3.1.

Random Error

First, we consider the dark noise (${\sigma }_{\text{dark}}$) generation in the polarization camera. The dark current is the technical term that defines the generation of electrons within the CCD even without absorption of photons. The magnitude of the dark current is closely associated with the operating temperature of the CCD and exposure durations. The magnitude of the dark current can be minimized by lowering the operating temperature of the CCD using a thermoelectric cooler although this feature is not available in the polarization camera described in this paper. Also, it is important to note that the CCD will not generate the same level of dark current in equivalent periods of time and this phenomenon gives rise to a dark noise. If not for the dark noise, the digital counts associated with the dark current can be easily removed from the raw image by subtracting a dark image taken with the camera shutter closed and with an exposure time matching the raw image. However, the dark noise, meaning the fluctuations in the amount of dark current for equivalent periods of time, has to be quantified to determine its contribution toward the error margins in the digital counts registered by the CCD. To quantify the dark noise, we followed the following steps. We took 10 dark images for durations of 1 to 12 s in 1-s interval. Then using the 10 dark images for each duration of exposure, the average dark counts over 978,120 pixels ($988\times 990\text{pixels}$) for each of the four polarization states on the CCD were calculated. For example, Table 1 shows the average dark counts registered by 978,120 pixels in 10 images of 1 s of exposure in each of the four states of polarization on the CCD. The four states of polarization are named P000, P045, P090, and P135 in the columns and the rows marked 1 to 10 represent the average counts registered by all the pixels in each state of polarization in the 10 sample dark images. In each column, it is apparent that the average dark counts over the 10 frames fluctuate for all four states of polarization. The penultimate row in Table 1 shows the mean of the average dark counts from the 10 images for each of the polarization states and the last row shows the standard deviation of the mean, which is used as a measure of the dark noise for that polarization state. Data pertaining to Table 1 were created for additional exposure times that lasted 2 to 12 s in intervals of 1 s and Table 2 shows the standard deviation or the dark noise determined for each of the other exposure time periods.

Table 1

Average dark counts registered by 10 frames of 1 s exposure by 978,120 pixels that represent each of the four states of the polarization in the CCD. The four states of polarization are named P000, P045, P090, and P135 in the columns and the 10 different exposure frames are listed along the rows. Similar tables were created for exposures lasting 1 to 12 s in 1-s interval with the camera shutter closed.

Table 2

Dark noise error (σdark) in terms of digital counts associated with the dark current measured by each of the four polarization states in the CCD. The four polarizations states composed of 978,120 pixels and are named P000, P045, P090, and P135 along the columns and the rows reflect the exposure times in seconds.

In Fig. 3, we have plotted the data from Table 2 against the related exposure times of the dark images for the four states of polarization. It is apparent from Fig. 3 that the dark noise counts fluctuate with exposure times in a sawtooth shaped pattern between dark noise counts levels of 10 and 45. It is also apparent that pixels in the CCD belonging to all four polarization states uniformly follow the same pattern showing a strong positive correlation and are confirmed by calculated linear Pearson correlation coefficients of 0.964, 0.964, and 0.999 to the pairs of data pertaining to (P000, P045), (P000, P090), (P000, P135), respectively. Here, we wish to note that the actual dark noise pertaining to subtracting a dark image from a raw image will be twice the dark noise derived from Fig. 3 for the corresponding exposure time. This accounts for the presence of dark noise in both the dark image used to remove the counts due to dark current as well as in the raw image.

Fig. 3

Plot of standard deviation values (or dark noise counts) from Table 2 against the related exposure times determined from the dark images for the four states of polarization. We also see that the dark noise counts fluctuate with exposure times in a sawtooth shaped pattern between dark noise counts levels of 10 and 45. The calculated linear Pearson correlation coefficients for pairs of data pertaining to (P000, P045), (P000, P090), and (P000, P135) are 0.964, 0.964, and 0.999, respectively.

Here, we wish to emphasize that in the real experiment, Tables 1 and 2 will be created for each individual pixel for a total of $1976\times 1980$ (or 3,912,480) pixels. Here, just for brevity and demonstrating the technique employed to determine the dark noise, the averages for all the pixels in each polarization state, comprising 978,120 pixels for each polarization state, were computed to create Tables 1 and 2. It is true that measuring the dark noise using the averages could be susceptible to bad pixels and illumination from cosmic rays and a measurement based on medians would be a better measure of the dark noise. From past experience, we see that the illumination of CCD pixels by cosmic rays is a rarity in laboratory or ground-based experiments although prevalent in space-based experiments. Since these averages would be determined through dark images on each individual pixel, we can easily observe any anomalies and exclude those individual pixels from analysis. Now, to quantify ${\sigma }_{\text{dark}}$ as percentage of the measured digital counts, suppose we calibrate the exposure times so that pixels fill up to half the full well depth of 32,768 counts [at 16-bit digitization the full well depth is $(2^{16})$ or 65,536 digital counts] for the raw images and assume ${\sigma }_{\text{dark}}$ to be twice the maximum from Fig. 3 or 90 digital counts accounting for ${\sigma }_{\text{dark}}$ present in both the raw and dark images will yield a measurement error 90/32,768 or $\pm 0.27\%$ of the measured digital counts. In order to minimize ${\sigma }_{\text{dark}}$, the measured digital counts of the raw images can be maximized through longer exposure times or repeated measurements.

Second, we consider the statistical noise (${\sigma }_{\text{statistical}}$) generation in the polarization camera. The statistical noise, also known as photon noise, results from the inherent statistical variation in the arrival rate of incident photons on to the CCD. The registered signal by the CCD is from converting photons to photoelectrons within the CCD. However, the magnitude of this registered signal is perturbed by the incident photons and follows a Poisson statistical distribution. The associated statistical error is then a function of the square root of the actual electrons that contribute toward the measured digital counts and is calculated in the following manner. For a CCD with 16-bit digitization and a full well depth of 300,000 electrons per pixel, the maximum digital counts that a pixel can register is 65,536 $(2^{16})$, which means 4.57 electrons generate 1 digital count. For example, for a pixel with 400 measured digital counts, the number of electrons required to generate 400 digital counts would be 1828 ($400\times 4.57$), which in turn generates a statistical noise level of 42.76 ($\surd (1828)$) electrons and corresponds to 9.36 ($42.76/4.57$) digital counts or 2.34% $[(9.36/400)\times 100\%]$ of the measured digital counts. In our experiments, we like to calibrate our exposures times to generate at least half the maximum digital (or threshold for saturation) counts totaling 32,768 in a pixel or obtain this number by adding several raw images. Then, ${\sigma }_{\text{statistical}}$, as a percentage of the measured digital counts for a pixel at half its full well depth of 32,768 digital counts, as per the example above, would be $\sim 85$ digital counts or $\pm 0.26\%$ of the measured digital counts.

Third, we consider the read noise (${\sigma }_{\text{read}}$) generation in the polarization camera that arises from reading an image from the CCD and converting in to a digital image file. ${\sigma }_{\text{read}}$ was quantified to be 0.01 digital counts (or $0.01\times 4.57=0.05$ electrons) by taking five zero exposure time dark frames (or $10\mu \mathrm{s}$, which is the shortest integration time allowed by the software), also called bias frames, and then determining $\frac{\sigma [\text{average}(\text{biasframes}1-5)-\text{biasframe}1]}{\sqrt{2}}$. Then, ${\sigma }_{\text{read}}$, as percentage of the measured digital counts for a pixel with half the well depth of 32,768 counts, is insignificant.

Finally, the total random error (${\sigma }_{\text{random}}$) due to ${\sigma }_{\text{dark}}$, ${\sigma }_{\text{statistical}}$, and ${\sigma }_{\text{read}}$ for each pixel will be determined from the expression $\surd ({\sigma }_{\text{dark}}^2+{\sigma }_{\text{statistical}}^2+{\sigma }_{\text{read}}^2)$ (or 0.27%, 0.26%, insignificant) and measures to be $\pm 0.37\%$ of the measured digital counts for a pixel at half its full well depth and satisfies our requirement for ${\sigma }_{\text{random}}$ to be $<1\%$ of the measured digital counts. This is because when the noise-to-signal ratio is $\le 1\%$, we can then measure the temperature and speed within $\pm 0.2\mathrm{MK}$ and $\pm 200\mathrm{km}{\mathrm{s}}^{-1}$ MK, respectively, as explained by Reginald and Davila.⁵

3.2.

Flat Fielding

Other noises associated with CCDs are sensitivity variations from photosite-to-photosite on the CCD detector or across the detector. Modern CCDs are uniform to better than 1% between neighboring photosites and uniform to better than 10% across the entire surface. Also, any sensitivity variation from pixel to pixel can be removed through flat fielding. For the polarization camera used in the ISCORE instrument, the flat fielding file has already been generated by the camera manufacturer and is built into the software package that runs the polarization camera and automatically corrects all images for flat fielding. The final processed image would then be obtained from subtracting the dark image from the raw image and dividing by the flat field image [(raw image—dark image)/flat fielding or (raw image/flat fielding—dark image/flat fielding)]. Here, through a crude experiment, we have explored the reliability of the manufacturer provided flat fielding file built into the operating software of the camera. To test this concept, we first illuminated the CCD in the polarization camera with white light that was unfocused, unpolarized, and passed through two diffusers and 10 images were captured with the built-in flat fielding corrector in the software switched off and we will call the average of these 10 images as ${FF}_{\mathrm{off},T_1}$, where $T_1$ is the integration time that allowed to record $\sim \mathrm{25,000}$ digital counts in majority of the pixels. Then we captured three dark images that matched the integration time $T_1$ with the white-light lamp and the built-in flat fielding corrector in the software switched off and we will call the average of these dark images as $D_{\mathrm{off},T_1}$. Then the gain (G) is obtained from $\frac{m}{{FF}_{\mathrm{off},T_1}-D_{\mathrm{off},T_1}}$, where $m$ is the average of the difference ${FF}_{\text{off},T_1}-D_{\text{off},T_1}$. Next, an image was taken by the polarization camera with the built-in flat fielding corrector in the software switched off and we call this $I_{\text{off},T_2}$, where $T_2$ is the integration time. Again, three dark images were taken with an integration time $T_2$ and the built-in flat fielding corrector in the software switched off and the average of these three we will call $D_{\text{off},T_2}$. Now the same image and the dark images were taken with an integration time of $T_2$, but this time with the built-in flat fielding corrector in the software switched on and we will call these $I_{\text{on},T_2}$ and $D_{\text{on},T_2}$, respectively. Now, Figs. 4(a)–4(c), show the following: $(I_{\text{off},T_2}-D_{\text{off},T_2})\times G$, $I_{\text{on},T_2}-D_{\text{on},T_2}$, and the difference between the two, respectively. The bottom image confirms that the majority of the differences are located close to the zero value as seen from the color coded bar that interprets the data and confirms that the laboratory generated flat fielding file does not qualitatively or quantitatively alter the final image from using the flat fielding file built-in to the operating software of the camera. However, individual pixels could degrade with time and optical systems in the ISCORE instrument can introduce sensitivity variation in the CCD, for example, vignetting. To account for these sensitivity variations, we will generate our own flat fielding file at the time and location of our experiment by taking multiple images through pointing the ISCORE instrument toward cloudless parts of the blue sky. The straight forward way to perform the comparison shown in Fig. 4 is to compare the flat fielding file created in the laboratory with the flat fielding file embedded in the camera software and both these were created without any lenses attached to the polarization camera. Unfortunately, we do not have access to the flat fielding file embedded in the camera software.

Fig. 4

(a) $(I_{\text{off},T_2}-D_{\text{off},T_2})\times G$, (b) $I_{\text{on},T_2}-D_{\text{on},T_2}$, and (c) the difference between the two. The panel (c) shows that the majority of the differences are located close to the zero value as seen from the color coded bar that interprets the data and confirms that the laboratory generated flat field file does not qualitatively or quantitatively alter the final image from using the flat field file built into the operating software of the camera. However, pixels could degrade with time and as a precaution, we will generate our own flat fielding file at the time and location of our experiment and will use that for image processing.

3.3.

Systematic Error

Here, we quantify the systematic error (${\sigma }_{\text{systematic}}$) associated with the extinction ratio of the micropolarizer array through a laboratory experiment by passing stabilized white light through two filters centered at 385.0 and 410.0 nm with FWHM of 5.0 nm, which will determine the temperature. The ISCORE experiment measures the temperature from the polarized intensity ratios from these two filters and the magnitude of the extinction ratios at wavelengths 385.0 and 410.0 nm will determine the magnitude of the associated systematic errors in the temperature measurements. Figure 5 shows the picture of the experimental setup. The optical components of the entire experimental were mounted on a stabilized optical table and the baffling around the optical elements (not shown in Fig. 5) ensured that light from the lamp that bounced off the walls outside of the optical elements did not reach the detector (diode or polarization camera). The entire experiment including the baffling was further enclosed inside a large light-tight black box that also included a light-tight lid (not shown in Fig. 5). The light-tightness to stray light was confirmed by first measuring the ambient light signal using the diode with all the lights in the laboratory including the lamp switched off. Then the ambient light signal was measured between the baffling and the walls of the black box with the lamp switched on and the diode reading in both cases were in the ${10}^{-12}$ Amperes (${10}^{-12}$ $\mathrm{A}=1\mathrm{pA}$) range and was not significant compared to the diode reading along the optical axis, which measured in the ${10}^{-6}$ Amperes (${10}^{-6}$ $\mathrm{A}=1\mu \mathrm{A}$) range.

Fig. 5

Picture of the experimental setup used to determine the extinction ratio of the micropolarizer array in the polarization camera. The entire experiment was conducted on an optical table. The experimental setup consisted of a stable white-light source. The light from the source passed through a bandpass filter centered at 410 nm with FWHM of 5 nm (or 385 nm with FWHM of 5 nm) on to the first linear polarizer (linear polarizer 1). The light from linear polarizer 1 then passed through a second linear polarizer (linear polarizer 2) to a diode (or polarization camera). The two linear polarizers, linear polarizers 1 and 2, were mounted on a computer controlled rotator that could be rotated in intervals of 1 deg. When the diode was used, the current was read using a Keithley electrometer and when the polarization camera was used, the images were read using the camera software. Not shown in the picture are the baffling and the light tight lid to the black box.

Then with the lamp on and light passing through the 410.0-nm filter and the Hammamatsu diode connected to a Keithley 6517B electrometer with measurement sensitivity in the pA range, diode readings were recorded by remotely turning linear polarizer 2 in intervals of 5 deg over a complete turn of 360 deg and the recorded readings are plotted in Fig. 6. Figure 6 was then used to determine the effective extinction ratio ($E_{1,2\mathrm{eff}}^{410\mathrm{nm}}$) of the combined linear polarizer 1 and linear polarizer 2, which was measured to be 0.04. With both linear polarizer 1 and linear polarizer 2 determined to be almost identical, it is reasonable to claim that each of the linear polarizers 1 and 2 has individual extinction ratios $E_1^{410\mathrm{nm}}$ and $E_2^{410\mathrm{nm}}$, respectively, and equal to $E_{1,2\mathrm{eff}}^{410\mathrm{nm}}/2$ or 0.02. Figure 7 is similar to Fig. 6 with the 410.0-nm filter replaced with the 385.0-nm filter and $E_1^{385\mathrm{nm}}$ and $E_2^{385\mathrm{nm}}$ equal to $E_{1,2\mathrm{eff}}^{385\mathrm{nm}}/2$ or 0.03.

Fig. 6

Diode current reading from the experimental setup shown in Fig. 5. Here, the lamp was turned on and light passed through the 410.0-nm filter and linear polarizer 1 before passing through linear polarizer 2, which was remotely turned in intervals of 5 deg over a complete turn of 360 deg. The effective extinction ratio $E_{1,2\mathrm{eff}}^{410\mathrm{nm}}$ of linear polarizers 1 and 2 was measured to be 0.04. With both linear polarizers 1 and 2 determined to be almost identical, it is reasonable to claim that each of the linear polarizers 1 and 2 has individual extinction ratios $E_1^{410\mathrm{nm}}$ and $E_2^{410\mathrm{nm}}$, respectively, and equal to $E_{1,2\mathrm{eff}}^{410\mathrm{nm}}/2$ or 0.02.

Fig. 7

Diode current reading from the experimental setup shown in Fig. 5. Here, the lamp was turned on and light passed through the 385.0-nm filter and linear polarizer 1 before passing through linear polarizer 2, which was remotely turned in intervals of 5 deg over a complete turn of 360 deg. The effective extinction ratio $E_{1,2\mathrm{eff}}^{385\mathrm{nm}}$ of linear polarizers 1 and 2 was measured to be 0.06. With both linear polarizers 1 and 2 determined to be almost identical, it is reasonable to claim that each of the linear polarizers 1 and 2 has individual extinction ratios $E_1^{385\mathrm{nm}}$ and $E_2^{385\mathrm{nm}}$, respectively, and equal to $E_{1,2\mathrm{eff}}^{385\mathrm{nm}}/2$ or 0.03.

Now linear polarizer 2 was removed and the diode was replaced with the polarization camera. This time linear polarizer 1 was remotely turned through 5-deg intervals over a complete turn through 360 deg and images from the polarization camera were captured using the camera control software. The exposure time was maintained at a constant value of $1000\mu \mathrm{s}$. Figure 8 shows the total digital counts ($I_0+I_{90}$) and ($I_{45}+I_{135}$) of the polarization states (P000+ P090) and (P045+ P135), respectively, which should be constant and the DOLP. According to Fig. 8, these three quantities are uniform throughout the rotation of linear polarizer 1 through 360 deg in intervals of 5 deg with slopes of $-4.44$, $-5.23$ and $3.79\times {10}^{-5}$, respectively, and confirms the near uniformity of the lamp throughout the experiment. Figure 9 is a repeat of Fig. 8 with the 410.0-nm filter replaced with the 385.0-nm filter and the exposure time was maintained at $2500\mu \mathrm{s}$. The DOLP plots in Figs. 8 and 9 show signs of a slight oscillation and we do not have any rational answer to this observation.

Fig. 8

Total digital counts ($I_0+I_{90}$) and ($I_{45}+I_{135}$) measured through the 410.0-nm filter for the polarization states (P000 + P090) and (P045 + P135), respectively, and the DOLP and the three quantities show near uniformity as linear polarizer 1 was turned through 360 deg in intervals of 5 deg with slopes of $-4.44$, $-5.23$, and $3.79\times {10}^{-5}$, respectively. This figure also confirms the stability of the lamp.

Fig. 9

Total digital counts ($I_0+I_{90}$) and ($I_{45}+I_{135}$) measured through the 385.0-nm filter for the polarization states (P000 + P090) and (P045 + P135), respectively, and the DOLP and the three quantities show near uniformity as linear polarizer 1 was turned through 360 deg in intervals of 5 deg with slopes of $-10.01$, $-8.85$, and $2.61\times {10}^{-5}$, respectively. This figure also confirms the stability of the lamp.

Figure 10 shows the total digital counts measured by the pixels in the four polarization states, P000, P045, P090, and P135 in the polarization camera with linear polarizer 2 removed and linear polarizer 1 rotated through 360 deg in intervals of 5 deg. The vertical lines are drawn through the peaks of the four different polarization states and Fig. 10 confirms that the polarization camera is operating as expected. For example, when the pixels of polarization state P090 peak, we expect the pixels of polarization state P000 to bottom out and the pixels of polarization states P045 and P135 to equal and represent one half the peak value of pixels of polarization state P090. Again, when the pixels of polarization state P045 peak, we expect the pixels of polarization state P135 to bottom out and the pixels of polarization states at P000 and P090 to equal and represent one half the peak value of pixels of polarization state P045. Figure 10 amply reflects this behavior. The horizontal lines in Fig. 10 show the maximum and the minimum for the four polarization states and based on these maximum and minimum values, the effective extinction ratios ($E_{1,\mathrm{PA}\mathrm{eff}}^{410\mathrm{nm}}$) due to linear polarizer 1 and PA in the polarization camera were measured to be 0.06. Figure 11 is analogous to Fig. 10 with the 410.0-nm filter replaced with the 385.0-nm filter.

Fig. 10

Total digital counts measured by the pixels in the four polarization states in the polarization camera when linear polarizer 1 was rotated through 360 deg in intervals of 5 deg. The vertical lines are drawn through the peaks of the different polarization states and the pattern confirms that the polarization camera is operating as expected. For example, when the pixels of polarization state P090 peak we expect the pixels of polarization state P000 to bottom out and the pixels of polarization states P045 and P135 to equal and represent one half the peak value of pixels of polarization state P090. Again, when the pixels of polarization state P045 peak we expect the pixels of polarization state P135 to bottom out and the pixels of polarization states P000 and P090 to equal and represent one half the peak value of pixels of polarization state P045 and this figure amply reflects this behavior. Based on the minimum and maximum digital counts measured by each of the four polarization states the effective extinction ratios ($E_{1,\mathrm{PA}\mathrm{eff}}^{410\mathrm{nm}}$) due to linear polarizer 1 and PA in the polarization camera was measured to be 0.06. In creating this plot the light from the lamp passed through the 410.0 nm filter.

Fig. 11

Total digital counts measured by the pixels in the four polarization states in the polarization camera when linear polarizer 1 was rotated through 360 deg in intervals of 5 deg. The vertical lines are drawn through the peaks of the different polarization states and the pattern confirms that the polarization camera is operating as expected. For example, when the pixels of polarization state P090 peak we expect the pixels of polarization state P000 to bottom out and the pixels of polarization states P045 and P135 to equal and represent one half the peak value of pixels of polarization state P090. Again when the pixels of polarization state P045 peak we expect the pixels of polarization state P135 to bottom out and the pixels of polarization states P000 and P090 to equal and represent one half the peak value of pixels of polarization state P045 and this figure amply reflects this behavior. Based on the minimum and maximum digital counts measured by each of the four polarization states, the effective extinction ratios ($E_{1,\mathrm{PA}\mathrm{eff}}^{385\mathrm{nm}}$) due to linear polarizer 1 and PA in the polarization camera were measured to be 0.05. In creating this plot, the light from the lamp passed through the 385.0-nm filter.

Now with the measured extinction ratios $E_{1,\mathrm{PA}\mathrm{eff}}^{\lambda }$ and $E_1^{\lambda }$, we will be able to determine the extinction ratio $E_{\mathrm{PA}}^{\lambda }$ from the following mathematical formulation, shown as a schematic diagram in Fig. 12. Then with the derived values for $E_{\mathrm{PA}}^{\lambda }$ at the two wavelength position 410.0 and 385.0 nm, we will be able to determine the associated systematic error in determining the electron temperature using the ISCORE instrument.

Fig. 12

Schematic diagram showing the formulation to determine the extinction ratio $E_{\mathrm{PA}}^{\lambda }$ of the PA. The extinction ratios $E_1^{\lambda }$ for the two wavelengths 410.0 and 385.0 nm were determined from Figs. 6 and 7, respectively, and the effective extinction ratios $E_{1,\mathrm{PA}\mathrm{eff}}^{\lambda }$ of linear polarizer 1 and PA for the same two wavelengths were determined from Figs. 10 and 11, respectively. This schematic diagram assumes light of intensity S passes along the axis of polarization ($Y$-axis) of linear polarizer 1 and based on its extinction ratio $E_1^{\lambda }$ light of intensity $\mathrm{S}{E}_1^{\lambda }$ will pass through the orthogonal $X$-axis in linear polarizer 1. If a polarization state $P^{\lambda }(\theta )$ of the PA is parallel to the $Y$-axis of linear polarizer 1 then polarization state $P^{\lambda }(\theta +90\mathrm{deg})$ will be parallel to the $X$-axis of linear polarizer 1. In this configuration, the schematic diagram shows the fractions of light intensity that gets distributed along $P^{\lambda }(\theta )$ and $P^{\lambda }(\theta +90\mathrm{deg})$ as a function of $E_1^{\lambda }$ and $E_{\mathrm{PA}}^{\lambda }$, where $E_{\mathrm{PA}}^{\lambda }$ is unknown and needs to be determined.

In Fig. 12, if S is the light of intensity passing through the axis of polarization ($Y$-axis) of linear polarizer 1, then the light passing along the $X$-axis of linear polarizer 1 will be $\mathrm{S}{E}_1^{\lambda }$, where $E_1^{\lambda }$ is the extinction ratio of linear polarizer 1. Then if $E_{\mathrm{PA}}^{\lambda }$ is the extinction ratio of the PA and if the polarization state $P^{\lambda }(\theta )$ is parallel to the $Y$-axis of linear polarizer 1, then the polarization state $P^{\lambda }(\theta +90\mathrm{deg})$ will be parallel to the $X$-axis in linear polarizer 1 and the intensity $I_0^{\lambda }$ and $I_{0+90\mathrm{deg}}^{\lambda }$ along $P^{\lambda }(\theta )$ and $P^{\lambda }(\theta +90\mathrm{deg})$, respectively, that will be directed toward the CCD detector is given by Eq. (5). Table 3 lists the values of $E_{\mathrm{PA}}^{\lambda }$ measured for the two wavelengths 385.0 and 410.0 nm from the laboratory tests conducted by the authors of this paper and at 550.0 nm from the manual made available by the manufacturer of the polarization camera, which are 0.02, 0.04, and 0.02, respectively. In Fig. 12, we have assumed that the incident light on the PA is transmitted on to the CCD detector without any transmission loss. Even if the transmission efficiency was less than 100%, this contribution would cancel off in Eq. (5) because both the numerator and denominator will be multiplied by the transmission efficiency coefficient of the PA:

Table 3

EPAλ derived from Eq. (5) for the two wavelengths 410.0 and 385.0 nm using E1λ from Figs. 6 and 7 and E1,PA effλ from Figs. 10 and 11. The polarization camera manual specifies an extinction ratio of 0.02 at 550.0 nm for the PA.

In creating Table 3, we have considered light passing from one axis to another through the extinction ratio process, as schematically shown in Fig. 12, although this process is not unique. A more rigorous form of conducting this measurement would be through matrix formulation of the measurement process based on the estimates of the Muller matrix of the polarizer coupled with the measurement matrix that will convert the Stokes vectors to the four channels in the CCD super pixel. However, we do not have the Muller matrix estimates of the polarizers used in this test from the original vendors and will consider implementing this process at a later time.

Now using the extinction ratios $E_{\mathrm{PA}}^{\lambda }$ from Table 3, we can estimate the systematic error in temperature measurements. To measure the temperature, we need to determine the “true” linearly polarized brightness ratio (${\mathrm{pB}}^{\mathrm{TR}}$), as defined by Eq. (4), and is given by Eq. (6):

However, with nonzero $E_{\mathrm{PA}}^{\lambda }$ values for the two wavelengths, the “measured” linearly polarized brightness ratio (${\mathrm{pB}}^{\mathrm{MR}}$) is given by Eq. (7), where we have used the same logical arguments shown in Fig. 12 to derive the “measured” ${(I_{\theta }^{\lambda })}^{\text{measured}}$ and ${(I_{(\theta +90\mathrm{deg})}^{\lambda })}^{\text{measured}}$ for the two polarization state pairs (P000, P090) and (P045, P135):

Now from Eq. (7), we can estimate ${\sigma }_{\text{systematic}}$ in the measured pB, which is given by Eq. (8) by using the values for $E_{\mathrm{PA}}^{\lambda }$ from Table 3. It was shown by Reginald et al.⁵ that an uncertainty of $\pm 1\%$ in “measured” ${\mathrm{pB}}^{\mathrm{MR}}$ can generate an uncertainty in the “measured” electron temperature by $\pm 0.2\mathrm{MK}$. Therefore, with a derived systematic error of $-4\%$ from Eq. (8), the measured electron temperature will be underestimated by 0.8 MK:

Equation (9) shows the “measured” total brightness ${\mathrm{tB}}^{\mathrm{MR}}$ using the analogy from Fig. 12 and it is apparent that ${\mathrm{tB}}^{\mathrm{MR}}$ is a function of both the polarized $I_p^{\lambda }$ and unpolarized $I_u^{\lambda }$ light, but is not a function of the extinction ratios $E_{\mathrm{PA}}^{\lambda }$. Figure 6 by Reginald et al.⁴ shows the “theoretically” modeled “temperature sensitive intensity ratio” (TSIR) plotted against electron temperature ($T$) in the corona based on photospheric light passing through the two color filters centered at 410.0 and 385.0 nm. TSIR, as defined in Eq. (9), is a special case of ${\mathrm{tB}}^{\mathrm{MR}}$ with $I_u$ equal to zero and this condition can be satisfied only in the low solar corona close to the solar limb, as evident in Fig. 1, where the linearly polarized K-corona brightness is much greater than the unpolarized F-corona brightness. Therefore, in the past ISCORE experiments, we limited our observations to low coronal regions close to the solar limb and assumed our measurements to satisfy the conditions to obtain TSIR and we interpreted the “measured” TSIR for electron temperature using Fig. 6 by Reginald et al.⁴ Here, we wish to note that ${\mathrm{tB}}^{\mathrm{MR}}$ cannot be used to determine the electron temperature since it also contains $I_u^{\lambda }$ and will need a separate $I_u^{\lambda }$ model or experimental results to correct ${\mathrm{tB}}^{\mathrm{MR}}$. On the other hand, TSIR, if limited to low solar coronal regions, is not subject to any systematic errors associated with $E_{\mathrm{PA}}^{\lambda }$ of the micropolarizer array in the polarization camera:

Figure 13 shows the theoretically modeled polarized brightness ratio ${\mathrm{pB}}^{\mathrm{Theo}.\mathrm{R}}$ plotted against assumed thermal electron temperatures ($T$) in the corona based on photospheric light passing through the two color filters centered at 410.0 and 385.0 nm. Now, Fig. 13 is similar to Fig. 6 by Reginald et al.⁴ and can be used to interpret the measured ${\mathrm{pB}}^{\mathrm{MR}}$ given by Eq. (7) for electron temperature.

Fig. 13

Theoretically modeled polarized brightness ratio $p{B}^{\mathrm{Theo}.\mathrm{R}}$ plotted against assumed thermal electron temperatures in the corona based on sun light passing through the same two color filters centered at 410.0 and 385.0 nm used in the laboratory tests. The ISCORE experiment will determine electron temperature by interpreting measured linearly polarized brightness ratio ${\mathrm{pB}}^{\mathrm{MR}}$ using this plot.

Table 4 lists in column 1 the possible measurable quantities—${\mathrm{pB}}^{\mathrm{MR}}$, ${\mathrm{tB}}^{\mathrm{MR}}$, and TSIR—that can be used to determine the electron temperature and their dependence on $I_p^{\lambda }$, $I_p^{\lambda }$, and $E_{\mathrm{PA}}^{\lambda }$ based on the type of camera used and whether a linear polarizer is used or not. By default, the polarization camera does have a built-in linear polarizer embedded on the CCD. The linear polarizer used with the regular (nonpolarization) camera is assumed to be an external linear polarizer with very high extinction ratio mounted on a rotator mechanism and images taken by rotating the linear polarizer through three well-defined angles.

Table 4

The possible measurable quantities using a polarization camera or a regular (nonpolarization camera) and their dependence on Ipλ, Ipλ, and EPAλ based on the type of camera used and whether a linear polarizer is used or not. By default, the polarization camera does use the micropolarizer array embedded on the camera. The linear polarizer used with the regular camera is assumed to be an external linear polarizer with very high extinction ratio mounted on a rotator mechanism and images taken by rotating the linear polarizer by three well-defined angles.

From Table 4, measuring TSIR, although it is not a function of $E_{\mathrm{PA}}^{\lambda }$, in both polarization and regular cameras, it assumes to be free of unpolarized light. As a result, TSIR can be used to determine the electron temperature only in the very low coronal region up to $1.5R_{\odot }$ from the sun center, where the linearly polarized K-coronal brightness is much greater than the unpolarized F-coronal brightness, as evident in Fig. 1. On the other hand, ${\mathrm{pB}}^{\mathrm{TR}}$ measured using a polarization camera, although it is a function of $E_{\mathrm{PA}}^{\lambda }$ and will generate a systematic error that requires accurate quantification, will be free of unpolarized light. While ${\mathrm{pB}}^{\mathrm{TR}}$ measured using a regular camera will not be a function of $E_{\mathrm{PA}}^{\lambda }$ but will require the use a polarization wheel mounted on a rotator mechanism to measure ${\mathrm{pB}}^{\mathrm{TR}}$, which runs contrary to our main objective of eliminating the need to turn a linear polarizer through three angles in order to preserve temporal resolution. As noted earlier, the quantity ${\mathrm{tB}}^{\mathrm{TR}}$ cannot be used for temperature interpretation because of the presence of unpolarized light. Another very important reason to use ${\mathrm{pB}}^{\mathrm{TR}}$ for temperature interpretation stems from the need to measure electron temperatures at coronal heights above $1.5R_{\odot }$ from the sun center. To elaborate, as evident in Fig. 1, the linearly polarized K-coronal brightness will match the unpolarized F-coronal brightness around $2R_{\odot }$ from the sun center and beyond that the F component will always be greater than the K component. Thus, in order to extend beyond the low coronal region for temperature measurements, we will either have to use a regular camera with a polarization wheel mounted on a rotator mechanism and lose temporal resolution or use a polarization camera and improve the temporal resolution but introduce a systematic error. However, if we can accurately measure this systematic error, then we will be able to use the polarization camera to measure the electron temperature from $(1-4)R_{\odot }$ from the sun center.

${\sigma }_{\text{systematic}}$ of 0.8 MK derived from Eq. (8) was from laboratory experiments. But this same systematic error could also be quantified through coronal observations made through the two color filters using the polarization camera during a total solar eclipse. To do so, first, from Table 4, we can measure electron temperature using TSIR at locations very close to solar limb and we will call it $T^{\mathrm{TSIR}}$. Second, at the same position as the first, we can also measure ${\mathrm{pB}}^{\mathrm{MR}}$ and derive the electron temperature and we will call it $T^{\mathrm{pB}}$, which will be offset by the systematic error. Therefore, the difference in temperature from ($T^{\mathrm{TSIR}}-T^{\mathrm{pB}}$) should match the systematic error shown in Eq. (8) and this verification process is summarized in Eq. (10):

In this paper, we have not shown the measurement of the extinction ratios of the two filters centered at 398.7 and 423.3 nm with FWHM of 5.0 nm that will be used to determine the speed. To determine the speed at a given super pixel, first, we have to determine the temperature for that super pixel. Then using the measured temperature for that super pixel as the model temperature for that super pixel, we have to create theoretically modeled polarized brightness ratio ${\mathrm{pB}}^{\mathrm{Theo}.\mathrm{R}}$ for assumed solar wind speeds in the corona based on sun light passing through the two speed sensitive color filters centered at 423.3 and 398.7 nm. This plot will have modeled values of ${\mathrm{pB}}^{423.3}/{\mathrm{pB}}^{398.7}$ along the ordinate and assumed wind speeds in the abscissa (for example, see Fig. 3 by Reginald and Davila⁵). Finally, the “measured” linearly polarized brightness ratio measured from the images taken through the 398.7 and 423.3 nm filters will be compared with the modeled ratio for the same two wavelengths to interpret wind speeds. This process for temperature and wind speed determination will be applied for each super pixel that is exposed to the solar corona.

3.4.

Total Error

We have shown how the ${\sigma }_{\text{random}}$ and ${\sigma }_{\text{systematic}}$ could be determined for each super pixel and will quantify the error margins associated with the electron temperature and speed measured for each super pixel.

3.5.

Other Camera Characteristics

From the characterization experiments conducted on the camera, we can now show by example to prove how well the polarization camera was able to reproduce a known physical parameter, as illustrated in Fig. 14. Figure 14 shows the four states of polarization—P000, P045, P090, and P135—which are separated from their immediate neighbors to the right and left by 45 deg with the polarization axis of linear polarizer 1 oriented at $\theta$ from the reference point, which is the direction of the P000 state. For each image taken by the polarization camera, the AOLP could be determined from Eq. (2). Now, if linear polarizer 1 was turned counterclockwise in 5-deg interval over a complete circle, then the successive measurements of AOLP also should change by 5 deg.

Fig. 14

Schematic diagram showing the four states of polarization, P000, P045, P090, and P135, which are separated from their immediate neighbors to the right and left by by 45 deg with the polarization axis of linear polarizer 1 oriented at $\theta$ from the reference point, which is the direction of the P000 state. For each image taken by the polarization camera, the AOLP could be determined from Eq. (2). Now, if linear polarizer 1 was turned counter clockwise in 5-deg interval over a complete circle, then the successive measurements of AOLP also should change by 5 deg.

Figures 15 and 16 show the derived AOLP using Eq. (2) for the two filters 410.0 and 380.0 nm, respectively, with linear polarizer 1 turned through 360 deg in intervals of 5 deg. The horizontal and vertical lines in Figs. 15 and 16 need to be compared with the similarly colored vertical lines in Figs. 10 and 11, respectively, which are drawn through the maximum points in the counts registered by the different polarization states. For example, in Fig. 16, linear polarizer 1 starts with its polarization axis oriented along the P090 polarization state, which is proven in Fig. 11 with polarization state P090 showing the maximum counts at the start of linear polarizer 1. In this case, the measured AOLP at the starting point of linear polarizer 1 could be labeled as 90 or 270 deg, as evident in Fig. 16, and we chose AOLP to be 270 deg at the starting point of linear polarizer 1. Then when linear polarizer 1 was rotated counterclockwise by 90 deg, the AOLP was zero, and in Fig. 11, the polarization axis of linear polarizer 1 coincided with the maximum point of P000 polarization state. The other vertical lines in Fig. 16 follow the same pattern, and the colors of these vertical lines have to be compared with the vertical lines of the same colors in Fig. 11 to verify the accuracy. In Fig. 15, linear polarizer 1 starts with $\theta$ in Fig. 14 at 320 deg and when the linear polarizer 1 turned through 40 deg, the polarization axis of linear polarizer 1 coincided with the P000 polarization state, as seen in Fig. 10. In conclusion, the AOLP values derived in Figs. 15 and 16 have a difference from their immediate neighbors to the left and right by $5\pm 0.29\mathrm{deg}$, which matches the rotation of linear polarizer 1 in 5-deg interval using a computer controlled rotator, although its accuracy was only verified visually. The uncertainty level of 0.29 deg matches with the uncertainty level of 0. 27 deg reported by Brock et al.¹⁰ in a similar experiment.

Fig. 15

Derived AOLP using Eq. (2) for the 410.0-nm filter with linear polarizer 1 turned through 360 deg in intervals of 5-deg interval. The starting position $\theta$ of linear polarizer 1 was at 320 deg, as illustrated in Fig. 14. The derived AOLP values have a difference from the immediate neighbors to the left and right by $5\pm 0.29\mathrm{deg}$. The accuracy of the rotation of linear polarizer 1 in 5-deg interval using a computer controlled software was only verified visually.

Fig. 16

Derived AOLP using Eq. (2) for the 385.0-nm filter with linear polarizer 1 turned through 360 deg in intervals of 5-deg interval. The starting position $\theta$ of linear polarizer 1 was at 270 deg, as illustrated in Fig. 14. The derived AOLP values have a difference from the immediate neighbors to the left and right by $5\pm 0.29\mathrm{deg}$. The accuracy of the rotation of linear polarizer 1 in 5-deg interval using a computer controlled software was only verified visually.

3.6.

Instrumental Polarization

Finally, we also have to account for the instrumental polarization introduced by the Schmidt–Cassegrain telescope used in the ISCORE instrument where even when a truly zero polarized external source is observed through the telescope and some nonzero polarization is detected at some orientation. Even with the optical components of a Schmidt–Cassegrain telescope being axisymmetric, such telescopes can introduce their own unique instrumental polarization due to the optical coatings on the primary and the secondary mirrors. Since the orientation of the instrumental polarization is fixed, its plane of polarization will also rotate with the telescope when the telescope is rotated. A commonly used method of testing for instrumental polarization is to observe a star which is known to be unpolarized and observe if any measurable polarization is detected. Such stars are usually referred to as unpolarized standard stars and a listing of such stars is presented by Serkowski.¹¹ Since the Schmidt–Cassegrain telescope used in the ISCORE experiment is an off-the-shelf telescope and the instrumental polarization and its orientation is unique to that telescope, we will measure the instrumental polarization by observing an unpolarized star through the four filters the night before the total solar eclipse of August 21, 2017. This way, the polarized camera mounted on the telescope will stay fixed in position and in orientation to the optical axis of the telescope through the measurements for instrumental polarization, flat fielding, and observation of the corona during the eclipse.