PHYSICAL REVIEW B 68, 155118 ~2003!

Probing method for the density of states in a photonic crystal with luminescent molecules

Xiao-Dong Liu,1,2 Xiang-Dong Zhang,3 Yi-Quan Wang,1 Bing-Ying Cheng,1 and Dao-Zhong Zhang1

1Laboratory of Optical Physics, Institute of Physics, Chinese Academy of Sciences, Beijing, 100080, China2Department of Electromechanical and Information Engineering, Dalian Nationalities University, Dalian 116600, China

3Department of Physics, Beijing Normal University, Beijing, 100875, China~Received 2 April 2003; revised manuscript received 25 June 2003; published 27 October 2003!

A decomposition scheme for probing the total density of states~DOS! in an arbitrary photonic crystal~PC!having luminescent molecules is suggested. Some number distributions of the luminescent molecules canpartially probe DOS information or even all the details of the total DOS by measuring the dynamic character-istics of the spontaneous decay of excited luminescent molecules in a PC. One can probe the total DOS directlywhen the ratio of the number densities of the homogeneously dispersed luminescent molecules in the scatterersand in the background material is equal to the ratio of the dielectric constants of the scatterer and thebackground material.

DOI: 10.1103/PhysRevB.68.155118 PACS number~s!: 42.70.Qs, 42.50.Ct, 33.50.Dq

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I. INTRODUCTION

In 1946, Purcell predicted, for the first time to our knowedges that an electromagnetic environment can modifyspontaneous emission spectrum of an excited atommolecule.1 At the end of the 1980s, the photonic crystal~PC!,which is an artificial periodic material, was proposed ascandidate to control the spontaneous emission of an excatom or molecule2 and to observe the strong localizationphotons.3 For these two fields of potential applicationsPCs, the fabrication of a PC, especially one working atnear-infrared frequencies of optical communications,been an active research goal for more than a decade,definitive successes have been achieved in the fabricaoptical characterization, and application of PCs, along wmuch technical progress in the microfabrication of semicductors in recent years.4

In the mean time, the theoretical research on controllthe quantum electrodynamic features of atoms or molecin a PC—for example, enhancing or suppressing the sponeous emission rates of a collection of excited atomsmolecules—has also made much progress such as finatom-photon bounding states, the oscillating behaviorspontaneous emission rates of an excited atom, the abnosuper-radiation rate, etc.5 Generally speaking, the modification of the spontaneous emission rate of an excited atomPC depends on its specific energy structure and intrinsiccay characteristics and also directly on the local densitystates~DOS! at the atom’s location in the PC~we call thetheory with this viewpoint the theory of local DOS,6,7 whichcan rightly interpret the modification of spontaneous emsion rates!. But many theoretical authors have been repsenting this modification using simply the total DOS.8 Theo-rists also have used the total DOS to corroborate experimthat showed a great reduction of the spontaneous emisspectrum of luminescent molecules in a PC wpseudogaps.9

It is now well known that, for the problem of spontaneoemission from an active atom or molecule, total DOS modgenerally fail to account for quantum optical phenomenasuch particles in three-dimensional photonic crystals. Si

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larly, total DOS models are only approximately valid to demonstrate the modified spontaneous emission spectra of alection of dependently emitting atoms or molecules wessentially random dipole orientations in a PC that has wpseudogaps but not appropriate in a PC with stropseudogaps and complete photonic band gaps~we call thismethod of interpreting the modification of the spontaneoemission rates thetheory of total DOS approximation!.10

Naturally, both the theory of the total DOS approximatioand the theory of the local DOS are correct; i.e., measurthe decay behavior of a single excited atom or moleculereflect only the local DOS in a PC—it cannot probe the toDOS in the PC.

This being the case, how can we directly probe the toDOS in a PC that has strong pseudogaps and completetonic band gaps by using the significant modification of tspontaneous emission spectra of excited atoms or molecuThis paper will demonstrate, for the first time to our knowedge, that one can directly probe some parts or evenwhole of the total DOS in an arbitrary PC by placing aappropriate number distribution of excited luminescent mecules in the PC and measuring the decay rates of thparticles. Using this probing method, we can make clmany experimental results with our theory of the total DO

II. THEORETICAL FORMALISM

Conventionally, for the problem of the spontaneous emsion of a luminescent molecule in PCs, the excited statecays to the ground state with the decay rate being11

G52pr l~u,r0 ,v!, ~1!

andr l(u,r0 ,v) is the generalized local DOS~GLDOS! de-fined as follows:7,11

r l~u,r0 ,v!5v0

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2vn,k!uu̇"En,k~r0!u2, ~2!

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LIU, ZHANG, WANG, CHENG, AND ZHANG PHYSICAL REVIEW B68, 155118 ~2003!

whereu and r0 are, respectively, the dipole matrix elemevector for the atomic or molecule transition and the positof the atomic or molecule; 1BZ denotes the first Brillouzone of the PC. Equation~1! shows that the decay constaof a luminescent molecule is proportional to the generalilocal DOS, which consists of a contribution of the atomdipole u, and is a natural reflection of light-atominteraction.11 However, given a collection of independentemitting atoms or molecules with essentially random diporientations in a small volume aroundr0 , the mean emissioncharacteristics of the system can be described, after aveing over all solid anglesV, as7

G52p1

4p^r l~u,r0 ,v!&V[C1r~r0 ,v!, ~3!

whereC15pu02v0

2/(3\«0v), in which u0 is the magnitudeof the dipole matrix element vector,\ is Plank constant,«0 isthe dielectric permittivity in vacuum,v0 is the atomic tran-sition frequency, andr(r0 ,v) is the common local density ostates~LDOS! defined as

r~r0 ,v!51

~2p!3 (nE

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~4!

which is only related to the macroscopic structure for eltromagnetic waves in a PC. Equations~3! and ~4! say thatone can directly describe the LDOS in a PC with the sptaneous emission feature of a small volume of luminescmaterial. This is just the main viewpoint of the so-calltheory of the local DOSinterpreting the modification ospontaneous emission rates.

It is well known that the total DOS is defined to be6

r~v!5V

~2p!3 (nE

BZd3kd~v2vn,k!, ~5!

whereV is the volume of the Wigner-Seitz cell,vn,k repre-sents the photonic dispersive relation. Further, thetheory ofthe total DOS approximationshows that, considering the relationship of the total DOS and the LDOS, the total DOSthe «~r !-weighted average of the LDOS over the WigneSeitz cell,12,13

r~v!5EWSC

d3r«~r !r~r ,v!, ~6!

where«~r ! is the distribution function for dielectric constanin the PC.

Equation~6! shows that the total DOS cannot really drectly describe the quantum optical phenomena of a lanumber of atoms or molecules in a PC with a lardielectric-constant contrast. But for the case of a smallelectric modulation in the PC, which implies a weak interation between the dielectric and the electromagnetic field,total DOS can provide a reasonable description of the fielany point in the PC. Clearly, a PC exhibiting a stropseudogap or a full photonic band gap~PBG! does not sat-isfy such a condition. Therefore, one can say that the t

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DOS can be used in the problem of spontaneous emissioa collection of dependently emitting atoms or molecules wessentially random dipole orientations in a photonic cryswith weak pseudogaps. This is just the main viewpoint ofso-calledtheory of total DOS approximation, interpreting themodification of the spontaneous emission rates, andtheory agrees appropriately with most of the experimenresults in recent years by using thetotal DOSapproximation.9,14–20

III. DECOMPOSITION AND PROBING OF THE TOTALDOS

In fact, according to Wanget al.,12 a small collection ofindependently emitting atoms~molecules! with random di-pole orientations in a small volume aroundr0 or other posi-tions having the same GLDOS, whose decay kinetics,scribed by the decay rateG, depend only on the GLDOSthere, have a very narrow lifetime distribution~an approxi-mately single lifetime decay! and therefore can only probthe local DOS of the place where the atoms~molecules! arelocated. As a result, when some atoms~molecules! are dis-persed at many places, they can probe the contribution othe LDOS of the places where they are located, and havwider distribution of lifetime, e.g., the decay rate will bdistributed across a wider range. Then, if the entire photocrystal is filled with the emitting atoms~molecules!, the re-sulting decay rate can reflect the contribution of all tLDOS in the PC, depending on the distribution of the emting atoms~molecules!, and the width of the responding distribution of lifetime will be several orders of magnitude fosome strongly scattering systems, such as those with a cplete PBG. Because the total DOS is just an embodimenthe «~r !-weighted average of all the LDOS over the PCkind of special distribution of emitting atoms~molecules!should be used, and therefore in these cases the resudecay rateG can reflect the total DOS exactly, otherwisecan not be directly connected with the total DOS. In othwords, in order to probe the total DOS directly—i.e., let tmeasured decay parameterG be proportional to the totaDOS—one may introduce an appropriate number distrition of luminescent molecules into the PC and measuredecay rates of these excited luminescent molecules.present, it is still difficult to put a single or several luminecent molecules at a specific place in a real macroscopicaand probe such a weak radiation fluorescence. In all ofexperiments, one mixes a large amount of luminescent mecules into a small volume or even into the whole PC, pviding the possibility for us to probe the detailed informatioof the total DOS, or even the total DOS itself.

First, we may introduce a number distribution function(r ), which describes the density distribution of luminescemolecules in the PC. Then the decay rateG, which will de-fine a wide lifetime distribution rather than an average sptaneous emission decay constant, may be then(r )-weightedaverage of that of a single molecule according to Eq.~3!,i.e.,10

G5C1

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PROBING METHOD FOR THE DENSITY OF STATES IN . . . PHYSICAL REVIEW B68, 155118 ~2003!

where N is the total number of the luminescent molecuwithin the PC, and PC represents the whole photonic crysThe physical reasonability of this formula has been justifiprimarily by the work of Wanget al.,12 which has studied thelifetime distribution function for different spreading configurations of the atoms~or molecules! including the cases ohomogeneous dispersion of the atoms~molecules! over thebackground medium, on one or two different spherical sfaces inside the dielectric globules, and over the wholeelectric globules, and provided trustworthy clarification fsubstantial discrepancies in the recent reporexperiments.12

Second, we introducenumber distribution functionsna,b(r ), which describe the number density distributionsthe scatterers and background material~matrix! in the PCand are the same in every Wigner-Seitz cell, as shown in1. We also assume that the molecules’ distribution issame for each unit cell and that the introduction of the lumnescent molecules does not affect the dielectric functiontribution of the PC. Then the decay rateG discussed abovemay be decomposed as

G5C1

N0H E

Scad3rna~r !r~r ,v!1E

Matd3rnb~r !r~r ,v!J ,

~8!

whereN0 is the number of luminescent molecules withinWigner-Seitz cell.10 Sca and Mat stand for the integral rgions in the scatterers and in the matrix of the Wigner-Scell, respectively.

Decomposing the total DOS into two partsrSca(v) andrMat(v),

r~v!5S «aESca

d3rr~r ,v!1«bEMat

d3rr~r ,v! D[rSca~v!1rMat~v! ~9!

FIG. 1. The schematic Wigner-Seitz cell of a photonic crysrepresented by the hexagonal region. In this figure, the spheresresent the scatterers.na(r ) and «a are the number distribution ofluorescent molecules and the dielectric constant in the scatterespectively.nb(r ) and«b are the number distribution of fluorescemolecules and the dielectric constant in the matrix, respectivel

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allows one to proberSca(v) of the total DOS when luminescent molecules are immersed homogeneously in every sterer and none in the matrix—i.e.,na(r )5na.0 andnb(r )50; then

G5C1

N0S na

«aD rSca~v!}rSca~v!. ~10!

Otherwise one can proberMat(v) of the total DOS whenluminescent molecules are immersed homogeneously inmatrix and none in the scatterers—i.e.,nb(r )5nb.0 andna(r )50; then

G5C1

N0S nb

«bD rMat~v!}rMat~v!. ~11!

Here, it must be noticed that the separation of the DOS itwo parts, as done in Eq.~9! is merely an intermediate numerical procedure, that is,rSca(v) is the average contribution of the LDOS in the scatterers, andrMat(v) is the aver-age contribution of the LDOS in the matrix, just as the toDOS r~v! is the n(r ) weighted average contribution of thLDOS throughout the whole of the Wigner-Seitz unit cellthe PC. Physically each term has no meaning, becausetotal density of state is not an additive physical quantity tcan be divided into several parts occupying different spregions. The total DOS is the count of photon in a certfrequency range, and each photon can extend over the wspace. It is inappropriate to say that how many photonsinside the scatterers, and how many photons lies insidematrix materials.

Now, there is a very good case for us to suggest a probmethod of the total DOS using the number distribution sisfying the relation

na

«a5

nb

«b5C2 . ~12!

According to Eqs.~10! and ~11!, one obtains

G5C1C2

N0r~v!}r~v!. ~13!

From Eqs.~12! and ~13!, we can say that experimentallarranging the number distribution of luminescent molecusatisfying Eq.~12! in a PC can really probe the PC’s totDOS directly. Naturally, different number distributions of luminescent molecules probe the superposition ofrSca(v) andrMat(v);

G5C1

N0F S na

«aD rSca~v!1S nb

«bD rMat~v!G . ~14!

IV. DISCUSSION

The spontaneous emission spectra of most organic dhave a linewidth up to 10%, hence they are very well suifor probing the total DOS and the local DOS in a PC in tcorresponding frequency region. Up to now, almost evexperiment uses organic dyes with wide spontaneous e

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LIU, ZHANG, WANG, CHENG, AND ZHANG PHYSICAL REVIEW B68, 155118 ~2003!

sion spectra to investigate the modulation of spontaneemission spectra of dye molecules in a PC, although onfew experiments have been reported in the lastyears.9,14–20 These reported experiments can be classiinto the following cases according to the number distributof the dye molecules in a PC

~a! The dye, such as Nile blue,9,18 Rhodamine 6G~R6G!,17 was adsorbed near the external surface of the T2spheres; or CdS nanocrystals were adsorbed near the exsurface of thePMMA spheres.20

~b! Rhodamine isothiocyanate dye was located onspherical surface inside the dielectric globule SiO2 .17

~c! A controlled amount of Coumarin 6 dye was addhomogeneously toPMMA beads during the bead synthesis19

~d! Kiton red dye in water was embedded in an ordecolloidal suspension of polystyrene spheres;14 a NBIA solu-tion in PMMA was impregnated in an opal PC with subsquent polymerization.15

In these important experiments, each of these PCsonly pseudogaps in the frequency region of the spontaneemission spectrum of the dye used, and has a high optransmittance at the frequency of the exciting laser. A signcant modulation of the spontaneous emission spectrum odye relative to its intrinsic spectrum was observed in allexperiments by measuring the optical transmittance speof the sample PC. These experimental results could be inpreted qualitatively using the relevant total DOS and thecal DOS because all of the sample PCs had small dielemodulations, or, say, small dielectric contrasts, and therehad weak pseudogaps.

Just as Ref. 13 pointed out, transmission experimentsphotonic band gap materials usually do not directly probethe details of the dispersion relationvn,k but rather measurethe number of states available for a given direction of progation. Integrating this number of states over all directiodescribes the average behavior of the structure under coeration, yielding the total DOS.13 According to our viewpointof probing the total DOS, none of the above-mentionedperiments probe the total DOS directly, even though thesearchers have measured modulations of the spontanemission spectra; for example, what Ref. 19 reported mreflect therSca(v) part, and what Refs. 14 and 15 reportmay reflect therMat(v) part of the total DOS in our decomposition scheme. To probe the total DOS directly, the numdistribution of the luminescent molecules in a PC sho

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satisfy Eq.~12!, which would be an easy task on the reportexperimental setup. In addition, the suggested method, uthis kind of number distribution, is valid for all the casesPCs discussed above—not only those with pseudogapsalso those having completely photonic band gaps.

If a number distributionna of luminescent molecules exists in the scatterers of only one or a few Wigner-Seitz ucells, the decay-time curve will still reflect therSca(v) partof the total DOS of the PC, although the total radiatipower may be too low to be measured. Consideringphysical reality, one should arrange the limited numberluminescent molecules near the center of the sample to athe edge effect of the PC. This discussion is also applicato the problem of probingrMat(v), but it is very difficult toprepare sample PCs of this kind, while it is relatively easyprepare a PC with the entire matrix containing the lumincent molecules.

Of course, the luminescent molecules mentioned abmay be any other materials having excited luminescmechanisms, such as light-emitting semiconductors, Ramactive materials, etc., and measurements should be madall directions—for example, using an integral-sphere tenology.

V. CONCLUSIONS

In summary, we have pointed out that some number dtributions of the luminescent molecules can partially proinformation or even all the details of the total density of tstates in the PC. Using our decomposition scheme, oneprobe the total density of the states in an arbitrary photocrystal when the ratio of the number densities of the lumnescent molecules in the scatterers and in the backgromaterial equals the ratio of the dielectric constants ofscatterer and the background material through measuringdecay-time distribution of excited luminescent moleculMost certainly, the physical reasonability of doing so neeto be justified further, especially by experiments.

ACKNOWLEDGMENTS

This work was supported the National Key Basic Rsearch Special Foundation of China under Grant N2001CB610402 and the National Natural Science Fountion of China under Grant No. 60078007.

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