Abstract

We present a fiber master oscillator power amplifier system designed to meet stringent requirements of large-scale laser facility front-end sources. A strictly single-mode beam without spatial phase, narrow-bandwidth nanosecond pulse amplification up to 20μJ is obtained with more than 40 dB optical signal-to-noise ratio, 30 dB polarization extinction ratio, and sub-nanosecond resolution temporal shaping. Experimental results are in excellent agreement with numerical simulations including amplified spontaneous emission. Such a system could advantageously replace cumbersome bulky solid systems.

© 2010 Optical Society of America

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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] [PubMed]

2010 (1)

2009 (1)

2008 (1)

2007 (3)

C. D. Brooks and F. Di Teodoro, “High peak power operation and harmonic generation of a single-polarization, Yb-doped photonic crystal fiber amplifier,” Opt. Commun. 280, 424–430 (2007).
[CrossRef]

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and T. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13, 537–545 (2007).
[CrossRef]

C. A. Haynam, P. J. Wegner, J. M. Auerbach, M. W. Bowers, S. N. Dixit, G. V. Erbert, G. M. Heestand, M. A. Henesian, M. R. Hermann, K. S. Jancaitis, K. R. Manes, C. D. Marshall, N. C. Mehta, J. Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C. D. Orth, R. Patterson, R. A. Sacks, M. J. Shaw, M. Spaeth, S. B. Sutton, W. H. Williams, C. C. Widmayer, R. K. White, S. T. Yang, and B. M. Wonterghem, “National Ignition Facility laser performance status,” Appl. Opt. 46, 3276–3303 (2007).
[CrossRef] [PubMed]

2006 (3)

2005 (1)

C. Cavailler, “Inertial fusion with the LMJ,” Plasma Phys. Controlled Fusion 47, B389–B403 (2005).
[CrossRef]

2004 (3)

2003 (2)

A. Jolly, J. F. Gleyze, J. Luce, H. Coïc, and G. Deschaseaux, “Front-end sources of the LIL-LMJ fusion lasers: progress report and prospects,” Opt. Eng. (Bellingham) 42, 1427–1438 (2003).
[CrossRef]

Y. Wang and H. Po, “Dynamic characteristics of double-clad fiber amplifiers for high-power pulse amplification,” J. Lightwave Technol. 21, 2262–2270 (2003).
[CrossRef]

1999 (1)

M. D. Martinez, K. M. Skulina, F. J. Deadrick, J. K. Crane, B. Moran, J. Braucht, B. Jones, S. Hawkins, R. Tilley, J. Crawford, D. Browning, and F. Penko, “Performance results of the high gain, Nd:glass, engineering prototype preamplifier module (PAM) for the National Ignition Facility (NIF),” Proc. SPIE 3611, 169–180 (1999).
[CrossRef]

1997 (1)

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33, 1049–1056 (1997).
[CrossRef]

1995 (1)

J. Lindl, “Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition and gain,” Phys. Plasmas 2, 3933–4025 (1995).
[CrossRef]

1991 (1)

B. Pedersen, A. Bjarklev, J. H. Povlsen, K. Dybdal, and C. C. Larsen, “The design of erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 1105–1112 (1991).
[CrossRef]

1963 (1)

L. M. Frantz and J. S. Nodvick, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34, 2346–2349 (1963).
[CrossRef]

Abramov, M.

V. Fomin, A. Mashkin, M. Abramov, A. Ferin, and V. Gapontsev, “3 kW Yb fiber lasers with a single-mode output,” at 2nd International Symposium on High-Power Fiber Lasers and Their Applications, St. Petersburg, Russia, 26–30 June 2006 (2006).

Alam, S.

Alam, S. U.

Auerbach, J. M.

Bjarklev, A.

B. Pedersen, A. Bjarklev, J. H. Povlsen, K. Dybdal, and C. C. Larsen, “The design of erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 1105–1112 (1991).
[CrossRef]

Bowers, M. W.

Braucht, J.

M. D. Martinez, K. M. Skulina, F. J. Deadrick, J. K. Crane, B. Moran, J. Braucht, B. Jones, S. Hawkins, R. Tilley, J. Crawford, D. Browning, and F. Penko, “Performance results of the high gain, Nd:glass, engineering prototype preamplifier module (PAM) for the National Ignition Facility (NIF),” Proc. SPIE 3611, 169–180 (1999).
[CrossRef]

Brooks, C. D.

C. D. Brooks and F. Di Teodoro, “High peak power operation and harmonic generation of a single-polarization, Yb-doped photonic crystal fiber amplifier,” Opt. Commun. 280, 424–430 (2007).
[CrossRef]

Browning, D.

M. D. Martinez, K. M. Skulina, F. J. Deadrick, J. K. Crane, B. Moran, J. Braucht, B. Jones, S. Hawkins, R. Tilley, J. Crawford, D. Browning, and F. Penko, “Performance results of the high gain, Nd:glass, engineering prototype preamplifier module (PAM) for the National Ignition Facility (NIF),” Proc. SPIE 3611, 169–180 (1999).
[CrossRef]

Browning, D. F.

P. J. Wisoff, M. W. Bowers, G. V. Erbert, D. F. Browning, and D. R. Jedlovec, “NIF injection laser system,” Proc. SPIE 5341, 146–155 (2004).
[CrossRef]

Cavailler, C.

C. Cavailler, “Inertial fusion with the LMJ,” Plasma Phys. Controlled Fusion 47, B389–B403 (2005).
[CrossRef]

Chen, K. K.

Codemar, C. A.

Coïc, H.

E. Hugonnot, J. Luce, and H. Coïc, “Optical parametric chirped-pulse amplifier and spatiotemporal shaping for a petawatt laser,” Appl. Opt. 45, 377–382 (2006).
[CrossRef] [PubMed]

A. Jolly, J. F. Gleyze, J. Luce, H. Coïc, and G. Deschaseaux, “Front-end sources of the LIL-LMJ fusion lasers: progress report and prospects,” Opt. Eng. (Bellingham) 42, 1427–1438 (2003).
[CrossRef]

Crane, J. K.

M. D. Martinez, K. M. Skulina, F. J. Deadrick, J. K. Crane, B. Moran, J. Braucht, B. Jones, S. Hawkins, R. Tilley, J. Crawford, D. Browning, and F. Penko, “Performance results of the high gain, Nd:glass, engineering prototype preamplifier module (PAM) for the National Ignition Facility (NIF),” Proc. SPIE 3611, 169–180 (1999).
[CrossRef]

Crawford, J.

M. D. Martinez, K. M. Skulina, F. J. Deadrick, J. K. Crane, B. Moran, J. Braucht, B. Jones, S. Hawkins, R. Tilley, J. Crawford, D. Browning, and F. Penko, “Performance results of the high gain, Nd:glass, engineering prototype preamplifier module (PAM) for the National Ignition Facility (NIF),” Proc. SPIE 3611, 169–180 (1999).
[CrossRef]

Deadrick, F. J.

M. D. Martinez, K. M. Skulina, F. J. Deadrick, J. K. Crane, B. Moran, J. Braucht, B. Jones, S. Hawkins, R. Tilley, J. Crawford, D. Browning, and F. Penko, “Performance results of the high gain, Nd:glass, engineering prototype preamplifier module (PAM) for the National Ignition Facility (NIF),” Proc. SPIE 3611, 169–180 (1999).
[CrossRef]

Deschaseaux, G.

A. Jolly, J. F. Gleyze, J. Luce, H. Coïc, and G. Deschaseaux, “Front-end sources of the LIL-LMJ fusion lasers: progress report and prospects,” Opt. Eng. (Bellingham) 42, 1427–1438 (2003).
[CrossRef]

Di Teodoro, F.

C. D. Brooks and F. Di Teodoro, “High peak power operation and harmonic generation of a single-polarization, Yb-doped photonic crystal fiber amplifier,” Opt. Commun. 280, 424–430 (2007).
[CrossRef]

Dixit, S. N.

Dybdal, K.

B. Pedersen, A. Bjarklev, J. H. Povlsen, K. Dybdal, and C. C. Larsen, “The design of erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 1105–1112 (1991).
[CrossRef]

Eberhardt, R.

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and T. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13, 537–545 (2007).
[CrossRef]

Erbert, G. V.

Ferin, A.

V. Fomin, A. Mashkin, M. Abramov, A. Ferin, and V. Gapontsev, “3 kW Yb fiber lasers with a single-mode output,” at 2nd International Symposium on High-Power Fiber Lasers and Their Applications, St. Petersburg, Russia, 26–30 June 2006 (2006).

Fomin, V.

V. Fomin, A. Mashkin, M. Abramov, A. Ferin, and V. Gapontsev, “3 kW Yb fiber lasers with a single-mode output,” at 2nd International Symposium on High-Power Fiber Lasers and Their Applications, St. Petersburg, Russia, 26–30 June 2006 (2006).

Frantz, L. M.

L. M. Frantz and J. S. Nodvick, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34, 2346–2349 (1963).
[CrossRef]

Gapontsev, V.

V. Fomin, A. Mashkin, M. Abramov, A. Ferin, and V. Gapontsev, “3 kW Yb fiber lasers with a single-mode output,” at 2nd International Symposium on High-Power Fiber Lasers and Their Applications, St. Petersburg, Russia, 26–30 June 2006 (2006).

Ghiringhelli, F.

Gleyze, J. F.

A. Jolly, J. F. Gleyze, J. Luce, H. Coïc, and G. Deschaseaux, “Front-end sources of the LIL-LMJ fusion lasers: progress report and prospects,” Opt. Eng. (Bellingham) 42, 1427–1438 (2003).
[CrossRef]

Hanna, D. C.

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33, 1049–1056 (1997).
[CrossRef]

Hawkins, S.

M. D. Martinez, K. M. Skulina, F. J. Deadrick, J. K. Crane, B. Moran, J. Braucht, B. Jones, S. Hawkins, R. Tilley, J. Crawford, D. Browning, and F. Penko, “Performance results of the high gain, Nd:glass, engineering prototype preamplifier module (PAM) for the National Ignition Facility (NIF),” Proc. SPIE 3611, 169–180 (1999).
[CrossRef]

Haynam, C. A.

Heestand, G. M.

Henesian, M. A.

Hermann, M. R.

Hickey, L. M. B.

Horak, P.

Hugonnot, E.

Jancaitis, K. S.

Jedlovec, D. R.

P. J. Wisoff, M. W. Bowers, G. V. Erbert, D. F. Browning, and D. R. Jedlovec, “NIF injection laser system,” Proc. SPIE 5341, 146–155 (2004).
[CrossRef]

Jeong, Y.

Jolly, A.

A. Jolly, J. F. Gleyze, J. Luce, H. Coïc, and G. Deschaseaux, “Front-end sources of the LIL-LMJ fusion lasers: progress report and prospects,” Opt. Eng. (Bellingham) 42, 1427–1438 (2003).
[CrossRef]

Jones, B.

M. D. Martinez, K. M. Skulina, F. J. Deadrick, J. K. Crane, B. Moran, J. Braucht, B. Jones, S. Hawkins, R. Tilley, J. Crawford, D. Browning, and F. Penko, “Performance results of the high gain, Nd:glass, engineering prototype preamplifier module (PAM) for the National Ignition Facility (NIF),” Proc. SPIE 3611, 169–180 (1999).
[CrossRef]

Klingebiel, S.

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and T. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13, 537–545 (2007).
[CrossRef]

Larsen, C. C.

B. Pedersen, A. Bjarklev, J. H. Povlsen, K. Dybdal, and C. C. Larsen, “The design of erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 1105–1112 (1991).
[CrossRef]

Limpert, J.

D. N. Schimpf, C. Ruchert, D. Nodop, J. Limpert, A. Tünnermann, and F. Salin, “Compensation of pulse-distortion in saturated laser amplifiers,” Opt. Express 16, 17637–17645 (2008).
[CrossRef] [PubMed]

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and T. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13, 537–545 (2007).
[CrossRef]

Lin, D.

Lindl, J.

J. Lindl, “Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition and gain,” Phys. Plasmas 2, 3933–4025 (1995).
[CrossRef]

Luce, J.

E. Hugonnot, J. Luce, and H. Coïc, “Optical parametric chirped-pulse amplifier and spatiotemporal shaping for a petawatt laser,” Appl. Opt. 45, 377–382 (2006).
[CrossRef] [PubMed]

A. Jolly, J. F. Gleyze, J. Luce, H. Coïc, and G. Deschaseaux, “Front-end sources of the LIL-LMJ fusion lasers: progress report and prospects,” Opt. Eng. (Bellingham) 42, 1427–1438 (2003).
[CrossRef]

Malinowski, A.

Manes, K. R.

Marciante, J. R.

Marshall, C. D.

Martinez, M. D.

M. D. Martinez, K. M. Skulina, F. J. Deadrick, J. K. Crane, B. Moran, J. Braucht, B. Jones, S. Hawkins, R. Tilley, J. Crawford, D. Browning, and F. Penko, “Performance results of the high gain, Nd:glass, engineering prototype preamplifier module (PAM) for the National Ignition Facility (NIF),” Proc. SPIE 3611, 169–180 (1999).
[CrossRef]

Mashkin, A.

V. Fomin, A. Mashkin, M. Abramov, A. Ferin, and V. Gapontsev, “3 kW Yb fiber lasers with a single-mode output,” at 2nd International Symposium on High-Power Fiber Lasers and Their Applications, St. Petersburg, Russia, 26–30 June 2006 (2006).

Mehta, N. C.

Menapace, J.

Moran, B.

M. D. Martinez, K. M. Skulina, F. J. Deadrick, J. K. Crane, B. Moran, J. Braucht, B. Jones, S. Hawkins, R. Tilley, J. Crawford, D. Browning, and F. Penko, “Performance results of the high gain, Nd:glass, engineering prototype preamplifier module (PAM) for the National Ignition Facility (NIF),” Proc. SPIE 3611, 169–180 (1999).
[CrossRef]

Moses, E.

Murray, J. R.

Nilsson, J.

Nodop, D.

Nodvick, J. S.

L. M. Frantz and J. S. Nodvick, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34, 2346–2349 (1963).
[CrossRef]

Nostrand, M. C.

Okishev, A. V.

Orth, C. D.

Paschotta, R.

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33, 1049–1056 (1997).
[CrossRef]

Patterson, R.

Payne, D.

Pedersen, B.

B. Pedersen, A. Bjarklev, J. H. Povlsen, K. Dybdal, and C. C. Larsen, “The design of erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 1105–1112 (1991).
[CrossRef]

Penko, F.

M. D. Martinez, K. M. Skulina, F. J. Deadrick, J. K. Crane, B. Moran, J. Braucht, B. Jones, S. Hawkins, R. Tilley, J. Crawford, D. Browning, and F. Penko, “Performance results of the high gain, Nd:glass, engineering prototype preamplifier module (PAM) for the National Ignition Facility (NIF),” Proc. SPIE 3611, 169–180 (1999).
[CrossRef]

Peschel, T.

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and T. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13, 537–545 (2007).
[CrossRef]

Po, H.

Povlsen, J. H.

B. Pedersen, A. Bjarklev, J. H. Povlsen, K. Dybdal, and C. C. Larsen, “The design of erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 1105–1112 (1991).
[CrossRef]

Richardson, D. J.

Röser, F.

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and T. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13, 537–545 (2007).
[CrossRef]

Ruchert, C.

Sacks, R. A.

Sahu, J.

Salin, F.

Schimpf, D. N.

Schreiber, T.

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and T. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13, 537–545 (2007).
[CrossRef]

Shaw, M. J.

Skulina, K. M.

M. D. Martinez, K. M. Skulina, F. J. Deadrick, J. K. Crane, B. Moran, J. Braucht, B. Jones, S. Hawkins, R. Tilley, J. Crawford, D. Browning, and F. Penko, “Performance results of the high gain, Nd:glass, engineering prototype preamplifier module (PAM) for the National Ignition Facility (NIF),” Proc. SPIE 3611, 169–180 (1999).
[CrossRef]

Spaeth, M.

Sutton, S. B.

Tilley, R.

M. D. Martinez, K. M. Skulina, F. J. Deadrick, J. K. Crane, B. Moran, J. Braucht, B. Jones, S. Hawkins, R. Tilley, J. Crawford, D. Browning, and F. Penko, “Performance results of the high gain, Nd:glass, engineering prototype preamplifier module (PAM) for the National Ignition Facility (NIF),” Proc. SPIE 3611, 169–180 (1999).
[CrossRef]

Tropper, A. C.

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33, 1049–1056 (1997).
[CrossRef]

Tünnermann, A.

Tünnermann, T.

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and T. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13, 537–545 (2007).
[CrossRef]

Vu, K. T.

Wang, Y.

Wegner, P. J.

White, R. K.

Widmayer, C. C.

Williams, W. H.

Wirth, C.

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and T. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13, 537–545 (2007).
[CrossRef]

Wisoff, P. J.

P. J. Wisoff, M. W. Bowers, G. V. Erbert, D. F. Browning, and D. R. Jedlovec, “NIF injection laser system,” Proc. SPIE 5341, 146–155 (2004).
[CrossRef]

Wonterghem, B. M.

Yang, S. T.

Zervas, M. N.

Zuegel, J.

Zuegel, J. D.

Appl. Opt. (4)

IEEE J. Quantum Electron. (1)

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33, 1049–1056 (1997).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Limpert, F. Röser, S. Klingebiel, T. Schreiber, C. Wirth, T. Peschel, R. Eberhardt, and T. Tünnermann, “The rising power of fiber lasers and amplifiers,” IEEE J. Sel. Top. Quantum Electron. 13, 537–545 (2007).
[CrossRef]

J. Appl. Phys. (1)

L. M. Frantz and J. S. Nodvick, “Theory of pulse propagation in a laser amplifier,” J. Appl. Phys. 34, 2346–2349 (1963).
[CrossRef]

J. Lightwave Technol. (2)

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Figures (9)

Fig. 1
Fig. 1

Experimental setup: AOM, acousto-optic modulator; EOM, electro-optic modulator; AWG, arbitrary waveform generator; PM, phase modulator; RF, radio frequency (2 GHz) driver; OI, optical isolator; YDFA1, YDFA2, and YDFA3—Yb-doped fiber amplifiers; F1, spectral filter at 1053 with 5 nm at 3 dB; F2, Bragg grating spectral filter at 1053 with 0.1 nm at 3 dB; L, lens; M, mirror; DM, dichroic mirror; BD, beam dumper; PBS, polarizing beam splitter; λ / 2 , half-wave plate; PD, photodiode; OSA, optical spectrum analyzer; Pol, polarimeter.

Fig. 2
Fig. 2

CW backward cladding-pumped amplifier geometry.

Fig. 3
Fig. 3

Fiber absorption and emission cross-sections used for simulations.

Fig. 4
Fig. 4

(a) Main amplifier gain at 1053 nm and (b) ASE induced OSNR function of fiber length for various pump powers. The input signal is 10 nJ, 10 kHz, and 10 ns square temporal profile at 1053 nm.

Fig. 5
Fig. 5

(a) Output pulse energy and (b) OSNR as functions of pump power for a 2.65 m long fiber. The input signal is 10 nJ, 10 kHz, and 10 ns square temporal profile at 1053 nm.

Fig. 6
Fig. 6

Optical spectra of the output amplified signal ( 20 μ J without ASE filter and 10 μ J with 10 nm bandwidth ASE filter).

Fig. 7
Fig. 7

(a) Distortion of pulse shape in the 2.65 m long fiber. Input signal is 10 nJ, 10 kHz, 10 ns at 1053 nm, and output signal energy is 20 μ J . (b) Pre-compensation of the gain saturation.

Fig. 8
Fig. 8

(a) Near-field image of the output beam. (b) Horizontal profile measured in the center of the near-field beam. (c) Far-field image of the output beam at the focus of an f = 1.2   m lens. (d) Horizontal profile of the far-field beam (thick line: experimental profile; thin line: diffraction limited profile).

Fig. 9
Fig. 9

Wave-front measurement of the output beam.

Equations (2)

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n 2 t = { σ a ( λ L ) n tot [ σ a ( λ L ) + σ e ( λ L ) ] n 2 } I L h ν L + { σ a ( λ P ) n tot [ σ a ( λ P ) + σ e ( λ P ) ] n 2 ( t , z ) } I P h ν P n 2 τ fluo + k { σ a ( λ k ) n tot [ σ a ( λ k ) + σ e ( λ k ) ] n 2 } I ASE + ( λ k ) + I ASE ( λ k ) h ν k ,
I L z + 1 v g I L t = { [ σ a ( λ L ) + σ e ( λ L ) ] n 2 σ a ( λ L ) n tot } Γ L I L I P z + 1 v g I P t = { [ σ a ( λ P ) + σ e ( λ P ) ] n 2 σ a ( λ P ) n tot } Γ P I P ± I ASE ± ( λ k ) z + 1 v g I ASE ± ( λ k ) t = { [ σ a ( λ k ) + σ e ( λ k ) ] n 2 σ a ( λ k ) n tot } Γ L I ASE ± ( λ k ) + 2 σ e ( λ k ) h ν k Δ ν k A core Γ L n 2 ,

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