Abstract

Malus’s law, when used to calculate the attenuation ratio of the combination of two imperfect polarizers (two-CIP), will introduce an error, especially near the crossed-axis orientation. In this paper, first, the Jones matrix of the imperfect polarizer is deduced and an exact algorithm of the attenuation ratio of the two-CIP is proposed as well as its monotonic attenuation interval. Experimental results confirm that our deduced expression is more accurate than Malus’s law. Then based on this algorithm, an attenuation-ratio expression of the combination of three imperfect polarizers (three-CIP) is presented. In this three-CIP model, it is found that when the electric field amplitude ratio of the imperfect polarizer is ϵ, the attenuation ratio can change from 1 to ϵ4 monotonically in a general model when P1 and P3 are rotated and P2 is fixed, which is proved by experiment. Finally, it is deduced that the combination of n imperfect polarizers (n-CIP) can obtain a minimum attenuation ratio of ϵ2(n1), which indicates the number of imperfect polarizers needed to achieve the required attenuation ratio.

© 2010 Optical Society of America

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References

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  1. M. Wu, C. Yang, X. Mao, X. Zhao, and B. Cai, “Novel MEMS variable optical attenuator,” Chin. Opt. Lett. 1, 139–141 (2003).
  2. A. Ioffe, “High-efficiency transmission neutron polarizer for high-resolution double crystal diffractometer,” Physica B 234–236, 1071–1073 (1997).
    [CrossRef]
  3. P. Böni, “Applications of remanent supermirror polarizers,” Physica B 267–268, 320–327 (1999).
    [CrossRef]
  4. T. Nakano, K. Baba, and M. Miyagi, “Insertion loss and extinction ratio of a surface plasmon-polariton polarizer: theoretical analysis,” J. Opt. Soc. Am. B 11, 2030–2035 (1994).
    [CrossRef]
  5. L. Pierantoni, “Analysis of an optical polarizer based on laminated-cover waveguide,” J. Lightwave Technol. 24, 1414–1424 (2006).
    [CrossRef]
  6. S.-M. F. Nee, “Characterization for imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 54–64 (1998).
    [CrossRef]
  7. S.-M. F. Nee, “Errors of Mueller matrix measurements with a partially polarized light source,” Appl. Opt. 45, 6497–6506 (2006).
    [CrossRef] [PubMed]
  8. S.-M. F. Nee, “Error analysis for Mueller matrix measurement,” J. Opt. Soc. Am. A 20, 1651–1657 (2003).
    [CrossRef]
  9. S.-M. F. Nee, T. Cole, C. Yoo, and D. Burge, “Characterization of infrared polarizers,” Proc. SPIE 3121, 213–224 (1997).
    [CrossRef]
  10. T. Sergan, M. Lavrentovich, J. Kelly, E. Gardner, and D. Hansen, “Measurement and modeling of optical performance of wire grids and liquid-crystal displays utilizing grid polarizers,” J. Opt. Soc. Am. A 19, 1872–1885 (2002).
    [CrossRef]
  11. X. Cai and F. Xu, “An combination of a half -wave plate and polarizers for precision controlled attenuator,” Chin. Laser A26, 47–51 (1999).
  12. G. E. Jellison, Jr., C. O. Griffiths, D. E. Holcomb, and C. M. Rouleau, “Transmission two-modulator generalized ellipsometry measurements,” Appl. Opt. 41, 6555–6566 (2002).
    [CrossRef] [PubMed]
  13. S. Bertucci, “Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).
    [CrossRef]
  14. S. F. Li, “Jones-matrix analysis with Pauli matrices: application to ellipsometry,” J. Opt. Soc. Am. A 17, 920–926 (2000).
    [CrossRef]
  15. S. Donati, “High-extinction coiled-fiber polarizers by careful control of interface reinjection,” IEEE Photonics Technol. Lett. 7, 1174–1176 (1995).
    [CrossRef]
  16. R. J. Brecha and L. M. Pedrotti, “Analysis of imperfect polarizer effects in magnetic rotation spectroscopy,” Opt. Express 5, 101–113 (1999).
    [CrossRef] [PubMed]

2006 (2)

2003 (2)

2002 (2)

2000 (1)

1999 (3)

R. J. Brecha and L. M. Pedrotti, “Analysis of imperfect polarizer effects in magnetic rotation spectroscopy,” Opt. Express 5, 101–113 (1999).
[CrossRef] [PubMed]

X. Cai and F. Xu, “An combination of a half -wave plate and polarizers for precision controlled attenuator,” Chin. Laser A26, 47–51 (1999).

P. Böni, “Applications of remanent supermirror polarizers,” Physica B 267–268, 320–327 (1999).
[CrossRef]

1998 (2)

S.-M. F. Nee, “Characterization for imperfect polarizers under imperfect conditions,” Appl. Opt. 37, 54–64 (1998).
[CrossRef]

S. Bertucci, “Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).
[CrossRef]

1997 (2)

S.-M. F. Nee, T. Cole, C. Yoo, and D. Burge, “Characterization of infrared polarizers,” Proc. SPIE 3121, 213–224 (1997).
[CrossRef]

A. Ioffe, “High-efficiency transmission neutron polarizer for high-resolution double crystal diffractometer,” Physica B 234–236, 1071–1073 (1997).
[CrossRef]

1995 (1)

S. Donati, “High-extinction coiled-fiber polarizers by careful control of interface reinjection,” IEEE Photonics Technol. Lett. 7, 1174–1176 (1995).
[CrossRef]

1994 (1)

Baba, K.

Bertucci, S.

S. Bertucci, “Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).
[CrossRef]

Böni, P.

P. Böni, “Applications of remanent supermirror polarizers,” Physica B 267–268, 320–327 (1999).
[CrossRef]

Brecha, R. J.

Burge, D.

S.-M. F. Nee, T. Cole, C. Yoo, and D. Burge, “Characterization of infrared polarizers,” Proc. SPIE 3121, 213–224 (1997).
[CrossRef]

Cai, B.

Cai, X.

X. Cai and F. Xu, “An combination of a half -wave plate and polarizers for precision controlled attenuator,” Chin. Laser A26, 47–51 (1999).

Cole, T.

S.-M. F. Nee, T. Cole, C. Yoo, and D. Burge, “Characterization of infrared polarizers,” Proc. SPIE 3121, 213–224 (1997).
[CrossRef]

Donati, S.

S. Donati, “High-extinction coiled-fiber polarizers by careful control of interface reinjection,” IEEE Photonics Technol. Lett. 7, 1174–1176 (1995).
[CrossRef]

Gardner, E.

Griffiths, C. O.

Hansen, D.

Holcomb, D. E.

Ioffe, A.

A. Ioffe, “High-efficiency transmission neutron polarizer for high-resolution double crystal diffractometer,” Physica B 234–236, 1071–1073 (1997).
[CrossRef]

Jellison, G. E.

Kelly, J.

Lavrentovich, M.

Li, S. F.

Mao, X.

Miyagi, M.

Nakano, T.

Nee, S.-M. F.

Pedrotti, L. M.

Pierantoni, L.

Rouleau, C. M.

Sergan, T.

Wu, M.

Xu, F.

X. Cai and F. Xu, “An combination of a half -wave plate and polarizers for precision controlled attenuator,” Chin. Laser A26, 47–51 (1999).

Yang, C.

Yoo, C.

S.-M. F. Nee, T. Cole, C. Yoo, and D. Burge, “Characterization of infrared polarizers,” Proc. SPIE 3121, 213–224 (1997).
[CrossRef]

Zhao, X.

Appl. Opt. (3)

Chin. Laser (1)

X. Cai and F. Xu, “An combination of a half -wave plate and polarizers for precision controlled attenuator,” Chin. Laser A26, 47–51 (1999).

Chin. Opt. Lett. (1)

IEEE Photonics Technol. Lett. (1)

S. Donati, “High-extinction coiled-fiber polarizers by careful control of interface reinjection,” IEEE Photonics Technol. Lett. 7, 1174–1176 (1995).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (3)

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

Opt. Express (1)

Physica B (2)

A. Ioffe, “High-efficiency transmission neutron polarizer for high-resolution double crystal diffractometer,” Physica B 234–236, 1071–1073 (1997).
[CrossRef]

P. Böni, “Applications of remanent supermirror polarizers,” Physica B 267–268, 320–327 (1999).
[CrossRef]

Proc. SPIE (1)

S.-M. F. Nee, T. Cole, C. Yoo, and D. Burge, “Characterization of infrared polarizers,” Proc. SPIE 3121, 213–224 (1997).
[CrossRef]

Thin Solid Films (1)

S. Bertucci, “Systematic errors in fixed polarizer, rotating polarizer, sample, fixed analyzer spectroscopic ellipsometry,” Thin Solid Films 313–314, 73–78 (1998).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

CIP model of two or three polarizers. P 1 , P 2 , and P 3 represent the polarizers aligned along the optical axis. U is the Jones matrix of the imperfect polarizer. E 1 , E 2 , and E 3 represent electric vectors after light passes through P 1 , P 2 , and P 3 . Light intensity after P 2 and P 3 is symbolized as I 2 and I 3 , whose values are determined by the variables enclosed in parentheses. θ 1 and θ 2 represent the angle enclosed by the transmission axes of the two polarizers. In the two-CIP, θ 1 , U 1 ( θ 1 , ϵ ) , E 2 ( θ 1 , ϵ ) , I 2 ( θ 1 , ϵ ) are simply expressed as θ, U, E 2 , I 2 .

Fig. 2
Fig. 2

Coordinate system of the imperfect polarizer’s electric vector. The x axis represents the transmission axis of P 1 , while y axis represents that of P 2 .

Fig. 3
Fig. 3

(a) Function surface I 2 θ . (b) The section plane of (a) at the place I 2 θ = 0 . When ϵ is in the range [ 10 5 2 , 10 1 ] , the zero value of I 2 θ is located near θ = 0 ° , and θ = 90 ° .

Fig. 4
Fig. 4

Deviation σ between T and the simplified attenuation ratio T when ϵ = 10 a and θ [ 0 ° , 360 ° ] .

Fig. 5
Fig. 5

When ϵ = 10 5 2 , the curve of output light intensity I 2 of the two-CIP.

Fig. 6
Fig. 6

If ϵ is a nonzero value, I 2 cannot reach the extrema at θ = 0 ° or θ = 90 ° . When ϵ = 10 5 2 , (a) shows one of the zero value of I 2 θ near θ = 0 ° ; (b) shows another near θ = 90 ° .

Fig. 7
Fig. 7

Deviation of the attenuation ratio between imperfect and perfect polarizers σ. (a) Theorical curve when ϵ = 10 5 2 and the experimental data. (b) Relation between σ and ϵ with different values of θ.

Fig. 8
Fig. 8

General form of the three-CIP.

Fig. 9
Fig. 9

Attenuation ratio of the three-CIP, with θ 1 and θ 2 as variables simultaneously.

Fig. 10
Fig. 10

Curve of I 3 ( θ 1 , θ 1 , 3 θ 1 , 10 5 2 ) relative to θ 1 when θ 1 , 3 = arctan ϵ , 30°, 60° and 90 ° + arctan ϵ .

Fig. 11
Fig. 11

When θ 1 , 3 = arctan 10 5 2 , the curve of (a) I 3 ( θ 1 , arctan ϵ θ 1 , 10 5 2 ) relative to θ 1 ; (b) I 3 ( θ 1 , arctan ϵ θ 1 , 10 5 2 ) θ 1 relative to θ 1 .

Tables (3)

Tables Icon

Table 1 Extremum Values of I 3 ( θ 1 , θ 1 , 3 θ 1 , 10 5 2 ) for Different θ 1 , 3

Tables Icon

Table 2 Extrema of I 3 ( θ 1 , arc tan 10 5 2 θ 1 , 10 5 2 )

Tables Icon

Table 3 Experimental Data with Three-CIP

Equations (30)

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A ξ = A 1 cos θ + B 1 sin θ ,
B η = ( A 1 sin θ + B 1 cos θ ) ϵ .
A 2 = A ξ cos θ B η sin θ ,
B 2 = A ξ sin θ + B η cos θ .
[ A 2 B 2 ] = [ cos θ sin θ sin θ cos θ ] [ cos θ sin θ ϵ sin θ ϵ cos θ ] [ A 1 B 1 ] = U [ A 1 B 1 ] .
U = [ cos 2 θ + ϵ sin 2 θ ( 1 ϵ ) sin θ cos θ ( 1 ϵ ) sin θ cos θ sin 2 θ + ϵ cos 2 θ ] .
E 1 = [ 1 ϵ ] .
E 2 = U E 1 = [ cos 2 θ + ϵ sin 2 θ + ϵ ( cos θ sin θ ϵ sin θ cos θ ) ( 1 ϵ ) cos θ sin θ + ϵ ( sin 2 θ + ϵ cos 2 θ ) ] .
I 2 = | E 2 | 2 = [ cos 2 θ + ϵ sin 2 θ + ϵ ( 1 ϵ ) cos θ sin θ ] 2 + [ ( 1 ϵ ) cos θ sin θ + ϵ ( sin 2 θ + ϵ cos 2 θ ) ] 2 = cos 2 θ + 2 ϵ cos θ sin θ + 2 ϵ 2 sin 2 θ 2 ϵ 3 cos θ sin θ + ϵ 4 cos 2 θ .
I 2 θ = ( 1 2 ϵ 2 + ϵ 4 ) sin 2 θ + 2 ϵ ( 1 ϵ 2 ) cos 2 θ .
θ = arctan ϵ or θ = π 2 + arctan ϵ .
E 2 max = [ 1 ϵ ] , I 2 max = 1 + ϵ 2 ( when θ = arctan ϵ ) ,
E 2 min = [ ϵ ϵ 2 ] , I 2 min = ϵ 2 + ϵ 4 ( when θ = π 2 + arctan ϵ ) .
T = I 2 I max = cos 2 θ + 2 ϵ cos θ sin θ + 2 ϵ 2 sin 2 θ 2 ϵ 3 cos θ sin θ + ϵ 4 cos 2 θ 1 + ϵ 2 .
T = cos 2 θ + 2 ϵ cos θ sin θ + 2 ϵ 2 sin 2 θ 1 + ϵ 2 .
σ = T T T = 2 ϵ 3 cos θ sin θ + ϵ 4 cos 2 θ cos 2 θ + 2 ϵ cos θ sin θ + 2 ϵ 2 sin 2 θ 2 ϵ 3 cos θ sin θ + ϵ 4 cos 2 θ .
I = I 0 cos 2 θ .
σ = T T 0 T .
E 2 ( π 2 + arctan ϵ , ϵ ) = [ ϵ ϵ 2 ] = ϵ E 1 .
E 3 ( π 2 + arctan ϵ , θ 2 , ϵ ) = ϵ [ cos 2 θ 2 + ϵ sin 2 θ 2 + ϵ ( cos θ 2 sin θ 2 ϵ sin θ 2 cos θ 2 ) ( 1 ϵ ) cos θ 2 sin θ 2 + ϵ ( sin 2 θ 2 + ϵ cos 2 θ 2 ) ] .
T 3 ( π 2 + arctan ϵ , θ 2 , ϵ ) = | E 3 ( π 2 + arctan ϵ , θ 2 , ϵ ) | 2 1 + ϵ 2 = { [ ( ϵ cos 2 θ 2 + ϵ 2 sin 2 θ 2 ) + ( ϵ 2 cos θ 2 sin θ 2 ϵ 3 cos θ 2 sin θ 2 ) ] 2 + [ ( ϵ cos θ 2 sin θ 2 ϵ 2 cos θ 2 sin θ 2 ) + ( ϵ 2 sin 2 θ 2 + ϵ 3 cos 2 θ 2 ) ] 2 } 1 1 + ϵ 2 = ϵ 2 T .
E n 1 ( θ 1 , , θ n 2 , ϵ ) = [ ϵ n 2 ϵ n 1 ] .
E n = U n 1 ( θ n 1 , ϵ ) × E n 1 ( θ 1 , , θ n 2 , ϵ ) .
E n = U n 1 ( θ n 1 , ϵ ) × U n 2 ( θ n 2 , ϵ ) × × U 1 ( θ 1 , ϵ ) × E 1 .
I n ( θ 1 , θ 2 , , θ n 1 , ϵ ) = | U n 1 ( θ n 1 , ϵ ) | 2 × | U n 2 ( θ n 1 , ϵ ) | 2 × × | U 1 ( θ 1 , ϵ ) | 2 × | E 1 | 2
T n = ϵ 2 ( n 2 ) T ,
T n , min = ϵ 2 ( n 1 ) .
E 3 ( θ 1 , θ 2 , ϵ ) = U ( θ 2 , ϵ ) × E ( θ 1 , ϵ ) = ( cos 2 θ 2 + ϵ sin 2 θ 2 ) ( cos 2 θ 1 + ϵ sin 2 θ 1 + ϵ cos θ 1 sin θ 1 ϵ 2 cos θ 1 sin θ 1 ) + ( cos θ 2 sin θ 2 ϵ cos θ 2 sin θ 2 ) ( cos θ 1 sin θ 1 ϵ cos θ 1 sin θ 1 + ϵ sin 2 θ 1 + ϵ 2 cos 2 θ 1 ) = ( cos θ 2 sin θ 2 ϵ cos θ 2 sin θ 2 ) ( cos 2 θ 1 + ϵ sin 2 θ 1 + ϵ cos θ 1 sin θ 1 ϵ 2 cos θ 1 sin θ 1 ) + ( sin 2 θ 2 + ϵ cos 2 θ 2 ) ( cos θ 1 sin θ 1 ϵ cos θ 1 sin θ 1 + ϵ sin 2 θ 1 + ϵ 2 cos 2 θ 1 ) ,
I 3 ( θ 1 , θ 2 , ϵ ) = | E 3 ( θ 1 , θ 2 , ϵ ) | 2 = [ ( cos 2 θ 2 + ϵ sin 2 θ 2 ) ( cos 2 θ 1 + ϵ sin 2 θ 1 + ϵ cos θ 1 sin θ 1 ϵ 2 cos θ 1 sin θ 1 ) + ( 1 ϵ ) cos θ 2 sin θ 2 ( cos θ 1 sin θ 1 ϵ cos θ 1 sin θ 1 + ϵ sin 2 θ 1 + ϵ 2 cos 2 θ 1 ) ] 2 + [ ( sin 2 θ 2 + ϵ cos 2 θ 2 ) ( cos θ 1 sin θ 1 ϵ cos θ 1 sin θ 1 + ϵ sin 2 θ 1 + ϵ 2 cos 2 θ 1 ) + ( 1 ϵ ) cos θ 2 sin θ 2 ( cos 2 θ 1 + ϵ sin 2 θ 1 + ϵ cos θ 1 sin θ 1 ϵ 2 cos θ 1 sin θ 1 ) ] 2 .
T 3 ( θ 1 , θ 2 , ϵ ) = I 3 ( θ 1 , θ 2 , ϵ ) 1 + ϵ 2 .

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