Abstract

A technique is demonstrated which efficiently transfers light between a tapered standard single-mode optical fiber and a high-Q, ultra-small mode volume, silicon photonic crystal resonant cavity. Cavity mode quality factors of 4.7×104 are measured, and a total fiber-to-cavity coupling efficiency of 44% is demonstrated. Using this efficient cavity input and output channel, the steady-state nonlinear absorption and dispersion of the photonic crystal cavity is studied. Optical bistability is observed for fiber input powers as low as 250 µW, corresponding to a dropped power of 100 µW and 3 fJ of stored cavity energy. A high-density effective free-carrier lifetime for these silicon photonic crystal resonators of ~0.5 ns is also estimated from power dependent loss and dispersion measurements.

© 2005 Optical Society of America

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References

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  1. K. Srinivasan, P. Barclay, O. Painter, J. Chen, A. X. Cho, and C. Gmachl, �??Experimental demonstration of a high quality factor photonic crystal microcavity,�?? Appl. Phys. Lett. 83, 1915�??1917 (2003).
    [CrossRef]
  2. 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]
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    [CrossRef]
  4. B. Lev, K. Srinivasan, P. E. Barclay, O. Painter, and H. Mabuchi, �??Feasibility of detecting single atoms using photonic bandgap cavities,�?? Nanotechnology 15, S556�??S561 (2004).
    [CrossRef]
  5. T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, O. Shchekin, and D. Deppe, �??Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,�?? Nature 432, 200�??203 (2004).
    [CrossRef] [PubMed]
  6. J. Reithmaier, G. Sek, A. Loffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. Keldysh, V. Kulakovskii, and T. Reinecke, �??Strong coupling in a single quantum dot-semiconductor microcavity system,�?? Nature 432, 197�??200 (2004).
    [CrossRef] [PubMed]
  7. E. Peter, P. Senellart, D. Martrou, A. Lemaitre, and J. Bloch, �??Exciton photon strong-coupling regime for a single quantum dot in a microcavity,�?? <a href=" http://arxiv.org/quant-ph/0411076 (2004)">http://arxiv.org/quant-ph/0411076 </a>
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    [CrossRef]
  10. J.-M. Gerard, �??Solid-state cavity-quantum electrodynamics with self-assembled quantum dots�?? in Single Quantum Dots: Fundamentals, Applications, and New Concepts, P. Michler ed. (Springer-Verlag, Germany, 2003), 269�??314.
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    [CrossRef]
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    [CrossRef]
  24. K. Srinivasan and O. Painter, �??Momentum Space Design of High-Q Photonic Crystal Nanocavities in Two-Dimensional Slab Waveguides,�?? Opt. Express 10, 670�??684 (2002), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670</a>
    [PubMed]
  25. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, �??Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,�?? Phys. Rev. Lett. 91, 043902 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  31. G. P. Agrawal and N. K. Dutta, Semiconductor Lasers (Van Nostrand Reinhold, New York, NY, 1993).
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    [CrossRef]
  33. R. A. Soref and B. R. Bennett, �??Electrooptical Effects in Silicon,�?? IEEE J. Quan. Elec. 23, 123�??129 (1987).
    [CrossRef]
  34. In using this approximate theory, in which regions of high two-photon absorbed power are correlated with high steady-state carrier density, we better approximate the cavity �??volume�?? of interest, and consequently the effective free-carrier lifetime better represents the average time a free-carrier stays in the region of the PC cavity mode.
  35. 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]
  36. V. R. Almeida and M. Lipson, �??Optical bistability on a silicon chip,�?? Opt. Lett. 29, 2387�??2389 (2004).
    [CrossRef] [PubMed]
  37. 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]
  38. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, San Diego, CA, 2003).
  39. 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]
  40. H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, �??Raman gain and nonlinear optical absorption measurements in a low-loss silicon waveguide,�?? Appl. Phys. Lett. 85, 2196�??2198 (2004).
    [CrossRef]
  41. P. E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, �??Probing the dispersive and spatial properties of planar photonic crystal waveguide modes via highly efficient coupling from optical fiber tapers,�?? Appl. Phys. Lett. 85, 4�??6 (2004).
    [CrossRef]
  42. H. M. Gibbs, Optical bistability: controlling light with light (Academic Press, Orlando, FL, 1985).
  43. Note that the sharp transition edge associated with optical bistability occurs at the cavity resonance wavelength when scanning from blue to red.
  44. 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]
  45. P.-T. Lee, J. R. Cao, S.-J. Choi, Z.-J.Wei, J. O�??Brien, and P. D. Dapkus, �??Operation of photonic crystal membrane lasers above room temperature,�?? Appl. Phys. Lett. 81, 3311�??3313 (2002).
    [CrossRef]
  46. A. Cutolo, M. Iodice, P. Spirito, and L. Zeni, �??Silicon Electro-Optic Modulator Based on a Three Terminal Device Integrated in a Low-Loss Single-Mode SOI Waveguide,�?? J. Lightwave Technol. 15, 505�??518 (1997).
    [CrossRef]
  47. G. Cocorullo and I. Rendina, �??Thermo-optical modulation at 1.5 m in silicon etalon,�?? IEE Electron Lett. 28, 83�??85 (1992).
    [CrossRef]
  48. M. Pelton, C. Santori, J. Vuckovic, B. Zhang, G. Solomon, J. Plant, and Y. Yamamoto, �??An Efficient Source of Single Photons: A Single Quantum Dot in a Micropost Microcavity,�?? Phys. Rev. Lett. 89, 299602 (2002).

Appl. Phys. Lett. (9)

K. Srinivasan, P. Barclay, O. Painter, J. Chen, A. X. Cho, and C. Gmachl, �??Experimental demonstration of a high quality factor photonic crystal microcavity,�?? Appl. Phys. Lett. 83, 1915�??1917 (2003).
[CrossRef]

M. F. Yanik, S. Fan, and M. Solja�?i�?, �??High-contrast all-optical bistable switching in photonic crystal microcavities,�?? Appl. Phys. Lett. 83, 2739�??2781 (2003).
[CrossRef]

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

T. Liang and H. Tsang, �??Role of free carriers from two photon absorption in Raman amplification in silion-oninsulator waveguides,�?? Appl. Phys. Lett. 84, 2745�??2757 (2004).
[CrossRef]

H. Rong, A. Liu, R. Nicolaescu, M. Paniccia, O. Cohen, and D. Hak, �??Raman gain and nonlinear optical absorption measurements in a low-loss silicon waveguide,�?? Appl. Phys. Lett. 85, 2196�??2198 (2004).
[CrossRef]

P. E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, �??Probing the dispersive and spatial properties of planar photonic crystal waveguide modes via highly efficient coupling from optical fiber tapers,�?? Appl. Phys. Lett. 85, 4�??6 (2004).
[CrossRef]

M. Lon�?ar, D. Nedeljkovi�?, T. Doll, J. Vu¡�?kovi�?, A. Scherer, and T. P. Pearsall, �??Waveguiding in planar photonic crystals,�?? Appl. Phys. Lett. 77, 1937�??1939 (2000).
[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]

P.-T. Lee, J. R. Cao, S.-J. Choi, Z.-J.Wei, J. O�??Brien, and P. D. Dapkus, �??Operation of photonic crystal membrane lasers above room temperature,�?? Appl. Phys. Lett. 81, 3311�??3313 (2002).
[CrossRef]

IEE Electron Lett. (1)

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

IEEE J. Quan. Elec. (1)

R. A. Soref and B. R. Bennett, �??Electrooptical Effects in Silicon,�?? IEEE J. Quan. Elec. 23, 123�??129 (1987).
[CrossRef]

IEICE Trans. Electron. (1)

K. Kanamoto, S. Lan, N. Ikeda, Y. Tanaka, Y. Sugimoto, K. Asakawa, and H. Ishikawa, �??Single Photonic-Crystal Defect Switch for All-Optical Ultrafast Operation Using Two Photon Absorption,�?? IEICE Trans. Electron. E87-C, 1142�??1146 (2004).

J. Lightwave Technol. (1)

A. Cutolo, M. Iodice, P. Spirito, and L. Zeni, �??Silicon Electro-Optic Modulator Based on a Three Terminal Device Integrated in a Low-Loss Single-Mode SOI Waveguide,�?? J. Lightwave Technol. 15, 505�??518 (1997).
[CrossRef]

J. Opt. Soc. AM. B (1)

Nanotechnology (1)

B. Lev, K. Srinivasan, P. E. Barclay, O. Painter, and H. Mabuchi, �??Feasibility of detecting single atoms using photonic bandgap cavities,�?? Nanotechnology 15, S556�??S561 (2004).
[CrossRef]

Nature (5)

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, O. Shchekin, and D. Deppe, �??Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,�?? Nature 432, 200�??203 (2004).
[CrossRef] [PubMed]

J. Reithmaier, G. Sek, A. Loffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. Keldysh, V. Kulakovskii, and T. Reinecke, �??Strong coupling in a single quantum dot-semiconductor microcavity system,�?? Nature 432, 197�??200 (2004).
[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]

E. Knill, R. Laflamme, and G. Millburn, �??A scheme for efficient quantum computation with linear optics,�?? Nature 409, 46�??52 (2001).
[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]

Opt. Express (7)

K. Srinivasan and O. Painter, �??Momentum Space Design of High-Q Photonic Crystal Nanocavities in Two-Dimensional Slab Waveguides,�?? Opt. Express 10, 670�??684 (2002), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670</a>
[PubMed]

S. J. McNab, N. Moll, and Y. A. Vlasov, �??Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides,�?? Opt. Express 11, 2927�??2939 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2927">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2927</a>
[CrossRef] [PubMed]

M. Notomi, A. Shinya, S. Mitsugi, and H.-Y. Ryu, �??Waveguides, resonators and their coupled elements in photonic crystal slabs,�?? Opt. Express 12, 1551�??1561 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1551">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1551</a>
[CrossRef] [PubMed]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. V. Campenhout, P. Bienstman, R. Baets, V. Wiaux, and S. Beckx, �??Basic structures for photonic integrated circuits in silicon-on-insulator,�?? Opt. Express 12, 1583�??1591 (2004) <a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1583">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1583</a>
[CrossRef] [PubMed]

A. R. Cowan, G. W. Rieger, and J. F. Young, �??Nonlinear transmission of 1.5 m pulses through single-mode silicon-on-insulator waveguide structures,�?? Opt. Expr. 12, 1611�??1621 (2004), <a href "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1611">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1611</a>
[CrossRef]

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]

Opt. Lett. (4)

Phys. Rev. (2)

A. R. Cowan and J. F. Young, �??Optical bistability involving photonic crystal microcavities and Fano line shapes,�?? Phys. Rev. E 68, 046606 (2003).
[CrossRef]

M. Solja�?i�?, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, �??Optimal bistable switching in nonlinear photonic crystals,�?? Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Phys. Rev. A (1)

A. Kiraz, M. Atatüre, and A. Imamo�?lu, �??Quantum-dot single-photon sources: Prospects for applications in linear optics quantum optics processing,�?? Phys. Rev. A 69, 032305 (2004).
[CrossRef]

Phys. Rev. B (1)

H. W. Tan, H. M. van Driel, S. L. Schweizer, R. B. Wehrspohn, and U. Gösele, �??Nonlinear optical tuning of a two-dimensional silicon photonic crystal,�?? Phys. Rev. B 70, 205110 (2004).
[CrossRef]

Phys. Rev. B. (1)

K. Srinivasan, P. E. Barclay, M. Borselli, and O. Painter, �??Optical-fiber based measurement of an ultra-small 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]

Phys. Rev. Lett. (4)

G. Brassard, N. Lutkenhaus, T. Mor, and B. Sanders, �??Limitations on practical quantum cryptography,�?? Phys. Rev. Lett. 85, 1330�??1333 (2000).
[CrossRef] [PubMed]

M. Pelton, C. Santori, J. Vuckovic, B. Zhang, G. Solomon, J. Plant, and Y. Yamamoto, �??An Efficient Source of Single Photons: A Single Quantum Dot in a Micropost Microcavity,�?? Phys. Rev. Lett. 89, 299602 (2002).

S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, �??Ideality in a Fiber-Taper-Coupled Microresonator System for Application to Cavity Quantum Electrodynamics,�?? Phys. Rev. Lett. 91, 043902 (2003).
[CrossRef] [PubMed]

M. Pelton, C. Santori, J. Vuckovic, B. Zhang, G. Solomon, J. Plant, and Y. Yamamoto, �??An Efficient Source of Single Photons: A Single Quantum Dot in a Micropost Microcavity,�?? Phys. Rev. Lett. 89, 299602 (2002).
[CrossRef]

Other (7)

G. P. Agrawal and N. K. Dutta, Semiconductor Lasers (Van Nostrand Reinhold, New York, NY, 1993).

J.-M. Gerard, �??Solid-state cavity-quantum electrodynamics with self-assembled quantum dots�?? in Single Quantum Dots: Fundamentals, Applications, and New Concepts, P. Michler ed. (Springer-Verlag, Germany, 2003), 269�??314.

E. Peter, P. Senellart, D. Martrou, A. Lemaitre, and J. Bloch, �??Exciton photon strong-coupling regime for a single quantum dot in a microcavity,�?? <a href=" http://arxiv.org/quant-ph/0411076 (2004)">http://arxiv.org/quant-ph/0411076 </a>

W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, San Diego, CA, 2003).

In using this approximate theory, in which regions of high two-photon absorbed power are correlated with high steady-state carrier density, we better approximate the cavity �??volume�?? of interest, and consequently the effective free-carrier lifetime better represents the average time a free-carrier stays in the region of the PC cavity mode.

H. M. Gibbs, Optical bistability: controlling light with light (Academic Press, Orlando, FL, 1985).

Note that the sharp transition edge associated with optical bistability occurs at the cavity resonance wavelength when scanning from blue to red.

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Figures (8)

Fig. 1.
Fig. 1.

(a) Schematic of the fiber taper to PC cavity coupling scheme. The blue arrow represents the input light, some of which is coupled contradirectionally into the PCWG. The green arrow represents the light reflected by the PC cavity and recollected in the backwards propagating fiber mode. The red colored region represents the cavity mode and its radiation pattern. (b) Illustration of the fiber-PC cavity coupling process. The dashed line represents the “local” band-edge frequency of the photonic crystal along the waveguide axis. The step discontinuity in the bandedge at the PCWG - PC cavity interface is due to a jump in the longitudinal () lattice constant (see Fig. 2). The parabolic “potential” is a result of the longitudinal grade in hole radius of the PC cavity. The bandwidth of the waveguide is represented by the gray shaded area. Coupling between the cavity mode of interest (frequency ω0) and the mode matched PCWG mode (ωWG0) is represented by γ0e, coupling to radiating PCWG modes is represented by γje >0, and intrinsic cavity loss is represented by γi . (c,d) Magnetic field profile, calculated using FDTD, of the high-Q PC cavity A20 mode and the fundamental TE1 PCWG mode, respectively.

Fig. 2.
Fig. 2.

SEM image of an integrated PCWG-PC cavity sample. The PC cavity and PCWG have lattice constants Λ~430 nm, Λ x ~430 nm, and Λ z ~550 nm. The surrounding silicon material has been removed to form a diagonal trench and isolated mesa structure to enable fiber taper probing.

Fig. 3.
Fig. 3.

(a) Illustration of the device and fiber taper orientation for (i) efficient PCWG mediated taper probing of the cavity, and (ii) direct taper probing of the cavity. (b) Normalized depth of the transmission resonance (Δ) at λ o ~1589.7, as a function of lateral taper displacement relative to the center of the PC cavity, during direct taper probing (taper in orientation (ii)).

Fig. 4.
Fig. 4.

(a) Measured reflected taper signal as a function of input wavelength (taper diameter d~1 µm, taper height g=0.80 µm). The sharp dip at λ~1589.7 nm, highlighted in panel (b), corresponds to coupling to the A20 cavity mode. (c) Maximum reflected signal (slightly detuned from the A20 resonance line), and resonance reflection contrast as a function of taper height. The dashed line at ΔR=0.6 shows the PCWG-cavity drop efficiency, which is independent of the fiber taper position for g≥0.8 µm.

Fig. 5.
Fig. 5.

(a) Measured cavity response as a function of input wavelength, for varying PCWG power (taper diameter d~1 µm, taper height g=0.80 µm).

Fig. 6.
Fig. 6.

(a) Power dropped (Pd ) into the cavity as a function of power in the PCWG (Pi ). The dashed line shows the expected result in absence of nonlinear cavity loss. (b) Resonance wavelength shift as a function of internal cavity energy. Solid blue lines in both Figs. show simulated results.

Fig. 7.
Fig. 7.

(a) Simulated effective quality factors for the different PC cavity loss channels as a function of power dropped into the cavity. (b) Contributions from the modeled dispersive processes to the PC cavity resonance wavelength shift as a function of power dropped into the cavity. (Simulation parameters: ηlin~0.40, Γth dT/dP abs=27 K/mW, τ-1~0.0067+(1.4×10-7)N 0.94 where N has units of cm-3 and τ has units of ns.)

Fig. 8.
Fig. 8.

Dependance of free-carrier lifetime on free-carrier density (red dots) as found by fitting Δλo (Pi ) and Pd (Pi ) with the constant material and modal parameter values of Table 1, and for effective PC cavity thermal resistance of Γth dT/dP abs=27 K/mW and linear absorption fraction ηlin=0.40. The solid blue line corresponds to a smooth curve fit to the point-by-point least-squared fit data given by τ-1~0.0067+(1.4×10-7)N 0.94, where N is in units of cm-3 and τ is in ns.

Tables (1)

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Table 1. Fixed parameters used in the model.

Equations (41)

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K γ 0 e γ i + j 0 γ j e ,
I γ 0 e Σ j γ j e ,
R o ( ω o ) = ( 1 K ) 2 ( 1 + K ) 2 .
Q i + P = 2 Q T 1 1 ± R o ( ω o ) = Q T ( 1 + K ) ,
η 0 = γ 0 e γ i + j γ j e = 1 1 + 1 K .
Q T Q i = 1 K ( I ( 1 + K ) ) = 1 η 0 I .
U = ( 1 R o ( ω o ) ) Q i + P ω o P i = 4 K ( 1 + K ) 2 I K ( 1 I ) I Q i ω o P i
γ i ( U ) = γ rad + γ lin + γ ̅ TPA ( U ) + γ ̅ FCA ( U ) .
U = 4 K ( U ) ( 1 + K ( U ) ) 2 Q i + p ( U ) ω o P i ,
K ( U ) = γ o e γ i ( U ) + j > 0 γ j e ,
ω o Q i + P ( U ) = γ i ( U ) + j > 0 γ j e .
γ TPA ( r ) = β ' ( r ) 1 2 ε o n 2 ( r ) E 2 ( r ) ,
γ ̅ TPA = γ TPA ( r ) n 2 ( r ) E 2 ( r ) d r n 2 ( r ) E 2 ( r ) d r = β ' ̅ U V TPA ,
β ' ̅ = β ' ( r ) n 4 ( r ) E 4 ( r ) d r n 4 ( r ) E 4 ( r ) d r
V TPA = ( n 2 ( r ) E 2 ( r ) d r ) 2 n 4 ( r ) E 4 ( r ) d r .
γ ̅ TPA = Γ TPA β ' Si U V TPA
Γ TPA = Si n 4 ( r ) E 4 ( r ) d r n 4 ( r ) E 4 ( r ) d r ,
γ FCA = σ ' ( r ) N ( r ) ,
N ( r ) = τ p TPA ( r ) 2 h ̅ ω o ,
p TPA ( r ) = 1 2 ε o n 2 ( r ) E 2 ( r ) γ TPA ( r ) .
γ ̅ FCA = τ 2 h ̅ ω o ( σ ' ( r ) 1 2 ε o n 2 ( r ) E 2 ( r ) γ TPA ( r ) ) n 2 ( r ) E 2 ( r ) d r n 2 ( r ) E 2 ( r ) d r .
γ ̅ FCA = Γ FCA ( τ σ ' Si β ' Si 2 h ̅ ω o U 2 V FCA 2 ) ,
Γ FCA = Si n 6 ( r ) E 6 ( r ) d r n 6 ( r ) E 6 ( r ) d r
V FCA 2 = ( n 2 ( r ) E 2 ( r ) d r ) 3 n 6 ( r ) E 6 ( r ) d r .
Δ ω o ( U ) ω o = Δ n ̅ ( U ) ,
Δ n ̅ ( U ) = ( Δ n ( r ) n ( r ) ) n 2 ( r ) E 2 ( r ) d r n 2 ( r ) E 2 ( r ) d r .
R o ( ω ) = 1 4 K ( U ) ( 1 + K ( U ) ) 2 ( δ ω 2 ) 2 ( ω ω o Δ ω o ( U ) ) 2 + ( δ ω ( U ) 2 ) 2 .
U = P d γ i + P = ( 1 R o ( ω ) ) Q i + P ( U ) ω o P i ,
Δ n Kerr ( r ) = n ' 2 ( r ) 1 2 ε o n 2 ( r ) E 2 ( r ) ,
Δ n ̅ Kerr ( U ) = Γ Kerr n Si ( n ' 2 , Si U V Kerr ) ,
Γ Kerr = Γ TPA
V Kerr = V TPA .
Δ n FCD ( r ) = ζ ( r ) N ( r ) ,
Δ n ̅ FCD ( U ) = Γ FCD n Si ( τ ζ Si β ' Si 2 h ̅ ω o U 2 V FCD 2 ) ,
Γ FCD = Γ FCA
V FCD = V FCA .
Δ n ̅ th = ( 1 n ( r ) d n d T ( r ) Δ T ( r ) ) n 2 ( r ) E 2 ( r ) d r n 2 ( r ) E 2 ( r ) d r .
Δ n ̅ th ( U ) = Γ th n Si ( d n Si d T d T d P abs P abs ( U ) )
P abs ( U ) = ( γ lin + γ ̅ TPA ( U ) + γ ̅ FCA ( U 2 ) ) U .
Q i + P ( P i ) = K ( Δ R o ( P i ) ) Q T ( P i = 0 ) η 0 ( P i = 0 ) .
Δ n FCD , Si = [ ζ e , Si N e + ( ζ h , Si N h ) 0.8 ] .

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