Power-law Schell-model Sources

Abstract A new type of Schell-model source is developed that has a spectral degree of coherence, or spatial power spectrum, which is described by a power-law function. These power-law sources generally produce cusped, or peaked far-zone spectral density patterns making them potentially useful in directed energy applications. The spectral degrees of coherence, spatial power spectra, and spatial coherence radii for power-law sources are derived and discussed. Two power-law sources are then synthesized in the laboratory using a liquid crystal spatial light modulator. The experimental spectral densities are compared to the corresponding theoretical predictions to serve as a proof of concept.

In this paper, a new class of partially coherent source is introduced and synthesized. These, so called power-law sources, possess spatial coherence functions which have, or produce spectral densities with, − dependence. These sources, like the sources in [3], can produce cusped spectral densities making them potentially useful in laser manufacturing, free-space optical communications, directed energy, et cetera. In addition, since many natural processes are well described by powerlaw formulas, these sources could potentially serve as simple models of light's interaction with power-law phenomena.
In the next section, power-law sources are analytically developed. Expressions for the spectral degrees of coherence , spatial power spectra, and spatial coherence radii are derived and discussed. In addition, the asymptotic behaviors of and as the power → ∞ are also determined. Lastly, in Section 3, power-law sources are physically realized in the laboratory. The experimental far-zone spectral densities are compared to the theoretical (also derived in Section 2) to validate the analysis.
E-mail address: milo.hyde@afit.edu. 1 The views expressed in this paper are those of the authors and do not reflect the official policy or position of the U.S. Air Force, the Department of Defense, or the U.S. Government.

Theory
From the work of Gori and Santarsiero [12], the sufficient condition for a genuine cross-spectral density function is where =x +ŷ , is an arbitrary kernel, and is a non-negative function. The dependence of the functions in (1) on radian frequency has been omitted for brevity. Letting ( , v) = ( ) exp (−j2 ⋅ v), where is a complex function, simplifies (1) to If the Fourier transform of exists, then (2) describes a Schell-model source [7,8,13] with equal to where is the spatial-domain form of the Wiener-Khinchin theorem [7,13]; thus, is the spatial power spectrum of the random source. Assuming that (and subsequently ) is rotationally invariant transforms (3) into a Fourier-Bessel integral, viz., where 0 is a zeroth-order Bessel function of the first kind and = | | has been introduced for notational convenience.

Power-law sources
Inspired by the von Kármán atmospheric turbulence power spectrum [14], let be where > 1 and > 0 is related to the spatial coherence radius of the source (discussed further in Section 2.3). Substituting (5) into (4) and evaluating the integral yields where is the Gamma function and is a modified Bessel function of the second kind [15].

Power-law sources
A source with a power-law spatial power spectrum can be derived by exploiting the symmetry of the Fourier-Bessel transform in (4). Let be Substituting (7) into (4) and evaluating the integral produces [15,16]

Spatial coherence radii
The spatial coherence radii can be found by evaluating where is the coherence area [13]. Substituting (5) into (9), evaluating the integral, and solving for yields The value of at can be found by substituting (10) into (5) and simplifying: For power-law sources, evaluating (9) directly is difficult. However, one can exploit Parseval's theorem [13] and arrive at a relation for much more easily, viz., The value of at is Clearly, from (10) and (12), the spatial coherence radii change with the power . It would be preferable if had the same physical meaning regardless of . This can be achieved by expressing and in terms of instead of : The asymptotic behaviors of as → ∞ are quite clear from (10) and (12), i.e., 0 and ∞, respectively. Interestingly, the behaviors of ( ) as → ∞ are equal for both the power-law and power-law sourcesboth are exp (−1∕2) ≈ 0.6065. Fig. 1(a) shows ( ) versus for both sources. The black dashed line is the asymptotic value exp (−1∕2). For > 4, the ( ) are practically identical. As might be expected from these results, the corresponding (and consequently the ) given in (14) are also very similar for > 4. This is verified in Fig. 1(b) which shows the sum of squared difference Δ for the two in (14), i.e., versus . The physical significance of this is discussed in the next section.

Propagation behavior
The behavior of a power-law Schell-model source after propagating a distance can be found by evaluating where = 2 ∕ , is the wavelength, and is given in (14). Considerable progress can be made by assuming that and by restricting the analysis to the behavior of the spectral density [7,8,13] versus . Substituting (17) into (16), setting 1 = 2 = , and carrying out the tedious but relatively straightforward mathematics yields The remaining integral (a Fourier-Bessel integral) must be computed numerically.

Approximate far-zone behavior
An approximate expression for the far-zone power-law source can be found by assuming that is much narrower than the spectral density . Under this condition, the source is a quasi-homogeneous source [7,8,13] and the generalized Van Cittert-Zernike theorem [13] can be used to predict the far-zone : wherẽis the Fourier transform of . The far-zone spectral density can be found quite easily from (19), i.e., Eq. (20) is used in Section 3 to validate the experimental results. Recall that the given in (14) are very similar for > 4; thus, under this condition, both power-law sources produce similar far-zone .  (18) numerically; the red traces in (f), (g), and (h) are the approximate far-zone given by (20). As expected, (20) is a very good approximation to (18) for small Fresnel numbers. Note that both sources produce cusped in the far zone-exploitable, in practice, using a lens.

Set-up
In this section, experimental results of power-law sources with given in (14) are presented. A schematic of the experimental set-up used to synthesize the sources is presented in Fig. 4. Light from a 632.8 nm helium-neon (HeNe) laser is expanded 20× before passing through a half-wave plate (HWP) and linear polarizer (LP). The HWP-LP combination serves to align the linearly polarized light exiting the laser with the control state of the spatial light modulator (SLM) -vertical in this case -and to control light power.
After passing through the HWP-LP, the light is incident on the SLM. The SLM used here is a Meadowlark Optics P512 which has a 512 × 512 liquid crystal pixel array with a 15 μm pitch. The light reflected from the SLM is diffracted into multiple orders. Partially coherent source instances are produced in the first diffraction order; thus, a spatial filter [composed of a 400 mm lens, an iris, and a 100 mm lens (L2)] is used to remove all orders other than desired first order.
Lastly, the random intensities are recorded by a camera located 40 cm beyond the focus of L2. The camera used here is a Lumenera  Lw135RM which has a 1392 × 1040 detector array with a 4.65 μm pitch. In addition to producing random field instances, the SLM also applies a 40 cm focus so that the intensities measured by the camera are, equivalently, the far-zone intensities.

Power-law sources & data processing
The power-law and power-law sources both had Gaussian-shaped [see (17)], where = 1 mm. The and for both sources were = 0.1 mm and = 1.5. The modified phase screen technique was used to synthesize power-law source realizations [18][19][20]. An example SLM command for generating a single instance of a power-law source is shown in Fig. 5. A detailed description of the screen (SLM command) synthesis process, including an illustration depicting the process, can be found in [20].
The experimental were formed by averaging 5,000 measured intensities. The camera continuously collected 66 ms exposures until the sequence of 5,000 random field instances was completed. The data collect for each power-law source took approximately 225 s. The raw experimental were then centered for ease of comparison with the theoretical derived from (20): where are given in (14).

Results
Figs. 6 and 7 shows the results for the power-law and powerlaw sources, respectively. The layout of both figures is the same:   or star shape is discernable around the origin in both figures. These are very minor experimental errors and likely caused by the spatial and phase discretization of the SLM.

Conclusion
In this paper, Schell-model sources which possessed power-law spectral degrees of coherence and spatial power spectra were developed. It was shown that these power-law sources can produce cusped, or peaked spectral densities making them potentially useful in laser manufacturing or directed energy applications. Expressions for , , and the spatial coherence radii were derived and discussed. Their asymptotic behaviors as the power → ∞ were also investigated. Lastly, power-law and power-law sources were physically realized in the laboratory using an SLM. The experimental were compared to the corresponding theoretical and found to be in excellent agreement.