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The Wigner distribution function applied to optical signals and systems
Volume 25, number 1
OPTICS COMMUNICATIONS
April 1978
THE WIGNER DISTRIBUTION FUNCTION APPLIED TO OPTICAL SIGNALS AND SYSTEMS M.J. BASTIAANS Eindhoven University of Technology, Eindhoven, The Netherlands
Department
of Electrical Engineering,
Received 21 November 1977 Revised manuscript received 18 January 1978
In this paper the Wigner distribution function has been introduced for optical signals and systems. The Wigner distribution function of an optical signal appears to be in close resemblance to the ray concept in geometrical optics. This resemblance reaches even farther: although derived from Fourier optics, the Wigner distribution functions of some elementary optical systems can directly be interpreted in terms of geometrical optics.
1. Introduction Optical system theory has been well developed in terms of Fourier optics. Among many other papers on this subject, we would like to refer to Butterweck [ 11. On the other hand, geometrical optics provides a welldeveloped optical system theory, too. It is a pity that there is no direct relationship between these two disciplines. The aim of this paper is to present a link between Fourier optics and geometrical optics. Starting with a Fourier-optical description of optical signals and systems, we introduce the Wigner distribution function [2,3,4] of a signal. We then notice a close resemblance between this Wigner distribution function and the ray concept in geometrical optics. The Wigner distribution function is not restricted to deterministic signals, but applies to partially coherent light as well. In the latter case, the Wigner distribution function is similar to the generalized radiance introduced by Walther [5,6,7]. Dealing with an optical system, we can formulate a relationship between the input and output signals in terms of their Wigner distribution functions. In this relationship the system is represented by a doubly Wigner distribution function. For some elementary optical systems, the descriptions of which are known in Fourier-optical as well as geometrical-optical terms we shall determine the double Wigner distributions functions. Although derived from Fourier optics, these 26
double Wigner distribution functions can directly be interpreted in terms of geometrical optics.
2. Basic theory; the Wigner distribution function Optical system theory has been well developed in terms of Fourier optics [ 11. In these terms, an optical signal can be described in the space domain *l by its complex amplitude *2 cp(r>, and, equivalently, in the (spatial) frequency domain by the Fourier transform *3 of the complex amplitude, defined by q(w) =ssdr)
exp(-iw
l
r)dr .
*l We shall consider two-dimensional signals, whose space coordinates are denoted by vectors like r = (x, y). ** Of course, the complex amplitude is a function of the temporal frequency variable, too. For the sake of convenience, we shall omit this frequency variable in the formulae, since in the present discussion the temporal frequency dependence is of no importance. *3 We shall denote (spatial) frequency coordinates by vectors like w = (u, u), and mean by expressions like w * r the inner product ux + uy. Throughout this paper, the Fourier transform of a function is denoted by the same symbol as the function itself, but marked by a dash on top of the symbol. Furthermore, expressions like J . ..& represent JJ ...dxdy. and, if not stated otherwise, all integrations extend from -- to +-.
OPTICS COMMUNICATIONS
Volume 25, number 1
Furthermore, an optical system that transforms a signal vi in the input plane into a signal p. in the output plane, can be represented in the space domain by its impulse response or point spread function g(t, P), and, equivalently, in the frequency domain by the “double Fourier transform” of the impulse response, defined by g(w, o) = b2 ssg(r,
p) exp(-i(w
l
r - w * p)) dr cl p .
We can then express the equivalent input-output lations for a system in terms of the superposition tegrals CP,(~)= Jg(r,
P) (Pi(P) dP ,
(2) rein-
(34
cPo(W)= JFCW, 0) cPi(O) do .
(3b)
We remind that the impulse response g(r, p) is the response of the system in the space domain to the input signal Cpi(r>= 6(r - P), whereas its double Fourier transform g(w, o) is the response in the frequency domain to the input signal Cpi(W)= 6(w - o). We now introduce the Wigner distribution function of a signal, defined in the space domain by F(r, w) = s cp(r + r’/2) cp*(f- r ‘/2) exp(-iw or, equivalently,
in the frequency
* r’) df’ ,
(4a)
April 1978
the pure frequency representation q(w). Furthermore, this simultaneous space-frequency description is in close resemblance to the ray concept in geometrical optics, where the position and direction of a ray are given simultaneously, too. We remark that the Wigner distribution function should not be confused with the ambiguity function [8,9], which is an intermediate signal description, as well. As an example we shall determine the Wigner distribution function of the “Gabor function” [4, lo] q(r)
=--&Aexp
+iw, * (rr,i2))
,
(6)
which function describes a ray, located at the space coordinate (position) ro, having a frequency (direction) wo. The Wigner distribution function of the Gabor function reads as
7
(7)
and, indeed, is concentrated at the space-frequency point (r,, wo). Dealing with systems, it is possible to express the Wigner distribution function F,(r, w) in the output plane in terms of the Wigner distribution function Fi(r, w) in the input plane, by combining relations (3) and (4). We thus find the input-output relation for a system in terms of the superposition integral [2]
domain by
(8)
F(r, w) =- 1 Aw + w’/2) Cp*(w- w’/2) exp(iw’ 46 s
l
r) dw’ .
where the function K(r, w, p, o) is completely determined by the system, according to the definition in the space domain
(4b) properties of the Wigner distribution function can be found elsewhere [2,3,4] ; we only mention its realness, and the relations l&)12
= b2
SF@,
w) dw ,
(54
=&J-j&t X exp(-i(w
r1/2, P + P’/2)g*(r-t’/2, * r’ - co * p’)) dr’dp’ ,
or the equivalent Iaw)12 = s F(r, w) dr .
W)
The Wigner distribution function F(r, w) represents the signal in the space and frequency domains simultaneously. It thus forms an intermediate signal description between the pure space representation cp(r> and
P - P’/2)
definition
in the frequency
@a> domain
K(r, w, P, 0) = +!JF(w+w’/2, 4lr2
o+ 0’/2)g*(w-w’/2,0-0’/2)
X exp(i(w’ * r - co’ p)) dw’do’ l
.
WI
27
Volume 25, number
1
OPTICS COMMUNICATIONS
We remark that the function K(r, w, p, o) is the response of the system in the space-frequency domain to the input signal Fi(r, w) = 6(r - p)6(w - CO).This input signal can be considered as a single ray, entering the system at the space coordinate r = p, and having a frequency w = o . Hence, we might call the function K(r, w, p, CO)the “ray response” or “ray spread function” of the system. We point out that there does not exist a physical signal cp,yielding the Wigner distribution function 6(r - p)6(w - co). With respect to this, it is a rather artificial Wigner distribution function. If we want to connect with it a physical signal, the function 6(r - p)6(w - co) must be considered as a symbolic representation of a function concentrated at the space-frequency point (p, a), for instance, the Wigner distribution function (7) of the Gabor function (6). Indeed, if we insert this Wigner distribution function in the superposition integral (8), the response equals K(r, w, ro, w,), under the condition that K(r, w, p,o) is sufficiently smooth with respect to p and CO.In fact, this symbolic meaning of the delta function already applied to the interpretation of 6(r - p) and 6(w - o) in the space and frequency domains, respectively. Relations (9) can be considered as the definitions of a “double Wigner distribution function”, which satisfies the property of realness. Physical constraints that can be imposed upon a system, can be expressed in terms of its double Wigner distribution function. Losslessness of a system [ 11, for instance, can thus be expressed as
. K(r, w, p, o) .I7
April 1978
ner distribution functions for some elementary systems [ 11, i.e. a spreadless system (especially a lens), a shift-invariant system (especially free space), a Fourier transformer and a magnifier. Furthermore, we determine the double Wigner distribution function of Bryngdahl’s coordinate transformer [ 1 I]. 3.1. Spreadless system: lens Let the spreadless system be represented q,(r) = McPi(r)
.
(12)
Then the double Wigner distribution K(r, w,p,o)
(10)
We finally mention the cascading of two systems that are represented by their double Wigner distribution functions KI (r, w, p, o) and K2(r, w, p, CO),respectively, It can readily be derived that the double Wigner distribution function of the overall system reads as
function
w - w) 6(r -0))
= (1/4n2)Fm(r,
reads as (13)
where F, (r, w) is defined by (4a) with m replacing cp. For a spreadless system, relation (8) reduces to F,(r, w) =
4$sFm(r,
w
- 0) Fi(r, 0) do
.
(14)
We note that relation (14) shows a convolution for the frequency variable w and a mere multiplication for the space variable r, as can be expected for a spreadless system. In the special case of a lens with a focal distance f, where m(r)=exp(-i$lr12),
(15)
the double Wigner distribution Kf(r,w,p,o)=6(r-p)6
drdw = 1 .
by
(
function w+$r-63
reduces to . )
(16)
3.2. Shift-invariant system; free space The frequency behaviour of a shift-invariant system is similar to the space behaviour of a spreadless system [ 11, Let the shift-invariant system be represented by cPr~(~) = h(w) cPi(w) * Then the double Wigner .distribution
(17) function
reads as
K*,t(r, w>Pa 0) K~(r,w,p,o)=F~(r-p,w)~(w-o), =
JJ K2(r,w,
~0, oo)KI(~o>
00,
P,O> dpo do,
.
(11)
In this section we shall determine 28
where Fh(r, w) is defined by (4b) with h replacing T. For a shift-invariant system, relation (8) reduces to F,(r,
3. Examples the double Wig-
(18)
W) = SF&
- P, w) Fi(p, w)
dp .
We note that relation (19) shows a convolution
(19) for
the space frequency invariant a distance
variable and a mere multiplication for the variable as can be expected for a shiftsystem. In the special case of free space over z, where
i;(w)=exp(-i&
(20)
lwl’),
the double Wigner distribution K,(r,w,p,o)=6
function
r-j+
reduces to
6(w-w).
(
1
(21)
3.3. Fourier transformer For a Fourier transformer reads as g(r, p) = &
April 1978
OPTICS COMMUNICATIONS
Volume 25, number 1
exp(-iL+
l
whose impulse response
Kp(‘, w,p,o)=W-
4. Partially coherent light In Fourier-optical terms, partially coherent light [ 12,131 can be described in the space domain by its (mutual) power spectrum *4 S(rl , r2), and, equivalently, in the frequency domain by the double Fourier transform of the power spectrum, defined by F(wr, w2) =
p> ,
the double Wigner distribution
We note that for low-frequency input signals (o x 0), relation (27) directly represents the coordinate transformation /3f = u(p).
(22) function
reduces to
~)qa+P).
(23)
We note that the space and frequency domains are interchanged, as can be expected for a Fourier transformer.
J.T
s(rl ,r2) exp(-i(wl
(28) * ‘1 - w2 . r2)) dr, dr2 .
We can apply the Wigner distribution function to partially coherent light as well, by replacing in the definitions (4a) and (4b) the products cp(r1)~*(r2) and $(wl)$*(w2) by the power spectrum s(rl, r2) and its double Fourier transform T(wl, w2), respectively. We thus have the equivalent definitions F(r, w) = 1 s(r t r’/2, r - r’/2) exp(-iw
*r’) dr’ , (29a)
3.4. Magnifier For a magnifier whose impulse response reads as &, P> = twtr
- P> 3
(24)
F(r, w) = 4$IK(wtw’/2,
w-w’/2)exp(iw’*r)dw’. (29b)
We note that the space and frequency domains are merely resealed, as can be expected for a magnifier.
These definition are similar to the definitions of the generalized radiance, as introduced by Walther [5,6,7]. As examples, we shall determine the Wigner distribution functions for incoherent light, spatially stationary light and quasihomogeneous light [ 141, respectively.
3.5. Coordinate transformer
4.1. Incoherent light
the double Wigner distribution K&w,
p,w) = S(tr-
reduces to
p) 6(w/t - Co).
For a coordinate transformer sponse reads as [ 111
(25)
whose impulse re-
Let incoherent
light be represented
S(f + i/2, r - r’/2) = p(r) 6(r’) . &, P) = 2$ exp(i+(P)
- ior * P> ,
the double Wigner distribution K,(r,w,p,o)=6(wiP+P)6(pr-u(p)-w), where a(x, r) = (6 J//&x, 6 $/Sy).
by (30)
(26) *4 As is the case with the complex amplitude of a determinis-
reduces to (27)
tic signal, the power spectrum of a stochastic signal is a function of a temporal frequency variable, too. For the sake of convenience we shall again omit this frequency variable in the formulae.
29
OPTICS COMMUNICATIONS
Volume 25. number 1
Then its Wigner distribution
function
reads as
April 1978
tively. Denoting these relationships
F(r, w) = P(r> *
(31)
by
[:]=M[:] 3
4.2. Spatially stationary ligh t
these systems are described by the matrices
The frequency behaviour of spatially stationary light is similar to the space behaviour of incoherent light, as can be concluded from the Van CittertZernike theorem. Let spatially stationary light be represented by
Mf=
F(w + w’/2, w - w’/2) = 4&4(w)s(w’) Then its Wigner distribution
function
.
Mo=[;;l”] , M,=[;$1, (32)
reads as
F(r, w) = q(w) .
(33)
4.3. Quasihomogeneous
light
Quasihomogeneous light [ 141 can locally be considered as spatially stationary, having, however, a slowly varying intensity. Let quasihomogeneous light be represented by
H.J. Butterweck, J. Opt. Sot. Am. 67 (1977) 60.
121 H. Mori, I. Oppenheim and J. Ross, Studies in statistical function
Wb) reads as (35)
5. Conclusion We conclude this paper with a few remarks. We already noted the close resemblance between the Wigner distribution function and the ray concept in geometrical optics. This resemblance reaches even farther. As a matter of fact, the relations (16), (2 l), (23) and (25) directly lead to the matrices that are commonly used in geometrical optics to describe the relationships between the position and directions of the incoming and outgoing rays for the systems: lens, free space, Fourier transformer and magnifier, respec-
30
respectively. Cascading such systems, we can find the overall properties, apart from using relation (1 l), by a simple matrix multiplication. We feel that the Wigner distribution function may be a powerful tool for an elegant optical system theory It provides a nice description of optical systems, and it is not dependent upon the state of coherence of optical signals: it applies to deterministic, completely coherent, completely incoherent and partially coherent light, as well.
[II
F(w + w’/2, w - w’/2) = ii(w) B(w’> .
F(r, w) = PW q(w) *
W’c,d)
(344
by
Then its Wigner distribution
Wa,b)
References
s(r + r’/2, r - r’/2) = p(r) q(r’) , or, equivalently,
[L,fy], M,=[k yz’k],
(36)
mechanics, Vol. 1, eds. J. de Boer and GE. Uhlenbeck (North-Holland, Amsterdam, 1962) p. 213. [31 H. Bremmer, Radio Sci. 8 (1973) 511. 141 N.G. de Bruijn, Nieuw Archief VOOIWiskunde 2 1 (3) (1973) 205. I51 A. Walther, J. Opt. Sot. Am. 58 (1968) 1256. [61 A. Walther, J. Opt. Sot. Am. 63 (1973) 1622. [71 E.W. Marchand and E. Wolf, J. Opt. Sot. Am. 64 (1974) 1273. 181 A. Papoulis, J. Opt. Sot. Am. 64 (1974) 779. 191 A. Papoulis, Signal analysis (McGraw-Hill, New York, 1977) p. 284. [lOI D. Gabor, J. Inst. Elec. Engrs. (London) 93 (III) (1946) 429. [ 111 0. Bryngdahl, J. Opt. Sot. Am. 64 (1974) 1092. [ 121 M.J. Beran and G.B. Parrent, Theory of partial coherence (Prentice-Hall, Englewood Cliffs, New Jersey, 1964). [ 131 A. Papoulis, Systems and transforms with applications in optics (McGraw-Hill, New York, 1968). [14] W.H. Carter and E. Wolf, J. Opt. Sot. Am. 67 (1977) 785.
OPTICS COMMUNICATIONS
April 1978
THE WIGNER DISTRIBUTION FUNCTION APPLIED TO OPTICAL SIGNALS AND SYSTEMS M.J. BASTIAANS Eindhoven University of Technology, Eindhoven, The Netherlands
Department
of Electrical Engineering,
Received 21 November 1977 Revised manuscript received 18 January 1978
In this paper the Wigner distribution function has been introduced for optical signals and systems. The Wigner distribution function of an optical signal appears to be in close resemblance to the ray concept in geometrical optics. This resemblance reaches even farther: although derived from Fourier optics, the Wigner distribution functions of some elementary optical systems can directly be interpreted in terms of geometrical optics.
1. Introduction Optical system theory has been well developed in terms of Fourier optics. Among many other papers on this subject, we would like to refer to Butterweck [ 11. On the other hand, geometrical optics provides a welldeveloped optical system theory, too. It is a pity that there is no direct relationship between these two disciplines. The aim of this paper is to present a link between Fourier optics and geometrical optics. Starting with a Fourier-optical description of optical signals and systems, we introduce the Wigner distribution function [2,3,4] of a signal. We then notice a close resemblance between this Wigner distribution function and the ray concept in geometrical optics. The Wigner distribution function is not restricted to deterministic signals, but applies to partially coherent light as well. In the latter case, the Wigner distribution function is similar to the generalized radiance introduced by Walther [5,6,7]. Dealing with an optical system, we can formulate a relationship between the input and output signals in terms of their Wigner distribution functions. In this relationship the system is represented by a doubly Wigner distribution function. For some elementary optical systems, the descriptions of which are known in Fourier-optical as well as geometrical-optical terms we shall determine the double Wigner distributions functions. Although derived from Fourier optics, these 26
double Wigner distribution functions can directly be interpreted in terms of geometrical optics.
2. Basic theory; the Wigner distribution function Optical system theory has been well developed in terms of Fourier optics [ 11. In these terms, an optical signal can be described in the space domain *l by its complex amplitude *2 cp(r>, and, equivalently, in the (spatial) frequency domain by the Fourier transform *3 of the complex amplitude, defined by q(w) =ssdr)
exp(-iw
l
r)dr .
*l We shall consider two-dimensional signals, whose space coordinates are denoted by vectors like r = (x, y). ** Of course, the complex amplitude is a function of the temporal frequency variable, too. For the sake of convenience, we shall omit this frequency variable in the formulae, since in the present discussion the temporal frequency dependence is of no importance. *3 We shall denote (spatial) frequency coordinates by vectors like w = (u, u), and mean by expressions like w * r the inner product ux + uy. Throughout this paper, the Fourier transform of a function is denoted by the same symbol as the function itself, but marked by a dash on top of the symbol. Furthermore, expressions like J . ..& represent JJ ...dxdy. and, if not stated otherwise, all integrations extend from -- to +-.
OPTICS COMMUNICATIONS
Volume 25, number 1
Furthermore, an optical system that transforms a signal vi in the input plane into a signal p. in the output plane, can be represented in the space domain by its impulse response or point spread function g(t, P), and, equivalently, in the frequency domain by the “double Fourier transform” of the impulse response, defined by g(w, o) = b2 ssg(r,
p) exp(-i(w
l
r - w * p)) dr cl p .
We can then express the equivalent input-output lations for a system in terms of the superposition tegrals CP,(~)= Jg(r,
P) (Pi(P) dP ,
(2) rein-
(34
cPo(W)= JFCW, 0) cPi(O) do .
(3b)
We remind that the impulse response g(r, p) is the response of the system in the space domain to the input signal Cpi(r>= 6(r - P), whereas its double Fourier transform g(w, o) is the response in the frequency domain to the input signal Cpi(W)= 6(w - o). We now introduce the Wigner distribution function of a signal, defined in the space domain by F(r, w) = s cp(r + r’/2) cp*(f- r ‘/2) exp(-iw or, equivalently,
in the frequency
* r’) df’ ,
(4a)
April 1978
the pure frequency representation q(w). Furthermore, this simultaneous space-frequency description is in close resemblance to the ray concept in geometrical optics, where the position and direction of a ray are given simultaneously, too. We remark that the Wigner distribution function should not be confused with the ambiguity function [8,9], which is an intermediate signal description, as well. As an example we shall determine the Wigner distribution function of the “Gabor function” [4, lo] q(r)
=--&Aexp
+iw, * (rr,i2))
,
(6)
which function describes a ray, located at the space coordinate (position) ro, having a frequency (direction) wo. The Wigner distribution function of the Gabor function reads as
7
(7)
and, indeed, is concentrated at the space-frequency point (r,, wo). Dealing with systems, it is possible to express the Wigner distribution function F,(r, w) in the output plane in terms of the Wigner distribution function Fi(r, w) in the input plane, by combining relations (3) and (4). We thus find the input-output relation for a system in terms of the superposition integral [2]
domain by
(8)
F(r, w) =- 1 Aw + w’/2) Cp*(w- w’/2) exp(iw’ 46 s
l
r) dw’ .
where the function K(r, w, p, o) is completely determined by the system, according to the definition in the space domain
(4b) properties of the Wigner distribution function can be found elsewhere [2,3,4] ; we only mention its realness, and the relations l&)12
= b2
SF@,
w) dw ,
(54
=&J-j&t X exp(-i(w
r1/2, P + P’/2)g*(r-t’/2, * r’ - co * p’)) dr’dp’ ,
or the equivalent Iaw)12 = s F(r, w) dr .
W)
The Wigner distribution function F(r, w) represents the signal in the space and frequency domains simultaneously. It thus forms an intermediate signal description between the pure space representation cp(r> and
P - P’/2)
definition
in the frequency
@a> domain
K(r, w, P, 0) = +!JF(w+w’/2, 4lr2
o+ 0’/2)g*(w-w’/2,0-0’/2)
X exp(i(w’ * r - co’ p)) dw’do’ l
.
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27
Volume 25, number
1
OPTICS COMMUNICATIONS
We remark that the function K(r, w, p, o) is the response of the system in the space-frequency domain to the input signal Fi(r, w) = 6(r - p)6(w - CO).This input signal can be considered as a single ray, entering the system at the space coordinate r = p, and having a frequency w = o . Hence, we might call the function K(r, w, p, CO)the “ray response” or “ray spread function” of the system. We point out that there does not exist a physical signal cp,yielding the Wigner distribution function 6(r - p)6(w - co). With respect to this, it is a rather artificial Wigner distribution function. If we want to connect with it a physical signal, the function 6(r - p)6(w - co) must be considered as a symbolic representation of a function concentrated at the space-frequency point (p, a), for instance, the Wigner distribution function (7) of the Gabor function (6). Indeed, if we insert this Wigner distribution function in the superposition integral (8), the response equals K(r, w, ro, w,), under the condition that K(r, w, p,o) is sufficiently smooth with respect to p and CO.In fact, this symbolic meaning of the delta function already applied to the interpretation of 6(r - p) and 6(w - o) in the space and frequency domains, respectively. Relations (9) can be considered as the definitions of a “double Wigner distribution function”, which satisfies the property of realness. Physical constraints that can be imposed upon a system, can be expressed in terms of its double Wigner distribution function. Losslessness of a system [ 11, for instance, can thus be expressed as
. K(r, w, p, o) .I7
April 1978
ner distribution functions for some elementary systems [ 11, i.e. a spreadless system (especially a lens), a shift-invariant system (especially free space), a Fourier transformer and a magnifier. Furthermore, we determine the double Wigner distribution function of Bryngdahl’s coordinate transformer [ 1 I]. 3.1. Spreadless system: lens Let the spreadless system be represented q,(r) = McPi(r)
.
(12)
Then the double Wigner distribution K(r, w,p,o)
(10)
We finally mention the cascading of two systems that are represented by their double Wigner distribution functions KI (r, w, p, o) and K2(r, w, p, CO),respectively, It can readily be derived that the double Wigner distribution function of the overall system reads as
function
w - w) 6(r -0))
= (1/4n2)Fm(r,
reads as (13)
where F, (r, w) is defined by (4a) with m replacing cp. For a spreadless system, relation (8) reduces to F,(r, w) =
4$sFm(r,
w
- 0) Fi(r, 0) do
.
(14)
We note that relation (14) shows a convolution for the frequency variable w and a mere multiplication for the space variable r, as can be expected for a spreadless system. In the special case of a lens with a focal distance f, where m(r)=exp(-i$lr12),
(15)
the double Wigner distribution Kf(r,w,p,o)=6(r-p)6
drdw = 1 .
by
(
function w+$r-63
reduces to . )
(16)
3.2. Shift-invariant system; free space The frequency behaviour of a shift-invariant system is similar to the space behaviour of a spreadless system [ 11, Let the shift-invariant system be represented by cPr~(~) = h(w) cPi(w) * Then the double Wigner .distribution
(17) function
reads as
K*,t(r, w>Pa 0) K~(r,w,p,o)=F~(r-p,w)~(w-o), =
JJ K2(r,w,
~0, oo)KI(~o>
00,
P,O> dpo do,
.
(11)
In this section we shall determine 28
where Fh(r, w) is defined by (4b) with h replacing T. For a shift-invariant system, relation (8) reduces to F,(r,
3. Examples the double Wig-
(18)
W) = SF&
- P, w) Fi(p, w)
dp .
We note that relation (19) shows a convolution
(19) for
the space frequency invariant a distance
variable and a mere multiplication for the variable as can be expected for a shiftsystem. In the special case of free space over z, where
i;(w)=exp(-i&
(20)
lwl’),
the double Wigner distribution K,(r,w,p,o)=6
function
r-j+
reduces to
6(w-w).
(
1
(21)
3.3. Fourier transformer For a Fourier transformer reads as g(r, p) = &
April 1978
OPTICS COMMUNICATIONS
Volume 25, number 1
exp(-iL+
l
whose impulse response
Kp(‘, w,p,o)=W-
4. Partially coherent light In Fourier-optical terms, partially coherent light [ 12,131 can be described in the space domain by its (mutual) power spectrum *4 S(rl , r2), and, equivalently, in the frequency domain by the double Fourier transform of the power spectrum, defined by F(wr, w2) =
p> ,
the double Wigner distribution
We note that for low-frequency input signals (o x 0), relation (27) directly represents the coordinate transformation /3f = u(p).
(22) function
reduces to
~)qa+P).
(23)
We note that the space and frequency domains are interchanged, as can be expected for a Fourier transformer.
J.T
s(rl ,r2) exp(-i(wl
(28) * ‘1 - w2 . r2)) dr, dr2 .
We can apply the Wigner distribution function to partially coherent light as well, by replacing in the definitions (4a) and (4b) the products cp(r1)~*(r2) and $(wl)$*(w2) by the power spectrum s(rl, r2) and its double Fourier transform T(wl, w2), respectively. We thus have the equivalent definitions F(r, w) = 1 s(r t r’/2, r - r’/2) exp(-iw
*r’) dr’ , (29a)
3.4. Magnifier For a magnifier whose impulse response reads as &, P> = twtr
- P> 3
(24)
F(r, w) = 4$IK(wtw’/2,
w-w’/2)exp(iw’*r)dw’. (29b)
We note that the space and frequency domains are merely resealed, as can be expected for a magnifier.
These definition are similar to the definitions of the generalized radiance, as introduced by Walther [5,6,7]. As examples, we shall determine the Wigner distribution functions for incoherent light, spatially stationary light and quasihomogeneous light [ 141, respectively.
3.5. Coordinate transformer
4.1. Incoherent light
the double Wigner distribution K&w,
p,w) = S(tr-
reduces to
p) 6(w/t - Co).
For a coordinate transformer sponse reads as [ 111
(25)
whose impulse re-
Let incoherent
light be represented
S(f + i/2, r - r’/2) = p(r) 6(r’) . &, P) = 2$ exp(i+(P)
- ior * P> ,
the double Wigner distribution K,(r,w,p,o)=6(wiP+P)6(pr-u(p)-w), where a(x, r) = (6 J//&x, 6 $/Sy).
by (30)
(26) *4 As is the case with the complex amplitude of a determinis-
reduces to (27)
tic signal, the power spectrum of a stochastic signal is a function of a temporal frequency variable, too. For the sake of convenience we shall again omit this frequency variable in the formulae.
29
OPTICS COMMUNICATIONS
Volume 25. number 1
Then its Wigner distribution
function
reads as
April 1978
tively. Denoting these relationships
F(r, w) = P(r> *
(31)
by
[:]=M[:] 3
4.2. Spatially stationary ligh t
these systems are described by the matrices
The frequency behaviour of spatially stationary light is similar to the space behaviour of incoherent light, as can be concluded from the Van CittertZernike theorem. Let spatially stationary light be represented by
Mf=
F(w + w’/2, w - w’/2) = 4&4(w)s(w’) Then its Wigner distribution
function
.
Mo=[;;l”] , M,=[;$1, (32)
reads as
F(r, w) = q(w) .
(33)
4.3. Quasihomogeneous
light
Quasihomogeneous light [ 141 can locally be considered as spatially stationary, having, however, a slowly varying intensity. Let quasihomogeneous light be represented by
H.J. Butterweck, J. Opt. Sot. Am. 67 (1977) 60.
121 H. Mori, I. Oppenheim and J. Ross, Studies in statistical function
Wb) reads as (35)
5. Conclusion We conclude this paper with a few remarks. We already noted the close resemblance between the Wigner distribution function and the ray concept in geometrical optics. This resemblance reaches even farther. As a matter of fact, the relations (16), (2 l), (23) and (25) directly lead to the matrices that are commonly used in geometrical optics to describe the relationships between the position and directions of the incoming and outgoing rays for the systems: lens, free space, Fourier transformer and magnifier, respec-
30
respectively. Cascading such systems, we can find the overall properties, apart from using relation (1 l), by a simple matrix multiplication. We feel that the Wigner distribution function may be a powerful tool for an elegant optical system theory It provides a nice description of optical systems, and it is not dependent upon the state of coherence of optical signals: it applies to deterministic, completely coherent, completely incoherent and partially coherent light, as well.
[II
F(w + w’/2, w - w’/2) = ii(w) B(w’> .
F(r, w) = PW q(w) *
W’c,d)
(344
by
Then its Wigner distribution
Wa,b)
References
s(r + r’/2, r - r’/2) = p(r) q(r’) , or, equivalently,
[L,fy], M,=[k yz’k],
(36)
mechanics, Vol. 1, eds. J. de Boer and GE. Uhlenbeck (North-Holland, Amsterdam, 1962) p. 213. [31 H. Bremmer, Radio Sci. 8 (1973) 511. 141 N.G. de Bruijn, Nieuw Archief VOOIWiskunde 2 1 (3) (1973) 205. I51 A. Walther, J. Opt. Sot. Am. 58 (1968) 1256. [61 A. Walther, J. Opt. Sot. Am. 63 (1973) 1622. [71 E.W. Marchand and E. Wolf, J. Opt. Sot. Am. 64 (1974) 1273. 181 A. Papoulis, J. Opt. Sot. Am. 64 (1974) 779. 191 A. Papoulis, Signal analysis (McGraw-Hill, New York, 1977) p. 284. [lOI D. Gabor, J. Inst. Elec. Engrs. (London) 93 (III) (1946) 429. [ 111 0. Bryngdahl, J. Opt. Sot. Am. 64 (1974) 1092. [ 121 M.J. Beran and G.B. Parrent, Theory of partial coherence (Prentice-Hall, Englewood Cliffs, New Jersey, 1964). [ 131 A. Papoulis, Systems and transforms with applications in optics (McGraw-Hill, New York, 1968). [14] W.H. Carter and E. Wolf, J. Opt. Sot. Am. 67 (1977) 785.