1. INTRODUCTION

Research into partially coherent sources or beams is quite popular because of their ability to maintain directionality while also being resistant to speckle or scintillation [1,2]. In addition, partially coherent sources are highly tailorable in shape, polarization, and coherence properties [3–6]. These attributes make them potentially useful in directed-energy, remote-sensing, optical-communication, and medical applications.

Research into tiled-aperture array technology has largely occurred in parallel with the coherence work outlined above. This research is motivated by the significant, potential improvements in system size, weight, and power afforded by fiber laser beam combining.

In general, beam combining in tiled-aperture arrays is accomplished in one of two ways—incoherently or coherently. Incoherent systems are simpler in design and, therefore, more technologically mature. The US Navy has invested heavily in this technology and has deployed an incoherently combined array [see the Laser Weapon System (LaWS) program] [7,8].

Coherently combined arrays hold the promise of much higher peak intensities than incoherent systems [9–12]. Since this scheme requires polarization control, path-length matching, and phase locking of multiple fiber lasers in a master-oscillator-power-amplifier configuration, the design of coherently combined arrays is much more complex than incoherent systems. As such, coherent beam combining is still largely in the research/laboratory phase of development [9,13–15].

Anticipating the possible benefits of combining coherence control with laser array technology, several authors have investigated partially coherent array sources and beams. For obvious reasons, a large majority of these studies are concerned with how these arrays perform in the presence of turbulence, both atmospheric and oceanic [16–21].

In order to arrive at analytical expressions for the average intensity, degree of coherence, scintillation index, etc., these studies use Gaussian Schell models, or their many variants, as the mutual coherence [or cross-spectral density (CSD)] functions of the array element fields [16,18–25]. Using Gaussian Schell models as the basis for the source statistics results in output statistics that are generally sums of Gaussian functions. These expressions can be hard to physically interpret and are typically explored computationally.

Here, we step back and examine the far-zone behavior of a tiled-aperture array fed by vector partially coherent sources. We present the general analytical far-zone spectral density with the assumptions that the array elements are circular and that the partially coherent sources feeding the array are wide-sense stationary with Schell-model CSD functions.

The spectral density obtained here is not a closed-form function; however, we express it as a convolution integral, which is easy to physically interpret. We discuss the physical nature of this expression in detail and further explore it with three examples. We then validate this far-zone spectral density expression by comparing and contrasting analytical predictions with Monte Carlo averages from wave-optics simulations. Lastly, we conclude this paper with a brief summary of contributions.

2. THEORY

Using Love’s equivalence theorem [26,27], image theory [28,29], and the physical optics approximation [30,31], the approximate far-zone electric field radiated from a planar, conformal tiled-aperture array takes the general form

The far-zone vector potential components $L_x$ and $L_y$ are

Because of our interest in directed-energy applications, we are primary concerned with the spatial distribution of power in the far-zone. This information is captured in the spectral density, namely,

In tiled-aperture arrays, $\lambda \ll d$, where $d$ is the diameter (or width) of an array element. A vast majority of the radiated power is, therefore, contained within a small angle centered on the $z$ axis. Applying what is equivalently the paraxial approximation (namely, $\cos \theta \approx 1$; $\sin \theta \approx \theta$, such that ${\sin }^2\theta \approx {\theta }^2\approx 0$; $r\approx z$; and $\widehat{\mathit{r}}=\mathit{r}/r\approx \widehat{\mathit{x}}x/z+\widehat{\mathit{y}}y/z+\widehat{\mathit{z}}$) to Eq. (3) yields

A. ${\mathsf{E}}^{\mathsf{src}}$ Cross-Spectral Density

To make further progress, an analytical model for the random field ${\mathit{E}}^{\mathrm{src}}$ must be chosen. Let a single realization of ${\mathit{E}}^{\mathrm{src}}$ be

The random complex transmittances, which make the source field in Eq. (6) stochastic, can arise physically from platform vibrations (or jitter) or from thermal gradients in the transmitter optics. They could also be applied intentionally, for example, to reduce turbulence-induced scintillation.

The expression for ${\mathit{E}}^{\mathrm{src}}$ given in Eq. (6) is very general. Note that each array element has a unique polarization state and spatial degree of coherence. The latter can be controlled to make an element mutually, spatially coherent (or incoherent) with any of the other array elements.

Taking the vector autocorrelation of Eq. (6) to form the CSD dyad (only the diagonal elements are required), where the $\omega$ dependence is assumed and, henceforth, suppressed, yields

B. Spectral Density $\mathsf{S}(\mathsf{\rho },\mathsf{z})$

Substituting Eq. (7) into Eq. (4), making the variable substitutions $\mathit{s}={\mathit{\rho }}_1^{\prime }$ and $\mathit{t}={\mathit{\rho }}_1^{\prime }-{\mathit{\rho }}_2^{\prime }$, and simplifying produces

Equation (8) is the inverse Fourier transform of the product of two functions; thus, by using the convolution theorem, Eq. (8) becomes

Evaluating ${\tilde{\mathrm{\Lambda }}}_{mn}$ and simplifying results in

C. Examples

In this section, we present three examples labeled Scenario 1, Scenario 2, and Scenario 3, respectively. The goal of this exercise is to reveal the physical nature of Eq. (11).

1. Scenario 1

First, assume that $C_{\alpha ,m}=C_{\alpha }$ and ${\mu }_{\alpha \alpha ,mn}={\mu }_{\alpha \alpha }$, then Eq. (11) simplifies to

The expression in braces is equivalent to the far-zone array pattern when the elements are fed by a common spatially coherent source (e.g., a laser). We refer to this expression throughout the remainder of this paper; therefore, we assign it the symbol $\mathcal{L}$ for convenience.

$\mathcal{L}$ has two scale sizes. The first is due to the $\mathrm{jinc}(x)$ function (also known as the element pattern), whose width is proportional to $\lambda z/d$; the second is due to the summation term (better known as the array factor), whose width is inversely proportional to the array’s full diameter, i.e., $\propto \lambda z/D$.

These dimensions, when compared to the widths of ${\mu }_{\alpha \alpha }$ (represented hereafter as ${\delta }_{\alpha \alpha }$), result in three general behaviors for $S(\mathit{\rho },z)$:

2. Scenario 2

For the next example, let $C_{\alpha ,m}=C_{\alpha }$ and ${\mu }_{\alpha \alpha ,mn}={\mu }_{\alpha \alpha }{\delta }_{mn}$, where ${\delta }_{mn}$ is the Kronecker delta function. Substituting these into Eq. (11) and simplifying produces

The quantity in braces is $N$ times the element pattern. As previously discussed, the width of the element pattern is proportional to $\lambda z/d$. Thus, how the ${\delta }_{\alpha \alpha }$ compare to $d$ determines the behavior of $S(\mathit{\rho },z)$:

3. Scenario 3

For the last example, assume that ${\mu }_{\alpha \alpha ,mn}$ are slow functions (${\tilde{\mu }}_{\alpha \alpha ,mn}$ are fast functions) compared to the other expressions in Eq. (11). Under these conditions, Eq. (11) simplifies to

Scenario 3A To explore this scenario further, assume that $N=2$, and $C_{x,1}=\exp (\text{j}\pi /4)/\sqrt{2}$, $C_{x,2}=\exp (-\text{j}\pi /4)/\sqrt{2}$, $C_{y,1}=\exp (-\text{j}\pi /4)/\sqrt{2}$, and $C_{y,2}=\exp (\text{j}\pi /4)/\sqrt{2}$ (i.e., right-hand circular polarization in element 1 and left-hand circular polarization in element 2). Substituting these assumptions into Eq. (14) and simplifying yields

Scenario 3B For a slightly more interesting example, let $N=2$, and $C_{x,1}=1$, $C_{x,2}=1/\sqrt{2}$, $C_{y,1}=0$, and $C_{y,2}=1/\sqrt{2}$ (i.e., horizontal linear polarization in element 1 and 45° linear polarization in element 2). Substituting these into Eq. (14) and simplifying yields

3. SIMULATIONS

Here, we present simulation results to validate Scenarios 1–3 discussed above—in particular, Eqs. (12), (13), (15), and (16) and their physical interpretations. Before presenting the results, we discuss the simulation setup.

A. Setup

The simulated array had seven circular elements arranged in a regular hexagon for Scenarios 1 and 2, and two circular elements arranged side-by-side on the $x$ axis for Scenario 3. Each circular element was $d=10\mathrm{cm}$ in diameter; they were spaced (center-to-center) $D=10.5\mathrm{cm}$ apart. The computational grid consisted of $N=1024$ points per side with $\mathrm{\Delta }=5\mathrm{mm}$ grid spacing.

For Scenarios 1 and 2, $C_{\alpha ,m}=C_{\alpha }=1/\sqrt{2}$ and ${\mu }_{\alpha \alpha ,mn}$ were Gaussian, namely,

Complex transmittance screens $t_{\alpha ,m}$ were generated using the Monte Carlo spectral method [35–38], viz.,

For all three scenarios, $t_x$ was assumed to be statistically independent of $t_y$. Recall that only the diagonal elements of the source field CSD dyad were required, i.e., the statistical relationship between the $x$ and $y$ components of the source field was irrelevant. For Scenarios 1 and 3, all of the element screens were perfectly correlated ($t_{\alpha ,m}=t_{\alpha }$), and, for Scenario 2, they were statistically independent.

Lastly, the simulated far-zone spectral densities were computed and then averaged over 2000 propagated instances of the array source field [see Eq. (6)]. The simulated propagation distance and wavelength were $z=10\mathrm{km}$ and $\lambda =632.8\mathrm{nm}$, respectively. All propagations were computed using fast Fourier transforms [39,40].

B. Results

This section presents results corresponding to Scenarios 1–3 discussed in Section C. In general, the Monte Carlo averages for the simulated far-zone spectral densities compare well to the theory. This comparison validates Eq. (11).

1. Scenario 1

Figure 1 shows the Scenario 1 simulation results. The figure is organized as follows: the left column subfigures [(a), (c), and (e)] show the theoretical spectral densities, and the right column subfigures [(b), (d), and (f)] show the simulated spectral densities. Proceeding down the rows are the results for ${\delta }_{xx}={\delta }_{yy}=31\mathrm{cm}$ [Figs. 1(a) and 1(b)], ${\delta }_{xx}={\delta }_{yy}=7.75\mathrm{cm}$ [Figs. 1(c) and 1(d)], and ${\delta }_{xx}={\delta }_{yy}=1.94\mathrm{cm}$ [Figs. 1(e) and 1(f)], respectively.

 

Fig. 1. Scenario 1 results: (a) $S^{\mathrm{thy}}$ and (b) $S^{\mathrm{sim}}$ with ${\delta }_{xx}={\delta }_{yy}=31\mathrm{cm}$; (c) $S^{\mathrm{thy}}$ and (d) $S^{\mathrm{sim}}$ with ${\delta }_{xx}={\delta }_{yy}=7.75\mathrm{cm}$ (see Visualization 1); (e) $S^{\mathrm{thy}}$ and (f) $S^{\mathrm{sim}}$ with ${\delta }_{xx}={\delta }_{yy}=1.94\mathrm{cm}$; (g) $y=0$ slices (a)–(f); and (h) $x=0$ slices (a)–(f).

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Lastly, Figs. 1(g) and 1(h) show the $y=0$ and $x=0$ slices through Figs. 1(a)–1(f), respectively. The spectral density results in Figs. 1(g) and 1(h) are plotted in decibels so that all results are clearly visible on a single plot. Note that the peak spectral densities in Figs. 1(a) and 1(b) are roughly 30 times larger than those in Figs. 1(e) and 1(f).

We also included a 30 s movie (Visualization 1) showing the formation of the far-zone spectral density result in Fig. 1(d). The movie shows the first 300 array and observation field realizations, as well as the running average of the “instantaneous” spectral densities.

Overall, the agreement between the simulated and theoretical spectral densities is very good. As predicted from Eq. (12), the far-zone spectral density has three semi-distinct behaviors, which are shown here.

In row 1 of Fig. 1, the $1/e$ point of $\mu$ (hereafter referred to as the coherence radius) is greater than the full diameter of the array—the array is essentially fed by a uniform coherent source—resulting in a pattern similar to $\mathcal{L}$.

In row 2, the coherence radius lies between the full array and element diameters $d$. The light emitted from each element is generally spatially coherent; however, the coherence radius is such that the interference of light emitted from different elements is only partially developed. The result is a blurred pattern that has approximately the same overall size as $\mathcal{L}$.

Lastly, in row 3, the coherence radius is smaller than $d$. The light emitted from each element is spatially incoherent, and the far-zone spectral density resembles $\tilde{\mu }$ (Gaussian in this case). Note that all information about the array (e.g., element shapes, diameters, spacings) is lost.

2. Scenario 2

Figure 2 shows the results for Scenario 2. The layout of the figure is exactly the same as Fig. 1. Like in the Scenario 1 results, we included a 30 s movie (Visualization 2) showing the formation of Fig. 2(d).

 

Fig. 2. Scenario 2 results: (a) $S^{\mathrm{thy}}$ and (b) $S^{\mathrm{sim}}$ with ${\delta }_{xx}={\delta }_{yy}=31\mathrm{cm}$; (c) $S^{\mathrm{thy}}$ and (d) $S^{\mathrm{sim}}$ with ${\delta }_{xx}={\delta }_{yy}=7.75\mathrm{cm}$ (see Visualization 2); (e) $S^{\mathrm{thy}}$ and (f) $S^{\mathrm{sim}}$ with ${\delta }_{xx}={\delta }_{yy}=1.94\mathrm{cm}$; (g) $y=0$ slices (a)–(f); and (h) $x=0$ slices (a)–(f).

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Again, the agreement between the theoretical and simulated spectral densities is very good with some minor differences in the diffraction rings in Figs. 2(b) and 2(d). Being equivalent to an array fed by seven mutually incoherent sources [recall the physical interpretation of Eq. (13)], the far-zone spectral density has two semi-distinct behaviors determined by the size of the coherence radius in comparison to $d$.

In rows 1 and 2 of Fig. 2, the coherence radius is greater than $d$. The light emitted from each array element is spatially coherent but mutually incoherent with respect to all other elements. The far-zone spectral density is approximately the sum of seven Airy patterns. In row 2, the diffraction ring has nearly disappeared because the coherence radius is approximately $d$; however, the peak spectral density is roughly the same as in row 1. This is quite different from the results in rows 1 and 2 of Fig. 1.

Lastly, in row 3 of Fig. 2, the coherence radius is less than $d$. The physical situation here is equivalent to the corresponding situation in Scenario 1, and the results are generally the same (note the Gaussian shape and peak spectral density values).

3. Scenario 3

Figure 3 shows the Scenario 3 results. Figures 3(a)–3(d) and 3(e)–3(h) pertain to Scenarios 3A [see Eq. (15)] and 3B [see Eq. (16)], respectively. Figures 3(a) and 3(e) show the theoretical spectral densities, Figs. 3(b) and 3(f) show the simulated spectral densities, Figs. 3(c) and 3(g) show the $y=0$ slices through $S^{\mathrm{thy}}$ and $S^{\mathrm{sim}}$ in decibels, and Figs. 3(d) and 3(h) show the $x=0$ slices through $S^{\mathrm{thy}}$ and $S^{\mathrm{sim}}$ in decibels.

 

Fig. 3. Scenario 3 results: Scenario 3A—(a) $S^{\mathrm{thy}}$ and (b) $S^{\mathrm{sim}}$ with ${\delta }_{xx}={\delta }_{yy}=31\mathrm{cm}$ (see Visualization 3); (c) $y=0$ slices (a) and (b); and (d) $x=0$ slices (a) and (b); Scenario 3B—(e) $S^{\mathrm{thy}}$ and (f) $S^{\mathrm{sim}}$ with ${\delta }_{xx}={\delta }_{yy}=31\mathrm{cm}$ (see Visualization 4); (g) $y=0$ slices (e) and (f); (h) $x=0$ slices (e) and (f).

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Like in the results of Scenarios 1 and 2, the agreement between simulation and theory is excellent. We included two 30 s movies—Visualization 3 and Visualization 4—showing the formations of Figs. 3(b) and 3(f), respectively.

Recall that Scenario 3A concerned a mutually coherent, two-element array with right-hand circular polarization transmitted from element 1 and left-hand circular polarization transmitted from element 2. Intuitively, the far-zone spectral density should be equivalent to the spatially incoherent result, and, indeed, the incoherent pattern is evident in the first two rows of Fig. 3.

Scenario 3B again concerned a mutually coherent, two-element array; however, this time horizontal linear polarization was transmitted from element 1, and 45° linear polarization was transmitted from element 2. In this case, we should observe an interference pattern in the far-zone spectral density because of the interference of the $x$ components of the element fields. This interference pattern is clearly visible in the last two rows of Fig. 3.

4. CONCLUSION

Here, we examined the far-zone behavior of a tiled-aperture array fed by vector partially coherent sources. As opposed to previous studies on partially coherent array beams, which used the popular Gaussian Schell-model family of sources, we derived a general form for the far-zone spectral density, assuming only that the array elements were circular and fed by optical fields with Schell-model CSD functions.

The far-zone spectral density was expressed as the sum of spatially coherent and incoherent terms. Both were convolution integrals, where the associated (coherent or incoherent) far-field array pattern was convolved with the Fourier transforms of the spatial cross-correlation functions. We explored the physical nature of this result with three examples. Lastly, we validated the far-zone spectral density expression by comparing and contrasting the Monte Carlo averages from wave-optics simulations with our analytical predictions. The simulation results were in excellent agreement with theory.

This work will find use in the prediction, simulation, and design of partially coherent beam arrays. These systems are now being developed for use in directed-energy, remote-sensing, optical-communications, and medical applications.