Abstract

We develop a more general methodology for a dual photoelastic modulator (PEM) system that is used for the determination of the Stokes parameters of an arbitrary light beam. This allows for a degree of arbitrariness in the setting of the retardation amplitudes of both PEMs, thus permitting a robust and effective optimization of the detection system. Various experimental issues are considered and a calibration procedure is described that ensures accurate measurement of the absolute values of the Stokes parameters. Measurements of the Faraday rotation of a CoPt multilayer film are provided as a demonstration of the sensitivity of the dual-PEM system.

© 2010 Optical Society of America

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  1. M. Billardon and J. Badoz, “Modulateur de birefringence,” C. R. Acad. Sci. Ser. B 262, 1672–1675 (1966).
  2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  3. D. H. Goldstein and E. Collett, Polarized Light, 2nd ed.(Marcel Dekker, 2003).
    [CrossRef]
  4. D. J. Diner, A. Davis, B. Hancock, G. Gutt, R. A. Chipman, and B. Cairns, “Dual-photoelastic-modulator-based polarimetric imaging concept for aerosol remote sensing,” Appl. Opt. 46, 8428–8445 (2007).
    [CrossRef] [PubMed]
  5. B. L. Wang, J. List, and R. Rockwell, “A Stokes polarimeter using two photoelastic modulators,” Proc. SPIE 4819, 1–8(2002).
    [CrossRef]
  6. G. E. Jellison and F. A. Modine, “Two-modulator generalized ellipsometry: theory,” Appl. Opt. 36, 8190–8198 (1997).
    [CrossRef]
  7. B. L. Wang, “Linear birefringence measurement instrument using two photoelastic modulators,” Opt. Eng. 41, 981–987(2002).
    [CrossRef]
  8. M. Kuldkepp, N. C. Hawkes, E. Rachlew, and B. Schunke, “Accurate polarization measurements with a dual photoelastic modulator,” Appl. Opt. 44, 5899–5904 (2005).
    [CrossRef] [PubMed]
  9. G. R. Boyer, B. F. Lamouroux, and B. S. Prade, “Automatic-measurement of the Stokes vector of light,” Appl. Opt. 18, 1217–1219 (1979).
    [CrossRef] [PubMed]
  10. E. Compain and B. Drevillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
    [CrossRef]
  11. Y. W. Liu, G. A. Jones, Y. Peng, and T. H. Shen, “Generalized theory and application of Stokes parameter measurements made with a single photoelastic modulator,” J. Appl. Phys. 100063537 (2006).
    [CrossRef]
  12. M. Mujat and A. Dogariu, “Real-time measurement of the polarization transfer function,” Appl. Opt. 40, 34–44 (2001).
    [CrossRef]
  13. B. L. Wang, “Measurement of circular and linear birefringence in chiral media and optical materials using the photoelastic modulator,” Proc. SPIE 3535, 294–302 (1999).
    [CrossRef]
  14. S. M. Jordan and J. S. S. Whiting, “Detecting two components of magnetization in magnetic layer structures by use of a photoelastic modulator,” Rev. Sci. Instrum. 67, 4286–4289(1996).
    [CrossRef]
  15. Y. Shindo, “Application of polarized modulation technique in polymer science,” Opt. Eng. 34, 3369–3384 (1995).
    [CrossRef]
  16. J. C. Canit and J. Badoz, “Photoelastic modulator for polarimetry and ellipsometry,” Appl. Opt. 23, 2861–2862(1984).
    [CrossRef] [PubMed]
  17. K. W. Hipps and G. A. Crosby, “Applications of the photo-elastic modulator to polarization spectroscopy,” J. Phys. Chem. 83, 555–562 (1979).
    [CrossRef]
  18. C. F. Wong, “Birefringence measurement using a photo-elastic modulator,” Appl. Opt. 18, 3996–3999 (1979).
    [CrossRef] [PubMed]
  19. F. A. Modine, G. E. Jellison, and G. R. Gruzalski, “Errors in ellipsometry measurements made with a photo-elastic modulator,” J. Opt. Soc. Am. 73, 892–900 (1983).
    [CrossRef]
  20. J. Badoz, M. P. Silverman, and J. C. Canit, “Wave-propagation through a medium with static and dynamic birefringence—theory of the photoelastic modulator,” J. Opt. Soc. Am. A 7, 672–682 (1990).
    [CrossRef]
  21. J. C. Kemp, G. D. Henson, C. T. Steiner, and E. R. Powell, “The optical polarization of the sun measured at a sensitivity of parts in 10-million,” Nature 326, 270–273 (1987).
    [CrossRef]
  22. S. N. Jasperson and S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
    [CrossRef]
  23. W. Guan, G. A. Jones, Y. W. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys. 103, 043104 (2008).
    [CrossRef]
  24. J. C. Kemp, “Piezo-optical birefringence modulators—new use for a long-known effect,” J. Opt. Soc. Am. 59, 950–954 (1969).

2008 (1)

W. Guan, G. A. Jones, Y. W. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys. 103, 043104 (2008).
[CrossRef]

2007 (1)

2006 (1)

Y. W. Liu, G. A. Jones, Y. Peng, and T. H. Shen, “Generalized theory and application of Stokes parameter measurements made with a single photoelastic modulator,” J. Appl. Phys. 100063537 (2006).
[CrossRef]

2005 (1)

2003 (1)

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

2002 (2)

B. L. Wang, “Linear birefringence measurement instrument using two photoelastic modulators,” Opt. Eng. 41, 981–987(2002).
[CrossRef]

B. L. Wang, J. List, and R. Rockwell, “A Stokes polarimeter using two photoelastic modulators,” Proc. SPIE 4819, 1–8(2002).
[CrossRef]

2001 (1)

1999 (2)

B. L. Wang, “Measurement of circular and linear birefringence in chiral media and optical materials using the photoelastic modulator,” Proc. SPIE 3535, 294–302 (1999).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

1998 (1)

E. Compain and B. Drevillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

1997 (1)

1996 (1)

S. M. Jordan and J. S. S. Whiting, “Detecting two components of magnetization in magnetic layer structures by use of a photoelastic modulator,” Rev. Sci. Instrum. 67, 4286–4289(1996).
[CrossRef]

1995 (1)

Y. Shindo, “Application of polarized modulation technique in polymer science,” Opt. Eng. 34, 3369–3384 (1995).
[CrossRef]

1990 (1)

1987 (1)

J. C. Kemp, G. D. Henson, C. T. Steiner, and E. R. Powell, “The optical polarization of the sun measured at a sensitivity of parts in 10-million,” Nature 326, 270–273 (1987).
[CrossRef]

1984 (1)

1983 (1)

1979 (3)

1969 (2)

S. N. Jasperson and S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

J. C. Kemp, “Piezo-optical birefringence modulators—new use for a long-known effect,” J. Opt. Soc. Am. 59, 950–954 (1969).

1966 (1)

M. Billardon and J. Badoz, “Modulateur de birefringence,” C. R. Acad. Sci. Ser. B 262, 1672–1675 (1966).

Badoz, J.

Billardon, M.

M. Billardon and J. Badoz, “Modulateur de birefringence,” C. R. Acad. Sci. Ser. B 262, 1672–1675 (1966).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Boyer, G. R.

Cairns, B.

Canit, J. C.

Chipman, R. A.

Collett, E.

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

Compain, E.

E. Compain and B. Drevillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

Crosby, G. A.

K. W. Hipps and G. A. Crosby, “Applications of the photo-elastic modulator to polarization spectroscopy,” J. Phys. Chem. 83, 555–562 (1979).
[CrossRef]

Davis, A.

Diner, D. J.

Dogariu, A.

Drevillon, B.

E. Compain and B. Drevillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

Goldstein, D. H.

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

Gruzalski, G. R.

Guan, W.

W. Guan, G. A. Jones, Y. W. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys. 103, 043104 (2008).
[CrossRef]

Gutt, G.

Hancock, B.

Hawkes, N. C.

Henson, G. D.

J. C. Kemp, G. D. Henson, C. T. Steiner, and E. R. Powell, “The optical polarization of the sun measured at a sensitivity of parts in 10-million,” Nature 326, 270–273 (1987).
[CrossRef]

Hipps, K. W.

K. W. Hipps and G. A. Crosby, “Applications of the photo-elastic modulator to polarization spectroscopy,” J. Phys. Chem. 83, 555–562 (1979).
[CrossRef]

Jasperson, S. N.

S. N. Jasperson and S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

Jellison, G. E.

Jones, G. A.

W. Guan, G. A. Jones, Y. W. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys. 103, 043104 (2008).
[CrossRef]

Y. W. Liu, G. A. Jones, Y. Peng, and T. H. Shen, “Generalized theory and application of Stokes parameter measurements made with a single photoelastic modulator,” J. Appl. Phys. 100063537 (2006).
[CrossRef]

Jordan, S. M.

S. M. Jordan and J. S. S. Whiting, “Detecting two components of magnetization in magnetic layer structures by use of a photoelastic modulator,” Rev. Sci. Instrum. 67, 4286–4289(1996).
[CrossRef]

Kemp, J. C.

J. C. Kemp, G. D. Henson, C. T. Steiner, and E. R. Powell, “The optical polarization of the sun measured at a sensitivity of parts in 10-million,” Nature 326, 270–273 (1987).
[CrossRef]

J. C. Kemp, “Piezo-optical birefringence modulators—new use for a long-known effect,” J. Opt. Soc. Am. 59, 950–954 (1969).

Kuldkepp, M.

Lamouroux, B. F.

List, J.

B. L. Wang, J. List, and R. Rockwell, “A Stokes polarimeter using two photoelastic modulators,” Proc. SPIE 4819, 1–8(2002).
[CrossRef]

Liu, Y. W.

W. Guan, G. A. Jones, Y. W. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys. 103, 043104 (2008).
[CrossRef]

Y. W. Liu, G. A. Jones, Y. Peng, and T. H. Shen, “Generalized theory and application of Stokes parameter measurements made with a single photoelastic modulator,” J. Appl. Phys. 100063537 (2006).
[CrossRef]

Modine, F. A.

Mujat, M.

Peng, Y.

Y. W. Liu, G. A. Jones, Y. Peng, and T. H. Shen, “Generalized theory and application of Stokes parameter measurements made with a single photoelastic modulator,” J. Appl. Phys. 100063537 (2006).
[CrossRef]

Powell, E. R.

J. C. Kemp, G. D. Henson, C. T. Steiner, and E. R. Powell, “The optical polarization of the sun measured at a sensitivity of parts in 10-million,” Nature 326, 270–273 (1987).
[CrossRef]

Prade, B. S.

Rachlew, E.

Rockwell, R.

B. L. Wang, J. List, and R. Rockwell, “A Stokes polarimeter using two photoelastic modulators,” Proc. SPIE 4819, 1–8(2002).
[CrossRef]

Schnatterly, S. E.

S. N. Jasperson and S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

Schunke, B.

Shen, T. H.

W. Guan, G. A. Jones, Y. W. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys. 103, 043104 (2008).
[CrossRef]

Y. W. Liu, G. A. Jones, Y. Peng, and T. H. Shen, “Generalized theory and application of Stokes parameter measurements made with a single photoelastic modulator,” J. Appl. Phys. 100063537 (2006).
[CrossRef]

Shindo, Y.

Y. Shindo, “Application of polarized modulation technique in polymer science,” Opt. Eng. 34, 3369–3384 (1995).
[CrossRef]

Silverman, M. P.

Steiner, C. T.

J. C. Kemp, G. D. Henson, C. T. Steiner, and E. R. Powell, “The optical polarization of the sun measured at a sensitivity of parts in 10-million,” Nature 326, 270–273 (1987).
[CrossRef]

Wang, B. L.

B. L. Wang, “Linear birefringence measurement instrument using two photoelastic modulators,” Opt. Eng. 41, 981–987(2002).
[CrossRef]

B. L. Wang, J. List, and R. Rockwell, “A Stokes polarimeter using two photoelastic modulators,” Proc. SPIE 4819, 1–8(2002).
[CrossRef]

B. L. Wang, “Measurement of circular and linear birefringence in chiral media and optical materials using the photoelastic modulator,” Proc. SPIE 3535, 294–302 (1999).
[CrossRef]

Whiting, J. S. S.

S. M. Jordan and J. S. S. Whiting, “Detecting two components of magnetization in magnetic layer structures by use of a photoelastic modulator,” Rev. Sci. Instrum. 67, 4286–4289(1996).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Wong, C. F.

Appl. Opt. (7)

C. R. Acad. Sci. Ser. B (1)

M. Billardon and J. Badoz, “Modulateur de birefringence,” C. R. Acad. Sci. Ser. B 262, 1672–1675 (1966).

J. Appl. Phys. (2)

Y. W. Liu, G. A. Jones, Y. Peng, and T. H. Shen, “Generalized theory and application of Stokes parameter measurements made with a single photoelastic modulator,” J. Appl. Phys. 100063537 (2006).
[CrossRef]

W. Guan, G. A. Jones, Y. W. Liu, and T. H. Shen, “The measurement of the Stokes parameters: a generalized methodology using a dual photoelastic modulator system,” J. Appl. Phys. 103, 043104 (2008).
[CrossRef]

J. Opt. Soc. Am. (2)

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

J. Phys. Chem. (1)

K. W. Hipps and G. A. Crosby, “Applications of the photo-elastic modulator to polarization spectroscopy,” J. Phys. Chem. 83, 555–562 (1979).
[CrossRef]

Nature (1)

J. C. Kemp, G. D. Henson, C. T. Steiner, and E. R. Powell, “The optical polarization of the sun measured at a sensitivity of parts in 10-million,” Nature 326, 270–273 (1987).
[CrossRef]

Opt. Eng. (2)

Y. Shindo, “Application of polarized modulation technique in polymer science,” Opt. Eng. 34, 3369–3384 (1995).
[CrossRef]

B. L. Wang, “Linear birefringence measurement instrument using two photoelastic modulators,” Opt. Eng. 41, 981–987(2002).
[CrossRef]

Proc. SPIE (2)

B. L. Wang, J. List, and R. Rockwell, “A Stokes polarimeter using two photoelastic modulators,” Proc. SPIE 4819, 1–8(2002).
[CrossRef]

B. L. Wang, “Measurement of circular and linear birefringence in chiral media and optical materials using the photoelastic modulator,” Proc. SPIE 3535, 294–302 (1999).
[CrossRef]

Rev. Sci. Instrum. (3)

S. M. Jordan and J. S. S. Whiting, “Detecting two components of magnetization in magnetic layer structures by use of a photoelastic modulator,” Rev. Sci. Instrum. 67, 4286–4289(1996).
[CrossRef]

S. N. Jasperson and S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969).
[CrossRef]

E. Compain and B. Drevillon, “High-frequency modulation of the four states of polarization of light with a single phase modulator,” Rev. Sci. Instrum. 69, 1574–1580 (1998).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

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

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the dual-PEM-based Stokes polarimeter.

Fig. 2
Fig. 2

Calibration results for J 0 ( δ 10 ) = J 0 ( δ 20 ) = 0 . (a) Angle α is set at 45 ° and β is set at two arbitrary angles for curves (1) and (2). (b) Angle β is set at 45 ° and α is set at 30 ° and 60 ° for curves (3) and (4), respectively.

Fig. 3
Fig. 3

Normalized “theoretical” signals derived from Eq. (1) plotted as a function of Δ 10 and Δ 20 for an arbitrary polarization state.

Fig. 4
Fig. 4

Tests for consistency for Δ 10 = Δ 20 = 0.486 λ . (a) Calculated value of k 8 / k 1 as a function of ellipticity angle. (b) Plot showing the experimental validity of Eq. (6).

Fig. 5
Fig. 5

Comparison of experimental validity in terms of Eq. (6) for (a) setting Δ 10 and Δ 20 to the nominal value of 0.383 λ and (b) ensuring the value of the zero-order Bessel function to be zero.

Fig. 6
Fig. 6

(a) Reduced Stokes parameters, (b) Faraday rotation angle, and (c) ellipticity angle of a CoPt multilayer measured with the calibrated dual-PEM system.

Tables (3)

Tables Icon

Table 1 Four Special Settings of the Polarizer Used to Determine the Relationships among k 1 , k 2 , k 3 , k 4 , and k 7

Tables Icon

Table 2 Normalized Calibration Constants Determined over Five Independent Sets of Measurements a

Tables Icon

Table 3 Comparison of Mean and Maximum Errors in the Degree and Angle of Polarization

Equations (10)

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I = 1 2 I + 1 2 Q cos 2 ( 2 α ) cos ( 2 β ) + 1 4 U sin ( 4 α ) cos ( 2 β ) + 1 2 sin ( 2 α ) cos ( 2 β ) [ Q sin ( 2 α ) U cos ( 2 α ) ] J 0 ( δ 10 ) + 1 2 sin ( 2 α ) sin ( 2 β ) [ Q cos ( 2 α ) + U sin ( 2 α ) ] J 0 ( δ 20 ) 1 2 cos ( 2 α ) sin ( 2 β ) [ Q sin ( 2 α ) U cos ( 2 α ) ] J 0 ( δ 10 ) J 0 ( δ 20 ) + { sin ( 2 α ) sin ( 2 β ) [ 1 J 0 ( δ 10 ) ] [ Q cos ( 2 α ) + U sin ( 2 α ) ] + U sin ( 2 β ) J 0 ( δ 10 ) } n J n ( δ 20 ) cos ( n Ω 2 t ) + [ sin ( 2 α ) cos ( 2 β ) cos ( 2 α ) sin ( 2 β ) J 0 ( δ 20 ) ] [ Q sin ( 2 α ) U cos ( 2 α ) ] n J n ( δ 10 ) cos ( n Ω 1 t ) sin ( 2 β ) [ Q sin ( 2 α ) U cos ( 2 α ) ] m 1 m 2 J m 1 ( δ 10 ) J m 2 ( δ 20 ) cos [ ( m 1 Ω 1 ± m 2 Ω 2 ) t ] cos ( 2 α ) sin ( 2 β ) [ Q sin ( 2 α ) U cos ( 2 α ) ] n 1 n 2 J n 1 ( δ 10 ) J n 2 ( δ 20 ) cos [ ( n 1 Ω 1 ± n 2 Ω 2 ) t ] + V sin ( 2 β ) J 0 ( δ 10 ) m J m ( δ 20 ) sin ( m Ω 2 t ) + V [ cos ( 2 α ) sin ( 2 β ) J 0 ( δ 20 ) sin ( 2 α ) cos ( 2 β ) ] m J m ( δ 10 ) sin ( m Ω 1 t ) + V cos ( 2 α ) sin ( 2 β ) m n J m ( δ 10 ) J n ( δ 20 ) sin [ ( m Ω 1 ± n Ω 2 ) t ] ± V sin ( 2 β ) n m J n ( δ 10 ) J m ( δ 20 ) sin [ ( n Ω 1 ± m Ω 2 ) t ] ,
I dc = v 1 I + v 2 Q + v 3 U , I Q U 1 = v 4 Q + v 5 U , I Q U 2 = v 6 Q + v 7 U , I V = v 8 V .
I dc = c dc S dc , I Q U 1 = c Q U 1 S Q U 1 , I Q U 2 = c Q U 2 S Q U 2 , I V = c V S V ,
( S dc S Q U 1 S Q U 2 S V ) = ( g 1 g 2 g 3 0 0 g 4 g 5 0 0 g 6 g 7 0 0 0 0 g 8 ) ( I Q U V ) = G ( I Q U V ) .
( I Q U V ) = ( k 1 k 2 k 3 0 0 k 4 k 5 0 0 k 6 k 7 0 0 0 0 k 8 ) ( S dc S Q U 1 S Q U 2 S V ) = K ( S dc S Q U 1 S Q U 2 S V ) ,
I 2 = Q 2 + U 2 + V 2 .
J 0 ( δ 10 ) = J 0 ( δ 20 ) = 0
k 2 k 1 = a + c 2 , k 3 k 1 = b + d 2 , k 4 k 1 = a c 2 , k 7 k 1 = b d 2 .
χ = 1 2 sin 1 ( V Q 2 + U 2 + V 2 ) .
Ψ = 1 2 tan 1 ( U / Q ) .

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