Abstract

Thin birefringent prisms placed near an image plane introduce sinusoidal fringes onto a 2D polarized scene making possible a snapshot imaging polarimeter, which encodes polarization information into the modulation of the fringes. This approach was introduced by Oka and Kaneko [Opt. Express 11, 1510 (2003)], who analyzed the instrument through the Mueller calculus. We show that the plane-wave assumption adopted in the Mueller theory can introduce unnecessary error in a polarimeter design. To directly take prism effects such as beam splitting and deviating into accounts we introduce a geometric imaging model, which allows for a versatile simulation of the birefringent prisms and provides a means for optimization. A calcite visible system is investigated as an example, which essentially shows how each design parameter affects the overall image quality and how to modify the polarimeter design to optimize overall performance. The approach is applicable to any prismatic imaging polarimeter with different prism materials and different working wavelengths.

© 2006 Optical Society of America

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References

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  1. K. Oka and T. Kaneko, "Compact complete imaging polarimeter using birefringent wedge prisms," Opt. Express 11, 1510-1518 (2003).
    [Crossref] [PubMed]
  2. M. C. Simon, "Ray tracing formulas for monoaxial optical components," Appl. Opt. 22, 354-360 (1983).
    [Crossref] [PubMed]
  3. M. C. Simon, "Wollaston prism with large split angle," Appl. Opt. 25, 369-376 (1986).
    [Crossref] [PubMed]
  4. M. Avendano-Alejo and O. Stavroudis, "Huygens's principle and rays in uniaxial anisotropic media II. Crystal axis with arbitrary orientation," J. Opt. Soc. Am. A. 19, 1674-1679 (2002).
    [Crossref]
  5. H. X. Ren, L. R. Liu, D. A. Liu, and Z. Song, "Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design," J. Mod. Opt. 52, 529-539 (2005).
    [Crossref]
  6. Q. T. Liang, "Simple ray tracing formulas for uniaxial optical crystals," Appl. Opt. 29, 1008-1010 (1990).
    [Crossref] [PubMed]
  7. D. Goldstein, Polarized Light, 2nd ed. (Dekker, 2003), Chap. 4, p. 60.
    [Crossref]

2005 (1)

H. X. Ren, L. R. Liu, D. A. Liu, and Z. Song, "Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design," J. Mod. Opt. 52, 529-539 (2005).
[Crossref]

2003 (1)

2002 (1)

M. Avendano-Alejo and O. Stavroudis, "Huygens's principle and rays in uniaxial anisotropic media II. Crystal axis with arbitrary orientation," J. Opt. Soc. Am. A. 19, 1674-1679 (2002).
[Crossref]

1990 (1)

1986 (1)

1983 (1)

Avendano-Alejo, M.

M. Avendano-Alejo and O. Stavroudis, "Huygens's principle and rays in uniaxial anisotropic media II. Crystal axis with arbitrary orientation," J. Opt. Soc. Am. A. 19, 1674-1679 (2002).
[Crossref]

Goldstein, D.

D. Goldstein, Polarized Light, 2nd ed. (Dekker, 2003), Chap. 4, p. 60.
[Crossref]

Kaneko, T.

Liang, Q. T.

Liu, D. A.

H. X. Ren, L. R. Liu, D. A. Liu, and Z. Song, "Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design," J. Mod. Opt. 52, 529-539 (2005).
[Crossref]

Liu, L. R.

H. X. Ren, L. R. Liu, D. A. Liu, and Z. Song, "Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design," J. Mod. Opt. 52, 529-539 (2005).
[Crossref]

Oka, K.

Ren, H. X.

H. X. Ren, L. R. Liu, D. A. Liu, and Z. Song, "Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design," J. Mod. Opt. 52, 529-539 (2005).
[Crossref]

Simon, M. C.

Song, Z.

H. X. Ren, L. R. Liu, D. A. Liu, and Z. Song, "Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design," J. Mod. Opt. 52, 529-539 (2005).
[Crossref]

Stavroudis, O.

M. Avendano-Alejo and O. Stavroudis, "Huygens's principle and rays in uniaxial anisotropic media II. Crystal axis with arbitrary orientation," J. Opt. Soc. Am. A. 19, 1674-1679 (2002).
[Crossref]

Appl. Opt. (3)

J. Mod. Opt. (1)

H. X. Ren, L. R. Liu, D. A. Liu, and Z. Song, "Double refraction and reflection of sequential crystal interfaces with arbitrary orientation of the optic axis and application to optimum design," J. Mod. Opt. 52, 529-539 (2005).
[Crossref]

J. Opt. Soc. Am. A. (1)

M. Avendano-Alejo and O. Stavroudis, "Huygens's principle and rays in uniaxial anisotropic media II. Crystal axis with arbitrary orientation," J. Opt. Soc. Am. A. 19, 1674-1679 (2002).
[Crossref]

Opt. Express (1)

Other (1)

D. Goldstein, Polarized Light, 2nd ed. (Dekker, 2003), Chap. 4, p. 60.
[Crossref]

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Figures (7)

Fig. 1
Fig. 1

(a) Schematic of the prismatic imaging polarimeter. (b) Layout of the wedged prisms PR 1 PR 4 , which are actually in tight contact. The double-arrow lines indicate the fast axes, which are at 0°, 90°, 45°, and - 45 ° with respect to the x axis. The solid lines represent the o light, and the dashed lines represent the e light. The circular shape sketches the image spot shape. The back surface of PR 4 is the image plane of interest.

Fig. 2
Fig. 2

BS and BD induced by a birefringent prism are shown exaggerated here for clarity. The splitting is a spatial translation of the e beam from the o beam. The deviation is an angular deflection between the incident and exit ray directions.

Fig. 3
Fig. 3

(Color online) (a) Spot diagrams (square grid) at z = 50.18 mm , z = 50.2491 mm , z = 50.33 mm , respectively. (b) Merit function of rms image spot radius versus z for Ray 1–Ray 4, respectively. The inset is the enlarged part at the bottom. For comparison, a similar curve has been plotted for the case of replacing the set of four calcite prisms with a glass plate of equal thickness and the index of refraction equal to the ordinary index of calcite.

Fig. 4
Fig. 4

(Color online) (a) Spot diagrams with the pupil diameter d p equal to 6, 20, and 30 mm , respectively, with diffraction effects ignored. (b) Rms image spot radius versus the pupil diameter d p . The system is at the optimum focus for the on-axis field.

Fig. 5
Fig. 5

(Color online) (a) Spot diagrams for different field points (at the optimum focus), i.e., (0, 0), (3°, 0), (7°, 0), (−7°, 0), (0, 7°), and (0, −7°) for z = 50.2491 mm and d p = 15 mm . (b) Merit function of spot radius versus z at different fields (by different line shapes), (0, 0), (7°, 0), (−7°, 0), (0, 7°), and (0, −7°), for Ray 1 and Ray 3.

Fig. 6
Fig. 6

(Color online) OPD mappings (normalized by λ) of the fringe components between all six ray configuration pairs.

Fig. 7
Fig. 7

(Color online) Spot diagrams of different apex angles of a calcite system both for the on-axis field and at optimized foci. (a) α = 1.5 ° , (b) α = 4.5 ° with rms spot size > 20 μm .

Tables (2)

Tables Icon

Table 1 Ray Configurations Ray 1–Ray 4 and the Complex Amplitudes of the E Fields

Tables Icon

Table 2 Fitted Slopes and Deviation Percentages of Different OPD Mappings in Fig. 5

Equations (13)

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I ( x , y ) = 1 2 S 0 ( x , y ) + 1 2 S 1 ( x , y ) cos ( 2 π U x ) + 1 4 S 2 ( x , y ) × { cos [ 2 π U ( x y ) ] cos [ 2 π U ( x + y ) ] } 1 4 S 3 ( x , y ) { sin [ 2 π U ( x y ) ] + sin [ 2 π U ( x + y ) ] } ,
n e ( θ ) = n o n e n o 2 sin 2 θ + n e 2 cos 2 θ ,
ξ x = cos α cos θ e + sin θ e sin α ( σ x sin θ e σ y cos θ e ) σ z 2 + ( σ x sin θ e σ y cos θ e ) 2 ,
ξ y = cos α sin θ e cos θ e sin α ( σ x sin θ e σ y cos θ e ) σ z 2 + ( σ x sin θ e σ y cos θ e ) 2 ,
ξ z = σ z sin α σ z 2 + ( σ x sin θ e σ y cos θ e ) 2 ,
I = | 1 2 E y ( t ) e i φ 1 1 2 E y ( t ) e i φ 2 + 1 2 E x ( t ) e i φ 3 + 1 2 E x ( t ) e i φ 4 | 2 = 1 4 { 2 [ | E y | 2 + | E x | 2 ] [ | E y | 2 e i ( φ 2 φ 1 ) + c.c . ] + [ | E x | 2 e i ( φ 4 φ 3 ) + c.c . ] + [ E x * E y e i ( φ 3 φ 1 ) + c.c . ] [ E x * E y e i ( φ 4 φ 2 ) + c.c . ] + [ E x * E y e i ( φ 4 φ 1 ) + c.c . ] [ E x * E y e i ( φ 3 φ 2 ) + c.c . ] } ,
| E x | 2 + | E y | 2 = S 0 ,   | E x | 2 = 1 2 ( S 0 + S 1 ) ,
| E y | 2 = 1 2 ( S 0 S 1 ) ,       E x * E y = 1 2 ( S 2 + i S 3 )
I = 1 2 S 0 1 4 ( S 0 S 1 ) cos ( φ 2 φ 1 ) + 1 4 ( S 0 + S 1 ) × cos ( φ 4 φ 3 ) + 1 4 Re { ( S 2 + i S 3 ) × [ e i ( φ 3 φ 1 ) e i ( φ 4 φ 2 ) + e i ( φ 4 φ 1 ) e i ( φ 3 φ 2 ) ] } ,
φ 2 ( x , y ) φ 1 ( x , y ) = φ 4 ( x , y ) φ 3 ( x , y ) = 2 π U x ,
φ 1 ( x , y ) φ 3 ( x , y ) = φ 2 ( x , y ) φ 4 ( x , y ) = 2 π U y ,
φ 4 ( x , y ) φ 1 ( x , y ) = 2 π U ( x y ) ,
φ 2 ( x , y ) φ 3 ( x , y ) = 2 π U ( x + y ) ,

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