Abstract

We report on an ultraviolet-enhanced supercontinuum generation in a uniform photonic crystal fiber pumped by a giant-chirped mode-locked Yb-doped fiber laser. We find theoretically and experimentally that the initial pluses with giant chirp leads more initial energy transferred to the dispersive waves in visible and ultraviolet wavelength. An extremely wide optical spectrum spanning from 370 nm to beyond 2400 nm with a broad 3 dB spectral bandwidth of 367 nm (from 431 nm to 798 nm) is obtained. Over 36% (350 mW) of the total output power locates in the visible and ultraviolet regime between 370 nm and 850 nm with a maximum spectral power density of 1.6 mW/nm at 550 nm.

© 2014 Optical Society of America

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References

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  1. R. Alfano and S. Shapiro, “Emission in the region 4000 to 7000 A via four-photon coupling in glass,” Phys. Rev. Lett. 24(11), 584–587 (1970).
    [Crossref]
  2. P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
    [Crossref] [PubMed]
  3. J. C. Knight, “Photonic crystal fibers and fiber lasers,” J. Opt. Soc. Am. B 24(8), 1661–1668 (2007).
    [Crossref]
  4. P. Cimalla, J. Walther, M. Mehner, M. Cuevas, and E. Koch, “Simultaneous dual-band optical coherence tomography in the spectral domain for high resolution in vivo imaging,” Opt. Express 17(22), 19486–19500 (2009).
    [Crossref] [PubMed]
  5. K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and Spectroscopy of Gold Nanoparticles Using Supercontinuum White Light Confocal Microscopy,” Phys. Rev. Lett. 93(3), 037401 (2004).
    [Crossref] [PubMed]
  6. Y. Sun, C. F. Booker, S. Kumari, R. N. Day, M. Davidson, and A. Periasamy, “Characterization of an orange acceptor fluorescent protein for sensitized spectral fluorescence resonance energy transfer microscopy using a white-light laser,” J. Biomed. Opt. 14(5), 054009 (2009).
    [Crossref] [PubMed]
  7. J. C. Travers, S. V. Popov, and J. R. Taylor, “Extended blue supercontinuum generation in cascaded holey fibers,” Opt. Lett. 30(23), 3132–3134 (2005).
    [Crossref] [PubMed]
  8. A. Kudlinski, A. K. George, J. C. Knight, J. C. Travers, A. B. Rulkov, S. V. Popov, and J. R. Taylor, “Zero-dispersion wavelength decreasing photonic crystal fibers for ultraviolet-extended supercontinuum generation,” Opt. Express 14(12), 5715–5722 (2006).
    [Crossref] [PubMed]
  9. J. M. Stone and J. C. Knight, “Visibly “white” light generation in uniform photonic crystal fiber using a microchip laser,” Opt. Express 16(4), 2670–2675 (2008).
    [Crossref] [PubMed]
  10. I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express 12(1), 124–135 (2004).
    [Crossref] [PubMed]
  11. C. Desem and P. L. Chu, “Effect of chirping on solution propagation in single-mode optical fibers,” Opt. Lett. 11(4), 248–250 (1986).
    [Crossref] [PubMed]
  12. Z. Zhu and T. Brown, “Effect of frequency chirping on supercontinuum generation in photonic crystal fibers,” Opt. Express 12(4), 689–694 (2004).
    [Crossref] [PubMed]
  13. X. Fu, L. Qian, S. Wen, and D. Fan, “Nonlinear chirped pulse propagation and supercontinuum generation in microstructured optical fibre,” J. Opt. A, Pure Appl. Opt. 6(11), 1012–1016 (2004).
    [Crossref]
  14. A. Fuerbach, C. Miese, W. Koehler, and M. Geissler, “Supercontinuum generation with a chirped-pulse oscillator,” Opt. Express 17(7), 5905–5911 (2009).
    [Crossref] [PubMed]
  15. R. Song, J. Hou, Z. F. Wang, R. Xiao, and Q. S. Lu, “Effect of initial chirp on near-infrared supercontinuum generation by a nanosecond pulse in a nonlinear fiber amplifier,” Chin. Phys. B. 22(8), 084206 (2013).
    [Crossref]
  16. H. Zhang, S. Yu, J. Zhang, and W. Gu, “Effect of frequency chirp on supercontinuum generation in photonic crystal fibers with two zero-dispersion wavelengths,” Opt. Express 15(3), 1147–1154 (2007).
    [Crossref] [PubMed]
  17. R. Driben and N. Zhavoronkov, “Supercontinuum spectrum control in microstructure fibers by initial chirp management,” Opt. Express 18(16), 16733–16738 (2010).
    [Crossref] [PubMed]
  18. Y. Kwon, L. A. Vazquez-Zuniga, S. Hong, H. Kim, and Y. Jeong, “Numerical Study on Fiber-Based Supercontinuum Generation in Anomalous Dispersion Pumping Regimes,” CLEO PR 2013, Kyoto, Japan, 30 Jun-4 Jul, 2013.
    [Crossref]
  19. J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation,” Opt. Express 17(24), 21497–21508 (2009).
    [Crossref] [PubMed]
  20. E. J. R. Kelleher, J. C. Travers, E. P. Ippen, Z. Sun, A. C. Ferrari, S. V. Popov, and J. R. Taylor, “Generation and direct measurement of giant chirp in a passively mode-locked laser,” Opt. Lett. 34(22), 3526–3528 (2009).
    [Crossref] [PubMed]
  21. P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41(1), 426–439 (1990).
    [Crossref] [PubMed]
  22. J. C. Travers, “Blue extension of optical fibre supercontinuum generation,” J. Opt. 12(11), 113001 (2010).
    [Crossref]
  23. T. Schreiber, J. Limpert, H. Zellmer, A. Tunnermann, and K. P. Hansen, “High average power supercontinuum generation in photonic crystal fibers,” Opt. Commun. 228(1-3), 71–78 (2003).
    [Crossref]

2013 (1)

R. Song, J. Hou, Z. F. Wang, R. Xiao, and Q. S. Lu, “Effect of initial chirp on near-infrared supercontinuum generation by a nanosecond pulse in a nonlinear fiber amplifier,” Chin. Phys. B. 22(8), 084206 (2013).
[Crossref]

2010 (2)

2009 (5)

2008 (1)

2007 (2)

2006 (1)

2005 (1)

2004 (4)

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and Spectroscopy of Gold Nanoparticles Using Supercontinuum White Light Confocal Microscopy,” Phys. Rev. Lett. 93(3), 037401 (2004).
[Crossref] [PubMed]

I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express 12(1), 124–135 (2004).
[Crossref] [PubMed]

Z. Zhu and T. Brown, “Effect of frequency chirping on supercontinuum generation in photonic crystal fibers,” Opt. Express 12(4), 689–694 (2004).
[Crossref] [PubMed]

X. Fu, L. Qian, S. Wen, and D. Fan, “Nonlinear chirped pulse propagation and supercontinuum generation in microstructured optical fibre,” J. Opt. A, Pure Appl. Opt. 6(11), 1012–1016 (2004).
[Crossref]

2003 (2)

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
[Crossref] [PubMed]

T. Schreiber, J. Limpert, H. Zellmer, A. Tunnermann, and K. P. Hansen, “High average power supercontinuum generation in photonic crystal fibers,” Opt. Commun. 228(1-3), 71–78 (2003).
[Crossref]

1990 (1)

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41(1), 426–439 (1990).
[Crossref] [PubMed]

1986 (1)

1970 (1)

R. Alfano and S. Shapiro, “Emission in the region 4000 to 7000 A via four-photon coupling in glass,” Phys. Rev. Lett. 24(11), 584–587 (1970).
[Crossref]

Akhmediev, N.

Alfano, R.

R. Alfano and S. Shapiro, “Emission in the region 4000 to 7000 A via four-photon coupling in glass,” Phys. Rev. Lett. 24(11), 584–587 (1970).
[Crossref]

Booker, C. F.

Y. Sun, C. F. Booker, S. Kumari, R. N. Day, M. Davidson, and A. Periasamy, “Characterization of an orange acceptor fluorescent protein for sensitized spectral fluorescence resonance energy transfer microscopy using a white-light laser,” J. Biomed. Opt. 14(5), 054009 (2009).
[Crossref] [PubMed]

Brown, T.

Chen, H. H.

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41(1), 426–439 (1990).
[Crossref] [PubMed]

Chu, P. L.

Cimalla, P.

Cristiani, I.

Cuevas, M.

Davidson, M.

Y. Sun, C. F. Booker, S. Kumari, R. N. Day, M. Davidson, and A. Periasamy, “Characterization of an orange acceptor fluorescent protein for sensitized spectral fluorescence resonance energy transfer microscopy using a white-light laser,” J. Biomed. Opt. 14(5), 054009 (2009).
[Crossref] [PubMed]

Day, R. N.

Y. Sun, C. F. Booker, S. Kumari, R. N. Day, M. Davidson, and A. Periasamy, “Characterization of an orange acceptor fluorescent protein for sensitized spectral fluorescence resonance energy transfer microscopy using a white-light laser,” J. Biomed. Opt. 14(5), 054009 (2009).
[Crossref] [PubMed]

Degiorgio, V.

Desem, C.

Dias, F.

Driben, R.

Dudley, J. M.

Fan, D.

X. Fu, L. Qian, S. Wen, and D. Fan, “Nonlinear chirped pulse propagation and supercontinuum generation in microstructured optical fibre,” J. Opt. A, Pure Appl. Opt. 6(11), 1012–1016 (2004).
[Crossref]

Ferrari, A. C.

Fu, X.

X. Fu, L. Qian, S. Wen, and D. Fan, “Nonlinear chirped pulse propagation and supercontinuum generation in microstructured optical fibre,” J. Opt. A, Pure Appl. Opt. 6(11), 1012–1016 (2004).
[Crossref]

Fuerbach, A.

Geissler, M.

Genty, G.

George, A. K.

Gu, W.

Hansen, K. P.

T. Schreiber, J. Limpert, H. Zellmer, A. Tunnermann, and K. P. Hansen, “High average power supercontinuum generation in photonic crystal fibers,” Opt. Commun. 228(1-3), 71–78 (2003).
[Crossref]

Hou, J.

R. Song, J. Hou, Z. F. Wang, R. Xiao, and Q. S. Lu, “Effect of initial chirp on near-infrared supercontinuum generation by a nanosecond pulse in a nonlinear fiber amplifier,” Chin. Phys. B. 22(8), 084206 (2013).
[Crossref]

Ippen, E. P.

Kalkbrenner, T.

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and Spectroscopy of Gold Nanoparticles Using Supercontinuum White Light Confocal Microscopy,” Phys. Rev. Lett. 93(3), 037401 (2004).
[Crossref] [PubMed]

Kelleher, E. J. R.

Kibler, B.

Knight, J. C.

Koch, E.

Koehler, W.

Kudlinski, A.

Kumari, S.

Y. Sun, C. F. Booker, S. Kumari, R. N. Day, M. Davidson, and A. Periasamy, “Characterization of an orange acceptor fluorescent protein for sensitized spectral fluorescence resonance energy transfer microscopy using a white-light laser,” J. Biomed. Opt. 14(5), 054009 (2009).
[Crossref] [PubMed]

Lee, Y. C.

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41(1), 426–439 (1990).
[Crossref] [PubMed]

Limpert, J.

T. Schreiber, J. Limpert, H. Zellmer, A. Tunnermann, and K. P. Hansen, “High average power supercontinuum generation in photonic crystal fibers,” Opt. Commun. 228(1-3), 71–78 (2003).
[Crossref]

Lindfors, K.

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and Spectroscopy of Gold Nanoparticles Using Supercontinuum White Light Confocal Microscopy,” Phys. Rev. Lett. 93(3), 037401 (2004).
[Crossref] [PubMed]

Lu, Q. S.

R. Song, J. Hou, Z. F. Wang, R. Xiao, and Q. S. Lu, “Effect of initial chirp on near-infrared supercontinuum generation by a nanosecond pulse in a nonlinear fiber amplifier,” Chin. Phys. B. 22(8), 084206 (2013).
[Crossref]

Mehner, M.

Miese, C.

Periasamy, A.

Y. Sun, C. F. Booker, S. Kumari, R. N. Day, M. Davidson, and A. Periasamy, “Characterization of an orange acceptor fluorescent protein for sensitized spectral fluorescence resonance energy transfer microscopy using a white-light laser,” J. Biomed. Opt. 14(5), 054009 (2009).
[Crossref] [PubMed]

Popov, S. V.

Qian, L.

X. Fu, L. Qian, S. Wen, and D. Fan, “Nonlinear chirped pulse propagation and supercontinuum generation in microstructured optical fibre,” J. Opt. A, Pure Appl. Opt. 6(11), 1012–1016 (2004).
[Crossref]

Rulkov, A. B.

Russell, P.

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
[Crossref] [PubMed]

Sandoghdar, V.

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and Spectroscopy of Gold Nanoparticles Using Supercontinuum White Light Confocal Microscopy,” Phys. Rev. Lett. 93(3), 037401 (2004).
[Crossref] [PubMed]

Schreiber, T.

T. Schreiber, J. Limpert, H. Zellmer, A. Tunnermann, and K. P. Hansen, “High average power supercontinuum generation in photonic crystal fibers,” Opt. Commun. 228(1-3), 71–78 (2003).
[Crossref]

Shapiro, S.

R. Alfano and S. Shapiro, “Emission in the region 4000 to 7000 A via four-photon coupling in glass,” Phys. Rev. Lett. 24(11), 584–587 (1970).
[Crossref]

Song, R.

R. Song, J. Hou, Z. F. Wang, R. Xiao, and Q. S. Lu, “Effect of initial chirp on near-infrared supercontinuum generation by a nanosecond pulse in a nonlinear fiber amplifier,” Chin. Phys. B. 22(8), 084206 (2013).
[Crossref]

Stoller, P.

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and Spectroscopy of Gold Nanoparticles Using Supercontinuum White Light Confocal Microscopy,” Phys. Rev. Lett. 93(3), 037401 (2004).
[Crossref] [PubMed]

Stone, J. M.

Sun, Y.

Y. Sun, C. F. Booker, S. Kumari, R. N. Day, M. Davidson, and A. Periasamy, “Characterization of an orange acceptor fluorescent protein for sensitized spectral fluorescence resonance energy transfer microscopy using a white-light laser,” J. Biomed. Opt. 14(5), 054009 (2009).
[Crossref] [PubMed]

Sun, Z.

Tartara, L.

Taylor, J. R.

Tediosi, R.

Travers, J. C.

Tunnermann, A.

T. Schreiber, J. Limpert, H. Zellmer, A. Tunnermann, and K. P. Hansen, “High average power supercontinuum generation in photonic crystal fibers,” Opt. Commun. 228(1-3), 71–78 (2003).
[Crossref]

Wai, P. K. A.

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41(1), 426–439 (1990).
[Crossref] [PubMed]

Walther, J.

Wang, Z. F.

R. Song, J. Hou, Z. F. Wang, R. Xiao, and Q. S. Lu, “Effect of initial chirp on near-infrared supercontinuum generation by a nanosecond pulse in a nonlinear fiber amplifier,” Chin. Phys. B. 22(8), 084206 (2013).
[Crossref]

Wen, S.

X. Fu, L. Qian, S. Wen, and D. Fan, “Nonlinear chirped pulse propagation and supercontinuum generation in microstructured optical fibre,” J. Opt. A, Pure Appl. Opt. 6(11), 1012–1016 (2004).
[Crossref]

Xiao, R.

R. Song, J. Hou, Z. F. Wang, R. Xiao, and Q. S. Lu, “Effect of initial chirp on near-infrared supercontinuum generation by a nanosecond pulse in a nonlinear fiber amplifier,” Chin. Phys. B. 22(8), 084206 (2013).
[Crossref]

Yu, S.

Zellmer, H.

T. Schreiber, J. Limpert, H. Zellmer, A. Tunnermann, and K. P. Hansen, “High average power supercontinuum generation in photonic crystal fibers,” Opt. Commun. 228(1-3), 71–78 (2003).
[Crossref]

Zhang, H.

Zhang, J.

Zhavoronkov, N.

Zhu, Z.

Chin. Phys. B. (1)

R. Song, J. Hou, Z. F. Wang, R. Xiao, and Q. S. Lu, “Effect of initial chirp on near-infrared supercontinuum generation by a nanosecond pulse in a nonlinear fiber amplifier,” Chin. Phys. B. 22(8), 084206 (2013).
[Crossref]

J. Biomed. Opt. (1)

Y. Sun, C. F. Booker, S. Kumari, R. N. Day, M. Davidson, and A. Periasamy, “Characterization of an orange acceptor fluorescent protein for sensitized spectral fluorescence resonance energy transfer microscopy using a white-light laser,” J. Biomed. Opt. 14(5), 054009 (2009).
[Crossref] [PubMed]

J. Opt. (1)

J. C. Travers, “Blue extension of optical fibre supercontinuum generation,” J. Opt. 12(11), 113001 (2010).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

X. Fu, L. Qian, S. Wen, and D. Fan, “Nonlinear chirped pulse propagation and supercontinuum generation in microstructured optical fibre,” J. Opt. A, Pure Appl. Opt. 6(11), 1012–1016 (2004).
[Crossref]

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

Opt. Commun. (1)

T. Schreiber, J. Limpert, H. Zellmer, A. Tunnermann, and K. P. Hansen, “High average power supercontinuum generation in photonic crystal fibers,” Opt. Commun. 228(1-3), 71–78 (2003).
[Crossref]

Opt. Express (9)

J. M. Dudley, G. Genty, F. Dias, B. Kibler, and N. Akhmediev, “Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation,” Opt. Express 17(24), 21497–21508 (2009).
[Crossref] [PubMed]

Z. Zhu and T. Brown, “Effect of frequency chirping on supercontinuum generation in photonic crystal fibers,” Opt. Express 12(4), 689–694 (2004).
[Crossref] [PubMed]

P. Cimalla, J. Walther, M. Mehner, M. Cuevas, and E. Koch, “Simultaneous dual-band optical coherence tomography in the spectral domain for high resolution in vivo imaging,” Opt. Express 17(22), 19486–19500 (2009).
[Crossref] [PubMed]

A. Kudlinski, A. K. George, J. C. Knight, J. C. Travers, A. B. Rulkov, S. V. Popov, and J. R. Taylor, “Zero-dispersion wavelength decreasing photonic crystal fibers for ultraviolet-extended supercontinuum generation,” Opt. Express 14(12), 5715–5722 (2006).
[Crossref] [PubMed]

J. M. Stone and J. C. Knight, “Visibly “white” light generation in uniform photonic crystal fiber using a microchip laser,” Opt. Express 16(4), 2670–2675 (2008).
[Crossref] [PubMed]

I. Cristiani, R. Tediosi, L. Tartara, and V. Degiorgio, “Dispersive wave generation by solitons in microstructured optical fibers,” Opt. Express 12(1), 124–135 (2004).
[Crossref] [PubMed]

A. Fuerbach, C. Miese, W. Koehler, and M. Geissler, “Supercontinuum generation with a chirped-pulse oscillator,” Opt. Express 17(7), 5905–5911 (2009).
[Crossref] [PubMed]

H. Zhang, S. Yu, J. Zhang, and W. Gu, “Effect of frequency chirp on supercontinuum generation in photonic crystal fibers with two zero-dispersion wavelengths,” Opt. Express 15(3), 1147–1154 (2007).
[Crossref] [PubMed]

R. Driben and N. Zhavoronkov, “Supercontinuum spectrum control in microstructure fibers by initial chirp management,” Opt. Express 18(16), 16733–16738 (2010).
[Crossref] [PubMed]

Opt. Lett. (3)

Phys. Rev. A (1)

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41(1), 426–439 (1990).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and Spectroscopy of Gold Nanoparticles Using Supercontinuum White Light Confocal Microscopy,” Phys. Rev. Lett. 93(3), 037401 (2004).
[Crossref] [PubMed]

R. Alfano and S. Shapiro, “Emission in the region 4000 to 7000 A via four-photon coupling in glass,” Phys. Rev. Lett. 24(11), 584–587 (1970).
[Crossref]

Science (1)

P. Russell, “Photonic crystal fibers,” Science 299(5605), 358–362 (2003).
[Crossref] [PubMed]

Other (1)

Y. Kwon, L. A. Vazquez-Zuniga, S. Hong, H. Kim, and Y. Jeong, “Numerical Study on Fiber-Based Supercontinuum Generation in Anomalous Dispersion Pumping Regimes,” CLEO PR 2013, Kyoto, Japan, 30 Jun-4 Jul, 2013.
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Comparison between spectra pumped by chirp-free pulses of C = 0 (black cure) and highly chirped pulses of C = 30 (red cure).
Fig. 2
Fig. 2 Schematic setup of the UV-enhanced SC generation all-fiber system. SMF, single mode fiber; WDM, 976/1064 nm wavelength division multiplexer coupler; PC, polarization controller.
Fig. 3
Fig. 3 Stable mode-locked pulse train of the oscillator at 198 kHz repetition rate.
Fig. 4
Fig. 4 Temporal (a) and spectral (b) intensity of the giant chirp oscillator.
Fig. 5
Fig. 5 Optical spectrum of the fiber power preamplifier (a) and amplifier (b) at maximum average output power. Insert, optical spectrum over a 60 nm bandwidth scale.
Fig. 6
Fig. 6 Calculated dispersion curve (black line) and experimental measurements of the dispersion (red line). The inset shows the SEM picture of the PCF.
Fig. 7
Fig. 7 SC output spectra at different power levels. (a) Traces were measured with optical spectrum analyzers Yokogawa AQ6373 (350 nm-1200 nm) and AQ6375 (1200 nm-2400 nm). (b) Trace was measured with optical spectrum analyzer Yokogawa AQ6370C (600 nm-1700 nm). (c) Spectral power density of SC in the visible wavelength regime.
Fig. 8
Fig. 8 Output powers of SC with different input pump powers.

Equations (4)

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

A z = m2 i m+1 β m m! m A τ m α 2 A+iγ(1+ i ω 0 τ )[ A( z,τ ) τ dτ'R( ττ' ) | A( z,τ' ) | 2 ].
A(0,τ)= P 0 sech( τ T 0 )exp( iC τ 2 2 T 0 2 ),
Δ ω MI = 2γP/| β 2 | ,
T MI =2π/(Δ ω MI ).

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