Abstract

A method is proposed for obtaining full-range sequential measurements of the slow axis angle and phase retardation of linear birefringent materials (LBMs) using a full-field heterodyne interferometer with a charge-coupled device (CCD) camera and an image processing algorithm based on a three-frame integrating-bucket method. The dynamic ranges of the principal axis and phase retardation measurements extend from 0° to 180° and from 0° to 360°, respectively. The proposed method not only enables full-range measurements of the slow axis angle to be obtained, but also allows a decision to be made as to whether the principal axis labeled by the manufacturer is the slow axis or the fast axis. The standard deviations of the slow axis angle and phase retardation measurements are found to be 0.14° and 0.27°, respectively. In addition, it is shown that the noises induced by environmental disturbances are reduced by elimination of the dc component of the output light intensity in the image processing algorithm. We also investigate the sensitivity of the measured error caused by the orientation of LBM.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2008 (1)

2007 (1)

J. F. Lin, T. T. Liao, Y. L. Lo, and S. Y. Lee, “The optical linear birefringence measurement using a Zeeman laser,” Opt. Commun. 274, 153-158 (2007).
[CrossRef]

2006 (3)

2005 (2)

W. C. Kuo, K. Y. Lia, and C. Chou, “Simultaneous measurement of phase retardation and fast-axis angle of phase retardation plate,” Jpn. J. Appl. Phys. 44, 1095-1100 (2005).
[CrossRef]

H. K. Teng, K. C. Lang, and C. C. Yen, “Optical polarization rotating technique for characterizing linear birefringence with full range,” Opt. Eng. 44, 123602 (2005).
[CrossRef]

2004 (5)

M. V. Aguanno, F. Lakestani, M. P. Whelan, and M. J. Connelly, “Single pixel carrier based approach for full field laser interferometry using a CMOS-DSP camera,” Proc. SPIE 5251, 304-312 (2004).
[CrossRef]

Y. L. Lo, C. H. Lai, J. F. Lin, and P. F. Hsu, “Simultaneous absolute measurements of principal angle and phase retardation with a new common-path heterodyne interferometer,” Appl. Opt. 43, 2013-2022 (2004).
[CrossRef] [PubMed]

Y. L. Lo, S. Y. Lee, and J. F. Lin, “The new circular polariscope and the Senarmont setup with electro-optic modulation for measuring the optical linear birefringent media properties,” Opt. Commun. 237, 267-273 (2004).
[CrossRef]

C. C. Montarou and T. K. Gaylord, “Two-wave-plate compensator for single-point retardation measurement,” Appl. Opt. 43, 6580-6595 (2004).
[CrossRef]

M. Mujat, E. Baleine, and A. Dogariu, “Interferometric imaging polarimeter,” J. Opt. Soc. Am. A 21, 2244-2249 (2004).
[CrossRef]

2003 (1)

2002 (1)

C. Chou and H. K. Teng, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. 41, 3140-3144 (2002).
[CrossRef]

2001 (1)

2000 (3)

1997 (1)

K. B. Rochford, A. H. Rose, and C. M. Wang, “NIST study investigates retardance uncertainty,” Laser Focus World 33, 223-227 (1997).

Aguanno, M. V.

M. V. Aguanno, F. Lakestani, M. P. Whelan, and M. J. Connelly, “Single pixel carrier based approach for full field laser interferometry using a CMOS-DSP camera,” Proc. SPIE 5251, 304-312 (2004).
[CrossRef]

Akiba, M.

M. Akiba, K. P. Chan, and N. Tanno, “Real-time, micrometer depth resolved imaging by low-coherence reflectometry and a two-dimensional heterodyne detection technique,” Jpn. J. Appl. Phys. Part 2 39, L1194-L1196 (2000).
[CrossRef]

Baleine, E.

Berezhna, S. Y.

Berezhnyy, I. V.

Boccara, A. C.

Chan, K. P.

M. Akiba, K. P. Chan, and N. Tanno, “Real-time, micrometer depth resolved imaging by low-coherence reflectometry and a two-dimensional heterodyne detection technique,” Jpn. J. Appl. Phys. Part 2 39, L1194-L1196 (2000).
[CrossRef]

Chatterjee, S.

Chih, H. W.

Chou, C.

W. C. Kuo, K. Y. Lia, and C. Chou, “Simultaneous measurement of phase retardation and fast-axis angle of phase retardation plate,” Jpn. J. Appl. Phys. 44, 1095-1100 (2005).
[CrossRef]

C. Chou and H. K. Teng, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. 41, 3140-3144 (2002).
[CrossRef]

Clerc, F. L.

Collot, L.

Connelly, M. J.

M. V. Aguanno, F. Lakestani, M. P. Whelan, and M. J. Connelly, “Single pixel carrier based approach for full field laser interferometry using a CMOS-DSP camera,” Proc. SPIE 5251, 304-312 (2004).
[CrossRef]

Dogariu, A.

Etzel, S. M.

Gaylord, T. K.

Gross, M.

Hsu, P. F.

Jeng, Y. T.

Kumar, Y. P.

Kuo, W. C.

W. C. Kuo, K. Y. Lia, and C. Chou, “Simultaneous measurement of phase retardation and fast-axis angle of phase retardation plate,” Jpn. J. Appl. Phys. 44, 1095-1100 (2005).
[CrossRef]

Lai, C. H.

Lakestani, F.

M. V. Aguanno, F. Lakestani, M. P. Whelan, and M. J. Connelly, “Single pixel carrier based approach for full field laser interferometry using a CMOS-DSP camera,” Proc. SPIE 5251, 304-312 (2004).
[CrossRef]

Lang, K. C.

H. K. Teng, K. C. Lang, and C. C. Yen, “Optical polarization rotating technique for characterizing linear birefringence with full range,” Opt. Eng. 44, 123602 (2005).
[CrossRef]

Lee, S. Y.

J. F. Lin, T. T. Liao, Y. L. Lo, and S. Y. Lee, “The optical linear birefringence measurement using a Zeeman laser,” Opt. Commun. 274, 153-158 (2007).
[CrossRef]

Y. L. Lo, S. Y. Lee, and J. F. Lin, “The new circular polariscope and the Senarmont setup with electro-optic modulation for measuring the optical linear birefringent media properties,” Opt. Commun. 237, 267-273 (2004).
[CrossRef]

Lia, K. Y.

W. C. Kuo, K. Y. Lia, and C. Chou, “Simultaneous measurement of phase retardation and fast-axis angle of phase retardation plate,” Jpn. J. Appl. Phys. 44, 1095-1100 (2005).
[CrossRef]

Liao, T. T.

J. F. Lin, T. T. Liao, Y. L. Lo, and S. Y. Lee, “The optical linear birefringence measurement using a Zeeman laser,” Opt. Commun. 274, 153-158 (2007).
[CrossRef]

Lin, J. F.

J. F. Lin, T. T. Liao, Y. L. Lo, and S. Y. Lee, “The optical linear birefringence measurement using a Zeeman laser,” Opt. Commun. 274, 153-158 (2007).
[CrossRef]

Y. L. Lo, S. Y. Lee, and J. F. Lin, “The new circular polariscope and the Senarmont setup with electro-optic modulation for measuring the optical linear birefringent media properties,” Opt. Commun. 237, 267-273 (2004).
[CrossRef]

Y. L. Lo, C. H. Lai, J. F. Lin, and P. F. Hsu, “Simultaneous absolute measurements of principal angle and phase retardation with a new common-path heterodyne interferometer,” Appl. Opt. 43, 2013-2022 (2004).
[CrossRef] [PubMed]

Lo, Y. L.

J. F. Lin, T. T. Liao, Y. L. Lo, and S. Y. Lee, “The optical linear birefringence measurement using a Zeeman laser,” Opt. Commun. 274, 153-158 (2007).
[CrossRef]

Y. L. Lo and T. C. Yu, “A polarimetric glucose sensor using a liquid-crystal polarization modulator driven by a sinusoidal signal,” Opt. Commun. 259, 40-48 (2006).
[CrossRef]

Y. T. Jeng and Y. L. Lo, “Heterodyne polariscope for sequential measurements of the complete optical parameters of a multiple-order wave plate,” Appl. Opt. 45, 1134-1141 (2006).
[CrossRef] [PubMed]

Y. L. Lo, H. W. Chih, C. Y. Yeh, and T. C. Yu, “Full-field heterodyne polariscope with an image signal processing method for principal axis and phase retardation measurements,” Appl. Opt. 45, 8006-8012 (2006).
[CrossRef] [PubMed]

Y. L. Lo, S. Y. Lee, and J. F. Lin, “The new circular polariscope and the Senarmont setup with electro-optic modulation for measuring the optical linear birefringent media properties,” Opt. Commun. 237, 267-273 (2004).
[CrossRef]

Y. L. Lo, C. H. Lai, J. F. Lin, and P. F. Hsu, “Simultaneous absolute measurements of principal angle and phase retardation with a new common-path heterodyne interferometer,” Appl. Opt. 43, 2013-2022 (2004).
[CrossRef] [PubMed]

Loriette, V.

Montarou, C. C.

Moreau, J.

Mujat, M.

Rochford, K. B.

K. B. Rochford, A. H. Rose, and C. M. Wang, “NIST study investigates retardance uncertainty,” Laser Focus World 33, 223-227 (1997).

Rose, A. H.

S. M. Etzel, A. H. Rose, and C. M. Wang, “Dispersion of the temperature dependence of the retardance in SiO2 and MgF2,” Appl. Opt. 39, 5796-5800 (2000).
[CrossRef]

K. B. Rochford, A. H. Rose, and C. M. Wang, “NIST study investigates retardance uncertainty,” Laser Focus World 33, 223-227 (1997).

Takashi, M.

Tanno, N.

M. Akiba, K. P. Chan, and N. Tanno, “Real-time, micrometer depth resolved imaging by low-coherence reflectometry and a two-dimensional heterodyne detection technique,” Jpn. J. Appl. Phys. Part 2 39, L1194-L1196 (2000).
[CrossRef]

Teng, H. K.

H. K. Teng, K. C. Lang, and C. C. Yen, “Optical polarization rotating technique for characterizing linear birefringence with full range,” Opt. Eng. 44, 123602 (2005).
[CrossRef]

C. Chou and H. K. Teng, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. 41, 3140-3144 (2002).
[CrossRef]

Wang, C. M.

S. M. Etzel, A. H. Rose, and C. M. Wang, “Dispersion of the temperature dependence of the retardance in SiO2 and MgF2,” Appl. Opt. 39, 5796-5800 (2000).
[CrossRef]

K. B. Rochford, A. H. Rose, and C. M. Wang, “NIST study investigates retardance uncertainty,” Laser Focus World 33, 223-227 (1997).

Whelan, M. P.

M. V. Aguanno, F. Lakestani, M. P. Whelan, and M. J. Connelly, “Single pixel carrier based approach for full field laser interferometry using a CMOS-DSP camera,” Proc. SPIE 5251, 304-312 (2004).
[CrossRef]

Yeh, C. Y.

Yen, C. C.

H. K. Teng, K. C. Lang, and C. C. Yen, “Optical polarization rotating technique for characterizing linear birefringence with full range,” Opt. Eng. 44, 123602 (2005).
[CrossRef]

Yu, T. C.

Y. L. Lo, H. W. Chih, C. Y. Yeh, and T. C. Yu, “Full-field heterodyne polariscope with an image signal processing method for principal axis and phase retardation measurements,” Appl. Opt. 45, 8006-8012 (2006).
[CrossRef] [PubMed]

Y. L. Lo and T. C. Yu, “A polarimetric glucose sensor using a liquid-crystal polarization modulator driven by a sinusoidal signal,” Opt. Commun. 259, 40-48 (2006).
[CrossRef]

Appl. Opt. (7)

J. Opt. Soc. Am. A (2)

Jpn. J. Appl. Phys. (2)

C. Chou and H. K. Teng, “Linear birefringence measurement with a differential-phase optical heterodyne polarimeter,” Jpn. J. Appl. Phys. 41, 3140-3144 (2002).
[CrossRef]

W. C. Kuo, K. Y. Lia, and C. Chou, “Simultaneous measurement of phase retardation and fast-axis angle of phase retardation plate,” Jpn. J. Appl. Phys. 44, 1095-1100 (2005).
[CrossRef]

Jpn. J. Appl. Phys. Part 2 (1)

M. Akiba, K. P. Chan, and N. Tanno, “Real-time, micrometer depth resolved imaging by low-coherence reflectometry and a two-dimensional heterodyne detection technique,” Jpn. J. Appl. Phys. Part 2 39, L1194-L1196 (2000).
[CrossRef]

Laser Focus World (1)

K. B. Rochford, A. H. Rose, and C. M. Wang, “NIST study investigates retardance uncertainty,” Laser Focus World 33, 223-227 (1997).

Opt. Commun. (3)

Y. L. Lo, S. Y. Lee, and J. F. Lin, “The new circular polariscope and the Senarmont setup with electro-optic modulation for measuring the optical linear birefringent media properties,” Opt. Commun. 237, 267-273 (2004).
[CrossRef]

Y. L. Lo and T. C. Yu, “A polarimetric glucose sensor using a liquid-crystal polarization modulator driven by a sinusoidal signal,” Opt. Commun. 259, 40-48 (2006).
[CrossRef]

J. F. Lin, T. T. Liao, Y. L. Lo, and S. Y. Lee, “The optical linear birefringence measurement using a Zeeman laser,” Opt. Commun. 274, 153-158 (2007).
[CrossRef]

Opt. Eng. (1)

H. K. Teng, K. C. Lang, and C. C. Yen, “Optical polarization rotating technique for characterizing linear birefringence with full range,” Opt. Eng. 44, 123602 (2005).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (1)

M. V. Aguanno, F. Lakestani, M. P. Whelan, and M. J. Connelly, “Single pixel carrier based approach for full field laser interferometry using a CMOS-DSP camera,” Proc. SPIE 5251, 304-312 (2004).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Full-field heterodyne interferometer for measurement of the slow axis angle of LBM.

Fig. 2
Fig. 2

Full-field heterodyne interferometer for measurement of the phase retardation of LBM.

Fig. 3
Fig. 3

Electronic block diagram showing the interface between the CCD and the CPLD [16].

Fig. 4
Fig. 4

Typical image acquired by the CCD camera during the measurement process.

Fig. 5
Fig. 5

Measured 2-D distributions of the quarter-wave plate slow axis angle: (a) sample = 0 ° , average = 0.07 ° ; (b) sample = 45 ° , average = 44.89 ° ; (c) sample = 90 ° , average = 90.13 ° ; and (d) sample = 135 ° , average = 134.82 ° .

Fig. 6
Fig. 6

Measured 2-D distributions of BC phase retardation: (a) sample = 45 ° , average = 44.76 ° ; (b) sample = 90 ° , average = 89.54 ° ; (c) sample = 135 ° , average = 134.59 ° ; (d) sample = 225 ° , average = 225.37 ° ; (e) sample = 270 ° , average = 269.28 ° ; and (f) sample = 315 ° , average = 314.89 ° .

Fig. 7
Fig. 7

Simulation result of the slow axis measurement error as Q 1 and Q 2 have 0.1 ° 0.5 ° misalignment.

Fig. 8
Fig. 8

(a)  Δ β as a function of α and β (with Δ Φ 2 = 0.28 ° ) and (b) alternative view angle.

Fig. 9
Fig. 9

Variation of Δ β with α in the range of α = 45 ° 135 ° and (b) variation of Δ β with α in the range of α = 45 ° 45 ° . Note that β = 90 ° , 120 ° , 150 ° , 180 ° in both cases.

Fig. 10
Fig. 10

Experimental results and simulated curve of the variation of Δ β ( β = 120 ° ) with α in the range from 90° to 135°.

Tables (1)

Tables Icon

Table 1 Experimental Results for Slow Axis Angle and Phase Retardation

Equations (12)

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[ LBM ] = [ cos ( β 2 ) - i cos ( 2 α ) sin ( β 2 ) - i sin ( 2 α ) s sin ( β 2 ) - i sin ( 2 α ) sin ( β 2 ) cos ( β 2 ) + i cos ( 2 α ) sin ( β 2 ) ] ,
E 1 = A Q 2 S Q 1 E O P E in = 1 2 [ 1 - 1 - 1 1 ] [ - i 0 0 1 ] [ LBM ] [ 1 0 0 - i ] [ cos ω t 2 i sin ω t 2 i sin ω t 2 cos ω t 2 ] [ E 0 0 ] e i ω 0 t ,
I 1 = I dc ( 1 + sin β sin ( ω t + 2 α ) ) = I dc + R 1 sin ( ω t + Φ 1 ) ,
I 2 = I dc - I dc cos ( β ) sin ω t + I dc sin ( 2 α ) sin ( β ) cos ω t .
S 1 ( x , y ) = 0 T / 4 I 1 ( x , y , t ) d t = T 4 I dc + R 1 ω sin Φ 1 + R 1 ω cos Φ 1 , S 2 ( x , y ) = T / 4 2 T / 4 I 1 ( x , y , t ) d t = T 4 I dc + R 1 ω cos Φ 1 - R 1 ω sin Φ 1 , S 3 ( x , y ) = 2 T / 4 3 T / 4 I 1 ( x , y , t ) d t = T 4 I dc - R 1 ω sin Φ 1 - R 1 ω cos Φ 1 .
α = 1 2 tan - 1 Φ 1 = 1 2 tan - 1 ( S 1 - S 2 S 2 - S 3 ) ,
S 1 = 0 T / 4 I 2 d t = T 4 I dc - I dc cos ( β ) ω + I dc sin ( 2 α ) sin ( β ) ω , S 2 = T / 4 2 T / 4 I 2 d t = T 4 I dc - I dc cos ( β ) ω - I dc sin ( 2 α ) sin ( β ) ω , S 3 = 2 T / 4 3 T / 4 I 2 d t = T 4 I d c + I d c cos ( β ) ω - I dc sin ( 2 α ) sin ( β ) ω .
β = tan - 1 ( sin ( β ) cos ( β ) ) = tan - 1 ( ( S 1 - S 2 ) / sin ( 2 α ) S 3 - S 2 ) .
I 1 = I dc , ε ( x , y ) ( 1 + sin β sin ( ω t + π 2 + 2 α ) ) = I dc , ε ( x , y ) + R 1 , ε ( x , y ) sin ( ω t + Φ 1 ) ,
I 2 = I dc , ε ( x , y ) - I dc , ε ( x , y ) cos ( β ) sin ω t + I dc , ε ( x , y ) sin ( 2 α ) sin ( β ) cos ω t .
β = tan - 1 ( ( S 1 - S 2 ) / sin ( 2 α ) - ( S 2 - S 3 ) ) = tan - 1 ( - tan Φ 2 sin 2 α ) ,
Δ β = - sin 2 α sin 2 2 α cos 2 Φ 2 + sin 2 Φ 2 Δ Φ 2 .

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