Numerical Investigation of Multiple Bound States inPhotonic Crystal Double Heterostructure Resonant

CavitiesAdam Mock, Ling Lu, Eui Hyun Hwang, John O’Brien and P. Daniel Dapkus

Viterbi School of EngineeringUniversity of Southern California

Los Angeles, CA 90089Email: amock@usc.edu

Abstract—Multiple bound states in photonic crystal doubleheterostructure resonant cavities are analyzed using the three-dimensional finite-difference time-domain method. Quality fac-tors and mode profiles are presented. A method for modediscrimination is discussed.

I. INTRODUCTION

Two dimensional photonic crystal waveguides (PCWG) andcavities are interesting candidates for future photonic inte-grated circuits due to their small size and versatility. Since theirinception [1] photonic crystal heterostructures have added anextra degree of freedom in designing PC devices. Over the pastfew years extremely high quality factor (Q) photonic crystaldouble heterostructure cavities (PCDH) have been proposedand demonstrated [2], [3], [4] with applications ranging fromlaser cavities and filters for integrated photonics to biologicalsensing. A schematic of a PCDH is shown in Fig. 1(a.) inwhich the lattice constant (a) of an otherwise uniform PCWGis increased along the x-direction by a few percent (a′).This perterbation drops the mode of the perterbed region intothe bandgap of the adjacent waveguide resulting in opticalconfinement along x. Confinement along y is due to Braggreflection in the PC cladding, and confinement along z is dueto total internal reflection of the high index slab.

The particular shape of of the PCDH makes them amenableto dense integration as the cavity region is naturally connectedto an output waveguide. Edge-emitting PCDH lasers have beendemonstrated with output powers reaching 100 µW [5].

The theoretical properties of PCDH have been investigatedby several researchers [2], [6], [7]. In [6] it was shownthat bound states form near stationary points in the PCWGdispersion diagram. If the perturbation increases (decreases)the lattice constant locally, the bound states form near localminima (maxima) in the PCWG dispersion diagram. In thisstudy we first show that the Q of the bound state associatedwith the fundamental TE-like PCWG mode drops approxi-mately exponentially with increasing perturbation. Second weshow that for sufficiently deep wells, second and third orderbound states are supported. This work is important both as anextension of the basic theory of PC heterostructures and as apossible route toward enhanced device operation.

Fig. 1. (a.) Schematic diagram of a PCDH. (b.) Q versus percent perturbationfor the first, second and third bound states. (c.) To scale diagram showingrelative depths of the various bound states.

II. CALCULATION METHOD AND RESULTS

We use the three-dimensional finite-difference time-domainmethod for the analysis of the PCDH cavities [8]. A typicalgeometry includes 20 uniform PCWG periods on each sideof the perturbation. The PCWG hole radii are r/a = 0.29and the slab thickness is d/a = 0.6 where a is the latticeconstant of the uniform PCWG. This geometry is discretizedon 950 × 340 × 200 spatial grid points with 14 layers ofperfectly matched layer absorbing boundary conditions onall boundaries. Due to the extremely high-Q nature of theseresonances and the low free spectral range due to the Fabry-Perot resonance originating in the PCWG sections [6], we runour simulation for 2×105 times steps with a broad band initialcondition which takes about 20 hours on 100 processors. Toestimate Q we perform a discrete Fourier transform on theresulting time sequence and use the Pade interpolation methodto overcome the limited resolution [9].

The dependence of Q of the PCDH bound state on percentincrease of the lattice constant along x is shown in Fig.1(b.). The roughly exponential drop in Q as a function ofperturbation is apparent. As the perturbation is made stronger,the spatial confinement increases. This causes the k-spacedistribution to broaden resulting in an increase of the mode’soverlap with the radiation light cone. Also shown in Fig. 1(b.)

Fig. 2. Left: Hz(x, y) at the midplane of the slab for the first, second andthird bound states. Right: Evelope function of |Hz(x, y = 0)| correspondingto the figures on the left.

are the Qs of the second and third order bound states. Oneinteresting observation is that for a given perturbation, thehighest order bound state has the highest Q. This is mostlikely due to the fact that it is the least tightly bound in thewell and thus has a narrower k-space distribution. The relativedepths of the bound state frequencies for a 20% perturbationare illustrated in Fig. 1(c.).

On the left in Fig. 2 we show a two-dimensional cut throughthe middle of the slab of the Hz(x, y) field component forthe first, second and third bound states. The first, second andthird bound states were obtained from 5%, 10% and 25%perturbations, respectively. On the right side of Fig. 2 we showthe envelope functions of the respective bound states whichwere obtained from the peaks amplitudes of |Hz(x, y)| alongthe center of the PCWG. The increase in the peaks and nodesis apparent as the order of the bound state increases. We alsopoint out that the first and third bound states are odd aboutx = 0 whereas the second bound state is even. The decreasein the field amplitudes near the center of the cavity for thesecond and third bound states may be useful for electricallypumped geometries with a high index and high conductivitypost directly beneath the cavity [10].

III. RESONANCE DISCRIMINATION

Due to the low free spectral range between resonances andthe high-Qs of the first and second bound states in the rangeof 5% to 10% perturbation, in order to achieve single modeoperation and high side mode suppression ratio, it would beadvantageous to be able to selectively feature one mode overthe other. We found that placing a small hole in the center ofthe PCWG along y at the maximum of the bound state alongx we wish to eliminate accomplishes this task. Fig. 3 shows

Fig. 3. Left: diagram of PCDH cavity featuring the (1) first and secondbound states, (2) first bound state, (3) second bound state. Right: numericallycalculated spectra for the three cavities.

three cavities: the first is a normal 10% PCDH. The second hasa hole at x = ±2.8a which is near the maxima of the secondbound state and near a null of the first bound state. The thirdhas a hole at x = 0 which is at the maximum of the first boundstate and at the null of the second bound state. The curves onthe right are numerically calculated spectra corresponding tothe three cavities. It is apparent that both the first and secondbound states are present in curve 1. The second bound state isabsent from curve 2, and the third bound state is absent fromcurve 3. The Q of the first bound state was decreased by 10%by the addition of the two holes, and the Q of the secondbound state was unaltered by the addition of the single holeat the center.

ACKNOWLEDGMENT

This study is based on research supported by the DefenseAdvanced Research Projects Agency (DARPA) under contractNo. F49620-02-1-0403 and by the National Science Foun-dation under grant ECS-0507270. Computation for the workdescribed in this Letter was supported, in part, by the Uni-versity of Southern California Center for High PerformanceComputing and Communications.

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