Abstract

We study the transmission properties of structures with one or two kinds of lossy single-negative (permittivity-negative and permeability-negative) material. Analytic results show that the transmission of the structure depends on the material absorption and reflection. In sharp contrast to lossy dielectrics, the reflection of the lossy single-negative material(s) can decrease as the dissipation coefficient increases. As a result, the transmission of the lossy single-negative material(s) will be nonmonotonic as the dissipation coefficient varies. In particular, the transmission can be enhanced even when the dissipation coefficient increases.

© 2009 Optical Society of America

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  1. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  2. A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
    [CrossRef]
  3. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  4. D. R. Fredkin and A. Ron, “Effectively left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753-1755 (2002).
    [CrossRef]
  5. H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
    [CrossRef]
  6. L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single negative materials,” Phys. Rev. B 70, 245102 (2004).
    [CrossRef]
  7. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
    [CrossRef] [PubMed]
  8. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335-338 (2005).
    [CrossRef] [PubMed]
  9. V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356-3358 (2005).
    [CrossRef]
  10. G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892-893 (2006).
    [CrossRef] [PubMed]
  11. M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12, 4072-4079 (2004).
    [CrossRef] [PubMed]
  12. 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]
  13. A. K. Popov and V. M. Shalaev, “Negative-index metamaterials: second-harmonic generation, Manley-Rowe relations and parametric amplification,” Appl. Phys. B 84, 131-137 (2006).
    [CrossRef]
  14. M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).
    [CrossRef]
  15. I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
    [CrossRef] [PubMed]
  16. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).
  17. E.D.Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

2007

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

2006

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
[CrossRef] [PubMed]

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892-893 (2006).
[CrossRef] [PubMed]

A. K. Popov and V. M. Shalaev, “Negative-index metamaterials: second-harmonic generation, Manley-Rowe relations and parametric amplification,” Appl. Phys. B 84, 131-137 (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]

2005

V. M. Shalaev, W. S. Cai, U. K. Chettiar, H. K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356-3358 (2005).
[CrossRef]

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

2004

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

M. P. Nezhad, K. Tetz, and Y. Fainman, “Gain assisted propagation of surface plasmon polaritons on planar metallic waveguides,” Opt. Express 12, 4072-4079 (2004).
[CrossRef] [PubMed]

2003

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

2002

D. R. Fredkin and A. Ron, “Effectively left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753-1755 (2002).
[CrossRef]

2001

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

2000

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Alù, A.

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Brueck, S. R. J.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Cai, W. S.

Chen, H.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Chettiar, U. K.

Dolling, G.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892-893 (2006).
[CrossRef] [PubMed]

Drachev, V. P.

Engheta, N.

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892-893 (2006).
[CrossRef] [PubMed]

Fainman, Y.

Fan, W.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Firsov, A. A.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Fredkin, D. R.

D. R. Fredkin and A. Ron, “Effectively left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753-1755 (2002).
[CrossRef]

Geim, A. K.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Gleeson, H. F.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Grigorenko, A. N.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Hooper, I. R.

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
[CrossRef] [PubMed]

Jiang, H. T.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Khrushchev, I. Y.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Kildishev, A. V.

Li, H. Q.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Linden, S.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892-893 (2006).
[CrossRef] [PubMed]

Malloy, K. J.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Nezhad, M. P.

Osgood, R. M.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Panoiu, N. C.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Petrovic, J.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Popov, A. K.

A. K. Popov and V. M. Shalaev, “Negative-index metamaterials: second-harmonic generation, Manley-Rowe relations and parametric amplification,” Appl. Phys. B 84, 131-137 (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]

Preist, T. W.

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
[CrossRef] [PubMed]

Ron, A.

D. R. Fredkin and A. Ron, “Effectively left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753-1755 (2002).
[CrossRef]

Sambles, J. R.

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
[CrossRef] [PubMed]

Sarychev, A. K.

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Shalaev, V. M.

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Soukoulis, C. M.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892-893 (2006).
[CrossRef] [PubMed]

Stockman, M. I.

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

Tetz, K.

Wang, L. G.

L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Wegener, M.

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892-893 (2006).
[CrossRef] [PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

Yuan, H. K.

Zhang, S.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

Zhang, Y.

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Zhang, Y. W.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Zhu, S. Y.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Zi, J.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Appl. Phys. B

A. K. Popov and V. M. Shalaev, “Negative-index metamaterials: second-harmonic generation, Manley-Rowe relations and parametric amplification,” Appl. Phys. B 84, 131-137 (2006).
[CrossRef]

Appl. Phys. Lett.

D. R. Fredkin and A. Ron, “Effectively left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753-1755 (2002).
[CrossRef]

IEEE Trans. Antennas Propag.

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Nature

A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature 438, 335-338 (2005).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. B

L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Phys. Rev. E

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, J. Zi, and S. Y. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Phys. Rev. Lett.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

M. I. Stockman, “Criterion for negative refraction with low optical losses from a fundamental principle of causality,” Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
[CrossRef] [PubMed]

Science

G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Simultaneous negative phase and group velocity of light in a metamaterial,” Science 312, 892-893 (2006).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[CrossRef] [PubMed]

Other

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980).

E.D.Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

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Figures (4)

Fig. 1
Fig. 1

Schematic of a monolayer of a lossy single-negative material in the air.

Fig. 2
Fig. 2

Transmittance (solid curve), reflectance (dashed), and absorbance (dotted) of the lossy ENM monolayer with thickness d 1 = 15 mm at ω 2 π = 0.8 GHz .

Fig. 3
Fig. 3

Transmittance (solid curve), reflectance (dashed), and absorbance (dotted) of the lossy ENM-MNM bilayer with γ e 2 π = 0.2 GHz at ω 2 π = 1 GHz . d 1 = d 2 = 10 mm .

Fig. 4
Fig. 4

Transmittance (solid curve), reflectance (dashed), and absorbance (dotted) of lossy ENM-MNM bilayer with d 2 at ω 2 π = 1.05 GHz . γ e 2 π = 0.01 GHz , γ m 2 π = 0.3 GHz , and d 1 = 15 mm .

Equations (29)

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

R = ( 1 n R ) 2 + n I 2 ( 1 + n R ) 2 + n I 2 ,
R 2 R 1 ( n 2 I n 1 I ) ( n 2 I + n 1 I ) ,
R = ρ A 2 exp ( 2 n 1 I β 1 ) + ρ B 2 exp ( 2 n 1 I β 1 ) + 2 ρ A ρ B cos ( ϕ B ϕ A 2 n 1 R β 1 ) exp ( 2 n 1 I β 1 ) + ρ A 2 ρ B 2 exp ( 2 n 1 I β 1 ) + 2 ρ A ρ B cos ( ϕ B + ϕ A 2 n 1 R β 1 ) ,
T = τ A 2 τ B 2 exp ( 2 n 1 I β 1 ) 1 + ρ A 2 ρ B 2 exp ( 4 n 1 I β 1 ) + 2 ρ A ρ B exp ( 2 n 1 I β 1 ) cos ( ϕ B + ϕ A 2 n 1 R β 1 ) ,
ρ A 2 = ρ B 2 = ( μ 1 n 1 R ) 2 + n 1 I 2 ( μ 1 + n 1 R ) 2 + n 1 I 2 ,
τ A 2 = 4 μ 1 2 ( μ 1 + n 1 R ) 2 + n 1 I 2 , τ B 2 = 4 ( n 1 R 2 + n 1 I 2 ) ( μ 1 + n 1 R ) 2 + n 1 I 2 ,
ϕ B = arctan ( 2 μ 1 n 1 I n 1 R 2 + n 1 I 2 μ 1 2 ) ,
ɛ 1 = 1 ω ep 2 ω 2 + i ω γ e , μ 1 = a ,
μ 2 = 1 ω m p 2 ω 2 + i ω γ m , ɛ 2 = b ,
ɛ 1 R = 1 ω ep 2 ω 2 + γ e 2 , ɛ 1 I = ω ep 2 γ e ω 3 + ω γ e 2 .
n 1 R = [ 1 2 μ 1 ( ɛ 1 R 2 + ɛ 1 I 2 + ɛ 1 R ) ] 1 2 ,
n 1 I = [ 1 2 μ 1 ( ɛ 1 R 2 + ɛ 1 I 2 ɛ 1 R ) ] 1 2 .
ρ A 2 = 1 ( μ 1 + n 1 R ) 2 ( μ 1 n 1 R ) 2 ( μ 1 + n 1 R ) 2 + n 1 I 2
ρ A 2 = 1 4 μ 1 n 1 R ( μ 1 + n 1 R ) 2 + n 1 I 2 ,
r = μ 1 n 0 cos θ 0 μ 0 n 1 cos θ 1 μ 1 n 0 cos θ 0 + μ 0 n 1 cos θ 1 ,
r = μ 0 n 1 cos θ 0 μ 1 n 0 cos θ 1 μ 0 n 1 cos θ 0 + μ 1 n 0 cos θ 1 ,
t = 2 μ 1 n 0 cos θ 0 μ 1 n 0 cos θ 0 + μ 0 n 1 cos θ 1 ,
t = 2 μ 1 n 0 cos θ 0 μ 0 n 1 cos θ 0 + μ 1 n 0 cos θ 1 ,
r A = μ 1 n 1 μ 1 + n 1 , r B = n 1 μ 1 μ 1 + n 1 , t A = 2 μ 1 μ 1 + n 1 , t B = 2 n 1 μ 1 + n 1 .
ρ A 2 = ( μ 1 n 1 R ) 2 + n 1 I 2 ( μ 1 + n 1 R ) 2 + n 1 I 2 , tan ϕ A = 2 μ 1 n 1 I n 1 R 2 + n 1 I 2 μ 1 2 ,
ρ B 2 = ( n 1 R μ 1 ) 2 + n 1 I 2 ( n 1 R + μ 1 ) 2 + n 1 I 2 , tan ϕ B = 2 μ 1 n 1 I n 1 R 2 + n 1 I 2 μ 1 2 ,
τ A 2 = 4 μ 1 2 ( μ 1 + n 1 R ) 2 + n 1 I 2 , tan χ A = μ 1 n 1 I μ 1 2 + μ 1 n 1 R ,
τ B 2 = 4 ( n 1 R 2 + n 1 I 2 ) ( μ 1 + n 1 R ) 2 + n 1 I 2 , tan χ B = μ 1 n 1 I n 1 R 2 + n 1 I 2 + μ 1 n 1 R ,
r = r A + r B exp ( 2 i δ 1 ) 1 + r A r B exp ( 2 i δ 1 ) = ρ exp ( i ϕ ) ,
t = t A t B exp ( i δ 1 ) 1 + r A r B exp ( 2 i δ 1 ) = τ exp ( i χ ) ,
d ( ɛ 1 R 2 + ɛ 1 I 2 + ɛ 1 R ) d γ e .
d a d γ e = 1 1 + ω ep 2 ( ω ep 2 2 ω 2 ) ω 2 ( ω 2 + γ e 2 ) γ e ω ep 2 ( 2 ω 2 ω ep 2 ) ω 2 ( ω 2 + γ e 2 ) 2 ,
d b d γ e = 2 ω ep 2 γ e ( ω 2 + γ e 2 ) 2 .
( ω ep 2 2 ω 2 ) ( γ e 2 3 ω 2 ) 2 ω 2 ( ω 2 + γ e 2 ) 0 .

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