## 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

A Fourier Transform Spectrometer (FTS) has many advantages over other forms of spectroscopic measurement, such as a diffraction grating spectrometer, in the MWIR. Predominantly, the FTS maintains the throughput (Jacquinot) and multiplex (Fellgett) advantages [1]. Yet aside from this, the FTS has another unique benefit for working with the channeled spectropolarimetric technique implemented by K. Oka, where the Stokes vector information is modulated onto the spectrum via several carrier frequencies [2]. These carrier frequencies can be observed as separate entities in the interferogram taken from an FTS, thereby providing a more direct means of acquiring the spectropolarimetric data [3].

First, we will describe the theoretical model behind the spectropolarimetric technique with an FTS. This will be followed with our laboratory results obtained by a commercial FTS in the MWIR using Yttrium Vanadate (YVO_{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

The basic setup of the Fourier Transform Spectropolarimeter (FTSP) is shown in Fig. 1. It consists of two high-order retarders R_{1} and R_{2} with thicknesses *d _{1}* and

*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.

To create a theoretical model of the above system, we utilize plane-wave propagation in a typical Twyman-Green interferometer to observe the functional form of the interferogram *I*(*σ*,*z*) [4]. Performing this analysis yields,

where S_{0}, S_{1}, S_{2}, and S_{3} are the Stokes parameters related to the intensity, linear 0° or 90°, linear +45° or -45°, and right or left circular polarizations, respectively. The phase terms *ϕ*, *ϕ*
_{2}, and *ϕ _{z}* are implicitly dependent upon wavenumber

*σ*(

*λ*) and are defined as

^{-1}where *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.

$$\frac{{S}_{2}}{8}\left[\mathrm{cos}\left({\varphi}_{z}+{\varphi}_{1}+{\varphi}_{2}\right)+\mathrm{cos}\left({\varphi}_{z}-{\varphi}_{1}-{\varphi}_{2}\right)-\mathrm{cos}\left({\varphi}_{z}+{\varphi}_{1}-{\varphi}_{2}\right)-\mathrm{cos}\left({\varphi}_{z}-{\varphi}_{1}+{\varphi}_{2}\right)\right]+$$

$$\frac{{S}_{3}}{8}\left[\mathrm{sin}\left({\varphi}_{z}+{\varphi}_{1}+{\varphi}_{2}\right)-\mathrm{sin}\left({\varphi}_{z}-{\varphi}_{1}-{\varphi}_{2}\right)-\mathrm{sin}\left({\varphi}_{z}+{\varphi}_{1}-{\varphi}_{2}\right)+\mathrm{sin}\left({\varphi}_{z}-{\varphi}_{1}+{\varphi}_{2}\right)\right]$$

From Eq. 5, there are seven distinct channels which contain the data related to the input polarization parameters S_{0}, S_{1}, S_{2}, and S_{3}. These channels can be seen superimposed onto real data from the instrument in Fig. 2. It should be noted that the data in this figure is Eq. 5 integrated over *σ* from 2000 cm^{-1} to approximately 4100 cm^{-1}.

Demodulation of the spectral Stokes vectors can now be performed by filtering the desired channels, followed by a Fourier transformation. Performing this on channels C_{0}, C_{2}, and C_{3} per Fig. 2 yields,

Hence, by measuring a known state of polarization (SOP), the modulating phase factors in C_{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

A method for calibration of the channeled spectropolarimeter is known as the “reference beam calibration technique” [5]. In this form of calibration, the phase factors exp(-*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,

In most cases, reference data collected with a linear polarizer oriented at 22.5° is sufficient for the reference data set. However, this is not sufficient if dichroism in the retarders is present, as will be described in §6. It should be noted that this method was chosen over the more sophisticated self-calibration technique since our retarders are not setup appropriately (in a compact enclosure and in contact) to make the necessary assumption that both retarders experience identical environmental changes. Additionally, the dichroism in the retarders also complicates the self-calibration technique by modulating S_{2} and S_{3} directly into one of the channels (for a 2:1 ratio). This is also highlighted in §6.

## 4. Experimental setup

The FTS equipment that is used for our experiments is an Oriel MIR8000 with an 80026 MCT detector and 80007 Silicon Carbide source. The crystal retarders are made from YVO_{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.

The use of the linear polarizer as a target in this instance enables the collection of well-known and stable reference data and input states. However, this generator lacks the ability to produce nonzero S_{3}. It is already well established that the system configuration can reconstruct this component [2]. Instead, this paper will use the fact that S_{3} is zero to examine phase errors due to the interferometer’s mechanical instabilities, which is described in §7.

For all of the experiments, data were averaged over 128 consecutive interferograms and obtained for the generating polarizer at orientations ranging from 0° to 180° in 10° increments. Reference data were acquired after each measurement so that thermal effects would not obscure the error analysis. Thermal calibration drift is described in detail in Ref. [5] and correction algorithms exist if certain assumptions about the system setup can be made.

## 5. Experimental results

Using the above model for our reconstructions, we obtained experimental data for the generated state of polarization (SOP) at various rotation angles of the generating polarizer (*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.

As can be noted from the reconstruction results, there are several abnormalities in the data. The first corresponds to an expected aliasing anomaly located at 4.28 μm from the sharp CO_{2} absorption line, which can always be reduced by nitrogen purging the instrument’s air path if required. Other problems with the data occurs around 4 μm and 4.8 μm that are due to absorption features in the YVO_{4} crystal (e.g. dichroism) that is not accounted for by the initial model. To see how the error can be explained by the dichroism, a transmission plot for our 2 mm thick YVO_{4} retarder for the fast and slow axes, as well as the transmission difference, can be seen in Fig. 6. The error observed in S_{1} follows the dichroism very closely, indicating that it is related to the observed phenomena in the reconstructed data.

## 6. Improved model

The model described in §2 works well for crystal materials such as calcite and quartz in the visible spectrum. However, when experiments are carried out in the infrared, dichroism can contribute significant error to the reconstructions. This is an issue since it creates polarization leakage between the Stokes parameters. To investigate this more closely, an improved model was produced by placing theoretical diattenuators behind both retarders per Fig. 7 below.

It should be noted that the position of the diattenuators in the system, either before or after the retarders, is of no consequence on-axis since the dichroism is aligned to the eigenvectors of the crystal. Using Muller calculus leads us to Eq. 14 below [7].

Analysis of this equation yields three key results:

- 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.

Therefore it can be presumed that error due to the dichroism can be compensated for if one knows T_{x1}, T_{y1}, T_{x2}, and T_{y2}, information which can be extracted from the data in Fig. 6.

#### 6.1 Improved calibration procedure

The modified calibration procedure that will be described here requires measuring the transmission of the fast and slow axes of the first retarder (R_{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,

Here, Eq. (14) has allowed us to write out the analytical form of the calibrated channels with the absorption terms, where *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,

Note that the above S_{1} is un-normalized. The final normalized output is then,

We now have a corrected measurement for S_{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,

where the prime is used to indicate that this is not the same S_{0} as in Eq. (17). Now, we only have to divide by *γ* to correct S_{2} and S_{3},

Hence the Stokes vectors S_{1}, S_{2}, and S_{3} can have the dichroic contribution from the retarders fully corrected with this method.

## 7. Interferometer instability

An explanation as to why data are present in the S_{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

If the center burst location of the interferogram differs *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

Division between the sample and reference data of channels C_{2} and C_{3} while taking the real and imaginary parts where appropriate yields

Hence, the Stokes components S_{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,

This is the square-root method alluded to earlier. Here *W _{R}(σ)* and

*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,

We now have a phase-corrected interferogram that is real and symmetric. This process will remove any phase errors associated with a displaced center-burst assuming the center burst begins close to zero OPD [8]. It also provides an additional benefit in that phase discrepancies between the reference and sample data caused by the temporal drift of the interferometer (usually thermal effects) are also corrected.

#### 7.2 Uneven interferogram sampling

It is known that rapid-scan interferometers, such as the one used for our experiments, can have significant errors introduced into spectra from uneven sampling of the interferogram [9]. This is caused by velocity jitter over the course of a scan; it typically contains both random and periodic components. Since the channeled spectropolarimetric technique is sensitive to sub-pixel misregistration between the sample and reference data, it can be a significant problem when trying to perform accurate reconstructions. Here, the misregistration occurs due to the primary assumption in the Fast Fourier Transform (FFT) algorithm: that all data are sampled uniformly.

If we configure the FFT algorithm with an arbitrary sampling function *s*(*z*), where *z* is in units of OPD, we have,

where *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.

As can be seen, the modulations in the dashed line are severely attenuated and distorted compared to the solid line. It must be emphasized here that such low-visibility spectral measurements from our instrument are rare; however they can occur during the course of data averaging hundreds of interferograms.

#### 7.3 Correction of nonuniformally sampled interferograms

Although it can not be tested experimentally with our current interferometer, it is envisioned that the best way to correct the remaining error due to nonuniform sampling in the data is to use a Nonuniform Fast Fourier Transform (NuFFT) or a spline-interpolation algorithm [11,12]. Unfortunately, this requires *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

and defining the error as the difference between the ideal sampling, *bs*(*z*) = 316.4×10^{-9}
*b* and Eq. (34) gives,

This yields the quadratic sampling error versus sample index (*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.

Simulating the interferometer with the same input spectrum as our experimental setup yields the following output results per Fig. 10.

The spectral reconstruction done with the NuFFT is superior to that of the FFT. Comparison between the NuFFT and the ideal spectra in a Stokes vector reconstruction yields a maximum error of approximately 1.6% in S_{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

All of the aforementioned issues play a roll in the errors contributed to the reconstruction. These phenomena typically manifest themselves as errors in the phase between the reference and sample data consisting of a small linear term plus some constant offset. To illustrate this effect, the residual phase from the 0° S_{1} reconstruction (before taking the real part of the Fourier transform) is shown in Fig. 11.

Such error causes some of the information about the Stokes vectors to be lost (in the case of S_{1}) or aliased (in the case of S_{2} and S_{3}) when the real or imaginary parts are extracted.

In order to alleviate the residual phase error, it was initially assumed that the function in Fig. 11 could be approximated by a static translation in phase between the reference and sample channels. Additionally, we are making the assumption that the temperature change of the retarders is negligible between sample and reference data, such that most of the phase error is due to the mechanical instabilities of the instrument. The static phase error can be modeled using the formulas of §6.1. The goal is to approximate this phase difference (*Δφ*) using only the reference and sample data provided. Beginning by dividing Eq. (25) by (23) and (26) by (24) yields,

where the additional subscripts, 0° and 45°, indicate which set of reference data are being used (e.g. the reference linear polarizer’s orientation). Hence, any phase error that exists between the reference and sample data is isolated in the C_{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.

Fortunately, this is always true due to the dichroism of the YVO_{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.

In the following equations, the < > operation indicates that the arithmetic mean has been taken and the subscripts *R* and *I* refer to the real and imaginary parts of the output, respectively. We first obtain for the S_{1} reconstruction,

Similarly, for S_{23},

This method works in general in the laboratory, although care in selecting the lower limit on *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

Using the improved model to calibrate the system reveals a marked improvement in the reconstruction. The new reconstruction can be seen in Fig. 12 along with its error in Fig. 13.

The abnormalities in the data around 4 μm and 4.8 μm are significantly diminished, reducing the error around the crystal’s dichroic regions by nearly 20 times. Additionally, the S_{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

From the above analysis and experimentation in the laboratory, it is apparent that there are several issues when operating a channeled spectropolarimeter in the infrared which contribute to an increase in error. These include:

- 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.

Other issues directly related to the use of an interferometer highlight the importance of taking into consideration issues with the mechanical instabilities of the system. Primarily, these can be addressed by double-sided interferogram phase correction (by using the magnitude spectrum) and compensating in some way for the nonuniform sampling (through interpolation or the NuFFT).

Ultimately the average absolute system error outside of the CO_{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.

Further improvements to the system that can be accomplished include moving to a 1:3 ratio between the retarder thicknesses such that all channels can be used in the reconstruction. Another improvement would be to mount the retarders close together in a small enclosure. These two changes would then enable the use of the self-calibration technique (or a modified version thereof that takes into account the dichroism), which would make the system calibration relatively insensitive to temperature changes in the retarders.

## 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. **24**, 1475–1477 (1999). [CrossRef]

**3. **T. Kusunoki and K. Oka, “Fourier spectroscopic measurement of polarization using birefringent retarders,” Jap. Soc. of Ap. Phys. **61**, 871 (2000).

**4. **M. Kudenov, N. Hagen, and 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 688401 (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, and I. Chapman*et al*, “Data processing pipeline for a time-sampled imaging Fourier transform spectrometer,” in *Imaging Spectrometry X*,
S. Shen and P. Lewis, eds., Proc. SPIE 560096 (2004).

**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. **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]