Abstract

For practical applications in quantum electrodynamics, it has been proposed to produce frequency tuning or Q-switching by dynamically changing the dielectric constant around a nano-cavity. Local changes in the dielectric constant of a photonic cavity with finite-lifetime, may affect not only the frequency of electromagnetic cavity modes but also their quality-factor (Q). Thus, it is important to have prediction capability regarding the combined effect of these changes. Here perturbation theory, usually applied to eigenmodes with real eigenvalues, is formulated in the complex domain, in which the eigen-frequency imaginary part is related to the Q-factor. Normalizable leaky modes, and bi-orthogonality in a finite volume are the basis for such a formulation. We introduce such capabilities by presenting semi-analytical expressions of first order perturbation analysis for a 3D cavity with radiation losses. The obtained results are in good agreement with numerical calculations.

© 2010 OSA

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  1. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
    [CrossRef]
  2. D. Englund, I. Fushman, and J. Vučković, “General recipe for designing photonic crystal cavities,” Opt. Express 13(16), 5961–5975 (2005).
    [CrossRef] [PubMed]
  3. 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(4), 041112 (2006).
    [CrossRef]
  4. M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. 73(5), 051803 (2006).
    [CrossRef]
  5. T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102(4), 043907 (2009).
    [CrossRef] [PubMed]
  6. M. Sandberg, C. M. Wilson, F. Persson, G. Johansson, V. Shumeiko, and P. Delsing, “In-situ frequency tuning of photons stored in a high Q microwave cavity,” arXiv:0801.2479v1 (2008)
  7. A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73(1), 013818 (2006).
    [CrossRef]
  8. A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
    [CrossRef]
  9. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGRAW-HILL, 1961).
  10. L. D. Landau, and E. M. Lifshitz, Quantum Mechanics Non-relativistic Theory, 2nd-ed (Addison-Wesley, 1965)
  11. L. I. Schiff, Quantum Mechanics, ed-3, (McGraw-Hill, 1968)
  12. R. L. Liboff, Introductory Quantum Mechanics, ed-2 (Addison-Wesley, 1992)
  13. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, (Princeton University Press, Princeton, NJ, 1995)
  14. M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystal Guiding (Cambridge University Press, Cambridge, England, 2009)
  15. P. Kristensen, T. Nielsen, J. Riishede, and J. Lagsgaard, “Light guiding in microstructures of purely imaginary refractive index contrasts,” Photonics Nanostruct. Fundam. Appl. 3(1), 1–9 (2005).
    [CrossRef]
  16. S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
    [CrossRef]
  17. P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion,” J. Phys. A 30(6), 2139–2151 (1997).
    [CrossRef]
  18. E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
    [CrossRef]
  19. P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in Open Systems via Bi-orthogonal Basis,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(5), 6101–6104 (1998).
    [CrossRef]
  20. A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
    [CrossRef] [PubMed]
  21. A. M. Brink and K. Young, “Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems,” J. Phys. Math. Gen. 34(12), 2607–2624 (2001).
    [CrossRef]
  22. O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton",” Phys. Rev. Lett. 102(16), 163902 (2009).
    [CrossRef] [PubMed]
  23. The Normalizable Leaky Modes may be regarded as a 3D extension of the quasi-normal modes [17–20].
  24. Discontinuity of the dielectric constant at surface S is followed by discontinuity of the electric field, therefore by integrating over S we actually mean integrating over a different surface which is outward remote from S but infinitesimally close to it where both the dielectric constant and the electric field are continuous.
  25. Bilinear Form is a function B:V×V→F mapping from two vectors of the vector space V into the scalar field F. Every inner product is by itself a bilinear map, but for example while every inner product is conjugate symmetric by definition, bilinear map doesn't have to be, and in the presented case it is just symmetric.
  26. We define 〈ψ|Θ^′|φ〉≡〈ψ|Θ^′φ〉, where Θ^′φ is the matrix by vector multiplication [see Eqs. (3) and (6)].
  27. Meep. http://ab-initio.mit.edu/wiki/index.php/Meep (13.10.2009), free finite-difference time-domain (FDTD) simulation software package developed at MIT to model electromagnetic systems.
  28. The absolute peak value in the computational region of Re{EIII} is roughly 23 times larger than that of Im{EIII}. Similarly the absolute peak value in the computational region of Re{wIII} is roughly 172 times larger than that of Im{wIII}.
  29. Tests for applicability to high Q systems suggested by an anonymous reviewer are gratefully acknowledged. Though the simulation resolution was decreased to 20 point per unit cell because of computational time considerations, the relative errors did not increased.
  30. In the examples studied here, even under a very large material modulation of ε′r = 0.15 (ε′r/εr0≅0.01) at every desired point in the volume V (including air holes), the possible values of Wr and Wi are limited to −0.422⋅10−3≤Wr≤2.341 and 9.615⋅10−3≤Wi≤0.0322 respectively.
  31. If, upon cavity design and mode calculation, Wrand Wi receive the adequate magnitude and sign, desired response in Δω or ΔQ can be obtained. This will be shown elsewhere (M. Shlafman and J. Salzman unpublished).

2009

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102(4), 043907 (2009).
[CrossRef] [PubMed]

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[CrossRef]

O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton",” Phys. Rev. Lett. 102(16), 163902 (2009).
[CrossRef] [PubMed]

2006

A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73(1), 013818 (2006).
[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(4), 041112 (2006).
[CrossRef]

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. 73(5), 051803 (2006).
[CrossRef]

2005

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

D. Englund, I. Fushman, and J. Vučković, “General recipe for designing photonic crystal cavities,” Opt. Express 13(16), 5961–5975 (2005).
[CrossRef] [PubMed]

P. Kristensen, T. Nielsen, J. Riishede, and J. Lagsgaard, “Light guiding in microstructures of purely imaginary refractive index contrasts,” Photonics Nanostruct. Fundam. Appl. 3(1), 1–9 (2005).
[CrossRef]

2004

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[CrossRef]

2003

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[CrossRef] [PubMed]

2001

A. M. Brink and K. Young, “Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems,” J. Phys. Math. Gen. 34(12), 2607–2624 (2001).
[CrossRef]

1998

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[CrossRef]

P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in Open Systems via Bi-orthogonal Basis,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(5), 6101–6104 (1998).
[CrossRef]

1997

P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion,” J. Phys. A 30(6), 2139–2151 (1997).
[CrossRef]

Akahane, Y.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

Asano, T.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

Bartal, G.

O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton",” Phys. Rev. Lett. 102(16), 163902 (2009).
[CrossRef] [PubMed]

Bertolotti, M.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[CrossRef] [PubMed]

Bloemer, M.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[CrossRef] [PubMed]

Botten, L.

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[CrossRef]

Bowden, C. M.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[CrossRef] [PubMed]

Brink, A. M.

A. M. Brink and K. Young, “Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems,” J. Phys. Math. Gen. 34(12), 2607–2624 (2001).
[CrossRef]

Centini, M.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[CrossRef] [PubMed]

Ching, E. S. C.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[CrossRef]

Christodoulides, D. N.

O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton",” Phys. Rev. Lett. 102(16), 163902 (2009).
[CrossRef] [PubMed]

Cohen, O.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[CrossRef]

D’Aguanno, G.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[CrossRef] [PubMed]

de Sterke, C.

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[CrossRef]

Dossou, K.

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[CrossRef]

Englund, D.

Fushman, I.

Greentree, A. D.

A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73(1), 013818 (2006).
[CrossRef]

Hollenberg, L. C. L.

A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73(1), 013818 (2006).
[CrossRef]

Jones, R.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[CrossRef]

Kristensen, P.

P. Kristensen, T. Nielsen, J. Riishede, and J. Lagsgaard, “Light guiding in microstructures of purely imaginary refractive index contrasts,” Photonics Nanostruct. Fundam. Appl. 3(1), 1–9 (2005).
[CrossRef]

Kuramochi, E.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102(4), 043907 (2009).
[CrossRef] [PubMed]

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(4), 041112 (2006).
[CrossRef]

Lagsgaard, J.

P. Kristensen, T. Nielsen, J. Riishede, and J. Lagsgaard, “Light guiding in microstructures of purely imaginary refractive index contrasts,” Photonics Nanostruct. Fundam. Appl. 3(1), 1–9 (2005).
[CrossRef]

Leung, P. T.

P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in Open Systems via Bi-orthogonal Basis,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(5), 6101–6104 (1998).
[CrossRef]

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[CrossRef]

P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion,” J. Phys. A 30(6), 2139–2151 (1997).
[CrossRef]

Liao, L.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[CrossRef]

Liu, A.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[CrossRef]

Maassen van den Brink, A.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[CrossRef]

Mahmoodian, S.

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[CrossRef]

Mattiucci, N.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[CrossRef] [PubMed]

McPhedran, R.

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[CrossRef]

Mitsugi, S.

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. 73(5), 051803 (2006).
[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(4), 041112 (2006).
[CrossRef]

Moiseyev, N.

O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton",” Phys. Rev. Lett. 102(16), 163902 (2009).
[CrossRef] [PubMed]

Nicolaescu, R.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[CrossRef]

Nielsen, T.

P. Kristensen, T. Nielsen, J. Riishede, and J. Lagsgaard, “Light guiding in microstructures of purely imaginary refractive index contrasts,” Photonics Nanostruct. Fundam. Appl. 3(1), 1–9 (2005).
[CrossRef]

Noda, S.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

Notomi, M.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102(4), 043907 (2009).
[CrossRef] [PubMed]

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. 73(5), 051803 (2006).
[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(4), 041112 (2006).
[CrossRef]

Paniccia, M.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[CrossRef]

Peleg, O.

O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton",” Phys. Rev. Lett. 102(16), 163902 (2009).
[CrossRef] [PubMed]

Poulton, C.

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[CrossRef]

Prawer, S.

A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73(1), 013818 (2006).
[CrossRef]

Riishede, J.

P. Kristensen, T. Nielsen, J. Riishede, and J. Lagsgaard, “Light guiding in microstructures of purely imaginary refractive index contrasts,” Photonics Nanostruct. Fundam. Appl. 3(1), 1–9 (2005).
[CrossRef]

Rubin, D.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[CrossRef]

Rubio, D. S.

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[CrossRef]

Salzman, J.

A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73(1), 013818 (2006).
[CrossRef]

Scalora, M.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[CrossRef] [PubMed]

Segev, M.

O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton",” Phys. Rev. Lett. 102(16), 163902 (2009).
[CrossRef] [PubMed]

Settimi, A.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[CrossRef] [PubMed]

Severini, S.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[CrossRef] [PubMed]

Shinya, A.

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(4), 041112 (2006).
[CrossRef]

Sibilia, C.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[CrossRef] [PubMed]

Song, B. S.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

Suen, W. M.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[CrossRef]

P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in Open Systems via Bi-orthogonal Basis,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(5), 6101–6104 (1998).
[CrossRef]

Sun, C. P.

P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in Open Systems via Bi-orthogonal Basis,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(5), 6101–6104 (1998).
[CrossRef]

Tanabe, T.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102(4), 043907 (2009).
[CrossRef] [PubMed]

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(4), 041112 (2006).
[CrossRef]

Taniyama, H.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102(4), 043907 (2009).
[CrossRef] [PubMed]

Tong, S. S.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[CrossRef]

P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion,” J. Phys. A 30(6), 2139–2151 (1997).
[CrossRef]

Vuckovic, J.

Watanabe, T.

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(4), 041112 (2006).
[CrossRef]

Young, K.

A. M. Brink and K. Young, “Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems,” J. Phys. Math. Gen. 34(12), 2607–2624 (2001).
[CrossRef]

P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in Open Systems via Bi-orthogonal Basis,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(5), 6101–6104 (1998).
[CrossRef]

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[CrossRef]

P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion,” J. Phys. A 30(6), 2139–2151 (1997).
[CrossRef]

Appl. Phys. Lett.

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(4), 041112 (2006).
[CrossRef]

J. Phys. A

P. T. Leung, S. S. Tong, and K. Young, “Two-component eigenfunction expansion for open systems described by the wave equation I: completeness of expansion,” J. Phys. A 30(6), 2139–2151 (1997).
[CrossRef]

J. Phys. Math. Gen.

A. M. Brink and K. Young, “Jordan blocks and generalized bi-orthogonal bases: realizations in open wave systems,” J. Phys. Math. Gen. 34(12), 2607–2624 (2001).
[CrossRef]

Nat. Mater.

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

Nature

A. Liu, R. Jones, L. Liao, D. S. Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 427 (2004).
[CrossRef]

Opt. Express

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P. Kristensen, T. Nielsen, J. Riishede, and J. Lagsgaard, “Light guiding in microstructures of purely imaginary refractive index contrasts,” Photonics Nanostruct. Fundam. Appl. 3(1), 1–9 (2005).
[CrossRef]

Phys. Rev.

M. Notomi and S. Mitsugi, “Wavelength conversion via dynamic refractive index tuning of a cavity,” Phys. Rev. 73(5), 051803 (2006).
[CrossRef]

Phys. Rev. A

S. Mahmoodian, R. McPhedran, C. de Sterke, K. Dossou, C. Poulton, and L. Botten, “Single and coupled degenerate defect modes in two-dimensional photonic crystal band gaps,” Phys. Rev. A 79(1), 013814 (2009).
[CrossRef]

A. D. Greentree, J. Salzman, S. Prawer, and L. C. L. Hollenberg, “Quantum gate for Q switching in monolithic photonic-band-gap cavities containing two-level atoms,” Phys. Rev. A 73(1), 013818 (2006).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

A. Settimi, S. Severini, N. Mattiucci, C. Sibilia, M. Centini, G. D’Aguanno, M. Bertolotti, M. Scalora, M. Bloemer, and C. M. Bowden, “Quasinormal-mode description of waves in one-dimensional photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(2), 026614 (2003).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics

P. T. Leung, W. M. Suen, C. P. Sun, and K. Young, “Waves in Open Systems via Bi-orthogonal Basis,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57(5), 6101–6104 (1998).
[CrossRef]

Phys. Rev. Lett.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102(4), 043907 (2009).
[CrossRef] [PubMed]

O. Peleg, M. Segev, G. Bartal, D. N. Christodoulides, and N. Moiseyev, “Nonlinear Waves in Subwavelength Waveguide Arrays: Evanescent Bands and the “Phoenix Soliton",” Phys. Rev. Lett. 102(16), 163902 (2009).
[CrossRef] [PubMed]

Rev. Mod. Phys.

E. S. C. Ching, P. T. Leung, A. Maassen van den Brink, W. M. Suen, S. S. Tong, and K. Young, “Quasinormal-mode expansion for waves in open systems,” Rev. Mod. Phys. 70(4), 1545–1554 (1998).
[CrossRef]

Other

The Normalizable Leaky Modes may be regarded as a 3D extension of the quasi-normal modes [17–20].

Discontinuity of the dielectric constant at surface S is followed by discontinuity of the electric field, therefore by integrating over S we actually mean integrating over a different surface which is outward remote from S but infinitesimally close to it where both the dielectric constant and the electric field are continuous.

Bilinear Form is a function B:V×V→F mapping from two vectors of the vector space V into the scalar field F. Every inner product is by itself a bilinear map, but for example while every inner product is conjugate symmetric by definition, bilinear map doesn't have to be, and in the presented case it is just symmetric.

We define 〈ψ|Θ^′|φ〉≡〈ψ|Θ^′φ〉, where Θ^′φ is the matrix by vector multiplication [see Eqs. (3) and (6)].

Meep. http://ab-initio.mit.edu/wiki/index.php/Meep (13.10.2009), free finite-difference time-domain (FDTD) simulation software package developed at MIT to model electromagnetic systems.

The absolute peak value in the computational region of Re{EIII} is roughly 23 times larger than that of Im{EIII}. Similarly the absolute peak value in the computational region of Re{wIII} is roughly 172 times larger than that of Im{wIII}.

Tests for applicability to high Q systems suggested by an anonymous reviewer are gratefully acknowledged. Though the simulation resolution was decreased to 20 point per unit cell because of computational time considerations, the relative errors did not increased.

In the examples studied here, even under a very large material modulation of ε′r = 0.15 (ε′r/εr0≅0.01) at every desired point in the volume V (including air holes), the possible values of Wr and Wi are limited to −0.422⋅10−3≤Wr≤2.341 and 9.615⋅10−3≤Wi≤0.0322 respectively.

If, upon cavity design and mode calculation, Wrand Wi receive the adequate magnitude and sign, desired response in Δω or ΔQ can be obtained. This will be shown elsewhere (M. Shlafman and J. Salzman unpublished).

M. Sandberg, C. M. Wilson, F. Persson, G. Johansson, V. Shumeiko, and P. Delsing, “In-situ frequency tuning of photons stored in a high Q microwave cavity,” arXiv:0801.2479v1 (2008)

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGRAW-HILL, 1961).

L. D. Landau, and E. M. Lifshitz, Quantum Mechanics Non-relativistic Theory, 2nd-ed (Addison-Wesley, 1965)

L. I. Schiff, Quantum Mechanics, ed-3, (McGraw-Hill, 1968)

R. L. Liboff, Introductory Quantum Mechanics, ed-2 (Addison-Wesley, 1992)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light, (Princeton University Press, Princeton, NJ, 1995)

M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystal Guiding (Cambridge University Press, Cambridge, England, 2009)

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Figures (2)

Fig. 1
Fig. 1

(a) Simulated structure. The Perfectly Matched Layer (PML) is marked by light blue dots region at the borders. Green dotted line marks region 1 of the perturbation. Purple dashed line marks region 2 of the perturbation. (b)-(e) Highest frequency normalized cavity mode | ϕ I I I ( 0 ) . (b) Re { E I I I } .(c) Im { E I I I } .(d) Re { w I I I } .(e) Im { w I I I } [28].

Fig. 2
Fig. 2

(a) The ratio f Q I I I ( W r , W i ) ( Δ Q I I I / Q I I I 0 ) in the wide range. (b) f Q I I I ( W r , W i ) in the relevant range with the specific perturbation marked on it. (c) f Q I I I ( W i ) with the four tested perturbations.

Tables (1)

Tables Icon

Table 1 Frequency and Q changes of the highest nanocavity mode obtained by an FDTD simulation and predicted by the Normalizable Leaky Modes and Conventional Electro-Magnetic perturbation treatments accompanied by the relative errors in comparison to the FDTD results.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

( a ) Θ ^ × ( 1 ε r ( r ) ) × ( b ) Θ ^ ( H ) = 1 c 2 2 t 2 H
H n | H k a l l s p a c e H n * H k d 3 r
( a ) | ϕ = ( E M ) ( b ) Θ ^ i ( 0 c 2 ε r ( r ) × × 0 )
Θ ^ | ϕ = i t | ϕ
S [ n o u t c t E + ( × E ) × n ^ ] d s = 0
ϕ n | ϕ k i V ( E n M k + E k M n ) d 3 r + i n o u t c S ( E n E k ) d s
Θ ^ = ( Θ ^ 0 + Θ ^ ) = i ( 0 c 2 ε r 0 ( r ) × × 0 ) + i ( 0 c 2 ε r ( r ) ε r 0 ( r ) 2 0 0 )
| ϕ n = | ϕ n ( 0 ) + k n ϕ k ( 0 ) | Θ ^ | ϕ n ( 0 ) ϕ k ( 0 ) | ϕ k ( 0 ) ( ω n ( 0 ) ω k ( 0 ) ) | ϕ k ( 0 )
Δ ω n = ϕ n ( 0 ) | Θ ^ | ϕ n ( 0 ) ϕ n ( 0 ) | ϕ n ( 0 )
Δ ω n = ( ω n ( 0 ) c ) 2 I V ε r ( r ) E n E n d 3 r
Δ ω n = 1 2 ω n 0 V ε r ( r ) E n E n d 3 r V ε r 0 ( r ) E n E n d 3 r + i n o u t 2 c ω n 0 S E n E n d s .
Δ Q n Q n 0 = [ Q n 0 W i 1 2 W r ( c 2 2 | ω n ( 0 ) | 1 ^ ) + Q n 0 W i W r ]

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