Design of triply-resonant microphotonic parametric oscillators based on Kerr nonlinearity

Xiaoge Zeng; Miloš A. Popović

doi:10.1364/OE.22.015837

Design of triply-resonant microphotonic parametric oscillators based on Kerr nonlinearity

Xiaoge Zeng and Miloš A. Popović

Optics Express
Vol. 22,
Issue 13,
pp. 15837-15867
(2014)
•https://doi.org/10.1364/OE.22.015837

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Xiaoge Zeng and Miloš A. Popović, "Design of triply-resonant microphotonic parametric oscillators based on Kerr nonlinearity," Opt. Express 22, 15837-15867 (2014)

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Related Topics
Table of Contents Category
- Nonlinear Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Nonlinear (130.4310)
- Nonlinear optics, four-wave mixing (190.4380)
- Parametric oscillators and amplifiers (190.4970)

About this Article
History
- Original Manuscript: April 3, 2014
- Revised Manuscript: May 30, 2014
- Manuscript Accepted: May 30, 2014
- Published: June 20, 2014

Accessible

Open Access

Abstract

We propose optimal designs for triply-resonant optical parametric oscillators (OPOs) based on degenerate four-wave mixing (FWM) in microcavities. We show that optimal designs in general call for different external coupling to pump and signal/idler resonances. We provide a number of normalized performance metrics including threshold pump power and maximum achievable conversion efficiency for OPOs with and without two-photon (TPA) and free-carrier absorption (FCA). We find that the maximum achievable conversion efficiency is bound to an upper limit by nonlinear and free-carrier losses independent of pump power, while linear losses only increase the pump power required to achieve a certain conversion efficiency. The results of this work suggest unique advantages in on-chip implementations that allow explicit engineering of resonances, mode field overlaps, dispersion, and wavelength-and mode-selective coupling. We provide universal design curves that yield optimum designs, and give example designs of microring-resonator-based OPOs in silicon at the wavelengths 1.55 μm (with TPA) and 2.3 μm (no TPA) as well as in silicon nitride (Si₃N₄) at 1.55 μm. For typical microcavity quality factor of 10⁶, we show that the oscillation threshold in excitation bus can be well into the sub-mW regime for silicon microrings and a few mW for silicon nitride microrings. The conversion efficiency can be a few percent when pumped at 10 times of the threshold. Next, based on our results, we suggest a family of synthetic “photonic molecule”-like, coupled-cavity systems to implement optimum FWM, where structure design for control of resonant wavelengths can be separated from that of optimizing nonlinear conversion efficiency, and where furthermore pump, signal, and idler coupling to bus waveguides can be controlled independently, using interferometric cavity supermode coupling as an example. Finally, consideration of these complex geometries calls for a generalization of the nonlinear figure of merit (NFOM) as a metric for performance in nonlinear photonic systems, and shows different efficiencies for single and multi-cavity geometries, as well as for standing and traveling wave excitations.

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References

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M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast, wide-fsr microring-resonator filter design and realization with frequency-shift compensation,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2005).
M. A. Popović, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett. 31, 2571–2573 (2006).
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2013 (1)

E. Engin, D. Bonneau, C. M. Natarajan, A. S. Clark, M. Tanner, R. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, “Photon pair generation in a silicon micro-ring resonator with reverse bias enhancement,” Opt. Express 21, 27826–27834 (2013).
[Crossref]

2012 (3)

A. R. Motamedi, A. H. Nejadmalayeri, A. Khilo, F. X. Kärtner, and E. P. Ippen, “Ultrafast nonlinear optical studies of silicon nanowaveguides,” Opt. Express 20, 4085–4101 (2012).
[Crossref] [PubMed]

S. Azzini, D. Grassani, M. J. Strain, M. Sorel, L. Helt, J. Sipe, M. Liscidini, M. Galli, and D. Bajoni, “Ultra-low power generation of twin photons in a compact silicon ring resonator,” Opt. Express 20, 23100–23107 (2012).
[Crossref] [PubMed]

M. Davanço, J. R. Ong, A. B. Shehata, A. Tosi, I. Agha, S. Assefa, F. Xia, W. M. Green, S. Mookherjea, and K. Srinivasan, “Telecommunications-band heralded single photons from a silicon nanophotonic chip,” Appl. Phys. Lett. 100, 261104 (2012).
[Crossref]

2011 (4)

D. M. Ramirez, A. W. Rodriguez, H. Hashemi, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Degenerate four-wave mixing in triply resonant kerr cavities,” Phys. Rev. A 83, 033834 (2011).
[Crossref]

D. A. Fishman, C. M. Cirloganu, S. Webster, L. A. Padilha, M. Monroe, D. J. Hagan, and E. W. Van Stryland, “Sensitive mid-infrared detection in wide-bandgap semiconductors using extreme non-degenerate two-photon absorption,” Nat. Photonics 5, 561–565 (2011).
[Crossref]

M. A. Foster, J. S. Levy, O. Kuzucu, K. Saha, M. Lipson, and A. L. Gaeta, “Silicon-based monolithic optical frequency comb source,” Opt. Express 19, 14233–14239 (2011).
[Crossref] [PubMed]

B. Kuyken, H. Ji, S. Clemmen, S. Selvaraja, H. Hu, M. Pu, M. Galili, P. Jeppesen, G. Morthier, S. Massar, L. K. Oxenløwe, G. Roelkens, and R. Baets, “Nonlinear properties of and nonlinear processing in hydrogenated amorphous silicon waveguides,” Opt. Express 19, B146–B153 (2011).
[Crossref]

2010 (1)

A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, “Ultrashort free-carrier lifetime in low-loss silicon nanowaveguides,” Opt. Express 18, 3582–3591 (2010).
[Crossref] [PubMed]

2009 (3)

H. Hashemi, A. W. Rodriguez, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Nonlinear harmonic generation and devices in doubly resonant kerr cavities,” Phys. Rev. A 79, 013812 (2009).
[Crossref]

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “Cmos-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2009).
[Crossref]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, “Cmos-compatible multiple-wavelength oscillator for on-chip optical interconnects,” Nat. Photonics 4, 37–40 (2009).
[Crossref]

2008 (3)

A. C. Turner, M. A. Foster, A. L. Gaeta, and M. Lipson, “Ultra-low power parametric frequency conversion in a silicon microring resonator,” Opt. Express 16, 4881–4887 (2008).
[Crossref] [PubMed]

W. D. Sacher and J. K. S. Poon, “Dynamics of microring resonator modulators,” Opt. Express 16, 15741–15753 (2008).
[Crossref] [PubMed]

2007 (4)

P. Dell’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007).
[Crossref]

Q. Lin, J. Zhang, G. Piredda, R. Boyd, P. Fauchet, and G. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 021111 (2007).
[Crossref]

A. Rodriguez, M. Soljačić, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doublyresonant cavities,” Opt. Express 15, 7303–7318 (2007).
[Crossref] [PubMed]

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007).
[Crossref] [PubMed]

2006 (4)

F. Ö. Ilday and F. X. Kärtner, “Cavity-enhanced optical parametric chirped-pulse amplification,” Opt. Lett. 31, 637–639 (2006).
[Crossref] [PubMed]

M. A. Popović, T. Barwicz, M. R. Watts, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “Multistage high-order microring-resonator add-drop filters,” Opt. Lett. 31, 2571–2573 (2006).
[Crossref]

C. Diederichs, J. Tignon, G. Dasbach, C. Ciuti, A. Lemaitre, J. Bloch, P. Roussignol, and C. Delalande, “Parametric oscillation in vertical triple microcavities,” Nature 440, 904–907 (2006).
[Crossref] [PubMed]

2005 (2)

D. Dimitropoulos, R. Jhaveri, R. Claps, J. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[Crossref]

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]

2004 (2)

T. Kippenberg, S. Spillane, and K. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-q toroid microcavity,” Phys. Rev. Lett. 93, 83904 (2004).
[Crossref]

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93, 233903 (2004).
[Crossref] [PubMed]

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[Crossref]

Adibi, A.

A. H. Atabaki and A. Adibi, “Ultra-compact coupled-resonator device for four-wave-mixing applications,” in “CLEO: Science and Innovations,” (Optical Society of America, 2011), p. CTuS6.

Agha, I.

Agrawal, G.

Q. Lin, J. Zhang, G. Piredda, R. Boyd, P. Fauchet, and G. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 021111 (2007).
[Crossref]

Agrawal, G. P.

Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: modeling and applications,” Opt. Express 15, 16604–16644 (2007).
[Crossref] [PubMed]

Arcizet, O.

Assefa, S.

Atabaki, A. H.

A. H. Atabaki and A. Adibi, “Ultra-compact coupled-resonator device for four-wave-mixing applications,” in “CLEO: Science and Innovations,” (Optical Society of America, 2011), p. CTuS6.

Azzini, S.

Baets, R.

Bajoni, D.

Barwicz, T.

M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast and wide-FSR microring-resonator filter design and realization with frequency-shift compensation,” in “Technical Digest of the Optical Fiber Communication Conference,” (Anaheim, CA, 2005). Paper OFK1.

M. A. Popović, T. Barwicz, P. T. Rakich, M. S. Dahlem, C. W. Holzwarth, F. Gan, L. Socci, M. R. Watts, H. I. Smith, F. X. Kärtner, and E. P. Ippen, “Experimental demonstration of loop-coupled microring resonators for optimally sharp optical filters,” in “Conference on Lasers and Electro-Optics,” (OSA, 2008).

M. A. Popović, T. Barwicz, F. Gan, M. S. Dahlem, C. W. Holzwarth, P. T. Rakich, H. I. Smith, E. P. Ippen, and F. X. Kärtner, “Transparent wavelength switching of resonant filters,” in “Conference on Lasers and Electro-Optics,” (Optical Society of America, 2007).

M. A. Popović, M. R. Watts, T. Barwicz, P. T. Rakich, L. Socci, E. P. Ippen, F. X. Kärtner, and H. I. Smith, “High-index-contrast, wide-fsr microring-resonator filter design and realization with frequency-shift compensation,” in “Optical Fiber Communication Conference,” (Optical Society of America, 2005).

Bloch, J.

Bonneau, D.

Boyd, R.

Q. Lin, J. Zhang, G. Piredda, R. Boyd, P. Fauchet, and G. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 021111 (2007).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear optics, 3rd ed. (Academic Press, 2008).

Chu, S.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “Cmos-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2009).
[Crossref]

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
[Crossref]

Cirloganu, C. M.

Ciuti, C.

Claps, R.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[Crossref]

Clark, A. S.

Clemmen, S.

Cohen, O.

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]

Dahlem, M. S.

M. S. Dahlem, C. W. Holzwarth, H. I. Smith, E. P. Ippen, and M. A. Popović, “Dynamical slow light cell based on controlled far-field interference of microring resonators,” in “Integrated Photonics Research, Silicon and Nanophotonics,” (OSA, 2010).
[Crossref]

Dasbach, G.

Dašic, M.

M. Dašić and M. A. Popović, “Minimum drop-loss design of microphotonic microring-resonator channel add-drop filters,” in “Telecommunications Forum (TELFOR), 2012 20th,” (IEEE, 2012), pp. 927–930.
[Crossref]

Davanço, M.

Delalande, C.

Dell’Haye, P.

Diederichs, C.

Dimitropoulos, D.

D. Dimitropoulos, R. Jhaveri, R. Claps, J. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
[Crossref]

Dorenbos, S. N.

Duchesne, D.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “Cmos-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2009).
[Crossref]

Engin, E.

Ezaki, M.

Fan, S.

Fang, A.

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]

Fauchet, P.

Q. Lin, J. Zhang, G. Piredda, R. Boyd, P. Fauchet, and G. Agrawal, “Dispersion of silicon nonlinearities in the near infrared region,” Appl. Phys. Lett. 91, 021111 (2007).
[Crossref]

Ferrera, M.

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Appl. Phys. Lett. (3)

D. Dimitropoulos, R. Jhaveri, R. Claps, J. Woo, and B. Jalali, “Lifetime of photogenerated carriers in silicon-on-insulator rib waveguides,” Appl. Phys. Lett. 86, 071115 (2005).
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J. Lightwave Technol. (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15, 998–1005 (1997).
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Nat. Photonics (3)

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. Little, and D. Moss, “Cmos-compatible integrated optical hyper-parametric oscillator,” Nat. Photonics 4, 41–45 (2009).
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Nature (3)

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).
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F. Ö. Ilday and F. X. Kärtner, “Cavity-enhanced optical parametric chirped-pulse amplification,” Opt. Lett. 31, 637–639 (2006).
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Phys. Rev. A (2)

Phys. Rev. Lett. (2)

T. Kippenberg, S. Spillane, and K. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-q toroid microcavity,” Phys. Rev. Lett. 93, 83904 (2004).
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Fig. 1 (a) Illustration of the micro-OPO model including a multimode resonator; (b) a traveling-wave resonant structure enables separated input and output ports; (c) example proposed multimode resonator based on 3 coupled microring cavities, showing an approach to unequal pump and signal/idler external coupling [22].

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Fig. 2 Normalized design curves for optimum OPO (using a “partial-TPA” model with pump-assisted TPA terms only and no FCA included): (a) maximum pump-to-signal/idler conversion efficiency versus pump power (normalized by oscillation threshold when loss due to TPA is ignored) and nonlinear loss sine [defined in Eq. (5)]; (b) corresponding optimum pump resonance coupling normalized by cavity intrinsic loss; (c) corresponding optimum ratio of signal/idler relative to pump resonance coupling.

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Fig. 3 Performance comparison of OPO designs with optimum unequal pump and signal/idler couplings and with optimized equal couplings (assuming no FCA): (a) power conversion efficiency; (b) optimum coupling values.

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Fig. 4 Normalized design curves for optimum OPO (I) using a “full-TPA” model (with all TPA terms but no FCA included) and (II) comparison of “partial-TPA” and “full-TPA” models (assuming no FCA): (a) maximum efficiency versus pump power and nonlinear loss sine, and corresponding (b) pump resonance coupling and (c) ratio of signal/idler relative to pump resonance coupling in (I) and signal/idler resonance coupling in (II). See Fig. 2 for parameter definitions.

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Fig. 5 Optimum OPO design curves for nonlinear media with and without TPA loss (assuming no FCA), representative of, e.g., of silicon nitride at 1550 nm and Si at 2.3 μm (linear), and Si at 1550 nm (σ₃ = 0.23).

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Fig. 6 Performance of silicon microcavity at 1550 nm resonance with various free-carrier lifetime and intrinsic cavity quality factors.

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Fig. 7 The OPO threshold vs (a) normalized free carrier lifetime and σ₃; (b) free carrier lifetime for silicon cavity resonant near 1550 nm with linear unloaded Q of 10⁶ and effective volume of 8.4 μm³.

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Fig. 8 Example microring cavity topology for illustration of effective figure of merit: (a) single-ring cavity with traveling-wave mode; (b) single-ring cavity with standing-wave mode; (c) triple-ring cavity with traveling-wave mode; (d) triple-ring cavity with standing-wave mode.

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Fig. 9 Mode fields of the pump, signal and idler resonances for the configurations (a)–(d) in Fig. 8 (color–coded intensity scales are different in single and triple-cavity cases in order to show the mode features clearly).

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Fig. 10 Mode overlap integrand for the FWM and various TPA coefficients for configurations (a)–(d) in Fig. 8. It shows that a vector of FOM is needed to account for the ratio of FWM relative to various TPA terms. Besides, different cavity topologies have different FOM (see Table 3).

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Fig. 11 Small signal gain and loss in an optical parametric oscillator based on degenerate four wave mixing.

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Tables (4)

Table 1 Third-order nonlinear properties of some common on-chip nonlinear material

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Table 2 Predicted performance of optical parametric oscillators based on some common on-chip nonlinear material in a single-ring cavity with traveling-wave mode

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Table 3 Comparison of FWM and TPA coefficients in various cavity topologies

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Table 4 Predicted performance of optical parametric oscillators based on 3-ring photonic molecule with traveling-wave mode

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Equations (77)

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\frac{d A_{s}}{d t} = - r_{s, tot} A_{s} - j ω_{s} β_{fwm, s} A_{p}^{2} A_{i}^{*}

\frac{d A_{p}}{d t} = - r_{p, tot} A_{p} - 2 j ω_{p} β_{fwm, p} A_{p}^{*} A_{s} A_{i} - j \sqrt{2 r_{p, ext}} S_{p, +}

\frac{d A_{i}}{d t} = - r_{i, tot} A_{i} - j ω_{i} β_{fwm, i} A_{p}^{2} A_{s}^{*}

S_{s, -} = - j \sqrt{2 r_{s, ext}} A_{s}

S_{p, -} = S_{p, +} - j \sqrt{2 r_{p, ext}} A_{p}

S_{i, -} = - j \sqrt{2 r_{i, ext}} A_{i}

β_{fwm, s} = β_{fwm, p}^{*} = β_{fwm, i} \equiv β_{fwm} .

\begin{array}{l} r_{s, tot} = r_{s, o} + r_{s, ext} + r_{FC} + ω_{s} (β_{tpa, ss} {| A_{s} |}^{2} + 2 β_{tpa, sp} {| A_{p} |}^{2} + 2 β_{tpa, si} {| A_{i} |}^{2}) \\ r_{p, tot} = r_{p, o} + r_{p, ext} + r_{FC} + ω_{p} (2 β_{tpa, sp} {| A_{s} |}^{2} + β_{tpa, pp} {| A_{p} |}^{2} + 2 β_{tpa, ip} {| A_{i} |}^{2}) \\ r_{i, tot} = r_{i, o} + r_{i, ext} + r_{FC} + ω_{i} (2 β_{tpa, si} {| A_{s} |}^{2} + 2 β_{tpa, ip} {| A_{p} |}^{2} + β_{tpa, ii} {| A_{i} |}^{2}) . \end{array}

\begin{array}{l} r_{FC} & = \frac{τ_{FC} σ_{a} v_{g}}{2 ℏ V_{eff}} (β_{tpa, ss} {| A_{s} |}^{4} + β_{tpa, pp} {| A_{p} |}^{4} + β_{tpa, ii} {| A_{i} |}^{4} \\ + 4 β_{tpa, sp} {| A_{s} |}^{2} {| A_{p} |}^{2} + 4 β_{tpa, ip} {| A_{i} |}^{2} {| A_{p} |}^{2} + 4 β_{tpa, si} {| A_{s} |}^{2} {| A_{i} |}^{2}) \end{array}

σ_{3} \equiv \frac{ℑ [χ^{(3)}]}{| χ^{(3)} |} = \frac{β_{tpa}}{β_{fwm}}

\frac{d B_{s}}{d τ} = - ρ_{s, tot} B_{s} - j 2 B_{p}^{2} B_{i}^{*}

\frac{d B_{p}}{d τ} = - ρ_{p, tot} B_{p} - j 4 B_{p}^{*} B_{s} B_{i} - j \sqrt{2 ρ_{p, ext}} T_{p, +}

\frac{d B_{i}}{d τ} = - ρ_{i, tot} B_{i} - j 2 B_{p}^{2} B_{s}^{*}

T_{s, -} = - j \sqrt{2 ρ_{s, ext}} B_{s}

T_{p, -} = T_{p, +} - j \sqrt{2 ρ_{p, ext}} B_{p}

T_{i, -} = - j \sqrt{2 ρ_{i, ext}} B_{i} .

τ \equiv r_{o} t

B_{k} \equiv \frac{A_{k}}{A_{o}}, with A_{o} \equiv \sqrt{\frac{2 r_{o}}{ω β_{fwm}}}

T_{k, \pm} \equiv \frac{S_{k, \pm}}{S_{o}}, with S_{o} \equiv \sqrt{\frac{2 r_{o}^{2}}{ω β_{fwm}}}

ρ_{s, tot} \equiv 1 + ρ_{s, ext} + 2 σ_{3} (d_{ss} {| B_{s} |}^{2} + 2 d_{sp} {| B_{p} |}^{2} + 2 d_{si} {| B_{i} |}^{2}) + ρ_{FC}

ρ_{p, tot} \equiv 1 + ρ_{p, ext} + 2 σ_{3} (2 d_{sp} {| B_{s} |}^{2} + d_{pp} {| B_{p} |}^{2} + 2 d_{ip} {| B_{i} |}^{2}) + ρ_{FC}

ρ_{i, tot} \equiv 1 + ρ_{i, ext} + 2 σ_{3} (2 d_{si} {| B_{s} |}^{2} + 2 d_{ip} {| B_{p} |}^{2} + d_{ii} {| B_{i} |}^{2}) + ρ_{FC} .

d_{mn} \equiv \frac{β_{tpa, mn}}{σ_{3} β_{fwm}}

ρ_{FC} \equiv \frac{r_{FC}}{r_{o}} = σ_{3} ρ_{FC}^{'} (d_{ss} {| B_{s} |}^{4} + d_{pp} {| B_{p} |}^{4} + d_{ii} {| B_{i} |}^{4} + 4 d_{sp} {| B_{s} |}^{2} {| B_{p} |}^{2} + 4 d_{ip} {| B_{i} |}^{2} {| B_{p} |}^{2} + 4 d_{si} {| B_{s} |}^{2} {| B_{i} |}^{2})

ρ_{FC}^{'} \equiv \frac{τ_{FC} σ_{a} v_{g}}{V_{eff}} \frac{β_{fwm}}{2 ℏ} \frac{4 r_{o}}{{(ω β_{fwm})}^{2}} = (\frac{σ_{a} n_{nl}^{2}}{ℏ ω n_{g} n_{2}}) \frac{τ_{FC}}{Q_{o}} .

η \equiv \frac{{| S_{s, -} |}^{2}}{{| S_{p, +} |}^{2}} .

B_{s} = - 2 j ρ_{s, tot}^{- 1} B_{p}^{2} B_{i}^{*}

B_{i} = - 2 j ρ_{i, tot}^{- 1} B_{p}^{2} B_{s}^{*}

T_{p, +} = j \frac{ρ_{p, tot} B_{p} + 4 j B_{p}^{*} B_{s} B_{i}}{\sqrt{2 ρ_{p, ext}}}

d_{ss} = d_{pp} = d_{ii} = d_{sp} = d_{si} = d_{ip} = 1 .

\begin{array}{l} ρ_{s, tot} = 1 + ρ_{s, ext} + 4 σ_{3} {| B_{p} |}^{2} \\ ρ_{p, tot} = 1 + ρ_{p, ext} + 2 σ_{3} {| B_{p} |}^{2} \\ ρ_{i, tot} = 1 + ρ_{s, ext} + 4 σ_{3} {| B_{p} |}^{2} \end{array}

{| B_{s} |}^{2} = {| B_{i} |}^{2} = 0 (below threshold)

{| B_{p} |}^{2} = \frac{(1 + ρ_{s, ext})}{2 (1 - 2 σ_{3})} (above threshold) .

P_{th, min} = \frac{1 - σ_{3}}{{(1 - 2 σ_{3})}^{2}} \frac{2 r_{o}^{2}}{ω β_{fwm}} = \frac{1 - σ_{3}}{{(1 - 2 σ_{3})}^{2}} P_{th, lin, min}

η \equiv \frac{{| S_{s, -} |}^{2}}{{| S_{p, +} |}^{2}} = \frac{2 r_{s, ext} {| A_{s} |}^{2}}{{| S_{p, +} |}^{2}} = \frac{2 ρ_{s, ext} {| B_{s} |}^{2}}{{| T_{p, +} |}^{2}} .

ρ_{p, ext, opt} = \frac{1 - 2 σ_{3}}{1 + ρ_{s, ext}} {| T_{p, +} |}^{2} .

{(1 + ρ_{s, ext})}^{2} (2 σ_{3} ρ_{s, ext} + 1 - σ_{3}) - {(1 - 2 σ_{3})}^{2} {| T_{p, +} |}^{2} = 0 .

\begin{matrix} ρ_{s, ext, opt} = \frac{1}{6 σ^{3}} (- 1 - 3 σ_{3} + {(D - E)}^{1 / 3} + {(D + E)}^{1 / 3}) \\ D \equiv {(3 σ_{3} - 1)}^{3} + 54 σ_{3}^{2} {(1 - 2 σ_{3})}^{2} {| T_{p, +} |}^{2} \\ E = 3 σ_{3} (1 - 2 σ_{3}) \sqrt{6 {| T_{p, +} |}^{2} {(3 σ_{3} - 1)}^{3} + D]} \end{matrix}

ρ_{s, ext, opt} = \sqrt{{| T_{p, +} |}^{2}} - 1

ρ_{p, ext, opt} = \sqrt{{| T_{p, +} |}^{2}} .

η_{max} ({| T_{p, +} |}^{2}, σ_{3} = 0) = \frac{{(\sqrt{{| T_{p, +} |}^{2}} - 1)}^{2}}{2 {| T_{p, +} |}^{2}}

{| T_{p, +}^{(ec)} |}^{2} = \frac{{(1 + ρ_{ext, opt})}^{3} {(1 + 2 ρ_{ext, opt})}^{2}}{ρ_{ext, opt} {(3 + 2 ρ_{ext, opt})}^{2}} .

η < \frac{1}{2} - σ_{3} .

P_{th, min}^{(ec)} = \frac{27 {(1 - σ_{3})}^{2}}{16 {(1 - 2 σ_{3})}^{3}} P_{th, lin, min} .

\begin{array}{l} ρ_{s, tot} = 1 + ρ_{s, ext} + 2 σ_{3} ({| B_{s} |}^{2} + 2 {| B_{p} |}^{2} + 2 {| B_{i} |}^{2}) \\ ρ_{p, tot} = 1 + ρ_{p, ext} + 2 σ_{3} (2 {| B_{s} |}^{2} + {| B_{p} |}^{2} + 2 {| B_{i} |}^{2}) \\ ρ_{i, tot} = 1 + ρ_{s, ext} + 2 σ_{3} (2 {| B_{s} |}^{2} + 2 {| B_{p} |}^{2} + {| B_{i} |}^{2}) \end{array}

{| B_{p} |}^{2} = \frac{(1 + ρ_{s, ext} + 6 σ_{3} {| B_{s} |}^{2})}{2 (1 - 2 σ_{3})}

\begin{array}{l} 4 {(1 - 2 σ_{3})}^{3} ρ_{p, ext} {| T_{p, +} |}^{2} = (6 σ_{3} {| B_{s} |}^{2} + 1 + ρ_{s, ext}) \\ \times {[(1 - 2 σ_{3}) (1 + ρ_{p, ext}) + σ_{3} (1 + ρ_{s, ext}) + 2 (2 - 5 σ_{3}^{2}) {| B_{s} |}^{2}]}^{2} \end{array}

σ_{3} ρ_{FC}^{'} B_{p}^{4} + (8 σ_{3} ρ_{FC}^{'} B_{s}^{2} + 4 σ_{3} - 2) B_{p}^{2} = - (6 σ_{3} ρ_{FC}^{'} B_{s}^{4} + 6 σ_{3} B_{s}^{2} + 1 + ρ_{s, ext})

{[(2 - 2 σ_{3}) B_{p}^{2} + (4 + 2 σ_{3}) B_{s}^{2} + ρ_{p, ext} - ρ_{s, ext}]}^{2} B_{p}^{2} = 2 ρ_{p, ext, opt} T_{p, +}^{2}

P_{th, min} = \frac{4 (1 - σ_{3})}{{[(1 - 2 σ_{3}) + \sqrt{{(1 - 2 σ_{3})}^{2} - σ_{3} ρ_{FC}^{'}}]}^{2}} P_{th, lin, min}

β_{fwm, s} = \frac{\frac{3}{16} ε_{0} \int d^{3} x (E_{s}^{*} \cdot {\bar{\bar{χ}}}^{(3)} \cdot E_{p}^{2} E_{i}^{*})}{\sqrt{\int d^{3} x (\frac{1}{2} ε {| E_{s} |}^{2}) \int d^{3} x (\frac{1}{2} ε {| E_{i} |}^{2}) \int d^{3} x (\frac{1}{2} ε {| E_{p} |}^{2})}} \equiv \frac{3 χ_{1111}^{(3)}}{4 n_{nl}^{4} ε_{0} V_{eff}}

V_{eff} \equiv \frac{χ_{1111}^{(3)} \sqrt{\int d^{3} x (ε {| E_{s} |}^{2}) \int d^{3} x (ε {| E_{i} |}^{2})} \int d^{3} x (ε {| E_{p} |}^{2})}{ε_{0}^{2} n_{n l}^{4} \int d^{3} x (E_{s}^{*} \cdot {\bar{\bar{χ}}}^{(3)} : E_{p}^{2} E_{i}^{*})} .

β_{tpa, sp} = \frac{\frac{3}{16} ε_{0} \int d^{3} x (E_{s}^{*} \cdot ℑ [{\bar{\bar{χ}}}^{(3)}] : E_{s} E_{p} E_{p}^{*})}{\int d^{3} x (\frac{1}{2} ε {| E_{s} |}^{2}) \int d^{3} x (\frac{1}{2} ε {| E_{p} |}^{2})} .

\frac{\partial N_{ν}}{\partial t} = G - \frac{N_{ν}}{τ_{ν}} + D_{ν} \nabla^{2} N_{ν} - s_{ν} μ_{ν} \nabla \cdot (N_{ν} E_{dc}) \equiv G - \frac{N_{ν}}{τ_{ν, eff}}

G = \frac{1}{2 ℏ ω} \frac{Δ E}{Δ t \cdot Δ V} = \frac{1}{4 ℏ ω} ℜ [E_{tot}^{*} \cdot J] = \frac{1}{4 ℏ ω} ℜ [j ω ε_{0} E_{tot}^{*} \cdot {\bar{\bar{χ}}}^{(3)} : E_{tot}^{3}]

N_{ν} = G τ_{ν, eff} .

α_{ν} = σ_{ν} N_{ν}

\begin{array}{l} r_{k, FC}^{ν} & = - \frac{j ω}{4} \frac{\int d^{3} x (E_{k}^{*} \cdot δ P_{k}^{(FCA, ν)})}{\int d^{3} x (\frac{1}{2} ε {| E_{k} |}^{2})} = \frac{ω}{4} \frac{\int d^{3} x (ε_{0} n_{nl} \frac{α_{ν}}{k_{0}} {| E_{k} |}^{2})}{\int d^{3} x (\frac{1}{2} ε {| E_{k} |}^{2})} \\ = \frac{ε_{0} n_{nl} ω σ_{ν}}{4 k_{0}} \frac{\int d^{3} x (G τ_{ν, eff} {| E_{k} |}^{2})}{\int d^{3} x (\frac{1}{2} ε {| E_{k} |}^{2})} = \frac{c ε_{0}^{2} n_{nl} σ_{ν}}{16 ℏ} \frac{\int d^{3} x (τ_{ν, eff} (E_{tot}^{*} \cdot ℑ [{\bar{\bar{χ}}}^{(3)}] : E_{tot}^{3}) {| E_{k} |}^{2})}{\int d^{3} x (\frac{1}{2} ε {| E_{k} |}^{2})} . \end{array}

\begin{array}{l} N_{ν} & = \frac{τ_{eff}}{ℏ V_{eff}} (β_{tpa, ss} {| A_{s} |}^{4} + β_{tpa, pp} {| A_{p} |}^{4} + β_{tpa, ii} {| A_{i} |}^{4} \\ + 4 β_{tpa, sp} {| A_{s} |}^{2} {| A_{p} |}^{2} + 4 β_{tpa, ip} {| A_{i} |}^{2} {| A_{p} |}^{2} + 4 β_{tpa, si} {| A_{s} |}^{2} {| A_{i} |}^{2}) . \end{array}

\begin{array}{l} r_{FC} & = \frac{α_{FC} v_{g}}{2} = \frac{σ_{a} N_{ν} v_{g}}{2} = \frac{τ_{eff} σ_{a} v_{g}}{2 ℏ V_{eff}} (β_{tpa, ss} {| A_{s} |}^{4} + β_{tpa, pp} {| A_{p} |}^{4} \\ + β_{tpa, ii} {| A_{i} |}^{4} + 4 β_{tpa, sp} {| A_{s} |}^{2} {| A_{p} |}^{2} + 4 β_{tpa, ip} {| A_{i} |}^{2} {| A_{p} |}^{2} + 4 β_{tpa, si} {| A_{s} |}^{2} {| A_{i} |}^{2}) \end{array}

T_{p, +} = j {(\sqrt{2 ρ_{p, ext}})}^{- 1} (ρ_{p, tot} + 8 ρ_{i, tot}^{- 1} | B_{s} |^{2} | B_{p} |^{2}) B_{p} .

{| T_{p, +} |}^{2} = \frac{ρ_{p, tot}^{2}}{2 ρ_{p, ext}} | B_{p} |^{2}

ρ_{p, ext} = 1 + 2 σ_{3} | B_{p} |^{2} + σ_{3} ρ_{F C}^{'} | B_{p} |^{4}

P_{th} = {(2 (1 + 2 σ_{3} | B_{p} |^{2} + σ_{3} ρ_{F C}^{'} | B_{p} |^{4}) | B_{p} |^{2})}_{min}

2 | B_{p} |^{2} = ρ_{s, tot} = 1 + ρ_{s, ext} + 4 σ_{3} | B_{p} |^{2} + σ_{3} ρ_{F C}^{'} | B_{p} |^{4}

| B_{p} |^{2} = \frac{(1 - 2 σ_{3}) - \sqrt{{(1 - 2 σ_{3})}^{2} - σ_{3} ρ_{F C}^{'} (1 + ρ_{s, ext})}}{σ_{3} ρ_{F C}^{'}}

ρ_{F C}^{'} \leq \frac{{(1 - 2 σ_{3})}^{3}}{σ_{3}}

P_{th} = \frac{4 (1 - σ_{3})}{{((1 - 2 σ_{3}) + \sqrt{{(1 - 2 σ_{3})}^{2} - σ_{3} ρ_{FC}^{'}})}^{2}}

| T_{p, +} |^{2} = \frac{ρ_{p, tot}^{3}}{4 (1 - σ_{3}) ρ_{ext}}

P_{th}^{'} = \frac{{(1 - σ_{3})}^{2}}{4 {(1 - 2 σ_{3})}^{3}} \frac{{(1 + ρ_{ext})}^{3}}{ρ_{ext}} |_{min} = \frac{27 {(1 - σ_{3})}^{2}}{16 {(1 - 2 σ_{3})}^{3}}

T_{p, +} = j {(\sqrt{2 ρ_{p, ext}})}^{- 1} (ρ_{p, tot} + 8 ρ_{i, tot}^{- 1} | B_{s} |^{2} | B_{p} |^{2}) B_{p} = A + B | B_{s} |^{2}

η = \frac{2 ρ_{s, ext} | B_{s} |^{2}}{| T_{p, +} |^{2}} = (\frac{2 ρ_{s, ext}}{B}) \frac{T_{p, +} - A}{| T_{p, +} |^{2}}

η = (\frac{2 ρ_{s, ext}}{| B |}) \frac{| T_{p, +} | - | A |}{| T_{p, +} |^{2}}

| T_{p, +} |^{2} = 4 A^{2} = \frac{2 ρ_{p, tot}^{2}}{ρ_{p, ext}} | B_{p} |^{2}

η_{max} = \frac{ρ_{s, ext}}{2 A B} = \frac{{(1 - 2 σ_{3})}^{2}}{2} \frac{ρ_{s, ext} ρ_{p, ext} ρ_{i, tot}}{ρ_{p, tot} {(1 + ρ_{s, ext})}^{2}} = (\frac{1}{2} - σ_{3}) \frac{ρ_{p, ext}}{ρ_{p, tot}} \frac{ρ_{s, ext} (1 + ρ_{i, ext})}{{(1 + ρ_{s, ext})}^{2}} < \frac{1}{2} - σ_{3} .

\frac{d B_{s}}{d τ} = - ρ_{s, tot} B_{s} + 4 ρ_{i, tot}^{- 1} | B_{p} |^{4} B_{s} \equiv - ρ_{loss} B_{s} + ρ_{gain} B_{s}

\frac{d B p}{d τ} = - ρ_{p, tot} B_{p} - 8 ρ_{i, tot}^{- 1} | B_{p} |^{2} B_{p} - j \sqrt{2 ρ_{p, ext} T_{p, +}}

Material	λ (μm)	n₂(10⁻⁵ $\frac{{cm}^{2}}{GW}$ )^a	β_TPA( $\frac{cm}{GW}$ )^a	NFOM	σ₃
c-Si	1.55 [29]	2.41	0.48	0.34	0.23
c-Si	2.3 [29]	1.0	≈ 0	∞	≈ 0
a-Si:H	1.55 [30]	16.6^b	0.49^b	2.2	0.036
Si₃N₄	1.55 [10]	0.24	≈ 0	∞	≈ 0

Material^a	λ (μm)	W(nm)×H(nm)×R_out(μm)^b	Q_o^c	V_eff (μm³)^d	β_fwm(10⁶J⁻¹)	P _th(mW)^e
c-Si	1.55	460 × 220 × 3	10⁶	2.1	29	0.055
c-Si	2.3	700 × 250 × 7	10⁶	10	2.5	0.16
a-Si:H	1.55	460 × 220 × 3	10⁶	2.1	186	0.004
Si₃N₄	1.55	1600 × 700 × 15	10⁶	84	0.22	2.8

Cavity Type^a	$\frac{β_{fwm}}{β_{fwm} (1 - ring, TW)}$ b	d_ss	d_pp	d_sp	d_si

1-ring (TW^c)	1	1	1	1	1
3-ring (TW^c)	$\frac{1}{4}$	$\frac{3}{2}$	2	1	$\frac{3}{2}$
1-ring (SW^c)	$\frac{1}{2}$	3	3	2	2
3-ring (SW^c)	$\frac{3}{8}$	$\frac{3}{2}$	2	1	$\frac{3}{2}$

Material	λ (μm)	V_eff(μm³)^a	β_fwm(10⁶J⁻¹)	P_th(mW)
c-Si	1.55	8.3	7.2	0.29
c-Si	2.3	40	0.63	0.65
a-Si:H	1.55	8.3	46	0.015
Si₃N₄	1.55	337	0.05	11

Material	λ (μm)	n₂(10⁻⁵ $\frac{{cm}^{2}}{GW}$ )^a	β_TPA( $\frac{cm}{GW}$ )^a	NFOM	σ₃
c-Si	1.55 [29]	2.41	0.48	0.34	0.23
c-Si	2.3 [29]	1.0	≈ 0	∞	≈ 0
a-Si:H	1.55 [30]	16.6^b	0.49^b	2.2	0.036
Si₃N₄	1.55 [10]	0.24	≈ 0	∞	≈ 0