Abstract

We present a first-principles method to compute radiation properties of ultra-high quality factor photonic crystal cavities. Our Frequency-domain Approach for Radiation (FAR) can compute the far-field radiation pattern and quality factor of cavity modes ∼ 100 times more rapidly than conventional finite-difference time domain calculations. We explain how the radiation pattern depends on the perturbation used to create the cavity and on the Bloch modes of the photonic crystal.

© 2012 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2010

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[CrossRef]

N. Tran, S. Combrié, P. Colman, A. De Rossi, and T. Mei, “Vertical high emission in photonic crystal nanocavities by band-folding design,” Phys. Rev. B 82, 075120 (2010).
[CrossRef]

D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vučković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett. 104, 073904 (2010).
[CrossRef] [PubMed]

S.L. Portalupi, M. Galli, C. Reardon, T. F. Krauss, L. O’Faolain, L. C. Andreani, and D. Gerace, “Planar photonic crystal cavities with far-field optimization for high coupling efficiency and quality factor,” Opt. Express 18, 16064–16073 (2010).
[CrossRef] [PubMed]

M. Galli, D. Gerace, K. Welna, T. F. Krauss, L. O’Faolain, and G. Guizzetti, “Low-power continuous-wave generation of visible harmonics in silicon photonic crystal nanocavities,” Opt. Express 18, 26613–26624 (2010).
[CrossRef] [PubMed]

2009

2008

2007

S. Tomljenovic Hanic, M. J. Steel, C. M. de Sterke, and D. J. Moss, “High-Q cavities in photosensitive photonic crystals,” Opt. Lett. 32, 542–544 (2007).
[CrossRef] [PubMed]

M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frédérick, P. J. Poole, and R. L. Williams “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76, 245104 (2007).
[CrossRef]

P. Chak, R. Iyer, J. S. Aitchison, and J. E. Sipe, “Hamiltonian formulation of coupled-mode theory in waveguiding structures,” Phys. Rev. E 75, 016608 (2007).
[CrossRef]

2006

2005

2004

A. Yoshie, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef] [PubMed]

2003

Y. Akahane, T. Asano, B.S. Song, and S. Noda,” Nature 425, 944–947 (2003).
[CrossRef] [PubMed]

2001

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

1997

V. A. Mandelshtam and H.S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem Phys.  107, 6756–6769 (1997).
[CrossRef]

S. L. Zhang, “GPBi-CG: Generalized product-type methods based on Bi-CG for solving nonsymmetric linear systems,” SIAM Journal on Scientific Computing 18, 537–551 (1997).
[CrossRef]

1987

Aitchison, J. S.

P. Chak, R. Iyer, J. S. Aitchison, and J. E. Sipe, “Hamiltonian formulation of coupled-mode theory in waveguiding structures,” Phys. Rev. E 75, 016608 (2007).
[CrossRef]

Akahane, Y.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[CrossRef]

Y. Akahane, T. Asano, B.S. Song, and S. Noda,” Nature 425, 944–947 (2003).
[CrossRef] [PubMed]

Andreani, L. C.

Asano, T.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[CrossRef]

Y. Akahane, T. Asano, B.S. Song, and S. Noda,” Nature 425, 944–947 (2003).
[CrossRef] [PubMed]

Bermel, P.

Bog, U.

Bulla, D. A. P.

Burr, G. W.

Chak, P.

P. Chak, R. Iyer, J. S. Aitchison, and J. E. Sipe, “Hamiltonian formulation of coupled-mode theory in waveguiding structures,” Phys. Rev. E 75, 016608 (2007).
[CrossRef]

Chaumet, P.C.

Choi, D. Y.

Colman, P.

N. Tran, S. Combrié, P. Colman, A. De Rossi, and T. Mei, “Vertical high emission in photonic crystal nanocavities by band-folding design,” Phys. Rev. B 82, 075120 (2010).
[CrossRef]

Combrié, S.

N. Tran, S. Combrié, P. Colman, A. De Rossi, and T. Mei, “Vertical high emission in photonic crystal nanocavities by band-folding design,” Phys. Rev. B 82, 075120 (2010).
[CrossRef]

Dalacu, D.

M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frédérick, P. J. Poole, and R. L. Williams “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76, 245104 (2007).
[CrossRef]

De Rossi, A.

N. Tran, S. Combrié, P. Colman, A. De Rossi, and T. Mei, “Vertical high emission in photonic crystal nanocavities by band-folding design,” Phys. Rev. B 82, 075120 (2010).
[CrossRef]

de Sterke, C. M.

Deppe, D. G.

A. Yoshie, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef] [PubMed]

Eggleton, B. J.

Ell, C.

A. Yoshie, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef] [PubMed]

Englund, D.

D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vučković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett. 104, 073904 (2010).
[CrossRef] [PubMed]

D. Englund, I. Fushman, and J. Vučković, “General recipe for designing photonic crystal cavities,” Opt. Express 13, 5961–5975 (2005).
[CrossRef] [PubMed]

Fan, S.

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

Faraon, A.

D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vučković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett. 104, 073904 (2010).
[CrossRef] [PubMed]

Farjadpour, A.

Frédérick, S.

M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frédérick, P. J. Poole, and R. L. Williams “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76, 245104 (2007).
[CrossRef]

Fushman, I.

Gai, X.

Galli, M.

Gerace, D.

Gibbs, H. M.

A. Yoshie, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef] [PubMed]

Grillet, C.

Guizzetti, G.

Hendrickson, J.

A. Yoshie, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef] [PubMed]

Hugonin, J. P.

Ibanescu, M.

Iyer, R.

P. Chak, R. Iyer, J. S. Aitchison, and J. E. Sipe, “Hamiltonian formulation of coupled-mode theory in waveguiding structures,” Phys. Rev. E 75, 016608 (2007).
[CrossRef]

Joannopoulos, J. D.

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31, 2972–2974 (2006).
[CrossRef] [PubMed]

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

Johnson, S. G.

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31, 2972–2974 (2006).
[CrossRef] [PubMed]

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

Karnutsch, C.

Khitrova, G.

A. Yoshie, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef] [PubMed]

Krauss, T. F.

Kuramochi, E.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006).
[CrossRef]

Lalanne, P.

Lecamp, G.

Lee, M. W.

Luther-Davies, B.

Madden, S.

Mägi, E. C.

Majumdar, A.

D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vučković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett. 104, 073904 (2010).
[CrossRef] [PubMed]

Mandelshtam, V. A.

V. A. Mandelshtam and H.S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem Phys.  107, 6756–6769 (1997).
[CrossRef]

Matsuo, S.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[CrossRef]

McCutcheon, M. W.

M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frédérick, P. J. Poole, and R. L. Williams “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76, 245104 (2007).
[CrossRef]

McPhedran, R. C.

Mei, T.

N. Tran, S. Combrié, P. Colman, A. De Rossi, and T. Mei, “Vertical high emission in photonic crystal nanocavities by band-folding design,” Phys. Rev. B 82, 075120 (2010).
[CrossRef]

Mekis, A.

S. G. Johnson, S. Fan, A. Mekis, and J. D. Joannopoulos, “Multipole-cancellation mechanism for high-Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3390 (2001).
[CrossRef]

Mitsugi, S.

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006).
[CrossRef]

Monat, C.

Moss, D. J.

Noda, S.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[CrossRef]

Y. Akahane, T. Asano, B.S. Song, and S. Noda,” Nature 425, 944–947 (2003).
[CrossRef] [PubMed]

Notomi, M.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[CrossRef]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006).
[CrossRef]

Nozaki, K.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[CrossRef]

O’Faolain, L.

Petroff, P.

D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vučković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett. 104, 073904 (2010).
[CrossRef] [PubMed]

Poole, P. J.

M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frédérick, P. J. Poole, and R. L. Williams “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76, 245104 (2007).
[CrossRef]

Portalupi, S.L.

Rahmani, A.

Reardon, C.

Rieger, G. W.

M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frédérick, P. J. Poole, and R. L. Williams “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76, 245104 (2007).
[CrossRef]

Rodriguez, A.

Roundy, D.

Rupper, G.

A. Yoshie, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef] [PubMed]

Sato, T.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[CrossRef]

Sauvan, C.

Shchekin, O. B.

A. Yoshie, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef] [PubMed]

Shinya, A.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[CrossRef]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006).
[CrossRef]

Sipe, J. E.

P. Chak, R. Iyer, J. S. Aitchison, and J. E. Sipe, “Hamiltonian formulation of coupled-mode theory in waveguiding structures,” Phys. Rev. E 75, 016608 (2007).
[CrossRef]

J. E. Sipe, “New Green-function formalism for surface optics,” J. Opt. Soc. Am. B 4, 481–489 (1987).
[CrossRef]

Smith, C. L. C.

Song, B. S.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4, 207–210 (2005).
[CrossRef]

Song, B.S.

Y. Akahane, T. Asano, B.S. Song, and S. Noda,” Nature 425, 944–947 (2003).
[CrossRef] [PubMed]

Steel, M. J.

Stoltz, N.

D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vučković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett. 104, 073904 (2010).
[CrossRef] [PubMed]

Tanabe, T.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[CrossRef]

E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. 88, 041112 (2006).
[CrossRef]

Taniyama, H.

K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
[CrossRef]

Taylor, H.S.

V. A. Mandelshtam and H.S. Taylor, “Harmonic inversion of time signals and its applications,” J. Chem Phys.  107, 6756–6769 (1997).
[CrossRef]

Toishi, M.

D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vučković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett. 104, 073904 (2010).
[CrossRef] [PubMed]

Tomljenovic Hanic, S.

Tomljenovic-Hanic, S.

Tran, N.

N. Tran, S. Combrié, P. Colman, A. De Rossi, and T. Mei, “Vertical high emission in photonic crystal nanocavities by band-folding design,” Phys. Rev. B 82, 075120 (2010).
[CrossRef]

Vuckovic, J.

D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vučković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett. 104, 073904 (2010).
[CrossRef] [PubMed]

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K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics 4, 477–483 (2010).
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Phys. Rev. B

M. W. McCutcheon, J. F. Young, G. W. Rieger, D. Dalacu, S. Frédérick, P. J. Poole, and R. L. Williams “Experimental demonstration of second-order processes in photonic crystal microcavities at submilliwatt excitation powers,” Phys. Rev. B 76, 245104 (2007).
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Figures (3)

Fig. 1
Fig. 1

Schematic of (a) photosensitive cavity with locally increased background index and of (b) fluid infiltrated cavity with increased hole index. Ã(r)Da(r) (arb. units) for (c) photosensitive cavity (d) fluid infiltrated cavity, both with a length of L = 4d. (e)–(f) Modulus of the Fourier transform of (c)–(d) respectively, with non-radiating components removed.

Fig. 2
Fig. 2

Quality factor versus cavity length for (a) the photosensitive cavity (Fig. 1(a)); (b), the fluid infiltrated cavity (Fig. 1(b)). Red symbols are computed using the FAR method, while blue ones are computed by FDTD.

Fig. 3
Fig. 3

Symmetric quadrants of far-field radiation (Sr) for (a),(b) cavities in Fig. 1(a) with Δnp = 0.02 and (c),(d) those in Fig. 1(b) with Δni = 0.2. Left frames are computed using the FAR method while right frames are computed using FDTD. Colors are as in Fig. 1(d). Angles ϕ and θ are azimuthal and declination angles respectively.

Equations (7)

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= 1 2 μ 0 d r B ( r ) B ( r ) + 1 2 ε 0 d r D ( r ) D ( r ) ε ( r ) .
D ( r , t ) = bound d k h ¯ ω k 2 a k e i ω k t D k ( r ) + c . c . ,
1 = d k d k [ h ¯ ω k δ ( k k ) + h ¯ ω k ω k d r γ ( r ) D k * ( r ) D k ( r ) ] a k a k
D a ( r ) = bound d k h ¯ ω k 2 v 0 ( k ) D k ( r ) .
P ( r ) = ε 0 [ ε ( r ) 1 ] d r G ( r r ; ω ) P ( r ) ,
P 1 rad ( r ) ε 0 ( ε ¯ ( r ) 1 ) d r G ( r r ; ω 0 ) P 1 rad ( r ) = A ˜ ( r ) D a ( r ) [ Γ ˜ ( r ) + ε ˜ ( r ) Γ ˜ ( r ) ε ¯ ( r ) ] D a ( r ) ,
e ± s ( κ ) = k 0 2 4 π ε 0 s ^ d z d R e i κ R e i w z P 1 rad ( R , z )

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