Optical Interferometry, 2e

When the first edition of "Optical Interferometry" was published, interferometry was regarded as a rather esoteric method of making measurements, largely confined to the laboratory. Today, however, besides its use in several fields of research, it has applications in fields as diverse as measurement of length and velocity, sensors for rotation, acceleration, vibration and electrical and magnetic fields, as well as in microscopy and nanotechnology. Most topics are discussed first at a level accessible to anyone with a basic knowledge of physical optics, then a more detailed treatment of the topic is undertaken, and finally each topic is supplemented by a reference list of more than 1000 selected original publications in total. * Historical development of interferometry * The laser as a light source * Two-beam interference * Techniques for frequency stabilization * Coherence * Electronic phase measurements * Multiple-beam interference * Quantum effects in optical interference * Extensive coverage of the applications of interferometry, such as measurements of length, optical testing, interference microscopy, interference spectroscopy, Fourier-transform spectroscopy, interferometric sensors, nonlinear interferometers, stellar interferometry, and studies of space-time and gravitation. Audience: The primary market for the book would be scientists and engineers interested in precision measurements of a range of physical quantities in industry as well as researchers and students in universities. A secondary market would be members of organizations such as the Optical Society of America, SPIE and IEEE who are interested in possible applications in their work.
Chapter One Interferometry: Its Development

Almost everyone has come across interference phenomena such as the vivid colors in a soap bubble, or in an oil slick on a wet road, or the colored fringes seen in a thin air film enclosed between two glass plates when they are brought into contact. The latter are commonly known as "Newton's rings," but were, in fact, first described by Boyle and, independently, by Hooke, in the latter half of the 17th century. Their observations can be called the starting point of optical interferometry.

The development of optical interferometry extends over more than 300 years and is closely linked with the history of wave optics. The aim of this chapter is to review briefly some of the significant stages of this development, so as to put in perspective the topics which will be discussed later, in more detail, in this book.

1.1 The Wave Theory of Light

We now know that the colored fringes seen by Boyle and Hooke were produced by the interference of the light waves reflected from the two surfaces of the film. However, though Hooke did put forward a wave theory of light to explain the fringes, and this theory was expanded and put into its present form by Huyghens in 1690, it made little progress because it was opposed by Newton, who believed that a wave theory could not explain either the rectilinear propagation of light or the phenomenon of polarization. As a result, the correct explanation had to wait for almost 150 years.

The barriers to the acceptance of the wave theory were first broken by Young, when, in his Bakerian lectures in 1801 and 1803, he stated the principle of interference and demonstrated that the summation of two rays of light could give rise to darkness. However, even this demonstration did not lead to immediate acceptance of the theory. Almost no one supported Young, and the discovery by Malus, in 1809, that light could be polarized by reflection shook even Young's confidence, as he, like Huyghens, thought that light was propagated as longitudinal waves. The turning point came with Fresnel's brilliant memoir on diffraction in 1818, which perfected the treatment of interference, and the discovery by Arago and Fresnel that two orthogonally polarized beams could not interfere. This led Young and Fresnel to the inevitable conclusion that light waves were transverse waves. Since only longitudinal waves can propagate in a fluid, Fresnel postulated that light waves were propagated through an elastic solid pervading all matter — the "luminiferous aether."

1.2 The Michelson–Morley Experiment

Most of the leading physicists of the 19th century supported the aether theory, even though it raised a number of questions which had no obvious answers. One such was the aberration of light discovered by Bradley in 1728, which indicated that the aether was stationary. However, theoretical calculations by Fresnel in 1818 showed that in a medium with a refractive index n moving with a velocity v, the aether should be carried along with a velocity v(1-1/n2). Fizeau therefore carried out an experiment in 1851 with an interferometer in which the two beams traversed two columns of running water, one beam always moving with the current, while the other moved against it. This experiment, which was repeated later by Jamin and by Michelson, showed a shift of the fringes of the expected magnitude. Based on these results, Maxwell predicted in 1880 that the movement of the earth through the aether should result in a change in the speed of light proportional to the square of the ratio of the speed of the earth to that of light. While Maxwell felt that this effect was too small to be detected experimentally, Michelson was confident that it could be observed by making use of the tremendous increase in accuracy obtained by interferometry. This led, in 1881, to Michelson's famous experiment which was designed to demonstrate the "aether drift." As things turned out, the null result obtained by Michelson led to the rejection of the concept of an aether and laid the foundations for the special theory of relativity [Shankland, 1973].

1.3 Measurement of the Metre

Other applications of interferometry followed in rapid succession. Thus, in 1896, Michelson carried out the first measurement of the length of the Pt–Ir bar which was the international prototype of the metre in terms of the wavelength of the red cadmium radiation. Although the idea of the wavelength of a monochromatic source as a natural standard of length had been suggested much earlier by Babinet and by Fizeau, it was Michelson's work which demonstrated its feasibility and led, in 1960, to the redefinition of the metre in terms of the wavelength of the orange radiation of 86Kr.

1.4 Optical Testing

Another major field of application of interferometry was opened up by Twyman in 1916, when he used a modified Michelson interferometer to test optical components. This interferometer was, in turn, adapted by Linnik [1933] to permit microscopic examination of reflecting surfaces. At the same time, interferometry became a valuable tool in studies of fluid flow and combustion.

1.5 Coherence

Studies by Michelson also revealed the connection between the visibility of the fringes in his interferometer and the dimensions and spectral purity of the source. Since any thermal source can be considered as made up of many elementary radiators (atoms) which are not synchronized, the interference pattern with such a source is obtained by adding the intensities in the interference patterns formed by these incoherent elementary radiators. The gradual transition from incoherent to coherent illumination with a thermal source was demonstrated as far back as 1869 by Verdet, who showed that light from two pinholes illuminated by the sun produced an interference pattern on a screen if their separation was less than 0.05 mm. However, the first quantitative concepts of coherence were formulated by von Laue only in 1907, and it was again only after a long delay that the foundations of modern coherence theory were laid in three papers by van Cittert [1934, 1939] and Zernike [1938]. These concepts were developed in more detail, 20 years later, by Hopkins [1951, 1953] and Wolf [1954, 1955]. The discovery by Brown and Twiss [1954] of intensity-correlation effects (fourth-order correlation) led to the formulation of a general description of higher-order coherence effects by Mandel and Wolf [1965].

1.6 Interference Spectroscopy

Towards the end of the 19th century, interference techniques found their way into high-resolution spectroscopy with the development of instruments such as the Fabry–Perot etalon, the Michelson echelon, and the Lummer–Gehrcke plate. However, as improved multilayer dielectric coatings, with high values of reflectance and negligible losses, became available, the greater light-gathering power (or etendue) of the Fabry–Perot interferometer led to its rapidly replacing the other two instruments.

At this stage, a completely new approach was opened up by the development of Fourier transform spectroscopy. The origins of this technique can be traced back to 1862, when Fizeau studied the effect of the separation of the plates on Newton's rings formed with sodium light. He found that the rings almost disappeared at a separation corresponding to the passage of 490 fringes, but reached maximum contrast once again when 980 fringes had passed, showing that the sodium line was a doublet. This method was taken a step further by Michelson in 1891, when he plotted the visibility of the fringes as a function of the optical path difference for a number of spectral lines. All of them, with the exception of the red cadmium line, exhibited a series of maxima and minima, indicating that they consisted of more than one component. A similar technique was also applied to far infrared spectroscopy (λ = 100 to 300 μm) by Rubens and Wood in 1911. However, the systematic development of Fourier-transform spectroscopy started with Fellgett [1951], who was the first to obtain a spectrum from a numerically Fourier-transformed interferogram and demonstrate the advantages of this technique. Subsequent improvements have brought this technique to the point where it reigns supreme at long wavelengths and is preferable to conventional dispersive instruments in the visible region for mapping complex spectra with the highest possible resolution.

1.7 The Laser

Throughout the first half of the 20th century, the most commonly used light source for interferometry was a pinhole illuminated by a mercury arc through a filter which isolated the green line (λ = 546 nm). Such a source gave only a very small amount of light with limited spatial and temporal coherence. The development of the laser made available, for the first time, an intense source of light with a remarkably high degree of spatial and temporal coherence and initiated a revolution in interferometry.

The origins of the laser can be traced back to 1917, when Einstein pointed out that atoms in a higher energy state, which normally radiate spontaneously, could also be stimulated to emit and revert to a lower energy state when irradiated by a wave of the correct frequency. The most remarkable feature of this process was that the emitted photon had the same frequency, polarization, and phase as the stimulating wave and propagated in the same direction.

Schawlow and Townes [1958] were the first to show that amplification by stimulated emission was possible in the visible region, and that a simple resonator consisting of two mirrors could be used for mode selection. The first practical laser was the pulsed ruby laser [Maiman, 1960]. Continuous laser action was achieved soon afterwards with the helium-neon laser, initially in the infrared [Javan, Bennett, and Herriott, 1961] and then in the visible region [White and Rigden, 1962].

The high degree of directionality and coherence of laser light were verified by Collins et al. [1960]. This was followed by the observation of interference fringes when the beams from two pulsed ruby lasers were superposed [Magyar and Mandel, 1963].

Lasers have removed most of the limitations of interferometry imposed by thermal sources and have made possible many new techniques.

1.8 Electronic Techniques

Another development which has revolutionized interferometry has been the increasing use of electronic techniques. This trend started with the use of photoelectric detectors with Fabry–Perot interferometers, but soon extended to such applications as fringe-counting interferometers for length measurements. Digital computers were first used in Fourier transform spectroscopy and have made it an extremely powerful tool. Digital systems, in conjunction with phase-shifting techniques, have also made possible direct measurements of the optical path difference at an array of points covering an interference pattern.

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Excerpted from Optical Interferometry by P. Hariharan Copyright © 2003 by Elsevier Science (USA). Excerpted by permission of ACADEMIC PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site. front cover......Page 1 copyright......Page 5 table of contents......Page 6 Preface to the First Edition......Page 14 Preface to the Second Edition......Page 16 1.1 The Wave Theory of Light......Page 18 1.3 Measurement of the Metre......Page 19 1.6 Interference Spectroscopy......Page 20 1.8 Electronic Techniques......Page 21 1.11 Nonlinear Interferometers......Page 22 1.12 Stellar Interferometry......Page 23 1.15 Future Directions......Page 24 2. Two-Beam Interference......Page 26 2.1 Complex Representation of Light Waves......Page 27 2.2 Interference of Two Monochromatic Waves......Page 28 2.3 Wavefront Division......Page 29 2.4 Amplitude Division......Page 31 2.5 Localization of Fringes......Page 35 2.6 Two-Beam Interferometers......Page 40 2.7 The Michelson Interferometer......Page 41 2.8 The Mach–Zehnder Interferometer......Page 43 2.9 The Sagnac Interferometer......Page 45 2.11 Channeled Spectra......Page 46 2.12 Achromatic Fringes......Page 48 2.14 Interferential Color Photography......Page 50 3.1 Quasi-Monochromatic Light......Page 52 3.2 Waves and Wave Groups......Page 53 3.3 Phase Velocity and Group Velocity......Page 54 3.4 The Mutual Coherence Function......Page 56 3.5 Spatial Coherence......Page 60 3.6 Temporal Coherence......Page 64 3.7 Coherence Time and Coherence Length......Page 65 3.8 Coherence in the Space-Frequency Domain......Page 66 3.10 Effects in Two-Beam Interferometers......Page 68 3.11 Source-Size Effects......Page 72 3.13 Spectral Changes Due to Coherence......Page 73 3.14 Polarization Effects......Page 74 4.1 Fringes in a Plane-Parallel Plate......Page 76 4.2 Fringes by Reflection......Page 80 4.3 Fringes of Equal Thickness......Page 81 4.4 Fringes of Equal Chromatic Order......Page 83 4.5 Fringes of Superposition......Page 85 4.6 Three-Beam Fringes......Page 90 4.7 Double-Passed Fringes......Page 93 5.1 Lasers for Interferometry......Page 96 5.2 Laser Modes......Page 97 5.3 Comparison of Laser Frequencies......Page 101 5.4 Frequency Stabilization......Page 103 5.5 Laser Beams......Page 108 6.2 Fringe Counting......Page 110 6.3 Heterodyne Interferometry......Page 112 6.4 Phase-Locked Interferometry......Page 113 6.5 Computer-Aided Fringe Analysis......Page 114 6.6 Phase-Shifting Interferometry......Page 115 6.7 Techniques of Phase Shifting......Page 119 6.8 Sinusoidal Phase Modulation......Page 121 7.2 End Standards......Page 122 7.3 The Integral Interference Order......Page 123 7.5 The Refractive Index of Air......Page 124 7.6 The International Prototype Metre......Page 126 7.8 Frequency Measurements......Page 128 7.9 The Definition of the Metre......Page 129 7.10 Length Measurements with Lasers......Page 130 7.12 Displacements......Page 133 7.13 Dynamic Angle Measurements......Page 135 8.1 The Fizeau Interferometer......Page 136 8.2 The Twyman–Green Interferometer......Page 138 8.4 Phase Unwrapping......Page 139 8.5 Analysis of Wavefront Aberrations......Page 140 8.6 Shearing Interferometers......Page 141 8.7 Grating Interferometers......Page 146 8.8 The Scatter-Plate Interferometer......Page 148 8.9 The Point-Diffraction Interferometer......Page 149 8.10 Computerized Test Methods......Page 150 8.11 Aspheric Surfaces......Page 153 8.12 Rough Surfaces......Page 155 8.13 The Optical Transfer Function......Page 156 9.2 Common Path Interference Microscopes......Page 160 9.3 Polarization Interferometers......Page 161 9.4 The Nomarski Interferometer......Page 164 9.5 Electronic Phase Measurements......Page 165 9.7 White-Light Interferometry......Page 168 10.1 Etendue of an Interferometer......Page 174 10.2 The Fabry–Perot Interferometer......Page 175 10.3 The Scanning FPI......Page 176 10.4 The Spherical-Mirror FPI......Page 178 10.5 The Multiple FPI......Page 179 10.7 Birefringent Filters......Page 180 10.8 Wavelength Meters......Page 182 10.10 Measurement of Laser Line Widths......Page 186 10.11 Laser Frequency Standards......Page 187 11.1 The Etendue and Multiplex Advantages......Page 190 11.2 Theory......Page 192 11.3 Resolution and Apodization......Page 195 11.4 Sampling......Page 196 11.5 Effect of Source and Detector Size......Page 197 11.8 Noise......Page 198 11.10 Interferometers for FTS......Page 199 11.12 Applications......Page 201 12.1 Rotation Sensors......Page 206 12.2 Fiber Interferometers......Page 208 12.3 Laser-Feedback Interferometers......Page 215 12.4 Doppler Interferometry......Page 218 12.5 Vibration Measurements......Page 221 12.6 Interferometric Magnetometers......Page 223 12.7 Adaptive Optics......Page 224 13.1 Second-Harmonic Interferometry......Page 226 13.3 Phase-Conjugating Mirrors......Page 230 13.4 Interferometers with Active Elements......Page 233 13.6 Measurement of Third-Order Susceptibility......Page 234 13.7 Optical Switches......Page 236 14.1 Michelson’s Stellar Interferometer......Page 238 14.2 The Intensity Interferometer......Page 240 14.3 Heterodyne Stellar Interferometry......Page 244 14.4 Long-Baseline Stellar Interferometers......Page 247 14.5 Stellar Speckle Interferometry......Page 249 14.6 Speckle Holography......Page 251 14.8 Astrometry......Page 252 14.10 Telescope Arrays......Page 253 15.1 The Michelson–Morley Experiment......Page 256 15.2 Gravitational Waves......Page 258 15.3 Gravitational Wave Detectors......Page 259 15.4 LIGO......Page 262 15.5 The Standard Quantum Limit......Page 263 15.6 Squeezed States of Light......Page 265 15.7 Interferometry Below the SQL......Page 268 16.1 Interference at the “Single-Photon” Level......Page 270 16.3 Sources of Nonclassical Light......Page 271 16.5 Interference with Single-Photon States......Page 273 16.6 The Geometric Phase......Page 274 16.7 Interference with Independent Sources......Page 277 16.8 Superposition States......Page 281 17.1 Nonclassical Fourth-Order Interference......Page 284 17.2 Interference in Separated Interferometers......Page 288 17.3 The Geometric Phase......Page 291 18.1 Tests of Bell’s Inequality......Page 294 18.2 Quantum Cryptography......Page 298 18.3 Beams from Two Down-Converters......Page 299 18.4 The Quantum Eraser......Page 301 18.5 Single-Photon Tunneling......Page 302 18.6 Conclusions......Page 305 A.1 The Fourier Transform......Page 306 A.2 Convolution and Correlation......Page 307 A.4 Random Functions......Page 308 Appendix B: The Fresnel–Kirchhoff Integral......Page 310 C.1 The Fresnel Formulas......Page 312 C.2 The Stokes Relations......Page 313 Appendix D: The Jones Calculus......Page 316 E.2 The Pancharatnam Phase......Page 318 F.1 The Off-Axis Hologram......Page 320 F.2 Computer-Generated Holograms......Page 322 G.1 Speckle Statistics......Page 324 G.2 Young’s Fringes......Page 325 Bibliography......Page 326 References......Page 328 Index......Page 360