## Fourier transform channeled spectropolarimetry in the MWIR

Optics Express, Vol. 15, Issue 20, pp. 12792-12805 (2007)

http://dx.doi.org/10.1364/OE.15.012792

Acrobat PDF (849 KB)

### 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 (YVO_{4}) 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

## 1. Introduction

2. K. Oka and T. Kato, “Spectroscopic Polarimetry with a Channeled Spectrum,” Opt. Lett. **24**, 1475–1477 (1999). [CrossRef]

_{4}) crystal retarders. Detailed error analyses are provided, and an improved system model is discussed. Lastly, the improved model is used to provide an enhanced calibration technique to remove the effect of the dichroism in the YVO

_{4}.

## 2. System model

_{1}and R

_{2}with thicknesses

*d*and

_{1}*d*followed by an analyzer. The orientation of the retarder’s fast axes relative to the transmission axis of the analyzer is 0° and 45° for R

_{2}_{1}and R

_{2}, respectively. The output is then sent into an interferometer and the data recorded.

*I*(

*σ*,

*z*) [4]. Performing this analysis yields,

*Δz*is the optical path difference (OPD) from the interferometer,

*B*(

*σ*) = (

*n*(

_{e}*σ*)-

*n*(

_{o}*σ*)) is the birefringence of the crystal, and

*d*,

_{1}*d*are the aforementioned retarder thicknesses. Expansion of Eq. 1 can be seen in Eq. 5. Here the constant offset has been suppressed since it’s independent of

_{2}*Δz*; it provides no viable spectral information in the Fourier transform.

_{0}, C

_{2}, and C

_{3}per Fig. 2 yields,

_{2}and C

_{3}can be calibrated out and the unknown values reconstructed. This model shows that we can perform reconstructions directly from the interferogram, which is two steps closer to the reconstruction than if we were using a diffraction grating. Here, the interferometer eliminates a wavelength to wave-number interpolation (when using a fast Fourier transform) in addition to an inverse Fourier transform to convert the spectrum from frequency space into the time domain.

## 3. Reference beam calibration

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]

*jϕ*

_{1}) and exp(-

*jϕ*

_{2}) found in channels C

_{2}and C

_{3}are measured by use of a known input SOP. These “reference data” are then divided by the unknown sample data per Eq. 9 through Eq. 13 below,

_{2}and S

_{3}directly into one of the channels (for a 2:1 ratio). This is also highlighted in §6.

## 4. Experimental setup

_{4}and are AR coated for 4 μm. The thickness ratio used is 2:1 for d

_{2}:d

_{1}where d

_{2}= 4 mm and d

_{1}= 2 mm. The experimental setup can be seen in Fig. 3; it consists of a wire-grid generating polarizer (G) to allow known SOP input for calibration and sample data measurements. This is followed by the two retarders (R

_{1}and R

_{2}), wire-grid analyzer (A), and finally the FTS.

_{3}. It is already well established that the system configuration can reconstruct this component [2

2. K. Oka and T. Kato, “Spectroscopic Polarimetry with a Channeled Spectrum,” Opt. Lett. **24**, 1475–1477 (1999). [CrossRef]

_{3}is zero to examine phase errors due to the interferometer’s mechanical instabilities, which is described in §7.

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]

## 5. Experimental results

*θ*in Fig. 3). In this case, all reconstructions were performed with double-sided interferograms that have been phase-corrected so that they are symmetric using the square-root phase correction technique [6]. The data obtained as a function of polarizer rotation angle and wavelength, portrayed in contour plots, can be seen below in Fig. 4 along with its error in Fig. 5. Here we note that the contours of the data in Fig. 4 should ideally consist of straight lines for S

_{1}and S

_{2}. Furthermore, if one wavelength is chosen and the Stokes vector visualized as a function of rotation angle, S

_{1}should appear cosinusoidal and S

_{2}should appear sinusoidal.

## 6. Improved model

- The S
_{0}and S_{1}components are mixing between each other in spectral locations where there are differences between T_{x1}and T_{y1}, as can be seen from the difference terms in the first and second lines. - Portions of S
_{2}and S_{3}are directly modulated by*φ*, and consequently their energy appears as an additional modulation in the C_{1}_{1}channel (see Fig. 3). This complicates the channel’s ability to reconstruct data accurately. Here we note that in a 1:3 ratio design (d_{2}:d_{1}),*these data would appear in an empty channel*, meaning all channels still remain usable. It would also preserve the self-calibration technique outlined by Oka without requiring further calibration steps. - S
_{2}and S_{3}can be reconstructed and calibrated in the traditional sense if one uses channel C_{3}or its conjugate (e.g. (*φ*+_{1}*φ*) or -(_{2}*φ*+_{1}*φ*)). However, error will still be present in the reconstruction of these channels due to the normalization of S_{2}_{2}and S_{3}to S_{0}. This is due to mixing between S_{1}and S_{0}when S_{1}is nonzero.

_{x1}, T

_{y1}, T

_{x2}, and T

_{y2}, information which can be extracted from the data in Fig. 6.

### 6.1 Improved calibration procedure

_{1}), in addition to collecting reference data with a linear polarizer at 0° and 45°. If we start by taking the sample data for S

_{0}and S

_{1}, along with the reference spectra taken at 0°, we have,

*γ*=

*T*

_{x1}+

*T*

_{y1}and

*ε*=

*T*

_{x1}-

*T*

_{y1}. We now wish to extract the original input, S

_{1}and S

_{0}. Solving for these terms yields,

_{1}is un-normalized. The final normalized output is then,

_{1}. Since S

_{2}and S

_{3}don’t suffer additional error from the dichroism (as long as channel C

_{3}is used for reconstruction in a 2:1 ratio system) except from the normalization to S

_{0}, we only need to correct the S

_{0}that’s seen in the S

_{2}and S

_{3}reference data. This is where the 45° reference data is used since S

_{1}is zero for this input, meaning that the S

_{0}component has the form,

_{0}as in Eq. (17). Now, we only have to divide by

*γ*to correct S

_{2}and S

_{3},

_{1}, S

_{2}, and S

_{3}can have the dichroic contribution from the retarders fully corrected with this method.

## 7. Interferometer instability

_{3}reconstruction, peaking on the order of 4% when it should be closer to zero, is likely due to mechanical instabilities in the interferometer. Here, two forms of interferometer instability will be analyzed, namely displacement of the center-burst from zero OPD and unevenly spaced sampling of the interferogram.

### 7.1 Displaced center-burst

*between*the reference and sample data by some OPD

*Δz*, then the reconstructed spectra will suffer a phase change of Δ

*ϕ*= 2

*π*Δ

*z*/

*λ*. This term will appear in the Fourier transform as an additional exponential phase factor, which will cause the real and imaginary components of the reconstruction to alias. Our reference and sample data Fourier transforms become

_{2}and C

_{3}while taking the real and imaginary parts where appropriate yields

_{1}, S

_{2}, and S

_{3}are multiplied by a sinusoidal error term proportional to the phase offset between the reference and sample data. Fortunately, this issue can be resolved relatively well by phase-correcting the interferogram (for double-sided interferograms) in the following way,

*W*and

_{R}(σ)*W*are the real and imaginary parts of the interferogram’s Fourier transform, respectively. Taking the absolute value of the spectrum will yield the magnitude spectrum |

_{I}(σ)*W(σ)*|, which contains no phase errors sans small noise non-linearity’s [1]. The symmetric interferogram,

*I’(z)*, can then be obtained by an inverse Fourier transform of the magnitude spectrum,

### 7.2 Uneven interferogram sampling

*s*(

*z*), where

*z*is in units of OPD, we have,

*N*is the total number of samples,

*I*is the interferogram, and

*U*is the FFT of

*I*. If

*s*(

*z*) is a constant, we have the ideal sampling situation (e.g.

*s*(

*z*) = 316.4 nm, the nominal value in our interferometer). However, if

*s*(

*z*) varies linearly, randomly, or otherwise, error can be introduced into the reconstructed spectrum. Such error usually manifests itself as a form of aliasing and affects data at higher frequencies more than lower [10]. For instance, real spectral data from our FTS using two consecutive measurements yields the two magnitude spectra in Fig. 8.

### 7.3 Correction of nonuniformally sampled interferograms

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]

*a priori*knowledge of how the interferogram has been sampled in order for the algorithm to convert it to a uniformly sampled dataset. But we can investigate the algorithm’s effectiveness of correcting a channeled interferogram’s nonuniform sampling in simulation. Here, we chose to investigate the NuFFT algorithm. Using an arbitrary sampling function defined as

*bs*(

*z*) = 316.4×10

^{-9}

*b*and Eq. (34) gives,

*b*) seen in Fig. 9. Here it can be seen that the error is large with a peak of 3.16 μm (10 pixels) at the center of the interferogram.

_{1}from channel C

_{2}, 0.4% in S

_{23}from channel C

_{1}and 3.6% in S

_{23}from channel C

_{3}. This confirms that the higher frequency data, as is seen in the outermost channel, is affected the most by the nonuniform sampling. It also emphasizes the algorithm’s ability to compensate for the nonuniform sampling given a channeled spectrum.

### 7.4 Further phase calibration

_{1}reconstruction (before taking the real part of the Fourier transform) is shown in Fig. 11.

_{1}) or aliased (in the case of S

_{2}and S

_{3}) when the real or imaginary parts are extracted.

*Δφ*) using only the reference and sample data provided. Beginning by dividing Eq. (25) by (23) and (26) by (24) yields,

_{2}channel for the S

_{1}reconstruction, while it is combined with the S

_{23}phase in the C

_{3}channel for the S

_{2}and S

_{3}reconstructions. Therefore, any imaginary part in the S

_{1}reconstruction can be attributed to an error resulting from a misregistration between the reference and sample data. This phase difference can be approximated as long as there is enough signal in the S

_{1}state during the measurement.

_{4}retarders. From Eq. (14), we see that the dichroic contribution from S

_{0}to S

_{1}is largest where (

*T*

_{x1}-

*T*

_{y1}) is largest. This occurs from 3.9–4.1 μm (2577–2419 cm

^{-1}) (see Fig. 6, right). As a result, the phase offset can be approximated by averaging the phase difference in this spectral region over wavelength and the conjugate of this phase can be applied to the entire spectrum to correct it. This dichroic contribution is extremely important for the 45° S

_{23}reference data, considering no signal in the S

_{1}channel is ideally present in the absence of the dichroism.

*R*and

*I*refer to the real and imaginary parts of the output, respectively. We first obtain for the S

_{1}reconstruction,

_{23},

*σ*in Eq. (39) and (43) must be taken to avoid the CO

_{2}absorption line for larger air paths (ours is ~1 m). This is important since the peak dichroic region of the YVO

_{4}is close to the CO

_{2}absorption band.

## 8. Results with improved calibration

_{3}reconstruction is improved by use of the modified calibration (§7.4). For these measurements, positioning error of the generating polarizer (+/- 0.10°) is expected to account for negligible error (< 0.01%) when S

_{1}or S

_{2}are at their maxima and 2% when at their minima.

## 9. Conclusion

- Aliasing effects. These will always limit the maximum achievable accuracy of this technique. This was seen to some extent in the CO
_{2}absorption line and can be a significant issue if*a priori*knowledge of aliasing effects cannot be obtained (e.g. via subtraction of an unpolarized reference spectrum). - Dichroism in the retarders. This phenomenon can create significant error; however the improved calibration technique can be utilized to compensate for its effect on the reconstructions.

_{2}absorption line, focusing on the regions with the largest dichroism (4.4–5.0 μm and 3.7–4.1 μm) over all the measurements was brought down from 8.72% for S

_{1}and 2.84% for S

_{2}to 1.07% for S

_{1}and 1.06% for S

_{2}. Additionally, the average absolute error of S

_{3}was brought down from 2.2% to 1.06%. Since our generating polarizer has an extinction ratio of around 100:1, this is essentially the limit of our equipment. This demonstrates that the Fourier transform spectropolarimetric technique shows significant promise in the infrared wavelengths for performing spectropolarimetric measurements.

## References and links

1. | P. Griffiths and J. D. Haseth, “Fourier Transform Infrared Spectrometry,” (John Wiley & Sons, Inc., NY, 1986). |

2. | K. Oka and T. Kato, “Spectroscopic Polarimetry with a Channeled Spectrum,” Opt. Lett. |

3. | T. Kusunoki and K. Oka, “Fourier spectroscopic measurement of polarization using birefringent retarders,” Jap. Soc. of Ap. Phys. |

4. | M. Kudenov, N. Hagen, and H. Luo, |

5. | A. Taniguchi, K. Oka, H. Okabe, and M. Hayakawa, “Stabilization of a channeled spectropolarimeter by self-calibration,” Opt. Lett. |

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), |

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, and I. Chapman |

10. | Vidi 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. |

12. | A. Dutt and V. Rokhlin, “Fast fourier transforms for nonequispaced data,” Siam J. Sci. Comput. |

**OCIS Codes**

(120.2130) Instrumentation, measurement, and metrology : Ellipsometry and polarimetry

(300.6340) Spectroscopy : Spectroscopy, infrared

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: May 22, 2007

Revised Manuscript: September 13, 2007

Manuscript Accepted: September 14, 2007

Published: September 21, 2007

**Citation**

Michael W. Kudenov, Nathan A. Hagen, Eustace L. Dereniak, and Grant R. Gerhart, "Fourier transform channeled spectropolarimetry in the MWIR," Opt. Express **15**, 12792-12805 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-12792

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### References

- P. Griffiths and J. D. Haseth, "Fourier Transform Infrared Spectrometry," (John Wiley and Sons, Inc., NY, 1986).
- K. Oka and T. Kato, "Spectroscopic Polarimetry with a Channeled Spectrum," Opt. Lett. 24, 1475-1477 (1999). [CrossRef]
- T. Kusunoki and K. Oka, "Fourier spectroscopic measurement of polarization using birefringent retarders," Jap. Soc. of Appl. Phys. 61, 871 (2000).
- 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).
- 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]
- 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).
- Q. H. Liu and N. Nguyen, "An accurate algorithm for nonuniform fast fourier transforms," IEE Mic. Guid. Wave Lett. 8, 18-20 (1998). [CrossRef]
- A. Dutt and V. Rokhlin, "Fast fourier transforms for nonequispaced data," Siam J. Sci. Comput. 14, 1368-1393 (1993). [CrossRef]

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