- Email: [email protected]
Gabor's signal expansion applied to partially coherent light
Optics Communications North-Holland
86 ( 1991)
14-l 8
OPTICS COMMUNICATIONS
Gabor’s signal expansion applied to partially coherent light Martin
J. Bastiaans
Technische UniversifeitEindhoven, FaculteitElektrotechniek,Postbus513, 5600 MB Eindhoven, Netherlands Received
4 April 199 1; revised manuscript
received
5 June 199 1
Gabor’s signal expansion is reviewed by applying it first to a coherent optical (or deterministic) signal; it is shown that such an optical signal can be expressed as a superposition of optical rays that appear at discrete positions and have discrete directions. The expansion is then applied to partially coherent light, yielding expansion coefftcients that express the correlations that exist between the different optical rays. It is shown how these expansion coefftcients are related to the mutual power spectrum of the partially coherent light and how they are propagated through linear systems. The special case of incoherent light is considered in more detail
1. Introduction In some recent papers [ 1,2], Gabor’s signal expansion [ 31 was applied to optical signals and systems. It was shown that Gabor’s signal expansion is strongly related to the ray concept in geometrical optics: an optical signal is expressed as a superposition of rays that appear at certain discrete positions and have certain discrete directions. Until now, however, Gabor’s signal expansion was applied only to completely coherent light; it is the aim of this paper to show how the expansion can be applied to partially coherent light. In section 2 we briefly review the concept of Gabor’s signal expansion, we show how the expansion coefficients can be determined and how they propagate through linear systems. The main work is done in section 3, where Gabor’s signal expansion is applied to partially coherent light; the case of completely incoherent light will get some special attention there. We conclude this introduction with some remarks about notation. _ For the sake of convenience, we shall use onedimensional space functions, only; the extension to two (or more) dimensions is straightforward. _ An asterisk is used to denote complex conjugation. - The spatial Fourier transform of a space func14
0030-4018/91/$
tion is denoted by the .same symbol as the function itself, but marked by a bar on top of the symbol; hence, p(u)=
s
v(x)exp(-iux)dx.
- Unless otherwise stated, all integrations and summations in this paper extend from -CC to + co. - The symbol 6, denotes the Kronecker delta: S,= 1, and &=O for k# 0. We have chosen the notation with only one subscript (which, by the way, is common usage in discrete-time signal processing, for instance) to stress the equivalence between the Kronecker delta 6, (which appears in a discrete treatment) and the Dirac delta function 6(x) (which appears in a continuous treatment ). - The function rect( t) denotes the rectangular function: rect(t)=l for -f
2. Gabor’s signal expansion In 1946 Gabor [ 31 suggested a signal description that is intermediate between the pure space description and the pure spatial-frequency description. His idea was to express a space signal q(x), say, as a superposition of properly shifted and modulated versions of an elementary signal g(x), say; hence 03.50 0 1991 Elsevier Science Publishers
B.V. All rights reserved.
cc &ml g(x-mX) WI”
P(X) =
exp(inUx)
,
(1)
where the space shift Xand the spatial-frequency shift U satisfy the relation UX=2a. Although Gabor restricted himself to an elementary signal that has a gaussian shape, his signal expansion ( 1) holds for rather arbitrarily shaped elementary signals. Gabor’s expansion coefficients a,,,,, can be determined easily, even in the case that the discrete set of elementary signals shifted and modulated g( x- mx) exp (inUx) is not orthogonal. We simply define a window function w(x) that is bi-orthonormal to the set of elementary signals in the sense
J
g(x) w*(x-mX)
exp( -inUx)
;
dx=&&
(2)
a way to derive such a window function has been described elsewhere [ 1,2,4]. With the help of the biorthonormal window function w(x), the expansion coefficients amn follow readily through the formula
amn=
s
15October I99 1
OPTICS COMMUNICATIONS
Volume 86. number 1
q(x) w*(x-mX)
exp( -inUx)
dx.
(3)
The integral that appears in the right-hand side of eq. (3) defines, in fact, the short-term (or windowed) Fourier transform [ 5 ] of the signal Q(x ) , and we conclude that Gabor’s expansion coefficients can be found by sampling [4] the windowed Fourier transform at the regular lattice (mX, n U). The reason why we called the function w(x) that is bi-orthonormal to the elementary signal g(x) a window function, may now be evident. Gabor’s expansion of a signal into a discrete set of properly shifted and modulated versions of an elementary signal has a conceptually clear meaning. For time signals it resembles the musical score in music, and any shifted and modulated elementary signal describes a musical note that appears at a certain moment with a certain pitch. For space signals it resembles the ray concept in geometrical optics, and the shifted and modulated elementary signal g(x- mX) exp( inUx) can be identified as an optical ray at the position x= mX with a direction (i.e., spatial frequency ) u = n V. Gabor’s signal expansion thus represents an optical signal as a superposition of rays that appear at discrete positions mX and have discrete directions nlJ. It is not difficult to derive how Gabor’s expansion
coefficients propagate through a linear system [ 21. Let us, for instance, represent such a system by means of the superposition integral
@o(u)=J h(u,X) Pi(X) cl-~>
(4)
in which the spatial Fourier transform of the output signal &(u) is expressed in terms of the input signal v,(x); the system is thus described by the kernel h (u, x). We now choose input and output elementary signals gi(X) and g,(x), and represent the input and the output signal by means of their Gabor expansions; note that the input and the output elementary signal need not be identical. We can then derive a relationship between the input and the output expansion coefficients a;,, and a& reading
(5) where the system coefficients c&,, are COIIIpletely determined by the system and the choice of the input and output elementary signals through the relationship 1 -- 211
Cklmn-
JJ
h(u,x)
w~(u--IU)gi(x-WZX)
xexp[i(kuX+nUx)]
tidu.
(6)
An interesting example of a linear system is the basic coherent-optical system consisting of a 4f-arrangement with rectangular apertures in the input plane and the Fourier plane [ 2 1. In the special case that the input-plane aperture truncates the signal to the space interval -fX
rect(u/U)
exp(iux)
.
(7)
The array of system coefficients I&,,,, (which we shall denote by ekl,,,n for this special case) is now given by --
eklmn -
1 2A
JJ
rect (x/X)
rect ( u/ U)
Xexp(iux)*z(u-IU)
g,(x-mX)
xexp[i(kuX+nUx)]
dxdu.
For a proper choice of the elementary ray ek/m,, can be strongly concentrated
(8) signals, the ar[ 21: the coef15
Volume 86, number
1
OPTICS COMMUNICATIONS
ficient eOOOO will be close to 1, whereas the other coefficients are close to 0. This is conceptually nice: roughly speaking, only the ray that enters the input plane at position x=0 having a direction u=O can pass both the input-plane aperture and the Fourierplane aperture; all other rays will be blocked. A more general system is the one whose kernel h ( U, x) takes the form h(u, x)=
CC h,,,rect(x/X-m) ,?I n
xrect(u/U-n)
exp(iux)
.
[ 81, be denoted by T(x,, x2, 0). The basic property of the power spectrum is that it is nonnegative definite hermitian [ 7,8]. Since in the present discussion the temporal-frequency dependence is of no importance, we shall - for the sake of convenience - omit the temporal-frequency variable w from the formulas. Extending now Gabor’s signal expansion from the one-dimensional case [for the signal p(x), cf. eqs. ( 1) and (3) ] to the two-dimensional case [for the power spectrum T(x,, x2) 1, we can construct a correlation matrix A by means of the definition
(9)
The array of system coefficients cklmn is now given by the four-dimensional discrete convolution
A m,n,m2n2 =
CM,,,,= L,LnL h the trated cients by the
* * * *eklmn .
T(x,,x2)
w*(x) -mlX)
w(x2-m2X)
(10)
case that the array ek[m,, is strongly concenaround the element e,,,,, the Gabor coeffiof the input and the output signal are related simple relation
a:,, = h,,,aL.
I5 October 199 1
Xexp[ -i(n,
Ux, -n2Ux2)]
dx, d-x2,
(12)
which is the two-dimensional analogue of eq. ( 3). We can then represent the power spectrum in the form
(11)
In the special case that h,, equals unity in the interval ( I m I Q M, I n I dN) and vanishes outside that interval, we have a?,,, ‘v am,, in the interval ( (m ( GM, (nl
Xexp[i(n,
Ux, +n2Ux2)]
,
which is the two-dimensional analogue of eq. ( 1). Clearly, the Gabor coefficient A mlnlmZn2express the correlation that exists between the two elementary signals g(x-m,X) exp(in,Ux) and g(x-m2X) Xexp(in2Ux). Properties of the array of Gabor coefficients A follow directly from the properties of the power spectrum r. The fact that the power spectrum is hermitian, T(x,, x2) =r*(x2, x1 ), reflects itself in the property A m,nlmZnZ--A*,,,,,,,,
3
whereas the nonnegative definiteness spectrum implies the property
(14a) of the power
3. Gabor’s signal expansion for partially coherent light Let partially coherent light be described as a temporally stationary stochastic process, and let its (mutual) power spectrum [ 6,7], or cross-spectral density 16
which holds for arbitrary coefficients a,,,,,. We conclude that the array of coefficients A m,n,m?.n2is nonnegative definite hermitian in the sense of eqs. ( 14). The cases of completely coherent light, completely
Volume 86, number
OPTICS
1
incoherent light, and spatially stationary light deserve special attention. For completely coherent light, the power spectrum factorizes in the form [ 7 ] nx,
>x2) =dx,
1 4*(x,)
and the array of Gabor coefficients rized form * A m,n,,n2n2=am,.,am,n,
takes the facto-
(16)
,
where a,,,,, are the Gabor coefficients of the deterministic function q(x) [cf. eqs. (1) and (3)]. It is easy to see that the formulas that arise in the case of completely coherent light are similar to the ones that arise for deterministic signals. Therefore, we shall not study the completely coherent case any further. For completely incoherent light, the power spectrum can be expressed in the form Qx, >x2) =.0(x,)
6(x, -x2)
,
(17)
where the “intensity” p(x) is real and nonnegative. (The constant X has been included to get an “intensity” function p that has the same dimension as the power spectrum ZY) The Gabor coefficients now take the form A m,n,m2n2
=x
P(X)
w*(x--m1X)
bor coefficients of the “intensity” mentary signal ]g(x) I*: p(x)=
(15)
>
I5 October 199 I
COMMUNICATIONS
,YC b,, ,n n
p(x),
with ele-
Ig(x-mX)]‘exp(inUx).
Spatially stationary light, also known as homogeneous light, for which the power spectrum T(x,, x2) depends only on the difference x, -x2 of the two spatial coordinates x, and x2, is dual to incoherent light, i.e., the space behaviour of incoherent light is similar to the spatial-frequency behaviour of spatially stationary light, and vice versa. Therefore, similar conclusions as the ones that have been drawn for incoherent light hold for spatially stationary light; we only have to interchange position with direction, x with U, space functions with their spatial Fourier transforms, and so on. It is not difficult to derive how the array of Gabor propagates through a linear coefficients A t~l~l??l*?l* system. If such a system propagates a deterministic signal according to eq. (5 ), then the relationship between the input and output Gabor coefficients A’IYllf7l~ZtlZ and A&,k2,2 reads
w(x-m2W
(21)
s
xexp[
-i(n,
-n2)Ux]
dx.
(18)
We observe that A m,n,m2n2 depends on the direction difference n, - n,, as can be expected from the Van Cittert-Zernike theorem. Furthermore, if we choose the elementary signal g(x) such that it vanishes outside the interval - 4X< x< f X, eq. ( 18 ) reduces to A mln,m*n*-b - wz,,“,-82 6m,-IT72> with
b,,n=x
(19a)
s
For the special system described relationship ( 11) we get A:,,,,,,,
Pi(x)=
dx=b*,,_, .
(19b)
We conclude that - there is no correlation between rays at different positions; - the correlation between rays at one position that have different directions, depends on the direction difference; - the correlation coefficients b,,, are just the Ga-
=h,,,,A’
by the input-output
m,n,m*n*h*mznz.
(22)
The case of completely incoherent light in the input plane deserves special attention. Let the input signal be incoherent, and let its “intensity” pi(x) be expressed by its Gabor expansion
P(X) Iw(x-mX)12
exp( -inUx)
(20)
CC &,, Ig,(x-mX) mn
I2 exp(inUx) .
(23)
If the system can be described by the input-output relationships ( 11) or (22), then the Gabor coefficients of the output signal read A”,,,,,,,,
= h,,,, b’fn,.n,-“2 h*m*n26ml--m*.
(24)
Note that the output signal is no longer completely incoherent but partially coherent, due to the fact that, unlike the property of complete coherence, the property of complete incoherence is in general not pre17
Volume 86, number
I
OPTICS
COMMUNICATIONS
served when light propagates through an optical system. Substituting eq. (24) into eq. ( 13) and putting X, =x2=x, yields the output intensity T,(x, x): T,(x, x)=
11 b&, Igo(x-mX) ,n n
I2 exp(inUx)
I5 October
I99 1
elsewhere. Note the difference between the incoherent case and the deterministic (or completely coherent) case, for which eq. (11) yields the result in the interval (ImJ
(25a) with
References
b?nn = b:,,, C h,,,.n+rr hk~ .
(25b)
II’
We observe that the input “intensity” p,(x) and the output intensity rO(x, X) have similar Gabor expansions [see relations (23) and (25a) 1; the relationship between their Gabor coefficients b’ and 6” is given by eq. (25b). In the special case, again, that h,,,, equals unity in the interval ( I m 1< M, I n 1
18
[I ] [2] [3] [4] [ 51
M.J. Bastiaans, Optik 57 ( 1980) 95. M.J. Bastiaans, Optica Acta 29 (1982) 1223. D. Gabor, J. Inst. Electr. Eng., Part 3, 93 ( 1946) 429. M.J. Bastiaans, Opt. Eng. 20 (1981) 594. L.R. Rabiner and R.W. Schafer, Digital processing of speech signals (Prentice-Hall, Englewood Cliffs, 1978) Ch. 6. [ 61 A. Papoulis, Systems and transforms with applications in optics (McGraw-Hill, New York, 1968) Ch. 10. [ 71 M.J. Bastiaans, Optica Acta 24 ( 1977) 261. [8] L. Mandel and E. Wolf, J. Opt. Sot. Am. 66 (1976) 529.
86 ( 1991)
14-l 8
OPTICS COMMUNICATIONS
Gabor’s signal expansion applied to partially coherent light Martin
J. Bastiaans
Technische UniversifeitEindhoven, FaculteitElektrotechniek,Postbus513, 5600 MB Eindhoven, Netherlands Received
4 April 199 1; revised manuscript
received
5 June 199 1
Gabor’s signal expansion is reviewed by applying it first to a coherent optical (or deterministic) signal; it is shown that such an optical signal can be expressed as a superposition of optical rays that appear at discrete positions and have discrete directions. The expansion is then applied to partially coherent light, yielding expansion coefftcients that express the correlations that exist between the different optical rays. It is shown how these expansion coefftcients are related to the mutual power spectrum of the partially coherent light and how they are propagated through linear systems. The special case of incoherent light is considered in more detail
1. Introduction In some recent papers [ 1,2], Gabor’s signal expansion [ 31 was applied to optical signals and systems. It was shown that Gabor’s signal expansion is strongly related to the ray concept in geometrical optics: an optical signal is expressed as a superposition of rays that appear at certain discrete positions and have certain discrete directions. Until now, however, Gabor’s signal expansion was applied only to completely coherent light; it is the aim of this paper to show how the expansion can be applied to partially coherent light. In section 2 we briefly review the concept of Gabor’s signal expansion, we show how the expansion coefficients can be determined and how they propagate through linear systems. The main work is done in section 3, where Gabor’s signal expansion is applied to partially coherent light; the case of completely incoherent light will get some special attention there. We conclude this introduction with some remarks about notation. _ For the sake of convenience, we shall use onedimensional space functions, only; the extension to two (or more) dimensions is straightforward. _ An asterisk is used to denote complex conjugation. - The spatial Fourier transform of a space func14
0030-4018/91/$
tion is denoted by the .same symbol as the function itself, but marked by a bar on top of the symbol; hence, p(u)=
s
v(x)exp(-iux)dx.
- Unless otherwise stated, all integrations and summations in this paper extend from -CC to + co. - The symbol 6, denotes the Kronecker delta: S,= 1, and &=O for k# 0. We have chosen the notation with only one subscript (which, by the way, is common usage in discrete-time signal processing, for instance) to stress the equivalence between the Kronecker delta 6, (which appears in a discrete treatment) and the Dirac delta function 6(x) (which appears in a continuous treatment ). - The function rect( t) denotes the rectangular function: rect(t)=l for -f
2. Gabor’s signal expansion In 1946 Gabor [ 31 suggested a signal description that is intermediate between the pure space description and the pure spatial-frequency description. His idea was to express a space signal q(x), say, as a superposition of properly shifted and modulated versions of an elementary signal g(x), say; hence 03.50 0 1991 Elsevier Science Publishers
B.V. All rights reserved.
cc &ml g(x-mX) WI”
P(X) =
exp(inUx)
,
(1)
where the space shift Xand the spatial-frequency shift U satisfy the relation UX=2a. Although Gabor restricted himself to an elementary signal that has a gaussian shape, his signal expansion ( 1) holds for rather arbitrarily shaped elementary signals. Gabor’s expansion coefficients a,,,,, can be determined easily, even in the case that the discrete set of elementary signals shifted and modulated g( x- mx) exp (inUx) is not orthogonal. We simply define a window function w(x) that is bi-orthonormal to the set of elementary signals in the sense
J
g(x) w*(x-mX)
exp( -inUx)
;
dx=&&
(2)
a way to derive such a window function has been described elsewhere [ 1,2,4]. With the help of the biorthonormal window function w(x), the expansion coefficients amn follow readily through the formula
amn=
s
15October I99 1
OPTICS COMMUNICATIONS
Volume 86. number 1
q(x) w*(x-mX)
exp( -inUx)
dx.
(3)
The integral that appears in the right-hand side of eq. (3) defines, in fact, the short-term (or windowed) Fourier transform [ 5 ] of the signal Q(x ) , and we conclude that Gabor’s expansion coefficients can be found by sampling [4] the windowed Fourier transform at the regular lattice (mX, n U). The reason why we called the function w(x) that is bi-orthonormal to the elementary signal g(x) a window function, may now be evident. Gabor’s expansion of a signal into a discrete set of properly shifted and modulated versions of an elementary signal has a conceptually clear meaning. For time signals it resembles the musical score in music, and any shifted and modulated elementary signal describes a musical note that appears at a certain moment with a certain pitch. For space signals it resembles the ray concept in geometrical optics, and the shifted and modulated elementary signal g(x- mX) exp( inUx) can be identified as an optical ray at the position x= mX with a direction (i.e., spatial frequency ) u = n V. Gabor’s signal expansion thus represents an optical signal as a superposition of rays that appear at discrete positions mX and have discrete directions nlJ. It is not difficult to derive how Gabor’s expansion
coefficients propagate through a linear system [ 21. Let us, for instance, represent such a system by means of the superposition integral
@o(u)=J h(u,X) Pi(X) cl-~>
(4)
in which the spatial Fourier transform of the output signal &(u) is expressed in terms of the input signal v,(x); the system is thus described by the kernel h (u, x). We now choose input and output elementary signals gi(X) and g,(x), and represent the input and the output signal by means of their Gabor expansions; note that the input and the output elementary signal need not be identical. We can then derive a relationship between the input and the output expansion coefficients a;,, and a& reading
(5) where the system coefficients c&,, are COIIIpletely determined by the system and the choice of the input and output elementary signals through the relationship 1 -- 211
Cklmn-
JJ
h(u,x)
w~(u--IU)gi(x-WZX)
xexp[i(kuX+nUx)]
tidu.
(6)
An interesting example of a linear system is the basic coherent-optical system consisting of a 4f-arrangement with rectangular apertures in the input plane and the Fourier plane [ 2 1. In the special case that the input-plane aperture truncates the signal to the space interval -fX
rect(u/U)
exp(iux)
.
(7)
The array of system coefficients I&,,,, (which we shall denote by ekl,,,n for this special case) is now given by --
eklmn -
1 2A
JJ
rect (x/X)
rect ( u/ U)
Xexp(iux)*z(u-IU)
g,(x-mX)
xexp[i(kuX+nUx)]
dxdu.
For a proper choice of the elementary ray ek/m,, can be strongly concentrated
(8) signals, the ar[ 21: the coef15
Volume 86, number
1
OPTICS COMMUNICATIONS
ficient eOOOO will be close to 1, whereas the other coefficients are close to 0. This is conceptually nice: roughly speaking, only the ray that enters the input plane at position x=0 having a direction u=O can pass both the input-plane aperture and the Fourierplane aperture; all other rays will be blocked. A more general system is the one whose kernel h ( U, x) takes the form h(u, x)=
CC h,,,rect(x/X-m) ,?I n
xrect(u/U-n)
exp(iux)
.
[ 81, be denoted by T(x,, x2, 0). The basic property of the power spectrum is that it is nonnegative definite hermitian [ 7,8]. Since in the present discussion the temporal-frequency dependence is of no importance, we shall - for the sake of convenience - omit the temporal-frequency variable w from the formulas. Extending now Gabor’s signal expansion from the one-dimensional case [for the signal p(x), cf. eqs. ( 1) and (3) ] to the two-dimensional case [for the power spectrum T(x,, x2) 1, we can construct a correlation matrix A by means of the definition
(9)
The array of system coefficients cklmn is now given by the four-dimensional discrete convolution
A m,n,m2n2 =
CM,,,,= L,LnL h the trated cients by the
* * * *eklmn .
T(x,,x2)
w*(x) -mlX)
w(x2-m2X)
(10)
case that the array ek[m,, is strongly concenaround the element e,,,,, the Gabor coeffiof the input and the output signal are related simple relation
a:,, = h,,,aL.
I5 October 199 1
Xexp[ -i(n,
Ux, -n2Ux2)]
dx, d-x2,
(12)
which is the two-dimensional analogue of eq. ( 3). We can then represent the power spectrum in the form
(11)
In the special case that h,, equals unity in the interval ( I m I Q M, I n I dN) and vanishes outside that interval, we have a?,,, ‘v am,, in the interval ( (m ( GM, (nl
Xexp[i(n,
Ux, +n2Ux2)]
,
which is the two-dimensional analogue of eq. ( 1). Clearly, the Gabor coefficient A mlnlmZn2express the correlation that exists between the two elementary signals g(x-m,X) exp(in,Ux) and g(x-m2X) Xexp(in2Ux). Properties of the array of Gabor coefficients A follow directly from the properties of the power spectrum r. The fact that the power spectrum is hermitian, T(x,, x2) =r*(x2, x1 ), reflects itself in the property A m,nlmZnZ--A*,,,,,,,,
3
whereas the nonnegative definiteness spectrum implies the property
(14a) of the power
3. Gabor’s signal expansion for partially coherent light Let partially coherent light be described as a temporally stationary stochastic process, and let its (mutual) power spectrum [ 6,7], or cross-spectral density 16
which holds for arbitrary coefficients a,,,,,. We conclude that the array of coefficients A m,n,m?.n2is nonnegative definite hermitian in the sense of eqs. ( 14). The cases of completely coherent light, completely
Volume 86, number
OPTICS
1
incoherent light, and spatially stationary light deserve special attention. For completely coherent light, the power spectrum factorizes in the form [ 7 ] nx,
>x2) =dx,
1 4*(x,)
and the array of Gabor coefficients rized form * A m,n,,n2n2=am,.,am,n,
takes the facto-
(16)
,
where a,,,,, are the Gabor coefficients of the deterministic function q(x) [cf. eqs. (1) and (3)]. It is easy to see that the formulas that arise in the case of completely coherent light are similar to the ones that arise for deterministic signals. Therefore, we shall not study the completely coherent case any further. For completely incoherent light, the power spectrum can be expressed in the form Qx, >x2) =.0(x,)
6(x, -x2)
,
(17)
where the “intensity” p(x) is real and nonnegative. (The constant X has been included to get an “intensity” function p that has the same dimension as the power spectrum ZY) The Gabor coefficients now take the form A m,n,m2n2
=x
P(X)
w*(x--m1X)
bor coefficients of the “intensity” mentary signal ]g(x) I*: p(x)=
(15)
>
I5 October 199 I
COMMUNICATIONS
,YC b,, ,n n
p(x),
with ele-
Ig(x-mX)]‘exp(inUx).
Spatially stationary light, also known as homogeneous light, for which the power spectrum T(x,, x2) depends only on the difference x, -x2 of the two spatial coordinates x, and x2, is dual to incoherent light, i.e., the space behaviour of incoherent light is similar to the spatial-frequency behaviour of spatially stationary light, and vice versa. Therefore, similar conclusions as the ones that have been drawn for incoherent light hold for spatially stationary light; we only have to interchange position with direction, x with U, space functions with their spatial Fourier transforms, and so on. It is not difficult to derive how the array of Gabor propagates through a linear coefficients A t~l~l??l*?l* system. If such a system propagates a deterministic signal according to eq. (5 ), then the relationship between the input and output Gabor coefficients A’IYllf7l~ZtlZ and A&,k2,2 reads
w(x-m2W
(21)
s
xexp[
-i(n,
-n2)Ux]
dx.
(18)
We observe that A m,n,m2n2 depends on the direction difference n, - n,, as can be expected from the Van Cittert-Zernike theorem. Furthermore, if we choose the elementary signal g(x) such that it vanishes outside the interval - 4X< x< f X, eq. ( 18 ) reduces to A mln,m*n*-b - wz,,“,-82 6m,-IT72> with
b,,n=x
(19a)
s
For the special system described relationship ( 11) we get A:,,,,,,,
Pi(x)=
dx=b*,,_, .
(19b)
We conclude that - there is no correlation between rays at different positions; - the correlation between rays at one position that have different directions, depends on the direction difference; - the correlation coefficients b,,, are just the Ga-
=h,,,,A’
by the input-output
m,n,m*n*h*mznz.
(22)
The case of completely incoherent light in the input plane deserves special attention. Let the input signal be incoherent, and let its “intensity” pi(x) be expressed by its Gabor expansion
P(X) Iw(x-mX)12
exp( -inUx)
(20)
CC &,, Ig,(x-mX) mn
I2 exp(inUx) .
(23)
If the system can be described by the input-output relationships ( 11) or (22), then the Gabor coefficients of the output signal read A”,,,,,,,,
= h,,,, b’fn,.n,-“2 h*m*n26ml--m*.
(24)
Note that the output signal is no longer completely incoherent but partially coherent, due to the fact that, unlike the property of complete coherence, the property of complete incoherence is in general not pre17
Volume 86, number
I
OPTICS
COMMUNICATIONS
served when light propagates through an optical system. Substituting eq. (24) into eq. ( 13) and putting X, =x2=x, yields the output intensity T,(x, x): T,(x, x)=
11 b&, Igo(x-mX) ,n n
I2 exp(inUx)
I5 October
I99 1
elsewhere. Note the difference between the incoherent case and the deterministic (or completely coherent) case, for which eq. (11) yields the result in the interval (ImJ
(25a) with
References
b?nn = b:,,, C h,,,.n+rr hk~ .
(25b)
II’
We observe that the input “intensity” p,(x) and the output intensity rO(x, X) have similar Gabor expansions [see relations (23) and (25a) 1; the relationship between their Gabor coefficients b’ and 6” is given by eq. (25b). In the special case, again, that h,,,, equals unity in the interval ( I m 1< M, I n 1
18
[I ] [2] [3] [4] [ 51
M.J. Bastiaans, Optik 57 ( 1980) 95. M.J. Bastiaans, Optica Acta 29 (1982) 1223. D. Gabor, J. Inst. Electr. Eng., Part 3, 93 ( 1946) 429. M.J. Bastiaans, Opt. Eng. 20 (1981) 594. L.R. Rabiner and R.W. Schafer, Digital processing of speech signals (Prentice-Hall, Englewood Cliffs, 1978) Ch. 6. [ 61 A. Papoulis, Systems and transforms with applications in optics (McGraw-Hill, New York, 1968) Ch. 10. [ 71 M.J. Bastiaans, Optica Acta 24 ( 1977) 261. [8] L. Mandel and E. Wolf, J. Opt. Sot. Am. 66 (1976) 529.