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Abstract
Polarization aberrations exist in almost all astronomical telescopes. Polarization aberrations would bring about asymmetric apodization in the exit pupil, leading to asymmetric PSFs. The shape of PSFs is critical to telescopes that are used to detect weak gravitational lensing (WGL) in the universe. In this paper, polarization aberrations and their connections with PSF ellipticity in an unobscured off-axis space telescope are analyzed. Together with the Jones pupil, cumulative diattenuation and retardance maps of the telescope are obtained via polarization ray tracing. Due to asymmetric apodization caused by polarization aberrations, the ellipticities of all four PSF components are found to be greater than zero. The PSF ellipticity of the telescope over the full FOV is obtained. Results show that polarization aberrations change PSF ellipticity in different degrees at different FOVs. The maximum variance of PSF ellipticity induced by polarization aberrations is 7.5e–3 and the average value is 2.7e–3. In addition, interpolation errors of PSF ellipticity would also be affected by polarization aberrations. It is found that there are 405 FOV points (about 4% of all FOV points involved in the calculations) whose variances of interpolation errors caused by polarization aberrations are greater than 1.4e–4. According to the results shown in this paper, polarization aberrations of telescopes play a significant role in WGL measurements.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Currently, a major challenge in cosmology is the understanding of the dual phenomena of dark matter and dark energy. The nature of dark matter and its relation to the baryonic matter comprising stars and galaxies remain as crucial questions in modern cosmology [1,2]. The weak gravitational lensing (WGL) of background galaxies by foreground galaxies has proven itself a powerful technique for studying dark matter and dark energy [3]. Due to WGL, the shapes of galaxy images appear be changed. Hence, it is an effective method for detecting WGL by measuring point spread function (PSF) ellipticity of galaxies via telescopes [4].
WGL measurements from the ground are fundamentally limited by the relatively large and variable PSFs introduced by the atmosphere [5]. By contrast, space telescopes have important advantages. Euclid space telescope [6], WFIRST [7] and Chinese Space Station Telescope (CSST) [8] are designed to observe WGL in the universe. The coherent distortions of galaxy images due to WGL are typically ∼1% in size [9]. Detecting this tiny signal is difficult because the image shapes are also changed an order of magnitude more by many other factors. These factors that dominate WGL detections can be divided into two major classes, i.e., statistical errors and systematic errors [10,11]. An important limitation of existing surveys has been their relatively small sky coverage, which results in an insufficient number of galaxies to average out their random shapes and orientations. However, several projects are attempting to overcome this fundamental limitation by significantly increasing the sky coverage. LSST [5], Euclid space telescope and CSST are all with big field of views (FOVs). A dramatic improvement in the statistical power of the data set is available. As the size of WGL surveys increases and the statistical errors keep going down, it becomes more and more important to similarly reduce systematic errors [12,13].
Since the galaxy shapes must be convolved by the PSF of telescopes before we observe them, optical properties of telescopes are important systematic errors in WGL measurements. More work needs to be done to improve the optics model of telescopes and apply it to both understand telescope performance and to gain better precision in PSF interpolation [5,13]. This step is crucial because the PSF is only known at star positions while corrections have to be performed at any position on the sky [13]. Hence, it is very necessary to fully understand the performance of telescopes [14]. Jarvis et al. [15] have analyzed the effects of low order wavefront aberrations on PSF ellipticity of telescopes. A physical model of PSF patterns including several primary aberrations has been built. It is shown that improvements in PSF estimation can be achieved by combining the purely empirical approach with a physical PSF model [12], especially for the cases in which too few stars are present in a single exposure. Several systematic errors including mirrors misalignment, wavelength response and surface perturbations of optical elements, telescope components such as the shutter and spider, and scattered light are simulated by Chang et al. [10]. Zeng et al. [16] have designed an off-axis space telescope for detecting WGL. In this unobscured telescope, quite smooth PSF ellipticity distribution over the full FOV is achieved, and the effects of geometric aberrations on PSF ellipticity are analyzed.
Evidently, identifying more systemic errors is helpful to build a physical PSF model of telescopes with higher precision. In the studies mentioned above, however, polarization aberrations of telescopes are not involved. Although off-axis telescopes have several important advantages over on-axis ones such as simpler PSFs, easy stray light control, low-scattering property, larger throughput, and so on [16–19], polarization aberrations of the former are usually greater than those of the later [20,21]. In this paper, we will focus our attentions on polarization aberrations and their connections with PSF ellipticity for an unobscured off-axis space telescope [8,16]. Galaxies with partly polarized light and polarization astigmatism have been studied by Lin et al. [22]. However, unpolarized light is more common in the universe. If only polarization astigmatism is considered, a radially symmetric PSF whose ellipticity is zero would be obtained for unpolarized sources [23,24]. Besides polarization astigmatism, in fact, polarization aberrations would induce several significant effects on the performance of telescopes including retardance, diatteuation, polarization crosstalk, amplitude and phase apodization. As a result, images of unpolarized light sources become asymmetric due to polarization aberrations.
Polarization aberrations indicate the variations of amplitude, phase and polarization associated with ray paths through optical systems [25–27]. As well as wavefront errors, polarization aberrations are inevitable for almost all astronomical telescopes, although wavefront errors are usually more important than polarization aberrations. Breckinridge et al. [23] have made important and detailed analyses on polarization aberrations in an on-axis telescope. However, systemic studies about the connections between polarization aberrations and the PSF ellipticity that is used in WGL measurements are missing.
In this paper, polarization aberrations in an unobscured off-axis space telescope that is designed to detect WGL will be obtained and their effects on PSF ellipticity will be analyzed systematically. The rest of this paper is arranged as follows. Major parameters and polarization aberrations of the off-axis telescope are shown in Section 2. In Section 3, both four polarization components and total PSF are obtained and their ellipticities are calculated and compared. PSF ellipticity variances in full FOV induced by polarization aberrations are obtained and ellipticity interpolation errors are analyzed in Section 4. Some conclusions are summarized in Section 5.
2. Polarization aberrations in an unobscured off-axis space telescope
2.1 Telescope optical layout
In this paper, an unobscured off-axis space telescope which is designed to observe WGL in the universe will be analyzed. The optical layout of the telescope is shown in Fig. 1. It is a 2 meter F/14 COOK type three-mirror anastigmatic (TMA) telescope with 1.1°×1° FOV [16,28]. There are three power mirrors, i.e., M1, M2, M3 and a plane mirror M4 in the telescope. The telescope is with a so big FOV that it has the ability to complete the survey of the sky in a short time and detect dark distant objects with high signal noise ratio [8]. The telescope is designed carefully and the average RMS wavefront error is only 21 nm [16]. Evidently, this is a diffraction limited system with competent imaging performance. The wavelength we analyzed is 632.8 nm and the coating of all mirrors is bare metal aluminum film whose refraction is $n = 1.45 + 7.54i$.
2.2 Diattenuation and retardance
Mirrors belong to weak polarization elements. According to Snell’s law, the parallel polarization component (P-polarized light) and vertical polarization component (S-polarized light) of light ray obliquely incident onto a mirror are characterized by
According to Eq. (1), it is easy to find that if refractive index n is constant, which is the common cases for telescopes when the coating of all mirrors is determined, reflective coefficients are dependent on angle of incidence $\theta$. As shown in Fig. 2, both the amplitude and phase of the two reflection coefficients vary when angles of incidence $\theta$ change.
Fig. 2. Reflection coefficients for (a) the amplitude and (b) the phase at different angles of incidence.
Diattenuation indicates that intensity transmittance of an element is a function of incident light polarization state [26]
In the universe, most stars and galaxies are unpolarized sources. Unpolarized incident light would become partly polarized light when it exits from telescopes due to diattenuation. Retardance characterizes the phase difference between two orthogonal polarization components
If unpolarized light is received by telescopes, the optical path differences between the two orthogonal polarization components may be different. As a result, the whole wavefront in the exit pupil plane of telescopes would become fairly complicated.
Different light rays suffer different angles of incidence when they pass through the telescope from the primary mirror to the exit pupil plane. Via polarization ray tracing [29,30], the cumulative diattenuation and retardance for the telescope at the central FOV of [0°, 0°] are obtained in the exit pupil plane. Results are shown in Fig. 3. In Fig. 3(a), the length of each line is proportional to the diattenuation magnitude and its orientation shows the axis of maximum transmission for a point in the exit pupil plane. The diattenuation map of the off-axis telescope is non-rotationally symmetric and the maximum diattenuation is about 0.008.
Fig. 3. The cumulative (a) diattenuation and (b) retardance map for the telescope at the central FOV of [0°, 0°] in the exit pupil plane.
In Fig. 3(b), the cumulative retardance of the telescope is shown. The length of each line is proportional to the value of retardance in radians and its orientation shows the fast axis. Retardance that results from geometric transformation has been removed [30]. As well as the diattenuation map shown in Fig. 3(a), retardance distribution is also non-rotationally symmetric. The maximum retardance is 0.0467 rad.
2.3 Jones pupil
In this paper, polarization aberrations are characterized by Jones matrices, which enable us to fully express polarization properties of any element except depolarization [31]. Mirrors in telescopes belong to ultra-smooth surfaces, whose depolarization can be ignored safely [25]. Each ray through an optical system has an associated Jones matrix. The polarization aberration function is a set of Jones matrices expressed as a function of pupil coordinates and object coordinates. The set of Jones matrices for a specified object point is called the Jones pupil, which has the form of a Jones matrix map over the exit pupil [23,32]. Jones pupil is represented by a set of 2×2 Jones matrices and contains complex components with amplitude and phases
Via polarization ray tracing, the Jones pupil of the off-axis telescope at the central [0°, 0°] FOV are obtained and shown in Fig. 4. On the whole, the Jones pupil is close to the identity matrix. Small deviations from the identity matrix occur due to polarization aberrations. The values of diagonal elements ${A_{XX}}$ and ${A_{YY}}$ are smaller than 1 because of reflection losses. The amplitudes of off-diagonal elements ${A_{XY}}$ and ${A_{YX}}$ indicate polarization coupling or polarization crosstalk, whose magnitudes are evidently smaller than those of the diagonal elements. The phases of the diagonal elements ${\varphi _{XX}}$ and ${\varphi _{YY}}$ are continuously changing. In contrast, the off-diagonal elements ${\varphi _{XY}}$ and ${\varphi _{YX}}$ change discontinuously because the phase of a complex number changes by π when amplitude passes through zero due to the Snell’s law. Typical Maltese cross pattern shown in on-axis telescopes [23,33] disappears in the off-axis telescope.
In Fig. 4, it can be seen that the maps of all Jones pupil elements are apodized to different degrees. The diagonal elements are slowly apodized while the off-diagonal elements are highly apodized. The key point is that all of the apodization is asymmetric. The field distribution in the Fraunhofer diffraction pattern is the Fourier transform of the field distribution across the exit pupil. Asymmertric aperture functions would result in asymmetric PSFs [34]. In this way, polarization aberrations will change PSF ellipticity. Hence, it is necessary to quantitatively analyze the connections between polarization aberrations and PSF ellipticity.
3. PSF ellipticity induced by polarization aberrations
3.1 PSF ellipticity
PSF ellipticity is a parameter to characterize the shape or spatial distribution of PSFs [2,4]. Let $I({x,y} )$ be the brightness distribution of an image, the center of the image is given by the weighted first order moments divided by the total intensity
The two component ellipticities ${\textbf {e}} = [{{e_1},\;{e_2}} ]$ are defined by
The magnitude of ellipticity e is [11]
3.2 Ellipticity of different PSF components
In conventional scalar image formation calculations, the amplitude response function is calculated as the Fourier transform of the exit pupil function. To evaluate the image formed by systems with polarization aberrations, McGuire and Chipman [35] introduced a Jones calculus version of the amplitude response function named amplitude response matrix (ARM)
Most stars observed in the UV, visible and IR are thermal emitters and their radiation is unpolarized. The PSF for an unpolarized light source is composed of four incoherent components, i.e., $\textrm{I = }{\textrm{I}_{XX}}\textrm{ + }{\textrm{I}_{XY}}\textrm{ + }{\textrm{I}_{YX}}\textrm{ + }{\textrm{I}_{YY}}$. As shown in Fig. 4, all Jones pupil elements are apodized asymmetrically to different degrees. Combining Eq. (8) with Eq. (10), we will show the ellipticity of each PSF component induced by polarization aberrations. In order to achieve high accuracy, all PSFs are oversampled in the image plane. In this paper, we sample every 0.27 micron to get a 1024 × 1024 matrix and compute ellipticity in a 67.9 microns (0.5″) radius circle centering its centroid.
3.2.1 JXX
As shown in Figs. 4(a) and 4(e), both the amplitude ${A_{XX}}$ and phase ${\varphi _{XX}}$ of ${J_{XX}}$ are slightly apodized asymmetrically. Combining the results shown in Fig. 4 with Eq. (10), the first PSF component ${I_{XX}}$ of the telescope is obtained, as shown in Fig. 5(a), in which all points are normalized by the maximum of ${I_{XX}}$. Evidently, ${I_{XX}}$ is similar to an Airy spot. Cross section curves in X axis and Y axis are shown in Fig. 5(b). The apodization of ${J_{XX}}$ is so gentle that the PSF curve in X axis is almost identical to that in Y axis. According to the definition of PSF ellipticity, as shown by Eqs. (5)–(8), the ellipticity of ${I_{XX}}$ is ${e_{XX}} = \textrm{8}\textrm{.0083e - 4}$. Of course, if there is no apodization in both ${A_{XX}}$ and ${\varphi _{XX}}$, ${e_{XX}} = \textrm{0}$.
3.2.2 JXY
As well as the calculation done for ${J_{XX}}$, the second PSF component ${I_{XY}}$ of the telescope is obtained, as shown in Figs. 6(a) and 6(b), in which all points are also normalized by the maximum of ${I_{XX}}$. The maximum of ${I_{XY}}$ is significantly smaller than the maximum of ${I_{XX}}$. As shown in Fig. 5(a) and Fig. 6(a), the spatial distribution of ${I_{XY}}$ become heavily asymmetric and far from an Airy spot. The reason is that significant asymmetric apodization occurs in ${\varphi _{XY}}$, as shown in Fig. 4(f). The ellipticity of ${I_{XY}}$ is ${e_{XY}} = \textrm{0}\textrm{.4612}$.
3.2.3 JYX and JYY
The case of ${I_{YX}}$ is similar to that of ${I_{XY}}$, as well as the situation between ${I_{YY}}$ and ${I_{XX}}$. ${I_{YX}}$ is shown in Fig. 7 and ${I_{YY}}$ is shown in Fig. 8. Respective ellipticity is ${e_{YX}} = \textrm{0}\textrm{.4613}$ and ${e_{YY}} = \textrm{0}\textrm{.001862}$.
3.2.4 Total PSF
For unpolarized sources, the aforementioned four PSF components are incoherent and total PSF is the sum of the four PSF components, i.e., $\textrm{I = }{\textrm{I}_{XX}}\textrm{ + }{\textrm{I}_{XY}}\textrm{ + }{\textrm{I}_{YX}}\textrm{ + }{\textrm{I}_{YY}}$. Combining the results shown in Figs. 5–8, the total PSF of the telescope is obtained and shown in Fig. 9. The ellipticity of the total PSF is $e = \textrm{0}\textrm{.001471}$.
It should be noted that the PSFs shown in Figs. 5–9 are directly obtained by Fourier transform of the Jones pupil shown in Fig. 4. Hence, wavefront errors of the telescope are not involved. In reality, however, both wavefront errors and polarization aberrations would change PSF ellipticity. What is more, wavefront errors play a more important role than polarization aberrations. In later sections, therefore, the telescope coated with bare metal aluminum film, in which both the two mentioned aberrations work, would be compared with that coated with ideal film, in which polarization aberrations are removed and only wavefront errors remain, to show the effects of polarization aberrations on PSF ellipticity.
4. PSF ellipticity interpolation errors
4.1 PSF ellipticity in full FOV
In order to accurately measure ellipticity variances of galactic images caused by WGL, the influences from our own PSF ellipticity of telescopes must be eliminated. To obtain the PSF ellipticity of telescopes at the FOV position where a galaxy image appears, it is usual to interpolate nearby stars whose ellipticities are known. Evidently, the PSF ellipticity distribution of telescopes in full FOV is critical. The smooth PSF ellipticity map of telescopes is helpful to reduce interpolation errors, which is key to increase the WGL detection precision. Figure 10(a) show the PSF ellipticity distribution of the off-axis telescope shown in Fig. 1 in the full FOV of 1.1° × 1° when ideal film is used. In this case, polarization aberrations are nonexistent and the telescope is only with wavefront errors. We can see that the PSF ellipticity map is quite smooth. In fact, we have also calculated the PSF ellipticity map in the same full FOV of a similar on-axis TMA telescope with the same F-number and clear aperture as the off-axis one shown in this paper. The results shown in Fig. 10(a) are much smoother than those of the on-axis telescope. This is one of the important superiorities of off-axis telescopes with filled pupil compared to on-axis telescopes.
Fig. 10. The PSF ellipticity map of the off-axis telescope with (a) ideal coating and (b) bare metal aluminum film, and (c) the differences of the PSF ellipticity of the telescope with the two kind of coatings.
For obtaining the effects of polarization aberrations on PSF ellipticity, four mirrors of the telescope are coated with bare metal aluminum film and all other parameters remain unchanged. The PSF ellipticity map in full FOV is shown in Fig. 10(b). Figure 10(c) shows the differences between Fig. 10(a) and Fig. 10(b). It can be seen that polarization aberrations will change the PSF ellipticity of the telescope. What is more, the differences induced by polarization aberrations are different at different FOVs. It should be noted that all results about the effects caused by polarization aberrations shown in Section 2 and 3 are at the central FOV. As shown in Fig. 10(c), the differences of PSF ellipticity at marginal FOVs are significantly greater than those at the central FOV. This is easy to understand because the light rays at marginal FOVs would suffer bigger angles of incidence than that at the central FOV. As shown in Fig. 2, the differences of both amplitude and phase of reflection coefficients between P-polarized light and S-polarized light increase along with bigger incident angles $\theta $ as long as $\theta < {83^ \circ }$. According to Fig. 10(c), the maximum variance of PSF ellipticity caused by polarization aberrations is 7.5e–3 and the average variance is 2.7e–3.
4.2 PSF ellipticity interpolation errors
As shown in Fig. 10, PSF ellipticity of the telescope varies along with the FOVs. To remove the effects of telescopes on WGL measurements, it is necessary to obtain the PSF ellipticity of telescopes at any position of the image plane in full FOV by interpolations from the known ellipticity of the chosen stars observed synchronously. PSF ellipticity interpolation errors directly determine the detection precision of WGL measurements [6]. According to the requirements of the telescope we are studying, the interpolation errors of PSF ellipticity in full FOV should be no bigger than 1.4e–4 [16]. For the telescope, the focal plane size of a single detector is 50 mm × 50 mm, and the sampling interval is 10 mm. The ellipticity values of 6 × 6 sampling points can be obtained each time, as shown in Fig. 11. From the 36 points we can get the PSF ellipticities at the position of the central 25 red points. Interpolation errors can be obtained by comparing the real ellipticity of the red dots with those obtained via interpolations.
For our telescope shown in Fig. 1, the image plane in full FOV is about 540mm × 490mm. Every area of 50 mm × 50 mm in the image plane is set as an interpolation unit. Every interpolation unit moves at the step of 5mm to cover the whole image plane. 25 interpolation errors at corresponding FOV points are generated each time. After going through the full FOV, the maxmium interpolation error at each FOV point is recorded. Cubic interpolations are performed for the PSF ellipticity maps shown in Fig. 10(a) and Fig. 10(b), respectively. Two PSF ellipticity interpolation error maps over the full FOV are obtained. The absolute values of the differences between the two interpolation error maps are calculated and shown in Fig. 12, which indicates the effects of polarization aberrations of the telescope on interpolation errors of PSF ellipticity. In Fig. 12, the maximum value is 8.38e–4 and mean value is 3.3e–5. There are 405 FOV points (about 4% of all FOV points) whose variances of interpolation errors induced by polarization aberrations are greater than 1.4e-4.
Fig. 12. The absolute values of the differences between the two interpolation error maps of the telescope coated with ideal coating and bare metal aluminum film. (a) Isometric view and (b) vertical view.
5. Discussions and conclusions
As well as wavefront errors, polarization aberrations are inevitable for almost all astronomical telescopes, although wavefront errors are usually more important than polarization aberrations. Due to polarization aberrations, both asymmetric amplitude and phase apodizations occur in the exit pupil, which would induce an asymmetric PSF. The shape of PSFs of telescopes is critical in WGL measurements. In this paper, we focus our attentions on polarization aberrations and their connections with PSF ellipticity in an unobscured off-axis space telescope, which is designed to detect WGL in the universe.
The cumulative diattenuation map, cumulative retardance map and Jones pupil of the off-axis telescope are obtained by polarization ray tracing. All these distributions characterizing polarization aberrations are non-rotationally symmetric, as shown in Figs. 3 and 4. Via Fourier transform, the ellipticity of each PSF component is found to be greater than zero and that of the total PSF is 0.001471, though it is at the central [0°, 0°] FOV. What is more, the PSF ellipticities of off-diagonal PSF components are bigger than 0.46. The PSF ellipticity of the telescope in full FOV is obtained. Results show that polarization aberrations change PSF ellipticity to different degrees at different FOVs. The maximum variance of PSF ellipticity is 7.5e–3 and the average value is 2.7e–3. The differences of PSF ellipticity caused by polarization aberrations at marginal FOVs are significantly greater than those at the central FOV.
The detection precision of WGL is closely related to the PSF ellipticity interpolation errors of telescopes, which are also affected by polarization aberrations. Via cubic interpolations, it is found that there are 405 FOV points (about 4% of all FOV points involved in the calculations) whose variances of interpolation errors induced by polarization aberrations are greater than 1.4e–4. Combining the results shown in Fig. 12 and Fig. 10, large variances of interpolation errors occur at the steep areas in the PSF ellipticity map, especially these near the FOVs of [−0.55°, 0.5°], [0.55°, 0.5°] and [0°, 0.25°].
According to the results shown in this paper, polarization aberrations of telescopes play a significant role in WGL measurements, especially for off-axis telescopes. Hence, polarization aberrations should be considered both in developing telescopes used to detect WGL in the universe and in building the physical PSF models of these telescopes.
Funding
National Natural Science Foundation of China (12003033, 61875190).
Disclosures
The authors declare no conflicts of interest.
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