.4 (see inset plate) shows the appearance of the phenomenon. To see the fringes outside S'y' and S'z' an intense source must be used. The photograph in Fig. 2.5 (see inset plate) was obtained with a gas laser in the Laboratoire Central de TBCcommunications (Central Telecommunications Laboratory). As the central region of the phenomenon is very intense the central maximum has to be masked along with the first fringes to avoid scattering of light in the emulsion. This scattering would spread through the image region thus obliterating the maxima far from the axes S'y' and S'z'.

FIGURE 2.4. Diffraction pattern produced by a square aperture.

FIGURE 2.5. Diffraction pattern produced by a square aperture. The photograph was taken with the use of a gas laser made at the Laboratoire Central de Telecommunications. The very bright central region was masked off to prevent fogging of

the emulsion.

FIGURE 2.13. Diffraction pattern produced by a circular aperture (the Airy spot).

FIGURE 2.14. Diffraction pattern produced by a circular aperture. The photograph was taken with the use of a gas laser. The very bright central region was masked off to prevent fogging of the emulsion.

DzjJruction by an Aperture of Simple Fonn 2.2 DMmctjo~by a narrow slit Let us suppose that one of the dimensions of the rectangular

aperture becomes very small in comparison with the other, so that, for example, q, 9 .,C The aperturt becomes a narrow slit extending in the direction Cq (Fig. 2.6). On the basis of (2.5) and (2.6) and for a given value of p:

A significant amount of light is diffracted only in the direction S'z' and along this axis the law for the distribution of intensity is

sin kvCo = kvc0

(

)

2

since the factor sin kuqo/kuq,-,, which is equal to 1 along S'z', is practically zero as soon as one leaves the axis S'z' whether above or below the axis. The curve in Fig. 2.2 therefore represents also the structure of the diffraction pattern for a narrow slit. In all the above we have always considered a point source at infinity. The source S is at the focus of an objective L (Fig. 2.7) and the objective 0 receives a beam of parallel rays. The image Sf is at the focus of the objective 0 and the digracted light only extends along S'z'.

Diffraction:Coherence in Optics

Let S1 be another point source on the line Sy parallel to the fine slit. Its geometric image is formed at S; and the diffracted light extends along S;z; parallel to S'z'. Let us assume that there are a whole series of incoherent point sources on Sy. Sy is then a fine luminous slit. For example, it may be the very fine straight filament of a lamp. The images of all these sources produce diffraction phenomena which are superimposed since the sources are incoherent. Thus a perfectly sharp effect can be observed in the plane n' if the point source S is replaced by a fine luminous slit. A whole series of fringes parallel to the slit source Sy can be observed. Figures 2.8, 2.9 and 2.10 summarize these results. The objective is masked by a narrow slit parallel to Cq. If the light source is a point, the diffracted light extends along S'z' (Fig.2.8). If the source

Dflraction by an Aperture of Simple Form

19

is a fine slit parallel to Cq (or S'yf),the superimposition in the vertical direction of effects analogous to that in Fig (2.8) gives riae to a system of diffraction fringes parallel to Sy' (Fig.2.9). Along a direction parallel to S'zf the variation in intensity is given by (2.7). If

the slit source Sy is not parallel to the slit placed in front of the objective, the fringes are just as sharp as in the case in Fig. (2.9) but they are inclined parallel to Sy.

2.3 DMtactlon by a circular aprtwe The aperture of the diaphragm D is circular (Fig. 2.11) and its centre is at C on the principal axis COS' of the objective 0.As before we consider a beam of rays parallel to CS' and normal to the plane of the diaphragm D. The source is at infinity and its image is formed at Sf. The phenomena observed have CS' as the axis of revolution. According to (1.19) the amplitude at P' is given by :

Let us put :

q = a cos 8

( = a sin 8

20

DtjJruct ion : Coherence in Optics

Then :

If a, is the radius of the aperture, the amplitude at P' is given by:

The origin of azimuths can be changed and cos(0-8') replaced with cos 8, whence :

If J , is the Bessel function of order zero:

and substituting in (211): a

f(a) = 2 4 Jo(kaa)n da Now, J , being the Bessel function of order 1, we have

Dt#Liruction by an Aperture of Simple Form

and substituting in (2.13):

and the intensity at P' is:

The intensity at a point on the diffraction pattern is proportional to the square of the free surfaa of the diaphragm. Let us put:

Z = kaoa

2 2. whence, neglecting the factor (na,) .

Figure 2.12 shows the curve for the variations in I, with respect to 2.

Diffraction : Coherence in Optics

The diffraction pattern or Airy spot is made up of a very bright central spot surrounded by rings which are alternately dark and bright. The maxima of the bright rings are much less intense than the central maximum and diminish rapidly. The dark rings correspond to the roots of J , ( Z ) = 0,that is to say to :

The maxima correspond to the following values :

with intensities (the intensity of the central maximum being unity)

Let us put a,/R = ub. Then:

The radius of the first dark ring of the diffraction spot is equal to:

Diffraction by an Aperture of Simple Form

and the angular radius is:

we consider objectives with the same aperture 2ab but any

diameter, the central disc and the diffraction rings will be the same for all these objectives. The following table gives some numerical values in microns for the radius of the first dark ring for apertures from 2ab = f to 2 ~ ; = 1/20. 2ab

=

1/3

116

1/10

1/20

Figure 2.13 (see inset plate) shows the centre and first rings of the Airy spot. Figure 2.14 (see inset plate) shows a very large number of rings. As the central maximum is very intense in comparison with the rings, the central region has to be masked with an opaque screen so that the distant rings are not obliterated by scattering in the emulsion. Figure 2.14 was obtained using a gas laser of the L.C.T. (Laboratoire Central de T~1ecommunications (central Telecommunications Laboratory)) as source. The asymptotic expansion for large values of Z If 2J,(Z)lZ is to be calculated for large values of 2,the following asymptotic expansion can be used :

2.4

JdZ)=

sin Z - cos Z

JnZ

with b

whence

The zero minima are given by tan Z = 1 and, if one is sufficiently far from the centre of the diffraction pattern ( Z B I), the distance between two consecutive zero minima is practically constant and equal to R.

24

Diffraction :Coherence in Optics

2.5 The distribution of luminous flux in the Airy spot The luminous flux incident on the objective 0 is spread out over the diffraction pattern in the plane x'. This flux can be determined from the amount of diffracted light obtained by rotating the curve in Fig. 2.12 about the ordinate axis. The flux distributed over a small annular element of surface 2np dp is equal to

Replacing I, by the value given by equation (2.19X the flux inside a circle of radius 2,apart from a constant factor, is given by:

Fi = 2nai

1JW) , z , 0

since from (2.20) p = Zlkab. Using the relation

J;(Z) =

- JIQz +J o ( Z )

we have

and

Then using the relation

and replacing J l ( Z ) by

whence

-Jb(Z) in equation (2.26):

.Dz~ractionby an Aperture of Simple Form

Now:

and we have :

whence, neglecting a constant factor :

lf Z tends to infinity, the total amount of light in the diffraction as J,(Z) and J , ( Z ) tend to zero. spot is given by The amount of light in the diffraction spot is proportional to the flux entering the objective. The intensity at a point in the image of an extended object, which is proportional to the flux entering the objective, is proportional to the free surface of the objective. The intensity at a point in the diffraction spot which is the image of a point source must not be confused with the intensity at a point in the image of an extended source. In the first case the intensity is proportional to the square of the free surface of the objective (Q 2.3) and in the second case it is proportional to the free surface of the objective. This is due to the integration which has to be carried out in passing from the image of a point to the image of an extended object. Taking the total flux to be unity, the flux inside a circle of radius Z (equation 2.27) is :

and the flux outside this circle is:

The tables for Bessel functions can be used to calculate the values of si and Fe. The following gives some numerical values.

26

Dlfliaction :Coherence in Optics

Despite the low intensity of the rings the total flux outside the central spot (of radius Z = 3-83) has a considerable value of about 16%. This flux behaves somewhat like parasitic light in that it hardly contributes at all to the formation of the useful image, and one can therefore see the interest in reducing the intensity of the diffraction rings (see Chapter VI, apodization).

CHAPTER I I I

The Fourier Transformation 3.1 The representation of a periodic f ' n by a Follrkr d m A function G(x) of the variable x, with period p (Fig. 3.1) can be represented by the sum of sinusoidal functions with periods p, p/2, p/3 ... that is with successive frequencies which are multiples of the fundamental frequency l/p. For example, the square wave (Fig. 3.2), which takes on alternate values of + 1 and - 1 and the period of which is p, can be represented by the series: ~ ( x= ) '[sin 7t

2nx

P

1 3

2ax

P

1

P

28

Dzyruct ion : Coherence in Optics

It is interesting to see how the various terms contribute to the more and more precise representation of the function. Curve (1) in Fig. 3.3 represents just the first term of (3.1) whilst curve (2) includes the first three terms of (3.1). The discontinuities are represented better as more and more high frequency terms are introduced.

21131' + 5f sin 5 P -.

Let us take as ordinates (Fig. 3.4) the coefficients of the various sinusoidal terms and as abscissae the frequencies. The set of discrete signals corresponding to the fundamental frequency p = l/p and its harmonics 3/p, 5 / p . . . represents the "spectrum" of G(x). The ordinates indicate the greater or lesser importance of any given frequency in the spectrum of G(x).

The Fourier integrJ The spectrum of the function G(x) can only be represented by a set of discrete signals for periodic functions, but the concept of a spectrum can be generalized for the case of non-periodic functions. Any function which is everywhere finite and integrable can be represented by the sum of an infinite number of sinusoidal components. Using the complex notation this is indicated by writing 3.2

The Fourier Trrmsformation coefficients of sinusoic~terms

the function G(x) in the form (p being the frequency lip):

with

+

and stipulating that the frequency p shall vary from - ~ x to , m. The function g(p) indicates the importance of the frequency p in the transformation of G(x). The relations (3.3) and (3.4) constitute the Fourier transformation. The two functions G(x) and g(x) play symmetric roles, one of the functions being the spectrum of the other. In the case of two variables, the Fourier transformation is written:

Let us reconsider the relation (1.221 which represents Huygens' principle for instruments with small apertures. This is identified with the Fourier transformation by putting :

30

D~ffraction: Coherence in Optics

with

v) = f(Ap, Av)

= f (u, v )

we find

where A and A' are two constants. If the amplitude distribution on the wave surface is F(q, c), the Fourier transformation (relation 3.7) can be used to calculate the spectrum f (a, v ) of F(q, that is, the diffraction pattern. Conversely, if the diffraction pattern f (u, v ) is known, the structure of the wave surface which gives rise to it can be calculated.

3.3 Some Fourier transform (a) Let us consider the "slit" function (Fig. 3.5) which is equal to 1/2C, or 0 depending on whether is less than or greater than I,.

then +oo

f (0) =

S

-00

F(&jkV< dr, =

sin kvc,

kvlo

The Fourier Trmformutiun

31

which is the result that was found in paragraph 2.2. The spectrum of the slit function is shown in Fig. 3.6 which is analogous to Fig. 2.2. If C,, diminishes (the slit becomes narrower), the spectrum spreads out since 1/2C, increases. (b) The "circle" function (Fig. 3.7) is defined by :

and we find (Fig. 3.8) :

with

This is the result obtained in paragraph 2.1.

32

Dzyritction :Coherence in Obtics

(c) The Gaussian function (Fig. 3.9)

F ( 0 = e - (=C2/a2) we have (Fig. 3.10)

The Fourier transfom of a Gaussian function is another Gaussian function. (d) Sinusoidal function F ( i ) (Fig. 3.1 1)

2nr F(() = sin -

P

where p is the period. F(() is zero if by the expression

1 ~ >1 c.,

The spectrum is given

shown in Fig. 3.12.

If the sinusoidal function F ( 0 contains many waves ( p small) one of the two terms on the right-hand side of (3.13) can be taken as zero when the other has a finite value. When 6 increases, the two curves (1) and (2) become even narrower (Figs. 3.13 and 3.14).

The Fourier Transformation

(continuous sinusoidal vibration) I

34

Diffraction : Coherence in Optics

(e) The function

has a spectrum given by

Figures 3.15 and 3.16 represent F ( [ ) and f (u) as 5, tends to infinity.

(f) The function F(C) of the form (Fig 3.17)

The Fourier Transformation

has a spectrum (Fig. 3.18) given by

sin koCo - Co sin kCo[(2Alp)- v ] f (v) = Co kvCo 2 ~ C , [ ( ~ J / P) vl 6 0 sin kc0 [ ( ~ A I P + ) vI -2 kt0 [(22lp)+ vl

(g) The function F(c) of the form :

2d F(() = cos2 P

(3.15)

36

Dt@ractioion :Coherence in Optics

has a spectrum given by

sin k v i , sin kC0[(2Vp)- v ] f ( 0 ) = i o ~ D L ,+-to 2 kloC(2~lp) -vl

If [,tends to infinity, the three terms of the right-hand side of (3.16) tend to ,c 5,/2 and 5,/2. Figures 3.19 and 3.20 represent F ( 0 and f (v) for this case. All the above.results can be interpreted as follows: if F(C) represents the variation in amplitude on the wave surface (one-dimensional phenomena), the spectra, that is the diffraction patterns, have an amplitude given by f (v).

3.4

Gmrvl )m)atia a m e c h g tbc wave

d a c e with

tk

ditfracth pttm (a) Dilation and contraction of the aperture of the diaphragm According to (1.22) the amplitude at a point in the diffraction pattern is given by :

Let us multiply the dimensions of the aperture in the directions Cq and Cc by m and n. The amplitude of the new diffraction pattern is given by: -

The Fourier Transformation

Let us put

q'=*mq

C'=q(

We find :

whence

The n e ~ diffraction . pattern can be deduced from the old one by dividing u and v by m and n. Any enlargement of the aperture in some direction results in the contraction of the diffraction pattern in this direction. (b) The aperture of the diaphragm has a centre .ofsymmetry Let us reconsider expression (1.22)or (3.17). When the diaphragm limiting the wave surface has a centre of symmetry:

Let us change the signs of u and v in (3.17)

and, changing the signs of q and C:

Dtrraction :Coherence in Optics

38 and from (3.21)

The diffraction pattern also bas a centre of symmetry.

(c) Translution, in its own plane, of the diaphragm D limiting the wave surfaee The diaphragm D undergoes any sort of translation CC, (Fig. 3.21). The principal axis COS' of the objective 0 intersects the plane of the diaphragm at C. In its original position the aperture was placed as shown in Fig. 1.11. D is shifted by CCl (Fig. 3.21) and the point C in Fig. 1.11 takes up the position C, in Fig. 3.21. We take a system of axes q ,C,C, passing through C, and parallel to the axes qC[. Let a, and /?, be the coordinates of C, with reference to qCc, and q,, the coordinates of the point hf of the aperture with reference to ~ t C l ( l .If the coordinates of M with reference to qC( are q and c, we have:

cI

When the point C, is at C (the aperture will be in the position indicated in Fig. 1.11) the amplitude at P' is given by (1.22):

If the aperture is in the position shown in Fig. 3.21 (shifted CC,), the change of variable (3.23) must be carried out. The function F(q, 0 gives the distribution of complex amplitude over the aperture. It is replaced by F(q C Hence the amplitude f,(u, v ) at Pf is:

,,

,+ vp, represents the path difference (CP')-(C,P') and 6 = uq ,+ vc, the path difference (C,P')- (MY).Hence clearly 6; = ua

The Fourier Transformation

and comparing (3.24) and (3.25)

fi(u,V ) = dkdi f(u, v)

(3.26)

whence

Ifi(% v)12

=

If

(u, 0)12

(3.27)

The shift in the aperture does not alter the intensity of the diffraction pattern. The diffraction pattern remains at rest in the plane x'. Only the phase has changed as a consequence of the factor emdi.

(d) Rotation of the diaphragm in its plane If the axes qC[ and y'S'z' are turned through the same angle the integral (1.22) remains unchanged. The diffraction pattern turns with the diaphragm.

40

Dtflruction : Coherence in Opt ies

(e) Displacement of point source In all the above we have always assumed that the point light

source S is on the principal axis of the objective 0 (Fig. 3.22). If the point source is displaced to S1 (in the focal plane of L),its image shifts to S, (in the focal plane of 0).Let u l and vl be the direction cosines of 0s; (or S,L) and u and v be those for OP'. u and v have to be replaced by u 1-u and v , - v, and the integral (1.22) becomes :

The point u , = y v 1 = v, the geometrical image of the source, is the centre of the diffraction pattern. The diffraction pattern follows the geometrical image of the source. In fact, when the source is displaced the angle at which the aperture is viewed from the point S' has to be taken into account. If the angular displacement of the point source is small, the effect is of the second order and can be neglected.

CHAPTER IV

Dlflraction by Several Apertures Diffraction by several apertures identical in shape and orientation This problem can be treated directly on the basis of the results of paragraph 3.4(c). The position of any aperture is characterized by a point such as C , (Fig. 3.21), and all the points C1, C2, C3.. . (Fig. 4.1) are situated in the same place in each aperture. The amplitude at P' due to the aperture No. 1 with centre C , is given by (3.26) : 4.1

fib,V ) = eiuif (u, v )

(4.1)

with 6; = u a l , + d l , where or, and are the coordinates of C, with reference to the axes qCC and f (u, v ) is the diffraction pattern due to one of the apertures when its characteristic point is at C:

Since the screen D is pierced by several apertures which are identical 41

42

Dzflraction : Coherence in Optics

and orientated in the same way, the amplitude at P' is obtained by summing the expressions (4.1) with f (u, v ) appearing as a factor. If there are N apertures we have :

fN(u,v ) = (gkai +gkdS + . ..)f(u,u) =f(u, v )

2 Bkd"4.3) n=l

and the intensity at P' :

the second factor on the right-hand side introduces sums such ~ vPn) sin2k(ua,, v/?,J2 and 2 cos k(ua,, vp,,) as C O Sk(~;, cos Nuam up,) 2 sin k(m, v&) sin k(uam+ vjl,). We can write :

+ + + +

+

+

+

The intensity is given by the product of the two terms: the first, If(y 0)12, shows the effect of diffraction and the second the effect of interference. The term 1f (u, v)12 represents the diffraction pattern of one of the apertures and the other term represents the interference between vibrations coming from the characteristic points C,, C2, C3... . In the case where the apertures are circular:

If the diffraction effects are neglected, the amplitude at P' can be calculated by replacing the apertures by point sources at the points C1, C2, C 3 . . .

d2 Diffraction by a large number of apertures, identical h shape d orientation, irregularly distributed over the diaphragm

If the apertures are very numerous and are distributed irregularly over the screen D, the terms of the double sum (4.5) have a large number of values between - 1 and + 1. The sums of the sines and cosines are zero except for the terms where m = n; these are each equal to unity and are equal in number to the number of apertures, that is N.The intensity at P' is given by

which is equal to N times the intensity due to a single aperture.

Dflaction by Several Apertures

.

'

43

In fact, observation of the pattern shows that the problem must be studied in more detail. The diffraction pattern produced by the set of apertures actually has a granular structure which is not by the above calculation. First, at the centre of the pattern at S' (Fig 3.21) where 6; = 0 for all the apertures, the amplitude f (u,v ) is equal to N and the intensity to lV2. Thus at the centre a small, very bright spot is observed. At a distance from the axis, at P', the phase differences 2 x 6 3 have values between 0 and 2n. Now consider another point close to P'. All the phase differences 211&/2 corresponding to the apertures alter. Even if all these changes are small, they can in the end produce large variations in the amplitude f(u, v). These fluctuations in intensity give the pattern its granular appearance. Let us consider a screen pierced by small circular holes which are identical and are distributed at random To study the intensity fluctuations, we must calculate the amplitude for different positions of the point P'. We could also consider a fixed point P' and replace the previous screen with a succession of different screens in which the holes are differently distributed. Of course all the holes have the same diameter and are always distributed at random. For each screen there is complex amplitude at P' which can be represented in the complex plane by a point. The straight line joining this point to the origin makes an angle with the real axis and this angle represents the phase of the resultant amplitude at P'. The distance p of this point from the origin is the modulus of the complex amplitude. In the complex plane we obtain a whole series of points, distributed at random, each of which corresponds to a screen. Let us find the probability that a screen will give a point in the cornplex plane such that p lies in the interval p and p +dp. This probability Pdp can be written 1

P d p = =e

-p2IN2np

dp =

pe - p 2 l ~dp

The mean value of p2 is the intensity at P'. We have :

Thus if a number of screens are used in succession, the mean value

44

Dzflraction : Coherence in Optics

of the intensity at a fixed point P at a distance from the centre is N times the intensity due to a single aperture. This is the result we found above. But this intensity represents a mean value and for each screen there is a certain intensity at P' which differs to a greater or lesser extent from the mean intensity. The difference can be denoted quantitatively by the deviation :

and we have :

From an analogous calculation to that for whence :

-

7 we find p4 = 2N2

The intensity fluctuations are of the same order of magnitude as the mean intensity. They explain the granular appearance of the pattern. As we have already said, this granular appearance disappears at the centre S where a small, brilliant spot with intensity N' can be observed. Experiment shows that this central spot is surrounded by a dark region which is not very extensive, after which comes the pattern which ~~ehave just studied, with a mean intensity equal to N. There are a large number of holes in a limited surface, that of the diaphragm, and consequently the distribution cannot be entirely "random". There is necessarily a certain order at a small distance around an aperture and it is this which produces the dark region very close to the central brilliant spot. Complementary screens. Babiaet's theorem Let us consider a screen D consisting of N small opaque screens of the same form as the above apertures and occupying the same positions. Figure 4.2 represents a screen Dl pierced by a number of apertures and Fig. 4.3 represents a screen D, consisting of small opaque regions. These two screens are complementary. Let us place the screen D, in front of the objective 0 ((Fig. 3.21) for example). The amplitude at P' is equal to the amplitude given by 4.3

D@ration by Seoeral Apertures

45

the objective 0 when there is no screen minus the amplitude which would be sent by the surfaces equal to those of the small opaque regions. Let f,(u, v ) be the amplitude of the diffraction pattern hen there is no diaphragm in front of the objective 0.The amplitude at Pf if the objective 0 is screened by the diaphragm D , (Fig. 4.2) is fDl(u, v), whence the amplitude at P' when the screen D 2is used is:

We know that the diffraction pattern is spread out more the smaller the aperture limiting the wave surface. Here the diffraction pattern corresponding to fD,(yv ) is the same as that of a single small aperture in the screen D , . It is therefore much more spread out than the diffraction pattern fo(u, v ) due to the whole of the free aperture of the objective 0.Thus at a distance from the centre of the diffraction pattern the term f'(u, v ) is practically negligible and we have :

whence

The diffraction patterns produced by two complementary screens are identical (Babinet's theorem). As we have just seen, this result is only valid at a distance from the diffraction pattern due to the free surface of the objective.

46

Diflmction : Coherence in Optics

Diffraction by two identical afts The screen D placed in front of the objective 0 (Fig. 4.4) is pierced by two identical slits parallel to Cq and symmetric .with respect to this axis. The width of each slit is 2c, and the distance between the centres is 2a. We assume that the height of the slits is large in comparison with their width 25, (narrow slits). In practice this means there will be no diffracted light except along 4.4

S'z'.

According to (4.3) the amplitude at P' is:

since from the definition for h; and 6; (§ 4.1)

and from (2.7) the diffraction pattern for a narrow slit parallel to Cq is:

sin kvCo

f (0) = kVCO

whence the amplitude at P' for the case of two slits is:

sin kvc, kvro

'kuo (Z +e-jkua)

D%fiaction by Several Apertures and the intensity at P', neglecting a constant factor, is :

sin kv6, Ip' = kvc,

(

) cos2kva 2

Neglecting the diffraction, I,. = cos2kva and the distribution of light along S'z' is the same as that observed in the case of Young's slits. Expression (4.11) shows that along S'z' we have Young's fringes modulated by the diffraction effect due to one or other of the two slits. The term

is represented by the dotted curve on Fig. 4.5. The curve shown by a continuous line shows the variations in I,. as a function of 1; (expression (4.1 1)). The zero minima due to the term cos2kva are given by @ being an integer):

The maxima correspond to

kva = pn

48

Dlflaction : Coherence in Optics

Since To is smaller than a the interference fringes are closer together than the diffraction fringes. In the case of very narrow slits the central maximum of the diffraction pattern is very much spread out : there will be more interference fringes in the central maximum the further apart the slits.

Diffraction by thne identical slits Let us consider three identical slits parallel to Cq and regularly spaced. The middle of the central slit is at C (Fig. 4.6). The distance separating the centres of two adjacent slits is equal to 2a. Writing the amplitude at P' in the form (4.10), we have:

4.5

since

sin 3kua 1+2cos2kva = . sin kva the intensity at P' is given by:

=

sin kvC, sin 3 kua kvi, '(sin kvn )2

(

)

(4.16)

Let us consider very narrow slits so that the diffraction is very spread out and let us investigate the intensity variations given by

Dirracttion by S e v k l Apertures the interference term

The maxima are given by

and are therefore in the same position as in the former case of two slits separated by the same interval 2a. Expression (4.16) sho~tsthat there are weak maxima between the previous maxima given by (4.17). Figure 4.7 shours the appearance of the fringes. The bright fringes given by (4.17) are called the principal maxima. The intermediate maxima are the secondary maxima.

Let us consider n narrow, parallel and equidistant slits. The distance between two adjacent centres is 2a. The figure is analogous to Fig. 4.6 adding a series of slits on each side of Cq. From the above the amplitude at P' is given by:

( I + &'2kva +e

-j2kva +d 4 k +~

jUva

+ ...)sink&,kvc,

50

Dlyraction :Coherence in Optics

Generalizing (4.15), the intensity at P' can be written :

sin kvr, I p = kvCo

(

)

2(

sin nkva sin kva

))

Figure 4.8 shows the evolution of the fringes for two slits (curve l), three slits (curve 2) and a large number of slits (curve 3). As the number of slits increases, the principal maxima become sharper and the secondary maxima become more and more numerous and

weaker and weaker. Let us consider a large number of slits and let us investigate the structure of a principal maximum. Inside the central maximum of the diffraction pattern of a slit (dotted lines in Fig. 4.8) we can put the diffraction term equal to 1 and write :

sin nkoa sin kva

Diffraction by Several Apertures

For the principal maxima we have (4.17) :

when studying the structure of a principal maximum, v remains very close to p(rl/2a) since the principal maxima are very narrowr. We can put NOW

all that is necessary to study the structure of the principal maximum under consideration is to vary w. We have: w is very small and

IP. =

(sinsinnk[-A/2)+ k[p(A/Z)+ wa] )I wa]

sin nkwa = (sin kwa

and

sin kwa

2

kwa

whence

1,. = n2(

sin nkwa nkwa )I

)I

(4.20)

52

Dzfiiaction : Coherence in Optics

A principal maximum thus has the structure of the diffraction pattern ~ i v e nby a slit of width Zna (Fig. 4.9). This is the diffraction pattern which would be given by the diaphragm D with all the opaque intervals separating the slits removed. The expression (4.21) shomrs that the distance of the first minimum from the centre of the pattern is equal to A/2na. This is a measure of the "width" of the principal maximum.

CHAPTER V

Extended Luminous Sources and Objects. Coherence the above we considered a 'point" source emitting monochromatic light. This idea1 case is never achieved in practice and we shall now investigate the influence of the dimensions of the source and the spectrum which it emits. This investigation introduces the concept of the coherence of vibrations, which is useful in understanding the mechanism of the formation of images of extended objects.

IN

ALL

5.1 Spatid coherence and he-coberence We shall first give an example derived from interference phenomena. Let us consider the diagram of the Michelson interferometer (Fig. 5.1). The ideal light source (point source emitting monochromatic light) is at the focus of an objective 0 and the interferometer is illuminated by a parallel beam. 53

54

Dz~raction:Coherence in Optics

The light reflected by the beam-divider G is normal to the mirror M , and the light which passes through G is normal to mirror M,. Let us incline mirror M, slightly in such a way that, due to reflection at the divider G, the image Mi of mirror M, makes a very small angle with M,.The effect is the same as if we observed fringes of equal width due to a film of air of variable thickness between M Iand Mi. In a region A where the thickness of air is e, the path difference is 6 = 2e. If 1 is the wavelength of the monochromatic light emitted by the point source S, the luminous intensity at A, apart from a constant, is :

(P = cos2 2ne = cos2 -

2

A

where rp is the phase difference corresponding to the path difference 6. Because the thickness of air e varies, the field of observation is covered with straight interference fringes parallel to the line of intersection of M I and Mi. Let I, be the intensity of the bright fringes and I , the intensity of the dark fringes. Let us define the contrast between the fringes, that is the visibility of the interference pattern, by the parameter:

The relation (5.1) shows that I, is zero each time that the thickness e is equal to an odd multiple of 114. The dark fringes are black and y = 1: the contrast is a maximum. In the theoretical scheme shown in Fig. 5.1 the contrast remains equal to one whatever the distance separating M I and Mi. In fact experiment shows that this is never so for the following two fundamental reasons: the source S always is slightly extended and it does not emit perfectly monochromatic light. We know, in fact, that the contrast of the fringes diminishes if the source S is enlarged. Let S, be a point on an extended source which has its centre at S. The distance from S , to S subtends an angle i at 0.For the point S1 the path difference at A is 6 = 2e cos i. As a result, the path difference S and the phase difference cp vary with the point

Extended Luminous Sources and Objects. Coherence

.

55

in question on the extended source. Consequently the intensity I produced at A by a point SI on the extended source varies with the position of this point. If for a given point on the source S a zero minimum is observed at A, the same will not be true for other points on the source. The variation in the phase difference p, results in a reduction in the contrast of the fringes. If the variations in p, are sufficiently small, they may cause only negligible variations in the intensity: there is spatial coherence. Large variations in cp can produce variations in I which are large enough for the contrast to fall almost to zero. The fringes are no longer visible, there is spatial incoherence. An intermediate case ~ o u l dindicate a degree of partial coherence. In addition, electromagnetic waves are not waves of unlimited length: they are emitted in wave trains. Because the lengths of the trains are finite, the atom does not emit a single frequency but a spectrum of frequencies which becomes broader as the lengths of the u7avetrains diminishes. This causes a reduction of the contrast of the fringes and the effect is said to be one of time-coherence or chromatic coherence. Thus the phenomena can be investigated either by introducing the length of the wave trains or by introducing the breadth of the frequency spectrum, that is the monochromatism of the source. Let us go back to the first point of view and study the previous experiment again, and take the source S to have a small diameter. Let us progressively move one of the mirrors of the interferometer so that the path difference is increased. It is found that the contrast of the fringes diminishes even if under the experimental conditions the diameter of the source is small enough not to influence the contrast of the fringes. This time the diminution in Lisibility is connected not with an increase in the diameter of the source, but with the finite length of the wave trains emitted by the atoms of the source. These trains of waves pass at random intervals and a very large number pass during the time of an observation. Each incident wave train such as A (Fig. 5.2) entering the interferometer is divided into two wave trains by the action of the beam-divider G. Let us adjust the difference between the two paths (1) and (2) so that it is greater than the length of the wave trains. When the wave train A; travelling in path (1) reaches P, the other is at A; if the mirror M, is the one which has been moved further away.

56

Difraction : Coherence in Optics

These two wave trains will not meet one another and cannot interfere. The wave trains which are superimposed at P come from two different wave trains emitted at different times, for example t and t+8. We shall consider first the case in Fig. 5.2 where 8 is much greater than the duration r of a wave train. During the time required to make an observation a large number of wave trains pass, and the times when they do so are randomly distributed. The phase difference between two wave trains takes on all possible values when we consider all the wave trains and there are no observable interference phenomena.

The situation is not the same if the path difference diminishes since then 8 decreases as well. When 8 is of the order of z the two wave trains resulting from the same incident wave train are partially superimposed. There is a "correlation" between these two wave trains and this correlation exists for all wave trains arriving at P. The interference fringes appear with low contrast of varying degree. There is partial coherence of the two beams (1) and (2) in the interferometer. When the path difference is sufficiently small for 9 to be much smaller than .r the fringes are of high contrast. The visibility of the fringes will depend on the degree of partial coherence, which in this case refers to time-coherence. Thus ure introduce a coherence time connected with T and a coherence length connected with the length of the wave trains.

Extended Luminous Sources and Objects. Coherence

57

If we now consider the second point of view we see from (5.1) that the intensity is a function of the wavelength A. For each wavelength there is a corresponding intensity for the phenomenon observed at A (Fig. 5.1). If for some wavelength I a zero minimum is observed at A, it will not be the same for other wavelengths and the contrast of the fringes is diminished. The visibility of the fringes is connected with the chromatism of the source, that is with a t ime-coherence, the coherence time being greater the smaller the range of frequencies. The problem arises in the same way for diffraction phenomena (Fig. 5.3). The wave trains leaving two points M I and M, on the wave surface at the same instant do not arrive at P' at the same time. If the path difference M ,H is greater than the length of the wave trains, the vibrations reaching P' from M I and M, are not coherent. If this is true for most of the elements of the uncovered wave surface I; diffraction fringes are not visible at P'. At the centre S', the geometric image of S, all the vibrations arrive in phase. Moving away from S' in the plane x', the vibrations emitted by different points on I: are no longer in phase. The coherence diminishes. Moving still further from S', the vibrations eventually become incoherent and the diffraction fringes are no longer visible. There is a ring of light with no periodic intensity variations.

5.2 The relation between the length of wave t r h a d the sjwdm1 width of the emitted radiations. Coherence lea@ a d coherence time Experiments have shown that the coherence of two beams liable to interfere is linked with the duration, and consequently the length, of the wave trains. But the length of the wave trains .determines

58

Dzffraction : Coherence in Optics

the quality of the radiation emitted by the atoms: a traifi of waves which is damped very slowly is approximately a sinusoidal oscilla- . tion and thus monochromatic. As we have seen, interference phenomena would then be perfectly visible even for large path differences (the dimensions of the source being assumed negligible). A rapidly attenuated wave train corresponds to an oscillation which is not simple harmonic and consequently to radiation which is not monochromatic. The visibility of the phenomena diminishes as the path difference increases. We are therefore lead to evaluating a coherence time connected with the duration of the wave trains and a length connected with the length of the wave trains. To d o this u7eshall find the relation between the length of the wave trains and the quality of the radiation emitted on the assumption that only a narrow spectrum of frequencies is involved. Let F(t) be the vibratory-motion at a given point at time t due to a single wave train. We assume that F = 0 for It1 2 7 where r is the duration of a ~ ~ a train. ve According to Fourier's theorem, any function which is everywhere finite and integrable can be represented by the sum of an infinite continuum of sinusoidal components. Thus the vibration F(t) due to a single wave train can be considered as the superposition of monochromatic vibrations with different frequencies :

where f (v) gives the amplitude distribution of the monochromatic components of F(t). The energy distribution of these components is given by 1f (v)I2. According to the properties of the Fourier transformation, we can w7rite:

f (v) =

dt

~(r)e-j~""

(594)

-m

so that f (v), the spectrum of F(t), can be calculated. As was originally assumed, f ( v ) differs from zero over only a narrow band of frequencies. Let us take first an example in which all the wave trains have the same duration r, during which F(t) is simple harmonic

Extended Luminous Sources and Objects. Coherence

59

with frequency vo :

F(t) = f0eiZzvot where fo is a constant. F(t) has the value given by (5.5) when t is between -r/2 and +r/2 and F = 0 if t is outside this interval, that is

F(t) = fod2zvot for

for

7

It1 I5 It1

>

C

The real part R[F(t)]of F(t) is represented in Fig. 5.4. From relation (5.4) the spectrum of F(t) is given by : + rl2

f (v) = fo

1

- r/2

- j ~ z ( v- vo)t

sin[Z(V - v,) 7] dr =for Z(V- v o ) ~ (5.6)

and the relative energy distribution in the spectrum by :

sin[n(v - vo)r] X ( V - vo)t The curve 1f (v)I2 as a function of v is shown in Fig. 5.5. The frequency interval between A and B is symmetric with respect to v, and is given by:

60

Diffraction:Coherence in Optics

Let us denote the width Av of the spectral band by half the interval AB. Then:

The width of the spectral band is of the order of the inverse of the duration of the wave train. The time z is called the coherence time. Let us put:

where c is the speed of light. From I = c/v we have:

and if ri, is the mean wavelength corresponding to v, :

The length I is called the coherence length. We know that with a point source of non-monochromatic light fringes can only be observed if the path difference S does not exceed a certain value. If the source emits in the small spectral interval AA, then to obtain interference fringes with good contrast the variation in the order of interference p = 6/A due to a change AA in the wavelength must be very much smaller than unity. We must

Extended Luminous Sources and Objects. Coherence

61

have :

Thus with a source emitting in a small spectral interval about a mean wavelength I , the difference between the paths of two interfering vibrations must be such that

Comparing (5.91, (5.10) and (5.1 1) we have :

S
F(t) - f , p d

.e -

(t"

(t

> 0)

(5.13)

is a constant and r is the decay constant. The real part R[F(t)] of F(t) is shown in Fig. 5.6. Let us investigate the frequency spectrum in the vicinity of the mean frequency v,. From the relation (5.4) we have : fb

whence

Let us put

The amplitude l/p off (v) is given by

Dtflraction : Coherence in Optics

Hence the energy distribution of frequencies in the vicinity of vo is

Let us denote the width Av of the spectral band by the width 2 2 corresponding to half the maximum ordinate for (Fig. 5.7). We have

whence

($) = 4n2r2 1 2

( V - Y ~ )= ~

Extended Luminous Sources and Objects. Coherence

63

and

We again find the previous result (5.8): the spectral width Av is inversely proportional to the damping constant r (coherence time), that is to the mean length of the wave trains (coherence length). Let us find an order of magnitude for the coherence length I = cr. From (5.91, (5.10) and (5.19) we have:

For a line of width A1 = 30 A we have

In the two examples above we assumed that all the wave trains emitted by the source were identical, which does not happen in practice. The atoms perturb one another and the wave trains are modified irregularly. Moreover, the spectral lines are widened by other phenomena such as the Doppler effect. Thus we can only define mean values for the duration of the wave trains and the spectral width Av. Nevertheless these mean values satisfy relations of the form of (5.8) or (5.19).

The vibration emitted by an atom. The variation in the complex. amplitude of tbe vibration during a wave train A monochromatic vibration with a frequency vo can be written in the form:

5.3

F(t) is, for example, the electric field and if the wave is propagated in a direction ox, the wave equation gives a solution for a in the form :

where v is the speed of propagation of the wave, and fo and 0

Difiraction : Coherence in Optics

64

are two constants. We then have the general expression for F(t):

The vibrations emitted by the atoms cannot be continuous sinusoidal vibrations (which last for ever). The vibrations of the atoms are, in fact, interrupted or perturbed by collisions, or damped as a result of loss of energy by radiation. We can assume at the outset that the vibration emitted by the atom has the form given by (5.22), but that it is interrupted after a time 7. The atom therefore emits a series of wave trains analogous to that in Fig. 5.6 which vary in a completely irregular fashion from one wave train to the next. If we assume that the vibration emitted by the atom is a damped harmonic vibration :

The phase B varies in a completely irregular fashion from one wave train to the next. In the general case a vibration can therefore be considered as comprising monochromatic vibrations of different frequencies. Each monochromatic component is represented by the expression (5.22) in the simplified form fo&2"" and the vibration can be written (5.3) : +a,

Introducing a mean frequency v, about which the frequencies v are distributed we have : +a0

Let us put:

a(t) is

in general complex, of the form A(t)P('),and we have

Extended Luminous Sources and Objects. Coherence

65

which represents the vibration emitted by an atom and due to a wrave train. The vibration (5.27) can be considered as a monochromatic vibration with a frequency corresponding to the mean frequency V, and a varying amplitude 4th where a(t) is the instantaneous complex amplitude of the vibration. As we assume that f (v), the spectrum of F(t1 only differs from zero for values of v close to v,, the integral (5.26) represents the superposition of harmonic components with frequencies v- v,, that is, low frequencies. If the interval Av in which the frequency differs from zero is small in comparison with the mean frequency v,, the term ej2z(v'-0* varies slowly with respect to d2"'" in the course of time. The emitted light is quasi-monochromatic. In the expression (5.27), the variations of a(t) with time are slow in cornparison with the variations of the oscillatory term e''2 zvot. Now we know that the quantity Av which represents the width of the emitted radiations is connected with the coherence time z, that is with the length of the wave trains. The interval Av is of the order of l/r. Let us take a radiation of spectral width A1 = 0.5 x lo-* A corresponding to a very narrow line. We have:

and for 1 = 0.5 p

since the period T of the vibrations is

we have :

Thus, in the time

.r occupied by a wave train, during which the complex amplitude a(t) varies relatively slowly, 360,000 periods of

the vibration are completed. It should, however, be noted that if the variations in a(t) are slow compared with the vibratory period, they could still be very fast for the means of observation.

Dif/raction:Coherence in Optics

66

L imaginary axis

t ' \

I

I

J V J

real axis

n TI U L I

!

I

' id1

(b)

(continuous sinusoidal vibration)

The complex amplitude a(t) of the different vibrations emitted by atoms will now be represented in the complex plane. In the case of a monochromatic vibration (5.22) the complex amplitude a(t) is represented by a point such as M (Fig. 5.8a). Figure 5.8b gives the spatial or temporal representation of the real part of F(t) and Fig. 5 . 8 ~shows the spectrum of F(t). If the vibration is of infinite duration, its spectrum consists of a single frequency v,. The same representation is used in Figs. 5.9 and 5.11. Figures 5.94 5.9b and 5 . 9 ~refer to the case of a vibration of type (5.22) but of limited duration. In the complex plane the amplitude is again represented by a point such as M during a vibration. The times r are usually extremely small in comparison with the time necessary to make an observation. Thus during the time for an observation, the complex amplitude a(t) will have a large number of values represented by the points M I , M,, M, . . . distributed at random (Fig. 5.10) and what is observed is the mean of the effects produced by the different values of a(t). Let us consider a vibration of the type (5.23); comparing (5.27) and (5.23), we have : = foe

- (tlr)&O

omitting 2xvox/v in the exponential. In the complex plane (Fig. 5.1 la), u(t) approaches 0 describing the straight line OM without undergoing rotation (0 is a constant for a wave train). If the atom emits a large number of wave trains, during the time for an observation there is a straight line such as OM representing

Extended Luminous Sources and Objects. Coherence

p imaginary axis

real axis

0 .

-

I

4

1

AV = -

rLndAbqulu: I

I

I

1 1

I . I

(b)

0

(sinusoidal vibration of limited duration)

axis

real axis

imaginary axis

M

real axis (a)

I.f(v)12

I

I

+ imaginary

0

67

(b) (damped sinusoidal vibration)

t

*v

68

Difiraction: Coherence in Optics

each wave train and these lines are randomly distributed in the complex plane. There is no relation between the phases of the various wave trains. The effect is as shown in Fig. 5.12. During the time for an observation all that can be observed is the average of the effects produced by all the values of a(t).

\

4

imaginary axis

real axis

If the Doppler effect is taken into account, there is an apparent variation of frequency 6v due to the speed v of the atom with respect to the observer. We have

From (5.27) the atom appears to emit a vibration : In the case of a damped vibration (5.28) we have: -(tmg(0+ 2uv.r)

4 0 =foe

(5.30)

The phase varies with time and in the complex plane we have the curve shown in Fig. 5.13. Finally in the general case where the vibration is perturbed by the adjacent atoms, the complex amplitude a(t) can vary in a complicated fashion and we have a curve analogous to that in Fig. 5.14. As in the previous cases the variations shown in Fig. 5.14 are produced in the course of the duration of a wave train. If, as is usually the case, the number of wave trains emitted during an observation is very large, we have a large number of curves similar

Extended Luminous Sources and Objects. Coherence

A

imaginary axis

4

69

imaginary axis

real

real axis

axis

to that in Fig. 5.14, but there is no relation between any of these curves. During the time for an observation, the amplitude a(t) varies very rapidly and all that can be observed is the average of the effects produced by the different values of a(t). The successive wave trains emitted by an atom. The case where a large number of wave trains are received during the observation time. The instantaneous vibration emitted by an atom can be characterized by a complex function a(t) which varies during a wave train as shown in Fig. 5.14. The times z are usually extremely small in comparison with the time necessary to make an observation. Thus during an observation an atom emits a large number of wave trains with unrelated phases. Let us assume that the atom emits wave trains F,(t), F,(t) . . . at times t,, t , . . . distributed at random (Fig. 5.15). The vibration F,(t) emitted at time t1 will be represented by Fl(t,-t,) at time to. Similar results are obtained for the other wave trains and the vibration at time to can be written: 5.4

The wave trains can be of any form and duration s,, r , .... The phase of each wave train varies (Fig. 5.13) and (5.14), and there is no relation between the phases of different wave trains. From (5.3) and (5.31) and replacing to by t, for each wave train we have : +a0 F ,(t - t ,) = f'(~)e'~"'(~-'~) dv (5.32)

I

Diffraction :Coherence in Optics

whence

-00

If we put: we have:

-00

~ ( v is ) the spectrum of V(t). Specifying a mean frequency v,, we can write (5.33) in the form:

Let us put :

ure have:

V ( t )= a(t)&2"v0t

(5.39)

This expression is identical to expression (5.27) but it refers to a large number of successive wave trains.

Extended Luminous Sources and Objects. Coherence

71

During the time for an observation, a(t)describes a large number of curves of the type shown in Fig. 5.14 and finally we obtain a very complicated curve. The observer only perceives the average of the effects produced by the rapid succession of values of a(t). If 2T is the time required for an observation, using the notation for either a(t) or V(t), we have : 1 +T a(t) = lim - a(t)dt = 0 T+m

2T

-T

since for any value of a(t) we can always find an opposite value in the complex plane. The receiver, whether it is the eye or a photoelectric cell, is only sensitive to the mean square of the amplitude: this is the quantity which measures the observed intensity. Corresponding to the instantaneous complex amplitude a(t) there is an energy la(t)I2, the square of the modulus of a(t). As far as the receiver is concerned, the intensity of the phenomenon is therefore proportional to the mean:

1 la(1)12 = a(t)a*(t) = 2T

+T

I

-T

la(t)12dt

I

1 +*

la(t)lz dl

(5.40)

-a,

5.5 Vibrations from two different atoms which radiate the same mean frequency v, (quasi-monochromatic light). The case where a large number of wave trains are received during the observation time. Let us consider two atoms M I and M, (Fig. 5.16) and investigate the vibrations emitted by these two atoms arriving at a point P. From (5.393 the vibrations Vl and V . emitted by the two atoms can be written:

The frequency v, is the same for V, and V2 since the two atoms are radiating with the same mean frequency. The vibrations V . and V2 are considered at the instant when they leave the two atoms. If 19, and 8, are the times taken by the vibrations to cover the

D~flraction:Coherence in Optics

paths M I P and M , P :

where v is the speed of light. The dispersion of the medium and the difference in the intensities at P due to the different distances M , P and M,P are neglected. The vibration emitted by M I is the same at P at time t as at M , at time t - el. It is represented by V1(t- 0) and a(t - 0,). The same is true for the vibration emitted by M, at P. The vibration at P at time t can therefore be written: (5.43) v(0 = vdt- el)+ v,(t- e,) The receiver is sensitive to the mean of [ V, + V2I2or the mean of

lal +a,12 taken over the time needed to make an observation. Thus the intensity at P is given, according to (5.4Q by:

I

=

[ a 1 ( t - e 1 ) + a 2 ( t - e 2 ) ] [a:(t -6,)+a;(r-e,)]

4%

Expanding this we have :

(5.44)

Extended Luminous Sources and Objects. Coherence

73

The origin of time can be changed in all the terms:

Let us put: We can write :

But a, and a, are complex and of the form a , = A,&@1

(6 5.3)

a2 = A,&@z

(5.47)

and we have :

where R[ ] indicates the real part of the expression in square brackets. Whence the intensity :

-

The terms a,aT and a,a: are a measure of the energy emitted by the atoms MI and M,. The last term on the right-hand side represents the correlation function of a, and a,. This is an interaction term which measures the coherence of the vibrations emitted by M I and M,. We shall transform the expression (5.49) by introducing the spectra vl(v) and v,(v) of the vibrations Vl(t) and V,(t). At the instant at which the vibrations leave the two atoms we have, from (5.35) and (5.34) :

Dlflraction :Coherence in Optics

74

and the spectra of M I and M, :

,,

where t t i , t,, ti are any times. Let us evaluate the vibrations at the moment at which they reach P. For a given frequency v, that is for a monochromatic component of the vibration, if the vibration is written &2"vt at MI, at P it will be given, from (5.22) and (5.42), by:

If each monochromatic component of frequency v has an amplitude ~ ( v ) ,the vibrations at P at time t can be written in the form given by expression (5.35) :

putting 6 = O2 - 0, as before we have :

specifying a mean frequency v,:

Extended Luminous Sources and Objects. Coherence

75

whence the complex amplitudes a,(t + 0) and a,(t) :

From (5.57), (5.58), (5.59) and (5.60): Parseval's theorem gives :

and :

Using relations (5.61) and(5.62) we can write :

and from (5.49), (5.63) and (5.64), the intensity at P is given, except for a constant, by :

Let us calculate the third term of the right-hand side of (5.49) and (5.65) which denotes the correlation between the vibrations emitted by the two atoms.

Dzyraction: Coherence in Optics

76

If N is the number of wave trains we have: N

N

If J , and Jk are the moduli ofh(v) and f k*(v) and am is the argument of the product f,(v) f k*(v), wc have : N

vl (v)v:(v) =

N

C C JnJk{cos[2n~(t,- t;) -a],

n= 1 m = l

Since we are dealing with two different atoms, the angles 2nv(t,,- tk) - a,,,,, have all possible values, and the sums of the sines and cosines are separately zero. We therefore have:

There is no correlation between the vibrations emitted by two different atoms and expression (5.49) reduces to :

-

-

I =a1a~+a2a~

(5.69)

or from (5.65) : +a0

The monochromatic components coming from two different atoms are incoherent and their intensities are additive. It is impossible to make vibrations from two different atoms interfere even if they emit the same frequency. A monochromatic filter only allowing the frequency v, to pass would give:

Vibrations with different frequencies from a single atom. The case where a large number of wave trains are received during the observation time. Let us now assume that the two vibrations arriving at P come from the same atom M (Fig. 5.1 7).

5.6

Extended Luminous Sources and Objects. Coherence

77

The vibration emitted by M is of the form:

if the atom emits a mean frequency v,. Some device, not shown in Fig. 5.17, divides the incident wave into two identical waves which travel along different paths (1) and (2).

If 8, and 8, are the times taken by the vibrations to complete the two paths, the vibration at P at time t can be written in the form (5.43) : The function V is the same for the two terms since the vibrations travelling along paths (1) and (2) come from the same atom. The wave trains travelling along (2) are merely displaced with respect to those travelling along (1). Let us consider two monochromatic filters, one placed in path (1) and passing the frequency v, the other placed in path (2) and passing frequency v'. We have :

the times t t2 .. . being the same in the two expressions since they are the instants at which the wave trains are emitted by the atom. The function fl represents the spectrum corresponding to the vibration emitted at time t 1 by the atom. If the paths (1) and (2) have the same transmission coefficient, the function f, is the same for the two paths. It has the value fl(v) corresponding to the frequency v for path (1) and the value f,(v') corresponding to the frequency v' for path (2). The same is true for the other

Diffraction :Coherence in Optics

78

functions f2, f3 . . . corresponding to various wave trains emitted by the atom. Using the previous notation, we have :

The angles vt, and v't, are still arbitrary and the sums of the sines and the cosines are zero so we have: Thus in expression (5.65) we still have: (5.74) 1 = 1v(v)1 Iv(v')l This expression shows that for ordinary observations taking a time which is very long in comparison with the duration of the wave trains, different frequencies coming from the same atom are incoherent and are additive for intensity. Different frequencies coming from the same atom therefore cannot give rise to interference phenomena. It should be noted that if observations are made for very short periods ( i t ) ,beats between the different frequencies could produce fluctuations in intensity I.

+

5.7 Vibrations with the same frequency from a single atom The two expressions (5.72) are identical. We have:

The times t,, and t,,, are still arbitrary and the sums of the sines and the cosines are zero, except for the terms where m = n :

The third term on the right-hand side of (5.65) now becomes:

ZR[

[ I e""v6(5 f ( ~ )')I dv] +m

+a,

v(v)~*(v)e""~~ dv] = ZR -00

-00

=2

1 cos 2nv0( C lfi(v)12)dv -m

n= 1

n= 1

Extended Luminous Sources and Objects. Coherence

79

as Vl = V,. Now if 6 represents the path difference between two vibrations reaching P (Fig. 5.17), one having covered path (1) and the other path (21, we have :

where I is the wavelength and c is the velocity of light in a vacuum. The expression (5.77) becomes :

Let us put : N

The intensity I at P, given by (5.65), becomes :

Vibrations of the same frequency emitted by the same atom are perfectly coherent. In monochromatic light we have:

which is the elementary classical formula. 5.8 The representation of the emission of an incoherent light source

Here we consider that the total number of wave trains arriving at the detector from the individual sources is very large during the observation time. A light source, even a "point" source in the optical sense, comprises a large number of atoms emitting wave trains at random times. The vibration emitted by a source at a given instant is the sum of the individual vibrations due to the different atoms. In the form given by expression (5.36) the vibration V ( t )represents a succession of wave trains emitted by an atom. Since all these wave trains are incoherent, it does not matter whether they are emitted by the same atom or by different atoms.

80

Dzffraction :Coherence in Optics

It is sufficient to consider in the expression (5.31) the superposition of a large number of terms at time to since a large number of atoms can emit at the same time, or at almost the same time. As a result, the vibration V ( t ) can represent the vibration emitted at a given instant by an extended source. In addition, since the wave trains are each of limited duration, V(t)as given by the expression (5.35) represents a non-monochromatic vibration. In consequence V ( t ) represents the vibration emitted by an extended, non-monochromatic source. Thus the emission from a source can be represented as follows: at a time t an amplitude and a phase can be defined. Using the complex representation, we can say that the vibration is of the form (5.39) Since the light is assumed to be quasi-monochromatic, the complex amplitude a(t) varies slowly in comparison with the oscillatory term e'2"vd,and the interval Av occupied by the frequency spectrum is small in comparison with the mean frequency v,. In most observations when we refer to the intensity of a source we do not introduce r since the time for an observation is large in comparison with 7. During time T the amplitude a(t) moves in a very complicated fashion in the complex plane and the intensity emitted by the source is represented by (5.40) : This is not true in the case of interference when we are considering wave trains passing a given point during small time intervals of the order of 7. Two interfering beams have travelled along paths which do not differ greatly: that is to say, the two wave trains arising from the same initial wave train pass a given point within a very small time interval 8. This interval remains constant during an observation. But since the difference between the amplitudes is not always the same for a given interval 8, the visibility of the fringes depends on the fluctuations in the pattern during the time 'of observation. Let us assume that the two vibrations which interfere have travelled paths differing by 6. This means that the complex amplitudes of these vibrations are given by the two values of a(t) which correspond to the times t and t+O, where 6 = d/c (c = velocity of light).

Extended Luminous Sources and Objects. Coherence

81

If 0 r, there is little difference between the amplitudes and this difference varies little with time. The interference phenomena are perfectly visible. If 0 % r, there is a large difference between the at times t and t + O and this difference changes considerably during the time for an observation. The interference effects are no longer visible. 5.9 The influence of time-coherence on dfffractim phenomena. The diffraction spot in qmsi-monochromatic ligM

Let us consider a point source of quasi-monochromatic light which gives a mean frequency v,. The emitted vibration can be expressed as a function of time t by (5.39):

We intend to calculate the structure of the image of the point source formed by a perfectly achromatic objective, for example, a curved mirror. We shall study the phenomena as a function of one variable by representing the distribution of amplitudes on the wave surface by the function F(c). The diffraction spot no longer has the structure given by the formulae in Chapter I1 since the light is not monochromatic. The finite length of the wave trains modifies the diffraction spot, which we shall now investigate. If the expression (5.84) represents the vibration at the time when it leaves some point M on the wave surface Z (Fig. 5.18), the vibration at P' at time t emitted by M can be written as follows, using the notation of paragraph 5.5 (5.55):

where 8 is the difference in the times for the vibration to travel to P' from M and C. Considering the effects along S ' i (Fig. 2.6) we have

8=

(CP')-(MY)V[ -C C

specifying a mean frequency v , and putting *

82

Dfiaction: Coherence in Optics

we have: if v remains close to v,, a(t+O) varies slowly compared with (0 is constant for a given point P'), and the source emits quasi-monochromatic light. In the case of strictly monochromatic light, the amplitude at P' will be given by ej2nvot

f (v) = J ~ ( ( ) dd5. ( ~ ~ - ~

The monochromatic vibration

must be replaced by the expression (5.88) and the amplitude at P' at time t becomes :

In fact the application of Huygens' principle to the case of nonmonochromatic light introduces a term which depends on the time derivative of the complex amplitude a(t). To avoid difficulty at the extremities of the wave trains, we replace the discontinuities with a regular decrease in a(t)extending over an interval of time At which is small in comparison with the duration of wave trains. Formula (5.90) then represents a sufficient approximation to Huygens' principle if the wave trains are fairly long compared with the period.

Extended Luminous Sources and Objects. Coherence

83

Taking the phase of the vibration arriving at P' from C as the phase origin, we have :

since from (1.18)

27~ lo

q = kovc and k , = -

and the intensity at P' is: we have

f (v, t )f *(v, 1) =

1-

a( 1

+~

) F ( ( ~ dCl ) $ ~ ~ ~ ~ ~

Let us put then we can write:

and

F(C 1 )F*(C2) = F(C ,IF*((

1

- ('1

~ince'themean only applies to the terms which vary as a function of time, we have:

and changing the notation 5, to [ let us put

J

D(r') = F(OF*(C- C') dr D([')is the autocorrelation function for F([). The intensity at P' is

84

Diffraction:Coherence in Optics

therefore :

In the case of strictly monochromatic light the wave trains are infinitely long and we have :

whence

The intensity at P' is the Fourier transform of the autocorrelation function for F([).

The diffraction pattern for a slit We have (Fig. 5.19):

F(C) = 1

for

F(O = 0

for

-f;o TO

From (5.96), as F ( l ) is real :

F ( 5 - r ) is identical to F(O but is displaced by C'. The function D ( f ) is therefore represented by the common part AB of the two slits of width 25, (Fig. 5.20) displaced by (' with respect to one another.

Extended Luminous Sources and Objects. Coherence

85

We have (Fig. 5.21)

o(c3 o(C3

= 2f;o

+l'

for for

=~ CO l' and substituting in (5.98) we obtain:

-2c0 < t;' < 0 (5.100) 0 < I' < 2C,

which is the diffraction pattern for a slit illuminated with rnonochromatic light. If the light is not monochromatic, that is if the wave trains are not infinitely long,

We assume that the light emitted by the source consists of limited sinusoidal wave trains of frequency v and duration r (8 5.2). If the complex amplitude a(t) is constant for each 'wave train during time 7 : a(t) = 1 for - 212 < t < r/2

+

a(t) = O

for

ItI > 212

Diffraction: Coherence in Optics

86

We have already seen ( 5 5.2) that in this case the amplitude of the spectrum emitted by the source is (equation 5.6):

v(v) =

sin n(v - v,)7 X(V

- v0)z

As a(t) is constant, the function A(vct/c)can be written:

The product a(t)a(t- vC'/c) can be represented by AB in Fig. 5.22. vc'/c is a time 8' which should be less than r if A(vC'/c) = A(0') is not to be zero (Fig. 5.23). A@') is the time-coherence factor. It is zero if lO'l z T since the wave trains are not superimposed and there is no correlation between them. We have (Fig. 5.23):

a(v!)

= A(0.) = 1

+ -8Z'

1--

for

8'

for

Z

-.c
0 < 8' < z

Putting [' = yc, (the displacement (' is less than or equal to 2c0; that is lrl 2), the functions D(c') and A(vC'/c) can be represented by :

+ 2Y

for

1- Y

for

D(y) = 1

- 2 i y < O

Extended Luminous Sources and Objects. Coherence

87

and similarly :

u5 o

A(Y)= I + y z

l-y%

CZ

for

-2
for

0
If 1 = cr is the length of the wave trains, the intensity given by (5.97) can be written :

Putting Z = k,vr,, we obtain :

Z

- sin 2Z z2

(J.104)

If AL is the spectral width emitted by the source (4 5.2) we have :

and (5.10-4) can be written :

sin Z

2z2

(5.105)

The factor

d

=

cosZZ

sin 2 2

z

-2zi

is shown by curve (1) in Fig. 5.24. It is always positive for Z > n/2. Expressions (5.104) and (5.105) demonstrate the influence of the length of the wave trains on the structure of the diffraction phenomena. If I + co (monochromatic light), (5.104) reduces to (5.101),that is to (sin z / z )which ~ is shown in Fig. 5.24 by curve (2). If the wave trains have a finite length, curve (2) multiplied by a suitable factor has to be added to curve (1). The maxima in I

D~j5raction: Coherence in Optics

very nearly correspond to the zero minima of (sin z / z )and ~ vice versa. The factor Itends to level off the fringes surrounding the central fringe. Take, for example, a line of spectral width Art = 30 A and wave length A, = 0-546p. We have

At Z = kn, corresponding to the zero maxima of the diffraction pattern for monochromatic light, we now have an intensity

If Z = n, we have I = 0.0006 instead of a zero minimum.

Note To calculate the image of a point source of non-monochromatic light we could sum directly the intensities of the diffraction patterns corresponding to the wavelengths of all the radiations emitted by the source. A coefficient taking into account the spectral distribution of the light energy is applied to each diffraction pattern. This metliod does not demonstrate clearly the influence of the length of the wave trains on the diffraction spot.

Extended Luminous Sources and Objects. Coherence

89

5.10 Spatidly coherent and incoherent objects If the elements of an object emit entirely incoherent vibrations, which are independent of one another, the object is said to be incoherent. The sun, the stars, the planets, landscapes and the various monochromatic radiations forming the spectrum of a source, are all examples of incoherent objects. In instruments using an auxiliary source, such as a microscope, a projector or a profile projector, all parts of the object are illuminated by vibrations emitted by the same source. If it is a point source, the illumination is coherent. Let us consider a monochromatic point source S placed at the focus of the objective lens 0 (Fig. 5.25) and let MI and M, be two points in a plane perpendicular to the beam of light. The illumination of the two points M Iand M , is perfectly coherent. In the case of Fig. 5.25 the optical paths from S to M I and M 2 are equal. The vibrations at M I and M, are identical. Let us imagine that the two points M , and M, are two small holes pierced in a diaphragm D. The holes M, and M, diffract perfectly coherent light and we can observe Young's fringes on a screen E placed after D. Let us consider another point source S, emitting the same wavelength. If the source S, is used alone, the two holes M I and M 2 are still perfectly coherent but they emit vibrations which are not in phase since the optical paths ( S , M , ) and (S,M,) are no longer equal. Young's fringes can sfill be observed on the screen E but they are displaced in the plane E with respect to the fringes produced by the source S acting alone. Suppose the two point sources S and S , are used simultaneously: the intensities of the two sets of fringes in the plane E are superposed since the sources S and S, are independent and therefore incoherent. The visibility of the fringes is decreased due to the relative displacement of the two systems of fringes. However, if the distance between S

90

Dtffraction: Coherence in Optics

and S, is smaller than a certain value d, the difference between the optical paths from S1 to MI and M,,(SIM,)-(S,M,), can be neglected in comparison with the wavelength 1(2110 for example). In this case the two systems practically coincide and the contrast is not appreciably reduced. The same is true for all point sources at distances from S which are less than d. We can say that in the case of an extended monochromatic source of diameter 2d, the elements of which are incoherent, the points M 1 and M, still behave like two coherent sources since they are capable of producing fringes on E which are as sharp as those obtained with a point source. There is spatial coherence. If the diameter of the source increases, the fringe systems produced by M I and M, overlap and the contrast is reduced. The two points M 1 and M, are partially coherent. If the diameter of the source is increased still further, the contrast continues to diminish and we approach complete incoherence. We shall investigate only the limiting cases of spatially coherent and incoherent illumination. Subsequently we shall study the influence of time-coherence (non-monochromatism of the source). 5.11 'Zbe image of an extended object inurnhated with spatially

incoherent light The object is in the plane n (Fig. 5.26) at the focus of the objective lens L, and is represented by the shaded area bounded by the contour A. This region is luminous (monochromatic) and the points in the plane n outside A are not luminous.

Extended Luminous Sources and Objects. Coherence

91

The luminosity of the object varies from one point to another and the vibrations emitted are incoherent. The image A' of the object A is formed in the focal plane R' of the objective lens 0. A diaphragm with an aperture of some sort is placed between L and 0 (this diaphragm is not shown in Fig. 5.26). We have the same conditions as before (Chapters 11, 111, IV). For each point on the object A the image in the plane n' is a diffraction spot which is characteristic of the shape of the aperture in the diaphragm covering L and 0. If the aperture is circular, the diffraction spot is the Airy disc. As all the points of the object are incoherent, to find the image of the extended object the intensities of all the diffraction spots have to be summed. We shall use the following notation: diffraction pattern (amplitude) (image of a point) (5 1.6): diffraction pat tern (intensity) (image of a point) :

&

intensity distribution in the geometrical image of the object:

intensity distribution in the image of the extended object:

If the objective 0 is limited by a circular aperture, f (u, v ) is given by expression (2.15) and I(u, v ) by (2.19). We wish to find the luminous intensity at a point P'. Let u, v be the direction cosines of OP' and u,, v , be those for the direction OP;, Pi being some point in the illuminated region. If we assume that Pi alone emits light, there will be a diffraction spot with Pi as centre. The intensity at P' due to this is:

Since all points in the geometrical image do not have the same intensity, expression (5.107) must be multiplied by O(ul, o,). The intensity at P' is then given by :

Dzflritctiun: Coherence in Optics

92

where the integral extends over the illuminated region of the geometrical image of the object. The intensity is represented by the convolution of the two functions 0 and I.

object of uniform intensity. The dipaction pattern contains a centre of symmetry The calculation of the intensity at a point P' is now simplified. From (5.108), apart from a constant, we have : Case of

mt

The integral on the right-hand side represents the "solid of diffraction" obtained in the following manner: lines perpendicular to the plane x' are drawn through every point of this plane. Lengths proportional to i(u,-u, v , - v ) are marked off from the plane n' along these perpendiculars. The ends of the lengths obtained in this way are on a surface which, together with the plane nt, encloses a certain volume. This is the volume of the "solid of diffraction". The perpendicular passing through P' is taken as the axis of the solid diffraction. This solid can easily be imagined by assuming that the only source of light is the point object of which P is the image. We assume that the diffraction spot surrounding P' has this point as a centre of symmetry. I(u, - u, v - a) represents the intensity at P' due to some point P; of the illuminated region. Since the point P' is a centre of symmetry, the intensity at P' due to Pi is the same as the intensity at Pi due to P'. Thus at P':

E(u,v) =

I(U-ul,

V - V ~ ) ~dvl U ~

(5.1 10)

I(u-u,, v-vl) represents the intensity at P; produced by P'. Thus the contribution at P due to some point P; in the illuminated region is given by the intensity diffracted to Pi by P'. The intensity at P' is found by summing all the intensities diffracted by P which reach points in the geometrically illuminated region of the image. Let us arrange the "solid of diffraction" in such a way that its axis passes through the point at which we wish to calculate the intensity, in this case the point P'. The intensity at P' is then proportional to the volume of the part of the solid of diffraction about P' enclosed by a right cylinder with the area inside the contour of the geometric image as base. This volume is represented by the integral (5.1 10).

Extended Luminous Sources and Objects. Coherence

93

Example: Let us cakulute the intensity E at the centre of the image of a small black disc on bright background. Case of an objective 1imited by a circular aperture. The intensity E is found by calculating the volume of the solid of diffraction corresponding to the luminous regions of the object. In this case this is the part of the solid of diffraction lying outside the volume enclosed by a right cylinder with a base equal to the small black disc. From (2.29) we have: where Z depends upon the radius of the black disc. If p is the (linear) radius of the disc, a; the aperture of the objective lens 0 (radius of the objective divided by focal length), from (2.20) we have:

At a distance from the image of the disc the intensity E' is a maximum and equal to one. If the disc is small in comparison with the diffraction spot, we can expand J,(Z) and J , ( Z ) :

whence

with

E' = 1 The contrast of the image of the small black disc can be defined by

If s = lrp2 is the surface area of the image of the black disc, we have :

Diffraction :Coherence in Optics

94

Tbe image d an extended object i l l d a t e d with spatidly coherent light Figure 5.27 shows the case of an extended object with spatially coherent illumination. The object is at x and is illuminated by a parallel beam. The objective lens 0 forms an image of the object n at n'. A point A on the object diffracts light which is focused by 0 to A' in the image plane. The image A' is a diffraction spot produced by the objective 0.The image at n of the extended object is the result of the sum of the amplitudes of all the diffraction effects produced by all the points on the object. The problem can therefore be treated in the same manner as before but the summation is carried out for amplitudes, not intensities. 5.12

We shall use the following notation : diffraction pattern (amplitude) (image of a point) (fj 1.6)f (u, v ) amplitude distribution in the geometrical image of the object 0)

amplitude distribution in the actual image of the extended object A(% v ) the parameters u and v are the direction cosines of the direction OP' (Fig. 5.28). From the above (§ 5.1 I), the amplitude at a point P' is given by and the intensity by:

Extended Luminous Sources and Objects. Coherence

95

In the case of coherent illumination, the calculation can be carried out using the method of double diffraction (Fig. 5.29). The object is in the plane n in front of an objective lens 0 and is illuminated by a parallel beam (coherent illumination). The diffraction phenomenon characteristic of the object is located at S the focus of 0.Up to now we have taken the function f (u, v ) to represent the amplitude of the diffraction pattern for the image of a point source. We shall keep this notation in the case of Fig. 5.29 and consequently the notation for the image has to be altered fof the double calculation. The function F(q, () (see 8 1.6) represents the object n and the function f (u, o) the diffraction phenomenon at S',

where u and v are the direction cosines of a direction through the optical centre of the objective lens 0 which defines the position of a point in the diffraction pattern where the amplitude is f(u, v). This is the same as placing the object in the plane of the diaphragm D in Fig. 1.8. A second lens L, placed in the plane of the diffraction phenomenon, forms an image of n at x'. It is assumed that the magnification is unity so that the same coordinates can be used in the planes n and n'. These coordinates will be q, c, which are different from those in the notation in formula (5.111). The diffraction pattern at S' is given by (1.22):

f (u, 0 ) = jJ F(q, ()dk(u*+vo dq d(

96

Dz~iaction: Coherence in Optics

The Fourier transform gives : - i k ( r ~ + v c )du

da

(5.1 14)

so that we can pass from the diffraction phenomenon f (u, v ) to the wave surface F(q, [), that is to the object. We can equally well pass from the diffraction phenomenon f (yv ) to the image of the object at n', and say that F(q, 5 ) also represents the structure of the image in the plane n' with the obvious condition that all vibrations diffracted by x are incident on rr'. The lens L must have an aperture such that this is true in practice. We can say in effect that if we move away from the centre S' by an amount x, for example, the diffraction becomes practically negligible. If the lens L has a diameter of

at least 2x, the whole of the diffracted light passes through L and the vibratory state at rr' is the same as at n. If the lens L stops the diffracted light given by f(yu) to any appreciable extent the vibratory state F'(q, 0 at n' is different from F(q, 0.Iff '(u, v ) represents the diffracted light passing through L, the image F'(q, () is obtained by calculating the following integral deduced from (5.1 14):

Let us take an example. The object placed in the plane q, (Fig. 5.29) is a sinusoidal test object with the stripes parallel to 0:

From (3.13) and (5.13) the diffraction pattern in the plane u, v is

Extended Luminous Sources and Objects. Coherence

97

given by:

If the diffraction is neglected, f (v) is made up of two terms with amplitudes proportional to j and -j. These two terms are in directions v = Alp and v = -Alp. Using the Fourier transformation (5.113), the distribution of amplitudes in the image can be calculated : F(O = je -ikVr -je -jkvC and replacing v by Alp and

-Alp :

whence, apart from a constant factor :

The object is perfectly reproduced by the instrument. This is obvious since the two spectra ji, and - j [ , pass through the lens L. If the diffraction is no longer neglected, the Fourier transform for (5.1 17) has to be calculated. As lens L does not have an infinite aperture, part of expression (5.117) does not pass through L and the image is no longer identical to the object.

CHAPTER VI

Diflraction Phenomena in Perfect Optical Instruments 6.1 Resolving power and the Emit of perception in optical instruments

6.1.I Resolving power and the limit of perception for an astronomical telescope Let us consider two stars S, and S, with the same intensity (Fig. 6.1). Their images formed by the objective lens 0 are two diffraction spots with their centres at S, and S2, the geometric images of S , and S,. These two images are observed by the eye by means of the eyepiece Oc. The objective 0 and eyepiece Oc make up an astronomical telescope. We assume that the aperture of the objective 0 is circular. If S;S; is large enough, the eye sees two distinct images. If the images S; and S; are too close together, the two diffraction patterns almost overlap and the eye sees effectively only one image. At what distance S;Sf2, called the limit

of resolution, can the eye no longer distinguish two images? To answer this question we have to know not only the structure of the diffraction pattern for each of the two images but also the nature of the detector, in this case the retina of the eye. However, under certain conditions the eye can be considered as a perfect instrument 98

Diffraction Phenomena in Perfect Optical Instruments

99

and the resolution is governed by diffraction together with only one consideration of a physiological nature : the minimum contrast perceptible to the eye. These conditions hold when the eye is operating with a pupil of about 0.6 mm diameter. We therefore assume that the exit pupil of the instrument (the image of the objective formed by the eyepiece) has a diameter of 0.6 mm. The diffraction patterns which form the images of the two stars are shown in Fig. 6.2 for an arbitrary separation S;S;. The resultant curve, the sum of the intensities, is shown in Fig. 6.3 on which theordinates have been reduced. If the two points S;, S; are not merged, a minimum of intensity I, can be observed between the two regions of maximum intensity I,. The eye therefore perceives a region of lower intensity at the centre of the pattern. The contrast is:

If y is greater than the minimum contrast perceptible to the perfect eye, that is about 0-02,the eye sees two separate images. If y < 0.02,

100

D%fiaction: Coherence in Optics

the eye can no longer distinguish two images. The limit of resolution is given by y = 0.02. This is true when the central maximum of one of the spots more or less coincides with the first zero minimum of the other spot (Fig. 6.4). From (2.22) the angle a = ~ ~ 6issgiven ; bv

where Za, is the diameter of the objective lens 0. For 2a0 = 70 cni and

1 = 0.6p. we have a = 0-2"

Formula (6.1) gives the resolving power for a perfect instrument with a magnification such that its exit pupil is equal to 0.6 mm. If it is important to separate the fine details of an object we must know the optical instrument's perception of details. Let us take as an object a small black disc on an incoherently illuminated bright background. The object is at infinity. If the diameter of the disc is smaller than that of the central maximum of the diffraction spot, its image with surface area s in the plane n' (Fig. 6.5) has a contrast :

If y < 0-02,the image is not perceived and the field of view appears uniformly bright. If y = 0-02, we are at the limit of perception. Introducing the angular radius 8 of the black disc and the diameter 2a, of the objective lens 0 we have:

Difraetion Phenomena in Perfect Optical Instruments

101

whence

For an objective of diameter 2a, = 70 cm and with L = 0 6 p formula (6.4) gives a limit of perception of 002". In fact the minimum contrast perceptible to the perfect eye varies a little with the nature of the object. The value 002 is a minimum value and in the case of a small disc the minimum perceptible contrast is more like 0-04. The limit of perception is then of the order of 0.03". Note In formulae (6.1) and (6.4) the values obtained depend on the diameter 2a0 of the objective lens. These formulae assume that the light beams are limited by .the objective: the optical elements placed after the objective do not act as diaphragms. In general the aperture stop mounted in the instrument limits the light beams. The image of the aperture stop in the image space is the exit pupil. Formulae (6.1) and (6.4) can also be applied to the image space, 2ao then being the diameter of the exit pupil of the telescope. The values obtained must then be divided by the magnification to find the angles a and 8 in the object space. 6.1.2 Resolving power and the limit of perception for a microscope Consider two luminous points A and B of equal intensity situated in the object plane examined by the microscope (Fig. 6.6). The latter consists of an objective 0 (circular aperture) and an eyepiece Oc. Let a. be the angle subtended at A by the radius of the objective and n be the refractive index of the object medium: the product n sin a, is the numerical aperture of the objective of the microscope. In actual fact the objective of a microscope is made up of a number of lenses and the angle a, is the angle subtended by the radius of the first lens (front lens) at A. As before, the images of A and B are two diffraction spots with their centres at A' and B', the geometric images of A and B. The limit of resolution is reached when the distance y' = A'B' between the centres of the two spots is given by (2.21):

102

Diffraction :Coherence in Optics

Microscope objectives always satisfy the sine law (Abbk law) so that we can write:

n y sin a, = n'y' sin a;

(6-6)

where y is the distance A B between the two points in the object plane and n' is the refractive index of the image space. The angle a; is always small in microscopy and of course n' = 1. We then have :

ny sin a, = y'ab =

1.222 2

whence

1*22A Y = 2n sin a,, Formula (67) is applicable if the magnification of the microscope is such that the exit pupil has a diameter of 0.6 mm. Let P be the power of the microscope in diopters and o the diameter (in metres) of the exit pupil. Then

P=

2n sin a, W

Now ,the magnification M is given by

M = 0*25rnxP whence

M=

n sin a, 2ct,

Dzrraction Phenomena in Perfect Optical Instruments

103

For an exit pupil of w = 00006 m we have:

n sin a, = 1000 n sin a. = 0°0012 For formula (6.7) to be applicable the magnification of the microscope must be a thousand times the numerical aperture. 6.1.3 Resolving power of a prism spectroscope The collimator (Fig. 6.7), formed by the narrow slit S placed perpendicular to the plane of the figure and at the focus of the objective lens O,,illuminates the prism P with parallel beams. The spectrum is observed in the plane rr' at the focus of the objective lens 0,.For a wavelength L the spectral line, which is the image of S, is at S;. For a different wavelength i + d l the image is at S;. The angle subtended at 0,by the distance S;S; is 0. The spectral lines S; and S; are diffraction patterns with a form which is determined by the way the beam entering the objective 0, is limited. In general the prism limits the beam, and S', and S, are then two diffraction spots produced by a rectangular aperture. Their centres are at the geometric images S\ and S; of S. As before we can assume that separation is achieved if the maximum of one of the spots coincides with the first zero minimum of the other. From (2.4) if D is the width of the beam limited by P and incident on 0,, we have:

Dzffraction: Coherence in Optics

104

The angle 0 corresponds to two images S; and S2 separated by a wavelength interval dL. Let us calculate 6 as a function of dl. We differentiate the prism formulae :

(

sin i = n sin r

{ sin if = n sin r'

(6.11)

and obtain :

cos i'di' = n cos r' dr'

+ sin r' dn

(6.12)

dr+dr' = 0 The first and third equations of (6.12) give :

sin r dn dr' = cos r n and substituting in the second : cos if di' = . .

.

sin A dn COS r

If e = BC is the (effective) base of the prism we have: e

sin A

AC --cos r

D cos i' cos r

and from (6.13):

di' =

sin A cos i' cos r

dn =

e

dn

Now considering diffraction, the two lines are separated if 0 = LID. As 0 = di' we have:

If R = A/dA is the resolving power of the spectroscope, we can

Diflract w n Phenomena in Perf et Optical Instruments

105

write :

Assuming that the prism is made of glass for which

we have

-

10- l o cm2,and the mean wavelength of sodium For flint glass B yellow lines is L = 0-5893 p

-

If the two sodium lines at an interval of d i = 6 A are to be separated the required resolving power is equal to 5893/6 1000, which gives a width e of the order of 1 cm. To increase e, that is to increase the resolving power, the number of prisms can be increased. With prism spectroscopes it is possible to reach R = 100,000, and larger values are obtained with diffraction gratings. Note that in spectrographs with large apertures the diffraction effects are less important than residual aberrations which generally limit the resolving power.

6.2.1 Description A screen pierced by a large number of fine parallel slits which are equal, equidistant and coplanar is called a plane transmission grating. The distance separating two homologous points of two adjacent slits is called the period of the grating. These gratings are made by cutting equidistant lines on a plate of glass with a diamond. In spectrography it is more usual to use a reflecting grating. To make these gratings, regular lines are cut on a metal surface, and these act as opaque bands. The grating can be ruled on a thin, opaque film of metal deposited on a plane sheet of glass by evaporation in a vacuum. Reflection gratings have the advantage that the light does not traverse the supporting glass as it does in the

'

106

Dzfiruction :Coherence in Optics

transmission grating, so that any inhomogeneities that may exist have no effect. In addition, reflection gratings can be used in the ultraviolet or infrared regions where a glass support would be unsuitable. A replica of a grating can be obtained by pouring a solution of collodion in ether onto the grating. When the ether has evaporated a transparent film in which the lines are reproduced can be removed from the grating. A transparent replica can be converted into an effective reflection grating by depositing onto it a thin layer of metal by evaporation in a vacuum. The gratings used in spectography usually have 500 lines per millimetre and the ruled

surface varies from 10 x 10 cm2 to 25 x 25 cm2. The spectra obtained with a reflection grating can be observed by means of the assembly shown schematically in Fig. 6.8. The plane grating R is illuminated by the collimator consisting of slit S and objective lens O1. The slit S is mounted parallel to the lines of the grating The light diffracted by and at the focus of the objective lens 0,. the grating is incident on the objective 0, and the spectra are observed in its focal plane n'. 6.2.2 Infiinite grating The grating is illuminated with monochromatic light. Let us consider a small portion of the grating and let S , I , and S,I, be two rays incident at two corresponding points I, and I, (Fig. 6.9) of

DzjJruction Phenomena in Perfeet Optical Instruments

107

two adjacent reflecting strips at an angle of incidence i. The reflecting strips separated. from one another by the non-reflecting regions act like the transparent slits in a transmission grating : they diffract the light. Let us consider the diffracted rays 1,s;and 1,s; at an angle 0 and draw through the corresponding point I, of another strip the plane waves Z and Z' corresponding to the directions S , I , and I,S;. From the corresponding point I, on the subsequent ruling drop perpendiculars 14H and 14Ht onto Z and X'. The path difference d between two contiguous rays S,I,S; and S,I,S; is

S = 14Hf- 14H

If 2a

= I,I,

is the period of the grating, or the grating constant :

S = 2a(sin 0 - sin i)

(6.17)

If the diffraction effects are neglected, the phenomena can be investigated by replacing the reflecting strips with points sources at I , , I 2 . . . (8 4.1). The rays diffracted by the reflecting strips are in phase in the direction 0 if

S = 2a(sin 8 - sin i ) = p l

(6.18)

Therefore the light has maximum values in directions such that a

sin8 = sin i + p -

A

2a

108

Diffraction : Coherence in Optics

If the beam is incident normally (i = 0) A

sin 8 = p ~ ; ; In the directions where cp is given by (6.19) we have light maxima which are monochromatic images of the slit S (Fig. 6.8). These images are the principal maxima considered in paragraph 4.6. If 6 is not equal to an integral number of wavelengths there is practically no diffracted light even if the difference 6 - p i is very small. if 6 = pA+l/2n, the path difference between the vibration diffracted by the (n+ 11th reflecting strip and that diffracted by the first is npl+;1/2, so that they cancel. The vibrations diffracted by the reflecting strips cancel in pairs and there is practically no light between the principal maxima. The preceding theory does not take account of the finite width of the grating. The position of the principal maxima can be determined without knowing their structure. Formula (6.18) shows that the number of observable principal maxima is limited since we must have sin 9 2 1. The number p therefore cannot exceed the values given by:

The number of observable principal maxima decreases as 2a becomes smaller.

6.2.3 The dispersion of a diffraction grating Formula (6.18) shows that the position of the principal maxima depends on I (except for p = 01, so there is dispersion. In white light and for a given value of p we obtain a spectrum which is a set of monochromatic images of the slit. These monochromatic images are the principal maxima corresponding to different wavelengths. When p = 0 all the monochromatic images of the slit are superimposed and there is a white image of the slit. Spectra can be observed on each side of this image (Fig. 6.10). The plane of Fig. 6.10 is the plane n' in Fig. 6.8. Let us consider two wavelengths L and A+ dl. The two monochromatic lines corresponding to these wavelengths are separated by an angle dB and from (6.18) : dA do = p 2a cos 8

Diffraction Phenomena in Perfect Optical Instruments

109

The angular dispersion of the grating is defined as the ratio :

d8p - sin 8 - sin i dA - 2a cos B - A cos 8

(6.22)

The dispersion increases with p. To separate two lines 1and 1 + d l it is advantageous to observe them in a high order spectrum if this is possible. spectra 2* order 1" order

spectra 1" order 2"6 order

Note Let us consider a spectrum of a given order p. From (6.18) and (6.22) we can see that if 1varies, 6 and dO/dd also vary. The dispersion is not constant along the spectrum. Let A, be a wavelength at the centre of the spectrum p under consideration. Let us take an angle of incidence i such that (6.18) :

sini = -

A0

p

~

From (6.18) we then have :

sin 6 = p

2a

For wavelengths not very different from 1, we can write:

The wavelength varies linearly as a function of 0: the spectrum is normal. 6.2.4 Superposition of the spectra of a grating Formula (6.18) shows that if i and 0 .have a given value, in the direction 0 there is a principal maximum p = 1 corresponding to

Dzffraciion :Coherence in Optics

110

the wavelength L1 given by

( p = 1) 1, = 2a(sin 8- sin i ) (6.26) In the same direction 0 there is a principal maximum of order p = 2 corresponding to another wavelength A,, such that:

(P = 2)

A1 1, = a(sin 8 - sin i ) = 2

The same is true for other orders. The spectra are thus superimposed unless I , is outside the visible spectrum and is absorbed by the optical elements of the apparatus. Let us calculate the spectral interval for which the spectra are not superimposed : in the direction 8 there is a maximum of order p for a wavelength A if: 2a L =(sin &sin P

i)

1

P o=-1 - 2a(sin 0 - sin i) There will also be a maximum of order p+ 1 in this direction 9 (Fig. 6.11) for the wavelength 1' if

2a -sin i) or of = P+I 2a(sin 0 - sin i) ~ + l There will be no superposition of the radiation in the interval

2'

= -(sin 0

1-1' for all orders up to p. The free interval in the spectra will be A-A' or Aa = a'-a: 4

A0 = And from (6.18)

1

2a(sin 0 - sin i)

Dzfraction Phenomena in Perfect Optical Instruments

111

6.2.5 Finite grating Paragraphs 6.2.2, 6.2.3 and 6.2.4 do not take account of the width of the grating. The principal maxima are infinitely narrow and their position can be calculated from the theory. As soon as the dimensions of the grating are introduced the principal maxima are no longer infinitely narrow and the resolving power of the grating has a finite value. We know (v.6)that when the diffracted beam does not diverge appreciably from the normal to the grating, the structure of the principal maximum is identical to the diffraction pattern produced by a slit of the same width as the grating. Let us take expression (4.19) and put v = 8. The product 2aO represents the path difference between the vibrations diffracted by two successive slits in the direction 0 when 0 is small. In the general case the path difference is 6 = 2a(sin 0- sin i) and formula (4.19) can be written :

sin nkS/2 = sin k 6 / 2 )

(

(6.30)

If we investigate the structure of the principal maximum, B hardly varies and we have :

+

+

S = 2a[sin(O w ) -sin i] z 2a[sin 9 - sin i w cos $1 (6.31) as w is small. All that is necessary to study the structure of the principal maximum in question is to vary w. As 0 corresponds to a principal maximum of order p we have

Za(sin 6 - sin i)

=

pl

whence

6 = pl+2aw cos 0 and substituting in (6.30)

1

I = [ sin(nkaw cos 8)

sin(kaw cos 0)

which can be written :

I = n2[ sin(nkaw cos O)]' nkaw cos 0

112

Dtflruction: Coherence in Optics

which reduces to (4.21) if 9 is small. The angular half-width of a principal maximum (Fig. 6.12) is equal to W

=

il

2na cos 0

-

;l

L cos 0

Now the distance dB separating two principal maxima with wavelengths 1and 1 dA is given by (6.22):

+

de = p d l

2a cos 0 If we assume that the two principal maxima are observed as separate when the maximum of one corresponds to the first zero minimum of the other, we have:

d0 2 w

whence

and the resolving power of the grating is:

Diffraction Phenomena in Perfect Optical Instruments

113

Since 0 here corresponds to a principal maximum of order p, we can write R in the form:

R=

L (sin B - sin i ) ;1

The resolving power of the grating depends only on the width L of the grating and the angles used. 6.2.6 Real gratings Formula (6.34) shows the structure of the principal maximum, that is the spectral line, but it does not tell us the relative intensities of the maxima. Formula (6.34) was obtained from (4.18) in which the diffraction term was made equal to one. If we wish to know the intensities of the principal maxima the variations of the diffraction term cannot be neglected. This term defines the envelope of the principal maxima, as shown by the dotted line in Fig. 4.8 (I), (2) and (3). But the diffraction term depends on the reflecting strip elements of the grating. Thus we must first know the form of the grating before we can calculate the relative intensities of the spectra.

By giving a particular form to the strips of the grating the energy can be concentrated into one spectrum so that the phenomena obtained are brighter. In an "echelette" grating the profile of each reflecting strip is, for example, that shown in Fig. 6.13. The profile is a right-angled triangle. Let us illuminate the grating with a beam of parallel rays normal to the faces A$,, A,&, A$, ... . If A,B1 = A,B, = .. . = h,the path difference between two

114

D~rraction:Coherence in Optics

successive rays (1) and (2) (8 = 0) is :

S = 4a sin ol Thus for a wavelength 1 in the direction 0 = 0 there is a spectrum of order p given by: 4a sin a = p l

Under these conditions the calculation of the diffraction term shows that the zero minima of this term coincide with all the principal maxima except for that corresponding to 0 = 0.The energy is concentrated in the corresponding principal maximum. In fact this calculation should only be considered as approximate, to show how part of the energy can be concentrated into one order. 6.2.7 The mounting of a grating The gratings can be used in different ways. We show Littrow's method for mounting a grating which is the one most often used (Fig. 6.14). The slit S is at the focus of the objective lens 0.The reflection irating R placed behind 0 is orientated in such a way that the spectrum under investigation is reflected onto the photographic plate P mounted close to S at the focus of 0.This is an autocollimating spectrograph.

6.2.8 Ghosts When ruling the gratings the number of lines corresponding to one turn of the guide screw is very large. If the guide screw is slightly eccentric the lines are no longer equidistant and the defect is reproduced in an identical manner for each turn. The homologous points of the reflecting rulings are no longer equidistant. They are displaced with respect to their theoretical positions and this affects

Diflruction Phenornem in Perfect Optical Instruments

1 15

the position of the diffracted wavelets and consequently the form of the surface envelope which is no longer plane. The wave surface diffracted in some direction has a periodic deformation. If the deformation is sinusoidal, the Fourier transformation easily shows that on each side of a principal maximum there will be two lines with the same wavelength as this maximum. These two lines are called "ghosts". If the wave surface has some periodic deformation it can be considered as the sum of sinusoidal deformations. For each sinusoidal component there are two ghosts and consequently the principal maximum is surrounded on both sides by a whole series of parasitic lines, the ghosts, all of the same wavelength. Naturally these ghosts affect the quality of the grating.

6.3 Phase contrast Objects characterized by variations in the refractive index or in the thickness are called "phase objects". These objects, which are perfectly transparent and do not show up against the surrounding field, are invisible using normal methods. Let us consider a small region A in a sheet of glass with parallel faces. The thickness of this region is e and its refractive index n differs from the index nf of the rest of the sheet (Fig. 6.15). The path difference between the ray (1) which passes through the region A and some ray (2) which passes to one side of it is 6 = (n- n')e. The object A is a dephasing object characterized by a path difference 6. In all that follows we assume that d and cp = 27c6/A are small. Objects which reflect can also be considered as phase objects. Let A be such an object (Fig. 6.16): irregularities in the surface height modify the optical paths travelled by rays (1) and (2) reflected at the surface A. The

116

Difraction : Coherence in Optics

path difference with respect to a plane reference surface Z parallel to the mean surface A is equal to 2e, where e is the difference in thickness of the object A in the regions M Iand M,. Objects of this type which are characterized only by variations in the optical path and not by variations in amplitude are not visible using the normal methods of observation.

Using the method of phase contrast it is possible to transform these phase variations in the object into amplitude variations in the image. The phase objects are then visible. As an example, consider the transparent object in Fig. 6.15. The object at r is illuminated with a parallel beam by means of the source S placed at the focus of a collimator lens C (Fig. 6.17). After passing through the object at x and an objective lens O,, the light beam is first converged to S', the image of the source S, then traverses the lens 0, and spreads over the plane r', the image of n. This beam is called the direct beam. The dephasing region in A diffracts light which falls on lenses 0, and 0,and is converged by them to form the image A'. The image A', which is viewed with an eyepiece Oc, results from interference between the direct beam which produces a coherent background and the diffracted light. This can be shown on a Fresnel diagram (Fig. 6.18). Let 0 be the origin. From this point draw a vector OM with a length proportional to the amplitude at the point of the object under consideration. The phase at this point is represented by the angle cp made by the vector OM with an axis Ox. As the object is transparent M lies on a circle with centre 0 and of, say, unit radius. Only the phase cp varies from point to point on the object. If all the light

Dzffruction Phenomena in Perfect Optical Instruments

117

8

diffracted by A is incident on 0, and 0 2 , the vibratory state at n' is the same as at n. The diagram in Fig. 6.18 thus represents the amplitudes and the phases in the image at x'. Let us split the vector OM into two component vectors OH and HM which are mutually perpendicular. Since cp is assumed to be small the point H is very nearly coincident with the point at which the circumference intersects with the axis Ox. The vibratory motion at a point

on n' is the resultant of two vibrations OH and HM. The amplitude OH is the amplitude for the regions where there is no dephasing area (cp = 0): this is the amplitude received at a' if there was no object, and it is therefore the amplitude of the direct light. The amplitude is the amplitude diffracted by the dephasing zone A in the object The amplitude H M would become zero if the object were not present (g, = 0). The intensity of the image I = O M 2 = OH2+ H M 2 is clearly the same in the image A' as in the rest of the field of observation and the object is not visible. Let us place a very thin transparent sheet Q at S' (Fig. 6.17). All the direct light passes through Q, but since the beam of diffracted light is very much spread- out in this region, the sheet Q has almost no effect on the diffracted light. Let us give the sheet Q an optical thickness such that the direct light passing through it is retarded by a quarter of a wavelength with respect to the vibrations of the diffracted light not passing through Q. In the Fresnel diagram the

118

DzJEraction : Coherence in Optics

effect is as if the origin were displaced to 0,.Under these conditions, the amplitude in the image A' becomes 0 2 H -H M = 1- cp and the intensity is I, = 1-2% neglecting rp2. Outside the image A', cp = 0 and the intensity I, = 1. The image A' thus becomes visible with a contrast

If the object A produces a phase advance of q((p > O), that is, a decrease in the optical path, the intensity at A' is smaller than the intensity of the rest of the field of observation, aad the phase contrast is negative. Instead of retarding the direct vibrations with respect to the diffracted vibrations, let us place a sheet Q at S' which advances the direct vibrations by a quarter of a wavelength. All we need to do is to choose a sheet Q which is thicker and retards the direct vibrations by 3L/4, which is equivalent to an advance of A14 with respect to the diffracted vibrations. In this case the origin is shifted to 0,and we have I, = 1+2q. The contrast is still 2rp but if A produces a phase advance cp (cp > 0) the intensity at A' is greater than that of the rest of the field, and the phase contrast is positive. The sheet Q used to modify the phase of the direct vibrations with respect to the diffracted vibrations is called a phase plate. Assuming that the minimum contrast perceptible to the eye is 0-02, the limiting phase difference which can be observed is given by 2 9 = 0.02 or a path difference at 10A. The sensitivity of the method can be greatly increased by using an absorbent phase plate. Let us denote the absorption of the phase plate by the factor N by which the intensity of the incident direct light is reduced. This is a phase plate with an optical density d given by

In the Fresnel diagram (Fig. 6.18) the effect is as if the origin were shifted to 0;such that

D%fraction Phenomena in Perfect Optical Instruments

1 19

The intensity of the image A' becomes:

and in the rest of the field of observation (cp = 0) we have I, = 1/N. The contrast of the image is:

fi.

In principle, if N = 2500 (d = 3.4) Thus it is multiplied by path differences of the order of 1 angstrom can be observed with a contrast of 0.1. Thus considerable sensitivity can be achieved with phase contrast provided the optical system itself is of good enough quality, and above all does not introduce too much stray light. The form of the image of an object can be found by using the method of double diffraction described in paragraph 5.12. If the object n (Fig. 6.17) produces phase variations represented by a function F(q, [), F is purely imaginary. We can write: assuming (p to be small. Whence the diffraction pattern at S' is

The first term on the right-hand side represents the diffraction pattern of the plane wave limited by the lens 0,if there were no phase object at x. The second term is due to the light diffracted by the phase variations of the object. The effect of the phase plate is to suppress the factor j, and to find the image at n' we apply the Fourier transformation to the expression :

Dark ground method Let us replace the phase plate by a small opaque sheet. The amplitude at A' (Fig. 6.17) is given by (p and it is zero outside the image A' because all the direct light is stopped by the opaque screen Q. The intensity at A' is thus equal to cp2 and it is zero in

6.4

120

Diffraction :Coherence in Optics

the rest of the field. The contrast is a maximum and always equal to unity. Unfortunately if rp is small, q2 is even smaller and the images have a very low intensity. Besides this, the slightest optical defects, dust for example, diffract a lot of light which veils the image. Phase contrast, which gives images with an intensity proportional to rp, is more advantageous in most cases. It should also be noted that the dark ground method often produces images which are difficult to interpret.

The ability of an optical instrument to separate the images of two point sources which are very close together is limited by the aperture of the instrument. If the two sources have the same intensity, the resolving power depends only on the size of the central diffraction spot, the intensity of the rings being negligible. But if one of the sources is much brighter than the other, the diffraction rings cannot be neglected and these often give a resolving power which differs from the usual value. We saw elsewhere that almost 20 % of the energy is to be found outside the central diffraction spot (5 2.5): this energy is not only lost from the point of view of the image but is also detrimental in that it produces stray light which veils the image. It is therefore of interest to reduce the diffraction rings as much as possible; this process is called "apodization" of the diffraction spot. In this way the visibility of a weak source close to an intense source is improved, and the same is true for objects with low contrast. The apodization method can be understood from the following example: let D be a circular diaphragm (Fig. 6.19) and let us observe the diffraction effects in

Diffractign Phenomena in perfect Optical Instruments

12 1

the plane n'. The image at S' is an Airy disc (Fig. 6.20 curve (1)). Let us place in front of the screen D a sheet of glass L with parallel faces (Fig. 6.19) such that the amplitude from the centre of the aperture to the edges follows a Gaussian distribution curve. The decrease in amplitude can be so arranged that the amplitude transmitted at the edges is negligible. The result is a Gaussian curve extending from zero (the centre) to infinity. Since the diffraction pattern at S' is the Fourier transformation of the amplitudes in the plane of the wave surface, the profile of the diffraction spot is also a Gaussian curve. We know in fact that the Fourier transform of a Gaussian curve is another Gaussian curve. The diffraction pattern has the form shown by curve (2) in Fig. 6.20. The rings disappear but the central spot is enlarged.

6.6 Filtering spatial frequencies (incoherent object)

The concept of resolving power given in paragraph 6.1 is very subjective and does not represent the properties of an optical instrument at all well. It is preferable to define the capabilities of an instrument in a different way. Consider an optical instrument, a telescope for example, used to observe an incoherent object. We shall choose an incoherent periodic object for which the intensity variations are given by

The variations in O(z) occur along Sz in Fig. 5.26. Figure 6.21

122

Dlflraction: Coherence in Optics

shows a representative curve. The object can be considered as the sum of sinusoidal variations with frequencies lip, 3/p, 5 / p . . . . The problem is to find how the different frequencies are transmitted by an optical instrument. Let us take the fundamental frequency. Using the Euler formula we can write:

Doing the same thing for all the other terms of (6.43) we obtain:

The image of the incoherent object O(z) produced by a lens can be calculated with the use of the following theorem which will not be proved : The Fourier transform of the image of an incoherent object is equal to the Fourier transform of the object multiplied by the Fourier transform of the image of an isolated point of light. To calculate the image of O(z) we must therefore know the Fourier transform for the image of a point produced by the lens. Let us suppose that the lens is covered by a slit. The Fourier transform for the image of a point is given by the function D in expression (5.98)

According to Fig. 5.21 Fig. 6.22 by putting

D(c)can be represented by curve A in

where R is the distance OS' in Fig. 5.26. The angle 2ab represents the aperture of the lens forming the image since 21, is the width of the slit covering the lens. Let us now calculate the Fourier transform for the object O(z). Taking a variable v' of the form v' = [/AR, the transform o(vr) of O(z)can be written : Myr)=

1O(z)d2"'" dz

(6.47)

v' has the dimensions of the inverse of a length. It is a spatial

frequency of the same form as l/p. From (6.45) :

The transform of O(z) is represented by the terms 1, -2j/x, +2j/n, ,. . . corresponding to the frequencies v' = 0, v' = - I/P, v" = l/p . . . (Fig. 6.23). These frequencies can be represented on Fig.6.22 by multiplying them by I since b' = c/R is an angle. We obtain Fig. 6.24. According to the theorem stated above, the Fourier transform of the image of O(z) is given by the product of the ordinates of the curve A and the ordinates corresponding to A/p, 3A/p, 5 A / p . . .. We can

DzJiraction: Coherence in Optics

now understand how frequencies are filtered by an optical instrument. In the case of Fig. 6.24, the frequencies l/p and 3/p are attenuated but transmitted by the instrument. The frequency 5 / p is cut off. The optical instrument behaves like a low-pass filter. The limiting case is reached when the lowest frequency l/p is such that L/p = 22;. The limiting frequency transmitted by the instrument is then

The function D(5') defines the form of the filter. When the lens is stopped with a circular aperture we obtain Fig. 6.25. The straight line A is replaced by a curve C,which is of calculable form. Defects

Diffraction Phenomena in Perfect Optical Instruments

125

in the optical instrument modify the curve C, which is called the response curve of the instrument. The properties of an optical instrument are therefore defined by its response curve and not by the very subjective concept of the resolving power. The response curve gives much more information since it shows how the different frequencies of the object are transmitted by the instrument.

CHAPTER VII

Difraction Phenomena in Real Optical Instruments 7.1 M m * in the case of a sphedd wave Let us consider a perfect optical instrument (Fig. 7.1), an objective lens 0 for example. The incident plane wave I: becomes a spherical wave Z' with its centre at S', the image of the point source at infinity. If the lens 0 is imperfect, the emergent wave I: is no longer spherical. The deviation A of the real wave Z" from the sphere Z' is a measure of the aberration of the objective lens 0. We can transfer this defect to the incident wave E and treat the problem as before. However, it is more logical and more general

to investigate the diffraction after 0 has been traversed, that is diffraction in the emergent wave. We shall assume at first that the real wave coincides with the spherical wave Z' (Fig. 7.2). Any point M on the wave surface diffracts vibrations (Huygens-Fresnel principle) and the vibratory state at a point P' in the plane n' is the result of interference between the vibrations from all points on L'. Taking the vibrations to be sinusoidal, the equation for the

Dz~ructctionPhenomena in Real Optical Instruments

127

propagation of spherical waves gives a vibration at P' of the form:

Let us put MP' = r. Using the imaginary notation (0 1.1) the vibration at P' can .be written:

whence the amplitude at P' due to the unbounded wave Z is :

where A is a constant. Figure 7.3 shows a perspective view of Fig 7.2. If R is the radius CS' = MS' of the spherical wave we can put r = R -b whence

-e - j b - e - j k ( R - 8 ) r

R-6

e-jkR

c=

R

dH

The expression for the amplitude becomes :

Formula (7.3) gives the amplitude at P', but not the correct phase for the vibration at this point. A rigorous calculation shows that the coeficient A is of the fonn j/A and the amplitude at P' is

Dzfrract ion: Coherence in Optics

written :

We can always assume that R is an integral multiple of the wavelength such that e-jkR

-1

Apart from a constant factor, the amplitude at P' can be written in the form :

If q is a unit vector along S'M we have : y', z' being the coordinates of P' and S'M we have

6 = j'y'

b', y' the direction cosines of

+y'z'

whence

If = f l y' = C/R, y' again.

= u .R,z' = v .R,

we find expression (1.19)

D firaction Phenomena in Real Optical htsbuments

129

Let us put S'P' = p. Changing the variables : a' cos 0 = /3'

p cos 8' = y'

sin 0 = y'

p sin 6' = z' in the case of a circular diaphragm with aperture a; the amplitude at P' is given by: a'

f(P) =

i

& 2% p ' p COW

- e.1 a da' d0

(7.8)

If a' = a/& a; = a,/R, a = p/R, we find expression (2.10) again.

7.2 Diffraction pattern when there is a focusing error The image of the source is at S' (Fig. 7.4) and, if the aperture is circular, an Airy disc is observed in the plane x' passing through S'. Let us shift the plane of observation from n' to xt' : the diffraction spot is modified. We wish to investigate the diffraction spot at S", that is for a focusing error of s. For the point S" there is a path difference of A between the wave surface Z (with centre S') and the reference sphere 2' with centre So.Taking expression (1.21), F(q, C) becomes F(Bt, y') and we have: If a' and E are small :

130

~@iractiun: Coherence in Optics

If e is taken as positive to the right of S, the amplitude at some point (in the plane Z'3 at a distance p from S" will be given by (7.8) 2%

f (p) =

11

p ' 2 / 2

eJk'~='a' da' d0

(7.10)

0 0

From the definition for a Bessel function of zero order Jo (equation 2.12) we have :

f ( p ) = 2nJ p"'2/2 Jo(kafp)a'da' 0

This integral can be calculated numerically or graphically. Let us calculatef (Oh that is the amplitude on the axis at S" as a function of the focusing error E. We have : ori,

f(0) = 2~

~ a a ' 2 J 2a'

da'

0

Putting

f

Intensity along the axis

apart from a constant, we have

40, = If ((91' =

(z, , ) sin @/2

When the focal plane is displaced, the diffraction pattern has a black centre at cb = 2n, 4n, etc. (Fig. 7.5).

D i m c t i o n Phenomena in Red Optical Instnunents

13 1

7.3 Rechion of focusing with an optid hstmment Let us observe a point of light S (Fig. 7.6). The observation is carried out with a view finder or microscope V and focusing is achieved by displacing V along its axis zz'. When the image appears sharp we say that we have focused longitudinally. Let us investigate the precision with which this focusing can be achieved. The

luminous point acting as object is at S (Fig. 7.7) and its image Sf is observed by means of an eyepiece Oe. Focusing is accomplished when the image S' js in the plane n' occupied by the cross-wires. Let us displace the object S by a small amount x so that it reaches So. The image is at Sf' but the eyepiece remains focused on Sf.

There is a focusing error of E. If this error is small the eye will not observe any difference between the image it sees at S' and the perfect image at S". Thus the position of S cannot be located with a precision greater than 2x since there is latitude of x on each side of S. x represents a distance such that for any position So at a distance from S less than or equal x, the eye can perceive practically no difference between the image it observes at S' and the perfect image (Airy disc at S"). The distance 2x represents the precision in focusing which we wish to calculate. The microscope may not be astigmatic for S and S', but this is not necessarily true for So and St'. If it is also anastigmatic for So and S" the microscope

132

Dzyraction : Coherence in Optics

satisfies Herschel's condition :

nx sin2 a0

= const.

In the image space of the objective lens 0 the angles are small and we can write Herschel's condition in the form: a0 nxsin2 = 6-ab2 2 4

Now from (7.9) e(ab2/2) represents the maximum path difference due to the focusing defect. It must not exceed a certain value or the eye immediately notices a difference between the image it sees at S' and the perfect image. The maximum value of the path difference eag12 characterizing the focusing error, that is the deformation of the diffraction spot, is given by Lord Rayleigh's rule. The eye sees no difference between the image Sf on which it is focused and the perfect image if the path difference &a: is less than or equal to 214. We therefore have :

whence

For an objective lens with a numerical aperture n sin a, = 1*3, we have 2x = 094 p for L = 0.6 p. Diffraction spot in the presence of aberrations Aberrations (spherical aberration, coma, astigmatism) produce deformation of the wave surface. The deviation A between the real wave surface Z" (Fig. 7.1) and a spherical reference surface Z'is a measure of the aberration of the lens. From (1.21) F(B: y') is of the form

7.4

F ( j f ,y') = PA and the diffraction pattern is given by the integral (1.22)

(7.18)

Difiraction Phenomena in Real Optical Instruments

133

In the case of spherical aberration of the third order, A is of the form A = ad4. If the spherical aberration is accompanied by a focusing error we have

In general aberrations cause a decrease in the central maximum of a diffraction pattern. The zero minima disappear and the diffraction fringes are brighter. The diffraction spot is enlarged and the contrast of the images diminishes.

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DITCHBURN R. W., Light, Blackie, London, 1963. DUPPIEUX P. M., La coh6mnce partielk et Ies fonctions de transmission, R m e dOptipe, 32, 129, 1953. DUFFIEUX P. M., L'intbgrale dp Fourier et ses applicutiorrr ti l'optique, Faculd des Sciences de Besanson. D w o m P., Sur la correspondence objet i m a w optique, Optim Acta, 1, 53, 1955. Duaao~mP., La correspondance objet image en optique, Publ. Sci. Univ. Alger B, 1, 33, 1955; Ibid., 2, 203, 1956. E m P., GREYD.S. and ROBINSON D. Z., Fourier treatment of optical processes, J. Opt. Soc. Amer., 42, 127-134, 1952. FLBURY P. and Mrrnosu J. P., Images Optkpes. Eyrolles, Paris, 1956. FRANCON M., Le microscope A wntraste de phase et le microscope interfbmtiel, Editions C. N. R. S., Paris 1954; and Handbuch der Physik, vol. 24, 1956, p. 346. FRANCON M., Modern Applicatiom of Physical Optics, John Wiley, New York, 1963.

GAMOH., Intensity matrix and degree of cohereace, J. Opt. Soc. Amer., 47, 976, 1957. HARDY A. C. and PERRINF. H., The Principles of Optics, McGraw-Hill, London, 1932. HOPIC~NS H. H., The concept of partial coherence in optics, Proc. Roy. Soc. A, BB, 263, 1951.

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Aberrations, effect on diffraction 132 Amy 1 Airy spot 22, 24 Apodization 120

Babinet's theorem 44

Coherence length 60 Coherence time 53,60 Correlation function 73

Huygens-Fresnel principle 7 Jllumiaation spatially coherent 94 spatially incoherent 90 Incoherent source 79 Intensity, luminous 12, 71 Luminous intensity 12, 71 Maxwell's equations 4

Dark ground method 119 Diffraction by circular aperture 19 by many slits 49 by narrow slit 17 in quasi-monochromatic light 84 by rectangular aperture 14 by several apertures 41,44 by two slits 46 Diffraction grating 105 dispersion 108 echelette 113 ghosts 114 mounting 114 resolving power 112 Diffraction patterns, properties of 36

Focusing error 129 Focusing, prm'sion of 13 1 Fourier integral 28 Fourier series 27 Fourier transformation 29 forms of 30 Fraunhofer diffraction 1 Fresnel diffraction 2

Optical instrument, as a filter 121 Parseval's theorem 75 Perception, lilnit of 100 Quasi-monochromatic source 8 1 Resolution, limit of 98 Resolving power microscope 101 prism spectroscope 103 telescope 98 Solid of diffraction 92 Spatial coherence 53 Time-coherence 53,60 Vibrations, complex representation 66-69

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