Abstract

When a birefringent slab is rotated between two crossed polarizers, the four maxima and four minima of the transmitted intensity occur, as known, at every π/4 radians. If the slab is also optically active, the maxima and minima are arranged in other patterns that may be grouped according to the number of extrema met in one complete turn of the slab: four or eight. All the possible patterns are classified and each is related to some peculiarity of the complex refractive index of the slab.

© 1980 Optical Society of America

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References

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  1. G. N. Ramachandran and S. Ramaseshan, “Crystal optics” in Handbuch der Physik (Springer-Verlag, Berlin, 1961), Bd.XXV/1.
  2. G. Castelnuovo, Lezioni di Geometria Analitica (Societá Editrice Dante Alighieri, Milano, 1961).
  3. S. Bhagavantam, Crystal Symmetry and Physical Properties (Academic, London and New York, 1966).

Bhagavantam, S.

S. Bhagavantam, Crystal Symmetry and Physical Properties (Academic, London and New York, 1966).

Castelnuovo, G.

G. Castelnuovo, Lezioni di Geometria Analitica (Societá Editrice Dante Alighieri, Milano, 1961).

Ramachandran, G. N.

G. N. Ramachandran and S. Ramaseshan, “Crystal optics” in Handbuch der Physik (Springer-Verlag, Berlin, 1961), Bd.XXV/1.

Ramaseshan, S.

G. N. Ramachandran and S. Ramaseshan, “Crystal optics” in Handbuch der Physik (Springer-Verlag, Berlin, 1961), Bd.XXV/1.

Other (3)

G. N. Ramachandran and S. Ramaseshan, “Crystal optics” in Handbuch der Physik (Springer-Verlag, Berlin, 1961), Bd.XXV/1.

G. Castelnuovo, Lezioni di Geometria Analitica (Societá Editrice Dante Alighieri, Milano, 1961).

S. Bhagavantam, Crystal Symmetry and Physical Properties (Academic, London and New York, 1966).

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Figures (1)

FIG. 1
FIG. 1

Azimuthal patterns of the positions of the maxima and minima of the light intensity transmitted by an absorbing anisotropic and optically active slab placed between two crossed polarizers in a parallel light beam, ν = linear birefringence; μ = linear dichroism; ψ = angle between the slower and the more absorbed linear polarization; ρ = rotatory power; σ = circular dichroism. When R2 < 1 the anisotropy prevails over gyrotropy; the contrary happens when R2 > 1 (see text). The special cases correspond to Eqs. (22) and (23) of the text.

Equations (31)

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d p / d z = i A ˆ p ,
A ˆ = ( a 11 a 12 a 21 a 22 ) = ( ( n + i m ) + ( ν + i μ cos 2 ψ ) i μ sin 2 ψ + i ( ρ + i σ ) i μ sin 2 ψ i ( ρ + i σ ) ( n + i m ) ( ν + i μ cos 2 ψ ) ) ,
α 1 = a 22 w , α 2 = a 11 + w ,
a 1 = h 1 ( w a 21 ) ,
a 2 = h 2 ( a 12 w ) ,
w = ( a 22 a 11 ) / 2 + [ ( a 22 a 11 ) 2 / 4 + a 12 a 21 ] 1 / 2 .
K ˆ = S ˆ ( e 1 0 0 e 2 ) S ˆ 1 ,
S ˆ = ( h 1 w h 1 a 21 h 2 a 12 h 2 w ) .
k 11 = ( w 2 e 1 + a 12 a 21 e 2 ) / υ , k 12 = ( e 2 e 1 ) w a 12 / υ , k 21 = ( e 2 e 1 ) w a 21 / υ , k 22 = ( w 2 e 2 + a 12 a 21 e 1 ) / υ , υ = w 2 + a 12 a 21 .
T ( θ ) = 1 + E 2 cos ( 2 θ + θ 2 ) + E 4 cos ( 4 θ + θ 4 ) ,
E 2 2 = 4 ( μ 2 ρ 2 + ν 2 σ 2 2 ν μ ρ σ cos 2 ψ ) / u 2 ,
E 4 2 = ( ν 4 + μ 4 + 2 ν 2 μ 2 cos 4 ψ ) / 4 u 2 ,
cos θ 2 = ( 2 μ ρ sin 2 ψ ) / u E 2 , sin θ 2 = 2 ( μ ρ cos 2 ψ ν σ ) / u E 2 , cos θ 4 = ( ν 2 + μ 2 cos 4 ψ ) / 2 u E 4 , sin θ 4 = ( μ 2 sin 4 ψ ) / 2 u E 4 ,
u = ( ν 2 + μ 2 ) / 2 + ρ 2 + σ 2 .
X = cos ( 2 θ + θ 4 / 2 ) , Y = sin ( 2 θ + θ 4 / 2 ) .
T ( X , Y ) = 1 + E 4 ( X 2 Y 2 + 4 b X 4 c Y ) ,
T ( X , Y ) = 4 E 4 ( X Y + c X + b Y ) = 0 ,
T ( X , Y ) = 4 E 4 ( Y 2 X 2 b X + c Y ) .
X 2 + Y 2 = 1 ,
b = ( E 2 / 4 E 4 ) cos ( θ 2 θ 4 / 2 ) , c = ( E 2 / 4 E 4 ) sin ( θ 2 θ 4 / 2 ) .
R 2 = ( b 2 / 3 + c 2 / 3 ) 3 < 1.
b c = 0.
ρ = σ = 0 ,
ν μ sin 2 ψ = 0 ,
( p 2 σ 2 ) cos 2 ψ = ρ σ ( ν 2 μ 2 ) ,
2 ν μ ρ σ cos 2 ψ = μ 2 ρ 2 + ν 2 σ 2 .
p = ( cos θ sin θ ) , q = ( sin θ cos θ ) .
q K ˆ p = ( 1 2 ) [ ( k 21 k 12 ) + ( k 22 k 11 ) sin 2 θ + ( k 21 + k 12 ) cos 2 θ ] .
( k 21 k 12 ) / 2 = H ( a 21 a 12 ) / 2 = H i ( ρ + i σ ) = H B 0 , ( k 21 + k 12 ) / 2 = H ( a 21 + a 12 ) / 2 = H i μ sin 2 ψ = H B C , ( k 22 k 11 ) / 2 = H ( a 22 a 11 ) / 2 = H ( ν + i μ cos 2 ψ ) = H B S ,
H = ( e 2 e 1 ) w / 2 υ .
| q K ˆ p | 2 = | f | 2 [ | B 0 | 2 + ( | B C | 2 + | B S | 2 ) / 2 + ( B 0 B C * + B 0 * B C ) cos 2 θ + ( B 0 B S * + B 0 * B S ) sin 2 θ + ( 1 2 ) ( | B C | 2 | B S | 2 ) cos 4 θ + ( 1 2 ) ( B S B C * + B S * B C ) sin 4 θ ] .