## 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 *S*
_{2} and *S*
_{3}, or the *S*
_{1} and *S*
_{2} Stokes parameters in white-light.

© 2009 OSA

## 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

*x*,

*y*are spatial coordinates in the scene,

*S*

_{0}is the total energy of the beam,

*S*

_{1}denotes preference for linear 0° over 90°,

*S*

_{2}for linear 45° over 135°, and

*S*

_{3}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 *μ*m) [6].

In this paper, we outline the theoretical and experimental development of a dispersion-compensated polarization Sagnac interferometer (DCPSI), enabling white-light operation of 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 SP_{1} and SP_{2}, generate a lateral shear ±45° to the *x*-axis, respectively. The input beam, after transmission through SP_{1}, 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 SP_{1} by 45°. Transmission through SP_{2} 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

_{1}and Ω

_{2}are the spatial carrier frequencies generated by SP

_{1}and SP

_{2}, respectively, and the Stokes parameters

*S*

_{0},

*S*

_{1},

*S*

_{2}, and

*S*

_{3}are implicitly dependent upon

*x*and

*y*. Consequently, the Stokes parameters

*S*

_{1}through

*S*

_{3}are amplitude modulated onto several carrier frequencies, while

*S*

_{0}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

*λ*is the wavelength of the incident light,

*f*is the focal length of the objective lens, and Δ

_{1}, Δ

_{2}are proportional to the thickness of SP

_{1}and SP

_{2}, respectively. The shearing distance (

*S*) is related to the thickness of the SP (

_{SP}*t*) bywhere

*n*,

_{e}*n*are the extraordinary and ordinary refractive indices of the SP, respectively [8].

_{o}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. 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

*OPD*

_{max}is the peak optical path difference in the system. For an

*OPD*

_{max}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.

## 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 .

The PSI consists of two mirrors, M_{1} and M_{2}, with a wire grid polarization beamsplitter (WGBS), a focal plane array (FPA), and an objective lens with focal length *f*
_{obj}. Two beams, with a shear *S _{PSI}*, are created when

*d*

_{1}≠

*d*

_{2}. If the distance,

*d*

_{1}, is offset by an amount

*α*, such that

*d*

_{1}=

*d*

_{2}+

*α*, the shear is

*x*-axis.The OPD between the two sheared beams exiting from a PSI or the simplified SPP is depicted in Fig. 4 , and is given bywhere

*S*is the shear generated by the SPP or PSI,

_{shear}*S*or

_{SPP}*S*, respectively [12].

_{PSI}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

*x*and

_{i}*y*are image-plane coordinates, and

_{i}*φ*,

_{1}*φ*, are the cumulative phases of each ray. Expansion of this expression yields

_{2}*E*,

_{x}*E*are now implicitly dependent upon

_{y}*x*and

_{i}*y*. The phase factors are

_{i}*φ*,

_{1}*φ*, yields

_{2}*S*

_{2}and

*S*

_{3}onto a carrier frequency, while

*S*

_{0}remains as an un-modulated component. The carrier frequency,

*U*, isFourier 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.

_{SPP}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, *φ _{1}* and

*φ*, which now depend on the interferometer’s beamsplitter to mirror separation difference,

_{2}*α*, in addition to an adjustment in the shear direction, now along +/−

*x*. The phase factors for the PSI are

_{p}*U*, isTherefore, 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. 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 ${S}_{shear}\propto \lambda \gamma $, where

_{PSI}*γ*is some optical thickness, analogous to

*α*or Δ.

## 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 *d _{1}* =

*d*and the inclusion of two identical gratings, G

_{2}_{1}and G

_{2}. A ray’s diffraction angle, after transmission through G

_{1}or G

_{2}, is calculated for normal incidence by

*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

*μ*m), small angle approximations can be used to simplify Eq. (18), yielding $\theta \approx m\lambda /d$.

In the DCPSI, the beam transmitted by the WGBS (now spectrally broadband) is diffracted by G_{1} into the 1 order. When these dispersed rays are incident on G_{2}, the diffraction angle, induced previously by G_{1}, is removed. The rays emerge parallel to the optical axis, but are offset by a distance proportional to -*λx _{o}*, where

*x*is some constant related to the DCPSI’s parameters. Conversely, the beam reflected by the WGBS is initially diffracted by G

_{o}_{2}. The dispersed rays are then diffracted to be parallel to the optical axis by G

_{1}, and exit the system offset by a distance proportional to +

*λx*. The functional form of the shear can be calculated by unfolding the optical layout of the DCPSI, as depicted in Fig. 6 .Assuming small angles and that G

_{o}_{1}and G

_{2}have an identical period, then the shear,

*S*, is

_{DCPSI}*a*,

*b*, and

*c*represent the distances between G

_{1}and M

_{1}, M

_{1}and M

_{2}, and M

_{2}and G

_{2}, respectively. Hence, the DCPSI can, to first order, generate a shear that is directly proportional to the wavelength. This makes the phase factors

*d*/

*λ*

_{1})sin(π/2), where

*λ*

_{1}is the minimum wavelength passed by the optical system and

*S*

_{0}

^{’}(

*m*),

*S*

_{2}

^{’}(

*m*), and

*S*

_{3}

^{’}(

*m*) are the Stokes parameters weighted by the diffraction efficiency (DE) of both gratings before integration over wavelength,

*λ*

_{1}and

*λ*

_{2}denote the minimum and maximum wavelengths passed by the optical system. The carrier frequency,

*U*, isTherefore, 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].

_{DCPSI}## 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 *θ _{G}*, 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 M

_{1}and M

_{2}are 25 mm diameter 1/10th wave optical flats. A collimating lens, with focal length

*f*= 50 mm (F/11), is used to image the object plane to infinity, while an objective lens, with focal length

_{c}*f*= 200 mm (F/4.5), is used for re-imaging. This gives a field of view of approximately +/− 1°.

_{obj}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 (*τ _{IR}*) and crossed polarizers (

*τ*), in addition to their combination (

_{CP}*τ*), is depicted in Fig. 8(a) . Lastly, the ruled diffraction gratings, G

_{CP}τ_{IR}_{1}and G

_{2}, are blazed for a first order (

*m*= 1) wavelength of

*λ*= 640 nm on a BK7 substrate with a period

_{B}*d*= 28.6

*μ*m. 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, *U _{DCPSI}*, 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 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 [

*S*

_{0},

*S*

_{1},

*S*

_{2},

*S*

_{3}]

^{T}= [1,0,1,0]

^{T}.

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

*A*,

*B*,

*ω*, and

*φ*are the offset, amplitude, frequency and phase of the fringes, spanning

*x*between

_{i}*x*

_{min}and

*x*

_{max}, for a given

*y*=

_{i}*y*

_{0}close to the optical axis.

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 *U _{PSI}* =

*U*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

_{DCPSI}^{−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.

#### 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

*δ*is the dirac-delta function and

*ξ*,

*η*are the Fourier transform variables along

*x*and

_{i}*y*, 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

_{i}*δ*(

*ξ*,

*η*) (channel C

_{0}) and

*δ*(

*ξ*-

*U*,

_{DCPSI}*η*) (channel C

_{1}) components yields

*S*

_{0}

^{’}Stokes parameter is obtained directly, while (

*S*

_{2}

^{’}(1) +

*jS*

_{3}

^{’}(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$\mathrm{exp}\left(j2\pi {U}_{DCPSI}{x}_{i}\right)$. The unknown sample data are divided by the measured reference data to solve for the incident Stokes parameters

*ℜ*and

*ℑ*indicate that either the real or imaginary parts of the expression have been taken.

#### 5.3 White light polarimetric reconstructions in S_{2} and S_{3}

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

*i*is an integer (1, 2, or 3) that indicates the Stokes parameter being analyzed. The calculated RMS errors for

*S*

_{2}and

*S*

_{3}for the data in Fig. 11 (a) are ${\epsilon}_{RMS,{S}_{2}}=0.0212$ and ${\epsilon}_{RMS,{S}_{3}}=0.0378$. Residual errors can likely be attributed to the broad bandwidth that the calibration covers [16].

#### 5.4 White light polarimetric reconstructions in S_{1} and S_{2}

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 (*S*
_{0}, *S*
_{1}, and *S*
_{2}). For the DCPSI, the extension is clear by use of the Mueller matrix for a QWR at 45°,

*S*

_{1}) into circular polarization (

*S*

_{3}) 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

*S*

_{1}

^{’}(

*m*) is analogues to

*S*

_{3}

*(*

^{’}*m*), and is defined as

*C*

_{0}and

*C*

_{1}, after extracting the channels from the Fourier transformation of Eq. (36), yields

*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 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 (

*S*

_{1}and

*S*

_{2}) are ${\epsilon}_{RMS,{S}_{1}}=0.0221$ and ${\epsilon}_{RMS,{S}_{2}}=0.0262$. 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
.The Stokes parameters, in addition to the degree of circular polarization (DOCP), are depicted in Fig. 13
, where$DOCP=\left|{S}_{3}^{}/{S}_{0}^{}\right|$.Next, measurements of a moving vehicle taken in *S*
_{1} and *S*
_{2} 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
.

## 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 *S*
_{2} and *S*
_{3} were ${\epsilon}_{RMS,{S}_{2}}=0.0212$ and ${\epsilon}_{RMS,{S}_{3}}=0.0378$, respectively. Additionally, inclusion of a QWR enables the sensor to measure the full linear polarization state of a scene. This implementation resulted in *S*
_{1} and *S*
_{2} RMS errors of ${\epsilon}_{RMS,{S}_{1}}=0.0221$ and ${\epsilon}_{RMS,{S}_{2}}=0.0262$, 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.

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