Contents9Interference 19.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2.1 Young’s Slits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2.2 Interference with an Extended Source: van Cittert-Zernike Theorem . 49.2.3 More General Formulation of Spatial Coherence; Lateral CoherenceLength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79.2.4 Generalization to two dimensions . . . . . . . . . . . . . . . . . . . . 89.2.5 Michelson Stellar Interferometer . . . . . . . . . . . . . . . . . . . . . 99.2.6 Temporal Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 109.2.7 Michelson Interferometer and Fourier-Transform Sp ectroscopy . . . . 119.2.8 Degree of Coherence; Relation to Theory of Random Processes . . . . 149.3 Radio Telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169.3.1 Two-Element Radio Interferometer . . . . . . . . . . . . . . . . . . . 179.3.2 Multiple Element Radio Interferometer . . . . . . . . . . . . . . . . . 189.3.3 Closure Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189.3.4 Angular Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199.4 Etalons and Fabry-Perot Interferometers . . . . . . . . . . . . . . . . . . . . 209.4.1 Multiple Beam Interferometry; Etalons . . . . . . . . . . . . . . . . . 209.4.2 Fabry-Perot Interferometer . . . . . . . . . . . . . . . . . . . . . . . . 259.4.3 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259.5T2 Laser Interferometer Gravitational Wave Detectors . . . . . . . . . . . 299.6T2 Intensit y Correlation and Photon Statistics. . . . . . . . . . . . . . . . 36iChapter 9InterferenceVersion 1009.1.K.pdf, 22 November 2008.Please send comments, suggestions, and errata via email to [email protected] or on paper toKip Thorne, 350-17 Caltech, Pasadena CA 91125Box 9.1Reader’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, as 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. 9.5, is a foundationfor the discussion of gravitational-wave detection in Chap. 26.• Nothing else in this book reliessubstantiallyonthischapter.9.1 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 of contributions from all points on a closed surfacesurrounding P.Thespatiallyvaryingfieldpatternresultingfromthissuperpositionofmanydifferent contributions was called diffraction.In this chapter, we continue our study of superposition, but for the more special casewhere only t wo 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 designedtomeasureorutilizethespatialandtemporalstructures 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 origin and independent of the means by which it is probed;and such descriptions lead us naturally to the fundamental concept of coherence (Sec. 9.2).The light from a distant, mono chromatic 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). W e shall see in Sec. 9.2 that forboth types of coherence there is a measurable quantity, called the degree of coherence,thatis the Fourier transform of either the angular intensity distribution or the spectrum of theradiation.Interspersed with our development of the theory o f coherence are a n application to thestellar interferometer (Sec. 9.2.5), 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 spectrometer (Sec.9.2.7), which use temporal coherence to measure electromagnetic sp ectra, e.g. the spectrumof the cosmic microwave background radiation(CMB).Afterdevelopingourfullformalismfor coherence, we shall go on in Sec. 9.3 to apply it to the op eration of radio telescopes,whichfunction by measuring the spatial coherence of the radiation field.In Sec. 9.4 we shall turn to multiple beam interferometry, in which incident radiation 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 radiation for a large number of reflections, is essentially a largescale etalon. These principles find exciting application in laser interferometer gravitational-wave detectors,discussedinSec.9.5. Inthesedevices,twoverylargeetalonsareusedtotrap laser radiation for a few tens of milliseconds, and the light beams emerging from thetwo etalons are then interfered with each other.Gravitational-wave-inducedchangesinthelengths of the etalons are monitored by observing time variations in the interference.Finally, in Sec. 9.6, we shall turn t o the intensity interferometer, which although it hasnot proved especially powerful in application, does illustrate some quite subtle issues ofphysics and, in particular, highlights the relationship between the classical and quantumtheories of light.9.2 Coherence9.2.1 Young’s SlitsThe most elementary example of interference is prov ided by Young’s slits. Suppose twolong, narrow, parallel slits are illuminated coherently by monochromatic light from a distantsource that lies on the perpendicular bisector of the line joining the slits (the optic axis),so an inciden t w avefron t reaches the slits …
View Full Document
Unlocking...