Abstract

The action of a polarizer is conventionally determined only for a normally incident plane wave. In this work polarization characteristics of polarizers are evaluated for a general incident wave front using the plane-wave spectral representation. It is shown that even an ideal polarizer will transmit part of a cross-polarized wave if it does not coincide with the wave front. Some interesting polarization effects are demonstrated by application of the theoretical analysis to polarized spherical waves.

© 1984 Optical Society of America

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References

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  1. Y. Fainman, J. Shamir, E. Lenz, “Coherent Light Scattering From Rough Conductors; Polarization Properties of Speckle Patterns,” in preparation.
  2. M. Born, E. Worf, Principles of Optics (Pergamon, London, 1970).
  3. J. D. Love, A. W. Snyder, “Generalized Fresnel’s Laws for a Curved Absorbing Interface,” J. Opt. Soc. Am. 65, 1072 (1975).
    [CrossRef]
  4. J. Ben-Uri, “Polarization and Interference in Optics V,” Optik 49, 375 (1978).
  5. M. V. R. K. Murty, R. P. Shukla, “Extinction of Light in Brewster Polarizers: A Shadowlike Phenomenon,” Appl. Opt. 22, 1094 (1983).
    [CrossRef] [PubMed]
  6. L. B. Felsen, N. Marcuvitz, “Radiation and Scattering of Waves,” (Prentice-Hall, Englewood Cliffs, N.J., 1973).
  7. R. H. Clarke, J. Brown, Diffraction Theory and Antennas (Ellis Horwood, Ltd., New York, 1980).
  8. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, London, 1966).
  9. E. V. Jull, Aperture Antennas and Diffraction Theory (Peter Peregrinus, London, 1981).
    [CrossRef]
  10. F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, London, 1979).
  11. T. A. Leonard, “Infrared Polarizer Selection,” Proc. Soc. Photo. Opt. Instrum. Eng., 288, 129 (1981).

1983 (1)

1981 (1)

T. A. Leonard, “Infrared Polarizer Selection,” Proc. Soc. Photo. Opt. Instrum. Eng., 288, 129 (1981).

1978 (1)

J. Ben-Uri, “Polarization and Interference in Optics V,” Optik 49, 375 (1978).

1975 (1)

Bass, F. G.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, London, 1979).

Ben-Uri, J.

J. Ben-Uri, “Polarization and Interference in Optics V,” Optik 49, 375 (1978).

Born, M.

M. Born, E. Worf, Principles of Optics (Pergamon, London, 1970).

Brown, J.

R. H. Clarke, J. Brown, Diffraction Theory and Antennas (Ellis Horwood, Ltd., New York, 1980).

Clarke, R. H.

R. H. Clarke, J. Brown, Diffraction Theory and Antennas (Ellis Horwood, Ltd., New York, 1980).

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, London, 1966).

Fainman, Y.

Y. Fainman, J. Shamir, E. Lenz, “Coherent Light Scattering From Rough Conductors; Polarization Properties of Speckle Patterns,” in preparation.

Felsen, L. B.

L. B. Felsen, N. Marcuvitz, “Radiation and Scattering of Waves,” (Prentice-Hall, Englewood Cliffs, N.J., 1973).

Fuks, I. M.

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, London, 1979).

Jull, E. V.

E. V. Jull, Aperture Antennas and Diffraction Theory (Peter Peregrinus, London, 1981).
[CrossRef]

Lenz, E.

Y. Fainman, J. Shamir, E. Lenz, “Coherent Light Scattering From Rough Conductors; Polarization Properties of Speckle Patterns,” in preparation.

Leonard, T. A.

T. A. Leonard, “Infrared Polarizer Selection,” Proc. Soc. Photo. Opt. Instrum. Eng., 288, 129 (1981).

Love, J. D.

Marcuvitz, N.

L. B. Felsen, N. Marcuvitz, “Radiation and Scattering of Waves,” (Prentice-Hall, Englewood Cliffs, N.J., 1973).

Murty, M. V. R. K.

Shamir, J.

Y. Fainman, J. Shamir, E. Lenz, “Coherent Light Scattering From Rough Conductors; Polarization Properties of Speckle Patterns,” in preparation.

Shukla, R. P.

Snyder, A. W.

Worf, E.

M. Born, E. Worf, Principles of Optics (Pergamon, London, 1970).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Optik (1)

J. Ben-Uri, “Polarization and Interference in Optics V,” Optik 49, 375 (1978).

Proc. Soc. Photo. Opt. Instrum. Eng. (1)

T. A. Leonard, “Infrared Polarizer Selection,” Proc. Soc. Photo. Opt. Instrum. Eng., 288, 129 (1981).

Other (7)

Y. Fainman, J. Shamir, E. Lenz, “Coherent Light Scattering From Rough Conductors; Polarization Properties of Speckle Patterns,” in preparation.

M. Born, E. Worf, Principles of Optics (Pergamon, London, 1970).

L. B. Felsen, N. Marcuvitz, “Radiation and Scattering of Waves,” (Prentice-Hall, Englewood Cliffs, N.J., 1973).

R. H. Clarke, J. Brown, Diffraction Theory and Antennas (Ellis Horwood, Ltd., New York, 1980).

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, London, 1966).

E. V. Jull, Aperture Antennas and Diffraction Theory (Peter Peregrinus, London, 1981).
[CrossRef]

F. G. Bass, I. M. Fuks, Wave Scattering from Statistically Rough Surfaces (Pergamon, London, 1979).

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Figures (4)

Fig. 1
Fig. 1

Planar interface between two media of refractive indices n1 and n2. N ^ is the normal to the interface while τ ^ is a unit vector defining the intersection of the interface with the plane of incidence.

Fig. 2
Fig. 2

Intensity distribution of a diverging polarized Gaussian beam transmitted through a crossed polarizer.

Fig. 3
Fig. 3

Brewster angle reflection of the beam in Fig. 2.

Fig. 4
Fig. 4

Definition of coordinate system. Ea (x,y) is the field distribution over the transversal aperture, and P is the observed field point.

Equations (70)

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p 2 + q 2 + m 2 = 1.
m 2 = 1 - p 2 - q 2 0.
U t ( p t , q t z ) = T ˜ U ( p , q ; z )
U r ( p r , q r ; z ) = R ˜ U ( p , q ; z )
e ^ k ^ × N ^ 1 - ( k ^ · N ^ ) 2 = k ^ × N ^ sin ϑ ,
e ^ k ^ × e ^ = - k ^ × ( k ^ × N ^ ) 1 - ( k ^ · N ^ ) 2 = k ^ × ( k ^ × N ^ ) sin ϑ = k ^ cos ϑ - N ^ sin ϑ .
τ ^ N ^ × e = [ N ^ × ( k ^ × N ^ ) 1 - ( k ^ · N ^ ) 2 = N ^ × ( k ^ × N ^ ) sin ϑ = 1 sin ϑ ( k ^ + N ^ cos ϑ ) .
k ^ t = - cos ϑ t N ^ + sin ϑ t τ ^ = - cos ϑ t N ^ + sin ϑ t sin ϑ ( k ^ + N ^ cos ϑ ) = n 1 n 2 k ^ + N ^ ( n 1 n 2 cos ϑ - cos ϑ t ) ,
k ^ r = cos ϑ N ^ + sin ϑ τ ^ = cos ϑ N ^ + k ^ + cos ϑ N ^ = k ^ + 2 cos ϑ N ^ ,
R ˜ = R R ˜ ;             T ˜ = T T ˜ ,
T ˜ = t e ^ t e ^ + t e ^ t e ^ ,
R ˜ = r e ^ r e ^ + r e ^ r e ^ ,
r = n 2 cos ϑ - n 1 1 - n 1 2 n 2 2 sin 2 ϑ n 2 cos ϑ + n 1 1 - n 1 2 n 2 2 sin 2 ϑ .
ϑ = ϑ B + Δ ϑ
r const sin Δ ϑ const [ ( N x - p B m B N z ) Δ p + ( N y - q B m B ) Δ q ] .
k ^ t = k ^
T = 1.
e ^ p = k ^ × ( P ^ × k ^ ) 1 - ( k ^ · P ^ ) 2 = P ^ - k ^ ( k ^ · P ^ ) 1 - ( k ^ · P ^ ) 2 ,
T ˜ = e ^ p e ^ p ,
J ( x , y ) = x ^ J x ( x , y ) = x ^ δ ( x - x 0 ) δ ( y - y 0 ) ,
H a ( x , y ) = y ^ δ ( x - x 0 ) δ ( y - y 0 )
G z ( p , q ) = 0 ;             G y ( p , q ) = 1 λ 2 exp [ j k ( p x 0 + q y 0 ) ] .
E ( x , y , z ) = const - [ ( 1 - p 2 ) x ^ - p q y ^ - p m z ^ ] exp ( - j k m z ) m · exp { - j k [ p ( x - x 0 ) + q ( y - y 0 ) ] } d p d q ,
E ( x , y , z ) = exp ( - j k r ) r * { - [ ( 1 - p 2 ) x ^ - p q y ^ - p m z ^ ] · exp { - j k [ p ( x - x 0 ) + q ( y - y 0 ) ] } d p d q } ,
r 2 = ( x - x 0 ) 2 + ( y - y 0 ) 2 + z 2 .
U ( p , q ; z ) = [ x ^ ( 1 - p 2 ) - y ^ p q - z ^ p m ] · exp ( - j k m z ) m exp [ j k ( p x 0 + q y 0 ) ] .
P ^ = y ^ ,
e ^ p = y ^ - k ^ q 1 - q 2 = y ^ ( 1 - q 2 ) - x ^ p q - z ^ m q 1 - q 2 .
U t ( p , q , z 0 ) = - ( y ^ - x ^ p q + z ^ m q 1 - q 2 ) · p q exp ( - j k m z 0 ) m exp [ j k ( p x 0 + q y 0 ) ] .
E t ( x , y , z 0 ) y ^ E y t ( x , y , z 0 ) = - y ^ x y FT { exp ( - j k m z 0 ) m exp [ j k ( p x 0 + q y 0 ) ] } ,
E y t ( x , y , z 0 ) = - x y [ exp ( - j k r 0 ) r 0 ] ,
E y t ( x , y , z 0 ) = ( x - x 0 ) ( y - y 0 ) [ exp ( - j k r 0 ) r 0 ( k 2 r 0 2 - 3 j k r 0 3 + 3 r 0 4 ) ] .
U r ( p r , q r ; z 0 ) = R ( r e ^ r e ^ + r e ^ r e ^ ) · U ( p , q ; z 0 ) .
U r ( p r , q r , z 0 ) = R [ const ( p B m B Δ p + q B m B Δ q ) × e ^ r e ^ + r e ^ r e ^ ] · U ( p , q ; z 0 ) ,
U r ( p , q , z 0 ) = const ( p B m B Δ p + q B m B Δ q ) exp ( - j k m r z 0 ) m r · exp [ j k ( p x 0 + q y 0 ) ] ,
E ( x , y , z 0 ) = exp ( - j k r 0 ) r 0 * ( const - a ( p , q ) ( p B m B Δ p + q B m B Δ q ) · exp { - j k [ p ( x - x 0 ) + q ( y - y 0 ) ] } d p d q ) ,
const - a ( Δ p , Δ q ) ( p B m B Δ p + q B m B Δ q ) exp { - j k [ Δ p ( x - x 0 ) + Δ q ( y - y 0 ) ] } d Δ p d Δ q ,
const ( p B m B x + q B m B y ) J 1 [ k Δ max ( x - x 0 ) 2 + ( y - y 0 ) 2 ] [ k Δ max ( x - x 0 ) 2 + ( y - y 0 ) 2 ] ,
E ( x , y , z 0 ) = const ( p B m B x + q B m B y ) { exp ( - j k r 0 ) r 0 * J 1 [ k Δ max ( x - x 0 ) 2 + ( y - y 0 ) 2 ] [ k Δ max ( x - x 0 ) 2 + ( y - y 0 ) 2 ] } .
E ( x , y , z 0 ) const [ p B m B ( x - x 0 ) + q B m B ( y - y 0 ) ] × exp ( - j k r 0 ) r 0 2 ( - j k - 1 r 0 ) ,
p B ( x - x 0 ) + q B ( y - y 0 ) = 0.
r = n 2 cos ϑ - n 1 cos ϑ t n 2 cos ϑ + n 1 cos ϑ t = n 2 ( k ^ · N ^ ) - n 1 1 - n 1 2 n 2 2 ( k ^ × N ^ ) 2 n 2 ( k ^ · N ^ ) + n 1 1 - n 1 2 n 2 2 ( k ^ × N ^ ) 2 ;
r = n 1 cos ϑ - n 2 cos ϑ t n 1 cos ϑ + n 2 cos ϑ t = n 1 ( k ^ · N ^ ) - n 2 1 - n 1 2 n 2 2 ( k ^ × N ^ ) 2 n 1 ( k ^ · N ^ ) + n 2 1 - n 1 2 n 2 2 ( k ^ × N ^ ) 2 ;
t = 2 n 1 cos ϑ n 2 cos ϑ + n 1 cos ϑ t = 2 n 1 ( k ^ · N ^ ) n 2 ( k ^ · N ^ ) + n 1 1 - n 1 2 n 2 2 ( k ^ × N ^ ) 2 ;
t = 2 n 1 cos ϑ n 1 cos ϑ + n 2 cos ϑ t = 2 n 1 ( k ^ · N ^ ) n 1 ( k ^ · N ^ ) + n 2 1 - n 1 2 n 2 2 ( k ^ × N ^ ) 2 .
cos ϑ = cos ϑ B cos Δ ϑ - sin ϑ B sin Δ ϑ cos ϑ B - sin ϑ B sin Δ ϑ ;
sin ϑ = sin ϑ B cos Δ ϑ + cos ϑ B sin Δ ϑ sin ϑ B + cos ϑ B sin Δ ϑ ;
sin 2 ϑ = sin 2 ϑ B cos 2 Δ ϑ + 2 sin ϑ B cos ϑ B sin Δ ϑ cos Δ ϑ + cos 2 ϑ B sin 2 Δ ϑ sin 2 ϑ B + sin 2 ϑ B sin Δ ϑ .
1 - n 1 2 n 2 2 sin 2 ϑ B - n 1 2 n 2 2 sin 2 ϑ B sin Δ ϑ = cos 2 ϑ B t - n 1 2 n 2 2 sin 2 ϑ B sin Δ ϑ = cos ϑ B t 1 - n 1 2 n 2 2 sin 2 ϑ B sin Δ ϑ cos 2 ϑ B t cos ϑ B t ( 1 - n 1 2 n 2 2 sin 2 ϑ B sin Δ ϑ 2 cos 2 ϑ B t ) .
r n 2 cos ϑ B - n 1 cos ϑ B t n 2 cos ϑ B - sin ϑ B sin Δ ϑ + n 1 cos ϑ B t ( 1 - n 1 2 n 2 2 sin 2 ϑ B sin Δ ϑ 2 cos 2 ϑ B t ) - n 2 sin ϑ B sin Δ ϑ + n 1 3 n 2 2 sin 2 ϑ B 2 cos ϑ B t · sin Δ ϑ n 2 cos ϑ B - sin ϑ B sin Δ + n 1 cos ϑ B t ( 1 - n 1 2 n 2 2 sin 2 ϑ B sin Δ ϑ 2 cos 2 ϑ B t ) .
r ( n 2 sin ϑ B + n 1 3 n 2 2 sin 2 ϑ B 2 cos ϑ B t ) n 2 sin ϑ B + n 1 cos ϑ B t sin Δ ϑ = const sin Δ ϑ .
p = p B + Δ p ;
q = q B + Δ q ;
m = m B + Δ m ;
p B 2 + q B 2 + m B 2 = 1 ,
( p B + Δ p ) 2 + ( q B + Δ q ) 2 + ( m B + Δ m ) 2 = 1 ,
2 p B Δ p + 2 q B Δ q + 2 m B Δ m 0.
Δ m - p B m B Δ p - q B m B Δ q .
cos ϑ = k ^ · N ^ ,
sin Δ ϑ 1 sin ϑ B ( N x Δ p + N y Δ q + N z Δ m ) .
sin Δ ϑ = 1 sin ϑ B [ ( N x - p B m B N z ) Δ p + ( N y - q B m B N z ) Δ q ] .
E ( x , y , z ) = η - { x ^ [ p q m G x ( p , q ) + 1 - p 2 m G y ( p , q ) ] - y ^ [ 1 - q 2 m G x ( p , q ) + p q m G y ( p , q ) ] + z ^ [ q G x ( p , q ) - p G y ( p , q ) ] } · exp [ - j k ( p x + q y + m z ) ] d p d q ,
G x ( p , q ) = 1 λ 2 - H a x ( x , y ) exp [ j k ( p x + q y ) ] d x d y ,
G y ( p , q ) = 1 λ 2 - H a y ( x , y ) exp [ j k ( p x + q y ) ] d x d y .
H a ( x , y ) = H a x ( x , y ) x ^ + H a y ( x , y ) y ^ .
N ^ × H = J ,
J ( x , y ) = J x ( x , y ) x ^ + J y ( x , y ) y ^
H a ( x , y ) = - J y ( x , y ) x ^ + J x ( x , y ) y ^
U ( p , q ; z ) = - E ( x , y , z ) exp [ j k ( p x + q y ) ] d x d y = k u ( p , q ) exp ( - j k m z ) m ,
u ( p , q ) = x ^ [ p q G x ( p , q ) + ( 1 - p 2 ) G y ( p , q ) ] - - y ^ [ ( 1 - q 2 ) G x ( p , q ) + p q G y ( p , q ) ] + z ^ [ q m G x ( p , q ) - p m G y ( p , q ) ]

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