Abstract

We study theoretically, numerically and experimentally the effect of self-phase modulation of ultrashort pulses with spectrally narrow phase features. We show that spectral enhancement and depletion is caused by changing the relative phase between the initial field and the nonlinearly generated components. Our theoretical results explain observations of supercontinuum enhancement by fiber Bragg gratings, and predict similar enhancements for spectrally shaped pulses in uniform fiber. As proof of principle, we demonstrate this effect in the laboratory using a femtosecond pulse shaper.

© 2006 Optical Society of America

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  1. R. R. Alfano, The supercontinuum laser source: fundamentals with updated references, 2nd ed. (Springer, New York, 2006).
  2. I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, "Ultrahigh-resolution optical coherence tomography using continuum generation in an air silica microstructure optical fiber," Opt. Lett. 26, 608-610 (2001).
    [CrossRef]
  3. A. S. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T.W. Hansch, "Direct Link between Microwave and Optical Frequencies with a 300 THz Femtosecond Laser Comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
    [CrossRef] [PubMed]
  4. K. Mori, T. Morioka, and M. Saruwatari, "Group-velocity dispersion measurement using supercontinuum picosecond pulses generated in an optical-fiber," Electron. Lett. 29, 987-989 (1993).
    [CrossRef]
  5. A. V. Husakou and J. Herrmann, "Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers," Phys. Rev. Lett. 87, 203901 (2001).
    [CrossRef] [PubMed]
  6. D. Türke, W. Wohlleben, J. Teipel, M. Motzkus, B. Kibler, J. Dudley, and H. Giessen, "Chirp-controlled soliton fission in tapered optical fibers," Appl. Phys. B 83, 37-42 (2006).
    [CrossRef]
  7. S. Xu, D. Reitze, and R. Windeler, "Controlling nonlinear processes in microstructured fibers using shaped pulses," Opt. Express 12, 4731-4741 (2004).
    [CrossRef] [PubMed]
  8. M. Tianprateep, J. Tada, T. Yamazaki, and F. Kannari, "Spectral-Shape-Controllable Supercontinuum Generation in Microstructured Fibers Using Adaptive Pulse Shaping Technique," Japanese J. of Appl. Phys. 43, 8059-8063 (2004).
    [CrossRef]
  9. P. S. Westbrook, J.W. Nicholson, K. S. Feder, and A. D. Yablon, "Improved supercontinuum generation through UV processing of highly nonlinear fibers," J. Lightwave Technol. 23, 13-18 (2005).
    [CrossRef]
  10. J. C. Travers, R. E. Kennedy, S. V. Popov, J. R. Taylor, H. Sabert, and B. Mangan, "Extended continuous-wave supercontinuum generation in a low-water-loss holey fiber," Opt. Lett. 30, 1938-1940 (2005).
    [CrossRef] [PubMed]
  11. F. Lu, Y. Deng, and W. Knox, "Generation of broadband femtosecond visible pulses in dispersion-micromanaged holey fibers," Opt. Lett. 30, 1566-1568 (2005).
    [CrossRef] [PubMed]
  12. P. S. Westbrook, J. W. Nicholson, K. S. Feder, Y. Li, and T. Brown, "Supercontinuum generation in a fibre grating," Appl. Phys. Lett. 85, 4600-4602 (2004).
    [CrossRef]
  13. K. Kim, S. A. Diddams, P. S. Westbrook, J. W. Nicholson, and K. S. Feder, "Improved stabilization of a 1.3 μm femtosecond optical frequency comb by use of a spectrally tailored continuum from a nonlinear fiber grating," Opt. Lett. 31, 277-279 (2006).
    [CrossRef] [PubMed]
  14. Y. Li, F. C. Salisbury, Z. Zhu, T. G. Brown, P. S. Westbrook, K. S. Feder, and R. S. Windeler, "Interaction of supercontinuum and Raman solitons with microstructure fiber gratings," Opt. Express 13, 998-1007 (2005).
    [CrossRef] [PubMed]
  15. P. S. Russell, "Bloch Wave Analysis of Dispersion and Pulse-Propagation in Pure Distributed Feedback Structures," J. Mod. Opt. 38, 1599-1619 (1991).
    [CrossRef]
  16. P. Westbrook and J. Nicholson, "Perturbative approach to continuum generation in a fiber Bragg grating," Opt. Express 14, 7610-7616 (2006).
    [CrossRef] [PubMed]
  17. C.-M. Chen and P. L. Kelley, "Nonlinear pulse compression in optical fibers: scaling laws and numerical analysis," J. Opt. Soc. Am. B 19, 1961-1967 (2002).
    [CrossRef]
  18. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
    [CrossRef] [PubMed]
  19. A. Präkelt, M. Wollenhaupt, C. Sarpe-Tudoran, A. Assion, and T. Baumerta, "Filling a spectral hole via self-phase modulation," Appl. Phys. Lett. 87, 121113 (2005).
    [CrossRef]
  20. A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1929 (2000).
    [CrossRef]

2006 (3)

2005 (5)

2004 (3)

P. S. Westbrook, J. W. Nicholson, K. S. Feder, Y. Li, and T. Brown, "Supercontinuum generation in a fibre grating," Appl. Phys. Lett. 85, 4600-4602 (2004).
[CrossRef]

S. Xu, D. Reitze, and R. Windeler, "Controlling nonlinear processes in microstructured fibers using shaped pulses," Opt. Express 12, 4731-4741 (2004).
[CrossRef] [PubMed]

M. Tianprateep, J. Tada, T. Yamazaki, and F. Kannari, "Spectral-Shape-Controllable Supercontinuum Generation in Microstructured Fibers Using Adaptive Pulse Shaping Technique," Japanese J. of Appl. Phys. 43, 8059-8063 (2004).
[CrossRef]

2002 (1)

2001 (2)

2000 (2)

A. S. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T.W. Hansch, "Direct Link between Microwave and Optical Frequencies with a 300 THz Femtosecond Laser Comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1929 (2000).
[CrossRef]

1996 (1)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

1993 (1)

K. Mori, T. Morioka, and M. Saruwatari, "Group-velocity dispersion measurement using supercontinuum picosecond pulses generated in an optical-fiber," Electron. Lett. 29, 987-989 (1993).
[CrossRef]

1991 (1)

P. S. Russell, "Bloch Wave Analysis of Dispersion and Pulse-Propagation in Pure Distributed Feedback Structures," J. Mod. Opt. 38, 1599-1619 (1991).
[CrossRef]

Assion, A.

A. Präkelt, M. Wollenhaupt, C. Sarpe-Tudoran, A. Assion, and T. Baumerta, "Filling a spectral hole via self-phase modulation," Appl. Phys. Lett. 87, 121113 (2005).
[CrossRef]

Baumerta, T.

A. Präkelt, M. Wollenhaupt, C. Sarpe-Tudoran, A. Assion, and T. Baumerta, "Filling a spectral hole via self-phase modulation," Appl. Phys. Lett. 87, 121113 (2005).
[CrossRef]

Brown, T.

P. S. Westbrook, J. W. Nicholson, K. S. Feder, Y. Li, and T. Brown, "Supercontinuum generation in a fibre grating," Appl. Phys. Lett. 85, 4600-4602 (2004).
[CrossRef]

Brown, T. G.

Chen, C.-M.

Chudoba, C.

Cundiff, S. T.

A. S. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T.W. Hansch, "Direct Link between Microwave and Optical Frequencies with a 300 THz Femtosecond Laser Comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

de Sterke, C. M.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Deng, Y.

Diddams, A. S.

A. S. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T.W. Hansch, "Direct Link between Microwave and Optical Frequencies with a 300 THz Femtosecond Laser Comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Diddams, S. A.

Dudley, J.

D. Türke, W. Wohlleben, J. Teipel, M. Motzkus, B. Kibler, J. Dudley, and H. Giessen, "Chirp-controlled soliton fission in tapered optical fibers," Appl. Phys. B 83, 37-42 (2006).
[CrossRef]

Eggleton, B. J.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Feder, K. S.

Fujimoto, J. G.

Ghanta, R. K.

Giessen, H.

D. Türke, W. Wohlleben, J. Teipel, M. Motzkus, B. Kibler, J. Dudley, and H. Giessen, "Chirp-controlled soliton fission in tapered optical fibers," Appl. Phys. B 83, 37-42 (2006).
[CrossRef]

Hall, J. L.

A. S. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T.W. Hansch, "Direct Link between Microwave and Optical Frequencies with a 300 THz Femtosecond Laser Comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Hansch, T.W.

A. S. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T.W. Hansch, "Direct Link between Microwave and Optical Frequencies with a 300 THz Femtosecond Laser Comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Hartl, I.

Herrmann, J.

A. V. Husakou and J. Herrmann, "Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers," Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef] [PubMed]

Holzwarth, R.

A. S. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T.W. Hansch, "Direct Link between Microwave and Optical Frequencies with a 300 THz Femtosecond Laser Comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Husakou, A. V.

A. V. Husakou and J. Herrmann, "Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers," Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef] [PubMed]

Jones, D. J.

A. S. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T.W. Hansch, "Direct Link between Microwave and Optical Frequencies with a 300 THz Femtosecond Laser Comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Kannari, F.

M. Tianprateep, J. Tada, T. Yamazaki, and F. Kannari, "Spectral-Shape-Controllable Supercontinuum Generation in Microstructured Fibers Using Adaptive Pulse Shaping Technique," Japanese J. of Appl. Phys. 43, 8059-8063 (2004).
[CrossRef]

Kelley, P. L.

Kennedy, R. E.

Kibler, B.

D. Türke, W. Wohlleben, J. Teipel, M. Motzkus, B. Kibler, J. Dudley, and H. Giessen, "Chirp-controlled soliton fission in tapered optical fibers," Appl. Phys. B 83, 37-42 (2006).
[CrossRef]

Kim, K.

Knox, W.

Ko, T. H.

Krug, P. A.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Li, X. D.

Li, Y.

Lu, F.

Mangan, B.

Mori, K.

K. Mori, T. Morioka, and M. Saruwatari, "Group-velocity dispersion measurement using supercontinuum picosecond pulses generated in an optical-fiber," Electron. Lett. 29, 987-989 (1993).
[CrossRef]

Morioka, T.

K. Mori, T. Morioka, and M. Saruwatari, "Group-velocity dispersion measurement using supercontinuum picosecond pulses generated in an optical-fiber," Electron. Lett. 29, 987-989 (1993).
[CrossRef]

Motzkus, M.

D. Türke, W. Wohlleben, J. Teipel, M. Motzkus, B. Kibler, J. Dudley, and H. Giessen, "Chirp-controlled soliton fission in tapered optical fibers," Appl. Phys. B 83, 37-42 (2006).
[CrossRef]

Nicholson, J.

Nicholson, J. W.

Nicholson, J.W.

Popov, S. V.

Präkelt, A.

A. Präkelt, M. Wollenhaupt, C. Sarpe-Tudoran, A. Assion, and T. Baumerta, "Filling a spectral hole via self-phase modulation," Appl. Phys. Lett. 87, 121113 (2005).
[CrossRef]

Ranka, J. K.

I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, "Ultrahigh-resolution optical coherence tomography using continuum generation in an air silica microstructure optical fiber," Opt. Lett. 26, 608-610 (2001).
[CrossRef]

A. S. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T.W. Hansch, "Direct Link between Microwave and Optical Frequencies with a 300 THz Femtosecond Laser Comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Reitze, D.

Russell, P. S.

P. S. Russell, "Bloch Wave Analysis of Dispersion and Pulse-Propagation in Pure Distributed Feedback Structures," J. Mod. Opt. 38, 1599-1619 (1991).
[CrossRef]

Sabert, H.

Salisbury, F. C.

Sarpe-Tudoran, C.

A. Präkelt, M. Wollenhaupt, C. Sarpe-Tudoran, A. Assion, and T. Baumerta, "Filling a spectral hole via self-phase modulation," Appl. Phys. Lett. 87, 121113 (2005).
[CrossRef]

Saruwatari, M.

K. Mori, T. Morioka, and M. Saruwatari, "Group-velocity dispersion measurement using supercontinuum picosecond pulses generated in an optical-fiber," Electron. Lett. 29, 987-989 (1993).
[CrossRef]

Sipe, J. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Slusher, R. E.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Tada, J.

M. Tianprateep, J. Tada, T. Yamazaki, and F. Kannari, "Spectral-Shape-Controllable Supercontinuum Generation in Microstructured Fibers Using Adaptive Pulse Shaping Technique," Japanese J. of Appl. Phys. 43, 8059-8063 (2004).
[CrossRef]

Taylor, J. R.

Teipel, J.

D. Türke, W. Wohlleben, J. Teipel, M. Motzkus, B. Kibler, J. Dudley, and H. Giessen, "Chirp-controlled soliton fission in tapered optical fibers," Appl. Phys. B 83, 37-42 (2006).
[CrossRef]

Tianprateep, M.

M. Tianprateep, J. Tada, T. Yamazaki, and F. Kannari, "Spectral-Shape-Controllable Supercontinuum Generation in Microstructured Fibers Using Adaptive Pulse Shaping Technique," Japanese J. of Appl. Phys. 43, 8059-8063 (2004).
[CrossRef]

Travers, J. C.

Türke, D.

D. Türke, W. Wohlleben, J. Teipel, M. Motzkus, B. Kibler, J. Dudley, and H. Giessen, "Chirp-controlled soliton fission in tapered optical fibers," Appl. Phys. B 83, 37-42 (2006).
[CrossRef]

Udem, T.

A. S. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T.W. Hansch, "Direct Link between Microwave and Optical Frequencies with a 300 THz Femtosecond Laser Comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Weiner, A. M.

A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1929 (2000).
[CrossRef]

Westbrook, P.

Westbrook, P. S.

Windeler, R.

Windeler, R. S.

Wohlleben, W.

D. Türke, W. Wohlleben, J. Teipel, M. Motzkus, B. Kibler, J. Dudley, and H. Giessen, "Chirp-controlled soliton fission in tapered optical fibers," Appl. Phys. B 83, 37-42 (2006).
[CrossRef]

Wollenhaupt, M.

A. Präkelt, M. Wollenhaupt, C. Sarpe-Tudoran, A. Assion, and T. Baumerta, "Filling a spectral hole via self-phase modulation," Appl. Phys. Lett. 87, 121113 (2005).
[CrossRef]

Xu, S.

Yablon, A. D.

Yamazaki, T.

M. Tianprateep, J. Tada, T. Yamazaki, and F. Kannari, "Spectral-Shape-Controllable Supercontinuum Generation in Microstructured Fibers Using Adaptive Pulse Shaping Technique," Japanese J. of Appl. Phys. 43, 8059-8063 (2004).
[CrossRef]

Ye, J.

A. S. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T.W. Hansch, "Direct Link between Microwave and Optical Frequencies with a 300 THz Femtosecond Laser Comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

Zhu, Z.

Appl. Phys. B (1)

D. Türke, W. Wohlleben, J. Teipel, M. Motzkus, B. Kibler, J. Dudley, and H. Giessen, "Chirp-controlled soliton fission in tapered optical fibers," Appl. Phys. B 83, 37-42 (2006).
[CrossRef]

Appl. Phys. Lett. (2)

P. S. Westbrook, J. W. Nicholson, K. S. Feder, Y. Li, and T. Brown, "Supercontinuum generation in a fibre grating," Appl. Phys. Lett. 85, 4600-4602 (2004).
[CrossRef]

A. Präkelt, M. Wollenhaupt, C. Sarpe-Tudoran, A. Assion, and T. Baumerta, "Filling a spectral hole via self-phase modulation," Appl. Phys. Lett. 87, 121113 (2005).
[CrossRef]

Electron. Lett. (1)

K. Mori, T. Morioka, and M. Saruwatari, "Group-velocity dispersion measurement using supercontinuum picosecond pulses generated in an optical-fiber," Electron. Lett. 29, 987-989 (1993).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

P. S. Russell, "Bloch Wave Analysis of Dispersion and Pulse-Propagation in Pure Distributed Feedback Structures," J. Mod. Opt. 38, 1599-1619 (1991).
[CrossRef]

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

Japanese J. of Appl. Phys. (1)

M. Tianprateep, J. Tada, T. Yamazaki, and F. Kannari, "Spectral-Shape-Controllable Supercontinuum Generation in Microstructured Fibers Using Adaptive Pulse Shaping Technique," Japanese J. of Appl. Phys. 43, 8059-8063 (2004).
[CrossRef]

Opt. Express (3)

Opt. Lett. (4)

Phys. Rev. Lett. (3)

A. S. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T.W. Hansch, "Direct Link between Microwave and Optical Frequencies with a 300 THz Femtosecond Laser Comb," Phys. Rev. Lett. 84, 5102-5105 (2000).
[CrossRef] [PubMed]

A. V. Husakou and J. Herrmann, "Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers," Phys. Rev. Lett. 87, 203901 (2001).
[CrossRef] [PubMed]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg Grating Solitons," Phys. Rev. Lett. 76, 1627-1630 (1996).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1929 (2000).
[CrossRef]

Other (1)

R. R. Alfano, The supercontinuum laser source: fundamentals with updated references, 2nd ed. (Springer, New York, 2006).

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Figures (6)

Fig. 1.
Fig. 1.

Spectral enhancement via narrow spectral phase features and nonlinear propagation; an ultrashort pulse (red) passes through a spectral filter whose phase response (green) is flat except for a narrowband feature. The pulse then propagates through a nonlinear medium, such as a small-core silica microstructured optical fibre. Upon propagation, the output acquires a spectral enhancement corresponding to the narrowband phase feature.

Fig. 2.
Fig. 2.

Phasor diagrams showing interference between strong linear and weak nonlinear components after a short propagation length; (a) self-phase modulation only, (b) short wavelength side of Bragg resonance and (c) long wavelength side.

Fig. 3.
Fig. 3.

Nonlinear spectral enhancement due to a narrowband 1-D photonic bandgap; (a) transformed wavenumber (blue solid) outside bandgap (black dotted); (b) relative spectral modulation resulting from a grating over the full 30 mm propagation length (red solid, Eq. (8) in text) and from a spectral filter of equivalent overall phase (blue dashed, Eq. (11) in text). Dashed black is a baseline, indicating the result in the absence of enhancement or suppression.

Fig. 4.
Fig. 4.

Exact (blue dashed) and approximate (red solid, Eq. (8) in text) pulse spectra under propagation through a 30 mm Bragg grating, shown on a linear scale with vertical offsets added for clarity. The initial pulse (black dotted) is repeated for comparison.

Fig. 5.
Fig. 5.

Experimental setup: BK7 glass in the Fourier plane of a 4-f femtosecond pulse shaper imparts spectrally narrow phase delays on Ti:Sapphire mode-locked laser pulses, which are then injected into nonlinear photonic crystal fibre (PCF). Rotating the glass about the horizontal transverse axis (dotted) affects the phase delay without changing its spectral extent.

Fig. 6.
Fig. 6.

(a) Output spectra: no phase mask (bottom), phase delay causing enhancement (middle) and depletion (top). Vertical offsets have been added for clarity. (b) Spectral intensity at phase delayed wavelength vs phase delay.

Equations (11)

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

A ( ω , z ) z = i β T ( ω ) A ( ω , z ) + i γ B ( ω , z ) ,
β T ( ω ) = β ( ω ) β F ( ω 0 ) ( ω ω 0 ) d β F d ω ω = ω 0 ,
A L ( ω , z ) = A ( ω , 0 ) exp [ i β T ( ω ) z ] ,
A ( ω , z ) A L ( ω , z ) + A NL ( ω , z ) ,
A NL ( ω , z ) z = i β T ( ω ) A NL ( ω , z ) + i γ B ( ω , 0 )
A NL ( ω , z ) = γ B ( ω , 0 ) [ e i β T ( ω ) z 1 ] β T ( ω )
A NL ( ω , z ) i γ B ( ω , 0 ) z .
A ( ω , z ) A ( ω , 0 ) 1 + B ( ω , 0 ) A ( ω , 0 ) 2 γ sin 2 [ β T ( ω ) z 2 ] β T ( ω )
A ( ω , 0 ) = A ( ω , 0 ) e i Δ ϕ ( ω ) ,
A ( ω , z ) = A ( ω , 0 ) e i Δ ϕ ( ω ) + i γ z B ( ω , 0 ) .
A ( ω , z ) A ( ω , 0 ) 1 + B ( ω , 0 ) A ( ω , 0 ) γ z sin Δ ϕ ( ω ) .

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