Abstract

For light propagation in stratified media the normal component of the Poynting vector is defined as an indefinite scalar product. The vanishing of this scalar product for two waves is regarded as proof of their mutual orthogonality. Orthogonality in this sense is an inherent property of optical eigenmodes with different real wave vectors. It is shown that the matrices D and P appearing in Berreman’s 4 × 4 matrix formalism are Hermitian and unitary, respectively, within this metric. By using the orthogonality property of optical eigenmodes, projection operators and a transformation matrix are constructed that can facilitate numerical calculations and analytical treatments. The equivalence of Berreman’s 4 × 4 matrix method with scattering matrix and transfer matrix formalisms is shown.

© 1989 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. W. Berreman, D. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid crystal films,” Phys. Rev. Lett. 25, 577–581 (1970); “Reflection and transmission by single-domain cholesteric liquid crystal films: theory and verification,” Mol. Cryst. Liq. Cryst. 11, 395–405 (1970).
    [CrossRef]
  2. A. Sugita, H. Takezoe, Y. Ouchi, A. Fukuda, E. Kuze, N. Goto, “Numerical calculations of optical eigenmodes in cholesteric liquid crystals by 4 × 4 matrix method,” Jpn. J. Appl. Phys. 21, 1543–1546 (1982).
    [CrossRef]
  3. D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4 matrix formulation,”J. Opt. Soc. Am. 62, 502–510 (1971).
    [CrossRef]
  4. K. Eidner, G. Mayer, M. Schmidt, H. Schmiedel, “Optics in stratified media–the use of optical eigenmodes of uniaxial crystals in the 4 × 4 matrix formalism,” Mol. Cryst. Liq. Cryst. (to be published).
  5. D. Y. K. Ko, J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863–1866 (1988).
    [CrossRef]
  6. K. Eidner, H. Schmiedel, “Generalized orthogonality criterion for optical eigenmodes,” submitted to Z. Naturforsch.
  7. M. A. Potier, “Sur le principe du retour des rayons et la reflexion cristalline,”J. Phys. 10, 349–357 (1891); F. Pockels, Lehrbuch der Kristalloptik (B. G. Teubner, Leipzig, Germany, 1906).
  8. A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals. Surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
    [CrossRef]
  9. M. Schmidt, H. Schmiedel, “The conservation of energy in Berreman’s 4 × 4-matrix formalism and conditions for the validity of numerical approximations,” Mol. Cryst. Liq. Cryst. (to be published).
  10. P. J. Lin-Chung, S. Teitler, “4 × 4 Matrix formalism for optics in stratified anisotropic media,” J. Opt. Soc. Am. A 1, 703–705 (1984).
    [CrossRef]
  11. K. Eidner, G. Mayer, R. Schuster, “Theoretical considerations on the determination of the tilt angle of a nematic liquid crystal at the surface of a substrate by polarization azimuth measurement,” Mol. Cryst. Liq. Cryst. 152, 447–452 (1987); “Determination of the surface anchoring energy of a nematic liquid crystal by polarization azimuth measurement,” Mol. Cryst. Liq. Cryst.159, 27–36 (1988).

1988 (1)

1987 (1)

K. Eidner, G. Mayer, R. Schuster, “Theoretical considerations on the determination of the tilt angle of a nematic liquid crystal at the surface of a substrate by polarization azimuth measurement,” Mol. Cryst. Liq. Cryst. 152, 447–452 (1987); “Determination of the surface anchoring energy of a nematic liquid crystal by polarization azimuth measurement,” Mol. Cryst. Liq. Cryst.159, 27–36 (1988).

1985 (1)

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals. Surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

1984 (1)

1982 (1)

A. Sugita, H. Takezoe, Y. Ouchi, A. Fukuda, E. Kuze, N. Goto, “Numerical calculations of optical eigenmodes in cholesteric liquid crystals by 4 × 4 matrix method,” Jpn. J. Appl. Phys. 21, 1543–1546 (1982).
[CrossRef]

1971 (1)

1970 (1)

D. W. Berreman, D. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid crystal films,” Phys. Rev. Lett. 25, 577–581 (1970); “Reflection and transmission by single-domain cholesteric liquid crystal films: theory and verification,” Mol. Cryst. Liq. Cryst. 11, 395–405 (1970).
[CrossRef]

1891 (1)

M. A. Potier, “Sur le principe du retour des rayons et la reflexion cristalline,”J. Phys. 10, 349–357 (1891); F. Pockels, Lehrbuch der Kristalloptik (B. G. Teubner, Leipzig, Germany, 1906).

Berreman, D. W.

D. W. Berreman, “Optics in stratified and anisotropic media: 4 × 4 matrix formulation,”J. Opt. Soc. Am. 62, 502–510 (1971).
[CrossRef]

D. W. Berreman, D. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid crystal films,” Phys. Rev. Lett. 25, 577–581 (1970); “Reflection and transmission by single-domain cholesteric liquid crystal films: theory and verification,” Mol. Cryst. Liq. Cryst. 11, 395–405 (1970).
[CrossRef]

Eidner, K.

K. Eidner, G. Mayer, R. Schuster, “Theoretical considerations on the determination of the tilt angle of a nematic liquid crystal at the surface of a substrate by polarization azimuth measurement,” Mol. Cryst. Liq. Cryst. 152, 447–452 (1987); “Determination of the surface anchoring energy of a nematic liquid crystal by polarization azimuth measurement,” Mol. Cryst. Liq. Cryst.159, 27–36 (1988).

K. Eidner, H. Schmiedel, “Generalized orthogonality criterion for optical eigenmodes,” submitted to Z. Naturforsch.

K. Eidner, G. Mayer, M. Schmidt, H. Schmiedel, “Optics in stratified media–the use of optical eigenmodes of uniaxial crystals in the 4 × 4 matrix formalism,” Mol. Cryst. Liq. Cryst. (to be published).

Fukuda, A.

A. Sugita, H. Takezoe, Y. Ouchi, A. Fukuda, E. Kuze, N. Goto, “Numerical calculations of optical eigenmodes in cholesteric liquid crystals by 4 × 4 matrix method,” Jpn. J. Appl. Phys. 21, 1543–1546 (1982).
[CrossRef]

Gaylord, T. K.

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals. Surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Goto, N.

A. Sugita, H. Takezoe, Y. Ouchi, A. Fukuda, E. Kuze, N. Goto, “Numerical calculations of optical eigenmodes in cholesteric liquid crystals by 4 × 4 matrix method,” Jpn. J. Appl. Phys. 21, 1543–1546 (1982).
[CrossRef]

Knoesen, A.

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals. Surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Ko, D. Y. K.

Kuze, E.

A. Sugita, H. Takezoe, Y. Ouchi, A. Fukuda, E. Kuze, N. Goto, “Numerical calculations of optical eigenmodes in cholesteric liquid crystals by 4 × 4 matrix method,” Jpn. J. Appl. Phys. 21, 1543–1546 (1982).
[CrossRef]

Lin-Chung, P. J.

Mayer, G.

K. Eidner, G. Mayer, R. Schuster, “Theoretical considerations on the determination of the tilt angle of a nematic liquid crystal at the surface of a substrate by polarization azimuth measurement,” Mol. Cryst. Liq. Cryst. 152, 447–452 (1987); “Determination of the surface anchoring energy of a nematic liquid crystal by polarization azimuth measurement,” Mol. Cryst. Liq. Cryst.159, 27–36 (1988).

K. Eidner, G. Mayer, M. Schmidt, H. Schmiedel, “Optics in stratified media–the use of optical eigenmodes of uniaxial crystals in the 4 × 4 matrix formalism,” Mol. Cryst. Liq. Cryst. (to be published).

Moharam, M. G.

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals. Surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

Ouchi, Y.

A. Sugita, H. Takezoe, Y. Ouchi, A. Fukuda, E. Kuze, N. Goto, “Numerical calculations of optical eigenmodes in cholesteric liquid crystals by 4 × 4 matrix method,” Jpn. J. Appl. Phys. 21, 1543–1546 (1982).
[CrossRef]

Potier, M. A.

M. A. Potier, “Sur le principe du retour des rayons et la reflexion cristalline,”J. Phys. 10, 349–357 (1891); F. Pockels, Lehrbuch der Kristalloptik (B. G. Teubner, Leipzig, Germany, 1906).

Sambles, J. R.

Scheffer, D. J.

D. W. Berreman, D. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid crystal films,” Phys. Rev. Lett. 25, 577–581 (1970); “Reflection and transmission by single-domain cholesteric liquid crystal films: theory and verification,” Mol. Cryst. Liq. Cryst. 11, 395–405 (1970).
[CrossRef]

Schmidt, M.

K. Eidner, G. Mayer, M. Schmidt, H. Schmiedel, “Optics in stratified media–the use of optical eigenmodes of uniaxial crystals in the 4 × 4 matrix formalism,” Mol. Cryst. Liq. Cryst. (to be published).

M. Schmidt, H. Schmiedel, “The conservation of energy in Berreman’s 4 × 4-matrix formalism and conditions for the validity of numerical approximations,” Mol. Cryst. Liq. Cryst. (to be published).

Schmiedel, H.

M. Schmidt, H. Schmiedel, “The conservation of energy in Berreman’s 4 × 4-matrix formalism and conditions for the validity of numerical approximations,” Mol. Cryst. Liq. Cryst. (to be published).

K. Eidner, G. Mayer, M. Schmidt, H. Schmiedel, “Optics in stratified media–the use of optical eigenmodes of uniaxial crystals in the 4 × 4 matrix formalism,” Mol. Cryst. Liq. Cryst. (to be published).

K. Eidner, H. Schmiedel, “Generalized orthogonality criterion for optical eigenmodes,” submitted to Z. Naturforsch.

Schuster, R.

K. Eidner, G. Mayer, R. Schuster, “Theoretical considerations on the determination of the tilt angle of a nematic liquid crystal at the surface of a substrate by polarization azimuth measurement,” Mol. Cryst. Liq. Cryst. 152, 447–452 (1987); “Determination of the surface anchoring energy of a nematic liquid crystal by polarization azimuth measurement,” Mol. Cryst. Liq. Cryst.159, 27–36 (1988).

Sugita, A.

A. Sugita, H. Takezoe, Y. Ouchi, A. Fukuda, E. Kuze, N. Goto, “Numerical calculations of optical eigenmodes in cholesteric liquid crystals by 4 × 4 matrix method,” Jpn. J. Appl. Phys. 21, 1543–1546 (1982).
[CrossRef]

Takezoe, H.

A. Sugita, H. Takezoe, Y. Ouchi, A. Fukuda, E. Kuze, N. Goto, “Numerical calculations of optical eigenmodes in cholesteric liquid crystals by 4 × 4 matrix method,” Jpn. J. Appl. Phys. 21, 1543–1546 (1982).
[CrossRef]

Teitler, S.

Appl. Phys. B (1)

A. Knoesen, M. G. Moharam, T. K. Gaylord, “Electromagnetic propagation at interfaces and in waveguides in uniaxial crystals. Surface impedance/admittance approach,” Appl. Phys. B 38, 171–178 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

J. Phys. (1)

M. A. Potier, “Sur le principe du retour des rayons et la reflexion cristalline,”J. Phys. 10, 349–357 (1891); F. Pockels, Lehrbuch der Kristalloptik (B. G. Teubner, Leipzig, Germany, 1906).

Jpn. J. Appl. Phys. (1)

A. Sugita, H. Takezoe, Y. Ouchi, A. Fukuda, E. Kuze, N. Goto, “Numerical calculations of optical eigenmodes in cholesteric liquid crystals by 4 × 4 matrix method,” Jpn. J. Appl. Phys. 21, 1543–1546 (1982).
[CrossRef]

Mol. Cryst. Liq. Cryst. (1)

K. Eidner, G. Mayer, R. Schuster, “Theoretical considerations on the determination of the tilt angle of a nematic liquid crystal at the surface of a substrate by polarization azimuth measurement,” Mol. Cryst. Liq. Cryst. 152, 447–452 (1987); “Determination of the surface anchoring energy of a nematic liquid crystal by polarization azimuth measurement,” Mol. Cryst. Liq. Cryst.159, 27–36 (1988).

Phys. Rev. Lett. (1)

D. W. Berreman, D. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid crystal films,” Phys. Rev. Lett. 25, 577–581 (1970); “Reflection and transmission by single-domain cholesteric liquid crystal films: theory and verification,” Mol. Cryst. Liq. Cryst. 11, 395–405 (1970).
[CrossRef]

Other (3)

K. Eidner, G. Mayer, M. Schmidt, H. Schmiedel, “Optics in stratified media–the use of optical eigenmodes of uniaxial crystals in the 4 × 4 matrix formalism,” Mol. Cryst. Liq. Cryst. (to be published).

K. Eidner, H. Schmiedel, “Generalized orthogonality criterion for optical eigenmodes,” submitted to Z. Naturforsch.

M. Schmidt, H. Schmiedel, “The conservation of energy in Berreman’s 4 × 4-matrix formalism and conditions for the validity of numerical approximations,” Mol. Cryst. Liq. Cryst. (to be published).

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.


Equations (32)

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

Ψ T = ( a E x , b H y , a E y , - b H x ) ,
Ψ n = P n Ψ n - 1 ,
z Ψ = i ω c D Ψ ,
D = [ ( - k x z x z z ) ( 1 - k x 2 z z ) ( - k x z y z z ) ( - k x k y z z ) ( x x - x z z x z z - k y 2 ) ( - k x x z z z ) ( x y - x z z y + k x k y z z ) ( - k y x z z z ) ( - k y z x z z ) ( - k y k x z z ) ( - k y z y z z ) ( 1 - k y 2 z z ) ( y x - y x z x + k y k x z z ) ( - k x y z z z ) ( y y - y z z y z z - k x 2 ) ( - k y y z z z ) ] P = exp [ i ( ω / c ) D d ] ,
Ψ N = ( n = N 1 P n ) Ψ 0 ,
Ψ N = R exp [ i ω c z 0 z N D ( z ) d z ] Ψ 0 .
D Ψ α ( z ) = k z α Ψ α ( z ) , P Ψ α ( z ) = exp [ i ( ω / c ) k z α d ] Ψ α ( z ) .
k α = ω c ( k x , k y , k z α ) .
D 11 = D 22 * ,             D 33 = D 44 * ,             D 41 = D 23 * , D 32 = D 14 * ,             D 42 = D 13 * ,             D 31 = D 24 * .
S α β = E x α * H y β + E x β H y α * - E y α * H x β - E y β H x α * ,
[ Ψ α , Ψ β ] 1 c S α β = Ψ α + Σ x Ψ β ,
Σ x = [ σ x 0 0 σ x ]             and σ x = [ 0 1 1 0 ] .
[ D Ψ α , Ψ β ] = [ Ψ α , D Ψ β ] ,
( D Ψ α ) + Σ x Ψ β = Ψ α + Σ x ( D Ψ β )
D + Σ x = Σ x D .
/ z ( Ψ + Σ x Ψ ) = i ( ω / c ) Ψ + ( Σ x D - D + Σ x ) Ψ = 0.
[ P Ψ α , P Ψ β ] = [ Ψ α , Ψ β ] ,
( P Ψ α ) + Σ x ( P Ψ β ) = Ψ α + Σ x Ψ β
P + Σ x P = Σ x ,
P - 1 = Σ x P + Σ x .
Ψ = α = 1 4 A α Ψ α ,
( Ψ α * Ψ α + Σ x N α ) Ψ = A α Ψ α ,
T α 1 N α Ψ α + Σ x ,
T Ψ = A ,
T - 1 = ( Ψ 1 , Ψ 2 , Ψ 3 , Ψ 4 ) .
T D T - 1 = K             with K α β = k z α δ α β
T P T - 1 = Φ             with Φ α β = exp [ i ( ω / c ) k z α d ] δ α β ,
P = ( T n - 1 Φ n T n ) ( T n - 1 - 1 Φ n - 1 T n - 1 ) .
P = ( Φ n T n T n - 1 - 1 ) ( Φ n - 1 T n - 1 T n - 2 - 1 ) ,    
M n = T n T n - 1 - 1
F = T N P T 0 - 1 .
F - 1 = ( T 0 Σ x T 0 + ) F + ( T N Σ x T N + ) - 1 ,

Metrics