EURASIP Journal on Applied Signal Processing 2005:10, 1535–1540 c ○ 2005 M. J. Bastiaans and T. Alieva Wigner Distribution Moments Measured as Intensity Moments in Separable First-Order Optical Systems

It is shown how all global Wigner distribution moments of arbitrary order can be measured as intensity moments in the output plane of an appropriate number of separable first-order optical systems (generally anamorphic ones). The minimum number of such systems that are needed for the determination of these moments is derived.


INTRODUCTION
After the introduction of the Wigner distribution (WD) [1] for the description of coherent and partially coherent optical fields [2], it became an important tool for optical signal/image analysis and beam characterization [3,4,5]. The WD completely describes the complex amplitude of a coherent optical field (up to a constant phase factor) or the mutual coherence function of a partially coherent field. As the WD of a two-dimensional optical field is a function of four variables, it is difficult to analyze. Therefore, the optical field is often represented not by the WD itself, but by its global moments. Beam characterization based on the second-order moments of the WD thus became the basis of an International Organization for Standardization standard [6].
Some of the WD moments can directly be determined from measurements of the intensity distributions in the image plane or the Fourier plane, but most of the moments cannot be determined in such an easy way. In order to calculate such moments, additional information is required. Since first-order optical systems [7]-also called ABCD systemsproduce affine transformations of the WD in phase space, the intensity distributions measured at the output of such systems can provide such additional information. The application of ABCD systems for the measurements of the second-This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. order WD moments has been reported in several publications [8,9,10,11,12,13].
It is the aim of this paper to show how all global WD moments can be measured as intensity moments only. We show that not only the second-order moments, but also all other moments of the four-dimensional WD can be obtained from measurements of only intensity distributions in an appropriate number of (generally anamorphic) separable first-order optical systems.

WIGNER DISTRIBUTION
Let partially coherent light be described by a temporally stationary stochastic process f (x, y; t); as far as the time dependence is concerned, the ensemble average of the product f (x 1 , y 1 ; t 1 ) f * (x 2 , y 2 ; t 2 ), where the asterisk denotes complex conjugation, is then only a function of the time difference t 1 − t 2 : The function γ(x 1 , x 2 ; y 1 , y 2 ; τ) is known as the mutual coherence function [14,15,16,17] of the stochastic process f (x, y; t). The mutual power spectrum [16,17] or crossspectral density function [18] Γ(x 1 , x 2 ; y 1 , y 2 ; ω) is defined as the temporal Fourier transform of the mutual coherence function: For x 1 = x 2 = x, y 1 = y 2 = y, the cross-spectral density function reduces to the (auto) power spectrum Γ(x, x; y, y; ω), which represents the intensity distribution of the light for the temporal frequency ω. Since in the present discussion the explicit temporal-frequency dependence is of no importance, we will, for the sake of convenience, omit the temporal-frequency variable ω from the formulas in the remainder of the paper. The Wigner distribution of partially coherent light is defined in terms of the cross-spectral density function by A distribution function according to definition (3) was first introduced in optics by Walther [19,20], who called it the generalized radiance. The WD W(x, u; y, v) represents partially coherent light in a combined space/spatial-frequency domain, the so-called phase space, where u is the spatialfrequency variable associated to the space variable x, and v the spatial-frequency variable associated to the space variable y.
In this paper we consider the normalized moments of the WD, where the normalization is with respect to the total energy E of the signal: These normalized moments µ pqrs of the WD are thus defined by Note that for q = s = 0 we have intensity moments, which can easily be measured: The WD moments µ pqrs provide valuable tools for the characterization of optical beams, see, for instance [21]. First-order moments yield the position of the beam (µ 1000 and µ 0010 ) and its direction (µ 0100 and µ 0001 ). Second-order moments give information about the spatial width of the beam (the shape µ 2000 and µ 0020 of the spatial ellipse and its orientation µ 1010 ) and the angular width in which the beam is radiating (the shape µ 0200 and µ 0002 of the spatial-frequency ellipse and its orientation µ 0101 ); moreover, they provide information about its curvature (µ 1100 and µ 0011 ) and its twist (µ 1001 and µ 0110 ). Many important beam characterizers, like the overall beam quality [12] µ 2000 µ 0200 − µ 2 1100 + µ 0020 µ 0002 − µ 2 are based on second-order moments. Higher-order moments are used, for instance, to characterize the beam's symmetry and its sharpness [21].

SEPARABLE FIRST-ORDER OPTICAL SYSTEMS
It is well known that the input-output relationship between the WD W in (x, u; y, v) at the input plane and the WD W out (x, u; y, v) at the output plane of a separable first-order optical system reads [3,4,5] W out (x, u; y, v) The coefficients a x , b x , c x , d x and a y , b y , c y , d y are the matrix entries of the symplectic ray transformation matrix [7] that relates the position x, y and direction u, v of an optical ray in the input and the output plane of the first-order optical system: For separable systems, symplecticity simply reads a x d x − b x c x = 1 and a y d y − b y c y = 1. Note that in a first-order optical system, with such a symplectic ray transformation matrix, the total energy E, see (4), is invariant.
As examples of first-order optical systems we mention the following in particular: (i) a section of free space in the paraxial approximation, or "parabolic" system [22] (with a = d = 1, c = 0, and b proportional to the propagation distance z), (ii) a fractional Fourier transform system [23], or "elliptic" system [22] (with a = d = cos α and b = −c = sin α), (iii) a "hyperbolic" system [22] (with a = d = cosh α and b = c = sinh α).
These three systems are characterized by one parameter.
Other one-parameter first-order optical systems are The latter systems however-like all systems for which the input and output planes are conjugate planes-cannot be used to determine the moments, as we will see later, because they have the property b ≡ 0.
The normalized moments µ out pqrs of the output WD W out (x, u; y, v) are related to the normalized moments and for the intensity moments in particular (i.e., q = s = 0) we have The remainder of this paper is based on (11), in which the output intensity moments µ out p0r0 are expressed in terms of the input moments µ pqrs and the system parameters a x , a y , b x , and b y . Note that only the parameters a and b enter this equation; the parameters c and d can be chosen freely, as long as the symplecticity condition a x d x − b x c x = a y d y − b y c y = 1 is satisfied.

First-order moments
For the first-order moments, the following two equations are relevant: µ out 0010 = a y µ 0010 + b y µ 0001 , which correspond to (11) with pqrs = 1000 and pqrs = 0010, respectively, and the four input moments µ 1000 , µ 0100 , µ 0010 , and µ 0001 can be determined by measuring the intensity moments µ out 1000 and µ out 0010 in the output planes of two systems with different values of a and b, see (12) and (13), respectively.
In the case of a fractional Fourier transform system we can choose, for instance, [24,25], the fractional angles α x = α y = 0 (leading to a x = a y = 1 and b x = b y = 0) and α x = α y = π/2 (leading to a x = a y = 0 and b x = b y = 1), but any other choice could be made as well, as long as it leads to four independent equations. In the case of free space propagation, we simply choose two different values of the propagation distance z, corresponding to two different values of b x and b y (with a x = a y = 1, of course).
Note that the two first-order optical systems can always be chosen such that they are isotropic, a x = a y = a i , b x = b y = b i , and so forth (i = 1, 2), with identical behavior in the x and the y direction.

Second-order moments
For the 3 + 4 + 3 = 10 second-order moments, the following equations are relevant: µ out 1010 = a x a y µ 1010 + a x b y µ 1001 + b x a y µ 0110 + b x b y µ 0101 , (15) µ out 0020 = a 2 y µ 0020 + 2a y b y µ 0011 + b 2 y µ 0002 , which equations correspond to (11) with pqrs = 2000, pqrs = 1010, and pqrs = 0020, respectively. The three input moments µ 2000 , µ 1100 , and µ 0200 can be determined by measuring the intensity moment µ out 2000 in the output planes of three systems with different values of a x and b x , see (14). Likewise, with the transversal coordinate x replaced by y, the three input moments µ 0020 , µ 0011 , and µ 0002 can be determined by measuring the intensity moment µ out 0020 in the output planes of three systems with different values of a y and b y , see (15). Note that we can choose a x = a y = a i and b x = b y = b i (i = 1, 2, 3) for these three systems, in which case we are obviously using isotropic systems.
The other four input moments µ 1010 , µ 1001 , µ 0110 , and µ 0101 follow from measuring the intensity moment µ out 1010 in the output planes of four different systems, see (15). However, if we would use only isotropic systems, like we could do for (14) and (16), (15) would reduce to µ out 1010 = a 2 µ 1010 + ab µ 1001 + µ 0110 + b 2 µ 0101 (17) and we can only determine the combination µ 1001 + µ 0110 . Hence, while three systems may be isotropic again-and, for instance, be identical to the ones that we used when we were dealing with (14) and (16)-at least one system should be anamorphic.
We conclude that all ten second-order moments can be determined from the knowledge of the output intensities of four first-order optical systems, where one of them has to be anamorphic. In the case of fractional Fourier transform systems we could choose, for instance [24,25], the fractional angles α x = α y = 0 (leading to a x = a y = 1 and b x = b y = 0), α x = α y = π/4 (leading to a x = a y = b x = b y = √ 2/2), α x = α y = π/2 (leading to a x = a y = 0 and b x = b y = 1), and the anamorphic combination α x = π/2π and α y = 0 (leading to a x = b y = 0 and a y = b x = 1). If we decide to determine the moments using free space propagation, we should be aware of the fact that an anamorphic free space system cannot be realized by mere free space, but can only be simulated by using a proper arrangement of cylindrical lenses.
Of course, optical schemes to determine all ten secondorder moments have been described before, see, for instance [8,9,11,12,13], but the way to determine these moments as presented in this paper is based on a general scheme that can also be used for the determination of arbitrary higher-order moments.

Higher-order moments
For higher-order moments we can proceed analogously. For the 4 + 6 + 6 + 4 = 20 third-order moments, the following equations are relevant: µ out 1020 = a x a 2 y µ 1020 + 2a x a y b y µ 1011 + a x b 2 y µ 1002 + b x a 2 y µ 0120 + 2b x a y b y µ 0111 + b x b 2 y µ 0102 , µ out 0030 = a 3 y µ 0030 + 3a 2 y b y µ 0021 + 3a y b 2 y µ 0012 + b 3 y µ 0003 . (21) Note again that these equations correspond to (11) with pqrs = 3000, pqrs = 2010, pqrs = 1020, and pqrs = 0030, respectively. The 20 third-order moments can be determined from the knowledge of the output intensities of six first-order optical systems, where two of them have to be anamorphic.
We consider in more detail how the third-order moments could be determined.
(i) The four input moments µ 3000 , µ 2100 , µ 1200 , and µ 0300 can be determined by measuring the intensity moment µ out 3000,i (i = 1, 2, 3, 4) in the output planes of four systems with different values of a x and b x , see (18). Likewise, with the transversal coordinate x replaced by y, the four input moments µ 0030 , µ 0021 , µ 0012 , and µ 0003 can be determined by measuring the intensity moment µ out 0030,i (i = 1, 2, 3, 4) in the output planes of four systems with different values of a y and b y , see (21). Note that we can choose a x = a y = a i and b x = b y = b i (i = 1, 2, 3, 4) for these four different systems, in which case we are obviously using isotropic systems. This then leads to the set of four equations based on (18) and a similar set of four equations (i = 1, 2, 3, 4) in the output planes of these four systems, see (19), while the two input moments µ 1020 and µ 0102 , together with the two moment combinations 2µ 1011 +µ 0120 and µ 1002 +2µ 0111 , follow from measuring the intensity moment µ out 1020,i (i = 1, 2, 3, 4), see (20). This leads to the set of four equations based on (19) and a similar set of four equations a 3 i µ 1020 + a 2 i b i 2µ 1011 + µ 0120 + a i b 2 i µ 1002 + 2µ 0111 + b 3 i µ 0102 = µ out 1020,i (i = 1, 2, 3, 4) based on (20). (iii) Twelve of the 20 input moments (together with four moment combinations) can thus be determined by using four isotropic systems. To determine the remaining eight moments, we need four more equations based on (19) and (20), for which we have to use two more systems (labeled i = 5 and i = 6), which should now  (27) The former system may be an anamorphic fractional Fourier transform system with fractional angles α x = π/2 and α y = 0 (and hence a x = b y = 0 and b x = a y = 1), while the latter may be an anamorphic fractional Fourier transform system with α x = 0 and α y = π/2 (and hence b x = a y = 0 and b y = a x = 1).
Altogether we have thus constructed 20 equations for the 20 third-order moments, using a total of six first-order systems: four isotropic systems where we measure the 16 output intensity moments µ 3000,i , µ 0030,i , µ 2010,i , and µ 1020,i (i = 1, 2, 3, 4), and two anamorphic systems where we measure the four output intensity moments µ out 2010,i and µ out 1020,i (i = 5, 6).
For the 5 + 8 + 9 + 8 + 5 = 35 fourth-order moments, the relevant equations follow from (11) with pqrs = 4000, pqrs = 3010, pqrs = 2020, pqrs = 1030, and pqrs = 0040, respectively. The 35 fourth-order moments can be determined from the knowledge of the output intensities of nine first-order optical systems spectra, where four of them have to be anamorphic. Constructing a measuring scheme along the lines described above for the second-order case and the third-order case, is rather straightforward.
To find the number of nth-order moments N, and the total number of first-order optical systems N t (with N a the number of anamorphic ones) that we need to determine these N moments, use can be made of the triangle presented in Table 1, which can easily be extended to higher order.
Note that N (the number of nth-order moments) is equal to the sum of the values in the nth row of the triangle, N = (n+1)(n+2)(n+3)/6; that N t (the total number of first-order optical systems) is equal to the highest value that appears in the nth row of the triangle, N t = (n + 2) 2 /4 for n = even, and N t = (n+3)(n+1)/4 for n = odd; that the number of isotropic systems is n + 1; and that N a (the number of anamorphic systems) follows from N a = N t − (n + 1).

CONCLUSIONS
We have shown how all global WD moments of arbitrary order can be measured as intensity moments in the output planes of an appropriate number of first-order optical systems (separable, but generally anamorphic ones), and we have derived the minimum number of such systems that are needed for the determination of these moments. The results followed directly from the general relationship (11) that expresses the intensity moments in the output plane of a separable first-order optical system in terms of the moments in the input plane and the system parameters a x , b x , c x , d x and a y , b y , c y , d y .