From vertical-cavities to hybrid metal/photonic-crystal
nanocavities: towards high-efficiency nanolasers
Se-Heon Kim,
1,2,
* Jingqing Huang,
1,2
and Axel Scherer
1,2
1
Department of Electrical Engineering, California Institute of Technology, Pasadena, California 91125, USA
2
Kavli Nanoscience Institute, California Institute of Technology, Pasadena, California 91125, USA
*Corresponding author: seheon@caltech.edu
Received September 1, 2011; revised December 1, 2011; accepted December 2, 2011;
posted December 2, 2011 (Doc. ID 153695); published March 9, 2012
We provide a numerical study showing that a bottom reflector is indispensable to achieve unidirectional emission
from a photonic-crystal (PhC) nanolaser. First, we study a PhC slab nanocavity suspended over a flat mirror formed
by a dielectric or metal substrate. We find that the laser
’
s vertical emission can be enhanced by more than a factor
of 6 compared with the device in the absence of the mirror. Then, we study the situation where the PhC nanocavity
is in contact with a flat metal surface. The underlying metal substrate may serve as both an electrical current
pathway and a heat sink, which would help achieve continuous-wave lasing operation at room temperature.
The design of the laser emitting at 1.3
μ
m reveals that a relatively high cavity
Q
of over 1000 is achievable assuming
room-temperature gold as a substrate. Furthermore, linearly polarized unidirectional vertical emission with the
radiation efficiency over 50% can be achieved. Finally, we discuss how this hybrid design relates to various plas-
monic cavities and propose a useful quantitative measure of the degree of the
“
plasmonic
”
character in a general
metallic nanocavity. © 2012 Optical Society of America
OCIS codes:
140.5960, 140.3945, 230.5298, 240.6680.
1. INTRODUCTION
Spontaneous emission of a dipole emitter can be altered by
the presence of a metallic or dielectric reflector. For instance,
the spontaneous emission may be completely inhibited inside
an appropriately designed optical cavity. Purcell provided the
first quantitative analysis on the emission dynamics of an
atomic dipole placed inside a cavity characterized by its qual-
ity factor (
Q
) and mode volume (
V
)[
1
]. It has been now well
established that the effect of the spontaneous emission
modification will be more pronounced with the higher
Q
∕
V
ratio [
2
].
On the other hand, researchers in the field of semiconduc-
tor lasers have searched various ways to achieve the so-called
“
thresholdless
”
laser in the context of spontaneous emission
control by some form of high
Q
∕
V
cavities [
3
]. The concept of
photonic-crystal (PhC) [
4
,
5
] has revolutionized the develop-
ment of such high
Q
∕
V
lasers, enabling further miniaturization
in device size and reduction in threshold power. The periodic
arrangement of dielectrics can result in a forbidden frequency
region within which any electromagnetic mode cannot propa-
gate, a property now called the
“
photonic band gap
”
(PBG) [
6
].
In fact, the presence of such an energy band gap in a one-
dimensional (1D) periodic structure (distributed Bragg
reflector, DBR) [
7
] has been known for some time and already
applied to the design of the vertical-cavity surface-emitting
laser (VCSEL) [
8
]. The artificial material possessing PBG en-
ables us to confine electromagnetic energy in a volume
smaller than the associated wavelength of light. The first na-
nocavity laser was achieved based on a thin dielectric mem-
brane with periodically arranged air-holes in two-dimensions
[
9
]. Laser gain was provided by multiple layers of quantum
wells embedded in the middle of the slab. Though
Q
of the
initial laser cavity was below 500, much progress has been
made toward higher
Q
and smaller
V
, and this has been a ma-
jor research topic until very recently [
10
–
12
]. However, the
importance of out-coupling efficiency in these wavelength-
scale emitters has for some time been neglected.
It is interesting to note that the basic architecture of the
PhC nanocavity design has not changed much since the ori-
ginal air-suspended thin-slab geometry [
13
]. The use of such
a thin slab appears to be indispensable to maximize the size of
the in-plane PBG [
14
]. For a high refractive index semiconduc-
tor slab, a typical thickness of the PhC slab is
0
.
5
a
, where
a
is
the lattice constant. Yet in reality, most PhC slab nanocavities
are suspended over a dielectric (or metallic) substrate, as de-
picted in Fig.
1(c)
. Thus, they are no longer isolated but ex-
perience feedback from their environment. It has been
pointed out that even cavities with
Q
in the range of 50 000
can
“
see
”
the underlying mirror surface, leading to severe
modification in the far-field radiation profile [
15
,
16
]. This phe-
nomenon, as is to be discussed in the following section, has a
strong analogy to the aforementioned cavity quantum electro-
dynamics (QED) example of the point dipole source near a
plane mirror [
17
]. We will show that the combination of a
highly reflective bottom mirror and a proper gap size between
the PhC slab and the mirror can lead to enhanced far-field di-
rectionality from the PhC cavity by more than a factor of 6 in
comparison with the PhC nanocavity in the absence of the bot-
tom reflector. This result should be of practical importance for
the various PhC based light emitters, including nanolasers and
single-photon sources [
16
,
18
,
19
].
Recently, the realization of an
electrically pumped
nanola-
ser has drawn the renewed attention of many researchers. The
first current-injection PhC nanolaser was demonstrated in
2004, in which a submicron-sized dielectric post was intro-
duced right below the laser cavity as a means to deliver
Kim
et al.
Vol. 29, No. 4 / April 2012 / J. Opt. Soc. Am. B 577
0740-3224/12/040577-12$15.00/0
© 2012 Optical Society of America
and confine electrical current [
20
,
21
]. However, in order to
obtain reasonably high
Q
for lasing, the post structure needs
to be made very thin (diameter
<
500
nm) and long (typically 1
μ
m). This could lead to unusually high electrical resistance of
over 1 k
Ω
, which has been a major bottleneck to achieve con-
tinuous-wave (CW) operation of a laser at room temperature
(RT). To circumvent this issue, laterally doped
p-i-n
struc-
tures were proposed recently [
22
–
24
]. CW lasing has been
achieved by external cooling at an ambient temperature of
∼
150
K[
23
]. Note that
optically pumped
RT-CW lasing opera-
tion of a PhC nanolaser was already demonstrated by Nozaki
et al.
[
25
]; however, realization of its electrical counterpart has
been a severe challenge.
Therefore, we believe now is the time to reconsider the PhC
slab design itself. The concept of using a bottom mirror can
naturally lead to the following question: What would happen if
we make the air-gap size zero by placing the whole PhC slab
directly on the mirror? We show that a flat metal substrate
[see Fig.
1(d)
] can be used effectively to achieve good vertical
confinement of the cavity mode [
26
]. Furthermore, metals are
very good conductors for both electrical current and heat dis-
sipation. This may imply that the aforementioned difficulties
in building current-injection PhC nanolasers could be miti-
gated using this hybrid metal-PhC slab design. We shall pro-
vide numerical simulation results on the metal-bonded PhC
nanocavities in Section
3
. Since the PhC structure is now
in contact with the metal substrate, we can expect certain
“
plasmonic
”
effects in this hybrid design. In fact, it may not
be easy to distinguish between
“
photonic
”
and
“
plasmonic
”
when both characters coexist in the same structure. There-
fore, we will discuss the relationship of this newly proposed
design to various
“
plasmonic
”
cavities. We will propose a
quantitative measure of the degree of the
“
plasmonic
”
charac-
ter,
“
plasmonicity
,
”
which would provide a useful guideline in
the design of
“
plasmonic
”
cavities.
2. A PHC NANOCAVITY NEARBY
A BOTTOM REFLECTOR
A. Cavity QED Analogy
Before going into the details of the hybrid metal-PhC nanocav-
ity, let us first consider a PhC nanocavity suspended over a flat
mirror as depicted in Fig.
2(a)
. The situation is directly ana-
logous to a point dipole emitter near a plane mirror, which is a
well-known cavity QED example [
17
]. Certainly, there are dif-
ferences between the PhC nanocavity and the dipole emitter.
The PhC nanocavity is much bigger in size than the atomic
dipole emitter. The radiation pattern from this hypothetical
emitter does not resemble that of the simple dipole emitter.
In general, the complete description requires higher-order
multipoles including both electric- and magnetic-multipoles
[
27
]. Thus, we can view this generalized light emitter as a
sum of pointlike multipole emitters. Similar modifications
in the decay rate and the radiation pattern from the PhC na-
nocavity are expected. In the absence of a bottom mirror, we
find that
Q
0
of the deformed hexapole mode (Fig.
2(b)
)is
∼
15
;
000
(subscript 0 for the cavity with no bottom reflector)
[
15
]. To make our analysis more analogous to [
17
], we trans-
late
Q
0
into the radiative decay rate using the relation
γ
0
ω
∕
Q
0
.
We have performed three-dimensional (3D) finite-
difference time-domain (FDTD) simulations to study how
γ
changes in the presence of a bottom reflector. In Fig.
2(d)
,
we plot normalized decay rate (
γ
∕
γ
0
) of the deformed hexa-
pole mode as a function of the air-gap size (
d
). Three different
types of the bottom reflector are considered in this study
—
perfect electric conductor (PEC), gold, and dielectrics.
First, let us focus on the ideal mirror case (PEC). As ex-
pected, we can observe modulations of
γ
as a function of
d
. Both enhanced decay (
γ
∕
γ
0
>
1
) and suppressed decay
(
γ
∕
γ
0
<
1
) can occur depending on a specific
d
value. Inter-
estingly, we have found that the modulating features appear to
have a certain periodicity. For example, the two consecutive
peak positions (one at
1
.
75
a
and the other at
3
.
5
a
) are sepa-
rated by
∼
0
.
5
λ
. Here, it should be noted that
1
a
is equal to
∼
0
.
29
λ
for the cavity displaced sufficiently far from the under-
lying mirror, where
λ
is the emission wavelength measured in
vacuum. The observed
∼
0
.
5
λ
periodicity reminds us of the
two-beam interference condition, where the bottom reflector
can reverse the downward propagating beam to the upward
direction to produce the interference. We will develop a more
rigorous model in Subsection
2.B
.
Second, the case of a gold reflector is considered. We use
the dielectric function of gold at RT for emission wavelength
of
∼
1
.
3
μ
m[
28
]. Slightly increased decay rates in comparison
with the previous ideal mirror case are partly due to the addi-
tional absorption and the slightly lower reflectivity of gold.
The absorption effect becomes more severe for smaller
d
. In-
deed, the decay rate becomes noticeably different from the
PEC case when
d<
0
.
5
a
.
Third, when we replace the gold mirror with a simple di-
electric substrate with a refractive index of 3.4, the periodic
modulation of
γ
becomes much weaker than the previous two
cases. This is not surprising, because the mirror reflectivity is
now only about 30%. For
d<
1
.
2
a
, we observe much more en-
hanced decay rate
—
at
d
0
.
5
a
,
γ
∕
γ
0
is about 38. Such a dra-
matically enhanced decay rate is mostly due to the enhanced
tunneling loss through the bottom substrate and the TE-TM
coupling loss in the horizontal direction [
15
]. In fact, the light
confinement mechanism in the in-plane directions is no longer
perfect as we break the vertical symmetry of the PhC slab [
14
].
In all three cases, we observe the breakdown of the PBG as
d
→
0
. However, it is much more severe in the case of a di-
electric mirror. We would like to note that
γ
∕
γ
0
converges
to 1 in the opposite limit of
d
→
∞
, though it has not been
shown explicitly in those plots. Finally, in Fig.
2(e)
, it can
be seen that the cavity resonance blue-shifts (
Δω
blue
)as
d
Fig. 1. (Color online) Evolution of PhC nanolaser: from VCSEL to
hybrid metal-PhC laser.
578
J. Opt. Soc. Am. B / Vol. 29, No. 4 / April 2012
Kim
et al.
decreases. Such energy-level shifts are also observed from the
aforementioned cavity QED example [
17
]. One may develop a
similar perturbative approach based on electric- and mag-
netic-multipoles to explain the observed
Δω
blue
. However, ar-
guments based on the electromagnetic variational theorem [
6
]
would suffice to explain differences in
Δω
blue
in the three dif-
ferent cases
—
the dielectric mirror case shows the smallest
Δω
blue
due to the more efficient overlap of the electric-field
energy with the dielectric mirror region.
B. Enhancing Energy Directionality: Planewave
Interference Model
As mentioned previously, the interference of electromagnetic
waves is mediated by the bottom mirror to produce the ob-
served decay rate modulation. Remember that the optical loss
of the nanocavity is closely related to its far-field radiation pat-
tern; therefore, we expect that the far-field radiation pattern
should undergo similar modifications. By using 3D FDTD, we
can directly simulate a far-field radiation pattern,
dP
θ
;
φ
∕
d
Ω
,
of the PhC nanocavity, which represents emitted power (
dP
)
within a unit solid angle (
d
Ω
). There exists a corresponding
wavevector component
k
for each angular direction
θ
;
φ
.
Thus, any radiation pattern can be decomposed in terms of
planewaves
P
k
dP
∕
d
Ω
k
. This planewave decomposition
provides an alternative to the multipole expansion method.
From now on, we will focus on planewaves with
k
x
0
and
k
y
0
, since we are interested in the vertical direction-
ality of laser emission.
Figure
3(b)
describes how the complex two-dimensional
(2D) slab nanocavity is simplified in the spirit of the plane-
wave decomposition. The perforated PhC membrane is ap-
proximated as a uniform dielectric slab with an effective
Fig. 2. (Color online) (a) A PhC nanocavity is suspended in air above a flat mirror (a bottom reflector). The radiative decay rate (
γ
) of the na-
nocavity mode can be tuned as a function of the air-gap size. (b) The design of the PhC nanocavity. Here, two air-holes facing each other are
enlarged by
Rp
0
.
05
a
. Other parameters are as follows: the slab thickness
T
0
.
9
a
, the modified hole radius
Rm
0
.
25
a
, and the back-
ground hole radius
R
0
.
25
a
. The lattice constant of the PhC is denoted as
“
a
”
throughout this paper. (c) Electric-field intensity distribution (
j
E
j
2
)
of the deformed hexapole mode detected in the middle of the slab (
z
0
). (d) Normalized decay rates (
γ
∕
γ
0
) of the deformed hexapole mode as a
function of the air-gap size. Perfect electric conductor (PEC), gold, and a dielectric of the same refractive index of 3.4 as the slab material are
considered as a bottom reflector. In the case of a gold mirror, we assume emission wavelength to be
∼
1
.
3
μ
m with
a
450
nm. Drude model
parameters are as follows:
ε
∞
10
.
48
,
ω
p
1
.
38
×
10
16
rad
∕
s, and
γ
m
1
.
18
×
10
14
rad
∕
s. (e) The resonance frequency also changes as we vary
the air-gap size.
Fig. 3. (Color online) (a) Vertical emission enhancement factor (
W
)
obtained by the planewave interference model. Both the air-gap size
and the slab thickness are varied, and the results of varying slab thick-
nesses were shown as multiple curves as a function of the air-gap. We
have assumed the effective refractive index of the slab (
n
eff
) to be 2.6,
which will result in
r
0
n
eff
−
1
∕
n
eff
1
≈
0
.
44
. (b) A schematic
of the model for the PhC nanocavity suspending over a bottom reflec-
tor. The perforated slab is replaced with a uniform dielectric slab with
n
eff
, and the underlying mirror is assumed to be PEC.
Kim
et al.
Vol. 29, No. 4 / April 2012 / J. Opt. Soc. Am. B 579
refractive index [
29
]. We will deal with the PEC mirror case
only, where any wave incident upon it will undergo a
π
-phase
shift. Detailed calculations have been described in our earlier
publication [
15
]; here we would like to summarize the essen-
tial ideas. First, the final result will depend only on the effec-
tive index of the slab,
n
eff
, the phase thickness of the slab,
φ
,
and the phase thickness of the air-gap,
φ
. Once
n
eff
is deter-
mined, we can derive coefficients of amplitude reflection and
transmission for a single dielectric interface such that
r
0
n
eff
−
1
∕
n
eff
1
and
t
0
1
−
r
2
0
1
∕
2
. Second, all of the sca-
lar waves will be treated as complex numbers in the form
exp
ik
z
z
−
i
ω
t
. Any wave traveling across a phase thickness
of
φ
0
will gain the same amount of phase. Third, we assume
that there are two
“
seed
”
waves, one on the top of the slab and
one at the bottom of the slab. Both waves are assumed to pro-
pagate in the opposite directions with the equal amplitude and
phase. This assumption can be justified by the fact that the
original PhC nanocavity mode is symmetric with respect to
the
z
0
plane. Fourth, the wave initially propagating in
the downward direction will be redirected upward by the bot-
tom mirror. In Fig.
3(b)
,
S
denotes the sum of all such waves
finally detected in the far-zone at
θ
0
. During this complex
process, a fraction of the energy can couple to the PhC nano-
cavity mode. If this coupling occurs, the resonant mode can
produce wavevector components with
k
x
≠
0
or
k
y
≠
0
. In this
model, we assume that this coupling process is negligible and
the in-plane wavevector components are conserved. This as-
sumption is analogous to the
weak-coupling regime
in the
cavity QED [
17
]
—
once a photon leaves the emitter (the
PhC resonant mode), it does not strongly interact with the ori-
ginal emitter. The photon will
“
see
”
only mirror boundaries of
an external cavity (the bottom mirror and the uniform dielec-
tric slab) whose
Q
is much smaller than that of the emitter. It
should be noted that similar approaches have been applied to
the calculation of spontaneous emission inhibition and en-
hancement of a dipole emitter in a cavity [
30
].
With all the assumptions above, the relative vertical en-
hancement can be written as
j
1
S
j
2
, which is normalized
to the result in the absence of the bottom reflector. Here is
the final result of the summation:
W
≡
j
1
S
j
2
1
t
2
0
e
i
φ
1
−
r
2
0
e
2
i
φ
r
0
−
e
−
2
i
φ
−
r
0
t
2
0
e
2
i
φ
2
:
(1)
We plot
W
as a function of the air-gap size (
φ
), where we
assume
n
eff
to be 2.6 in consideration of the effective surface
coverage ratio of
∼
1
−
2
π
R
2
∕
3
p
a
2
for the PhC slab with
R
0
.
35
a
. Figure
3
shows the results for various
φ
values
(the slab thickness). When
φ
satisfies the
“
slab resonance
”
condition (
φ
m
×
π
, where
m
is an integer), the slab will
be
“
transparent
”
for any wave incident upon it. Under this con-
dition, the redirected wave from the bottom mirror does not
“
see
”
the slab, which results in the simple two-beam interfer-
ence.
W
modulates between 0 and 4 by varying the air-gap
size. The effective thickness of a PhC slab is typically chosen
to be around
“
T
≈
λ
∕
2
n
eff
”
in order to maximize the PBG
[
14
], which happens to be near the slab resonance condition.
This implies that near 100% transmission is possible by slightly
tuning the emission wavelength toward the exact slab reso-
nance condition [
29
]. Then, the subsequent optimization of
the air-gap size can bring up the vertical emission intensity
by up to a factor of 4. Here, it is worth mentioning that the
enhancement factor is a quantity proportional to the decay
rate. Thus,
W
max
4
>
1
1
does not violate the conserva-
tion of energy
—
most of the electromagnetic energy,
U
t
,
is stored in the cavity, and it will experience exponential de-
cay in time such that
U
t
U
t
0
exp
γ
t
−
t
0
, where the de-
cay rate,
γ
, is proportional to
W
. Conversely, if the air-gap size
is chosen to be
1
.
5
π
, then, for an ideal 1D system, we can com-
pletely quench the radiation to the vertical direction, resulting
in infinitely large
Q
. Similar unbounded
Q
behavior has been
found from the second
Γ
-point band-edge state in a 2D hon-
eycomb-lattice PhC slab [
31
] and the
Γ
-point band-edge state
in a 2D triangular-lattice PhC slab [
32
], with and without using
a bottom reflector, respectively.
Now let us turn to our original problem and find the con-
dition that maximizes
W
. Surprisingly,
W
can increase up to
about 9 by tuning the slab thickness near the
“
slab antireso-
nance
”
[
φ
m
0
.
5
×
π
], as shown in Fig.
3(a)
[
15
]. This
phenomenon can be understood by the fact that the slab be-
comes highly reflective (reflectance can be as high as 70% due
to the absence of the slab resonance) [
29
]. Thus, the slab and
the bottom mirror system constitutes a certain vertical reso-
nator, which enhances the density of photon modes near
k
x
k
y
0
point. In the following subsection, through rigorous 3D
FDTD simulations, we will show that more than sixfold ver-
tical emission enhancement can be achieved in the deformed
hexapole mode example.
C. Enhancing Energy Directionality: FDTD
The planewave interference model predicts that vertical emis-
sion can be enhanced by more than a factor of 4. Here, we
would like to test the validity of the model by using rigorous
FDTD simulation. In our previous paper, we have adopted the
near- to far-field transformation algorithm for more efficient
and fast simulation [
15
]. However, for more accurate results,
we will perform direct FDTD simulations in a very large com-
putational domain with
L
x
×
L
y
×
L
z
≥
6
λ
×
6
λ
×
6
λ
.We
take the same deformed hexapole mode shown in Fig.
2
as
an example and assume a PEC bottom mirror. Note that
we will vary only the air-gap size while we keep the slab thick-
ness constant at
0
.
9
a
. First, far-field emission profiles,
f
θ
;
φ
,
are obtained by detecting the radial component of the Poynt-
ing vectors over a hemispherical surface whose radius is lar-
ger than
3
λ
(see Fig.
4
). These far-field patterns are then
normalized by the original far-field pattern,
f
0
θ
;
φ
, obtained
in the absence of the bottom reflector (see Fig.
4
). By varying
the gap size, it is found that the internal electromagnetic en-
ergy,
U
t
, does not show noticeable change within the FDTD
time needed for the far-field simulation. Thus, errors asso-
ciated with small variations in
U
t
will be less than 1%.
We present normalized far-field patterns,
~
f
θ
;
φ
≡
f
∕
f
0
,in
Fig.
5(a)
, where we use a simple mapping defined by
x
θ
cos
φ
and
y
θ
sin
φ
. Thus, the center corresponds to
θ
0
and the angle
θ
will be proportional to the radial distance
from the origin. Clearly, we can have more than sixfold en-
hancement by choosing the air-gap to be near
1
λ
.
In Fig.
5(b)
, we plot
~
f
θ
0
as a function of air-gap size.
The three solid lines are from the previous planewave inter-
ference model, for
φ
1
.
1
π
,
1
.
2
π
, and
1
.
3
π
, respectively. Gen-
erally, the curve for
φ
1
.
3
π
shows good agreement with the
580
J. Opt. Soc. Am. B / Vol. 29, No. 4 / April 2012
Kim
et al.
FDTD results. Interestingly, the model also predicts the asym-
metrical behavior around
1
.
0
λ
, as is confirmed by the FDTD
result
—
the enhancement value decreases more slowly on the
left side than on the right side. Thus, both the FDTD result and
the planewave interference model confirm that the vertical
emission can be enhanced by more than a factor of 6 by em-
ploying a highly reflective mirror at the bottom.
D. Applications
These findings will be of significant importance to the design
of various PhC based light emitters, such as nanolasers and
single-photon sources [
16
,
18
,
19
]. The emission wavelength
is a rather fixed property by the active material embedded
in the slab. In most wafer designs, the final selective wet-
etching process will leave a flat surface of a dielectric sub-
strate, which will serve as a good bottom mirror, as we have
shown in Fig.
2
. Furthermore, in GaAs material systems, we
may include a GaAs/AlAs DBR whose resultant reflectivity
will be more than 98%. To optimize the vertical directionality,
the following two criteria should be considered for the wafer
design: (1) The thickness of the sacrificial layer (which is to
become the air-gap) should be designed to be equal to the
emission wavelength, and (2) the PhC slab thickness should
be chosen to be near the slab antiresonance. However, in con-
sideration of the PBG size, the slab thickness may be limited
by
1
.
0
a
(
φ
max
≈
1
.
3
π
). It should be noted that the enhanced
directional emission has been verified from InP/InGaAsP
PhC nanolasers by direct measurement of far-field emission
profiles [
33
].
3. A PHC NANOCAVITY ON A METAL
SUBSTRATE
As mentioned previously, the planewave interference model
may not be applicable to the case where the air-gap size is
below
λ
∕
2
. However, we have seen already what would hap-
pen in the limit of zero air-gap from Figs.
2(d)
and
2(e)
. In the
case of the dielectric mirror, we cannot find any mode whose
Q
is higher than 30. However, a relatively high-
Q
(
>
1
;
000
)
mode is found by assuming the PEC mirror. The case of
the gold mirror is not as good as the case of the PEC mir-
ror
—
Q
∼
150
has been obtained. Though this
Q
value is al-
ready comparable with typical
Q
values from previously
reported metallic nanocavities [
34
–
38
], we will propose a
method to bring it up to several thousands. In this section,
we will explore another opportunity with this new metal-
PhC nanocavity design [Fig.
6(a)
] for a practical nanolaser.
The fact that a metal substrate is in direct contact with the
PhC slab structure may mitigate aforementioned difficulties
in building electrically pumped nanolasers, namely, excess
electrical and thermal resistance. Moreover, the metal may
work as a good bottom reflector so that the vertical direc-
tional emission could be enhanced. This hybrid metal-PhC
design will also be an important building block in the field
of plasmonics in the context of building efficient plasmonic
lasers [
39
].
A.
Q
,
V
, and Purcell Factor
F
p
Before beginning with our discussions on the result in Fig.
6
,
let us clarify the definitions of
Q
,
V
, and
F
p
and several energy
related quantities, especially in the context where dispersive
metals are involved.
Q
tot
is defined through the decay rate of the total electro-
magnetic energy contained in the cavity,
U
EM
t
, such that
U
EM
t
U
EM
0
exp
−
ω
Q
tot
t
:
(2)
Here,
U
EM
t
is the sum of the electric-field energy,
U
E
t
, and
the magnetic-field energy,
U
M
t
, stored in the PhC nanocavity
mode.
U
E
t
and
U
M
t
can be defined by the energy density
functions,
u
E
r
;t
and
u
M
r
;t
, respectively.
Fig. 4. (Color online) FDTD simulated far-field emission profiles
from the deformed hexapole mode shown in Fig.
2
. Far-field patterns
detected over the hemispherical surface are transformed into the 2D
plane by using a simple mapping defined by
x
θ
cos
φ
and
y
θ
sin
φ
. Numbers represent the air-gap size normalized to the
emission wavelength of the reference cavity (
∞
) in the absence of
the mirror.
Fig. 5. (Color online) (a) The far-field emission profiles shown in
Fig.
4
are normalized by the reference far-field pattern (
∞
in Fig.
4
),
where white regions denote values
>
6
.
33
. (b) We extract
θ
0
com-
ponents from the normalized far-field patterns and plot them together
with the theoretical curves obtained by the planewave interference
model.
Kim
et al.
Vol. 29, No. 4 / April 2012 / J. Opt. Soc. Am. B 581
U
E
t
≡
Z
V
d
3
r
u
E
r
;t
;
(3)
U
M
t
≡
Z
V
d
3
r
u
M
r
;t
.
(4)
In the case where dispersive media are involved in the energy
calculations, special care must be taken. In this paper, we will
use the following definitions for
u
E
and
u
M
, as has been noted
by Chang and Chuang [
40
,
41
].
u
E
r
;t
≡
ε
0
2
Re
d
ωε
d
ω
h
E
r
;t
·
E
r
;t
i
T
;
(5)
u
M
r
;t
≡
ε
0
2
Re
ε
ω
h
E
r
;t
·
E
r
;t
i
T
.
(6)
Here, the bracket
h i
T
denotes time average over one optical
period. If
r
lies in a normal dielectric medium of
ε
d
, then the
above expressions will take the familiar forms of
u
E
r
;t
u
M
r
;t
ε
0
ε
d
∕
2
h
E
r
;t
·
E
r
;t
i
T
, where the equality be-
tween the electric-field energy and the magnetic-field energy
comes from the fact that the resonant mode is harmonically
oscillating in time [
42
].
If
r
lies in a Drude medium where
ε
m
ω
ε
∞
−
ω
2
p
∕
ω
2
i
γ
m
ω
, we obtain
u
E
r
;t
ε
0
2
ε
∞
ω
2
p
ω
2
−
γ
2
m
ω
2
γ
2
m
2
h
E
r
;t
·
E
r
;t
i
T
;
(7)
u
M
r
;t
ε
0
2
ε
∞
−
ω
2
p
ω
2
γ
2
m
h
E
r
;t
·
E
r
;t
i
T
.
(8)
Throughout this paper, we adopt the cavity QED definition
for the mode volume
V
, which, using the above definition of
U
EM
, can be written as
Fig. 6. (Color online) (a) A PhC nanocavity is brought into contact with the underlying metal substrate. We assume realistic optical constants of
gold at RT, which is implemented using the single-pole Drude model in FDTD. (b) FDTD simulated electric-field intensity profiles (
j
E
j
2
) of the dipole
mode when the slab thickness is 606 nm. Other structural parameters are as follows:
Rm
0
.
25
a
,
R
0
.
25
a
,
Rp
0
.
05
a
, and
a
315
nm. (c) Op-
tical properties of the dipole mode. Quality factor (
Q
), effective mode volume (
V
), and Purcell factor (
F
p
) derived from
Q
and
V
are plotted as a
function of the slab thickness. Here, slightly different lattice constants (
a
) have been used for different slab thicknesses to keep the emission
wavelength a constant at
∼
1
.
3
μ
m. (d) The total electromagnetic energy contained in the cavity dissipates into two independent loss channels,
one in the form of propagating radiation in air, and the other in the form of absorption in metal. Here, radiation efficiency refers to the fraction of
total dissipation into the radiation.
582
J. Opt. Soc. Am. B / Vol. 29, No. 4 / April 2012
Kim
et al.
V
≡
U
EM
t
max
f
u
E
r
max
;t
u
M
r
max
;t
g
;
(9)
where
r
max
maximizes the sum of
u
E
and
u
M
in the
denominator.
Using
Q
tot
(Eq. (
9
)) and
V
(Eq. (
9
)), Purcell
’
s figure of merit
can be written as [
2
]
F
p
≡
3
Q
tot
4
π
2
V
λ
n
3
;
(10)
where a single emitter is assumed to be located in a medium of
a refractive index
n
.
Now consider the dipole mode in the PhC cavity in contact
with the gold substrate as shown in Figs.
6(a)
and
6(b)
.As
mentioned previously,
Q
tot
cannot be larger than 200 in the
case of the deformed hexapole mode in a thin slab (slab thick-
ness
T
0
.
9
a
). Traditionally, the PhC slab thickness is not
chosen to be much larger than
1
.
0
a
, because the in-plane
PBG begins to close at this value. However, Tandaechanurat
et al.
have shown recently that relatively high
Q
can be ob-
tained from the PhC dipole mode suspended in air after the
PBG closure [
43
]. They have found that
Q
tot
can increase
up to
∼
10
;
000
at
T
∼
1
.
4
a
. In Fig.
6(c)
, we plot
Q
tot
and
V
as a function of the slab thickness. We have increased the slab
thickness up to
6
a
∼
1
;
800
nm. Surprisingly,
Q
tot
shows a
rather monotonic increase, and a value of 3000 can be ob-
tained when
T>
1
;
500
nm.
Q
tot
may be further improved
by inserting a thin low refractive index layer between the
PhC slab and the bottom mirror [
44
], which will reduce the
optical overlap with the underlying gold. What is interesting
here is that
Q
tot
seems to increase indefinitely with increasing
T
, although the in-plane PBG is completely closed. This unu-
sual behavior may be understood based on the waveguide dis-
persion along the
z
direction (
ω
−
k
z
diagram)
—
the structure
can be viewed as a PhC fiber with a finite length [
6
,
45
]. In the
case of a thin slab,
k
z
is not a well-defined quantum number.
However, as
T
increases,
k
z
of the fundamental mode can be
more and more precisely defined in accordance to the uncer-
tainty relation between
Δ
z
and
Δ
k
z
. In fact,
k
z
of the funda-
mental slab mode will converge to zero in the limit of infinite
T
. This
k
z
0
point corresponds to the ideal 2D limit in the
ω
−
k
z
diagram. Thus, the mode can be more and more con-
fined as
T
increases. It should be noted that, with a certain
large
T
, radiation loss will occur mostly in the horizontal di-
rections (
x
−
y
plane), rather than in the vertical direction (
z
)
[
46
]. This is due to the presence of the zero group velocity
dispersion at
k
z
0
point. The slow group velocity mode will
effectively reduce the scattering loss at the top surface of the
PhC slab [
45
]. It turns out that this vertical confinement me-
chanism works much more effectively than the horizontal
confinement mechanism by the PhC mirror after
T>
1
a
,
hence more radiation in the horizontal direction. For practical
applications, however, we may redirect the horizontal propa-
gating wave
’
s energy into the vertical direction by employing
grating couplers [
47
].
On the other hand,
V
tends to increase almost linearly with
the slab thickness. It should be noted that
V
∼
1
.
0
λ
∕
n
3
at
T
∼
720
nm, which is comparable with that of the widely used
L3 nanocavity [about
1
.
2
λ
∕
n
3
][
11
]. We have also estimated
the maximum achievable spontaneous emission rate enhance-
ment through the Purcell factor
F
p
[
2
]. It is shown that,
theoretically,
F
p
of more than 100 can be achieved, a reason-
ably high value within the weak-coupling regime of the cavity
QED [
48
].
B. Threshold Gain
Q
tot
tells us how much threshold gain will be required to
achieve lasing. According to the recent formulation by Chang
and Chuang [
40
,
41
], threshold gain,
g
th
, is given by
g
th
1
Γ
E
·
Q
tot
·
2
π
n
g;a
λ
;
(11)
where
n
g;a
is the material group index defined as
∂
ω
n
a
ω
∕
∂
ω
, and
n
a
ω
is the refractive index of the active
region.
Γ
E
is the energy confinement factor defined as
Γ
E
R
V
a
d
3
r
u
E
r
;t
u
M
r
;t
R
V
d
3
r
u
E
r
;t
u
M
r
;t
:
(12)
Here, the above volume integration for
V
a
is taken over the
volume of the active region, and
u
E
and
u
M
are defined in
Eqs. (
5
) and (
6
). Strictly speaking, in situations involving dis-
persive media,
Γ
E
is a quantity that depends on time. How-
ever, in most cases, we can safely assume
Γ
E
as a constant
for a given resonant mode.
We take the PhC dipole mode as an example, whose elec-
tric-field intensity profiles are shown in Fig.
6(b)
. We choose
the slab thickness to be 606 nm, roughly
2
a
. We find that
Q
tot
and resonant wavelength are 1032 and 1307 nm, respectively.
Assuming the appropriate number of quantum wells as the
gain material,
Γ
E
∼
10
%
is not so difficult to achieve [
46
], from
which
g
th
is estimated to be
∼
1000
cm
−
1
. This
g
th
value is
achievable by employing conventional InP/InGaAsP quantum
wells [
49
].
C. Radiation Efficiency
Because of the presence of the absorbing metal layer near the
PhC cavity, energy contained in the cavity will be lost by two
independent mechanisms: absorption loss in the metal
(
∼
1
∕
Q
abs
) and radiation loss into air (
∼
1
∕
Q
rad
). Therefore,
the total
Q
can be decomposed into the radiation
Q
and
the absorption
Q
in the following manner.
1
Q
tot
1
Q
rad
1
Q
abs
.
(13)
We define the radiation efficiency,
η
rad
, as the ratio of total
radiated power over the total dissipated power. This can be
written in terms of quality factors in the following way.
η
rad
≡
1
∕
Q
rad
1
∕
Q
tot
1
−
1
∕
Q
abs
1
∕
Q
tot
.
(14)
Here,
Q
tot
can be easily estimated through field decay in the
time-domain using Eq. (
2
). Estimating
Q
abs
or
Q
rad
requires
additional volume integration or surface integration. For ex-
ample, to calculate absorbed power in the Drude metal, we
should use the following volume integration [
42
]:
Kim
et al.
Vol. 29, No. 4 / April 2012 / J. Opt. Soc. Am. B 583
P
abs
t
≡
Z
V
d
3
r
ωε
0
Im
ε
m
ω
h
E
r
;t
·
E
r
;t
i
T
.
(15)
Then,
Q
abs
can be calculated by
Q
abs
ω
U
EM
t
P
abs
t
.
(16)
On the other hand,
Q
rad
can be calculated by
Q
rad
ω
U
EM
t
P
rad
t
;
(17)
where
P
rad
t
is given by
P
rad
t
≡
I
S
d
2
r
·
h
E
r
;t
×
H
r
;t
i
T
.
(18)
However, this type of surface integration usually requires a
much larger computational domain since the surface of the
integration should be located sufficiently far from the near-
zone of the PhC nanocavity mode. Thus, we have adopted
Eq. (
16
) for the calculation of the radiation efficiency.
As expected,
η
rad
is quite low (below 10%) when
T
≤
300
nm. However, we can bring up
η
rad
by simply increas-
ing the slab thickness. It is found that
η
rad
becomes about 50%
at
T
600
nm. From this slab thickness, we can argue that
the device
’
s vertical radiation efficiency will begin to compete
with the traditional PhC slab cavity suspended in air, by which
at most 50% of photons can be collected from the top (without
the bottom mirror). In any case, such
η
rad
values will be much
higher than that of other classes of metallic nanocavities,
where
Q
tot
tends to be limited by
Q
abs
[
34
,
36
].
D. Far-Field Emission
Far-field directionality can be tuned by varying the slab thick-
ness. Similar systematic optimization of the vertical collection
efficiency has been reported for the nanowire cavity sitting on
a flat metal surface [
50
]. For such a simple cavity geometry, a
Fabry
–
Pérot model can be used to optimize the far-field direc-
tionality. However, our PhC nanocavity involves complex
geometrical features, together with the zero group velocity
dispersion along the
z
direction, making it difficult for us
to develop a simple model as has been done for the PhC na-
nocavity suspended over the bottom mirror. Therefore, we
have used 3D FDTD and the near- to far-field transformation
algorithm developed in our previous work [
15
]. Fig.
7
shows
the evolution of far-field patterns as a function of the slab
thickness. The result of
T
600
nm looks most promising;
about 50% of photons emitted to the top surface will be col-
lected within
30
°. Figure
7(b)
shows that the vertical
emission is linearly polarized along the
x
direction, whose di-
rection of polarization has been determined by the two en-
larged air-hole positions [see Figs.
2(b)
and
6(b)
].
4. DEGREE OF PLASMONIC EFFECTS:
PLASMONICITY
As discussed in the previous section, the effect of the thick
slab on the vertical confinement mechanism can be under-
stood from the viewpoint of a PhC fiber. On the other hand,
the effect of the bottom mirror may be understood by the
method of image charges [
42
]. In the case of a PEC mirror,
this scheme works perfectly
—
the fundamental resonant
mode (even) in a PhC slab with thickness
T
is equivalent
to the first-order mode (odd) in a PhC slab with thickness
2
T
[
6
]. However, in the case of a realistic metal mirror, the
situation is not so simple. The existence of the evanescent
field of the cavity mode in the metal does not preclude the
possibility of interesting
plasmonic
effects within such a
structure.
So far, interesting aspects particular to surface plasmon po-
lariton (SPP) modes have been emphasized by many research-
ers. Two representative examples are the extremely large
local electric-fields in a metal nanoantenna (hot spots) [
51
]
and various metamaterial engineering, such as negative re-
fractive index [
52
]. However, we would like to emphasize that
almost same performance results have been demonstrated
from other engineered structures consisting purely of lossless
dielectric media [
53
,
54
]. Recently, Ishizaki and Noda have em-
phasized the similarity between the SPP wave and the surface
state of a 3D PhC possessing a 3D PBG [
55
].
In fact, what makes metal
metallic
is
the presence of nega-
tive permittivity
, rather than the metallic absorption originat-
ing from the imaginary part of the permittivity. One can easily
show that any medium with
ε
ω
<
0
cannot support any pro-
pagating planewave solution [
42
]. This result reminds us of
light propagation behavior within the PBG. This observation
suggests to us another viewpoint for metal
—
metal as a
natural
3D PBG material. In principle, we can carve and mold
metal into an arbitrary geometrical shape. Interestingly, it has
been known that even a tiny section (dimension
<
10
nm) of
metal does not loose its bulk optical properties [
56
], which is
in contrast to the case of an
artificial
3D PhC; at least several
lattice periods are needed to function as a PBG material [
57
].
Another aspect of metal is that conduction electrons are
strongly coupled to the electromagnetic fields. In fact,
Fig. 7. (Color online) Here, we assume RT gold substrate and ana-
lyze the same dipole mode shown in Fig.
6(b)
. (a) Far-field emission
profiles by varying the slab thickness from 500 nm to 900 nm. (b) Po-
larization resolved far-field pattern for
T
600
nm. In the simulation,
microscopic linear polarizers, one polarized along
x
direction and the
other along
y
direction, are assumed to scan over the hemispherical
surface to measure
j
E
x
j
2
and
j
E
y
j
2
, respectively.
584
J. Opt. Soc. Am. B / Vol. 29, No. 4 / April 2012
Kim
et al.