1.

Introduction

MezzoCielo represents a revolutionary concept of global field of view (FoV) optical instruments: it aims to simultaneously survey the whole sky observable from a single site on Earth, exhibiting, at the same time, a relatively large equivalent aperture. Unlike devices devoted to a similar goal, MezzoCielo employs 1-m aperture lenses to achieve a high equivalent collecting surface while maintaining a relatively small size.

The underlying optical design of the telescope is that of a refractive monocentric device, conceived on the idea of a central spherical lens focusing light upon a concentric focal surface on which several cameras, equipped with field lenses and CMOS detectors, are placed. This design is optically effective because it allows increasing the system image quality by reducing to zero all the Seidel aberrations, except the spherical one. In addition, the latter is made constant along each of the lines of sight by projecting a virtual stop at the center of the monocentric lens, making in this way the illumination uniform across the entire FoV.¹ A schematic depiction of the MezzoCielo system cross-section is illustrated in Fig. 1.

Fig. 1

Example of the MezzoCielo system optical layout, considering limited airmass elevations, such as 15 or 30 deg above the horizon.

As stated in Ref. 1, refractive monocentric devices employing bulk glass spherical lens are known, but usually, they are characterized by extremely small apertures. Indeed, realizing a glass sphere having aperture of at least 1 m is not an easy task, particularly in the context of achieving an optical instrument characterized by sufficiently high performance to be employed for scientific purposes. Therefore, two improvements have been hypothesized to solve the technological difficulties and simplify the design:

Regarding the first point, reducing the amount of glass bulk, generating a two refractive index central system composed by an outer shell of glass, encompassing a inner cavity filled with liquid, is a viable solution for large size systems from a mechanical stand-point.¹ However, such solution is also favorable in terms of optical performance: a large sphere of glass would exhibit a significant spherical aberration, which could be lowered employing, as bulk substrate, an optical material with index slightly smaller than the glass one. Such material should be characterized, in addition to a low refractive index, also by a high-transparency in the visible range (in which MezzoCielo operates), thermal and chemical stability and no toxicity, to give some examples.

In Ref. 2, all the details about the selection and the characterization of the optical fluids are reported. Here, the main results related to the most promising one, the commercially known FC-72, are summarized in Table 1.

Table 1

FC-72 main properties at T=20°C.

Regarding the second point, instead, the lenses have to be supported by a frame which could be circumscribed by a sphere, assuring that the glass elements are equal to each other and arranged in such a manner that the light beam, entering the telescope through one of the lenses, can exit at least through a specular/symmetrical one. Such characteristics suggest the employment, among the possible ones, of a platonic solid configuration, in which the lenses represent the solid faces and the supporting frame the edges. Indeed, except for the tetrahedron, all the other platonic solids, namely cube, octahedron, dodecahedron, and icosahedron, satisfy the abovementioned requirements. Note that other geometries are eligible for the MezzoCielo application, such as the fullerene one, but that provided by the platonic solids is characterized by the required regularity and design simplicity.

Now, it is crucial to select the most suitable shape to maximize the collecting area, minimizing the overall size. Given a lens of diameter $d$, the equivalent collecting surface for the several platonic solids presents size, $D_{\mathrm{eq}}$, given by

Setting $L=D$ for simplicity and considering the filling of the sphere with FC-72, the ratio $D_{\mathrm{eq},a}/D$ for a dodecahedron and a icosahedron having 1 m aperture faces is 1.2 and 0.91, respectively. For the dodecahedron, the equivalent size is greater than the telescope physical one: with 1-m diameter lenses, the structure would have a size of around 2 m, with 2.5 m of equivalent aperture. A dodecahedral structure is hence selected for the MezzoCielo application. An overall (likely, but not unique) representation of the MezzoCielo assembly is shown in Fig. 2.

Fig. 2

Visual depiction of a possible layout for MezzoCielo.

In Fig. 2, the monocentric dodecahedral structure is easily recognizable, as well as the pentagonal shape of the lenses, the supporting spider and the focal surface completely filled with cameras. Other configurations are clearly possible.

In the next sections, the main aspects concerning the thermo-mechanical sizing of the MezzoCielo structure and its optical components will be described: temperature and gravitational loads are deemed critical parameters in terms of telescope performance. The facility altitude above the sea level, namely the atmospheric pressure, has, instead, a negligible impact upon the telescope from a mechanical stand-point: according to Pascal’s law, the variation of external pressure around a sphere filled with fluid produces an identical change of the inner pressure (admitting a certain deformability of the shell) and the two effects tend to compensate each other, maintaining unchanged the spherical shape. The analysis presented in this work take into account atmospheric pressure at sea level; however, once the facility site will be selected, a proper investigation should be conducted.

2.

Thermo-Mechanical Sizing

For the dodecahedral configuration described in Sec. 1, previous studies³ have analytically shown that, to achieve a maximum glass deflection (with respect to the reference sphere) not greater than $20\mu \mathrm{m}$, the circular simply supported lens of the assembly, made of BK7, analyzed under the hypothesis of thin plate theory,⁴ must present a thickness of at least 120 mm for a 2000-mm diameter sphere. Subsequent finite element analysis (FEA) has actually demonstrated that, for such geometry, load and constraint conditions, the maximum displacement at the lens center is around $10\mu \mathrm{m}$: indeed, the analytical estimation slightly oversizes stresses and strains because the thin plate theory does not consider the actual geometry of the meniscus lens, nor stresses and strains along the axis normal to the lens plane. Moreover, as the outcome of the study performed in Ref. 3, it has been verified that the glass exhibits a linear elastic behavior in the context of the amount of stresses and deformations to which it is subjected. These results, in particular, have been obtained for the lowermost lens of the telescope, located at the bottom of the structure and, as such, subjected to the highest hydrostatic loads. In addition, given the axial symmetric nature of the loads, such a lens deflects in a spherical way, with equal meridional and sagittal radius of curvature.

Given these premises, it is chosen to consider, as the sizing starting point, a dodecahedron inscribed in a 1120-mm radius sphere, comprising 12 identical meniscus BK7 lenses 120-mm thick with clear aperture of 1000 mm each, enclosing a 1000-mm radius sphere filled with FC-72 fluid. A representation of such geometry is sketched in Fig. 3.

Fig. 3

Schematic representation of the telescope dodecahedral structure. The lenses are not depicted, but they occupy the 12 faces of the solid and are limited by the two sphere of radius $R_1$ and $R_2$.

In Fig. 3, $R_1$ represents the radius of the sphere circumscribed to the dodecahedral structure, $R_2$ the sphere radius containing the fluid, and $n_1$ and $n_2$ the glass and fluid refractive index, respectively. The lenses are placed upon the twelve pentagonal faces of the solid, comprised between the two spheres of radius $R_1$ and $R_2$. A more accurate representation of the lenses themselves is given in Fig. 4.

Fig. 4

Realistic representation of the telescope assembly. Because the instrument is, geometrically, a dodecahedron, the lenses could be either pentagonal or circular in shape. The pentagon outer diameter $d_p$ and surface are 24% and 16%, respectively, greater than the circle ones: (a) pentagonal shape lenses and (b) circular shape lenses.

Because we are dealing with a dodecahedron, the lenses could have either a pentagonal shape or a circular one. In both cases, the supporting frame must occupy the edges of the pentagonal faces. In this work, the thermo-mechanical behavior of both configurations has been investigated.

2.1.

Radius of Curvature Change Due to Gravitational Loads

To mitigate the geometrical aberrations coming from the lens deformations under the gravitational loads, it is necessary to know the amount of displacement the lenses are subjected to in terms of radius of curvature. To not being limited by the thin plate approximation and given the complex nature of the problem in terms of geometry and loads, it has been chosen to perform finite element analysis for all the lenses of the assembly to estimate stresses, strains, and displacements, relying to analytical calculations only to compute the radius of curvature change.

The telescope will have a face pointing at the local zenith and, therefore, one toward the nadir. Hence, it is possible to recognize an uppermost lens and a lowermost one. In terms of thermo-mechanical behavior, other two lens typologies are significant: the lateral low glass elements (five lenses), adjacent to the lowermost lens and pointing 26.55 deg below the horizon, and the lateral high ones (five lenses), adjacent to the uppermost one and pointing 26.55 deg above the horizon. Each kind of glass element, given its placement within the dodecahedral structure, is characterized by a certain type of load:

• 1. The lowermost lens is subjected to a hydrostatic load given by the weight of a fluid column of length $2R_2$. The related pressure, the highest among those beared by the lenses, is distributed in an axial-symmetrical way. The lens weight is added to the hydrostatic force, having the same direction, and the lens radius of curvature tends to decrease;

• 2. Each of the five lateral lenses placed in the dodecahedron lower hemisphere forms with the horizontal plane (where lies the lowermost lens); the dihedral angle of 116.55 deg, namely they are inclined of 26.55 deg with respect to the gravity vector. As such, they are subjected to their own weight, with the gravity vector rotated of 63.45 deg (toward the lowermost lens) with respect to the normal to the lens plane, and to the hydrostatic pressure, acting normally to the plane of the lens and linearly variable along the lens diameter. For these lenses, the hydrostatic force and the lens weight component normal to the lens plane present the same direction and the radius of curvature hence reduces. Given the not axial symmetric loads, the radius of curvature in the meridional plane of the lens will be different from that in the sagittal one, producing a cylindrical deformation (at least in a first-order approximation);

• 3. For the five lateral lenses located in the higher hemisphere, the same considerations described above for the lower hemisphere lenses apply, but hydrostatic force and lens weight exhibit opposite directions and they tend to compensate each other;

• 4. The uppermost lens is subjected to its own weight, which tends to increase the radius of curvature. The load is again axial symmetric and the deformation follows a spherical pattern.

In Fig. 5, the displacements along the direction parallel to the lens optical axis ($Z$ axis) are depicted for all the optical elements of the assembly, considered with both pentagonal and circular shape (the lens lies in the $X\text{–}Y$ plane). Such displacements are computed with respect to an unloaded lens, characterized by an ideal meniscus shape geometry with outer and inner radius, respectively, of $R_1=1120\mathrm{mm}$ and $R_2=1000\mathrm{mm}$. The lenses are simply supported along the edges.

Fig. 5

$Z$ displacements of the MezzoCielo lenses subjected to gravitational loads. From the top: uppermost, upper hemisphere, lower hemisphere, and lowermost lenses. (a) Circular lenses and (b) pentagonal lenses.

In the same Fig. 5, the positive sign denotes outward displacements, whereas the negative one inward displacements, namely toward the sphere center. The numerical analysis described in these figures (and in the following as well) have been carried out using the AutoCad tool Fusion360.

In Tables 2 and 3, instead, the lenses maximum displacement and the corresponding radius changes in the meridional and sagittal plane with respect to the nominal value of $R_1=1120\mathrm{mm}$ are reported for circular and pentagonal lenses, respectively. These results have been achieved for lenses simply supported along the edges. Moreover, both shapes have been defined on the basis of considerations reported in Sec. 2.3.

Table 2

Maximum displacements and radius changes for circular lenses with only gravitational loads acting.

	Max displacement (μm)	ΔR1 – meridional (μm)	ΔR1 – sagittal (μm)
Lowermost lens	+9.6	−85	−85
Lateral low lenses	+3.9	−26	−32
Lateral high lenses	−2.2	+9	+12
Uppermost lens	−2.7	+8	+8

Table 3

Maximum displacements and radius changes for pentagonal lenses with only gravitational loads acting.

	Max displacement (μm)	ΔR1 – meridional (μm)	ΔR1 – sagittal (μm)
Lowermost lens	+7.1	−59	−59
Lateral low lenses	+2.4	−18	−21
Lateral high lenses	−1.1	+8	+9
Uppermost lens	−1.5	+8	+8

The plus sign in the second column of Tables 2 and 3 indicates outward displacement, while that in the third and fourth ones, a radius increase.

In summary, Fig. 5 and Tables 2 and 3 suggest that the pentagonal lens, given the same boundary conditions, exhibits lesser deformations with respect to the circular one. This could be due to the fact that the supports along the long edges of the pentagonal shape are closer together than those for the circular one and so allow smaller deflections in one meridian. Now, the radius of curvature tolerances for precise optics can be as small as 0.01% of the radius itself, namely, in this case, around $100\mu \mathrm{m}$. Observing the expected variation of the radius of curvature in Tables 2 and 3, it can be noted that such deviations are slightly lesser than the construction tolerances: this suggests that the gravitational loads do not produce an appreciable modification of the lens shape, that is, they do not substantially affect the telescope sphericity.

2.3.

Frame Sizing

We will now focus upon the sizing of the frame supporting the lenses and on the coupling between such elements.

The starting point is the frame material choice. The lattice has to provide the required rigidity to reduce the lenses displacements, without reaching the yielding strength limit; moreover, its material has to exhibit a coefficient of thermal expansion (CTE) similar to the BK7 one to minimize thermal stresses, has to be compatible with fluorine fluids, affordable and feasible in large quantities. Such characteristics lead to the choice of metallic materials. Among them, it has been selected Kovar, a iron-nickel-cobalt alloy, which exhibits the same elasticity, strength, and possibility of mass production of steel, but with a CTE significantly lesser.

The sizing of the Kovar frame is based on the following considerations, which represent constraints and requirements:

• 1. The frame will occupy the edges of a dodecahedron inscribed in a 1120-mm radius sphere. Hence, pentagonal faces have to be realized with edges of 799.281 mm;

• 2. Because the frame has to support the lenses, the lattice is conceived as a sequence of L-shape beams arranged along the edges of the platonic solid. Moreover, to assure a proper interface between the pentagonal support of the single lens, each of the L-shape beam has to exhibit, along the edge direction, an inclination of 26.55 deg toward the center of the solid itself;

• 3. The flanges of the beams are in contact with the fluid. To minimize their deflections, allowing at the same time a good interface between lenses and frame, it is chosen to consider 20-mm long, 10-mm thick flanges. From construction science, such dimensions should ensure a flange deflection less than $2\mu \mathrm{m}$ (it should be noticed that such value was obtained considering the flange itself clamped at one end—that in contact with the pentagonal frame—and subjected to the lens weight, modeled as a distributed load along the flange free edge). The 1000-mm clear aperture of the lenses is therefore surrounded by the frame flanges;

• 4. The height of the $L$-shape beam is constrained by the overall geometry. To allow a clear aperture of 1000 mm and to realize a dodecahedral lattice fitting exactly within a 1120-mm radius sphere, the frame $L$-beams should be no more than 45 mm high, with webs of 35 mm;

• 5. To improve the quality of the coupling frame-lenses in terms of displacements, a 10-deg taper angle is foreseen.

Strictly speaking, such considerations apply for a frame supporting lenses of pentagonal shape. For circular lenses, the frame cannot be simply seen as an assembly of $L$-beams and the area it covers is around 2.3 times greater than that requested for pentagonal shape lenses.

Regarding the lens geometry, the outer radius has to be 1120 mm, with thickness of 120 mm and 1000 mm clear aperture. The contact with the frame will take place along the entire length of the frame webs and flanges. Given the need for this type of coupling, it is possible to estimate that the pentagonal shape lens, occupying around 90% of the dodecahedral face surface, exhibits a collecting area around 1.17 times greater than the circular one (which occupies 77% of the entire face). For both lenses, sharp edges should be smoothed out creating chambers: it is immediate to observe that such operation is much more critical for the pentagonal lens, which requires, in addition to those of the circular lens, also the beveling of all the five sharp corners to avoid stress concentrations. The size of the beveling has been firstly chosen arbitrarily, and then, after several finite element simulations, it has been selected the one assuring minimum stress. The lenses lateral sides exhibit a 10-deg taper angle. The front and rear views of the single lens assembly are depicted in Fig. 6 for both pentagonal and circular glass elements.

Fig. 6

Front (top) and rear (bottom) view of the MezzoCielo lens assembly: (a) circular lenses and (b) pentagonal lenses.

In terms of thermal design, it is desirable that the metallic frame temperature exhibits the same variation law of the glass shell. The values of $\tau$ and $T_p$ in Eq. (7) have hence been used to estimate the required frame thermal resistance. After some calculations, we found that for the frame to show a thermal gradient similar to the glass one, it has to be covered with an insulating material having thermal conductivity of around $0.025\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$: a polyurethane layer 7.5-mm thick for the pentagonal shape lens and 5-mm thick for the circular one should assure the required thermal balance. The equivalent thermal resistance given by the frame and lenses ones, arranged parallel to each other, is consistent with the glass shell total resistance employed in Eq. (7).

Finally, regarding the coupling between lenses and frame, it is chosen to employ adhesive bonding. Indeed, adhesives distribute stresses more uniformly than mechanical fasteners, are more flexible, lighter in weight, lower in cost, and easier to assemble than those made by mechanical methods.⁸ For the side bonding, the one bearing the most load, the two components epoxies Milbond is employed as structural adhesive. The axial joint, instead, has to provide primarily the watertightness of the system; therefore, it must not chemically react with the fluid, along with a reasonable degree of rigidity and strength. Even though the most employed sealants are the silicone based adhesives, the latter are not suitable for the MezzoCielo application due to their low joint strength. A suitable alternative could be then represented by the Scotch-Weld Epoxy Adhesive 2214 Regular, a one part epoxy produced by 3M Company, which combines high rigidity, strength, and sealing properties.

A first sizing of the adhesive layers could be done in terms of athermalization. Indeed, as suggested in Ref. 8, if the bonding layer has a particular thickness, the assembly will, to a first-order approximation, be athermal in the radial and axial direction. This minimizes stresses build-up within the opto-mechanical components which are caused by differential expansion or contraction of the lens, mounting and adhesive under temperature changes. This thickness can be computed with

Eq. (12)

$$t_a=l\frac{(1-{\nu }_a)({\mathrm{CTE}}_m-{\mathrm{CTE}}_g)}{{\mathrm{CTE}}_a-{\mathrm{CTE}}_m-{\nu }_a({\mathrm{CTE}}_g-{\mathrm{CTE}}_a)}.$$

In Eq. (12), ${\nu }_a$ is the adhesive Poisson’s ratio and the subscripts $a$, $m$, and $g$ refer to adhesive, metal, and glass, respectively. With Eq. (12), we can compute the structural adhesive thickness substituting to the length $l$ the lens radius $d/2$ and the sealant thickness if $l=t$, being $t$ the lens thickness. Strictly speaking, Eq. (12) applies to annular adhesive layer placed radially along the lens circumference: the employment of this formula to size the axially located layer of sealant in terms of athermalization should be considered a first-order approximation to be verified through the numerical simulations.

In Table 7, the materials taken into account for the several components of the telescope assembly are listed along with their main properties, namely Young’s module $E$, bending, yield and shear strength for glass, metal, and adhesives, respectively, thermal conductivity $k$ and CTE. Such properties can been retrieved in Refs. 5, 8, and 9. In particular, the bending strength of a glass is a complex function of both chemical composition and surface finish. In the context of this work, for BK7, it was chosen to consider one of the lowest values of strength reported by Schott in Ref. 10, namely the 50% probability failure breakage bending strength for glass with “coarse” surface finish.

Table 7

Telescope assembly materials main properties.

	Material	E (MPa)	Strength (MPa)	k (W m−1 K−1)	CTE (K−1)
Lenses	BK7	82,000	48	1.2	$7.1·{10}^{-6}$
Frame	Kovar	138,000	345	17	$6.5·{10}^{-6}$
Adhesive	Milbond epoxy	592	17.7	1.36	$62·{10}^{-6}$
Sealant	2214 reg. epoxy	5170	20.7	0.4	$49·{10}^{-6}$
Insulating	Polyurethane	/	/	0.025	/

By virtue of the CTE values in Table 7, we obtain, on the basis of Eq. (12), a thickness of 2 mm for the structural adhesive and 0.5 mm for the sealant. Given the frame dimensions and the forces at stake, it is chosen to employ a 20-mm wide structural adhesive layer and a 18-mm wide sealant layer. In this way, almost the entire lens-frame interface surface would be covered with adhesive, except for few millimeters, useful to allow the expansion under the gravitational loads. The side view of the single lens assembly with a zoom detailing the arrangement of lens, frame, structural adhesive, and sealant is reported in Fig. 7.

Fig. 7

Side view of the MezzoCielo lens assembly and its main components.

As defined in geometry, materials, and thermo-mechanical loads, it is now possible to perform finite element analysis of the pentagonal and circular assemblies to verify the analytical results and the actual system feasibility.

3.

Results and Discussion

In this section, the main results of the finite element analysis are presented for all the telescope lenses, considered both pentagonal and circular in shape, in terms of temperatures, safety factors, and displacement distribution. The gravitational loads taken into account descend from the mass values reported in Table 8 for all the main components of the MezzoCielo system. In this table, the fluid mass is given by the product between the telescope inner volume, $V_i=(4/3)\pi R_2^3$, and the fluid density, ${\rho }_f=1670\mathrm{kg}/{\mathrm{m}}^3$; the telescope mass is, instead, computed as the sum of the mass of the 12 lens-frame systems and that of the fluid.

Table 8

Mass of MezzoCielo telescope and its main elements.

For a thermal description of the systems under analysis, we can refer to Fig. 8, in which the outer surface of the lenses assemblies is depicted for both pentagonal and circular glass elements.

Fig. 8

Temperature distributions upon the outer surface of the lenses assemblies: (a) circular lens and (b) pentagonal lens.

The results in Fig. 8 have been achieved considering a steady-state situation, characterized by a (constant) outer temperature of $-5°\mathrm{C}$ in natural convection regime and assemblies inner surface at the constant temperature of 20°C (not shown in Fig. 8). The numerical results confirm that regardless of the lens shape, the lowest temperature reached by the optical device is around 14°C and, above all, that such temperature is the same for both frame and glass components: the temperature differences are so minimized and stresses and strains as well.

Regarding the safety factors, the latter are computed using the following formula:

Table 9

Safety coefficient and allowable stress for the MezzoCielo materials.

The maximum stress, instead, is computed in terms of Von Mises stress for the metallic frame and maximum principal stress for glass and adhesives. Several simulations have demonstrated that the most critical situation (characterized by the highest stresses, namely the lowest safety factors) occurs when a combination of thermal and gravitational loads is taken into account. In Fig. 9, it is reported the safety factors distribution for the lowermost lens, subjected, in addition to its own weight, to the highest hydrostatic pressure and to the thermal loads descending from the configuration described in Fig. 8. Along the pentagonal frame edges, a fixed constraint (with no displacements and rotations possibility in the $X,Y,Z$ directions) has been applied.

Fig. 9

Safety factors distribution for the lenses assemblies: (a) circular lens and (b) pentagonal lens.

As it can be noticed by observing Fig. 9, the safety factors are higher than 14 for each component of both assemblies: this assures the design robustness and a promising result in the context of the telescope actual realization.

The displacement distribution is, instead, shown in Fig. 10 for all the lenses, both pentagonal and circular in shape. Loads and constraints are those described above for the safety factors simulation: a temperature difference of about 6°C across each assembly, gravitational loads modulated as a function of the assembly inclination with respect to the gravity vector and fixed frame edges are therefore the boundary conditions of the analysis.

Fig. 10

$Z$ displacements of the MezzoCielo lenses subjected to thermal and gravitational loads. From the top: uppermost, upper hemisphere, lower hemisphere, and lowermost lenses. (a) circular lenses and (b) pentagonal lenses.

In the same Fig. 10, the lens displacements are computed considering the glass elements bonded to the metallic frame, which is not shown; the positive sign denotes outward displacements, the negative one inward displacements, and toward the telescope center.

A comparison between Figs. 5 and 10 clearly shows that the temperature influence is much more pronounced with respect to the gravitational loads one: indeed, the overall displacements reported in Fig. 10 not only present higher absolute values but also for all the lenses, regardless of their location in the telescope assembly, they exhibit an inward shift with approximately the same pattern, namely a spherical one, with maximum values at the center and minimum ones at the edges; the cylindrical deformation, typical of the lateral lenses subjected to the mechanical loads alone, is hence strongly reduced. In addition, Tables 5 and 6 suggest that the analytical results slightly oversize displacements and radius changes as well. Given the same load and constraint conditions, the pentagonal lenses displace less than the circular ones, probably as a consequence of their supports arrangement, which tends to keep unchanged their position and the related deformation.

4.

Stress and Birefringence

In this section, the birefringence analysis of the MezzoCielo glass elements is performed: the objective is the quantification of birefringence in terms of refractive index change and phase retardation along the principal stresses directions. The latter quantity, in particular, will be employed to compute the Mueller matrix for each lens, with which it will be possible to characterize the polarization type induced by these relatively large glass elements.

The birefringence phenomenon happens when the refractive index of a glass, normally isotropic, becomes locally anisotropic due to the presence of stress. This means that, along the directions parallel and perpendicular to that of the stress, a difference in the values of $n$ is established, which, in turn, produces phase retardation and optical path difference proportionally to the stress itself.¹¹

The birefringence analysis follows the procedure employed in Ref. 12. Therefore, from the FEA executed in Fusion360, the stress tensor for each node of each lens has been estimated

The stress tensor and corresponding principal stresses and directions have hence been estimated for each lens of the assembly, considering both gravitational and thermal loads acting. In particular, it has been verified that the eigenvector associated to the third principal stress is parallel to the sphere normal (actually, this fact is strictly true for the lenses central portion, the one mostly interested by the light passing, while along the edges and the vertices—for the pentagons—such statement represents an approximation, because the third principal direction can deviate from the sphere normal up to 20 deg). Therefore, at least in the first approximation, we can compute the amount of birefringence as

In Tables 10 and 11, a summary of the maximum values of principal stresses, refractive index difference, and phase retardation for the circular and pentagonal lenses, computed for glass thickness of 120 mm and $\lambda =580.7\mathrm{nm}$, is reported. A visual representation of the amount of birefringence and phase retardation is, instead, depicted in Figs. 11 and 12, respectively, for all the lenses of the assembly, both pentagonal and circular in shape.

Table 10

Principal stresses, refractive index difference, and phase retardation max values for the circular lenses at Te=−5°C.

Table 11

Principal stresses, refractive index difference, and phase retardation max values for the pentagonal lenses at Te=−5°C.

Fig. 11

Refractive index difference between the first and second principal directions for the MezzoCielo lenses. From the top: uppermost, upper hemisphere, lower hemisphere, and lowermost lenses. (a) Circular lenses and (b) pentagonal lenses.

Fig. 12

Phase retardation (in rad) between the first and second principal directions for the MezzoCielo lenses. From the top: uppermost, upper hemisphere, lower hemisphere, and lowermost lenses. (a) Circular lenses and (b) pentagonal lenses.

As we can see from the above figures and tables, birefringence and phase difference increase passing from the lens center to the edges, as a consequence of the fact that the first principal stress values are the highest in the latter regions. The related optical path difference goes from 0 to around $\lambda /2$ passing from the lenses center to the edges. Moreover, the birefringence, proportional to the difference between the first and second principal stresses, results always positive: this means that the refractive index along the first principal direction is greater than that parallel to the second principal one, and the latter therefore represents the propagation fast axis. On the basis of the Mueller theory,¹³ which provides the most general description of the interaction between light and optical elements in terms of polarization, each lens can be seen as a wave-plate, namely as a phase retarder which can be considered linear (at least in a first-order approximation). Therefore, the entity of the phase retardation (and hence of the polarization) induced between the principal directions can be described by the following matrix:

However, because the light must cross two specular glass elements, rotated of $\pi$ rad relative to each other in the dodecahedral structure, the overall Mueller matrix becomes $M_{\delta }(\theta )M_{\delta }(\theta +\pi )$: the total retardation tends hence to zero and the light can cross the instrument without significant change in the polarization status.

5.

Conclusions

In this paper, the thermo-mechanical sizing of the MezzoCielo telescope structure has been described.

On the basis of geometrical and optical considerations, employed as starting point, the telescope optimal design has been chosen to be a dodecahedral one. The foreseen gravitational loads have been used to compute the expected lenses radius of curvature changes, and it has been verified that the desired radius tolerances (for high-quality optics) are greater than the load-induced displacements. Such considerations apply to both pentagonal and circular lenses.

In addition, the (supposed) environmental conditions during the winter activity have been chosen to size the telescope frame with the objective to minimize the thermal gradients between the instrument components (glass optics, metallic frame, and adhesives). Indeed, several finite element analysis have shown that the combination of both thermal and mechanical loads is much more severe, in terms of stresses, with respect to the only gravitational ones: safety factors and displacements have been therefore estimated in such worst-case scenario. The results are promising because they confirm the frame robustness and rigidity.

Eventually, the principal stresses within the lenses (both circular and pentagonal in shape) have been estimated and employed to compute the amount of birefringence (in terms of refractive index difference and phase retardation). The result was the achievement of a retardation map for each lens: on the basis of the Mueller theory, it has been deduced that the phase retardation (comprises between 0 and $\lambda /2$ for the single lens) tends to zero when two specular lenses are crossed in sequence.

In summary, it has been verified that the pentagonal lenses exhibit a lesser amount of displacements and stresses with respect to the circular ones, given the same boundary conditions; moreover, their birefringence is located mainly in a limited area around the edges, while the circular lenses birefringence occupies a significant portion of the clear aperture; finally, employing pentagonal lenses would allow reducing the obstruction produced by the metallic frame, for the benefit of the amount of gathered light and, therefore, limiting magnitude of the observable objects and resolution.

These advantages are counterbalanced by the increased manufacturing complexity and cost related to the realization of pentagonal lenses respect to “more standard” circular ones. Indeed, the abovementioned perks can be obtained only for pentagonal lenses properly made, namely characterized by a suitable degree of beveling, in particular near the vertices and the adjacent sharp edges.

The final choice regarding the MezzoCielo lenses shape should be made after the execution of tests on optical and mechanical bench, prepared to validate the results presented in this work.