Abstract

In this paper the stimulated Raman scattering (SRS) effect in high-power fiber amplifiers seeded by the narrow-band filtered superfluorescent source (SFS) is firstly analyzed both theoretically and experimentally. Spectral models for the formation of the SFS and the spectral evolution in high-power fiber amplifiers seeded by filtered SFS are proposed. It is found that the SRS effect in high-power fiber amplifiers depends on the spectral width of the filtered SFS seed. The theoretical predictions are in qualitative agreements with the experimental results.

© 2016 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref]
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2015 (8)

X. Jin, X. Wang, J. Xu, P. Zhou, and Z. Liu, “High-power thulium-doped all-fiber superfluorescent source with ultranarrow linewidth,” IEEE Photonics J. 7(1), 1600206 (2015).
[Crossref]

P. Ma, R. Tao, X. Wang, P. Zhou, and Z. Liu, “High-power narrow-band and polarization-maintained all fiber superfluorescent source,” IEEE Photonics Technol. Lett. 27(8), 879–882 (2015).
[Crossref]

W. Liu, W. Kuang, L. Huang, and P. Zhou, “Modeling of the spectral properties of CW Yb-doped fiber amplifier and experimental validation,” Laser Phys. Lett. 12(4), 045104 (2015).
[Crossref]

A. Rosales-Garcia, H. Tobioka, K. Abedin, H. Dong, Z. Várallyay, Á. Szabó, T. Taunay, S. Sullivan, and C. Headley, “2.1 kW single mode continuous wave monolithic fiber laser,” Proc. SPIE 9344, 93441G (2015).
[Crossref]

F. Beier, M. Strecker, J. Nold, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “6.8 kW peak power quasi-continuous wave tandem-pumped Ytterbium amplifier at 1071nm,” Proc. SPIE 9344, 93441H (2015).
[Crossref]

J. Xu, L. Huang, J. Leng, H. Xiao, S. Guo, P. Zhou, and J. Chen, “1.01 kW superfluorescent source in all-fiberized MOPA configuration,” Opt. Express 23(5), 5485–5490 (2015).
[Crossref] [PubMed]

P. Belden, D. Chen, and F. D. Teodoro, “Watt-level, gigahertz-linewidth difference-frequency generation in PPLN pumped by an nanosecond-pulse fiber laser source,” Opt. Lett. 40(6), 958–961 (2015).
[Crossref] [PubMed]

J. Xu, W. Liu, J. Leng, H. Xiao, S. Guo, P. Zhou, and J. Chen, “Power scaling of narrowband high-power all-fiber superfluorescent fiber source to 1.87 kW,” Opt. Lett. 40(13), 2973–2976 (2015).
[Crossref] [PubMed]

2014 (2)

J. Liu, K. Liu, F. Tan, and P. Wang, “High-power thulium-doped all-fiber superfluorescent sources,” IEEE J. Sel. Top. Quantum Electron. 20(5), 497–502 (2014).
[Crossref]

T. Schreiber, A. Liem, E. Freier, C. Matzdorf, R. Eberhardt, C. Jauregui, J. Limpert, and A. Tünnermanna, “Analysis of stimulated Raman scattering in cw kW fiber oscillators,” Proc. SPIE 8961, 89611T (2014).

2013 (1)

2012 (1)

2011 (3)

2010 (3)

2009 (2)

C. Jauregui, J. Limpert, and A. Tünnermann, “Derivation of Raman treshold formulas for CW double-clad fiber amplifiers,” Opt. Express 17(10), 8476–8490 (2009).
[Crossref] [PubMed]

O. Andrusyak, V. Smirnov, G. Venus, V. Rotar, and L. Glebov, “Spectral combining and coherent coupling of lasers by volume Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 15(2), 344–353 (2009).
[Crossref]

2008 (1)

2007 (1)

P. Wang, J. K. Sahu, and W. A. Clarkson, “Power scaling of ytterbium-doped fiber superfluorescent sources,” IEEE J. Sel. Top. Quantum Electron. 13(3), 580–587 (2007).
[Crossref]

2006 (1)

1997 (1)

A. Hardy and R. Oron, “Signal amplification in strongly pumped fiber amplifiers,” IEEE J. Quantum Electron. 33(3), 307–313 (1997).
[Crossref]

1991 (1)

C. R. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. 9(2), 271–283 (1991).
[Crossref]

1986 (1)

C. H. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,” J. Lightwave Technol. 4(3), 288–297 (1986).
[Crossref]

Abedin, K.

A. Rosales-Garcia, H. Tobioka, K. Abedin, H. Dong, Z. Várallyay, Á. Szabó, T. Taunay, S. Sullivan, and C. Headley, “2.1 kW single mode continuous wave monolithic fiber laser,” Proc. SPIE 9344, 93441G (2015).
[Crossref]

Andrusyak, O.

O. Andrusyak, V. Smirnov, G. Venus, V. Rotar, and L. Glebov, “Spectral combining and coherent coupling of lasers by volume Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 15(2), 344–353 (2009).
[Crossref]

Babin, S. A.

Barty, C. P. J.

Beach, R. J.

Bednyakova, A. E.

Beier, F.

F. Beier, M. Strecker, J. Nold, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “6.8 kW peak power quasi-continuous wave tandem-pumped Ytterbium amplifier at 1071nm,” Proc. SPIE 9344, 93441H (2015).
[Crossref]

Belden, P.

Chen, D.

Chen, J.

Churkin, D. V.

Clarkson, W. A.

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B 27(11), B63–B92 (2010).
[Crossref]

P. Wang, J. K. Sahu, and W. A. Clarkson, “Power scaling of ytterbium-doped fiber superfluorescent sources,” IEEE J. Sel. Top. Quantum Electron. 13(3), 580–587 (2007).
[Crossref]

Dawson, J. W.

Desurvire, E.

C. R. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. 9(2), 271–283 (1991).
[Crossref]

Dong, H.

A. Rosales-Garcia, H. Tobioka, K. Abedin, H. Dong, Z. Várallyay, Á. Szabó, T. Taunay, S. Sullivan, and C. Headley, “2.1 kW single mode continuous wave monolithic fiber laser,” Proc. SPIE 9344, 93441G (2015).
[Crossref]

Eberhardt, R.

F. Beier, M. Strecker, J. Nold, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “6.8 kW peak power quasi-continuous wave tandem-pumped Ytterbium amplifier at 1071nm,” Proc. SPIE 9344, 93441H (2015).
[Crossref]

T. Schreiber, A. Liem, E. Freier, C. Matzdorf, R. Eberhardt, C. Jauregui, J. Limpert, and A. Tünnermanna, “Analysis of stimulated Raman scattering in cw kW fiber oscillators,” Proc. SPIE 8961, 89611T (2014).

C. Wirth, O. Schmidt, I. Tsybin, T. Schreiber, R. Eberhardt, J. Limpert, A. Tünnermann, K. Ludewigt, M. Gowin, E. ten Have, and M. Jung, “High average power spectral beam combining of four fiber amplifiers to 8.2 kW,” Opt. Lett. 36(16), 3118–3120 (2011).
[Crossref] [PubMed]

O. Schmidt, M. Rekas, C. Wirth, J. Rothhardt, S. Rhein, A. Kliner, M. Strecker, T. Schreiber, J. Limpert, R. Eberhardt, and A. Tünnermann, “High power narrow-band fiber-based ASE source,” Opt. Express 19(5), 4421–4427 (2011).
[Crossref] [PubMed]

Fedoruk, M. P.

Fotiadi, A. A.

Freier, E.

T. Schreiber, A. Liem, E. Freier, C. Matzdorf, R. Eberhardt, C. Jauregui, J. Limpert, and A. Tünnermanna, “Analysis of stimulated Raman scattering in cw kW fiber oscillators,” Proc. SPIE 8961, 89611T (2014).

Giles, C. R.

C. R. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. 9(2), 271–283 (1991).
[Crossref]

Glebov, L.

O. Andrusyak, V. Smirnov, G. Venus, V. Rotar, and L. Glebov, “Spectral combining and coherent coupling of lasers by volume Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 15(2), 344–353 (2009).
[Crossref]

Gorbunov, O. A.

Gowin, M.

Guo, S.

Haarlammert, N.

F. Beier, M. Strecker, J. Nold, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “6.8 kW peak power quasi-continuous wave tandem-pumped Ytterbium amplifier at 1071nm,” Proc. SPIE 9344, 93441H (2015).
[Crossref]

Hardy, A.

A. Hardy and R. Oron, “Signal amplification in strongly pumped fiber amplifiers,” IEEE J. Quantum Electron. 33(3), 307–313 (1997).
[Crossref]

Headley, C.

A. Rosales-Garcia, H. Tobioka, K. Abedin, H. Dong, Z. Várallyay, Á. Szabó, T. Taunay, S. Sullivan, and C. Headley, “2.1 kW single mode continuous wave monolithic fiber laser,” Proc. SPIE 9344, 93441G (2015).
[Crossref]

Heebner, J. E.

Henderson, A.

Henry, C. H.

C. H. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,” J. Lightwave Technol. 4(3), 288–297 (1986).
[Crossref]

Hsu, W. L.

Huang, L.

J. Xu, L. Huang, J. Leng, H. Xiao, S. Guo, P. Zhou, and J. Chen, “1.01 kW superfluorescent source in all-fiberized MOPA configuration,” Opt. Express 23(5), 5485–5490 (2015).
[Crossref] [PubMed]

W. Liu, W. Kuang, L. Huang, and P. Zhou, “Modeling of the spectral properties of CW Yb-doped fiber amplifier and experimental validation,” Laser Phys. Lett. 12(4), 045104 (2015).
[Crossref]

Jauregui, C.

T. Schreiber, A. Liem, E. Freier, C. Matzdorf, R. Eberhardt, C. Jauregui, J. Limpert, and A. Tünnermanna, “Analysis of stimulated Raman scattering in cw kW fiber oscillators,” Proc. SPIE 8961, 89611T (2014).

C. Jauregui, J. Limpert, and A. Tünnermann, “Derivation of Raman treshold formulas for CW double-clad fiber amplifiers,” Opt. Express 17(10), 8476–8490 (2009).
[Crossref] [PubMed]

Jin, X.

X. Jin, X. Wang, J. Xu, P. Zhou, and Z. Liu, “High-power thulium-doped all-fiber superfluorescent source with ultranarrow linewidth,” IEEE Photonics J. 7(1), 1600206 (2015).
[Crossref]

Jung, M.

Kablukov, S. I.

Kliner, A.

Koplow, J. P.

Kuang, W.

W. Liu, W. Kuang, L. Huang, and P. Zhou, “Modeling of the spectral properties of CW Yb-doped fiber amplifier and experimental validation,” Laser Phys. Lett. 12(4), 045104 (2015).
[Crossref]

Kurkov, A. S.

Kuznetsov, A. G.

Latkin, A. I.

Leng, J.

Liem, A.

T. Schreiber, A. Liem, E. Freier, C. Matzdorf, R. Eberhardt, C. Jauregui, J. Limpert, and A. Tünnermanna, “Analysis of stimulated Raman scattering in cw kW fiber oscillators,” Proc. SPIE 8961, 89611T (2014).

Limpert, J.

Liu, J.

J. Liu, K. Liu, F. Tan, and P. Wang, “High-power thulium-doped all-fiber superfluorescent sources,” IEEE J. Sel. Top. Quantum Electron. 20(5), 497–502 (2014).
[Crossref]

Liu, K.

J. Liu, K. Liu, F. Tan, and P. Wang, “High-power thulium-doped all-fiber superfluorescent sources,” IEEE J. Sel. Top. Quantum Electron. 20(5), 497–502 (2014).
[Crossref]

Liu, W.

W. Liu, W. Kuang, L. Huang, and P. Zhou, “Modeling of the spectral properties of CW Yb-doped fiber amplifier and experimental validation,” Laser Phys. Lett. 12(4), 045104 (2015).
[Crossref]

J. Xu, W. Liu, J. Leng, H. Xiao, S. Guo, P. Zhou, and J. Chen, “Power scaling of narrowband high-power all-fiber superfluorescent fiber source to 1.87 kW,” Opt. Lett. 40(13), 2973–2976 (2015).
[Crossref] [PubMed]

Liu, Z.

P. Ma, R. Tao, X. Wang, P. Zhou, and Z. Liu, “High-power narrow-band and polarization-maintained all fiber superfluorescent source,” IEEE Photonics Technol. Lett. 27(8), 879–882 (2015).
[Crossref]

X. Jin, X. Wang, J. Xu, P. Zhou, and Z. Liu, “High-power thulium-doped all-fiber superfluorescent source with ultranarrow linewidth,” IEEE Photonics J. 7(1), 1600206 (2015).
[Crossref]

Ludewigt, K.

Ma, P.

P. Ma, R. Tao, X. Wang, P. Zhou, and Z. Liu, “High-power narrow-band and polarization-maintained all fiber superfluorescent source,” IEEE Photonics Technol. Lett. 27(8), 879–882 (2015).
[Crossref]

Matzdorf, C.

T. Schreiber, A. Liem, E. Freier, C. Matzdorf, R. Eberhardt, C. Jauregui, J. Limpert, and A. Tünnermanna, “Analysis of stimulated Raman scattering in cw kW fiber oscillators,” Proc. SPIE 8961, 89611T (2014).

Messerly, M. J.

Moore, S. W.

Nilsson, J.

Nold, J.

F. Beier, M. Strecker, J. Nold, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “6.8 kW peak power quasi-continuous wave tandem-pumped Ytterbium amplifier at 1071nm,” Proc. SPIE 9344, 93441H (2015).
[Crossref]

Oron, R.

A. Hardy and R. Oron, “Signal amplification in strongly pumped fiber amplifiers,” IEEE J. Quantum Electron. 33(3), 307–313 (1997).
[Crossref]

Pax, P. H.

Podivilov, E. V.

Politko, M. O.

Rekas, M.

Rhein, S.

Richardson, D. J.

Rosales-Garcia, A.

A. Rosales-Garcia, H. Tobioka, K. Abedin, H. Dong, Z. Várallyay, Á. Szabó, T. Taunay, S. Sullivan, and C. Headley, “2.1 kW single mode continuous wave monolithic fiber laser,” Proc. SPIE 9344, 93441G (2015).
[Crossref]

Rotar, V.

O. Andrusyak, V. Smirnov, G. Venus, V. Rotar, and L. Glebov, “Spectral combining and coherent coupling of lasers by volume Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 15(2), 344–353 (2009).
[Crossref]

Rothhardt, J.

Sahu, J. K.

P. Wang, J. K. Sahu, and W. A. Clarkson, “Power scaling of ytterbium-doped fiber superfluorescent sources,” IEEE J. Sel. Top. Quantum Electron. 13(3), 580–587 (2007).
[Crossref]

Schmidt, O.

Schreiber, T.

F. Beier, M. Strecker, J. Nold, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “6.8 kW peak power quasi-continuous wave tandem-pumped Ytterbium amplifier at 1071nm,” Proc. SPIE 9344, 93441H (2015).
[Crossref]

T. Schreiber, A. Liem, E. Freier, C. Matzdorf, R. Eberhardt, C. Jauregui, J. Limpert, and A. Tünnermanna, “Analysis of stimulated Raman scattering in cw kW fiber oscillators,” Proc. SPIE 8961, 89611T (2014).

C. Wirth, O. Schmidt, I. Tsybin, T. Schreiber, R. Eberhardt, J. Limpert, A. Tünnermann, K. Ludewigt, M. Gowin, E. ten Have, and M. Jung, “High average power spectral beam combining of four fiber amplifiers to 8.2 kW,” Opt. Lett. 36(16), 3118–3120 (2011).
[Crossref] [PubMed]

O. Schmidt, M. Rekas, C. Wirth, J. Rothhardt, S. Rhein, A. Kliner, M. Strecker, T. Schreiber, J. Limpert, R. Eberhardt, and A. Tünnermann, “High power narrow-band fiber-based ASE source,” Opt. Express 19(5), 4421–4427 (2011).
[Crossref] [PubMed]

Schroder, K. L.

Sholokhov, E.

Shverdin, M. Y.

Siders, C. W.

Smirnov, S. V.

Smirnov, V.

O. Andrusyak, V. Smirnov, G. Venus, V. Rotar, and L. Glebov, “Spectral combining and coherent coupling of lasers by volume Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 15(2), 344–353 (2009).
[Crossref]

Soh, D. B. S.

Sridharan, A. K.

Stafford, R.

Stappaerts, E. A.

Strecker, M.

F. Beier, M. Strecker, J. Nold, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “6.8 kW peak power quasi-continuous wave tandem-pumped Ytterbium amplifier at 1071nm,” Proc. SPIE 9344, 93441H (2015).
[Crossref]

O. Schmidt, M. Rekas, C. Wirth, J. Rothhardt, S. Rhein, A. Kliner, M. Strecker, T. Schreiber, J. Limpert, R. Eberhardt, and A. Tünnermann, “High power narrow-band fiber-based ASE source,” Opt. Express 19(5), 4421–4427 (2011).
[Crossref] [PubMed]

Sullivan, S.

A. Rosales-Garcia, H. Tobioka, K. Abedin, H. Dong, Z. Várallyay, Á. Szabó, T. Taunay, S. Sullivan, and C. Headley, “2.1 kW single mode continuous wave monolithic fiber laser,” Proc. SPIE 9344, 93441G (2015).
[Crossref]

Szabó, Á.

A. Rosales-Garcia, H. Tobioka, K. Abedin, H. Dong, Z. Várallyay, Á. Szabó, T. Taunay, S. Sullivan, and C. Headley, “2.1 kW single mode continuous wave monolithic fiber laser,” Proc. SPIE 9344, 93441G (2015).
[Crossref]

Tan, F.

J. Liu, K. Liu, F. Tan, and P. Wang, “High-power thulium-doped all-fiber superfluorescent sources,” IEEE J. Sel. Top. Quantum Electron. 20(5), 497–502 (2014).
[Crossref]

Tao, R.

P. Ma, R. Tao, X. Wang, P. Zhou, and Z. Liu, “High-power narrow-band and polarization-maintained all fiber superfluorescent source,” IEEE Photonics Technol. Lett. 27(8), 879–882 (2015).
[Crossref]

Taunay, T.

A. Rosales-Garcia, H. Tobioka, K. Abedin, H. Dong, Z. Várallyay, Á. Szabó, T. Taunay, S. Sullivan, and C. Headley, “2.1 kW single mode continuous wave monolithic fiber laser,” Proc. SPIE 9344, 93441G (2015).
[Crossref]

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Teodoro, F. D.

Tobioka, H.

A. Rosales-Garcia, H. Tobioka, K. Abedin, H. Dong, Z. Várallyay, Á. Szabó, T. Taunay, S. Sullivan, and C. Headley, “2.1 kW single mode continuous wave monolithic fiber laser,” Proc. SPIE 9344, 93441G (2015).
[Crossref]

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Turitsyn, S. K.

Várallyay, Z.

A. Rosales-Garcia, H. Tobioka, K. Abedin, H. Dong, Z. Várallyay, Á. Szabó, T. Taunay, S. Sullivan, and C. Headley, “2.1 kW single mode continuous wave monolithic fiber laser,” Proc. SPIE 9344, 93441G (2015).
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[Crossref]

J. Liu, K. Liu, F. Tan, and P. Wang, “High-power thulium-doped all-fiber superfluorescent sources,” IEEE J. Sel. Top. Quantum Electron. 20(5), 497–502 (2014).
[Crossref]

O. Andrusyak, V. Smirnov, G. Venus, V. Rotar, and L. Glebov, “Spectral combining and coherent coupling of lasers by volume Bragg gratings,” IEEE J. Sel. Top. Quantum Electron. 15(2), 344–353 (2009).
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X. Jin, X. Wang, J. Xu, P. Zhou, and Z. Liu, “High-power thulium-doped all-fiber superfluorescent source with ultranarrow linewidth,” IEEE Photonics J. 7(1), 1600206 (2015).
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P. Ma, R. Tao, X. Wang, P. Zhou, and Z. Liu, “High-power narrow-band and polarization-maintained all fiber superfluorescent source,” IEEE Photonics Technol. Lett. 27(8), 879–882 (2015).
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Proc. SPIE (3)

T. Schreiber, A. Liem, E. Freier, C. Matzdorf, R. Eberhardt, C. Jauregui, J. Limpert, and A. Tünnermanna, “Analysis of stimulated Raman scattering in cw kW fiber oscillators,” Proc. SPIE 8961, 89611T (2014).

A. Rosales-Garcia, H. Tobioka, K. Abedin, H. Dong, Z. Várallyay, Á. Szabó, T. Taunay, S. Sullivan, and C. Headley, “2.1 kW single mode continuous wave monolithic fiber laser,” Proc. SPIE 9344, 93441G (2015).
[Crossref]

F. Beier, M. Strecker, J. Nold, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “6.8 kW peak power quasi-continuous wave tandem-pumped Ytterbium amplifier at 1071nm,” Proc. SPIE 9344, 93441H (2015).
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Other (1)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2012).

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Figures (10)

Fig. 1
Fig. 1 The simulated temporal evolution and corresponding optical spectrum for the broadband SFS with the theoretical model; (a) the temporal evolution of SFS with the time window of 10ns; (b) the corresponding optical spectrum; (c) the PDF of the output power.
Fig. 2
Fig. 2 The simulated temporal evolution and corresponding optical spectra for the fiber amplifier at 12 W and 1.3 kW; (a) the temporal evolution with the time window of 10 ns; (b) the corresponding optical spectra.
Fig. 3
Fig. 3 The simulated optical spectra and the corresponding Raman ratios for seeds with different spectral widths; (a) the simulated optical spectra; (b) the corresponding Raman ratios versus the output power.
Fig. 4
Fig. 4 The experimental set-up of the filtered SFS seed.
Fig. 5
Fig. 5 The output spectra of the SFS before and after filtering; (a) before filtering; (b) FWHM of ~0.55 nm; (c) FWHM of ~0.23 nm; (d) FWHM of ~0.15 nm.
Fig. 6
Fig. 6 The experimental set-up of the fiber amplifier.
Fig. 7
Fig. 7 The output powers vary with the launched pump powers for the three seeds.
Fig. 8
Fig. 8 The measured optical spectra for the fiber amplifier at different power levels seeded by the three seeds; (a) FWHM of ~0.55 nm; (b) FWHM of ~0.23 nm; (c) FWHM of ~0.15 nm.
Fig. 9
Fig. 9 The experimental and simulated Raman ratios versus output powers for the three seeds.
Fig. 10
Fig. 10 The theoretical temporal characteristics of the three filtered seeds from the same broadband SFS; (a) FWHM of ~0.55 nm; (b) FWHM of ~0.23 nm; (c) FWHM of ~0.15 nm.

Equations (7)

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A ˜ (z,ω) z = 1 2 ( g(ω)α(ω) ) A ˜ (z,ω)+i n=1 3 β n n! ω n A ˜ (z,ω) +iγ( 1+ ω ω 0 )F{ A(z,t)R(t) | A(z,t) | 2 }+ f SE (z,ω)
d P p (z) dz = Γ p { σ a ( ω p ) N 0 ( σ a ( ω p )+ σ e ( ω p ) ) N 2 } P p (z) α p P p (z)
N 2 N 0 = Γ p ω p A σ a ( ω p ) P p + 1 2π T m A Γ s (ω) ω σ a (ω) | A ˜ (z,ω) | 2 dω Γ p ω p A ( σ a ( ω p )+ σ e ( ω p ) ) P p + 1 τ + 1 2π T m A Γ s (ω) ω ( σ a (ω)+ σ e (ω) ) | A ˜ (z,ω) | 2 dω
{ f SE ( z,ω ) f SE ( z , ω ) =2 D FF ( z,ω )δ( z z )δ( ω ω ) f SE (z,ω) =0 2 D FF (z,ω)= ω 3 π c 2 n(ω)g(z,ω) n sp
A + z + β 1 A + t + i β 2 2 2 A + t 2 iγ( | A + | 2 +2 | A | 2 ) A + = ( g s α s ) 2 A + A z + β 1 A t + i β 2 2 2 A t 2 iγ( | A | 2 +2 | A + | 2 ) A = ( g s α s ) 2 A
A ˜ (z,ω) z = 1 2 ( g(ω)α(ω) ) A ˜ (z,ω)+ f SE (z,ω)
A out ( ω )= A in ( ω )× R( ω )

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