Alignment Sensitivity of Reflective Optical Elements and Analysis Of

Alignment sensitivity of reflective optical elements and analysis of automated alignment methods Chris van Ewijk November 2017 Contents 1 Introduction 3 2 Introduction to optical aberrations 5 2.1 Zernike polynomials . 7 2.2 Common aberrations . 8 2.2.1 Piston . 8 2.2.2 Defocus . 9 2.2.3 Wavefront tilt . 10 2.2.4 Spherical aberration . 11 2.2.5 Coma . 12 2.2.6 Astigmatism . 13 2.2.7 Field curvature . 14 2.2.8 Distortion . 15 3 Alignment sensitivity analysis of standard mirror shapes 17 3.1 Zemax sensitivity analysis . 17 3.2 Results . 18 3.2.1 Parabolic mirror . 18 3.2.2 Spherical mirror . 20 3.2.3 Elliptical mirror . 22 3.2.4 Fold mirror . 24 3.3 Discussion . 26 4 Sensitivity table method 27 4.1 Testing method of the sensitivity table . 28 4.2 Results of the sensitivity table method . 28 4.3 Discussion . 29 5 Merit function regression method 30 5.1 Testing method of the merit function regression . 30 5.2 Results of the merit function regression method . 31 5.3 Discussion . 33 1 6 Differential Wavefront Sampling Method 34 6.1 Computation method . 35 6.2 Measurements and control uncertainties of DWS . 36 6.3 DWS for off-axis systems . 37 6.4 Discussion . 37 7 Conclusions 39 Appendices 41 A Detailed lens data of the OGSE Re-Imager 41 B Merit function used in the sensitivity table method 42 C Merit function regression method 43 2 1 Introduction The mission of SRON is to bring about breakthroughs in international space research. Therefore the institute develops pioneering technology and advanced space instruments, which typically operate in vacuum and cryogenic environments. Imaging space instruments and their testing setups have to be aligned in order to create high quality images. Here, alignment is the tweaking of optical elements to reduce the optical aberrations in the signal. The alignment of these space instruments and their testing systems is usually done at room temperature. However, when cooling down to cryogenic temperatures, the elements of a system typically shrink and thus do not maintain their original alignment positions. Triggered by ESA's tender "Novel in-vacuum alignment and assembly tech- nologies for optical assemblies" and experience at SRON with optical alignment in cryogenic conditions, a new method has been proposed by Huisman and Eggens [6] to perform optical alignment with sub-micrometer accuracy in vacuum and cryogenic environments. This method can increase the accuracy of the alignment as it can account for typical changes in alignment observed when cooling down from room temperature to cryogenic temperatures. The method also makes alignment at the operating wavelengths realistic, if this is not possible under atmospheric pressure. The method utilizes piëzoelements for simultaneous positioning, sensing and control over the friction forces between the mirror and its support structure. After alignment of the optical system, the same piëzoelements can be used for fixation of the mirror in the cryogenic environment. The combination of actuation, position sensing and the control over the friction forces creates the possibility for automated alignment in a closed loop configura- tion when using a wavefront sensor to measure system aberrations. Simultane- ous to our research, three students from the Windesheim University of applied sciences have been working on the hardware of the proposed method. Computer aided alignment methods using the relation between optical aberrations and alignment state have been developed to make alignment more ef- ficient. One of those methods is the sensitivity table (ST) method [15], where the alignment state of an optical system is determined by a linear sensitivity table and measurements of the wavefront. In order to overcome the non-linearity of aberrations the merit function regression method (MFR) was proposed [8], which determines the alignment state of an optical system by simulation of the optical system in an attempt to reproduce the measured wavefront. When aligning multi-element systems Lee et Al. [11] showed that there exists a cou- pling effect between optical elements. They introduced the differential wavefront sampling (DWS) method [12] for alignment of centered multi-element systems, which uses second derivative information of the wavefront. In this thesis a basic understanding of the most important optical aberrations 3 is provided, whereafter the sensitivity of standard reflective optical elements to displacements and tilts is investigated to create a general idea of the important degrees of freedom for each element separately. We only focus on the aberrations induced by misalignment of the optical elements. Surface shape errors such as radius of curvature and surface roughness are ignored. Several automated alignment methods are discussed for a multi-element system and where possible tested by simulation in the lens design software Zemax. The final goal of the thesis is to determine under which assumptions and limitations each automated alignment method can be used, in order to determine the functionality in our application. A recent example of an optical system for which the automated alignment in vacuum and cryogenic environments is applicable is the Safari OGSE Re-Imager. The optical layout of the system is given in figure 1.1 and the detailed lens data is given in the appendix A.1. This optical system is used to focus the incoming light rays from a point source via two off-axis parabolic mirrors and two fold mirrors onto the entrance focal plane of the Safari instrument for the SPICA space telescope. Therefore it is important that the Safari OGSE Re-Imager does not induce aberrations in the signal. Since a detailed description of the system in the lens design software Zemax was already present, and the system is relevant to our research, the OGSE Re-Imager is used in the simulations done throughout this thesis. Figure 1.1: The optical design of the Safari OGSE Re-Imager by M. Ferlet. 4 2 Introduction to optical aberrations The general concept of an optical system of two lenses and an aperture stop is shown in figure 2.1. The rays leave the point object and propagate through the system to focus on the image point. The limiting diameter which determines the amount of light that reaches the image plane is called the aperture stop. The entrance pupil is the optical image of the aperture stop, as seen from the object. The corresponding image of the aperture as seen through the image plane is called the exit pupil. The quality of an optical image can be defined by the wavefront aberrations of the system at the exit pupil. For an aberration free image the wavefront at the exit pupil must be spherical, with its center of curvature at the image plane. In real applications the diffraction limit plays an important role in the image quality, yet it is system and wavelength dependent. Therefore we ignore the diffraction limit in our analysis, where it should be taken as a criteria when aligning an real optical system. Figure 2.1: Entrance and exit pupils of an optical system consisting of two lenses and an aperture stop. The reference spherical wavefront is shown in black, whereas the aberrated wavefront is shown in red [4]. In figure 2.2 the marginal and chief rays are displayed. The ray leaving the 5 origin of the object and passing through the maximum aperture of the system is known as the marginal ray. The chief ray originates from an off-axis point in the object and passes through the center of the system. We also consider paraxial rays, which are rays that make a small angle to the optical axis of the system and lay close to that axis throughout the system. Figure 2.2: A simple lens system and aperture stop with illustrations of the marginal and chief rays [2]. The wavefront aberration function W (x; y) describes the optical path differ- ence between the aberrated wavefront and the ideal spherical reference wavefront. Aberrations of a rotationally symmetric system can be expressed by expanding the wave aberration function in a power series of pupil and field coordinates, ρ, θ and H: I J K WIJK ) H ρ cos (θ) 2 4 3 W (H; ρ, θ) = W000 + W020ρ + W111Hρ cos(θ) + W040ρ + W131Hρ cos(θ) 2 2 2 2 2 3 + W222H ρ cos (θ) + W220H ρ + W311H ρ cos(θ) + O(6) (2.1) where H is the normalized field coordinate and ρ and θ denote the normalized pupil coordinates as shown in figure 2.3. The other terms in the power series, for example W123, are forbidden by the rotational symmetry. In our study the higher order terms (O(6)) are neglected as their influence on the optical image is negligible. The coefficients of the lower order terms are defined as: W000: Piston W131: Coma W020: Defocus W222: Astigmatism W111: Wavefront tilt W220: Field curvature W040: Spherical aberration W311: Distortion The magnitude of a certain aberration is given by the value of the corresponding coefficient WIJK which is a function of the parameters of the optical 6 element such as surface roughness and radius of curvature. Defocus and wavefront tilt are called first order aberrations, whereas spherical, coma, astigmatism, field curvature and distortion are third order aberrations. In section 2.2 these common aberrations and their dependence on the pupil coordinates are discussed. Figure 2.3: Normalized field coordinate H and pupil coordinates ρx = ρ sin(θ) and ρy = ρ cos(θ) normalized over the exit pupil. [7]. 2.1 Zernike polynomials Zernike polynomials are a set of circular symmetric orthogonal basis functions defined over a unit circle, named after their inventor Frits Zernike [13].

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