Methods and Applications of Snapshot Spectropolarimetry Nathan Hagena , Ann M. Lockea , Derek S. Sabatkeab , Eustace L. Dereniaka , David T. Sassc a Optical b Ball

Sciences Center, University of Arizona, Tucson, AZ 85721 Aerospace and Technologies Corp., Boulder, CO 80301 c U. S. Army TACOM, Warren, MI 48397 ABSTRACT

We present adaptations of the channelled spectropolarimetry technique, a method which allows both spectral and polarization information to be captured in a single integration period. The first adaptation uses a mathematical decomposition of the system matrix, which is then modified for imaging spectropolarimetry; the second adaptation is applied first to a single-point and then to an imaging system, for which we also show applications and measurements from experimental work. Keywords: polarimetry, spectropolarimetry, imaging spectrometry, CTIS

1. BACKGROUND

Unpolarized Input

Channelled spectropolarimetry (CHSP) was developed in 19981, 2 to take advantage of interferometric techniques to obtain the state of polarization (SOP) of an input spectrum in a single snapshot (a single detector integration period). Since its inception, CHSP has undergone continuing development, both for improvements to the method and for measurement applications,3, 4 and has been adapted by our own research group for use with our snapshot imaging spectrometer.5 The advantages of the CHSP technique are its speed and compactness, and there are several reconstruction techniques which can be used for a given instrument.

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Figure 1. A schematic of a CHSP device (adapted from ref. 2).

The hardware configuration of a CHSP device is illustrated above in Fig. 1, comprising two high-order retarders, R1 and R2 , oriented respectively at 0◦ and 45◦ to horizontal, an analyzer A at 0◦ , followed by the spectrometer. With this configuration, an unpolarized input spectrum reaches the spectrometer essentially unmodified. With a polarized input, however (shown in the left of the figure, with a solid line for the beam intensity and dashed lines for the intensity of the individual polarization components), the spectrum is modulated at high frequencies, as is indicated in the resulting Fourier Transform. This is referred to as a channelled spectrum, often seen in the field of Fourier Transform Spectrometry. Further author information: (Send correspondence to E.L.D.) E-mail: [email protected]

Polarization: Measurement, Analysis, and Remote Sensing VI, edited by Dennis H. Goldstein, David B. Chenault, Proceedings of SPIE Vol. 5432 (SPIE, Bellingham, WA, 2004) 0277-786X/04/$15 · doi: 10.1117/12.548475

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The electric field vector for light passing through the system can be most conveniently represented in terms of the Stokes vector representation s = (s0 s1 s2 s3 )T . Using Mueller calculus, we can then calculate the measured intensity resulting from an input SOP by cascading Mueller matrices for each of the optical elements, giving a signal on the spectrometer of I(σ) =

iδ2 1 1 + e−iδ2 ) 2 s0 + 4 s1 (e + 18 [(s2 − is3 )ei(δ2 +δ1 ) +

(s2 + is3 )e−i(δ2 +δ1 ) + (−s2 − is3 )ei(δ2 −δ1 ) + (−s2 + is3 )e−i(δ2 −δ1 ) ] , (1)

recalling that s0 , s1 , s2 , s3 , δ1 , and δ2 are all functions of wavenumber σ (≡ 1/λ). For quartz retarders, δ1 and δ2 are roughly linear functions of wavenumber (see Fig. 5), so that the seven phase terms in the equation above represent seven channels into which the polarization information is encoded (or, equivalently, four frequencies at which the spectrum is modulated). The configuration of retarders and analyzer shown thus induce interference fringes into the spectrum (Fig. 2), where the frequency of modulation carries information about the distribution of each of the Stokes components across the spectrum. If we now calculate the autocorrelation of this power spectrum, C(OPD) = F −1 {I(σ)}, we can see the resulting 7-channel distribution, shown in Fig. 3. With appropriate choices of δ1 and δ2 , it is possible to separate the Stokes component terms into independent channels, much in the same way that communications systems using a single carrier reserve bandwidth regions for each independent communication channel, referred to as sideband modulation.

Figure 2. The measured channelled spectrum for a compound input polarization state.

Figure 3. The magnitude of the channelled spectrum autocorrelation function |C(OPD|) for the input polarization state.

Fig. 3 gives the autocorrelation function for the given input spectrum, and superimposes windows to indicate the channels into which the signal is distributed. The figure shows the case for the choice δ2 = 3δ1 , a choice which allows separation of channels but which also results in the introduction of a pair of empty channels in the OPD-domain. Our current preference is for a choice of δ1 = 2δ2 , which allows for a more efficient use of the available spectral bandwidth.4, 6 As long as none of the Stokes component spectra si (σ) exceed the bandwidth of a given channel, then each of the Stokes components can be extracted separately via masking and shifting in the Fourier domain. The spectral resolution of the system may then be characterized by the OPD-bandwidth provided to each channel. For quartz retarders, the birefringence at visible wavelengths is approximately 1% of the crystal’s physical thickness. For the two visible-spectrum experiments discussed below, a pair of quartz retarders with thicknesses of 1.84 and 5.52 mm was used, giving a retardance of δ1 ≈ 18.4 µm for the orthogonal polarizations passing through the thinner retarder. This gives the width of each individual channel.

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Figure 4. The measured birefringence of the quartz retarders used in the visible spectrum CHSP experiments.

Figure 5. The measured retardance for the 5.52 mm thick retarder used in the visible spectrum CHSP experiments.

2. FOURIER METHOD The Fourier method for reconstructing the input SOP for a given channelled spectrum first involves representing the spectrum in a wavenumber basis, which is preferred because the retardances giving modulation of the spectrum (1) are nearly linear with wavenumber but not with wavelength. Thus, in a wavenumber basis the modulation frequencies are nearly constant across the spectrum, allowing for easy adaptation of Fourier methods. Moving from a wavelength to wavenumber basis, however, involves more than just renumerating the abscissa; it also involves an implicit nonuniform stretching (or compressing) of the spectral bands covered by the spectrometer. While grating spectrometers generally permit uniform sampling with respect to λ, with the change to a wavenumber basis, the spectrum is no longer uniformly sampled (the blue end of the spectrum contains fewer samples per unit ∆σ than does the red end of the spectrum), making the use of standard Fourier Transform routines difficult. Thus, prior to transformation it is necessary to interpolate the wavenumber-sampled spectrum onto a uniform data set. Moreover, the nonuniform stretching of the bands requires correction in order to maintain throughout the transformation the same energy within each band. Thus, we can write the necessary normalization as  


1 −1 (2) = I(σ) 2 dσ . P = I(λ) dλ = I(σ) d σ σ Multiplying each spectral band in I(σ) by −1/σ 2 correctly renormalizes the sampled spectrum. The next step is the calibration of the system retarders in order to obtain the quantities δ1 (σ) and δ2 (σ). Using the shifting property of the Fourier Transform, if we premultiply the input channelled spectrum by the retardation phase factor appropriate to the channel we wish to reconstruct, then we can shift that channel in the Fourier domain to center at DC. Thus, for example, to reconstruct s1 (σ), we can take the input spectrum I(σ) and multiply by the phase factor e−iδ2 . The autocorrelation of the resulting “demodulated” channelled spectrum shows that the s1 lobe in the OPD-spectrum has been shifted to the center. We can then construct a windowing function to mask off the central channel and Forward Transform the windowed result, in this case giving 14 s1 (σ). Due to the use of Fourier methods in the reconstruction, artifacts are produced at the ends of the spectral region, where there is a discontinuity in the data as a result of the finite size of the detection range. To eliminate the distortion in the reconstructed Stokes spectra, it is therefore necessary to truncate the data at the low-λ and high-λ ends. In the non-imaging experiment using this technique, complete elimination of the artifacts resulted in a tolerable loss of 10% of the spectral range. Another difficulty which must be treated in the reconstruction process is that of the transfer function of the spectrometer. The response of the spectrometer decreases with higher modulation frequencies in the spectrum, an effect which is analogous to the MTF of an optical imaging system describing the attenuation of system response with increasing spatial frequency, and so can be referred to as spectral blurring. The effect of blurring

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in the reconstruction is a nonuniform attenuation of the polarization components. To compensate for this, we need to account and correct for the transfer function. Since the s0 Stokes component is encoded in a channel centered at DC whereas the other Stokes components appear at higher frequencies, the spectrometer’s transfer function would tend to preferentially suppress the polarized components, resulting in reconstructions which would underestimate the DOP of the input. Additionally, this process can be turned around and used to calibrate any spectrometer system. That is, by placing a combination of polarizer, high-order retarder, and analyzer in front of any given spectrometer, by which it is possible to generate a chirped channelled spectrum in the wavelength domain, it is a straightforward process to use the modulation amplitudes to derive the transfer function of the spectrometer. This is a method which is analogous but simpler than previous methods using white-light interferometry to introduce channelled spectra to perform the transfer function measurement.7

3. LINEAR OPERATOR METHOD While perhaps less familiar than the Fourier reconstruction technique, the calibration and reconstruction in a CHSP system can benefit from a more general linear operator framework. In this latter treatment, all non-ideal (but linear) effects, such as the imperfect and spectrally-varying resolution in the spectrometer, and dispersion in the retarder materials, can be treated in a unified fashion. If the spectrometer’s band sensitivity functions are uniform translates (shift invariant), the effects of blurring of the detected spectrum due to imperfect resolution can be understood using results from the study of linear shift-invariant (LSI) systems.8 In this case the spectrometer behaves as if the input spectrum is imaged through an LSI system and sampled, and the resolution of the system may be described by an overarching transfer function in the OPD-domain. Even if the assumption of shift-invariance is not a good one, the LSI system viewpoint remains useful for a conceptual understanding of blurring effects. In the linear operator model, the input spectral polarization state and output data are viewed as vectors, and the system is modelled as a linear operator H which maps between them, g = Hs(σ) .

(3)

The image vector g is a discrete column vector of data acquired by the spectrometer, with each element corresponding to a spectral band or pixel. For purposes of data processing the spectral polarization state is also given a discrete representation, obtained by concatenating samples of the four Stokes component spectra into a single column vector. The only assumption being made in this process is that the transformation is linear, which leaves the approach sufficiently general to take account of non-ideal effects to which real systems are subject. Calibration of the system amounts to experimental estimation of a matrix Hij representing the linear operator H. Reconstruction is obtained by pseudoinverting the matrix, constraining the result with an appropriate choice of object space. The system matrix Hij is estimated by recording the output data vectors for a set of known inputs. Arranging the inputs sk as columns of a matrix Qjk , and the recorded data vectors gk as corresponding columns of a matrix Gik , (3) becomes (4) Gik = Hij Qjk , and Hij may be estimated as

Hij = Gik Q+ kj ,

(5)

where the superscripted + represents the Moore-Penrose pseudoinverse.9, 10 Since the matrix contains experimental (noisy) data, the threshold applied to singular values in the pseudoinversion must be meaningfully set. Singular value decomposition provides the best procedure for this, as it allows for inspection of the singular value spectrum and of the corresponding singular vectors, giving a direct reference for determination of the threshold. Leaving the threshold definition to standard routines will likely result in having noisy singular vectors built into the pseudoinverse. Calibration states were prepared by feeding light of a narrow spectral bandwidth from a monochromator through a spectral polarization state generator (SPSG). Measurements were made for 179 different wavelengths 170

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at each polarization setting (we chose 6 different settings for each measurement). With approximately 1000 bands in the spectrometer, the calibration comprises 179 × 1, 000 × 6 ≈ 1, 000, 000 individual measurements, a considerable volume of data. Nonetheless the system matrix contains 1000 × 4000 elements (that is, it maps a single spectrum of 1000 bands into the four Stokes spectra, each containing 1000 bands), leaving the matrix estimation (5) severely under-determined. We can trust, however, that the estimated matrix will accurately predict the system’s output for any input that is adequately modelled as a linear combination of the input calibration states. This representation requirement serves as a guideline for selection of calibration states, the spectral content of which consists of sharp, narrowband spikes, while the object states of interest presumably have smooth, OPD-bandlimited Stokes spectra. Pseudoinversion is not so simple as calling a generally available (or “canned”) matrix analysis routine on the system matrix. The pseudoinverse depends on the choice of what vector space should serve as object space. Using a canned routine without explicitly enforcing a choice of object space amounts to an implicit choice of a space in which the Stokes component spectra are not subject to any OPD bandlimit. Such a choice for object space allows spurious high frequency (sharp) detail in the reconstruction. There is not a single right choice for object space, though the amplitude modulation analogy mentioned in the Fourier Method section suggests a truncated Fourier basis. That is, each Stokes component spectrum is expanded on a Fourier basis (consisting of sines and cosines) with frequencies ranging from DC up to a specified OPD bandlimit. Any other linearly independent set of expansion states could be used to specify the object space, such as orthogonal polynomials or signatures derived from physical phenomenology. However, the reconstructions used here all employed a truncated Fourier basis. A canned pseudoinverse routine may be coerced into imposing an object space constraint. This is accomplished by applying a coordinate system transformation and truncation to the matrix before the routine is called, and converting back to the original system afterward. The coordinate transformation is applied to object space; none is required for image space. This truncated coordinate system represents an object space of low dimension; if it is chosen suitably, the inversion problem is overdetermined in this space, and high frequency artifacts cease to be a concern. In order to apply this approach to the truncated Fourier basis, two additional conversion steps are included in the coordinate transformation. Since the amplitude modulation analogy is conveniently represented in terms of wavenumber, we include a conversion from distributions over wavelength (which the spectrometer is calibrated to measure) to distributions over wavenumber. This conversion amounts to multiplication of a factor of λ2 into each spectrum (this is the same conversion as mentioned in (2)). In the wavenumber domain our sample spacing is non-uniform. Therefore we further weight each spectral sample by a factor of the square root of its bin width, so that the usual dot product properly represents an integral scalar product.

4. EXPERIMENTAL RESULTS The first experiment at our lab to adapt CHSP methods used a non-imaging system in visible light. The experimental layout is shown in Fig. 6, involving a broadband source with a configurable polarization output, a sample to be measured, and the CHSP itself. The initial experiments used the linear operator reconstruction method, and were intended to provide a proof-of-principle that the reconstruction method works, and that results correspond with more traditional measurement techniques. To perform a comparison between the CHSP and standard methods, a rotating-compensator spectropolarimeter (RCSP) was constructed and calibrated, using the same spectrometer but different polarization-analyzing optics. In order to generate a complex input polarization state, the two instruments were used to measure stress-birefringence in a sample of ordinary plastic (PMMA). Fig. 7 shows the resulting reconstructions of the two instruments for the output SOP of the sample at the maximum stress applied. With the goal of adapting the CHSP instrument to our CTIS imaging spectrometer design, we decided to temporarily take a step back from the linear operator reconstruction method and use Fourier techniques, intending to return to the linear model once the system is better understood. The Fourier method was then applied to the same instrument and again compared against the RCSP reconstructions. The results for input polarization states of vertically-polarized and right-circularly polarized are shown in Fig. 8 (in these figures Proc. of SPIE Vol. 5432

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Figure 6. A diagram of the experimental layout, using a broad spectrum polarization state generator (PSG), the sample, and a channelled spectropolarimeter (CHSP). 1

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Figure 7. Measured stokes spectra for the stress-birefringence experiment (Linear Operator method). Note that the scale of the s1 plot is slightly magnified. The error bars shown are ±2σ uncertainties in the RCSP data.

no sample was being used to modify the generated polarization state). There are some obvious discrepancies between the two reconstructions, particularly at the short-λ end of the spectrum. These are the earlier-mentioned artifacts resulting from use of the Fourier domain techniques; truncating the ends of the spectrum eliminates the effects of the artifacts on the remaining data. Additionally, the RCP-input reconstruction shows substantial discrepancies between the two instruments, the investigation of which is the subject of ongoing research, but which seems primarily to be the result of calibration difficulties with the broadband retarders being used in our RCSP instrument rather than due to problems in the CHSP. Finally, in tandem with the visible-spectrum experiments above, a short-wave infrared (λ = 1.25 µm − −1.99 µm) CTIS was built incorporating a CHSP to produce a snapshot imaging spectropolarimeter. The instrument contains a 801 × 512 pixel PtSi FPA, and sapphire retarders of ∼4 mm and ∼8 mm, into a CTIS spectrometer. The advantage of choosing sapphire as a retarder material is that in the SWIR wavelength range above it possesses a low dispersion of birefringence. Combining multiple retarder materials to produce a retarder (much in the way that achromatic retarders are made) with even lower dispersion of birefringence

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s3 s2 and s3 for RCSP and CHSP

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} }

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Figure 8. RCSP and CHSP reconstructions for the visible-spectrum CHSP using Fourier methods: (a) vertically-polarized input, and (b) RCP input (s1 = solid, s2 = dashed, s3 = dotted). The edge artifacts at the short-λ end of the spectrum appear only in the CHSP data.

Figure 9. The complete SWIR CTIS hardware incorporating CHSP.

produces a result which is quite thick and expensive to manufacture. With these components, the assembled spectropolarimeter becomes capable of 10 nm resolution for 54 × 46 pixels of image spatial resolution.11, 12 During system calibration, we found that the dichroic polarizer in the calibration system was only usable for wavelengths shorter than 1.7 µm, after which it begins to go transparent. This resulted in a loss of spectral range over which it was possible to perform reliable Stokes reconstruction. Moreover, with the PtSi FPA it proved impossible to get sufficiently dense sampling of the input spectrum to produce reconstructed Stokes spectra without appreciable aliasing. The result was strong spurious “ringing” in the reconstructed Stokes components. Continued work on this system will involve replacement of the current FPA with a larger array in order to achieve improved spatial and spectral sampling.

5. CONCLUSION We have shown that there are a couple of approaches to reconstructing the input SOP using a CHSP instrument. Ongoing efforts include understanding the observed discrepancies between the standard techniques and CHSP, and we are developing methods which can minimize the error in CHSP reconstructions. And, in light of the difficulties which we have experienced in inserting CHSP into our SWIR CTIS instrument, we are currently working on developing a visible-light CTIS incorporating CHSP; the availability of advanced FPAs for visiblespectrum measurements should ensure a more robust and accurate imaging spectropolarimeter.

REFERENCES ¯ 1. T. Kat¯o, K. Oka, T. Tanaka, and Y. Otsuka, “Measurement of the spectral distribution of polarization based on frequency domain interferometry,” Proceedings of the Hokkaido branch of the Japan Society of Applied Proc. of SPIE Vol. 5432

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Physics 34, p. 41, 1998 [in Japanese]. Þ ãˆ, Ç S4, Ù? Ä, ℠Å], 6Å ™ÕGŠGê+Ç %»F.:(K—h§¡7, ã34 »Ö•Q .BQöƒ¦ ƒ¦ÂsÝ*»Ö•Q .B, #, 1998), p.41. 2. K. Oka and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. 24(21), pp. 1475– 1477, 1999. 3. K. Oka, “Singleshot spectroscopic polarimetry using channeled spectrum,” in Advanced Materials and Devices for Sensing and Imaging, J. Yao and Y. Ishii, eds., Proc. SPIE 4919, pp. 167–175, 2002. 4. D. Sabatke, A. Locke, E. L. Dereniak, M. Descour, J. Garcia, T. Hamilton, and R. W. McMillan, “Snapshot imaging spectropolarimeter,” Opt. Eng. 41(5), pp. 1048–1054, 2002. 5. M. Descour and E. Dereniak, “Computed-tomography imaging spectrometer: experimental calibration and reconstruction results,” Applied Optics 34(22), pp. 4817–4826, 1995. 6. K. Oka and T. Kato, “Static spectroscopic ellipsometer based on optical frequency-domain interferometry,” in Polarization Analysis, Measurement, and Remote Sensing IV, D. H. Goldstein, D. B. Chenault, W. G. Egan, and M. J. Duggin, eds., Proc. SPIE 4481, pp. 137–140, 2001. 7. V. N. Kumar and D. N. Rao, “Interferometric measurement of the modulation transfer function of a spectrometer by using spectral modulations,” Applied Optics 38(4), pp. 660–665, 1999. 8. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics, Wiley, New York, 1978. 9. H. H. Barrett and K. Meyers, Foundations of Image Science, Wiley, New York, 2003. 10. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, Cambridge U.P., Cambridge, 2 ed., 1995. 11. A. M. Locke, Design and Analysis of a Snapshot Imaging Spectropolarimeter. PhD thesis, University of Arizona, Tucson, Arizona, 2003. 12. A. M. Locke, D. Salyer, D. S. Sabatke, and E. L. Dereniak, “Design of a SWIR computed tomographic imaging channeled spectropolarimeter,” in Polarization Science and Remote Sensing, J. A. Shaw and J. S. Tyo, eds., Proc. SPIE 5158, pp. 12–23, 2003.

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