Abstract

We consider theoretically light propagation at oblique incidence in absorbing cholesteric liquid crystals in the Bragg mode. Using 2 × 2 matrix procedure, we analyze the eigenmodes in terms of the forward- and backward-propagating eigenwaves inside the isotropic medium bounding the cholesteric liquid crystal. The nature of the eigenmodes in the medium changes continuously from the circular to the linear state with increase in the angle of incidence. Using the appropriate eigenmodes, we studied anomalous transmission in first and second Bragg orders in these systems and its relevance to the reported experimental results. We find some interesting polarization features that arise as a result of the absorption in the cholesteric medium.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. A. Belyakov, V. D. Dmitrienko, “Theory of the optical properties of cholesteric liquid crystals,” Sov. Phys. Solid State 15, 1811–1815 (1974).
  2. C. Oldano, E. Miraldi, P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
    [CrossRef]
  3. C. Oldano, “Many-wave approximation for light propagation in cholesteric liquid crystals,” Phys. Rev. A 31, 1014–1021 (1985).
    [CrossRef] [PubMed]
  4. H. L. Ong, “Wave propagation in cholesteric and chiral smectic C liquid crystals: exact and generalized geometrical-optics approximation,” Phys. Rev. A 37, 3520–3529 (1988).
    [CrossRef] [PubMed]
  5. H. Takezoe, Y. Ouchi, M. Hara, A. Fukuda, E. Kuze, “Experimental studies of reflection spectra in monodomain cholesteric liquid crystal cell: total reflection, subsidiary oscillation and its beats or swell structure,” Jpn. J. Appl. Phys. 22, 1080–1091 (1983).
    [CrossRef]
  6. R. Nityananda, U. D. Kini, S. Chandrasekhar, K. A. Suresh, “Anomalous transmission (Borrmann effect) in absorbing cholesteric liquid crystals,” in Proceedings of the International Liquid Crystal Conference, Bangalore, December 3–8, 1973, Pramana Suppl.1, 325–340 (1975).
  7. S. Chandrasekhar, G. S. Ranganath, K. A. Suresh, “Dynamical theory of reflection from cholesteric liquid crystals,” in Proceedings of the International Liquid Crystal Conference, Bangalore, December 3–8, 1973, Pramana Suppl.1, 341–352 (1975).
  8. K. A. Suresh, “An experimental study of anomalous transmission (Borrmann effect) in absorbing cholesteric liquid crystals,” Mol. Cryst. Liquid Cryst. 35, 267–273 (1976).
    [CrossRef]
  9. S. Endo, T. Kuribara, T. Akahane, “A study of the anomalous transmission (Borrmann effect) on obliquely incident light in an absorbing single-domain cholesteric liquid crystal,” Jpn. J. Appl. Phys. 22, L499–L501 (1983).
    [CrossRef]
  10. V. D. Dmitrienko, V. A. Belyakov, “Higher orders of selective reflection of light by cholesteric liquid crystals,” Sov. Phys. Solid State 15, 2365–2366 (1974).
  11. D. W. Berreman, T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid crystal films,” Phys. Rev. Lett. 25, 577–581 (1970).
    [CrossRef]
  12. H. Takezoe, K. Hashimoto, Y. Ouchi, M. Hara, A. Fukuda, E. Kuze, “Experimental study on higher order reflection by monodomain cholesteric liquid crystal,” Mol. Cryst. Liquid Cryst. 101, 329–340 (1983).
    [CrossRef]
  13. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,”J. Opt. Soc. Am. 62, 502–510 (1972).
    [CrossRef]
  14. C. Oldano, “Electromagnetic wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989).
    [CrossRef]
  15. P. B. Sunil Kumar, G. S. Ranganath, “Structure and optical behavior of cholesteric soliton lattices,”J. Phys. II France 3, 1497–1510 (1993).
    [CrossRef]
  16. E. Miraldi, C. Oldano, P. I. Taverona, L. Trossi, “Optical properties of cholesteric liquid crystal at oblique incidence,” Mol. Cryst. Liquid Cryst. 103, 155–176 (1983).
    [CrossRef]
  17. R. W. James, The Optical Principles of the Diffraction of X-rays (Bell, London, 1967).

1993 (1)

P. B. Sunil Kumar, G. S. Ranganath, “Structure and optical behavior of cholesteric soliton lattices,”J. Phys. II France 3, 1497–1510 (1993).
[CrossRef]

1989 (1)

C. Oldano, “Electromagnetic wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989).
[CrossRef]

1988 (1)

H. L. Ong, “Wave propagation in cholesteric and chiral smectic C liquid crystals: exact and generalized geometrical-optics approximation,” Phys. Rev. A 37, 3520–3529 (1988).
[CrossRef] [PubMed]

1985 (1)

C. Oldano, “Many-wave approximation for light propagation in cholesteric liquid crystals,” Phys. Rev. A 31, 1014–1021 (1985).
[CrossRef] [PubMed]

1983 (5)

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

S. Endo, T. Kuribara, T. Akahane, “A study of the anomalous transmission (Borrmann effect) on obliquely incident light in an absorbing single-domain cholesteric liquid crystal,” Jpn. J. Appl. Phys. 22, L499–L501 (1983).
[CrossRef]

H. Takezoe, Y. Ouchi, M. Hara, A. Fukuda, E. Kuze, “Experimental studies of reflection spectra in monodomain cholesteric liquid crystal cell: total reflection, subsidiary oscillation and its beats or swell structure,” Jpn. J. Appl. Phys. 22, 1080–1091 (1983).
[CrossRef]

C. Oldano, E. Miraldi, P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[CrossRef]

E. Miraldi, C. Oldano, P. I. Taverona, L. Trossi, “Optical properties of cholesteric liquid crystal at oblique incidence,” Mol. Cryst. Liquid Cryst. 103, 155–176 (1983).
[CrossRef]

1976 (1)

K. A. Suresh, “An experimental study of anomalous transmission (Borrmann effect) in absorbing cholesteric liquid crystals,” Mol. Cryst. Liquid Cryst. 35, 267–273 (1976).
[CrossRef]

1974 (2)

V. A. Belyakov, V. D. Dmitrienko, “Theory of the optical properties of cholesteric liquid crystals,” Sov. Phys. Solid State 15, 1811–1815 (1974).

V. D. Dmitrienko, V. A. Belyakov, “Higher orders of selective reflection of light by cholesteric liquid crystals,” Sov. Phys. Solid State 15, 2365–2366 (1974).

1972 (1)

1970 (1)

D. W. Berreman, T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid crystal films,” Phys. Rev. Lett. 25, 577–581 (1970).
[CrossRef]

Akahane, T.

S. Endo, T. Kuribara, T. Akahane, “A study of the anomalous transmission (Borrmann effect) on obliquely incident light in an absorbing single-domain cholesteric liquid crystal,” Jpn. J. Appl. Phys. 22, L499–L501 (1983).
[CrossRef]

Belyakov, V. A.

V. D. Dmitrienko, V. A. Belyakov, “Higher orders of selective reflection of light by cholesteric liquid crystals,” Sov. Phys. Solid State 15, 2365–2366 (1974).

V. A. Belyakov, V. D. Dmitrienko, “Theory of the optical properties of cholesteric liquid crystals,” Sov. Phys. Solid State 15, 1811–1815 (1974).

Berreman, D. W.

D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4-matrix formulation,”J. Opt. Soc. Am. 62, 502–510 (1972).
[CrossRef]

D. W. Berreman, T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid crystal films,” Phys. Rev. Lett. 25, 577–581 (1970).
[CrossRef]

Chandrasekhar, S.

R. Nityananda, U. D. Kini, S. Chandrasekhar, K. A. Suresh, “Anomalous transmission (Borrmann effect) in absorbing cholesteric liquid crystals,” in Proceedings of the International Liquid Crystal Conference, Bangalore, December 3–8, 1973, Pramana Suppl.1, 325–340 (1975).

S. Chandrasekhar, G. S. Ranganath, K. A. Suresh, “Dynamical theory of reflection from cholesteric liquid crystals,” in Proceedings of the International Liquid Crystal Conference, Bangalore, December 3–8, 1973, Pramana Suppl.1, 341–352 (1975).

Dmitrienko, V. D.

V. A. Belyakov, V. D. Dmitrienko, “Theory of the optical properties of cholesteric liquid crystals,” Sov. Phys. Solid State 15, 1811–1815 (1974).

V. D. Dmitrienko, V. A. Belyakov, “Higher orders of selective reflection of light by cholesteric liquid crystals,” Sov. Phys. Solid State 15, 2365–2366 (1974).

Endo, S.

S. Endo, T. Kuribara, T. Akahane, “A study of the anomalous transmission (Borrmann effect) on obliquely incident light in an absorbing single-domain cholesteric liquid crystal,” Jpn. J. Appl. Phys. 22, L499–L501 (1983).
[CrossRef]

Fukuda, A.

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

H. Takezoe, Y. Ouchi, M. Hara, A. Fukuda, E. Kuze, “Experimental studies of reflection spectra in monodomain cholesteric liquid crystal cell: total reflection, subsidiary oscillation and its beats or swell structure,” Jpn. J. Appl. Phys. 22, 1080–1091 (1983).
[CrossRef]

Hara, M.

H. Takezoe, Y. Ouchi, M. Hara, A. Fukuda, E. Kuze, “Experimental studies of reflection spectra in monodomain cholesteric liquid crystal cell: total reflection, subsidiary oscillation and its beats or swell structure,” Jpn. J. Appl. Phys. 22, 1080–1091 (1983).
[CrossRef]

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

Hashimoto, K.

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

James, R. W.

R. W. James, The Optical Principles of the Diffraction of X-rays (Bell, London, 1967).

Kini, U. D.

R. Nityananda, U. D. Kini, S. Chandrasekhar, K. A. Suresh, “Anomalous transmission (Borrmann effect) in absorbing cholesteric liquid crystals,” in Proceedings of the International Liquid Crystal Conference, Bangalore, December 3–8, 1973, Pramana Suppl.1, 325–340 (1975).

Kuribara, T.

S. Endo, T. Kuribara, T. Akahane, “A study of the anomalous transmission (Borrmann effect) on obliquely incident light in an absorbing single-domain cholesteric liquid crystal,” Jpn. J. Appl. Phys. 22, L499–L501 (1983).
[CrossRef]

Kuze, E.

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

H. Takezoe, Y. Ouchi, M. Hara, A. Fukuda, E. Kuze, “Experimental studies of reflection spectra in monodomain cholesteric liquid crystal cell: total reflection, subsidiary oscillation and its beats or swell structure,” Jpn. J. Appl. Phys. 22, 1080–1091 (1983).
[CrossRef]

Miraldi, E.

C. Oldano, E. Miraldi, P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[CrossRef]

E. Miraldi, C. Oldano, P. I. Taverona, L. Trossi, “Optical properties of cholesteric liquid crystal at oblique incidence,” Mol. Cryst. Liquid Cryst. 103, 155–176 (1983).
[CrossRef]

Nityananda, R.

R. Nityananda, U. D. Kini, S. Chandrasekhar, K. A. Suresh, “Anomalous transmission (Borrmann effect) in absorbing cholesteric liquid crystals,” in Proceedings of the International Liquid Crystal Conference, Bangalore, December 3–8, 1973, Pramana Suppl.1, 325–340 (1975).

Oldano, C.

C. Oldano, “Electromagnetic wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989).
[CrossRef]

C. Oldano, “Many-wave approximation for light propagation in cholesteric liquid crystals,” Phys. Rev. A 31, 1014–1021 (1985).
[CrossRef] [PubMed]

C. Oldano, E. Miraldi, P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[CrossRef]

E. Miraldi, C. Oldano, P. I. Taverona, L. Trossi, “Optical properties of cholesteric liquid crystal at oblique incidence,” Mol. Cryst. Liquid Cryst. 103, 155–176 (1983).
[CrossRef]

Ong, H. L.

H. L. Ong, “Wave propagation in cholesteric and chiral smectic C liquid crystals: exact and generalized geometrical-optics approximation,” Phys. Rev. A 37, 3520–3529 (1988).
[CrossRef] [PubMed]

Ouchi, Y.

H. Takezoe, Y. Ouchi, M. Hara, A. Fukuda, E. Kuze, “Experimental studies of reflection spectra in monodomain cholesteric liquid crystal cell: total reflection, subsidiary oscillation and its beats or swell structure,” Jpn. J. Appl. Phys. 22, 1080–1091 (1983).
[CrossRef]

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

Ranganath, G. S.

P. B. Sunil Kumar, G. S. Ranganath, “Structure and optical behavior of cholesteric soliton lattices,”J. Phys. II France 3, 1497–1510 (1993).
[CrossRef]

S. Chandrasekhar, G. S. Ranganath, K. A. Suresh, “Dynamical theory of reflection from cholesteric liquid crystals,” in Proceedings of the International Liquid Crystal Conference, Bangalore, December 3–8, 1973, Pramana Suppl.1, 341–352 (1975).

Scheffer, T. J.

D. W. Berreman, T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid crystal films,” Phys. Rev. Lett. 25, 577–581 (1970).
[CrossRef]

Sunil Kumar, P. B.

P. B. Sunil Kumar, G. S. Ranganath, “Structure and optical behavior of cholesteric soliton lattices,”J. Phys. II France 3, 1497–1510 (1993).
[CrossRef]

Suresh, K. A.

K. A. Suresh, “An experimental study of anomalous transmission (Borrmann effect) in absorbing cholesteric liquid crystals,” Mol. Cryst. Liquid Cryst. 35, 267–273 (1976).
[CrossRef]

S. Chandrasekhar, G. S. Ranganath, K. A. Suresh, “Dynamical theory of reflection from cholesteric liquid crystals,” in Proceedings of the International Liquid Crystal Conference, Bangalore, December 3–8, 1973, Pramana Suppl.1, 341–352 (1975).

R. Nityananda, U. D. Kini, S. Chandrasekhar, K. A. Suresh, “Anomalous transmission (Borrmann effect) in absorbing cholesteric liquid crystals,” in Proceedings of the International Liquid Crystal Conference, Bangalore, December 3–8, 1973, Pramana Suppl.1, 325–340 (1975).

Takezoe, H.

H. Takezoe, Y. Ouchi, M. Hara, A. Fukuda, E. Kuze, “Experimental studies of reflection spectra in monodomain cholesteric liquid crystal cell: total reflection, subsidiary oscillation and its beats or swell structure,” Jpn. J. Appl. Phys. 22, 1080–1091 (1983).
[CrossRef]

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

Taverona, P. I.

E. Miraldi, C. Oldano, P. I. Taverona, L. Trossi, “Optical properties of cholesteric liquid crystal at oblique incidence,” Mol. Cryst. Liquid Cryst. 103, 155–176 (1983).
[CrossRef]

Trossi, L.

E. Miraldi, C. Oldano, P. I. Taverona, L. Trossi, “Optical properties of cholesteric liquid crystal at oblique incidence,” Mol. Cryst. Liquid Cryst. 103, 155–176 (1983).
[CrossRef]

Valabrega, P. T.

C. Oldano, E. Miraldi, P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. II France (1)

P. B. Sunil Kumar, G. S. Ranganath, “Structure and optical behavior of cholesteric soliton lattices,”J. Phys. II France 3, 1497–1510 (1993).
[CrossRef]

Jpn. J. Appl. Phys. (2)

H. Takezoe, Y. Ouchi, M. Hara, A. Fukuda, E. Kuze, “Experimental studies of reflection spectra in monodomain cholesteric liquid crystal cell: total reflection, subsidiary oscillation and its beats or swell structure,” Jpn. J. Appl. Phys. 22, 1080–1091 (1983).
[CrossRef]

S. Endo, T. Kuribara, T. Akahane, “A study of the anomalous transmission (Borrmann effect) on obliquely incident light in an absorbing single-domain cholesteric liquid crystal,” Jpn. J. Appl. Phys. 22, L499–L501 (1983).
[CrossRef]

Mol. Cryst. Liquid Cryst. (3)

K. A. Suresh, “An experimental study of anomalous transmission (Borrmann effect) in absorbing cholesteric liquid crystals,” Mol. Cryst. Liquid Cryst. 35, 267–273 (1976).
[CrossRef]

E. Miraldi, C. Oldano, P. I. Taverona, L. Trossi, “Optical properties of cholesteric liquid crystal at oblique incidence,” Mol. Cryst. Liquid Cryst. 103, 155–176 (1983).
[CrossRef]

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

Phys. Rev. A (4)

C. Oldano, “Electromagnetic wave propagation in anisotropic stratified media,” Phys. Rev. A 40, 6014–6020 (1989).
[CrossRef]

C. Oldano, E. Miraldi, P. T. Valabrega, “Dispersion relation for propagation of light in cholesteric liquid crystals,” Phys. Rev. A 27, 3291–3299 (1983).
[CrossRef]

C. Oldano, “Many-wave approximation for light propagation in cholesteric liquid crystals,” Phys. Rev. A 31, 1014–1021 (1985).
[CrossRef] [PubMed]

H. L. Ong, “Wave propagation in cholesteric and chiral smectic C liquid crystals: exact and generalized geometrical-optics approximation,” Phys. Rev. A 37, 3520–3529 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

D. W. Berreman, T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid crystal films,” Phys. Rev. Lett. 25, 577–581 (1970).
[CrossRef]

Sov. Phys. Solid State (2)

V. A. Belyakov, V. D. Dmitrienko, “Theory of the optical properties of cholesteric liquid crystals,” Sov. Phys. Solid State 15, 1811–1815 (1974).

V. D. Dmitrienko, V. A. Belyakov, “Higher orders of selective reflection of light by cholesteric liquid crystals,” Sov. Phys. Solid State 15, 2365–2366 (1974).

Other (3)

R. Nityananda, U. D. Kini, S. Chandrasekhar, K. A. Suresh, “Anomalous transmission (Borrmann effect) in absorbing cholesteric liquid crystals,” in Proceedings of the International Liquid Crystal Conference, Bangalore, December 3–8, 1973, Pramana Suppl.1, 325–340 (1975).

S. Chandrasekhar, G. S. Ranganath, K. A. Suresh, “Dynamical theory of reflection from cholesteric liquid crystals,” in Proceedings of the International Liquid Crystal Conference, Bangalore, December 3–8, 1973, Pramana Suppl.1, 341–352 (1975).

R. W. James, The Optical Principles of the Diffraction of X-rays (Bell, London, 1967).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Reflectance of TM–TM (solid curve), TE–TM (short-dashed curve), TE–TE (long-dashed curve) waves for an absorbing cholesteric at θ = 60°.

Fig. 2
Fig. 2

Transmittance of TM–TM (solid curve), TE–TM (short-dashed curve), and TE–TE (long-dashed curve) waves for an absorbing cholesteric at θ = 60°.

Fig. 3
Fig. 3

Peak reflectance (Rp) for TM–TM (solid curve), TM–TE (short-dashed curve), and TE–TE (long-dashed curve) as a function of α (in degrees), the angle between the plane of incidence and the director of the cholesteric layer at the boundary.

Fig. 4
Fig. 4

Intensity of the E field of the two nonpropagating modes in the non-Bragg region for nonabsorbing (long-dashed curve) cholesteric as a function of number of pitches (in this case the two modes are attenuated by the same amount) and the intensity of the E field for the same two modes (solid and short-dashed curves) in the same region for the absorbing cholesteric medium at θ = 60°.

Fig. 5
Fig. 5

Transmittance of TM (solid curve) wave for an absorbing cholesteric at θ = 60° in the second-order Bragg reflection. The dashed vertical lines indicate the region of anomalous transmission. Here the transmittance for the TE wave is almost zero.

Fig. 6
Fig. 6

Reflectance of TM–TM (solid curve), TE–TM (short-dashed curve), TE–TE (long-dashed curve) waves for an absorbing cholesteric at θ = 60° in the second-order Bragg reflection.

Fig. 7
Fig. 7

Transmittance of TM wave (solid curve) at θ = 60°. The dashed line represents the transmittance of the same wave through a medium that has an absorption coefficient given by [ Im ( ɛ 1 ) + Im ( ɛ 2 ) ] / 2 for the same θ.

Equations (12)

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

ɛ ^ = [ ɛ + δ cos 2 β z δ sin 2 β z 0 δ sin 2 β z ɛ - δ cos 2 β z 0 0 0 ɛ 2 ] .
ψ ( z ) z = i ω c Δ ( z ) ψ ( z ) ,
ψ ( z ) = [ E x H y E y - H x ] .
ψ ( z + P ) = F ( z , P ) ψ ( z ) .
ξ j ( z 1 ) = exp [ i ω c Δ ( z ) h ] ξ j ( z ) .
Ψ ( z ) = T ϕ I ( z ) ,
ϕ I ( z ) = [ i 1 i 2 r 1 r 2 ] ,
ψ ( z + n P ) = T ϕ I ( z + n P ) ,
ϕ I ( z + n P ) = [ t 1 t 2 0 0 ] .
ψ ( z ) = F - 1 ( z , n P ) ψ ( z + n P ) .
ϕ I ( z ) = C ϕ I ( z + n P ) ,
[ t 1 t 2 ] = D 1 [ i 1 i 2 ] ,             [ r 1 r 2 ] = D 2 [ i 1 i 2 ] .

Metrics