We analyzed [1] the coherent combining of multiple lasers that are not phase locked as well as the case of phase matched beams. Goodno and Rothenberg concentrate their comments [2] on a beam combining architecture where a master oscillator power amplifier (MOPA) configuration maintains phase coherence between beams from fiber lasers [their Fig. 1(b)]. Our analysis considers other classes of lasers where the beams to be combined are not phase locked [their Fig. 1(a)]. Although phase-matched MOPA systems using fiber amplifiers are currently of most interest for directed energy applications, other high power laser systems may emerge that will not be easily phase matched. Examples include the DARPA high energy liquid laser area defense system [3], free-electron lasers [4], or possibly even a diode-pumped alkali laser (DPAL) [5] or an exciplex pumped alkali laser (XPAL) [6]. We want to point out that in Section 3 of our paper we also specifically consider the phase-matched situation. The results in Section 3, specifically regarding the “phase-matched monochromatic” beam, do not appear to have been discussed by Goodno and Rothenberg. Our results demonstrate that even when phase locking is achieved in the near field, coherent combining performs comparably to incoherent combining when the transverse coherence length (Fried parameter) $r_0=1.67{(C_n^2{k}_0^{2}z)}^{-3/5}$ nears the dimension of the individual tile size. Figure 1 displays the average on-axis intensity resulting from the propagation of phase-locked tiles (blue) and incoherent tiles (red) as a function of the ratio of the tile dimension to the transverse coherence length. Using the notation and definitions from our paper, this ratio is denoted as $a/r_0$. The propagation distance and beam director geometry in Fig. 1 are consistent with our paper, specifically $z=5\mathrm{km}$ and $a=2\mathrm{cm}$. As expected, there is a steep decline in the averaged on-axis intensity in the vicinity of $a/r_0\cong 1$.

 

Fig. 1. Average on-axis intensity as a function of the ratio of the adjacent beam centroid separation to the transverse coherence length.

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Our analysis is also completely consistent with the experimental results reported by Vorontsov et al. [7]. They show that coherent combining works in non-deep turbulence. The largest value of $C_n^2$ that they used was $2.7\times {10}^{-15}{\mathrm{m}}^{-2/3}$. At this level of turbulence the Fried parameter at their stated wavelength of operation of 1.064 μm and range of operation of 7 km is 34 mm. Even at this low value of turbulence they reported that coherent combining was showing a degraded performance. Our analysis uses a much higher level of turbulence of ${10}^{-13}{\mathrm{m}}^{-2/3}$ and a Fried parameter of 5 mm. As laser powers used in coherent combining increase, optical damage to collimating apertures also necessitates that these apertures be made larger, and consequently placed further apart. This reduces the ability of coherent combining to work in deep turbulence. In our experiment on test ranges at the Townes Institute Science and Technology Experimentation Facility at the Kennedy Space Center we routinely see turbulence levels of ${10}^{-13}{\mathrm{m}}^{-2/3}$. Turbulence levels near the sea surface can easily be as large as ${10}^{-12}{\mathrm{m}}^{-2/3}$ [8]. This deep turbulence scenario is especially significant for directed energy applications involving near-horizontal paths, as might occur in Navy defense scenarios. Higher altitude or slant path scenarios would likely involve much larger Fried parameters and coherent combining will be more successful.

In regard to Ref.  [8] of Goodno and Rothenberg in Ref. [2], the authors are correct that we did not state that our model also implicitly amplitude locks the tiles. Mathematically, this is an obvious conclusion. Since our results in Section 3 are only concerned with the long time averaged quantities, the amplitude fluctuations do not play a significant role because they are averaged out.

In the final paragraph of Section 3 of the comment, the authors mention that we stated that there was a potential for adaptive optics (AO). Then they claim that “if indeed AO can improve performance then in principle a similar improvement could be realized for the same beam director without AO using a higher channel count CBC [coherent beam combination] MOPA array, with individual tiles appropriately sized relative to the transverse Fried coherence length.” The CBC MOPA array will work if the size of the individual laser apertures to be combined is less than the Fried parameter (lateral coherence length). However, each tile must be large enough to handle the laser power without optical damage. This effectively places a limit on the strength of turbulence in which the CBC MOPA array is successful because the CBC MOPA array only predistorts on a grid with resolution determined by subaperture spacing. In contrast, an adaptive optics system with many actuators can, in principle, pre-distort a wavefront with resolution greater than the Fried parameter. Arrays of fiber lasers that are phase locked can be coherently combined only if the piston, tip-tilt, and power amplitude of each subaperture are sufficient to allow the stochastic parallel gradient descent or some other means of correction of laser fluence on a distant target. This is possible, as demonstrated by Vorontsov et al., in turbulence that is not deep, if the target is cooperative, and if each subaperture is not larger than the Fried parameter. Unfortunately, real targets do not have embedded retroreflectors to facilitate this. Of course, the overall combined beam from the subapertures could be corrected with external adaptive optics, as would be necessary for incoherently combined beams, but this also requires information about the laser fluence on a remote target. However, AO will not work in any case for distances much larger than the Rayleigh range.