Optical Signal Processing

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1 Optical Signal Processing ANTHONY VANDERLUGT North Carolina State University Raleigh, North Carolina A Wiley-Interscience Publication John Wiley & Sons, Inc. New York / Chichester / Brisbane / Toronto / Singapore

2 Contents Chapter 1. Basic Signal Parameters Introduction Characterization of a General Signal By Bandwidth By Time By Sample Interval By Number of Samples By Number of Amplitude Levels or Signal Features By Degrees of Freedom The Sample Function Examples of Signals Spatial Signals 8 Chapter 2. Geometrical Optics Introduction 2.2 Refractive Index and Optical Path 2.3 Basic Laws of Geometrical Optics Law of Reflection Law of Refraction Fermat's Principle The Critical Angle 2.4 Refraction by Prisms Minimum Deviation Angle Dispersion by a Prism Beam Magnification by a Prism Counter-Rotating Prisms The Wobble Plate 2.5 The Lens Formulas The Sign Convention Refraction at a Curved Surface

3 xiv CONTENTS The Refraction Equation for Combined Surfaces The Condenser Lens Configuration The Collimating Lens Configuration Principal Planes Thin-Lens Systems Afocal or Telescopic Configurations The General Imaging Condition Ray Tracing Lateral Magnification The Principal Pupil Ray The Optical Invariant Magnification Revisited Spatial Resolution Space Bandwidth Product Matching the Information Capacity of System Components Classification of Lenses and Systems The Coddington Shape Factor The Coddington Position Factor Aberrations Spherical Aberration Coma Astigmatism Curvature of Field Distortion Splitting the Lens 65 Chapter 3. Physical Optics Introduction The Fresnel Transform Convolution and Impulse Response Diffraction by Two Sources Fresnel Zones, Chirp Functions, and Holography The Fresnel Transform of a Slit The Fourier Transform The Fourier Transform of a Periodic Function The Fourier Transform for Nonperiodic Signals The Fourier Transform in Optics Examples of Fourier Transforms Fourier Transforms of Aperture Functions A Partitioned Aperture Function A Periodic Signal 103

4 XV 3.5 The Inverse Fourier Transform Bandlimited Signals Rayleigh-Resolution Criterion Abbe's Resolution Criterion The Sample Function, Sampling Theorem, and Decomposition Extended Fourier-Transform Analysis The Basic Elements of an Optical System Operational Notation A Basic Optical System Cascaded Optical Systems The Scale of the Fourier Transform Maximum Information Capacity and Optimum Packing Density Maximum Information Capacity Optimum Packing Density Convergent Illumination The Chirp-Z Transform System Coherence Spatial Coherence Temporal Coherence Spatial and Temporal Coherence 140 er 4. Spectrum Analysis Introduction Light Sources Spatial Light Modulators Light Valve Spatial Light Modulators Optically Addressed Electro-Optic Spatial Light Modulators Liquid-Crystal Spatial Light Modulators Magneto-Optic Spatial Light Modulators The Detection Process in the Fourier Domain A Special Photodetector Array Spectral Responsivity and Typical Power Levels The Number of Photodetector Elements Array Geometry Readout Rate Blooming and Electrical Crosstalk Linearity and Uniformity of Response System Performance Parameters Total Spatial Frequency Bandwidth Sidelobe Control and Crosstalk 160

5 XVI CONTENTS Frequency Resolution/Photodetector Spacing Array Spacing and Number of Photodetector Elements Dynamic Range Intermodulation Products Signal-to-Noise Ratio and the Minimum Signal Level Integration Time/Bandwidth Example Quantum Noise Limit Raster-Format Spectrum Analyzer The Recording Format The Two-Dimensional Spectrum Analyzer Illustration of Raster-Format Spectra Summary of the Main Design Concepts 194 Chapter 5. Spatial Filtering Introduction Some Fundamentals of Signal Processing Linear, Space-Invariant Systems Parseval's Theorem Correlation Input/Output Spectral Densities Matched Filtering Inverse Filtering Spatial Filters Binary Spatial Filters Binary Filters for Signal Detection or Excision Other Applications of Binary Filters Magnitude Spatial Filters Phase Spatial Filters Real-Valued Spatial Filters Experimental Examples The Spatial Carrier Frequency Filter Interferometric Methods for Constructing Filters Limitations of the Mach-Zehnder Interferometer The Rayleigh Interferometer The Minimum-Aperture Interferometer Information Processing Arbitrary Reference Function Bandwidth Considerations 237

6 CONTENTS xvii 5.14 Multiplexed Filters Computer Generated Filters Reference Function Optical Processors 240 Chapter 6. Spatial Filtering Systems Introduction Optical Signal Processor and Filter Generator The Light Source The Spatial Light Modulator The Fourier Transform Lens The Filter Plane The Imaging Lens The Readout Module The Thresholding Operation The Importance of Nonoverlapping Signals On-Chip Processing Constant False-Alarm Rate The Reference-to-Signal-Beam Ratio Orientation and Scale-Searching Operations The Orientation Search The Scale Search Methods for Handling Nonuniform Noise Spectral Densities Dual Frequency-Plane Processing Transposed Processing for Adaptive Filtering Other Applications for Optical Spatial Filtering Target Recognition Motion Analysis Frame Alignment and Stereo Compilation The Effects of Small Displacements of Spatial Filters Lateral Displacement Longitudinal Displacements Random Motion of the Filter 284 Chapter 7. Acousto-Optic Devices Introduction Acousto-Optic Cell Spatial Light Modulators Raman-Nath Mode The Bragg Mode Diffraction Angles, Spatial Frequencies, and Temporal Frequencies The Time Bandwidth Product 296

7 xviii CONTENTS 7.3 Dynamic Transfer Relationships Diffraction Efficiency Input/Output Relationships Time Delays and Notation Phase-Modulation Notation Sign Notation Conjugate Relationships Visualization of the Acousto-Optic Interaction Applications of Acousto-Optic Devices Acousto-Optic Modulation Acousto-Optic Beam Deflectors 309 Chapter 8. Acousto-Optic Power Spectrum Analyzers Introduction A Basic Spectrum Analyzer The Illumination Subsystem A Raman-Nath-Mode Spectrum Analyzer A Bragg-Mode Spectrum Analyzer The Generalization to Arbitrary Signals Aperture Weighting for Sidelobe Control Resolution Dynamic Range and Signal-to-Noise Ratio Spur-Free Dynamic Range Intermodulation Products Due to Acousto-Optic Cells Signal Compression Scattered Light Photodetector Geometric Considerations Example The Signal-to-Noise Ratio Radiometers Summary of the Main Design Concepts 365 Chapter 9. Heterodyne Systems Introduction The Interference Between Two Waves Spatial Interference Temporal and Spatial Interference Overlapping Waves and Photodetector Size Optimum Photodetector Size for Plane-Wave Interference 375

8 CONTENTS xix Optimum Photodetector Size for a Two-Dimensional Chirp Optimum Photodetector Size for a One-Dimensional Chirp Optimum Photodetector Size for a General Signal The Optical Radio Direct Detection Heterodyne Detection A Generalized Heterodyne System 393 Chapter 10. Heterodyne Spectrum Analysis Introduction Basic Theory Spatial and Temporal Frequencies: The Mixed Transform The cw Signal A Short Pulse The Evolving Pulse The Distributed Local Oscillator The Ideal Reference Signal The Mixed Transform of the Reference Signal Photodetector Geometry and Bandwidth The Bandpass Filter Shape Crosstalk Resolution, Accuracy, and Photodetector Size Temporal Frequencies of the Reference Bias Term Dynamic Range Comparison of the Heterodyne and Power Spectrum Analyzer Performance Both Systems Thermal-Noise Limited Both Systems Shot-Noise Limited Power Spectrum Analyzer Thermal-Noise Limited; Heterodyne Spectrum Analyzer Shot-Noise Limited Power Spectrum Analyzer Using a CCD Array Hybrid Heterodyne Spectrum Analyzer 430 Chapter 11. Decimated Arrays and Cross-Spectrum Analysis Introduction Background for the Heterodyne Spectrum Analyzer 433

9 XX CONTENTS 11.3 Photodetector Geometry and Detection Scheme The Reference and Scanning Functions Signal-to-Noise Ratio and Dynamic Range Improved Reference Waveform The Cross-Spectrum Analyzer Cross-Spectrum Analysis with Spatial Heterodyning Cross-Spectrum Analysis with Temporal Heterodyning 448 Chapter 12. The Heterodyne Transform and Signal Excision Introduction The Heterodyne Transform The Temporal Frequency Range of the Baseband Terms Probing Arbitrary Three-Dimensional Fields Signal Excision Arbitrary Filter Function 472 Chapter 13. Space-Integrating Correlators Introduction Reference-Function Correlators Real-Valued Impulse Responses Complex-Valued Impulse Responses A Wavefront View of Matched Filtering The Photodetector Bandwidth Correlation in the Presence of Doppler Frequency Shifts Programmable Matched Filter Multichannel Operation Heterodyne/Homodyne Detection Homodyne Detection in the Fourier Domain Heterodyne Detection Carrier Frequency Requirements Illumination Requirements Integrate and Dump Some More Configurations 500 Chapter 14. Time-Integrating Systems Introduction Spectrum Analysis Requirements on the Reference Signals 506

10 CONTENTS The Basic Operation of the Spectrum Analyzer The Key Features of the Time-Integrating Spectrum Analyzer Time-Integrating Correlation Time-Integrating Correlator Due to Montgomery Time-Integrating Correlator Due to Sprague and Koliopoulos Electronic Reference Correlator Comparison of Features Integrated Optical Systems 530 Chapter 15. Two-Dimensional Processing Introduction Triple-Product Processing Crossed Acousto-Optic Cell Geometry The Bispectrum Spectrum Analysis Real-Time Raster-Format Spectrum Analysis Frequency Resolution Experimental Results Ambiguity Function Generation Ambiguity Function for a cw Signal Ambiguity Function for a Short-Pulse Signal Ambiguity Function for an Infinite Time Duration Chirp Signal Wigner-Ville Distributions Range and Doppler Signal Processing Optical Transversal Processor for Notch Filtering Sampled Time Analysis Continuous-Time Analysis A Frequency Plane Implementation Phased Array Processing 566 Appendix I 570 Appendix II 571 References 574 Bibliography 582 Index 587

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