Instrumental Polarization of Telescopes on Alt-Azimuth and Equatorial Mounts

Transcription

1 Project Documentation Report #0009 Revision #A Instrumental Polarization of Telescopes on Alt-Azimuth and Equatorial Mounts Christoph U. Keller 4 December 2002 Advanced Technology Solar Telescope 950 N. Cherry Avenue Tucson, AZ Phone atst@nso.edu Fax

2 Instrumental Polarization of Telescopes on Alt-Azimuth and Equatorial Mounts Revision Summary: 1. Date: 4 December 2002 Revision: Revision A Changes: Initial release 2. Date: Revision: Changes: 3. Date: Revision: Changes: RPT-0009 Rev A Page i

3 Table Of Contents 1. Introduction Polarized Ray-Tracing Mueller Matrices Gregorian Focus Coudé Focus for alt-azimuth mount Coudé Focus for equatorial mount Conclusions References... 7 RPT-0009 Rev A Page ii

4 1. INTRODUCTION While polarimeters are often considered to be pure post-focus instruments on nighttime telescopes and most existing solar telescopes, accurate solar polarimetry, as required by the scientific goals of ATST, mandates that polarimetry post-focus instrumentation and the telescope be designed as a system. All optical elements, and in particular mirrors with oblique reflections inside the telescope, change the state of polarization or even produce polarization. This artificially introduced polarization is often many orders of magnitude larger than the expected solar signal. To measure solar polarization accurately, it is important to minimize the instrumental polarization before the polarization is measured. Although it can be determined quite accurately, instrumental polarization often couples with other instrumental effects such as unknown offsets and non-linearities in detectors. As a rule of thumb, it is very hard to measure polarization accurately at levels below one percent of the instrumental polarization. Modern observations of polarized light from the Sun often work at that level, which requires instrumental polarization levels below the 1% level as stated in the Science Requirements Document. The following sections provide quantitative predictions of the Mueller matrices for an off-axis telescope on an alt-azimuth mount and on two equatorial mount configurations. Section 2 provides a short overview of the methods used, while section 3 presents the Mueller matrices as a function of hour angle for a solar declination of degrees and latitude of the telescope location of 29 degrees. These are the same values as those in the document on image rotation of an alt-azimuth telescope [1]. Section 4 concludes with a discussion of alt-azimuth vs. equatorial mount options. 2. POLARIZED RAY-TRACING The influence of optical elements on the polarization of light is conveniently written in a matrix formalism involving the Stokes vector to describe polarized light and the so-called Mueller matrices to describe the optical elements. The Stokes vector I Q I = U V describes polarized light, where I is the intensity, Q the difference of the linearly polarized components at 0 and 90 degrees, U the difference of the linearly polarized components at 45 and 135 degrees, and V the difference of left and right circularly polarized light. The Mueller matrices are real 4 by 4 matrices that operate on Stokes vectors, i.e. I' = M I where I is the input Stokes vector, M the Mueller matrix, and I the output Stokes vector. An optical element with a diagonal Mueller matrix will not mix the various components of the Stokes vector. Any off-diagonal element, however, describes the mixing (or cross-talk) of polarization components in the Stokes vector. In the following the term instrumental polarization will always refer to these off-diagonal elements. RPT-0009 Rev A Page 1 of 7

5 The true instrumental polarization of ATST will be determined by the optical properties of the actual coating, which can only be predicted to within certain accuracy. It therefore does not make sense to try to predict the polarization performance of ATST to better than about a factor of 2. Here we use the current f/2 primary off-axis telescope configuration with an f/69 beam in the coudé room and assume that all mirrors have pure aluminum coatings. Furthermore, we only consider a wavelength of 400 nm where aluminum has an index of refraction with a real part of n=0.4 and an imaginary part of k=4.45. In principle, one could determine the Mueller matrix for each point in the point-spread function (PSF) for all possible locations within the field-of-view. This is a very demanding task and also requires knowledge of the optical aberrations of the system. This is not needed at the present time. Therefore, a much simpler approach has been chosen where the Mueller matrix is calculated and averaged over the PSF for the center of the field-of-view. The Mueller matrix corresponding to measurements that average over the PSF is the (incoherent) superposition of the Mueller matrices of the individual rays. This is a direct consequence of energy conservation between pupil plane and image plane. Optical aberrations and diffraction are therefore neglected. All calculations have been performed with PolTrace, a JAVA-based program developed for calculating polarization optics. It can import ZEMAX files. The Mueller matrices in the following sections do not take into account the general absorption upon reflections because solar observations are almost always relative and not absolute measurements. However, the full effect of reflection on a metal surface onto polarization is considered here. The coordinate system for the linear polarization has been chosen such that Stokes Q is parallel to the direction of geographic north on the solar disk. Instrumental polarization typically increases with the square of the inclination on the reflective surface. Beams at the outer edge of mirrors will therefore show considerably more instrumental polarization as compared to beams closer to the optical axis. To take this effect into account, 81 beams uniformly distributed in the pupil plane have been traced through the configuration and their respective Mueller matrices have been averaged. 3. MUELLER MATRICES 3.1 GREGORIAN FOCUS The first question to address concerns the Mueller matrices in the Gregorian focus as a function of the hour angle. For a polarization coordinate system that is fixed to the telescope, the Mueller matrix in the Gregorian focus is given by Of course, this matrix is independent of the hour angle. When looking at a polarization coordinate system that rotates with the solar image, the Mueller matrix is still independent of the hour angle (and the declination) for an equatorially mounted telescope. However, for an alt-azimuth mounted telescope, the RPT-0009 Rev A Page 2 of 7

6 Mueller matrix changes with the hour angle as shown in Fig. 1. The changing Mueller matrix is given by the Mueller matrix above being rotated with the solar image. Fig. 1 Mueller matrix of the alt-azimuth mounted off-axis telescope in the Gregorian focus in a polarization coordinate system that rotates with the solar image. The equivalent figure for the equatorially mounted off-axis telescope would show no change in time. 3.2 COUDÉ FOCUS FOR ALT-AZIMUTH MOUNT The Mueller matrix in the coudé focus is considerably more complex because it involves four more mirrors that rotate at different rates and a final rotation of the coudé platform. For these calculations, it is assumed that the instrument is mounted vertically in the center of the platform. While it is likely that real RPT-0009 Rev A Page 3 of 7

7 instruments will be feed through additional mirror, it is outside of the scope of this note to look at the various options to feed instruments. Furthermore, the orientation of the instrument is another degree of freedom that can be used to minimize the instrumental polarization. Again, such an optimization is outside of the scope of this report. Fig. 2 Mueller matrix as a function of hour angle for the alt-azimuth mounted telescope in the coudé focus. Figure 2 shows the variation of the instrumental polarization as a function of hour angle. As expected, the variation is most pronounced around local noon. RPT-0009 Rev A Page 4 of 7

8 3.3 COUDÉ FOCUS FOR EQUATORIAL MOUNT Two different configurations of the equatorial mount were considered. The first configuration uses a single mirror with a large inclination of the incoming and outgoing beams to transfer the beam from the RA axis to the vertical axis. The second configuration uses two mirrors with much smaller inclinations to transfer the beam from the RA axis to the vertical axis. Fig. 3 Mueller matrix for the equatorially mounted telescope with a single mirror transferring the beam from the RA axis to the vertical. Figure 3 shows the resulting Mueller matrix for the first configuration. The coudé table is uniformly rotating throughout the day to compensate for image rotation. The absolute rotation angle includes the RPT-0009 Rev A Page 5 of 7

9 declination to provide the same image orientation throughout the year. Most of the matrix element variations are due to the rotation of the coudé table with respect to the final, highly inclined mirror. Fig. 4 Mueller matrix of the equatorially mounted telescope with two reflections to go from the RA axis to the vertical. The second configuration replaces this highly inclined reflection with two much less inclined reflections, which leads to the Mueller matrix variations shown in Fig. 4. At first, it might be surprising to see that this equatorial configuration performs much better than the alt-azimuth configuration. However, one needs to realize that the first 45-degree reflection to make the beam parallel to the declination axis is almost completely compensated for by the two successive mirrors that make the beam parallel to the RA axis. This advantage has been discussed extensively for the German Gregory-Coudé telescope [2]. A similar compensation is achieved with the alt-azimuth mount when pointing at the horizon. RPT-0009 Rev A Page 6 of 7

10 4. CONCLUSIONS Table 1 summarizes the values of the relevant Mueller matrix elements that show the largest deviation from their ideal values (1 for the diagonal and 0 for the off-diagonal values) for the various mount configurations. We only consider the coudé location here since the difference between the alt-azimuth and the equatorial mount in the Gregorian focus is a rotation of the polarization reference frame. Table 1 Comparison of maximum Mueller matrix element deviations in the coudé focus for the various mount configurations and the requirements from the ATST Science Requirements Document. Q,U to Q,U V to V I to Q,U I to V Q,U to U,Q Q,U to V Alt-azimuth Equatorial Equatorial Requirements none none <0.01 <0.01 <0.05 <0.05 The equatorial configuration 1 with the highly inclined reflection is significantly worse than the other two configurations. The second equatorial configuration, however, is at least a factor of two better than the alt-azimuth mount configuration. A comparison with the requirements from the ATST Science Requirements Document indicates that none of the solutions fulfill all the requirements in the coudé focus. To fulfill the requirements, the polarization needs to be measured in the Gregorian focus, which can be done with all three mount configurations. 5. REFERENCES 1) Hubbard R., ATST Technical Note TN-0012, Image Rotation at the Gregorian and Coudé Positions for an Alt-Azimuth Telescope, October ) Sánchez Almeida J., Martínez Pillet V., Wittmann A.D., The instrumental polarization of a Gregory-Coudé telescope, Solar Physics 134, 1-13, RPT-0009 Rev A Page 7 of 7