Characterization of Optical Aberrations Induced by Thermal Gradients and Vibrations via Zernike and Legendre Polynomials

For every astronomical instrument, the operating conditions are undoubtedly different from those defined in a setup experiment. Besides environmental conditions, the drives, the electronic cabinets containing heaters and fans introduce disturbances that must be taken into account already in the preliminary design phase. Such disturbances can be identified as being mostly of two types: heat sources/sinks or cooling systems responsible for heat transfer via conduction, radiation, free and forced convection on one side and random and periodic vibrations on the other. For this reason, a key role already from the very beginning of the design process is played by integrated model merging the outcomes based on a Finite Element Model from thermo-structural and modal analysis into the optical model to estimate the aberrations. The current paper presents the status of such model, capable of analyzing the deformed surfaces deriving from both thermo-structural and vibrational analyses and measuring their effect in terms of optical aberrations by fitting them by Zernike and Legendre polynomial fitting respectively for circular and rectangular apertures. The independent contribution of each aberration is satisfied by the orthogonality of the polynomials and mesh uniformity.


Introduction
as thermo-structural static, dynamic and fluodynamic analyses with the use of implicit optimization routines has drastically reduced the designing phase times, cutting down as well the manufacturing costs by eliminating the need to build prototypes and tests, clearly derived from space applications [1].
Moreover the last decade has witnessed an increasing number of integrated toolboxes not only for space or air-borne but also for ground-based astronomical instrumentation, as for instance the SMI (Structural Modeling Interface Tool), applied to VLTI (Very Large Telescope Interferometer) design [2] or IMAT (I-DEAS Matlab Toolkit) to import nodal model of the structure from FEM into Matlab/Simulink used for the GSMT (Giant Segmented Mirror Telescope), a 30 m class telescope [3]: the aim is the investigation of how cross-coupled perturbations affect the optical performance.In particular wind buffeting and wind force power spectral density, reconstructed via empirical formulas from the wind data, are crucial ingredients for a proper harmonic response analysis.
Even for smaller modules or subunits of instruments coupled to such large structures, we need an analogous procedure passing in both directions the structural data from a FEM code to an integrating platform, in most of the cases MATLAB/Simulink, prone to a correct state-space representation, but can also be Python based if it's only needed to establish a dialogue with ray tracing program [4].
Also in our analysis, like in the abovementioned tools, for the specific target of retrieving the Zernike functions to investigate the optical aberrations, it was not necessary to use a dedicated commercial software.Moreover with our implemented tool it is possible automatically to retrieve the nodal displacements from the opticalsurfaces within an harmonic analysis launched in FEM software (ANSYS) and to post-process them isolating in the power spectra in specific amplitudes and phases of interest.

FEM Analyses: Thermo-Structural and Harmonic Response
The model under study is a Schwarzschild collimator (SC), setting the proper size of the entrance beam for an optical polarimeter [5].It has a primary mirror (M1) bonded onto three pads located at the end of three flexure arms, which can be manufactured via EDM technology [6].Three spokes having a rectangular cross section 10 × 5.5 mm, optimized to reduce the primary mirror obscuration, link the M1 cell to the top ring, holding the secondary mirror (M2).The mirrors, M1 with an outer diameter Ø104 and central bore of Ø18 mm, and M2of Ø16 mm are made of Zerodur, and the supporting structure plus the top ring of Invar 32.Both materials in facts have a very close coefficient of thermal expansion.The primary mirror cell is bolted to a central plate of 40 mm thickness.
Two different scenarios have been simulated: -A structural analysis with gravity load applied (LC1), at zenith angle z = 75˚, α = 45˚ with α representing the field rotation angle, plus a thermal gradient from 15˚C to 22˚C, uniformly spanned over the entire assembly.The components of the gravity acceleration are the following: sin cos A motor torque of 0.63 Nm is applied to the top ring, mimicking the focuser presence.As external constraints the rim of the central plate has been bonded on three orthogonal planes.
-Modal analysis restricted to the frequency range between 20 and 50 Hz (LC2), including the first 2 eigenmodes, embedded into a harmonic response analysis, where a constant damping ratio ξ = 0.02 has been imposed.Two periodic loads, one due to the presence of a rotary stage motor, the other one associated to a quarter wave retarder plate insertion mechanism have been taken into account.They're acting along the global z direction, have a phase shift Δφ = 45˚, and their expression is: ( ) Additionally a preload F b = 500 N of two roller bearings allowing the rotation of the polarization subunit has been also considered (Figure 1).
The nodal displacements over each optical surface are rearranged into the pseudo inverse matrix A + and by applying the least mean square error the sag and tip-tilt, namely the six rigid body motion values, can be easily retrieved [5]: where , , , , , , , , ,  r x y z α β γ = .The requirement to fulfill for the tilt angles are Δα, Δβ ≤ 0.1 mrad.Requirements for the yaw angle are much less stringent, and can be 1 mrad, anyway still under definition.Tolerances for the linear displacements are Δx, Δy ≤ 50 µ, Δz ≤ 1 mm.As visible in Table 1, the largest values are reached on M2 because of the high focuser torque.
In the second load case the rotation values are quite modest, if not negligible, and instead a sag effect of M2 towards M1 is more evident, mainly related to the first mode of vibration of the spiders.Spectral distribution of amplitude and phase, for displacement, velocity and acceleration have been determined with peaks corresponding to the first two eigenmodes of 28 and 48 Hz of M1 component (see Figure 2 and Figure 3, Table 2).

Zernike Polynomials Fitting
The deformed optical surfaces are consequently fitted via circular Zernike polynomials that, based on the Noll convention [7], are defined as: The advantage of using Zernike polynomials over circular and annular apertures is their orthogonality that renders it possible to decompose any wavefront into a sum of independent aberrations.Although orthogonality is fully accomplished only for continuous data [7], whereas for FEM data is partially lost, we assume anyway that the fitting procedure is suitable because of the homogeneous mesh.Any wavefront can be expressed as an infinite series of Zernike polynomials with coefficients a k given by:   It's relevant for our purpose to see how precise the approximation is up to a certain order, usually set to 38, since most of the meaningful aberrations induced by thermal loads and vibrations are within the exploited range.This leads to the estimation of the residual error indicated as ( ) , r W ρ θ .The accuracy is strongly dependent on the grid spacing [8]: a trade-off fitting to both M1and M2 meshes has been found to be (ρ,θ) = 40 × 360.The rms has been used as a proper indicator of convergence versus the increase of the polynomial order, and by the minimal rms it's possible to estimate the limit of the meaningful primary and secondary aberrations, over which their contribution can be neglected.
The algorithm developed in Python for the determination of the Zernike coefficients converts the nodal Cartesian coordinates of the finite element model to the normalized polar coordinates, applies to them a reverse rigid motion to get back the original reference frame position [9], employing the following equation: where R zyz is the Euler rotation matrix, the deformation values, and u δ are the first three component of the rigid body motion vector.It then projects the coordinates of the meshed data points onto the 40 × 360 uniform grid by a linear interpolation.A polar mask is applied if a central hole is present filtering out the non-existent nodes on the grid.The polynomial coefficients illustrated in Equation ( 5) are computed by means of the covariance matrix, defined in the following fashion: Premultiplying both members by the inverse of the covariance matrix, the coefficients relative to the desired order are obtained.For the sake of clarity in Figure 4 contour plots are illustrated relative to a synthetic surface with 512 × 512 grid where random deformations are applied with a maximum absolute error ε max = 1e−04.As it can be seen in Figure 5 minimal rms is reached at different orders of Zernike polynomials for the two mirrors, depending on the load case: for M2 in both scenarios the rms is smaller than for M1 and for each optical component rms(LC2) < rms(LC1): 3.6545E−07 mm for LC1, 1.73E−08 mm for LC2.In summary, concerning M1 aberrations, under LC1 they can be represented by the first eleven polynomials, namely up to the primary tetra-foil, under LC2 by the first sixteen (primary penta-foil); concerning M2, when LC1 occurs it presents primary trefoil, in LC2 primary astigmatism.

Legendre Polynomial Fitting
In the case of rectangular apertures the Zernike polynomials could still be used but their orthogonality is not valid anymore [10].For this reason 2D Legendre polynomials have been calculated and by least square method the coefficients have been determined as in the Zernike case.The wavefront equation in Cartesian coordinates is given by [11]: The best tradeoff number of grid elements selected is 30 × 30, to provide an enough accurate mesh sampling, consisting of 341 nodes for the square entrance surface of the linear polarizing prism.The algorithm is much more sensitive to grid spacing than the Zernike polynomials and it stops after the 5 th order, i.e. 36 iterations.The The augmented development of commercial finite element software with the various simulation packages such How to cite this paper: Di Varano, I. (2016) Characterization of Optical Aberrations Induced by Thermal Gradients and Vibrations via Zernike and Legendre Polynomials.Optics and Photonics Journal, 6, 113-123.http://dx.doi.org/10.4236/opj.2016.66014

Figure 1 .
Figure 1.Loads distribution with application points and surface relative to the local coordinate system (mainly M1 and M2).
are the radial and azimuthal factors:

Figure 2 .
Figure 2. Spectral distribution of the amplitudes vs frequency in the operating range between 20 and 50 Hz for M1.

Figure 3 .
Figure 3. Spectral distribution of phase angle vs frequency in the 20 -50 Hz range for M1.

Figure 4 .
Figure 4. First sixteen circular Zernike polynomials, fitting a randomly deformed circular aperture with a grid sampling of 512 × 512 elements.They are numbered following the Noll notation [7], and in particular can be identified: the piston (N = 1), A-and B-tilt for N = 2, 3, focus (N = 5), pri-coma (N = 9) up to the pri-pentafoil-A, which corresponds to r 5 cos(5θ).

Figure 5 .
Figure 5.All four contour plots are relative to the min rms: in particular the top left one concerns the Z 11 for M1 for LC1, top right Z 7 for M2 and LC1, bottom left Z 16 for M1 at load case 2, bottom right Z 6 at load case 2. Load case 2 taken into consideration is relative to 28 Hz, so called first step of the harmonic response analysis.

Figure 6 .
Figure 6.The behaviour of rms error with the variation of n, m orders for LC1 (left) and LC2 (right) for component KALC1.

Figure 7 .
Figure 7.Each quadrant shows surface plots of the LSF of the above described Legendre polynomials fitting the deformed surface of component KALC1 and the generic Legendre polynomial of the same order for a quick shape identification: upper left and right plots are relative to LC1, lower left-right to the vibration analysis at different orders.

Table 1 .
Rigid body motion values for the mirrors M1 and M2 belonging to the SC, relative to the first loads scenario.

Table 2 .
Rigid body motion values for M1 and M2 relative to the harmonic case scenario relative to the first eigenfrequency of 28 Hz (see also Figure2).