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Design of channeled spectropolarimeters
Nathan Hagen
Department of Optical Engineering, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi 321-0954, Japan (nh@hagenlab.org)
Received 10 February 2022; revised 23 March 2022; accepted 24 March 2022; posted 28 March 2022; published 13 April 2022

I present design and tolerancing guidelines for constructing channeled spectropolarimeter systems employing
high-order retarders. The discussion includes how to select appropriate retarder thicknesses, how to accuratelyalign the elements, how to tolerance the retarders, and how to analyze the effect of different polarizer types on thesystem performance. © 2022 Optica Publishing Grouphttps://doi.org/10.1364/AO.455828

1. INTRODUCTION
Although a number of publications on calibration techniques,reconstruction methods, and example applications using
channeled spectropolarimetry (CHSP) have been published,
information on the procedure for designing these channeledpolarimeters is incomplete and is scattered across many different
publications in the literature. In the discussion below, we step
through the process and provide expressions for designing thevarious elements of each system, and we explain the various
trade-offs.
Channeled spectropolarimetry takes advantage of the wavelength dependence of retardance in thick wave plates to encode
the input light polarization state into intensity modulations in
the light spectrum [1]. Using a standard dispersive spectrometer to detect the resulting channeled spectrum, this allows thesimultaneous measurement of the spectrally resolved Stokes
vector in a snapshot (single measurement frame). Figure 1 showsa typical CHSP system layout.
If we represent the Mueller matrices of CHSP systemcomponents using R(α, δ) for a linear retarder with fast axis
oriented at angle α with retardance δ(σ ), and P(α) for alinear polarizer with transmission axis at angle α, then the
Mueller calculus of the system will take an input Stokes vector
sin = (s 0 , s 1 , s 2 , s 3 )> and produce an output spectrum
Iout (σ ) = d · P(0◦ ) · R(45◦ , δ2 ) · R(0◦ , δ1 ) · sin= 12 s 0 + 14 s 1 (e iδ2 + e −iδ2 )+ 81 [(s 2 − is 3 )e i(δ2 +δ1 ) + (s 2 + is 3 )e −i(δ2 +δ1 )+ (−s 2 − is 3 )e i(δ2 −δ1 ) + (−s 2 + is 3 )e −i(δ2 −δ1 ) ] ,(1)
where d = (1, 0, 0, 0) is the detection row vector. The
superscript > indicates a transpose operation. In Eq. (1), theretardances δi and Stokes vector elements s i are functions ofwavenumber σ = 1/λ. With this expression, we see that the1559-128X/22/123381-09 Journal © 2022 Optica Publishing Group

Stokes vector components generate modulations in the intensity
spectrum, encoded into seven heterodyne frequencies. With
suitable choices for the thicknesses of the two retarders, if wetake the Fourier transform of I (σ ), we obtain a seven-channel
distribution such as the one shown in Fig. 2(b) [2]. By filtering
the Fourier domain (OPD) spectrum and extracting the sevenchannels individually, we can reconstruct the individual Stokes
component spectra s i (σ ).
Figure 1 shows a configuration in which the two retarders aredesigned with relative thicknesses of 1:2 (i.e., R 2 has twice thethickness of R 1 ). Another possible setup uses a thickness ratioof 3:1, which creates nine channels in the OPD domain, rather
than seven, and thus is limited to 7/9ths the resolving power. A
benefit of the 3:1 configuration is that it prevents signal leaking
from the baseband into the first heterodyne band, and it also
creates two nominally empty channels that contain useful calibration data if the system’s retarders have significant dichroism,
or if the retarders are misaligned [3,4].
Because the heterodyne measurement technique relies on thedata being bandlimited to within the width of a single Fourierdomain channel, it is helpful to have a spectrum that contains
light over as broad a range as possible. For the spectrum shownin Fig. 2(a), the falloff in signal at the edges of the spectrum is
not due to the application of an apodizing filter but rather due tothe weak output of the halogen light source at UV wavelengths
and to the falloff in sensitivity of the spectrometer at the longest
wavelengths.
Figure 2 gives an example channeled spectrum measured
in the visible to near-infrared range (0.36−1.045 µm).
Figures 2(b)–2(d) show the Fourier-domain representationof the spectrum, the reconstructed Stokes spectra s i (σ ), andthe normalized Stokes spectra, s˜i (σ ). (We use a tilde to indicate
normalization.) Since the input spectrum is uniformly linearlypolarized at 22.5◦ , s 1 (σ ) and s 2 (σ ) coincide, and s 3 (σ ) = 0.
The normalized Stokes spectra are each drawn such that the
curve represents the mean value, and the region surrounding each curve the ±1 standard deviation uncertainty region
for the spectra. Figures 2(e) and 2(f ) shows the repeatability

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Fig. 1. Channeled spectropolarimeter layout, showing the input light passing through a pair of high-order retarders R 1 (0◦ ) and R 2 (45◦ ) and a linear polarizer P (0◦ ) to encode the polarization state into spectrum modulation frequencies. The two (upper and lower) rows of spectra indicate thedifference in behavior for two different spectrally resolved Stokes parameters as they propagate through the system. The upper row show an unpolarized spectrum passing through the system and the Fourier transform of the resulting measured spectrum. The lower row shows a similar set for a fullypolarized spectrum.

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 2. (a) Experimentally measured channeled spectrum, using quartz wave plates of thickness t1 = 3.89 mm and t2 = 7.77 mm. (b) Absolutevalue of the corresponding Fourier-domain spectrum, showing the seven channels C −3 through C +3 , centered at OPD values√ of 0, ±39.3 µm,±78.6 µm, and ±117.9 µm. (c) Reconstructed Stokes parameter spectra for a 22.5◦ input state, i.e., s 1 (σ ) = s 2 (σ ) = s 0 (σ )/ 2. (d) Reconstructednormalized Stokes vectors. The curves in (c),(d) are shown overlaid on their uncertainty regions (±1 standard deviation error). (e) RMS errors(standard deviation values at each wavenumber) for the spectrum reconstructed in Fig. 2 (c). (f ) Zoom-in view of the RMS errors, showing only thewavenumber region 1.1–2 µm.

Research Article
standard deviation at each wavelength, and we can see that themeasurement repeatability is best near the center of the spectral range, where the light intensity is strongest. In the central
region, we can obtain repeatability of better than 0.0005 in thenormalized parameters without difficulty, but in the UV range(σ > 2.5 µm) the repeatability rises above 0.01.
2. CHANNELED SPECTROPOLARIMETERALIGNMENT AND SETUP
A number of papers have been published relating to the alignment errors in a channeled spectropolarimeter system and how
to compensate for them [4–9]. However, these works do not
explain why they do not fix these errors during the alignment
process in the first place, so that the reconstruction algorithms
need not deal with the extra complexity. It is, in fact, simple toalign polarization components to accuracies of better than 0.1◦
in azimuth angle, as long as one relies on the optical signal fordoing the alignment rather than relying on the precision of themechanical mounts [10].
For aligning channeled spectropolarimeter components,
an easy method is to keep the CHSP polarizer P (0◦ ) in place
while placing an orthogonally oriented polarizer P (90◦ ) at thelight source side of the optical train. Between the two crossed
polarizers, one can align a retarder to 0◦ or 90◦ by detecting thelocation at which the transmitted signal is darkest, or to ±45◦ bydetecting the azimuth position at which the transmitted signal isbrightest—when integrating the spectrum over all wavelengths.
The choice of polarizer types can also affect system performance. Figure 3 shows the transmission spectra for three different
polarizer types: a calcite Glan–Thompson polarizer, a dichroic
glass polarizer (embedded with silver nanorods), and a standard
dichroic polymer sheet polarizer. Over the full spectral rangeof this system (0.36−1.045 µm), the Glan–Thompson has
clearly the best performance, as it has the highest transmittanceand maintains a high extinction ratio over the entire range. The
silver nanorod polarizer also maintains a high extinction ratio
but suffers from low transmission for λ < 600 nm. The polymer
polarizer has less of a problem with transmission loss in the visible wavelength range, but its extinction ratio declines quickly
for wavelengths above 700 nm. In addition, the polymer used
for these polarizers can sometimes contain significant amountsof retardance, which is problematic if placed on the source sideof the optical train. (It is not generally a significant problem
if placed as the last polarization element before the detector.)

Fig. 3. Transmission spectra of three polarizer types: a calcite Glan–
Thompson polarizer, a dichroic glass polarizer (embedded with silver
nanorods), and a visible-range dichroic polymer sheet polarizer.

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There also exist broadband wire-grid polarizers that are designed
to operate over this wide wavelength range, but these are notavailable for me to test here.
A weakness of using Glan–Thompson polarizers is thatthe elements themselves are quite large, limiting the system’s
compactness and permitting only a narrow field of view. Thisis generally not a problem for point-measurement systems,
but it may be for channeled spectropolarimeters used within
imaging systems. While the nanorod polarizers are far thinner,
they suffer from a significant backreflection—a problem sharedwith wiregrid polarizers—unless they are designed with goodantireflection (AR) properties [11]. If using uncoated retarders,or if the AR coatings on the retarders are suboptimal over someof the spectral range, then this backreflection can cause spurious
signals in the system, producing errors in the reconstructed spectrum. This problem is amplified when using a backreflecting
polarizer to adjust the polarization state of the input spectrum,such as when calibrating or when testing measurement accuracy.
In this case the backreflection from the CHSP polarizer willreach the backside of the illumination-side polarizer and make asecond pass through the retarders. Thus, backreflecting polarizers are particularly problematic in pairs. While the effect can be
mitigated by tilting the nanorod polarizer so that the light is nolonger normally incident, this only slightly reduces the spurious
signal.
3. ESTIMATING THE EFFECTIVE RESOLUTION
OF THE SPECTROMETER
The first step in designing a channeled spectropolarimetry system is to determine the usable spectral range of the spectrometerand the spectrometer’s resolution. For the discussion below,
we will assume the use of a grating spectrometer, but the results
are easily adjusted to wavenumber-sampling systems such as
Fourier transform spectrometers. For the channeled spectropolarimeter in our lab, the minimum and maximum wavelengths
λmin and λmax are determined by weak output of the halogen
light source at UV wavelengths and the poor response of theUSB4000 spectrometer used for detection. Thus our usefulspectral range is approximately 400–1035 nm.
The spectrometer sensor has N = 3648 pixels and a full
spectral range of 345.6–1045 nm, so that the wavelength
sampling rate is 0.192 nm/pixel. The 400–1035 nm useful spectral range, however, is somewhat smaller, so that the
total number of spectral samples is reduced to N = 3312.
As is typical of spectrometer measurements, N is muchlarger than the optical resolution of the system, in order toprevent aliasing. For this spectrometer, the manufacturer’s
stated resolution is R λ = 1.01 nm, so that we can estimate
its resolving power (spectral range divided by resolution) as
RP = (1035−400)/1.01 = 629.
One can also perform the same calculations in the wavenumber domain by dividing the spectrometer’s wavenumber range
by the “average” wavenumber resolution. The tricky point hereis that because of the inverse relationship between the wavelength and wavenumber, one must use the geometric mean of
the minimum and maximum wavenumber resolution values inorder to calculate this average. The minimum and maximum
wavenumbers bracketing the measured spectrum are3384

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σmin = 1/1.045 µm = 0.966 µm−1(2)σmax = 1/0.400 µm = 2.500 µm−1 ,

(3)

and

so that the useful wavenumber range is 1.534 µm−1 . While
the spectral resolution is approximately uniform in the wavelength domain, it is decidedly nonuniform in the wavenumber
domain, with the coarsest resolution occurring at the shortwavelength end of the spectrum. The resolutions at the two endsof the spectrum, and their geometric mean, are obtained as
R σ,max =Rλ1.01 nm= 6.31 × 10−3 µm−1 ,=λ2min (400 nm)2

(4)

R σ,min =Rλ1.01 nm== 9.43 × 104 µm−1 ,λ2max (1035 nm)2

(5)

and
R σ,avg =p

R σ,min R σ,max = 2.44 × 10−3 µm−1 .

(6)

The resolving power in the wavenumber domain is therefore
obtained asRPσ =σmax − σmin
= 632.
R σ,avg(7)

For a Stokes channeled spectropolarimeter design having
seven channels in the OPD domain, this means that the numberof resolvable points within each channel will be no more than632/7 ≈ 90.
When we take the Fourier transform of our data, the spectrum abscissa becomes the retardance in OPD. The resolvable
OPD domain range is roughly equal to the smallest resolvable
increment in the wavenumber spectrum: R σ,avg . Therefore theusable OPD domain range for our example setup is
OPDmax =1= 410 µm
R σ,avg(8)

so that the useful OPD domain will extend from negative to positive values between ±205 µm. As we will see from experimental
measurements, this gives roughly the correct value but is somewhat optimistic.

4. MEASURING THE SPECTROMETERRESOLUTION
To measure the spectrometer’s modulation contrast at eachwavelength, one can insert a series of thick retarders between
a pair of polarizers and measure the contrast in the resulting
channeled spectrum. Figure 4(a) shows the spectrum measured before and after inserting a 180λ thick retarder. For ameasurement with a high modulation contrast, the fringes in
the channeled spectrum will extend from zero all the way up to
the unmodulated spectrum value as they do in the NIR regionof Fig. 4(a). For low contrast measurements, the modulation
amplitude decreases, so that the fringes get smaller and eventually converge onto the value halfway between fully dark andfully bright (i.e., half the value of the unmodulated spectrum).
From the channeled spectrum shown in Fig. 4(a), we can see thatthe modulation contrast is closer to 1 than to 0 for most of thespectral range [12].
If we perform similar measurements for a series of different
retarder thicknesses, we can calculate the modulation contrast
from the channeled spectrum, producing the contrast spectra
shown in Fig. 4(b). As we expect, the contrast is much worse in
the short wavelength end of the spectrum because the modulation frequencies are higher there. For wavelengths around
400 nm, the modulation contrast falls below 0.1 when using
retardances of about 270λ or more. If we use this value of 0.1
contrast as a rough cutoff for the resolution of our system atthe shortest wavelengths, then the outermost CHSP channels
must not exceed 270λ of retardance. For a seven-channel CHSP,
this means that we can select a total retardance in our system ofδ1 + δ2 = (6/7) × 270λ = 231λ. This value is 6/7ths of theresolution limit because the channels themselves will have awidth, and the retardance value indicates the OPD for the channel center and not the channel edge. One can see the differencein Fig. 2(b), where the leftmost channel is centered at about
120 µm but the outer edge of the channel extends to about
140 µm.
Another option is to design the 1:2 configuration such thatthe outermost channel (i.e., channel C +3 in Fig. 2) is outside
the resolvable region, and only the inner two channels are used.
By itself, this does not cause any problems for the system, andit allows a 50% increase in the resolution of the reconstructed
spectra. The primary trade-off with this option is that the twochannels will no longer contain sufficient redundant information to allow the dynamic calibration technique to be used

Fig. 4. (Left) DC spectrum and the channeled spectrum obtained with a 180λ retarder. (Right) Estimated modulation contrast at eachwavelength, for different retarder thicknesses from 30λ to 340λ.

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Vol. 61, No. 12 / 20 April 2022 / Applied Optics[13,14]. In this case, one must instead resort to periodic recalibrations of the system and try to minimize thermal changes.
However, this configuration allows one to choose retardances ofδ1 = 115λ and δ2 = 231λ for the above example system.
Since the reference wavelength for the retarders is the He–Ne
laser wavelength of 632.8 nm, OPDmax = 270λ corresponds
to an OPD value of 171 µm. This is somewhat smaller thanour earlier estimate for the maximum OPD of 205 µm (inSection 3). The difference in the two values is equivalent tochoosing a different cutoff point for the contrast-limited
resolution.
Note that while the thing that we are measuring here is effectively a modulation transfer function (MTF) cutoff point for thesystem, an MTF is generally only defined for shift-invariant systems and not for situations such as experienced here, where thewavenumber resolution varies across the spectrum.
5. SELECTING THE PHYSICAL THICKNESS OF
THE RETARDERS
Once we know the OPD range for the system, we can design
the retarders themselves. There are three different forms of theretardance equations, with each being useful in different ways,
given byδ(λ) = 2πβt/λ [units : radians],(9)δ = B(λref )t/λref[units : waves],(10)[“OPD”, units : µm],

(11)

δopd = βtwhere β is the wave plate material’s first-order “effective birefringence” coefficient [15]
 
dB
(12)
β = B(σ̄ ) + σ̄dσ σ̄for birefringence B = n e − n o , and σ̄ is the central wavenumberσ̄ = 21 (σmin + σmax ). The subscript σ̄ on the gradient termin Eq. (12) means that the gradient should be evaluated at
the central wavenumber. The He–Ne wavelength 632.8 nm is
the most common choice for λref . In Eq. (10), we must use thebirefringence at the reference wavelength B(λref ) rather thanβ because expressing the retardance in waves assumes a single
wavelength and not a spectrum.
Knowing the desired OPDmax value from Section 3 orSection 4, we can use Eq. (11) to calculate the corresponding
retarder physical thickness (which can be converted to units ofwaves, if the manufacturer prefers) byt = δopd /β.

(13)Unfortunately, β is generally not a quantity that can belooked up directly in references, since Eq. (12) shows that it
depends on both the birefringence and the gradient of thebirefringence, each evaluated at the middle wavenumber inthe system’s spectrum. Thus, it varies with the chosen spectral
range. For those working in roughly the same spectral range as
the experimental system shown in Section 4, Ref. [15] shows

3385

that for most common retarder materials (ADP, MgF2 , αBBO,
quartz, sapphire, and calcite) β has a value about 10% larger
than the nominal birefringence alone. This allows for a quick
estimate of the thickness.
For a 1:2 ratio of retarder thicknesses in the CHSP, a single
channel width will be 2/7ths of the maximum OPD. Thus, fora system with OPDmax = 171 µm, the OPD corresponding
to a single channel (and to that of the thinnest retarder) will beδ1 = 27 (171 µm) = 48.9 µm. For a quartz retarder operatingover the example spectral range, β = 0.00998—a number thatis conveniently close to 0.01, so one can simply say that the R 1
and R 2 should be given thicknesses of 4.89 mm and 9.78 mm.
6. TOLERANCING THE WAVE PLATE
THICKNESSES
Although Dubreuil et al. analyze wave plate thickness errors for
CHSP systems, they do so in the context of calculating errors
in the locations of the retarder peaks rather than errors in theratios between the various retarder thicknesses [16]. As wehave seen in Section 4, the system’s resolvable range is actuallya fuzzy boundary that leaves a good deal of flexibility for the
thicknesses. What is more serious, however, is an error in therelative ratio of retarder thicknesses. Thus, although one might
be tempted to specify the thickness tolerances of the retardersas, say, t1 = 5 mm ± 0.05 mm and t2 = 10 mm ± 0.05 mm,tighter tolerances should be placed on the thickness ratio
directly rather than on each retarder individually. Therefore, abetter specification would be
t1 = 5 mm ± 0.1 mm,(14)

t2 = t1 × (2 ± 0.005).

(15)

The reason for tolerancing the ratio much more tightly than
the nominal thickness is that the OPD position of some of
the channels are determined by taking sums and differences
between the retarder OPDs. An error in the thickness ratios
will cause a relative shift in the position of these sum/difference
channels relative to channels whose position is determined
entirely by an individual retarder. To deal with this, we can either
allow each of the channels to have unique widths, or we canreduce the bandwidth of each channel to prevent overlapping,effectively reducing the resolution in the reconstructed spectra.
The one causes increased complexity in the reconstruction
algolrithm, while the other sacrifices performance.
Errors in thickness ratios cause much more difficulty for
Mueller spectropolarimeters. In the latter case, errors in the
thickness ratios not only cause channels to shift in position, butsince there are often multiple pathways through the system thatlead to the same OPD value, ratio errors will cause one set of data
to shift with respect to another overlapped set of data within asingle channel [17]. Figure 5 shows an example of this type ofproblem, in which the OPD for channel C 1 is generated by light
modulated by either δ1 or δ2 − δ1 . If δ2 is exactly twice δ1 , thenthese two pathways will produce exactly the same OPD value,
but if not, then the two will be shifted slightly with respect toone another. This is difficult to correct. However, one approachto relaxing the tolerances on the retarder thicknesses of a Mueller

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Fig. 6. Error factor in the amplitude of the reconstructed spectra
due to a shift by opd = 1/σrange (solid curve) and by opd = 1/(2σrange )(dashed curve).

Fig. 5. Set of channels in the OPD domain for a Mueller matrix
channeled spectropolarimeter employing a 1:2:5:10 ratio of retarder
thicknesses. The heights of the individual channels reflect the channel’stransmission factor. The six highlighted channels (in red, green, andblue) indicate channels in which multiple pathways through the set ofretarders produce the same OPD value. The inset shows a zoomed-in
view of channel C 1 generated by two pathways, given by OPD valuesof either δ1 or δ2 − δ1 , when δ2 is not exactly equal to 2δ1 .

spectropolarimeter is to choose a 2:1:7:15 ratio of thicknesses.
In this design, there will be only one pathway to any given channel, so that a relative shift within overlapping data will not occur.
The trade-off is that the number of channels increases to 51, sacrificing resolution in the reconstructed Mueller spectra. Anotherapproach to mitigating this shifted-overlapping-data problem is
to use two-channel detection at the spectrometer—an approach
discussed in Section 8 below.
In order to decide on a particular value for the thickness
ratio tolerance for a Mueller spectropolarimeter design, we cananalyze the effect that this error would have on the reconstructed
spectra if we simply ignored it. Consider one path through theretarders generating an OPD that we call the nominal OPD of
the channel, while the OPD value of a second path through thesystem leads to the same nominal value but is subject to a small
shift due to manufacturing error. In this case, a worst-case effectis that the data from path 1 is absent and the data from path
2 is the entire data present in the channel, due to the specificpolarization properties of the object under measurement. If
the wavenumbers in our spectrum span 1 µm−1 to 2.5 µm−1 ,then the full wavenumber range is σrange = 1.5 µm−1 , so thatan OPD shift of 1/σrange = 0.667 µm will cause an amplitudechange corresponding to a half-wave cosine (Fig. 6, solid curve).
This is obviously a severe error: although the data at the center ofthe spectrum is largely unaffected, data at the edges of the spectrum are reduced to zero. If the path #2 OPD shift is half this(i.e., 0.333 µm), the smaller shift induces an error factor thattakes the shape of a wider cosine wave (Fig. 6, dashed curve). In
this case, the maximum error factor (at the edge of the spectrum)
is now only 0.707.
The OPD shift of data from one path relative to another, due
to manufacturing tolerances, is what we can call the OPD error
opd . As this error decreases in size below 0.333 µm, so too does

the magnitude of error. If we are willing to tolerate a maximum
amplitude error of 1% at the edges of the spectral range (so there
will be <1% error everywhere else in the spectrum), then theallowable OPD error will beopd <

arccos (0.99)
4×.
2π
σrange(16)

For our example system, with σrange = 1.5 µm−1 , this comesto an OPD error tolerance of opd = 0.060 µm = λref /10. Fora quartz retarder in this spectral range, where β = 0.00998, thisOPD error corresponds to a retarder thickness tolerance oft = opd /β = 6.02 µm.

(17)This is not at all a difficult tolerance for a manufacturer to test
for, if an optical setup designed for testing linear retardance isavailable. What is unusual about this tolerance, however, is that
this is not a tolerance on the retardance modulo λ/2 but rather atolerance on the absolute retardance value, and this latter is not
something that manufacturers typically test for.
In order to test for the absolute retardance, one can takeadvantage of the fact that the absolute retardance of the overallsystem can be loose, while the relative retardances should be kept
tight. Thus, when testing these retarders, one can make an initial
retarder at the loose tolerances and then check all subsequent
retarders against it. If the pair of retarders are affixed to one
another at orthogonal azimuth angles (to cancel the effect of one
another) and placed between crossed polarizers, then one should
see no transmitted light, regardless of the angle at which the pairare rotated. If there is a small difference in thickness between thetwo retarders, then the cancellation will not be complete and
one will see modulation in the transmitted light as a function of
the rotation angle. Figure 7 shows the result of doing this test,by placing two 60λ retarders between crossed polarizers. If thetwo retarders are aligned to one another, then the modulations
are maximized when the retarder is oriented at 45◦ to the crossed
polarizers as shown in the black curve of the figure. If the retarders are at orthogonal angles to one another, then a matched pair
will produce no modulations in the spectrum (i.e., a retardance
difference of zero), even when placed at 45◦ to the crossed polarizers. This is the result shown with the red curve of Fig. 7. If thispair of retarders were instead a 60λ wave plate and a 60.5λ wave

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7. RETARDER AR COATING SPECIFICATIONS

Fig. 7. Spectra obtained when two 60λ retarders are placed in(black) parallel orientation (i.e., both at 45◦ azimuth), and (red)orthogonal orientation (i.e., one at +45◦ and one at −45◦ azimuth).
The baseline spectrum (blue curve) shows the unmodulated distribution that occurs when the final polarizer is removed. Note that the twomodulated curves (black and red) use the left and right axes, respectively, and that the right-hand side scale is much smaller. If we integrate
across all wavelengths, we find that the parallel-config spectrum isabout 83× brighter than the cross-config spectrum.

plate, then orienting them orthogonally to one another should
generate the same effect as a single λ/2 retarder.
While the red curve in Fig. 7 should be zero in a good system,
we see that the unexpected spectrum takes on the same shapeas the baseband spectrum (the DC channel, blue curve) but
about 184× lower in amplitude. Since the leakage spectrum(red curve) has the same shape as the baseline spectrum (bluecurve), we can conclude that there is no significant error in theretardance ratio of the two, since such an error would produce a
low-frequency modulation in the leakage spectrum. The leakage
spectrum also shows small-amplitude modulations at the sum
frequency corresponding to a 120λ wave plate—a phenomenon
we discuss in Section 7 below.
This same approach to calculating thickness tolerances canalso be used to specify tolerances for the spatial nonuniformity across the retarders, the lowest-order form of which is“wedge”—in which the two faces of the retarder are not quite
parallel to one another. Variation in the retarder thickness acrossits face will cause a continuous range of OPD values for lightpropagating through different locations in the face of the wave
plate. One approach to tolerancing these would be to specify
that the peak-to-valley thickness range should be less than somevalue, such as the λ/10 tolerance we used above for the overall
thickness.
While a spatial uniformity of λ/10 is not a relaxed tolerance,
it is also not a difficult tolerance to meet for a manufacturer witha setup allowing mapping of polarization signatures across theface of the sample [18]. Baur and Kraemer report measurementsof 0.008 waves of retardance uniformity on a 105 mm diameter
plate of 4.2 mm thick quartz [19]. This is an unusually large
diameter wave plate, making it challenging to manufacture.
Most wave plates are well under 25 mm diameter in size, where
uniformity is easier to achieve. Baur and Kraemer also explainthat their measured retardance uniformity value cannot be
explained entirely by thickness variation. At this level of precision, there are likely material variations within the volume of thecrystal, including residual stress birefringence, that also play arole in affecting retardance uniformity.

If the channeled spectropolarimeter retarders are used overa wide spectral band, a good AR coating will require a nonstandard coating design. High-quality AR coatings can be
designed to operate over the entire 400–1000 nm band and canalso be designed to work for both the ordinary and extraordinaryrefractive indices of the material, but the large number of coating layers needed for this may make such a coating prohibitive
for users. A useful compromise is to apply a standard λ/4 layer
coating of MgF2 on both sides of the retarder. While this does
not reduce the surface reflectivity below 1%, it is an easy coating for manufacturers to apply and still provides a significant
reduction in the reflectivity [20].
In order to understand whether one can get by without an
AR coating, or whether a high-quality coating is needed for agiven system, we can analyze the effect that these reflections
have on measurements. Figure 7 showed the spectrum of light
passing through two uncoated 60λ retarders oriented in parallel and crossed orientations. When this retarder pair is placed
between crossed polarizers, the orthogonally oriented retarders should maintain light extinction at the second polarizer
regardless of the retarder pair’s relative orientation with respect
to the polarizers. (During this check, the retarder pair must be
maintained at orthogonal angles to one another while rotating
together.) The red curve of Fig. 7, however, shows that there is asignificant residual signal that is not extinguished and contains amodulation equivalent to a 120λ retarder.
The various paths that transmitted light can experience in asystem subject to double reflections are shown in Fig. 8. If thetwo retarders are aligned in parallel, then δ1 = δ2 in our experimental setup, giving the “parallel” results shown in the figure.
If aligned orthogonal to one another, then the δ1 = −δ2 , giving
the “crossed” results. As we see, double reflections in the system
can generate doubled-frequency terms when the retarders are
orthogonally aligned.
Seeking to verify that the unexpected signal in Fig. 7 may be
due to double reflections in the uncoated retarders, we can calculate the expected amplitude from the Fresnel coefficients. For
typical quartz, the ordinary and extraordinary refractive indicesvary within [21]n o (400 nm) = 1.5577,

n o (1000 nm) = 1.5350,(18)

Fig. 8. Seven different paths that transmitted light can take inthe presence of double reflections and their consequent retardancesums. The example retardance sums assume δ1 and δ2 are both 60λretardance.

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Vol. 61, No. 12 / 20 April 2022 / Applied Optics

n e (400 nm) = 1.5673,

n e (1000 nm) = 1.5438,(19)

so that the normal-incidence Fresnel reflectivities at each surfaceareρo (400 nm) = 0.0475,

ρo (1000 nm) = 0.0445,(20)ρe (400 nm) = 0.0488,

ρe (1000 nm) = 0.0457.

(21)

The remnant of light produced after a double reflection isρ 2 , or about 0.002 for both ordinary and extraordinary light.
This may seem a negligible amount, but we can also see from
Fig. 8 that there are four separate pathways leading to a doubled
frequency, so we can expect the fraction of light intensity modulated at 120λ to be about 4ρ 2 ≈ 0.0088. From the ratio of theintegrated spectra of the parallel and crossed configurations in
Fig. 8, the ratio of intensities is about 1/83 ≈ 0.012, or somewhat higher than what we calculate from the Fresnel reflections
alone.
A second effect that we might see with uncoated retarders
is that the transmission for the ordinary and extraordinary
beams will be slightly different. For quartz refractive indices,
this amounts to a small difference in transmittance of about0.001 per surface. With four retarder surfaces in the channeledspectropolarimeter, we might expect this to amplify by a factorof 4, but since the crossed configuration causes a swap of theordinary and extraordinary beams passing through the retarderpair, the effective transmittance will be very nearly identical forthe two beams. As a result, diattenuation at the retarder faces isunlikely to produce the unwanted signal of Fig. 7.
The reason for the factor of 2 difference between calculations
and experiment is unclear, but one thing we can note in Fig. 7 isthat the crossed configuration signal exhibits a smaller modulation contrast than the parallel configuration signal does—its
DC spectrum is larger relative to the modulation amplitude.
This may indicate that the light arises from backreflections
elsewhere in the system (such as at the polarizers, focusing lens,
fiber input to the spectrometer), or from bulk scattering fromwithin the retarder crystals themselves. When performing thisexperiment with different retarders, I also found that the ratio ofintensities varies significantly with the retarders used, indicatingthat the problem likely stems from something beyond just the
Fresnel reflections alone.
Such an unanticipated signal as the red curve of Fig. 7 isobviously problematic for accurate spectral reconstruction.
The unextinguished DC spectrum component will change thebaseband spectrum relative to the s 1 , s 2 , and s 3 spectra, and thiswill cause error in the normalized spectra if not calibrated for.
The doubled-frequency component is also a problem, as it shifts
the data into a different channel, and the shifted data cannot
easily be separated from the other data that is there.
8. DESIGN ISSUES FOR TWO-CHANNEL
DETECTION
Dubreuil et al. presented an adaptation of the standard channeled spectropolarimeter design in which the detection pathis split into two with a Wollaston prism [22]. This can be amajor advantage for calibration and for reducing the effective

bandwidth needed in the heterodyne channels. However, some
additional care may be needed in the optical design when using
these splitters over spectral bands as wide as 400–1000 nm dueto the wavelength dependence of the prism’s splitting angle.
To give an example, a calcite Wollaston prism with a nominal
splitting angle of φ = 20◦ (at λ = 633 nm) will use a pair ofcalcite prisms with wedge angles of α = 44.5◦ . Adopting thefollowing approximate expressions for the angle of the rays
emerging from the Wollaston prism [23],

 

no
sin α
,
(22)
θo = arcsin n e α − arcsin
ne
 



ne
θe = arcsin n o arcsin
sin α − α ,(23)noallows us to determine the splitting angle dispersion from therefractive indices’ dispersion. For the 20◦ split calcite Wollaston
prism, the ordinary and extraordinary ray angles vary as
θo (400 nm) = 10.13◦ ,θe (400 nm) = −11.45◦ ,θo (1000 nm) = 9.00◦ ,(24)

θe (1000 nm) = −10.04◦ . (25)Therefore, the optics of the two spectrometers being used forsuch a system should be able to accept a range of angles between±0.57◦ and ±0.70◦ for ordinary and extraordinary light acrossthe full 400–1035 nm spectral range [24]. While these are
not large angles, they are still large enough that they must be
accounted for in the design, in order to prevent excessive vignetting for the shortest and longest wavelengths at the slit. For
example, one should make sure that the spectrometer entrance
slits have their long dimension orthogonal to the angle-splitting
direction.
Although one can try using a different material for the
Wollaston prism, the most commonly available materials(quartz, αBBO) have a similar dispersion profile in this spectral
range, and so there is not much benefit to be had by switchingmaterials. Reducing the nominal splitting angle will also reduce
the splitting angle dispersion but causes crowding of the spectrometer optics. One may also consider using a cube polarizing
beam splitter, though it requires more care in calibrating its
spectrally dependent polarization properties than a Wollaston
prism does.
9. CONCLUSION
Starting with the spectral range of the measurement system,
we have shown how to determine the appropriate OPD cutoff
values from which one can calculate the appropriate retarder
thicknesses for a channeled spectropolarimetry system. Because
these are custom-made components, manufacturers will needguidance on the necessary tolerances, and we have shown thatwhile the overall thickness tolerances can be very loose, thetolerances of the thickness ratios between the various wave plates
must be tightly maintained. This is not a standard part of manufacturers’ testing procedures, so we have presented guidelines toshow a simple method by which one can test these parts to highprecision. We also discussed the requirements for specifying

Research Article
AR coatings for these wave plates, with trade-offs between astandard coating that is easily available to manufacturers, but
having some loss in performance, or a more expensive wideband
coating that must be custom-made for these parts. Finally, we
showed that the two-channel detection approach is viable if we
ensure that the spectrometers’ exit slits are not vignetted by the
Wollaston prism’s splitting angle dispersion.
Funding. Japan Society for the Promotion of Science (20K04516).
Disclosures. The author declares no conflict of interest.
Data availability. Data underlying the results presented in this paper are
not publicly available at this time but may be obtained from the author upon
reasonable request.

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