Abstract

A theoretical treatment of the reflection and transmission properties of a stack of birefringent plates, surrounded by semi-infinite birefringent media, is presented. The orientation of the principal dielectric axes for each plate is restricted to the case for which one principal axis is perpendicular to the plane of incidence. The analysis treats only incident waves having electric fields polarized parallel to the plane of incidence. For numerical illustration, the effects due to slight misalignment of the optic axis in a calcite Berek rotary compensator are examined.

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  1. When the electric-field vector of a cw wave propagating in an anistropic medium is expressed in the form (1), the quantity no+, which can properly be called a refractive index, depends on material parameters, the direction of propagation, and the polarization. Since the refractive index does depend upon wave properties, as well as material properties, we identify the refractive index with a specific traveling wave when the medium is given; for example, in the zeroth, or incident, medium we call n0+ the refractive index of the incident wave.
  2. G. N. Ramachandran and S. Ramaseshan in Handbuch der Physik, S. Flugge, Ed. (Springer-Verlag, Berlin, 1961), Vol. 25, Ch. 1, p. 111. This reference reviews many topics in the optics of anisotropic media.
  3. The derivation of Eq. (2) is considered later.
  4. H. Schopper, Z. Physik 132, 146 (1952). Schopper's work has been translated into English in condensed form by O. S. Heavens, see Ref. 15, pp. 92–95.
  5. A. B. Winterbottom, Kgl. Norske Videnskab. Selskabs Skrifter 1, 27, 37 (1955).
  6. D. A. Holmes, J. Opt. Soc. Am. 54, 1340 (1964); 55, 209 (1965).
  7. A. M. Goncharenko and F. I. Federov, Opt. Spectry. 13, 48 (1962).
  8. P. H. Berning in Physics of Thin Films Vol. 1, G. Hass, editor (Academic Press, New York and London, 1963), Ch. 2, p. 71.
  9. As originally submitted, this work treated only a single plate. We are indebted to an anonymous referee for suggesting that we extend our coverage to p plates, or the multilayer problem.
  10. In Eq. (4) and all subsequent equations we suppress the factor exp(i2πƒt).
  11. The factor Cp+1+ is introduced for convenience and will be discussed in greater detail later.
  12. Our classifications, "positively" and "negatively" traveling stem from the fact that, when θ0+=0 (normal incidence), Sj+ points in the positive z direction while Sj- points in the negative z direction.
  13. A more complete solution for hjg will be derived later.
  14. B. Salzberg, J. Opt. Soc. Am. 40, 465 (1950).
  15. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, London, 1955), pp. 69 et seq.; and in Physics of Thin Films Vol. 2, G. Hass and R. E. Thun, Eds. (Academic Press, New York and London, 1964).
  16. M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959) pp. 50 et. seq.
  17. A. Vašček, Optics of Thin Films (North-Holland Publishing Co., Amsterdam and Interscience Publishers Inc., New York, 1960), Ch. 4.
  18. F. Partovi, J. Opt. Soc. Am. 52, 918 (1962).
  19. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill Book Co., New York, 1960), pp. 97–100. See also Ref. 2, p. 56.
  20. ε0j and εg0j are the values of ε0 and εg0 in the jth medium.
  21. The simplest way to make (33) applicable to an isotropic medium is first to set φ=0 and then to set εα = εγ.
  22. D. A. Holmes, Optics of a Birefringent Plate with Applications to Ellipsometry, Ph.D. thesis, Carnegie Institute of Technology, May 1965, Appendix A, pp. 73–74.
  23. The numerical values for the principal refractive indices were taken from American Institute of Physics Handbook, 2d ed. (McGraw-Hill Book Company, Inc., New York, 1963), Calcite (λ=8010Å), p. 6–18; Rutile (λ=5770Å), p. 6–33.
  24. Since media 0 and 2 are isotropic and identical, |T0,2|2 represents the power transmittance.
  25. Rotary compensators were discussed from the electromagnetic standpoint in Ref. 6, assuming perfect alignment of the principal dielectric axis.
  26. We believe that φ1= 5′ is reasonable as a maximum misalignment angle for the following reason. We have privately communicated with Crystal Optics, 3959 North Lincoln Avenue, Chicago 13, Illinois and have learned that, in the best-quality calcite Glan-Thompson prism polarizers made by them, the alignment of the optic axis is guaranteed within ±5 min of arc.

Berning, P. H.

P. H. Berning in Physics of Thin Films Vol. 1, G. Hass, editor (Academic Press, New York and London, 1963), Ch. 2, p. 71.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959) pp. 50 et. seq.

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill Book Co., New York, 1960), pp. 97–100. See also Ref. 2, p. 56.

Federov, F. I.

A. M. Goncharenko and F. I. Federov, Opt. Spectry. 13, 48 (1962).

Goncharenko, A. M.

A. M. Goncharenko and F. I. Federov, Opt. Spectry. 13, 48 (1962).

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, London, 1955), pp. 69 et seq.; and in Physics of Thin Films Vol. 2, G. Hass and R. E. Thun, Eds. (Academic Press, New York and London, 1964).

Holmes, D. A.

D. A. Holmes, J. Opt. Soc. Am. 54, 1340 (1964); 55, 209 (1965).

D. A. Holmes, Optics of a Birefringent Plate with Applications to Ellipsometry, Ph.D. thesis, Carnegie Institute of Technology, May 1965, Appendix A, pp. 73–74.

Partovi, F.

F. Partovi, J. Opt. Soc. Am. 52, 918 (1962).

Ramachandran, G. N.

G. N. Ramachandran and S. Ramaseshan in Handbuch der Physik, S. Flugge, Ed. (Springer-Verlag, Berlin, 1961), Vol. 25, Ch. 1, p. 111. This reference reviews many topics in the optics of anisotropic media.

Ramaseshan, S.

G. N. Ramachandran and S. Ramaseshan in Handbuch der Physik, S. Flugge, Ed. (Springer-Verlag, Berlin, 1961), Vol. 25, Ch. 1, p. 111. This reference reviews many topics in the optics of anisotropic media.

Salzberg, B.

B. Salzberg, J. Opt. Soc. Am. 40, 465 (1950).

Schopper, H.

H. Schopper, Z. Physik 132, 146 (1952). Schopper's work has been translated into English in condensed form by O. S. Heavens, see Ref. 15, pp. 92–95.

Vašcek, A.

A. Vašček, Optics of Thin Films (North-Holland Publishing Co., Amsterdam and Interscience Publishers Inc., New York, 1960), Ch. 4.

Winterbottom, A. B.

A. B. Winterbottom, Kgl. Norske Videnskab. Selskabs Skrifter 1, 27, 37 (1955).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959) pp. 50 et. seq.

Other (26)

When the electric-field vector of a cw wave propagating in an anistropic medium is expressed in the form (1), the quantity no+, which can properly be called a refractive index, depends on material parameters, the direction of propagation, and the polarization. Since the refractive index does depend upon wave properties, as well as material properties, we identify the refractive index with a specific traveling wave when the medium is given; for example, in the zeroth, or incident, medium we call n0+ the refractive index of the incident wave.

G. N. Ramachandran and S. Ramaseshan in Handbuch der Physik, S. Flugge, Ed. (Springer-Verlag, Berlin, 1961), Vol. 25, Ch. 1, p. 111. This reference reviews many topics in the optics of anisotropic media.

The derivation of Eq. (2) is considered later.

H. Schopper, Z. Physik 132, 146 (1952). Schopper's work has been translated into English in condensed form by O. S. Heavens, see Ref. 15, pp. 92–95.

A. B. Winterbottom, Kgl. Norske Videnskab. Selskabs Skrifter 1, 27, 37 (1955).

D. A. Holmes, J. Opt. Soc. Am. 54, 1340 (1964); 55, 209 (1965).

A. M. Goncharenko and F. I. Federov, Opt. Spectry. 13, 48 (1962).

P. H. Berning in Physics of Thin Films Vol. 1, G. Hass, editor (Academic Press, New York and London, 1963), Ch. 2, p. 71.

As originally submitted, this work treated only a single plate. We are indebted to an anonymous referee for suggesting that we extend our coverage to p plates, or the multilayer problem.

In Eq. (4) and all subsequent equations we suppress the factor exp(i2πƒt).

The factor Cp+1+ is introduced for convenience and will be discussed in greater detail later.

Our classifications, "positively" and "negatively" traveling stem from the fact that, when θ0+=0 (normal incidence), Sj+ points in the positive z direction while Sj- points in the negative z direction.

A more complete solution for hjg will be derived later.

B. Salzberg, J. Opt. Soc. Am. 40, 465 (1950).

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths Scientific Publications, London, 1955), pp. 69 et seq.; and in Physics of Thin Films Vol. 2, G. Hass and R. E. Thun, Eds. (Academic Press, New York and London, 1964).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, London, 1959) pp. 50 et. seq.

A. Vašček, Optics of Thin Films (North-Holland Publishing Co., Amsterdam and Interscience Publishers Inc., New York, 1960), Ch. 4.

F. Partovi, J. Opt. Soc. Am. 52, 918 (1962).

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill Book Co., New York, 1960), pp. 97–100. See also Ref. 2, p. 56.

ε0j and εg0j are the values of ε0 and εg0 in the jth medium.

The simplest way to make (33) applicable to an isotropic medium is first to set φ=0 and then to set εα = εγ.

D. A. Holmes, Optics of a Birefringent Plate with Applications to Ellipsometry, Ph.D. thesis, Carnegie Institute of Technology, May 1965, Appendix A, pp. 73–74.

The numerical values for the principal refractive indices were taken from American Institute of Physics Handbook, 2d ed. (McGraw-Hill Book Company, Inc., New York, 1963), Calcite (λ=8010Å), p. 6–18; Rutile (λ=5770Å), p. 6–33.

Since media 0 and 2 are isotropic and identical, |T0,2|2 represents the power transmittance.

Rotary compensators were discussed from the electromagnetic standpoint in Ref. 6, assuming perfect alignment of the principal dielectric axis.

We believe that φ1= 5′ is reasonable as a maximum misalignment angle for the following reason. We have privately communicated with Crystal Optics, 3959 North Lincoln Avenue, Chicago 13, Illinois and have learned that, in the best-quality calcite Glan-Thompson prism polarizers made by them, the alignment of the optic axis is guaranteed within ±5 min of arc.

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