Abstract

We derive an equation that describes the nonlinear operation of a Fabry–Perot resonator with a large group index waveguide. Specifically, a silicon photonic crystal microcavity with two-photon-excited free carrier nonlinearity and Kerr nonlinearity is assumed. The equation clearly explains the bistability of the device and the reduction of the required pump energy for a specific nonlinear phase shift at an appropriate phase detuning from the resonance. We present a simple procedure to predict the required optical pump energy for the modulation and the resulting modulation depth by use of the equation and the device parameters.

© 2006 Optical Society of America

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

2006 (1)

2005 (4)

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, Opt. Lett. 30, 2575 (2005).
[CrossRef] [PubMed]

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, Nature 433, 725 (2005).
[CrossRef] [PubMed]

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, Appl. Phys. Lett. 87, 151112 (2005).
[CrossRef]

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, Nat. Mater. 4, 207 (2005).
[CrossRef]

2004 (5)

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, Phys. Rev. B 70, 205110 (2004).
[CrossRef]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, Nature 427, 615 (2004).
[CrossRef] [PubMed]

W. Nakagawa and Y. Fainman, IEEE J. Quantum Electron. 10, 478 (2004).
[CrossRef]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, Nature 431, 1081 (2004).
[CrossRef] [PubMed]

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H.-Y. Ryu, Opt. Express 12, 1551 (2004).
[CrossRef] [PubMed]

2003 (1)

M. Dinu, F. Quochi, and H. Garcia, Appl. Phys. Lett. 82, 2954 (2003).
[CrossRef]

2001 (1)

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

1982 (1)

1974 (1)

Akahane, Y.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, Nat. Mater. 4, 207 (2005).
[CrossRef]

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, Nature 431, 1081 (2004).
[CrossRef] [PubMed]

Asano, T.

T. Uesugi, B.-S. Song, T. Asano, and S. Noda, Opt. Express 14, 377 (2006).
[CrossRef] [PubMed]

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, Nat. Mater. 4, 207 (2005).
[CrossRef]

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, Nature 431, 1081 (2004).
[CrossRef] [PubMed]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 1992).

Chodorow, M.

Cohen, O.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, Nature 433, 725 (2005).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, Nature 427, 615 (2004).
[CrossRef] [PubMed]

Dinu, M.

M. Dinu, F. Quochi, and H. Garcia, Appl. Phys. Lett. 82, 2954 (2003).
[CrossRef]

Fainman, Y.

W. Nakagawa and Y. Fainman, IEEE J. Quantum Electron. 10, 478 (2004).
[CrossRef]

H.-C. Kim, K. Ikeda, and Y. Fainman, "Tunable transmission resonant filter and modulator with vertical gratings," submitted to J. Lightwave Technol.

Fang, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, Nature 433, 725 (2005).
[CrossRef] [PubMed]

Garcia, H.

M. Dinu, F. Quochi, and H. Garcia, Appl. Phys. Lett. 82, 2954 (2003).
[CrossRef]

Gosele, U.

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Hak, D.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, Nature 433, 725 (2005).
[CrossRef] [PubMed]

Ikeda, K.

H.-C. Kim, K. Ikeda, and Y. Fainman, "Tunable transmission resonant filter and modulator with vertical gratings," submitted to J. Lightwave Technol.

Jones, R.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, Nature 433, 725 (2005).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, Nature 427, 615 (2004).
[CrossRef] [PubMed]

Kim, H.-C.

H.-C. Kim, K. Ikeda, and Y. Fainman, "Tunable transmission resonant filter and modulator with vertical gratings," submitted to J. Lightwave Technol.

Kogelnik, H.

Kuramochi, E.

Liao, L.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, Nature 427, 615 (2004).
[CrossRef] [PubMed]

Lipson, M.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, Nature 431, 1081 (2004).
[CrossRef] [PubMed]

Liu, A.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, Nature 433, 725 (2005).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, Nature 427, 615 (2004).
[CrossRef] [PubMed]

Mitsugi, S.

Nakagawa, W.

W. Nakagawa and Y. Fainman, IEEE J. Quantum Electron. 10, 478 (2004).
[CrossRef]

Nicolaescu, R.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, Nature 427, 615 (2004).
[CrossRef] [PubMed]

Noda, S.

T. Uesugi, B.-S. Song, T. Asano, and S. Noda, Opt. Express 14, 377 (2006).
[CrossRef] [PubMed]

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, Nat. Mater. 4, 207 (2005).
[CrossRef]

Notomi, M.

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, Appl. Phys. Lett. 87, 151112 (2005).
[CrossRef]

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, Opt. Lett. 30, 2575 (2005).
[CrossRef] [PubMed]

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H.-Y. Ryu, Opt. Express 12, 1551 (2004).
[CrossRef] [PubMed]

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Panepucci, R. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, Nature 431, 1081 (2004).
[CrossRef] [PubMed]

Paniccia, M.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, Nature 433, 725 (2005).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, Nature 427, 615 (2004).
[CrossRef] [PubMed]

Quochi, F.

M. Dinu, F. Quochi, and H. Garcia, Appl. Phys. Lett. 82, 2954 (2003).
[CrossRef]

Rong, H.

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, Nature 433, 725 (2005).
[CrossRef] [PubMed]

Rubin, D.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, Nature 427, 615 (2004).
[CrossRef] [PubMed]

Ryu, H.-Y.

Samara-Rubio, D.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, Nature 427, 615 (2004).
[CrossRef] [PubMed]

Schweizer, S. L.

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Shaw, H. J.

Shinya, A.

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, Opt. Lett. 30, 2575 (2005).
[CrossRef] [PubMed]

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, Appl. Phys. Lett. 87, 151112 (2005).
[CrossRef]

M. Notomi, A. Shinya, S. Mitsugi, E. Kuramochi, and H.-Y. Ryu, Opt. Express 12, 1551 (2004).
[CrossRef] [PubMed]

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Song, B.-S.

T. Uesugi, B.-S. Song, T. Asano, and S. Noda, Opt. Express 14, 377 (2006).
[CrossRef] [PubMed]

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, Nat. Mater. 4, 207 (2005).
[CrossRef]

Stokes, L. F.

Takahashi, C.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Takahashi, J.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Tan, H. W.

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Tanabe, T.

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, Appl. Phys. Lett. 87, 151112 (2005).
[CrossRef]

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, Opt. Lett. 30, 2575 (2005).
[CrossRef] [PubMed]

Uesugi, T.

van Driel, H. M.

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Weber, H. P.

Wehrspohn, R. B.

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Yamada, K.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Yokohama, I.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Appl. Phys. Lett. (2)

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, Appl. Phys. Lett. 87, 151112 (2005).
[CrossRef]

M. Dinu, F. Quochi, and H. Garcia, Appl. Phys. Lett. 82, 2954 (2003).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. Nakagawa and Y. Fainman, IEEE J. Quantum Electron. 10, 478 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

Nat. Mater. (1)

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, Nat. Mater. 4, 207 (2005).
[CrossRef]

Nature (3)

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, Nature 427, 615 (2004).
[CrossRef] [PubMed]

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, Nature 433, 725 (2005).
[CrossRef] [PubMed]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, Nature 431, 1081 (2004).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. B (1)

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gosele, Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Other (2)

R. W. Boyd, Nonlinear Optics (Academic, 1992).

H.-C. Kim, K. Ikeda, and Y. Fainman, "Tunable transmission resonant filter and modulator with vertical gratings," submitted to J. Lightwave Technol.

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram describing the analytic model for Fabry–Perot resonator in a large group index waveguide.

Fig. 2
Fig. 2

Schematic plots of the LHS and RHS of Eq. (5). The circles are solutions for various input intensities and phase detunings.

Fig. 3
Fig. 3

Plots of the solutions of Eq. (5) for data from Ref. [4]. (a) Plots of LHS and RHS of Eq. (5) with relation to power inside the cavity for various input peak powers and wavelength detunings. (b) Nonlinear phase shift and (c) modulation depth in relation to the input peak power for various wavelength detunings.

Equations (10)

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n = ( n 0 j n 0 ) + n d I 2 + ( n 2 j n 2 ) I ,
n d = e 2 2 n 0 ω s 2 m * ϵ 0 τ p π 4 ln 2 β 2 ω p ,
k = k 0 n g = k 0 n cos θ ,
k = k 0 cos θ ( n 0 + n d I 2 + n 2 I ) j ( α + β I 2 cos θ ) κ j μ ,
I ( z ) a 2
= e 2 ξ ( z ) + R 2 e 4 ξ ( L ) e 2 ξ ( z ) + 2 R e 2 ξ ( L ) cos 2 [ ϕ ( L ) ϕ ( z ) ] 1 2 R 2 e 2 ξ ( L ) cos 2 ϕ ( L ) + R 4 e 4 ξ ( L ) ( 1 S 2 R 2 ) ,
ϕ ( z ) = 0 z κ ( z ) d z = k 0 n 0 z cos θ + k 0 cos θ [ n d 0 z I 2 ( z ) d z + n 2 0 z I ( z ) d z ] ,
ξ ( z ) = 0 z μ ( z ) d z = α z 2 cos θ + β 2 cos θ 0 z I ( z ) d z .
ϕ ( L ) = k 0 n 0 L cos θ + k 0 n 2 L eff 1 cos θ I p + k 0 n d L eff 2 cos θ I p 2 ,
ξ ( L ) = α L 2 cos θ + β 2 cos θ I p L eff 1 ,

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