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

S(x,y)=[S0(x,y)S1(x,y)S2(x,y)S3(x,y)]=[I0(x,y)+I90(x,y)I0(x,y)I90(x,y)I45(x,y)I135(x,y)IR(x,y)IL(x,y)]
where 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 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 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

I(x,y)=12S0+12S1cos[2πΩ2y]+14|S23|cos[2π(Ω1+Ω2)xarg(S23)]14|S23|cos[2π(Ω1+Ω2)y+arg(S23)]
with
S23=S2+jS3,
where Ω1 and Ω2 are the spatial carrier frequencies generated by SP1 and SP2, 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
Ω1=Δ1λfandΩ2=Δ2λf,
where λ is the wavelength of the incident light, f is the focal length of the objective lens, and Δ1, Δ2 are proportional to the thickness of SP1 and SP2, respectively. The shearing distance (SSP) is related to the thickness of the SP (t) by
SSP=2Δ=2ne2no2ne2+no2t,
where ne, no are the extraordinary and ordinary refractive indices of the SP, respectively [8].

 

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 f obj. The combination of both Savart plates generates four sheared beams, separated by a distance 2Δ. 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.

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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

B=0.6λ2OPDmax,
where 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 .

 

Fig. 2 One configuration for a polarization Sagnac interferometer. The distance between the WGBS and mirrors, M1 and M2, is denoted d 1 and d 2, respectively. A shear (SPSI) is produced when d 1d 2. The case d 1 > d 2 is illustrated, with d 1 = d 2 + α.

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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 f obj. Two beams, with a shear SPSI, are created when d 1d 2. If the distance, d 1, is offset by an amount α, such that d 1 = d 2 + α, the shear is

SPSI=2α
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.The OPD between the two sheared beams exiting from a PSI or the simplified SPP is depicted in Fig. 4 , and is given by
OPD=Sshearsin(θ)Sshearθ,
where Sshear is the shear generated by the SPP or PSI, SSPP or SPSI, respectively [12].

 

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

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

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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

ISPP(xi,yi)=|12Ex(xi,yi,t)ejϕ1+12Ey(xi,yi,t)ejϕ2|2,
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
ISPP(xi,yi)=12{(ExEx*+ExEy*)+(ExEy*+Ex*Ey)cos(ϕ1ϕ2)+j(ExEy*+Ex*Ey)sin(ϕ1ϕ2)},
where Ex, Ey are now implicitly dependent upon xi and yi. The phase factors are
ϕ1=2πΔλfobjxiandϕ2=2πΔλfobjyi.
The Stokes parameters are defined from the components of the electric field as
[S0S1S2S3]=[ExEx*+EyEy*ExEx*EyEy*ExEy*+Ex*Eyj(ExEy*Ex*Ey)].
Re-expressing Eq. (10) by use of the Stokes parameter definitions and φ1, φ2, yields
ISPP(xi,yi)=12[S0+S2cos(2πλfobjΔ(xiyi))S3sin(2πλfobjΔ(xiyi))].
Consequently, the simplified SPP modulates S 2 and S 3 onto a carrier frequency, while S 0 remains as an un-modulated component. The carrier frequency, USPP, is
USPP=2Δλfobj.
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

ϕ1=2πλfobjα2xiandϕ2=2πλfobjα2xi.
Use of these expressions yields the intensity pattern for the PSI
IPSI(xi,yi)=12[S0+S2cos(2πλfobj2αxi)S3sin(2πλfobj2αxi)].
The carrier frequency, UPSI, is
UPSI=2αλfobj.
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. 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 Sshearλγ, where 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 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(θ)=mλd,
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 θmλ/d.

 

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.

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

SDCPSI=2mλd(a+b+c)
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
ϕ1=2πfobjmd(a+b+c)xiandϕ2=2πfobjmd(a+b+c)xi.
Using these phase factors, the intensity on the FPA is
IDCPSI(xi,yi)=12m=0d/λ1S0'(m)+12m=1d/λ1[S2'(m)cos(2πfobj2md(a+b+c)xi)S3'(m)sin(2πfobj2md(a+b+c)xi)]
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/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,
S0'(m)=λ1λ2DE2(λ,m)S0(λ)dλ
S2'(m)=λ1λ2DE2(λ,m)S2(λ)dλ
S3'(m)=λ1λ2DE2(λ,m)S3(λ)dλ
where λ 1 and λ 2 denote the minimum and maximum wavelengths passed by the optical system. The carrier frequency, UDCPSI, is
UDCPSI=2mdfobj(a+b+c).
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].

 

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.

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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.

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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].

 

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.

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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 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.

 

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.

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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

I(xi,yi)=A+Bcos(ωxi+ϕ),{xminxixmaxyi=y0}
where A, B, ω, and φ are the offset, amplitude, frequency and phase of the fringes, spanning xi between x min and x max, for a given yi = 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 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.

 

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.

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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

F[IDCPSI(xi,yi)]=S0'(1)2δ(ξ,η)+14(S2'(1)+jS3'(1))δ(ξUDCPSI,η)14(S2'(1)jS3'(1))δ(ξ+UDCPSI,η),
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
F1[C0]=S0'(1)2
F1[C1]=14(S2'(1)+jS3'(1))exp(j2πUDCPSIxi)
Therefore, the 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 isolateexp(j2πUDCPSIxi). The unknown sample data are divided by the measured reference data to solve for the incident Stokes parameters
S0,sample(xi,yi)=|F(C0,sample)|
S2,sample(xi,yi)=[F(C2,sample)F(C2,reference,45°)|F(C0,reference,45°)|S0,sample]
S3,sample(xi,yi)=[(C2,sample)F(C2,reference,45°)|F(C0,reference,45°)|S0,sample],
where and 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

εRMS,Si=1Nn=1N(Si,Meas(n)Si,PG(n))2,
where 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 εRMS,S2=0.0212 and εRMS,S3=0.0378. 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.

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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 (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°,

MQWR,45°=[1000000100100100]
Multiplication of this matrix by an arbitrary incident Stokes vector yields
Sout=MQWR,45°[S0S1S2S3]T=[S0S3S2S1]T.
Therefore, the QWR converts any incident linear horizontal or vertical polarization states (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
IDCPSI(xi,yi)=12m=0d/λ1S0'(m)+12m=1d/λ1[S2'(m)cos(2πfobj2md(a+b+c)xi)S1'(m)sin(2πfobj2md(a+b+c)xi)],
where S 1 (m) is analogues to S 3 (m), and is defined as
S1'(m)=λ1λ2DE2(λ,m)S1(λ)dλ.
Inverse Fourier transformation of channels C 0 and C 1, after extracting the channels from the Fourier transformation of Eq. (36), yields
F1[C0]=S0'(1)2
F1[C1]=14(S2'(1)+jS1'(1))exp(j2πUDCPSIxi),
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
DOLP=S12+S22S0
ϕ=12atan(S2S1).
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 εRMS,S1=0.0221 and εRMS,S2=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 , whereDOCP=|S3/S0|.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 .

 

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 S 2 and S 3 over the surface of the window.

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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: S 0, DOCP, S 2/S 0, and S 3/S 0.

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Fig. 14 Raw image of a moving vehicle. The QWR is inserted in front of the DCPSI to measure S 1 and S 2.

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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: S 0, DOLP, S 2/S 0, and S 1/S 0.

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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 εRMS,S2=0.0212 and εRMS,S3=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 εRMS,S1=0.0221 and εRMS,S2=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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9. G. Zhan, K. Oka, T. Ishigaki, and N. Baba, “Static Fourier-transform spectrometer based on Savart polariscope,” Proc. SPIE 4480, 198–203 ( 2002). [CrossRef]  

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.

13. J. C. Wyant, “OTF measurements with a white light source: an interferometric technique,” Appl. Opt. 14(7), 1613–1615 ( 1975). [CrossRef]   [PubMed]  

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). [CrossRef]   [PubMed]  

16. M. W. Kudenov, J. L. Pezzaniti, and G. R. Gerhart, “Microbolometer-infrared imaging Stokes polarimeter,” Opt. Eng. 48(6), 063201 ( 2009). [CrossRef]  

17. F. Snik, T. Karalidi, and C. U. Keller, “Spectral modulation for full linear polarimetry,” Appl. Opt. 48(7), 1337–1346 ( 2009). [CrossRef]   [PubMed]  

References

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  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).
    [CrossRef] [PubMed]
  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).
    [CrossRef] [PubMed]
  4. K. Oka and T. Kaneko, “Compact complete imaging polarimeter using birefringent wedge prisms,” Opt. Express 11(13), 1510–1519 (2003).
    [CrossRef] [PubMed]
  5. K. Oka and N. Saito, “Snapshot complete imaging polarimeter using Savart plates,” Proc. SPIE 6295, 629508 (2006).
    [CrossRef]
  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).
    [CrossRef] [PubMed]
  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).
    [CrossRef] [PubMed]
  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).
    [CrossRef]
  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.
  13. J. C. Wyant, “OTF measurements with a white light source: an interferometric technique,” Appl. Opt. 14(7), 1613–1615 (1975).
    [CrossRef] [PubMed]
  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).
    [CrossRef] [PubMed]
  16. M. W. Kudenov, J. L. Pezzaniti, and G. R. Gerhart, “Microbolometer-infrared imaging Stokes polarimeter,” Opt. Eng. 48(6), 063201 (2009).
    [CrossRef]
  17. F. Snik, T. Karalidi, and C. U. Keller, “Spectral modulation for full linear polarimetry,” Appl. Opt. 48(7), 1337–1346 (2009).
    [CrossRef] [PubMed]

2009 (2)

M. W. Kudenov, J. L. Pezzaniti, and G. R. Gerhart, “Microbolometer-infrared imaging Stokes polarimeter,” Opt. Eng. 48(6), 063201 (2009).
[CrossRef]

F. Snik, T. Karalidi, and C. U. Keller, “Spectral modulation for full linear polarimetry,” Appl. Opt. 48(7), 1337–1346 (2009).
[CrossRef] [PubMed]

2008 (2)

2006 (2)

2003 (1)

2002 (1)

G. Zhan, K. Oka, T. Ishigaki, and N. Baba, “Static Fourier-transform spectrometer based on Savart polariscope,” Proc. SPIE 4480, 198–203 (2002).
[CrossRef]

1999 (1)

1994 (1)

1975 (1)

Baba, N.

G. Zhan, K. Oka, T. Ishigaki, and N. Baba, “Static Fourier-transform spectrometer based on Savart polariscope,” Proc. SPIE 4480, 198–203 (2002).
[CrossRef]

Chenault, D. B.

DeHoog, E.

Dereniak, E. L.

Gerhart, G. R.

Goldstein, D. L.

Graf, U. U.

Ishigaki, T.

G. Zhan, K. Oka, T. Ishigaki, and N. Baba, “Static Fourier-transform spectrometer based on Savart polariscope,” Proc. SPIE 4480, 198–203 (2002).
[CrossRef]

Jaffe, D. T.

Kaneko, T.

Karalidi, T.

Kato, T.

Keller, C. U.

Kim, E. J.

Kudenov, M.

Kudenov, M. W.

Lacy, J. H.

Ling, H.

Luo, H.

Moore, J. T.

Oka, K.

Pezzaniti, J. L.

M. W. Kudenov, J. L. Pezzaniti, and G. R. Gerhart, “Microbolometer-infrared imaging Stokes polarimeter,” Opt. Eng. 48(6), 063201 (2009).
[CrossRef]

Pezzaniti, L.

Rebeiz, G.

Saito, N.

K. Oka and N. Saito, “Snapshot complete imaging polarimeter using Savart plates,” Proc. SPIE 6295, 629508 (2006).
[CrossRef]

Schiewgerling, J.

Shaw, J. A.

Snik, F.

Tyo, J. S.

Wyant, J. C.

Zhan, G.

G. Zhan, K. Oka, T. Ishigaki, and N. Baba, “Static Fourier-transform spectrometer based on Savart polariscope,” Proc. SPIE 4480, 198–203 (2002).
[CrossRef]

Appl. Opt. (5)

Opt. Eng. (1)

M. W. Kudenov, J. L. Pezzaniti, and G. R. Gerhart, “Microbolometer-infrared imaging Stokes polarimeter,” Opt. Eng. 48(6), 063201 (2009).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (2)

K. Oka and N. Saito, “Snapshot complete imaging polarimeter using Savart plates,” Proc. SPIE 6295, 629508 (2006).
[CrossRef]

G. Zhan, K. Oka, T. Ishigaki, and N. Baba, “Static Fourier-transform spectrometer based on Savart polariscope,” Proc. SPIE 4480, 198–203 (2002).
[CrossRef]

Other (6)

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).

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

R. S. Sirohi, and M. P. Kothiyal, Optical components, Systems, and Measurement Techniques, (Marcel Dekker, 1991), p. 68.

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.”

M. H. Smith, J. B. Woodruff, and J. D. Howe, “Beam wander considerations in imaging polarimetry,” in Proc. SPIE 3754, 50–54 (1999).

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

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Figures (15)

Fig. 1
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 f obj. The combination of both Savart plates generates four sheared beams, separated by a distance 2 Δ . 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.

Fig. 2
Fig. 2

One configuration for a polarization Sagnac interferometer. The distance between the WGBS and mirrors, M1 and M2, is denoted d 1 and d 2, respectively. A shear (SPSI ) is produced when d 1d 2. The case d 1 > d 2 is illustrated, with d 1 = d 2 + α.

Fig. 3
Fig. 3

A simplified SPP with a single Savart plate.

Fig. 4
Fig. 4

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

Fig. 5
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.

Fig. 6
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.

Fig. 7
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.

Fig. 8
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.

Fig. 9
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.

Fig. 10
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.

Fig. 11
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.

Fig. 12
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 S 2 and S 3 over the surface of the window.

Fig. 13
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: S 0, DOCP, S 2/S 0, and S 3/S 0.

Fig. 14
Fig. 14

Raw image of a moving vehicle. The QWR is inserted in front of the DCPSI to measure S 1 and S 2.

Fig. 15
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: S 0, DOLP, S 2/S 0, and S 1/S 0.

Equations (41)

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S ( x , y ) = [ S 0 ( x , y ) S 1 ( x , y ) S 2 ( x , y ) S 3 ( x , y ) ] = [ I 0 ( x , y ) + I 90 ( x , y ) I 0 ( x , y ) I 90 ( x , y ) I 45 ( x , y ) I 135 ( x , y ) I R ( x , y ) I L ( x , y ) ]
I ( x , y ) = 1 2 S 0 + 1 2 S 1 cos [ 2 π Ω 2 y ] + 1 4 | S 23 | cos [ 2 π ( Ω 1 + Ω 2 ) x arg ( S 23 ) ] 1 4 | S 23 | cos [ 2 π ( Ω 1 + Ω 2 ) y + arg ( S 23 ) ]
S 23 = S 2 + j S 3 ,
Ω 1 = Δ 1 λ f and Ω 2 = Δ 2 λ f ,
S S P = 2 Δ = 2 n e 2 n o 2 n e 2 + n o 2 t ,
B = 0.6 λ 2 O P D max ,
S P S I = 2 α
O P D = S s h e a r sin ( θ ) S s h e a r θ ,
I S P P ( x i , y i ) = | 1 2 E x ( x i , y i , t ) e j ϕ 1 + 1 2 E y ( x i , y i , t ) e j ϕ 2 | 2 ,
I S P P ( x i , y i ) = 1 2 { ( E x E x * + E x E y * ) + ( E x E y * + E x * E y ) cos ( ϕ 1 ϕ 2 ) + j ( E x E y * + E x * E y ) sin ( ϕ 1 ϕ 2 ) } ,
ϕ 1 = 2 π Δ λ f o b j x i and ϕ 2 = 2 π Δ λ f o b j y i .
[ S 0 S 1 S 2 S 3 ] = [ E x E x * + E y E y * E x E x * E y E y * E x E y * + E x * E y j ( E x E y * E x * E y ) ] .
I S P P ( x i , y i ) = 1 2 [ S 0 + S 2 cos ( 2 π λ f o b j Δ ( x i y i ) ) S 3 sin ( 2 π λ f o b j Δ ( x i y i ) ) ] .
U S P P = 2 Δ λ f o b j .
ϕ 1 = 2 π λ f o b j α 2 x i and ϕ 2 = 2 π λ f o b j α 2 x i .
I P S I ( x i , y i ) = 1 2 [ S 0 + S 2 cos ( 2 π λ f o b j 2 α x i ) S 3 sin ( 2 π λ f o b j 2 α x i ) ] .
U P S I = 2 α λ f o b j .
sin ( θ ) = m λ d ,
S D C P S I = 2 m λ d ( a + b + c )
ϕ 1 = 2 π f o b j m d ( a + b + c ) x i and ϕ 2 = 2 π f o b j m d ( a + b + c ) x i .
I D C P S I ( x i , y i ) = 1 2 m = 0 d / λ 1 S 0 ' ( m ) + 1 2 m = 1 d / λ 1 [ S 2 ' ( m ) cos ( 2 π f o b j 2 m d ( a + b + c ) x i ) S 3 ' ( m ) sin ( 2 π f o b j 2 m d ( a + b + c ) x i ) ]
S 0 ' ( m ) = λ 1 λ 2 D E 2 ( λ , m ) S 0 ( λ ) d λ
S 2 ' ( m ) = λ 1 λ 2 D E 2 ( λ , m ) S 2 ( λ ) d λ
S 3 ' ( m ) = λ 1 λ 2 D E 2 ( λ , m ) S 3 ( λ ) d λ
U D C P S I = 2 m d f o b j ( a + b + c ) .
I ( x i , y i ) = A + B cos ( ω x i + ϕ ) , { x min x i x max y i = y 0 }
F [ I D C P S I ( x i , y i ) ] = S 0 ' ( 1 ) 2 δ ( ξ , η ) + 1 4 ( S 2 ' ( 1 ) + j S 3 ' ( 1 ) ) δ ( ξ U D C P S I , η ) 1 4 ( S 2 ' ( 1 ) j S 3 ' ( 1 ) ) δ ( ξ + U D C P S I , η ) ,
F 1 [ C 0 ] = S 0 ' ( 1 ) 2
F 1 [ C 1 ] = 1 4 ( S 2 ' ( 1 ) + j S 3 ' ( 1 ) ) exp ( j 2 π U D C P S I x i )
S 0 , s a m p l e ( x i , y i ) = | F ( C 0 , s a m p l e ) |
S 2 , s a m p l e ( x i , y i ) = [ F ( C 2 , s a m p l e ) F ( C 2 , r e f e r e n c e , 45 ° ) | F ( C 0 , r e f e r e n c e , 45 ° ) | S 0 , s a m p l e ]
S 3 , s a m p l e ( x i , y i ) = [ ( C 2 , s a m p l e ) F ( C 2 , r e f e r e n c e , 45 ° ) | F ( C 0 , r e f e r e n c e , 45 ° ) | S 0 , s a m p l e ] ,
ε R M S , S i = 1 N n = 1 N ( S i , M e a s ( n ) S i , P G ( n ) ) 2 ,
M Q W R , 45 ° = [ 1 0 0 0 0 0 0 1 0 0 1 0 0 1 0 0 ]
S o u t = M Q W R , 45 ° [ S 0 S 1 S 2 S 3 ] T = [ S 0 S 3 S 2 S 1 ] T .
I D C P S I ( x i , y i ) = 1 2 m = 0 d / λ 1 S 0 ' ( m ) + 1 2 m = 1 d / λ 1 [ S 2 ' ( m ) cos ( 2 π f o b j 2 m d ( a + b + c ) x i ) S 1 ' ( m ) sin ( 2 π f o b j 2 m d ( a + b + c ) x i ) ] ,
S 1 ' ( m ) = λ 1 λ 2 D E 2 ( λ , m ) S 1 ( λ ) d λ .
F 1 [ C 0 ] = S 0 ' ( 1 ) 2
F 1 [ C 1 ] = 1 4 ( S 2 ' ( 1 ) + j S 1 ' ( 1 ) ) exp ( j 2 π U D C P S I x i ) ,
D O L P = S 1 2 + S 2 2 S 0
ϕ = 1 2 a tan ( S 2 S 1 ) .

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