Abstract

Cascaded third-harmonic generation (THG) of femtosecond pulses in a two-dimensional periodically poled lithium niobate is demonstrated experimentally. The poling pattern satisfies quasi-phase-matching conditions for both second-harmonic generation (SHG) and the following sum-frequency generation. The pulse-front tilt in noncollinear interactions compensates for group-velocity mismatch among fundamental, second-harmonic (SH), and third-harmonic (TH) pulses, thereby achieving broad acceptance bandwidths in SHG and THG. A 2mm thick device generates SH and TH pulses of 106 and 110fs, respectively, from fundamental pulses with temporal duration of 118fs and a 1568nm center wavelength. The frequency conversion properties are studied theoretically, and the optimal focusing condition for maximum conversion efficiency is derived.

© 2007 Optical Society of America

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References

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  1. E. Sidick, A. Knoesen, and A. Dienes, "Ultrashort-pulse second-harmonic generation. I. Transform-limited fundamental pulses," J. Opt. Soc. Am. B 12, 1704-1712 (1995).
    [CrossRef]
  2. E. Sidick, A. Dienes, and A. Knoesen, "Ultrashort-pulse second-harmonic generation. II. Non-transform-limited fundamental pulses," J. Opt. Soc. Am. B 12, 1713-1722 (1995).
    [CrossRef]
  3. M. A. Arbore, O. Marco, and M. M. Fejer, "Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings," Opt. Lett. 22, 865-867 (1997).
    [CrossRef] [PubMed]
  4. M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, "Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped period-poled lithium niobate," Opt. Lett. 22, 1341-1343 (1997).
    [CrossRef]
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    [CrossRef]
  6. G. Imeshev, M. A. Arbore, S. Kasriel, and M. M. Fejer, "Pulse shaping and compression by second-harmonic generation with quasi-phase-matching gratings in the presence of arbitrary dispersion," J. Opt. Soc. Am. B 17, 1420-1437 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, and N. B. Ming, "Third-harmonic generation in a general two-component quasi-periodic optical superlattice," Opt. Lett. 26, 899-901 (2001).
    [CrossRef]
  21. M. L. Sundheimer, A. Villeneuve, G. I. Stegeman, and J. D. Bierlein, "Simultaneous generation of red, green, and blue light in a segmented KTP waveguide using a single source," Electron. Lett. 30, 975-976 (1994).
    [CrossRef]
  22. P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, "Simultaneous generation of red, green, and blue light in room temperature periodically poled lithium niobate waveguides using single source," Electron. Lett. 31, 1350-1351 (1995).
    [CrossRef]
  23. G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, "Simultaneously efficient blue and red light generations in aperiodically poled LiTaO3," Appl. Phys. Lett. 78, 3006-3008 (2001).
    [CrossRef]
  24. M. Marangoni, M. Lobino, and R. Ramponi, "Simultaneously phase-matched second- and third-harmonic generation from 1.55μm radiation in annealed proton-exchanged periodically poled lithium niobate waveguides," Opt. Lett. 31, 2707-2709 (2006).
    [CrossRef] [PubMed]
  25. V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).
    [CrossRef]
  26. S. Saltiel and Y. S. Kivshar, "Phase matching in nonlinear χ(2) photonic crystals," Opt. Lett. 25, 1204-1206 (2000).
    [CrossRef]
  27. N. G. R. Broderick, R. T. Bratfalean, T. M. Monro, D. J. Richardson, and C. M. de Sterke, "Temperature and wavelength tuning of second-, third-, and fourth-harmonic generation in a two-dimensional hexagonally poled nonlinear crystal," J. Opt. Soc. Am. B 19, 2263-2272 (2002).
    [CrossRef]
  28. A. H. Norton and C. M. de Sterke, "Optimal poling of nonlinear photonic crystals for frequency conversion," Opt. Lett. 28, 188-190 (2003).
    [CrossRef] [PubMed]
  29. A. H. Norton and C. M. de Sterke, "Two-dimensional poling patterns for third and fourth harmonic generation," Opt. Express 11, 1008-1014 (2003).
    [CrossRef] [PubMed]
  30. N. Fujioka, S. Ashihara, H. Ono, T. Shimura, and K. Kuroda, "Group-velocity-mismatch compensation in cascaded third-harmonic generation with two-dimensional quasi-phase-matching gratings," Opt. Lett. 31, 2780-2782 (2006).
    [CrossRef] [PubMed]
  31. S. M. Saltiel and Y. S. Kivshar, "Group-velocity-matched multistep cascading in nonlinear photonic crystals," Opt. Lett. 31, 3321-3323 (2006).
    [CrossRef] [PubMed]
  32. G. J. Edwards and M. Lawrence, "A temperature-dependent dispersion equation for congruently grown lithium niobate," Opt. Quantum Electron. 16, 373-375 (1984).
    [CrossRef]
  33. P. Baum, S. Lochbrunner, and E. Riedle, "Generation of tunable 7-fs ultraviolet pulses: achromatic phase matching and chirp management," Appl. Phys. B 79, 1027-1032 (2004).
    [CrossRef]
  34. G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3639 (1968).
    [CrossRef]

2006 (3)

2005 (2)

2004 (3)

I. A. Begishev, M. Kalashnikov, V. Karpov, P. Nickles, H. Schönnagel, I. A. Kulagin, and T. Usmanov, "Limitation of second-harmonic generation of femtosecond Ti:sapphire laser pulses," J. Opt. Soc. Am. B 21, 318-322 (2004).
[CrossRef]

S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, "Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate," Appl. Phys. Lett. 84, 1055-1057 (2004).
[CrossRef]

P. Baum, S. Lochbrunner, and E. Riedle, "Generation of tunable 7-fs ultraviolet pulses: achromatic phase matching and chirp management," Appl. Phys. B 79, 1027-1032 (2004).
[CrossRef]

2003 (3)

2002 (3)

2001 (2)

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, "Simultaneously efficient blue and red light generations in aperiodically poled LiTaO3," Appl. Phys. Lett. 78, 3006-3008 (2001).
[CrossRef]

C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, and N. B. Ming, "Third-harmonic generation in a general two-component quasi-periodic optical superlattice," Opt. Lett. 26, 899-901 (2001).
[CrossRef]

2000 (3)

1998 (2)

V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).
[CrossRef]

K. Mori, Y. Tamaki, M. Obara, and K. Midorikawa, "Second-harmonic generation of femtosecond high-intensity Ti:sapphire laser pulses," J. Appl. Phys. 83, 2915-2919 (1998).
[CrossRef]

1997 (3)

1996 (1)

1995 (3)

E. Sidick, A. Knoesen, and A. Dienes, "Ultrashort-pulse second-harmonic generation. I. Transform-limited fundamental pulses," J. Opt. Soc. Am. B 12, 1704-1712 (1995).
[CrossRef]

E. Sidick, A. Dienes, and A. Knoesen, "Ultrashort-pulse second-harmonic generation. II. Non-transform-limited fundamental pulses," J. Opt. Soc. Am. B 12, 1713-1722 (1995).
[CrossRef]

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, "Simultaneous generation of red, green, and blue light in room temperature periodically poled lithium niobate waveguides using single source," Electron. Lett. 31, 1350-1351 (1995).
[CrossRef]

1994 (1)

M. L. Sundheimer, A. Villeneuve, G. I. Stegeman, and J. D. Bierlein, "Simultaneous generation of red, green, and blue light in a segmented KTP waveguide using a single source," Electron. Lett. 30, 975-976 (1994).
[CrossRef]

1990 (1)

G. Szabó and Z. Bor, "Broadband frequency doubler for femtosecond pulses," Appl. Phys. B 50, 51-54 (1990).
[CrossRef]

1989 (1)

O. E. Martinez, "Achromatic phase matching for second harmonic generation of femtosecond pulses," IEEE J. Quantum Electron. 25, 2464-2468 (1989).
[CrossRef]

1984 (1)

G. J. Edwards and M. Lawrence, "A temperature-dependent dispersion equation for congruently grown lithium niobate," Opt. Quantum Electron. 16, 373-375 (1984).
[CrossRef]

1968 (1)

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3639 (1968).
[CrossRef]

Appl. Phys. B (2)

G. Szabó and Z. Bor, "Broadband frequency doubler for femtosecond pulses," Appl. Phys. B 50, 51-54 (1990).
[CrossRef]

P. Baum, S. Lochbrunner, and E. Riedle, "Generation of tunable 7-fs ultraviolet pulses: achromatic phase matching and chirp management," Appl. Phys. B 79, 1027-1032 (2004).
[CrossRef]

Appl. Phys. Lett. (2)

S. Ashihara, T. Shimura, K. Kuroda, N. E. Yu, S. Kurimura, K. Kitamura, M. Cha, and T. Taira, "Optical pulse compression using cascaded quadratic nonlinearities in periodically poled lithium niobate," Appl. Phys. Lett. 84, 1055-1057 (2004).
[CrossRef]

G. Z. Luo, S. N. Zhu, J. L. He, Y. Y. Zhu, H. T. Wang, Z. W. Liu, C. Zhang, and N. B. Ming, "Simultaneously efficient blue and red light generations in aperiodically poled LiTaO3," Appl. Phys. Lett. 78, 3006-3008 (2001).
[CrossRef]

Electron. Lett. (2)

M. L. Sundheimer, A. Villeneuve, G. I. Stegeman, and J. D. Bierlein, "Simultaneous generation of red, green, and blue light in a segmented KTP waveguide using a single source," Electron. Lett. 30, 975-976 (1994).
[CrossRef]

P. Baldi, C. G. Trevino-Palacios, G. I. Stegeman, M. P. De Micheli, D. B. Ostrowsky, D. Delacourt, and M. Papuchon, "Simultaneous generation of red, green, and blue light in room temperature periodically poled lithium niobate waveguides using single source," Electron. Lett. 31, 1350-1351 (1995).
[CrossRef]

IEEE J. Quantum Electron. (1)

O. E. Martinez, "Achromatic phase matching for second harmonic generation of femtosecond pulses," IEEE J. Quantum Electron. 25, 2464-2468 (1989).
[CrossRef]

J. Appl. Phys. (2)

K. Mori, Y. Tamaki, M. Obara, and K. Midorikawa, "Second-harmonic generation of femtosecond high-intensity Ti:sapphire laser pulses," J. Appl. Phys. 83, 2915-2919 (1998).
[CrossRef]

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3639 (1968).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

F. W. Wise, L. Qian, and X. Liu, "Applications of cascaded quadratic nonlinearities to femtosecond pulse generation," J. Nonlinear Opt. Phys. Mater. 11, 317-338 (2002).
[CrossRef]

J. Opt. Soc. Am. B (11)

G. Imeshev, M. A. Arbore, M. M. Fejer, A. Galvanauskas, M. Fermann, and D. Harter, "Ultrashort-pulse second-harmonic generation with longitudinally nonuniform quasi-phase-matching gratings: pulse compression and shaping," J. Opt. Soc. Am. B 17, 304-318 (2000).
[CrossRef]

E. Sidick, A. Knoesen, and A. Dienes, "Ultrashort-pulse second-harmonic generation. I. Transform-limited fundamental pulses," J. Opt. Soc. Am. B 12, 1704-1712 (1995).
[CrossRef]

E. Sidick, A. Dienes, and A. Knoesen, "Ultrashort-pulse second-harmonic generation. II. Non-transform-limited fundamental pulses," J. Opt. Soc. Am. B 12, 1713-1722 (1995).
[CrossRef]

T. Ditmire, A. M. Rubenchik, D. Eimerl, and M. D. Perry, "Effects of cubic nonlinearity on frequency doubling of high-power laser pulses," J. Opt. Soc. Am. B 13, 649-655 (1996).
[CrossRef]

S. Ashihara, T. Shimura, and K. Kuroda, "Group-velocity matched second-harmonic generation in tilted quasi-phase-matched gratings," J. Opt. Soc. Am. B 20, 853-856 (2003).
[CrossRef]

G. Imeshev, M. A. Arbore, S. Kasriel, and M. M. Fejer, "Pulse shaping and compression by second-harmonic generation with quasi-phase-matching gratings in the presence of arbitrary dispersion," J. Opt. Soc. Am. B 17, 1420-1437 (2000).
[CrossRef]

N. G. R. Broderick, R. T. Bratfalean, T. M. Monro, D. J. Richardson, and C. M. de Sterke, "Temperature and wavelength tuning of second-, third-, and fourth-harmonic generation in a two-dimensional hexagonally poled nonlinear crystal," J. Opt. Soc. Am. B 19, 2263-2272 (2002).
[CrossRef]

S. Ashihara, J. Nishina, T. Shimura, and K. Kuroda, "Soliton compression of femtosecond pulses in quadratic media," J. Opt. Soc. Am. B 19, 2505-2510 (2002).
[CrossRef]

I. A. Begishev, M. Kalashnikov, V. Karpov, P. Nickles, H. Schönnagel, I. A. Kulagin, and T. Usmanov, "Limitation of second-harmonic generation of femtosecond Ti:sapphire laser pulses," J. Opt. Soc. Am. B 21, 318-322 (2004).
[CrossRef]

N. Fujioka, S. Ashihara, H. Ono, T. Shimura, and K. Kuroda, "Group-velocity-matched noncollinear second-harmonic generation in quasi-phase matching," J. Opt. Soc. Am. B 22, 1283-1289 (2005).
[CrossRef]

A. M. Schober, M. Charbonneau-Lefort, and M. M. Fejer, "Broadband quasi-phase-matched second-harmonic generation of ultrashort optical pulses with spectral angular dispersion," J. Opt. Soc. Am. B 22, 1699-1713 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (8)

A. H. Norton and C. M. de Sterke, "Optimal poling of nonlinear photonic crystals for frequency conversion," Opt. Lett. 28, 188-190 (2003).
[CrossRef] [PubMed]

M. Marangoni, M. Lobino, and R. Ramponi, "Simultaneously phase-matched second- and third-harmonic generation from 1.55μm radiation in annealed proton-exchanged periodically poled lithium niobate waveguides," Opt. Lett. 31, 2707-2709 (2006).
[CrossRef] [PubMed]

N. Fujioka, S. Ashihara, H. Ono, T. Shimura, and K. Kuroda, "Group-velocity-mismatch compensation in cascaded third-harmonic generation with two-dimensional quasi-phase-matching gratings," Opt. Lett. 31, 2780-2782 (2006).
[CrossRef] [PubMed]

S. M. Saltiel and Y. S. Kivshar, "Group-velocity-matched multistep cascading in nonlinear photonic crystals," Opt. Lett. 31, 3321-3323 (2006).
[CrossRef] [PubMed]

M. A. Arbore, O. Marco, and M. M. Fejer, "Pulse compression during second-harmonic generation in aperiodic quasi-phase-matching gratings," Opt. Lett. 22, 865-867 (1997).
[CrossRef] [PubMed]

M. A. Arbore, A. Galvanauskas, D. Harter, M. H. Chou, and M. M. Fejer, "Engineerable compression of ultrashort pulses by use of second-harmonic generation in chirped period-poled lithium niobate," Opt. Lett. 22, 1341-1343 (1997).
[CrossRef]

S. Saltiel and Y. S. Kivshar, "Phase matching in nonlinear χ(2) photonic crystals," Opt. Lett. 25, 1204-1206 (2000).
[CrossRef]

C. Zhang, H. Wei, Y. Y. Zhu, H. T. Wang, S. N. Zhu, and N. B. Ming, "Third-harmonic generation in a general two-component quasi-periodic optical superlattice," Opt. Lett. 26, 899-901 (2001).
[CrossRef]

Opt. Quantum Electron. (1)

G. J. Edwards and M. Lawrence, "A temperature-dependent dispersion equation for congruently grown lithium niobate," Opt. Quantum Electron. 16, 373-375 (1984).
[CrossRef]

Phys. Rev. Lett. (1)

V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).
[CrossRef]

Science (1)

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, "Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice," Science 278, 843-846 (1997).
[CrossRef]

Other (1)

S. M. Saltiel, A. A. Sukhorukov, and Y. S. Kivshar, "Multistep parametric processes in nonlinear optics," in Progress in Optics, Vol. XLVII (Elsevier, 2005), pp. 1-74.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

(a) Phase-matching diagram of cascaded THG for a frequency ω component in the z x plane, where k ( j ω ) ( j = 1 , 2 , 3 ) is a wave vector of the fundamental, SH, and TH pulses, respectively. Grating vectors K S and K T are QPM vectors for SHG and SFG processes, respectively. (b) Time-domain picture of the compensation for GV mismatch among fundamental, SH, and TH pulses in 2D QPM gratings. The thick solid outline, the thin dashed outline, and the gray shaded area of ellipse indicate the fundamental, SH, and TH envelopes, respectively, u i is the GV of the pulse and v i is the normal speed of the pulse front, where i = 1 , 2 , 3 denotes the fundamental, SH, and TH pulses, respectively. The pulse fronts of three pulses travel with the same speed in a direction normal to the pulse front, while they slide in a direction parallel to the pulse front.

Fig. 2
Fig. 2

(a) Photograph of the + Z surface of the fabricated 2D PPLN device. The domain boundaries were observed after the sample was etched in an acid. Long side of the parallelogram (the unit poling pattern) is aligned parallel to the Y axis of the LN crystal. (b) Arrangement of the fabricated device in the cascaded THG experiment.

Fig. 3
Fig. 3

Experimental setup: half-wave plate (HWP); polarizer (Pol.); beam splitter (BS); mirrors (M); diffraction gratings (G); vertical-focusing cylindrical lens (CL); optical parametric amplifier (OPA). Each of the telescopes consists of two cylindrical lenses for horizontal focusing.

Fig. 4
Fig. 4

SH and the TH power measured by varying the center wavelength of a fundamental pulse, while holding the input power constant at 530 μ W . Both the SH power and the TH power are maximized at 1568 nm .

Fig. 5
Fig. 5

(a) Power spectrum of an SH pulse and (b) TH pulse, measured for an input peak intensity of 10 GW cm 2 .

Fig. 6
Fig. 6

(a) Intensity autocorrelation trace of an SH pulse and (b) intensity cross-correlation trace of a TH pulse and an original fundamental pulse. The temporal duration in FWHM is estimated as 106 fs for an SH pulse and 110 fs for a TH pulse. The measurements are performed after the second diffraction grating corrects the spectral angular dispersions.

Fig. 7
Fig. 7

(a) Measured SH and TH power against the input fundamental power (or peak intensity) at a low-excitation regime and (b) at higher excitations. (c) Corresponding SHG/THG efficiency for low-excitation regime and (d) at higher excitations.

Fig. 8
Fig. 8

(a) Calculated SHG efficiency ( h 2 w 0 ) against beam size w 0 of a fundamental pulse. (b) Calculated THG efficiency ( h 3 w 0 2 ) against beam size w 0 of a fundamental pulse. The SHG and THG efficiencies are maximized at beam sizes of w 0 10 and 24 μ m , respectively.

Fig. 9
Fig. 9

Frequency-domain beam profiles of SH pulses, calculated for fundamental beam sizes w 0 of (a) 10 μ m , (b) 24 μ m , (c) 100 μ m , and (d) 1000 μ m (solid curves). The dashed curves are the beam profiles of SH pulses, calculated with the assumptions of no spatial walk-off, no beam diffraction, and no GVD.

Fig. 10
Fig. 10

Frequency-domain beam profiles of TH pulses, calculated for fundamental beam sizes w 0 of (a) 10 μ m , (b) 24 μ m , (c) 100 μ m , and (d) 1000 μ m (solid curves). The dashed curves are the beam profiles of TH pulses, calculated with the assumptions of no spatial walk-off, no beam diffraction, and no GVD.

Tables (1)

Tables Icon

Table 1 Spectral Angular Dispersion Parameter γ 1 , Aperture Length L a , 2 , L a , 3 , and Rayleigh Lengths L R , 1 , L R , 2 and L R , 3 Are Listed for Different Beam Diameters of w 0 = 10 , 100, and 1000 μ m

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

k ( 2 ω 0 ) = 2 k ( ω 0 ) + K 2 ,
k ( 3 ω 0 ) = 3 k ( ω 0 ) + K 3 .
d k ( ω ) d ω ω 0 = d k ( ω ) d ω 2 ω 0 = d k ( ω ) d ω 3 ω 0 .
d k d ω r t = 0 .
d k d ω u = 1 .
d k ( ω ) d ω = d k ( ω ) d ω e ( ζ k α j k ) k ( ω ) d α j k d ω e ( ζ k α j k + π 2 ) ,
tan ρ j = k ( ω ) ( d k ( ω ) d ω ) 1 d α j k d ω .
E j ( z , x , t ) = E 0 B j ( z , x , t ) exp ( i ω j t i k z , j z i k x , j x ) ,
2 B 1 z 2 2 i k 1 B 1 z + 2 B 1 x 2 2 i k 1 u 1 B 1 t ( 1 u 1 2 + k 1 β 1 ) 2 B 1 t 2 = 0 ,
2 B j z 2 2 i k z , j B j z + 2 B j x 2 2 i k x , j B j x 2 i k j u j B j t ( 1 u j 2 + k j β j ) 2 B j t 2 = { 4 ω 1 2 E 0 c 2 d ( z , x ) B 1 2 exp ( i Δ k z , 2 z + i Δ k x , 2 x ) , ( j = 2 ) 18 ω 1 2 E 0 c 2 d ( z , x ) B 1 B 2 exp ( i Δ k z , 3 z + i Δ k x , 3 x ) , ( j = 3 ) ,
B 1 z ¯ + i 1 2 L L R , 1 2 B 1 x ¯ 2 i L L D , 1 2 B 1 τ ¯ 2 i γ 1 L L R , 1 2 B 1 x ¯ τ ¯ = 0 ,
B j z ¯ ( L L g , j + γ 1 L L a , j ) B i τ ¯ + L L a , j B j x ¯ + i 1 2 L L R , j 2 B j x ¯ 2 i L L D , j 2 B j τ ¯ 2 i γ 1 L L R , j 2 B j x ¯ τ ¯ = { i c 2 B 1 2 , ( j = 2 ) i c 3 B 1 B 2 , ( j = 3 ) .
B j ( z ¯ , x ¯ , τ ¯ ) = B ̂ j ( z ¯ , ξ ¯ j , Ω ¯ j ) exp ( i Ω ¯ j τ ¯ i ξ ¯ j x ¯ ) d ξ ¯ j d Ω ¯ j ,
B ̂ j ( z ¯ , ξ ¯ j , Ω ¯ j ) = 1 2 π B j ( z ¯ , x ¯ , τ ¯ ) exp ( i Ω ¯ j τ ¯ + i ξ ¯ j x ¯ ) d x ¯ d τ ¯ ,
B ̂ 1 z ¯ i 1 2 L L R , 1 ξ ¯ 1 2 B ̂ 1 + i L L D , 1 Ω ¯ 1 2 B ̂ 1 i γ 1 L L R , 1 Ω ¯ 1 ξ ¯ 1 B ̂ 1 = 0 ,
B ̂ j z ¯ i ( L L g , j + γ 1 L L a , j ) Ω ¯ j B ̂ j i L L a , j ξ ¯ j B ̂ j i 1 2 L L R , j ξ ¯ j 2 B ̂ j + i L L D , j Ω ¯ j 2 B ̂ j i γ 1 L L R , j Ω ¯ j ξ ¯ j B ̂ j = i c j F ̂ j ( z ¯ , ξ ¯ j , Ω ¯ j ) ,
( j = 2 , 3 ) ,
F ̂ 2 ( z ¯ , ξ ¯ 2 , Ω ¯ 2 ) = 1 2 π B ̂ 1 ( z ¯ , ξ ¯ , Ω ¯ ) B ̂ 1 ( z ¯ , ξ ¯ 2 ξ ¯ , Ω ¯ 2 Ω ¯ ) d ξ ¯ d Ω ¯ ,
F ̂ 3 ( z ¯ , ξ ¯ 3 , Ω ¯ 3 ) = 1 2 π B ̂ 1 ( z ¯ , ξ ¯ 3 ξ ¯ , Ω ¯ 3 Ω ¯ ) B ̂ 2 ( z ¯ , ξ ¯ , Ω ¯ ) d ξ ¯ d Ω ¯ .
B ̂ 1 ( z ¯ , ξ ¯ 1 , Ω ¯ 1 ) = B ̂ 0 ( ξ ¯ 1 , Ω ¯ 1 ) exp ( i K 1 ( ξ ¯ 1 , Ω ¯ 1 ) ( z ¯ z ¯ 0 ) ) ,
B ̂ j ( z ¯ , ξ ¯ j , Ω ¯ j ) = G ̂ j ( z ¯ , ξ ¯ j , Ω ¯ j ) exp ( i K j ( ξ ¯ j , Ω ¯ j ) ( z ¯ z ¯ 0 ) ) .
K 1 ( ξ ¯ 1 , Ω ¯ 1 ) = 1 2 L L R , 1 ξ ¯ 1 2 + L L D , 1 Ω ¯ 1 2 γ 1 L L R , 1 Ω ¯ 1 ξ ¯ 1 ,
K j ( ξ ¯ j , Ω ¯ j ) = ( L L g , j + γ 1 L L a , j ) Ω ¯ j L L a , j ξ ¯ j 1 2 L L R , j ξ ¯ j 2 + L L D , j Ω ¯ j 2 γ 1 L L R , j Ω ¯ j ξ ¯ j ,
G ̂ j ( z ¯ , ξ ¯ j , Ω ¯ j ) = i c j 1 2 z ¯ F ̂ j ( z ¯ , ξ ¯ j , Ω ¯ j ) exp ( i K j ( ξ ¯ j , Ω ¯ j ) z ¯ ) d z ¯ .
B 0 ( x ¯ , τ ¯ ) = exp ( x ¯ 2 2 τ ¯ 2 2 ) ,
B ̂ 0 ( ξ ¯ 1 , Ω ¯ 1 ) = exp ( ξ ¯ 1 2 2 Ω ¯ 1 2 2 ) ,
G ̂ 2 ( z ¯ , ξ ¯ 2 , Ω ¯ 2 ) = i c 2 2 exp ( ξ ¯ 2 2 4 Ω ¯ 2 2 4 ) I 2 ( z ¯ , ξ ¯ 2 , Ω ¯ 2 ) ,
I 2 ( z ¯ , ξ ¯ 2 , Ω ¯ 2 ) = 1 π d ξ ¯ d Ω ¯ exp ( ( ξ ¯ ξ ¯ 2 2 ) 2 ( Ω ¯ Ω ¯ 2 2 ) 2 ) 1 2 z ¯ d z ¯ exp ( i ( K 2 ( ξ ¯ 2 , Ω ¯ 2 ) K 1 ( ξ ¯ , Ω ¯ ) K 1 ( ξ ¯ 2 ξ ¯ , Ω ¯ 2 Ω ¯ ) ) z ¯ ) .
G ̂ 3 ( z ¯ = 1 2 , ξ ¯ 3 , Ω ¯ 3 ) = c 2 c 3 6 exp ( ξ ¯ 3 2 6 Ω ¯ 3 2 6 ) I 3 ( z ¯ = 1 2 , ξ ¯ 3 , Ω ¯ 3 ) ,
I 3 ( z ¯ = 1 2 , ξ ¯ 3 , Ω ¯ 3 ) = 3 2 π d ξ ¯ d Ω ¯ exp ( 3 4 ( ξ ¯ 2 3 ξ ¯ 3 ) 2 3 4 ( Ω ¯ 2 3 Ω ¯ 3 ) 2 ) 1 2 1 2 d z ¯ I 2 ( z ¯ , ξ ¯ , Ω ¯ ) exp ( i ( K 3 ( ξ ¯ 3 , Ω ¯ 3 ) K 2 ( ξ ¯ , Ω ¯ ) K 1 ( ξ ¯ 3 ξ ¯ , Ω ¯ 3 Ω ¯ ) ) z ¯ ) .
η 2 = U 2 U 1 = n 2 n 1 B ̂ 2 ( z ¯ = 1 2 , ξ ¯ 2 , Ω ¯ 2 ) 2 d ξ ¯ 2 d Ω ¯ 2 B ̂ 0 ( ξ ¯ 1 , Ω ¯ 1 ) 2 d ξ ¯ 1 d Ω ¯ 1 ,
η 3 = U 3 U 1 = n 3 n 1 B ̂ 3 ( z ¯ = 1 2 , ξ ¯ 3 , Ω ¯ 3 ) 2 d ξ ¯ 3 d Ω ¯ 3 B ̂ 0 ( ξ ¯ 1 , Ω ¯ 1 ) 2 d ξ ¯ 1 d Ω ¯ 1 ,
η 20 = n 2 n 1 c 2 2 2 = 4 π d S 2 ε 0 c n 1 2 n 2 λ 1 2 U 1 L 2 w 0 σ y τ 0 ,
η 30 = n 3 n 1 c 2 2 c 3 2 12 = 48 π 2 d S 2 d T 2 ε 0 2 c 2 n 1 3 n 2 2 n 3 λ 1 4 U 1 2 L 4 w 0 2 σ y 2 τ 0 2 .
η 2 = η 20 h 2 ,
h 2 = 1 2 π exp ( Ω ¯ 2 2 2 ξ ¯ 2 2 2 ) I 2 ( z = 1 2 , ξ 2 , Ω 2 ) 2 d Ω ¯ 2 d ξ ¯ 2 ,
η 3 = η 30 h 3 .
h 3 = 1 3 π exp ( Ω ¯ 3 2 3 ξ ¯ 3 2 3 ) I 3 ( z = 1 2 , ξ 3 , Ω 3 ) 2 d Ω ¯ 3 d ξ ¯ 3 .

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