Giant frequency-selective near-field heat transfer in PT -symmetric structures
Chinmay Khandekar,1 Weiliang Jin,1 Owen D. Miller,2 Adi Pick,3 and Alejandro W. Rodriguez1
arXiv:1511.04492v1 [cond-mat.mes-hall] 14 Nov 2015
2
1
Department of Electrical Engineering, Princeton University, Princeton, NJ 08544, USA
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
3
Department of Physics, Harvard University, Cambridge, MA 02138, USA
We demonstrate the possibility of achieving giant, amplified spontaneous energy transfer (ASET) between
active and passive bodies—dimers consisting of ion-doped metallic spheres separated by small gaps. We show
that while ASET mediated by wavelength-scale resonances is often comparable or smaller than far-field emission, subwavelength resonances at low frequencies can, under specific circumstances, i.e. special configurations
of dimers operating close to or under PT symmetry, enable enhanced amplified spontaneous emission (ASE)
into the passive object rather than into the far field, providing optimal conditions for achieving large heat transfer below lasing. These predictions pave the way for further studies of efficient and optically tunable thermal
extraction or cooling of microstructured surfaces.
Radiative heat transfer between nearby objects can be many
orders of magnitude larger in the near field (sub-micron separations) than in the far field [1–3], a consequence of the coupling between evanscent (surface-localized) waves [4, 5]. In
this paper, we investigate heat exchange between active and
passive bodies—the amplified spontaneous emission (ASE)
from a gain medium absorbed by a nearby passive object—
and demonstrate orders of magnitude enhancements compared to far-field emission or transfer between passive structures. We consider an illustrative geometry, a metallic dimer
consisting of two spheres separated by a gap close to or under
parity-time (PT ) symmetry, ǫ(x) = ǫ(−x)∗ , that supports
highly subwavelength resonances and yields amplified spontaneous energy transfer (ASET) ≫ ASE well below the lasing threshold (LT). We analyze the complex relationship between heat transfer, far-field emission and dimer parameters,
and find that limits to ASET are not uniquely optimal, occurring for multiple configurations of sphere radii, permittivities
and separations, in contrast to heat transfer between passive
objects which is often a monotonic function of object separations. We find that ASE from wavelength-scale resonances
is primarily dominated by far-field emission but that, not surprisingly, the nearby presence of a lossy sphere near an active
(but non-lasing) sphere can induce or suppress lasing, a phenomenon recently observed experimentally and termed “lossinduced lasing” which, though unique to non-Hermitian systems [6], does not require the existence of exceptional points
(EP) as recently suggested [7]. We show that with proper design, metallic nanospheres doped with active emitters can exhibit large ASET well below the LT, paving the way for realizing efficient thermal extraction (heating or cooling) from
nanoscale objects.
Recent approaches to tailoring incoherent emission from
nanostructured surfaces have begun to explore situations that
deviate from the usual linear and passive materials of the
past [8–13]. The majority of these works have primarily focused on ways to control far-field emission, e.g. the lasing
properties of active materials [6]. Here, we consider a different subset of such systems: structured active–passive bodies
that exchange heat among one another more efficiently than
they do into the far field. Our predictions below extend re-
cent progress in understanding and tailoring heat exchange
between structured materials, which thus far include doped
semiconductors [14], phase-change materials [15, 16], and
metallic gratings [17–19]. Active control of near field heat
exchange offers a growing number of applications, from heat
flux control [20, 21] and solid-state cooling [21] to thermal
diodes [22, 23]. Our work extends these recent ideas to situations involving systems undergoing gain-induced amplification.
The starting point of our analysis is the well-known linear
fluctuational electrodynamics framework established by Rytov, Polder, and van Hove [24, 25]. In particular, given two
bodies held at temperatures T1 and T2 , and separated by a distance d, the power or heat transfer from 1 → 2 is given by [4]:
P (T1 , T2 ) =
Z
0
∞
[Θ(ω, T1 ) − Θ(ω, T2 )]Φ12 (ω)
dω
2π
(1)
where Θ(ω, T ) is the mean energy of a Planck oscillator at frequency ω and temperature T , and Φ12 (ω) denotes the spectral
radiative heat flux, or the absorbed power in object 2 due to
spatially incoherent dipole currents in 1. Such an expression
is often derived by application of the fluctuation-dissipation
theorem (FDT), which relates the spectral density of current
fluctuations in the system to dissipation [4]:
hJi (x, ω), Jj∗ (x′ , ω ′ )i =
4
ωǫ0 Im ǫ(x, ω)δ(x − x′ )Θ(ω, T )δ(ω − ω ′ )δij ,
π
(2)
where Ji denotes the current density in the ith direction, ǫ0
and ǫ(x, ω) are the vacuum and relative permittivities at position x, and h· · · i denotes a thermodynamic ensemble-average.
Extensions of the FDT above to situations involving active
media require macroscopic descriptions of their dielectric response. Below, we consider an atomically doped gain medium
that, ignoring stimulated emission or nonlinear effects arising above threshold [26], can be accurately modelled (under
the stationary-inversion approximation) by a simple two-level
Lorentzian gain profile of the atomic populations n1 and n2 ,
2
-4
-3
-2
-1
0
1
2
-2
3
-1
10-1
0
5 4
4
3
2
1
PT
0.32
0 0
-3
10-3
-4
Distance, d/
10-2
-1
Re ε
10-1
10-2
10
-4
10-2
-3
0
104
-3
10-1
100
101
Frequency ωR/c
9
10 11
12
(i)
(ii)
Broken PT
EP
10-2
| (ω)|
-2
Broken PT
(-)
0(+)
0.28
-2
8
0.28
0.32
-1
10-1
7
0.3
-4
10-3
6
(iii)
0.3
-2 Mode
-3 splitting
5
PT
EP
(-)
-2
10
-1
10-1
10-2
Frequency, ωR/c
10-1
GL
100
10-4
10
LL
-2
10
-1
100
Frequency, ωR/c
101
FIG. 1. Far-field flux Φ0 (ω) and flux-transfer Φ12 (ω) spectra of dimers consisting of two spheres of radii R and gain/loss permittivities ǫG,L
separated by a distance d, as a function of (a) Im ǫG for fixed Re ǫG,L = −1.522 and either d → ∞ (top) or d/R = 0.3 (middle/bottom),
(b) Re ǫ for fixed Im ǫG = − Im ǫL = −0.05 and d/R = 0.3, and (c) d/R under (top) or close to (bottom) PT symmetry, corresponding to
Re ǫG,L = −1.522, Im ǫG = −0.05 and values of Im ǫL = 0.05 or Im ǫL = 0.055, respectively. White dashed lines indicate the operating
parameters (cross sections of the contour plots) explored in (d), which shows Φ12 (solid lines) and Φ0 (dashed lines) for PT -symmetric dimers
(red lines) with Re ǫG,L = −1.522, Im ǫG = − Im ǫL = −0.05, and d/R = 0.3, comparing them to the corresponding flux rates of passive
dimers (blue lines) with Im ǫG,L = 0.05. The contours in (c) focus on one of multiple small regions of separations where the dimers are
found to support subwavelength resonances of frequency ω0 ≪ c/R. Here, the latter appears at d ≈ 0.32R and splits at a critical distance dc ,
an exceptional point marked in the plots, below which the modes ω0± move apart (dotted white lines). The plots in (e) illustrate the distance
dependence of the flux rates in this subwavelength regime by examining the spectra at various d/R ≈ {0.3056, 0.302, 0.3017}, marked by
the white dots (i), (ii), and (iii), respectively, in (c).
resulting in the following effective permittivity:
ǫ(ω) = ǫr (ω) +
4πg 2
γ⊥ D 0
~γ⊥ ω − ω21 + iγ⊥
(3)
where ǫr denotes the permittivity of the background
medium [27] and the second term describes the gain profile,
which depends on the “lasing” frequency ω21 , polarization
decay rate γ⊥ , coupling strength g, and population inversion
D0 = n2 − n1 associated with the 2 → 1 transition. Detailedbalance and thermodynamic considerations lead to a modified
version of the FDT [26, 28, 29] involving an effective Planck
distribution Θ(ω21 , T ) = −n2 /D0 , in which case the system
exhibits a negative effective temperature under n2 > n1 [29].
Note that even though Θ < 0 under population inversion, the
radiation from such a medium is positive definitive since the
corresponding Im ǫg < 0. It follows from (1) that the heat
transfer from an active to a passive body flows from the former
to the latter, i.e. T < 0 reservoirs always transfer energy [26].
Although planar geometries offer analytical insights and
computational ease, their closed nature make them far from
ideal for analyzing ASET, as described in Appendix A. Instead, we consider an illustrative open geometry consisting of
two spheres separated by vacuum. One important consideration is the LT of such a dimer system, which we quantify (neglecting stimulated emission) from the far-field flux Φ0 . Here,
we compute both quantities by exploiting a semi-analytical
method based on Mie-series expansion of scattered waves,
described in Appendix B. The flux rates associated with two
spheres of radius R1,2 and permittivities ǫ1,2 separated by distance d are given by:
R1 X
1
1
Φ12 (ω) =
Im
Im
R2
xqν (R2 )
xpℓ (R1 )
m,ℓ,ν
q,p=±
Φ0 (ω) =
2k03 R12
π
′
z (1) (k R )Dℓqp 2
ℓ
1 1
p
νm ×
|x (R2 )|2 , (4)
zν(1) (k0 R2 ) ℓ
X p
ℓqp 2
ℓqp 2
yℓ (R1 ) |Dνm
| + |Cνm
| , (5)
m,l,ν
q,p=±
′
ℓqq
ℓqq
where Cνm
and Dνm
are so-called Mie coefficients [30],
(1)
(1)
(1)
(1)
x+
ν (r) = k0 rζν (k1 r)zν (k0 r) − k1 rζν (k0 r)zν (k1 r)
yν+ (r) = lim R2 Im[zν(3) (k0 R)ζν(3)∗ (k0 R)]
R→∞
× Im[zν(1) (k1 r)ζν(1)∗ (k1 r)],
(p)
+
−
+
x−
are
ν (r) = xν (r|ζ ↔ z), yν (r) = yν (r|ζ ↔ z), zℓ
spherical Bessel (p = 1) and Hankel (p = 3) functions of
√
(p)
(p)
d
[xzℓ (x)], and ki = ω ǫi /c. Equaorder ℓ, ζℓ (x) = x1 dx
tion (5) appears to be new, but we have checked its valid-
3
ity against brute-force numerics [31] as well as known semianalytical expressions in the limit (d → ∞) of an isolated
sphere [30].
Figure 1 explores radiative features of dimers comprising
spheres of constant (dispersionless) dielectric permittivities
ǫG,L and equal radii R. The contour plots explore variations
in the flux rates of dimers supporting surface-plasmon resonances close to or under PT symmetry, with respect to (a)
Im ǫG , (b) Re ǫ, and (c) d/R. The top contour in Fig. 1(a)
shows Φ0 from an isolated sphere of Re ǫ = −1.522 as a
function of gain permittivity Im ǫG , illustrating the appearance of Mie resonances and consequently, ASE peaks occurring at k0 R & 1. As expected, the LTs (white circles indicate
a select few) associated with each resonance occur at those
values of gain where Φ0 → ∞ and mode bandwidths → 0, decreasing with increasing k0 R (smaller radiative losses). Note
that such divergences are obscured in the contour plot by our
finite numerical resolution, which sets an upper bound on Φ0 .
The middle contour plot in Fig. 1(a) shows that a passive
sphere with Im ǫL = 0.05 in proximity to the gain sphere
(d/R = 0.3) causes the Mie resonances to couple and split,
leading to dramatic changes in the corresponding LTs. Noticeably, while the presence of the lossy sphere introduces additional dissipative channels, in some cases it can nevertheless enhance ASE (decreasing LTs) by suppressing radiative
losses, i.e. evanescent scattering can decrease rather than increase radiation from the gain sphere [7]. Evanescent fields
also lead to significant Φ12 , depicted in the bottom contour
plot.
Figure 1(d) shows slices (white dashed lines) of the contour
maps of Fig. 1(a), comparing flux rates from PT -symmetric
dimers with Im ǫG = − Im ǫL = −0.05 against those
from passive dimers with Im ǫG,L = 0.05 (blue lines), and
demonstrating orders of magnitude enhancements in the former stemming from significant loss cancellation. Noticeably,
while the large radiative components of Mie resonances at intermediate and large frequencies k0 R & 1 result in Φ12 ∼ Φ0 ,
the saturating and dominant contribution of evanescent fields
in the long wavelength regime cause Φ0 → 0 and Φ12 ≫ 1
to asymptote to a constant as ω → 0. The latter is dominated
by surface-plasmon resonances arising at Re ǫ ≈ −2 and depending sensitively on d/R. This is illustrated in Fig. 1(b),
which shows the evolution of Mie resonances and appearance
of subwavelength resonances for dimers with d/R = 0.3 as
Re ǫ is varied. As expected, in the limit as d → {0, ∞} these
resonances congregate at values of Re ǫ ≈ {−2, −1}, corresponding to well-known conditions for existence of plasmonic
resonances in isolated spheres and plates, respectively.
A peculiar feature of the dimer system that is otherwise absent at far-away separations (d/R ≫ 1) is the presence of
additional plasmonic resonances in the subwavelength regime
k0 R . 1, whose frequencies ω0 depend sensitively on dimer
parameters. Such resonances lead to peaks in both Φ12 and
Φ0 , the former of which is more prominent upon inspection
of Fig. 1(d). The origin and properties of these subwavelength
resonances provide a basis to understand and quantify the pos-
sible relative enhancements in ASET versus ASE (an important figure of merit for achieving large ASET below threshold)
that can be achieved in the subwavelength regime. In fact, as
we show further below, when the dimer is close to or under
PT symmetry, Φ12 can be many times larger than Φ0 (even
diverging under special parameters) in the quasistatic regime
k0 R ≪ 1. Figure 1(c) shows the flux spectra of the same
dimer of Fig. 1(b) but at a fixed value of Re ǫ = −1.522
and with respect to variations in d/R, both under (top) or
close to (bottom) PT symmetry. Noticeably, the subwavelength resonance appears at a finite separation d ≈ 0.317R
and ω0 ≈ 0.25c/R and then splits into two resonances at a
critical separation dc ≈ 0.306R, with ω0± moving farther apart
as d decreases. Note that while ω0− → 0, both branches move
at the same rate (somewhat obscured by the logarithmic scaling of the plot). The critical separation dc is known as an EP,
defined by the coalescence of two or more resonances possessing equal loss rates [32], a unique feature of non-Hermitian
systems that is further delineated by the simple coupled-mode
theory (CMT) analysis in Appendix C. The dotted white line
in Fig. 1(c) follows ω0 as d is varied, illustrating the EP and
splitting behavior. Interestingly, it is precisely under the ultralarge red shift ω0− → 0 that Φ12 , Φ0 → ∞. This is illustrated in Fig. 1(e), which shows the flux spectra at multiple
separations [white circles in Fig. 1(c)] at or below dc , demonstrating huge enhancements in both flux rates as ω0− → 0.
In contrast, small deviations from the PT condition result in
an abrupt suppression of Φ12 and Φ0 in the quasistatic domain. Note that a similar divergence (lasing phenomenon)
occurs at ω0+ . 0.2c/R, albeit in systems close to (but not
exactly under) PT symmetry. Similar EPs and splittings arise
at many other separations (not shown), the number of which
increase with decreasing d/R. The sensitive quasistatic divergence of flux rates in the vicinity of EPs can be qualitatively
explained from the fact that dimers experience lower dissipative and radiative losses as ω → 0 under PT symmetry and
resonance conditions, which we now quantify by exploiting a
simple quasistatic analysis [33].
12
eff
eff
and
In the quasistatic regime, Φ12 = πl
6 Im α1 Im α2
4
3
eff
Φ0 = π (k0 R) Im α1 can be expressed in terms of each
spheres’ effective anisotropic polarizability, with parallel (k)
and perpendicular (⊥) components given by [33]:
α
αeff
⊥,1/2 = α1/2
2α
1 − l2/1
1 + l2/1
3
3
eff
=
α
,
α
1/2
k,1/2
1 − α1l6α2
1 − 4αl16α2
(6)
−ǫ0
denotes the vacuum polarizability of the
where αi = ǫǫii+2ǫ
0
d
denotes their
isolated spheres in units of 4πR3 and l = 2 + R
center-center distance in units of R.
As is well known [34, 35], Φ12 , Φ0 → ∞ in the far-field
dipolar limit l ≫ 1 under the resonance condition, Re ǫ = −2
and Im ǫ → 0 [36]. At smaller separations l & 2, this condition is modified due to changes in the effective polarizability of each sphere, depending sensitively on the existence and
amount of gain. For instance, Im αeff → ∞ in passive dimers
with α = α1 = α2 at specific l3 = − Re α (⊥ compo-
4
nent) and l3 = 2 Re α (k component) under the condition
of zero loss, Im ǫ → 0, and for Re ǫ 6= 2. In contrast,
gain–loss dimers exhibit diverging flux rates when the effective loss rate of the combined system goes to zero under a
modified resonance condition, satisfied for instance (though
not uniquely)
α∗2 at special
√ under PT symmetry α = α1 =
∗
3
l = {|α|, 2|α|}. (In dimers with α1 6= α2 , the zero-loss
Re α2
Re α1
condition becomes Im
α1 = − Im α2 . Note also that while
this analysis suggests a unique l at which Φ12 → ∞, in the
realistic finite dimer above, we find that there exist multiple
such critical separations arising from the presence of several
EPs.) It follows that for a finite loss rate Im α 6= 0, the emission from a gain–loss dimer can be made arbitrarily larger
than that of a passive dimer. Deviations from zero-loss conditions in both systems also lead to different limits: for small
α 2
but finite Im α ≪ | Re α|, the passive Φ12 ∼ ( Re
Im α ) ; in
contrast, under an equally small breaking of PT symmetry
α 2 Im α 2
α = α1 = α∗2 + iδ, Φ12 ∼ ( Im
Re α ) ( δ ) . Considering
the typically large loss rates of metals near the plasma frequency, i.e. Im α/ Re α ∼ 1, it is clear that one can achieve
significantly larger enhancement factors in active versus passive dimers, even under relatively large δ. In conclusion,
our analysis thus far demonstrates that gain–loss dimers in
the subwavelength regime can act as resonant dipoles with
negligible loss rates, leading to efficient radiation and internal energy transfer between the gain/loss subparts. From a
practical perspective, the existence of highly subwavelength
resonances provides a mechanism for tuning Φ12 /Φ0 below
threshold while still allowing significant external (radiative)
coupling to the dimer (which could prove helpful under optical pumping [37]). Indeed, as illustrated in Fig. 1(e) and
from our quasistatic analysis, it follows that the ratio of ASET
(R/d)6
to ASE, ΦΦ12
∼ (k
3 , favoring absorption to radiation as
0
0 R)
k0 R → 0. Ultimately, the ability to achieve gain at low frequencies is highly constrained by material and size considerations.
We now consider a practical realization of the dimer consisting of ion-doped metallic spheres, shown schematically
on the inset of Fig. 2, supporting Re ǫr ≈ −2 near the
atomic transition frequency ω21 . While there are many potential candidates, including various choices of metal-doped
oxides and chalcogenides [38], for illustration, we consider
a medium consisting of (3wt%) Al-doped zinc oxide that is
further doped with 3-level Erbium (Er3+ ) ions, in which case
ω21 lies in the near-infrared [39]. The gain profile of the ions
and zinc-oxide permittivity are well described by (3), with
ω21 = 1.22 × 1015 rad/s, γ⊥ ≈ 0.01ω21, and
ǫr (ω) = ǫ∞ −
ωp2
f1 ω12
+ 2
ω(ω + iΓp ) ω1 − ω 2 − iωΓ1
(7)
where ǫ∞ = 2.852, f1 = 0.596, ωp = 1.88ω21 ,
Γp = 8.8 × 10−3 ωp , ω1 = 5.45ω21 , and Γ1 = 0.0179ω1
[37, 38, 40]. These parameters dictate dimer sizes and configurations needed to operate in the subwavelength regime.
Figure 2(a) shows Φ12 (red line) and Φ0 (blue line)
for one possible dimer configuration, corresponding to
FIG. 2. (a) Far-field flux Φ0 (ω) (blue line) and flux-transfer Φ12 (ω)
(red line) spectra of a dimer consisting of two Al-doped zincoxide spheres of radii R = 0.2c/ω21 and separated by a distance
d/R = 0.5, one of which is doped with erbium (Er3+ ) ions with
transition frequency ω21 (inset) and pumped to a population inversion D0 = 0.165 (~γ⊥ /4π 2 g 2 ). Also shown is the far-field emission Φ0 (d → ∞) of the isolated gain sphere (green line). The top
inset shows the peak ratio max Φ12 / max Φ0 with respect to changes
in R, keeping d/R and D0 fixed. Contour plots in (b) illustrate variations in Φ0 (left/middle) and Φ12 (right) with respect to D0 , with
white dashed lines indicating operating parameters in (a).
R = 0.2c/ω21 ≈ 50nm, d/R = 0.5, and population inversion D0 = 0.165 (~γ⊥ /4πg 2 ), demonstrating orders of magnitude larger ASET compared to ASE within the gain bandwidth. Noticeably, the emission from an isolated sphere under
the same gain parameters (green line) is significantly smaller,
evidence of the decreased LT induced by the presence of the
lossy sphere. The flux spectra of this system are explored in
Fig. 2(b) with respect to changes in D0 , illustrating the appearance of multiple peaks and large Φ12 ≫ 1. As expected,
the LT corresponding to the first peak occurs close to the PT
symmetry condition, at values of D0 ≈ Im ǫL ≈ 0.167 that
compensate for the zinc-oxide loss. The existence of multiple
peaks can be explained from the sensitive dependence of subwavelength resonances on Re ǫr , which varies (albeit slowly)
within the gain bandwidth. The white dashed lines in the contours denote the operating parameters of Fig. 2(a), confirming
that the system lies below the LT. Based on our analysis above,
smaller dimers lead to larger ΦΦ12
∼ (k0 R)−3 at fixed d/R, as
0
illustrated on the top inset of Fig. 2(a).
Our predictions shed light on considerations needed to
achieve large ASET between macroscopic objects, leaving
open many important questions for future work. From a de-
5
sign perspective, one might ask what kinds of extended microstructures can enhance ASET while minimizing ASE. One
possibility is to exploit finite clusters of metallic nanoparticles operating in the subwavelength regime, following a similar approach for reaching limits to heat transfer between extended, passive structures [36]. Moreover, although accurate below threshold, our theoretical analysis ignores radiative feedback on the gain medium due to noise, nor does it
consider specific pump mechanisms, which will necessarily
affect power requirements and ASET predictions [41, 42],
especially near threshold. To answer such questions, future
analyses based on full solution of the Maxwell–Bloch equations [37, 43] or variants thereof [40, 44] are needed.
Acknowledgments.— We would like to thank Steven G.
Johnson and Zin Lin for useful discussions. This work was
partially supported by the Army Research Office through
the Institute for Soldier Nanotechnologies under Contract no.
W911NF-13-D-0001 and National Science Foundation under
Grant no. DMR-1454836.
Appendix A: Heat transfer between plates
at a rate [5, 24],
Φ12 (ω, kk ) ≈
γ −γ
X Im(rq ) Im(rq )e−2 Im(γ0 )d
1 q q 2
,
1 − r r e−2 Im(γ0 )d 2
q=s,p
(A1)
1 2
ǫ γ −γ
where rjs = γ00 +γjj and rjp = ǫjj γ00 +γjj are the Fresnel reflection coefficients at the interface between vacuum and the
dielectric media, for s and p polarizations, respectively, defined in terms of the wavevectors kj = kk r̂ + γj ẑ, with
√
|k0 | = ω/c and |kj |2 = kk2 + γj2 = ǫj ω/c. The same
expressions are applicable in the case of gain, albeit with a
subtle caveat: the derivation of Fresnel coefficients requires
special care
q since the sign of the perpendicular wavevector
γj = ± ǫj ω 2 /c2 − kk2 needs to be chosen correctly inside
the gain medium [49]. Here, we make the physically motivated choice that yields decaying surface waves inside the
semi-infinite gain medium.
In the extreme near field, kk ≫ ω/c (under passive or gain
media), the Fresnel coefficients can be further simplified and
lead to an approximate expression for the flux,
4ǫ′′1 ǫ′′2 e2kk d
|(ǫ1 + 1)(ǫ2 + 1) − (ǫ1 − 1)(ǫ2 − 1)e−2kk d |2
4ǫ′′1 ǫ′′2
= 2
|ǫ1 − 1||ǫ22 − 1||y − y1 |2
Φ12 (ω, kk ) ≈
z
T2, ε2
ε=1
T1, ε1
k0
γ0
d
r
β
FIG. 3. Schematic of two semi-infinite plates of permittivities ǫ1 and
ǫ2 maintained at temperatures T1 and T2 , respectively, and separated
by a vacuum gap d. Fourier decomposition of scattered waves with
respect to parallel kk and perpendicular γ wavevectors simplifies calculations of energy transfer in this geometry.
Radiative heat exchange between passive semi-infinite
plates has been thoroughly studied in the past [45–48]; here,
we consider the possibility of optical gain in one of the plates.
In particular, we show that the existence of non-radiative
states causes Φ12 → ∞ under PT symmetry, ǫ(x) = ǫ(−x)∗ ,
i.e. purely evanescent plasmons lead to lasing even under an
infinitesimally small amount of gain. Such a divergence can
also be readily understood from a simple CMT analysis, described in Appendix C.
For simplicity, we omit the frequency dependence in the
complex dielectric functions ǫj of the two plates (j = 1, 2),
shown schematically in Fig. 3 along with our chosen coordinate convention. Due to the translational symmetry of the
system, it is natural to express the heat flux in a Fourier basis of propagating transverse waves kk R[4], in which case the
flux is given by an integral Φ(ω) = Φ(ω, kk )kk dkk . In
the near field, kk > ω/c, the main contributions to the integrand come from evanescent waves which exchange energy
where we have defined ekk d =
the denominator of (B1),
q
(ǫ1 −1)(ǫ2 −1)
(ǫ1 +1)(ǫ2 +1) y.
Expanding
1 2
1
y − = |y|2 + 2
y
|y|
s
(ǫ1 + 1)∗ (ǫ2 + 1)∗ (ǫ1 − 1)(ǫ2 − 1)
− 2 Re
(ǫ1 − 1)∗ (ǫ2 − 1)∗ (ǫ1 + 1)(ǫ2 + 1)
it becomes clear that the flux is maximized under the conditions |y|2 = 1 and ǫ = −1, which are none other than conditions for the excitation of surface modes, i.e. metal-insulatormetal surface polaritons [50]. While the third term is independent of kk , at the plasmonic excitation frequency the denominator goes to zero under the condition ǫ1 (ω) = ǫ∗2 (ω),
leading to diverging flux. Note that the flux between two passive plates diverges in the limit of zero loss, Im ǫ → 0+ [36],
which in the case of active–passive plates is achieved whenever the loss in one plate is compensated by the gain in the
other. Such a condition is satisfied under PT symmetry at
special kk , or equivalently, for plasmons exhibiting the proper
coupling rates, a rate matching condition that is further described via a CMT analysis in Appendix C.
Appendix B: Heat transfer and radiation from spheres
The calculation of heat transfer between two spheres was
only recently carried out using both semi-analytical [47] and
6
spherical vector harmonics [52],
(1)
Vℓm (θ, φ) = r̂Yℓm
FIG. 4. Schematic of dimer system consisting of two spheres of permittivities ǫ1 and ǫ2 and radii R1 and R2 , maintained at temperatures
T1 and T2 , respectively, and separated by a gap d. Calculation of heat
flux and radiation in this system is performed by a Mie-series expansion of scattered fields, at a point x = x1 = x2 in medium 0, with
xi denoting the position relative to the center of sphere i.
brute-force methods [31]. Here, we extend these studies to
consider radiation from one of the spheres (in the presence
of the other) and the possibility of gain. Due to the spherical
symmetry of each object, it is natural to consider scattering
in this system by employing field expansions in terms of Mie
series [30]. Figure 4 shows a schematic of the system, consisting of two vacuum-separated spheres of radii Ri and dielectric
permittivities ǫi , separated by surface–surface distance d.
The thermal flux through a surface S in
medium 0 from dipoles x′1
∈
V1 is given by
H
H R
2
ǫ1
3 ′
∗
Re S hE∗ × Hi = ω Im
d
x
Im
1 G × (∇× G)·dS,
π
S V1
where G(x, x′1 ) is the Dyadic Green’s function (GF), or the
electric field due to a dipole source at x′1 evaluated at a
point x = x1 = x2 in medium 0, with xi denoting the
position relative to the center of sphere i, and where we have
employed the FDT above to express the flux as a sum of
contributions from individual (spatially uncorrelated) dipoles.
Integrating over V1 and over either the surface S : |x2 | → R2
circumscribing sphere 2 (as derived previously in Ref. 47)
or a far-away surface S : |x| → ∞, and exploiting the GF
expansion below yields (4) and (5) above, respectively.
When expressed in a basis of Mie modes, the GF from a
dipole at a position x′1 ∈ V1 evaluated at x is given by [47]:
G(x, x′1 )
= ik0
ℓ,ν=N
m=N
X
(−1)m
X
q,q′ =±
ℓ,ν=(1,m)
m=−N
(1)q′
Mℓ,−m (k1 x′1 ) ⊗
i
h
ℓqq′
ℓqq′
(3)q
Cνm
M(3)q
νm (k0 x1 ) + Dνm Mνm (k0 x2 ) , (B1)
√
where ki = ǫi ω/c, ℓ ∈ Z+ , |m| ≤ ℓ, N denotes the maxℓqq′
ℓqq′
imum Mie order, Cνm
and Dνm
are standard Mie coeffi(p)±
cients [30, 51], and Mℓm denote spherical vector waves,
(p)+
(p)
(2)
Mℓm (kx) = zℓ (kr)Vℓm (θ, φ),
(p)−
(p)
(B2)
(3)
Mℓm (kx) = ζℓ (kr)Vℓm (θ, φ)
+
(p)
(p)
zℓ (kr) p
kr
(1)
ℓ(ℓ + 1)Vℓm (θ, φ),
(B3)
where zℓ are spherical Bessel (p = 1) and Hankel (p = 3)
(p)
(p)
(p)
d
[xzℓ (x)], and Vℓm are
functions of order ℓ, ζℓ (x) = x1 dx
(B4)
∂Yℓm
1
m
(2)
−φ̂
Vℓm (θ, φ) = p
+ iθ̂
Yℓm
∂θ
sin θ
ℓ(ℓ + 1)
(B5)
∂Y
m
ℓm
θ̂
+ iφ̂
Yℓm . (B6)
V(3) (θ, φ) = p
∂θ
sin θ
ℓ(ℓ + 1)
1
The advantages of employing spherical vector waves and harmonics in this geometry comes from the following useful orthogonality relations [47]:
I
(p)
(p′ )∗
Vℓm · Vℓ′ m′ = δℓℓ′ δpp′ δmm′
S
I
I
(2)
(3)∗
(p)
(p)∗
dΩ Vℓm × Vℓ′ m′ · r̂ = − dΩ Vℓm × Vℓ′ m′ · r̂
S
S
= δℓℓ′ δmm′
Z
dx′ Mℓm (kx′ ) · Mℓ′ m′ (kx′ )
Z
dx′ Mℓm (kx′ ) · Mℓ′ m′ (kx′ )
Vi
Vi
(1)+
(1)+∗
h
i δ ′δ ′
ℓℓ mm
(1)
(1)∗
,
= Ri2 Im ki∗ zℓ (ki R)ζℓ (ki R) 2
k0 Im ǫi
(1)−
(1)−∗
h
i δ ′δ ′
ℓℓ mm
(1)∗
(1)
= Ri2 Im ki∗ zℓ (ki R)ζℓ (ki R) 2
,
k0 Im ǫi
which greatly simplify the calculation of flux described above,
requiring integration over a surface S in medium 0.
Appendix C: Heat transfer in PT resonators
In this section, we employ a simple CMT to analyze ASET
and ASE between two resonant objects mediated by coupled
resonances. Consider a system comprising two coupled resonators of equal frequency ω0 , decay rates γ1 , γ2 , and coupling rate κ (determined by the overlap of evanescent fields of
each mode and hence dependent exponentially on object separations [53]). Here, γ denotes the total decay rate of the mode,
which includes both internal dissipation or gain γd as well as
external radiation γr . The coupled-mode equations describing
the resonator fields a1 , a2 are given by [53–55]:
p
2γ1 ξ1
p
ȧ2 = (−iω0 − γ2 )a2 + iκa1 + 2γ2 ξ2
ȧ1 = (−iω0 − γ1 )a1 + iκa2 +
(C1)
(C2)
where ξj denote fluctuating noise sources satisfying
hξj∗ (ω)ξj (ω ′ )i = Θ(ω0 , Tj )δ(ω − ω ′ ), as determined from the
FDT above. (Note that modes exhibiting net gain have γ < 0
and hence Θ < 0.) The flux transfer Φ12 and the emission
7
spectrum Φ0 of the gain resonator are then given by:
4κ2 γ1 γ2
,
D(ω)
4γ1r γ1 [(ω − ω0 )2 + γ22 ]
Φ0 (ω) =
D(ω)
Φ12 (ω) =
(C3)
(C4)
where D(ω) = (γ1 +γ2 )2 (ω−ω0 )2 +[γ1 γ2 +κ2 −(ω−ω0)2 ]2 .
It follows that under passivity, γ1d , γ2d > 0, Φ12 ≤ 1 and
Φ0 ≤ γ1 /γ1 are both positive and maximized under the ratematching condition, κ2 = γ1 γ2 (equivalently, at a particular
separation). In contrast, when the gain in one of the resonances overcomes loss, γ1 < 0, both Φ12 and Φ0 become
negative and one finds that the on-resonant Φ12 , Φ0 → ∞
under the PT symmetry condition, γ1 = −γ2 , and in the
limit as κ2 → −γ1 γ2 . Such a divergence occurs when
there is zero net dissipation in the system which implies
that the system has reached the lasing threshold. In particular, the complex eigenfrequencies
of the coupled system
√
i(γ1 +γ2 )
2
2
2
ω± = ω0 −
± κ − Γ , where Γ = γ1 −γ
2
2 . In the
weak coupling regime κ ≪ |Γ|, the imaginary part of the frequencies differ, approaching one another as κ increases and
eventually coalescing at an exceptional point, κEP = |Γ|, beyond which the real parts bifurcate. (Note that when Γ < 0,
the coupled resonances are already past the LT.) Under total
loss compensation, Γ = 0, both modes lie exactly on the real
axis (Im ω± = 0) and consequently Φ0 , Φ12 → ∞.
It follows from (C3) and (C4) that ASET is significantly
(ω0 )
κ2
= γ1r
large compared to ASE when the ratio ΦΦ12
γ2 ≫ 1.
0 (ω0 )
This can arise in finite dimer-like structures at short separations (large κ), or when the radiative loss rate of the gain mode
γ1r or overall loss rate of the passive mode γ2 are sufficiently
small. In the sphere dimer above, such conditions can be fulfilled to different degrees, depending in a complicated way on
geometry as well as material parameters. In the absence of radiative decay channels, e.g. plasmons in semi-infinite slabs or
quasistatic modes in dimers, loss compensation and hence lasing occurs under the PT -symmetry condition, γ1d = −γ2d .
(Note that because planar metal-insulator-metal modes do not
experience radiative losses, any finite amount of gain necessarily implies that the system is lasing.)
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