Abstract

An etched calcite square-wave retarder is designed, fabricated, and demonstrated as an illuminator for an interlaced polarization computer-generated hologram (PCGH). The calcite square-wave retarder enables alternating columns of orthogonal linear polarizations to illuminate the interlaced PCGH. Together, these components produce a speckled, tangentially polarized PCGH diffraction pattern with a measured ratio of polarization of 84% and a degree of linear polarization of 0.81. An experimental alignment tolerance analysis is also reported.

© 2011 Optical Society of America

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References

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    [CrossRef] [PubMed]
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2010 (1)

2009 (1)

2006 (1)

W. J. Dallas, Digital Holography and 3-Dimensional Display (Springer, 2006).

2005 (1)

U. Schnars and W. Jueptner, Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

2003 (2)

J. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, andM. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE 4984, 210–218(2003).
[CrossRef]

D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003).
[CrossRef]

2002 (1)

1996 (1)

1995 (1)

1993 (1)

J. Hossfeld, D. Columbus, H. Sprave, T. Tschudi, and W. Dultz, “Polarizing computer-generated holograms,” Opt. Eng. 32, 1835–1838 (1993).
[CrossRef]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

1967 (1)

Bagwell, B.

J. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, andM. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE 4984, 210–218(2003).
[CrossRef]

Cheng, C. C.

Chipman, R. A.

Columbus, D.

J. Hossfeld, D. Columbus, H. Sprave, T. Tschudi, and W. Dultz, “Polarizing computer-generated holograms,” Opt. Eng. 32, 1835–1838 (1993).
[CrossRef]

Dallas, W.

Dallas, W. J.

W. J. Dallas, Digital Holography and 3-Dimensional Display (Springer, 2006).

Descour, M.

J. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, andM. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE 4984, 210–218(2003).
[CrossRef]

Dultz, W.

J. Hossfeld, D. Columbus, H. Sprave, T. Tschudi, and W. Dultz, “Polarizing computer-generated holograms,” Opt. Eng. 32, 1835–1838 (1993).
[CrossRef]

Fainman, Y.

Fischer, P.

Ford, E.

Ford, J. E.

Fratz, M.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

Giel, D. M.

Goldstein, D.

D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003).
[CrossRef]

Hansen, D.

Hossfeld, J.

J. Hossfeld, D. Columbus, H. Sprave, T. Tschudi, and W. Dultz, “Polarizing computer-generated holograms,” Opt. Eng. 32, 1835–1838 (1993).
[CrossRef]

Jabbour, G.

J. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, andM. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE 4984, 210–218(2003).
[CrossRef]

Jueptner, W.

U. Schnars and W. Jueptner, Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

Khulbe, P.

Kimbrough, B.

J. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, andM. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE 4984, 210–218(2003).
[CrossRef]

Lohmann, A. W.

Matsubara, I.

McClain, S.

Milster, T. D.

Noble, H.

Paris, D. P.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

Scherer, A.

Schnars, U.

U. Schnars and W. Jueptner, Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

Smith, M. H.

Sprave, H.

J. Hossfeld, D. Columbus, H. Sprave, T. Tschudi, and W. Dultz, “Polarizing computer-generated holograms,” Opt. Eng. 32, 1835–1838 (1993).
[CrossRef]

Sun, P. C.

Tamkin, J.

J. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, andM. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE 4984, 210–218(2003).
[CrossRef]

Tschudi, T.

J. Hossfeld, D. Columbus, H. Sprave, T. Tschudi, and W. Dultz, “Polarizing computer-generated holograms,” Opt. Eng. 32, 1835–1838 (1993).
[CrossRef]

Tsuji, T.

T. Tsuji, “Illumination optical system, exposure apparatus, and device manufacturing method with modified illumination generator,” U.S. patent 7,265,816 (4 September 2007).

Tyan, R. C.

Unno, Y.

Xu, F.

Appl. Opt. (3)

Opt. Eng. (1)

J. Hossfeld, D. Columbus, H. Sprave, T. Tschudi, and W. Dultz, “Polarizing computer-generated holograms,” Opt. Eng. 32, 1835–1838 (1993).
[CrossRef]

Opt. Lett. (3)

Optik (Jena) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

Proc. SPIE (1)

J. Tamkin, B. Bagwell, B. Kimbrough, G. Jabbour, andM. Descour, “High speed gray scale laser direct write technology for micro-optic fabrication,” Proc. SPIE 4984, 210–218(2003).
[CrossRef]

Other (4)

D. Goldstein, Polarized Light, 2nd ed. (Marcel Dekker, 2003).
[CrossRef]

T. Tsuji, “Illumination optical system, exposure apparatus, and device manufacturing method with modified illumination generator,” U.S. patent 7,265,816 (4 September 2007).

W. J. Dallas, Digital Holography and 3-Dimensional Display (Springer, 2006).

U. Schnars and W. Jueptner, Digital Hologram Recording, Numerical Reconstruction, and Related Techniques (Springer-Verlag, 2005).

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Figures (11)

Fig. 1
Fig. 1

Interlaced x- and y-polarized components add to form a tangentially polarized annulus. Overlap of these two coherent states produces the ideal tangentially polarized ring U 0 with a continuously varying polarization orientation.

Fig. 2
Fig. 2

A polarization compensator system (enclosed in the dashed line region) provides an arbitrary polarization state incident on the etched calcite square-wave retarder. A 4 f optical system images the square-wave retarder to illuminate the interlaced CGH, providing alternating stripes of x and y polarization. A CCD camera is used at the back focal plane of an imaging lens to view the reconstructed image.

Fig. 3
Fig. 3

The calcite square-wave retarder generates alternating stripes of orthogonal linear polarizations when illuminated with a linearly polarized plane wave oriented at 45 ° with respect to the crystal's fast axis. Grooves are etched into the substrate to the base thickness level and are filled with index matching oil to produce half-wave plate ridges, which rotate the incident linear polarization by 90 ° .

Fig. 4
Fig. 4

An SEM cross section of the fabricated calcite for the square-wave retarder. The grooves are smooth and uniform, although there is some rounding of the corners.

Fig. 5
Fig. 5

Retardance of a 1 mm wide horizontal cross section at the center of the calcite square-wave retarder sample was measured at 632.8 nm using the Mueller matrix imaging polarimeter. Retardance associated with the half-wave plate ridges varies between approximately 160 ° and the ideal 180 ° in the center of the sample.

Fig. 6
Fig. 6

The measured Stokes vector for the PCGH reconstruction shows that the polarization is linearly tangential with some circular polarization dispersed throughout the annulus. S 3 shows that the presence of circular polarization is most notable in the synthesis regions. S 1 , S 2 , and S 3 are normalized to S 0 , which remains unnormalized.

Fig. 7
Fig. 7

The orientation φ of the major axis of the polarization ellipse around the annulus is calculated from the normalized Stokes vector. The measured polarization orientation of the PCGH reconstruction is tangential. Its orientation differs by several degrees from the desired orientation at the sides of the annulus.

Fig. 8
Fig. 8

The pixel-by-pixel degree of linear polarization (DoLP) for the PCGH reconstruction. The average DoLP over the annulus is 0.81. The synthesis regions have a lower DoLP than the vertical and horizontal regions of the annulus because the DoLP in these regions is dependent on a high degree of speckle correlation from the two orthogonal polarizations.

Fig. 9
Fig. 9

Pixel-by-pixel ratio of tangential polarization (RoP) to flux for the PCGH reconstruction. The RoP is noticeably lower in the 45 ° , 135 ° , 225 ° , and 315 ° regions of the annulus and in the dim regions between speckles. Average RoP around the ring is 84%.

Fig. 10
Fig. 10

Simulation of square-wave retarder output. (a) In an ideal optical system, the polarization orientation exiting the calcite square-wave retarder alternates between x and y according to the grooves and ridge stripes; (b) an extraordinary axis orientation error of 10 ° with respect to the etched stripes causes an orientation error of 20 ° in the y -polarized component exiting the calcite.

Fig. 11
Fig. 11

Properties of residual wedge in the calcite. (a) MMIP measurement of retardance as a function of position in an ungrooved substrate exhibiting a small residual wedge; and (b) conceptual simulation illustrating transmitted polarization from a square-wave retarder in the presence of a 0.3 ° residual wedge.

Tables (1)

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Table 1 RoP and DoLP Are Measured as a Function of the Horizontal Translation of the Calcite and Calcite Rotation about the Optical Axis

Equations (5)

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U x 0 = A x 0 e i ϕ x 0 = g s ( r ) sin θ ,
U y 0 = A y 0 e i ϕ y 0 = g s ( r ) cos θ ,
S 0 = I ( 0 ° ) + I ( 90 ° ) S 1 = I ( 0 ° ) I ( 90 ° ) S 2 = I ( 45 ° ) I ( 135 ° ) S 3 = 1 p 2 [ I ( 45 ° , 90 ° ) I ( 135 ° , 90 ° ) ] ,
φ = 1 2 tan 1 S 2 S 1 .
RoP = I T S 0 = 1 2 ( 1 + S 1 S 0 cos 2 θ + S 2 S 0 sin 2 θ ) .

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