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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 6 — Mar. 24, 2003
  • pp: 579–593
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Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals

Kartik Srinivasan and Oskar Painter  »View Author Affiliations


Optics Express, Vol. 11, Issue 6, pp. 579-593 (2003)
http://dx.doi.org/10.1364/OE.11.000579


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Abstract

Building upon the results of recent work [1], we use momentum space design rules to investigate high quality factor (Q) optical cavities in standard and compressed hexagonal lattice photonic crystal (PC) slab waveguides. Beginning with the standard hexagonal lattice, the results of a symmetry analysis are used to determine a cavity geometry that produces a mode whose symmetry immediately leads to a reduction in vertical radiation loss from the PC slab. The Q is improved further by a tailoring of the defect geometry in Fourier space so as to limit coupling between the dominant Fourier components of the defect mode and those momentum components that radiate. Numerical investigations using the finite-difference time-domain (FDTD) method show significant improvement using these methods, with total Q values exceeding 105. We also consider defect cavities in a compressed hexagonal lattice, where the lattice compression is used to modify the in-plane bandstructure of the PC lattice, creating new (frequency) degeneracies and modifying the dominant Fourier components found in the defect modes. High Q cavities in this new lattice geometry are designed using the momentum space design techniques outlined above. FDTD simulations of these structures yield Q values in excess of 105 with mode volumes of approximately 0.35 cubic half-wavelengths in vacuum.

© 2003 Optical Society of America

1. Introduction

High quality factor (Q) photonic crystal (PC) microcavities are potentially important devices for both lightwave technology and studies in quantum optics. Their ability to confine light to a single resonant mode within an extremely small volume has opened up potential applications for low threshold light sources, ultra-high density planar lightwave circuits, and experiments examining electron-photon interactions in quantum optics. The host geometry for such cavities has typically been a two-dimensional (2D) PC slab waveguide (WG) [2

2. 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]

, 3

3. 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]

], largely as a result of the maturity of planar fabrication technology. The control exercised by current processing techniques has been evidenced in the ability to integrate PC microresonators with WGs [4

4. 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]

, 5

5. 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]

], and in the construction of defect cavity lasers with prescribed emission properties [6

6. 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]

].

The PC optical microcavities studied in [7

7. 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]

] trapped light to a volume of ≈ 2.5(λ/2)3 in the material 1, nearing the theoretical limit of (λ/2)3. The measured Q values were less than 1500, however, and this number must be increased by roughly an order of magnitude or more for PC slab WG microcavities to be favorably compared with their competitors in the applications previously mentioned. Refinements in design [8

8. J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E65 (2002).

] and fabrication [9

9. 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 further improved the performance of these devices, resulting in predicted Q values of up to 3×104 (the highest predicted Q for the designs in [10

10. 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]

] was 2×104) and measured Q values of up to 2800. At the same time, other types of defect modes have been studied, including a monopole mode in the hexagonal lattice [11

11. H. Park, J. Hwang, J. Huh, H. Ryu, Y. Lee, and J. Hwang, “Nondegenerate monopole-mode two-dimensional photonic band gap laser,” Appl. Phys. Lett. 79, 3032–3034 (2001). [CrossRef]

, 12

12. J. Huh, J.-K. Hwang, H.-Y. Ryu, and Y.-H. Lee, “Nondegenerate monopole mode of single defect two-dimensional triangular photonic band-gap cavity,” J. Appl. Phys. 92, 654–659 (2002). [CrossRef]

] and different types of defect modes in the square lattice [13

13. H.-Y. Ryu, S.-H. Kim, H.-G. Park, J.-K. Hwang, Y.-H. Lee, and J.-S. Kim, “Square-lattice photonic band-gap single-cell laser operating in the lowest-order whispering gallery mode,” Appl. Phys. Lett. 80, 3883–3885 (2002). [CrossRef]

]. The predicted Q values in these structures were also on the order of 104, with a measured Q for the monopole mode of 1, 900.

2. Summary of momentum space design rules

Defect cavities in 2D PC slab WGs confine light through two methods, standard total internal reflection (TIR) in the vertical (waveguiding) direction, and distributed Bragg reflection (DBR) in-plane (Fig. 1). In-plane losses are thus determined by the number of periods in the host photonic lattice and the width and angular extent of the in-plane guided mode bandgap. Vertical confinement is dictated by the condition that the inplane momentum, k⊥, be sufficiently large to support guiding. To see this, recall that the energy-momentum dispersion relationship for the air cladding of the PC slab WG is given by (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. Examining this equation, we see that k2=(ω/c)2 defines a cone in (kx , ky , ω) space, referred to as the “light cone” of the WG cladding, as illustrated in Fig. 1. At a given ω, for a mode to be guided vertically, k >ω/c. This is our fundamental guideline in designing cavities that reduce vertical radiation loss.

Before proceeding, let us discuss how this simple rule relates to other methods that have been used to design high-Q PC defect cavities. In addition to our earlier work [1

1. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002). [CrossRef] [PubMed]

], in Ref. [14

14. J. Vučković, M. Lončcar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in Photonic Crystal Microcavities,” IEEE J. Quan. Elec. 38, 850–856 (2002). [CrossRef]

], the authors note that a reduction of power within the cladding light cone is a signature for structures with a high vertical Q (Q ). The authors tune the geometry of their particular defect geometry to maximize Q , and show that this is reflected by the elimination of small momentum components in the Fourier spectrum of their mode. In Ref. [15

15. 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 band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001). [CrossRef]

], Johnson and co-workers examine the multipole expansion of the radiation field, and demonstrate that a reduction in vertical radiation loss is a result of cancellation of the lower-order components in this expansion. Both in that work and in Ref. [16

16. H. Benisty, D. Labilloy, C. Weisbuch, C. Smith, T. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000). [CrossRef]

], delocalization of the field (in-plane in the case of the former work and vertical in the latter) has also been used as a means for improving Q , as broadening of the mode in real space corresponds to more localized dominant Fourier components in momentum space, thus reducing the presence of power within the cladding light cone. It is important to note that (as discussed in Ref. [1

1. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002). [CrossRef] [PubMed]

]) methods other than strict delocalization of the field can be used to improve Q . In particular, in that earlier work, we use a tailoring of the defect geometry to improve Q from 69, 000 to 110, 000 without an appreciable increase in the real-space extent of the field. As was the case in Ref. [1

1. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002). [CrossRef] [PubMed]

], the approach that we discuss in this article combines several different techniques for improving Q , not all of which rely on delocalization of the field.

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

Our first step in limiting the presence of small in-plane momentum components is to use symmetry to enforce specific boundary conditions on the Fourier space representation of the mode. In particular, we choose modes whose symmetry is odd about mirror planes normal to the dominant Fourier components of the mode. Within the vertical mirror plane of the slab WG (coordinates r ) the fundamental even (TE-like) 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, which corresponds to having the in-plane electric field components spatially odd in these directions. In Fourier space, this choice of symmetry is equivalent to eliminating these in-plane electric field polarizations at k =0 (DC). This elimination of DC momentum components is the first step in reducing vertical radiation loss, and serves as our criterion for choosing the desired symmetry for our defect mode (note that this use of symmetry to eliminate lossy momentum components can also be viewed as a cancellation of lower-order multipole radiation components, as described in [15

15. 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 band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001). [CrossRef]

]). In order to determine which modes are consistent with the symmetry criterion, we use an approximate group theory analysis (described in [6

6. 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]

, 17

17. 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 classify the symmetries of donor and acceptor modes in the lattice under consideration. The process by which this is done is outlined in the following section.

To further improve both in-plane and out-of-plane performance, we consider the mode coupling that is introduced by a defect to a perfect (unperturbed) photonic crystal in Fourier space (a more complete analysis is given in Ref. [18

18. O. Painter, K. Srinivasan, and P. E. Barclay, “A Wannier-like Equation for the Resonant Optical Modes of Locally Perturbed Photonic Crystals,” submitted to Phys. Rev. B, December 2002.

]). Considering the PC slab as an approximately 2D system, and focusing on the fundamental TE-like (even) modes of the slab, reduces the problem to an effective scalar field theory in which the magnetic field is given by H(r)≈ẑH(r ), with r labeling the coordinates within the horizontal plane of the slab (to simplify notation, from here on we drop the ⊥ label from the in-plane coordinates). The scalar field eigenoperator for the magnetic field in this quasi-2D approximation is given by,

ˆHTE=(ηo+Δη)·(ηo+Δη)2.
(1)

η o represents the inverse of the the square of the refractive index of the unperturbed photonic crystal, 1/n2D2(r), and Δη is the localized perturbation to 1/n2D2(r). The mixing of the Bloch modes of the PC due to the presence of the defect perturbation, ^H =-∇(Δη)·∇-(Δη)∇2, can be shown to be given by [18

18. O. Painter, K. Srinivasan, and P. E. Barclay, “A Wannier-like Equation for the Resonant Optical Modes of Locally Perturbed Photonic Crystals,” submitted to Phys. Rev. B, December 2002.

]:

Hl,kˆHHl,k=Gk(Δη˜kKl,lkkG+Δη˜k(ik)·Ll,lkkG)δkk+G,k,
(2)

where Δ͠ηk″ is the k″th Fourier coefficient of Δη(r), l and k label the band index and crystal momentum of the H l,k Bloch wave, and the G are reciprocal lattice vectors. As shown in Ref. [18

18. O. Painter, K. Srinivasan, and P. E. Barclay, “A Wannier-like Equation for the Resonant Optical Modes of Locally Perturbed Photonic Crystals,” submitted to Phys. Rev. B, December 2002.

], K l′,l(k′, k,G) and L l′,l(k′, k,G) are scalar and vector coupling matrix elements, respectively, which depend upon the Bloch waves.

From Eq. (2), it is clear that the Fourier Transform (FT) of the dielectric perturbation, Δ͠η(k), is a key quantity in determining the coupling of different Bloch modes of the unperturbed crystal (modulo a reciprocal lattice vector). By tailoring this quantity appropriately, we can thus limit couplings that lead to in-plane and vertical leakage. In particular, we seek to eliminate couplings between a mode’s dominant Fourier components and regions of momentum space that are known to radiate, such as the interior of the light cone. Such a tailoring was implemented in Ref. [1

1. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002). [CrossRef] [PubMed]

] for square photonic lattice defect geometries; we now consider its implementation for standard and compressed hexagonal lattice designs.

3. High-Q defect modes in a hexagonal lattice

For the sake of cogency and completeness, it is worth repeating the results of Ref. [1

1. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002). [CrossRef] [PubMed]

] for standard hexagonal lattices. In subsection 3.1, we outline the symmetry classifications for donor and acceptor modes in a hexagonal lattice, and proceed to pick a mode of symmetry consistent with our momentum space design rules of Section 2. A simple defect cavity consistent with this symmetry is simulated through three-dimensional (3D) FDTD methods and results are given. Using this structure as a starting point, subsection 3.2 considers modifications to the dielectric lattice to reduce lossy coupling in Fourier space, and 3D FDTD simulations results are presented.

3.1 Summary of previous results

The real and reciprocal space depictions of a hexagonal PC lattice are given in Fig. 2(a), while its in-plane bandstructure is given in Fig. 2(b). Defect modes are typically 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. Joannopoulos, “Donor and acceptor modes in photonic band-structure,” Phys. Rev. Lett. 67, 3380–3383 (1991). [CrossRef] [PubMed]

], depending upon whether the defect creates modes from the conduction band-edge (the X-point in the hexagonal lattice) or the valence band-edge (the J-point in the hexagonal lattice), respectively. 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. 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 in Section 2.

The high symmetry points in the hexagonal lattice, as indicated in Fig. 2(a), are points a (C 6υ symmetry), b (C 2υ symmetry), and c ( C3υ,συ symmetry). We consider modes formed at points a and b (modes formed from point c are not of the requisite symmetry and dominant Fourier components, as seen from [17

17. 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).

]), including reduced symmetry modes formed at point a, where the reduction of symmetry from C 6υ to C 2υ is accomplished by choosing a defect that breaks the symmetry of the lattice and is consistent with C 2υ. Duplicating the results given in Ref. [1

1. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002). [CrossRef] [PubMed]

], we present Table 1 for donor modes and Table 2 for acceptor modes. These tables provide the labeling scheme for the C 6υ and C 2υ 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 [19

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

]).

Fig. 2. (a) Real and reciprocal space lattices of a standard 2D hexagonal lattice. Refer to Table 5 for identification of key geometrical quantities. (b) Fundamental TE-like (even) guided mode bandstructure for hexagonal lattice calculated using a 2D plane-wave expansion method with an effective index for the vertical guiding; r/a=0.36, n slab=n eff=2.65.

As discussed in Ref. [1

1. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002). [CrossRef] [PubMed]

], none of the donor modes presented in the tables above are consistent with our symmetry criteria for reducing vertical radiation losses. Out of the C 6υ acceptor modes in Table 2, the BA2a,a1 mode satisfies the symmetry criteria. For reference, the approximate form for the BA2a,a1 mode is listed below [6

6. 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(kJ1·ra)+cos(kJ3·ra)+cos(kJ5·ra)),
(3)

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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To create the BA2a,a1 mode, a central hole (about point a) was 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 BA2a,a1 are given in Table 3 for two different pairs of values (r, r′). The dominant Fourier components are seen to be ±{kJ1,kJ3,kJ5} , 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, say, the x-dipole donor mode, as discussed in Ref. [1

1. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002). [CrossRef] [PubMed]

]), it is still significant. 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).

Table 3. Characteristics of the BA2a,a1 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).

3.2 Tailoring of the defect geometry

When comparing defect modes of a square lattice with those of a hexagonal lattice in the context of forming high-Q microcavities, there are a number of salient points that merit consideration. The first is that the square lattice designs adopted in Ref. [1

1. K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002). [CrossRef] [PubMed]

] provided a natural “geometric” advantage in that Δ͠η(k ) (even in the simplest case of two reduced size air holes) was automatically zero at the dominant Fourier components (kx =0, ky =±π/a), thereby reducing coupling between those components and small momentum components that radiate. Furthermore, these dominant Fourier components were in directions orthogonal to the available mirror planes of the system, maximizing the symmetry-based reduction of small momentum components as discussed in Section 2. In the hexagonal lattice, it is difficult to obtain a similar set of circumstances. The only mode consistent with the symmetry criteria is the BA2a,a1 mode, but defects that create such a mode have Δ͠η(k ) that is non-zero at the mode’s dominant Fourier components ( ±{kJ1,kJ3,kJ5} ). Conversely, a mode such as BA2b,d1 , formed by a defect such as two reduced size holes at (0,±a√3/2), could have Δ͠η(k )=0 at its dominant Fourier components ( ±{kX2,kX3} ), but these Fourier components are oriented along directions that are not orthogonal to the available mirror planes of the system.

Despite these obstacles, it is certainly possible to design high-Q defect cavities in a hexagonal lattice. One advantage of the hexagonal lattice is that it exhibits a relatively large and complete in-plane bandgap for TE-like modes due to its nearly circular first Brillouin zone (IBZ) boundary. This essentially guarantees the ability to achieve high in-plane Q provided that the mode is suitably positioned within the bandgap, and that a sufficient number of periods of the photonic lattice are used (it is still important not to entirely neglect in-plane considerations in cavity designs as the mode volume can be affected significantly). To address vertical radiation losses, the defect geometry can be tailored to reduce couplings to the light cone, even though Δ͠η(k ) does not necessarily have the automatic zeros it had in the case of the square lattice. Examining such tailorings is the focus of this section.

Table 4. FDTD simulation results for graded hexagonal lattice geometries (images are for the first PC cavity listed below; d/a=0.75 in all designs).

Fig. 3. (a) Δ͠η(k ) for single enlarged hole design in hexagonal lattice (r/a=0.30, r′/a=0.45). (b) Δ͠η(k ) for graded hexagonal lattice design shown in Table 4.

As a final example, we consider adjusting the first level of confinement to reduce the mode volume. Starting with our original graded cavity design (the first design of Table 4), the size of the holes adjacent to the central defect are increased to a value of (r/a) nn =0.355. The results are for the most part intermediate to the first two examples, with Q =8×105 and Q =1.07×105. One important exception is that V eff=0.24 is actually much smaller than both of the original designs. Upon further consideration, this result is not too surprising; the smaller mode volume and the relatively large Q are a result of the stronger yet more extended central perturbation to the photonic lattice.

4. Defect modes in a compressed hexagonal lattice

4.1 Preliminary analysis

We would like to create a mode whose dominant Fourier components are orthogonal to σ x and σ y . Such a mode would have dominant Fourier components ±kX1 and/or ±kJ2 . Let us begin by considering acceptor modes. By compressing the lattice in the ŷ-direction, so that the spacing between two adjacent rows of holes is less than its usual value (changing it from a√3/2 to γa√3/2, where γ is the compression factor), we intuitively expect the position of the band edges in that direction of Fourier space (corresponding to ±kX1 ) to increase in frequency, perhaps to the point where the valence band-edge at X 1 is nearly degenerate with the valence band-edge at the J-points. Of course, this qualitative justification leaves many questions unanswered (such as the position of the band-edges at the other high symmetry points in the lattice). To properly answer these questions, we formulate a symmetry analysis of defect modes in compressed hexagonal lattices, using the methods of [6

6. 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]

, 17

17. 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).

].
Fig. 4. (a) Real and reciprocal space lattices of a compressed 2D hexagonal lattice. Refer to Table 5 for more identification of key geometrical quantities; (b) Fundamental TE-like (even) guided mode bandstructure for a compressed hexagonal lattice, calculated using a 2D plane-wave expansion method with an effective index for the vertical guiding; r/a=0.35, n slab=n eff=2.65, γ=0.7.

Consider the real and reciprocal space representations of the compressed hexagonal lattice as illustrated in Fig. 4(a). Compression has reduced the point group symmetry of the lattice to C 2υ, and the irreducible Brillouin zone (IrBZ) is no longer a 30°-60°-90° triangle, but is now a quadrilateral, traced between Γ-X 1-J 1-X 2-J 2-Γ. The modifications in various geometrical quantities associated with the real and reciprocal space compressed lattice are given in Table 5. Note that, in particular, the group of the wavevector ℊok at the X and J points has been reduced in symmetry, and that [kX1][kX2] (the kji are still equal in magnitude). Furthermore, kX1 now approaches |k J |. Indeed, for a compression factor γ=1/√3, the vectors coincide and the resulting lattice is in fact square. For compression factors between 0.8 and 1/√3, the vectors are still quite close in magnitude, and we qualitatively expect that the lowest frequency band (the valence band) will be very nearly degenerate at the X 1 and J points. It is in this way that the compressed hexagonal lattices considered in this section are intermediate to the hexagonal and square lattices. In using the compressed hexagonal lattice we hope to take advantage of the large in-plane bandgap of the hexagonal lattice and the favorable symmetry of the square lattice.

Table 5. Key geometrical quantities associated with the standard and compressed hexagonal lattices.

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Using the 2D plane wave expansion method with an effective index to account for vertical waveguiding ([10

10. 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]

]), we arrive at the bandstructure shown in Fig. 4(b). The compression ratio (γ) has been set at a value of 0.7 for this calculation. We see that the valence band is nearly degenerate at points X 1, J 1, and J 2, and thus, we expect an acceptor mode to be formed by mixing the valence band modes formed at all of these points in Fourier space. Following the symmetry analysis techniques originally described in [6

6. 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 determine approximate forms for valence band modes at these points. Grouping all of them together, we arrive at the following expressions for modes formed about the high symmetry point a shown in Fig. 4(a):

VBa=ẑ(cos(kX1·ra)eikJ1·ra+eikJ3·raeikJ4·ra+eikJ6·raeikJ2·raeikJ5·ra)
(4)

PA2=(2000001100011000001100011),PB2=(0000001100011000001100011).
(5)

Before moving on to discuss FDTD simulations, for the sake of completeness, let us briefly consider donor modes in this lattice. Such modes will be formed from the conduction band-edge located at point X 2 in Fig. 4(b). Using a symmetry analysis similar to that described above, we determine the conduction band modes for the a and b high symmetry points:

CBa=ẑ(sin(kX2·ra)sin(kX3·ra)),CBb=ẑ(cos(kX2·rb)cos(kX3·rb)),
(6)

where rb=ra-b.

The representation of the CBa basis under C 2υ (the defect symmetry), labeled S a,d1, is given by S a,d1=B 1B 2, while the representation of the CBb basis under C 2υ, labeled S b,d1 is given by S b,d1=A 1A 2. Projecting the CBa and CBb bases onto the irreducible representations above, we get

BB1a,d1=ẑ(sin(kX2·ra)sin(kX3·ra)),
BB2a,d1=ẑ(sin(kX2·ra)+sin(kX3·ra)),
BA1b,d1=ẑ(cos(kX2·rb)cos(kX3·rb)),
BA2b,d1=ẑ(cos(kX2·ra)+cos(kX3·rb)),
(7)

as approximate forms for the donor modes at points a and b.

Fig. 5. Modal characteristics of a simple defect mode in a compressed hexagonal lattice (d/a=0.75).

4.2 FDTD results

The procedure followed is the same as what has been done in the square and hexagonal lattices, namely, we modify the lattice (and therefore Δ͠η(k )) to reduce couplings between the mode’s dominant Fourier components (in this case, {±kX1,±kJ2} ) and the light cone. We do so by starting with a defect consisting of the four enlarged holes surrounding the b-point (we choose r′/a=(r/a)c=0.30), and then parabolically decreasing the hole radius as we move away from the defect center (down to a value of (r/a) e =0.225 at the edge of the crystal). The resulting lattice is shown in Table 6 (only the central region has been shown; in total there are 10 periods of air holes in and 8 periods in ŷ surrounding the defect center), along with the magnetic field amplitude and Fourier transformed electric field components for the defect mode. FDTD calculations predict Q =1.5×105, Q =7.5×105, Qtot =1.3×105, and Veff =0.35 for this design.

Table 6. FDTD simulation results for graded compressed hexagonal lattice geometries.

5. Summary

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

Footnotes

1 This value is quoted in terms of cubic-half wavelengths in the material. Throughout this paper we use the convention that mode volume is to be written in terms of cubic-half wavelengths in vacuum. For comparison, the mode volume in terms of vacuum wavelengths of the referenced defect cavity is 0.064.

References and links

1.

K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002). [CrossRef] [PubMed]

2.

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]

3.

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]

4.

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]

5.

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]

6.

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]

7.

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]

8.

J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E65 (2002).

9.

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]

10.

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]

11.

H. Park, J. Hwang, J. Huh, H. Ryu, Y. Lee, and J. Hwang, “Nondegenerate monopole-mode two-dimensional photonic band gap laser,” Appl. Phys. Lett. 79, 3032–3034 (2001). [CrossRef]

12.

J. Huh, J.-K. Hwang, H.-Y. Ryu, and Y.-H. Lee, “Nondegenerate monopole mode of single defect two-dimensional triangular photonic band-gap cavity,” J. Appl. Phys. 92, 654–659 (2002). [CrossRef]

13.

H.-Y. Ryu, S.-H. Kim, H.-G. Park, J.-K. Hwang, Y.-H. Lee, and J.-S. Kim, “Square-lattice photonic band-gap single-cell laser operating in the lowest-order whispering gallery mode,” Appl. Phys. Lett. 80, 3883–3885 (2002). [CrossRef]

14.

J. Vučković, M. Lončcar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in Photonic Crystal Microcavities,” IEEE J. Quan. Elec. 38, 850–856 (2002). [CrossRef]

15.

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 band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001). [CrossRef]

16.

H. Benisty, D. Labilloy, C. Weisbuch, C. Smith, T. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,” Appl. Phys. Lett. 76, 532–534 (2000). [CrossRef]

17.

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).

18.

O. Painter, K. Srinivasan, and P. E. Barclay, “A Wannier-like Equation for the Resonant Optical Modes of Locally Perturbed Photonic Crystals,” submitted to Phys. Rev. B, December 2002.

19.

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

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]

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

ToC Category:
Research Papers

History
Original Manuscript: January 6, 2003
Revised Manuscript: March 5, 2003
Published: March 24, 2003

Citation
Kartik Srinivasan and Oskar Painter, "Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals," Opt. Express 11, 579-593 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-6-579


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References

  1. K. Srinivasan and O. Painter, �??Momentum space design of high-Q photonic crystal optical cavities,�?? Opt. Express 10, 670�??684 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670</a> [CrossRef] [PubMed]
  2. 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]
  3. 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]
  4. 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]
  5. C. Smith, R. De la Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. Krauss, U. Oesterle, and R. Houdre, �??Coupled guide and cavity in a two-dimensional photonic crystal,�?? Appl. Phys. Lett. 78, 1487�??1489 (2001). [CrossRef]
  6. 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]
  7. 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]
  8. J. Vu¡ckovic, M. Lon¡car, H. Mabuchi, and A. Scherer, �??Design of photonic crystal microcavities for cavity QED,�?? Phys. Rev. E 65 (2002).
  9. 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]
  10. O. Painter, J. Vu¡ckovi´c, 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]
  11. H. Park, J. Hwang, J. Huh, H. Ryu, Y. Lee, and J. Hwang, �??Nondegenerate monopole-mode two-dimensional photonic band gap laser,�?? Appl. Phys. Lett. 79, 3032�??3034 (2001). [CrossRef]
  12. J. Huh, J.-K. Hwang, H.-Y. Ryu, and Y.-H. Lee, �??Nondegenerate monopole mode of single defect two-dimensional triangular photonic band-gap cavity,�?? J. Appl. Phys. 92, 654�??659 (2002). [CrossRef]
  13. H.-Y. Ryu, S.-H. Kim, H.-G. Park, J.-K. Hwang, Y.-H. Lee, and J.-S. Kim, �??Square-lattice photonic band-gap single-cell laser operating in the lowest-order whispering gallery mode,�?? Appl. Phys. Lett. 80, 3883�??3885 (2002). [CrossRef]
  14. J. Vu¡ckovic, M. Lon¡car, H. Mabuchi, and A. Scherer, �??Optimization of the Q factor in Photonic Crystal Microcavities,�?? IEEE J. Quantum Electron. 38, 850�??856 (2002). [CrossRef]
  15. 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 band gap,�?? Appl. Phys. Lett. 78, 3388�??3390 (2001). [CrossRef]
  16. H. Benisty, D. Labilloy, C. Weisbuch, C. Smith, T. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, �??Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,�?? Appl. Phys. Lett. 76, 532�??534 (2000). [CrossRef]
  17. 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).
  18. O. Painter, K. Srinivasan, and P. E. Barclay, �??A Wannier-like Equation for the Resonant Optical Modes of Locally Perturbed Photonic Crystals,�?? submitted to Phys. Rev. B, December 2002.
  19. M. Tinkham, Group Theory and Quantum Mechanics, International Series in Pure and Applied Physics (McGaw-Hill, Inc., New York, NY, 1964).
  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]

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