Abstract

We present a technique for large-scale optimization of optical microcavities based on the frequency-averaged local density of states (LDOS), which circumvents computational difficulties posed by previous eigenproblem-based formulations and allows us to perform full topology optimization of three-dimensional (3d) leaky cavity modes. We present theoretical results for both 2d and fully 3d computations in which every pixel of the design pattern is a degree of freedom (“topology optimization”), e.g. for lithographic patterning of dielectric slabs in 3d. More importantly, we argue that such optimization techniques can be applied to design cavities for which (unlike silicon-slab single-mode cavities) hand designs are difficult or unavailable, and in particular we design minimal-volume multi-mode cavities (e.g. for nonlinear frequency-conversion applications).

© 2013 Optical Society of America

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2013 (3)

B. Zhen, S.-L. Chua, J. Lee, A. W. Rodriguez, X. Liang, S. G. Johnson, J. D. Joannopoulos, M. Soljačić, and O. Shapira, “Enabling enhanced emission and low-threshold lasing of organic molecules using special Fano resonances of macroscopic photonic crystals,” Proc. Natl. Acad. Sci. U. S. A.110, 13711–13716 (2013).
[CrossRef] [PubMed]

B. Osting and M. I. Weinstein, “Long-lived scattering resonances and Bragg structures,” SIAM J. Appl. Math.73, 827–852 (2013).
[CrossRef]

L. Li, M. Trusheim, O. Gaathon, K. Kisslinger, C.-J. Cheng, M. Lu, D. Su, X. Yao, H.-C. Huang, I. Bayn, A. Wolcott, R. M. Osgood, and D. Englund, “Reactive ion etching: Optimized diamond membrane fabrication for transmission electron microscopy,” J. Vac. Sci. Technol. B31,06FF01 (2013).
[CrossRef]

2012 (7)

A. Faraon, C. Santori, Z. Huang, V. M. Acosta, and R. G. Beausoleil, “Coupling of nitrogen-vacancy centers to photonic crystal cavities in monocrystalline diamond,” Phys. Rev. Lett.109,033604 (2012).
[CrossRef] [PubMed]

M. Nomura, “GaAs-based air-slot photonic crystal nanocavity for optomechanical oscillators,” Opt. Express20, 5204–5212 (2012).
[CrossRef] [PubMed]

Z.-F. Bi, A. W. Rodriguez, H. Hashemi, D. Duchesne, M. Loncar, K.-M. Wang, and S. G. Johnson, “High-efficiency second-harmonic generation in doubly-resonant χ(2)microring resonators,” Opt. Express20, 7526–7543 (2012).
[CrossRef] [PubMed]

C. Van Vlack and S. Hughes, “Finite-difference time-domain technique as an efficient tool for calculating the regularized Green function: applications to the local-field problem in quantum optics for inhomogeneous lossy materials,” Opt. Lett.37, 2880–2882 (2012).
[CrossRef] [PubMed]

A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express20, 21558–21575 (2012).
[CrossRef] [PubMed]

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers,” J. Comput. Phys.231, 3406–3431 (2012).
[CrossRef]

H. Hashemi, C. W. Qiu, A. P. McCauley, J. D. Joannopoulos, and S. G. Johnson, “A diameter–bandwidth product limitation of isolated-object cloaking,” Phys. Rev. A86,013804 (2012).
[CrossRef]

2011 (7)

E. Andreassen, A. Clausen, M. Schevenels, B. Lazarov, and O. Sigmund, “Efficient topology optimization in MATLAB using 88 lines of code,” Struct. Multidiscip. Optim.43, 1–16 (2011).
[CrossRef]

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photonics Rev.5, 308–321 (2011).
[CrossRef]

F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B28, 387–397 (2011).
[CrossRef]

J. Lu, S. Boyd, and J. Vuckovic, “Inverse design of a three-dimensional nanophotonic resonator,” Opt. Express19, 10563–10570 (2011).
[CrossRef] [PubMed]

F. Wang, B. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscip. Optim.43, 767–784 (2011).
[CrossRef]

Z. M. Meng, F. Qin, Y. Liu, and Z. Y. Li, “High-Q microcavities in low-index one-dimensional photonic crystal slabs based on modal gap confinement,” J. Appl. Phys.109,043107 (2011).
[CrossRef]

S. Kita, K. Nozaki, S. Hachuda, H. Watanabe, Y. Saito, S. Otsuka, T. Nakada, Y. Arita, and T. Baba, “Photonic Crystal Point-Shift Nanolasers With and Without Nanoslots Design, Fabrication, Lasing, and Sensing Characteristics,” IEEE J. Sel. Top. Quantum Electron.17, 1632–1647 (2011).
[CrossRef]

2010 (11)

S. Xu, Y. Cai, and G. Cheng, “Volume preserving nonlinear density filter based on heaviside functions,” Struct. Multidiscip. Optim.41, 495–505 (2010).
[CrossRef]

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization for unconstrained simulation-based problems,” Oper. Res.58, 161–178 (2010).
[CrossRef]

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U. S. A.107, 17491–17496 (2010).
[CrossRef] [PubMed]

J. Lu and J. Vuckovic, “Inverse design of nanophotonic structures using complementary convex optimization,” Opt. Express18, 3793–3804 (2010).
[CrossRef] [PubMed]

M. Nomura, K. Tanabe, S. Iwamoto, and Y. Arakawa, “High-Q design of semiconductor-based ultrasmall photonic crystal nanocavity,” Opt. Express18, 8144–8150 (2010).
[CrossRef] [PubMed]

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of light trapping in grating structures,” Opt. Express18, A366–A380 (2010).
[CrossRef] [PubMed]

Y. Li, J. Zheng, J. Gao, J. Shu, M. S. Aras, and C. W. Wong, “Design of dispersive optomechanical coupling and cooling in ultrahigh-Q/V slot-type photonic crystal cavities,” Opt. Express18, 23844–23856 (2010).
[CrossRef] [PubMed]

A. F. Koenderink, “On the use of Purcell factors for plasmon antennas,” Opt. Lett.35, 4208–4210 (2010).
[CrossRef] [PubMed]

S. Lin, E. Schonbrun, and K. Crozier, “Optical manipulation with planar silicon microring resonators,” Nano Lett.10, 2408–2411 (2010).
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N. A. P. Nicorovici, R. C. McPhedran, and L. C. Botten, “Relative local density of states for homogeneous lossy materials,” Physica B405, 2915–2919 (2010).
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2009 (3)

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2004 (5)

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2001 (1)

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Figures (12)

Fig. 1
Fig. 1

Starting with the naive objective of maximizing a microcavity’s Purcell factor Q/V, we perform a sequence of transformations of the problem in order to make it well posed and tractable. Here, we give a schematic diagram of each transformation, along with the corresponding section of the paper in which they are discussed.

Fig. 2
Fig. 2

Contour integration path. The frequency-averaged LDOS is the path integral along arc A1 in the limit of an infinite-radius arc. By choosing the proper window/weight function W (ω) for optimizing LDOS in a desired bandwidth, the contribution along arcs A2 and A3 can be made negligible compared to A1. Therefore, the residues at poles ω̄k enclosed by this contour can be used to approximate the averaged LDOS.

Fig. 3
Fig. 3

For 2d cavity optimization, we start in Secs. 8.1–8.3 by optimizing over every pixel in the interior of the computational domain as indicated in (a). This leads to cavities that utilize bandgap structures to confine light with arbitrary Q, regardless of V, limited only by the size of the domain. In order to investigate Q vs. V tradeoffs analogous to those in 3d slabs, in Sec. 8.4 we limit the degrees of freedom to a thin strip (b), which imposes intrinsic radiation losses (perpendicular to the strip) and forces the optimization to sacrifice V in order to increase Q. A full 3d optimization is considered in Sec. 8.5.

Fig. 4
Fig. 4

2d TM optimization from PhC cavity initial guess (Sec. 8.1).

Fig. 5
Fig. 5

2d TM optimization from vacuum initial guess (Sec. 8.1). Q=1.30×109 and V = 0.075(λ/n)2.

Fig. 6
Fig. 6

2d TE optimization for êx polarization from Sec. 8.2.1. The structure has Q=5.16×108 and V = 0.092(λ/n)2 obtained from vacuum initial guess.

Fig. 7
Fig. 7

Optimized doubly-degenerate cavity from Sec. 8.2.3, generated by maximizing the minimum LDOS over all in-plane polarizations. Starting from a vacuum initial guess, the optimization discovers a structure with C3v (3-fold) rotational symmetry (top), which supports doubly-degenerate [74] TE modes whose Hz fields are plotted at bottom (blue/white/red = negative/zero/positive). (The rectangular FDFD grid slightly breaks the three-fold symmetry and the degeneracy, but the results converge to exact C3v symmetry and degeneracy as resolution increases.)

Fig. 8
Fig. 8

Two-frequency cavity optimization from Sec. 8.3, for either TM (top) or TE (bottom) polarizations. Left: microcavities which maximize the minimum LDOS at two frequencies ω̃1 and ω̃2 = 2ω̃1, e.g. for intra-cavity second-harmonic generation applications [79, 81]. Confinement of such integer-multiple frequencies is physically enabled by the fact the concentric Bragg-onion “1d” bandgaps tend to occur at integer-multiple frequencies [14]. A more challenging case is a two-frequency cavity for ω̃2 = 1.5ω̃1, resulting in the more complicated structures shown at right. (All structures were optimized from vacuum initial guesses.)

Fig. 9
Fig. 9

2D TE optimization for thin strips with fixed width (geometry sketched in Fig. 3b). Fig. (a): Qrad vs. for 2d thin stirps with same width (λ) but different length (d = λ, 2λ, 3λ, 5λ). As is increased in the optimization, higher Qrad are obtained until Qrad is limited by the degrees of freedom. As the degrees of freedom increase, Qrad first gets bigger, but becomes saturated at some level around 107 due to numerical precision. Fig. (b): An optimized 2d thin-strip structure with width λ and length d = 5λ.

Fig. 10
Fig. 10

We wish to optimize a microcavity in an air-membrane Si slab in Sec. 8.5, with the effective computational domain depicted in (a), where the degrees of freedom are every pixel in the 2d pattern of the slab cross-section for a fixed thickness. Since all 2d single-polarization optimizations found structures with two mirror symmetry planes, we can reduce the computational domain to 1/8 the volume (b) by imposing these mirror symmetries along with the vertical mirror symmetry.

Fig. 11
Fig. 11

Optimized pattern for a 3d slab from vacuum initial guess (Sec. 8.5.1) with dimensions 3λ-3λ-0.19λ: Q=30000 and V = 0.06(λ/n)3.

Fig. 12
Fig. 12

3d slab structure after manually removing tiny features in Fig. 11 (Sec. 8.5.2): Q = 10000 and V = 0.06(λ/n)3.

Tables (1)

Tables Icon

Table 1: Comparison of Q and V for structures from our result and the literature.

Equations (35)

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× 1 μ ( x ) × E n ( x ) = ω n 2 ε ( x ) E n ( x )
Q = Re [ ω n ] 2 Im [ ω n ] .
V = ε ( x ) | E n ( x ) | 2 d x max { ε ( x ) | E n ( x ) | 2 } ,
3 Q 4 π 2 V ( λ n ) 3 .
1 Q total = 1 Q rad + 1 Q roughness + 1 Q absorption .
min V s . t . Q Q ˜ ,
P j ( ω , x ) = 1 2 Re [ J * ( x ) E ( x ) ] d x ,
( ε , ω ) E ( x ) = i ω J ( x ) ( ε , ω ) = × 1 μ ( x ) × ε ( x ) ω 2 J ( x ) = δ ( x x ) e ^ j .
LDOS j ( ω , x ) = 12 π P j ( ω , x ) = 6 π Re [ J * ( x ) E ( x ) d x ] ,
LDOS j ( ω , x ) = 6 π ω ε ( x ) Q V ,
L = LDOS ( ω ) W ( ω ) d ω .
f ( ω , x ) = 6 π J * ( x ) E ( x , ω ) d x .
L = LDOS ( ω ) W ( ω ) d ω = Re [ f ( ω ) ] W ( ω ) d ω = Re [ p . v . f ( ω ) W ( ω ) d ω ] .
A 1 + A 2 + A 3 f ( ω ) W ( ω ) d ω = 2 π i k Res [ f ( ω ) W ( ω ) , ω ¯ k ] .
A 2 f ( ω ) W ( ω ) d ω = 1 2 2 π i W ( 0 ) lim ω 0 ω f ( ω ) .
L = Re [ 2 π i k Res [ f ( ω ) W ( ω ) , ω ¯ k ] ( A 2 + A 3 f ( ω ) W ( ω ) d ω ) ] Re [ 2 π i k Res [ f ( ω ) W ( ω ) , ω ¯ k ] ] .
( × 1 μ ( x ) × ε ( x ) ( ω + i Γ ) 2 ) E ( x , ω + i Γ ) = i ( ω + i Γ ) J ( x ) ( × 1 μ ( x ) ( 1 + i Γ ω ) × ε ( x ) ω 2 ( 1 + i Γ ω ) ) E ( x , ω + i Γ ) = i ω J ( x ) ( × 1 μ ( x ) ( 1 + i 2 Q ) × ε ( x ) ( 1 + i 2 Q ) ω 2 ) E ( x , ω + i Γ ) = i ω J ( x ) .
˜ ( ε , ω ) = × 1 μ ( x ) ( 1 + i 2 Q ) × ε ( x ) ( 1 + i 2 Q ) ω 2 = × 1 μ ˜ ( x ) × ε ˜ ( x ) ω 2 .
L 1 = LDOS ( ω ) Γ ˜ / π ( ω ω ˜ ) 2 + Γ ˜ 2 = Re [ f ( ω ˜ + i Γ ˜ ) ] ,
W ( ω ) = 2 Γ ˜ 3 / π ( ( ω ω ˜ ) 2 + Γ ˜ 2 ) 2 ,
L = LDOS ( ω ) W ( ω ) d ω Re [ f ( ω ˜ + i Γ ˜ ) i Γ ˜ f ( ω ˜ + i Γ ˜ ) ] ,
f ( ω , x ) = f ( ω , x ) ω + i 12 π ε ( x ) E T ( x , ω ) E ( x , ω ) d x .
f ( ω ˜ + i Γ ˜ ) i Γ ˜ f ( ω ˜ + i Γ ˜ ) = ω ˜ ω ˜ + i Γ ˜ ( 6 π ) e ^ j * E ( x , ω ˜ + i Γ ˜ ) + 12 π Γ ˜ ε ( x ) E T ( x , ω ˜ + i Γ ˜ ) E ( x , ω ˜ + i Γ ˜ ) d x .
max { designs } L = LDOS ( ω ) W ( ω ) d ω .
max { designs } L = Re [ = f ( ω ˜ + i Γ ˜ ) i Γ ˜ f ( ω ˜ + i Γ ˜ ) ] .
˜ ( ε , ω ˜ ) A ( x , ω ˜ + i Γ ˜ ) = ε ( x ) E ( x , ω ˜ + i Γ ˜ )
ε k = ( i + 1 Q ˜ ) 2 π ω ˜ E T ( x k , ω ˜ + i Γ ˜ ) E ( x k , ω ˜ + i Γ ˜ ) + 12 π ω ˜ 3 Q ˜ ( 1 + i 2 Q ˜ ) A T ( x k , ω ˜ + i Γ ˜ ) E ( x k , ω ˜ + i Γ ˜ ) .
( ε , ω ) E ( x , ω ) ω + ( ε , ω ) ω E ( x , ω ) = i J ( x ) E ( x , ω ) ω = 1 ( i J ( x ) ( ε , ω ) ω E ( x , ω ) ) = 1 ( i J ( x ) + 2 ω ε ( x ) E ( x , ω ) ) .
f ( ω , x ) = 6 π J * ( x ) E ( x , ω ) ω d x = 6 π [ 1 ω J * ( x ) 1 ( i ω J ( x ) ) d x + 2 ω J * ( x ) 1 ε ( x ) E ( x , ω ) d x ] = 6 π 1 ω J * ( x ) E ( x , ω ) d x + i 12 π ( 1 ( i ω J ( x ) ) ) T ε ( x ) E ( x , ω ) d x = f ( ω , x ) ω + i 12 π ε ( x ) E T ( x , ω ) E ( x , ω ) d x .
= f ( ω ˜ + i Γ ˜ ) = i Γ ˜ f ( ω ˜ + i Γ ˜ ) = f ( ω ˜ + i Γ ˜ ) i Γ ˜ ( f ( ω ˜ + i Γ ˜ , x ) ω ˜ + i Γ ˜ + i 12 π ε ( x ) E T ( x , ω ˜ + i Γ ˜ ) E ( x , ω ˜ + i Γ ˜ ) d x ) = ω ˜ ω ˜ + i Γ ˜ f ( ω ˜ + i Γ ˜ ) + 12 π Γ ˜ ε ( x ) E T ( x , ω ˜ + i Γ ˜ ) E ( x , ω ˜ + i Γ ˜ ) d x . = ω ˜ ω ˜ + i Γ ˜ ( 6 π ) e ^ j * E ( x , ω ˜ + i Γ ˜ ) + 12 π Γ ˜ ε ( x ) E T ( x , ω ˜ + i Γ ˜ ) E ( x , ω ˜ + i Γ ˜ ) d x .
˜ ( ε , ω ˜ ) E ( x , ω ˜ + i Γ ˜ ) ε k + ˜ ( ε , ω ˜ ) ε k E ( x , ω ˜ + i Γ ˜ ) = 0 E ( x , ω ˜ + i Γ ˜ ) ε k = ˜ 1 [ ω ˜ 2 ( 1 + i 2 Q ˜ ) δ ( x x k ) E ( x , ω ˜ + i Γ ˜ ) ] .
J * ( x ) E ( x , ω ˜ + i Γ ˜ ) d x ε k = J * ( x ) E ( x , ω ˜ + i Γ ˜ ) ε k d x = ω ˜ 2 ( 1 + i 2 Q ˜ ) ( ˜ 1 J ( x ) ) T δ ( x x k ) E ( x , ω ˜ + i Γ ˜ ) d x = i ω ˜ ( 1 + i 2 Q ˜ ) E T ( x , ω ˜ + i Γ ˜ ) δ ( x x k ) E ( x , ω ˜ + i Γ ˜ ) d x = i ω ˜ ( 1 + i 2 Q ˜ ) E T ( x k , ω ˜ + i Γ ˜ ) E ( x k , ω ˜ + i Γ ˜ ) ,
ε ( x ) E T E d x ε k = 2 ε ( x ) E t E ( x , ω ˜ + i Γ ˜ ) ε k d x + δ ( x x k ) E T ( x , ω ˜ + i Γ ˜ ) E ( x , ω ˜ + i Γ ˜ ) d x = 2 ε ( x ) E T ˜ 1 [ ω ˜ 2 ( 1 + i 2 Q ˜ ) δ ( x x k ) E ( x , ω ˜ + i Γ ˜ ) ] + E T ( x k , ω ˜ + i Γ ˜ ) E ( x k , ω ˜ + i Γ ˜ ) = 2 ω ˜ 2 ( 1 + i 2 Q ˜ ) ( ˜ 1 [ ε ( x ) E ( x , ω ˜ + i Γ ˜ ) ] ) T δ ( x x k ) E ( x , ω ˜ + i Γ ˜ ) d x + E T ( x k , ω ˜ + i Γ ˜ ) E ( x k , ω ˜ + i Γ ˜ ) .
ε k = ( i + 1 Q ˜ ) 6 π ω ˜ E T ( x k , ω ˜ + i Γ ˜ ) E ( x k , ω ˜ + i Γ ˜ ) + 12 π ω ˜ 3 Q ˜ ( 1 + i 2 Q ˜ ) A T ( x k , ω ˜ + i Γ ˜ ) E ( x k , ω ˜ + i Γ ˜ ) ,
˜ ( ε , ω ˜ ) A ( x , ω ˜ + i Γ ˜ ) = ε ( x ) E ( x , ω ˜ + i Γ ˜ ) .

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