4,

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DEPARTMENT OF THE AIR FORCE

AIR UNIVERSITY

-AIR FORCE INSTITUTE OF TECHNOLOGY

Wright-Patterson Air Force Base, Ohio

QUA=.TT~ XNE 2

AF1T/GE/ENG/95D-25

Characterization of Atmospheric Turbulence

Over Long Horizontal Paths Using

Optical Slope Measurements

THESISEric Eugene Silbaugh

Captain, USAF

AFIT/GFENG/95D-25

Approved for public release; distribution unlimited

The views expressed in this thesis are those of the author and do not reflect the official policy or position

of the Department of Defense or the U. S. Government.

AFIT/GE/ENG/95D-25

Characterization of Atmospheric Turbulence Over Long Horizontal Paths

Using Optical Slope Measurements

THESIS

Presented to the Faculty of the School of Engineering

of the Air Force Institute of Technology

Air University

In Partial Fulfillment of the

Requirements for the Degree of

Master of Science in Electrical Engineering

Eric Eugene Silbaugh, BEE

Captain, USAF

December 1995

Approved for public release; distribution unlimited

Acknowledgements

Many people besides the author have contributed to the success of this thesis. Special thanks

are due to my advisor, Dr. Byron Welsh, for his guidance and encouragement throughout my time at

AFIT. Dr. Welsh's copious knowledge of optical systems operating in turbulence was crucial to my

success. It was a pleasure working with you. Maj Mike Roggemann freely shared his knowledge of the

otfs im3 code and kept me straight on the physics of atmospheric turbulence. Thanks to Maj Bob Mills

for providing encouragement and many interesting discussions on new and bizarre communications

methods. My classmates in the comm/radar program provided many enlightening discussions, much

needed sanity checks, and the occasional comic relief. Last, but not least, thanks to my Creator for this

opportunity to learn a little more about one small aspect of His universe.

Eric Eugene Silbaugh

ii

Table of Contents

Page

Acknowledgements .. .. .. .. ... ... ... ... ... ... ... ... .. .......

List of Figures. .. .. .. ... ... .... ... ... ... ... ... ... ...... vi

List of Tables. .. .. .. .. .... ... ... ... ... ... ... ... ... ..... ix

Abstract. .. .. .. .... ... ... ... ... ... ... ... ... ... ...... x

Preface .. .. .. .. ... ... ... ... ... .... ... ... ... ... .. .....

I. Introduction. .. .. .. .. ... ... ... .... ... ... ... ... ..... 1-1

1.1 Background .. .. .. .. ... ... ... ... ... ... ... ... 1-1

1.2 Previous experiments .. .. .. ... ... ... ... ... ... ... 1-2

1.3 The ABLE ACE experiment. .. .. .. .. ... ... ... ... ... 1-2

1.4 Problem description and scope .. .. .. .. .. .... ... ... ... 1-4

1.5 Summary .. .. .. .. ... ... .... ... ... ... ... ... 1-5

II. Theory .. .. .. ... ... ... ... ... ... ... ... .... ... ... 2-1

2.1 Atmospheric Turbulence .. .. .. ... ... ... .... ... ... 2-1

2.2 Phase structure functions .. .. .. ... ... ... ... ... ..... 2-2

2.3 Hartmann wavefront sensor .. .. .. .. ... ... .... ... ... 2-3

2.4 Slope structure functions .. .. .. ... ... ... ... .... ... 2-4

2.4.1 Self slope structure function. .. .. .. ... ... ... ... 2-5

2.4.2 Cross slope structure function .. .. .. ... ... ... ... 2-9

2.5 Slope structure function properties .. .. .. .. ... ... ... .... 2-12

Page

UTI. Statistical methods and simulation results. .. .. .. .... ... ... ... ... 3-1

3.1 Hartmann WFS slope measurements .. .. .. .. ... ... ... ... 3-1

3.2 Structure function estimation. .. .. .. .... ... ... ... ... 3-3

3.3 WFS image processing. .. .. .. ... .... ... ... ... ... 3-4

3.4 Statistical analysis .. .. .. .. ... ... ... ... ... ... ... 3-5

3.5 Homogeneity, isotropy, and fit to theory .. .. .. .. .... ... ... 3-8

3.6 Simulation results .. .. .. .. ... ... ... ... ... ... ... 3-9

3.6.1 Homogeneity .. .. .. ... ... ... ... ... ..... 3-10

3.6.2 Isotropy .. .. .. .. ... ... ... .... ... ...... 3-10

3.6.3 Effect of aberrations .. .. .. .. ... ... .... ...... 3-10

3.6.4 Effect of average removal .. .. .. ... ... ... ..... 3-17

3.7 Observations. .. .. .. ... .... ... ... ... ... ..... 3-24

IV. ABLE ACE Data Analysis. .. .. .. .. ... ... ... ... ... ... .... 4-1

4.1 ABLE ACE data. .. .. .. .... ... ... ... ... ... ... 4-1

4.2 The Science3 data set .. .. .. .. ... .... ... ... ... ... 4-1

4.2.1 Global statistics. .. .. .. ... ... ... ... ....... 4-2

4.2.2 Testing for homogeneity .. .. .. .. ... ... ... ..... 4-2

4.2.3 Testing for isotropy. .. .. .. .... ... ... ... ... 4-6

4.3 Summary. .. .. .. ... ... ... ... ... ... ... ..... 4-15

V. Conclusions and Recommendations. .. .. .. ... ... ... ... .... ... 5-1

5.1 Results and conclusions .. .. .. .. ... ... ... .... ...... 5-1

5.2 Suggestions for further work. .. .. ... ... ... ... ... ... 5-2

Appendix A. ABLE ACE data analysis. .. .. .. ... ... ... ... ... ... A-i

A. 1 Osan2 analysis .. .. .. ... ... ... ... ... ... ....... A-2

A.2 Science2 analysis .. .. .. .. ... .... ... ... ... ...... A-9

A.3 Yokota,3-1 data analysis. .. .. .. ... ... ... ... ... .... A- 16

A.4 Yokota3-2 data analysis. .. .. .. ... ... ... ... ... .... A-23

iv

Page

Appendix B. Computer code user guide .. .. .. .. ... ... ... ... ...... B-i

B. 1 IDL code .. .. .. ... ... ... ... ... ... ... ... ... B-i

B. 1.1 ACEMIAKE.PRO .. .. .. ... ... ... ... ... ... B-i

B.1.2 SLOPES.PRO .. .. .. .. ... ... .... ... ...... B-i

B.2 MATLAB code .. .. .. .. .... ... ... ... ... ... ... B-i

B.2.1 DS-GEN.M .. .. .. ... ... ... ... ... ... ... B-2

B.2.2 DSXX.M. .. .. .. .... ... ... ... ... ...... B-2

B.2.3 DS-ARG.M .. .. .. .. .... ... ... ... ... ... B-2

B.2.4 ACEFILES.M .. .. .. .. ... ... .... ... ...... B-2

B.2.5 ACEREAD.M...............................B-2

B.2.6 ACEVIEWM.............................. B-2

B.2.7 DSX.M..................................B-2

B.2.8 DSYM..................................B-2

B.2.9 DSN45.M.................................B-2

B.2.10 DSP45.M.................................B-2

B.2.11 DSTRUCT.M.............................. B-3

B.2.12 DSTAR.M.................................B-3

B.2.13 WFS-STAT.M...............................B-3

B.2.14 DVIEW.M................................ B-3

B.2.15 DPICKM.................................B-3

B.2.16 SPCAVG.M............................... B-3

B.2.17 MSFIT.M.................................B-3

Bibliography............................................... BIB-i

Vita................................................... VITA-i

v

List of Figures

Figure Page

2.1. Normalized x-slope structure function for x, y, and xy shifts as a function of

subaperture separation .. .. .. .. ... ... ... ... .... ... ...... 2-13

3.1. Hartmann wave front sensor diagram. .. .. .. .. ... ... .... ... ... 3-2

3.2. Homogeneity and isotropy .. .. .. .. ... ... ... .... ... ... ... 3-4

3.3. WFS subaperture grid thresholding .. .. .. .. ... ... ... ... ... ... 3-6

3.4. Slope differences .. .. .. ... ... .... ... ... ... ... ... ... 3-7

3.5. Simulation, slope average. .. .. .. ... .... ... ... ... ... .... 3-11

3.6. Simulation, slope variance. .. .. .. ... ... ... ... ... ... ..... 3-12

3.7. Homogeneity, simulated x-slopes .. .. .. ... ... .... ... ... .... 3-13

3.8. Homogeneity, simulated y-siopes .. .. .. ... ... .... ... ... .... 3-14

3.9. Isotropy, simulated x-slopes .. .. .. .. .... ... ... ... ... ..... 3-15

3.10. Isotropy, simulated y-slopes .. .. .. ... ... ... ... ... ... ..... 3-16

3.11. Simulation, aberrated slope average. .. .. ... ... ... ... ... ...... 3-18

3.12. Simulation, aberrated slope variance .. .. .. .. ... .... ... ... .... 3-19

3.13. Homogeneity, aberrated x-slopes. .. .. .. .. ... ... ... .... ...... 3-20

3.14. Isotropy, aberrated x-slopes .. .. .. ... ... ... ... ... ... ..... 3-21

3.15. Homogeneity, average removed x-slopes. .. .. .. .. .... ... ... .... 3-22

3.16. Isotropy, average removed x-slopes. .. .. ... ... ... ... ... ..... 3-23

4.1. Science3 slope average .. .. .. .. ... ... ... ... ... ... ... ... 4-3

4.2. Osan2 slope average .. .. .. .. ... ... ... ... ... .... ... ... 4-4

4.3. Science3 slope variance .. .. .. ... ... ... ... ... ... ... ..... 4-5

4.4. Science3: x-slope homogeneity .. .. .. ... ... ... ... ... ... ... 4-7

4.5. Science3: y-slope homogeneity .. .. .. ... ... ... ... ... ....... 4-8

4.6. Science3: average removed x-slope homogeneity .. .. .. .. .. .... ...... 4-9

vi

Figure Page

4.7. Science3: average removed y-slope homogeneity .. .. .. .. ... ... ...... 4-10

4.8. Science3: x-slope isotropy. .. .. .. .. ... ... .... ... ... ...... 4-11

4.9. Science3: y-slope isotropy. .. .. .. ... ... ... ... ... ... ...... 4-12

4.10. Science3: average removed x-slope isotropy. .. .. .. ... ... ... ..... 4-13

4.11. Science3: average removed y-slope isotropy. .. .. .. ... ... ... ..... 4-14

A. 1. Osan2: slope average. .. .. ... ... ... ... ... ... ... ... ... A-3

A.2. Osan2: slope variance .. .. .. .. ... .... ... ... ... ... ...... A-4

A.3. Osan2: average removed x-slope homogeneity .. .. .. .. ... ... ... ... A-5

AA4 Osan2: average removed y-slope homogeneity .. .. .. .. ... ... ... ... A-6

A.5. Osan2: average removed x-slope isotropy. .. .. .. ... .... ... ...... A-7

A.6. Osan2: average removed y-slope isotropy. .. .. .. .... ... ... ...... A-8

A.7. Science2: slope average .. .. .. .. .... ... ... ... ... ... .... A-10

A.8. Science2: slope variance .. .. .. .. ... ... ... ... ... ... ..... A-il

A.9. Science2: average removed x-slope homogeneity. .. .. .. .. ... ... .... A- 12

A. 10. Science2: average removed y-siope homogeneity. .. .. .. .. ... ... .... A- 13

A. 11. Science2: average removed x-slope isotropy .. .. .. .. ... ... .... .. A-14

A. 12. Science2: average removed y-slope isotropy .. .. .. .. ... ... .... .. A- 15

A. 13. Yokota3: slope average. .. .. ... ... ... ... ... ... ... ..... A- 17

A. 14. Yokota3: slope variance .. .. .. .. ... .... ... ... ... ... .... A-18

A. 15. Yokota3: average removed x-slope homogeneity .. .. .. .. ... ... ..... A- 19

A. 16. Yokota3: average removed y-slope homogeneity .. .. .. .. ... ... ..... A-20

A. 17. Yokota3: average removed x-slope isotropy. .. .. .. ... .... ... .... A-21

A. 18. Yokota3: average removed y-slope isotropy. .. .. ... ... ... ... .... A-22

A. 19. Yokota3-2: slope average. .. .. ... ... ... ... ... ... ... .... A-24

A.20. Yokota3-2: slope variance. .. .. .. ... ... ... ... ... ... ..... A-25

A.21. Yokota3-2: average removed x-slope homogeneity .. .. .. .. ... ... .... A-26

A.22. Yokota3-2: average removed y-slope homogeneity .. .. .. .. ... ... .... A-27

vii

Figure Page

A.23. Yokota3-2: average removed x-slope isotropy. .. .. .. . .. . .. . ... ... A-28

A.24. Yokota3-2: average removed y-slope isotropy .. .. .. .. . ... ... ..... A-29

viii

List of Tables

Table Page

2. 1. Mean-square tilt .. .. .. .. ... ... ... ... ... .... ... ...... 2-11

4. 1. ABLE ACE data sets. .. .. .. .. ... ... ... ... ... ... ... ... 4-1

4.2. Recovered r, values. .. .. .. .. .... ... ... ... ... ... ..... 4-15

ix

AFIT/GE/ENG/95D-25

Abstract

Atmospheric turbulence has long been recognized as one of the fundamental factors affecting

optical systems operating through the atmosphere. Turbulence over vertical paths has been well

characterized, both theoretically and experimentally. Much less is known about turbulence over long,

horizontal paths. Perturbations of the wave front phase can be measured using a Hartmann wave

front sensor (H-WFS). Theoretical expressions for the tilt-removed structure function of the H-WFS

slope measurements were derived and evaluated using quadrature. These slope structure functions are

functions of the phase structure function. The Kolmogorov turbulence model was assumed. Simulated

H-WFS slope measurements were generated using an adaptive optics simulation code. The effects

of deterministic aberrations on the structure function estimator were illustrated using the simulated

slope measurements. Average slope removal was shown to be effective in removing the effects of

aberrations. Five H-WFS measurement data sets from the ABLE ACE database were analyzed for

homogeneity, isotropy, and fit to theory. The ABLE ACE experiment recorded H-WFS measurements

over horizontal paths 20-200 km in length. Results indicate that the turbulence encountered appears to

be homogeneous and isotropic and seems to fit the Kolmogorov turbulence model.

x

Characterization of Atmospheric Turbulence Over Long Horizontal Paths

Using Optical Slope Measurements

I. Introduction

Atmospheric turbulence has long been recognized as one of the most significant factors limiting

the performance of optical systems. Even the best astronomical observatories, with telescopes meters in

diameter, obtain resolutions equivalent to telescopes only ten to twenty centimeters in diameter because

of atmospheric turbulence [17]. Thus, characterization of atmospheric turbulence is of paramount

importance in understanding its effects on optical systems.

1.1 Background

Understanding atmospheric turbulence is critical for several technologies which support USAF

missions. Ground-based telescopes are used to image satellites and orbital debris. The resolution of

these telescopes can be improved using adaptive optics which compensate for the effects of turbulence.

However, many adaptive techniques require stochastic models of turbulence. Using incorrect models

can artificially degrade the performance of an adaptive optical system [21].

High energy airborne lasers involve laser beams propagated over long, horizontal paths in the

troposphere and stratosphere. Scintillation of the optical beam causes the spot on the target to break

up reducing the available power. Adaptive optics may help compensate for some of the scintillation.

Understanding the statistics of atmospheric turbulence will help determine the feasibility and operational

limits of this concept, and the role adaptive optics can play in overcoming some of these limits [25].

Finally, air-to-air optical communication has the potential benefits of long range, low probability

of intercept (LPI), and high data rates. Predicting system performance, detector design, and estimating

the benefits of forward error correction schemes are highly dependent on an accurate knowledge of

turbulence statistics [4, 18, 1].

Vertical paths through the atmosphere have been studied intensively and sophisticated turbulence

models have been developed because ground-based telescopes have been the primary interest [1, 10,

13, 17]. Balloon-born thermosondes, radar sounders, and sonar profilometers are all well established

1-1

methods for measuring atmospheric turbulence [13]. Unfortunately, these methods are most useful

over vertical paths.

Long, horizontal paths are important for new applications such as airborne laser weapons and air-

to-air optical communications [18, 25]. However, whether atmospheric turbulence over long, horizontal

paths conforms to the same theoretical models as turbulence over vertical paths is not well understood.

A few experiments have been conducted to measure atmospheric turbulence over horizontal paths.

1.2 Previous experiments

Wright Laboratory performed a series of experiments, called HAVE LACE, during 1985. The

goal of these experiments was to determine the feasibility of air-to-air optical communications by

measuring the scintillation statistics of a laser beam propagated between two aircraft [5]. Unfortunately,

these scintillation statistics tell us little about the phase perturbations of the wave front. It is the phase

perturbations which ultimately produce scintillations. Thus, the HAVE LACE measurements are not

useful for directly characterizing the effects of atmospheric turbulence on the phase of an optical wave

front.

Phillips Laboratory performed a series of experiments in 1993, called ABLEX, to characterize

turbulence in the upper atmosphere. ABLEX used a temperature probe mounted on the nose of an

aircraft to provide an indirect measurement of turbulence along the aircraft's path [3]. Calibration of

this temperature sensor is difficult and the subject of ongoing research [19]. To characterize horizontal

paths, these temperature measurements must be integrated along the aircraft's path while accounting

for variations in the the aircraft's velocity.

1.3 The ABLE ACE experiment

ABLE ACE is a series of experiments performed by the Phillips Laboratory during January and

March of 1995. The goal of ABLE ACE is to measure laser beam propagation characteristics over long,

near-horizontal paths above and around the tropopause. The effects of atmospheric turbulence on the

amplitude and phase characteristics of a laser beam are of primary interest. Predicting the performance

of optical systems depends on knowledge of these characteristics [3, 25].

1-2

During the experiment, measurements were made on laser beams propagating between two

aircraft flying in the stratosphere separated by up to hundreds of kilometers. The laser beam propagates

over a long horizontal path from the transmitter aircraft to the receiver aircraft which carries a telescope

that collects a portion of the laser beam. As the laser beam propagates between the two aircraft through

the atmosphere the wave front is perturbed. These wave front perturbations are a direct effect of the

turbulence encountered by the laser beam along the propagation path. Measuring and analyzing these

perturbations should enable us to characterize upper atmosphere turbulence and validate theoretical

models.

Measurements of wave front phase perturbations made using a Shack-Hartmann wave front

sensor (H-WFS) are the focus of this thesis. Collecting H-WFS slope measurements aboard a moving

aircraft requires a tracking system that keeps the receiving telescope pointed at the transmitting laser.

This tracking system attempts to remove the overall tilt from the wave front as it tracks the transmitter.

Tracking errors produce overall tilts not induced by atmospheric turbulence. Tracking errors can also

be produced by random jolts and vibrations within the aircraft. Any observed overall tilt must be

removed in the data analysis because of this removal and corruption of the overall tilt.

A H-WFS cannot sense the wave front phase directly. Instead, it produces a set of wave front

slope measurements and the wave front phase must be inferred from the slope measurements. The

standard approach is to use a phase reconstruction algorithm to recover the phase from the phase slope

measurements. Phase reconstruction is widely used in adaptive optics systems and has been shown

to work well in laboratory measurements of turbulence [12]. Unfortunately, phase reconstruction

algorithms are sensitive to errors in the slope measurements. Any aberrations or tilt not due to

turbulence will be reconstructed. Measurement noise will produce errors the reconstructed phase.

The primary statistical measure of atmospheric turbulence is the phase structure function, which

can be calculated from the recovered phase. The phase structure function describes the perturbations

induced on the wave front phase by atmospheric turbulence. A large body of theory using the phase

structure function to describe turbulence is available.

Researchers at the USAF Phillips Laboratory have attempted to use phase reconstruction tech-

niques to analyze atmospheric turbulence using the ABLE ACE H-WFS data [2]. The researchers

used tilt-removed H-WFS slope measurements and a well-known least-squares phase reconstruction

1-3

algorithm [16] to recover a set of coefficients for the Zemike polynomials [14] which describe the wave

front phase. These results did not fit any known model for atmospheric turbulence.

The researcher's next approach was to use a minimum-variance phase reconstruction algo-

rithm [24] known to produce better results than the least-squares reconstructor under noisy condi-

tions [16]. The minimum-variance reconstructor uses knowledge about the statistics of the measure-

ment noise and atmospheric turbulence to improve its performance. The Zemike coefficients obtained

using this method fit the Kolmogorov turbulence model. However, it was unclear if the fit to the

Kolmogorov model was because the turbulence was indeed Kolmogorov or was an artifact of the

assumptions used in the minimum-variance reconstructor.

Other problems encountered were with obscured and unreliable subapertures in the H-WFS.

Special efforts had to be made in the phase reconstruction algorithms to account for these obscured and

unreliable subapertures.

These difficulties provided the impetus to consider using the slope measurements themselves to

characterize atmospheric turbulence. Using the slope measurements directly allows us to avoid phase

reconstruction with its heavy computational load and sensitivity to measurement noise and aberrations.

1.4 Problem description and scope

Characterizing atmospheric turbulence over long horizontal paths using H-WFS slope measure-

ments directly is the goal of this thesis. The ABEL ACE H-WFS measurements will be used. Since

turbulence is random this characterization must be through the turbulence's statistical properties. Struc-

ture functions are one of the most powerful and widely used statistical quantities for characterizing

atmospheric turbulence. In this thesis the structure function of the H-WFS slope measurements was

used to avoid phase reconstruction. The analysis steps used in this thesis were:

" Derive and evaluate a theoretical expression for the structure function of the H-WFS slopes.

" Estimate slope structure functions from the H-WFS slope measurements.

" Analyze the slope structure functions to determine the homogeneity and isotropy of the measured

turbulence.

* Compare the estimated and theoretical slope structure functions to see how well they fit.

1-4

1.5 Summary

Chapter I provides the derivation and evaluation of the theoretical slope structure function

using widely accepted models of atmospheric turbulence. The H-WFS measurement processing and

estimation of the slope structure function are described in Chapter m. Chapter IV identifies the

ABLE ACE data sets analyzed and presents an analysis of the homogeneity and isotropy of upper

atmosphere turbulence. The fit between estimated and theoretical slope structure functions is also

detailed. Chapter V presents conclusions and provides suggestions for further research.

1-5

II. Theory

The difficulties experienced by researchers at the USAF Phillips Laboratory in using standard

wave front reconstruction techniques motivates us to examine the slope measurements directly, bypass-

ing the computationally intense phase reconstruction step. This chapter begins with a brief description

of atmospheric turbulence and how a Hartmann wave front sensor (H-WFS) is modeled mathematically.

The slope structure function is then defined and manipulated into a form which depends on the phase

structure function. Last, the theoretical expressions obtained are evaluated using numerical quadrature.

2.1 Atmospheric Turbulence

Atmospheric turbulence is fundamentally generated by differential heating of the Earth's atmo-

sphere. Large eddies of warm air mix with colder air, transferring energy from larger eddies to smaller

eddies in the process. These smaller eddies transfer energy to still smaller eddies until viscous effects

dominate. At this point, the flow becomes laminar and energy is dissipated as heat. Eddies with

different temperatures have different densities and therefore different indices of refraction. It is these

random index of refraction variations which corrupt an optical wave front as it propagates through the

atmosphere.

Analyses by Kolmogorov [11] and Tatarski [22] are the seminal theoretical works on the statistics

of atmospheric turbulence. Kolmogorov derived a three dimensional power spectral density (PSD) for

the index of refraction variations in the atmosphere. Tatarski, in turn, derived the structure function

for index of refraction variations. Key assumptions in Kolmogorov's and Tatarski's derivations are

the homogeneity and isotropy of the atmospheric turbulence. Homogeneity means the turbulence's

PSD does not depend upon location in the atmosphere; the turbulence is wide-sense stationary (WSS).

Isotropy means the turbulence's PSD is independent of the direction of propagation through the

atmosphere.

The Kolmogorov PSD is proportional to a simple 11/3 power law within a region called the

inertial subrange defined by the inner scale and the outer scale. The inner scale is the smallest eddy size

below which the turbulence becomes laminar and kinetic energy is dissipated as heat. The outer scale

denotes the size of the largest eddies which form in the atmosphere. Clearly, eddy size is limited by

the finite size of the atmosphere and the assumptions of homogeneity and isotropy will eventually be

2-1

violated. Fortunately, the range between the inner and outer scales covers several orders of magnitude

in many situations. Also, many optical propagation calculations are insensitive to the exact nature of

the PSD beyond the inner and outer scales.

Mathematically, Kolmogorov PSD implies infinite energy, which is not physically possible.

While the Kolmogorov PSD may not be strictly physically valid it is the most widely used power

spectrum. The Kolmogorov PSD is also known to accurately model the turbulence seen on vertical

paths. Von Karman [23] proposed a PSD which has finite energy and falls off rapidly outside the

inner and outer scales. Recently, Stribling [21] has investigated turbulence power spectrums based on

general power laws with exponents between 3 and 4.

2.2 Phase structure functions

The wave front phase of an optical beam is perturbed by random refractive index variations as

it travels through atmospheric turbulence. These perturbations are assumed to have zero mean. The

effect of random refractive index variations on an optical beam are most commonly described using

the phase structure function, defined as,

Do(if1 , r2) = - (2.1)

where O(rj is a single realization of the wave front phase, i 1 and i2 are two coordinate positions, and

(.) denotes the ensemble average. This is actually the general form for any structure function [8].

If the random refractive index variations are homogeneous the phase structure function will be

a function only of the difference between the two coordinates, if1 - if2. If the the refractive index

variations are also isotropic the phase structure function will depend only on the magnitude of the

separation, r = IF, - i 21.

Fried [7] was able to derive a functional form for the phase structure function by applying the

index of refraction structure function, derived from Kolmogorov turbulence theory, to an optical wave

propagating through the atmosphere. His expression for plane wave propagation,

DO(r) = 6.88 (r )5/3, (2.2)

2-2

accounts for variations of the turbulence strength along the propagation path length since r,, the

'coherence diameter' or 'seeing parameter', is defined as

6.8 8)2 3/5(23'= [2.91 fo dz z (2.3)

where C2 (z) is the Kolmogorov structure constant along the path of propagation, L is the path length,

and A is the wavelength.

2.3 Hartmann wavefront sensor

One method of measuring the wave front phase is to use a Hartmann wave front sensor (H-WFS).

An H-WFS cannot sense the phase directly; it produces slope measurements which are the average

wave front phase gradient over the subapertures of the sensor [16, 24],

n jdifWn (rj(V(f.)

= - Jdf (VWn (r d (i), (2.4)

where s' is the slope in the ith direction at the nth subaperture, 0(rl is the wave front phase, W,, (r-) is

the subaperture weighting function, ' is a two-dimensional vector in the plane transverse to the wave

front propagation, di is the unit vector in the ith direction, V is the gradient operator, and represents

the vector dot product. The indices i and j can take on the values x or y since the H-WFS produces

slope measurements in both the x and x directions.

The H-WFS used in the ABEL ACE experiments had square subapertures so we can write the

subaperture weighting function as,

Wn (? = W, (x, y) = -rect (X rect (Y -Yn) (2.5)

where d is the distance across the subaperture edge and the point (x,, y,) is the center of the nth

subaperture, and

X-X,) 1 ,Ix-x I< (2.6)rect d 1 0 ,else 2

2-3

Note that W, (r) is normalized to have unit volume. The gradient of the subaperture weighting function

then becomes,

VW (il=VW.(x,y) = - ,--W(xY) + d -W(xY) (2.7)

- -- [X(x-x,+2) -6 X-n d)]rect(y- )

-f y~ ( - yn+ d) _ (y - _d ) ] rect (X Xn)

where 6(.) is the unit impulse function.

2.4 Slope structure functions

The phase structure function statistically describes the random perturbations of the wave front

perturbations. Since the H-WFS slope measurements are a linear operation on the phase we expect the

slope structure function to be a function of the phase structure function. Combining the definition of a

structure function and the wave front sensor slopes produces a slope structure function,

D'i (n, k) = - 8W

-K[Jdi;l (VWn(iF1 ) -di) 0(rl - Jdi2 (VWk(i;2) d, s(F 2) . (2.8)

Here the notation D ~(n, k) represents the slope structure function in the ith and jth directions for the

nth and kth subapertures at the points i 1 and F2. The H-WFS produces slope measurements in the x

and y directions; these directions are the values i and j may take.

Collecting WFS slope measurements aboard a moving aircraft requires a tracking system that

keeps the receiving telescope pointed at the transmitting laser. This tracking system attempts to remove

the overall tilt from the wave front as it tracks the transmitter. The tracking system is also noisy.

Tracking errors, and random jolts and vibrations within the aircraft, produce overall tilts not induced

by atmospheric turbulence. Because of this removal and corruption of the overall tilt, any observed

overall tilt will be removed in the data analysis. Thus, we must determine the effect of removing the

overall tilt from the slope structure function.

2-4

The tilt removed phase can be represented as, O(r- - f. f, where 6 is a vector representing the

overall wavefront tilt. Substituting the tilt removed phase into Eq. (2.8) produces,

D'j(n, k) K[Jd;' (VW(T 1 ) . O¢(F) - JdF (VWkA(2 ) "di) O(i2)

-(i,. di_ i. j)12) (2.9)

The first two terms are the wave front sensor slopes and the last term is the difference of the overall

tilts in the ith and jth directions. When the direction vectors are identical the overall tilts exactly

cancel, and the slope structure function is insensitive to overall tilt. This insensitivity to overall tilt is

analogous to the insensitivity of the phase structure function to overall piston.

However, when the direction vectors are different the overall tilts will not, in general, cancel and

the slope structure function will be sensitive to tilt removal. Thus, we must consider two cases in this

derivation. The case where the direction vectors are identical (D" and DYY) is referred to as the self

slope structure function; the case where the direction vectors differ (Dr' and DY') is referred to as the

cross slope structure function.

2.4.1 Self slope structure function. Consider the self slope structure function for the x-slopes

first. Expanding Eq. (2.9), canceling the tilt, and moving the expectation operator inside the integrations

produces:

Dx+(n, k) = Jfd'di'' (vwn(F1) . a) (VWn(f). a) ()

- 2fdf 2 (vwn(F1) . d) (VWn(i 2) . dx) (2.10)

Expanding the phase correlations using the identity1 1 1 _

1 (0 2 (F))) + I (02(2))) - (rl, 2),

where D,(T,F2) is the phase structure function defined in Eq. (2.1), allows us to write the slope

structure function in terms of the phase structure function. The correlations ( F2(f1)) and (2 (F2))

are constants because the phase, O(r, is assumed to be homogeneous (WSS). Since VWn(rl is an

2-5

odd function and the integrals have symmetric limits these terms will vanish. Thus, the slope structure

function becomes,

1

+ !JfJdF2d (VW(F) - 4x) (VWk,(i) -d.) D(F, F)

Using Eq. (2.8), the sifting property of the impulse function, and the homogeneity and isotropy of the

phase structure function we can write,

D" (n,2k) 1 -Jd yidy; [2D,(Iy, - y1) - DO(I - d, y - y'l) - DO(Id, yi - Yfl)]

xrect (Y Yn) rect (Y Yn

W d dy2 y' [2D(Iy 2 -YJ)- DoD(I -d,y 2 -Y21)- D (Id, yl Y1)]

x rect (Y2 d ) rect (Y' Yk)

+-Jfdydy 2 [2DO(Ix -xk, yj-Y21) Do(Ixn -xk-d, y -Y21)

-Do(xn - xkd, y- Y21)] rect (Y1 d-Yn) rect (Y2 d Yk)

Here the shorthand notation Ix, Y = - + y2 is used to simplify the arguments of the phase structure

functions. Using the variable substitutions

U=y -y 1 u 1

U Y2-Y 2 (Y + Y

U=Y1-Y2 u'= (yl+y2)

2-6

in these three integrals, respectively, produces

D,"x(n, k) I.Jdu [2DO(Iul) - D,(I - d,u) - DO(ld,u)]

x Jdu'rect (U + U/2 - y. rect U - u/2 - yn)

jdure u'+ u/2 - Yk) rc(u' -u/2 - Yk'\

T 4jdu [2D(lX - XkUl) - D,5(Ix - Xk - d,uI) - DO(JXn Xk + duI)]

x duIrect (U' +u/2 - y'e(u'-u/2 -Ykd drec d

Recognizing the fdu' terms as a correlation, which can be written as

jdu rect ( U2-y,, rect(u-u/ -yk) d-U Yn+kl U Y+Yi<d d 1 0 ,else

=r [U -(Yn - Yk)]d

the self slope structure function reduces to:

Dx(nk) 1 ]du [2DO(Jul) - Do(I - d,ul) - DO(ldul)] tri

+ 4 du [2DO(Ixn - xk,uI) - Do (Ix.,, Xk - d, u)

-DO(Ixl- k +d, ul)]tri[U- (Yn - Yk)] (2.11)d

Using this same procedure on the y-slope self structure function produces:

D~Iv(rk) -1 ]du [2DO(Iul) - DO(,u, -dl) - D¢(,u, dl)] tri dD~v (, k)d4

+-T ]du [2DO (u,y. - Ykl) - Do(Ju,Y - Yk -dl)

-Do(Iu,y. - yk + dl)] tri [ - (X,- .k) (2.12)

d

2-7

Up to this point in the derivation the only assumptions made were the homogeneity and isotropy

of the phase structure function, and the zero mean characteristics of the wave front phase. In particular,

nothing has been assumed about the form of the phase structure function. Any appropriate phase

structure function (Kolmogorov, Von Karman, general power law) may be used. The simplest and most

widely used form is the Kolmogorov structure function which Fried writes as [71,

D ,(r) 6.88 -5 /3

Substituting this phase structure function into the x-slope self structure function produces,

D,(nk) = 6.88 r05 /3d [2jdu - I - du - Id, U1, / 3 d

+ -r 5/3 du - - Ixn - d, ,

-IX-,- +d,U/3]tri [ U - (y - )] . (2.13)

Normalizing the integrals by using du = d d$ produces the x-slope self structure function

D:(n,k) = -6.88d-1 / 3 5 ' {2/d [11, i1'5 - ii / 3 ] tri[{]

fd Xnd 1Xk ,5/ + - + 1, 5/ 2 d , /

xtri[{ (Yn - Yk)] (2.14)

where the explicit dependence on the turbulence parameter, r., and the WFS subaperture dimension.

d, have been moved outside the integrals. The same procedure produces the y-slope self structure

function

DYY(n,k) = -6.88r- 5 3' 1 {2jd [iE 15/3' - iEK5 3] tri( c)

-dA+ d A+ 1,5 d ,

x×tri [ (Xn - Xk)] (2.15)

2-8

Notice that D" (n, k) and Dv(n, k) have exactly the same functional forms except for the shift

direction dependence in the tri(.) function. This result is not surprising and is a direct result of the

isotropy of the phase structure function and the symmetry of the subaperture weighting function. The

self slope structure functions, D"' (n, k) and DY '(n, k), will be evaluated numerically in Section 2.5

to determine their behavior. The next challenge is to determine the cross slope structure function.

2.4.2 Cross slope structure function. Recall that the cross slope structure function is sensitive

to tilt removal. Expanding Eq. (2.9) for the xy slopes and moving the expectation operator inside the

integrations produces,

Dx(n, k) = Jf d iS' (vw(F) -&) (Vw(f;) .

+ Jfd;'dF,' (VWk(f') .d) (VWk(i;2) - dy) (O(f;2)0(fV)

- 2 Jfdf~jdi (VWn(F;1) -d.) (VWk(f; 2) d) (O(F;1>0( 2))

+ K[i. -_i. dX] 2). (2.16)

The first three terms are due only to the wavefront phase and thus compose the non-tilt-removed cross

slope structure function. The last three terms arise from the tilt removal. With the procedure used for

the self slope structure functions the first three terms can be immediately written in terms of the phase

2-9

structure function. The cross slope structure function becomes,

D"'(n, k)

- 2 Jdu [2DO(Iul) - DO(I - d, ul) - DO(Id, ul)] tri

1 Jdu [2D,(Iul) - DO(I - d,u) - Do(Id, ul)] tri(d)2d4(U

4 dudv [D,(IU - x, + d/2, v - yk + d/21) + DO(Iu - xn - d/2, v -Yk -d/21)

- Do(ju - x, + d/2,v - Yk - d/21) - DO(Iu - x - d/2, v - yk + d/21)]

xrect (U Xk) rect (V Yn)

+ ( d -d' -it dx]) .(2.17)

The fourth and fifth terms of the cross slope structure function require calculation of the correlation

between the wavefront's tilt and phase. Heidbreder [9] shows that these correlations can be written as,

O(X., )( y)=- ffududv P(u, v)Do(Iu - x, v -Y (2.18)

2J]u2dudv P(u, v)

O(x'y)} = fvdudvP(u'v)D¢(u-xv-Y) (2.19)

2 Jfv2dudv P(u, v)

where P(x, y) is the pupil function. Restricting P(x, y) to be symmetric, inserting Eqs. (2.18) and

(2.19) into the fourth and fifth terms of Eq. (2.17), expanding, grouping terms, and recognizing a

common denominator produces,

_ f(u-v)dudv P(u, v)dyD¢(lU-xn+d/2, v-yl - DO(Iu-xn-d/2,v-y)}rect(Y n

d2J.u2dudv P(u, v)

(2.20)

2-10

and

Jju-v)dudvP(u~v)Jdx2 {DOt(u-X2,v-Yk+d/2l) - Dp(ju-X2 ,V-Yk-d/2j)}rect (X2 - k

d2ffu2dudv P(u, v)

(2.21)

which depend only on the pupil function, P(x, y), and the phase structure function, Do (r).

Assuming the tilts in the x and y directions are independent and zero mean, the last term of

Eq. 2.17 can be expanded as,

Kdd.2 + K i y2which is the sum of the mean-square tilts in the x and y directions. Heidbreder shows that the

mean-square tilt can be written as

-X) Juv dudv P(u)P(v)DO(U - V)K (d)2 (( ) 2Jfuv dudvP(u)P(v) (2.22)

Because the pupil function is restricted to be symmetric, and the phase structure function is assumed

to be homogeneous and isotropic, the mean-square tilt in the x and y directions are identical.

Pupil I Mean-square tilt -

Square 6.88(0.8598)D-1 /3 r 5/3

Circular 6.88(0.8281)D-1/3r-5 /3

Table 2.1 Mean-square tilt for square and circular pupils using Kolmogorov structure function. D is

the side length or diameter of the pupil, respectively, and ro is Fried's strength parameter.

2-11

Combining Eqs. (2.17), (2.20), (2.21), and (2.22) produces

D"Y (n, k)

= Jdu [2DO(IuI)-DO(- d,u)-D(ld,u)]tri

1 f du [2D4(u)- D(I -d, ul) - D¢(Id, ul)] tri (d)4 dudv [D,(Iu - xn + d/2, v -yk + d/21) + DO(u - x,, - d/2, v -yk - d/21)

- De(Iu - x + d/2, v- Yk - d/21) - DO(tu - xn - d/2, v - Y + d/21)]

x rect ( U d Xk) rect (V dYn)

+ -fu-v)dudvP(u, v)Jdyi{D(Iu-xn +d/2, v-y 1 )-D(Iu-xn-d/2, v - y)}recty- Yn)

+ d2ffu2 dudv P(u, v)

J -(uv)dudvP(u, v) dz{Dh(u-x2, v-yk+d/21)-D¢(lu-x 2, V-yk-d/21)}rect(X2 -Xk)

d2ffu2dudv P(u, v)

f fuv dudv P(u)P(v)DO (u - v)

Jfuv dudvP(u)P(v)

Note that an explicit functional form of the phase structure function has not been assumed; only

homogeneity and isotropy. Any appropriate structure function may be used at this point.

Heidbreder has computed the mean-square tilt (Eq. 2.22) for both square and circular pupils

assuming the Kolmogorov phase structure function. Heidbreder's results are summarized in Table 2.1.

2.5 Slope structure function properties

Now that we have the self slope structure function we can examine its properties. The self slope

structure function is only a function of the subaperture separation. However, there are an enormous

number of possible separations in many directions. Concentrating on shifts in the x, y, and +450

directions will provide a representative sample of all possible shifts.

The self slope structure function consists of two scaling parameters which depend on the subaper-

ture dimensions and the turbulence strength and a dimensionless integral which depends on differences

2-12

10

9C

0 8

--

6-

x X-shifts4-_ XY-shifts

x Y-shiftsE0 3

z2

0510 15 20 25 30Separation (subapertures)

Figure 2.1 Normalized x-slope structure function for x, y, and xy shifts as a function of subapertureseparation

of phase structure functions for various subaperture separations. Because the integral is dimensionless

we need only evaluate it once for the separations of interest.

The dimensionless integral portion of Eq. (2.14) was evaluated using numerical quadrature for

shifts in the x, y, and ±45' shifts. The results are shown in Fig. 2.1 (the xy shifts correspond to the

±45' directions. We can see that the slope structure function vanishes for zero shifts, as it should,

and that all shift directions asymptotically tend toward the the same slope. The actual value of the

slope structure function depends on the shift direction. These curves are also valid for the y-slope self

structure function, DYY, except the curve for the x and y shifts must be interchanged.

2-13

III. Statistical methods and simulation results

As we have seen in Chapter II, the WFS slope structure function is a function of the phase

structure function, for which there exists a highly developed theory. Our goal is to use H-WFS slope

measurements to estimate the slope structure functions and test for homogeneity, isotropy, and fit to

the theory developed in Chapter II. In this chapter, the statistical methods used to analyze the H-WFS

slope measurements are presented. The performance of these methods using simulated WFS slope

measurements is also presented.

The steps used in analyzing the ABLE ACE WFS measurements were:

" choose a data set and segment the WFS images into contiguous sets of frames,

" process the WFS images to extract the slope measurements,

" estimate the slope structure functions,

" examine the slope structure functions for homogeneity and isotropy,

" and compare the estimated and theoretical slope structure functions.

In the next two sections we will briefly describe how a Hartmann WFS operates and define the

slope structure function estimator. The following sections will examine each of the above steps in

detail and the chapter will close with some comments on tradeoffs and limitations.

3.1 Hartmann WFS slope measurements

A Hartmann WFS consists of an array of lenslets and a detector array in the focal plane of the

lenslet array. The incoming wave front is spatially segmented by the lenslet array; each segment is

called a subaperture. A slice through this configuration is depicted in Fig. 3.1(a). The lenslet array in

the H-WFS used in the ABEL ACE experiment was a square 21 x 21 subaperture grid and the detector

was a CCD camera having a 512 x 512 pixel array.

Figure 3.1(b) depicts a single subaperture. Each lenslet produces a spot on the detector array

whose location is proportional to the average tilt, 0, of the wave front incident on the subaperture.

Spot location, x, is defined as the centroid of the pixel intensities within the spot's main lobe. Given a

3-1

Incident HartmannWave Front Wave Front

Sensor f

/E

(a) (b)

Figure 3.1 Diagram of (a) Hartmann wave front sensor and (b) a single subaperture.

subaperture size d the wave front tilt can be written as,

S=tan- (f) , (3.1)

where x is the spot location in pixels and f is the focal length of an individual lenslet.

Each subaperture subtends 21 pixels per side in the pixel array and the telescope pupil was

imaged onto the lenslet array such that each subaperture corresponds to a distance of three centimeters

in the pupil. The main lobe of the diffraction pattern is four pixels wide from peak to first null. Thus, a

shift of four pixels corresponds to a AO of one wavelength [2]. Using Fraunhoffer diffraction theory [8]

the angle, 9, can be written as,

o 9 , (3.2)4d

where the subaperture width, d, is 3 cm, the laser wavelength, A, is 0.532 /zm, and small angle

approximations are used. The slope, s (in rad/m), is calculated using

(0d) 2f 7

d (3.3)

This conversion factor is used to convert from spot centroid locations, x (in pixels), to wave front

slopes, s (in radians/m).

3-2

The timing used to capture the H-WFS images is also important. The atmosphere is effectively

'frozen' during each measurement since the aircraft flew at approximately 200 m/s. Each WFS image

is captured using a single laser pulse 25 ns in duration. The CCD camera frame rate was 1/3 Hz

so the aircraft covered approximately 67 m meters between frames. Each frame is thus affected

by independent realizations of atmospheric turbulence. These qualities simplify the corresponding

statistical calculations.

3.2 Structure function estimation

The slope structure function is the main statistic we will use to characterize the turbulence

encountered by the laser beam. An estimate of the slope structure function can be found by computing

the mean square difference of the H-WFS slope measurements between any two subapertures within

the pupil of the telescope. This operation corresponds to,

1M j

D," (n, k) = [s' - sj]2 (3.4)

where D (n, k) is the estimated slope structure function, M is the number of frames (realizations) in

a data set, and s' and si are the slope measurements at the nth and kth subapertures in the ith and

jth directions, respectively. The i and j directions are unit vectors in the x and y directions in a plane

perpendicular to the direction of the wave front propagation. The indices i and j may take values of x

or y depending on the direction of the slope measurement. The mean is calculated over the ensemble

of frames in a particular data set.

When estimating the slope structure function, only slope measurements from the chosen sub-

aperture pairs and slope directions are averaged across the ensemble of frames. Thus, no assumptions

are made about the homogeneity or isotropy of the turbulence. However, without these assumptions

the slope structure function must be estimated separately for each possible pair of subapertures and four

possible slope direction combinations; an enormous number of possibilities. To keep the computations

manageable, yet still have a representative sample, the analysis was confined to subaperture separations

in the x, y, and ±450 directions. Section 3.4 describes how these calculations were implemented.

Not assuming homogeneity or isotropy allows us to test the resulting estimated slope structure

functions for these very qualities by examining the behavior of the slope structure function. In

3-3

(a) (b)

Figure 3.2 The phase structure function will have the same value for each vector separation shown

if the turbulence is (a) homogeneous and (b) isotropic.

Chapter II we found that the slope structure function is a function of the phase structure function. Thus,

the behavior of the slope structure function depends on the behavior of the phase structure function.

We can represent any subaperture pair as a vector whose length corresponds to the separation

between the subapertures and whose orientation corresponds to the direction in which the subapertures

are separated. If the measured turbulence is homogeneous, the phase structure function will depend

only on the vector separation between two subapertures and not on the location within the pupil. Thus,

the value of the phase structure function corresponding to each of the vectors in Fig. 3.2(a) will be the

same. If the measured turbulence is isotropic the values of the phase structure function corresponding

of each the vectors in Fig. 3.2(b) will be equal. Isotropic turbulence depends on the magnitude of the

separation and not on the orientation.

Now that we know what the WFS sensor measures, and how to estimate the slope structure

function, we can process the WFS images to retrieve the slope measurements.

3.3 WFS image processing

The first step is to select a set of WFS images from among the thousands available in several

data sets. We would like as many contiguous frames of data as possible to provide a large ensemble

to average across. However, there is a tradeoff between ensemble length and the stationarity of the

turbulence. We must also account for interruptions in data collection.

3-4

The stationarity of the data was primarily gauged by visually assessing the average strength and

characteristics of the WFS sensor spot scintillation. Data sets were made as long as possible while

keeping the scintillation characteristics similar throughout the data set.

Extracting the WFS slopes requires centering the subaperture grid over the WFS spots as shown

in Figure 3.3(a). Unfortunately, the initial grid placement could not be automated because the WFS

spots did not appear in the same locations across the data sets. Also, the WFS spots sometimes

wandered slightly within a data set. Next, the subapertures not containing a well-formed spot were

rejected. This rejection was accomplished by comparing the mean-square error between the magnitude

of the subaperture spot's fast Fourier transform (FFT) and magnitude of the FFT an ideal spot to a fixed

threshold. Both FF1 magnitudes were normalized to have unity peak magnitude. A typical result of

this threshold rejection scheme is shown in Figure 3.3(b).

This threshold scheme worked well because the spot position is encoded in the FFT phase, which

we ignore. The FFT magnitude is only sensitive to the spot's shape. Subapertures containing only

noise and those with highly aberrated spots have large mean-square errors and were rejected.

Once the subaperture grid was defined, the average of the spot centroids was used to place the

subaperture grid. This fine centering step ensured that nearly all the overall tilt was removed. Finally,

the x and y spot centroid locations within each subaperture was extracted, scaled using Eq.( 3.3), and

saved in two separate matrices. A mask matrix defining the valid subapertures was also extracted and

saved.

The centering operations remove the overall tilt from the H-WFS slopes. As noted earlier, any

overall tilt cannot be trusted because it is corrupted by the telescope tracking mechanism and aircraft

vibrations. Now we can compute the statistical quantities of interest.

3.4 Statistical analysis

The first statistical quantities computed are the average and variance of the slopes at each

subaperture, averaging across the ensemble. Perturbations caused by turbulence are zero mean, so we

expect the slope averages to be zero across the aperture. Any non-zero average slope corresponds to a

deterministic or slowly varying aberration. In fact, possible aberrations were noticed during the WFS

image processing, which made it imperative to closely examine the average slopes.

3-5

I- 1 110 o ~ '

(a)

474a

a2

(b)

Figure 3.3 WFTS subaperture grid (a) initial placement and (b) after shape thresholding.

3-6

(a) (b) (c)

Figure 3.4 Slope differences showing (a) matrix overlap, (b) mask overlap, and (c) difference matrix

Perturbations due to turbulence will not cancel out in the second moment calculation. Removing

the average slope should remove most of the effects of any deterministic aberrations. Thus, the slope

variance will be primarily a function of the turbulence strength. How uniform the variance is across

the aperture gives a rough indication of the homogeneity of the turbulence. Homogeneous turbulence

should produce equal variances at each point in the aperture. The value of the variance also serves as

a crude measure of the turbulence strength.

Once these global statistics have been calculated, the slope structure function is estimated using

Eq. (3.4). As noted at the beginning of this chapter, this estimator makes no assumptions about

homogeneity or isotropy. Thus, there are an enormous number of possible separations and directions to

consider. Confining the computations to the x, y, and ±45' directions and even subaperture separations

provides a representative sample while keeping the computations manageable.

The differencing operation in the slope structure function estimator (see Eq.( 3.4)) can be

computed for each valid subaperture simultaneously by differencing shifted versions of the matrices

containing the slopes. Figure 3.4(a) shows a representation of two slope matrices shifted by four

subapertures in the +45' direction. The overlap shown by the shaded region contains the oniy valid

separations for this shift value.

Both points in the overlap region may not be within the region defined by the subaperture

mask, which changes for each frame. We must also compute the subaperture mask corresponding to

the overlap region. The overlap mask is calculated by simply multiplying the shifted mask matrices

since the mask matrices consist of ones for valid subapertures and zeros for invalid subapertures. The

resulting valid region in the mask matrix is shown by the shaded region in Fig. 3.4(b). This mask

3-7

is multiplied with the differenced slope matrix and centered in a matrix the same size as the H-WFS

subaperture grid. The resulting matrix is shown in Fig. 3.4(c). The shaded region contains all the valid

slope differences for that particular shift direction and separation. Each subaperture position within

the matrix corresponds to the center point between the differenced slope locations and contains the

differenced slopes.

The shifted mask, the squared slope differences, and the second moment of the squared slope

differences are accumulated in running sums. The mask matrix tracks how many measurements

are accumulated at each subaperture location. Dividing the accumulated slope differences by the

corresponding mask sum produces the structure function estimate at a particular subaperture location.

The accumulated second moment of the squared slope differences allows us to calculate the

variance of the squared slope difference's. Using the Tchebycheff inequality allows us to compute

a confidence interval for our slope structure function estimate [15]. A 95% confidence interval is

calculated using

C195 = :4.47 (3.5)

where o-2 is the variance and n is the number of frames in the ensemble. Using the Tchebycheff

inequality produces a conservative estimate of the actual confidence interval since no assumptions

about the probability distribution function of the structure function are necessary.

The slope structure function estimates are calculated using both the raw slopes and the slopes

obtained by subtracting the average slope from each frame. Comparing the raw and average removed

structure functions allows us to gauge the effects of removing deterministic aberrations. Now that the

estimated slope structure functions are in hand, we can test our results.

3.5 Homogeneity, isotropy, and fit to theory

Before comparing the estimated slope structure functions with the theoretical model developed

in Chapter II we must determine if the measured turbulence is actually homogeneous and isotropic. We

can examine the variance of the slopes to get a global sense for the homogeneity. The slope variances

should be approximately equal across the pupil if the turbulence is homogeneous. How similar the

slope structure functions are at various points in the aperture, for a particular shift direction, indicate

the homogeneity of the turbulence.

3-8

Determining the isotropy of the turbulence is more complicated. From the theoretical results in

Chapter II (Fig. 2.1) the structure functions for different shift directions will not be identical and cannot

be directly compared. The estimated slope structure functions from Eq.( 3.4) can be least-mean-square

fit to the normalized theoretical slope structure functions using Eq.( 3.6). Since the strength of the

turbulence and subaperture size are multiplicative factors we need only estimate a scale parameter. The

mean-square fit is computed using

d-1/3ro 5 / 3 = (NNT)-NTD, (3.6)

where N is the vector containing the theoretically computed normalized slope structure function and

D is the vector containing the estimated slope structure function [20].

The coherence diameter, r, can be calculated since the subaperture width, d, is known. Co-

herence diameters can be compared for various subaperture locations and shift directions to give an

indication of homogeneity and isotropy. The root-mean-square error between the estimated and scaled

theoretical slope structure functions gives an indication of how well the turbulence is modeled by the

Kolmogorov model.

3.6 Simulation results

Before applying the statistical methods described above to real data, we should determine how

well these methods perform. Do determine performance we need an ensemble of simulated WFS slope

measurements with known statistical properties. An adaptive optics simulation code available at ART,

otfsim3, was used to create such an ensemble [16]. The otfsim3 code contains a phase screen

generator which has been thoroughly tested and is known to produce phase screens with Kolmogorov

statistics [16]. The code also simulates a Hartmann WFS.

The phase screen generator and H-WFS simulation modules were used to produce an ensemble

of 200 frames of H-WFS measurements for a grid 22 subapertures on a side. Each subaperture side

was 4.5 centimeters long when projected into the pupil. A wavelength of 0.523Atm was used for the

H-WFS. This wavelength is the laser wavelength used in the ABLE ACE experiments. High light

levels were used so the simulated measurements will have very little noise. The r, value used to in

generating the ensemble was 10 centimeters.

3-9

Phase screens were generated, the overall tilt was removed, and the resulting phase screen was

sent to the H-WFS module. The resulting x and y slope measurements, as well as a subaperture mask,

were extracted in the same format as the slope measurements from the real data. Since the simulation

code provides slope measurements, the statistical methods were directly applied as described above.

Figures 3.5 and 3.6 show the x and y slope average and variance, respectively. As expected, the

average slopes are nearly zero mean. The slope variance is fairly uniform across the pupil.

3.6.1 Homogeneity. The next step is to compute the structure functions. Figures 3.7 and 3.8

show the x and y slope structure functions value versus subaperture separation for various locations

within the pupil. Each curve plotted with a dotted line corresponds to a structure function computed

at a particular location within the pupil of the telescope. The confidence intervals shown correspond

to one of the selected structure functions. The fairly large confidence intervals arise from the finite

ensemble used. As expected, for homogeneous turbulence the structure functions cluster together.

3.6.2 Isotropy. Since the simulated turbulence is homogeneous, we can average the structure

functions from various locations within the pupil together. The resulting spatially averaged structure

function will have many more samples than the non-spatially averaged structure functions and should

have much tighter confidence intervals.

Figures 3.9 and 3.10 show the spatially averaged x and y slope structure functions, respectively,

for the various separation directions. The dotted line and circles show the estimated structure function.

The corresponding theoretical slope structure function, least-squares fit to the estimated structure

function, is shown with the solid line.

As expected, the confidence intervals for the spatially averaged structure function are much

tighter. The root-mean-square (RMS) error between the estimated and scaled theoretical structure

functions is also very small. The value of r, was recovered to within two significant figures from

the scale factor in all cases. The methods work for perfect slope measurements; but what if there are

deterministic aberrations?

3.6.3 Effect of aberrations. Deterministic aberrations in the slope measurements can arise

from uncompensated aberrations in the optical train used to direct the light from the telescope pupil to

the wave front sensor. Such an aberration may influence our structure function estimates.

3-10

4,

01

-2,

-4,

2015 10'< l5 20

10 10 1

Y (subapertures) 0 0 5 X (subapertures)

4,

01

CfO

-2,

-4,

20 1

10 1510

Y (subapertures) 0 0 5 X (uaetrs

(b)

Figure 3.5 Simulation: (a) x-slope average, (sx), and (b) y-slope average, (sy).

3-11

700,

650,

S600,

550,

x 500,

450,

400,

20

Y (subapertures) 0 0 5 X (subapertures)

(a)

700,

650,C'J

S600,

550,

500,

450,

400

20

Y (subapertures) 0 0 5 X (subapertures)

(b)

Figure 3.6 Simulation: (a) x-slope variance, ((S,) 2) , and (b) y-slope variance, ((sv2)

3-12

2000 2000

1800- 1800-

1600- 1600-

1400- 1400-

1200- 1200-

X" 10001000-

800- 800-

400- 400-

201200 '

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8X separation (mn) Y separation (in)

(a) (b)2000 2000

1800- 1800-

1600- 1600-

1400- 1400-N E

S1200- 1200-

,1000- - .- ~1000 ""

800-80

600-60

400 400

200# 200f

0 0.2 0.4 0.6 0.8 1 0 '02 0.4 0.6 0.8XY separation (in) XY separation (in)

(c) (d)

Figure 3.7 Structure function homogeneity, simulation: D"~ for (a) x, (b) y, (c) +450, and (d) -45'separations.

3-13

2000 2000

1800- 1800-

1600- 1600-

1400- 1400-

S1200 - 1200-

1000 - 1000

600 1,' --. -600

200- 200':

_0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8X separation (in) Y separaion (in)

(a)(b2000 2000

1800- 1800-

1600- 1600-

1400- 1400-

S1200- 1200-

71000 / - 1000 -

800 800

600- 0

200 *200

06 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8XY separation (in) XY separaion (in)

(c) (d)

Figure 3.8 Structure function homogeneity, simulation: Dy'v for (a) x, (b) y, (c) +450, and (d) -450separations.

3-14

1200- 1200-

1000- 1000-

xx 600 600 Siuato

0 Simulation -Theoretical400 ~~~~~- Theoretical40-RVSerr-3.

RMVS error- 45.8 Ro -0.119Ro -0.119

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8X separation (in) Y separation (in)

(a) (b)

1200- 1200-

1000- 1000-

Ow 600- x 600-0 Simulation a 0 Simulation

- Thereticl- heoretical40040

FIVMS error - 77.5 CMS error-iGORo =0.119 Ro =0.121

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8XY separation (mn) XY separation (in)

(c) (d)

Figure 3.9 Structure function isotropy, simulation: spatially averaged D" for (a) x, (b) y, (c) +450,

and (d) -450 separations.

3-15

1200- 1200-

1000- 1000-

c'J4E 800 8001 0 Simulation

600 600-- Theoretical

0 Simulation oRMS error =39.4

-theoretical Ro -0.1 1

400 ~RMS error =44.5 40

Ro -O0.11

200- 200

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8X aeparation (in) Y separation (in)

(a) (b)

1200- 1200-

1000- 1000-

S800 800

'. 600 "'600O0 Simulation 00 Simulation

- Theoretical I- Theoretical

40 400lt'o

40-RMS error -29.8 40A MS error - 51.4Ro -0.117 rRo -0.117

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8XY separation (in) XY separation (in)

(c) (d)

Figure 3. 10 Structure function isotropy, simulation: spatially averaged DY'Y for (a) x, (b) y, (c) +450,

and (d) -450 separations.

3-16

To gauge the effect of deterministic aberrations, a simulated aberration was added to the simulated

slopes. The aberration used was a plane tilted in the x or y directions added to the x and y slopes,

respectively. The form of this simulated aberration was suggested by the shape of average slope

measurements seen in several ABEL ACE data sets.

Again, the average and variance of the slopes were calculated. Figures 3.11 and 3.12 show the

x and y slope average and variance, respectively. The deterministic aberration is recovered very well

by the slope average calculation. The slope variance is nearly identical to the unaberrated variance.

These results suggest that we can recover a deterministic aberration using the slope average. But what

effect does this aberration have on our structure functions?

Figure 3.13 shows the x slope structure functions for the various shift directions from selected

locations within the pupil. Others follow by symmetry. Homogeneity is not destroyed by this particular

aberration. The shape of the structure functions is distorted, especially at the larger separations.

Figure 3.14 shows the spatially averaged x-slope structure functions for the various shift direc-

tions. Large RMS errors occur between the estimated and scaled theoretical structure functions. The

recovered r, values are approximately two centimeters low. The fit between estimate and theory in

Fig. 3.14(b) is an artifact of the particular aberration used and will not in general happen. Can we

improve the performance of our methods by removing the average slope before estimating the structure

functions?

3.6.4 Effect of average removal. The average slope calculation closely estimated the

aberration added to the simulated slope measurements. Thus, we should be able to correct for a

deterministic aberration by subtracting the average slope from each frame of WFS slope measurements

before estimating the slope structure functions.

Figure 3.15 shows the average removed x slope structure functions for the various separation

directions from selected locations within the pupil. The structure functions are now nearly identical to

the results from the unaberrated slope measurements.

Figure 3.16 shows the average removed, spatially averaged x slope structure functions for the

various shift directions. The shape of the structure functions again matches the theoretical slope

structure functions and the RMS error has been reduced. The recovered r, values are nearly identical

to those recovered from the unaberrated slope measurements.

3-17

20,

10,

01

-10,

-20,A

20 15

2010 1510

Y (subapertures) X (subapertures)0 0

20,

-10,

-20,

-20

15 2010 15

10

Y (subapertures) 00 5 X (subapertures)

(b)

Figure 3.11 Simulation with aberration: (a) x-slope average, (sx), and (b) y-slope average, (sqr).

3-18

700,

650,c'e;a 600,

S550,

xCj 500,

450,

400,

20

Y (subapertures) 0 0 X (uaetrs

700,

650,

S600,

550,

~?500,

450

400

20

Y (subapertures) 00 5 X (subapertures)

(b)

Figure 3.12 Simulation with aberration: (a) x-slope variance, ((sx)), and (b) y-slope variance,

((svn))

3-19

2000- 2000-

c 1500 C L. 15002 E

0 1000 1000

500 500-

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8X separation (in) Y separation (in)

(a) (b)250 2500

2000- 2000-

S100100

50 50

0 0.2 0.4 0.6 0.8 1 0t- 0.2 0.4 0.6 0.8XY separation (in) XY separation (in)

(c) (d)

Figure 3.13 Structure function homogeneity, simulation: D" for (a) x, (b) y, (c) +450, and (d)-45' separations. Aberrated slope measurements.

3-20

2000 2000

1800- 1800- 0 Simulation

1600-1600-- Theoretical

RMS error =38.9

1400- 1400- Ro -0.119

E 1200- 1200-

1000- 0o Simulation 10

800 ~ ~ Theoretical 80

60-RMS error - 1.13e+03 600Ro - 0.0912

400 - 400-

0 0.2 0.4 0.6 0.83 1 0 0.2 0.4 0.6 0.8X separation (m) Y separaion (in)

(a) (b)2000 2000

1800- 80

1600- 1600-

1400- 1400(N CNE 1200 E 1200-a

800 0~~~~ Simulation80 e0 Smlto60- Theoretical 60- Theoretical

40RMS error - 567 40RMS error - 569Ro - 0.0973 Ro - 0.0989

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8XY separation (mn) XY separation (m)

(c) (d)

Figure 3.14 Structure function isotropy, simulation: D" for (a) x, (b) y, (c) +45', and (d) -45'separations. Aberrated slope measurements.

3-21

2000 2000

1800- 1600-

1600- 1600-

1400- 1400-

c'JJ

1000- 1000-

800-

200-;

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8X separation (in) Y separation (in)

(a) (b)2000 2000

1800- 1800-

1600- 1600-

1400- 1400-

800- 800

1200 E 100 -

400-400

200-a 200

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8XY separation (mn) XY separation (in)

(c) (d)

Figure 3.15 Structure function homogeneity, simulation: D"7 for (a) x, (b) y, (c) +450, an~d (d)-45' separations. Average slope removed.

3-22

1200- 1200-

1000 1000.

xx 600 ~'600a 0~~~ Simulation 00 Smlto

40 Theoretical - Theoretical4000

RMS error - 44.4 RMS error =37.2

Ro -0.119 Ro - 0.119

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8X separation (in) Y separation (mn)

(a) (b)

1200- 1200-

1000- 1000

w 600 Ow 60000 Simulation 00 Simulation

-Theoreficall - Theoretical400 400

RMVS error =75.9 RMVS error -103Ro -O0.11 Ro - 0.121

0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8XY separation (in) XY separation (in)

(c) (d)

Figure 3.16 Structure function isotropy, simulation: D " for (a) x, (b) y, (c) +45', and (d) -. 45'

separations. Average slope removed.

3-23

3.7 Observations

Using the H-WFS slope measurements directly to sense atmospheric turbulence appears to be

feasible. The slope structure function confines us to a discrete set of separations, defined by the

subaperture grid. Phase reconstruction would be more flexible since we would not be confined to the

subaperture locations. However, the computational burden would be much higher.

Removing the overall tilt is a tradeoff. The overall tilt contains approximately 87% of the total

power in the turbulence, so the remaining higher order modes are correspondingly weak [6]. The

counterpoint is that removing the overall tilt removes exactly those modes which are corrupted by the

telescope tracking and aircraft motion. Also, the self slope structure function is insensitive to overall

tilt so we cannot learn anything about overall tilt anyway.

Based on performance with simulated slope measurements we can use the structure function

estimator to recover the slope structure function. Removing the average slope from the slope measure-

ments was shown to be an important step. The average slopes must be removed before computing the

slope structure functions. Average removal allows us to remove the effects of deterministic aberrations

in our slope measurements. Now we can analyze the actual ABLE ACE data.

3-24

IV ABLE ACE Data Analysis

This chapter presents the results of the calculations outlined in Chapter I. The results are

analyzed to determine the homogeneity and isotropy of the turbulence, and to compare the results with

the theory developed in Chapter II. The first section describes the parameters of each of the ABLE

ACE data sets analyzed. The following sections present the results and analysis of the statistical and

slope structure function calculations.

4.1 ABLE ACE data sets

The ABLE ACE experiment gathered data over a three month period at various locations within

the continental United States and overseas. The primary purpose was to gather data on turbulence in the

stratosphere; thus, the aircraft altitudes ranged between 35,00-45,000 feet. Many different separations

between the aircraft were used, ranging from twenty kilometers to over one hundred kilometers.

The data sets analyzed were chosen to span the range of aircraft separations, provide a sampling

of the various geographic locations, and to produce fairly long ensembles of frames for the statistical

calculations. Even so, these results barely scratch the surface of the gigabytes of data available.

Table 4.1 provides the aircraft separation, altitude, and number of frames in each data set.

Several thousand WFS frames, representing several hours of data collection, were examined to select

the hundreds of frames used. The data sets in Table 4.1 were analyzed using the procedures outlined in

Chapter II. The results for the Science3 data set are presented in the remainder of this chapter.

4.2 The Science3 data set

The Science3 data set was chosen for a detailed presentation because the characteristics and trends

of this data set are representative. The analysis for the remaining data sets is found in Appendix A.

Data Set Separation I Altitude Framesj

Yokota3-1 22-27 km 44,500 ft 112Science3 58-59km - 40,000 ft 114

Yokota3-2 59-63 km 43,000 ft 150Osan2 103-105 km 41,500 ft 71

Science2 107-110 km 41,500 ft 206

Table 4.1 Parameters of the ABLE ACE data sets analyzed.

4-1

A few results from other data sets are presented when those results are exceptions to the general trends

illustrated by the Science3 data set.

One of the major difficulties encountered in analyzing the results are gaps in the structure function

calculations produced by the central obscuration of the telescope. These gaps arise since both slope

measurements differenced in the structure function calculations are not always within the pupil of the

telescope. Another difficulty is the finite ensemble length. The short ensembles available, coupled

with the fact that all subaperture locations are not always valid, can produce statistically unreliable

points. Structure function values calculated using fewer than 75% of the possible ensemble samples

are removed during the analysis. Once the unreliable structure function estimates are removed we can

examine the global slope statistics.

4.2.1 Global statistics. The mean and variance of the slope measurements were computed

in an effort to discover any deterministic aberrations and the strength of the turbulence across the

telescopes pupil. The x and y slope averages for the Science3 data set are shown in Fig. 4.1.

Atmospheric turbulence should be zero mean. The Science3 slope averages are not zero mean;

which suggests some deterministic aberration is present. The aberrations for the Science3 data set are

not particularly large. A data set with a much larger and more consistent aberration was the Osan2 data

set. The slope averages for the Osan2 data set are shown in Fig. 4.2. The aberration in this data set

was noticed during the WFS image processing and provided the impetus to analyze all the WFS slope

measurements for any deterministic aberrations.

The slope variances for the Science3 data set are shown in Fig. 4.3. The slope variances are

fairly uniform across the telescope's pupil. Homogeneous turbulence would produce slope variances

which are uniform across the telescope's pupil. Thus, a fairly uniform slope variance across the pupil

suggests that the turbulence encountered was indeed homogeneous. This characteristic held for all

the data sets analyzed. Homogeneity of the turbulence can also be tested by examining the estimated

structure functions.

4.2.2 Testing for homogeneity. Once the global statistics were known the self slope structure

functions were computed. Gaps in the structure functions caused by the central obscuration require us

to plot structure functions from several locations to observe the overall behavior. Confidence intervals

4-2

150,

100,

S50,

'o- -50,

-100,

-150

2015 20

10 1501

1000

555

10

-50,

-100

2015 20

10 1510

Y (subapertures) 0 0 5 (subapertures)

(b)

Figure 4.1 Science3: (a) x-slope average, (s'), and (b) y-slope average, (sy).

4-3

150,

100,

50,

Co-50,

-150

2015 2

10 1510

Y (subapertures) 0 0 5 X (uaetrs

(a)

150,

100,

50,

a) -50,

-100,

-150,

2015 2

Y (subapertures) 00X (subapertures)

(b)

Figure 4.2 Osan2: (a) x-slope average, (sx), and (b) y-slope average, (sx).

4-4

5000,

C 4500,E

S4000,

3500,

A3000,x a

S2500,

2000

1500

20

15 20

10 15

Y (subapertures) 5 X (subapertures)0 0

(a)

5000,

4500,CN-E 4000,

S3500,

S3000,

>a2500,

'~2000,

1500,

20

15 2

1010 15

Y (subapertures) 00 5 X (subapertures)

(b)

Figure 4.3 Science3: (a) x-slope variance, ((sX) 2) , and (b) y-slope variance, ((Sy) 2)

4-5

for several structure functions are shown to give a feeling for how the confidence intervals behave.

The many outliers and relatively large confidence intervals arise primarily from the limited ensemble

length.

Figures 4.4 and 4.5 show the self slope structure functions without average removal. Structure

functions from many locations in the pupil have been plotted. The circles show the estimated structure

function values. The dotted lines show the various structure functions. Solid vertical lines show the

95% confidence intervals. The structure functions do cluster together slightly. Nevertheless, there

are still many outliers, and the confidence intervals are very large compared to the estimated structure

function values.

Figures 4.7 and 4.6 show the self slope structure functions with average removal. As shown in

Chapter III removing the average slope can compensate for deterministic aberrations which may be

present. Such aberrations will affect the estimated structure functions. With average slope removal

the structure function clustering is more apparent. The confidence intervals have also been markedly

reduced; although, they are still large.

Overall, the structure functions tend to cluster together. But with confidence intervals nearly

the size of the structure function values. How closely the structure functions cluster together is an

indication of the homogeneity of the turbulence. Combining these observations with the uniformity of

the global slope variances tends to support, but does not confirm, the homogeneity of the atmospheric

turbulence encountered.

4.2.3 Testing for isotropy. We cannot directly compare the self slope structure functions for

various separation directions since the structure functions are not identical. Providing the turbulence is

homogeneous, we can spatially average the slope structure functions. Spatial averaging will effectively

increase the size of our ensemble and provide a better estimate of the slope structure function. We can

then least-squares fit the estimated and theoretical structure functions. If the turbulence is isotropic,

the r, values recovered from the least-square fit scale factor should be nearly the same value.

Figures 4.8 and 4.9 show the spatially averaged self slope structure functions without average

removal. Figures 4.10 and 4.11 show the spatially averaged self slope structure functions with without

average removal.

4-6

16000 1 11 i 16000 il/1 i

140 ~~~I \N 14000ii

It I' q' if '4l~12000 : 1 i ' - /12000 I k I V

10000 S/~ ~ &y .10000 , \

8000 ::1It ~ 8000 ji p. , i

600 i UyV'Ui i 6000 .

4000 ~ it 1 4000 ~~, . ..

:2000 A2000 ril I

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6X separation (in) Y separation (in)

(a) (b)

18000 1 it " I I 18000- iff

qI I A L.l

All ~I It it/ii1

1 0 0 0 - 1 1' \ ki i / : i i 1 0 0 0 - 14 'ii~ h Aii I Ini IA i~

il Ii It I i. li iii ~.ji t if, I j III

12000 ~ i14000 IlINi~il

1fill ill 11 1 Itil ~ ii, ii ' 1

800 ii , i00

600 I ii~/,' -

~ 8000 ~;lII '.~ VI' 000 Aiii/

6000 4!i 000 - '

Ii i i 2000 ~ /

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0. 1 0 .2 0.3 0.4 0.5 0.6Y separation (in) XY separation (in)

(c) (d)

Figure 4.4 Structure function homogeneity: D"X for (a) x, (b) y, (c) +450, and (d) -450 separations.

Average slopes were not removed.

4-7

A. V1 A ' , I 'J18000 18000 II~ ~ I

1606000C01',;1600 1 tl I II i I AlIit f I

16000 "It II I I 1 16000 An Al 1 AL1-~ Ol

12 00 J l-100 If\.. .I I"" hll 1 1 '" I I

140001000 i I

120000 A I1 1 1 I i ll , I ItI I I

I I I , ,. I , .

,I I IlI I fill II

"8000 V 8000

I /000IA l / 6000 l~Aj

40001 / ,/i , . . l 4000 eII .i iI

2000 l 2000 ' r i

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6X separation (m) Y separation (m)

(a) (b)

] , /'] ,! / .. . . .. .

18000 f:IiI 1 lt A II I1 - I• 18000 iii I 1 It

16000 t' l 16000 II'I I A \ I

14000 It, 14000 if Ill ,lit .. . I

14000- , I I II / • -' -A -Il /

l '. 1 l I \ Il I

I1/ I , Il... .. . . .. ,,A I/ ,1

a/" ' "". / 12000 It I v. .... .

10000 i 10-. • 11, .it A /1 ' L If V Al

o 8000 .... 0 8000 1 I- I ,

6000 i' "000, / , .. .i,,,,-l-II,,__ j , .......

4000 4 000 ,: / ! ,"/li I "- -- Ill

0.1 0.2 0.3 0.4 0.5 0'6 0 0.1 0.2 0.3 0.4 0.5 0.6XY separation (m) XY separation (m)

(c) (d)

Figure 4.5 Structure function homogeneity: Dvy for (a) x, (b) y, (c) +450, and (d) -450 separations.Average slopes were not removed.

4-8

I II

128000 800012000

If I

10000 2 10001

62000- 12000-~ -

40000 6 1000

2a 000 I\ 8 000

'A,

-1 ri I

12000 'A 12000w

10 0 0.1 0.2 0. 1000 0. . . . . .XY seaato (m Y seaaio m

(c (d)/

-45 seaatos Avrg slopes removed.

~ / / 1-It I4-9~

12000 9 12000 -

\ r

10000 a 10000 \\ i

r" 8000 E 8000 / ', 7'

w 6000 I / 6000 .... t, ",,,,

4000 4000 al. - \.,

2000 , ' , , 2000

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6X separation (m) Y separation (m)

(a) (b)

12000 12000 -

10000 10000 "

=!= /jo / a

o 8000 E 8000

, -- -., .d ,', % ",- "

4000 i - 4000 "a ' -' '\"

2000 2000 - / '

0--- 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6XY separation (m) XY separation (m)

(c) (d)

Figure 4.7 Structure function homogeneity: DLv for (a) x, (b) y, (c) +45', and (d) -45o separations.

Average slopes removed.

4-10

18000- 18000-

16000- 16000-

14000- 14000-

ci. 12000-100

10000 '' - 10000-

0 8000-0 8000-

o0 Experimental6000- ~~~- Theoretical600 Thoeia

4000 4000RMS error - 4.25e+03RMS error - 4.78e+03 Ro =0.0253

2000 ~ Ro - 0.0275 20

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6X separation (mn) Y separation (in)

(a) (b)

18000- 18000-

16000- 16000-

14000- 14000-

12000-. 12000 -

8000 80

0 Exerimntal0 Experimental

60-- Theoretical 6000- - Theoretical

RIMS error - 1 .84e+03RMVS error - 3.56e+03 0pr.t

40-R-0.254000 Ro - 0.0252

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6XY separation (in) XY separation (in)

(c) (d)

Figure 4.8 Structure function isotropy: spatially averaged DII for (a) x, (b) y, (c) +450, and (d)45' separations. Average slopes not removed.

4-11

18000- 0 Experimental 18000-

16000- ~ - Theoretical 100

RMS error - 3.13e+03

14000- Ro -0.0318 14000-

12000 -120

-10000- 10000

a 8000 4 -- 8000 0 Eprmna

6000- IVIS error - 9.61 e+034000-4000Ro - 0.0266

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6X separation (in) Y separation (in)

(a) (b)18000- 0 Experimental 18000- 0 Experimental

1600- - heoeticl 1600-- Theoretical

16000 16000 - .1 e03MVS error - 3.05e+03

14000- Ro -0.0305 14000- o-007

N- 12000- c- 12000-E E

10000 a

8000-1000

6000 8000

6000 20

0 0.1 0.2 0.3 0.4 0.5 0.6 0) 0.1 0.2 0.3 0.4 0.5 0.6XY separation (in) XY separation (in)

(c) (d)

Figure 4.9 Structure function isotropy: spatially averaged Dvv for (a) x, (b) Y, (c) +450, and (d)-. 450 separations. Average slopes not removed.

4-12

0 Experimental 0 Experimental12000- - Theoretical 12000- - Theoretical

RIMS error-i 1.55e+03 RMVS error - 1 .56e+03

10000- Ro -0.0402 m 10000- Ro -0.0378 m

8000-2 8000-

6000-x 6000-

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6X separation (in) Y separation (in)

(a) (b)

0 Experimental12000- 12000- - Theoretical

RMS error - 1 .45e+03

10000- 10000-R 8- 0.041 m

E 8000- 8000-

6000 6000

oExperimental4000F - Theoretical 40

RIMS error - 2.04e+032000 Ro80- 0.0363 m200

011 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0 .4 0.5 0.6XY separation (in) XY separation (in)

(c) (d)

Figure 4.10 Structure function isotropy: spatially averaged D" for (a) x, (b) y, (c) +45', and (d)-45' separations. Average slopes removed.

4-13

o Experimental12000 - Theoretical 12000•

RMS error - 2.39e+03

10000 Ro - 0.046 m 10000

8000 E 8000/

6000 a 6000

~- Theoretical4000] . -T-... 4000 o Eprmna

2000 000 -RMS error - 3.09e+0320002000Ro - 0.0416 m

0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6X separation (m) Y separation (m)

(a) (b)

0 Experimental 0 Experimental12000 Theoretical 12000 - Theoretical

RMS error - 1.15e+03 RMS error - 959

10000 Ro -0.0416 m 10000 Ro - 0.0442 m

8000- 8000

6000- 6000

4000 4000

2000 2000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6XY separation (m) XY separation (m)

(c) (d)

Figure 4.11 Structure function isotropy: spatially averaged DYY for (a) x, (b) y, (c) +45', and (d)-450 separations. Average slopes removed.

4-14

Data SetsSlopes Direction Yokota3-1 Science3 Yokota3-2 Osan2 Science2

x 0.045 0.040 0.041 0.037 0.037x y 0.044 0.038 0.039 0.038 0.042

+450 0.044 0.036 0.040 0.039 0.040-450 0.045 0.042 0.040 0.040 0.040

x 0.045 0.046 0.034 0.040 0.045y y 0.044 0.042 0.040 0.044 0.039

+450 0.045 0.042 0.040 0.044 0.038

-450 0.047 0.044 0.041 0.042 0.042

Table 4.2 Recovered r, values (in meters) for average slope removed structure functions.

As expected, the confidence intervals for the estimated structure functions are markedly reduced.

This is due to the increase in the effective ensemble size from the spatial averaging. Overall, the structure

functions have the functional forms predicted by theory. The RMS errors for the average slope removed

structure functions are in general lower than the non-average removed structure functions. However,

there are some exceptions. Average removal had much greater success with data sets such as Osan2

which had much larger average slopes than the Science3 data set.

Recovered r, values for all the data sets are shown in Table 4.2. The measurements for the

data sets were collected at various geographical locations and path lengths. The recovered r, values

are remarkably constant over the various path lengths. There seems to be a slight, possibly illusory,

decrease in the average r, values as the path lengths increase. However, such a trend is highly dependent

on the actual strength of the turbulence encountered along a given path. Another unknown is the effect

of boundary layer turbulence on the optical beam. Such an effect should be fairly constant since the

aircraft velocity was kept constant during data collection. Any effects of boundary layer turbulence

should reduce the r, value.

4.3 Summary

Five ABLE ACE H-WFS data sets were analyzed using the slope structure function estimator

and compared to theory. Tests for homogeneity and isotropy of the turbulence encountered suggest,

but do not prove, that the turbulence was homogeneous and isotropic. Knowing if the turbulence

has these properties is critical since these are fundamental assumptions made by theoretical models of

4-15

atmospheric turbulence. The behavior of the estimated slope structure functions follows that of the

theoretical slope structure functions derived using the Kolmogorov model for atmospheric turbulence.

4-16

V Conclusions and Recommendations

In this chapter the major results and conclusions from this thesis are highlighted. Recommenda-

tions for extending the concepts and methods used in this thesis in future work are also presented.

5.1 Results and conclusions

The theoretical results presented in Chapter II provide expressions for the tilt-removed slope

structure functions for the in terms of the phase structure function. This derivation assumes only

homogeneity, isotropy, and zero mean characteristics of the wavefront phase. Thus, the large body

of theoretical work for the phase structure function is available for use. There are many different

formulations for the phase structure function which can be used ranging from the simple Kolmogorv

model to the von Karman model to the general power law model developed by Stribling [21].

The self slope structure function has been evaluated for the Kolmogorov turbulence model using

numerical quadrature. The symmetry inherent in the H-WFS geometry allowed the integrations to be

reduced to single dimensional integrals from the original four-fold integrations and normalized. This

normalization dramatically reduced the computations necessary to evaluate the integrals.

In Chapter I a method for estimating the slope structure function, without assuming homo-

geneity and isotropy, was presented. This estimator allows us to check any assumptions about the ho-

mogeneity and isotropy of our data. Using simulated slope measurements known to have Kolmogorov

turbulence statistics shows that the estimator can recover the slope structure function. Deterministic

aberrations were shown to have a large effect on the estimator's performance. Average slope removal

was shown to be effective in removing aberrations similar to those found in several of the ABLE ACE

data sets. The r, values were also recovered to within two significant figures.

Many thousands of ABEL ACE WFS image frames were processed to find the several hundred

frames which make up the ensembles analyzed in Chapter IV. Due to the limited ensemble length the

results shown in the tests for homogeneity are not conclusive. Combining the structure function results

and the slope variance calculations suggests that the turbulence encountered probably is homogeneous.

The spatially averaged structure functions tended to follow the behavior expected based on the

theoretical slope structure functions. Of course, spatial averaging is only valid if the turbulence truly

is homogeneous. The significance of the results from the ABLE ACE data sets is difficult to determine

5-1

since the effects of the boundary layer turbulence on the slope measurements cannot be separated from

the atmospheric turbulence effects. Recovered r, values for all path lengths were approximately 0.04

meters.

One of the advantages of using the H-WFS slope measurements directly are that the computation-

ally intense step of phase reconstruction is bypassed. Also, any noise enhancement and assumptions

about the turbulence characteristics are also avoided. One of the limitations of this method is that it can

only sense the perturbations of orders higher than the overall tilt. Unfortunately, the overall tilt of the

wave front phase contains 90% of the energy in the perturbations [7]. This lack of sensitivity can be an

advantage if the overall tilt cannot be trusted. This technique should be useful for real-time turbulence

sensing.

5.2 Suggestions for further work

Some extensions and further work which would be useful are:

" Evaluate the theoretical slope structure functions for the von Karman and generalized power

laws. These evaluations would provide information on how the slope structure function behavior

changes with different turbulence statistics.

" Evaluate the behavior of the structure function estimator with noisy slope measurements. Know-

ing how sensitive this estimator is to noise would be useful in assessing the reliability of results.

" Apply these methods to other data sets or more of the ABLE ACE data. The ABEL ACE data

was collected under challenging conditions and contained several unknown sources of noise or

distortion. The measurements recently collected by researchers at AFIT over a long, horizontal

path between mountain tops in Hawaii would be another good data set.

" Determining the effects of boundary layer turbulence on the optical beam would be beneficial.

Such an analysis would help determine how much of the wave front phase distortions are due to

boundary layer versus atmospheric turbulence.

Characterizing atmospheric turbulence using H-WFS slope measurements directly is feasible.

The calculations involved are much less computationally intense than the traditional phase reconstruc-

tion methods. The self slope structure function is insensitive to overall tilt and cannot be sensed by this

5-2

method. Recovering r, values in real-time is a possibility. Such r, values could be used to adaptively

optimize the performance of optical systems. Overall, another tool is available in our tool box for

characterizing atmospheric turbulence.

5-3

Appendix A. ABLE ACE data analysis

The Science3 data set was displayed in Chapter IV in full. This appendix displays the analysis

for the remaining ABLE ACE data sets. The global average and variances for the x and y slopes

are displayed first. Next plots showing the homogeneity of the average removed x and y slopes are

displayed. Last, plots showing the isotropy and fit to theory are displayed.

A-1

A.] Osan2 analysis

A-2

150,

100,

S 50,

0lx c

-500

-1505

Y (subapertures) 00 5 X (subapertures)

(b)

-5A03

60020

50005

2000

22015 25

Y (subapertures) 0 0 X (subapertures)

(b)

Fiue . Oa2 ()x-lp vrane (s)),ad7b000oe ainc,((, 2

60A04

- -r - --

. I i I-

'

12000 I / 12000-

10000-" 10000\ Ii

8000 8 /" I000 :

, 6000 , 6000

4000 40 0'.../ -'

2000 1/ ,0 ,

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6X separation (in) Y separation (in)

(a) (b)

12000 , 12000

10000" " "' 10000 '

+ ..... .... °. .... r . .Il ' l '

4oo " , - -, v"-1\ / / " /

E 8000 -', -8000 /1' /

,,! . I, \,j /"- --- "

~, 6000 6000--7'-' eo '',

4000 ' ''- / ~ ~4000 ? - /1 '

, . ,, /\

2000 2000

!'/ : -. ,e :1: ll .. .I ,,

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6XY separation (m) XY separation (m)

(c) (d)

Figure A.3 Structure function homogeneity: D"-slopes shifted in for (a) x, (b) y, (c) +45', and (d)

-45' separations. Average slopes removed.

A-5

12000' 12000// , -I

I a j

10000 , 10000 /

,' a, I" ' ' I-i

/8000 28000a / a

6000 / 6000

4000 4 , 4000 / / "

, II

I,, 8 o? i-/,. ' a.., 6 0

2000 ' 2000 /,' ' . . i C ,,

0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6

X separation (m) Y separation (m)

(a) (b)

12000 12000/

10000 - 10000- Ia I . I

/ . / I

8000 E 8000 : " " " Ia, , I ,

-a-I /?-

0.4 0 a a 0 0" 0i a , 0

801 . EW 6000 / '' /, a I,..

Fu 600 S , (b , - c) ,5,, (d - ,

2000 t . \ , , / 2 000 / " ' ' < , '-- ... a<,'

a .,

) (d)i \ . \

Average slopes removed.

A-6

1200 Epermeta 12000- - Theoretical-Theoretical 0 Exeimna

RMVS error -7.31 e+03 RMVS error-i 1.68e+0310000- Ro -0.037 10000- Ro -0.0378

C'J C'JE 8000- E 8000-

~'6000-~ 6000-

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0. 06X separation (in) Y separaion (in)

(a) (b)

0 Experimental 0 Experimental12000- - Theoretical 12000- - Theoretical

RMS error - 936 RMVS error - 1 .26e+0310000- Ro -0.0392 10000- Ro - 0.0396

C2L 8000 E? 8000-

S6000 60

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6XV separation (in) XV separation (mn)

(c) (d)

Figure A.5 Structure function isotropy: spatially averaged D " for (a) x, (b) y, (c) +45', and (d)-450 separations. Average slopes removed.

A-7

0 Experimental 0 Experimental12000- - Theoretica 12000- - Theoretical

FIMIS error-i 1.63e+03 RMVS error-i 1.09e+0310000- Ro - 0.0404 10000- Ro - 0.0435

8000 E 8000

~6000 ~W6000

4000 40

0 0.1 0.2 0. 3 0.4 0.5 0.6 0) 0.1 0.2 0.3 0.4 0.5 0.6X separation (in) Y separation (in)

(a) (b)

1200o Experimental 1200o Experimental12000 Theoretical 100- Theoretical

AMVS error - 914 RMVS error - 74110000- Ro - 0.0439 10000- Ro -0.0424

E 8000- E 8000-

60000 6000-

4000 400

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6XY separation (in) XY separation (in)

(c) (d)

Figure A.6 Structure functionn isotropy: spatially averaged D.YY for (a) x, (b) y, (c) +450, and (d)-450 separations. Average slopes removed.

A-8

A.2 Science2 analysis

A-9

150,

x c

220

15

Y (subapertures) 5 0 0 5 X (subaportures)

150,

100,

S 50,

ctn -50,

-100,

-150,

20

15 20

10 1510

Y (subapertures) 5 X (subapertures)

(b)

Figure A.7 Science2: (a) x-slope average, (sx), and (b) y-slope average, (sv).

A-10

xC

22015 15

Y (subapertures) 0 0 5 X (subapertures)

(a)7000,

50020

20

15

Y (subapertures) 0 0 5 X (subapertures)

(b)

Figure A.8 Science2: (a) x-slope variance, ((S,) 2), and (b) y-slope variance, ((Sy) 2)

A-11

12000-J, 12000-

10000 : - 10000

~6000 -7 ~x~6000 /I

4004000 a-

2002000 /

0 0.1 0.2 0.3 0.4 0.5 0.6 0F 0.1 0.2 0.3 0.4 0.5 0.6X separation (in) Y separation (in)

(a) (b)

12000- 12000 I

10000 I10000-

C\ N

., 6000 / >'/ '6000 /-

400 1 - \

2000 ~ 5 12000

0 .1 0. 03 .4 0.5 0.6 0 0.1 0.2 0.3 04 0.5 0.6

Figure A.9 Structure function homogeneity: D" -slopes shifted in for (a) x, (b) y, (c) +450, and (d)-. 45'0eaains vrg slopes removed.

A- 12

, I

10000 10000 ,'; "

8000 I 80001, I 1000 -

60 "

10000 4" /000'

2 1000 2000

I I

e~i " , II ,"/ I"

12000 ' 2000 /"--"000 "' I /

C- I NI

I \,

-.6000 .. 000 / :I /

4000 • 1 4000 - / /

- , .' /,-/• 'I

1000 2000 ' " '/

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6

X separation (i) /Y separation (i)

(a) (b)

Fi - - 1 _ , . , and d -

12000.. ., .| '' , , 12000 , .., -, ',.,,>. ,; - .T / , 1 , , .....

10000/ . //_.., -'

ti. Aveag slpe removed...

, -. . ' " - 1. A / . I,i

N I .2' 000 >' :, •I : ,', 2000 # ' ' / . ..

, 3': , II" -4000 3 . , 'I40 i , - .. .

2000 --1%' 2000 '-° ' ; ' '

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6

XY separation (in) XY separation (in)

(c) (d)

Figure A.10 Structure function homogeneity: D ' for (a) x, (b) y, (c) +450, and (d) -450 separa-tions. Average slopes removed.

A-13

1200- 1200-0 Experimental1200012000- Theoretical

RMS error -1 .46e+0310000- 10000- Ro -0.0424

E 8000- E 8000-

6' 000- 6'000-

0 Experimental-Theoretical

4004000 z.RMS error - 3.83e+03 U

2000- ~ Ro - 0.0366 0

01 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6X separation (mn) Y separation (in)

(a) (b)

1200 Experimentali200 0 Experimental

1200- Theoretical 100- Theoretical

RMS error - 1.21 e+03 RMS error- 74410000- Ro - 0.0395 10000- Ro - 0.04

('4j

E 8000 2 8000-

~'6000 - 6000

4000 4000

0 0.1 0.2 0.3 0.'4 0.5 0.86 0.1 0.2 0.3 0.4 0.5 0. 6XV separation (in) XV separation (in)

(c) (d)

Figure A.1 1 Structure function isotropy: spatially averaged D " for (a) x, (b) y, (c) +450, and (d)-450 separations. Average slopes removed.

A-14

0 Experimental120

12000- - Theoretical 100

RMS error - 1 .24e+0310000- Ro -0.0453 10000-

8000- E 8000-

6000- 6000

4 000 -40

4000 - 2... 000 - Theoretical

0 0.1 0.2 0.3 0.4 0.5 0.6 _0 0.1 0.2 0.3 0.4 0.5 0.6X separation (in) Y aeparation (in)

(a) (b)

1200- 1200-a Experimental1200012000- Theoretical

10000- MS error - 736100010000- Ro -0U416

8000E 00

: 6000 6000-

4000 0 ~ Experimental 4000-

200RMS error - 3,77e+03 2000-

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6XY aeparation (mn) XY separation (mn)

(c) (d)

Figure A. 12 Structure functionn isotropy: spatially averaged D*,Y for (a) x, (b) y, (c) +450, and (d)-45' separations. Average slopes removed.

A-15

A.3 Yokota3-1 data analysis

A-16

10205015

x, a

202015 25

Y (subapertures) 5 0 0 5 X (uaetrs

(b)

-10017

7000,

2020

15 25

Y (subapertures) 5 0 0 5 X (subapertures)

(a)7000,

S6000,

C 5000,

>. C

22015 15

Y (subapertures) 00 5 X (subapertures)

(b)

Figure A. 14 Yokota3: (a) x-slope variance, ((s,)2), and (b) y-slope variance, (S) 2 )

A-18

12000 I'12000-

10000 I10000-

1 41

6 000 60

12000 12t 000

10000 10000

~ 0 0 0. 800.3 04 . 00 0. . 03 '4 . .

1200 1200 I- lk

4000 6~ 000 /, ''

2000 I~i' -2000 ,~

0 0.1 0.2 0.3 0 .4 0.5 0.6 0 .1 0.2 0.3 0.4 0.5 0.6XY separation (in) XY separation (in)

(c) (d)

Figure A. 15 Structure function homogeneity: D"-slopes shifted in for (a) x, (b) y, (c) +45', and(d) -45C' separations. Average slopes removed.

A- 19

12000' 12000

, I

10000 10000 /

I ~ii/N - ! i

t I .

E 8000 E I 8000 //" 6oo7 1, , // / oo 7 - " -

I" 6000 6000-A ".= ' ' " I ,,

I 4- / III ' I\ .

4000' - / -- 4000 /r

t- -I L I"I" .- - -

2000 41/ 2000i,, . . . ,"'. , .

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6X separation (m) Y separation (m)

(a) (b)

12000' 12000-

" - I j II- -

10000II I 10000

6 /

Noo I ~ I Noo

RZO 6ooo0 - / " '" I 6000 , I,a IZ 'I a I /- ( I /

I I - I i

( I " I T I

4w6000 ' r ' / " &J 6000 , - 'I,

2000 2000 - - - , \

0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6XY separation (m) XY separation (m)

(c) (d)

Figure A. 16 Structure function homogeneity: DYY for (a) x, (b) y, (c) +45), and (d) -45" separa-tions. Average slopes removed.

A-20

o Experimental 0 Experimental12000 - Theoretical 12000- - Theoretical

RIMS error - 2.01le+03 RMVS error-i 1.54e+0310000- Ro - 0.045 10000. Ro - 0.0436

r'j 'S8000- E 8000-

'X 6000- a 6000

4004000 ~ - j

0 0 .1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6

X separation (in) Y separation (in)

(a) (b)

0 Experimental o Exerietal

12000- - Theoretical 12000 0 Therietal

RMVS error -906 RIMS error - 1 .23e+0310000- Ro 0.043; 10000- Ro -0.0445

C'aS8000- 8000-

~a6000-~ 6000

!4000 40

0 0.1 0.2 0.3 0.4 0.5 0.6 _0 0.1 0.2 0.3 0.4 0.5 0.6XY separation (in) XY separation (in)

(c) (d)

Figure A. 17 Structure function isotropy: spatially averaged Dx' for (a) x, (b) y, (c) +45', and (d)-45' separations. Average slopes removed.

A-21

o200 Experimental 12000- 0 Experimental12000- Theoretical 1 - Theoretical

100-RMS error - 2.12e+03 1000 MS error - 4.89e+0310000 Ro - 0.045 100 Ro - 0.0443

E. 800 8000-

8 000- 00

4000 4000

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6X separation (ml Y separation (in)

(a) (b)

120-0 Experimental o200 Experimental1200- Theoretical 120 - Theoretical

RMS error - 1.53e+03 RMS error-I 1.37e+0310000- Ro - 0.0454 10000- Ro - 0.0472

CY

E 8000 E 8000-

6000 ..j60004000 4000 - --

-0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6XY separation (in) XY separation (in)

(c) (d)

Figure A. 18 Structure functionn isotropy: spatially averaged DY'Y for (a) x, (b) y, (c) +450, and (d)-450 separations. Average slopes removed.

A-22

A.4 Yokota3-2 data analysis

A-23

150,

100,

50,

-0,

22015 "__15

Y (subapertures) 0 0 X (subapertures)

(b)

Fiue .9 ooa32 ()x-lp aeag,(s) nd()150oeaerg, s)

1A-24

7000,

x C

2000

20015 25

Y (subapertures) 5 0 0 5 X (subapertures)

7000,

50020

>115

202

12000-100

10000- 'U00

S8000 8000'f / y' / ~

m 6000 / .- V' ' 6000 ' ~

4004000 I

, -8 - s-- -___

0F 0.1 0.2 0.3 0.4 0.5 060 .1 0.2 0.3 0.47 0.5 0.6X separation (in) Y separation (mn)

(a) (b)

12000- 12000-

10000 1 / 10000 1

8000 '8// 000 V ~Xl, 1~ 75~~,

w 6000 IT5N. 6000 /7.

4000 /~ -V4000 fl

200 2000~~

0 0.1 -0.2 0.3 0.4 0.5 0.6 060 0.1 0.2 0.3 0.4 0.5 0.6XY separation (in) XY separation (in)

(c) (d)

Figure A.21 Structure function homogeneity: D"-slopes shifted in for (a) x, (b) y, (c) +450, and(d) -. 45' separations. Average slopes removed.

A-26

12000- 1 1200

10000- 10000-

8000E 80 .

6000- 6000 I -s.

4000.2000

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6

X separation (in) Y separation (in)

(a) (b)

12000- 12000-

10000 I 10000-

E 8000 /E 8000 .. i

40 6000 ' -' 6000 'Y '4A

lit

It,

2000 *'~i I'2000 /

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6XY separation (in) XY separation (in)

(c) (d)

Figure A.22 Structure function homogeneity: Dvv for (a) x, (b) y, (c) +45', and (d) -450 separa-tions. Average slopes removed.

A-27

0 Experimental 0 Experimental12000- - Theoretical 12000- - Theoretical

RMVS error-i 1.48e+03 RMVS error-i 1.88e+0310000- Ro - 0.0405 10000- Ro - 0.0387

E 8000 8 8000-

6' 000-~ 6000

0C -0

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6X separation (m) Y separation (in)

(a) (b)

12000- 0 Experimental 12000- 0 Experimental- Theoretical -1- Theoretical

100-RMVS error - 1.4e+03 100-RMVS error-i 1.54e+0310000Ro -0.0395 100Ro -0.0404

E8000 8000

S6000 60

00

0 0.1 0.2 0.'3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6XY separation (in) XY separation (m)

(c) (d)

Figure A.23 Structure function isotropy: spatially averaged D"7 for (a) x, (b) y, (c) +450, and (d)-450 separations. Average slopes removed.

A-28

1:00- 0 Experimental 12000- Experimental100 - Theoretical - Theoretical

100-RMS error - 1 .05e+04 1000 MS error - 4.03e+031000 P-0.0337 1000040

8000-~ 6000o

6000 ,00

4000- 00

0 0.1 0.2 0.3 0.4 0.5 0.86 0.1 0.2 0.3 0.4 0.5 0.6X separation (ml Y separation (m)

(a) (b)

120-0 Experimental 120-0 Experimental12000 - Theoretical 100- Theoretical

PMS error-i 1.84e+03 PMS error - 1.09e+0310000- R0 - 0.0402 10000- P0 - 0.0408

6 000 8ZO600

2 000 20

0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6XY separation (in) XY separation (in)

(C) (d)

Figure A.24 Structure functionn isotropy: spatially averaged D~v for (a) x, (b) y, (c) +450, and (d)-.. 45' separations. Average slopes removed.

A-29

Appendix B. Computer code user guide

This appendix is intended to be a short high level overview of the computer code developed for

analyzing the H-WFS slope measurements. A brief description of each program or module is provided.

As always, the actual code takes precedence over any comment or description.

B.1 IDL code

The H-WFS images are processed in the Interactive Data Language (IDL) environment. The

main programs used are described, along with the inputs and outputs.

B.1.1 ACEMAKE.PRO. Compiles the various IDL programs required to process the H-WFS

images. Executed by typing @acemake at the IDL command prompt.

B.1.2 SLOPES.PRO. The main command interpreter for processing the H-WFS images. Ex-

ecuted by typing slopes, s=source, d=destination, th=threshold. The parameter source

is a string containing the full path to the directory containing the H-WFS images. This directory must

contain only ABLE ACE image files or the program will abort. The parameter destination is a

string containing the full path to the directory where the H-WFS slopes will be stored. The parameter

threshold is a number setting the threshold at which to display the H-WFS images.

Comments in the slopes. pro file contain descriptions of the various commands and their

actions for processing the H-WFS images. One file is created for each H-WFS image; its file name

will the be same as the image file. Each H-WFS slope file contains four 21x21 matrices in unformatted

FORTRAN-77 format. The x slope, y slope, peak intensity, and mask matrices are saved in that order.

The x slope, y slope, and intensity matrices are real and the mask matrix is integer.

B.2 MATLAB code

All statistical and structure function calculations were done using the MATLAB environment.

Essentially, the MATLAB code is a family of functions and scripts which are called from the MATLAB

command line with the proper parameters. A brief description of the inputs, outputs, and action of

each function are provided. Detailed comments giving the exact input and output parameters and any

restrictions are found in comments in the MATLAB M-files. These comments can be viewed by typing

B-1

help function at the MATLAB command line, where function is the name of the function in

question.

B.2.1 DSGEN.M. Computes the normalized theoretical D" structure function for x, y and

-45' separations. The structure functions are returned in vectors. Uses the dsxx. m function.

B.2.2 DSXX.M. Computes the a value of the theoretical D x structure function. The input

parameters are the x and y subaperture separations, ro, and the subaperture size. Returns a single

value. Uses the ds arg, m function.

B.2.3 DSARG.M. Evaluates the normalized integrals for the D"X structure function.

B.2.4 ACEFILES.M. Reads the H-WFS slope filenames from a directory. The full pathname

must be provided. Only H-WFS slope files must be in the given directory or the function will abort.

The file names are returned in the rows of a matrix.

B.2.5 ACEREAD.M. Reads an H-WFS slope file and returns the x slope, y slope, amplitude,

and mask matrices. The full path must be specified.

B.2.6 ACEVIEW.M. Displays a matrix using the 'hot' colormap. Useful for debugging.

B.2.7 DSX.M. Computes structure functions for the x separations. Inputs are a mask matrix

and two slope matrices. Outputs a structure function and mask matrix.

B.2.8 DSY.M. Computes structure functions for the y separations. Inputs are a mask matrix

and two slope matrices. Outputs a structure function and mask matrix.

B.2.9 DSN45.M. Computes structure functions for the -45' separations. Inputs are a mask

matrix and two slope matrices. Outputs a structure function and mask matrix.

B.2.10 DSP45.M. Computes structure functions for the +45' separations. Inputs are a

mask matrix and two slope matrices. Outputs a structure function and mask matrix.

B-2

B.2.11 DSTRUCT.M. A script which computes the structure functions for all slope and

separation combinations. Accumulates running sums of the structure function, squared structure

function, and mask matrices. These sums are used to compute the structure function, structure function

variance, and sum matrices. The results are saved in .mat files. Inputs are a string containing the path

to the directory containing the H-WFS slope files.

B.2.12 DSTAR.M. Removes the average slopes and computes the structure functions for

all slope and separation combinations. Accumulates running sums of the structure function, squared

structure function, and mask matrices. These sums are used to compute the structure function, structure

function variance, and sum matrices. The results are saved in .mat files. Inputs are a string containing

the path to the directory containing the H-WFS slope files.

B.2.13 WFSSTAT.M. Computes the slope average, variance, and sum at each point in the

aperture. Writes the results to . mat files. Inputs are a string containing the path to the directory

containing the H-WFS slope files.

B.2.14 DVIEW.M. Displays the submatrices of a structure function matrix in increasing

subaperture separation. Inputs are a structure function matrix. Useful for debugging.

B.2.15 DPICK.M. Picks the structure function values corresponding to a particular anchor

point and returns those values in a column. Inputs are are a set of vectors listing the x-y coordinates of

the structure function and a structure function matrix.

B.2.16 SPCAVG.M. Spatially averages the structure function and computes confidence

intervals. Inputs are a set of vectors listing the x-y locations to be averaged; and the structure function,

variance, and sum matrices. Outputs are two vectors; one containing the spatially averaged structure

function, the other containing the corresponding confidence intervals.

B.2.17 MSFIT.M. Computes the mean-square fit scale parameter and the RMS error. The

scale parameter scales the normalized structure function to provide the best mean-square fit between

the estimated structure function and theory. The RMS error is the root-mean-square error between

B-3

the estimated and the scaled theoretical structure functions. Inputs are the estimated and theoretical

structure functions, in vectors. Outputs are the scale factor and the RMS error.

B-4

Bibliography

1. Brookner, E. "Atmosphere propagation and communication channel model for laser wave-lengths," IEEE Transaction on Communications Technology, COM-18(4):396-416 (August1970).

2. Butts, R. R. "Personnal communication." USAF Phillips Laboratory, Kirtland AFB, NM, July1995.

3. Butts, R. R. and L. D. Weaver. "ABLEX high-altitude laser propagation experiment." Laser BeamPropagation and Control, Proc. SPIE 2120, edited by H Weichel and L. F DeSandre. 30-42.1994.

4. Cloutier, M. A. Using Error Detection and Correction Coding for a Turbulent Atmospheric Opti-cal Comminacations Link. MS thesis, AFIT/GEO/ENG/90D-3, Graduate School of Engineering,Air Force Institute of Technology (AETC), Wright-Patterson AFB OH, 1990.

5. Feldmann, R. J. Airborne Laser Communications Scintillation Measurements. MS thesis,AFJT/GEO/ENG/87J-2, Graduate School of Engineering, Air Force Institute of Technology(AETC), Wright-Patterson AFB OH, 1987.

6. Fried, D. L. "Aperture averaging of scintillation," Journal of the Optical Society of America,169-174 (February 1967).

7. Fried, D. L. "Optical heterodyne detection of an atmospherically distorted signal wavefront,"IEEE Proceedings, 57-67 (January 1967).

8. Goodman, J. W. Introduction to Fourier Optics. New York: McGraw-Hill, 1968.

9. Heidbreder, G. R. "Image degredation with random wavefront tilt compensation," IEEE Trans.on Antennas and Propagation, 90-98 (January 1967).

10. Hoversten, E. V., et al. "Communication theory for the turbulent atmosphere," IEEE Proceedings,58(10):180-204 (1970).

11. Kolmogorov, A. Turbulence, Classical Papers on Statistical Theory edited by S.K. Friedlanderand L. Topper, New York: Wiley-Interscience, 1961.

12. Magee, E. P. and B. M. Welsh. "Characterization of laboratory generated turbulence by opticalphase measurements," Optical Engineering, 3810-3817 (November 1994).

13. Markson, R. M., et al. Atmospheric Turbulence Measurements in Support of Adaptive OpticsTechnology. Technical Report RADC-TR-89-289, Rome Air Development Center, 1989.

14. Noll, R. J. "Zernike polynomials and atmospheric turbulence," JOSA, 66:207-211 (1976).

15. Papoulis, A. Probability, Random Variables, and Stochastic Processes (3rd Edition). New York:McGraw-Hill, Inc., 1991.

16. Roggemann, M. C. "Optical performance of fully and partially compensated adaptive optics sys-tems using least-squares and minimum variance phase reconstructors," Computers and ElectricalEngineering, 451-466 (June 1992).

17. Roggemann, M. C. and D. W. Tyler. "Unconventional astronomical imaging," Optics and Pho-tonics News, 17-21 (March 1992).

BIB-1

18. Seeger, R. N. Characterization of an Air-to-Air Optical Heterodyne Communication System. MSthesis, AF1T/GE/ENG/90D-55, Graduate School of Engineering, Air Force Institute of Technol-ogy (AETC), Wright-Patterson AFB OH, 1990.

19. Spreen, D. E. and C. B. Hogge. "Characterizing high-altitude horizontal path optical propagation."Laser Beam Propagation and Control, Proc SPIE 2120, edited by H Weichel and L. F. DeSandre.2-9. 1994.

20. Strang, G. Linear Algebra and its Applications (3rd Edition). Fort Worth, TX: Harcourt BraceJovanovich, Inc., 1988.

21. Stribling, B. E. Laser Beam Propagation in Non-Kolmogorov Atmospheric Turbulence. MS the-sis, AFIT/GEO/ENG/94D-04, Graduate School of Engineering, Air Force Institute of Technology(AETC), Wright-Patterson AFB OH, 1994.

22. Tatarski, V. I. Wave Propagation in a Turbulent Medium. New York: McGraw-Hill BookCompany, 1961.

23. von Karman, T. "Progress in the statistical theory of turbulence." Turbulence, Classic Paperson Statistical Theory edited by S.K. Friedlander and L. Topper, 162-174, New York: Wiley-Interscience, 1961.

24. Wallner, Edward P. "Optimal wave front correction using slope measurements," Journal of theOptical Society of America, 1771-1776 (December 1983).

25. Washburn, D. ABLE ACE: ABL Extended Atmospheric Characterization Experiment. TechnicalReport, Kirtland AFB, NM 87117-5776: USAF Phillips Laboratory, 1994.

BIB-2

Vita

Eric Eugene Silbaugh le graduated from

the Alliance Christian Academy, Elk River, MN high school in June 1985. Eric graduated from

the University of Minnesota, Minneapolis, MN in December 1989 with a Bachelos of Electrical

Engineering. Eric was commissioned a Second Licutenant in the USAF through the. ROTC prorm

upon graduation. Second Lieutenant Silbaugh entered active duty in October, 1990.

After attending Basic Communications Officer Course at Kessler AFB. MS Lieutenant Silbaugh

was assigned to the 485th Engineering Installation Group, Griffiss AFB, NY. While at the 485th EIG

Lieutenant Silbaugh worked as a wideband systems engineer mid also as the Syste"s Te!cco_umica-

dons Engineering Manager for Columbus AFB, MS. Captain Silbaugh was then selected to attend the

Air Force Institute of Technology (.Affl) in May, 1994.

Captain Silbaugh graduated from AFIT in December. 1995 with a Masters of Science in Elec-

trical Engineering degree, specializing in communications ard rd systems and opk"cl ifefin

processing. Captain Silbaugh has been assigned to the Air Force C4 Agency at Scott AFB. 1L

I

VTA-l

REPORT DOCUMENTATION PAGE Form Noroved0MB No 07')4-C 88

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ret Se' _orat Ic faws- r-a1 3te 224 aCn ,A :22-.32 aa : ,r e! . rape ro Reductor, Proect 07C4-1 S). A r '

1. AGENCY USE ONLY (Leave olank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVEREDDecember 1995 Master's Thesis

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

CHARACTERIZATION OF ATMOSPHERIC TURBULENCE OVERLONG HORIZONTAL PATHS USING OPTICAL SLOPEMEASUREMENTS

6. AUTHOR(S)

Eric Eugene Silbaugh, Capt, USAF

j7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION! REPORT NUMBER

Air Force Institute of Technology, WPAFB OH 45433-7765AFIT/GE/ENG/95D-25

9 SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSORING/MONITORINGAGENCY REPORT NUMBER

Dr. Russell ButtsPL/LIAE3550 Aberdeen SWKirtland AFB, NM 87117-5776

11. SUPPLEMENTARY NOTES

12a. DISTRIBUTION/ AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Approved for Public Release; Distribution Unlimited

13. ABSTRACT (Maximum 200 words)

Atmospheric turbulence has long been recognized as one of the fundamental factors affecting optical systemsoperating through the atmosphere. Turbulence over vertical paths has been well characterized, both theoreticallyand experimentally. Much less is known about turbulence over long, horizontal paths. Perturbations of the wavefront phase can be measured using a Hartmann wave front sensor (H-WFS). Theoretical expressions for thetilt-removed structure function of the H-WFS slope measurements were derived and evaluated using quadrature.These slope structure functions are functions of the phase structure function. The Kolmogorov turbulence modelwas assumed. Simulated H-WFS slope measurements were generated using an adaptive optics simulation code.The effects of deterministic aberrations on the structure function estimator were illustrated using the simulatedslope measurements. Average slope removal was shown to be effective in removing the effects of aberrations.Five H-WFS measurement data sets from the ABLE ACE database were analyzed for homogeneity, isotropy,

and fit to theory. The ABLE ACE experiment recorded H-WFS measurements over horizontal paths 20-200 kmin length. Results indicate that the turbulence encountered appears to be homogeneous and isotropic and seemsto fit the Kolmogorov turbulence model.

14. SUBJECT TERMS 15. NUMBER OF PAGES

atmospheric turbulence, wave front sensing, Hartmann-type wave front sensor, 110structure function, optical communication 16. PRICE CODE

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACTOF REPORT OF THIS PAGE OF ABSTRACTUNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL

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