Abstract

The loci of polarization states for which either the ellipticity alone or the azimuth alone remains invariant upon passing through an optical system are introduced. The cartesian equations of these two loci are derived in the complex plane in which the polarization states are represented. The equations are quartic and are conveniently expressed in terms of the elements of the Jones. matrix of the optical system. As an exple the loci are determined for a system composed of a π/4 rotator followed by a quarter-wave retarder.

© 1973 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Clark Jones, J. Opt. Soc. Am. 32, 486 (1942).
    [CrossRef]
  2. H. de Lang, Phillips Res. Rep. 8, 1 (1967).
  3. R. M. A. Azzam, N. M. Bashara, J. Opt. Soc. Am. 62, 222 (1972).
    [CrossRef]
  4. R. Clark Jones, J. Opt. Soc. Am. 37, 110 (1947).
    [CrossRef]
  5. The elements of the Jones matrix in the example of a π/4 rotator and QWP of Eq. (16) satisfy Eq. (21) after multiplication by exp(iπ/4).
  6. The special case when Eq. (14) is factored in the form QaQb = 0, where Qa and Qb are two quadratics, gives rise to a locus composed of two conic sections.

1972 (1)

1967 (1)

H. de Lang, Phillips Res. Rep. 8, 1 (1967).

1947 (1)

1942 (1)

J. Opt. Soc. Am. (3)

Phillips Res. Rep. (1)

H. de Lang, Phillips Res. Rep. 8, 1 (1967).

Other (2)

The elements of the Jones matrix in the example of a π/4 rotator and QWP of Eq. (16) satisfy Eq. (21) after multiplication by exp(iπ/4).

The special case when Eq. (14) is factored in the form QaQb = 0, where Qa and Qb are two quadratics, gives rise to a locus composed of two conic sections.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Introduces the locus ɛi of the incident invariant-ellipticity states (IES).

Fig. 2
Fig. 2

The loci ɛi and a i of the incident IES and IAS, respectively, intersect at two points E1 and E2, representing the two eigenpolarizations of the optical system. The optical system maps ɛi and a i into ɛ0 and a 0 which represent the loci of the IES and IAS at the output of the system, respectively. Obviously E1 and E2 remain invariant and thus belong to all four loci ɛi, ɛ0, a i, and a 0.

Fig. 3
Fig. 3

The locus ɛi of the incident IES for a system composed of a π/4 rotator followed by a QW retarder. Also shown are the limiting circles γ1, γ2; Γ1 and Γ2 (see Fig. 1). The points r and l coincide with the origin and the point at infinity of the complex plane, respectively. ɛi is given by Eq. (19b).

Fig. 4
Fig. 4

The locus a i of the incident IAS for a system of a π/4 rotator followed by a QW retarder has two separate branches a i ( 1 ) and a i ( 2 ). a i is defined by Eq. (20).

Fig. 5
Fig. 5

The intersection of the loci ɛi, ɛ0; a i and a 0 for the system of Figs. 3 and 4. Note that only E1 (0.366, 0.366) and E2 (−1.366, −1.366) are common to all four loci and hence represent the true eigenpolarizations of the optical system. S1 and S2 are common to ɛi and a i, but they are mapped by the system into S1′ and S2′, respectively. They represent incident vibrations whose ellipticities are preserved but whose major axes are rotated by π/2 after propagation through the system.

Fig. 6
Fig. 6

Shows the set of four vectors A, B, C, and D [corresponding to the four complex coefficients of the polarization transfer function in Eq. (5)] in the case of a system exhibiting pure phase anisotropy and whose Jones matrix is unitary. These vectors determine the loci ɛi and a i of the incident IES and IAS according to Eqs. (10)(13).

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

x 2 + ( y - csc 2 ) 2 = cot 2 2 ,
sin 2 = 2 y / ( 1 + x 2 + y 2 ) , = [ 2 Im ( χ ¯ ) ] / ( 1 + χ ¯ 2 ) ,
( x + cot 2 θ ) 2 + y 2 = csc 2 2 θ ,
tan 2 θ = 2 x / ( 1 - x 2 - y 2 ) , = [ 2 Re ( χ ¯ ) ] / ( 1 - χ ¯ 2 ) ,
ξ ¯ = ( A χ ¯ + B ) / ( C χ ¯ + D ) ,
Im ( χ ¯ ) / ( 1 + χ ¯ 2 ) = Im ( ξ ¯ ) / ( 1 + ξ ¯ 2 ) ,
Re ( χ ¯ ) / ( 1 - χ ¯ 2 ) = Re ( ξ ¯ ) / ( 1 - ξ ¯ 2 ) ,
χ ¯ = x + j y ,
A = a 1 + j a 2 , B = b 1 + j b 2 , C = c 1 + j c 2 , D = d 1 + j d 2 ,
( x 2 + y 2 + 1 ) Q 1 - y Q 2 = 0 ,
( x 2 + y 2 - 1 ) Q 3 - x Q 4 = 0.
Q 1 = ( C × A ) ( x 2 + y 2 ) - ( B × C - D × A ) x - ( B · C - D · A ) y + D × B ,
Q 2 = ( A · A + C · C ) ( x 2 + y 2 ) + 2 ( A · B + C · D ) x + 2 ( A × B + C × D ) y + ( B · B + D · D ) ; Q 3 = ( C · A ) ( x 2 + y 2 ) + ( B · C + D · A ) x - ( B × C + D × A ) y + D · B , Q 4 = ( A · A - C · C ) ( x 2 + y 2 ) + 2 ( A · B - C · D ) x + 2 ( A × B - C × D ) y + ( B · B - D · D ) ,
λ 1 x 4 + λ 2 y 4 + + λ 14 y + λ 15 = 0.
χ ¯ = ( D ξ ¯ - B ) / ( - C ξ ¯ + A ) ,
A = - B = - j , C = D = 1 ,
Cut 91             Q 1 = 1 - x 2 - y 2 , Q 2 = 2 ( 1 + x 2 + y 2 ) , Q 3 = 2 y , Q 4 = - 4 x ,
( x 2 + y 2 + 1 ) ( x 2 + y 2 + 2 y - 1 ) = 0 ,
x 2 + y 2 + 1 = 0
x 2 + y 2 + 2 y - 1 = 0 ,
- 0.414 tan 0.414.
y 3 + y x 2 + 2 x 2 - y = 0 ,
x 2 = y ( 1 - y ) ( 1 + y ) / ( 2 + y ) .
χ ¯ ( E 1 ) = ( 0.366 , 0.366 ) , χ ¯ ( S 1 ) = ( - 0.366 , 0.366 ) , χ ¯ ( S 2 ) = ( 1.366 , - 1.366 ) , χ ¯ ( E 2 ) = ( - 1.366 , - 1.366 ) ,
D = A * , C = - B * .
A · B + C · D = 0 , A × B + C × D = 0.
Q 2 = ( A 2 + B 2 ) ( x 2 + y 2 + 1 ) .
( x 2 + y 2 + 1 ) [ Q 1 - ( A 2 + B 2 ) y ] = 0 ,
x 2 + y 2 + 1 = 0
Q 1 - ( A 2 + B 2 ) y = 0.

Metrics