Abstract

A large-ratio stretcher for ultra-short pulses is proposed based on photonic crystal fiber (PCF). Through proper design of the PCF structure, we obtain over 300-nm wavelength range with flattened dispersion characteristics. Analysis indicates that 1-km of such fiber can broaden over 10,000 times for ultra-short pulses with <100-fs pulse-width. Distortion due to dispersion and nonlinearity is negligible.

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  1. P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24(2), 398–403 (1988).
    [CrossRef]
  2. G. Cheriaux, P. Rousseau, F. Salin, J. P. Chambaret, B. Walker, and L. F. Dimauro, “Aberration-free stretcher design for ultrashort-pulse amplification,” Opt. Lett. 21(6), 414–416 (1996).
    [CrossRef] [PubMed]
  3. D. Du, J. Squier, S. Kane, G. Korn, G. Mourou, C. Bogusch, and C. T. Cotton, “Terawatt Ti:sapphire laser with a spherical reflective-optic pulse expander,” Opt. Lett. 20(20), 2114–2116 (1995).
    [CrossRef] [PubMed]
  4. A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. QE-28(4), 908–920 (1992).
    [CrossRef]
  5. H. Takada, M. Kakehata, and K. Torizuka, “Large-ratio stretch and recompression of sub-10-fs pulse utilizing dispersion managed devices and a spatial light modulator,” Appl. Phys. B 74(9), s253–s257 (2002).
    [CrossRef]
  6. Y. Nabekawa, Y. Shimizu, and K. Midorikawa, “Sub-20-fs terawatt-class laser system with a mirrorless regenerative amplifier and an adaptive phase controller,” Opt. Lett. 27(14), 1265–1267 (2002).
    [CrossRef]
  7. G. Chériaux, O. Albert, V. Wänman, J. P. Chambaret, C. Félix, and G. Mourou, “Temporal control of amplified femtosecond pulses with a deformable mirror in a stretcher,” Opt. Lett. 26(3), 169–171 (2001).
    [CrossRef]
  8. O. E. Martinez, “3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in the 1.3-1.6mm region,” IEEE J. Quantum Electron. 23(1), 59–64 (1987).
    [CrossRef]
  9. B. E. Lemoff and C. P. J. Barty, “Quintic-phase-limited, spatially uniform expansion and recompression of ultrashort optical pulses,” Opt. Lett. 18(19), 1651–1653 (1993).
    [CrossRef] [PubMed]
  10. G. Cheriaux, P. Rousseau, F. Salin, J. P. Chambaret, B. Walker, and L. F. Dimauro, “Aberration-free stretcher design for ultrashort-pulse amplification,” Opt. Lett. 21(6), 414–416 (1996).
    [CrossRef] [PubMed]
  11. L. Kuznetsova and F. W. Wise, “Scaling of femtosecond Yb-doped fiber amplifiers to tens of microjoule pulse energy via nonlinear chirped pulse amplification,” Opt. Lett. 32(18), 2671–2673 (2007).
    [CrossRef] [PubMed]
  12. I. Hartl, T. R. Schibli, A. Marcinkevicius, D. C. Yost, D. D. Hudson, M. E. Fermann, and J. Ye, “Cavity-enhanced similariton Yb-fiber laser frequency comb: 3×1014W/cm2 peak intensity at 136 MHz,” Opt. Lett. 32(19), 2870–2872 (2007).
    [CrossRef] [PubMed]
  13. L. Grüner-Nielsen, D. Jakobsen, K. G. Jespersen, and B. Pálsdóttir, “A stretcher fiber for use in fs chirped pulse Yb amplifiers,” Opt. Express 18(4), 3768–3773 (2010).
    [CrossRef] [PubMed]
  14. T. A. Birks, J. C. Knight, and P. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Express 22(13), 961–963 (1997).
  15. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21(19), 1547–1549 (1996).
    [CrossRef] [PubMed]
  16. J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
    [CrossRef] [PubMed]
  17. L. Farr, J. C. Knight, B. J. Mangan, and P. J. Roberts, “Low loss photonic crystal fiber,” in European Conference on Optical Communication (Copenhagen, 2002), post-deadline paper PD13, (2002).
  18. K. Saitoh and M. Koshiba, “Empirical relations for simple design of photonic crystal fibers,” Opt. Express 13(1), 267–274 (2005).
    [CrossRef] [PubMed]
  19. K. Saitoh, T. Fujisawa, T. Kirihara, and M. Koshiba, “Approximate empirical relations for nonlinear photonic crystal fibers,” Opt. Express 14(14), 6572–6582 (2006).
    [CrossRef] [PubMed]
  20. A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. S. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 25(18), 1325–1327 (2000).
    [CrossRef]
  21. N. G. R. Broderick, T. M. Monro, P. J. Bennett, and D. J. Richardson, “Nonlinearity in holey optical fibers: measurement and future opportunities,” Opt. Lett. 24(20), 1395–1397 (1999).
    [CrossRef]
  22. M. Midrio, M. P. Singh, and C. G. Someda, “The space filling mode of holey fibers: an analytical vectorial solution,” J. Lightwave Technol. 18(7), 1031–1037 (2000).
    [CrossRef]
  23. P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Loss in solid-core photonic crystal fibers due to interface roughness scattering,” Opt. Express 13(20), 7779–7793 (2005).
    [CrossRef] [PubMed]
  24. N. A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express 7, 341–348 (2002).
  25. K. R. Khan, T. X. Wu, D. N. Christodoulides, and G. I. Stegeman, “Soliton switching and multi-frequency generation in a nonlinear photonic crystal fiber coupler,” Opt. Express 16(13), 9417–9428 (2008).
    [CrossRef] [PubMed]
  26. Z. Zhu and T. G. Brown, “Analysis of the space filling modes of photonic crystal fibers,” Opt. Express 8(10), 547–554 (2001).
    [CrossRef] [PubMed]
  27. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).
  28. A. Efimov and A. J. Taylor, “Supercontinuum generation and soliton timing jitter in SF6 soft glass photonic crystal fibers,” Opt. Express 16(8), 5942–5953 (2008).
    [CrossRef] [PubMed]
  29. M. Erkintalo, G. Genty, and J. M. Dudley, “Rogue-wave-like characteristics in femtosecond supercontinuum generation,” Opt. Lett. 34(16), 2468–2470 (2009).
    [CrossRef] [PubMed]
  30. A. Peleg, “Energy exchange in fast optical soliton collisions as a random cascade model,” Phys. Lett. A 373(31), 2734–2738 (2009).
    [CrossRef]

2010

2009

M. Erkintalo, G. Genty, and J. M. Dudley, “Rogue-wave-like characteristics in femtosecond supercontinuum generation,” Opt. Lett. 34(16), 2468–2470 (2009).
[CrossRef] [PubMed]

A. Peleg, “Energy exchange in fast optical soliton collisions as a random cascade model,” Phys. Lett. A 373(31), 2734–2738 (2009).
[CrossRef]

2008

2007

2006

2005

2002

N. A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express 7, 341–348 (2002).

H. Takada, M. Kakehata, and K. Torizuka, “Large-ratio stretch and recompression of sub-10-fs pulse utilizing dispersion managed devices and a spatial light modulator,” Appl. Phys. B 74(9), s253–s257 (2002).
[CrossRef]

Y. Nabekawa, Y. Shimizu, and K. Midorikawa, “Sub-20-fs terawatt-class laser system with a mirrorless regenerative amplifier and an adaptive phase controller,” Opt. Lett. 27(14), 1265–1267 (2002).
[CrossRef]

2001

2000

1999

1998

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[CrossRef] [PubMed]

1997

T. A. Birks, J. C. Knight, and P. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Express 22(13), 961–963 (1997).

1996

1995

1993

1992

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. QE-28(4), 908–920 (1992).
[CrossRef]

1988

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24(2), 398–403 (1988).
[CrossRef]

1987

O. E. Martinez, “3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in the 1.3-1.6mm region,” IEEE J. Quantum Electron. 23(1), 59–64 (1987).
[CrossRef]

Albert, O.

Arriaga, J.

Atkin, D. M.

Bado, P.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24(2), 398–403 (1988).
[CrossRef]

Barty, C. P. J.

Bennett, P. J.

Birks, T. A.

Bogusch, C.

Broderick, N. G. R.

Broeng, J.

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[CrossRef] [PubMed]

Brown, T. G.

Chambaret, J. P.

Cheriaux, G.

Chériaux, G.

Christodoulides, D. N.

Cotton, C. T.

Couny, F.

Dimauro, L. F.

Du, D.

Dudley, J. M.

Efimov, A.

Erkintalo, M.

Félix, C.

Fermann, M. E.

Fujisawa, T.

Genty, G.

Grüner-Nielsen, L.

Hartl, I.

Hudson, D. D.

Jakobsen, D.

Jespersen, K. G.

Kakehata, M.

H. Takada, M. Kakehata, and K. Torizuka, “Large-ratio stretch and recompression of sub-10-fs pulse utilizing dispersion managed devices and a spatial light modulator,” Appl. Phys. B 74(9), s253–s257 (2002).
[CrossRef]

Kane, S.

Khan, K. R.

Kirihara, T.

Knight, J. C.

Korn, G.

Koshiba, M.

Kuznetsova, L.

Leaird, D. E.

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. QE-28(4), 908–920 (1992).
[CrossRef]

Lemoff, B. E.

Maine, P.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24(2), 398–403 (1988).
[CrossRef]

Mangan, B. J.

Marcinkevicius, A.

Martinez, O. E.

O. E. Martinez, “3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in the 1.3-1.6mm region,” IEEE J. Quantum Electron. 23(1), 59–64 (1987).
[CrossRef]

Midorikawa, K.

Midrio, M.

Monro, T. M.

Mortensen, N. A.

N. A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express 7, 341–348 (2002).

Mourou, G.

Nabekawa, Y.

Ortigosa-Blanch, A.

Pálsdóttir, B.

Patel, J. S.

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. QE-28(4), 908–920 (1992).
[CrossRef]

Peleg, A.

A. Peleg, “Energy exchange in fast optical soliton collisions as a random cascade model,” Phys. Lett. A 373(31), 2734–2738 (2009).
[CrossRef]

Pessot, M.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24(2), 398–403 (1988).
[CrossRef]

Richardson, D. J.

Roberts, P. J.

Rousseau, P.

Russell, P. J.

T. A. Birks, J. C. Knight, and P. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Express 22(13), 961–963 (1997).

Russell, P. S. J.

Russell, P. St. J.

Sabert, H.

Saitoh, K.

Salin, F.

Schibli, T. R.

Shimizu, Y.

Singh, M. P.

Someda, C. G.

Squier, J.

Stegeman, G. I.

Strickland, D.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24(2), 398–403 (1988).
[CrossRef]

Takada, H.

H. Takada, M. Kakehata, and K. Torizuka, “Large-ratio stretch and recompression of sub-10-fs pulse utilizing dispersion managed devices and a spatial light modulator,” Appl. Phys. B 74(9), s253–s257 (2002).
[CrossRef]

Taylor, A. J.

Torizuka, K.

H. Takada, M. Kakehata, and K. Torizuka, “Large-ratio stretch and recompression of sub-10-fs pulse utilizing dispersion managed devices and a spatial light modulator,” Appl. Phys. B 74(9), s253–s257 (2002).
[CrossRef]

Wadsworth, W. J.

Walker, B.

Wänman, V.

Weiner, A. M.

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. QE-28(4), 908–920 (1992).
[CrossRef]

Wise, F. W.

Wu, T. X.

Wullert, J. R.

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. QE-28(4), 908–920 (1992).
[CrossRef]

Ye, J.

Yost, D. C.

Zhu, Z.

Appl. Phys. B

H. Takada, M. Kakehata, and K. Torizuka, “Large-ratio stretch and recompression of sub-10-fs pulse utilizing dispersion managed devices and a spatial light modulator,” Appl. Phys. B 74(9), s253–s257 (2002).
[CrossRef]

IEEE J. Quantum Electron.

P. Maine, D. Strickland, P. Bado, M. Pessot, and G. Mourou, “Generation of ultrahigh peak power pulses by chirped pulse amplification,” IEEE J. Quantum Electron. 24(2), 398–403 (1988).
[CrossRef]

O. E. Martinez, “3000 times grating compressor with positive group velocity dispersion: application to fiber compensation in the 1.3-1.6mm region,” IEEE J. Quantum Electron. 23(1), 59–64 (1987).
[CrossRef]

A. M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable shaping of femtosecond optical pulses by use of 128-element liquid crystal phase modulator,” IEEE J. Quantum Electron. QE-28(4), 908–920 (1992).
[CrossRef]

J. Lightwave Technol.

Opt. Express

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Loss in solid-core photonic crystal fibers due to interface roughness scattering,” Opt. Express 13(20), 7779–7793 (2005).
[CrossRef] [PubMed]

N. A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express 7, 341–348 (2002).

K. R. Khan, T. X. Wu, D. N. Christodoulides, and G. I. Stegeman, “Soliton switching and multi-frequency generation in a nonlinear photonic crystal fiber coupler,” Opt. Express 16(13), 9417–9428 (2008).
[CrossRef] [PubMed]

Z. Zhu and T. G. Brown, “Analysis of the space filling modes of photonic crystal fibers,” Opt. Express 8(10), 547–554 (2001).
[CrossRef] [PubMed]

A. Efimov and A. J. Taylor, “Supercontinuum generation and soliton timing jitter in SF6 soft glass photonic crystal fibers,” Opt. Express 16(8), 5942–5953 (2008).
[CrossRef] [PubMed]

L. Grüner-Nielsen, D. Jakobsen, K. G. Jespersen, and B. Pálsdóttir, “A stretcher fiber for use in fs chirped pulse Yb amplifiers,” Opt. Express 18(4), 3768–3773 (2010).
[CrossRef] [PubMed]

T. A. Birks, J. C. Knight, and P. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Express 22(13), 961–963 (1997).

K. Saitoh and M. Koshiba, “Empirical relations for simple design of photonic crystal fibers,” Opt. Express 13(1), 267–274 (2005).
[CrossRef] [PubMed]

K. Saitoh, T. Fujisawa, T. Kirihara, and M. Koshiba, “Approximate empirical relations for nonlinear photonic crystal fibers,” Opt. Express 14(14), 6572–6582 (2006).
[CrossRef] [PubMed]

Opt. Lett.

A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. S. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. 25(18), 1325–1327 (2000).
[CrossRef]

N. G. R. Broderick, T. M. Monro, P. J. Bennett, and D. J. Richardson, “Nonlinearity in holey optical fibers: measurement and future opportunities,” Opt. Lett. 24(20), 1395–1397 (1999).
[CrossRef]

J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21(19), 1547–1549 (1996).
[CrossRef] [PubMed]

B. E. Lemoff and C. P. J. Barty, “Quintic-phase-limited, spatially uniform expansion and recompression of ultrashort optical pulses,” Opt. Lett. 18(19), 1651–1653 (1993).
[CrossRef] [PubMed]

G. Cheriaux, P. Rousseau, F. Salin, J. P. Chambaret, B. Walker, and L. F. Dimauro, “Aberration-free stretcher design for ultrashort-pulse amplification,” Opt. Lett. 21(6), 414–416 (1996).
[CrossRef] [PubMed]

L. Kuznetsova and F. W. Wise, “Scaling of femtosecond Yb-doped fiber amplifiers to tens of microjoule pulse energy via nonlinear chirped pulse amplification,” Opt. Lett. 32(18), 2671–2673 (2007).
[CrossRef] [PubMed]

I. Hartl, T. R. Schibli, A. Marcinkevicius, D. C. Yost, D. D. Hudson, M. E. Fermann, and J. Ye, “Cavity-enhanced similariton Yb-fiber laser frequency comb: 3×1014W/cm2 peak intensity at 136 MHz,” Opt. Lett. 32(19), 2870–2872 (2007).
[CrossRef] [PubMed]

G. Cheriaux, P. Rousseau, F. Salin, J. P. Chambaret, B. Walker, and L. F. Dimauro, “Aberration-free stretcher design for ultrashort-pulse amplification,” Opt. Lett. 21(6), 414–416 (1996).
[CrossRef] [PubMed]

D. Du, J. Squier, S. Kane, G. Korn, G. Mourou, C. Bogusch, and C. T. Cotton, “Terawatt Ti:sapphire laser with a spherical reflective-optic pulse expander,” Opt. Lett. 20(20), 2114–2116 (1995).
[CrossRef] [PubMed]

Y. Nabekawa, Y. Shimizu, and K. Midorikawa, “Sub-20-fs terawatt-class laser system with a mirrorless regenerative amplifier and an adaptive phase controller,” Opt. Lett. 27(14), 1265–1267 (2002).
[CrossRef]

G. Chériaux, O. Albert, V. Wänman, J. P. Chambaret, C. Félix, and G. Mourou, “Temporal control of amplified femtosecond pulses with a deformable mirror in a stretcher,” Opt. Lett. 26(3), 169–171 (2001).
[CrossRef]

M. Erkintalo, G. Genty, and J. M. Dudley, “Rogue-wave-like characteristics in femtosecond supercontinuum generation,” Opt. Lett. 34(16), 2468–2470 (2009).
[CrossRef] [PubMed]

Phys. Lett. A

A. Peleg, “Energy exchange in fast optical soliton collisions as a random cascade model,” Phys. Lett. A 373(31), 2734–2738 (2009).
[CrossRef]

Science

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[CrossRef] [PubMed]

Other

L. Farr, J. C. Knight, B. J. Mangan, and P. J. Roberts, “Low loss photonic crystal fiber,” in European Conference on Optical Communication (Copenhagen, 2002), post-deadline paper PD13, (2002).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).

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Figures (5)

Fig. 1
Fig. 1

The cross section of triangular lattice of the PCF

Fig. 2
Fig. 2

Simulation results of effective dispersion for the proposed PCF with (a) different radius of air holes and (b) different lattice pitch. The black curve corresponds to the material dispersion of pure silica.

Fig. 3
Fig. 3

Dispersion parameter (a)β2 and (b)β3 versus wavelength for the optimal PCF

Fig. 4
Fig. 4

Stretching ratio as a function of the pulse width ΔT after 1-km PCF, inserted (a) ΔT=40fs and (b) ΔT=90fs

Fig. 5
Fig. 5

Waveforms for different nonlinear coefficient γ values (ΔT=60fs)

Equations (12)

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

E z ( ρ = R ) = H z ( ρ = R ) 0
H z ( ρ , θ , z ) = H l ( ρ ) e i l θ e γ z E z ( ρ , θ , z ) = E l ( ρ ) e i l θ e γ z
I 2 ( w , r ) I 1 ( w , r ) = 1 w r w r 2 ( 1 + n si 2 ( λ ) n a i r 2 ) g ( u ) w r [ 1 4 ( 1 n s i 2 ( λ ) n a i r 2 ) 2 g 2 ( u ) + f ( w , u ) n a i r 2 ] 1 / 2
g ( u ) = 1 ω r J 0 ( u r ) Y 1 ( u R ) Y 0 ( u r ) J 1 ( u R ) J 1 ( u r ) Y 1 ( u R ) Y 1 ( u r ) J 1 ( u R ) 1 u 2 r 2
f ( w , u ) = 1 r 4 ( 1 u 2 + 1 w 2 ) ( n s i 2 u 2 + n a i r 2 w 2 )
w 2 = ω 2 ( n e f f 2 n a i r 2 ) / c 2
u 2 = ω 2 ( n s i 2 n e f f 2 ) / c 2
D = λ c       d 2 n e f f ' d λ 2 & β 2 = λ 2 2 π c D
γ ( λ ) = 2 π λ s n 2 ( x , y ) I 2 ( x , y , λ ) d x d y ( s I ( x , y , λ ) d x d y ) 2 = 2 π λ n 2 ( λ ) A e f f ( λ )
A e f f ( λ ) = ( | I ( x , y , λ | 2 d x d y ) 2 | I ( x , y , λ | 4 d x d y = π R 2 = π a e f f 2 ln V N L
V N L = V 1 k 0 2 n c o 2 n 2 R 2 | A | 2 / 8 Z 0
A z = α 2 i 2 β 2 2 A T 2 + 1 6 β 3 3 A T 3 + i γ | A | 2 A

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