Abstract

The influence of strong loss peaks on the dispersion (through the Kramers-Kronig relations) of a nonlinear waveguide is investigated theoretically. It is found specifically for degenerate four-wave mixing in a poly(methyl methacrylate) microstructured polymer optical fiber that the loss-induced dispersion significantly modifies the wavelengths for which there is phase-match. Depending on the pump wavelength, the waveguide dispersion, and the loss peaks, it is possible for the output spectrum to either be unaffected by the loss-induced dispersion modulation, or to show an increase in the efficiency of nonlinear spectral broadening, compared to the expected efficiency when ignoring the loss-induced dispersion modulation.

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, Burlington, MA, USA, 2007).
  2. P. D. Rasmussen, J. Laegsgaard, and O. Bang, “Degenerate four wave mixing in solid core photonic bandgap fibers,” Opt. Express 16(6), 4059–4068 (2008).
    [CrossRef] [PubMed]
  3. S.-J. Im, A. Husakou, and J. Herrmann, “Guiding properties and dispersion control of kagome lattice hollow-core photonic crystal fibers,” Opt. Express 17(15), 13050–13058 (2009).
    [CrossRef] [PubMed]
  4. T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, “Chalcogenide holey fibres,” Electron. Lett. 36(24), 1998–2000 (2000).
    [CrossRef]
  5. M. van Eijkelenborg, M. Large, A. Argyros, J. Zagari, S. Manos, N. Issa, I. Bassett, S. Fleming, R. McPhedran, C. M. de Sterke, and N. A. Nicorovici, “Microstructured polymer optical fibre,” Opt. Express 9(7), 319–327 (2001).
    [CrossRef] [PubMed]
  6. M. J. Large, L. Poladian, G. W. Barton, and M. A. van Eijkelenborg, Microstructured Polymer Optical Fibres (Springer, 2008).
  7. B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., New York, 1991).
  8. M. H. Frosz, T. Sørensen, and O. Bang, “Nanoengineering of photonic crystal fibers for supercontinuum spectral shaping,” J. Opt. Soc. Am. B 23(8), 1692–1699 (2006).
    [CrossRef]
  9. G. M. Gehring, R. W. Boyd, A. L. Gaeta, D. J. Gauthier, and A. E. Willner, “Fiber-Based Slow-Light Technologies,” J. Lightwave Technol. 26(23), 3752–3762 (2008).
    [CrossRef]
  10. T. Kaino, “Absorption losses of low-loss plastic optical fibers,” Jpn. J. Appl. Phys. Part 1 - Regul,” Pap. Short Notes Rev. Pap. 24, 1661–1665 (1985).
  11. J. Zagari, A. Argyros, N. A. Issa, G. Barton, G. Henry, M. C. J. Large, L. Poladian, and M. A. van Eijkelenborg, “Small-core single-mode microstructured polymer optical fiber with large external diameter,” Opt. Lett. 29(8), 818–820 (2004).
    [CrossRef] [PubMed]
  12. D. Morichère, M. L. Dumont, Y. Levy, G. Gadret, and F. Kajzar, “Nonlinear properties of poled polymer films: SHG and electro-optic measurements,” in Nonlinear Optical Properties of Organic Materials IV, (SPIE, 1991), 214–225.
  13. F. Kajzar, “Third Harmonic Generation,” in Characterization techniques and tabulations for organic nonlinear optical materials, M. G. Kuzyk and C. W. Dirk, eds. (Marcel Dekker, Inc., 1998).
  14. M. H. Frosz, K. Nielsen, P. Hlubina, A. Stefani, and O. Bang, “Dispersion-engineered and highly nonlinear microstructured polymer optical fibres,” Proceedings of the SPIE - The International Society for Optical Engineering 7357, 735705 (735709 pp.) (2009).
  15. A. Sherman, (personal communication, 2009).
  16. A. Sherman, E. Benkler, and H. R. Telle, “Small third-order optical-nonlinearity detection free of laser parameters,” Opt. Lett. 34(1), 49–51 (2009).
    [CrossRef]
  17. Data kindly provided by Optical Fibre Technology Centre, University of Sydney, Australia.
  18. COMSOL, Multiphysics 3.4 (2007), http://www.comsol.com .
  19. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
    [CrossRef]
  20. S. Coen, A. H. L. Chau, R. Leonhardt, J. D. Harvey, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Supercontinuum generation by stimulated Raman scattering and parametric four-wave mixing in photonic crystal fibers,” J. Opt. Soc. Am. B 19(4), 753–764 (2002).
    [CrossRef]
  21. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C + + : The Art of Scientific Computing, 2nd ed. (Cambridge University Press, Cambridge, 2002).

2009

2008

2006

M. H. Frosz, T. Sørensen, and O. Bang, “Nanoengineering of photonic crystal fibers for supercontinuum spectral shaping,” J. Opt. Soc. Am. B 23(8), 1692–1699 (2006).
[CrossRef]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[CrossRef]

2004

2002

2001

2000

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, “Chalcogenide holey fibres,” Electron. Lett. 36(24), 1998–2000 (2000).
[CrossRef]

1985

T. Kaino, “Absorption losses of low-loss plastic optical fibers,” Jpn. J. Appl. Phys. Part 1 - Regul,” Pap. Short Notes Rev. Pap. 24, 1661–1665 (1985).

Argyros, A.

Bang, O.

Barton, G.

Bassett, I.

Benkler, E.

Boyd, R. W.

Broderick, N. G. R.

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, “Chalcogenide holey fibres,” Electron. Lett. 36(24), 1998–2000 (2000).
[CrossRef]

Chau, A. H. L.

Coen, S.

de Sterke, C. M.

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[CrossRef]

Fleming, S.

Frosz, M. H.

Gaeta, A. L.

Gauthier, D. J.

Gehring, G. M.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[CrossRef]

Harvey, J. D.

Henry, G.

Herrmann, J.

Hewak, D. W.

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, “Chalcogenide holey fibres,” Electron. Lett. 36(24), 1998–2000 (2000).
[CrossRef]

Husakou, A.

Im, S.-J.

Issa, N.

Issa, N. A.

Kaino, T.

T. Kaino, “Absorption losses of low-loss plastic optical fibers,” Jpn. J. Appl. Phys. Part 1 - Regul,” Pap. Short Notes Rev. Pap. 24, 1661–1665 (1985).

Knight, J. C.

Laegsgaard, J.

Large, M.

Large, M. C. J.

Leonhardt, R.

Manos, S.

McPhedran, R.

Monro, T. M.

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, “Chalcogenide holey fibres,” Electron. Lett. 36(24), 1998–2000 (2000).
[CrossRef]

Nicorovici, N. A.

Poladian, L.

Rasmussen, P. D.

Richardson, D. J.

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, “Chalcogenide holey fibres,” Electron. Lett. 36(24), 1998–2000 (2000).
[CrossRef]

Russell, P. S. J.

Sherman, A.

Sørensen, T.

Telle, H. R.

van Eijkelenborg, M.

van Eijkelenborg, M. A.

Wadsworth, W. J.

West, Y. D.

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, “Chalcogenide holey fibres,” Electron. Lett. 36(24), 1998–2000 (2000).
[CrossRef]

Willner, A. E.

Zagari, J.

Electron. Lett.

T. M. Monro, Y. D. West, D. W. Hewak, N. G. R. Broderick, and D. J. Richardson, “Chalcogenide holey fibres,” Electron. Lett. 36(24), 1998–2000 (2000).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Pap. Short Notes Rev. Pap.

T. Kaino, “Absorption losses of low-loss plastic optical fibers,” Jpn. J. Appl. Phys. Part 1 - Regul,” Pap. Short Notes Rev. Pap. 24, 1661–1665 (1985).

Rev. Mod. Phys.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78(4), 1135–1184 (2006).
[CrossRef]

Other

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, Burlington, MA, USA, 2007).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C + + : The Art of Scientific Computing, 2nd ed. (Cambridge University Press, Cambridge, 2002).

M. J. Large, L. Poladian, G. W. Barton, and M. A. van Eijkelenborg, Microstructured Polymer Optical Fibres (Springer, 2008).

B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, Inc., New York, 1991).

D. Morichère, M. L. Dumont, Y. Levy, G. Gadret, and F. Kajzar, “Nonlinear properties of poled polymer films: SHG and electro-optic measurements,” in Nonlinear Optical Properties of Organic Materials IV, (SPIE, 1991), 214–225.

F. Kajzar, “Third Harmonic Generation,” in Characterization techniques and tabulations for organic nonlinear optical materials, M. G. Kuzyk and C. W. Dirk, eds. (Marcel Dekker, Inc., 1998).

M. H. Frosz, K. Nielsen, P. Hlubina, A. Stefani, and O. Bang, “Dispersion-engineered and highly nonlinear microstructured polymer optical fibres,” Proceedings of the SPIE - The International Society for Optical Engineering 7357, 735705 (735709 pp.) (2009).

A. Sherman, (personal communication, 2009).

Data kindly provided by Optical Fibre Technology Centre, University of Sydney, Australia.

COMSOL, Multiphysics 3.4 (2007), http://www.comsol.com .

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Figures (10)

Fig. 1
Fig. 1

The measured loss in bulk PMMA from Ref [10]. (blue, dash-dotted) and the measured loss in an mPOF (red, solid). The losses for the mPOF fibre could not be measured in wavelength regions with very high loss (around 1180 nm, 1400 nm, and above 1600 nm). A fit to the loss peaks between 850 and 1850 nm is also shown (green, dotted), made from a sum of 7 functions with the form given by Eqn. (7).

Fig. 2
Fig. 2

The calculated change in refractive index due to loss in bulk PMMA, calculated from the Kramers-Kronig relation.

Fig. 3
Fig. 3

Calculated dispersion in PMMA mPOF with Λ = 2.0 µm, d/Λ = 0.6, without (blue, solid) and with (green, dashed) loss-modulated dispersion. The zero-dispersion wavelength for the mPOF without loss-modulated dispersion is 1040 nm.

Fig. 4
Fig. 4

Wavelengths λ aS and λ S with maximum gain for FWM as a function of pump wavelength λ p when the influence of loss peaks on dispersion is either neglected (blue dots connected by lines) or included (green dots). The pump peak power P0 = 300 W and the nonlinear parameter of the fiber γ = 0.067 (W∙m)−1.

Fig. 5
Fig. 5

Wavelengths λ aS and λ S with maximum gain for FWM as a function of pump wavelength λ p and pump peak power P 0 when the influence of loss peaks on dispersion is either neglected (blue dots connected by lines) or included (green dots). The nonlinear parameter of the fiber is γ = 0.067 (W∙m)−1.

Fig. 6
Fig. 6

Solid, blue: dispersion from full finite-element method calculation including loss-modified refractive index (identical to green, dashed plot in Fig. 3). Green, dashed: dispersion when calculating the propagation constant using a perturbation of the loss-free mode-calculation, Eq. (12).

Fig. 7
Fig. 7

Simulated spectra at different lengths along the fibre up to 0.8 m, when both losses and loss-modulated dispersion are neglected.

Fig. 8
Fig. 8

Simulated spectra after 0.6 m of propagation, pump wavelength λp = 1064 nm, when only loss is included (green, solid) and when both loss and loss-modulated dispersion are included (red, dash-dotted). Compared to when loss is neglected (Fig. 7), it is seen that the peaks shift closer to the pump.

Fig. 9
Fig. 9

Simulated spectra after 0.6 m of propagation, when only loss is included (green, solid) and when both loss and loss-modulated dispersion are included (red, dash-dotted), for a pump wavelength of 1040 nm.

Fig. 10
Fig. 10

The power gain g P = 2g when loss-modulated dispersion is either neglected (left, blue) or included (right, green) for various input peak powers P 0, shown together with the loss α(ν) (black). The pump wavelength is 1040 nm, which is the same as the zero-dispersion wavelength.

Tables (1)

Tables Icon

Table 1 The fitting parameters used with Eqn. (7) to obtain the fitted loss profile shown in Fig. 1.

Equations (12)

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n2,PMMAn2,silica=χPMMA(3)χsilica(3)·nsilicanPMMA=7×1014esu3.1×1014esu·1.451.482.
α(ν)=2πνn0c0χ(ν),  χ(ν)1.
χ(ν)=2π0sχ(s)s2ν2ds,
n(ν)n0=χ(ν)2n0,  χ(ν)1.
n02(λ)=A0+A1λ2+A2λ2+A3λ4+A4λ6+A5λ8,
χ(ν)=χ0ν02νΔν(ν02ν2)2+(νΔν)2.
αm(ν)=2πχ0ν02Δνn0c0ν2(ν02ν2)2+(νΔν)2.
κ(Ω,ωp)=2γP0+Δβ=2γP0+β(ωpΩ)+β(ωp+Ω)2β(ωp),
g(Ω,ωp)=[γP0]2[κ(Ω,ωp)/2]2.
A˜z=i{β(ω)β0β1[ωω0]}A˜(z,ω)α(ω)2A˜(z,ω)+iγ{A(z,t)|A(z,t)2|},
D^(ω)=m=2βm[ωω0]mm!,
neffwithloss(ω)neffwithloss(ω)+Δn(ω),

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