Abstract

A complete Fourier Transform Spectropolarimeter in the MWIR is demonstrated. The channeled spectral technique, originally developed by K. Oka, is implemented with the use of two Yttrium Vanadate (YVO4) crystal retarders. A basic mathematical model for the system is presented, showing that all the Stokes parameters are directly present in the interferogram. Theoretical results are compared with real data from the system, an improved model is provided to simulate the effects of absorption within the crystal, and a modified calibration technique is introduced to account for this absorption. Lastly, effects due to interferometer instabilities on the reconstructions, including nonuniform sampling and interferogram translations, are investigated and techniques are employed to mitigate them.

© 2007 Optical Society of America

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References

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  1. P. Griffiths and J. D. Haseth, "Fourier Transform Infrared Spectrometry," (John Wiley and Sons, Inc., NY, 1986).
  2. K. Oka and T. Kato, "Spectroscopic Polarimetry with a Channeled Spectrum," Opt. Lett. 24, 1475-1477 (1999).
    [CrossRef]
  3. T. Kusunoki and K. Oka, "Fourier spectroscopic measurement of polarization using birefringent retarders," Jap. Soc. of Appl. Phys. 61, 871 (2000).
  4. M. Kudenov, N. Hagen, H. Luo,  et al, "Polarization acquisition using a commercial Fourier transform spectrometer in the MWIR," in Infrared Detectors and Focal Plane Arrays VII, E. Dereniak and R. Sampson, eds., Proc. SPIE 68, 8401 (2006).
  5. A. Taniguchi, K. Oka, H. Okabe, and M. Hayakawa, "Stabilization of a channeled spectropolarimeter by self-calibration," Opt. Lett. 31, 3279-3281 (2006).
    [CrossRef] [PubMed]
  6. J. Connes, "Aspen International Conference on Fourier Spectroscopy," G. A. Vanasse, A. T. Stair, Jr., and D. J. Baker, eds. (Air Force Cambridge Labs Report, No. 114), 83 (1970).
  7. Dennis Goldstein, Polarized Light (Marcel Dekker, NY, 2003).
  8. R. Bell, Introductory Fourier Transform Spectroscopy, (Academic Press, 1972).
  9. D. Naylor, T. Fulton, P. Davis, I. Chapman, et al, "Data processing pipeline for a time-sampled imaging Fourier transform spectrometer," Proc. SPIE 5546, 61-72 (2004).
  10. V. Saptari, "Fourier-transform spectroscopy instrumentation engineering," (SPIE Press, Bellingham, WA, 2004).
  11. Q. H. Liu and N. Nguyen, "An accurate algorithm for nonuniform fast fourier transforms," IEE Mic. Guid. Wave Lett. 8, 18-20 (1998).
    [CrossRef]
  12. A. Dutt and V. Rokhlin, "Fast fourier transforms for nonequispaced data," Siam J. Sci. Comput. 14, 1368-1393 (1993).
    [CrossRef]

2006

2000

T. Kusunoki and K. Oka, "Fourier spectroscopic measurement of polarization using birefringent retarders," Jap. Soc. of Appl. Phys. 61, 871 (2000).

1999

1998

Q. H. Liu and N. Nguyen, "An accurate algorithm for nonuniform fast fourier transforms," IEE Mic. Guid. Wave Lett. 8, 18-20 (1998).
[CrossRef]

1993

A. Dutt and V. Rokhlin, "Fast fourier transforms for nonequispaced data," Siam J. Sci. Comput. 14, 1368-1393 (1993).
[CrossRef]

Dutt, A.

A. Dutt and V. Rokhlin, "Fast fourier transforms for nonequispaced data," Siam J. Sci. Comput. 14, 1368-1393 (1993).
[CrossRef]

Hayakawa, M.

Kato, T.

Kusunoki, T.

T. Kusunoki and K. Oka, "Fourier spectroscopic measurement of polarization using birefringent retarders," Jap. Soc. of Appl. Phys. 61, 871 (2000).

Liu, Q. H.

Q. H. Liu and N. Nguyen, "An accurate algorithm for nonuniform fast fourier transforms," IEE Mic. Guid. Wave Lett. 8, 18-20 (1998).
[CrossRef]

Nguyen, N.

Q. H. Liu and N. Nguyen, "An accurate algorithm for nonuniform fast fourier transforms," IEE Mic. Guid. Wave Lett. 8, 18-20 (1998).
[CrossRef]

Oka, K.

Okabe, H.

Rokhlin, V.

A. Dutt and V. Rokhlin, "Fast fourier transforms for nonequispaced data," Siam J. Sci. Comput. 14, 1368-1393 (1993).
[CrossRef]

Taniguchi, A.

IEE Mic. Guid. Wave Lett.

Q. H. Liu and N. Nguyen, "An accurate algorithm for nonuniform fast fourier transforms," IEE Mic. Guid. Wave Lett. 8, 18-20 (1998).
[CrossRef]

Jap. Soc. of Ap. Phys.

T. Kusunoki and K. Oka, "Fourier spectroscopic measurement of polarization using birefringent retarders," Jap. Soc. of Appl. Phys. 61, 871 (2000).

Opt. Lett.

Siam J. Sci. Comput.

A. Dutt and V. Rokhlin, "Fast fourier transforms for nonequispaced data," Siam J. Sci. Comput. 14, 1368-1393 (1993).
[CrossRef]

Other

J. Connes, "Aspen International Conference on Fourier Spectroscopy," G. A. Vanasse, A. T. Stair, Jr., and D. J. Baker, eds. (Air Force Cambridge Labs Report, No. 114), 83 (1970).

Dennis Goldstein, Polarized Light (Marcel Dekker, NY, 2003).

R. Bell, Introductory Fourier Transform Spectroscopy, (Academic Press, 1972).

D. Naylor, T. Fulton, P. Davis, I. Chapman, et al, "Data processing pipeline for a time-sampled imaging Fourier transform spectrometer," Proc. SPIE 5546, 61-72 (2004).

V. Saptari, "Fourier-transform spectroscopy instrumentation engineering," (SPIE Press, Bellingham, WA, 2004).

P. Griffiths and J. D. Haseth, "Fourier Transform Infrared Spectrometry," (John Wiley and Sons, Inc., NY, 1986).

M. Kudenov, N. Hagen, H. Luo,  et al, "Polarization acquisition using a commercial Fourier transform spectrometer in the MWIR," in Infrared Detectors and Focal Plane Arrays VII, E. Dereniak and R. Sampson, eds., Proc. SPIE 68, 8401 (2006).

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Figures (13)

Fig. 1.
Fig. 1.

Basic FTSP block diagram.

Fig. 2.
Fig. 2.

The 7 channels in the interferogram are separated in OPD space by the retardances φ1 and φ2 . Shown is a real channeled interferogram with the functional forms of each channel listed, along with the relative channel widths indicated by the boxes. Spacing between each channel is for a 2:1 thickness ratio (d2:d1) using our setup, which will be described in §4.

Fig. 3.
Fig. 3.

The experimental setup consists of the MWIR source, generating polarizer G, the two retarders R1 and R2, the Analyzer A, and the FTS with the MCT detector.

Fig. 4.
Fig. 4.

Contour plots of the initial reconstruction results for the normalized Stokes parameters S1, S2, and S3 using the model per §2 with channels C0, C2, and C3. The spectral resolution is approximately 72 cm-1.

Fig. 5.
Fig. 5.

Contour plots of the percent error in the reconstruction for S1 and S2. The relative percent error is undefined for S2 at 0°, 90°, and 180°. Therefore it has been set to zero for these values.

Fig. 6.
Fig. 6.

(Left) Percent transmission of the AR coated 2 mm retarder and a 0.73 mm non-AR coated YVO4 retarder. The absorption feature is present in both. (Right) Percent difference between the fast and slow axis transmission for the AR coated and non-AR coated samples, as well as a scaled overlay of the S1 error at 40° to visualize how well its features correspond to the dichroism.

Fig. 7.
Fig. 7.

Improved system model layout. Partial polarizers P1 and P2 with transmission attenuation coefficients Tx1, Tx2, Ty1, and Ty2 have been included to simulate the effects of the varying transmission on the reconstructed results.

Fig. 8.
Fig. 8.

Two consecutive spectral measurements taken with our FTS of a linear polarizer at 22.5°. The solid (blue) line indicates a relatively evenly sampled interferogram, whereas the dashed (green) line is indicative of uneven sampling occurring.

Fig. 9.
Fig. 9.

Quadratic sampling error used for the simulated nonuniformally sampled interferogram.

Fig. 10.
Fig. 10.

Ideal channeled spectra and the spectra reconstructed after nonuniform sampling with (left) the normal FFT (which assumes uniform spacing) and (right) the NuFFT (setup with an over-sampling factor of m = 2 and q = 8).

Fig. 11.
Fig. 11.

Residual phase in the S1 reconstruction. It consists of a linear term in addition to a constant offset.

Fig. 12.
Fig. 12.

Contour plots of the reconstruction results for the normalized Stokes parameters S1, S2, and S3 using the improved model per. §6 and §7.4. The spectral resolution is approximately 72 cm-1.

Fig. 13.
Fig. 13.

Contour plots of the percent error in the improved reconstructions. Large error is still present near the CO2 absorption line.

Equations (46)

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I ( σ , z ) ( 1 + cos ( ϕ z ) ) 2 [ S 0 + S 1 cos ( ϕ 2 ) + S 2 sin ( ϕ 1 ) sin ( ϕ 2 ) S 3 cos ( ϕ 1 ) sin ( ϕ 2 ) ]
ϕ 1 ( σ ) = 2 πB ( σ ) d 1 σ
ϕ 2 ( σ ) = 2 πB ( σ ) d 2 σ
ϕ z ( σ ) = 2 π Δ
I ( σ , z ) S 0 2 cos ( ϕ z ) + S 1 4 ( cos ( ϕ z + ϕ 2 ) + cos ( ϕ z ϕ 2 ) ) +
S 2 8 [ cos ( ϕ z + ϕ 1 + ϕ 2 ) + cos ( ϕ z ϕ 1 ϕ 2 ) cos ( ϕ z + ϕ 1 ϕ 2 ) cos ( ϕ z ϕ 1 + ϕ 2 ) ] +
S 3 8 [ sin ( ϕ z + ϕ 1 + ϕ 2 ) sin ( ϕ z ϕ 1 ϕ 2 ) sin ( ϕ z + ϕ 1 ϕ 2 ) + sin ( ϕ z ϕ 1 + ϕ 2 ) ]
( C 0 ) = 1 2 S 0 ( σ )
( C 2 ) = 1 4 S 1 ( σ ) exp ( j ϕ 2 )
( C 3 ) = 1 8 ( S 2 ( σ ) j S 3 ( σ ) ) exp ( j ϕ 1 ) exp ( j ϕ 2 )
S 0 , reference ( σ ) = ( C 0 , reference , 22.5 ° )
S 0 , sample ( σ ) = ( C 0 , sample )
S 1 , sample ( σ ) = [ ( C 2 , sample ) ( C 2 , reference , 22.5 ° ) S 0 , reference S 0 , sample ]
S 2 , sample ( σ ) = [ ( C 3 , sample ) ( C 3 , reference , 22.5 ° ) S 0 , reference S 0 , sample ]
S 3 , sample ( σ ) = [ ( C 3 , sample ) ( C 3 , reference , 22.5 ° ) S 0 , reference S 0 , sample ]
I ( σ ) ( 1 + cos ( ϕ x ) ) 2 [ S 0 4 ( 1 2 ( T x 1 + T y 1 ) ( T x 2 + T y 2 ) + T x 2 T y 2 ( T x 1 T y 1 ) cos ( ϕ 2 ) ) + S 1 4 ( 1 2 ( T x 1 T y 1 ) ( T x 2 + T y 2 ) + T x 2 T y 2 ( T x 1 + T y 1 ) cos ( ϕ 2 ) ) + S 2 2 ( 1 2 T x 1 T y 1 ( T x 2 T y 2 ) cos ( ϕ 1 ) + T x 1 T y 1 T x 2 T y 2 sin ( ϕ 1 ) sin ( ϕ 2 ) ) + S 3 2 ( 1 2 T x 1 T y 1 ( T x 2 T y 2 ) sin ( ϕ 1 ) + T x 1 T y 1 T x 2 T y 2 cos ( ϕ 1 ) sin ( ϕ 2 ) ) ]
S 0 , sample ( σ ) = ( C 0 , sample ) = ( S 0 γ + S 1 ε )
S 1 , sample ( σ ) = ( [ ( C 2 , sample ) ( C 2 , reference , 0 ° ) ( C 0 , reference , 0 ° ) ( C 0 , sample ) ] ) = S 0 ε + S 1 γ S 0 γ + S 1 ε
S 0 = S 0 , sample ε S 1 , sample γ ( ε 2 γ 2 )
S 1 = S 0 , sample ε γ S 1 , sample ( ε 2 γ 2 )
S 1 , corrected = S 1 S 0
S 0 , reference ( σ ) = ( C 0 , reference , 45 ° ) = S 0 γ
S 2 , corrected = [ ( C 3 , sample ) ( C 3 , reference , 45 ° ) S 0 , reference ( σ ) S 0 ( σ ) ] 1 γ
S 3 , corrected = [ ( C 3 , sample ) ( C 3 , reference , 45 ° ) S 0 , reference ( σ ) S 0 ( σ ) ] 1 γ
( C 2 , reference ) = 1 4 S 1 ( σ ) exp ( j ϕ 2 ( σ ) )
( C 3 , reference ) = 1 8 ( S 2 ( σ ) j S 3 ( σ ) ) exp ( j ϕ 1 ( σ ) ) exp ( j ϕ 2 ( σ ) )
( C 2 , sample ) = 1 4 S 1 ( σ ) exp ( j ϕ 2 ( σ ) ) exp ( j Δ ϕ )
( C 3 , sample ) = 1 8 ( S 2 ( σ ) j S 3 ( σ ) ) exp ( j ϕ 1 ( σ ) ) exp ( j ϕ 2 ( σ ) ) exp ( j Δ ϕ )
[ ( C 2 , sample ) ( C 2 , reference ) ] = S 1 ( σ ) cos ( Δ ϕ )
[ ( C 3 , sample ) ( C 3 , reference ) ] = S 2 ( σ ) cos ( Δ ϕ )
[ ( C 3 , sample ) ( C 3 , reference ) ] = S 3 ( σ ) sin ( Δ ϕ )
W ' ( σ ) = [ I ( σ , z ) ] = W R ( σ ) + j W I ( σ )
W ( σ ) = W ' ( σ ) = ( W R 2 ( σ ) + W I 2 ( σ ) ) 1 2
I ' ( z ) = 1 [ W ( σ ) ]
U ( k ) = b = N 2 N 2 1 I ( b s ( z ) ) exp ( j 2 πbk N )
bs ( z ) = 316.4 × 10 9 [ b 10 ( 1 4 b 2 N 2 ) ] , N 2 b N 2 1
ε = 316.4 × 10 9 [ 10 ( 1 4 b 2 N 2 ) ] , N 2 b N 2 1
( C 2 , sample ) ( C 2 , reference , 0 ° ) = S 1 ( σ ) exp ( j Δ ϕ 0 ° )
( C 3 , sample ) ( C 3 , reference , 45 ° ) = ( S 2 ( σ ) j S 3 ( σ ) ) e j Δ ϕ 45 ° = S 23 ( σ ) exp ( j ϕ 23 ) exp ( j Δ ϕ 45 ° )
S 1 ( σ ) = ( C 2 , sample ) ( C 2 , reference , 0 ° ) = S 1 , R , 0 ° ( σ ) + j S 1 , I , 0 ° ( σ )
ϕ 0 a tan ( S 1 , I , 0 ° ( σ ) S 1 , R , 0 ° ( σ ) ) , ( 2419 σ 2577 cm 1 )
S 1 ( σ ) = S 1 ( σ ) exp ( j ϕ 0 )
S 1 ( σ ) = ( C 2 , sample ) ( C 2 , reference , 45 ° ) = S 1 , R , 45 ° ( σ ) + j S 1 , I , 45 ° ( σ )
S 23 ( σ ) = ( C 3 , sample ) ( C 3 , reference , 45 ° )
ϕ 45 = a tan ( S 1 , I , 45 ° ( σ ) S 1 , R , 45 ° ( σ ) ) , ( 2419 σ 2577 cm 1 )
S 23 ( σ ) = S 23 ( σ ) exp ( j ϕ 45 )

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