Fourier Optics An Introduction (Second

A straightforward introduction to the Fourier principles behind modern optics, this text is appropriate for advanced undergraduate and graduate students. Topics include the Fraunhofer diffraction, Fourier series and periodic structures, Fourier transforms, optical imaging and processing, image reconstruction from projections (medical imaging), and interferometry and radiation sources. Solutions. 1989 edition.

E. G. Steward: Professor Emeritus of Physics and Molecular Medicine City University, London

Preface to the Dover edition

5

(2)

Preface

7

(4)

Preface to the first edition

11

(2)

J.B.J. Fourier 1768-1830

13

(2)

1 PRELIMINARIES

1.1 Introduction

15

(5)

1.2 Coherence and light sources

20

(3)

1.2.1 Temporal coherence

21

(1)

1.2.2 Spatial coherence

21

(2)

1.3 Optical image formation

23

(7)

1.3.1 Fraunhofer diffraction

25

(3)

1.3.2 Lens aperture

28

(2)

1.4 Interference by division of amplitude

30

(2)

Problems

32

(2)

2 FRAUNHOFER DIFFRACTION

2.1 Introduction

34

(1)

2.2 Single-slit pattern

34

(4)

2.3 Circular aperture

38

(5)

2.4 Double aperture

43

(3)

2.4.1 Two slits

43

(3)

2.4.2 Two circular apertures

46

(1)

2.5 N-slit grating

46

(4)

2.6 2-dimensional gratings

50

(1)

2.7 Crystals as 3-dimensional gratings

51

(4)

Problems

55

(2)

3 FOURIER SERIES AND PERIODIC STRUCTURES

3.1 Introduction

57

(1)

3.2 Fourier series

58

(2)

3.3 Determining Fourier coefficients: even functions

60

(3)

3.4 Optical and crystal diffraction gratings: physical interpretation of Fourier terms

63

(2)

3.4.1 Optical diffraction

63

(1)

3.4.2 Crystal diffraction

64

(1)

3.5 Fourier series: general formulations

65

(5)

3.5.1 The sine and cosine series

65

(1)

3.5.2 Exponential notation

66

(3)

3.5.3 Space and time

69

(1)

Problems

70

(1)

4 FOURIER TRANSFORMS, CONVOLUTION AND CORRELATION

4.1 Introduction

71

(1)

4.2 The Fourier transform and single-slit diffraction

72

(7)

4.3 The grating pattern as a product of transforms

79

(3)

4.4 Convolution

82

(4)

4.4.1 Introduction

82

(3)

4.4.2 The grating as a convolution

85

(1)

4.5 The convolution theorem and diffraction

86

(2)

4.6 Fourier transforms and light waves

88

(3)

4.7 Correlation

91

(6)

4.7.1 Autocorrelation theorem (Wiener-Khintchine theorem)

95

(2)

Problems

97

(2)

5 OPTICAL IMAGING AND PROCESSING

5.1 Introduction

99

(3)

5.2 Incoherent optical imaging

102

(4)

5.2.1 Determination of transfer functions

104

(2)

5.3 Coherent optical imaging

106

(12)

5.3.1 Periodic objects

107

(3)

5.3.2 Non-periodic objects

110

(1)

5.3.3 Illustrations: optical transforms

110

(8)

5.4 Holography

118

(4)

5.5 Optical processing

122

(14)

5.5.1 Coherent processing

122

(9)

5.5.2 Incoherent processing

131

(5)

Problems

136

(1)

6 IMAGE RECONSTRUCTION FROM PROJECTIONS (MEDICAL IMAGING)

6.1 introduction

137

(2)

6.2 X-ray computed tomography

139

(13)

6.2.1 Introduction

139

(4)

6.2.2 Reconstruction by simple back-projection

143

(2)

6.2.3 Iterative reconstruction

145

(1)

6.2.4 Analytical methods of reconstruction

146

(6)

6.3 Other modalities

152

(9)

6.3.1 Emission computed tomography

152

(1)

6.3.2 Magnetic resonance imaging (I.S. Mackenzie)

153

(5)

6.3.3 Ultrasonic computed tomography (J.P. Weight)

158

(3)

7 INTERFEROMETRY AND RADIATION SOURCES

7.1 Introduction

161

(1)

7.2 Michelson's stellar interferometer

162

(7)

7.2.1 Introduction

162

(3)

7.2.2 Fringe visibility aspects

165

(4)

7.3 Michelson's spectral interferometer

169

(7)

7.3.1 Introduction

169

(5)

7.3.2 Fringe visibility and spectral distribution

174

(2)

7.4 Partial coherence, correlation, and visibility

176

(7)

7.4.1 Spatial coherence

180

(1)

7.4.2 Temporal coherence

181

(2)

7.5 Fourier transform spectroscopy

183

(6)

7.6 Applications in astronomy

189

(11)

7.6.1 Aperture synthesis

189

(9)

7.6.2 The intensity interferometer

198

(2)

Problems

200

(1)

APPENDICES

A The scalar-wave description of electromagnetic waves

201

(8)

B The Stokes treatment of reflection and refraction

209

(2)

C Circular-aperture Fraunhofer diffraction pattern

211

(3)

D The Fourier transform of a linear array of N equally spaced δ-functions

214

(2)

E Diffraction of X-rays by crystals. The equivalence of the Laue conditions and the Bragg reflection concept

216

(5)

F Summary notes on Chapters 1-5

221

(5)

G The length of wavetrains and their spectral width

226

(3)

H Maximum entropy data processing (S.F. Gull)

229

(2)

I Applications in Communications (J.D. Weaver)

231

(6)

J The impulse response of the eye: a diffraction analogy (Janet E. Wolf)

237

(6)

K The electromagnetic spectrum. Approximate range of principal, named regions

243

(1)

L Useful formulae

244

(1)

References

245

(6)

Bibliography

251

(4)

Solutions to problems

255

(6)

Index

261

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Fourier Optics An Introduction (Second Edition)

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