Polarization Accuracy Verification of the Chromospheric LAyer SpectroPolarimeter

Abstract

We have developed an advanced UV spectropolarimeter called Chromospheric LAyer SpectroPolarimeter (CLASP2), aimed at achieving very high accuracy measurements (<0.1% at $3\sigma$) of the linear ($Q/I$ and $U/I$) and circular ($V/I$) polarizations of the Mg ii h and k lines (280 nm). CLASP2 was launched on board a NASA sounding rocket on April 11, 2019. It successfully detected the full Stokes vector in an active-region plage and in the quiet Sun near the limb across the Mg ii h and k lines for the first time. To verify the polarization characteristics of CLASP2, the response matrix is estimated by combining the results obtained from the preflight calibration on the ground, with the results of the inflight calibration acquired at the solar-disk center. We find that the response matrix of CLASP2 in the Mg ii h and k lines is notably close to an ideal response matrix, i.e., the scale factor and the crosstalk terms are close to 1 and 0, respectively. Moreover, the uncertainty of each Stokes parameter estimated by the repeatability of the measurements is verified to be within the required tolerance. Based on our investigation, we conclude that CLASP2 achieves $0.1\%$ polarization accuracy at a $3\sigma$ level.

Appendices

Appendix A: Error Budgets for the CLASP2 Instrument

CLASP2 aims to achieve a polarization accuracy of $0.1\%$ at the $3\sigma$ level. The error budgets of CLASP2 are investigated to verify the performance of the instrument.

1.1 A.1 Instrumental Polarization by the CLASP2 Telescope

CLASP2 reused the classical Cassegrain telescope from the CLASP1 experiment, which has a symmetrical structure. Ishikawa et al. (2014) reported from the polarization ray tracing that the instrumental polarization caused by offaxis incidence at $\pm 200''$ in CLASP1 was negligibly small. Based on this result, we assume that the influence of the offaxis incidence for CLASP2 is negligible, because the CLASP2 FOV is half that of CLASP1.

Meanwhile, for CLASP2, we recoated the primary mirror with a dual-bandpass “cold-mirror” coating (Yoshida et al., 2018). This allowed us to successfully achieve high reflectivity not only in the spectral window of Mg ii h and k lines (280 nm), but also in the Ly$\alpha$ line (121.6 nm) for the SJ. However, the nonuniformity of this new coating can cause instrumental polarization. To evaluate the coating performance, we measured the reflectivity with $p$- and $s$-polarized beams around 280 nm of ten witness samples (1-inch flat mirrors), which are evenly deployed over the effective area of the primary mirror and coated simultaneously with the primary mirror (Song et al., 2017b).

The reflectivity of the witness samples is verified to be uniform over the ten witness samples within $\pm 2\%$. In addition, the reflectivity between the $p$- and $s$-polarized Mg ii beams are confirmed to be the same within 0.1%. Ishikawa et al. (2014) evaluated that the coating nonuniformity within $\pm 2\%$ can suppress the instrumental polarization down to $10^{-3}\%$, which is negligibly small compared to other terms in Table 6.

Table 6 Error budgets of CLASP2.

1.2 A.2 Spurious Polarization by Photon Noise of CLASP2 Targets

The photon noise can limit the polarization sensitivity of CLASP2. Therefore, it is important to estimate the photon noise for each of the scientific targets of CLASP2: plage and quiet Sun near the limb. To this end, we measured the typical Mg ii intensity in a plage ($1.0\times 10^{17}$ [$\mbox{photon}\,\mbox{cm}^{-2}\,\mbox{s}^{-1}\,\mbox{sr}^{-1}$ Å⁻¹]) and a quiet Sun ($2.0\times 10^{16}$ [$\mbox{photon}\,\mbox{cm}^{-2}\,\mbox{s}^{-1}\,\mbox{sr}^{-1}$ Å⁻¹]) by using the Mg ii spectral data obtained from IRIS (De Pontieu et al., 2014a). Moreover, the number of photons are estimated by using the parameters of CLASP2 (spatial plate scale: $0.''55$/pixel, spectral plate scale: 0.005 nm/pixel, slit width: $0.''55$, and photon throughput for all the optical components: $1.8\%$) as well as the scientific requirements for each target of a plage, i.e., 0.01 nm spectral resolution (2 pixels), $2\,\mbox{--}\,3''$ spatial-resolution (4 pixels), and 2.5-minute observations (754 exposures) and a quiet Sun, i.e., 0.02 nm spectral resolution (4 pixels), $10\,\mbox{--}\,11''$ spatial resolutions (20 pixels), and 2.3-minutes observation (690 exposures)). Subsequently, the expected total number of photons ($N_{tot}$) are $8.5 \times 10^{7}$ (plage) and $1.5 \times 10^{8}$ (quiet Sun).

The spurious polarization caused by the photon noise is evaluated by

$$ \Delta S_{PN}'=\frac{1}{a_{1}\sqrt{N_{tot}}}, $$

(36)

where $a_{1}=0.505$ is a modulation coefficient when the phase retardation of the CLASP2 waveplate is $234^{\circ}$ (see Section 3.2). Therefore, the anticipated spurious polarizations by the photon noise are $0.021\%$ at the plage and $0.018\%$ at the quiet Sun near the limb.

1.3 A.3 Spurious Polarization Caused by the Readout Noise of the CLASP2 Spectropolarimeter Camera

The readout noise ($\sigma$) of the CLASP2 spectropolarimeter camera is about 5.1 photons/exposure from one exposure. Based on the measured readout noise of the spectropolarimeter camera, the estimated spurious polarization caused by the readout noise is

$$ \Delta S_{RN}'= \frac{\sqrt{n \times s_{\lambda} \times s_{d}}\sigma}{a_{1} N_{tot}} ,$$

(37)

where $N_{tot}$ and $a_{1}$ are identical to the parameters explained in Appendix A.2. $n$ and $s$ are the total number of exposures and the number of pixels summed along the slit, respectively ($n=754$ exposures and $s_{\lambda}=2$ pixels and $s_{d}=4$ pixels at the plage and $n=690$ exposures and $s_{\lambda}=4$ pixels and $s_{d}=20$ pixels at the quiet Sun). The measured spurious polarizations with these parameters are $0.0005\%$ (plage) and $0.0007\%$ (quiet Sun).

Appendix B: Imperfection of the Light-Source System

2.1 B.1 Measurement Accuracy of the Phase Retardation for the Linear Polarizer and Misalignment Between the Waveplate and the Linear Polarizer

The demodulated polarization signals of $q'$ and $u'$ should show opposite signs between $+V$ and $-V$ inputs (Equation 17). However, as discussed in Section 5.3, $V \rightarrow Q$ and $U$ crosstalks have the same sign and are greater than 0.01 (maximum signal is about 0.029 at $+V$ input of both channels) on both channels.

Next, we investigate whether this behavior can be explained by the imperfection of the light source that is used for the polarization calibration. Note that the phase retardation of the quarter waveplate $\delta _{1/4}$ of the light source may not be exactly $90^{\circ}$ and misalignment of the principal axis for the quarter waveplate installed inside the light-source chamber may exist. The output beam from the light-source system can be expressed as $[I, Q, U, V]^{\top}=[1, \cos 2^{2}\chi +\sin 2^{2}\chi \cos \delta _{1/4}, \cos 2\chi \sin 2\chi (1-\cos \delta _{1/4}), \sin 2\chi \sin \delta _{1/4}]^{ \top}$ in the Stokes-vector form. Here, $\chi$ is the angle of the principal axis for the quarter waveplate with respect to the X-axis of the CLASP2 coordinate system. In the ideal case, $\chi =135^{\circ}$ and $\chi =45^{\circ}$ for $+V$ and $-V$ inputs, respectively. Note that the principal axis of the polarizer is set to the X-axis of the CLASP2 coordinate system and the linear polarization imparted to the quarter waveplate is $[1,1,0,0]^{\top}$.

Figure 10 shows the polarization states of the light source as a function of $\delta _{1/4}$ and $\chi$. This figure clearly shows that the residual $Q$ and $U$ can be caused by $\delta _{1/4} \ne 90^{\circ}$ and $\chi \ne 135^{\circ}$ or $\chi \ne 45^{\circ}$, respectively. The residual $Q$ signal can be explained by $\delta _{1/4} \approx 89^{\circ}$ (the diamond symbol in Figure 10). The phase retardation is wavelength dependent and such a deviation by $1^{\circ}$ is possible considering the measurement accuracy of the phase retardation. Similarly, the misalignment of the principal axis for the quarter waveplate by $\approx 0.5^{\circ}$ is possible, which is comparable to the worst-case accuracy of the angle of the principal axis of the waveplate. On the other hand, the change in the $V$ signal caused by such imperfection of the light source is small enough to be negligible, $\approx 10^{-4}$.

Figure 10

Variations of Stokes parameters $Q$, $U$, and $V$ of the light source depending on the phase retardations (top) and the angles of the principal axis (bottom) of a quarter waveplate installed inside the light-source chamber. The black lines are for the configuration of $-V$ input (the nominal angle of the principal axis of the quarter waveplate $\chi$ is $45^{\circ}$), and the red dashed lines are for the configuration of $+V$ input ($\chi = 135^{\circ}$). The black diamond and the asterisk represent the average signals of the residual Stokes $Q$ and $U$ parameters measured from $\pm V$ inputs.

Based on these findings, the change of the response matrices due to the imperfection of the light source for a circularly polarized input is investigated. We find from Figure 10 that the degree of polarization of the $V$ input is approximately 0.9998 when $\delta _{1/4} \ne 90^{\circ}$, $\chi \ne 135^{\circ}$, or $\chi \ne 45^{\circ}$. This uncertainty of the $V$ input beam mainly affects the accuracy of the scale factor of $\boldsymbol{x}_{44}$ and the crosstalks of $\boldsymbol{x}_{14}$, $\boldsymbol{x}_{24}$, and $\boldsymbol{x}_{34}$. Assuming that the degree of polarization of the $V$ input beam is 0.9998, the representative response matrices (Equations 26 and 27) can be rewritten as follows:

$$ \boldsymbol{X}_{1}= \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & -0.00846 & -0.00550 & -0.00025 \\ 0 & +0.98549 & -0.03445 & +0.00682 \\ 0 & +0.03418 & +0.98621 & -0.00111 \\ 0 & +0.00743 & +0.00186 & +{\mathbf{0.98667}} \end{array}\displaystyle \right ) $$

(38)

$$ \boldsymbol{X}_{2}= \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & +0.00692 & -0.00704 & -0.00636 \\ 0 & +0.99088 & -0.00350 & +0.00693 \\ 0 & +0.00314 & +0.99119 & -0.00168 \\ 0 & -0.00143 & +0.00219 & +{\mathbf{0.98675}} \end{array}\displaystyle \right ). $$

(39)

We find that $\boldsymbol{x}_{44}$ (bold text) of the renewed response matrices increases by $\approx 0.0002$, but other elements remain the same as before. If we assume that the maximum degree of polarization in the solar observations is $3\%$, the change of $\boldsymbol{x}_{44}$ results in the scale error of $0.0006\%$, which is negligible since the error is sufficiently small.

2.2 B.2 Contamination of the $v$ Signal in $\pm Q$ and $\pm U$ Inputs

We raised the possibility that the linear polarization beam from the light source was not completely linearly polarized, and we estimated the contamination of the circular-polarization signal ($\Delta v$) for the input beam under such an assumption. If $v$ is not zero ($v\neq 0$), the demodulated polarization signals of $v'$ (see Equation 14) for $\pm Q$ and $\pm U$ inputs are given by:

$$ [v']_{\pm Q} = \frac{x_{41}+qx_{42}+ux_{43}+vx_{44}}{1+qx_{12}+ux_{13}+vx_{14}}\sim \frac{\pm x_{42}+\Delta v}{1\pm x_{12}} $$

(40)

with $(1,\mathrm{q},\mathrm{u},\mathrm{v})^{\top}=(1,\pm 1,0,v\neq 0)^{\top}$ and

$$ [v']_{\pm U} = \frac{x_{41}+qx_{42}+ux_{43}+vx_{44}}{1+qx_{12}+ux_{13}+vx_{14}}\sim \frac{\pm x_{43}+\Delta v}{1\pm x_{13}} $$

(41)

with $(1,\mathrm{q},\mathrm{u},\mathrm{v})^{\top}=(1,0,\pm 1,\mathrm{v}\neq 0)^{\top}$. Subsequently, the contaminated $v$ signals are estimated by

$$ \Delta v = \frac{1}{2}([v']_{+Q} (1+ \boldsymbol{x}_{12})-[v']_{-Q}(1- \boldsymbol{x}_{12})) $$

(42)

and

$$ \Delta v = \frac{1}{2}([v']_{+U} (1+ \boldsymbol{x}_{13})-[v']_{-U}(1- \boldsymbol{x}_{13})). $$

(43)

The average $v$-contamination measured from Equations 42 and 43 is about 0.0035 on both channels. Therefore, it is confirmed that $v$-contamination may exist in the linear-polarization beam emitted from the light source. However, the influence of $v$-contamination can be neglected when determining each element of the response matrix. This is because the $v$-contamination is completely canceled out while solving the respective simultaneous equations shown in $[v']_{\pm Q}$ and $[v']_{\pm U}$ to estimate the elements of $\boldsymbol{x}_{42}$ and $\boldsymbol{x}_{43}$ (Equations 40 and 41).

2.3 B.3 Polarization Degree of the Light Source

The polarization degree of our light source used in the preflight calibration is determined by the extinction ratio ($r_{e}$) of the linear polarizer installed inside the light-source chamber. Before the integration of the light source, in 2018, we measured an extinction ratio of 400, leading to the polarization degree of the light source as $p = \frac{r_{e}-1}{r_{e}+1}$ $\approx 0.995$. However, the extinction ratio was measured to be $r_{e} > 2000$ (corresponding to $p \approx 0.999$) by the vendor in 2013. There are two possibilities for this difference. The first is the aging of the linear polarizer, and the second is caused by our experimental environment. The measurement of $r_{e}$ can be significantly influenced by the scattered light in a laboratory, which leads to underestimation of $r_{e}$.

Next, we investigated the change of the response matrix depending on the polarization degree of the light source. Assuming that the polarization degree of the input beam is 0.995, the representative response matrices, as shown in Equations 26 and 27, are changed as follows:

$$ \boldsymbol{X}_{1}= \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & -0.00850 & -0.00553 & -0.00026 \\ 0 & +{\mathbf{0.99044}} & -0.03445 & +0.00682 \\ 0 & +0.03418 & +{\mathbf{0.99112}} & -0.00111 \\ 0 & +0.00743 & +0.00186 & +{\mathbf{0.99175}} \end{array}\displaystyle \right ) $$

(44)

$$ \boldsymbol{X}_{2}= \left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1 & +0.00696 & -0.00705 & -0.00636 \\ 0 & +{\mathbf{0.99586}} & -0.00350 & +0.00693 \\ 0 & +0.00314 & +{\mathbf{0.99617}} & -0.00168 \\ 0 & -0.00143 & +0.00219 & +{\mathbf{0.99291}} \end{array}\displaystyle \right ). $$

(45)

From Equations 44 and 45, the values of the diagonal (bold text) of the renewed response matrices increases by $\approx 0.005$ to close to unity, but other elements of the crosstalk remain the same as before. The change of the diagonal component results in the scale error of 0.015% if the degree of the polarization is $3\%$, which is the maximum value expected for the solar observations (see Section 3.4). This error is sufficiently small that using the vendor supplied $r_{e}$ is sufficient and any reduction of $r_{e}$ from degradation or the experimental environment is neglected.