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 (3)

2007 (12)

P. S. Light, F. Benabid, F. Couny, M. Maric, and A. N. Luiten, “Electromagnetically induced transparency in Rb-filled coated hollow-core photonic crystal fiber,” Opt. Lett. 32(10), 1323–1325 (2007), http://ol.osa. org/abstract.cfm?URI=ol-32-10-1323.
[CrossRef] [PubMed]

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]

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
[CrossRef] [PubMed]

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Optical solitons in a silicon waveguide,” Opt. Express 15, 7682–7688 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-12-7682.
[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 Nonlin-earity 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]

J. Laegsgaard, “Mode profile dispersion in the generalised nonlinear Schrüdinger equation,” Opt. Express 15, 16,110–16,123 (2007), http://www.opticsexpress.org/abstract.cfm?URI= oe-15-24-16110.
[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,” Nature photon. 1, 115 (2007).
[CrossRef]

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), http://www.opticsexpress. org/abstract.cfm?URI=oe-15-26-17891.
[CrossRef]

2006 (8)

T. Fujisawa and M. Koshiba, “Guided Modes of Nonlinear Slot Waveguides,” IEEE Photon. technol. lett. 18, 1530 (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]

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

J. Y. Y. Leong, P. Petropoulos, J. H. V. Price, H. Ebendorff-Heidepriem, S. Asimakis, R. C. Moore, K. E. Framp-ton, V. Finazzi, X. Feng, T. M. Monro, and D. J. Richardson, “High-Nonlinearity Dispersion-Shifted Lead-Silicate Holey Fibers for Efficient 1-μm Pumped Supercontinuum Generation,” IEEE J. Lightwave Technol. 24, 183 (2006), http://jlt.osa.org/abstract.cfm?URI=JLT-24-1-183.
[CrossRef]

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(2), 023603 (pages 4) (2006), http://link.aps.org/abstract/PRL/v97/e023603.
[CrossRef] [PubMed]

F. M. Cox, A. Argyros, and M. C. J. Large, “Liquid-filled hollow core microstructured polymer optical fiber,” Opt. Express 14, 4135–4140 (2006), http://www.opticsexpress.org/abstract.cfm?URI= oe-14-9-4135.
[CrossRef] [PubMed]

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), http://www.opticsexpress.org/abstract. cfm?URI=oe-14-15-6800.
[CrossRef] [PubMed]

M. Nagel, A. Marchewka, and H. Kurz, “Low-index discontinuity terahertz waveguides,” Opt. Express 14, 9944–9954 (2006), http://www.opticsexpress.org/abstract.cfm?URI=oe-14-21-9944.
[CrossRef] [PubMed]

2005 (9)

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]

A. Fuerbach, P. Steinvurzel, J. Bolger, and B. Eggleton, “Nonlinear pulse propagation at zero dispersion avelength in anti-resonant photonic crystal fibers,” Opt. Express 13, 2977–2987 (2005), http://www. opticsexpress.org/abstract.cfm?URI=oe-13-8-2977.
[CrossRef] [PubMed]

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.
[CrossRef] [PubMed]

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), http://www.opticsexpress.org/ abstract.cfm?URI=oe-13-15-5694.
[CrossRef] [PubMed]

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]

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]

S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant Optical Interactions with Molecules Confined in Photonic Band-Gap Fibers,” Phys. Rev. Lett. 94, 093902 (2005), http://link.aps.org/ abstract/PRL/v94/e093902.
[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,” App. Phys. B-Lasers and Optics 81, 363–367 (2005).
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S. Konar, S. Jana, and M. Mishra, “Induced focusing and all optical switching in cubic quintic nonlinear media,” Opt. Commun 255, 114–129 (2005).
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2004 (15)

Q. Lin and G. P. Agrawal, “Vector Theory of Cross-Phase Modulation: Role of Nonlinear Polarization Rotation,” IEEE J. Quantum Electron. 40, 958 (2004).
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M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036,604 (2004).
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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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C. A. Barrios, “High-Performance All-Optical Silicon Microswitch,” Electron Lett. 40, 862 (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]

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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O. Boyraz, T. Indukuri, and B. Jalali, “Self-Pahse-Modulation Induced Spectral Broadening in Silicon Waveguides,” Opt. Express 12, 829 (2004).
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M. Steel, “Reflection symmetry and mode transversality in microstructured fibers,” Opt. Express 12, 1497–1509 (2004), http://www.opticsexpress.org/abstract.cfm?URI=oe-12-8-1497.
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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).
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O. Boyraz, P. Koonath, V. Raghunathan, and B. Jalali, “All Optical Switching and Continuum Generation in Silicon Waveguides,” Opt. Express 12, 4094 (2004).
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H. Ebendorff-Heidepriem, P. Petropoulos, S. Asimakis, V. Finazzi, R. Moore, K. Frampton, F. Koizumi, D. Richardson, and T. Monro, “Bismuth glass holey fibers with high nonlinearity,” Opt. Express 12, 5082–5087 (2004), http://www.opticsexpress.org/abstract.cfm?URI=oe-12-21-5082.
[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(24), 2867–2869 (2004).
[CrossRef]

2003 (3)

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).
[CrossRef] [PubMed]

M. Dinu, F. Quochi, and H. Garcia, “Third-Order Nonlinearities in Silicon at Telecom Wavelengths,” App. Phys. Lett. 82, 2954 (2003).
[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).
[CrossRef]

2002 (5)

2001 (3)

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

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. Quantum Electron. 37, 398 (2001).
[CrossRef]

2000 (1)

A. L. Gaeta, “Catastrophic Collapse of Ultrashort Pulses,” Phys. Rev. Lett. 84, 3582 (2000).
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1997 (2)

T. Brabec and F. Krausz, “Nonlinear Optical Pulse Ropagation in the Single-Cycle Regime,” Phys. Rev. Lett. 78, 3282 (1997).
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M. R. P. et al, “Dependence of Rotational and Vibrational Raman Scattering on Focusing Geometry,” IEEE. J. Quantum Electron. 33, 938 (1997).
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1991 (2)

1990 (1)

1989 (2)

G. I. Stegeman and R. H. Stolen, “Waveguides and fibers for nonlinear optics,” J. Opt. Soc. Am. B 6, 652–662 (1989), http://josab.osa.org/abstract.cfm?URI=josab-6-4-652.
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K. J. Blow and D. Wood, “Theoretical Description of Transient Stimulated Raman Scattering in Optical Fibers,” IEEE J. of Quantum Electron. 25, 2665 (1989).
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1987 (1)

Abbott, D.

Afshar V., S.

S. Afshar V. and T. M. Monro, “Kerr Nonlinearity in Small Core Optical Fibres and Nanowires: A Generalised Model, and Application to Microstructured Fibres”, Joint OECC/ACOFT 2008 conference, IEEE conference proceeding.

Afshar. V., S.

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) http://www.opticsexpress.org/ abstract.cfm?URI=oe-16-12-8845.
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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), http://www.opticsexpress. org/abstract.cfm?URI=oe-15-26-17891.
[CrossRef]

Agrawal, G. P.

J. Zhang, Q. Lin, G. Piredda, R. W. Boyd, G. P. Agrawal, and P. M. Fauchet, “Optical solitons in a silicon waveguide,” Opt. Express 15, 7682–7688 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-12-7682.
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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).
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Agrawal, P.

P. Agrawal, Nonlinear Fiber Optics (Academic press, 2007).

Almeida, V.

Almeida, V. R.

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J. Y. Y. Leong, P. Petropoulos, J. H. V. Price, H. Ebendorff-Heidepriem, S. Asimakis, R. C. Moore, K. E. Framp-ton, V. Finazzi, X. Feng, T. M. Monro, and D. J. Richardson, “High-Nonlinearity Dispersion-Shifted Lead-Silicate Holey Fibers for Efficient 1-μm Pumped Supercontinuum Generation,” IEEE J. Lightwave Technol. 24, 183 (2006), http://jlt.osa.org/abstract.cfm?URI=JLT-24-1-183.
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H. Ebendorff-Heidepriem, P. Petropoulos, S. Asimakis, V. Finazzi, R. Moore, K. Frampton, F. Koizumi, D. Richardson, and T. Monro, “Bismuth glass holey fibers with high nonlinearity,” Opt. Express 12, 5082–5087 (2004), http://www.opticsexpress.org/abstract.cfm?URI=oe-12-21-5082.
[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]

Atakaramians, S.

Auguste, J.-L.

Barrios, C.

Barrios, C. A.

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]

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]

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

Benabid, F.

P. S. Light, F. Benabid, F. Couny, M. Maric, and A. N. Luiten, “Electromagnetically induced transparency in Rb-filled coated hollow-core photonic crystal fiber,” Opt. Lett. 32(10), 1323–1325 (2007), http://ol.osa. org/abstract.cfm?URI=ol-32-10-1323.
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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,” Nature photon. 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, P. Light, F. Couny, and P. Russell, “Electromagnetically-induced transparency grid in acetylene-filled hollow-core PCF,” Opt. Express 13, 5694–5703 (2005), http://www.opticsexpress.org/ abstract.cfm?URI=oe-13-15-5694.
[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]

Bhagwat, A. R.

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(2), 023603 (pages 4) (2006), http://link.aps.org/abstract/PRL/v97/e023603.
[CrossRef] [PubMed]

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).
[CrossRef]

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]

Blondy, J.-M.

Blow, K. J.

K. J. Blow and D. Wood, “Theoretical Description of Transient Stimulated Raman Scattering in Optical Fibers,” IEEE J. of Quantum Electron. 25, 2665 (1989).
[CrossRef]

Bolger, J.

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).
[CrossRef]

Boyd, R. W.

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]

Butcher, P. N.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, Cambridge CB2 1RP UK, 1990).

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,” App. Phys. B-Lasers and Optics 81, 363–367 (2005).
[CrossRef]

Carlsten, J. L.

Chang, G.

Chen, X.

Chernikov, S. V.

Chinaud, J.

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).
[CrossRef]

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,” Nature photon. 1, 115 (2007).
[CrossRef]

Cotter, D.

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, Cambridge CB2 1RP UK, 1990).

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,” Nature photon. 1, 115 (2007).
[CrossRef]

P. S. Light, F. Benabid, F. Couny, M. Maric, and A. N. Luiten, “Electromagnetically induced transparency in Rb-filled coated hollow-core photonic crystal fiber,” Opt. Lett. 32(10), 1323–1325 (2007), http://ol.osa. org/abstract.cfm?URI=ol-32-10-1323.
[CrossRef] [PubMed]

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), http://www.opticsexpress.org/ abstract.cfm?URI=oe-13-15-5694.
[CrossRef] [PubMed]

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]

Coutaz, J.-L.

Cox, F. M.

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,” Nature photon. 1, 115 (2007).
[CrossRef]

Dadap, J. L.

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,” IEEE Journal of lightwave technology Accepted (2008).

Delaye, P.

Dinu, M.

M. Dinu, F. Quochi, and H. Garcia, “Third-Order Nonlinearities in Silicon at Telecom Wavelengths,” App. Phys. Lett. 82, 2954 (2003).
[CrossRef]

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,” App. Phys. B-Lasers and Optics 81, 363–367 (2005).
[CrossRef]

Dudley, J. M.

Dulkeith, E.

Ebendorff-Heidepriem, H.

J. Y. Y. Leong, P. Petropoulos, J. H. V. Price, H. Ebendorff-Heidepriem, S. Asimakis, R. C. Moore, K. E. Framp-ton, V. Finazzi, X. Feng, T. M. Monro, and D. J. Richardson, “High-Nonlinearity Dispersion-Shifted Lead-Silicate Holey Fibers for Efficient 1-μm Pumped Supercontinuum Generation,” IEEE J. Lightwave Technol. 24, 183 (2006), http://jlt.osa.org/abstract.cfm?URI=JLT-24-1-183.
[CrossRef]

H. Ebendorff-Heidepriem, P. Petropoulos, S. Asimakis, V. Finazzi, R. Moore, K. Frampton, F. Koizumi, D. Richardson, and T. Monro, “Bismuth glass holey fibers with high nonlinearity,” Opt. Express 12, 5082–5087 (2004), http://www.opticsexpress.org/abstract.cfm?URI=oe-12-21-5082.
[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]

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,” IEEE Journal of lightwave technology Accepted (2008).

Eggleton, B.

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 Nonlin-earity in Sub-Wavelength Diameter As2Se3 Chalcogenide Fiber Tapers,” Opt. Express 15, 10,324 (2007).
[CrossRef]

Fauchet, P. M.

Feng, X.

J. Y. Y. Leong, P. Petropoulos, J. H. V. Price, H. Ebendorff-Heidepriem, S. Asimakis, R. C. Moore, K. E. Framp-ton, V. Finazzi, X. Feng, T. M. Monro, and D. J. Richardson, “High-Nonlinearity Dispersion-Shifted Lead-Silicate Holey Fibers for Efficient 1-μm Pumped Supercontinuum Generation,” IEEE J. Lightwave Technol. 24, 183 (2006), http://jlt.osa.org/abstract.cfm?URI=JLT-24-1-183.
[CrossRef]

Février, S.

Finazzi, V.

J. Y. Y. Leong, P. Petropoulos, J. H. V. Price, H. Ebendorff-Heidepriem, S. Asimakis, R. C. Moore, K. E. Framp-ton, V. Finazzi, X. Feng, T. M. Monro, and D. J. Richardson, “High-Nonlinearity Dispersion-Shifted Lead-Silicate Holey Fibers for Efficient 1-μm Pumped Supercontinuum Generation,” IEEE J. Lightwave Technol. 24, 183 (2006), http://jlt.osa.org/abstract.cfm?URI=JLT-24-1-183.
[CrossRef]

H. Ebendorff-Heidepriem, P. Petropoulos, S. Asimakis, V. Finazzi, R. Moore, K. Frampton, F. Koizumi, D. Richardson, and T. Monro, “Bismuth glass holey fibers with high nonlinearity,” Opt. Express 12, 5082–5087 (2004), http://www.opticsexpress.org/abstract.cfm?URI=oe-12-21-5082.
[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]

Fischer, B. M.

Flytzanis, C.

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,” App. Phys. B-Lasers and Optics 81, 363–367 (2005).
[CrossRef]

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(24), 2867–2869 (2004).
[CrossRef]

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

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,” Nature photon. 1, 115 (2007).
[CrossRef]

Frampton, K.

Framp-ton, K. E.

J. Y. Y. Leong, P. Petropoulos, J. H. V. Price, H. Ebendorff-Heidepriem, S. Asimakis, R. C. Moore, K. E. Framp-ton, V. Finazzi, X. Feng, T. M. Monro, and D. J. Richardson, “High-Nonlinearity Dispersion-Shifted Lead-Silicate Holey Fibers for Efficient 1-μm Pumped Supercontinuum Generation,” IEEE J. Lightwave Technol. 24, 183 (2006), http://jlt.osa.org/abstract.cfm?URI=JLT-24-1-183.
[CrossRef]

Freude, W.

Frey, R.

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 Nonlin-earity in Sub-Wavelength Diameter As2Se3 Chalcogenide Fiber Tapers,” Opt. Express 15, 10,324 (2007).
[CrossRef]

Fuerbach, A.

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,” App. Phys. B-Lasers and Optics 81, 363–367 (2005).
[CrossRef]

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(24), 2867–2869 (2004).
[CrossRef]

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

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]

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]

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(2), 023603 (pages 4) (2006), http://link.aps.org/abstract/PRL/v97/e023603.
[CrossRef] [PubMed]

S. Ghosh, J. E. Sharping, D. G. Ouzounov, and A. L. Gaeta, “Resonant Optical Interactions with Molecules Confined in Photonic Band-Gap Fibers,” Phys. Rev. Lett. 94, 093902 (2005), http://link.aps.org/ abstract/PRL/v94/e093902.
[CrossRef] [PubMed]

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

Garcia, H.

M. Dinu, F. Quochi, and H. Garcia, “Third-Order Nonlinearities in Silicon at Telecom Wavelengths,” App. Phys. Lett. 82, 2954 (2003).
[CrossRef]

Gatz, S.

Genty, G.

Geraghty, D. F.

Ghosh, S.

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M. Kolesik and J. V. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E 70, 036,604 (2004).
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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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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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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,” IEEE Journal of lightwave technology Accepted (2008).

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App. Phys. B-Lasers and Optics (1)

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,” App. Phys. B-Lasers and Optics 81, 363–367 (2005).
[CrossRef]

App. Phys. Lett. (1)

M. Dinu, F. Quochi, and H. Garcia, “Third-Order Nonlinearities in Silicon at Telecom Wavelengths,” App. Phys. Lett. 82, 2954 (2003).
[CrossRef]

Appl. Phys. B (2)

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. 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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Electron Lett. (1)

C. A. Barrios, “High-Performance All-Optical Silicon Microswitch,” Electron Lett. 40, 862 (2004).
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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)

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× 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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