Chapter 8InterferenceVersion 0408.1.K.pdf, 17 November 2004Please send comments, suggestions, and errata via email to [email protected] and [email protected], or on paper to Kip Thorne, 130-33 Caltech, Pasadena CA 911258.1 OverviewIn the last chapter, we considered superpositions of waves that pass through a (typicallylarge) aperture. The foundation for our analysis was an expression for the field at a chosenpoint P as a sum of contributions from all points on a closed surface surrounding P. The spa-tially varying field pattern resulting from this superposition of many different contributionswas called diffraction.In this chapter, we continue our discussion of the effects of superposition, but for themore special case where only two or at most several discrete beams are being superposed.For this special case one uses the term interference rather than diffraction. Interferenceis important in a wide variety of practical instruments designed to measure or utilize thespatial and temporal structures of electromagnetic radiation. However interference is not justof practical importance. Attempting to understand it forces us to devise ways of describingthe radiation field that are independent of the field’s origin and independent of the meansby which it is probed; and such descriptions lead us naturally to the 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 see in Sec. 8.2 that for bothtypes of coherence there is a measurable quantity, called the degree of coherence, that isthe Fourier transform of either the angular intensity distribution or the spectrum of theradiation.Interspersed with our development of the theory of coherence are applications to thestellar interferometer by which Michelson measured the diameters of Jupiter’s moons andseveral bright stars using spatial coherence (Sec. 8.2.5), and a Michelson interferometer12aaFFmaxFminFmax + Fminarg ( )(c)(b)(a)Fmax Fmin=θθθ−FθFig. 8.1: a) Young’s Slits. b) Interference fringes observed from a point source on the optic axis[Eq. (8.2)]. c) Interference fringes observed from an extended source.(and its practical implementation in a Fourier Transform Spectroscope) that uses temporalcoherence to measure electromagnetic spectra, e.g. the spectrum of the cosmic microwavebackground (CMB); Sec. 8.2.7. After developing our full formalism for coherence, we go onin Sec. 8.3 to apply it to the operation of radio telescopes, which function by measuring thespatial coherence of the radiation field.In Sec. 8.4 we turn to multiple beam interferometry, in which incident radiation is splitmany times into several different paths and then recombined. A simple example is a Fabry-Perot etalon made from two parallel, highly reflecting surfaces. A cavity resonator (e.g. ina laser), which traps radiation for a large number of reflections, is essentially a large scaleetalon. These principles find exciting application in laser interferometer gravitational-wavedetectors, discussed in Sec. 8.5. In these devices, two very large etalons are used to trap laserradiation for a few tens of milliseconds, and the light beams emerging from the two etalonsare then interfered with each other. Gravitational-wave-induced changes in the lengths ofthe etalons are monitored by observing time variations in the interference.Finally, in Sec. 8.6, we turn to the intensity interferometer, which although it has notproved especially powerful in application, does illustrate some quite subtle issues of physicsand, in particular, highlights the relationship between the classical and quantum theories oflight.8.2 Coherence.8.2.1 Young’s SlitsThe most elementary example of interference is provided 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 that an incident wavefront reaches the slits simultaneously [Fig. 8.1(a)]. This situationcan be regarded as having only one lateral dimension. The waves from the slits (effectively,two one-dimensional beams) fall onto a screen in the distant, Fraunhofer region, and there3they interfere. The Fraunhofer interference pattern observed at a point P, whose position isspecified using polar coordinates r, θ, is proportional to the spatial Fourier transform of thetransmission function [Eq. (7.11)]. If the slits are very narrow, we can regard the transmissionfunction as two δ-functions, separated by the slit spacing a, and its Fourier transform willbeψP(θ) ∝ e−ikaθ/2+ eikaθ/2∝ coskaθ2. (8.1)(That we can sum the wave fields from the two slits in this manner is a direct consequenceof the linearity of the underlying wave equation.) The energy flux (energy per unit timecrossing a unit area) at P will beFP(θ) ∝ |ψ|2c ∝ cos2(kaθ/2); (8.2)cf. Fig. 8.1(b). The alternating regions of dark and bright illumination in this flux distribu-tion are known as interference fringes. Notice that the flux falls to zero between the brightfringes. This will be so even if (as is always the case in practice) the field is very slightlynon-monochromatic, i.e. even if the field hitting the slits has the form ei[ωot+δφ(t)], whereωo= c/k and δφ(t) is a phase that varies randomly on a timescale extremely long comparedto 1/ωo.1Notice also that there are many fringes, symmetrically disposed with respect tothe optic axis. [If we were to take account of the finite width w a of the two slits, then wewould find, by contrast with Eq. (8.2) that the actual number of fringes is finite, in fact oforder a/w; cf. Fig. 7.4 and associated discussion.] This type of interferometry is sometimesknown as interference by division of the wave front.This Young’s slits experiment is, of course, familiar from quantum mechanics where itis often used as a striking example of the non-particulate behavior of electrons.2Just asfor electrons, so also for photons, it is possible to produce interference fringes even if onlyone photon is in the apparatus at any time, as was
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