1. science
2. physics
3. optics

Sample Publications
[1] Jun Tan*, Ming Lu, Aaron Stein, and Wei Jiang, "High purity transmission of a slow-light odd mode
in a photonic crystal waveguide," Opt. Lett. (accepted).
[2] Manjit Chahal*, George K. Celler, Yogesh Jaluria, and Wei Jiang, "Thermo-optic characteristics
andswitching power limit of slow-light photonic crystal structures on a silicon-on-insulator platform,"
Opt. Express, vol. 20, 4225 (2012).
[3] Ryan A. Integlia*, Lianghong Yin*, Duo Ding, David Z. Pan, Douglas M. Gill, and Wei Jiang,
“Parallel-Coupled Dual Racetrack Silicon Micro-Resonators for Quadrature Amplitude Modulation,”
Opt. Express, vol. 19, 14892 (2011).
[4] Weiwei Song*, Ryan A. Integlia*, and Wei Jiang, "Slow light loss due to roughness in photonic
crystal waveguides: An analytic approach", Physical Review B vol. 82, 235306 (2010).
[5] Ryan A. Integlia*, Weiwei Song*, Jun Tan*, and Wei Jiang, "Longitudinal and Angular Dispersions
in Photonic Crystals: A Synergistic Perspective on Slow Light and Superprism Effects," Journal of
Nanoscience and Nanotechnology, vol. 10, 1596-1605 (2010).
[6] Wei Jiang, and Ray T. Chen, “Symmetry induced singularities of the dispersion surface curvature and
high sensitivities of a photonic crystal,” Phys. Rev. B vol. 77, 075104 (2008).
(* indicates students or postdoctoral researcher supervised by me.)
August 15, 2012 / Vol. 37, No. 16 / OPTICS LETTERS
1
High-purity transmission of a slow light odd mode
in a photonic crystal waveguide
Jun Tan,1 Ming Lu,2 Aaron Stein,2 and Wei Jiang1,3,*
1
3
Department of Electrical and Computer Engineering, Rutgers University, Piscataway, New Jersey 08854, USA
2
Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973, USA
Institute for Advanced Materials, Devices, and Nanotechnology, Rutgers University, Piscataway, New Jersey 08854, USA
*Corresponding author: [email protected]
Received May 4, 2012; revised June 5, 2012; accepted June 8, 2012;
posted June 11, 2012 (Doc. ID 166625); published 0 MONTH 0000
We demonstrate a novel scheme to control the excitation symmetry for an odd mode in a photonic crystal waveguide
and investigate the spectral signature of this slow light mode. An odd-mode Mach–Zehnder coupler is introduced to
transform mode symmetry and excite a high-purity odd mode with 20 dB signal contrast over the background.
Assisted by a mixed-mode Mach–Zehnder coupler, slow light mode beating can be observed and is utilized to
determine the group index of this odd mode. With slow light enhancement, this odd mode can help enable novel
miniaturized devices such as one-way waveguides. © 2012 Optical Society of America
OCIS codes: 130.5296, 230.5298.
Photonic crystal waveguides (PCWs) [1–5] can modify
light propagation and dispersion characteristics through
their periodic structures and thus have important applications in communications and sensing. Particularly, the
slow light effect in a PCW can significantly enhance
light–matter interaction [6–8], as demonstrated in significant reduction of interaction lengths for PCW-based
modulators and switches [9–11]. To date, most of the
PCW research has been focused on the TE-like mode
with even symmetry. However, a PCW often has an
odd TE-like mode inside the photonic bandgap exhibiting
the slow light effect as well. This odd mode can potentially open up the opportunities for mode-symmetrybased novel devices, such as one-way waveguides that
exploit indirect interband photonic transitions between
even and odd modes [12]. The slow light effect in PCWs
can help reduce the interaction length for such transitions, enabling ultracompact devices. To utilize this odd
mode in any device, it is crucial to control its excitation
symmetry and understand its slow light spectral characteristics. Normally, this odd mode does not exhibit itself
evidently in the PCW transmission spectrum because its
odd symmetry prohibits its excitation by the fundamental
even mode of a conventional waveguide typically used at
input. Symmetry-breaking structure imperfections sometimes may induce some coupling to this odd mode, causing a decrease of PCW transmission in the odd mode
band [13,14]. Here we demonstrate a novel scheme to
control the excitation symmetry for high-purity transmission of this odd mode and investigate the spectral signatures under various excitation symmetries.
Consider a W1 PCW formed on a silicon-on-insulator
(SOI) wafer by removing a row of air holes in a hexagonal
lattice with lattice constant a 400 nm, hole radius
r 0.325a, and Si slab thickness t 260 nm. The band
diagram in Fig. 1(a) is calculated by three-dimensional
(3D) plane wave expansion [15] (with >1 μm top/bottom
claddings and six rows of holes per side). Below the lightline (for the oxide bottom cladding), the even TE-like
mode has a flat dispersion relation with group index ng >
50 and a narrow bandwidth (<4 nm). In contrast, below
the lightline, the odd TE-like mode has a much wider
bandwidth, ∼20 nm, with ng down to ∼15. Such a mod0146-9592/12/160001-03$15.00/0
erate ng range is favorable for many applications as various types of losses are reduced at lower ng [5,15–17].
Furthermore, the dispersion relation of the TM-like
guided mode usually crosses that of the even mode
[5], as seen in Fig. 1(a). But the TM-like mode does not
cross the odd mode in the region below the lightline in
Fig. 1(a). For ωa ∕ 2πc 0.28–0.286, only the odd mode is
below the lightline.
Systematic simulations show that as the hole radius increases, the odd-mode band edge moves up faster than
the TM cutoff, as shown in Fig. 1(b). For a sufficiently
large r, the TM cutoff is below the odd-mode band edge;
thus the two modes do not cross each other below the
lightline, helping avoid their intercoupling due to asymmetric top and bottom claddings. However, as r increases, the transmission bandwidth bounded by the
band edge and the cutoff decreases for both the even
and odd modes, as shown in Fig. 1(b). Hence, this work
focuses on the intermediate r case shown in Fig. 1(a),
which shows a sufficient clearance between the oddmode band edge and the TM cutoff, and a sufficiently
wide bandwidth.
Excitation of this odd PCW mode is usually deterred by
the opposite symmetry of the fundamental even mode
of a Si waveguide. To solve this problem, we employ a
Fig. 1. (Color online) PCW photonic band structures. (a) Band
diagram for r 0.325a. The dark grey region indicates the lower photonic band. H z field profiles for even and odd modes at
k π ∕ a are shown in the insets (PCW axis along y); (b) variation of the band edge and cutoff of even (blue) and odd (green)
TE-like modes with hole radius. For each TE-like mode, the
lower line (solid) gives the band edge; the upper line (dashed)
gives the cutoff frequency where a mode crosses the lightline.
The TM cutoff is also shown.
© 2012 Optical Society of America
2
OPTICS LETTERS / Vol. 37, No. 16 / August 15, 2012
Fig. 2. (Color online) FDTD simulation results; (a) schematic
of the MZC structure. The right arm has two extra waveguide
segments (in orange) with a combined length of Δlπ . The input and output E x field profiles (cross section) are shown in the
insets (200 nm per division on axes); (b) PCW coupling efficiency. Insets: coupled E x field patterns at 1390 nm. Light
(in odd or even mode) enters from a Si waveguide at the bottom
of each figure and into a PCW upward.
two-step approach. First, a Mach–Zehnder coupler
(MZC) whose two arms have a phase difference of π is
utilized to transform mode symmetry and excite an
odd mode in a wide (multimode) Si wire waveguide; then
this odd mode is coupled to the odd mode of the PCW. To
create π phase difference in this odd-mode MZC, its two
arms can be designed to have a length difference of
Δlπ λ ∕ 2neff , where neff is the effective index of the
Si waveguide. Finite difference time-domain (FDTD) simulation has been performed to confirm that such a MZC
produces an odd mode in a wide output waveguide, as
shown in Fig. 2(a). The input and output waveguide
widths are 400 and 700 nm, respectively. The coupling
between the odd mode of a Si wire waveguide (700 nm
wide) and that of the PCW is also simulated. Simulation
results in Fig. 2(b) show coupling efficiencies up to ∼84%
(∼0.75 dB) for the odd mode. The field pattern in
Fig. 2(b), left inset, confirms that the coupled PCW mode
is an odd mode. The fundamental even mode of a Si wire
waveguide couples into the PCW with inconsequential
change of coupling efficiency for the spectral range in
Fig. 2(b). The field pattern in Fig. 2(b), right inset, indicates that the coupled mode has even symmetry. Indeed,
this mode is an even TE-like mode above the lightline.
The Ex field has been shown in Fig. 2 for direct comparison with the modes of the conventional Si waveguide,
whose TE modes are commonly visualized by E x (note
Ex and H z have the same symmetry with respect to x).
The PCW structure is fabricated on a SOI wafer with a
2 μm buried oxide layer and a 260 nm top Si layer according to the parameters used in Fig. 1(a). The structure is
patterned by a JEOL JBX-6300FS high-resolution e-beam
lithography system, operating at 100 keV, on a 100 nm
thick layer of ZEP 520A e-beam resist. Then the pattern
is transferred to the Si layer by an Oxford Plasmalab
100 ICP etcher. Figure 3 is a scanning electron microscope
(SEM) image of the fabricated structure. Two MZCs with a
10 μm bending radius are connected through 700 nm wide
Si waveguides of 1 μm length to both ends of the PCW.
To measure transmission spectra, light from a superluminescent LED with a spectral range of about 80 nm
is coupled to the TE mode of Si access waveguides
(tapered to 4 μm at chip edges) via lensed fibers. A polarizer is used at the output end to block TM polarization.
Fig. 3. SEM image of a PCW with odd-mode MZCs. Inset:
close-up view of the coupling region at one end of the PCW.
The PCW insertion loss is measured with reference to a Si
wire waveguide. Figure 4(a) shows the spectrum of a
PCW with odd-mode MZCs. A substantial transmission
bandwidth is observed, approximately 22 nm at 10 dB
below the peak. The contrast between the transmitted
mode and background is >20 dB. The peak insertion loss
is about −4 dB. Separate measurements show that each
MZC contributes ∼1 dB. Thus the loss due to the PCW
is estimated at ∼2 dB. For comparison, the spectrum
of a directly coupled PCW without MZCs is shown in
Fig. 4(b). The transmission is due to the leaky even
TE-like mode as simulated in Fig. 2(b). Figure 4(b) also
shows the PCW transmission with MZCs whose two arms
have a length difference Δl deliberately designed to be
50% greater than Δlπ . Such a mixed-mode MZC offers
a symmetry configuration that can excite a mixture
of even and odd modes according to I ∝ 1 ∕ 2
1 cos2πneff Δl ∕ λ. As such, the background transmission due to the even mode rises. In the odd-mode band,
the mixed-mode spectrum oscillates strongly due to the
beating of two modes. Figures 4(a) and 4(b) illustrate
that distinctive spectral signatures can be observed with
controlled excitation symmetries.
The mode-beating pattern of the mixed-mode spectrum contains important information of the odd mode.
The beating period is related to the group indices of even
and odd modes through Δλ λ2 ∕ ng;odd − ng;even L,
where L is the PCW length. Simulation indicates that
ng;even is virtually a constant ∼5 in the odd-mode band.
Thus the chirped beating periods are due to the dispersion of ng;odd . We have calculated Δng ng;odd − ng;even
from the mixed-mode spectrum and plotted it in Fig. 4(c).
The peak spacing and valley spacing of the spectrum give
two sets of Δng data, plotted by circles and crosses
Fig. 4. (Color online) Transmission spectra for 20 μm long
PCWs: (a) with odd-mode MZCs; (b) direct transmission without MZC and transmission with mixed-mode MZCs; (c) Δng obtained from the mixed-mode spectrum; the solid line delineates
the trend; (d) Fourier transform of the transmission spectrum of
another directly coupled PCW (the peak position gives ng;even ).
August 15, 2012 / Vol. 37, No. 16 / OPTICS LETTERS
respectively. They agree with each other, as expected.
Note that the Δng value obtained from two adjacent
peaks (valleys) is assigned to the midpoint wavelength
in between. Further, ng;even 4.9 is obtained in Fig. 4(d)
through the Fourier transform [18] of the transmission
spectrum of another directly coupled PCW with more obvious spectral ripples. Note that the Fourier frequency f λ
is just the inverse of the spectral oscillation period δλ,
thus ng;even f λ × λ2 ∕ 2L. Based on Figs. 4(c) and 4(d),
we find ng;odd Δng ng;even in the range of 14 to 29.
Note that the Fabry–Perot (F-P) oscillation amplitude
in Fig. 4(a) is relatively weak. In contrast, the modebeating amplitude of the mixed-mode spectrum in
Fig. 4(b) is much higher and more robust against noise,
which facilitates the evaluation of ng;odd . Also note that in
Fig. 4(a), the background transmission increases discernibly beyond 1430 nm due to the dispersive effect in the
odd-mode MZC, which modifies the phase shift difference between the two arms as λ deviates far from the
designed value (1390 nm). The TM-like mode (guided
for λ > 1.45 μm) may also contribute to the background
at long wavelengths. However, these effects are much
weaker for 1380–1415 nm.
Although this work focuses on PCWs on a SOI chip, the
MZC and the mode-beating-based ng;odd measurement
method can be adapted to the cases of air-bridge or
oxide-covered PCWs and coupled-cavity PCWs, where interesting anomalous propagation related to an odd mode
has been observed [19]. It would be interesting also to
explore a refined design to optimize the bandwidth
and the slow-down of light together for this odd mode.
Detailed discussion of these possibilities is beyond the
scope of this work. The odd-mode wavelength can also
be shifted to ∼1550 nm or other values (depending on
specific applications) by changing the lattice constant.
In a SOI PCW, there is some coupling between the TElike guided modes and the TM-like photonic crystal bulk
modes due to asymmetric top/bottom claddings. Prior
work on the even mode has demonstrated that reducing
ng can reduce the loss due to such coupling [5]. This odd
mode has a much lower ng , ∼14, than the normal even
mode (ng ∼ 50) below the lightline. This helps to reduce
the coupling to the TM-like bulk modes. For many PCW
devices operating at a short length <80 μm [9,10], the
propagation loss of the odd mode is expected to be reasonable. Lastly, the understanding of the slow light and
mode-beating characteristics of this odd mode, as well
as the controlled excitation and ng;odd characterization
schemes developed here, can facilitate the development
of mode-symmetry-based novel devices, such as one-way
waveguides that involve active transition and passive
conversion between even and odd modes [12]. Slow light
can help reduce device interaction length. Note that previously demonstrated conventional waveguide mode
converters employed branching waveguides [20,21] or
multimode interference couplers [22]. Photonic-crystalbased mode converters have also been designed [23].
Here, the odd-mode MZC is focused on transforming
mode symmetry to attain a high-purity odd mode, and the
mixed-mode MZC offers a symmetry configuration for coherent mixing of even and odd modes, which enables
ng;odd measurement through slow-light mode beating.
3
As a side note, beating between two degenerate modes
in a periodically patterned microring resonator has
recently been observed, but the resonant wavelength
spacing is not affected by beating [24].
In summary, we have experimentally demonstrated the
control of excitation symmetry for an odd TE-like mode
in a PCW. An odd-mode MZC is utilized to selectively excite the odd mode with a contrast >20 dB over the background. Assisted by a mixed-mode MZC, slow light mode
beating is observed and is utilized to measure the group
index of this odd mode.
This work is supported in part by AFOSR Grant
No. FA9550-10-C-0049. This research is carried out in part
at the Center for Functional Nanomaterials, Brookhaven
National Laboratory, which is supported by the U.S.
Department of Energy, Office of Basic Energy Sciences,
under Contract No. DE-AC02-98CH10886.
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Thermo-optic characteristics and switching
power limit of slow-light photonic crystal
structures on a silicon-on-insulator platform
Manjit Chahal,1 George K. Celler,2,3 Yogesh Jaluria,4 and Wei Jiang1,3,*
1
3
Department of Electrical and Computer Engineering, Rutgers University, Piscataway, NJ 08854, USA
2
Department of Materials Science and Engineering, Rutgers University, Piscataway, NJ 08854, USA
Institute for Advanced Materials, Devices, and Nanotechnology, Rutgers University, Piscataway, NJ 08854, USA
4
Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08854, USA
*
[email protected]
Abstract: Employing a semi-analytic approach, we study the influence of
key structural and optical parameters on the thermo-optic characteristics of
photonic crystal waveguide (PCW) structures on a silicon-on-insulator
(SOI) platform. The power consumption and spatial temperature profile of
such structures are given as explicit functions of various structural, thermal
and optical parameters, offering physical insight not available in finiteelement simulations. Agreement with finite-element simulations and
experiments is demonstrated. Thermal enhancement of the air-bridge
structure is analyzed. The practical limit of thermo-optic switching power in
slow light PCWs is discussed, and the scaling with key parameters is
analyzed. Optical switching with sub-milliwatt power is shown viable.
©2012 Optical Society of America
OCIS codes: (230.5298) Photonic crystals; (130.4815) Optical switching devices; (130.5296)
Photonic crystal waveguides; (130.4110) Modulators.
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#159828 - $15.00 USD
(C) 2012 OSA
Received 15 Dec 2011; revised 26 Jan 2012; accepted 27 Jan 2012; published 6 Feb 2012
13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 4225
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1. Introduction
Silicon photonics benefits from the wealth of experience and the infrastructure of the Si
electronics industry, and the compatibility of Si photonic circuits with CMOS electronics
allows for mass production of low cost integrated photonic/electronic circuits [1,2]. Siliconon-insulator (SOI) substrates [3] are an attractive medium for making silicon photonic
integrated circuits (PICs). For Si PICs, switching and modulation devices are indispensable
components. Thermo-optic effect is one of the preferred options for optical switching in
compact SOI photonic devices. Photonic crystal structures can be incorporated into these
devices to help shrink the interaction length based on the slow light effect [4]. In recent
research, thermo-optic switching and modulation in ultra-compact photonic crystal structures
have been studied [5–10]. However, the performance of these structures varies widely. For
example, the power consumption of most of these structures ranges from 2mW to tens of
milliwatts. As these structures vary widely in their size, design, and the group velocity of
light, it is not always clear what physical factors are crucial to their performance. A theory
that can describe the thermo-optic characteristics of an SOI photonic crystal structure as
explicit functions of various parameters is desired. Note that the understanding of such
thermo-optic characteristics could also help control the thermo-optic effect in photonic crystal
electro-optic modulators and other active devices [11–13].
A photonic crystal thermo-optic device on an SOI chip comprises structural components
whose scales differ by orders of magnitude, such as small holes of ~200nm in diameter and
thick substrates of hundreds of microns. Simulations of such a multi-scale structure can be
time-consuming and challenging. Such simulations may be performed for a small number of
structures. However they are not efficient for systematically studying a large ensemble of
structures in which many parameters such as the hole diameter and the buried-oxide thickness
vary over a large range. Here we develop an efficient and accurate approach to analyze the
thermo-optic characteristics of an SOI photonic crystal structure. The effective thermal
conductivity κeff for a silicon photonic crystal slab is determined through the lateral thermal
spreading length. Physical properties such as the spatial temperature profile and the power
consumption required to induce a π phase shift can be described semi-analytically based on a
quasi-1D model with numerically determined κeff. The results agree well with 3D simulations
based on the finite element method (FEM). The theoretical results also explain the low
switching power observed in an air-bridge structure [6]. The analytic formulas offer insight
into the key factors governing the thermo-optic characteristics of SOI photonic crystal
structures.
2. Analysis of SOI photonic crystal thermo-optic structures
Figure 1 illustrates two common configurations of active photonic crystal waveguide (PCW)
structures on an SOI wafer. A heat source of width W and length L is assumed to be
embedded in the top silicon layer. Such a heat source can be formed by a lightly doped (e.g.
#159828 - $15.00 USD
(C) 2012 OSA
Received 15 Dec 2011; revised 26 Jan 2012; accepted 27 Jan 2012; published 6 Feb 2012
13 February 2012 / Vol. 20, No. 4 / OPTICS EXPRESS 4226
~1014cm−3) Si strip surrounded by a relatively highly doped (e.g. ~1017cm−3) silicon on both
sides [6]. Passing current laterally through this structure produces concentrated ohmic heating
in the center strip.
(a)
(b)
Heat
source
Si
air holes
Si
oxide
z
substrate
y
x
Fig. 1. Configurations of Si active PCW structures. (a) SOI; (b) Air-bridge (membrane).
The heat conduction process in a photonic crystal slab can be effectively modeled by that
of an equivalent hole-free homogeneous slab with an effective thermal conductivity κeff. This
is valid because the temperature varies spatially on a scale much larger than the typical
photonic crystal lattice constant a. To determine κeff, the heat transfer process is simulated
using the finite element method for one period of the PCW structure, as shown in the inset of
Fig. 2. The thicknesses of the top Si layer and buried oxide layer are tSi = 250 nm and tox = 2
µm respectively. The hexagonal lattice has a lattice constant a = 400nm. The simulations
indicate that the vertical temperature variation in the top Si layer and the in-plane temperature
variation in each unit cell are small. The temperature of the top Si layer varies significantly
only along the x axis, as plotted in Fig. 2. Outside the heater (centered at x = 0), it closely
follows an exponential form
T ( x) ≈ exp  − ( x − W / 2 ) / X spr ( r )  , for x > W / 2,
(1)
Fig. 2. Temperature profiles in the top Si layer of a PCW (center: x = 0) for various hole radii
and in a homogenized slab with κeff(r). Inset: 3D temperature profile in a PCW with r/a = 0.25.
One period of the PCW along the y axis is shown.
where Xspr(r) is the thermal spreading length. For an unpatterned SOI structure, it is given by
[14,15]
X spr = X Si = [tSi toxκ Si / κ ox ]1/ 2 ,
(2)
where κSi and κox are the thermal conductivities of silicon and SiO2 respectively (values from
Ref [14].).
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For a photonic crystal slab, Xspr(r) depends on the hole radius r and it can be obtained
from an exponential fit of the lateral temperature profile in the slab. The effective thermal
conductivity of a Si photonic crystal slab can then be calculated from
2
(r ) / (tSi tox ).
κ eff (r ) = κ ox X spr
(3)
The values of κeff/κSi and Xspr determined from the plots are given in Table 1. To further
verify the results, homogenized slab structures with the tabulated κeff(r) are simulated, with all
other parameters unchanged. The lateral temperature profile in the homogenized slab is
generally in good agreement (within 6%) with that of the original photonic crystal slab, as
shown in Fig. 2.
Table 1. Values of Xspr(r) and κeff(r) for Various Hole Sizes
r/a
Xspr (µm)
κeff / κSi
0.25
6.3
0.68
0.275
6.0
0.61
0.3
5.7
0.55
0.325
5.3
0.48
0.35
4.9
0.41
For an SOI structure, the heat conduction can generally be described by a quasi-1D model
predicated on the vertical heat conduction in the buried oxide [14,15]. Note that the thermal
spreading increases the effective heat flux cross-section to Aeff = L[W + 2Xspr]. For the
photonic crystal structure in Fig. 1(a), this model yields
Q = κ ox L[W + 2 X spr (r )](∆Tox / tox ),
(4)
where Q is the heat transfer rate (equal to the heating power in steady state) and ∆Tox the
temperature difference between the top and bottom of the oxide at x = 0. To verify Eq. (4), 3D
FEM steady-state simulations are performed for an SOI chip having a homogenized top layer
with κeff (Fig. 3 inset). The absence of small holes significantly mitigates the difficulty in
mesh generation for multi-scale structures, and reduces the simulation time significantly.
Fig. 3. ∆Tox/Q vs. waveguide length L (for tox=2µm). Inset: 3D temperature distribution in a
chip obtained from finite element simulation for a PCW structure having tSi=250nm,
W=400nm, tox=2µm, κeff(r=0.25a) on a 200µm×200µm substrate with a thickness of 100µm.
Due to the small thermal conductivity and natural convection coefficient of air [16], the
heat dissipation from the top and side surfaces of the chip is negligible, hence adiabatic
boundary conditions are used for the top and side surfaces [7,17]. The bottom surface is kept
at 300 K. The simulated ∆Tox per unit heating power Q and the results based on Eq. (4) agree
well (within 3%), as shown in Fig. 3 for various lengths of the heat source.
3. Thermo-optic characteristics and switching power for SOI and air-bridge structures
To study the thermo-optic characteristics, we note that the phase shift induced in a PCW is
given by [4]
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∆φ = 2π∆nσ Lng / (nλ ),
(5)
where ng is the group index of the mode, λ the wavelength, and σ the fraction of the mode
energy stored in the region where the refractive index change ∆n = (dn/dT)∆T occurs. By
virtue of Eqs. (4) and (5), the power required to induce a phase shift of π for a structure in Fig.
1(a) is given by
Qπ = nλκ ox [W + 2 X spr (r )] / [2σ tox ng (dn / dT )].
(6)
Because X spr ~ tox , the power Qπ actually scales as 1 / tox for heater width W<<Xspr.
Figure 4 shows the results for σ = 0.9, λ = 1.55µm and dn/dT = 1.86 × 10−4 K−1 with different
values of oxide layer thickness. For ng = 60, r/a = 0.25 and tox = 2µm, Qπ is less than 2.5mW.
10
200
150
6
100
L (µm)
Q , SOI (mW)
π
8
4
50
1 µm
2
2 µm
5 µm
0
0
50
ng
0
150
100
Fig. 4. Qπ vs. group index (ng) for r/a = 0.25 for an SOI PCW structure (for various tox), and the
estimated 3dB-propagation-loss length for a membrane PCW structure. Inset: The thermal
enhancement for a membrane PCW with tox = 2µm.
This approach can also be applied to an air-bridge (membrane) structure shown in Fig.
1(b). Here the heat conduction consists of two steps in series: (1) the lateral heat conduction in
the suspended membrane; and (2) the quasi-1D heat conduction in the SOI region. Based on
the continuity of heat flux, one readily finds for the left (or right) half membrane
Qmembrane / 2 = κ eff LtSi
(∆T )membrane − (∆T )edge
Wmembrane / 2
= κ ox LX Si
(∆T )edge
tox
,
(7)
where Wmembrane is the membrane width, XSi is given by Eq. (2), (∆T)membrane is the membrane
temperature rise evaluated at the PCW core and (∆T)edge at the membrane edge. Eliminating
(∆T)edge, we find
Qmembrane = κ ox L(2 X Si )
κ eff (r ) X Si
(∆T )membrane
,
tox
κ eff (r ) X Si + κ SiWmembrane / 2
(8)
For the same power Q, the membrane structure may enhance the temperature rise by a
factor
(∆T )membrane X spr (r ) κ eff (r ) X Si + κ SiWmembrane / 2
.
≈
(∆T ) SOI
X Si
κ eff (r ) X Si
(9)
Correspondingly, Qπ of the membrane structure is reduced by this factor. The
enhancement factors obtained from Eq. (9) agree very well (within 6%) with the simulation
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results, as shown in Fig. 4 inset. Based on Fig. 4, the attainable power consumption for a Si
air-bridge PCW thermo-optic Mach-Zehnder switch is estimated between 1~2mW for ng~60
and tox = 2µm, which agrees well with the experimental result [6].
4. Discussions
The scaling of the thermo-optic characteristics of an SOI photonic crystal structure with
various parameters is of significant interest in device design. The analytic formulas enable us
to study such scaling over a wide parameter range. The heater location is an important factor
in determining the power consumption. Here we consider two options: in the PCW core [6], at
the lateral edge of the PCW [8]. The temperature profile given in Eq. (1) shows that the
temperature rise in the silicon layer decreases exponentially with the lateral distance from the
heater. Compared to a heater embedded exactly in the PCW core, a heat source located at ∆x
= 6µm from the core has an efficiency reduction by exp(−6µm/Xspr)≈0.3~0.4 for r/a =
0.25~0.35. The buried oxide thickness is another crucial factor. Generally, a thicker oxide is
preferred for lower power consumption according to Eqs. (6) and (8). However, the thermal
time constant of an SOI chip increases with the oxide thickness. Therefore, some trade-off
must be made in realistic device design to balance power consumption and speed. For the
membrane structure, the enhancement factor in Eq. (9) is found to weaken the scaling of Qπ
with tox due to XSi ~ tox . Thus, Qπ scales slower than tox −1/ 2 , particularly for a large Wmembrane.
Ultimately, the reduction of Qπ based on the slow light effect is limited by optical loss, which
increases with ng. The optical loss of a PCW can be attributed to a number of factors, such as
random variation of hole positions due to fabrication tolerances, sidewall roughness, and the
input/output coupling. The random variation of the hole positions in fabricated PCWs can be
controlled to be within a small range (<1nm) with high-end e-beam lithography tools [18];
and the corresponding loss is usually small. Sidewall roughness of the holes depends on the
lithography tool, resist, and etching process and is more difficult to control. Such roughness
could induce substantial loss at large ng. The estimated PCW length for 3dB propagation loss
is plotted against ng in Fig. 4 based on theoretical calculations with experimentally achievable
rms roughness σ = 3nm and correlation length lc = 40nm [19]. To further address the effect of
the input/output coupling loss, we consider two prior experiments [6,20]. In an earlier
experiment [6], the insertion loss of well-fabricated PCWs is about 10~13dB at ng~110 for L
= 50µm and 250µm and shows weak dependence on the PCW lengths. This indicates that
most of the observed loss is due to input/output coupling [6]. A more recent experiment based
on group index tapering has shown that the coupling loss can be significantly reduced
throughout the spectrum of the defect-mode, including the slow light region near the band
edge [20]. To summarize, with the best fabrication tools and best design, optical loss due to
random hole position variation and input/output coupling can be very small, but the roughness
induced loss [19] (especially the backscattering loss, which scales roughly as ng2) will be a
primary limiting factor. Hence the roughness-induced loss (including backscattering and outof-plane scattering loss) is considered in Fig. 4 to explore the limit of Qπ in connection with
ng. Considering all the factors discussed above, a practical lower limit of Qπ is estimated on
the order of 0.5mW for a reasonable tox~5µm, L~10µm, and ng~110. Our calculation also
shows that for ng~60, Qπ already enters the sub-milliwatt regime for the tox~5µm case.
It should be noted that this theory indicates that many factors are insignificant. For
example, Qπ is insensitive to the choice of the heater width W as long as W<<2Xspr(~12µm).
Also, Qπ varies only ~20% for the typical radius range of r/a = 0.25~0.35. Note that typical
silicon photonic crystal waveguides used for the 1550nm communications window have a =
380nm to 440nm. As the lattice constant is much smaller than the scale of temperature
variation (a<<Xspr), this approach works well for this range of a. For a given lattice structure,
when a and r vary simultaneously while maintaining a fixed ratio of r/a, Xspr is essentially
invariant. Note that the power Qπ given above is for switching and steadily holding a state.
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This is pertinent for most optical switching applications that require holding a switching state
steadily over an extended period. The thermal time constant of an SOI structure is
τ ~ tox2 ρox cox / κ ox (~µs for tox = 1~2µm), where ρox is the density and cox the specific heat
capacity of SiO2. Our simulations confirm that τ is relatively insensitive to the details of a
photonic crystal structure. Although the heating transient can be shortened [7,14], the overall
performance of a switch over an extended period is limited primarily by Qπ and τ given above.
The effect of the temperature drop in the substrate is less than 10% for all cases we simulated.
Note that κeff used in this work is obtained based on the structured “porosity” of materials
within the framework of classical heat transfer theory, neglecting quantum mechanical effects
such as phonon scattering in a periodic structure [21]. When quantum effects are considered,
most formulas in this work remain useful, except κeff values from quantum mechanical
calculations will be used.
5. Summary
In conclusion, the thermo-optic characteristics of active photonic crystal structures on an SOI
platform are investigated semi-analytically. The power consumption Qπ and spatial
temperature profile are given as explicit functions of structural, thermal, and optical
parameters. The results agree well with FEM simulations and also explain the low switching
power in air-bridge structures. The scaling of Qπ with key physical parameters is analyzed.
The practical limit of Qπ is estimated on the sub-milliwatt level considering all key factors.
Acknowledgments
We are grateful to Dr. S. R. McAfee for helpful discussions. This work is supported in part by
AFOSR MURI Grant No. FA9550-08-1-0394 (G. Pomrenke) and a Rutgers ECE Graduate
Fellowship (for M.C.).
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Parallel-coupled dual racetrack silicon microresonators for quadrature amplitude modulation
Ryan A. Integlia,1 Lianghong Yin,1 Duo Ding,3 David Z. Pan,3 Douglas M. Gill,4
and Wei Jiang1,2,*
2
1
Department of Electrical and Computer Engineering, Rutgers University, Piscataway, NJ 08854, USA
Institute for Advanced Materials, Devices, and Nanotechnology, Rutgers University, Piscataway, NJ 08854, USA
3
Department of Electrical and Computer Engineering, University of Texas at Austin,
Austin, TX 78712, USA
4
Alcatel-Lucent Bell Labs, 600 Mountain Avenue, Murray Hill, NJ 07974, USA
* [email protected]
Abstract: A parallel-coupled dual racetrack silicon micro-resonator
structure is proposed and analyzed for M-ary quadrature amplitude
modulation. The over-coupled, critically coupled, and under-coupled
scenarios are systematically studied. Simulations indicate that only the
over-coupled structures can generate arbitrary M-ary quadrature signals.
Analytic study shows that the large dynamic range of amplitude and phase
of a modulated over-coupled structure stems from the strong cross-coupling
between two resonators, which can be understood through a delicate
balance between the direct sum and the “interaction” terms. Potential
asymmetries in the coupling constants and quality factors of the resonators
are systematically studied. Compensations for these asymmetries by phase
adjustment are shown feasible.
© 2011 Optical Society of America
OCIS Codes: (230.5750) Resonators; (230.4110) Modulators; (130.3120) Integrated optics
devices.
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1. Introduction
Advanced optical modulation formats could offer significant advantages for optical
communications [1,2]. For example, quadrature phase-shift keying provides higher spectral
efficiency, better tolerance to fiber nonlinearity and chromatic dispersion, and enhanced
receiver sensitivity compared to on-off keying. Traditional lithium niobate (LiNbO3)
modulators can be used for such modulation. However, LiNbO3 modulators are relatively
large in size. For a general M-ary modulation format that requires a large number of optical
modulator components along with their driving signal circuitries, the overall size of the entire
modulator is rather cumbersome. Recent breakthroughs in silicon photonics [3,4], particularly
silicon based optical modulators [5,6], have fundamentally changed the landscape of
modulator technology. Notably, micro-resonator based silicon modulators [6–10] constitute
an ideal candidate for optical modulation due to their compact size, low power consumption,
and ease of monolithic integration with driving circuitries on the same silicon chip. Most
research on silicon microring modulators employed intensity modulation in binary formats.
Recently, microring resonator based modulators for differential binary phase-shift-keying and
differential quadrature phase-shift keying (QPSK) have been proposed, and satisfactory
performances have been predicted [11,12]. Another work employed the anti-crossing between
paired amplitude and phase resonators and demonstrated enhanced sensitivity to the input
drive signal [13]. A high-Q microring quadrature modulator incorporating dual 2 × 2 MachZehnder interferometers has also been recently proposed with beneficial performance [14].
We propose a novel parallel-coupled dual racetrack micro-resonator structure, illustrated
in Fig. 1(a), for phase-shift keying and M-ary quadrature amplitude modulation (QAM). Two
identical racetrack resonators are symmetrically side-coupled in parallel to a through
waveguide in the center. The modulator can be fabricated on a silicon-on-insulator (SOI)
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wafer. The carriers can be injected or depleted from the racetrack resonators using a pin diode
[15] or metal-oxide-semiconductor capacitor [16] embedded in a silicon waveguide. The
plasma dispersion effect [17] of the injected carriers causes a change of refractive index, Δn1,
Δn3, in each racetrack resonator, which modifies the cross-coupled resonances of the two
racetrack resonators. By carefully choosing the voltage signals applied to each resonator, the
amplitude and phase of output optical signal can be controlled to generate arbitrary M-ary
quadrature signals.
x
a1
b1
a2
b2
a3
b3
L
Eout
3-waveguide
coupler
1
Intensity
Ein
z
(b)
0.9
0.8
-2
Phase
(a)
-4
(c)
-6
-8
1.54
1.545
1.55 1.555 1.56
wavelength( m)
1.565
Fig. 1. Parallel-coupled dual racetrack resonators. (a) schematic of the structure, and typical
spectra for an over-coupled structure: (b) output intensity and (c) phase in radians.
A distinctive feature of the proposed structure is that the coherent cross-coupling between
the two racetrack resonators mediated by the center waveguide drastically modifies the
amplitude/phase characteristics of resonance. This enables M-ary quadrature signal
generations including quadrature phase shift keying (QPSK). The outcome of the crosscoupling of the resonances is fairly complex. However, our analysis shows that it can be
understood through the direct sum and coherent “interaction” of the optical characteristics of
two individual resonators as presented in Sec. 2.4. The structure of this paper is organized as
follows. First the cross-coupling between the racetrack resonators is analyzed and the output
transfer function of the proposed structure is presented. The critical coupling condition is
obtained. Systematic studies of the over-coupled, critically coupled, and under-coupled
scenarios for the parallel-coupled racetrack resonator structure indicate that strong overcoupling case is desired for arbitrary M-ary quadrature signal generation. The interaction
between the resonances of two racetracks is analyzed, and its critical role in M-ary quadrature
signal generation is presented. The effects of asymmetries in the coupling strengths and
quality factors of resonators are systematically studied, and phase compensations for such
asymmetries are presented. Lastly, the electrical aspects of the proposed modulators are
briefly discussed, followed by a conclusion.
2. Principles of parallel-coupled racetrack resonators
2.1 Cross-coupling analysis and output transfer function
The coupling between the two racetrack resonators and the through-waveguide in Fig. 1(a)
can be described by multi-waveguide coupling theory [18–20]. Assume the fields in three
identical single-mode waveguides have slowly varying envelopes un(z)
En  x, y, z   M n ( x, y ) exp(i  z )un ( z ), n  1, 2, 3,
(1)
where Mn(x,y) is the lateral mode profile, β is the propagation constant along the waveguide
axis z for an isolated waveguide. For the parallel coupled racetrack resonator structure in Fig.
1, the input fields and output fields of the coupling segments are given by
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E(nin )  x, y , 0   M n ( x, y )un (0)  M n ( x, y )an ,
E(nout )  x, y, L   M n ( x, y ) exp(i  L)un ( L)  M n ( x, y )bn ,
(2)
where an and bn are the normalized input and output complex amplitudes, respectively. The
solution of the coupled mode equations yields [20]
 b1 
c1  1 / 2 c2
b   exp(i  L)  c
2c1
2
 2

 b3 
 c1  1 / 2 c2
c1  1 / 2   a1 
c2   a2  ,
c1  1 / 2   a3 
(3)
where
c1  12 cos( 2 L), c2 
1
2
i sin( 2 L).
The strength of the cross-coupling between the two racetrack resonators mediated by the
through waveguide is given by |c11/2|. In addition, light propagation along a racetrack gives
rise to the following relations
a1  1 exp(i1 )b1 ,
a3  3 exp(i 3 )b3 ,
(4)
where the amplitude attenuation along a racetrack is given by ηn<1, and the phase shift is
given by θn. Assuming a unity input amplitude a2 = 1, the output amplitude b2 can be solved
from Eqs. (3) and (4)
Eout  b2 
ei   1/ 2  c1  (u1  u3 )  2c1u1u3 
,
1/ 2  c1  (u1  u3 )  u1u3
(5)
where  = βL, and
un 
1
 1, n  1, 3,
(6)
e
n
Because of the symmetry of the structure shown in Fig. 1(a), the output amplitude, Eq. (5),
only involves terms symmetric with respect to an interchange of Δu1 and Δu3. As such, the
symmetry of the structure can be utilized to help simplify the understanding of the device
principles, as noted in the study of other devices [21]. Detailed analysis of a modulated
symmetric dual racetrack resonator structure will be given in the following sections.
i  i n
2.2 Critical coupling condition and vanishing amplitude for a modulated over-coupled
structure
The critical coupling condition [22] can be obtained by setting b2 = 0 in Eq. (5). For
symmetric parallel-coupled racetracks without modulation ( u1  u3 ), one readily shows
that the critical coupling condition for such a parallel-coupled dual racetrack structure is given
by
1  2c1  cos 2 L.
(7)
The asymmetric cases will be discussed in a later section.
For modulated racetracks, the phase shift θn in each ring will be a linear function of the
refractive index changes, Δnn, due to carrier injection or depletion in the respective racetrack
resonator. Therefore the output amplitude b2 depends on Δnn through the phase shift terms. To
understand the modulation characteristics, it is helpful to rewrite the output amplitude in the
following form
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

(2c1  1)
b2  ei  1 
(8)
.
1/
2
c
(1/
u
1
/
u
)
1






1
1
3


As c1 is a real number, for a modulated symmetric (η1 = η3) dual-racetrack structure, the
output amplitude can vanish only if u1  u3 * . Indeed, one can show that even if the critical
coupling condition is not satisfied in absence of modulation, the modulated amplitude can still
vanish under the following modulation condition
  1  2m1   , and   3  2m3   ,
(9a)
 c (1/ 12  1) 
cos   1 1  1
,
c1  1/ 2 

(9b)
where m1 and m3 are two integers. For real nonzero Δθ, this requires
1  2c1  cos 2 L,
(10)
which corresponds to over-coupling in comparison to Eq. (8). The spectra of an over-coupled
dual racetrack structure (without modulation) are illustrated in Fig. 1(b) and (c).
2.3 Arbitrary M-ary quadrature signal generation capability
For intensity and phase modulation, the refractive index of the silicon waveguides in each
racetrack is varied on the order of 0.001. Such an amount of Δn can be achieved with carrier
concentration changes ΔNe, ΔNh~3 × 1017cm3 according to the well-known plasma dispersion
relation reported in [17].
2
1
0.8
-3
n3 (10 )
1
0.6
0
0.4
-1
-2
-2
(b)
0.2
-1
0
-3
n1 (10 )
1
2
0
2
3
2
1
-3
n3 (10 )
(a)
1
0
0
-1
-1
-2
-2
-2
-3
-1
0
1
2
-3
 n1 (10 )
Fig. 2. Intensity (a) and phase (b) variations under refractive index modulation for the parallelcoupled dual racetrack resonators. r1 = r3 = 3μm, L = 3μm, η1 = η3 = 0.994, c1 = 0.4243. The
intensity vanishes at two points (Δn1,Δn3) = ( ± 3.5 × 104,  3.5 × 104). The color code for
the phase is in radians in (b).
Figure 2 depicts the simulated intensity and phase variations as a function of refractive
index variations n1 and n3 at the resonant wavelength for an over-coupled structure. The
structure parameters are r1 = r3 = 3μm, L = 3μm, η1 = η3 = 0.994, c1 = 0.4243. Note that
compact silicon racetrack resonators have been systematically characterized recently [23]. It
was shown that the coupling strength and quality factor can be varied over large ranges by
changing the Si pedestal layer thickness of the rib waveguide and modifying the gap width
between the waveguide and the resonator. The parameters used here are in accordance with
the ranges given Ref [23]. Evidently, the intensity vanishes at two points (Δn1,Δn3) = ( ± 3.5 ×
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104,  3.5 × 104), in accordance with the analytic results given in Eq. (9)b). In all phase
plots starting from Fig. 2, the overall constant phase factor ei in b2 is omitted to better
illustrate the symmetry of the modulated output. On a side note, if η1 and η3 decrease
simultaneously (η1 = η3), the two “eyes” on the diagonal of Fig. 2(a) widen and the phase
contours in Fig. 2(b) expand accordingly.
To visualize the complex amplitude, Eout(Δn1,Δn3), for M-ary signal generations, the
ensemble of complex b2 values for all values of n1 and n3 are mapped onto the complex
plane of the normalized output electric field. Each blue point in Fig. 3(a) gives the amplitude
and phase of the output signal for a particular pair of Δn1, Δn3 values in the aforementioned
range. Evidently, the ensemble of blue points covers most part of the unit circle (the symbol
space), therefore, allowing for the access of a wide range of amplitude and phase values. A
close examination of Fig. 2 indicates that the intensity and phase varies widely in the second
and fourth quadrants where Δn1 and Δn3 have opposite signs, which corresponds to a pushpull configuration. In contrast, the intensity and phase are much less sensitive to Δn1 and Δn3
when they have the same sign. Indeed, our simulations indicate that the push-pull
configuration is usually responsible for over 80% of coverage on the complex E plane. Hence
a push-pull modulation configuration is preferred for such a parallel-coupled dual-racetrack
structure.
Fig. 3. Mapping of the normalized complex output field amplitude Eout on the complex plane
for refractive index Δn1, Δn3 varying in the range of 0.002 ~0.002. (a)-(c) for parallel-coupled
dual racetrack resonators; (d)-(f) for two uncoupled racetrack resonators in series. Evidently,
only case (a) is suitable for arbitrary M-ary quadrature signal generation. Constellations for
QPSK (brown circles) and 16-QAM (red squares) modulation formats are illustrated in (a).
2.4 The cross-coupling of two racetrack resonances: direct sum and “interaction”
It should be noted that the broad coverage inside the unit circle observed in Fig. 3(a) is a
signature of the strong cross-coupling between the two racetrack resonators mediated by the
center waveguide. To illustrate this point, the simulated typical coverage of a critically
coupled case and an under-coupled case is shown in Fig. 3(b) and (c), respectively, for
parallel-coupled dual racetrack resonators. In addition, the simulated typical coverage for two
uncoupled racetrack resonator in series is plotted in Fig. 3(d)-(f). None of the cases illustrated
in Fig. 3(b)-(f) has adequate coverage for arbitrary M-ary quadrature signal generation.
The cross-coupling present in the parallel coupled racetrack resonators helps only the
over-coupling case to achieve sufficient coverage over all four quadrants inside the unit circle.
It can be shown that such a behavior stems from a delicate balance between the direct sum
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term Δu1 + Δu3 and the “interaction” term Δu1Δu3 on both the numerator and denominator in
Eq. (5). Based on their definitions un  (1  ei  in n ) / ei  in  n , Δun can be regarded as the
normalized change of the field amplitude after one round-trip propagation in a racetrack. Here
the initial field amplitude is unity, and the amplitude change is normalized by the final field
amplitude ei  i n  n . For a racetrack without modulation (Δn1 = Δn3 = 0), ui is small (on the
order of 1η1) near resonance, and Δu1 and Δu3 are in phase. Therefore, we find
| u1u3 || 1/ 2  c1  (u1  u3 ) |
because 1η1<2(1/2c1) according to the strong coupling condition. The dominance of the
direct sum term in Eq. (5) yields an output amplitude close to 1. With sufficient modulation
in a push-pull configuration, Δun can gain large imaginary parts (Im(Δun)~Δθn, up to ± 0.09 at
Δnn = 0.001) with opposite signs whereas their real parts remain small. Therefore, the product
term exceeds the sum by a large margin, |Δu1Δu3|>>|Δu1 + Δu3| such that
1 / 2  c1  (u1  u3 ) and u1u3 in Eq. (5) become comparable. Now the output amplitude
can take virtually any value. Particularly, the two terms in the numerator can exactly cancel
each other so that the output amplitude vanishes. Hence the large dynamic range of
| 1 / 2  c1  (u1  u3 ) / u1u3 | in the over-coupling case causes the output amplitude given
by Eq. (5) to vary widely, traversing a large fraction of the area in the unit circle. Thus the
output amplitude and phase have a large dynamic range. In contrast, for an under-coupling
case, it is straightforward to show that, in general,
| 1/ 2  c1  (u1  u3 ) || u1 u3 | .
The dominance of the “interaction” term limits the accessible area in the unit circle.
3. Asymmetry effect in parallel-coupled dual racetrack resonators
As two racetrack resonators are involved in this structure, their asymmetry due to fabrication
imperfections can be a major concern for practical applications. Note that the relatively long
straight segments of racetracks ensure that the cross-coupling between the two resonators is
insensitive to small misalignment between the left and right racetracks. As two racetracks can
be patterned in one e-beam lithography process with a typical positioning accuracy of 20nm
or better, the misalignment is estimated less than 1% for a coupling length L>2μm. Optical
path differences between the two racetracks can usually be compensated by a proper DC bias
or by additional thermo-optic heaters [24,25]. However, the asymmetries in quality factors
and coupling ratios cannot be directly compensated as easily. Therefore, their impacts on the
device performance must be evaluated.
3.1 Asymmetric coupling
For three parallel waveguides with asymmetric coupling constants, the coupled mode
equations can be written as
 u1 ( z )   0
d 
u2 ( z )   i 12
dz 
u3 ( z )   0
12
0
 23
0   u1 ( z ) 
 23  u2 ( z )  ,
(11)
0  u3 ( z ) 
where the coupling constants between waveguide pairs (1,2) and (2,3) are κ12 and κ23,
respectively. To solve such a set of differential equation, dzd [um ]  i[ mn ][un ] , the coupling
matrix is decomposed into the following form [ mn ]  X X  , where Λ is a diagonal matrix
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whose diagonal elements are the eigenvalues of the matrix [ mn ] , the columns of X are the
eigenvectors of [ mn ] , and XX+ = I. The original equation can then be integrated according to
[um ( z )]  exp(i[ mn ]z )[un (0)]  X exp(iz ) X  [un (0)].
Thus the solution of Eq. (11) is given by
2
2
 u1 ( z )   cos(  z ) 1  3
u ( z )   
i sin(  z ) 1
 2  
u3 ( z )  cos(  z ) 1 3  1 3
where
   122   232 ,
1  12 /   ,
i sin(  z ) 1
cos(  z )
i sin(  z ) 3
and
(12)
cos(  z ) 1 3  1 3   u1 (0) 


i sin(  z ) 3
 u2 (0)  , (13)
cos(  z ) 32  12  u3 (0) 
3   23 /   .
In
a
symmetric
case,
   212  2 23  2 , 1  3  1 / 2 , Eq. (13) returns to Eq. (3). The output
amplitude b2 can be solved in a procedure similar to that given for the symmetric case. After
lengthy calculations, the final result is surprisingly simple


1  cos(  L)
b2  ei  1 
,
2
2
[1  cos(  L)]( 1 /  u1  3 /  u3 )  1 

(14)
where Δun are defined the same way as in the symmetric case. Comparing Eq. (14) and Eq.
(8), it is evident that all asymmetry effects can be effectively factored into the term
12 /  u1  32 /  u3 
 2 ei  i 3
122 ei  i 1
 232
.
2
i  i
  1  e 1   1  ei  i 3
3
1
1
3
(15)
Fig. 4. Effect of asymmetric coupling constants. (a) Required phase compensation in each
racetrack for up to 50% asymmetry in the coupling ratios. The characteristics of the
asymmetric dual racetrack structure for the worst case scenario (κ23/κ12 = 1.5) are illustrated in
(b)-(d). (b) Output spectrum without modulation; (c) Intensity variation with index modulation;
(d) Mapping of the output field on the complex plane. All parameters are the same as those
used in Fig. 2 except κ23/κ12 is varied.
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As a consequence, for reasonable asymmetries in the coupling constants and resonator
quality factors, there exists a pair of phases Δθ1 and Δθ3 such that the output amplitude b2
vanishes. The required phase variations are plotted against the asymmetric coupling ratio,
κ23/κ12, in Fig. 4(a) for up to 50% asymmetry. As Δθ1 and Δθ3 generally have opposite signs,
we plot Δθ1 and Δθ3 to better illustrate the deviation from symmetry. Note that Δθ1 = Δθ3 is
required for b2 = 0 in a symmetric structure (κ23/κ12 = 1), according to Eq. (9). The difference
between Δθ1 and Δθ3 becomes larger as the asymmetry increases.
Figure 4(a) shows that although it is not easy to directly compensate the asymmetric
coupling constants themselves, asymmetric phase shifts (through different DC biases applied
to the two resonators) can be introduced to recover the low intensity states (b2~0). The unmodulated output spectrum for the worst case (κ23/κ12 = 1.5) is illustrated in Fig. 4(b) and
shows no anomaly. However, the intensity variation upon refractive index modulation shows
obvious distortion from the symmetric case. Nonetheless, two features remain: (1) there are
two points with relatively small index changes ( ± 2.2 × 104,  5.4 × 104) where the
intensity vanishes; (2) the intensity varies significantly in the push-pull configuration and
much less otherwise. The coverage on the complex E plane is slightly enhanced, although a
small hole exists at a large amplitude value, which may limit the maximum accessible
amplitude to 0.78 for a generic M-ary modulation format.
3.2 Asymmetric quality factors
The effects of asymmetric quality factors are illustrated in Fig. 5. The required phase shifts,
Δθ1 and Δθ3, for vanishing b2, are plotted against the ratio of the quality factors in Fig. 5(a).
The unloaded quality factor Q1 is fixed at its original value ~2.5 × 104.
Fig. 5. Effect of asymmetric quality factors. (a) Required phase compensation in each racetrack
for asymmetry in the quality factors. The characteristics of the asymmetric dual racetrack
structure for the worst case scenario (Q3 = 0.5Q1) are illustrated in (b)-(d). (b) Output
spectrum; (c) Intensity variation with index modulations; (d) Mapping of the output field on
the complex plane. All parameters are the same as those used in Fig. 2 except η3 is varied to
yield different Q3.
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Note that Δθ1 = Δθ3 for the case of Q3/Q1 = 1 in accordance to the symmetric case. The
un-modulated output spectrum for the worst case (Q3/Q1 = 0.5) is illustrated in Fig. 5(b). A
small yet noticeable spike appears at the resonance due to the asymmetric quality factors of
the two racetrack resonators. The modulated intensity variation upon refractive index
modulation depicted in Fig. 5(c) shows less severe distortion compared to the distortion
observed in the Fig. 4(c). Again, two features remain: (1) there are two points with relatively
small index changes ( ± 4.4 × 104,  4.1 × 104) where the intensity vanishes; (2) the
intensity varies significantly in the push-pull configuration and much less otherwise. The
coverage on the complex E plane slightly deteriorates. There exists a small hole, which may
limit the maximum accessible amplitude to 0.74 for a generic M-ary modulation format.
Note that the evolution from symmetry to the worst case asymmetry is gradual. For
example, the two “eyes” in Fig. 4(c) gradually narrow as the asymmetry in the coupling
constant worsens. Also, the centers of the “eyes” rotate clockwise around the origin (Δn1 =
Δn3 = 0). As the asymmetry in the quality factors worsens, the “eye” centers do not narrow or
rotate substantially although there are some deformations.
Overall, the asymmetry analysis presented above show that substantial asymmetries in
coupling constants and quality factors of the two racetrack resonators can be compensated by
refractive index changes on the order of 4 × 104, which can be readily provided with a lowpower heater or a small change of the DC bias. Fundamentally, such compensations are
possible because all these asymmetries enter the output amplitude, Eq. (14), through the term
given in Eq. (15). For structures with asymmetric η’s or Q’s, asymmetric phase shifts can
restore the value of the term given in Eq. (15) to a corresponding symmetric structure.
Specifically, to achieve vanishing output intensity under modulation, a structure with 50%
asymmetry in the coupling constant requires (Δn1,Δn3) = ( ± 2.2 × 104,  5.4 × 104) whereas
a symmetric structure requires (Δn1,Δn3) = ( ± 3.5 × 104,  3.5 × 104). The difference
between |Δn1| and |Δn3| in the asymmetric case is used to restore Eq. (15) to the value of the
symmetric case such that b2 = 0.
4. Discussion
In general, an encoder is needed to convert an original M-ary digital signal into the driving
signal for the modulator. Consider the case of a QPSK signal with four symbols shown in Fig.
3(a). The encoder will have a two-bit input and two output ports. Each output port has four
output voltage levels. The design of such an encoder and its supporting circuitries has been
well studied in the state-of-the-art high-speed data conversion systems [26] and CMOS VLSI
[27]. Under the given specifications (resolution, signal-to-noise ratio, bandwidth, driving
power, etc.), this encoder can be easily architected and implemented as a high-speed digitalto-analog data converter, which can be fabricated economically using the silicon-on-insulator
technology together with the dual racetrack resonator modulator. Note that a conventional
nested Mach-Zehnder QPSK [1] modulator needs two output voltage levels for each port. The
additional voltage levels required for the proposed QPSK modulator will somewhat increase
the size of the driving circuitry. However, electronic devices such as transistors are generally
significantly smaller than photonic devices. Therefore, the enlargement of the driving
circuitry is usually negligible compared to the significant space saving offered by changing
from a bulky nested Mach-Zehnder modulator to the proposed dual racetrack resonators.
Driving voltages and power consumption are important issues for silicon modulators used
in optical interconnects [28,29]. For a nested Mach-Zehnder QPSK modulator which is biased
across the minimum point of the transfer curve, a lower driving voltage and lower RF power
consumption can be achieved at the expense of a lower maximum output intensity (which
entails a trade-off with the detector sensitivity or the input laser power). For the proposed
parallel-coupled dual racetrack modulator, a similar power reduction scheme is possible. For
simplicity, we consider silicon racetrack resonators with embedded MOS capacitors, whose
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index change is approximately linearly dependent on the voltage. As illustrated in Fig. 6(a),
the driving power can be significantly lower at lower output intensity. Asymmetries of the
coupling constants and quality factors could entail extra power penalty but the power remains
reasonable. According to Fig. 6, if the asymmetry is large, electrical power penalty is
significantly lower when the modulator operates at a lower output intensity level. Therefore,
for a modulator that happens to have a large asymmetry due to imperfection in fabrication, the
balance of the power trade-off may tip towards enhancing the detector sensitivity.
0.8
0.8
(b)
0.6
0.4
0.2
1
1.1
1.3
1.5
Max. Intensity
Max. Intensity
(a)
0.6
0.4
0.2
0.50
0.75
1.00
2.00
0
-2
10
0
-1
0
1
-2
-1
0
1
10
10
10
10
10
10
10
Normalized electrical power
Normalized electrical power
Fig. 6. Output intensity as a function of the driving power for parallel-coupled dual racetrack
modulators with varying degrees of asymmetry. (a) For various κ23/κ12 values and (b) For
various Q3/Q1 values. The output intensity is normalized by the input intensity. The driving
power is normalized by the power level corresponding to the case that each racetrack is driven
to Δn = 0.001.
5. Conclusion
In summary, we have proposed and analyzed a parallel-coupled dual racetrack microresonator modulator for arbitrary M-ary quadrature signal generation. The critical coupling
condition is obtained for such a structure. The intensity and phase modulations are obtained
by varying the refractive indices of the silicon waveguides in the two parallel-coupled
resonators. It is shown that a push-pull configuration effectively modulates the intensity and
phase. The coverage of the complex plane of the output field Eout is systematically studied for
over-coupling, critical-coupling, and under-coupling scenarios, and is compared to the
corresponding scenarios of two uncoupled racetrack-resonators in series. It is found that only
the over-coupling scenario of a parallel-coupled dual racetrack resonator structure results in
adequate coverage for arbitrary M-ary quadrature signal generation. The interaction between
the parallel-coupled racetrack resonators is key to the coverage of the complex E plane. In an
over-coupled dual racetrack structure, a delicate balance is achieved between the direct sum
and the interaction of the two racetrack resonances, which results in a large dynamic range of
the output amplitude and phase. Particularly, the modulated intensity can reach zero in a pushpull configuration although the intensity of the un-modulated over-coupled racetrack
resonators do not vanish at any wavelength. The effects of asymmetries in the coupling
constants and quality factors are systematically studied. Despite the distortion of the intensity
and phase mapping, small refractive index changes, which can be readily obtained with a
reasonable thermal or electrical bias, can be used to compensate the asymmetry. The coverage
of the complex E plane remains sufficient despite asymmetries.
Acknowledgments
The authors are grateful to Zhong Shi, Qianfan Xu, Lin Zhang, Rene-Jean Essiambre, and
Ying Qian for helpful discussions. This work is supported in part by AFOSR Grants No.
FA9550-10-C-0049 and No. FA9550-08-1-0394 (G. Pomrenke). R. A. I. acknowledges the
partial support of a NSF IGERT Traineeship.
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PHYSICAL REVIEW B 82, 235306 共2010兲
Slow light loss due to roughness in photonic crystal waveguides: An analytic approach
Weiwei Song, Ryan A. Integlia, and Wei Jiang*
Department of Electrical and Computer Engineering, and Institute for Advanced Materials, Devices and Nanotechnology,
Rutgers University, Piscataway, New Jersey 08854, USA
共Received 6 November 2010; published 2 December 2010兲
We analytically study roughness-induced scattering loss in a photonic crystal waveguide 共PCW兲. A crosssectional eigenmode orthogonality relation is derived for a one-dimensional 共1D兲-periodic system, which
allows us to significantly simplify the coupled mode theory in the fixed eigenmode basis. Assisted by this
simplification, analytic loss formulas can be obtained with reasonable assumptions despite the complexity of
PCW mode fields. We introduce the radiation and backscattering loss factors ␣1 and ␣2 such that the loss
coefficient ␣ can be written as ␣ = ␣1ng + ␣2n2g 共ng is the group index兲. By finding analytic formulas for ␣1 and
␣2, and examining their ratio, we show why the backscattering loss generally dominates the radiation loss for
ng ⬎ 10. The interplay between certain mode-field characteristics, such as the spatial phase, and structure
roughness is found crucial in the loss-generation process. The loss contribution from each row of holes is
analyzed. The theoretical loss results agree well with experiments. Combined with systematic simulations of
loss dependences on key structure parameters, the insight gained in this analytic study helps identify promising
pathways to reducing the slow light loss. The cross-sectional eigenmode orthogonality may be applicable to
other 1D-periodic systems such as electrons in a polymer chain or a nanowire.
DOI: 10.1103/PhysRevB.82.235306
PACS number共s兲: 42.70.Qs, 42.25.Fx, 42.79.Gn, 42.82.Et
I. INTRODUCTION
Photonic crystal waveguides 共PCWs兲 can slow down light
significantly, which has important applications such as optical switching and modulation1–3 and all optical storage.4
However, significant optical loss in the slow light regime
stymies further advance in this field. Roughness-induced loss
has been previously investigated.5–15 The scattering from a
single sidewall irregularity was theoretically studied at first.6
Random sidewall roughness with spatial correlation was later
introduced to account for loss characteristics in real photonic
crystal waveguide structures.9,10 Although the scaling of
slow light loss with respect to the group velocity, vg, has
been examined,5,8–16 it has been difficult to reach a conclusive answer. Theory predicted 1 / vg scaling for the radiation
loss and 1 / v2g scaling for the backscattering loss8,9 in the
absence of multiple scattering. Experimental studies, however, often fitted the loss data with a simple power law v−g ␯,
where ␯ was found to vary widely.10,11,17 To explain these
variations, theory should provide a global picture of how the
backscattering and radiation losses 共and their relative
strength兲 vary with a wide range of structure and roughness
parameters commonly found in experiments. More importantly, theory should provide pertinent insight into the lossgeneration process and suggest promising pathways to loss
reduction.
In this work, we develop a theoretical framework for calculating PCW scattering loss based on the coupled mode
theory in the fixed eigenmode basis. Here we will prove an
interesting cross-sectional eigenmode orthogonality relation,
which allows us to significantly simplify the coupled mode
theory in the fixed eigenmode basis. Assisted by this simplification, analytic loss formulas can be obtained with reasonable assumptions despite the complexity of PCW mode
fields. We will introduce the radiation and backscattering loss
factors ␣1 and ␣2, such that the loss coefficient ␣ can be
expressed as ␣ = ␣1ng + ␣2n2g, where ng is the group index. By
1098-0121/2010/82共23兲/235306共7兲
finding analytic formulas for ␣1 and ␣2, and examining their
ratio, we show why the backscattering loss dominates the
radiation loss under fairly general conditions. The analytic
study provides further insight into the underpinning physics,
such as how the mode-field characteristics 共e.g., spatial
phase兲 interact with roughness to produce loss. The dependences of loss on the structure/roughness parameters are
simulated to corroborate the analytic results. Unlike numerical studies that are limited to several instances of structures
with specific structure/roughness parameters, this analytic
study reveals general loss characteristics and fresh insight
into the loss-generation process, helping identify new pathways to loss reduction.
This paper is organized as follows. In Sec. II, we will
present our scattering loss theory. An interesting eigenmode
orthogonality relation will be derived and will be utilized to
simplify the coupled mode theory in the fixed eigenmode
basis. The backscattering and radiation losses will be calculated for the air-bridge type of photonic crystal waveguides,
and the loss contribution from each row of holes will be
analyzed. In Sec. III, we will present analytic formulas of the
backscattering and radiation loss factors and give a general
proof of the dominance of the backscattering loss for ng
⬎ 10. The interplay between the mode-field characteristics
共e.g., spatial phase兲 and the roughness will be analyzed. In
Sec. IV, we will systematically study the loss dependences
on the structure and roughness parameters such as the hole
diameter, the waveguide width, and the correlation length.
Strategies of reducing the roughness-induced loss will be
discussed. The theoretical results are found to agree well
with experiments. Section V presents our conclusions.
II. SCATTERING LOSS THEORY
A. Coupled mode theory and mode orthogonality in
a PCW crossection
The coupled mode theory of a photonic crystal waveguide
can be written concisely with Dirac notation. This particular
235306-1
©2010 The American Physical Society
PHYSICAL REVIEW B 82, 235306 共2010兲
SONG, INTEGLIA, AND JIANG
form of coupled mode theory was first developed by Johnson
et al.18 for taper transitions in photonic crystals, and was
later applied to the disorder-induced scattering problem.12
The theory can use the fixed eigenmode basis or the instantaneous eigenmode basis. It has the advantage of giving clear
dependence of mode coupling on the group velocity through
the mode normalization factor. In this theory, the Maxwell’s
equations are rewritten as18
⳵
Aˆ兩␺典 = − i Bˆ兩␺典,
⳵z
共1a兲
冉
冊
0
␻␧ − ␻−1ⵜt ⫻ ␮−1ⵜt ⫻
,
Aˆ =
−1
0
␻␮ − ␻ ⵜt ⫻ ␧−1ⵜt ⫻
共1b兲
Bˆ =
冉
0
− zˆ ⫻
zˆ ⫻
0
冊
兩␺典 =
,
冋 册
Et共x兲
,
Ht共x兲
Et ⬅
冉冊
Ex
,
Ey
共1c兲
Ht ⬅
冉 冊
Hx
,
Hy
where ␧共X兲 is the dielectric function, ␮ the permeability. The
eigenmodes, 兩␺␤典 = ei␤z兩␤典, satisfy
冉
冊
ˆ 兩␤典 ⬅ Aˆ + i ⳵ Bˆ 兩␤典 = ␤Bˆ兩␤典.
C
⳵z
共2兲
Here we consider guided and radiation modes with real ␤.
The inner product is defined as
具␺兩Bˆ兩␺⬘典 = zˆ ·
冕
Eⴱt
⫻ Ht⬘ + Et⬘ ⫻
Hⴱt ⬘dxdy.
共4兲
For a PCW periodic along z, solid-state theory suggests that
the eigenstate orthogonality can be obtained only by further
integration along z,
冕
ei共␤⬘−␤兲z具␤兩Bˆ兩␤⬘典dz = ␩␤␦␤␤⬘ .
⳵
共Aˆ + ⌬Aˆ兲兩␺典 = − i Bˆ兩␺典,
⳵z
共6兲
共7兲
where 兩␺典 = 兺ncn共z兲ei␤nz兩n典, and 兩n典 are the eigenmodes of the
unperturbed system. The coupled mode theory generally requires to use 具m兩 to select cm for a particular mode from Eq.
共7兲. If the conventional orthogonality relation, Eq. 共5兲, is
applied, the evaluation of 兰具m兩 ⳵⳵z Bˆ兩␺典dz will be problematic
because cm共z兲 depends on z. With the orthogonality relation,
Eq. 共4兲, however, it is straightforward to show that the coupling coefficients are governed by the following equation:
⳵ cm
= 共i/␩m兲 兺 ei共␤n−␤m兲z具m兩⌬Aˆ兩n典cn .
⳵z
n
共5兲
Such an orthogonality relation cannot be directly used in a
rigorous PCW coupled mode theory because the modal coupling coefficients also have z dependence and will appear in
the above integral. To overcome this problem, a complicated
virtual coordinate theory was previously developed.18
Here we show that Eq. 共4兲 still holds for a PCW in any z
section. By partial integration, one can readily show
ˆ 兩␤典兲ⴱ + i ⳵ 具␤兩Bˆ兩␤⬘典. Therefore,
具␤兩Cˆ兩␤⬘典 = 共具␤⬘兩C
⳵z
⳵
具␤兩Bˆ兩␤⬘典.
⳵z
This is a differential equation of 具␤兩Bˆ兩␤⬘典z with a solution
However,
具␤兩Bˆ兩␤⬘典z+a
具␤兩Bˆ兩␤⬘典z = e−i共␤⬘−␤兲z具␤兩Bˆ兩␤⬘典z=0.
ˆ
= 具␤兩B兩␤⬘典z according to Bloch theorem. Therefore,
具␤兩Bˆ兩␤⬘典 = ␩␤␦␤,␤⬘−共2n␲/a兲, which gives Eq. 共4兲 for ␤ and ␤⬘ in
the first Brillouin zone.
The orthogonality Eq. 共4兲 for a photonic crystal waveguide is an interesting result. According to the Bloch theorem, the eigenstate orthogonality in a generic onedimensional 共1D兲-periodic system should be obtained by
integrating 兰␺ⴱb␺adz = 0 along the periodicity direction 共z in
this case兲. However, the above proof has shown that if there
are multiple eigenstates with different on-axis wave vectors
at a given frequency 共or photon energy兲, they must be orthogonal by integrating 兰␺ⴱb␺adxdy in any cross section perpendicular to the periodicity axis. Note an equivalent form of
this orthogonality was proved in a different theoretical
framework based on the Lorentz reciprocity,20 which is limited to electromagnetic wave. The proof given here is generally valid for any scalar or vector wave satisfying Eq. 共2兲.
Therefore, the orthogonality relation presented here may be
potentially applicable to other 1D-periodic systems, such as
electrons in a polymer chain or a nanowire.
The coupled mode theory in the fixed eigenmode basis
can now be established easily based on Eq. 共4兲 for a photonic
crystal waveguide. With a potential perturbation ⌬Aˆ, the
mode equation becomes
共3兲
A rigorous formulation of the coupled mode theory must be
established upon a complete set of orthogonal modes.18,19
For an ordinary waveguide, whose structure is invariant
along z, it is straightforward to show that any two eigenmodes at a given frequency ␻ must be orthogonal19
具␤兩Bˆ兩␤⬘典 = ␩␤␦␤␤⬘ .
共␤⬘ − ␤兲具␤兩Bˆ兩␤⬘典 = i
共8兲
We should emphasize that although it appears similar to the
equation for a conventional waveguide homogenous along z,
this simplified Eq. 共8兲 can be rigorously established for a
PCW only with the help of Eq. 共4兲. This simplification enabled by the cross-sectional orthogonality relation, Eq. 共4兲, is
the main improvement for the coupled mode theory used in
this work. This simplification allows us to derive analytic
loss formulas that can be calculated almost by hand, as we
shall see in Sec. III, and provides a clearer physical picture.
235306-2
PHYSICAL REVIEW B 82, 235306 共2010兲
SLOW LIGHT LOSS DUE TO ROUGHNESS IN PHOTONIC…
B. Separate calculation of backscattering loss
and radiation loss
-3
-2
The scattering loss can be introduced through a random
potential ⌬Aˆ due to dielectric perturbation ⌬␧ and ⌬共␧−1兲.
For a frequency range with a single guided mode 兩␤典, the
perturbed mode is given by
i␤z
兩␺典 = c␤共z兲e 兩␤典 + c−␤共z兲e
−i␤z
兩− ␤典 + 兺 ck共z兲e
ikzz
-1
兩k典.
cm = 共i/␩m兲
冕冕冕
ei共␤−␤m兲z共⌬Aˆ兲m␤dxdydz,
ⴱ ˆ
where 共⌬Aˆ兲m␤ ⬅ ␾m
⌬A␾␤, ␾m = 具x 兩 m典. The loss coefficient is
given by the conservation of power flux19
冋
册
␣ = 共1/Lz兲 具兩c−␤兩2典 + 兺 具兩ck兩2典兩␩k/␩␤兩 ,
k
共10兲
where the ensemble average 具 · 典 over the random roughness
has been applied. To show explicit dependence on the group
velocity vg,␤ of mode ␤, we introduce U␤ ⬅ 41 兩␩␤ / vg,␤兩, the
time averaged mode energy per unit length along the z axis.
For a radiation mode 兩k典, we define Uk ⬅ 41 兩␩k / vgz,k兩, where
vgz,k is the z component of vg,k. Then the ␩m terms in Eq. 共10兲
can be replaced by Um and vg. Assuming that the sidewall
roughness of different holes is uncorrelated,9 the ensemble
averaged ␣ of a PCW is a sum of the ensemble averaged loss
contribution from each hole. For roughness-related calculation, it is more convenient to use the polar coordinates 共r , ␪兲
in each hole in place of 共x , z兲. After some calculations, we
find
␣ = ␣1ng + ␣2n2g ,
共11a兲
␣1 = 共1/a兲 兺 兺 I共k, ␤,nx兲共c/vgz,k兲兩Uk/U␤兩,
共11b兲
␣2 = 共1/a兲 兺 I共− ␤, ␤,nx兲,
共11c兲
k
nx
nx
where nx and nz are the indices of holes along x and z, respectively 共see Fig. 1兲. The PCW has a lattice constant a,
mean hole radius r0, and slab thickness tslab. The integral for
the nxth hole is
I共m, ␤,nx兲 ⬅ 共r0tslab/4Umc兲2
⫻
冕
⍀n
x
respectively. Note that the coordinates 共r , ␪兲 are centered in
each cell ⍀nx.
Now the loss coefficients can be numerically calculated
using Eqs. 共11兲 and 共12兲. Instead of directly calculating the
loss coefficient ␣, we will calculate the radiation and backscattering loss factors ␣1 and ␣2. Note that ␣ diverges as the
frequency approaches the band edge whereas ␣1 and ␣2 are
slowly varying functions even near the band edge. Thus the
calculation of ␣1 and ␣2 generally leads to significantly more
stable numerical results than directly calculating ␣.
Here we consider the TE guided modes 共i.e., electric field
primarily in the xz plane兲 of a Si air-bridge PCW. The guided
modes can be obtained by a preconditioned eigensolver21
with a tensorial average of the dielectric constant near
interfaces.22 The perturbation potential is evaluated using the
continuous components on interfaces.23
The radiation modes are calculated by considering the
PCW supercell delineated in dashed lines in Fig. 1 共the one
used in actual calculation is much longer along x兲 as one
period of a two-dimensional grating in the x-z plane. The
mode field for a given plane-wave incident upon the PCW
top surface can be obtained by any grating diffraction
theory.24,25 Due to the artificial x periodicity imposed by the
grating theory, this treatment is equivalent to calculating the
radiation loss for an array of parallel PCWs. For a sufficiently large spacing between waveguides, the radiation
losses of adjacent waveguides are independent of each other
for weak scattering. Figure 2共a兲 clearly shows that only the
first two rows 共nx = ⫾ 1, ⫾2兲 contribute significantly to the
radiation loss. For each row, data plotted in symbols and
lines are obtained by two supercell sizes differing by 50%.
Their small differences of ␣1 confirm that adjacent
waveguides do not affect each other. The backscattering loss
shows even stronger dominance by the first row 关Fig. 2共b兲兴.
Obviously, this can be attributed to the fact that the scattering
matrix elements 具−␤兩⌬Aˆ兩␤典 and 具k兩⌬Aˆ兩␤典 involve ␾␤共x兲,
which decays very fast with x.
III. ANALYTIC FORMULAS FOR BACKSCATTERING
AND RADIATION LOSSES
x
共12兲
where ⌬Aˆm␤,nx共␪兲 = 共1 / tslab兲兰⌬Aˆm␤ 兩r=r+dy. A typical autocor0
relation function is given by 具⌬r共␪⬘兲⌬r共␪兲典 = ␴2e−兩␪−␪⬘兩r0/lc,
where ␴ and lc are the rms roughness and correlation length,
x
FIG. 1. In-plane view of a line-defect waveguide in a photonic
crystal slab.
ⴱ
ˆ
ei共␤−␤m兲r0共sin ␪−sin ␪⬘兲⌬Aˆm
␤,n 共␪⬘兲⌬Am␤,nx共␪兲
⫻具⌬r共␪⬘兲⌬r共␪兲典d␪⬘d␪ ,
nx 3
k
where 兩k典 are radiation modes, and cm共z兲, m = ⫾ ␤, k are the
coupling amplitudes. With Eq. 共4兲, it is straightforward to
solve the coupled mode equations to the first order. For unity
input, the output amplitudes are given by
z
1
2
共9兲
Eeff
nz
Interestingly, the factors ␣1 and ␣2 roughly have the same
order of magnitude in Fig. 2. As a consequence, the backscattering loss 共␣2n2g兲 dominates the radiation loss 共␣1ng兲,
which can be seen from their ratio
235306-3
PHYSICAL REVIEW B 82, 235306 共2010兲
SONG, INTEGLIA, AND JIANG
100
(a)
0.2
ng
50
g
row ±1
row ±2
row ±3
total
n
α1 (dB/cm)
0.3
0.1
0
α2 (dB/cm)
0
0.6
row ±1
row ±2
row ±3
total
estimate
(b)
0.4
0.2
0
0.265
0.27
0.275
ωa/2π
␣2 ⬇ 2Nx,back␲共n21 − n22兲2共k20␴2lcr0/aw2d兲兩eef f,␤兩4I0共2␣xr0兲,
共15兲
0.28
共for ng ⬎ 10兲.
Numerical simulations of a few other PCW structures
showed similar dominance.9,26 Mathematically, the n2g term
surely dominates the ng term in Eq. 共11a兲 for a sufficiently
large ng. But the ng threshold for the onset of this dominance
depends on ␣1 and ␣2 and could be too large to be observed
共e.g., ng ⬎ 1000兲. To ascertain the universal dominance of
backscattering in practically observable ng ranges and to explore the underpinning mechanism of this dominance, an
analytic study is needed. Moreover, such a study may offer
insight into the interaction between the mode field and
roughness.
We have performed analytic calculation of the factor ␣2
with some simple reasonable assumptions. As a first step, we
assume a guided mode field of the form E␤ ⬃ e−␣xx/2ei␤z. After some calculation, we find
I共− ␤, ␤,nx兲 = 共␻r0tslab/4Umc兲 共⌬␧12兲 兩Eef f,nx兩 ␴ Iang ,
2
Iang =
冕
2
3
␣1 ⬇ 2Nx,rad共n21 − n22兲2nsub
共k40␴2lcr0tslab/awd兲
⫻兩eef f,␤¯eef f,k兩2I0共␣xr0兲,
共16兲
4 2
ei共2␤r0兲共sin ␪−sin ␪⬘兲−␣xr0共2+cos ␪+cos ␪⬘兲e−兩␪−␪⬘兩r0/lcd␪⬘d␪ ,
共13兲
where ⌬␧12 is the dielectric constant difference, Eef f,nx is the
effective field at the hole’s inner edge 共␪ = ␲ in Fig. 1兲. Typically, the correlation length lc is small. For e−r0/lc Ⰶ 1, ␣xlc
Ⰶ 1, and 2␤lc Ⰶ 1, one finds
Iang ⬇ 共4␲lc/r0兲I0共2␣xr0兲,
and 2Nx,back is the effecwhere k0 = 2␲ / ␭,
tive number of rows of holes contributing to backscattering.
For numerical estimate, we assume Nx,back ⬇ 1, wd = w0
⬅ 冑3a. In addition, Eef f,␤ is obtained by averaging 兩E␤兩2 at
the inner hole edge across the slab thickness. We find that
eef f,␤ typically varies around 0.3–0.4 in the slow light regime. The decay constant ␣x ⬇ 0.77共2␲ / a兲 is obtained by
fitting the mode energy against x near the band edge. Note
I0共2␣xr0兲 is a slowly varying function for this parameter
range of interest. Figure 2共b兲 shows that Eq. 共15兲 gives a
reasonable estimate of the order of magnitude of ␣2 and its
trend. There is an overestimate of two to three times because
we have neglected the following factors: 共a兲 the vector nature
of the field; 共b兲 the high-G Fourier components; and 共c兲 the
variation in the field along y.
For the radiation modes, considering two polarizations
共␮ = 1 , 2兲 and two propagation directions 共sz = ⫾ z兲, the sum
L xL y
over k in Eq. 共11b兲 becomes 兺k → 兺␮,sz 共2␲兲2 兰dkxdky, where
Lx and Ly are the transverse dimensions of the normalization
volume. Note the final result of ␣1 is independent of LxLy
because I共k , ␤ , nx兲 · 兩Uk兩 ⬃ 共LxLy兲−1 in Eq. 共11b兲. One can then
show that
n21 − n22 = ⌬␧12 / ␧0,
FIG. 2. 共Color online兲 Loss factors as a function of frequency
and the contribution from each pair of rows of holes. 共a兲 Radiation
loss factor ␣1; and 共b兲 backscattering factor ␣2 and the analytic
estimate. PCW parameters: ␣ = 430 nm, r0 / a = 0.25, tslab = 200 nm,
␴ = 3 nm, and lc = 40 nm.
␣2n2g
⬃ ng Ⰷ 1
␣ 1n g
second scenario, the phases of ␾␤共x兲 and ␾−ⴱ ␤共x兲 almost exactly cancel each other in 具−␤兩⌬Aˆ兩␤典 and become irrelevant.
When these conditions are not satisfied, the spatial phase
variations tend to reduce Iang below the value given in Eq.
共14兲.
For a guided mode, we can define a modal field ampli2
tude, ¯Esp,␤, by U␤ = ␧0¯Esp,
␤wdtslab / 2 and normalize the effective field as eef f,␤ = Eef f,␤ / ¯Esp,␤. Then combining Eqs. 共11c兲,
共13兲, and 共14兲, we obtain
共14兲
where I0共x兲 = I0共x兲exp共−x兲 and I0 is the modified Bessel
function of the first kind. One can show that Eq. 共14兲 still
holds for a more general form of the field E␤
⬃ e−␣xx/2兺GuGei共␤+G兲z under two scenarios: 共1兲 the mode is
dominated by Fourier terms satisfying Glc Ⰶ 1 so that the
phase of each eiGz varies little within one correlation length
and 共2兲 near the band edge where ␾␤共x兲 ⬇ ␾−␤共x兲. For the
where ¯eef f,k is the normalized field amplitude at the hole
inner edge averaged over all k states, nsub = 1 is the substrate
refractive index, and 2Nx,rad is the effective number of rows
of holes contributing to radiation loss. Comparing Eq. 共15兲
and Eq. 共16兲, we find
3
¯ ef f,k兩2 I0共␣xr0兲
k20wdtslab 兩e
␣1 Nx,radnsub
.
⬇
·
·
␣2
␲Nx,back
兩eef f,␤兩2 I0共2␣xr0兲
共17兲
With Nx,rad, Nx,back = 1 ⬃ 2, wd = w0, tslab ⬃ 220 nm, ␣x
⬃ 0.5共2␲ / a兲, and normalized fields eef f,␤, ¯eef f,k ⬃ 0.5, each
ratio in Eq. 共17兲 is on the order of unity. This equation therefore predicts that ␣1 and ␣2 are generally on the same order.
Therefore, this analytic study explains why the backscattering generally dominates, ␣2n2g Ⰷ ␣1ng, in the slow light regime ng ⱖ 10. Note that Eqs. 共15兲 and 共16兲 contain no fastvarying functions, which implies that ␣1 and ␣2 should be
fairly insensitive to most structure parameters for a typical
PCW.
235306-4
PHYSICAL REVIEW B 82, 235306 共2010兲
SLOW LIGHT LOSS DUE TO ROUGHNESS IN PHOTONIC…
ij
0.3
ε
0.2
(b)
0.3
0.2
0.1
0.1
30
Ngrid
40
0
20
lc (nm)
40
1
0.5
0.9
1
Wd/W0
1.1
0.23
0.25
0.27
0.29
10
exp. − α/n
g
1
exp. − α/n2
g
10
0
2
(d)
0.5
0
0.8
g
2
theory − α/ng
0
1
(c)
2
theory − α/n
g
20
10
α (dB/cm)
α2 (dB/cm)
0
10
0.4
2
0.475
0.425
0.375
g
(a)
α/n ; α/n2 (dB/cm)
<ε>
α (dB/cm)
|∆α2|/α2
0.4
10
0
−1
0.265
r/a
FIG. 3. 共Color online兲 Variation in ␣2 with 共a兲 grid size per edge
of the unit cell, with tensorial average 具␧典ij and without 共up to 30%
oscillation兲; 共b兲 correlation length lc, for modes at different ␤a / 2␲;
共c兲 PCW width; and 共d兲 hole radius. PCW parameters: a = 420 nm,
r0 / a = 0.25, tslab = 220 nm, ␴ = 3 nm, and lc = 40 nm.
Note that prior scattering loss formulas still involve the
photonic crystal mode field and the Green’s function,9 which
must be obtained through further computation. Our analytic
loss formulas, Eqs. 共11a兲, 共15兲, and 共16兲, do not have these
terms, and can be evaluated almost by hand. More importantly, the ratio of ␣1 and ␣2 derived from these formulas, as
presented in Eq. 共17兲, gives a general mathematical proof of
the dominance of the backscattering loss over the radiation
loss, along with a predicted dominance threshold ng ⬃ 10.
Prior numerical studies discovered this dominance in a limited number of structures with specific parameters.9,26 However, the generality of the dominance and its threshold ng
were not clearly determined in numerical studies.
IV. DISCUSSION
A. Loss dependence on structure and roughness parameters
and loss reduction strategy
As the backscattering loss dominates, we focus on the
dependences of ␣2 on several key roughness/structure parameters. Note that the tensorial average of the dielectric
function near interfaces is found to significantly improve the
convergence with the spatial grid size, as shown in Fig. 3共a兲.
This allows us to study small structure parameter changes.
First, we examine the limitation of the preceding analytic
results due to the assumption of small lc. The dependences of
␣2 on lc for various normalized ␤ values are plotted in Fig.
3共b兲. For guided modes near the band edge 共␤a / 2␲ ⬃ 0.5兲,
␣2共lc兲 is almost perfectly linear. As discussed above, this
linearity predicted in Eq. 共15兲 is due to ␾␤共x兲 ⬇ ␾−␤共x兲 near
the band edge, which causes phase cancellation in
具−␤兩⌬Aˆ兩␤典. Away from the band edge, the phase variation
causes the integral Iang to become sublinear at large lc values
关but Eqs. 共15兲 and 共16兲 remain useful as estimates兴, which is
also confirmed in Fig. 3共b兲. Second, the dependence on the
waveguide width is studied in Fig. 3共c兲. The loss factor ␣2
could be reduced by a factor about 5 from wd = 0.83w0 to
0.27
ωa/2π
0.275
FIG. 4. 共Color online兲 Comparison with experimental results in
Ref. 10. The experimental spectrum is shifted to align the band edge
with the theory.
1.1w0 near the mode edge. Third, in most experimental
works, the air hole diameter and slab thickness usually
spread over certain ranges 共e.g., r0 / a : 0.23– 0.29 and
tslab : 0.19– 0.25 ␮m兲 and the exact values may vary due to
uncertainties in fabrication processes. Our simulations show
that ␣2 varies insignificantly over the typical ranges of a, r0,
and tslab. The variation of ␣2共␤a / 2␲ ⬃ 0.5兲 is plotted against
r0 / a in Fig. 3共d兲.
The analytic and computational studies offer insight into
the loss mechanism and point to promising pathways to loss
reduction. First, among four essential geometric parameters
共r, a, tslab, and wd兲, wd appears to be the only one that allows
for substantial loss reduction. Second, the spatial phase
analysis in the derivation of Eq. 共15兲 suggests that designing
guided modes with accentuated high-wave-number Fourier
components might help reduce the loss due to random roughness. But the eigenfrequency and other deterministic characteristics of such a mode also tend to be sensitive to the variations in structure parameters 共mean value兲. Thus, ingenious
designs are needed to account for both statistical and deterministic properties. Third, manipulating the polarization,
through introducing anisotropic materials, for example,
could yield loss much lower than that predicted in Eq. 共15兲,
which neglects the polarization. Lastly, Eqs. 共15兲 and 共16兲
and the spatial phase analysis may offer new insight into the
mode shaping effect.27
B. Comparison with experiments
In Fig. 4, we compare with experimental results from Ref.
10 using ␴ = 3 nm and lc = 40 nm suggested therein. Evi˜
dently, our theory agrees well with experiments for ␻
⬍ 0.273, including the upswing of the ␣ / n2g共⬇␣2兲 curve near
the band edge. This can be partially explained by the fact that
the integral of the guided mode intensity 兩Eb共x兲兩2 over the
hole surface increases with the group index.27 However, a
full explanation must be based on the characteristics of the
random potential matrix element 具−␤兩⌬Aˆ兩␤典. As discussed
above, the phase cancellation in 具−␤兩⌬Aˆ兩␤典 causes an increase in Iang and ␣2 near the band edge. Due to the interplay
235306-5
PHYSICAL REVIEW B 82, 235306 共2010兲
SONG, INTEGLIA, AND JIANG
between the spatial phase of the mode and the roughness,
this upswing is stronger for larger correlation lengths. Because ␣1 and ␣2 are not constant in general, a simple power
law fitting ␣ ⬃ ng␯ of experimental data would unlikely give
consistent ␯ values, which agrees with the findings of Ref.
27. Note that if the coupling loss28,29 is included, the loss-ng
relation could even become sublinear 共or logarithmic兲, espe˜ = ␻a / 2␲c = 0.273, the
cially for short waveguides. Above ␻
localized band tail states21,30 of the second guided mode
˜ ⬇ 0.281兲 introduce in the experimental spec共band edge ␻
trum a broad resonance accompanied by a “softened” vg at
the nominal band edge.31 This effect is beyond the scope of
this work. Fortunately, this effect can be avoided by designing the second mode above the useful spectral range of the
first mode. Below a sufficiently small vg, multiple scattering
occurs for the first mode, accompanied by undesirably high
loss.11,26,31–33 The studies presented here could help reduce
scattering losses and delay the onset of this regime.
In this work, we have considered loss introduced by
guided and radiation modes with real ␤ values. In a nonperturbed photonic crystal structure 共including a PCW兲, modes
with complex ␤ values generally arise locally near the end
faces and affect the end-face coupling loss34 but not the
propagation loss of a truly guided mode. The propagation
loss is generally more important for a sufficiently long photonic crystal waveguide. Also within the photonic band gap
of a PCW, those modes with complex ␤ values usually do
not carry away energy themselves and thus may not introduce propagation loss directly. Some higher order 共multiple兲
scattering processes in a PCW with random perturbations
*Electronic address: [email protected]
1 Y.
A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab,
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Y. Q. Jiang, W. Jiang, L. Gu, X. Chen, and R. T. Chen, Appl.
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3 L. L. Gu, W. Jiang, X. Chen, L. Wang, and R. T. Chen, Appl.
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J. T. Mok and B. J. Eggleton, Nature 共London兲 433, 811 共2005兲.
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S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, Appl. Phys. B: Lasers Opt. 81,
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useful 共relatively low loss兲 spectral ranges of photonic crystal waveguides, as discussed in the comparison with experimental data above.
V. CONCLUSION
In summary, analytic formulas, Eqs. 共11a兲, 共15兲, and 共16兲,
of the PCW scattering losses can be obtained despite the
complexity of the PCW mode fields. With these formulas,
the loss of a typical photonic crystal waveguide can be estimated almost by hand. The analytic study reveals that the
interplay between the mode characteristics and the structure
roughness may hold the key to loss reduction. These results
are corroborated by systematic simulations with varying
structure parameters. As a byproduct, the cross-sectional
eigenmode orthogonality relation for a 1D periodic system
may be applicable to other problems, such as electrons in a
polymer chain or a nanowire.
ACKNOWLEDGMENTS
We are grateful to David Vanderbilt, Steven G. Johnson,
Chee Wei Wong, Philippe Lalanne, Stephen Hughes, Eiichi
Kuramochi, Fabian Pease, Leonard C. Feldman, George K.
Celler, and George Sigel for helpful discussions. This work
is supported by AFOSR MURI under Grant No. FA9550-081-0394 共G. Pomrenke兲.
B 78, 245108 共2008兲.
Le Thomas, H. Zhang, J. Jagerska, V. Zabelin, R. Houdre, I.
Sagnes, and A. Talneau, Phys. Rev. B 80, 125332 共2009兲.
15
A. Petrov, M. Krause, and M. Eich, Opt. Express 17, 8676
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16 J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, Opt.
Express 18, 15484 共2010兲.
17
L. O’Faolain, T. P. White, D. O’Brien, X. Yuan, M. D. Settle, and
T. F. Krauss, Opt. Express 15, 13129 共2007兲.
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S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu,
E. Lidorikis, and J. D. Joannopoulos, Phys. Rev. E 66, 066608
共2002兲.
19 D. Marcuse, Theory of Dielectric Optical Waveguides 共Academic Press, San Diego, 1991兲.
20 G. Lecamp, J. P. Hugonin, and P. Lalanne, Opt. Express 15,
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21
W. Jiang and C. D. Gong, Phys. Rev. B 60, 12015 共1999兲.
22 S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173
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S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. D.
Joannopoulos, and Y. Fink, Phys. Rev. E 65, 066611 共2002兲.
24 W. Jiang and R. T. Chen, J. Opt. Soc. Am. A 23, 2192 共2006兲.
25
We also used the finite difference time-domain technique but
found that it required very long simulation time to obtain the
frequency resolutions needed.
14 N.
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PHYSICAL REVIEW B 82, 235306 共2010兲
SLOW LIGHT LOSS DUE TO ROUGHNESS IN PHOTONIC…
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S. Mazoyer, J. P. Hugonin, and P. Lalanne, Phys. Rev. Lett. 103,
063903 共2009兲.
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M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L.
O’Faolain, and T. F. Krauss, Phys. Rev. B 80, 195305 共2009兲.
28 R. A. Integlia, W. Song, J. Tan, and W. Jiang, J. Nanosci. Nanotechnol. 10, 1596 共2010兲.
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Y. A. Vlasov and S. J. McNab, Opt. Lett. 31, 50 共2006兲.
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S. John, Phys. Rev. Lett. 58, 2486 共1987兲.
31 M. Patterson, S. Hughes, S. Combrie, N. V.-Quynh Tran, A. De
Rossi, R. Gabet, and Y. Jaouen, Phys. Rev. Lett. 102, 253903
共2009兲.
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S. Mazoyer, P. Lalanne, J. C. Rodier, J. P. Hugonin, M. Spasenović, L. Kuipers, D. M. Beggs, and T. F. Krauss, Opt. Express
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235306-7
Copyright © 2010 American Scientific Publishers
All rights reserved
Printed in the United States of America
Journal of
Nanoscience and Nanotechnology
Vol. 10, 1596–1605, 2010
Longitudinal and Angular Dispersions in
Photonic Crystals: A Synergistic Perspective on
Slow Light and Superprism Effects
Ryan A. Integlia, Weiwei Song, Jun Tan, and Wei Jiang∗
Department of Electrical and Computer Engineering, and Institute for Advanced Materials, Devices, and Nanotechnology,
Rutgers University, 94 Brett Road, Piscataway, NJ 08854, USA
REVIEW
The slow-light effect and the superprism effect are two important effects in photonic crystal structures. In this paper, we will review some of our recent works on the fundamental physics and device
applications of these two effects. We
will present
synergisticto:
perspective that examines these two
Delivered
bya Ingenta
effects as a whole. Apparently,
theUniversity,
slow light effect
is due to
the dispersion of a photonic crystal
Rice
Fondren
Library
along the direction of light propagation,IPnamely
the longitudinal direction, and the superprism effect
: 128.42.154.22
is related to angular dispersion. However,
a deep
analysis
will show that the superprism effect has
Sat, 19 Mar
2011
15:22:48
an elusive dependence on the longitudinal dispersion as well. Some subtle connections and distinctions between the slow-light effect and the superprism effect will be revealed through our physical
analysis. This allows us to treat these two effects under a common theoretical framework. As an
example, we will apply this framework to make a direct comparison of the slow-light optical phase
array approach and the superprism approach to beam steering applications. Dispersive effects are
frequently accompanied by high optical loss and/or narrow bandwidths. We will discuss these issues
for both longitudinal and angular dispersions. For the slow light effect, we will give a simple proof
of the scaling of fabrication-imperfection related random scattering losses in a slow-light photonic
crystal waveguide. Similar to the bandwidth-delay product for the longitudinal dispersion, we will
introduce a simple, yet fundamental, limit that governs the bandwidth and angular sensitivities of
the superprism effect. We will also discuss the application of the slow-light effect to making compact
silicon optical modulators and switches.
Keywords: Photonic Crystals, Slow Light, Superprism, Dispersion, Optical Loss, Bandwidth,
Silicon Photonics.
CONTENTS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. The Slow Light Effect and the Longitudinal Dispersion . . . . . .
2.1. The Origin of the Slow Light
Effect in Photonic Crystals . . . . . . . . . . . . . . . . . . . . . . . .
2.2. Applications of the Slow Light Effect . . . . . . . . . . . . . . . .
2.3. Loss-Limited Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. The Superprism Effect and the Angular Dispersion . . . . . . . . . .
3.1. Dispersion Surface Curvature and
Angular Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. Slow-Light Induced Strong Angular Dispersion . . . . . . . . .
3.3. “Pure” Angular Dispersion Effect:
Bandwidth Limited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Longitudinal Dispersion versus Angular Dispersion:
A Direct Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
∗
Author to whom correspondence should be addressed.
This is an invited review paper.
1596
J. Nanosci. Nanotechnol. 2010, Vol. 10, No. 3
1. INTRODUCTION
1596
1598
1598
1598
1600
1601
1601
1602
1603
1603
1604
1604
1604
Photonic crystals possess a wide range of extraordinary
properties that are absent in conventional materials. First
and foremost, the periodic structure of a photonic crystal causes photonic bands and bandgaps to form on the
frequency spectrum of photons. Therefore, photonic crystals with photonic bandgaps can serve as “perfect mirrors”
to confine light in small dimensions, forming ultracompact waveguides and cavities. On the other hand, there are
other technologies that can also provide tight confinement
of light. Compared to these alternative technologies, the
uniqueness of photonic crystal-based waveguides and cavities often comes from the fact that the periodic structure
of a photonic crystal provides some additional distinctive
opportunities to modify the spectral property of light, leading to many dispersive effects with a wide range of applications. For example, in a photonic crystal waveguide, the
1533-4880/2010/10/1596/010
doi:10.1166/jnn.2010.2039
Integlia et al.
Longitudinal and Angular Dispersions in Photonic Crystals
dispersion relation, k, generally has a portion where
its slope tends to zero, implying a vanishing group velocity. Such a slow light effect, together with the tight light
confinement provided by the photonic bandgap, leads to
extraordinary enhancement of phase shift and time delay
in such a waveguide.1 We shall emphasize that some of
these dispersion effects can be significant in their own
right, without the presence of tight light confinement. For
example, the superprism effect2 causes the beam propagation angle inside a photonic crystal to be highly sensitive
to the wavelength of light. Essentially, this effect amounts
to significantly enhanced angular dispersion. In the superprism effect, the light propagation is not confined. Moreover, we will show that the superprism effect does not
necessarily appear near a bandgap (or at a bandedge). It
can indeed appear in the midst of a photonic band, thanks
Ryan Integlia received his M.S. in 2008 jointly from Rutgers Civil and Environmental
Engineering Department and Electrical and Computer Engineering Department studying the
subject of wireless sensor networks. He is currently researching the subjects of the slow light
effect and the superprism effect. He received his bachelor’s degree in 2001 from Rutgers
University’s Electrical and computer Engineering Department, graduating with high honors.
Ryan Integlia was also a recipient of the NSF’s IGERT Traineeship, the Center for Advanced
Infrastructure and Technology Fellowship, among other honors.
Computer Engineering of Rutgers University and working in the group of Professor Wei
Jiang. His research involves silicon photonic crystal structures and devices, with a focus
on the simulation and characterization. In 2005, Weiwei graduated from Nanjing University
with a B.S. in Condensed Matter Physics.
Jun Tan received his B.S. in 2005 from Nanjing University, P. R. China, and his M.S. in
2008 from Shanghai Institute of Technical Physics, Chinese Academy of Sciences. Currently, he is pursuing his Ph.D. in the Department of Electrical and Computer Engineering,
Rutgers University. In Dr. Jiang’s group, his research is focused on device fabrication and
characterization.
Wei Jiang received the B.S. degree in physics from Nanjing University, Nanjing, China,
in 1996, and the M.A. degree in physics and the Ph.D. degree in electrical and computer
engineering from the University of Texas, Austin, in 2000 and 2005, respectively. He held
research positions with Omega Optics, Inc., Austin, Texas from 2004 to 2007. Since September 2007, he has been an assistant professor in the department of electrical and computer
engineering of Rutgers University, Piscataway, NJ. His doctoral research made a contribution to the fundamental understanding of the wave coupling, transmission, and refraction
at a surface of a periodic lattice. At Omega Optics, he led a research project to the successful demonstration of the first high-speed photonic crystal modulator. He also recognized
a scaling law for the current density of high speed silicon electro-optic devices. His current research interests include photonic crystals, silicon photonics, optical interconnects,
and beam steering. He received Ben Streetman Prize of the University of Texas at Austin
in 2005.
J. Nanosci. Nanotechnol. 10, 1596–1605, 2010
1597
REVIEW
Delivered by Ingenta to:
Rice University, Fondren Library
IP his
: 128.42.154.22
Weiwei Song received
M.S. in 2008 from the Department of Physics of Nanjing UniverSat,
19
Marpursuing
2011 15:22:48
sity, China. He is currently
his Ph.D. degree in the Department of Electrical and
REVIEW
Longitudinal and Angular Dispersions in Photonic Crystals
Integlia et al.
to the high symmetry of a photonic crystal structure. The
typically occurs along certain high symmetry axes of a
photonic crystal.
presence of a photonic bandgap is not a necessary condi(2) For a photonic crystal waveguide (PCW) composed of
tion for the superprism effect.
a line-defect, generally the waveguide is already aligned
In this paper, we will review some of our recent thewith a high symmetry axis of the photonic crystal latoretical and experimental works concerning these dispertice (for example, the K axis of a hexagonal lattice9 ).
sive effects. In these works, our emphasis was placed on
The original lattice periodicity remains along the lonfinding the general, quantitative physical laws governing
gitudinal direction of the waveguide. This results in a
these effects. For example, in the first superprism experione-dimensional (1D) photonic band structure with a maxment, it was found that the photonic crystal could enhance
imum or minimum (or other types of extrema) at the 1D
the angular dispersion or sensitivity by 500 times;2 later
Brillouin zone (BZ) boundary = /a, where is the
numerical simulations and experimental works reported
propagation constant of the photonic crystal waveguide in
varying enhancement factors.3–7 However, a rigorous, comquestion. Since the dispersion relation is generally a
pact mathematical form of the physical law that can
smooth function, an extremum ensures vg = d/dk = 0 at
express this enhancement factor in terms of the photonic
the BZ boundary. Therefore, the periodicity along the lonband parameters is missing. As such, while we can easily
gitudinal direction dictates that a vanishing group velocity
obtain an instance of high sensitivity structures, we can
must exist in such a photonic crystal waveguide.
not systematically predict the trend of such an effect and
we do not know whether there is a quantitative upper limit
of the enhancement factor. Our works were devoted
to elu-by Ingenta
2.2. Applications
of the Slow Light Effect
Delivered
to:
cidating such issues.
Rice University, Fondren Library
The slow group velocity of light renders the phase shift
While the slow light effect is obviously due toIP
the: 128.42.154.22
disin a photonic crystal structure more sensitive to refractive
persion of a photonic crystal along the direction
of
light
Sat, 19 Mar 2011
15:22:48
index changes.1 Generally, as the refractive index changes,
propagation, namely the longitudinal direction; the superthe dispersion relation of a photonic crystal or a PCW will
prism effect, apparently related to angular dispersion, will
be shifted by a certain amount = n/n along
be shown to have an elusive dependence on the longithe frequency axis. Here is the frequency of light, n is
tudinal dispersion as well. Some subtle connections and
the refractive index, and is a factor typically on the
distinctions between the slow-light effect and the superorder of unity. In many cases, we are interested in a small
prism effect will be revealed through our physical analfrequency range where can be regarded as a constant. In
ysis. This allows us to examine these two effects under
the
case of a PCW, the factor can be interpreted as the
a common theoretical framework based on photonic crysfraction
of the mode-energy in the waveguide core region.
tal dispersion function, k. An example will be used to
For
a
given
wavelength, the propagation constant changes
illustrate the value of this synergistic theoretical perspecas
=
/v
g . Therefore, the phase shift induced by a
tive on the slow-light effect and superprism effect, two
refractive
index
change of n is given by1
seemingly distinctive phenomena in photonic crystals. Dispersive effects are frequently accompanied by high loss
and/or narrow bandwidth. We will discuss these issues
for both longitudinal and angular dispersions. Similar to
the bandwidth-delay product for the longitudinal dispersion, we will introduce a simple, yet fundamental, limit
that governs the bandwidth and sensitivities of the angular
dispersion.
2. THE SLOW LIGHT EFFECT AND
THE LONGITUDINAL DISPERSION
2.1. The Origin of the Slow Light
Effect in Photonic Crystals
The group velocity of light can be slowed down in various
types of photonic crystal structures, especially when the
wavelength of light approaches a bandedge. Two common
cases shall be considered.
(1) For a “bulk” photonic crystal, such a bandedge typically appears around some high symmetry points in reciprocal space.8 As such, in real space, slow light propagation
1598
ng n 2L
(1)
n
where ng = c/vg is the group index. Evidently, a slow
group velocity (or a high ng ) enhances the phase shift
significantly.
To exploit such a significant slow-light enhancement,
a number of physical mechanisms10–14 have been employed
to change the refractive index and actively tune the
phase shift in a photonic crystal waveguide. Here we
briefly review our works on thermo-optic and electrooptic tuning of the phase shift for optical modulation
and switching applications. Many common semiconductor materials, such as silicon and GaAs, have appreciable
thermo-optic coefficients (dn/dT > 10−4 /K). As such, they
are suitable for making thermo-optically tunable slowlight photonic crystal devices. In these devices, thermal
expansion also contributes to the tuning of the phase
shift. In many cases, these two effects add upon each
other to produce a larger phase shift. We have demonstrated thermo-optic tuning in a photonic crystal waveguide Mach-Zehnder interferometer with an interaction
= L =
J. Nanosci. Nanotechnol. 10, 1596–1605, 2010
Integlia et al.
Longitudinal and Angular Dispersions in Photonic Crystals
(a)
(b)
(c)
(d)
Fig. 1.
ferometer with one active tuning arm on a silicon-on-insulator wafer. (b) cross-sectional schematic of an active arm with thermo-optic tuning (inset:
micrograph of a thermo-optic device—top view); (c) cross-sectional schematic of an active arm with an embedded silicon p–i–n diode for electro-optic
modulation (inset: micrograph of an electro-optic device—top view); (d) cross-sectional schematic of an active arm with an embedded silicon MOS
capacitor for electro-optic modulation.
n = −88 × 10−22 Ne + 85 × 10−18 Nh 08 (2)
where Ne and Nh are electron and hole concentration changes, respectively. A refractive index change
up to n ∼ 10−3 can be obtained with Ne = Nh ∼
3 × 1017 cm−3 . We demonstrated the first high-speed photonic crystal waveguide modulator on silicon in 2007.11
A schematic of the active arm of the device is shown in
Figure 1(c). The device was made on a silicon-on-insulator
wafer through a series of micro- and nano-fabrication
processes, including e-beam lithography, photolithography,
dry and wet etching, ion implantation, and metal liftoff. The device had a measured modulation bandwidth
in excess of 1 GHz, with the lowest driving voltage for
high-speed silicon modulators at the time of publication. It
J. Nanosci. Nanotechnol. 10, 1596–1605, 2010
should be noted that the introduction of air holes does not
significantly increase the electrical resistance of silicon.
In Figure 2, we plot the electrical resistance of a photonic
crystal waveguide made in a silicon slab for varying air
hole sizes and varying number of rows of air holes. In the
simulation, the electrical contact pads were assumed to be
placed at a fixed separation about 10 m along the sides
of the photonic crystal waveguide. The contact resistance
Resistance/R0
length of 80 m.10 A photonic crystal waveguide MachZehnder interferometer with one active arm is schematically illustrated in Figure 1(a). A close-up view of the
arm with thermo-optic tuning is shown in Figure 1(b). The
device was patterned on a silicon-on-insulator (SOI) wafer
using a combination of e-beam nanolithography and photolithography. The switching rise time and fall time were
measured to be 19.6 s and 11.4 s, respectively.
Alternatively, we can electro-optically change the
refractive index by carrier injection into silicon. Soref and
Bennett found the following relation between the refractive
index of silicon and the carrier concentrations for wavelengths near 1.55 m15
2.6
0.05
2.4
0.1
2.2
0.2
0.3
2
0.4
1.8
1.6
1.4
1.2
1
0
5
10
15
20
N
Fig. 2. Electrical resistances for a photonic crystal waveguide in a silicon slab. The horizontal axis indicates the total number of rows of air
holes between two electrodes separated by 10 m. The different curves
correspond to different values of r/a (005 ∼ 04), where r is the radius
of the air holes and a is the lattice constant (∼400 nm). The resistance
values are normalized by the original slab resistance, R0 .
1599
REVIEW
Delivered by Ingenta to:
Rice University, Fondren Library
IP : 128.42.154.22
Sat,
19 Mar 2011 (a)
15:22:48
Photonic crystal waveguide based Mach-Zehnder modulators/switches.
A generic photonic crystal waveguide based Mach-Zehnder inter-
Longitudinal and Angular Dispersions in Photonic Crystals
Integlia et al.
can generally be neglected. Evidently, the electrical resistance may increase about 2.6 times for a large hole radius
and for a large number of holes, but the value generally
remains on the same order of magnitude as the original
silicon slab. In our experiments, we also did not observe
an order-of-magnitude change of the resistance after the
photonic crystal structures were etched in a silicon slab.
The resistance values in Figure 2 were computed by 2D
finite element method for the DC case.
In our 2007 work, we also derived the AC injection
current density for a Si modulator based on a forwardbiased p–i–n diode11 12
REVIEW
j = 2qwi Nf
(3)
loss to increase significantly. To understand the slow light
effect, a close examination of the accompanying optical
loss is warranted.
The total insertion loss of a photonic crystal waveguide
is given by
LossdB = 10 log10 C1 + 10 log10 C2 − L
(4)
where C1 and C2 are the coupling effeciencies at the input
and output end of the photonic crystal waveguide, and is the propagation loss coefficient (in the unit of dB/cm).
Note that in our definition, 0 < C1 < 1, 0 < C2 < 1, > 0.
The loss coefficient can be expressed as
= 1 ng + 2 n2g + · · ·
(5)
where the first term can be attributed to absorption and
where wi is the intrinsic region width of the p–i–n
out-of-plane scattering by random imperfections in the
diode, and f is the modulation frequency. Combining Eqs.
photonic crystal waveguide, and the second term can be
(2) and (3), we obtained a minimum AC current density
attributed primarily to back-scattering (due to random
of 104 A/cm2 for high speed (>1 GHz) modulation in a
imperfections) into the reverse propagating mode with an
Delivered
to:
typical SOI waveguide. In addition, we showed
that dueby Ingenta
identical group index.
Rice
University,
Fondren
Library
to the non-ideal diode I–V relation I ∼ expqV /2kB T at
Several works21 22 have discussed the scaling of scatIP
:
128.42.154.22
high injection, it is possible to limit (or “lock”) the injected
tering loss theoretically. Here we give a proof of Eq. (5)
17 19−3
Sat,
carrier concentrations to around N ∼ 3 × 10
cm Mar
for 2011
that15:22:48
does not invoke the detailed solution of the waveguide
a diode with proper doping levels and under normal forequation. We consider the scattering process due to ranward bias conditions. This ensures that the silicon modudom imperfections in a photonic crystal waveguide. For
lator naturally works under the most desired electro-optic
any scattering event of interest, the initial state must be
state. In a follow-up work, we predicted that an RF power
a guided mode, which we assume has a propagation conconsumption of less than 50 mW is possible for 10 GHz
stant . The final state can be a guided mode or a radiation
silicon modulators.12 Subsequently, IBM demonstrated a
mode. For a line-defect waveguide formed in a photonic
10 GHz silicon modulator with 50 mW RF driving power
crystal slab, the radiation modes propagate out of the
in the forward bias mode;16 MIT Lincoln Laboratory also
plane. Assume the scattering amplitude between an arbireported similar power consumption for 10 GHz silicon
trary initial state and a final state k is Tk . In any physical
situation, the incoming light is always a wave-packet with
modulators.17 These results affirmed the value of Eq. (3)
a continuous distribution of values, although often the in designing high speed silicon modulators.
values are within a narrow range centered around 0 . The
We also developed a metal-oxide-semiconductor (MOS)
scattering coefficient for such a wave-packet is roughly
type photonic crystal waveguide modulator, as illustrated in
Figure 1(d). A MOS capacitor can be embedded into a slot
∼
d Tk 2 f − i S
eff
photonic crystal waveguide, where the slot is filled with
k
oxide.18 In such a waveguide, there exist two enhancement
effects: the slow light effect, and the field boost inside the
∼ d dk3 Tk 2 f − i low-dielectric slot due to the continuity of surface-normal
displacement vector component. Our most recent results
+ CB d d T 2 f − i (6)
show that such a configuration can help reduce the power
consumption of a silicon MOS modulator.19
where k represents a final radiation mode, represents
a final guided mode, and CB is a constant. The factor
As mentioned earlier, the slow light effect can also occur
f − i ensures that the frequencies of the initial and
in a “bulk” photonic crystal without intentionally introducfinal states are the same (energy conservation). The 1/vg ing line-defects. Such a configuration has been explored in
factor will arise naturally from each integration of an arbiphotonic crystal slabs made of conventional electro-optic
trary function with respect to or materials such as LiNbO3 .20
d 1 =
f d = f d
f d (7)
2.3. Loss-Limited Effect
d v
g
The practical application of the slow-light effect is primarily limited by optical loss. For most practical applications,
group velocity values of 100 or less have been currently
considered. Further slowing down light causes the optical
1600
Therefore, we find
Seff ∼ T1 /vg + T2 /vg vg = T1 /vg + T2 /vg2 (8)
J. Nanosci. Nanotechnol. 10, 1596–1605, 2010
Integlia et al.
Longitudinal and Angular Dispersions in Photonic Crystals
3. THE SUPERPRISM EFFECT AND
THE ANGULAR DISPERSION
When a light beam is incident upon a photonic crystal surface, the refraction angle inside the photonic crystal could
be 500 times more sensitive to the wavelength perturbation
J. Nanosci. Nanotechnol. 10, 1596–1605, 2010
kx ky = 0
This equation, which gives ky as an implicit function of
kx , can be reformulated into an explicit form
ky = kx (10)
1601
REVIEW
than in a conventional medium.2 This so called superprism
where T1 and T2 are some constants. The second line in
effect is a manifestation of the strong angular dispersion
Eq. (8) follows from = −, according to energy conof a photonic crystal.
servation in typical photonic crystal waveguides. Note that
Significant progress has been made in investigata similar factor 1/vg k may arise from the integration
ing the superprism effect in the first ten years after
dk3 as well. However, the group velocity, vg k , of a
its initial discovery. A number of experiments have
radiation mode never vanishes. Therefore, this factor has
demonstrated the potential of the superprism effect in
no significance here and is absorbed into T1 . Thus, Eq. (5)
wavelength division multiplexing, beam steering, and
is proved.
sensing applications.3–5 7 28 29 Nonetheless, many fundaThe above derivation clearly shows that the out-of-plane
mental questions remained unanswered: (1) How to express
scattering has only one ng factor because only the initial
the angular dispersion or angular sensitivity of a photonic
state is a slow-light state, whereas the backscattering procrystal in terms of basic parameters of a photonic band
cess has a n2g factor because both the initial and final states
structure as we have seen in the slow-light effect? (2) Is
are slow-light states. We would be tempted to assume
there an ultimate limit of the angular sensitivity of a phothat the value of the second integral in Eq. (6) is much
tonic crystal? (3) If there is a limit, what are the limiting
smaller than that of the first, because there are a large numfactors?
ber of radiation modes that satisfy the energy conservaTo build a foundation for the solution of these problems,
tion whereas only one backward guided mode does so. In
we
developed a rigorous theoretical framework to comother words, the total “scattering cross-section” of all radipute
the transmission and reflection coefficients for refracation modes could be much larger than that Delivered
of the back-by Ingenta
to: a photonic crystal surface in a 2005 work.30
tion
across
ward guided mode. However, a general,Rice
rigorous
proof is Fondren Library
University,
A parallel work was reported by a group at the Univerneeded.
IP : 128.42.154.22
sity of Toronto in the same period.31 32 Subsequently, we
Experimentally, quantitative evaluation ofSat,
these19scaling
Mar 2011
15:22:48
developed
the first theory to systematically address the
laws has been elusively difficult and the reported loss
aforementioned
general questions in a 2008 work.33 While
23–26
−1/2
−2
dependences
vary between vg
and vg , as discussed
the key parameter for tuning the longitudinal dispersion
in Ref. [27]. It should be clarified that because the scatteris the group velocity, a new parameter, the curvature of
ing events occur statistically uniformly over a given disthe dispersion surface, must be introduced to describe the
tance, the scattering loss should have the general form
angular dispersion. Here the dispersion surface refers to
−
−
= −1/L log Tprop ∼ vg , not Tprop ∼ vg . On the other
the constant-frequency surface in reciprocal space. This
hand, the coupling loss coefficients should have the form
curvature can be calculated directly from the dispersion
C1 , C2 ∼ vg , where = 1 for a normally (abruptly) termirelation k, which also represents the photonic band
nated photonic crystal waveguide (see Eq. (15) and related
structure. With this theory, we can now directly express the
discussions in Section 3.1). Therefore, for a normally tersensitivity of the superprism effect in terms of k and
minated photonic crystal waveguide, the total insertion loss
explore the fundamental limiting factors of the superprism
is given by
effect.
2
In this section, we will briefly introduce our theoretLoss(dB) ≈ B0 − 10 log10 ng − 1 ng + 2 ng L (9)
ical framework for the superprism effect. Then we will
where B0 is a constant that gives the “baseline” insertion
separately discuss two types of superprism effects: the
loss.
slow-light induced angular dispersion effect and the “pure”
Here we list several issues in characterizing the optiangular dispersion effect. We will see some critical scalcal loss in the slow light regime against Eqs. (4) and (5),
ings and limiting factors for these two types of effects.
or (9): (1) the unknown relative magnitudes of 1 and
2 ; (2) the difficulties of separating the coupling loss and
3.1. Dispersion Surface Curvature and
propagation loss in experiments; (3) the proper application
Angular Sensitivity
of the scaling laws of the propagation and coupling losses.
For convenience, we will consider a 2D photonic crystal
It is our feeling that systematic and careful experimental
and the TM mode of light (magnetic field in the plane).
studies over a wide range of waveguide parameters must be
However, our discussion is applicable to other cases. First,
performed before a conclusive statement can be put forth
we introduce the concept of the dispersion surface curvaregarding the optical loss in the slow-light regime.
ture. The dispersion surface at an arbitrary circular frequency, 0 = 2c/0 , can be described by
Longitudinal and Angular Dispersions in Photonic Crystals
where we have omitted the parameter 0 . Generally, the
curvature of a curve on the kx − ky plane is given by33
dky 2 3/2
d 2 ky
1
+
≡
(11)
dkx2
dkx
Integlia et al.
However, it turns out that the optical transmission across
the photonic crystal surface is given by30
T ∝ t2 em vg cos (15)
where em is the cell-averaged mode energy density and
t is the complex coupling amplitude of the mode in question. Evidently, while larger values of ng or 1/ cos will
help enhance the angular dispersion and angular sensitivities, they will also inevitably suppress the optical transmission. Therefore, this type of enhancement will eventually
be limited by the maximal optical loss that can be tolerated in a particular application. Note that light propagation
inside the photonic crystal may further induce significant
1
(12)
nI sin = kx0 + u sin − v cos optical loss in the slow-light regime, in addition to the
c
surface transmission loss given in Eq. (15). The total loss
where is the incident angle, nI is the refractive index
may be handled by a theory similar to the discussion folof the incident medium, is the direction of the group
lowing Eq. (5). On the other hand, enhancing the angular
velocity (i.e., the beam direction) with respect to the sursensitivity through large values will not entail high optiface normal, u and v are local Cartesian coordinates
Deliveredinby Ingenta
cal loss, to:
therefore is highly preferred in a wide range of
the disper- Fondren Library
the neighborhood of a point, k0 = kx0 kRice
y0 , onUniversity,
applications.
sion surface. Here the local u-axis is parallel to the
IP group
: 128.42.154.22
velocity, and v-axis is tangential to the dispersion
surface.
Sat, 19
Mar 2011
3.2.15:22:48
Slow-Light Induced Strong Angular Dispersion
It can be shown from Eq. (12) that the sensitivity of
the beam angle to wavelength change (i.e., the angular
Although Eqs. (13) and (14) obviously indicate a linear
dispersion) is given by33
dependence of the angular dispersion and angular sensitiv ity on the group index, a casual numeric analysis without
d 2
=
(13)
the knowledge of Eqs. (13) and (14) would yield a decep d 2 cos ng sin − nI sin tively stronger enhancement in the slow-light regime. ConIn addition, the sensitivity to refractive index perturbation
sider the following approximate mode dispersion near a
is given by
photonic bandedge
d sin = 0 − bkx2 + ky2 (16)
(14)
ng dn = − c cos n
REVIEW
where the derivatives can be calculated from the function
given in Eq. (10).
To derive the relation between the curvature of the dispersion surface and the angular dispersion of the photonic crystal, we consider the conservation of tangential
wavevector component across the photonic crystal surface
for a configuration depicted in Figure 3
a
a
k
In Eqs. (13) and (14), the quantities, , 1/ cos (note
tan = sin / cos ), and ng are the only three factors that
can grow several orders of magnitude compared to a conventional medium, which can result in significant enhancements of angular dispersion/sensitivities as observed in
prior superprism experiments.
where 0 is the bandedge frequency and b is a constant.
The two key parameters for the slow-light effect and
the superprism effect have the following frequency dependence near the bandedge
ng = c/2 b − 0 (17)
= b/ − 0 Evidently, the group index and the curvature diverge at the
same rate as approaches the bandedge 0 . It is straightforward to show
d
1
∼
d − 0
d
1
∼
dna
− 0
Fig. 3.
1602
Schematic of a simple configuration for the superprism effect.
(18)
A straightforward numeric calculation should find that
near the bandedge, when the group index increases
10 times, the angular dispersion and angular sensitivities
would increase 100 times. Thus, if we did not have the
knowledge of the analytic form of Eqs. (13) and (14),
we would be tempted to conclude d/d ∝ n2g and
J. Nanosci. Nanotechnol. 10, 1596–1605, 2010
Integlia et al.
Longitudinal and Angular Dispersions in Photonic Crystals
For example, a sensitivity of 100
/nm can be sustained
over a bandwidth less than 1.8 nm, and a sensitivity of
1000
/nm can be sustained over a bandwidth less than
0.18 nm. For a laser having 1pm linewidth (∼100 MHz),
these two cases may allow for 1800 and 180 wavelength
tuning points, respectively, which are reasonable for practical applications. These performance parameters are possible with the existing laser technologies. Lastly, we should
keep in mind that not all applications require a wide bandwidth. There are some applications that can benefit from a
high angular dispersion/sensitivity in a narrow bandwidth.
3.3. “Pure” Angular Dispersion Effect:
A detailed theoretical analysis based on group theory
Bandwidth Limited
shows that such types of “pure” angular dispersion origWe have found that the dispersion surface can exhibit
inates fundamentally from symmetry induced degeneracy
ultra-high curvature values in the vicinity of certain highin photonic band structures. Furthermore, such types of
symmetry points in the Brillouin zone (BZ) without the
symmetry-induced enhancement of angular sensitivities
presence of the slow-light effect.33 An example is the K
can only occur in 2D and 3D photonic crystals, but not in
point of a triangular lattice. Some photonic bands have a
1D photonic crystals. Discovering such a crystal-symmetry
double degeneracy at this point. Approaching such a douinduced effect
Delivered by Ingenta
to: exemplifies the effectiveness of utilizing the
bly degenerate point, the curvature of the dispersion sursolid-state
physics paradigm to shed new light on the study
Rice University, Fondren Library
face tends to infinity whereas the group velocity approaches
of
periodic
which is the central theme
IP : 128.42.154.22 dielectric structures,
34
a non-zero constant. Therefore, we can enjoy the benefit
of
photonic
crystal
research.
In
passing, we note that the
19 Mar
15:22:48
of the high dispersion and high sensitivity,Sat,
according
to 2011
rigorous theory for computing the transmission of each
Eqs. (13) and (14), without worrying about the high optical
photonic crystal mode30 has been extended to gratings,35
loss that would occur in the slow-light case.
which can be regarded as monolayer photonic crystals, and
In this case, the scalings of the group index and curva3D photonic crystals.36 Formulas similar to Eq. (15) can
ture differ from the slow-light induced superprism effect,
be used to assess the optical loss in the 1D and 3D cases
as well.
ng → const
Before we conclude this section, we would like to men(19)
→ 1/ − 0 tion that there are a number of applications for the superprism effect. Different applications may have additional
where 0 is the frequency of the doubly degenerate point,
limiting factors specific to themselves. For example, for
where the curvature is singular. Interestingly, according
the widely studied wavelength demultiplexer application,
to Eqs. (13) and (14), the overall scaling of the angular
the beam width divergence is an additional limiting factor
dispersion and angular sensitivities with frequency perturspecific to this application. Fortunately, recent works7 37
bation, = − 0 , in this case remain the same as in
have demonstrated a promising method of overcoming this
Eq. (18). However, the optical loss is almost constant in
limit. Note that this factor is important only for those
this case, independent of the angular dispersion values.
applications that require narrow beam width (or spot size)
With the optical loss no longer being a limiting facat the receiving end of the superprism. If a sufficiently
tor, the angular dispersion and angular sensitivities can
large beam width (hence a small lateral spread of the
now be enhanced to much higher values until it encounters
wavevector) is used, then this factor is less important.
some other limits. Now we present a fundamental angular
sensitivity-bandwidth limit similar to the bandwidth-delay
limit for the longitudinal dispersion. Assume the angular
4. LONGITUDINAL DISPERSION VERSUS
dispersion is maintained at a value above d/d0 over
ANGULAR DISPERSION: A DIRECT
a spectral range of BW (unit: nm). Because the maximum
COMPARISON
beam steering range can not exceed 180
(no backward
propagation is physically possible), we find
The analyses in two preceding sections show that the two
key parameters, vg and , of the slow-light effect and the
d/d0 × BW < 180
(20)
superprism effect are entirely determined by the dispersion function kx ky . In other words, we may say that
This relation is the angular dispersion correspondent of
these
two effects are manifestations of the longitudinal and
the bandwidth-delay product.
angular
characteristics of the dispersion function. In this
Therefore, the maximum bandwidth for a sustained high
section,
we will unveil some further connections between
sensitivity d/d0 is
these two effects through an application example. Here
we choose the beam steering application, which intends
BW < 180
/d/d0
(21)
d/dna ∝ n2g in this particular case. However, the above
analysis shows that the proper dependence should be
d/d ∝ ng and d/dna ∝ ng because ng and diverge
at the same rate. Although the above analysis is based
on a specific photonic band structure described by Eq.
(16), a general asymptotic analysis, given in Section IIA of
Ref. [33], indicates that such a slow-light induced strong
angular dispersion due to equal diverging rate of ng and exists in a wider range of slow-light scenarios.
1603
REVIEW
J. Nanosci. Nanotechnol. 10, 1596–1605, 2010
Longitudinal and Angular Dispersions in Photonic Crystals
Integlia et al.
REVIEW
To simplify the comparison with the superprism effect,
we assume = a , and n = na . Then the ratio of the beam
angle changes in the two cases is given by
SP n
= sin 2d g SP (26)
L ng SL SL
where SP denotes the superprism effect, and SL denotes the
slow-light effect. In most optical phase arrays, the waveguide spacing, d, is on the order of the wavelength, . If
we assume sin > 01 and note the 2 factor in Eq. (26),
these two factors have an overall contribution on the order
Fig. 4. Schematic of a one-dimensional optical phase array composed
of unity. Therefore, the difference between SP and SL
of photonic crystal waveguide phase shifters.
primarily comes from the terms, (/L) and (ng SP /ng SL .
to manipulate the direction of a laser beam by changing
Note the curvature also has the dimension of length.
the refractive indices of materials in certain device strucAs a numeric example, we consider a silicon photonic
tures. Two approaches are considered: (1) an optical phase
crystal waveguide with ng SL ∼ 30, and n ∼ 10−3 . This
array38 (OPA) composed of a 1D array of slow-light phogenerally requires a waveguide length on the order of
tonic crystal waveguides as shown in Figure 4; (2) a superL = 100 m to achieve a phase shift of 2 at = 155 m.
prism composed of a 2D photonic crystal. Note
that toby Ingenta
Note thatto:
to extend the waveguide length far beyond this
Delivered
steer the output beam in free space, the superprism
device Fondren
value to Library
achieve larger will cause multiple side-lobes
Rice University,
can not have a flat output surface parallel to the input
surand is not desired for many practical OPA beam steering
IP : 128.42.154.22
face. In this example, we assume the outputSat,
surface
has
a 2011
applications.
19
Mar
15:22:48On the other hand, it is relatively easy to get
semi-circular configuration for simplicity.3
100 m in a properly designed photonic crystal superFirst, we re-write Eqs. (1) and (14) in the following
prism. For example, we find 2/ > 103 for a hexagonal
forms:
photonic crystal with ng ∼ 7.33 In this particular example,
ng 2
the ratio in Eq. (26) is around 3 sin . For moderate val
= L
n
(22a)
ues, the beam steering efficiencies due to the two effects are
n roughly on the same order of magnitude. A more detailed
ng 2
investigation is beyond the scope of this work.
= tan (22b)
na n a
The second equation follows from
= a
na k
na
5. CONCLUSION
(23)
where a measures the fraction of mode energy located
in the medium a. Note this relation, Eq. (23), is essentially the same as that used in an early derivation of
the slow-light enhancement of the phase sensitivity in a
photonic crystal waveguide.1 Therefore, it is not surprising to see that Eqs. (22a) and (22b) share similar factors
/n2/, which come from /n. In the case of a
photonic crystal waveguide, n refers to the refractive index
of the waveguide core, and denotes the fraction of the
mode-energy in the core region.1
More interestingly, a direct comparison of the steering angle sensitivity between the two approaches can be
obtained from Eqs. (22a) and (22b). For an optical phase
array, the far-field beam angle relates to the phase difference, , between adjacent array elements as follows
(24)
2 d
Therefore, the beam steering sensitivity for a slow-light
based optical phase array is given as
1 ng L
=
n
(25)
cos n d
sin =
1604
In this paper, we have discussed the slow-light effect and
the superprism effect in one synergistic perspective based
on dispersions. We give rigorous analysis of the phase shift
sensitivities, angular sensitivities, optical loss, and bandwidth for these effects in fairly general situations. Particularly, a rigorous proof of the scaling of the scattering
loss in the slow light regime is given. The general relations and trends that we have obtained regarding these
important parameters provide an important guide for further experiments to verify these effects and to explore new
applications.
Acknowledgments: This work is supported by AFOSR
MURI (Grant No. FA9550-08-1-0394 supervised by
Gernot Pomrenke) and Air Force Research Laboratory
(Grant No. FA8650-06-C-5403 supervised by Robert
L. Nelson).
References and Notes
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J. D. Joannopoulos, J. Opt. Soc. Am. B 19, 2052 (2002).
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T. Sato, and S. Kawakami, Phys. Rev. B 58, 10096 (1998).
J. Nanosci. Nanotechnol. 10, 1596–1605, 2010
Integlia et al.
Longitudinal and Angular Dispersions in Photonic Crystals
Received: 1 February 2009. Accepted: 31 March 2009.
J. Nanosci. Nanotechnol. 10, 1596–1605, 2010
1605
REVIEW
3. L. J. Wu, M. Mazilu, T. Karle, and T. F. Krauss, IEEE J. Quantum
21. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, Phys. Rev. Lett.
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M. Soljacic, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos,
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Proceedings of 2003 5th International Conference on Transparent
Fedeli, Opt. Express 12, 5690 (2004).
Optical Networks (2003), p. 103.
6. J. J. Baumberg, N. M. B. Perney, M. C. Netti, M. D. C. Charlton,
23. E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, and
M. Zoorob, and G. J. Parker, Appl. Phys. Lett. 85, 354 (2004).
L. Ramunno, Phys. Rev. B 72, 161318 (2005).
7. B. Momeni, J. D. Huang, M. Soltani, M. Askari, S. Mohammadi,
24. L. O’Faolain, T. P. White, D. O’Brien, X. Yuan, M. D. Settle, and
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PHYSICAL REVIEW B 77, 075104 共2008兲
Symmetry-induced singularities of the dispersion surface curvature and high sensitivities
of a photonic crystal
Wei Jiang1,2,* and Ray T. Chen1
1Department
of Electrical and Computer Engineering and Microelectronics Research Center, University of Texas,
Austin, Texas 78758, USA
2Department of Electrical and Computer Engineering and Institute for Advanced Materials, Devices, and Nanotechnology,
Rutgers University, Piscataway, New Jersey 08854, USA
共Received 11 October 2007; revised manuscript received 17 December 2007; published 6 February 2008兲
We rigorously analyze the dispersion function and the curvature of the dispersion surface of a photonic
crystal to explore the fundamental limit of its angular sensitivities. With insight gained from group theory, we
find that symmetry induced degeneracy gives rise to a singular dispersion surface curvature and a nonvanishing
group velocity simultaneously. Near such a singularity, high angular sensitivities can be achieved at low optical
loss. This phenomenon exists generally in most common two-dimensional and three-dimensional photonic
crystal lattices, although it occurs only for certain photonic bands as dictated by symmetry. This symmetryinduced effect is absent in one-dimensional crystals. Rigorous formulas of the sensitivities of the light beam
directions to wavelength and refractive index changes are derived. Individual contributions of the dispersion
surface curvature and group velocity to these sensitivities are separated. In the absence of the Van Hove
d␪
singularity, a singular dispersion surface curvature gives rise to ultrahigh dispersion 兩 d␭ 兩 ⬎ 103 deg/ nm and
d␪
4
refractive index sensitivity 兩 dna 兩 ⬎ 10 deg without compromising optical transmission. The angular dispersion
value is significantly larger than those previously reported for the superprism effect and is not due to slow
group velocity. We also discuss how various parameters intrinsic and extrinsic to a photonic crystal may
suppress or enhance the angular sensitivities according to the rigorous formulas we obtain.
DOI: 10.1103/PhysRevB.77.075104
PACS number共s兲: 42.70.Qs, 07.07.Df, 42.65.⫺k, 61.50.Ah
I. INTRODUCTION
Photonic crystals provide high optical sensitivities not
achievable in conventional media. The high dispersion and
slow group velocity of photonic crystal waveguides help significantly shorten the interaction length for optical
modulation.1–5 Critical to this advance is a clear-cut expression that gives the enhancement of the nonlinear phase sensitivity of a photonic crystal waveguide mode in terms of the
group velocity.1 On the other hand, the high anisotropy and
angular dispersion of photonic crystals were found to cause
beam directions to have 500-fold higher sensitivities to
wavelength variation, which was named the superprism
effect.5 Such high wavelength sensitivities are frequently accompanied by high sensitivities to refractive index
perturbations.6,7 These high sensitivities have aroused wide
interest for potential applications in fiber optic communication, sensing, and nonlinear optics.5–17 However, a general,
quantitative relation between the anisotropy and these sensitivities of photonic crystals is needed to determine the ultimate limits of the sensitivities before we can fully uncover
the potential of these sensitive effects for a wide range of
important applications. For example, the sensitivity of the
superprism effect is often enhanced near a band edge at the
cost of a low optical transmission due to the slow group
velocity. Whether this sets a fundamental limit of the maximum achievable sensitivity for practical applications is an
interesting question to explore.
In this work, we rigorously show that high angular sensitivities to wavelength and refractive index perturbations can
be achieved in the vicinity of a singular dispersion surface
1098-0121/2008/77共7兲/075104共8兲
curvature or the vicinity of a vanishing group velocity. Explicit analytical expressions are given to separate the effects
of the curvature and group velocity. Of particular interest is
the case where a singular curvature appears together with a
nonvanishing group velocity 共and therefore high transmission兲 owing to symmetry-induced mode degeneracy. Such a
case is predicted prevalent in high symmetry twodimensional 共2D兲 and 3D photonic crystal lattices, but absent
in 1D photonic crystals. Van Hove singularities, the singularities of the density of states due to a vanishing group velocity, have been proven to be an insightful concept for understanding some interesting effects in photonic crystals.18
The singularities of the dispersion surface curvature discussed here may occur at nonvanishing group velocities,
where Van Hove singularities are absent. Therefore this type
of singularity can lead us to some different effects or new
functional regimes of photonic crystals.
This paper is organized as follows. In Sec. II A, we will
first examine some analytic properties of the photonic crystal
dispersion function. This allows us to identify the correlation
and decorrelation of a singular curvature and a vanishing
group velocity, which highly depends on lattice symmetry
and mode degeneracy. An example based on an approximate
model is presented in Sec. II B. In Sec. III, we give key
rigorous formulas of the angular sensitivities of photonic
crystals. An example is used to illustrate the individual control of the curvature and group velocity so as to achieve large
sensitivities at low optical loss. Section IV discusses the contributions of intrinsic and extrinsic parameters to photonic
crystal sensitivities, the effect of dimensionality and lattice
types, and the difference from Van Hove singularities.
075104-1
©2008 The American Physical Society
PHYSICAL REVIEW B 77, 075104 共2008兲
WEI JIANG AND RAY T. CHEN
II. CURVATURE OF THE DISPERSION SURFACE AND
GROUP VELOCITY
␨=
A. Some key analytic properties and symmetry considerations
For simplicity, we illustrate our ideas with the TM polarization 共magnetic field in the plane兲 of a 2D photonic crystal.
The field equation, according to Bloch theorem, can be written as
− 共k + G兲2E共G兲 + ␻2 兺 ␧共G − G⬘兲E共G⬘兲 = 0,
共2兲
j=0
where the v axis of this local coordinate system is tangent to
the dispersion contour at an arbitrary k0, i.e., du / dv = 0; and
␥ is the angle between the u and x axes. Also, the u axis is
parallel to the normal vector of the dispersion contour. Then
the dispersion surface curvature,
d 2u
␨⬅
dv2
冋 冉 冊册
du
2
3/2
,
1+
dv
is simplified to
␨=
d 2u
.
dv2
It is a simple exercise to show 共see Appendix A兲
冉 冊
du
⳵u
=
dv
⳵v
=−
␻
f1
= 0,
⳵ f 0/ ⳵ u
␻
2f 2
,
⳵ f 0/ ⳵ u
共3b兲
共4a兲
⳵␻
f1
=−
⬅ 0.
⳵v
⳵ f 0/⳵␻
共4b兲
Comparing Eqs. 共3b兲 and 共4a兲, it is apparent that in the
neighborhood of certain kc where 共⳵ f 0 / ⳵u兲kc = 0, a large curvature ␨ and a slow group velocity vg would appear simultaneously. For convenience of discussion, we introduce the
group index ng = c / vg. The large values of ␨ and ng are generally correlated to each other through a common factor
共⳵ f 0 / ⳵u兲−1 in this neighborhood.
However, physically it is often undesirable to have a slow
group velocity because it may cause high optical loss 共details
discussed later兲. Further inspection of Eq. 共4a兲 indicates that
a simultaneously vanishing ⳵ f 0 / ⳵␻ could break the aforementioned correlation between ␨ and ng and give an arbitrarily large ␨ without entailing a vanishingly small group
velocity. At first glance, to simultaneously achieve ⳵ f 0 / ⳵␻
= 0 and ⳵ f 0 / ⳵u = 0 may require a photonic crystal to have
specially designed structure parameters. However, we note
that since the group velocity cannot exceed c due to causality, a vanishing ⳵ f 0 / ⳵␻ always leads to a vanishing ⳵ f 0 / ⳵u
according to Eq. 共4a兲, though not conversely. To trace the
origin of ⳵ f 0 / ⳵␻ = 0, we note an identity at v = 0,
⬁
= 兺 f j共u, ␻兲v j = 0,
=−
⳵ f 0/ ⳵ u
⳵␻
,
=−
⳵u
⳵ f 0/⳵␻
vg =
G⬘
D共kx0 + u cos ␥ − v sin ␥,ky0 + u sin ␥ + v cos ␥, ␻兲
⳵ 2u
⳵v2
at k0 共i.e., u = v = 0兲, hence f 1共0 , ␻兲 = 0 in this coordinate system. The group velocity components are given by
共1兲
where G and G⬘ are reciprocal lattice vectors, k is the wave
vector, ␧ is the dielectric constant, and E is the electric field
component normal to the plane. We have assumed the speed
of light c = 1 for convenience. Generally, the zeros of the
secular determinant D共kx , ky , ␻兲 of the eigenvalue problem
Eq. 共1兲 give the frequency ␻ as an implicit function of k.
Starting from this implicit dispersion function instead of an
explicit ␻共k兲 is essential to linking the mode degeneracy to a
special type of curvature singularities of the dispersion surface.
To study the curvature, it is necessary to find the secondorder expansion of D around an arbitrary point k0. A full
Taylor expansion of D in terms of ⌬kx, ⌬ky involves many
terms. We choose a local coordinate system 共u , v兲 that gives
a simpler expansion and a clear physical picture.19
Consider the expansion of the determinant D共kx , ky , ␻兲 for
a fixed frequency ␻,
冉 冊
f 0共u, ␻兲 ⬅ D共kx0 + u cos ␥,ky0 + u sin ␥, ␻兲 = 0.
Now one readily shows that ⳵ f 0 / ⳵␻ = 0 共hence a singular ␨ at
vg ⫽ 0兲 can be achieved if the dispersion function D has a
degenerate mode at k0, i.e.,
D共kx0,ky0, ␻兲 = c0共␻ − ␻0兲n ,
where the degree of degeneracy n ⬎ 1.
It is well-known that degenerate eigenmodes will appear
at certain high symmetry points of the Brillouin zone 共BZ兲
where the associated little groups have at least one irreducible representation 共IRREP兲 whose dimension is two or
higher;20,21 and the dimensionality of irreducible representations is known to play important roles in determining the
physical properties of some photonic crystal structures of
wide interests.20–22 The K point of a triangular lattice is associated with a little group C3v, which has two 1D irreducible representations A1, A2, and one 2D irreducible representation E.20 Therefore the above analysis predicts that a
singularity of ␨, together with a nonvanishing vg, may appear
at K for certain photonic bands.
B. Example and some approximate forms of the curvature on
the BZ boundary
共3a兲
As an example, we study the dispersion surface curvature
for a triangular lattice with parameters na = 3.8, nb = 1.33, and
075104-2
PHYSICAL REVIEW B 77, 075104 共2008兲
SYMMETRY-INDUCED SINGULARITIES OF THE…
only, we consider coordinates 共u , v兲 with the origin fixed at
the K point and ␥ = 0, which allows for a simple analytic
expression of ␨ on the BZ boundary. The determinant D of
Eq. 共1兲 for the three leading Fourier components is given by
(a)
ω A1
ωE
D=
冨
␳20 − k2
␳21
␳21
␳21
␳20 − k21
␳21
␳21
␳21
␳20
−
k22
冨
= 0,
where k = kc + uex + vey, k1 = k + b1, and k2 = k + b2. We compute f 0共u , ␻兲, and f 2共u , ␻兲 from the secular determinant D
according to Eq. 共2兲,
(b)
v
y
na
g
(c)
k0
q0
f 2共u, ␻兲 = − 关g0g1 + g1g2 + g0g2 + g1b22 − 3␳41兴,
共5b兲
where
g2 = g0. At K, we have
g1 = ␳20 − 共kxc + u − b1x兲2,
and
f 0共0, ␻兲 = 共g0 − ␳21兲2共g0 + 2␳21兲 = 0.
x
α
共5a兲
g0 = ␳20 − 共kxc + u兲2 − b22 / 4,
nb
θ
f 0共u, ␻兲 = g0g1g2 − ␳41共g0 + g1 + g2兲 + 2␳61 ,
The first factor clearly indicates a doubly degenerate root,
nI
2
␻1,2
= b22/关3␧共0兲 − 3␧共b1兲兴 = ␻E2 ,
FIG. 1. 共Color online兲 A triangular lattice. 共a兲 Photonic bands
for a structure: na = 3.8, nb = 1.33; and r = 0.3a; lines: PWE method;
circles: DPT method. The two lowest bands have IRREPs E and A1
at K. 共b兲 Schematic of a typical experimental configuration. 共c兲 A
typical dispersion surface for the lower branch of the first band.
r = 0.3a. Intuitively, the dispersion contour depicted in Fig.
1共c兲 shrinks as ␻ approaches ␻E; its curvature ␨ 共roughly the
inverse of the radius兲 grows toward infinity for smaller and
small contours. A close-up examination of the ␻共k兲 curves in
Fig. 1共a兲 shows that their slopes approach a nonvanishing
value as ␻ approaches ␻E; this indicates nonvanishing vg
values. These intuitive pictures facilitate the qualitative understanding of the phenomena in this particular example.
However, to prove that these effects mathematically follow
from the theory given in the preceding section and therefore
are a particular instance of a general effect proposed herein
requires more detailed study.
In this section we employ a degenerate perturbation technique 共DPT兲 involving three dominant Fourier components23
to analytically compute ␨ for this 2D triangular lattice. The
numerical form of this DPT has been studied in detail.23 Here
we find that this DPT turns out to give some heuristic analytic forms of the curvature. As a by-product, we also find
some interesting analytic forms of other physical quantities
such as the frequencies at the band edge of high symmetry
points. Note that this approximate DPT theory is used in this
section 共Sec. II B兲 only.
We have verified that this DPT method agrees well with
the rigorous plane wave expansion 共PWE兲 method for eigenfrequencies of the lowest band, as shown in Fig. 1共a兲. We
define ␳20 = ␻2␧共0兲, ␳21 = ␻2␧共b1兲 = ␻2␧共b2兲 = ␻2␧共b3兲, where
b1, b2, and b3 are the three shortest reciprocal lattice vectors
and ␧共G兲 can be given in Bessel functions.20 For this section
at the K point. The other root is ␻23 = b22 / 关3␧共0兲 + 6␧共b1兲兴.
Note ␧共b1兲 ⬍ 0 in this case.
The curvature ␨ is computed via Eq. 共3b兲 along MK for
part of the first band below ␻E,
␨=−
1 ˜f + b22g1
,
u ˜f + b2g
共6兲
2 0
where ˜f 共u , ␻兲 = g0g1 + g1g2 + g0g2 − 3␳41. One readily verifies
that g0 = g1 = ␳21 for the degenerate mode at ␻E 共where u = 0兲.
Therefore, on the MK line, the curvature has the asymptotic
form ␨ → −1 / u as u → 0. Furthermore, we note
⳵ f 0/⳵u = − u共f˜ + b22g0兲;
⳵ f 0/⳵␻ = 共2/␻兲关␳20˜f − 2␳41共g0 + g1 + g2 − 3␳21兲兴.
One readily verifies ⳵ f 0 / ⳵␻ → const⫻ u as u → 0. Hence the
cancellation according to Eq. 共4a兲 gives a nonvanishing vg at
K. Thus, within the DPT framework, a singularity of ␨ with a
finite ng is analytically verified for this example.
Figure 2 shows that the values of ␨ and ng obtained from
the DPT method agree well with the PWE method. The logarithmic plot also reveals that the variation of ␨ follows the
group index ng along most of the MK line except near K
共kxa = 1 / 3兲, where a varnishing ⳵ f 0 / ⳵␻ breaks the correlation
between a singular ␨ and a singular ng, as predicted. The
curvature in the entire BZ is plotted in Fig. 2 共inset兲 for the
lower branch 共below E兲 of the first band. The large ␨ values
on the BZ boundaries reflect the typical contour shape depicted in Fig. 1共c兲, where the dispersion contour bends
abruptly across the BZ boundary. The curvature becomes
singular at high symmetry points M 共kxa = 0兲 and K. The
normalized curvature 2␲␨ / ␭ for an ordinary medium is
around unity, but it can be several orders of magnitude
075104-3
PHYSICAL REVIEW B 77, 075104 共2008兲
WEI JIANG AND RAY T. CHEN
nI⌬␻ sin ␣ = kx0 + ⌬u cos ␥ − ⌬v sin ␥
around u = v = 0. Note in the above equation, terms proportional to ⌬␥ vanish at u = v = 0 and should not appear. By
virtue of Eq. 共8兲, one readily shows
冋
冉 冊 册
−1
⳵u
⌬v
=
nI sin ␣ −
⌬␻ sin ␥
⳵␻
cos ␥ .
共9兲
v
Note the higher order terms omitted in Eq. 共8兲 have vanishing contributions in Eq. 共9兲 in the limit ⌬␻ → 0; hence they
are not shown. The derivative 共⳵u / ⳵␻兲v is just the group
index, ng 共note c = 1 in this paper兲. Also, by definition of the
curvature, we have 兩⌬␪ / ⌬v兩 = 兩␨兩. Hence, by virtue of Eq. 共9兲,
the wavelength sensitivity is given by
冏 冏冏 冏冏
冏
d␪
d␪ dv
␨
=
=
共nI sin ␣ − ng sin ␪兲 . 共10兲
d␻
dv d␻
cos ␪
FIG. 2. 共Color online兲 The dispersion surface curvature and
group index on the MK line for the lower branch of the first band.
2␲兩␨兩
The inset plots log10共 ␭ + 1兲 in the 1 BZ for this branch.
Note ␪ = ␲2 − ␥ 关note Figs. 1共b兲 and 1共c兲 illustrate a case of
␪ ⬍ 0, ␣ ⬍ 0兴.
Now we study the perturbation of refractive index. A perturbation of ⌬na at a fixed ␻ results in
higher in photonic crystals. Note the DPT is used to give an
intuitive expression of ␨ for this example, it will not be employed in the derivation or calculation in the rest of this
work.
⌬v
⌬u
cos ␥ =
sin ␥
⌬na
⌬na
according to Eq. 共7兲. The refractive index sensitivity can be
⌬
d␪
= ⌬⌬␪v ⌬nva , which gives
calculated from dn
a
冏 冏 冏 冋 冉 冊 册冏
III. ANGULAR SENSITIVITIES OF A PHOTONIC
CRYSTAL
d␪
␨
⳵u
=
sin ␪
dna
cos ␪
⳵na
A. Derivation of the sensitivity formulas
The response of the mode energy flux direction 共i.e., beam
direction兲 of a photonic crystal to a small perturbation of
wavelength or refractive index is of paramount interest to
many applications. To study this optical response, we need to
find the relationship between ␨ and a measurable 共extrinsic兲
quantity in a typical experimental configuration illustrated in
Fig. 1共b兲. The coupling condition for the input surface is
given by
nI␻ sin ␣ = kx0 + u cos ␥ − v sin ␥ ,
⌬u =
冉 冊 冉 冊
⳵u
⳵␻
⌬␻ +
v
⳵u
⳵v
␻
⌬v + ¯ =
冉 冊
⳵u
⳵␻
v
⌬␻ + ¯ ,
共8兲
where we have omitted terms of the order ⌬␻2, ⌬v2, ⌬␻⌬v,
and higher. The second term in Eq. 共8兲 vanishes because we
have 共⳵u / ⳵v兲␻ = 0 along a dispersion contour 共constant-␻
contour兲. We also note that according to Eq. 共7兲, a perturbation ⌬␻ with a fixed value of ␣ leads to
.
共11兲
Using the Jacobian determinants, one readily shows that
冉 冊
⳵u
⳵na
=−
␻,v
冉 冊冉 冊
⳵␻
⳵na
u,v
⳵u
⳵␻
,
na,v
where the second term on the right side is just the group
index, ng 共note c = 1兲. Now we find that the angular sensitivity to a refractive index perturbation 共for a fixed incident
angle and fixed wavelength兲 is given by
冏 冏冏
共7兲
where ␥ is the angle of the group velocity at k0 = kx0ex
+ ky0ey with respect to the input interface. Here k0 is an arbitrary point on an arbitrary dispersion contour. Consider
cases where the incident angle, ␣, is fixed and one varies the
wavelength or the refractive index of one constituent material 共for example, na兲.
First, we analyze the wavelength perturbation. We notice
␻,v
冉 冊 冏
d␪
⳵␻
= − ␨ tan ␪
dna
⳵na
ng .
共11⬘兲
k
One can compute 共⳵␻ / ⳵na兲u,v ⬅ 共⳵␻ / ⳵na兲k easily by varying
na in the photonic band calculation. Note that except for
angles ␪, ␣, and nI, other quantities in Eqs. 共10兲 and 共11⬘兲 are
intrinsic properties of a photonic crystal, independent of the
choice of the coordinate system and the crystallographic orientation of the input surface. According to Eqs. 共10兲 and
共11⬘兲, the sensitivities to wavelength and index perturbations
can be enhanced by a large dispersion surface curvature or
by a large group index. A key feature of Eqs. 共10兲 and 共11⬘兲
is that high sensitivities to wavelength and refractive index
perturbations are usually correlated, through common terms
␨ and ng.
B. Individual control of the curvature and the group velocity
However, it turns out that a large value of ng results in low
transmission, and therefore enhancing ␨ is practically a better
075104-4
PHYSICAL REVIEW B 77, 075104 共2008兲
Angle θ (deg)
θ
vg /c
FIG. 3. 共Color online兲 Normalized transmission and sensitivities
for modes near the K-valley of the second band, ␻A1a / 2␲c
⬃ 0.248, ␣ = 20.8°, n1 = na.
approach to ultrahigh sensitivities. According to our previous
theory on surface coupling,24 the normalized transmission is
given by the ratio of the surface-normal component of the
Poynting vector Sy ⬃ ␧emvg cos ␪,
T ⬃ 兩t兩2␧emvg cos ␪ ,
共12兲
where ␧em is the mode energy density and t is the complex
coupling amplitude24 of the mode in question. Typically, the
wavelength or refractive index varies over an ultranarrow
range 共less than ⬃1%兲 in high sensitivity cases, and 兩t兩 and
␧em generally vary insignificantly across this range according
to our computation. For the modes in the nondegenerate
K-valley 共IRREP A1兲 in Fig. 1共a兲, the group velocity vg vanishes as the wavelength approaches the band edge at ␻A1.
Thus the normalized transmission T is low, and T largely
follows the trend of small vg as shown in Fig. 3共b兲 in accord␪
dance to Eq. 共12兲. High sensitivities, 兩 dn
兩, shown in Fig. 3共a兲
a
are achieved at the cost of low transmission. In contrast, the
symmetry-induced degeneracy limits vg to a nonvanishing
value around the degenerate K point 共IRREP E兲 at ␻E. Therefore the transmission remains high for most of the spectrum
as depicted in Fig. 4共b兲.
To recapitulate, we note that the sensitivities to wavelength and refractive index can be significantly enhanced if
any of the three terms, ␨, ng, and 1 / cos ␪, is large in Eqs.
共10兲 and 共11⬘兲. Unfortunately, the transmission given in Eq.
共12兲 is inversely proportional to the last two terms, leaving ␨
the only desired route for high sensitivities at low optical
loss.
For a numerical example, we set the transmission threshd␪
old at 50%. This gives the maximum achievable 兩 d␭
兩⬃3
d
␪
⫻ 103 deg/ nm and 兩 dna 兩 ⬃ 3.5⫻ 104 deg near the degenerate
K point at ␻E in the first band, according to Fig. 4共a兲.
Clearly, orders of magnitude higher sensitivities can be
achieved in a photonic crystal without severely suppressing
optical transmission. Here we are more interested in index
sensitivities for sensing and nonlinear optics applications.
These applications do not require the beam center shift to be
much larger than the beam width, and therefore are not lim-
dθ/dn
(b)
T
vg/c
vg /c
Transmission
T
(b)
θ
(a)
|dθ /dn| (deg)
dθ/dn
vg / c
Transmission
(a)
|dθ/dn| (deg)
Angle θ (deg)
SYMMETRY-INDUCED SINGULARITIES OF THE…
FIG. 4. 共Color online兲 Normalized transmission and sensitivities
for modes near the degenerate ␻E mode of the first band,
␻Ea / 2␲c ⬃ 0.187, ␣ = 27.9°, nI = na.
ited by some issues found in wavelength demultiplexing
applications.10 Generally, it is easy to detect a minimum lateral shift 10 ␮m of the beam center on the exit end of a
photonic crystal. Then a photonic crystal sensor only needs a
length ⬍50 ␮m to resolve a refractive index change of
d␪
d␪
⌬na ⬃ 0.001 with a sensitivity 兩 dn
兩 ⬃ 104 deg. A high 兩 dn
兩
a
a
value may also significantly enhance certain nonlinear optical effects such as all optical switching and beam-steering7
or deflection based Q-switching, where a small ⌬na can be
generated by a pump-control beam or by the signal beam
itself. Detailed discussion on applications is beyond the
scope of this paper.
IV. DISCUSSIONS
A. Intrinsic and extrinsic parameters
All quantities in Eqs. 共10兲 and 共11⬘兲 can be easily calculated in any coordinate system. For example, 共⳵␻ / ⳵na兲u,v
⬅ 共⳵␻ / ⳵na兲k, the latter can be computed in a regular 共kx , ky兲
coordinate system whose origin is at the BZ center, ⌫. Also,
d 2k
dk 2 3/2
and ng
the well-known formulas, ␨ ⬅ dk 2y / 关1 + 共 dkyx 兲 兴
x
= c / 兩ⵜk␻兩, can be employed to calculate ␨ and ng in 共kx , ky兲
coordinates. Indeed, it is straightforward to see that these
intrinsic quantities 关兩␨兩, ng, and 共⳵␻ / ⳵na兲k兴 do not rely on the
choice of the coordinate systems. Note this statement is valid
only for those coordinate systems that can be related to
共kx , ky兲 through a Euclidian transformation, which is sufficient for all practical purposes. As long as the values of ␨,
共⳵␻ / ⳵na兲u,v, and ng are computed for each k point separately,
we will not compromise the rigor of Eqs. 共10兲 and 共11⬘兲.
More discussion on understanding the rigor of our method is
presented in Appendix B.
We call ␪ and ␣ extrinsic parameters because they are
related to how a photonic crystal is coupled on the surface.
The angle ␪ should be determined as the angle of the group
velocity at the coupled k0 point with respect to the normal of
the photonic crystal surface. Therefore if we rotate the x,y
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PHYSICAL REVIEW B 77, 075104 共2008兲
WEI JIANG AND RAY T. CHEN
axes 共but not lattice axes intrinsic to the photonic crystal兲,
the angle ␪ should not change.
Some further analysis helps understand the effects of intrinsic and extrinsic parameters. Once a coupling configuration 共including surface orientation, incident angle, and wavelength兲 is given, the sensitivity values given in Eqs. 共10兲 and
共11⬘兲 are determined, independent of the choice of the coordinate system. On the other hand, consider two experiments
for the same photonic crystal lattice: 共1兲 light impinges on a
surface having Miller indices 共h1h2兲 = 共10兲 at an incident
angle ␣1; and 共2兲 light impinges on a surface having 共h1h2兲
= 共23兲 at an incident angle ␣2. By coincidence we may
couple to the same 共physical兲 k0 point on the dispersion surface in these two experiments. Although the intrinsic parameters ␨, 共⳵␻ / ⳵na兲u,v, and ng are the same, the different extrinsic parameters cause entirely different angular sensitivity
values. In this sense, these angular sensitivities themselves
are also extrinsic quantities, which depend on the external
coupling conditions.
The separation of extrinsic parameters and intrinsic parameters also allows us to see some interesting effects. Generally, a large intrinsic parameter ␨ at a photonic band of
interest means that a photonic crystal is potentially highly
sensitive to both wavelength and refractive index perturbations. However, the external coupling conditions, described
by the extrinsic parameters, could modify or even suppress
certain sensitivities actually observed. For example, if the
term nI sin ␣ − ng sin ␪ in Eq. 共10兲 vanishes under a given
coupling configuration 共or close to zero over a range of coupling parameters兲, it is possible to produce a device that has
a very high refractive index sensitivity and a relatively low
wavelength sensitivity in certain parameter ranges. This may
help enhance the bandwidth of certain devices. Further study
is needed to explore this possibility.
B. Effects of dimensionality and lattice types
The theory developed here can be extended to treat other
common lattices in 2D and 3D. We note that a similar singularity of ␨ can occur for a square lattice, where the corresponding BZ corner 共M point兲 retains the C4v symmetry and
has one 2D irreducible representation. Therefore an ultralarge curvature can appear with a nonvanishing vg for the
two most common 2D lattice types. Similar analysis can be
applied to the TE polarization. It can be proven from group
theory that symmetry-induced singularities of the dispersion
surface curvature at vg ⫽ 0 can also exist for most common
3D lattices, such as simple-cubic, face-centered-cubic, and
body-centered-cubic lattices. The analysis will be similar in
spirit though more complicated in form because a surface in
3D has two principal curvatures.19 For many practical scenarios in 3D, effective 2D dispersion surfaces may be used,5
then the above 2D analysis remains applicable.
The phenomena discussed here do not exist in 1D photonic crystals because all 1D point groups are Abelian and
have no degeneracy. For a 1D grating of the same period a,
d␪
1
its angular dispersion25 d␭
⬃ a cos
is much lower
␪
共⬍0.2 deg/ nm兲 for any reasonable value of ␪. The preceding
d␪
d␪
values of the 2D lattice are obtained at 兩␪兩
, dn
high d␭
a
⬍ 70° with negligible contribution from the cos1 ␪ factor in
Eqs. 共10兲 and 共11⬘兲.
Note that our photonic crystal surface coupling theory24
has been extended to compute mode transmissions for 3D
photonic crystals26 and 1D gratings.27 Also note that in grating diffraction, the diffracted beam angle refers to the angle
outside the grating. For the original superprism effect, the
beam angle sensitivity refers to the angle inside the photonic
crystal. For some integrated devices,15 beam angles outside
the photonic crystal need not be sensitive and the sensitivity
is employed through the sensitive shift of the output position
on the exit surface. If the output angle sensitivity is exploited, then the output surface must not be parallel to the
input surface.12,14
C. Van Hove singularities and some other issues
We shall mention that Van Hove envisioned that a minimum of one band and a maximum of another may “contact”
each other, and the Van Hove singularity will be weakened or
suppressed by “compensation.”28 In our case, it is straightforward to prove that the Van Hove singularity is virtually
absent29 at ␻E due to a nonvanishing vg, but the curvature ␨
exhibits a singularity. Note that one type of extreme anisotropy near the ⌫ point was found to be associated with a
divergent Van Hove singularity for a 2D square lattice.30 The
curvature singularity discussed here comes with a suppressed
Van Hove singularity, and is a different, general phenomenon. Our Eq. 共10兲 shows dd␻␪ ⬀ ␨ and may also shed new light
on some issues for ␨ = 0.
Note that usually the dielectric function of a photonic
crystal is local and is considered accurately known. Hence
additional boundary conditions31 are not needed for a semiinfinite photonic crystal even when multiple modes appear.
Other theoretical works on photonic crystal surface coupling
also do not invoke additional boundary conditions.32
In summary, we rigorously analyze the dispersion surface
curvature in 2D photonic crystals and discuss its relation to
angular sensitivities of a photonic crystal. The individual
contributions of the curvature and group index to the angular
sensitivities are separated. Furthermore, assisted by group
theory, we analytically show that symmetry induced degeneracy allows for high sensitivities without compromising optical transmission in 2D and 3D photonic crystals, leading to
promising applications including sensing and nonlinear optics. We also discuss the possibility of maximizing the refractive index sensitivity while suppressing the wavelength sensitivity.
ACKNOWLEDGMENTS
We thank Nicholas X. Fang, R. L. Nelson, J. Haus, and
Tao Ling for helpful scientific-technical discussions. We are
also grateful to many colleagues and friends for kind encouragement and support. This work was supported by Air Force
Research Laboratory under Grant No. FA8650-06-C-5403.
Partial support from AFOSR 共Grant No. FA9550-05-C-0171兲
and NASA 共Grant No. NNX07CA84P兲 is acknowledged.
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PHYSICAL REVIEW B 77, 075104 共2008兲
SYMMETRY-INDUCED SINGULARITIES OF THE…
APPENDIX A: DERIVATIVES IN A LOCAL
COORDINATE SYSTEM
APPENDIX B: RIGOROUS NATURE OF THE
SENSITIVITY FORMULAS
By virtue of Eq. 共2兲, the total derivative of D with respect
to v along a constant-␻ contour is given by
To understand the rigorous nature of Eqs. 共10兲 and 共11⬘兲,
we point out some key features of our derivation. First, the
origin of the local coordinate system is an arbitrary point k0
共not necessarily a high symmetry point兲. Second, in deriving
these relations, we have not assumed a finite Fourier series.
Third, the local expansion with respect to u , v is rigorous at
k0 only. For another point k0⬘, no matter how close it is to k0,
another local expansion with another set of local coordinates
共u⬘ , v⬘兲 must be used. In this way, the rigor of Eqs. 共10兲 and
共11⬘兲 is not compromised. Lastly, to compute the beam angle
change due to a finite change of refractive index, the rigorous
way is to integrate ⳵␪ / ⳵na given by Eq. 共11⬘兲 over the finite
span of ⌬na.
The preceding procedure that involves local coordinates
共u , v兲, although useful in understanding the rigor of Eqs. 共10兲
and 共11⬘兲, is somewhat complicated in practical calculations.
Fortunately, we have simplified the sensitivity forms such
that all quantities in Eqs. 共10兲 and 共11⬘兲 can be easily calculated in any coordinate systems. The details have been discussed in Sec. IV A. Note that, in contrast, Eq. 共11兲 involves
共⳵u / ⳵na兲␻,v, which is much less intuitive for direct computation in any coordinate systems. As long as the values of ␨,
共⳵␻ / ⳵na兲u,v, and ng are computed for each k point individually, we will not compromise the rigor of Eqs. 共10兲 and 共11⬘兲.
Lastly, even though the equations given are rigorous, the
values of quantities such as ␻ and cos ␪ entering Eqs. 共10兲
and 共11⬘兲 are usually approximated in numerical calculations
due to a finite cutoff of the series used in computing ␻ and
the cosine function. Nonetheless, rigorous equations with
simple forms like Eqs. 共10兲 and 共11⬘兲 serve at least two important purposes. First, they single out a few key factors
共e.g., ␨, ng兲 that affect sensitivities. Such physical insight
helps us easily identify the high sensitivity regimes of interest and avoid an aimless search in a large design space. Second, it provides a priori information of various quantities
near numeric singularities 共e.g., singular ␨兲. Generally such a
priori information is invaluable in numeric calculations and
helps us design numeric schemes that are highly accurate,
reliable, and efficient near singularities.33 We have employed
at least 121 Fourier components to ensure better than 1%
convergence for sensitivities.34
冏 冏
dD
0=
dv
⬁
␻
=兺
j=0
⬁
⳵ f j du j
v + 兺 f j共u, ␻兲j v j−1 .
⳵u dv
j=1
共A1兲
At the origin of this local coordinate system 共u , v兲, we
have v = 0. Hence Eq. 共A1兲 is reduced to
⳵ f 0 du
+ f 1 = 0,
⳵u dv
共A2兲
from which we obtain Eq. 共3a兲. Further differentiation of Eq.
共A1兲 with respect to v yields
0=
冏 冏 兺冋 冉 冊
d 2D
dv2
⬁
+兺
j=1
⬁
=
␻
j=0
⳵2 f j du
⳵u2 dv
2
vj +
⳵ f j d 2u j
v
⳵u dv2
册
⬁
⳵ f j du j−1
⳵ f j du j−1
jv + 兺
jv
⳵u dv
j=1 ⳵u dv
⬁
+ 兺 f j共u, ␻兲j共j − 1兲v j−2 .
共A3兲
j=2
Here the first two sums come from the differentiation of the
first sum in Eq. 共A1兲, the last two sums from the second sum
in Eq. 共A1兲. At the origin of this local coordinate system
共u , v兲, we have v = 0, and du / dv = 0. Hence Eq. 共A3兲 is reduced to
0=
⳵ f 0 d 2u
+ 2f 2 ,
⳵u dv2
共A4兲
from which we obtain Eq. 共3b兲. By considering the total
derivatives of D with respect to u and v along constant-v and
constant-u lines respectively, Eqs. 共4a兲 and 共4b兲 can be
proved similarly.
*[email protected]
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Note many relations regarding the local tangent vector, normal
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