THE PRINCIPLES AND METHODS OF GEOMETRICAL OPTICS

By JOSHUA MILLER

CONTENTS 1 INTRODUCTION REFERENCES 2 WAVES IN RANDOM MEDIA 21 MAXWELL’S EQUATIONS IN RANDOM MEDIA 22 DESCRIBING RANDOM MEDIA 221 STATIONARY RANDOM PROCESSES 222 ENSEMBLE AVERAGES 223 THE SPATIAL COVARIANCE DESCRIPTION 224 THE STRUCTURE-FUNCTION DESCRIPTION 225 THE WAVENUMBER-SPECTRUM DESCRIPTION 226 THE WAVENUMBER SPECTRUM OF IRREGULARITIES 227 DESCRIBING ANISOTROPIC IRREGULARITIES 228 INHOMOGENEOUS RANDOM MEDIA 23 TROPOSPHERIC PARAMETERS 231 REFRACTIVE-INDEX EXPRESSIONS N NEAR THE SURFACE 234 INNER-SCALE MEASUREMENTS 235 ANISOTROPY AND THE OUTER SCALE LENGTH 24 IONOSPHERIC PROPERTIES 241 THE AMBIENT IONOSPHERE 242 OBSERVATIONS OF IONOSPHERIC IRREGULARITIES 243 A DESCRIPTION OF PLASMA IRREGULARITIES 25 PROBLEMS 3 GEOMETRICAL OPTICS EXPRESSIONS 31 SOLUTIONS OF THE EIKONAL EQUATION 32 NOMINAL RAY PATHS 33 THE SIGNAL PHASE 34 THE SIGNAL AMPLITUDE 35 THE ANGLE OF ARRIVAL 36 VALIDITY CONDITIONS 361 THE SMOOTH-MEDIUM CONDITION 362 THE CAUSTIC CONDITION 363 THE DI?RACTION CONDITION 37 PROBLEMS 4 THE SINGLE-PATH PHASE VARIANCE 41 TERRESTRIAL LINKS 411 MICROWAVE PHASE MEASUREMENTS 412 OPTICAL PHASE MEASUREMENTS 413 A BASIC EXPRESSION FOR THE PHASE VARIANCE 414 FREQUENCY SCALING 415 PHASE-VARIANCE ESTIMATES 416 ANISOTROPIC MEDIA AND DISTANCE SCALING 417 PHASE TRENDS AND THE SAMPLE-LENGTH E?ECT 418 RECEIVER-APERTURE AVERAGING 42 SATELLITE SIGNALS 421 GPS RANGE ERRORS 422 THE IONOSPHERIC IN?UENCE 43 PROBLEMS 5 THE PHASE STRUCTURE FUNCTION 51 TERRESTRIAL LINKS 511 SPHERICAL WAVES ON HORIZONTAL PATHS 512 PLANE WAVES AND COLLIMATED BEAMS 513 THE E?ECT OF ATMOSPHERIC ANISOTROPY 514 BEAM WAVES ON HORIZONTAL PATHS 52 MICROWAVE INTERFEROMETRY 521 D?(?) FOR ATMOSPHERIC TRANSMISSION 522 SEPARATION SCALING FOR THE 2/3 LAW 523 THE IN?UENCE OF THE SAMPLE LENGTH 524 MICROWAVE-INTERFEROMETER MEASUREMENTS 53 OPTICAL INTERFEROMETRY 531 OPTICAL INTERFEROMETER INSTALLATIONS 532 LIMITATIONS ON THE ACCURACY OF OPTICAL INTERFEROMETERS 54 PROBLEMS 6 THE TEMPORAL VARIATION OF PHASE 61 ATMOSPHERIC VARIABILITY 611 ATMOSPHERIC WIND FIELDS 612 TAYLOR’S FROZEN-RANDOM-MEDIUM HYPOTHESIS 613 THE LOCALLY FROZEN RANDOM-MEDIUM APPROXIMATION 614 MORE GENERAL DESCRIPTIONS 62 THE SINGLE-PATH VARIABILITY OF PHASE 621 AUTOCORRELATION OF PHASE 622 THE TEMPORAL STRUCTURE FUNCTION 623 THE PHASE POWER SPECTRUM 624 WIND-SPEED VARIATIONS 625 THE IN?UENCE OF PHASE TRENDS 626 THE ALLAN VARIANCE 63 THE PHASE DI?ERENCE FOR SPHERICAL WAVES 631 THE AUTOCORRELATION OF THE PHASE DI?ERENCE 632 THE POWER SPECTRUM OF THE PHASE DI?ERENCE 64 THE PHASE DI?ERENCE FOR PLANE WAVES 641 THE AUTOCORRELATION OF THE PHASE DI?ERENCE 642 PHASE-DI?ERENCE POWER SPECTRA 65 ASTRONOMICAL INTERFEROMETRY 651 THE TIME-SHIFTED PHASE STRUCTURE FUNCTION 652 THE AUTOCORRELATION OF THE PHASE DI?ERENCE 653 THE ALLAN VARIANCE FOR INTERFEROMETRIC DATA 654 THE FREQUENCY STABILITY OF ARRIVING SIGNALS 655 INTERFEROMETRIC POWER-SPECTRUM MEASUREMENTS 66 PROBLEMS 7 ANGLE-OF-ARRIVAL FLUCTUATIONS 71 MEASUREMENT CONSIDERATIONS 711 INTERFEROMETERS 712 CENTROID TRACKERS 72 TERRESTRIAL LINKS 721 PLANE WAVES 722 SPHERICAL WAVES 723 APERTURE AVERAGING 724 BEAM WAVES 73 OPTICAL ASTRONOMICAL SIGNALS 731 ANGULAR POSITION ERRORS 732 THE SPATIAL CORRELATION OF ANGULAR ERRORS 733 THE ANGULAR-ERROR CORRELATION FOR ADJACENT STARS 734 ANGULAR-ERROR AVERAGING BY EXTENDED SOURCES 735 THE IN?UENCE OF ANISOTROPY 74 MICROWAVE TRACKING OF SATELLITES 75 RADIO ASTRONOMY 751 IONOSPHERIC IN?UENCES 752 TROPOSPHERIC IN?UENCES 76 PROBLEMS 8 PHASE DISTRIBUTIONS 81 THE SINGLE-PATH PHASE DISTRIBUTION 811 ?? GAUSSIAN 812 ?? NON-GAUSSIAN 813 THE E?ECT OF PHASE TRENDS 814 EXPERIMENTAL RESULTS 82 THE PHASE DI?ERENCE DISTRIBUTION 821 THE PREDICTED DISTRIBUTION 822 EXPERIMENTAL CON?RMATIONS 83 THE ANGLE-OF-ARRIVAL DISTRIBUTION 84 TEMPORAL DISTRIBUTIONS 85 PROBLEM 9 FIELD-STRENGTH MOMENTS 91 THE AVERAGE FIELD STRENGTH 92 THE MUTUAL COHERENCE FUNCTION 93 FREQUENCY COHERENCE 94 SHORTCOMINGS 95 PROBLEMS