5032

Research Article

Vol. 59, No. 16 / 1 June 2020 / Applied Optics

Compensating for nonlinear dispersion inchanneled spectropolarimetry
Nathan Hagen,1, *Prathan Buranasiri,2 AND Yukitoshi Otani1

1

Utsunomiya University, Department of Optical Engineering, Center for Optical Research and Engineering (CORE), 7-1-2 Yoto, Utsunomiya,Tochigi 321-0954, Japan2
Department of Physics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand*Corresponding author: nh@hagenlab.org
Received 17 January 2020; revised 13 April 2020; accepted 28 April 2020; posted 29 April 2020 (Doc. ID 388473); published 28 May 2020

All common waveplate materials exhibit nonlinear dispersion of retardance, producing an unwanted chirp in theinterference fringes that channeled spectropolarimeters use for heterodyning polarization data. After showing
how to quantify this nonlinearity, we survey the common waveplate materials and find that MgF2 has significantly
lower nonlinearity than any other available material. We also quantify the degree of crosstalk caused by dispersion
nonlinearity and show that, unlike in linear dispersion, the degree of crosstalk depends on the sequence of how
the phase calibration is implemented. Regardless of how the calibrated phases have been obtained, shifting eachchannel to baseband prior to windowing minimizes crosstalk error. © 2020 Optical Society of America
https://doi.org/10.1364/AO.388473

1. INTRODUCTION
Channeled spectropolarimetry (CHSP) takes advantage of the
wavelength dependence of thick retarders to encode the input
light polarization state into intensity modulations in the light
spectrum [1,2]. Using a standard dispersive spectrometer todetect the resulting channeled spectrum, this allows the simultaneous measurement of the spectrally resolved Stokes vector ina snapshot (single measurement frame). Figure 1 shows a typical
CHSP system layout.
If we represent the Mueller matrices of the system components using R(α, δ) for a linear retarder with the fast axis
oriented at angle α with retardance δ(σ ), and P(α) for a linear polarizer with the transmission axis at angle α, then the
Mueller calculus of the system will take an input Stokes vector
sin = (s0 , s1 , s2 , s3 )T and produce an output spectrum [3]:Iout (σ ) = d · P(0◦ ) · R(45◦ , δ2 ) · R(0◦ , δ1 ) · sin= 21 s 0 + 14 s1 (e iδ2 + e −iδ2 )+ 81 [(s2 − is3 )e i(δ2 +δ1 ) + (s2 + is3 )e −i(δ2 +δ1 )+(−s2 − is3 )e i(δ2 −δ1 ) + (−s2 + is3 )e −i(δ2 −δ1 ) ] ,(1)
where d = (1, 0, 0, 0) is the detection row vector. In Eq. (1),the retardances δi and Stokes vector elements s i are both functions of wavenumber σ = 1/λ. With this expression, we seethat the four Stokes components are each encoded into theintensity spectrum, where the data are modulated into sevenheterodyne frequencies. With suitable choices [4] for the thicknesses of the two retarders, if we take the Fourier Transform ofI (σ ), we obtain a seven-channel distribution such as the one
1559-128X/20/165032-09 Journal © 2020 Optical Society of America

shown in Fig. 2(b). By filtering the Fourier domain [optical
path difference (OPD)] spectrum and extracting the sevenchannels individually, we can reconstruct the individual Stokes
component spectra si (σ ).
Figure 2 gives an example channeled spectrum measured in
the visible to near-infrared range (0.45–0.95 µm). Figure 2(b)shows a Fourier-domain representation of the same spectrum,
while 2(c) gives the calibrated nonlinear phase and 2(d) thereconstructed Stokes spectra. Since the input spectrum is uniformly linearly polarized at 22.5◦ , s1 (σ ) and s2 (σ ) coincide, ands3 (σ ) = 0. At the full scale shown in Fig. 2(d), the error is too
small to be visible in all but the s3 spectrum, where we can see
a small fluctuation away from exactly zero. There is no visible
difference between the compensated and uncompensated spectra in this simple example, since the input spectra are so tightly
bandlimited.
In the discussion below, we show that this straightforward
interpretation of Eq. (1) is more complex than it looks due tothe fact that the retardance of standard waveplates is a nonlinear
function of wavenumber. While this has generally been passed
over silently in the literature, the nonlinearity causes a chirp inthe modulation frequencies that can lead to serious reconstruction errors if not correctly compensated for. For example, weshow below that if the reconstruction steps are not followed ina specific sequence, then the reconstructed Stokes spectra areprone to crosstalk errors.
A number of researchers have presented methods for calibrating the CHSP phases, such as reference beam calibration [5,6],
self-calibration [7], and phase re-arrangement [8]. However,where previous researchers discuss how to calibrate the phases

Research Article

Vol. 59, No. 16 / 1 June 2020 / Applied Optics

5033

channel positions (the heterodyne frequencies in the OPD
domain). In order to locate the channel positions precisely, weneed to separate the linear components of the retardance fromthe constant and nonlinear components.
The wavenumber-dependent expression for the retardance inradians is given by
Fig. 1. Channeled spectropolarimeter layout, showing the input
light passing through a pair of high-order retarders R 1 (0◦ ) andR 2 (45◦ ), and a linear polarizer P (0◦ ) to encode the polarization state
into spectrum modulation frequencies.

δi (σ ), here we discuss the meaning of the obtained phase functions, how their nonlinearity can cause an unexpected increasein crosstalk, and how to implement the calibration to minimize crosstalk error. We also provide simulations of different
waveplate materials in order to evaluate quantitatively whichmaterials provide the smallest nonlinearity for a given spectral
range.
2. LOCATING THE OPD CHANNEL CENTERS
The seven channels of data are centered at heterodyne frequencies determined by the OPDs of the retarders. Previous
researchers have almost universally written the channel OPDs
in the form of δ2 − δ1 , δ2 , and δ2 + δ1 since these are the phasefactors appearing in the complex exponential terms of the measurement Eq. (1). However, these retardances are spectrally
dependent and so describe a range of OPDs rather than a single
OPD value: they are a shorthand notation for describing theδ(σ ) = 2πt B(σ )σ(2)

for a plate of thickness t, birefringence B(σ ) = n e (σ ) − n o (σ ),and at wavenumber σ . If we take the Taylor expansion of thisexpression about the central wavenumber,
1σ̄ = (σmin + σmax ),2
we obtain
δ(σ ) = δ (0) + δ (1) (σ − σ̄ ) + δ (2) (σ − σ̄ )2 + ... ,

(3)

where
δ (0) = 2πt B(σ̄ )σ̄ = 2πtb (0) σ̄ ,

  
dB
σ̄ = 2πt(b (0) + b (1) σ̄ )σ̄ ,δ (1) = 2πt B(σ̄ ) + σ̄d σ σ̄
 

 
dB
σ̄ d 2B
δ (2) = 2πt+σ̄ = 2πt(b (1) + b (2) σ̄ )σ̄ 2 ,d σ σ̄
2 d σ 2 σ̄and where subscript σ̄ means “evaluate σ at σ̄ ,” and we havedefined the birefringence coefficients b (i) for the ith-order

(a)

(b)

(c)

(d)

Fig. 2. (a) Experimentally measured channeled spectrum for a 22.5◦ linearly polarized input, using quartz waveplates of thicknesses t1 = 3.89 mm
and t2 = 7.77 mm. (b) Absolute value 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. The dashed curves indicate how the spectrum should appear if the retardance were apurely linear function of wavenumber. (c) First-order birefringence spectrum b (1) (σ − σ̄ ) for quartz (dashed line) and the exact birefringence
B(σ )√(solid curve), as expressed in Eq. (4). (d) Reconstructed Stokes parameter spectra for a 22.5◦ input state, i.e., s1 (σ ) = s2 (σ ) = s 0 (σ )/ 2.

5034

Research Article

Vol. 59, No. 16 / 1 June 2020 / Applied Optics

expansion term. From these expressions, we see that the zerothorder retardance coefficient δ (0) is given by the birefringenceb (0) at the center of the spectral range. The value of δ (0) changes
only the initial phase of the modulation, and so is not of muchinterest for CHSP. Conversely, the first-order coefficient δ (1)produces an ideal linear phase across the spectrum, given bya sum of the central birefringence b (0) and the central groupbirefringence b (1) .
Even though the birefringence of a material may be nearlya linear function of wavenumber, the retardance contains anadditional wavenumber factor that generates nonlinearity. To
provide a clear distinction between the birefringence proper andthis augmented quantity, it is useful to define a set of “effective
birefringence” coefficients βi given by
β (0) (σ ) = b (0) ,β (1) (σ ) = b (0) + b (1) σ̄ ,β (2) (σ ) = (b (1) + b (2) σ̄ )σ̄ ,β (3) (σ ) = (b (2) + b (3) σ̄ )σ̄ 2 ,(4)

where the first-order coefficient β (1) is a weighted sum of thezeroth- and first-order birefringence coefficients b (0) and b (1) ,while the second-order coefficient β (2) contains a term that also
depends on b (1) . Thus, any birefringence dispersion at all will
introduce nonlinear modulation (chirping) because the secondorder retardance coefficient δ (2) includes the birefringence
gradient.
A waveplate material having a dispersion of birefringenceb (1) = −b (0) /σ̄ will be achromatic: β (1) = 0. This is a desirable
condition for most conventional polarimeters, [9] but undesirable for CHSP, which utilizes the linear dispersion of retardance,β (1) 6 = 0, to encode the Stokes vector spectra. For channeledsystems, the concern is to minimize the higher-order dispersion
terms, and we can note that materials satisfying a similar relation of b (2) = −b (1) /σ̄ will give zero second-order nonlinear
dispersion (β (2) = 0). As we will see below, materials closer tosatisfying this condition provide better measurement quality.
Next, we express the heterodyne frequencies in OPD (µmunits) as a product of the retarder thicknesses and the effective
birefringence, i.e., tβ (1) etc. With these definitions, we canprovide formulas to calculate the retardance nonlinearity for agiven material and spectral range. Because 2πtβ (1) σ represents
an ideal linear phase, it also gives us the location of the channel
centers (the heterodyne frequencies) in the OPD domain. If we
express the retardance coefficients in µm units as we need to for
the Fourier-domain representation of the channeled spectrum,
then the channel centers are given simply by
C 0 center OPD:
C ±1 center OPD:
C ±2 center OPD:
C ±3 center OPD:

0,±(t2 − t1 )β (1) ,±t2 β (1) ,±(t2 + t1 )β (1) ,(5)where t1 and t2 are the physical thicknesses of retarders R 1 andR 2 , respectively.

3. MODELING BIREFRINGENCE DISPERSIONNONLINEARITY
The channeled spectrum shown in Fig. 2 simulates the measurement of a 22.5◦ linearly polarized input spectrum. For this
input, the data in each of channels C 1 , C 2 , and C 3 will have thesame profile, but with C 2 having twice the amplitude of theother two channels. However, we can see in Fig. 2(c) that thepeak in C 3 is broader and lower than that of C 1 . Two effects
produce these differences. First, the spectrometer’s modulationtransfer function (MTF) produces a lower fringe contrast at
higher frequencies, reducing the amplitudes of the signal there.
Second, the nonlinear chirp in the retardances causes a broadening of the signals that increases with increasing frequency(increasing OPD).
In order to illustrate the effect of nonlinear retardance on
measurements, we consider a material whose retardance ispurely quadratic, i.e., δ(σ ) = a σ 2 , where a determines theamplitude of the quadratic phase. For a CHSP that generates
only a single pair of sidebands, i.e., a system having only one
high-order retarder rather than two,I (σ ) = 1 + cos[δ(σ )],(6)

so that2

2

I (σ ) = 1 + 21 (e iδ + e −iδ ) = 1 + 21 (e ia σ + e −ia σ ),(7)

the Fourier transform of which produces a DC peak and two
chirp phases. When the spectral fringe modulation rate at
the long-wavelength end of the spectrum is lower than in the
short-wavelength end, the sideband peaks in the OPD spectrum
spread out, as in Fig. 2(b).
Polynomial fitting to the quartz birefringence dispersion
curve for the 0.4–1.0 µm spectral range gives the coefficientsb (0) = 0.0091271,b (1) = 0.0005144 µm−1 ,b (2) = 0.0000750 µm−2 ,so that the effective birefringence values are
β (0) = 0.0091271,
β (1) = 0.0091271 + (0.0005144 µm−1 × σ̄ )= 0.0100273,
β (2) = [0.0005144 µm−1 + (0.0000750 µm−2 × σ̄ )] × σ̄
= 0.0011299
for central wavenumber σ̄ = 1.75 µm−1 .
For this wavelength range in quartz, we can see that the terms
rapidly decrease in amplitude with increasing order of approximation. Whereas the linear term alone gives an effective birefringence at the central wavenumber of β (1) = 0.01003, adding thequadratic term changes the effective birefringence toβ (1) + β (2) (σmin − σ̄ ) = 0.00918(8)

Research Article

Vol. 59, No. 16 / 1 June 2020 / Applied Optics

(a)

5035

(b)

Fig. 3. (a) Birefringence spectra B(σ ) for waveplate materials in the 0.4–1.0 µm spectral range. (b) Normalized birefringence spectra B(σ )/B(σ̄ )for central wavenumber σ̄ = 1.75 µm−1 .

and
β

(1)

(2)

+ β (σmax − σ̄ ) = 0.01087(9)

at the infrared and blue ends of the spectrum, respectively—changes of ±9.2%. These values can also be estimated directlyfrom the measured Fourier-domain spectrum. Looking closely
at Fig. 2(b), we can measure the width at the base of the DC
peak to be about 7.8 µm, whereas the width of the C 2 peak is
about 20.1 µm. Thus, although the uniform polarization statemeans that the C 2 peak shape should be a scaled copy of the DC
peak, it has been broadened by 12.3 µm. Since the C 2 peak iscentered at 78.6 µm, this corresponds to a broadening of 15.6%(±7.8%) in the range of frequencies about the peak center, inrough agreement with the range given by Eqs. (8) and (9).
This analysis shows that we can correctly quantify the effect,
and that the broadening is a percentage of the underlying
OPD—doubling the thickness of a retarder will double themodulation frequency, and also double the broadening of thepeak in OPD units. This can cause large errors in the reconstructed spectra if the nonlinearity is not properly calibrated andcorrected.
One way to minimize the effect of dispersion nonlinearity on
the system is to select a retarder material that has minimal dispersion nonlinearity. Figure 3(a) shows the birefringence spectra
for common waveplate materials in the VIS–NIR spectral range(400–1000 nm). Because the shape of each curve is difficult tosee at the full scale, Fig. 3(b) shows the same data normalized
by the zeroth-order coefficient b (0) . An ideal material shouldhave a constant birefringence (a horizontal line when plottedversus wavelength), and we can see from the figure that among
all of the materials MgF2 has the lowest normalized slope, so thatMgF2 retarders will have the smallest dispersion nonlinearity fora given OPD.
Other materials whose birefringence is shown in Fig. 3 andtabulated in Table 1 are the three most common waveplatematerials—quartz (α-SiO2 ), calcite (CaCO3 ), and BBO(barium borate, α-BaB2 O4 )—as well as less common materials: sapphire (α-Al2 O3 ), yttrium vanadate (YVO4 ), ADP[ammonium dihydrogen phosphate, (NH4 )(H2 PO4 )], andrutile (TiO2 ). Materials TiO2 and YVO4 are not shown in Fig. 3
because they have significant absorption within the spectral
region. Muscovite mica (KAl2 (AlSi3 O10 )(FOH)2 ) is not shown
here because micas typically exhibit strong variability fromsample to sample.

To better quantify the differences among materials, we cancalculate the effective birefringence coefficients βi via Eq. (4),using the known birefringence spectra B(λ) for each material.
The results are shown in Table 1 together with the β (2)/β (1)ratio. For most spectra, this ratio gives a good measure of thedegree of nonlinearity. However, when any higher-order coefficients (e.g., β (3) ) are larger than the lower-order coefficients,
then using β (2) alone to compare to β (1) is not sufficient, such as
for the ADP spectrum in the UV–VIS spectral range. Whereas
Fig. 3(b) implies that ADP has quite a high degree of nonlinearity in this spectral range, the magnitude of the (β (2)/β (1) ) ratio
for ADP is actually the smallest among the materials considered.
The third-order coefficient, however, is quite large, and in thiscase dominates the nonlinearity.
As we should expect, Table 1 shows that the nonlinearity
increases as we move deeper into the ultraviolet, and relaxes
as we move deeper into the infrared. Among all the materials
considered here, MgF2 exhibits the lowest nonlinearity for allof the spectral ranges shown. This remains true up until close to
5 µm, where MgF2 begins to exhibit absorption. Unfortunately,
the tabulated data also tell us that no material has a β (2) coefficient so small that the nonlinearity can be ignored. The smallest(β (2) /β (1) ) ratio shown is for MgF2 in the NIR, in which thesecond-order coefficient is 2% that of the linear coefficient.
In order to see whether this 2% ratio is too large, we can takeas an example the system illustrated in Fig. 2, whose spectrum issampled at 2590 points from 0.45 µm to 0.95 µm. This spectrum thus spans a wavenumber range of 1σ = 1.17 µm−1 ,so that the distance between OPD-domain samples is
1OPD = 1/σrange = 0.85 µm. If channel C 1 at 39.3 µm(as indicated in the figure) were to be shifted by 0.85 µm—a
distance of one point in the OPD domain—this corresponds
to a relative shift of 0.85/39.3 = 0.022. This is approximately
the value of the OPD shift induced by the nonlinearity of MgF2 .
A shift by a single data point in the OPD domain, however,corresponds to a full sinusoidal modulation in the wavenumber
domain—an extreme degree of error. In order to minimize thiserror, we can restrict the allowed shift to a tenth of that amount.
Thus, by this criterion, we can say that a 2% nonlinearity is
about 10 times too large to ignore.
Among the three most common waveplate materials (quartz,
calcite, and BBO), quartz and BBO are almost identical innonlinearity, while calcite is significantly worse. Quartz’birefringence is the smallest of the three, making it easier to5036

Research Article

Vol. 59, No. 16 / 1 June 2020 / Applied Optics

Table 1. Effective Birefringence Coefficients for Common Waveplate Materials Across Several Spectral Ranges, asaCalculated by Eq. (4)Material

β (0)

β (1)

β (2)

β (3)

(β (2) /β (1) )−0.00594
0.00031−0.00986
0.00060−0.00044−0.01966−0.023
0.052
0.107
0.109
0.120
0.154

−0.00460
0.00033
0.00067−0.01109−0.00053−0.02338
0.53709

0.047
0.064
0.134
0.138
0.138
0.191
0.860

0.00075−0.01160−0.00211
0.00318−0.06898−0.15484

0.159
0.264
0.355
0.374
0.466
0.615

0.00020−0.00526
0.00033−0.00902−0.00018−0.00689
0.022440.04898

0.020
0.031
0.040
0.052
0.054
−0.0880.167
0.245

VIS-NIR (0.4–1 µm) Range
ADP
MgF2
α BBO
QuartzSapphire
Calcite−0.04554
0.01182−0.11857
0.00913−0.00809−0.17292−0.05285
0.01237−0.13048
0.00998−0.00876−0.19531

ADP
MgF2
Quartzα BBO
Sapphire
CalciteTiO2−0.04602
0.01186
0.00919−0.11946−0.00815−0.17462
0.31027−0.05331
0.012480.01017−0.13295−0.00893−0.20040
0.47979

MgF2
ADP
Sapphire
Quartzα BBO
Calcite0.01233−0.05001−0.00882
0.00999−0.13080−0.19753

0.01382−0.06117−0.01114
0.01272−0.17173−0.28304

0.001230.00065−0.01398
0.00108−0.00105−0.03006
VIS (0.45–0.65 µm) Range−0.00248
0.000800.00136−0.01837−0.00124−0.03833
0.41248
UV (0.28–0.4 µm) Range
0.00220−0.01614−0.00395
0.00476−0.07994−0.17402
NIR (0.7–1.05 µm) Range

MgF2
α BBO
QuartzCalciteSapphire
ADP
YVO4
TiO2

0.01165−0.11476
0.00887−0.16620−0.00792−0.04196
0.212880.26994

0.01203−0.12359
0.00943−0.18060−0.00822−0.05369
0.237970.31287

0.00024−0.00385
0.00038−0.00943−0.00045
0.00471
0.03969
0.07650aBirefringence data for each material obtained from Refs. [10–16].

manufacture waveplates designed to operate over wide spectral
ranges, so that the waveplate thicknesses are on the order ofmillimeters. The high birefringence of BBO and calcite makes
waveplates of these materials easier to manufacture when operating over much narrower spectral ranges. (Note, however, thatBBO is slightly hygroscopic and so can become opaque after
exposure to humid air.)
It is not well known that high-order waveplates of the typeused in CHSP are actually not unusual. The reason for theirprevalence indicates why a waveplate thickness of a few millimeters is a convenient choice. Optical component manufacturers
often offer “multi-order waveplates” for use as quarter-wave
plates or half-wave plates, and these are actually thick retardersthat have been polished to give the correct retardance modulo
2π. They are sometimes given narrowband transmission coatings, but if the coatings are broadband AR coatings, then thesame waveplates easily function for CHSP systems. If we compare quartz to an approximately equivalent calcite waveplate, acrystal thickness of say 5 mm is much easier to work with than
100 µm thickness because it makes the manufacturing processeasier—the crystal is stronger when thicker, the mounts areeasier to make and easier to use, etc.
4. IMPACT OF DISPERSION NONLINEARITY ON
MEASUREMENTS
Because the retardance nonlinearity produces a spreading of
the data in the OPD domain, data from one channel can beblurred into an adjacent channel, causing crosstalk, as shownin Fig. 4. This crosstalk error becomes worse for data farther
from the origin, since the OPD spreading is proportional to thenominal OPD value. The data in Fig. 4 consider a system where
the OPD spectrum extends to ±138 µm, with the seven independent data channels C i separated by an OPD of 39.3 µm, sothat each channel is also 39.3 µm wide. As we found in Eq. (8),quartz retarders induce a ±9.2% broadening due to retardance
dispersion nonlinearity, so that data inner edge of channel C 1
will experience broadening of ±0.092 × 19.6 µm = ±1.8 µm.
At the boundary between channels C 1 and C 2 , the broadening
triples to ±5.4 µm, and between C 2 and C 3 to ±9.0 µm. This

Research Article

Vol. 59, No. 16 / 1 June 2020 / Applied Optics

5037

Fig. 4. Example OPD spectrum, showing only channels C 0 through C 2 at their approximate positions. At the border between C 0 and C 1 , the data
are blurred by ±1.8 µm, whereas the border between C 1 and C 2 occurs at an OPD value three times larger, so that the blurring there is ±5.4 µm.

explains why each of the measured peaks in the OPD spectrumof Fig. 5(b) becomes broader and flatter as we move farther fromthe origin. The broadening also means that any data at the rightedge of channel C 1 and the left edge of channel C 2 get blurred
into the adjacent channel [as illustrated in Fig. 5(c)], potentiallycausing crosstalk artifacts, and likewise for data at the border
between C 2 and C 3 .
In order to minimize the blurring between channels, it is possible to compensate for the nonlinearity by pre-multiplying themeasurement spectrum by the inverse of the channel’s modulation in order to shift it to the origin—a kind of homodyne
detection. Thus, in order to move channel C 1 to the origin,
we multiply the measurement spectrum I (σ ) by C 1 ’s modulation phase e −i(δ2 −δ1 ) . This modulation phase includes theretardance spectrum’s nonlinear OPD, so that the data at thecenter of the shifted C 1 channel are unblurred, and the dataat the edges of C 1 are blurred by only ±1.8 µm. If we apply awindow function at the baseband of this shifted spectrum, we
can extract C 1 with minimal crosstalk effects. We can follow asimilar procedure for channels C 2 and C 3 : we pre-multiply I (σ )by e −iδ2 or e −i(δ2 +δ1 ) , respectively, and then extract them fromthe baseband channel. In each case, the crosstalk effect at theedges of the channels has been reduced to the minimum amountof ±1.8 µm.
Since the blurring at the channel edges can be reduced
to 1.8 µm but cannot be reduced to zero, it is necessary to
reduce the bandwidth of the channels from 39.3 µm to39.3 µm − 3.8 µm = 35.5 µm in order to prevent crosstalk
between channels. If we do not compensate for the nonlinearity by pre-shifting the spectrum before windowing, the OPD
blurring is not minimized, and the channel bandwidth must be
even narrower or else the reconstruction error will be prone tocrosstalk error. Narrower channels certainly remove crosstalkerror but also reduce resolution in the reconstructed Stokes
spectra.
Figure 5 shows a simulation measurement spectrum todemonstrate the benefits of compensating for the nonlinearity
prior to windowing. (While this demonstration can also bedone with experimental data, it is difficult to shape the intensity
spectrum and Stokes spectra to generate the desired bandwidth
constraints needed to clearly demonstrate the crosstalk effects.)
In 5(b), the Fourier-transformed original spectrum is shown asa black curve, while the pre-shifted spectra for centering C 1 , C 2 ,and C 3 at the origin are shown in blue, red, and green respectively. In the shifted spectra, we can see that each channel’s data
are most compact after shifting to baseband, while the channels
at negative OPD are moved increasingly far from the origin andexperience an increased broadening and chirping.

The optimal reconstruction procedure thus involves three orfour steps:
1. Take the Fourier transform of the original spectrum, window the baseband region (for extracting C 0 ), and inverse
transform to extract the s 0 (σ ) spectrum.
2. Premultiply the measurement spectrum by −[δ2 (σ ) −δ1 (σ )], the inverse of the nonlinear phase for channel C 1 ,and take the Fourier transform. The result is the blue curve
shown in Fig. 5(b). Next, window the baseband region to
extract C 1 , and inverse transform to obtain s2 (σ ) and s3 (σ )spectra from the real and imaginary parts.
3. Follow the same steps as #2, but premultiply by −δ2 (σ ).
This centers channel C 2 at baseband [the red curve in
Fig. 5(b)], and windowing the baseband region followed byinverse transform produces the s1 (σ ) spectrum.
4. Channel C 3 is not required for reconstruction, but can beused in combination with C 1 if desired. For this channel,
we premultiply by −[δ2 (σ ) + δ1 (σ )] [obtaining the green
curve in Fig. 5(b)], extract the baseband region of the resulting Fourier-domain spectrum, and inverse transform toobtain s2 (σ ) and s3 (σ ).
While it is possible to extract channels C 0 through C 3 fromthe initial measurement spectrum without premultiplication,
doing so makes the reconstruction prone to crosstalk between
channels due to the nonlinearity, and channels C 2 and C 3 suffer
more from this than channels C 0 and C 1 do.
While the reconstruction steps described above do not
say how to obtain the nonlinear phases δ1 (σ ) and δ2 (σ ),Refs. [3,7,17] provide details on static and dynamic phase calibration methods to estimate them. The methods described there
explain how to calibrate the (static) nonlinear phase of eachchannel by using the measurement of a 22.5◦ linearly polarized
input spectrum, [5] and by dynamically adapting those phases
during measurements to compensate for small changes such as
those due to temperature changes in the waveplates.
Figures 5(c) and 5(d) show reconstructed Stokes spectra that
demonstrate the difference between the nonlinearity compensated approach and the uncompensated approach. The input
intensity spectrum is bandlimited due to its smooth Gaussian
profile, while the Stokes spectra contain low-frequency modulations so that they too are bandlimited to the channel OPD of
39.3 µm. When nonlinearity compensation is used, the bandlimiting means that very little ringing is induced at the edges ofthe spectrum [magnified in the zoomed region of Fig. 5(c)]. In
the uncompensated spectra, however, we can see the error in theuncompensated spectra becomes severe for wavenumbers above
about 2.2 µm−1 .

5038

Vol. 59, No. 16 / 1 June 2020 / Applied Optics

Research Article

Fig. 5. Simulated measurement to demonstrate the effect of changing the order of operations during reconstruction. (a) Channeled spectropolarimeter OPD spectrum. (b) Original OPD spectrum [black curve], and the three shifted spectra [blue, red, and green curves, respectively], showing
how the shifting centers each channel at the origin. (c) Reconstructed Stokes spectra for the shift-first method (“nonlinearity compensated”), showing
the spectra reconstructed with channels C 1 and C 2 [solid curves] and spectra reconstructed with C 3 [dotted curves]. (d) Reconstructed Stokes spectra
for the window-first method (“uncompensated”). In the spectra plots, the gray box indicates the range of the zoomed region shown at right. Dashedcurves indicate the true Stokes spectra.

Also, because channels C 1 and C 3 both contain informationfor estimating the normalized Stokes spectra s2 (σ ) and s3 (σ ),we can do the reconstruction separately for each and compare.
Because C 3 is farther from the origin, we can expect the nonlinearity effects to be more severe there. In Figs. 5(c) and 5(d),the reconstructed spectra s2 (σ ) and s3 (σ ) using channel C 1 areshown in solid curves, while those using C 3 [s 20 (σ ) and s 30 (σ ),indicated with primes] are shown as dotted curves. The true
input spectra are displayed as dashed curves. In the center of thespectrum, where the signal-to-noise ratio in the data is high,the difference between the two methods is negligible for thisspectrum. However, at the ends of the spectral range, where the
SNR drops, the reconstruction shows ringing artifacts, withthe artifacts being larger and affecting more of the spectrum forthe uncompensated reconstruction.
Reference [6] shows that in the case when the input intensity
spectrum is not bandlimited within the limits of a single channel, then avoiding C 1 and using C 3 in its place can provide better

data, due to the reduced crosstalk from the DC peak. Thus, theuser should trade off the needs to avoid crosstalk due to violationof the bandlimit with crosstalk induced by dispersion nonlinearity. However, if both C 1 and C 3 contain negligible crosstalkerror, the data from both channels can be combined together toreduce noise in the reconstructed s1 and s3 spectra.
In Fig. 5(d), we can see that the uncompensated Stokes spectra at the high wavenumber end (short wavelengths) containmuch larger errors than the data at the low wavenumber end(long wavelengths). This difference is produced by heterogeneous wavenumber-sampling by the spectrometer. Although
the data are re-sampled onto a uniform grid in wavenumber
space prior to Fourier transformation, the raw data are uniformly sampled in wavelength space, which means that the
sampling rate at the short-λ end of the spectrum approaches
the Nyquist limit while the sampling rate at the long-λ end of
the spectrum remains far below the Nyquist limit. As a result,the short-λ data suffer from larger crosstalk error.

Research Article

Vol. 59, No. 16 / 1 June 2020 / Applied Optics

Table 2. RMS Errors Corresponding to theReconstructions Shown in Fig. 5
Normalized
Stokes Spectrum

ChannelUseds1 (σ )s2 (σ )s3 (σ )
s 20 (σ )s 30 (σ )

C2
C1
C1
C3
C3

RMS Error
Compensated Uncompensated
0.006
0.016
0.029
0.008
0.020

0.029
0.022
0.030
0.107
0.117

In order to calculate the degree of error quantitatively, wetake the root-mean-square error between the true and estimatednormalized Stokes spectra for the two methods, giving theresults shown in Table 2. Almost all of the difference between thecompensated and uncompensated reconstruction approaches islocated at the ends of the spectrum. While the improvement for
using the nonlinearity-compensated approach is small for the
spectra obtained from the innermost channel C 1 , the improvement increases for s1 (σ ) obtained from C 2 , and for s2 (σ ) ands3 (σ ) obtained from C 2 . (Note that measuring the error fromthe normalized Stokes spectra rather than the standard Stokes
spectra amplifies the noise in the two ends of the spectrum,
where the spectral intensity is low.)
5. CONCLUSION
The period of the fringes in a channeled spectrum is determined
not just by the thickness of the material and its birefringencebut also by its dispersion of birefringence. If the birefringence gradient has the same sign as the birefringence itself,
the fringe period is shorter than for an equivalent material
with no birefringence dispersion. If it has the opposite sign,
the fringe period lengthens. In addition, any dispersion ofbirefringence also produces nonlinearity in the dispersion ofretardance. Table 1 shows that among the possible waveplatematerials, MgF2 has the lowest nonlinearity, followed by BBO
and quartz. However, while using MgF2 waveplates will reduce
the nonlinearity, it cannot reduce it to the point where it can beneglected.
The birefringence for quartz at σ = 1.75 µm−1
(λ = 0.571 µm) is 0.0091, so the OPD of quartz retarders
will be 0.91% of the waveplate thickness. This is the zerothorder coefficient for the effective birefringence, β (0) , over the
spectral range. For channeled spectra, the fringe frequency isgiven by a combination of the birefringence and also the dispersion of birefringence, so that the fringe frequency is given
by β (1) ≈ 0.010 for the visible spectral range. As a result, thechanneled spectrum OPD will be about 10% higher than the
monochromatic OPD, at 1.0% of the crystal thickness. Thisdifference between β (0) and β (1) needs to be taken into accountwhen designing a CHSP system. It is important to keep in mind,
however, that different sources in the literature give substantiallydifferent values for quartz birefringence, and even more so forthe dispersion of birefringence, so it is best to measure the value
experimentally rather than rely on theory for the exact curves.
Discussions of channeled spectropolarimetric measurement in previous publications have largely assumed linearity5039

of the retardances and have left aside the issue of what happens to the measurements when nonlinearity is present. Since
the channel windowing procedure is normally insensitive tothe exact positions of the channel boundaries, errors arisingfrom ignoring the nonlinearity might not at first be apparent.
However, if the Stokes spectra have significant data near the
OPD channel edges then the channel windowing procedure
can induce significant error if performed prior to compensation of the nonlinearity, as shown in Fig. 5. The impact of thecrosstalk error on the reconstructed spectrum thus depends
on how much of the spectrum lies near the channel boundaries. Even with nonlinearity compensation, however, there
remains a small amount of nonlinearity-induced blurringof the signal at the edges of the baseband channel. This canbe reduced or eliminated by apodizing the amplitudes near
the channel edges, though this has the effect of smoothing
the reconstructed spectra. For quartz waveplates operatingover the 0.45–1.0 µm range, this apodization amounts toa loss of about 10% of channel bandwidth. In general, theuse of shaped (non-rectangular) window functions such as
Hamming, Hann, Blackmun, etc., are useful only for reducing
the error caused by working with spectra that are not sufficiently
bandlimited.
Funding. King Mongkut Institute of Technology (Academic
Melting Pot Fellowship 2019).
Disclosures. The authors declare no conflicts of interest.
REFERENCES
1. K. Oka and T. Kato, “Spectroscopic polarimetry with a channeledspectrum,” Opt. Lett. 24, 1475–1477 (1999).
2. N. Hagen, K. Oka, and E. L. Dereniak, “Snapshot Mueller matrix
spectropolarimeter,” Opt. Lett. 32, 2100–2102 (2007).
3. A. Taniguchi, H. Okabe, H. Naito, N. Nakatsuka, and K. Oka,
“Stabilized channeled spectropolarimeter using integrated calcite
prisms,” Proc. SPIE 5888, 588811 (2005).
4. K. Oka, “Singleshot spectroscopic polarimetry using channeledspectrum,” Proc. SPIE 4919, 167–175 (2002).
5. M. W. Kudenov, N. A. Hagen, E. L. Dereniak, and G. R. Gerhart,“Fourier transform channeled spectropolarimetry in the MWIR,” Opt.
Express 15, 12792–12805 (2007).
6. J. Craven and M. W. Kudenov, “False signature reduction inchanneled spectropolarimetry,” Opt. Eng. 49, 053602 (2010).
7. A. Taniguchi, K. Oka, H. Okabe, and M. Hayakawa, “Stabilization ofa channeled spectropolarimeter by self-calibration,” Opt. Lett. 31,
3279–3281 (2006).
8. Q. Li, C. Zhang, T. Yan, and Y. Wei, “Polarization state demodulationof channeled imaging spectropolarimeter by phase rearrangement
calibration method,” Opt. Comm. 379, 54–63 (2016).
9. C. D. West and A. S. Makas, “The spectral dispersion of birefringence, especially of birefringent plastic sheets,” J. Opt. Soc. Am. 39,
791–794 (1949).
10. J. R. DeVore, “Refractive indices of rutile and sphalerite,” J. Opt. Soc.
Am. 41, 416–419 (1951).
11. I. H. Malitson, “Refraction and dispersion of synthetic sapphire,”
J. Opt. Soc. Am. 52, 1377–1379 (1962).
12. F. Zernike, Jr., “Refractive indices of ammonium dihydrogen phosphate and potassium dihydrogen phosphate between 2000 Å and1.5 µ,” J. Opt. Soc. Am. 54, 1215–1222 (1964).
13. M. J. Dodge, “Refractive properties of magnesium fluoride,” Appl.
Opt. 23, 1980–1984 (1984).

5040

Vol. 59, No. 16 / 1 June 2020 / Applied Optics

14. G. Ghosh, “Dispersion-equation coefficients for the refractive index
and birefringence of calcite and quartz crystals,” Opt. Commun. 163,
95–102 (1999).
15. H. S. Shi, G. Zhang, and H. Y. Shen, “Measurement of principal
refractive indices and the thermal refractive index coefficients ofyttrium vanadate,” J. Synth. Cryst. 30, 85–88 (2001).
16. G. Tamošauskas, G. Beresnevičius, D. Gadonas, and A. Dubietis,“Transmittance and phase matching of BBO crystal in the 3–5 µm

Research Article
range and its application for the characterization of mid-infrared laser
pulses,” Opt. Mater. Exp. 8, 1410–1418 (2018).
17. B. D. Chrysler, Y. Otani, and N. Hagen, “Dynamic calibration of achanneled spectropolarimeter for extended temperature stability,”
Proc. SPIE 11132, 111320P (2019).