Abstract

The choice of optimum phase-matching conditions for noncollinear optical parametric amplifiers is usually made on the basis of the linear spectral dispersion characteristics of the anisotropic nonlinear crystal. However, for high-peak-power operation, where pump depletion is involved, it is shown that the tolerance of the parametric gain with regard to k-vector mismatch is to change the optimum phase-matching parameters. Our calculations show that, with the revised parameters, an enhancement in peak power approaching 50% could be achieved.

© 2007 Optical Society of America

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    [CrossRef]

2007

2006

2005

2002

2001

X. Yang, Z. Xu, Z. Zhang, Y. Leng, J. Peng, J. Wang, S. Jin, W. Zhang, and R. Li, "Dependence of spectrum on pump-signal angle in BBO-I noncollinear optical-parametric chirped-pulse amplification," Appl. Phys. B 73, 219-222 (2001).
[CrossRef]

M. Zavelani-Rossi, G. Cerullo, S. De Silvestri, L. Gallmann, N. Matuschek, G. Steinmeyer, U. Keller, G. Angelow, V. Scheuer, and T. Tschudi, "Pulse compression over a 170-THz bandwidth in the visible by use of only chirped mirrors," Opt. Lett. 26, 1155-1157 (2001).
[CrossRef]

2000

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Sporlein, and W. Zinth, "Generation of 10 to 50fs pulses tunable through all of the visible and the NIR," Appl. Phys. B 71, 457-465 (2000).
[CrossRef]

T. Kobayashi and A. Shirakawa, "Tunable visible and near-infrared pulse generator in a 5fs regime," Appl. Phys. B 70, 239-246 (2000).
[CrossRef]

1999

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, "Sub-5-fs visible pulse generation by pulse-front-matched noncollinear optical parametric amplification," Appl. Phys. Lett. 74, 2268-2270 (1999).
[CrossRef]

1998

1997

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, "The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers," Opt. Commun. 144, 125-133 (1997).
[CrossRef]

1996

1995

1994

S. Takeuchi and T. Kobayashi, "Broadband near-infrared pulse generation in KTiOPO4," J. Appl. Phys. 75, 2757-2760 (1994).
[CrossRef]

T. J. Driscoll, G. M. Gale, and F. Hache, "Ti:sapphire second-harmonic-pumped visible range femtosecond optical parametric oscillator," Opt. Commun. 110, 638-644 (1994).
[CrossRef]

1993

1992

A. Dubietis, G. Jonusauskas, and A. Piskarskas, "Powerful femtosecond pulse generation by chirped and stretched pulse parametric amplification in BBO crystal," Opt. Commun. 88, 437-440 (1992).
[CrossRef]

1987

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical mechanical, and thermal properties of barium borate," J. Appl. Phys. 62, 1968-1983 (1987).
[CrossRef]

1986

K. Kato, "Second-harmonic generation to 2048Å in B-Ba2O4," IEEE J. Quantum Electron. 22, 1013-1014 (1986).
[CrossRef]

1979

R. A. Baumgartner and R. L. Byer, "Optical parametric amplification," IEEE J. Quantum Electron. 15, 432-444 (1979).
[CrossRef]

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interaction of light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Aguergaray, C.

Amthor, K.-U.

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, "Laser-plasma acceleration of quasi-monoenergetic protons from microstructuered targets," Nature 439, 445-448 (2006).
[CrossRef] [PubMed]

Andersen, T. V.

Andreoni, A.

Angelow, G.

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interaction of light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Baltuska, A.

Baumgartner, R. A.

R. A. Baumgartner and R. L. Byer, "Optical parametric amplification," IEEE J. Quantum Electron. 15, 432-444 (1979).
[CrossRef]

Beutter, M.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Sporlein, and W. Zinth, "Generation of 10 to 50fs pulses tunable through all of the visible and the NIR," Appl. Phys. B 71, 457-465 (2000).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interaction of light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Bruchmann, C.

Butcher, P. B.

P. B. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, 1990).

Byer, R. L.

R. A. Baumgartner and R. L. Byer, "Optical parametric amplification," IEEE J. Quantum Electron. 15, 432-444 (1979).
[CrossRef]

Cavallari, M.

Cerullo, G.

Charukhchev, A. V.

Chen, C.

Collier, J. L.

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, "The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers," Opt. Commun. 144, 125-133 (1997).
[CrossRef]

Cormier, E.

Cotter, D.

P. B. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge U. Press, 1990).

Danielius, R.

Davis, L.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical mechanical, and thermal properties of barium borate," J. Appl. Phys. 62, 1968-1983 (1987).
[CrossRef]

De Silvestri, S.

Descamps, D.

Di Trapani, P.

Driscoll, T. J.

G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, "Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator," Opt. Lett. 20, 1562-1564 (1995).
[CrossRef] [PubMed]

T. J. Driscoll, G. M. Gale, and F. Hache, "Ti:sapphire second-harmonic-pumped visible range femtosecond optical parametric oscillator," Opt. Commun. 110, 638-644 (1994).
[CrossRef]

Dubietis, A.

E. Zeromski, A. Dubietis, G. Tamosauskas, and A. Piskarskas, "Gain bandwidth broadening of the continuum-seeded optical parametric amplifier by use of two pump beams," Opt. Commun. 203, 435-440 (2002).

A. Dubietis, R. Danielius, G. Tamosauskas, and A. Piskarskas, "Combining effect in a multiple-beam-pumped optical parametric amplifier," J. Opt. Soc. Am. B 15, 1135-1139 (1998).
[CrossRef]

P. Di Trapani, A. Andreoni, C. Solcia, P. Foggi, R. Danielius, A. Dubietis, and A. Piskarskas, "Matching of group velocities in three-wave parametric interaction with femtosecond pulses and application to traveling-wave generators," J. Opt. Soc. Am. B 12, 2237-2244 (1995).

A. Dubietis, G. Jonusauskas, and A. Piskarskas, "Powerful femtosecond pulse generation by chirped and stretched pulse parametric amplification in BBO crystal," Opt. Commun. 88, 437-440 (1992).
[CrossRef]

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interaction of light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Eimerl, D.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical mechanical, and thermal properties of barium borate," J. Appl. Phys. 62, 1968-1983 (1987).
[CrossRef]

Esirkepov, T.

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, "Laser-plasma acceleration of quasi-monoenergetic protons from microstructuered targets," Nature 439, 445-448 (2006).
[CrossRef] [PubMed]

Foggi, P.

Freidman, G. I.

Fuji, T.

Gale, G. M.

G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, "Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator," Opt. Lett. 20, 1562-1564 (1995).
[CrossRef] [PubMed]

T. J. Driscoll, G. M. Gale, and F. Hache, "Ti:sapphire second-harmonic-pumped visible range femtosecond optical parametric oscillator," Opt. Commun. 110, 638-644 (1994).
[CrossRef]

Gallmann, L.

Garanin, S. G.

Gerke, R. R.

Ginzburg, V. N.

Graham, E. K.

D. Eimerl, L. Davis, S. Velsko, E. K. Graham, and A. Zalkin, "Optical mechanical, and thermal properties of barium borate," J. Appl. Phys. 62, 1968-1983 (1987).
[CrossRef]

Hache, F.

G. M. Gale, M. Cavallari, T. J. Driscoll, and F. Hache, "Sub-20-fs tunable pulses in the visible from an 82-MHz optical parametric oscillator," Opt. Lett. 20, 1562-1564 (1995).
[CrossRef] [PubMed]

T. J. Driscoll, G. M. Gale, and F. Hache, "Ti:sapphire second-harmonic-pumped visible range femtosecond optical parametric oscillator," Opt. Commun. 110, 638-644 (1994).
[CrossRef]

Huang, J. Y.

Jäckel, O.

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, "Laser-plasma acceleration of quasi-monoenergetic protons from microstructuered targets," Nature 439, 445-448 (2006).
[CrossRef] [PubMed]

Jin, S.

X. Yang, Z. Xu, Z. Zhang, Y. Leng, J. Peng, J. Wang, S. Jin, W. Zhang, and R. Li, "Dependence of spectrum on pump-signal angle in BBO-I noncollinear optical-parametric chirped-pulse amplification," Appl. Phys. B 73, 219-222 (2001).
[CrossRef]

Jonusauskas, G.

A. Dubietis, G. Jonusauskas, and A. Piskarskas, "Powerful femtosecond pulse generation by chirped and stretched pulse parametric amplification in BBO crystal," Opt. Commun. 88, 437-440 (1992).
[CrossRef]

Katin, E. V.

Kato, K.

K. Kato, "Second-harmonic generation to 2048Å in B-Ba2O4," IEEE J. Quantum Electron. 22, 1013-1014 (1986).
[CrossRef]

Keller, U.

Khazanov, E. A.

Kirsanov, A. V.

Kobayashi, T.

A. Baltuska, T. Fuji, and T. Kobayashi, "Visible pulse compression to 4fs by optical parametric amplification and programmable dispersion control," Opt. Lett. 27, 306-308 (2002).
[CrossRef]

T. Kobayashi and A. Shirakawa, "Tunable visible and near-infrared pulse generator in a 5fs regime," Appl. Phys. B 70, 239-246 (2000).
[CrossRef]

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, "Sub-5-fs visible pulse generation by pulse-front-matched noncollinear optical parametric amplification," Appl. Phys. Lett. 74, 2268-2270 (1999).
[CrossRef]

A. Shirakawa and T. Kobayashi, "Noncollinearly phase-matched femtosecond optical parametric amplification with a 2000cm−1 bandwidth," Appl. Phys. Lett. 72, 147-149 (1998).
[CrossRef]

S. Takeuchi and T. Kobayashi, "Broadband near-infrared pulse generation in KTiOPO4," J. Appl. Phys. 75, 2757-2760 (1994).
[CrossRef]

Langley, A. J.

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, "The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers," Opt. Commun. 144, 125-133 (1997).
[CrossRef]

Ledingham, K. W. D.

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, "Laser-plasma acceleration of quasi-monoenergetic protons from microstructuered targets," Nature 439, 445-448 (2006).
[CrossRef] [PubMed]

Leng, Y.

X. Yang, Z. Xu, Z. Zhang, Y. Leng, J. Peng, J. Wang, S. Jin, W. Zhang, and R. Li, "Dependence of spectrum on pump-signal angle in BBO-I noncollinear optical-parametric chirped-pulse amplification," Appl. Phys. B 73, 219-222 (2001).
[CrossRef]

Li, R.

X. Yang, Z. Xu, Z. Zhang, Y. Leng, J. Peng, J. Wang, S. Jin, W. Zhang, and R. Li, "Dependence of spectrum on pump-signal angle in BBO-I noncollinear optical-parametric chirped-pulse amplification," Appl. Phys. B 73, 219-222 (2001).
[CrossRef]

Liesfeld, B.

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, "Laser-plasma acceleration of quasi-monoenergetic protons from microstructuered targets," Nature 439, 445-448 (2006).
[CrossRef] [PubMed]

Limpert, J.

Lochbrunner, S.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Sporlein, and W. Zinth, "Generation of 10 to 50fs pulses tunable through all of the visible and the NIR," Appl. Phys. B 71, 457-465 (2000).
[CrossRef]

Lozhkarev, V. V.

Luchinin, G. A.

Mal'shakov, A. N.

Manek-Hönniger, I.

Martyanov, M. A.

Matousek, P.

I. N. Ross, P. Matousek, G. H. C. New, and K. Osvay, "Analysis and optimization of optical parametric chirped pulse amplification," J. Opt. Soc. Am. B 19, 2945-2956 (2002).
[CrossRef]

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, "The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers," Opt. Commun. 144, 125-133 (1997).
[CrossRef]

Matuschek, N.

Montant, S.

New, G. H. C.

Nisoli, M.

Norris, T. B.

Osvay, K.

Palashov, O. V.

Peng, J.

X. Yang, Z. Xu, Z. Zhang, Y. Leng, J. Peng, J. Wang, S. Jin, W. Zhang, and R. Li, "Dependence of spectrum on pump-signal angle in BBO-I noncollinear optical-parametric chirped-pulse amplification," Appl. Phys. B 73, 219-222 (2001).
[CrossRef]

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, "Interaction of light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

Petit, S.

Pfotenhauer, S.

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, "Laser-plasma acceleration of quasi-monoenergetic protons from microstructuered targets," Nature 439, 445-448 (2006).
[CrossRef] [PubMed]

Piel, J.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Sporlein, and W. Zinth, "Generation of 10 to 50fs pulses tunable through all of the visible and the NIR," Appl. Phys. B 71, 457-465 (2000).
[CrossRef]

Piskarskas, A.

Poteomkin, A. K.

Riedle, E.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Sporlein, and W. Zinth, "Generation of 10 to 50fs pulses tunable through all of the visible and the NIR," Appl. Phys. B 71, 457-465 (2000).
[CrossRef]

Röser, F.

J. Limpert, F. Röser, T. Schreiber, and A. Tünnermann, "High-power ultrafast fiber laser systems," IEEE J. Sel. Top. Quantum Electron. 12, 233-244 (2006).
[CrossRef]

Ross, I. N.

I. N. Ross, P. Matousek, G. H. C. New, and K. Osvay, "Analysis and optimization of optical parametric chirped pulse amplification," J. Opt. Soc. Am. B 19, 2945-2956 (2002).
[CrossRef]

I. N. Ross, P. Matousek, M. Towrie, A. J. Langley, and J. L. Collier, "The prospects for ultrashort pulse duration and ultrahigh intensity using optical parametric chirped pulse amplifiers," Opt. Commun. 144, 125-133 (1997).
[CrossRef]

Rothhardt, J.

Rukavishnikov, N. N.

Sakane, I.

A. Shirakawa, I. Sakane, M. Takasaka, and T. Kobayashi, "Sub-5-fs visible pulse generation by pulse-front-matched noncollinear optical parametric amplification," Appl. Phys. Lett. 74, 2268-2270 (1999).
[CrossRef]

Salin, F.

Sauerbrey, R.

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, "Laser-plasma acceleration of quasi-monoenergetic protons from microstructuered targets," Nature 439, 445-448 (2006).
[CrossRef] [PubMed]

Schenkl, S.

E. Riedle, M. Beutter, S. Lochbrunner, J. Piel, S. Schenkl, S. Sporlein, and W. Zinth, "Generation of 10 to 50fs pulses tunable through all of the visible and the NIR," Appl. Phys. B 71, 457-465 (2000).
[CrossRef]

Scheuer, V.

Schimpf, D. N.

Schmidt, O.

Schreiber, T.

Schwoerer, H.

H. Schwoerer, S. Pfotenhauer, O. Jäckel, K.-U. Amthor, B. Liesfeld, W. Ziegler, R. Sauerbrey, K. W. D. Ledingham, and T. Esirkepov, "Laser-plasma acceleration of quasi-monoenergetic protons from microstructuered targets," Nature 439, 445-448 (2006).
[CrossRef] [PubMed]

Sergeev, A. M.

Shaykin, A. A.

Shen, Y. R.

Shirakawa, A.

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[CrossRef]

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X. Yang, Z. Xu, Z. Zhang, Y. Leng, J. Peng, J. Wang, S. Jin, W. Zhang, and R. Li, "Dependence of spectrum on pump-signal angle in BBO-I noncollinear optical-parametric chirped-pulse amplification," Appl. Phys. B 73, 219-222 (2001).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Full gain bandwidth at half-maximum as a function of signal frequency for collinear type-1 phase-matching in BBO at a pump wavelength of 515 nm ( 582 THz ) . The crystal length is 1 mm . The initial signal intensity is 1 MW cm 2 and the pump intensity is 100 GW cm 2 . A wide flat-top input signal spectrum is assumed.

Fig. 2
Fig. 2

k-vector triangle visualizing perfect type-I phase-matching in uniaxial crystals.

Fig. 3
Fig. 3

Type-1 phase-matching angle Θ as a function of signal frequency ν s for different pump-tilt angles in BBO. The black and gray solid curves show the spectral behavior of the signal and idler, respectively. The black dashed curves represent the continuous extension of the signal at pump-signal angle α into the idler spectral region, and the corresponding gray dashed curve is the extension of the idler into the signal region. The dotted curve illustrates the location of broadband as well as perfect phase-matching versus signal frequency ν s . The pump is situated at a wavelength of 515 nm ( 582 THz ) .

Fig. 4
Fig. 4

(a) Spectral behavior of the phase-matching angle, (b) k-vector mismatch versus frequency, (c) corresponding gain spectrum including (solid curve) and without (dotted curve) pump depletion, (d) resulting temporal profile. The pump frequency is at 582 THz , the pump-tilt angle is 1.4 ° , and the phase-matching angle Θ is adjusted for perfect phase-match for a signal frequency of 375 THz and its value is 23.3 ° .

Fig. 5
Fig. 5

(a) Spectral behavior of the phase-matching angle, (b) k-vector mismatch versus frequency, (c) corresponding gain spectrum including (solid curve) and without (dotted curve) pump depletion, (d) resulting temporal profile. The pump frequency is at 582 THz , the pump-tilt angle is 2.0 ° , and the phase-matching angle Θ is adjusted for perfect phase-match for a signal frequency of 375 THz and its value is 23.8 ° .

Fig. 6
Fig. 6

(a) Spectral behavior of the phase-matching angle, (b) k-vector mismatch versus frequency, (c) corresponding gain spectrum including (solid curve) and without (dotted curve) pump depletion, (d) resulting temporal profile. The pump frequency is at 582 THz , the pump-tilt angle is 2.6 ° , and the phase-matching angle Θ is adjusted for perfect phase-match for a signal frequency of 375 THz and its value is 24.6 ° .

Fig. 7
Fig. 7

(a) Spectral behavior of the phase-matching angle, (b) k-vector mismatch versus frequency, (c) corresponding gain spectrum including (solid curve) and without (dotted curve) pump depletion, (d) resulting temporal profile. The pump frequency is at 582 THz , the pump-tilt angle is 3.2 ° , and the phase-matching angle Θ is adjusted for perfect phase-match for a signal frequency of 375 THz and its value is 25.5 ° .

Fig. 8
Fig. 8

Estimated effective peak power (figure of merit for the temporal pulse envelope) as a function of pump-tilt angle ( α ) for perfectly phase-matched signal frequency at 375 THz (solid curve). The location of the ultrabroadband point that is expected from the analysis of the linear system is marked with the dotted line.

Fig. 9
Fig. 9

Estimated effective peak power (figure of merit for the temporal pulse envelope) as a function of pump-tilt angle ( α ) and perfectly phase-matched signal frequency ( ν s PM ) .

Fig. 10
Fig. 10

Optimal point of operation. (a) Spectral behavior of the phase-matching angle, (b) k-vector mismatch versus frequency, (c) corresponding gain spectrum including (solid curve) and without (dotted curve) pump depletion, (d) resulting temporal profile. The pump frequency is at 582 THz , the pump-tilt angle is 2.2 ° , and the phase-matching angle Θ is adjusted for perfect phase-match for a signal frequency of 400 THz and its value is 24 ° .

Equations (31)

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ω p = ω s + ω i .
1 2 A ( z ) e exp ( i ( k z ω t ) ) + c.c. ,
d A i d z = i ω i d eff n i c A s * A p e i Δ k z ,
d A s dz = i ω s d eff n s c A i * A p e i Δ k z ,
d A p d z = i ω p d eff n p c A i A s e i Δ k z ,
I p ( z ) = I total ω p [ b + ( f b ) sn 2 ( q b γ ( z + z 0 ) , m ) ] ,
u = 0 sn ( u , m ) d t 1 ( 1 t 2 ) ( 1 m t 2 ) ,
m = f b q b .
γ = d eff 2 I total ε 0 c 3 ω i ω s ω p n i n s n p ,
I i ( z ) = I i ( 0 ) + ω i ω p [ I p ( 0 ) I p ( z ) ] ,
I s ( z ) = I s ( 0 ) + ω s ω p [ I p ( 0 ) I p ( z ) ] .
G = I s ( z ) I s ( 0 ) = ω s ω p [ I p ( 0 ) I p ( z ) ] I s ( 0 ) + 1 .
b = u 2 u 2 4 v ,
f = I p ( 0 ) I total ω p ,
q = u 2 + u 2 4 v .
z 0 = K ( m ) ( q b γ ) ,
K ( m ) = 0 1 d t 1 ( 1 t 2 ) ( 1 m t 2 ) ,
I s ( z ) = I s ( 0 ) [ 1 + Γ 2 g 2 sin h 2 ( g z ) ] ,
I i ( z ) = I s ( 0 ) ω i ω s Γ 2 g 2 sin h 2 ( g z ) ,
Δ k = k p k s k i ,
Δ k = k p cos α k s k i cos Ω ,
Δ k = k p sin α k i sin Ω ,
k p = k s cos ( α ) + k i cos ( Ω α ) .
k s sin ( α ) = k i sin ( Ω α ) .
k s ( ω s ) = ω s c n o ( ω s ) ,
k i ( ω p ω s ) = ( ω p ω s ) c n o ( ω p ω s ) ,
k p ( ω p , Θ ) = ω p c n eo ( ω p , Θ ) = ω p c n o ( ω p ) n e ( ω p ) n o 2 ( ω p ) sin 2 Θ + n e 2 ( ω p ) cos 2 Θ .
( k s ω s ) cos Ω bb = ( k i ω i ) ,
sin ( α ) k i = sin ( π Ω ) k s 2 + k i 2 + 2 k s k i cos ( Ω ) .
( 2 k s ω s 2 ) cos Ω magic + ( 2 k i ω i 2 ) = k i ( Ω ω s ) 2 .
Δ k = k p cos α k s k p 2 cos α 2 k p 2 + k i 2 ,

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