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Page 1: White light Sagnac interferometer for snapshot linear polarimetric imaging

White light Sagnac interferometer for snapshot

linear polarimetric imaging

Michael W. Kudenov,1 Matthew E. L. Jungwirth,

1 Eustace L. Dereniak,

1 and

Grant R. Gerhart2

1College of Optical Science, The University of Arizona, 1630 E. University Blvd., Tucson, AZ 85721 2U.S. Army TACOM, 6501 E. 11 Miles Rd., Warren, MI 48397

Abstract: The theoretical and experimental demonstration of a dispersion-

compensated polarization Sagnac interferometer (DCPSI) is presented. An

application of the system is demonstrated by substituting the uniaxial

crystal-based Savart plate (SP) in K. Oka’s original snapshot polarimeter

implementation with a DCPSI. The DCPSI enables the generation of an

achromatic fringe field in white-light, yielding significantly more radiative

throughput than the original quasi-monochromatic SP polarimeter.

Additionally, this interferometric approach offers an alternative to the

crystal SP, enabling the use of standard reflective or transmissive materials.

Advantages are anticipated to be greatest in the thermal infrared, where

uniaxial crystals are rare and the at-sensor radiance is often low when

compared to the visible spectrum. First, the theoretical operating principles

of the Savart plate polarimeter and a standard polarization Sagnac

interferometer polarimeter are provided. This is followed by the theoretical

and experimental development of the DCPSI, created through the use of

two blazed diffraction gratings. Outdoor testing of the DCPSI is also

performed, demonstrating the ability to detect either the S2 and S3, or the S1

and S2 Stokes parameters in white-light.

©2009 Optical Society of America

OCIS codes: (110.3175) Interferometric imaging; (110.5405) Polarimetric imaging; (120.5410)

Polarimetry

References and links

1. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote

sensing applications,” Appl. Opt. 45(22), 5453–5469 (2006).

2. M. H. Smith, J. B. Woodruff, and J. D. Howe, “Beam wander considerations in imaging polarimetry,” in Proc.

SPIE 3754, 50–54 (1999).

3. K. Oka, and T. Kato, “Spectroscopic polarimetry with a channeled spectrum,” Opt. Lett. 24(21), 1475–1477

(1999).

4. K. Oka, and T. Kaneko, “Compact complete imaging polarimeter using birefringent wedge prisms,” Opt. Express

11(13), 1510–1519 (2003).

5. K. Oka, and N. Saito, “Snapshot complete imaging polarimeter using Savart plates,” Proc. SPIE 6295, 629508

(2006).

6. M. W. Kudenov, L. Pezzaniti, E. L. Dereniak, and G. R. Gerhart, “Prismatic imaging polarimeter calibration for

the infrared spectral region,” Opt. Express 16(18), 13720–13737 (2008).

7. H. Luo, K. Oka, E. DeHoog, M. Kudenov, J. Schiewgerling, and E. L. Dereniak, “Compact and miniature

snapshot imaging polarimeter,” Appl. Opt. 47(24), 4413–4417 (2008).

8. D. Malacara, Optical Shop Testing (John Wiley & Sons, Inc., New York, 1992), p. 100.

9. G. Zhan, K. Oka, T. Ishigaki, and N. Baba, “Static Fourier-transform spectrometer based on Savart polariscope,”

Proc. SPIE 4480, 198–203 (2002).

10. R. Suda, N. Saito, and K. Oka, “Imaging Polarimetry by Use of Double Sagnac Interferometers,” in Extended

Abstracts of the 69th Autumn Meeting of the Japan Society of Applied Physics (Japan Society of Applied

Physics, Tokyo), p. 877 (2008).

11. M. Born, and E. Wolf, Principles of Optics (Cambridge University Press, 1999), p. 831.

12. R. S. Sirohi, and M. P. Kothiyal, Optical components, Systems, and Measurement Techniques, (Marcel Dekker,

1991), p. 68.

#116682 - $15.00 USD Received 3 Sep 2009; revised 7 Oct 2009; accepted 19 Oct 2009; published 24 Nov 2009

(C) 2009 OSA 7 December 2009 / Vol. 17, No. 25 / OPTICS EXPRESS 22520

Page 2: White light Sagnac interferometer for snapshot linear polarimetric imaging

13. J. C. Wyant, “OTF measurements with a white light source: an interferometric technique,” Appl. Opt. 14(7),

1613–1615 (1975).

14. M. W. Kudenov, College of Optical Sciences, The University of Arizona, 1630 E. University Blvd. #94, Tucson,

AZ 85721, M. E. L. Jungwirth, E. L. Dereniak, and G. R. Gerhart are preparing a manuscript to be called, “White

light Sagnac interferometer for snapshot multispectral imaging.”

15. U. U. Graf, D. T. Jaffe, E. J. Kim, J. H. Lacy, H. Ling, J. T. Moore, and G. Rebeiz, “Fabrication and evaluation

of an etched infrared diffraction grating,” Appl. Opt. 33(1), 96–102 (1994).

16. M. W. Kudenov, J. L. Pezzaniti, and G. R. Gerhart, “Microbolometer-infrared imaging Stokes polarimeter,” Opt.

Eng. 48(6), 063201 (2009).

17. F. Snik, T. Karalidi, and C. U. Keller, “Spectral modulation for full linear polarimetry,” Appl. Opt. 48(7), 1337–

1346 (2009).

1. Introduction

Instantaneous acquisition of the Stokes polarization parameters is of great interest in many

areas of remote sensing [1]. A Stokes imaging polarimeter is capable of obtaining either the

partial or complete polarization state of a scene via four Stokes parameters. These parameters

express the state of polarization in a 4x1 matrix, defined as

( )

( )( )( )( )

( ) ( )( ) ( )( ) ( )( ) ( )

0 0 90

1 0 90

2 45 135

3

, , ,

, , ,,

, , ,

, , ,R L

S x y I x y I x y

S x y I x y I x yx y

S x y I x y I x y

S x y I x y I x y

+ − = = −

S (1)

where x, y are spatial coordinates in the scene, S0 is the total energy of the beam, S1 denotes

preference for linear 0° over 90°, S2 for linear 45° over 135°, and S3 for circular right over left

polarization states. Each is defined by an addition or subtraction of intensity measurements

that represent different analyzer states, and complete characterization requires at least four

such states be measured. These measurements can be acquired over time by use of a rotating

element (division of time) with a single imaging lens and focal plane array (FPA). Such an

implementation yields a compact and relatively inexpensive instrument. However, these

sensors are susceptible to misregistration errors caused by platform or scene motion, as in

many applications, Stokes parameters are acquired from moving platforms

To remedy concerns regarding temporal misregistration, the instrument must acquire

multiple analyzer measurements in parallel. Typically, the intensity measurements needed for

Stokes parameter calculations are taken through different imaging systems. Consequently,

differences in distortion, focal length, and intensity necessitate sophisticated image

registration algorithms [2]. Alternatively, these intensity measurements can be taken in

parallel by amplitude modulating the Stokes parameters onto various interferometrically

generated carrier frequencies, referred to here as a “fringe polarimeter” (FP).

Amplitude modulation of the spatially-dependent Stokes parameters onto spatial carrier

frequencies was first demonstrated by K. Oka [3]. Oka established that the complete Stokes

vector can be encoded onto various interference fringes with the use of optimized Wollaston

prisms, located in an intermediate image plane [4]. Alternatively, the same effect can be

achieved using two Savart plates located within a collimated space of an optical system [5].

These systems provide the advantages of being snapshot, while also offering inherent image

registration since the Stokes parameters are encoded on coincident fringe fields. However,

one remaining concern is that the carrier frequency’s visibility decreases as the coherence

length of the incident illumination is decreased. Consequently, high-visibility fringes are

generated only when the light is quasi-monochromatic, leading to a reduced signal-to-noise

ratio in remote sensing applications. This is especially a concern for operation of the sensor in

the thermal infrared (3-12 µm) [6].

In this paper, we outline the theoretical and experimental development of a dispersion-

compensated polarization Sagnac interferometer (DCPSI), enabling white-light operation of

#116682 - $15.00 USD Received 3 Sep 2009; revised 7 Oct 2009; accepted 19 Oct 2009; published 24 Nov 2009

(C) 2009 OSA 7 December 2009 / Vol. 17, No. 25 / OPTICS EXPRESS 22521

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an FP. In section 2, the theoretical background of a FP, based on using Savart plates (SP), is

provided. In section 3, a FP based on a polarization Sagnac interferometer (PSI) is

demonstrated to theoretically produce the same effect as a SP-based FP. In section 4,

diffraction gratings are introduced to the PSI, creating the DCPSI. This is followed by the

theoretical calculation of the fringe pattern. Lastly, in section 5, experimental laboratory and

outdoor data of a DCPSI in white-light are provided. It is also demonstrated that the addition

of a quarter-wave retarder enables the DCPSI to measure the full linear polarization content of

a scene.

2. Savart plate polarimeter

The operating principle for the PSI can be considered an analog of the Savart plate

polarimeter (SPP) [5, 7]. A diagram of the SPP is provided in Fig. 1. Two Savart plates,

denoted SP1 and SP2, generate a lateral shear ±45° to the x-axis, respectively. The input beam,

after transmission through SP1, is sheared into two orthogonally polarized beams. A half-

wave plate (HWP), oriented at 22.5°, rotates the polarization state of the two beams exiting

SP1 by 45°. Transmission through SP2 shears each of the two beams a second time, producing

four components. These are incident on the analyzer (A), which consists of a linear polarizer

oriented with its transmission axis parallel to the x-axis. Once the four beams are combined by

the objective lens, they produce spatially-varying interference fringes on the FPA. The

intensity pattern has the form

( ) [ ] ( ) ( )

( ) ( )

0 1 2 23 1 2 23

23 1 2 23

1 1 1, cos 2 cos 2 arg

2 2 4

1 cos 2 arg

4

I x y S S y S x S

S y S

π π

π

= + Ω + Ω +Ω −

− Ω +Ω +

(2)

with

23 2 3

,S S jS= + (3)

where Ω1 and Ω2 are the spatial carrier frequencies generated by SP1 and SP2, respectively,

and the Stokes parameters S0, S1, S2, and S3 are implicitly dependent upon x and y.

Consequently, the Stokes parameters S1 through S3 are amplitude modulated onto several

carrier frequencies, while S0 remains as an un-modulated component. Extraction of the

spatially varying polarimetric information is accomplished using Fourier filtration techniques

[4]. The carrier frequencies relate to the shear by

1 2

1 2 and ,f fλ λ

∆ ∆Ω = Ω = (4)

where λ is the wavelength of the incident light, f is the focal length of the objective lens, and

∆1, ∆2 are proportional to the thickness of SP1 and SP2, respectively. The shearing distance

(SSP) is related to the thickness of the SP (t) by

2 2

2 22 2 ,e o

SP

e o

n nS t

n n

−= ∆ =

+ (5)

where ne, no are the extraordinary and ordinary refractive indices of the SP, respectively [8].

One significant drawback of the SPP is that the carrier frequency, Ω, is inversely

dependent upon wavelength, λ, as illustrated in Eq. (4). It is this dispersion in the carrier

frequency that necessitates the use of a narrow bandpass filter (BPF) in the sensor, such that a

long coherence length can be maintained. This enables high visibility fringes to be preserved

for large optical path differences (OPD), which are typically on the order of +/− 40 waves.

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For example, assuming a uniform (rectangular) spectral distribution, the maximum spectral

bandwidth, B, that the BPF can transmit before fringe visibility decreases to 50% can be

calculated by

2

max

0.6 ,BOPD

λ= (6)

where OPDmax is the peak optical path difference in the system. For an OPDmax of 40 waves at

λ = 550 nm, the maximum bandwidth allowable is B = 8.25 nm. Narrow bandwidth operation

often has detrimental effects on the signal to noise ratio (SNR) in the measured Stokes

parameters, especially when operating in the thermal infrared (TIR) [6]. To rectify the low

SNR, a larger bandwidth must be obtained, implying compensation for the dispersion within

the carrier frequency.

Fig. 1. Schematic of the Savart plate polarimeter (SPP). Two Savart plates, SP1 and SP2, reside

in front of an objective lens with focal length fobj. The combination of both Savart plates

generates four sheared beams, separated by a distance 2∆ . A depiction of the initial beam,

as well as the beams generated after transmission through SP1 and SP2, is also provided in the

x, y plane (denoted xp, yp). A narrow bandpass filter is used to maintain a high fringe visibility.

3. Sagnac interferometer as a Savart plate

A Sagnac interferometer with polarization optics can duplicate the shearing properties of a

Savart plate (SP) [9, 10]. One configuration for a polarization Sagnac interferometer (PSI) is

depicted in Fig. 2.

Fig. 2. One configuration for a polarization Sagnac interferometer. The distance between the

WGBS and mirrors, M1 and M2, is denoted d1 and d2, respectively. A shear (SPSI) is produced

when d1 ≠ d2. The case d1 > d2 is illustrated, with d1 = d2 + α.

#116682 - $15.00 USD Received 3 Sep 2009; revised 7 Oct 2009; accepted 19 Oct 2009; published 24 Nov 2009

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The PSI consists of two mirrors, M1 and M2, with a wire grid polarization beamsplitter

(WGBS), a focal plane array (FPA), and an objective lens with focal length fobj. Two beams,

with a shear SPSI, are created when d1 ≠ d2. If the distance, d1, is offset by an amount α, such

that d1 = d2 + α, the shear is

2PSI

S α= (7)

Therefore, the shear produced by a PSI per Eq. (7) is analogous to the shear generated by a

SP, observed previously in Eq. (5). Note that the shear generated by both the SP and PSI is

essentially achromatic; it is independent of the wavelength of the incident illumination

[11].The functional form of the intensity pattern for a PSI and SP is also similar. First,

consider the single PSI, illustrated previously in Fig. 2, with the WGBS oriented at 0° and a

linear polarizer (LP) oriented at 45° as the analyzer (A). Also, consider the simplified SPP

with a single Savart plate per Fig. 3, in which the SP produces a shear 45° with respect to the

x-axis.

Fig. 3. A simplified SPP with a single Savart plate.

The OPD between the two sheared beams exiting from a PSI or the simplified SPP is depicted

in Fig. 4, and is given by

( )sin ,shear shear

OPD S Sθ θ= ≈ (8)

where Sshear is the shear generated by the SPP or PSI, SSPP or SPSI, respectively [12].

Fig. 4. OPD generated by a shearing distance, Sshear, for a PSI and simplified SPP.

When the two beams are combined by the lens, they produce interference fringes on the

FPA. For the simplified SPP, the interference of the two sheared rays can be expressed as

( ) ( ) ( )1 2

2

1 1, , , , , ,

2 2

j j

SPP i i x i i y i iI x y E x y t e E x y t eφ φ− −= + (9)

where < > represents the time average, xi and yi are image-plane coordinates, and φ1, φ2, are

the cumulative phases of each ray. Expansion of this expression yields

#116682 - $15.00 USD Received 3 Sep 2009; revised 7 Oct 2009; accepted 19 Oct 2009; published 24 Nov 2009

(C) 2009 OSA 7 December 2009 / Vol. 17, No. 25 / OPTICS EXPRESS 22524

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( )( ) ( ) ( )

( ) ( )

* * * *

1 2

* *

1 2

cos1, ,

2 sin

x x x y x y x y

SPP i i

x y x y

E E E E E E E EI x y

j E E E E

φ φ

φ φ

+ + + − + =

− + −

(10)

where Ex, Ey are now implicitly dependent upon xi and yi. The phase factors are

1 2

2 2 and .i i

obj obj

x yf f

π πφ φ

λ λ∆ ∆

= = − (11)

The Stokes parameters are defined from the components of the electric field as

( )

* *

0* *

1

* *2

* *3

.

x x y y

x x y y

x y x y

x y x y

E E E ES

E E E ES

S E E E E

Sj E E E E

+ − = + −

(12)

Re-expressing Eq. (10) by use of the Stokes parameter definitions and φ1, φ2, yields

( ) ( ) ( )0 2 3

1 2 2, cos sin .

2SPP i i i i i i

obj obj

I x y S S x y S x yf f

π πλ λ

= + ∆ − − ∆ −

(13)

Consequently, the simplified SPP modulates S2 and S3 onto a carrier frequency, while S0

remains as an un-modulated component. The carrier frequency, USPP, is

2

.SPP

obj

Ufλ∆

= (14)

Fourier filtering can then be used to calibrate and reconstruct the spatially-dependent Stokes

parameters over the image plane. This procedure is detailed in section 5.2.

A similar procedure can be conducted to calculate the intensity at the image plane for the

PSI, and is directly analogous to the procedure depicted above for the simplified SPP. The

difference involves the cumulative phase factors, φ1 and φ2, which now depend on the

interferometer’s beamsplitter to mirror separation difference, α, in addition to an adjustment

in the shear direction, now along +/− xp. The phase factors for the PSI are

1 2

2 2 and .

2 2i i

obj obj

x xf f

π α π αφ φ

λ λ= = − (15)

Use of these expressions yields the intensity pattern for the PSI

( ) 0 2 3

1 2 2, cos 2 sin 2 .

2PSI i i i i

obj obj

I x y S S x S xf f

π πα α

λ λ

= + −

(16)

The carrier frequency, UPSI, is

2

.PSI

obj

Uf

αλ

= (17)

Therefore, the carrier frequency for both the PSI and the single Savart plate SPP is directly

proportional to the shear and inversely proportional to the focal length of the objective lens.

#116682 - $15.00 USD Received 3 Sep 2009; revised 7 Oct 2009; accepted 19 Oct 2009; published 24 Nov 2009

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Additionally, the carrier frequencies have dispersion, due to an inverse proportionality to the

wavelength of the incident light. Again, dispersion in the carrier frequency is the reason for

the narrow bandwidth limitation of the FP technique, regardless of whether it is implemented

with a SP or PSI; as the bandwidth increases, the visibility of the fringes decreases at

increasing values of OPD. A method to remove the carrier frequency’s dispersion is to make

the shear directly proportional to the wavelength, such that shear

S λγ∝ , where γ is some

optical thickness, analogous to α or ∆.

4. Dispersion compensation in the Sagnac interferometer

Compensation of the dispersion in the PSI’s carrier frequency can be realized with the

introduction of two blazed diffraction gratings. Diffraction gratings are well known for their

ability to generate white-light interference fringes [13]. The optical layout now takes the form

of the dispersion-compensated PSI (DCPSI) depicted in Fig. 5, and is the PSI observed

previously in Fig. 2 with d1 = d2 and the inclusion of two identical gratings, G1 and G2. A

ray’s diffraction angle, after transmission through G1 or G2, is calculated for normal incidence

by

( )sin ,m

d

λθ = (18)

where θ is the diffraction angle of the ray as measured from the grating’s normal, m is the

order of diffraction, and d is the period of the grating. Since d is typically large (>≈30 µm),

small angle approximations can be used to simplify Eq. (18), yielding m dθ λ≈ .

Fig. 5. DCPSI with blazed diffraction gratings, G1 and G2, positioned at each output of the

WGBS. To remove the achromatic shear, the distance between the WGBS and mirrors, M1 and

M2, depicted previously in Fig. 2, is d1 = d2. Inclusion of the gratings generates a shear that is

directly proportional to the wavelength.

In the DCPSI, the beam transmitted by the WGBS (now spectrally broadband) is

diffracted by G1 into the 1 order. When these dispersed rays are incident on G2, the diffraction

angle, induced previously by G1, is removed. The rays emerge parallel to the optical axis, but

are offset by a distance proportional to -λxo, where xo is some constant related to the DCPSI’s

parameters. Conversely, the beam reflected by the WGBS is initially diffracted by G2. The

dispersed rays are then diffracted to be parallel to the optical axis by G1, and exit the system

offset by a distance proportional to + λxo. The functional form of the shear can be calculated

by unfolding the optical layout of the DCPSI, as depicted in Fig. 6.

#116682 - $15.00 USD Received 3 Sep 2009; revised 7 Oct 2009; accepted 19 Oct 2009; published 24 Nov 2009

(C) 2009 OSA 7 December 2009 / Vol. 17, No. 25 / OPTICS EXPRESS 22526

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Fig. 6. DCPSI with an unfolded optical layout. Light traveling from left to right was initially

transmitted through the WGBS, while light traveling from right to left was initially reflected

from the WGBS.

Assuming small angles and that G1 and G2 have an identical period, then the shear, SDCPSI, is

( )2DCPSI

mS a b c

d

λ= + + (19)

where a, b, and c represent the distances between G1 and M1, M1 and M2, and M2 and G2,

respectively. Hence, the DCPSI can, to first order, generate a shear that is directly

proportional to the wavelength. This makes the phase factors

( ) ( )1 2

2 2 and .i i

obj obj

m ma b c x a b c x

f d f d

π πφ φ= + + = − + + (20)

Using these phase factors, the intensity on the FPA is

( ) ( )( ) ( )

( ) ( )

1 1

'

2/ /

'

0

0 1'

3

2 2cos

1 1,

2 2 2 2 sin

id d

obj

DCPSI i i

m m

i

obj

mS m a b c x

f dI x y S m

mS m a b c x

f d

λ λ

π

π= =

+ +

= +

− + +

∑ ∑ (21)

where the total intensity pattern is a summation from the minimum to maximum order of a

blazed grating, such that the maximum achievable order is (d/λ1)sin(π/2), where λ1 is the

minimum wavelength passed by the optical system and S0’(m), S2

’(m), and S3

’(m) are the

Stokes parameters weighted by the diffraction efficiency (DE) of both gratings before

integration over wavelength,

( ) ( ) ( )2

1

' 2

0 0,S m DE m S d

λ

λ

λ λ λ= ∫ (22)

( ) ( ) ( )2

1

' 2

2 2,S m DE m S d

λ

λ

λ λ λ= ∫ (23)

( ) ( ) ( )2

1

' 2

3 3,S m DE m S d

λ

λ

λ λ λ= ∫ (24)

where λ1 and λ2 denote the minimum and maximum wavelengths passed by the optical

system. The carrier frequency, UDCPSI, is

#116682 - $15.00 USD Received 3 Sep 2009; revised 7 Oct 2009; accepted 19 Oct 2009; published 24 Nov 2009

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( )2.DCPSI

obj

mU a b c

df= + + (25)

Therefore, insertion of the blazed diffraction gratings enables the dispersion, observed

previously in Eqs. (14) and 17, to be removed from the carrier frequency. This also creates the

possibility of generating a multispectral imager by modulating different spectral bands onto

unique carrier frequencies. This topic is beyond the scope of the current paper, and its

discussion is reserved for a future publication [14].

5. Experimental verification of the DCPSI

To verify the operating principles of the DCPSI, the experimental setup depicted in Fig. 7 was

implemented. Spatially and temporally incoherent light is configured by aiming the output of

a fiber-light, sourced by a tungsten-halogen lamp, onto a diffuser. A dichroic polymer linear

polarizer at 45°, followed by a polymer achromatic quarter-wave retarder (QWR) oriented at

θG, is used as the polarization generator (PG) to create known input states for calibration and

testing. The WGBS has a clear aperture of 21 mm and consists of anti-reflection coated

aluminum wires, enabling operation from 400 to 700 nm. Mirrors M1 and M2 are 25 mm

diameter 1/10th wave optical flats. A collimating lens, with focal length fc = 50 mm (F/11), is

used to image the object plane to infinity, while an objective lens, with focal length fobj = 200

mm (F/4.5), is used for re-imaging. This gives a field of view of approximately +/− 1°.

Fig. 7. Experimental setup for laboratory testing of the DCPSI using white light. The distances

a, b, and c are 14.4 mm, 57.2 mm, and 12.9 mm, respectively.

To ensure that the incident illumination is maintained within the spectral operating region

of the polarizers and QWR, an IR blocking filter is included. The transmission of the filter

(τIR) and crossed polarizers (τCP), in addition to their combination (τCPτIR), is depicted in Fig.

8(a). Lastly, the ruled diffraction gratings, G1 and G2, are blazed for a first order (m = 1)

wavelength of λB = 640 nm on a BK7 substrate with a period d = 28.6 µm. A depiction of the

ideal theoretical diffraction efficiencies for these gratings is provided in Fig. 8(b), and

illustrates the m = 0, 1, and 2 diffraction orders. While ruled transmission gratings are not

available in the TIR, blazed gratings can be made in Silicon by lithography [15].

5.1 Carrier frequency dispersion

To quantify the dispersion in the carrier frequency, UDCPSI, a fiber bundle was used to pass

light from a monochrometer to a plane close to the object plane, per Fig. 9. Positioning the

fiber bundle away from the object plane by approximately 5 mm keeps it out of focus, such

#116682 - $15.00 USD Received 3 Sep 2009; revised 7 Oct 2009; accepted 19 Oct 2009; published 24 Nov 2009

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that the fibers are not imaged directly. For this experiment, the fringe frequency was

measured at various quasi-monochromatic wavelengths with an the incident Stokes vector of

[S0, S1, S2, S3]T = [1,0,1,0]

T.

Fig. 8. (a) Transmission percentages vs. wavelength for the IR blocking filter, crossed linear

polarizers (to demonstrate their effective spectral region of operation), and the multiplication of

the two curves. (b) Ideal theoretical diffraction efficiencies vs. wavelength for gratings G1 and

G2 illustrating the m = 0, 1, and 2 diffraction orders.

To produce this polarization state, the QWR is removed from the PG and the linear

polarizer is oriented at 45°. The fringe frequency was measured by using a least-squares

fitting procedure based on the equation

( ) ( ) min max

0

, cos , i

i i i

i

x x xI x y A B x

y yω φ

≤ ≤ = + +

= (26)

where A, B, ω, and φ are the offset, amplitude, frequency and phase of the fringes, spanning xi

between xmin and xmax, for a given yi = y0 close to the optical axis.

Fig. 9. Monochrometer configuration for sending light into the DCPSI for verification of the

fringe frequency. The bandwidth of the light exiting the monochrometer was approximately 8.9

nm using a 2 mm exit slit.

Measuring the fringe frequency from 460 to 700 nm, in 10 nm increments, yields the

results depicted in Fig. 10. The fringe’s carrier frequency measured from the DCPSI is

depicted alongside the theoretical carrier frequency for a PSI. The PSI’s shear was selected

such that UPSI = UDCPSI at a wavelength of 600 nm. Using the theoretical data, the carrier

frequency’s total peak-to-peak variation for the uncompensated PSI was calculated to be

11.31E3 m−1

. Conversely, the total variation in the carrier frequency of the DCPSI was

measured to be 23.61 m−1

. Consequently, the total variation in the carrier frequency for the

DCPSI is over two orders of magnitude lower than in the uncompensated PSI.

#116682 - $15.00 USD Received 3 Sep 2009; revised 7 Oct 2009; accepted 19 Oct 2009; published 24 Nov 2009

(C) 2009 OSA 7 December 2009 / Vol. 17, No. 25 / OPTICS EXPRESS 22529

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Fig. 10. Measured carrier frequency vs. wavelength for the DCPSI (solid dark-gray line) and

the theoretical carrier frequency for a PSI (dotted light-gray line). The PSI’s carrier frequency

was set to equal the DCPSI’s carrier frequency at a wavelength of 600 nm.

5.2 DCPSI calibration

In order to calibrate and reconstruct data from the instrument, data processing is accomplished

in the Fourier domain [6]. Taking the Fourier transform of the intensity pattern (Eq. (21),

assuming the m = 1 diffraction order is dominant, yields

( ) ( ) ( ) ( ) ( )( ) ( )

( ) ( )( ) ( )

'

0 ' '

2 3

' '

2 3

1 1, , 1 1 ,

2 4

1 1 1 , ,

4

DCPSI i i DCPSI

DCPSI

SI x y S jS U

S jS U

δ ξ η δ ξ η

δ ξ η

= + + −

− − +

F

(27)

where δ is the dirac-delta function and ξ, η are the Fourier transform variables along xi and yi,

respectively. Extraction of the Stokes parameters is accomplished by taking an inverse Fourier

transform of the filtered carrier frequencies, or “channels.” Filtration and inverse Fourier

transformation of the δ(ξ, η) (channel C0) and δ(ξ - UDCPSI, η) (channel C1) components yields

[ ] ( )'

01

0

1

2

SC

− =F (28)

[ ] ( ) ( )( ) ( )1 ' '

1 2 3

11 1 exp 2

4DCPSI iC S jS j U xπ− = +F (29)

Therefore, the S0’ Stokes parameter is obtained directly, while (S2

’(1) + jS3

’(1)) requires

demodulation of the exponential phase factor. Demodulation is accomplished using the

reference beam calibration technique [6], in which the phase factor is measured for a known

Stokes vector over a uniformly illuminated scene. Reference data are obtained for a polarizer

oriented at 45° to isolate ( )exp 2DCPSI i

j U xπ . The unknown sample data are divided by the

measured reference data to solve for the incident Stokes parameters

( ) ( )0, 0,,

sample i i sampleS x y C= F (30)

( )( )

( )( )0, ,452,

2,

0,2, ,45

,referencesample

sample i i

samplereference

CCS x y

SC

°

°

= ℜ

FF

F (31)

#116682 - $15.00 USD Received 3 Sep 2009; revised 7 Oct 2009; accepted 19 Oct 2009; published 24 Nov 2009

(C) 2009 OSA 7 December 2009 / Vol. 17, No. 25 / OPTICS EXPRESS 22530

Page 12: White light Sagnac interferometer for snapshot linear polarimetric imaging

( )( )

( )( )0, ,452,

3,

0,2, ,45

, ,referencesample

sample i i

samplereference

CCS x y

SC

°

°

= ℑ

FF

F (32)

where ℜ and ℑ indicate that either the real or imaginary parts of the expression have been

taken.

5.3 White light polarimetric reconstructions in S2 and S3

To validate the accuracy of the measured Stokes parameters, a PG was implemented that

consisted of a LP at 45° followed by a rotating QWR, as depicted previously in Fig. 7.

Rotating the QWR from 0° to 180° in 10° increments yielded the data depicted in Fig. 11 (a).

The RMS error between the measured data and the output of the PG is calculated by

( ) ( )( )2

, , ,

1

1,

i

N

RMS S i Meas i PG

n

S n S nN

ε=

= −∑ (33)

where i is an integer (1, 2, or 3) that indicates the Stokes parameter being analyzed. The

calculated RMS errors for S2 and S3 for the data in Fig. 11 (a) are 2, 0.0212RMS Sε = and

3, 0.0378RMS Sε = . Residual errors can likely be attributed to the broad bandwidth that the

calibration covers [16].

Fig. 11. (a) Reconstructed Stokes parameters for a PG consisting of a LP at 45° followed by a

rotating QWR. (b) Reconstructed Stokes parameters for a rotating LP followed by a QWR

oriented at 45°, thereby forming an imaging polarimeter capable of full linear polarization

measurements.

5.4 White light polarimetric reconstructions in S1 and S2

An additional configuration for the polarimeter can be considered a new extension derived

from F. Snik et. al. [17]. Snik demonstrated that use of a QWR oriented at 45° in front of a

simplified channeled spectropolarimeter (CP) can be used to measure linear polarization (S0,

S1, and S2). For the DCPSI, the extension is clear by use of the Mueller matrix for a QWR at

45°,

,45

1 0 0 0

0 0 0 1

0 0 1 0

0 1 0 0

QWR °

− =

M (34)

Multiplication of this matrix by an arbitrary incident Stokes vector yields

#116682 - $15.00 USD Received 3 Sep 2009; revised 7 Oct 2009; accepted 19 Oct 2009; published 24 Nov 2009

(C) 2009 OSA 7 December 2009 / Vol. 17, No. 25 / OPTICS EXPRESS 22531

Page 13: White light Sagnac interferometer for snapshot linear polarimetric imaging

[ ] [ ],45 0 1 2 3 0 3 2 1.

T T

out QWRS S S S S S S S°= = −S M (35)

Therefore, the QWR converts any incident linear horizontal or vertical polarization states (S1)

into circular polarization (S3) and vice versa. Consequently, by including the QWR as part of

the DCPSI (depicted in Fig. 7 as part of the PG), the intensity pattern becomes

( ) ( )( ) ( )

( ) ( )

1 1

'

2/ /

'

0

0 1'

1

2 2cos

1 1, ,

2 2 2 2 sin

id d

obj

DCPSI i i

m m

i

obj

mS m a b c x

f dI x y S m

mS m a b c x

f d

λ λ

π

π= =

+ +

= +

− + +

∑ ∑ (36)

where S1’(m) is analogues to S3

’(m), and is defined as

( ) ( ) ( )2

1

' 2

1 1, .S m DE m S d

λ

λ

λ λ λ= ∫ (37)

Inverse Fourier transformation of channels C0 and C1, after extracting the channels from the

Fourier transformation of Eq. (36), yields

[ ] ( )'

01

0

1

2

SC

− =F (38)

[ ] ( ) ( )( ) ( )1 ' '

1 2 1

11 1 exp 2 ,

4DCPSI iC S jS j U xπ− = +F (39)

which again assumes that the m = 1 diffraction order is dominant. This is a straightforward

and valuable extension, primarily because a full linear polarization measurement, namely the

degree of linear polarization (DOLP) and its orientation, can be calculated by a single DCPSI.

The DOLP and orientation are defined as

2 2

1 2

0

S SDOLP

S

+= (40)

2

1

1a tan .

2

S

=

(41)

To verify reconstruction accuracy when measuring linear polarization states, the generating

LP was rotated in front of the DCPSI with the QWR oriented at 45°. These data are portrayed

in Fig. 11(b). The RMS error between the measured and theoretical Stokes parameters (S1 and

S2) are 1, 0.0221RMS Sε = and

2, 0.0262RMS Sε = . Again, residual error is likely due to the broad

bandwidth of the calibration.

5.5 Outdoor measurements

Outdoor measurements with the DCPSI were conducted to further demonstrate the feasibility

of the sensor implementation. First, a measurement was taken of the sky’s reflection viewed

off several large windows. These windows, present in larger buildings, are typically tempered.

This process can yield appreciable amounts of stress birefringence, and consequently they

serve as a source of elliptically or circularly polarized light. The raw, unprocessed image is

depicted in Fig. 12.

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Fig. 12. Raw image of several large windows viewing the sky in reflection through the DCPSI.

Fringes are observed where polarized light is present. The fringes change in phase and

amplitude due to the varying amounts of S2 and S3 over the surface of the window.

The Stokes parameters, in addition to the degree of circular polarization (DOCP), are depicted

in Fig. 13, where 3 0DOCP S S= .

Fig. 13. Processed polarization data of the large windows, calculated from the data in Fig. 12.

From the image in the first row and first column and moving clockwise: S0, DOCP, S2/S0, and

S3/S0.

Next, measurements of a moving vehicle taken in S1 and S2 were performed, requiring the

inclusion of the QWR, oriented at 45°, in front of the DCPSI. The raw, unprocessed image of

a vehicle is depicted in Fig. 14, while the processed Stokes parameters and DOLP are

portrayed in Fig. 15.

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Page 15: White light Sagnac interferometer for snapshot linear polarimetric imaging

Fig. 14. Raw image of a moving vehicle. The QWR is inserted in front of the DCPSI to

measure S1 and S2.

Fig. 15. Processed polarization data of the vehicle, calculated from the data in Fig. 14. From

the image in the first row and first column and moving clockwise: S0, DOLP, S2/S0, and S1/S0.

6. Conclusion

A PSI with diffraction gratings was demonstrated to produce an achromatic fringe field,

enabling white-light operation of a fringe polarimeter (FP). The measured RMS errors for S2

and S3 were 2, 0.0212RMS Sε = and

3, 0.0378RMS Sε = , respectively. Additionally, inclusion of a

QWR enables the sensor to measure the full linear polarization state of a scene. This

implementation resulted in S1 and S2 RMS errors of 1, 0.0221RMS Sε = and

2, 0.0262RMS Sε = ,

respectively. Consequently, a snapshot white-light FP is realizable by use of a DCPSI, thus

avoiding the narrow-bandwidth limitation associated with uniaxial crystal-based FPs.

#116682 - $15.00 USD Received 3 Sep 2009; revised 7 Oct 2009; accepted 19 Oct 2009; published 24 Nov 2009

(C) 2009 OSA 7 December 2009 / Vol. 17, No. 25 / OPTICS EXPRESS 22534