Bootstrap beacon creation for overcoming the. effects of beacon anisoplanitism in laser beam. projection system

Transcription

1 Bootstrap beacon creation for overcoming the effects of beacon anisoplanitism in laser beam projection system Aleksandr V. Sergeyev, Piotr Piatrou and Michael C. Roggemann Department of Electrical and Computer Engineering, Michigan Technological University, 1400 Townsend Drive, Houghton, Michigan

2 In this paper we address the problem of using adaptive optics to deliver power from an airborne laser platform to a ground target through atmospheric turbulence under conditions of strong scintillation and anisoplanatism. We explore three options for creating a beacon for use in adaptive optics beam control: scattering laser energy from the target, using a single uncompensated Rayleigh beacon, and using a series of compensated Rayleigh beacons. Our work demonstrates that a using a series of compensated Rayleigh beacons provides the best beam compensation. Keywords: artificial laser beacon, turbulence, scintillation, anisoplanitism, partial compensation. c 2007 Optical Society of America 2

3 1. Introduction Due partly to the success and maturity of astronomical adaptive optical systems, interest in developing adaptive optical systems for directed energy applications has developed. The key goal of any adaptive optical system is to compensate for the wavefront errors induced by atmospheric turbulence. In the present case, we examine adaptive optical solutions to compensate for turbulence effects in order to deliver a sufficient level of laser energy from an airborne platform to ground-based receivers. To overcome the degradation in beam quality caused by atmospheric turbulence, it is generally necessary to measure and compensate for the phase distortions in the wavefront. A fundamental problem these systems face is the lack of a suitable beacon to probe the optical effects of the atmospheric turbulence along the path. In astronomy a natural star can serve as a beacon, and in other cases a high altitude artificial beacon can be created. However, in most non-astronomy cases there is no suitable beacon for wavefront sensing available, and as a result, a beacon must be created artificially. Due to the desired engagement geometries and turbulence strength along the optical path of interest, simple strategies for creating a beacon, such as scattering light from a beacon laser off the target, can result in beacons which are scintillated, 5 and more importantly, significantly larger that the isoplanatic patch in the scene. In this work we consider a down-looking scenario, in which the aircraft carrying the laser is flying at various altitudes and pointing the laser beam at a target located at ground level with a fixed slant path distance of 5000 m.it is important to note that we use conventional AO methods: Shack-Hartman WFS, LS reconstractor that are intended for use in 3

4 weak turbulence strength conditions to make compensation in strong turbulence. We present three techniques for beacon creation in a down-looking scenario, and explore beam control performance tradeoffs: 1. Scattering an initially focused laser beam from the surface of the target. 2. Generating a Rayleigh beacon part way to the target. 3. Applying a novel bootstrap technique by placing a series of compensated Rayleigh beacons between the aperture and the target. In the first technique the whole volume of the atmosphere between the transmitter and the target is probed by scattering the beacon laser from the target. In this case, the beacon is created by the laser beam, propagated through the turbulence, and scattered from the surface of the target. The second strategy uses a single Rayleigh beacon which is created at some intermediate distance between the transmitter and the target. This method allows compensation for part of the atmospheric path, which in some cases provides improved performance. Lastly, we present a bootstrap beacon generation technique, in which a series of compensated beacons is created along the optical path, with the goal of providing a physically smaller beacon at the target plane. The first beacon is an uncompensated Rayleigh beacon generated relatively close to the transmitter. We note, that due to the fact that the first beacon is generated close to the receiver and relatively weakly distorted by the turbulence, it is statable for wavefront sensing. The back-scattered field from the first beacon carries information about wavefront errors induced by part of the turbulent atmosphere. This information is used by the adaptive optical system to precompensate the next beacon 4

5 to be generated at some greater distance from the aperture. The bootstrapping procedure continues until the beacon reaches the target. In all cases there is no tracking information available due to the reciprocity of the optical path, 15 and this information must be obtained from some other aspect of the scene or target. We conjecture that a tracker based on a block-matching algorithm using an image of the scene, 10, 11 may provide sufficiently accurate tracking information. The beacon created using the first approach is generally larger than the isoplanatic angle for the path, and due to the scattering from the surface of the target, will also be corrupted by the coherent laser speckle effect. 12 The second approach will generally provide a beam with smaller angular extent compared to the case of scattering from the surface of the target, even though only part of the atmosphere along the path was probed by the light returned from the beacon. Compensation performance of a single Rayleigh beacon as a function of the position of the beacon is investigated in this work. In order to investigate compensation performance of a single Rayleigh beacon as a function of the position of the beacon along the optical path we perform a set of simulations, generating uncompensated Rayleigh beacons at distance 2.75 km, 3.25 km, 3.75 km, and 4.25 km in front of the aperture. It was found that the compensation performance depends on the position of the generated beacon along the optical path. This dependence is especially pronounced in the case where the Rayleigh beacon is generated at the lower altitudes probing the strongest turbulent layers of the atmosphere. The bootstrap technique probes the whole volume of the atmosphere along the path, and provides a suitably small beacon at a useful distance from the aperture. Compensation performance of the bootstrap technique was explored as a function of the number of beacons gener- 5

6 ated along the optical path. We test developed AO system with four different sets of beacon distributions: 1) [4.25 km, 5 km]; 2) [3.75 km, 4.25 km, 5 km]; 3) [2.75 km, 3.75 km, 4.25 km, 5 km], and 4) [2.75 km, 3.25 km, 3.75 km, 4.25 km, 5 km], where the numbers in square brackets give the distance of the beacons in kilometers from the transmitting laser. To represent the turbulent structure of the atmosphere, the Hufnagel-Valley 57 turbulence model was used. 1 To study the performance of the techniques presented here we perform the tests under different turbulence conditions by placing the platform with the transmitting laser at different altitudes: 1000, 500 and 300 meters, while maintaining a constant distance to the target of 5 km, representing the cases of mild, moderate and strong turbulence, respectively. The performance is evaluated by using the average Strehl ratio and the radially averaged intensity of the beam falling on the target plane. Simulation results show that under most turbulence conditions of practical interest the bootstrap technique provides better compensation performance compared to other two techniques. In order to explore the upper bound of performance for the case of a beacon in the target plane an ideal point-like beacon placed at the target plane was used. To obtain the upper bound of performance for single Rayleigh beacon generation technique we performed a set of simulations for a five kilometer path length, placing the ideal pointlike Rayleigh beacon at the distance 2.75 km, 3.25 km, 3.75 km,and 4.25 km in front of the aperture. Power requirements for Rayleigh beacon generation were also investigated in this work. Calculations presented here are based on computing the number of expected photodetection events in a circular subaperture of the WFS from Rayleigh backscat- 6

7 ter given initial power of the source. The functional dependence of the number of photodetection events per WFS subaperture per integration time as a function of the laser pulse energy was derived. This functional dependence can be used to find the optimal laser pulse energy based on the desired photo return. We found that a laser capable of generating on the order of 3 J of laser pulse energy per pulse is suitable for the laser projection system described in this work. We also note that calculations for Rayleigh backscatter lead us to believe that for the lasers with 3 J of laser pulse energy per pulse and higher the high signal-to-noise(snr) ratio is obtainable and therefore wavefront sensor SNR can be ignored. The remainder of this paper is organized as follows. In the next section we discuss theoretical considerations for beam projection through the turbulence, specifically examining a down-looking scenario. The simulation developed to study the performance of artificially created beacon using strategies described above is presented in Section 3. Results are presented in Section 4, power requirements for Rayleigh beacon generation are shown in Section 5, and conclusions are drawn in Section Theoretical background Light passing through the turbulent atmosphere becomes distorted. This distortions are caused by variations in the index of refraction along the optical path of the beam and wave propagation mechanism. These variations are caused by turbulenceinduced temperature fluctuations in the atmosphere resulting in density changes. In this section we describe the theoretical characteristics of a laser beam arriving at the target plane. The impact of turbulence on an optical beam of a given path through the 7

8 atmosphere is commonly characterized by the parameters: r o, θ 0, σχ 2 ρ I, and ρ 2 s. The Fried parameter r o is the aperture size beyond which further increases in its diameter result in no further increase in the resolution of an imaging system. 1 The isoplanatic angle θ 0 defines the maximum angle between two optical paths for which the two paths may be regarded as having approximately the same turbulence distortions. 5 The Rytov variance σχ 2 is the variance of the log-amplitude fluctuation of the field in the plane of the receiving optical system, and is a measure of whether the effects of the turbulence on a particular system is dominated by phase effects. For a a converging spherical wave centered at the target plane z = 0 the parameters r o, θ 0, and σχ 2 can be calculated using the following formulas. 5 [ ] L 3/5 r 0 = sec(φ)k 2 dzcn(z)(z/l) 2 5/3 (1) 0 [ ] L 3/5 θ 0 = 2.914k 2 [sec(φ)] 8/3 dzcn(z)z 2 5/3 (2) 0 L σχ 2 = 0.563k 7/6 [sec(φ)] 11/6 dzcn(z)z 2 5/6 (z/l) 5/6 (3) 0 where the wave number is given by k = 2π/λ, λ is the wavelength, h is the altitude of the propagation path, and φ is the zenith angle. The structure constant Cn(z) 2 characterizes the strength of the index of refraction fluctuations. The Hufnagel-Valley profile for the Cn(z), 2 which we used here, is given by. 1 C 2 n(z) = (υ/27) 2 z 10 e z/ e z/ Ae z/100 (4) where A and υ are free parameters. 4 The parameter A sets the strength of the turbulence near the ground level and υ represents the high altitude wind speed. Typical 8

9 values for the A and υ are m 2/3 and 21 m/s, respectively. Beam spreading and beam wander are the beam counterparts to the image blurring and centroid jitter. Turbulence scales that are large with respect to the beam size cause tilt, while turbulence scales that are small relative to the beam size cause beam broadening. As a result, a long exposure of the beam would result in the superposition of many realizations of the random wander of the broadened beam, which is an important consideration for beam pointing and tracking. However, the short-term broadening is important for pulse propagation and high-energy laser systems which have accurate trackers. The mean square short-term beam radius of an initially collimated beacon laser beam focused on the surface of the target ρ 2 s is given by 5 [ ( ) ] ρ 2 s = 4L2 (kd) + 4L2 ρ0 1/3 6/ (5) 2 (kρ 0 ) 2 D where the transverse correlation length of the field ρ 0 is related to the Fried parameter r 0 by r 0 = 2.1ρ 0. The isoplanatic angle projected to the target plane has radius Lθ 0 /2, which is referred to as the isoplanatic patch radius ρ I ρ I = Lθ 0 2 (6) We evaluate beam parameters r 0, θ 0, σχ, 2 ρ I, and ρ 2 s as as a function of the altitude z, for wavelength λ = 1.06µm, transmitting lens diameter of D = 0.5 m. The result of evaluating these parameters for the geometry of interest is shown in Figs 5, 6, 7, and 8. The structure constant Cn(z) 2 shown in Fig. 4 for low altitudes takes values close to m 2/3, representing fairly strong turbulence conditions. Fig. 5 shows that r 0 varies from 10 cm to 1 m depending on the altitude. Fig. 6 shows that for our geometry the isoplanatic angle is of the order of 1 µrad if the propagation takes 9

10 place at low altitudes and reaches 65 µrad at the altitude of 5000 m. Also, from Fig. 7, it can be seen that the short term RMS beam radius, representing the root mean square instantaneous spot radius in the target plane after passing through the atmosphere and focused on the target, at altitudes of the laser platform below 700 m will be bigger than the isoplanatic patch radius. This leads us to conclude that beacon anisoplanatism will effect on the performance for beacons created by scattering light from the target. 23 In addition, in Fig. 9 we present isoplanatic patch radius for different altitudes of the laser platform as a function of the slant range. Data presented in Fig. 9 is used to find the correct position of the beacon in the bootstrap technique so that the footprint of the last beacon generated at the target plane stays within the isoplanatic patch and therefore reduce the effect of beacon anisoplanitism. Fig. 8 demonstrates that significant fluctuations of the field amplitude are expected, especially at low altitudes. As a result, we conclude that scintillation will be nonnegligible over many paths of practical length for beam projection systems, and that simulations are an appropriate means of modelling atmospheric optics effects and the performance of strategies for mitigating turbulence effects. It is evident that there are certain difficulties and limitations for artificial laser beacon generation for wavefront sensing and beam control. In the case when the beacon is created by scattering an initially focused beam from the surface of the target, the footprint of the beacon laser in the scene is considerably larger than the isoplanatic patch. As a result, light scattered from a surface in the scene will propagate through many different and only weakly correlated atmospheric paths on its way back to the aperture. The turbulence induced abberations from all these paths arrive superimposed at the aperture and 10

11 hence make computing a useful set of deformable mirror commands based on a wave front sensor measurements of this field exceedingly difficult. An alternative method is to generate an uncompensated Rayleigh beacon at some intermediate distance between the transmitter and the target, by scattering light from atomic, molecular, and aerosol content in the atmosphere. This method allows compensation for only part of the atmospheric path, that is, the part between the beacon and the transmitter. It was shown in 6 that generation of a Rayleigh beacon can be effective, and under some conditions provide results close to those obtained in the case of an ideal point source beacon in the target plane. For the down-looking geometry examined here the approach of using a single Rayleigh beacon may be not as effective due to the fact that the strongest turbulence is located near the ground. A novel bootstrap technique for artificial beacon creation, which allows dynamic wavefront compensation, involves formation of a series of compensated beacons at increasing distances from the aperture. The bootstrap strategy probes the whole volume of the atmosphere using multiple step precompensation of the beacons, that should help achieve a smaller footprint of light distribution in the target plane, reduce effects of scintillation, and beacon anisoplanatism present in other approaches for beacon creation. In the next section we describe the simulation approach used to compare the performance of the compensation techniques discussed above. 3. Simulation approach The main body of the simulation used in all three approaches is a three way propagator that is summarized by the following steps: 11

12 1. An artificial laser beacon is propagated through the atmosphere from the laser aperture to the beacon plane. 2. Light is scattered from the surface of the beacon plane, which is modeled as an incoherent scatterer. 3. Scattered light propagated back through the atmosphere and intercepted by the aperture is used to form wave front sensor measurements, which are used to compute deformable mirror commands. 4. A compensated outgoing beam is reflected from the surface of corrected deformable mirror model and propagated through the atmosphere back to the target plane, where performance metrics are computed. The key parameters of all the simulations whose results will be presented below are summarized next. The system working wavelength is 1.06 µm; and the transmitter/receiver telescope diameter is 50 cm. The distance to the target was fixed at 5 km, and different propagation angles were specified by changing the aircraft altitude. Scalar diffraction theory was used to model light propagation. The atmospheric turbulent volume as well as the optical elements were modeled as multiple thin phase screens for which the mathematical relationship between the incident field U i ( x,z n ) and the field U t ( x,z n ) after a screen can be described as U t ( x,z n ) = T( x,z n )U i ( x,z n ), (7) where x is a two-component coordinate vector in the phase screen plane, z n is the position of the n th screen along the propagation path, T( x,z n ) = exp(jφ( x)) is a pure 12

13 phase transmission function of the screen describing either a random turbulenceinduced phase perturbation or the phase mask associated with an optical element. Screen-to-screen optical field propagation was performed in Fourier domain: U( x,z n+1 ) = F 1 {F[T( x,z n )U( x,z n )]H( f,z n+1 z n )}, x = (x,y), (8) where H( f,z n+1 z n ) = exp { j2π (z } n+1 z n ) (1 λ 2 f λ 2 ) 1/2 (9) is the angular-spectrum propagation transfer function for the distance (z n+1 z n ), f is a spatial frequency domain vector corresponding to x, and F and F 1 stand for direct and inverse Fourier transforms. To appropriately sample the quadratic phase term in the transmittance functions T l ( x) = exp ( j 2π λ x 2 ), (10) 2f associated with either the lens responsible for beam focusing or for the WFS lenslet, the sampling criterion should satisfy the condition x i 2λf 3D, (11) where f and D stand for focal distance and diameter of a lens, respectively. Hence, we choose for the final sampling period same in all phase screens x min [ l turb 1 lturb,..., 3 P3, 2λf 1,..., 2λf ] K, (12) 3D 1 3D K where the minimum is taken over all P phase screens and all K lenses present in the optical system. 13

14 Given x, the size of sampling grid N necessary to represent all the frequencies present in the angular spectrum propagator H( f, z n ) is found from the criterion N 3λ ( x) max (z 2 i+1 z i ), (13) i where index i runs over positions of all the atmospheric phase screens and optical elements. According to the criteria given in Eqs. 12 and 13 a square computational grid with sampling period x = mm and size N = 430 was used to appropriately sample each phase screen and the angular spectrum transfer function given in Eqn. 9. Unphysical wrap-around error, which arises from light scattered at wide angles from the rough surface of the target, was strongly attenuated by a set of filter masks placed on to each turbulence layer, with mask transparency function equal to unity within the angular extent of the transmitter/receiver aperture and a Gaussian roll-off outside. The random phase screens were generated using technique developed by Cochran. 9 The parameter r 0 is required by the phase screen generator, and needs to be calculated for each turbulent layer as a function of the altitude. In this simulation the atmospheric path was modelled with 10 equally spaced, different strength phase screens, with the first phase screen placed in the aperture plane and the last one 500 m away from the target. The Fried parameter r 0 was calculated for each phase screen using Eqn. 1. We note that integration of the structure constant Cn 2 in the kernel of the Eqn. 1 is in the direction pointing from the target to the aperture. The laser beacon is propagated from the laser aperture through all the phase screens between the aperture and the beacon plane. The beacon light is then scattered from the surface 14

15 of the target, for the target plane compensation case, or from the atmosphere in the case of the Rayleigh beacon and bootstrap approaches. Scattering from the target plane and the atmosphere was modeled by repeatedly multiplying the phase of the incident field on the surface by a random phase uniformly distributed on ( π, π), propagating this scattered field back to the aperture, and accumulating the resulting intensities in the wave front sensor detector plane. This approach models the incoherent nature of the scattered field. 12 The number of random scattering phases used here was N sp = 100. The focal length of the transmitting lens was chosen to match the propagation path length between the output lens and the required position of the beacon, so that in the absence of the turbulence, a diffraction-limited spot would appear if a collimated beam were passed through the lens toward the required location of the beacon. In the case of delivering the compensated beacon to the target location the focal length of the lens was set to the total distance between the transmitting laser and the target. In the bootstrap case, we use a lens with variable focal length in order to deliver the precompensated beacon to the current required position. The expression variable focal length is used here to describe the adjustment required to mach the focal length of the lens and the propagation distance to the current position of the generated beacon. The all three approaches were executed for 100 independent realizations of the atmosphere, resulting intensity patterns were accumulated and averaged to obtain the final results presented below. In all three techniques we assumed that the sum of the round trip propagation time and the time required to compute deformable mirror commands was shorter than the time required for the turbulence to change significantly. That assumption allowed us to use the same phase screens for 15

16 the outgoing beacon illumination laser, the returning scattered light, and the outgoing compensated laser beam. Next we describe simulation models of the tilt removal system,the Shack-Hartman wavefront sensor, the deformable mirror model, and the least-square wavefront reconstructor used in this work. 3.A. Maximum Tracker-Tilt Correction Model Wavefront tilt shifts the center of the focused image by an amount directly proportional to the amount of wavefront tilt. Usually tilt correction is achieved separately from the correction procedure for higher order of the turbulence-induced abberations. In our simulation we exclude the tilt from our consideration assuming that an independent beam steering system that stabilizes the spot position already operates on the laser platform, and focuss on tilt-removed or short-exposure performance of the laser beam control system. We do not make any specific assumptions about principle of operation of this independent tilt control system or its key characteristics such as residual error. On the other hand, the work of such a system should be imitated during the simulation of the compensation techniques described in this work. We do this imitation by using an iterative target-in-the-loop algorithm that measures the position of the target intensity distribution maximum and adds tilt correction to the DM such that to enforce the maximum to be at the origin. Two features of this algorithm should be emphasized: 1) it is not realistic because the information about target intensity distribution maximum position is generally not available, so the algorithm can be used only as imitation of a real tilt control system; 2) the residual tilt error of this algorithm corresponds to just a fraction of a pixel in the target plane, 16

17 i.e. it is much smaller than the residual error of any real tilt control system, so the short-exposure performance results of our simulations represent the upper limit for the overall beam control system performance. 3.B. Shack-Hartman Wavefront Sensor The wavefront sensor measures the lateral displacement of the centroid of the intensity spot in the focal plane. When an incoming wavefront is planar, all images are centered on the optical axis of the lenslets. When aberrations are present, the images are displaced from their nominal positions. Displacements of image centroid in two orthogonal directions are proportional to the average wavefront slopes over the subapertures. Shack-Hartmann wave front sensor is represented by its lenslet transmission function T wfs = 1 d 2 S i=1 rect( x x i ) exp{ j k x x i 2 }, (14) d 2f l where S is the number of subapertures of the wavefront sensor, rect( x) is a 2-D rectangular function, f l is the focal length of a lenslet, x i is the center of the i th subaperture and d is the length of the subaperture. The WFS readout s proportional the local wave front tilts was modeled as an output of a four-pixel CCD cells centered around each subaperture using the standard quad rule: s x = (I 1 + I 2 I 3 I 4 ) (I 1 + I 2 + I 3 + I 4 ), (15) s y = (I 2 + I 3 I 1 I 4 ) (I 1 + I 2 + I 3 + I 4 ), where s x, s y are x- and y-components of the readout from a single subaperture, {I i } 4 i=1 17

18 are the intensities in each of four pixels in the quad cell. The local wave front tilts can be found from the readout given by Eq. 15 after calibration process associated with the DM-to-WFS interaction matrix computation as is described in the next section. A wave-optics model of the Hartman sensor was used 7 with subaperture sides in the pupil of length 3.8 cm, yielding a total of 137 subapertures in the pupil. The physical side length of the lenslets modeled was 180 µm, and the focal length of the lenslets was 7 mm. 3.C. Deformable Mirror Model A single DM used for phase correction is optically conjugated to the exit pupil of the projector telescope. The linear model of the DM mirror surface is given by: 1 φ dm ( x) = M ˆ ur i ( x) (16) i where ˆ u is the control signal applied to the ith actuator of the DM and {r i ( x)} M i=1 are the DM influence functions. A continuous face sheet DM was assumed for our simulation, with influence functions approximated by bilinear splines: [ ] [ ] x xi y yi r i ( x) = tri tri d d (17) where tri(x) = 1 x if x 1 0 otherwise placed over a square grid x i N i=1 with grid size equal to the inter-actuator spacing. The deformable mirror was modeled using a Cartesian array of actuators with bilinear spline influence functions separated by 3.8 cm, yielding a total of 164 active actuators 18

19 inside the pupil. The base width of the influence functions was twice of their separation distance which leads to 7.6 cm. Fried geometry of the WFS and DM mutual layout is shown in Fig D. Least-Squares Wavefront Reconstructor In this work we used the least-squares wavefront reconstructor to map the measured wavefront slopes s given by Egn. 15 into DM commands. This is motivated by the desire to use a reconstruction algorithm that does not rely on knowledge of turbulence statistics often unavailable in air-to-ground applications. A least-squares estimator computes the optimal set of DM control commands u as ˆ u = arg min u s G u 2, (18) where G is the DM-to-WFS influence matrix that gives the WFS readout in response to a command applied to the DM. The G-matrix can be found by the WFS/DM calibration process when a unit amplitude commands are sent sequentially to a single DM actuator and the corresponding WFS readouts, that are equal to the corresponding columns of the G-matrix, are recorded. This process was modeled using the angular spectrum propagation technique by applying the phase screen T i ( x) = T wfs ( x)r i ( x), i = 1,...,M (19) to a plane wave, propagating the result to the WFS focal plane and forming WFS readouts by Eq. 15. Here, T wfs, r i are the WFS transmittance and i th DM influence functions given by Eqs. 14 and 17, respectively. 19

20 4. Results In this section we evaluate compensation performance of developed AO system using proposed in this work beacon generation techniques: 1) bootstrap technique; 2) target plane compensation technique; and 3) single Rayleigh beacon generation technique. First, we present the result of beacon creation using target plane compensation technique and compare it with the uncompensated beacon generated at the target plane. Laser platform altitude for this case was placed at 300 m representing strong turbulence strength conditions. Fig. 11 show short-exposure target plane intensities of the uncompensated and target plane compensated beacons. The circles drown on both figures represent an isoplanatic patch. From Fig. 11(a) we observe that the generated beacon is strongly corrupted by atmospheric turbulence. Also, due to the tilt the centroid of beacon is shifted from the origin. Inspection of Fig. 11(b) shows that the generated beacon using target plane compensation technique is significantly bigger than the isoplanatic patch which is consistent with theoretical calculations presented in Fig 7 and, as a result, the effect of the beacon anisoplanitism will affect the compensation performance. In order to reduce anisoplanatic nature of the generated at the target plane beacon the dynamic beacon creation technique was implemented. In Fig. 12 we demonstrate the beacon formation process along the optical path using bootstrap compensation technique. Laser platform was placed at 300 m to simulate strong turbulence strength conditions. Fig 12(a-e) show five beacons created at [2.75km, 3.25km, 3.75km, 4.25km, 5km] from the aperture. Circles drown on the top represent isoplanatic patch radii for the current position of the beacon. Fig 12(f) 20

21 shows the final beacon generated at the target plane. Only the first uncompensated beacon shown in Fig 12(a) is not within the isoplanatic patch but the rest 4 beacons and, what is the most important, the final beacon generated at the target plane are indeed within the isoplanatic patch that should significatively reduce the effect of beacon anisoplanitism. As we mentioned earlier in this paper, it was of our interest to study the compensation performance of a single Rayleigh beacon as a function of the position of the generated beacon along the optical path and compare its compensation performance with the results obtained from the target plane and the bootstrap compensation techniques. To fully investigate the performance of a single Rayleigh beacon technique we tested the AO system under different turbulence strength conditions. We performed a set of simulations, generating an uncompensated Rayleigh beacon at different locations along the optical path. Next, we summarize and analyze the compensation performance of the beacon generation techniques presented in this work. In Table 1 we present Strehl ratios for various beam compensation scenarios as a function of the laser platform altitude. To represent mild, moderate, and strong turbulence strength conditions we placed the laser platform at the altitude 1000, 500, and 300 meters respectively, and a constant 5000 slant range between the transmitter an the target for all cases. In order to explore the upper bound on performance of the simulated AO system under different turbulence strength conditions, we performed a set of simulations for a five kilometer path length, placing an ideal pointlike beacon at the target location. A Rayleigh beacons were generated at the distances of 2750 m, 3250 m, 3750 m, and 4250 m from the laser platform. In the bootstrap technique 4 sets of beacons were used with the following distributions along the op- 21

22 tical path: [4.25km, 5km], [3.25km, 4.25km, 5km], [3.25km, 3.75km, 4.25km, 5km], and [2.75km, 3.25km, 3.75km, 4.25km, 5km]. In the target plane compensation technique the beacon was generated directly at the target location probing whole volume of the turbulent atmosphere in one step. Based on the data presented in Table 1 we make a few conclusions. It was found that under different turbulence strength conditions the compensation performance of a single Rayleigh beacon depends on the distance of the generated beacon from the aperture. A single Rayleigh beacon generated at the farthest distance from the transmitting aperture probes the strongest turbulent layers and, as a result, provides better compensation performance. For example, for the Rayleigh beacon generated at the distance 2750 m from the transmitter under mild, moderate, and strong turbulence strength conditions, Strehl ratios are 0.66, 0.34, and 0.14, respectively. On the other hand, for the Rayleigh beacon generated at the distance 4250 m from the transmitter under mild, moderate, and strong turbulence strength conditions, Strehl ratios are 0.77, 0.5, and Comparing all three techniques we observe that the bootstrap technique in all cases provides the best compensation performance. The advantage of the bootstrap technique is especially pronounced under strong turbulence strength condition providing about 30 percent of improvement over the target plane compensation technique and about 32 present over the best case of a single Rayleigh compensation technique. This improvement, as expected, is not that significant under mild turbulence strength conditions which is consistent with the theoretical calculations presented in Fig 7 showing that at 1000 m altitude of the laser platform the beacon generated directly on the target plane is within the isoplanatic patch and therefore the effect of the beacon anisoplanitism is 22

23 negligible. Confidence in the results was obtained computing the standard divination of the strehl ratios for each case and was always within range. 5. Power Requirements for Rayleigh Beacon Generation For practical applications, it is important to estimate the power required to generate a single Rayleigh beacon at some distance from the transmitting laser source which will allow wavefront sensor measurements to be made with satisfactory signal-tonoise ratio. In this section I present the outline of calculations conducted in order to estimate the power requirements for a single Rayleigh beacon as a function of the laser altitude and the slant range between the transmitting laser and generated beacon. To fully understand the result presented in this section some physical understanding of the nature of scattering is required. 5.A. Types of Scattering Scattering at the atomic level occurs when a photon of incident radiation is annihilated while a photon of scattered radiation is created. There are two types of scattering that may occur: elastic and inelastic. If the frequency of the scattered radiation is the same as the incident radiation the scattering is defined as elastic. If the quantum state of an atom is changed in the process, then the emitted radiation is changed in frequency, and the scattering is said to be inelastic. Scattering from larger particles in the atmosphere is elastic. Different modes of scattering can be summarized in four main groups: 1. Rayleigh scattering, in which radiation scattered from the atoms or molecules 23

24 does not experience the change in a frequency. 2. Raman scattering, in which scattered radiation has change in frequency due to the change in the vibrational or rotational quantum states of the atoms or molecules. 3. Mie scattering, in which radiation is scattered from small particles or aerosols of size comparable to the wavelength of the incident radiation, with no change in frequency. 4. Resonance scattering, in which radiation matched in frequency to a specific atomic transition is scattered, with no change in frequency. 5.B. A Lower Bound on Power Requirements Raman scattering is very weak. Typically, only one photon out of 10 7 is Raman scattered. Resonance scattering requires tuning the laser on the frequency closely comparable to the internal rotational or vibrational frequency present in a specific atom or molecule. The presence of dust, fog, haze, and clouds causes the Mie scattering, which occurs mostly in the lower atmosphere(below 35 kilometers), and varies unpredictably. It should be noted that the Mie scattering cross-section can have a large value, producing a strong backscatter. In practice, it is important to produce a stable, reliable beacon. The generation technique for a Rayleigh beacon should not rely on surrounding atomic and molecular content, and unpredictable events such as presence of clouds, haze, dust or fog. Therefore, it is reasonable to count on the always present, elastic Rayleigh scattering. As a result, I based the 24

25 calculations for power requirements for a Rayleigh beacon generation only on elastic Rayleigh scattering excluding impact of other types of scattering. The result of our calculation forms a lower bound on the required power for Rayleigh beacon generation, and occasionally the presence of other types of scattering may improve the final result by boosting the scattered energy. 5.C. Power Requirements Calculation The process of using a light beam to probe a medium using backscattered energy as a function of range is known as LIDAR (light detection and ranging). The LIDAR equation 13 defines the energy detected at the receiver because of the scattering process as a light propagates through the media. To calculate the power backscattered into receiver we start by defining the angular scattering cross-section per molecule for Rayleigh scattering σ(θ) as: σ(θ) = σ cos 2 (θ) 2 (20) where σ 0 = 4π2 (n 0 1) 2 N 2 0λ 4 (21) λ is the wavelength, n is the refractive index of the air, N is the number density of molecules and the subscript 0 denotes sea-level values. For backscattering θ = π. The scattering coefficient for losses from a beam as a function of an altitude h is given by 3 β(h) = β 0 ρ(h) ρ 0 (22) 25

26 where ρ is the mass density of the air, h is altitude and scattering coefficient for losses from a beam at sea level β 0 = 8π 3 N 0σ 0 (23) The effect of air density variations for altitudes ranging from 0 and 5 kilometers are studied here. The transmission efficiency η of a beam projected at the zenith angle ψ to the range z is calculated using z ] η(z,ψ) = exp [ β 0 ρ 0 dhρ(h) 0 (24) The power P(z,t) at the range z is P(z,t) = P 0 (t z/c)η(z,ψ) (25) where P 0 is the power transmitted by the laser. The power P r backscattered into the receiver is P r (t) = 0 dzp 0 (t 2z/c)η 2 (z,ψ)β r (z,ψ) (26) where β r (z,ψ) = σ 0 N 0 A r z 2 ρ(z cos ψ) ρ 0 (27) where A r is the area of the receiving aperture. Finally, the energy received from scattering with the range gate z is 3 E = 2Z/c+ z/2 2Z/c z/2 dtp r (t) = 0 2Z/c+ z/2 dz dtp 0 (t 2z/c)η 2 (z,ψ)β r (z,ψ) (28) 2Z/c z/2 where Z is the mean range associated with the range gate. The physical meaning of the laser beacon range gate is visualized in figure

27 The angular size of a beacon formed at the focus of a converging beam is determined by the depth of the beam covered by the range gate in the receiver. 5.D. Power Optimization for Rayleigh Beacon Generation To minimize the energy requirements the range gate of the receiver should be chosen to use the greatest possible scattering depth. It was shown 13 that the maximum scattering depth is calculated using z = 2 αz 2 D p [ D 2 p (z α) 2 ] (29) where α is the angular size of the beacon and D p is the diameter of the laser projection aperture. The optimum angular size of the beacon is determined by the subaperture size of the wavefront sensor, or by the value of Fried parameter r 0, whichever is smaller. The angular size of a beacon as a function of r 0 can be expressed as: α = 2.44 λ r 0 (30) when α is limited by the turbulence, the maximum value of z may be approximated as z 4.88λz2 D p r 0 (31) To maximize the probability that incident photon will be scattered and captured by the receiver the receiver gate range should be adjusted appropriately. The optimal receiver gate ranges were calculated as a function of the beacon position along the optical path using Eqn 31 and are presented in the Table 2. 27

28 5.E. Simulation Details and Results The theoretical considerations discussed above are conveniently embedded in the widely available MATLAB tool box AOTOOLS 3 that was used to evaluate power requirements for Rayleigh beacon generation. As it was mentioned earlier in this section the laser energy delivered to the beacon location causes inelastic Rayleigh scattering from atoms and molecules. Backscattered photons are detected as photo events by the CCD. The number of photo-detection events (PDE) is directly related to the laser pulse energy. We investigate the dependence of the backscattered photo events detected by the CCD on the laser pulse energy of the transmitting laser. It should be noted that the total energy per pulse is the integration of the power over the duration of range gate. Reference information for the typical lasers used for Rayleigh beacon generation in lidar systems and their values can be found in. 13 In our simulation the following parameters of the AO system were used. The laser operated at the wavelength of 1.06 µm. In this simulation the laser pulse energy range was restricted between 0.1 and 15 joules with the pulse duration equal to 1µsec, that corresponds to 300 meters in length, but calculations can be easily extended for the lasers with different power. Optical efficiency of the system output, defined as the fraction of laser photons that exit the transmitter, was idealized and set to unity. Optical efficiency of the system input, defined as the fraction of backscattered photons entering the aperture that reach CCD, was considered lossless and set to unity. Quantum efficiency, that is the fraction of photons striking the CCD that result in a photo detection event was set to 0.8. The receiving aperture diameter was 0.5 meters. The receiver 28

29 gate range was adjusted for different slant ranges to its optimal value according to the Table 2. As it was mentioned earlier in this section, our goal was to estimate the power requirements for a single Rayleigh beacon generation as a function of the laser altitude and the slant range between a transmitting laser and generated beacon. It was shown in Eqn. 24, that the transmission efficiency of the atmosphere is a function of a zenith angle and the initial altitude of the laser platform. Therefore for the different initial positions of the laser platform, the outgoing laser radiation will experience different transmission of the atmosphere. To investigate this effect for the wide spectrrange of zenith angles we consider three cases by placing the laser platform at 3000, 1500 and 500 meters above the sea level. Figures 13, 14, and 15 demonstrate functional dependence of number of photo detection events per WFS subaperture per laser pulse for different altitudes of the laser platform on the transmitting laser pulse energy. It was also of interest, to investigate the number of photo detection events at the receiver for the same laser pulse energy but different positions of the beacon along the optical path. To investigate effect of the slant range Rayleigh beacons were generated at 4500, 3000, 1500, and 500 meters from the aperture. Optimal integration time for each slant range was chosen in accordance to Table 2. The number of PDE as a function of the laser pulse energy for different positions of the beacon is emphasized in Figures 13, 14, and 15 with different line styles. It is interesting to note that for beacons generated at 500 and 3000 meters it takes the same laser pulse energy to observe a similar number of PDE. In my opinion, a reasonable explanation for this effect would be a trade off between number density of the molecules at lower altitude 29

30 and the optimal receiver gate range, but more work is required to fully understand this. The accuracy of wavefront sensing depends on the subaperture slope measurements which are corrupted by shot noise. Shot noise is characterized by a standard deviation given by 1 σ n = 0.74πη ( K w ) 1/2 d, (32) where η is a parameter accounting for imperfections of the detector array used in focal plane of the WFS, Kw is the average number of photon events per WFS slope measurement, and d is the subaperture side length. The factor η is greater than or equal to unity, and η is unity only in the ideal case of a perfect detector array. For the computational results of this work I assume η = 1.2. We calculated the standard deviation of shot noise based on the photo-detection events. The calculations were performed for a Rayleigh beacon generated at the distance 4500 meters from the transmitter as a function of the laser pulse energy. As it was mentioned earlier, the outgoing laser radiation experiences different transmission through the atmosphere for the different altitudes of the laser platform. To investigate this effect we consider three cases by placing laser platform at 3000, 1500 and 500 meters above the sea level. The result of the calculations are shown in Fig 16 Inspection of Fig. 16 shows that if a laser with pulse energy less then 1 J is used to generate an artificial laser beacon, slope measurements will be corrupted by a shot noise with a standard deviation greater than Another words, the laser with pulse energy less then 1 J is not capable of providing signal-to-noise (SNR) ratio for which shot noise can be ignored. On the other hand, the beacon generated by the laser with 30

31 pulse energy greater then 3 J can generate SNR at which level it can be safe to ignore the effect of shot noise. The results presented in this work provide the future user with a convenient and efficient tool in identifying the tradeoffs in the laser power the efficiency of the receiver or otherwise. This tool is helpful in finding the optimal laser pulse energy based on the required photon return. 6. Conclusions Three techniques for artificial laser beacon creation in look down, shoot down scenario for various turbulence conditions were explored in this work. Performance of three techniques: scattering from the surface of the target, generating Rayleigh beacon at some distance from the target, and dynamic bootstrap beacon creation technique for wave front sensing and deformable mirror control were compared with each other and also with an ideal case of a point source beacon located at the target plane. Under different turbulence conditions it was found that novel bootstrap technique provides higher Strehl ratio compare to the other compensation techniques examined in this work. We have only examined least squares phase reconstruction approach, but it is likely that wave control improvements would result from use of more advanced, branch point reconstructor. We conjecture that it might be possible to use presented bootstrap technique in conjunction with the approach based on the contrast optimization. Presented here novel bootstrap technique involves generation of multiple Rayleigh beacons along the propagation path and the power requirements for making a single Rayleigh beacon is required to understand the performance tradeoffs and 31

32 therefore have been investigated as a function of the laser altitude and the slant range between a transmitting laser and generated beacon. Results for power requirements for Rayleigh beacon generation presented here are based on computing the number of expected photodetection events in a circular aperture from Rayleigh backscatter giving initial power of the source. We found that a laser capable of generating on the order of 3 joules of laser pulse energy per pulse is suitable for the laser projection system described in this work. There are practical challenges arising with implementation of the presented in this work approaches for a beacon generation. First of all, creating a beacon using atmospheric backscatter is energy inefficient operation, and as a result required beacon laser powers may be exceedingly high. Secondly, the usage of high power pulse lasers create additional complications related to the hardware requirements. Also propagation of high-energy laser beams through the atmosphere at a laser wavelength of λ = 1.06 µm is limited by thermal blooming due to aerosol absorption. Finally, in both strategies for creating the artificial beacon: scattering light from a surface of the target or in the scene, and use of atmospheric backscatter such as in case of a Rayleigh beacon, there is no tracking information available from the beacons, and hance tracking information must be obtained from some other aspect of the scene or target. 32

33 Acknowledgements This research was supported by the Air Force Office of Scientific Research under a Multiple University Research Initiative lead by the University of California at Los Angeles, contract number F