Abstract

Integrated chip-scale optical systems are an attractive platform for the implementation of non-linear optical interactions as they promise compact robust devices that operate reliably with lower power consumption compared to analogs based on bulk nonlinear crystals. The use of guided modes to facilitate nonlinear parametric interactions between optical fields, as opposed to bulk beams, has certain implications on optical parametric oscillations, the most important of which are additional methods for achieving phase synchronism and reduced threshold power due to the tight confinement associated with the guided modes. This work presents a theoretical investigation on the use of polarization dependent mode dispersion in guided wave structures as a means to achieve non-linear parametric oscillations from continuous wave sources with outputs in the mid-infrared region of the spectrum. An Al2O3/GaP/Al2O3 waveguide system is investigated and shown to produce parametric oscillations at 3µm to 5µm from 1µm to 2µm input waves utilizing 0.14µm to 0.30µm GaP cores. The threshold power is shown to be 320 × less than that obtainable using more traditional quasi-phase matched bulk crystals over the same wavelength range.

© 2010 OSA

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2010 (1)

V. Tassev, D. Bliss, C. Lynch, C. Yapp, W. Goodhue, and K. Termkoa, “Low pressure-temperature-gas flow HVPE growth of GaP for nonlinear optical frequency conversion devices,” J. Cryst. Growth 312(8), 1146–1149 (2010).
[CrossRef]

2009 (1)

E. Guillotel, M. Ravaro, F. Ghiglieno, C. Langlois, C. Ricolleau, S. Ducci, I. Favero, and G. Leo, “Parametric amplification in GaAs/AlOx waveguide,” Appl. Phys. Lett. 94(17), 171110 (2009).
[CrossRef]

2008 (1)

S. Radic, “Parametric amplification and processing in optical fibers,” Laser Photon. Rev. 2(6), 498–513 (2008).
[CrossRef]

2007 (4)

A. Berrou, J.-M. Melkonian, M. Raybaut, A. Godard, E. Rosencher, and M. Lefebre, “Specific architectures for optical parametric oscillators,” C. R. Phys. 8(10), 1162–1173 (2007).
[CrossRef]

W. Denzer, G. Hancock, A. Hutchinson, M. Munday, R. Peverall, and G. A. D. Ritchie, “Mid-infrared generation and spectroscopy with a PPLN ridge waveguide,” Appl. Phys. B 86(3), 437–441 (2007).
[CrossRef]

M. Fiorentino, S. M. Spillane, R. G. Beausoleil, T. D. Roberts, P. Battle, and M. W. Munro, “Spontaneous parametric down-conversion in periodically poled KTP waveguides and bulk crystals,” Opt. Express 15(12), 7479–7488 (2007).
[CrossRef] [PubMed]

S. M. Spillane, M. Fiorentino, and R. G. Beausoleil, “Spontaneous parametric down conversion in a nanophotonic waveguide,” Opt. Express 15(14), 8770–8780 (2007).
[CrossRef] [PubMed]

2006 (2)

2004 (1)

S. V. Rao, K. Moutzouris, and M. Ebrahimzadeh, “Nonlinear frequency conversion in semiconductor optical waveguides using birefringent, modal, and quasi-phase-matching techniques,” J. Opt. A 6(6), 569–584 (2004).

2003 (1)

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74(1), 1–18 (2003).
[CrossRef]

2002 (2)

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[CrossRef]

M. W. Todd, R. A. Provencal, T. G. Owano, B. A. Paldus, A. Kachanov, K. L. Vodopyanov, M. Hunter, S. L. Coy, J. I. Steinfeld, and J. T. Arnold, “Application of mid-infrared cavity-ringdown spectroscopy to trace explosives vapor detection using a broadly tunable (6–8μm) optical parametric oscillator,” Appl. Phys. B 75(2–3), 367–376 (2002).
[CrossRef]

2001 (1)

J. Seres, “Dispersion of second-order nonlinear optical coefficients,” Appl. Phys. B 73(7), 705–709 (2001).
[CrossRef]

1998 (1)

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391(6666), 463–466 (1998).
[CrossRef]

1996 (1)

1974 (1)

D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar-dielectric waveguides,” IEEE Trans. Microw. Theory Tech. 22(12), 995–1000 (1974).
[CrossRef]

1968 (1)

G. D. Boyd and D. A. Kleiman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39(8), 3597–3639 (1968).
[CrossRef]

Alexander, J. I.

Andrekson, P. A.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[CrossRef]

Arnold, J. T.

M. W. Todd, R. A. Provencal, T. G. Owano, B. A. Paldus, A. Kachanov, K. L. Vodopyanov, M. Hunter, S. L. Coy, J. I. Steinfeld, and J. T. Arnold, “Application of mid-infrared cavity-ringdown spectroscopy to trace explosives vapor detection using a broadly tunable (6–8μm) optical parametric oscillator,” Appl. Phys. B 75(2–3), 367–376 (2002).
[CrossRef]

Battle, P.

Beausoleil, R. G.

Berger, V.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391(6666), 463–466 (1998).
[CrossRef]

Berrou, A.

A. Berrou, J.-M. Melkonian, M. Raybaut, A. Godard, E. Rosencher, and M. Lefebre, “Specific architectures for optical parametric oscillators,” C. R. Phys. 8(10), 1162–1173 (2007).
[CrossRef]

Bliss, D.

V. Tassev, D. Bliss, C. Lynch, C. Yapp, W. Goodhue, and K. Termkoa, “Low pressure-temperature-gas flow HVPE growth of GaP for nonlinear optical frequency conversion devices,” J. Cryst. Growth 312(8), 1146–1149 (2010).
[CrossRef]

P. S. Kuo, K. L. Vodopyanov, M. M. Fejer, D. M. Simanovskii, X. Yu, J. S. Harris, D. Bliss, and D. Weyburne, “Optical parametric generation of a mid-infrared continuum in orientation-patterned GaAs,” Opt. Lett. 31(1), 71–73 (2006).
[CrossRef] [PubMed]

Bosenberg, W. R.

Boyd, G. D.

G. D. Boyd and D. A. Kleiman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39(8), 3597–3639 (1968).
[CrossRef]

Bravetti, P.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391(6666), 463–466 (1998).
[CrossRef]

Byer, R. L.

Cerullo, G.

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74(1), 1–18 (2003).
[CrossRef]

Coleman, P. D.

D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar-dielectric waveguides,” IEEE Trans. Microw. Theory Tech. 22(12), 995–1000 (1974).
[CrossRef]

Coy, S. L.

M. W. Todd, R. A. Provencal, T. G. Owano, B. A. Paldus, A. Kachanov, K. L. Vodopyanov, M. Hunter, S. L. Coy, J. I. Steinfeld, and J. T. Arnold, “Application of mid-infrared cavity-ringdown spectroscopy to trace explosives vapor detection using a broadly tunable (6–8μm) optical parametric oscillator,” Appl. Phys. B 75(2–3), 367–376 (2002).
[CrossRef]

De Silvestri, S.

G. Cerullo and S. De Silvestri, “Ultrafast optical parametric amplifiers,” Rev. Sci. Instrum. 74(1), 1–18 (2003).
[CrossRef]

Denzer, W.

W. Denzer, G. Hancock, A. Hutchinson, M. Munday, R. Peverall, and G. A. D. Ritchie, “Mid-infrared generation and spectroscopy with a PPLN ridge waveguide,” Appl. Phys. B 86(3), 437–441 (2007).
[CrossRef]

Drobshoff, A.

Ducci, S.

E. Guillotel, M. Ravaro, F. Ghiglieno, C. Langlois, C. Ricolleau, S. Ducci, I. Favero, and G. Leo, “Parametric amplification in GaAs/AlOx waveguide,” Appl. Phys. Lett. 94(17), 171110 (2009).
[CrossRef]

Ebrahimzadeh, M.

S. V. Rao, K. Moutzouris, and M. Ebrahimzadeh, “Nonlinear frequency conversion in semiconductor optical waveguides using birefringent, modal, and quasi-phase-matching techniques,” J. Opt. A 6(6), 569–584 (2004).

Fan, S.

Favero, I.

E. Guillotel, M. Ravaro, F. Ghiglieno, C. Langlois, C. Ricolleau, S. Ducci, I. Favero, and G. Leo, “Parametric amplification in GaAs/AlOx waveguide,” Appl. Phys. Lett. 94(17), 171110 (2009).
[CrossRef]

Fejer, M. M.

Fiore, A.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391(6666), 463–466 (1998).
[CrossRef]

Fiorentino, M.

Ghiglieno, F.

E. Guillotel, M. Ravaro, F. Ghiglieno, C. Langlois, C. Ricolleau, S. Ducci, I. Favero, and G. Leo, “Parametric amplification in GaAs/AlOx waveguide,” Appl. Phys. Lett. 94(17), 171110 (2009).
[CrossRef]

Godard, A.

A. Berrou, J.-M. Melkonian, M. Raybaut, A. Godard, E. Rosencher, and M. Lefebre, “Specific architectures for optical parametric oscillators,” C. R. Phys. 8(10), 1162–1173 (2007).
[CrossRef]

Goodhue, W.

V. Tassev, D. Bliss, C. Lynch, C. Yapp, W. Goodhue, and K. Termkoa, “Low pressure-temperature-gas flow HVPE growth of GaP for nonlinear optical frequency conversion devices,” J. Cryst. Growth 312(8), 1146–1149 (2010).
[CrossRef]

Guillotel, E.

E. Guillotel, M. Ravaro, F. Ghiglieno, C. Langlois, C. Ricolleau, S. Ducci, I. Favero, and G. Leo, “Parametric amplification in GaAs/AlOx waveguide,” Appl. Phys. Lett. 94(17), 171110 (2009).
[CrossRef]

Hancock, G.

W. Denzer, G. Hancock, A. Hutchinson, M. Munday, R. Peverall, and G. A. D. Ritchie, “Mid-infrared generation and spectroscopy with a PPLN ridge waveguide,” Appl. Phys. B 86(3), 437–441 (2007).
[CrossRef]

Hansryd, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[CrossRef]

Harris, J. S.

Hedekvist, P.-O.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[CrossRef]

Hunter, M.

M. W. Todd, R. A. Provencal, T. G. Owano, B. A. Paldus, A. Kachanov, K. L. Vodopyanov, M. Hunter, S. L. Coy, J. I. Steinfeld, and J. T. Arnold, “Application of mid-infrared cavity-ringdown spectroscopy to trace explosives vapor detection using a broadly tunable (6–8μm) optical parametric oscillator,” Appl. Phys. B 75(2–3), 367–376 (2002).
[CrossRef]

Huo, Y.

Hutchinson, A.

W. Denzer, G. Hancock, A. Hutchinson, M. Munday, R. Peverall, and G. A. D. Ritchie, “Mid-infrared generation and spectroscopy with a PPLN ridge waveguide,” Appl. Phys. B 86(3), 437–441 (2007).
[CrossRef]

Kachanov, A.

M. W. Todd, R. A. Provencal, T. G. Owano, B. A. Paldus, A. Kachanov, K. L. Vodopyanov, M. Hunter, S. L. Coy, J. I. Steinfeld, and J. T. Arnold, “Application of mid-infrared cavity-ringdown spectroscopy to trace explosives vapor detection using a broadly tunable (6–8μm) optical parametric oscillator,” Appl. Phys. B 75(2–3), 367–376 (2002).
[CrossRef]

Kleiman, D. A.

G. D. Boyd and D. A. Kleiman, “Parametric interaction of focused Gaussian light beams,” J. Appl. Phys. 39(8), 3597–3639 (1968).
[CrossRef]

Kuo, P. S.

Langlois, C.

E. Guillotel, M. Ravaro, F. Ghiglieno, C. Langlois, C. Ricolleau, S. Ducci, I. Favero, and G. Leo, “Parametric amplification in GaAs/AlOx waveguide,” Appl. Phys. Lett. 94(17), 171110 (2009).
[CrossRef]

Lefebre, M.

A. Berrou, J.-M. Melkonian, M. Raybaut, A. Godard, E. Rosencher, and M. Lefebre, “Specific architectures for optical parametric oscillators,” C. R. Phys. 8(10), 1162–1173 (2007).
[CrossRef]

Leo, G.

E. Guillotel, M. Ravaro, F. Ghiglieno, C. Langlois, C. Ricolleau, S. Ducci, I. Favero, and G. Leo, “Parametric amplification in GaAs/AlOx waveguide,” Appl. Phys. Lett. 94(17), 171110 (2009).
[CrossRef]

Li, J.

J. Hansryd, P. A. Andrekson, M. Westlund, J. Li, and P.-O. Hedekvist, “Fiber-based optical parametric amplifiers and their applications,” IEEE J. Sel. Top. Quantum Electron. 8(3), 506–520 (2002).
[CrossRef]

Lynch, C.

V. Tassev, D. Bliss, C. Lynch, C. Yapp, W. Goodhue, and K. Termkoa, “Low pressure-temperature-gas flow HVPE growth of GaP for nonlinear optical frequency conversion devices,” J. Cryst. Growth 312(8), 1146–1149 (2010).
[CrossRef]

Melkonian, J.-M.

A. Berrou, J.-M. Melkonian, M. Raybaut, A. Godard, E. Rosencher, and M. Lefebre, “Specific architectures for optical parametric oscillators,” C. R. Phys. 8(10), 1162–1173 (2007).
[CrossRef]

Moutzouris, K.

S. V. Rao, K. Moutzouris, and M. Ebrahimzadeh, “Nonlinear frequency conversion in semiconductor optical waveguides using birefringent, modal, and quasi-phase-matching techniques,” J. Opt. A 6(6), 569–584 (2004).

Munday, M.

W. Denzer, G. Hancock, A. Hutchinson, M. Munday, R. Peverall, and G. A. D. Ritchie, “Mid-infrared generation and spectroscopy with a PPLN ridge waveguide,” Appl. Phys. B 86(3), 437–441 (2007).
[CrossRef]

Munro, M. W.

Myers, L. E.

Nagle, J.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391(6666), 463–466 (1998).
[CrossRef]

Owano, T. G.

M. W. Todd, R. A. Provencal, T. G. Owano, B. A. Paldus, A. Kachanov, K. L. Vodopyanov, M. Hunter, S. L. Coy, J. I. Steinfeld, and J. T. Arnold, “Application of mid-infrared cavity-ringdown spectroscopy to trace explosives vapor detection using a broadly tunable (6–8μm) optical parametric oscillator,” Appl. Phys. B 75(2–3), 367–376 (2002).
[CrossRef]

Paldus, B. A.

M. W. Todd, R. A. Provencal, T. G. Owano, B. A. Paldus, A. Kachanov, K. L. Vodopyanov, M. Hunter, S. L. Coy, J. I. Steinfeld, and J. T. Arnold, “Application of mid-infrared cavity-ringdown spectroscopy to trace explosives vapor detection using a broadly tunable (6–8μm) optical parametric oscillator,” Appl. Phys. B 75(2–3), 367–376 (2002).
[CrossRef]

Peverall, R.

W. Denzer, G. Hancock, A. Hutchinson, M. Munday, R. Peverall, and G. A. D. Ritchie, “Mid-infrared generation and spectroscopy with a PPLN ridge waveguide,” Appl. Phys. B 86(3), 437–441 (2007).
[CrossRef]

Provencal, R. A.

M. W. Todd, R. A. Provencal, T. G. Owano, B. A. Paldus, A. Kachanov, K. L. Vodopyanov, M. Hunter, S. L. Coy, J. I. Steinfeld, and J. T. Arnold, “Application of mid-infrared cavity-ringdown spectroscopy to trace explosives vapor detection using a broadly tunable (6–8μm) optical parametric oscillator,” Appl. Phys. B 75(2–3), 367–376 (2002).
[CrossRef]

Radic, S.

S. Radic, “Parametric amplification and processing in optical fibers,” Laser Photon. Rev. 2(6), 498–513 (2008).
[CrossRef]

Rao, S. V.

S. V. Rao, K. Moutzouris, and M. Ebrahimzadeh, “Nonlinear frequency conversion in semiconductor optical waveguides using birefringent, modal, and quasi-phase-matching techniques,” J. Opt. A 6(6), 569–584 (2004).

Ravaro, M.

E. Guillotel, M. Ravaro, F. Ghiglieno, C. Langlois, C. Ricolleau, S. Ducci, I. Favero, and G. Leo, “Parametric amplification in GaAs/AlOx waveguide,” Appl. Phys. Lett. 94(17), 171110 (2009).
[CrossRef]

Raybaut, M.

A. Berrou, J.-M. Melkonian, M. Raybaut, A. Godard, E. Rosencher, and M. Lefebre, “Specific architectures for optical parametric oscillators,” C. R. Phys. 8(10), 1162–1173 (2007).
[CrossRef]

Ricolleau, C.

E. Guillotel, M. Ravaro, F. Ghiglieno, C. Langlois, C. Ricolleau, S. Ducci, I. Favero, and G. Leo, “Parametric amplification in GaAs/AlOx waveguide,” Appl. Phys. Lett. 94(17), 171110 (2009).
[CrossRef]

Ritchie, G. A. D.

W. Denzer, G. Hancock, A. Hutchinson, M. Munday, R. Peverall, and G. A. D. Ritchie, “Mid-infrared generation and spectroscopy with a PPLN ridge waveguide,” Appl. Phys. B 86(3), 437–441 (2007).
[CrossRef]

Roberts, T. D.

Rosencher, E.

A. Berrou, J.-M. Melkonian, M. Raybaut, A. Godard, E. Rosencher, and M. Lefebre, “Specific architectures for optical parametric oscillators,” C. R. Phys. 8(10), 1162–1173 (2007).
[CrossRef]

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391(6666), 463–466 (1998).
[CrossRef]

Scaccabarozzi, L.

Seres, J.

J. Seres, “Dispersion of second-order nonlinear optical coefficients,” Appl. Phys. B 73(7), 705–709 (2001).
[CrossRef]

Simanovskii, D. M.

Spillane, S. M.

Steinfeld, J. I.

M. W. Todd, R. A. Provencal, T. G. Owano, B. A. Paldus, A. Kachanov, K. L. Vodopyanov, M. Hunter, S. L. Coy, J. I. Steinfeld, and J. T. Arnold, “Application of mid-infrared cavity-ringdown spectroscopy to trace explosives vapor detection using a broadly tunable (6–8μm) optical parametric oscillator,” Appl. Phys. B 75(2–3), 367–376 (2002).
[CrossRef]

Tassev, V.

V. Tassev, D. Bliss, C. Lynch, C. Yapp, W. Goodhue, and K. Termkoa, “Low pressure-temperature-gas flow HVPE growth of GaP for nonlinear optical frequency conversion devices,” J. Cryst. Growth 312(8), 1146–1149 (2010).
[CrossRef]

Termkoa, K.

V. Tassev, D. Bliss, C. Lynch, C. Yapp, W. Goodhue, and K. Termkoa, “Low pressure-temperature-gas flow HVPE growth of GaP for nonlinear optical frequency conversion devices,” J. Cryst. Growth 312(8), 1146–1149 (2010).
[CrossRef]

Thompson, D. E.

D. E. Thompson and P. D. Coleman, “Step-tunable far infrared radiation by phase matched mixing in planar-dielectric waveguides,” IEEE Trans. Microw. Theory Tech. 22(12), 995–1000 (1974).
[CrossRef]

Todd, M. W.

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Figures (6)

Fig. 1
Fig. 1

(Color) Symmetric (left) and asymmetric (right) waveguides. The width in the lateral direction d is large enough so that modal indices and fields can be estimated assuming planar waveguide geometry.

Fig. 2
Fig. 2

(Color) Effective index for fundamental TM 0 (black solid) and TE 0 (black long-dash) modes at pump wavelength and linear combination of effective indices for signal and idler TE 0 modes with values of the parameter a of 0.2 (green dots), 0.3 (green dot-dash), and 0.5 (green short dash), as functions of the film thickness.

Fig. 3
Fig. 3

(Color) The idler (red) and signal (blue) wavelengths as a function of the film thickness for the pump wavelength of 1.0μm (solid), 1.5μm (long-dash), and 2.0μm (short-dash). The phase synchronism is assumed between the fundamental TM 0 mode at the pump wavelength and TE 0 idler and signal modes in a symmetric waveguide.

Fig. 4
Fig. 4

(Color) The idler (red) and signal (blue) wavelengths as a function of the film thickness for the pump wavelength of 1.0μm (solid), 1.5μm (long-dash), and 2.0μm (short-dash). The phase synchronism is assumed between the fundamental TM 0 mode at the pump wavelength and TE 0 idler and signal modes in an asymmetric waveguide.

Fig. 5
Fig. 5

Overlap of pump, signal, and idler waves in a bulk crystal (left) and in a waveguide (right). In the case of bulk crystals the cross section of each beam at its waist as well as the normalized area are scaled directly proportional to the nonlinear medium length. The guided modes remain tightly confined regardless of the length of the nonlinear medium.

Fig. 6
Fig. 6

(Color) Strength of electric field as a function of a coordinate across the waveguide for pump (black solid), signal (red short-dash), and idler (blue long-dash) guided modes. The pump wave is a TM 0 mode, while signal and idler waves are TE 0 modes.

Equations (25)

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1 λ p = 1 λ s + 1 λ i .
n p λ p = n s λ s + n i λ i ,
n p = ( 1 a ) n s + a n i .
n T M * ( t ) | λ = λ p = ( 1 a ) n T E * ( t ) | λ = λ p / ( 1 a ) + a n T E * ( t ) | λ = λ p / a .
2 π t λ n g 2 n Q * 2 = atan ( n g 2 Q n c 2 Q n Q * 2 n c 2 n g 2 n Q * 2 ) + atan ( n g 2 Q n s 2 Q n Q * 2 n s 2 n g 2 n Q * 2 ) + M π ,
n g ( λ ) = 1 + 1.390 λ 2 λ 2 0.172 2 + 4.131 λ 2 λ 2 0.234 2 + 2.570 λ 2 λ 2 0.345 2 + 2.056 λ 2 λ 2 27.52 2 ,
n s ( λ ) = 1 + 1.4313493 λ 2 λ 2 0.0726631 2 + 0.65054713 λ 2 λ 2 0.1193242 2 + 5.3414021 λ 2 λ 2 18.028251 2 .
P p N L = 2 ε 0 d e f f E s E i ;
P s N L = 2 ε 0 d e f f E p E i * ;
P i N L = 2 ε 0 d e f f E p E s * .
d Q P M = η Q P M d e f f = 2 π d e f f .
4 π 2 n s n i λ s λ i d Q P M 2 L 2 | E p   t h | 2 = δ s δ i ,
8 π 2 n p n s n i λ s λ i μ 0 ε 0 η Q P M 2 d e f f 2 L 2 P t h A = δ s δ i .
P p N L E p d x d y = P s N L E s d x d y = P i N L E i d x d y = 2 ε 0 d e f f E p E s E i d x d y .
1 A n o r m = E p E s E i d x d y E p 2 d x d y E s 2 d x d y E i 2 d x d y .
8 π 2 n p n s n i λ s λ i μ 0 ε 0 η Q P M 2 d e f f 2 L 2 P t h A n o r m = δ s δ i .
E p , s , i ( x , y ) = E 0 p , s , i exp ( x 2 + y 2 w p , s , i 2 ) .
A n o r m = π 8 ( w s w i w p + w p w i w s + w s w p w i ) 2 .
A n o r m = L 16 ξ λ p λ s λ i n p n s n i ( n p λ p + n s λ s + n i λ i ) 2 .
A n o r m = L 4 ξ λ s λ i λ p n p n s n i .
8 π 2 n p * n s * n i * λ s λ i μ 0 ε 0 η G W 2 d e f f 2 L 2 P t h A w   n o r m = δ s δ i ,
1 A w   n o r m = N L E p E s E i d x d y E p 2 d x d y E s 2 d x d y E y 2 d x d y .
1 W x = N L E p E s E i d x E p 2 d x E s 2 d x E y 2 d x ,
1 W y = N L F p F s F i d y F p 2 d y F s 2 d y F y 2 d y .
P t h   G W P t h   b = ( η Q P M η G W ) 2 W x W y A n o r m = 9 32 d W x A n o r m .

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