Strategy for Realizing High-Precision VUV Spectro-Polarimeter

Abstract

Spectro-polarimetric observations in the vacuum ultraviolet (VUV) range are currently the only means to measure magnetic fields in the upper chromosphere and transition region of the solar atmosphere. The Chromospheric Lyman-Alpha Spectro-Polarimeter (CLASP) aims to measure linear polarization at the hydrogen Lyman-α line (121.6 nm). This measurement requires a polarization sensitivity better than 0.1 %, which is unprecedented in the VUV range. We here present a strategy with which to realize such high-precision spectro-polarimetry. This involves the optimization of instrument design, testing of optical components, extensive analyses of polarization errors, polarization calibration of the instrument, and calibration with onboard data. We expect that this strategy will aid the development of other advanced high-precision polarimeters in the UV as well as in other wavelength ranges.

Appendices

Appendix A: Spurious Polarization

1.1 A.1 Radiometry and Photon Noise

Photon noise is an important factor that limits the polarization sensitivity, and therefore an accurate estimation of the instrument throughput is required early in the program. We have estimated and measured the throughput for all the optical components. We create, maintain, and update the photon throughput budget (Table 6). The reflectivity and transmissivity of all the optical components are obtained with the actual measurements (at the UVSOR synchrotron facility) of the flight components, the witness samples coated simultaneously with the flight optics, or the prototype components from the same vendors except for the grating and CCD. The grating reflectivity (diffraction efficiency at +/− first order) was measured by the supplier with unpolarized Lyman-α light. Information on the quantum efficiency of the flight CCD with Lumogen-E coating was provided by the vendor. The efficiencies of the flight grating and the CCD will be measured by the CLASP project prior to integration, and the values in the throughput table will be replaced with the measured ones. The degradation of throughput by 50 % caused by molecular contamination to critical optical surfaces is taken into account in the photon throughput budget.

Table 6 Estimated CLASP throughput at Lyman-α.

To estimate the number of photons detected by the spectro-polarimeter (SP) cameras, we employed the spectrum of the Lyman-α emission line observed in the HRTS sounding rocket experiment Ishikawa et al. (2013). Although absorption by the geo-corona occurs in the wavelength range of 0.003 nm and is not negligible, it affects only 1 or 2 pixels in the dispersion direction. With the set of the relevant CLASP parameters (slit width: 1.45 arcsec; spatial scale along the slit: 1.11 arcsec/pix; and spectral sampling: 0.0048 nm/pix) and with the photon throughput budget, we estimate the total number of photons N collected with nine-pixel binning for an observation time of 220 s.

The spurious polarization caused by the photon noise ($\sqrt{N}$) is evaluated from Equation (1) as follows:

$$ \Delta\biggl(\frac{Q}{I} \biggr)_{\mathrm{photon}}=a^{-1} \times\frac{\sqrt {N}}{N}=\frac{1}{a\sqrt{N}}, $$

(3)

where a is the modulation coefficient. We have evaluated the flight waveplate at the UVSOR synchrotron facility and found the phase retardation to be 180^∘−2.5^∘/+0^∘ at the Lyman-α line center, resulting in a=0.64 (see Section 3.2).

1.2 A.2 Readout Noise of CCD Camera

We evaluate the spurious polarization caused by the readout noise of the SP CCD camera. As σ _s is the readout noise from one exposure, from Equation (1), the spurious polarization caused by the CCD readout noise with a total number of pixels used for demodulation n is

$$ \Delta\biggl(\frac{Q}{I} \biggr)_{\mathrm{CCD}}=\frac{\sqrt{n}\sigma_s}{aN}, $$

(4)

where N and a are identical to those in Equation (3), n=c [total number of exposure]×b [total number of summed pixels along the slit]. With b=9 pixels, c=733 (=[total observing time]/[exposure duration]=220/0.3) and σ _s =25 [photons/exposure] (from CCD readout noise of 25 electrons (specification), which approximately corresponds to 25 Lyman-α photons with the Lumogen-E coating), the spurious polarization is estimated to be 0.007 %.

1.3 A.3 Spurious Polarization Caused by Fluctuation of Exposure Durations

The strobe command to instruct the SP camera to take a single image is generated by the master clock in the waveplate driver electronics and is timewise accurate. Uniform rotation of the waveplate is guaranteed with the master clock as well. Each SP camera is supposed to take c=733 images during the observation. As we perform the demodulation using all the acquired data, the spurious polarization caused by the fluctuation of exposure duration ΔT is estimated to be

$$\begin{aligned} \Delta\biggl(\frac{Q}{I} \biggr)_{\mathrm{exp}} =&a^{-1} \frac{(|\Delta T_1|+|\Delta T_2|+|\Delta T_3|+|\Delta T_4|)+\cdots}{cT} \\ \approx&\frac{\sqrt{c}\Delta T}{acT}=\frac{\Delta T}{a\sqrt {c}T}=10^{-4}~\%, \end{aligned}$$

(5)

where T is the exposure time of 0.3 s. We assume that the random fluctuation of the exposure strobe signal has a standard deviation of ΔT=5 μs, which is much larger than the real fluctuation.

1.4 A.4 Spurious Polarization Caused by Time Variation of Source Intensity

1.4.1 A.4.1 Temporal Variation of Lyman-α Intensity

We have estimated the temporal variation of the Lyman-α intensity using the data set obtained in the VAULT sounding rocket experiment (Trujillo Bueno, Štěpán, and Casini, 2011). With this rocket experiment, high-resolution Lyman-α images were taken with a spatial sampling of 0.125 arcsec × 0.11 arcsec and with a temporal cadence of 17 s. To simulate the CLASP data, we degrade the VAULT data into that with a pixel sampling of 1.1 arcsec×1.1 arcsec and create the 0.3 s cadence data by linearly interpolating two successive VAULT data. Then, the standard deviations of intensity variation σ _t for the 60 CLASP pixels during the CLASP exposure time of 0.3 s are estimated to be 0.030 [%/exposure] in the bright quiet region and 0.034 [%/exposure] in the dark quiet region. Note that the primary scientific target of the CLASP experiment is a quiet-sun region, as discussed by (Štěpán et al., 2012).

1.4.2 A.4.2 Estimation of Spurious Polarization

Here, we assume the single-channel demodulation scheme (Section 3.3), which is vulnerable to the time variation of source intensity, to estimate the spurious polarization in the worst-case demodulation. From Equation (1), spurious polarization caused by the intensity change is given by

$$ \Delta\biggl(\frac{Q}{I} \biggr)_{\mathrm{source}}=a^{-1} \frac {(-d_{t_1t_2}+d_{t_3t_4}) +(-d_{t_5t_6}+d_{t_7t_8})+\cdots }{(D_{t_1}+D_{t_2}+D_{t_3}+D_{t_4})+(D_{t_5}+D_{t_6}+D_{t_7}+D_{t_8})+\cdots}, $$

(6)

where $d_{t_{i}t_{i+1}}$ with i=1,2,… is the intensity change between two successive data, and the subscript that represents a channel is eliminated. It is worth noting that spurious polarization does not occur if the intensity linearly changes with time (i.e. $d_{t_{i}t_{i+1}}=\mbox{constant}$).

Assuming that the intensity variation $d_{t_{i}t_{i+1}}/D_{t_{i}}$ is random with a standard deviation of σ _t, the spurious polarization caused by the intensity variation results in

$$ \Delta\biggl(\frac{Q}{I} \biggr)_{\mathrm{source}} \approx\frac{\sqrt {c/2}\sigma_\mathrm{t}}{ac}=\frac{1}{a\sqrt{2c}}\sigma_\mathrm{t}, $$

(7)

where c is the number of exposures. For the very conservative estimation, by considering a single demodulation from four successive data (i.e. c=4) without spatial summation along the slit, the spurious polarization is estimated to be 0.018 % in the dark quiet region.

1.5 A.5 Spurious Polarization Caused by Pointing Jitter and Drift

Image motion at the slit position due to the pointing jitter and the drift of the vehicle causes the change in intensity, resulting in spurious polarization. Here, we consider a maximum of 5 arcsec drift within the target and peak-to-peak jitter of 0.5 arcsec within a 1 s window, which are the requirements to the Solar Pointing Attitude Rocket Control System (SPARCS) for CLASP (e.g., NASA Sounding Rocket Program Handbook available from the web page of NASA Sounding Rocket Program Office¹).

The spatial variation of the Lyman-α intensity is σ _x =5.1 %/pixel in the bright quiet region and σ _x =3.0 %/pixel in the dark quiet region using the data set obtained in the VAULT sounding rocket experiment. With Δl being the image shift per unit time and unit SP camera pixel size, the intensity change during one exposure of T=0.3 s at one σ level is expressed by

$$ \sigma_\mathrm{t}=T\Delta l\sigma_x. $$

(8)

Substituting σ _t into Equation (7), the spurious polarization caused by the image shift is given by

$$ \Delta(Q/I)_{\mathrm{drift}}=\frac{T\Delta l\sigma_x}{a\sqrt{2c}}. $$

(9)

The observation duration for the science target is 220 s, and the highest drift rate Δl is 0.023 [arcsec s⁻¹]=0.020 [pixel s⁻¹]. Using Equation (9), the spurious polarization caused by the pointing drift is estimated to be 0.017 %. This estimate is conservative because a single-channel single-dataset (c=4) demodulation scheme is employed.

We assume that the pointing jitter is represented by the sinusoidal variation with an amplitude of 0.25 arcsec and estimate the spurious polarization after the single-channel demodulation from the entire data set for 200 s. The amount of spurious polarization depends on the period of the sinusoidal variation, and we find a maximum spurious polarization of 0.015 %.

Combining the above two errors, we estimate the spurious polarization caused by the pointing draft and jitter to be $\sqrt{(0.017~\% )^{2}+(0.015~\%)^{2}}=0.023~\%$.

1.6 A.6 Spurious Polarization Caused by Waveplate Rotation

Since the waveplate is located upstream of the slit, image wobbling occurs at the slit position with the rotation of the waveplate because of 1) misalignment between the rotation axis and the waveplate normal (tilt of the waveplate) and 2) the waveplate wedge. The image has a circular motion with dx=Acosϕ, where ϕ is the rotation angle of the waveplate. The amplitude A is estimated to be 0.05 and 0.03 SP pixels, respectively, because of the misalignment and wedge based on the measurements of the flight component. Intensity variation at the slit position is expressed as

$$ I(\phi)=I_{0}+ \biggl(\frac{\mathrm{d}I}{\mathrm{d}x} \biggr) \mathrm{d}x=I_0+ \biggl(\frac {\mathrm{d}I}{\mathrm{d}x} \biggr)A\cos\phi. $$

(10)

Performing demodulation using the four successive data, the polarization signal becomes

$$\begin{aligned} \Delta\biggl(\frac{Q}{I} \biggr)_{t_{1}-t_{4}} =& \frac{1}{a} \cdot\frac {D_{t_1}-D_{t_2}-D_{t_3}+D_{t_4}}{D_{t_1}+D_{t_2}+D_{t_3}+D_{t_4}} \\ \approx& \frac{A}{a} \biggl(\frac{\mathrm{d}I}{\mathrm{d}x} \biggr) \frac{\cos\phi_{1}-\cos \phi_{2}-\cos\phi_{3}+\cos\phi_{4}}{4I_{0}}, \end{aligned}$$

(11)

where we neglect the effect of the CCD integration with exposure duration of T for simplicity. Similarly, we derive the Stokes signal using data between t ₉ and t ₁₂, during which the waveplate angle differs by 180^∘ from those between t ₁ and t ₄,

$$\begin{aligned} \Delta\biggl(\frac{Q}{I} \biggr)_{t_{9}-t_{12}} \approx& \frac {A}{a} \biggl(\frac{\mathrm{d}I}{\mathrm{d}x} \biggr)\frac{\cos\phi_{9}-\cos\phi_{10}-\cos \phi_{11}+\cos\phi_{12}}{4I_{0}} \\ =& \frac{A}{a} \biggl(\frac{\mathrm{d}I}{\mathrm{d}x} \biggr)\frac{-\cos\phi_{1}+\cos\phi _{2}+\cos\phi_{3}-\cos\phi_{4}}{4I_{0}} \\ =& -\Delta\biggl(\frac{Q}{I} \biggr)_{t_{1}-t_{4}}. \end{aligned}$$

(12)

This indicates that the use of 16 successive images corresponding to one rotation of the waveplate will cancel the image motion caused by the waveplate rotation, resulting in the complete cancellation of the spurious polarization.

Appendix B: Scale Error

2.1 B.1 Scale Error Caused by the Change of Retardation due to Temperature Change

Simple thermal calculation indicates a temperature increase of the waveplate during the flight by approximately 10 K because of the continuous illumination of visible (and infrared) sunlight, although it is significantly obscured by the cold-mirror coating of the primary mirror. We measure the change of phase retardation of 6^∘ associated with the temperature increase of 12 K at the UVSOR synchrotron facility using the prototype of the MgF₂ waveplate.

When the phase retardation δ of the waveplate deviates from 180^∘, the observed signals in channel 2 are given by (cf. Table 2 in Section 3.2)

$D_{t_{1}}$	$D_{t_{2}}$	$D_{t_{3}}$	$D_{t_{4}}$
I+a 1 Q+b 1 U−c 1 V	I−a 2 Q+b 1 U−c 2 V	I−a 2 Q−b 1 U−c 2 V	I+a 1 Q−b 1 U−c 1 V

where

$$\begin{aligned} a_{1} =&\frac{1}{22.5^{\circ}}\int^{22.5^{\circ}}_{0^{\circ}} \frac {1}{2}\bigl[(1+\cos4\phi)+\cos\delta(1-\cos4\phi)\bigr]\,\mathrm{d}\phi, \\ a_{2} =&\frac{1}{22.5^{\circ}}\int^{45.0^{\circ}}_{22.5^{\circ}} \frac {1}{2}\bigl[(1+\cos4\phi)+\cos\delta(1-\cos4\phi)\bigr]\,\mathrm{d}\phi, \\ b_{1} =&\frac{1}{22.5^{\circ}}\int^{22.5^{\circ}}_{0^{\circ}} \frac {1}{2}(1-\cos\delta)\sin4\phi \,\mathrm{d}\phi, \\ c_{1} =&\frac{1}{22.5^{\circ}}\int^{22.5^{\circ}}_{0^{\circ}} \sin\delta\sin2\phi \,\mathrm{d}\phi, \\ c_{2} =&\frac{1}{22.5^{\circ}}\int^{45.0^{\circ}}_{22.5^{\circ}} \sin\delta\sin2\phi \,\mathrm{d}\phi. \end{aligned}$$

By performing the demodulation using these four successive data [e.g., Equation (1)], Stokes Q/I is derived as

$$\begin{aligned} \biggl(\frac{Q}{I} \biggr)' =& a^{-1} \frac {D_{t_1}-D_{t_2}-D_{t_3}+D_{t_4}}{D_{t_1}+D_{t_2}+D_{t_3}+D_{t_4}} \\ \approx& \frac{a_{1}+a_{2}}{2a} \biggl(\frac{Q}{I} \biggr)-\frac {c_{1}-c_{2}}{2a} \biggl(\frac{V}{I} \biggr) \\ \approx& \frac{1-\cos\delta}{2} \biggl(\frac{Q}{I} \biggr), \end{aligned}$$

(13)

where we neglect the Stokes V contribution because it is expected to be negligibly small and cancels out with the demodulation using data from one waveplate rotation. From this equation, we find a scale error of 0.3 % for δ=180^∘±6^∘, which is expected because of the temperature change.

2.2 B.2 Scale Error Caused by the Nonuniform Rotation Speed of the Waveplate Motor

When the rotation speed of the waveplate motor is not uniform, the modulation coefficient $a_{t_{i}}$ at each exposure differs from a=0.64, resulting in a scale error. Using the flight drive electronics with high-fidelity servo-logic and mechanical motor system, we found that the deviation from the ideal rotation in terms of rotation angle is essentially random (i.e., no drift or secular deviation from the ideal rotation angle). From our measurement, we find the scale error to be lower than 0.02 % with the demodulation using the 16 images corresponding to one waveplate rotation. Although there is also a fixed periodic pattern over one waveplate rotation, this effect is negligibly small and the scale error will be significantly reduced when we use the entire data set for the demodulation.

2.3 B.3 Scale Error Caused by the Nonlinearity of the CCD Camera Gain

If the relation between the number of incoming photons and the CCD camera output signal (digital number) is not linear, the amplitude of the linear polarization signal will not be precisely determined. We measured the linearity of CCD gain using the commercial CCD with the same readout specification as the flight CCD cameras and found the scale error to be lower than 1 %, which is regarded as the maximum error for the CCD device. Since we allocate 5 % for the scale error caused by the nonlinearity of the CCD camera (CCD device together with its circuitry), the maximum error caused by the flight camera circuitry is 4 % (conservatively in linear sum), which is the specification for the camera electronics.

2.4 B.4 Scale Error Caused by Scattered Visible Light

Since the grating may be one of the main sources of (specular and non-specular) scattered visible light, we measured a commercial holographic grating to obtain a crude order-of-magnitude estimate in the early phase of the CLASP program. The grating was made by the vendor who will provide the flight grating, and the groove shape is similar to that of the flight grating. On the basis of this measured value and information on the instrument structures, we used LightTools software to estimate the total scattered visible light as I _sca=4 % with respect to the Lyman-α line core intensity.

Assuming that the scattered visible light is unpolarized, the linear polarization signal in the Lyman-α line $Q_{\mathrm{Ly\alpha }}/I_{\mathrm{Ly}\alpha}$ becomes Q _Lyα /(I _sca+I _Lyα ) because of visible-light contamination, and the linear polarization is underestimated by I _sca/(I _sca+I _Lyα )≈I _sca/I _Lyα , resulting in a scale error of 4 %. After the integration of the telescope and spectro-polarimeter, we will perform an end-to-end test for the scattered visible light by feeding sunlight into the telescope using a heliostat (built for the Solar-B (Hinode) project) in the clean room of the Advanced Technology Center of the National Astronomical Observatory of Japan.