Abstract

A novel method for the spectropolarimetric measurement of materials using the channeled spectrum is described. A pair of high order retarders are incorporated into the illuminating optics of a broadband spectropolarimeter, so that the sample under measurement is illuminated with the light that is modulated in the spectral-dependence of its polarization. The Fourier analysis of the channeled spectrum obtained from the spectropolarimeter allows determining the four spectrally-resolved polarimetric parameters of the sample simultaneously. This approach has a feature that it requires neither mechanically- nor electrically-controllable components for polarization modulation, similar to the previous method for the channeled spectropolarimetry in which the high-order retarders are placed in the receiving optics. The new method can offer the same information about the sample as has been obtained by the previous method, provided that all the optical components satisfy the principle of reciprocity. Furthermore, the new method has an additional advantage over the previous method that it is less susceptible to the sample-induced fluctuations of the wavefront or ray-direction. The effectiveness of this method is experimentally demonstrated with the measurement of a birefringent sample.

© 2007 Optical Society of America

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References

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  1. T. Kato, K. Oka, S. Tanaka, and K. Ohtsuka, in Extended Abstracts of the 34th Meeting of the Hokkaido Chapter of the Japan Society of Applied Physics, (Hokkaido Chapter of the Japan Society of Applied Physics, Hokkaido, 1998), p. 41(in Japanese).
    [PubMed]
  2. K. Oka and T. Kato, "Spectroscopic Polarimetry with a Channeled Spectrum," Opt. Lett. 24, 1475-1477 (1999).
    [CrossRef]
  3. P. L. Kebabian, "Polarimetric spectral intensity modulation spectropolarimeter," US Patent 6,490,043 (2002).
  4. S. H. Jones, F. J. Iannarilli, and P. L. Kebabian, "Realization of quantitative-grade fieldable snapshot imaging spectropolarimeter," Opt. Express 12, 6559-6573 (2004).
    [CrossRef] [PubMed]
  5. R.W. Collins, "Automatic Rotating Element Ellipsometry : Calibration, Operation, and Real-Time Applications," Rev. Sci. Instrum. 61, 2029-2062 (1990).
    [CrossRef]
  6. G. E. Jellison, Jr., and F. A. Modine, "Optical Constants for Silicon at 300 ad 10K Determined Form 1.64 to 4.73eV by Ellipsometry," J. Appl. Phys. 53, 3745-3753 (1982).
    [CrossRef]
  7. K. Oka and T. Kato, "Static Spectroscopic Ellipsometer Based on Optical Frequency-Domain Interferometry," in Polarization Analysis and Measurement IV, D. H. Goldstein, D. B. Chenault, W. G. Egan, and M. J. Duggin, eds., Proc. SPIE 4481, 137-140 (2001).
    [CrossRef]
  8. D. Sabatke, A. Locke, E. I. Dereniak, M. Descour, J. Garcia, T. Hamilton, and R. W. McMillan, "Snapshot Imaging Spectropolarimeter," Opt. Eng. 41, 1048-1054 (2002).
    [CrossRef]
  9. E. L’opez-Lago and R. de la Fuente, "Measurement of the polarization dynamics of ultrashort pulses by using nonlinear phase modulation and channeled spectroscopic polarimetry," J. Opt. A: Pure Appl. Opt. 7, 400-403 (2005).
    [CrossRef]
  10. E. Kim, D. Dave, and T. E. Milner, "Fiber-optic Spectral Polarimeter Using a broadband swept laser source," Opt. Commun. 249, 351-356 (2005).
    [CrossRef]
  11. A. Taniguchi, K. Oka, H. Okabe, and M. Hayakawa, "Stabilization of the channeled spectropolarimeter by self calibration," Opt. Lett. 31, 3279-3281 (2006).
    [CrossRef] [PubMed]
  12. H. Okabe, K. Matoba, M. Hayakawa, A. Taniguchi, K. Oka, H. Naito, and N. Nakatsuka, "New configuration of channeled spectropolarimeter for snapshot polarimetric measurement of materials," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies II, A. Duparre, B. Singh, and Z.-H. Gu, eds., Proc. SPIE 5878, 58780H (2005).
    [CrossRef]
  13. J. W. Hovenier, "Structure of a general pure Mueller matrix," Appl. Opt. 33, 8318-8324 (1994).
    [CrossRef] [PubMed]
  14. O. Hunderi, "Rotating depolarizer ellipsometry," Appl. Opt. 16, 3012-3015 (1977).
    [CrossRef] [PubMed]
  15. H. R. Philipp, "Silicon Dioxide (SiO2), Type α (Crystalline)," in Handbook of optical constants of solids, E. D. Palic, eds. (Academic Press, San Diego, CA, 1985), pp.719-747.

2006 (1)

2005 (2)

E. L’opez-Lago and R. de la Fuente, "Measurement of the polarization dynamics of ultrashort pulses by using nonlinear phase modulation and channeled spectroscopic polarimetry," J. Opt. A: Pure Appl. Opt. 7, 400-403 (2005).
[CrossRef]

E. Kim, D. Dave, and T. E. Milner, "Fiber-optic Spectral Polarimeter Using a broadband swept laser source," Opt. Commun. 249, 351-356 (2005).
[CrossRef]

2004 (1)

2002 (1)

D. Sabatke, A. Locke, E. I. Dereniak, M. Descour, J. Garcia, T. Hamilton, and R. W. McMillan, "Snapshot Imaging Spectropolarimeter," Opt. Eng. 41, 1048-1054 (2002).
[CrossRef]

1999 (1)

1994 (1)

1990 (1)

R.W. Collins, "Automatic Rotating Element Ellipsometry : Calibration, Operation, and Real-Time Applications," Rev. Sci. Instrum. 61, 2029-2062 (1990).
[CrossRef]

1982 (1)

G. E. Jellison, Jr., and F. A. Modine, "Optical Constants for Silicon at 300 ad 10K Determined Form 1.64 to 4.73eV by Ellipsometry," J. Appl. Phys. 53, 3745-3753 (1982).
[CrossRef]

1977 (1)

Collins, R.W.

R.W. Collins, "Automatic Rotating Element Ellipsometry : Calibration, Operation, and Real-Time Applications," Rev. Sci. Instrum. 61, 2029-2062 (1990).
[CrossRef]

Dave, D.

E. Kim, D. Dave, and T. E. Milner, "Fiber-optic Spectral Polarimeter Using a broadband swept laser source," Opt. Commun. 249, 351-356 (2005).
[CrossRef]

Dereniak, E. I.

D. Sabatke, A. Locke, E. I. Dereniak, M. Descour, J. Garcia, T. Hamilton, and R. W. McMillan, "Snapshot Imaging Spectropolarimeter," Opt. Eng. 41, 1048-1054 (2002).
[CrossRef]

Descour, M.

D. Sabatke, A. Locke, E. I. Dereniak, M. Descour, J. Garcia, T. Hamilton, and R. W. McMillan, "Snapshot Imaging Spectropolarimeter," Opt. Eng. 41, 1048-1054 (2002).
[CrossRef]

Garcia, J.

D. Sabatke, A. Locke, E. I. Dereniak, M. Descour, J. Garcia, T. Hamilton, and R. W. McMillan, "Snapshot Imaging Spectropolarimeter," Opt. Eng. 41, 1048-1054 (2002).
[CrossRef]

Hamilton, T.

D. Sabatke, A. Locke, E. I. Dereniak, M. Descour, J. Garcia, T. Hamilton, and R. W. McMillan, "Snapshot Imaging Spectropolarimeter," Opt. Eng. 41, 1048-1054 (2002).
[CrossRef]

Hayakawa, M.

Hovenier, J. W.

Hunderi, O.

Iannarilli, F. J.

Jellison, G. E.

G. E. Jellison, Jr., and F. A. Modine, "Optical Constants for Silicon at 300 ad 10K Determined Form 1.64 to 4.73eV by Ellipsometry," J. Appl. Phys. 53, 3745-3753 (1982).
[CrossRef]

Jones, S. H.

Kato, T.

Kebabian, P. L.

Kim, E.

E. Kim, D. Dave, and T. E. Milner, "Fiber-optic Spectral Polarimeter Using a broadband swept laser source," Opt. Commun. 249, 351-356 (2005).
[CrossRef]

Locke, A.

D. Sabatke, A. Locke, E. I. Dereniak, M. Descour, J. Garcia, T. Hamilton, and R. W. McMillan, "Snapshot Imaging Spectropolarimeter," Opt. Eng. 41, 1048-1054 (2002).
[CrossRef]

McMillan, R. W.

D. Sabatke, A. Locke, E. I. Dereniak, M. Descour, J. Garcia, T. Hamilton, and R. W. McMillan, "Snapshot Imaging Spectropolarimeter," Opt. Eng. 41, 1048-1054 (2002).
[CrossRef]

Milner, T. E.

E. Kim, D. Dave, and T. E. Milner, "Fiber-optic Spectral Polarimeter Using a broadband swept laser source," Opt. Commun. 249, 351-356 (2005).
[CrossRef]

Modine, F. A.

G. E. Jellison, Jr., and F. A. Modine, "Optical Constants for Silicon at 300 ad 10K Determined Form 1.64 to 4.73eV by Ellipsometry," J. Appl. Phys. 53, 3745-3753 (1982).
[CrossRef]

Oka, K.

Okabe, H.

Sabatke, D.

D. Sabatke, A. Locke, E. I. Dereniak, M. Descour, J. Garcia, T. Hamilton, and R. W. McMillan, "Snapshot Imaging Spectropolarimeter," Opt. Eng. 41, 1048-1054 (2002).
[CrossRef]

Taniguchi, A.

Appl. Opt. (2)

J. Appl. Phys. (1)

G. E. Jellison, Jr., and F. A. Modine, "Optical Constants for Silicon at 300 ad 10K Determined Form 1.64 to 4.73eV by Ellipsometry," J. Appl. Phys. 53, 3745-3753 (1982).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

E. L’opez-Lago and R. de la Fuente, "Measurement of the polarization dynamics of ultrashort pulses by using nonlinear phase modulation and channeled spectroscopic polarimetry," J. Opt. A: Pure Appl. Opt. 7, 400-403 (2005).
[CrossRef]

Opt. Commun. (1)

E. Kim, D. Dave, and T. E. Milner, "Fiber-optic Spectral Polarimeter Using a broadband swept laser source," Opt. Commun. 249, 351-356 (2005).
[CrossRef]

Opt. Eng. (1)

D. Sabatke, A. Locke, E. I. Dereniak, M. Descour, J. Garcia, T. Hamilton, and R. W. McMillan, "Snapshot Imaging Spectropolarimeter," Opt. Eng. 41, 1048-1054 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Rev. Sci. Instrum. (1)

R.W. Collins, "Automatic Rotating Element Ellipsometry : Calibration, Operation, and Real-Time Applications," Rev. Sci. Instrum. 61, 2029-2062 (1990).
[CrossRef]

Other (5)

H. Okabe, K. Matoba, M. Hayakawa, A. Taniguchi, K. Oka, H. Naito, and N. Nakatsuka, "New configuration of channeled spectropolarimeter for snapshot polarimetric measurement of materials," in Advanced Characterization Techniques for Optics, Semiconductors, and Nanotechnologies II, A. Duparre, B. Singh, and Z.-H. Gu, eds., Proc. SPIE 5878, 58780H (2005).
[CrossRef]

H. R. Philipp, "Silicon Dioxide (SiO2), Type α (Crystalline)," in Handbook of optical constants of solids, E. D. Palic, eds. (Academic Press, San Diego, CA, 1985), pp.719-747.

K. Oka and T. Kato, "Static Spectroscopic Ellipsometer Based on Optical Frequency-Domain Interferometry," in Polarization Analysis and Measurement IV, D. H. Goldstein, D. B. Chenault, W. G. Egan, and M. J. Duggin, eds., Proc. SPIE 4481, 137-140 (2001).
[CrossRef]

T. Kato, K. Oka, S. Tanaka, and K. Ohtsuka, in Extended Abstracts of the 34th Meeting of the Hokkaido Chapter of the Japan Society of Applied Physics, (Hokkaido Chapter of the Japan Society of Applied Physics, Hokkaido, 1998), p. 41(in Japanese).
[PubMed]

P. L. Kebabian, "Polarimetric spectral intensity modulation spectropolarimeter," US Patent 6,490,043 (2002).

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Figures (11)

Fig. 1.
Fig. 1.

Conceptual scheme of the CSPSG.

Fig. 2.
Fig. 2.

Schematic drawings of the four spectrally-resolved Stokes parameters of the light emerging from the CSPSG.

Fig. 3.
Fig. 3.

Conceptual scheme of the spectropolarimeter using the CSPSG (SP-CSPSG). Measurable elements in the Mueller matrix of the S-PSA Block are indicated by the open circles.

Fig. 4.
Fig. 4.

Conceptual schemes of (a) the CSPSA and (b) the spectropolarimeter using the CSPSA (SP-CSPSA). Measurable elements in the Mueller matrix of the PSG-S block are indicated by the open circles in (b).

Fig. 5.
Fig. 5.

Effects of the ray-bending introduced by the sample. The directions of the light ray passing through R1 and R2 change between the measurement and the calibration in the SP-CSPSA (a), but do not change in the SP-CSPSG (b).

Fig. 6.
Fig. 6.

Two other causes for the fluctuations of the light introduced by the sample in the SP-CSPSA.

Fig. 7.
Fig. 7.

Channeled spectrum P(σ) obtained by the SP-CSPSG.

Fig. 8.
Fig. 8.

Measured azimuth of a thin quartz plate (TQP) plotted as a function of the wavenumber.

Fig. 9.
Fig. 9.

Retardations of a thin quartz plate (TQP) plotted as a function of the wavenumber. Solid and broken curves are respectively obtained by the SP-CSPSG and the SP-CSPSA, and the dotted curve is the theoretical value based on the Sellmeier’s equation.

Fig. 10.
Fig. 10.

Azimuths of a TQP measured by the SP-CSPSG. The orientations of the TQP were changed by a rotary hollow stage, and the azimuths were measured at every orientations.

Fig. 11.
Fig. 11.

Fluctuations of the measured retardation of a TQP under the ray-direction variations. The three graphs are respectively taken at wavenumbers (a) 1.4 × 104 cm-1, (b) 1.6 × 104 cm-1, and (c) 1.8 × 104 cm-1.

Equations (43)

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S G ( σ ) = S 0 G ( σ ) S 1 G ( σ ) S 2 G ( σ ) S 3 G ( σ ) = 1 2 P o ( σ ) 1 cos ϕ 1 ( σ ) sin ϕ 1 ( σ ) sin ϕ 2 ( σ ) sin ϕ 1 ( σ ) cos ϕ 2 ( σ ) ,
ϕ j ( σ ) = 2 π B ( σ ) D j σ = 2 π L j σ + Φ j ( σ ) ,
L j = 1 2 π j σ 0 = D j ( B ( σ 0 ) + dB σ 0 σ 0 ) ,
Φ j ( σ ) = { ϕ j ( σ 0 ) 2 π L j σ 0 } + 1 2 d 2 ϕ j 2 σ 0 ( σ σ 0 ) 2 + ,
S G ( σ ) = 1 2 P o ( σ ) 1 cos { 2 π L 1 σ + Φ 1 ( σ ) } 1 2 [ cos { 2 π L σ + Φ ( σ ) } cos { 2 πL + σ + Φ + ( σ ) } ] 1 2 sin { 2 π L σ + Φ ( σ ) } sin { 2 πL + σ + Φ + ( σ ) } ] ,
L = L 1 L 2 ,
L + = L 1 + L 2 ,
Φ ( σ ) = Φ 1 ( σ ) Φ 2 ( σ ) ,
Φ + ( σ ) = Φ 1 ( σ ) + Φ 2 ( σ ) ,
P ( σ ) = 1 0 0 0 M a ( σ ) S G ( σ )
= 1 0 0 0 m 00 a ( σ ) m 01 a ( σ ) m 02 a ( σ ) m 03 a ( σ ) S 0 G ( σ ) S 1 G ( σ ) S 2 G ( σ ) S 3 G ( σ )
= m 00 a ( σ ) S 0 G ( σ ) + m 01 a ( σ ) S 1 G ( σ ) + m 02 a ( σ ) S 2 G ( σ ) + m 03 a ( σ ) S 3 G ( σ ) .
P ( σ ) = P 0 ( σ ) [ 1 2 m 00 a ( σ ) + 1 4 m 0 ( 23 ) a ( σ ) cos [ ϕ 1 ( σ ) ϕ 2 ( σ ) + arg { m 0 ( 23 ) a ( σ ) } ]
+ 1 2 m 01 a ( σ ) cos ϕ 1 ( σ ) 1 4 m 0 ( 23 ) a ( σ ) cos [ ϕ 1 ( σ ) + ϕ 2 ( σ ) arg { m 0 ( 23 ) a ( σ ) } ] ] ,
m 0 ( 23 ) a ( σ ) = m 02 a ( σ ) im 03 a ( σ ) .
F 0 ( σ ) = [ 1 2 ] P 0 ( σ ) m 00 a ( σ ) ,
F ( σ ) = [ 1 8 e i { ϕ 1 ( σ ) ϕ 2 ( σ ) } ] P 0 ( σ ) { m 02 a ( σ ) m 03 a ( σ ) } ,
F 1 ( σ ) = [ 1 4 e i ϕ 1 ( σ ) ] P 0 ( σ ) m 01 a ( σ ) ,
F + ( σ ) = [ 1 8 e i { ϕ 1 ( σ ) + ϕ 2 ( σ ) } ] P 0 ( σ ) { m 02 a ( σ ) + m 03 a ( σ ) } .
F 0 ( σ ) = [ 1 2 ] 1 2 P 0 ( σ ) ,
F ( σ ) = [ 1 8 e i { ϕ 1 ( σ ) ϕ 2 ( σ ) } ] 1 2 P 0 ( σ ) sin 2 θ ,
F 1 ( σ ) = [ 1 4 e i ϕ 1 ( σ ) ] 1 2 P 0 ( σ ) cos 2 θ ,
F + ( σ ) = [ 1 8 e i { ϕ 1 ( σ ) + ϕ 2 ( σ ) } ] 1 2 P 0 ( σ ) sin 2 θ ,
m 00 a ( σ ) = 1 ,
m 01 a ( σ ) = cos 2 α sin 2 α [ 1 cos δ ( σ ) ] ,
m 02 a ( σ ) = sin 2 2 α + cos 2 2 α cos δ ( σ ) ,
m 03 a ( σ ) = cos 2 α sin δ ( σ ) .
α = 1 2 tan 1 ( m 01 a ( σ ) m 00 a ( σ ) m 02 a ( σ ) ) ,
δ ( σ ) = tan 1 ( m 03 a ( σ ) cos 2 α m 02 a ( σ ) m 00 a ( σ ) sin 2 2 α ) .
P ( σ ) = 1 2 S 0 A ( σ ) + 1 4 S 23 A ( σ ) cos [ ϕ 2 ( σ ) ϕ 1 ( σ ) + arg { S 23 A ( σ ) } ]
+ 1 2 S 1 A ( σ ) cos ϕ 2 ( σ ) 1 4 S 23 A ( σ ) cos [ ϕ 2 ( σ ) + ϕ 1 ( σ ) arg { S 23 A ( σ ) } ] ,
S 23 A ( σ ) = S 2 A ( σ ) + i S 3 A ( σ ) .
S 0 A S 1 A S 2 A S 3 A = M b ( σ ) 1 0 0 0 P 0 ( σ ) = m 00 b ( σ ) m 10 b ( σ ) m 20 b ( σ ) m 30 b ( σ ) P 0 ( σ ) 0 0 0
= P 0 ( σ ) m 00 b ( σ ) P 0 ( σ ) m 10 b ( σ ) P 0 ( σ ) m 20 b ( σ ) P 0 ( σ ) m 30 b ( σ ) ,
M a ( σ ) = Δ 3 M ˜ b ( σ ) Δ 3 ,
m 00 a ( σ ) = m 00 b ( σ ) ,
m 01 a ( σ ) = m 10 b ( σ ) ,
m 02 a ( σ ) = m 20 b ( σ ) ,
m 03 a ( σ ) = m 30 b ( σ ) .
m 00 b ( σ ) = 1 ,
m 01 b ( σ ) = cos 2 α sin 2 α [ 1 cos δ ( σ ) ] ,
m 02 b ( σ ) = sin 2 2 α cos 2 2 α cos δ ( σ ) ,
m 03 b ( σ ) = cos 2 α sin δ ( σ ) .

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