Abstract

A new heterodyne polariscope for sequential measurements of the complete optical parameters of linearly birefringent materials is proposed. A multiple-order crystalline quartz quarter-wave plate used as a sample was tested in two sequential setups. In the first setup we used an electro-optic modulator to modulate the circular heterodyne polariscope and then applied a phase-locking technique to measure the principal axis angle precisely. In the second setup, removing the first quarter-wave plate, resulted in a linear heterodyne polariscope, and again we used the phase-locking technique to extract the apparent retardance. Furthermore, by tilting the sample and placing a material of known thickness into the second setup, we determined the order, thickness, and refractive indices (ne  and  no) of a multiple-order wave plate by using the new algorithm. The proposed method has average absolute errors of 0.2167° and 0.15% with respect to the principal axis angle and the apparent retardance, respectively. The order, thickness, and refractive indices are also in good agreement with the known sample data. In contrast to the conventional measurement schemes that could not measure more than two parameters, the proposed heterodyne polariscope uniquely measures six parameters.

© 2006 Optical Society of America

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  1. H. B. Serreze and R. B. Goldner, "A phase-sensitive technique for measuring small birefringence changes," Rev. Sci. Instrum. 45, pp. 1613-1614 (1974).
    [CrossRef]
  2. Y. Shindo and H. Hanabusa, "Highly sensitive instrument for measuring optical birefringence," Polym. Commun. 24, 240-244 (1983).
  3. M. H. Chiu, C. D. Chen, and D. C. Su, "Method for determining the fast axis and phase retardation of a wave plate," J. Opt. Soc. Am. A 13, 1924-1929 (1996).
    [CrossRef]
  4. B. D. Cameron and G. L. Cóte, "Noninvasive glucose sensing utilizing a digital closed-loop polarimetric approach," IEEE Trans. Biomed. Eng. 44, 1221-1227 (1997).
    [CrossRef] [PubMed]
  5. A. Márquez, M. Yamauchi, J. A. Davis, and D. J. Franich, "Phase measurement of a twist nematic liquid crystal spatial light modulator with a common-path interferometer," Opt. Commun. 190, 129-133 (2001).
    [CrossRef]
  6. S. Ohkubo and N. Umeda, "Near-field scanning optical microscope based on fast birefringence measurements," Sens. Mater. 13, 433-443 (2001).
  7. Y. L. Lo and P. F. Hsu, "Birefringence measurements by an electro-optic modulator using a new heterodyne scheme," Opt. Eng. 41, 2764-2767 (2002).
    [CrossRef]
  8. R. P. Shukla, G. M. Perera, M. C. George, and P. Venkateswarlu, "Measurement of birefringence of optical materials using a wedged plate interferometer," Opt. Commun. 78, 7-12 (1990).
    [CrossRef]
  9. 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]
  10. Y. L. Lo, J. F. Lin, and S. Y. Lee, "Polariscope for the simultaneous measurements of the principal axis and phase retardation using two phase-locked extractions," Appl. Opt. 43, 6248-6254 (2004).
    [CrossRef] [PubMed]
  11. Y. C. Huang, C. Chou, and M. Chang, "Direct measurement of refractive indices of a linear birefringent retardation plate," Opt. Commun. 133, 11-16 (1997).
    [CrossRef]
  12. M. H. Chiu, J. Y. Lee, and D. C. Su, "Complex refractive-index measurement based on Fresnel's equations and the uses of heterodyne interferometry," Appl. Opt. 38, 4047-4052 (1999).
    [CrossRef]
  13. C. C. Hsu and D. C. Su, "Method for determining the optical axis and (ne, no) of birefringent crystal," Appl. Opt. 41, 3936-3940 (2002).
    [CrossRef] [PubMed]
  14. E. Hecht, Optics (Addison-Wesley, 2002), p. 343.
  15. X. Zhu, "Explicit Jones transformation matrix for a tilted birefringent plate with its optic axis parallel to the plate surface," Appl. Opt. 33, 3502-3506 (1994).
    [CrossRef] [PubMed]
  16. K. B. Rochford, A. H. Rose, and C. M. Wang, "NIST study investigates retardance uncertainty," Laser Focus World 33, 223-227 (1997).
  17. E. D. Palik, Handbook of Optical Constants of Solids III (Academic, 1998), p. 729.
  18. R. Guenther, Modern Optics (McGraw-Hill, 1995), p. 530.

2004 (2)

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, J. F. Lin, and S. Y. Lee, "Polariscope for the simultaneous measurements of the principal axis and phase retardation using two phase-locked extractions," Appl. Opt. 43, 6248-6254 (2004).
[CrossRef] [PubMed]

2002 (2)

C. C. Hsu and D. C. Su, "Method for determining the optical axis and (ne, no) of birefringent crystal," Appl. Opt. 41, 3936-3940 (2002).
[CrossRef] [PubMed]

Y. L. Lo and P. F. Hsu, "Birefringence measurements by an electro-optic modulator using a new heterodyne scheme," Opt. Eng. 41, 2764-2767 (2002).
[CrossRef]

2001 (2)

A. Márquez, M. Yamauchi, J. A. Davis, and D. J. Franich, "Phase measurement of a twist nematic liquid crystal spatial light modulator with a common-path interferometer," Opt. Commun. 190, 129-133 (2001).
[CrossRef]

S. Ohkubo and N. Umeda, "Near-field scanning optical microscope based on fast birefringence measurements," Sens. Mater. 13, 433-443 (2001).

1999 (1)

1997 (3)

Y. C. Huang, C. Chou, and M. Chang, "Direct measurement of refractive indices of a linear birefringent retardation plate," Opt. Commun. 133, 11-16 (1997).
[CrossRef]

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

B. D. Cameron and G. L. Cóte, "Noninvasive glucose sensing utilizing a digital closed-loop polarimetric approach," IEEE Trans. Biomed. Eng. 44, 1221-1227 (1997).
[CrossRef] [PubMed]

1996 (1)

1994 (1)

1990 (1)

R. P. Shukla, G. M. Perera, M. C. George, and P. Venkateswarlu, "Measurement of birefringence of optical materials using a wedged plate interferometer," Opt. Commun. 78, 7-12 (1990).
[CrossRef]

1983 (1)

Y. Shindo and H. Hanabusa, "Highly sensitive instrument for measuring optical birefringence," Polym. Commun. 24, 240-244 (1983).

1974 (1)

H. B. Serreze and R. B. Goldner, "A phase-sensitive technique for measuring small birefringence changes," Rev. Sci. Instrum. 45, pp. 1613-1614 (1974).
[CrossRef]

Cameron, B. D.

B. D. Cameron and G. L. Cóte, "Noninvasive glucose sensing utilizing a digital closed-loop polarimetric approach," IEEE Trans. Biomed. Eng. 44, 1221-1227 (1997).
[CrossRef] [PubMed]

Chang, M.

Y. C. Huang, C. Chou, and M. Chang, "Direct measurement of refractive indices of a linear birefringent retardation plate," Opt. Commun. 133, 11-16 (1997).
[CrossRef]

Chen, C. D.

Chiu, M. H.

Chou, C.

Y. C. Huang, C. Chou, and M. Chang, "Direct measurement of refractive indices of a linear birefringent retardation plate," Opt. Commun. 133, 11-16 (1997).
[CrossRef]

Cóte, G. L.

B. D. Cameron and G. L. Cóte, "Noninvasive glucose sensing utilizing a digital closed-loop polarimetric approach," IEEE Trans. Biomed. Eng. 44, 1221-1227 (1997).
[CrossRef] [PubMed]

Davis, J. A.

A. Márquez, M. Yamauchi, J. A. Davis, and D. J. Franich, "Phase measurement of a twist nematic liquid crystal spatial light modulator with a common-path interferometer," Opt. Commun. 190, 129-133 (2001).
[CrossRef]

Franich, D. J.

A. Márquez, M. Yamauchi, J. A. Davis, and D. J. Franich, "Phase measurement of a twist nematic liquid crystal spatial light modulator with a common-path interferometer," Opt. Commun. 190, 129-133 (2001).
[CrossRef]

George, M. C.

R. P. Shukla, G. M. Perera, M. C. George, and P. Venkateswarlu, "Measurement of birefringence of optical materials using a wedged plate interferometer," Opt. Commun. 78, 7-12 (1990).
[CrossRef]

Goldner, R. B.

H. B. Serreze and R. B. Goldner, "A phase-sensitive technique for measuring small birefringence changes," Rev. Sci. Instrum. 45, pp. 1613-1614 (1974).
[CrossRef]

Guenther, R.

R. Guenther, Modern Optics (McGraw-Hill, 1995), p. 530.

Hanabusa, H.

Y. Shindo and H. Hanabusa, "Highly sensitive instrument for measuring optical birefringence," Polym. Commun. 24, 240-244 (1983).

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 2002), p. 343.

Hsu, C. C.

Hsu, P. F.

Y. L. Lo and P. F. Hsu, "Birefringence measurements by an electro-optic modulator using a new heterodyne scheme," Opt. Eng. 41, 2764-2767 (2002).
[CrossRef]

Huang, Y. C.

Y. C. Huang, C. Chou, and M. Chang, "Direct measurement of refractive indices of a linear birefringent retardation plate," Opt. Commun. 133, 11-16 (1997).
[CrossRef]

Lee, J. Y.

Lee, S. Y.

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, J. F. Lin, and S. Y. Lee, "Polariscope for the simultaneous measurements of the principal axis and phase retardation using two phase-locked extractions," Appl. Opt. 43, 6248-6254 (2004).
[CrossRef] [PubMed]

Lin, J. F.

Y. L. Lo, J. F. Lin, and S. Y. Lee, "Polariscope for the simultaneous measurements of the principal axis and phase retardation using two phase-locked extractions," Appl. Opt. 43, 6248-6254 (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]

Lo, Y. L.

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, J. F. Lin, and S. Y. Lee, "Polariscope for the simultaneous measurements of the principal axis and phase retardation using two phase-locked extractions," Appl. Opt. 43, 6248-6254 (2004).
[CrossRef] [PubMed]

Y. L. Lo and P. F. Hsu, "Birefringence measurements by an electro-optic modulator using a new heterodyne scheme," Opt. Eng. 41, 2764-2767 (2002).
[CrossRef]

Márquez, A.

A. Márquez, M. Yamauchi, J. A. Davis, and D. J. Franich, "Phase measurement of a twist nematic liquid crystal spatial light modulator with a common-path interferometer," Opt. Commun. 190, 129-133 (2001).
[CrossRef]

Ohkubo, S.

S. Ohkubo and N. Umeda, "Near-field scanning optical microscope based on fast birefringence measurements," Sens. Mater. 13, 433-443 (2001).

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids III (Academic, 1998), p. 729.

Perera, G. M.

R. P. Shukla, G. M. Perera, M. C. George, and P. Venkateswarlu, "Measurement of birefringence of optical materials using a wedged plate interferometer," Opt. Commun. 78, 7-12 (1990).
[CrossRef]

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.

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

Serreze, H. B.

H. B. Serreze and R. B. Goldner, "A phase-sensitive technique for measuring small birefringence changes," Rev. Sci. Instrum. 45, pp. 1613-1614 (1974).
[CrossRef]

Shindo, Y.

Y. Shindo and H. Hanabusa, "Highly sensitive instrument for measuring optical birefringence," Polym. Commun. 24, 240-244 (1983).

Shukla, R. P.

R. P. Shukla, G. M. Perera, M. C. George, and P. Venkateswarlu, "Measurement of birefringence of optical materials using a wedged plate interferometer," Opt. Commun. 78, 7-12 (1990).
[CrossRef]

Su, D. C.

Umeda, N.

S. Ohkubo and N. Umeda, "Near-field scanning optical microscope based on fast birefringence measurements," Sens. Mater. 13, 433-443 (2001).

Venkateswarlu, P.

R. P. Shukla, G. M. Perera, M. C. George, and P. Venkateswarlu, "Measurement of birefringence of optical materials using a wedged plate interferometer," Opt. Commun. 78, 7-12 (1990).
[CrossRef]

Wang, C. M.

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

Yamauchi, M.

A. Márquez, M. Yamauchi, J. A. Davis, and D. J. Franich, "Phase measurement of a twist nematic liquid crystal spatial light modulator with a common-path interferometer," Opt. Commun. 190, 129-133 (2001).
[CrossRef]

Zhu, X.

Appl. Opt. (4)

IEEE Trans. Biomed. Eng. (1)

B. D. Cameron and G. L. Cóte, "Noninvasive glucose sensing utilizing a digital closed-loop polarimetric approach," IEEE Trans. Biomed. Eng. 44, 1221-1227 (1997).
[CrossRef] [PubMed]

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

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. (4)

A. Márquez, M. Yamauchi, J. A. Davis, and D. J. Franich, "Phase measurement of a twist nematic liquid crystal spatial light modulator with a common-path interferometer," Opt. Commun. 190, 129-133 (2001).
[CrossRef]

R. P. Shukla, G. M. Perera, M. C. George, and P. Venkateswarlu, "Measurement of birefringence of optical materials using a wedged plate interferometer," Opt. Commun. 78, 7-12 (1990).
[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. C. Huang, C. Chou, and M. Chang, "Direct measurement of refractive indices of a linear birefringent retardation plate," Opt. Commun. 133, 11-16 (1997).
[CrossRef]

Opt. Eng. (1)

Y. L. Lo and P. F. Hsu, "Birefringence measurements by an electro-optic modulator using a new heterodyne scheme," Opt. Eng. 41, 2764-2767 (2002).
[CrossRef]

Polym. Commun. (1)

Y. Shindo and H. Hanabusa, "Highly sensitive instrument for measuring optical birefringence," Polym. Commun. 24, 240-244 (1983).

Rev. Sci. Instrum. (1)

H. B. Serreze and R. B. Goldner, "A phase-sensitive technique for measuring small birefringence changes," Rev. Sci. Instrum. 45, pp. 1613-1614 (1974).
[CrossRef]

Sens. Mater. (1)

S. Ohkubo and N. Umeda, "Near-field scanning optical microscope based on fast birefringence measurements," Sens. Mater. 13, 433-443 (2001).

Other (3)

E. Hecht, Optics (Addison-Wesley, 2002), p. 343.

E. D. Palik, Handbook of Optical Constants of Solids III (Academic, 1998), p. 729.

R. Guenther, Modern Optics (McGraw-Hill, 1995), p. 530.

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Figures (9)

Fig. 1
Fig. 1

Setup for the phase-locked methodology.

Fig. 2
Fig. 2

Schematic diagram for tilting a sample where the principle axis is oriented in the y direction.

Fig. 3
Fig. 3

Relationship between cos θ i and cos θ o.

Fig. 4
Fig. 4

Simulation of phase retardation with the simplified and theoretical formulas when ϕ = 90°.

Fig. 5
Fig. 5

Setup for the phase-locked methodology.

Fig. 6
Fig. 6

Schematic diagram for tilting a sample where the principle axis is oriented in the x direction.

Fig. 7
Fig. 7

Simulation of phase retardation with the simplified and theoretical formulas when ϕ = 0°.

Fig. 8
Fig. 8

Experimental results for the principal-axis angle and the apparent retardance.

Fig. 9
Fig. 9

Flow chart; for measurements with the new heterodyne periscope.

Equations (23)

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E 1 = A ( 0 ° ) Q 2 ( 45 ° ) S ( α , β ) Q 1 ( 45 ° ) EO ( 90 ° , ω t ) P ( 45 ° ) E in = [ 1 0 0 0 ] 1 2 [ 1 i 1 i 1 i 1 i ] [ cos ( β / 2 ) + i cos ( 2 α ) sin ( β / 2 ) i sin ( 2 α ) sin ( β / 2 ) i sin ( 2 α ) sin ( β / 2 ) cos ( β / 2 ) i cos ( 2 α ) sin ( β / 2 ) ] 1 2 [ 1 - i 1 + i 1 + i 1 - i ] [ exp [ i ( ω t / 2 ) ] 0 0 exp [ i ( ω t / 2 ) ] ] [ 1 2 1 2 1 2 1 2 ] [ 0 E 0 ] exp ( i ω 0 t ) ,
I 1 = I dc [ 1 sin β sin ( ω t + π 2 2 α ) ] = I dc R 1   sin ( ω t + Φ 1 ) ,
E 2 = A ( 0 ° ) Q 2 ( 45 ° ) S ( α , β ) EO ( 90 ° , ω t ) P ( 45 ° ) E i n = [ 1 0 0 0 ] 1 2 [ 1 i 1 i 1 i 1 i ] [ cos ( β / 2 ) + i cos ( 2 α ) sin ( β / 2 ) i sin ( 2 α ) sin ( β / 2 ) i sin ( 2 α ) sin ( β / 2 ) cos ( β / 2 ) i cos ( 2 α ) sin ( β / 2 ) ] [ exp [ i ( ω t / 2 ) ] 0 0 exp [ i ( ω t / 2 ) ] ] [ 1 2 1 2 1 2 1 2 ] [ 0 E 0 ] exp ( i ω 0 t )
I 2 = I dc { 1 + [ ( sin β cos 2 α ) 2 + ( cos β ) 2 ] 1 / 2 sin [ ω t - tan 1 ( cos 2 α tan β ) ] }
= I dc + R 2 sin ( ω t Φ 2 ) ,
α = π 4 Φ 1 2 ,
β = tan 1 [ tan ( Φ 2 ) cos 2 α ] .
( β + 2 m π ) = 2 π λ ( n e n o ) d ,
β 1 + 2 m π = 2 π λ d ( n e n o ) [ 1 ( sin 2 θ 1 ) / n o     2 ] 1 / 2 { 1 sin 2 θ 1 cos 2 ϕ n e     2 [ 1 [ ( 1 / n o     2 ) ( 1 / n e     2 ) ] sin 2 θ 1 cos 2 ϕ ] } ,
β 2 + 2 m π = 2 π λ d ( n e n o ) [ 1 ( sin 2 θ 2 ) / ( n o     2 ) ] 1 / 2 { 1 sin 2 θ 2 cos 2 ϕ n e     2 [ 1 [ ( 1 / n o     2 ) ( 1 / n e     2 ) ] sin 2 θ 2 cos 2 ϕ ] } .
cos θ o = ( 1 sin 2 θ 1 , 2 n o     2 ) 1 / 2 ,
( β 1 + 2 m π ) = 2 π λ ( n e n o ) d / cos θ 1 ,
( β 2 + 2 m π ) = 2 π λ ( n e n o ) d / cos θ 2 .
m = β 1 β 2 ( 1 / cos θ 1 1 / cos θ 2 ) β 2 π ( 1 / cos θ 1 1 / cos θ 2 ) .
I 3 = I dc + R 3   sin ( ω t Φ 3 ) ,
I 4 = I dc + R 4   sin ( ω t Φ 4 ) ,
( β + 2 m π + β 1 + 2 m 1 π ) = 2 π λ ( n e n o ) ( d + d 1 ) ,
( β + 2 m π + β 2 + 2 m 2 π ) = 2 π λ ( n e n o ) ( d + d 2 ) ,
( n e n o ) = λ 2 π ( d 1 d 2 ) ( β 1 β 2 ) .
d = ( β + 2 m π ) λ 2 π ( n e n o ) .
n eff ( θ ) = [ ( n e n o ) 2 ( n e sin θ ) 2 + ( n o cos θ ) 2 ] 1 / 2 ,
β″ + 2 m π = 2 π λ d [ n eff ( θ i ) n ο ] [ 1 ( sin 2 θ i ) / ( n o     2 ) ] 1 / 2 { 1 sin 2 θ i cos 2 ϕ n e     2 [ 1 ( 1 / n o     2 ) ( 1 / n e     2 ) ] sin 2 θ i cos 2 ϕ } .
( β″ + 2 m π ) = 2 π λ [ n eff ( θ ) n o ] d / cos θ .

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