Abstract

The field-of-view dependence of polarizing beam-splitter cubes has been studied to characterize their behavior in imaging systems such as optical computers and optical correlators and in other applications that involve noncollimated light. Significant polarization aberration is present in polarizing beamsplitter cubes for two reasons: (1) the s- and p-component orientations, which define the polarizing axes, at the beam-splitting interface vary with the direction of propagation, and (2) the performance of the coating is a function of the angle of incidence. We describe the polarization aberration of a polarizing beam-splitter cube in terms of its diattenuation (polarizing efficiency). We use an imaging polarimeter to measure six figures of merit for three polarizing beam-splitter cubes demonstrating typical polarization aberrations. Finally, we derive the Mueller matrix for a polarizing beam-splitter cube in terms of the s and p transmittance and reflectance and the phase retardances, the parameters generally calculated with thin-film analysis programs.

© 1994 Optical Society of America

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    [CrossRef] [PubMed]
  2. F. B. McCormick, M. E. Prise, “Optical circuitry for free-space interconnections,” Appl. Opt. 29, 2013–2018 (1990).
    [CrossRef] [PubMed]
  3. M. E. Prise, M. M. Downs, F. B. McCormick, S. J. Walker, N. Streibl, “Design of an optical digital computer,” in Optical Bistability IV, W. J. Firth et al., eds. (Les Editions de Physique, Paris, 1988), pp. C2–C15.
  4. M. E. Prise, R. E. LaMarche, N. C. Craft, M. M. Downs, S. J. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Optical systems using arrays of symmetric self-electrooptic effect devices,” in Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper MQ3, p. 38.
  5. M. E. Prise, N. Streibl, M. M. Down, “Optical considerations in the design of a digital computer,” Opt. Quantum Electron. 20, 49–77 (1988).
    [CrossRef]
  6. A. Dickinson, M. E. Prise, “Free-space optical interconnection scheme,” Appl. Opt. 29, 2001–2005 (1990).
    [CrossRef] [PubMed]
  7. J. L. Pezzaniti, R. A. Chipman, “Imaging polarimeters for optical system metrology,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray, R. A. Chipman, J. W. Morris, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1317, 280–294(1990).
  8. S. M. MacNeille, “Beam splitter,” U.S. patent2,403,731 (9July1946).
  9. L. Songer, “The design and fabrication of a thin-film polarizer,” Opt. Spectra 12, 45–50 (1978).
  10. D. Blanc, P. H. Lissberger, A. Roy, “The design, preparation and optical measurement of thin-film polarizers,” Thin Solid Films 57, 191–198 (1979).
    [CrossRef]
  11. R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).
  12. R. P. Netterfield, “Practical thin-film polarizing beam splitters,” Opt. Acta 24, 69–79 (1977).
    [CrossRef]
  13. J. Mouchart, J. Begel, E. Duda, “Modified MacNeille cube polarizer for a wide angular field,” Appl. Opt. 28, 2847–2853 (1989).
    [CrossRef] [PubMed]
  14. R. A. Chipman, “Precision polarimetry of polarization components,” in Polarization and Analysis and Measurement, D. Goldstein, R. A. Chipman, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1746, 49–60 (1992).
  15. R. M. A. Azzam, W. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (North-Holland, Amsterdam, 1977), Appendix A, pp.490–492.

1990 (2)

1989 (2)

1988 (1)

M. E. Prise, N. Streibl, M. M. Down, “Optical considerations in the design of a digital computer,” Opt. Quantum Electron. 20, 49–77 (1988).
[CrossRef]

1979 (1)

D. Blanc, P. H. Lissberger, A. Roy, “The design, preparation and optical measurement of thin-film polarizers,” Thin Solid Films 57, 191–198 (1979).
[CrossRef]

1978 (1)

L. Songer, “The design and fabrication of a thin-film polarizer,” Opt. Spectra 12, 45–50 (1978).

1977 (1)

R. P. Netterfield, “Practical thin-film polarizing beam splitters,” Opt. Acta 24, 69–79 (1977).
[CrossRef]

1947 (1)

Azzam, R. M. A.

R. M. A. Azzam, W. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (North-Holland, Amsterdam, 1977), Appendix A, pp.490–492.

Banning, M.

Bashara, W. M.

R. M. A. Azzam, W. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (North-Holland, Amsterdam, 1977), Appendix A, pp.490–492.

Begel, J.

Blanc, D.

D. Blanc, P. H. Lissberger, A. Roy, “The design, preparation and optical measurement of thin-film polarizers,” Thin Solid Films 57, 191–198 (1979).
[CrossRef]

Chipman, R. A.

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

J. L. Pezzaniti, R. A. Chipman, “Imaging polarimeters for optical system metrology,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray, R. A. Chipman, J. W. Morris, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1317, 280–294(1990).

R. A. Chipman, “Precision polarimetry of polarization components,” in Polarization and Analysis and Measurement, D. Goldstein, R. A. Chipman, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1746, 49–60 (1992).

Chirovsky, L. M. F.

M. E. Prise, R. E. LaMarche, N. C. Craft, M. M. Downs, S. J. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Optical systems using arrays of symmetric self-electrooptic effect devices,” in Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper MQ3, p. 38.

Craft, N. C.

M. E. Prise, R. E. LaMarche, N. C. Craft, M. M. Downs, S. J. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Optical systems using arrays of symmetric self-electrooptic effect devices,” in Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper MQ3, p. 38.

D’Asaro, L. A.

M. E. Prise, R. E. LaMarche, N. C. Craft, M. M. Downs, S. J. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Optical systems using arrays of symmetric self-electrooptic effect devices,” in Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper MQ3, p. 38.

Dickinson, A.

Down, M. M.

M. E. Prise, N. Streibl, M. M. Down, “Optical considerations in the design of a digital computer,” Opt. Quantum Electron. 20, 49–77 (1988).
[CrossRef]

Downs, M. M.

M. E. Prise, R. E. LaMarche, N. C. Craft, M. M. Downs, S. J. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Optical systems using arrays of symmetric self-electrooptic effect devices,” in Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper MQ3, p. 38.

M. E. Prise, M. M. Downs, F. B. McCormick, S. J. Walker, N. Streibl, “Design of an optical digital computer,” in Optical Bistability IV, W. J. Firth et al., eds. (Les Editions de Physique, Paris, 1988), pp. C2–C15.

Duda, E.

LaMarche, R. E.

M. E. Prise, R. E. LaMarche, N. C. Craft, M. M. Downs, S. J. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Optical systems using arrays of symmetric self-electrooptic effect devices,” in Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper MQ3, p. 38.

Lissberger, P. H.

D. Blanc, P. H. Lissberger, A. Roy, “The design, preparation and optical measurement of thin-film polarizers,” Thin Solid Films 57, 191–198 (1979).
[CrossRef]

MacNeille, S. M.

S. M. MacNeille, “Beam splitter,” U.S. patent2,403,731 (9July1946).

McCormick, F. B.

F. B. McCormick, M. E. Prise, “Optical circuitry for free-space interconnections,” Appl. Opt. 29, 2013–2018 (1990).
[CrossRef] [PubMed]

M. E. Prise, M. M. Downs, F. B. McCormick, S. J. Walker, N. Streibl, “Design of an optical digital computer,” in Optical Bistability IV, W. J. Firth et al., eds. (Les Editions de Physique, Paris, 1988), pp. C2–C15.

Mouchart, J.

Netterfield, R. P.

R. P. Netterfield, “Practical thin-film polarizing beam splitters,” Opt. Acta 24, 69–79 (1977).
[CrossRef]

Pezzaniti, J. L.

J. L. Pezzaniti, R. A. Chipman, “Imaging polarimeters for optical system metrology,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray, R. A. Chipman, J. W. Morris, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1317, 280–294(1990).

Prise, M. E.

A. Dickinson, M. E. Prise, “Free-space optical interconnection scheme,” Appl. Opt. 29, 2001–2005 (1990).
[CrossRef] [PubMed]

F. B. McCormick, M. E. Prise, “Optical circuitry for free-space interconnections,” Appl. Opt. 29, 2013–2018 (1990).
[CrossRef] [PubMed]

M. E. Prise, N. Streibl, M. M. Down, “Optical considerations in the design of a digital computer,” Opt. Quantum Electron. 20, 49–77 (1988).
[CrossRef]

M. E. Prise, M. M. Downs, F. B. McCormick, S. J. Walker, N. Streibl, “Design of an optical digital computer,” in Optical Bistability IV, W. J. Firth et al., eds. (Les Editions de Physique, Paris, 1988), pp. C2–C15.

M. E. Prise, R. E. LaMarche, N. C. Craft, M. M. Downs, S. J. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Optical systems using arrays of symmetric self-electrooptic effect devices,” in Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper MQ3, p. 38.

Roy, A.

D. Blanc, P. H. Lissberger, A. Roy, “The design, preparation and optical measurement of thin-film polarizers,” Thin Solid Films 57, 191–198 (1979).
[CrossRef]

Songer, L.

L. Songer, “The design and fabrication of a thin-film polarizer,” Opt. Spectra 12, 45–50 (1978).

Streibl, N.

M. E. Prise, N. Streibl, M. M. Down, “Optical considerations in the design of a digital computer,” Opt. Quantum Electron. 20, 49–77 (1988).
[CrossRef]

M. E. Prise, M. M. Downs, F. B. McCormick, S. J. Walker, N. Streibl, “Design of an optical digital computer,” in Optical Bistability IV, W. J. Firth et al., eds. (Les Editions de Physique, Paris, 1988), pp. C2–C15.

Walker, S. J.

M. E. Prise, M. M. Downs, F. B. McCormick, S. J. Walker, N. Streibl, “Design of an optical digital computer,” in Optical Bistability IV, W. J. Firth et al., eds. (Les Editions de Physique, Paris, 1988), pp. C2–C15.

M. E. Prise, R. E. LaMarche, N. C. Craft, M. M. Downs, S. J. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Optical systems using arrays of symmetric self-electrooptic effect devices,” in Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper MQ3, p. 38.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

Opt. Acta (1)

R. P. Netterfield, “Practical thin-film polarizing beam splitters,” Opt. Acta 24, 69–79 (1977).
[CrossRef]

Opt. Eng. (1)

R. A. Chipman, “Polarization analysis of optical systems,” Opt. Eng. 28, 90–99 (1989).

Opt. Quantum Electron. (1)

M. E. Prise, N. Streibl, M. M. Down, “Optical considerations in the design of a digital computer,” Opt. Quantum Electron. 20, 49–77 (1988).
[CrossRef]

Opt. Spectra (1)

L. Songer, “The design and fabrication of a thin-film polarizer,” Opt. Spectra 12, 45–50 (1978).

Thin Solid Films (1)

D. Blanc, P. H. Lissberger, A. Roy, “The design, preparation and optical measurement of thin-film polarizers,” Thin Solid Films 57, 191–198 (1979).
[CrossRef]

Other (6)

J. L. Pezzaniti, R. A. Chipman, “Imaging polarimeters for optical system metrology,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray, R. A. Chipman, J. W. Morris, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1317, 280–294(1990).

S. M. MacNeille, “Beam splitter,” U.S. patent2,403,731 (9July1946).

M. E. Prise, M. M. Downs, F. B. McCormick, S. J. Walker, N. Streibl, “Design of an optical digital computer,” in Optical Bistability IV, W. J. Firth et al., eds. (Les Editions de Physique, Paris, 1988), pp. C2–C15.

M. E. Prise, R. E. LaMarche, N. C. Craft, M. M. Downs, S. J. Walker, L. M. F. Chirovsky, L. A. D’Asaro, “Optical systems using arrays of symmetric self-electrooptic effect devices,” in Annual Meeting, Vol. 18 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), paper MQ3, p. 38.

R. A. Chipman, “Precision polarimetry of polarization components,” in Polarization and Analysis and Measurement, D. Goldstein, R. A. Chipman, eds. Proc. Soc. Photo-Opt. Instrum. Eng.1746, 49–60 (1992).

R. M. A. Azzam, W. M. Bashara, Ellipsometry and Polarized Light, 1st ed. (North-Holland, Amsterdam, 1977), Appendix A, pp.490–492.

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Figures (10)

Fig. 1
Fig. 1

Diagram of a polarizing beam-splitter cube.

Fig. 2
Fig. 2

Geometry of a polarizing beam-splitter cube. Side (a, b, c, d) is the entrance face. Face (a, d, f, g) is the beam-splitting interface. The transmitted beam exits the face (f, e, g) while the reflected light exits at face (b, c, f, g). An incident ray is identified in air by θx and θy.

Fig. 3
Fig. 3

Relative orientations of relevant planes and directions. The beam-splitting interface is oriented the same as in Fig. 2. x ˆ, P ˆ, and p ˆ, shown as dashed lines, are positioned on the opposite side of the beam-splitting interface, k ˆ is perpendicular to the plane of polarization, ŝ and p ˆ lie in the plane of polarization. Ŝ and P ˆ are projections of ŝ and p ˆ onto the x ˆ y ˆ plane.

Fig. 4
Fig. 4

Orientation of the Ŝ and P ˆ directions at the beam-splitting interface, plotted as a function of the angle of incidence on the cube face. The relative lengths of the Ŝ and P ˆ vectors illustrate the variation of the angle of incidence at the beam-splitting interface. The large circles indicate the zones where the angles of incidence are 5° and 10°. Incident rays for which θy > 0° involve a counterclockwise rotation of the Ŝ and P ˆ orientations of linear diattenuation, and rays for which θy < 90° involves a clockwise rotation, ϕ indicates the amount of this rotation.

Fig. 5
Fig. 5

Refracted electric-field vector Ex projected to the P ˆ and Ŝ directions.

Fig. 6
Fig. 6

General form of the linear diattenuation observed for polarizing beam splitters with the angle of incidence. The orientation varies systematically from the top of the pupil to the bottom and is independent of coating design. The magnitude of linear diattenuation, depicted by the lengths of the arrows, varies according to the individual polarizing beam-splitter coating design.

Fig. 7
Fig. 7

Configuration of the imaging polarimeter used to measure three polarizing beam-splitter cubes at 850 nm over a ±10° field of view. The source is an 850-nm laser diode. The dashed curve shows the path of the rim ray. PBS, polarizing beam splitter.

Fig. 8
Fig. 8

Figures of merit for cube 1, an 830-nm polarizing beam-splitter cube, displayed as contour plots. The measurement wavelength was 850 nm. The polarizing beam splitter would have (a) and (b) equal to 1 and (c)–(f) equal to zero. The angular dependence of the s and p transmittance and reflectance is found along the θx axis in Figs. 5(a)5(d). The orientation of diattenuation rotates for angles of incidence in which θy ≠ 0°. The rotation of diattenuation results in coupling between the incident polarization state and the orthogonal polarization state in both the transmitted (e) and reflected (f) beams.

Fig. 9
Fig. 9

Measured figures of merit of cube 2, an 850-nm polarizing beam-splitter cube measured at 850 nm. The p transmittance and reflectance have a strong angular dependence, shown in Figs. 6(a) and 6(c), along the θx axis. The s transmittance and reflectance have a very weak angular dependence, seen in Figs. 6(b) and 6(d). Rotation of the diattenuation vector causes coupling of up to 3% in transmission and 7% in reflection.

Fig. 10
Fig. 10

Angular performance of cube 3, a polarizing beam splitter specifically designed to operate at 850 nm over a 7° field of view. Both s and p transmittance and reflectance remain fairly constant over the design range, as can be seen along the θx axis in Figs. 7(a)7(d). Coupling resulting from the inherent geometry of the cube is present and can reach up to 8% in transmission and reflection.

Tables (2)

Tables Icon

Table 1 Definitions of Amplitude Transmission and Reflection Coefficientsa

Tables Icon

Table 2 Figures of Merit with the Corresponding Mueller Matrix Expressions and Amplitude Coefficients of a Polarizing Beam-Splitter Cubea

Equations (24)

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D = I max I min I max + I min , 0 D 1 .
D R = I max r I min r I max r + I min r .
D T = I max t I min t I max t + I min t .
D T = ( I max t I min t ) ( I max t + I min t ) p ˆ , D R = ( I max r I min r ) ( I max r + I min r ) s ˆ ,
θ x = n c θ x , θ y = n c θ y ,
k ˆ = [ sin θ x , sin θ y , ( 1 sin 2 θ x sin 2 θ y ) 1 / 2 ] .
k ˆ = [ sin θ x , sin θ y , ( 1 sin 2 θ x sin 2 θ y ) 1 / 2 ] .
s ˆ = n ˆ × k ˆ | n ˆ × k ˆ | , p ˆ = s ˆ × k ˆ ,
P ˆ = cos ϕ x ˆ + sin ϕ y ˆ , S ˆ = sin ϕ x ˆ + cos ϕ y ˆ ,
ϕ ( θ x , θ y ) = θ y ( 1 θ x ) = θ y n c ( 1 θ x n c ) ,
ϕ ( θ x , θ y ) = θ y n c ( 1 + θ x n c ) .
I ( θ x , θ y ) = I 0 cos 2 ϕ ( θ x , θ y ) ,
D T ( i , λ ) = ( I max t I min t ) ( I max t + I min t ) = [ ρ p , t 2 ( i , λ ) ρ s , t 2 ( i , λ ) ] [ ρ p , t 2 ( i , λ ) + ρ s , t 2 ( i , λ ) ] .
D R ( i , λ ) = ( I max r I min r ) ( I max r + I min r ) = [ ρ s , r 2 ( i , λ ) ρ p , r 2 ( i , λ ) ] [ ρ s , r 2 ( i , λ ) + ρ p , r 2 ( i , λ ) ] .
M T ( θ x , θ y , λ ) = [ m 11 t m 12 t m 13 t m 14 t m 21 t m 22 t m 23 t m 24 t m 31 t m 32 t m 33 t m 34 t m 41 t m 42 t m 43 t m 44 t ] ,
M R ( θ x , θ y , λ ) = [ m 11 r m 12 r m 13 r m 14 r m 21 r m 22 r m 23 r m 24 r m 31 r m 32 r m 33 r m 34 r m 41 r m 42 r m 43 r m 44 r ] .
M T = 1 2 [ 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ] , M R = 1 2 [ 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ] .
i ( θ x , θ y ) = arccos { 1 2 [ 1 θ x n c 1 2 ( θ x 2 n c 2 + θ y 2 n c 2 ) ] } ,
δ T = δ p , t δ s , t ,
M T = ( ρ p , t 2 + ρ s , t 2 ) 2 × [ 1 D T 0 0 D T 1 0 0 0 0 2 1 D T 2 cos δ T 2 1 D T 2 sin δ T 0 0 2 1 D T 2 sin δ T 2 1 D T 2 cos δ T ] .
M T ( ϕ ) = R ( ϕ ) M T R ( ϕ ) ,
R ( ϕ ) = [ 1 0 0 0 0 cos 2 ϕ sin ϕ 0 0 sin 2 ϕ cos ϕ 0 0 0 0 1 ] .
M T ( θ x , θ y , λ ) = ( ρ p , t 2 + ρ s , t 2 ) 2 × [ 1 D T cos 2 ϕ D T sin 2 ϕ 0 D T cos 2 ϕ cos 2 2 ϕ + 2 1 D T 2 sin 2 2 ϕ cos δ T cos 2 ϕ sin 2 ϕ ( 1 2 1 D T 2 cos δ T ) 2 1 D T 2 sin 2 ϕ sin δ T D T sin 2 ϕ cos 2 ϕ sin 2 ϕ ( 1 2 1 D T 2 cos δ T ) sin 2 2 ϕ + 2 1 D T 2 cos 2 2 ϕ cos δ T 2 1 D T 2 cos 2 ϕ sin δ T 0 2 1 D T 2 sin 2 ϕ sin δ T 2 1 D T 2 cos 2 ϕ sin δ T 2 1 D T 2 cos δ T ] .
M R ( θ x , θ y , λ ) = ( ρ s , r 2 + ρ p , r 2 ) 2 × [ 1 D R cos 2 ϕ D R sin 2 ϕ 0 D R cos 2 ϕ cos 2 2 ϕ + 2 1 D R 2 sin 2 2 ϕ cos δ R cos 2 ϕ sin 2 ϕ ( 1 2 1 D R 2 cos δ R ) 2 1 D R 2 sin 2 ϕ sin δ R D R sin 2 ϕ cos 2 ϕ sin 2 ϕ ( 1 2 1 D R 2 cos δ R ) sin 2 2 ϕ 2 1 D R 2 cos 2 2 ϕ cos δ R 2 1 D R 2 cos 2 ϕ sin δ R 0 2 1 D R 2 sin 2 ϕ sin δ R 2 1 D R 2 cos 2 ϕ sin δ R 2 1 D R 2 cos δ R ] .

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