Demonstrational Optics Part 2: Coherent and Statistical Optics

Oleg Marchenko St. Petersburg State University St. Petersburg, Russia

Sergei Kazantsev Paris Observatory Paris, France

and

Laurentius Windholz Technical University of Graz Graz, Austria

Springer

Contents

Preface xiii Introduction xv

1. INTERFERENCE OF LIGHT WAVES 1

1 Classical interference schemes 2 1.1 YOUNG'S double slit experiment 5 1.2 The LLOYD mirror 8 1.3 FRESNEL'S mirrors 10 1.4 FRESNEL'S bi-prism. 11

2 Interference with glass plates 12 2.1 Plane parallel plate 12 2.2 Interference patterns 16 2.3 Air wedge 19 2.4 NEWTON'S rings 22 2.5 A plane parallel plate in a divergent pencil of beams

- Testing the quality of a glass surface 25 2.6 The MICHELSON interferometer 30

3 Multiple-beam interference 34 3.1 The LUMMER-GEHRCKE interferometer 35

4 The FABRY-PEROT interferometer 41 4.1 Resolving power and free spectral ränge 43 4.1.1 Shape of the interference fringes for high values of

% 43 4.1.2 The resolving power 46 4.1.3 Free spectral ränge 48 4.1.4 Example 1: The interference filter 50

VI Contents

4.1.5 Example 2: Resolution of the Zeeman structure of a spectral line 51

5 Optical resonator of a laser 55

6 Scanning FABRY-PEROT interferometer 58

2. DIFFRACTION OF LIGHT 69

1 FRESNEL diffraction 70 1.1 Circular aperture 70 1.2 Zone construction of FRESNEL 71 1.3 A useful relation for FRESNEL diffraction 76 1.4 Poisson's spot 77 1.5 FRESNEL'S zone plate 79 1.6 FRESNEL diffraction at a straight edge and at a slit 80 1.7 FRESNEL integrals 82 1.8 FRESNEL diffraction by a slit 86

2 FRAUNHOFER diffraction 87 2.1 FRAUNHOFER diffraction by a slit 88 2.2 The FRAUNHOFER diffraction integral 89 2.2.1 Rectangular aperture 93 2.3 Circular aperture 95 2.4 Geometrical optics and diffraction 98

3 FRAUNHOFER diffraction in optical instruments 99 3.1 Amplitude diffraction grating 99 3.2 Reflection grating 106

4 Resolving power of optical instruments 108 4.1 Resolving power of a diffraction grating 109 4.2 Resolving power of a telescope 112

3. FOURIER OPTICS 121

1 Properties of FOURIER spectra in optics 121 1.1 Spatial frequencies 121 1.2 Image construction with parallel rays 126 1.3 Theorems of two-dimensional FOURIER analysis 128

2 Isoplanatic linear Systems 133 2.1 Transfer function for FRESNEL diffraction 136 2.2 Transparency function of a thin positive lens 138 2.3 FOURIER spectrum implemented by a thin positive

lens and by a double slit 139

Contents vn

2.4 Image construction in general 142

3 Spatial filtration 144 3.1 The resolving power of a microscope 146 3.2 The phase contrast ZERNIKE microscope 149

4. HISTORY OF QUANTA 157

1 Black body radiation 157

2 PLANCK's law of radiation 160

3 Formulae for equilibrium radiation 167

4 EINSTEIN's hypothesis oflight quanta 171

5 Photoeffect 171

6 Spontaneous and induced radiation 175 6.1 Population 178

5. SHOTNOISE 187

1 Instantaneous intensity 187

2 Quantum efficiency 189

3 The random experiment 190 3.1 The random number generator 191 3.2 The Statistical trial 191 3.3 The uniform distribution of random value 192

4 Statistics of the number of photoelectrons 193 4.1 The counting time 193 4.2 A Computer model of probabilities P(N) 194 4.3 The PoiSSON random process 196

5 Detection of low intensities 199 5.1 Shot noise of a photodetector 199 5.2 Photomultiplier 200 5.3 Temporal resolution 201 5.4 A Computer model of photocurrent pulses 202

6 PoiSSON's statistics with the concept of photons 203 6.1 Probability waves 203 6.2 A Computer model for space distributions of quanta

206

7 Interference of single photons 208 7.1 The YOUNG interferometer under weak intensity 208 7.2 Interference of single photons with a MICHELSON

interferometer 210

Contents

7.3 Photons and classical mechanics 212

WHITE GAUSSIAN LIGHT 223

1 Fluctuations 223 1.1 The stationarity property 225 1.2 The probability density function 226 1.3 The Statistical ensemble 228 1.4 Stationarity and ergodicity of stochastic fields 230 1.5 The concept of correlation 233

2 The quadrature components 238 2.1 Probability distributions of the quadrature components

240 2.2 The probability distribution of the instantaneous

intensity 241

3 Computer model for the GAUSSian light 244 3.1 Method of polar coordinates 244 3.2 Simulating distributions of quadrature components

and intensity 244

CORRELATION OF LIGHT FIELDS 249

1 Visibility and complex degree of coherence 250

2 General form of the correlation of fields 256

3 Spatial correlation of the fleld 258 3.1 Typical cases of spatial correlation 261 3.1.1 Rectangular aperture and slit 261 3.1.2 A circular aperture 263 3.1.3 Two radiating disks 265 3.2 Demonstration of spatial correlation 266 3.2.1 Visibility of fringes obtained with the YOUNG double-

slit interferometer 266 3.2.2 The MlCHELSON stellar interferometer 268 3.3 A labor model of MICHELSON'S stellar interferometer

270 3.3.1 A "star" 270 3.3.2 A "double star" 270

4 Temporal correlation of the light field 272 4.1 Typical cases of temporal correlation 275 4.1.1 The LoRENTZian spectrum 276 4.1.2 The DOPPLER spectrum 277 4.1.3 A double spectral line 278

Contents ix

4.1.4 Visibility of the yellow double-line of mercury 280

5 The spatial mode 283 5.1 Coherence conditions and interference 286 5.2 The spatial mode of a black body source 290

6 Computer Simulation of field correlation 291 6.1 A slit 292 6.2 A disk 293 6.3 A DOPPLER spectral line profile 295 6.4 A double-ÜOPPLER contour 296

8. CORRELATION OF LIGHT INTENSITY 307

1 Correlation functions of intensity 307

2 Photocurrent statistics 310 2.1 The MANDEL formula 310 2.2 A Computer model of inertialess detection 312 2.3 A Computer model of inertial detection 315

3 The statistics of thermal radiation 317 3.1 An explicit form of the probabilities (P(N)) 317 3.2 Statistics of quanta within one spatial mode 320

4 Optical beats experiment 322 4.1 Spectrum of intensity fluctuations 324 4.2 A Computer model of the optical beat experiment 327

5 HANBURY-BROWN and TwiSS experiments 330 5.1 A laboratory experiment in the optical frequency

ränge 331 5.2 Correlation function of photocurrents 332 5.3 The signal-to-noise ratio of the intensity interferometer

335 5.4 The stellar interferometer of intensities 337 5.5 A Computer model of the HANBURY-BROWN - TwiSS

experiment 339

6 Correlation of pseudo-GAUSSian light 341

References 351

Index 353