Design of photonic band gap nanocavities for stimulated Raman amplification and lasing in monolithic silicon

Xiaodong Yang; Chee Wei Wong

doi:10.1364/OPEX.13.004723

Design of photonic band gap nanocavities for stimulated Raman amplification and lasing in monolithic silicon

Xiaodong Yang and Chee Wei Wong

Optics Express
Vol. 13,
Issue 12,
pp. 4723-4730
(2005)
•https://doi.org/10.1364/OPEX.13.004723

Get Citation
Copy Citation Text
Xiaodong Yang and Chee Wei Wong, "Design of photonic band gap nanocavities for stimulated Raman amplification and lasing in monolithic silicon," Opt. Express 13, 4723-4730 (2005)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (380 KB)
Set citation alerts
Save article

Related Topics
Table of Contents Category
- Research Papers
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Lasers, Raman (140.3550)
- Semiconductor lasers (140.5960)
- Nonlinear optics, integrated optics (190.4390)
- Micro-optical devices (230.3990)
- Resonators (230.5750)

About this Article
History
- Original Manuscript: May 9, 2005
- Revised Manuscript: June 5, 2005
- Manuscript Accepted: June 6, 2005
- Published: June 13, 2005

Accessible

Open Access

Abstract

The concept and design of L5 photonic band gap nanocavities in two-dimensional photonic crystal slabs for enhancement of stimulated Raman amplification and lasing in monolithic silicon is suggested for the first time. Specific high quality factor and small modal volume nanocavities are designed which supports the required pump-Stokes modal spacing in silicon, with ultralow lasing thresholds.

Full Article | PDF Article

OSA Recommended Articles

Coupled-mode theory for stimulated Raman scattering in high-Q/Vm silicon photonic band gap defect cavity lasers

Xiaodong Yang and Chee Wei Wong
Opt. Express 15(8) 4763-4780 (2007)

Coupled photonic crystal micro-cavities with ultra-low threshold power for stimulated Raman scattering

Qiang Liu, Zhengbiao Ouyang, and Sacharia Albin
Opt. Express 19(5) 4795-4804 (2011)

Implementing a Raman silicon nanocavity laser for integrated optical circuits by using a (100) SOI wafer with a 45-degree-rotated top silicon layer

Yukiko Yamauchi, Makoto Okano, Hiroaki Shishido, Susumu Noda, and Yasushi Takahashi
OSA Continuum 2(7) 2098-2112 (2019)

References

View by:
Article Order
|
Year
|
Author
|
Publication

L. Pavesi and D. J. Lockwood, Silicon Photonics, (Springer-verlag, New York, 2004); G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (John Wiley, West Sussex, 2004).
K. Wada, K., H. C. Luan, D. R. C. Lim, and L. C. Kimerling., “On-chip interconnection beyond semiconductor roadmap: silicon microphotonics,” Proc. SPIE 4870, 437–443 (2002).
[Crossref]
V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature431, 1081–1084 (2004); V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Letters29, 2387–2389 (2004).
[Crossref] [PubMed]
R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express 11, 1731–1739 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-15-1731
[Crossref] [PubMed]
R. L. Espinola, J. I. Dadap, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Raman amplification in ultrasmall silicon-on-insulator wire waveguides,” Opt. Express 12, 3713–3718 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-16-3713
[Crossref] [PubMed]
T. K. Liang and H. K. Tsang, “Efficient Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett. 85, 3343–3345 (2004).
[Crossref]
A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 12, 4261–4268 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-18-4261
[Crossref] [PubMed]
Q. Xu, V. R. Almeida, and M. Lipson, “Time-resolved study of Raman gain in highly confined silicon-on-insulator waveguides,” Opt. Express 12, 4437–4442 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4437
[Crossref] [PubMed]
O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5269
[Crossref] [PubMed]
R. Jones, H. Rong, A. Liu, A. W. Fang, M. J. Paniccia, D. Hak, and O. Cohen, “Net continuous wave optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express13, 519–525 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-2-519 ; H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature433, 292–294 (2005); H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature433, 725–728 (2005).
[Crossref] [PubMed]
O. Boyraz and B. Jalali, “Demonstration of directly modulated silicon Raman laser,” Opt. Express 13, 796–800 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-796
[Crossref] [PubMed]
R. Jones, A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Lossless optical modulation in a silicon waveguide using stimulated Raman scattering,” Opt. Express 13, 1716–1723 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-5-1716
[Crossref] [PubMed]
K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670
[PubMed]
Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[Crossref] [PubMed]
H. Ryu, M. Notomi, G. Kim, and Y. Lee, “High quality-factor whispering-gallery mode in the photonic crystal hexagonal disk cavity,” Opt. Express 12, 1708–1719 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1708
[Crossref] [PubMed]
Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express 12, 3988–3995 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-3988
[Crossref] [PubMed]
B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Materials4, 207–210 (2005); B. S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science300, 1537 (2003).
[Crossref]
P. Koonath, T. Indukuri, and B. Jalali, “Vertically-coupled microdisk resonators realized using three-dimensional sculpting in silicon,” Appl. Phys. Lett. 85, 1018–1020 (2003).
[Crossref]
M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693–3695 (2004).
[Crossref]
K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref] [PubMed]
H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447(2004).
[Crossref] [PubMed]
T. Yoshie, A. Scherer, 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]
M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nature materials 3, 211–219 (2004).
[Crossref] [PubMed]
J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, “Coupled resonator optical waveguides: toward the slowing and storage of light,” Optics & Photonics News 16 (2), 36–40, (2005).
[Crossref]
P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express 13, 801–820 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-801
[Crossref] [PubMed]
M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13, 2678–2687 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2678
[Crossref] [PubMed]
M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]
T. J. Kippenberg, S. M. Spillane, D. K. Armani, and K. J. Vahala, “Ultralow-threshold microcavity Raman laser on a microelectronic chip,” Optics Letters 29, 1224–1226 (2004).
[Crossref] [PubMed]
H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 µm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]
X. Yang, J. Yan, and C. W. Wong, “Design and fabrication of L5 photonic band gap nanocavities for stimulated Raman amplification in monolithic silicon,” CLEO/QELS, CMU2, Baltimore, Maryland (2005).
Y. R. Shen, The Principles of Nonlinear Optics, (Wiley, Hoboken, New Jersey, 2003); Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev.137 (6A), A1787 (1965).
F. X. Kärtner, D. J. Dougherty, H. A. Haus, and E. P. Ippen, “Raman noise and soliton squeezing,” J. Opt. Soc. Am. B.11, 1267 (1994); R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B.6, 1159 (1989).
[Crossref]
K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quan. Elect. 25, 2665 (1989).
[Crossref]
A. Höök, “Influence of stimulated Raman scattering on cross-phase modulation between waves in optical fibers,” Opt. Lett. 17, 115 (1992).
[Crossref] [PubMed]
E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fiberes,” IEEE J. of Quan. Elect. 26 (10), 1815 (1990).
[Crossref]
R. G. Zaporozhchenko, S. Ya. Kilin, and A. G. Smirnov, “Stimulated Raman scattering of light in a photonic crystal,” Quan. Elect. 30, 997 (2000).
[Crossref]
X. Chen, Nicolae C. Panoiu, and R. M. Osgood, Microelectronics Sciences Laboratories, Columbia University, New York, NY 10027, (personal communication, 2005).
V. E. Perlin and H. G. Winful, “Stimulated Raman Scattering in nonlinear periodic structures,” Phys. Rev. A64, 043804 (2001); H. G. Winful, V. E. Perlin, and M. Franke, “Stimulated Raman and Brillouin Scattering in nonlinear periodic structures,” in Proc. of Nonlinear Optics: Materials, Fundamentals, and Applications, Kaua’i-Lihue, Hawaii (2000).
[Crossref]
M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express 12, 5703–5710 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-23-5703
[Crossref] [PubMed]
C. Sauvan, P. Lalanne, and J.P. Hugonin, “Slow-wave effect and mode-profile matching in Photonic Crystal microcavities,” Phys. Rev. B. (to be published), http://arxiv.org/abs/cond-mat/0502664.
A. B. Matsko, A. A. Savchenkov, R. J. Letargat, V. S. Ilchenko, and L. Maleki, “On cavity modification of stimulated Raman scattering,” J. Opt. B: Quantum Semiclass. Opt. 5, 272–278 (2003).
[Crossref]
H.-B. Lin and A. J. Campillo, “cw nonlinear optics in droplet microcavities displaying enhanced gain,” Phys. Rev. Lett.73, 2440 (1994); H.-B. Lin and A. J. Campillo, “Microcavity enhanced Raman gain,” Opt. Comm.133, 287 (1997).
[Crossref] [PubMed]
H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 66 (10), 4801 (1989).
[Crossref]
H. M. Lai, P. T. Leung, and K. Young, “Electromagnetic decay into a narrow resonance in an optical cavity,” Phys. Rev. A. 37, 1597 (1988).
[Crossref] [PubMed]
B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded Raman laser,” Optics Lett. 28 (17), 1507 (2003).
[Crossref]
Y. Wu, X. Yang, and P. T. Leung, “Theory of microcavity-enhanced Raman gain,” Opt. Lett.24, 345 (1999); Y. Wu and P. T. Leung, “Lasing threshold for whispering-gallery-mode microsphere lasers,” Phys. Rev. A60, 630 (1999).
[Crossref]
For a one-dimensional cavity, the enhancement reduces to the Yokoyama-Brorson factor (Ref. [43]) as suggested in Ref. [46].
S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173
[Crossref] [PubMed]
A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House Publishers, 2000); RSoft FullWave commercial software used.
S. G. Johnson, MIT, Cambridge, MA, personal communication, 2005; V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys.107, 6756 (1997); Erratum, ibid.109, 4128 (1998).

2005 (5)

J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, “Coupled resonator optical waveguides: toward the slowing and storage of light,” Optics & Photonics News 16 (2), 36–40, (2005).
[Crossref]

O. Boyraz and B. Jalali, “Demonstration of directly modulated silicon Raman laser,” Opt. Express 13, 796–800 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-796
[Crossref] [PubMed]

P. E. Barclay, K. Srinivasan, and O. Painter, “Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and fiber taper,” Opt. Express 13, 801–820 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-801
[Crossref] [PubMed]

R. Jones, A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Lossless optical modulation in a silicon waveguide using stimulated Raman scattering,” Opt. Express 13, 1716–1723 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-5-1716
[Crossref] [PubMed]

M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express 13, 2678–2687 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2678
[Crossref] [PubMed]

2004 (13)

H. Ryu, M. Notomi, G. Kim, and Y. Lee, “High quality-factor whispering-gallery mode in the photonic crystal hexagonal disk cavity,” Opt. Express 12, 1708–1719 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1708
[Crossref] [PubMed]

R. L. Espinola, J. I. Dadap, R. M. Osgood, S. J. McNab, and Y. A. Vlasov, “Raman amplification in ultrasmall silicon-on-insulator wire waveguides,” Opt. Express 12, 3713–3718 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-16-3713
[Crossref] [PubMed]

Z. Zhang and M. Qiu, “Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs,” Opt. Express 12, 3988–3995 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-17-3988
[Crossref] [PubMed]

A. Liu, H. Rong, M. Paniccia, O. Cohen, and D. Hak, “Net optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express 12, 4261–4268 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-18-4261
[Crossref] [PubMed]

Q. Xu, V. R. Almeida, and M. Lipson, “Time-resolved study of Raman gain in highly confined silicon-on-insulator waveguides,” Opt. Express 12, 4437–4442 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-19-4437
[Crossref] [PubMed]

O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5269
[Crossref] [PubMed]

M. Krause, H. Renner, and E. Brinkmeyer, “Analysis of Raman lasing characteristics in silicon-on-insulator waveguides,” Opt. Express 12, 5703–5710 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-23-5703
[Crossref] [PubMed]

T. J. Kippenberg, S. M. Spillane, D. K. Armani, and K. J. Vahala, “Ultralow-threshold microcavity Raman laser on a microelectronic chip,” Optics Letters 29, 1224–1226 (2004).
[Crossref] [PubMed]

T. K. Liang and H. K. Tsang, “Efficient Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett. 85, 3343–3345 (2004).
[Crossref]

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693–3695 (2004).
[Crossref]

H. G. Park, S. H. Kim, S. H. Kwon, Y. G. Ju, J. K. Yang, J. H. Baek, S. B. Kim, and Y. H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447(2004).
[Crossref] [PubMed]

T. Yoshie, A. Scherer, 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]

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nature materials 3, 211–219 (2004).
[Crossref] [PubMed]

2003 (6)

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref] [PubMed]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[Crossref] [PubMed]

P. Koonath, T. Indukuri, and B. Jalali, “Vertically-coupled microdisk resonators realized using three-dimensional sculpting in silicon,” Appl. Phys. Lett. 85, 1018–1020 (2003).
[Crossref]

A. B. Matsko, A. A. Savchenkov, R. J. Letargat, V. S. Ilchenko, and L. Maleki, “On cavity modification of stimulated Raman scattering,” J. Opt. B: Quantum Semiclass. Opt. 5, 272–278 (2003).
[Crossref]

B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded Raman laser,” Optics Lett. 28 (17), 1507 (2003).
[Crossref]

R. Claps, D. Dimitropoulos, V. Raghunathan, Y. Han, and B. Jalali, “Observation of stimulated Raman amplification in silicon waveguides,” Opt. Express 11, 1731–1739 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-15-1731
[Crossref] [PubMed]

2002 (4)

K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Express 10, 670–684 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670
[PubMed]

H. K. Tsang, C. S. Wong, T. K. Liang, I. E. Day, S. W. Roberts, A. Harpin, J. Drake, and M. Asghari, “Optical dispersion, two-photon absorption and self-phase modulation in silicon waveguides at 1.5 µm wavelength,” Appl. Phys. Lett. 80, 416–418 (2002).
[Crossref]

M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]

K. Wada, K., H. C. Luan, D. R. C. Lim, and L. C. Kimerling., “On-chip interconnection beyond semiconductor roadmap: silicon microphotonics,” Proc. SPIE 4870, 437–443 (2002).
[Crossref]

2001 (1)

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173
[Crossref] [PubMed]

2000 (1)

R. G. Zaporozhchenko, S. Ya. Kilin, and A. G. Smirnov, “Stimulated Raman scattering of light in a photonic crystal,” Quan. Elect. 30, 997 (2000).
[Crossref]

1992 (1)

A. Höök, “Influence of stimulated Raman scattering on cross-phase modulation between waves in optical fibers,” Opt. Lett. 17, 115 (1992).
[Crossref] [PubMed]

1990 (1)

E. Golovchenko, P. V. Mamyshev, A. N. Pilipetskii, and E. M. Dianov, “Mutual influence of the parametric effects and stimulated Raman scattering in optical fiberes,” IEEE J. of Quan. Elect. 26 (10), 1815 (1990).
[Crossref]

1989 (2)

H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 66 (10), 4801 (1989).
[Crossref]

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quan. Elect. 25, 2665 (1989).
[Crossref]

1988 (1)

H. M. Lai, P. T. Leung, and K. Young, “Electromagnetic decay into a narrow resonance in an optical cavity,” Phys. Rev. A. 37, 1597 (1988).
[Crossref] [PubMed]

Akahane, Y.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[Crossref] [PubMed]

B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nature Materials4, 207–210 (2005); B. S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science300, 1537 (2003).
[Crossref]

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature431, 1081–1084 (2004); V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Letters29, 2387–2389 (2004).
[Crossref] [PubMed]

Armani, D. K.

Asano, T.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[Crossref] [PubMed]

Asghari, M.

Baek, J. H.

Barclay, P. E.

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693–3695 (2004).
[Crossref]

Barrios, C. A.

Bloembergen, N.

Y. R. Shen, The Principles of Nonlinear Optics, (Wiley, Hoboken, New Jersey, 2003); Y. R. Shen and N. Bloembergen, “Theory of stimulated Brillouin and Raman scattering,” Phys. Rev.137 (6A), A1787 (1965).

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quan. Elect. 25, 2665 (1989).
[Crossref]

Borselli, M.

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693–3695 (2004).
[Crossref]

Boyraz, O.

O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5269
[Crossref] [PubMed]

Brinkmeyer, E.

Brorson, S. D.

H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 66 (10), 4801 (1989).
[Crossref]

Campillo, A. J.

H.-B. Lin and A. J. Campillo, “cw nonlinear optics in droplet microcavities displaying enhanced gain,” Phys. Rev. Lett.73, 2440 (1994); H.-B. Lin and A. J. Campillo, “Microcavity enhanced Raman gain,” Opt. Comm.133, 287 (1997).
[Crossref] [PubMed]

Chen, X.

X. Chen, Nicolae C. Panoiu, and R. M. Osgood, Microelectronics Sciences Laboratories, Columbia University, New York, NY 10027, (personal communication, 2005).

Claps, R.

Cohen, O.

R. Jones, H. Rong, A. Liu, A. W. Fang, M. J. Paniccia, D. Hak, and O. Cohen, “Net continuous wave optical gain in a low loss silicon-on-insulator waveguide by stimulated Raman scattering,” Opt. Express13, 519–525 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-2-519 ; H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, R. Nicolaescu, A. Fang, and M. Paniccia, “An all-silicon Raman laser,” Nature433, 292–294 (2005); H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “A continuous-wave Raman silicon laser,” Nature433, 725–728 (2005).
[Crossref] [PubMed]

Dadap, J. I.

Day, I. E.

Deppe, D. G.

Dianov, E. M.

Dimitropoulos, D.

Dougherty, D. J.

F. X. Kärtner, D. J. Dougherty, H. A. Haus, and E. P. Ippen, “Raman noise and soliton squeezing,” J. Opt. Soc. Am. B.11, 1267 (1994); R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B.6, 1159 (1989).
[Crossref]

Drake, J.

Ell, C.

Espinola, R. L.

Fang, A.

Fang, A. W.

Franke, M.

V. E. Perlin and H. G. Winful, “Stimulated Raman Scattering in nonlinear periodic structures,” Phys. Rev. A64, 043804 (2001); H. G. Winful, V. E. Perlin, and M. Franke, “Stimulated Raman and Brillouin Scattering in nonlinear periodic structures,” in Proc. of Nonlinear Optics: Materials, Fundamentals, and Applications, Kaua’i-Lihue, Hawaii (2000).
[Crossref]

Gibbs, H. M.

Golovchenko, E.

Gordon, J. P.

Hagness, S.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House Publishers, 2000); RSoft FullWave commercial software used.

Hak, D.

Han, Y.

Harpin, A.

Haus, H. A.

Hendrickson, J.

Höök, A.

A. Höök, “Influence of stimulated Raman scattering on cross-phase modulation between waves in optical fibers,” Opt. Lett. 17, 115 (1992).
[Crossref] [PubMed]

Hugonin, J.P.

C. Sauvan, P. Lalanne, and J.P. Hugonin, “Slow-wave effect and mode-profile matching in Photonic Crystal microcavities,” Phys. Rev. B. (to be published), http://arxiv.org/abs/cond-mat/0502664.

Ilchenko, V. S.

Indukuri, T.

P. Koonath, T. Indukuri, and B. Jalali, “Vertically-coupled microdisk resonators realized using three-dimensional sculpting in silicon,” Appl. Phys. Lett. 85, 1018–1020 (2003).
[Crossref]

Ippen, E. P.

Jalali, B.

O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5269
[Crossref] [PubMed]

P. Koonath, T. Indukuri, and B. Jalali, “Vertically-coupled microdisk resonators realized using three-dimensional sculpting in silicon,” Appl. Phys. Lett. 85, 1018–1020 (2003).
[Crossref]

Joannopoulos, J. D.

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nature materials 3, 211–219 (2004).
[Crossref] [PubMed]

Johnson, S. G.

S. G. Johnson, MIT, Cambridge, MA, personal communication, 2005; V. A. Mandelshtam and H. S. Taylor, “Harmonic inversion of time signals,” J. Chem. Phys.107, 6756 (1997); Erratum, ibid.109, 4128 (1998).

Jones, R.

Ju, Y. G.

Kärtner, F. X.

Khitrova, G.

Kilin, S. Ya.

R. G. Zaporozhchenko, S. Ya. Kilin, and A. G. Smirnov, “Stimulated Raman scattering of light in a photonic crystal,” Quan. Elect. 30, 997 (2000).
[Crossref]

Kim, G.

Kim, S. B.

Kim, S. H.

Kimerling, L. C.

K. Wada, K., H. C. Luan, D. R. C. Lim, and L. C. Kimerling., “On-chip interconnection beyond semiconductor roadmap: silicon microphotonics,” Proc. SPIE 4870, 437–443 (2002).
[Crossref]

Kippenberg, T. J.

B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded Raman laser,” Optics Lett. 28 (17), 1507 (2003).
[Crossref]

M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]

Kira, G.

Knights, A. P.

L. Pavesi and D. J. Lockwood, Silicon Photonics, (Springer-verlag, New York, 2004); G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (John Wiley, West Sussex, 2004).

Koonath, P.

P. Koonath, T. Indukuri, and B. Jalali, “Vertically-coupled microdisk resonators realized using three-dimensional sculpting in silicon,” Appl. Phys. Lett. 85, 1018–1020 (2003).
[Crossref]

Krause, M.

Kuramochi, E.

Kwon, S. H.

Lai, H. M.

H. M. Lai, P. T. Leung, and K. Young, “Electromagnetic decay into a narrow resonance in an optical cavity,” Phys. Rev. A. 37, 1597 (1988).
[Crossref] [PubMed]

Lalanne, P.

C. Sauvan, P. Lalanne, and J.P. Hugonin, “Slow-wave effect and mode-profile matching in Photonic Crystal microcavities,” Phys. Rev. B. (to be published), http://arxiv.org/abs/cond-mat/0502664.

Lee, Y.

Lee, Y. H.

Letargat, R. J.

Leung, P. T.

H. M. Lai, P. T. Leung, and K. Young, “Electromagnetic decay into a narrow resonance in an optical cavity,” Phys. Rev. A. 37, 1597 (1988).
[Crossref] [PubMed]

Y. Wu, X. Yang, and P. T. Leung, “Theory of microcavity-enhanced Raman gain,” Opt. Lett.24, 345 (1999); Y. Wu and P. T. Leung, “Lasing threshold for whispering-gallery-mode microsphere lasers,” Phys. Rev. A60, 630 (1999).
[Crossref]

Liang, T. K.

T. K. Liang and H. K. Tsang, “Efficient Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett. 85, 3343–3345 (2004).
[Crossref]

Lim, D. R. C.

K. Wada, K., H. C. Luan, D. R. C. Lim, and L. C. Kimerling., “On-chip interconnection beyond semiconductor roadmap: silicon microphotonics,” Proc. SPIE 4870, 437–443 (2002).
[Crossref]

Lin, H.-B.

Lipson, M.

Liu, A.

Lockwood, D. J.

L. Pavesi and D. J. Lockwood, Silicon Photonics, (Springer-verlag, New York, 2004); G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (John Wiley, West Sussex, 2004).

Luan, K., H. C.

K. Wada, K., H. C. Luan, D. R. C. Lim, and L. C. Kimerling., “On-chip interconnection beyond semiconductor roadmap: silicon microphotonics,” Proc. SPIE 4870, 437–443 (2002).
[Crossref]

Maleki, L.

Mamyshev, P. V.

Mandelshtam, V. A.

Matsko, A. B.

McNab, S. J.

Min, B.

B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded Raman laser,” Optics Lett. 28 (17), 1507 (2003).
[Crossref]

Mitsugi, S.

Nicolaescu, R.

Noda, S.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[Crossref] [PubMed]

Notomi, M.

Osgood, R. M.

X. Chen, Nicolae C. Panoiu, and R. M. Osgood, Microelectronics Sciences Laboratories, Columbia University, New York, NY 10027, (personal communication, 2005).

Painter, O.

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693–3695 (2004).
[Crossref]

Paloczi, G. T.

Panepucci, R. R.

Paniccia, M.

Paniccia, M. J.

Panoiu, Nicolae C.

X. Chen, Nicolae C. Panoiu, and R. M. Osgood, Microelectronics Sciences Laboratories, Columbia University, New York, NY 10027, (personal communication, 2005).

Park, H. G.

Pavesi, L.

L. Pavesi and D. J. Lockwood, Silicon Photonics, (Springer-verlag, New York, 2004); G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (John Wiley, West Sussex, 2004).

Perlin, V. E.

Pilipetskii, A. N.

Poon, J. K. S.

Qiu, M.

Raghunathan, V.

Reed, G. T.

L. Pavesi and D. J. Lockwood, Silicon Photonics, (Springer-verlag, New York, 2004); G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (John Wiley, West Sussex, 2004).

Renner, H.

Roberts, S. W.

Rong, H.

Rupper, G.

Ryu, H.

Sauvan, C.

C. Sauvan, P. Lalanne, and J.P. Hugonin, “Slow-wave effect and mode-profile matching in Photonic Crystal microcavities,” Phys. Rev. B. (to be published), http://arxiv.org/abs/cond-mat/0502664.

Savchenkov, A. A.

Scherer, A.

Scheuer, J.

Shchekin, O. B.

Shen, Y. R.

Shinya, A.

Smirnov, A. G.

R. G. Zaporozhchenko, S. Ya. Kilin, and A. G. Smirnov, “Stimulated Raman scattering of light in a photonic crystal,” Quan. Elect. 30, 997 (2000).
[Crossref]

Soljacic, M.

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nature materials 3, 211–219 (2004).
[Crossref] [PubMed]

Song, B. S.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[Crossref] [PubMed]

Spillane, M.

M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]

Spillane, S. M.

Srinivasan, K.

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693–3695 (2004).
[Crossref]

Stolen, R. H.

Taflove, A.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House Publishers, 2000); RSoft FullWave commercial software used.

Tanabe, T.

Taylor, H. S.

Tomlinson, W. J.

Tsang, H. K.

T. K. Liang and H. K. Tsang, “Efficient Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett. 85, 3343–3345 (2004).
[Crossref]

Vahala, K. J.

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref] [PubMed]

B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded Raman laser,” Optics Lett. 28 (17), 1507 (2003).
[Crossref]

M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]

Vlasov, Y. A.

Wada, K.

K. Wada, K., H. C. Luan, D. R. C. Lim, and L. C. Kimerling., “On-chip interconnection beyond semiconductor roadmap: silicon microphotonics,” Proc. SPIE 4870, 437–443 (2002).
[Crossref]

Winful, H. G.

Wong, C. S.

Wong, C. W.

X. Yang, J. Yan, and C. W. Wong, “Design and fabrication of L5 photonic band gap nanocavities for stimulated Raman amplification in monolithic silicon,” CLEO/QELS, CMU2, Baltimore, Maryland (2005).

Wood, D.

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quan. Elect. 25, 2665 (1989).
[Crossref]

Wu, Y.

Xu, Q.

Yan, J.

Yang, J. K.

Yang, X.

Yariv, A.

Yokoyama, H.

H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 66 (10), 4801 (1989).
[Crossref]

Yoshie, T.

Young, K.

H. M. Lai, P. T. Leung, and K. Young, “Electromagnetic decay into a narrow resonance in an optical cavity,” Phys. Rev. A. 37, 1597 (1988).
[Crossref] [PubMed]

Zaporozhchenko, R. G.

R. G. Zaporozhchenko, S. Ya. Kilin, and A. G. Smirnov, “Stimulated Raman scattering of light in a photonic crystal,” Quan. Elect. 30, 997 (2000).
[Crossref]

Zhang, Z.

Appl. Phys. Lett. (4)

T. K. Liang and H. K. Tsang, “Efficient Raman amplification in silicon-on-insulator waveguides,” Appl. Phys. Lett. 85, 3343–3345 (2004).
[Crossref]

P. Koonath, T. Indukuri, and B. Jalali, “Vertically-coupled microdisk resonators realized using three-dimensional sculpting in silicon,” Appl. Phys. Lett. 85, 1018–1020 (2003).
[Crossref]

M. Borselli, K. Srinivasan, P. E. Barclay, and O. Painter, “Rayleigh scattering, mode coupling, and optical loss in silicon microdisks,” Appl. Phys. Lett. 85, 3693–3695 (2004).
[Crossref]

IEEE J. of Quan. Elect. (1)

IEEE J. Quan. Elect. (1)

K. J. Blow and D. Wood, “Theoretical description of transient stimulated Raman scattering in optical fibers,” IEEE J. Quan. Elect. 25, 2665 (1989).
[Crossref]

J. Appl. Phys. (1)

H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 66 (10), 4801 (1989).
[Crossref]

J. Opt. B: Quantum Semiclass. Opt. (1)

Nature (4)

M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold Raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002).
[Crossref] [PubMed]

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[Crossref] [PubMed]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[Crossref] [PubMed]

Nature materials (1)

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nature materials 3, 211–219 (2004).
[Crossref] [PubMed]

Opt. Express (14)

O. Boyraz and B. Jalali, “Demonstration of a silicon Raman laser,” Opt. Express 12, 5269–5273 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5269
[Crossref] [PubMed]

Opt. Lett. (1)

A. Höök, “Influence of stimulated Raman scattering on cross-phase modulation between waves in optical fibers,” Opt. Lett. 17, 115 (1992).
[Crossref] [PubMed]

Optics & Photonics News (1)

Optics Lett. (1)

B. Min, T. J. Kippenberg, and K. J. Vahala, “Compact, fiber-compatible, cascaded Raman laser,” Optics Lett. 28 (17), 1507 (2003).
[Crossref]

Optics Letters (1)

Phys. Rev. A. (1)

H. M. Lai, P. T. Leung, and K. Young, “Electromagnetic decay into a narrow resonance in an optical cavity,” Phys. Rev. A. 37, 1597 (1988).
[Crossref] [PubMed]

Proc. SPIE (1)

K. Wada, K., H. C. Luan, D. R. C. Lim, and L. C. Kimerling., “On-chip interconnection beyond semiconductor roadmap: silicon microphotonics,” Proc. SPIE 4870, 437–443 (2002).
[Crossref]

Quan. Elect. (1)

R. G. Zaporozhchenko, S. Ya. Kilin, and A. G. Smirnov, “Stimulated Raman scattering of light in a photonic crystal,” Quan. Elect. 30, 997 (2000).
[Crossref]

Science (1)

Other (15)

X. Chen, Nicolae C. Panoiu, and R. M. Osgood, Microelectronics Sciences Laboratories, Columbia University, New York, NY 10027, (personal communication, 2005).

C. Sauvan, P. Lalanne, and J.P. Hugonin, “Slow-wave effect and mode-profile matching in Photonic Crystal microcavities,” Phys. Rev. B. (to be published), http://arxiv.org/abs/cond-mat/0502664.

For a one-dimensional cavity, the enhancement reduces to the Yokoyama-Brorson factor (Ref. [43]) as suggested in Ref. [46].

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House Publishers, 2000); RSoft FullWave commercial software used.

L. Pavesi and D. J. Lockwood, Silicon Photonics, (Springer-verlag, New York, 2004); G. T. Reed and A. P. Knights, Silicon Photonics: An Introduction (John Wiley, West Sussex, 2004).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

Click here to see a list of articles that cite this paper

Fig. 1. (a) Resonant frequencies of the pump and Stokes modes within the photonic band gap. (b) Q_pump and Q_Stokes as a function of shift S₁ .

Download Full Size | PPT Slide | PDF

Fig. 2. The electric field profile (E_y ) (a) and 2D FT spectrum (b) of pump mode.

Download Full Size | PPT Slide | PDF

Fig. 3. The electric field profile (E_y ) (a) and 2D FT spectrum (b) of Stokes mode.

Download Full Size | PPT Slide | PDF

Fig. 4. SEM picture of the PhC L5 nanocavity for Raman lasing in silicon.

Download Full Size | PPT Slide | PDF

Tables (1)

Table 1. Design summary of photonic crystal L5 nanocavity for Raman lasing in silicon.

View Table

Equations (6)

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

\pm \frac{\partial A_{p}^{\pm}}{\partial z} + \frac{1}{v_{p}} \frac{\partial A_{p}^{\pm}}{\partial t} = {- \frac{g_{p}}{2} ({∣ A_{S}^{+} ∣}^{2} + {∣ A_{S}^{-} ∣}^{2}) + i γ_{p} [{∣ A_{p}^{\pm} ∣}^{2} + 2 ({∣ A_{p}^{\mp} ∣}^{2} + {∣ A_{S}^{+} ∣}^{2} + {∣ A_{S}^{-} ∣}^{2})] - \frac{α_{p}}{2}

- β [{∣ A_{p}^{\pm} ∣}^{2} + 2 ({∣ A_{p}^{\mp} ∣}^{2} + {∣ A_{S}^{+} ∣}^{2} + {∣ A_{S}^{-} ∣}^{2})] - \overset{̅}{φ} λ_{p}^{2} {\overset{̅}{N}}_{eff}} A_{p}^{\pm}

\pm \frac{\partial A_{S}^{\pm}}{\partial z} + \frac{1}{v_{s}} \frac{\partial A_{S}^{\pm}}{\partial t} = {\frac{g_{s}}{2} ({∣ A_{p}^{+} ∣}^{2} + {∣ A_{p}^{-} ∣}^{2}) + i γ_{s} [{∣ A_{S}^{\pm} ∣}^{2} + 2 ({∣ A_{S}^{\mp} ∣}^{2} + {∣ A_{p}^{+} ∣}^{2} + {∣ A_{p}^{-} ∣}^{2})] - \frac{α_{s}}{2}

- β [{∣ A_{S}^{\pm} ∣}^{2} + 2 ({∣ A_{S}^{\mp} ∣}^{2} + {∣ A_{p}^{+} ∣}^{2} + {∣ A_{p}^{-} ∣}^{2})] - \overset{̅}{φ} λ_{S}^{2} {\overset{̅}{N}}_{eff}} A_{S}^{\pm}

+ i κ A_{S}^{\mp} + i δ β A_{S}^{\pm}

P_{threshold} = \frac{π^{2} n_{s} n_{p}}{g_{s} ξ λ_{s} λ_{p}} \frac{V_{m}}{Q_{s} Q_{p}}

S₁ (x a)	a(nm)	λ_pump (nm)	Q_pump	λ_Stokes (nm)	Q_Stokes
0	456	1592	560	1735.6	20693
0.05	414	1456.4	739	1575.8	19036
0.07	395	1395.7	863	1505	20843
0.10	376	1333.1	1030	1432.4	24642
0.15	342	1215.2	1550	1297.1	42445