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

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References

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  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).

Appl. Phys. Lett. (2)

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]

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]

IEEE J. Quantum Electron. (1)

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]

Nature (1)

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

Nature Mater. (1)

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

Opt. Express (4)

Phys. Rev. E (2)

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]

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).

Phys. Rev. Lett. (1)

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]

Science (1)

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]

Other (2)

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

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

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Figures (6)

Fig. 1.
Fig. 1.

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

Fig. 2.
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.

Fig. 3.
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.

Fig. 4.
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.

Fig. 5.
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.

Fig. 6.
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.

Tables (1)

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

Equations (52)

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Q ω U P
U = 1 2 ( ε E 2 + μ H 2 ) dV
P = P + P
1 Q = 1 Q + 1 Q
P = 0 π 2 0 π 2 dθdϕ sin ( θ ) K θ ϕ
K k x k y = η k z 2 2 λ 2 k 2 [ 1 η 2 FT 2 ( E z ) 2 + FT 2 ( H z ) 2 ]
P η 2 λ 2 k k k d k x d k y k 2 k z [ 1 η 2 FT 2 ( E z ) 2 + FT 2 ( H z ) 2 ]
ε ( r ) = G ε G e i G r
R mj = ma x ̂ + ja y ̂
G ql = 2 πq a x ̂ + 2 πl a y ̂
R mj = ma ( x ̂ + y ̂ 3 ) 2 + ja ( x ̂ y ̂ 3 ) 2
G ql = 4 π a 3 q ( x ̂ 3 + y ̂ ) 2 + 4 π a 3 l ( x ̂ 3 + y ̂ ) 2 ,
E k = e i k r G A k , G e i G r
H z x y ~ G k 0 Δ k 2 k 0 + Δ k 2 ( A k , G e i k r + A k . G e i k r ) e i G r
H z x y = d k x d k y A ( k x k y ) e i k r ,
FT 2 ( H z ) = k x 0 , k y 0 sign ( k x 0 ) exp ( ( k x k x 0 ) ( σ x 2 ) 2 ( k y k y 0 ) ( σ y 2 ) 2 ) ,
FT 2 ( H z ) = k x 0 , k y 0 exp ( ( k y k y 0 ) 2 ( σ y 2 ) 2 Rect ( k x k x 0 , Δ k x ) ) ,
μ 0 2 H c t 2 = ω c 2 μ 0 H c = × 1 ε c × H c
μ 0 2 H w t 2 = ω w 2 μ 0 H w = × 1 ε w × H w
H c = k c k u k ( r ) e i ( k . r ω k t )
H c u k 0 ( r ) e i ( k 0 r ω w t ) k c k e i ( ( k k 0 ) r ( ω k ω w ) t ) = H w H e ,
2 H c t 2 = ω c 2 H c = k c k u k ( r ) e i ( k 0 r ω w t ) [ ω w 2 + α k k 0 2 ]
H w k c k e i ( ( k k 0 ) r ( ω k ω w ) t ) [ ω w 2 + α k k 0 2 ] = ω w 2 H w H e + α H w 2 H e
x [ 1 ε w ( 1 ε e 1 ) x H c ] + y [ 1 ε w ( 1 ε e 1 ) y H c ] μ 0 ( ω w 2 ω c 2 ) H c 0
ε c ( x ) x H c C + 1 ε w x H c
ja a 2 ja + a 2 ( ε h + ( ε l ε h ) Rect ja r j ) E c 2 dx = ja a 2 ja + a 2 ε c ( x ) E c 2 dx ,
ω c 2 μ 0 H c = ( 1 ε c H c )
ω w 2 μ 0 H w = ( 1 ε w H w )
ω w 2 μ 0 H e H w ω c 2 μ 0 H c = μ 0 H c ( ω w 2 ω c 2 ) 0
= . ( 1 ε c H c ) H e . ( 1 ε w H w )
. ( 1 ε pert H c )
1 ε pert H c = × ξ
1 ε pert = × ξ . H c * H c 2
K θ ϕ = η 8 λ 2 ( N θ + L ϕ η 2 + N ϕ + L θ η 2 ) ,
N θ = ( N x cos ϕ + N y sin ϕ ) cos θ
N ϕ = N x sin ϕ + N y cos ϕ ,
N x = F T 2 ( H y ) k
N y = F T 2 ( H x ) k
L x = F T 2 ( E y ) k
L y = F T 2 ( E x ) k
k = k x r 0 y r 0 = k sin θ ( cos ϕ x ̂ + sin ϕ y ̂ )
k z = k cos θ ,
FT 2 ( f ( x , y ) ) = d x d y f x y e i k x y
= d x d y f x y e i ( k x x + k y y )
N θ = k z k k ( k x FT 2 ( H y ) + k y FT 2 ( H x ) )
= i k z k k FT 2 ( H y x H x y )
= k z c ε 0 k k FT 2 ( E z )
K( k x , k y )= η 8 λ 2 k || 2 [ 1 η 2 | k z F T 2 ( E z )+iF T 2 ( E z z ) | 2 +| k z F T 2 ( H z )+iF T 2 ( H z z ) | 2 ]
K k x k y = η k z 2 2 λ 2 k 2 [ 1 η 2 FT 2 ( E z ) 2 + FT 2 ( H z ) 2 ]
P = 0 π 2 0 2 π dθdϕ sin ( θ ) K θ ϕ
= k k d k x d k y k k K k x k y J k x k y
= η 2 λ 2 k k k d k x d k y k 2 k z [ 1 η 2 FT 2 ( E z ) 2 + FT 2 ( H z ) 2 ]

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