Abstract

Polarization properties of the corner-cube retroreflector are discussed theoretically by use of ray tracing and analytical geometry. The Jones matrices and eigenpolarizations of the six propagation trips of the corner-cube retroreflector are derived. An experiment is also set up for the determination of the linear eigenpolarizations and the output states of polarization for incident linearly polarized light. The experimental results are consistent with theoretical expectations.

© 1997 Optical Society of America

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References

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  1. E. R. Peck, “Polarization properties of corner reflectors and cavities,” J. Opt. Soc. Am. 52, 253–257 (1962).
    [Crossref]
  2. P. Rabinowitz, S. F. Jacobs, T. Shultz, G. Gould, “Cube-corner Fabry–Perot interferometer,” J. Opt. Soc. Am. 52, 452–453 (1962).
    [Crossref]
  3. P. I. Lamekin, “Intrinsic polarization states of corner reflectors,” Sov. J. Opt. Tech. 54, 658–661 (1987).
  4. M. A. Acharekar, “Derivation of internal incidence angles and coordinate transformations between internal reflections for corner reflectors at normal incidence,” Opt. Eng. 23, 669–674 (1984).
    [Crossref]
  5. R. R. Hodgson, R. A. Chipman, “Measurement of corner cube polarization,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray, R. A. Chipman, J. W. Morris, eds., Proc. SPIE1317, 436–447 (1990).
  6. J. R. Mayer, “Polarization optics design for a laser tracking triangulation instrument based on dual-axis scanning and a retroreflective target,” Opt. Eng. 32, 3316–3326 (1993).
    [Crossref]
  7. M. S. Scholl, “Ray trace through a corner-cube retroreflector with complex reflection coefficients,” J. Opt. Soc. Am. A 12, 1589–1592 (1995).
    [Crossref]
  8. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  9. D. Korsch, Reflective Optics (Academic, New York, 1991).
  10. Edmund Scientific, Barrington, New Jersey.
  11. Meadowlark Optics, Longmont, Colorado.
  12. R. M. A. Azzam, N. M. Bashara, “Loci of invariant-azimuth and invariant-ellipticity polarization states of an optical system,” Appl. Opt. 12, 62–67 (1973).
    [Crossref] [PubMed]
  13. R. M. A. Azzam, N. M. Bashara, “Parametric equation of the locus of invariant-ellipticity states of an optical system,” Appl. Opt. 12, 2545–2547 (1973).

1995 (1)

1993 (1)

J. R. Mayer, “Polarization optics design for a laser tracking triangulation instrument based on dual-axis scanning and a retroreflective target,” Opt. Eng. 32, 3316–3326 (1993).
[Crossref]

1987 (1)

P. I. Lamekin, “Intrinsic polarization states of corner reflectors,” Sov. J. Opt. Tech. 54, 658–661 (1987).

1984 (1)

M. A. Acharekar, “Derivation of internal incidence angles and coordinate transformations between internal reflections for corner reflectors at normal incidence,” Opt. Eng. 23, 669–674 (1984).
[Crossref]

1973 (2)

1962 (2)

Acharekar, M. A.

M. A. Acharekar, “Derivation of internal incidence angles and coordinate transformations between internal reflections for corner reflectors at normal incidence,” Opt. Eng. 23, 669–674 (1984).
[Crossref]

Azzam, R. M. A.

Bashara, N. M.

Chipman, R. A.

R. R. Hodgson, R. A. Chipman, “Measurement of corner cube polarization,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray, R. A. Chipman, J. W. Morris, eds., Proc. SPIE1317, 436–447 (1990).

Gould, G.

Hodgson, R. R.

R. R. Hodgson, R. A. Chipman, “Measurement of corner cube polarization,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray, R. A. Chipman, J. W. Morris, eds., Proc. SPIE1317, 436–447 (1990).

Jacobs, S. F.

Korsch, D.

D. Korsch, Reflective Optics (Academic, New York, 1991).

Lamekin, P. I.

P. I. Lamekin, “Intrinsic polarization states of corner reflectors,” Sov. J. Opt. Tech. 54, 658–661 (1987).

Mayer, J. R.

J. R. Mayer, “Polarization optics design for a laser tracking triangulation instrument based on dual-axis scanning and a retroreflective target,” Opt. Eng. 32, 3316–3326 (1993).
[Crossref]

Peck, E. R.

Rabinowitz, P.

Scholl, M. S.

Shultz, T.

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

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

Opt. Eng. (2)

M. A. Acharekar, “Derivation of internal incidence angles and coordinate transformations between internal reflections for corner reflectors at normal incidence,” Opt. Eng. 23, 669–674 (1984).
[Crossref]

J. R. Mayer, “Polarization optics design for a laser tracking triangulation instrument based on dual-axis scanning and a retroreflective target,” Opt. Eng. 32, 3316–3326 (1993).
[Crossref]

Sov. J. Opt. Tech. (1)

P. I. Lamekin, “Intrinsic polarization states of corner reflectors,” Sov. J. Opt. Tech. 54, 658–661 (1987).

Other (5)

R. R. Hodgson, R. A. Chipman, “Measurement of corner cube polarization,” in Polarimetry: Radar, Infrared, Visible, Ultraviolet, and X-Ray, R. A. Chipman, J. W. Morris, eds., Proc. SPIE1317, 436–447 (1990).

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

D. Korsch, Reflective Optics (Academic, New York, 1991).

Edmund Scientific, Barrington, New Jersey.

Meadowlark Optics, Longmont, Colorado.

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Figures (8)

Fig. 1
Fig. 1

Geometry of the corner-cube retroreflector.

Fig. 2
Fig. 2

Definition of the external reference p- and s-polarization directions for the input and the output light. Edge AB of the base of the corner cube is used as a reference.

Fig. 3
Fig. 3

Propagation vectors for the incident and reflected light from a planer surface. N is the surface normal.

Fig. 4
Fig. 4

Light ray vectors for the ACOABOBCO propagation trip.

Fig. 5
Fig. 5

Experimental setup for the determination of the linear eigenpolarizations of a corner-cube retroreflector.

Fig. 6
Fig. 6

Polarization-state detector that uses two liquid crystal (LC) retarders and an analyzer. F1 is the fast axis for LC retarder 1; F2 is the fast axis for LC retarder 2; and P A is the transmission axis of the analyzer.

Fig. 7
Fig. 7

Measured normalized Stokes parameters of the output light of a corner cube when the incident light is linearly polarized. The propagation trip is ACO → ABO → BCO. Solid curves are theoretically calculated from Eq. (23) and the circles are experimental results.

Fig. 8
Fig. 8

Measured normalized Stokes parameters of the output light of a corner cube when the incident light is linearly polarized. The propagation trip is ACOBCOABO. Solid curves are theoretically calculated from Eq. (31) and the circles are experimental results.

Tables (8)

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Table 1 Coordinates of Points of Intersection with Each Surface of a Corner Cube, a = 10 cm

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Table 2 Unit Vectors Along the Segmented Light Path

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Table 3 Angles Between the Surface Normals and n1

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Table 4 Angles for Surface Normals Relative to n6

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Table 5 Angles of Rotation of Coordinates (deg)

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Table 6 Jones Matrices, Eigenvectors, and Eigenvalues for the Counterclockwise Propagation Trips of a Corner-Cube Retroreflector

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Table 7 Jones Matrices, Eigenvectors, and Eigenvalues for the Clockwise Propagation Trips of a Corner-Cube Retroreflector

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Table 8 Experimental Results for the Determination of the Linear Eigenpolarizations of a Corner Cube

Equations (41)

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X=0, N1=i
Z=0, N2=-k
Y=0, N3=-j
X-Y-Z+a=0,
N4=-i+j+k/31/2;
L2=L1-2N cos ϕ,
n=L1×L2/L1×L2
ϕ=cos-113=54.7356°
L6=O1O5/O1O5=-0.7715, -0.6172, -0.1543.
n6=L0×L6L0×L6=-0.2673, 0.5345, -0.8018.
n1=0.7071, 0.7071, 0.0000.
n2=L1×L2L1×L2=0.0000, 0.7071, -0.7071.
n3=L2×L3L2×L3=-0.7071, 0.7071, 0.0000.
n4=L3×L4L3×L4=-0.7071, 0.0000, -0.7071.
n5=0.7071, 0.7071, 0.0000.
Ti=Rpiexpjδpi00Rsiexpjδsi, i=1, 2, 3.
Ti=rpi00rsi, i=1, 2, 3,
J123=R-120°T3R60°T2R-60°T1R-60°,
Rθ=cos θsin θ-sin θcos θ.
J123=J11J12J21J22,
J11=rp2rp1 cos2 60°-rs1 sin2 60°-rp3 cos2 60°+rs3 sin2 60°-rp1+rs1×rp3+rs3rs2 sin2 60° cos2 60°,
J12=rp2rp1+rs1rp3 cos2 60°-rs3 sin2 60°×sin 60° cos 60°--rp1 sin2 60°+rs1 cos2 60°rp3+rs3rs2 sin 60° cos 60°,
J21=-rs1 sin2 60°+rp1 cos2 60°×rp3+rs3rp2 sin 60° cos 60°--rp3 sin2 60°+rs3 cos2 60°rp1+rs1rs2 sin 60° cos60°,
J22=rs2rs1 cos2 60°-rp1 sin2 60°-rs3 cos2 60°+rp3 sin2 60°-rp1+rs1×rp3+rs3rp2 sin2 60° cos2 60°.
J11=-116rp3+15rprs2-3rsrp2+3rs3,
J12=316rp3+rsrp2-rprs2-rs3,
J21=316rp3+rsrp2-rprs2-rs3,
J22=-1163rp3- 3rprs2+15rsrp2+rs3.
J123=-0.0266+0.9586i0.2282+0.1685i0.2282+0.1685i-0.9083+0.3075i.
Ee1=10.2434, Ee2=1-4.1076,
Ve1=0.0290+0.9996i, Ve2=-0.9638+0.2665i.
n=-L1×L2/L1×L2.
J123=R-120°T3R60°T2R-60°T1R-60°,
J231=R120°T1R60°T3R-60°T2R180°,
J312=R0°T2R0°T1R-60°T3R60°,
J321=R-60°T1R-60°T2R60° T3R-120°,
J132=R180°T2R-60°T3R60°T1R120°,
J213=R60°T3R-60°T1R60° T2R0°,
=Vmax-Vmin/Vmax+Vmin,
E=1-X,
X=i sinδ12-cosδ12expiδ2cosδ12-i sinδ12expiδ2.

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