Abstract

Nonlinear metamaterials have potentially interesting applications in highly efficient wave-mixing and parametric processes, owing to their ability to combine enhanced nonlinearities with exotic and configurable linear properties. However, the strong dispersion and unconventional configurations typically associated with metamaterials place strong demands on phase matching in such structures. In this paper, we present an overview of potential phase matching solutions for wave-mixing processes in nonlinear metamaterials. Broadly speaking, we divide the phase matching solutions into conventional techniques (anomalous dispersion, birefringence, and quasi-phase matching) and metamaterial-inspired techniques (negative-index and index-near-zero phase matching), offering numerical and experimental examples where possible. We find that not only is phase matching feasible in metamaterials, but metamaterials can support a wide range of phase matching configurations that are otherwise impossible in natural materials. These configurations have their most compelling applications in those devices where at least one of the interacting waves is counter-propagating, such as the mirror-less optical parametric oscillator and the nonlinear optical mirror.

© 2011 OSA

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2011 (4)

D. Huang, A. Rose, E. Poutrina, S. Larouche, and D. R. Smith, “Wave mixing in nonlinear magnetic metacrystal,” Appl. Phys. Lett.98, 204102 (2011).
[CrossRef]

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 366–371 (2011).
[CrossRef]

A. Rose, D. Huang, and D. R. Smith, “Controlling the second harmonic in a phase-matched negative-index metamaterial,” Phys. Rev. Lett.107, 063902 (2011).
[CrossRef] [PubMed]

A. Rose and D. R. Smith, “Broadly tunable quasi-phase-matching in nonlinear metamaterials,” Phys. Rev. A84, 013823 (2011).
[CrossRef]

2010 (4)

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E82, 036608 (2010).
[CrossRef]

D. Huang, E. Poutrina, and D. R. Smith, “Analysis of the power dependent tuning of a varactor-loaded metamaterial at microwave frequencies,” Appl. Phys. Lett.96, 104104 (2010).
[CrossRef]

A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching of momentum and energy in nonlinear optical processes,” Nat. Photonics4, 570–575 (2010).
[CrossRef]

E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” New J. Phys.12, 093010 (2010).
[CrossRef]

2009 (3)

F. Niesler, N. Feth, S. Linden, J. Niegemann, J. Gieseler, K. Busch, and M. Wegener, “Second-harmonic generation from split-ring resonators on a gaas substrate,” Opt. Lett.34, 1997–1999 (2009).
[CrossRef] [PubMed]

T. Driscoll, H. T. Kim, B. G. Chae, B. J. Kim, Y. W. Lee, N. M. Jokerst, S. Palit, D. R. Smith, M. Di Ventra, and D. N. Basov, “Memory metamaterials,” Science325, 1518–1521 (2009).
[CrossRef] [PubMed]

N. M. Litchinitser and V. Shalaev, “Metamaterials: Loss as a route to transparency,” Nat. Photonics3, 75 (2009).
[CrossRef]

2008 (2)

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics2, 295–298 (2008).
[CrossRef]

M. A. Castellanos-Beltran, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Amplification and squeezing of quantum noise with a tunable josephson metamaterial,” Nat. Phys.4, 929–931 (2008).
[CrossRef]

2007 (3)

D. A. Powell, I. V. Shadrivov, Y. S. Kivshar, and M. V. Gorkunov, “Self-tuning mechanisms of nonlinear split-ring resonators,” Appl. Phys. Lett.91, 144107 (2007).
[CrossRef]

A. Degiron, J. J. Mock, and D. R. Smith, “Modulating and tuning the response of metamaterials at the unit cell level,” Opt. Express15, 1115–1127 (2007).
[CrossRef] [PubMed]

C. Canalias and V. Pasiskevicius, “Mirrorless optical parametric oscillator,” Nat. Photonics1, 459–462 (2007).
[CrossRef]

2006 (7)

M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science313, 502–504 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science314, 977–980 (2006).
[CrossRef] [PubMed]

H.-T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature444, 597–600 (2006).
[CrossRef] [PubMed]

A. Popov and V. Shalaev, “Negative-index metamaterials: second-harmonic generation, manley-rowe relations and parametric amplification,” Appl. Phys. B84, 131–137 (2006).
[CrossRef]

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B23, 529–534 (2006).
[CrossRef]

A. K. Popov and V. M. Shalaev, “Compensating losses in negative-index metamaterials by optical parametric amplification,” Opt. Lett.31, 2169–2171 (2006).
[CrossRef] [PubMed]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ɛ-near-zero materials,” Phys. Rev. Lett.97, 157403 (2006).
[CrossRef] [PubMed]

2005 (2)

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E71, 036609 (2005).
[CrossRef]

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E71, 036617 (2005).
[CrossRef]

2004 (2)

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B69, 165112 (2004).
[CrossRef]

R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E70, 046608 (2004).
[CrossRef]

2003 (2)

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90, 077405 (2003).
[CrossRef] [PubMed]

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett.91, 037401 (2003).
[CrossRef] [PubMed]

2002 (1)

I. Shoji, T. Kondo, and R. Ito, “Second-order nonlinear susceptibilities of various dielectric and semiconductor materials,” Opt. Quantum Electron.34, 797–833 (2002).
[CrossRef]

2001 (1)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292, 77–79 (2001).
[CrossRef] [PubMed]

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).
[CrossRef] [PubMed]

1999 (2)

C. Conti, G. Assanto, and S. Trillo, “Cavityless oscillation through backward quasi-phase-matched second-harmonic generation,” Opt. Lett.24, 1139–1141 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech.47, 2075–2084 (1999).
[CrossRef]

1998 (1)

Y. Ding, J. Kang, and J. Khurgin, “Theory of backward second-harmonic and third-harmonic generation using laser pulses in quasi-phase-matched second-order nonlinear medium,” IEEE J. Quantum Electron.34, 966–974 (1998).
[CrossRef]

1997 (1)

1996 (1)

Y. Ding and J. Khurgin, “Backward optical parametric oscillators and amplifiers,” IEEE J. Quantum Electron.32, 1574–1582 (1996).
[CrossRef]

1995 (1)

1992 (1)

M. Fejer, G. Magel, D. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron.28, 2631–2654 (1992).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

1969 (1)

J. G. Meadors, “Steady-state theory of backward-traveling-wave parametric interactions,” J. Appl. Phys.40, 2510–2512 (1969).
[CrossRef]

1966 (1)

S. E. Harris, “Proposed backward wave oscillation in the infrared,” Appl. Phys. Lett.9, 114–166 (1966).
[CrossRef]

1963 (1)

P. A. Franken and J. F. Ward, “Optical harmonics and nonlinear phenomena,” Rev. Mod. Phys.35, 23–39 (1963).
[CrossRef]

1962 (1)

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev.127, 1918–1939 (1962).
[CrossRef]

1961 (1)

P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics,” Phys. Rev. Lett.7, 118–119 (1961).
[CrossRef]

Agranovich, V. M.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B69, 165112 (2004).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev.127, 1918–1939 (1962).
[CrossRef]

Assanto, G.

Averitt, R. D.

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics2, 295–298 (2008).
[CrossRef]

H.-T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature444, 597–600 (2006).
[CrossRef] [PubMed]

Azad, A. K.

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics2, 295–298 (2008).
[CrossRef]

Bahabad, A.

A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching of momentum and energy in nonlinear optical processes,” Nat. Photonics4, 570–575 (2010).
[CrossRef]

Basov, D. N.

T. Driscoll, H. T. Kim, B. G. Chae, B. J. Kim, Y. W. Lee, N. M. Jokerst, S. Palit, D. R. Smith, M. Di Ventra, and D. N. Basov, “Memory metamaterials,” Science325, 1518–1521 (2009).
[CrossRef] [PubMed]

Baughman, R. H.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B69, 165112 (2004).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev.127, 1918–1939 (1962).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic Press, 2008).

Burns, W. K.

Busch, K.

Byer, R.

M. Fejer, G. Magel, D. Jundt, and R. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron.28, 2631–2654 (1992).
[CrossRef]

Cahill, P. A.

Canalias, C.

C. Canalias and V. Pasiskevicius, “Mirrorless optical parametric oscillator,” Nat. Photonics1, 459–462 (2007).
[CrossRef]

Castellanos-Beltran, M. A.

M. A. Castellanos-Beltran, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Amplification and squeezing of quantum noise with a tunable josephson metamaterial,” Nat. Phys.4, 929–931 (2008).
[CrossRef]

Chae, B. G.

T. Driscoll, H. T. Kim, B. G. Chae, B. J. Kim, Y. W. Lee, N. M. Jokerst, S. Palit, D. R. Smith, M. Di Ventra, and D. N. Basov, “Memory metamaterials,” Science325, 1518–1521 (2009).
[CrossRef] [PubMed]

Chen, H.-T.

H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics2, 295–298 (2008).
[CrossRef]

H.-T. Chen, W. J. Padilla, J. M. O. Zide, A. C. Gossard, A. J. Taylor, and R. D. Averitt, “Active terahertz metamaterial devices,” Nature444, 597–600 (2006).
[CrossRef] [PubMed]

Choi, M.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 366–371 (2011).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Ciattoni, A.

M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second and third harmonic generation in a ɛ-near-zero nanolayer,” arXiv:1107.2354 (2011).

Conti, C.

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science314, 977–980 (2006).
[CrossRef] [PubMed]

de Ceglia, D.

M. A. Vincenti, D. de Ceglia, A. Ciattoni, and M. Scalora, “Singularity-driven second and third harmonic generation in a ɛ-near-zero nanolayer,” arXiv:1107.2354 (2011).

Degiron, A.

Di Ventra, M.

T. Driscoll, H. T. Kim, B. G. Chae, B. J. Kim, Y. W. Lee, N. M. Jokerst, S. Palit, D. R. Smith, M. Di Ventra, and D. N. Basov, “Memory metamaterials,” Science325, 1518–1521 (2009).
[CrossRef] [PubMed]

Ding, Y.

Y. Ding, J. Kang, and J. Khurgin, “Theory of backward second-harmonic and third-harmonic generation using laser pulses in quasi-phase-matched second-order nonlinear medium,” IEEE J. Quantum Electron.34, 966–974 (1998).
[CrossRef]

Y. Ding and J. Khurgin, “Backward optical parametric oscillators and amplifiers,” IEEE J. Quantum Electron.32, 1574–1582 (1996).
[CrossRef]

Ding, Y. J.

Dmitriev, V. G.

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, 1997).

Driscoll, T.

T. Driscoll, H. T. Kim, B. G. Chae, B. J. Kim, Y. W. Lee, N. M. Jokerst, S. Palit, D. R. Smith, M. Di Ventra, and D. N. Basov, “Memory metamaterials,” Science325, 1518–1521 (2009).
[CrossRef] [PubMed]

Ducuing, J.

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D. A. Powell, I. V. Shadrivov, Y. S. Kivshar, and M. V. Gorkunov, “Self-tuning mechanisms of nonlinear split-ring resonators,” Appl. Phys. Lett.91, 144107 (2007).
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[CrossRef]

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[CrossRef] [PubMed]

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N. M. Litchinitser and V. Shalaev, “Metamaterials: Loss as a route to transparency,” Nat. Photonics3, 75 (2009).
[CrossRef]

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[CrossRef]

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[CrossRef]

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I. Shoji, T. Kondo, and R. Ito, “Second-order nonlinear susceptibilities of various dielectric and semiconductor materials,” Opt. Quantum Electron.34, 797–833 (2002).
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H.-T. Chen, J. F. O’Hara, A. K. Azad, A. J. Taylor, R. D. Averitt, D. B. Shrekenhamer, and W. J. Padilla, “Experimental demonstration of frequency-agile terahertz metamaterials,” Nat. Photonics2, 295–298 (2008).
[CrossRef]

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M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ɛ-near-zero materials,” Phys. Rev. Lett.97, 157403 (2006).
[CrossRef] [PubMed]

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Smith, D. R.

A. Rose and D. R. Smith, “Broadly tunable quasi-phase-matching in nonlinear metamaterials,” Phys. Rev. A84, 013823 (2011).
[CrossRef]

A. Rose, D. Huang, and D. R. Smith, “Controlling the second harmonic in a phase-matched negative-index metamaterial,” Phys. Rev. Lett.107, 063902 (2011).
[CrossRef] [PubMed]

D. Huang, A. Rose, E. Poutrina, S. Larouche, and D. R. Smith, “Wave mixing in nonlinear magnetic metacrystal,” Appl. Phys. Lett.98, 204102 (2011).
[CrossRef]

D. Huang, E. Poutrina, and D. R. Smith, “Analysis of the power dependent tuning of a varactor-loaded metamaterial at microwave frequencies,” Appl. Phys. Lett.96, 104104 (2010).
[CrossRef]

E. Poutrina, D. Huang, and D. R. Smith, “Analysis of nonlinear electromagnetic metamaterials,” New J. Phys.12, 093010 (2010).
[CrossRef]

A. Rose, S. Larouche, D. Huang, E. Poutrina, and D. R. Smith, “Nonlinear parameter retrieval from three- and four-wave mixing in metamaterials,” Phys. Rev. E82, 036608 (2010).
[CrossRef]

T. Driscoll, H. T. Kim, B. G. Chae, B. J. Kim, Y. W. Lee, N. M. Jokerst, S. Palit, D. R. Smith, M. Di Ventra, and D. N. Basov, “Memory metamaterials,” Science325, 1518–1521 (2009).
[CrossRef] [PubMed]

A. Degiron, J. J. Mock, and D. R. Smith, “Modulating and tuning the response of metamaterials at the unit cell level,” Opt. Express15, 1115–1127 (2007).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science314, 977–980 (2006).
[CrossRef] [PubMed]

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E71, 036617 (2005).
[CrossRef]

D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E71, 036609 (2005).
[CrossRef]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90, 077405 (2003).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science292, 77–79 (2001).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).
[CrossRef] [PubMed]

A. Rose and D. R. Smith, “A quantitative study of the enhancement of bulk nonlinearities in metamaterials,” (submitted to Phys. Rev. A).

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D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E71, 036617 (2005).
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D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science314, 977–980 (2006).
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D. R. Smith, J. J. Mock, A. F. Starr, and D. Schurig, “Gradient index metamaterials,” Phys. Rev. E71, 036609 (2005).
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J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech.47, 2075–2084 (1999).
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Figures (3)

Fig. 1
Fig. 1

A nonlinear metamaterial consisting of overlapping silver bars embedded in a nonlinear dielectric, designed to operate as a birefringence phase matched MOPO. (a) Schematic of a single layer of the metamaterial, showing the propagation direction and angle relative to the crystal axes. (b) The basic unit-cell of the metamaterial. (c) The retrieved extraordinary and ordinary indices of refraction. (d) Plot of the phase matched signal and idler frequencies as a function of angle. (e) Real and imaginary parts of the retrieved nonlinear coefficient.

Fig. 2
Fig. 2

(a) Diagram of the periodic poling technique employed in Ref. [37] for the varactor-loaded split-ring resonator medium, whereby the phase of the nonlinear coefficient is periodically flipped by reversing the orientation of the nonlinear element in each individual unit-cell. (b) Schematic of tunable parallel-I QPM difference frequency generation in an active metamaterial Bragg cell. An external stimulus is used to produce a periodic variation in the linear properties of the metamaterial with a tunable period Λ.

Fig. 3
Fig. 3

(a) Plot of the calculated coherence lengths for parallel-I (dashed purple) and anti-parallel-I (solid orange) second-harmonic generation in a negative-index waveguide. The index of refraction at the fundamental frequency is included (dotted black), showing the negative-index and index-near-zero regimes (vertical dashed lines). (b) Schematic of the nonlinear optical mirror effect in a negative-index metamaterial. (c) Schematic of simultaneous QPM in a periodically poled index-near-zero metamaterial. Si refers to the direction of energy flow of the ith wave.

Tables (1)

Tables Icon

Table 1 Directions of Propagation and Corresponding Phase Matching Condition for the Four Co-linear Three-Wave Mixing Configurations

Equations (8)

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

ω 1 + ω 2 = ω 3
k 1 + k 2 = k 3 ,
μ ( ω ) = μ 0 ( 1 + F ω 2 ω 0 2 ω 2 i γ ω )
1 + F ω 1 2 ω 0 2 ω 1 2 1 + F ω 3 2 ω 0 2 ω 3 2 = ɛ 3 ɛ 1 .
F = δ ɛ 10 + 8 δ ɛ ,
1 ( n i e ( θ ) ) 2 = sin ( θ ) 2 ( n ¯ i e ) 2 + cos ( θ ) 2 ( n i o ) 2 ,
k ( ω ) = ω μ ( ɛ b ( 1 ω c 2 ω 2 ) ) ,
μ = μ 0 ( 1 + F ω 2 ω 0 2 i γ ω ω 2 ) .

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