## Abstract

We discuss achievement of a long-standing technology goal: the first practical realization of a quantitative-grade, field-worthy snapshot imaging spectropolarimeter. The instrument employs *Polarimetric Spectral Intensity Modulation* (PSIM), a technique that enables full Stokes instantaneous “snapshot” spectropolarimetry with perfect channel registration. This is achieved with conventional single beam optics and a single focal plane array (FPA). Simultaneity and perfect registration are obtained by encoding the polarimetry onto the spectrum via a novel optical arrangement which enables sensing from moving platforms against dynamic scenes. PSIM is feasible across the electro-optical sensing range (UV-LWIR). We present measurement results from a prototype sensor that operates in the visible and near infrared regime (450–900 nm). We discuss in some detail the calibration and Stokes spectrum inversion algorithms that are presently achieving 0.5% polarimetric accuracy.

© 2004 Optical Society of America

## 1. Introduction

The fusion of polarimetric and spectral imaging into a single sensor has clear benefits and broad application to many remote sensing problems. However, polarimetric remote sensing outside the laboratory has, to date, been plagued by spatial and temporal misregistration artifacts masquerading as false polarization[20][21][16][17][18]. The technique we describe here overcomes these prior-art shortcomings without introducing new burdens such as alignment and device challenges.

In this paper we describe *Polarimetric Spectral Intensity Modulation* (PSIM)[4][5] technique and present measurement results from a prototype sensor operating in the visible and near infrared regime (450–900 nm). We discuss in some detail the calibration and Stokes spectrum inversion algorithms that are presently achieving 0.5% polarimetric accuracy. Oka and Kato [6] independently report the PSIM technique and described a wideband Fourier inversion method.

## 2. PSIM sensor principle of operation

#### 2.1. Overview

The PSIM technique employs a novel configuration of two *stationary* birefringent crystals, followed by a *stationary* polarizer, packaged in a *polarization module*. When placed optically upstream from the spectrometer dispersing element (in our design, outside the spectrometer before the slit), this polarization module produces polarization-dependent interference fringes of the measured intensity spectrum (cf. Fig. 1). There are no moving parts. The full Stokes vector spectrum can subsequently be retrieved from the modulated intensity spectrum, based on the fringe patterns. Both 1D (line-imaging) and 2D imaging spectrometers can be employed. In a single “snapshot” (integration period), the focal plane in the slit-based line-imaging spectrometer records spatial information along one dimension (columns) and modulated spectra along the other dimension (rows).

We have demonstrated the PSIM technique in the laboratory at both visible and LWIR (8–12 *µ*m) wavelengths. An example of the PSIM spectrum from a laboratory demonstration is also shown within Fig. 1. The incident light was essentially 100% polarized. Note the 0 and 90 degree cases are of opposite phase. The modulation effect is sinusoidal in *wavenumber*, hence the chirped appearance in wavelength. The 45 and 135 degree cases contain other frequency components, as can be seen from the shallower nulls at some wavelengths. In short, this modulation is produced by a wavelength dependent interference phenomenon of the fast and slow waves in two birefringent crystals. We show later how the spectrum can be demodulated to retrieve the full Stokes spectrum.

## 2.2. Mueller matrix derivation

The Mueller calculus is a convenient means of describing the polarization state of a single beam of light as it propagates through a system [8]. The system Mueller matrix describing on-axis propagation through the PSIM polarization module per se is simply

which is evaluated using standard forms for the component matrices. However, since the photon detector measures only intensity, only the first row of **M**
_{sys}
is relevant for describing the detected spectral intensity *𝓘*(*ν*). If the sum and difference retardances of the two crystals are denoted ∑*ϕ* and Δ*ϕ*, respectively, then the following expression for *𝓘*(*ν*) can be written:

$$=\frac{1}{2}[I,Q\mathrm{cos}{\varphi}_{2},\frac{1}{2}U\left(\mathrm{cos}\Delta \varphi -\mathrm{cos}\Sigma \varphi \right),\frac{1}{2}V\left(\mathrm{sin}\Sigma \varphi -\mathrm{sin}\Delta \varphi \right)]$$

where,

*ν*=Optical frequency (Hz)

*c*=Speed of Light

*𝓘*(*ν*)=Detected spectral intensity

**m**
_{sys,1}(*ν*)=First row of system Mueller matrix, **M**
_{sys}

s(*ν*)=Incident Stokes vector, [*I*
*Q*
*U*
*V*]
^{T}

Δ*ϕ*=*ϕ*
_{1}-*ϕ*
_{2}

∑*ϕ*=*ϕ*
_{1}+*ϕ*
_{2}

*ϕ*
_{1}=2*πν*(*n*_{e}
-*n*_{o}
)*ℓ*
_{1}/*c*

*ϕ*
_{2}=2*πν*(*n*_{e}
-*n*_{o}
)*ℓ*
_{2}/*c*

*n*_{o}
,*n*_{e}
=Ordinary and extraordinary indices of refraction

*ℓ*
_{1},*ℓ*
_{2}=Crystal lengths.

and presuming our canonical crystal and polarizer orientations.

## 2.3. Practical system model

Equations 2 indicates an idealized sensor system model but does not include the effects of retardance spectral variation, spatial and spectral blurs, and finite pixel subtense. The inclusion of these effects is essential to obtaining useful, calibrated spectropolarimetry. Thus, a more accurate version of Eq. 2 is

where *λ*
_{∘} is the center wavelength of the pixel, *𝓘*_{raw}
(*λ*
_{∘}) is the raw uncorrected digitized pixel reading, *R*(*λ*) is the pixel responsivity in A/D counts per radiance unit, *C*(*λ*
_{∘}) is the offset of the pixel *λ*
_{∘} in A/D counts, *K*(*λ*) is the spectral blur function, and Δ*n*(*λ*) is the birefringence and the integration is over the spectral blur subtended by a single pixel.

We assume detector responsivity is constant across a particular pixel’s subtense. For moderately high fringe densities, this assumption could lead to notable error, if indeed a deep PSIM fringe suffered unanticipated differential weighting across a pixel’s spatially varying responsivity. This might be of particular concern for FPA detectors employing microlenses, but to date we have not observed such potential artifacts. Thus, the flat-fielded PSIM-modulated pixel measurement *L*(*λ*
_{∘}) in radiance units is:

## 2.3.1. Spectral blur kernel model

We assume a gaussian system spectral blur that includes the combined effects of spectrometer slit function and the point spread function of the optics. It is *exclusive* of spectral averaging effect due to the finite width of the pixel. One is led to the following model for *L*(*x*_{o}
,*y*_{o}
), the amount of signal measured by an *entire* pixel centered at (*x*_{o}
,*y*_{o}
) with spectral integer index *x*_{o}
, from gaussian system blurring of dispersed light source *s* (*λ*):

with *K*(*x*) the spectral blur kernel function for varying spectral index *x*, taken as a continuous variable. The function *λ* (*x*) describes the spatial position to wavelength mapping. If a diffraction grating is used as the dispersive element, then the mapping is nominally a linear one (i.e. *λ* (*x*)=*mx*+*λ*_{b}
). However, to obtain well calibrated results, one must make use of slightly more elaborate models which are described in the following sections. The normalized blur parameter σ expresses the gaussian system blur width in pixel integer index units. Therefore, *K*(*x*), and not the gaussian kernel, is the appropriate blur function *K* to use in the high-fidelity system model (Eq. 4). *K*(*x*) is mirror-symmetric and normalized to unit area. The spectral blur kernel *K*(*x*) is approximately gaussian for moderate normalized blur parameter values, and approaches the ideal tophat averaging response for negligible system blur.

## 3. Calibration for Stokes inversion

#### 3.1. Calibration sequence for estimating system model

For the PSIM Mueller matrix system equation (Eq. 2) and the non-ideal forward model (Eq. 4), the system parameters can be sufficiently well known or calibrated to permit its mathematical inversion to estimate the incident Stokes spectrum. We factor the forward system model estimation problem into the following calibration sequence, designed to estimate the component terms of the high-fidelity system model (Eq. 4).

## 3.1.1. Global spectral calibration

Global spectral calibration fits the observed dispersion of gas-lamp atomic emission lines to a low-order polynomial model, globally-applicable across the pixel array. This wavelength-to-pixel mapping model *λ* (*x*_{o}
,*y*_{o}
) is parameterized by a quadratic polynomial in *x*_{o}
and linear term in *y*_{o}
(accounting for any slant due to slight rotational misalignment of the grating in its mount), where *x*_{o}
is the spectral pixel integer index on the FPA, i.e. along the spectral dispersion dimension, and *y*_{o}
is the spatial along-slit dimension integer index. In addition, the calibration determines the blur-width parameter σ of the spectral blur kernel *K*(*λ*). To motivate the importance of the spectral blur estimation, a σ value that is 50% too large will result in about 3–4% typical error. For a 10% error in σ, the typical relative prediction error is <1%, although it occasionally will peak to about 5% near the fringe nulls.

## 3.1.2. Spectroradiometric non-uniformity calibration (NUC)

Spectroradiometric Non-Uniformity Calibration (NUC), also known as flat-fielding, relates detector digital counts to absolute radiance units, and also equalizes non-uniformity among pixel dark current offset and responsivity. We assume and operate the sensor in a linear photosignal response range. NUC is performed by viewing a Labsphere URS-600 Uniform Radiance Standard at various known illumination levels. The result is the pair of *C*(*λ*
_{∘}) and *R*(*λ*
_{∘}) values for each pixel.

## 3.1.3. Floating retardance and spectral spline calibration

Up to this point in the calibration sequence, there remains the ascertainment of the Mueller matrix **m**
_{sys}
_{,1} row term to achieve a fully-determined system model (Eq. 4). The principal parameters of **m**
_{sys,1}, by means of Eq. 2, are the PSIM polarization module waveplate lengths *ℓ*
_{1} and *ℓ*
_{2} as well as their spectral birefringence (*n*_{e}
(*λ*)-*n*_{o}
(λ)). However, there are factors that complicate such a simple reckoning of **m**
_{sys,1}. Furthermore, Eq. 2 for **m**
_{sys,1} only applies to ideal axial propagation through the PSIM polarization module, although by design, deviations from this have been rendered a negligible factor. Remaining factors include stray birefringence and waveplate retardance temperature sensitivity. The significance of these are gauged against the *noise-equivalent retardance* Δ
_{ne}
, defined as the change in optical path difference (OPD), (*n*_{e}
-*n*_{o}
)*ℓ*, that causes the relative fringe intensity registered by a pixel to shift by 1/*SNR*, where SNR is signal-to-noise ratio. For our prototype sensor, at a 0.5% level of significance or SNR of 200:1, Δ
_{ne}
is conservatively 1 nm.

The light path through the sensor, up to the polarizer in the polarization module encounters optical elements that may contribute stray birefringence, thus impacting the polarimetric fringe structure. The residual stress birefringence of typical optical glasses (Schott, Ohara), in terms of OPD, is <10 nm/cm of glass thickness for the fine annealed grade. Our 13 element lens has a total glass thickness of 2.7 cm, corresponding to a worst case retardance of 27 nm (0.06λ at 450 nm). Furthermore, it is highly unlikely that the random rotational alignment of the optical elements would combine in such a fashion. Not surprisingly, we have not discerned an observable effect and it thus appears that the added expense of precision annealing (<4 nm/cm) is unjustified. The temperature coefficients of expansion and refractive index for quartz shift the respective nominal values of *ℓ*
_{1}, *ℓ*
_{2} and [*n*_{e}
(*λ*)-*n*_{o}
(*λ*)]. For example, a 1°*C* change shifts the OPD of a 1 mm length quartz crystal by about 1 nm, comparable to Δ
_{ne}
.

Floating retardance and spectral spline calibration is a joint estimation procedure to refine the wavelength-to-pixel map, and to estimate the deviation in waveplate spectral retardances *ℓ*
_{1}Δ*n*(*λ*
_{∘}) and *ℓ*
_{2}Δ*n*(*λ*
_{∘}) from their nominally-known values. This is performed by viewing the scene through a linear polarizer. This permits estimation of the **m**
_{sys,1}(*λ* ; *ℓ*
_{1},*ℓ*
_{2},Δ*n*(*λ*
_{∘})) in Eq. 4, expressed in the functional form of Eq. 2. The floating-retardance calibration is more fully described in §4.3.

Regarding knowledge of waveplate retardances, temperature changes that occur during the course of a typical extended measurement exercise, particularly outdoors, are sufficient to render earlier retardance calibrations inapplicable. There are two general solutions, 1) temperature stabilization of the polarization module, and 2) periodic retardance calibration. Retardance calibration can be accomplished by viewing light passed through a linear polarizer or, in some instances, by viewing scene-intrinsic polarization.

One solution to refining the spectral calibration is to use the zero-crossings of the PSIM fringes themselves, generated while viewing a polarized source. To these we fit a spline to represent the wavelength-to-pixel mapping. To use the fringes in this manner requires their accurate prediction, which in turn requires knowledge of the waveplate spectral retardance, which varies with temperature. Fringe zero-crossings can be unambiguously identified from the gas lamp calibration data, nominal birefringence, and crystal thickness values. An estimate is then made of the deviation from nominal retardance, *δ* (*λ*).

## 4. Stokes inversion

#### 4.1. Linear spectrum signal model

The forward model for the PSIM-modulated data, **d**, within a spectral analysis window centered at pixel *x*_{o}
is:

where **d** is a data vector of length 2*N*+1 samples, in radiance units ensuing from the flat-fielded measurements (Eq. 4):

**M**
_{1}(*δ* ; *x*_{o}
,*y*_{o}
) is the forward system matrix for the spectral samples centered about pixel (*x*_{o}
,*y*_{o}
), and s_{1}(*x*_{o}
,*y*_{o}
) is the incident Stokes vector for wavelength *λ* (*x*_{o}
,*y*_{o}
) at the *center* of the pixel. The linear-spectrum model assumes that s_{1}(*x*_{o}
,*y*_{o}
) exhibits linear variation with wavelength *within* a 2*N*+1 sample-length analysis window. An inversion can be performed at each pixel, yielding a smoothing effect similar to a boxcar moving average filter with an impulse response of the same extent as the PSIM analysis window. Thus, although an inversion is computed at each pixel, the effective resolution is on the order of the window length.

Within an analysis window, the incident Stokes spectrum s (*x*-*x*_{o}
) is presumed to exhibit linear dependence on wavelength, expressed by a spectral constant (pedestal) and a spectral linear ramp signal for each Stokes component:

where *x* is the continuous wavelength variable normalized in pixel units. For modeling purposes which will be made clear, we stack the coefficients of the above Stokes linear spectrum model into an *augmented* Stokes vector:

The corresponding system matrix is:

where

and

*x*_{o}
=FPA detector index in dispersion dimension (indexing wavelength),

*i*=Relative pixel index within analysis window: *i*∊[-*N*,*N*]

*y*_{o}
=Pixel index along the spatial (slit) dimension,

*δ*=Deviation from the nominal spectral retardance of the waveplates.

The PSIM kernel functions are:

in accordance with Eq. 4 and Eq. 2. The material properties are

*ℓ*
_{1},*ℓ*
_{2}=Waveplate thicknesses

Δ*n*(*x*_{o}
,*y*_{o}
)=Pixel-indexed spectral birefringence.

The spectral blur kernel, *K*(*x*), is given by Eq. 6. The values of Δ*n*(*x*_{o}
,*y*_{o}
), *ℓ*
_{1}, and *ℓ*
_{2} are taken to be their nominal values, based on the published values for the spectral birefringence of the waveplate material and the specified or measured thicknesses of the waveplates. The signed multiplicative deviation of Δ*n*(*x*_{o}
,*y*_{o}
)*ℓ*_{x}
from the nominal value is termed the floating retardance perturbation and denoted by *δ* (*λ*). It is spectrally dependent, thus in turn dependent on (*x*_{o}
,*y*_{o}
). We will have more to say about it later (§4.3). All spectral integrations extend over an interval in which the spectral blur kernel takes on significant values. In practice, ±(1/2+3σ) seems to be sufficient. It is conceivable that, in some practical systems, σ, and hence *K*(*x*), may require characterization as functions of *x* and *y*. For instance, it is conceivable that the point spread function of the collection optics could vary significantly over the operating band.

It is well worth noting that the system matrix elements above, corresponding to the linear signal components, must include the linearly-varying signal term (*x*+*i*) *within* (inside) the spectral integral. It does not suffice to employ a staircase approximation, where for example the *mQ*
_{1} column within Eq. 11 would be formed by multiplying the *mQ*
_{0} column by a zero-mean linear ramp vector. The linear-spectrum model is motivated by the failure of the *constant-spectrum* model which, when employed against sources exhibiting even modest spectral variation such as a linear trend, induces objectionable ripple artifacts in the inverted Stokes spectrum.

## 4.2. Linear spectrum Stokes inversion

With *N*≥4, the linear-spectrum forward model of Eq. 7 and Eq. 11 forms an overdetermined system that we invert in a least squares sense using the pseudoinverse ${\mathbf{M}}_{1}^{+}$(*δ* ; *x*_{o}
,*y*_{o}
), which results from the singular value decomposition of **M**
_{1} [9]. The system matrix for a properly designed instrument is always well conditioned so all singular values are retained in the inversion.

The inverted augmented Stokes vector corresponding to the data and *δ* is computed as follows:

We report the Stokes vector for the analysis window as:

that is, we report the scalar intensities of the spectrally-constant Stokes signal terms. The values of *Î*_{1}, *Q̂*_{1}, etc. (*cf*. Eq. 10) represent the slopes of the linear ramp terms (*x*-*x*_{o}
). These slope values are presently discarded. Since the linear ramp atoms are zero-mean over the analysis window, *[Î*_{0}
*Q̂*_{0}
*Û*_{0}
*V̂*_{0}] thus represents the mean value of the Stokes vector within the analysis window.

For a linear-in-wavelength dispersive spectrometer, the PSIM fringes are *chirped* as shown in Fig. 1. This corresponds to the fringe density decreasing by a factor of two between the ends of an octave spectral range. Consequently, our inversion technique similarly increases the window length of the data vector **d** by a factor of two while traversing from the short to the long wavelength end of the range, in essence yielding a *constant Q* smoothing effect.

## 4.3. Floating-retardance calibration for δ

The predominant reason for the need to include *δ* in the formulation is the temperature sensitivity of the retardance (*cf*. §3.1.3). Ideally from the signal processing standpoint, the polarization module would be temperature stabilized to eliminate the need to track *δ* in the analysis. Currently, the value of *δ* is estimated using a highly *linearly* polarized input to the sensor of sufficiently high SNR. To find the optimum value of *δ*, the following problem is solved:

where

where **X**
^{+} denotes the pseudoinverse of **X**, **M**
_{1} is the system matrix, Eq. 11, for the linear-spectrum model, and the diagonal matrix C is included to incorporate the knowledge that the *V* Stokes component is zero:

For our present configuration with the material and dimensional properties of the waveplates being sufficiently well known, the search bounds on *δ* need only account for temperature drift of retardance and is, in fact, able to encompass a large range of temperatures. The Stokes inversion algorithm using this method to find the actual retardance has been named *floating retardance inversion*.

## 4.4. Enhanced Stokes inversion strategies

We have achieved nearly noise-limited spectropolarimetry for 200:1 SNR measurements against sources exhibiting moderate spectral variation. Our linear-spectrum model presentation (§4.1) is cast in the generalized form of the over-complete spectral dictionary model, which we believe is capable of addressing more complex signals, perhaps including those with absorption or emission spectra. This might be achieved by expanding the contents of the spectral dictionary, and applying it against a single analysis window of size 2*N* +1 to encompass the entire sensor spectral range. This is not inconceivable and such *superresolution* performance is routine in the mature field of autoregressive (AR) spectral estimation.

## 4.4.1. Over-complete spectral dictionary model

The linear spectrum model is a particular instance of the more general spectral dictionary signal model. The linear-spectrum signal dictionary **z**
_{1}(*x*) is comprised of pedestal and linear ramp spectral functions, which we term signal elements *z*
_{{I,Q,U,V}p}(*x*):

$${\mathbf{s}}_{1}=\left[{I}_{0}{\phantom{\rule{.2em}{0ex}}Q}_{0}\phantom{\rule{.2em}{0ex}}{U}_{0}\phantom{\rule{.2em}{0ex}}{V}_{0}{\phantom{\rule{.2em}{0ex}}I}_{1}{\phantom{\rule{.2em}{0ex}}Q}_{1}\phantom{\rule{.2em}{0ex}}{U}_{1}\phantom{\rule{.2em}{0ex}}{V}_{1}\right]$$

$${\mathbf{z}}_{1}\left(x\right)=\mathrm{diag}\left[1\phantom{\rule{.2em}{0ex}}1\phantom{\rule{.2em}{0ex}}1\phantom{\rule{.2em}{0ex}}1\phantom{\rule{.2em}{0ex}}x\phantom{\rule{.2em}{0ex}}x\phantom{\rule{.2em}{0ex}}x\phantom{\rule{.2em}{0ex}}x\right]$$

To represent more complex spectral behavior, the system model can be further augmented with additional elements, expanding the signal dictionary in an obvious manner from 2, to *P* spectral signal elements per Stokes component, **z**
_{P}
(*x*). Consequently, the augmented Stokes vector s
_{P}
now contains 4*P* terms, the system matrix, **M**
_{P}
, in turn expands its number of columns, with elements expressed more generally as:

Otherwise, the forward model form and its interpretation remain the same.

The linear-spectrum model employs signal elements forming a first-order Taylor series expansion in spectral variable *x*. To handle increasingly complex spectral structure, one might intuitively expand the number of signal elements in the Stokes signal dictionary **z**
_{P}
(*x*), for example by adding Taylor series terms. We are motivated to include signal elements to represent spectral resonance shapes directly because their representation using higher-order Taylor series terms may not converge quickly.

However, for given analysis window, if we increase the number of signal elements such that 4*P*>(2*N*+1), the system becomes under-determined. Moreover, depending on the signal elements, even if (2*N*+1)≥4*P*, **M**
_{P} may yet remain under-determined due to rank deficiency. Rather than increasing the analysis window size 2*N*+1 to compensate, we can purposely seek the under-determined Stokes inversion s
_{P}
which fits the signal **d** against an *over-complete* spectral dictionary **M**
_{P}
.

## 4.4.2. Over-complete Stokes inversion via basis pursuit

For an *over-determined* system, the pseudoinverse ${\mathbf{M}}_{P}^{+}$
produces the solution ŝ
_{P}
which minimizes *L*
_{2} norm error, i.e., the sum of squared errors *ε*^{T}*ε* between the predicted and actual data, where:

The pseudoinverse can also be employed against an *under-determined* system, such as the overcomplete spectral dictionary, **M**
_{P}
. In such case, there are arbitrarily many possible solution vectors ŝ
_{P}
=${\mathbf{M}}_{P}^{+}$
**d**, and the pseudoinverse returns the *minimal-length vector* ŝ
_{P}
that satisfies **M**
_{P}
ŝ
_{P}
=**d**. The terminology for such an under-determined problem is *best-basis* matching [10], even though the returned solution ŝP may retrieve a non-orthogonal vector set.

The pseudoinverse is not the best algorithm for basis matching. The Basis Pursuit algorithm finds the minimal *L*
_{1} norm (rather than *L*
_{2}) ŝ
_{P}
satisfying **M**
_{P}
ŝ
_{P}
=**d**, casting the problem as a linear programming (LP) exercise [10]. This algorithm has better sparsity-preserving characteristics, meaning that the retrieved ŝ
_{P}
assigns non-zero weights to the fewest possible signal elements in the dictionary, rather than attributes or diffuses the energy in **d** across many elements. Remarkably, if a signal possesses a sufficiently sparse representation, the *L*
_{1} optimization of basis pursuit necessarily yields this unique solution [11].

## 4.4.3. Populating the spectral dictionary with signal elements

A looming question regarding the practicality of the overcomplete spectral dictionary model is how to populate the signal dictionary **z**
_{P}
(*x*) with a reasonable number of signal elements that nevertheless span the space of potential Stokes spectra. The notion of populating **z**
_{P}
(*x*) with replicas of resonance signal templates for all potential *λ* translations of spectral location and dilations of spectral width is daunting. Fortunately, by appealing to design of steerable *framelets*, such a brute-force approach appears unnecessary.

For a given signal element, *z*(*x*), a steerable frame is that set of vectors or functions whose linear combinations represent the possible shifts and dilations of *z*(*x*). As a simple example, the signal function cos (*ωt* +*β*), for *any* shift *β*, can be succinctly represented by a linear combination of two functions, cos (*ωt*) and sin (*ωt*). For a given class of signal elements, the existence of a steerable frame therefore enables a compact signal dictionary to implicitly represent all possible signal shifts and dilations.

The Taylor series elements by nature exhibit *λ* shift and dilation invariance, but are, however, quite simple compared to signal elements that would represent spectral resonance features, e.g., those ensuing from the Drude-Lorentz model for refractive index, manifested through Fresnel reflectance [14]. At issue is whether we can find a steerable framelet representation for such physically-motivated signal elements of our choosing. Recent work by Daubechies *et. al.* indicates that this may indeed be achievable [15].

## 5. Measurement results

In this section, we describe some of our measurement results. Figure 2 shows a schematic layout of one of our prototype sensors. The polarization module (i.e. W1, W2, and P) is located in a collimated region of the beam which is critically important so as not to induce substantially more complicated propagation behavior in the crystals. The source of the complication is that the effective index of the extraordinary ray is a function of the angle between the wave normal and the optic axis[8][7]. Maintaining all ray angles of incidence less than about 5° appears to be sufficient to allow us to neglect this effect. The spectrometer uses a convex grating design and is based on an Offner relay. The Offner arrangement places all optics on a common center of curvature which results in vanishing Seidel aberrations. However, when the inner mirror is made diffractive, the symmetry is slightly disturbed. The resulting aberrations can be well-corrected by a small positional adjustment to one of the outer mirrors. In particular, smile and keystone distortion can be reduced to <0.1% over an 18 mm slit.

An alternative calibration approach to detailed modeling of the system matrix is to attempt to measure the system Mueller matrix using inputs of well known polarization state. Conceptually, this approach is far simpler but, unfortunately, the difficulty lies in the generation of precise states, particularly over the full operating band and at enough states to robustly characterize the response. Many polarization components (e.g. polarizers and waveplates) exhibit substantial non-ideal and poorly specified behavior, including non-uniformity across their aperture and in-accurate orientation[19]. One means of generating a relatively well known polarization state is to view a collimated, unpolarized beam through a tilted dielectric plate. The Fresnel transmittances, depending only on the index of refraction, can easily be computed and thus the DoP. A continuously variable DoP can be generated by varying the angle of incidence at the plate. For most glasses (*n*≈1.5), maximum DoPs are ≈20–30% at an angle of incidence of 65°. Higher DoPs can be generated by stacking multiple plates.

Figure 5 shows our measurement results using a Pyrex plate (*n*=1.478 at 500 nm). There error is less than 1% over much of the range. These results were obtained using the HyperSpectral Polarimeter for Aerosol Retrievals (HySPAR) which is shown in Fig. 3. This instrument has a 120° field of view and operates over 450–900 nm. The instrument was developed to perform multi-angle spectropolarimetric measurements for retrieval of aerosol microphysical parameters. It has been shown that the addition of polarimetric information can improve accuracy over intensity-only methods[3][2][1]. Figure 6 shows the result of a measurement in which HySPAR was mounted in an uplooking configuration, spatially imaging in the solar principal plane (i.e. the plane defined by the sun, observer, and zenith). Of particular note is the ability to discern clouds in the data. In all but the most forward scattering directions, the white clouds are evident against the blue Rayleigh sky. The strong depolarizing effect of the clouds can also be seen in the degree of polarization image. In addition to the aforementioned benefits of snapshot polarimetry, the dense spatial sampling of HySPAR should allow a robust cloud screening algorithm to be developed.

## 6. Conclusion

We have described the patented Polarimetric Spectral Intensity Modulation (PSIM) technique for performing imaging spectropolarimetry in a single snapshot. Single snapshot polarimetry is important because conventional polarimeters that perform time sequential snapshots through polarizers or waveplates oriented at different rotation angles are vulnerable to scene changes between snapshots that result in artifacts that can be falsely interpreted as a polarization signal. The PSIM technique overcomes this drawback as well as spatial registration problems that are common in approaches that use multiple beams. The technique relies on accurate modeling of the sensor response. In particular, accurate characterization of the wavelength to pixel mapping is critical to obtaining calibrated spectropolarimetry. We described a mapping technique that makes use of the spectral fringes of a polarized source and spline interpolation to significantly improve the broadband mapping accuracy as compared to gas calibration lamp techniques. Our Stokes inversion method is based on a singular value decomposition and subsequent inversion of the system matrix. To date, our model assumes that the underlying signal spectrum may have a linear (in wavelength) variation over the analysis window. We also described a potential method for performing inversions in the presence of higher order spectral features. We presented measurement results from our prototype instruments that show calibrated, wide angle imaging performance. The results of a measurement validation method based on the Fresnel transmittance of a tilted dielectric plate were shown to yield a degree of polarization accuracy better than 0.5% over much of the 450–750 nm band. We also presented initial sky measurement results from the HyperSpectral Polarimeter for Aerosol Retrievals (HySPAR). In addition to the single snapshot advantages of this instrument, its dense spatial coverage may yield robust cloud screening algorithms.

## Acknowledgments

This effort was partially supported by contract NAS1-02115 under the Atmospheric Sciences Division, NASA Langley Research Center.

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