Abstract

Detailed similarities between polarization states of light and ballistic charge carriers in graphene are derived. Based on these, the optical equivalent of quantum wavefunctions, the Dirac equation, and the effect of an electrostatic potential are found, and the quantum analogs of the refractive index of light and of the optical composition law of reflection coefficients are obtained. The differences between the behaviors of quantum wavefunctions in graphene and electromagnetic fields, due to the chiral symmetry of ballistic charge carriers that cannot be mimicked in classical polarization optics, are also evidenced.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2009 (7)

N. M. R. Peres, “The transport properties of graphene,” J. Phys. Condens. Matter 21, 323201 (2009).
[CrossRef] [PubMed]

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).
[CrossRef]

S. Ghosh and M. Sharma, “Electron optics with magnetic vector potential barriers in graphene,” J. Phys. Condens. Matter 21, 292204 (2009).
[CrossRef] [PubMed]

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydło, “Quantum Goos–Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

H. Kawaguchi, “Polarization-bistable vertical-cavity surface-emitting lasers: application for optical bit memory,” Opto-Electron. Rev. 17, 265–274 (2009).
[CrossRef]

D. Dragoman, “Evidence against Klein paradox in graphene,” Phys. Scr. 79, 015003 (2009).
[CrossRef]

A. Shytov, M. Rudner, N. Gu, M. Katsnelson, and L. Levitov, “Atomic collapse, Lorentz boosts, Klein scattering, and other quantum-relativistic phenomena in graphene,” Solid State Commun. 149, 1087–1093 (2009).
[CrossRef]

2008 (1)

A. V. Shytov, M. S. Rudner, and L. S. Levitov, “Klein backscattering and Fabry–Perot interference in graphene heterojunctions,” Phys. Rev. Lett. 101, 156804 (2008).
[CrossRef] [PubMed]

2007 (3)

D. Dragoman and M. Dragoman, “Giant thermoelectric effect in graphene,” Appl. Phys. Lett. 91, 203116 (2007).
[CrossRef]

V. V. Cheianov, V. Fal’ko, and B. L. Altshuler, “The focusing of electron flow and a Veselago lens in graphene p-n junctions,” Science 315, 1252–1255 (2007).
[CrossRef] [PubMed]

J. Cserti, A. Pályi, and C. Péterfalvi, “Caustics due to a negative refractive index in circular graphene p-n junctions,” Phys. Rev. Lett. 99, 246801 (2007).
[CrossRef]

2006 (1)

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nat. Phys. 2, 620–625 (2006).
[CrossRef]

2004 (3)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
[CrossRef] [PubMed]

R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717–754 (2004).
[CrossRef]

S. Başkal, E. Georgieva, Y. S. Kim, and M. E. Noz, “Lorentz group in classical ray optics,” J. Opt. B: Quantum Semiclassical Opt. 6, S455–S472 (2004).
[CrossRef]

2003 (3)

S. Jiao, W. Yu, G. Stoica, and L. V. Wang, “Contrast mechanisms in polarization-sensitive Mueller-matrix optical coherence tomography and application in burn imaging,” Appl. Opt. 42, 5191–5197 (2003).
[CrossRef] [PubMed]

B. Bêche and E. Gaviot, “Matrix formalism to enhance the concept of effective dielectric constant,” Opt. Commun. 219, 15–19 (2003).
[CrossRef]

A. Y.-G. Fuh, C.-R. Lee, and K.-T. Cheng, “Fast optical recording of polarization holographic grating based on an azo-dye-doped polymer-ball-type polymer-dispersed liquid crystal film,” Jpn. J. Appl. Phys., Part 1 42, 4406–4410 (2003).
[CrossRef]

2001 (1)

1999 (4)

O. V. Angelsky, N. N. Dominikov, P. P. Maksimyak, and T. Tudor, “Experimental revealing of polarization waves,” Appl. Opt. 38, 3112–3117 (1999).
[CrossRef]

D. Han, Y. S. Kim, and M. E. Noz, “Wigner rotations and Iwasawa decompositions in polarization optics,” Phys. Rev. E 60, 1036–1041 (1999).
[CrossRef]

J. J. Monzón and L. L. Sánchez-Soto, “Fully relativisticlike formulation of multilayer optics,” J. Opt. Soc. Am. A 16, 2013–2018 (1999).
[CrossRef]

D. Dragoman and M. Dragoman, “Optical analogue structures to mesoscopic devices,” Prog. Quantum Electron. 23, 131–188 (1999).
[CrossRef]

1998 (2)

H. Kuratsuji and S. Kakigi, “Maxwell–Schrödinger equation for polarized light and evolution of the Stokes parameters,” Phys. Rev. Lett. 80, 1888–1891 (1998).
[CrossRef]

J.-M. Vigoureux and D. van Labeke, “A geometric phase in optical multilayers,” J. Mod. Opt. 45, 2409–2416 (1998).
[CrossRef]

1997 (3)

Ph. Grossel and J.-M. Vigoureux, “Calculation of wave functions and of energy levels: application to multiple quantum wells and continuous potential,” Phys. Rev. A 55, 796–799 (1997).
[CrossRef]

D. Han, Y. S. Kim, and M. E. Noz, “Jones-matrix formalism as a representation of the Lorentz group,” J. Opt. Soc. Am. A 14, 2290–2298 (1997).
[CrossRef]

D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
[CrossRef]

1995 (1)

1992 (1)

1991 (1)

G. N. Henderson, T. K. Gaylord, and E. N. Glytsis, “Ballistic electron transport in semiconductor heterostructures and its analogies in electromagnetic propagation in general dielectrics,” Proc. IEEE 79, 1643–1659 (1991).
[CrossRef]

1990 (1)

S. Datta and B. Das, “Electronic analog of the electro-optic modulator,” Appl. Phys. Lett. 56, 665–667 (1990).
[CrossRef]

Akhmerov, A. R.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydło, “Quantum Goos–Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

Altshuler, B. L.

V. V. Cheianov, V. Fal’ko, and B. L. Altshuler, “The focusing of electron flow and a Veselago lens in graphene p-n junctions,” Science 315, 1252–1255 (2007).
[CrossRef] [PubMed]

Angelsky, O. V.

Baskal, S.

S. Başkal, E. Georgieva, Y. S. Kim, and M. E. Noz, “Lorentz group in classical ray optics,” J. Opt. B: Quantum Semiclassical Opt. 6, S455–S472 (2004).
[CrossRef]

Bêche, B.

B. Bêche and E. Gaviot, “Matrix formalism to enhance the concept of effective dielectric constant,” Opt. Commun. 219, 15–19 (2003).
[CrossRef]

Beenakker, C. W. J.

C. W. J. Beenakker, R. A. Sepkhanov, A. R. Akhmerov, and J. Tworzydło, “Quantum Goos–Hänchen effect in graphene,” Phys. Rev. Lett. 102, 146804 (2009).
[CrossRef] [PubMed]

Biener, G.

Bomzon, Z.

Castro Neto, A. H.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).
[CrossRef]

Cheianov, V. V.

V. V. Cheianov, V. Fal’ko, and B. L. Altshuler, “The focusing of electron flow and a Veselago lens in graphene p-n junctions,” Science 315, 1252–1255 (2007).
[CrossRef] [PubMed]

Cheng, K. -T.

A. Y.-G. Fuh, C.-R. Lee, and K.-T. Cheng, “Fast optical recording of polarization holographic grating based on an azo-dye-doped polymer-ball-type polymer-dispersed liquid crystal film,” Jpn. J. Appl. Phys., Part 1 42, 4406–4410 (2003).
[CrossRef]

Cserti, J.

J. Cserti, A. Pályi, and C. Péterfalvi, “Caustics due to a negative refractive index in circular graphene p-n junctions,” Phys. Rev. Lett. 99, 246801 (2007).
[CrossRef]

Das, B.

S. Datta and B. Das, “Electronic analog of the electro-optic modulator,” Appl. Phys. Lett. 56, 665–667 (1990).
[CrossRef]

Datta, S.

S. Datta and B. Das, “Electronic analog of the electro-optic modulator,” Appl. Phys. Lett. 56, 665–667 (1990).
[CrossRef]

Dominikov, N. N.

Dragoman, D.

D. Dragoman, “Evidence against Klein paradox in graphene,” Phys. Scr. 79, 015003 (2009).
[CrossRef]

D. Dragoman and M. Dragoman, “Giant thermoelectric effect in graphene,” Appl. Phys. Lett. 91, 203116 (2007).
[CrossRef]

D. Dragoman and M. Dragoman, “Optical analogue structures to mesoscopic devices,” Prog. Quantum Electron. 23, 131–188 (1999).
[CrossRef]

D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, 2004).

Dragoman, M.

D. Dragoman and M. Dragoman, “Giant thermoelectric effect in graphene,” Appl. Phys. Lett. 91, 203116 (2007).
[CrossRef]

D. Dragoman and M. Dragoman, “Optical analogue structures to mesoscopic devices,” Prog. Quantum Electron. 23, 131–188 (1999).
[CrossRef]

D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, 2004).

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
[CrossRef] [PubMed]

Fal’ko, V.

V. V. Cheianov, V. Fal’ko, and B. L. Altshuler, “The focusing of electron flow and a Veselago lens in graphene p-n junctions,” Science 315, 1252–1255 (2007).
[CrossRef] [PubMed]

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
[CrossRef] [PubMed]

Fuh, A. Y.-G.

A. Y.-G. Fuh, C.-R. Lee, and K.-T. Cheng, “Fast optical recording of polarization holographic grating based on an azo-dye-doped polymer-ball-type polymer-dispersed liquid crystal film,” Jpn. J. Appl. Phys., Part 1 42, 4406–4410 (2003).
[CrossRef]

Fujiwara, H.

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (Wiley, 2007).

Gaviot, E.

B. Bêche and E. Gaviot, “Matrix formalism to enhance the concept of effective dielectric constant,” Opt. Commun. 219, 15–19 (2003).
[CrossRef]

Gaylord, T. K.

G. N. Henderson, T. K. Gaylord, and E. N. Glytsis, “Ballistic electron transport in semiconductor heterostructures and its analogies in electromagnetic propagation in general dielectrics,” Proc. IEEE 79, 1643–1659 (1991).
[CrossRef]

Geim, A. K.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).
[CrossRef]

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nat. Phys. 2, 620–625 (2006).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
[CrossRef] [PubMed]

Georgieva, E.

S. Başkal, E. Georgieva, Y. S. Kim, and M. E. Noz, “Lorentz group in classical ray optics,” J. Opt. B: Quantum Semiclassical Opt. 6, S455–S472 (2004).
[CrossRef]

Ghosh, S.

S. Ghosh and M. Sharma, “Electron optics with magnetic vector potential barriers in graphene,” J. Phys. Condens. Matter 21, 292204 (2009).
[CrossRef] [PubMed]

Glytsis, E. N.

G. N. Henderson, T. K. Gaylord, and E. N. Glytsis, “Ballistic electron transport in semiconductor heterostructures and its analogies in electromagnetic propagation in general dielectrics,” Proc. IEEE 79, 1643–1659 (1991).
[CrossRef]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
[CrossRef] [PubMed]

Grossel, Ph.

Ph. Grossel and J.-M. Vigoureux, “Calculation of wave functions and of energy levels: application to multiple quantum wells and continuous potential,” Phys. Rev. A 55, 796–799 (1997).
[CrossRef]

Gu, N.

A. Shytov, M. Rudner, N. Gu, M. Katsnelson, and L. Levitov, “Atomic collapse, Lorentz boosts, Klein scattering, and other quantum-relativistic phenomena in graphene,” Solid State Commun. 149, 1087–1093 (2009).
[CrossRef]

Guinea, F.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).
[CrossRef]

Han, D.

D. Han, Y. S. Kim, and M. E. Noz, “Wigner rotations and Iwasawa decompositions in polarization optics,” Phys. Rev. E 60, 1036–1041 (1999).
[CrossRef]

D. Han, Y. S. Kim, and M. E. Noz, “Jones-matrix formalism as a representation of the Lorentz group,” J. Opt. Soc. Am. A 14, 2290–2298 (1997).
[CrossRef]

D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
[CrossRef]

Hasman, E.

Henderson, G. N.

G. N. Henderson, T. K. Gaylord, and E. N. Glytsis, “Ballistic electron transport in semiconductor heterostructures and its analogies in electromagnetic propagation in general dielectrics,” Proc. IEEE 79, 1643–1659 (1991).
[CrossRef]

Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
[CrossRef] [PubMed]

Jiao, S.

Kakigi, S.

H. Kuratsuji and S. Kakigi, “Maxwell–Schrödinger equation for polarized light and evolution of the Stokes parameters,” Phys. Rev. Lett. 80, 1888–1891 (1998).
[CrossRef]

Katsnelson, M.

A. Shytov, M. Rudner, N. Gu, M. Katsnelson, and L. Levitov, “Atomic collapse, Lorentz boosts, Klein scattering, and other quantum-relativistic phenomena in graphene,” Solid State Commun. 149, 1087–1093 (2009).
[CrossRef]

Katsnelson, M. I.

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nat. Phys. 2, 620–625 (2006).
[CrossRef]

Kawaguchi, H.

H. Kawaguchi, “Polarization-bistable vertical-cavity surface-emitting lasers: application for optical bit memory,” Opto-Electron. Rev. 17, 265–274 (2009).
[CrossRef]

Kim, Y. S.

S. Başkal, E. Georgieva, Y. S. Kim, and M. E. Noz, “Lorentz group in classical ray optics,” J. Opt. B: Quantum Semiclassical Opt. 6, S455–S472 (2004).
[CrossRef]

D. Han, Y. S. Kim, and M. E. Noz, “Wigner rotations and Iwasawa decompositions in polarization optics,” Phys. Rev. E 60, 1036–1041 (1999).
[CrossRef]

D. Han, Y. S. Kim, and M. E. Noz, “Jones-matrix formalism as a representation of the Lorentz group,” J. Opt. Soc. Am. A 14, 2290–2298 (1997).
[CrossRef]

D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
[CrossRef]

Kleiner, V.

Kuratsuji, H.

H. Kuratsuji and S. Kakigi, “Maxwell–Schrödinger equation for polarized light and evolution of the Stokes parameters,” Phys. Rev. Lett. 80, 1888–1891 (1998).
[CrossRef]

Lee, C. -R.

A. Y.-G. Fuh, C.-R. Lee, and K.-T. Cheng, “Fast optical recording of polarization holographic grating based on an azo-dye-doped polymer-ball-type polymer-dispersed liquid crystal film,” Jpn. J. Appl. Phys., Part 1 42, 4406–4410 (2003).
[CrossRef]

Levitov, L.

A. Shytov, M. Rudner, N. Gu, M. Katsnelson, and L. Levitov, “Atomic collapse, Lorentz boosts, Klein scattering, and other quantum-relativistic phenomena in graphene,” Solid State Commun. 149, 1087–1093 (2009).
[CrossRef]

Levitov, L. S.

A. V. Shytov, M. S. Rudner, and L. S. Levitov, “Klein backscattering and Fabry–Perot interference in graphene heterojunctions,” Phys. Rev. Lett. 101, 156804 (2008).
[CrossRef] [PubMed]

Maksimyak, P. P.

Monzón, J. J.

Morozov, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
[CrossRef] [PubMed]

Novoselov, K. S.

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).
[CrossRef]

M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, “Chiral tunnelling and the Klein paradox in graphene,” Nat. Phys. 2, 620–625 (2006).
[CrossRef]

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
[CrossRef] [PubMed]

Noz, M. E.

S. Başkal, E. Georgieva, Y. S. Kim, and M. E. Noz, “Lorentz group in classical ray optics,” J. Opt. B: Quantum Semiclassical Opt. 6, S455–S472 (2004).
[CrossRef]

D. Han, Y. S. Kim, and M. E. Noz, “Wigner rotations and Iwasawa decompositions in polarization optics,” Phys. Rev. E 60, 1036–1041 (1999).
[CrossRef]

D. Han, Y. S. Kim, and M. E. Noz, “Jones-matrix formalism as a representation of the Lorentz group,” J. Opt. Soc. Am. A 14, 2290–2298 (1997).
[CrossRef]

D. Han, Y. S. Kim, and M. E. Noz, “Stokes parameters as a Minkowskian four-vector,” Phys. Rev. E 56, 6065–6076 (1997).
[CrossRef]

Pályi, A.

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K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbon films,” Science 306, 666–669 (2004).
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D. Dragoman and M. Dragoman, Quantum-Classical Analogies (Springer, 2004).

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Figures (2)

Fig. 1
Fig. 1

Refraction of a ballistic wavefunction at an interface between an ungated and a gated region: geometry (top) and potential energy profile (bottom).

Fig. 2
Fig. 2

The electromagnetic wave refraction at an abrupt interface (left) should be replaced with the configuration at right in order to obtain the same reflection coefficient as for ballistic charge carriers in graphene.

Equations (30)

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v F σ k ( ψ 1 ψ 2 ) = v F ( 0 k x i k y k x + i k y 0 ) ( ψ 1 ψ 2 ) = ( E V ) ( ψ 1 ψ 2 ) ,
ψ = ( ψ 1 ψ 2 ) = 1 2 ( 1 s   exp ( i φ ) ) exp ( i k x x + i k y y ) ,
J = ( E x E y ) = 1 2 ( 1 exp ( i θ ) ) ,
( ψ + T ) ψ = ( ψ T ) ψ + = 0
M θ 1 2 ( 1 1 ) = ( 1 0 0 exp ( i θ ) ) 1 2 ( 1 1 ) = 1 2 ( 1 exp ( i θ ) ) .
i λ π d d z ( E x E y ) = H opt ( E x E y ) = ( ε ̂ r n 0 2 ) ( E x E y ) = ( α β + i γ β i γ α ) ( E x E y ) ,
Δ ε ̂ = ( 0 β + i γ β i γ 0 ) .
ψ 1 ( x , y ) = exp ( i k y y ) × { exp ( i k 1 x ) + r 12   exp ( i k 1 x ) ,     x < 0 t 12   exp ( i k 2 x ) ,     x 0 , }
ψ 2 ( x , y ) = exp ( i k y y ) × { s 1 [ exp ( i k 1 x + i φ 1 ) r 12   exp ( i k 1 x i φ 1 ) ] ,     x < 0 s 2 t 12   exp ( i k 2 x + i φ 2 ) ,     x 0 , }
r 12 = s 1   exp ( i φ 1 ) s 2   exp ( i φ 2 ) s 1   exp ( i φ 1 ) + s 2   exp ( i φ 2 ) = s 1 ( k x 1 + i k y ) / k F 1 s 2 ( k x 2 + i k y ) / k F 2 s 1 ( k x 1 i k y ) / k F 1 + s 2 ( k x 2 + i k y ) / k F 2 ,
t 12 = 2 s 1   cos   φ 1 s 1   exp ( i φ 1 ) + s 2   exp ( i φ 2 ) = 2 s 1 k x 1 / k F 1 s 1 ( k x 1 i k y ) / k F 1 + s 2 ( k x 2 + i k y ) / k F 2 ,
r 12 em = n 1 n 2 n 1 + n 2 ,     t 12 em = 2 n 1 n 1 + n 2 .
M = ( 1 0 0 sgn ( s 2 / s 1 ) exp [ i ( φ 2 φ 1 ) ] ) ,
M ret = ( 1 0 0 exp ( i Δ θ ) ) ,
k x 1   tan   φ 1 = k x 2   tan   φ 2
( E V 1 v F ) sin   φ 1 = ( E V 2 v F ) sin   φ 2 .
ψ + = 1 2 ( 1 exp ( i φ ) ) exp ( i k x x ) ,
ψ = 1 2 ( 1 exp ( i ϕ ) ) exp ( i k x x )
r 12 = exp ( i φ 1 ) exp ( i φ 2 ) exp ( i φ 1 ) + s 2   exp ( i φ 2 ) ,
t 12 = 2   cos   φ 1 exp ( i φ 1 ) + exp ( i φ 2 ) .
ψ 1 ( x , y ) = exp ( i k y y ) × { exp ( i k 1 x ) + r   exp ( i k 1 x ) ,     x < x 1 A   exp ( i k 2 x ) + B   exp ( i k 2 x ) ,     x 1 x < x 2 t   exp ( i k 3 x ) ,     x x 2 , }
ψ 2 ( x , y ) = exp ( i k y y ) × { s 1 [ exp ( i k 1 x + i φ 1 ) r   exp ( i k 1 x i φ 1 ) ] ,     x < x 1 s 2 [ A   exp ( i k 2 x + i φ 2 ) B   exp ( i k 2 x i φ 2 ) ] ,     x 1 x < x 2 s 3 t   exp ( i k 3 x + i φ 3 ) ,     x x 2 , }
r 23 = s 2   exp ( i φ 2 ) s 3   exp ( i φ 3 ) s 2   exp ( i φ 2 ) + s 3   exp ( i φ 3 ) ,
r 13 = r 12 + r 23   exp ( 2 i α 2 ) exp ( i Φ ) 1 + r 12 r 23   exp ( 2 i α 2 ) exp ( i Φ ) exp ( 2 i α 1 ) ,
Φ = Arg [ s 1   exp ( i φ 1 ) + s 2   exp ( i φ 2 ) s 1   exp ( i φ 1 ) + s 2   exp ( i φ 2 ) ] ,
r em = r 12 em + r 23 em   exp ( i 2 β 2 ) 1 + r 12 em r 23 em   exp ( 2 i β 2 ) exp ( i 2 β 1 ) ,
R = R 1 R 2 = R 1 + R 2 1 + R ¯ 1 R 2
v = v 1 v 2 = v 1 + v 2 1 + v 1 v 2 / c 2 ,
R 1 R 2 = ( R 2 R 1 ) exp ( i 2 ϑ ) ,
ϑ = Arg ( 1 + R 1 R ¯ 2 1 + R ¯ 1 R 2 ) .

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