Abstract

The polarization properties of solid-state cube-corner retroreflectors, which are uncoated glass or metal-coated glass on the reflecting surfaces, are examined analytically and compared. Experimental verifications are presented for the case of linearly polarized incident light. When the uncoated cube-corner is used in a heterodyne interferometer, the polarization properties reveal that the axial orientation of the corner reflector has an effect on the strength of the beat signal and the nonlinearity error. A theoretical analysis of the effect is presented together with experimental results.

© 1996 Optical Society of America

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References

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  1. Y. Ranimura, K. Toyodar, “Linewidth measuring instrument for micropatterns like VLSI,” Bull. Natl. Res. Lab. Metrol. Jpn. (Jan.1989), pp. 105–112.
  2. N. Bobroff, “Residual errors in laser interferometry from air turbulence and non-linearity,” Appl. Opt. 26, 2676–2682 (1987).
    [CrossRef] [PubMed]
  3. E. R. Peck, “Polarization properties of corner reflectors and cavities,” J. Opt. Soc. Am. 38, 253–257 (1962).
    [CrossRef]
  4. M. A. Player, “Polarization properties of a cube-corner reflector,” J. Mod. Opt. 35, 1813–1820 (1988).
    [CrossRef]
  5. J. R. 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]
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).
  7. G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart, & Winston, New York, 1975).
  8. R. R. Baldwin, G. B. Gordon, A. F. Rude, “Remote laser interferometry,” Hewlett-Packard J. 23, 14–20 (1971).
  9. R. C. Quenelle, L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J.3–13 (Apr.1983).
  10. W. Hou, G. Wilkening, “Investigation and compensation of the non-linearity of heterodyne interferometers,” Prec. Eng. 14, 91–98 (1992).
    [CrossRef]
  11. R. Thalmann, W. Hou, “Limitations of interpolation accuracy in heterodyne interferometry,” in Proceedings of the Seventh Engineering Seminar, Kobe (Butterworth-Heinmann, Boston, 1993) pp. 11–23.

1993 (1)

J. R. 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]

1992 (1)

W. Hou, G. Wilkening, “Investigation and compensation of the non-linearity of heterodyne interferometers,” Prec. Eng. 14, 91–98 (1992).
[CrossRef]

1989 (1)

Y. Ranimura, K. Toyodar, “Linewidth measuring instrument for micropatterns like VLSI,” Bull. Natl. Res. Lab. Metrol. Jpn. (Jan.1989), pp. 105–112.

1988 (1)

M. A. Player, “Polarization properties of a cube-corner reflector,” J. Mod. Opt. 35, 1813–1820 (1988).
[CrossRef]

1987 (1)

1983 (1)

R. C. Quenelle, L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J.3–13 (Apr.1983).

1971 (1)

R. R. Baldwin, G. B. Gordon, A. F. Rude, “Remote laser interferometry,” Hewlett-Packard J. 23, 14–20 (1971).

1962 (1)

Baldwin, R. R.

R. R. Baldwin, G. B. Gordon, A. F. Rude, “Remote laser interferometry,” Hewlett-Packard J. 23, 14–20 (1971).

Bobroff, N.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Fowles, G. R.

G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart, & Winston, New York, 1975).

Gordon, G. B.

R. R. Baldwin, G. B. Gordon, A. F. Rude, “Remote laser interferometry,” Hewlett-Packard J. 23, 14–20 (1971).

Hou, W.

W. Hou, G. Wilkening, “Investigation and compensation of the non-linearity of heterodyne interferometers,” Prec. Eng. 14, 91–98 (1992).
[CrossRef]

R. Thalmann, W. Hou, “Limitations of interpolation accuracy in heterodyne interferometry,” in Proceedings of the Seventh Engineering Seminar, Kobe (Butterworth-Heinmann, Boston, 1993) pp. 11–23.

Mayer, J. R. R.

J. R. 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.

Player, M. A.

M. A. Player, “Polarization properties of a cube-corner reflector,” J. Mod. Opt. 35, 1813–1820 (1988).
[CrossRef]

Quenelle, R. C.

R. C. Quenelle, L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J.3–13 (Apr.1983).

Ranimura, Y.

Y. Ranimura, K. Toyodar, “Linewidth measuring instrument for micropatterns like VLSI,” Bull. Natl. Res. Lab. Metrol. Jpn. (Jan.1989), pp. 105–112.

Rude, A. F.

R. R. Baldwin, G. B. Gordon, A. F. Rude, “Remote laser interferometry,” Hewlett-Packard J. 23, 14–20 (1971).

Thalmann, R.

R. Thalmann, W. Hou, “Limitations of interpolation accuracy in heterodyne interferometry,” in Proceedings of the Seventh Engineering Seminar, Kobe (Butterworth-Heinmann, Boston, 1993) pp. 11–23.

Toyodar, K.

Y. Ranimura, K. Toyodar, “Linewidth measuring instrument for micropatterns like VLSI,” Bull. Natl. Res. Lab. Metrol. Jpn. (Jan.1989), pp. 105–112.

Wilkening, G.

W. Hou, G. Wilkening, “Investigation and compensation of the non-linearity of heterodyne interferometers,” Prec. Eng. 14, 91–98 (1992).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

Wuerz, L. J.

R. C. Quenelle, L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J.3–13 (Apr.1983).

Appl. Opt. (1)

Bull. Natl. Res. Lab. Metrol. Jpn. (1)

Y. Ranimura, K. Toyodar, “Linewidth measuring instrument for micropatterns like VLSI,” Bull. Natl. Res. Lab. Metrol. Jpn. (Jan.1989), pp. 105–112.

Hewlett-Packard J. (2)

R. R. Baldwin, G. B. Gordon, A. F. Rude, “Remote laser interferometry,” Hewlett-Packard J. 23, 14–20 (1971).

R. C. Quenelle, L. J. Wuerz, “A new microcomputer-controlled laser dimensional measurement and analysis system,” Hewlett-Packard J.3–13 (Apr.1983).

J. Mod. Opt. (1)

M. A. Player, “Polarization properties of a cube-corner reflector,” J. Mod. Opt. 35, 1813–1820 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

J. R. 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]

Prec. Eng. (1)

W. Hou, G. Wilkening, “Investigation and compensation of the non-linearity of heterodyne interferometers,” Prec. Eng. 14, 91–98 (1992).
[CrossRef]

Other (3)

R. Thalmann, W. Hou, “Limitations of interpolation accuracy in heterodyne interferometry,” in Proceedings of the Seventh Engineering Seminar, Kobe (Butterworth-Heinmann, Boston, 1993) pp. 11–23.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980).

G. R. Fowles, Introduction to Modern Optics (Holt, Rinehart, & Winston, New York, 1975).

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Figures (18)

Fig. 1
Fig. 1

Reflection of light at the interface plane.

Fig. 2
Fig. 2

Front view of the cube-corner reflector. The light has two types of reflection that depend on the incident position. Path 1 and 2 represent clockwise and counterclockwise reflections, respectively. The direction of incident polarization and the coordinate axes at the entrance plane of the cube-corner is shown.

Fig. 3
Fig. 3

Three reflection planes and related notation.

Fig. 4
Fig. 4

(a) Variation of the copolarized (CP) and orthogonally polarized (OP) intensity when reflected in the (a) clockwise (thick curve) or (b) counterclockwise (thin curve) inside the uncoated cube-corner as a function of the angle α in Fig 2.

Fig. 5
Fig. 5

(a) Variation of the copolarized (CP) and orthogonally polarized (OP) intensity when reflected (a) clockwise (thick curve) or (b) counterclockwise (thin curve) inside the aluminium-coated cube-corner as a function of the angle α in Fig. 2.

Fig. 6
Fig. 6

(a) Variation of the copolarized (CP) and orthogonally polarized (OP) intensity when reflected (a) clockwise (thick curve) or (b) counterclockwise (thin curve) inside the silver-coated cube-corner as a function of the angle α in Fig. 2.

Fig. 7
Fig. 7

Experimental setup used to determine the polarization property of the the uncoated cube-corner with the linearly polarized expanded beam.

Fig. 8
Fig. 8

Intensity distribution of the orthogonally polarized component of the reflected beam when the linearly polarized beam is incident upon the entire entrance plane of the uncoated cube-corner.

Fig. 9
Fig. 9

Experimental used to determine the polarization property of cube-corners with the linearly polarized beam spot.

Fig. 10
Fig. 10

Variation of the copolarized (CP) and orthogonally polarized (OP) intensity reflected from the uncoated cube-corner with the linearly polarized incident beam (spot size, 7 mm). Solid curve, theoretical; points, experimental data.

Fig. 11
Fig. 11

Variation of the copolarized (CP) and orthogonally polarized (OP) intensity reflected from the aluminium-coated cube-corner with the linearly polarized incident beam (spot size, 7 mm). Solid curve, theoretical; points, experimental data.

Fig. 12
Fig. 12

Variation of the copolarized (CP) and orthogonally polarized (OP) intensity reflected from the silver-coated cube-corner with the linearly polarized incident beam (spot size, 7 mm). Solid curve, theoretical; points, experimental data.

Fig. 13
Fig. 13

Variation of the copolarized (CP) and orthogonally polarized (OP) intensity reflected from the aluminium-coated cube-corner with the linearly polarized incident beam (spot size, 3 mm). Dotted curve, theoretical; circles, experimental data.

Fig. 14
Fig. 14

Three-dimensional plot of the beat signal strength as a function of the orientation of the two uncoated cube-corners in the reference and the signal arms.

Fig. 15
Fig. 15

Optimal incident point and polarization for maximizing the strength of the beat signal with the incident light of linear polarization in the case of uncoated cube-corners.

Fig. 16
Fig. 16

Maximum length-measuring error that is due to nonlinearity as the orientation of one uncoated cube-corner is changed while the other uncoated cube-corner remains fixed in the orientation maximizing the its signal strength in the heterodyne interferometer.

Fig. 17
Fig. 17

Experimental setup to measure the strength of the beat signal and phase error caused by nonlinearity as the orientation of the uncoated cube-corner changes. CC1, CC2, cube-corners; PBS1, PBS2, polarizing beam splitters; D1, D2, P-I-N detectors.

Fig. 18
Fig. 18

Variation of the beat intensity and phase error caused by nonlinearity as a function of the rotational angle of the uncoated cube-corner.

Equations (32)

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r s = cos θ ( n 2 sin 2 θ ) 1 / 2 cos θ + ( n 2 sin 2 θ ) 1 / 2 ,
r p = n 2 cos θ ( n 2 sin 2 θ ) 1 / 2 n 2 cos θ + ( n 2 sin 2 θ ) 1 / 2 .
s ˆ 1 = s ˆ 1 = k ˆ 0 × M ˆ 1 = 1 2 ( 0 , 1 , 1 ) ,
p ˆ 1 = k ˆ 0 × s ˆ 1 = 1 6 ( 2 , 1 , 1 ) ,
k ˆ 1 = k ˆ 0 2 ( k ˆ 0 · M ˆ 1 ) M ˆ 1 = 1 3 ( 1 , 1 , 1 ) ,
p ˆ 1 = k ˆ 1 × s ˆ 1 = 1 6 ( 2 , 1 , 1 ) ,
s ˆ 2 = s ˆ 2 = k ˆ 1 × M ˆ 2 = 1 2 ( 1 , 1 , 0 ) ,
p ˆ 2 = k ˆ 1 × s ˆ 2 = 1 6 ( 1 , 1 , 2 ) ,
k ˆ 2 = k ˆ 1 2 ( k ˆ 1 · M ˆ 2 ) M ˆ 2 = 1 3 ( 1 , 1 , 1 ) ,
p ˆ 2 = k ˆ 2 × s ˆ 2 = 1 6 ( 1 , 1 , 2 ) ,
s ˆ 3 = s ˆ 3 = k ˆ 2 × M ˆ 3 = 1 2 ( 1 , 0 , 1 ) ,
p ˆ 3 = k ˆ 2 × s ˆ 3 = 1 6 ( 1 , 2 , 1 ) ,
k ˆ 3 = k ˆ 2 2 ( k ˆ 2 · M ˆ 3 ) M ˆ 3 = 1 3 ( 1 , 1 , 1 ) ,
p ˆ 3 = k ˆ 3 × s ˆ 3 = 1 6 ( 1 , 2 , 1 ) .
E 1 = a r s s ˆ 1 + b r p p ˆ 1 = s ˆ 2 ( a r s s ˆ 1 · s ˆ 2 + b r p p ˆ 1 · s ˆ 2 ) + p ˆ 2 ( a r s s ˆ 1 · p ˆ 2 + b r p p ˆ 1 · p ˆ 2 ) = s ˆ 2 ( 1 2 a r s + 3 2 b r p ) + p ˆ 2 ( 3 2 a r s + 1 2 b r p ) .
E 2 = s ˆ 2 ( 1 2 a r s + 3 2 b r p ) r s + p ˆ 2 ( 3 2 a r s + 1 2 b r p ) r p = s ˆ 3 ( a 4 r s 2 + 3 b + 3 a 4 r s r p 3 b 4 r p 2 ) + p ˆ 3 ( 3 a 4 r s 2 + 3 b 3 a 4 r s r p + b 4 r p 2 ) ,
E 3 = s ˆ 3 ( 1 4 r s 2 + 3 b + 3 a 4 r s r p 3 b 4 r p 2 ) r s + p ˆ 3 ( 3 a 4 r s 2 + 3 b 3 a 4 r s r p + b 4 r p 2 ) r p .
E 3 = s ˆ 1 [ a 8 ( r s 3 + 6 r s r p 2 3 r s r p 2 ) + 3 b 4 ( r s 2 r p + 2 r s r p 2 + r p 3 ) ] + p ˆ 1 * [ 3 a 8 ( r s 3 + 2 r s 2 r p + r s r p 2 ) b 8 ( 3 r s 2 r p 6 r s r p 2 r p 3 ) ] .
[ E 3 s E 3 p ] = [ 1 2 3 2 3 2 1 2 ] [ r s 0 0 r p ] [ 1 2 3 2 3 2 1 2 ] [ r s 0 0 r p ] × [ 1 2 3 2 3 2 1 2 ] [ r s 0 0 r p ] [ a b ] = [ c 11 c 12 c 21 c 22 ] [ a b ] ,
c 11 = 1 8 ( r s 3 + 6 r s 2 r p 3 r s r p 2 ) ,
c 12 = 3 8 r p ( r s + r p ) 2 ,
c 21 = 3 8 r s ( r s + r p ) 2 ,
c 22 = 1 8 ( r p 3 + 6 r p 2 r s 3 r p r s 2 ) ,
[ E 3 s E 3 p ] = [ 1 2 3 2 3 2 1 2 ] [ r s 0 0 r p ] [ 1 2 3 2 3 2 1 2 ] [ r s 0 0 r p ] × [ 1 2 3 2 3 2 1 2 ] [ r s 0 0 r p ] [ a b ] = [ c 11 c 12 c 21 c 22 ] [ a b ]
[ E 3 s E 3 p * ] = [ c 11 c 12 c 21 c 22 ] [ cos α sin α ] .
[ E 3 , | | E 3 , ] = [ cos α sin α sin α cos α ] [ c 11 c 12 c 21 c 22 ] [ cos α sin α ] .
[ E 3 , | | E 3 , ] = [ cos α sin α sin α cos α ] [ c 11 c 12 c 21 c 22 ] [ cos α sin α ] .
E 1 = x ˆ cos ( 2 π f 1 t ) ,
E 2 = y ˆ cos ( 2 π f 2 t ) .
E 1 = x ˆ a 1 cos ( 2 π f 1 t + ψ 1 ) + y ˆ b 1 cos ( 2 π f 1 t + ψ 1 + π ) , E 2 = x ˆ b 2 cos ( 2 π f 2 t + ψ 2 + π 2 ) + y ˆ a 2 cos ( 2 π f 2 t + ψ 2 ) ,
a 1 cos ( 2 π f 1 t + ψ 1 ) + a 2 cos ( 2 π f 2 t + ψ 2 ) ,
I beat | a 1 exp ( 2 π if 1 t + i ψ 1 ) + a 2 exp ( 2 π if 2 t + i ψ 2 ) | 2 , = | a 1 | 2 + | a 2 | 2 + 2 | a 1 | | a 2 | cos ( 2 π Δ f + Δ ψ ) ,

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