Abstract

The pump-induced degradation of the temporal contrast of laser pulses amplified by optical parametric chirped-pulse amplification (OPCPA) is studied analytically. In OPCPA systems, the temporal fluctuations of the pump pulse are coupled to the spectrum of the chirped signal by the instantaneous parametric gain and lead to a reduction in the temporal contrast of the recompressed amplified signal. The intensity and shape of the induced temporal pedestal depend on the pump fluctuations and the parametric amplifier operating regime. General equations describing the contrast degradation are derived and applied to the case of sinusoidal pump-intensity modulation and pump amplified spontaneous emission. The reduction of the contrast in the amplified pulse is quantified analytically and via simulations for an OPCPA system.

© 2007 Optical Society of America

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  1. D. Umstadter, "Review of physics and applications of relativistic plasmas driven by ultra-intense lasers," Phys. Plasmas 8, 1774-1785 (2001).
    [CrossRef]
  2. S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovksy, "Generation and characterization of the highest laser intensities (1022 W/cm2)," Opt. Lett. 29, 2837-2839 (2004).
    [CrossRef]
  3. J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I. Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453-482 (2006).
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    [CrossRef]
  5. V. Bagnoud, J. D. Zuegel, N. Forget, and C. Le Blanc, "High-dynamic-range temporal measurements of short pulses amplified by OPCPA," Opt. Express 15, 5504-5511 (2007).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  8. A. Dubietis, R. Butkus, and A. P. Piskarskas, "Trends in chirped pulse optical parametric amplification," IEEE J. Sel. Top. Quantum Electron. 12, 163-172 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
  10. V. Bagnoud, I. A. Begishev, M. J. Guardalben, J. Puth, and J. D. Zuegel, "5-Hz, >250-mJ optical parametric chirped-pulse amplifier at 1053 nm," Opt. Lett. 30, 1843-1845 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  17. J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA Laser Facility," J. Phys. IV 133, 75-80 (2006).
    [CrossRef]
  18. Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, "Prepulse-free petawatt laser for a fast ignitor," IEEE J. Quantum Electron. 40, 281-293 (2004).
    [CrossRef]
  19. C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, B. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, "Vulcan petawatt--an ultra-high-intensity interaction facility," Nucl. Fusion 44, S239-S246 (2004).
    [CrossRef]
  20. N. Forget, A. Cotel, E. Brambrink, P. Audebert, C. Le Blanc, A. Jullien, O. Albert, and G. Chériaux, "Pump-noise transfer in optical parametric chirped-pulse amplification," Opt. Lett. 30, 2921-2923 (2005).
    [CrossRef] [PubMed]
  21. I. N. Ross, G. H. C. New, and P. K. Bates, "Contrast limitation due to pump noise in an optical parametric chirped pulse amplification system," Opt. Commun. 273, 510-514 (2007).
    [CrossRef]
  22. C. Dorrer, A. V. Okishev, I. A. Begishev, J. D. Zuegel, V. I. Smirnov, and L. B. Glebov, "Optical parametric chirped-pulse-amplification contrast enhancement by regenerative pump spectral filtering," Opt. Lett. 32, 2378-2380 (2007).
    [CrossRef] [PubMed]

2007 (4)

2006 (5)

V. V. Lozhkarev, G. I. Freidman, V. N. Ginzburg, E. V. Katin, E. A. Khazanov, A. V. Kirsanov, G. A. Luchinin, A. N. Mal'shakov, M. A. Martyanov, O. V. Palashov, A. K. Poteomkin, A. M. Sergeev, A. A. Shaykin, I. V. Yakovlev, S. G. Garanin, S. A. Sukharev, N. N. Rukavishnikov, A. V. Charukhchev, R. R. Gerke, and V. E. Yashin, "200 TW fs laser based on optical parametric chirped pulse amplification," Opt. Express 14, 446-454 (2006).
[CrossRef] [PubMed]

O. V. Chekhlov, J. L. Collier, I. N. Ross, P. K. Bates, M. Notley, C. Hernandez-Gomez, W. Shaikh, C. N. Danson, D. Neely, P. Matousek, S. Hancock, and L. Cardoso, "35 J broadband femtosecond optical parametric chirped pulse amplification system," Opt. Lett. 31, 3665-3667 (2006).
[CrossRef] [PubMed]

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA Laser Facility," J. Phys. IV 133, 75-80 (2006).
[CrossRef]

A. Dubietis, R. Butkus, and A. P. Piskarskas, "Trends in chirped pulse optical parametric amplification," IEEE J. Sel. Top. Quantum Electron. 12, 163-172 (2006).
[CrossRef]

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I. Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453-482 (2006).

2005 (5)

2004 (3)

Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, "Prepulse-free petawatt laser for a fast ignitor," IEEE J. Quantum Electron. 40, 281-293 (2004).
[CrossRef]

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, B. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, "Vulcan petawatt--an ultra-high-intensity interaction facility," Nucl. Fusion 44, S239-S246 (2004).
[CrossRef]

S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovksy, "Generation and characterization of the highest laser intensities (1022 W/cm2)," Opt. Lett. 29, 2837-2839 (2004).
[CrossRef]

2003 (1)

2001 (1)

D. Umstadter, "Review of physics and applications of relativistic plasmas driven by ultra-intense lasers," Phys. Plasmas 8, 1774-1785 (2001).
[CrossRef]

1998 (1)

M. Nantel, J. Itatani, A.-C. Tien, J. Faure, D. Kaplan, M. Bouvier, T. Buma, P. Van Rompay, J. A. Nees, P. P. Pronko, D. Umstadter, and G. A. Mourou, "Temporal contrast in Ti:sapphire lasers: characterization and control," IEEE J. Sel. Top. Quantum Electron. 4, 449-458 (1998).
[CrossRef]

1997 (1)

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]

1992 (1)

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]

Fusion Sci. Technol. (1)

J. D. Zuegel, S. Borneis, C. Barty, B. LeGarrec, C. Danson, N. Miyanaga, P. K. Rambo, C. LeBlanc, T. J. Kessler, A. W. Schmid, L. J. Waxer, J. H. Kelly, B. Kruschwitz, R. Jungquist, E. Moses, J. Britten, I. Jovanovic, J. Dawson, and N. Blanchot, "Laser challenges for fast ignition," Fusion Sci. Technol. 49, 453-482 (2006).

IEEE J. Quantum Electron. (1)

Y. Kitagawa, H. Fujita, R. Kodama, H. Yoshida, S. Matsuo, T. Jitsuno, T. Kawasaki, H. Kitamura, T. Kanabe, S. Sakabe, K. Shigemori, N. Miyanaga, and Y. Izawa, "Prepulse-free petawatt laser for a fast ignitor," IEEE J. Quantum Electron. 40, 281-293 (2004).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

M. Nantel, J. Itatani, A.-C. Tien, J. Faure, D. Kaplan, M. Bouvier, T. Buma, P. Van Rompay, J. A. Nees, P. P. Pronko, D. Umstadter, and G. A. Mourou, "Temporal contrast in Ti:sapphire lasers: characterization and control," IEEE J. Sel. Top. Quantum Electron. 4, 449-458 (1998).
[CrossRef]

A. Dubietis, R. Butkus, and A. P. Piskarskas, "Trends in chirped pulse optical parametric amplification," IEEE J. Sel. Top. Quantum Electron. 12, 163-172 (2006).
[CrossRef]

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

J. Phys. IV (1)

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA Laser Facility," J. Phys. IV 133, 75-80 (2006).
[CrossRef]

Nucl. Fusion (1)

C. N. Danson, P. A. Brummitt, R. J. Clarke, J. L. Collier, B. Fell, A. J. Frackiewicz, S. Hancock, S. Hawkes, C. Hernandez-Gomez, P. Holligan, M. H. R. Hutchinson, A. Kidd, W. J. Lester, I. O. Musgrave, D. Neely, D. R. Neville, P. A. Norreys, D. A. Pepler, C. J. Reason, W. Shaikh, T. B. Winstone, R. W. W. Wyatt, and B. E. Wyborn, "Vulcan petawatt--an ultra-high-intensity interaction facility," Nucl. Fusion 44, S239-S246 (2004).
[CrossRef]

Opt. Commun. (3)

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]

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]

I. N. Ross, G. H. C. New, and P. K. Bates, "Contrast limitation due to pump noise in an optical parametric chirped pulse amplification system," Opt. Commun. 273, 510-514 (2007).
[CrossRef]

Opt. Express (3)

Opt. Lett. (8)

S.-W. Bahk, P. Rousseau, T. A. Planchon, V. Chvykov, G. Kalintchenko, A. Maksimchuk, G. A. Mourou, and V. Yanovksy, "Generation and characterization of the highest laser intensities (1022 W/cm2)," Opt. Lett. 29, 2837-2839 (2004).
[CrossRef]

H. Yoshida, E. Ishii, R. Kodama, H. Fujita, Y. Kitagawa, Y. Izawa, and T. Yamanaka, "High-power and high-contrast optical parametric chirped pulse amplification by β-BaB2O4 crystal," Opt. Lett. 28, 257-259 (2003).
[CrossRef] [PubMed]

V. Bagnoud, I. A. Begishev, M. J. Guardalben, J. Puth, and J. D. Zuegel, "5-Hz, >250-mJ optical parametric chirped-pulse amplifier at 1053 nm," Opt. Lett. 30, 1843-1845 (2005).
[CrossRef] [PubMed]

N. Ishii, L. Turi, V. S. Yakovlev, T. Fuji, F. Krausz, A. Baltuska, R. Butkus, G. Geitas, V. Smilgevicius, R. Danielius, and A. Piskarskas, "Multimillijoule chirped parametric amplification of few-cycle pulses," Opt. Lett. 30, 567-569 (2005).
[CrossRef] [PubMed]

I. Jovanovic, C. G. Brown, C. A. Ebbers, C. P. J. Barty, N. Forget, and C. Le Blanc, "Generation of high-contrast millijoule pulses by optical parametic chirped-pulse amplification in periodically poled KTiOPO4," Opt. Lett. 30, 1036-1038 (2005).
[CrossRef] [PubMed]

O. V. Chekhlov, J. L. Collier, I. N. Ross, P. K. Bates, M. Notley, C. Hernandez-Gomez, W. Shaikh, C. N. Danson, D. Neely, P. Matousek, S. Hancock, and L. Cardoso, "35 J broadband femtosecond optical parametric chirped pulse amplification system," Opt. Lett. 31, 3665-3667 (2006).
[CrossRef] [PubMed]

N. Forget, A. Cotel, E. Brambrink, P. Audebert, C. Le Blanc, A. Jullien, O. Albert, and G. Chériaux, "Pump-noise transfer in optical parametric chirped-pulse amplification," Opt. Lett. 30, 2921-2923 (2005).
[CrossRef] [PubMed]

C. Dorrer, A. V. Okishev, I. A. Begishev, J. D. Zuegel, V. I. Smirnov, and L. B. Glebov, "Optical parametric chirped-pulse-amplification contrast enhancement by regenerative pump spectral filtering," Opt. Lett. 32, 2378-2380 (2007).
[CrossRef] [PubMed]

Phys. Plasmas (1)

D. Umstadter, "Review of physics and applications of relativistic plasmas driven by ultra-intense lasers," Phys. Plasmas 8, 1774-1785 (2001).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematic of an OPCPA system. Pump-intensity modulation gets transferred onto the spectrum of the chirped signal in the optical parametric amplifier (OPA). The modulation of the spectrum of the recompressed signal induces contrast-reducing temporal features on the recompressed signal.

Fig. 2
Fig. 2

Representation of the transfer function between output signal intensity and pump intensity around the operating point of a parametric amplifier in the (a) unsaturated and (b) saturated regimes. At point A, there is a linear relation between pump intensity modulation and amplified signal intensity modulation. At point B, there is a quadratic relation between pump-intensity and amplified signal intensity modulation.

Fig. 3
Fig. 3

Transfer function of the OPCPA preamplifier for a signal intensity equal to 0.1 W cm 2 . Points A and B identify the linear and quadratic regimes of operation, respectively.

Fig. 4
Fig. 4

Intensity of the recompressed signal for a sinusoidal modulation of the pump intensity with a period equal to 30 ps . The first column corresponds to an amplifier run in the linear modulation regime for a modulation amplitude (a) 10 3 , (b) 10 2 , and (c) 10 1 . The second column corresponds to an amplifier run in the quadratic modulation regime for a modulation amplitude (d) 10 3 , (e) 10 2 , and (f) 10 1 . Arrows point to the features predicted analytically.

Fig. 5
Fig. 5

Close-up on the temporal intensity of the pump for ASE with a 0.14 nm FWHM Gaussian spectrum and a fractional energy equal to (a) 10 5 , (b) 10 4 , and (c) 10 3 , and (d) for ASE with a 0.03 nm FWHM Gaussian spectrum and a fractional energy equal to 10 3 .

Fig. 6
Fig. 6

Intensity of the recompressed signal for ASE with a 0.14 nm FWHM Gaussian spectrum. The first column corresponds to an amplifier run in the linear modulation regime when the fractional ASE energy is equal to (a) 10 5 , (b) 10 4 , and (c) 10 3 . The second column corresponds to an amplifier run in the quadratic modulation regime when the fractional ASE energy is equal to (d) 10 5 , (e) 10 4 , and (f) 10 3 . In each case, the simulated intensity is plotted with a continuous line, and the intensity predicted analytically is plotted with round markers.

Fig. 7
Fig. 7

(a) and (d): Intensity of the recompressed signal for ASE with a 0.03 nm FWHM Gaussian spectrum and a fractional ASE energy equal to 10 3 in the linear and quadratic modulation regimes, respectively. (b) and (e): Intensity of the recompressed signal for ASE with a 0.14 nm FWHM Gaussian spectrum filtered by a 0.20 nm FWHM, 20th-order super-Gaussian filter and a fractional ASE energy equal to 10 3 in the linear and quadratic modulation regimes, respectively. (c) and (f): Intensity of the recompressed signal for ASE with a 0.14 nm FWHM Gaussian spectrum centered 0.07 nm away from the central wavelength of the pump pulse, and a fractional ASE energy equal to 10 3 in the linear and quadratic modulation regimes, respectively. In each case, the simulated intensity is plotted with a continuous curve, and the intensity predicted analytically is plotted with round markers.

Equations (47)

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E signal , 1 ( t ) = ( 1 φ ) E ̃ signal , 0 ( t φ ) exp ( i t 2 2 φ ) ,
I signal , 2 ( t ) = f [ I signal , 1 ( t ) , I pump ( t ) ] .
f [ I signal , 1 ( φ ω ) , I pump ( φ ω ) ] = f [ I ̃ signal , 0 ( ω ) φ , I pump ( φ ω ) ] .
f [ I ̃ signal , 0 ( ω ) φ , I pump ( φ ω ) ] = f [ I ̃ signal , 0 ( ω ) φ , I pump ( 0 ) ( φ ω ) ] + δ I pump ( φ ω ) f I pump [ I ̃ signal , 0 ( ω ) φ , I pump ( 0 ) ( φ ω ) ] .
I ̃ signal , 3 ( ω ) = φ f [ I ̃ signal , 0 ( ω ) φ , I pump ( φ ω ) ] ,
I ̃ signal , 3 ( 0 ) ( ω ) = φ f [ I ̃ signal , 0 ( ω ) φ , I pump ( 0 ) ( φ ω ) ] ,
f ( 1 ) = f I pump [ I ̃ signal , 0 ( ω ) φ , I pump ( 0 ) ( φ ω ) ] .
E ̃ signal , 3 ( ω ) = I ̃ signal , 3 ( 0 ) ( ω ) + φ f ( 1 ) δ I pump ( φ ω ) exp [ i φ residual ( ω ) ] .
E ̃ signal , 3 ( ω ) = I ̃ signal , 3 ( 0 ) ( ω ) exp [ i φ residual ( ω ) ] [ 1 + φ f ( 1 ) δ I pump ( φ ω ) 2 I ̃ signal , 3 ( 0 ) ( ω ) ] .
E ̃ signal , 3 ( ω ) = E ̃ signal , 3 ( 0 ) ( ω ) [ 1 + f ( 1 ) δ I pump ( φ ω ) 2 I signal , 2 ] .
E signal , 3 ( t ) = E signal , 3 ( 0 ) ( t ) + f ( 1 , N ) 2 φ I pump ( 0 ) E signal , 3 ( 0 ) ( t ) δ I ̃ pump ( t φ ) .
I signal , 3 ( t ) = I signal , 3 ( 0 ) ( t ) + f ( 1 , N ) 2 4 [ φ I pump ( 0 ) ] 2 E signal , 3 ( 0 ) ( t ) δ I ̃ pump ( t φ ) 2 ,
f [ I ̃ signal , 0 ( ω ) φ , I pump ( φ ω ) ] = f [ I ̃ signal , 0 ( ω ) φ , I pump ( 0 ) ( φ ω ) ] + 1 2 [ δ I pump ( φ ω ) ] 2 × 2 f I pump 2 [ I ̃ signal , 0 ( ω ) φ , I pump ( 0 ) ( φ ω ) ] .
f ( 2 ) = 2 f I pump 2 [ I ̃ signal , 0 ( ω ) φ , I pump ( 0 ) ( φ ω ) ] .
E ̃ signal , 3 ( ω ) = E ̃ signal , 3 ( 0 ) ( ω ) [ 1 + f ( 2 ) δ I pump 2 ( φ ω ) 4 I signal , 2 ] .
E signal , 3 ( t ) = E signal , 3 ( 0 ) ( t ) + f ( 2 , N ) 4 [ φ I pump ( 0 ) ] 2 E signal , 3 ( 0 ) ( t ) δ I ̃ pump ( t φ ) δ I ̃ pump ( t φ ) ,
I signal , 3 ( t ) = I signal , 3 ( 0 ) ( t ) + f ( 2 , N ) 2 16 [ φ I pump ( 0 ) ] 4 E signal , 3 ( t ) δ I ̃ pump ( t φ ) δ I ̃ pump ( t φ ) 2 .
δ I ̃ pump ( ω ) = α I pump ( 0 ) 2 [ δ ( ω 2 π β ) + δ ( ω + 2 π β ) ] .
E signal , 3 ( t ) = E signal , 3 ( 0 ) ( t ) + α f ( 1 , N ) 4 [ E signal , 3 ( 0 ) ( t 2 π φ β ) + E signal , 3 ( 0 ) ( t + 2 π φ β ) ] .
I signal , 3 ( t ) = I signal , 3 ( 0 ) ( t ) + α 2 f ( 1 , N ) 2 16 [ I signal , 3 ( 0 ) ( t 2 π φ β ) + I signal , 3 ( 0 ) ( t + 2 π φ β ) ] .
E signal , 3 ( t ) = E signal , 3 ( 0 ) ( t ) + f ( 2 , N ) α 2 16 [ E signal , 3 ( 0 ) ( t 4 π φ β ) + 2 E signal , 3 ( 0 ) ( t ) + E signal , 3 ( 0 ) ( t + 4 π φ β ) ] .
I signal , 3 ( t ) = I signal , 3 ( 0 ) ( t ) + f ( 2 , N ) 2 α 4 256 [ I signal , 3 ( 0 ) ( t 4 π φ β ) + I signal , 3 ( 0 ) ( t + 4 π φ β ) ] .
E pump ( t ) = E pump ( 0 ) ( t ) + E ASE ( t ) .
δ I ̃ pump ( ω ) = I pump ( 0 ) [ E ̃ ASE , T ( ω ) + E ̃ ASE , T * ( ω ) ] .
I signal , 3 ( t ) = I signal , 3 ( 0 ) ( t ) + f ( 1 , N ) 2 4 ε pump I signal , 3 ( 0 ) ( t φ ω ) [ I ̃ ASE , T ( ω ) + I ̃ ASE , T ( ω ) ] d ω ,
I signal , 3 ( t ) = I signal , 3 ( 0 ) ( t ) + f ( 1 , N ) 2 ε signal ( 0 ) 4 φ ε pump [ I ̃ ASE , T ( t φ ) + I ̃ ASE , T ( t φ ) ] .
ε pedestal ε signal ( 0 ) = f ( 1 , N ) 2 2 ε ASE , T ε pump .
I signal , 3 ( t ) = I signal , 3 ( 0 ) ( t ) + f ( 2 , N ) 2 8 ( φ ε pump ) 2 I signal , 3 ( 0 ) ( t ) [ I ̃ ASE , T ( t φ ) + I ̃ ASE , T ( t φ ) ] [ I ̃ ASE , T ( t φ ) + I ̃ ASE , T ( t φ ) ] .
I signal , 3 ( t ) = I signal , 3 ( 0 ) ( t ) + f ( 2 , N ) 2 ε signal ( 0 ) 8 φ 2 ε pump 2 × [ I ̃ ASE , T ( t φ ) + I ̃ ASE , T ( t φ ) ] [ I ̃ ASE , T ( t φ ) + I ̃ ASE , T ( t φ ) ] .
ε pedestal ε signal ( 0 ) = f ( 2 , N ) 2 ε ASE , T 2 2 ε pump 2 .
A = E signal , 3 ( 0 ) ( φ ω ) E ̃ ASE , T ( t φ ω ) d ω ,
B = E signal , 3 ( 0 ) ( φ ω ) E ̃ ASE , T * ( t φ + ω ) d ω .
A 2 = E signal , 3 ( 0 ) ( φ ω ) E signal , 3 ( 0 ) * ( φ ω ) E ̃ ASE , T ( t φ ω ) E ̃ ASE , T * ( t φ ω ) d ω d ω .
A 2 = I signal , 3 ( 0 ) ( φ ω ) I ̃ ASE , T ( t φ ω ) d ω T .
B 2 = I signal , 3 ( 0 ) ( φ ω ) I ̃ ASE , T ( t φ + ω ) d ω T .
I signal , 3 ( t ) = I signal , 3 ( 0 ) ( t ) + f ( 1 , N ) 2 4 ε pump I signal , 3 ( 0 ) ( t φ ω ) [ I ̃ ASE , T ( ω ) + I ̃ ASE , T ( ω ) ] d ω .
E signal , 3 ( 0 ) ( t ) δ I ̃ pump ( t φ ) δ I ̃ pump ( t φ ) [ φ 2 I pump ( 0 ) ] = A + 2 B + C ,
A = E signal , 3 ( 0 ) ( t ) E ̃ ASE , T ( t φ ) E ̃ ASE , T ( t φ ) φ 2 ,
B = E signal , 3 ( 0 ) ( t ) E ̃ ASE , T ( t φ ) E ̃ ASE , T ( t φ ) φ 2 ,
C = E signal , 3 ( 0 ) ( t ) E ̃ ASE , T ( t φ ) E ̃ ASE , T ( t φ ) φ 2 .
A 2 = E signal , 3 ( 0 ) ( t φ ω ) E signal , 3 ( 0 ) * ( t φ Ω ) × E ̃ ASE , T ( ω ω ) E ̃ ASE , T * ( Ω Ω ) × E ̃ ASE , T ( ω ) E ̃ ASE , T * ( Ω ) d ω d ω d Ω d Ω .
A 2 = 2 I signal , 3 ( 0 ) ( t ) I ̃ ASE , T ( t φ ) I ̃ ASE , T ( t φ ) ( φ T ) 2 .
C 2 = 2 I signal , 3 ( 0 ) ( t ) I ̃ ASE , T ( t φ ) I ̃ ASE , T ( t φ ) ( φ T ) 2 .
B 2 = E signal , 3 ( 0 ) ( t φ ω ) E signal , 3 ( 0 ) * ( t φ Ω ) E ̃ ASE , T ( ω ω ) E ̃ ASE , T * ( Ω Ω ) E ̃ ASE , T * ( ω ) E ̃ ASE , T ( Ω ) d ω d ω d Ω d Ω .
I signal , 3 ( 0 ) ( φ ω ) I ̃ ASE , T ( ω ) I ̃ ASE , T ( ω ) ( φ T ) 2 ,
B 2 = I signal , 3 ( 0 ) ( t ) I ̃ ASE , T ( t φ ) I ̃ ASE , T ( t φ ) ( φ T ) 2 .
I signal , 3 ( t ) = I signal , 3 ( 0 ) ( t ) + f ( 2 , N ) 2 8 ( φ ε pump ) 2 I signal , 3 ( 0 ) ( t ) [ I ̃ ASE , T ( t φ ) + I ̃ ASE , T ( t φ ) ] [ I ̃ ASE , T ( t φ ) + I ̃ ASE , T ( t φ ) ] .

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