Chapter 8InterferenceVersion 0208.1, 20 November 2002Please 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 shall see in Sec. 8.2 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.After developing a formalism to describe coherence, we shall go on in Sec. 8.3 to applyit to the operation of radio telescopes, which function by measuring the spatial coherenceof the radiation field. Temporal coherence is probed by a Michelson interferometer, or its12aaFFmaxFminFmax + 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.c) Interference fringes observed from an extended source.practical implementation, a Fourier transform spectroscope, which we discuss in Sec. 8.2.In Sec. 8.4 we shall turn to multiple beam interferometry, in which incident radiationis split many times into several different paths and then recombined. A simple example isa Fabry-Perot etalon, which is essentially two parallel, highly reflecting surfaces. A cavityresonator (e.g. in a laser), which traps radiation for a large number of reflections, is essen-tially a large scale etalon. These principles find exciting application in laser interferometergravitational-wave detectors, discussed in Sec. 8.5. In these devices, two very large etalons areused to trap laser radiation for a few tens of milliseconds, and the light beams emerging fromthe two etalons are then interfered with each other. Gravitational-wave-induced changes inthe lengths of the etalons are monitored by observing time variations in the interference.Finally, in Sec. 8.6, we shall 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 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 therethey 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. If the slits are very narrow, we can regard the transmission function3as two δ-functions, separated by the slit spacing a, and its Fourier transform will beψP(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, thenwe would find, by contrast with Eq. (8.2) that the actual number of fringes is finite, in factof order a/w.) This type of interferometry is sometimes known as interference by divisionof 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 demonstrated in a famous experimentperformed by G. I. Taylor in 1909. However, our concerns are with the classical limit whenmany photons are present simultaneously and their fields can be described by Maxwell’sequations. In the next subsection we shall depart from the usual quantum mechanicaltreatment by asking what happens to the fringes when the
View Full Document
Unlocking...