Decay of excited molecules in absorbing planar cavities

PHYSICAL REVIEW A

VOLUME 56, NUMBER 5

NOVEMBER 1997

Decay of excited molecules in absorbing planar cavities M. S. Tomasˇ1 and Z. Lenac2 1

Rudjer Bosˇkovic´ Institute, P.O. Box 1016, 10001 Zagreb, Croatia Pedagogical Faculty, University of Rijeka, 51000 Rijeka, Croatia ~Received 16 June 1997!

2

The decay rate of an excited molecule ~atom! embedded in a dispersive and absorbing planar cavity is derived by using a recently obtained compact form of the Green’s function for a multilayer. As a by-product, a hint is provided for a straightforward extension of the results obtained for lossless cavities of other shapes to the corresponding absorbing cavities. The decay rate in an absorbing cavity consists of the spontaneous emission rate and of the nonradiative rates caused by the near-field interaction of the molecule with the cavity medium and, for nearby molecules, with the cavity mirrors. Only the spontaneous emission rate is satisfactorily described in the macroscopic approach adopted. The theory is applied to an analysis of the effects of the weak cavity absorption on the decay rate in a dielectric microcavity formed by two metallic mirrors. As expected, dissipation in the cavity medium spoils the conditions for controlled spontaneuos emission and strongly suppresses the intensity of spontaneous emission. However, its effect on the spontaneous emission rate is much less pronounced. @S1050-2947~97!03411-2# PACS number~s!: 42.50.Lc, 42.60.Da, 33.50.2j

I. INTRODUCTION

The development of various light-emitting microdevices based on controlled spontaneous emission ~SE! has raised the problem of a theoretical description of spontaneous relaxation of excited molecules ~atoms! in realistic, i.e., dispersive and absorbing cavities. Macroscopic electrodynamics is clearly a natural framework in which to deal with such systems. However, it is only very recently that, based on the result of the canonical field quantization for a microscopic model of a dielectric @1,2#, a scheme has been proposed for quantization of the macroscopic field in dispersive and absorbing inhomogeneous systems @3–5#. So far, this program has been fulfilled only for waves propagating along the normal to simple multilayers, i.e., effectively one-dimensional systems @4,5#. Since the coupling of the molecule to all possible waves supported by the system has to be considered to describe its decay correctly, the appropriate macroscopic approach to this goal is therefore quantum-mechanical linearresponse theory in conjuction with the fluctuation-dissipation theorem, familiar from the theory of spontaneous emission in the presence of ~absorbing! boundaries @6,7#, or the classical theory developed in the context of molecular fluorescence and energy transfer at interfaces @8–11#. At zero temperature these methods give the same result for the ~normalized! molecular decay rate and can be easily extended to fully absorptive systems including also the cavity interior @12–14#. So far, only the SE rate @12# and the total decay rate @14,15# of the molecule in an infinite absorbing cavity ~medium! have been considered in more detail. In our previous work @13# we pointed out that a straightforward extension of a classical result for the molecular decay rate in a transparent layer @10,11# should describe the molecular decay rate in a fully absorbing multilayered system. In the subsequent development, however, we have restricted our attention to a transparent cavity case. Recently, Lee and Yamanishi @16# presented a theory on the SE rate in absorbing inhomoge1050-2947/97/56~5!/4197~10!/$10.00

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neous dielectrics based on a photon Green’s-function formalism. In addition to not accounting for the near-field interactions of the molecule, their approach rests heavily on the calculation of a Green’s function of the system. In a very recent related work, Nha and Jhe @17# generalized the theory of molecular decay by Wylie and Sipe @7# to finite but transparent ~empty! planar cavites. In this paper, using a recently obtained compact form of the Green’s function for a multilayer @18#, we reconsider and complement our approach in @13# and develop a theory of the molecular decay in an absorbing planar cavity or, generally, a multilayer. In this way, we simultaneously extend the theory of Wylie and Sipe @7# to finite and absorbing cavities, as well as the theory of Barnett et al. @12,14# to finite planar cavities. The paper is organized as follows. In Sec. II we establish a relationship between the classical theory of @13# and the QED approach described and define the normalized rate. Throughout the paper we exploit this equivalence of the ~normalized! rate in the two approaches and use ~mainly! the classical language to identify various contributions to the total rate. In order to clarify the method of calculation employed in Sec. IV, in Sec. III we briefly rederive a few basic results concerning the decay of a molecule embedded in an absorbing dielectric host. In Sec. IV we derive the molecular decay rate in absorbing planar cavities and discuss the relationship between this result and the results obtained previously. As an application of the theory, in Sec. V we consider the decay of an excited molecule embedded in a dielectric microcavity formed by two metallic mirrors under the circumstances of controlled SE and discuss the effects of cavity absorbtion on various contributions to the total molecular decay rate. Our conclusions are summarized in Sec. VI. II. PRELIMINARIES

Consider an excited molecule at a position r0 in a cavity. In the classical approach, the molecule is simulated by a 4197

© 1997 The American Physical Society

M. S. TOMASˇ AND Z. LENAC

4198

point dipole p oscillating at the frequency of the transition v. The associated current density j(r,t)52i v pd (r 2r0 )exp(2ivt) gives rise to the electric field in the system of the form E(r,t)5E(r,r0 ; v )exp(2ivt). The molecular decay rate G is then related via G5W/\ v to the power W ~ r0 ! 5

v Imp* •E~ r0 ,r0 ; v ! 2

~2.1!

lost by the dipole in supporting its own field. Introducing the Green’s function of the system through @19# E~ r,r0 ; v ! 5

v2 I G~ r,r0 ; v ! •p, c2

~2.2!

we have

v2 I ~ r ,r ; v ! •p, p* •ImG G ~ r0 ! 5 0 0 2\c 2

~2.3!

I (r ,r ; v ) is the diagonal where we have assumed that G 0 0 dyadic. The quantum-mechanical decay rate is obtained in the usual way starting from the molecule- ~total! field interaction ˆ 52pˆ•Eˆ(r ), usHamiltonian of the standard form @14# H int 0 ing the Fermi golden rule, and employing the fluctuationdissipation theorem. At zero temperature, one arrives at Eq. ~2.3! with p→2p f i , where p f i is the corresponding transition dipole matrix element @20#. Therefore, with this replacement in mind, we refer to Eq. ~2.3! as the QED rate as well. In general, G consists of the rate G S associated with losses W S owing to the quasistatic interaction of the molecule with its environment and the rate G T related to molecular losses W T due to its transverse ~retarded! interaction with the surroundings. The quantities W S (G S ) and W T (G T ) can be obtained separately from the above formulas through the quasistatic and transverse parts, respectively, of the total dipole field ~Green’s function!. As usual, we refer to G T as the spontaneous emission rate G SE and define the normalized molecular decay rate Gˆ with respect to the SE rate in the 0 corresponding infinite cavity G SE , i.e., Gˆ 5G/G 0T 5W/W 0T . Clearly, both the classical and the QED approach lead to the same result for Gˆ given by Gˆ ~ r0 ! 5

I ~ r ,r ; v ! •pˆ pˆ•ImG 0 0 I 0 ~ r ,r ; v ! •pˆ pˆ•ImG T 0 0

,

~2.4!

where now pˆ describes the direction of the transition. Note that eventual local-field corrections do not affect Gˆ for centrosymmetric cavity media and in this work, therefore, we ignore the difference between the local field actually acting on the molecule and the macroscopic field used in the above derivation. A cavity represents an inhomogeneous system that can be described by the position-dependent complex dielectric function «~r,v! defined in a piecewise fashion. One usually I ! in such solves for the full dipole field E ~Green’s function G a system. The dipole quasistatic field ES is then obtained by

56

letting c→` in this solution, i.e., ES 5limc→` E. Obviously, the difference ET 5E2ES represents the dipole transverse ~retarded! field. Indeed, since Gauss’s law holds for both E and ES , the field ET obeys the generalized transversality condition “•« ~ r, v ! ET ~ r,r0 ; v ! 50,

~2.5!

as appropriate for inhomogeneous systems @21#. According I to Eq. ~2.2!, the Green’s function for the transverse field G T is therefore given by @18# 2 2 I ~ r,r ; v ! 5G I ~ r,r ; v ! 2 c lim v G I ~ r,r ; v ! , G T 0 0 0 v 2 c→` c 2 ~2.6!

where the last term represents the Green’s function for the I comprises the contriI . By definition, G quasistatic field G S T butions of all polaritonic ~retarded! modes in the system. III. DECAY RATE IN AN INFINITE CAVITY

I 0 for an infinite medium is obThe Green’s function G tained by a straightforward generalization of the free-space Green’s function and reads @22#

F

ˆˆ I ˜R ! 2 4 p IId ~ R! I 0 ~ r,r ; v ! 5 1 3RR2I ~ 12ik G 0 2 3 ˜ k R 3

G

I ˆˆ ˜2 I2RR e i ˜k R , 1k R

~3.1!

ˆ 5R/R, and where II is the unit dyadic, R5r2r0 , R ˜ ˜8 ~ v ! 1ik ˜9 ~ v ! [ A« ~ v ! v 5 @ h ~ v ! 1i k ~ v !# v , k ~ v ! 5k c c ~3.2! with h and k being, respectively, the refractive index and the I 0 (r ,r ; v ) is extinction coefficient of the medium. Since G 0 0 ˆ to the isotropic tensor, Eq. ~3.1! has to be averaged over R ˆR ˆ & 5(1/3)II, one find its limit for R→0 correctly. With ^ R therefore has

F

˜

G

ikR I 0 ~ r ,r ; v ! 5 lim 2 4 p d ~ R! 1 2 e II. G 0 0 2 ˜ 3k 3 R R→0

~3.3!

In the spirit of the macroscopic field approach, we remove the singularity that appears in the quasistatic component of the dipole field by letting d R→0 (R)→1/V m , where V m 5(4 p /3)R 3m is an appropriately chosen spherical volume around the molecule. It is clear from Eq. ~3.1! that making this substitution is equivalent to averaging the quasistatic dipole field ~Green’s function! over V m , as suggested by Barnett et al. @14#. The total dipole power loss W 0 is therefore given through Eq. ~2.1! by

DECAY OF EXCITED MOLECULES IN ABSORBING . . .

56

F

W 0 5W 0S 1W 0T [Im 2

G

1 v u pu 2 v 4 u pu 2 , 3 1h~ v ! « ~ v ! 2R m 3c 3 ~3.4!

where the first term describes the molecular quasistatic ~nearfield! loss in the medium, whereas the second one gives the familiar transverse ~far-field! loss of the molecule, i.e., the spontaneous emission power. The quantity R m can be considered as an effective molecule-medium distance @14#; this interpretation is also suggested by a very recent microscopic result for the molecular near-field loss @15#. Clearly, W 0S is due to the excitation of longitudinal modes in the medium and vanishes for nonabsorbing (« 9 50) media. Since pˆ and ˆ enter symmetrically into the expression for W 0 , note that R Eq. ~3.4! could have been obtained by averaging W 0 over pˆ I 0 over R ˆ . This method of calculating instead of averaging G 0 W will prove useful in Sec. IV. We stress that the need of averaging the Green’s function ˆ concerns its quasistatic ~longitudinal! component and over R therefore the above procedure is unnecessary if solely the SE rate is to be obtained. Indeed, the application of the recipe ~2.6! to Eq. ~3.1! leads to ˆˆ I ˜R ! e i ˜k R 21 # I 0 ~ r,r ; v ! 5 1 3RR2I @~ 12ik G 0 T 2 3 ˜ k R 1

4199

II2R ˆR ˆ R

˜

e i k R.

b l 5 A˜ k 2l 2k 2 5 b 8l 1i b 9l ,

4 v 3u pf iu 2 , 3\c 3

~4.1!

The Green’s-function element relating an observation point r in the cavity to a source point r0 in the cavity then reads

(

0,z, z 0 ,d j

~3.5! I 8 ~ r,r ; v ! 5 i G 0 q 2p

E

d 2k e ib jd j b j q Dqj

, 3 @ E. q j ~ k, v ;z ! Eq j ~ 2k, v ;z 0 ! u ~ z2z 0 !

~3.6!

. 1E, q j ~ k, v ;z ! Eq j ~ 2k, v ;z 0 ! u ~ z 0 2z !#

Through Eq. ~2.3! this gives for the SE rate in absorbing media 0 G SE 5h~ v !

b 8l >0, b 9l >0.

I 8 ~ r,r ; v ! , I ~ r,r ; v ! 52 4 p zˆzˆd ~ r2r ! 1 G G 0 0 0 q ˜ q5p,s k 2j

Expanding this for small R, one directly finds ˜8 ~ v ! 2 II. I 0 ~ r ,r ; v ! 5k ImG T 0 0 3

FIG. 1. System considered schematically. « l 5n 2l are the dielectric functions of the layers.

3e ik• ~ r2 r0 ! .

~4.2!

Here r[( r,z), j p 51, j s 521, and ~3.7!

as obtained by Barnett et al. @12#. IV. DECAY RATE IN A PLANAR CAVITY

Consider an absorbing planar cavity ( j) formed by two generally multilayered lossy mirrors with an excited molecule embedded in it, as depicted in Fig. 1. A detailed derivation of a convenient plane-wave expansion of the Green’s function for such a multilayer was presented in Ref. @18#. Here we quote only the Green’s function element relevant to the present problem.

D q j 512r qj2 r qj1 e 2i b j d  ,

with r qj6 [r qj/n(0) being the reflection coefficients of the upper ~lower! cavity mirror. These coefficients obey the usual . recurrence relations @18#. The functions E, q j and Eq j describe the z dependence of the electric field in the cavity of a q 5p polarized or a q5s polarized plane wave of unit strength incident on the system from its upper ~downward! and lower ~upward! side, respectively. They are given by 7

7

2i b j z ib jz ˆ7 E" 1r qj7 eˆ6 , q j ~ k, v ;z ! 5eq j ~ k ! e q j ~ k! e

z 2 [z,

A. Green’s function

Denoting the ~conserved! wave vector parallel to the system surfaces by k5(k x ,k y ), we write the wave vector of an upward ~downward! propagating wave in an lth layer as ˆ K6 l 5k6 b l z, where

~4.3!

eˆ7 p j ~ k! 5

1 ~ 6 b j kˆ1kzˆ! , ˜ kj

z 1 [d j 2z

ˆ ˆ eˆ7 s j ~ k ! 5k3z,

~4.4!

M. S. TOMASˇ AND Z. LENAC

4200

where eˆ7 q j are the orthonormal ~complex! polarization vectors associated with the downward ~upward! propagating wave in the cavity. B. Molecular decay rate

We insert the Green’s function ~4.2! into Eq. ~2.1! @or Eq. ~2.3!# and let r→r0 , remembering, however, that the terms in W corresponding to the infinite-cavity case must be averaged over pˆ. Then, as before, the singular term in the Green’s function gives the molecular quasistatic bulk loss W 0S . One can therefore write ¯ ~ z !, W ~ z 0 ! 5W 0S 1W 0

~4.5!

¯ is given through the nonsingular part of the Green’s where W function and, in most cases, gives the only contribution to the molecular loss. Omitting hereafter the index j denoting the cavity, we have 3

¯ ~ z ! 5W 0 Re W 0 T

4p

E

d 2k ˜ k 8b

(

q5 p,s

jq

e ibd Dq

ˆ , ˆ 3E. q ~ k, v ;z 0 ! •p Eq ~ 2k, v ;z 0 ! •p,

~4.6!

where W 0T is classically given by Eq. ~3.4! or quantum me0 of Eq. ~3.7!. Developing E’s along kˆ, chanically by \ v G SE ˆz, and kˆ3zˆ and performing the angular integration (kˆ 5coswxˆ1sinwyˆ), we obtain ¯ ~ z ! 5W 0 Re 3 W 0 T 4

E

`

dkk ˜ k 8b

0

HF

1 b p 2i b z 2 0 ! e ~ 12r 2 2 ˜ D k

1

Ds

~ 11r s2 e

2 2i b z 0

! (11r s1 e

1 2i b z 0

G

p

) pˆ2i

J

where pˆi and pˆ' describe the orientation of the transition dipole relative to the cavity mirrors. To extract the remaining ¯ 0 from W ¯ , we set r q 50 in Eq. ~4.7!. infinite-cavity part W 6 This gives

E

`

0

dk k ˜ k 8b

FS D

~4.9!

where W sc comes from the molecular ~dipole! field scattered from the cavity walls and is obtained by substracting Eq. ~4.8! from Eq. ~4.7!. This finally leads to the normalized molecular decay rate Gˆ 5W/W 0T in the cavity: Gˆ ~ z 0 ! 5Gˆ 0S 111Gˆ sc~ z 0 ! ,

~4.10a!

where

S D

c 3k~ v ! Gˆ 0S ~ z 0 ! 5 2 u «~ v !u v R m

3 Gˆ sc~ z 0 ! 5Re 4

1 2k 2 1 p 2i b z 2 p 0 !~ 11r e 2i b z 0 ! p ˆ'2 , ~4.7! 1 e ~ 11r 2 1 2 ˜ D k

¯ 0 5W 0 Re 3 W T 4

¯ ~ z ! 5W 0 1W sc~ z ! , W 0 0 T

3

~4.10b!

is the normalized molecular decay rate due to the quasistatic losses in the bulk of the cavity @14# and

p

1

1

explicitly. This can be easily verified by noting that, in this case, the operation Re cuts the above integral at ˜ k. Equation ~4.7! is of a form well known from the theory of molecular fluorescence and energy transfer in transparent layers @10,11#. In @13# we obtained it by making a direct generalization of these classical results for simple systems and conjectured that it also described the molecular power loss in an arbitrary absorbing multilayered system. The present derivation justifies this approach provided, however, that the infinite-cavity part of the total loss is handled more carefully. Actually, according to the above discussion, Eq. ~4.7! correctly describes the molecular loss not only for transparent cavities but ~i! for a freely rotating ~unoriented! ¯ over pˆ is needed, and ~ii! if molecule, where averaging W ¯ solely the SE power is considered, in which case W ¯ 2limc→` W is to be calculated. A generally valid result is, however,

2

p 2i b z 0 3 ~ 12r 1 e !

56

G

b2 2k 2 2 ˆ 2i 1 11 p pˆ' . ˜ ˜ k2 k2

~4.8!

Of course, upon prescribed averaging @^ pˆ2i & 52/3 and ^ pˆ'2 & ¯ 0 becomes W 0 , as expected. Note that 51/3#, the quantity W T 0 ¯ W , as it stands, diverges for absorbing cavities owing to the appearance of a nonphysical quasistatic contibution at the upper limit of integration in Eq. ~4.8!. Therefore, taking its ¯ 0 2lim ¯0 transverse part W c→` W also leads to the correct re˜ is real! cavity, however, W ¯ 0 5W 0 sult. For a transparent ~k T

E

`

0

dk k ˜ k 8b

HF

1

1

p 2i b z 0 2r 1 e !1

1

G

Ds

1r s1 e 2i b z 0 ! pˆ2i 1 1

p 2i b z 0 ˆ 2 1r 1 e ! p'

J

b2 1 p p 2i b d p 2i b z 2 0 r 1e 2r 2 e ~ 2r 2 2 ˜ k Dp 2

~ 2r s2 r s1 e 2i b d 1r s2 e 2i b z 0

2k 2 1 p p 2i b d p 2i b z 2 0 r 1e 1r 2 e ~ 2r 2 2 ˜ k Dp ~4.10c!

is the normalized total cavity-induced decay rate of the molecule. Clearly, W sc5W 0T Gˆ sc fully describes the effect of field confinement in the cavity on the decay rate as well as molecular losses due to its interaction with the cavity mirrors. The corresponding quasistatic losses W sc S are obtained from Eq. ~4.10c! in the c→` limit. As seen, these losses are due to the excitation of p-polarized waves with the large wave vectors k ( b l 5ik) and, according to Eq. ~4.1!, cannot be explicitly separated from the retarded losses in W sc. Therefore, further extraction of the SE contribution to Gˆ can be

56

DECAY OF EXCITED MOLECULES IN ABSORBING . . .

done only numerically for realistic cavities. This can be performed, e.g., by calculating the transverse part of Eq. ~4.10! numerically:

v c Gˆ SE~ z 0 ! 511Gˆ sc~ z 0 ! 2 3 lim 3 Gˆ sc~ z 0 ! , v c→` c 3

3

~4.11!

where the last term on the right-hand side represents the sc 0 normalized molecular quasistatic rate Gˆ sc S 5W S /W T ~see the Appendix!. Alternatively, one may simply omit G 0S and cut the integral in Eq. ~4.10c! at a wave vector K that is larger than the wave vector of any polaritonic mode in the system. We note, however, that W sc S is significant only for molecules ˜8 z ,1) to a lossy mirror. It has been discussed in a close (k 0 number of papers concerning molecular energy transfer at interfaces, e.g., in the context of molecular fluorescence near a mirror @10,11# or in surface physics @23#, and its proper account demands a microscopic description of the corresponding mirror ~interface!. For molecules at larger distances from the mirrors, W sc S may, in a very good approximation, be neglected and for a lossless system it vanishes. Equation ~4.10! is the main result of this work. It extends the early classical results for the molecular decay rate in a transparent layers @10,11# to fully absorbing multilayered systems. Also, it provides a generalization of the corresponding QED result ~r q1 50 and «51! derived by Wylie and Sipe @7# to finite and absorbing cavities as well as that (r q6 50) of Barnett et al. @14# to finite planar cavities. To make contact with other QED theories, we return to Eq. ~4.7! and, as in @13#, use the identity 1 2 1 2 D q 5 @~ 12r q2 e 2i b z 0 !~ 11r q1 e 2i b z 0 ! 1 ~ 11r q2 e 2i b z 0 ! 2 1

3 ~ 12r q1 e 2i b z 0 !#

6

q 2 24 b 9 z I6 12i Imr q6 e 2i b z q ~ k,z ! 512 u r 6 u e

ˆ ~ z ! 5 3 Re W 0 4 1

E

`

0

dk k ˜ k 8b

HF

G

b i F * ~ k,z 0 ! 1F s ~ k,z 0 ! pˆ2i 2 p ˜ k

J

2k 2 ' F p ~ k,z 0 ! pˆ'2 , ˜ k2

F p ~ k,z 0 ! 5

1 p 2i b z 2 0 u 2 1 $ 6→7 % # , e @ I 1 ~ k,z 0 ! u 12r 2 2 u D pu 2 p

F'p ~ k,z 0 ! 5

1 1 p 2i b z 2 0 u 2 1 $ 6→7 % # , 2 @ I p ~ k,z 0 ! u 11r 2 e 2 u D pu

i

F s ~ k,z 0 ! 5

where

1 2 @ I 1 ~ k,z 0 ! u 11r s2 e 2i b z 0 u 2 1 $ 6→7 % # , 2 u D su 2 s ~4.12!

6

~4.13!

and the symbol $ 6→7 % denotes the expression in the corresponding bracket with changed subscripts and superscripts. Since for a transparent cavity we have, for example,

H

12 u r q6 u 2 , 1 6 1 Re I q ~ k,z 0 ! 5 3 6 b ubu 2 Imr q6 e 22 b 9 z 0 ,

˜ k