Abstract

Dispersive compression of chirped few-picosecond pulses at the microjoule level in a hollow-core photonic bandgap fiber is studied numerically. The performance of ideal parabolic input pulses is compared to pulses from a narrowband picosecond oscillator broadened by self-phase modulation during amplification. It is shown that the parabolic pulses are superior for compression of high-quality femtosecond pulses up to the few-megawatts level. With peak powers of 510MW or higher, there is no significant difference in power scaling and pulse quality between the two pulse types for comparable values of power, duration, and bandwidth. The same conclusion is found for the peak power and energy of solitons formed beyond the point of maximal compression. Long-pass filtering of these solitons is shown to be a promising route to clean solitonlike output pulses with peak powers of several MW.

© 2009 Optical Society of America

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References

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  1. C. J. S. De Matos, J. R. Taylor, T. P. Hansen, K. P. Hansen, and J. Broeng, “All-fiber chirped pulse amplification using highly-dispersive air-core photonic bandgap fiber,” Opt. Express 11, 2832-2837 (2003).
    [CrossRef] [PubMed]
  2. J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, and A. Tunnermann, “All fiber chirped-pulse amplification system based on compression in air-guiding photonic bandgap fiber,” Opt. Express 11, 3332-3337 (2003).
    [CrossRef] [PubMed]
  3. C. K. Nielsen, K. G. Jespersen, and S. R. Keiding, “A 158 fs5.3 nj fiber-laser system at 1 μm using photonic bandgap fibers for dispersion control and pulse compression,” Opt. Express 14, 6063-6068 (2006).
    [CrossRef] [PubMed]
  4. C. Billet, J. M. Dudley, N. Joly, and J. C. Knight, “Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm,” Opt. Express 13, 3236-3241 (2005).
    [CrossRef] [PubMed]
  5. J. Lægsgaard and P. J. Roberts, “Dispersive pulse compression in hollow-core photonic bandgap fibers,” Opt. Express 16, 9628-9644 (2008).
    [CrossRef] [PubMed]
  6. D. G. Ouzounov, C. J. Hensley, A. L. Gaeta, N. Venkateraman, M. T. Gallagher, and K. W. Koch, “Soliton pulse compression in photonic band-gap fibers,” Opt. Express 13, 6153-6159 (2005).
    [CrossRef] [PubMed]
  7. F. Gerome, K. Cook, A. K. George, W. J. Wadsworth, and J. C. Knight, “Delivery of sub-100 fs pulses through 8 m of hollow-core fiber using soliton compression,” Opt. Express 15, 7126-7131 (2007).
    [CrossRef] [PubMed]
  8. F. Gérôme, J. Dupriez, J. C. Knight, and W. J. Wadsworth, “High power tunable femtosecond soliton source using hollow-core photonic bandgap fiber, and its use for frequency doubling,” Opt. Express 16, 2381-2386 (2008).
    [CrossRef] [PubMed]
  9. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010-6013 (2000).
    [CrossRef] [PubMed]
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    [CrossRef]
  11. C. K. Nielsen, B. Ortac, T. Schreiber, J. Limpert, R. Hohmuth, W. Richter, and A. Tünnermann, “Self-starting self-similar all-polarization maintaining yb-doped fiber laser,” Opt. Express 13, 9346-9351 (2005).
    [CrossRef] [PubMed]
  12. F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58-73 (2008).
    [CrossRef]
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    [CrossRef]
  14. D. Turchinovich, X. Liu, and J. Laegsgaard, “Monolithic all-pm femtosecond yb-fiber laser stabilized with a narrow-band fiber Bragg grating and pulse-compressed in a hollow-core photonic crystal fiber,” Opt. Express 16, 14004-14014 (2008).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  20. A. V. Gorbach and D. V. Skryabin, “Soliton self-frequency shift, non-solitonic radiation and self-induced transparency in air-core fibers,” Opt. Express 16, 4858-4865 (2008).
    [CrossRef] [PubMed]
  21. D. N. Papadopoulos, Y. Zaouter, M. Hanna, F. Druon, E. Mottay, E. Cormier, and P. Georges, “Generation of 63 fs4.1 mW peak power pulses from a parabolic fiber amplifier operated beyond the gain bandwidth limit,” Opt. Lett. 32, 2520-2522 (2007).
    [CrossRef] [PubMed]
  22. T. Schreiber, C. K. Nielsen, B. Ortac, J. Limpert, and A. Tünnermann, “Microjoule-level all-polarization-maintaining femtosecond fiber source,” Opt. Lett. 31, 574-576 (2006).
    [CrossRef] [PubMed]
  23. P. Dupriez, C. Finot, A. Malinowski, J. K. Sahu, J. Nilsson, D. J. Richardson, K. G. Wilcox, H. D. Foreman, and A. C. Tropper, “High-power, high repetition rate picosecond and femtosecond sources based on yb-doped fiber amplification of VECSELs,” Opt. Express 14, 9611-9616 (2006).
    [CrossRef] [PubMed]
  24. B.-W. Liu, M.-L. Hu, X.-H. Fang, Y.-Z. Wu, Y.-J. Song, L. Chai, C.-Y. Wang, and A. M. Zheltikov, “High-power wavelength-tunable photonic-crystal-fiber-based oscillator-amplifier-frequency-shifter femtosecond laser system and its applications for material microprocessing,” Laser Phys. Lett. 6, 44-48 (2009).
    [CrossRef]
  25. J. Lægsgaard has prepared a paper to be called “Soliton formation in hollow-core photonic bandgap fibers.”

2009 (1)

B.-W. Liu, M.-L. Hu, X.-H. Fang, Y.-Z. Wu, Y.-J. Song, L. Chai, C.-Y. Wang, and A. M. Zheltikov, “High-power wavelength-tunable photonic-crystal-fiber-based oscillator-amplifier-frequency-shifter femtosecond laser system and its applications for material microprocessing,” Laser Phys. Lett. 6, 44-48 (2009).
[CrossRef]

2008 (6)

2007 (3)

2006 (4)

2005 (3)

2003 (2)

2001 (1)

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

2000 (1)

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

1998 (1)

1997 (1)

Agrawal, G. P.

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

Billet, C.

Broeng, J.

Chai, L.

B.-W. Liu, M.-L. Hu, X.-H. Fang, Y.-Z. Wu, Y.-J. Song, L. Chai, C.-Y. Wang, and A. M. Zheltikov, “High-power wavelength-tunable photonic-crystal-fiber-based oscillator-amplifier-frequency-shifter femtosecond laser system and its applications for material microprocessing,” Laser Phys. Lett. 6, 44-48 (2009).
[CrossRef]

Chong, A.

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58-73 (2008).
[CrossRef]

Cook, K.

Cormier, E.

De Matos, C. J. S.

Druon, F.

Dudley, J. M.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597-603 (2007).
[CrossRef]

C. Billet, J. M. Dudley, N. Joly, and J. C. Knight, “Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm,” Opt. Express 13, 3236-3241 (2005).
[CrossRef] [PubMed]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Dupriez, J.

Dupriez, P.

Fang, X.-H.

B.-W. Liu, M.-L. Hu, X.-H. Fang, Y.-Z. Wu, Y.-J. Song, L. Chai, C.-Y. Wang, and A. M. Zheltikov, “High-power wavelength-tunable photonic-crystal-fiber-based oscillator-amplifier-frequency-shifter femtosecond laser system and its applications for material microprocessing,” Laser Phys. Lett. 6, 44-48 (2009).
[CrossRef]

Fermann, M. E.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Finot, C.

Foreman, H. D.

Franco, M. A.

Gaeta, A. L.

Gallagher, M. T.

George, A. K.

Georges, P.

Gerome, F.

Gérôme, F.

Gorbach, A. V.

Grillon, G.

Hanna, M.

Hansen, K. P.

Hansen, T. P.

Harvey, J. D.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Hensley, C. J.

Hohmuth, R.

Hu, M.-L.

B.-W. Liu, M.-L. Hu, X.-H. Fang, Y.-Z. Wu, Y.-J. Song, L. Chai, C.-Y. Wang, and A. M. Zheltikov, “High-power wavelength-tunable photonic-crystal-fiber-based oscillator-amplifier-frequency-shifter femtosecond laser system and its applications for material microprocessing,” Laser Phys. Lett. 6, 44-48 (2009).
[CrossRef]

Jespersen, K. G.

Joly, N.

Keiding, S. R.

Knight, J. C.

Koch, K. W.

Kruglov, V. I.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Lægsgaard, J.

J. Lægsgaard and P. J. Roberts, “Dispersive pulse compression in hollow-core photonic bandgap fibers,” Opt. Express 16, 9628-9644 (2008).
[CrossRef] [PubMed]

J. Lægsgaard, “Control of fiber laser mode-locking by narrow-band Bragg gratings,” J. Phys. B 41, 095401 (2008).
[CrossRef]

Laegsgaard, J.

Lægsgaard, J.

J. Lægsgaard has prepared a paper to be called “Soliton formation in hollow-core photonic bandgap fibers.”

Limpert, J.

Liu, B.-W.

B.-W. Liu, M.-L. Hu, X.-H. Fang, Y.-Z. Wu, Y.-J. Song, L. Chai, C.-Y. Wang, and A. M. Zheltikov, “High-power wavelength-tunable photonic-crystal-fiber-based oscillator-amplifier-frequency-shifter femtosecond laser system and its applications for material microprocessing,” Laser Phys. Lett. 6, 44-48 (2009).
[CrossRef]

Liu, X.

Malinowski, A.

Millot, G.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597-603 (2007).
[CrossRef]

Mlejnek, M.

Moloney, J. V.

Mottay, E.

Mysyrowicz, A.

Nibbering, E. T. J.

Nielsen, C. K.

Nilsson, J.

Nolte, S.

Ortac, B.

Ouzounov, D. G.

Papadopoulos, D. N.

Parmigiani, F.

Petropoulos, P.

Prade, B. S.

Renninger, W. H.

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58-73 (2008).
[CrossRef]

Richardson, D. J.

Richter, W.

Roberts, P. J.

Sahu, J. K.

Schreiber, T.

Skryabin, D. V.

Song, Y.-J.

B.-W. Liu, M.-L. Hu, X.-H. Fang, Y.-Z. Wu, Y.-J. Song, L. Chai, C.-Y. Wang, and A. M. Zheltikov, “High-power wavelength-tunable photonic-crystal-fiber-based oscillator-amplifier-frequency-shifter femtosecond laser system and its applications for material microprocessing,” Laser Phys. Lett. 6, 44-48 (2009).
[CrossRef]

Taylor, J. R.

Thomsen, B. C.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Tropper, A. C.

Tunnermann, A.

Tünnermann, A.

Turchinovich, D.

Venkateraman, N.

Wadsworth, W. J.

Wang, C.-Y.

B.-W. Liu, M.-L. Hu, X.-H. Fang, Y.-Z. Wu, Y.-J. Song, L. Chai, C.-Y. Wang, and A. M. Zheltikov, “High-power wavelength-tunable photonic-crystal-fiber-based oscillator-amplifier-frequency-shifter femtosecond laser system and its applications for material microprocessing,” Laser Phys. Lett. 6, 44-48 (2009).
[CrossRef]

Wilcox, K. G.

Wise, F. W.

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58-73 (2008).
[CrossRef]

Wright, E. M.

Wu, Y.-Z.

B.-W. Liu, M.-L. Hu, X.-H. Fang, Y.-Z. Wu, Y.-J. Song, L. Chai, C.-Y. Wang, and A. M. Zheltikov, “High-power wavelength-tunable photonic-crystal-fiber-based oscillator-amplifier-frequency-shifter femtosecond laser system and its applications for material microprocessing,” Laser Phys. Lett. 6, 44-48 (2009).
[CrossRef]

Zaouter, Y.

Zellmer, H.

Zheltikov, A. M.

B.-W. Liu, M.-L. Hu, X.-H. Fang, Y.-Z. Wu, Y.-J. Song, L. Chai, C.-Y. Wang, and A. M. Zheltikov, “High-power wavelength-tunable photonic-crystal-fiber-based oscillator-amplifier-frequency-shifter femtosecond laser system and its applications for material microprocessing,” Laser Phys. Lett. 6, 44-48 (2009).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. B (1)

J. Lægsgaard, “Control of fiber laser mode-locking by narrow-band Bragg gratings,” J. Phys. B 41, 095401 (2008).
[CrossRef]

Laser Photonics Rev. (1)

F. W. Wise, A. Chong, and W. H. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58-73 (2008).
[CrossRef]

Laser Phys. Lett. (1)

B.-W. Liu, M.-L. Hu, X.-H. Fang, Y.-Z. Wu, Y.-J. Song, L. Chai, C.-Y. Wang, and A. M. Zheltikov, “High-power wavelength-tunable photonic-crystal-fiber-based oscillator-amplifier-frequency-shifter femtosecond laser system and its applications for material microprocessing,” Laser Phys. Lett. 6, 44-48 (2009).
[CrossRef]

Nat. Phys. (1)

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597-603 (2007).
[CrossRef]

Opt. Express (13)

C. J. S. De Matos, J. R. Taylor, T. P. Hansen, K. P. Hansen, and J. Broeng, “All-fiber chirped pulse amplification using highly-dispersive air-core photonic bandgap fiber,” Opt. Express 11, 2832-2837 (2003).
[CrossRef] [PubMed]

J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, and A. Tunnermann, “All fiber chirped-pulse amplification system based on compression in air-guiding photonic bandgap fiber,” Opt. Express 11, 3332-3337 (2003).
[CrossRef] [PubMed]

C. Billet, J. M. Dudley, N. Joly, and J. C. Knight, “Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm,” Opt. Express 13, 3236-3241 (2005).
[CrossRef] [PubMed]

D. G. Ouzounov, C. J. Hensley, A. L. Gaeta, N. Venkateraman, M. T. Gallagher, and K. W. Koch, “Soliton pulse compression in photonic band-gap fibers,” Opt. Express 13, 6153-6159 (2005).
[CrossRef] [PubMed]

C. K. Nielsen, B. Ortac, T. Schreiber, J. Limpert, R. Hohmuth, W. Richter, and A. Tünnermann, “Self-starting self-similar all-polarization maintaining yb-doped fiber laser,” Opt. Express 13, 9346-9351 (2005).
[CrossRef] [PubMed]

C. Finot, F. Parmigiani, P. Petropoulos, and D. J. Richardson, “Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime,” Opt. Express 14, 3161-3170 (2006).
[CrossRef] [PubMed]

C. K. Nielsen, K. G. Jespersen, and S. R. Keiding, “A 158 fs5.3 nj fiber-laser system at 1 μm using photonic bandgap fibers for dispersion control and pulse compression,” Opt. Express 14, 6063-6068 (2006).
[CrossRef] [PubMed]

P. Dupriez, C. Finot, A. Malinowski, J. K. Sahu, J. Nilsson, D. J. Richardson, K. G. Wilcox, H. D. Foreman, and A. C. Tropper, “High-power, high repetition rate picosecond and femtosecond sources based on yb-doped fiber amplification of VECSELs,” Opt. Express 14, 9611-9616 (2006).
[CrossRef] [PubMed]

F. Gerome, K. Cook, A. K. George, W. J. Wadsworth, and J. C. Knight, “Delivery of sub-100 fs pulses through 8 m of hollow-core fiber using soliton compression,” Opt. Express 15, 7126-7131 (2007).
[CrossRef] [PubMed]

F. Gérôme, J. Dupriez, J. C. Knight, and W. J. Wadsworth, “High power tunable femtosecond soliton source using hollow-core photonic bandgap fiber, and its use for frequency doubling,” Opt. Express 16, 2381-2386 (2008).
[CrossRef] [PubMed]

A. V. Gorbach and D. V. Skryabin, “Soliton self-frequency shift, non-solitonic radiation and self-induced transparency in air-core fibers,” Opt. Express 16, 4858-4865 (2008).
[CrossRef] [PubMed]

J. Lægsgaard and P. J. Roberts, “Dispersive pulse compression in hollow-core photonic bandgap fibers,” Opt. Express 16, 9628-9644 (2008).
[CrossRef] [PubMed]

D. Turchinovich, X. Liu, and J. Laegsgaard, “Monolithic all-pm femtosecond yb-fiber laser stabilized with a narrow-band fiber Bragg grating and pulse-compressed in a hollow-core photonic crystal fiber,” Opt. Express 16, 14004-14014 (2008).
[CrossRef] [PubMed]

Opt. Lett. (3)

Phys. Rev. Lett. (1)

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Other (3)

“JCMwave GmbH, www.jcmwave.com.”

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

J. Lægsgaard has prepared a paper to be called “Soliton formation in hollow-core photonic bandgap fibers.”

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Figures (13)

Fig. 1
Fig. 1

Schematic of the fiber laser layout studied in the present work. The gain section of the oscillator is modeled as a point amplifier, as described in the text.

Fig. 2
Fig. 2

Dispersion curve for the HC-PBG fiber investigated in the present work. Inset shows the modeled fiber structure.

Fig. 3
Fig. 3

(a) Spectra and (b) temporal profiles of the MOPA pulse labeled M5 in Table 1, compared to parabolic pulses with either the peak curvature (P1) or the main peak pulse energy (P2) matched to M5. Only the P1 spectrum is shown, because the P1 and P2 spectra are very similar.

Fig. 4
Fig. 4

(a) Pulse quality Q versus compressed-pulse peak power for MOPA pulses of bandwidth 13 15 nm , compared to ideal parabolic pulses as described in the text. (b) FWHM ( t FWHM ) of the main peak at maximal compressed power.

Fig. 5
Fig. 5

(a) Pulse shape at maximal compression for MOPA pulses M1 and M5, with power normalized to the maximal input power, P 0 . (b) Same as (a), for parabolic pulses of type P1 with the same input power as in (a).

Fig. 6
Fig. 6

Maximal pulse power, P max as a function of propagation distance, z for the M5 MOPA pulse and the corresponding P1 and P2 parabolic pulses with matching peak power.

Fig. 7
Fig. 7

Spectrograms for the MOPA pulse labeled M5 in Table 1.

Fig. 8
Fig. 8

Spectrograms for the P1 input pulse matched to the peak power of the M5 MOPA pulse.

Fig. 9
Fig. 9

Maximal peak power versus bandwidth for MOPA pulses of 1100 nJ and 1800 nJ pulse energy.

Fig. 10
Fig. 10

Spectra at maximal peak power for the MOPA pulses M5 and M10.

Fig. 11
Fig. 11

Long-wavelength spectra of the pulses M5, M9, and M10 after 2 m of propagation.

Fig. 12
Fig. 12

Maximal peak power of long-pass filtered solitons as a function of propagation distance, z, for broadband and narrowband MOPA input pulses with pulse energy 1100 nJ (M5, M10) and 1800 nJ (M7, M15).

Fig. 13
Fig. 13

Energy in long-pass filtered soliton, E sol as a function of bandwidth for MOPA input pulses with 1100 nJ and 1800 nJ pulse energy.

Tables (1)

Tables Icon

Table 1 Amplifier Parameters and Output Pulse Parameters for the Various MOPA Designs Investigated

Equations (11)

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

E ( t ) = E 0 [ 1 ( t t 0 ) 2 ] exp ( i ( C t 2 + ω 0 t ) ) ,
t < t 0 ; E ( t ) = 0 , t > t 0 , C = ω 0 2 W 8 π c t 0 .
G ( ω ) = exp [ L G ( N 2 σ e ( ω ) N 1 σ a ( ω ) ) ] ,
N 2 = N t 2 + E ¯ p E sat ,
A ( z , ω ) z = i ω c exp ( i β ( ω ) z ) ν = 1 2 n 2 ( ν ) [ A eff ( ν ) ( ω ) ] 1 4 × 1 ( 2 π ) 2 d ω 1 2 A ̂ ( ν ) ( z , ω 1 ) A ̂ ( ν ) ( z , ω 2 ) A ̂ ( ν ) * ( z , ω 1 + ω 2 ω ) R ν ( ω ω 1 ) ,
A ̂ ( ν ) ( z , ω ) = A ̃ ( z , ω ) [ A eff ( ν ) ( ω ) ] 1 4 ,
A eff ( ν ) = μ 0 [ Re d r e × h * z ̂ ] 2 ε 0 n ν 2 ν d r e ( r ) 4 .
E ( r , t ) = 1 2 π d ω A ( z , ω ) e ( r , ω ) exp ( i ( β ( ω ) z ω t ) ) ,
R ν ( t ) = δ ( t ) + f ν τ 1 ν 2 + τ 2 ν 2 τ 1 ν τ 2 ν 2 sin ( t τ 1 ν ) exp ( t τ 2 ν ) ,
F ( λ ) = 1 exp [ ( λ f λ ) Δ λ ] + 1 ,
E sol = τ P max 2 1.76 ,

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