Abstract

A 4×4-matrix technique was recently introduced by Teitler and Henvis for finding propagation and reflection by stratified anisotropic media. It is more general than the 2×2-matrix technique developed by Jones and by Abelès and is applicable to problems involving media of low optical symmetry. A little later, we developed a 4×4 differential-matrix technique in order to solve the problem of reflection and transmission by cholesteric liquid crystals and other liquid crystals with continuously varying but planar ordering. Our technique is mathematically equivalent to that of Teitler and Henvis, but we used a somewhat different approach. We start with a 6×6-matrix representation of Maxwell’s equations that can include Faraday rotation and optical activity. From this, we derive expressions for 16 differential-matrix elements so that a wide variety of specific problems can be attacked without repeating a large amount of tedious algebra. The 4×4-matrix technique is particularly well suited for solving complicated reflection and transmission problems on a computer. It also serves as an illuminating alternative way to rederive closed solutions to a number of less-complicated classical problems. Teitler and Henvis described a method of solving some of these problems, briefly in their paper. We give solutions to several such problems and add a solution to the Oseen-DeVries optical model of a cholesteric liquid crystal, to illustrate the power and simplicity of the 4×4-matrix technique.

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  1. R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941); 31, 500 (1941) and succeeding papers.
  2. F. Abelès, Ann. Phys. (Paris) 5, 598 (1950). [See also M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Sec. 1.6.]
  3. S. Teitler and B. W. Henvis, J. Opt. Soc. Am. 60, 830, (1970).
  4. D. W. Berreman and T. J. Scheffer, Phys. Rev. Letters 25, 577 (1970); Mol. Cryst. Liquid Cryst. 11, 395 (1970).
  5. Reflectance of obliquely incident light by semi-infinite samples was computed for one particular model of a cholesteric liquid crystal for the first time a little earlier, using a different technique, by D. Taupin, J. Phys. (Paris) 30, 32 (1969).
  6. Partial analytic solutions for semi-infinite samples from a fourth-order wave equation were reported by R. Dreher, G. Meier, and A. Saupe, Mol. Cryst. Liquid Cryst. 13, 17 (1971).
  7. D. W. Berreman and T. J. Scheffer, Phys. Rev. A 5, 1397 (1972).
  8. Paul Drude, The Theory of Optics, translated by C. R. Mann and R. A. Millikan (Longmans, Green, N. Y., 1922), Ch. VI, Eqs. (7) and (8).
  9. M. Born, Optik (Springer, Berlin, 1933).
  10. C. W. Oseen, Trans. Faraday Soc. 29, 833 (1933).
  11. HI. deVries, Acta Cryst. 4, 219 (1951).
  12. A. S. Marathay, J. Opt. Soc, Am, 61, 1363 (1971).
  13. See, for example, W. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), Eqs. 11–47 and 11–49.

Abelès, F.

F. Abelès, Ann. Phys. (Paris) 5, 598 (1950). [See also M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Sec. 1.6.]

Berreman, D. W.

D. W. Berreman and T. J. Scheffer, Phys. Rev. Letters 25, 577 (1970); Mol. Cryst. Liquid Cryst. 11, 395 (1970).

D. W. Berreman and T. J. Scheffer, Phys. Rev. A 5, 1397 (1972).

Born, M.

M. Born, Optik (Springer, Berlin, 1933).

deVries, HI.

HI. deVries, Acta Cryst. 4, 219 (1951).

Dreher, R.

Partial analytic solutions for semi-infinite samples from a fourth-order wave equation were reported by R. Dreher, G. Meier, and A. Saupe, Mol. Cryst. Liquid Cryst. 13, 17 (1971).

Drude, Paul

Paul Drude, The Theory of Optics, translated by C. R. Mann and R. A. Millikan (Longmans, Green, N. Y., 1922), Ch. VI, Eqs. (7) and (8).

Henvis, B. W.

S. Teitler and B. W. Henvis, J. Opt. Soc. Am. 60, 830, (1970).

Jones, R. C.

R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941); 31, 500 (1941) and succeeding papers.

Marathay, A. S.

A. S. Marathay, J. Opt. Soc, Am, 61, 1363 (1971).

Meier, G.

Partial analytic solutions for semi-infinite samples from a fourth-order wave equation were reported by R. Dreher, G. Meier, and A. Saupe, Mol. Cryst. Liquid Cryst. 13, 17 (1971).

Oseen, W.

C. W. Oseen, Trans. Faraday Soc. 29, 833 (1933).

Panofsky, W.

See, for example, W. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), Eqs. 11–47 and 11–49.

Phillips, M.

See, for example, W. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), Eqs. 11–47 and 11–49.

Saupe, A.

Partial analytic solutions for semi-infinite samples from a fourth-order wave equation were reported by R. Dreher, G. Meier, and A. Saupe, Mol. Cryst. Liquid Cryst. 13, 17 (1971).

Scheffer, T. J.

D. W. Berreman and T. J. Scheffer, Phys. Rev. Letters 25, 577 (1970); Mol. Cryst. Liquid Cryst. 11, 395 (1970).

D. W. Berreman and T. J. Scheffer, Phys. Rev. A 5, 1397 (1972).

Taupin, D.

Reflectance of obliquely incident light by semi-infinite samples was computed for one particular model of a cholesteric liquid crystal for the first time a little earlier, using a different technique, by D. Taupin, J. Phys. (Paris) 30, 32 (1969).

Teitler, S.

S. Teitler and B. W. Henvis, J. Opt. Soc. Am. 60, 830, (1970).

Other (13)

R. C. Jones, J. Opt. Soc. Am. 31, 488 (1941); 31, 500 (1941) and succeeding papers.

F. Abelès, Ann. Phys. (Paris) 5, 598 (1950). [See also M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, New York, 1964), Sec. 1.6.]

S. Teitler and B. W. Henvis, J. Opt. Soc. Am. 60, 830, (1970).

D. W. Berreman and T. J. Scheffer, Phys. Rev. Letters 25, 577 (1970); Mol. Cryst. Liquid Cryst. 11, 395 (1970).

Reflectance of obliquely incident light by semi-infinite samples was computed for one particular model of a cholesteric liquid crystal for the first time a little earlier, using a different technique, by D. Taupin, J. Phys. (Paris) 30, 32 (1969).

Partial analytic solutions for semi-infinite samples from a fourth-order wave equation were reported by R. Dreher, G. Meier, and A. Saupe, Mol. Cryst. Liquid Cryst. 13, 17 (1971).

D. W. Berreman and T. J. Scheffer, Phys. Rev. A 5, 1397 (1972).

Paul Drude, The Theory of Optics, translated by C. R. Mann and R. A. Millikan (Longmans, Green, N. Y., 1922), Ch. VI, Eqs. (7) and (8).

M. Born, Optik (Springer, Berlin, 1933).

C. W. Oseen, Trans. Faraday Soc. 29, 833 (1933).

HI. deVries, Acta Cryst. 4, 219 (1951).

A. S. Marathay, J. Opt. Soc, Am, 61, 1363 (1971).

See, for example, W. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison-Wesley, Reading, Mass., 1955), Eqs. 11–47 and 11–49.

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