Abstract

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.

1. INTRODUCTION

The development of active phase-locking techniques, most notably, locking of optical coherence by single-detector electronic-frequency tagging (LOCSET) [13] 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 [1,69]. 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 [5,1218], 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 [19] techniques. The latter is quite interesting because it provides estimates of the local (referred to as telescope in Tyler’s work [19]), atmosphere, and target (or speckle) phases, as opposed to the former, which only provides telescope and atmosphere phases [20].

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 [21]. The benefits of this are quite obvious.

Unfortunately, the direct-solve approach discussed by Tyler does not accurately estimate several low-order array aberrations [19]. Of these, the one that most negatively impacts OPA performance is the array tilt aberration [22].

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 [19]. 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 [1,69], OPAs will likely require a local phasing loop in addition to the TIL phasing system [11]. 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. [26].

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—λ/d<θ0, where d is the diameter of an array element, λ is the wavelength, and θ0 is the isoplanatic angle [2630].

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 λ/d (near diffraction-limited performance) [31]. 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.

2. THEORY

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 d; they are center-to-center spaced D apart. The array is circumscribed by a circle of radius R7=D+d/2, which is the equivalent clear-aperture radius.

 figure: Fig. 1.

Fig. 1. Seven-element hexagonal array composed of identical circular elements of diameter d and center-to-center spacing D. R7 is the radius of the circle that circumscribes the array.

Download Full Size | PPT Slide | PDF

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,

RN=LD+d2,
where N is the number of array elements and L is the number of loops or rings in the hexagonal array. N and L are related by [32]
N=1+6l=1Ll.

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.,

ϕ=2πRW,[ϕ1ϕN]=2π[2D3x12Dy12D3xN2DyN][WxWy],
where xi and yi are the x and y coordinates of the ith element center, respectively, and Wx and Wy are the amounts of x and y array tilt in waves, respectively [22]. The atmospheric phase averaged over element i is
ϕi=1π(d/2)2circ(|ρρi|d/2)ϕ(ρ)d2ρ,
where ρi=x^xi+y^yi, ρ=x^x+y^y, ϕ is an instance of atmospheric phase, and circ is the circle function defined by Goodman [33].

The variance of Wx and Wy are of interest here; thus, taking the autocovariance of Eq. (3) yields

ϕϕ=(2π)2RWWR,
where superscript denotes the Hermitian transpose, and · is the average over the ensemble. Note that ϕ=0 because ϕ=0. Equation (5) can easily be inverted to yield the autocovariance matrix of x and y array tilts:
WW=1(2π)2(RR)1RϕϕR(RR)1.

The autocovariance matrix of ϕ is the main quantity to be derived in Eq. (6). The ith row, jth column of that matrix takes the form

ϕiϕj*=1A2Pi(ρ1)Pj*(ρ2)ϕ(ρ1)ϕ*(ρ2)d2ρ1d2ρ2,
where, for convenience, Pi and Pj have been introduced to represent the ith and jth element circle functions, respectively [see Eq. (4)], and A=π(d/2)2. The moment in the integrand of Eq. (7) is the autocovariance (or, equivalently in this case, the autocorrelation) of the atmospheric phase Γϕ [27]. Assuming that the atmospheric phase is statistically homogeneous and isotropic [27,28,34] and making the variable substitutions s=ρ1 and t=ρ1ρ2 transforms Eq. (7) to
ϕiϕj*=1A2Γϕ(t)Pi(s)Pj*(st)d2sd2t.

Equation (8) is most easily evaluated in the spatial frequency domain; thus, defining the Fourier transform pair

P˜i(f)=Pi(s)exp(j2πf·s)d2s,Pi(s)=P˜i(f)exp(j2πf·s)d2f,
substituting these into Eq. (8), and simplifying yields
ϕiϕj*=1A2P˜i(f)ϕϕ(f)P˜j*(f)d2f,
where ϕϕ is the atmospheric phase power spectrum.

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 L0 has a significant effect on the variance of low-order aberrations [26,35], the von Kármán phase spectrum is used [37,38]:

ϕϕ(f)=5π8/311[245Γ(65)]5/6Γ(17/6)Γ(1/6)r05/3(f2+f02)11/60.023r05/3(f2+f02)11/6,
where f0=1/L0 and r0 is the atmospheric coherence diameter (Fried’s parameter) [2628,34]. Substituting Eq. (11), P˜i, and P˜j* into Eq. (10) and simplifying produces
ϕiϕj*=0.046Af011/3r05/30dffJ12(f)J0(ρijd/2f)×[1+(1πdf0)2f2]11/6,
where J0 and J1 are zero- and first-order Bessel functions of the first kind and ρij=|ρiρj|.

Equation (12) can be evaluated using Mellin transforms. This analysis yields two Taylor series—one applicable when i=j, the other when ij—which rapidly converge when L0RN [26]. When i=j, Eq. (12) evaluates to

|ϕi|2=0.0863(L0r0)5/3F32(32,1;16,3,2;π2d2L02)1.0324(dr0)5/3F21(73;176,236;π2d2L02),
where Fba is the generalized hypergeometric function [26,28,39]. When ij, Eq. (12) evaluates to
ϕiϕj*=0.0863(L0r0)5/3n=0(πρij/L0)2n(1)n(1/6)n×F32(32,n+1;n+16,3,2;π2d2L02)3.4419(ρijr0)5/3n=0(πρij/L0)2n(1)n(11/6)n×F23(32,n56,n56;3,2;d2ρij2),
where (a)n is the Pochhammer symbol [26,28].

Equations (13) and (14) are the main results of this section. As a realistic example, let d=r0=0.1m, L0=100m, d/D=0.95, and N=7. Evaluating Eqs. (13) and (14) and substituting the results into Eq. (6) produces

WW[0.0852000.0284].
Note that |Wx|2=3|Wy|2 due to the geometry of the array, in particular, the 3 in the denominators of R’s first column [see Eq. (3)]. Because of this and WxWy*=0, all results presented henceforth will show only |Wy|2.

Returning briefly to the above example, the standard deviation of the y 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 |Wy|2, via ϕiϕj*, 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 λ/d, where dλ [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 [42]. This quantity is derived in the next section.

Before proceeding to this derivation, it is interesting to note that the differential piston variance [36] can be obtained from Eqs. (13) and (14). In the context of this work, the differential piston variance is

Dij=|ϕiϕj|2=2(ϕi2ϕiϕj),
where the latter equality uses the fact that ϕ is real. The integrals in this expression converge for infinite outer scale; thus, with minor manipulations to Eq. (3) and taking the limit as L0 in Eq. (16) [via Eqs. (13) and (14)], one can obtain the array tilt variance from Dij 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 L0) 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:

ϕi,tr=ϕia2Z2,ia3Z3,i.
Here, ϕi is given in Eq. (4), the subscript “tr” stands for tilt removed, Z2,i and Z3,i are Zernike polynomials 2 and 3 (x and y tilt) spatially averaged over element i, and a2 and a3 are their respective weights, i.e.,
ak=W(ρ)Zk(ρ)ϕ(ρ)d2ρ,W(ρ)={1πRN2ρ<RN0ρ>RN,
where k=2,3. The Z2,i and Z3,i integrals evaluate to 2xi/RN and 2yi/RN, respectively (details shown in Appendix B). Taking the autocovariance (or equivalently, the autocorrelation) of Eq. (17) produces
ϕi,trϕj,tr*=ϕiϕj*+4RN2(xixj|a2|2+yiyj|a3|2+xiyja2a3*+xjyia2*a3)2RN(xia2ϕj*+yia3ϕj*+xja2*ϕi+yja3*ϕi),
where ϕiϕj* is given in Eqs. (13) and (14).

The moments in Eq. (19) are of the form

akal*=W(ρ1)Zk(ρ1)W*(ρ2)Zl*(ρ2)×Γϕ(ρ1ρ2)d2ρ1d2ρ2,akϕi*=1AW(ρ1)Zk(ρ1)Pi*(ρ2)×Γϕ(ρ1ρ2)d2ρ1d2ρ2,
where k,l=2,3. Substituting in the spatial Fourier transform for W(ρ)Zk(ρ) [24,26], transforming the integrals into polar coordinates, and evaluating the integrals over the angle in transform space produces
|ak|2=0.092πRN2r05/3f011/30dffJ22(f)×[1+(12πf0RN)2f2]11/6,[a2ϕi*a3ϕi*]=[cosθisinθi]0.092(d/2)πRNr05/3f011/30dff×J1(d/2RNf)J1(ρiRNf)J2(f)[1+(12πf0RN)2f2]11/6,
where J2 is a second-order Bessel function of the first kind, akal*=0 when kl, and θi=tan1(yi/xi).

Like Eq. (12), both integrals in Eq. (21) can be evaluated using Mellin transforms, resulting in Taylor series, which rapidly converge when L0RN [26]. The single-axis Zernike tilt variance, including outer scale [the first moment in Eq. (21)], was evaluated in Refs. [26,35]:

|a2|2=|a3|2=|aT|2=1.4251(RNr0)5/3F32(73,116;56,176,296;4π2RN2L02)2.5557(RNL0)1/3(RNr0)5/3F32(52,2;5,3,76;4π2RN2L02).
The last moment in Eq. (21) is
[a2ϕi*a3ϕi*]=2RN[xiyi]IiIi=1.5645(RNr0)5/3×m=0n=0(d2RN)2m(ρiRN)2n(16)m+n(116)m+n(2)m(1)m(2)n(1)n×F21(116;mn+56,mn+176;π2RN2L02)2.5557(RNL0)1/3(RNr0)5/3×m=0n=0(πd2L0)2m(πρiL0)2n(2)m+n(2)m(1)m(2)n(1)n(76)m+n×F21(m+n+2;m+n+76,3;π2RN2L02).

With the above analysis, Eq. (19) simplifies to

ϕi,trϕj,tr*=ϕiϕj*+4RN2(xixj+yiyj)(|aT|2IiIj).
Evaluating this expression and substituting the results into Eq. (6) yields the Zernike-tilt-removed array tilt variance.

Returning to the above example, where d=r0=0.1m, L0=100m, d/D=0.95, and N=7, the Zernike-tilt-removed array tilt variance is |Wy,tr|2=9.9985×105waves2, 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

U(ρ)=i=1NPi(ρ)exp(jϕi),ϕi=2π2D3xiWx+2π2DyiWy.
The mean intensity in the target plane, assuming that the field is focused on the target and all aberrations except array tilt are perfectly corrected, is
I(ρ,z)=[Aλz]2jinc2(kzd2ρ)×i=1Nj=1Nexp[j(ϕiϕj)]exp[jkz(ρiρj)·ρ],
where λ is the wavelength, k=2π/λ, and jinc(x)=2J1(x)/x.

The moment in Eq. (26) is related to the joint characteristic function of Wx and Wy [34]. Assuming that Wx and Wy are independent, zero-mean, Gaussian random variables, the mean intensity simplifies to

I(ρ,z)=[Aλz]2jinc2(kzd2ρ)×i=1Nj=1Nexp(8π2D2ρij2|Wy|2)exp(jkzρij·ρ),
where |Wx|2=3|Wy|2 has been used to arrive at this expression.

The second moment of intensity takes the form

I2(ρ,z)=[Aλz]4jinc4(kzd2ρ)×i=1Nj=1Nm=1Nn=1Nexp[j(ϕiϕj+ϕmϕn)]×exp(jkzρij·ρ)exp(jkzρmn·ρ).
Again, assuming that Wx and Wy are independent, zero-mean, Gaussian random variables, the above expression simplifies to
I2(ρ,z)=[Aλz]4jinc4(kzd2ρ)×i=1Nj=1Nm=1Nn=1Nexp(8π2D2ρijmn2|Wy|2)×exp(jkzρijmn·ρ),
where ρijmn=ρij+ρmn=ρiρj+ρmρn and ρijmn=|ρijmn|. The target-plane intensity variance is
σI2(ρ,z)=I2(ρ,z)[I(ρ,z)]2.
Figure 2 shows the mean target-plane intensities using |Wy|2 and |Wy,tr|2 [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 Iic. Thus, ideal OPA performance is an on-axis value of N=7. The white circle on both figures marks the edge of the Airy disk for a single array element with λ=1μm. Note that OPA performance in Fig. 2(a) is only marginally better than an equivalent (d=0.1m, d/D=0.95, and N=7) incoherently combined array. The on-axis intensity statistics (I±σI)/Iic are 1.3662±1.8672 and 6.9063±0.092735 for |Wy|2 and |Wy,tr|2, respectively.

 figure: Fig. 2.

Fig. 2. Mean target-plane intensities using (a) |Wy|2=0.0284waves2 and (b) |Wy,tr|2=9.9985×105waves2, respectively. These array tilt variances were computed assuming that d=r0=0.1m, L0=100m, d/D=0.95, and N=7. The white circle in both images marks the edge of the Airy disk for a single array element with λ=1μm.

Download Full Size | PPT Slide | PDF

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.

3. VALIDATION

A. Simulation Setup

Monte Carlo simulations were performed varying d/r0, d/D, and N to validate the array tilt variance expressions derived above: d/r0 varied from 2 to 0.5 in seven steps, d/D varied from 1 to 0.25 in ten steps, N=7,19,37, and d=0.1m. For each of the seven d/r0 values, 2000 atmospheric phase screens were synthesized using the Fourier transform method augmented with subharmonics [37]. The von Kármán phase power spectrum [see Eq. (11)] with an L0=10m 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 d/r0 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 WtrWtr was computed over the 2000 Wx,tr and Wy,tr instances from step 4.

Steps 1–5 were repeated for each of the ten d/D and three N values. Steps 1 and 2 were skipped when computing the “full” array tilt variance.

B. Results

Figures 35 show the results for the 7, 19, and 37 element arrays, respectively. The three figures are composed of two subfigures, which show |Wy|2 and |Wy,tr|2 versus d/D in (a) and (b), respectively. On each subfigure, the results for the seven d/r0 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.

 figure: Fig. 3.

Fig. 3. Array tilt variances for an N=7 element array plotted versus fill factor d/D—(a) “full” array tilt variance |Wy|2 and (b) Zernike-tilt-removed array tilt variance |Wy,tr|2. The solid traces are the theoretical variances; the symbols are the variances obtained from the simulation. d/r0 values are differentiated by the color of the trace or symbol.

Download Full Size | PPT Slide | PDF

 figure: Fig. 4.

Fig. 4. Array tilt variances for an N=19 element array plotted versus fill factor d/D—(a) “full” array tilt variance |Wy|2 and (b) Zernike-tilt-removed array tilt variance |Wy,tr|2. The solid traces are the theoretical variances; the symbols are the variances obtained from the simulation. d/r0 values are differentiated by the color of the trace or symbol.

Download Full Size | PPT Slide | PDF

 figure: Fig. 5.

Fig. 5. Array tilt variances for an N=37 element array plotted versus fill factor d/D—(a) “full” array tilt variance |Wy|2 and (b) Zernike-tilt-removed array tilt variance |Wy,tr|2. The solid traces are the theoretical variances; the symbols are the variances obtained from the simulation. d/r0 values are differentiated by the color of the trace or symbol.

Download Full Size | PPT Slide | PDF

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 d/r0 increases. This result is intuitive and not surprising. Second, and most interestingly, the array tilt variance generally increases as fill factor d/D decreases, yet decreases as array element number N increases. This is a counterintuitive result—array tilt either increases or decreases depending on the manner in which RN 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 |Wy,tr|2 results [Figs. 4(b) and 5(b)], where |Wy,tr|2 minima occur at d/D approximately equal to 0.6 and 0.5, respectively.

4. DISCUSSION

It must be stated that phase differences between array elements greater than 2π 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 2π) 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 (Wx,Wy)=(0.5,0),(0,0.5),(0.5,0), and (0,0.5). Array tilts that lie outside this region alias (or wrap) back into the diamond. As an example, array tilts (0.5,0),(0,0.5),(0.5,0), and (0,0.5) produce the same array tilt pattern and are effectively equivalent. All of these can be corrected by adding or subtracting 0.5 waves of x or y 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-2π, or wrapped, phases. These difficulties are caused by the “wrapping function,” which performs a nonlinear operation. Since phase differences between elements can exceed 2π here, the analysis presented in this paper likely over-predicts the atmospheric array tilt variance.

Evidence supporting this hypothesis can be found in Ref. [44]. There, the authors computed through simulation the probability distribution of array tilts, assuming the element phases were independent and uniformly distributed between π and π. 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 d/r01, the array tilt variances reported here are larger than in Ref. [44].

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.

5. CONCLUSION

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 [19]. 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 are

ϕTIL=ϕtele+ϕatm+ϕtar,at,ϕloc=ϕtele,
where the telescope and atmospheric phases are
ϕtele=ϕtele,ho+ϕtele,at,ϕatm=ϕatm,ho+ϕatm,at,
and “ho” and “at” stand for higher order and array tilt, respectively. The target-induced array tilt ϕtar,at is included in ϕTIL because it corrupts the TIL telescope and atmospheric phase estimates [19]. The higher-order target phases are estimated correctly and are not included in ϕTIL because they are not used to phase the array on the target [19]. Array pistons are neglected in this analysis.

Taking the difference of ϕloc and ϕTIL yields

ϕlocϕTIL=ϕatm,hoϕatm,atϕtar,at.
Applying the array tilt estimator discussed in Ref. [44] to Eq. (A3) produces
Wat(ϕlocϕTIL)=ϕatm,atϕtar,at.
Note that ϕatm,at and ϕtar,at cannot be individually estimated.

The phase command, or correction, is found by adding Eq. (A4) to ϕTIL,

ϕcmd=ϕTIL+Wat(ϕlocϕTIL)=ϕtele+ϕatm,ho.
The residual phase, or error after correction, is
ϕres=ϕtrueϕcmd=ϕtele+ϕatmϕteleϕatm,ho=ϕatm,at.
Thus, the local-loop method for estimating array tilt corrects everything except atmospheric-induced array tilt (and an insignificant array piston, which is ignored).

APPENDIX B: Z2 AND Z3 AVERAGED OVER AN ARRAY ELEMENT

Zernike polynomials 2 and 3 spatially averaged over array element i are

Zk,i=1APi(ρ)Zk(ρ)d2ρ=πRN2APi(ρ)W(ρ)Zk(ρ)d2ρ.
Substituting in the spatial Fourier transform for W(ρ)Zk(ρ) [24,26], transforming the integrals into polar coordinates, and evaluating the integral over the angle in transform space produces
[Z2,iZ3,i]=4RNd/2[cosθisinθi]0dffJ1(2πfd2)J1(2πfρi)×J2(2πfRN).
The integral in the above expression is evaluated in Refs. [26,39]. Substituting in the particulars from Eq. (B2) and simplifying yields
[Z2,iZ3,i]=2ρiRN[cosθisinθi]m=0n=0(1)m(1)nm!n!(d2RN)2m×(ρiRN)2nΓ(m+n+2)Γ(mn+1)Γ(m+2)Γ(n+2).
The first gamma function in the denominator is infinite for all values of m,n0; thus, only the m=n=0 term of the double sum is nonzero:
[Z2,iZ3,i]=2ρiRN[cosθisinθi]=2RN[xiyi].

Acknowledgment

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.

REFERENCES AND NOTES

1. A. Brignon, ed., Coherent Laser Beam Combining (Wiley-VCH, 2013).

2. B. N. Pulford, “LOCSET phase locking: operation, diagnostics, and applications,” Ph.D. thesis (The University of New Mexico, 2011).

3. T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 14, 12188–12195 (2006). [CrossRef]  

4. M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009). [CrossRef]  

5. M. Vorontsov, G. Filimonov, V. Ovchinnikov, E. Polnau, S. Lachinova, T. Weyrauch, and J. Mangano, “Comparative efficiency analysis of fiber-array and conventional beam director systems in volume turbulence,” Appl. Opt. 55, 4170–4185 (2016). [CrossRef]  

6. G. D. Goodno and J. E. Rothenberg, “Atmospheric propagation and combining of high power lasers: comment,” Appl. Opt. 55, 8335–8337 (2016). [CrossRef]  

7. M. A. Vorontsov and T. Weyrauch, “High-power lasers for directed-energy applications: comment,” Appl. Opt. 55, 9950–9953 (2016). [CrossRef]  

8. S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014). [CrossRef]  

9. A. Flores, T. Ehrehreich, R. Holten, B. Anderson, and I. Dajani, “Multi-kW coherent combining of fiber lasers seeded with pseudo random phase modulated light,” Proc. SPIE 9728, 97281Y (2016). [CrossRef]  

10. P. Sprangle, B. Hafizi, A. Ting, and R. Fischer, “High-power lasers for directed-energy applications,” Appl. Opt. 54, F201–F209 (2015). [CrossRef]  

11. W. Nelson, P. Sprangle, and C. C. Davis, “Atmospheric propagation and combining of high-power lasers,” Appl. Opt. 55, 1757–1764 (2016). [CrossRef]  

12. M. A. Vorontsov and S. L. Lachinova, “Laser beam projection with adaptive array of fiber collimators. I. Basic considerations for analysis,” J. Opt. Soc. Am. A 25, 1949–1959 (2008). [CrossRef]  

13. S. L. Lachinova and M. A. Vorontsov, “Laser beam projection with adaptive array of fiber collimators. II. Analysis of atmospheric compensation efficiency,” J. Opt. Soc. Am. A 25, 1960–1973 (2008). [CrossRef]  

14. M. A. Vorontsov, “Speckle effects in target-in-the-loop laser beam projection systems,” Adv. Opt. Technol. 2, 369–395 (2013). [CrossRef]  

15. T. Weyrauch, M. Vorontsov, J. Mangano, V. Ovchinnikov, D. Bricker, E. Polnau, and A. Rostov, “Deep turbulence effects mitigation with coherent combining of 21 laser beams over 7 km,” Opt. Lett. 41, 840–843 (2016). [CrossRef]  

16. V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009). [CrossRef]  

17. Y. Ma, P. Zhou, R. Tao, L. Si, and Z. Liu, “Target-in-the-loop coherent beam combination of 100 W level fiber laser array based on an extended target with a scattering surface,” Opt. Lett. 38, 1019–1021 (2013). [CrossRef]  

18. R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011). [CrossRef]  

19. G. A. Tyler, “Accommodation of speckle in object-based phasing,” J. Opt. Soc. Am. A 29, 722–733 (2012). [CrossRef]  

20. In the case of SPGD, the metric that is maximized or minimized to yield phase estimates is typically based on the statistics of the received speckle pattern [1, 14]. The target-induced or speckle phases are not directly estimated.

21. J. F. Riker, G. A. Tyler, and J. L. Vaughn, “Speckle imaging from an array,” Proc. SPIE 9982, 99820J (2016). [CrossRef]  

22. M. W. Hyde, J. E. Wyman, and G. A. Tyler, “Rigorous investigation of the array-tilt aberration for hexagonal, optical phased arrays,” Appl. Opt. 53, 2416–2424 (2014). [CrossRef]  

23. V. N. Mahajan, Optical Imaging and Aberrations, Part III: Wavefront Analysis (SPIE, 2013).

24. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976). [CrossRef]  

25. M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012). [CrossRef]  

26. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE, 2007).

27. M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).

28. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

29. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998).

30. R. K. Tyson, Principles of Adaptive Optics, 4th ed. (CRC Press, 2015).

31. Under normal operating conditions (i.e., d<r0), an initially uncorrected array will produce a focused spot on the target with a diameter proportional to λz/d, where z is the distance to the target. The angle this spot subtends when viewed from the array is λ/d. Since the array uses the scattered return from this spot to phase on the target, θ0 must be greater than λ/d, otherwise array performance is limited by θ0 and not diffraction.

32. R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High-Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).

33. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

34. J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015).

35. D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A 8, 1568–1573 (1991). [CrossRef]  

36. J. P. Bos, V. S. Rao Gudimetla, and J. D. Schmidt, “Differential piston phase variance in non-Kolmogorov atmospheres,” J. Opt. Soc. Am. A 34, 1433–1440 (2017). [CrossRef]  

37. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

38. J. P. Bos, M. C. Roggemann, and V. S. Rao Gudimetla, “Anisotropic non-Kolmogorov turbulence phase screens with variable orientation,” Appl. Opt. 54, 2039–2045 (2015). [CrossRef]  

39. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

40. M. W. Hyde IV, J. E. McCrae, and G. A. Tyler, “Target-based coherent beam combining of an optical phased array fed by a broadband laser source,” J. Mod. Opt. 64, 2149–2156 (2017). [CrossRef]  

41. M. W. Hyde, G. A. Tyler, and C. R. Garcia, “Target-in-the-loop phasing of a fiber laser array fed by a linewidth-broadened master oscillator,” Proc. SPIE 10192, 101920K (2017). [CrossRef]  

42. The small difference between Zernike tilt and gradient tilt—measured by a tracking system employing a centroid sensor—is ignored [26].

43. J. Wyman and M. W. Hyde, “Detection and correction of stair mode across an optical phased array,” in IEEE Aerospace Conference (2014), pp. 1–10.

44. J. E. McCrae and S. T. Fiorino, “Simulation of array tilt effects in laser phased arrays,” in IEEE Aerospace Conference (2016), pp. 1–7.

References

  • View by:
  • |
  • |
  • |

  1. A. Brignon, ed., Coherent Laser Beam Combining (Wiley-VCH, 2013).
  2. B. N. Pulford, “LOCSET phase locking: operation, diagnostics, and applications,” Ph.D. thesis (The University of New Mexico, 2011).
  3. T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 14, 12188–12195 (2006).
    [Crossref]
  4. M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
    [Crossref]
  5. M. Vorontsov, G. Filimonov, V. Ovchinnikov, E. Polnau, S. Lachinova, T. Weyrauch, and J. Mangano, “Comparative efficiency analysis of fiber-array and conventional beam director systems in volume turbulence,” Appl. Opt. 55, 4170–4185 (2016).
    [Crossref]
  6. G. D. Goodno and J. E. Rothenberg, “Atmospheric propagation and combining of high power lasers: comment,” Appl. Opt. 55, 8335–8337 (2016).
    [Crossref]
  7. M. A. Vorontsov and T. Weyrauch, “High-power lasers for directed-energy applications: comment,” Appl. Opt. 55, 9950–9953 (2016).
    [Crossref]
  8. S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
    [Crossref]
  9. A. Flores, T. Ehrehreich, R. Holten, B. Anderson, and I. Dajani, “Multi-kW coherent combining of fiber lasers seeded with pseudo random phase modulated light,” Proc. SPIE 9728, 97281Y (2016).
    [Crossref]
  10. P. Sprangle, B. Hafizi, A. Ting, and R. Fischer, “High-power lasers for directed-energy applications,” Appl. Opt. 54, F201–F209 (2015).
    [Crossref]
  11. W. Nelson, P. Sprangle, and C. C. Davis, “Atmospheric propagation and combining of high-power lasers,” Appl. Opt. 55, 1757–1764 (2016).
    [Crossref]
  12. M. A. Vorontsov and S. L. Lachinova, “Laser beam projection with adaptive array of fiber collimators. I. Basic considerations for analysis,” J. Opt. Soc. Am. A 25, 1949–1959 (2008).
    [Crossref]
  13. S. L. Lachinova and M. A. Vorontsov, “Laser beam projection with adaptive array of fiber collimators. II. Analysis of atmospheric compensation efficiency,” J. Opt. Soc. Am. A 25, 1960–1973 (2008).
    [Crossref]
  14. M. A. Vorontsov, “Speckle effects in target-in-the-loop laser beam projection systems,” Adv. Opt. Technol. 2, 369–395 (2013).
    [Crossref]
  15. T. Weyrauch, M. Vorontsov, J. Mangano, V. Ovchinnikov, D. Bricker, E. Polnau, and A. Rostov, “Deep turbulence effects mitigation with coherent combining of 21 laser beams over 7 km,” Opt. Lett. 41, 840–843 (2016).
    [Crossref]
  16. V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
    [Crossref]
  17. Y. Ma, P. Zhou, R. Tao, L. Si, and Z. Liu, “Target-in-the-loop coherent beam combination of 100 W level fiber laser array based on an extended target with a scattering surface,” Opt. Lett. 38, 1019–1021 (2013).
    [Crossref]
  18. R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
    [Crossref]
  19. G. A. Tyler, “Accommodation of speckle in object-based phasing,” J. Opt. Soc. Am. A 29, 722–733 (2012).
    [Crossref]
  20. In the case of SPGD, the metric that is maximized or minimized to yield phase estimates is typically based on the statistics of the received speckle pattern [1, 14]. The target-induced or speckle phases are not directly estimated.
  21. J. F. Riker, G. A. Tyler, and J. L. Vaughn, “Speckle imaging from an array,” Proc. SPIE 9982, 99820J (2016).
    [Crossref]
  22. M. W. Hyde, J. E. Wyman, and G. A. Tyler, “Rigorous investigation of the array-tilt aberration for hexagonal, optical phased arrays,” Appl. Opt. 53, 2416–2424 (2014).
    [Crossref]
  23. V. N. Mahajan, Optical Imaging and Aberrations, Part III: Wavefront Analysis (SPIE, 2013).
  24. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [Crossref]
  25. M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012).
    [Crossref]
  26. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE, 2007).
  27. M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).
  28. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).
  29. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998).
  30. R. K. Tyson, Principles of Adaptive Optics, 4th ed. (CRC Press, 2015).
  31. Under normal operating conditions (i.e., d<r0), an initially uncorrected array will produce a focused spot on the target with a diameter proportional to λz/d, where z is the distance to the target. The angle this spot subtends when viewed from the array is λ/d. Since the array uses the scattered return from this spot to phase on the target, θ0 must be greater than λ/d, otherwise array performance is limited by θ0 and not diffraction.
  32. R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High-Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).
  33. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).
  34. J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015).
  35. D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A 8, 1568–1573 (1991).
    [Crossref]
  36. J. P. Bos, V. S. Rao Gudimetla, and J. D. Schmidt, “Differential piston phase variance in non-Kolmogorov atmospheres,” J. Opt. Soc. Am. A 34, 1433–1440 (2017).
    [Crossref]
  37. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).
  38. J. P. Bos, M. C. Roggemann, and V. S. Rao Gudimetla, “Anisotropic non-Kolmogorov turbulence phase screens with variable orientation,” Appl. Opt. 54, 2039–2045 (2015).
    [Crossref]
  39. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).
  40. M. W. Hyde, J. E. McCrae, and G. A. Tyler, “Target-based coherent beam combining of an optical phased array fed by a broadband laser source,” J. Mod. Opt. 64, 2149–2156 (2017).
    [Crossref]
  41. M. W. Hyde, G. A. Tyler, and C. R. Garcia, “Target-in-the-loop phasing of a fiber laser array fed by a linewidth-broadened master oscillator,” Proc. SPIE 10192, 101920K (2017).
    [Crossref]
  42. The small difference between Zernike tilt and gradient tilt—measured by a tracking system employing a centroid sensor—is ignored [26].
  43. J. Wyman and M. W. Hyde, “Detection and correction of stair mode across an optical phased array,” in IEEE Aerospace Conference (2014), pp. 1–10.
  44. J. E. McCrae and S. T. Fiorino, “Simulation of array tilt effects in laser phased arrays,” in IEEE Aerospace Conference (2016), pp. 1–7.

2017 (3)

J. P. Bos, V. S. Rao Gudimetla, and J. D. Schmidt, “Differential piston phase variance in non-Kolmogorov atmospheres,” J. Opt. Soc. Am. A 34, 1433–1440 (2017).
[Crossref]

M. W. Hyde, J. E. McCrae, and G. A. Tyler, “Target-based coherent beam combining of an optical phased array fed by a broadband laser source,” J. Mod. Opt. 64, 2149–2156 (2017).
[Crossref]

M. W. Hyde, G. A. Tyler, and C. R. Garcia, “Target-in-the-loop phasing of a fiber laser array fed by a linewidth-broadened master oscillator,” Proc. SPIE 10192, 101920K (2017).
[Crossref]

2016 (7)

2015 (2)

2014 (2)

M. W. Hyde, J. E. Wyman, and G. A. Tyler, “Rigorous investigation of the array-tilt aberration for hexagonal, optical phased arrays,” Appl. Opt. 53, 2416–2424 (2014).
[Crossref]

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

2013 (2)

2012 (2)

G. A. Tyler, “Accommodation of speckle in object-based phasing,” J. Opt. Soc. Am. A 29, 722–733 (2012).
[Crossref]

M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012).
[Crossref]

2011 (1)

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

2009 (2)

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[Crossref]

2008 (2)

2006 (1)

1991 (1)

1976 (1)

Adams, L. N.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Anderson, B.

A. Flores, T. Ehrehreich, R. Holten, B. Anderson, and I. Dajani, “Multi-kW coherent combining of fiber lasers seeded with pseudo random phase modulated light,” Proc. SPIE 9728, 97281Y (2016).
[Crossref]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

Aschenbach, K.

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[Crossref]

Bennal, B.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Berdine, R. W.

R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High-Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).

Beresnev, L. A.

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[Crossref]

Bos, J. P.

Bourdon, P.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Bricker, D.

Canat, G.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Carhart, G. W.

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[Crossref]

Dajani, I.

A. Flores, T. Ehrehreich, R. Holten, B. Anderson, and I. Dajani, “Multi-kW coherent combining of fiber lasers seeded with pseudo random phase modulated light,” Proc. SPIE 9728, 97281Y (2016).
[Crossref]

Davis, C. C.

Dong, X.

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

Ehrehreich, T.

A. Flores, T. Ehrehreich, R. Holten, B. Anderson, and I. Dajani, “Multi-kW coherent combining of fiber lasers seeded with pseudo random phase modulated light,” Proc. SPIE 9728, 97281Y (2016).
[Crossref]

Filimonov, G.

Fiorino, S. T.

J. E. McCrae and S. T. Fiorino, “Simulation of array tilt effects in laser phased arrays,” in IEEE Aerospace Conference (2016), pp. 1–7.

Fischer, R.

Flores, A.

A. Flores, T. Ehrehreich, R. Holten, B. Anderson, and I. Dajani, “Multi-kW coherent combining of fiber lasers seeded with pseudo random phase modulated light,” Proc. SPIE 9728, 97281Y (2016).
[Crossref]

Garcia, C. R.

M. W. Hyde, G. A. Tyler, and C. R. Garcia, “Target-in-the-loop phasing of a fiber laser array fed by a linewidth-broadened master oscillator,” Proc. SPIE 10192, 101920K (2017).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015).

Goodno, G. D.

G. D. Goodno and J. E. Rothenberg, “Atmospheric propagation and combining of high power lasers: comment,” Appl. Opt. 55, 8335–8337 (2016).
[Crossref]

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Goular, D.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Hafizi, B.

Hardy, J. W.

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998).

Ho, J. G.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Holten, R.

A. Flores, T. Ehrehreich, R. Holten, B. Anderson, and I. Dajani, “Multi-kW coherent combining of fiber lasers seeded with pseudo random phase modulated light,” Proc. SPIE 9728, 97281Y (2016).
[Crossref]

Hyde, M. W.

M. W. Hyde, J. E. McCrae, and G. A. Tyler, “Target-based coherent beam combining of an optical phased array fed by a broadband laser source,” J. Mod. Opt. 64, 2149–2156 (2017).
[Crossref]

M. W. Hyde, G. A. Tyler, and C. R. Garcia, “Target-in-the-loop phasing of a fiber laser array fed by a linewidth-broadened master oscillator,” Proc. SPIE 10192, 101920K (2017).
[Crossref]

M. W. Hyde, J. E. Wyman, and G. A. Tyler, “Rigorous investigation of the array-tilt aberration for hexagonal, optical phased arrays,” Appl. Opt. 53, 2416–2424 (2014).
[Crossref]

M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012).
[Crossref]

J. Wyman and M. W. Hyde, “Detection and correction of stair mode across an optical phased array,” in IEEE Aerospace Conference (2014), pp. 1–10.

Jaouen, Y.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Johnson, A. M.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Jolivet, V.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Lachinova, S.

Lachinova, S. L.

Liu, L.

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[Crossref]

Liu, Z.

Y. Ma, P. Zhou, R. Tao, L. Si, and Z. Liu, “Target-in-the-loop coherent beam combination of 100 W level fiber laser array based on an extended target with a scattering surface,” Opt. Lett. 38, 1019–1021 (2013).
[Crossref]

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

Lombard, L.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Ma, Y.

Y. Ma, P. Zhou, R. Tao, L. Si, and Z. Liu, “Target-in-the-loop coherent beam combination of 100 W level fiber laser array based on an extended target with a scattering surface,” Opt. Lett. 38, 1019–1021 (2013).
[Crossref]

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

Machan, J. P.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations, Part III: Wavefront Analysis (SPIE, 2013).

Mangano, J.

McCrae, J. E.

M. W. Hyde, J. E. McCrae, and G. A. Tyler, “Target-based coherent beam combining of an optical phased array fed by a broadband laser source,” J. Mod. Opt. 64, 2149–2156 (2017).
[Crossref]

J. E. McCrae and S. T. Fiorino, “Simulation of array tilt effects in laser phased arrays,” in IEEE Aerospace Conference (2016), pp. 1–7.

McNaught, S. J.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Moreau, B.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Motes, R. A.

R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High-Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).

Nelson, W.

Noll, R. J.

Ovchinnikov, V.

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

Polnau, E.

Pourtal, E.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Pulford, B. N.

B. N. Pulford, “LOCSET phase locking: operation, diagnostics, and applications,” Ph.D. thesis (The University of New Mexico, 2011).

Rao Gudimetla, V. S.

Riker, J. F.

J. F. Riker, G. A. Tyler, and J. L. Vaughn, “Speckle imaging from an array,” Proc. SPIE 9982, 99820J (2016).
[Crossref]

Roggemann, M. C.

Rostov, A.

Rothenberg, J. E.

G. D. Goodno and J. E. Rothenberg, “Atmospheric propagation and combining of high power lasers: comment,” Appl. Opt. 55, 8335–8337 (2016).
[Crossref]

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE, 2007).

Schmidt, J. D.

Shakir, S. A.

R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High-Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).

Shay, T. M.

Shih, C. C.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Shimabukuro, D. M.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Si, L.

Y. Ma, P. Zhou, R. Tao, L. Si, and Z. Liu, “Target-in-the-loop coherent beam combination of 100 W level fiber laser array based on an extended target with a scattering surface,” Opt. Lett. 38, 1019–1021 (2013).
[Crossref]

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

Spencer, M. F.

M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012).
[Crossref]

Sprangle, P.

Tao, R.

Y. Ma, P. Zhou, R. Tao, L. Si, and Z. Liu, “Target-in-the-loop coherent beam combination of 100 W level fiber laser array based on an extended target with a scattering surface,” Opt. Lett. 38, 1019–1021 (2013).
[Crossref]

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

Thielen, P. A.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Ting, A.

Tyler, G. A.

M. W. Hyde, J. E. McCrae, and G. A. Tyler, “Target-based coherent beam combining of an optical phased array fed by a broadband laser source,” J. Mod. Opt. 64, 2149–2156 (2017).
[Crossref]

M. W. Hyde, G. A. Tyler, and C. R. Garcia, “Target-in-the-loop phasing of a fiber laser array fed by a linewidth-broadened master oscillator,” Proc. SPIE 10192, 101920K (2017).
[Crossref]

J. F. Riker, G. A. Tyler, and J. L. Vaughn, “Speckle imaging from an array,” Proc. SPIE 9982, 99820J (2016).
[Crossref]

M. W. Hyde, J. E. Wyman, and G. A. Tyler, “Rigorous investigation of the array-tilt aberration for hexagonal, optical phased arrays,” Appl. Opt. 53, 2416–2424 (2014).
[Crossref]

G. A. Tyler, “Accommodation of speckle in object-based phasing,” J. Opt. Soc. Am. A 29, 722–733 (2012).
[Crossref]

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics, 4th ed. (CRC Press, 2015).

Vasseur, O.

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

Vaughn, J. L.

J. F. Riker, G. A. Tyler, and J. L. Vaughn, “Speckle imaging from an array,” Proc. SPIE 9982, 99820J (2016).
[Crossref]

Vorontsov, M.

Vorontsov, M. A.

Wacks, M. P.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Weber, M. E.

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

Welsh, B. M.

M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).

Weyrauch, T.

Winker, D. M.

Wyman, J.

J. Wyman and M. W. Hyde, “Detection and correction of stair mode across an optical phased array,” in IEEE Aerospace Conference (2014), pp. 1–10.

Wyman, J. E.

Zhou, P.

Y. Ma, P. Zhou, R. Tao, L. Si, and Z. Liu, “Target-in-the-loop coherent beam combination of 100 W level fiber laser array based on an extended target with a scattering surface,” Opt. Lett. 38, 1019–1021 (2013).
[Crossref]

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

Adv. Opt. Technol. (1)

M. A. Vorontsov, “Speckle effects in target-in-the-loop laser beam projection systems,” Adv. Opt. Technol. 2, 369–395 (2013).
[Crossref]

Appl. Opt. (7)

Appl. Phys. B (1)

R. Tao, Y. Ma, L. Si, X. Dong, P. Zhou, and Z. Liu, “Target-in-the-loop high-power adaptive phase-locked fiber laser array using single-frequency dithering technique,” Appl. Phys. B 105, 285–291 (2011).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (3)

S. J. McNaught, P. A. Thielen, L. N. Adams, J. G. Ho, A. M. Johnson, J. P. Machan, J. E. Rothenberg, C. C. Shih, D. M. Shimabukuro, M. P. Wacks, M. E. Weber, and G. D. Goodno, “Scalable coherent combining of kilowatt fiber amplifiers into a 2.4-kW beam,” IEEE J. Sel. Top. Quantum Electron. 20, 174–181 (2014).
[Crossref]

M. A. Vorontsov, T. Weyrauch, L. A. Beresnev, G. W. Carhart, L. Liu, and K. Aschenbach, “Adaptive array of phase-locked fiber collimators: analysis and experimental demonstration,” IEEE J. Sel. Top. Quantum Electron. 15, 269–280 (2009).
[Crossref]

V. Jolivet, P. Bourdon, B. Bennal, L. Lombard, D. Goular, E. Pourtal, G. Canat, Y. Jaouen, B. Moreau, and O. Vasseur, “Beam shaping of single-mode and multimode fiber amplifier arrays for propagation through atmospheric turbulence,” IEEE J. Sel. Top. Quantum Electron. 15, 257–268 (2009).
[Crossref]

J. Mod. Opt. (1)

M. W. Hyde, J. E. McCrae, and G. A. Tyler, “Target-based coherent beam combining of an optical phased array fed by a broadband laser source,” J. Mod. Opt. 64, 2149–2156 (2017).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (4)

A. Flores, T. Ehrehreich, R. Holten, B. Anderson, and I. Dajani, “Multi-kW coherent combining of fiber lasers seeded with pseudo random phase modulated light,” Proc. SPIE 9728, 97281Y (2016).
[Crossref]

M. F. Spencer and M. W. Hyde, “An investigation of stair mode in optical phased arrays using tiled apertures,” Proc. SPIE 8520, 852006 (2012).
[Crossref]

M. W. Hyde, G. A. Tyler, and C. R. Garcia, “Target-in-the-loop phasing of a fiber laser array fed by a linewidth-broadened master oscillator,” Proc. SPIE 10192, 101920K (2017).
[Crossref]

J. F. Riker, G. A. Tyler, and J. L. Vaughn, “Speckle imaging from an array,” Proc. SPIE 9982, 99820J (2016).
[Crossref]

Other (18)

The small difference between Zernike tilt and gradient tilt—measured by a tracking system employing a centroid sensor—is ignored [26].

J. Wyman and M. W. Hyde, “Detection and correction of stair mode across an optical phased array,” in IEEE Aerospace Conference (2014), pp. 1–10.

J. E. McCrae and S. T. Fiorino, “Simulation of array tilt effects in laser phased arrays,” in IEEE Aerospace Conference (2016), pp. 1–7.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed. (SPIE, 2007).

M. C. Roggemann and B. M. Welsh, Imaging Through Turbulence (CRC Press, 1996).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998).

R. K. Tyson, Principles of Adaptive Optics, 4th ed. (CRC Press, 2015).

Under normal operating conditions (i.e., d<r0), an initially uncorrected array will produce a focused spot on the target with a diameter proportional to λz/d, where z is the distance to the target. The angle this spot subtends when viewed from the array is λ/d. Since the array uses the scattered return from this spot to phase on the target, θ0 must be greater than λ/d, otherwise array performance is limited by θ0 and not diffraction.

R. A. Motes, S. A. Shakir, and R. W. Berdine, Introduction to High-Power Fiber Lasers, 2nd ed. (Directed Energy Professional Society, 2013).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

J. W. Goodman, Statistical Optics, 2nd ed. (Wiley, 2015).

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, 2007).

In the case of SPGD, the metric that is maximized or minimized to yield phase estimates is typically based on the statistics of the received speckle pattern [1, 14]. The target-induced or speckle phases are not directly estimated.

V. N. Mahajan, Optical Imaging and Aberrations, Part III: Wavefront Analysis (SPIE, 2013).

A. Brignon, ed., Coherent Laser Beam Combining (Wiley-VCH, 2013).

B. N. Pulford, “LOCSET phase locking: operation, diagnostics, and applications,” Ph.D. thesis (The University of New Mexico, 2011).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1. Seven-element hexagonal array composed of identical circular elements of diameter d and center-to-center spacing D . R 7 is the radius of the circle that circumscribes the array.
Fig. 2.
Fig. 2. Mean target-plane intensities using (a)  | W y | 2 = 0.0284 waves 2 and (b)  | W y , tr | 2 = 9.9985 × 10 5 waves 2 , respectively. These array tilt variances were computed assuming that d = r 0 = 0.1 m , L 0 = 100 m , d / D = 0.95 , and N = 7 . The white circle in both images marks the edge of the Airy disk for a single array element with λ = 1 μm .
Fig. 3.
Fig. 3. Array tilt variances for an N = 7 element array plotted versus fill factor d / D —(a) “full” array tilt variance | W y | 2 and (b) Zernike-tilt-removed array tilt variance | W y , tr | 2 . The solid traces are the theoretical variances; the symbols are the variances obtained from the simulation. d / r 0 values are differentiated by the color of the trace or symbol.
Fig. 4.
Fig. 4. Array tilt variances for an N = 19 element array plotted versus fill factor d / D —(a) “full” array tilt variance | W y | 2 and (b) Zernike-tilt-removed array tilt variance | W y , tr | 2 . The solid traces are the theoretical variances; the symbols are the variances obtained from the simulation. d / r 0 values are differentiated by the color of the trace or symbol.
Fig. 5.
Fig. 5. Array tilt variances for an N = 37 element array plotted versus fill factor d / D —(a) “full” array tilt variance | W y | 2 and (b) Zernike-tilt-removed array tilt variance | W y , tr | 2 . The solid traces are the theoretical variances; the symbols are the variances obtained from the simulation. d / r 0 values are differentiated by the color of the trace or symbol.

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

R N = L D + d 2 ,
N = 1 + 6 l = 1 L l .
ϕ = 2 π R W , [ ϕ 1 ϕ N ] = 2 π [ 2 D 3 x 1 2 D y 1 2 D 3 x N 2 D y N ] [ W x W y ] ,
ϕ i = 1 π ( d / 2 ) 2 circ ( | ρ ρ i | d / 2 ) ϕ ( ρ ) d 2 ρ ,
ϕ ϕ = ( 2 π ) 2 R W W R ,
W W = 1 ( 2 π ) 2 ( R R ) 1 R ϕ ϕ R ( R R ) 1 .
ϕ i ϕ j * = 1 A 2 P i ( ρ 1 ) P j * ( ρ 2 ) ϕ ( ρ 1 ) ϕ * ( ρ 2 ) d 2 ρ 1 d 2 ρ 2 ,
ϕ i ϕ j * = 1 A 2 Γ ϕ ( t ) P i ( s ) P j * ( s t ) d 2 s d 2 t .
P ˜ i ( f ) = P i ( s ) exp ( j 2 π f · s ) d 2 s , P i ( s ) = P ˜ i ( f ) exp ( j 2 π f · s ) d 2 f ,
ϕ i ϕ j * = 1 A 2 P ˜ i ( f ) ϕ ϕ ( f ) P ˜ j * ( f ) d 2 f ,
ϕ ϕ ( f ) = 5 π 8 / 3 11 [ 24 5 Γ ( 6 5 ) ] 5 / 6 Γ ( 17 / 6 ) Γ ( 1 / 6 ) r 0 5 / 3 ( f 2 + f 0 2 ) 11 / 6 0.023 r 0 5 / 3 ( f 2 + f 0 2 ) 11 / 6 ,
ϕ i ϕ j * = 0.046 A f 0 11 / 3 r 0 5 / 3 0 d f f J 1 2 ( f ) J 0 ( ρ i j d / 2 f ) × [ 1 + ( 1 π d f 0 ) 2 f 2 ] 11 / 6 ,
| ϕ i | 2 = 0.0863 ( L 0 r 0 ) 5 / 3 F 3 2 ( 3 2 , 1 ; 1 6 , 3 , 2 ; π 2 d 2 L 0 2 ) 1.0324 ( d r 0 ) 5 / 3 F 2 1 ( 7 3 ; 17 6 , 23 6 ; π 2 d 2 L 0 2 ) ,
ϕ i ϕ j * = 0.0863 ( L 0 r 0 ) 5 / 3 n = 0 ( π ρ i j / L 0 ) 2 n ( 1 ) n ( 1 / 6 ) n × F 3 2 ( 3 2 , n + 1 ; n + 1 6 , 3 , 2 ; π 2 d 2 L 0 2 ) 3.4419 ( ρ i j r 0 ) 5 / 3 n = 0 ( π ρ i j / L 0 ) 2 n ( 1 ) n ( 11 / 6 ) n × F 2 3 ( 3 2 , n 5 6 , n 5 6 ; 3 , 2 ; d 2 ρ i j 2 ) ,
W W [ 0.0852 0 0 0.0284 ] .
D i j = | ϕ i ϕ j | 2 = 2 ( ϕ i 2 ϕ i ϕ j ) ,
ϕ i , tr = ϕ i a 2 Z 2 , i a 3 Z 3 , i .
a k = W ( ρ ) Z k ( ρ ) ϕ ( ρ ) d 2 ρ , W ( ρ ) = { 1 π R N 2 ρ < R N 0 ρ > R N ,
ϕ i , tr ϕ j , tr * = ϕ i ϕ j * + 4 R N 2 ( x i x j | a 2 | 2 + y i y j | a 3 | 2 + x i y j a 2 a 3 * + x j y i a 2 * a 3 ) 2 R N ( x i a 2 ϕ j * + y i a 3 ϕ j * + x j a 2 * ϕ i + y j a 3 * ϕ i ) ,
a k a l * = W ( ρ 1 ) Z k ( ρ 1 ) W * ( ρ 2 ) Z l * ( ρ 2 ) × Γ ϕ ( ρ 1 ρ 2 ) d 2 ρ 1 d 2 ρ 2 , a k ϕ i * = 1 A W ( ρ 1 ) Z k ( ρ 1 ) P i * ( ρ 2 ) × Γ ϕ ( ρ 1 ρ 2 ) d 2 ρ 1 d 2 ρ 2 ,
| a k | 2 = 0.092 π R N 2 r 0 5 / 3 f 0 11 / 3 0 d f f J 2 2 ( f ) × [ 1 + ( 1 2 π f 0 R N ) 2 f 2 ] 11 / 6 , [ a 2 ϕ i * a 3 ϕ i * ] = [ cos θ i sin θ i ] 0.092 ( d / 2 ) π R N r 0 5 / 3 f 0 11 / 3 0 d f f × J 1 ( d / 2 R N f ) J 1 ( ρ i R N f ) J 2 ( f ) [ 1 + ( 1 2 π f 0 R N ) 2 f 2 ] 11 / 6 ,
| a 2 | 2 = | a 3 | 2 = | a T | 2 = 1.4251 ( R N r 0 ) 5 / 3 F 3 2 ( 7 3 , 11 6 ; 5 6 , 17 6 , 29 6 ; 4 π 2 R N 2 L 0 2 ) 2.5557 ( R N L 0 ) 1 / 3 ( R N r 0 ) 5 / 3 F 3 2 ( 5 2 , 2 ; 5 , 3 , 7 6 ; 4 π 2 R N 2 L 0 2 ) .
[ a 2 ϕ i * a 3 ϕ i * ] = 2 R N [ x i y i ] I i I i = 1.5645 ( R N r 0 ) 5 / 3 × m = 0 n = 0 ( d 2 R N ) 2 m ( ρ i R N ) 2 n ( 1 6 ) m + n ( 11 6 ) m + n ( 2 ) m ( 1 ) m ( 2 ) n ( 1 ) n × F 2 1 ( 11 6 ; m n + 5 6 , m n + 17 6 ; π 2 R N 2 L 0 2 ) 2.5557 ( R N L 0 ) 1 / 3 ( R N r 0 ) 5 / 3 × m = 0 n = 0 ( π d 2 L 0 ) 2 m ( π ρ i L 0 ) 2 n ( 2 ) m + n ( 2 ) m ( 1 ) m ( 2 ) n ( 1 ) n ( 7 6 ) m + n × F 2 1 ( m + n + 2 ; m + n + 7 6 , 3 ; π 2 R N 2 L 0 2 ) .
ϕ i , tr ϕ j , tr * = ϕ i ϕ j * + 4 R N 2 ( x i x j + y i y j ) ( | a T | 2 I i I j ) .
U ( ρ ) = i = 1 N P i ( ρ ) exp ( j ϕ i ) , ϕ i = 2 π 2 D 3 x i W x + 2 π 2 D y i W y .
I ( ρ , z ) = [ A λ z ] 2 jinc 2 ( k z d 2 ρ ) × i = 1 N j = 1 N exp [ j ( ϕ i ϕ j ) ] exp [ j k z ( ρ i ρ j ) · ρ ] ,
I ( ρ , z ) = [ A λ z ] 2 jinc 2 ( k z d 2 ρ ) × i = 1 N j = 1 N exp ( 8 π 2 D 2 ρ i j 2 | W y | 2 ) exp ( j k z ρ i j · ρ ) ,
I 2 ( ρ , z ) = [ A λ z ] 4 jinc 4 ( k z d 2 ρ ) × i = 1 N j = 1 N m = 1 N n = 1 N exp [ j ( ϕ i ϕ j + ϕ m ϕ n ) ] × exp ( j k z ρ i j · ρ ) exp ( j k z ρ m n · ρ ) .
I 2 ( ρ , z ) = [ A λ z ] 4 jinc 4 ( k z d 2 ρ ) × i = 1 N j = 1 N m = 1 N n = 1 N exp ( 8 π 2 D 2 ρ i j m n 2 | W y | 2 ) × exp ( j k z ρ i j m n · ρ ) ,
σ I 2 ( ρ , z ) = I 2 ( ρ , z ) [ I ( ρ , z ) ] 2 .
ϕ TIL = ϕ tele + ϕ atm + ϕ tar , at , ϕ loc = ϕ tele ,
ϕ tele = ϕ tele , ho + ϕ tele , at , ϕ atm = ϕ atm , ho + ϕ atm , at ,
ϕ loc ϕ TIL = ϕ atm , ho ϕ atm , at ϕ tar , at .
W at ( ϕ loc ϕ TIL ) = ϕ atm , at ϕ tar , at .
ϕ cmd = ϕ TIL + W at ( ϕ loc ϕ TIL ) = ϕ tele + ϕ atm , ho .
ϕ res = ϕ true ϕ cmd = ϕ tele + ϕ atm ϕ tele ϕ atm , ho = ϕ atm , at .
Z k , i = 1 A P i ( ρ ) Z k ( ρ ) d 2 ρ = π R N 2 A P i ( ρ ) W ( ρ ) Z k ( ρ ) d 2 ρ .
[ Z 2 , i Z 3 , i ] = 4 R N d / 2 [ cos θ i sin θ i ] 0 d f f J 1 ( 2 π f d 2 ) J 1 ( 2 π f ρ i ) × J 2 ( 2 π f R N ) .
[ Z 2 , i Z 3 , i ] = 2 ρ i R N [ cos θ i sin θ i ] m = 0 n = 0 ( 1 ) m ( 1 ) n m ! n ! ( d 2 R N ) 2 m × ( ρ i R N ) 2 n Γ ( m + n + 2 ) Γ ( m n + 1 ) Γ ( m + 2 ) Γ ( n + 2 ) .
[ Z 2 , i Z 3 , i ] = 2 ρ i R N [ cos θ i sin θ i ] = 2 R N [ x i y i ] .

Metrics