1

Spontaneous Emission within a Photonic Atom: RadiativeDecay Rates and Spectroscopy of Levitated Microspheres

STEPHEN ARNOLD *Microparticle Photophysics Lab (MP3L)**Polytechnic University, Brooklyn, NY 11201

1. Introduction

Several years ago I proposed as a graduate project that a student investigateintermolecular energy transfer within an isolated aerosol particle ~10 µm in radius. I was

interested in how a photo-excited molecule disperses its energy among other moleculeswhen the container's size is just a few wavelengths of light. Intermolecular energy transferis an important intermediary in photo-chemistry. In the photosynthetic system a nearest

neighbor (< 0.5 nm) resonance mechanism enables energy absorbed from the Sun by achlorophyll molecule to hop to neighboring chlorophyll molecules as it searches for areaction center. Since there are typically a couple of hundred light harvesting moleculesfor each reaction center, without energy transfer in a leaf, we would likely starve. I

wondered whether the size of the container could make a difference. To most, it must haveseemed a crazy idea, which would yield nothing new. After all, the range of the longestsubstantial energy transfer, as discovered by the Nobelist Jean Perrin1 and describedQuantum Mechanically by Theodore Förster,2 is only ~ 5 nm, and our vessel would be

2,000x this size. However, spectral evidence within fluorescence taken from hydrosolpolymer micro-particles at Richard Chang's lab (Yale) pointed to possible couplingbetween an excited molecule and a micro-particle.3 I felt that such an interaction could

modify chemical physics within the micro-structure. Little did I know that the resultsobtained by us and others in the aerosol photo-physics community would distinguish thelowly aerosol particle as a high-tech item, a Photonic Atom,4 with a wide range ofapplications. Optical filters with unprecedented spectral purity, ultra-low threshold micro-

lasers and micro-scale chemical sensors are pouring out of research labs. I also neverimagined that the research would become an interdisciplinary collage spanning areas fromChemical Physics and Photonics to Quantum Electrodynamics.

To investigate intermolecular energy transfer you optically excite a molecule known

as a donor and look for the transfer of its excited energy to another molecule known as anacceptor. You know that energy transfer has taken place if the acceptor gives off itscharacteristic fluorescence. If it does not, only the donor will fluoresce with its own

2

particular color. The ratio of acceptor to donor fluorescence measures the amount of

transfer.To make the measurement one simply has to isolate the microdroplet after filling it

with donors and acceptors, excite the donors with an external laser beam, and measure theemitted spectrum. We constructed a dynamic levitator-trap in which one could contain a

charged microdroplet while irradiating it and measuring its spectrum.5,6 This apparatuslevitated a particle by using an electrostatic force to balance the particle's weight, just asRobert Millikan7 had done decades before. However, Millikan's scheme alone does notprovide a trap, the particle can drift. To overcome this drift, the apparatus superimposed

an oscillating field onto the electrostatic levitation field. Wolfgang Paul8 had shown thatsuch a dynamic field could trap atomic ions. By shaking a charged micro-particle with aharmonic field having a gradient, it is driven to a spatial null point in this dynamic field.

For our apparatus this null point is at the center of the trap. In its modern version thelevitator-trap sits on the stage of a microscope, which enables fluorescence images andspectra to be acquired (Fig.1).

V cos(ωt)ac

CCD

Microscope

Filter

Spectrometer

CCD Detector

Laser

V /2dc

V /2dc

Picopipette

Solvent Reservoir

Controllercharge pulse

He-Cd

442 nm

Thermoelectric Cooler

yz

Fig.1 Aerosol Particle Microscope-Spectrometer

3

The first recorded spectrum sent a wave of surprise through the laboratory. We had

planned a series of experiments at varying concentrations, starting on the dilute side. At aparticular dilute concentration (10-6M), the average distance between donors and acceptorswas 75 nm (~15 x the Förster range), and the transfer from a donor to all availableacceptors was expected to be less than 0.2%. However, the ratio of spectrally integrated

acceptor to donor luminescence was found to be ~10%. As the concentration wasincreased it became clear that we were dealing with a phenomenon much different thanconventional energy transfer. Förster transfer in a dilute solution is proportional to theconcentration. However the transfer taking place in the droplet revealed a region more

than two orders of magnitude wide where there was essentially no concentrationdependence (Fig.2).9

[A] (moles/liter)

0.1

10

1

10-4-810 -710 -610 -510

R(%)

Microparticle

Förster

Fig.2 Ratio of Spectrally Integrated Acceptor to Donor Fluorescence vs. Acceptor Concentration forFörster transfer taken in a centimeter sized cuvette, and for microparticle transfer take using twodifferent donor molecules.

Furthermore, in both the donor and acceptor portions of the spectrum spikes were seenwhich are never in evidence in experiments on a centimeter scale test-tube (Fig.3). Thesespikes were most pronounced in the acceptor portion of the spectrum. The results in Figs.2 and 3 modified our entire view of the energy transfer process.

An excited molecule behaves like a sub-nanoscopic dipole transmitter. Close to it thefield is intense and drops off like 1/r3 . This is the region where Förster transfer is

4

Wavelength (nm)535 545 555 565 575

40

20

60

80

100

(R6G)ε

Fluo

resc

ence

(A

rb. U

nits

)

Fig.3 The acceptor portion of the fluorescence spectrum

dominant. The field induces oscillation in the electron cloud of a nearby acceptormolecule (e.g. induced dipole), and this coupling can eventually lead to energy transfer, aslong as energy is conserved. The transition to lower energy by the donor must beaccompanied by a transition to a corresponding higher energy by the acceptor. Although

no photon is involved, the donor-acceptor pair qualifies for Förster transfer when theemission spectrum of the donor overlaps the absorption spectrum of the acceptor. Thetransfer rate goes as the product of the induced dipole and the field inducing it, andtherefore drops of as 1/r6. For the exchange of energy to occur the acceptor must be close

enough for the probability of transfer to overwhelm the natural probability for the donor tofluoresce. The typical range of transfer is ~5 nm, far too short to account for the effectwhich we see at 10-6M. Although no photons are involved in the Förster mechanism,

photon transfer can occur further out (~1 wavelength away), but its contribution hasalways been considered to be trivially small.

An organic molecule near room temperature has an optical absorption cross section atits maximum which is only ~0.1 nm on a side. The probability for such a molecule to

absorb a photon from a donor, which is hundreds of nanometers away, is less than 10-7.Even by considering all of the available acceptors, which could absorb the photon, wewould still be shy of our measured effect by orders of magnitude. So the direct transfer bya photon cannot produce a 10% effect. The only way that something like this could

happen is if the emitted photon were to return many times to the same region. Since spikesin the spectrum represented electromagnetic modes of the entire particle, the scenario

5

which evolved was that the energy transfer was associated with donor emission

preferentially into global electromagnetic modes of the particle. Although such modes arenormally described by their wave character, where they are known as Mie resonances orWhispering Gallery Modes, H.M. Nussenzveig10 of the Universidade Federal do Rio deJaneiro has given the modes a geometrical interpretation. The photon can be thought of as

circumnavigating the interior while being confined by total internal reflection (Fig.4shows a particular example).

R

Fig.4 Photonic Atom model

We will call these states Photonic Atom modes, because of their resemblance toorbits of electrons in atoms. The anomalous concentration dependence (Fig.2) can beexplained in terms of absorption of the orbiting photons by acceptors. The rate for this

absorption Γt competes with Γu, the sum of all other rates by which the photon can be lost.

The transfer rate is proportional to the absorption cross section σa of an acceptor, the

acceptor density ρa, and the speed of light in the material, c/n; Γt = σa ρac/n. Thus the

probability for transfer,

Pt =Γ t

Γ t + Γu=

1

1+ Γun

σaρac

. (1)

Eqn.1 has similar behavior to the data in Fig.2. It saturates at large molecular densities,and is in proportion to concentration at low molecular densities. The rate of all otherlosses can be obtained by fitting Eqn.1 to the data in Fig.2 (after multiplying the

expression on the right by 10%). For the pairs of dyes used in our experiments, thespectrally averaged absorption in the overlap region between the emission of the donorand the absorption of the acceptor are nearly the same, σa ~10-16 cm2. The fit that is

demonstrated by the solid curve in Fig.2 requires a value of Γu of 5x108 sec-1. Photons

6

remain within the microsphere for an average time Γu-1 = 2 nanoseconds, time enough for

light to travel 0.6 m in free space, or 60,000x the microsphere radius! This extraordinaryconfinement is the key to virtually all of the physics that we will discuss in what follows.

A microparticle is a compact optical oscillator with a very large quality factor. Prof.von Baltz in his chapter on photonic structures defines the quality factor Q in a particular

way. Here is another one. It is 2π times N1/e, the number of amplitude oscillations beforethe energy damps to 1/e of its original value. For the 10µm sphere used in our energytransfer experiments N1/e= 1.1x10 6 and Q is 6.9x106. To give you some idea of just howlarge this Q is, let's calculate the time a bell would ring out the A above middle C (440

Hz) if it had such a Q. The number of oscillations is Q/(2π), so a bell would ring for6.9x106/(2πx440) seconds. That's ~2500 sec or 0.7 hour. It is useful to note that largerspheres of quartz (diam.~1mm) have demonstrated Q's as large as 1010. 11

But, why such large Q's. Why not? Fig.4 implies complete confinement. However, thecontainment cannot be complete. As a structure is reduced in size diffraction eliminatesour pristine picture of a trapped polygon of light. Diffraction gives breadth to our lightrays and makes reflections less certain, leading to leakage. The only way to understand

these Q's and their consequences is to go beyond geometrical optics. Prof. von Baltz hasgiven a tutorial into the electromagnetic theory (EMT) of photonic structures, however Ifind in my teaching of interdisciplinary physics that chemistry students are more familiarwith Quantum Mechanics (QM) than EMT. Since QM is a common "thread" we will

reduce the EMT problem of confinement by a sphere, to a Quantum analog. We willattempt to solve the Q problem using as few special functions as possible. Following this,we will address the question of whether confinement can alter emission rates.

2. Photonic Atom Physics 101

Consider a dielectric sphere having no excess charge (i.e. ∇.E =0). The field within

such a particle must satisfy a vector Helmholtz equation of the form

∇2E + k2E = 0 (2)

where k = ωn(r)/c and n(r) is a radially dependent refractive index. In spherical

coordinates it is convenient to express the Laplacian in terms of the angular momentum

operator, ) L = −ir x∇ ,

1r

∂2(r∂r2

– L2

r2E + k2E =0 (3)

7

The angular momentum operator follows the well-known commutation relationship

[) L 2 ,

) L ] = 0. In fact it commutes with all of the other functions and operators in Eqn.3

(i.e. as long as the dielectric constant only has a radial dependence). This suggests asolution of the field of the form

E = Lψ. (4)

where ψ is a scalar function.12 With this form Eqn.3 is rewritten as

L 1r

∂2(rψ)∂r2

–L2ψr2

+ k2ψ =0 (5)

So our field will satisfy the vector wave equation so long as the scalar function ψsatisfies

∂2(rψ)∂r2

–L2(rψ)

r2+ k2(rψ) =0 . (6)

Spherical symmetry allows ψ to be separated into a product of radial and angular

function of the form ψ = [ψ r(r)/r] Ylm . The radial function is quite natural when one

considers that waves in the far field should have an amplitude that drops off like 1/r. Since

) L 2Yl,m = l(l + 1)Yl,m , Eqn.5 is easily re-written in the form of a Schrodinger Eqn.,

d2ψr

dr2+ {k0

2 - [k02(1-n2(r)) + l(l 1)/r2]}ψr = 0. (7)

Here an addition and subtraction of k02 (i.e. k0= ω/c) in the second term allows one to

identify an equivalent energy k02 and 1-D effective potential 13

Veff = k02(1-n2(r)) + l(l +1)/r2. (8)

The 1st term in the effective potential is negative and acts to confine light (photons). Theother term, the centripetal part, is repulsive. The combination produces a well in which

photons can be confined. The well is reminiscent of the potential which confines αparticles in a nucleus. As an example in Fig.5, we have plotted the potential for l =23,

refractive index n =1.414, and k0= k0,1 =10 5 cm-1 ( i.e. λ = 630 nm), for a particle having

a radius R=1.942 µm (k0,1R = 19.42). At this value of ko energy is seen to buildup in the

well; a particle irradiated at k0,1 = 10 5 cm-1 will be in resonance with the light source.

8

l(l+1)/r2

Veffl = 23, n= 1.414R = 1.942 µm,

[Ψr /r]2

r/R

(1-n2)ko2

1.0 2.0

R

rmin r2

R

l(l+1)/r2

(k0,1)2

(k0,2)2

Fig.5 Effective potential in Eqn.7

We also indicate another solution at a larger k 0,2 = 1.16 x 10 5

cm-1. To be precise, this

second order solution should not be plotted using the same potential, however there is onlya slight difference in the depth of the well associated with this resonance. In either case

this very small particle only will resonate, for l=23, at two frequencies designated bymode orders s =1 and s =2. The order of a resonance corresponds to the number of peaks

in the radial intensity.A closer examination of the radial wave equation shows that resonances of the sort

shown in Fig.5 occur at a particular values of k0R for a given l and refractive index n.Since koR, which is known as "optical size" X, is the circumference of the sphere divided

by the wavelength, a resonance can be viewed geometrically as a wave which wraps itself

around the circumference of the sphere and returns in phase. This picture is similar to ourearly views of the atom, and reinforces the term "photonic atom".

One can find the X values that correspond to resonances using an algebraic equationderived by Lam et al.14 It is an asymptotic result that is good to O(1/l). For first order

modes with fields having the form in Eqn.4, the result is

nX = L + (1.855)L1/3 – nn2 –1 1/2

+(1.033)L–1/3 –1.855 n3/3

n2 –1 3/2L–2/3

(8a)

9

where L = Sqrt[l(l+1)].

A classical particle at an effective energy (k0,1)2 can be considered at bouncing back

and forth between the classical turning points at r = rmin and r = R (see Fig.5). Beyond theclassical barrier at r = R is a region which extends to r = r2 in which the state function isseen to fall off in an "exponential fashion". r2 can be obtained from the effective potential

by setting n=1, and equating the potential to (k0,1)2; the result is r2 = L/k 0,1. From an EMT

standpoint this is the region of the evanescent field. It is a part of the mode just outside thesphere, and its existence is a clue to coupling energy into the mode. By overlapping this

evanescent field with the evanescent field surrounding the core of a fiber, one canstimulate resonances of the sphere. This allows the combination (i.e. fiber + sphere) to actas a photonic sensor.15

The clue to understanding how long the sphere will "ring" after putting energy in a

certain mode requires taking a closer look at the tunneling process. The Q for the firstorder mode in Fig.5 is only ~524. As we will see, in what follows, as the sphere increasesin size the Q grows exponentially.

The Q of a mode is associated with the photon lifetime τ, the time for the energy to

"ring down" to (1/e)x100%. The number of oscillations in this time is τω/(2π), so Q =

2π N1/e = τω or ω/Γ, where Γ is the rate of decay. Considering the photon which is

attempting to tunnel as bouncing back and forth radially between the classical turningpoints rmin and R (Figs.5 and 6), the rate of decay is the product of an attempt frequency

ν times the tunneling probability, P ; Γ= ν P. The attempt rate can be obtained

kinematically by considering the geometrically in Fig.6.

R

rmin

Fig.6 Internal reflection at a spherical surface.

The photon is incident at the spherical surface each time it travels through one leg ofthe polygon (Fig.6). This takes a time 2(R2- rmin

2)1/2/(c/n), giving a rateν = c/[2n(R2- rmin

2)1/2]. Of course, rmin , the minimum "classical" turning point radius must

be obtained from the effective potential (Eqn.8). By equating the equivalent energy ko2 to

the effective potential we find rmin = L/(kon). Together then

10

ν = c/{2n[R2 - (L/(k0n))2]1/2}. (9)

Now we have to get at the probability for tunneling.The probability for tunneling can be derived by analogy with Quantum Mechanics. In

QM the tunneling probability through a radially dependent potential is

P = e–2 (2m/h2) V(r) – E 1/2 dr (10)R

r2

For our analogous Schrodinger equation ( 2 m /h2)= 1, and E = k02. The tunneling

probability on each bounce is then

P = e–2 Veff – ko2 1/2

drR

r2

= e–2 Lr

2– ko

2 1/2dr (11)

R

L/k0

So the rate of tunneling Γ = νP, is

Γ = c2n R2 – L

kon2

e–2 Lr

2– ko

21/2

dr (12)R

L/k0

Finally the Q of the resonance is the angular frequency of the resonance, ω = koc, divided

by dissipation rate Γ. This ratio can be expressed in terms of L, and the optical size X

=k0R.

Q L,X = 2 (nX) 2– L 2 e 2 Ly

2– 1

1/2dy (13)

X

L

Eqn.13 is easily integrated

Q L,X = 2 (nX) 2– L 2 • exp 2 L lnL+ L2–X2

X– L2–X2 (14)

Now we can calculate the intrinsic Q vs. angular momentum L for our energy transfer

particles (glycerol, n=1.47, Fig.7). The most surprising result is that the intrinsic Q for aparticle of 10µm radius, in a first order mode, is ~1024. This is a colossal number. Thecorresponding photon lifetime τ =Q/ω, for visible light (e.g. 6x1014 Hz) is ~ 2.65 x 108 sec,

or 8.4 years. Obviously the calculation does not compare well with reality. Our

measurements place the Q at 6.9x106. The disparity is due to the molecular underpinning

11

of our sample. Molecules absorb light and scatter light. Absorption can be included by

introducing a round trip loss for the light circumnavigating the particle. The loss forglycerol reduces the Q to 109. The remaining two order of magnitude disparity which isleft is thought to be due to surface roughness produced from the basic moleculardiscreteness and thermally induced capillary ripples.

.

104

106

108

1010

1012

1014

1016

1018

1020

1022

1024

40 60 80 100 120 140

Q

l

n =1.47TE Modess =1

diameter(µm)20µm10µm 15µm

Fig.7 Intrinsic Q's for the first order (s=1) TE modes of a sphere with n=1.47. The trianglesrepresent exact calculations using special functions. The solid line is the result of applying Eqn.13.

What we have learned so far is that the confinement of a photon in a photonic atommode is considerably longer than an excited state lifetime. Thus an excited atom ormolecule "feels" its new environment throughout its lifetime. This alters both the rate of

decay, and the spectrum of emission.So far we have dealt with one type of mode as defined by Eqn.4. This form of

solution will only have angular vector components(i.e. θ and φ). The electric field is

tangent at the surface, It is known as a TE mode (transverse electric). There is a second

12

type of mode, known as transverse magnetic (TM). It has electric field components that

are both radial and tangential, however it has similar spectral properties and Q's.

3. Spontaneous Emission in Microspheres: Lifetime Effects and CavityQuantum Electrodynamics

One of the most interesting aspects of a micro-structure which can confine photons isthe manner in which spontaneous emission is modified. Not only does the emissionspectrum become structured (Fig.3), but the emission rate is also changed. These effects

are due to altering the density of photon states from that of an extended medium and"squeezing" zero point energy into a small volume. The latter creates large quantumfluctuations at modal frequencies. The subject has become known as Cavity Quantum

Electrodynamics (CQED).16 Its signature is a pronounced size dependence of thespontaneous emission rate. Not only is this effect a harbinger for new technology, such as"threshold-less" lasers, it also provides a tool for learning about the dynamics of amolecular emitter. In what follows we review CQED for a microsphere.

CQED within a microsphere is best demonstrated by data taken on the fluorescenceemission rate from a typical dye in a suspended liquid droplet. Fig.8 shows thefluorescence emission rate enhancement over bulk for rhodamine 6G dye in a glycerolmicrodroplet vs. droplet diameter.

droplet diameter (µm)

0

4

8

12

16

20

2 4 6 8 10 12

Γc fΓo

Fig. 8 Experimental data on the relative emission rate from rhodamine 6-G in a glycerol droplet vs.droplet diameter. The solid line is a fit to CQED theory.

13

The measurements were taken at room temperature,17 with the microdroplet isolated in an

electrodynamic levitator-trap5 set-up for nanosecond lifetime experiments. Although thefluorescence decays measured on the droplet are inhomogeneous (i.e. a sum ofexponentials) due to varying spatial and spectral positions of molecules, the data in Fig.8is of the fastest component for each diameter. These data were obtained by applying a

Laplace inversion technique to the fluorescence decay curves. As one can see, the decayrate begins to increase just below a size of 8µm. As the diameter is reduced to 4µm, therate is found to have increased over the bulk rate by a remarkable 1300%. Understandingthis effect requires figuring out how zero point energy is spectrally redistributed as the

particle size is changed. As a result we will be able to determine the spectral linewidth ofthe dye molecule.

The zero point energy density is proportional to the photon density of states. Whereas

the density of states in a continuous medium increases monotonically with frequency, thedensity of states in a confined domain is redistributed spectrally and spatially. In anenclosure the frequency spectrum of the zero point energy density becomes bunchedaround discrete frequencies, corresponding to the long-lived modes. Near room

temperature molecular excited states are rapidly dephased by molecular solvent collisions.In this limit, coherent effects such as Rabi flopping have no time to build up, and theexcited state decay rate is expected to be governed by a Fermi Golden Rule,

Γc(ω) =

2πh2 | M |2 ρc(ω) , (14)

where M is the matrix element of the molecule-field interaction and ρc(ω) volume

averaged density of states in the micro-droplet "cavity". Yokoyama and Brorson18 haveshown that the application of this equation to an emission process associated with aspectrum h(ω), gives a rate of decay relative to that in bulk of

ΓcΓ0

=h(ω) ρc(ω) dω

o

∞

h(ω) ρ0(ω) dω0

∞ , (15)

where Γ0 and ρ

0 are the decay rate and density of states in bulk, respectively. Within the

particle, the density of states is redistributed according to the sum rule

ρc(ω)dω ≈∆ω

ρ0(ω)dω∆ω

, (16)

14

where ∆ω in the inter-resonance separation (a.k.a "free spectral range").19 Fig.9 shows a

pictorial representation of all of the relevant spectra associated with molecules havingoptimal spectral coupling. Note that we have separated the density of states within theparticle ρc into resonant and background contributions;

ρc = ρr + ρb . (17)

The resonant contribution is made up of a sum of modes, i.e. ρr (ω) = ρr,s(ω)s∑ , for

which the strength of each is controlled by a second sum rule, 18

ρr,s(ω) dω ≈ Ds/Vp, (18)

h(ω)∆ω

ρ(ω)o

ρ(ω)b

ω

ρ ρb

+r

ρ =c

Fig. 9 Relevant spectra for evaluating Eqn. 9 for optimal spectral coupling.

where Ds is the mode degeneracy and Vp is the particle volume.

By choosing a normalized Lorentzian line shape for h(ω), i.e. h(ω) =

(Γh/2π)/(ω2+Γh2/4), and centering it on a resonance (Fig.9), we are now in a position to

evaluate Eqn.15. for optimal volume averaged effect (i.e. the fastest decay rate, consistentwith experiment).

We assume that the resonant modes are much narrower than the homogeneous

line width and use the periodicity of the modes to replace the integration over allfrequency by integration over one free spectral range, i.e. the range of integration becomes

ωs

− ∆ω / 2< ω < ωs

+ ∆ω / 2. Using Eqn.18 the integral over the resonant contribution

15

in the numerator of Eqn.15, h( ω)ρr,s(ω)d∫ ω = 2Ds /(πΓhVp) . The non-resonant

contribution to the numerator is obtained by determining ρb. Here we are aided by the sum

rule in Eqn.16. The background density of states is the average level left behind after"carving out" the resonant contribution, i.e. ρb = ρo -Ds/Vp∆ω. Together, the numerator in

Eqn.15 becomes h(ω)ρc(ω) d∫ ω = ρ0

+D

s

Vp∆ω

(2∆ωπΓ

h

− 1). The denominator in

Eqn.15 is h(ω)ρ0(ω)d∫ ω = ρ

0(ω ) , and the quotient

ΓcΓo

= 1 +Ds

ρ0Vp∆ω2 ∆ωπ Γh

–1 (19)

In arriving at Eqn.19 we have effectively assumed that 2∆ω/πΓh ≥ 1.

Since the free spectral range ∆ω increases with decreasing size, the enhancement

<Γc>/Γο −1 has a built in threshold in size. It occurs when 2(∆ω)t/ πΓh = 1, where the

threshold "free spectra range" (∆ω)t ≈ c/nRt with n equal to the refractive index. So the

homogeneous linewidth of the dye molecule can be determined from the threshold size,i.e.

Γh = 2c/(π n Rt). (20)

For rhodamine 6G in glycerol the threshold radius by visual inspection of Fig.8 is 4 µm,

from which the homogeneous linewidth is found to be 160 cm-1.

The degeneracy of a whispering gallery mode Ds = 2l +1, where l is the angular

momentum quantum number of the mode. The angular momentum quantum number l isessentially equal to the number of interior wavelengths which can wrapped around thecircumference, and therefore for l >>1, Ds is proportional to the radius R. Since the free

spectral range between whispering gallery modes in a dielectric particle of refractive index

n, ∆ω ≈ c/(n R), and the volume is proportional to R3, Ds /(ρ0Vp∆ω)∝ 1/R. Therefore

the enhancement, < Γc > / Γ0

− 1, from Eqn.13 is seen to contain a sum of terms which

vary as 1/R and (1/R)2.The spatial distribution of excited molecules within a spherical particle irradiated by a

plane wave is highly nonuniform, and the enhancement will be dependent on the radialposition of the excited states. To account for this we replace Ds /(ρ

0Vp∆ω) in Eqn.13 by

f/R, where f depends on the radial distribution of the excited states. On this basis the sizedependence of the enhancement takes the form

16

< Γc >

Γ0

≈ 1+f

R

Rt

R−1

(21)

Eqn.21 provides a very good fit to the experimental data in Fig.10, as shown by the solidcurve in that figure. For this fit f = 20µm and the value of R t is determined from theenhancement threshold (i.e. Rt = 4µm).

The determination of homogeneous linewidth is essential to understanding thedynamics of an excited molecule in a dielectric. Room temperature values have beendetermined in the past on very few molecules by instrumentally intensive femtosecondhole burning experiments.20 The description of the CQED effect using Eqn.20 and 21

provides a new avenue for obtaining this important parameter.21

3. Spontaneous Emission in Microspheres: Spectroscopy and theRadiation Reaction Model

The effect of optical confinement is particularly pronounced in emission spectra.Fig.10 shows an example of a fluorescence from a glycerol particle (R = 5.85 µm)

containing a particular dye.22

0

2000

4000

6000

8000

1 10 4

Inte

nsit

y [a

rb. un

its]

560 565 570 575 580 585 590

Wavelength [nm]

Bulk

Microparticle

Fig.10. Emission spectum taken in a levitator trap of a glycerol microsphere 5.85µm inradius, and doped with 10-6M DiI(3).

17

The spectrum of the dye in bulk, over this limited region, is also shown in the figure. This

spectrum has essentially no detail. However, in the microdroplet the fluorescencespectrum is imprinted by the resonant modes of the droplet. No background subtractionhas been done on this spectrum. At wavelengths corresponding to resonance, the emissionclimbs by nearly 6x the apparent background. The contrast in this spectrum is much

greater than we observed in the energy transfer experiments (Fig.3). The reason, in part,is due to the smaller particle used in Fig.10 (5.85µm vs. 10 µm). However the major causeis the result of molecular segregation at the surface. Although the acceptor dye in theenergy transfer experiments is the ionic dye rhodamine 6G, a molecule known to be

soluble in glycerol, the dye used in Fig.10 is DiI(3). This dye has two long non-polaraliphatic tails attached at either end, and a charge near its fluorescent chromaphore. Thealiphatic tails remain just outside the particle, while the charged chromaphore system is

pulled into the particle's polar region. Consequently DiI(3) stradles the surface of theglycerol as shown in Fig.11. Fluorescence is emitted principally from an excited electronassocoiated with the conjugated chain of three carbons, (i.e. like a particle in a box) andconsequently is polarized parallel to the chain.

N

N+

ClO4-

Surface

airglycerol

Fi.g.11.DiI(3):1,1'-dioctadecyl-3,3,3',3'-tetramethylindocarbocyanine perchlorate

This places DiI(3) within the region where the photonic atom mode resides. This isevident from the approximate geometrical picture of a typical mode (Fig. 4), and would be

supported by a complete wave description (e.g. Fig.5).Spontaneous emission may be considered to be emission triggered by quantum

fluctuations associated with zero point energy. The greater number of modes per unit

frequency, the greater the spectral density of zero point energy, and consequently the

18

greater rate of emission. At a resonance frequency of a sphere, more than one mode is

represented. In fact there are 2l+1 modes asssociated with the azimuthal degeneracy of thespherical harmonics. For the particle sizes we dealt with in our energy transferexperiments, this degeneracy is greater than 100. Thus the Fermi Golden rule (Eqn.14)accounts, in a qualitative way, for the pronounced peaks seen in Fig.10. This is not the

only point of view which one can take for describing emission. Fluctuations requiredissipation, which is associated with reaction forces on the emitter in the process ofemitting. One can describe the spectrum in Fig.10 by representing the excited molecule as

an oscillating dipole, and considering its perturbation from retro-reflections from themicrosphere surface. As a result of this perturbation the decay rate of the dipole ismodified. The spectrum of this decay rate can then be compared with experiment.

Imagine an emitter as a charge q oscillating in front of a surface (Fig.12).

Fig.12 An oscillating dipole and its emitted field reflected by a mirror

The dynamics of the dipole positioned at rd is described by

+ γ0 +ω02 = q2

mn•Es , (22)

where q is the magnitude of oscillating charge, m the mass, and ) n is a unit vector in the

direction of the dipole (i.e. ) n = /µ). In order to see the effect which this coherent

scattered field Es has on the dynamics, it is useful to look at the characteristic equation,

associated with Eqn.22, in the frequency domain. This is most readily accomplished by

representing µ as µ0e-iωt and Es as

t T (rd, rd, ω). =

t T ( rd, rd, ω).n µ, where

t T (rd, rd,

ω) is the Green's function dyadic in which both the "source" and field points are at rd. The

resulting characteristic equation for ω is

19

– ω2 +ω02 – iω γ0 =

q2

mn•T•n . (23)

For convenience n•T•n will be defined alternatively as Tn . Tn is complex, since in

general Es will not be in phase with . Consequently, we will represent Tn as

Tnr + i Tni. Eqn.23 can be rewritten in a rather revealing way,

– ω2 + ω02–

q2

m Tnr – iω γ0 +q2

mωTni = 0 (24)

Eqn. 24 suggests the scattered field can not only shift the frequency of the oscillator, butchange its rate of decay as well. We are particularly interested in the latter. For a typical

chromophore the emission frequency, ω0, is considerably larger than the fluorescence

decay rate, γ0, or the absolute interaction strength q2|Tn|/m. As a consequence, ω may be

taken to be ω0 in estimating the effective decay rate γ. From Eqn.24,

γ ≈ γ0 +q2

m

Im[n•T(rd, rd, ω)•n]ω0

. (25)

This simple formula displays two important aspects of the physics. First, the rate of decayis potentially sensitive to the orientation of the dipole (i.e. perpendicular or parallel to theparticle surface), as denoted by sandwiching the dyadic between the orientational unit

vector n. Second, the component of the scattered field 900 out of phase with the dipole

oscillation controls the modification of the decay rate, through the imaginary part of thescattered field dyadic. Neither should come as a great surprise. Clearly orientation mustenter. If the scattered field has no component along the dipole, it can produce no

interaction. By analogy, the effect of the imaginary part is similar to the case of pushing achild on a swing. If you want the swing to lose or gain energy in the most optimal way,

you push +/- 90 degrees out of phase with its displacement, and in phase or 180o out ofphase with its motion (i.e. velocity).

Eqn.25 accounts for the total rate of decay. For dielectrics with low loss andmolecules having high fluorescence quantum efficiencies, each of which can beexperimentally arranged, this rate will be equivalent to the radiative rate.23 In an extended

medium of refractive index, n, the radiative decay rate for an oscillating charge inGaussian units is

20

γrn = 23

nq2k3

mω . (26)

Using this expression for γ0 in Eqn.25, we find the decay rate, relative to the radiative rate

in an extended medium, to be

γγrn

≈ 1 +3

2 k3nIm[n•T(rd, rd, ω)•n]. (27)

Eqn.27 is a calculation of the total rate of decay. This includes both the radiative and

nonradiative rate. Our interest is principally in the radiative contribution. Fortunately thenonradiative contribution due to interactions with the microdroplet will be minimal in ourexperiments since the dielectrics we will be dealing with are nearly transparent; theimaginary part of the refractive index is less than 10-7.

The dyadic t T contains all of the information about the interaction of the dipole with

the sphere, including resonances. The disturbance emanating from the donor is received

back at the donor as a field E(rd) = t T (rd, rd, ω). . The only way to get at this dyadic is to

solve the boundary value problem for the geometry shown below.

x

z

y

x

rd

Fig.13 Set-up for boundary value problem, for determining t T (rd, rd, ω).

We will not display the resultant dyadic,24 but we will show how well the basic radiation

reaction model works.We have applied Eqn.27 to an emission spectrum taken from DiI on a somewhat

larger particle of glycerol (Fig.14 A).

21

572 574 576 578 580 582 584

Wavelength (nm)

Wavelength (nm)

30

35

40

45

50

55

60 572 574 576 578 580 582 584

0

0.5

1

1.5

2

2.5

3

0

0.5

1

1.5

γ/γ0

A)

B)

C)

TE89

5

TE93

4

TM885

TM92

4

F(A.U.)

γ/γ0

Fig.14 (A) Experimental Fluorescence Spectrum of a DiI(3) doped glycerol particle (10-6 M).Semiclassical calculation for a dipole tangent (B) or perpendicular (C) to the phase boundary in aparticle 7.3656 µm in radius.

Figs. 14(B) and 14(C) show spectra calculated using Eqn.27, for a dipole oriented

parallel or perpendicular to the surface, respectively.25 Clearly, all resonances are

22

accounted for by using only one adjustable parameter, the droplet size(a = 7.3656 µm).

Whereas only TM modes can be stimulated by the perpendicularly oriented dipole, bothTE and TM modes are excited by the dipole oriented tangentially to the surface. Thechoice between these two scenarios is clear: the dipole is tangent to the surface. Thisconclusions is consistent with our picture based on the physico-chemical interactions at a

polar interface.

4. Summary, Conclusions and Future Directions

In this brief course we have reviewed a number of phenomena associated withfluorescence emanating from an isolated and confined microsphere. The major resultsinvolve effects which optical confinement has on the dynamics of an excited state. These

effects are brought about from the modification of the photon density of states. Thedensity of states is changed from that in an extended medium, both spatially andspectrally. The spatial modification within a sphere results in photonic atom modes (a.k.a.whispering gallery modes). These long-lived modes in the presence of donor and acceptor

molecules become a major influence on intermolecular energy transfer, with the photon asan intermediary. As the microsphere is reduced in size we see a pronounced increase inthe rate of decay. Excited molecules within the mode volume emit rapidly into photonicatom modes. This effect is influenced by the dynamics of the excited state. In fact the size

dependence can be used to determine the homogeneous linewidth. Finally we have made aconnection between two approached to viewing the physics of spontaneous emission[CQED and radiation reaction (RR)], and have used the radiation reaction from the

microsphere to calculate emission spectra. The RR description allowed us to take a firststep in characterizing the patterning of the surface. We showed that one could use thefluorescence spectrum coupled with the RR formalism to determine the orientation ofmolecular species at the phase boundary. Although all of the experimental results were

obtained by isolating the microsphere through levitation, and interacting with it using laserbeams travelling through air, it is also possible through recent breakthroughs in photonicsto bring the research "down to earth".

One can stimulate photonic atom modes with the particles sitting near the core of an

optical fiber. This truly photonic approach can alternately be described as "evanescentcoupling" or "photonic tunneling".

Fig. 15 shows the basic scheme as it was first presented15 and recently illustrated26

23

Fig. 15 Basic scheme for coupling energy into a microsphere from an eroded optical fiber (Top).Light is scattered by a microsphere seated on the eroded fiber as seen from above, as the laser istuned to resonance(Bottom).26

The cladding material of a single mode optical fiber is eroded almost down to its core.This exposes the evanescent field of the fiber. The evanescent field extends into the waterabove the fiber. A sphere is immersed in the water and touches the eroded surface. At

resonance energy is transferred between the fiber and the sphere. This results in scatteringfrom the sphere, with dips seen in transmission through the fiber.27,28 With the spheredoped with active ions or molecules a number of inventions are possible which can utilizefluorescence and photonic modes.

An important application which utilizes the CQED effect involves miniature lasers.Normally lasers require an input threshold power to get started. This is because laseraction begins on spontaneous emission. But in traditional centimeter scale lasers, only atiny fraction of spontaneous photons- fewer than one in a thousand- go into a lasing mode.

Thus considerable power is needed to ensure that an adequate number of photons will beavailable to start the laser going. Here is where shrinking dimensions and the CQED effecthelp out. Reducing the size of the laser cavity limits the number of available modes. In

principle the cavity can be made small enough that the emission spectrum of the lasermaterial will overlap only one resonance frequency. The tiny size of the cavity thenprovides enhanced quantum fluctuations and rapid emission at this frequency. As a resultthe power needed to start the laser can be severely reduced.

24

Although only elastically scattered light was viewed in the first microsphere-fiber

coupling experiment, recently researchers have observed lasing from dopedmicrospheres.29 Just as the sphere can be stimulated by the giuded wave using "opticaltunneling", the laser emission tunnels back into the fiber, to complete the laser. It appearsthat the fiber-sphere system has become a new configuration for quantum optics and

electro-optics.There is much more to say on the subject of applications. Instead of saying it here I

refer the reader to a recent paper which I wrote during the time I spent in the inspiringsurroundings of Erice.24

Acknowledgements

I thank Rino DiBartolo for his kind invitation to lecture at "his" school, and forintroducing me to the land of his birth.

This was the first time I have attempted to make a course out of my research. By

learning from those who have done this in the past years, I gained a great deal of insightinto the technique. I am particularly indebted to Eric Mazur and Ralph von Baltz for theirencouragement.

Finally I would like to thank the National Science Foundation, AFOSR, and NASA

for supporting my interest in Photonic Atoms over the years.

25

*arnold@photon.poly.edu

**http://www.poly.edu/microparticle

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(1992).5 . S. Arnold and L.M. Folan, Rev. Sci. Inst. 57, 2250(1986).6 . S. Arnold, S. Holler, and N.L. Goddard, Opt.Lett. 22, 1452(1997).7 . R.A. Millikan, The Electron, (The University of Chicago Press, Chicago, US, 1917).8 . W. Paul, Rev. Mod. Phys. 62, 531(1990).9 . L. M. Folan, S. Arnold, and S.D. Druger, Chem.Phys.Lett. 118,322 (1985).10 . H.M. Nussenzveig, Diffraction Effects in Semiclassical Scattering, Cambridge; New York :

Cambridge University Press, 199211 . M. Gorodetsky, A. Savchenko, and V. Ilchenko, Opt. Lett., 21, 453(1996).12. This form of solution will only have angular vector components(i.e. θ and φ). The electric field

is tangent at the surface, It is known as a TE mode (transverse electric). There is a second typeof mode, known as transverse magnetic (TM). It has electric field components which are bothradial and tangential, however it has similar spectral properties and Q's.

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American, April 1993, p.54-6217. M.D. Barnes, W.B. Whitten,, S. Arnold and J.M. Ramsey, J. Chem. Phys. 97, 7842(1992).18. H. Yokoyama and S.D. Brorson, J. Appl. Phys. 66, 4801 (1989).19. S.C. Ching, H.M. Lai, and K. Young, J. Opt. Soc. Am.B 4, 1995 (1987).20. C. J. Bardeen and C.V. Shank, Chem.Phys.Lett. 203, 535(1993).21. S. Arnold, Chem.Phys.Lett. 106, 8280(1997).22. S. Arnold and N.L. Goddard, J.Chem.Phys. 111, 10407(1999).23. Y.S. Kim, P.T. Leung, and T.F. George, Surface Sci. 195, 1(1988).

24. H. Chew, J. Chem. Phys. 87, 1355(1987).25. S. Holler, N.L. Goddard and S. Arnold, J. Chem. Phys. 108, 6545(1998).26. S. Arnold, "Microspheres, Photonic Atoms, and the Physics of Nothing", Ameroican Scientist,

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813(1995).28. G. Griffel, S. Arnold, D. Taskent, and A.Serpengüzel, John Connolly, and Nancy

Morris., Opt.Lett. 21, 695 (1996).29. M, Cai, O. Painter, K.J. Vahala, and P.C. Sercel, Opt.Lett. 25, 1430(2000).