Abstract

The polarization properties of single-corner reflectors and of certain types of cavities using corner reflectors are investigated and described. States of polarization are found which remain invariant under transmission. Particular attention is given to the lossless case of total internal reflection.

© 1962 Optical Society of America

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References

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  1. E. R. Peck, J. Opt. Soc. Am. 38, 1015 (1948).
    [Crossref] [PubMed]

1948 (1)

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Figures (5)

Fig. 1
Fig. 1

Coordinate axes at a reflecting surface.

Fig. 2
Fig. 2

Face of a corner reflector with reference frame in the sextant of incidence.

Fig. 3
Fig. 3

Eigenstates of polarization for a single, totally reflecting corner, with refractive index μ=1.517.

Fig. 4
Fig. 4

Cavities using corner reflectors.

Fig. 5
Fig. 5

Prism faces as viewed each from the other.

Equations (38)

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tan ( ϕ / 2 ) = ( μ 2 sin 2 α - 1 ) 1 2 / μ cos α ; tan ( χ / 2 ) = μ 2 tan ( ϕ / 2 ) .
p 1 = 1 2 p 2 - 1 2 3 q 2 q 1 = 1 2 3 p 2 + 1 2 q 2 p 2 = 1 2 p 3 + 1 2 3 q 3 q 2 = - 1 2 3 p 3 + 1 2 q 3 p 3 = 1 2 p 1 + 1 2 3 q 1 q 3 = 1 2 3 p 1 - 1 2 q 1 .
E = ( c 11 u + c 12 v ) p 1 + ( c 21 u + c 22 v ) q 1
( u , v ) = ( c i j ) ( u v ) .
c 11 = a 8 [ ( a + b ) 2 + 4 b ( a - b ) ] c 12 = 1 8 3 b ( a + b ) 2 c 21 = 1 8 3 a ( a + b ) 2 c 22 = b 8 [ 4 a ( a - b ) - ( a + b ) 2 ] .
( c i j ) ( u v ) = s ( u , v ) ,
| c 11 - s c 12 c 21 c 22 - s | = 0
s = 1 2 T ± 1 2 ( T 2 - 4 D ) 1 2 .
v / u = ( s - c 11 ) / c 12 ,
v u = - ( c 11 - c 22 ) 2 c 12 ± 1 2 c 12 [ ( c 11 - c 22 ) 2 + 4 c 12 c 21 ] 1 2 .
v / u = v 0 e i β ,             v 0 0
tan 2 θ = 2 v 0 cos β / 1 - v 0 2 .
e 2 = 2 ( 1 - m 2 ) 1 2 1 + ( 1 - m 2 ) 1 2 ,
m 2 4 sin 2 β ( v 0 + 1 / v 0 ) 2 .
( e i ϕ + e i χ ) n = 2 n cos n ( δ 2 ) e i n ( ϕ + χ ) / 2 ( e i ϕ - e i χ ) n = ( 2 i ) n sin n ( δ 2 ) e i n ( ϕ + χ ) / 2 ,
δ ϕ - χ .
s = e i σ
s = 1 2 T [ 1 ± i [ ( 4 D / T 2 ) - 1 ] 1 2 ]
v u = - ( c 11 - c 22 ) 2 c 12 { 1 ± [ 1 + 4 c 12 c 21 ( c 11 - c 22 ) 2 ] 1 2 } .
4 c 12 c 21 / ( c 11 - c 22 ) 2 = 2 c 12 / ( c 11 - c 22 ) 2 .
σ = τ ± Cos - 1 ( T 2 / 4 D ) 1 2 .
e i 2 τ = - c 21 / c 12 * = - c 12 / c 21 * .
β 2 = β 1 ± π .
v 02 = 1 / v 01 .
D = - a 3 b 3 T = 1 8 ( a - b ) [ ( a + b ) 2 + 8 a b ] c 11 - c 22 = 1 8 ( a + b ) 3 .
σ = 3 2 ( ϕ + χ ) - π 2 ± Cos - 1 { 1 2 | sin δ 2 | ( cos 2 δ 2 + 2 ) } .
v u = - 1 3 cos δ 2 [ 1 ( 1 + 3 sec 2 δ 2 ) 1 2 ] e i δ / 2 .
tan 2 θ = 3 ,
c 11 = 1 64 a ( a 5 + 15 a 4 b + 42 a 3 b 2 - 18 a 2 b 3 + 21 a b 4 + 3 b 5 ) c 12 = 3 64 b ( a + b ) 2 ( a - b ) [ ( a + b ) 2 + 8 a b ] c 21 = 3 64 a ( a + b ) 2 ( a - b ) [ ( a + b ) 2 + 8 a b ] c 22 = 1 64 b ( 3 a 5 + 21 a 4 b - 18 a 3 b 2 + 42 a 2 b 3 + 15 a b 4 + b 5 ) .
D = a 6 b 6 T = 1 64 ( a + b ) 4 [ ( a + b ) 2 + 12 a b ] - 2 a 3 b 3 c 11 - c 22 = 1 64 ( a + b ) 3 ( a - b ) [ ( a + b ) 2 + 8 a b ] .
σ = 3 ( ϕ + χ ) ± Cos - 1 [ 1 2 cos 4 δ 2 ( cos 2 δ 2 + 3 ) - 1 ] .
( v u ) = - 1 3 cos δ 2 [ 1 ± ( 1 + 3 sec 2 δ 2 ) 1 2 ] e i δ / 2 .
[ 1 2 cos 4 δ 2 ( cos 2 δ 2 + 3 ) - 1 ] / sin δ 2 < 0.
c 11 = 1 16 a 2 ( a 4 + 6 a 3 b + 12 a 2 b 2 - 6 a b 3 + 3 b 4 ) c 12 = c 21 = 3 16 a b ( a + b ) 3 ( a - b ) c 22 = 1 16 b 2 ( 3 a 4 - 6 a 3 b + 12 a 2 b 2 + 6 a b 3 + b 4 ) .
D = a 6 b 6 T = 1 16 [ ( a + b ) 6 - 32 a 3 b 3 ] c 11 - c 22 = 1 16 ( a + b ) 3 ( a - b ) [ ( a + b ) 2 + 2 a b ] .
σ = 3 ( ϕ + χ ) ± Cos - 1 ( 2 cos 6 δ 2 - 1 ) .
( v u ) = - 1 3 ( 2 cos 2 δ 2 + 1 ) × [ 1 ± [ 1 + 3 / ( 2 cos 2 δ 2 + 1 ) 2 ] 1 2 ] .
tan 2 θ = 3 / ( 2 cos 2 δ 2 + 1 ) .