Abstract

Supercontinuum generation using picosecond pulses pumped into cobweb photonic crystal fibers is investigated. Dispersion profiles are calculated for several fiber designs and used to analytically investigate the influence of the fiber structural parameters (core size and wall thickness) on the location of the Stokes and anti-Stokes bands and gain bandwidth. An analysis shows that the Raman effect is responsible for reducing the four-wave mixing gain and a slight reduction in the corresponding frequency shift from the pump, when the frequency shift is much larger than the Raman shift. Using numerical simulations we find that four-wave mixing is the dominant physical mechanism for the pumping scheme considered, and that there is a trade-off between the spectral width and the spectral flatness of the supercontinuum. The balance of this trade-off is determined by nanometer-scale design of the fiber structural parameters. It is also shown that the relatively high loss of the nonlinear fiber does not significantly affect the supercontinuum generation.

© 2006 Optical Society of America

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    [CrossRef]
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2005 (2)

2004 (3)

2003 (8)

O. V. Sinkin, R. Holzlöhner, J. Zweck, and C. R. Menyuk, "Optimization of the split-step Fourier method in modeling optical-fiber communications systems," J. Lightwave Technol. 21, 61-68 (2003).
[CrossRef]

F. Vanholsbeeck, P. Emplit, and S. Coen, "Complete experimental characterization of the influence of parametric four-wave mixing on stimulated Raman gain," Opt. Lett. 28, 1960-1962 (2003).
[CrossRef] [PubMed]

N. I. Nikolov, T. Sørensen, O. Bang, and A. Bjarklev, "Improving efficiency of supercontinuum generation in photonic crystal fibers by direct degenerate four-wave mixing," J. Opt. Soc. Am. B 20, 2329-2337 (2003).
[CrossRef]

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

W. Shuang-Chun, S. Wen-Hua, Z. Hua, F. Xi-Quan, Q. Lie-Jia, and F. Dian-Yuan, "Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres," Chin. Phys. Lett. 20, 852-854, doi:10.1088/0256-307X/20/6/321 (2003).
[CrossRef]

F. Biancalana, D. V. Skryabin, and P. St. J. Russell, "Four-wave mixing instabilities in photonic-crystal and tapered fibers," Phys. Rev. E 68, 046603, doi:10.1103/PhysRevE.68.046603 (2003).
[CrossRef]

C. A. De Francisco, B. V. Borges, and M. A. Romero, "A semivectorial method for the modeling of photonic crystal fibers," Microwave Opt. Technol. Lett. 38, 418-421 (2003).
[CrossRef]

T. Schreiber, J. Limpert, H. Zellmer, A. Tünnermann, and K. P. Hansen, "High average power supercontinuum generation in photonic crystal fibers," Opt. Commun. 228, 71-78, doi:10.1016/j.optcom.2003.09.091 (2003).
[CrossRef]

2002 (4)

2001 (2)

2000 (1)

1999 (1)

P. V. Kelkar, F. Coppinger, A. S. Bhusan, and B. Jalali, "Time-domain optical sensing, "Electron. Lett. 35, 1661-1662 (1999).
[CrossRef]

1998 (1)

A. Freiberg, J. A. Jackson, S. Lin, and N. W. Woodbury, "Subpicosecond pump-supercontinuum probe spectroscopy of LH2 photosynthetic antenna," J. Phys. Chem. A 102, 4372-4380 (1998).
[CrossRef]

1989 (1)

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

Alfano, R. R.

R. R. Alfano, ed., The Supercontinuum Laser Source (Springer-Verlag, 1989).

Apolonski, A.

Bang, O.

Bhusan, A. S.

P. V. Kelkar, F. Coppinger, A. S. Bhusan, and B. Jalali, "Time-domain optical sensing, "Electron. Lett. 35, 1661-1662 (1999).
[CrossRef]

Biancalana, F.

W. J. Wadsworth, N. Joly, J. C. Knight, T. A. Birks, F. Biancalana, and P. St. J. Russell, "Supercontinuum and four-wave mixing with Q-switched pulses in endlessly single-mode photonic crystal fibres," Opt. Express 12, 299-309 (2004).
[CrossRef] [PubMed]

F. Biancalana, D. V. Skryabin, and P. St. J. Russell, "Four-wave mixing instabilities in photonic-crystal and tapered fibers," Phys. Rev. E 68, 046603, doi:10.1103/PhysRevE.68.046603 (2003).
[CrossRef]

Birks, T. A.

Bjarklev, A.

N. I. Nikolov, T. Sørensen, O. Bang, and A. Bjarklev, "Improving efficiency of supercontinuum generation in photonic crystal fibers by direct degenerate four-wave mixing," J. Opt. Soc. Am. B 20, 2329-2337 (2003).
[CrossRef]

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, "Highly nonlinear photonic crystal fiber with zero-dispersion at 1.55μm," in Optical Fiber Communication Conference (Optical Society of America, 2002), pp. FA91-FA93.

Blow, K. J.

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

Borges, B. V.

C. A. De Francisco, B. V. Borges, and M. A. Romero, "A semivectorial method for the modeling of photonic crystal fibers," Microwave Opt. Technol. Lett. 38, 418-421 (2003).
[CrossRef]

Broeng, J.

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, "Highly nonlinear photonic crystal fiber with zero-dispersion at 1.55μm," in Optical Fiber Communication Conference (Optical Society of America, 2002), pp. FA91-FA93.

Chau, A. H. L.

Chudoba, C.

Coen, S.

Coker, A.

Coppinger, F.

P. V. Kelkar, F. Coppinger, A. S. Bhusan, and B. Jalali, "Time-domain optical sensing, "Electron. Lett. 35, 1661-1662 (1999).
[CrossRef]

De Francisco, C. A.

C. A. De Francisco, B. V. Borges, and M. A. Romero, "A semivectorial method for the modeling of photonic crystal fibers," Microwave Opt. Technol. Lett. 38, 418-421 (2003).
[CrossRef]

Dian-Yuan, F.

W. Shuang-Chun, S. Wen-Hua, Z. Hua, F. Xi-Quan, Q. Lie-Jia, and F. Dian-Yuan, "Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres," Chin. Phys. Lett. 20, 852-854, doi:10.1088/0256-307X/20/6/321 (2003).
[CrossRef]

Drexler, W.

Dudley, J. M.

Eggleton, B. J.

Emplit, P.

Falk, P.

Fercher, A. F.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Fiorentino, M.

Freiberg, A.

A. Freiberg, J. A. Jackson, S. Lin, and N. W. Woodbury, "Subpicosecond pump-supercontinuum probe spectroscopy of LH2 photosynthetic antenna," J. Phys. Chem. A 102, 4372-4380 (1998).
[CrossRef]

Frosz, M. H.

Fujimoto, J. G.

Genty, G.

Ghanta, R. K.

Grossard, N.

Hansen, K. P.

T. Schreiber, J. Limpert, H. Zellmer, A. Tünnermann, and K. P. Hansen, "High average power supercontinuum generation in photonic crystal fibers," Opt. Commun. 228, 71-78, doi:10.1016/j.optcom.2003.09.091 (2003).
[CrossRef]

K. P. Hansen, Crystal Fiber A/S (personal communication, 2006).

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, "Highly nonlinear photonic crystal fiber with zero-dispersion at 1.55μm," in Optical Fiber Communication Conference (Optical Society of America, 2002), pp. FA91-FA93.

Hartl, I.

Harvey, J. D.

S. Coen, D. A. Wardle, and J. D. Harvey, "Observation of non-phase-matched parametric amplification in resonant nonlinear optics," Phys. Rev. Lett. 89, 273901/1-4, doi:10.1103/PhysRevLett.89.273901 (2002).
[CrossRef]

S. Coen, A. H. L. Chau, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, "Supercontinuum generation by stimulated Raman scattering and parametric four-wave mixing in photonic crystal fibers," J. Opt. Soc. Am. B 19, 753-764 (2002).
[CrossRef]

Hitzenberger, C. K.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Holzlöhner, R.

Hua, Z.

W. Shuang-Chun, S. Wen-Hua, Z. Hua, F. Xi-Quan, Q. Lie-Jia, and F. Dian-Yuan, "Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres," Chin. Phys. Lett. 20, 852-854, doi:10.1088/0256-307X/20/6/321 (2003).
[CrossRef]

Jackson, J. A.

A. Freiberg, J. A. Jackson, S. Lin, and N. W. Woodbury, "Subpicosecond pump-supercontinuum probe spectroscopy of LH2 photosynthetic antenna," J. Phys. Chem. A 102, 4372-4380 (1998).
[CrossRef]

Jacobsen, C.

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, "Highly nonlinear photonic crystal fiber with zero-dispersion at 1.55μm," in Optical Fiber Communication Conference (Optical Society of America, 2002), pp. FA91-FA93.

Jalali, B.

P. V. Kelkar, F. Coppinger, A. S. Bhusan, and B. Jalali, "Time-domain optical sensing, "Electron. Lett. 35, 1661-1662 (1999).
[CrossRef]

Jensen, J. R.

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, "Highly nonlinear photonic crystal fiber with zero-dispersion at 1.55μm," in Optical Fiber Communication Conference (Optical Society of America, 2002), pp. FA91-FA93.

Joly, N.

Kaivola, M.

Kelkar, P. V.

P. V. Kelkar, F. Coppinger, A. S. Bhusan, and B. Jalali, "Time-domain optical sensing, "Electron. Lett. 35, 1661-1662 (1999).
[CrossRef]

Knight, J. C.

Ko, T. H.

Kumar, P.

Lasser, T.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Lehtonen, M.

Leonhardt, R.

Leon-Saval, S. G.

Li, X. D.

Lie-Jia, Q.

W. Shuang-Chun, S. Wen-Hua, Z. Hua, F. Xi-Quan, Q. Lie-Jia, and F. Dian-Yuan, "Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres," Chin. Phys. Lett. 20, 852-854, doi:10.1088/0256-307X/20/6/321 (2003).
[CrossRef]

Limpert, J.

T. Schreiber, J. Limpert, H. Zellmer, A. Tünnermann, and K. P. Hansen, "High average power supercontinuum generation in photonic crystal fibers," Opt. Commun. 228, 71-78, doi:10.1016/j.optcom.2003.09.091 (2003).
[CrossRef]

Lin, S.

A. Freiberg, J. A. Jackson, S. Lin, and N. W. Woodbury, "Subpicosecond pump-supercontinuum probe spectroscopy of LH2 photosynthetic antenna," J. Phys. Chem. A 102, 4372-4380 (1998).
[CrossRef]

Ludvigsen, H.

Maillotte, H.

Mason, M. W.

Menyuk, C. R.

Nikolov, N. I.

Petersson, A.

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, "Highly nonlinear photonic crystal fiber with zero-dispersion at 1.55μm," in Optical Fiber Communication Conference (Optical Society of America, 2002), pp. FA91-FA93.

Povazay, B.

Provino, L.

Ranka, J. K.

Riishede, J.

J. Riishede, "Modelling photonic crystal fibres with the finite difference method," Ph.D. dissertation (Research Center COM, Technical University of Denmark, 2005).

Romero, M. A.

C. A. De Francisco, B. V. Borges, and M. A. Romero, "A semivectorial method for the modeling of photonic crystal fibers," Microwave Opt. Technol. Lett. 38, 418-421 (2003).
[CrossRef]

Russell, P. St. J.

Schreiber, T.

T. Schreiber, J. Limpert, H. Zellmer, A. Tünnermann, and K. P. Hansen, "High average power supercontinuum generation in photonic crystal fibers," Opt. Commun. 228, 71-78, doi:10.1016/j.optcom.2003.09.091 (2003).
[CrossRef]

Sharping, J. E.

Shuang-Chun, W.

W. Shuang-Chun, S. Wen-Hua, Z. Hua, F. Xi-Quan, Q. Lie-Jia, and F. Dian-Yuan, "Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres," Chin. Phys. Lett. 20, 852-854, doi:10.1088/0256-307X/20/6/321 (2003).
[CrossRef]

Simonsen, H. R.

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, "Highly nonlinear photonic crystal fiber with zero-dispersion at 1.55μm," in Optical Fiber Communication Conference (Optical Society of America, 2002), pp. FA91-FA93.

Sinkin, O. V.

Skovgaard, P. M. W.

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, "Highly nonlinear photonic crystal fiber with zero-dispersion at 1.55μm," in Optical Fiber Communication Conference (Optical Society of America, 2002), pp. FA91-FA93.

Skryabin, D. V.

F. Biancalana, D. V. Skryabin, and P. St. J. Russell, "Four-wave mixing instabilities in photonic-crystal and tapered fibers," Phys. Rev. E 68, 046603, doi:10.1103/PhysRevE.68.046603 (2003).
[CrossRef]

Sørensen, T.

N. I. Nikolov, T. Sørensen, O. Bang, and A. Bjarklev, "Improving efficiency of supercontinuum generation in photonic crystal fibers by direct degenerate four-wave mixing," J. Opt. Soc. Am. B 20, 2329-2337 (2003).
[CrossRef]

M. H. Frosz, T. Sørensen, and O. Bang, "Nano-engineering of photonic crystal fibers for Supercontinuum generation," in Photonic Crystals and Fibers, W. Urbanczyk, B. Jaskorzynska, and P. St. J. Russell, eds., Proc. SPIE 5950, 185-192 (2005).

Tünnermann, A.

T. Schreiber, J. Limpert, H. Zellmer, A. Tünnermann, and K. P. Hansen, "High average power supercontinuum generation in photonic crystal fibers," Opt. Commun. 228, 71-78, doi:10.1016/j.optcom.2003.09.091 (2003).
[CrossRef]

Unterhuber, A.

Vanholsbeeck, F.

Wadsworth, W. J.

Wardle, D. A.

S. Coen, D. A. Wardle, and J. D. Harvey, "Observation of non-phase-matched parametric amplification in resonant nonlinear optics," Phys. Rev. Lett. 89, 273901/1-4, doi:10.1103/PhysRevLett.89.273901 (2002).
[CrossRef]

Wen-Hua, S.

W. Shuang-Chun, S. Wen-Hua, Z. Hua, F. Xi-Quan, Q. Lie-Jia, and F. Dian-Yuan, "Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres," Chin. Phys. Lett. 20, 852-854, doi:10.1088/0256-307X/20/6/321 (2003).
[CrossRef]

Windeler, R. S.

Wirideler, R. S.

Wood, D.

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

Woodbury, N. W.

A. Freiberg, J. A. Jackson, S. Lin, and N. W. Woodbury, "Subpicosecond pump-supercontinuum probe spectroscopy of LH2 photosynthetic antenna," J. Phys. Chem. A 102, 4372-4380 (1998).
[CrossRef]

Xi-Quan, F.

W. Shuang-Chun, S. Wen-Hua, Z. Hua, F. Xi-Quan, Q. Lie-Jia, and F. Dian-Yuan, "Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres," Chin. Phys. Lett. 20, 852-854, doi:10.1088/0256-307X/20/6/321 (2003).
[CrossRef]

Zellmer, H.

T. Schreiber, J. Limpert, H. Zellmer, A. Tünnermann, and K. P. Hansen, "High average power supercontinuum generation in photonic crystal fibers," Opt. Commun. 228, 71-78, doi:10.1016/j.optcom.2003.09.091 (2003).
[CrossRef]

Zweck, J.

Chin. Phys. Lett. (1)

W. Shuang-Chun, S. Wen-Hua, Z. Hua, F. Xi-Quan, Q. Lie-Jia, and F. Dian-Yuan, "Influence of higher-order dispersions and Raman delayed response on modulation instability in microstructured fibres," Chin. Phys. Lett. 20, 852-854, doi:10.1088/0256-307X/20/6/321 (2003).
[CrossRef]

Electron. Lett. (1)

P. V. Kelkar, F. Coppinger, A. S. Bhusan, and B. Jalali, "Time-domain optical sensing, "Electron. Lett. 35, 1661-1662 (1999).
[CrossRef]

IEEE J. Quantum Electron. (1)

K. J. Blow and D. Wood, "Theoretical description of transient stimulated Raman scattering in optical fibers," IEEE J. Quantum Electron. 25, 2665-2673 (1989).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (4)

J. Phys. Chem. A (1)

A. Freiberg, J. A. Jackson, S. Lin, and N. W. Woodbury, "Subpicosecond pump-supercontinuum probe spectroscopy of LH2 photosynthetic antenna," J. Phys. Chem. A 102, 4372-4380 (1998).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

C. A. De Francisco, B. V. Borges, and M. A. Romero, "A semivectorial method for the modeling of photonic crystal fibers," Microwave Opt. Technol. Lett. 38, 418-421 (2003).
[CrossRef]

Opt. Commun. (1)

T. Schreiber, J. Limpert, H. Zellmer, A. Tünnermann, and K. P. Hansen, "High average power supercontinuum generation in photonic crystal fibers," Opt. Commun. 228, 71-78, doi:10.1016/j.optcom.2003.09.091 (2003).
[CrossRef]

Opt. Express (5)

Opt. Lett. (4)

Phys. Rev. E (1)

F. Biancalana, D. V. Skryabin, and P. St. J. Russell, "Four-wave mixing instabilities in photonic-crystal and tapered fibers," Phys. Rev. E 68, 046603, doi:10.1103/PhysRevE.68.046603 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

S. Coen, D. A. Wardle, and J. D. Harvey, "Observation of non-phase-matched parametric amplification in resonant nonlinear optics," Phys. Rev. Lett. 89, 273901/1-4, doi:10.1103/PhysRevLett.89.273901 (2002).
[CrossRef]

Rep. Prog. Phys. (1)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography--principles and applications," Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Other (7)

R. R. Alfano, ed., The Supercontinuum Laser Source (Springer-Verlag, 1989).

K. P. Hansen, J. R. Jensen, C. Jacobsen, H. R. Simonsen, J. Broeng, P. M. W. Skovgaard, A. Petersson, and A. Bjarklev, "Highly nonlinear photonic crystal fiber with zero-dispersion at 1.55μm," in Optical Fiber Communication Conference (Optical Society of America, 2002), pp. FA91-FA93.

M. H. Frosz, T. Sørensen, and O. Bang, "Nano-engineering of photonic crystal fibers for Supercontinuum generation," in Photonic Crystals and Fibers, W. Urbanczyk, B. Jaskorzynska, and P. St. J. Russell, eds., Proc. SPIE 5950, 185-192 (2005).

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

J. Riishede, "Modelling photonic crystal fibres with the finite difference method," Ph.D. dissertation (Research Center COM, Technical University of Denmark, 2005).

Overview of the BlazePhotonics nonlinear line, http://www.crystal-fibre.com/products/nonlinear.shtm(2005).

K. P. Hansen, Crystal Fiber A/S (personal communication, 2006).

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Figures (9)

Fig. 1
Fig. 1

Examples of three different dispersion curves, calculated for a cobweb structure (see inset) with wall thickness w = 130 nm , pitch Λ = 8.53 μ m , and core size d ranging from 1400 to 1475 nm .

Fig. 2
Fig. 2

Real (solid curve) and imaginary (dashed curve) part of h ̃ ( Ω ) , as given by Eq. (7). The peak of the imaginary part occurs at Ω = Ω R = 2 π × 13.2 THz .

Fig. 3
Fig. 3

Top: Phase mismatch κ for cobweb PCFs with Λ = 8.53 μ m , w = 130 nm , and d = 1400 nm (solid curve), 1450 nm (dashed curve), and 1475 nm (dashed–dotted curve); the pump wavelength is 647 nm , γ = 0.15 ( Wm ) 1 (the slight variation of γ with core size is neglected in this figure), and P 0 = 400 W . The upper and lower horizontal lines at κ = ± 2 γ P 0 ( 1 f R ) = ± 98.4 m 1 indicate the borders of the gain region. For d = 1400 nm , β ¯ 2 < 0 ; for d = 1450 nm , β ¯ 2 0 ; and for d = 1475 nm , β 2 > 0 . Bottom: Corresponding parametric gain given by Eq. (9).

Fig. 4
Fig. 4

Dispersion parameters β ¯ 2 (top) and β ¯ 4 (bottom) calculated for a wide range of two of the cobweb-fiber structural parameters: core size d and wall thickness w are given in the inset.

Fig. 5
Fig. 5

Stokes (top) and anti-Stokes (bottom) gain bandwidth as a function of core size d and wall thickness w. The vertical lines indicate the core size at which β ¯ 2 = 0 for the various wall thicknesses.

Fig. 6
Fig. 6

Stokes (top) and anti-Stokes (bottom) wavelengths for various values of the core size d and the wall thickness w. The vertical lines indicate the core size at which β ¯ 2 = 0 for the various wall thicknesses. The dashed horizontal lines indicate the pump wavelength.

Fig. 7
Fig. 7

Calculated spectra for d = 1425 nm core (top), 1475 nm core (bottom) and w = 130 nm , at z = 0.6 m (solid, lowest curve), z = 0.9 m (dotted curve), and z = 1.2 m (solid, highest curve). The parametric gain g is included (dashed curves) in arbitrary units to show how well the locations of the Stokes and anti-Stokes bands are predicted.

Fig. 8
Fig. 8

Calculated output spectra after 1.2 m of propagation in three different fibers. The wall thickness w = 130 nm .

Fig. 9
Fig. 9

Calculated output spectra after 1.2 m of propagation in the d = 1450 nm , w = 130 nm fiber, without loss in the simulation (solid curve) and including a wavelength-independent loss of 300 dB km (dotted curve).

Equations (19)

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A z = i m 2 i m β ¯ m m ! m A T m + i γ A ( z , T ) R ( T ) A ( z , T T ) 2 d T ,
β ¯ m = β m ( ω 0 ) = ( d m β d ω m ) ω = ω 0 ,
R ( t ) = ( 1 f R ) δ ( t ) + f R h ( t ) = ( 1 f R ) δ ( t ) + f R τ 1 2 + τ 2 2 τ 1 τ 2 2 exp ( t τ 2 ) sin ( t τ 1 ) Θ ( t ) ,
a ( z , T ) = a 1 exp [ i ( K z Ω T ) ] + a 2 exp [ i ( K z Ω T ) ] ,
K = m = 1 β 2 m + 1 ( 2 m + 1 ) ! Ω 2 m + 1 ± [ m = 1 β 2 m ( 2 m ) ! Ω 2 m + 2 γ P 0 ( 1 f R + f R h ̃ ) ] m = 1 β 2 m ( 2 m ) ! Ω 2 m ,
h ̃ ( Ω ) = τ 1 2 + τ 2 2 τ 2 2 τ 1 2 ( i + τ 2 Ω ) 2 .
g ( Ω ) = [ m = 1 β 2 m ( 2 m ) ! Ω 2 m + 2 γ P 0 ( 1 f R ) ] m = 1 β 2 m ( 2 m ) ! Ω 2 m , Ω Ω R ,
m = 1 β 2 m ( 2 m ) ! Ω 0 2 m = γ P 0 ( 1 f R ) , Ω Ω R .
g ( Ω ) = [ γ P 0 ( 1 f R ) ] 2 ( κ 2 ) 2 , Ω Ω R ,
κ = 2 γ P 0 ( 1 f R ) + 2 m = 1 β 2 m ( 2 m ) ! Ω 2 m , Ω Ω R .
β 2 ( ω ) = β ¯ 2 + β ¯ 3 [ ω ω 0 ] + 1 2 β ¯ 4 [ ω ω 0 ] 2 + 1 6 β ¯ 5 [ ω ω 0 ] 3 +
b 2 + b 3 [ ω ω 0 ] + + 1 ( M 2 ) ! b M [ ω ω 0 ] M ,
b ( ω ) = b 0 + b 1 [ ω ω 0 ] + 1 2 b 2 [ ω ω 0 ] 2 + + 1 M ! b M [ ω ω 0 ] M
A z = i m 2 i m β ¯ m m ! m A T m α 2 A + i γ [ 1 + i ω 0 T ] [ A ( z , T ) R ( T ) A ( z , T T ) 2 d T ] .
P ( z ) = A ̃ ( z , ω ) 2 ω d ω ,
S ( λ ) = c λ 2 S ( ν ) = c λ 2 f rep A ̃ ( ν ) 2 ,
g max = 2 γ P 0 ( 1 f R ) , Ω Ω R ,
Ω max 2 = 2 γ P 0 ( 1 f R ) β 2 , Ω Ω R .
Ω max 4 = 24 γ P 0 ( 1 f R ) β 4 , Ω Ω R ,

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