Abstract

Mueller matrices for waves reflected and transmitted through chiral materials are derived in terms of elements of the like- and cross-linearly polarized reflection and transmission coefficients. The diagonal-like-polarized reflection and transmission coefficients are insensitive to the chiral properties of the material, while the off-diagonal cross-polarized terms are proportional to the chiral parameter to first order. Therefore only the eight quasi-off-diagonal elements of the Mueller matrix are proportional to the chiral parameters. Since the cross-polarized coefficients are in time quadrature with respect to the like-polarized coefficients, two pairs of these quasi-off-diagonal elements are most sensitive to chirality. The waveguide modes that propagate along a chiral slab are derived. Applications are considered for the optimum detection of chiral materials.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. M. P. Silverman, "Reflection and refraction at the surface of a chiral medium: comparison of gyrotropic constitutive relations invariant or noninvariant under a duality transformation: errata," J. Opt. Soc. Am. A 4, 1145 (1987).
    [CrossRef]
  4. A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, 1994), pp. 313-318.
  5. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Vittanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).
  6. D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, "Periodic chiral structures," IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
    [CrossRef]
  7. A. Lakhtakia, V. V. Varadan, and V. K. Varadan, "Scattering by periodic achiral-chiral interfaces," J. Opt. Soc. Am. A 6, 1675-1681 (1989).
    [CrossRef]
  8. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time Harmonic Electromagentic Fields in Chiral Media, Lecture Notes Phys. Series Vol. 335 (Springer-Verlag, 1989).
  9. C. F. Bohren, "Light scattering by an optically active sphere," Chem. Phys. Lett. 29, 458-462 (1974).
    [CrossRef]
  10. C. F. Bohren, "Scattering of electromagnetic waves by an optically active spherical shell," J. Chem. Phys. 62, 1566-1571 (1975).
    [CrossRef]
  11. P. E. Crittenden and E. Bahar, "A modal solution for reflection and transmission at a chiral-chiral interface," Can. J. Phys. 83, 1267-1290 (2005).
    [CrossRef]
  12. L. Barron, Molecular Light Scattering and Optical Activity, 2nd ed. (Cambridge U. Press, 2004).
    [CrossRef]
  13. E. Bahar, "Application of Mueller matrix and near field measurements to detect and identify trace species in drugs and threat agents," Proc. SPIE 5993, 59930G (2005).
    [CrossRef]
  14. E. Georgieva, "Jones and Mueller matrices for specular reflection from a chiral medium: determination of the basic chiral parameters using the elements of the Mueller matrix and experimental configurations to measure the basic chiral parameters," Appl. Opt. 30, 5081-5088 (1991).
    [CrossRef] [PubMed]
  15. S. Manhas, M. K. Swami, P. Puddhwant, N. Ghosh, P. K. Gupa, and K. Singh, "Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry," Opt. Express 16, 190-202 (2006).
    [CrossRef]
  16. W. N. Herman, "Polarization eccentricity of the transverse field for modes in chiral core planar waveguides," J. Opt. Soc. Am. A 18, 2806-2818 (2001).
    [CrossRef]
  17. E. Bahar, R. D. Kubik, and D. R. Alexander, "Use of a new polarimetric optical bistatic scatterometer to measure the transmission and reflection Mueller matrix for arbitrary incident and scatter directions," in Proceedings of the 1992 Scientific Conference on Obscuration and Aerosol Research (Army Research Office, 1993), pp. 61-75.
  18. R. D. Kubik and E. Bahar, "Measurements for a polarimetric optical bistatic scatterometer," in Proceedings of the Combined Optical-Microwave Earth and Atmosphere Sensing (CO-MEAS'93) (IEEE, 1993), pp. 173-176.
    [CrossRef]
  19. G. V. Eleftheriades and K. G. Balmain, Negative Refraction Metamaterials, Fundamental Principles and Applications (IEEE-Wiley Interscience, 2005). This book contains contributions and references from numerous authors.
    [CrossRef]
  20. A. Alu and N. Engheta, "Guided modes in a waveguide filled with a pair of single-negative (SNG), double negative (DNG) and/or double positive (DPS) layers," IEEE Trans. Microwave Theory Tech. MTT 52, 199-210 (2004).
    [CrossRef]

2006 (1)

S. Manhas, M. K. Swami, P. Puddhwant, N. Ghosh, P. K. Gupa, and K. Singh, "Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry," Opt. Express 16, 190-202 (2006).
[CrossRef]

2005 (2)

P. E. Crittenden and E. Bahar, "A modal solution for reflection and transmission at a chiral-chiral interface," Can. J. Phys. 83, 1267-1290 (2005).
[CrossRef]

E. Bahar, "Application of Mueller matrix and near field measurements to detect and identify trace species in drugs and threat agents," Proc. SPIE 5993, 59930G (2005).
[CrossRef]

2004 (1)

A. Alu and N. Engheta, "Guided modes in a waveguide filled with a pair of single-negative (SNG), double negative (DNG) and/or double positive (DPS) layers," IEEE Trans. Microwave Theory Tech. MTT 52, 199-210 (2004).
[CrossRef]

2001 (1)

1999 (1)

1991 (1)

1989 (2)

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, "Periodic chiral structures," IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

A. Lakhtakia, V. V. Varadan, and V. K. Varadan, "Scattering by periodic achiral-chiral interfaces," J. Opt. Soc. Am. A 6, 1675-1681 (1989).
[CrossRef]

1987 (1)

1986 (1)

1975 (1)

C. F. Bohren, "Scattering of electromagnetic waves by an optically active spherical shell," J. Chem. Phys. 62, 1566-1571 (1975).
[CrossRef]

1974 (1)

C. F. Bohren, "Light scattering by an optically active sphere," Chem. Phys. Lett. 29, 458-462 (1974).
[CrossRef]

Alexander, D. R.

E. Bahar, R. D. Kubik, and D. R. Alexander, "Use of a new polarimetric optical bistatic scatterometer to measure the transmission and reflection Mueller matrix for arbitrary incident and scatter directions," in Proceedings of the 1992 Scientific Conference on Obscuration and Aerosol Research (Army Research Office, 1993), pp. 61-75.

Alu, A.

A. Alu and N. Engheta, "Guided modes in a waveguide filled with a pair of single-negative (SNG), double negative (DNG) and/or double positive (DPS) layers," IEEE Trans. Microwave Theory Tech. MTT 52, 199-210 (2004).
[CrossRef]

Bahar, E.

P. E. Crittenden and E. Bahar, "A modal solution for reflection and transmission at a chiral-chiral interface," Can. J. Phys. 83, 1267-1290 (2005).
[CrossRef]

E. Bahar, "Application of Mueller matrix and near field measurements to detect and identify trace species in drugs and threat agents," Proc. SPIE 5993, 59930G (2005).
[CrossRef]

E. Bahar, R. D. Kubik, and D. R. Alexander, "Use of a new polarimetric optical bistatic scatterometer to measure the transmission and reflection Mueller matrix for arbitrary incident and scatter directions," in Proceedings of the 1992 Scientific Conference on Obscuration and Aerosol Research (Army Research Office, 1993), pp. 61-75.

R. D. Kubik and E. Bahar, "Measurements for a polarimetric optical bistatic scatterometer," in Proceedings of the Combined Optical-Microwave Earth and Atmosphere Sensing (CO-MEAS'93) (IEEE, 1993), pp. 173-176.
[CrossRef]

Balmain, K. G.

G. V. Eleftheriades and K. G. Balmain, Negative Refraction Metamaterials, Fundamental Principles and Applications (IEEE-Wiley Interscience, 2005). This book contains contributions and references from numerous authors.
[CrossRef]

Barron, L.

L. Barron, Molecular Light Scattering and Optical Activity, 2nd ed. (Cambridge U. Press, 2004).
[CrossRef]

Bohren, C. F.

C. F. Bohren, "Scattering of electromagnetic waves by an optically active spherical shell," J. Chem. Phys. 62, 1566-1571 (1975).
[CrossRef]

C. F. Bohren, "Light scattering by an optically active sphere," Chem. Phys. Lett. 29, 458-462 (1974).
[CrossRef]

Carrieri, A. H.

Crittenden, P. E.

P. E. Crittenden and E. Bahar, "A modal solution for reflection and transmission at a chiral-chiral interface," Can. J. Phys. 83, 1267-1290 (2005).
[CrossRef]

Eleftheriades, G. V.

G. V. Eleftheriades and K. G. Balmain, Negative Refraction Metamaterials, Fundamental Principles and Applications (IEEE-Wiley Interscience, 2005). This book contains contributions and references from numerous authors.
[CrossRef]

Engheta, N.

A. Alu and N. Engheta, "Guided modes in a waveguide filled with a pair of single-negative (SNG), double negative (DNG) and/or double positive (DPS) layers," IEEE Trans. Microwave Theory Tech. MTT 52, 199-210 (2004).
[CrossRef]

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, "Periodic chiral structures," IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

Georgieva, E.

Ghosh, N.

S. Manhas, M. K. Swami, P. Puddhwant, N. Ghosh, P. K. Gupa, and K. Singh, "Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry," Opt. Express 16, 190-202 (2006).
[CrossRef]

Gupa, P. K.

S. Manhas, M. K. Swami, P. Puddhwant, N. Ghosh, P. K. Gupa, and K. Singh, "Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry," Opt. Express 16, 190-202 (2006).
[CrossRef]

Herman, W. N.

Jaggard, D. L.

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, "Periodic chiral structures," IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

Kim, Y.

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, "Periodic chiral structures," IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

Kowarz, M. W.

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, "Periodic chiral structures," IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

Kubik, R. D.

E. Bahar, R. D. Kubik, and D. R. Alexander, "Use of a new polarimetric optical bistatic scatterometer to measure the transmission and reflection Mueller matrix for arbitrary incident and scatter directions," in Proceedings of the 1992 Scientific Conference on Obscuration and Aerosol Research (Army Research Office, 1993), pp. 61-75.

R. D. Kubik and E. Bahar, "Measurements for a polarimetric optical bistatic scatterometer," in Proceedings of the Combined Optical-Microwave Earth and Atmosphere Sensing (CO-MEAS'93) (IEEE, 1993), pp. 173-176.
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, V. V. Varadan, and V. K. Varadan, "Scattering by periodic achiral-chiral interfaces," J. Opt. Soc. Am. A 6, 1675-1681 (1989).
[CrossRef]

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time Harmonic Electromagentic Fields in Chiral Media, Lecture Notes Phys. Series Vol. 335 (Springer-Verlag, 1989).

A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, 1994), pp. 313-318.

Lindell, I. V.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Vittanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

Liu, J. C.

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, "Periodic chiral structures," IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

Manhas, S.

S. Manhas, M. K. Swami, P. Puddhwant, N. Ghosh, P. K. Gupa, and K. Singh, "Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry," Opt. Express 16, 190-202 (2006).
[CrossRef]

Pelet, P.

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, "Periodic chiral structures," IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

Puddhwant, P.

S. Manhas, M. K. Swami, P. Puddhwant, N. Ghosh, P. K. Gupa, and K. Singh, "Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry," Opt. Express 16, 190-202 (2006).
[CrossRef]

Sihvola, A. H.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Vittanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

Silverman, M. P.

Singh, K.

S. Manhas, M. K. Swami, P. Puddhwant, N. Ghosh, P. K. Gupa, and K. Singh, "Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry," Opt. Express 16, 190-202 (2006).
[CrossRef]

Swami, M. K.

S. Manhas, M. K. Swami, P. Puddhwant, N. Ghosh, P. K. Gupa, and K. Singh, "Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry," Opt. Express 16, 190-202 (2006).
[CrossRef]

Tretyakov, S. A.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Vittanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

Varadan, V. K.

A. Lakhtakia, V. V. Varadan, and V. K. Varadan, "Scattering by periodic achiral-chiral interfaces," J. Opt. Soc. Am. A 6, 1675-1681 (1989).
[CrossRef]

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time Harmonic Electromagentic Fields in Chiral Media, Lecture Notes Phys. Series Vol. 335 (Springer-Verlag, 1989).

Varadan, V. V.

A. Lakhtakia, V. V. Varadan, and V. K. Varadan, "Scattering by periodic achiral-chiral interfaces," J. Opt. Soc. Am. A 6, 1675-1681 (1989).
[CrossRef]

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time Harmonic Electromagentic Fields in Chiral Media, Lecture Notes Phys. Series Vol. 335 (Springer-Verlag, 1989).

Vittanen, A. J.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Vittanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

Appl. Opt. (2)

Can. J. Phys. (1)

P. E. Crittenden and E. Bahar, "A modal solution for reflection and transmission at a chiral-chiral interface," Can. J. Phys. 83, 1267-1290 (2005).
[CrossRef]

Chem. Phys. Lett. (1)

C. F. Bohren, "Light scattering by an optically active sphere," Chem. Phys. Lett. 29, 458-462 (1974).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, and Y. Kim, "Periodic chiral structures," IEEE Trans. Antennas Propag. 37, 1447-1452 (1989).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. Alu and N. Engheta, "Guided modes in a waveguide filled with a pair of single-negative (SNG), double negative (DNG) and/or double positive (DPS) layers," IEEE Trans. Microwave Theory Tech. MTT 52, 199-210 (2004).
[CrossRef]

J. Chem. Phys. (1)

C. F. Bohren, "Scattering of electromagnetic waves by an optically active spherical shell," J. Chem. Phys. 62, 1566-1571 (1975).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Express (1)

S. Manhas, M. K. Swami, P. Puddhwant, N. Ghosh, P. K. Gupa, and K. Singh, "Mueller matrix approach for determination of optical rotation in chiral turbid media in backscattering geometry," Opt. Express 16, 190-202 (2006).
[CrossRef]

Proc. SPIE (1)

E. Bahar, "Application of Mueller matrix and near field measurements to detect and identify trace species in drugs and threat agents," Proc. SPIE 5993, 59930G (2005).
[CrossRef]

Other (7)

L. Barron, Molecular Light Scattering and Optical Activity, 2nd ed. (Cambridge U. Press, 2004).
[CrossRef]

E. Bahar, R. D. Kubik, and D. R. Alexander, "Use of a new polarimetric optical bistatic scatterometer to measure the transmission and reflection Mueller matrix for arbitrary incident and scatter directions," in Proceedings of the 1992 Scientific Conference on Obscuration and Aerosol Research (Army Research Office, 1993), pp. 61-75.

R. D. Kubik and E. Bahar, "Measurements for a polarimetric optical bistatic scatterometer," in Proceedings of the Combined Optical-Microwave Earth and Atmosphere Sensing (CO-MEAS'93) (IEEE, 1993), pp. 173-176.
[CrossRef]

G. V. Eleftheriades and K. G. Balmain, Negative Refraction Metamaterials, Fundamental Principles and Applications (IEEE-Wiley Interscience, 2005). This book contains contributions and references from numerous authors.
[CrossRef]

A. Lakhtakia, Beltrami Fields in Chiral Media (World Scientific, 1994), pp. 313-318.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Vittanen, Electromagnetic Waves in Chiral and Bi-isotropic Media (Artech House, 1994).

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time Harmonic Electromagentic Fields in Chiral Media, Lecture Notes Phys. Series Vol. 335 (Springer-Verlag, 1989).

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Figures (2)

Fig. 1
Fig. 1

Single free-space chiral interface.

Fig. 2
Fig. 2

Chiral slab in free space or bounded by two different media.

Equations (99)

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D = ε ( E + β × E ) ,
B = μ ( H + β × H ) .
E r c = [ E r R E r L ] = [ R R R R R L R L R R L L ] [ E i R E i L ] = R c E i c ,
E t c = [ E t R E t L ] = [ T R R T R L T L R T L L ] [ E i R E i L ] = T c E i c ,
E C = [ E R E L ] = [ 1 j 1 j ] [ E V E H ] = A E L .
E L = A 1 E C = 1 2 [ 1 1 j j ] E C .
R L = [ R V V R V H R H V R H H ] = A 1 R C A ,
T L = [ T V V T V H T H V T H H ] = A 1 T C A .
R C = R C O + β 1 R C O , T C = T C O + β 1 T C O ,
R 10 C O = [ R 10 R R 0 0 R 10 L L ] = 1 2 k 1 T 01 H H T 10 V V tan 2 θ 1 [ 1 0 0 1 ] ,
T 01 H T 10 V = T 01 V T 10 H = 4 cos θ 0 cos θ 1 ( Y 0 cos θ 0 + Y 1 cos θ 1 ) ( Z 0 cos θ 0 + Z 1 cos θ 1 ) .
R 10 R L = R 10 R L = 0 , R 10 R R = R 10 L L .
R L 0 = [ R R R + R R L + R L L + R L R j ( R R R R R L R L L + R L R ) j ( R R R + R R L R L L R L R ) R R R R R L + R L L R L R ] .
R L 0 = [ R V V 0 0 R H H ] .
R R R + R R L = R L L + R L R = R V V ,
R R R R R L = R L L R L R = R H H ,
R 10 V V = Z 0 cos Z 1 cos θ 1 Z 0 cos θ 0 + Z 1 cos θ 1 , R 10 H H = Y 0 cos θ 0 Y 1 cos θ 1 Y 0 cos θ 0 + Y 1 cos θ 1 ,
Y 0 = ε 0 μ 0 = 1 Z 1 , Y 1 = ε 1 μ 1 = 1 Z 1 .
k 0 sin θ 0 = k 1 sin θ 1 ,
k 0 sin θ 0 = γ 1 R sin θ 1 R = γ 1 L sin θ 1 L = k 1 sin θ 1 ,
γ 1 R = k 1 1 k 1 β 1 , γ 1 L = k 1 1 + k 1 β 1 .
β 1 A 1 R C 0 A = j β 1 [ 0 R L L R R R 0 ] = [ 0 R V H R H V 0 ] .
R V H = R H V .
M = [ M T L M T R M B L M B R ] ,
M T L = 1 2 [ R H H 2 + R V V 2 R H H 2 R V V 2 R H H 2 R V V 2 R H H 2 + R V V 2 ] = [ m 11 m 12 m 21 m 22 ] ,
M B R = [ Re [ R V V R H H * ] Im [ R V V R H H * ] Im [ R V V R H H * ] Re [ R V V R H H * ] ] = [ m 33 m 34 m 43 m 44 ] ,
M B L = [ Re [ ( R H H R V V ) R H V * ] Re [ ( R H H + R V V ) R H V * ] Im [ ( R H H + R V V ) R H V * ] Im [ ( R H H R V V ) R H V * ] ] = [ m 31 m 32 m 41 m 42 ] ,
M T R = [ Re [ ( R H H R V V ) R H V * ] Im [ ( R H H + R V V ) R H V * ] Re [ ( R H H + R V V ) R H V * ] Im [ ( R H H R V V ) R H V * ] ] = [ m 13 m 14 m 23 m 24 ] .
T C = 1 2 Y 0 [ ( Y 0 + Y 1 ) ( 1 + R R R ) + ( Y 0 Y 1 ) R L R ( Y 0 Y 1 ) ( 1 + R L L ) + ( Y 0 + Y 1 ) R R L ( Y 0 Y 1 ) ( 1 + R R R ) + ( Y 0 + Y 1 ) R L R ( Y 0 + Y 1 ) ( 1 + R L L ) + ( Y 0 Y 1 ) R R L ] = [ T R R T R L T L R T L L ] .
T L = 1 2 [ T R R + T R L + T L L + T L R j ( T R R T R L T L L + T L R ) j ( T R R + T R L T L L T L R ) T L L T R L + T L L T L R ] .
T L = 1 2 [ 1 + R R R + R L R + 1 + R L L + R R L j ( 1 + R R R + R L R [ 1 + R L L + R R L ] ) Y 1 Y 0 ( 1 + R R R R L R [ 1 + R L L R R L ] ) Y 1 Y o ( 1 + R R R R L R + 1 + R L L R R L ) ] .
T L 0 = [ 1 + R V V 0 0 Y 1 Y 0 ( 1 + R H H ) ] .
1 + R 10 V V = 2 Z 0 cos θ 0 Z 0 cos θ 0 + Z 1 cos θ 1 1 + R 10 H H = 2 Y 0 cos θ 0 Y 0 cos θ 0 + Y 1 cos θ 1 .
T C = [ T R R T R L T L R T L L ] = 1 2 Y 0 [ ( Y 0 + Y 1 ) R R R ( Y 0 Y 1 ) R L L ( Y 0 Y 1 ) R R R ( Y 0 + Y 1 ) R L L ] .
T C = R L L 2 Y 0 [ ( Y 0 + Y 1 ) ( Y 0 Y 1 ) ( Y 0 Y 1 ) ( Y 0 + Y 1 ) ] .
β T L = j β 1 R L L [ 0 1 Y 1 Y 0 0 ] = [ 0 T V H T H V 0 ] .
T V H = T H V ( Z 1 Z 0 ) = j β 1 R L L = R V H .
R = R 10 + T 01 ( I P L R 01 P L R 01 ) 1 P L R 01 P L T 10 ,
T = T 01 [ I P L R 01 P L R 01 ] 1 P L T 10 ,
P L = exp ( j k 1 h cos θ 1 ) [ 1 0 0 1 ] = exp ( j k 1 h cos θ 1 ) I ,
P C = [ exp ( j γ 1 R h cos θ 1 R ) 0 0 exp ( j γ 1 L h cos θ 1 L ) ] .
R C = R 10 C + T 01 C [ I P C R 01 C P C R 01 C ] 1 P C R 01 C P C T 10 C ,
T C = T 01 C [ I P C R 01 C P C R 01 C ] 1 P C T 01 C .
R L = A 1 R C A , T L = A 1 T C A .
I = A A 1
R L = R 10 L + T 01 L [ I P T R 01 L P T R 01 L ] 1 P T R 01 L P T T 10 L ,
T L = T 01 L [ I P T R 01 L P T R 01 L ] 1 P T T 10 L .
P T = A 1 P C A = A 1 [ e j u 1 R h 0 0 e j u 1 L h ] A ,
u 1 R = [ ( γ 1 R ) 2 k 0 2 sin 2 θ 0 ] 1 2 , u 1 L = [ ( γ 1 L ) k 0 2 sin 2 θ 0 ] 1 2 .
u 1 R = k 1 cos θ 1 + k 1 2 β 1 cos θ 1 , u 1 L = k 1 cos θ 1 k 1 2 β 1 cos θ 1 .
P T = exp ( j k 1 cos θ 1 h ) { cos ( k 1 2 β 1 h cos θ 1 ) [ 1 0 0 1 ] + sin ( k 1 2 β 1 h cos θ 1 ) [ 0 1 1 0 ] } P L + P β ,
P β = k 1 2 β 1 h cos θ 1 [ 0 1 1 0 ] exp ( j k 1 cos θ 1 h ) .
P T R L = ( P L + P β ) ( R 0 + R β ) P L R 0 + P L R β + P β R 0 ,
P L R β = exp ( j k 1 cos θ 1 h ) [ 0 R V H R H V 0 ] ,
P β R 0 = exp ( j k 1 cos θ 1 h ) sin ( k 1 2 β 1 h cos θ 1 ) [ 0 1 1 0 ] [ R V V 0 0 R H H ] exp ( j k 1 cos θ 1 h ) ( k 1 2 β 1 h cos θ 1 ) [ 0 R H H R V V 0 ] .
M H V = R H V R V V ( k 1 2 β 1 h cos θ 1 ) ,
M V H = R V H + R H H ( k 1 2 β 1 h cos θ 1 ) .
P L R 10 β + P β R 10 0 = exp ( i k 1 cos θ 1 h ) [ 0 M 10 V H M 10 H V 0 ] .
P T R 01 L P T R 01 L = exp ( 2 j k 1 cos θ 1 h ) [ R 01 0 + M 01 ] [ R 01 0 + M 01 ]
exp ( 2 j k 1 cos θ 1 h ) ( ( R 01 0 ) 2 + R 01 0 M 01 + M 01 R 01 0 )
= exp ( 2 k 1 cos θ 1 h ) [ [ ( R 01 V V ) 2 0 0 ( R 01 H H ) 2 ] + ( R 01 V V + R 01 H H ) [ 0 M 01 V H M 01 H V 0 ] ] .
R L = R 10 L + T 01 L [ I exp ( 2 j k 1 cos θ 1 h ) ( R 01 0 ) 2 + ( R 01 V V + R 01 H H ) M 01 ] 1 exp ( 2 j k 1 cos θ 1 h ) [ R 01 0 + M 01 ] [ T 10 0 + N 10 ] ,
T L = T 01 L [ I exp ( 2 j k 1 cos θ 1 h ) ( R 01 0 ) 2 + ( R 01 V V + R 01 H H ) M 01 ] 1 exp ( j k 1 cos θ 1 h ) [ T 10 0 + N 10 ] ,
P L T 10 β + P β T 10 0 = exp ( j k 1 cos θ 1 h ) [ 0 N 10 V H N 10 H V 0 ] .
( 1 + R V V ) = T V V
R H V = Z 1 Z 0 T H V
( 1 + R H H ) = Z 1 Z 0 T H H
R V H = T V H
[ I P C R 01 C P C R 01 C ] ,
[ I P T R 01 L P T R 01 L ] .
I exp ( j 2 k 1 h cos θ 1 ) [ ( R 01 0 ) 2 + ( R 01 V V + R 01 H H ) M 01 ] ,
( 1 exp ( j 2 k 1 h cos θ 1 ) ( R 01 V V ) 2 ) ( 1 exp ( j 2 k 1 h cos θ 1 ) ( R 01 H H ) 2 ) = [ ( R 01 V V + R 01 H H ) 2 M 01 V H M 01 H V ] exp ( j 4 k 1 h cos θ 1 ) .
u 1 = k 1 cos θ 1 = u 1 0 + u 1 β 1 .
u 1 0 V = ( n π j ln R 01 V V ) h , u 1 0 H = ( n π j ln R 01 H H ) h ,
ln R 01 P P = ln ( R 01 P P e j ϕ 01 V V ) = ln R 01 V V + j ϕ 01 V V .
[ exp { i 2 ( u 1 0 V + u 1 V β 1 ) } ( R 01 V V ) 2 ] [ exp { i 2 ( u 1 0 H + u 1 H β 1 ) ( R 01 H H ) 2 ] = ( R 01 V V + R 01 H H ) 2 M 01 V H M 01 H = ( R 01 V V R 01 H H ) 2 [ exp ( i 2 u 1 V β 1 ) 1 ] [ exp ( i 2 u 1 H β 1 ) 1 ] = ( R 01 V V + R 01 H H ) 2 M 01 V H M 01 H V .
u 1 V u 1 H = ( R 01 V V + R 01 H H R 01 V V R 01 H H ) 2 M 01 V H M 01 H V β 1 2 .
u 1 V = j ( R 01 V V + R 01 H H ) R 01 V V R 01 H H M 01 H V β 1 , u 1 H = j ( R 01 V V + R 01 H H ) R 01 V V R 01 H H M 01 V H β 1 ,
u 1 V = u 1 0 V + β 1 u V = ( n π j ln R 01 V V ) h j β 1 [ ( R 01 V V + R 01 H H R 01 V V R 01 H H ) [ R V V ( k 1 2 h cos θ 1 ) R H V β 1 ] ]
u 1 H = u 1 O H + β 1 u H = ( n π j ln R 01 H H ) h + j β 1 [ ( R 01 V V + R 01 H H R 01 V V R 01 H H ) [ R 01 H H ( k 1 2 h cos θ 1 ) R 01 V H β 1 ] ] .
u 1 = u 1 V = n π h = u 1 H = ( m π + π ) h = n π h , n = 1 , 2 , 3 .
v 1 R = γ 1 R sin θ 1 R , v 1 L = γ 1 L sin θ 1 L , u 1 R = u 1 L = γ 1 R cos θ 1 R = γ 1 L cos θ 1 L = ( n π h ) = u .
( 2 π h ) > k 1 1 k 1 β 1 > ( π h ) > k 1 1 + k 1 β 1 ,
R = R 10 L + T 01 V [ 1 P T R 21 L P T R 01 L ] 1 P T R 21 L P T T 10 ,
T = T 21 L [ 1 P T R 01 L P T R 21 L ] 1 P T T 10 L .
R 0 R R D = [ ( w 0 + w 0 ) 2 ( w o ξ r w 1 + ) ( w 0 η r w 1 + ) ] ,
R 0 L R D = 2 s 0 ( ξ r η r ) w 1 + ,
T 0 R R D = 4 s 0 ( t 0 + t 1 ) ( 1 + η r ) ,
T 0 L R D = 4 s 0 ( t 0 s 1 ) ( 1 η r ) ,
D = ( w 0 + w 1 ) 2 ( w 0 + + ξ r w 1 + ) ( w 0 + + η r w 1 + ) ,
s n = u 1 n γ 1 n , t n = u 2 n γ 2 n ,
w n ± = s n ± t n .
η r = η 0 η 1 , η n = ( μ n ε n ) 1 2 ,
γ 1 = k ( 1 k β ) , γ 2 = k ( 1 + k β ) ,
u n = ( γ n 2 q 2 ) 1 2 .
R 0 L L ( γ 1 , γ 2 ) = R 0 R R ( γ 2 , γ 1 ) ,
R 0 R L ( γ 1 , γ 2 ) = R 0 L R ( γ 2 , γ 1 ) ,
T 0 L L ( γ 1 , γ 2 ) = T 0 R R ( γ 2 , γ 1 ) ,
T 0 R L ( γ 1 , γ 2 ) = T 0 L R ( γ 2 , γ 1 ) .

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