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Polarimetry has widespread applications within atmospheric sensing, telecommunications, biomedical imaging, and target detection. Several existing methods of imaging polarimetry trade off the sensor’s spatial resolution for polarimetric resolution, and often have some form of spatial registration error. To mitigate these issues, we have developed a system using oriented polymer-based organic photovoltaics (OPVs) that can preferentially absorb linearly polarized light. Additionally, the OPV cells can be made semitransparent, enabling multiple detectors to be cascaded along the same optical axis. Since each device performs a partial polarization measurement of the same incident beam, high temporal resolution is maintained with the potential for inherent spatial registration. In this paper, a Mueller matrix model of the stacked OPV design is provided. Based on this model, a calibration technique is developed and presented. This calibration technique and model are validated with experimental data, taken with a cascaded three cell OPV Stokes polarimeter, capable of measuring incident linear polarization states. Our results indicate polarization measurement error of 1.2% RMS and an average absolute radiometric accuracy of 2.2% for the demonstrated polarimeter.
© 2016 Optical Society of America
1. Introduction
Polarimetry has applications in many disciplines, such as ellipsometry [1], remote sensing [2], atmospheric sensing [3], target identification [4], manufacturing and quality control [5], telecommunications [6], and biomedical imaging [7]. Current methods of imaging polarimetry, such as channeled (CH) [8] and division of focal plane (DoF) polarimeters [9], usually limit spatial resolution, while others, like division of aperture (DoA) or division of amplitude polarimeters (DoAM), have higher complexity and size [10–12]. Compactness is realized in CH and DoF polarimeters by using a spatially modulated birefringence pattern or a focal plane array-mounted polarization mask, respectively [9, 13]. A significant drawback for both techniques relates to the lateral spatial multiplexing of the intensity measurements, which limits the maximum lateral spatial resolution. To eliminate this drawback, Azzam proposed a design that employs a cascaded array of detectors, where each preceding detector was positioned to reflect unabsorbed light towards a proceeding detector [14]. While this approach eliminated lateral multiplexing by cascading the measurements longitudinally along the optical axis, light must enter the polarimeter across a narrow range of angles. Due to this, and other geometric constraints, Azzam’s embodiment would be impractical for forming dense imaging arrays.
In this paper, an optoelectronic approach to implement what we refer to as an intrinsic coincident polarimeter (ICP) is described. The polarimeter is based on semitransparent, strain-aligned polymer semiconductor-based organic photovoltaics (OPVs). Unlike previous intrinsically polarization sensitive detectors [15, 16] our approach implements semi-transparent devices by exploiting oriented polymer semiconductor films [17, 18]. Through this semi-transparency, the issue of instantaneous and spatially-coincident spatial sampling is resolved in a device that has the potential of becoming monolithic. Experimentally, 3 semi-transparent OPVs were cascaded and aligned, such that each OPV can measure a different, known linear combination of the first three Stokes parameters. Based on the OPVs’ outputs and these known combinations, the first three (linear) Stokes parameters can be computed. This paper is organized as follows: the OPV’s fabrication techniques are overviewed in Section 2, followed by the polarimeter’s model in Section 3. Sections 4 and 5 detailed the radiometric calibration and experimental setup of the free-space polarimeter. In Section 6, the model validation procedure is detailed, and a few concluding remarks are provided in Section 7.
2. Detector considerations and fabrication
In polymer semiconductors, the primary optical transition dipole moment is typically aligned along the polymer backbone [19, 20]. This unique property results in anisotropic absorption when the polymer backbone is preferentially oriented along one axis in the plane of the film, as illustrated in Fig. 1(a) and Fig. 2. This property has been exploited to fabricate polarization sensitive organic photovoltaic (OPV) devices [17, 21], which consist of polymer-fullerene bulk-heterojunction (BHJ) active layers. In the demonstration by Awartani et al. [17], the polarization sensitivity was achieved by employing a strain-alignment approach that allows for fine control of the level of polymer alignment and thus the magnitude of the polarization response. Applying uniaxial strain to the P3HT:PCBM [poly(3-hexylthiophene):Phenyl-C61-butryc acid methyl ester] active layer, plastically deforms the film and the polymer backbones align to the direction of strain, resulting in a long-range oriented polymer film. When the polymer is aligned in plane, the devices exhibit larger opto-electronic response (higher photocurrent) when incident light is linearly polarized parallel to the strain’s direction. The ability to have a well-defined polarization sensitivity is important to optimize the polarimeter’s condition number [22]; thus, we employed this strain alignment approach to fabricate our polarization sensitive OPV devices. In addition to polarization sensitivity, the devices are made semitransparent, which allows for a vertically stacked OPV design enabling coincident detection [23].
Fig. 1 A cross-sectional view of the OPV device structure with strain alignment direction indicated. (b) Plot of transmission versus wavelength, for the device under linear polarization states that is polarized either parallel (Para,) or perpendicular (Perp, ) to the strain direction. The transmittance of the gold electrode with MoOx layer (Elect.) is also provided. (c) The I-V curve of the device in (a) under polarized illumination with broad spectrum light at approximately 50 mW/cm2. This is compared to an unstrained OPV cell (O).
Fig. 2 A schematic representation of the polarimeter model configured to measure a laser’s polarization state. The stack consists of strain orient OPV cells oriented at 0° (OPV1), and 45° (OPV2), follows by an unstrained active layer device (OPV3). Inset (a) shows the strain-aligned structure of the polymer chains, which result in preferential absorption of light at different states of polarization, while (b) depicts the isotropic distribution in an unstrained photoconductive film.
The OPV cells were fabricated using a bottom-up approach on a patterned indium tin oxide (ITO) coated glass substrate. The process began by spin casting an ethoxylated polyethylinimine (PEIE) solution onto ITO. The PEIE decreases the work function of ITO [22], allowing for an inverted OPV configuration where the ITO serves as the cathode. After spin casting PEIE, the substrates were thermally annealed at 100 °C for 10 minutes. The strain aligned P3HT:PCBM photoactive layer is then transfer printed onto the PEIE layer. This process has been previously described in detail [17]. The film is then thermally annealed at 135 °C for 10 minutes. Finally a semitransparent 10-nm Au film was deposited onto the P3HT:PCBM layer resulting in an active detector area of 0.25 cm2. Beyond this, the semitransparent OPV devices were also encapsulated, to increase lifetime, by drop casting a UV curable epoxy on the cell (Epo-Tek OG142-87) followed by attaching a glass cover slip and curing with UV light.
The magnitude of the responsivity anisotropy has previously been shown to increase with the magnitude of the applied strain due to the increasing alignment of the polymer backbones [17, 20]. A detailed analysis of the polarized OPV cell performance has shown that the internal quantum efficiency of the device remains similar to an unstrained device and is independent of the incident polarization state [24]. Thus the strained devices have a similar efficiency of collecting photogenerated carriers as an archetypal processed counterpart and the anisotropic performance is attributed primarily to the difference in light absorption in the cell. In the devices demonstrated here, the P3HT:PCBM films are strained by 30%. The diattenuation is observed by comparing the OPV’s transmittance under orthogonal linear polarizations, depicted in Fig. 1(b). The greatest transmittance is found for incident light perpendicular to the strain-alignment direction, indicative of weaker absorption of the P3HT:PCBM layer. Meanwhile, anisotropy is also observed in the photogenerated current. The current-voltage characteristics of a 30% strain-aligned and unstrained device, under illumination by polarized light parallel and perpendicular to the strain-alignment axis, is given in Fig. 1(c). It should be noted that unstrained devices are independent of the linear polarization orientation.
3. Polarimetric model
The ICP is modeled as three semitransparent detectors cascaded along the same optical axis. As illustrated in Fig. 2, two devices (OPV1 and OPV2) are made polarization sensitive while the third (OPV3) is unstrained (polarization insensitive).
OPV1 and OPV2 were aligned with their strain axes at 0° and 45° to the x-axis, respectively. It should be mentioned that this configuration was chosen by minimizing the polarimeter’s condition number, based on the model that will be described in this section.
The polarimeter was modeled using Mueller calculus. Due to their behavior as a weak polarizer, and the inherent retardance within the strained polymer layer, the OPVs were modeled as a diattenuator in series with a parallel retarder. The polarization state of light, transmitted through a single OPV cell, can be described by
where ST and SI are the Stokes vectors of the transmitted and incident light, respectively, MD and MR are Mueller matrices for a diattenuator and retarder, respectively, and R is the rotation matrix.A diattenuator’s Mueller matrix can be expressed as
where the diattenuation DT and variable ET are defined as, such that Tx and Ty are the transmittances of the x and y eigenvectors, respectively. Meanwhile, the rotation matrix is defined aswhere θ is the orientation angle measured relative to the x-axis. Meanwhile, the Mueller matrix of a general retarder is defined aswhere ϕ is the retardance.While Eq. (1) can be used to calculate the OPV’s transmitted polarization state, it must be modified to relate it to the polarization-induced photogenerated current. The Mueller matrix for absorption in the OPV was calculated per conservation of energy such that
where T, A, and R is the OPV’s transmittance, absorptance, and reflectance, respectively. For the purposes of our model, we assumed that the cell’s reflectance R = 0. Ultimately, the reflectance was measured to be close to the expected 4 percent for a standard air-glass interface. However, reflected light did not show polarization dependencies at near normal incidence; therefore, while this lost energy influences the overall magnitude of the transmitted or absorbed power, it does not influence the polarization properties. Thus, R is inherently included in our radiometric calibration and there is no need to model it polarimetrically. With this assumption, the absorption Mueller matrices of each OPV were modeled per Eq. (2), where Eqs. (3) and (4) were modified such that where Ax and Ay are the absorptance of the x and y eigenvectors, respectively, and are given as The Mueller matrix of an OPV can be expressed as where MT and MA are the OPV’s transmission and absorption Mueller matrices, respectively. For the ICP based on three OPVs, the Mueller matrix for each detector can be calculated by considering its absorption properties in combination with the preceding cells’ transmission matrices as where i is an integer (1-3) that denotes the ith OPV. Based on the aforementioned Mueller matrices, the polarimeter’s measurement matrix (W) can be computed from the OPVs’ analyzer vectors (i.e. the first rows of MOPV1, MOPV2, and MOPV3) [25]. It should be noted that the OPVs’ absorption Mueller matrices are independent of retardance, since the diattenuation and retardance eigenvectors are parallel. However, subsequent ‘down-stream’ cells are not parallel and thus the cells’ retardance must be included in transmission. A detailed discussion on W is provided in the following section.4. Radiometric and polarometric calibration
The primary goal of radiometric calibration is to ensure that all of the polarimeter’s detectors produce the same electrical output given the same optical input. This also implies linearizing the detector’s responsivity. Most inorganic photodetectors, at normal incidence, respond only to the incident light’s S0 component, such that their electrical response is directly proportional to its total intensity [26]. This allows the radiometric and polarimetric calibration procedures to be performed separately on these devices, greatly simplifying their characterization and calibration. Conversely, for OPV detectors, it is not possible to physically separate their responsivity and polarimetric response functions, due to their intrinsic polarization sensitivity. For this case, the radiometric and polarimetric calibrations must be conducted simultaneously. Based on measurements of our OPVs, we modeled a linear responsivity. A depiction of the electrical current versus incident optical power, in response to one incident polarization state, is illustrated in Fig. 3(a)-3(c) for OPV1, OPV2, and OPV3. Provided also is the coefficient of determination, which in the worst case (OPV2) is 0.995.
Fig. 3 Responsivity for one incident polarization state for (a) OPV1, (b) OPV2, and (c) OPV3 when in devices are in the cascaded configuration illustrated in Fig. 2. The coefficient of determination, R2, for the linear fit is provided for each OPV.
The OPV’s electrical current was modeled using the responsivity as
where P is the photogenerated electrical current, Φ is the incident flux (watts), η is the responsivity (amps/watt), and β is the electrical (or radiometric) offset. Since we are using visible light and dark subtraction, the contribution from the offset β is zero. Thus, to calibrate the system, we used a linear operator model [25] that included an additional term for the responsivity, such thatwhere P is a vector of electrical current measurements (one from each OPV), η is a matrix of responsivities, W is the measurement matrix, and Sin is the incident Stokes vector. The input Stokes vectors can be calculated from any power matrix P by,Thus, calibration focuses on quantifying the measurement matrix , where for our 3-cell linear polarimeter
where η1, η2, and η3 are the responsivities for OPV1, OPV2, and OPV3 while the coefficients of W denote the linear analyzer vectors, modeled from Eqs. (14)-(16). To experimentally measure the ICP was illuminated with Q known Stokes vectors. The measured current, produced by each OPV, was used to determine one row of . Thus, the calibration procedure can be described bywhere S0,q, S1,q, S2,q, and S3,q are Stokes parameters of calibration light incident into the system and the subscript j is the row number of W from which the coefficients wj1, wj2, and wj3 were extracted. It is to be noted that the different polarization states of light, which were used to illuminate the system, consisted of linear states and that the S3 component was always zero.5. Calibration measurement setup
Figure 4 depicts the experimental setup. A linearly polarized laser diode from Thorlabs (DJ532-40), with a nominal lasing wavelength of 532 nm, was used as the light source. Laser light, polarized parallel to the x-axis, first strikes an uncoated glass window, which was introduced into the system to reflect approximately 8% of the incident light into an integrating sphere and radiometer. This measurement ensured that all fluctuations of the incident laser power were recorded, and in later stages, corrected during data processing. Meanwhile, transmitted light propagated to a Glan-Thompson clean-up polarizer (LP1) with a transmission axis parallel to the x-axis. A rotatable half wave plate (HWP) followed the polarizer, enabling the generation of known linear polarization states for calibration and validation. After the waveplate, light then transmitted through the three OPV cells. Alignment of the OPV’s diattenuation axis, with respect to the x-axis, is identical to that previously described in Fig. 2. The laser spot size is within the OPV cell area, and the detector photocurrent is constant with changes of the laser position within the cell area.
Fig. 4 (a) Schematic of the experimental setup. Acronyms: LP1 is a linear polarizer at 0° to the x-axis; HWP is a rotatable half wave plate; OPV1-3 are the organic photovoltaic detectors, arranged as: OPV1 at 0°, OPV2 at 45°, and OPV3 (unstrained). (b) Image of free-space polarimeter [18].
To acquire calibration and validation data, the system was illuminated with three different laser powers: 5.24 mW (P1), 5.84 mW (P2), and 6.81 mW (P3). For each incident laser power, the HWP was rotated from 0° to 90° in 5° increments, thereby generating a total of Q = 57 unique Stokes vectors. Under each illumination state, the OPV cells generated a photocurrent and the I-V characteristics of each OPV was acquired using a Semiconductor Parameter Analyzer (SPA). Photocurrents were extracted from the I-V curves at a bias voltage of −0.4 V, which were subsequently used for calibration and validation. Figure 5 depicts the measured electrical output power of the three OPVs at different laser powers and incident polarization states.
Fig. 5 Current versus HWP orientation from (a) OPV1, (b) OPV2, and (c) OPV3, as measured by the SPA at the three optical powers P1 (5.24 mW), P2 (5.84 mW), and P3 (6.81 mW) as compared to the current calculated from the fitting procedure (P1F, P2F, and P3F).
To fit the data, a Nelder-Mead minimization function was used to determine the elements of W and the responsivities η, such that the least square difference, provided by the best coefficients, was minimized. Figure 5 depicts the fitted results (P1F, P2F, and P3F) as compared to the values measured by the SPA (P1, P2, and P3). The elements of were determined as
These results were calculated by directly fitting the coefficients in W and η directly to the data. Consequently, this matrix was not derived by fitting the various parameters, known or otherwise, to the polarimeter model established previously in Eqs. (14)-(16). We will demonstrate this fitting method in the next section.
6. Model validation
After determining the polarimetric and radiometric calibration, validation experiments were performed. It should be noted that of the measured Stokes vectors, 10% were chosen at random for validation while 90% were used to compute the measurement matrix per Eq. (22). Table 1 provides the data that was used to validate the polarimeter’s accuracy. Provided is the laser power (PL) at the input of LP1 and the orientation of LP1’s transmission axis. Additionally, the power of the calibrated S0 component is provided and compared to the value measured from the radiometer after LP1. Finally, the RMS error is provided between the normalized S1/S0 and S2/S0 components, calculated as
where subscript T and M indicate theoretical or measured values, respectively. From these results, the average absolute percent error in S0 is 2.2% while the RMS error of the normalized Stokes parameters is 1.2%. Likely sources of remaining error include alignment error in LP1 (estimated at +/− 0.25 degrees), noise, and our assumption that there is no S3 in the system.
Table 1. Theoretical and measured Stokes vectors. Laser power (PL) is measured at the input of LP1. Orientation angle of LP1 is given, along with percentage errors on the measured S0 component and RMS error of the normalized Stokes parameters.
To determine how S3 may interfere with the linear measurements, a parameterized model based on Eqs. (14)-(16) was fit to the data. This allowed the measurement matrix to be determined that has dimensions , instead of per our direct fit. Before fitting, each cell had its maximum transmittance measured separately (e.g., Tx = Ix/Iin, where Ix and Iin are the transmitted and incident x polarized optical powers, respectively) and transmission axes aligned to LP1. With these a priori inputs, the fitting algorithm was constrained such that the remaining parameters included: Ty1, Ty2, η1, η2, η3, ϕ1, and ϕ2. Final values of the aforementioned variables, obtained upon completion of the fitting procedure, are tabulated in Table 2.
Table 2. Final values of the fitting variables obtained using the parametric fit.
The measurement matrix, determined by fitting these parameters to the SPA’s measured current per Fig. 5, was calculated to be
The RMS error between the measurement matrix, as determined by the previous fitting procedure, was calculated and compared to our theoretical model. The RMS error was calculated bywhere w represents coefficients of W in Eq. (20), denotes coefficients of W in Eq. (24), and subscripts j and i are integers for the row and column indicies, respectively. Since Eq. (20) was assumed a linear polarimeter, its last column was set to zero during the RMS error calculation (e.g., w14 = w24 = w34 = 0). Performing this calculation yields an RMS error of 1.4%. Thus, it was concluded that the polarimetric model was providing an accurate representation of the OPV cells. Electrical output powers, simulated using Eq. (24), are compared to the measured data in Fig. 6. The calculated RMS error between the data and fit is 0.00266. From Eq. (24), it is also notable that there is a small (average 3%) S3 response in the second and third OPVs. This is due to the stress birefringence contained within the strained polymer film. Ultimately, to resolve issues with this residual retardance, orthogonal (but equal in magnitude) film retarders would need to be inserted in between the detectors. Alternatively, a four-detector polarimeter would be needed to fully resolve the matrix W and remove ambiguities associated with circularly polarized light propagating between the OPV cells.Fig. 6 The measured current versus HWP orientation angle, for incident optical powers P1 (5.24 mW), P2 (5.84 mW), and P3 (6.81 mW) for (a) OPV1, (b) OPV2, and (c) OPV3. In this case, the fitted optical powers P1F, P2F, and P3F are calculated using the parametric model.
7. Conclusion
The design, calibration, and experimental validation of an intrinsic coincident linear polarimeter was presented. Coincident measurement of the three optical powers, needed to calculate a linear Stokes vector, was demonstrated by the use of semi-transparent polarization sensitive OPVs. High temporal resolution is feasible due to the simultaneous data acquisition from all the OPVs. Calibration and validation processes, conducted with different intensities of input light, demonstrated that the aforementioned calibration procedure can account for both radiometric and polarimetric responses. It has been concluded, with experimental validation, that this new polarimeter design can measure incident Stokes vectors with an average error of 1.2% in the normalized Stokes parameters, and to within 2.2% for absolute radiometric power in S0. The demonstrated intrinsic coincident polarimeter could be made into an imaging array by fabricating the detectors as a monolithic stack and integrating them into a sensor array, similar to previous demonstrations of organic image sensors [27, 28].
Acknowledgments
This material is based on work supported by the National Science Foundation under Grant No. ECCS-1407885.
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