This paper investigates atmospheric array tilt and its effect on target-in-the-loop optical phased array (OPA) performance. Assuming a direct-solve, piston-only-phase-compensation OPA, two expressions for the atmospheric array tilt variance are derived using Mellin transform techniques. The first—the “full” array tilt variance—is germane when the OPA is sensitive to atmospheric tilt and is shown to significantly impact OPA target-plane intensity. The second—the Zernike-tilt-removed array tilt variance—is relevant when a separate system compensates for atmospheric tilt (the more likely scenario) and is shown to negligibly affect OPA performance. To show how atmospheric array tilt errors affect target-plane intensity, moments of the far-zone (or focused) array intensity, as functions of the array tilt variance, are derived and discussed. Lastly, Monte Carlo simulation results are presented to validate the theoretical array tilt variance expressions.
The development of active phase-locking techniques, most notably, locking of optical coherence by single-detector electronic-frequency tagging (LOCSET) [1–3] and stochastic parallel gradient descent (SPGD) [1,4,5], have made the coherent combination of multiple fiber lasers possible. Originally demonstrated using narrow-linewidth, low-power lasers, these techniques and other similar active coherent beam combining (CBC) methods have recently been extended to phase high-power (multi-kilowatt), broadband (tens of gigahertz) fiber lasers [1,6–9]. This latter achievement clears a major engineering hurdle (namely, the excitation of stimulated Brillouin scattering in high-power, narrow-linewidth fiber lasers) to fielding these systems for directed energy applications [10,11].
With a few notable exceptions [5,12–18], a majority of the published CBC research to date has focused on locally phasing multiple fiber lasers. To successfully deploy a coherently combined fiber laser array [hereafter, referred to as an optical phased array (OPA)] as a directed energy weapon requires target-in-the-loop (or target-based) phasing, where, in addition to local phase errors, atmospheric-turbulence-induced and target-induced (i.e., speckle) phase errors must be sensed and corrected.
Target-in-the-loop (TIL) phasing approaches can generally be organized into two groups—iterative, or “hill-climbing” (of which, SPGD and LOCSET are two examples) and direct-solve  techniques. The latter is quite interesting because it provides estimates of the local (referred to as telescope in Tyler’s work ), atmosphere, and target (or speckle) phases, as opposed to the former, which only provides telescope and atmosphere phases .
As discussed in Refs. [19,21], based on the geometrical configuration of the array transmitters and receivers, the speckle phase estimates are obtained on a synthetic aperture that is approximately twice as large as the array. Since the speckle phase estimates are, in essence, the pupil plane field absent atmospheric degradation, they can be used to produce a high-resolution image (synthetic aperture image) of the target, free of atmospheric distortion . The benefits of this are quite obvious.
Unfortunately, the direct-solve approach discussed by Tyler does not accurately estimate several low-order array aberrations . Of these, the one that most negatively impacts OPA performance is the array tilt aberration .
Array tilt is the discrete representation of the aberration tilt, i.e., Zernike polynomials 2 and 3 [23,24]. It can be visualized as a stair-step phase ramp across the array and has the effect of steering the main lobe off axis, transferring power into the grating lobes, and consequently, decreasing on-axis intensity. Since the direct-solve approach does not accurately estimate array tilt, it must be sensed and corrected by some other means.
One approach that has been proposed is the aptly named array tilt imager [22,25], where a dedicated system images the spot on the target and estimates the amount of array tilt using pattern recognition techniques. While simple in concept, this approach has many drawbacks. Foremost among them is the size of the entrance pupil, which, at a minimum, must be as large as an array transmitter to resolve target spot features. When one considers the effects of turbulence and speckle, the pupil size will likely need to be much larger, and the imaging system will possibly require adaptive optics to function effectively.
Another idea that has been investigated, but not published, is to combine the TIL phase estimates with those of a local phasing loop. An approximate linear analysis of this technique is presented in Appendix A.
The local-loop method for correcting array tilt is built upon the following idea: The local phasing system correctly estimates all telescope aberrations, including telescope array tilt. The TIL phasing system correctly estimates the higher-order (everything except array tilt) atmospheric aberrations . After combining the local-loop and TIL estimates, all telescope and higher-order atmospheric aberrations can be corrected. This leaves only atmospheric-induced array tilt (and array piston, which does not affect the target-plane intensity) uncorrected in the target plane.
Because of the stringent optical-path-length-matching requirements (fractions of a millimeter) for phasing broadband, high-power fiber lasers [1,6–9], OPAs will likely require a local phasing loop in addition to the TIL phasing system . Thus, this local-loop approach to estimating and correcting array tilt does not add to system complexity and, in this regard, is far superior to the array tilt imager discussed above. However, since this approach does not correct atmospheric-induced array tilt (see Appendix A and above), this quantity must be small for the local-loop method to be effective.
For this reason, the purpose of this paper is to investigate and quantify the amount of atmospheric array tilt. In the next section, two expressions for the atmospheric array tilt variance (the “full” and Zernike-tilt-removed array tilt variances) are derived using the Mellin transform techniques developed in Ref. .
The analysis assumes an array with identical circular elements arranged in a regular hexagon, as this is the most commonly proposed OPA geometry, and an OPA, which employs piston-only phase compensation. The analysis also assumes weak atmospheric turbulence and isoplanatic conditions—, where is the diameter of an array element, is the wavelength, and is the isoplanatic angle [26–30].
The former (weak turbulence) implies negligible scintillation, which ensures that phase-only compensation is sufficient to phase the array on target. The latter (isoplanatic conditions) ensures that an array (with compensation) can focus to a spot whose angular subtense is approximately (near diffraction-limited performance) . It also permits the atmosphere to be modeled as a single phase screen located in the array plane.
To quantify the impact of atmospheric array tilt on OPA performance, the mean and variance of the far-zone (or equivalently, focused) OPA intensity, as functions of the array tilt variance, are derived and discussed. Lastly, Section 3 presents Monte Carlo simulation results to validate the theoretical array tilt variances derived in Section 2.
Figure 1 shows the array geometry that is used in this work. The figure depicts a seven-element hexagonal array of identical circular elements. Each element has a diameter ; they are center-to-center spaced apart. The array is circumscribed by a circle of radius , which is the equivalent clear-aperture radius.
The seven-element array depicted in Fig. 1 is only a convenient representative case of the arrays analyzed in this work. In the analysis to follow, no limitations are placed on array size, number of elements, or fill factor. The only requirement is that the array be composed of identical circular elements arranged in a regular hexagon.
Because of this, it is necessary to introduce a generalized clear-aperture radius expression,32]
A. Array Tilt Variance
The amount of array tilt contained within an instance of the atmosphere can be obtained by first spatially averaging the atmospheric phase over each array element and then least-squares fitting the average phases to the array tilt aberration, i.e.,22]. The atmospheric phase averaged over element is 33].
The variance of and are of interest here; thus, taking the autocovariance of Eq. (3) yields5) can easily be inverted to yield the autocovariance matrix of and array tilts:
The autocovariance matrix of is the main quantity to be derived in Eq. (6). The th row, th column of that matrix takes the form4)], and . The moment in the integrand of Eq. (7) is the autocovariance (or, equivalently in this case, the autocorrelation) of the atmospheric phase . Assuming that the atmospheric phase is statistically homogeneous and isotropic [27,28,34] and making the variable substitutions and transforms Eq. (7) to
Equation (8) is most easily evaluated in the spatial frequency domain; thus, defining the Fourier transform pair8), and simplifying yields
It is well known that Eq. (10) does not converge when one uses the Kolmogorov spectrum [24,26,35,36]. Because of this and the fact that outer scale has a significant effect on the variance of low-order aberrations [26,35], the von Kármán phase spectrum is used [37,38]:26–28,34]. Substituting Eq. (11), , and into Eq. (10) and simplifying produces 26,28,39]. When , Eq. (12) evaluates to 26,28].3)]. Because of this and , all results presented henceforth will show only .
Returning briefly to the above example, the standard deviation of the array tilt is 0.1685 waves. The full impact of this array tilt error will be shown later. Nonetheless, this is a large amount of phase error and, on average, would devastatingly impact OPA performance.
A critical realization is that , via , contains atmospheric tilt, or Zernike tilt. Zernike tilt causes the array spot to wander but otherwise does not change the target-plane intensity. It, by itself, does not cause array tilt; however, if the OPA senses and then attempts to correct Zernike tilt, array tilt is induced. Thus, the 0.1685 wave error is accurate if the OPA is sensitive to Zernike tilt in addition to the other higher-order aberrations that comprise array tilt.
This scenario is unlikely for two reasons. First, there is evidence that OPAs, employing either iterative or direct-solve TIL phasing schemes, are insensitive to Zernike tilt, i.e., they phase on the target near the initial hit point independent of location [40,41]. This physically makes sense considering that the received speckle field—used to phase the array on target—generally looks the same regardless of target hit point, viz., the speckle field is statistically homogeneous. Although the OPA is likely to phase wherever it strikes the target, the hit point will wander as the atmospheric tilt changes, resulting in a significant decrease in performance.
Second, even if the OPA were to sense atmospheric tilt, beam steering using OPAs is limited to very small angles—approximately , where [22,25,32]. This makes OPA tilt correction inefficient and impractical. For OPAs using piston-only phase compensation, like the ones considered here, a tracking system in combination with a beam director will likely be necessary to compensate for Zernike tilt and maintain the target hit point.
If ones assumes that a separate system will compensate for atmospheric tilt, the true quantity of interest is the Zernike-tilt-removed array tilt variance . This quantity is derived in the next section.
Before proceeding to this derivation, it is interesting to note that the differential piston variance  can be obtained from Eqs. (13) and (14). In the context of this work, the differential piston variance is3) and taking the limit as in Eq. (16) [via Eqs. (13) and (14)], one can obtain the array tilt variance from using the Kolmogorov phase power spectrum. This analysis is not presented here, as it yields similar results to the phase autocovariance approach (assuming a large but finite value of ) presented above.
B. Zernike-Tilt-Removed Array Tilt Variance
The procedure for deriving the Zernike-tilt-removed array tilt variance is the same as above, except Zernike tilt is first removed from the atmospheric phase instance:4), the subscript “tr” stands for tilt removed, and are Zernike polynomials 2 and 3 ( and tilt) spatially averaged over element , and and are their respective weights, i.e., B). Taking the autocovariance (or equivalently, the autocorrelation) of Eq. (17) produces 13) and (14).
The moments in Eq. (19) are of the form24,26], transforming the integrals into polar coordinates, and evaluating the integrals over the angle in transform space produces
Like Eq. (12), both integrals in Eq. (21) can be evaluated using Mellin transforms, resulting in Taylor series, which rapidly converge when . The single-axis Zernike tilt variance, including outer scale [the first moment in Eq. (21)], was evaluated in Refs. [26,35]:21) is
With the above analysis, Eq. (19) simplifies to6) yields the Zernike-tilt-removed array tilt variance.
Returning to the above example, where , , , and , the Zernike-tilt-removed array tilt variance is , which equates to a 0.01 wave error. This amount of phase error has a negligible impact on OPA performance and will likely be insignificant when compared to other phase errors such as fitting error, temporal error, etc [29,30]. These results and those of the prior section are analyzed further below.
C. OPA Performance
Here, the mean and variance of the target-plane intensity, as functions of the array tilt variance, are derived. The source-plane array field takes the form
The second moment of intensity takes the form2 shows the mean target-plane intensities using and [Fig. 2(a) and 2(b), respectively] computed above. The mean intensity values in both figures have been “normalized” by the on-axis intensity value for an incoherently combined array . Thus, ideal OPA performance is an on-axis value of . The white circle on both figures marks the edge of the Airy disk for a single array element with . Note that OPA performance in Fig. 2(a) is only marginally better than an equivalent (, , and ) incoherently combined array. The on-axis intensity statistics are and for and , respectively.
These results visually show the impact of atmospheric array tilt errors on target-plane intensity (i.e., OPA performance). It should be stated that these results are optimistic, in the sense that neither includes phase errors like fitting error or temporal error [29,30]. Nevertheless, they clearly and not surprisingly emphasize the need for tilt compensation and, most importantly for this work, show that the turbulence-induced array tilt (assuming atmospheric tilt is corrected) has a trivial impact on OPA performance.
A. Simulation Setup
Monte Carlo simulations were performed varying , , and to validate the array tilt variance expressions derived above: varied from 2 to 0.5 in seven steps, varied from 1 to 0.25 in ten steps, , and . For each of the seven values, 2000 atmospheric phase screens were synthesized using the Fourier transform method augmented with subharmonics . The von Kármán phase power spectrum [see Eq. (11)] with an was used to synthesize the screens. Each screen was discretized using 2048 points on a side with a sample spacing equal to 2.4414 mm.
The simulation procedure for computing the Zernike-tilt-removed array tilt variance was as follows. For each of the 2000 screens comprising each of the seven values,
- 1. the amount of Zernike tilt present in the screen was estimated over the clear-aperture area of the array;
- 2. the estimated Zernike tilt from step 1 was removed from the screen;
- 3. the phase encircled by each array element was spatially averaged;
- 4. the average phases from step 3 were least-squares fitted to the array tilt aberration via Eq. (3);
- 5. the covariance matrix was computed over the 2000 and instances from step 4.
Steps 1–5 were repeated for each of the ten and three values. Steps 1 and 2 were skipped when computing the “full” array tilt variance.
Figures 3–5 show the results for the 7, 19, and 37 element arrays, respectively. The three figures are composed of two subfigures, which show and versus in (a) and (b), respectively. On each subfigure, the results for the seven values—differentiated by color—are plotted on a log scale. The solid traces are the theoretical array tilt variances; the symbols are the variances obtained from the simulation.
The agreement between the theoretical and simulated array tilt variances is excellent. These results validate the theory presented in Section 2. Two general trends are evident in these results. First, the array tilt variance increases as increases. This result is intuitive and not surprising. Second, and most interestingly, the array tilt variance generally increases as fill factor decreases, yet decreases as array element number increases. This is a counterintuitive result—array tilt either increases or decreases depending on the manner in which grows—and shows that the relationship between array tilt and array size is multifaceted. This complex relationship is on further display in the 19 and 37 element results [Figs. 4(b) and 5(b)], where minima occur at approximately equal to 0.6 and 0.5, respectively.
It must be stated that phase differences between array elements greater than cannot be sensed. This could result in erroneous array tilt estimates because aberrations that couple into array tilt (which produce phase differences between elements greater than ) will alias to incorrect, lower array tilt values. This “phase aliasing” and its impact on array tilt correction were studied in Refs. [22,43]. It was shown that unique array tilt patterns occur within a diamond-shaped region with vertices located at , and . Array tilts that lie outside this region alias (or wrap) back into the diamond. As an example, array tilts , and produce the same array tilt pattern and are effectively equivalent. All of these can be corrected by adding or subtracting 0.5 waves of or array tilt.
Although Refs. [22,43] showed that phase aliasing does not affect array tilt correction, it does impact the array tilt statistics presented in this paper. Here, continuous, or unwrapped, phases are used to avoid the significant mathematical complications that arise from using modulo-, or wrapped, phases. These difficulties are caused by the “wrapping function,” which performs a nonlinear operation. Since phase differences between elements can exceed here, the analysis presented in this paper likely over-predicts the atmospheric array tilt variance.
Evidence supporting this hypothesis can be found in Ref. . There, the authors computed through simulation the probability distribution of array tilts, assuming the element phases were independent and uniformly distributed between and . Starting from wrapped phases, they estimated the array tilts both linearly [as done here, see Eq. (3)] and nonlinearly (directly from the complex phasors). For the corresponding cases, i.e., Fig. 3(a) when , the array tilt variances reported here are larger than in Ref. .
Although the approach employed here might over-predict the array tilt variance, this work’s key finding regarding atmospheric-induced array tilt’s negligible impact on OPA performance—assuming Zernike tilt is corrected—is still valid.
This paper quantified the amount of array tilt present in atmospheric turbulence and investigated its effect on OPA performance. Two forms of the atmospheric array tilt variance were derived—the “full” and Zernike-tilt-removed array tilt variances.
The “full” array tilt variance included atmospheric tilt and was appropriate if the OPA was sensitive to Zernike tilt. The amount of associated phase error was shown to be large, so much so that OPA performance was only marginally better than an equivalent incoherently combined array.
The Zernike-tilt-removed array tilt variance did not include atmospheric tilt and was relevant to scenarios in which a tracking system compensated for tilt (very likely in practice). The corresponding amount of phase error was found to be small and likely insignificant compared to other errors, like fitting and temporal phase errors [29,30].
To show how these two array tilt errors affected OPA performance, the mean and variance of the target-plane (far-zone) array intensity, as functions of the array tilt variance, were derived and discussed. Lastly, Monte Carlo simulation results were presented to validate the theoretical array tilt variance expressions derived in Section 2. The simulated results were in excellent agreement with the theoretical array tilt variances.
This work is germane to the design of OPAs thatutilize direct-solve TIL phasing methods. These methods cannot accurately estimate the array tilt aberration . Recent unpublished work has developed a technique to correct array tilt by combining the TIL phase estimates with those of a local phasing loop. This method does not correct atmospheric-induced array tilt and therefore requires this phase error to be small to work effectively.
This work definitively shows that if atmospheric tilt is corrected, atmospheric-induced array tilt does not significantly affect OPA performance. This research addresses the primary concern regarding adoption of the local-loop method to correct array tilt for direct-solve TIL OPAs.
APPENDIX A: LINEAR ANALYSIS OF LOCAL-LOOP ARRAY TILT ESTIMATOR
The TIL and local-loop phase estimates are19]. The higher-order target phases are estimated correctly and are not included in because they are not used to phase the array on the target . Array pistons are neglected in this analysis.
Taking the difference of and yields44] to Eq. (A3) produces
The phase command, or correction, is found by adding Eq. (A4) to ,
APPENDIX B: AND AVERAGED OVER AN ARRAY ELEMENT
Zernike polynomials 2 and 3 spatially averaged over array element are24,26], transforming the integrals into polar coordinates, and evaluating the integral over the angle in transform space produces 26,39]. Substituting in the particulars from Eq. (B2) and simplifying yields
I thank Dr. Jack E. McCrae for many helpful discussions on the nature of array tilt and for his feedback on this paper. The views expressed in this paper are those of the author and do not reflect the official policy or position of the U.S. Air Force, the Department of Defense, or the U.S. Government.
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