Abstract

The concept of retardance is critically analyzed for ray paths through optical systems described by a three-by-three polarization ray-tracing matrix. Algorithms are presented to separate the effects of retardance from geometric transformations. The geometric transformation described by a “parallel transport matrix” characterizes nonpolarizing propagation through an optical system, and also provides a proper relationship between sets of local coordinates along the ray path. The proper retardance is calculated by removing this geometric transformation from the three-by-three polarization ray-tracing matrix. Two rays with different ray paths through an optical system can have the same polarization ray-tracing matrix but different retardances. The retardance and diattenuation of an aluminum-coated three fold-mirror system are analyzed as an example.

© 2011 Optical Society of America

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References

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2011 (1)

1998 (1)

1995 (1)

1994 (1)

1991 (1)

1988 (2)

R. A. Chipman, “Polarization analysis of optical systems,” Proc. SPIE 891, 10–31 (1988).

R. Bhandari and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
[CrossRef] [PubMed]

1987 (1)

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407(1987).
[CrossRef]

1977 (1)

J. M. Leinaas and J. Myrheim, “On the theory of identical particles,” Il Nuovo Cimento B 37, 1–23 (1977).
[CrossRef]

1956 (2)

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. A 44, 247–262(1956).

R. C. Jones, “A new calculus for the treatment of optical systems. VIII. Electromagnetic theory,” J. Opt. Soc. Am. 46, 126–131 (1956).
[CrossRef]

1948 (1)

1947 (2)

1942 (1)

1941 (3)

Berry, M. V.

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407(1987).
[CrossRef]

Bhandari, R.

R. Bhandari and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
[CrossRef] [PubMed]

Bone, D.

Booker, G. R.

Chipman, R.

Chipman, R. A.

R. A. Chipman, “Polarization analysis of optical systems,” Proc. SPIE 891, 10–31 (1988).

Collaro, A.

Crabtree, K.

Ferreiro, M. S.

Fowles, G. R.

G. R. Fowles, Introduction to Modern Optics (Dover, 1975), p. 53.

Franceschetti, G.

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 2002), pp. 352–357.

Jones, R. C.

Laczik, Z.

Leinaas, J. M.

J. M. Leinaas and J. Myrheim, “On the theory of identical particles,” Il Nuovo Cimento B 37, 1–23 (1977).
[CrossRef]

Lu, S.

Macleod, A.

A. Macleod, “Phase matters,” oemagazine June/July 2005, 29–31 (2005).

Myrheim, J.

J. M. Leinaas and J. Myrheim, “On the theory of identical particles,” Il Nuovo Cimento B 37, 1–23 (1977).
[CrossRef]

Palmieri, F.

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. A 44, 247–262(1956).

Samuel, J.

R. Bhandari and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
[CrossRef] [PubMed]

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard Univ. Press, 1962), pp. 95–98.

W. A. Shurcliff, Polarized Light (Harvard Univ. Press, 1962), pp. 87–88.

Török, P.

Varga, P.

Yun, G.

Appl. Opt. (2)

Il Nuovo Cimento B (1)

J. M. Leinaas and J. Myrheim, “On the theory of identical particles,” Il Nuovo Cimento B 37, 1–23 (1977).
[CrossRef]

J. Mod. Opt. (1)

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34, 1401–1407(1987).
[CrossRef]

J. Opt. Soc. Am. (8)

J. Opt. Soc. Am. A (3)

Phys. Rev. Lett. (1)

R. Bhandari and J. Samuel, “Observation of topological phase by use of a laser interferometer,” Phys. Rev. Lett. 60, 1211–1213 (1988).
[CrossRef] [PubMed]

Proc. Indian Acad. Sci. A (1)

S. Pancharatnam, “Generalized theory of interference, and its applications,” Proc. Indian Acad. Sci. A 44, 247–262(1956).

Proc. SPIE (1)

R. A. Chipman, “Polarization analysis of optical systems,” Proc. SPIE 891, 10–31 (1988).

Other (5)

W. A. Shurcliff, Polarized Light (Harvard Univ. Press, 1962), pp. 87–88.

E. Hecht, Optics (Addison-Wesley, 2002), pp. 352–357.

W. A. Shurcliff, Polarized Light (Harvard Univ. Press, 1962), pp. 95–98.

G. R. Fowles, Introduction to Modern Optics (Dover, 1975), p. 53.

A. Macleod, “Phase matters,” oemagazine June/July 2005, 29–31 (2005).

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Figures (7)

Fig. 1
Fig. 1

Polarimeter measuring a sample retarder with the PSA (a) aligned with the polarization state generator (PSG) and (b) rotated to an arbitrary orientation. By rotating the PSA, the exiting local coordinates for the Jones matrix are also rotated. The measured retardance of the sample now includes a “circular retardance” component of 2 θ as well as the proper retardance.

Fig. 2
Fig. 2

(a) Evolution of a local coordinate pair { x ^ A , y ^ A } (solid green arrows) through a system of three fold- mirrors. The exiting local coordinates (dashed red arrows) undergo a 90 ° rotation from the initial local coordinates (solid green). (b) Three fold-mirror system. When a collimated beam enters the system along the z axis, the beam exits along the z axis.

Fig. 3
Fig. 3

Incident, reflected, and transmitted local coordinates calculated from parallel transport matrices. { x ^ L , 0 , y ^ L , 0 , k ^ 0 } are the right-handed incident local coordinates, { x ^ L , r , 1 , y ^ L , r , 1 , k ^ r , 1 } are the left-handed reflected local coordinates, and { x ^ L , t , 1 , y ^ L , t , 1 , k ^ t , 1 } are the right-handed transmitted local coordinates.

Fig. 4
Fig. 4

Two-mirror system. The red solid arrows show the s vector at the first mirror and its geometric transformation along each ray segment using Q. The blue dashed arrows show the p vector in object space and its geometric transformations.

Fig. 5
Fig. 5

Ideal reflection at normal incidence with the incident and exiting right-handed local coordinates, { x ^ L , 0 , y ^ L , 0 } and { x ^ L , 1 , y ^ L , 1 } . In this particular choice of local coordinates, the x ^ L vector was flipped after the reflection.

Fig. 6
Fig. 6

Aluminum-coated three fold-mirror system.

Fig. 7
Fig. 7

Local coordinate transformation using Q q for the three fold-mirror system. Incident x ^ L , 0 state (solid red arrows) exits as x ^ polarized and incident y ^ L , 0 state (dashed blue arrow) exits as y ^ polarized after three reflections due to the geometric transformation.

Tables (1)

Tables Icon

Table 1 Propagation Vectors P q and Q q for a Ray Propagating Through the Aluminum-Coated Three Fold-Mirror System

Equations (35)

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P Total = q = N , 1 1 P q = P N P N 1 P q P 2 P 1 .
Q Total = q = N , 1 1 Q q = Q N Q N 1 Q q Q 2 Q 1 .
M Total = Q Total 1 P Total
M Total = M Total , R M Total , D = M Total , D M Total , R ,
v 1 , v 2 , k ^ 0 ,
λ 1 , λ 2 , λ 3 .
δ = arg ( λ 2 ) arg ( λ 1 ) ,
J ( θ ) = R ( θ ) I = ( cos θ sin θ sin θ cos θ ) ( 1 0 0 1 ) = ( cos θ sin θ sin θ cos θ ) ,
λ 1 = exp ( i θ ) , λ 2 = exp ( i θ ) ,
w 1 = ( 1 i ) , w 2 = ( 1 i ) .
δ = arg ( λ 1 ) arg ( λ 2 ) = 2 θ ,
x ^ L , 0 = k ^ 0 × k ^ 1 | k ^ 0 × k ^ 1 | , y ^ L , 0 = k ^ 0 × x ^ L , 0 .
Q Total · x ^ L , 0 = x ^ L , N Q Total 1 · x ^ L , N = x ^ L , 0 Q Total · y ^ L , 0 = y ^ L , N Q Total 1 · y ^ L , N = y ^ L , 0 Q Total · k ^ 0 = k ^ N Q Total 1 · k ^ N = k ^ 0 ,
P = ( 1 0 0 0 1 0 0 0 1 ) .
( 1 0 0 0 1 0 0 0 1 ) ( 1 i 0 ) = e i π ( 1 i 0 ) ,
( 1 0 0 0 1 0 0 0 1 ) ( cos θ sin θ 0 ) = e i π ( cos θ sin θ 0 ) ,
J f = ( 1 0 0 1 ) .
Q = ( 1 0 0 0 1 0 0 0 1 ) .
P Total = ( 0 0.549 + 0.705 i 0 0.365 + 0.788 i 0 0 0 0 1 ) , Q Total = ( 0 1 0 1 0 0 0 0 1 ) .
{ x ^ L , 0 , y ^ L , 0 , k ^ 0 } = { y ^ , x ^ , z ^ } ,
{ x ^ L , 3 , y ^ L , 3 , k ^ 3 } = { x ^ , y ^ , z ^ } ,
M Total = Q Total 1 · P Total = ( 0.365 + 0.788 i 0 0 0 0.549 + 0.705 i 0 0 0 1 ) .
λ 1 = 0.868 e i 2.005 , λ 2 = 0.8938 e i 2.232 , λ 3 = 1 ,
v 1 = { 1 , 0 , 0 } , v 2 = { 0 , 1 , 0 } , v 3 = k ^ 0 = { 0 , 0 , 1 } .
δ = arg ( λ 2 ) arg ( λ 1 ) = 0.227 = 13.0 ° ,
λ 1 = 0.945 e i 0.455 , λ 2 = 0.972 e i 2.914 δ = arg ( λ 2 ) arg ( λ 1 ) = 3.369 = 193.0 ° ,
Q q = 1 A 2 ( A 2 B + ( 1 B ) D 2 F [ ( 1 B ) D G ] D [ ( 1 B ) C + H ] F [ ( 1 B ) D + L ] A 2 B + ( 1 B ) F 2 C [ ( 1 B ) F H ] D [ ( 1 B ) C L ] C [ ( 1 B ) F + G ] A 2 B + ( 1 B ) C 2 ) ,
A = Norm [ k ^ q 1 × k ^ q ] , B = k ^ q 1 · k ^ q , { C , D , F } = k ^ q 1 × k ^ q , G = A 1 B 2 A 2 F 2 C 2 + F 2 , H = A 1 B 2 A 2 C 2 C 2 + D 2 , L = A 1 B 2 A 2 D 2 D 2 + F 2 .
Q q = ( 1 2 ( k x , q 1 k x , q ) 2 A B A C A B A 1 2 ( k y , q 1 k y , q ) 2 A D A C A D A 1 2 ( k z , q 1 k z , q ) 2 A ) ,
A = Norm [ k ^ q 1 k ^ q ] , B = 2 ( k x , q 1 k x , q ) ( k y , q 1 k y , q ) , C = 2 ( k x , q 1 k x , q ) ( k z , q 1 k z , q ) , D = 2 ( k y , q 1 k y , q ) ( k z , q 1 k z , q ) .
S R = U · M Total · U = ( J 0 0 0 0 1 ) .
1 H ( k x 2 cos θ + k y 2 k x ( cos θ k y ) H k x sin θ k x k y ( cos θ 1 ) k x 2 + k y 2 cos θ H k y sin θ H k x sin θ H k y sin θ H cos θ ) ,
δ = 2 cos 1 ( | tr J + det J | det J | tr J | 2 tr ( J J ) + 2 | det J | ) .
v 1 = U · w 1 , v 2 = U · w 2 , k ^ 0 ,
v 1 = Q · v 1 = Q · U · w 1 , v 2 = Q · v 2 = Q · U · w 2 , k ^ N .

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