## General recipe for designing photonic crystal cavities

Optics Express, Vol. 13, Issue 16, pp. 5961-5975 (2005)

http://dx.doi.org/10.1364/OPEX.13.005961

Acrobat PDF (1083 KB)

### Abstract

We describe a general recipe for designing high-quality factor (Q) photonic crystal cavities with small mode volumes. We first derive a simple expression for out-of-plane losses in terms of the *k*-space distribution of the cavity mode. Using this, we select a field that will result in a high *Q*. We then derive an analytical relation between the cavity field and the dielectric constant along a high symmetry direction, and use it to confine our desired mode. By employing this inverse problem approach, we are able to design photonic crystal cavities with *Q* > 4 ∙ 10^{6} and mode volumes V ~ (λ/*n*)^{3}. Our approach completely eliminates parameter space searches in photonic crystal cavity design, and allows rapid optimization of a large range of photonic crystal cavities. Finally, we study the limit of the out-of-plane cavity *Q* and mode volume ratio.

© 2005 Optical Society of America

## 1. Introduction

1. S. Johnson, S. Fan, A. Mekis, and J. 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]

3. J. Vucπković, M. Loncπar, H. Mabuchi, and A. Scherer, “Optimization of Q factor in microcavities based on freestanding membranes,” IEEE J. Quantum Electron. **38**, 850–856 (2002). [CrossRef]

4. Y. Akahane, T. Asano, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944–947 (2003). [CrossRef] [PubMed]

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

6. 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]

7. P. Lalanne, S. Mias, and J. P. Hugonin, “Two physical mechanisms for boosting the quality factor to cavity volume ratio of photonic crystal microcavities,” Opt. Express **12**, 458–467 (2004). [CrossRef] [PubMed]

8. H.-Y. Ryu, M. Notomi, G.-H. Kim, and Y.-H. Lee, “High quality-factor whispering-gallery mode in the photonic crystal hexagonal disk cavity,” Opt. Express **12**, 1708–1719 (2004). [CrossRef] [PubMed]

9. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express **13**, 1202–1214 (2005). [CrossRef] [PubMed]

10. J. M. Geremia, J. Williams, and H. Mabuchi, “An inverse-problem approach to designing photonic crystals for cavity QED,” Phys. Rev. E **66**, 066606 (2002). [CrossRef]

11. B. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double heterostructure nanocavity,” Nature Mater. **4**, 207 (2005). [CrossRef]

*Q*and mode volume. In Section 3 we estimate the optimum

*k*-space distribution of the cavity mode field. In Section 4, we will finally address the question of finding PhC cavities with maximum possible figures of merit for various applications. In this process, we start from the optimum

*k*-space distribution of the cavity field; then we derive an approximate analytical relation between the cavity mode and the dielectric constant along a direction of high symmetry, and use it to create a cavity that supports the selected high-

*Q*mode in a single step. Thus, we eliminate the need for trial and error or other parameter search processes, that are typically used in PhC cavity designs. Furthermore, we study the limit of out-of-plane

*Q*factor for a given

*V*of the cavity mode with a particular field pattern.

## 2. Simplified relation between *Q* of a cavity mode and its *k*-space Distribution

*Q*measures how well the cavity confines light and is defined as

*P*, the far-field radiation intensity.

*Q*

_{∥}arbitrarily large) by addition of PhC layers. On the other hand, out-of-plane confinement, which dictates

*Q*

_{⊥}, depends on the modal k-distribution that is not confined by TIR. This distribution is highly sensitive to the exact mode pattern and must be optimized by careful tuning of the PhC defect. Assuming that the cavity mode is well inside the photonic band gap,

*Q*

_{⊥}gives the upper limit for the total

*Q*-factor of the cavity mode.

3. J. Vucπković, M. Loncπar, H. Mabuchi, and A. Scherer, “Optimization of Q factor in microcavities based on freestanding membranes,” IEEE J. Quantum Electron. **38**, 850–856 (2002). [CrossRef]

*S*above the PhC slab contains the complete information about the out-of-plane radiation losses of the mode, and thus about

*Q*

_{⊥}(Fig. 1).

*S*is:

*K*(

*θ,ϕ*) is the radiated power per unit solid angle. In the appendix, we derive a very simple form for

*K*in terms of 2D Fourier Transforms (FTs) of

*H*

_{z}and

*E*

_{z}at the surface

*S*, after expressing the angles

*θ, ϕ*in terms of

*k*

_{x}and

*k*

_{y}:

*λ*is the mode wavelength in air,

*k*= 2

*π*/

*λ*, and

_{∥}= (

*k*

_{x}

*,k*

_{y}) =

*k*(sin

*θ*cos

*ϕ*,sin

*θ*sin

*ϕ*) and

*k*

_{z}=

*k*cos(

*θ*) denote the in-plane and out-of-plane

*k*-components, respectively. In Cartesian coordinates, the radiated power (5) can thus be re-written as the integral over the light cone,

*k*

_{∥}<

*k*. Substituting (6) into (5) gives

*Q*for a given mode. In the following sections, when considering the qualitative behavior of (7), we will restrict ourselves to TE-like modes, described at the slab center by the triad (

*E*

_{x}

*,E*

_{y}

*,H*

_{z}), that have

*H*

_{z}even in at least one dimension

*x*or

*y*. For such modes, the term |

*FT*

_{2}(

*H*

_{z})|

^{2}in (7) just above the slab is dominant, and |

*FT*

_{2}(

*E*

_{z})|

^{2}can be neglected in predicting the general trend of Q.

*Q*/

*V*; for nonlinear optical effects

*Q*

^{2}/

*V*; while for the strong coupling regime of cavity QED, maximizing ratios

*g*/κ ~

*Q*/√

*V*and

*g*/

*γ*~ 1/√

*V*is important. In these expressions,

*V*is the cavity mode volume:

*V*= (∫

*ε*(

^{2}

*d*

^{3}

*ε*(

^{2}),

*g*is the emitter-cavity field coupling, and

*κ*and

*γ*are the cavity field and emitter dipole decay rates, respectively [2].

## 3. Optimum *k*-space field distribution of the cavity mode field

*x,y*). The forbidden energy bands are the source of DBR confinement.

*ε*(

*ε*(

*k*

_{x}

*, k*

_{y}) plane and are defined by

*πm*for integer

*m*. The real and reciprocal lattice vectors for the square and hexagonal lattices with periodicity

*a*are:

*m, j, q*and

*l*are integers. The electromagnetic field corresponding to a particular wave vector

*E*

_{k→}. In Fig. 3, we plot the dispersion of a waveguide in the Γ

*J*direction of a hexagonal lattice PhC.

*H*=

*a*

_{k0}

*H*

_{k0}+

*a*

_{-k0}

*H*

_{-k0}. (Here we focus on TE-like PhC modes, and discuss primarily

*H*

_{z}, although similar relations can be written for all other field components). Imperfect mirrors introduce a phase shift upon reflection; moreover, the reduction of the distance between the mirrors (shortening of the cavity) broadens the distribution of

*k*vectors in the mode to some width ∆

*k*. The

*H*

_{z}field component of the cavity (i.e., a closed waveguide) mode can then be approximated as:

*H*

_{z}can be made at the surface

*S*directly above the PhC slab (see Fig. 1), which is relevant for calculation of radiation losses. The Fourier transform of the above equation gives the

*k*-space distribution of the cavity mode, with coefficients

*A*

_{k→ ,G→}and

*A*

_{-k→ ,G→}. The distribution peaks are positioned at ±

k →

_{o}±

*, with widths directly proportional to ∆*G →

*k*. The

*k*-space distribution is mapped to other points in Fourier space by the reciprocal lattice vectors

3. J. Vucπković, M. Loncπar, H. Mabuchi, and A. Scherer, “Optimization of Q factor in microcavities based on freestanding membranes,” IEEE J. Quantum Electron. **38**, 850–856 (2002). [CrossRef]

_{0}should be positioned at the edge of the first Brillouin zone, which is the region in

*k*-space that cannot be mapped into the light cone by any reciprocal lattice vector

*j*

_{x}

*,j*

_{y}∈ 0,1. Clearly, |

*j*

_{x}| = |

*j*

_{y}| = 1 is a better choice for

_{0}, since it defines the edge point of the 1st Brillouin zone which is farthest from the light cone. Thus, modes centered at this point and

*k*-space broadened due to confinement, will radiate the least. Similarly, the optimum

_{0}for the cavities resonating in the Γ

*J*direction of the hexagonal lattice is

4. Y. Akahane, T. Asano, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944–947 (2003). [CrossRef] [PubMed]

*X*direction is

**38**, 850–856 (2002). [CrossRef]

_{0}at the edge of the first Brillouin zone has been made, the summation over

*Q*factor allows us to investigate the theoretical limits of this parameter and its relation to the mode volume of the cavity. Second, it allows us to quantify the effect of our perturbation on the optimization of

*Q*using only one or two layers of the computational field and almost negligible computational time compared to standard numerical methods. We applied Eq. (7

7. P. Lalanne, S. Mias, and J. P. Hugonin, “Two physical mechanisms for boosting the quality factor to cavity volume ratio of photonic crystal microcavities,” Opt. Express **12**, 458–467 (2004). [CrossRef] [PubMed]

**38**, 850–856 (2002). [CrossRef]

4. Y. Akahane, T. Asano, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944–947 (2003). [CrossRef] [PubMed]

*Q*using Eq. (7) at S, as well as full first-principle FDTD simulations are shown in Fig. 4. A good match is observed. Therefore, our expression (7) is a valid measure of the radiative properties of the cavity and can be used to theoretically approach the design problem; we can also use this form to speed up the optimization of the cavity parameters. The discrepancy between Eq. (7) and FDTD is primarily due to discretization errors.

## 4. Inverse problem approach to designing PhC cavities

*FT*

_{2}(

*Q*/

*V*for structures of varying mode volume. Then we present two approaches for analytically estimating the PhC structure

*ε*(

*k*-space distribution

*FT*

_{2}(

*B*

_{eo}

*,B*

_{oe}, and

*B*

_{ee}(Fig. 3(b)) for which we can approximate the trend of the radiation (7) by considering only

*H*

_{z}at the surface

*S*just above the PhC slab. Moreover, to make a rough estimate of the cavity dielectric constant distribution from the desired

*H*

_{z}field on

*S*, we approximate that

*H*

_{z}at

*S*is close to

*H*

_{z}at the slab center.

### 4.1. General trend of Q/V

*Q*/

*V*for a cavity with varying mode volume. Here, we assume that a structure has been found to support the desired field

*H*

_{z}.

*Q*using Eq. (1). All that is required of the cavity field is that its FT at the surface S above the slab be distributed around the four points

*B*

_{oe}. The Fourier Transform of the

*H*

_{z}field is then given by

*σ*

_{x}and

*σ*

_{y}denote the modal width in real space. The mode and its FT are shown in Fig. 5 (d-e). We use Eq. 7 without the

*E*

_{z}terms to estimate the trend in

*Q*, as described above. As the mode volume grows, the radiation inside the light cone shrinks exponentially. This results in an exponential increase in

*Q*. This relationship is shown in Fig. 5(g) for field

*B*

_{oe}at frequency

*a*/

*λ*= 0.248. At the same time, the mode volume grows linearly with

*σ*

_{x}. The growth of

*Q*/

*V*is therefore dominantly exponental , and we can find the optimal

*Q*for a particular choice of mode volume (i.e.

*σ*

_{x}) of the Gaussian mode cavity.

*Qs*can be reached with large mode volumes and there does not seem to be an upper bound on

*Q*

_{⊥}. As the mode volume of the Gaussian cavity increases, the radiative Fourier components vanish exponentially, but are never zero. A complete lack of Fourier components in the light cone should result in the highest possible Q. As an example of such a field, we propose a mode with a sinc envelope in x and a Gaussian one in

*y*. The FT of this mode in Fig. 5(b) is described by

*Rect*(

*k*

_{x},∆

*k*

_{x}) is a rectangular function of width ∆

*k*

_{x}and centered at

*k*

_{x}.

*x*whose width is inversely proportional to the width of the rectangle

*Rect*(

*k*

_{x},∆

*k*

_{x}). To our knowledge, this target field has not been previously considered in PhC cavity design. This field is shown in Fig. 5(a-c). Though it has no out-of-plane losses, this field drops off as

11. B. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double heterostructure nanocavity,” Nature Mater. **4**, 207 (2005). [CrossRef]

*Q*/

*V*increases exponentially with mode size, these large

*Q*s are not surprising. On the other hand, direct measurements on fabricated PhC cavities indicate that

*Q*s are bounded to currently ~ 10

^{4}by material absorption and surface roughness [13

13. A. Badolato, K. Hennessy, M. Atature, J. Dreiser, E. Hu, P. Petroff, and A. Imamoglu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science **308** (5725), 1158–61 (2005). [CrossRef] [PubMed]

14. D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucπković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. **95**, 013904(2005), arxiv/quant-ph/0501091 (2005). [CrossRef] [PubMed]

^{5}[9

9. Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express **13**, 1202–1214 (2005). [CrossRef] [PubMed]

11. B. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double heterostructure nanocavity,” Nature Mater. **4**, 207 (2005). [CrossRef]

*H*

_{z}can be confined for this purpose so that

*K*(

*k*

_{x}

*, k*

_{y}) dominates losses at the origin in

*k*-space, as shown for instance in Fig. 5(h-i) for the confined mode pattern Bee of Fig. 3.

### 4.2. Estimating Photonic Crystal Design from k-space Field Distribution

*ε*(

_{c}. These methods directly calculate the dielectric profile from the desired field distribution, without any dynamic tuning of PhC parameters, and are thus computationally fast. We focus on TE-like modes, since they see a large bandgap and exhibit electric-field maxima at the slab center. For TE-like modes,

_{c}=

*H*

_{c}

*ẑ*at the center of the slab, and

_{c}≈

*H*

_{c}

*ẑ*at the surface. First, we relate

_{c}to one of the allowed waveguide fields

_{w}. The fields

*H*

_{c}and

*H*

_{w}at the center of the PhC slab (

*z*= 0) are solutions to the homogeneous wave equation with the corresponding refractive indices ec and

*ε*

_{w}, respectively.

_{c}and ω

_{w}are the frequencies of the cavity and waveguide fields. We expand the cavity mode into waveguide Bloch modes:

*u*

_{k}is the periodic part of the Bloch wave. Assuming that the cavity field is composed of the waveguide modes with

_{0}, we can approximate

*u*

_{k}(

*u*

_{k0}(

*H*

_{w}=

*u*

_{k0}(

*e*

^{i(k→0∙r→ - ωwt}and the cavity field envelope

*H*

_{e}= ∑

_{k→}

*c*

_{k}

*e*

^{((k→ - k→0)∙r→ -(ωk - ωw)t}

_{0}|

^{2}and α ≪ 1 (i.e., the band is nearly flat).Differentiating(19) in time twice gives:

^{2}

*H*

_{e}, ω

_{c}≈ ω

_{w}, i.e., the cavity resonance is very close to the frequency of the dominant waveguide mode. The condition α ≪ 1, also implies that ω

_{k}≈ ω

_{w}, i.e.

*H*

_{e}≈ ∑

_{k→}

*c*

_{k}

*e*

^{i(k→ - k→0)∙ r→}.

#### 4.2.1. Estimating Photonic Crystal Design from *k*-space field Distribution: Approach 1

*ε*

_{c}as

*ε*

_{c}=

*ε*

_{w}

*ε*

_{e}, where

*ε*

_{e}is the unknown envelope.

*H*

_{c}is a solution of Eq. (16) and each waveguide mode

*u*

_{k}(

*e*

^{i(k→∙r→-ωkt}satisfies Eq. (17). From previous arguments, for

_{c}≈ ω

_{w}≈ ω

_{k}. Thus, a linear superposition of waveguide modes ∑

_{k→}

*c*

_{k}

*e*

^{i(k→ -ωkt)}=

*H*

_{e}

*H*

_{w}=

*H*

_{c}also satisfies Eq. (17), i.e. the cavity mode is also a solution of Eq. (17) for a slowly varying envelope. We assume that the mode is TE-like, so that the

*H*field only has a

*z*component at the center of the slab. Then inserting

*H*

_{c}from (19) into Eq.(16) and Eq. (17) and subtracting the two equations with

*ε*

_{c}=

*ε*

_{w}

*ε*

_{e}, yields a partial differential equation for

*ε*

_{c}(

*x,y*):

*B*

_{oe}symmetry in Fig. 3(b). These modes are even in

*ŷ*so the partial derivatives

*y*in (21) vanish at

*y*= 0. The resulting simplified first-order differential equation in 1/

*ε*

_{e}can then be solved directly near

*y*= 0, and the solution is

*C*is an arbitrary constant of integration. In our analysis

*ε*

_{w}corresponds to removing holes along one line (

*x*-axis) in the PhC lattice. The cavity is created by introducing holes into this waveguide, which means that

*C*> 0, this leads to the following result for the cavity dielectric constant near

*y*= 0:

*C*is a positive constant of integration, and

*H*

_{c}=

*H*

_{w}

*H*

_{e}, where

*H*

_{w}is the known waveguide field and

*H*

_{e}is the desired field envelope.

*C*can be chosen by fixing the value of

*ε*

_{c}at some x, leading to a particular solution for

*ε*

_{c}. In our cavity designs we chose

*C*such that the value of

*ε*

_{c}is close to

*ε*

_{w}at the cavity center. To implement this design in a practical structure, we need to approximate this continuous

*ε*

_{c}by means of a binary function with low and high-index materials

*ε*

_{l}and

*ε*

_{h}, respectively. We do this by finding, in every period

*j*, the air hole radius

*r*

_{j}that gives the same field-weighted averaged index on the

*x*-axis:

_{c}is estimated from a linear superposition of waveguide modes as

_{c}∝ ∇ ×

_{c}. We assume that the holes are centered at the positions of the unperturbed hexagonal lattice PhC holes.

*r*

_{j}thus give the required index profile along the

*x*symmetry axis. The exact shape of the holes in 3D is secondary – we choose cylindrical holes for convenience. Furthermore, we are free to preserve the original hexagonal crystal structure of the PhC far away from the cavity where the field is vanishing.

*B*

_{oe}of Fig. 3(b) confined in a line-defect of a hexagonal PhC. The calculated dielectric structures and FDTD simulated fields inside them are shown in Fig. 6. The FT fields on S also show a close match and very little power radiated inside the light cone (Fig.6(c,f)). This results in very large

*Q*values, estimated from

*Q*

_{⊥}to limit computational constraints. These estimates were done in two ways, using first principles FDTD simulations [2], and direct integration of lossy components by Eq. (7

7. P. Lalanne, S. Mias, and J. P. Hugonin, “Two physical mechanisms for boosting the quality factor to cavity volume ratio of photonic crystal microcavities,” Opt. Express **12**, 458–467 (2004). [CrossRef] [PubMed]

*σ*

_{x}/

*a*≈ 1.6, which according to the plot in Fig. 5 g., puts us at the attainable limit of

*Q*

_{⊥}at this mode volume.

*Q*

_{⊥}correctly estimates

*Q*by noting that

*Q*

_{⊥}did not change appreciably as the number of PhC periods in the

*x*- and

*y*- directions,

*N*

_{x}and

*N*

_{y}, was increased: for the Gaussian-type (sinc-type) mode, increasing the simulation size from

*N*

_{x}= 13,

*N*

_{y}= 13 (

*N*

_{x}= 21,

*N*

_{y}= 9) PhC periods to

*N*

_{x}= 25,

*N*

_{y}= 13 (

*N*

_{x}= 33,

*N*

_{y}= 13) changed quality factors from

*Q*

_{∥}= 22 ∙ 10

^{3},

*Q*

_{⊥}= 1.4 ∙ 10

^{6}(

*Q*

_{∥}= 17 ∙ 10

^{3},

*Q*

_{⊥}= 4.2 ∙ 10

^{6}) to

*Q*

_{∥}= 180 ∙ 10

^{3},

*Q*

_{⊥}= 1.48 ∙ 10

^{6}(

*Q*

_{∥}= 260 ∙ 10

^{3},

*Q*

_{⊥}= 4.0 ∙ 10

^{6}). (The number of PhC periods in the x-direction in which the holes are modulated to introduce a cavity is 9 and 29 for Gaussian and sinc cavity, respectively, while both cavities consist of only one line of defect holes in the y-direction) Thus, with enough periods, the quality factors would be limited to

*Q*

_{⊥}, as summarized in the table. In the calculation of

*Q*, the vertically emitted power 〈

*P*

_{∥}〉 was estimated from the fields a distance ~ 0.25 ∙ λ above the PhC surface. Note that the frequencies

*a*/λ closely match those of the original waveguide field

*B*

_{oe}(

*a*/

*λ*

_{cav}=0.251), validating the assumption in the derivation.

#### 4.2.2. Estimating Photonic Crystal Design from *k*-space Field Distribution: Approach 2

*ε*

_{c}(

*x,y*) that is valid in the whole PhC plane (instead of the center line only). Again, begin with the cavity field

*ẑH*

_{c}consisting of the product of the waveguide field and a slowly varying envelope,

*H*

_{c}=

*H*

_{w}

*H*

_{e}, and treat the cavity dielectric constant as:

*H*

_{e}, subtracting from the first, and recalling that ω

_{c}~ ω

_{w}yields

*H*

_{e}. This relation is a quasilinear partial differential equation in 1/

*ε*

_{pert}. With boundary conditions that can be estimated from the original waveguide field, this equation can in principle be solved for

*ε*

_{c}(e.g., [15]).

*ε*

_{pert}by assuming a vector function

## 5. Conclusions

*Q*> 10

^{6}and near-minimal mode volume ~ (λ/

*n*)

^{3}. These values follow our theoretically estimated value of

*Q*

_{⊥}/

*V*for the cavity with the Gaussian field envelope, which means that we were able to find the maximal Q for the given mode volume V under our assumptions. Our approach is analytical, and the results are obtained within a single computational step. We first derive a simple expression of the modal out-of-plane radiative loss and demonstrate its utility by the straightforward calculation of

*Q*factors on several cavity designs. Based on this radiation expression, the recipe begins with choosing the FT mode pattern that gives the desired radiation losses. For high-

*Q*cavities with minimal radiative loss inside the light cone, we show that the transform of the mode should be centered at the extremes of the Brillouin Zone, as far removed from the light cone as possible. Next we proved that for a cavity mode with a Gaussian envelope,

*Q*/

*V*grows exponentially with mode volume

*V*, while the cavity with the sinc field envelope should lead to even higher Q’s by completely eliminating the Fourier components in the light cone. Finally, we derived approximate solutions to the inverse problem of designing a cavity that supports a desired cavity mode. This approach yields very simple design guides that lead to very large

*Q*/

*V*. Since it eliminates the need for lengthy trial-and-error optimization, our recipe enables rapid and efficient design of a wide range of PhC cavities.

## A. Derivation of Cavity Radiative Loss

*K*(

*θ,ϕ*) can be expressed in terms of the radiation vectors

*r, θ, ϕ*):

*L*

_{θ}and

*L*

_{ϕ}. As described in Reference [3

**38**, 850–856 (2002). [CrossRef]

*x*and

*y*) field components at the surface

*S*(Fig. 1):

*k*= 2

*π*/λ, λ is the mode wavelength in air, and

*k*

_{∥}=

*k*sin

*θ*.

*f*(

*x,y*) is

*H*

_{x}

*,H*

_{y}

*,E*

_{x}, and

*E*

_{y}. This expression is in general difficult to track analytically. We will now use Maxwell’s relations to express (30) in terms of only two scalars,

*H*

_{z}and

*E*

_{z}.

*g*,

*N*

_{θ}as

*S*, it also follows that

*E*

_{z}

*,H*

_{z}∝ exp(

*ik*

_{z}

*Z*) for propagating waves inside the light cone (which are the only ones that determine

*P*), implying that

*H*

_{z}. This allows further simplification of the previous expression to

*J*(

*k*

_{x}

*,k*

_{y}) is the Jacobian resulting from the change of coordinates from (

*θ, ϕ*) to (

*k*

_{x}

*,k*

_{y}).

## Acknowledgments

## References and links

1. | S. Johnson, S. Fan, A. Mekis, and J. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. |

2. | J. Vuπković, M. Loncπar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E |

3. | J. Vucπković, M. Loncπar, H. Mabuchi, and A. Scherer, “Optimization of Q factor in microcavities based on freestanding membranes,” IEEE J. Quantum Electron. |

4. | Y. Akahane, T. Asano, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature |

5. | K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express |

6. | 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. |

7. | P. Lalanne, S. Mias, and J. P. Hugonin, “Two physical mechanisms for boosting the quality factor to cavity volume ratio of photonic crystal microcavities,” Opt. Express |

8. | H.-Y. Ryu, M. Notomi, G.-H. Kim, and Y.-H. Lee, “High quality-factor whispering-gallery mode in the photonic crystal hexagonal disk cavity,” Opt. Express |

9. | Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “Fine-tuned high-Q photonic-crystal nanocavity,” Opt. Express |

10. | J. M. Geremia, J. Williams, and H. Mabuchi, “An inverse-problem approach to designing photonic crystals for cavity QED,” Phys. Rev. E |

11. | B. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double heterostructure nanocavity,” Nature Mater. |

12. | A. Yariv and P. Yeh, |

13. | A. Badolato, K. Hennessy, M. Atature, J. Dreiser, E. Hu, P. Petroff, and A. Imamoglu, “Deterministic coupling of single quantum dots to single nanocavity modes,” Science |

14. | D. Englund, D. Fattal, E. Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vucπković, “Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,” Phys. Rev. Lett. |

15. | R. Haberman, |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(130.3120) Integrated optics : Integrated optics devices

(140.3410) Lasers and laser optics : Laser resonators

(140.5960) Lasers and laser optics : Semiconductor lasers

(230.5750) Optical devices : Resonators

(230.6080) Optical devices : Sources

(250.5300) Optoelectronics : Photonic integrated circuits

(260.5740) Physical optics : Resonance

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 20, 2005

Revised Manuscript: July 20, 2005

Published: August 8, 2005

**Citation**

Dirk Englund, Ilya Fushman, and Jelena Vu?kovi?, "General recipe for designing photonic crystal cavities," Opt. Express **13**, 5961-5975 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-16-5961

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### References

- S. Johnson, S. Fan, A. Mekis, and J. 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]
- J. Vu?kovi?, M. Lon?ar, H. Mabuchi, and A. Scherer, �Design of photonic crystal microcavities for cavity QED,� Phys. Rev. E 65, 016,608 (2002).
- J. Vu?kovi? , M. Lon?ar, H. Mabuchi, and A. Scherer, �Optimization of Q factor in microcavities based on freestanding membranes,� IEEE J. Quantum Electron. 38, 850�856 (2002). [CrossRef]
- Y. Akahane, T. Asano, and S. Noda, �High-Q photonic nanocavity in a two-dimensional photonic crystal,� Nature 425, 944�947 (2003). [CrossRef] [PubMed]
- K. Srinivasan and O. Painter, �Momentum space design of high-Q photonic crystal optical cavities,� 10, 670�684 (2002). [PubMed]
- 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]
- P. Lalanne, S. Mias, and J. P. Hugonin, �Two physical mechanisms for boosting the quality factor to cavity volume ratio of photonic crystal microcavities,� Opt. Express 12, 458�467 (2004). [CrossRef] [PubMed]
- H.-Y. Ryu, M. Notomi, G.-H. Kim, and Y.-H. Lee, �High quality-factor whispering-gallery mode in the photonic crystal hexagonal disk cavity,� Opt. Express 12, 1708�1719 (2004). [CrossRef] [PubMed]
- Y. Akahane, T. Asano, B.-S. Song, and S. Noda, �Fine-tuned high-Q photonic-crystal nanocavity,� 13, 1202�1214 (2005). [CrossRef] [PubMed]
- J. M. Geremia, J. Williams, and H. Mabuchi, �An inverse-problem approach to designing photonic crystals for cavity QED,� Phys. Rev. E 66, 066606 (2002). [CrossRef]
- B. Song, S. Noda, T. Asano, and Y. Akahane, �Ultra-high-Q photonic double heterostructure nanocavity,� Nature Mater. 4, 207 (2005). [CrossRef]
- A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley and Sons, 2003).
- A. Badolato, K. Hennessy, M. Atature, J. Dreiser, E. Hu, P. Petroff, and A. Imamoglu, �Deterministic coupling of single quantum dots to single nanocavity modes,� Science 308(5725), 1158�61 (2005). [CrossRef] [PubMed]
- D. Englund, D. Fattal, E.Waks, G. Solomon, B. Zhang, T. Nakaoka, Y. Arakawa, Y. Yamamoto, and J. Vu?kovi? , �Controlling the spontaneous emission rate of single quantum dots in a two-dimensional photonic crystal,� Phys. Rev. Lett. 95, 013904(2005), arxiv/quant-ph/0501091 (2005). [CrossRef] [PubMed]
- R. Haberman, Elementary Applied Partial Differential Equations (Prentice-Hall, 1987).

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