Contents8 Interference 18.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.1 Young’s Slits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2.2 Interference with an Extended Source: van Cittert-Zernike Theorem . 48.2.3 More General Formulation of Spatial Coherence; Lateral CoherenceLength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78.2.4 Generalization to two dimensions . . . . . . . . . . . . . . . . . . . . 88.2.5 Michelson Stellar Interferometer . . . . . . . . . . . . . . . . . . . . . 98.2.6 Temp oral Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 108.2.7 Michelson Interferometer and Fourier-Transform Spectroscopy . . . . 118.2.8 Degree of Coherence; Relation to Theory of Random Processes . . . . 148.3 Radio Telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168.3.1 Two-Element Radio Interferometer . . . . . . . . . . . . . . . . . . . 178.3.2 Multiple Element Radio Interferometer . . . . . . . . . . . . . . . . . 188.3.3 Closure Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.3.4 Angular Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.4 Etalons and Fabry-Perot Interferometers . . . . . . . . . . . . . . . . . . . . 208.4.1 Multiple Beam Interferometry; Etalons . . . . . . . . . . . . . . . . . 208.4.2 Fabry-Perot Interferometer . . . . . . . . . . . . . . . . . . . . . . . . 258.4.3 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.5T2 Laser Interferometer Gravitational Wave Detectors . . . . . . . . . . . 298.6T2 Intensity Correlation and Photon Statistics. . . . . . . . . . . . . . . . 36iChapter 8InterferenceVersion 0808.1.K.pdf, 19 November 2008Please send comments, suggestions, and errata via email to [email protected] and [email protected], or on paper to Kip Thorne, 130-33 Caltech, Pasadena CA 91125Box 8.1Reader’s Guide• This chapter depends substantially on– Secs. 7.2, 7.3 and 7.5.5 of Chap. 7– The Wiener-Khintchine theorem for random processes, Sec. 5.3.3 of Chap. 5.• The concept of coherence length or coherence time, a s developed in this chapter,will be used in Chaps. 8, 14, 15 and 22 of this book.• Interferometry as developed in this chapter, especially in Sec. 8.5, is a foundationfor t he discussion of gravitational-wave detection in Chap. 26.• Nothing else in this book relies substantially on this chapter.8.1 OverviewIn the last chapter, we considered superpositions of waves that pass through a (typicallylarge) aperture. The foundation for our analysis was the Helmholtz-Kirchoff expression forthe field at a chosen point P as a sum o f contributions from all points on a closed surfacesurrounding P. The spatially varying field pattern resulting from this superposition of manydifferent contributions was called diffra c tion .In this chapter, we continue our study of superposition, but for the more special casewhere only two or at most several discrete beams are being superposed. For this special12case one uses the term interference rather than diffraction. Interference is important in awide variety of practical instruments designed to measure or utilize the spatial and tempora lstructures of electromagnetic radiation. However interference is not just of practical impor-tance. Attempting to understand it forces us to devise ways of describing the radiation fieldthat are independent of the field’s or ig in and independent of the means by which it is probed;and such descriptions lead us naturally to the fundamental concept of coherence (Sec. 8.2).The light from a distant, monochromatic point source is effectively a plane wave; we callit “perfectly coherent” radiation. In fact, there are two different types of coherence present:lateral or spatial coherence (coherence in the angular structure of the radiation field), andtemporal or longitudinal coherence (coherence in the field’s temporal structure, which clearlymust imply something also about its frequency structure). We shall see in Sec. 8.2 that f orboth types o f coherence there is a measurable quantity, called the degree of coherence, thatis the Fourier transform of either the a ngular intensity distribution or the spectrum of theradiation.Interspersed with our development of t he theory of coherence are an a pplication to thestellar interferometer (Sec. 8.2.5 ) , by which Michelson measured the diameters of Jupiter’smoons and several bright stars using spatial coherence; and applications to a Michelsoninterferometer and its practical implementation in a Fourier-transform spectrom eter (Sec.8.2.7), which use temporal coherence to measure electromagnetic spectra, e.g. the spectrumof the cosmic microwave background radiation (CMB). After developing our full formalismfor coherence, we shall go on in Sec. 8.3 to apply it to the operation of radi o telescopes, whichfunction by measuring the spatial coherence of the radiation field.In Sec. 8.4 we shall turn to multiple beam interferometry, in which incident radiatio n issplit many times into several different paths and then recombined. A simple example is aFabry-Perot etalon made from two parallel, highly reflecting surfaces. A cavity resonator(e.g. in a laser), which traps ra diation for a large number of reflections, is essentially a largescale etalon. These principles find exciting application in laser interferometer gravitational-wave detectors, discussed in Sec. 8.5. In these devices, two very large etalons are used totrap laser radiation for a few tens of milliseconds, and the light beams emerging from thetwo etalons are then interfered with each other. Gravitational-wave-induced changes in thelengths of the etalons are monitored by observing t ime variations in the interference.Finally, in Sec. 8.6, we shall turn to the intensity interferometer, which although it hasnot proved especially powerful in application, does illustrate some quite subtle issues o fphysics …
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