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.

© 1966 Optical Society of America

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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 n0+, 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.
    [CrossRef]
  5. A. B. Winterbottom, Kgl. Norske Videnskab. Selskabs Skrifter 1, 27, 37 (1955).
  6. D. A. Holmes, J. Opt. Soc. Am. 54, 1340 (1964); J. Opt. Soc. Am. 55, 209 (1965).
    [CrossRef]
  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 pplates, or the multilayer problem.
  10. In Eq. (4) and all subsequent equations we suppress the factor exp (i 2πft).
  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).
    [CrossRef]
  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. Vađîèek, 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).
    [CrossRef]
  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. ∊0j and ∊90j are the values of ∊0and ∊90in the j th medium.
  21. The simplest way to make (33) applicable to an isotropic medium is first to set ϕ= 0 and then to set ∊α= ∊γ.
  22. D. A. Holmes, Optics of a Birefringent Plate with Applications to Ellipsometry, Ph.D. thesis, Carnegie Institute of Technology, May1965, 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 = 8010 Å), p. 6–18; Rutile (λ = 5770 Å), 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.

1964 (1)

1962 (2)

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

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

1955 (1)

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

1952 (1)

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.
[CrossRef]

1950 (1)

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); J. Opt. Soc. Am. 55, 209 (1965).
[CrossRef]

D. A. Holmes, Optics of a Birefringent Plate with Applications to Ellipsometry, Ph.D. thesis, Carnegie Institute of Technology, May1965, Appendix A, pp. 73–74.

Partovi, F.

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.

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.
[CrossRef]

Vadîèek, A.

A. Vađîèek, 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.

J. Opt. Soc. Am. (3)

Kgl. Norske Videnskab. Selskabs Skrifter (1)

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

Opt. Spectry. (1)

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

Z. Physik (1)

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.
[CrossRef]

Other (20)

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 pplates, or the multilayer problem.

In Eq. (4) and all subsequent equations we suppress the factor exp (i 2πft).

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.

When the electric-field vector of a cw wave propagating in an anistropic medium is expressed in the form (1), the quantity n0+, 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.

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

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

∊0j and ∊90j are the values of ∊0and ∊90in the j th medium.

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

D. A. Holmes, Optics of a Birefringent Plate with Applications to Ellipsometry, Ph.D. thesis, Carnegie Institute of Technology, May1965, 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 = 8010 Å), p. 6–18; Rutile (λ = 5770 Å), 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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Figures (5)

Fig. 1
Fig. 1

Diagram of p birefringent plates sandwiched together and surrounded on both sides by semi-infinite birefringent media. The plates are assumed to have perfectly flat and parallel surfaces and to have thicknesses d j , j = 1 , 2 , 3 , , p. All media are homogeneous, not optically active and not magnetic. An xyz right-handed coordinate system is defined such that the xy plane coincides with the interface between the zeroth and first media. The positive y axis points out of the plane of the paper. For each medium, one of the principal dielectric axes is parallel to the y axis. The symbols αj and γj denote the (positive) directions of the other two principal dielectric axes. The angle ϕj is the angle between the positive αj direction and the positive x direction. All physically distinct orientations of the αjγj principal axes can be obtained by restricting ϕj to the range 0 ≦ ϕj < π. Along a principal axis αj, the principal refractive index is nαj, while along γj the principal index is nγj. An electromagnetic wave is incident in medium zero with the plane of incidence parallel to the xz plane. The quantities θj+ and θj are angles between unit wave-normal vectors and the positive z direction. Note that 0° ≦ θj+ ≦ 90° and 90° ≦ θj ≦ 180°, in our convention. The intensity of the incident wave is sufficiently low so that nonlinear effects are negligible in all media.

Fig. 2
Fig. 2

Calculated values of power transmittance |T0,2|2 vs d1/λ for a single birefringent plate immersed in air. The values of the principal refractive indices of the plate, nα1 = 1.64869 and nγ1 = 1.48215, correspond to calcite. The solid curve is for ϕ1 = 0° and the dashed curve is for ϕ1 = 30°. The angle of incidence was θ0+ = 45° for both curves. When ϕ1 = 0°, we found that n1+ = n1 = 1.6123; θ1+ = 180° − θ1 = 26.013°; h1+ = 1.876; r 0 , 1 = r 1 , 0 = r 1 , 2 = 0.1403; t0,1 = 0.8597; t1,0 = t1,2 = 1.1403. When ϕ1 = 30°, we found that n1+ = 1.6474; n1 = 1.5427; θ1+ = 25.416°; θ1 = 152.369°; h1+ = 1.808; r 0 , 1 = r 1 , 0 = r 1 , 2 = 0.1221 ;t0,1 = 0.8779; t1,0 = t1,2 = 1.1221.

Fig. 3
Fig. 3

Calculated electric-field amplitude reflectances as a function of angle of incidence θ0+. The “smooth” curves are the interface reflectance | r 0 , 1 | describing reflection from an air–rutile interface. The solid curve is for ϕ1 = 0°, while the dashed curve is for ϕ1 = 45°. The principal refractive indices of rutile were taken as nα1 = 2.921 and nγ1 = 2.623. Note that | r 0 , 1 | goes to zero at the Brewster angle (near 70°) and that the Brewster angle depends on the orientation angle ϕ1. When the angle of incidence is less than (greater than) the Brewster angle, then r 0 , 1 is negative (positive). The “humped” curves are |R0,2| for a rutile plate of thickness d1/λ = 5 (arbitrarily chosen) immersed in air. Again, the solid curve is for ϕ1 = 0°, while the dashed curve is for ϕ1 = 45°. The departure of |R0,2| from | r 0 , 1 | vividly shows the influence of multiple internal reflections within the plate. Note that |R0,2| is more sensitive to the change in ϕ1 than is | r 0 , 1 |.

Fig. 4
Fig. 4

Calculated values of the refractive indices n1+ and n1 vs angle of incidence θ0+, for a rutile plate surrounded by air. For curve 1, ϕ1 = 0° and n1+ = n1. The curves identified by the number 2 are for ϕ1 = 45°; here the upper curve is n1+ and the lower curve is n1. For curve 3, ϕ1 = 90° and again n1+ = n1.

Fig. 5
Fig. 5

Effect on a Berek compensator’s phase difference owing to slight misalignment of the optic axis. To understand this figure, let us make the following definitions: Δe is the exact (wave-optics) value of the phase difference when ϕ1 = 0; Δe is the exact value of phase difference when ϕ1 = 5 min of arc; Δα is the approximate (geometrical-optics) value of the phase difference when ϕ1 = 0, and Δα is the approximate phase difference when ϕ1 = 5 min of arc. Note that Δα would correspond to the usual calculated calibration of the compensator. The solid curve is Δe − Δα (degrees, ordinate) vs angle of incidence θ0+ (degrees, upper abscissa values) and represents a duplicate of the solid curve in Fig. 2(a) of Ref. 6. The dashed curve is Δe − Δα vs θ0+. The middle abscissa scale gives selected values of Δα in degrees. The lower abscissa scale gives values of Δα in degrees. The optical constants and thickness of the calcite plate are the same as used in Ref. 6.

Equations (51)

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0 + = E 0 + exp { i 2 π f t i ( 2 π n 0 + / λ ) S 0 + · r } ,
n 0 + = n α 0 n γ 0 [ n α 0 2 sin 2 ( θ 0 + ϕ 0 ) + n γ 0 2 cos 2 ( θ 0 + ϕ 0 ) ] 1 2 .
0 = E 0 exp { i 2 π f t i ( 2 π n 0 / λ ) S 0 · r } ;
p + 1 + = E p 1 + C + p + 1 exp { i ( 2 π n p + 1 + / λ ) S p + 1 + · r } ,
C p + 1 + = exp { i ( 2 π n p + 1 + cos θ p + 1 + / λ ) k = 1 p d k } .
S j g = U x sin θ j g + U z cos θ j g , g = + , ,
j g = E j g C j g exp { i ( 2 π n j g / λ ) S j g · r } ,
C j g = exp { i ( 2 π n j g cos θ j g / λ ) k = 1 j 1 d k } .
E j g = U x E x , j g + U z E z , j g .
E x , j + / B j + + E x , j / B j = E x , j + 1 + + E x , j + 1 ,
B j g exp ( i 2 π n j g d j cos θ j g / λ ) .
n j g sin θ j g = n 0 + sin θ 0 + ,
( f μ 0 λ ) H y , j g = h j g E x , j g ,
h j g n j g { cos θ j g ( E z , j g / E x , j g ) sin θ j g } .
h j + E x , j + / B j + + h j E x , j / B j = h j + 1 + E x , j + 1 + + h j + 1 E x , j + 1 .
T p , p + 1 ; T p 1 , p + 1 ; ; T j , p + 1 ; ; T 0 , p + 1 ,
R p , p + 1 ; R p 1 , p + 1 ; ; R j , p + 1 ; ; R 0 , p + 1 .
T p , p + 1 = t p , p + 1 , R p , p + 1 = r p , p + 1 ,
T j , p + 1 = t j , j + 1 T j + 1 , p + 1 B j + 1 + r j + 1 , j R j + 1 , p + 1 B j + 1 ,
R j , p + 1 = r j , j + 1 B j + 1 + + R j + 1 , p + 1 ( r j , j + 1 + t j + 1 , j ) B j + 1 B j + 1 + r j + 1 , j R j + 1 , p + 1 B j + 1 .
r j , j + 1 = ( h j + h j + 1 + ) / ( h j + 1 + h j ) ,
r j + 1 , j = ( h j + 1 h j ) / ( h j h j + 1 + ) ,
t j , j + 1 = ( h j h j + ) / ( h j h j + 1 + ) ,
t j + 1 , j = ( h j + 1 + h j + 1 ) / ( h j + 1 + h j ) .
E x , p + / E x , p + 1 + = B p + / T p , p + 1 ,
E x , p / E x , p + 1 = B p R p , p + 1 / T p , p + 1 .
E x , p 1 + / E x , p + 1 + = B p 1 + / T p 1 , p + 1 ,
E x , p 1 / E x , p + 1 + = B p 1 R p 1 , p + 1 / T p 1 , p + 1 .
E x , j + / E x , 0 + = B j + T 0 , p + 1 / T j , p + 1 ,
E x , j / E x , 0 + = B j R j , p + 1 T 0 , p + 1 / T j , p + 1 .
[ E x , 0 + E x , 0 ] = ( j = 0 j = p t j , j + 1 1 ) M 0 , 1 M 1 , 2 M j , j + 1 M p , p + 1 [ E x , p + 1 + 0 ] ,
M j , j + 1 = [ B j + r j + 1 , j B j + r j , j + 1 B j { r j , j + 1 + t j + 1 , j } B j ] .
[ E x , j + E x , j [ = ( 1 / t j , j + 1 ) M j , j + 1 [ E x , j + 1 + E x , j + 1 ] .
= n 2 , α = n α 2 , γ = n γ 2 , N = ( n 0 + sin θ 0 + ) 2 , 0 = α + N [ 1 ( α / γ ) ] , 90 = γ + N [ 1 ( γ / α ) ] .
( α S α 2 + γ S γ 2 ) = α γ ,
S α = sin ( θ ϕ ) , S γ = cos ( θ ϕ ) .
cos [ 2 θ 2 ψ ] = α γ N ( α + γ ) [ ( 0 γ cos ϕ ) 2 + ( 90 α sin ϕ ) 2 ] 1 2 ,
2 ψ = arctan { 2 ( γ α ) N 0 γ cot ϕ + 90 γ tan ϕ } .
[ γ cos 2 ϕ + α sin 2 ϕ ] 2 · n 2 = 0 γ 2 cos 2 ϕ + 90 α 2 sin 2 ϕ ± | ( γ α ) sin 2 ϕ | · [ a γ N ( γ cos 2 ϕ + α sin 2 ϕ N ) ] 1 2 .
D = K 0 [ E S ( S · E ) ] ,
E γ / E α = ( S α 2 α ) / S α S γ ,
E z / E x = { ( E γ / E α ) tan ϕ } / { 1 + ( E γ / E α ) tan ϕ } .
h n = γ cos 2 ( θ ϕ ) + α sin 2 ( θ ϕ ) γ cos ( θ ϕ ) cos ϕ α sin ( θ ϕ ) sin ϕ .
h j + = h j .
h j + h j = sin θ j + sin θ j · cos θ j 1 2 A j sin θ j cos θ j + 1 2 A j sin θ j + ,
A j = ( α j γ j ) sin 2 ϕ j / ( γ j cos 2 ϕ j + α j sin 2 ϕ j ) .
A j = sin ( θ j + + θ j ) / ( sin θ j + sin θ j ) .
r j , j + 1 = r j + 1 , j = ( h j + h j + 1 + ) / ( h j + + h j + 1 + ) ,
t j , j + 1 = 2 h j + / ( h j + + h j + 1 + ) ,
t j + 1 , j = 2 h + / ( h j + 1 + + h j + ) .
r j , j + 1 + t j + 1 , j = r j + 1 , j + t j , j + 1 = 1.