Stanislav Gordeyev is an Associate Professor at the Department of Aerospace and Mechanical Engineering at the University of Notre Dame. His expertise includes the investigation of aero-optical distortions caused by compressible turbulent flows around airborne systems.

Eric Jumper is the Roth-Gibson Professor of Aerospace and Mechanical Engineering at the University of Notre Dame. Although he has performed research in a wide range of topics, his present principal research focus is on the understanding of aero-optical phenomena.

Matthew Whitely, PhD, is Vice President and Senior Scientist at MZA Associates Corporation in Dayton, Ohio. His research includes aero-optical beam control and sensing of atmospheric turbulence for laser propagation.

Acknowledgements ix

1 Introduction 1

1.1 Motivation for Revisiting Aero-Optics 2

2 Fundamentals 7

2.1 Wavefronts and Index of Refraction 7

2.2 Huygens’ Principle 8

2.3 Basic Equations and Optical Path Difference 11

2.4 Linking Equation 15

2.5 Image at a Focal Plane (Far-field Propagation) 17

2.6 Far-field Intensity in the Presence of Near-field Distortions 20

2.6.1 Temporal Intensity Variation 25

2.7 Wavefront Components 26

3 Measuring Wavefronts 31

3.1 Interferometry Methods 31

3.2 Wavefront Curvature Methods 33

3.3 Gradient-based Wavefront Sensors 35

3.3.1 Shack-Hartmann Wavefront Sensor 37

3.3.1.1 Wavefront Reconstruction Algorithm 41

3.3.2 Malley Probe 43

3.3.3 SABT Sensor 46

3.4 Typical Optical Set-Ups 47

4 Data Reduction and Interpretation 55

4.1 Statistical Analysis 56

4.1.1 Temporal and Spatial OPD rms 56

4.1.2 Histograms and Higher-Moment Statistics 58

4.2 Spectral Analysis 60

4.2.1 Relation between the Deflection Angle Spectrum and the Wavefront Statistics 62

4.2.2 Dispersion Analysis 63

4.3 Modal Analysis 65

4.3.1 Zernike Functions 66

4.3.2 Proper Orthogonal Decomposition (POD) 70

4.3.2.1 Direct Method 73

4.3.2.2 Snapshot Method 74

4.3.3 Dynamic Mode Decomposition (DMD) 82

4.4 Cross-correlation-based Techniques 85

4.4.1 Local Convective Speeds 85

4.4.2 Multi-point Malley Probe Analysis 89

4.4.3 Spatially Varying 2-D Convective Velocity 92

5 Aperture Effects 97

6 Typical Aero-Optical Flows 105

6.1 Scaling Arguments 105

6.2 Free Shear Layers 106

6.2.1 Shear-Layer Physics 106

6.2.2 Aero-Optical Effects 109

6.2.3 Historical Shear Layer Measurements in AEDC 110

6.2.4 Weakly Compressible Model 114

6.3 Boundary Layers 118

6.3.1 Model of Aero-optical Distortions for Boundary Layers with Adiabatic Walls 122

6.3.2 Angular Dependence 130

6.3.3 Finite Aperture Effects 132

6.3.4 Nonadiabatic Wall Boundary Layers 133

6.3.5 Instantaneous Far-Field Intensity Drop-Outs 142

6.3.5.1 Absolute SR Threshold 147

6.3.5.2 Relative Intensity Variation 150

6.4 Turrets 152

6.4.1 AAOL 154

6.4.2 Flow Topology and Dynamics 159

6.4.3 Steady-lensing Effects at Forward-looking Angles 167

6.4.4 Aero-optical Environment at Back-looking Angles 169

6.4.5 Shock-effects at Transonic Speeds 172

7 Aero-Optical Jitter 179

7.1 Local and Global Jitter 180

7.1.1 Local Jitter 180

7.1.2 Global Jitter 180

7.2 Subaperture Effects 183

7.3 Techniques to Remove the Mechanically Induced Jitter 183

7.3.1 Cross-correlation Techniques 184

7.3.2 Large-Aperture Experiments 186

7.3.3 Stitching Method 187

8 Applications to Adaptive Optics 195

8.1 Beam-Control Components 195

8.2 How Much Correction Is Needed 198

8.3 Flow-Control Mitigation 198

8.3.1 Non-Flow-Control Mitigation 199

8.3.2 Some Qualities of Separated Shear Layers 200

8.3.3 Using the POD Analysis to Develop Requirements 204

8.4 Proper Number of Wavefront Sensor Subapertures to Actuator Ratio 208

8.4.1 Numerical Simulation 209

8.4.2 Simulation Results 211

8.4.3 Conclusion from the Simulation Results 213

9 Adaptive Optics for Aero-Optical Compensation 217

9.1 Analogies from Free-Stream Turbulence Compensation 217

9.1.1 Statistical Optics Theoretical Considerations 217

9.1.2 Power-Law Observations from Aero-Optical Wavefront Data 219

9.2 Compensation Scaling Laws for Aero-Optics 222

9.2.1 Adaptive Optics Control Law and Error Rejection Transfer Function 222

9.2.2 Asymptotic Results for Aero-Optics Compensation 223

9.2.3 Aero-Optics Compensation Frequency 225

9.2.4 Relation of Aero-Optics Scaling Laws to Free-Stream Turbulence 227

9.3 Spatial and Temporal Limitations of Adaptive Optics 228

9.3.1 Framework for Analysis of Aero-Optical Compensation 228

9.3.2 Deformable Mirror Fitting Error for Aero-Optical POD Modes 229

9.3.3 Decomposition of Correctable and Uncorrectable Power Spectrum 233

9.3.3.1 DM Sensitivity Transfer Function 233

9.3.4 Closed-Loop Residual Wavefront Error 234

9.3.5 Effect of Latency in Aero-Optics Compensation 236

9.4 Application to System Performance Modeling 238

9.4.1 Scaling of Aero-Optical Statistics to Flight Conditions 239

9.4.2 Joint Variations in Adaptive Optics Bandwidth and Actuator Density 239

9.4.3 Relative Impact of Aero-Optics with Other Propagation and System Effects 244

9.4.3.1 Comparing Aero-Optics to Free-Stream Turbulence Propagation 245

9.4.3.2 Comparing Aero-Optics to System Optical Jitter 246

9.4.4 Tracker Performance Degradations Related to Aero-Optics 248

9.4.4.1 Track Sensor Aero-Optical Imaging Resolution Degradation 249

9.4.4.2 Illuminator Propagation and Active Imaging through Aero-Optics 250

10 Concluding Remarks 255

References 259

Index 271

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