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 49931-1295

avsergue@mtu.edu, ppiatrou@mtu.edu, mroggema@mtu.edu

1

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

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

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 gen-

erated 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

to be generated at some greater distance from the aperture. The bootstrapping pro-

cedure 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 informa-

tion 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 func-

tion 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 un-

compensated 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

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 tech-

niques 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 point-

like 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

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 perfor-

mance 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 6.

2. 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 turbulence-

induced 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

atmosphere is commonly characterized by the parameters: ro, θ0, σ2χ ρI , and 〈ρ2

s〉. The

Fried parameter ro 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 θ0 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 σ2χ 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 ro, θ0, and σ2χ can

be calculated using the following formulas.5

r0 =

[

0.423 sec(φ)k2

∫ L

0

dzC2

n(z)(z/L)5/3

]−3/5

(1)

θ0 =

[

2.914k2 [sec(φ)]8/3

∫ L

0

dzC2

n(z)z5/3

]−3/5

(2)

σ2

χ = 0.563k7/6 [sec(φ)]11/6

∫ L

0

dzC2

n(z)z5/6(z/L)5/6 (3)

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 C2n(z)

characterizes the strength of the index of refraction fluctuations. The Hufnagel-Valley

profile for the C2n(z), which we used here, is given by.1

C2

n(z) = 5.94 × 10−53(υ/27)2z10e−z/1000 + 2.7 × 10−16e−z/1500 + Ae−z/100 (4)

where A and υ are free parameters.4 The parameter A sets the strength of the tur-

bulence near the ground level and υ represents the high altitude wind speed. Typical

8

values for the A and υ are 1.7 × 10−14m−2/3 and 21 m/s, respectively. Beam spread-

ing 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 〈ρ2s〉 is given by5

〈ρ2

s〉 =4L2

(kD)2+

4L2

(kρ0)2

[

1 − 0.62(ρ0

D

)1/3]6/5

(5)

where the transverse correlation length of the field ρ0 is related to the Fried parameter

r0 by r0 = 2.1ρ0. The isoplanatic angle projected to the target plane has radius Lθ0/2,

which is referred to as the isoplanatic patch radius ρI

ρI =Lθ0

2(6)

We evaluate beam parameters r0, θ0, σ2χ, ρI , and 〈ρ2

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 C2n(z) shown in Fig. 4 for low altitudes takes values

close to 10−14 m−2/3, representing fairly strong turbulence conditions. Fig. 5 shows

that r0 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

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 scatte-

ring 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 non-

negligible 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

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 be-

tween 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 in6 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 tech-

nique 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 us-

ing 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:

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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 de-

formable 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 transmit-

ter/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 tur-

bulent volume as well as the optical elements were modeled as multiple thin phase

screens for which the mathematical relationship between the incident field Ui(~x, zn)

and the field Ut(~x, zn) after a screen can be described as

Ut(~x, zn) = T (~x, zn)Ui(~x, zn), (7)

where ~x is a two-component coordinate vector in the phase screen plane, zn is the

position of the nth screen along the propagation path, T (~x, zn) = exp(jφ(~x)) is a pure

12

phase transmission function of the screen describing either a random turbulence-

induced phase perturbation or the phase mask associated with an optical element.

Screen-to-screen optical field propagation was performed in Fourier domain:

U(~x, zn+1) = F−1{F [T (~x, zn)U(~x, zn)]H(~f, zn+1 − zn)}, ~x = (x, y), (8)

where

H(~f, zn+1 − zn) = exp

{

j2π(zn+1 − zn)

λ(1 − λ2|~f |2)1/2

}

(9)

is the angular-spectrum propagation transfer function for the distance (zn+1 − zn), ~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

Tl(~x) = exp

(

j2π

λ

|~x|2

2f

)

, (10)

associated with either the lens responsible for beam focusing or for the WFS lenslet,

the sampling criterion should satisfy the condition

△xi ≤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

[lturb1

3, ...,

lturbP

3,

2λf1

3D1

, ...,2λfK

3DK

]

, (12)

where the minimum is taken over all P phase screens and all K lenses present in the

optical system.

13

Given △x, the size of sampling grid N necessary to represent all the frequencies

present in the angular spectrum propagator H(~f,△zn) is found from the criterion

N ≥3λ

(△x)2max

i(zi+1 − zi), (13)

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 = 0.0019 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 r0 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 atmo-

spheric 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 r0 was calculated for each phase screen using

Eqn. 1. We note that integration of the structure constant C2n in the kernel of the

Eqn. 1 is in the direction pointing from the target to the aperture. The laser bea-

con 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

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 inco-

herent nature of the scattered field.12 The number of random scattering phases used

here was Nsp = 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 ap-

pear 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 av-

eraged 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

the outgoing beacon illumination laser, the returning scattered light, and the outgo-

ing 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 propor-

tional 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 inde-

pendent 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 al-

gorithm should be emphasized: 1) it is not realistic because the information about

target intensity distribution maximum position is generally not available, so the al-

gorithm 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

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 inten-

sity 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 im-

ages 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 func-

tion

Twfs =1

d2

S∑

i=1

rect(~x− ~xi

d) exp{−j

k

2fl

|~x− ~xi|2}, (14)

where S is the number of subapertures of the wavefront sensor, rect(~x) is a 2-D

rectangular function, fl is the focal length of a lenslet, ~xi is the center of the ith

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:

sx =(I1 + I2 − I3 − I4)

(I1 + I2 + I3 + I4), (15)

sy =(I2 + I3 − I1 − I4)

(I1 + I2 + I3 + I4),

where sx, sy are x- and y-components of the readout from a single subaperture, {Ii}4i=1

17

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 used7 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

φ̂dm(~x) =M∑

i

~̂uri(~x) (16)

where ~̂u is the control signal applied to the ith actuator of the DM and {ri(~x)}Mi=1

are the DM influence functions. A continuous face sheet DM was assumed for our

simulation, with influence functions approximated by bilinear splines:

ri(~x) = tri[x− xi

d

]tri[y − yi

d

](17)

where

tri(x) =

1 − |x| if |x| ≤ 1

0 otherwise

placed over a square grid −→x iNi=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

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. 10.

3.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

~̂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 cali-

bration 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

Ti(~x) = Twfs(~x)ri(~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, Twfs, ri are the WFS transmittance and ith DM influence

functions given by Eqs. 14 and 17, respectively.

19

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 tech-

nique 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 turbu-

lence 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 com-

pensation 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 simu-

late 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

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 compensa-

tion 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 con-

ditions. 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 perfor-

mance 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 point-

like 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

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 turbu-

lence strength conditions, Strehl ratios are 0.77, 0.5, and 0.29. 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

negligible. Confidence in the results was obtained computing the standard divination

of the strehl ratios for each case and was always within 0.01 − 0.013 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-to-

noise 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

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 107 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

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

equation13 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) = σ0

1 + cos2(θ)

2(20)

where

σ0 =4π2(n0 − 1)2

N20λ

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 by3

β(h) = β0

ρ(h)

ρ0

(22)

25

where ρ is the mass density of the air, h is altitude and scattering coefficient for losses

from a beam at sea level

β0 =8π

3N0σ0 (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

η(z, ψ) = exp[−β0ρ0

∫ z

0

dhρ(h)]

(24)

The power P (z, t) at the range z is

P (z, t) = P0(t− z/c)η(z, ψ) (25)

where P0 is the power transmitted by the laser. The power Pr backscattered into the

receiver is

Pr(t) =∫

∞

0

dzP0(t− 2z/c)η2(z, ψ)βr(z, ψ) (26)

where

βr(z, ψ) = σ0N0

Ar

z2

ρ(z cosψ)

ρ0

(27)

where Ar is the area of the receiving aperture.

Finally, the energy received from scattering with the range gate △z is3

E =∫

2Z/c+△z/2

2Z/c−△z/2

dtPr(t) =∫

∞

0

dz∫

2Z/c+△z/2

2Z/c−△z/2

dtP0(t− 2z/c)η2(z, ψ)βr(z, ψ) (28)

where Z is the mean range associated with the range gate. The physical meaning of

the laser beacon range gate is visualized in figure 17.

26

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 shown13 that the maximum

scattering depth is calculated using

△z =2△αz2Dp[

D2p − (z△α)2

] (29)

where △α is the angular size of the beacon and Dp is the diameter of the laser projec-

tion aperture. The optimum angular size of the beacon is determined by the subaper-

ture size of the wavefront sensor, or by the value of Fried parameter r0, whichever is

smaller. The angular size of a beacon as a function of r0 can be expressed as:

△α = 2.44λ

r0(30)

when △α is limited by the turbulence, the maximum value of △z may be approxi-

mated as

△z ≈4.88λz2

Dpr0(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

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 simula-

tion 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 aper-

ture 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

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 re-

quirements 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 re-

ceiver 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

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 measure-

ments which are corrupted by shot noise. Shot noise is characterized by a standard

deviation given by1

σn =0.74πη

(K̄w)1/2d, (32)

where η is a parameter accounting for imperfections of the detector array used in

focal plane of the WFS, K̄w 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 0.15. 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

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 opti-

mization. Presented here novel bootstrap technique involves generation of multiple

Rayleigh beacons along the propagation path and the power requirements for mak-

ing a single Rayleigh beacon is required to understand the performance tradeoffs and

31

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 implemen-

tation 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 re-

quirements. 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 ab-

sorption. 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

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 F49620 − 02 − 01 − 0319.

33

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36

Command

processor

for DM

AO system located

on the transmitter

side.

Slant range

5000m

Altitude

3000m

Beam

splitter

Target

DM

WFS

Compensated Beacon

at the Target Plane

Target Plane

Fig. 1. Creating an artificial beacon by scattering the laser beam from the

surface of the target.

37

Command

processor

for DM

AO system located

on the transmitter

side.

Slant range

5000m

Altitude

3000m

Beam

splitter

Target

Uncompensated

Rayleigh

Beacon

DM

WFS

Compensated Beacon

at the Target Plane

Target Plane

Fig. 2. Generating a Rayleigh beacon part way to the target.

38

Command

processor

for DM

AO system located

on the transmitter

side.

Altitude

3000m

Beam

splitter

Target

Uncompensated Rayleigh

Beacon

First Dynamically Precompensated

Beacon

Second Dynamically Precompensated

Beacon

Dynamically Compensated

Beacon at the Target

DM

WFS

Target Plane

Fig. 3. Novel “Bootstrap” beacon generation technique.

39

0 1000 2000 3000 4000 5000

10−16

10−15

10−14

10−13

Altitude, h(meters)

Cn2 (h

),m

−2/

3

Cn2 for HV57

Fig. 4. Structure constant C2n as a function of altitude.

40

0 1000 2000 3000 4000 50000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Altitude,h(meters)

r 0,met

ers

r0(h)

Fig. 5. Fried parameter r0 as a function of altitude.

41

0 1000 2000 3000 4000 50000

1

2

3

4

5

6

7x 10

−5

Altitude,h(meters)

θ 0(rad

)

θ0(h)

Fig. 6. Isoplanatic angle θ0 as a function of altitude.

42

0 1000 2000 3000 4000 50000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Altitude, h(meters)

[<ρ s2 >

]1/2 a

nd ρ

I, met

ers

[<ρs2>]1/2 and ρ

I as a function of h

[<ρs2>]1/2

ρI

Fig. 7. Isoplanatic patch radius ρI and Mean square instantaneous spot radius

in the target plane 〈ρ2s〉 as a function of altitude.

43

0 1000 2000 3000 4000 50000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Altitude,h(meters)

σ2 χ

σ2χ(h)

Fig. 8. Rytov variance σ2χ as a function of altitude.

44

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Slant distance from receiver L, m

Isop

lana

tic p

atch

rad

ius

ρ I, m

ρI(L)

Altitude 200 m

Altitude 300 m

Altitude 500 m

Altitude 1000 m

Altitude 2000 m

Fig. 9. Isoplanatic patch radius ρI in the target plane for different altitudes of

the laser platform as a function of the slant range.

45

−1 −0.5 0 0.5 1

x 10−3

−1

−0.5

0

0.5

1

x 10−3

x, m

y, m

Fig. 10. Fried geometry of the WFS and DM mutual layout. Crosses in the

vertices of the square grid represent DM actuators location and crosses in the

centers represent WFS subapertures locations.

46

x, m

y, m

Uncompensated

−0.05 0 0.05 −0.025 0.025

−0.05

−0.025

0

0.025

0.05

(a)

x, m

y. m

Compensated

−0.05 0 0.05 −0.025 0.025

−0.05

−0.025

0

0.025

0.05

(b)

Fig. 11. a)Uncompensated beacon created at the target plane. b)Beacon cre-ated using target plane compensation technique. Laser platform altitude was300 m

47

x, m

y, m

Beacon generated at 2.75 km

−0.02 −0.01 0 0.01 0.02

−0.02

−0.01

0

0.01

0.02

x, m

y. m

Beacon generated at 3.25 km

−0.02 −0.01 0 0.01 0.02

−0.02

−0.01

0

0.01

0.02

(a) (b)

x, m

y, m

Beacon generated at 3.75 km

−0.02 −0.01 0 0.01 0.02

−0.02

−0.01

0

0.01

0.02

x, m

y, m

Beacon generated at 4.25 km

−0.02 −0.01 0 0.01 0.02

−0.02

−0.01

0

0.01

0.02

(c) (d)

x, m

y, m

Beacon generated at 5 km

−0.02 −0.01 0 0.01 0.02

−0.02

−0.01

0

0.01

0.02

x, m

y, m

Final beacon generated at the target

−0.02 −0.01 0 0.01 0.02

−0.02

−0.01

0

0.01

0.02

(e) (f)

Fig. 12. Beacons formation process along the optical path using bootstrapcompensation technique under strong turbulence strength conditions. Laserplatform altitude was 300 m. Figures a) − e) show five beacons created at[2.75km, 3.25km, 3.75km, 4.25km, 5.0km] from the aperture. Circles drown onthe top represent isoplanatic patch radii for the current location of the beacon.f shows the final beacon generated at the target plane.

48

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4x 10

6

Laser pulse energy(Joules)

Num

ber

of p

hoto

dect

ion

even

ts(P

DE

) pe

r W

FS

sub

aper

ture

# PDE(Laser Pulse Energy), Laser Altitude 3000 m

Beacon at 4500 mBeacon at 3000 mBeacon at 1500 mBeacon at 500 m

Fig. 13. Number of photo-detection events per WFS subaperture per laserpulse for different altitudes of the laser platform as a function of the laserpulse energy. Altitude of the laser platform is 3000 meters.

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4x 10

6

Laser pulse energy(Joules)

Num

ber

of p

hoto

dect

ion

even

ts(P

DE

) pe

r W

FS

sub

aper

ture

# PDE(Laser Pulse Energy), Laser Altitude 1500 m

Beacon at 4500 mBeacon at 3000 mBeacon at 1500 mBeacon at 500 m

Fig. 14. Number of photo-detection events per WFS subaperture per laserpulse for different altitudes of the laser platform as a function of the laserpulse energy. Altitude of the laser platform is 1500 meters.

49

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4x 10

6

Laser pulse energy(Joules)

Num

ber

of p

hoto

dect

ion

even

ts(P

DE

) pe

r W

FS

sub

aper

ture

# PDE(Laser Pulse Energy), Laser Altitude 500 m

Beacon at 4500 mBeacon at 3000 mBeacon at 1500 mBeacon at 500 m

Fig. 15. Number of photo-detection events per WFS subaperture per laserpulse for different altitudes of the laser platform as a function of the laserpulse energy. Altitude of the laser platform is 500 meters.

50

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Laser pulse energy(Joules)

σ nσ

n as a Function of the Laser Pulse Energy , Slant Range is 4500 m

Laser altitude 3000 mLaser altitude 1500 mLaser altitude 500 m

Fig. 16. The standard deviation of shot noise for Rayleigh beacon generated atthe distance 4500 meters from the transmitter as a function of the laser pulseenergy. − denotes the case of the laser platform located at the altitude 3000meters; .− denotes the case of the laser platform located at the altitude 1500meters, and −− denotes the case of the laser platform located at the altitude500 meters

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Converging

laser beam

Laser beacon

range gateTransmitting

aperture

Focal distance

Fig. 17. Geometry of Rayleigh beacon.

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0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Altitude h, meters

Atmospheric density versus altitude

Den

sity

of t

he a

ir ρ,

kg/

m3

Fig. 18. Density of the air as a function of the altitude.

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Compensation Technique Altitude300 500 1000

Point-like beacon 0.44 0.74 0.89at the target planeBootstrap[4.25 5.0] 0.36 0.63 0.86[3.25 4.25 5.0] 0.36 0.64 0.86[3.25 3.75 4.25 5.0] 0.38 0.65 0.87[2.75 3.25 3.75 4.25 5.0] 0.39 0.65 0.87Target Plane 0.31 0.60 0.84Rayleigh[2.75] 0.14 0.34 0.66[3.25] 0.18 0.40 0.68[3.75] 0.22 0.44 0.73[4.25] 0.29 0.5 0.77Uncompensated 0.05 0.21 0.54

Table 1. Strehl ratios for various beam compensation scenarios as a func-tion of the laser platform altitude. A Rayleigh beacon were generated at thedistances 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 followingdistributions along the optical 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].Standard deviation of the strehl ratios calculated for each computation waswithin 0.01 − 0.013 range.

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Distance from the aperture(in meters) 4500 4000 3000 2000 1500 1000 500Receiver Gate range(in microseconds) 23.4 18.4 10.3 4.6 2.6 1.2 0.3

Table 2. Optimal receiver gate range as a function of the beacon position alongthe optical path

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