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

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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.
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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. 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
>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 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 eΓ
and eΓ
[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 eΓ
, the renor-
malized partition of decay rates is given by:
Γleak =eΓ
·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=2oµ2k3
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].
o=1
4p2Γo)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=ωoωpl .
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)·ekδjk
iΓo
2(6)
= ∆total
jk iΓtotal
jk
2(7)
total
jk and Γtotal
jk represent entries of N×Nmatrices.
Each component of matrix ∆jk =<total
jk ) represents
the virtual photon exchange between two dipoles at po-
sition rjand rkin presence of metal nanostructure and
matrix Γjk =2=jk) 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(rrj).(9)
where Iis a unit dyad, the wave number k=ω
c, and
δ(rrj) represents the point source. This gives us the
dyadics for direct interaction among the point-dipoles:
Go(ri,rj;ω) = (I+55
k2)g(||rirj||) (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= ∆JiΓJ
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.
~J= 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) ρ0.
Case II: ρnc 6= 0; (a) ρ= constant and (b) ρ0.
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 ρ0 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 ρconstant.
2. Constant area density : ρ0
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 d5.5 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 ρ0 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 eΓ/, 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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Radius of metal particle (nm)
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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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