sample publications - rutgers university

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.)

High-purity transmission of a slow light odd modein a photonic crystal waveguide

Jun Tan,1 Ming Lu,2 Aaron Stein,2 and Wei Jiang1,3,*1Department of Electrical and Computer Engineering, Rutgers University, Piscataway, New Jersey 08854, USA

2Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973, USA3Institute 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 oddmode in a photonic crystal waveguideand investigate the spectral signature of this slow light mode. An odd-modeMach–Zehnder coupler is introduced totransform 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 todetermine the group index of this odd mode. With slow light enhancement, this odd mode can help enable novelminiaturized devices such as one-way waveguides. © 2012 Optical Society of AmericaOCIS codes: 130.5296, 230.5298.

Photonic crystal waveguides (PCWs) [1–5] can modifylight propagation and dispersion characteristics throughtheir periodic structures and thus have important appli-cations in communications and sensing. Particularly, theslow light effect in a PCW can significantly enhancelight–matter interaction [6–8], as demonstrated in signif-icant reduction of interaction lengths for PCW-basedmodulators and switches [9–11]. To date, most of thePCW research has been focused on the TE-like modewith even symmetry. However, a PCW often has anodd TE-like mode inside the photonic bandgap exhibitingthe slow light effect as well. This odd mode can poten-tially open up the opportunities for mode-symmetry-based novel devices, such as one-way waveguides thatexploit indirect interband photonic transitions betweeneven and odd modes [12]. The slow light effect in PCWscan help reduce the interaction length for such transi-tions, enabling ultracompact devices. To utilize this oddmode in any device, it is crucial to control its excitationsymmetry and understand its slow light spectral charac-teristics. Normally, this odd mode does not exhibit itselfevidently in the PCW transmission spectrum because itsodd symmetry prohibits its excitation by the fundamentaleven mode of a conventional waveguide typically used atinput. Symmetry-breaking structure imperfections some-times may induce some coupling to this odd mode, caus-ing a decrease of PCW transmission in the odd modeband [13,14]. Here we demonstrate a novel scheme tocontrol the excitation symmetry for high-purity transmis-sion of this odd mode and investigate the spectral signa-tures 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 hexagonallattice with lattice constant a � 400 nm, hole radiusr � 0.325a, and Si slab thickness t � 260 nm. The banddiagram in Fig. 1(a) is calculated by three-dimensional(3D) plane wave expansion [15] (with>1 μm top/bottomcladdings and six rows of holes per side). Below the light-line (for the oxide bottom cladding), the even TE-likemode has a flat dispersion relation with group index ng >50 and a narrow bandwidth (<4 nm). In contrast, belowthe lightline, the odd TE-like mode has a much widerbandwidth, ∼20 nm, with ng down to ∼15. Such a mod-

erate ng range is favorable for many applications as var-ious types of losses are reduced at lower ng [5,15–17].Furthermore, the dispersion relation of the TM-likeguided mode usually crosses that of the even mode[5], as seen in Fig. 1(a). But the TM-like mode does notcross the odd mode in the region below the lightline inFig. 1(a). For ωa ∕ 2πc � 0.28–0.286, only the odd mode isbelow the lightline.

Systematic simulations show that as the hole radius in-creases, the odd-mode band edge moves up faster thanthe TM cutoff, as shown in Fig. 1(b). For a sufficientlylarge r, the TM cutoff is below the odd-mode band edge;thus the two modes do not cross each other below thelightline, helping avoid their intercoupling due to asym-metric top and bottom claddings. However, as r in-creases, the transmission bandwidth bounded by theband edge and the cutoff decreases for both the evenand odd modes, as shown in Fig. 1(b). Hence, this workfocuses on the intermediate r case shown in Fig. 1(a),which shows a sufficient clearance between the odd-mode band edge and the TM cutoff, and a sufficientlywide bandwidth.

Excitation of this odd PCWmode is usually deterred bythe opposite symmetry of the fundamental even modeof a Si waveguide. To solve this problem, we employ a

Fig. 1. (Color online) PCW photonic band structures. (a) Banddiagram for r � 0.325a. The dark grey region indicates the low-er photonic band. Hz field profiles for even and odd modes atk � π ∕ a are shown in the insets (PCW axis along y); (b) varia-tion of the band edge and cutoff of even (blue) and odd (green)TE-like modes with hole radius. For each TE-like mode, thelower 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.

August 15, 2012 / Vol. 37, No. 16 / OPTICS LETTERS 1

0146-9592/12/160001-03$15.00/0 © 2012 Optical Society of America

two-step approach. First, a Mach–Zehnder coupler(MZC) whose two arms have a phase difference of π isutilized to transform mode symmetry and excite anodd mode in a wide (multimode) Si wire waveguide; thenthis odd mode is coupled to the odd mode of the PCW. Tocreate π phase difference in this odd-mode MZC, its twoarms can be designed to have a length difference of�Δl�π � λ ∕ 2neff , where neff is the effective index of theSi waveguide. Finite difference time-domain (FDTD) si-mulation has been performed to confirm that such a MZCproduces an odd mode in a wide output waveguide, asshown in Fig. 2(a). The input and output waveguidewidths are 400 and 700 nm, respectively. The couplingbetween the odd mode of a Si wire waveguide (700 nmwide) and that of the PCW is also simulated. Simulationresults in Fig. 2(b) show coupling efficiencies up to∼84%(∼0.75 dB) for the odd mode. The field pattern inFig. 2(b), left inset, confirms that the coupled PCW modeis an odd mode. The fundamental even mode of a Si wirewaveguide couples into the PCW with inconsequentialchange of coupling efficiency for the spectral range inFig. 2(b). The field pattern in Fig. 2(b), right inset, indi-cates 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 compar-ison with the modes of the conventional Si waveguide,whose TE modes are commonly visualized by Ex (noteEx and Hz 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 accord-ing to the parameters used in Fig. 1(a). The structure ispatterned by a JEOL JBX-6300FS high-resolution e-beamlithography system, operating at 100 keV, on a 100 nmthick layer of ZEP 520A e-beam resist. Then the patternis transferred to the Si layer by an Oxford Plasmalab100 ICP etcher. Figure 3 is a scanning electronmicroscope(SEM) image of the fabricated structure. TwoMZCswith a10 μmbending radius are connected through 700 nmwideSi waveguides of 1 μm length to both ends of the PCW.To measure transmission spectra, light from a super-

luminescent LED with a spectral range of about 80 nmis coupled to the TE mode of Si access waveguides(tapered to 4 μm at chip edges) via lensed fibers. A po-larizer is used at the output end to block TM polarization.

The PCW insertion loss is measured with reference to a Siwire waveguide. Figure 4(a) shows the spectrum of aPCW with odd-mode MZCs. A substantial transmissionbandwidth is observed, approximately 22 nm at 10 dBbelow the peak. The contrast between the transmittedmode and background is>20 dB. The peak insertion lossis about −4 dB. Separate measurements show that eachMZC contributes ∼1 dB. Thus the loss due to the PCWis estimated at ∼2 dB. For comparison, the spectrumof a directly coupled PCW without MZCs is shown inFig. 4(b). The transmission is due to the leaky evenTE-like mode as simulated in Fig. 2(b). Figure 4(b) alsoshows the PCW transmission with MZCs whose two armshave a length difference Δl deliberately designed to be50% greater than �Δl�π . Such a mixed-mode MZC offersa symmetry configuration that can excite a mixtureof even and odd modes according to I� ∝ �1 ∕ 2��1� cos�2πneffΔl ∕ λ��. As such, the background transmis-sion due to the even mode rises. In the odd-mode band,the mixed-mode spectrum oscillates strongly due to thebeating of two modes. Figures 4(a) and 4(b) illustratethat distinctive spectral signatures can be observed withcontrolled excitation symmetries.

The mode-beating pattern of the mixed-mode spec-trum contains important information of the odd mode.The beating period is related to the group indices of evenand odd modes through Δλ � λ2 ∕ �ng;odd − ng;even�L,where L is the PCW length. Simulation indicates thatng;even is virtually a constant �∼5� in the odd-mode band.Thus the chirped beating periods are due to the disper-sion of ng;odd. We have calculated Δng � ng;odd − ng;evenfrom the mixed-mode spectrum and plotted it in Fig. 4(c).The peak spacing and valley spacing of the spectrum givetwo sets of Δng data, plotted by circles and crosses

Fig. 2. (Color online) FDTD simulation results; (a) schematicof the MZC structure. The right arm has two extra waveguidesegments (in orange) with a combined length of �Δl�π . The in-put and output Ex field profiles (cross section) are shown in theinsets (200 nm per division on axes); (b) PCW coupling effi-ciency. Insets: coupled Ex field patterns at 1390 nm. Light(in odd or even mode) enters from a Si waveguide at the bottomof each figure and into a PCW upward.

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.

Fig. 4. (Color online) Transmission spectra for 20 μm longPCWs: (a) with odd-mode MZCs; (b) direct transmission with-out MZC and transmission with mixed-mode MZCs; (c)Δng ob-tained from the mixed-mode spectrum; the solid line delineatesthe trend; (d) Fourier transform of the transmission spectrum ofanother directly coupled PCW (the peak position gives ng;even).

2 OPTICS LETTERS / Vol. 37, No. 16 / August 15, 2012

respectively. They agree with each other, as expected.Note that the Δng value obtained from two adjacentpeaks (valleys) is assigned to the midpoint wavelengthin between. Further, ng;even � 4.9 is obtained in Fig. 4(d)through the Fourier transform [18] of the transmissionspectrum of another directly coupled PCWwith more ob-vious spectral ripples. Note that the Fourier frequency f λis 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 amplitudein Fig. 4(a) is relatively weak. In contrast, the mode-beating amplitude of the mixed-mode spectrum inFig. 4(b) is much higher and more robust against noise,which facilitates the evaluation of ng;odd. Also note that inFig. 4(a), the background transmission increases discern-ibly beyond 1430 nm due to the dispersive effect in theodd-mode MZC, which modifies the phase shift differ-ence between the two arms as λ deviates far from thedesigned value (1390 nm). The TM-like mode (guidedfor λ > 1.45 μm) may also contribute to the backgroundat long wavelengths. However, these effects are muchweaker for 1380–1415 nm.Although this work focuses on PCWs on a SOI chip, the

MZC and the mode-beating-based ng;odd measurementmethod can be adapted to the cases of air-bridge oroxide-covered PCWs and coupled-cavity PCWs, where in-teresting anomalous propagation related to an odd modehas been observed [19]. It would be interesting also toexplore a refined design to optimize the bandwidthand the slow-down of light together for this odd mode.Detailed discussion of these possibilities is beyond thescope of this work. The odd-mode wavelength can alsobe shifted to ∼1550 nm or other values (depending onspecific applications) by changing the lattice constant.In a SOI PCW, there is some coupling between the TE-like guided modes and the TM-like photonic crystal bulkmodes due to asymmetric top/bottom claddings. Priorwork on the even mode has demonstrated that reducingng can reduce the loss due to such coupling [5]. This oddmode has a much lower ng, ∼14, than the normal evenmode (ng ∼ 50) below the lightline. This helps to reducethe coupling to the TM-like bulk modes. For many PCWdevices operating at a short length <80 μm [9,10], thepropagation loss of the odd mode is expected to be rea-sonable. Lastly, the understanding of the slow light andmode-beating characteristics of this odd mode, as wellas the controlled excitation and ng;odd characterizationschemes developed here, can facilitate the developmentof mode-symmetry-based novel devices, such as one-waywaveguides that involve active transition and passiveconversion between even and odd modes [12]. Slow lightcan help reduce device interaction length. Note that pre-viously demonstrated conventional waveguide modeconverters employed branching waveguides [20,21] ormultimode interference couplers [22]. Photonic-crystal-based mode converters have also been designed [23].Here, the odd-mode MZC is focused on transformingmode symmetry to attain a high-purity odd mode, and themixed-mode MZC offers a symmetry configuration for co-herent mixing of even and odd modes, which enablesng;odd measurement through slow-light mode beating.

As a side note, beating between two degenerate modesin a periodically patterned microring resonator hasrecently been observed, but the resonant wavelengthspacing is not affected by beating [24].

In summary, we have experimentally demonstrated thecontrol of excitation symmetry for an odd TE-like modein a PCW. An odd-mode MZC is utilized to selectively ex-cite the odd mode with a contrast >20 dB over the back-ground. Assisted by a mixed-mode MZC, slow light modebeating is observed and is utilized to measure the groupindex of this odd mode.

This work is supported in part by AFOSR GrantNo. FA9550-10-C-0049. This research is carried out in partat the Center for Functional Nanomaterials, BrookhavenNational 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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August 15, 2012 / Vol. 37, No. 16 / OPTICS LETTERS 3

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 Jiang

1,3,*

1Department of Electrical and Computer Engineering, Rutgers University, Piscataway, NJ 08854, USA 2Department of Materials Science and Engineering, Rutgers University, Piscataway, NJ 08854, USA

3Institute for Advanced Materials, Devices, and Nanotechnology, Rutgers University, Piscataway, NJ 08854, USA 4Department 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 finite-

element 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 Received 15 Dec 2011; revised 26 Jan 2012; accepted 27 Jan 2012; published 6 Feb 2012(C) 2012 OSA 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]. Silicon-

on-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.


~1014

cm−3

) Si strip surrounded by a relatively highly doped (e.g. ~1017

cm−3

) silicon on both

sides [6]. Passing current laterally through this structure produces concentrated ohmic heating

in the center strip.

(a) (b)

z

xy

Si

substrate

oxide

air holes

Heat

sourceSi

(a) (b)

z

xy

Si

substrate

oxide

air holes

Heat

sourceSi

(a) (b)

z

xy

Si

substrate

oxide

air holes

Heat

sourceSi

z

xy

z

xy

SiSi

substrate

oxide

air holes

Heat

sourceSi

substrate

oxide

air holes

Heat

sourceSi

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

( ) ( )( ) exp / 2 / , for 2,sprT x x W X r x W / ≈ − − > (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]

1/2

[ / ] ,spr Si Si ox Si ox

X X t t κ κ= = (2)

where κSi and κox are the thermal conductivities of silicon and SiO2 respectively (values from

Ref [14].).


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

( ) ( ) / ( ).eff ox spr Si ox

r X r t tκ κ= (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 0.25 0.275 0.3 0.325 0.35

Xspr (µm) 6.3 6.0 5.7 5.3 4.9

κeff / κSi 0.68 0.61 0.55 0.48 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

[ 2 ( )]( / ),ox spr ox ox

Q L W X r T tκ= + ∆ (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]


2 / ( ),g

n Ln nφ π σ λ∆ = ∆ (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

[ 2 ( )] / [2 ( / )].ox spr ox g

Q n W X r t n dn dTπ λκ σ= + (6)

Because ~spr ox

X t , the power Qπ actually scales as 1/ox

t 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.

0 50 100 1500

2

4

6

8

10

ng

Qπ,

SO

I (m

W)

0 50 100 1500

50

100

150

200

L (µ

m)

1µm

2µm

5µm

0 50 100 1500

2

4

6

8

10

ng

Qπ,

SO

I (m

W)

0 50 100 1500

50

100

150

200

L (µ

m)

1µm

2µm

5µm

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

( ) ( ) ( )

/ 2 ,/ 2

membrane edge edge

membrane eff Si ox Si

membrane ox

T T TQ Lt LX

W tκ κ

∆ − ∆ ∆= = (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

( )( )

(2 ) ,( ) / 2

eff Simembrane

membrane ox Si

ox eff Si Si membrane

r XTQ L X

t r X W

κκ

κ κ∆

=+

(8)

For the same power Q, the membrane structure may enhance the temperature rise by a

factor

( ) ( ) / 2( )

.( ) ( )

spr eff Si Si membranemembrane

SOI Si eff Si

X r r X WT

T X r X

κ κ

κ

+∆≈

∆ (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


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 ~ oxt . Thus, Qπ scales slower than 1/2

oxt

− , 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 out-

of-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.


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 2~ /ox ox ox ox

t cτ ρ κ (~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.).


Parallel-coupled dual racetrack silicon micro-resonators 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,*

1Department of Electrical and Computer Engineering, Rutgers University, Piscataway, NJ 08854, USA 2Institute for Advanced Materials, Devices, and Nanotechnology, Rutgers University, Piscataway, NJ 08854, USA

3Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, TX 78712, USA

4Alcatel-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.

References and Links

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#144846 - $15.00 USD Received 28 Mar 2011; revised 14 Jun 2011; accepted 15 Jun 2011; published 19 Jul 2011(C) 2011 OSA 1 August 2011 / Vol. 19, No. 16 / OPTICS EXPRESS 14892

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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 Mach-Zehnder 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)


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.

0.8

0.9

1

Inte

nsity

1.54 1.545 1.55 1.555 1.56 1.565

-8

-6

-4

-2

wavelength( m)

Pha

se

(c)

(b)

Ein Eout

a2

a3

a1 b1

b2

b3 3-waveguide coupler

L

(a)x

z

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 cross-coupling 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 over-coupling 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)

, , ( , ) exp( ) ( ), 1, 2, 3,n n nx y z x y i z u z n E M (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


( )

( )

, ,0 ( , ) (0) ( , ) ,

, , ( , ) exp( ) ( ) ( , ) ,

inn n n n n

outn n n n n

x y x y u x y a

x y L x y i L u L x y b

E M M

E M M (2)

where an and bn are the normalized input and output complex amplitudes, respectively. The solution of the coupled mode equations yields [20]

1 1 2 1 1

2 2 1 2 2

3 1 2 1 3

1/ 2 1/ 2

exp( ) 2 ,

1/ 2 1/ 2

b c c c a

b i L c c c a

b c c c a

(3)

where

1 11 22 2

cos( 2 ), sin( 2 ).c L c i L

The strength of the cross-coupling between the two racetrack resonators mediated by the through waveguide is given by |c11/2|. In addition, light propagation along a racetrack gives rise to the following relations

1 1 1 1

3 3 3 3

exp( ) ,

exp( ) ,

a i b

a i b

(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)

1 1 3 1 1 3

21 1 3 1 3

1/ 2 ( ) 2,

1/ 2 ( )

i

out

e c u u c u uE b

c u u u u

(5)

where = βL, and

1

1, 1, 3,nn i i

n

u ne (6)

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.

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 ( 1 3u u ), one readily shows

that the critical coupling condition for such a parallel-coupled dual racetrack structure is given by

1 12 cos 2 .c 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


12

1 1 3

(2 1)1 .

1/ 2 (1/ 1/ ) 1i c

b ec u u

(8)

As c1 is a real number, for a modulated symmetric (η1 = η3) dual-racetrack structure, the output amplitude can vanish only if 1 3 *u u . 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 1 3 32 , and 2 ,m m (9a)

2

1 11

1

(1/ 1)cos 1 ,

1/ 2

c

c

(9b)

where m1 and m3 are two integers. For real nonzero Δθ, this requires

1 12 cos 2 ,c 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 × 1017cm3 according to the well-known plasma dispersion relation reported in [17].

n1 (10-3)

n 3

(10-3

)

-2 -1 0 1 2-2

-1

0

1

2

0

0.2

0.4

0.6

0.8

1

n1 (10-3)

n 3

(10-3

)

-2 -1 0 1 2-2

-1

0

1

2

-3

-2

-1

0

1

2

3

(a) (b)

n1 (10-3)

n 3

(10-3

)

-2 -1 0 1 2-2

-1

0

1

2

0

0.2

0.4

0.6

0.8

1

n1 (10-3)

n 3

(10-3

)

-2 -1 0 1 2-2

-1

0

1

2

-3

-2

-1

0

1

2

3

(a) (b)

Fig. 2. Intensity (a) and phase (b) variations under refractive index modulation for the parallel-coupled 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 × 104, 3.5 × 104). 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 1n and 3n 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 ×


104, 3.5 × 104), 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 1n and 3n 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 push-pull 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


term Δu1 + Δu3 and the “interaction” term Δu1Δu3 on both the numerator and denominator in

Eq. (5). Based on their definitions (1 ) /n ni i i in n nu e e , 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 ni ine . For a racetrack without modulation (Δn1 = Δn3 = 0), iu is small (on the

order of 1η1) near resonance, and Δu1 and Δu3 are in phase. Therefore, we find

1 3 1 1 3| | | 1 / 2 ( ) |u u c u u

because 1η1<2(1/2c1) 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 1 31/ 2 ( )c u u and 1 3u u 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 1 31 / 2 ( )c u u / 1 3u u | 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 1 3 1 3| 1 / 2 ( ) | | | .c u u u u

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

1 12 1

2 12 23 2

3 23 3

( ) 0 0 ( )

( ) 0 ( ) ,

( ) 0 0 ( )

u z u zd

u z i u zdz

u z u z

(11)

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, [ ] [ ][ ]d

m mn ndz u i u , the coupling

matrix is decomposed into the following form [ ]mn X X , where Λ is a diagonal matrix


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

[ ( )] exp( [ ] )[ (0)] exp( ) [ (0)].m mn n nu z i z u X i z X u (12)

Thus the solution of Eq. (11) is given by

2 21 1 3 1 1 3 1 3 1

2 1 3 22 2

3 1 3 1 3 3 3 1 3

( ) cos( ) sin( ) cos( ) (0)

( ) sin( ) cos( ) sin( ) (0) ,

( ) cos( ) sin( ) cos( ) (0)

u z z i z z u

u z i z z i z u

u z z i z z u

(13)

where 2 212 23 , 1 12 / , and 3 23 / . In a symmetric case,

12 232 2 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

2 2 21 1 3 3

1 cos( )1 ,

[1 cos( )]( / / ) 1i L

b eL u u

(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

31

1 3

222 2 23 312 11 1 3 3 2 2

1 3

/ / .1 1

i ii i

i i i i

eeu u

e e

(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.


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 un-modulated 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 × 104, 5.4 × 104) 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.


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 × 104, 4.1 × 104) 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 × 104, which can be readily provided with a low-power 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 × 104, 5.4 × 104) whereas

a symmetric structure requires (Δn1,Δn3) = ( ± 3.5 × 104, 3.5 × 104). 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 digital-to-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


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.

10-2

10-1

100

101

0

0.2

0.4

0.6

0.8

Normalized electrical power

Ma

x. I

nte

nsi

ty

11.1

1.31.5

10-2

10-1

100

101

0

0.2

0.4

0.6

0.8

Normalized electrical power

Ma

x. I

nte

nsi

ty

0.500.75

1.002.00

(a) (b)

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 micro-resonator 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 push-pull 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.


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�Received 6 November 2010; published 2 December 2010�

We analytically study roughness-induced scattering loss in a photonic crystal waveguide �PCW�. A cross-sectional eigenmode orthogonality relation is derived for a one-dimensional �1D�-periodic system, whichallows us to significantly simplify the coupled mode theory in the fixed eigenmode basis. Assisted by thissimplification, analytic loss formulas can be obtained with reasonable assumptions despite the complexity ofPCW mode fields. We introduce the radiation and backscattering loss factors �1 and �2 such that the losscoefficient � can be written as �=�1ng+�2ng

2 �ng is the group index�. By finding analytic formulas for �1 and�2, and examining their ratio, we show why the backscattering loss generally dominates the radiation loss forng�10. The interplay between certain mode-field characteristics, such as the spatial phase, and structureroughness is found crucial in the loss-generation process. The loss contribution from each row of holes isanalyzed. The theoretical loss results agree well with experiments. Combined with systematic simulations ofloss dependences on key structure parameters, the insight gained in this analytic study helps identify promisingpathways to reducing the slow light loss. The cross-sectional eigenmode orthogonality may be applicable toother 1D-periodic systems such as electrons in a polymer chain or a nanowire.

DOI: 10.1103/PhysRevB.82.235306 PACS number�s�: 42.70.Qs, 42.25.Fx, 42.79.Gn, 42.82.Et

I. INTRODUCTION

Photonic crystal waveguides �PCWs� can slow down lightsignificantly, which has important applications such as opti-cal switching and modulation1–3 and all optical storage.4

However, significant optical loss in the slow light regimestymies further advance in this field. Roughness-induced losshas been previously investigated.5–15 The scattering from asingle sidewall irregularity was theoretically studied at first.6

Random sidewall roughness with spatial correlation was laterintroduced to account for loss characteristics in real photoniccrystal waveguide structures.9,10 Although the scaling ofslow light loss with respect to the group velocity, vg, hasbeen examined,5,8–16 it has been difficult to reach a conclu-sive answer. Theory predicted 1 /vg scaling for the radiationloss and 1 /vg

2 scaling for the backscattering loss8,9 in theabsence of multiple scattering. Experimental studies, how-ever, often fitted the loss data with a simple power law vg

−�,where � was found to vary widely.10,11,17 To explain thesevariations, theory should provide a global picture of how thebackscattering and radiation losses �and their relativestrength� vary with a wide range of structure and roughnessparameters commonly found in experiments. More impor-tantly, theory should provide pertinent insight into the loss-generation process and suggest promising pathways to lossreduction.

In this work, we develop a theoretical framework for cal-culating PCW scattering loss based on the coupled modetheory in the fixed eigenmode basis. Here we will prove aninteresting cross-sectional eigenmode orthogonality relation,which allows us to significantly simplify the coupled modetheory in the fixed eigenmode basis. Assisted by this simpli-fication, analytic loss formulas can be obtained with reason-able assumptions despite the complexity of PCW modefields. We will introduce the radiation and backscattering lossfactors �1 and �2, such that the loss coefficient � can beexpressed as �=�1ng+�2ng

2, where ng is the group index. By

finding analytic formulas for �1 and �2, and examining theirratio, we show why the backscattering loss dominates theradiation loss under fairly general conditions. The analyticstudy provides further insight into the underpinning physics,such as how the mode-field characteristics �e.g., spatialphase� interact with roughness to produce loss. The depen-dences of loss on the structure/roughness parameters aresimulated to corroborate the analytic results. Unlike numeri-cal studies that are limited to several instances of structureswith specific structure/roughness parameters, this analyticstudy reveals general loss characteristics and fresh insightinto the loss-generation process, helping identify new path-ways to loss reduction.

This paper is organized as follows. In Sec. II, we willpresent our scattering loss theory. An interesting eigenmodeorthogonality relation will be derived and will be utilized tosimplify the coupled mode theory in the fixed eigenmodebasis. The backscattering and radiation losses will be calcu-lated for the air-bridge type of photonic crystal waveguides,and the loss contribution from each row of holes will beanalyzed. In Sec. III, we will present analytic formulas of thebackscattering and radiation loss factors and give a generalproof of the dominance of the backscattering loss for ng�10. The interplay between the mode-field characteristics�e.g., spatial phase� and the roughness will be analyzed. InSec. IV, we will systematically study the loss dependenceson the structure and roughness parameters such as the holediameter, the waveguide width, and the correlation length.Strategies of reducing the roughness-induced loss will bediscussed. The theoretical results are found to agree wellwith experiments. Section V presents our conclusions.

II. SCATTERING LOSS THEORY

A. Coupled mode theory and mode orthogonality ina PCW crossection

The coupled mode theory of a photonic crystal waveguidecan be written concisely with Dirac notation. This particular

PHYSICAL REVIEW B 82, 235306 �2010�

1098-0121/2010/82�23�/235306�7� ©2010 The American Physical Society235306-1

http://dx.doi.org/10.1103/PhysRevB.82.235306

form of coupled mode theory was first developed by Johnsonet al.18 for taper transitions in photonic crystals, and waslater applied to the disorder-induced scattering problem.12

The theory can use the fixed eigenmode basis or the instan-taneous eigenmode basis. It has the advantage of giving cleardependence of mode coupling on the group velocity throughthe mode normalization factor. In this theory, the Maxwell’sequations are rewritten as18

A�� = − i�

�zB�� , �1a�

A = �� − �−1�t � �−1�t� 0

0 �� − �−1�t � �−1�t�� ,

�1b�

B = � 0 − z�

z� 0�, �� = �Et�x�

Ht�x� �, Et �Ex

Ey� ,

�1c�

Ht �Hx

Hy� ,

where ��X� is the dielectric function, � the permeability. Theeigenmodes, ��=eiz��, satisfy

C�� A + i�

�zB�� = B�� . �2�

Here we consider guided and radiation modes with real .The inner product is defined as

��B�� = z ·� Et� � Ht� + Et� � Ht

��dxdy . �3�

A rigorous formulation of the coupled mode theory must beestablished upon a complete set of orthogonal modes.18,19

For an ordinary waveguide, whose structure is invariantalong z, it is straightforward to show that any two eigen-modes at a given frequency � must be orthogonal19

�B�� = ��. �4�

For a PCW periodic along z, solid-state theory suggests thatthe eigenstate orthogonality can be obtained only by furtherintegration along z,

� ei��−�z�B��dz = ��. �5�

Such an orthogonality relation cannot be directly used in arigorous PCW coupled mode theory because the modal cou-pling coefficients also have z dependence and will appear inthe above integral. To overcome this problem, a complicatedvirtual coordinate theory was previously developed.18

Here we show that Eq. �4� still holds for a PCW in any zsection. By partial integration, one can readily show

�C��= ��C��+ i ��z �B��. Therefore,

�� − ��B�� = i�

�z�B�� . �6�

This is a differential equation of �B��z with a solution

�B��z=e−i��−�z�B��z=0. However, �B��z+a

= �B��z according to Bloch theorem. Therefore,

�B��=�,�−�2n�/a�, which gives Eq. �4� for and � inthe first Brillouin zone.

The orthogonality Eq. �4� for a photonic crystal wave-guide is an interesting result. According to the Bloch theo-rem, the eigenstate orthogonality in a generic one-dimensional �1D�-periodic system should be obtained byintegrating ��b

��adz=0 along the periodicity direction �z inthis case�. However, the above proof has shown that if thereare multiple eigenstates with different on-axis wave vectorsat a given frequency �or photon energy�, they must be or-thogonal by integrating ��b

��adxdy in any cross section per-pendicular to the periodicity axis. Note an equivalent form ofthis orthogonality was proved in a different theoreticalframework based on the Lorentz reciprocity,20 which is lim-ited to electromagnetic wave. The proof given here is gener-ally valid for any scalar or vector wave satisfying Eq. �2�.Therefore, the orthogonality relation presented here may bepotentially applicable to other 1D-periodic systems, such aselectrons in a polymer chain or a nanowire.

The coupled mode theory in the fixed eigenmode basiscan now be established easily based on Eq. �4� for a photonic

crystal waveguide. With a potential perturbation A, themode equation becomes

�A + A�� = − i�

�zB�� , �7�

where ��= ncn�z�einz�n�, and �n� are the eigenmodes of theunperturbed system. The coupled mode theory generally re-quires to use m� to select cm for a particular mode from Eq.�7�. If the conventional orthogonality relation, Eq. �5�, is

applied, the evaluation of �m� ��z B��dz will be problematic

because cm�z� depends on z. With the orthogonality relation,Eq. �4�, however, it is straightforward to show that the cou-pling coefficients are governed by the following equation:

�cm

�z= �i/m�

n

ei�n−m�zm� A�n�cn. �8�

We should emphasize that although it appears similar to theequation for a conventional waveguide homogenous along z,this simplified Eq. �8� can be rigorously established for aPCW only with the help of Eq. �4�. This simplification en-abled by the cross-sectional orthogonality relation, Eq. �4�, isthe main improvement for the coupled mode theory used inthis work. This simplification allows us to derive analyticloss formulas that can be calculated almost by hand, as weshall see in Sec. III, and provides a clearer physical picture.

SONG, INTEGLIA, AND JIANG PHYSICAL REVIEW B 82, 235306 �2010�

235306-2

B. Separate calculation of backscattering lossand radiation loss

The scattering loss can be introduced through a random

potential A due to dielectric perturbation � and ��−1�.For a frequency range with a single guided mode ��, theperturbed mode is given by

�� = c�z�eiz�� + c−�z�e−iz�− � + k

ck�z�eikzz�k� .

�9�

where �k� are radiation modes, and cm�z�, m= �, k are thecoupling amplitudes. With Eq. �4�, it is straightforward tosolve the coupled mode equations to the first order. For unityinput, the output amplitudes are given by

cm = �i/m�� ei�−m�z� A�mdxdydz ,

where � A�m�m� A�, �m= x �m�. The loss coefficient is

given by the conservation of power flux19

� = �1/Lz��c−�2� + k

�ck�2��k/�� , �10�

where the ensemble average · � over the random roughnesshas been applied. To show explicit dependence on the groupvelocity vg, of mode , we introduce U 1

4 � /vg,�, thetime averaged mode energy per unit length along the z axis.For a radiation mode �k�, we define Uk 1

4 �k /vgz,k�, wherevgz,k is the z component of vg,k. Then the m terms in Eq. �10�can be replaced by Um and vg. Assuming that the sidewallroughness of different holes is uncorrelated,9 the ensembleaveraged � of a PCW is a sum of the ensemble averaged losscontribution from each hole. For roughness-related calcula-tion, it is more convenient to use the polar coordinates �r ,��in each hole in place of �x ,z�. After some calculations, wefind

� = �1ng + �2ng2, �11a�

�1 = �1/a� k

nx

I�k,,nx��c/vgz,k��Uk/U� , �11b�

�2 = �1/a� nx

I�− ,,nx� , �11c�

where nx and nz are the indices of holes along x and z, re-spectively �see Fig. 1�. The PCW has a lattice constant a,mean hole radius r0, and slab thickness tslab. The integral forthe nxth hole is

I�m,,nx� �r0tslab/4Umc�2

��nxei�−m�r0�sin �−sin �� Am,nx

� �� Am,nx��

� r�� r��d��d� , �12�

where Am,nx��= �1 / tslab�� Am �r=r0

+dy. A typical autocor-

relation function is given by r�� r��=�2e−��−��r0/lc,where � and lc are the rms roughness and correlation length,

respectively. Note that the coordinates �r ,�� are centered ineach cell �nx

.Now the loss coefficients can be numerically calculated

using Eqs. �11� and �12�. Instead of directly calculating theloss coefficient �, we will calculate the radiation and back-scattering loss factors �1 and �2. Note that � diverges as thefrequency approaches the band edge whereas �1 and �2 areslowly varying functions even near the band edge. Thus thecalculation of �1 and �2 generally leads to significantly morestable numerical results than directly calculating �.

Here we consider the TE guided modes �i.e., electric fieldprimarily in the xz plane� of a Si air-bridge PCW. The guidedmodes can be obtained by a preconditioned eigensolver21

with a tensorial average of the dielectric constant nearinterfaces.22 The perturbation potential is evaluated using thecontinuous components on interfaces.23

The radiation modes are calculated by considering thePCW supercell delineated in dashed lines in Fig. 1 �the oneused in actual calculation is much longer along x� as oneperiod of a two-dimensional grating in the x-z plane. Themode field for a given plane-wave incident upon the PCWtop surface can be obtained by any grating diffractiontheory.24,25 Due to the artificial x periodicity imposed by thegrating theory, this treatment is equivalent to calculating theradiation loss for an array of parallel PCWs. For a suffi-ciently large spacing between waveguides, the radiationlosses of adjacent waveguides are independent of each otherfor weak scattering. Figure 2�a� clearly shows that only thefirst two rows �nx= �1, �2� contribute significantly to theradiation loss. For each row, data plotted in symbols andlines are obtained by two supercell sizes differing by 50%.Their small differences of �1 confirm that adjacentwaveguides do not affect each other. The backscattering lossshows even stronger dominance by the first row �Fig. 2�b��.Obviously, this can be attributed to the fact that the scattering

matrix elements −� A�� and k� A�� involve ��x�,which decays very fast with x.

III. ANALYTIC FORMULAS FOR BACKSCATTERINGAND RADIATION LOSSES

Interestingly, the factors �1 and �2 roughly have the sameorder of magnitude in Fig. 2. As a consequence, the back-scattering loss ��2ng

2� dominates the radiation loss ��1ng�,which can be seen from their ratio

x

z

�1

2

3

-3

-2

-1nz

nx

Eeff

FIG. 1. In-plane view of a line-defect waveguide in a photoniccrystal slab.

SLOW LIGHT LOSS DUE TO ROUGHNESS IN PHOTONIC… PHYSICAL REVIEW B 82, 235306 �2010�

235306-3

�2ng2

�1ng� ng � 1 �for ng � 10� .

Numerical simulations of a few other PCW structuresshowed similar dominance.9,26 Mathematically, the ng

2 termsurely dominates the ng term in Eq. �11a� for a sufficientlylarge ng. But the ng threshold for the onset of this dominancedepends on �1 and �2 and could be too large to be observed�e.g., ng�1000�. To ascertain the universal dominance ofbackscattering in practically observable ng ranges and to ex-plore the underpinning mechanism of this dominance, ananalytic study is needed. Moreover, such a study may offerinsight into the interaction between the mode field androughness.

We have performed analytic calculation of the factor �2with some simple reasonable assumptions. As a first step, weassume a guided mode field of the form E�e−�xx/2eiz. Af-ter some calculation, we find

I�− ,,nx� = ��r0tslab/4Umc�2� �12�2�Eef f ,nx�4�2Iang,

Iang =� ei�2r0��sin �−sin ��−�xr0�2+cos �+cos ��e−��−��r0/lcd��d� ,

�13�

where �12 is the dielectric constant difference, Eef f ,nxis the

effective field at the hole’s inner edge ��=� in Fig. 1�. Typi-cally, the correlation length lc is small. For e−r0/lc �1, �xlc�1, and 2lc�1, one finds

Iang � �4�lc/r0�I0�2�xr0� , �14�

where I0�x�= I0�x�exp�−x� and I0 is the modified Besselfunction of the first kind. One can show that Eq. �14� stillholds for a more general form of the field E

�e−�xx/2 GuGei�+G�z under two scenarios: �1� the mode isdominated by Fourier terms satisfying Glc�1 so that thephase of each eiGz varies little within one correlation lengthand �2� near the band edge where ��x��−�x�. For the

second scenario, the phases of ��x� and �−� �x� almost ex-

actly cancel each other in −� A�� and become irrelevant.When these conditions are not satisfied, the spatial phasevariations tend to reduce Iang below the value given in Eq.�14�.

For a guided mode, we can define a modal field ampli-

tude, Esp,, by U=�0Esp,2 wdtslab /2 and normalize the effec-

tive field as eef f ,=Eef f , / Esp,. Then combining Eqs. �11c�,�13�, and �14�, we obtain

�2 � 2Nx,back��n12 − n2

2�2�k02�2lcr0/awd

2��eef f ,�4I0�2�xr0� ,

�15�

where k0=2� /�, n12−n2

2= �12 /�0, and 2Nx,back is the effec-tive number of rows of holes contributing to backscattering.For numerical estimate, we assume Nx,back�1, wd=w0�3a. In addition, Eef f , is obtained by averaging �E�2 atthe inner hole edge across the slab thickness. We find thateef f , typically varies around 0.3–0.4 in the slow light re-gime. The decay constant �x�0.77�2� /a� is obtained byfitting the mode energy against x near the band edge. NoteI0�2�xr0� is a slowly varying function for this parameterrange of interest. Figure 2�b� shows that Eq. �15� gives areasonable estimate of the order of magnitude of �2 and itstrend. There is an overestimate of two to three times becausewe have neglected the following factors: �a� the vector natureof the field; �b� the high-G Fourier components; and �c� thevariation in the field along y.

For the radiation modes, considering two polarizations��=1,2� and two propagation directions �sz= �z�, the sumover k in Eq. �11b� becomes k→ �,sz

LxLy

�2��2 �dkxdky, whereLx and Ly are the transverse dimensions of the normalizationvolume. Note the final result of �1 is independent of LxLybecause I�k , ,nx� · �Uk��LxLy�−1 in Eq. �11b�. One can thenshow that

�1 � 2Nx,rad�n12 − n2

2�2nsub3 �k0

4�2lcr0tslab/awd�

��eef f ,eef f ,k�2I0��xr0� , �16�

where eef f ,k is the normalized field amplitude at the holeinner edge averaged over all k states, nsub=1 is the substraterefractive index, and 2Nx,rad is the effective number of rowsof holes contributing to radiation loss. Comparing Eq. �15�and Eq. �16�, we find

�1

�2�

Nx,radnsub3 k0

2wdtslab

�Nx,back·

�eef f ,k�2

�eef f ,�2·

I0��xr0�I0�2�xr0�

. �17�

With Nx,rad, Nx,back=1�2, wd=w0, tslab�220 nm, �x�0.5�2� /a�, and normalized fields eef f ,, eef f ,k�0.5, eachratio in Eq. �17� is on the order of unity. This equation there-fore predicts that �1 and �2 are generally on the same order.Therefore, this analytic study explains why the backscatter-ing generally dominates, �2ng

2��1ng, in the slow light re-gime ng�10. Note that Eqs. �15� and �16� contain no fast-varying functions, which implies that �1 and �2 should befairly insensitive to most structure parameters for a typicalPCW.

0

0.1

0.2

0.3

α 1(d

B/c

m)

row ±1

row ±2

row ±3

total

0

50

100

ng

0.265 0.27 0.275 0.280

0.2

0.4

0.6

ωa/2π

α 2(d

B/c

m)

row ±1

row ±2

row ±3

total

estimate

(b)

(a)

ng

FIG. 2. �Color online� Loss factors as a function of frequencyand the contribution from each pair of rows of holes. �a� Radiationloss factor �1; and �b� backscattering factor �2 and the analyticestimate. PCW parameters: �=430 nm, r0 /a=0.25, tslab=200 nm,�=3 nm, and lc=40 nm.


235306-4

Note that prior scattering loss formulas still involve thephotonic crystal mode field and the Green’s function,9 whichmust be obtained through further computation. Our analyticloss formulas, Eqs. �11a�, �15�, and �16�, do not have theseterms, and can be evaluated almost by hand. More impor-tantly, the ratio of �1 and �2 derived from these formulas, aspresented in Eq. �17�, gives a general mathematical proof ofthe dominance of the backscattering loss over the radiationloss, along with a predicted dominance threshold ng�10.Prior numerical studies discovered this dominance in a lim-ited number of structures with specific parameters.9,26 How-ever, the generality of the dominance and its threshold ngwere not clearly determined in numerical studies.

IV. DISCUSSION

A. Loss dependence on structure and roughness parametersand loss reduction strategy

As the backscattering loss dominates, we focus on thedependences of �2 on several key roughness/structure pa-rameters. Note that the tensorial average of the dielectricfunction near interfaces is found to significantly improve theconvergence with the spatial grid size, as shown in Fig. 3�a�.This allows us to study small structure parameter changes.First, we examine the limitation of the preceding analyticresults due to the assumption of small lc. The dependences of�2 on lc for various normalized values are plotted in Fig.3�b�. For guided modes near the band edge �a /2��0.5�,�2�lc� is almost perfectly linear. As discussed above, thislinearity predicted in Eq. �15� is due to ��x��−�x� nearthe band edge, which causes phase cancellation in

−� A��. Away from the band edge, the phase variationcauses the integral Iang to become sublinear at large lc values�but Eqs. �15� and �16� remain useful as estimates�, which isalso confirmed in Fig. 3�b�. Second, the dependence on thewaveguide width is studied in Fig. 3�c�. The loss factor �2could be reduced by a factor about 5 from wd=0.83w0 to

1.1w0 near the mode edge. Third, in most experimentalworks, the air hole diameter and slab thickness usuallyspread over certain ranges �e.g., r0 /a :0.23–0.29 andtslab :0.19–0.25 �m� and the exact values may vary due touncertainties in fabrication processes. Our simulations showthat �2 varies insignificantly over the typical ranges of a, r0,and tslab. The variation of �2�a /2��0.5� is plotted againstr0 /a in Fig. 3�d�.

The analytic and computational studies offer insight intothe loss mechanism and point to promising pathways to lossreduction. First, among four essential geometric parameters�r, a, tslab, and wd�, wd appears to be the only one that allowsfor substantial loss reduction. Second, the spatial phaseanalysis in the derivation of Eq. �15� suggests that designingguided modes with accentuated high-wave-number Fouriercomponents might help reduce the loss due to random rough-ness. But the eigenfrequency and other deterministic charac-teristics of such a mode also tend to be sensitive to the varia-tions in structure parameters �mean value�. Thus, ingeniousdesigns are needed to account for both statistical and deter-ministic properties. Third, manipulating the polarization,through introducing anisotropic materials, for example,could yield loss much lower than that predicted in Eq. �15�,which neglects the polarization. Lastly, Eqs. �15� and �16�and the spatial phase analysis may offer new insight into themode 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 � /ng

2��2� curve nearthe band edge. This can be partially explained by the fact thatthe integral of the guided mode intensity �Eb�x��2 over thehole surface increases with the group index.27 However, afull explanation must be based on the characteristics of the

random potential matrix element −� A��. As discussed

above, the phase cancellation in −� A�� causes an in-crease in Iang and �2 near the band edge. Due to the interplay

10 20 30 400

0.1

0.2

0.3

0.4

Ngrid

|∆α 2|/α

2

(a)<ε>ij

ε

0 20 400

0.1

0.2

0.3

0.4

lc

(nm)

α2

(dB/cm

)

(b)0.4750.4250.375

0.8 0.9 1 1.10

0.5

1

Wd/W

0

α 2(d

B/c

m)

(c)

0.23 0.25 0.27 0.290

0.5

1

r/a

α2

(dB/cm

)

(d)

FIG. 3. �Color online� Variation in �2 with �a� grid size per edgeof the unit cell, with tensorial average ��ij and without �up to 30%oscillation�; �b� correlation length lc, for modes at different a /2�;�c� PCW width; and �d� hole radius. PCW parameters: a=420 nm,r0 /a=0.25, tslab=220 nm, �=3 nm, and lc=40 nm.

0.265 0.27 0.275

10−1

100

101

102

ωa/2π

α/n g;

α/n g2

(dB

/cm

)

theory − α/ng

theory − α/ng2

exp. − α/ng

exp. − α/ng2

FIG. 4. �Color online� Comparison with experimental results inRef. 10. The experimental spectrum is shifted to align the band edgewith the theory.


235306-5

between the spatial phase of the mode and the roughness,this upswing is stronger for larger correlation lengths. Be-cause �1 and �2 are not constant in general, a simple powerlaw fitting ��ng

� of experimental data would unlikely giveconsistent � values, which agrees with the findings of Ref.27. Note that if the coupling loss28,29 is included, the loss-ngrelation could even become sublinear �or logarithmic�, espe-cially for short waveguides. Above �=�a /2�c=0.273, thelocalized band tail states21,30 of the second guided mode�band edge ��0.281� introduce in the experimental spec-trum a broad resonance accompanied by a “softened” vg atthe nominal band edge.31 This effect is beyond the scope ofthis work. Fortunately, this effect can be avoided by design-ing the second mode above the useful spectral range of thefirst mode. Below a sufficiently small vg, multiple scatteringoccurs for the first mode, accompanied by undesirably highloss.11,26,31–33 The studies presented here could help reducescattering losses and delay the onset of this regime.

In this work, we have considered loss introduced byguided and radiation modes with real values. In a nonper-turbed photonic crystal structure �including a PCW�, modeswith complex values generally arise locally near the endfaces and affect the end-face coupling loss34 but not thepropagation loss of a truly guided mode. The propagationloss is generally more important for a sufficiently long pho-tonic crystal waveguide. Also within the photonic band gapof a PCW, those modes with complex values usually donot carry away energy themselves and thus may not intro-duce propagation loss directly. Some higher order �multiple�scattering processes in a PCW with random perturbations

may involve these modes as an intermediate step. These mul-tiple scattering processes are usually negligible in practicallyuseful �relatively low loss� spectral ranges of photonic crys-tal waveguides, as discussed in the comparison with experi-mental data above.

V. CONCLUSION

In summary, analytic formulas, Eqs. �11a�, �15�, and �16�,of the PCW scattering losses can be obtained despite thecomplexity of the PCW mode fields. With these formulas,the loss of a typical photonic crystal waveguide can be esti-mated almost by hand. The analytic study reveals that theinterplay between the mode characteristics and the structureroughness may hold the key to loss reduction. These resultsare corroborated by systematic simulations with varyingstructure parameters. As a byproduct, the cross-sectionaleigenmode orthogonality relation for a 1D periodic systemmay be applicable to other problems, such as electrons in apolymer chain or a nanowire.

ACKNOWLEDGMENTS

We are grateful to David Vanderbilt, Steven G. Johnson,Chee Wei Wong, Philippe Lalanne, Stephen Hughes, EiichiKuramochi, Fabian Pease, Leonard C. Feldman, George K.Celler, and George Sigel for helpful discussions. This workis supported by AFOSR MURI under Grant No. FA9550-08-1-0394 �G. Pomrenke�.

*Electronic address: [email protected] Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab,

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EVIEW

Copyright © 2010 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofNanoscience and Nanotechnology

Vol. 10, 1596–1605, 2010

Longitudinal and Angular Dispersions inPhotonic 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

The slow-light effect and the superprism effect are two important effects in photonic crystal struc-tures. In this paper, we will review some of our recent works on the fundamental physics and deviceapplications of these two effects. We will present a synergistic perspective that examines these twoeffects as a whole. Apparently, the slow light effect is due to the dispersion of a photonic crystalalong the direction of light propagation, namely the longitudinal direction, and the superprism effectis related to angular dispersion. However, a deep analysis will show that the superprism effect hasan elusive dependence on the longitudinal dispersion as well. Some subtle connections and distinc-tions between the slow-light effect and the superprism effect will be revealed through our physicalanalysis. This allows us to treat these two effects under a common theoretical framework. As anexample, we will apply this framework to make a direct comparison of the slow-light optical phasearray approach and the superprism approach to beam steering applications. Dispersive effects arefrequently accompanied by high optical loss and/or narrow bandwidths. We will discuss these issuesfor both longitudinal and angular dispersions. For the slow light effect, we will give a simple proofof the scaling of fabrication-imperfection related random scattering losses in a slow-light photoniccrystal waveguide. Similar to the bandwidth-delay product for the longitudinal dispersion, we willintroduce a simple, yet fundamental, limit that governs the bandwidth and angular sensitivities ofthe superprism effect. We will also discuss the application of the slow-light effect to making compactsilicon optical modulators and switches.

Keywords: Photonic Crystals, Slow Light, Superprism, Dispersion, Optical Loss, Bandwidth,Silicon Photonics.

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15962. The Slow Light Effect and the Longitudinal Dispersion . . . . . . 1598

2.1. The Origin of the Slow LightEffect in Photonic Crystals . . . . . . . . . . . . . . . . . . . . . . . . 1598

2.2. Applications of the Slow Light Effect . . . . . . . . . . . . . . . . 15982.3. Loss-Limited Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1600

3. The Superprism Effect and the Angular Dispersion . . . . . . . . . . 16013.1. Dispersion Surface Curvature and

Angular Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16013.2. Slow-Light Induced Strong Angular Dispersion . . . . . . . . . 16023.3. “Pure” Angular Dispersion Effect:

Bandwidth Limited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16034. Longitudinal Dispersion versus Angular Dispersion:

A Direct Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16035. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604References and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604

∗Author to whom correspondence should be addressed.�This is an invited review paper.

1. INTRODUCTION

Photonic crystals possess a wide range of extraordinaryproperties that are absent in conventional materials. Firstand foremost, the periodic structure of a photonic crys-tal causes photonic bands and bandgaps to form on thefrequency spectrum of photons. Therefore, photonic crys-tals with photonic bandgaps can serve as “perfect mirrors”to confine light in small dimensions, forming ultracom-pact waveguides and cavities. On the other hand, there areother technologies that can also provide tight confinementof light. Compared to these alternative technologies, theuniqueness of photonic crystal-based waveguides and cav-ities often comes from the fact that the periodic structureof a photonic crystal provides some additional distinctiveopportunities to modify the spectral property of light, lead-ing to many dispersive effects with a wide range of appli-cations. For example, in a photonic crystal waveguide, the

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Integlia et al. Longitudinal and Angular Dispersions in Photonic Crystals

dispersion relation, ��k�, generally has a portion whereits slope tends to zero, implying a vanishing group veloc-ity. Such a slow light effect, together with the tight lightconfinement provided by the photonic bandgap, leads toextraordinary enhancement of phase shift and time delayin such a waveguide.1 We shall emphasize that some ofthese dispersion effects can be significant in their ownright, without the presence of tight light confinement. For

Ryan Integlia received his M.S. in 2008 jointly from Rutgers Civil and EnvironmentalEngineering Department and Electrical and Computer Engineering Department studying thesubject of wireless sensor networks. He is currently researching the subjects of the slow lighteffect and the superprism effect. He received his bachelor’s degree in 2001 from RutgersUniversity’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 AdvancedInfrastructure and Technology Fellowship, among other honors.

Weiwei Song received his M.S. in 2008 from the Department of Physics of Nanjing Univer-sity, China. He is currently pursuing his Ph.D. degree in the Department of Electrical andComputer Engineering of Rutgers University and working in the group of Professor WeiJiang. His research involves silicon photonic crystal structures and devices, with a focuson the simulation and characterization. In 2005, Weiwei graduated from Nanjing Universitywith 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. in2008 from Shanghai Institute of Technical Physics, Chinese Academy of Sciences. Cur-rently, 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 andcharacterization.

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 computerengineering from the University of Texas, Austin, in 2000 and 2005, respectively. He heldresearch positions with Omega Optics, Inc., Austin, Texas from 2004 to 2007. Since Septem-ber 2007, he has been an assistant professor in the department of electrical and computerengineering of Rutgers University, Piscataway, NJ. His doctoral research made a contribu-tion to the fundamental understanding of the wave coupling, transmission, and refractionat a surface of a periodic lattice. At Omega Optics, he led a research project to the suc-cessful demonstration of the first high-speed photonic crystal modulator. He also recognizeda scaling law for the current density of high speed silicon electro-optic devices. His cur-rent research interests include photonic crystals, silicon photonics, optical interconnects,and beam steering. He received Ben Streetman Prize of the University of Texas at Austinin 2005.

example, the superprism effect2 causes the beam propaga-tion angle inside a photonic crystal to be highly sensitiveto the wavelength of light. Essentially, this effect amountsto significantly enhanced angular dispersion. In the super-prism effect, the light propagation is not confined. More-over, we will show that the superprism effect does notnecessarily appear near a bandgap (or at a bandedge). Itcan indeed appear in the midst of a photonic band, thanks

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Longitudinal and Angular Dispersions in Photonic Crystals Integlia et al.

to the high symmetry of a photonic crystal structure. Thepresence of a photonic bandgap is not a necessary condi-tion for the superprism effect.In this paper, we will review some of our recent the-

oretical and experimental works concerning these disper-sive effects. In these works, our emphasis was placed onfinding the general, quantitative physical laws governingthese effects. For example, in the first superprism experi-ment, it was found that the photonic crystal could enhancethe angular dispersion or sensitivity by 500 times;2 laternumerical simulations and experimental works reportedvarying enhancement factors.3–7 However, a rigorous, com-pact mathematical form of the physical law that canexpress this enhancement factor in terms of the photonicband parameters is missing. As such, while we can easilyobtain an instance of high sensitivity structures, we cannot systematically predict the trend of such an effect andwe do not know whether there is a quantitative upper limitof the enhancement factor. Our works were devoted to elu-cidating such issues.While the slow light effect is obviously due to the dis-

persion of a photonic crystal along the direction of lightpropagation, namely the longitudinal direction; the super-prism effect, apparently related to angular dispersion, willbe shown to have an elusive dependence on the longi-tudinal dispersion as well. Some subtle connections anddistinctions between the slow-light effect and the super-prism effect will be revealed through our physical anal-ysis. This allows us to examine these two effects undera common theoretical framework based on photonic crys-tal dispersion function, ��k�. An example will be used toillustrate the value of this synergistic theoretical perspec-tive on the slow-light effect and superprism effect, twoseemingly distinctive phenomena in photonic crystals. Dis-persive effects are frequently accompanied by high lossand/or narrow bandwidth. We will discuss these issuesfor both longitudinal and angular dispersions. Similar tothe bandwidth-delay product for the longitudinal disper-sion, we will introduce a simple, yet fundamental, limitthat governs the bandwidth and sensitivities of the angulardispersion.

2. THE SLOW LIGHT EFFECT ANDTHE LONGITUDINAL DISPERSION

2.1. The Origin of the Slow LightEffect in Photonic Crystals

The group velocity of light can be slowed down in varioustypes of photonic crystal structures, especially when thewavelength of light approaches a bandedge. Two commoncases shall be considered.(1) For a “bulk” photonic crystal, such a bandedge typi-cally appears around some high symmetry points in recip-rocal space.8 As such, in real space, slow light propagation

typically occurs along certain high symmetry axes of aphotonic crystal.(2) For a photonic crystal waveguide (PCW) composed ofa line-defect, generally the waveguide is already alignedwith a high symmetry axis of the photonic crystal lat-tice (for example, the �K axis of a hexagonal lattice9).The original lattice periodicity remains along the lon-gitudinal direction of the waveguide. This results in aone-dimensional (1D) photonic band structure with a max-imum or minimum (or other types of extrema) at the 1DBrillouin zone (BZ) boundary � = �/a, where � is thepropagation constant of the photonic crystal waveguide inquestion. Since the dispersion relation �� is generally asmooth function, an extremum ensures vg = d�/dk= 0 atthe BZ boundary. Therefore, the periodicity along the lon-gitudinal direction dictates that a vanishing group velocitymust exist in such a photonic crystal waveguide.

2.2. Applications of the Slow Light Effect

The slow group velocity of light renders the phase shiftin a photonic crystal structure more sensitive to refractiveindex changes.1 Generally, as the refractive index changes,the dispersion relation of a photonic crystal or a PCW willbe shifted by a certain amount �� = ��n/n� alongthe frequency axis. Here � is the frequency of light, n isthe refractive index, and is a factor typically on theorder of unity. In many cases, we are interested in a smallfrequency range where can be regarded as a constant. Inthe case of a PCW, the factor can be interpreted as thefraction of the mode-energy in the waveguide core region.For a given wavelength, the propagation constant changesas �� = ��/vg . Therefore, the phase shift induced by arefractive index change of �n is given by1

�= ��L= ng�n

n

2�L

�(1)

where ng = c/vg is the group index. Evidently, a slowgroup velocity (or a high ng) enhances the phase shiftsignificantly.To exploit such a significant slow-light enhancement,

a number of physical mechanisms10–14 have been employedto change the refractive index and actively tune thephase shift in a photonic crystal waveguide. Here webriefly review our works on thermo-optic and electro-optic tuning of the phase shift for optical modulationand switching applications. Many common semiconduc-tor materials, such as silicon and GaAs, have appreciablethermo-optic coefficients (dn/dT> 10−4/K). As such, theyare suitable for making thermo-optically tunable slow-light photonic crystal devices. In these devices, thermalexpansion also contributes to the tuning of the phaseshift. In many cases, these two effects add upon eachother to produce a larger phase shift. We have demon-strated thermo-optic tuning in a photonic crystal waveg-uide Mach-Zehnder interferometer with an interaction

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(a)

(c) (d)

(b)

Fig. 1. Photonic crystal waveguide based Mach-Zehnder modulators/switches. (a) A generic photonic crystal waveguide based Mach-Zehnder inter-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-opticmodulation (inset: micrograph of an electro-optic device—top view); (d) cross-sectional schematic of an active arm with an embedded silicon MOScapacitor for electro-optic modulation.

length of 80 �m.10 A photonic crystal waveguide Mach-Zehnder interferometer with one active arm is schemat-ically illustrated in Figure 1(a). A close-up view of thearm with thermo-optic tuning is shown in Figure 1(b). Thedevice was patterned on a silicon-on-insulator (SOI) waferusing a combination of e-beam nanolithography and pho-tolithography. The switching rise time and fall time weremeasured 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 andBennett found the following relation between the refractiveindex of silicon and the carrier concentrations for wave-lengths near 1.55 �m15

�n=− 8�8×10−22�Ne+8�5×10−18��Nh�0�8� (2)

where �Ne and �Nh are electron and hole concen-tration changes, respectively. A refractive index changeup to �n∼ 10−3 can be obtained with �Ne = �Nh ∼3× 1017 cm−3. We demonstrated the first high-speed pho-tonic crystal waveguide modulator on silicon in 2007.11

A schematic of the active arm of the device is shown inFigure 1(c). The device was made on a silicon-on-insulatorwafer through a series of micro- and nano-fabricationprocesses, including e-beam lithography, photolithography,dry and wet etching, ion implantation, and metal lift-off. The device had a measured modulation bandwidthin excess of 1 GHz, with the lowest driving voltage forhigh-speed silicon modulators at the time of publication. It

should be noted that the introduction of air holes does notsignificantly increase the electrical resistance of silicon.In Figure 2, we plot the electrical resistance of a photoniccrystal waveguide made in a silicon slab for varying airhole sizes and varying number of rows of air holes. In thesimulation, the electrical contact pads were assumed to beplaced at a fixed separation about 10 �m along the sidesof the photonic crystal waveguide. The contact resistance

0 5 10 15 20

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

N

Res

ista

nce/

R0

0.05

0.1

0.2

0.3

0.4

Fig. 2. Electrical resistances for a photonic crystal waveguide in a sil-icon slab. The horizontal axis indicates the total number of rows of airholes between two electrodes separated by 10 �m. The different curvescorrespond to different values of r/a (0�05∼ 0�4), where r is the radiusof the air holes and a is the lattice constant (∼400 nm). The resistancevalues are normalized by the original slab resistance, R0.



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can generally be neglected. Evidently, the electrical resis-tance may increase about 2.6 times for a large hole radiusand for a large number of holes, but the value generallyremains on the same order of magnitude as the originalsilicon slab. In our experiments, we also did not observean order-of-magnitude change of the resistance after thephotonic crystal structures were etched in a silicon slab.The resistance values in Figure 2 were computed by 2Dfinite 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 forward-biased p–i–n diode11�12

j = 2qwi�Nf (3)

where wi is the intrinsic region width of the p–i–ndiode, and f is the modulation frequency. Combining Eqs.(2) and (3), we obtained a minimum AC current densityof 104 A/cm2 for high speed (>1 GHz) modulation in atypical SOI waveguide. In addition, we showed that dueto the non-ideal diode I–V relation I ∼ exp�qV /2kBT � athigh injection, it is possible to limit (or “lock”) the injectedcarrier concentrations to around �N ∼ 3× 1017 cm−3 fora diode with proper doping levels and under normal for-ward bias conditions. This ensures that the silicon modu-lator naturally works under the most desired electro-opticstate. In a follow-up work, we predicted that an RF powerconsumption of less than 50 mW is possible for 10 GHzsilicon modulators.12 Subsequently, IBM demonstrated a10 GHz silicon modulator with 50 mW RF driving powerin the forward bias mode;16 MIT Lincoln Laboratory alsoreported similar power consumption for 10 GHz siliconmodulators.17 These results affirmed the value of Eq. (3)in designing high speed silicon modulators.We also developed a metal-oxide-semiconductor (MOS)

type photonic crystal waveguide modulator, as illustrated inFigure 1(d). A MOS capacitor can be embedded into a slotphotonic crystal waveguide, where the slot is filled withoxide.18 In such a waveguide, there exist two enhancementeffects: the slow light effect, and the field boost inside thelow-dielectric slot due to the continuity of surface-normaldisplacement vector component. Our most recent resultsshow that such a configuration can help reduce the powerconsumption of a silicon MOS modulator.19

As mentioned earlier, the slow light effect can also occurin a “bulk” photonic crystal without intentionally introduc-ing line-defects. Such a configuration has been explored inphotonic crystal slabs made of conventional electro-opticmaterials such as LiNbO3.

20

2.3. Loss-Limited Effect

The practical application of the slow-light effect is primar-ily limited by optical loss. For most practical applications,group velocity values of 100 or less have been currentlyconsidered. Further slowing down light causes the optical

loss to increase significantly. To understand the slow lighteffect, a close examination of the accompanying opticalloss is warranted.The total insertion loss of a photonic crystal waveguide

is given by

Loss�dB�= 10 log10C1+10 log10C2−�L (4)

where C1 and C2 are the coupling effeciencies at the inputand 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

�= �1ng +�2n2g +· · · (5)

where the first term can be attributed to absorption andout-of-plane scattering by random imperfections in thephotonic crystal waveguide, and the second term can beattributed primarily to back-scattering (due to randomimperfections) into the reverse propagating mode with anidentical group index.Several works21�22 have discussed the scaling of scat-

tering loss theoretically. Here we give a proof of Eq. (5)that does not invoke the detailed solution of the waveguideequation. We consider the scattering process due to ran-dom imperfections in a photonic crystal waveguide. Forany scattering event of interest, the initial state must bea guided mode, which we assume has a propagation con-stant �. The final state can be a guided mode or a radiationmode. For a line-defect waveguide formed in a photoniccrystal slab, the radiation modes propagate out of theplane. Assume the scattering amplitude between an arbi-trary initial state � and a final state k is Tk�. In any physicalsituation, the incoming light is always a wave-packet witha continuous distribution of � values, although often the �values are within a narrow range centered around �0. Thescattering coefficient for such a wave-packet is roughly

Seff ∼∫d�

∑k

�Tk��2��f −�i�

∼∫d�

∫dk′3�Tk′��2��f −�i�

+CB

∫d�

∫d�′�T�′��2��f −�i� (6)

where k′ represents a final radiation mode, �′ representsa final guided mode, and CB is a constant. The factor��f −�i� ensures that the frequencies of the initial andfinal states are the same (energy conservation). The �1/vg�factor will arise naturally from each integration of an arbi-trary function with respect to � or �′

∫f ��d�=

∫f ��d�

∣∣∣∣d�d�∣∣∣∣= 1

vg

∫f ��d�� (7)

Therefore, we find

Seff ∼ �T1/vg��+ �T2/ vg��vg��′��

= �T1/vg��+ �T2/v2g�� (8)



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where �T1 and �T2 are some constants. The second line inEq. (8) follows from �′ = −�, according to energy con-servation in typical photonic crystal waveguides. Note thata similar factor 1/vg�k

′� may arise from the integration∫dk′3 as well. However, the group velocity, vg�k

′�, of aradiation mode never vanishes. Therefore, this factor hasno significance here and is absorbed into �T1. Thus, Eq. (5)is proved.The above derivation clearly shows that the out-of-plane

scattering has only one ng factor because only the initialstate is a slow-light state, whereas the backscattering pro-cess has a n2

g factor because both the initial and final statesare slow-light states. We would be tempted to assumethat the value of the second integral in Eq. (6) is muchsmaller than that of the first, because there are a large num-ber of radiation modes that satisfy the energy conserva-tion whereas only one backward guided mode does so. Inother words, the total “scattering cross-section” of all radi-ation modes could be much larger than that of the back-ward guided mode. However, a general, rigorous proof isneeded.Experimentally, quantitative evaluation of these scaling

laws has been elusively difficult and the reported lossdependences23–26 vary between v−1/2

g and v−2g , as discussed

in Ref. [27]. It should be clarified that because the scatter-ing events occur statistically uniformly over a given dis-tance, the scattering loss should have the general form� = −�1/L� logTprop ∼ v−�

g , not Tprop ∼ v−�g . On the other

hand, the coupling loss coefficients should have the formC1, C2 ∼ v�g , where �= 1 for a normally (abruptly) termi-nated photonic crystal waveguide (see Eq. (15) and relateddiscussions in Section 3.1). Therefore, for a normally ter-minated photonic crystal waveguide, the total insertion lossis given by

Loss(dB)≈ B0−10 log10 ng − ��1ng +�2n2g�L (9)

where B0 is a constant that gives the “baseline” insertionloss.Here we list several issues in characterizing the opti-

cal loss in the slow light regime against Eqs. (4) and (5),or (9): (1) the unknown relative magnitudes of �1 and�2; (2) the difficulties of separating the coupling loss andpropagation loss in experiments; (3) the proper applicationof the scaling laws of the propagation and coupling losses.It is our feeling that systematic and careful experimentalstudies over a wide range of waveguide parameters must beperformed before a conclusive statement can be put forthregarding the optical loss in the slow-light regime.

3. THE SUPERPRISM EFFECT ANDTHE ANGULAR DISPERSION

When a light beam is incident upon a photonic crystal sur-face, the refraction angle inside the photonic crystal couldbe 500 times more sensitive to the wavelength perturbation

than in a conventional medium.2 This so called superprismeffect is a manifestation of the strong angular dispersionof a photonic crystal.Significant progress has been made in investigat-

ing the superprism effect in the first ten years afterits initial discovery. A number of experiments havedemonstrated the potential of the superprism effect inwavelength division multiplexing, beam steering, andsensing applications.3–5�7�28�29 Nonetheless, many funda-mental questions remained unanswered: (1) How to expressthe angular dispersion or angular sensitivity of a photoniccrystal in terms of basic parameters of a photonic bandstructure as we have seen in the slow-light effect? (2) Isthere an ultimate limit of the angular sensitivity of a pho-tonic crystal? (3) If there is a limit, what are the limitingfactors?To build a foundation for the solution of these problems,

we developed a rigorous theoretical framework to com-pute the transmission and reflection coefficients for refrac-tion across a photonic crystal surface in a 2005 work.30

A parallel work was reported by a group at the Univer-sity of Toronto in the same period.31�32 Subsequently, wedeveloped the first theory to systematically address theaforementioned general questions in a 2008 work.33 Whilethe key parameter for tuning the longitudinal dispersionis the group velocity, a new parameter, the curvature ofthe dispersion surface, must be introduced to describe theangular dispersion. Here the dispersion surface refers tothe constant-frequency surface in reciprocal space. Thiscurvature can be calculated directly from the dispersionrelation ��k�, which also represents the photonic bandstructure. With this theory, we can now directly express thesensitivity of the superprism effect in terms of ��k� andexplore the fundamental limiting factors of the superprismeffect.In this section, we will briefly introduce our theoret-

ical framework for the superprism effect. Then we willseparately discuss two types of superprism effects: theslow-light induced angular dispersion effect and the “pure”angular dispersion effect. We will see some critical scal-ings and limiting factors for these two types of effects.

3.1. Dispersion Surface Curvature andAngular Sensitivity

For convenience, we will consider a 2D photonic crystaland the TM mode of light (magnetic field in the plane).However, our discussion is applicable to other cases. First,we introduce the concept of the dispersion surface curva-ture. The dispersion surface at an arbitrary circular fre-quency, �0 = 2�c/�0, can be described by

��kx� ky�= �0

This equation, which gives ky as an implicit function ofkx, can be reformulated into an explicit form

ky = ��kx� (10)



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where we have omitted the parameter �0. Generally, thecurvature of a curve on the kx −ky plane is given by33

� ≡ d2ky

dk2x

/[1+

(dky

dkx

)2]3/2

(11)

where the derivatives can be calculated from the functiongiven in Eq. (10).To derive the relation between the curvature of the dis-

persion surface and the angular dispersion of the pho-tonic crystal, we consider the conservation of tangentialwavevector component across the photonic crystal surfacefor a configuration depicted in Figure 3

1cnI� sin�= kx0+u sin �−v cos� (12)

where � is the incident angle, nI is the refractive indexof the incident medium, � is the direction of the groupvelocity (i.e., the beam direction) with respect to the sur-face normal, u and v are local Cartesian coordinates inthe neighborhood of a point, k0 = �kx0� ky0�, on the disper-sion surface. Here the local u-axis is parallel to the groupvelocity, and v-axis is tangential to the dispersion surface.It can be shown from Eq. (12) that the sensitivity of

the beam angle to wavelength change (i.e., the angulardispersion) is given by33∣∣∣∣d�d�

∣∣∣∣=∣∣∣∣ 2��

�2 cos��ng sin �−nI sin��

∣∣∣∣ (13)

In addition, the sensitivity to refractive index perturbationis given by ∣∣∣∣ d�dna

∣∣∣∣=∣∣∣∣−�

c

sin �cos�

(��

�na

)k

ng

∣∣∣∣ (14)

In Eqs. (13) and (14), the quantities, � , 1/ cos� (notetan � = sin �/ cos�), and ng are the only three factors thatcan grow several orders of magnitude compared to a con-ventional medium, which can result in significant enhance-ments of angular dispersion/sensitivities as observed inprior superprism experiments.

Fig. 3. Schematic of a simple configuration for the superprism effect.

However, it turns out that the optical transmission acrossthe photonic crystal surface is given by30

T ∝ �t�2�emvg cos� (15)

where �em is the cell-averaged mode energy density andt is the complex coupling amplitude of the mode in ques-tion. Evidently, while larger values of ng or 1/ cos� willhelp enhance the angular dispersion and angular sensitivi-ties, they will also inevitably suppress the optical transmis-sion. Therefore, this type of enhancement will eventuallybe limited by the maximal optical loss that can be toler-ated in a particular application. Note that light propagationinside the photonic crystal may further induce significantoptical loss in the slow-light regime, in addition to thesurface transmission loss given in Eq. (15). The total lossmay be handled by a theory similar to the discussion fol-lowing Eq. (5). On the other hand, enhancing the angularsensitivity through large � values will not entail high opti-cal loss, therefore is highly preferred in a wide range ofapplications.

3.2. Slow-Light Induced Strong Angular Dispersion

Although Eqs. (13) and (14) obviously indicate a lineardependence of the angular dispersion and angular sensitiv-ity on the group index, a casual numeric analysis withoutthe knowledge of Eqs. (13) and (14) would yield a decep-tively stronger enhancement in the slow-light regime. Con-sider the following approximate mode dispersion near aphotonic bandedge

�= �0−b�k2x +k2y� (16)

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 depen-dence near the bandedge

ng = c/ 2√b��−�0��

� =√b/��−�0�

(17)

Evidently, the group index and the curvature diverge at thesame rate as � approaches the bandedge �0. It is straight-forward to show

d�

d�∼ 1

�−�0

d�

dna

∼ 1�−�0

(18)

A straightforward numeric calculation should find thatnear the bandedge, when the group index increases10 times, the angular dispersion and angular sensitivitieswould increase 100 times. Thus, if we did not have theknowledge of the analytic form of Eqs. (13) and (14),we would be tempted to conclude d�/d� ∝ n2

g and



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d�/dna ∝ n2g in this particular case. However, the above

analysis shows that the proper dependence should bed�/d�∝ ng� and d�/dna ∝ ng� because ng and � divergeat the same rate. Although the above analysis is basedon a specific photonic band structure described by Eq.(16), a general asymptotic analysis, given in Section IIA ofRef. [33], indicates that such a slow-light induced strongangular dispersion due to equal diverging rate of ng and �exists in a wider range of slow-light scenarios.

3.3. “Pure” Angular Dispersion Effect:Bandwidth Limited

We have found that the dispersion surface can exhibitultra-high curvature values in the vicinity of certain high-symmetry points in the Brillouin zone (BZ) without thepresence of the slow-light effect.33 An example is the Kpoint of a triangular lattice. Some photonic bands have adouble degeneracy at this point. Approaching such a dou-bly degenerate point, the curvature of the dispersion sur-face tends to infinity whereas the group velocity approachesa non-zero constant. Therefore, we can enjoy the benefitof the high dispersion and high sensitivity, according toEqs. (13) and (14), without worrying about the high opticalloss that would occur in the slow-light case.In this case, the scalings of the group index and curva-

ture differ from the slow-light induced superprism effect,

ng → const

� → 1/��−�0�(19)

where �0 is the frequency of the doubly degenerate point,where the curvature is singular. Interestingly, accordingto Eqs. (13) and (14), the overall scaling of the angulardispersion and angular sensitivities with frequency pertur-bation, �� = �−�0, in this case remain the same as inEq. (18). However, the optical loss is almost constant inthis case, independent of the angular dispersion values.With the optical loss no longer being a limiting fac-

tor, the angular dispersion and angular sensitivities cannow be enhanced to much higher values until it encounterssome other limits. Now we present a fundamental angularsensitivity-bandwidth limit similar to the bandwidth-delaylimit for the longitudinal dispersion. Assume the angulardispersion is maintained at a value above �d�/d��0 overa spectral range of BW (unit: nm). Because the maximumbeam steering range can not exceed 180 (no backwardpropagation is physically possible), we find

�d�/d��0×BW < 180 (20)

This relation is the angular dispersion correspondent ofthe bandwidth-delay product.Therefore, the maximum bandwidth for a sustained high

sensitivity �d�/d��0 is

BW < 180/�d�/d��0 (21)

For example, a sensitivity of 100/nm can be sustainedover a bandwidth less than 1.8 nm, and a sensitivity of1000/nm can be sustained over a bandwidth less than0.18 nm. For a laser having 1pm linewidth (∼100 MHz),these two cases may allow for 1800 and 180 wavelengthtuning points, respectively, which are reasonable for practi-cal applications. These performance parameters are possi-ble with the existing laser technologies. Lastly, we shouldkeep in mind that not all applications require a wide band-width. There are some applications that can benefit from ahigh angular dispersion/sensitivity in a narrow bandwidth.A detailed theoretical analysis based on group theory

shows that such types of “pure” angular dispersion orig-inates fundamentally from symmetry induced degeneracyin photonic band structures. Furthermore, such types ofsymmetry-induced enhancement of angular sensitivitiescan only occur in 2D and 3D photonic crystals, but not in1D photonic crystals. Discovering such a crystal-symmetryinduced effect exemplifies the effectiveness of utilizing thesolid-state physics paradigm to shed new light on the studyof periodic dielectric structures, which is the central themeof photonic crystal research.34 In passing, we note that therigorous theory for computing the transmission of eachphotonic crystal mode30 has been extended to gratings,35

which can be regarded as monolayer photonic crystals, and3D photonic crystals.36 Formulas similar to Eq. (15) canbe used to assess the optical loss in the 1D and 3D casesas well.Before we conclude this section, we would like to men-

tion that there are a number of applications for the super-prism effect. Different applications may have additionallimiting factors specific to themselves. For example, forthe widely studied wavelength demultiplexer application,the beam width divergence is an additional limiting factorspecific to this application. Fortunately, recent works7�37

have demonstrated a promising method of overcoming thislimit. Note that this factor is important only for thoseapplications that require narrow beam width (or spot size)at the receiving end of the superprism. If a sufficientlylarge beam width (hence a small lateral spread of thewavevector) is used, then this factor is less important.

4. LONGITUDINAL DISPERSION VERSUSANGULAR DISPERSION: A DIRECTCOMPARISON

The analyses in two preceding sections show that the twokey parameters, vg and � , of the slow-light effect and thesuperprism effect are entirely determined by the disper-sion function ��kx� ky�. In other words, we may say thatthese two effects are manifestations of the longitudinal andangular characteristics of the dispersion function. In thissection, we will unveil some further connections betweenthese two effects through an application example. Herewe choose the beam steering application, which intends



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EVIEW


Fig. 4. Schematic of a one-dimensional optical phase array composedof photonic crystal waveguide phase shifters.

to manipulate the direction of a laser beam by changingthe refractive indices of materials in certain device struc-tures. Two approaches are considered: (1) an optical phasearray38 (OPA) composed of a 1D array of slow-light pho-tonic crystal waveguides as shown in Figure 4; (2) a super-prism composed of a 2D photonic crystal. Note that tosteer the output beam in free space, the superprism devicecan not have a flat output surface parallel to the input sur-face. In this example, we assume the output surface has asemi-circular configuration for simplicity.3

First, we re-write Eqs. (1) and (14) in the followingforms:

�= L

(ng

n

2��

)�n (22a)

�� =∣∣∣∣� tan �

(ng

na

2��

)�na

∣∣∣∣ (22b)

The second equation follows from(��

�na

)k

= a

�

na

(23)

where a measures the fraction of mode energy locatedin the medium a. Note this relation, Eq. (23), is essen-tially the same as that used in an early derivation ofthe slow-light enhancement of the phase sensitivity in aphotonic crystal waveguide.1 Therefore, it is not surpris-ing to see that Eqs. (22a) and (22b) share similar factors�/n��2�/��, which come from ��/�n. In the case of aphotonic crystal waveguide, n refers to the refractive indexof the waveguide core, and denotes the fraction of themode-energy in the core region.1

More interestingly, a direct comparison of the steer-ing angle sensitivity between the two approaches can beobtained from Eqs. (22a) and (22b). For an optical phasearray, the far-field beam angle � relates to the phase dif-ference, �, between adjacent array elements as follows

sin � = �

2��

d(24)

Therefore, the beam steering sensitivity for a slow-lightbased optical phase array is given as

�� = 1cos�

ng

n

L

d�n (25)

To simplify the comparison with the superprism effect,we assume = a, and n= na. Then the ratio of the beamangle changes in the two cases is given by∣∣∣∣��SP��SL

∣∣∣∣=∣∣∣∣sin �

(2�d�

)�

L

ng�SP

ng�SL

∣∣∣∣ (26)

where SP denotes the superprism effect, and SL denotes theslow-light effect. In most optical phase arrays, the waveg-uide spacing, d, is on the order of the wavelength, �. Ifwe assume sin � > 0�1 and note the 2� factor in Eq. (26),these two factors have an overall contribution on the orderof unity. Therefore, the difference between ��SP and ��SLprimarily comes from the terms, (�/L) and (ng�SP /ng�SL�.Note the curvature � also has the dimension of length.As a numeric example, we consider a silicon photonic

crystal waveguide with ng�SL ∼ 30, and �n ∼ 10−3. Thisgenerally requires a waveguide length on the order ofL= 100 �m to achieve a phase shift of 2� at �= 1�55 �m.Note that to extend the waveguide length far beyond thisvalue to achieve larger � will cause multiple side-lobesand is not desired for many practical OPA beam steeringapplications. On the other hand, it is relatively easy to get� � 100 �m in a properly designed photonic crystal super-prism. For example, we find 2��/� > 103 for a hexagonalphotonic crystal with ng ∼ 7.33 In this particular example,the ratio in Eq. (26) is around 3 sin �. For moderate � val-ues, the beam steering efficiencies due to the two effects areroughly on the same order of magnitude. A more detailedinvestigation is beyond the scope of this work.

5. CONCLUSION

In this paper, we have discussed the slow-light effect andthe superprism effect in one synergistic perspective basedon dispersions. We give rigorous analysis of the phase shiftsensitivities, angular sensitivities, optical loss, and band-width for these effects in fairly general situations. Partic-ularly, a rigorous proof of the scaling of the scatteringloss in the slow light regime is given. The general rela-tions and trends that we have obtained regarding theseimportant parameters provide an important guide for fur-ther experiments to verify these effects and to explore newapplications.

Acknowledgments: This work is supported by AFOSRMURI (Grant No. FA9550-08-1-0394 supervised byGernot Pomrenke) and Air Force Research Laboratory(Grant No. FA8650-06-C-5403 supervised by RobertL. Nelson).

References and Notes

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3. L. J. Wu, M. Mazilu, T. Karle, and T. F. Krauss, IEEE J. QuantumElectron. 38, 915 (2002).

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6. J. J. Baumberg, N. M. B. Perney, M. C. Netti, M. D. C. Charlton,M. Zoorob, and G. J. Parker, Appl. Phys. Lett. 85, 354 (2004).

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8. K. Sakoda, Optical Properties of Photonic Crystals, Springer, Berlin(2001).

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10. L. L. Gu, W. Jiang, X. N. Chen, and R. T. Chen, IEEE PhotonicsTechnol. Lett. 19, 342 (2007).

11. L. L. Gu, W. Jiang, X. N. Chen, L. Wang, and R. T. Chen, Appl.Phys. Lett. 90, 071105 (2007).

12. W. Jiang, L. Gu, X. Chen, and R. T. Chen, Solid-State Electronics51, 1278 (2007).

13. T. F. Krauss, J. Phys. D-Appl. Phys. 40, 2666 (2007).14. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, Nature

438, 65 (2005).15. R. A. Soref and B. R. Bennett, IEEE J. Quantum Electron. 23, 123

(1987).16. W. M. J. Green, M. J. Rooks, L. Sekaric, and Y. A. Vlasov, Opt.

Express 15, 17264 (2007).17. S. J. Spector, Technical Digest of CLEO, San Jose, CA (2008),

p. CFH4.18. X. N. Chen, W. Jiang, J. Q. Chen, L. L. Gu, and R. T. Chen, Appl.

Phys. Lett. 91, 091111 (2007).19. X. Chen, Y.-S. Chen, Y. Zhao, W. Jiang, and R. T. Chen, Opt. Lett.

34, 602 (2009).20. M. Roussey, M. P. Bernal, N. Courjal, D. Van Labeke, F. I. Baida,

and R. Salut, Appl. Phys. Lett. 89, 241110 (2006).

21. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, Phys. Rev. Lett.94, 033903 (2005).

22. S. G. Johnson, M. L. Povinelli, P. Bienstman, M. Skorobogatiy,M. Soljacic, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos,Proceedings of 2003 5th International Conference on TransparentOptical Networks (2003), p. 103.

23. E. Kuramochi, M. Notomi, S. Hughes, A. Shinya, T. Watanabe, andL. Ramunno, Phys. Rev. B 72, 161318 (2005).

24. L. O’Faolain, T. P. White, D. O’Brien, X. Yuan, M. D. Settle, andT. F. Krauss, Opt. Express 15, 13129 (2007).

25. Y. Tanaka, Y. Sugimoto, N. Ikeda, H. Nakamura, K. Asakawa,K. Inoue, and S. G. Johnson, Electron. Lett. 40, 174 (2004).

26. Y. A. Vlasov and S. J. McNab, Opt. Lett. 31, 50 (2006).27. R. J. P. Engelen, D. Mori, T. Baba, and L. Kuipers, Phys. Rev. Lett.

101, 103901 (2008).28. N. C. Panoiu, M. Bahl, and R. M. Osgood, Opt. Lett. 28, 2503

(2003).29. L. Wang, W. Jiang, X. Chen, L. Gu, J. Chen, and R. T. Chen, J.

Appl. Phys. 101, 114316 (2007).30. W. Jiang, R. T. Chen, and X. J. Lu, Phys. Rev. B 71, 245115

(2005).31. E. Istrate, A. A. Green, and E. H. Sargent, Phys. Rev. B 71, 195122

(2005).32. E. Istrate and E. H. Sargent, Rev. Mod. Phys. 78, 455 (2006).33. W. Jiang and R. T. Chen, Phys. Rev. B 77, 075104 (2008).34. J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn,

Photonic Crystals: Molding the Flow of Light, Princeton UniversityPress, Princeton (2008).

35. W. Jiang and R. T. Chen, J. Opt. Soc. Am. A 23, 2192 (2006).36. X. N. Chen, W. Jiang, J. Q. Chen, and R. T. Chen, J. Lightwave

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(2005).

Received: 1 February 2009. Accepted: 31 March 2009.


Symmetry-induced singularities of the dispersion surface curvature and high sensitivitiesof 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

�Received 11 October 2007; revised manuscript received 17 December 2007; published 6 February 2008�

We rigorously analyze the dispersion function and the curvature of the dispersion surface of a photoniccrystal to explore the fundamental limit of its angular sensitivities. With insight gained from group theory, wefind that symmetry induced degeneracy gives rise to a singular dispersion surface curvature and a nonvanishinggroup velocity simultaneously. Near such a singularity, high angular sensitivities can be achieved at low opticalloss. This phenomenon exists generally in most common two-dimensional and three-dimensional photoniccrystal lattices, although it occurs only for certain photonic bands as dictated by symmetry. This symmetry-induced effect is absent in one-dimensional crystals. Rigorous formulas of the sensitivities of the light beamdirections to wavelength and refractive index changes are derived. Individual contributions of the dispersionsurface curvature and group velocity to these sensitivities are separated. In the absence of the Van Hovesingularity, a singular dispersion surface curvature gives rise to ultrahigh dispersion � d�

d� ��103 deg /nm andrefractive index sensitivity � d�

dna��104 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 slowgroup velocity. We also discuss how various parameters intrinsic and extrinsic to a photonic crystal maysuppress or enhance the angular sensitivities according to the rigorous formulas we obtain.

DOI: 10.1103/PhysRevB.77.075104 PACS number�s�: 42.70.Qs, 07.07.Df, 42.65.�k, 61.50.Ah

I. INTRODUCTION

Photonic crystals provide high optical sensitivities notachievable in conventional media. The high dispersion andslow group velocity of photonic crystal waveguides help sig-nificantly shorten the interaction length for opticalmodulation.1–5 Critical to this advance is a clear-cut expres-sion that gives the enhancement of the nonlinear phase sen-sitivity of a photonic crystal waveguide mode in terms of thegroup velocity.1 On the other hand, the high anisotropy andangular dispersion of photonic crystals were found to causebeam directions to have 500-fold higher sensitivities towavelength variation, which was named the superprismeffect.5 Such high wavelength sensitivities are frequently ac-companied by high sensitivities to refractive indexperturbations.6,7 These high sensitivities have aroused wideinterest for potential applications in fiber optic communica-tion, sensing, and nonlinear optics.5–17 However, a general,quantitative relation between the anisotropy and these sensi-tivities of photonic crystals is needed to determine the ulti-mate limits of the sensitivities before we can fully uncoverthe potential of these sensitive effects for a wide range ofimportant applications. For example, the sensitivity of thesuperprism effect is often enhanced near a band edge at thecost of a low optical transmission due to the slow groupvelocity. Whether this sets a fundamental limit of the maxi-mum achievable sensitivity for practical applications is aninteresting question to explore.

In this work, we rigorously show that high angular sensi-tivities to wavelength and refractive index perturbations canbe achieved in the vicinity of a singular dispersion surface

curvature or the vicinity of a vanishing group velocity. Ex-plicit analytical expressions are given to separate the effectsof the curvature and group velocity. Of particular interest isthe case where a singular curvature appears together with anonvanishing group velocity �and therefore high transmis-sion� owing to symmetry-induced mode degeneracy. Such acase is predicted prevalent in high symmetry two-dimensional �2D� and 3D photonic crystal lattices, but absentin 1D photonic crystals. Van Hove singularities, the singu-larities of the density of states due to a vanishing group ve-locity, have been proven to be an insightful concept for un-derstanding some interesting effects in photonic crystals.18

The singularities of the dispersion surface curvature dis-cussed here may occur at nonvanishing group velocities,where Van Hove singularities are absent. Therefore this typeof singularity can lead us to some different effects or newfunctional regimes of photonic crystals.

This paper is organized as follows. In Sec. II A, we willfirst examine some analytic properties of the photonic crystaldispersion function. This allows us to identify the correlationand decorrelation of a singular curvature and a vanishinggroup velocity, which highly depends on lattice symmetryand mode degeneracy. An example based on an approximatemodel is presented in Sec. II B. In Sec. III, we give keyrigorous formulas of the angular sensitivities of photoniccrystals. An example is used to illustrate the individual con-trol of the curvature and group velocity so as to achieve largesensitivities at low optical loss. Section IV discusses the con-tributions of intrinsic and extrinsic parameters to photoniccrystal sensitivities, the effect of dimensionality and latticetypes, and the difference from Van Hove singularities.

PHYSICAL REVIEW B 77, 075104 �2008�

1098-0121/2008/77�7�/075104�8� ©2008 The American Physical Society075104-1


II. CURVATURE OF THE DISPERSION SURFACE ANDGROUP VELOCITY

A. Some key analytic properties and symmetry considerations

For simplicity, we illustrate our ideas with the TM polar-ization �magnetic field in the plane� of a 2D photonic crystal.The field equation, according to Bloch theorem, can be writ-ten as

− �k + G�2E�G� + �2�G�

��G − G��E�G�� = 0, �1�

where G and G� are reciprocal lattice vectors, k is the wavevector, � is the dielectric constant, and E is the electric fieldcomponent normal to the plane. We have assumed the speedof light c=1 for convenience. Generally, the zeros of thesecular determinant D�kx ,ky ,�� of the eigenvalue problemEq. �1� give the frequency � as an implicit function of k.Starting from this implicit dispersion function instead of anexplicit ��k� is essential to linking the mode degeneracy to aspecial type of curvature singularities of the dispersion sur-face.

To study the curvature, it is necessary to find the second-order expansion of D around an arbitrary point k0. A fullTaylor expansion of D in terms of �kx, �ky involves manyterms. We choose a local coordinate system �u ,v� that givesa simpler expansion and a clear physical picture.19

Consider the expansion of the determinant D�kx ,ky ,�� fora fixed frequency �,

D�kx0 + u cos � − v sin �,ky0 + u sin � + v cos �,��

= �j=0

f j�u,��v j = 0, �2�

where the v axis of this local coordinate system is tangent tothe 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 isparallel to the normal vector of the dispersion contour. Thenthe dispersion surface curvature,

�

d2u

dv2

�1 + �du

dv�23/2

,

is simplified to

=d2u

dv2 .

It is a simple exercise to show �see Appendix A�

du

dv= � �u

�v�

�

= −f1

�f0/�u= 0, �3a�

= � �2u

�v2��

= −2f2

�f0/�u, �3b�

at k0 �i.e., u=v=0�, hence f1�0,��=0 in this coordinate sys-tem. The group velocity components are given by

vg =��

�u= −

�f0/�u

�f0/��, �4a�

��

�v= −

f1

�f0/�� 0. �4b�

Comparing Eqs. �3b� and �4a�, it is apparent that in theneighborhood of certain kc where ��f0 /�u�kc

=0, a large cur-vature and a slow group velocity vg would appear simulta-neously. For convenience of discussion, we introduce thegroup index ng=c /vg. The large values of and ng are gen-erally correlated to each other through a common factor��f0 /�u�−1 in this neighborhood.

However, physically it is often undesirable to have a slowgroup velocity because it may cause high optical loss �detailsdiscussed later�. Further inspection of Eq. �4a� indicates thata simultaneously vanishing �f0 /�� could break the afore-mentioned correlation between and ng and give an arbi-trarily large without entailing a vanishingly small groupvelocity. At first glance, to simultaneously achieve �f0 /��=0 and �f0 /�u=0 may require a photonic crystal to havespecially designed structure parameters. However, we notethat since the group velocity cannot exceed c due to causal-ity, a vanishing �f0 /�� always leads to a vanishing �f0 /�uaccording to Eq. �4a�, though not conversely. To trace theorigin of �f0 /��=0, we note an identity at v=0,

f0�u,�� D�kx0 + u cos �,ky0 + u sin �,�� = 0.

Now one readily shows that �f0 /��=0 �hence a singular atvg�0� can be achieved if the dispersion function D has adegenerate mode at k0, i.e.,

D�kx0,ky0,�� = c0�� − �0�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 �BZ�where the associated little groups have at least one irreduc-ible representation �IRREP� whose dimension is two orhigher;20,21 and the dimensionality of irreducible representa-tions is known to play important roles in determining thephysical properties of some photonic crystal structures ofwide interests.20–22 The K point of a triangular lattice is as-sociated with a little group C3v, which has two 1D irreduc-ible representations A1, A2, and one 2D irreducible represen-tation E.20 Therefore the above analysis predicts that asingularity of , together with a nonvanishing vg, may appearat K for certain photonic bands.

B. Example and some approximate forms of the curvature onthe BZ boundary

As an example, we study the dispersion surface curvaturefor a triangular lattice with parameters na=3.8, nb=1.33, and

WEI JIANG AND RAY T. CHEN PHYSICAL REVIEW B 77, 075104 �2008�

075104-2

r=0.3a. Intuitively, the dispersion contour depicted in Fig.1�c� shrinks as � approaches �E; its curvature �roughly theinverse of the radius� grows toward infinity for smaller andsmall contours. A close-up examination of the ��k� curves inFig. 1�a� shows that their slopes approach a nonvanishingvalue as � approaches �E; this indicates nonvanishing vgvalues. These intuitive pictures facilitate the qualitative un-derstanding of the phenomena in this particular example.However, to prove that these effects mathematically followfrom the theory given in the preceding section and thereforeare a particular instance of a general effect proposed hereinrequires more detailed study.

In this section we employ a degenerate perturbation tech-nique �DPT� involving three dominant Fourier components23

to analytically compute for this 2D triangular lattice. Thenumerical form of this DPT has been studied in detail.23 Herewe find that this DPT turns out to give some heuristic ana-lytic forms of the curvature. As a by-product, we also findsome interesting analytic forms of other physical quantitiessuch as the frequencies at the band edge of high symmetrypoints. Note that this approximate DPT theory is used in thissection �Sec. II B� only.

We have verified that this DPT method agrees well withthe rigorous plane wave expansion �PWE� method for eigen-frequencies of the lowest band, as shown in Fig. 1�a�. Wedefine �0

2=�2��0�, �12=�2��b1�=�2��b2�=�2��b3�, where

b1, b2, and b3 are the three shortest reciprocal lattice vectorsand ��G� can be given in Bessel functions.20 For this section

only, we consider coordinates �u ,v� with the origin fixed atthe K point and �=0, which allows for a simple analyticexpression of on the BZ boundary. The determinant D ofEq. �1� for the three leading Fourier components is given by

D = �02 − k2 �1

2 �12

�12 �0

2 − k12 �1

2

�12 �1

2 �02 − k2

2 = 0,

where k=kc+uex+vey, k1=k+b1, and k2=k+b2. We com-pute f0�u ,��, and f2�u ,�� from the secular determinant Daccording to Eq. �2�,

f0�u,�� = g0g1g2 − �14�g0 + g1 + g2� + 2�1

6, �5a�

f2�u,�� = − �g0g1 + g1g2 + g0g2 + g1b22 − 3�1

4� , �5b�

where g0=�02− �kxc+u�2−b2

2 /4, g1=�02− �kxc+u−b1x�2, and

g2=g0. At K, we have

f0�0,�� = �g0 − �12�2�g0 + 2�1

2� = 0.

The first factor clearly indicates a doubly degenerate root,

�1,22 = b2

2/�3��0� − 3��b1�� = �E2 ,

at the K point. The other root is �32=b2

2 / �3��0�+6��b1��.Note ��b1��0 in this case.

The curvature is computed via Eq. �3b� along MK forpart of the first band below �E,

= −1

u

f + b22g1

f + b22g0

, �6�

where f�u ,��=g0g1+g1g2+g0g2−3�14. One readily verifies

that g0=g1=�12 for the degenerate mode at �E �where u=0�.

Therefore, on the MK line, the curvature has the asymptoticform →−1 /u as u→0. Furthermore, we note

�f0/�u = − u� f + b22g0�;

�f0/�� = �2/��02 f − 2�1

4�g0 + g1 + g2 − 3�12�� .

One readily verifies �f0 /��→const u as u→0. Hence thecancellation according to Eq. �4a� gives a nonvanishing vg atK. Thus, within the DPT framework, a singularity of with afinite ng is analytically verified for this example.

Figure 2 shows that the values of and ng obtained fromthe DPT method agree well with the PWE method. The loga-rithmic plot also reveals that the variation of follows thegroup index ng along most of the MK line except near K�kxa=1 /3�, where a varnishing �f0 /�� breaks the correlationbetween a singular and a singular ng, as predicted. Thecurvature in the entire BZ is plotted in Fig. 2 �inset� for thelower branch �below E� of the first band. The large valueson the BZ boundaries reflect the typical contour shape de-picted in Fig. 1�c�, where the dispersion contour bendsabruptly across the BZ boundary. The curvature becomessingular at high symmetry points M �kxa=0� and K. Thenormalized curvature 2� /� for an ordinary medium isaround unity, but it can be several orders of magnitude

x

αq0 nI

nb

na

y(b)

θθθθ v g

(a)

ωE

1Aω

(c)

k0

FIG. 1. �Color online� A triangular lattice. �a� Photonic bandsfor 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. �b� Schematic of a typical experimental configuration. �c� Atypical dispersion surface for the lower branch of the first band.

SYMMETRY-INDUCED SINGULARITIES OF THE… PHYSICAL REVIEW B 77, 075104 �2008�

075104-3

higher in photonic crystals. Note the DPT is used to give anintuitive expression of for this example, it will not be em-ployed in the derivation or calculation in the rest of thiswork.

III. ANGULAR SENSITIVITIES OF A PHOTONICCRYSTAL

A. Derivation of the sensitivity formulas

The response of the mode energy flux direction �i.e., beamdirection� of a photonic crystal to a small perturbation ofwavelength or refractive index is of paramount interest tomany applications. To study this optical response, we need tofind the relationship between and a measurable �extrinsic�quantity in a typical experimental configuration illustrated inFig. 1�b�. The coupling condition for the input surface isgiven by

nI� sin � = kx0 + u cos � − v sin � , �7�

where � is the angle of the group velocity at k0=kx0ex+ky0ey with respect to the input interface. Here k0 is an ar-bitrary point on an arbitrary dispersion contour. Considercases where the incident angle, �, is fixed and one varies thewavelength or the refractive index of one constituent mate-rial �for example, na�.

First, we analyze the wavelength perturbation. We notice

�u = � �u

��

v�� + � �u

�v�

�

�v + ¯ = � �u

��

v�� + ¯ ,

�8�

where we have omitted terms of the order ��2, �v2, ��v,and higher. The second term in Eq. �8� vanishes because wehave ��u /�v��=0 along a dispersion contour �constant-�contour�. We also note that according to Eq. �7�, a perturba-tion �� with a fixed value of � leads to

nI�� sin � = kx0 + �u cos � − �v sin �

around u=v=0. Note in the above equation, terms propor-tional to �� vanish at u=v=0 and should not appear. Byvirtue of Eq. �8�, one readily shows

�v��

=− 1

sin ��nI sin � − � �u

��

vcos � . �9�

Note the higher order terms omitted in Eq. �8� have vanish-ing contributions in Eq. �9� in the limit ��→0; hence theyare not shown. The derivative ��u /��v is just the groupindex, ng �note c=1 in this paper�. Also, by definition of thecurvature, we have �� /�v�= ��. Hence, by virtue of Eq. �9�,the wavelength sensitivity is given by

d�

d� = d�

dv

dvd� =

cos ��nI sin � − ng sin �� . �10�

Note �= �2 −� �note Figs. 1�b� and 1�c� illustrate a case of

��0, ��0�.Now we study the perturbation of refractive index. A per-

turbation of �na at a fixed � results in

�u

�nacos � =

�v�na

sin �

according to Eq. �7�. The refractive index sensitivity can becalculated from d�

dna= ��

�v�v�na

, which gives

d�

dna =

cos ��sin �� u

�na�

�,v . �11�

Using the Jacobian determinants, one readily shows that

� �u

�na�

�,v= − � ��

�na�

u,v� �u

��

na,v,

where the second term on the right side is just the groupindex, ng �note c=1�. Now we find that the angular sensitiv-ity to a refractive index perturbation �for a fixed incidentangle and fixed wavelength� is given by

d�

dna = − tan ��

�na�

kng . �11��

One can compute �� /�na�u,v�� /�na�k easily by varyingna in the photonic band calculation. Note that except forangles �, �, and nI, other quantities in Eqs. �10� and �11�� areintrinsic properties of a photonic crystal, independent of thechoice of the coordinate system and the crystallographic ori-entation of the input surface. According to Eqs. �10� and�11��, the sensitivities to wavelength and index perturbationscan be enhanced by a large dispersion surface curvature orby a large group index. A key feature of Eqs. �10� and �11��is that high sensitivities to wavelength and refractive indexperturbations 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 lowtransmission, and therefore enhancing is practically a better

FIG. 2. �Color online� The dispersion surface curvature andgroup index on the MK line for the lower branch of the first band.The inset plots log10� 2��

� +1� in the 1 BZ for this branch.


075104-4

approach to ultrahigh sensitivities. According to our previoustheory on surface coupling,24 the normalized transmission isgiven by the ratio of the surface-normal component of thePoynting vector Sy��emvg cos �,

T � �t�2�emvg cos � , �12�

where �em is the mode energy density and t is the complexcoupling amplitude24 of the mode in question. Typically, thewavelength or refractive index varies over an ultranarrowrange �less than �1%� in high sensitivity cases, and �t� and�em generally vary insignificantly across this range accordingto our computation. For the modes in the nondegenerateK-valley �IRREP A1� in Fig. 1�a�, the group velocity vg van-ishes as the wavelength approaches the band edge at �A1

.Thus the normalized transmission T is low, and T largelyfollows the trend of small vg as shown in Fig. 3�b� in accor-dance to Eq. �12�. High sensitivities, � d�

dna�, shown in Fig. 3�a�

are achieved at the cost of low transmission. In contrast, thesymmetry-induced degeneracy limits vg to a nonvanishingvalue around the degenerate K point �IRREP E� at �E. There-fore the transmission remains high for most of the spectrumas depicted in Fig. 4�b�.

To recapitulate, we note that the sensitivities to wave-length and refractive index can be significantly enhanced ifany of the three terms, , ng, and 1 /cos �, is large in Eqs.�10� and �11��. Unfortunately, the transmission given in Eq.�12� is inversely proportional to the last two terms, leaving the only desired route for high sensitivities at low opticalloss.

For a numerical example, we set the transmission thresh-old at 50%. This gives the maximum achievable � d�

d� ��3 103 deg /nm and � d�

dna��3.5 104 deg near the degenerate

K point at �E in the first band, according to Fig. 4�a�.Clearly, orders of magnitude higher sensitivities can beachieved in a photonic crystal without severely suppressingoptical transmission. Here we are more interested in indexsensitivities for sensing and nonlinear optics applications.These applications do not require the beam center shift to bemuch larger than the beam width, and therefore are not lim-

ited by some issues found in wavelength demultiplexingapplications.10 Generally, it is easy to detect a minimum lat-eral shift 10 �m of the beam center on the exit end of aphotonic crystal. Then a photonic crystal sensor only needs alength �50 �m to resolve a refractive index change of�na�0.001 with a sensitivity � d�

dna��104 deg. A high � d�

dna�

value may also significantly enhance certain nonlinear opti-cal effects such as all optical switching and beam-steering7

or deflection based Q-switching, where a small �na can begenerated by a pump-control beam or by the signal beamitself. Detailed discussion on applications is beyond thescope of this paper.

IV. DISCUSSIONS

A. Intrinsic and extrinsic parameters

All quantities in Eqs. �10� and �11�� can be easily calcu-lated in any coordinate system. For example, �� /�na�u,v�� /�na�k, the latter can be computed in a regular �kx ,ky�coordinate system whose origin is at the BZ center, �. Also,

the well-known formulas, �d2ky

dkx2 /�1+ � dky

dkx�2�3/2

and ng

=c / ��k��, can be employed to calculate and ng in �kx ,ky�coordinates. Indeed, it is straightforward to see that theseintrinsic quantities ��, ng, and �� /�na�k� do not rely on thechoice of the coordinate systems. Note this statement is validonly for those coordinate systems that can be related to�kx ,ky� through a Euclidian transformation, which is suffi-cient for all practical purposes. As long as the values of ,�� /�na�u,v, and ng are computed for each k point separately,we will not compromise the rigor of Eqs. �10� and �11��.More discussion on understanding the rigor of our method ispresented in Appendix B.

We call � and � extrinsic parameters because they arerelated to how a photonic crystal is coupled on the surface.The angle � should be determined as the angle of the groupvelocity at the coupled k0 point with respect to the normal ofthe photonic crystal surface. Therefore if we rotate the x,y

|dθ /dn |(deg)v

g /c

Tra

nsm

issi

onA

ngle

θ(d

eg) dθ/dn

θ

T

vg /c

(a)

(b)

FIG. 3. �Color online� Normalized transmission and sensitivitiesfor modes near the K-valley of the second band, �A1

a /2�c�0.248, �=20.8°, n1=na.

|dθ/dn |(deg)

vg /cT

rans

mis

sion

Ang

leθ

(deg

)

dθ/dn

θ

vg/c

T

(a)

(b)

FIG. 4. �Color online� Normalized transmission and sensitivitiesfor modes near the degenerate �E mode of the first band,�Ea /2�c�0.187, �=27.9°, nI=na.


075104-5

axes �but not lattice axes intrinsic to the photonic crystal�,the angle � should not change.

Some further analysis helps understand the effects of in-trinsic and extrinsic parameters. Once a coupling configura-tion �including surface orientation, incident angle, and wave-length� is given, the sensitivity values given in Eqs. �10� and�11�� are determined, independent of the choice of the coor-dinate system. On the other hand, consider two experimentsfor the same photonic crystal lattice: �1� light impinges on asurface having Miller indices �h1h2�= �10� at an incidentangle �1; and �2� light impinges on a surface having �h1h2�= �23� at an incident angle �2. By coincidence we maycouple to the same �physical� k0 point on the dispersion sur-face in these two experiments. Although the intrinsic param-eters , �� /�na�u,v, and ng are the same, the different extrin-sic parameters cause entirely different angular sensitivityvalues. In this sense, these angular sensitivities themselvesare also extrinsic quantities, which depend on the externalcoupling conditions.

The separation of extrinsic parameters and intrinsic pa-rameters also allows us to see some interesting effects. Gen-erally, a large intrinsic parameter at a photonic band ofinterest means that a photonic crystal is potentially highlysensitive to both wavelength and refractive index perturba-tions. However, the external coupling conditions, describedby the extrinsic parameters, could modify or even suppresscertain sensitivities actually observed. For example, if theterm nI sin �−ng sin � in Eq. �10� vanishes under a givencoupling configuration �or close to zero over a range of cou-pling parameters�, it is possible to produce a device that hasa very high refractive index sensitivity and a relatively lowwavelength sensitivity in certain parameter ranges. This mayhelp enhance the bandwidth of certain devices. Further studyis needed to explore this possibility.

B. Effects of dimensionality and lattice types

The theory developed here can be extended to treat othercommon lattices in 2D and 3D. We note that a similar sin-gularity of can occur for a square lattice, where the corre-sponding BZ corner �M point� retains the C4v symmetry andhas one 2D irreducible representation. Therefore an ul-tralarge curvature can appear with a nonvanishing vg for thetwo most common 2D lattice types. Similar analysis can beapplied to the TE polarization. It can be proven from grouptheory that symmetry-induced singularities of the dispersionsurface curvature at vg�0 can also exist for most common3D lattices, such as simple-cubic, face-centered-cubic, andbody-centered-cubic lattices. The analysis will be similar inspirit though more complicated in form because a surface in3D has two principal curvatures.19 For many practical sce-narios 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 photo-

nic crystals because all 1D point groups are Abelian andhave no degeneracy. For a 1D grating of the same period a,its angular dispersion25 d�

d� � 1a cos � is much lower

��0.2 deg /nm� for any reasonable value of �. The precedinghigh d�

d� , d�dna

values of the 2D lattice are obtained at ��

�70° with negligible contribution from the 1cos � factor in

Eqs. �10� and �11��.Note that our photonic crystal surface coupling theory24

has been extended to compute mode transmissions for 3Dphotonic crystals26 and 1D gratings.27 Also note that in grat-ing diffraction, the diffracted beam angle refers to the angleoutside the grating. For the original superprism effect, thebeam angle sensitivity refers to the angle inside the photoniccrystal. For some integrated devices,15 beam angles outsidethe photonic crystal need not be sensitive and the sensitivityis employed through the sensitive shift of the output positionon the exit surface. If the output angle sensitivity is ex-ploited, then the output surface must not be parallel to theinput surface.12,14

C. Van Hove singularities and some other issues

We shall mention that Van Hove envisioned that a mini-mum of one band and a maximum of another may “contact”each other, and the Van Hove singularity will be weakened orsuppressed by “compensation.”28 In our case, it is straight-forward to prove that the Van Hove singularity is virtuallyabsent29 at �E due to a nonvanishing vg, but the curvature exhibits a singularity. Note that one type of extreme aniso-tropy near the � point was found to be associated with adivergent Van Hove singularity for a 2D square lattice.30 Thecurvature singularity discussed here comes with a suppressedVan Hove singularity, and is a different, general phenom-enon. Our Eq. �10� shows d�

d� � and may also shed new lighton some issues for =0.

Note that usually the dielectric function of a photoniccrystal is local and is considered accurately known. Henceadditional boundary conditions31 are not needed for a semi-infinite photonic crystal even when multiple modes appear.Other theoretical works on photonic crystal surface couplingalso do not invoke additional boundary conditions.32

In summary, we rigorously analyze the dispersion surfacecurvature in 2D photonic crystals and discuss its relation toangular sensitivities of a photonic crystal. The individualcontributions of the curvature and group index to the angularsensitivities are separated. Furthermore, assisted by grouptheory, we analytically show that symmetry induced degen-eracy allows for high sensitivities without compromising op-tical transmission in 2D and 3D photonic crystals, leading topromising applications including sensing and nonlinear op-tics. We also discuss the possibility of maximizing the refrac-tive index sensitivity while suppressing the wavelength sen-sitivity.

ACKNOWLEDGMENTS

We thank Nicholas X. Fang, R. L. Nelson, J. Haus, andTao Ling for helpful scientific-technical discussions. We arealso grateful to many colleagues and friends for kind encour-agement and support. This work was supported by Air ForceResearch Laboratory under Grant No. FA8650-06-C-5403.Partial support from AFOSR �Grant No. FA9550-05-C-0171�and NASA �Grant No. NNX07CA84P� is acknowledged.


075104-6

APPENDIX A: DERIVATIVES IN A LOCALCOORDINATE SYSTEM

By virtue of Eq. �2�, the total derivative of D with respectto v along a constant-� contour is given by

0 = dD

dv

�

= �j=0

�f j

�u

du

dvv j + �

j=1

f j�u,��jv j−1. �A1�

At the origin of this local coordinate system �u ,v�, wehave v=0. Hence Eq. �A1� is reduced to

�f0

�u

du

dv+ f1 = 0, �A2�

from which we obtain Eq. �3a�. Further differentiation of Eq.�A1� with respect to v yields

0 = d2D

dv2 �

= �j=0

� �2f j

�u2 �du

dv�2

v j +�f j

�u

d2u

dv2v j+ �

j=1

�f j

�u

du

dvjv j−1 + �

j=1

�f j

�u

du

dvjv j−1

+ �j=2

f j�u,��j�j − 1�v j−2. �A3�

Here the first two sums come from the differentiation of thefirst sum in Eq. �A1�, the last two sums from the second sumin Eq. �A1�. At the origin of this local coordinate system�u ,v�, we have v=0, and du /dv=0. Hence Eq. �A3� is re-duced to

0 =�f0

�u

d2u

dv2 + 2f2, �A4�

from which we obtain Eq. �3b�. By considering the totalderivatives of D with respect to u and v along constant-v andconstant-u lines respectively, Eqs. �4a� and �4b� can beproved similarly.

APPENDIX B: RIGOROUS NATURE OF THESENSITIVITY FORMULAS

To understand the rigorous nature of Eqs. �10� and �11��,we point out some key features of our derivation. First, theorigin of the local coordinate system is an arbitrary point k0�not necessarily a high symmetry point�. Second, in derivingthese relations, we have not assumed a finite Fourier series.Third, the local expansion with respect to u ,v is rigorous atk0 only. For another point k0�, no matter how close it is to k0,another local expansion with another set of local coordinates�u� ,v�� must be used. In this way, the rigor of Eqs. �10� and�11�� is not compromised. Lastly, to compute the beam anglechange due to a finite change of refractive index, the rigorousway is to integrate �� /�na given by Eq. �11�� over the finitespan of �na.

The preceding procedure that involves local coordinates�u ,v�, although useful in understanding the rigor of Eqs. �10�and �11��, is somewhat complicated in practical calculations.Fortunately, we have simplified the sensitivity forms suchthat all quantities in Eqs. �10� and �11�� can be easily calcu-lated in any coordinate systems. The details have been dis-cussed in Sec. IV A. Note that, in contrast, Eq. �11� involves��u /�na��,v, which is much less intuitive for direct computa-tion in any coordinate systems. As long as the values of ,�� /�na�u,v, and ng are computed for each k point individu-ally, we will not compromise the rigor of Eqs. �10� and �11��.Lastly, even though the equations given are rigorous, thevalues of quantities such as � and cos � entering Eqs. �10�and �11�� are usually approximated in numerical calculationsdue to a finite cutoff of the series used in computing � andthe cosine function. Nonetheless, rigorous equations withsimple forms like Eqs. �10� and �11�� serve at least two im-portant purposes. First, they single out a few key factors�e.g., , ng� that affect sensitivities. Such physical insighthelps us easily identify the high sensitivity regimes of inter-est and avoid an aimless search in a large design space. Sec-ond, it provides a priori information of various quantitiesnear numeric singularities �e.g., singular �. Generally such apriori information is invaluable in numeric calculations andhelps us design numeric schemes that are highly accurate,reliable, and efficient near singularities.33 We have employedat least 121 Fourier components to ensure better than 1%convergence for sensitivities.34

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singularity at �E; it can be removed by redefining ��E�.30 Y. A. Urzhumov and G. Shvets, Phys. Rev. E 72, 026608 �2005�.31 V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial

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34 This work was first presented by W. Jiang, in Air Force Workshopon Electro-optic Beam Steering �Dayton, Ohio, 2007�.


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