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Compact and miniature snapshot imaging polarimeter

Haitao Luo,1Kazuhiko Oka,2Edward DeHoog,3Michael Kudenov,1

James Schiewgerling,1and Eustace L. Dereniak1

1College of Optical Science, University of Arizona, 1630 East University Boulevard, Tucson, Arizona 85721

2Division of Applied Physics, Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan

3Biomedical Engineering Program, University of Arizona, 1657 East Helen Street, Tucson, Arizona 85721

Received 7 May 2008; accepted 19 June 2008;

posted 9 July 2008 (Doc. ID 95874); published 19 August 2008

We present and demonstrate a compact and miniature snapshot imaging polarimeter camera; it is an-

ticipated that such a camera can be scaled down to less than 1.5 cm. Two Savart plates are used at the

pupil planeto generatemultiplefringesto encodethefull Stokesvector in asingle image. A geometricray

model is developed to explain the system. The numerical simulation based on this model is presented.

Finally, the validity of the device is demonstrated by showing experimental results.

Society of America

OCIS codes:

120.2130, 120.5410, 260.5430.

© 2008 Optical

1.

In remote sensing, bioscience, or other scientific

areas, a compact or miniature imaging polarimeter

that can be used to measure the state of polarization

(SOP) of an object would be a powerful tool. For ex-

ample, a surveillance imaging polarimeter would be

valuable on small unmanned air vehicles. Likewise,

an endoscopic imaging polarimeter could be easily in-

serted into a body cavity. These are challenging tasks

for conventional polarimeters that use several rotat-

ing polarization elements to extract the complete

Stokes vector of an object, as they are sensitive to vi-

bration and contain more mechanical complexity.

One technique that uses microretarder and polarizer

arrays on the image plane has proved to achieve a

compact device [1,2] athough the sensitivity to fabri-

cation errors and noise limits the accuracy. Another

strategy demonstrated in monochromatic applica-

tions was the use of cascading birefringent prisms

[3]. The fabrication difficulty of the tiny prisms is

a drawback. Oka and Saito invented a Savart plate

(SP) snapshot imaging polarimeter (SIP) whose prin-

ciple can be analogous to a prism SIP [4,5]. The dif-

ference between them is that the SPs are placed at

Introduction

the aperture plane rather than the image plane as

is the prism SIP, which makes device assembly much

easier. Principally, this device was based on a pair of

optical Fourier transforms produced by a 4f imaging

system. The SPs inserted at the first Fourier plane

create polarization-dependent shearing on the wave-

front; the second optical Fourier transform converts

the wavefront shearing into linear phases that final-

ly produce sinusoidal fringes that encode the Stokes

vector. By implementing a standard Fourier analysis,

the Stokes vector can be decoupled and reconstructed

simultaneously. This snapshot ability eliminates the

need for any movable part of the system.

Conceptually, the 4f length of the system is neces-

sary to produce double Fourier transforms. A rela-

tively lengthy system results. However, a geometric

model of the system would prove that an essential re-

duction can be achieved. In addition, the SP’s innate

propertytocreatethefringefrequencywillallowusto

further miniaturize the polarimeter to a centimeter.

Here we first introduce the layout of the compact

and miniature SIP. A geometric model to replace

the original Fourier-transform model will be devel-

oped to prove the new scheme. We then illustrate

the unique condition of the system to miniaturize

the device. A ministructure will be simulated by

means of a ray tracing technique to demonstrate

0003-6935/08/244413-05$15.00/0

© 2008 Optical Society of America

20 August 2008 / Vol. 47, No. 24 / APPLIED OPTICS4413

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the validity of the miniature SIP. Finally, a portable

SIP is fabricated and demonstrated by showing the

experimental results.

2.

Figure 1 illustrates the optical layout of the minia-

ture SIP, which contains two SPs that sandwich a

half-wave plate (HWP) oriented at 22:5°and an ana-

lyzer oriented at 45°. The whole as a group is placed

at the aperture plane of a single camera. The camera

is equipped with a bandpass filter for quasi-

monochromatic applications. Compared to the origi-

nal 4f system, the current structure eliminates any

unwanted space between the two elements in front of

the camera. It also benefits from the flexible zoom

ability of a single camera. These two changes make

the device compact and cost-effective.

The double-Fourier-transform model cannot be

used to explain the current configuration. Instead, a

geometric ray model was developed to prove the de-

vice’s principle. First, the SP can be considered as a

ray shearing element in which two orthogonal polar-

ization rays, the ordinary (o) and extraordinary (e),

experience parallel shearing from each other after

propagating through the SP [6]. This shearing is cru-

cial in that it creates a geometric optical path differ-

ence (OPD) between the o ray and the e ray that

varies linearly with incident angle θ as illustrated

by inset I in Fig. 1. With the imaging lens, the inci-

dent angle can be interpreted by the coordinates at

the image plane. The combination of the two SPs and

Layout and Geometric Modeling

HWP generates four output rays from a common in-

cident ray, with equal and diagonal shearing as illu-

strated in inset II in Fig. 1. Note that the HWP at

22:5°rotates the electric field vector by 45°, enabling

the second SP to shear the rays that exit from the

first SP. At the end, the imaging lens recombines

the four rays and the rays mutually interfere at the

image plane. The resultant interferogram can be de-

tected by the insertion of an analyzer at 45°.

The total irradiance at the image plane can be

written as a sum of the four rays and their mutual

interference:

Iðxi;yiÞ ¼

?????

1

2Eyðxi;yi;tÞe−iϕ1−1

þ1

2Eyðxi;yi;tÞe−iϕ2

2Exðxi;yi;tÞe−iϕ3þ1

2Exðxi;yi;tÞe−iϕ4

????

2?

ð1Þ

;

where the bracket represents the time average, xi

and yiare the coordinates of the image plane, each

term within the brackets represents the electric field

of a ray after it passes the analyzer, and ϕ1through

ϕ4denote the accumulative phases of the individual

ray paths. By unfolding Eq. (1) with proper arrange-

ments, we can obtain

Fig. 1.

surfaces, depict the optic axes of the calcite plates. Inset I, the OPD formation of a SP between two orthogonal polarization rays for

a skewed incident ray. Inset II, the four emerging rays off the back surface of SP 2. The e and o denote the polarization consequences

through the system.

(Color online) Optical layout of a miniature SIP. The double ended arrows, each tilting at 45°with respect to the edge of the

4414APPLIED OPTICS / Vol. 47, No. 24 / 20 August 2008

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I ¼1

4f2½hjEyj2i þ hjExj2i? − ½hjEyj2ieiðϕ2−ϕ1Þþ c:c:?

þ ½hjExj2ieiðϕ4−ϕ3Þþ c:c:? þ ½hE?

− ½hE?

− ½hE?

where variables xi, yi, and t are suppressed. Under

ideal conditions, ϕ1through ϕ4can be written as spa-

tially linear functions as

xEyieiðϕ3−ϕ1Þþ c:c:?

xEyieiðϕ4−ϕ1Þþ c:c:?

xEyieiðϕ4−ϕ2Þþ c:c:? þ ½hE?

xEyieiðϕ3−ϕ2Þþ c:c:?g;

ð2Þ

ϕ1ðxi;yiÞ ¼ 0;

ϕ3ðxi;yiÞ ¼ 2π2Δ

ϕ2ðxi;yiÞ ¼ 2πΔ

ϕ4ðx;yÞ ¼ 2πΔ

λfðxiþ yiÞ;

λfxi;

λfðxi? yiÞ;

ð3Þ

where λ is the wavelength and

shearing distance generated by a SP. It is readily

known that the following equations are valid [7]:

ffiffiffi

2

p

Δ represents the

hjExj2i þ hjEyj2i ¼ S0;

hjEyj2i ¼1

hjExj2i ¼1

xEyi ¼1

2ðS0þ S1Þ;

2ðS0− S1Þ;

hE?

2ðS2þ iS3Þ:

ð4Þ

By plugging Eqs. (3) and (4) into Eq. (2), we can final-

ly arrive at the intensity pattern as

Iðxi;yiÞ ¼1

2S0þ1

þ1

−1

4jS23ji · cos½2πð2ΩÞyiþ argðS23Þ?

Ω ¼Δ

2S1· cosð2πΩðxiþ yiÞÞ

4jS23j · cos½2πð2ΩÞxi− argðS23Þ?

S23¼ S2þ iS3;

λf:

ð5Þ

Note that in Eq. (5) the Stokes parameters S1

through S3are modulated by various carrier fre-

quencies, but S0remains a DC component. This fact

allows us to decouple them in the frequency domain

by implanting a Fourier transform and then recover-

ing them simultaneously by inverse Fourier transfor-

mation [3].

This model convinces us that the new system

formulates a linear combination of individual Stokes

parameters in an image but in a much more compact

layout. We need to mention that this model ignores

any diffraction and aberration effects that can poten-

tially affect the polarimeter’s performance. An

advanced aberration model will be developed else-

where to address these profound issues [8].

3.

In addition to the system reduction, a unique prop-

erty of the SP can make an additional contribution

to the miniaturizatioin of the device. The fringe fre-

Condition of Miniaturization

quencies in Eq. (5), which we call the carrier frequen-

cies (CFs), are proportional to Ω ¼Δ

the shearing distance

2

Δ ¼ 0:075tðμmÞ, where t is

the thickness in millimeters of the SP [6]. At a fixed

wavelength and CFs (Ω), t is proportional to f, the

focal length of the imaging lens. This becomes an im-

portant feature in that the total length of the polari-

meter can be scaled by a common factor. If a

miniature lens can be implemented, the whole device

can be scaled down significantly. We use a cell phone

camera as an example with f ¼ 5mm and pixel spa-

cing of 4:75μm, λ ¼ 0:55μm, and Ω ¼ 4 pixels/fringe.

The required thickness t then equals 1:54mm and

the total length of the polarimeter is shorter than

1:5cm, which to our knowledge is the smallest ima-

ging polarimeter to date. If we change the calcite to

other higher birefringence materials, the size of the

polarimeter can be made even smaller. Compared

with other polarization elements, the plane-parallel

feature of the SPs makes it more cost-effective in

terms of fabrication and assembly.

λf. For a calcite SP,

ffiffiffip

4.

Imaging Polarimeter

As a theoretical demonstration, the above-mentioned

parameters of a miniature system are inserted into

Zemax with a 5mm diameter pupil. The image spot

diagrams and ray shearing diagrams from this simu-

lation are shown in Fig. 2. Two different field points

are sampled, and the object is at a distance of 1m for

each field point. The image quality looks impressive

Ray Tracing Analysis of a Miniature Snapshot

Fig. 2.

iature SIP at best focus. The imaging lens was modeled as a para-

xial lens. The object is 1 m away, and two field points (0°, 0°) and

(25°, 25°) are sampled. The off-axis spot diagram displays anisotro-

py. (b) The ray shearing diagrams at the back surface of SP 2 for

the (0°, 0°) and (25°, 25°) field points, respectively. For visual pur-

poses, one incident ray is shown. The dotted rectangle is drawn to

illustrate the diagonal shearing among the four rays.

(Color online) (a) Simulated image spot diagrams of a min-

20 August 2008 / Vol. 47, No. 24 / APPLIED OPTICS 4415

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for both the on-axis and the off-axis cases where the

spot sizes of the image are close to the diffraction lim-

it. This result diminishes the concern that the SPs

could introduce excessive aberrations when imaging

a finite conjugate object. In Fig. 2(b), the on-axis ray

shearing diagrams are consistent with expectation,

i.e., four rays are sheared equally and diagonally

with values of

2

Δ ¼ 113μm. Conversely, in the

off-axis case, the four rays are not sheared ideally

and their shearing distances are not equal to each

other. Both are due to the retardance isotropy of

the SP itself. This deviation deteriorates from an

on-axis to an off-axis case, and it could undermine

the sinusoidal nature of the fringes, producing errors

in the reconstruction. Fortunately, our simulation in-

dicates that such a deviation deteriorates slowly

from the on-axis case to the off-axis case, and we de-

termined that these errors can be calibrated by a re-

ference image (see Section 5). Moreover, we also

determined through simulation that the miniature

SIP also works well over a large depth-of-focus range

and a wide field of view.

ffiffiffip

5.

For a proof-of-concept demonstration, we built a com-

pact system with commercially available elements,

all of which are 1 in. (2.54 cm) diameter and mounted

independently. The SPs were manufactured by Karl

Lambrecht Corporation, Chicago, Illinois. We used a

75mm focal-length lens, which extended the total

length of the system by a factor of 15 from its min-

iature counterpart. However, it still remains a porta-

ble size as seen in the photo of the polarimeter

in Fig. 3.

Figure 4(a) shows a sample raw image obtained

with the compact SIP; Fig. 4(b) shows the recon-

structed Stokes images. A uniform polarized image

is formed with a linear polarizer at 22:5°(S1¼

0:707, S2¼ 0:707, S3¼ 0) to be used as a reference

to calibrate the system. The camera’s view angle is

approximately 60 deg with respect to the ground,

and the object, a car, is approximately 30m away.

In Fig. 4(a), the clear fringes seen across the image

indicate the existence of polarization signals. This is

due to the skewed Sun-object-camera angle and less

scattering surfaces that are inside the scene. There is

a change in the fringe pattern on various surfaces

Experimental Demonstration

(e.g., the window) of the car, indicating that different

SOPs are reflected. The reconstructed Stokes images

in Fig. 4(b) also demonstrate this by clearly identify-

ing the shape of the car in different Stokes para-

meters. We can see that the circular polarization

(in S3) is weaker than the linear polarization (S1

and S2). The region around the broken ground (below

the car in the image) looks noisy, because it contains

a higher frequency content that the processing algo-

rithm cannot fully recover [3].

It is worth mentioning that, compared with a min-

iature SIP, the enlarged system suffers from larger

aberrations. Aberrations degrade the camera perfor-

mance in two ways: (1) introducing errors in the re-

construction, especially for off-axis fields and (2)

washing out the fringe contrast [8]. With the current

Fig. 3.

from a miniature SIP. The CCD camera has a pixel spacing of

4:75μm.

Demonstrated compact SIP with a scaling factor of 15Fig. 4.

bandwidth filter is used in front of the polarimeter. (b) Recon-

structed Stokes images.

(a) Raw image obtained with the compact SIP. A 3nm

4416APPLIED OPTICS / Vol. 47, No. 24 / 20 August 2008

Page 5

encouraging results, it is expected that better perfor-

mance can be achieved with a miniature SIP.

6.

In conclusion, we have demonstrated a compact

snapshot imaging polarimeter by use of a group of

functional polarization elements at the aperture

plane. This approach is superior for making a minia-

ture SIP that could open many applications for which

polarization information is important. The simple

configuration, easy alignment, and cost-effectiveness

make this technique competitive. A proof-of-concept

device has been fabricated to demonstrate the min-

iature SIP by showing both the numerical simulation

and the real experimental results.

Conclusions

References

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1486–1492 (2000).

3. K. Oka and T. Kaneko, “Compact complete imaging polari-

meter using birefringent wedge prisms,” Opt. Express 11,

1510–1519 (2003)

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20 August 2008 / Vol. 47, No. 24 / APPLIED OPTICS4417