Download Design of Photonic Crystal Cavities for Extreme Light Concentration

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pubs.acs.org/journal/apchd5
Design of Photonic Crystal Cavities for Extreme Light Concentration
Shuren Hu† and Sharon M. Weiss*,†,‡
†
Department of Physics and Astronomy and ‡Department of Electrical Engineering and Computer Science, Vanderbilt University,
Nashville, Tennessee 37235, United States
S Supporting Information
*
ABSTRACT: The fields of photonic crystals and plasmonics
have taken two different approaches to increasing light−matter
interaction. Photonic crystal cavities increase temporal
confinement of light in a material, as represented by their
high quality factor, while plasmonic structures increase spatial
confinement, as represented by their low mode volume.
However, the inability to simultaneously attain extreme
temporal and spatial confinement of light remains a barrier
to realizing ultimate control of light in a material and
maximum performance in photonic devices. Here, by
engineering the photonic crystal unit cell to incorporate deep subwavelength dielectric inclusions, we show that it is possible
in a single structure to achieve a mode volume commensurate with plasmonic elements while maintaining a quality factor that is
characteristic of traditional photonic crystal cavities. Manipulating the geometric design of the unit cell leads to precise control of
the band structure and mode distribution in the photonic crystal. With a dielectric bow-tie unit cell, photonic crystals can achieve
a mode volume as small as 0.0005 (λ/n)3 with a quality factor as large as 1.75 × 106. Our results demonstrate that there exists a
promising alternative to lossy metals for extreme light concentration and manipulation.
KEYWORDS: optical resonator, light−matter interaction, photonic crystal, plasmonics, Purcell factor, silicon
T
limitations of plasmonic elements and the modal volume
limitations of traditional dielectric cavities. Our de novo
photonic crystal design method relies on understanding how
geometrical changes in a single unit cell can be leveraged to
control the optical mode profile of the photonic crystal. We
focus on changing the optical band structure by engineering the
geometrical shape of the unit cell, contrary to the conventional
methods of changing the lattice constant or scaling factor of the
unit cell. We introduce the concept of antislots in the photonic
crystal lattice, in which light can be confined in nanoscale
dielectric inclusions. We further show that by considering the
orthogonal nature of electromagnetic boundary conditions, we
can progressively squeeze light within the photonic crystal unit
cell by creating an interlocked array of antislots and slots.
Finally, we demonstrate the ability to form a cavity with
geometrically modified unit cells that supports a record Q/Vm
ratio of 109.
he interaction of light with matter is the foundation for
the fundamental processes of emission, absorption,
transmission, and reflection. The ability to control how light
interacts with matter is therefore the key to achieving
technological breakthroughs across a wide range of photonic
and optoelectronic applications including smaller and lower
power light sources,1−4 more efficient solar cells,5−8 faster and
lower power optical data processing,9−12 enhanced nonlinear
process,13,14 stronger optomechanical coupling,15 and more
sensitive detectors for ultralow concentrations of chemical and
biological molecules.16−18 Plasmonic and metal-based metamaterial structures capable of strong light−matter interaction
based on concentrating light into deep-subwavelength volumes
(e.g., mode volume = Vm ≈ 10−3 (λ/n)3) were long thought to
be the most promising avenue for interfacing nanophotonic
structures with electronics and advancing technologies related
to generating, guiding, modulating, and detecting light.6
However, ohmic losses (i.e., quality factor = Q ≈ 10−100)
have precluded the realization of practical devices to date19,20
despite attempts to mitigate losses through the use of hybrid
plasmonic elements,21 the incorporation of gain,22 and the
fabrication of alternative plasmonic materials20 or all-dielectric
metamaterials23 and metasurfaces.24 Low-loss (i.e., Q ≈ 105−
106), all-dielectric cavities, on the other hand, have been the
basis for practical devices, despite their largely diffractionlimited mode volume (typically Vm ≈ (λ/2n)3, with slotted
cavities approaching Vm ≈ 0.01 (λ/n)3).25−27
In this article, we report on the design of an all-dielectric
photonic crystal structure that can overcome both the loss
© 2016 American Chemical Society
■
THEORETICAL BACKGROUND
Engineering the Optical Field Distribution within the
Unit Cell: Breaking Rotational Symmetry. To demonstrate
how modifying a photonic crystal unit cell can modify the band
structure and mode profile, we consider a building block unit
cell of a one-dimensional (1D) photonic crystal waveguide
consisting of a circular air hole with a narrow dielectric beam
Received: March 30, 2016
Published: August 22, 2016
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Figure 1. Mode change due to breaking the rotational symmetry of the photonic crystal unit cell. The unit cell consists of a circular air hole in a
dielectric material with a thin dielectric beam spanning the diameter of the air hole. (a) Dielectric beam parallel to the direction of light propagation
(R0). (b) Dielectric beam perpendicular to the direction of light propagation (R90). (c) Band diagram for 1D silicon photonic crystal waveguides
with R0 or R90 unit cells displaying mode change due to the breaking of rotational symmetry within the unit cell. The width of the Si beam (ws),
diameter of the circular holes (d), width of the waveguide (w), and photonic crystal lattice spacing (a) are 50, 300, 700, and 400 nm, respectively. (d)
Electric field energy distribution for R0 and R90 unit cells at the band edge. The white traces show the 1D (horizontal) energy profile at the middle
of the respective unit cell. (e) Electric field distribution for R0 and R90 unit cells at the band edge (k = 0.5).
Figure 2. Explanation of antislot effect and comparison to slot effect. (a) Schematic of a photonic crystal waveguide formed with a R90 unit cell,
termed an antislot waveguide. (b) Simulated profiles of E, D, and electric field energy density of a silicon antislot waveguide assuming the same
dimensions as given in Figure 1. (c) Schematic line profiles of the dielectric constant (ε), electric field amplitude (E), electric displacement field
amplitude (D), and electric field energy density across the antislot along the x-axis as indicated by the dashed line in (a). (d) Schematic of a slot
waveguide. (e) Simulated profiles of E, D, and electric field energy density of a silicon slot waveguide assuming the widths of the waveguide and slot
are 500 and 50 nm, respectively. (f) Schematic line profiles of ε, E, D, and electric field energy density across the slot waveguide along the y-axis as
indicated by the dashed line in (d).
including free-space coupling using guided resonances, can be
considered as well.28 The addition of the dielectric beam in the
unit cell breaks the rotational symmetry. The rotation angle
here is defined as the angle between the long axis of the beam
spanning its diameter, as depicted in Figure 1. Here, light
propagates along the waveguide in the direction labeled kx in
Figure 1, but alternate configurations including two-dimensional photonic crystals, and alternate excitation approaches
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The traditional slot effect, on the other hand, squeezes light
into nanoscale low-permittivity regions surrounded by higher
permittivity media, which is the opposite of the antislot effect.
To explain the slot effect, electromagnetic boundary conditions
for the normal components of the electromagnetic field are
considered. As given in eq 3, considering that there is no
surface charge, the normal component of the electric
displacement field is continuous across the interface between
two different materials, where Dn,,low and Dn,,high are the
intensities of the normal components of the electric displacement field in the media of lower and higher permittivity,
respectively, on opposite sides of the interface. The
corresponding normal component of the electric field, En, is
therefore discontinuous across the interface, as shown in eq 4.
and direction of light propagation. Therefore, when the beam is
parallel to the light propagation direction, the rotation angle is
0°, or R0 (Figure 1a), and when the beam is perpendicular to
the light propagation direction, the rotation angle is 90°, or R90
(Figure 1b). As a result of breaking the rotational symmetry of
the unit cell, the optical mode changes according to the
rotational angle, as shown in Figure 1c (simulation details for
all calculations can be found in Methods). The higher
frequency air band that gives maximum mode confinement
within the air holes reduces in frequency as the beam is rotated
from 0° to 90°, while the lower frequency dielectric band
remains largely unchanged as a function of the beam rotation
angle because there is little effective difference between the unit
cells with different beam rotation angles when the mode is
primarily confined in the dielectric regions between the
photonic crystal lattice holes. The electric field energy density
profile at the air band edge for both R0 and R90 unit cells
(Figure 1d) provides insight on why the rotation angle of the
unit cell controls the position of the air band. For the R0 unit
cell, the energy is almost uniformly distributed throughout the
air hole with almost no energy localized within the dielectric
beam. In contrast, for the R90 unit cell, the majority of the field
is localized within the narrow dielectric beam, increasing the
effective permittivity of the air hole and therefore decreasing
the air band edge frequency. The position of the air band can
be nearly continuously tuned between the R0 position and R90
position by changing the rotation angle between 0° and 90°.
Exploring Electromagnetic Boundary Conditions:
Antislot Effect. The phenomenon of squeezing light into
nanoscale regions of high-permittivity dielectric materials
(Figure 1d) is named here the antislot effect because it is
opposite of the well-known slot effect.29 Understanding the
complementary nature of the slot and antislot effects can lead
to new approaches for manipulating the mode profile and
energy distribution of light in photonic structures. Here, we
provide a more comprehensive explanation of the newly coined
antislot effect and its relationship to the slot effect.
As shown in eq 1, electromagnetic boundary conditions
dictate that the tangential component of the electric field, Et, is
continuous across an interface between two materials. In this
equation, Et,,high and Et,,low are the intensities of the tangential
components of the electric field in the media of higher and
lower permittivity, respectively, on opposite sides of the
interface. Accordingly, the corresponding tangential component
of the electric displacement field, Dt = εEt, is discontinuous
across the interface, as shown in eq 2.
Et,high = Et,low
Dt,high = Dantislot =
Dn,low = Dn,high
En,low = Eslot =
ϵlow
Dt,low
ϵhigh
ϵlow
En,high
(4)
As a result of these boundary conditions, and as illustrated in
Figure 2d−f, when a material of lower permittivity is squeezed
into a nanoscale slot between two higher permittivity regions,
the electric displacement field (TE polarization) in this slot has
the same intensity as that in the surrounding higher permittivity
regions (eq 3), while the electric field intensity in the slot is
enhanced by a factor of εhigh/εlow (eq 4). The electric field
energy density inside the low-permittivity slot is therefore also
enhanced by a factor of εhigh/εlow due to the enhancement of
the electric field. We note that the slot effect leads to the lower
electric field intensity and lower energy density in the dielectric
beam in the R0 unit cell compared to the surrounding air
regions inside the lattice hole for TE-polarized light, as well as
the slight energy density enhancements that arise at edges of
the lattice holes for both the R0 and R90 unit cells (Figure
1d,e).
Unlike the slot effect, which is supported using a traditional
waveguide geometry, the antislot effect displayed in the R90
unit cell can only be supported when incorporated as part of a
photonic crystal or engineered material that supports light
propagation based on the designed periodicity of the arrayed
unit cells. For the antislot effect, light is confined in a thin
dielectric beam that is positioned with its long axis
perpendicular to the propagation direction of light to satisfy
the electromagnetic boundary condition shown in eq 2. Hence,
an antislot waveguide cannot be formed in the same way as a
slot waveguide. For a guided mode to interact with an antislot,
the light confinement must be provided by incorporating the
antislot into another structure that can support the necessary
waveguide confinement. For example, when a 1D photonic
crystal is used to provide confinement for a guided mode, the
peaks of the electric field are localized within the lattice holes
for the air band edge mode (k = π/a). The antislot effect can
then be supported in the 1D photonic crystal when thin
dielectric beams are placed at the peak locations of the electric
field inside the photonic crystal lattice air holes, where the
strongest electromagnetic energy can be squeezed into the
dielectric beam.
(1)
ϵhigh
(3)
(2)
Considering a 1D photonic crystal with a R90 unit cell and
light polarized in-plane (TE polarization), as illustrated in
Figure 2a, the electric displacement field in the highpermittivity dielectric beams is enhanced by a factor of εhigh/
εlow compared to the surrounding lower permittivity air regions
inside the lattice holes (eq 2), while the electric field intensity
in the beams is the same as that of the surrounding regions
inside the air hole (eq 1). Importantly, the electric field energy
1
density, uE = 2 DE , inside the dielectric beams is also enhanced
by a factor of εhigh/εlow due to the enhancement of the electric
displacement field, as shown in Figure 2b,c.
■
PHOTONIC CRYSTAL UNIT CELL DESIGN
Progressively Squeezing Light into a Point-like Space:
Interlocked Design. Due to the orthogonal nature of the
electromagnetic boundary conditions that give rise to the slot
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Figure 3. Nanofocusing of light into ultrasmall mode volumes by progressively interlocking antislot and slot designs. (a) Top view (x−y plane) of
the proposed unit cells (air holes in silicon) with increasing numbers of incorporated silicon antislots (A) and air slots (S) from left to right. The R90
until cell with one antislot is shown schematically for reference. The propagation direction of light is in the x-direction. In the limit of an infinite
number of interlocked, orthogonal antislots and slots, the geometry approaches that of a bow-tie shape. (b and c) Energy density cross sections (y−z
and z−x planes, respectively) through the center of the unit cells shown in (a). The line traces in (c) show the electric field energy density of each
unit cell (Figure S2 shows these line traces on a log scale). All color maps are scaled according to the minimum and maximum values of each
individual unit cell.
dimension of the antislots and slots reduces to a deep
subwavelength feature size, the enhancement factor of the
energy density must be related to an effective index: for
antislots, the effective index decreases from the bulk silicon
refractive index with decreasing feature size, and for slots, the
effective index increases from the refractive index of air with
decreasing feature size. Therefore, the energy density does not
approach infinity as the number of interlocked antislots and
slots increases. Details regarding the importance of the
interlocked configuration for light confinement are provided
in the Supporting Information Figures S4 and S5.
In the limit of an infinite number of interlocked antislots and
slots added to the unit cell, the geometry approaches that of a
bow-tie shape. The advantages of the bow-tie unit cell design
are twofold. First, the design provides extreme subwavelength
confinement of light with a mode volume that is on par with
plasmonic bow-tie structures, as discussed in the next section.
The energy density in the center of the bow-tie structure is
1500 times higher than the energy density in the center of a
zeroth-order unit cell (Figure 3c line trace). Second, a highquality bow-tie structure is easier to fabricate with high fidelity
compared to a high-order interlocked geometry. Importantly,
unlike plasmonic bow-ties, all-dielectric bow-ties support the
and antislot effects, light can be progressively squeezed in both
the propagation direction (i.e., x-direction) and the perpendicular in-plane direction (i.e., y-direction) by introducing a series
of interlocked antislots and slots, respectively. Figure 3 shows
unit cells containing an increasing number of interlocked
antislots and slots. Detailed dimensions of these simulated unit
cells can be found in Figure S1. The nomenclature for these
interlocked antislot−slot designs follows the number of
antislots and slots introduced within the unit cell. Accordingly,
the zeroth-order structure is absent of antislots (A) and slots
(S), the first-order structure contains a single antislot, and
subsequent orders are given by AS (second order), ASA (third
order), ASAS (fourth order), etc., as labeled in Figure 3. Light
is localized within the last antislot or slot introduced, and
therefore the optical energy is progressively squeezed into
smaller and smaller mode volumes. The energy density cross
sections shown in Figure 3b and c indicate that the mode
profile along the y-direction is squeezed only when a slot is
introduced, while the mode profile along the x-direction is
squeezed only when an antislot is added. Magnetic field
distributions, which complement the electric field distributions
in antislot and higher order interlocked unit cells, are shown in
Supporting Information Figure S3. We note that as the critical
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allow control over the band structure. The bow-tie photonic
crystal cavity possesses an ultrahigh Q of 1.76 × 106 at a
resonance wavelength of 1496 nm with an ultrasmall Vm of 5.6
× 10−4 (λ/n)3. Since the cavity is designed using slotterminated bow-tie unit cells, the optical energy is highly
localized in the air gap between the bow-tie tips, as shown in
Figure S6, and the refractive index n given in the mode volume
of the bow-tie photonic crystal cavity is close to that of air. The
mode volume is calculated as follows:
design freedom and fabrication tolerance to allow extreme light
localization in either an air gap or a dielectric connection
between the two sides of the bow-tie, while maintaining a high
Q. If there is an air gap between the tips such that the
interlocked geometry ends with a slot, then energy is confined
to that air gap. Alternatively, if there is no gap between the tips,
corresponding to an interlocked geometry ending with an
antislot, then energy is confined within the tiny region of
dielectric material spanning the tips.
High-Q Photonic Crystal Cavity with Deep Subwavelength Confinement. Finally, we demonstrate the capabilities
of the proposed bow-tie photonic crystal unit cell to form a
cavity with deep subwavelength confinement at the level of
plasmonic elements and ultrahigh Q on par with photonic
crystal structures. Such a combination of ultralow Vm and
ultrahigh Q was previously thought to be impossible for an alldielectric structure.6,19 We select the bow-tie unit cell for this
demonstration and spatially vary the rotation angle of the unit
cell to create a cavity. Figure 4 shows the band edge (k = 0.5)
Vm =
∫ ϵ |E|2 dV
max(ϵ |E|2 )
(5)
The mode profile shown in Figure 5b along with a
superimposed outline of the dielectric structure reveals that
the electric field energy is concentrated between the bow-tie
tips, similar to the case for a plasmonic bow-tie hot spot. By
plotting the energy distribution on a log scale in Figure 5c, it is
clear that the mode decays gradually into the mirror segments,
giving a Gaussian-shaped electric field profile (Figure 5d) that
minimizes radiation losses. The Fourier transformed field
distribution shown in Figure 5e confirms that the field
components are localized at k= ±π/a with minimal extension
into the light cone. The electric field in the cavity center is
more than 103 times stronger than that in a planar waveguide
(E0), and therefore the electric field energy density is more than
106 times larger than the input waveguide energy density, as
confirmed in the electric field energy profile shown in Figure 5f.
This is the highest electromagnetic energy enhancement factor
reported to date, and it is at least 2 orders of magnitude larger
than what is achieved in a plasmonic bow-tie cavity.6,13 We note
that optimization of the design parameters, such as the angle of
the bow-tie tip, the rotation step of tapering segments, and the
number of mirror segments, may lead to bow-tie photonic
crystals with an even higher Q/Vm ratio.
Figure 4. Band diagrams of dielectric bow-tie unit cell with different
rotation angles. The air band of the R90-bow-tie unit cell falls in the
middle of the band gap of the R0-bow-tie unit cell. It is therefore
possible to design a cavity to confine the band edge mode of a R90
unit cell by using R0 unit cells. Rotating the bow-tie provides a natural
way to taper the cavity from R0 to R90 and back to R0.
■
CONCLUSION
This work shows that it is possible for all-dielectric cavities to
support a deep subwavelength confined optical mode along
with an ultrahigh quality factor. Extreme light localization in an
all-dielectric, high-Q photonic crystal was achieved by
exploiting the orthogonality of electromagnetic boundary
conditions to squeeze the electric energy density in both inplane directions via iterative application of the slot and antislot
effects. A 1D photonic crystal cavity with bow-tie-shaped unit
cells in which the bow-tie-shaped dielectric inclusions represent
the geometry approximating an infinite number of interlocked
slots and antislots was shown to support a mode volume as
small as that of a plasmonic resonator (Vm = 0.0005 (λ/n)3)
with temporal confinement as long as that of a dielectric
resonant cavity (Q ≈ 1.75 × 106). We believe that the design
method presented here and the achievement of such
unparalleled characteristics in an all-dielectric material is an
important milestone that could inspire new photonic device
designs that support record performance across a broad range
of optical applications.
frequencies of dielectric bow-tie unit cells with different
rotation angles. Similar to the band diagram shown in Figure
1 for R0 and R90 unit cells, the air band of the R90-bow-tie lies
in the middle of the R0-bow-tie band gap.
Accordingly, we place a R90-bow-tie unit cell in the center of
the cavity and use R0-bow-tie unit cells to create mirrors on
either side of the cavity. As is the case for all 1D photonic
crystal cavities, the Q-factor is governed in large part by the
selected band gap tapering from the cavity to mirror unit cells
and the number of mirror unit cells.30,31 For simplicity, we
choose to transition between the R0-bow-tie mirror segments
and the cavity center R90-bow-tie unit cell by rotating the
intermediary bow-tie unit cells by 5 deg/step in the taper
segments, as schematically illustrated in Figure 5a. Ten R0bow-tie unit cells are placed at each end of the photonic crystal,
serving as mirrors at the end of the cavity. The taper segments
between the mirrors and cavity increase the mode volume;
however, in order to support a high Q-factor resonance, it is
essential to form a Gaussian mode profile that minimizes
losses.31 There are other approaches beyond unit cell rotation
that could be employed to form bow-tie photonic crystal
cavities, such as modulating the hole radius, waveguide width,
or bow-tie dimensions since these unit cell modifications also
■
METHODS
The numerical simulations presented here were carried out
with three-dimensional FDTD analysis using both the open
source package MEEP and commercial software, Lumerical
FDTD (Lumerical Solutions Inc.). The band structures and
mode profiles of the unit cells in Figure 1 and Figure 3 were
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Figure 5. 1D photonic crystal cavity design based on a bow-tie unit cell. (a) The photonic crystal cavity is formed by gradually rotating a silicon bowtie unit cell from the R0-bow-tie orientation in the mirror segments to the R90-bow-tie orientation in the center of the cavity (5 degree rotation per
unit cell). The calculated Q and Vm are 1.76 × 106 and 0.0005 (λ/n)3, respectively. (b) Linear and (c) log plot of mode profile with dielectric
structure superimposed. (d) Electric field profile along a horizontal slice through the middle of the photonic crystal showing a gradual modulation of
the electric field from cavity center to mirror edges. (e) 2D Fourier transform of the electric field distribution in (d), demonstrating good in-plane
confinement of the mode. (f) Electric field energy profile along a vertical slice through the center of the bow-tie photonic crystal cavity showing
extreme energy localization. The electric field intensities in (d) and (f) are normalized to the maximum electric field intensity in a ridge waveguide
(E0).
■
calculated in MEEP. The mesh size was chosen to be a uniform
5 nm per grid with one exception. The mode profile of the
bow-tie unit cell in Figure 3 was calculated with a 2.5 nm mesh
grid in order to obtain improved resolution at the tip. The
calculated mode intensities were normalized to the total optical
energy in the unit cell for comparison.
The cavity mode profile in Figure 5 was calculated using
Lumerical FDTD. The photonic crystal lattice holes were
defined as mesh regions with a grid size of 2 nm, while the
remainder of the simulation space is meshed by the Lumerical
default conformal mesh. Symmetrical boundary conditions
along the x- and z-directions were applied to reduce the
calculation time. The source was defined as a magnetic dipole
Hz at the (200 nm, 0 nm, 0 nm) coordinates. To calculate the
electric field enhancement in the cavity, a control simulation
was calculated on a waveguide structure with the same width
and height as the nanobeam cavity. The simulation
configurations (mesh size, simulation time, source, and
boundary conditions) for the control run are kept the same
as the cavity mode calculations.
ASSOCIATED CONTENT
S Supporting Information
*
The Supporting Information is available free of charge on the
ACS Publications website at DOI: 10.1021/acsphotonics.6b00219.
Design dimensions of interlocked unit cells, comparison
of field enhancement between interlocked and noninterlocked antislot/slot configuration, scaling of mode
volume with antislot width, electric field energy density
comparison of different orders of interlocked antislot/
slot structures, dielectric and mode profiles of the
photonic crystal bow-tie unit cell (PDF)
■
AUTHOR INFORMATION
Corresponding Author
*E-mail (S. M. Weiss): [email protected].
Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS
This work was supported by the National Science Foundation
(ECCS1407777). The authors thank G. Gaur, K. J. Miller, Y.
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(22) Xiao, S. M.; Drachev, V. P.; Kildishev, A. V.; Ni, X.; Chettiar, U.
K.; Yuan, H. K.; Shalaev, V. M. Loss-free and active optical negativeindex metamaterials. Nature 2010, 466, 735−738.
(23) Moitra, P.; Yang, Y.; Anderson, Z.; Kravchenko, I. I.; Briggs, D.
P.; Valentine, J. Realization of an all-dielectric zero-index optical
metamaterial. Nat. Photonics 2013, 7, 791−795.
(24) Lin, D. M.; Fan, P. Y.; Hasman, E.; Brongersma, M. L. Dielectric
gradient metasurface optical elements. Science 2014, 345, 298−302.
(25) Joannopoulos, J. D.; Johnson, S. G.; Winn, J. N.; Meade, R. D.
Photonic Crystals: Molding the Flow of Light, 2nd ed.; Princeton
University Press, 2008; pp 217−218.
(26) Ryckman, J. D.; Weiss, S. M. Low mode volume slotted
photonic crystal single nanobeam cavity. Appl. Phys. Lett. 2012, 101,
071104.
(27) Seidler, P.; Lister, K.; Drechsler, U.; Hofrichter, J.; Stoferle, T.
Slotted photonic crystal nanobeam cavity with an ultrahigh quality
factor-to-mode volume ratio. Opt. Express 2013, 21, 32468−32483.
(28) Fan, S.; Joannopoulos, J. D. Analysis of guided resonances in
photonic crystal slabs. Phys. Rev. B: Condens. Matter Mater. Phys. 2002,
65, 235112.
(29) Almeida, V. R.; Xu, Q. F.; Barrios, C. A.; Lipson, M. Guiding
and confining light in void nanostructure. Opt. Lett. 2004, 29, 1209−
1211.
(30) Kuramochi, E.; Taniyam, H.; Tanabe, T.; Kawasaki, K.; Roh, Y.;
Notomi, M. Ultrahigh-Q one-dimensional photonic crystal nanocavities with modulated mode-gap barriers on SiO2 claddings and on
air claddings. Opt. Express 2010, 18, 15859−15869.
(31) Quan, Q.; Loncar, M. Deterministic design of wavelength scale,
ultra-high Q photonic crystal nanobeam cavities. Opt. Express 2011,
19, 18529−18542.
Yang, W. Li, W. M. J. Green, M. H. Khater, C. Xiong, B. Peng,
and J. S. Orcutt for helpful discussions. This work was
conducted in part using the resources of the Advanced
Computing Center for Research and Education at Vanderbilt
University, Nashville, TN, and computational resources at the
IBM T. J. Watson Research Center at Yorktown Heights, NY.
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REFERENCES
(1) Noda, S.; Fujita, M.; Asano, T. Spontaneous-emission control by
photonic crystals and nanocavities. Nat. Photonics 2007, 1, 449−458.
(2) Oulton, R. F.; Sorger, V. J.; Zentgraf, T.; Ma, R.; Gladden, C.;
Dai, L.; Bartal, G.; Zhang, X. Plasmon lasers at deep subwavelength
scale. Nature 2009, 461, 629−632.
(3) Matsuo, S.; Shinya, A.; Kakitsuka, T.; Nozaki, K.; Segawa, T.;
Sato, T.; Kawaguchi, Y.; Notomi, M. High-speed ultracompact buried
heterostructure photonic-crystal laser with 13 fJ of energy consumed
per bit transmitted. Nat. Photonics 2010, 4, 648−654.
(4) Takahashi, Y.; Inui, Y.; Chihara, M.; Asano, T.; Terawaki, R.;
Noda, S. A micrometre-scale Raman silicon laser with a microwatt
threshold. Nature 2013, 498, 470−474.
(5) Atwater, H. A.; Polman, A. Plasmonics for improved photovoltaic
devices. Nat. Mater. 2010, 9, 205−213.
(6) Schuller, J. A.; Barnard, E. S.; Cai, W.; Jun, Y. C.; White, J. S.;
Brongersma, M. L. Plasmonics for extreme light concentration and
manipulation. Nat. Mater. 2010, 9, 193−204.
(7) Brongersma, M. L.; Cui, Y.; Fan, S. Light management for
photovoltaics using high-index nanostructures. Nat. Mater. 2014, 13,
451−460.
(8) Green, M. A.; Pillai, S. Harnessing plasmonics for solar cells. Nat.
Photonics 2012, 6, 130−132.
(9) Xu, Q. F.; Schmidt, B.; Pradhan, S.; Lipson, M. Micrometre-scale
silicon electro-optic modulator. Nature 2005, 435, 325−327.
(10) Reed, G. T.; Mashanovich, G.; Gardes, F. Y.; Thomson, D. J.
Silicon optical modulators. Nat. Photonics 2010, 4, 518−526.
(11) Nozaki, K.; Tanabe, T.; Shinya, A.; Matsuo, S.; Sato, T.;
Taniyama, H.; Notomi, M. Sub-femtojoule all-optical switching using a
photonic-crystal nanocavity. Nat. Photonics 2010, 4, 477−483.
(12) Melikyan, A.; Alloatti, L.; Muslija, A.; Hillerkuss, D.; Schindler,
P. C.; Li, J.; Palmer, R.; Korn, D.; Muehlbrandt, S.; Van Thourhout,
D.; Chen, B.; Dinu, R.; Sommer, M.; Koos, C.; Kohl, M.; Freude, W.;
Leuthold, J. High-speed plasmonic phase modulators. Nat. Photonics
2014, 8, 229−233.
(13) Kim, S.; Jin, J.; Kim, Y. J.; Park, I. Y.; Kim, Y.; Kim, S. W. Highharmonic generation by resonant plasmon field enhancement. Nature
2008, 453, 757−760.
(14) Nozaki, K.; Shinya, A.; Matsuo, S.; Suzaki, Y.; Segawa, T.; Sato,
T.; Kawaguchi, Y.; Takahashi, R.; Notomi, M. Ultralow-power alloptical RAM based on nanocavities. Nat. Photonics 2012, 6, 248−252.
(15) Eichenfield, M.; Chan, J.; Camacho, R. M.; Vahala, K. J.; Painter,
O. Optomechanical crystals. Nature 2009, 462, 78−82.
(16) Brolo, A. G. Plasmonics for future biosensors. Nat. Photonics
2012, 6, 709−713.
(17) Chakravarty, S.; Lai, W. C.; Zou, Y.; Drabkin, H. A.; Gemmill, R.
M.; Simon, G. R.; Chin, S. H.; Chen, R. T. Multiplexed specific labelfree detection of NCI-H358 lung cancer cell line lysates with silicon
based photonic crystal microcavity biosensors. Biosens. Bioelectron.
2013, 43, 50−55.
(18) Hu, S.; Zhao, Y.; Qin, K.; Retterer, S. T.; Kravchenko, I. I.;
Weiss, S. M. Enhancing the Sensitivity of Label-Free Silicon Photonic
Biosensors through Increased Probe Molecule Density. ACS Photonics
2014, 1, 590−597.
(19) Khurgin, J. B. How to deal with the loss in plasmonics and
metamaterials. Nat. Nanotechnol. 2015, 10, 2−6.
(20) Boltasseva, A.; Atwater, H. A. Low-Loss Plasmonic Metamaterials. Science 2011, 331, 290−291.
(21) Oulton, R. F.; Sorger, V. J.; Genov, D. A.; Pile, D. F. P.; Zhang,
X. A hybrid plasmonic waveguide for subwavelength confinement and
long-range propagation. Nat. Photonics 2008, 2, 496−500.
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DOI: 10.1021/acsphotonics.6b00219
ACS Photonics 2016, 3, 1647−1653