Abstract

We computationally investigate supercontinuum generation in an As2S3 solid core photonic crystal fiber (PCF) with a hexagonal cladding of air holes. We study the effect of varying the system (fiber and input pulse) parameters on the output bandwidth. We find that there is significant variation of the measured bandwidth with small changes in the system parameters due to the complex structure of the supercontinuum spectral output. This variation implies that one cannot accurately calculate the experimentally-expected bandwidth from a single numerical simulation. We propose the use of a smoothed and ensemble-averaged bandwidth that is expected to be a better predictor of the bandwidth of the supercontinuum spectra that would be produced in experimental systems. We show that the fluctuations are considerably reduced, allowing us to calculate the bandwidth more accurately. Using this smoothed and ensemble averaged bandwidth, we maximize the output bandwidth with a pump wavelength of 2.8 μm and obtain a supercontinuum spectrum that extends from 2.5 μm to 6.2 μm with an uncertainty of ± 0.5 μm. The optimized bandwidth is consistent with prior work, but with a significantly increased accuracy.

© 2010 Optical Society of America

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References

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  1. J. K. Ranka, R. S. Windeler, and A. J. Stentz, "Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm," Opt. Lett. 25, 25-27 (2000).
    [CrossRef]
  2. J. M. Dudley, and J. R. Taylor, "Ten years of nonlinear optics in photonic crystal fibre," Nat. Photonics 3, 85-90 (2009).
    [CrossRef]
  3. X. Feng, A. Mairaj, D. Hewak, and T. Monro, "Nonsilica Glasses for Holey Fibers," J. Lightwave Technol. 23, 2046-2054 (2005).
    [CrossRef]
  4. P. Rolfe, "In vivo near-infrared spectroscopy," Annu. Rev. Biomed. Eng. 2, 715-754 (2000).
    [CrossRef]
  5. R. Holzwarth, T. Udem, T. Hänsch, J. Knight, W. Wadsworth, and P. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
    [CrossRef] [PubMed]
  6. 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]
  7. D. I. Yeom, E. C. Mgi, M. R. E. Lamont, M. A. F. Roelens, L. Fu, and B. J. Eggleton, "Low-threshold supercontinuum generation in highly nonlinear chalcogenide nanowires," Opt. Lett. 33, 660-662 (2008).
    [CrossRef] [PubMed]
  8. P. Domachuk, N. A. Wolchover, M. Cronin-Golomb, A. Wang, A. K. George, C. M. Cordeiro, J. C. Knight, and F. G. Omenetto, "Over 4000 nm bandwidth of mid-IR supercontinuum generation in sub-centimeter segments of highly nonlinear tellurite PCFs," Opt. Express 16, 7161-7168 (2008).
    [CrossRef] [PubMed]
  9. M. R. Lamont, B. Luther-Davies, D. Choi, S. Madden, and B. J. Eggleton, "Supercontinuum generation in dispersion engineered highly nonlinear (γ = 10 W/m) As2S3 chalcogenide planar waveguide," Opt. Express 16, 14938-14944 (2008).
    [CrossRef] [PubMed]
  10. J. H. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, F. Vittoria, J. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, "Non-silica Microstructured Optical Fibers For Mid-IR Supercontinuum Generation From 2 μm-5 μm," IEEE J. Sel. Top. Quantum Electron. 13(3), 738-749 (2007).
    [CrossRef]
  11. J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, "Maximizing the bandwidth of supercontinuum generation in As2Se3 chalcogenide fibers," Opt. Express 18, 6722-6739 (2010).
    [CrossRef] [PubMed]
  12. B. Ung, and M. Skorobogatiy, "Chalcogenide microporous fibers for linear and nonlinear applications in the mid-infrared," Opt. Express 18, 8647-8659 (2010).
    [CrossRef] [PubMed]
  13. W. Q. Zhang, S. V. Afshar, and T. M. Monro, "A genetic algorithm based approach to fiber design for high coherence and large bandwidth supercontinuum generation," Opt. Express 17, 19311-19327 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  16. R. J. Weiblen, J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, "Maximizing the Supercontinuum Bandwidth in As2S3 Chalcogenide Photonic Crystal Fibers," in Proc. Conference on Lasers and Electro-Optics (CLEO), San Jose, CA, paper CTuX7, (2010).
  17. J. S. Sanghera, L. Brandon Shaw, and I. D. Aggarwal, "Chalcogenide Glass-Fiber-Based Mid-IR Sources and Applications," IEEE J. Sel. Top. Quantum Electron. 15, 114-119 (2009).
    [CrossRef]
  18. J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
    [CrossRef]
  19. J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, "Computational study of 3-5 μm source created by using supercontinuum generation in As2S3 chalcogenide fibers with a pump at 2 μm," Opt. Lett. 35, 2907-2909 (2010).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]

2010 (3)

2009 (3)

J. M. Dudley, and J. R. Taylor, "Ten years of nonlinear optics in photonic crystal fibre," Nat. Photonics 3, 85-90 (2009).
[CrossRef]

J. S. Sanghera, L. Brandon Shaw, and I. D. Aggarwal, "Chalcogenide Glass-Fiber-Based Mid-IR Sources and Applications," IEEE J. Sel. Top. Quantum Electron. 15, 114-119 (2009).
[CrossRef]

W. Q. Zhang, S. V. Afshar, and T. M. Monro, "A genetic algorithm based approach to fiber design for high coherence and large bandwidth supercontinuum generation," Opt. Express 17, 19311-19327 (2009).
[CrossRef]

2008 (3)

2007 (1)

J. H. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, F. Vittoria, J. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, "Non-silica Microstructured Optical Fibers For Mid-IR Supercontinuum Generation From 2 μm-5 μm," IEEE J. Sel. Top. Quantum Electron. 13(3), 738-749 (2007).
[CrossRef]

2006 (2)

2005 (1)

2003 (1)

2002 (1)

2001 (1)

2000 (3)

J. K. Ranka, R. S. Windeler, and A. J. Stentz, "Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm," Opt. Lett. 25, 25-27 (2000).
[CrossRef]

P. Rolfe, "In vivo near-infrared spectroscopy," Annu. Rev. Biomed. Eng. 2, 715-754 (2000).
[CrossRef]

R. Holzwarth, T. Udem, T. Hänsch, J. Knight, W. Wadsworth, and P. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
[CrossRef] [PubMed]

1986 (1)

Afshar, S. V.

Aggarwal, I. D.

Brambilla, G.

J. H. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, F. Vittoria, J. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, "Non-silica Microstructured Optical Fibers For Mid-IR Supercontinuum Generation From 2 μm-5 μm," IEEE J. Sel. Top. Quantum Electron. 13(3), 738-749 (2007).
[CrossRef]

Brandon Shaw, L.

J. S. Sanghera, L. Brandon Shaw, and I. D. Aggarwal, "Chalcogenide Glass-Fiber-Based Mid-IR Sources and Applications," IEEE J. Sel. Top. Quantum Electron. 15, 114-119 (2009).
[CrossRef]

Choi, D.

Chudoba, C.

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

J. Dudley, and S. Coen, "Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers," Opt. Lett. 27, 1180-1182 (2002).
[CrossRef]

Cordeiro, C. M.

Cronin-Golomb, M.

Domachuk, P.

Dudley, J.

Dudley, J. M.

J. M. Dudley, and J. R. Taylor, "Ten years of nonlinear optics in photonic crystal fibre," Nat. Photonics 3, 85-90 (2009).
[CrossRef]

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Ebendorff-Heidepriem, H.

J. H. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, F. Vittoria, J. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, "Non-silica Microstructured Optical Fibers For Mid-IR Supercontinuum Generation From 2 μm-5 μm," IEEE J. Sel. Top. Quantum Electron. 13(3), 738-749 (2007).
[CrossRef]

Efimov, A.

Eggleton, B. J.

Feng, X.

J. H. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, F. Vittoria, J. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, "Non-silica Microstructured Optical Fibers For Mid-IR Supercontinuum Generation From 2 μm-5 μm," IEEE J. Sel. Top. Quantum Electron. 13(3), 738-749 (2007).
[CrossRef]

X. Feng, A. Mairaj, D. Hewak, and T. Monro, "Nonsilica Glasses for Holey Fibers," J. Lightwave Technol. 23, 2046-2054 (2005).
[CrossRef]

Flanagan, J. C.

J. H. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, F. Vittoria, J. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, "Non-silica Microstructured Optical Fibers For Mid-IR Supercontinuum Generation From 2 μm-5 μm," IEEE J. Sel. Top. Quantum Electron. 13(3), 738-749 (2007).
[CrossRef]

Fu, L.

Fujimoto, J. G.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

George, A. K.

Ghanta, R. K.

Gordon, J. P.

Hänsch, T.

R. Holzwarth, T. Udem, T. Hänsch, J. Knight, W. Wadsworth, and P. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
[CrossRef] [PubMed]

Hartl, I.

Hewak, D.

Holzlhner, R.

Holzwarth, R.

R. Holzwarth, T. Udem, T. Hänsch, J. Knight, W. Wadsworth, and P. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
[CrossRef] [PubMed]

Hu, J.

Joly, N. Y.

Knight, J.

R. Holzwarth, T. Udem, T. Hänsch, J. Knight, W. Wadsworth, and P. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
[CrossRef] [PubMed]

Knight, J. C.

Ko, T. H.

Kumar, V. V.

Lamont, M. R.

Lamont, M. R. E.

Leong, J. Y.

J. H. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, F. Vittoria, J. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, "Non-silica Microstructured Optical Fibers For Mid-IR Supercontinuum Generation From 2 μm-5 μm," IEEE J. Sel. Top. Quantum Electron. 13(3), 738-749 (2007).
[CrossRef]

Li, X. D.

Luther-Davies, B.

Madden, S.

Mairaj, A.

Menyuk, C. R.

Mgi, E. C.

Monro, T.

Monro, T. M.

W. Q. Zhang, S. V. Afshar, and T. M. Monro, "A genetic algorithm based approach to fiber design for high coherence and large bandwidth supercontinuum generation," Opt. Express 17, 19311-19327 (2009).
[CrossRef]

J. H. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, F. Vittoria, J. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, "Non-silica Microstructured Optical Fibers For Mid-IR Supercontinuum Generation From 2 μm-5 μm," IEEE J. Sel. Top. Quantum Electron. 13(3), 738-749 (2007).
[CrossRef]

Omenetto, F. G.

Petropoulos, P.

J. H. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, F. Vittoria, J. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, "Non-silica Microstructured Optical Fibers For Mid-IR Supercontinuum Generation From 2 μm-5 μm," IEEE J. Sel. Top. Quantum Electron. 13(3), 738-749 (2007).
[CrossRef]

Poletti, F.

J. H. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, F. Vittoria, J. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, "Non-silica Microstructured Optical Fibers For Mid-IR Supercontinuum Generation From 2 μm-5 μm," IEEE J. Sel. Top. Quantum Electron. 13(3), 738-749 (2007).
[CrossRef]

Price, J. H.

J. H. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, F. Vittoria, J. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, "Non-silica Microstructured Optical Fibers For Mid-IR Supercontinuum Generation From 2 μm-5 μm," IEEE J. Sel. Top. Quantum Electron. 13(3), 738-749 (2007).
[CrossRef]

Ranka, J. K.

Richardson, D. J.

J. H. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, F. Vittoria, J. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, "Non-silica Microstructured Optical Fibers For Mid-IR Supercontinuum Generation From 2 μm-5 μm," IEEE J. Sel. Top. Quantum Electron. 13(3), 738-749 (2007).
[CrossRef]

Roelens, M. A. F.

Rolfe, P.

P. Rolfe, "In vivo near-infrared spectroscopy," Annu. Rev. Biomed. Eng. 2, 715-754 (2000).
[CrossRef]

Ross, M.

Russell, P.

R. Holzwarth, T. Udem, T. Hänsch, J. Knight, W. Wadsworth, and P. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
[CrossRef] [PubMed]

Russell, P. S.

Sanghera, J. S.

Shaw, L. B.

Sinkin, O. V.

Skorobogatiy, M.

Stentz, A. J.

Taylor, A. J.

Taylor, J. R.

J. M. Dudley, and J. R. Taylor, "Ten years of nonlinear optics in photonic crystal fibre," Nat. Photonics 3, 85-90 (2009).
[CrossRef]

Udem, T.

R. Holzwarth, T. Udem, T. Hänsch, J. Knight, W. Wadsworth, and P. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
[CrossRef] [PubMed]

Ung, B.

Vittoria, F.

J. H. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, F. Vittoria, J. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, "Non-silica Microstructured Optical Fibers For Mid-IR Supercontinuum Generation From 2 μm-5 μm," IEEE J. Sel. Top. Quantum Electron. 13(3), 738-749 (2007).
[CrossRef]

Wadsworth, W.

R. Holzwarth, T. Udem, T. Hänsch, J. Knight, W. Wadsworth, and P. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
[CrossRef] [PubMed]

Wang, A.

Wehner, M. R.

Windeler, R. S.

Wolchover, N. A.

Yeom, D. I.

Zhang, W. Q.

Zweck, J.

Annu. Rev. Biomed. Eng. (1)

P. Rolfe, "In vivo near-infrared spectroscopy," Annu. Rev. Biomed. Eng. 2, 715-754 (2000).
[CrossRef]

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

J. H. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, F. Vittoria, J. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, "Non-silica Microstructured Optical Fibers For Mid-IR Supercontinuum Generation From 2 μm-5 μm," IEEE J. Sel. Top. Quantum Electron. 13(3), 738-749 (2007).
[CrossRef]

J. S. Sanghera, L. Brandon Shaw, and I. D. Aggarwal, "Chalcogenide Glass-Fiber-Based Mid-IR Sources and Applications," IEEE J. Sel. Top. Quantum Electron. 15, 114-119 (2009).
[CrossRef]

J. Lightwave Technol. (2)

Nat. Photonics (1)

J. M. Dudley, and J. R. Taylor, "Ten years of nonlinear optics in photonic crystal fibre," Nat. Photonics 3, 85-90 (2009).
[CrossRef]

Opt. Express (6)

F. G. Omenetto, N. A. Wolchover, M. R. Wehner, M. Ross, A. Efimov, A. J. Taylor, V. V. Kumar, A. K. George, J. C. Knight, N. Y. Joly, and P. S. Russell, "Spectrally smooth supercontinuum from 350 nm to 3 μm in subcentimeter lengths of soft-glass photonic crystal fibers," Opt. Express 14, 4928-4934 (2006).
[CrossRef] [PubMed]

P. Domachuk, N. A. Wolchover, M. Cronin-Golomb, A. Wang, A. K. George, C. M. Cordeiro, J. C. Knight, and F. G. Omenetto, "Over 4000 nm bandwidth of mid-IR supercontinuum generation in sub-centimeter segments of highly nonlinear tellurite PCFs," Opt. Express 16, 7161-7168 (2008).
[CrossRef] [PubMed]

M. R. Lamont, B. Luther-Davies, D. Choi, S. Madden, and B. J. Eggleton, "Supercontinuum generation in dispersion engineered highly nonlinear (γ = 10 W/m) As2S3 chalcogenide planar waveguide," Opt. Express 16, 14938-14944 (2008).
[CrossRef] [PubMed]

W. Q. Zhang, S. V. Afshar, and T. M. Monro, "A genetic algorithm based approach to fiber design for high coherence and large bandwidth supercontinuum generation," Opt. Express 17, 19311-19327 (2009).
[CrossRef]

J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, "Maximizing the bandwidth of supercontinuum generation in As2Se3 chalcogenide fibers," Opt. Express 18, 6722-6739 (2010).
[CrossRef] [PubMed]

B. Ung, and M. Skorobogatiy, "Chalcogenide microporous fibers for linear and nonlinear applications in the mid-infrared," Opt. Express 18, 8647-8659 (2010).
[CrossRef] [PubMed]

Opt. Lett. (6)

Phys. Rev. Lett. (1)

R. Holzwarth, T. Udem, T. Hänsch, J. Knight, W. Wadsworth, and P. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78, 1135-1184 (2006).
[CrossRef]

Other (1)

R. J. Weiblen, J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, "Maximizing the Supercontinuum Bandwidth in As2S3 Chalcogenide Photonic Crystal Fibers," in Proc. Conference on Lasers and Electro-Optics (CLEO), San Jose, CA, paper CTuX7, (2010).

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Figures (6)

Fig. 1
Fig. 1

Hexagonal PCF air-hole geometry.

Fig. 2
Fig. 2

Chromatic dispersion (a) and loss (b) for hexagonal PCFs with pitches of 2.5 μm (blue), 3.0 μm (red), 3.5 μm (green), and 4.0 μm (cyan).

Fig. 3
Fig. 3

(a) Example spectra from an OSA with filter widths of λw = 10 nm (top), λw = 50 nm (middle), and λw = 100 nm (lower). (b) Power spectral density (PSD) in arbitrary units (a. u.) for the same example spectrum (blue solid) and equivalent rectangular spectrum (green dashed) that are defined in the text.

Fig. 4
Fig. 4

(a) Bandwidth versus (a) input pulse width, and (b) input pulse peak power, for air-hole pitches of 2.5 μm (blue), 3.0 μm (red), 3.5 μm (green), and 4.0 μm (cyan). In (a) the peak power is 1500 W, while in (b) the pulse width FWHM is 1500 fs. The bandwidth is calculated for different pulse widths in steps of 1 fs. The peak power is varied in steps of 1 W.

Fig. 5
Fig. 5

(a) Calculated bandwidth (solid) and ensemble-averaged bandwidth for a 10% parameter variation (dashed) vs. pulse width for a pitch of 3.5 μm. (b) Calculated bandwidth (solid) and ensemble-averaged bandwidth for a 10% parameter variation (dashed) vs. peak input power for a pitch of 3.5 μm.

Fig. 6
Fig. 6

(a) Bandwidth found by the averaging procedure of Section 4. The averaged bandwidth is plotted versus (a) input pulse width, and (b) input pulse peak power, for air-hole pitches of 2.5 μm (blue), 3.0 μm (red), 3.5 μm (green), and 4.0 μm (cyan). In (a) the peak power is 1500 W, while in (b) the pulse width is 1500 fs. The bandwidth is calculated for different pulse widths in steps of 1 fs. The peak power is varied in steps of 1 W.

Equations (2)

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A ( z , t ) z i IFT { [ a ( ω ) 2 + β ( ω 0 + Ω ) β ( ω 0 ) Ω β 1 ( ω 0 ) ] A ˜ ( z , Ω ) } = i γ ( 1 + i ω 0 t ) [ A ( z , t ) t R ( t t ) | A ( z , t ) | 2 d t ] ,
S f ( λ ) = λ Λ λ + Λ exp [ ( λ v λ w ) 4 ] S ( ν ) d ν ,

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