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Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 15 — Jul. 29, 2002
  • pp: 670–684
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Momentum space design of high-Q photonic crystal optical cavities

Kartik Srinivasan and Oskar Painter  »View Author Affiliations


Optics Express, Vol. 10, Issue 15, pp. 670-684 (2002)
http://dx.doi.org/10.1364/OE.10.000670


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Abstract

The design of high quality factor (Q) optical cavities in two dimensional photonic crystal (PC) slab waveguides based upon a momentum space picture is presented. The results of a symmetry analysis of defect modes in hexagonal and square host photonic lattices are used to determine cavity geometries that produce modes which by their very symmetry reduce the vertical radiation loss from the PC slab. Further improvements in the Q are achieved through tailoring of the defect geometry in Fourier space to limit coupling between the dominant momentum components of a given defect mode and those momentum components which are either not reflected by the PC mirror or which lie within the radiation cone of the cladding surrounding the PC slab. Numerical investigations using the finite-difference time-domain (FDTD) method predict that radiation losses can be significantly suppressed through these methods, culminating with a graded square lattice design whose total Q approaches 105 with a mode volume of approximately 0.25 cubic half-wavelengths in vacuum.

© 2002 Optical Society of America

1 Introduction

While the PC optical microcavities studied in [14

14. O. J. Painter, A. Husain, A. Scherer, J. D. O’Brien, I. Kim, and P. D. Dapkus, “Room Temperature Photonic Crystal Defect Lasers at Near-Infrared Wavelengths in InGaAsP,” J. Lightwave Tech. 17, 2082–2088 (1999). [CrossRef]

] had very small mode volumes and loss properties sufficient to sustain lasing, the measured Q values were still less than 1500. In particular, the radiation losses were predominantly out-of-plane, while the in-plane losses were in comparison small [15

15. O. Painter, J. Vučković, and A. Scherer, “Defect Modes of a Two-Dimensional Photonic Crystal in an Optically Thin Dielectric Slab,” J. Opt. Soc. Am. B 16, 275–285 (1999). [CrossRef]

]. Although refinements in design [16

16. J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65 (2002).

] and fabrication [17

17. T. Yoshie, J. Vučković, A. Scherer, H. Chen, and D. Deppe, “High quality two-dimensional photonic crystal slab cavities,” Appl. Phys. Lett. 79, 4289–4291 (2001). [CrossRef]

] have since increased the total measured Q to as high as 2800, most of the potential applications previously mentioned require Q values on the order of 104 or 105 in order for PC slab WG microcavities to show marked improvement from what is available through non-PC designs.

Fig. 1. 2D hexagonal PC slab waveguide structure and cladding light cone.

2 Momentum Space Consideration of Vertical Radiation Loss

The optical cavities studied here are comprised of defects situated in 2D PC slab WGs (Figure 1). As a result, the in-plane confinement of the defect mode is governed by the Distributed Bragg Reflection (DBR) of the surrounding photonic lattice. Leakage of light in the plane of the PC slab WG from the cavity is thus determined by the number of periods of the host lattice surrounding the defect and the width and angular extent of the in-plane guided mode bandgap. The vertical confinement, on the other hand, is due to standard waveguiding by total internal reflection. Vertical radiation loss occurs when the magnitude of the in-plane momentum component, k , is inappropriate to support guiding. More concretely, we note that the energy-momentum dispersion relationship for a homogenous dielectric cladding (refractive index n) of the PC slab WG is (n ω/c)2 = k2 + kz2, where ω is the angular frequency, kz is the momentum normal to the slab, and c is the speed of light. For an air clad PC WG as studied here, k2 = (ω/c)2 defines a cone in (kx ,ky ,ω) space, commonly referred to as the “light cone” (Figure 1). Modes that radiate vertically will have small in-plane momentum components that lie within the light cone of the cladding. This simple rule serves as our fundamental guideline in designing cavities that limit vertical radiation loss. In particular, we seek out structures that support resonant modes whose in-plane momentum components are primarily situated outside of the cladding light cone.

Before discussing methods to improve the vertical loss properties of PC defect cavities, it is instructive to consider the characteristics of the previously studied[15

15. O. Painter, J. Vučković, and A. Scherer, “Defect Modes of a Two-Dimensional Photonic Crystal in an Optically Thin Dielectric Slab,” J. Opt. Soc. Am. B 16, 275–285 (1999). [CrossRef]

] dipole-like defect modes in a hexagonal lattice PC. Consider the x-dipole donor mode produced by a symmetric defect consisting of the removal of a single air hole from a hexagonal lattice of air holes in a 2D slab WG[15

15. O. Painter, J. Vučković, and A. Scherer, “Defect Modes of a Two-Dimensional Photonic Crystal in an Optically Thin Dielectric Slab,” J. Opt. Soc. Am. B 16, 275–285 (1999). [CrossRef]

] (illustrated in cross-section in Figure 1). Following the symmetry analysis presented in reference [13

13. O. Painter, K. Srinivasan, J. D. O’Brien, A. Scherer, and P. D. Dapkus, “Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides,” J. Opt. A 3, S161–S170 (2001). [CrossRef]

], we see that this mode is composed of dominant Fourier components directed along ±{k x1, k x2, k x3}, where the k x directions are shown in the hexagonal PC reciprocal space lattice of Figure 2(a). The 2D spatial Fourier Transform (FT) of the x-dipole field component E x at the middle of the PC slab WG is given in Figure 3(a). It shows the E-field to be primarily composed of momentum components located about the X points, with ±k x1 as the strongest components. Note that the field has a significant number of momentum components lying within the light cone, the boundary of which is shown in Figure 3(a) as a dashed white circle. These low momentum components radiate and are the cause of the mode’s relatively low effective vertical Q-factor (Q ≈ 1000).

Fig. 2. Real and reciprocal space lattices of (a) a 2D hexagonal lattice, and(b) a 2D square lattice. For the hexagonal lattice: |a1| = |a2| = a, |G1| = |G2| = 4π/√3a, |kx| = 2π/√3a, |k J| = 4π/3a. For the square lattice: |a1| = |a2| = a, |G1| =|G2| = 2π/a, |k x| = π/a, |k M| = √2π/a.

3 Symmetry Analysis of Defect Modes in Hexagonal and Square Lattices

Fig. 3. Spatial FT of x-dipole donor mode in a hexagonal lattice (r/a = 0.30) with a central missing air hole. (a) in 2D, (b) along the ky direction with kx = 0.

The defect modes of a PC cavity are generally classified into donor and acceptor type modes [20

20. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopou-los, “Donor and acceptor modes in photonic band-structure,” Phys. Rev. Lett. 67, 3380–3383 (1991). [CrossRef] [PubMed]

], based upon whether the defect creates modes from the conduction band-edge (donor modes) or valence band-edge (acceptor modes). For the hexagonal lattice, whose real and reciprocal space depictions are given in Figure 2(a), the valence band-edge is at the J-point and the conduction band-edge is at the X-point (Figure 4(a)), while the square lattice of Figure 2(b) has its valence band-edge at the M-point and conduction band-edge at the X-point (Figure 4(b)). The dominant Fourier components and symmetry of a defect mode are determined by the type of mode (donor or acceptor) under consideration, the symmetry of the surrounding PC lattice, and the point group symmetry of the defect. The use of such an analysis to produce approximate forms for the modes in hexagonal and square lattice PC defect cavities is the focus of other recent articles [13

13. O. Painter, K. Srinivasan, J. D. O’Brien, A. Scherer, and P. D. Dapkus, “Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides,” J. Opt. A 3, S161–S170 (2001). [CrossRef]

, 18

18. O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” submitted to Phys. Rev. B (2002).

], and as such, we primarily incorporate the results of these works and describe their implications towards the design of high-Q defect resonators. The course of study is the following: we use the results of [13

13. O. Painter, K. Srinivasan, J. D. O’Brien, A. Scherer, and P. D. Dapkus, “Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides,” J. Opt. A 3, S161–S170 (2001). [CrossRef]

] and [18

18. O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” submitted to Phys. Rev. B (2002).

] to determine the symmetry and dominant Fourier components for the available donor and acceptor type modes formed at different high symmetry points within hexagonal and square lattice PC’s. Candidate modes for high-Q resonators are then chosen from these sets of available modes based upon the criteria placed on the mode’s momentum components as described above. Within the mirror plane of the slab WG (coordinates r ) the fundamental even modes are described by the field components E x, E y, and B z. Since the magnetic field is exactly scalar within this mirror plane, the criterion reduces to looking for modes in which the magnetic field pattern is spatially even in the directions of its dominant Fourier components. This is equivalent to having the in-plane electric field components spatially odd in these directions.

Fig. 4. Fundamental TE-like (even) guided mode bandstructure for hexagonal and square lattices, calculated using a 2D plane-wave expansion method with an effective index for the vertical guiding: (a) hexagonal lattice with r/a = 0.36, n slab = n eff = 2.65, (b) square lattice with r/a = 0.40, n slab = n eff = 2.65.

3.1 Hexagonal Lattice

For a hexagonal lattice, the high symmetry points about which a defect may be formed are points a (C 6v symmetry), b (C 2v symmetry), and c (C 3vv symmetry) shown in Figure 2(a). In this paper we consider donor and acceptor modes formed only at points a and b as from the analysis presented in reference [18

18. O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” submitted to Phys. Rev. B (2002).

] those centered at point c do not contribute modes with the requisite symmetry and dominant Fourier components. We also examine reduced symmetry modes formed at point a where the reduction of symmetry from C 6v to C 2v is accomplished by choosing a defect that breaks the symmetry of the lattice and is compatible with C 2v. Based upon the analysis of [13

13. O. Painter, K. Srinivasan, J. D. O’Brien, A. Scherer, and P. D. Dapkus, “Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides,” J. Opt. A 3, S161–S170 (2001). [CrossRef]

, 18

18. O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” submitted to Phys. Rev. B (2002).

], we create Table 1 for donor modes and Table 2 for acceptor modes. These tables provide the labeling scheme for the C 6v and C 2v modes, the dominant Fourier components of the modes, and their transformation properties about the available mirror planes (the mirror plane properties are represented by their character values[21

21. M. Tinkham, Group Theory and Quantum Mechanics, International Series in Pure and Applied Physics (McGaw-Hill, Inc., New York, NY, 1964).

]).

Donor modes of C 6v symmetry, formed at point a in the lattice, have their dominant Fourier components in the ±{k X1, k X2, k X3} directions, and we thus require that σd ≠ - 1, where the σdi are the mirror planes labeled in Figure 2(a). However, σd ≠ -1 for the modes listed in Table 1. Reducing the symmetry of the mode to C 2v (through a modified defect at point a or re-centering to point b) results in modes with dominant Fourier components that are not orthogonal to the available mirror planes, or as in the case of the B b,d1 B1 mode, incorrect spatial symmetry.

Out of the C 6v acceptor modes in Table 2, the B a,a1 A2 mode satisfies the symmetry criteria. The B b,a1 A2 mode produced at position b does not quite satisfy our criteria, as two of the three pairs of dominant Fourier components (±{k J1, k J3}) are not orthogonal to the mirror planes; however, distortions of the lattice that preferentially select for the ±k J5 Fourier components over ±k J1 and ±k J3 can be made so that the symmetry condition is satisfied. Such lattice distortions are addressed in a future article. As a reference, the approximate form for the B a,a1 A2 mode is listed below [13

13. O. Painter, K. Srinivasan, J. D. O’Brien, A. Scherer, and P. D. Dapkus, “Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides,” J. Opt. A 3, S161–S170 (2001). [CrossRef]

]:

BA2a,a1=ẑ(cos(kJ1ra)+cos(kJ3ra)+cos(kJ5ra)),
(1)

where ra denotes in-plane coordinates referenced to point a.

Table 1. Symmetry classification and dominant Fourier components for the B-field of conduction band donor modes in a hexagonal lattice.

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Table 2. Symmetry classification and dominant Fourier components for the B-field of valence band acceptor modes in a hexagonal lattice.

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3.2 Square Lattice

The square lattice of air holes in a dielectric slab, whose real and reciprocal space representations are shown in Figure 2(b), and whose TE-mode bandstructure is depicted in Figure 4(b), also provides a photonic lattice from which low-loss defect modes can be formed. Defects in a square lattice may be centered at the C 4v symmetry points d and f, or the C 2v symmetry point e. Again, following the analysis of [18

18. O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” submitted to Phys. Rev. B (2002).

], we produce Tables 3 and 4 for the square lattice defect modes.

Based on their properties under mirror reflection, the B f,d1 A2,B f,d1 B2, and B e,d1 A2 donor modes all meet the symmetry condition we have placed on the modes. By suitable modification of the defect geometry, the symmetry of modes formed at points d and f can be reduced to C 2v,σv or C 2v,σd, where the subscript σv denotes symmetry with respect to the (σx ,σy ) mirror planes and the subscript σd refers to the (σx',σy') mirror planes (Figure 2(b)). The modes at f continue to satisfy the symmetry criteria under C 2v,σd, but no longer do so under C 2v,σd, as the σd mirror planes are not orthogonal to the modes’ dominant Fourier components.

The acceptor states formed from the valence band edge at the M-point are analyzed in a similar fashion, and in this case, the modes at points d and f are candidates. The reduced symmetry C 2v,σv modes at points d and f are ruled out, while the C 2v,σd modes at these two high symmetry points remain on the list. As a reference, the approximate forms for the candidate donor and acceptor modes are given in Table 5 below.

Table 3. Symmetry classification and dominant Fourier components for the B-field of conduction band donor modes in a square lattice.

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Table 4. Symmetry classification and dominant Fourier components for the B-field of valence band acceptor modes in a square lattice.

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Table 5. Candidate donor and acceptor modes in a square lattice.

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4 Initial FDTD Simulation Results

4.1 Hexagonal Lattice

The B a,a1 A2 mode, our candidate mode for study, is formed by enlarging holes in a manner consistent with the C 6v symmetry of the lattice, so that an acceptor mode is formed. We choose the defect geometry shown in Table 6, where the central hole (about point a) is enlarged from radius r to r′. The defect is surrounded by a total of 8 periods of the hexagonal lattice in the -direction and 12 periods in the ŷ-direction. The magnetic field amplitude and momentum space electric field components x and y of mode B a,a1 A2 are given in Table 6 for two different pairs of values (r, r′). The dominant Fourier components are seen to be ±{k J1,k J3,k J5}, as predicted by the symmetry analysis. Examining x and y, it is also clear that, although the power within the light cone has been reduced in comparison to the x-dipole donor mode, it is still significant. This fact is evidenced in Q which, at 4, 900 for r/a = 0.35 and r′/a = 0.45, is larger than that obtained for the x-dipole mode. By reducing the frequency, and consequently the radius of the light cone, the PC cavity with r/a = 0.30 and r′/a = 0.45 has an improved vertical Q of 8, 800 (although its in-plane Q has degraded due to a reduction in the in-plane bandgap for smaller lattice hole radii).

One complication in the hexagonal lattice (as opposed to the square lattice designs studied below) is the number of dominant Fourier components that must be accounted for when trying to maximize Q. As we alluded to earlier, modifications to the host lattice itself, rather than the defect geometry alone, provide an interesting method to reduce the number of components and further improve the Q of the structure. The detailed discussion of these ideas is left to an upcoming paper. From this point on, we will focus on square lattice designs, but it is important to note that many of the ideas described below are equally applicable to hexagonal lattice structures.

4.2 Square Lattice

We choose the B e,d1 A2 mode as our candidate for study. This mode, centered in the dielectric at point e in the lattice, is appealing in that it has Fourier components primarily situated at ±k X1, while the other modes of correct symmetry have a larger number of Fourier components. This simplifies the design considerations of Section 5. To create the mode, we consider the structure depicted in Table 7. Defining point e as the origin, (0, 0), we see that the structure consists of a standard square lattice of air holes in which the two holes centered at (0, ±a/2) are decreased in size so as to create a donor mode of A 2 symmetry. In the FDTD simulations, the structure consists of 12 rows and 8 columns of air holes surrounding the defect holes.

Table 6. Characteristics of the B a,a1 A2 resonant mode in a hexagonal lattice (images are for a PC cavity with r/a = 0.35, r′/a = 0.45, d/a = 0.75, and n slab = 3.4).

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Starting with r/a = 0.30, r′/a = 0.28, and d/a = 0.75, we produce a mode with normalized frequency ωn = ao = 0.264. The magnetic field amplitude and 2D spatial FTs (x and y) of the mode are given in Table 7. As the magnitude of y is negligible in comparison to that of y, the mode is predominantly made up of components centered at ±k X1, as predicted. The effective vertical Q of this mode is approximately 54, 000, easily exceeding the values obtained in [13

13. O. Painter, K. Srinivasan, J. D. O’Brien, A. Scherer, and P. D. Dapkus, “Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides,” J. Opt. A 3, S161–S170 (2001). [CrossRef]

] for a mode of even symmetry. The small Q (17,400) is a result of the weak defect perturbation and extended nature of the cavity mode (V eff = 0.43). Improving the localization of the mode by lowering r′/a of the defect to 0.25 improves Q to a value of 60,000 and lowers V eff by a factor of almost two. Surprisingly, Q has also increased from 54,000 to 69,000 despite the stronger localization of the mode and its expected broadening in Fourier space. This rather counter-intuitive result indicates that a more detailed study of the effects of the defect geometry on cavity loss is required. This is the focus of the following section.

5 Momentum Space Design of the Defect Geometry in a Square Lattice

The results given thus far indicate that improving the loss properties of the defect mode resonators requires isolation of the mode’s momentum components to regions outside the light cone to maintain a high Q⊥, and to those regions for which the in-plane bandgap is substantial for a high Q . To determine how to tailor the defect geometries to accomplish these goals, we consider a simple model to illustrate the couplings induced in Fourier space between the dominant momentum components of a given defect mode and those modes which radiate. We employ a two-step process where in the first step, the approximate form of the defect mode is taken based on symmetry arguments, as outlined in Section 3, with the allowance for finite k-space bandwidths in the dominant Fourier components due to the localization of the defect mode. We then consider couplings of this approximate symmetry mode to other modes of the PC slab WG through the dielectric perturbation Δη(r), where η = 1/ϵ is the inverse of the dielectric profile of the lattice. The most important mode couplings from the perspective of increasing the Q are those between the dominant Fourier components and “leaky cavity modes”. The leaky cavity modes consist of vertical radiation modes and guided modes of the PC slab WG which are not reflected by the PC and thus leak in-plane. Δη] induces the change δH d(r) in the defect mode, and this change is written as a superposition over the set of nearly (frequency) degenerate guided and radiation modes of the PC slab WG. The coupling amplitude between the symmetry mode composed of the dominant Fourier components, Hod (r), and a leaky cavity mode, Holcm(r), of the unperturbed PC slab is given by the following matrix element:

Table 7. Characteristics of the B e,d1 A2 resonant mode in a square lattice (images are for a PC cavity with r/a = 0.30, r′/a = 0.28, d/a = 0.75, and n slab = 3.4).

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d3r(Holcm(r))*(×(Δη(r)×Hod(r)))~d2k(2π)4(B˜z,olcm)*([Δη˜(k2B˜z,od)]
+[(kxΔη˜)(kxB˜z,od)]+[(kyΔη˜)(kyB˜z,od)])
(2)

where ∗ denotes convolution. In converting from the real space integral to momentum space, we have neglected the variation of η(r) and Δη(r) in the -direction, so that Hod (r) ≈ Bz,od(r⊥) (TE-like mode). From this equation, it is clear that the FT of the dielectric perturbation, Δη˜ (k⊥), is the key quantity that couples Fourier components between the basis modes of the system. By tailoring this quantity appropriately, we can thus limit couplings that lead to in-plane and vertical leakage. Such a tailoring can be implemented for both the square and hexagonal lattice designs; however, due to the relative simplicity of the Fourier space representation of the B e,d1 A2 defect mode in the square lattice, we focus on it in this paper. The implementation of Fourier space design rules in standard and distorted hexagonal lattices is discussed in future work.

Our candidate mode, B e,d1 A2 , has dominant in-plane Fourier components at ±k X1. We must therefore modify the defect so that Δη˜ does not couple the ±k X1 momentum components to those regions in k-space which are “leaky”. In order to reduce radiation normal to the PC slab through coupling to the light cone, the amplitude of Δη̃ in the neighborhood of ky = ±π/a should be minimized. In addition, for the square lattice designs investigated here the bandgap between the conduction band-edge at the X-point and the valence band-edge at the M-point is at best very narrow, consequently, we look to reduce coupling between neighborhoods surrounding the X- and M-poits.

Fig. 5. Illustration showing the mode coupling for the B e,d1 A2, mode in k-space through the Δη̃ perturbation.

This implies that it will also be necessary to reduce the amplitude of Δη̃ in the region about kx = ±π/a.

The crux of the argument described above is depicted in Figure 5, where lossy couplings are illustrated for the upper region of k-space (the negative ky region will behave identically in this case). Here we have assumed that the defect mode frequency lies below the conduction band-edge at the X-point but slightly within the valence band near the M-point, resulting in an annular region of k-space about the M-point which is strongly coupled to. With reference to this simple schematic, the Fourier components of Δη̃ that lead to radiation losses from the defect cavity are approximately:

Δη˜(kx(k1c+Δx),ky±kX1(k1c+Δy))couplingtolightcone,
Δη˜kx±kX2Δx,kyΔy)couplingtoleakyMpoint.
(3)

where klc is the radius of the light cone, and Δx and Δy, are the widths of the dominant Fourier peaks in the x- and y-directions, respectively.

Before attempting any design modifications, we first consider the simple defect geometry studied in Section 4, where the holes located at (0, ±a/2) were reduced from the standard hole radius r to a radius r′. The perturbation Δη is given by the difference in η with and without the defect holes, and thus simply consists of a pair of annuli, each of width (r - r′), centered at (0, ±a/2). The 2D spatial FT of this function can be obtained analytically[22

22. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, Germany, 2001).

], and is separable into the form

Δη˜(k)=F(k;r,r)cos(kya2),
(4)

where F(k ;r,r′) is a function of the magnitude of the in-plane momentum, with r and r′ as parameters. This function, along with one-dimensional (1D) slices along the kx and ky axes, is shown in figure 6 (the figure shown is actually the direct FT of the structure used in FDTD calculations, to take into account any staircasing effects in the rendering of the holes; however, the difference between it and the analytic function are insignificant.). We notice, as is clear from examining eq. (4), that Δη̃ = 0 at ky = ±π/a. Our choice of defect was thus a fortuitous one, as the zero amplitude of Δη̃ at the X points eliminated coupling between the dominant Fourier components of the B e,d1 A2 mode and DC. Of course, a localized defect mode has a finite bandwidth in Fourier space about its dominant momentum components, and the light cone encompassing the radiation modes is of finite radius as well. As a result it is desirable to minimize the Fourier components of the dielectric perturbation over an extended region about ky = ±π/a.

Fig. 6. Δη̃(k ) for dielectric structure of Table 7.

It is also necessary to modify the dielectric to improve Q . The most straightforward way to immediately do so is to increase the r/a of the host PC, as that will provide a bandgap for an increased range of momentum values. This also tends to decrease Q , as the increased r/a will produce a mode of higher frequency, resulting in a cladding light cone of increased radius encompassing a larger range of momentum values. Fortunately, this does not necessarily have to hold for a general defect geometry. In particular, the hole radius can be kept relatively small in the region where the mode is primarily located, but can be graded outside this region to increase the in-plane reflectivity. The choice of grading can be determined by considering the need to limit the in-plane momentum components of the mode to regions in which the bandgap is substantial (Note that for the simple two-hole design considered in Section 4, Δη̃ is quite large in this region of momentum space about kx = ±π/a). The benefit of this approach is that it does not necessarily result in increased vertical radiation loss, thus allowing for both a large Q and Q .

Fig. 7. Properties of the graded square lattice.

Consider the graded lattice shown in figure 7(a). The standard defect holes at (0, ±a/2) have r/a = 0.23, while their immediate neighbors have r/a = 0.253. The hole radii are then increased parabolically outwards for 5 periods in the -direction and 7 periods in the ŷ-direction, after which they are held constant. The nature of this grading is shown in figure 7(a), where the r/a profiles are given for slices along y = a/2 and x = 0. Along these axes the maximum value r/a attains is 0.31, but along the diagonal directions r/a grows to be as large as 0.35. The dielectric perturbation, which now consists of a series of annuli of decreasing width from the center to the edges, has a FT given in Figure 7(b). Examining both the 2D image and the 1D line scans of the FT, we see that our grading has greatly diminished the amplitude of Δη̃ in the regions surrounding ky = ±π/a and kx = ±π/a.

The FDTD simulations of the defect mode of this structure largely confirm the ideas described thus far. Q has increased to over 110,000, while Q has improved even further to approximately 470,000, giving an overall Q tot ≈ 89,000. The magnetic field amplitude and FT of the in-plane electric field components in Table 8 support these results. In particular, consider the line scan of x along the ky axis. It shows that the grading has met with success, as power has largely been eliminated within the light cone. This point is particularly striking when contrasted with the corresponding image shown in Figure 3(b) for the low Q x-dipole mode we took as our baseline. Note that Δη̃(kx = 0, ky = ±π/a) is identically zero regardless of the grade, due to the position of the defect holes with respect to the center of the defect, whereas Δη̃(kx = ±π/a, ky = 0) is not automatically zero. It may be advantageous to identically zero Δη̃/(kx = ±π/a, ky = 0) as this will allow for the formation of a more localized mode that is still of high Q . Such a mode would be centered at the f-point of the square lattice, and would either be the B f,d1 A2 or B f,d1 B2 mode.

Table 8. Field characteristics of graded square lattice shown in figure 7(a).

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It should again be emphasized that the increased Q for these graded lattice designs is not solely the result of real-space delocalization of the mode. It is instead largely due to the aforementioned reduction of amplitude for those Fourier components of the dielectric perturbation that couple the dominant momentum components of the defect mode to those which radiate. Of course, real-space localization plays a role in determining the spread in k-space of the dominant Fourier components of the mode, and if this spread exceeds the size of the region about ±k X1 that Δη̃ has been flattened, vertical radiation will result. An increase in the slab thickness also effects the performance of the structure. It causes a decrease in the frequency of the mode, thus increasing Q . It also slightly reduces the size of the in-plane bandgap, decreasing Q . This is in fact seen in the results of FDTD simulations compiled in Table 8.

Finally, we note that the criteria for choosing the geometries presented in this paper were entirely based on Q considerations, and optimization of the lattice grading to further increase Q can still be made. Changes may also be made to improve other aspects of the design. In particular, reducing the mode volume may be of importance to applications in quantum optics, while reducing the complexity of the design (in terms of the number and size of holes comprising the defect) may be of interest from a fabrication standpoint. The approach to such designs can be aided through the Fourier space consideration of the dielectric perturbation as has been described in this section. Doing so will elucidate the potential lossy couplings that occur when the defect mode is formed, and will help determine whether a given structure is able to sustain a high-Q mode.

6 Summary

The design of high-Q defect modes in a 2D PC slab WG has been developed through use of momentum space methods. Starting with the fundamental criterion that the reduction of vertical radiation losses requires an elimination of momentum components within the light cone of the slab waveguide, we proceed to present methods by which this is accomplished. The first is through a judicious choice of the mode’s symmetry so that it is odd about mirror planes orthogonal to the mode’s dominant Fourier components. To determine the precise nature of the symmetry for such modes in square and hexagonal lattices, we refer to the symmetry analysis of [13

13. O. Painter, K. Srinivasan, J. D. O’Brien, A. Scherer, and P. D. Dapkus, “Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides,” J. Opt. A 3, S161–S170 (2001). [CrossRef]

, 18

18. O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” submitted to Phys. Rev. B (2002).

], from which we produce a set of candidate modes that satisfy this momentum space criteria. Although symmetry alone can reduce vertical radiation loss, further modifications of the defect geometry based upon Fourier space considerations can be used to increase Q even further. Tailoring the lattice to avoid momentum space couplings which lead to in-plane and vertical radiation losses, we present graded square lattice structures for which Q exceeds 105 while maintaining Q in the 3 – 5 × 105 range, demonstrating the possibility of producing high-Q modes in a planar PC slab WG by using these techniques.

K. Srinivasan thanks the Hertz Foundation for its financial support.

References and links

1.

H. Yokoyama, “Physics and Device Application of Optical Microcavities,” Science 256, 66–70 (1992). [CrossRef] [PubMed]

2.

J. L. Jewell, J. P. Harbison, A. Scherer, Y. H. Lee, and L. T. Florez, “Vertical-Cavity Surface-Emitting Lasers: Design, Growth, Fabrication, Characterization,” IEEE J. Quan. Elec. 27, 1332–1346 (1991). [CrossRef]

3.

S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A. Logan, “Whispering-gallery mode lasers,” Appl. Phys. Lett. 60, 289–291 (1992). [CrossRef]

4.

S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using spherical dielectric microcavity,” Nature 415, 621–623 (2002). [CrossRef] [PubMed]

5.

B. E. Little, S. T. Chu, W. Pan, D. Ripin, T. Kaneko, Y. Kokubun, and E. Ippen, “Vertically Coupled Glass Microring Resonator Channel Dropping Filters,” IEEE Photonics Tech. Lett. 11, 215–217 (1999). [CrossRef]

6.

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608–610 (2000). [CrossRef] [PubMed]

7.

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Channel drop filters in photonic crystals,” Opt. Express 3, 4–11 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-1-4. [CrossRef] [PubMed]

8.

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode-coupling semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999). [CrossRef]

9.

D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Roberts, “Photonic bandstructure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996). [CrossRef]

10.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejaki, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999). [CrossRef]

11.

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal andanalysis,” Opt. Lett. 24, 711–713 (1999). [CrossRef]

12.

C. Smith, R. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. Krauss, U. Oesterlé, and R. Houdré, “Coupled guide and cavity in a two-dimensional photonic crystal,” Appl. Phys. Lett. 78, 1487–1489 (2001). [CrossRef]

13.

O. Painter, K. Srinivasan, J. D. O’Brien, A. Scherer, and P. D. Dapkus, “Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides,” J. Opt. A 3, S161–S170 (2001). [CrossRef]

14.

O. J. Painter, A. Husain, A. Scherer, J. D. O’Brien, I. Kim, and P. D. Dapkus, “Room Temperature Photonic Crystal Defect Lasers at Near-Infrared Wavelengths in InGaAsP,” J. Lightwave Tech. 17, 2082–2088 (1999). [CrossRef]

15.

O. Painter, J. Vučković, and A. Scherer, “Defect Modes of a Two-Dimensional Photonic Crystal in an Optically Thin Dielectric Slab,” J. Opt. Soc. Am. B 16, 275–285 (1999). [CrossRef]

16.

J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65 (2002).

17.

T. Yoshie, J. Vučković, A. Scherer, H. Chen, and D. Deppe, “High quality two-dimensional photonic crystal slab cavities,” Appl. Phys. Lett. 79, 4289–4291 (2001). [CrossRef]

18.

O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” submitted to Phys. Rev. B (2002).

19.

This can be viewed in the far-field as elimination of lower-order multi-pole radiation components[23].

20.

E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopou-los, “Donor and acceptor modes in photonic band-structure,” Phys. Rev. Lett. 67, 3380–3383 (1991). [CrossRef] [PubMed]

21.

M. Tinkham, Group Theory and Quantum Mechanics, International Series in Pure and Applied Physics (McGaw-Hill, Inc., New York, NY, 1964).

22.

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, Germany, 2001).

23.

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic bandgap,” Appl. Phys. Lett. 78, 3388–3390 (2001). [CrossRef]

OCIS Codes
(140.5960) Lasers and laser optics : Semiconductor lasers
(230.5750) Optical devices : Resonators

ToC Category:
Research Papers

History
Original Manuscript: June 3, 2002
Revised Manuscript: July 17, 2002
Published: July 29, 2002

Citation
Kartik Srinivasan and Oskar Painter, "Momentum space design of high-Q photonic crystal optical cavities," Opt. Express 10, 670-684 (2002)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-15-670


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References

  1. H. Yokoyama, ???Physics and Device Application of Optical Microcavities,??? Science 256, 66???70 (1992) [CrossRef] [PubMed]
  2. J. L. Jewell, J. P. Harbison, A. Scherer, Y. H. Lee, andL. T. Florez, ???Vertical-Cavity Surface-Emitting Lasers: Design, Growth, Fabrication, Characterization,??? IEEE J. Quantum Electron. 27, 1332???1346 (1991). [CrossRef]
  3. S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R . A. Logan, ???Whispering-gallery mode lasers,??? Appl. Phys. Lett. 60, 289???291 (1992). [CrossRef]
  4. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, ???Ultralow-threshold Raman laser using spherical dielectric microcavity,??? Nature 415, 621???623 (2002). [CrossRef] [PubMed]
  5. B. E. Little, S. T. Chu, W. Pan, D. Ripin, T. Kaneko, Y. Kokubun, and E. Ippen, ???Vertically Coupled Gl ass Microring Resonator Channel Dropping Filters,??? IEEE Photonics Tech. Lett. 11, 215???217 (1999). [CrossRef]
  6. S. Noda, A. Chutinan, and M. Imada, ???Trapping and emission of photons by a single defect in a photonic bandgap structure,??? Nature 407, 608???610 (2000). [CrossRef] [PubMed]
  7. S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, ???Channel drop filters in photonic crystals,??? Opt. Express 3, 4???11 (1998), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-1-4">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-1-4</a>. [CrossRef] [PubMed]
  8. G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, ???Nonlinear optics of normalmode-coupling semiconductor microcavities,??? Rev. Mod. Phys. 71, 1591???1639 (1999). [CrossRef]
  9. D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Roberts, ???Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,??? J. Mod. Opt. 43, 1035???1053 (1996). [CrossRef]
  10. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejaki, ???Guided modes in photonic crystal slabs,??? Phys. Rev. B 60, 5751???5758 (1999). [CrossRef]
  11. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, ???Coupled-resonator optical waveguide: a proposal and analysis,??? Opt. Lett. 24, 711???713 (1999). [CrossRef]
  12. C. Smith, R. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. Krauss, U. Oesterle, andR . Houdre, ???Coupled guide and cavity in a two-dimensional photonic crystal,??? Appl. Phys. Lett. 78, 1487???1489 (2001). [CrossRef]
  13. O. Painter, K. Srinivasan, J. D. O???Brien, A. Scherer, and P. D. Dapkus, ???Tailoring of the resonant mode properties of optical nanocavities in two-dimensional photonic crystal slab waveguides,??? J. Opt. A 3, S161???S170 (2001). [CrossRef]
  14. O. J. Painter, A. Husain, A. Scherer, J.D. O???Brien, I.Kim, and P.D.Dapkus, ???RoomTemperature Photonic Crystal Defect Lasers at Near-Infrared Wavelengths in InGaAsP,??? J. Lightwave Tech. 17, 2082???2088 (1999). [CrossRef]
  15. O. Painter, J. Vu¡ckovic, andA . Scherer, ???Defect Modes of a Two-Dimensional Photonic Crystal in an Optically Thin Dielectric Slab,??? J. Opt. Soc. Am. B 16, 275???285 (1999). [CrossRef]
  16. J. Vu¡ckovic, M. Lon¡car, H. Mabuchi, and A . Scherer, ???Design of photonic crystal microcavities for cavity QED,??? Phys. Rev. E 65 (2002).
  17. T. Yoshie, J. Vu¡ckovic, A. Scherer, H. Chen, and D. Deppe, ???High quality two-dimensional photonic crystal slab cavities,??? Appl. Phys. Lett. 79, 4289???4291 (2001). [CrossRef]
  18. O. Painter andK . Srinivasan, ???Localizedd efect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,??? submitted to Phys. Rev. B (2002).
  19. This can be viewed in the far-field as elimination of lower-order multi-pole radiation components [23].
  20. E. Yablonovitch, T. J. Gmitter, R. D. Meade, A. M. Rappe, K. D. Brommer, and J. D. Joannopoulos, ???Donor and acceptor modes in photonic band-structure,??? Phys. Rev. Lett. 67, 3380???3383 (1991). [CrossRef] [PubMed]
  21. M. Tinkham, Group Theory and Quantum Mechanics, International Series in Pure and Applied Physics (McGaw-Hill, Inc., New York, NY, 1964).
  22. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, Germany, 2001).
  23. S. G. Johnson, S. Fan, A. Mekis, andJ. D. Joannopoulos, ???Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic bandgap,??? Appl. Phys. Lett. 78, 3388???3390 (2001). [CrossRef]

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