Abstract

Direct time-domain observations are reported of a low-power, self-induced modulation of the transmitted optical power through a high-Q silicon microdisk resonator. Above a threshold input power of 60 μW the transmission versus wavelength deviates from a simple optical bistability behavior, and the transmission intensity becomes highly oscillatory in nature. The transmission oscillations are seen to consist of a train of sharp transmission dips of width approximately 100 ns and period close to 1 μs. A model of the system is developed incorporating thermal and free-carrier dynamics, and is compared to the observed behavior. Good agreement is found, and the self-induced optical modulation is attributed to a nonlinear interaction between competing free-carrier and phonon populations within the microdisk.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  27. Here we assume that the coupling to each of the standing-wave modes is identical. In general, the coupling can be different, although experimentally we have noticed only small differences in coupling.
  28. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, "Influence of nonlinear absorption on Raman amplification in silicon waveguides," Opt. Express 12, 2774-2780 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2774">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2774</a>.
    [CrossRef] [PubMed]
  29. Note that the confinement factor and effective mode volume for the two standing-wave modes are identical, hence we drop the c/s subscript.
  30. For TPA with the standing wave modes one has an additional term dependent upon the product UcUs, with cross-confinement factor ? c/s,TPA and cross-mode volume 3Vc/s,TPA pre-factors. For FCA, described below, one cannot write the total absorption just in terms of products of powers of the cavity energies, but rather the mode amplitudes themselves must be explicitly used.
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  32. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Perturbation theory for Maxwell's equations with shifting material boundaries," Phys. Rev. E 65, 066611 (2002).
    [CrossRef]
  33. A variable order Adams-Bashforth-Moulton predictor-corrector method was used.
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    [CrossRef]
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    [CrossRef]
  40. Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
    [CrossRef] [PubMed]
  41. Luxtera, <a href="http://www.luxtera.com/news press.htm#081505">http://www.luxtera.com/news press.htm#081505</a>.
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    [CrossRef] [PubMed]
  43. A. Liu, R. Jones, L. Liao, D. Samara-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, 615-618 (2004).
    [CrossRef] [PubMed]
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  45. B.-S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nature Materials 4, 207-210 (2005).
    [CrossRef]

Appl. Phys. Lett. (6)

T. Tanabe, M. Notomi, S. Mitsugi, A. Shinya, and E. Kuramochi, "All-optical switches on a silicon chip realized using photonic-crystal nanocavities," Appl. Phys. Lett. 87, 151112-1-151112-3 (2005).
[CrossRef]

S. L. McCall, "Instability and regenerative pulsation phenomena in Fabry-Perot nonlinear optic media devices," Appl. Phys. Lett. 32, 284-286 (1978).
[CrossRef]

J. L. Jewell, H. M. Gibbs, S. S. Tarng, A. C. Gossard, and W. Weigmann, "Regenerative pulsations from an intrinsic bistable device," Appl. Phys. Lett. 40, 291-293 (1982).
[CrossRef]

R. K. Jain and D. G. Steel, "Degenerate four-wave mixing of 10.6µm radiation in Hg1-xCdxTe," Appl. Phys. Lett. 37, 1-3 (1980).
[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]

M. Dinu, F. Quochi, and H. Garcia, "Third-order nonlinearities in silicon at telecom wavelengths," Appl. Phys. Lett. 82, 2954-2956 (2003).
[CrossRef]

Electron. Lett. (1)

G. Cocorullo and I.Rendina, "Thermo-optical modulation at 1.5µm in silicon etalon," Electron. Lett. 28, 83-85 (1992).
[CrossRef]

Frontiers in Optics 2005 (1)

T. J. Johnson, M. Borselli, and O. Painter, "Self-generated optical modulation in a high-Q SOI microdisk resonator," In Frontiers in Optics 2005/Laser Science XXI, (OSA, Washington, DC, 2005).

IEEE J. Quantum Electron. (2)

R. A. Soref and J. P. Lorenzo, "All-Silicon Active and Passive Guided-Wave Components for ?=1.3 and 1.6µm," IEEE J. Quantum Electron. 22, 873-879 (1986).
[CrossRef]

R. A. Soref and B. R. Bennett, "Electrooptical Effects in Silicon," IEEE J. Quantum Electron. 23, 123-129 (1987).
[CrossRef]

IEEE Trans. Electron. Dev. (1)

D. K. Schroder, R. N. Thomas, and J. C. Swartz, "Free Carrier Absorption in Silicon," IEEE Trans. Electron. Dev. 25, 254-261 (1978).
[CrossRef]

IPR and NIS on CD-ROM 2005 (1)

M. Borselli, T. J. Johnson, and O. Painter, "Nonlinear Optics in High-Q SOI Optical Microcavities," In Integrated Photonics Research and Applications/Nanophotonics for Information Systems Topical Meetings on CD-ROM, (OSA, Washington, DC, 2005).
[PubMed]

J. Microelectromec. Syst. (1)

K. Aubin, M. Zalalutdinov, T. Alan, R. Reichenbach, R. Rand, A. Zehnder, J. Parpia, and H. Craighead, "Limit Cycle Oscillations in CW Laser Driven NEMS," J. Microelectromec. Syst. 13, 1018-1026 (2004).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. (1)

H. M. Gibbs, J. L. Jewell, J. V. Moloney, K. Tai, S. Tarng, D. A. Weinberger, A. C. Gossard, S. L. McCall, A. Passner, and W. Weigmann, "Optical Bistability, Regenerative Pulsations, and Transverse Effects in Room-Temperature GaAs-AlGaAs Superlattice etalons," J. Phys. (Paris) 44, 195-204 (1983).
[CrossRef]

Laser Phys. (1)

M. Gorodetsky and V. Ilchenko, "Thermal Nonlinear Effects in Optical Whispering Gallery Microresonators," Laser Phys. 2, 1004-1009 (1992).

Nature (4)

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, "Micrometre-scale silicon electro-optic modulator," Nature 435, 325-327 (2005).
[CrossRef] [PubMed]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431, 1081-1084 (2004).
[CrossRef] [PubMed]

A. Liu, R. Jones, L. Liao, D. Samara-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, 615-618 (2004).
[CrossRef] [PubMed]

H. Rong, R. Jones, A. Liu, O. Cohen, D. Hak, A. Fang, and M. Paniccia, "A continuous-wave Raman silicon laser," Nature 433, 725-728 (2005).
[CrossRef] [PubMed]

Nature (London) (1)

K. J. Vahala, "Optical Microcavities," Nature (London) 424, 839-846 (2003).
[CrossRef]

Nature Materials (1)

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nature Materials 4, 207-210 (2005).
[CrossRef]

Opt. Express (6)

R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, "Influence of nonlinear absorption on Raman amplification in silicon waveguides," Opt. Express 12, 2774-2780 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2774">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2774</a>.
[CrossRef] [PubMed]

T. Carmon, L. Yang, and K. J. Vahala, "Dynamical thermal behavior and thermal self-stability of microcavities," Opt. Express 12, 4742-4750 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4742">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4742</a>.
[CrossRef] [PubMed]

M. Borselli, T. J. Johnson, and O. Painter, "Beyond the Rayleigh scattering limit in high-Q silicon microdisks: theory and experiment," Opt. Express 13, 1515-1530 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-5-1515">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-5-1515</a>.
[CrossRef] [PubMed]

G. Priem, P. Dumon, W. Bogaerts, D. V. Thourhout, G. Morthier, and R. Baets, "Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures," Opt. Express 13, 9623-9628 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-23-9623">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-23-9623</a>.
[CrossRef] [PubMed]

P. E. Barclay, K. Srinivasan, and O. Painter, "Nonlinear response of silicon photonic crystal microresonators excited via an integrated waveguide and a fiber taper," Opt. Express 13, 801-820 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-801">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-3-801</a>.
[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), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2678">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2678</a>.
[CrossRef] [PubMed]

Opt. Lett. (5)

Phys. Rev. B (1)

K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, "Optical-fiber-based measurement of an ultrasmall volume, high-Q photonic crystal microcavity," Phys. Rev. B 70, 081306(R) (2004).
[CrossRef]

Phys. Rev. E (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Perturbation theory for Maxwell's equations with shifting material boundaries," Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Other (9)

A variable order Adams-Bashforth-Moulton predictor-corrector method was used.

Handbook of optical constants of solids, E. Palick, ed., (Academic Press, Boston, MA, 1985).

S. Sze, Physics of Semiconductor Devices, 2nd ed. (John Wiley and Sons, New York, New York, 1981).

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd ed. (Springer-Verlag New York, New York, NY, 2003).

H. M. Gibbs, Optical Bistability:Controlling Light with Light (Academic Press, San Diego, CA, 1985).

Note that the confinement factor and effective mode volume for the two standing-wave modes are identical, hence we drop the c/s subscript.

For TPA with the standing wave modes one has an additional term dependent upon the product UcUs, with cross-confinement factor ? c/s,TPA and cross-mode volume 3Vc/s,TPA pre-factors. For FCA, described below, one cannot write the total absorption just in terms of products of powers of the cavity energies, but rather the mode amplitudes themselves must be explicitly used.

Here we assume that the coupling to each of the standing-wave modes is identical. In general, the coupling can be different, although experimentally we have noticed only small differences in coupling.

Luxtera, <a href="http://www.luxtera.com/news press.htm#081505">http://www.luxtera.com/news press.htm#081505</a>.

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Figures (5)

Fig. 1.
Fig. 1.

(a) Scanning electron microscope image of Si microdisk under study. (b) Optical image (top-view) of microdisk with side-coupled optical fiber taper waveguide.

Fig. 2.
Fig. 2.

(a) Normalized optical transmission spectrum of a silicon microdisk WGM resonance at 0.5,μW input power (i) and 35μW input power (ii). (b) Normalized transmission with 480,μW input power. (inset) Power spectrum of transmission at input wavelength λl = 1454.56 nm, indicated by a green star in (b).

Fig. 3.
Fig. 3.

(a) Time-domain behavior of the transmitted optical power. (b) Dependence of the time-domain behavior upon input laser wavelength. (c) Detail of the transmission oscillation (boxed region in (a)).

Fig. 4.
Fig. 4.

Picture of the various physical processes involved in the nonlinear model of the Si microdisk considered here. (a) A scanning electron micrography of a representative SOI microdisk resonator. As discussed below, heat flows by conduction through the SiO2 pedestal. (b) Square-magnitude of the electric field for the WGM under consideration as calculated by finite-element method. High intensity fields are found in the red regions. High field strengths in the silicon disk (the white box delineates the disk) generate free-carriers via two-photon absorption (TPA). (c) Schematic depiction of dominant processes in the Si microdisk: TPA, TPA-generated free-carrier density (e-, h+ denoting electrons and holes, respectively), free-carrier absorption, and surface-state absorption. (d) Schematic of the dispersive effects of heat and free-carriers on the WGM resonance wavelength.

Fig. 5.
Fig. 5.

Comparison between model and measurement. The shaded regions (i), (ii), (iii), and (iv) correspond to different phases of the dynamics as described in the text. (a) Comparison of the modeled and measured time-dependent normalized optical transmission. (b) Normalized excursion of the modeled optical cavity energy, free-carrier density, and differential microdisk temperature. The normalization for a function f(t) is calculated as f ( t ) min ( f ( t ) ) range ( f ( t ) ) . . The differential temperature covers the range ΔT = 1.9 - 2.4K, the free-carrier density covers the range N = 1 × 1014 - 0.9 × 1017cm-3, and the stored optical cavity energy ranges from U = 0.8 - 29fJ (c) Resonance frequency shift (in units of γβ), broken into thermal and free-carrier contributions. The dashed line indicates the pump laser wavelength (λ l ). The insets within each of (a), (b), and (c) corresponds to a narrow time-slice about the transmission dip.

Tables (1)

Tables Icon

Table 1. Parameters used in the Si microdisk model.

Equations (27)

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d a c dt = ( γ c 2 + i ( Δ ω 0 + γ β 2 ) ) a c + κ s
d a s dt = ( γ s 2 + i ( Δ ω 0 γ β 2 ) ) a s + κ s ,
U s = a s 2
U c = a c 2 .
γ c / s = γ c / s , 0 e + j > 0 γ c / s , j e + γ c / s , rad + γ c / s , lin + γ ¯ c / s , TPA + γ ¯ c / s , FCA .
γ ¯ c / s , TPA ( t ) = Γ TPA β Si c 2 V TPA n g 2 U c / s ( t )
Γ TPA = Si n 4 ( r ) E c / s ( r ) 4 d r n 4 ( r ) E c / s ( r ) 4 d r ,
V TPA = ( n 2 ( r ) E c / s ( r ) 2 d r ) 2 n 4 ( r ) E c / s ( r ) 4 d r .
P abs , TPA ( t ) = Γ TPA β Si c 2 V TPA n g 2 ( U c 2 ( t ) + U s 2 ( t ) ) .
γ c / s , FCA ( r , t ) = σ Si ( r ) c n g N ( r , t ) ,
γ ¯ c / s , FCA ( t ) = σ Si c n g N ( t ) ¯ ,
N ( t ) ¯ = N ( r , t ) n ( r ) 2 | E ( r ) | 2 d r n ( r ) 2 | E ( r ) | 2 d r .
P abs ( t ) = ( γ c , lin + γ ¯ c , TPA + γ ¯ c , FCA ) U c ( t ) + ( γ s , lin + γ ¯ s , TPA + γ ¯ s , FCA ) U s ( t ) .
Δ ω 0 ( t ) ω 0 = ( Δ n ( r , t ) n ( r ) ) ¯ ,
( Δ n ( t ) n ) ¯ = ( Δ n ( r , t ) n ( r ) ) n 2 ( r ) E ( r ) 2 d r n 2 ( r ) E ( r ) 2 d r .
Δ n ( r , t ) n ( r ) = 1 n ( r ) dn ( r ) dT Δ T ( r , t ) + 1 n ( r ) dn ( r ) dN N ( r , t ) .
Δ ω 0 ( t ) ω 0 = ( 1 n Si dn Si dT Δ T ( t ) ¯ + 1 n Si dn Si dN N ( t ) ¯ ) ,
Δ T ( t ) ¯ = Δ T ( r , t ) n 2 ( r ) E ( r ) 2 d r n 2 ( r ) E ( r ) 2 d r .
d Δ T ( t ) ¯ dt = γ th Δ T ( t ) ¯ + Γ disk ρ Si c p , Si V disk P abs ( t ) .
Γ disk = Si n ( r ) 2 E ( r ) 2 d r n ( r ) 2 E ( r ) 2 d r ,
γ th = k ρ Si c p , Si V disk ,
N ( r , t ) t = γ ( r ) N ( r , t ) + · ( D ( r ) N ( r , t ) ) + G ( r , t ) .
d N ( t ) ¯ dt = γ ( r ) N ( r , t ) ¯ + · ( D ( r ) N ( r , t ) ) ¯ + G ( t ) ¯ .
G ( r , t ) ¯ = Γ FCA β Si c 2 2 h ̅ ω p n g 2 V FCA 2 ( U c ( t ) 2 + U s ( t ) 2 ) ,
Γ FCA = Si n 6 ( r ) E c / s ( r ) 6 d r n 6 ( r ) E c / s ( r ) 6 d r ,
V FCA 2 = ( n 2 ( r ) E c / s ( r ) 2 d r ) 3 n 6 ( r ) E c / s ( r ) 6 d r .
d N ( t ) ¯ dt = γ′ fc N ( t ) ¯ + Γ FCA β Si c 2 2 h ¯ ω p n g 2 V FCA 2 + ( U c ( t ) 2 + U s ( t ) 2 ) .

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