Abstract

In this paper we develop an analytical model for the soliton self-frequency shift, which includes second- and third-order dispersion, self-steepening, the full Raman term, and, for the first time to our best knowledge, the effect of two-photon absorption (TPA). We show that TPA can have a significant effect on soliton dynamics in soft-glass materials such as chalcogenides, by severely depleting a soliton and thereby limiting the achievable redshift. Based on the model, we derive a nonlinear loss length after which the redshift is effectively halted by TPA, which proves to be a useful design tool.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. T. Soifer and J. L. Pipher, “Instrumentation for infrared astronomy,” Ann. Rev. Astron. Astrophys. 16, 335–369 (1978).
    [Crossref]
  2. J. D. Monnie, “Optical interferometry in astronomy,” Rep. Prog. Phys. 66, 789–857 (2003).
    [Crossref]
  3. P. Rolfe, “In vivo near-infrared spectroscopy,” Annu. Rev. Biomed. Eng. 02, 715–754 (2000).
    [Crossref]
  4. B. Guo, Y. Wang, C. Peng, H. L. Zhang, G. P. Luo, H. Q. Le, C. Gmachl, D. L. Sivco, M. L. Peabody, and A. Y. Cho, “Laser-based mid-infrared reflectance imaging of biological tissues,” Opt. Express 12, 208–219 (2004).
    [Crossref]
  5. H. B. Gray, Chemical Bonds: An Introduction to Atomic and Molecular Structure (University Science, 1994).
  6. P. Werle, F. Slemr, K. Maurer, R. Kormann, R. Mucke, and B. Janker, “Near- and mid-infrared laser-optical sensors for gas analysis,” Opt. Lasers Eng. 37, 101–114 (2002).
    [Crossref]
  7. Federation of American Scientists, “AN/AAQ-24 Directional Infrared Countermeasures (DIRCM),” http://www.fas.org/man/dod-101/sys/ac/equip/an-aaq-24.htm .
  8. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
    [Crossref]
  9. N. Savage, “Supercontinuum sources,” Nat. Photon. 3, 114–115 (2009).
    [Crossref]
  10. C. Xia, M. Kumar, O. P. Kulkarni, M. N. Islam, F. L. Terry, M. J. Freeman, M. Poulain, and G. Maz, “Mid-infrared supercontinuum generation to 4.5 μm in ZBLAN fluoride fibers by nanosecond diode pumping,” Opt. Lett. 31, 2553–2555 (2006).
    [Crossref]
  11. G. Qin, X. Yan, C. Kito, M. Liao, C. Chaudhari, T. Suzuki, and Y. Ohishi, “Ultrabroadband supercontinuum generation from ultraviolet to 6.28 μm in a fluoride fiber,” Appl. Phys. Lett. 95, 161103 (2009).
    [Crossref]
  12. J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
    [Crossref]
  13. P. Domachuk, N. A. Wolchover, M. Cronin-Golomb, A. Wang, A. K. George, C. M. B. 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]
  14. D. Buccoliero, H. Steffensen, O. Bang, H. Ebendorff-Heidepriem, and T. M. Monro, “Thulium pumped high power supercontinuum in loss-determined optimum lengths of tellurite photonic crystal fiber,” Appl. Phys. Lett. 97, 061106 (2010).
    [Crossref]
  15. D.-I. Yeom, E. C. Mägi, 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]
  16. D. D. Hudson, S. A. Dekker, E. C. Mägi, A. C. Judge, S. D. Jackson, E. Li, J. S. Sanghera, L. B. Shaw, I. D. Aggarwal, and B. J. Eggleton, “Octave spanning supercontinuum in an As2S3 taper using ultralow pump pulse energy,” Opt. Lett. 36, 1122–1124 (2011).
    [Crossref]
  17. T. M. Monro and H. Ebendorff-Heidepriem, “Progress in microstrusctured optical fibers,” Ann. Rev. Mater. Res. 36, 467–495 (2006).
    [Crossref]
  18. P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
    [Crossref]
  19. W. Q. Zhang, S. Afshar V., 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]
  20. T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibers,” Opt. Lett. 25, 1415–1417 (2000).
    [Crossref]
  21. 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, 5715–5722 (2006).
    [Crossref]
  22. S. T. Sørensen, A. Judge, C. L. Thomsen, and O. Bang, “Optimum tapers for increasing the power in the blue-edge of a supercontinuum—group-acceleration matching,” Opt. Lett. 36, 816–818 (2011).
    [Crossref]
  23. C. Agger, S. T. Sørensen, C. L. Thomsen, S. R. Keiding, and O. Bang, “Nonlinear soliton matching between optical fibers,” Opt. Lett. 36, 2596–2598 (2011).
    [Crossref]
  24. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
    [Crossref]
  25. E. N. Tsoy and C. M. de Sterke, “Dynamics of ultrashort pulses near zero dispersion wavelength,” J. Opt. Soc. Am. B 23, 2425–2433 (2006).
    [Crossref]
  26. A. M. Zheltikov, “Perturbative analytical treatment of adiabatically moderated soliton self-frequency shift,” Phys. Rev. E 75, 037603 (2007).
    [Crossref]
  27. A. C. Judge, O. Bang, B. J. Eggleton, B. T. Kuhlmey, E. C. Mägi, R. Pant, and C. Martijn de Sterke, “Optimization of the soliton self-frequency shift in a tapered photonic crystal fiber,” J. Opt. Soc. Am. B 26, 2064–2071 (2009).
    [Crossref]
  28. A. C. Judge, O. Bang, and C. Martijn de Sterke, “Theory of dispersive wave frequency shift via trapping by a soliton in an axially nonuniform optical fiber,” J. Opt. Soc. Am. B 27, 2195–2202 (2010).
    [Crossref]
  29. Z. Chen, A. J. Taylor, and A. Efimov, “Soliton dynamics in non-uniform fiber tapers: analytical description through an improved moment method,” J. Opt. Soc. Am. B 27, 1022–1030 (2010).
    [Crossref]
  30. L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32, 391–393 (2007).
    [Crossref]
  31. H. Nguyen, K. Finsterbusch, D. Moss, and B. Eggleton, “Dispersion in nonlinear figure of merit of As2Se3 chalcogenide fibre,” Electron. Lett. 42, 571–572 (2006).
    [Crossref]
  32. A. I. Maimistov, “Evolution of single waves close to solitons of Schrödinger nonlinear equation,” J. Exp. Theor. Phys. 77, 727–731 (1993) [Zh. Eksp. Teor. Fiz. 104, 3620–3629 (1993)].
  33. W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
    [Crossref]
  34. Q. Kong, Q. Wang, O. Bang, and W. Krolikowski, “Analytical theory for dark nonlocal solitons,” Opt. Lett. 35, 2152–2154 (2010).
    [Crossref]
  35. E. N. Tsoy, A. Ankiewicz, and N. Akhmediev, “Dynamical models for dissipative localized waves of the complex Ginzburg–Landau equation,” Phys. Rev. E 73, 036621 (2006).
    [Crossref]
  36. C. Petersen, S. Dupont, C. Agger, J. Thøgersen, O. Bang, and S. Keiding, “Stimulated Raman scattering in soft glass fluoride fibers,” J. Opt. Soc. Am. B 28, 2310–2313 (2011).
    [Crossref]
  37. P. Liu, W. L. Smith, H. Lotem, J. H. Bechtel, N. Bloembergen, and R. S. Adhav, “Absolute two-photon absorption coefficients at 355 and 266 nm,” Phys. Rev. B 17, 4620–4632 (1978).
    [Crossref]
  38. 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]

2011 (4)

2010 (5)

2009 (4)

2008 (2)

2007 (3)

J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[Crossref]

L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32, 391–393 (2007).
[Crossref]

A. M. Zheltikov, “Perturbative analytical treatment of adiabatically moderated soliton self-frequency shift,” Phys. Rev. E 75, 037603 (2007).
[Crossref]

2006 (7)

E. N. Tsoy and C. M. de Sterke, “Dynamics of ultrashort pulses near zero dispersion wavelength,” J. Opt. Soc. Am. B 23, 2425–2433 (2006).
[Crossref]

E. N. Tsoy, A. Ankiewicz, and N. Akhmediev, “Dynamical models for dissipative localized waves of the complex Ginzburg–Landau equation,” Phys. Rev. E 73, 036621 (2006).
[Crossref]

H. Nguyen, K. Finsterbusch, D. Moss, and B. Eggleton, “Dispersion in nonlinear figure of merit of As2Se3 chalcogenide fibre,” Electron. Lett. 42, 571–572 (2006).
[Crossref]

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, 5715–5722 (2006).
[Crossref]

T. M. Monro and H. Ebendorff-Heidepriem, “Progress in microstrusctured optical fibers,” Ann. Rev. Mater. Res. 36, 467–495 (2006).
[Crossref]

C. Xia, M. Kumar, O. P. Kulkarni, M. N. Islam, F. L. Terry, M. J. Freeman, M. Poulain, and G. Maz, “Mid-infrared supercontinuum generation to 4.5 μm in ZBLAN fluoride fibers by nanosecond diode pumping,” Opt. Lett. 31, 2553–2555 (2006).
[Crossref]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

2004 (2)

B. Guo, Y. Wang, C. Peng, H. L. Zhang, G. P. Luo, H. Q. Le, C. Gmachl, D. L. Sivco, M. L. Peabody, and A. Y. Cho, “Laser-based mid-infrared reflectance imaging of biological tissues,” Opt. Express 12, 208–219 (2004).
[Crossref]

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

2003 (2)

J. D. Monnie, “Optical interferometry in astronomy,” Rep. Prog. Phys. 66, 789–857 (2003).
[Crossref]

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

2002 (1)

P. Werle, F. Slemr, K. Maurer, R. Kormann, R. Mucke, and B. Janker, “Near- and mid-infrared laser-optical sensors for gas analysis,” Opt. Lasers Eng. 37, 101–114 (2002).
[Crossref]

2000 (2)

1993 (1)

A. I. Maimistov, “Evolution of single waves close to solitons of Schrödinger nonlinear equation,” J. Exp. Theor. Phys. 77, 727–731 (1993) [Zh. Eksp. Teor. Fiz. 104, 3620–3629 (1993)].

1986 (1)

1978 (2)

P. Liu, W. L. Smith, H. Lotem, J. H. Bechtel, N. Bloembergen, and R. S. Adhav, “Absolute two-photon absorption coefficients at 355 and 266 nm,” Phys. Rev. B 17, 4620–4632 (1978).
[Crossref]

B. T. Soifer and J. L. Pipher, “Instrumentation for infrared astronomy,” Ann. Rev. Astron. Astrophys. 16, 335–369 (1978).
[Crossref]

Adhav, R. S.

P. Liu, W. L. Smith, H. Lotem, J. H. Bechtel, N. Bloembergen, and R. S. Adhav, “Absolute two-photon absorption coefficients at 355 and 266 nm,” Phys. Rev. B 17, 4620–4632 (1978).
[Crossref]

Afshar V., S.

Aggarwal, I. D.

Agger, C.

Agrawal, G. P.

Akhmediev, N.

E. N. Tsoy, A. Ankiewicz, and N. Akhmediev, “Dynamical models for dissipative localized waves of the complex Ginzburg–Landau equation,” Phys. Rev. E 73, 036621 (2006).
[Crossref]

Ankiewicz, A.

E. N. Tsoy, A. Ankiewicz, and N. Akhmediev, “Dynamical models for dissipative localized waves of the complex Ginzburg–Landau equation,” Phys. Rev. E 73, 036621 (2006).
[Crossref]

Bang, O.

C. Petersen, S. Dupont, C. Agger, J. Thøgersen, O. Bang, and S. Keiding, “Stimulated Raman scattering in soft glass fluoride fibers,” J. Opt. Soc. Am. B 28, 2310–2313 (2011).
[Crossref]

C. Agger, S. T. Sørensen, C. L. Thomsen, S. R. Keiding, and O. Bang, “Nonlinear soliton matching between optical fibers,” Opt. Lett. 36, 2596–2598 (2011).
[Crossref]

S. T. Sørensen, A. Judge, C. L. Thomsen, and O. Bang, “Optimum tapers for increasing the power in the blue-edge of a supercontinuum—group-acceleration matching,” Opt. Lett. 36, 816–818 (2011).
[Crossref]

D. Buccoliero, H. Steffensen, O. Bang, H. Ebendorff-Heidepriem, and T. M. Monro, “Thulium pumped high power supercontinuum in loss-determined optimum lengths of tellurite photonic crystal fiber,” Appl. Phys. Lett. 97, 061106 (2010).
[Crossref]

Q. Kong, Q. Wang, O. Bang, and W. Krolikowski, “Analytical theory for dark nonlocal solitons,” Opt. Lett. 35, 2152–2154 (2010).
[Crossref]

A. C. Judge, O. Bang, and C. Martijn de Sterke, “Theory of dispersive wave frequency shift via trapping by a soliton in an axially nonuniform optical fiber,” J. Opt. Soc. Am. B 27, 2195–2202 (2010).
[Crossref]

A. C. Judge, O. Bang, B. J. Eggleton, B. T. Kuhlmey, E. C. Mägi, R. Pant, and C. Martijn de Sterke, “Optimization of the soliton self-frequency shift in a tapered photonic crystal fiber,” J. Opt. Soc. Am. B 26, 2064–2071 (2009).
[Crossref]

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

Bechtel, J. H.

P. Liu, W. L. Smith, H. Lotem, J. H. Bechtel, N. Bloembergen, and R. S. Adhav, “Absolute two-photon absorption coefficients at 355 and 266 nm,” Phys. Rev. B 17, 4620–4632 (1978).
[Crossref]

Birks, T. A.

Bloembergen, N.

P. Liu, W. L. Smith, H. Lotem, J. H. Bechtel, N. Bloembergen, and R. S. Adhav, “Absolute two-photon absorption coefficients at 355 and 266 nm,” Phys. Rev. B 17, 4620–4632 (1978).
[Crossref]

Brambilla, G.

J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[Crossref]

Buccoliero, D.

D. Buccoliero, H. Steffensen, O. Bang, H. Ebendorff-Heidepriem, and T. M. Monro, “Thulium pumped high power supercontinuum in loss-determined optimum lengths of tellurite photonic crystal fiber,” Appl. Phys. Lett. 97, 061106 (2010).
[Crossref]

Chaudhari, C.

G. Qin, X. Yan, C. Kito, M. Liao, C. Chaudhari, T. Suzuki, and Y. Ohishi, “Ultrabroadband supercontinuum generation from ultraviolet to 6.28 μm in a fluoride fiber,” Appl. Phys. Lett. 95, 161103 (2009).
[Crossref]

Chen, Z.

Cho, A. Y.

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

Cordeiro, C. M. B.

Cronin-Golomb, M.

de Sterke, C. M.

de Sterke, C. Martijn

Dekker, S. A.

Domachuk, P.

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

Dupont, S.

Ebendorff-Heidepriem, H.

D. Buccoliero, H. Steffensen, O. Bang, H. Ebendorff-Heidepriem, and T. M. Monro, “Thulium pumped high power supercontinuum in loss-determined optimum lengths of tellurite photonic crystal fiber,” Appl. Phys. Lett. 97, 061106 (2010).
[Crossref]

J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[Crossref]

T. M. Monro and H. Ebendorff-Heidepriem, “Progress in microstrusctured optical fibers,” Ann. Rev. Mater. Res. 36, 467–495 (2006).
[Crossref]

Edmundson, D.

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

Efimov, A.

Eggleton, B.

H. Nguyen, K. Finsterbusch, D. Moss, and B. Eggleton, “Dispersion in nonlinear figure of merit of As2Se3 chalcogenide fibre,” Electron. Lett. 42, 571–572 (2006).
[Crossref]

Eggleton, B. J.

Feng, X.

J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[Crossref]

Finazzi, V.

J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[Crossref]

Finsterbusch, K.

H. Nguyen, K. Finsterbusch, D. Moss, and B. Eggleton, “Dispersion in nonlinear figure of merit of As2Se3 chalcogenide fibre,” Electron. Lett. 42, 571–572 (2006).
[Crossref]

Flanagan, J. C.

J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[Crossref]

Freeman, M. J.

Fu, L.

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.

Gmachl, C.

Gordon, J. P.

Gray, H. B.

H. B. Gray, Chemical Bonds: An Introduction to Atomic and Molecular Structure (University Science, 1994).

Guo, B.

Horak, P.

J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[Crossref]

Hu, J.

Hudson, D. D.

Islam, M. N.

Jackson, S. D.

Janker, B.

P. Werle, F. Slemr, K. Maurer, R. Kormann, R. Mucke, and B. Janker, “Near- and mid-infrared laser-optical sensors for gas analysis,” Opt. Lasers Eng. 37, 101–114 (2002).
[Crossref]

Judge, A.

Judge, A. C.

Keiding, S.

Keiding, S. R.

Kito, C.

G. Qin, X. Yan, C. Kito, M. Liao, C. Chaudhari, T. Suzuki, and Y. Ohishi, “Ultrabroadband supercontinuum generation from ultraviolet to 6.28 μm in a fluoride fiber,” Appl. Phys. Lett. 95, 161103 (2009).
[Crossref]

Knight, J. C.

Kong, Q.

Kormann, R.

P. Werle, F. Slemr, K. Maurer, R. Kormann, R. Mucke, and B. Janker, “Near- and mid-infrared laser-optical sensors for gas analysis,” Opt. Lasers Eng. 37, 101–114 (2002).
[Crossref]

Krolikowski, W.

Q. Kong, Q. Wang, O. Bang, and W. Krolikowski, “Analytical theory for dark nonlocal solitons,” Opt. Lett. 35, 2152–2154 (2010).
[Crossref]

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

Kudlinski, A.

Kuhlmey, B. T.

Kulkarni, O. P.

Kumar, M.

Lamont, M. R. E.

Le, H. Q.

Leong, J. Y. Y.

J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[Crossref]

Li, E.

Liao, M.

G. Qin, X. Yan, C. Kito, M. Liao, C. Chaudhari, T. Suzuki, and Y. Ohishi, “Ultrabroadband supercontinuum generation from ultraviolet to 6.28 μm in a fluoride fiber,” Appl. Phys. Lett. 95, 161103 (2009).
[Crossref]

Lin, Q.

Liu, P.

P. Liu, W. L. Smith, H. Lotem, J. H. Bechtel, N. Bloembergen, and R. S. Adhav, “Absolute two-photon absorption coefficients at 355 and 266 nm,” Phys. Rev. B 17, 4620–4632 (1978).
[Crossref]

Lotem, H.

P. Liu, W. L. Smith, H. Lotem, J. H. Bechtel, N. Bloembergen, and R. S. Adhav, “Absolute two-photon absorption coefficients at 355 and 266 nm,” Phys. Rev. B 17, 4620–4632 (1978).
[Crossref]

Luo, G. P.

Mägi, E. C.

Maimistov, A. I.

A. I. Maimistov, “Evolution of single waves close to solitons of Schrödinger nonlinear equation,” J. Exp. Theor. Phys. 77, 727–731 (1993) [Zh. Eksp. Teor. Fiz. 104, 3620–3629 (1993)].

Maurer, K.

P. Werle, F. Slemr, K. Maurer, R. Kormann, R. Mucke, and B. Janker, “Near- and mid-infrared laser-optical sensors for gas analysis,” Opt. Lasers Eng. 37, 101–114 (2002).
[Crossref]

Maz, G.

Menyuk, C. R.

Monnie, J. D.

J. D. Monnie, “Optical interferometry in astronomy,” Rep. Prog. Phys. 66, 789–857 (2003).
[Crossref]

Monro, T. M.

D. Buccoliero, H. Steffensen, O. Bang, H. Ebendorff-Heidepriem, and T. M. Monro, “Thulium pumped high power supercontinuum in loss-determined optimum lengths of tellurite photonic crystal fiber,” Appl. Phys. Lett. 97, 061106 (2010).
[Crossref]

W. Q. Zhang, S. Afshar V., 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. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[Crossref]

T. M. Monro and H. Ebendorff-Heidepriem, “Progress in microstrusctured optical fibers,” Ann. Rev. Mater. Res. 36, 467–495 (2006).
[Crossref]

Moss, D.

H. Nguyen, K. Finsterbusch, D. Moss, and B. Eggleton, “Dispersion in nonlinear figure of merit of As2Se3 chalcogenide fibre,” Electron. Lett. 42, 571–572 (2006).
[Crossref]

Mucke, R.

P. Werle, F. Slemr, K. Maurer, R. Kormann, R. Mucke, and B. Janker, “Near- and mid-infrared laser-optical sensors for gas analysis,” Opt. Lasers Eng. 37, 101–114 (2002).
[Crossref]

Neshev, D.

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

Nguyen, H.

H. Nguyen, K. Finsterbusch, D. Moss, and B. Eggleton, “Dispersion in nonlinear figure of merit of As2Se3 chalcogenide fibre,” Electron. Lett. 42, 571–572 (2006).
[Crossref]

Nikolov, N.

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

Ohishi, Y.

G. Qin, X. Yan, C. Kito, M. Liao, C. Chaudhari, T. Suzuki, and Y. Ohishi, “Ultrabroadband supercontinuum generation from ultraviolet to 6.28 μm in a fluoride fiber,” Appl. Phys. Lett. 95, 161103 (2009).
[Crossref]

Omenetto, F. G.

Pant, R.

Peabody, M. L.

Peng, C.

Petersen, C.

Petropoulos, P.

J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[Crossref]

Pipher, J. L.

B. T. Soifer and J. L. Pipher, “Instrumentation for infrared astronomy,” Ann. Rev. Astron. Astrophys. 16, 335–369 (1978).
[Crossref]

Poletti, F.

J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[Crossref]

Popov, S. V.

Poulain, M.

Price, J. H. V.

J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[Crossref]

Qin, G.

G. Qin, X. Yan, C. Kito, M. Liao, C. Chaudhari, T. Suzuki, and Y. Ohishi, “Ultrabroadband supercontinuum generation from ultraviolet to 6.28 μm in a fluoride fiber,” Appl. Phys. Lett. 95, 161103 (2009).
[Crossref]

Rasmussen, J.

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

Richardson, D. J.

J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[Crossref]

Roelens, M. A. F.

Rolfe, P.

P. Rolfe, “In vivo near-infrared spectroscopy,” Annu. Rev. Biomed. Eng. 02, 715–754 (2000).
[Crossref]

Rulkov, A. B.

Russell, P.

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

Sanghera, J. S.

Savage, N.

N. Savage, “Supercontinuum sources,” Nat. Photon. 3, 114–115 (2009).
[Crossref]

Shaw, L. B.

Sivco, D. L.

Slemr, F.

P. Werle, F. Slemr, K. Maurer, R. Kormann, R. Mucke, and B. Janker, “Near- and mid-infrared laser-optical sensors for gas analysis,” Opt. Lasers Eng. 37, 101–114 (2002).
[Crossref]

Smith, W. L.

P. Liu, W. L. Smith, H. Lotem, J. H. Bechtel, N. Bloembergen, and R. S. Adhav, “Absolute two-photon absorption coefficients at 355 and 266 nm,” Phys. Rev. B 17, 4620–4632 (1978).
[Crossref]

Soifer, B. T.

B. T. Soifer and J. L. Pipher, “Instrumentation for infrared astronomy,” Ann. Rev. Astron. Astrophys. 16, 335–369 (1978).
[Crossref]

Sørensen, S. T.

St. J. Russell, P.

Steffensen, H.

D. Buccoliero, H. Steffensen, O. Bang, H. Ebendorff-Heidepriem, and T. M. Monro, “Thulium pumped high power supercontinuum in loss-determined optimum lengths of tellurite photonic crystal fiber,” Appl. Phys. Lett. 97, 061106 (2010).
[Crossref]

Suzuki, T.

G. Qin, X. Yan, C. Kito, M. Liao, C. Chaudhari, T. Suzuki, and Y. Ohishi, “Ultrabroadband supercontinuum generation from ultraviolet to 6.28 μm in a fluoride fiber,” Appl. Phys. Lett. 95, 161103 (2009).
[Crossref]

Taylor, A. J.

Taylor, J. R.

Terry, F. L.

Thøgersen, J.

Thomsen, C. L.

Travers, J. C.

Tsoy, E. N.

E. N. Tsoy and C. M. de Sterke, “Dynamics of ultrashort pulses near zero dispersion wavelength,” J. Opt. Soc. Am. B 23, 2425–2433 (2006).
[Crossref]

E. N. Tsoy, A. Ankiewicz, and N. Akhmediev, “Dynamical models for dissipative localized waves of the complex Ginzburg–Landau equation,” Phys. Rev. E 73, 036621 (2006).
[Crossref]

Wadsworth, W. J.

Wang, A.

Wang, Q.

Wang, Y.

Werle, P.

P. Werle, F. Slemr, K. Maurer, R. Kormann, R. Mucke, and B. Janker, “Near- and mid-infrared laser-optical sensors for gas analysis,” Opt. Lasers Eng. 37, 101–114 (2002).
[Crossref]

Wolchover, N. A.

Wyller, J.

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

Xia, C.

Yan, X.

G. Qin, X. Yan, C. Kito, M. Liao, C. Chaudhari, T. Suzuki, and Y. Ohishi, “Ultrabroadband supercontinuum generation from ultraviolet to 6.28 μm in a fluoride fiber,” Appl. Phys. Lett. 95, 161103 (2009).
[Crossref]

Yeom, D.-I.

Yin, L.

Zhang, H. L.

Zhang, W. Q.

Zheltikov, A. M.

A. M. Zheltikov, “Perturbative analytical treatment of adiabatically moderated soliton self-frequency shift,” Phys. Rev. E 75, 037603 (2007).
[Crossref]

Ann. Rev. Astron. Astrophys. (1)

B. T. Soifer and J. L. Pipher, “Instrumentation for infrared astronomy,” Ann. Rev. Astron. Astrophys. 16, 335–369 (1978).
[Crossref]

Ann. Rev. Mater. Res. (1)

T. M. Monro and H. Ebendorff-Heidepriem, “Progress in microstrusctured optical fibers,” Ann. Rev. Mater. Res. 36, 467–495 (2006).
[Crossref]

Annu. Rev. Biomed. Eng. (1)

P. Rolfe, “In vivo near-infrared spectroscopy,” Annu. Rev. Biomed. Eng. 02, 715–754 (2000).
[Crossref]

Appl. Phys. Lett. (2)

G. Qin, X. Yan, C. Kito, M. Liao, C. Chaudhari, T. Suzuki, and Y. Ohishi, “Ultrabroadband supercontinuum generation from ultraviolet to 6.28 μm in a fluoride fiber,” Appl. Phys. Lett. 95, 161103 (2009).
[Crossref]

D. Buccoliero, H. Steffensen, O. Bang, H. Ebendorff-Heidepriem, and T. M. Monro, “Thulium pumped high power supercontinuum in loss-determined optimum lengths of tellurite photonic crystal fiber,” Appl. Phys. Lett. 97, 061106 (2010).
[Crossref]

Electron. Lett. (1)

H. Nguyen, K. Finsterbusch, D. Moss, and B. Eggleton, “Dispersion in nonlinear figure of merit of As2Se3 chalcogenide fibre,” Electron. Lett. 42, 571–572 (2006).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

J. H. V. Price, T. M. Monro, H. Ebendorff-Heidepriem, F. Poletti, P. Horak, V. Finazzi, J. Y. Y. Leong, P. Petropoulos, J. C. Flanagan, G. Brambilla, X. Feng, and D. J. Richardson, “Mid-IR supercontinuum generation from nonsilica microstructured optical fibers,” IEEE J. Sel. Top. Quantum Electron. 13, 738–749 (2007).
[Crossref]

J. Exp. Theor. Phys. (1)

A. I. Maimistov, “Evolution of single waves close to solitons of Schrödinger nonlinear equation,” J. Exp. Theor. Phys. 77, 727–731 (1993) [Zh. Eksp. Teor. Fiz. 104, 3620–3629 (1993)].

J. Opt. B (1)

W. Krolikowski, O. Bang, N. Nikolov, D. Neshev, J. Wyller, J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in nonlocal nonlinear media,” J. Opt. B 6, S288–S294 (2004).
[Crossref]

J. Opt. Soc. Am. B (5)

Nat. Photon. (1)

N. Savage, “Supercontinuum sources,” Nat. Photon. 3, 114–115 (2009).
[Crossref]

Opt. Express (5)

Opt. Lasers Eng. (1)

P. Werle, F. Slemr, K. Maurer, R. Kormann, R. Mucke, and B. Janker, “Near- and mid-infrared laser-optical sensors for gas analysis,” Opt. Lasers Eng. 37, 101–114 (2002).
[Crossref]

Opt. Lett. (9)

C. Xia, M. Kumar, O. P. Kulkarni, M. N. Islam, F. L. Terry, M. J. Freeman, M. Poulain, and G. Maz, “Mid-infrared supercontinuum generation to 4.5 μm in ZBLAN fluoride fibers by nanosecond diode pumping,” Opt. Lett. 31, 2553–2555 (2006).
[Crossref]

T. A. Birks, W. J. Wadsworth, and P. St. J. Russell, “Supercontinuum generation in tapered fibers,” Opt. Lett. 25, 1415–1417 (2000).
[Crossref]

D.-I. Yeom, E. C. Mägi, 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]

D. D. Hudson, S. A. Dekker, E. C. Mägi, A. C. Judge, S. D. Jackson, E. Li, J. S. Sanghera, L. B. Shaw, I. D. Aggarwal, and B. J. Eggleton, “Octave spanning supercontinuum in an As2S3 taper using ultralow pump pulse energy,” Opt. Lett. 36, 1122–1124 (2011).
[Crossref]

S. T. Sørensen, A. Judge, C. L. Thomsen, and O. Bang, “Optimum tapers for increasing the power in the blue-edge of a supercontinuum—group-acceleration matching,” Opt. Lett. 36, 816–818 (2011).
[Crossref]

C. Agger, S. T. Sørensen, C. L. Thomsen, S. R. Keiding, and O. Bang, “Nonlinear soliton matching between optical fibers,” Opt. Lett. 36, 2596–2598 (2011).
[Crossref]

J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986).
[Crossref]

L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32, 391–393 (2007).
[Crossref]

Q. Kong, Q. Wang, O. Bang, and W. Krolikowski, “Analytical theory for dark nonlocal solitons,” Opt. Lett. 35, 2152–2154 (2010).
[Crossref]

Phys. Rev. B (1)

P. Liu, W. L. Smith, H. Lotem, J. H. Bechtel, N. Bloembergen, and R. S. Adhav, “Absolute two-photon absorption coefficients at 355 and 266 nm,” Phys. Rev. B 17, 4620–4632 (1978).
[Crossref]

Phys. Rev. E (2)

A. M. Zheltikov, “Perturbative analytical treatment of adiabatically moderated soliton self-frequency shift,” Phys. Rev. E 75, 037603 (2007).
[Crossref]

E. N. Tsoy, A. Ankiewicz, and N. Akhmediev, “Dynamical models for dissipative localized waves of the complex Ginzburg–Landau equation,” Phys. Rev. E 73, 036621 (2006).
[Crossref]

Rep. Prog. Phys. (1)

J. D. Monnie, “Optical interferometry in astronomy,” Rep. Prog. Phys. 66, 789–857 (2003).
[Crossref]

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]

Science (1)

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

Other (2)

H. B. Gray, Chemical Bonds: An Introduction to Atomic and Molecular Structure (University Science, 1994).

Federation of American Scientists, “AN/AAQ-24 Directional Infrared Countermeasures (DIRCM),” http://www.fas.org/man/dod-101/sys/ac/equip/an-aaq-24.htm .

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

Center wavelength (top) and peak power (bottom) as a function of propagation distance for (a) T=100fs and (b) T=25fs pulses. It shows that there is a good agreement between our model (solid) and the full simulation of Eq. (1) (dashed) and the model of Chen et al. (dashed–dotted).

Fig. 2.
Fig. 2.

Center wavelength (top) and peak power (bottom) as a function of propagation distance for a T=10fs pulse. Because of the limit of the choice of a soliton ansatz for short pulses, there is a greater disagreement between the full simulation of Eq. (1) (dashed) to both our model (solid) and the model of Chen et al. (dashed–dotted).

Fig. 3.
Fig. 3.

Center wavelength (top) and peak power (bottom) for our model (solid) and simulation of Eq. (1) (dashed) for a T=50fs pulse with three different values of the TPA. The vertical lines in the lower plot mark the nonlinear loss length as given by Eq. (9).

Fig. 4.
Fig. 4.

Evolution of center wavelength (top) and peak power (bottom) for a chalcogenide fiber for cases where TPA is (a) neglected and (b) included as found by our model (solid) and the simulation of Eq. (1) (dashed). The nonlinear loss length is marked by the vertical lines in (b).

Fig. 5.
Fig. 5.

(a) An and (b) Bn functions used in Eq. (B3).

Fig. 6.
Fig. 6.

Value of the integrals over (a) An and (b) Bn functions from Eq. (B3) as a function of pulse width. The integral over An has been multiplied with fRT to show that the integral goes toward TR/(fRT) when the pulse width is increased.

Tables (1)

Tables Icon

Table 1. Used Values of the Nonlinearity and Two-Photon Absorption for a As2Se3 Chalcogenide Fiber by Nguyen et al. [31]

Equations (63)

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

zu(z,t)+α2u(z,t)im2imβmm!tmu(z,t)=in0inγnn!tn{u(z,t)R(t)|u(z,tt)|2dt},
γ0=ω0n2cAeff+iβTPA2Aeff,
u(z,t)=P(z)sech(ttc(z)T(z))exp[iΦ(z)ib(z)(ttc(z))iμ(z)(ttc(z))2],
Q=|u|2dt,
PM=12(utu*u*tu)dt,
I1=t|u|2dt,
I2=(ttc)2|u|2dt,
I3=(ttc)(u*tuutu*)dt,
zb815(γ0r+bγ1r)PTRT2+43μγ1rP43μ(γ0i+bγ1i)PTR45γ1iPT24(π2923)μ2γ1iPT2,
ztcbβ2+12β3[π23μ2T2+13T2+b2]+γ1rP23(γ0i+bγ1i)PTR2(π2923)μγ1iPT2,
zT2μT(β2+bβ3)+8π2γ1rPTRT+4π2(γ0i+bγ1i)PT(4312π2)μγ1iPTTR,
zPαP2μP(β2+bβ3)(8π2+815)γ1rP2TRT2(43+4π2)(γ0i+bγ1i)P212π2μγ1iP2TR,
zμ(2π21T42μ2)(β2+bβ3)+2π2(γ0r+bγ1r)PT2(8154π2)μγ1rPTRT27615π2γ1iPTRT4,
TR=fRthR(t)dt,
zb815γ0rPTRT2+43μγ1rP43μγ0iPTR45γ1iPT24(π2923)μ2γ1iPT2,
ztc12β3[π23μ2T2+13T2]+γ1rP23γ0iPTR2(π2923)μγ1iPT2,
zT2μTβ2+8π2γ1rPTRT+4π2γ0iPT(4312π2)μγ1iPTTR,
zPαP2μPβ2(8π2+815)γ1rP2TRT2(43+4π2)γ0iP212π2μγ1iP2TR,
zμ(2π21T42μ2)β2+2π2γ0rPT2(8154π2)μγ1rPTRT27615π2γ1iPTRT4.
zP=αP(43+4π2)γ0iP2,
zT=8π2γ1rPTRT+4π2γ0iPT,
zb=815γ0rPTRT2.
LLoss=1αln[e+(43+4π2)γ0iP0α1+(43+4π2)γ0iP0α],
hR(t)=τ12+τ22τ1τ22exp(tτ2)sin(tτ1),
zb=815(γ0r+bγ1r)PTfRhR(t)A1(tT)dt+43μγ1rP[1fR+fRhR(t)B1(tT)dt]43μ(γ0i+bγ1i)PTfRhR(t)A2(tT)dt45γ1iPT2[1fR+fRhR(t)B2(tT)dt]4(π2923)μ2γ1iPT2[1fR+fRhR(t)B3(tT)dt],
ztc=b(β2+bβ3)+12β3[π23μ2T2+13T2b2]+γ1rP[1fR+fRhR(t)B4(tT)dt]23(γ0i+bγ1i)PTfRhR(t)A2(tT)dt2(π2923)μγ1iPT2[1fR+fRhR(t)B3(tT)dt],
zT=2μT(β2+bβ3)+8π2γ1rPfRhR(t)A3(tT)dt+4π2(γ0i+bγ1i)PT[1fR+fRhR(t)B5(tT)dt](4312π2)μγ1iPT2fRhR(t)A4(tT)dt,
zP=αP2μP(β2+bβ3)(8π2+815)γ1rP2TfRhR(t)A5(tT)dt(43+4π2)(γ0i+bγ1i)P2[1fR+fRhR(t)B6(tT)dt]12π2μγ1iP2TfRhR(t)A6(tT)dt,
zμ=(β2+bβ3)(2π21T42μ2)+2π2(γ0r+bγ1r)PT2[1fR+fRhR(t)B7(tT)dt](8154π2)μγ1rPTfRhR(t)A7(tT)dt7615π2γ1iPT3fRhR(t)A8(tT)dt.
A1(x)=158csch4(x)[4x+2xcosh(2x)3sinh(2x)],
A2(x)=3xcsch3(x)[xcosh(x)sinh(x)],
A3(x)=14csch4(x)[(2x3+6x)cosh(2x)+9x2sinh(2x)4x3+6x],
A4(x)=3π29xcsch3(x)[(3x3+π2x)cosh(x)(6x2+π2)sinh(x)],
A5(x)=158π2+120csch4(x)[(18x23π2)sinh(2x)(4x3+12x2π2x)cosh(2x)8x3+12x+4π2x],
A6(x)=1304csch5(x)[(68π2x2180x2+75π2450)cosh(3x)(158π2x720x)sinh(3x)+(412π2x22700x275π2+450)cosh(x)(306π2x2520x)sinh(x)],
A7(x)=158π260csch4(x)[16x3+4π2x6x+(8x3+2π2x+6x)cosh(2x)(18x2+3π2)sinh(2x)],
A8(x)=45152csch5(x)[8xsinh(3x)+28xsinh(x)(2x2+5)cosh(3x)(30x25)cosh(x)],
B1(x)=32csch4(x)[(6x21)cosh(2x)8xsinh(2x)+12x2+1],
B2(x)=512csch5(x)[7sinh(3x)+27sinh(x)3xcosh(3x)45xcosh(x)],
B3(x)=3π26csch3(x)[(4x3+π2x)cosh(x)(6x2+π2)sinh(x)],
B4(x)=136csch5(x)[(18x2+61)sinh(3x)+(54x2+297)sinh(x)48xcosh(3x)432xcosh(x)],
B5(x)=csch3(x)[3x2sinh(x)2x3cosh(x)],
B6(x)=3π2+3csch3(x)[(3x2π2)sinh(x)(2x3π2x)cosh(x)],
B7(x)=32csch4(x)[4xsinh(2x)(2x2+1)cosh(2x)4x2+1],
A5(x)=15π2+15[π215A1(x)+A3(x)],
A8(x)=1π26[π2A2(x)6A6(x)],
B6(x)=1π2+3[9B5(x)+(π26)B3(x)],
B7(x)=2B2(x)+2π212π2B3(x)3B4(x)+12π2B5(x).
zb=815(γ0r+bγ1r)PTfRhR(t)A1(tT)dt+43μγ1rP[1fR+fRhR(t)B1(tT)dt],
ztc=b(β2+bβ3)+12β3[π23μ2T2+13T2b2]+γ1rP[1fR+fRhR(t)B4(tT)dt],
zT=2μT(β2+bβ3)+8π2γ1rPfRhR(t)A3(tT)dt,
zP=αP2μP(β2+bβ3)(8π2+815)γ1rP2TfRhR(t)A5(tT)dt,
zμ=(β2+bβ3)(2π21T42μ2)+2π2(γ0r+bγ1r)PT2[1fR+fRhR(t)B7(tT)dt](8154π2)μγ1rPTfRhR(t)A7(tT)dt.
E=2PT,
C=μ2T2.
zb=415(γ0r+bγ1r)ET2fRhR(t)A1(tT)dt+13γ1rECT3[1fR+fRhR(t)B1(tT)dt],
ztc=b(β2+bβ3)+12β3[π212C2T2+13T2b2]+12γ1rET[1fR+fRhR(t)B4(tT)dt],
zT=(β2+bβ3)CT+4π2γ1rETfRhR(t)A3(tT)dt,
zE=αE415γ1rE2T2fRhR(t)A1(tT)dt,
zC=(C2+4π2)(β2+bβ3)T2+2π2(γ0r+bγ1r)ET[1fR+fRhR(t)B7dt]+1504π215π2γ1rECT2fRhR(t)A9(tT)dt,
A9(x)=11504π2[120A3(x)(4π230)A7(x)],
A3(x)=A9(x)=A1(x)=158csch4(x)[4x+2xcosh(2x)3sinh(2x)],
B1(x)=B4(x)=1.

Metrics