Abstract

Many corner cube prisms, or retroreflectors, employ total internal reflection (TIR) via uncoated rear surfaces. The different elliptical polarization states emerging from the six unique paths through the corner cube complicate the far-field diffraction pattern by introducing various phase delays between the six paths. In this paper, we present a computational framework to evaluate polarization through TIR corner cubes for arbitrary incidence angles and input polarization states, presenting example output for key normal-incidence conditions. We also describe a method to produce far-field diffraction patterns resulting from the polarization analysis, presenting representative images—broken into orthogonal polarizations—and characterizing key cases. Laboratory confirmation is also presented for both polarization states and far-field diffraction patterns.

© 2013 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. M. S. Scholl, “Ray trace through a corner-cube retroreflector with complex reflection coefficients,” J. Opt. Soc. Am. A 12, 1589–1592 (1995).
    [CrossRef]
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    [CrossRef]
  6. D. A. Arnold, “Method of calculating retroreflector-array transfer functions,” Special Report 382 (Smithsonian Astrophysical Observatory, Massachusetts, 1979).
  7. M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetallized faces,” Opt. Spektrosk. 107, 213–218(2009).
    [CrossRef]
  8. A. L. Sokolov and V. V. Murashkin, “Diffraction polarization optical elements with radial symmetry,” Opt. Spectrosc. 111, 859–865 (2011).
    [CrossRef]
  9. T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
    [CrossRef]
  10. S. D. Goodrow and T. W. Murphy, “Effects of thermal gradients in total internal reflection corner cubes,” Appl. Opt.51, 8793–8799 (2012).
    [CrossRef]
  11. T. W. Murphy, E. G. Adelberger, J. B. R. Battat, C. D. Hoyle, R. J. McMillan, E. L. Michelsen, R. L. Samad, C. W. Stubbs, and H. E. Swanson, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
    [CrossRef]
  12. http://physics.ucsd.edu/~tmurphy/papers/ccr-sim/ccr-sim.html .

2012

2011

A. L. Sokolov and V. V. Murashkin, “Diffraction polarization optical elements with radial symmetry,” Opt. Spectrosc. 111, 859–865 (2011).
[CrossRef]

2010

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, C. D. Hoyle, R. J. McMillan, E. L. Michelsen, R. L. Samad, C. W. Stubbs, and H. E. Swanson, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[CrossRef]

2009

M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetallized faces,” Opt. Spektrosk. 107, 213–218(2009).
[CrossRef]

2008

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
[CrossRef]

1997

1995

1990

R. R. Hodgson and R. A. Chipman, “Measurement of corner cube polarization,” Proc. SPIE 1317, 436–447(1990).
[CrossRef]

1971

1962

Adelberger, E. G.

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, C. D. Hoyle, R. J. McMillan, E. L. Michelsen, R. L. Samad, C. W. Stubbs, and H. E. Swanson, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[CrossRef]

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
[CrossRef]

Alley, C. O.

Arnold, D. A.

D. A. Arnold, “Method of calculating retroreflector-array transfer functions,” Special Report 382 (Smithsonian Astrophysical Observatory, Massachusetts, 1979).

Azzam, R. M. A.

Battat, J. B. R.

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, C. D. Hoyle, R. J. McMillan, E. L. Michelsen, R. L. Samad, C. W. Stubbs, and H. E. Swanson, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[CrossRef]

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
[CrossRef]

Carey, L. N.

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
[CrossRef]

Chang, R. F.

Chipman, R. A.

R. R. Hodgson and R. A. Chipman, “Measurement of corner cube polarization,” Proc. SPIE 1317, 436–447(1990).
[CrossRef]

Currie, D. G.

Goodrow, S. D.

Hodgson, R. R.

R. R. Hodgson and R. A. Chipman, “Measurement of corner cube polarization,” Proc. SPIE 1317, 436–447(1990).
[CrossRef]

Hoyle, C. D.

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, C. D. Hoyle, R. J. McMillan, E. L. Michelsen, R. L. Samad, C. W. Stubbs, and H. E. Swanson, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[CrossRef]

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
[CrossRef]

LeBlanc, P.

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
[CrossRef]

Liu, J.

McMillan, R. J.

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, C. D. Hoyle, R. J. McMillan, E. L. Michelsen, R. L. Samad, C. W. Stubbs, and H. E. Swanson, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[CrossRef]

Michelsen, E. L.

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, C. D. Hoyle, R. J. McMillan, E. L. Michelsen, R. L. Samad, C. W. Stubbs, and H. E. Swanson, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[CrossRef]

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
[CrossRef]

Murashkin, V. V.

A. L. Sokolov and V. V. Murashkin, “Diffraction polarization optical elements with radial symmetry,” Opt. Spectrosc. 111, 859–865 (2011).
[CrossRef]

Murphy, T. W.

S. D. Goodrow and T. W. Murphy, “Effects of thermal gradients in total internal reflection corner cubes,” Appl. Opt.51, 8793–8799 (2012).
[CrossRef]

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, C. D. Hoyle, R. J. McMillan, E. L. Michelsen, R. L. Samad, C. W. Stubbs, and H. E. Swanson, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[CrossRef]

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
[CrossRef]

Nordtvedt, K.

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
[CrossRef]

Orin, A. E.

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
[CrossRef]

Peck, E. R.

Pittman, M. E.

Sadovnikov, M. A.

M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetallized faces,” Opt. Spektrosk. 107, 213–218(2009).
[CrossRef]

Samad, R. L.

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, C. D. Hoyle, R. J. McMillan, E. L. Michelsen, R. L. Samad, C. W. Stubbs, and H. E. Swanson, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[CrossRef]

Scholl, M. S.

Sokolov, A. L.

A. L. Sokolov and V. V. Murashkin, “Diffraction polarization optical elements with radial symmetry,” Opt. Spectrosc. 111, 859–865 (2011).
[CrossRef]

M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetallized faces,” Opt. Spektrosk. 107, 213–218(2009).
[CrossRef]

Strasburg, J. D.

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
[CrossRef]

Stubbs, C. W.

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, C. D. Hoyle, R. J. McMillan, E. L. Michelsen, R. L. Samad, C. W. Stubbs, and H. E. Swanson, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[CrossRef]

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
[CrossRef]

Swanson, H. E.

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, C. D. Hoyle, R. J. McMillan, E. L. Michelsen, R. L. Samad, C. W. Stubbs, and H. E. Swanson, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[CrossRef]

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
[CrossRef]

Williams, E.

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
[CrossRef]

Appl. Opt.

Icarus

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, C. D. Hoyle, R. J. McMillan, E. L. Michelsen, R. L. Samad, C. W. Stubbs, and H. E. Swanson, “Long-term degradation of optical devices on the Moon,” Icarus 208, 31–35 (2010).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Spectrosc.

A. L. Sokolov and V. V. Murashkin, “Diffraction polarization optical elements with radial symmetry,” Opt. Spectrosc. 111, 859–865 (2011).
[CrossRef]

Opt. Spektrosk.

M. A. Sadovnikov and A. L. Sokolov, “Spatial polarization structure of radiation formed by a retroreflector with nonmetallized faces,” Opt. Spektrosk. 107, 213–218(2009).
[CrossRef]

Proc. SPIE

R. R. Hodgson and R. A. Chipman, “Measurement of corner cube polarization,” Proc. SPIE 1317, 436–447(1990).
[CrossRef]

Publ. Astron. Soc. Pac.

T. W. Murphy, E. G. Adelberger, J. B. R. Battat, L. N. Carey, C. D. Hoyle, P. LeBlanc, E. L. Michelsen, K. Nordtvedt, A. E. Orin, J. D. Strasburg, C. W. Stubbs, H. E. Swanson, and E. Williams, “The apache point observatory lunar laser-ranging operation: instrument description and first detections,” Publ. Astron. Soc. Pac. 120, 20–37 (2008).
[CrossRef]

Other

D. A. Arnold, “Method of calculating retroreflector-array transfer functions,” Special Report 382 (Smithsonian Astrophysical Observatory, Massachusetts, 1979).

http://physics.ucsd.edu/~tmurphy/papers/ccr-sim/ccr-sim.html .

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Figures (11)

Fig. 1.
Fig. 1.

Corner cube geometry and global coordinate system. The three back faces are labeled A, B, and C. Dotted lines represent reflections of the real edges. Three-letter sequences placed on each “wedge” identify the exit location for the six unique paths through the CCR (the corresponding input wedge is diametrically opposite). The view at right is along the global y axis, with face C exposed to view. In units of the circular radius, r , h = 2 , e = 3 / 2 , and c = 1 / 2 . The distance t is arbitrary, representing the height of the uninterrupted cylinder around the CCR.

Fig. 2.
Fig. 2.

Coordinate system for representation of elliptical polarization, looking along the propagation direction (note central × denoting k ^ going into the page). By the conventional definition, the electric field vector pictured rotates in a left-handed sense, when looking toward the light source.

Fig. 3.
Fig. 3.

Output polarization states at normal incidence for linear input polarization rotated in 15° increments. The input polarization is depicted at lower left, with a cross to indicate light traveling into the page. The output polarization is drawn in the wedge from which it emerges as it would be oriented in the frame looking at the CCR face. Light output emerges from the page in this view (indicated by the dot in the center of each ellipse) so that right-handed polarization states show clockwise rotation. The pattern in the rightmost frame matches that in the leftmost frame with a 120° rotation.

Fig. 4.
Fig. 4.

Output polarization states at normal incidence for circular input polarization. Parameters and conventions are as described for Fig. 3. The input light is depicted traveling into the page so that left-handed polarization is seen at left and right-handed at right.

Fig. 5.
Fig. 5.

Experimental polarization results, plotted following conventions in Figs. 3 and 4. At left is linear polarization matching the leftmost panel in Fig. 3, and at right is right-handed polarization input. Slight irregularities are discussed in the text, but the overall agreement with theoretical expectations is good.

Fig. 6.
Fig. 6.

Normal-incidence far-field diffraction patterns for five orientations of linear polarization input in 15° increments, paralleling the sequence in Fig. 3. The top row is total irradiance, indicating input polarization direction in the lower-left corner of each panel; the middle row is the polarization component parallel to the input polarization (indicated in the lower left of each panel); the bottom row is the orthogonal polarization component (also indicated in the lower left of each panel). At right is the experimental result corresponding to the first column in the set of model results at left. Each frame is 50 λ / 7 D radians across. Intensities are normalized to the same value in all frames.

Fig. 7.
Fig. 7.

Orthogonal cuts (dashed and dotted) through the normal-incidence far-field diffraction pattern for the TIR CCR under linear input polarization, showing the similarity of the central peak to the scaled Airy function (solid). The cuts correspond to the upper-left panel of Fig. 6.

Fig. 8.
Fig. 8.

Orientation scheme and aperture shapes for the diffraction patterns to follow. Normal incidence is at the top left, with each tile representing a 5° step along the positive- x axis to the right and along the negative- y axis in the down direction. The horizontal–vertical basis vectors ( s ^ 0 and p ^ 0 ) are placed at the azimuthal position of the observer, vertical pointing toward the aperture. Black lines represent the refracted appearance of real edges, while gray lines are the reflected edges.

Fig. 9.
Fig. 9.

Diffraction patterns for horizontal linear input polarization at a range of viewing angles. In each grid, the top-left panel is at normal incidence, following the orientation scheme depicted in Fig. 8. At left is the total intensity, followed by the horizontal and vertical polarization components. Intensities are normalized to a common maximum within each of the three sets, but the intensity of the vertical polarization panel has been scaled by a factor of 2.57 relative to the other two in order to show details in these intrinsically dimmer patterns.

Fig. 10.
Fig. 10.

Diffraction patterns for left-handed circular input polarization, following the conventions of Figs. 8 and 9. The horizontal and vertical polarization decomposition does not in this case break cleanly into distinct patterns, as was the case for linear input polarization, since the central spot in this case is circularly polarized and thus shares equally in the two components.

Fig. 11.
Fig. 11.

Comparison of experimental results (top) to the simulation results (bottom), stretched to emphasize faint structure. At left is total intensity given horizontal polarization input light, followed by horizontal and vertical polarization output patterns. Each frame is 64 λ / 3 D across.

Tables (2)

Tables Icon

Table 1. Rotation Sequences in Degrees

Tables Icon

Table 2. Example Electric Field Parametersa

Equations (19)

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

n ^ A = 1 6 ( 1 3 2 ) n ^ B = 1 6 ( 2 0 2 ) n ^ C = 1 6 ( 1 3 2 ) ,
s ^ 0 = sin A , cos A , 0 p ^ 0 = s ^ 0 × k ^ 0 ,
s ^ = k ^ × n ^ | k ^ × n ^ | p ^ = s ^ × k ^ ,
α = a tan 2 ( s ^ · v ^ , s ^ · u ^ ) ,
E⃗ = ( E u cos ( ω t + δ u ) E v cos ( ω t + δ v ) ) ,
( E s cos ( ω t + δ s ) E p cos ( ω t + δ p ) ) = ( cos α sin α sin α cos α ) ( E u cos ( ω t + δ u ) E v cos ( ω t + δ v ) ) .
E s cos δ s = E u cos δ u cos α + E v cos δ v sin α E s sin δ s = E u sin δ u cos α + E v sin δ v sin α E p cos δ p = E v cos δ v cos α E u cos δ u sin α E p sin δ p = E v sin δ v cos α E u sin δ u sin α .
δ s = a tan 2 ( E s sin δ s , E s cos δ s ) δ p = a tan 2 ( E p sin δ p , E p cos δ p ) ,
δ s δ s + Δ δ s δ p δ p + Δ δ p ,
Δ δ s = 2 tan 1 ( n 2 sin 2 θ 1 n cos θ ) Δ δ p = 2 tan 1 ( n n 2 sin 2 θ 1 cos θ )
T = F · R ( α 4 ) · P · R ( α 3 ) · P · R ( α 2 ) · P · R ( α 1 ) ,
R ( α ) = ( cos α sin α sin α cos α ) .
P = ( e i Δ δ s 0 0 e i Δ δ p ) ,
F = ( 1 0 0 1 )
T ACB = ( 0.655 e 2.78 i 0.755 e 2.24 i 0.755 e 1.51 i 0.655 e 2.16 i ) .
tan 2 ω t = P y 2 sin 2 δ P x 2 + P y 2 cos 2 δ ,
x 1 = P x cos ω t y 1 = P y cos ( ω t + δ ) x 2 = P x cos ( ω t + π 2 ) y 2 = P y cos ( ω t + π 2 + δ )
I ( χ , η ) = | aperture S ( u , v ) exp [ i ϕ ( u , v ) ] exp [ i k ( χ u + η v ) ] d u d v | 2 ,
I ( 0 , 0 ) = | π R 2 6 n = 1 6 S n e i ϕ n | 2 0.264 π 2 R 4 ,

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