Shaping the Far-zone Intensity, Degree of Polarization, Angle of Polarization, and Ellipticity Angle Using Vector Schell-model Sources

This paper presents a method to control both the shape and polarization of a beam in the far field using a vector Schell-model source. Given a desired far-zone beam shape and polarization, and applying Fourier and statistical optics theory, we derive the requisite second-order moments of said source, discuss what aspects of the far-zone beam can be controlled, and develop a step-by-step procedure for synthesizing the required random vector field instances. We validate this approach with Monte-Carlo wave-optics simulations. The results are found to be in very good agreement with the desired far-zone beam characteristics. The beam-shaping technique developed in this paper will find use in optical trapping, optical communications, directed energy, remote sensing, and medical applications.

The primary objective of these efforts has been to generate the desired source, i.e., control the field's shape, polarization, or coherence, in the source plane. Few studies have dealt with controlling these parameters at another axial location-at the focus of a lens, for instance. This ability is crucial for all the applications listed above. Of the studies that focused on controlling beam parameters at the focus of a lens (or equivalently, in the far zone of the source), all have dealt with scalar partially coherent sources and only controlled beam shape [24][25][26][27].
In this paper, we extend the prior scalar work to control both beam shape and polarization in the far zone of the source. Using Fourier and statistical optics theory, we first derive expressions for the spatial correlation functions (or via Fourier transforms, the spatial power spectra) of a vector Schell-model source that radiates a beam with desired, or designed, far-zone shape and polarization properties. We discuss what aspects of the resulting beam can be controlled, and develop a procedure (or recipe) to synthesize the required vector field instances.
Next, we validate the analysis and synthesis procedure with Monte-Carlo wave-optics simulations. As a proof of concept, we synthesize vector Schell-model sources that radiate far-zone beams with shapes and polarization parameters that are complex grayscale images. We quantitatively assess the results by computing the correlation coefficients versus trial number to study the convergence of the random vector field instances to the desired shape and polarization images.
Lastly, we conclude with a summary of the work presented in the paper and a brief list of potential applications.
behavior of the random vector fields, and lastly, show how to generate vector field realizations with desired statistics. In the latter half of this section, we delve into the mathematical particulars of controlling the far-zone beam's shape and polarization. We derive expressions for the required statistics of the vector Schell-model source, and discuss what aspects of the beam can be controlled. We close this section with an enumerated procedure, which concisely summarizes the theory presented herein and provides a recipe for generating optical fields that, in the far field, result in a beam with desired complex shape and polarization characteristics.

Preliminaries Assumptions
The goal here is to design a vector partially coherent source that produces a far-zone beam with desired shape (or intensity) and polarization properties. To this end, we make two assumptions that significantly simplify the analysis, while not overly restricting the applicability. The first is that the shapes of the field's x and y components (also called the component spectral densities, S x and S y , respectively) are equal in the source plane. This has the physical consequence that the polarization state across the source is uniform [4,28,29]. The second assumption is that the vector spatial correlation functions are much narrower (are fast functions) compared to the associated component spectral densities. This assumption is known in the literature as the quasi-homogeneous approximation, and the partially coherent source is referred to as a quasi-homogeneous electromagnetic source [4,30]. The cross-spectral density (CSD) matrix W for such a source is where W ij is the i j , th th element of S W, i is the spectral density of the i th field component, µ ij is the cross-correlation function between the i th and j th field components, and = i j x y , , . The transverse position vectors 1 and 2 are = + x y x y 1,2 1,2 1,2 .

Stochastic vector field
Let an instance of a random optical vector field be where C i is the complex amplitude and T i is the stochastic complex transmittance screen for the i th component of the field, respectively. A is the amplitude function (beam shape) of the source; recall that Taking the vector autocorrelation of (2) to form W [4,28,29] produces where is the average taken over all complex screen realizations. By comparing this result to (1), it is clear that Note that C C arg( ) x y is generally a free parameter and comes into play when dealing with circular polarization. We return to these expressions later in the paper.

Far-zone W( , )
To control the intensity and polarization in the far zone, we require the far-zone, single-point W, i.e., W evaluated at = = 1 2 . This can be found by propagating the W ij in (3) to the far field (a four-dimensional spatial Fourier transform [28,31]) and evaluating the resulting expression at = = 1 2 . This expression takes the form where = k 2 / , is the wavelength, and z is the distance to the farfield observation plane. Making the variable transformations = s 1 and = t 1 2 The amplitude function integrals are equal to the autocorrelation of A, represented hereafter as t ( ). Recall that A was assumed to vary much more slowly (be much broader) than µ ij -the source was assumed to be an electromagnetic quasi-homogeneous source. This means that also varies much more slowly than µ ij , such that where ij , by the Wiener-Khinchin theorem [28,29,31], is the spatial cross-power spectrum of the i th and j th field components.
Eq. (7) is a specialization of the generalized van Cittert-Zernike theorem [28,31] and has been used in the past to generate a scalar Schell-model source which produces any desired shape in the far field [24][25][26][27]. Here, we use it to generate a vector Schell-model source that radiates a beam with a desired shape and polarization characteristics.
In contrast to the prior scalar work where any shape could be produced, here, the correlation functions of the x and y components of the field (µ xx and µ yy , respectively) ultimately determine the vector crosscorrelation function µ xy . This fundamentally limits what we can control. We discuss this in more detail in the following sections.

T T ,
x y , and µ xy To produce an instance of a vector source, given in (2), that radiates a beam with the far-zone, single-point W ij given in (7), one must generate T x and T y with the proper statistics. The most common way of achieving this is to spatially filter two, two-dimensional arrays (one representing T x , the other T y ) of circular complex, Gaussian random numbers [4,15,17,18,[32][33][34]. The x and y spatial filters are related to the corresponding vector spatial correlation functions, namely, µ xx and µ yy . For computational efficiency, it is best to perform the filtering in the spectral domain using the convolution theorem.
To produce a source that radiates a beam in a general polarization state, µ 0 xy , which means that T x and T y must be generated from correlated Gaussian random numbers. To see how the statistics of T x and T y affect µ xy , we note that a realization of discrete T x (or T y ) can be produced by [15,24,26,32] [ , ] [ , ] 2 exp j 2 exp j 2 , x m n x xx x y x y (8) where k l , are discrete spatial indices, m n , are discrete spatial frequency indices, N N , x y are the numbers of grid points in the x y , directions, x y y are the lengths of the grid in the x y , directions in meters, and is the grid spacing. In (8), r x is an × N N y x grid of zero-mean, unit-variance circular complex Gaussian random numbers and xx is the spatial power spectrum of the x field component, i.e., the Fourier transform of µ xx . Note that (8) is the inverse Fourier transform of the product of two Fourier transforms (r x and xx ), and is physically equivalent to filtering white noise. Eq. (8) is in the form of a discrete inverse Fourier transform, and therefore, we can use the fast Fourier transform algorithm to quickly realize T x .
The moment µ xy is formed by taking the cross-correlation of (8) with T y , namely, [ , ] [ , ] exp j exp j exp j exp j . x y x y x y x y The moment = r m n r m n m m n n where is the correlation coefficient between the r x and r y random numbers and n [ ] is the discrete Dirac delta function. This simplifies (9) to [ , ] [ , ] ) .
Recall from (4) 2 . For this to be true, xx yy in (10) must equal xy . In practice, this means that the "self" power spectra set the cross-power spectrum, and subsequently, we can only simultaneously control two of the four polarization parameters, e.g., the intensity plus the degree of polarization or the intensity plus the ellipticity angle. We discuss this further below.
Before proceeding to the next topic, we substitute the above result into (7) as it will be useful in the analysis to follow: For convenience, we let transforming (11) to

Far-zone polarization control
Here, in the second half of Section "Theory", we present the analytical details of controlling the far-zone beam's shape and polarization.
We also discuss what characteristics of the far-zone beam can be controlled. We begin by introducing the far-zone Stokes and Poincaré sphere parameters, and derive expressions for them in terms of the selfpower spectra introduced above.

Polarization parameters
With ( where S 0 is the total average intensity, 0 1 is the degree of polarization, < /2 /2 is the angle of polarization, and /4 /4 is the ellipticity angle [4,29,35]. For brevity, we drop the functional dependencies of the Stokes parameters, Poincaré sphere parameters, and CSD matrix elements. Henceforth, their dependence on and z is assumed and suppressed. Substituting (13) In this form, it is clear that only two polarization parameters can be controlled at a time. It turns out that only S 0 and one other parameter can be controlled-the others , , and are dependent on each other. In the next three sections, we derive equations for xx and yy in terms of S 0 and S , 0 and , and S 0 and , respectively.
Solving the S 0 equation for yy , substituting the resulting expression into , and solving for xx produces Since both xx and yy must be real and positive, . Thus, 's minimum value is set by . This means that < 1 or cannot be controlled.
Both roots in (18) are physical. The "+" root of xx (hereafter referred to as just the "+" root) corresponds to the case when > > S , 0 xx yy 1 , and the beam is polarized predominately in the horizontal direction. The " " root of xx (hereafter referred to as just the " " root) corresponds to the opposite case- , and the beam is polarized predominately in the vertical direction. Since S 1 is squared in (16), the root choice is irrelevant and both produce the desired S 0 and . Note that the root choice does affect the associated , but we are not concerned with that quantity here.
Solving the S 0 equation for yy , substituting the resulting expression into tan(2 ), and solving for xx produces cos (2 ), and = C cos( ) There are several aspects of (21) that warrant discussion. If = 0, then = = S /2 xx yy 0 . This situation physically corresponds to unpolarized light, and is undefined. Thus, > 0 or cannot be controlled.
A similar thing happens when = ± m /2 x y , where m is an odd integer. Here again, = = S /2 xx yy 0 ; however, in this situation, the resulting beam is circularly polarized. When it comes to controlling , circular polarization is no different than random polarization, and ± m /2 x y or cannot be controlled. Lastly, inspection of (19) reveals that the signs of xx yy and C determine the range of physically meaningful (or possible) values of . Fig. 1 shows how these signs affect the range of . The figure depicts in the S 1 -S 2 plane. The polarization angle is physically limited to ( /2, /2]. For < | | /4, the beam is more horizontally polarized than vertically polarized. This corresponds to positive > S , xx yy 1 , and the "+" root in (21). For > | | /4, the opposite is true: the beam is more vertically polarized than horizontally polarized, < < S 0, xx yy 1 , and the " " root in (21) is applicable. These are labeled in Fig. 1 as well as the special angles = ± /4.
The sign of C -C sgn( ), where x sgn( ) is the signum function-determines whether is restricted to the upper or lower quadrant of the right-half S 1 -S 2 plane. When = C sgn( ) 1, the physically possible values of are between [0, /2]. This corresponds to the green region in Fig. 1.
can take on the values in the blue region, namely, ( /2, 0]. The utility of Fig. 1 is best illustrated through an example. Let = 0.5 and = 2 /3 x y . We want to generate a vector Schellmodel source that radiates a beam with an S 0 and that are grayscale images, which are arbitrarily scaled. The S 0 image scale is irrelevant; however, the image must be mapped to a set of values that are physically possible given C sgn( ). Here, = C sgn( ) 1, the blue region in Fig. 1 Like the S 0 and section above, both the "+" and " " roots in (24) are physical and correspond to the same scenarios described therein. Since S 1 is squared in (22), the root choice is again irrelevant and both produce the desired S 0 and . Exactly the same as when controlling S 0 and > , 0 or cannot be controlled.
= 0 corresponds to unpolarized light, and has no physical meaning.
Lastly, since xx and yy must be real and positive,    To illustrate the utility of Fig. 2, we refer back to the example discussed in the S 0 and section. Let = 0.5 and = 2 /3 x y . Our goal is to generate a vector Schell-model source that radiates a beam with an S 0 and that are grayscale images, which are arbitrarily scaled. Again, the scale of the S 0 image is irrelevant. The image must be mapped to a set of values that are physically possible given S sgn( ). Here, = S sgn( ) 1, the blue region in Fig. 2 is applicable, and the image values should be mapped to S [ |arcsin( )/2|, 0]. As previously discussed, the root choice in (24) is irrelevant.

Theory summary
In summary, to produce a vector Schell-model source that radiates a beam with a desired shape and polarization properties.

Choose C C
, , x y , and A ( ). To some extent, these parameters can be used to control the beam shape and polarization in the source plane. Recall that C , arg( ) x , and C arg( ) y affect the polarization state in the far zone. 2. Choose the desired S 0 and , , or images. 3. Use (18), (21), or (24) (whichever is applicable) to find xx and yy .
where the superscripts "r" and "i" stand for real and imaginary parts, respectively. 5. Use (8) to generate instances of T x and T y . Recall that xx and yy are related to xx and yy by (12). 6. Use (2) to generate a vector Schell-model source field realization. 7. Synthesize field realization using spatial light modulators [2,3,15,[19][20][21].
We demonstrate and validate the use of the above procedure via simulation in the next section.

Validation
Here, we perform Monte-Carlo wave-optics simulations to validate the analysis of the previous section. Before proceeding to the results, we discuss the simulation particulars so that the interested reader can reproduce our results, or perform a similar simulation for their own purposes.

Simulation description
For these simulations, we used computational grids that were = = N N 1024 x y points on a side with grid spacings = 15 µm. The simulated source plane field was where = D 7.68 mm and x rect( ) was [36] The rectangular shape and D were chosen to correspond with a popular model of liquid crystal spatial light modulator [37]. The simulated wavelength was = 632.8 nm.
The T x and T y were generated following the procedure in Section "Theory summary". The desired S 0 was the Celtic cross image shown in Figs. 3-5(a); the desired , , and were the Air Force Institute of Technology logo shown in Figs. 3-5(c), respectively. Recall that to control < , 1 and to control or > , 0. Here, we arbitrarily chose = 0.5. We generated 20,000 realizations of the vector field in (26) and propagated each to the far field using fast Fourier transforms [38,39]. We then computed the far-zone, single-point CSD matrix elements and, from these, computed the Stokes and Poincaré sphere parameters using (14). We lastly compared the simulated S 0 and , , or to the desired images. We performed the simulations using MATLAB® version R2017a; the scripts (.m files) are included as supplementary materials. The agreement between the desired images and the simulated results is very good. The results in Figs. 3-5 validate the theoretical analysis presented in Section "Theory".

Results and discussion
The simulated S 0 and , , or converge to their asymptotic values within approximately 1000 trials. The in Figs. 3-5(e) asymptote at approximately the same level, i.e., between 0.97 and 0.984. These numbers are not likely to appreciably increase, even with running many more trials, because of a theoretical assumption we made in Section "Theory".
Recall (6), which is rewritten below for the reader's convenience: In the analysis, we assumed that the partially coherent source was a quasi-homogeneous electromagnetic source. In other words, we assumed that (the autocorrelation of the source's shape) was much broader than µ ij , such that could be evaluated at = t 0 and removed from the integral, leaving S ij being (approximately) proportional to the spatial cross-power spectrum ij (or equivalently, the spatial Fourier transform of µ ij ). This assumption was necessary to derive the closedform expressions later in Section "Theory" that comprised the main contributions of this paper.
Although it is not possible to derive the expressions presented in the latter half of Section "Theory" and include source shape, we can gain a physical understanding of how source shape affects S 0 and , , or by examining (28) more closely. Eq. (28) is the Fourier transform of the product of two functions. By the convolution theorem, (28) is equivalent to the convolution of ij with the Fourier transform of (hereafter, ). Thus, the true far-zone S ij is a spatially filtered version of ij ; the filter is . For broad, or slowly varying (as assumed in Section "Theory"), is narrow or fast, and in the asymptotic limit, S ij is proportional to ij . For narrow or fast , is broad or slow, and S ij is proportional to in that asymptotic limit. The simulated results include the effects of source shape [recall the simulated source field in (26)]. Thus, the above discussion explains the results in Figs. 3-5(e). It also explains the minor qualitative differences in Figs. 3-5(a) and (b), and (c) and (d). More importantly, the above discussion provides the user with a physical understanding of the actual, true performance of the beam shaping technique developed in this paper.

Conclusion
In this paper, we developed a method using a vector Schell-model source to control the far-zone beam shape and polarization. This research extended prior scalar Schell-model source work which only controlled beam shape.
By applying Fourier and statistical optics theory, we derived expressions for the vector power spectra, necessary to generate a vector Schell-model source that radiates a beam with designer far-zone shape and polarization properties. We discussed what aspects of the far-zone beam-S , , 0 , and -can be controlled. We also developed a stepby-step procedure that described how to synthesize random vector field instances with the proper statistics.
Lastly, we presented Monte-Carlo simulation results to validate our analysis. We successfully demonstrated the concept by generating vector Schell-model sources that radiated beams with shapes (S 0 ) and Poincaré sphere parameters ( , , or ) that were complex grayscale images.
The beam-shaping method introduced in this paper will be useful in optical trapping, optical communications, directed energy, remote sensing, and medical applications.