## Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals

Optics Express, Vol. 11, Issue 6, pp. 579-593 (2003)

http://dx.doi.org/10.1364/OE.11.000579

Acrobat PDF (987 KB)

### Abstract

Building upon the results of recent work [^{5}. 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 10^{5} with mode volumes of approximately 0.35 cubic half-wavelengths in vacuum.

© 2003 Optical Society of America

## 1. Introduction

*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. 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]

^{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] 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]

*Q*values of up to 3×10

^{4}(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]

^{4}) 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. 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]

*Q*values in these structures were also on the order of 10

^{4}, with a measured

*Q*for the monopole mode of 1, 900.

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

*Q*PC slab optical cavities, and considered their application in the design of various defect geometries. The essence of the design is a Fourier space approach to remove those momentum space components that radiate. Using both symmetry and a tailoring of the defect geometry to reduce the presence of these lossy Fourier space components, we were able to design a square lattice defect cavity with

*Q*≈105 and modal volume

*V*

_{eff}≈0.25(λ/2)

^{3}in

*vacuum*. The work described in this paper is largely an application of the momentum space design rules developed in the earlier work, but now in regards to defect cavities in standard and modified hexagonal lattices. The use of a compressed hexagonal lattice introduces additional degeneracies amongst the satellite extrema of the bandstructure, thus providing the additional design flexibility required to efficiently localized defect modes both vertically and in the plane of the dielectric slab. We begin in Section 2 with a summary of the momentum space design rules previously developed 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]

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

*Q*cavities in a modified hexagonal lattice, where the lattice is compressed in the

*ŷ*-direction (that is, the spacing between two rows of holes is changed from

*a*√3/2 to γ

*a*√3/2, where

*a*is the original lattice spacing and γ<1 is the lattice compression factor). Subsection

*4.1*discusses the effects of lattice compression on the in-plane photonic bandstructure, and on the symmetry analysis of defect modes within the lattice. The symmetry analysis is used to choose a mode whose Fourier amplitude is zero at

**k**=0 (DC), and in subsection

*4.2*FDTD results for defect cavities consistent with this symmetry are given.

## 2. Summary of momentum space design rules

*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}=

*k*

_{z}is the momentum normal to the slab, and

*c*is the speed of light. Examining this equation, we see that

*c*)

^{2}defines a cone in (

*k*

_{x},

*k*

_{y}, 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.

**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]

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]

**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,

_{o}represents the inverse of the the square of the refractive index of the unperturbed photonic crystal, 1/

**r**), and Δη is the localized perturbation to 1/

**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]:

_{k}″ 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],

*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.

**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

**10**, 670–684 (2002). [CrossRef] [PubMed]

## 3. High-Q defect modes in a hexagonal lattice

**10**, 670–684 (2002). [CrossRef] [PubMed]

*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

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]

*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.

*a*(

*C*

_{6υ}symmetry),

*b*(

*C*

_{2υ}symmetry), and

*c*(

*a*and

*b*(modes formed from point

*c*are not of the requisite symmetry and dominant Fourier components, as seen from [17]), 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

**10**, 670–684 (2002). [CrossRef] [PubMed]

*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]).

**10**, 670–684 (2002). [CrossRef] [PubMed]

*C*

_{6υ}acceptor modes in Table 2, the

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]

*a*.

**10**, 670–684 (2002). [CrossRef] [PubMed]

**3**, S161–S170 (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]

*Q*values

*Q*

_{⊥}and

*Q*

_{‖}can be calculated. A normalized slab thickness of

*d/a*=0.75 is typically used in order to ensure a single vertical mode of the PC slab waveguide. Variations in the slab thickness will of course affect both

*Q*

_{‖}and

*Q*

_{⊥}; in general, we would expect

*Q*

_{⊥}to increase and

*Q*

_{‖}to decrease as

*d/a*increases. The increase in

*Q*

_{⊥}is due to the reduced frequency for a defect mode in a thicker slab, as this causes a decrease in the size of the light cone, which determines the degree of vertical radiation loss. The decrease in

*Q*

_{‖}is predicted because the in-plane bandgap shrinks in size as the slab thickness increases.

*a*) was enlarged from radius

*r*to

*r*′. The defect is surrounded by a total of 8 periods of the hexagonal lattice in the

*x̂*-direction and 12 periods in the

*ŷ*-direction. The magnetic field amplitude and momentum space electric field components

**Ẽ**

_{x}and

**Ẽ**

_{y}of mode

*r*,

*r*′). The dominant Fourier components are seen to be

**Ẽ**

_{x}and

**Ẽ**

_{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

**10**, 670–684 (2002). [CrossRef] [PubMed]

*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).

### 3.2 Tailoring of the defect geometry

*Q*microcavities, there are a number of salient points that merit consideration. The first is that the square lattice designs adopted in Ref. [1

**10**, 670–684 (2002). [CrossRef] [PubMed]

**k**

_{⊥}) (even in the simplest case of two reduced size air holes) was automatically zero at the dominant Fourier components (

*k*

_{x}=0,

*k*

_{y}=±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

**k**

_{⊥}) that is non-zero at the mode’s dominant Fourier components (

*a*√3/2), could have Δ͠η(

**k**

_{⊥})=0 at its dominant Fourier components (

*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.

**10**, 670–684 (2002). [CrossRef] [PubMed]

*r/a*)

_{c}=0.35) followed by a relatively large decrease in hole radius ((

*r/a*)

_{nn}=0.325) for the nearest neighbor holes. The hole radii are then parabolically decreased in moving radially outwards (down to (

*r/a*)

_{e}=0.225 at the edge of the crystal), forming the second level of confinement. The effect this has on Δ͠η(

**k**

_{⊥}) is evident in Fig. 3(a)–(b), where we plot this function for the single enlarged hole design of the previous section and for the graded lattice design just described. It is clear that Δ͠η(

**k**

_{⊥}) has been dramatically reduced at

*Q*values and mode volume, as listed in Table 4, are

*Q*

_{⊥}=1.8×10

^{5},

*Q*

_{‖}=4×10

^{5}, and

*V*

_{eff}=0.49. As previously mentioned,

*Q*

_{‖}could be made larger by simply increasing the number of periods in the photonic lattice; however, this will not have an appreciable effect on the mode volume, which is somewhat large in this case.

*Q*

_{⊥}, we would like to modify it so as to reduce the mode volume, which, at

*V*

_{eff}=0.49, is roughly twice that which we had for square lattice designs[1

**10**, 670–684 (2002). [CrossRef] [PubMed]

*Q*has increased considerably, to a value of

*Q*

_{‖}=1.54×10

^{6}, and the mode volume has decreased to

*V*

_{eff}=0.34, but at the expense of a decreased vertical

*Q*, now at

*Q*

_{⊥}=76, 000. The decreased

*Q*

_{⊥}is the result of a number of factors. The improved in-plane localization widens the mode in Fourier space, broadening the dominant Fourier components to the extent that they extend into the cladding light cone. The modified grade also changes the magnitude of Δ͠η(

**k**

_{⊥}) at

*r/a*)

_{nn}=0.355. The results are for the most part intermediate to the first two examples, with

*Q*

_{‖}=8×10

^{5}and

*Q*

_{⊥}=1.07×10

^{5}. 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

*b*-point in Fig. 2(a). From the standpoint of designing a high-

*Q*mode, the donor and acceptor modes formed at this point do not meet our symmetry criteria, as the dominant Fourier components of the modes (as listed in Tables 1 and 2) are not orthogonal to the available mirror planes (σ

_{x}and σ

_{y}for the

*C*

_{2υ}symmetry found at the

*b*-point). This is a reflection of the fact that the

### 4.1 Preliminary analysis

_{x}and σ

_{y}. Such a mode would have dominant Fourier components

*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 γ

*a*√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

*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

**3**, S161–S170 (2001). [CrossRef]

*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 ℊ

_{ok}at the

*X*and

*J*points has been reduced in symmetry, and that

**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.

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]

*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

**3**, S161–S170 (2001). [CrossRef]

*b*(found by taking

**b**) differ from these only by constant phase factors and hence the modes above can be used for investigations about

*b*as well. Both the

*a*and

*b*points have

*C*

_{2υ}symmetry, and the representation of the

*VB*

_{a}basis under

*C*

_{2υ}, labeled

*S*

^{a,a1}, is given by

*S*

^{a,a1}=3

*A*

_{2}⊕2

*B*

_{2}, where

*A*

_{2}and

*B*

_{2}label irreducible representations (IRREPs) of

*C*

_{2υ}. In our previous analyses, we were able to take such a representation and use projection operators on the basis functions to get approximate forms for the localized modes. In this case, we have no such luxury, as there is no way to distinguish between the modes of the different

*A*

_{2}(or

*B*

_{2}) subspaces without some additional physical knowledge of the system. The best we can do is to form one projection operator for a composite

*A*

_{2}subspace and another for a composite

*B*

_{2}subspace. Doing so yields the following matrices, where the rows and columns are ordered in accordance with that which was chosen for the

*VB*

_{a}modes above:

*A*

_{2}modes can potentially be formed from any of the degenerate band-edge points

*B*

_{2}modes do not include

*A*

_{2}modes which only contain

*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:

**b**.

*CB*

_{a}basis under

*C*

_{2υ}(the defect symmetry), labeled

*S*

^{a,d1}, is given by

*S*

^{a,d1}=

*B*

_{1}⊕

*B*

_{2}, while the representation of the

*CB*

_{b}basis under

*C*

_{2υ}, labeled

*S*

^{b,d1}is given by

*S*

^{b,d1}=

*A*

_{1}⊕

*A*

_{2}. Projecting the

*CB*

_{a}and

*CB*

_{b}bases onto the irreducible representations above, we get

*a*and

*b*.

### 4.2 FDTD results

*A*

_{2}symmetry mode in the compressed hexagonal lattice, centered about the

*b*-point, whose dominant Fourier components are situated at

*b*-point in a compressed hexagonal lattice with compression factor γ=0.7. FDTD simulations of such a design (choosing, for example,

*r/a*=0.30 and

*r*′/

*a*=0.35), give the magnetic field amplitude and Fourier transformed dominant electric field components shown in Fig. 5(b)–(d). We see that our defect geometry has produced a mode with dominant Fourier components centered at

*Q*mode.

**k**

_{⊥})) to reduce couplings between the mode’s dominant Fourier components (in this case,

*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

*x̂*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*

_{⊥}=1.5×10

^{5},

*Q*

_{‖}=7.5×10

^{5},

*Q*

_{tot}=1.3×10

^{5}, and

*V*

_{eff}=0.35 for this design.

*Q*s, while keeping the modal volume reasonably small (although this value is still larger than our previous designs). Improvements can still be made; for example, simulation results indicate that there are still momentum components present within the light cone of

**Ẽ**

_{y}; hence a further tailoring of the lattice in the

*x̂*-direction (

**Ẽ**

_{y}has its dominant Fourier components along

*Q*

_{⊥}, though potentially at the expense of a larger mode volume.

## 5. Summary

*Q*defect modes in a 2D PC slab WG through Fourier space methods, as discussed in Ref. [1

**10**, 670–684 (2002). [CrossRef] [PubMed]

*Qs*in excess of 10

^{5}. As this mode is centered about an air hole, we seek out a mode centered about a dielectric region for applications where modal overlap with the dielectric is an important asset. To be consistent with our symmetry criterion, which precluded using such a mode in the standard hexagonal lattice, we consider a compression of the lattice in the

*ŷ*-direction as a method for creating new band-edge degeneracies which then alter the dominant Fourier components present in the defect modes. The symmetry-based analysis [6

**3**, S161–S170 (2001). [CrossRef]

*Q*modes in square and hexagonal lattices is applied to this new lattice, and FDTD simulations of defect geometries tailored to reduce lossy momentum space couplings predict

*Q*′

*s*exceeding 10

^{5}with a modal volume

*V*

_{eff}≈0.35. Including the results of Ref. [1

**10**, 670–684 (2002). [CrossRef] [PubMed]

*Q*cavities in three different lattices, indicating the generality of this Fourier space-based approach.

## 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 |

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. |

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 |

4. | S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature |

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. |

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 |

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. |

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. |

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 |

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. |

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. |

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. |

14. | J. Vučković, M. Lončcar, H. Mabuchi, and A. Scherer, “Optimization of the Q factor in Photonic Crystal Microcavities,” IEEE J. Quan. Elec. |

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. |

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. |

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, |

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. |

**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

- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- J. Vu¡ckovic, M. Lon¡car, H. Mabuchi, and A. Scherer, �??Design of photonic crystal microcavities for cavity QED,�?? Phys. Rev. E 65 (2002).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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).
- 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.
- M. Tinkham, Group Theory and Quantum Mechanics, International Series in Pure and Applied Physics (McGaw-Hill, Inc., New York, NY, 1964).
- 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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