_{1}

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 mod el 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.

The augmented development of commercial finite element software with the various simulation packages such 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 [

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 [

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 [

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.

The model under study is a Schwarzschild collimator (SC), setting the proper size of the entrance beam for an optical polarimeter [

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:

- 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 (

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 [

where

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

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

The deformed optical surfaces are consequently fitted via circular Zernike polynomials that, based on the Noll convention [

where

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 [_{k} given by:

LC1 | Δx [mm] | Δy [mm] | Δz [mm] | Δα [arcsec] | Δβ [arcsec] | Δγ [arcsec] |
---|---|---|---|---|---|---|

M1_surf | −2.0093e−06 | −6.6036e−05 | −5.32124805e−05 | 0.173133 | 0.022460 | −0.397047 |

M2_surf | −1.0873e−05 | −3.9889849e−03 | 3.03661364e−04 | −3.902581 | 0.010332 | 277.508638 |

LC2 | Δx [mm] | Δy [mm] | Δz [mm] | Δα [arcsec] | Δβ [arcsec] | Δγ [arcsec] |
---|---|---|---|---|---|---|

M1_surf | −3.8777e−04 | 1.3668e−05 | 6.2259e−04 | −0.38530 | 1.44259e−03 | 0.09973 |

M2_surf | −4.7142e−04 | −4.4495e−05 | 1.0464e−03 | 0.03972 | 4.7655e−04 | −0.2272 |

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

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 [

where R_{zyz} is the Euler rotation matrix,

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 _{max} = 1e−04. As it can be seen in

In the case of rectangular apertures the Zernike polynomials could still be used but their orthogonality is not valid anymore [

with

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

rms is not lowering down during the simulation but instead presents oscillations. Under the LC1 rms_{min} = 7.47793971e−06 and is reached at n = 1, m = 1 (_{min} = 1.65e−07 and next ones n = 0, m = 3 and n = 2, m = 3.

The final goal, i.e. to seek how perturbation of the wavefront affects image quality, is reached by exporting the Zernike coefficients into Zemax: it consists in inserting them into the so called Extra Data Editor (EDE) immediately above the corresponding lens as a fictitious surface. The maximum number of coefficient allowed is 37 [_{37} have been implemented respectively for M1 and M2. An offset of 0.05 mm in front of the mirrors has been considered, implying a change in thickness in the lens data editor. The spot size and shape visibly change leading to an increase of the radius from to 83.7 to 108.7 µ and a drastic reduction of the Airy disk diameter from 69.08 down to 13.01 µ (

In the present work, it has been demonstrated how it’s possible to combine both mechanical FEM analysis and ray tracing software to optimize a set of design parameters for an astronomical instrument subjected to thermal loads and vibrations. The separate outcomes from a thermo-structural static and modal harmonic analyses have been post-processed into Zernike polynomials for circular apertures and Legendre polynomials for the rectangular ones to preliminarily disentangle the optical aberrations they’re affected by. The meaningful surfaces we concentrated on are the primary, secondary mirror collimating the beam from Cassegrain focus with f#11.0 to a 12 mm pencil and the entrance surface to a polarizing beam splitter. In agreement with literature [

The next step to implement will be to develop a fluid analysis to estimate by the temperature distribution also the variation of the refractive index for air and glasses, in order to investigate which wavefront aberrations are introduced and more precisely also at what distance from the surface they’re located.

Igor Di Varano, (2016) Characterization of Optical Aberrations Induced by Thermal Gradients and Vibrations via Zernike and Legendre Polynomials. Optics and Photonics Journal,06,113-123. doi: 10.4236/opj.2016.66014