AN ELEMENTARY TREATISE ON GEOMETRICAL OPTICS
CONTENTS 1 INTRODUCTION REFERENCES 2 WAVES IN RANDOM MEDIA 2 1 MAXWELL’S EQUATIONS IN RANDOM MEDIA 2 2 DESCRIBING RANDOM MEDIA 2 2 1 STATIONARY RANDOM PROCESSES 2 2 2 ENSEMBLE AVERAGES 2 2 3 THE SPATIAL COVARIANCE DESCRIPTION 2 2 4 THE STRUCTURE-FUNCTION DESCRIPTION 2 2 5 THE WAVENUMBER-SPECTRUM DESCRIPTION 2 2 6 THE WAVENUMBER SPECTRUM OF IRREGULARITIES 2 2 7 DESCRIBING ANISOTROPIC IRREGULARITIES 2 2 8 INHOMOGENEOUS RANDOM MEDIA 2 3 TROPOSPHERIC PARAMETERS 2 3 1 REFRACTIVE-INDEX EXPRESSIONS N NEAR THE SURFACE 2 3 4 INNER-SCALE MEASUREMENTS 2 3 5 ANISOTROPY AND THE OUTER SCALE LENGTH 2 4 IONOSPHERIC PROPERTIES 2 4 1 THE AMBIENT IONOSPHERE 2 4 2 OBSERVATIONS OF IONOSPHERIC IRREGULARITIES 2 4 3 A DESCRIPTION OF PLASMA IRREGULARITIES CONTENTS 2 5 PROBLEMS 3 GEOMETRICAL OPTICS EXPRESSIONS 3 1 SOLUTIONS OF THE EIKONAL EQUATION 3 2 NOMINAL RAY PATHS 3 3 THE SIGNAL PHASE 3 4 THE SIGNAL AMPLITUDE 3 5 THE ANGLE OF ARRIVAL 3 6 VALIDITY CONDITIONS 3 6 1 THE SMOOTH-MEDIUM CONDITION 3 6 2 THE CAUSTIC CONDITION 3 6 3 THE DI?RACTION CONDITION 3 7 PROBLEMS 4 THE SINGLE-PATH PHASE VARIANCE 4 1 TERRESTRIAL LINKS 4 1 1 MICROWAVE PHASE MEASUREMENTS 4 1 2 OPTICAL PHASE MEASUREMENTS 4 1 3 A BASIC EXPRESSION FOR THE PHASE VARIANCE 4 1 4 FREQUENCY SCALING 4 1 5 PHASE-VARIANCE ESTIMATES 4 1 6 ANISOTROPIC MEDIA AND DISTANCE SCALING 4 1 7 PHASE TRENDS AND THE SAMPLE-LENGTH E?ECT 4 1 8 RECEIVER-APERTURE AVERAGING 4 2 SATELLITE SIGNALS 4 2 1 GPS RANGE ERRORS 4 2 2 THE IONOSPHERIC IN?UENCE 4 3 PROBLEMS 5 THE PHASE STRUCTURE FUNCTION 5 1 TERRESTRIAL LINKS 5 1 1 SPHERICAL WAVES ON HORIZONTAL PATHS 5 1 2 PLANE WAVES AND COLLIMATED BEAMS 5 1 3 THE E?ECT OF ATMOSPHERIC ANISOTROPY 5 1 4 BEAM WAVES ON HORIZONTAL PATHS 5 2 MICROWAVE INTERFEROMETRY 5 2 1 D?(?) FOR ATMOSPHERIC TRANSMISSION 5 2 2 SEPARATION SCALING FOR THE 2/3 LAW 5 2 3 THE IN?UENCE OF THE SAMPLE LENGTH 5 2 4 MICROWAVE-INTERFEROMETER MEASUREMENTS 5 3 OPTICAL INTERFEROMETRY 5 3 1 OPTICAL INTERFEROMETER INSTALLATIONS CONTENTS 5 3 2 LIMITATIONS ON THE ACCURACY OF OPTICAL INTERFEROMETERS 5 4 PROBLEMS 6 THE TEMPORAL VARIATION OF PHASE 6 1 ATMOSPHERIC VARIABILITY 6 1 1 ATMOSPHERIC WIND FIELDS 6 1 2 TAYLOR’S FROZEN-RANDOM-MEDIUM HYPOTHESIS 6 1 3 THE LOCALLY FROZEN RANDOM-MEDIUM APPROXIMATION 6 1 4 MORE GENERAL DESCRIPTIONS 6 2 THE SINGLE-PATH VARIABILITY OF PHASE 6 2 1 AUTOCORRELATION OF PHASE 6 2 2 THE TEMPORAL STRUCTURE FUNCTION 6 2 3 THE PHASE POWER SPECTRUM 6 2 4 WIND-SPEED VARIATIONS 6 2 5 THE IN?UENCE OF PHASE TRENDS 6 2 6 THE ALLAN VARIANCE 6 3 THE PHASE DI?ERENCE FOR SPHERICAL WAVES 6 3 1 THE AUTOCORRELATION OF THE PHASE DI?ERENCE 6 3 2 THE POWER SPECTRUM OF THE PHASE DI?ERENCE 6 4 THE PHASE DI?ERENCE FOR PLANE WAVES 6 4 1 THE AUTOCORRELATION OF THE PHASE DI?ERENCE 6 4 2 PHASE-DI?ERENCE POWER SPECTRA 6 5 ASTRONOMICAL INTERFEROMETRY 6 5 1 THE TIME-SHIFTED PHASE STRUCTURE FUNCTION 6 5 2 THE AUTOCORRELATION OF THE PHASE DI?ERENCE 6 5 3 THE ALLAN VARIANCE FOR INTERFEROMETRIC DATA 6 5 4 THE FREQUENCY STABILITY OF ARRIVING SIGNALS 6 5 5 INTERFEROMETRIC POWER-SPECTRUM MEASUREMENTS 6 6 PROBLEMS 7 ANGLE-OF-ARRIVAL FLUCTUATIONS 7 1 MEASUREMENT CONSIDERATIONS 7 1 1 INTERFEROMETERS 7 1 2 CENTROID TRACKERS 7 2 TERRESTRIAL LINKS 7 2 1 PLANE WAVES 7 2 2 SPHERICAL WAVES 7 2 3 APERTURE AVERAGING 7 2 4 BEAM WAVES 7 3 OPTICAL ASTRONOMICAL SIGNALS 7 3 1 ANGULAR POSITION ERRORS 7 3 2 THE SPATIAL CORRELATION OF ANGULAR ERRORS CONTENTS 7 3 3 THE ANGULAR-ERROR CORRELATION FOR ADJACENT STARS 7 3 4 ANGULAR-ERROR AVERAGING BY EXTENDED SOURCES 7 3 5 THE IN?UENCE OF ANISOTROPY 7 4 MICROWAVE TRACKING OF SATELLITES 7 5 RADIO ASTRONOMY 7 5 1 IONOSPHERIC IN?UENCES 7 5 2 TROPOSPHERIC IN?UENCES 7 6 PROBLEMS 8 PHASE DISTRIBUTIONS 8 1 THE SINGLE-PATH PHASE DISTRIBUTION 8 1 1 ?? GAUSSIAN 8 1 2 ?? NON-GAUSSIAN 8 1 3 THE E?ECT OF PHASE TRENDS 8 1 4 EXPERIMENTAL RESULTS 8 2 THE PHASE DI?ERENCE DISTRIBUTION 8 2 1 THE PREDICTED DISTRIBUTION 8 2 2 EXPERIMENTAL CON?RMATIONS 8 3 THE ANGLE-OF-ARRIVAL DISTRIBUTION 8 4 TEMPORAL DISTRIBUTIONS 8 5 PROBLEM 9 FIELD-STRENGTH MOMENTS 9 1 THE AVERAGE FIELD STRENGTH 9 2 THE MUTUAL COHERENCE FUNCTION 9 3 FREQUENCY COHERENCE 9 4 SHORTCOMINGS 9 5 PROBLEMS