Abstract

We present a theoretical study of second-harmonic generation (SHG) in tapered optical fibers in the continuous wave and monochromatic regimes. The mechanism is based on modal phase matching, where the taper is used to sweep the phase mismatch across the phase-matched condition during the propagation. Self- and cross-phase modulations are taken into account. We report general results about the conversion efficiency and demonstrate that this SHG scheme is not only robust with respect to phase modulation, but can even benefit from it. We also show that the conversion bandwidth is easily controlled by the taper design and can be realistically extended to tens of nanometers.

© 2010 Optical Society of America

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References

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  1. W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer, 2006).
  2. Y. Fujii, B. S. Kawasaki, K. O. Hill, and D. C. Johnson, “Sum-frequency light generation in optical fibers,” Opt. Lett. 5, 48–50 (1980).
    [CrossRef] [PubMed]
  3. M. Farries, P. Russell, M. Fermann, and D. Payne, “Second-harmonic generation in an optical fibre by self-written χ(2) grating,” Electron. Lett. 23, 322–324 (1987).
    [CrossRef]
  4. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).
  5. R. Kashyap, “Phase-matched second-harmonic generation in periodically poled optical fibers,” Appl. Phys. Lett. 58, 1233–1235 (1991).
    [CrossRef]
  6. V. Pruneri, G. Bonfrate, P. G. Kazansky, D. J. Richardson, N. G. Broderick, J. P. de Sandro, C. Simonneau, P. Vidakovic, and J. A. Levenson, “Greater than 20%-efficient frequency doubling of 1532-nm nanosecond pulses in quasi-phase-matched germanosilicate optical fibers,” Opt. Lett. 24, 208–210 (1999).
    [CrossRef]
  7. A. Canagasabey, C. Corbari, A. V. Gladyshev, F. Liegeois, S. Guillemet, Y. Hernandez, M. V. Yashkov, A. Kosolapov, E. M. Dianov, M. Ibsen, and P. G. Kazansky, “High-average-power second-harmonic generation from periodically poled silica fibers,” Opt. Lett. 34, 2483–2485 (2009).
    [CrossRef]
  8. T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
    [CrossRef]
  9. K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
    [CrossRef]
  10. A. Canagasabey, M. Ibsen, K. Gallo, A. V. Gladyshev, E. M. Dianov, C. Corbari, and P. G. Kazansky, “Aperiodically poled silica fibers for bandwidth control of quasi-phase-matched second-harmonic generation,” Opt. Lett. 35, 724–726 (2010).
    [CrossRef]
  11. M. Fermann, L. Li, and M. Farries, “Frequency-doubling by modal phase matching in poled optical fibres,” Electron. Lett. 24, 894–895 (1988).
    [CrossRef]
  12. V. Grubsky and J. Feinberg, “Phase-matched third-harmonic uv generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007).
    [CrossRef]
  13. A. Podlipensky, J. Lange, G. Seifert, H. Graener, and I. Cravetchi, “Second-harmonic generation from ellipsoidal silver nanoparticles embedded in silica glass,” Opt. Lett. 28, 716–718 (2003).
    [CrossRef]
  14. J. I. Dadap, J. Shan, and T. F. Heinz, “Theory of optical second-harmonic generation from a sphere of centrosymmetric material: small-particle limit,” J. Opt. Soc. Am. B 21, 1328–1347 (2004).
    [CrossRef]
  15. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  16. D. Harter and D. Brown, “Effects of higher order nonlinearities on second-order frequency mixing,” IEEE J. Quantum Electron. 18, 1146–1151 (1982).
    [CrossRef]
  17. S. Trillo and S. Wabnitz, “Nonlinear parametric mixing instabilities induced by self-phase and cross-phase modulation,” Opt. Lett. 17, 1572–1574 (1992).
    [CrossRef]
  18. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  19. W. Choe, P. P. Banerjee, and F. C. Caimi, “Second-harmonic generation in an optical medium with second- and third-order nonlinear susceptibilities,” J. Opt. Soc. Am. B 8, 1013–1022 (1991).
    [CrossRef]
  20. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974).
  21. K. Petermann, “Fundamental mode microbending loss in graded-index and W fibres,” Opt. Quantum Electron. 9, 167–175 (1977).
    [CrossRef]
  22. R. A. Myers, N. Mukherjee, and S. R. J. Brueck, “Large second-order nonlinearity in poled fused silica,” Opt. Lett. 16, 1732–1734 (1991).
    [CrossRef]
  23. T. Birks and Y. Li, “The shape of fiber tapers,” J. Lightwave Technol. 10, 432–438 (1992).
    [CrossRef]
  24. A. Kudlinski and A. Mussot, “Visible cw-pumped supercontinuum,” Opt. Lett. 33, 2407–2409 (2008).
    [CrossRef]
  25. A. Bétourné, Y. Quiquempois, G. Bouwmans, and M. Douay, “Design of a photonic crystal fiber for phase-matched frequency doubling or tripling,” Opt. Express 16, 14255–14262 (2008).
    [CrossRef]
  26. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  27. K. Lai, S. G. Leon-Saval, A. Witkowska, W. J. Wadsworth, and T. A. Birks, “Wavelength-independent all-fiber mode converters,” Opt. Lett. 32, 328–330 (2007).
    [CrossRef]

2010 (1)

2009 (1)

2008 (2)

2007 (2)

K. Lai, S. G. Leon-Saval, A. Witkowska, W. J. Wadsworth, and T. A. Birks, “Wavelength-independent all-fiber mode converters,” Opt. Lett. 32, 328–330 (2007).
[CrossRef]

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic uv generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007).
[CrossRef]

2004 (1)

2003 (1)

1999 (1)

1997 (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

1994 (1)

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

1992 (2)

1991 (3)

1990 (1)

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

1988 (1)

M. Fermann, L. Li, and M. Farries, “Frequency-doubling by modal phase matching in poled optical fibres,” Electron. Lett. 24, 894–895 (1988).
[CrossRef]

1987 (1)

M. Farries, P. Russell, M. Fermann, and D. Payne, “Second-harmonic generation in an optical fibre by self-written χ(2) grating,” Electron. Lett. 23, 322–324 (1987).
[CrossRef]

1982 (1)

D. Harter and D. Brown, “Effects of higher order nonlinearities on second-order frequency mixing,” IEEE J. Quantum Electron. 18, 1146–1151 (1982).
[CrossRef]

1980 (1)

1977 (1)

K. Petermann, “Fundamental mode microbending loss in graded-index and W fibres,” Opt. Quantum Electron. 9, 167–175 (1977).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Banerjee, P. P.

Bétourné, A.

Birks, T.

T. Birks and Y. Li, “The shape of fiber tapers,” J. Lightwave Technol. 10, 432–438 (1992).
[CrossRef]

Birks, T. A.

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Bonfrate, G.

Bouwmans, G.

Broderick, N. G.

Brown, D.

D. Harter and D. Brown, “Effects of higher order nonlinearities on second-order frequency mixing,” IEEE J. Quantum Electron. 18, 1146–1151 (1982).
[CrossRef]

Brueck, S. R. J.

Caimi, F. C.

Canagasabey, A.

Choe, W.

Corbari, C.

Cravetchi, I.

Dadap, J. I.

de Sandro, J. P.

Dianov, E. M.

Douay, M.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

Farries, M.

M. Fermann, L. Li, and M. Farries, “Frequency-doubling by modal phase matching in poled optical fibres,” Electron. Lett. 24, 894–895 (1988).
[CrossRef]

M. Farries, P. Russell, M. Fermann, and D. Payne, “Second-harmonic generation in an optical fibre by self-written χ(2) grating,” Electron. Lett. 23, 322–324 (1987).
[CrossRef]

Feinberg, J.

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic uv generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007).
[CrossRef]

Fermann, M.

M. Fermann, L. Li, and M. Farries, “Frequency-doubling by modal phase matching in poled optical fibres,” Electron. Lett. 24, 894–895 (1988).
[CrossRef]

M. Farries, P. Russell, M. Fermann, and D. Payne, “Second-harmonic generation in an optical fibre by self-written χ(2) grating,” Electron. Lett. 23, 322–324 (1987).
[CrossRef]

Fujii, Y.

Gallo, K.

Gladyshev, A. V.

Graener, H.

Grubsky, V.

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic uv generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007).
[CrossRef]

Guillemet, S.

Harter, D.

D. Harter and D. Brown, “Effects of higher order nonlinearities on second-order frequency mixing,” IEEE J. Quantum Electron. 18, 1146–1151 (1982).
[CrossRef]

Heinz, T. F.

Hernandez, Y.

Hill, K. O.

Ibsen, M.

Johnson, D. C.

Kashyap, R.

R. Kashyap, “Phase-matched second-harmonic generation in periodically poled optical fibers,” Appl. Phys. Lett. 58, 1233–1235 (1991).
[CrossRef]

Kato, M.

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

Kawasaki, B. S.

Kazansky, P. G.

Koechner, W.

W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer, 2006).

Kosolapov, A.

Kudlinski, A.

Lai, K.

Lange, J.

Leon-Saval, S. G.

Levenson, J. A.

Li, L.

M. Fermann, L. Li, and M. Farries, “Frequency-doubling by modal phase matching in poled optical fibres,” Electron. Lett. 24, 894–895 (1988).
[CrossRef]

Li, Y.

T. Birks and Y. Li, “The shape of fiber tapers,” J. Lightwave Technol. 10, 432–438 (1992).
[CrossRef]

Liegeois, F.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974).

Mizuuchi, K.

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

Mukherjee, N.

Mussot, A.

Myers, R. A.

Nishihara, H.

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

Payne, D.

M. Farries, P. Russell, M. Fermann, and D. Payne, “Second-harmonic generation in an optical fibre by self-written χ(2) grating,” Electron. Lett. 23, 322–324 (1987).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Petermann, K.

K. Petermann, “Fundamental mode microbending loss in graded-index and W fibres,” Opt. Quantum Electron. 9, 167–175 (1977).
[CrossRef]

Podlipensky, A.

Pruneri, V.

Quiquempois, Y.

Richardson, D. J.

Russell, P.

M. Farries, P. Russell, M. Fermann, and D. Payne, “Second-harmonic generation in an optical fibre by self-written χ(2) grating,” Electron. Lett. 23, 322–324 (1987).
[CrossRef]

Sato, H.

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

Seifert, G.

Shan, J.

Simonneau, C.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Suhara, T.

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

Trillo, S.

Vidakovic, P.

Wabnitz, S.

Wadsworth, W. J.

Witkowska, A.

Yamamoto, K.

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

Yashkov, M. V.

Appl. Phys. Lett. (1)

R. Kashyap, “Phase-matched second-harmonic generation in periodically poled optical fibers,” Appl. Phys. Lett. 58, 1233–1235 (1991).
[CrossRef]

Electron. Lett. (2)

M. Farries, P. Russell, M. Fermann, and D. Payne, “Second-harmonic generation in an optical fibre by self-written χ(2) grating,” Electron. Lett. 23, 322–324 (1987).
[CrossRef]

M. Fermann, L. Li, and M. Farries, “Frequency-doubling by modal phase matching in poled optical fibres,” Electron. Lett. 24, 894–895 (1988).
[CrossRef]

IEEE J. Quantum Electron. (3)

T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings,” IEEE J. Quantum Electron. 26, 1265–1276 (1990).
[CrossRef]

K. Mizuuchi, K. Yamamoto, M. Kato, and H. Sato, “Broadening of the phase-matching bandwidth in quasi-phase-matched second-harmonic generation,” IEEE J. Quantum Electron. 30, 1596–1604 (1994).
[CrossRef]

D. Harter and D. Brown, “Effects of higher order nonlinearities on second-order frequency mixing,” IEEE J. Quantum Electron. 18, 1146–1151 (1982).
[CrossRef]

J. Lightwave Technol. (2)

T. Birks and Y. Li, “The shape of fiber tapers,” J. Lightwave Technol. 10, 432–438 (1992).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

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

Opt. Commun. (1)

V. Grubsky and J. Feinberg, “Phase-matched third-harmonic uv generation using low-order modes in a glass micro-fiber,” Opt. Commun. 274, 447–450 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (9)

A. Kudlinski and A. Mussot, “Visible cw-pumped supercontinuum,” Opt. Lett. 33, 2407–2409 (2008).
[CrossRef]

A. Canagasabey, C. Corbari, A. V. Gladyshev, F. Liegeois, S. Guillemet, Y. Hernandez, M. V. Yashkov, A. Kosolapov, E. M. Dianov, M. Ibsen, and P. G. Kazansky, “High-average-power second-harmonic generation from periodically poled silica fibers,” Opt. Lett. 34, 2483–2485 (2009).
[CrossRef]

A. Canagasabey, M. Ibsen, K. Gallo, A. V. Gladyshev, E. M. Dianov, C. Corbari, and P. G. Kazansky, “Aperiodically poled silica fibers for bandwidth control of quasi-phase-matched second-harmonic generation,” Opt. Lett. 35, 724–726 (2010).
[CrossRef]

K. Lai, S. G. Leon-Saval, A. Witkowska, W. J. Wadsworth, and T. A. Birks, “Wavelength-independent all-fiber mode converters,” Opt. Lett. 32, 328–330 (2007).
[CrossRef]

Y. Fujii, B. S. Kawasaki, K. O. Hill, and D. C. Johnson, “Sum-frequency light generation in optical fibers,” Opt. Lett. 5, 48–50 (1980).
[CrossRef] [PubMed]

R. A. Myers, N. Mukherjee, and S. R. J. Brueck, “Large second-order nonlinearity in poled fused silica,” Opt. Lett. 16, 1732–1734 (1991).
[CrossRef]

S. Trillo and S. Wabnitz, “Nonlinear parametric mixing instabilities induced by self-phase and cross-phase modulation,” Opt. Lett. 17, 1572–1574 (1992).
[CrossRef]

V. Pruneri, G. Bonfrate, P. G. Kazansky, D. J. Richardson, N. G. Broderick, J. P. de Sandro, C. Simonneau, P. Vidakovic, and J. A. Levenson, “Greater than 20%-efficient frequency doubling of 1532-nm nanosecond pulses in quasi-phase-matched germanosilicate optical fibers,” Opt. Lett. 24, 208–210 (1999).
[CrossRef]

A. Podlipensky, J. Lange, G. Seifert, H. Graener, and I. Cravetchi, “Second-harmonic generation from ellipsoidal silver nanoparticles embedded in silica glass,” Opt. Lett. 28, 716–718 (2003).
[CrossRef]

Opt. Quantum Electron. (1)

K. Petermann, “Fundamental mode microbending loss in graded-index and W fibres,” Opt. Quantum Electron. 9, 167–175 (1977).
[CrossRef]

Phys. Rev. (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Other (4)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

W. Koechner, Solid-State Laser Engineering, 6th ed. (Springer, 2006).

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Figures (9)

Fig. 1
Fig. 1

Fundamental and SH wave normalized powers as functions of the position along the fiber taper for z ̃ 0 = 100 , and (a) | κ ̃ | = 0.1 , (b) | κ ̃ | = 1 , and (c) | κ ̃ | = 10 .

Fig. 2
Fig. 2

Evolutions of (a) the relative phase θ ( z ̃ ) and of (b) the fundamental and SH powers along the fiber sample for z ̃ 0 = L / L shg = 100 and | κ ̃ | = 0.2 . Inset (a): θ ( z ̃ ) over the whole fiber length.

Fig. 3
Fig. 3

Conversion efficiency as a function of the modulus of the phase mismatch slope.

Fig. 4
Fig. 4

Fundamental and SH wave normalized powers as functions of the position along the fiber taper for z ̃ 0 = 100 and γ ̃ f ranging from 0.1 to 40. (a)–(d) κ ̃ = 1 and (e)–(h) κ ̃ = 1 .

Fig. 5
Fig. 5

Conversion efficiency as a function of γ ̃ f / | κ ̃ | for several values of | κ ̃ | , with z ̃ 0 = L / L shg = 100 . Solid lines are for κ ̃ < 0 , and dashed lines are for κ ̃ > 0 .

Fig. 6
Fig. 6

Effective phase mismatch Δ k eff and its two contributions 2 γ ̃ f Δ I and κ ̃ ( z ̃ ) as functions of the position along the fiber taper for z ̃ 0 = L / L shg = 100 and γ ̃ f = 10 . In (a) κ ̃ = 1 , and in (b) κ ̃ = 1 . In (b) we have added κ ̃ ( z ̃ ) to highlight the balancing of both terms of Eq. (18).

Fig. 7
Fig. 7

(a) Normalized field distributions ψ f ( r ) and ψ sh ( r ) corresponding to, respectively, the LP 01 mode at λ = 1064   nm and the LP 02 mode at λ = 532   nm , with r being the radial position with respect to the center of the fiber. Inset: Intensity distributions | ψ f ( r ) | 2 and | ψ sh ( r ) | 2 . (b) L shg , given by Eq. (10), as a function of the geometry of χ e ( 2 ) , for a fundamental wave input power of 1 W. The upper curve corresponds to a ring geometry with inner radius R and outer radius r fiber , and the lower curve corresponds to a step geometry with outer radius R. The optimum value of R opt = 0.7 μ m gives a minimum L shg = 14   cm . The case of a homogeneous χ e ( 2 ) = 1   pm / V over the whole fiber cross section corresponds to the upper curve at R = 0 μ m and to the asymptotic value of lower curve at R , almost already reached at R 4 μ m .

Fig. 8
Fig. 8

Oscillation period L c of the signal power as a function of the relative deviation Δ a = ( a a ideal ) / a ideal of the core radius from the ideal (i.e., corresponding to perfect phase matching) value. Calculations are performed based on the fiber design described in the text.

Fig. 9
Fig. 9

Conversion efficiency as a function of the fundamental wavelength λ pump for three tapers with the same phase mismatch slope κ ̃ = 10 , but with different normalized lengths L ̃ = L / L shg and different diameter variations Δ D . Solid line: L ̃ = 140 , Δ D = 1.6 % ; dashed line: L ̃ = 280 , Δ D = 3.2 % ; dotted line: L ̃ = 420 , Δ D = 4.8 % . No phase modulation is assumed.

Equations (37)

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[ Δ + ω i 2 c 2 n z 2 ( r , ω i ) β i l 2 ( z ) ] ψ i l ( r , θ , z ) = 0.
E f ( r , θ , z ; ω ) = 1 2 { ψ f ( r , θ ) A f ( z ) exp ( α f z ) exp [ i ϕ f ( z ) ] + c .c . } ,
E sh ( r , θ , z ; 2 ω ) = 1 2 { ψ sh ( r , θ ) A sh ( z ) exp ( α sh z ) exp [ i ϕ sh ( z ) ] + c .c . } ,
ϕ i ( z ) = 0 z β i ( ξ ) d ξ .
d A f d z = α f A f + i ( γ f f | A f | 2 + 2 γ f sh | A sh | 2 ) A f + i ω f 2 n f c χ ̃ e ( 2 ) A sh A f e i Δ ϕ ,
d A sh d z = α sh A sh + i ( γ sh sh | A sh | 2 + 2 γ sh f | A f | 2 ) A sh + i ω f 2 n sh c χ ̃ e ( 2 ) A f 2 e i Δ ϕ ,
β i = 2 π λ i n i ,
Δ ϕ ( z ) = 0 z κ ( ξ ) d ξ .
γ i j = n 2 ω i c A eff i j ,
χ ̃ e ( 2 ) = χ e ( 2 ) ψ f 2 ψ sh r d r d θ ,
A eff i j = 1 | ψ i | 2 | ψ j | 2 r d r d θ .
L shg = [ ω f 2 n c χ ̃ e ( 2 ) A f ( 0 ) ] 1 ,
L pm f = [ γ f | A f ( 0 ) | 2 ] 1 ,
L pm sh = [ γ sh | A f ( 0 ) | 2 ] 1 .
d A ̃ f d z ̃ = i γ ̃ f ( | A ̃ f | 2 + 2 | A ̃ sh | 2 ) A ̃ f + i A ̃ sh A ̃ f e i Δ ϕ ,
d A ̃ sh d z ̃ = i γ ̃ sh ( | A ̃ sh | 2 + 2 | A ̃ f | 2 ) A ̃ sh + i A ̃ f 2 e i Δ ϕ ,
A ̃ i = A i / A f ( 0 ) ,
z ̃ = z / L shg ,
κ ̃ ( z ̃ ) = κ ( z ̃ ) L shg ,
Δ ϕ ( z ̃ ) = 0 z ̃ κ ̃ ( ξ ) d ξ ,
γ ̃ f = L shg / L pm f ,
γ ̃ sh = L shg / L pm sh .
A ̃ sh = a sh e i θ sh ,
A ̃ f = a f e i θ f ,
θ = θ sh 2 θ f Δ ϕ ,
d a f d z ̃ = a sh a f   sin   θ ,
d a sh d z ̃ = a f 2   sin   θ ,
d θ d z ̃ = Δ k eff + ( a f 2 a sh 2 a sh ) cos   θ ,
Δ k eff = κ ̃ + γ ̃ sh ( a sh 2 + 2 a f 2 ) 2 γ ̃ f ( 2 a sh 2 + a f 2 ) .
κ ̃ ( z ̃ ) = κ ̃ ( 2 z ̃ z ̃ 0 ) .
κ ̃ ( z ̃ ) = 0.
Δ k eff = κ ̃ ( z ̃ ) + 2 γ ̃ f Δ I ,
z ̃ pm = z ̃ 0 2 + γ ̃ f κ ̃ ,
Δ I ( z ̃ ) = κ ̃ 2 γ ̃ f ( 2 z ̃ z ̃ 0 ) ,     | z ̃ z ̃ 0 / 2 | < γ ̃ f / κ ̃ ,
= 1 ,     z ̃ < z ̃ 0 / 2 + γ ̃ f / κ ̃ ,
= 1 ,     z ̃ > z ̃ 0 / 2 γ ̃ f / κ ̃ .
Δ κ ̃ L ̃ fluct . 2 κ ̃ ,

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