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Quantum optics and multiple scattering in dielectrics

Wubs, M.

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Citation for published version (APA):Wubs, M. (2003). Quantum optics and multiple scattering in dielectrics. Enschede: PrintPartners Ipskamp B.V.

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Download date: 23 Jan 2020

Chapterr 1

Introduction :: quantum optics of photonicc media

1.11 Light in free space and in dielectrics

Wee can see things because these things emit or scatter light that is subsequently absorbed in ourr eyes. Emission, absorption and scattering of light are examples of interactions of light withh matter. According to classical optics, light is a name for those electromagnetic waves thatt happen to be visible for the human eye. Electromagnetic waves can be described ass the solutions of Maxwell's equations. In free space, solutions of Maxwell's equations correspondcorrespond to plane waves that move in straight lines. In many experiments, especially thosee involving interference, light appears to be a wave indeed. In other experiments, light cann best be understood as a stream of particles (photons). This two-sided nature of light cann nowadays be described consistently in one theory, namely in the quantum theory of lightt (or quantum electrodynamics). The subject of this thesis is the quantum theory of lightt in media other than free space. The rest of this section describes some properties of opticall media that one should wish to include in such a theory.

Whenn the field interacts with matter, the properties of the electromagnetic field are changedd [1]. In air, for instance, light travels just a bit slower than in free space, because thee interactions with the air molecules take some time. The factor 1.0003 by which the lightt is slower is the refractive index of air. If one looks even more precisely, one finds that airr shows frequency dispersion: different colors move at different speeds. Violet light has a refractivee index of 1.000298 and red light is faster with a refractive index of 1.000291 [2].

Anotherr process that occurs in air but not in free space is light scattering. Air molecules andd small dust particles scatter red light with long wavelengths less effectively out of sight thann blue and violet light with shorter wavelengths. This explains why the sun appears red att sunset, when light waves travel a long way through air before reaching our eyes. Light scatteringg by air molecules is also the mechanism that explains why at daytime the sky is blue,, because it is scattered light that we see. Light with shorter wavelengths is scattered moree easily by air molecules towards our eyes.

11 1

12 2 Introduction:: quantum optics of photonic media

Lightt can propagate forever in free space without disappearing, but in a medium it is possiblee that light gets absorbed. Absorption means that light becomes something else than light,, heat for example. Light absorption is more important in fluid or solid media than inn gases: sun glasses are better light absorbers than air. Sometimes the word absorption iss used for scattering, but for this thesis it is important to distinguish the two processes. Bothh scattering and absorption can be the cause of extinction, which is loss of light in the originall direction.

Inn general, a medium can either be described microscopically, in terms of the indi-viduall atoms or molecules, or macroscopically, with the use of a dielectric function (the squaree of the refractive index). In this thesis, the medium is described macroscopically. Withh "macroscopic" it is meant that the dielectric function is assumed constant on the atomicc scale but it it is allowed to vary strongly on the wavelength scale of light, as is thee case in photonic media. The medium forms a modified optical environment for "guest atoms"" that do not belong to the medium. Only these atoms are described microscopi-cally.. Guest atoms atoms will interact with light and they wil l probe the modified optical propertiess [1].

Dispersion,, absorption and scattering of light in a medium can be described theore-ticallyy by adding terms in the Maxwell equations. These extra terms model the pieces off matter from which the medium is built up. The modified Maxwell equations have differentt light waves as their solutions. Not only a light wave is different, but also a photonn in a medium is different from a photon in free space. In this thesis it is studied in particularr how the free-space quantum theory of light can be modified in order to describe photonss in inhomogeneous media (including photonic media) or in media that show light extinction.. Furthermore, calculations are presented that show strong modifications of the opticall properties of media in which light strongly scatters off many scatterers.

1.22 Classical and quantum optics

Inn the past centuries scientist have debated fiercely whether light is a stream of particles or aa wave. Newton developed a corpuscular theory of light (1704) while Huygens worked on hiss wave theory (1690). The phenomenon that two light beams can interfere destructively andd constructively, can best be understood if light is a wave. Therefore, the light-as-a-wave picturee had the best cards for a long time after Fresnel's work on diffraction (1816) and its experimentall verification [2]. Around 1900, it had become even almost impossible to think off light in terms of particles, because of the enormous success of the classical "Maxwell" wavee theory of light [3]. And even today, classical electrodynamics suffices to explain the phenomenaa in many branches of optics [4].

However,, observations of the photo-electric effect defied explanation around 1900: lightt can induce an electrical current in some metals, but only if the frequency of the light exceedss a critical value. In 1905, Einstein gave a new argument for the light-as-a-particle picturee by his explanation of this effect [5]: the electrons interact with particles of light calledd photons. The energy Eu of a photon equals faü, where h is Plank's constant divided byy 2ir. Only a photon with energy high enough to overcome the binding potential of an electron,, can kick it out of its bound state and make it detectable as an electrical current.

1.33 Elements of quantum optics in dielectrics 13 3

Forr this revolutionary explanation of the photo-electric effect Einstein received the Nobel Prizee of 1921 [3].

Present-dayy quantum optics describes how light can have the interference properties off waves as described by Maxwell's equations, even so for light beams containing only one photon.photon. The first experimental proof of the particle-nature of light emitted by individual atomss came relatively recently, in 1986 [6]: light was measured that was spontaneously emittedd by a gas of atoms. It was found that light emitted by a single atom was detected in onlyy one of two detectors, with equal probability. If the emitted light had been a classical wave,, then one would sometimes have measured light in both detectors at the same time. Thee measurements proved that each single atom emitted only a single particle.

Nott only the existence of photons but also the fact that they can be entangled is an importantt difference between classical and quantum optics. Entanglement surely is one of thee mind-boggling aspects of quantum physics. An entangled system of two parts can not bee treated as two separate objects with well-defined individual properties that already exist "outt there" prior to measurement. A measurement on one part of the entangled system changess the state of the whole system, including the other part. Entanglement is subtle: neitherr the entanglement nor the measurement is some kind of interaction between the two parts.. This last statement is supported by the fact that the entanglement is not influenced byy the distance between the parts. Furthermore, a measurement of only one part of an entangledd system never tells you that it indeed was entangled with something else; only correlationn measurements of both parts tell you so.

Theree is a by now standard technique to entangle photons. It is based on the process off parametric down-conversion in nonlinear crystals, which gives pairs of photons with entangledd momenta [7]. The electromagnetic field can also be entangled with an atom. Impressivee examples can be found in cavity quantum electrodynamics, where the final statee of an atom that flies through a micro-cavity depends noticeably on the presence or absencee of a single photon in the cavity [8].

Manyy fundamental tests of quantum mechanics have been performed in the optical domain.. For example, the famous Einstein-Podolsky-Rosen Gedanken experiments have beenn realized with pairs of photons with entangled polarisation directions. The measure-mentss proved that quantum mechanical entanglement indeed exists [9].

Quantumm mechanics is nowadays reconsidered and further developed with the new perspectivee called "quantum information theory" [7], which encompasses research in quan-tumm computation, quantum communication and quantum cryptography. In all these areas, thee idea is to put quantum entanglement to use, either for a fast and new way of computing orr for inherently safe information transfer. This also drives the field of quantum optics, becausee photons would be suitable information carriers. Even without these (future) ap-plicationss in mind, it is interesting to study the quantum theory of light in media other than freee space.

1.33 Elements of quantum optics in dielectrics

Ann inhomogeneous dielectric where dispersion and absorption can be neglected, can be describedd by a position-dependent dielectric function e(r). The harmonic solutions of

14 4 Introduction:: quantum optics of photonic media

thee Maxwell equations are called optical mode functions. The mode functions f A of the dielectricc are solutions of the wave equation

- VV x V x fA(r) + e(r)(ux/c)2fx(r) = 0. (1.1)

Everyy mode is assumed to have a different label A. Light in a certain mode will forever stay inn that mode, unless light interacts with matter not accounted for in the dielectric function. Thee latter more interesting situation presents itself when guest atoms are introduced in thee dielectric that can scatter light out of one mode into another. The free-space mode functionss are transverse plane waves (proportional to exp(ik r)), which are the solutions off equation (1.1) in the special case that e equals unity everywhere. These plane waves cann be characterized by a wavelength, a direction, and a transverse polarization direction.

Inn classical optics, the amplitude and phase of the electric field are determined by a sett of complex numbers, one for each mode. These numbers are the mode amplitudes. If theree is no light, all mode amplitudes are zero; a larger absolute value of a particular mode amplitudee means that there is more light in that mode.

Inn ordinary quantum mechanics, the position and momentum of a particle become op-eratorss that do not commute. These and other operators work on the quantum mechanical statee vectors of the particles. In quantum optics, the electric and magnetic fields also be-comee non-commuting operators. Again, there is a state vector and it describes the quantum statee of light. The electric-field operator has the form [10-14]

E ( r ' É ) = * E V l ^^ h e - ^ ' f A ( r) - 4 e ^ ' f j ; ( r ) (1.2) )

Thiss mode expansion of the electric-field operator is presented already in this introductory chapter,, in order to show how quantum optics describes light both in terms of waves and off particles. The optical modes in quantum optics are the same harmonic solutions f̂ off the Maxwell equations as in classical optics. The difference with classical optics is thatt mode amplitudes have now been replaced by the creation and annihilation operators a\a\ and ax in Eq. (1.2). The energy operator of a particular mode is hiü\(axa\ + 1/2). Thee spectrum corresponding to a particular mode A is a harmonic-oscillator spectrum with energyy separation Fvjj\\ the ground (or vacuum) state |0) of the electromagnetic field is the statee with no photons present in any mode; the j t n excited state of mode A corresponds to jj photons in that mode. Photons in a dielectric are different from free space because they aree the elementary excitations of different waves than in free space.

Thee terms in Eq. (1.2) featuring annihilation operators taken together are called the positive-frequencyy part E^+^ of the electric field; the creation-operator part is called E^-). Inn terms of these two quantities, the light intensity operator is given by [15]

J(r,, t) = 2e0c E(->(r, t) E(+>(r, *). (1.3)

AA measured light intensity (in [J.m~2 .s~1]) corresponds to the expectation value of this operatorr with respect to the quantum state of light. The detector position r need not be inn the far field. It can be near or even inside the inhomogeneous dielectric (at a position wheree e equals unity).

1.44 Spontaneous emission 15 5

1.44 Spontaneous emission Spontaneouss emission is responsible for most of the light around us. Excited atoms will falll back to a lower energy level by emitting one or more photons. The term "sponta-neouss emission" was invented for this process when only the inadequate classical electro-magneticc theory was around. In classical optics, if no light is present near the atom, the electricc field is zero. An absent electromagnetic field can not be the external cause of the emission,, which explains the word "spontaneous". Einstein derived spontaneous-emission ratess from classical theory by using energy-balance arguments, but a full understanding of spontaneouss emission requires quantum optics.

Inn quantum optics, atoms are surrounded by an omnipresent electromagnetic field. Thee field is always there, even when it is in its lowest energy state. In this vacuum state, noo photons are present and no light can be detected. However, the electromagnetic field fluctuatess even when in the vacuum state. Similar ground-state fluctuations occur for the positionn and momentum of an electron in a hydrogen atom: both the electromagnetic field operatorss and the atomic operators obey their respective Heisenberg uncertainly relations. Ann excited atom feels the fluctuations of the ground-state electromagnetic field. These fluctuationss cause the atom to decay "spontaneously" to a lower state. In a semiclassical picture,, exactly half of the spontaneous-decay rate of an atom can be attributed to these fieldd fluctuations; the other half of the decay rate is caused by the electric field that the atomm exerts on itself, the so-called radiation reaction [16].

Thee process of spontaneous emission starts with an atom in an excited state and the fieldd in the vacuum state. After the emission, the atom is in a lower-energy state and thee field can be in any one-photon state that gives the same total energy before and after thee emission. Usually, spontaneous-emission rates T or inverse decay times r _ 1 can be calculatedd with Fermi's golden rule:

rr = ~ = (%) E K/I^AFN)! 2 S(Ui - UJj). (1.4) VV / ƒ

Thee initial state \i) and the final states | ƒ) are states of both the atom and the field. The deltaa function in the rule (1.4) ensures energy conservation. If one assumes the electric-dipolee interaction between the atom and the field, one finds the free-space decay rate T00 = n2Q3/(3irfreQC3), where Q is the relevant atomic transition frequency and /i is the magnitudee of the atomic transition dipole. In an inhomogeneous medium, the decay-rate inn terms of the optical modes becomes

rr = - r ^ V > - f A ( r ) | 2 < 5 K - f ) ) . (1.5)

Thiss formula is presented here to show that the spontaneous-decay rate in inhomogeneous dielectricss depends not only on the atomic transition frequency, but also on its dipole ori-entationn and position in the medium. When averaged over all possible orientations of the atomicc dipole moment, the position-dependent spontaneous-emission rate (1.5) becomes proportionall to the local optical density of states (LDOS) at frequency Ct. The LDOS is a classicall quantity and it depends on the properties of the medium only.

16 6 Introduction:: quantum optics of photonic media

1.55 Photonic media and photonic band gaps Somee media are designed to modify light propagation very effectively without absorbing them.. These are the so-called photonic media. In order to make them, in the first place thee right building blocks are needed, which must scatter light well. Secondly, the typical distancee between neighboring scatterers must be of the order of the wavelength of light. Opticall modes in a photonic medium differ very much from the free-space plane waves. Thee waves can only be calculated by taking the multiple scattering of light into account.

Fromm quantum mechanics it is known that electronic band gaps exist in semiconduc-torss or metals, because electrons Bragg-reflect off a periodic potential formed by the ions. Itt was proposed in 1987 to fabricate the optical analogy: a photonic band gap should exist inn some thee-dimensional periodic dielectric structures with a periodicity on the wave-lengthh scale of light. These structures are also known as photonic crystals [17,18].

Itt would be very interesting to have a photonic crystal with a photonic band gap: light off frequency inside the band gap will not be able to traverse the crystal in any direction andd wil l be reflected. Even more interestingly, spontaneous emission by an atom inside the crystall would be completely suppressed. The inhibition of spontaneous emission can be illustratedd with Eq. (1.5): if modes A with eigenfrequencies UJ\ equal to the atomic transi-tionn frequency J7 do not exist, then the spontaneous-emission rate T is zero. Inhibition of spontaneouss emission has been observed before in a medium other than a photonic crystal, namelyy in a cavity small compared to the optical wavelength [19]. There is no such size limitationn for photonic crystals.

Inn model calculations of light emission in band-gap materials, many more unusual quantumm optical phenomena showed up. To name a few phenomena (without explaining themm here), there are predictions of anomalous Lamb shifts, localization of superradiance, oscillatoryy spontaneous emission rates, and fractionally populated excited states in the steadyy state [20-28]. For some of the phenomena, the frequency of light has to be inside thee band gap, while for others it has to be just on one of the band-gap edges. Predictions off band-edge effects strongly depend on the shape and analytic behavior of the assumed opticall density of states as a function of frequency [29,30]. A photonic band-gap crystal wouldd be a fascinating new playground for quantum electrodynamics.

Thee material requirements for obtaining a band gap for light are demanding. Usually, photonicc crystals are two-component dielectrics with refractive indices ni and n^. To beginn with, the material must be strongly photonic, meaning that the refractive index must varyy on the wavelength scale of light and the refractive-index contrast ni/ri2 must be high. Furthermore,, the volume fraction of the high-index material should be small [24,31] and thee crystals preferably should have many unit cells in all three directions. Light should not getget lost by light absorption, if only because a band gap can not be defined in that case [32], AA state-of-the-art fabrication technique of three-dimensional photonic crystals is based onn self-organization of colloidal spheres. Figure 1.1 shows a so-called inverted opal or air-spheree crystal [33,34] of titania and air. Already a fivefold reduction of spontaneous-emissionn rates has recently been reported [35] for crystals as in figure 1.1. The search for aa full photonic band gap continues until today [35-37].

Photonicc crystals with refractive-index variations in only one or two dimensions do nott exhibit full photonic band gaps, but they also strongly modify the propagation of

1.66 Multiple light scattering 17 7

Figuree 1.1: Scanning-electronn microscopy picture of the (lll)-surfacee of a three-dimensional fee air-spheree crystal of TiCb (anatase) and air. The widthh of the picture corresponds to 10 pm. Thee TiÜ2/air refractive index contrast is (2.77 0.4). The air-spheres overlap each otherr partly. The black spots in the incom-pletee air spheres at the surface are the "doors" off these spheres to the complete air spheres lyingg one level deeper. (Picture kindly pro-videdd by L. Bechger.)

light.. One-dimensional photonic crystals are also known as Bragg mirrors. Spontaneous-emissionn rates inside Bragg mirrors can depend strongly on the atomic position in a unit cell.. Two-dimensional photonic crystals have periodic refractive-index variations in a plane.. Light propagation in the in-plane directions can be fully inhibited so that light iss forced to propagate in the third dimension only. This is how photonic-crystal fibres workk [38].

Photonicc crystals are studied both for fundamental reasons as a new playground for quantumm optics and for possible applications in optical communication and opto-electro-nicss [39]. In both cases, it is important to be able to calculate the optical properties of the realizedd or imagined structures.

1.66 Multiple light scattering

Lightt propagation in photonic media is in the multiple-scattering regime, and in the follow-ingg it will be explained what this means. If a plane wave of light scatters off two particles, ass depicted in figure 1.2, then according to Huygens' principle each particle acts as a sec-ondaryy source of light and sends out waves, which for simplicity were drawn spherical in thee figure. If the particles do not scatter strongly or if they are far enough apart, then the totall scattered wave is just the sum of the two waves emitted by the individual particles. Thiss is called the single-particle scattering regime. Light scattering by air is in this regime. Onn the other hand, if light is scattered well by the individual scatterers and if they are close too each other, then the light scattered by the one becomes a non-negligible source of light forr the other, and vice versa. Indeed, light can go back and forth many times between the twoo scatterers. The total scattered wave is no longer simply the sum of the two waves that thee particles would have emitted if they had been alone. This is the multiple-scattering regime.. Light scattering in photonic media belongs to this regime.

Thee (multiple) scattering properties of a medium can be inferred from the Green func-tionn G(r, r', t) of that medium: given a light pulse Ep(r , to) at some initial time to before thee pulse has scattered at all, the light pulse (including scattered light) at a later time at a

18 8 Introduction:: quantum optics of photonic media

Figuree 1.2: AA plane wave (denoted by parallel lines) scatteredd off two particles, which sub-sequentlyy act as secondary sources of light.. In the single-particle scattering regime,, the total scattered wave is the summ of the waves scattered by individ-uall particles alone. In the multiple-scatteringg regime, it is important that the particless scatter the waves scattered by thee other (and so on). Then the two par-ticless must be considered as one more complicatedd composite scatterer.

positionn r is given by

E p ( r , 0== / d r ' G ( r , r ' , t - f o ) - Ep ( r ' , i o ) - (1.6)

Thiss equation shows how the Green function propagates the initial field to a scattered field elsewheree in space and later in time. All information of light propagation in the medium mustt be included in G for this relation to hold. Because of the property (1.6), one can say thatt the Green function is the solution of the scattering problem.

Greenn functions and optical mode functions f A can be calculated with multiple-scatte-ringg theory. Starting with the Green function of a known medium, free space for example, ass well as with the properties of the extra particles that scatter light, mode functions and thee Green function of the total system can be calculated. Usually, it is easier to calculate thee frequency-dependent Green function rather than the time-dependent one. The two are relatedd by a Fourier transformation. However, some interesting properties of a multiply-scatteringg medium can be found from the frequency-dependent Green function, so that the oftenn difficult inverse Fourier transformation is not needed. For example, the local density off states (LDOS), introduced in section 1.4, can be directly determined from the imaginary partt of the Green function G(r, r, w).

Thee optical modes of an infinite photonic crystal are the solutions f\ of the wave equationn (1.1), where the dielectric function e(r) has discrete time-translation symmetry inn all three directions. Because of the symmetry of the problem, the optical modes are Blochh modes which can be labelled by an incoming wave vector k and a band index n. Thiss is very much analogous to the more familiar electronic band-structure problem. The opticall problem is less straightforward because the mode functions are vector functions andd because the optical periodic potential is frequency-dependent. Methods for calculating bandd structures were recently reviewed in [40].

Ass was discussed in section 1.4, orientational averages of spontaneous-emission rates aree proportional to the local optical density of states. The LDOS inside a photonic crystal

1.77 True modes of a beam splitter 19 9

cann not be inferred given its band structure, except for frequencies inside a band gap: inn that case the LDOS simply vanishes everywhere in the unit cell. In other situations, calculationss of the LDOS are numerically involved and are therefore restricted to specific positionss in the unit cell [41,42].

Reall photonic crystals have finite sizes and experiments always involve crystal bound-aries.. The calculation of optical modes and other properties of finite crystals requires dif-ferentt methods. The reduced symmetry makes it natural to work in real space rather than inn reciprocal space. Usually, the dielectric environment is discretized and Maxwell's equa-tionss are solved with finite-difference frequency-domain (FDFD) or time-domain (FDTD) techniquess [40]. In some special cases light propagation can be analyzed further, for ex-amplee for photonic crystals consisting of a finite number of parallel cylinders [43,44].

1.77 Tru e modes of a beam splitter Thee equation (1.5) shows the position-dependent spontaneous-emission rate in terms of thee true modes of a medium. In section 1.6 it was stated that these true modes can be calculatedd with multiple-scattering theory. In the following, the true modes of a relatively simplee scatterer, a beam splitter, wil l be calculated. This example introduces the so-called T-matrixx formalism of multiple-scattering theory. It is a problem in classical scattering theoryy to find the relation between the true modes and the free-space modes. However, scatteringg theory will also give relations between the operators of the true modes and the free-spacee operators [11].

Thee textbook problem about a beam splitter is how to relate light in the output chan-nelss to the input-ports [15,45]. Input-output formalisms have been set up for more com-plicatedd quantum scattering situations as well [46,47] and these formalisms are based on formall quantum scattering theory, such as given in [48,49]. Input-output formalisms are usefull for describing non-stationary situations. The typical example is a pulse of light that enters,, traverses and leaves a medium. On the other hand, the true-mode formalism is usefull for calculating stationary properties such as spontaneous-emission rates inside the medium. .

Thee present discussion of the true modes of a beam splitter starts with an evenn sim-plerr situation, namely with free space. Consider light propagation in one dimension only. Assumee further that the polarization direction of the light is fixed, so that light can be de-scribedd as a scalar wave. Then light propagates in straight lines forever, either to the left or too the right, as sketched in figure 1.3(a). The optical modes in free space are plane waves, becausee plane waves are the solutions of the wave equation for light in free space:

[d2/dx22 + (uk/c)2] e = 0. (1.7)

Thee plane wave elkx corresponds to light propagating to the right, with wave vector k and frequencyy u^; light in mode e~lkx propagates to the left. The one-dimensional analogy forr the electric-field operator (1.2) is

E(x,t)E(x,t) = ij^k J 2^T0^ieikX + ak>2e~ikX] e~lUJkt ~ H x - } - (L8)

200 Introduction: quantum optics of photonic media

(a) ) (b) )

—— 1 * - 22 4 - —— 2

(c) ) (d) )

„ „

Figuree 1.3: Sketchess of optical modes in one dimension. In figure (a), the arrow to the right denotes the plane-wavee mode exp(ifca;) and the other arrow to the mode exp(—ikx). In figure (b), each mode of figurefigure (a) is split into two new modes, corresponding to light propagating to the left or to the right off the imaginary cut at position xo- There are two input channels (1 and 2) and two output channels (33 and 4). In figure (c) and (d), a beam splitter is placed at position xo that reflects part of the light.. Figure (c) shows the "true" mode T/>I . which comes in from the left and is partially reflected andd partially transmitted. Similarly, the true mode fa represented in figure (d) corresponds to light comingg from the right.

Forr later use, the free-space Green function go will be introduced as the solution of the samee wave equation (1.7), but with a delta-function source at the right-hand side of the equation: :

[d2/dx22 + (cok/c)2] g0(x,x',Lu) = 5{x - x% (1.9)

wheree the right-hand side is a Dirac delta function. It can be checked that go{x, x', u) is equall to ex.p(iu>+\x — x'\/c)/[2i(Lu/c)], where u+ equals (w + irj) and rj is infinitesimally smalll and positive.

Supposee now mat in free space one would like to measure light in a plane wave-mode, eitiierr at a position to the left of some arbitrary position xo, or to the right of it. Suppose thatt me detectors are sensitive to die direction in which the photon is heading. The situation iss sketched in figure 1.3(b): mere are four modes corresponding to light moving to the left orr right, at the left or right of x0. The first mode function is a plane wave moving to the right,, proportional to exp(ifcx), when x is smaller than XQ and it is zero for larger x. This modee function can be summarized as 9(XQ — x)\/2exp(ikx); the factor y/2 gives the new modee function the same norm as the mode exp(ikx) on the whole real axis. The other threee mode functions can be written down analogously. The description of light in terms off the four half-space modes is equivalent to me two full-space plane-wave modes that

1.77 True modes of a beam splitter 21 1

wee started with. In the electric-field operator (1.8), the only thing that changes in the new representationn is the quantity between square brackets, which now becomes

y/2y/2 [d(x0 - x)(bk>1eikx + bkAe~ikx) + 9(x - x0)(bkae-ikx + bk,3e

ikx)] . (1.10)

Noww add a beam splitter at position XQ. The presence of the beam splitter can be modelledd as a delta-function potential of strength V in the wave equation for the optical modes.. The wave equation changes from Eq. (1.7) into

{d 2/da;22 + [(uk/c)2 - V6(x - x0)] } l M * 0 = °- C1-11)

Thee harmonic solutions of this equation no longer are plane waves, but the unknown mode functionss -0k, I and ipkt2, corresponding to light coming from the left or right, respectively. Ass an Ansatz, suppose that the unknown modes have the form

Mx)Mx) = ékx+gQ{x,xQ,u)T{uj)ékx°. (1.12a)

lfe',2(* )) = e~ik'x + g0(x,x0,u)')T(u>')e-ik'x°1 (1.12b)

wheree the Green function (1.9) shows up. (The reason why the unknown modes are sup-posedd to have this form will be given in chapter 2.) The new terms at the right-hand sides cann be read from right to left: at position XQ, the plane wave scatters as described by the unknownn quantity T{UJ). Finally, the free-space Green function describes how scattered lightt propagates away from the beam splitter. The unknown quantity T(UJ) is called the T-matrixx of the beam splitter. With the use of Eqs. (1.7), (1.11) and the definition (1.9) off the free-space Green function, one can check that the Ansatz (1.12a) indeed gives two exactt solutions of the wave equation (1.11), if

T(u>)T(u>) = V/[l-go(x0lzo,u>)V]. (1.13)

Thee T-matrix contains powers of the potential V up to infinite order, as one can see by expandingg (1 — goV)'1 into an infinite series. The T-matrix can thus be seen as an exact summationn of an infinitely long perturbation expansion in terms of the potential V. The modee function ip\ (x) corresponds to a plane wave coming from the left that is partially reflectedd and partially transmitted, as in figure 1.3(c). Figure 1.3(d) depicts the mode functionn ip2 that corresponds to a plane wave coming from the right.

Thee free-space plane-wave modes are independent, in the sense that their inner prod-uctuct ƒ dx[exp(ikx)]* exp(ik'x) is zero for k unequal to k'. Physically, this means that in freee space light in one mode wil l never jump to another plane-wave mode. From now on assumee that x0 = 0. Now consider the inner product of the two true modes ip\ and ip2 of thee beam splitter:

/

oo o

dxipli{x)ipdxipli{x)ipkk>A>Axx)) ( U 4 ) o o

== 27r6{k - k') [l + Re(W) + \W\2] + 2>nö(k + k')Rt{W),

wheree W is an abbreviation for T/[2i(u>/c)]. The wave vectors k and k' are both positive, soo that the term proportional to6(k + k') does not contribute to the inner product. Then it

22 2 Introduction:: quantum optics of photonic media

followss that the inner product (1.14) of the true modes is equal to the inner product of the free-spacee plane wave modes if [Re(W) + \W\2] is equal to zero, in other words if

ImrM=-M.. (1.15) 2(io/c) 2(io/c)

Thiss nonlinear relation of the T-matrix is called the optical theorem of the beam splitter. Thee theorem only holds if the potential V representing the beam splitter is a real quantity. Thiss statement can be checked by inserting the form (1.13) of the T-matrix into the optical theoremm (1.15). A potential V with an imaginary part would model loss or gain of light in thee beam splitter. Thus, the true modes are orthonormal if the beam splitter scatters light elastically. .

InIn the presence of the beam splitter, the electric-field operator (1.8) can be expanded inn terms of the true modes instead. All that changes in Eq. (1.8) is the term between the squaree brackets, which becomes

[ck,iipk,i{x)[ck,iipk,i{x) + cfci2^k,2(aO]» (1-16)

wheree ck,i and cki2 are annihilation operators corresponding to the true modes. For each modee separately, the commutation relations of the creation and annihilation operators are justt like in the standard quantum mechanical treatment of the harmonic oscillator: [c, c*] = 1;; operators that belong to different modes commute.

Onee could measure light to the left or to the right of XQ = 0, and heading towards or awayy from the beam splitter. The electric-field operators (1.16) in terms of the true modes andd (1.10) in terms of the half-space modes should therefore be equivalent. By taking the innerr products with ^k'\ and tpk'2 of the assumed equality, one finds that the equivalence onlyy holds if

c*,ii = [bk,i + (l+W*)bkt3 + W*bkt4]/y/2, (1.17a)

Cfc,22 = [bk,2 + Wbki3 + {l + W)bkt4]/y/2. (1.17b)

Onee can check that these relations between operators in different mode expansions can onlyy be consistent with the standard commutation relations of both the operators c and b inn each expansion, if the optical theorem (1.15) holds.

Thee example of the beam splitter served to show that an optical theorem holds if lightt is scattered elastically; that the theorem guarantees that true modes have the same orthonormalityy relations as the original modes without the scatterers. Finally, the impor-tancee of the optical theorem in quantum optics is that it ensures that mode operators of true modess can consistently be written in terms of the original operators, as shown above.

Iff the optical theorem does not hold, in the case of absorption or gain in the beam splitter,, then the quantum optical formalism has to be modified. The Kramers-Kronig re-lationss tell that in general, the dielectric function is a complex quantity as a function of frequencyy [50,51]. So one can always find frequencies for which a beam splitter absorbs aa fraction of the light. Quantum optical descriptions of light propagation in absorbing dielectricss rely on the so-called fluctuation-dissipation theorem: wherever there is absorp-tion,, there are also sources of quantum noise. These noise sources show up in opera-torr relations like (1.17a) and (1.17b) so as to keep the relations consistent when there is losss [52].

1.88 Outlook into this thesis 23 3

1.88 Outlook into this thesis Inn the chapters 2 and 3 of this thesis, it is shown how a layered dielectric (or dielectric mirror)) can be modelled as a crystal of infinitely thin planes. Multiple-scattering theory iss used to calculate the propagating and guided modes of this finite one-dimensional pho-tonicc crystal. The formalism allows a relatively easy calculation of the Green function off such a structure. It is studied how the spontaneous-emission rate of a radiating atom dependss on the atomic position and dipole orientation. Chapter 2 deals with scalar waves. Thee T matrix formalism of multiple-scattering theory is introduced. In chapter 3 the for-malismm is generalized to vector waves in order to study the influence of dipole orientation onn spontaneous-emission rates. The free-space Green function must be modified ("regu-larized")) in order to set up the T-matrix formalism for vector waves.

Thee subject of chapter 4 is the quantum optical description of light in inhomogeneous dielectrics,, and the interaction of guest atoms with light. The chapter is quite formal and thee dielectric function e(r) is left arbitrary. Material dispersion and loss of the dielectric aree neglected. Models of finite or infinite photonic crystals with a periodic variation of the refractivee index n(r) belong to this class. Starting from a minimal-coupling Lagrangian, aa Hamiltonian is derived with multipolar interaction between light and the guest atoms. Speciall attention is paid to the derivation of Maxwell's equations after choosing a suitable gaugee in which all (static and retarded) interactions between atoms are mediated by the electromagneticc field. In the dipole approximation, it is found that a dipole couples to a fieldfield that at first sight is neither the electric nor the displacement field. Results of this chap-terr justify the calculations of spontaneous-emission rates of chapters 2 and 3. Furthermore, thee results obtained here are the starting point for chapter 5.

Single-atomm decay rates change in the presence of a dielectric, but also multi-atom processess such as superradiance will be modified. This is the subject of chapter 5. The modee functions and Green function of the dielectric are assumed to be known, determined perhapss with the use of multiple-scattering theory as in chapters 2 and 3. The T-matrix formalismm of multiple-scattering theory is applied here to study the single- and multi-atomm processes of guest atoms in the dielectric. Spontaneous-emission rates and elastic scatteringg of single atoms are considered. The strength of the formalism lies in the fact thatt results can readily be generalized to more than one guest atom. This is shown in the canonicall example of two-atom superradiance in an inhomogeneous dielectric.

Finally,, in chapter 6, the effects of material dispersion and absorption on spontaneous-emissionn rates in a homogeneous dielectric are considered. In a damped-polariton model forr the dielectric, light is coupled to a material resonance, which in turn is coupled to a continuumm into which electromagnetic energy can dissipate (or scatter, if the electromag-neticc field serves as the continuum). From the microscopic model, a complex dielectric functionn e(uj) is obtained that satisfies the Kramers-Kronig relations. Different dielec-tricc functions can be obtained, depending on the interactions between light and matter andd between matter and continuum. Maxwell field operators are obtained in a convenient wayy and their form justifies more phenomenological approaches. The formalism is used to studyy time-dependent spontaneous-emission rates near material resonances, in a frequency rangee where the optical density of states changes rapidly.