## Abstract

The calibration of a complete Stokes birefringent prismatic imaging polarimeter (BPIP) in the MWIR is demonstrated. The BPIP technique, originally developed by K. Oka, is implemented with a set of four Yttrium Vanadate (YVO_{4}) crystal prisms. A mathematical model for the polarimeter is presented in which diattenuation due to Fresnel effects and dichroism in the crystal are included. An improved polarimetric calibration technique is introduced to remove the diattenuation effects, along with the relative radiometric calibration required for the BPIP operating with a thermal background and large detector offsets. Data demonstrating emission polarization are presented from various blackbodies, which are compared to data from our Fourier transform infrared spectropolarimeter.

© 2008 Optical Society of America

## 1. Introduction

Polarimetry data is a viable asset for remote sensing, biomedical, and industrial applications. Several techniques exist for designing an optical system to produce polarimetry data products. Each has advantages and disadvantages related to cost, optical complexity, and temporal and spatial registration. The rotating retarder polarimeter is a common design, used to modulate the Stokes vectors onto a temporally varying intensity signal. With one moving part and a single imaging lens that’s common to all measurements, it benefits by having low system complexity and relatively simple spatial registration. However, it suffers from poor temporal registration for rapidly changing scenes - a major concern for remote sensing applications on moving platforms.

Several design options exist to remedy this temporal registration issue in which four polarization states are measured in parallel. They are divided into four categories: division of aperture, division of amplitude, division of focal plane, and coboresighted. Polarimeters in these categories either separate one image of a scene into multiple subimages and onto a single focal plane array (FPA), use beam splitters to divide the incident light from the scene into multiple FPA/lens combinations, place polarization analyzers directly on top of the FPA’s pixels, or they consist of multiple FPA/lens combinations mounted side by side, respectively. Yet unlike the rotating retarder polarimeter, multiple lenses with differing aberrations (primarily distortion) must be used. This increases cost and system complexity, and leads to more involved spatial registration concerns [1].

The prismatic polarimeter demonstrated here, as first proposed by K. Oka [2], modulates the complete Stokes vectors of a scene onto multiple spatial carrier frequencies via four birefringent prisms. Prisms of this kind are readily available and since it uses one FPA and two lenses, it provides a high temporal registration polarimeter at a lower cost than the aforementioned designs. Image registration is also inherent because all four Stokes components are recorded simultaneously on coincident fringe fields. These advantages often come by trading off spatial resolution and maximum operating spectral bandwidth, where the latter typically diminishes the data’s signal to noise ratio. In this paper, §2 provides the prismatic polarimeter model in the absence of dichroism. §3 derives an improved model applicable to the infrared, and new calibration procedures based on this model are developed in §4. Radiometric calibration of the instrument is described in §5, followed by a discussion of the experimental setup and results in §6 and §7, respectively.

## 2. Visible system model

The primary component of the BPIP is comprised of four birefringent prisms. As can be seen in Fig. 1, the first pair of prisms, P_{1} and P_{2}, varies in thickness along *y* with fast axis orientations of 0° and 90° with respect to the *x* axis, respectively. The second pair of prisms, P_{3} and P_{4}, varies in thickness along *x* with fast axis orientations of 45° and 135°, respectively. An analyzer (A) follows the group with its transmission axis at 0°. This enables each prism pair (P_{1} and P_{2} or P_{3} and P4) to form a spatially varying retardance as a function of *x* and *y*, where the optical path difference (OPD) between the orthogonally polarized components is zero at the center with maximum (positive) and minimum (negative) values at either edge. When a polarized scene is imaged onto the prisms, interference fringes are developed. These fringes, along with the spatial information of the scene, are then relayed onto a focal plane array (FPA) as illustrated in Fig. 2.

The fundamental model of the prism polarimeter utilizes thin prisms. With the use of Mueller calculus, the intensity pattern behind the analyzer becomes, [2]

$$-\frac{1}{4}\mid {S}_{23}(x,y)\mid \mathrm{cos}\left(2\pi U\left(x+y\right)-\mathrm{arg}\left[{S}_{23}(x,y)\right]\right)$$

with

where *B*=(*n _{e}*-

*n*) is the birefringence,

_{o}*λ*is the operating wavelength,

*β*is the prism angle (see Fig. 1),

*U*is the carrier frequency, and

*S*

_{0}(

*x*,

*y*),

*S*

_{1}(

*x*,

*y*), …

*S*

_{3}(

*x*,

*y*) are the spatially dependent Stokes vectors. It is important to note that the carrier frequency’s inverse proportionality to wavelength Eq. 2b is where this technique acquires its narrow bandwidth limitation [2]. In broadband use,

*I*(

*x*,

*y*) becomes a superposition of fringe patterns with different carrier frequencies, thereby decreasing the visibility of the modulated Stokes vectors. For now, assuming a monochromatic source, Fourier transformation of

*I*(

*x*,

*y*) yields seven carrier frequencies per Fig. 3. Reconstruction of the spatially varying Stokes vector requires filtration of the desired channel, followed by a Fourier transformation. Performing this on channels C

_{0}, C

_{1}, and C

_{2}yields,

Calibration is conducted by the reference beam technique, similar to Ref. [3,4], in which the modulating phase factors exp(*j*2*πUx*) and exp(*j*2*πUy*) are measured and removed from the Fourier transforms. To measure these phase factors a diffuser, blackbody, or integrating sphere is viewed by the instrument through a linear polarizer to obtain a uniformly illuminated and polarized scene.

Reference data are obtained for polarizer transmission axis orientations of 0° to isolate exp(*j*2*π Ux*) in Eq. 4 and 45° to isolate exp(*j*2*πUx*)exp(*j*2*πUy*) in Eq. 5. These reference data are divided by the unknown sample data per Eqs. 6–9 below,

This method works well to calibrate and model the performance in the visible spectrum, where absorption in the birefringent material comprising the prisms is negligible. However in the infrared, absorption in materials is common, and differential absorption between the fast and slow axes of birefringent materials (dichroism) is often a concern. Since dichroism causes the crystal to behave as a diattenuator, it can produce error if it’s not accounted for in the calibration procedure.

Moreover, when modeling birefringent materials, it is often wise to include the diattenuation created by the difference in the Fresnel transmission coefficients seen by rays entering the ordinary or extraordinary axes. This is made more pertinent for the BPIP since the prisms are not typically anti-reflection (AR) coated. Since the eigenvectors of the diattenuation created by the dichroism and the Fresnel coefficients commute, both effects can be simultaneously incorporated into the calibration procedure.

## 3. Infrared system model

In order to account for the dichroism and the impact of Fresnel losses in the prisms, the general Mueller matrix for a linear diattenuator is implemented at the front and rear interfaces of each prism. Additionally, one of these matrices is used at the front of each prism to allow for modeling the dichroism, while the prisms themselves are modeled as a spatially varying retarder. Note that placement of the diattenuation matrices before or after each of the prism retardance matrices makes no difference since the dichroism and Fresnel losses are aligned to the retardance axes. A general diattenuator can be written as,

where *T _{x}*,

*T*are the transmission ratios in the

_{y}*x*and

*y*directions,

*θ*is the angle at which the diattenuator is oriented, and

*R*(

*θ*) is the Mueller rotation matrix,

Modeling each prism as a spatially varying retarder yields,

where *δ _{n}*(

*x*,

*y*) indicates the retardance and

*θ*the orientation of the

_{n}*n*

^{th}prism. The spatially varying retardances

*δ*

_{1}(

*x*,

*y*) through

*δ*

_{4}(

*x*,

*y*) are defined as,

$${\delta}_{3}(x,y)=\frac{2\pi B}{\lambda}\left[{d}_{T}+\mathrm{tan}\left(\beta \right)\left(\frac{{d}_{x}}{2}+x\right)\right]{\delta}_{4}(x,y)=\frac{2\pi B}{\lambda}\left[{d}_{T}+\mathrm{tan}\left(\beta \right)\left(\frac{{d}_{x}}{2}-x\right)\right]$$

where *d _{x}*,

*d*are the prism’s

_{y}*x*and

*y*side lengths, respectively,

*d*the thickness of the terminus (Fig. 1), and

_{T}*B*is the birefringence.

Diattenuation matrices are now configured for each prism. Beginning with Fresnel losses, normal incidence is assumed since chief ray angles are less than 10°. With the fast-axis oriented at 0° (parallel to the *x* axis) the Fresnel transmission coefficients for crystal-air interfaces are established as,

where the subscript *Fca* refers to the Fresnel losses at the crystal-air interface. The same can be done with the crystal-glue interfaces,

where, similarly, the subscript *Fcg* refers to the Fresnel losses at the crystal-glue interface. For the absorption contribution of the ordinary (*α _{o}*) and extraordinary (

*α*) axes, we have,

_{e}where *d _{zn}* is the thickness of the

*n*

^{th}prism as a function of

*x*and

*y*,

$${d}_{z3}={d}_{T}+\left(\frac{{d}_{x}}{2}+x\right)\mathrm{tan}\left(\beta \right)\phantom{\rule{.5em}{0ex}}{d}_{z4}={d}_{T}+\left(\frac{{d}_{x}}{2}-x\right)\mathrm{tan}\left(\beta \right)$$

Multiplying the sequence of diattenuation matrices with that of the prism retardance matrices yields the total system Mueller matrix *M _{sys}*,

$${M}_{D}({T}_{x,\mathrm{Fcg}},{T}_{y,\mathrm{Fcg}},45\xb0)\xb7{M}_{p3}\xb7{M}_{D}({T}_{x3,\alpha},45\xb0)\xb7{M}_{D}({T}_{x,\mathrm{Fcg}},{T}_{y,\mathit{Fcg}},45\xb0)\xb7\dots \text{}$$

$${M}_{D}({T}_{x,\mathrm{Fcg}},{T}_{y,\mathrm{Fcg}},90\xb0)\xb7{M}_{p2}\xb7{M}_{D}({T}_{x2,\alpha}{,T}_{y2,\alpha},90\xb0)\xb7{M}_{D}({T}_{x,\mathrm{Fcg}},{T}_{y,\mathrm{Fcg}},90\xb0)\xb7\dots \text{}$$

$${M}_{D}({T}_{x,\mathrm{Fcg}},{T}_{y,\mathrm{Fcg}},0\xb0)\xb7{M}_{p1}\xb7{M}_{D}({T}_{x1,\alpha},0\xb0)\xb7{M}_{D}({T}_{x,\mathrm{Fca}},{T}_{y,\mathrm{Fca}},0\xb0)$$

where *A* is the ideal Mueller matrix for a linear horizontal polarizer (*T _{x}*=1,

*T*=0, and

_{y}*θ*=0° per Eq. (10)). Recalling that these diattenuation matrices commute with respect to their associated retardance matrices enables the simplification of this expression by grouping the Fresnel transmittances and absorption losses together as,

where *T _{xn}* and

*T*are implicit functions of

_{yn}*x*and

*y*. This allows one to write,

$${M}_{D}({T}_{x2},{T}_{y2},90\xb0)\xb7{M}_{p2}\xb7{M}_{D}({T}_{x1},{T}_{y1},0\xb0)\xb7{M}_{p1}$$

Expansion of this expression yields the following output for the intensity when multiplied by an arbitrary Stokes vector ([*S*
_{0}, *S*
_{1}, *S*
_{2}, *S*
_{3}]^{T}),

$$\left[\frac{1}{8}\left({T}_{x4}{T}_{y3}+{T}_{x3}{T}_{y4}\right)\left({T}_{x1}{T}_{y2}-{T}_{x2}{T}_{y1}\right)+\frac{1}{4}\left({T}_{x1}{T}_{y2}+{T}_{x2}{T}_{y1}\right)\sqrt{{T}_{x3}{T}_{y3}{T}_{x4}{T}_{y4}}\mathrm{cos}\left({\delta}_{3}-{\delta}_{4}\right)\right]{S}_{1}+$$

$$\left[\frac{1}{4}\sqrt{{T}_{x1}{T}_{y1}{T}_{x2}{T}_{y2}}(\left({T}_{x3}{T}_{y4}-{T}_{x4}{T}_{y3}\right)\mathrm{cos}\left({\delta}_{1}+{\delta}_{2}\right)+2\sqrt{{T}_{x3}{T}_{y3}{T}_{x4}{T}_{y4}}\mathrm{sin}\left({\delta}_{1}-{\delta}_{2}\right)\mathrm{sin}\left({\delta}_{3}-{\delta}_{4}\right))\right]{S}_{2}+$$

$$\left[\frac{1}{4}\sqrt{{T}_{x1}{T}_{y1}{T}_{x2}{T}_{y2}}\left(\left({T}_{x3}{T}_{y4}-{T}_{x4}{T}_{y3}\right)\mathrm{sin}\left({\delta}_{1}-{\delta}_{2}\right)-2\sqrt{{T}_{x3}{T}_{y3}{T}_{x4}{T}_{y4}}\mathrm{cos}\left({\delta}_{1}-{\delta}_{2}\right)\mathrm{sin}\left({\delta}_{3}-{\delta}_{4}\right)\right)\right]{S}_{3}$$

Taking the Fourier transform of Eq. (21), filtering the appropriate frequency components, and inserting the appropriate channels into Eqs. (6)–(9) yields the following instrumental outputs for an arbitrary input,

There are several important aspects regarding Eqs. (21)–(25),

1. In Eq. (23) it can be observed that when *T _{x1}*

*T*

^{y2}equals

*T*

_{x2}

*T*

_{y1},

*S*

*1*,

*yields the normalized input Stokes component*

_{sample}*S*

_{1}/

*S*

_{0}. Conversely, when dichroism and Fresnel effects cause these terms to differ, a fraction of the energy from

*S*

_{0}emerges in the numerator due to the nonzero (

*T*

_{x1}*T*

_{y2}-

*T*

_{x2}

*T*

_{y1}) term. The reason for this is seen in Eq. 21, in which a portion of

*S*

_{0}is directly modulated by cos(

*δ*

_{]}-

*δ*

_{4}). Likewise, energy from

*S*

_{1}is affected in a similar manner to influence

*S*

_{0}in the denominator.

2. In Eqs. (24) and (25) similar trends are observed. Again, if *T*
_{x1}
*T*
_{y2} equals *T*
_{x2}
*T*
_{y1} then *S*
_{2,sample} and *S*
_{3,sample} yield their correct normalized inputs. Conversely, when *T*
_{x1}
*T*
_{y2} and *T*
_{x2}
*T*
_{y1} differ, error is introduced into *S*
_{2,sample} and *S*
_{3,sample} at image locations where *S*
_{1} is present; a result of the nonzero (*T*
_{x1}
*T*
_{y2} - *T*
_{x2}
*T*
_{y1}) term in the denominator.

3. From Eqs. (22)–(25) it’s apparent that only the dichroism and Fresnel effects from prisms 1 and 2 contribute error to the polarimetric measurements. This illustrates that the transmission differences of prisms 3 and 4 are removed during the normalization procedure inherent to the reference beam calibration technique.

4. In Eq. (21) S_{2} and S_{3} are directly modulated by cos(*δ _{1}* -

*δ*) or sin(

_{2}*δ*-

_{1}*δ*). Consequently their energy appears as an additional modulation in a new (previously empty) channel at (ξ=0, η=+/-

_{2}*U*) (see Fig. 3 for reference). Since this modulation doesn’t alias into any useful channels, it has no affect on the reconstructed data.

In order to remove the influence of the prism’s diattenuation on the measured Stokes parameters, a modified calibration procedure must be implemented from that depicted previously in §2.

## 4. Infrared polarimetric calibration procedure

For a more adequate calibration of the instrument, the effect that the dichroism and Fresnel losses have on the reconstruction of the incident polarization state must be removed. Rewriting Eqs. (22) and (23) for convenience yields,

where *γ*=(*T*
_{x1}
*T*
_{y2}+*T*
_{x2}
*T*
_{y1}) and *ε*=(*T*
_{x1}
*T*
_{y2}-*T*
_{x2}
*T*
_{y1}) [4]. 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,

which yields a corrected normalized measurement for *S*
_{1}. Additionally, Eq. (28) provides a corrected estimate for *S*
_{0} in the sample data. Since *S*
_{2} and *S*
_{3} incur no error when normalized to the sample *S*
_{0} per Eq. (28), correction is only needed for *S*
_{0,reference,45°} in the 45° *S*
_{2} and *S*
_{3} reference data. Since *S*
_{1} is zero for this input, the *S*
_{0,reference,45°} component has the form,

where the prime is used to indicate that this is not the same *S*
_{0} as in Eq. (26) or (27). To remove the dichroic and Fresnel contribution from *S*
_{2} and *S*
_{3}, we divide by *γ* and Eq. 28 in the reference beam calibration,

Thus the Stokes vectors *S*
_{1}, *S*
_{2}, and *S*
_{3} can have the dichroic and Fresnel contribution from the prisms fully corrected with this method.

In order to implement this calibration procedure it’s necessary to know the precise spatial distribution of the transmission components, specifically corresponding to *T*
_{x1}, *T*
_{y1}, *T*
_{x2}, and *T*
_{y2}. Even before the prisms are cemented together, this is an unlikely situation, and becomes extremely challenging after they are bonded. Fitting the model to calculated parameters is possible; yet it would be more precise if one can acquire the data regarding *ε* and *γ* within the polarimeter. Fortunately, this is made possible after a few approximations.

## 4.1 S_{1} approximation

Since measurement of *ε* and *γ* inside the polarimeter is not easily feasible, Eq. 30 can be rearranged as,

This rearrangement is beneficial since the ratio *ε*/*γ* can be approximated from a measurement of unpolarized light, which provides an interferogram that contains the combined diattenuation of prisms 1 and 2,

Therefore measurement of *S*
_{1} with normalization to *S*
_{0} yields *ε*/*γ*. However, this state will be reconstructed *without* correction for the dichroic contribution and must use the original calibration scheme. Using Eq. (7) to reconstruct the polarization state measured from the unpolarized source yields an output of,

If we make the assumption that *ε* is small, then *ε*
^{2}→0 and we can write,

Hence for small ε, measurement of unpolarized light can provide the needed information to yield correction for the dichroism and Fresnel losses in S1.

## 4.2 S_{2} and S_{3} calibration

Another aspect to the required approximations involves division by *γ* in Eq. (32) and (33). Recall that this 1/*γ* expression exists to remove the linear dependence on *γ* seen in *S*
_{0,reference,45°}(*x*,*y*). Since we only have a ratio of *ε*/*γ*, dividing by *γ* isn’t feasible; therefore an approximation for *S*
_{0} in Eq. (28) must be produced that leaves a first order linear dependence upon *γ*. Through some trial and error, an expression was developed that acquires this property,

To demonstrate the dependence upon *γ* in *S*
_{0,approx}, Eq. (26) and (27) can be substituted into this new expression,

For small *ε* this equation can be approximated with the same justification seen previously. When *ε*
^{2}→0,

Therefore division by *γ* in Eq. (32) and (33) is unnecessary since *γ* in *S*
_{0,reference,45°}(*x*,*y*) is removed to first order. This makes the calibration procedure for *S*
_{2} and *S*
_{3},

## 4.3 Approximation validation

To verify this approach’s validity in simulation, we begin by assuming a realistic Yttrium Vanadate (YVO_{4}) prism configuration seen in Fig. 4. Recall that prisms 1 and 2 are solely responsible for the dichroic error, such that prisms 3 and 4 can be neglected in this analysis. For proper simulation of the dichroism, the absorption coefficient for YVO_{4} was measured from several flat samples on our Fourier transform spectrometer, and can be seen in Fig. 5 for the ordinary and extraordinary axes [6].

To investigate the maximum error in the approximations, a region on the edge of the prism (far away from its center) will be simulated. This corresponds to the maximum absorption difference between orthogonal polarizations, as the diattenuation neglecting Fresnel losses is zero in the center due to symmetry. Performing this analysis by calculating Eq. 36 and Eq. 37 and comparing them yields the result in Fig. 6. The calculated percent error between the exact expression *εγ*(*ε*
^{2}+*γ*
^{2}) and its approximation *ε*/*γ*, given reasonable estimates of the Fresnel and absorption losses, peaks at 0.14% at 4 µm. This much error in *ε*/*γ* translates into an error in *S*
_{1,corrected} of 0.23%, 0.02%, and 0.007% for an input *S*
_{1} of 0.02, 0.20, and 0.50, respectively. This is a significant improvement, as errors for these inputs before dichroic correction are 181%, 17.2%, and 5.3%, respectively. Consequently, the improved calibration provides more than 2 orders of magnitude less error over the previous calibration technique.

Using the same approach, the error in *S*
_{2} and *S*
_{3} due to the approximation made for the reference data can be simulated. Setting [*S*
_{0}, *S*
_{1}, *S*
_{2}, *S*
_{3}]^{T}=[1,1,0,0]^{T} and again choosing the edge of the prism in Fig. 4, we can compare the exact calibration procedure, *S*
_{0,reference,45°}(*x*,*y*)/*γ*
*S*
_{0}(*x*,*y*), to the approximate calibration, *S*
_{0,reference,45°}(*x*,*y*)/*S*
_{0,approx}(*x*,*y*), and to the calibration that neglects diattenuation effects, *S*
_{0,reference,45°}(*x*,*y*)/*S*
_{0,sample}(*x*,*y*). The percent error from these calculations can be seen in Fig. 7. Since the reference beam calibration is directly proportional to *S*
_{0,reference,45°}(*x*,*y*)/*S*
_{0,approx}(*x*,*y*) or *S*
_{0,reference,45°}(*x*,*y*)/*S*
_{0,sample}(*x*,*y*), error in the reconstructions are identical to the errors observed in Fig. 7. Consequently, the approximation results in a peak error in *S*
_{2} and *S*
_{3} of 0.14% at 4 µm and 3.6% if no correction is made. Hence these are valid approximations to make, vastly simplifying the calibration procedure.

## 5. Radiometric calibration

Similar to other polarimeters operating in the infrared, detector offsets and emission from the optics must be accounted for with the BPIP. Even though its calibration differs significantly from a conventional polarimeter, the experimental procedures used for nullifying the offset are similar [1].

The polarimetric calibration setup used to obtain reference data can be seen in Fig. 8. Emission at temperature *T* and reflection from an area blackbody, *S*
_{0,bb,T} and *S*
* _{0,bb,R}*, respectively, is used as the source and imaged through a generating polarizer that’s employed to create the 0° and 45° reference data. The generator emits energy (

*S*) and reflects sources from the surrounding environment (

_{0,e}*S*). The optics also emit energy (

_{0,r}*S*,

_{0,objective}*S*, and

_{0,prism}*S*) in addition to the FPA’s analog to digital conversion offset (

_{0,R&F}*S*). It should be noted that the reflected term is included for the blackbody since, in practice, these never have an ideal emissivity of 1.0.

_{0,FPA}First the offset from the reference data must be removed. The total offset including the FPA’s contribution is,

where,

Acquiring a reference measurement of the blackbody through the generator yields,

$$+\left[\left({S}_{0,\mathrm{bb},T}+{S}_{0,\mathrm{bb},R}+{S}_{0,r}+{S}_{0,e}+{S}_{0,\mathrm{objective}}\right)\epsilon +{S}_{1,\mathrm{bb},T}\gamma \right]\mathrm{cos}\left(2\pi \mathrm{Ux}\right)$$

$$+{S}_{2,\mathrm{bb},T}\left[\mathrm{cos}\left(2\pi U\left(x+y\right)\right)-\mathrm{cos}\left(2\pi U\left(x-y\right)\right)\right]$$

In a procedure similar to [7] and [8], the reference data are taken at two different blackbody temperatures *T _{1}* and

*T*. Subtraction of the two data sets removes the offsets from

_{2}*S*,

_{0,e}*S*,

_{0,r}*S*

_{0,objective},

*S*

_{0,optics}, and

*S*. This results in an intensity pattern that’s proportional to the difference in the blackbody emission terms, ${S}_{0,\mathrm{bb},{T}_{1}}$ and ${S}_{0,\mathrm{bb},{T}_{2}},$

_{0,bb,R}$$\left[\left({S}_{0,\mathrm{bb},{T}_{2}}-{S}_{0,\mathrm{bb},{T}_{1}}\right)\epsilon +\left({S}_{1,\mathrm{bb},{T}_{2}}-{S}_{1,\mathrm{bb},{T}_{1}}\right)\gamma \right]\mathrm{cos}\left(2\pi \mathrm{Ux}\right)+$$

$$\left({S}_{2,\mathrm{bb},{T}_{2}}-{S}_{2,\mathrm{bb},{T}_{1}}\right)\left[\mathrm{cos}\left(2\pi U\left(x+y\right)\right)-\mathrm{cos}\left(2\pi U\left(x-y\right)\right)\right]$$

Since *S _{1}*,

*,*

_{bb}*and*

_{T}*S*increase linearly with respect to

_{2,bb,T}*S*, the normalized Stokes vector from the difference of the two temperatures is equal to the original normalized inputs,

_{0,bb,T}Therefore we have preserved the necessary relationship for the calibration while removing the offset terms from the reference data.

Similarly, the offsets must be isolated from the sample data taken after the generator and blackbody are removed. These data will contain the offset related to *S _{0,optics}* with an additional term,

*S*, which is analogous to

_{0,sample}*S*but with an arbitrary spatial distribution. The sample interferogram is,

_{0,bb,T}$$\left[\epsilon \left({S}_{0,\mathrm{sample}}+{S}_{0,\mathrm{objective}}\right)+\gamma {S}_{1,\mathrm{sample}}\right]\mathrm{cos}\left(2\pi \mathrm{Ux}\right)$$

$$+\mid {S}_{23,\mathrm{sample}}\mid \left[\mathrm{cos}\left(2\pi U\left(x+y\right)+\mathrm{arg}\left[{S}_{23,\mathrm{sample}}\right]\right)-\mathrm{cos}\left(2\pi U\left(x-y\right)-\mathrm{arg}\left[{S}_{23,\mathrm{sample}}\right]\right)\right]$$

To avoid normalization of the reconstructed Stokes vector to a value larger than *S*
_{0,sample}, we must measure *S*
_{0,optics} and *S*
_{0,objective}, using an appropriate experimental procedure, and remove them from *I _{sample}*.

The predominant method for estimating these offset terms involves imaging an area blackbody at several known temperatures. Plotting the detector output at each pixel as a function of *T*
^{4} and extrapolating to a temperature of *T*=0 K yields the total offset [7]. However, there are certain complications with this technique due to the BPIP’s narrow bandwidth of 50 nm (centered at 4.5 µm). Using an f/2.3 objective lens and InSb FPA, we obtain a theoretical noise equivalent temperature difference (NEΔT) of 0.37 K at 300 K, 0.16 K at 333 K, and 0.05 K at 393 K [9]. Therefore high blackbody temperatures (greater than 333 K) are required to obtain signal to noise ratios greater than 10; a temperature that our calibrated area blackbody isn’t capable of. Therefore, given this limitation, an alternative method is required.

An alternative for calculating the offset is to image a cold area blackbody placed directly in front of the main objective. If the instrument images a temperature close to liquid nitrogen (77 K), then we obtain a band-averaged irradiance onto the FPA of 446E-18 W/m2 over 4.475–4.525 µm. Assuming the BPIP is viewing a blackbody with an emissivity of 0.985 in a 24 °C background, we obtain an FPA flux of 2.88E-12 W. Therefore the total irradiance from a cold blackbody in this waveband is almost equal to the theoretical noise equivalent power (NEP) of 2.79E-12 W. Consequently * _{S0,optics}* and

*will be the predominant components of the recorded signal. To implement this technique, a blackbody in the shape of a cone was prepared from fabric, where the conical shape effectively increases the fabric’s emissivity [10]. Soaking this cone in liquid nitrogen and placing it in front of the polarimeter provides a black scene to view for the brief time required to obtain an offset measurement. This yields an output of,*

_{S0,objective}Subtracting Eq. 49 from Eq. 48 yields the sample data sans the offsets,

$$+\mid {S}_{23,\mathrm{sample}}\mid \left[\mathrm{cos}\left(2\pi U\left(x+y\right)+\mathrm{arg}\left[{S}_{23,\mathrm{sample}}\right]\right)-\mathrm{cos}\left(2\pi U\left(x-y\right)-\mathrm{arg}\left[{S}_{23,\mathrm{sample}}\right]\right)\right]$$

Consequently, by use of Eq. 46 and 50, the offsets can be removed from the reference and sample data.

## 6. Experimental setup

The experimental configuration for the polarimeter can be seen in Fig. 9, where a set of YVO_{4} prisms (*β*=3.2°) and an analyzer are located near the image plane of an objective lens. A second lens relays the objective’s intermediate image onto a 320×256 InSb FPA. A bandpass filter (50 nm bandwidth at 4.5 µm) is included to maintain fringe visibility in the highest OPD regions of the prism, where the OPD is zero in the center and increases to +/- 40 waves at either edge. This means fringe visibility goes to zero for a bandpass of 112 nm at 4.5 µm (using OPD=*λ*
^{2}/Δ*λ*); hence fringes are visible over the entire prism given our 50 nm bandwidth.

In order to verify the accuracy of the data, an area blackbody was rotated in front of the instrument at various orientations (*θ*), as indicated in Fig. 10, from 0° to 80° in 10° increments. It consists of a flat aluminum plate that’s painted with high temperature Krylon black spray paint (emissivity estimated at 0.93) and attached to an electric hotplate. This produces a scene in which variations in the *s* and *p* Fresnel transmission coefficients (τ* _{p}*, τ

*) cause the emitted light’s degree of linear polarization to gradually increase for increasing*

_{s}*θ*[11]. The same measurement is also performed on our non-imaging MWIR spectropolarimeter to validate the BPIP measurement.

Another source used in our experiments is a spherical incandescent vanity light bulb that’s also painted with Krylon high temperature black spray paint. Spherical emitters like this produce a continuously rising degree of linearly polarized emission as one views increasingly oblique surfaces of the sphere [12]. This is due to the changing normal vector of the spherical surface as seen by the optical system, where the angle of incidence within the plane of incidence varies from 0° to 90°. The orientation of the linearly polarized states also changes uniformly across the surface due to the normal vector’s changing projection onto the *xy* plane.

## 7. Experimental results

The raw data and the degree of linear polarization (DOLP) for the tilted blackbody plate can be seen in Fig. 11 for orientations (*θ*) of 0°, 60°, and 80°. On-axis DOLP values from these data are 0.0016, 0.0967, and 0.2841, respectively. The temperature of the blackbody’s surface is approximately 212 °C in a room temperature (24 °C) environment. Using complex index of refraction data (*n*,*k*) for ultra flat black Krylon spray paint calculated at 10 µm in Ref. [11], it’s possible to demonstrate that the obtained values are theoretically reasonable for a similar material. For an (*n*,*k*)=(1.28, 0.30) we obtain comparable DOLP’s for the same orientations of 0°, 60°, and 80°, of 0, 0.0858, and 0.2250, respectively. Discrepancies between the theoretical and measured values are primarily due to the different waveband and type of paint used in Ref [11].

A critical aspect of these data are that fringes exist in the raw data when the area blackbody is viewed at normal incidence (e.g. an unpolarized source); a direct result of the dichroism and Fresnel losses. As was mentioned previously in §4.1 and §4.2, measurement of an unpolarized source and its subsequent reconstruction via Eq. (7) provides the approximate measurement of *ε*/*γ*. To generate unpolarized light for this purpose, the area blackbody is oriented at *θ*=0° and viewed by the instrument. However it should be noted that polarization exists in the blackbody off-axis due to the angle that the chief ray makes with the plate’s surface normal within the plane of incidence. Given a field of view of 17°, the maximum angle of incidence for the chief ray is 8.5°. Using an (*n*,*k*) of (1.28, 0.30), this yields a DOLP of 1.0E-3. Since this value is at least an order of magnitude smaller than the expected contribution from the dichroism and Fresnel effects, it is neglected in our analysis.

Also important is that to obtain fringes for an unpolarized input such that *ε*/*γ* can be approximated, the camera non-uniformity correction (NUC) must be executed before the analyzing polarizer is inserted into the optical path. This ensures any fringes that are present while looking at the source during the NUC calibration procedure aren’t interpreted as part of the offset, and consequently removed from the image. Implementation of this NUC procedure and use of Eq. 7 yields the estimated spatial distribution of *ε*/*γ* per Fig. 12 alongside the expected *ε*/*γ* from the theoretical model.

From the theoretical model of the dichroism, the anticipated spatial distribution of *ε*/*γ* forms an inclined plane (Fig 12, right). This slope is visible in the measured data, and can be seen more clearly in Fig. 13 in which a cross section of *ε*/*γ* was extracted from the region under the dashed lines per Fig. 12. The error between the slope of the fitted line (*m _{fit}*=-4.04E-5 units/pixel) to that of the simulation (

*m*=-2.85E-5 units/pixel) is 42%. Likely reasons for this discrepancy are higher order effects; namely polarization aberrations from the fast objective lens (f/2.3), multiple reflections from the non-AR coated prism interfaces, and the effect of a relatively high numerical aperture (NA=0.2174) beam focusing through prisms that aren’t infinitely thin.

_{sim}## 7.1 DOLP verification

Since (*n*,*k*) of the high temperature Krylon paint on the rotating hotplate is unknown, we make use of our Fourier transform channeled spectropolarimeter (FTSP) to verify the accuracy of the BPIP reconstructions. For an accurate comparison between the two instruments, only pixels close to the optical axis of the BPIP can be used. This is made clear in Fig. 11 (*θ*=60°), where the DOLP of the blackbody increases to the left of the center and decreases to the right of it. This gradient across the sample is strictly a field of view (FOV) issue related to the angle of incidence that the chief ray has with respect to the tilted plate’s surface normal, e.g. the effective *θ* is grater than 60° to the left of center and less than 60° to the right of it. Since our FTSP has a non-imaging (single pixel) detector, only the pixel on the optical axis of the BPIP has an equivalent chief ray. Therefore only the center pixel is used for verification.

A comparison of the BPIP to the FTSP data can be seen in Fig. 14. Typical error of the FTSP is on the order of 1% for linear polarization measurements [4]. The RMS error between the BPIP and FTSP measurements, ${\epsilon}_{\mathrm{RMS}}=\sqrt{\frac{1}{N}\sum _{1}^{N}{(x\left(n\right)-x\text{'}\left(n\right))}^{2}}$, is 0.0073, meaning there’s greater than 99% agreement between the two instruments. Without the improved calibration, it’s estimated that agreement would be 97% at 4.5 µm and 85% at 4 µm, where dichroic effects are significantly larger. Again, these metrics are for measurements made on-axis. Off-axis performance is anticipated to have similar accuracy except a 15 pixel wide border encapsulating the edges of the image; these contain aliasing errors due to the Fourier domain data reduction process.

## 7.2 Spherical source

Data were also obtained from the spherical emissive source. An unprocessed image of the light bulb can be seen in Fig. 15.

The normalized Stokes components for this source can be seen in Fig. 16.

As expected, positive and negative *S*
_{1} components are located on the left/right and top/bottom portions of the sphere, corresponding to linear horizontal and linear vertical polarization states, respectively. For *S*
_{2}, the emission pattern seen in the *S*
_{1} image is rotated by 45° and for *S*
_{3} no significant signature is detected. Using the above data, the DOLP, orientation, and ellipticity are calculated per Fig. 17.

Here we see a nearly uniform DOLP around the perimeter of the light bulb with some minor fluctuations, especially near the bottom where the surface is coolest. This is due to uneven heating of the surface by the filament inside the bulb. The orientation angle demonstrates a continuously varying distribution, with reliable performance close to the singularity in the center. Lastly, as expected, the ellipticity angle is essentially zero for the polarized emission, with destabilization of the value in the middle close to the singularity.

## 8. Conclusion

The BPIP technique has been demonstrated in the MWIR using YVO_{4} prisms, where polarized emission from various blackbodies was successfully observed. General calibration techniques have been developed that enable accurate use of the BPIP method when operating in the infrared. These include accounting for radiometric offsets and dichroism inherent to many infrared birefringent materials, both of which are common issues within this spectral region. Additionally, diattenuation due to differential Fresnel losses between the ordinary and extraordinary rays have been included in the calibration; a facet that can also be useful when calibrating a BPIP in the visible spectrum. The measurements between the BPIP and our FTSP show an agreement greater than 99% for the sample tested. Future work will focus on characterization of the higher order effects seen in the estimate of *ε*/*γ*, which are likely due to polarization aberrations from the objective lens, multiple reflections from the non-AR coated prism interfaces, and the high NA beam that’s focused through the thick prisms.

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