2.1.

Description of Scenarios

The analysis uses three typical, tactical scenarios for evaluating CBC and IBC. The turbulence being stronger near the surface impacts performance depending on the scenario geometry. The scenarios use the generic names; air-to-surface (A2S), surface-to-air (S2A), and surface-to-surface (S2S). The description of each, listed in Table 1, provides enough information to perform the paper’s detailed analysis and to relate each to an individual service’s tactical engagement. A2S represents an airborne laser moving toward and firing upon a stationary ground target. S2A is the reciprocal of A2S, and represents a stationary ground laser firing at an aircraft. S2S represents a ship-to-ship horizontal path engagement. The complete design space, used in conjunction with the scenario definitions found in Table 1, is listed in Table 2. The analysis to follow will vary the parameters listed Table 2, individually and in combinations, to show the effects on performance. Here, we provide a complete listing for convenient reference.

Table 1

Tactical scenario details.

Table 2

System design space.

3.1.

Subaperture Piston and Tilt Errors

The focused irradiance pattern at the target plane is developed by modeling a coherent array with each subaperture field being uniform and containing terms to account for subaperture tilt and piston. The modulus squared of the combined fields can then model either a coherent array, by using little or no piston variations, or an incoherent array by choosing a high enough piston variation. The analysis assumes that the tilt and piston variations are equal in magnitude and pair-wise independent. This is reproduced from Butts.¹⁸

Define a uniform field across each individual subaperture as

The total focused field of the array in the target plane (neglecting leading phase terms) is then²²

This analytical formulation provides an envelope of performance that provides intuition before proceeding to a wave optics simulation. A plot of BPE using Eq. (3) is instructive in understanding the tradeoffs between coherent and incoherent combining for the same pupil geometry. Figure 2(a) is a 3D plot of BPE for a 5 cm circular bucket with the beam focused at 7.5 km. The independent axes show the effect due to increases in subaperture RMS tilt jitter and subaperture RMS piston jitter as defined in Eq. (3), ${\sigma }_T$, ${\sigma }_p$, respectively. The plot on the top right [Fig. 2(b)] is for the case of subaperture tilt jitter only, while the plot on the bottom right [Fig. 2(c)] is for subaperture piston jitter only. Considering each effect individually, BPE approaches zero as subaperture tilt RMS is increased, yet the increase in piston RMS only shows that BPE asymptotes to the $n$-subaperture incoherent performance (IBC). The latter is intuitively pleasing since we would expect the subaperture beamlets to be completely incoherent with each other when the RMS disturbance approaches that for a uniform, random distribution over $\pm \pi$. The analytical model of the array for the range of 7.5 km and a 5 cm bucket shows CBC provides more energy on target than IBC. The assumption that this is strictly true for all ranges and bucket sizes though is not correct and leads to a key design tradeoff for a multiaperture system. This topic is addressed in detail in the subsequent sections.

Fig. 2

Plot of BPE for the system defined in Fig. 1 focused at 7.5 km on a bucket with a 5 cm diameter. (a) The independent axes show the reduction in BPE for increasing subaperture tilt and piston RMS as defined in Eq. (1). (b) The subaperture tilt error reduces the PIB toward zero, while (c) the subaperture piston error only asymptotes to the completely incoherent beam combining level.

3.2.

Comparison of Uniform and Truncated Gaussian Performance

A plot of the ensemble averaged, integrated power as a function of bucket radius illustrates the tradeoff between CBC and IBC in more detail. The curves within Fig. 3 show this dependency for the same conditions of Fig. 2 where the array is focused onto a 5 cm diameter bucket at 7.5 km. Since Eq. (3) assumed a uniform field over each subaperture, the plot shows the integrated irradiance for both uniform and an optimally-truncated Gaussian waveform (i.e., 2.25 times the beam waist).²³ Both contain the same power out of the exit pupil. The solid lines are for the coherent case and the dotted curves are for incoherent combining. The first point is that the uniform wave provides more energy on target within the smaller buckets (note, describing this in terms of angular extent takes the range consideration out of the description) than the Gaussian waveform for CBC. It is only when bucket radii become relatively large that a Gaussian waveform surpasses integrated power of a uniform waveform for CBC. This point is moot, though, since IBC surpasses the integrated power of CBC for either Gaussian or uniform waveforms. Clearly, as the bucket size for BPE approaches the central lobe for a single subaperture, the integrated power of an array with Gaussian beams will exceed that of a uniform beam whether one considers the case for CBC or IBC. But, for these bucket sizes, the comparison between CBC and IBC breaks down since the aperture is not a free parameter in the analysis. The consideration of bucket sizes for BPE that approach the central lobe of a single subaperture implies that the wrong aperture geometry has been defined for the system. The conclusion: range and bucket size are key parameters that must be part of the definition of the aperture geometry. Once determined, CBC will provide the better BPE over IBC, regardless of beam shape. The analytical treatment using a uniform field is only a gauge for the best possible BPE since a uniform field (CBC or IBC) provides the higher power densities for the smaller bucket sizes. In the simulations that follow, a Gaussian beam is used due to the practicalities of waveguide lasers.

Fig. 3

Plot of integrated BPE of the system defined in Fig. 1 as a function of bucket radius for a beam focused at 7.5 km. This plot compares the integrated BPE for a uniform field and an optimally truncated Gaussian field over each subaperture. The solid curves are for CBC, the dashed are for IBC. Note that for the smaller bucket sizes, the uniform field over the subaperture diffracts more energy on axis. Not until a large radius bucket does the Gaussian field exceed the uniform in energy density. In the limit, the integrated energy curves come together as the bucket radius increases. These curves justify the use of a uniform field analysis as the envelope of far field performance for bucket radii that is a few multiples of the vacuum far field spot.

The consideration to optimize far field irradiance over a given angular extent in the far field opens up a new design dimension for DE systems not available in a monolithic beam design. The analysis within this work assumes that energy coupling into the target is limited to the first few radii of the coherent central lobe of the full aperture.

3.3.

CBC and IBC versus Bucket Size and Range

The relationship between PIB and BPE is just the normalizing factor of total power out of the aperture, which for this case is 60 kW. What follows is given in terms of PIB to show the marginal differences between CBC and IBC as the angular extent of the bucket is made large. The comparison of PIB for a fixed radius bucket as the range increases is presented in Fig. 4. This figure contains PIB($z$) for two bucket sizes and for both cases CBC and IBC. The bucket sizes are 5 and 10 cm and the PIB is for the vacuum case. Since the bucket size is fixed, as the range increases the angular radius decreases and eventually only encloses the central lobe of the coherent full aperture. This is manifest in the CBC curve for the 5 cm bucket size at 6 km where the curve levels off. The PIB level at this inflection is dependent on the ‘fill factor,’ $F_f$, of the array. For the uniform subaperture field, the fill factor is defined as the ratio of the total subaperture area divided by the total area that encloses the subapertures.² The fill factor for the system in Fig. 1 is calculated in Eq. (4) as,

Fig. 4

Plot of PIB(z) of CBC and IBC. The comparison shows that as the bucket encircles multiple radii of the CBC far field central spot, IBC can have higher PIB. The plateauing of the CBC case for the 5 cm bucket at 6 km arises from the fill factor of the array.

PIB differences, CBC-IBC, for the two bucket sizes, shown in Fig. 5, more readily displays the performance difference between the two methods of beam combining. CBC is providing more power in the defined bucket than IBC when the curves are positive. For ranges where the angular radius of the bucket is relatively small, the CBC captures more PIB than IBC. In general, these are typical bucket sizes well below the central lobe of a single subaperture. As discussed in Sec. 3.2, for large angular bucket sizes, these curves clearly show that IBC captures more power than CBC. But, again, to conclude that IBC is better than CBC for the close in ranges is to miss the point that a CBC design for this range and bucket size can be defined to outperform IBC. Also, the differences are marginal for the shorter ranges and CBC clearly provides performance out to significantly further distances.

Fig. 5

Difference of CBC minus IBC PIB(z) presented in Fig. 4. These differences (CBC–IBC) are for the two bucket sizes, 5 cm and 10 cm. When the curves become positive, CBC encircles more power than IBC. This occurs when the angular extent of the bucket is only a few radii of the central spot of the full coherent aperture. As the angular bucket size becomes exceedingly large, then the PIB of IBC exceeds that of the CBC.

3.4.

Exit Pupil Scaling

The previous discussion established design considerations for identical geometries given target ranges and vulnerable area diameters. The following establishes a framework to understand the design trade-offs for changes to the exit pupil geometries. The PIB for various configurations of the exit pupil are shown in Fig. 6. All configurations transmit the same amount of power. The solid curve is the encircled power for the IBC case for six subapertures. The dotted curve just above the solid line is the encircled power for the CBC. These represent the baseline design of Fig. 1 and are the same curves for uniform subaperture illumination as shown in Fig. 3. If the fill factor is changed by reducing the subaperture diameter and then using two rings to fill the 30 cm exit pupil, the first plateau rises indicating the higher power densities due to the increased fill factor. If instead, the same fill factor is maintained, but the diameter of the exit pupil is scaled down by a factor of 3, the encircled power is now spread out to the wings lowering the power density in the center (shown as the dashed line). (Not shown is the IBC case for the 18 subapertures. This curve would have fallen well below the 6-subaperture IBC case.)

Fig. 6

Encircled energy for various exit pupil geometries in vacuum. All configurations transmit the same amount of power. The solid line curve is the PIB for the IBC case. The 6 element CBC case as introduced in Fig. 1 is the dotted curve just above the IBC case. If the number of subapertures is increased over the same exit pupil diameter, the power density is increased as shown by the short-dotted line. Keeping the fill factor the same, but decreasing the exit pupil diameter by a factor of 3, changes the PIB curve to the dashed line.

This single plot displays the key tradeoffs for exit pupil geometries when comparing CBC and IBC. The comparison clearly shows that fill factor raises the power density near the center of the optical axis while keeping the same fill factor but reducing the size of the exit pupil diameter scales the power density by the same ratio as the change of diameters, but to larger angles in the far field. Note that the two 6 element arrays plateau at the same level, but the smaller aperture does so at a larger radius.

3.5.

Aperture Mismatch

The architecture presented in Fig. 1 uses a full aperture tracker to provide tilt corrections for the subaperture pointers. This is proposed for design economy and simplicity. However, since the subapertures are smaller than the whole aperture used by the tracker and since they are displaced from the centerline of the tracker line-of-sight (LOS), their pointing will be incorrect to some degree. The magnitude of this error is dependent upon the tracker error rejection bandwidth, turbulence strength, and the effective crosswind. Reference 13 provides a complete development of this error with further study in Refs. 14 and 15. The analysis in this work considers the three scenarios defined in Sec. 2.1. Recall, they are described as air-to-surface (A2S), surface-to-air (S2A), and surface-to-surface (S2S). The curves presented in Fig. 7 are for the case of no AO and plot RMS tilt error for an individual subaperture as a function of tracker error rejection bandwidth. The errors decrease initially as the tracker error rejection bandwidth increases. For the geometry described in the paper, the errors reach near minimum value at a bandwidth of around 50 Hz. Any further increase in the error rejection bandwidth does not reduce the tilt error appreciably since the correlation of tilt between the mismatched pupils is lower for the higher frequencies. Note with no AO, the magnitude of the minimum error can be quite high for the S2A and S2S scenarios. For these scenarios the error ranges from $0.4\lambda /d$ to $\lambda /d$, where $d$ is the diameter of a subaperture. In contrast, the A2S scenario, however, shows a much smaller error of $0.13\lambda /d$ with no AO. The difference in pupil mismatch tilt error between the A2S and the S2A is due to the turbulence strength near the aperture being lower for the A2S scenario. As the beam propagates to the target, any phase effects that would cause a tilt have less path length to manifest as an error. When an AO subsystem corrects for turbulence across the full aperture the tilt error due to pupil mismatch is mitigated significantly. With AO, the minimum error in all 3 scenarios is reduced to approximately $0.1\lambda /d$. Thus, as a cautionary note, it would not be prudent to implement a candidate design as presented in Fig. 1 for S2A or S2S, for either CBC or IBC, without due consideration of the impact of this error in pointing.

Fig. 7

Subaperture pointing tilt error due to full aperture tracker without AO. The tilt error is a function of both the turbulence and the tracker’s error rejection bandwidth. This tilt error is mitigated significantly when an AO subsystem corrects for turbulence across the full aperture. For all systems analyzed in this work, the error with AO is reduced to less than $1/10$ $\lambda /d$, where, $d$ is the diameter of the subaperture.

4.1.

Wave Optics Setup

The subaperture waveforms in the simulation are modeled as truncated Gaussian beams since this is expected to well model the likely beam shape from a waveguide source. Each Gaussian beam (also referred to as a ‘beamlet’) is truncated at $d=2.25w_0$, where $w_0$ is the $1/e^2$ irradiance radius of the Gaussian beamlet and $d$ is the diameter of the truncating aperture. The transmitted power at the exit pupil is set after truncation to be the same in all cases. Such truncation produces maximum on-axis intensity for a single laser source.²² For the case of coherent combination, the optimal sizing would be about $d=2.13w_0$.¹⁹ To allow direct comparison of performance, the analysis uses the same size, $2.25w_0$, for all analysis of CBC and IBC. It should be noted that performance on-axis for CBC could be slightly increased by optimally sizing the beamlets for the CBC case. Performance could also be slightly increased for the cases with tilt jitter, ${\sigma }_T$, if the beamlets were resized according to Yura’s equations (see Ref. 24). Additionally, the truncation to maximize PIB for a certain target size could differ from that to maximize on-axis intensity. The analysis for optimizing PIB as a function of subaperture field is left as a follow on report. In this work, however, the authors do not expect these possible changes in truncation and subaperture fields would alter the conclusions of this paper in any way.

The wave optics simulation includes all effects from exit aperture to the target. A point source is used as a target to simplify the analysis of coherent versus incoherent beam combining. The simulation adds random disturbances to model residual errors that would arise from the system’s local tilt loop for the full aperture, and the tilt and phasing loops of the multilaser source. These disturbances are all considered pairwise independent. Each is modeled as a random variable with an underlying white, Gaussian distribution, with the RMS chosen to vary the magnitude of the disturbance. In addition to these residuals, the beam quality of each subaperture field can be degraded to model less than perfect sources. These disturbances and the effects they model are summarized as follows:

The system degradations described above are used to model realistic systems and to evaluate the sensitivity of the system to these residuals. The wave optics simulation with the defined residuals then provides an evaluation of BPE. This metric is evaluated for three different scenarios, each for two different atmospheric conditions. The performance of a fully-coherent, idealized laser array would be simulated with the residual disturbance parameter set with ${\mathrm{LOS}}_T=0$, ${\sigma }_p=0$, ${\sigma }_T=0$, and $\mathrm{BQ}=1$.

Figure 8 is an instance of the subapertures’ fields, each with either random piston [Fig. 8(a)], random tilt [Fig. 8(b)], or random higher-order phase error to model a reduction in beam quality [Fig. 8(c)].

Fig. 8

The OPD of the beamlets for incoherence, tilt jitter error, and imperfect beam quality: (a) shows the OPD for incoherent combination (${\sigma }_p=0.4\lambda$) for a single time instance, (b) shows the OPD with $0.4\lambda /d$ differential jitter (${\sigma }_T$), and (c) shows the OPD for poor beam quality ($\mathrm{BQ}=2.20$).

The wave optics model also includes control loops for full-aperture tracking and adaptive optics. The full transmitting aperture is used to collect target radiance and is spectrally shared with a single, full aperture tracker and adaptive optics (AO) system. The tracker controls a single fast steering mirror (FSM), which corrects all beamlets using the measured, full aperture tilt of the line-of-sight (LOS) to the target. The tracker has two bandwidths: 160 Hz error rejection for track only mode and 320 Hz when the AO subsystem is closed loop. Both loops have a nominal 6 db of overshoot. The higher tracker bandwidth was used to accommodate the higher frame rates used for the AO subsystem.

The AO subsystem uses full aperture measurements with a Shack-Hartmann wavefront sensor (WFS) and a single deformable mirror (DM) to correct for higher order atmospheric disturbances. The DM has a $16\times 16$ square grid of actuators truncated by the 32 cm diameter aperture. The AO error rejection bandwidth is 200 Hz with 6 db of over shoot. The analysis to follow will report results with and without the AO in operation. A list of pertinent parameters for the laser aperture and beam control subsystem are given in Table 2. Figure 9 shows, qualitatively, the effect that AO can have on both CBC and IBC far-field spots for the surface to air case at 7.5 km range with no subaperture phase errors or platform jitter. The left column is CBC, while the right is IBC. The first row shows the far-field spots with no turbulence, while the second row includes turbulence, visibly degrading both CBC and IBC spots. The last row includes both turbulence and AO, showing that AO does reduce turbulence degradations.

Fig. 9

The far-field spots for the surface to air engagement after propagating 7.5 km. (a) The spot for coherent beam combination with no turbulence is tightly-focused with a bright central lobe and distinct side lobes. (b) For incoherent combination with no turbulence, the spot is Gaussian and much less intense than the coherent spot. (c) and (d) When moderate atmospheric turbulence is included, the far-field CBC and IBC spots are noticeably distorted. (e) and (f) Adaptive optics can correct some of the turbulence, increasing intensity for both CBC and IBC. The scale is the same for all plots. Simulation conditions: surface to air engagement, 7.5 km range, 5 cm bucket, moderate turbulence (when included), 60 kW array, ${\mathrm{LOS}}_T=0$, ${\sigma }_T=0$, ${\sigma }_p=0$, $\mathrm{BQ}=1.0$.

4.2.

Description of Atmospheric Conditions

The atmospheres chosen for this work are based on parameter and particle distributions constructed by the laser environmental effects definitions and reference (LEEDR) software package using the ExPERT database. The ExPERT database contains world-wide, climatological data recorded over approximately the last thirty years for 573 worldwide sites. LEEDR uses the observations recorded at these sites to estimate the average transmission within the boundary layer for each location. After forming a cumulative distribution of transmissions at the 1.07 μm wavelength, we selected a value close to the worldwide median for tactical engagements. The surface conditions associated with this site provided a consistent parameter set to model the most probable atmospheric condition. The site associated with the median transmission was Kandahar, Afghanistan.

LEEDR provided high-fidelity profiles of absorption, scattering, and turbulence for the selected atmospheres at 1.07 μm. Figure 10 shows the two components of atmospheric extinction, absorption and scattering, for the two selected profiles above Kandahar, one for 0600 and the other for 1500 local time. As the surface temperature rises in the afternoon, the boundary layer height and thickness increases accompanied by a corresponding increase in turbulence. LEEDR modifies the lower portion of the Hufnagel-Valley (HV) profile to account for the change in turbulence strength as the surface temperature increases. This modification is based on the McClung-Tunick model developed by the Air Force Institute of Technology,^25,26 which more accurately accounts for turbulence in the first 50 m above the surface. Above 50 m, the turbulence profile blends with the Hufnegel-Valley $5/7$ profile. Figure 11 shows the turbulence profiles used for the two different times. Note that below 50 m altitude at 0600, the predicted turbulence levels are lower than the HV $5/7$ profile. The results presented in Sec. 5 use the atmospheric conditions for Kandahar for 0600 and 1500 local to represent a moderate and a high turbulence case, respectively.

Fig. 10

The total (molecular and aerosol) absorption and scattering profiles from LEEDR at 1.07 μm for Kandahar, Afghanistan in the summer at 0600 and 1500 local time. Note that the boundary height rises in the afternoon.

Fig. 11

The Hufnagel-Valley and modified Hufnagel-Valley turbulence profiles used in our evaluation. The difference in the first 50 m above the surface accounts for surface heating during the day. The modification is based on research into atmospheric modeling at the Air Force Institute of Technology.

4.3.

Numerical Considerations

The wave optics simulation is the most appropriate method for analysis beyond what can be done with theory, but requires due consideration in generating reliable results. Recognizing that ${\mathrm{Cn}}^2$ is only a parameterization of the variability of a given atmosphere, a performance data point for PIB consists of an average over thirty experiments of the same scenario with each experiment using a different random realization of the atmosphere for the given ${\mathrm{Cn}}^2$. The PIB calculated from a single experiment is the average PIB over the last 50 ms of the time simulation. Choosing the last 50 ms of PIB in the experiment avoids transients due to the tracker and AO responses and ensures the thermal model of the beam-atmosphere interaction has reached steady-state. Figure 12 displays the variability in PIB (dots) over different atmospheric realizations (i.e., for 30 different experiments) for the surface to air engagement. The solid line is the running average of the PIB over the experiment number. The dotted curves enclosing the running mean of PIB show the 95% confidence interval. Table 3 shows the percentages for 95% confidence intervals for the three scenarios. The limits shown bracket the confidence intervals calculated for the three scenarios for the moderate and high turbulence atmospheres. The conditions for the two intervals are listed in the table. The greatest variability occurs for the S2S scenario with a confidence interval just under 10%. Most others remain around 5%. The results reported for all cases in the remaining sections can be considered to have a 95% confidence interval for BPE when $+/-10\%$ of the values reported is applied.

Fig. 12

The dots show the time-averaged PIB over 50 ms of the steady-state condition of a single experiment. An experiment is defined as a simulation involving a single, random realization of atmospheric turbulence for a specific ${\mathrm{Cn}}^2$ profile. A single PIB data point is obtained by averaging the PIB from all 30 experiments. All experiments use the same specification of atmosphere (moderate or high turbulence), engagement, AO or not, ${\mathrm{LOS}}_T$, ${\sigma }_p$, ${\sigma }_T$, and BQ. The solid line is the running mean of PIB over the 30 experiments. The 95% confidence interval for 5 cm diameter PIB converges to $+/-5.18\%$ (dashed lines). Simulation conditions: S2A scenario, 7.5 km range, 5 cm bucket, moderate turbulence, 60 kW array, ${\mathrm{LOS}}_T=0$, ${\sigma }_T=0$, ${\sigma }_p=0.4\lambda$, $\mathrm{BQ}=1.0$, and no AO.

Table 3

Ninety-five percent confidence interval bounds (%) for the 7.5 km slant range.

5.1.

Impact of Subaperture Piston and Tilt Error, ${\sigma }_p$ and ${\sigma }_T$

The performance of CBC has been so far reported without residual errors that would be present in the phasing and pointing loops due to finite bandwidths. The next few sections present a sensitivity study to the magnitude of these residuals, starting with the subaperture phase and tilt errors. The theoretical curves are plotted with use of Eq. (3) adjusted for atmospheric extinction over the 7.5 km path.

Figure 14 separately shows the sensitivity to subaperture tilt jitter, ${\sigma }_T$, and piston jitter, ${\sigma }_p$. As a point of reference, consider the A2S scenario. BPE is reduced by $\sim 10\%$ when all subapertures have a residual tilt jitter of $0.2\lambda /d$ RMS. This is considered significant and would require care in design, but not necessarily a stressing design requirement. Note the increased sensitivity to piston jitter versus subpaperture tilt jitter, but even with large increases in piston jitter, the BPE asymptotes to a level defined by the IBC of the array. This level can be seen in the figure to be a BPE of about 15% occurring at an RMS value of $0.28\lambda$. For the A2S CBC scenario, keeping the reduction in BPE less than $\sim 10\%$ due to piston jitter alone requires an RMS value below $0.1\lambda$. Given published results for phasing multiple sources report $0.03\lambda$ RMS or better in a laboratory environment this requirement is not considered stressing.⁸

Fig. 14

Theory and simulation BPE results for 7.5 km slant range, moderate turbulence, and a 5 cm bucket versus (a) ${\sigma }_T$ and (b) ${\sigma }_p$ with ${\mathrm{LOS}}_T=0$, $\mathrm{BQ}=1$, and no AO. Note that the theory curve has been adjusted down for atmospheric extinction.

Since the BPE plot versus piston RMS jitter in Fig. 14(b) spans values equivalent to an incoherent array, a comparison of IBC to CBC can be made by comparing the values to the right of $0.28\lambda$ RMS to those left of this point for CBC.

5.2.

Impact of Beam Quality, BQ

The analysis included beam quality residual errors to specifically determine if its effects would be masked by moderate or high turbulence. The analysis that follows shows that beam quality directly impacts BPE even in strong turbulence and a reduction in beam quality is not masked with other beam spreading effects in a root-summed-squared (RSS) relationship. Figure 15 shows the dramatic drop in performance for both CBC [Fig. 15(a)] and IBC [Fig. 15(b)] for all three engagements as beam quality degrades for the case of a 7.5 km slant range, a 5 cm diameter bucket, and moderate turbulence.

Fig. 15

BPE for a 5 cm bucket is significantly reduced due to changes in subaperture beam quality. Results for CBC are shown in (a) and those for IBC are in (b) (note the scale change between the two figures). Simulation conditions: 7.5 km range, 5 cm bucket, moderate turbulence, 60 kW array, ${\mathrm{LOS}}_T=0$, ${\sigma }_T=0$, ${\sigma }_p=0$, and no AO.

Figure 15(a) shows BPE drops by 70% for the CBC air to surface case (30.0 cm $r_0$) as beam quality is reduced from 1.0 to 2.2. For the surface to surface engagement (4.28 cm $r_0$), the same change in beam quality reduces BPE by 54%. Figure 15(b) shows similar trends for IBC. Again, poor beam quality for IBC results in reduced performance for all engagements. Since BPE for IBC is lower than CBC for the better beam quality values, the reduction is less dramatic but still measureable.

5.3.

Impact of Platform Jitter, ${\mathsf{LOS}}_T$

Residual error in platform jitter is one of the major contributors to loss of energy on target in all laser systems.⁵ The effects of subaperture jitter, ${\sigma }_T$, have already been presented, this section presents the impact when all beamlets are moving in unison, i.e., as coherent tilt jitter, previously defined as full aperture tilt jitter, ${\mathrm{LOS}}_T$. Since a poorer BQ will cause a spreading of the far field beam, BQ is varied in conjunction with full aperture tilt to investigate the performance combination of these two effects.

Figure 16(a) shows the BPE with ${\mathrm{LOS}}_T=5\mathrm{urad}$ (RMS) for all three engagements and $\mathrm{BQ}=1.0$, 1.22, and 2.20. These results are compared to zero ${\mathrm{LOS}}_T$ in Fig. 15 reprinted to the left of Fig. 16(a) for easy reference. It can be seen that ${\mathrm{LOS}}_T$ significantly reduces BPE from 22% to 12% for the A2S scenario with $\mathrm{BQ}=1$. S2A and S2S result show similar drops in performance due to ${\mathrm{LOS}}_T$ and all scenarios show similar percentage drops in BPE with BQ reductions as seen without a full aperture tilt (Fig. 15). It should be noted that the jitter being $1.6\lambda /D$ is high and demonstrates that BQ cannot be relaxed without reducing performance even in the presence of large tilt RMS’s.

Fig. 16

BPE for CBC with (a) 5 urad RMS platform jitter, when compared with (b) no platform jitter, shows that platform jitter does significantly degrade performance of the coherent array. Simulation conditions: 7.5 km range, 5 cm bucket, moderate turbulence, 60 kW array, ${\mathrm{LOS}}_T=1.66\lambda /D$ (left only), ${\sigma }_T=0$, ${\sigma }_p=0$, and no AO.

The same comparison is made in Fig. 17 for the IBC case. For the IBC case the impact of ${\mathrm{LOS}}_T$ jitter is reduced since the diffraction angle is much larger, approximately three times that of the CBC. Both simulations were run under the same conditions of moderate turbulence.

Fig. 17

BPE for IBC with (a) 5 urad RMS platform jitter, compared with (b) no platform jitter, shows that platform jitter degrades performance of IBC noticeably less than that of the coherent array. Simulation conditions: 7.5 km range, 5 cm bucket, moderate turbulence, 60 kW array, ${\mathrm{LOS}}_T=1.66\lambda /D$ (left only), ${\sigma }_T=0$, ${\sigma }_p=0.4\lambda$, and no AO.

Figure 18 completes the sensitivity for IBC. The turbulence is high, the full aperture pointing error is high with ${\text{LOS}}_T=1.66\lambda /D$, the subaperture pointing error is moderate with, ${\sigma }_T=0.2\lambda /d$, and ${\sigma }_p$ is near zero at $0.01\lambda$. The high turbulence dominates the reduction in performance, but all effects contribute to a reduction in performance which is considered not viable (BPE $<10\%$) for all scenarios except A2S. However, even under these conditions, a reduction in beam quality is still measureable for A2S and S2A. Note these results do not use AO, and BPE is now measured over a 10 cm bucket. The fact that AO is not employed and the larger, 10 cm bucket is used, favors a reduction in sensitivity to BQ changes, yet the results still show an impact on performance for these two scenarios. For the A2S engagement, reducing BQ from 1.0 to 2.2 reduces BPE by 52%. These results are presented for completeness, but it is questionable that a system with such a low BPE would be considered useful once it falls well below 10%. Therefore, for any tactical laser system, beam quality will be a major driver of overall system performance.

Fig. 18

BPE of IBC system under multiple effects and with a 10 cm bucket to show that BQ still has a measureable impact on performance. Performance decreases significantly due to poor beam quality for both A2S and S2A. Simulation conditions: 7.5 km range, 10 cm bucket, high turbulence, 60 kW array, ${\mathrm{LOS}}_T=1.66\lambda /D$, ${\sigma }_T=0.2\lambda /d$, ${\sigma }_p=0.01\lambda$, and no AO.

5.4.

Impact of Adaptive Optics

Simulation results show the addition of an AO system significantly improves the performance of both CBC and IBC. The design used in the analysis has an actuator spacing of 2.0 cm in exit pupil space. Using a Fried geometry, this has approximately 5 actuators across each 10.0 cm diameter beamlet. Figure 19 presents AO performance for coherent combination for the moderate turbulence case as subaperture tilt [Fig. 19(a)] and piston jitter [Fig. 19(b)] are added. The A2S scenario has the highest BPE before AO is applied and, as expected, its performance changes the least with the addition of AO.

Fig. 19

BPE both with and without adaptive optics for varying (a) tilt and (b) piston errors, moderate turbulence, 200 Hz AO error rejection (see Table 5 for Greenwood frequency). Adaptive optics improves performance for both CBC and IBC for all scenarios. Simulation conditions: 7.5 km range, 5 cm bucket, moderate turbulence, 60 kW array, ${\mathrm{LOS}}_T=0$, ${\sigma }_T$ and ${\sigma }_p$ as shown, and $\mathrm{BQ}=1.0$.

The addition of AO in the A2S scenario has little impact on performance improvement since the turbulence strength; in this case, being stronger closer to the target has little impact on irradiance in the target plane. Note, however, for the other scenarios that AO consistently improves the performance of CBC for both tilt and piston jitter. In these cases the performance increase is due to the correction of the higher order atmospheric effects and pupil mismatch.

With AO employed, the phasing loops are the only difference between CBC and IBC and, thus, the sensitivity plot of Fig. 19(b) can be used to compare CBC and IBC with AO. Referencing this figure, it can be seen that piston jitter at $0.4\lambda$ can be directly compared to CBC-performance points to the left of $0.28\lambda$. Since the AO subsystem does not correct for residual subaperture piston jitter (thus, the downward trend in Fig. 19(b), this is the only difference between IBC and CBC. With this mind, the lower IBC performance can be understood in terms of adaptive optics where the individual beamlets are subapertures of a full aperture AO system, but in this case, subaperture tilt is only corrected. Hardy remarks that piston errors between subapertures in an AO subsystem are more detrimental than subaperture tilts in AO corrections.²⁷ The higher order phase across the full aperture of CBC and IBC, including tilt errors due to pupil mismatch, are corrected up to the random piston over each beamlet. Considering the individual beamlets of the IBC as a sparse AO subsystem the observation from Hardy provides another way to understand the performance difference between CBC and IBC when AO is used. Figure 20 summarizes the performance increase with the inclusion of AO. As explained, AO improves both S2A and S2S significantly in moderate turbulence with little impact on A2S.

Fig. 20

The percentage increase in BPE with the inclusion of AO versus engagement and BQ for CBC and the moderate turbulence case. AO greatly improves performance for the S2A and S2S engagements. Simulation conditions: 7.5 km range, 5 cm bucket, moderate turbulence, 60 kW array, ${\mathrm{LOS}}_T=0$, ${\sigma }_T=0$, and ${\sigma }_p=0$.

Figure 21 presents the impact of AO on BPE for the high turbulence profile. The turbulence profiles were previously defined in Figs. 10 and 11. The results presented used the larger bucket at 10 cm to show that CBC still out performs IBC even for high turbulence and for a moderately sized target area. Recall that as the angular extent of the bucket increases, IBC performance will approach CBC. The S2S scenario for a range of 7.5 km and with high turbulence is out of reach with conventional AO. S2A shows significant improvement since the turbulence, weighted closer to the exit pupil of the transmitter, can be corrected more easily than the other scenarios. Conversely, the turbulence being weighted further from the transmitting aperture for the A2S shows little change in performance.

Fig. 21

BPE under high turbulence for CBC and IBC with and without adaptive optics (AO with a 200 Hz error rejection, see Table 5 for Greenwood frequencies). Adaptive optics improves performance for both CBC and IBC for scenarios with high turbulence except for the S2S scenario. The values shown for S2S are statistically zero for the confidence levels determined for this analysis. Simulation conditions: 7.5 km range, 10 cm bucket, high turbulence, 60 kW array, ${\mathrm{LOS}}_T=0$, ${\sigma }_T=0$, ${\sigma }_p=0$ or $0.4\lambda$, and $\mathrm{BQ}=1.0$.

5.5.

Impact of Thermal Blooming

Thermal blooming is the nonlinear interaction of the beam intensity with the atmosphere. The localized heating of the air, as the beam transits to the target, changes the index of refraction and causes a distortion of the beam along its path. The salient features of tilt and astigmatism in the far field spot can only be partially corrected for with an AO subsystem. The increased density of the lower atmosphere makes this effect more severe as the path becomes more horizontal. In this section the sensitivity to thermal blooming is investigated. All results presented previously included this effect inherent to the atmosphere. In the results to follow the comparison is made with thermal blooming turned off and on within the simulation.

The magnitude of thermal blooming has traditionally been qualitatively defined with the distortion number, $N_d$. Definitions of $N_d$ vary in the literature depending on the author’s preference as to which effects are most important to model. The definition chosen for this work is described in Ref. 5. The integral for $N_d$ is defined in Eq. (5). This equation is based on a monolithic beam, thus, it is presented here with the cautionary note that its fidelity in representing a multilaser beam has not yet been evaluated. In addition to $N_d$, another parameter that gauges the effect of thermal blooming is critical power. This is the level of power above which the effect of thermal blooming dominates transmission and any further increase in power could cause even less power to arrive on target.⁵

The distortion numbers and critical powers for the three scenarios and the two power levels used in this work are shown in Table 7. Figure 22(a) and 22(b) shows the performance decrease in BPE for CBC and IBC for a 60 and 150 kW array, respectively. The 60 kW power level shows minimal impact on BPE for all scenarios due to thermal blooming. The largest magnitude in reduction due to thermal blooming occurs for the A2S scenario at the 150 kW level with a reduction in BPE from 24% to 16%. This dramatic decrease in performance is most likely due to the following. The reduction of the turbulence strength with an increase in altitude favors the A2S scenario. But, unlike the turbulence, the absorption within the boundary layer is fairly constant in altitude, thus, the A2S beam, less spread than in the other two scenarios, will bloom more due to the beam’s higher power density. The S2A and the S2S have been degraded already by the turbulence and, thus, show much lower reduction due to thermal blooming.

Table 7

Nd and critical power (kW) levels for the three scenarios at 7.5 km slant range.

Fig. 22

BPE with and without the effects of thermal blooming on CBC and IBC. (a) The 60 kW system and (b) the 150 kW system. For the A2S CBC case, thermal blooming reduced BPE from 24 to 22 for the 60 kW array and from 24 to 16 for 150 kW array (all numbers percentages). Simulation conditions: are 7.5 km range, 5 cm bucket, moderate turbulence, ${\mathrm{LOS}}_T=0$, ${\sigma }_T=0$, ${\sigma }_p=0$ or $0.4\lambda$, $\mathrm{BQ}=1.0$, and no AO.

The distortion numbers and critical powers calculated for these scenarios are not consistent with the simulation results. A full explanation is still being explored, but two comments are appropriate. First, the starting performance for all IBC scenarios are low (10% or lower) and adding the additional effect of thermal blooming has low impact. Second, it appears that use of the Eq. (5) over estimates the reduction in performance for a laser array.

5.6.

Analysis of System Performance

This section selects nominal values for the set of parameters used in the previous sensitivity studies and reports the performance for the three scenarios as a function of range, both with and without AO included. This last look at performance is restricted to the CBC since it has been shown to be an upper bound for IBC for all previously presented results.

Figure 23 presents performance for the three scenarios as a function of range with all disturbances included and set to reasonable values listed in Table 8. A far field pattern provided to the right of Fig. 23 shows a BPE of $\sim 10\%$. This is an average of far field performance for the A2S scenario at a range of 7.5 km. Given a 10% BPE as a minimally accepted performance, the maximum ranges for the three scenarios are presented in Table 9.

Fig. 23

CBC BPE performance for all disturbances set to nominal values listed in Table 8. The far-field spot in the upper right is for the A2S case with AO closed loop. The circle represents the 5 cm bucket that captures $\sim 10\%$ of the transmitted power for the 7.5 km slant range.

Table 8

Nominal system parameters.

Table 9

Maximum range for CBC with a BPE of at least 10%.