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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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Chapterr 3

Spontaneouss emission of vector wavess in crystals of plane scatterers s

TheThe concept of a plane scatterer that was developed in the previous chapter for scalar waveswaves is generalized to Maxwell vector waves. T-matrices can be defined after a Green functionfunction regularization. Optical modes and Green functions are determined and dif-ferencesferences with scalar waves are stressed. The theory is used to calculate position- and orientation-dependentorientation-dependent spontaneous-emission rates and radiative line shifts.

3.11 Multiple-scattering theory for vector waves

Inn section 2.1 a general introduction and motivation to study optical properties of crystals off plane scatterers was given. Since chapter 2 dealt with scalar waves only and since lightt is a vector wave, it is essential to study how the concept of a plane scatterer can be generalizedd to vector waves. This is the subject of the present chapter.

Thee general multiple-scattering theory as introduced in section 2.2 for scalar waves describedd by the Helmholtz equation, can be generalized to vector waves that satisfy Maxwell'ss equations. The notation inevitably becomes more involved: scalar quantities suchh as frequency UJ or components of vectors or matrices such as the wave vector compo-nentt kz will be written in standard font; vectors will be written in bold standard font, for examplee the electric field "Eko-"; finally, matrices or tensors such as the free-space Green functionn "Go" will also be written bold but in a sans serif Tont.

Thee wave equation for the electric field Eo(r, ui) in vacuum is

j ( w / c ) 2 h - V x V x } E o ( r a j )) = 0. (3.1)

Thee symbol I denotes the identity operator in three-dimensional space. The solutions of Eq.. (3.1) are plane waves with wave vector k and polarization direction normal to k.

47 7

488 Spontaneous emission of vector waves in crystals of plane scatterers

Withh the wave equation (3.1) the free-space Green tensor (or dyadic Green function) is associatedd that satisfies

{ ( ^ / c ) 2 l - - V xx V x }G 0( r , r ' ,^) = 53 ( r - r / ) l . (3.2)

Noww it wil l be convenient to consider Eq. (3.1) as the real-space representation of an abstractt tensor operator L(u>) acting on the vector field E0(w), so that (3.1) in abstract notationn becomes

L(o;) .EoMM = 0. (3.3)

Inn the same notation the dyadic Green function GQ(UJ) satisfies

L{ÜÜ)L{ÜÜ) G o M = 1 ® I (3.4)

Thee identity operator in real space we denote by 11 and it has the property {r|ll|r' ) = 6633(r(r - r ' ); confusion with the unit tensor I should not arise; the ® denotes the tensor product. .

Inn the presence of an inhomogeneous dispersive linear dielectric, the wave equation forr the electric field is modified as follows:

{{UJ/C){{UJ/C)22\\ - V x V x }E(r,tj ) = - [(e{r,uj) - l)(w/c)2] E( r ,u)

== V(r,u;).E(r,u;), (3.5)

wheree in the last equality the frequency-dependent optical potential V was defined in terms off the dielectric function £(r, u). The electric field Ein(o;) is modified into E(CJ), and the twoo fields are related through the Lippmann-Schwinger equation

E(w)) = Ein(u;) + G0(cu) V(u) E(u;). (3.6)

Thee field E(u>) that satisfies Eq. (3.6) is also a solution of Eq. (3.5). The solution of Eq.. (3.6) can be found iteratively in higher and higher orders of the optical potential V:

EE = Ei n + G0 V Ei n + Go V G0 V Ei n + . . .. (3.7)

Soo we have a multiple-scattering series for the electric field in terms of the potential, both whenn the potential consists of many scatterers and when it models just a single scatterer. A (dyadic)) T-matrix can be defined that sums up how the electric field depends on incoming fieldfield and scatterer:

E(cu)) = Ein(u;) + G oM T(w) Ein(w). (3.8)

Thiss is a formal definition of the T-matrix as a 3 x 3 tensor, and by combining Eqs. (3.7) andd (3.8), the formal solution for the T-matrix is

T MM = VM [I - GoM V(cu)]-1. (3.9)

Thee scattering problem is solved exactly once the T-matrix is known. Ass we saw for scalar waves, there may exist electric field modes that are bound to

thee scatterer. Such bound modes are solutions of the Lippmann-Schwinger equation in the absencee of an incident field:

E{LO)E{LO) = Go(tj) V(w) E(CJ). (3.10)

3.22 Plane scatterers for vector waves 49 9

Withh the help of the formal definition (3.9) of the T matrix, we can rewrite this homoge-neouss equation as

V(u;)-T -1(u;)'E(u;)) = 0. (3.11)

Thiss equation shows that nontrivial bound solutions of the electric field will correspond too the poles of the T-matrix. So the T-matrix not only solves the scattering problem for incidentt fields but also contains all information about bound modes.

Inn the presence of the dielectric the Green function also changes from Go to G and thee latter satisfies the following equation

[L(w)) - V M ] G(u;) = 11 Cx) I, (3.12)

Thee solution for the Green function analogous to Eq. (3.6) for the electric field is the three-dimensionall Dyson-Schwinger equation

G MM = GoM + GoM V M GM- (3.13)

Itt can be verified that a solution of (3.13) also is a solution of equation (3.12). The problem howw to find such a solution is solved once the T-matrix (3.9) is determined, because an iterationn of Eq. (3.13) analogous to the series expansion (3.7) for the electric field shows thatt the Green function can also be expressed in terms of the T-matrix:

G(u>)) = G o M + G o M T M G o M - (3-14)

Thee equations (3.13) and (3.14) also hold when the total potential V(w) is a sum of single-scattererr potentials VQ(u;). By iterating one finds that the total T-matrix for an arbitrary numberr N of these scatterers is

T ( N ) = ^ V aa + ^ ^ V , , - G o - V a + ^ ^ ^ V v G o - V , . G o ' V a + . . . . (3.15) aa ,.j a -y rf tv

Again,, it is much more convenient to do the expansion in terms of the single-plane T-matrices: :

T ( N ) =ZT -- + E ET ^ G ° - T - + EEET'>-Go-T3-Go-Ta + .... (3.16)

Thee equations of multiple-scattering theory for vector waves were introduced, in the no-tationn that will be used throughout this chapter. The analogies with and differences from scalarr waves can be found when comparing with section 2.2.

3.22 Plane scatterers for vector waves

Thee elements of the scattering theory are the potential V, the free-space Green function Go, thee T-matrix T, and the incoming electric field Ejn. Here we will determine the specific formm that these elements take for dielectrics that can be described as a collection of parallel planes.. First the free-space Green function is determined, followed by the single-plane T-matrixx and finally the T-matrix for an arbitrary number of planes.

50 0 Spontaneouss emission of vector waves in crystals of plane scatterers

3.2.11 Dyadic Green function in plane representation

AA solution for the dyadic Green function can be found in three-dimensional Fourier space andd by translational invariance of free space we have (k|G0(u;)|k'} = (27r)3^3(k — k')Go(k.u;).. The Green function G0(k. UÜ) that we thus defined satisfies

{{ [(u/c)2 - A'2] I + A'2kk}-G 0(k.^) = I. (3.17)

Here,, k denotes a unit vector in the direction of the wave vector k. Equation (3.17) is a 3 x 3 matrixx equation whose representation diagonalizes in the polarization basis {k.ai,a2} withh the longitudinal direction k and two orthogonal transverse directions <7i.2. Note that thiss representation is co-rotating with the wave vector. The solution of (3.17) is

GSfc(k.u;)) = (C/OJ)\ (3.18)

wheree j denotes o or a2. All six non-diagonal elements of the Green tensor are zero in thiss representation.

Thee spatial Fourier representation is not what we need. As in chapter 2, it is conve-nientt to work in the "plane representation": in two-dimensional Fourier space in the direc-tionss parallel to the planes and in real space in the z-direction perpendicular to the planes. Forr the polarization representation we choose the orthonormal basis {ik , vk , z} . Here, z iss the unit vector in the z-direction; Vk is the unit vector in the direction of the projection of thee wave vector k on the plane, so that the wave vector k has a wave vector component kn inn the Vk-direction and its full representation is (0. k\\. kz); the Sk-polarization direction iss orthogonal to the optical plane that is spanned by the other two basis vectors. Then the operatorr L(u;) has the form <k||, z|L(u;)|k'||. z') = (27r)2J2(k|| -k ' | | )<$(z-2')L (k |h z,u) andd the operator L(ky. z. UJ) has the matrix representation

(w/c)22

L(k|, .2.w)== I 0 {u>/c)'2 + d'j -ik\\dz I . (3.19)

Thee wave vector ky is a two-dimensional in-plane wave vector. The Green function in the samee representation becomes (ky. z|G0(^')|k'n- z') — (27r)2^2{k| | -k'||)Go(k||. z, Z',UJ),

andd the matrix equation (3.17) becomes a system of differential equations in the plane representation: :

// riSS f~1SV (~1S2

L(k,| .u; )) cls cf GI* \=8{Z-Z')\ 0 1 0 ) . (3.20) \\ /-1Z8 /~1ZV S1ZZ

\\ ^ 0 ^ 0 ^ 0

Thee GQ'} are the components of Go and the arguments (k||, z,z', UJ) were dropped for brevity.. By choosing this representation, the matrix elements of G0 only depend on the magnitudee and not on the orientation of kn. Evidently, all components involving an s-label


aree zero except for the ss-component that has an equation that is uncoupled from the rest. Actually,, GQS satisfies the differential equation for the Green function of scalar waves that wee encountered in the previous chapter. So we have

G^(kG^(kttz,z\u>)z,z\u>) = ^ — - . (3.21)

wheree kz is not a new independent variable different from k|j, but rather an abbreviation for

J(tjj/c)J(tjj/c)22 — k2. The remaining differential equations are coupled and can also be solved.

Itt turns out that GQV is proportional to GQS that is already known. Once GQ" is known, GQ1'' immediately follows. When solving for GQZ, the equation for GQZ is satisfied by thee solution for component GQV that we just found, so that the two these components are equal.. The last nonzero component GQZ can then be expressed in terms of the others and thee results are

Gnk||,*,*v ) )

GGzz00

vv(k^z,z'.uj) (k^z,z'.uj)

(?o2(k||,, z,z',oO

Thee above method of solving differential equations does not give a clue as to what the valuee of the sign-function for z equal to z' should be. It appears that there is another way too obtain the results above, one that does give the value of sign(O), namely by using the Fourierr relation between the Green functions in the two representations:

G0(k|hz.z',u;)) = i - dkz Go(k|,.^.£j) elk^z-z']. (3.23)

Thee integration can only be performed in a representation that does not co-rotate with kz. Thee basis of Eq. (3.18) is not adequate, but again the basis {sk. Vk- z} suits well. Again wee find the Green function components of Eq. (3.22), with the additional information that integrandss asymmetric in the variable kz lead to a sign-function that is zero when z equals z'.z'. The free-space dyadic Green function in the plane representation is now determined.

3.2.22 Attempt to define a T-matri x Thee T-matrix of a plane scatterer for vector waves can be found by solving the appropriate Lippmann-Schwingerr (LS) equation (3.6), just as was done in the previous chapter for scalarr waves. A plane wave incident from z = — oc with wave vector k and arbitrary amplitudee EQ and transverse polarization vector cr = (as. av. oz) is scattered by a plane att z — za. Because of the symmetry in the in-plane directions, it is convenient to choose thee plane representation for the LS equation. In terms of the Dirac-notation, the electric

kk22 eikt\z-z\

He) He) kuk.. kuk..

2tk2tkz z

ikikss\z-z'\ \z-z'\

(UJ/C)(UJ/C)22 2ik, stgn(--

Ak,\zAk,\z — z

(u/c)(u/c)22 2ik: ++ (c/u;)2S(z-z'). (3.22) )


fieldfield is a "ket" and the plane representation is found by taking the inner product of equation (3.6)) for the electric field with the "bra" (k||. z\, and by inserting the unit operator

(2 )22 /dV,|d*MkV*'><k'ih*' l (3.24)

att the positions of the dots in the representation-independent equation (3.6). The incident fieldfield takes the form Ek<r,in(k||. z- w) = -^o k exp(ikzz). The solution of the LS equation correspondingg to this incident field is the field Ek(T(u;) and note that it is labelled by thee wave vector and polarization of the incoming field from which it originates. The planee itself is described by the optical potential V(2,w) = - [s(z.u;) - 1] (LU/C)2\ = V(LÜ)Ó(ZV(LÜ)Ó(Z — za)\. Then the Lippmann-Schwinger equation in the mixed representation becomes s

Ek C T ( k | ( . ^ )) = £0 o ' k e ^2 + f dc/Go(k|| .2.s ' .^) -V(z ' .^)-Ek f f (k ||.2, .u,0-

JJ — OC

(3.25) ) Thee integral is very simple because the planes are assumed to be infinitely thin, and we get

Ek < T(k| | .2.w)-y(w)Go(k|,12.2a.u;)-Ek £ r(k| | .2r t .^)) = £0<Tkf^ 2. (3.26)

Thee analogous equation for scalar waves could be solved at this point by putting the posi-tionn z in this equation equal to za, solving for Ek<T(k||. za. UJ) and putting back this result inn the above equation to obtain an expression for Ek(T(k||. za.uj). However, unlike the scalarr Green function go, the Green tensor Go is not defined when the positions z and za

aree identical, because of the delta function in the component GQZ [Eq. (3.22)]. Therefore itt is impossible with this Green tensor to set up a scattering theory for vector waves scat-teredd by infinitely thin planes. In the next section we propose a regularization of the Green functionn in order to overcome the problem encountered above. We could just neglect the deltaa function, as is sometimes done in other calculations [77], but as we shall see later in subsectionn 3.2.4, this procedure does not give the correct results in our case.

3.2.33 Regularization of the Green function

Whenn studying multiple scattering of a wave by objects much smaller than the wavelength, thee assumption that the scatterers are infinitely thin or small can simplify the analysis considerably.. Often it is the only route to analytical results for a truly multiply-scattering system.. Thus, spatially extended physical objects are modelled as mathematical objects withh zero volume. But this simplification comes at a price. One may run into the kind of problemm encountered in the previous section, that a Green function does not have a finite valuee precisely at the position of a zero-volume-scatterer. This technical complication can bee remedied by a procedure called "regularization".

Whenn modelling scatterers not as infinitely thin planes but as mathematical points, thee above problem of unphysical infinities shows up both for scalar and for vector waves. Forr point scatterers the problem has been studied extensively and several regularization schemess have been proposed (see [69] and references therein). In a regularization pro-ceduree usually some cutoff parameter is introduced that modifies the behavior of Green


functionss at distances much smaller than optical wavelengths and infinities are thus re-moved. .

Afterr a successful regularization, two situations can present themselves. In the first situation,, the optical properties are not affected by the procedure, in the sense that the valuee of the cutoff parameter can be chosen infinitely large in the final stage. In this first case,, the mathematical problem of an infinity could be solved with mathematics and the scatteringg theory is called "renormalizable".

Inn the second situation, the cutoff parameter can not be sent to infinity. It should then bee possible to express the finite cutoff parameter in terms of physical quantities. But then itt also works the other way around: physical properties depend on the cutoff parameter. In thiss second situation, the theory is not renormalizable and the physical parameters together withh the cutoff parameter form an overdetermined set. The latter situation was found for pointt scatterers for vector waves [69]. A T-matrix for a point scatterer could only be definedd when two regularization parameters were introduced, namely a longitudinal and a transversee cutoff parameter. The longitudinal cutoff is related to a static quantity, namely thee natural size of the scatterer; the transverse cutoff can be expressed either in terms off dynamic quantities (for example: resonance frequencies) or static quantities, the latter choicee depending on the physical scatterer that is modelled as a point. So much for general remarkss about Green function regularization.

Inn the rest of this subsection, we regularize the dyadic Green function with compo-nentss as in Eq. (3.22). We propose the same regularization procedure as was done for point scattering,, namely a high-momentum cutoff in three-dimensional Fourier space: instead off the free-space Green function G0(k. UJ) of Eq. (3.18) we use the regularized free-space Greenn function Go(k.u;), and define the latter in terms of the former as

G0(k.o;)) = —^-—Go(k.o,-) . (3.27)

Here,, the cutoff momentum A is assumed to be much larger than any optical momentum, so thatt at optical wavelengths G0 ^ Go- Now we are interested what the effect of this cutoff wil ll be in the plane representation. For the unregularized Green function, we showed that thee plane representation could be found either by solving differential equations (3.20) or byy an inverse Fourier transform (3.23). The latter method is the simplest for finding the regularizedd Green function in the plane representation, and we find:

ikikzz\z-zi\\z-zi\ g -

-- + -A 22 /etKz\z-

hhz,zz,zuuu>)u>) = A 2 + ( u ; / c )2 [—^ 2Ai i

rQQ V « , | | , * , A l , u / , — 2

kk k c2

G^(k||,z,2i,u;)) = —^- |— sign(2-2i)G^s(k||,2,zi,u;)

Gnk,,,*,*!,* )) = GS*(k,|, *,*!,* ) k?,Ck?,C22 _ A2 e-A, | |z-* ,|

Ggz(k||.*,*1?u;)) - -j;rG*0*(khz.zl.u>)+ 2 ( u ; / c ) 2A (3-28)


Inn these equations, A|| is short-hand notation for J A2 + k?,. (Again, the sign-function is zeroo when its argument is.) All components of the regularized Green tensor consist of two parts:: an oscillating and a decaying part, as a function of \z - z\ |. The decay is very fast becausee the decay length A - 1 is much smaller than the optical wavelength A = 2TTC/LO.

Forr X\z - z\ | > 1 and A » A, the regularized Green function approaches the unregularized one.. If one would take the limit of A —> oc, then all the components in (3.28) approach the unregularizedd components of equation (3.22), and in particular the limit of the last term in GQ22 gives the delta function that made the regularization procedure necessary. However, wee keep A finite for the moment. Unlike GQZ, the component GQZ does not have a term proportionall to S(z - z\), but it has a term that grows with A instead. This enables us to sett up a theory of scattering by vector waves from plane scatterers.

3.2.44 T-matri x of a plane for vector waves

Inn subsection 3.2.2, we stumbled on a problem to define the T-matrix of a plane for vector waves.. After completing the regularization of the Green function in subsection 3.2.3, we noww go back to the Lippmann-Schwinger equation (3.25) and replace the Green function byy the regularized one:

Ekor(k||.z.a;)) =E0(Tkeik* z+ f dz' G0(k„ . z. Z'.LU) V(z'.u;) - Ek(T(k,|. Z\LÜ). JJ — oc

(3.29) ) Noww take z = za in equation (3.29) and write it out in components:

ELr\ELr\ ( °* \ , / Gg- 0 0 \ / E \ £k«rr = M a« K + V ^ M 0 G^' 0 ^ .(3.30) EElala I \ °* ) \ 0 0 G*0* J V E^ J

Here,, GQS = GQS (k |, zn. zQ. u) and similarly for the other components. The off-diagonal elementss of the Green tensor are all zero when the position z is equal to za. Hence the equationn can be solved for every component separately. If we insert this result into the Lippmann-Schwingerr equation for general 2, we find

Eko-(k||,2.u;)) = Eka.,in(k||,2,cu) + G0(k||, z. za.u}) ) Ek<T.in(k||, 2a,u), (3.31) )

wheree the T-matrix for scattering from a plane by arbitrarily polarized light is given by

t(ki,,ü, )) = 00 v ^ l 0 1-V(W)GS--

\\ 0 0 v(u!K I \\ l-V(w)Ggs /

(3.32) )

Thiss T-matrix is what we were looking for. Now that it is known we can calculate trans-missionn and reflection and local densities of states as well as other physical quantities.

Thee scattering of the s-polarization component of the light can be considered inde-pendentlyy from the v and z directions, according to Eq. (3.31). It can be verified with

3.22 Plane scatterers for vector waves 55

Eqs.. (3.28, 3.32) that since A » {u/c), the matrix component fs" for all practical pur-posess is equal to the T-matrix for scalar waves, and the same holds for the Green tensor componentt G™: the regularization was not necessary for .s-polarized light and fortunately itt does not affect the scattering properties of s-polarized light either.

Thee need for regularization did show up in the description of scattering of />-polarized lightt and it is interesting to analyze how the cutoff influences the scattering properties. Assumee that p-polarized light is impinging on a plane, with amplitude E0, wave vector k, andd the polarization state

&& = p= {kz/k)vk - (fr||/fc)z. (3-33)

Writtenn out explicitly, the incoming field is

/ £ k . i n ( k | | ^ ) \\ / 0 \ Ek < T > i n(k | , .2^)== Ek< in(k,|,*,ü,) U E 0 kz/k r'** = . (3.34)

VV E^.Jk^z^) / V -h/k ) Thee planes do not mix the s and p-polarization so that the incoming-plus-scattered field doess not have an ,s-component either. For the nonzero components we find:

// E°{z) \ ( E?n{z) + G^(z. za)f^'Eln(zn) + Gl*t"E?n{za) \ \\ Ez(z) ) V E?n{z) + Gl^z.za)f

vvEya{za)+G^{z,zn)T»Ern{za) ) ' (3.35) )

wheree the labels (k||. <r) were dropped for readability. Now for distances far enough from thee plane so that K\z - za\ > 1, the term G ^ T " falls off as A - 1 exp(-A|s - 2Q|)and GQGQZZTTZZZZ as exp(-A|c - za\), so that for optical purposes these terms can be neglected. Forr finite very large A we arrive at the following effective description:

// E°(z)-E?n(z)\ ( Gtr 0 0 E*(z)-EyE*(z)-Eynn(z)(z) U o <??;< GF

VV E*{z)-E?n(z) J \ 0 G(f Gïf (3.36) )

wheree the ,s.s-component of the T-matrix is equal to V{^) [1 - V(U>)GQS]~ , and analo-gouslyy for the Ti'-component. The Green functions have arguments (ky. z. za.uj). In this effectivee description, - where the T-matrix is denoted by T rather than T - the cutoff pa-rameterrameter A does not occur anymore. The cutoff was necessary in order to set up a scattering theoryy and it shows up in the elements of the scattering theory such as the T-matrix (3.32) andd the regularized Green function (3.28). But it does not show up in the electric field and preciselyy this enables us to arrive at the effective description. Note also that in the effective descriptionn the value of GQZ has become irrelevant.

Cann the delta function in G^z (see Eq. (3.22)) just be left out of the theory, as is sometimess done in other work [77], and produce the same T-matrix? The answer is No. Leavingg out the delta function in the Lippmann-Schwinger equation (3.25) will result in a nonzeroo Tzz, in contrast with Eq. (3.36). Furthermore, the T matrix would be such that the transmittedd part of an incoming wave would not be parallel to this incoming wave, which iss unphysical. The conclusion is that a regularization of the Green function was necessary, evenn when in the end the scattering theory is renormalizable, in the sense described in sectionn 3.2.3.

TTHH* * 0 0 {) )

0 0 y r r r

{) )

0 \ \ 0 0

o/ /

// £fn(~«) ^ i n ( - ) )

VV E?n(zn)


3.2.55 Transmission and energy conservation

Thee transmission of light through the plane can be found by choosing z > za in Eq. (3.36). Thee transmitted wave can be expressed in terms of the incoming wave as E ^ k y ,z.u>) — r(k||.u;)) Eko-.infkii- Z.UJ), with the transmission matrix

r(k||.u;)) - 0 Trr(kn.u,') 0 (3.37)

l -VMGJ J ' ( fr | | .2 a.So.u; ) ) forr j = s.v. whichh has nonzero elements Tjj(k||.u,')

Furthermore,, r ; i ! (k| j .^) = Gzv{k\^zn.zQ'.uj)Tvv(kll.u}). With the help of Eqs. (3.22, 3.34,, 3.36) one can find that both for .s-polarized and for p-polarized light, the transmitted electricc field is a polarization-dependent scalar times the incoming electric field vector. Inn other words, after transmission the direction is unchanged but the amplitude can be differentt from the incoming wave.

Noww we can ask which forms the T-matrix can take such that energy is conserved in thee scattering process. This has been analyzed for scalar waves before and since s-waves mapp on scalar waves, we know the optical theorem for the ss-component of the T matrix:

Imm T" (k,| .<*;) = - - l- LILJJ- . (3.3 8)

Thee most general T-matrix satisfying this requirement has the form

r - ( k | | ,W )) = - [ ^ ( k , , . ^ - i/{2kg)]~l , (3.39)

withh the optical potential Fs(k||.u;) being a real-valued function. Energyy conservation of light implies a conservation of the z-component of the Poynt-

ingg vector before and after the plane. For reflection and transmission of p-polarized light, onlyy the matrix element Tvv is important and again we are interested in the form that this matrixx element can take. An incoming plane p-wave with polarization (3.33) gives the electricc field (3.36) and with a Maxwell equation the accompanying magnetic field can alsoo be found. The Poynting vector is then [45]

S(r,, t) = -?— Re [E* (r, t) x B(r. t)]. (3.40) Z7T}1Z7T}10 0

Firstt transform the equations to frequency space. At the side of the plane where light is reflectedd (z < za), the Poynting vector is then proportional to 1 - (kzc/uj)2\Tvv\2/4. Thiss should be equal to 11 — ikz{c/u)2Tvv/2\2, the expression that one finds for the other side.. From energy conservation one can obtain the optical theorem for the scattering of p-polarizedd light by a plane:

Imr ' " (k I I .W )) = ^ ï | T - ( k | | . W ) | 2 . (3.41)


Thiss expression differs from the optical theorem for s-polarized light. Also, the most generall solution of the optical theorem is different:

- l l

(3.42) )

wheree the optical potential Fp(k||,u;) is real.

3.2.66 T-matri x for N planes Thee Green function and the T-matrix of a single plane are known now, and with this a multiple-scatteringg theory can be set up. Assume now that there are N plane scatterers placedd at arbitrary positions. Assume them to be parallel, so that s- and p-polarized light do nott mix in the scattering process. Consequently, an iV-plane T-matrix can be determined forr s- and p-polarized light separately. The iV-plane T-matrix for s-waves is equal to the N-planee T-matrix for scalar waves, which was derived in the previous chapter. Here, we cann concentrate on the TV-plane T-matrix of p-polarized light. We repeat the formal sum (3.16)) for a T matrix of N scatterers T WW = E T « + E E T ^ G ° ' T Q + E E E T ^ G ( | - T ^ G I ) J ^ - - ' (3-43)

Inn Eq. (3.43) the Green functions are always sandwiched between T-matrices of planes att different positions. The values of Go(k||, zg. za.uj) with zj ^ zn are finite. There-fore,, once the single-plane T matrix is defined with the help of a regularization, it is not necessaryy to perform another regularization in order to find the N-plane T-matrix. One couldd argue that for consistency all Green functions in (3.43) should be replaced by the regularizedd ones, for without the regularization the T matrices would not even be defined. Inn practice, however, the differences between G0 and Go can be neglected, because the planesplanes are assumed to be at optical distances apart such that A\za — ztj\ ^> 1 even for the planesplanes a and (3 closest to each other.

Withh the explicit form for the Green tensor and the T-matrices known, the series (3.43) cann be summed exactly, each component separately and analogously to the scalar case in thee previous chapter. This gives the central result of this section, the A7-plane T-matrix for scatteringg by vector waves:

T t A ° MM = 7 2^ /d 2 k |1 S |k||â)(k||.2,|T^ )(k ||.^ ) (3.44)

Thee only two nonzero spatial components of the 3 x 3 T matrices T^ ^ are

r^'(JV)(k||,u;)) = T™(kûj)[t -V^.^Tîkû;)} (3.45)

TTvvvv(la\.u)(la\.u) = -F - ^ k i i . u ; ) --ik; ik;

2(u,/r)2 2


Thee A72 matrix elements (£>J)QJ are defined as (1 - öa,j)GJ0j(k\\. zrt. ZJ.UJ), forj — s. v.

Thee calculation T boils down to the inversion of an TV x N matrix for the two transverse polarizationn directions separately. The .s.s-component is identical to the T-matrix T( jV ) for scalarr waves.

Thee trick of the analytical inversion of Ts s ( A^ that was presented in appendix A andd used in the previous chapter can be accomplished for j ,t't'-(Ar) as well: u = us ~ -2ik-2ikzz/T/Tssss{\a{\a}V}Vu)u) should be replaced by vp = ~2i{uj/c)2/[kzT

vv{^. u)]. The deriva-tionn will not be repeated here, but we will give some results. In the first place, the Bloch wavee vector Kp for p-polarized light reflected by planes separated by a distance a is given byy arccos{Cp)/«, with

^^ {ikz[kz\Tt'''\2 + 2^jc)2lmT^}+2kz(uj/c)2RQT^\ , „

LLpp = cos{kza) + i J J — —— sin hö . VV A:2(ReT"")2+[2(o; /r)+^Imr" ' ]2 J

(3.46) ) Inn general, Cp is a complex constant. However, if there is no light absorption in the planes soo that the optical theorem (3.41) holds, then the imaginary part of Cp becomes identically zero.. (For .s-polarization we saw the same thing happening: without absorption, Cs is real.)) For jy-polarized light with wave vector and frequency such that \CP\ > 1, the Bloch wavee vector is complex. The system of planes has a stop band for this light, meaning that thee light will be 100% reflected when falling on a semi-infinite system of planes. If the opticall theorem indeed holds, then the single-plane T-matrix is of the form (3.42) and the expressionn (3.46) can be simplified to give

__ , , kzc2FJ'k\\,{jj)

CCpp = cos(M) - £2 " sm(kza). (3.47)

3.2.77 A model for the optical potential

Thee most general T-matrix for p-waves (3.42) features an as yet unspecified optical po-tentiall that should be real when energy is conserved but for the rest it can be an arbitrary functionn of the in-plane wave vector and frequency. Until now we assumed that the optical potentiall F(k||,u;) was only frequency-dependent and we called it -V{UJ), thereby ne-glectingg spatial dispersion and anisotropy that would show up as a A;||- and k|| -dependence, respectively.. Both were neglected as early as in the wave equation (3.5).

Inn chapter 2 plane scatterers were introduced as a simplified model for dielectric slabs off finite thickness d and nondispersive dielectric function E{UJ) = e. Recall that the optical potentiall for the plane scatterer in thiss model is obtained via the limiting process of making thee dielectric slab thinner and increasing the polarizability e-I while keeping their product constantt and equal to an "effective thickness". Now for vector waves we use the same proceduree and for both s- and p-polarized light we find the optical potential F(k||. to) = —V{UJ)—V{UJ) — Deff(uj/c)2, with the effective thickness F>eff equal to d[e{uj) - 1].

Onee can hope that transmission and reflection properties of a plane scatterer and a finitefinite dielectric slab do not differ much. In this respect, p-polarized light is different from 6-polarizedd light. When light in a medium with refractive index n\ meets an interface with aa second medium with refractive index n% > n\, then there is a typical angle of incidence,

3.33 Optical modes and omnidirectional mirrors 59 9

calledd the Brewster angle, for which all p-polarized light is reflected. (For s-polarized light theree is no such angle.) The Brewster angle OB depends on the refractive-index contrast in thee following way:

99BB = t an- 1 (n2 /m). (3.48)

Inn the limiting procedure for going from a finite slab-in-air to an infinitely thin plane-in-air,, the dielectric contrast y/e/1 is going to infinity and consequently the Brewster angle becomess 90° in that limit, so that effectively the Brewster effect becomes invisible. There-fore,, in our limiting procedure, a plane scatterer will not have a Brewster angle at the same anglee as the finite dielectric slab that one tries to model with it.

3.33 Optical modes and omnidirectional mirror s

3.3.11 Propagating modes Thee optical modes are the harmonic solutions of the wave equation (3.5). These modes are alsoo the solutions of the Lippmann-Schwinger equation, which is solved once the T-matrix off the total scattering system is found:

EkCT(k||.2.^)) = ^ 0 c r k e ^5 + ^ G o { k | | . ^ ^ . ^ 0 - T ^) ( k | | . ^ ) - o -k £ o c ^ ^- (3.49) a,3 a,3

Thee modes that correspond to an incoming plane wave (E0 0) are the propagating modes (or:: radiative modes) and they are labelled by the incoming wave vector k and polarization o-k.. The s-polarized modes [with <xk = (1.0.0)] are identical to the modes for scalar wavess derived in the previous chapter. The p-polarized modes [<rk = (0. kz/k, —k\\/k)] aree the new ones.

Modee functions are complex functions of position and mode profiles are their absolute valuess squared. Mode profiles for «-polarized light are identical to the mode profiles of scalarr waves that were discussed in chapter 2. We choose the light as incoming from the left.. If in particular we choose light of frequency such that a/A = 0.5, then the mode pro-filess of .s-polarized light will correspond to the scalar mode profile of the same frequency inn figure 2.2. For perpendicularly incident light, there is no difference between s- and p-polarization.. In figure 3.1 the mode profiles for s- and p-polarized light inside a ten-plane crystall are compared both for an incoming angle of 30° and for 60°. In figure 3.1(a) we seee that at an angle of 30° the mode profiles corresponding to both polarizations do not differr much yet. Both modes decay rapidly inside the crystal structure and are reflected (almost)) completely. Only for the s-wave the polarization directions of the incoming and thee reflected wave are equal, so that the amplitude of its mode profile at the left side of the crystall is four times the amplitude of the incoming electric field. The Bloch wave vectors aree imaginary for both polarizations.

Thee situation is different at an incoming angle of 60° as shown in figure 3.1(b): there thee mode profile of the .s-polarized light again rapidly decays inside the crystal (and the correspondingg Bloch wave vector will again be imaginary), whereas the p-polarized light


at at

UJ J

(a) )

•II (I 1 • • i i i i

nn J n i A

ILLL L vrvL I ^ 00 5 10

Positionn z (units of a) 10 0

Positionn z (units of a)

Figuree 3.1: Squaress of absolute values of mode functions for s-polarized (solid lines) and p-polarized light (dashedd lines), as a function of position. The light is scattered by a crystal of ten planes with Dea = 0.46a,, separated by a distance o. Both modes correspond to light incoming from the left with a/AA = 0.5. Figure (a): 0i n = 30°; figure (b): 0in = 60°.

cann propagate inside the crystal (so the Bloch wave vector is real) and the light is trans-mittedd almost completely. For this frequency and incoming angle, the crystal is a good polarizationn filter.

Notee in figures 3.1(a,b) that the mode profiles of the s-polarized waves are continuous whereass p-polarized waves form more irregular patterns and are discontinuous at the po-sitionss of the planes. This reflects the boundary conditions that are automatically satisfied byy the modes that we find by multiple-scattering theory: the tangential components of the electricc fields must be continuous and the normal components show a jump at a dielectric interface.. Now the electric field of s-polarized light only has a tangential component and p-polarizedd light consists of both a tangential and a normal component. This explains the differencess in the mode profiles for s- and p-waves.

Onee can also compare reflection by the ten-plane Bragg mirror as a function of fre-quencyy for the two polarization directions. This is plotted in figure 3.2. For light incident perpendicularlyy to the planes, both transverse polarization vectors are equivalent and ac-cordinglyy in figure 3.2(a), s- and p-polarized light show identical reflection characteristics ass a function of frequency. This reflection plot is identical to the one for scalar waves in thee previous chapter. Differences between the two polarizations do appear for non-normal incidence.. In figure 3.2(b) the incident angle is 30° and in figures (c)-(d) the incident angle iss increased further. The red edges of the stop bands for s-polarized light move to higher frequenciess and the widths of the stop bands become larger. For p-polarized light on the otherr hand, the red edges of the stop bands also shift to the blue, and faster so for larger incidentt angles. In contrast, the stop bands for p-light become narrower while moving to higherr frequencies. In figure (d) for an incident angle of 80°, the first stop band for p-polarizedd light has even moved out of the picture, while the s-stop band has become very broad.. It is harder for the ten-plane crystal to stop p-polarized light than to stop s-polarized


-(b ) )

--

I I AAAAAJJ J

11 1

1/ / ; ;

iii i

>> i

II!! iii II

H.1ÏIMM ni l

i i I I

11 • I1» » Hi i

i i i i 4 4 1 1 I I 1 1 t t t t 1 1 t t

0.25 5 0.500 0.75 a/A A

1.00 0

a/A, , a/A. .

Figuree 3.2: Reflectionn off a ten-plane crystal, as a function of a/A, for s-polarized light (solid lines) and p-polarizedd light (dashed lines). Incoming angles are 0°, 30°, 60° and 80°. The planes have effective thicknesss Deff = 0.46a and they are separated by a distance a.

light. . Inn the previous chapter we showed that a ten-plane crystal can act as an omnidirec-

tionall mirror for scalar waves. But is this crystal also an omnidirectional mirror for vector waves?? For s-polarized light we know that there are frequency intervals in which light comingg from all directions is reflected, because the s-waves map on the scalar waves. So heree we need to analyze whether there are also such frequency intervals for p-waves and iff so, whether any s- and p-frequency intervals have an overlap. Only in the latter case can wee speak of an omnidirectional mirror for light.

Itt is the Bloch wave vector that distinguishes between light that can propagate inside a a crystall and light that feels a stop band: a real Bloch wave vector corresponds to propagat-ingg light and a complex Bloch wave vector to light that is reflected off the crystal. In our formalism,, the Bloch wave vectors are the arc cosines of the constants Cs and Cp. Now forr light of a frequency corresponding to a/A = 0.5 and planes with Dea = 0.46a we knoww that s-waves are reflected omnidirectionally. In figure 3.3 we plot both constants for


Figuree 3.3: Constantss Cs and Cp as a function of anglee of the incoming light, for the pa-rameterss a/A = 0.5 and Deff = 0.46a. Regionss where — 1 < C < 1 corre-spondd to propagating waves inside the crystal. .

00 20 40 60 80

Angle e

thiss frequency, as a function of angle of the incident light. Unlike for s-waves, for p-waves theree are incident angles larger than the critical angle 9C > 55° for which the values of Cp

aree between -1 and 1. Light incident with these large angles can propagate inside the crys-tall and therefore the crystal is not an omnidirectional mirror for this frequency. Actually, thiss information could already be read off from the mode profile in figure 3.1 (b). However, thee conclusion does not only hold for the specific parameter values that we chose: if we choosee Z?eff larger then 8C moves up to larger angles, but it can be shown by expanding Eq.. (3.47) around 0in = 90° that for every finite Deg and a/A there always is a finite intervall of angles corresponding to propagating p-polarized light. So strictly speaking, omnidirectionall mirrors for vector waves can not be made with equidistant identical plane scatterers,, but it is possible to block incident light from almost all directions.

3.3.22 Guided modes

Wee have found the propagating modes of vector waves near plane scatterers but these do nott necessarily form the complete set of modes. As pointed out in section 3.1, a scattering problemm can also have bound modes that do not correspond to incoming light. By sol-vingg the Lippmann-Schwinger equation without an incoming field, modes of scalar waves boundd or guided by one or more plane scatterers were found in section 2.4.2. Later on, inn section 2.5, it was shown that with each guided mode a nonzero local density of states iss associated. One can also turn it around and try and find guided modes by looking for valuesvalues of fey > OJ/C corresponding to a nonzero density of states. It is the latter approach thatt we follow here for vector waves, because it directly allows us to use the effective descriptionn (3.36) for the T-matrix.

Forr vector waves, the local density of states is a tensor proportional to the imaginary partt of the Green tensor G(r, r, ui). In planar geometries the latter can best be found as an integrall over the Green tensor in the plane representation:

Q(r,r ,w)) = -^—2 ƒ d\G(kh Z,Z,LO). (3.50)

Clearly,, the local density of states can only be nonzero if the imaginary part of the in-tegrandd in (3.50) is nonzero. A guided mode manifests itself when this integrand has a

• •

• •

• •

_ _ • •

11 • 1 • 1 • 1

s s / /

CC / fir. fir.

• • « ^ ^^ **

\c c ^^««s s

11 . 1 . 1 . 1


nonzeroo imaginary part for a certain A-y > UJ/C. For the crystals of plane scatterers the Greenn tensor directly follows from the Dyson-Schwinger equation:

N N

Gik^z.z'.aj)Gik^z.z'.aj) =G0(kt z.z'.^)+ Y, G0(k| , .5.5a.u;)T^)(k| | .u;) .Go(k| | .^.V.u;).

(3.51) ) Al ll the components of Gy(k| , z, z'', a;) become real quantities for k\\ > uj/c. This means thatt there is no density of states corresponding to guided modes in free space, and this comescomes as no surprise. Therefore, equation (3.51) tells us that the imaginary part of the Greenn tensor G can only be nonzero when at least one of the two spatial components of the N-planee T matrix "P has a pole. These components T B (k| , OJ) and T (kj . co) aree given in equation (3.45).

Firstt look for the poles of the component T ^ (k| , u,'). This is easy, because this componentt is identical to the iV-plane T matrix T^ for scalar waves, as presented in thee previous chapter. So we find that there are at most TV guided modes corresponding to s-polarizeds-polarized light in a crystal of N planes. The dispersion relations of these guided modes aree given in figure 2.4.

Aree there also guided modes corresponding to p-polarized light in a crystal of N planes?? This question will be addressed in some detail now. We need to find the poles off the component T^p (k|.u;). They occur when the determinant det[(Tin '^N))~1] is equall to zero. An expression for this determinant can be found if one replaces vs by up

(ass defined in section 3.2.6) in the expression A. 14 for the corresponding determinant for s-polarizedd light. The result is that for p-polarized light a guided mode exists when the followingg equation is satisfied:

2(u;/r)'22 sin[(Ar + 1)A>] + [KF - 2{UJ/C)2}(-*' ! sm{NKpa) = 0. (3.52)

wheree n is defined as — iJ(^j/c)2 — k'2. The Bloch wave vector Kp is still defined as a- 1

timess the arc cosine of the constant Cv (Eq. (3.47)), but in terms of H it reads

KCKC22F F CCpp — cosh («a) + ——— sinh(Ka). (3.53)

ZuJ ZuJ

Equationn (3.52) should lead to the dispersion relations U/(K) for the guided modes, if they exist.. When increasing the frequency, new guided modes appear that at first are only just capturedd by the structure so that K = 0+ . As we did for s-polarized guided modes, we lookk for the guided modes in this small-K limit. Note that for K very small and positive, thee constant Cp can be written as 1 + (\K)2/2, up to second order in K, with \ defined as a-y/ll + Fc2/(2auj2). We can estimate the Bloch wave vector for small K as follows:

Cpp = 1 + (x«)2/ 2 + O{K3) = cosh(xK) + O{K3) => Kp = ix^/a (3.54)

Itt follows that for small and positive n, solutions of equation (3.52) will only exist when sinh[(iVV + 1)XK] equals sinh(ArxK), or equivalently x = 0. If for the optical potential wee take F = Deff (UJ/C)2, then we find that there are guided modes for p-polarized light


wheneverr l-\-Deff/(2a) — 0. So in a crystal of N planes with a positive effective thickness, theree are no guided modes corresponding to p-polarized light. This is in agreement with thee result in [65] for infinite crystals.

Inn conclusion, all guided modes in the finite crystal of plane scatterers correspond to .s-polarizedd light and their properties have already been studied in the previous chapter. For dielectricc slabs of finite thickness it is known that .s-waves can be guided more easily than p-p-waveswaves [60]. The non-existence of guided p-waves for plane scatterers is a manifestation off this difference between the two polarization directions.

3.44 Spontaneous emission

3.4.11 General theory applied to planes Thee spontaneous emission rate T of an atom embedded in an inhomogeneous dielectric is givenn by [11]

T(ft.K.Ü)T(ft.K.Ü) = ix V - ^ - | / i • E ?(R) |2 S(io, - O). (3.55)

Thiss can be found by applying Fermi's golden rule. The spontaneous-emission rate (3.55) dependss on the atomic transition frequency Q, on the ^-coordinate of its position R = (x.y,(x.y, z) and on the magnitude and orientation of its dipole moment fi. The E; are the normall mode solutions of the wave equation (3.5) for the electric field. The spontaneous-emissionn rate can alternatively be expressed in terms of the Green function of the medium. AA derivation will be presented in section 5.3.1, but the rate to be found in Eq. (5.36) will alreadyy be used now:

T(ii.T(ii. R. fi) = - -Im [u • G(R, R. £l)-u\. (3.56)

Here,, G is the dyadic Green function of the electric-field wave equation (3.5). For systems consistingg of dielectric layers (not necessarily plane scatterers), it is easiest to first calculatee the Green function in the plane representation G(k||. z. z< Q). We only need to Fourier transformm this Green function to real space as in Eq. (3.50) in order to find the Green functionn of Eq. (3.56) that determines the spontaneous-emission rates. A slight complicationn is that until now we chose a representation for G(k||, 2, z, Q.) which is co-rotating withh the incoming wave vector ky that we now want to integrate over. We need a fixed basiss {x. y, z} and we can choose it such that the atomic dipole becomes (fix, 0, fiz) in thee new representation. We perform the two-dimensional integral ƒ d ky by writing it in

polarr coordinates J^° dk\\k\\ fQ n dky. By doing the angular integral first, only diagonal

elementss of the dyadic Green function survive. The total spontaneous emission-rate is the summ of two contributions, the perpendicular and the parallel decay rate, which are given

3.44 Spontaneous emission 65 5

by,, respectively,

TTzz{zA2){zA2) = - 3 c f ^ 2 I m / dkfyG^ik^.z.zM) (3.57)

Thee parallel decay-rate has a contribution both from s- and p-polarized light (Gss and Gvv, respectively)) whereas the perpendicular decay-rate only has a p-polarized decay channel (throughh Gzz). When the dipole is oriented parallel to the plane, the perpendicular decay-ratee is zero, and vice versa. The spontaneous-emission rates in Eq. (3.57) and (3.58) are integralss over all possible lengths of the in-plane wave vector. Both rates can be subdivided intoo a propagating (or radiative) rate corresponding to the integration of k\\ from 0 to u?/c, andd a guided rate which is the integral from LO/C to infinity.

Moree precisely, we can subdivide the parallel decay-rate into three parts: an s-polarized radiativee rate (sr), a p-polarized radiative rate (pr) and an s-polarized guided rate (sg). Theree are no guided modes for p-polarized light and as a consequence the Green tensor GGzzzz has no imaginary part (zero density of states) for k\\ > u>/c. The perpendicular decay-ratee Tz is therefore purely radiative:

3cr00 /// 22 fjj/c

TTSSJ(Z,UJ)J(Z,UJ) = Imj dfc,|A:||G"(fc||,*,*,fi), (3.59a)

TTppxx

rr(z.n)(z.n) = — ï c r \ — ) I m / dkfyGvv{k\bz.z,n), (3.59b)

oo „p / \ 2 poo

TTss/{zM)/{zM) = ——^ ~ Im / dkfyGss(kl{,z,z,a). (3.59c) 1U1U \ n ) j^ic

3cT3cT00 ( IL

/c /c

22 ,^jc

TTzz{z.tt){z.tt) = ?r[ ) I m / dfc||fc||G"(fc||,z.z,n). (3.59d)

Withh all the spontaneous-emission rates spelled out now, let us first study spontaneous-emissionn rates near a single plane.

3.4.22 Spontaneous emission near one plane scatterer

Thee Green function in the plane representation can be calculated directly using the single-planee T-matrix T of section 3.2.6 in the Dyson-Schwinger equation

G(k||,2,2',u>)) = Go(k^z,z\uj)+Go(k\\,z,za,uj)-T(k\\,u)-Go(k\\,za,z',u)). (3.60)

Thiss Green function can be used in the expressions of the previous section 3.4.1 to obtain spontaneous-emissionn rates near a single plane.

Whatt happens to the spontaneous-emission rates very close to the plane? (By "close" itt is meant: at distances small compared to the wavelength of light, but still large com-paredd to atomic distances where local-field effects come into play that we do not consider


Figuree 3.4: Spontaneous-emissionn rates of dipoles withh zero distance to the single plane scatterer,, as a function of the dimen-sionlesss parameter £ = 7rDeff/A . Solid line:: the rate Tx, which is the sum of a ratee into radiative (dashed line) and into guidedd modes (dotted line). The rate Tz

iss the dash-dotted line.

00 2

\ \

here.)) In the limit that the atomic position z becomes equal to the plane position za, the spontaneous-emissionn rates for the two orientations can be calculated analytically. Let Too be the spontaneous-emission rate in vacuum, equal to fi2Q3/(STTHEOC3). For a dipole perpendicularr to the planes we find

TTzz(z(zaa,, Q) = T0 (2 + A ) - 3-T0 (l + 1 ) ??*f®, (3.61)

wheree the dimensionless parameter £ is defined as irDeff/\. The three contributions to thee parallel decay-rate at the position of the plane can also be expressed in terms of the parameterr £ alone:

TTssxx

rr(z(zaa,Q),Q) = ^ r 0 [ l - £ a r c t a n ( l / 0 ], (3.62a)

rr ppxx

rr(z(zaa,Q),Q) = ^ r 0

r-»(za,n)) = ^ r 0 . 0.62c)

Inn figure 3.4 the relative rates are plotted as a function of £. The results can be checked inn two limiting cases: the weakest test is the limit of invisible planes where Deg = 0 and thereforee £ = 0. Then both Tz and Tx approach the free-space value, as they should. The otherr test is the limit of a perfect mirror, when £ = De = oo. This limit is not visible inn the figure, but we find r2/To = 2 and TX/TQ = 0. To be precise, we first have taken thee limit of the perfect mirror and then come closer with the atom to the mirror. In doing so,, the contribution to the spontaneous-emission rate of the guided waves goes to zero. Takingg the limits in the reverse order would give the unphysical result that the contribution off the guided waves diverges when the positions of the atom and mirror coincide. The spontaneous-emissionn rates for atoms near perfect mirrors agree with the values in the literaturee [16,78]. These results can be explained in terms of constructive or destructive interferencee of the light emitted by the atom and its perfect mirror image.

u u

NN /

arctan( ) ) (3.62b) )

3.44 Spontaneous emission 67 7

2.5 5

2.0 0

1.5 5

1.0 0

0.5 5

0.0 0

11 1

(a) )

ii i

\ \

I I

11 1 '

JJ ; 11 1 1

-2 2 Positionn (units of a) Positionn (units of a)

Figuree 3.5: Spontaneous-emissionn rates of dipoles near a single partially transmitting plane scatterer, relative to F0.. Solid lines correspond to total spontaneous-emission rates rx. for dipoles parallel to the plane, dottedd lines are radiative contributions to Tx, and the dashed lines denote Tz. In (a), the effective thicknesss Deff of the plane equals 0.46a and in (b) Dea = 10a.

Inn our model, effects on spontaneous emission become more pronounced when the pa-rameterr £ is larger and this can be accomplished by choosing a higher frequency. However, thiss only makes sense physically in the frequency regime where we can neglect dispersion inn the dielectric function e(ui) of the dielectric slabs that are being modelled. Otherwise, thee dispersion shows up in the effective thickness Defi = d[e(oj) — 1], with the effect thatt there is a frequency for which the parameter £ is maximal. Dielectrics do not make perfectt mirrors for high frequencies. Theoretically, this point is important for a consistent descriptionn of quantum optics in dielectrics [79].

Inn figure 3.5(a), spontaneous-emission rates as a function of position are plotted for Defii = 0.46a. The values for z = 0 are finite in both cases (although not shown in figurefigure 3.5) and indeed correspond to Eqs. (3.61) and (3.62a). For both dipole orientations, farr away from the plane the rate approaches the free-space value. Close to the plane, Tzz is increased; the Tx is also larger than To, but this decay-rate consists of a rate into propagatingg modes that is less than To and a guided-mode-rate. The contributions of radiativee and guided s-waves for an atom with fj, = /ix are the same as for scalar waves withh "scalar dipole moment" fi, but since the total decay-rate T0 for vector waves is larger thann for scalar waves, the relative contributions of ,s-waves to r /To are smaller for vector wavess (by a factor 3/4). In figure 3.5(b), the same rates are plotted, but now for a much betterr reflecting plane with an effective thickness of Deff = 10a. There the limiting values off spontaneous-emission rates near the plane are approached: Tz climbs up but does not reachh the value of 2r0 yet, whereas the propagating part of Tx is practically zero when Deff equalss 10a. The partial emission-rate into the guided mode has a much larger amplitude nearr the plane but falls off much more rapidly away from the plane.


3.4.33 Spontaneous emission near a ten-plane scatterer InIn our formalism we can study spontaneous-emission rates in the vicinity of an arbitrary numberr of planes. Here we choose to present results for N — 10, as we did for scalar waves.. The spontaneous-emission rates calculated in this section are based on Eqs. (3.59a-3.59d).. The following Dyson-Schwinger equation gives the relevant Green functions in termss of the T-matrix of section 3.2.6:

N N

G(k||,z,z',u;)) = G0(k||,2.2/,u;)+ ^ Goik^z.Za^yT^ik^^-Goik^.za.z'.uj). a.0=1 a.0=1

(3.63) ) Thee first thing that one should like to do in a formalism for vector waves is to compare the totall emission rates for dipoles parallel (Tx, solid lines) with dipoles perpendicular (Tz, dashedd lines) to the planes. In figure 3.6(a-d) orientation-dependent spontaneous-emission ratess are plotted for several frequencies. For clarity in the pictures, the positions of the planess at 1.2,.... 10 are not shown as vertical lines this time. The most striking difference iss that Tx becomes very spiky near the planes because of the guided modes to which only parallell dipoles can couple in the vicinity of the planes. We can also see that Tz, which iss purely radiative, on average increases due to the presence of the planes whereas the radiativee part of Tx on average decreases near the planes. In the previous section we saw thee same trend near a single plane where in the limit of a perfect mirror rx /To —> 0 andd Tz/T0 —• 2. Figure 3.1 showed that the optical modes of p-polarized light have discontinuitiess at the plane positions. There are also discontinuities in the spontaneous-emissionn rates, but these discontinuities are too small to be visible in figure 3.6. They are smalll because the discontinuities per mode are averaged in the emission rate.

Thee dotted lines in figure 3.6 are the radiative parts of Tx. These are similar to the radiativee rates for scalar waves, but not identical since in r x not only s-polarized light but alsoo p-polarized light contributes. In particular, far away from the planes, the emission ratee of dipoles parallel to the planes consists of 75 percent «-polarized and 25 percent p-polarizedd light.

Forr scalar waves the ten-plane structure can act an omnidirectional mirror, whereas in sectionn 3.3.1 it was found that it is not an omnidirectional mirror for vector waves. Correspondingly,, the radiative LDOS for scalar waves at a/A = 0.5 dropped down to (almost) zeroo inside the ten-plane omni-directional mirror, whereas the emission rates in figure 3.6(c)) show that the radiative LDOS for vector waves stays nonzero inside the crystal. In thee inner unit cells, dipoles parallel to the planes emit predominantly guided light, but the smalll amount of light that leaves the structure is strongly p-polarized. This is the case aroundd a/A = 0.5 only, where s-polarized light is omnidirectionally reflected; at higher andd lower frequencies the emitted light can have both polarization directions.

3.55 Radiative line shifts

Nott only spontaneous-decay rates change inside a dielectric medium, but there are also radiativee shifts of the atomic transition frequencies. Both effects will be derived in detail

3.55 Radiative line shifts 69 9

-i-i 1 r

w~-w~-_ll i L

00 5 10 15

Positionn (units of a)

00 5 10 15

Positionn (units of a)

(c) )

wv\4 4 pf^'nil l (d) )

-\-\ ' 1 ' 1 ' r

Jj^W s s

10 0 15 5 10 0 15 5

Positionn (units of a) Positionn (units of a)

Figuree 3.6: Spontaneous-emissionn rates Fx (solid line) and Tz (dashed) near a ten-plane crystal. The dotted line iss the radiative part of Tx. The figures (a)-(d) correspond to four frequencies: (a) a/A = 0.2; (b) a/AA = 0.4; (c) a/A = 0.5; (d) a/A = 0.6.

inn section 5.3.1. As given in Eq. (5.38), the atomic transition frequency in a medium will bee shifted with respect to the value ft in free space by an amount

A(R,n) ) /'" " heocheoc2 2 Re/ii • [G(R, R, ft) - G0(R, R. ft)] • ft, (3.64) )

whichh depends both on position and frequency. In a T-matrix formalism, the Green functionn is the sum of the free-space Green function and the term Gn • T • Go, see Eq. (3.63). Linee shifts are therefore determined by the latter term only. Proceeding much the same wayy as for the decay rates in section 3.4.1, the line shift at position R = (x,y, z) in a crystall of plane scatterers is found to be

A(R,ft) ) /3cTo o VV 2ft

Re e 2 ^ [ s ++ v)+ M2 (3.65) )


11 2 3 Distancee (units of a)

11 2 3 Distancee (units of a)

Figuree 3.7: Linee shifts of the optical transition frequency relative to the free-space value Q. Shifts are given as a functionn of the distance of the atom to a single plane, both for atoms with dipole moments pointing parallell (solid lines) and perpendicular (dashed lines) to the plane. Shifts are scaled to the free-space decayy rate IV In (a), Dea = 0.46a and in (b) Deff = 10a. In both figures, a/A = 0.5.

wheree the integrals Is, ƒ,, and Iz are given by

/•OO O

II ss = V / dkfyGs0s(kll.z,za,n)T^(kll,n)Gs

0s(kll, Zf,,z,Sl), (3.66)

a/33 J° rOC rOC

II vv = V / dkfyGlvm.z,za.Ü)Tr0{khÜ)Glv{khzp,z^), (3.67) a/33 Jo

/•OO O

II zz = V / dkfyGzov(khz,za,n)Tr0(khn)Gv

oz(khz0,z,Ü). (3.68)

a/33 J°

Inn these integrals, the magnitude ku of the wave vector parallel to the planes goes from zeroo to infinity. The matrix elements T^Jk\\. u>) in the integral Is have poles on the real-fc|lfc|l axis corresponding to the guided modes. The real part of Is can be evaluated by taking thee Cauchy principal-value integral at these poles. Although there are no guided modes forr p-polarized light, there is a nonzero contribution to the integral Iz from in-plane wave vectorss larger than fi/c. There was no such contribution to the spontaneous-emission rate TTzz in Eq. (3.59d). The difference stems from the fact that the decay rate is an on-shell quantity,, whereas line shifts A(fi ) have off-shell contributions. In other words, unlike the ratee r(Q), the shift A(Q) also depends on optical modes with eigenfrequencies different fromm £7. Actually, it is a weighted sum over all optical modes.

Inn figure 3.7(a), line shifts are presented as a function of distance to a single plane, for dipoless pointing parallel and perpendicular to the plane. Away from the plane, the shifts showw damped oscillatory behavior. The magnitudes are very small, typically ten percent orr less of the free-space decay rate. Close to the planes, the off-shell contributions of largee wave vectors become important. For both dipole orientations, the line shifts even

3.66 Conclusions and outlook 71 1

divergee when approaching the plane. At distances on the order of 0.1 A and smaller, the perpendicularr dipoles feel larger frequency shifts than the parallel ones. On the other hand,, far away from the plane the shifts are larger for parallel dipoles. Figure 3.7(b) shows shiftss near a plane with much larger effective thickness. The amplitudes of the shifts are increasedd as compared to figure 3.7(a), but the overall picture has not changed.

Inn order to avoid the infinite line shifts at the position of the plane, the cutoff parameter AA could have been kept finite. Then the discontinuities in modes and in Green functions wouldd have become smooth. By keeping A > il/c finite, only the line shifts at distances off typically A" 1 or less are modified and become finite. The emission rates presented in thiss chapter already were finite and they will not be affected either at distances greater than A - 11 when a finite cutoff is chosen to keep line shifts finite.

3.66 Conclusions and outlook

Inn this chapter a multiple-scattering theory was set up for the scattering of vector waves by parallell planes. The Green function had to be regularized and this was done by introducing aa high-momentum cutoff. In the end, the regularization parameter was sent to infinity. An effectivee scattering theory for plane scatterers emerged with a nonzero T-matrix that no longerr depends on the cutoff. This is in contrast with point scatterers where two cutoffs mustt be kept finite in order to define a nonzero T matrix [69]. The only reason to keep a finitefinite cutoff for plane scatterers would be to avoid diverging line shifts at the positions of thee planes.

Thee planar symmetry of the crystal of parallel planes enabled us to separate ,s- and p-polarizedd modes. The scattering by a nonabsorbing plane scatterer satisfies a separate opticall theorem for each polarization. The radiative and guided modes of s-polarized lightt could be mapped onto modes for scalar waves. The .s-polarized light has continuous modee functions, whereas the mode function of the /^-polarized light is discontinuous at thee positions of the planes. As a consequence, the p-polarization contributions to the spontaneouss emission rate show discontinuities as well. Spontaneous emission rates were calculated,, both for dipoles parallel and perpendicular to the planes.

Onee could wish to create a crystal such that lOO/f of the spontaneously emitted light iss guided light. In our model of identical equidistant plane scatterers this will not happen. Iff there was only s-polarized light, then it would be possible, as it was for scalar waves [figuree 2.7(a)]. Deep inside an omnidirectional mirror all spontaneously emitted light iss guided light. For vector waves, however, equidistant plane scatterers can not be an omnidirectionall mirror.

Omnidirectionall mirrors consisting of dielectric layers do exist [59,60,80). The oc-curencee of omnidirectional reflection is promoted by the fact that a single dielectric layer andd a multilayer stack both reflect p-polarized light at the Brewster angle [55]. This is to bee contrasted with our plane scatterer model where the Brewster angle is shifted away to 90°.. Therefore, the Brewster effect does help slabs but does not help planes to create the omnidirectionall reflector.

Whatt use can plane scatterers be in the future? One obvious idea is to study crystals builtt up of non-parallel planes. This turns out to be problematic, however, not so much

722 ^ Spontaneous emission of vector waves in crystals of plane scatterers

becausee the planes have an overlap of measure zero, but rather because the scattering of eachh plane as a whole must be described in its own plane representation. For that reason onee can not treat the individual planes as wholes. Instead, one is forced to think of the planess as being built up of infinitely many line scatterers or point scatterers.

Albeitt for parallel planes, the plane-scattering formalism can be used to study much moree than was presented here. With the knowledge of the complete Green function, one couldd calculate near-field and far-field spectra. One could study the effect of more than onee plane in a unit cell. Furthermore, the planes need not be identical and in fact the result (3.44)) for the T-matrix also holds for planes chosen at arbitrary positions with different effectivee thicknesses. The model can therefore be used to study the effect of disorder, bothh in the positions and in the effective thicknesses of individual planes. Disorder can be increasedd further to study random dielectrics rather than crystals of plane scatterers.

Crystalss of plane scatterers can also be used as a model system to study the modifi-cationn of several quantum optical processes of embedded atoms. Transient effects in the spontaneous-emissionn rates are a first example, thereby generalizing work done on a one-dimensionall cavity formed by two planes [81]. It is also interesting to study (two-atom) superradiancee modified by the crystal (see chapter 5). Or one could generalize the formal-ismm by allowing light absorption or gain in the planes, by choosing the effective thickness complexx and frequency-dependent. In that case, a quantum optical description would re-quiree the identification of quantum noise operators (see chapter 6) for the planes. Also, Casimirr forces can be calculated between an arbitrary number of imperfect mirrors. For twoo passive mirrors in one dimension, these forces are always attractive [82]. For planes withh optical gain, the Casimir force might change sign.

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