Abstract

When the real parts of the permittivity and the permeability dyadics of a structurally chiral, magnetic-dielectric material are reversed in sign, the circular Bragg phenomenon displayed by the material is proved here to suffer a change which indicates that the structural handedness has been, in effect, reversed. Additionally, reflection and transmission coefficients suffer phase reversal.

© 2003 Optical Society of America

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References

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  1. N. Kato, “The significance of Ewald’s dynamical theory of diffraction,” in P.P. Ewald and His Dynamical Theory of X-ray Diffraction (D. W. J. Cruickshank, H. J. Juretschke, and N. Kato, eds) (Oxford University Press, Oxford, UK, 1992), pp. 3–23.
  2. H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, Bristol, UK, 2001), pp. 185–208.
  3. I. J. Hodgkinson and Q. h. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1997), pp. 302–322.
  4. S. D. Jacobs (ed), Selected Papers on Liquid Crystals for Optics (SPIEO ptical Engineering Press, Bellingham, WA, USA, 1992).
  5. V. C. Venugopal and A. Lakhtakia, “Sculptured thin films: Conception, optical properties, and applications,” in Electromagnetic Fields in Unconventional Materials and Structures (O. N. Singh and A. Lakhtakia, eds) (Wiley, New York, NY, USA, 2000), pp. 151–216.
  6. J. Wang, A. Lakhtakia, and J. B. Geddes III, “Multiple Bragg regimes exhibited by a chiral sculptured thin film half-space on axial excitation,” Optik 113, 213–222 (2002).
    [Crossref]
  7. A. Lakhtakia, “Sculptured thin films: accomplishments and emerging uses,” Mater. Sci. Engg. C 19, 427–434 (2002).
    [Crossref]
  8. J. B. Geddes III and A. Lakhtakia, “Reflection and transmission of optical narrow-extent pulses by axially excited chiral sculptured thin films,” Eur. Phys. J. Appl. Phys.13, 3–14 (2001); corrections: 16, 247 (2001).
    [Crossref]
  9. http://www.esm.psu.edu/HTMLs/Faculty/Lakhtakia/TimeBragg/TD Bragg.html
  10. H. Takezoe, K. Hashimoto, Y. Ouchi, M. Hara, A. Fukuda, and E. Kuze, “Experimental study on higher order reflection by monodomain cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 101, 329–340 (1983).
    [Crossref]
  11. V. C. Venugopal and A. Lakhtakia, “Electromagnetic plane-wave response characteristics of nonaxially excited slabs of dielectric thin-film helicoidal bianisotropic mediums,” Proc. R. Soc. Lond. A 456, 125–161 (2000).
    [Crossref]
  12. A. Lakhtakia and W. S. Weiglhofer, “Further results on light propagation in helicoidal bianisotropic mediums: oblique propagation,” Proc. R. Soc. Lond. A453, 93–105 (1997); corrections: 454, 3275 (1998).
    [Crossref]
  13. F. Brochard and P.G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. (Paris) 31, 691–708 (1970).
    [Crossref]
  14. A. Lakhtakia, “Reversal of circular Bragg phenomenon in ferrocholesteric materials with negative real permittivities and permeabilities,” Adv. Mater. 14, 447–449 (2002).
    [Crossref]
  15. V. Ponsinet, P. Fabre, M. Veyssie, and L. Auvray, “A small-angle neutron-scattering study of the ferrosmectic phase,” J. Phys. II (Paris) 3, 1021–1039 (1993).
  16. J. Pendry, “Electromagnetic materials enter the negative age,” Physics World14 (9), 47–51 (2001), September issue.
  17. A. Lakhtakia, M. W. McCall, and W. S. Weiglhofer, “Brief overview of recent developments on negative phase-velocity mediums (alias left-handed materials),” Arch. Elektr. Über. 56, 407–410 (2002).
  18. M. Schubert and C. M. Herzinger, “Ellipsometry on anisotropic materials: Bragg conditions and phonons in dielectric helical thin films,” Phys. Stat. Sol. (a) 188, 1563–1575 (2001).
    [Crossref]
  19. F. de Fornel, Evanescent Waves (Springer, Berlin, Germany, 2001), pp. 12–18.
  20. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  21. S. F. Nagle, A. Lakhtakia, and W. Thompson, Jr., “Modal structures for axial wave propagation in a continuously twisted structurally chiral medium (CTSCM),” J. Acoust. Soc. Am. 97, 42–50 (1995).
    [Crossref]
  22. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, NY, USA, 1953), Sec. 4.3.
  23. A. Lakhtakia, “On planewave remittances and Goos-Hänchen shifts of planar slabs with negative real permittivity and permeability,” Electromagnetics 23, 71–75 (2003).
    [Crossref]

2003 (1)

A. Lakhtakia, “On planewave remittances and Goos-Hänchen shifts of planar slabs with negative real permittivity and permeability,” Electromagnetics 23, 71–75 (2003).
[Crossref]

2002 (4)

J. Wang, A. Lakhtakia, and J. B. Geddes III, “Multiple Bragg regimes exhibited by a chiral sculptured thin film half-space on axial excitation,” Optik 113, 213–222 (2002).
[Crossref]

A. Lakhtakia, “Sculptured thin films: accomplishments and emerging uses,” Mater. Sci. Engg. C 19, 427–434 (2002).
[Crossref]

A. Lakhtakia, “Reversal of circular Bragg phenomenon in ferrocholesteric materials with negative real permittivities and permeabilities,” Adv. Mater. 14, 447–449 (2002).
[Crossref]

A. Lakhtakia, M. W. McCall, and W. S. Weiglhofer, “Brief overview of recent developments on negative phase-velocity mediums (alias left-handed materials),” Arch. Elektr. Über. 56, 407–410 (2002).

2001 (1)

M. Schubert and C. M. Herzinger, “Ellipsometry on anisotropic materials: Bragg conditions and phonons in dielectric helical thin films,” Phys. Stat. Sol. (a) 188, 1563–1575 (2001).
[Crossref]

2000 (1)

V. C. Venugopal and A. Lakhtakia, “Electromagnetic plane-wave response characteristics of nonaxially excited slabs of dielectric thin-film helicoidal bianisotropic mediums,” Proc. R. Soc. Lond. A 456, 125–161 (2000).
[Crossref]

1995 (1)

S. F. Nagle, A. Lakhtakia, and W. Thompson, Jr., “Modal structures for axial wave propagation in a continuously twisted structurally chiral medium (CTSCM),” J. Acoust. Soc. Am. 97, 42–50 (1995).
[Crossref]

1993 (1)

V. Ponsinet, P. Fabre, M. Veyssie, and L. Auvray, “A small-angle neutron-scattering study of the ferrosmectic phase,” J. Phys. II (Paris) 3, 1021–1039 (1993).

1983 (1)

H. Takezoe, K. Hashimoto, Y. Ouchi, M. Hara, A. Fukuda, and E. Kuze, “Experimental study on higher order reflection by monodomain cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 101, 329–340 (1983).
[Crossref]

1970 (1)

F. Brochard and P.G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. (Paris) 31, 691–708 (1970).
[Crossref]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Auvray, L.

V. Ponsinet, P. Fabre, M. Veyssie, and L. Auvray, “A small-angle neutron-scattering study of the ferrosmectic phase,” J. Phys. II (Paris) 3, 1021–1039 (1993).

Brochard, F.

F. Brochard and P.G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. (Paris) 31, 691–708 (1970).
[Crossref]

de Fornel, F.

F. de Fornel, Evanescent Waves (Springer, Berlin, Germany, 2001), pp. 12–18.

de Gennes, P.G.

F. Brochard and P.G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. (Paris) 31, 691–708 (1970).
[Crossref]

Fabre, P.

V. Ponsinet, P. Fabre, M. Veyssie, and L. Auvray, “A small-angle neutron-scattering study of the ferrosmectic phase,” J. Phys. II (Paris) 3, 1021–1039 (1993).

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, NY, USA, 1953), Sec. 4.3.

Fukuda, A.

H. Takezoe, K. Hashimoto, Y. Ouchi, M. Hara, A. Fukuda, and E. Kuze, “Experimental study on higher order reflection by monodomain cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 101, 329–340 (1983).
[Crossref]

Geddes III, J. B.

J. Wang, A. Lakhtakia, and J. B. Geddes III, “Multiple Bragg regimes exhibited by a chiral sculptured thin film half-space on axial excitation,” Optik 113, 213–222 (2002).
[Crossref]

J. B. Geddes III and A. Lakhtakia, “Reflection and transmission of optical narrow-extent pulses by axially excited chiral sculptured thin films,” Eur. Phys. J. Appl. Phys.13, 3–14 (2001); corrections: 16, 247 (2001).
[Crossref]

Hara, M.

H. Takezoe, K. Hashimoto, Y. Ouchi, M. Hara, A. Fukuda, and E. Kuze, “Experimental study on higher order reflection by monodomain cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 101, 329–340 (1983).
[Crossref]

Hashimoto, K.

H. Takezoe, K. Hashimoto, Y. Ouchi, M. Hara, A. Fukuda, and E. Kuze, “Experimental study on higher order reflection by monodomain cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 101, 329–340 (1983).
[Crossref]

Herzinger, C. M.

M. Schubert and C. M. Herzinger, “Ellipsometry on anisotropic materials: Bragg conditions and phonons in dielectric helical thin films,” Phys. Stat. Sol. (a) 188, 1563–1575 (2001).
[Crossref]

Hodgkinson, I. J.

I. J. Hodgkinson and Q. h. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1997), pp. 302–322.

Kato, N.

N. Kato, “The significance of Ewald’s dynamical theory of diffraction,” in P.P. Ewald and His Dynamical Theory of X-ray Diffraction (D. W. J. Cruickshank, H. J. Juretschke, and N. Kato, eds) (Oxford University Press, Oxford, UK, 1992), pp. 3–23.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kuze, E.

H. Takezoe, K. Hashimoto, Y. Ouchi, M. Hara, A. Fukuda, and E. Kuze, “Experimental study on higher order reflection by monodomain cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 101, 329–340 (1983).
[Crossref]

Lakhtakia, A.

A. Lakhtakia, “On planewave remittances and Goos-Hänchen shifts of planar slabs with negative real permittivity and permeability,” Electromagnetics 23, 71–75 (2003).
[Crossref]

A. Lakhtakia, “Sculptured thin films: accomplishments and emerging uses,” Mater. Sci. Engg. C 19, 427–434 (2002).
[Crossref]

J. Wang, A. Lakhtakia, and J. B. Geddes III, “Multiple Bragg regimes exhibited by a chiral sculptured thin film half-space on axial excitation,” Optik 113, 213–222 (2002).
[Crossref]

A. Lakhtakia, M. W. McCall, and W. S. Weiglhofer, “Brief overview of recent developments on negative phase-velocity mediums (alias left-handed materials),” Arch. Elektr. Über. 56, 407–410 (2002).

A. Lakhtakia, “Reversal of circular Bragg phenomenon in ferrocholesteric materials with negative real permittivities and permeabilities,” Adv. Mater. 14, 447–449 (2002).
[Crossref]

V. C. Venugopal and A. Lakhtakia, “Electromagnetic plane-wave response characteristics of nonaxially excited slabs of dielectric thin-film helicoidal bianisotropic mediums,” Proc. R. Soc. Lond. A 456, 125–161 (2000).
[Crossref]

S. F. Nagle, A. Lakhtakia, and W. Thompson, Jr., “Modal structures for axial wave propagation in a continuously twisted structurally chiral medium (CTSCM),” J. Acoust. Soc. Am. 97, 42–50 (1995).
[Crossref]

A. Lakhtakia and W. S. Weiglhofer, “Further results on light propagation in helicoidal bianisotropic mediums: oblique propagation,” Proc. R. Soc. Lond. A453, 93–105 (1997); corrections: 454, 3275 (1998).
[Crossref]

V. C. Venugopal and A. Lakhtakia, “Sculptured thin films: Conception, optical properties, and applications,” in Electromagnetic Fields in Unconventional Materials and Structures (O. N. Singh and A. Lakhtakia, eds) (Wiley, New York, NY, USA, 2000), pp. 151–216.

J. B. Geddes III and A. Lakhtakia, “Reflection and transmission of optical narrow-extent pulses by axially excited chiral sculptured thin films,” Eur. Phys. J. Appl. Phys.13, 3–14 (2001); corrections: 16, 247 (2001).
[Crossref]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, Bristol, UK, 2001), pp. 185–208.

McCall, M. W.

A. Lakhtakia, M. W. McCall, and W. S. Weiglhofer, “Brief overview of recent developments on negative phase-velocity mediums (alias left-handed materials),” Arch. Elektr. Über. 56, 407–410 (2002).

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, NY, USA, 1953), Sec. 4.3.

Nagle, S. F.

S. F. Nagle, A. Lakhtakia, and W. Thompson, Jr., “Modal structures for axial wave propagation in a continuously twisted structurally chiral medium (CTSCM),” J. Acoust. Soc. Am. 97, 42–50 (1995).
[Crossref]

Ouchi, Y.

H. Takezoe, K. Hashimoto, Y. Ouchi, M. Hara, A. Fukuda, and E. Kuze, “Experimental study on higher order reflection by monodomain cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 101, 329–340 (1983).
[Crossref]

Pendry, J.

J. Pendry, “Electromagnetic materials enter the negative age,” Physics World14 (9), 47–51 (2001), September issue.

Ponsinet, V.

V. Ponsinet, P. Fabre, M. Veyssie, and L. Auvray, “A small-angle neutron-scattering study of the ferrosmectic phase,” J. Phys. II (Paris) 3, 1021–1039 (1993).

Schubert, M.

M. Schubert and C. M. Herzinger, “Ellipsometry on anisotropic materials: Bragg conditions and phonons in dielectric helical thin films,” Phys. Stat. Sol. (a) 188, 1563–1575 (2001).
[Crossref]

Takezoe, H.

H. Takezoe, K. Hashimoto, Y. Ouchi, M. Hara, A. Fukuda, and E. Kuze, “Experimental study on higher order reflection by monodomain cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 101, 329–340 (1983).
[Crossref]

Thompson, Jr., W.

S. F. Nagle, A. Lakhtakia, and W. Thompson, Jr., “Modal structures for axial wave propagation in a continuously twisted structurally chiral medium (CTSCM),” J. Acoust. Soc. Am. 97, 42–50 (1995).
[Crossref]

Venugopal, V. C.

V. C. Venugopal and A. Lakhtakia, “Electromagnetic plane-wave response characteristics of nonaxially excited slabs of dielectric thin-film helicoidal bianisotropic mediums,” Proc. R. Soc. Lond. A 456, 125–161 (2000).
[Crossref]

V. C. Venugopal and A. Lakhtakia, “Sculptured thin films: Conception, optical properties, and applications,” in Electromagnetic Fields in Unconventional Materials and Structures (O. N. Singh and A. Lakhtakia, eds) (Wiley, New York, NY, USA, 2000), pp. 151–216.

Veyssie, M.

V. Ponsinet, P. Fabre, M. Veyssie, and L. Auvray, “A small-angle neutron-scattering study of the ferrosmectic phase,” J. Phys. II (Paris) 3, 1021–1039 (1993).

Wang, J.

J. Wang, A. Lakhtakia, and J. B. Geddes III, “Multiple Bragg regimes exhibited by a chiral sculptured thin film half-space on axial excitation,” Optik 113, 213–222 (2002).
[Crossref]

Weiglhofer, W. S.

A. Lakhtakia, M. W. McCall, and W. S. Weiglhofer, “Brief overview of recent developments on negative phase-velocity mediums (alias left-handed materials),” Arch. Elektr. Über. 56, 407–410 (2002).

A. Lakhtakia and W. S. Weiglhofer, “Further results on light propagation in helicoidal bianisotropic mediums: oblique propagation,” Proc. R. Soc. Lond. A453, 93–105 (1997); corrections: 454, 3275 (1998).
[Crossref]

Wu, Q. h.

I. J. Hodgkinson and Q. h. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1997), pp. 302–322.

Adv. Mater. (1)

A. Lakhtakia, “Reversal of circular Bragg phenomenon in ferrocholesteric materials with negative real permittivities and permeabilities,” Adv. Mater. 14, 447–449 (2002).
[Crossref]

Arch. Elektr. Über. (1)

A. Lakhtakia, M. W. McCall, and W. S. Weiglhofer, “Brief overview of recent developments on negative phase-velocity mediums (alias left-handed materials),” Arch. Elektr. Über. 56, 407–410 (2002).

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Electromagnetics (1)

A. Lakhtakia, “On planewave remittances and Goos-Hänchen shifts of planar slabs with negative real permittivity and permeability,” Electromagnetics 23, 71–75 (2003).
[Crossref]

J. Acoust. Soc. Am. (1)

S. F. Nagle, A. Lakhtakia, and W. Thompson, Jr., “Modal structures for axial wave propagation in a continuously twisted structurally chiral medium (CTSCM),” J. Acoust. Soc. Am. 97, 42–50 (1995).
[Crossref]

J. Phys. (Paris) (1)

F. Brochard and P.G. de Gennes, “Theory of magnetic suspensions in liquid crystals,” J. Phys. (Paris) 31, 691–708 (1970).
[Crossref]

J. Phys. II (Paris) (1)

V. Ponsinet, P. Fabre, M. Veyssie, and L. Auvray, “A small-angle neutron-scattering study of the ferrosmectic phase,” J. Phys. II (Paris) 3, 1021–1039 (1993).

Mater. Sci. Engg. C (1)

A. Lakhtakia, “Sculptured thin films: accomplishments and emerging uses,” Mater. Sci. Engg. C 19, 427–434 (2002).
[Crossref]

Mol. Cryst. Liq. Cryst. (1)

H. Takezoe, K. Hashimoto, Y. Ouchi, M. Hara, A. Fukuda, and E. Kuze, “Experimental study on higher order reflection by monodomain cholesteric liquid crystals,” Mol. Cryst. Liq. Cryst. 101, 329–340 (1983).
[Crossref]

Optik (1)

J. Wang, A. Lakhtakia, and J. B. Geddes III, “Multiple Bragg regimes exhibited by a chiral sculptured thin film half-space on axial excitation,” Optik 113, 213–222 (2002).
[Crossref]

Phys. Stat. Sol. (a) (1)

M. Schubert and C. M. Herzinger, “Ellipsometry on anisotropic materials: Bragg conditions and phonons in dielectric helical thin films,” Phys. Stat. Sol. (a) 188, 1563–1575 (2001).
[Crossref]

Proc. R. Soc. Lond. A (1)

V. C. Venugopal and A. Lakhtakia, “Electromagnetic plane-wave response characteristics of nonaxially excited slabs of dielectric thin-film helicoidal bianisotropic mediums,” Proc. R. Soc. Lond. A 456, 125–161 (2000).
[Crossref]

Other (11)

A. Lakhtakia and W. S. Weiglhofer, “Further results on light propagation in helicoidal bianisotropic mediums: oblique propagation,” Proc. R. Soc. Lond. A453, 93–105 (1997); corrections: 454, 3275 (1998).
[Crossref]

J. B. Geddes III and A. Lakhtakia, “Reflection and transmission of optical narrow-extent pulses by axially excited chiral sculptured thin films,” Eur. Phys. J. Appl. Phys.13, 3–14 (2001); corrections: 16, 247 (2001).
[Crossref]

http://www.esm.psu.edu/HTMLs/Faculty/Lakhtakia/TimeBragg/TD Bragg.html

N. Kato, “The significance of Ewald’s dynamical theory of diffraction,” in P.P. Ewald and His Dynamical Theory of X-ray Diffraction (D. W. J. Cruickshank, H. J. Juretschke, and N. Kato, eds) (Oxford University Press, Oxford, UK, 1992), pp. 3–23.

H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, Bristol, UK, 2001), pp. 185–208.

I. J. Hodgkinson and Q. h. Wu, Birefringent Thin Films and Polarizing Elements (World Scientific, Singapore, 1997), pp. 302–322.

S. D. Jacobs (ed), Selected Papers on Liquid Crystals for Optics (SPIEO ptical Engineering Press, Bellingham, WA, USA, 1992).

V. C. Venugopal and A. Lakhtakia, “Sculptured thin films: Conception, optical properties, and applications,” in Electromagnetic Fields in Unconventional Materials and Structures (O. N. Singh and A. Lakhtakia, eds) (Wiley, New York, NY, USA, 2000), pp. 151–216.

F. de Fornel, Evanescent Waves (Springer, Berlin, Germany, 2001), pp. 12–18.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, NY, USA, 1953), Sec. 4.3.

J. Pendry, “Electromagnetic materials enter the negative age,” Physics World14 (9), 47–51 (2001), September issue.

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Figures (1)

Fig. 1.
Fig. 1.

Computed reflectances of a chiral ferrosmectic slab in free space, for Ω=140 nm, L=60Ω, and χ=30°. Case (i): h=1, ψ=35°, ∊a=2.7(1+iδ), ∊ b =3.3(1+iδ), ∊ c =3(1+iδ), µa=1.1(1+iδµ), µ b =1.4(1+iδµ), µ c =1.2(1+iδµ), δ=2δµ=2× 10-3. Case (ii): Same as (i) except h=-1 and ψ=-35°. Case (iii): Same as (i) except that ψ=215° and the real parts of ∊ a,b,c and μ a,b,c are negative.

Equations (27)

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

¯ ¯ ( r ) = 0 S ¯ ¯ z S ¯ ¯ y [ a u z u z + b u x u x + c u y u y ] S ¯ ¯ y T S ¯ ¯ z T μ ¯ ¯ ( r ) = μ 0 S ¯ ¯ z S ¯ ¯ y [ μ a u z u z + μ b u x u x + μ c u y u y ] S ¯ ¯ y T S ¯ ¯ z T } , 0 z L ,
E ( r ) = E ˜ ( z ) exp [ ( x cos ψ + y sin ψ ) ] H ( r ) = H ˜ ( z ) exp [ ( x cos ψ + y sin ψ ) ] } , < z < ,
d dz [ f ¯ ( z ) ] = i [ P ¯ ¯ ( z ) ] [ f ¯ ( z ) ] , 0 < z < L .
[ P ¯ ¯ ( z ) ] = [ P ¯ ¯ 0 ( z ) ] + κ k 0 [ P ¯ ¯ 1 ( z ) ] + ( κ k 0 ) 2 [ P ¯ ¯ 2 ( z ) ] ,
[ P ¯ ¯ 0 ( z ) ] = ω { [ 0 0 0 μ 0 μ c + μ ˜ d 2 0 0 μ 0 μ c + μ ˜ d 2 0 0 0 c + ˜ d 2 0 0 0 c + ˜ d 2 0 0 0 ]
+ [ 0 0 μ 0 μ c μ ˜ d 2 sin 2 ζ μ 0 μ c μ ˜ d 2 cos 2 ζ 0 0 μ 0 μ c μ ˜ d 2 cos 2 ζ μ 0 μ c μ ˜ d 2 sin 2 ζ 0 c ˜ d 2 sin 2 ζ 0 c ˜ d 2 cos 2 ζ 0 0 0 c ˜ d 2 cos 2 ζ 0 c ˜ d 2 sin 2 ζ 0 0 ] } ,
[ P ¯ ¯ 1 ( z ) ] = ( k 0 sin χ cos χ ) ×
{ ˜ d ( a b ) a b [ cos ζ cos ψ 0 0 0 0 sin ζ sin ψ 0 0 0 0 sin ζ sin ψ 0 0 0 0 cos ζ cos ψ ]
+ μ ˜ d ( μ a μ b ) μ a μ b [ sin ζ sin ψ 0 0 0 0 cos ζ cos ψ 0 0 0 0 cos ζ cos ψ 0 0 0 0 sin ζ sin ψ ]
+ ˜ d μ ˜ d ( a μ b b μ a ) a b μ a μ b [ 0 sin ζ cos ψ 0 0 cos ζ sin ψ 0 0 0 0 0 0 sin ζ cos ψ 0 0 cos ζ sin ψ 0 ] } ,
[ P ¯ ¯ 2 ( z ) ] = ω ×
[ 0 0 μ 0 ˜ d a b cos ψ sin ψ μ 0 ˜ d a b cos 2 ψ 0 0 μ 0 ˜ d a b sin 2 ψ μ 0 ˜ d a b cos ψ sin ψ 0 μ ˜ d μ a μ b cos ψ sin ψ 0 μ ˜ d μ a μ b cos 2 ψ 0 0 0 μ ˜ d μ a μ b sin 2 ψ 0 μ ˜ d μ a μ b cos ψ sin ψ 0 0 ] .
[ f ¯ ( L ) ] = [ M ¯ ¯ ] [ f ¯ ( 0 ) ] ,
e inc ( r ) = [ ( i s p + ) 2 a L ( i s + p + ) 2 a R ] e i k 0 z cos θ , z 0 ,
e ref ( r ) = [ ( i s p ) 2 r L + ( i s + p ) 2 r R ] e i k 0 z cos θ , z 0 ,
e tr ( r ) = [ ( i s p + ) 2 t L ( i s + p + ) 2 t R ] e i k 0 ( z L ) cos θ , z L ,
[ r L r R ] = [ r LL r LR r RL r RR ] [ a L a R ] , [ t L t R ] = [ t LL t LR t RL t RR ] [ a L a R ] .
{ Re [ ¯ ¯ ( r ) ] Re [ ¯ ¯ ( r ) ] , Re [ μ ¯ ¯ ( r ) ] Re [ μ ¯ ¯ ( r ) ] , 0 z L } .
z [ 0 , L ] , [ P ¯ ¯ ( z ; ¯ ¯ ( r ) , μ ¯ ¯ ( r ) ; h , ψ ) ] = [ R ¯ ¯ ] [ P ¯ ¯ ( z ; ¯ ¯ ( r ) , μ ¯ ¯ ( r ) ; h , ψ ) ] [ R ¯ ¯ ]
= [ P ¯ ¯ ( z ; ¯ ¯ * ( r ) , μ ¯ ¯ * ( r ) ; h , π + ψ ) ] * ,
z 0 , L , [ f ¯ ( z ; ¯ ¯ ( r ) , μ ¯ ¯ ( r ) ; h , ψ ) ] = [ R ¯ ¯ ] [ f ¯ ( z ; ¯ ¯ ( r ) , μ ¯ ¯ ( r ) ; h , ψ ) ]
= [ f ¯ ( z ; ¯ ¯ * ( r ) , μ ¯ ¯ * ( r ) ; h , π + ψ ) ] * .
{ h h , ψ ψ } { a L a R , r L r R , t L t R } ;
{ Re [ a , b , c ] Re [ a , b , c ] Re [ μ a , b , c ] Re [ μ a , b , c ] ψ π + ψ } { a L a R * , a R a L * r L r R * , r R r L * t L t R * , t R t L * } .
{ h h , ψ ψ } { r LL r RR , r LR r RL t LL t RR , t LR t RL } ,
{ Re [ a , b , c ] Re [ a , b , c ] Re [ μ a , b , c ] Re [ μ a , b , c ] ψ π + ψ }
{ r LL r RR * , r RR r LL * , r LR r RL * , r RL r LR * t LL t RR * , t RR t LL * , t LR t RL * , t RL t LR * } .

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