Effect of dielectric constant tuning on a photonic cavity frequency and Q-factor |
Optics Express, Vol. 18, Issue 15, pp. 15907-15916 (2010)
http://dx.doi.org/10.1364/OE.18.015907
Acrobat PDF (1032 KB)
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
1. Introduction
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]
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]
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]
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]
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]
2. Normalizable leaky modes perturbation theory foundation
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
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]
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]
3. NLM perturbation theory and complex frequency predictability
3.1 NLM perturbation theory development
3.2 Complex frequency change
3.3 Calculation of the change in Q
4. Comparing present formulation with conventional electromagnetic perturbation and numerical simulations
4.1 Introducing the example
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.
4.2 Results
4.3 Discussion
5. Parameter space of perturbative changes in Q-factor
6. Conclusion
References and links
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 |
26. | We define |
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 |
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 |
31. | If, upon cavity design and mode calculation, |
OCIS Codes
(000.3860) General : Mathematical methods in physics
(000.4430) General : Numerical approximation and analysis
(050.1755) Diffraction and gratings : Computational electromagnetic methods
(050.5298) Diffraction and gratings : Photonic crystals
ToC Category:
Photonic Crystals
History
Original Manuscript: March 16, 2010
Revised Manuscript: May 6, 2010
Manuscript Accepted: May 10, 2010
Published: July 13, 2010
Citation
Michael Shlafman, Igal Bayn, and Joseph Salzman, "Effect of dielectric constant tuning on a photonic cavity frequency and Q-factor," Opt. Express 18, 15907-15916 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-15-15907
Sort: Year | Journal | Reset
References
- 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]
- 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]
- 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. 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)
- 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]
- 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]
- 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)
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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).
Cited By |
Alert me when this paper is cited |
OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.
« Previous Article | Next Article »
OSA is a member of CrossRef.