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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 16 — Aug. 8, 2005
  • pp: 5961–5975
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General recipe for designing photonic crystal cavities

Dirk Englund, Ilya Fushman, and Jelena Vuckovic  »View Author Affiliations


Optics Express, Vol. 13, Issue 16, pp. 5961-5975 (2005)
http://dx.doi.org/10.1364/OPEX.13.005961


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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 ∙ 106 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

One of the most interesting applications of photonic crystals (PhCs) is the localization of light to very small mode volumes – below a cubic optical wavelength. The principal confinement mechanism is Distributed Bragg Reflection (DBR), in contrast to, e.g., microspheres or microdisks, which rely solely on Total Internal Reflection (TIR). However, since three-dimensional (3D) PhCs (employing DBR confinement in all 3D in space) have not been perfected yet, the mainstream of the PhC research has addressed 2D PhCs of finite depth, employing DBR in 2D and TIR in the remaining 1D. Such structures are also more compatible with the present microfabrication and planar integration techniques. These cavities can still have small mode volumes, but the absence of full 3D confinement by DBR makes the problem of the high-quality factor (high-Q) cavity design much more challenging. The main problem is out-of-plane loss by imperfect TIR, which becomes particularly severe in the smallest volume cavities.

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

In order to simplify PhC cavity optimization, we derive an analytical relation between the near-field pattern of the cavity mode and its quality factor in this section. Q measures how well the cavity confines light and is defined as

QωUP
(1)

where ω is the angular frequency of the confined mode. The mode energy is

U=12(εE2+μH2)dV
(2)

The difficulty lies in calculating P, the far-field radiation intensity.

Following our prior work [2

2. J. Vuπković, M. Loncπar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016,608 (2002).

], we consider the in-plane and out-of-plane mode loss mechanisms in two-dimensional photonic crystals of finite depth separately:

P=P+P
(3)

or

1Q=1Q+1Q
(4)

Given a PhC cavity or waveguide, we can compute the near-field using Finite Difference Time Domain (FDTD) analysis. As described in Reference [3

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]

], the near-field pattern at a surface 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).

Fig. 1. Estimating the radiated power and Q from the known near field at the surface S

The total time-averaged power radiated into the half-space above the surface S is:

P=0π20π2dθdϕsin(θ)Kθϕ
(5)

where 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 Hz and Ez at the surface S, after expressing the angles θ, ϕ in terms of kx and ky :

Kkxky=ηkz22λ2k2[1η2FT2(Ez)2+FT2(Hz)2]
(6)

Here, ημoεo λ is the mode wavelength in air, k = 2π/λ, and k = (kx,ky ) = k(sinθcosϕ,sinθsinϕ) and kz = kcos(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

Pη2λ2kkkdkxdkyk2kz[1η2FT2(Ez)2+FT2(Hz)2]
(7)

This is the simplified expression we were seeking. It gives the out-of-plane radiation loss as the light cone integral of the simple radiation term (6), evaluated above the PhC slab. Substituting Eq. (2) and Eq. (7) into Eq. (1) thus yields a straightforward calculation of the 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 (Ex,Ey,Hz ), that have Hz even in at least one dimension x or y. For such modes, the term |FT 2(Hz )|2 in (7) just above the slab is dominant, and |FT 2(Ez )|2 can be neglected in predicting the general trend of Q.

The figure of merit for cavity design depends on the application: for spontaneous emission rate enhancement through the Purcell effect, one desires maximal 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(r )|E (r )|2 d 3 r /max(ε(r )|E (r )|2), g is the emitter-cavity field coupling, and κ and γ are the cavity field and emitter dipole decay rates, respectively [2

2. J. Vuπković, M. Loncπar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016,608 (2002).

].

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

The photonic crystals considered here are made by periodically modulating the refractive index of a thin semiconductor slab (see Fig. 2). The periodic modulation introduces an energy band-structure for light in two dimensions r = (x,y). The forbidden energy bands are the source of DBR confinement.

The periodic refractive index ε(r ) = ε(r + R ) can be expanded in a Fourier sum over spatial frequency components in the periodic plane:

ε(r)=GεGeiGr
(8)

Here G are the reciprocal lattice vectors in the (kx, ky ) plane and are defined by GR = 2πm for integer m. The real and reciprocal lattice vectors for the square and hexagonal lattices with periodicity a are:

Square Lattice:

Rmj=max̂+jaŷ
Gql=2πqax̂+2πlaŷ
(9)
Rmj=ma(x̂+ŷ3)2+ja(x̂ŷ3)2
(10)

Hexagonal Lattice:

Fig. 2. Left: TE-like mode band diagram for the square lattice photonic crystal. The inset on the right shows the reciprocal lattice (orange circles), the first Brillouin zone (blue) and the irreducible Brillouin zone (green) with the high symmetry points. The inset on the left shows the square lattice PhC slab and relevant parameters: periodicity a, hole radius r, and slab thickness d. The parameters used in the simulation were r/a = 0.4,d/a = 0.55, and n = 3.6. Right: Band diagram for TE-like modes of the hexagonal lattice PhC. The inset on the right shows the reciprocal lattice (orange), the Brillouin zone (blue), and the first irreducible Brillouin zone (green) for the hexagonal lattice photonic crystal. The inset on the left shows the hexagonal lattice PhC slab. The parameters used for the simualtion were r/a = 0.3,d/a = 0.65,n = 3.6. A discretization of 20 points per period a was used for both diagrams.
Gql=4πa3q(x̂3+ŷ)2+4πa3l(x̂3+ŷ)2,

where m, j, q and l are integers. The electromagnetic field corresponding to a particular wave vector k inside such a periodic medium can be expressed as [12

12. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley and Sons, 2003).

]:

Ek=eikrGAk,GeiGr
(11)

By introducing linear defects into a PhC lattice, waveguides can be formed. Waveguide modes have the discrete translational symmetry of the particular direction in the PhC lattice and can therefore be expanded in Bloch states Ek . In Fig. 3, we plot the dispersion of a waveguide in the ΓJ direction of a hexagonal lattice PhC.

Cavity modes can then be formed by closing a portion of a waveguide, i.e., by introducing mirrors to confine a portion of the waveguide mode. In case of perfect mirrors, the standing wave is described by H = a k0 H k0 + a -k0 H -k0. (Here we focus on TE-like PhC modes, and discuss primarily Hz , 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 Hz field component of the cavity (i.e., a closed waveguide) mode can then be approximated as:

Hzxy~Gk0Δk2k0+Δk2(Ak,Geikr+Ak.Geikr)eiGr
(12)

A similar expansion of Hz 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 ±ko ± G, with widths directly proportional to ∆k. The k-space distribution is mapped to other points in Fourier space by the reciprocal lattice vectors G . To reduce radiative losses, the mapping of components into the light cone should be minimized [3

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]

]. Therefore, the center of the mode distribution k 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 G (see Fig. 2). For example, this region corresponds to k0=±jxπak̂x±jyπak̂y for the square lattice, where jx,jy ∈ 0,1. Clearly, |jx | = |jy | = 1 is a better choice for k 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 k 0 for the cavities resonating in the ΓJ direction of the hexagonal lattice is k0=±πak̂x (as it is for the cavity from Ref [4

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]

], and for the ΓX direction is k0=±2πa3k̂y (as it is for the cavity from Refs. [2

2. J. Vuπković, M. Loncπar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016,608 (2002).

] and [3

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]

]).

Fig. 3. Left: Band diagram for hexagonal waveguide in ΓJ direction, with r/a = 0.3, d/a = 0.65, n = 3.6. The bandgap (wedged between the gray regions) contains three modes. Mode Boo can be pulled inside the bandgap by additional neighbor hole tuning. Right: Bz of confined modes of hexagonal waveguide. The modes are indexed by the B-field’s even (“e”) or odd (“o”) parities in the x and y directions, respectively. The confined cavity modes Boo, Bee , and Beo required additional structure perturbations for shifting into the bandgap. This was done by changing the diameters of neighboring holes.

Assuming that the optimum choice of k 0 at the edge of the first Brillouin zone has been made, the summation over G can be neglected in Eq. (12), because it only gives additional Fourier components which are even further away from the light cone and do not contribute to the calculation of the radiation losses. In that case, we can express the field distribution as:

Hzxy=dkxdkyA(kxky)eikr,
(13)

There are two main applications of Eq. (7). First, this formulation of the cavity 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]

) to cavities obtained from an iterative parameter space search. These cavities were previously studied in [3

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]

] and [4

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]

]. The results for 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.

Fig. 4. Comparison of Q factors derived from Eq. (7) (squares) to those calculated with FDTD (circles). Left: cavity made by removing three holes along the ΓJ direction confining the Boe mode. The Q factor is tuned by shifting the holes closest to the defect as shown by the red arrow. The x-axis gives the shift as a fraction of the periodicity a. Right: the X dipole cavity described in [3]. The Q factor is tuned by stretching the center line of holes in the ΓX direction, as shown by the arrow. The x-axis gives the dislocation in terms of the periodicity a.

4. Inverse problem approach to designing PhC cavities

In the inverse approach, we begin with a desired in-plane Fourier decomposition of the resonant mode, FT 2(H(r)), chosen again to minimize radiation losses given by Eq. (7). The difficulty lies with designing a structure that supports the field which is approximately equal to the target field, H(r).

In this section, we first estimate the general behavior of Q/V for structures of varying mode volume. Then we present two approaches for analytically estimating the PhC structure ε(r) from the desired k-space distribution FT 2(H(H)). As mentioned in Section 2, we restrict the analysis to TE-like modes Beo,Boe , and Bee (Fig. 3(b)) for which we can approximate the trend of the radiation (7) by considering only Hz at the surface S just above the PhC slab. Moreover, to make a rough estimate of the cavity dielectric constant distribution from the desired Hz field on S, we approximate that Hz at S is close to Hz at the slab center.

4.1. General trend of Q/V

The simplification described above allows us to study the general behavior of Q/V for a cavity with varying mode volume. Here, we assume that a structure has been found to support the desired field Hz .

We again start from the expression for radiated power, Eq. (7), and calculateQ 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 kx0=±πa,ky0=±2πa3, to minimize the components inside the light cone. As an example, we choose a field with a Gaussian envelope in Fourier space, as described in Sec. 3. For now, let us consider mode symmetry Boe . The Fourier Transform of the Hz field is then given by

FT2(Hz)=kx0,ky0sign(kx0)exp((kxkx0)(σx2)2(kyky0)(σy2)2),
(14)

According to Fig.5(g), very large 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

FT2(Hz)=kx0,ky0exp((kyky0)2(σy2)2Rect(kxkx0,Δkx)),
(15)

where Rect(kx ,∆kx ) is a rectangular function of width ∆kx and centered at kx .

The Fourier-transform implies that the cavity mode is described by a sinc function in x whose width is inversely proportional to the width of the rectangle Rect(kx ,∆kx ). 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 1r and therefore requires a larger structure than the Gaussian field for confinement.

Over the past years, many new designs with ever-higher theoretical quality factors have been suggested [11

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

]. In light of our result that Q/V increases exponentially with mode size, these large Qs are not surprising. On the other hand, direct measurements on fabricated PhC cavities indicate that Qs are bounded to currently ~ 104 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

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]

], while indirect measurements using waveguide coupling indicate values up to 6∙105[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

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

].

It is interesting to note that Eq. (6) also allows one to calculate the field required to radiate with a desired radiation distribution. For example, many applications require radiation with a strong vertical component; waveguide modes with even Hz can be confined for this purpose so that K(kx, ky ) 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.

Fig. 5. Idealized cavity modes at the surface S above the PhC slab; all with mode volume ~ (λ/n)3. (a-c) Mode with sinc and Gaussian envelopes in x and y, respectively: Hz (x,y), FT 2(Hz ), and K(kx,ky ) inside the light cone; (d-g) Mode Boe with Gaussian envelopes in the x and y directions : Hz (x,y), FT 2(Hz ), K(kx,ky ), and Q (σx /a) as well as Q /V . Q was calculated using Eq. (7) and Ez was neglected.; (h-i) Mode Bee with Gaussian envelopes in x and y can be confined to radiate preferentially upward.

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

Now we introduce two analytical ways of estimating the dielectric structure ε(H) that supports a cavity field that is approximately equal to the desired field Hc . 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, Hc = Hc at the center of the slab, and HcHc at the surface. First, we relate Hc to one of the allowed waveguide fields Hw . The fields Hc and Hw at the center of the PhC slab (z = 0) are solutions to the homogeneous wave equation with the corresponding refractive indices ec and εw , respectively.

μ02Hct2=ωc2μ0Hc=×1εc×Hc
(16)
μ02Hwt2=ωw2μ0Hw=×1εw×Hw
(17)

Here ωc and ωw are the frequencies of the cavity and waveguide fields. We expand the cavity mode into waveguide Bloch modes:

Hc=kckuk(r)ei(k.rωkt)
(18)

where uk is the periodic part of the Bloch wave. Assuming that the cavity field is composed of the waveguide modes with kk 0, we can approximate uk (r) ≈ u k0(r), which leads to the slowly varying envelope approximation:

Hcuk0(r)ei(k0rωwt)kckei((kk0)r(ωkωw)t)=HwHe,
(19)

where the waveguide mode Hw = u k0(r)e i(k0r - ωwt and the cavity field envelope He = ∑k→ ck e ((k - k0)∙r -(ωk - ωw)t

The cavity and waveguide fields FT-distributions are concentrated at the edge of the Brillouin zone, where ωk2ωw2 + α|k - k 0|2 and α ≪ 1 (i.e., the band is nearly flat).Differentiating(19) in time twice gives:

2Hct2=ωc2Hc=kckuk(r)ei(k0rωwt)[ωw2+αkk02]
Hwkckei((kk0)r(ωkωw)t)[ωw2+αkk02]=ωw2HwHe+αHw2He
(20)

Thus, for α ≪ 1 and finite ∇2 He , ωc ≈ ωw, i.e., the cavity resonance is very close to the frequency of the dominant waveguide mode. The condition α ≪ 1, also implies that ωk ≈ ωw, i.e. He ≈ ∑k ck e i(k - k0)∙ r.

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

x[1εw(1εe1)xHc]+y[1εw(1εe1)yHc]μ0(ωw2ωc2)Hc0
(21)

For this approach, we consider waveguide modes with Boe 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/εe can then be solved directly near y = 0, and the solution is 1εw(1εe1)xHcC, where C is an arbitrary constant of integration. In our analysis εw 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 1εe1>0 The solution holds when we take the absolute value of both its sides, and for C > 0, this leads to the following result for the cavity dielectric constant near y = 0:

εc(x)xHcC+1εwxHc
(22)

C is a positive constant of integration, and Hc = HwHe , where Hw is the known waveguide field and He is the desired field envelope. C can be chosen by fixing the value of εc at some x, leading to a particular solution for εc . In our cavity designs we chose C such that the value of εc is close to εw at the cavity center. To implement this design in a practical structure, we need to approximate this continuous εc by means of a binary function with low and high-index materials εl and εh , respectively. We do this by finding, in every period j, the air hole radius rj that gives the same field-weighted averaged index on the x-axis:

jaa2ja+a2(εh+(εlεh)Rectjarj)Ec2dx=jaa2ja+a2εc(x)Ec2dx,
(23)

where Ec is estimated from a linear superposition of waveguide modes as Ec ∝ ∇ × Hc . We assume that the holes are centered at the positions of the unperturbed hexagonal lattice PhC holes.

The radii rj 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.

To illustrate the power of this inverse approach, we now design PhC cavities that support the Gaussian and sinc-type modes of Eq. (14), (15). In each case, we start with the waveguide field Boe 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

2. J. Vuπković, M. Loncπar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016,608 (2002).

], 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]

). The results are listed in Table 1 and show an improvement of roughly three orders of magnitude over the unmodified structure of Fig. 4 with as small increase in mode volume. Furthermore, a fit of the resulting field pattern to a Gaussian envelope multiplied by a Sine, yielded a value of σx /a ≈ 1.6, which according to the plot in Fig. 5 g., puts us at the attainable limit of Q at this mode volume.

In our FDTD simulations, we verified that Q correctly estimates Q by noting that Q did not change appreciably as the number of PhC periods in the x- and y- directions, Nx and Ny , was increased: for the Gaussian-type (sinc-type) mode, increasing the simulation size from Nx = 13,Ny = 13 (Nx = 21,Ny = 9) PhC periods to Nx = 25,Ny = 13 (Nx = 33,Ny = 13) changed quality factors from Q = 22 ∙ 103, Q = 1.4 ∙ 106 (Q = 17 ∙ 103,Q = 4.2 ∙ 106) to Q = 180 ∙ 103, Q = 1.48 ∙ 106 (Q = 260 ∙ 103, Q = 4.0 ∙ 106). (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 Boe (a/λcav =0.251), validating the assumption in the derivation.

Table 1. Q values of structures derived with inverse-approach 1

table-icon
View This Table
Fig. 6. FDTD simulations for the derived Gaussian cavity (a-c) and the derived sinc cavity (d-f). Gaussian: (a) Bz ; (b) |E|; (c) FT pattern of Bz taken above the PhC slab (blue) and target pattern (red). Sinc: (d) Bz ; (e) |E|; (f) FT pattern of Bz taken above the PhC slab (blue) and target pattern (red) (The target FT for the sinc cavity appears jagged due to sampling, since the function was expressed with the resolution of the simulations). The cavities were simulated with a discretization of 20 points per period a, PhC slab hole radius r = 0.3a, slab thickness of 0.6a and refractive index 3.6. Starting at the center, the defect hole radii in units of periodicity a are: (0,0,0.025,0.05,0.075,0.1,0.075,0.075,0.1,0.125,0.125,0.125,0.1,0.125,0.15,0.3,0.3) for the sinc cavity, and (0.025,0.025,0.05,0.1,0.225) for the Gaussian cavity.

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

We now derive a closed-form expression for εc (x,y) that is valid in the whole PhC plane (instead of the center line only). Again, begin with the cavity field H(r) = ẑHc consisting of the product of the waveguide field and a slowly varying envelope, Hc = HwHe , and treat the cavity dielectric constant as: 1εc=1εpert+1εw. In the PhC plane, Eq. (16, 17) for a TE-like mode can be rewritten as

ωc2μ0Hc=(1εcHc)
(24)
ωw2μ0Hw=(1εwHw)
(25)

Multiplying the last equation by He , subtracting from the first, and recalling that ωc ~ ωw yields

ωw2μ0HeHwωc2μ0Hc=μ0Hc(ωw2ωc2)0
=.(1εcHc)He.(1εwHw)
(26)
.(1εpertHc)
(27)

where the last line results after some algebra and dropping spatial derivatives of the slowly varying envelope He . This relation is a quasilinear partial differential equation in 1/εpert . With boundary conditions that can be estimated from the original waveguide field, this equation can in principle be solved for εc (e.g., [15

15. R. Haberman, Elementary Applied Partial Differential Equations (Prentice-Hall, 1987).

]).

Alternatively, one can find a formal solution for εpert by assuming a vector function ξ(r) chosen to satisfy the boundary conditions, so that

1εpertHc=×ξ
(28)
1εpert=×ξ.Hc*Hc2
(29)

or

This gives is the formal solution of the full dielectric constant εc=(εpert1+εw1)1 1 in the plane of the photonic crystal.

5. Conclusions

A. Derivation of Cavity Radiative Loss

The radiated power per unit solid angle K(θ,ϕ) can be expressed in terms of the radiation vectors N and L in spherical polar coordinates (r, θ, ϕ):

Kθϕ=η8λ2(Nθ+Lϕη2+Nϕ+Lθη2),
(30)

where η=μ0ε0. The radiation vectors in spherical polar coordinates can be expressed from their components in Cartesian coordinates:

Nθ=(Nxcosϕ+Nysinϕ)cosθ
Nϕ=Nxsinϕ+Nycosϕ,
(31)

and similarly for Lθ and Lϕ . As described in Reference [3

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]

], the radiation vectors in Cartesian coordinates are proportional to 2D Fourier transforms of the parallel (x and y) field components at the surface S (Fig. 1):

Nx=FT2(Hy)k
Ny=FT2(Hx)k
Lx=FT2(Ey)k
Ly=FT2(Ex)k
k=kxr0yr0=ksinθ(cosϕx̂+sinϕŷ)
kz=kcosθ,
(32)

where k = 2π/λ, λ is the mode wavelength in air, and k = ksinθ.

Here the 2D Fourier Transform of the function f(x,y) is

FT2(f(x,y))=dxdyfxyeikxy
(33)
=dxdyfxyei(kxx+kyy)
(34)

Substitution of expressions (32) and (33) into (30) now yields an expression for the radiated power (5) in terms of the FTs of the four scalars Hx,Hy,Ex , and Ey . 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, Hz and Ez .

Noting that for a bounded function g, FT2(gx)=ikxFT2(g)(similarlyforgy), we can re-write Nθ as

Nθ=kzkk(kxFT2(Hy)+kyFT2(Hx))
=ikzkkFT2(HyxHxy)
=kzcε0kkFT2(Ez)
(35)

where the last step follows from Maxwell’s Eq. ×H=εoEt=εoE. Similarly, we find Lθ=kzcμ0kFT2(Hz). From H = 0 and E = 0 at the surface S, it also follows that Nϕ=ikFT2(Hyz)andLϕ=ikFT2(Hzz). Substituting these expressions into Eq. (30) yields

K(kx,ky)=η8λ2k||2[1η2|kzFT2(Ez)+iFT2(Ezz)|2+|kzFT2(Hz)+iFT2(Hzz)|2]
(36)

Kkxky=ηkz22λ2k2[1η2FT2(Ez)2+FT2(Hz)2]
(37)

Substituting this result back into the expression for total radiated power (5) gives the required result (7):

P=0π202πdθdϕsin(θ)Kθϕ
=kkdkxdkykkKkxkyJkxky
=η2λ2kkkdkxdkyk2kz[1η2FT2(Ez)2+FT2(Hz)2]
(38)

Above, J(kx,ky ) is the Jacobian resulting from the change of coordinates from (θ, ϕ) to (kx,ky ).

Acknowledgments

This work has been supported by the MURI Center for photonic quantum information systems (ARO/ARDA Program DAAD19-03-1-0199). Dirk Englund has also been supported by the NDSEG fellowship, and Ilya Fushman by the NIH training grant

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. 78, 3388–3390 (2001). [CrossRef]

2.

J. Vuπković, M. Loncπar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavities for cavity QED,” Phys. Rev. E 65, 016,608 (2002).

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]

12.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley and Sons, 2003).

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]

15.

R. Haberman, Elementary Applied Partial Differential Equations (Prentice-Hall, 1987).

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

  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]
  2. 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).
  3. 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]
  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,� 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,� 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]
  12. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley and Sons, 2003).
  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. 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]
  15. R. Haberman, Elementary Applied Partial Differential Equations (Prentice-Hall, 1987).

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