Modeling and optimization for a prismatic snapshot
imaging polarimeter
Haitao Luo, Kazuhiko Oka, Nathan Hagen, Tomasz Tkaczyk, and Eustace L. Dereniak

Thin birefringent prisms placed near an image plane introduce sinusoidal fringes onto a 2D polarized
scene making possible a snapshot imaging polarimeter, which encodes polarization information into themodulation of the fringes. This approach was introduced by Oka and Kaneko [Opt. Express 11, 1510
(2003)], who analyzed the instrument through the Mueller calculus. We show that the plane-wave
assumption adopted in the Mueller theory can introduce unnecessary error in a polarimeter design. To
directly take prism effects such as beam splitting and deviating into accounts we introduce a geometric
imaging model, which allows for a versatile simulation of the birefringent prisms and provides a meansfor optimization. A calcite visible system is investigated as an example, which essentially shows how each
design parameter affects the overall image quality and how to modify the polarimeter design to optimize
overall performance. The approach is applicable to any prismatic imaging polarimeter with different
prism materials and different working wavelengths. © 2006 Optical Society of America
OCIS codes: 120.2130, 120.5410, 260.5430.

1. Introduction

A conventional polarimeter is usually equipped withseveral rotating polarization elements to extract thecomplete Stokes vectors of an object. This common
method suffers from vibration noise and unwanted size
and mechanical complexity. Oka and Kaneko1 recently
demonstrated a snapshot imaging polarimetry technique, which replaces the traditional polarization optics with four sequential birefringent prisms (Fig. 1). If
we treat the wavefront incident on the prisms as planewaves and consider the prisms as being thin enough toignore the beam-splitting (BS) and beam deviation(BD) effects induced on the wavefronts (Fig. 2 illustrates these effects), then a straightforward application of Mueller calculus gives the image intensity
I共x, y兲 as

H. Luo (htluo@email.arizona.edu), N. Hagen, T. Tkaczyk, andE. L. Dereniak are with the College of Optical Science, Universityof Arizona, 1630 E. University Boulevard, Tucson, Arizona 85721.
K. Oka (oka@eng.hokudai.ac.jp) is with the Division of Applied
Physics, Graduate School of Engineering, Hokkaido University,
Sapporo 060-8628, Japan.
Received 23 June 2006; accepted 24 July 2006; posted 31 July
2006 (Doc. ID 72256).
0003-6935/06/338400-10$15.00/0© 2006 Optical Society of America
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1

1

1

I共x, y兲 ⫽ 2 S0共x, y兲 ⫹ 2 S1共x, y兲cos共2␲Ux兲 ⫹ 4 S2共x, y兲⫻ 兵cos关2␲U共x ⫺ y兲兴 ⫺ cos关2␲U共x ⫹ y兲兴其⫺ 14 S3共x, y兲兵sin关2␲U共x ⫺ y兲兴⫹ sin关2␲U共x ⫹ y兲兴其,(1)where S0, S1, S2, and S3 represent the spatially dependent Stokes parameters of the incident beam andU denotes the fringe spatial frequency. If we take theFourier transform of I共x, y兲, we find that the four
Stokes parameters are separated into individual channels in the frequency domain and thus can be reconstructed simultaneously from a single measurement
image.
However, a closer look at this approach showsthat some assumptions can frustrate optimal design of the instrument limiting its performance. In
particular, there are three effects ignored in theabove approach, which are important for understanding the limitations of the instrument: (1) The
presence of the objective lens means that each pixelon the detector measures the light integrated acrossa range of angles of incidence onto the prisms. In ageometric model, this can be approximated by sampling the pupil and tracing rays from the pupil
plane to the image plane. (2) A plane wave enteringa birefringent plate is sheared such that the o ande waves propagate at slightly different angles.
When they exit the plate, the o and e waves are
displaced relative to one another, producing a BS

Fig. 1. (a) Schematic of the prismatic imaging polarimeter.
(b) Layout of the wedged prisms PR1–PR4, which are actually intight contact. The double-arrow lines indicate the fast axes, which
are at 0°, 90°, 45°, and ⫺45° with respect to the x axis. The solidlines represent the o light, and the dashed lines represent the elight. The circular shape sketches the image spot shape. The backsurface of PR4 is the image plane of interest.

effect. (3) Since the prisms are not plane parallel
plates, a ray exits the prism in a direction, which is
not parallel to the ray incident on the prism. This
deflection angle is dependent on the index of refraction of the prism, and therefore differs for the o ande waves, producing the BD effect (Fig. 2). Further
complicating this effect, depending on the angle ofincidence and the angle of the crystal axis orientation, the deflection of the e wave varies as the angle
of incidence varies. The BS and BD effects bothdegrade the image, the first by enlarging the spot
size (reducing resolution and fringe contrast) andthe latter by introducing distortion. Both are amplified in systems with a small f number.
To address these problems, a more complete imaging model is needed. A complete vectorial diffraction
theory may solve the problem, because all three of theeffects mentioned can be included in the vectorialmodel for the intensity pattern and the interferogram
can be straightly observed. However, the deformationof the wavefront in the birefringent prisms and thetilted interfaces render the calculation complicated.
The geometric ray-tracing method is a good compromise for most imaging systems, offering easy accessfor aberration analysis as well as describing the raypath. Algorithms for ray tracing in a uniaxial material have been greatly developed in recent years.2–5
The majority of them are based on Huygen’s principles and feature considerable complexity. A relatively
simple approach, developed by Liang6 by making useof phase-matching conditions requires no additional
assumptions and is the method we make use of here.
In this paper, we first build up a geometric imaging
model, based on fringe decomposition, capable of analyzing the off-axis rays passing through the birefringent prisms as well as the prism BS and BD effects.
We then simulate a system of calcite prisms andillustrate the BS and BD effects in the presence of alens to show that, for prism apex angles less than 2°,
the Mueller calculus is sufficient, but for larger apex
angles, the geometric model provides a way to optimize the system design without introducing unnecessary error. With the necessary background now
provided, we outline a procedure for designing andoptimizing a prismatic polarimeter.
2. Analysis Model
A. Beam Splitting and Beam Deviation of
Birefringent Prisms

To consider a beam incident on a uniaxial medium,
we can first split the beam into two components, one
whose E field is normal to the optic axis of the crystal(the ordinary light), and one whose E field lies in theplane containing both the optic axis and the wave
vector kជ e (the extraordinary light). The refractive index of o light is uniform across the medium, whereasthat of the e light obeys
ne共␪兲 ⫽

Fig. 2. BS and BD induced by a birefringent prism are shown
exaggerated here for clarity. The splitting is a spatial translationof the e beam from the o beam. The deviation is an angular deflection between the incident and exit ray directions.

none

冑no2 sin2 ␪ ⫹ ne2 cos2 ␪,(2)where ␪ is the angle between the e wave vector kជ e andthe optic axis. no and ne are the refractive indicesalong the principal axes of the crystal. Note that ␪ isdependent on the angle of incidence. The difficulty intreating the e wave results from the fact that theជ are notwave vector kជ e and the Poynting vector Seជparallel. ke complies with Snell’s law at the interface
ជ , the detected signal, does not. The direcwhereas Seជ ,tion cosines ␰x, ␰y, and ␰z of the Poynting vector Sewith respect to the coordinate system 共x, y, z兲 defined
20 November 2006 兾 Vol. 45, No. 33 兾 APPLIED OPTICS

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Table 1. Ray Configurations Ray 1–Ray 4 and the Complex Amplitudes of the E Fields

Ray 1

Ray 2

Ray 3

Ray 4

PrismsStatusAmplitude

StatusAmplitude

StatusAmplitude

StatusAmplitude

PR1
PR2
PR3
PR4
Field at Z=

o
e
e
o

Ey共t兲
Ey共t兲
Ey共t兲兾冑2
Ey共t兲兾冑2

o
e
o
e

Ey共t兲
Ey共t兲
Ey共t兲兾冑2
Ey共t兲兾冑2

e
o
e
o

Ex共t兲
Ex共t兲
Ex共t兲兾冑2
Ex共t兲兾冑2

e
o
o
e

Ex共t兲
Ex共t兲
Ex共t兲兾冑2
Ex共t兲兾冑2

1
E 共t兲e⫺i␸1共x, y兲
2 y

⫺

1
E 共t兲e⫺i␸2共x, y兲
2 yby the air– crystal plane interface are␰x ⫽ cos ␣ cos ␪e ⫹␰y ⫽ cos ␣ sin ␪e ⫺␰z ⫽

冑␴z2 ⫹ 共␴x sin ␪e ⫺ ␴y cos ␪e兲2

cos ␪e sin ␣共␴x sin ␪e ⫺ ␴y cos ␪e兲␴z sin ␣冑␴z2 ⫹ 共␴x sin ␪e ⫺ ␴y cos ␪e兲2

,

,

,

(3)

where ␣ is the dispersion angle, ␴x, ␴y, ␴z are thedirection cosines of the optic axis (also with respect tothe air– crystal plane interface) and ␪e is the angle
between kជ e and the x axis.6 Used together with Snell’slaw, these formulas make the tracing of e rays
through the birefringent materials possible.
However, the sequential arrangement of prisms
greatly complicates the situation compared to a single plane interface. Figure 1(b) illustrates the prism
layout. PR1 and PR2 are mounted oppositely with
their acute dihedral edges parallel to the y axis. Thisis arranged to create a retardance, which varies linearly along y. Similarly, PR3 and PR4 have their acutedihedral edge parallel to the x axis creating a retardance varying linearly along the x axis. The fast axesof the four prisms are oriented at 0°, 90°, 45°, and⫺45°, respectively (relative to the x axis). A ray incident on PR1 is split into o兾e rays, and due to theorthogonal orientation of the optic axes of PR1 andPR2, upon entering prism PR2, the two rays exchange
their o兾e status. Since the optic axis of PR3 is oriented
at 45° relative to that of PR2, each o and e ray incident
onto the PR2–PR3 interface is split into equal o and erays in the new medium and finally switch their o兾estatus upon entering PR4. Table 1 summarizes theprocess of tracing these four rays through the prisms,
denoted by Rays 1– 4.
Each of the four rays can be considered as forming
one of four independent images summed together at
the focal plane array (FPA). Displacement between thefour images leads to a loss of resolution due to BS and
BD effects. A second problem is a loss of phase purity,
which we call phase hybridization. The expansion andnonuniform shift of each image spot results in a phasemixture within adjacent image points and thus reducing the visibility of fringes and introducing errors in
8402

1
E 共t兲e⫺i␸4共x, y兲
2 x

the image reconstruction. Both problems are amplified
if the lens is not properly designed.

sin ␪e sin ␣共␴x sin ␪e ⫺ ␴y cos ␪e兲冑␴z2 ⫹ 共␴x sin ␪e ⫺ ␴y cos ␪e兲2

1
E 共t兲e⫺i␸3共x, y兲
2 x

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B.

Fringe Decomposition

Besides the regular imaging process, the formationof well-shaped fringes is also vital to system operation. Mueller calculus only predicts the overall intensity pattern when the BS and BD effects are notpresent [Eq. (1)] and provides a poor guide to exploring how these effects modify the measurement. To
modify the setup for ray-tracing purposes, the fringe
pattern has to be decomposed. Table 1 indicates thatthe optical path differences (OPDs) between each pairof ray configurations may contribute to the fringes.
For instance, Ray 1 and Ray 2 have the same o-lightpath in PR1 and PR2 but disjunct optical paths in PR3
and PR4 so the OPD between them should vary along
the x axis and so should their interference. Similarly,
ray configurations 1 and 3, 1 and 4, 2 and 3, 2 and 4,
and 3 and 4 may produce five independent sets offringes, along y, x ⫺ y, x ⫹ y, y, and x directions,
respectively. To see how these come about, we startfrom an incident ray with its electric field decomposed sequentially through the prisms in terms of the
optic axis orientation inside each prism, and then we
get four linearly polarized rays (i.e., Ray 1–Ray 4).
The components of the fields along the axis of theanalyzer are transmitted. The details of each ray configuration is tabulated in Table 1, where Ex(t) andEy(t) represent the time-dependent complex amplitudes of the incident light field along the x and yaxes, respectively. ␸1, ␸2, ␸3, and ␸4 denote the cumulative change in phase experienced by Ray 1–Ray 4 asthey pass through the prisms, and the differences
共␸i ⫺ ␸j兲 between each pair of them are the OPDs. The
total irradiance on the image plane, I ⫽ |E|2, istherefore

冓冏

1
1
1
Ey共t兲e⫺i␸1 ⫺ Ey共t兲e⫺i␸2 ⫹ Ex共t兲e⫺i␸3
2
2
2
2
1
⫹ Ex共t兲e⫺i␸421⫽ 兵2关具ⱍEyⱍ2典 ⫹ 具ⱍExⱍ2典兴 ⫺ 关具ⱍEyⱍ2典ei共␸2⫺␸1兲 ⫹ c.c.兴
4

I⫽

冏冔

⫹ 关具ⱍExⱍ2典ei共␸4⫺␸3兲 ⫹ c.c.兴 ⫹ 关具Ex*Ey典ei共␸3⫺␸1兲 ⫹ c.c.兴⫺ 关具Ex*Ey典ei共␸4⫺␸2兲 ⫹ c.c.兴 ⫹ 关具Ex*Ey典ei共␸4⫺␸1兲 ⫹ c.c.兴⫺ 关具Ex*Ey典ei共␸3⫺␸2兲 ⫹ c.c.兴其,(4)where the angular brackets are time averages, theasterisk denotes the complex conjugate, and c.c.
stands for the complex conjugate of the preceding
term. Each bracketed term except the first (the intensity background) forms a fringe due to the spatial variation of the phase terms. Recalling thedefinition of Stokes parameters7 and making thesubstitutions,具ⱍExⱍ2典 ⫹ 具ⱍEyⱍ2典 ⫽ S0,

1
具ⱍExⱍ2典 ⫽ 共S0 ⫹ S1兲,2

1
1
具ⱍEyⱍ2典 ⫽ 共S0 ⫺ S1兲, 具Ex*Ey典 ⫽ 共S2 ⫹ iS3兲2
2
into Eq. (4), we get

I⫽

1
1
1
S ⫺ S ⫺ S1兲cos共␸2 ⫺ ␸1兲 ⫹ 共S0 ⫹ S1兲
2 0 4共 0
41⫻ cos共␸4 ⫺ ␸3兲 ⫹ Re兵共S2 ⫹ iS3兲4
⫻关ei共␸3⫺␸1兲 ⫺ ei共␸4⫺␸2兲 ⫹ ei共␸4⫺␸1兲 ⫺ ei共␸3⫺␸2兲兴其,(5)where Re兵·其 takes the real part of a complex number.
If we temporarily go ahead and adopt the on-axis
plane-wave assumption, then the phase relations between ␸1 and ␸4 can be written as␸2共x, y兲 ⫺ ␸1共x, y兲 ⫽ ␸4共x, y兲 ⫺ ␸3共x, y兲 ⫽ 2␲Ux,␸1共x, y兲 ⫺ ␸3共x, y兲 ⫽ ␸2共x, y兲 ⫺ ␸4共x, y兲 ⫽ 2␲Uy,
␸4共x, y兲 ⫺ ␸1共x, y兲 ⫽ 2␲U共x ⫺ y兲,
␸2共x, y兲 ⫺ ␸3共x, y兲 ⫽ 2␲U共x ⫹ y兲,(6)where U ⫽ 2共no ⫺ ne兲tan共␪兲兾␭. By plugging Eqs. (6)into Eq. (5), the intensity pattern simplifies to theform given in Eq. (1), verifying the assumptions
taken in the Mueller calculus approach. In reality,
however, the OPDs are not exactly linear due to thereasons discussed in Section 2. Considerable deviations from linearity are possible, in which case, thecancellation of S0 between the second and thirdterms in Eq. (5) cannot occur, and errors will leak into
the S1 component if we continue to use Eq. (1). We
must really calculate ␸1–␸4 to know the practical
OPDs.
The advantages of the model used here are (a) theentire image pattern is determined by the four individual ray configurations so that it is now possible touse ray tracing to evaluate such effects as BS, BD,phase hybridization, etc.; (b) complicated imaging
systems can be modeled with little increase in difficulty for the analysis; and (c) the use of ray-tracing
tools provides methods for other high-level optimiza-tion such as maximizing energy transmission andachromatizing for broadband applications.
3. Simulation Results and Optimization
A.

Optimum Focus

Since multiple parameters (e.g., pupil size, field coordinates) are involved in the discussion, we will firstinvestigate the system with a given pupil for the
on-axis field and then find a proper pupil size andapply the optimum condition to a large field of view(FOV). Throughout this paper, we assume the scene
is at infinity and model the lens as being aberration
free, 50 mm focal length, and 15 mm in diameter sothat the diffraction limited spot on the FPA is ⬃2.5
␮m in radius. The simulation model contains calcite
prisms, each with an apex angle ␪ of 1.5°, a clear
aperture of 12 mm ⫻ 12 mm, and a thickness of 500
␮m at the base. The operating wavelength is taken
to be 0.633 ␮m. The distance z⬘ from the lens to theback surface of PR4 is defined as the image distance.
Unless stated otherwise, all the following discussions
will refer to the image plane. The on-axis field
共hx, hy兲 ⫽ 共0, 0兲 is selected as the initial setup for
optimization and the rms spot radius versus z= isdefined as the merit function. The objective for imaging optimization is to minimize the image spot size
below the detector resolution. This is also necessary
to diminish the phase hybridization.
Figure 3(a) shows three typical spot diagrams atvarious z⬘ of 50.18, 50.2491, and 50.33 mm, respectively. Because of the BS effect, one image spot ismade up of four independent spots, colored by red,
yellow, blue, and green for Ray 1–Ray 4, respectively.
These four image spots are shifted relative to eachother. Figure 3 also shows an obvious asymmetry in
the image spot shapes incurred by the physical asymmetry of the prisms. However, the dominant factor
in determining the image spot size and its shape is
the prism position z= as illustrated in Fig. 3(b). Atz⬘ ⫽ 50.2491 mm, the rms spot size reaches its smallest radius of 1.8 ␮m, which is well below the detector
resolution for most detector arrays. A lucky feature of
Fig. 3(b) is that all four ray configurations show very
similar changes as a function of z⬘ (the same minimum and same slopes). Hence a trade-off in optimizing the four images is avoided. Additionally, thecentroid of the image spots, at the best focus, isslightly shifted (by less than 2 ␮m) from the paraxial
image point, indicating a small BD effect for the onaxis field.
B.

Pupil Size Dependence

The pupil size that defines the marginal ray angle isanother key factor to characterize the system. It is
anticipated that the edge rays through the pupil will
suffer more BS and BD inside the prisms than thecentral rays. By fixing the other system parameters,
we can obtain the pupil size dependence on the merit
function, the result of which is shown in Fig. 4 (foran on-axis field and z⬘ ⫽ 50.2491 mm). Figure 4(a)20 November 2006 兾 Vol. 45, No. 33 兾 APPLIED OPTICS

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Fig. 3. (Color online) (a) Spot diagrams (square grid) at z⬘ ⫽ 50.18 mm, z⬘ ⫽ 50.2491 mm, z⬘ ⫽ 50.33 mm, respectively. (b) Merit functionof rms image spot radius versus z⬘ for Ray 1–Ray 4, respectively. The inset is the enlarged part at the bottom. For comparison, a similar
curve has been plotted for the case of replacing the set of four calcite prisms with a glass plate of equal thickness and the index of refraction
equal to the ordinary index of calcite.

shows the spot diagrams for the pupil diameters 6,
20, and 30 mm. If diffraction effects are ignored, thena smaller image spot corresponds to a smaller pupil.
The spot asymmetry is also worsened with increasing pupil size, whereas the image shift is not significantly changed. If diffraction effects are includedin the simulation, then competition happens between the diffraction limit and the geometric image spot size; a bigger pupil reduces the size of Airy
disk but increases the BS and BD effects. We userd ⫽ 0.61 ␭兾NA to approximate the diffraction limit(though real diffraction should consider the prism
geometry) and take the maximum of rd and the raytracing results. Figure 4(b) plots the final spot radius
versus the pupil diameter dp; the optimum image spot
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APPLIED OPTICS 兾 Vol. 45, No. 33 兾 20 November 2006

size occurs for a pupil diameter of 8–15 mm (we keepusing dp ⫽ 15 mm in the following calculation for a
maximum throughput). Note that the optimum focus
position will vary with different choices of pupil diameter.
C.

Field Dependence

Due to the lens, different field angles will experience
different BS and BD inside the prisms. Object points
at larger field angles are imaged farther off axis
where more BS and BD occur in the prisms. Retaining the specifications used in Fig. 4, Fig. 5(a) illustrates the sensitivity of the spot size on the field
angles. Looking at the first row of Fig. 5(a), we seethat the image spot becomes larger as the field angle

Fig. 4. (Color online) (a) Spot diagrams with the pupil diameter dp equal to 6, 20, and 30 mm, respectively, with diffraction effects ignored.
(b) Rms image spot radius versus the pupil diameter dp. The system is at the optimum focus for the on-axis field.

increases in the x direction. The next row of three
examples gives results for field angles symmetricalabout the z axis. The difference in spot shape can be
attributed to the asymmetry of the prisms again.
Compared to the on-axis field, the off-axis fields havecomatic image spots. The spot radius increases to a
maximum of ⬃10 ␮m at the maximum field angle
given by hx ⫽ 7°, hy ⫽ 0°. This value is still reasonable
for typical applications. Figure 5(b) illustrates thevariation in image spot size with focus position [asin Fig. 3(b)], for all of the six field angles listed, andfor both of ray configurations, 1 and 3 (Rays 2 and
4 are not shown because they correspond very
closely to 1 and 3). Again, we see that all configurations share the same optimum focus position,
eliminating any competition among different angles
inside the FOV. In other words, it is only necessary
to consider the on-axis field position to locate theoptimum focus.

In addition to blurring, image distortion is also a
concern. Because of the nonuniform BS and BD effects across the FOV, the image spots of different fieldcoordinates will be unequally shifted. To estimate thedistortion, we average the image centroids of Ray
1–Ray 4, with an equal weight for each, to obtain thereal image position. The intersection point of the chiefray (the chief ray of the optical system with no prismspresent) with the image plane is used as the reference
image point. The shift between the real image position
and the reference point indicates the distortion. At the
optimum focus 共z⬘ ⫽ 50.2491 mm兲, with dp ⫽ 15 mm
and across a FOV of ⫾7°, the maximum distortion isless than 4.5 ␮m (occurring at the edge of the FOV),
which is not a problem for the polarimeter design
considered here.
Finally, we can note that in each of the spot diagrams of Figs. 3(a), 4(a), and 5(a), Ray 1 and Ray 2
are always closer to each other than are Ray 3 and
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Fig. 5. (Color online) (a) Spot diagrams for different field points (at the optimum focus), i.e., (0, 0), (3°, 0), (7°, 0), (⫺7°, 0), (0, 7°), and(0, ⫺7°) for z⬘ ⫽ 50.2491 mm and dp ⫽ 15 mm. (b) Merit function of spot radius versus z⬘ at different fields (by different line shapes),(0, 0), (7°, 0), (⫺7°, 0), (0, 7°), and (0, ⫺7°), for Ray 1 and Ray 3.

Ray 4. Since Rays 1 and 2 originate from the same
ray path in PR1 and PR2, we can conclude that PR1
and PR2 generate the majority of the BS and BD
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effects seen at the image. Thus when optimizing thesystem, we find that minimizing the apex angle usedin prisms PR1 and PR2 is more important than min-

imizing the angle used in PR3 and PR4. This information is especially attractive in the mid-to-longinfrared, where it is necessary to use a much larger
prism angle.
D. Optical Path Difference

As discussed in Section 1, the spatial linearity of thephase is essential for the device operation, whereas
the BS and BD effects and the off-axis rays will introduce errors in the phase mappings eventuallycausing errors in the reconstructed polarimetric images. To evaluate the phase errors, we trace throughall the ray configurations to obtain ␸1–␸4. In the simulation, we uniformly sample the FOV by a squaregrid, and for each field point, we average the phase␸i 共i ⫽ 1, 2, 3, 4 for Rays 1, 2, 3, 4兲 across the pupil,
resulting in the mean phase ␸៮ i. The optimum focus isagain used, since only if the image spot is confinedwithin a single detector, can the phase mixture fromadjacent image spots be ignored. Also, equal weight isassumed across the pupil and other specifications are
kept as those in Fig. 5(b) with z⬘ ⫽ 50.2491 mm. The
subtraction between each pair of ␸៮ i and ␸៮ j gives thereal OPDs, which are normalized by ␭ and shown in
Fig. 6.
It is seen that, under the modeled conditions, thephase linearity in all six mappings generally remains. Only a small curvature occurs at the edges of
the image simply because of the enhanced BS and BD
effects at larger field angles. Linearity guarantees theFig. 6. (Color online) OPD mappings (normalized by ␭) of the fringe components between all six ray configuration pairs.
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Table 2. Fitted Slopes and Deviation Percentages of Different OPD
Mappings in Fig. 5
U21
Value (mm⫺1)
Deviation %

U43

U41

U23

U13

U24

14.1156 14.1108 14.1075 14.1138 14.1104 14.1033
0.4
0.3
0.3
0.3
0.3
0.3

validity of Eq. (1) and the reconstruction algorithm.
However, small variations are observed unpredictably
distributed across the image, which slightly weakens
the linearity uniformity. These variations can introduce small errors in the reconstructed Stokes images.
To compare with the ideal linear phase, we select thecentral lines in each plot, and fit the data with anapproximate slope Uij, where the subscripts denotes
the pair of ray configurations. The theoretical value is
Utheory ⫽ 2共no ⫺ ne兲tan共␪兲兾␭ ⫽ 14.065 mm⫺1 for thissystem. The phase deviation percentage defined by
Deviation% ⫽ |Uij ⫺ Utheory|兾Utheory ⫻ 100% is calculated and tabulated in Table 2, which shows smallvalues in the deviation for all six mappings. Importantly, since U21 and U43 are almost equal, then共␸2 ⫺ ␸1兲 and 共␸4 ⫺ ␸3兲 are also equal so that the S0
component of the second and third terms in Eq. (5)will very nearly cancel, which is one of the conditions
necessary to go from Eq. (5) back to Eq. (1) (i.e.,
validating Mueller theory results). The difference between U13 and U24, no matter how great, will notaffect measurement, because the y-dependent modulation is not used in the image reconstruction (it isautomatically filtered out in the reconstruction).1 As
for U41 and U23, a small phase deviation (as defined
above) in them can be compensated for in the calibration, which is an approach that will be reported
elsewhere.
E.

Large Apex Angle

The apex angles ␪1,2 and ␪3,4 of the prism are parameters for the system optimization. Since the spatial
frequency U ⫽ 2共no ⫺ ne兲tan共␪兲兾␭ of fringes characterizes the bandwidth of each signal channel [Eq. (1)],increasing the apex angle of prism will widen the
bandwidth. A large apex angle can also compensate
for small values of birefringence in the prisms. However, all the above results imply that the increase in
the apex angle will be limited by the aberrations
introduced due to these prisms. A larger apex angle
causing more refraction will generate more BS and
BD effects. Figure 7 illustrates such an effect by making a simple comparison between apex angle values of
1.5° and 4.5° in a calcite system (both for the on-axis
field and at the optimized focus). The minimum image spot, ⬎20 ␮m in the latter system of ␪ ⫽ 4.5°,
would result in a significant loss of resolution for a
detector with a pixel pitch of less than ⬃20 ␮m.
4. Approach for Designing a Prismatic Imaging
Polarimeter

The previous sections have discussed how the resolution at the image plane depends on the system parameters for the polarimeter. The geometric model makes
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APPLIED OPTICS 兾 Vol. 45, No. 33 兾 20 November 2006

Fig. 7. (Color online) Spot diagrams of different apex angles ofa calcite system both for the on-axis field and at optimized foci.
(a) ␣ ⫽ 1.5°, (b) ␣ ⫽ 4.5° with rms spot size ⬎20 ␮m.

clear that polarimetry errors in the instrument will bedominated by the increase in the spot size rather thanin distortion. This permits a straightforward approach
to using the geometric model to design the polarimeter:(1) The basic system requirement is a minimum
spatial resolution in the reconstructed Stokes images
defining the required bandwidth U for the Stokes
images.
(2) Select a detector and lens to achieve a desired
FOV and pupil diameter.
(3) Select a birefringent material and use ␪ ⫽arctan共U␭兾2⌬n兲 to obtain the prisms’ apex angles.
(4) Using the geometric model, estimate the blurring at the image to locate the cutoff frequency in themeasurement (where the modulation transfer function falls below the noise spectrum).
(5) If the cutoff frequency is less than 1.5U, thenthe Stokes images will suffer significant error requiring a modification of the system parameters. When
this happens, we can select a material with a higher
birefringence, reduce the FOV, or increase the system f兾#.
When trading off FOV versus f兾#, it is useful toconsider that, for typical systems, the design is much
more sensitive to the f兾# than the FOV.
(6) Finally, it is also necessary to verify that the
distortion present in the system is small enough toprevent excessive aliasing artifacts due to deviationfrom phase linearity.
5. Conclusions

To study the BS and BD effects of the birefringent
prisms on the performance of a prismatic imaging
polarimeter, a useful approach is to construct ageometric imaging model within an optical design
software package. This enables a detailed simulation and analysis of the imaging performance. Previous publication of the results from the prismaticpolarimeter1 showed reconstruction artifacts, whichraises the question of whether the BS and BD effectsof the prisms prevented accurate system performance. We have been able to show that for a standard
imaging system the use of calcite birefringent prisms
will produce little degradation in the reconstructed
Stokes images. However, an imaging polarimeter requiring larger apex angles may experience serious
reconstruction error, for which the geometric modelprovides a convenient means of redesigning the system to produce Stokes images at the desired resolution.
References
1. K. Oka and T. Kaneko, “Compact complete imaging polarimeterusing birefringent wedge prisms,” Opt. Express 11, 1510 –1518(2003).
2. M. C. Simon, “Ray tracing formulas for monoaxial optical components,” Appl. Opt. 22, 354 –360 (1983).
3. M. C. Simon, “Wollaston prism with large split angle,” Appl.
Opt. 25, 369 –376 (1986).

4. M. Avendano-Alejo and O. Stavroudis, “Huygens’s principle
and rays in uniaxial anisotropic media II. Crystal axis with
arbitrary orientation,” J. Opt. Soc. Am. A. 19, 1674 –1679(2002).
5. H. X. Ren, L. R. Liu, D. A. Liu, and Z. Song, “Double refraction
and reflection of sequential crystal interfaces with arbitrary
orientation of the optic axis and application to optimum design,”
J. Mod. Opt. 52, 529 –539 (2005).
6. Q. T. Liang, “Simple ray tracing formulas for uniaxial opticalcrystals,” Appl. Opt. 29, 1008 –1010 (1990).
7. D. Goldstein, Polarized Light, 2nd ed. (Dekker, 2003), Chap. 4,
p. 60.

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