1. Introduction

Measurements of the spectral content and polarization state of light are important in many fields, including atomic and chemical physics [1, 2], materials identification and characterization [3], astronomy [4], remote sensing [5, 6], biomedical imaging [7, 8], and stress analysis [9, 10]. A variety of measurement methods have been developed. Many are described in review articles [11, 12, 13, 14, 15], which also provide substantial bibliographies.

The raw data acquired in these measurements usually consist of a discrete sequence of detector readings. In some cases a detector array is incorporated into the system, which is designed so that each detector has a different spectral or polarization sensitivity. In others a single detector is read sequentially in time, while fore optics are arranged to vary the system’s spectral or polarization sensitivity. Time-sequential systems may perform well in applications featuring a polarization state and spectrum which is constant in time. However their limited data acquisition rates make them undesirable for applications in which the characteristics of the light under analysis vary rapidly. Such situations call for snapshot techniques, that is measurements in which data is captured in a single integration time.

Interest has arisen recently in a clever snapshot technique, which we term channeled spectropolarimetry (SP). Channeled SP allows determination of the spectral dependence of the Stokes vector in a snapshot mode with no moving parts. Recent publications [16, 17, 18, 19] on the technique have been primarily of a conceptual or proof-of-principle character. Very few quantitative experimental results have been shown, and description of calibration and reconstruction procedures is lacking. This paper addresses the gap by describing an approach to calibration and reconstruction based on a linear operator model of the system.

In this model the input spectral polarization state and output data are viewed as vectors, and the system is modeled as a linear operator which maps between them. As it assumes only linearity, it is 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 representing the linear operator. A method of linear reconstruction is obtained by pseudoinverting the matrix, constraining the result with an appropriate choice of object space. The mathematical aspects of the linear operator model are interesting in their own right, and are described in a separate article [20]. In this paper we concentrate on their practical application to calibration and reconstruction. We present experimental results, including a demonstration of time-resolved spectropolarimetry using stress-induced birefringence in a transparent material. This experiment’s data is given a geometrical interpretation on the Poincaré sphere, in which the retardance and fast axis orientation of the sample appear as angles in a three-dimensional construction.

2. Background

Figure 1 illustrates the principle of operation of a channeled SP. The radiation under analysis is passed through two thick (high order) retarders and a linear analyzer, and the spectrum of the exiting light is recorded by a spectrometer. The fast axis of the first retarder is aligned with the transmission axis of the analyzer, and the second retarder is oriented with its fast axis at 45° to the analyzer’s axis.

We denote the wavenumber distribution of the incident radiation’s Stokes vector by

The recorded spectrum is a linear superposition of the Stokes component spectra s_k(σ), in which the coefficients are sinusoidal terms depending on the retardances of the retarders. Since each retardance is nominally proportional to wavenumber σ, the spectra are modulated. With proper choice of modulation frequencies (i.e. proper choice of retarder thicknesses) they can be separated in the Fourier domain. This technique is a form of amplitude modulation, and is analogous to methods in widespread use in radio communications.

 

Fig. 1. The channeled spectropolarimeter (after Oka and Kato [18]). The complicated spectrum recorded at the output of the polarization optics is formed by a superposition of the Stokes component spectra modulating carriers. With proper choice of carrier frequencies, the Stokes components can be isolated in the Fourier domain.

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A brief review of the insights to be had from the amplitude modulation analogy is worthwhile. Including negative frequencies, the channeled SP creates seven channels in the h domain (where h denotes the variable conjugate to wavenumber σ under the Fourier transform). As shown in Fig. 1, a channel containing the (Fourier transform of the) s ₀ Stokes component lies centered at zero frequency (DC). Channels for s ₁ and the complex combinations s ₂±is ₃ are located at carrier frequencies determined by the optical path difference (OPD) between ordinary and extraordinary rays for each retarder. The OPD is the product of the retarder’s thickness d and the birefringence Δn of the material (assuming it is cut with the optic axis in the face of the retarder). The sign and magnitude of the carriers differ between channels, as is indicated by the orientation and height of lobes in Fig. 1.

Choosing the retarder OPDs in a 1:2 ratio results in tightly packed, uniformly spaced channels in the h domain. An h bandwidth of d ₁Δn ₁ (the OPD of the thinner retarder) is allotted to each channel. So long as none of the Stokes component spectra s_k(σ) exceed this bandwidth, the channels will be separated. The incident spectral polarization state might then be reconstructed by masking and shifting in the Fourier domain. The spectral resolution of the system may be characterized by the h bandwidth provided to each channel.

The amplitude modulation analogy describes the principle of operation of the system, and may be carried so far as to treat reconstruction procedures and resolution. Nonetheless, calibration and reconstruction benefit from a more general linear operator framework, in which non-ideal (but linear) effects may be readily treated. Such effects include imperfect and spectrally-varying resolution in the spectrometer, and dispersion in the retarder materials. Dispersion in the birefringence Δn of the thick retarders may be interpreted as causing the carrier frequencies for each SP channel to vary across the spectrum. That is, the carriers are chirped.

If the spectrometer’s band sensitivity functions are uniform translates, the effects of blurring due to imperfect resolution can be understood using results from the study of linear shift-invariant (LSI) systems [21]. 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 h domain. Even if the band sensitivities are not technically uniform translates, the LSI system viewpoint is useful for a conceptual understanding of blurring effects. Since the s ₀ Stokes component is encoded in a channel at DC whereas the other Stokes components appear at higher frequencies, the spectrometer’s transfer function would tend to suppress the polarized components. If uncorrected, it may cause reconstructions to underestimate the degree of polarization of the input spectral polarization state.

3. Hardware

Our experiments feature three simple systems: a prototype channeled SP; a rotating compensator, fixed analyzer (RCFA) polarimeter; and a spectral polarization state generator (SPSG). The prototype channeled SP accepts a collimated beam at its input. An achromatic doublet following the retarders and analyzer focuses the beam onto a multimode fiber which leads to a commercial grating spectrometer. The system features quartz retarders of approximately 5.5 and 1.84 mm thicknesses (a 3:1 ratio which predates our preference for a 1:2 ratio). The 3:1 ratio results in the introduction of two empty channels into the h domain structure of the data set, but the principles of operation of the system are otherwise unchanged.

The chosen retarder thicknesses result in a ΔσΔh product for each channel of about 18. That is, we can expect to resolve about 18 spectral bands in each reconstructed Stokes component, and require about 9 times this number of bands in the spectrometer. The grating spectrometer used in our experiments has a wavelength resolution of about 1 nm full width at half maximum. This about satisfies the resolution requirement in the blue portion of the spectrum and comfortably exceeds it in the red. Even in the red however, account must be taken of the spectrometer’s resolution since some contrast in spectral detail is still lost to blurring.

To enable separate reference measurements for comparison with the channeled SP output, an RCFA system was constructed. It features a rotatable, (nominally) achromatic quarter wave plate and fixed linear analyzer, followed by an achromatic doublet which couples the beam into a fiber leading to the grating spectrometer. A single grating spectrometer is used for both SPs by swapping the fiber between systems as necessary. Basic reconstructions of data from the RCFA system may be carried out relatively easily, since they require knowledge of only a small set of system parameters which may be determined in tabletop experiments. These include the retarder’s retardance and fast axis orientation (and their potential variation with wavelength), as well as the overall spectral transmittance profile.

We characterized the quarter-wave retarder with a technique illustrated in Fig. 2. The retarder under test is placed between two aligned polarizers in a beam of white light, and the transmitted spectrum is recorded as the retarder is rotated. (Unless otherwise noted, the axis of rotation for rotating optical elements is understood in this article to be parallel to the beam.) The retardance and fast axis orientation for each wavelength in the spectral data set are then determined from a least-squares fit using a Mueller matrix model. This procedure yields retardance as a function of wavelength. The fast axis orientation was also treated as if it varied with wavelength in our Mueller matrix models for the retarders, however the reason for the observed variation is presumably due to small misaligments between the two elements from which each of the achromatic retarders is constructed [22]. An improved model for the achromatic retarders would improve the absolute accuracy of our systems. The agreement between measurements made with the channeled and RCFA instruments should not be affected, since calibration of the channeled SP traces to a similar model for the SPSG.

 

Fig. 2. Retarder characterization system.

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To measure a spectral polarization state with the RCFA SP, several spectra are captured, each with the retarder rotated to a different orientation. A system matrix is computed for each wavelength using the characterized parameters. The system matrix is pseudoinverted and multiplied into the data to obtain a reconstruction [14, 23]. In reconstructions for our experiments, an error propagation was carried out through this process. Uncertainties were estimated for each of the parameters entering into the reconstruction, including retardance, fast axis orientation, transmittance and spectrometer data. Following standard practice [24], these were propagated through the reconstruction using numerical derivatives and summed in quadrature to obtain uncertainties in each Stokes component across the spectrum.

The SPSG is essentially the rotating compensator, fixed analyzer system run in reverse. Light is carried to the system with a multimode optical fiber and collimated with an achromatic doublet. The collimated beam passes through a linear polarizer and an achromatic quarter-wave plate. The spectral content of the output is controlled by the choice of source, which may be for example an LED, a tungsten-halogen lamp, or a lamp filtered by a monochromator. The polarization content is controlled by varying the orientation of the retarder (and if necessary the polarizer). The SPSG was characterized in a manner similar to the RCFA SP, allowing calculation of the output spectral polarization state via the Mueller calculus.

Our characterization of these three systems leaves overall multiplicative factors undetermined, so we have not attempted to calibrate to absolute units. Accordingly, we report results normalized by an overall factor, chosen to make total power of the beam (i.e. the integral under s ₀) equal to unity. This is radiometric normalization, but not polarimetric normalization.

4. Calibration

We consider the spectral polarization state input to the system and the data output by the spectrometer to be vectors, and treat the channeled SP as a linear operator H which maps between them. Neglecting noise, the system is described by

In keeping with optical engineering terminology, we will refer to the vector space from which the system’s inputs are drawn as object space, and that to which the outputs belong as image space. The image vector g is a discrete column vector of data acquired by the spectrometer, each element corresponding to a spectral band or pixel. A dark pedestal (with contributions from dark current and stray light) is assumed to have been subtracted. 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. It is convenient to choose the sample locations at the same wavelengths reported by the spectrometer. With discrete representations of object and image space, the system operator becomes a matrix. Our approach to calibration is simply to estimate this matrix experimentally.

The system matrix is estimated by recording the output data vectors for a set of known inputs. Arranging the inputs s_k as columns of a matrix Q, and the recorded data vectors g_k as corresponding columns of a matrix G, Eq. 2 becomes

and H may be estimated as

where a superscripted + represents the Moore-Penrose pseudoinverse [25, 26]. Since the matrix contains experimental (noisy) data, the threshold applied to singular values in the pseudoinversion must be meaningfully set. It is advisable to compute the pseudoin-verse by means of singular value decomposition, and inspect both the singular value spectrum and the singular vectors in determining the threshold, since if the choice is completely neglected a pseudoinverse routine may default to a very small value based on machine precision. In this case noisy singular vectors are built into the pseudoinverse.

Calibration states were prepared by feeding light of narrow spectral bandwidth from a monochromator through the SPSG. Their polarization and spectral content were controlled via the SPSG retarder’s orientation and the monochromator’s center wave-length setting respectively. Many states were generated in sequence by looping through wavelength and retarder positions. At least four retarder orientations are required to span the four dimensions of the Stokes vector, and we used six to provide moderate redundancy. These were chosen to correspond to polarization states which are widely dispersed across the surface of the Poincaré sphere [23, 27, 28, 29]. Measurements were made for 179 different wavelengths at each retarder setting.

With approximately 1000 bands in the spectrometer, the calibration comprises 179, 000 individual measurements, a considerable volume of data. Nonetheless the system matrix contains 1000×4000 elements, leaving the matrix estimation (Eq. 4) severely under-determined. We can trust however, that the estimated matrix will accurately predict the system’s output for any input that is adequately modeled as a linear combination of the input calibration states. This representation requirement serves as a guideline for selection of calibration states, and explains why 179 wavelengths were used when the system is only designed for about 18 bands of spectral resolution in the final output. The spectral content of the calibration states consisted of sharp, narrow-band spikes, while the object states of interest presumably have smooth, h band limited Stokes spectra.

Another fine point in the calibration has to do with dual use of the grating spectrometer for measurements in object space and image space. In object space, the spectrometer is used to characterize the spectral profile of the SPSG’s output, facilitating calculation of the generated state. This use relies on calibration of the pixel-to-wavelength mapping and spectral responsivity of the spectrometer. Data on the mapping was provided by the manufacturer (based presumably on identification of spectral lines from a vapor discharge source), and the spectral responsivity was calibrated by reference to a calibrated incandescent lamp. In image space, the spectrometer serves as the detector in the instrument under calibration, and direct use of the mapping and responsivity calibration is unnecessary since the detector’s characterization is embodied in the system matrix Ĥ.

Now the catch. We have explained the importance of calibrating the channeled SP to account for spectral resolution in the spectrometer. However, we have used the spectrometer to characterize the input calibration states, leaving us in the position of employing the spectrometer as its own reference. This apparent dilemma is resolved by the recognition that in object space we require calibrated data but low spectral resolution, so we can get by without characterizing and correcting the spectrometer’s resolution. In image space we need to at least partially retain fine spectral detail, but the recorded data need not be calibrated. It is inconsequential whether the calibration matrix Ĥ attributes blurring effects correctly to spectral resolution, erroneously to polarization sensitivity, or to a combination of the two, as long as the representation requirement is met.

5. Reconstruction

Our basic approach to reconstruction is to pseudoinvert the calibration matrix, and carry out the reconstruction as

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 [20]. Using a canned routine without taking measures to stipulate the object space amounts to an implicit choice of a space in which the Stokes component spectra are not subject to any h band limit. Such a choice for object space leads to an under-determined inverse problem, and allows spurious high frequency (sharp) detail in the reconstruction.

There is not a single right choice for object space, though the amplitude modulation analogy 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 h band limit. 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 [17]. However, the reconstructions presented in this article all employ 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. The 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.

 

Fig. 3. Measurements of a spectral polarization state generated with light from a yellow LED in the state generator. The overall normalization is determined so that the s0 curve has unit area.

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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. 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.

6. Experimental results

A measured spectral polarization state, generated using a yellow LED as the source in the SPSG, is shown in Fig. 3. The SPSG retarder was oriented with its fast axis at 75° to horizontal and the analyzer’s axis was horizontal, resulting in elliptical polarization. Each Stokes component spectrum is plotted on separate axes. The channeled spectropolarimeter reconstruction is overlayed with the predicted SPSG output and RCFA spectropolarimeter measurement. The RCFA spectropolarimeter results are plotted on a gray strip whose width represents the estimated (plus or minus one standard deviation) uncertainty in the measurement. Agreement among the measurements is very good.

The data shown in Fig. 4 comes from a similar measurement, except that white light is used in the SPSG. The source is a tungsten-halogen lamp with a red-suppression filter. The SPSG retarder is oriented with its fast axis at 45° to horizontal to produce nominally right circularly polarized light.

 

Fig. 4. Measurements of a spectral polarization state generated with white light in the state generator. The overall normalization is determined so that the s0 curve has unit area. Note that each vertical axis has a different scale.

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Agreement among the measurements is not as good in this case, and warrants some discussion. Of particular concern is the ringing in the channeled spectropolarimeter reconstruction, which increases in severity toward the edges of the spectral range. Such ringing is anticipated when broad λ (or σ) spectra which do not fall to zero within the system’s spectral range are present [19, 20]. With a pseudoinverse reconstruction using a Fourier basis, the reconstruction entails a truncated Fourier series representation of the spectral polarization state. The Fourier series may be evaluated outside the spectral range of the instrument, in which case it generates the periodic continuation of the measurement results.

Since the Stokes component spectra in question do not have the same height at the two ends of the range there is a discontinuity in the periodic continuation. Upon truncation of the Fourier series, deviation at the edges of the spectral band and ringing result. These errors may be alleviated somewhat by the choice of a different object space, or by apodization [19, 30] in the reconstruction. Apodization refers to multiplication of a window function which falls to zero at the edges of the system’s spectral range into the raw data to prevent discontinuities in the periodic continuation. The window function is divided back out of each reconstructed Stokes component spectrum. However this step may aggravate the discrepancies at the edges of the spectrum, where the division is nearly by zero.

Another set of measurements was performed to demonstrate time-resolved spectropolarimetry, and motivate a potential practical application to time-resolved stress analysis [31]. For these measurements stressed plexiglass was used as the retarder in a setup similar to the SPSG. Linearly polarized, white light was focused onto the plexiglass (so as to illuminate only a region of fairly uniform stress), collimated, and analyzed with the channeled SP. A lever arm attached to the plexiglass was used to vary the stress. Figure 5 shows a measured spectral polarization state; its variation in time is indicated by the animation. Apodization was used in the reconstructions, and the data shown spans only the 0.45–0.65 µm wavelength band. The channeled SP’s acquisition rate is limited by the integration and read-out times of the spectrometer. In this experiment measurements were made at a rate of about 3 Hz, which was determined by the integration time necessary to produce nearly full-range readings.

 

Fig. 5. (116 KB) Measurements of a spectral polarization state generated by linearly polarizing light from a white LED and passing it through plastic with stress-induced birefringence. Time variation of the state (contained in the animation) was introduced by varying the stress. The overall normalization is determined on the basis of total optical power.

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Plots of the Stokes spectra may not be the most convenient format for presentation and interpretation of spectropolarimetry data. The Poincaré sphere affords an alternative which is particularly interesting for the stress birefringence demonstration. At each wavelength, the measured polarization state (neglecting the randomly polarized component and normalizing) is represented by a point on the surface of the Poincaré sphere. Together these points form a path on the surface of the sphere, which is sometimes termed a Stokes snake [32]. The snake representing the linearly polarized input is compressed to a point on the Poincaré sphere’s equator. The effect of a retarder is to rotate an input state on the Poincaré sphere about an axis in the equator [33]. Since the retardance in this experiment varies with wavelength, the Stokes snake is rotated and stretched into a circular arc by the stress birefringence. Figure 6 illustrates this behavior. The angular displacement and extent of this arc increase as the stress is increased, as illustrated in the animation.

Figure 7 shows the Stokes snake for a similar channeled SP data set. The plane of a best-fit circular arc is shown as well. The plane’s intersection with the Poincaré sphere’s equator indicates the orientation of the incident linear polarization, at least to within two choices. The normal to the plane which passes through the center of the Poincaré sphere is the rotation axis. It intersects the equator at the two linear polarization states aligned with the fast and slow axes. The retardance (modulo 2π) at a given wavelength is indicated by the angle between incident and final polarization states in the plane of the Stokes snake. With phase unwrapping, this construction allows extraction of the retardance variation as a function of wavelength. Resolving the ambiguity in identification of the fast and slow axes is a matter of requiring retardance to increase with wavenumber. When the rotation axis is chosen so that the snake traces a right-handed rotation from red to blue, the positive end of the rotation axis indicates the orientation of the retarder’s slow axis. In the context of two-dimensional photoelasticity [9], the difference between the principal stresses and the orientation of the principal axes may be determined from a single channeled SP data set if the plexiglass’ thickness and stress-optic coefficient are known.

 

Fig. 6. (108 KB) Animation of the time-varying spectral polarization state plotted on the Poincaré sphere.

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Fig. 7. The Stokes snake for the channeled SP data from the stress-induced bire-fringence experiment corresponds to a circular arc on the Poincaré sphere.

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7. Summary

We have described a practical approach to calibration and reconstruction in channeled spectropolarimetry, and demonstrated it with a hardware prototype. We use a linear operator (matrix) model of the system capable of describing non-ideal effects, such as dispersion in the thick retarders and spectral blur in the spectrometer. This matrix is experimentally estimated in the calibration. Reconstructions are carried out using its pseudoinverse, which is computed under a constraint to a chosen object space. Our results compare favorably with spectral polarization state measurements performed with a rotating-compensator polarimeter, and with the expected output of our polarization state generator. Application of the system to stress analysis demonstrates its snapshot capability, and provides an interesting example of interpretation and exploitation of spectropolarimetric data.