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Role of non-classical paths of a photon in the density of optical states

Kritika Jain and Murugesan Venkatapathi∗

Computational and Statistical Physics Laboratory, Indian Institute of Science, Bangalore, 560012

The modiﬁed density of optical states due to a weak coupling with external cavities or other

resonant matter (Purcell eﬀect), can also be recast as the eﬀect of coherent superposition of the

classical paths of the photon. When the coupling is stronger, the quantum interference of additional

paths representing the possible re-absorption of the photon from the excited proximal matter, also

plays a signiﬁcant role in the radiative decay of the emitter. The eﬀect of these additional non-

classical paths of the photon on the density of radiative and non-radiative states, is included using

a proposed simple modiﬁcation to the weak coupling model. This eﬀect is especially evident in

the anomalous enhancements of spontaneous emission due to extremely small fully absorbing metal

nanoparticles less than 10 nm in dimensions. Incorporating multiple emitters and such very small

metal nanoparticles coupled to each other, the large contribution of non-classical paths to radiative

states in such bulk materials is elucidated.

The role of vacuum modes on spontaneous emission

of photons has been well elucidated and the reversible

exchange of a photon between a mode of vacuum and

an atom inside a micro-cavity is observable [1–4]. This

Rabi oscillation of the excitation between the cavity and

a resonant atom is the sign of a strongly coupled atom-

vacuum system, and the coupling strength inferred by

the ratio of the frequency of oscillation and the decay

rate of the cavity is large (Ω/Γ >1). Even in the ab-

sence of the cavity i.e. weak coupling of the emitter with

vacuum, when a proximal resonant object is introduced,

such Rabi oscillations with the object can emerge indi-

cating a strong coupling with matter [5–7]. The increase

in decay rates due to the presence of the proximal ob-

ject, is derived using the number of additional modes

available for the spontaneous emission [8]. Notably, the

partition of optical states into the radiative and non-

radiative parts reﬂected the classical absorption and scat-

tering properties of this object [9, 10]. We showed that

this conventional partition results in signiﬁcant anoma-

lies when the emitter is strongly coupled to absorbing

matter [11, 12]. This anomaly is also related to the

large unexpected gains of emission in surface-enhanced-

Raman-spectroscopy (SERS) where a strongly absorbing

resonant metal structure increases the exciting radiation

in the near-ﬁeld by orders of magnitude, but surpris-

ingly without apparent absorption of the emitted pho-

tons. This remarkable divergence of SERS from ﬁrst

principle theoretical predictions has been widening for

four decades, during which the reported SERS enhance-

ments have grown from 104to 1014 [13–15]. Meanwhile,

anomalous enhancements of spontaneous emission near

fully absorbing metal nanoparticles less than 10 nm in

dimensions, have also been reported [16–19].

This work has two parts. We begin with a phe-

nomenological description of the strong coupling regime

of the emitter and a nanostructure, to deduce a use-

ful renormalization of the conventional partition of op-

tical states into radiative and non-radiative parts [11].

∗murugesh@iisc.ac.in

(a)

(b)

(a)

FIG. 1. Coupling of emitter ewith object Oand resulting

coherent paths of the photon to a point P; Ω is the frequency

of emitter-object Rabi oscillations, Γ and Γoare the decay

rates from the object and the emitter in free-space respec-

tively (a) Interference with classical paths represents addi-

tional decay from object (b) Interference with non-classical

path B (in blue) represents coherent decay of the emitter and

object.

This re-normalized partition of states is especially suited

for studying bulk materials with multiple emitters and

nanostructures strongly interacting with each other, as

studied in this work. A ﬁrst principles model of the

non-Markovian interaction between a single nanoparticle

and an emitter will be presented elsewhere, to validate

this convenient re-normalization. In the second and more

signiﬁcant part of this paper, we elucidate the eﬀect of

this re-normalization on the density of radiative states

in bulk-materials using detailed evaluations with many

coupled emitters and metal particles.

First, we recast the modiﬁed spontaneous emission due

to a body as a quantum interference of all the possi-

ble paths of the emitted photon [20–22]. This equiv-

alent description allows us to highlight that the con-

ventional partition of optical states into radiative and

non-radiative parts is valid only for an emitter that is

weakly coupled to the proximal object. We consider the

eﬀect of non-classical paths of a photon in the strong-

coupling regime, which include the re-absorption of a

photon by the emitter that would have otherwise dis-

sipated in the object. Interestingly, this permits a fully

absorbing object to increase the eﬃciency of spontaneous

arXiv:2104.00932v1 [physics.optics] 2 Apr 2021

2

emission. The quantum interference of these additional

non-classical paths and the resulting eﬀects on the radia-

tive and non-radiative decay channels can be explained

succinctly using ﬁgure 1.

When the frequency of Rabi oscillations with the ob-

ject is signiﬁcantly less than the decay rate of the emitter

in free-space i.e. Ω Γo, the interaction is incoherent

and the emitter is uncoupled, and we observe no eﬀect

of the object on the emission. When the resonant object

is relatively close to the emitter, the increase in coupling

strength [23] and a reversal of the above inequality to Γ

Ω>Γo, renders the classical paths AA’ through the ob-

ject and the direct free-space path A shown in ﬁgure 1a,

indistinguishable. It is well known that the interference

of these paths over all points P given by the superposi-

tion of the scattered ﬁeld from the object and incident

ﬁeld from the emitter, provides us the additional radia-

tive and non-radiative optical states due to the object

[24]. But note that this path A’ is relevant only when

the decay from the object is signiﬁcantly faster than the

Rabi oscillations (Γ Ω). The above condition signiﬁes

the weak coupling of the emitter with the object.

However, when the emitter is strongly coupled to the

object (Γ Ω), paths A’ become irrelevant as Rabi oscil-

lations are much faster than the decay in the object. The

dominant path of the photon through the object is now

path B, and its interference with A as shown in ﬁgure

1b should be of primary interest. The crucial diﬀerence

in this path is that there is no decay in the metal and

hence no absorption. In case the probabilities of both

these paths are comparable (Γ ∼Ω), the two cases of

interference (1a and 1b) can be averaged with the corre-

sponding probabilities of these mutually exclusive paths

given by 1 −e−Γ

Ωand e−Γ

Ω[11].

Let Γr

oand Γnr

obe the known radiative and non-

radiative decay rates of the isolated emitter adding to

Γo. Γrand Γnr be the corresponding decay rates of the

object adding to the total metallic contribution Γ. The

total radiative and non-radiative rates of the system are

a sum of the free-space and metallic components. Due

to the strong coupling and the large probability of path

B, the decay rates Γrand Γnr of the object evaluated

using only paths A’ have to be renormalized. The co-

herent decay of the emitter and object through path B,

where dissipation is absent, carries more signiﬁcance for

the dipole mode of the object that represents its coupling

to vacuum modes [11]. The substitution of the paths A’

with path B, replaces non-radiative decay of the object’s

dipole mode numbered ‘1’ with radiative decay. Using

the probability of this path B given by e−Γ

Ω, the renor-

malized partition of decay rates is given by:

Γleak =e−Γ

Ω·Γnr

1(1)

The eﬀective decay rates are:

Γr

eff = Γr

o+ Γr+ Γleak

Γnr

eff = Γnr

o+ Γnr −Γleak (2)

Now we brieﬂy revisit the conventional evaluation of

the above decay rates and Rabi frequencies for a single

emitter, before we proceed to multiple interacting emit-

ters and metal nanoparticles. The cumulative eﬀect of in-

terference of the paths of a photon over all spatial points

P i.e. superposition of scattered ﬁeld of object and inci-

dent ﬁeld of emitter, and the increase in the density of

optical states, can be evaluated using the self-interaction

of the emitter due to the presence of the object. The ad-

ditional self-energy of the emitter at rodue to a nanos-

tructure is:

Σ(ω) = −2πq2ω

mc2eo·G(ro,ro;ω)·eo(3)

and the above can be integrated over polarization vec-

tors eo, and over frequency ωif required. Green dyad

Grepresents the additional self-interaction due to the

nanostructure. Here qis the oscillating charge, mis

its mass, and cis speed of light. The increase in de-

cay rate due to the structure is given by imaginary part

i.e. Γ = −2=(Σ) where the reduced Planck’s constant

was divided out of the self-energy in equation(3). This

model represents a dipole approximation of a two-level

emitter in the weak vacuum-coupling regime, and it uses

the Fermi golden rule to relate the decay rates to the

density of optical states [25, 26]. It is convenient to drop

charge, mass and amplitude of the oscillator and nor-

malize all self-energy components by Γr

0for evaluations,

where Γr

0=2√oµ2k3

3~and µis the electric dipole mo-

ment of the emitter; k,oand ~are the wave number,

free-space permittivity and reduced Planck’s constant.

This widely used conventional description of the self-

interaction in equation (3) includes an implicit rotating

wave approximation (|<(Σ)| ω). In the supplementary

[27], we conﬁrm that this conventional approximation

does not alter the conclusions for the cases discussed in

this paper. The non-classical path B and its interference

can also make dynamics of the emission non-Markovian

i.e. exponentially damped oscillatory decay for a single

excitation, which manifests as a multi-exponential decay

in ensembles [28, 29]. Since the coupling strengths are

moderate in our examples here, we need not normalize

the total decay rates for these marginal eﬀects. The real

part of self-energy in equation (3) represents the cou-

pling strength ‘g’ and also energy split ∆Ebetween the

two modes of the strongly coupled oscillators [30, 31]:

Ω=2g= 2|<(Σ)|=∆E

~(4)

Note that Rabi frequency Ω used here should be dis-

tinguished from the generalized Rabi frequency Ωowhich

includes the eﬀects of dephasing, both due to asymmetry

in damping of the two oscillators and detuning of reso-

nance frequencies of the two strongly coupled oscillators

[32–36].

Ωo=1

4pΩ2−(Γ −Γo)2+ 4δ2(5)

3

(a)

(b)

Molecule

Emission

Metal Surface

FIG. 2. (a) Enhancements in the proposed theory and the

weak-coupling approximation for an emitter near a resonant

plane metal surface (relative permittivity ≈ −1). Minimum

(Q/Qo) is given by increase in quantum eﬃciency deﬁned in

equation (14) and the maximum (QΓ/QoΓo) represents in-

crease in photon counts due to continuous excitation (b) The

strong-coupling model extended to multiple emitters.

Where the detuning of the emitter from the plasmon

resonance of the object is δ=ωo−ωpl .

In ﬁgure 2a we show the eﬀect of the above modiﬁed

partition of optical states, given by equation 2, for spon-

taneous emission very near a strongly absorbing surface

[11]. For larger separations on the order of a wavelength

that are not captured in this ﬁgure, the strong and weak

coupling predictions of quantum eﬃciency are indistin-

guishable and display an oscillatory behavior. But very

proximal to the surface, only the predictions with the

non-classical paths B through the metal yield emission

gains large enough to multiply with the enhancement in

near-ﬁeld excitation intensity (∼105) and produce the

observed large gains greater than 1010 in SERS.

The signiﬁcant question of interest here is if this coher-

ence of non-classical paths survive when multiple dipole

emitters are strongly coupled to metal nanoparticles.

This is practically signiﬁcant as bulk materials have many

emitters and even single emitters like quantum dots have

a ﬁnite size eﬀect on the coupling [12]. We are espe-

cially interested in the extremely small metal nanoparti-

cles which have a negligible scattering eﬃciency and are

fully absorbing; see the appendix for the variation of the

strength of coupling with size of metal nanoparticles. In

the rest of this paper we use a model of many strongly

coupled dipole emitters and metal particles, both with

and without the non-classical paths described above, to

elucidate the density of optical states in such materials.

A coupled system of Lorentz dipole oscillators was used

to model an excitation of one quantum of energy shared

among Nemitters, along with coupled metal nanoparti-

cles (ﬁgure 2b). It represents weak excitations given by

superpositions of any one excited emitter among the N

emitters i.e. the W1,N −1states [37, 38], as required for

the study here. This model was ﬁrst treated analytically

under long-wavelength approximations for a spherical

metal particle [30], was extended to retarded waves [39],

and also to arbitrary geometries without long-wavelength

approximations [40]. The pair-wise self-energy contribu-

tion of Ncoupled Lorentz dipole oscillators proximal to

metal nanostructure is:

Σtotal

jk (ω) = −2πq2ω

mc2ej·G(rj,rk;ω)·ek−δjk

iΓo

2(6)

= ∆total

jk −iΓtotal

jk

2(7)

∆total

jk and Γtotal

jk represent entries of N×Nmatrices.

Each component of matrix ∆jk =<(Σtotal

jk ) represents

the virtual photon exchange between two dipoles at po-

sition rjand rkin presence of metal nanostructure and

matrix Γjk =−2=(Σjk) represents eﬀects of the coupling

on decay rates of the two dipole emitters. Green dyadic

Grepresents interaction between the two dipole emitters

in the presence of the metal nanostructure.

The self-energy matrix evaluated in equation (6) is also

further decomposed into its metallic contribution Σ and

the contribution due to direct interaction among the N

coupled dipoles in the absence of metal nanostructures

Σo.

Σtotal

jk (ω) = Σo

jk (ω)+Σjk(ω) (8)

where Gmreplaces Gin equation (6) and the second

term with Γois ignored, to evaluate Σjk(ω). Gm, the

metallic contribution of green dyadic is calculated by sub-

tracting Gorepresenting free-space interaction between

the dipoles in the absence of metal nanostructures, from

the total green dyadic G.Gois calculated using the

solutions of point source in a homogeneous background:

5 × 5 ×Go(r,rj;ω)−k2Go(r,rj;ω) = Iδ(r−rj).(9)

where Iis a unit dyad, the wave number k=√ω

c, and

δ(r−rj) represents the point source. This gives us the

dyadics for direct interaction among the point-dipoles:

Go(ri,rj;ω) = (I+55

k2)g(||ri−rj||) (10)

where g(r) = eikr

4πr The details of calculation of Gfor an

arbitrary structure using such dipole granules was shown

4

elsewhere [40]. The total decay rate Γ in the metal is de-

composed into its radiative (Γr) and non-radiative (Γnr)

by a factorization of the dyads Gusing real and imagi-

nary parts of polarizability of dipole granules, which rep-

resent the metal nanoparticle in this volume integral ap-

proach.

The eigenstates of the coupled many emitter-metal sys-

tem are calculated using:

Σtotal|Ji= ∆J−iΓJ

2|Ji(11)

Speciﬁcally, eigenvectors Jrepresent one of Nmodes

of emission here. The imaginary part of an eigenvalue of

Σtotal represents decay rate of the mode while the real

part of the eigenvalue represents the energy shift. The

energy shifts ∆EJof a collective mode can also be eval-

uated.

~ΩJ= 2∆EJ= 2~|∆J|(12)

Note that |Jiis not an eigenstate of Σ that represents

only the metallic contribution. We evaluate contributions

of the metal to the energy shifts and decay rates of a

mode using an entry-wise decomposition of Σjk for all ‘j’

and ‘k’, into ∆jk −iΓjk/2 and corresponding expectations

hJ|∆|Jiand hJ|Γ|Ji. The strength of coupling between

emitters through the metal is given by Kjk = 2|∆jk |/Γjk

and its expectation hJ|K|Jiaveraged over |Jirepresents

the signiﬁcance of non-classical metallic paths for the sys-

tem (ﬁgure 3b). Similarly, the strong-coupling correction

to the partition of the decay rate is given by entries of a

matrix Γleak

jk calculated as:

Γleak

jk =e−

Γjk

2∆jk ·Γnr(1)

jk (13)

Here the superscript (1) refers to the dipole mode con-

tribution of the metal nanoparticles. The mode-wise

Γleak is calculated by the expectation hJ|Γleak|Ji. The

eﬀective decay rates of a mode remain as given in equa-

tion (2). The quantum eﬃciency of a mode and the ex-

pected quantum eﬃciency are given by:

QJ=Γr

eff

Γr

eff + Γnr

eff

Q=1

N

N

X

J=1

QJ(14)

For our analysis of results, let ρbe a measure of cou-

pling of emitters with metal nanoparticles, and it is de-

ﬁned as:

ρ=nnc +nc

lnc +lc+nnc +nc

(15)

where nrepresents the number of paths through the

metal and lrepresents paths only through the emitters;

subscripts cand nc indicate classical and non-classical

paths respectively. ρnc is a measure of strong coupling

of the emitters with metal nanoparticles through Rabi

oscillations while ρcis a measure of its weak coupling, and

(a)

(b)

FIG. 3. (a) Strong-coupling of emitters and a metal parti-

cle: non-classical paths Aithrough other emitters as well as

paths Bithrough Rabi oscillation with the metal particle are

included. (b) Coupling strength of emitters with metal given

by the expected ratio of Rabi frequency and decay rate.

they include only nnc or ncrespectively in the numerator

of equation (15). In the weak coupling regime ρnc = 0,

while in strong coupling regime both ρcand ρnc are non-

zero. We investigate two conditions in each of the two

cases. Case I: ρnc = 0; (a) ρ= constant and (b) ρ−→ 0.

Case II: ρnc 6= 0; (a) ρ= constant and (b) ρ−→ 0.

In both cases distinguished in ﬁgure 3a, condition (a)

represents an increase in the number of emitters within

a constant area, while condition (b) represents a case of

increase in the number of emitters with a constant area

density. See appendix for a description of the geometries

used in these cases.

In case I, the emitters are strongly coupled among

themselves [41] but the weak coupling approximation

with the metal particle is used. This leads to a predicted

quenching of emission and a reduction in quantum eﬃ-

ciency as shown in ﬁgure 4a. The predicted quenching is

similar to the case of single emitter weakly coupled to the

metal nanoparticle, and the evaluated coupling strengths

shown in ﬁgure 3b indicate the breakdown of this approx-

imation. This quenching is constant when the number of

emitters increase in a constant area as ρ=ρcis constant.

When the system expands with a constant area density

of emitters, this quenching decreases as does the frac-

5

FIG. 4. Modiﬁed emission due to gold nanoparticles 1.9 nm

in radii at λ= 560 nm in free-space and Qo=0.33; additional

information provided in [27] (I) quenching when only classical

paths of the metal are included (II) enhancements when non-

classical paths of metal are also included.

tion of paths through the metal particle. When some of

the emitters are replaced by metal particles in the model

(to reach a ratio of 1:6 for number of metal particles and

emitters), quenching of the weak coupling approximation

is regained even when this system has now an extended

area with many coupled emitters and metal particles.

In case II, emitters are strongly coupled among them-

selves as well as with a metal particle. There exists non-

classical paths among emitters as well as between emit-

ters and the metal nanoparticle, and this leads to en-

hancements in the quantum eﬃciency (ﬁgure 4b). This

enhancement is constant as the number of emitters in-

crease in a constant area, as both ρcand ρnc remain

roughly constant. When the system is expanded with a

constant area density of emitters ρnc −→ 0 due to weaker

couplings, as does ρc. The former results in a loss of co-

herence of non-classical paths B through the metal par-

ticle, and the lost enhancement is only regained when

some of the emitters are replaced by metal nanoparticles

ﬁxing a ratio of 1:6 for the metal nanoparticles and the

emitters.

From the above two cases studied, we can infer

that the coherence of non-classical paths among many

strongly coupled emitters and metal nanoparticles is

sustained independent of the number of emitters and

metal particles. This eﬀect diminishes only when a large

fraction of emitters are weakly coupled to the metal

nanoparticle i.e. when separations between the emitters

and a lone metal particle increase. Considering the low

dissipative loss of these very small metal particles they

are expected to be much more eﬀective in enhancement

of spontaneous emission, compared to the larger metal

particles required in the weak coupling regime. Further,

the emerging coherence in the dynamics of emission in

such materials can be gainfully exploited for applications

other than light generation [42].

ACKNOWLEDGMENTS

K.J. and M.V. thank the department of Computational

& Data Sciences, Indian Institute of Science for its gen-

erous support.

Appendix A: Details of simulations

Here we explain additional details of geometries corre-

sponding to ρ=Constant and ρ→0 used for simula-

tions in ﬁgure 3 and 4 of the main paper. The emission

wavelength of emitters is 560 nm in free-space and the

refractive index of surrounding medium is 1.5. Qo=0.33

was assumed without loss of generality. We used gold

nanoparticles of diameter 3.8 nm coupled to many dipole

emitters.

1. Constant area : ρ=constant

Here we consider N-1 dipole emitters around a multi-

pole metal nanoparticle of diameter 3.8 nm (consisting

of 552 dipoles) at an average distance of h= 3.5 nm from

the surface of the nanoparticle as shown in ﬁgure S1. The

point dipole emitters are uniformly distributed in a ﬁxed

area of shell of 2 nm. We compute the decay rates with

increasing number of emitters in the shell to observe the

eﬀect of non-classical interactions with metal nanoparti-

cle, both in weak and strong coupling regime. The quan-

tum eﬃciency is roughly constant with the increase of

number of emitters as shown in ﬁgures 4a (quenching by

weak coupling) and 4b (enhancement by strong coupling)

of the main paper. Note that both non-classical paths

only through other emitters lnc, and coherent classical

paths through metal ncincrease as a factorial of N. But

so does the non-classical paths through metal nanopar-

ticle nnc due to Rabi oscillations among the metal and

6

FIG. A1. This geometry represents emitters distributed in

constant area (A 1). The golden sphere is a gold nanoparti-

cle of radius 1.9 nm. Blue spheres represents dipole emitters

where h is average distance between emitters and the nanopar-

ticle.

FIG. A2. This geometry represents emitters distributed in

constant area density (A2). The golden sphere is a gold

nanoparticle of radius 1.9 nm. Blue spheres represents dipole

emitters where h is distance between nearest emitters and the

nanoparticle.

emitters, in case II. The latter is possible as the strong-

coupling of all emitters with the metal is ensured in this

geometry. While the classical direct paths lcincrease

only linearly with N. This ensures that the fraction of

metallic paths ρ≈constant.

2. Constant area density : ρ→0

Here we consider N-1 dipole emitters around a multi-

pole metal nanoparticle (consisting of 552 dipole grains)

of diameter 3.8 nm, where distances among emitters are

ﬁxed so that number of emitters per unit area i.e. area

density is constant (see ﬁgure S2). The ﬁrst emitter is

placed at a distance of h=3.5 nm from the surface of the

metal nanoparticle and then more emitters are added on

lattice sites which are at d≈5.5 nm apart from each

other. The lattice sites are located in concentric circles

around metal nanoparticle and distance between them is

chosen so that ﬁrst circle around metal contains exactly

6 emitters which may represent a hexagonal lattice. Note

that the non-classical paths only through other emitters

lnc increase as a factorial of N, while in case II the coher-

ent non-classical paths nnc through Rabi oscillation with

metal marginally increases with Nto a constant, beyond

which emitters are not coupled strongly enough to the

metal. The classical paths ncdue to a weak coupling

increase as a factorial of Ninitially, but as the couplings

reduce further it converges to a constant when the far-

ther emitters are not coherently coupled to the metal

particle. While the classical direct paths of an emitter to

metal lcincreases only linearly with N. This results in

the fraction of metallic paths ρ→0 as Nincreases.

The quantum eﬃciency increases towards the free-

space value (Q/Qo=1) with number of emitters added

as shown in ﬁgure 4a (for weak coupling) and the en-

hancement decreases in 4b (strong coupling) towards the

free-space value. We also show that this quantum eﬃ-

ciency again increases when some of the random emit-

ters are replaced by metal nanoparticles on same lattice

sites so that overall metal to lattice sites ratio is 1:6. So

green squares in ﬁgures 4 of the main paper show the

qualitative behaviour of this model (B) on adding more

metal nanoparticles to replace the emitters. The green

squares show simulations for 224 lattice sites out of which

32 are metal nanoparticles. Note that we used dipole

metal nanoparticles to model this special case because of

computational complexity of the multipole nanoparticle.

Each data point in the ﬁgures involve a number of sim-

ulations of many random conﬁguration of polarizations

and permuted positions, numbering greater than Nuntil

the relative variation in the expected value is small.

Appendix B: Coupling strengths and size of gold

nanoparticles

A larger factor e−Γ/Ω, determines the degree of di-

vergence of observations from the predictions of weak

matter-coupling approximation. Variation of this expo-

nent in a logarithmic scale are plotted below for a ﬁxed

small distance of 3 nm and for a ‘relative’ distance ﬁxed

as the radius ‘R’ of metal particle. These ﬁgures B1 and

B2 provide insight into the strong-coupling eﬀects and

the size of metal particles. All the above cases represent

gold nanoparticles with a surrounding medium of refrac-

tive index 1.5, and at a free-space wavelength of 560 nm.

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0 5 10 15 20 25 30 35 40

Radius of metal particle (nm)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

< / >

Surface to emitter distance is 3 nm

FIG. B1.

0 2 4 6 8 10 12 14 16 18 20 22 24 26

Radius of metal particle (nm)

1

1.1

1.2

1.3

1.4

1.5

< / >

Surface to emitter distance is R nm

FIG. B2.

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