Abstract

The propagation of pulses through waveguides with sub-wavelength features, inhomogeneous transverse structure, and high index contrast cannot be described accurately using existing models in the presence of nonlinear effects. Here we report the development of a generalised full vectorial model of nonlinear pulse propagation and demonstrate that, unlike the standard pulse propagation formulation, the z-component of guided modes plays a key role for these new structures, and results in generalised definitions of the nonlinear coefficient γ, Aeff , and mode orthognality. While new definitions reduce to standard definitions in some limits, significant differences are predicted, including a factor of ~2 higher value for γ, for emerging waveguides and microstructured fibers.

© 2009 Optical Society of America

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2008

2007

G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, "Nonlinear Envelope Equation Modeling of Sub-Cycle Dynamics and Harmonic Generation in Nonlinear Waveguides," Opt. Express 15, 5382 (2007).
[CrossRef] [PubMed]

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, "All-Optical Regeneration on Silicon Chip," Opt. Express 15, 7802 (2007).
[CrossRef] [PubMed]

Q. Lin, O. J. Painter, and G. P. Agrawal, "Nonlinear Optical Phenomena in Silicon Waveguides: Modeling and Applications," Opt. Express 15, 16,604 (2007).
[CrossRef]

J. C. Knight and D. V. Skryabin, "Nonlinear Waveguide Optics and Photonic Crystal Fibers," Opt. Express 15, 15,365 (2007).
[CrossRef]

E. C. Magi, L. B. Fu, H. C. Nguyen, M. R. E. Lamont, D. I. Yeom, and B. J. Eggleton, "Enhanced Kerr Nonlinearity in Sub-Wavelength Diameter As2Se3 Chalcogenide Fiber Tapers," Opt. Express 15, 10,324 (2007).
[CrossRef]

M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, "Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides," Opt. Express 15, 12,949-12,958 (2007).
[CrossRef]

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, "Field Enhancement within an Optical Fibre with a Subwavelength Air Core," Nat. Photonics 1, 115 (2007).
[CrossRef]

2006

P. Mullner and R. Hainberger, "Structural Optimization of Silicon-on-Insulator Slot Waveguides," IEEE Photon. Technol. Lett. 18, 2557 (2006).
[CrossRef]

X. Chen, N. C. Panoiu, and R. M. Osgood, "Theory of Raman-Mediated Pulsed Amplification in Silicon-Wire Waveguides," IEEE J. Quantum Electron. 42, 160 (2006).
[CrossRef]

T. Fujisawa and M. Koshiba, "Guided Modes of Nonlinear Slot Waveguides," IEEE Photon. Technol. Lett. 18, 1530 (2006).
[CrossRef]

2005

S. Konar, S. Jana, and M. Mishra, "Induced focusing and all optical switching in cubic quintic nonlinear media," Opt. Commun 255, 114-129 (2005).
[CrossRef]

B. Kibler, J. M. Dudley, and S. Coen, "Supercontinuum Generation and Nonlinear Pulse Propagation in Photonic Crystal Fiber: Influence of the Frequency-Dependent Effective Mode Area," Appl. Phys. B 81, 337-342 (2005).
[CrossRef]

M. Foster, J. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B-Lasers Opt. 81, 363-367 (2005).
[CrossRef]

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, "Compact, Stable and Efficient All-Fibre Gas Cells using Hollow-Core Photonic Crystal Fibres," Nature 434, 488 (2005).
[CrossRef] [PubMed]

A. Zheltikov, "Gaussian-Mode Analysis of Waveguide-Enhanced Kerr-Type Nonlinearity of Optical Fibers and Photonic Wires," J. Opt. Soc. Am. B 22, 1100 (2005).
[CrossRef]

2004

O. Boyraz, T. Indukuri, and B. Jalali, "Self-Pahse-Modulation Induced Spectral Broadening in Silicon Waveguides," Opt. Express 12, 829 (2004).
[CrossRef] [PubMed]

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, "Guiding and Confining Light in Void Nanostructure," Opt. Lett. 29, 1209 (2004).
[CrossRef] [PubMed]

M. Foster, K. Moll, and A. Gaeta, "Optimal waveguide dimensions for nonlinear interactions," Opt. Express 12, 2880-2887 (2004).
[CrossRef] [PubMed]

Q. Xu, V. R. Almeida, R. R. Panepucci, and M. Lipson, "Experimental Demonstration of Guiding and Confining Light in Nanometer-Size Low-Refractive-Index Material," Opt. Lett. 29, 1626 (2004).
[CrossRef] [PubMed]

O. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, "All Optical Switching and Continuum Generation in Silicon Waveguides," Opt. Express 12, 4094 (2004).
[CrossRef] [PubMed]

H. Ebendorff-Heidepriem, P. Petropoulos, S. Asimakis, V. Finazzi, R. C. Moore, K. Frampton, F. Koizumi, D. J. Richardson, and T. M. Monro, "Bismuth Glass Holey Fibers with High Nonlinearity," Opt. Express 12, 5082 (2004).
[CrossRef] [PubMed]

V. Almeida, C. Barrios, R. Panepucci, M. Lipson, M. Foster, D. Ouzounov, and A. Gaeta, "All-optical switching on a silicon chip," Opt. Lett. 29, 2867-2869 (2004).
[CrossRef]

C. A. Barrios, "High-Performance All-Optical Silicon Microswitch," Electron Lett. 40, 862 (2004).
[CrossRef]

F. Benabid, G. Bouwmans, J. C. Knight, and P. S. J. Russell, "Ultrahigh Efficiency Laser Wavelength Conversion in a Gas-Filled Hollow Core Photonic Crystal Fiber by Pure Stimulated Rotational Raman Scattering in Molecular Hydrogen," Phys. Rev. Lett. 93, 123,903-1 (2004).
[CrossRef]

M. Kolesik and J. V. Moloney, "Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations," Phys. Rev. E 70, 036,604 (2004).
[CrossRef]

M. Kolesik, E. M. Wright, and J. V. Moloney, "Simulation of Femtosecond Pulse Propagation in Sub-Micron Diameter Tapered Fibers," Appl. Phys. B 79, 293 (2004).
[CrossRef]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-Optical Control of Light on a Silicon Chip," Nature 43, 1081 (2004).
[CrossRef]

Q. Lin and G. P. Agrawal, "Vector Theory of Cross-Phase Modulation: Role of Nonlinear Polarization Rotation," IEEE J. Quantum Electron. 40, 958 (2004).
[CrossRef] [PubMed]

2003

M. Dinu, F. Quochi, and H. Garcia, "Third-Order Nonlinearities in Silicon at TelecomWavelengths," App. Phys. Lett. 82, 2954 (2003).

F. Biancalana, D. V. Skryabin, and P. S. J. Russell, "Four-Wave Mixing Instabilities in Photonic-Crystal and Tapered Fibers," Phys. Rev. E 68, 046,603-1 (2003).

G. Chang, T. B. Norris, and H. G. Winful, "Optimization of Supercontinuum Generation in Photonic Crystal Fibers for Pulse Compression," Opt. Lett. 28, 546 (2003).
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2002

2001

M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, "Symmetry and Degeneracy in Microstructured Optical Fibers," Opt. Lett. 26, 488 (2001).
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N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, "Comparison between Theory and Experiment of Nonlinear Propagation for a-Few-Cycle and Ultrabroadband Optical Pulses in a Fused-Silica Fiber," IEEE J. of Quantum Electron. 37, 398 (2001).
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F. D. Tomasi, "Stimulated Rotational and Vibrational Raman Scattering by Elliptical Polarized Pump Radiation," Phys. Rev. A 64, 023,812 (2001).

2000

A. L. Gaeta, "Catastrophic Collapse of Ultrashort Pulses," Phys. Rev. Lett. 84, 3582 (2000).
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1997

T. Brabec and F. Krausz, "Nonlinear Optical Pulse Ropagation in the Single-Cycle Regime," Phys. Rev. Lett. 78, 3282 (1997).
[CrossRef]

M. R. Perrone, V. Piccinno, G. de Nunzio, and V. Nassisi, "Dependence of Rotational and Vibrational Raman Scattering on Focusing Geometry," IEEE. J. Quantum Electron. 33, 938-944 (1997).
[CrossRef]

1990

1989

K. J. Blow and D. Wood, "Theoretical Description of Transient Stimulated Raman Scattering in Optical Fibers," IEEE J. Quantum Electron. 25, 2665 (1989).
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Agrawal, G. P.

Q. Lin, O. J. Painter, and G. P. Agrawal, "Nonlinear Optical Phenomena in Silicon Waveguides: Modeling and Applications," Opt. Express 15, 16,604 (2007).
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Q. Lin and G. P. Agrawal, "Vector Theory of Cross-Phase Modulation: Role of Nonlinear Polarization Rotation," IEEE J. Quantum Electron. 40, 958 (2004).
[CrossRef] [PubMed]

Almeida, V.

Almeida, V. R.

Asimakis, S.

Barrios, C.

Barrios, C. A.

V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, "Guiding and Confining Light in Void Nanostructure," Opt. Lett. 29, 1209 (2004).
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C. A. Barrios, "High-Performance All-Optical Silicon Microswitch," Electron Lett. 40, 862 (2004).
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V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-Optical Control of Light on a Silicon Chip," Nature 43, 1081 (2004).
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Benabid, F.

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, "Field Enhancement within an Optical Fibre with a Subwavelength Air Core," Nat. Photonics 1, 115 (2007).
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F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, "Compact, Stable and Efficient All-Fibre Gas Cells using Hollow-Core Photonic Crystal Fibres," Nature 434, 488 (2005).
[CrossRef] [PubMed]

F. Benabid, G. Bouwmans, J. C. Knight, and P. S. J. Russell, "Ultrahigh Efficiency Laser Wavelength Conversion in a Gas-Filled Hollow Core Photonic Crystal Fiber by Pure Stimulated Rotational Raman Scattering in Molecular Hydrogen," Phys. Rev. Lett. 93, 123,903-1 (2004).
[CrossRef]

Biancalana, F.

F. Biancalana, D. V. Skryabin, and P. S. J. Russell, "Four-Wave Mixing Instabilities in Photonic-Crystal and Tapered Fibers," Phys. Rev. E 68, 046,603-1 (2003).

Birks, T. A.

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, "Compact, Stable and Efficient All-Fibre Gas Cells using Hollow-Core Photonic Crystal Fibres," Nature 434, 488 (2005).
[CrossRef] [PubMed]

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 (1989).
[CrossRef]

Botten, L. C.

Bouwmans, G.

F. Benabid, G. Bouwmans, J. C. Knight, and P. S. J. Russell, "Ultrahigh Efficiency Laser Wavelength Conversion in a Gas-Filled Hollow Core Photonic Crystal Fiber by Pure Stimulated Rotational Raman Scattering in Molecular Hydrogen," Phys. Rev. Lett. 93, 123,903-1 (2004).
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Boyraz, O.

Brabec, T.

T. Brabec and F. Krausz, "Nonlinear Optical Pulse Ropagation in the Single-Cycle Regime," Phys. Rev. Lett. 78, 3282 (1997).
[CrossRef]

Cao, Q.

M. Foster, J. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B-Lasers Opt. 81, 363-367 (2005).
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Carlsten, J. L.

Chang, G.

Chen, X.

Chernikov, S. V.

Chou, C.-Y.

Coen, S.

B. Kibler, J. M. Dudley, and S. Coen, "Supercontinuum Generation and Nonlinear Pulse Propagation in Photonic Crystal Fiber: Influence of the Frequency-Dependent Effective Mode Area," Appl. Phys. B 81, 337-342 (2005).
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J. M. Dudley and S. Coen, "Coherence Properties of Supercontinuum Spectra Generated in Photonic Crystal and Tapered Optical Fibers," Opt. Lett. 27, 1180 (2002).
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Cordeiro, C. M. B.

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, "Field Enhancement within an Optical Fibre with a Subwavelength Air Core," Nat. Photonics 1, 115 (2007).
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Couny, F.

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, "Field Enhancement within an Optical Fibre with a Subwavelength Air Core," Nat. Photonics 1, 115 (2007).
[CrossRef]

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, "Compact, Stable and Efficient All-Fibre Gas Cells using Hollow-Core Photonic Crystal Fibres," Nature 434, 488 (2005).
[CrossRef] [PubMed]

Cruz, C. H. B.

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, "Field Enhancement within an Optical Fibre with a Subwavelength Air Core," Nat. Photonics 1, 115 (2007).
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Dadap, J. L.

de Nunzio, G.

M. R. Perrone, V. Piccinno, G. de Nunzio, and V. Nassisi, "Dependence of Rotational and Vibrational Raman Scattering on Focusing Geometry," IEEE. J. Quantum Electron. 33, 938-944 (1997).
[CrossRef]

de Sterke, C. M.

Debs, J. E.

J. E. Debs, H. Ebendorff-Heidepriem, J. S. Quinton, and T. M. Monro, "A Fundamental Study Into the Surface Functionalization of Soft Glass Microstructured Optical Fibers Via Silane Coupling Agents," J. Lightwave Technol., Accepted (2008).

Dinu, M.

M. Dinu, F. Quochi, and H. Garcia, "Third-Order Nonlinearities in Silicon at TelecomWavelengths," App. Phys. Lett. 82, 2954 (2003).

Dudley, J.

M. Foster, J. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B-Lasers Opt. 81, 363-367 (2005).
[CrossRef]

Dudley, J. M.

Dulkeith, E.

Ebendorff-Heidepriem, H.

J. E. Debs, H. Ebendorff-Heidepriem, J. S. Quinton, and T. M. Monro, "A Fundamental Study Into the Surface Functionalization of Soft Glass Microstructured Optical Fibers Via Silane Coupling Agents," J. Lightwave Technol., Accepted (2008).

H. Ebendorff-Heidepriem, P. Petropoulos, S. Asimakis, V. Finazzi, R. C. Moore, K. Frampton, F. Koizumi, D. J. Richardson, and T. M. Monro, "Bismuth Glass Holey Fibers with High Nonlinearity," Opt. Express 12, 5082 (2004).
[CrossRef] [PubMed]

Eggleton, B. J.

E. C. Magi, L. B. Fu, H. C. Nguyen, M. R. E. Lamont, D. I. Yeom, and B. J. Eggleton, "Enhanced Kerr Nonlinearity in Sub-Wavelength Diameter As2Se3 Chalcogenide Fiber Tapers," Opt. Express 15, 10,324 (2007).
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Finazzi, V.

Foster, M.

M. Foster, J. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B-Lasers Opt. 81, 363-367 (2005).
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M. Foster, K. Moll, and A. Gaeta, "Optimal waveguide dimensions for nonlinear interactions," Opt. Express 12, 2880-2887 (2004).
[CrossRef] [PubMed]

V. Almeida, C. Barrios, R. Panepucci, M. Lipson, M. Foster, D. Ouzounov, and A. Gaeta, "All-optical switching on a silicon chip," Opt. Lett. 29, 2867-2869 (2004).
[CrossRef]

Foster, M. A.

Fragnito, H. L.

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, "Field Enhancement within an Optical Fibre with a Subwavelength Air Core," Nat. Photonics 1, 115 (2007).
[CrossRef]

Frampton, K.

Fu, L. B.

E. C. Magi, L. B. Fu, H. C. Nguyen, M. R. E. Lamont, D. I. Yeom, and B. J. Eggleton, "Enhanced Kerr Nonlinearity in Sub-Wavelength Diameter As2Se3 Chalcogenide Fiber Tapers," Opt. Express 15, 10,324 (2007).
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Fujisawa, T.

T. Fujisawa and M. Koshiba, "Guided Modes of Nonlinear Slot Waveguides," IEEE Photon. Technol. Lett. 18, 1530 (2006).
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Gaeta, A.

M. Foster, J. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B-Lasers Opt. 81, 363-367 (2005).
[CrossRef]

M. Foster, K. Moll, and A. Gaeta, "Optimal waveguide dimensions for nonlinear interactions," Opt. Express 12, 2880-2887 (2004).
[CrossRef] [PubMed]

V. Almeida, C. Barrios, R. Panepucci, M. Lipson, M. Foster, D. Ouzounov, and A. Gaeta, "All-optical switching on a silicon chip," Opt. Lett. 29, 2867-2869 (2004).
[CrossRef]

Gaeta, A. L.

M. A. Foster, A. C. Turner,M. Lipson, and A. L. Gaeta, "Nonlinear Optics in Photonic Nanowires," Opt. Express 16, 1300 (2008).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, "Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides," Opt. Express 15, 12,949-12,958 (2007).
[CrossRef]

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, "All-Optical Regeneration on Silicon Chip," Opt. Express 15, 7802 (2007).
[CrossRef] [PubMed]

A. L. Gaeta, "Catastrophic Collapse of Ultrashort Pulses," Phys. Rev. Lett. 84, 3582 (2000).
[CrossRef]

Garcia, H.

M. Dinu, F. Quochi, and H. Garcia, "Third-Order Nonlinearities in Silicon at TelecomWavelengths," App. Phys. Lett. 82, 2954 (2003).

Genty, G.

Geraghty, D. F.

Green, W. M. J.

Hainberger, R.

P. Mullner and R. Hainberger, "Structural Optimization of Silicon-on-Insulator Slot Waveguides," IEEE Photon. Technol. Lett. 18, 2557 (2006).
[CrossRef]

Hsieh, I.-W.

Indukuri, T.

Jalali, B.

Jana, S.

S. Konar, S. Jana, and M. Mishra, "Induced focusing and all optical switching in cubic quintic nonlinear media," Opt. Commun 255, 114-129 (2005).
[CrossRef]

Jr, R. M. O.

Karasawa, N.

N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, "Comparison between Theory and Experiment of Nonlinear Propagation for a-Few-Cycle and Ultrabroadband Optical Pulses in a Fused-Silica Fiber," IEEE J. of Quantum Electron. 37, 398 (2001).
[CrossRef]

Kibler, B.

G. Genty, P. Kinsler, B. Kibler, and J. M. Dudley, "Nonlinear Envelope Equation Modeling of Sub-Cycle Dynamics and Harmonic Generation in Nonlinear Waveguides," Opt. Express 15, 5382 (2007).
[CrossRef] [PubMed]

M. Foster, J. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B-Lasers Opt. 81, 363-367 (2005).
[CrossRef]

B. Kibler, J. M. Dudley, and S. Coen, "Supercontinuum Generation and Nonlinear Pulse Propagation in Photonic Crystal Fiber: Influence of the Frequency-Dependent Effective Mode Area," Appl. Phys. B 81, 337-342 (2005).
[CrossRef]

Kinsler, P.

Knight, J. C.

J. C. Knight and D. V. Skryabin, "Nonlinear Waveguide Optics and Photonic Crystal Fibers," Opt. Express 15, 15,365 (2007).
[CrossRef]

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, "Field Enhancement within an Optical Fibre with a Subwavelength Air Core," Nat. Photonics 1, 115 (2007).
[CrossRef]

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, "Compact, Stable and Efficient All-Fibre Gas Cells using Hollow-Core Photonic Crystal Fibres," Nature 434, 488 (2005).
[CrossRef] [PubMed]

F. Benabid, G. Bouwmans, J. C. Knight, and P. S. J. Russell, "Ultrahigh Efficiency Laser Wavelength Conversion in a Gas-Filled Hollow Core Photonic Crystal Fiber by Pure Stimulated Rotational Raman Scattering in Molecular Hydrogen," Phys. Rev. Lett. 93, 123,903-1 (2004).
[CrossRef]

Koizumi, F.

Kolesik, M.

M. Kolesik, E. M. Wright, and J. V. Moloney, "Simulation of Femtosecond Pulse Propagation in Sub-Micron Diameter Tapered Fibers," Appl. Phys. B 79, 293 (2004).
[CrossRef]

M. Kolesik and J. V. Moloney, "Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations," Phys. Rev. E 70, 036,604 (2004).
[CrossRef]

M. Kolesik, J. V. Moloney, and M. Mlejnek, "Unidirectional Optical Pulse Propagation Equation," Phys. Rev. Lett. 89, 283,902-1 (2002).
[CrossRef]

Konar, S.

S. Konar, S. Jana, and M. Mishra, "Induced focusing and all optical switching in cubic quintic nonlinear media," Opt. Commun 255, 114-129 (2005).
[CrossRef]

Koonath, P.

Koshiba, M.

T. Fujisawa and M. Koshiba, "Guided Modes of Nonlinear Slot Waveguides," IEEE Photon. Technol. Lett. 18, 1530 (2006).
[CrossRef]

Krausz, F.

T. Brabec and F. Krausz, "Nonlinear Optical Pulse Ropagation in the Single-Cycle Regime," Phys. Rev. Lett. 78, 3282 (1997).
[CrossRef]

Kuhlmey, B. T.

Lamont, M. R. E.

E. C. Magi, L. B. Fu, H. C. Nguyen, M. R. E. Lamont, D. I. Yeom, and B. J. Eggleton, "Enhanced Kerr Nonlinearity in Sub-Wavelength Diameter As2Se3 Chalcogenide Fiber Tapers," Opt. Express 15, 10,324 (2007).
[CrossRef]

Lee, D.

M. Foster, J. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B-Lasers Opt. 81, 363-367 (2005).
[CrossRef]

Lin, Q.

Q. Lin, O. J. Painter, and G. P. Agrawal, "Nonlinear Optical Phenomena in Silicon Waveguides: Modeling and Applications," Opt. Express 15, 16,604 (2007).
[CrossRef]

Q. Lin and G. P. Agrawal, "Vector Theory of Cross-Phase Modulation: Role of Nonlinear Polarization Rotation," IEEE J. Quantum Electron. 40, 958 (2004).
[CrossRef] [PubMed]

Lipson, M.

Liu, X.

Magi, E. C.

E. C. Magi, L. B. Fu, H. C. Nguyen, M. R. E. Lamont, D. I. Yeom, and B. J. Eggleton, "Enhanced Kerr Nonlinearity in Sub-Wavelength Diameter As2Se3 Chalcogenide Fiber Tapers," Opt. Express 15, 10,324 (2007).
[CrossRef]

Maier, S. A.

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, "Field Enhancement within an Optical Fibre with a Subwavelength Air Core," Nat. Photonics 1, 115 (2007).
[CrossRef]

Mamyshev, P. V.

Maystre, D.

McNab, S. J.

McPhedran, R. C.

Meng, L. S.

Mishra, M.

S. Konar, S. Jana, and M. Mishra, "Induced focusing and all optical switching in cubic quintic nonlinear media," Opt. Commun 255, 114-129 (2005).
[CrossRef]

Mlejnek, M.

M. Kolesik, J. V. Moloney, and M. Mlejnek, "Unidirectional Optical Pulse Propagation Equation," Phys. Rev. Lett. 89, 283,902-1 (2002).
[CrossRef]

Moll, K.

Moloney, J. V.

M. Kolesik, E. M. Wright, and J. V. Moloney, "Simulation of Femtosecond Pulse Propagation in Sub-Micron Diameter Tapered Fibers," Appl. Phys. B 79, 293 (2004).
[CrossRef]

M. Kolesik and J. V. Moloney, "Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations," Phys. Rev. E 70, 036,604 (2004).
[CrossRef]

M. Kolesik, J. V. Moloney, and M. Mlejnek, "Unidirectional Optical Pulse Propagation Equation," Phys. Rev. Lett. 89, 283,902-1 (2002).
[CrossRef]

Monro, T. M.

J. E. Debs, H. Ebendorff-Heidepriem, J. S. Quinton, and T. M. Monro, "A Fundamental Study Into the Surface Functionalization of Soft Glass Microstructured Optical Fibers Via Silane Coupling Agents," J. Lightwave Technol., Accepted (2008).

H. Ebendorff-Heidepriem, P. Petropoulos, S. Asimakis, V. Finazzi, R. C. Moore, K. Frampton, F. Koizumi, D. J. Richardson, and T. M. Monro, "Bismuth Glass Holey Fibers with High Nonlinearity," Opt. Express 12, 5082 (2004).
[CrossRef] [PubMed]

Moore, R. C.

Morita, R.

N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, "Comparison between Theory and Experiment of Nonlinear Propagation for a-Few-Cycle and Ultrabroadband Optical Pulses in a Fused-Silica Fiber," IEEE J. of Quantum Electron. 37, 398 (2001).
[CrossRef]

Mullner, P.

P. Mullner and R. Hainberger, "Structural Optimization of Silicon-on-Insulator Slot Waveguides," IEEE Photon. Technol. Lett. 18, 2557 (2006).
[CrossRef]

Nakagawa, N.

N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, "Comparison between Theory and Experiment of Nonlinear Propagation for a-Few-Cycle and Ultrabroadband Optical Pulses in a Fused-Silica Fiber," IEEE J. of Quantum Electron. 37, 398 (2001).
[CrossRef]

Nakamura, S.

N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, "Comparison between Theory and Experiment of Nonlinear Propagation for a-Few-Cycle and Ultrabroadband Optical Pulses in a Fused-Silica Fiber," IEEE J. of Quantum Electron. 37, 398 (2001).
[CrossRef]

Nassisi, V.

M. R. Perrone, V. Piccinno, G. de Nunzio, and V. Nassisi, "Dependence of Rotational and Vibrational Raman Scattering on Focusing Geometry," IEEE. J. Quantum Electron. 33, 938-944 (1997).
[CrossRef]

Nguyen, H. C.

E. C. Magi, L. B. Fu, H. C. Nguyen, M. R. E. Lamont, D. I. Yeom, and B. J. Eggleton, "Enhanced Kerr Nonlinearity in Sub-Wavelength Diameter As2Se3 Chalcogenide Fiber Tapers," Opt. Express 15, 10,324 (2007).
[CrossRef]

Norris, T. B.

Osgood, R. M.

X. Chen, N. C. Panoiu, and R. M. Osgood, "Theory of Raman-Mediated Pulsed Amplification in Silicon-Wire Waveguides," IEEE J. Quantum Electron. 42, 160 (2006).
[CrossRef]

Ouzounov, D.

Painter, O. J.

Q. Lin, O. J. Painter, and G. P. Agrawal, "Nonlinear Optical Phenomena in Silicon Waveguides: Modeling and Applications," Opt. Express 15, 16,604 (2007).
[CrossRef]

Panepucci, R.

Panepucci, R. R.

Panoiu, N. C.

Perrone, M. R.

M. R. Perrone, V. Piccinno, G. de Nunzio, and V. Nassisi, "Dependence of Rotational and Vibrational Raman Scattering on Focusing Geometry," IEEE. J. Quantum Electron. 33, 938-944 (1997).
[CrossRef]

Petropoulos, P.

Piccinno, V.

M. R. Perrone, V. Piccinno, G. de Nunzio, and V. Nassisi, "Dependence of Rotational and Vibrational Raman Scattering on Focusing Geometry," IEEE. J. Quantum Electron. 33, 938-944 (1997).
[CrossRef]

Quinton, J. S.

J. E. Debs, H. Ebendorff-Heidepriem, J. S. Quinton, and T. M. Monro, "A Fundamental Study Into the Surface Functionalization of Soft Glass Microstructured Optical Fibers Via Silane Coupling Agents," J. Lightwave Technol., Accepted (2008).

Quochi, F.

M. Dinu, F. Quochi, and H. Garcia, "Third-Order Nonlinearities in Silicon at TelecomWavelengths," App. Phys. Lett. 82, 2954 (2003).

Raghunathan, V.

Renversez, G.

Richardson, D. J.

Roos, P. A.

Russell, P. S. J.

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, "Compact, Stable and Efficient All-Fibre Gas Cells using Hollow-Core Photonic Crystal Fibres," Nature 434, 488 (2005).
[CrossRef] [PubMed]

F. Benabid, G. Bouwmans, J. C. Knight, and P. S. J. Russell, "Ultrahigh Efficiency Laser Wavelength Conversion in a Gas-Filled Hollow Core Photonic Crystal Fiber by Pure Stimulated Rotational Raman Scattering in Molecular Hydrogen," Phys. Rev. Lett. 93, 123,903-1 (2004).
[CrossRef]

F. Biancalana, D. V. Skryabin, and P. S. J. Russell, "Four-Wave Mixing Instabilities in Photonic-Crystal and Tapered Fibers," Phys. Rev. E 68, 046,603-1 (2003).

Salem, R.

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, "All-Optical Regeneration on Silicon Chip," Opt. Express 15, 7802 (2007).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, "Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides," Opt. Express 15, 12,949-12,958 (2007).
[CrossRef]

Sekaric, L.

Shibata, M.

N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, "Comparison between Theory and Experiment of Nonlinear Propagation for a-Few-Cycle and Ultrabroadband Optical Pulses in a Fused-Silica Fiber," IEEE J. of Quantum Electron. 37, 398 (2001).
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N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, "Comparison between Theory and Experiment of Nonlinear Propagation for a-Few-Cycle and Ultrabroadband Optical Pulses in a Fused-Silica Fiber," IEEE J. of Quantum Electron. 37, 398 (2001).
[CrossRef]

Skryabin, D. V.

J. C. Knight and D. V. Skryabin, "Nonlinear Waveguide Optics and Photonic Crystal Fibers," Opt. Express 15, 15,365 (2007).
[CrossRef]

F. Biancalana, D. V. Skryabin, and P. S. J. Russell, "Four-Wave Mixing Instabilities in Photonic-Crystal and Tapered Fibers," Phys. Rev. E 68, 046,603-1 (2003).

Steel, M. J.

Tomasi, F. D.

F. D. Tomasi, "Stimulated Rotational and Vibrational Raman Scattering by Elliptical Polarized Pump Radiation," Phys. Rev. A 64, 023,812 (2001).

Trebino, R.

M. Foster, J. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B-Lasers Opt. 81, 363-367 (2005).
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Turner, A. C.

Vlasov, Y. A.

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Wiederhecker, G. S.

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, "Field Enhancement within an Optical Fibre with a Subwavelength Air Core," Nat. Photonics 1, 115 (2007).
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Wood, D.

K. J. Blow and D. Wood, "Theoretical Description of Transient Stimulated Raman Scattering in Optical Fibers," IEEE J. Quantum Electron. 25, 2665 (1989).
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Wright, E. M.

M. Kolesik, E. M. Wright, and J. V. Moloney, "Simulation of Femtosecond Pulse Propagation in Sub-Micron Diameter Tapered Fibers," Appl. Phys. B 79, 293 (2004).
[CrossRef]

Xia, F.

Xu, Q.

Yamashita, M.

N. Karasawa, S. Nakamura, N. Nakagawa, M. Shibata, R. Morita, H. Shigekawa, and M. Yamashita, "Comparison between Theory and Experiment of Nonlinear Propagation for a-Few-Cycle and Ultrabroadband Optical Pulses in a Fused-Silica Fiber," IEEE J. of Quantum Electron. 37, 398 (2001).
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Yeom, D. I.

E. C. Magi, L. B. Fu, H. C. Nguyen, M. R. E. Lamont, D. I. Yeom, and B. J. Eggleton, "Enhanced Kerr Nonlinearity in Sub-Wavelength Diameter As2Se3 Chalcogenide Fiber Tapers," Opt. Express 15, 10,324 (2007).
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App. Phys. Lett.

M. Dinu, F. Quochi, and H. Garcia, "Third-Order Nonlinearities in Silicon at TelecomWavelengths," App. Phys. Lett. 82, 2954 (2003).

Appl. Phys. B

M. Kolesik, E. M. Wright, and J. V. Moloney, "Simulation of Femtosecond Pulse Propagation in Sub-Micron Diameter Tapered Fibers," Appl. Phys. B 79, 293 (2004).
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B. Kibler, J. M. Dudley, and S. Coen, "Supercontinuum Generation and Nonlinear Pulse Propagation in Photonic Crystal Fiber: Influence of the Frequency-Dependent Effective Mode Area," Appl. Phys. B 81, 337-342 (2005).
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Appl. Phys. B-Lasers Opt.

M. Foster, J. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B-Lasers Opt. 81, 363-367 (2005).
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J. Opt. Soc. Am. B

Nat. Photonics

G. S. Wiederhecker, C. M. B. Cordeiro, F. Couny, F. Benabid, S. A. Maier, J. C. Knight, C. H. B. Cruz, and H. L. Fragnito, "Field Enhancement within an Optical Fibre with a Subwavelength Air Core," Nat. Photonics 1, 115 (2007).
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Nature

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. S. J. Russell, "Compact, Stable and Efficient All-Fibre Gas Cells using Hollow-Core Photonic Crystal Fibres," Nature 434, 488 (2005).
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R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, "All-Optical Regeneration on Silicon Chip," Opt. Express 15, 7802 (2007).
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J. L. Dadap, N. C. Panoiu, X. Chen, I.-W. Hsieh, X. Liu, C.-Y. Chou, E. Dulkeith, S. J. McNab, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. O. Jr., "Nonlinear-Optical Phase Modification in Dispersion-Engineered Si Photonic Wires," Opt. Express 16, 1280 (2008).
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M. A. Foster, A. C. Turner,M. Lipson, and A. L. Gaeta, "Nonlinear Optics in Photonic Nanowires," Opt. Express 16, 1300 (2008).
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H. Ebendorff-Heidepriem, P. Petropoulos, S. Asimakis, V. Finazzi, R. C. Moore, K. Frampton, F. Koizumi, D. J. Richardson, and T. M. Monro, "Bismuth Glass Holey Fibers with High Nonlinearity," Opt. Express 12, 5082 (2004).
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M. A. Foster, A. C. Turner, R. Salem, M. Lipson, and A. L. Gaeta, "Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides," Opt. Express 15, 12,949-12,958 (2007).
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Opt. Lett.

Phys. Rev. A

F. D. Tomasi, "Stimulated Rotational and Vibrational Raman Scattering by Elliptical Polarized Pump Radiation," Phys. Rev. A 64, 023,812 (2001).

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F. Benabid, P. Light, F. Couny, and P. Russell, "Electromagnetically-induced transparency grid in acetylene-filled hollow-core PCF," Opt. Express 13, 5694-5703 (2005). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-13-15-5694.
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S. Ghosh, A. R. Bhagwat, C. K. Renshaw, S. Goh, A. L. Gaeta, and B. J. Kirby, "Low-Light-Level Optical Interactions with Rubidium Vapor in a Photonic Band-Gap Fiber," Phys. Rev. Lett. 97, 023603 (2006). URL http://link.aps.org/abstract/PRL/v97/e023603.
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C. Koos, L. Jacome, C. Poulton, J. Leuthold, and W. Freude, "Nonlinear silicon-on-insulator waveguides for all-optical signal processing," Opt. Express 15, 5976-5990 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-10-5976.
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A. Fuerbach, P. Steinvurzel, J. Bolger, and B. Eggleton, "Nonlinear pulse propagation at zero dispersion wavelength in anti-resonant photonic crystal fibers," Opt. Express 13, 2977-2987 (2005). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-13-8-2977.
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S. Yiou, P. Delaye, A. Rouvie, J. Chinaud, R. Frey, G. Roosen, P. Viale, S. Février, P. Roy, J.-L. Auguste, and J.-M. Blondy, "Stimulated Raman scattering in an ethanol core microstructured optical fiber," Opt. Express 13, 4786-4791 (2005). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-13-12-4786.
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F. M. Cox, A. Argyros, and M. C. J. Large, "Liquid-filled hollow core microstructured polymer optical fiber," Opt. Express 14, 4135-4140 (2006). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-14-9-4135.
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R. Zhang, J. Teipel, and H. Giessen, "Theoretical design of a liquid-core photonic crystal fiber for supercontinuum generation," Opt. Express 14, 6800-6812 (2006). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-14-15-6800
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S. Afshar. V., S. C. Warren-Smith, and T. M. Monro, "Enhancement of fluorescence-based sensing using microstructured optical fibres," Opt. Express 15,17,891-17,901 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-26-17891.
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S. Atakaramians, S. Afshar. V., B. M . Fischer, D . Abbott, and T. M. Monro, "Porous fibers: a novel approach to low loss THz waveguides," Opt. Express 16, 8845-8854 (2008). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-16-12-8845.
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Figures (6)

Fig. 1.
Fig. 1.

Three definitions of γ, black is γ V based on VNSE, blue is γ F reported by Foster et. al., and red is γ S by Agrawal, as a function of core diameter and for three different materials silica ( n = 1.45, n 2 = 2.6×10-20 m 2/W), bismuth (n = 2.05, n 2 = 3.2×10-19 m 2/W), and silicon (n = 3.45, n 2 = 4.5×10-18 m 2/W ). The wavelength is λ= 800 nm, and the cladding is air with n = 1.0 in a) and b) and is silica, n = 1.45 in c). Plus signs and solid lines show the actual calculated data and the lines of best fit, respectively.

Fig. 2.
Fig. 2.

Transversality versus core diameter for two glasses silica (n = 1.45 ) and bismuth (n = 2.05 ) and silicon (n = 3.45 ). The structure is a simple rod in the air for the glasses and a rod within the substrate of silica for silicon.

Fig. 3.
Fig. 3.

2D plot of Ez (a,c) and E r 2 + E θ 2 (b,d) for two step index rods with core diameters 0.4 μm (a,b) and 1.8 μm (c,d) at the wavelength 1550 nm.The material is Bismuth with refractive index of n = 2.05.

Fig. 4.
Fig. 4.

Ratio of nonlinear coefficients γμν (1) and γν , see Eq. (40) and (39), as a function of core diameter for step index rods with host materials Silica (n = 1.45), Bismuth (n = 2.05), and Silicon (n = 3.45). The cladding material for glasses is air and for silicon is silica. Signs and the solid lines are the calculated data and lines of best fit, respectively.

Fig. 5.
Fig. 5.

a) γ as a function of core diameter of a step index rod for different wavelengths. The solid lines are γ V and the dashed lines are γ A. In b) the maximum of γ in a) have been plotted as a function of wavelength. The host material is Bismuth (n = 2.05).

Fig. 6.
Fig. 6.

γ V (solid lines) and γ S (dashed lines) vs wavelength for different core diameters for a step index rod with host material of Bismuth glass (n = 2.05).

Equations (76)

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

× E ˜ r ω = i μ 0 ω H ˜ r ω
× H ˜ r ω = i ε 0 ω E ˜ r ω P ˜ r ω ,
F ( r , t ) = 1 2 π F ˜ r ω e iωt ,
× E ˜ r ω = i μ 0 ω H ˜ r ω
× H ˜ r ω = i ε 0 n 2 r ω E ˜ r ω P ˜ NL r ω ,
e ̂ ν = e ν ( x , y , ω 0 ) N ν e i β ν z
h ̂ ν = h ν ( x , y , ω 0 ) N ν e i β ν z ,
e μ ( x , y , ω ) × h ν * x y ω . z ̂ dA = N μ δ μν
N μ = 1 2 e μ ( x , y , ω ) × h μ * x y ω . z ̂ dA .
F c = E ˜ 0 × H ˜ * + E ˜ * × H ˜ 0 ,
z = F C . z ̂ dA = . F C dA ,
. F C = i μ 0 ( ω ω 0 ) H ˜ * . H ˜ 0 i ε 0 [ ω n 2 r ω ω 0 n 2 ( r , ω 0 ) ] E ˜ * + E ˜ 0 + E ˜ 0 P ˜ NL * r ω .
E ˜ r ω = μ a ˜ μ z ω e μ ( x , y , ω 0 ) N μ e i β μ z + a ˜ μ z ω e μ ( x , y , ω 0 ) N μ e i β μ z + Radiation Modes ,
H ˜ r ω = μ a ˜ μ z ω h μ ( x , y , ω 0 ) N μ e i β μ z + a ˜ μ z ω h μ ( x , y , ω 0 ) N μ e i β μ z + Radiation Modes .
E ˜ 0 ( r , ω 0 ) = e ν ( x , y , ω 0 ) N ν e i β ν z
H ˜ 0 ( r , ω 0 ) = h ν ( x , y , ω 0 ) N ν e i β ν z ,
z a ˜ ν z ω = 1 4 μ [ A νμ + B νμ ] a ˜ μ e i β ν z 4 N ν e ν * . P ˜ NL r ω dA .
A νμ = i μ 0 e i ( β ν β μ ) z N ν N μ ( ω ω 0 ) h μ . h ν * dA
B νμ = i ε 0 e i ( β ν β μ ) z N ν N μ [ ω n 2 x y ω ω 0 n 2 ( x , y , ω 0 ) ] e μ . e ν * dA .
z a ˜ ν z ω = i n = 1 ( Δ ω ) n n ! β ν ( n ) a ˜ ν + i μ ν n ( Δ ω ) n n ! β νµ ( n ) a ˜ μ
e i β ν z 4 N ν e ν * . P ˜ NL r ω dA
β ν ( 1 ) = 1 4 N ν [ μ 0 h ν 2 + ε 0 ω ( ω n 2 ) ω = ω 0 e ν 2 ] dA
β ν ( n ) = n ω n β ν 1
β νμ ( 1 ) = e i ( β ν β μ ) z 4 N ν N μ [ μ 0 h µ . h ν * + ε 0 ω ( ω n 2 ) ω = ω 0 e μ . e ν * ] dA
β νμ ( n ) = n ω n β νμ 1 .
a ν z t 1 / 2 [ a ν z t e i ω 0 t + c . c . ]
P NL r t = 1 / 2 [ P NL z t e i ω 0 t + c . c . ] ,
z a ν z t = i n = 1 ( i / t ) n n ! β ν ( n ) a v + i μ ν n = 1 ( i / t ) n n ! β νμ ( n ) a μ
e i β ν z 4 N ν ( 1 + τ shock / t ) e ν * . P NL r t dA ,
P ( 3 ) r t = ( 3 / 4 ) ε 0 χ ( 3 ) ( ω 0 ; ω 0 , ω 0 , ω 0 ) E r t E r t E * r t ,
χ ijkl ( 3 ) = χ xxyy ( 3 ) χ ij χ kl + χ xyxy ( 3 ) χ ik χ jl + χ xyyx ( 3 ) χ il χ jk ,
χ xxxx = χ yyyy = χ zzzz = χ xxyy ( 3 ) + χ xyxy ( 3 ) + χ xyyx ( 3 ) ,
P i ( 3 ) r t = ( 3 / 4 ) ε 0 [ j χ xxyy ( 3 ) E j 2 E i + j χ xyxy ( 3 ) E j 2 E i + j χ xyyx ( 3 ) ( E j ) 2 E i * ] ,
P ( 3 ) r t = ( 1 / 2 ) ε 10 χ xxxx ( 3 ) [ ( E . E * ) E + ( 1 / 2 ) ( E . E ) E * ] .
χ ijkl ( 3 ) = χ xxyy ( 3 ) δ ij δ kl + χ xyxy ( 3 ) δ ik δ jl + χ xyyx ( 3 ) δ il δ jk + χ d δ ijkl ,
χ ijkl ( 3 ) = χ xxxx [ ρ 3 ( δ ij δ kl + δ ik δ jl + δ il δ jk ) + ( 1 ρ ) δ ijkl ] ,
P ( 3 ) r t = ( ρ / 2 ) ε 0 χ xxxx ( 3 ) [ ( E . E * ) E + ( 1 / 2 ) ( E . E ) E * ] + ( 3 / 4 ) ε 0 ( 1 ρ ) χ xxxx ( 3 ) E . E . E * ,
( 1 / N ν ) e i β ν z e ν * P NL r t = ( 1 / 2 ) ε 0 χ xxxx ( 3 ) μ , η , ζ
[ ( 1 / N μ N η N ζ N ν ) a μ a η * a ζ ( e μ . e η * ) ( e v * . e ζ ) e i ( β ν β μ + β η β ζ ) z
+ [ ( 1 / 2 N μ N η N ζ N ν ) a μ a η a ζ * ( e μ . e η * ) ( e v * . e ζ * ) e i ( β ν β μ β η + β ζ ) z ]
( 1 / N ν ) e i β ν z e ν * P NL r t = ( 3 / 4 ) ε 0 χ xxxx ( 3 ) ×
{ ( a ν 2 a ν 3 N ν 2 ) [ 2 e ν 2 + e ν 2 2 ]
+ ( μ ν 2 a ν a μ 2 3 N ν 2 N μ 2 ) [ e ν · e μ * 2 + e ν · e μ 2 + e ν 2 e μ 2 ]
+ μ ν ( a μ * a ν 2 3 N ν 3 N μ ) [ 2 e ν 2 ( e μ * · e ν ) + ( e ν ) 2 ( e μ * · e ν * ) ] e i ( β μ β ν ) z
+ μ ν ( 2 a μ a ν 2 3 N ν 3 N μ ) [ 2 e ν 2 ( e μ · e ν * ) + ( e ν * ) 2 ( e μ ·e ν ) ] e i ( β ν β μ ) z
+ μ ν ( a μ 2 a μ 3 N μ 3 N ν ) [ 2 e μ 2 ( e μ · e ν * ) + ( e μ ) 2 ( e μ * · e ν * ) ] e i ( β ν β μ ) z
+ μ ν ( a μ 2 a ν * 3 N ν 2 N μ 2 ) [ 2 ( e μ · e v * ) 2 + ( e μ ) 2 ( e ν ) 2 ] e 2 i ( β ν β μ ) z
+ μ η ζ ν other phase terms } ·
( e ̂ × h ̂ * ) . z ̂ = ( e ̂ t × h ̂ t * ) . z ̂ ,
h ̂ t = ( ε 0 μ 0 ) 1 / 2 1 k z ̂ × [ β e ̂ t + i t e ̂ z ] ,
( e ̂ ν × h ̂ μ * ) . z ̂ dA = ( ε 0 μ 0 ) 1 / 2 1 k ( e ̂ νt × { z ̂ × [ β e ̂ μt * i t e μz * ] } ) . z ̂ dA ,
= ( ε 0 μ 0 ) 1 / 2 1 k ( β e ̂ νt e ̂ μt * i e ̂ νt · t e μz * ] } ) dA
e ̂ νt . e ̂ μt * . dA = ( μ 0 ε 0 ) 1 / 2 k β δ νμ + ( i β ) ( e ̂ νt t e ̂ μz * ) dA ,
z a ν z t = i n ( i / t ) n n ! β ν ( n ) a ν +
ik 4 ( ε 0 μ 0 ) ( 1 + τ shock / t ) { 1 3 N ν 2 a ν 2 a ν n 2 x y n 2 x y [ 2 e ν 4 + e ν 2 2 ] dA
+ 2 3 N ν N μ a μ 2 a ν n 2 x y n 2 x y [ e ν . e μ * 2 + e ν . e μ 2 + e ν 2 e μ 2 ] dA } ,
z a ν z t = i n ( i / t ) n n ! β ν ( n ) a ν
( 1 + τ shock / t ) [ i γ ν a ν 2 a ν + i γ μν a μ 2 a ν ] .
γ ν = k ( ε 0 μ 0 ) n 2 x y n 2 x y [ 2 e ν 4 + e ν 2 2 ] dA 3 ( e ν × h ν * ) . z ̂ dA 2 ,
γ μν = γ μν ( 1 ) + γ μν ( 2 ) = k ( ε 0 μ 0 ) [ ( 2 n 2 x y n 2 x y [ e ν . e μ * 2 + e ν . e μ 2 ] dA 3 ( e μ × h μ * ) . z ̂ dA ( e ν × h ν * ) . z ̂ dA +
2 n 2 x y n 2 x y [ 2 e μ 2 + e ν 2 dA ] 3 ( e μ × h μ * ) . z ̂ dA ( e ν × h ν * ) . z ̂ dA ] .
A eff = ( e ν × h ν * ) . z ̂ dA 2 ( e ν × h ν * ) z ̂ 2 dA
γ ν = 2 π λ n 2 ̄ A eff
n 2 ̄ = k ( ε 0 μ 0 ) n 2 x y n 2 x y [ 2 e ν 4 + e ν 2 2 ] dA 3 ( e ν × h ν * ) . z ̂ 2 dA ,
A eff = [ β e t 2 + i ( e t . t e z ) ] . dA 2 β e t 2 + i ( e t . t e z ) 2 dA .
A eff = e t 2 dA 2 e t 4 dA ,
z a ν z t = i n = 1 ( i / t ) n n ! β ν ( n ) a ν + i n = 1 ( i / t ) n n ! β νμ ( n ) a μ
ik 4 ( ε 0 μ 0 ) ( 1 + τ shock / t ) ×
{ 1 3 N ν 2 a v 2 a ν n 2 x y n 2 x y [ 2 e ν 4 + e ν 2 2 ] dA
+ 2 3 N ν N μ a μ 2 a ν n 2 x y n 2 x y [ e ν · e μ * 2 + e ν ·e μ 2 + e ν 2 e μ 2 ] dA
+ 1 3 N ν 3 N μ a μ * a ν 2 n 2 x y n 2 x y [ 2 e ν 2 ( e μ * · e ν ) + ( e ν ) 2 ( e μ * · e ν * ) ] dA
+ 1 3 N ν 3 N μ a μ a ν 2 n 2 x y n 2 x y [ 2 e ν 2 ( e μ · e ν * ) + ( e ν * ) 2 ( e μ · e ν ) ] dA
+ 1 3 N μ 3 N ν a μ 2 a μ n 2 x y n 2 x y [ 2 e μ 2 ( e μ ·e ν * ) + ( e μ * ) 2 ( e μ * e ν * ) ] dA
+ 1 3 N ν 2 N μ 2 a μ * a ν * n 2 x y n 2 x y [ 2 ( e μ e ν * ) 2 + ( e μ ) 2 ( e ν ) 2 ] dA } ,
γ S = ( 2 π n 2 / λ ) F 4 dA ( F 2 dA ) 2
γ F = ( 2 π / λ ) n 2 [ ( e ν × h ν * ) . z ̂ ] 2 dA [ ( e ν × h ν * ) . z ̂ dA ] 2 .

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