2100

OPTICS LETTERS / Vol. 32, No. 15 / August 1, 2007

Snapshot Mueller matrix spectropolarimeter
Nathan Hagen,1,* Kazuhiko Oka,2 and Eustace L. Dereniak1
1

College of Optical Sciences, University of Arizona, 1630 East University Boulevard, Tucson, Arizona 85721, USA
2
Graduate School of Engineering, Division of Applied Physics, Hokkaido University,
Sapporo 060-8628, Japan*Corresponding author: nhagen@optics.arizona.edu
Received April 9, 2007; revised May 17, 2007; accepted May 24, 2007;
posted May 29, 2007 (Doc. ID 81899); published July 19, 2007
We present a new snapshot technique for performing spectrally resolved Mueller matrix polarimetry. The
basic approach is an extension of the channeled spectropolarimetry technique, employing frequency-domaininterferometry to encode polarization information into modulation of the spectrum. © 2007 Optical Societyof America
OCIS codes: 120.2130, 120.5410, 260.5430, 300.6190.

Mueller matrix polarimeters measure the complete
polarization properties of a sample in the context of
the Stokes vector formalism. Many such instruments
are designed for use with quasi-monochromatic illumination, for reasons of calibration simplicity andspeed [1]. When spectrally resolved information is
also needed, a common approach is to use the same
instrumentation and simply step through the various
wavelength bands [2]. This trades off temporal resolution for chromatic information and introduces theinconvenience of using scanning hardware (such as
rotatable mounts).
We present a snapshot method for performing spectrally resolved Mueller matrix measurements of asample, which avoids these difficulties. By use of
frequency-domain interferometry, instead of a temporal resolution trade-off, we can trade with spectral
resolution to achieve a complete measurement. By“snapshot” we do not mean to imply fast, since monochromatic systems are typically far faster. Rather, wemean that the complete measurements can be taken
in a single integration period of a single detector array, with no moving or active parts in the system.
The resulting instrument is potentially very compact.
The basic layout of the snapshot Mueller matrix
spectropolarimeter is shown in Fig. 1. The device concept extends the method of channeled spectropolarimetry [3] to not only analyze polarized light, butalso to generate it [4,5]. The sample is thus illuminated by a polarization state that varies spectrally,
allowing a simultaneous measurement of the output
polarization states of the sample for many input polarization states. If the Mueller matrix of the sample
varies only slowly with wavelength, then the measurements will not overlap and can be accurately recovered.
Using the Mueller calculus to model the propagation of polarization states through the system, wefirst write sin共␴兲 as the Stokes vector for the spectrumof light incident on the first polarizer (␴ ⬅ 1 / ␭ is thewavenumber). Rn共␪n , ␦n兲 represents the Mueller matrix for the nth retarder element, with retardance ␦nand fast axis orientation angle ␪n. Likewise, Pn共␪兲 isthe matrix for the nth polarizer, with transmission
axis at angle ␪. M is the Mueller matrix of the sample
itself. (Note that the quantities ␦n and mij are func0146-9592/07/152100-3/$15.00

tions of wavenumber ␴; to clarify the expressions, theexplicit dependence will not be shown.) The light infront of the spectrometer is therefore given by
sout共␴兲 = P2共0°兲R4共45°, ␦4兲R3共0°, ␦3兲M
⫻ R2共0°, ␦2兲R1共45°, ␦1兲P1共0°兲sin共␴兲.

共1兲

Performing the matrix multiplications of Eq. (1), we
obtain an irradiance I共␴兲 = sout,0共␴兲 at the spectrometer ofI共␴兲 =

sin,0共␴兲
4

共m00 + m01 cos ␦1 + m02 cos ␦1 sin ␦2兲

+ m03 cos ␦2 sin ␦1 + m10 cos ␦4+ m11 cos ␦1 cos ␦4 + m12 sin ␦1 sin ␦2 cos ␦4+ m13 sin ␦1 cos ␦2 cos ␦4 + m20 sin ␦3 sin ␦4+ m21 cos ␦1 sin ␦3 sin ␦4+ m22 sin ␦1 sin ␦2 sin ␦3 sin ␦4+ m23 sin ␦1 cos ␦2 sin ␦3 sin ␦4− m30 cos ␦3 sin ␦4 − m31 cos ␦1 cos ␦3 sin ␦4− m32 sin ␦1 sin ␦2 cos ␦3 sin ␦4共− m33 sin ␦1 cos ␦2 cos ␦3 sin ␦4兲.
Using dn for the thickness of retarder Rn and Bn共␴兲
for its birefringence, we can write the retardanceterms as ␦n共␴兲 = 2␲dn␴Bn共␴兲. To obtain the Mueller
matrix elements from the measured spectrum I共␴兲,
we first take its Fourier transform. With the properchoice of retarder thicknesses, the various matrix el-

Fig. 1. (Color online) Basic layout of the snapshot Mueller
matrix spectropolarimeter. Retarders 1 and 4 have theirfast axes oriented at 45°, retarders 2 and 3 at 0°. Polarizers
1 and 2 both have their transmission axes oriented at 0°.
CHSP, channeled spectropolarimeter.
© 2007 Optical Society of America

August 1, 2007 / Vol. 32, No. 15 / OPTICS LETTERS

ements are separated into independent channels in
the Fourier domain. Inspired by the dual-rotatingretarder Mueller matrix polarimeter [6], we choose a
5:1 ratio of thicknesses for the pair of analyzing retarders to the generating pair to give a compact result. Further selecting a 2:1 ratio of thicknesses forthe retarders within each pair gives a set of thicknesses that we designate as a 1-2-5-10 configuration,
for the relative thicknesses of R1 through R4.
The optical path difference (OPD) in waves, for retarder Rn, is defined as OPDn共␴兲 = 共1 / 2␲兲␦n共␴兲; themean OPD is then OPDn共␴0兲 for ␴0, the mean wavenumber in the spectrum. The OPDs of the channels
in the Fourier domain are illustrated in Fig. 2. The
Mueller matrix elements encoded within each channel are listed in Table 1.
Extracting the Mueller matrix involves windowing
a channel (or a set of channels) in the Fourier domain
and performing an inverse transform to obtain the
spectrally resolved elements. Table 2 lists the steps
needed to recover all 16 matrix elements. Figure 3
shows a simulated measurement of a polymer achromatic quarter-wave retarder (based on measurements in our lab of one such retarder from Meadowlark Optics Inc.). This element provides aninteresting object for spectropolarimetry because of
the nonlinear spectral variation of both retardance ␦and fast-axis orientation angle ␪. The simulation
uses a blackbody spectral source for the 400– 700 nm
range and calcite retarders with 1, 2, 5, and 10 times
the thickness of 0.265 mm. The birefringence is derived from the standard Sellmeier coefficients for theordinary and extraordinary rays in quartz crystal.
The detector is assumed to have 2000 spectral resolution elements, digitized to 10 bit range and corrupted by Poisson noise.
A practical concern for implementing this instrument is whether it is sensitive to misalignment andmanufacturing tolerances. To analyze these issues,
we ran the previously described simulation for perturbed system parameters ˜␪n = ␪n + ⌬␪ and ˜␦n = ␦n + ⌬␦,with azimuth error ⌬␪ and retarder thickness error⌬␦. The calibration procedure for a channeled spectropolarimeter generally involves assuming the azimuth angles to be without error and using a reference spectrum measurement to calibrate theretardances ␦n共␴兲. In a Mueller matrix spectropolarimeter, a no-sample measurement likewise allows

Table 1. Fourier-Domain Channels Cn Encoding the
Mueller Matrix Elements for the 1-2-5-10
Configurationa
OPD
共Cn兲

Channel Content ⫻共64/ sin,0兲

0
±1
±2
±3
±4
±5
±6
±7
±8
±9
±10
±11
±12
±13
±14
±15
±16
±18

16m00
8m01 + 4m02 ± 4im03−m22 ± im23 ⫿ im32 − m33−4m02 ⫿ 4im03
2m21 + m22 ⫿ im23 ± 2im31 ± im32 + m33
4m20 ± 4im30
2m21 + m22 ± im23 ± 2im31 ± im32 − m33−2m12 ± 2im13−m22 ⫿ im23 ⫿ im32 + m33
4m11 + 2m12 ⫿ 2im13
8m10
4m11 + 2m12 ± 2im13
m22 ⫿ im23 ⫿ im32 − m33−2m12 ⫿ 2im13−2m21 − m22 ± im23 ± 2im31 ± im32 + m33−4m20 ± 4im30−2m21 − m22 ⫿ im23 ± 2im31 ± im32 − m33
m22 ± im23 ⫿ im32 + m33a

The OPD numbers listed for the channels are given in terms of
multiples of OPD1共␴0兲, the mean OPD of the thinnest retarder.

calibration of the ␦n共␴兲. For azimuth errors, the approach needs to be more complex, but the effect of ignoring them is simply that azimuth errors in systemelements cannot be differentiated from changes inthe sample. Thus, for example, an azimuth error of
0.3° of a polarizer can appear as a 0.3° azimuthchange in the sample.
The primary restriction on the kind of measurement that can be made accurately with this approachis that the spectral variation of all Mueller matrix elements must vary slowly enough that none of thechannels overlap in the Fourier domain. The width ofeach channel is the mean OPD of the thinnest retarder. The effective Nyquist limit of the Mueller matrix elements is thus given not by the sampling rateof the spectrometer but by the sampling rate of asingle channel. If the retarder thicknesses are chosensuch that the Nyquist limit of the spectrometer coinTable 2. Spectrally Resolved Mueller Matrix
Elements mij„␴… Obtained by Operating on SpectralDomain Channels cna
m00共␴兲 = 4c0
m01共␴兲 = 8共c1 + c3兲
m02共␴兲 = −16 Re兵c3其
m03共␴兲 = 16 Re兵c1其
m10共␴兲 = 8c10
m11共␴兲 = 16共c7 + c9兲
m12共␴兲 = −32 Re兵c7其
m13共␴兲 = 32 Im兵c7其aFig. 2. Fourier domain, 37 channels Cn.

2101

m20共␴兲 = 16 Re兵c5其
m21共␴兲 = 32 Re兵c2 + c4其
m22共␴兲 = −32 Re兵c2 + c8其m23共␴兲 = 32 Im兵c2 − c8其m30共␴兲 = 16 Re兵c5其
m31共␴兲 = 16 Im兵c2 + c4 + c6 + c8其
m32共␴兲 = −32 Im兵c2 + c8其m33共␴兲 = 32 Re兵c8 − c2其These are given in terms of the Fourier-domain channels bycn共␴兲 = 关1 / sin,0共␴兲兴F−1兵w共OPD兲Cn共OPD兲其, for windowing function
w.

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OPTICS LETTERS / Vol. 32, No. 15 / August 1, 2007

plus the DC channel). This can place stringent requirements on readily available spectrometers to produce a useful spectral resolution for the Mueller matrix. For example, the highest-resolution visible-light
spectrometer available through Ocean Optics Inc.
has 2000 resolution elements, with which Mueller
matrix elements can thus be measured with
2000/ 37⬃ 54 spectral resolution elements.
Because of its snapshot capability, this techniquecould be employed in some applications where moving parts are difficult to use and the trade-off in spectral resolution is acceptable, though it is limited tomeasuring samples whose Mueller matrix elements
vary only slowly with wavelength (i.e., each Mueller
matrix element’s spectral dependence must be bandlimited to within the width of a single channel).
Examples include in situ ellipsometry of thin films,
in vitro biomedical imaging, or underwater measurement of samples.
Polarimetry is a relatively small field, in part because the instrumentation required for polarization
measurements is too complicated for casual use. A
compact snapshot instrument may allow access to users whose budgetary and time constraints have prevented them from making use of polarization in theirresearch.

Fig. 3. Simulated measurement of an achromatic polymer
retarder using a snapshot Mueller matrix polarimeter. Upper figure: the Fourier-domain representation of the measured spectrum. Lower figure: the retardance ␦共␴兲 and orientation angle ␪共␴兲 of the sample reconstructed from themeasured Mueller matrix elements. The error bars are obtained from the standard deviation of data taken over 100
instances of Poisson noise.

cides with the edge of the highest-frequency channel,
the analysis implies a loss in spectral resolution by afactor of 37 (for 18 positive and 18 negative channels

This work was supported in part by Department of
Defense contract DAAE07-02-C-L011.
References
1. D. H. Goldstein, Appl. Opt. 31, 6676 (1992).
2. L. Jin, K. Takizawa, Y. Otani, and N. Umeda, Opt. Rev.
12, 281 (2005).
3. K. Oka and T. Kato, Opt. Lett. 24, 1475 (1999).
4. T. Wakayama, Y. Otani, and N. Umeda, Proc. SPIE
5888 (2005).
5. H. Okabe, M. Hayakawa, H. Naito, A. Taniguchi, andK. Oka, Opt. Express 15, 3093 (2007).
6. R. M. A. Azzam, Opt. Lett. 2, 148 (1978).

May 15, 2013 / Vol. 38, No. 10 / OPTICS LETTERS

1675

Snapshot Mueller matrix spectropolarimeter: erratum
Nathan Hagen,1 Kazuhiko Oka,2 and Eustace L. Dereniak1
1

College of Optical Sciences, University of Arizona, 1630 E. University Boulevard, Tucson, Arizona 85721, USA
2

Division of Applied Physics, Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan*Corresponding author: nhagen@optics.arizona.edu
Received April 17, 2013;
posted April 17, 2013 (Doc. ID 187686); published May 10, 2013

We correct an error in one of the equations in our original manuscript. © 2013 Optical Society of America
OCIS codes: (120.2130) Ellipsometry and polarimetry; (120.5410) Polarimetry; (260.5430) Polarization; (300.6190)
Spectrometers.
http://dx.doi.org/10.1364/OL.38.001675

The second equation in our original manuscript [1]contains a typo in which m02 sin δ1 sin δ2 appears as
m02 cos δ1 sin δ2 . Thus, the correct form of the equation
should beIσ 

sin;0 σm00  m01 cos δ1  m02 sin δ1 sin δ2
4
 m03 cos δ2 sin δ1  m10 cos δ4

The remainder of the manuscript is unaffected by thechange.
Our thanks to Andrey Alenin for pointing out the error.
Reference
1. N. Hagen, K. Oka, and E. L. Dereniak, Opt. Lett. 32, 2100
(2007).

 m11 cos δ1 cos δ4  m12 sin δ1 sin δ2 cos δ4
 m13 sin δ1 cos δ2 cos δ4  m20 sin δ3 sin δ4
 m21 cos δ1 sin δ3 sin δ4
 m22 sin δ1 sin δ2 sin δ3 sin δ4
 m23 sin δ1 cos δ2 sin δ3 sin δ4− m30 cos δ3 sin δ4 − m31 cos δ1 cos δ3 sin δ4− m32 sin δ1 sin δ2 cos δ3 sin δ4− m33 sin δ1 cos δ2 cos δ3 sin δ4 .

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