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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 6 — Mar. 25, 2013
  • pp: 7065–7081
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Opto-thermal analysis of a lightweighted mirror for solar telescope

Ravinder K. Banyal, B. Ravindra, and S. Chatterjee  »View Author Affiliations


Optics Express, Vol. 21, Issue 6, pp. 7065-7081 (2013)
http://dx.doi.org/10.1364/OE.21.007065


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Abstract

In this paper, an opto-thermal analysis of a moderately heated lightweighted solar telescope mirror is carried out using 3D finite element analysis (FEA). A physically realistic heat transfer model is developed to account for the radiative heating and energy exchange of the mirror with surroundings. The numerical simulations show the non-uniform temperature distribution and associated thermo-elastic distortions of the mirror blank clearly mimicking the underlying discrete geometry of the lightweighted substrate. The computed mechanical deformation data is analyzed with surface polynomials and the optical quality of the mirror is evaluated with the help of a ray-tracing software. The thermal print-through distortions are further shown to contribute to optical figure changes and mid-spatial frequency errors of the mirror surface. A comparative study presented for three commonly used substrate materials, namely, Zerodur, Pyrex and Silicon Carbide (SiC) is relevant to vast area of large optics requirements in ground and space applications.

© 2013 OSA

1. Introduction

Evidently, one of the most challenging tasks is to carefully handle the excessive heat generated by large radiation flux incident on the primary mirror. Equally important is the choice of high quality, thermally stable and rigid material capable of maintaining the optical figure over a specified range of observing conditions. The thermal response of the solar mirror is influenced by radiative heating of the mirror caused by direct light absorption and the temperature imbalance with the surroundings. Mirror heating adversely affects the final image quality of the telescope in two ways, as is explained below.

First and foremost, it is hard for a massive mirror having large heat capacity and high thermal resistance to attain a temperature balance with the surrounding air. Under such conditions, the temperature difference between the mirror surface and the ambient air leads to refractive index fluctuations along the beam path. This ‘mirror seeing’ produces wavefront aberrations responsible for the image blurring. To preserve the image quality of the telescope, the effect of mirror seeing has to be minimized. This means the temperature difference between the mirror surface and the ambient air should not exceed beyond ±2° C [9

9. P. Emde, J. Kühn, U. Weis, and T. Bornkessel, “Thermal design features of the solar telescope GREGOR,” Proc. SPIE 5495, 238–246 (2004) [CrossRef] .

]. More crucially, the temperature uniformity across the mirror surface has to be maintained within ±0.5° C. Several approaches which include, air conditioning [10

10. A. Miyashita, R. Ogasawara, G. Macaraya, and N. Itoh, “Temperature control for the primary mirror of Subaru Telescope using the data from ‘Forecast for Mauna Kea observatories’,” Publ. Nat. Astron. Observ. Jpn 7, 25–31 (2003).

], fanning the optical surface [11

11. C. M. Lowne, “An investigation of the effects of mirror temperature upon telescope seeing,” Mon. Not. R. Astron. Soc. 188, 249–259 (1979).

], ventilating mirror interior, resistive heating [12

12. B. Bohannan, E. T. Pearson, and D. Hagelbarger, “Thermal control of classical astronomical primary mirrors,” Proc. SPIE 4003, 406–416 (2000) [CrossRef] .

, 13

13. R. J. S. Greenhalgh, L. M. Stepp, and E. R. Hansen, “Gemini primary mirror thermal management system,” Proc. SPIE 2199, 911–921 (1994) [CrossRef] .

] have been proposed to regulate the mirror temperature for ensuring high quality astronomical observations. Laminar air flow across the mirror homogenizes the temperature by disintegrating the ‘thermal plumes’ closer to the surface. These methods also speed up the thermalisation process by improving the convective heat transfer rate between the substrate and the surrounding air.

Secondly, a poorly conducting mirror is differentially heated to create temperature inhomogeneities within the substrate volume. Thermally induced material expansion or contraction can significantly alters the optical quality of the telescope mirror. This problem can be eliminated by choosing mirror substrates made of ultra low expansion (ULE) material. In fact, the demand for high quality astronomical optics has led to the development of many attractive low expansion materials such as Zerodur (Schott), fused quartz, Cervit, ULE (Corning), Clearceram (Ohara) and Astro-Sitall etc. These ULE materials usually have high heat capacity and low thermal conductivity, implying they can retain a significant amount of heat, leaving the mirror prone to undesirable seeing effects and surface buckling under high thermal loads.

The severity of mirror heating problem scales adversely with the aperture size. To reduce the overall weight and large thermal inertia, the most preferred choice is to use lightweighted mirror structures that are easy to mount and do not deform significantly under gravity. Besides, the reduced mass also facilitates efficient and faster cooling of the mirror in harsh thermal conditions. For space applications, reducing the weight of the mirror has greater premium on the overall cost reduction of the payload.

2. Mathematical formulation

In order to study the primary mirror heating and the associated thermal stress/strain fields affecting the surface figure, each physical process should be identified. In the following we give a brief mathematical formulation of the heat transfer problem along with appropriate prescription for boundary and initial conditions.

2.1. Heat transfer

In a simplified form, the telescope mirror can be considered as a 3D circular disk whose front face is exposed to intense solar irradiance I(W/m2). The thin metallic coating on the faceplate absorbs about 10%–15% of radiant flux which is converted into heat. The temperature distribution inside the mirror changes as the heat conducts from front surface to the rear side. The mirror also exchanges heat (via convection and radiation) with ambient air, which is at a temperature Ta. A heat transfer equation can be derived based on the principle of conservation of energy, solution of which gives the temperature distribution T(x,y,z,t). The 2nd order partial differential equation relevant to the current studies for the conductive heat transfer in a solid can be expressed as [20

20. F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer (Wiley, 2001).

]
x(KTx)+y(KTy)+z(KTz)=Tt+q(x,y,z;t)
(1)
where T is the temperature, K = k/(ρCp) is the thermal diffusivity of the system, Cp is the heat capacity, k is the thermal conductivity, ρ is the density and q = Q/(ρCp), where Q(x, y, z; t) represents the source term for heat. In the present case there is no internal heat source, i.e. Q(x, y, z; t) = 0 everywhere and at all instants of time. The heat flux enter the system through the surfaces, a fact which must be accommodated in the boundary condition.

For solving the temperature T(x,y,z,t) we must assign:
  • – the initial condition
    T(x,y,z;t=0)=Ti(x,y,x)and
    (2)
  • – the boundary condition that the net incident flux, which is balance between the incident and emitted fluxes, is transmitted into the substrate and accounts for the temperature gradients on the domain surface Ω. This condition is described by,
    k(Tn)|Ω=[q0h(TTa)εsσR(T4Ta4)]|Ω
    (3)
    where n is surface normal vector at any point on Ω. In Eq. (3) the l.h.s. denotes the heat inflow due to temperature gradient on the surface, q0 is incident flux due to solar radiation, h (TTa) describes the convective heat flow to surroundings, where Ta(t) is ambient temperature and h is the heat transfer coefficient. The term εsσR(T4Ta4) describes the radiative exchange with surroundings, where εs is the emissivity of the surface and σR is Stefan–Boltzmann constant.

2.2. Model prescription for initial temperature, heat flux and ambient temperature

2.2.1. Initial temperature Ti(x, y, z)

Due to poor thermal conductivity and asymmetric heating, the initial temperature distribution Ti(x, y, z) inside the mirror cannot be assumed to be uniform, i.e., independent of (x, y, z). In order to establish a definitive temperature field at t = 0, simulations were carried out by subjecting the mirror to several day-time heating and night-time cooling cycles [21

21. R. K. Banyal and B. Ravindra, “Thermal characteristics of a classical solar telescope primary mirror,” New Astron. 16, 328–336 (2011) [CrossRef] .

]. The resulting temperature distribution was used as initial Ti for all subsequent studies.

2.2.2. Surface boundary conditions for inward heat flux q0

The boundary prescription for inward heat flux q0 in Eq. (3) depends on the solar irradiance I(W/m2) that varies between the sun-rise and the sun-set as [21

21. R. K. Banyal and B. Ravindra, “Thermal characteristics of a classical solar telescope primary mirror,” New Astron. 16, 328–336 (2011) [CrossRef] .

]
q0=γIand
(4)
I=I0[sin(φ)sin(δ)+cos(φ)cos(δ)cos(H)]
(5)
where γ is absorption coefficient (fraction of radiant energy that is absorbed and converted into heat) of the metallic film on the reflecting surface, I0(W/m2) is irradiance amplitude when sun is at zenith, δ (rad) is declination angle of the sun, φ (rad) is latitude of the place and H = (π/12)t is solar hour angle and t is the local time.

2.2.3. A simple model for ambient temperature Ta

In Eq. (3), the heat loss or gain by free convection and radiation is driven by ambient temperature Ta. For convenience, we can split the ambient temperature into two parts, i.e., Ta(t) = TD(t) + TN(t), where TD(t) and TN(t) describe the day-time ambient heating and night-time cooling, respectively. For cloud-free sky conditions, these two terms can be approximated using a simple model given by Gottsche and Olesen [22

22. F. M. Gottsche and F. S. Olesen, “Modelling of diurnal cycles of brightness temperature extracted from METEOSAT data,” Remote Sens. Environ. 76, 337–348 (2001) [CrossRef] .

]
TD(t)=Tr+T0cos[πω(tτm)]fort<ts
(6)
TN(t)=Tr+Td+{T0cos[πω(tτm)]Td}exp[(tts)/κ]fortts
(7)
We use Eq. (6) and Eq. (7) as input to our FEA model to simulate a range of ambient temperature conditions typically existing at observatory locations. One such realization for 10°C ≤ Ta(t) ≤ 20°C and solar flux I(W/m2) is shown in Fig. 1. The assumed values for various parameters in Eq. (6) and Eq. (7) used to obtain a smoothly varying temperature curve in Fig. 1 are [21

21. R. K. Banyal and B. Ravindra, “Thermal characteristics of a classical solar telescope primary mirror,” New Astron. 16, 328–336 (2011) [CrossRef] .

]: T0 = 6.8°C; Tr = 13.4°C; Td = −3.5°C; ω = 12hrs (the half period oscillation); τm = 14 : 00 hrs (the time when temperature maxima is reached); ts = 18 : 00 hrs (the time when temperature attenuation begins); κ = 3.5 hrs (the attenuation constant). The ambient temperature Ta shown in Fig. 1 continues to rise well past the solar noon. This is due to thermal time lag in terrestrial environment that exists between the peak of solar flux and the peak of ambient temperature.

Fig. 1 Typical variations in ambient temperature (red curve, scale: vertical–left) and solar flux reaching the mirror surface (blue curve, scale: vertical–right). The peaks of solar flux and ambient temperature are shifted by 2 hrs due to thermal time lag.

2.3. Structural deformation

The heat conducted to different parts of the mirror builds up thermal strain which results in structural deformation. For isotropic linear elastic solid, the thermal strain depends on coefficient of thermal expansion α(K−1), the instantaneous temperature T(t) and the stress–free reference temperature Tref as:
εth=α(TTref)
(8)
If u, v, and w are the thermally induced deformation components in x, y and z direction, then the thermal strain can be completely specified in terms of u, v, and w and their derivatives. For small displacements, the constitutive equations relating the components of normal and shear strain to the deformation derivatives can be expressed as [23

23. M. H. Sadd, Elasticity: Theory, Applications, and Numerics (Academic Press, 2009), Chap. 2.

]:
εxx=ux;εxy=εyx=12(uy+vx)εyy=vy;εyz=εzy=12(vz+wy)εzz=wz;εzx=εxz=12(wx+uz)
(9)
The displacements at the bottom boundary of the mirror were kept fully constrained, i.e., u = v = w = 0, to avoid the rigid body motion.

3. The mirror geometry

Among several possibilities of lightweighted geometry, the open-back mirror configuration is highly suitable for solar telescopes [24

24. M. Süß, T. Volkmer, and P. Eisenträger, “GREGOR M1 mirror and cell design: Effect of different mirror substrate on the telescope design,” Proc. SPIE 7736, 77391l (2010) [CrossRef] .

]. It consists of a thin faceplate fused to an array of ribs formed by scooped-out material from the back. The pocket geometry can be circular, triangular, hexagonal, square or any other combination of these shapes. The resulting structure has side walls/ribs of certain thickness that supports the thin reflecting faceplate of the mirror from behind. Besides the mechanical support, the rib walls also provide the thermal flow path for heat to conduct away from the faceplate. The presence of discrete grid makes the heat removal rate to vary across the faceplate thus creating temperature impressions resembling the cellular geometry of the lightweighted structure.

Figure 2(a) shows the schematic of the parabolic mirror (conic constant =−1, focal length 4m) geometry chosen for the present studies. A similar design is also under consideration for the proposed NLST by India. The diameter of the mirror is 2m and the diameter of the inner hole is 0.42m. The honeycomb bottom comprising about 708 hexagonal cells is enclosed by 20 mm thick outer rim. The side length L of the hex-cell (inset image in Fig. 2(b)) is 34.6 mm and the wall/rib thickness t is 8 mm. The diameter D of the inscribed circle is 60 mm while the wall/rib depth varies from 148 mm for the outermost cells to 85 mm for the one at the center. The faceplate thickness is 12 mm. Compared to a solid geometry, the overall mass of the current lightweighted mirror is reduced by ≃ 68%. The main reason for choosing these hex-cell dimensions is because their structural stability has already been verified for the Gregor 1.5 m mirror telescope [25

25. T. Westerhoff, M. Schäfer, A. Thomas, M. Weissenburger, T. Werner, and A. Werz, “Manufacturing of the ZERODUR 1.5-m primary mirror for the solar telescope GREGOR as preparation of light weighting of blanks up to 4-m diameter,” Proc. SPIE 7739, 77390M–77390M-9 (2010) [CrossRef] .

]. The mirror has a 6-fold symmetry about the optical axis as seen in the CAD design shown in Fig. 2(b). The symmetry property is utilized to reduce the model size and computational requirements by selecting only one section of the mirror in the FEA studies.

Fig. 2 Mirror geometry: (a) The side view and (b) the CAD model of the mirror with open back lightweighted honeycomb structure and 12 mm thick faceplate on top. Mirror has 6-fold symmetry about the z-axis. The faceplate is shown placed only on one of the 6 sectors of the mirror.

4. The FEA results

The governing equations and associated boundary conditions for the heat transfer and the structural deformation problem are outlined in Section 2. The analytical solution for Eqs. (1)(3) and Eqs. (8)(9) exists only for simple and regular shapes. In most practical cases where object geometry is highly complex, boundary conditions are complicated, material properties are temperature dependent and various physical parameters are coupled, problem then has to be solved by numerical methods. We used COMSOL Multiphysics as our FEA tool to solve the time-dependent 3D heat transfer and structural problem. The segregated solver in COMSOL takes advantage of the direct coupling between heat transfer and structural model. That is, for each time step (Δt = 100 sec), the heat transfer module first solves for the temperature distribution of the mirror over 24 hours. Each temperature field is then passed onto the the structural analysis module to perform the stress-strain analysis. No data transformation was necessary since a common mesh is applicable to both thermal as well as structural models in COMSOL.

A 3D CAD model of the mirror conforming to optical prescriptions is shown in Fig. 2. The reflecting surface of the faceplate was modelled using a 100 nm thick aluminium coating on the faceplate. The FEA model was run on the 1/6th axisymmetric section of the mirror that was partitioned into 100644 tetrahedral mesh elements. The top surface of the mirror was continuously heated by time varying irradiated solar flux plotted in Fig. 1. The thermal and structural response was evaluated at three different ranges of ambient temperature Ta i.e., 0°C–10°C, 10°C–20°C and 20°C–30°C. The front, back and the outer rim of the mirror were subjected to convective cooling with the heat transfer coefficient h = 5 W/m2 representing natural cooling and h = 15 W/m2 representing a moderately forced cooling. In addition, surface-to-ambient radiation condition was specified for the faceplate. Also the symmetry boundary condition i.e., k(∇T · n) = 0, was imposed on the side edges of the mirror. In the FEA model, the mirror heating was ‘on’ (Eq. (4)) between sunrise (6:00 hrs) and sunset (18:00 hrs) and ‘off’ for rest of the time. The standard values of the material parameters used in FEA studies are listed in Table 1[21

21. R. K. Banyal and B. Ravindra, “Thermal characteristics of a classical solar telescope primary mirror,” New Astron. 16, 328–336 (2011) [CrossRef] .

].

Table 1. Values of material properties used in FEA studies

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4.1. Thermal print-through and temperature nonuniformities

Fig. 3 The impact of daytime heating on the Zerodur mirror with ambient temperature varying between 20–30°C and h = 5 W/m2. (a) Temperature print-through pattern on the mirror faceplate subjected to daytime radiative heating. (b) Temperature variations along the radial line AA′ of the mirror faceplate at different times.

The maximal departure of ΔTmax, δTmax and ξTmax for an ambient temperature range Ta : 20°C–30°C for Zerodur, Pyrex and SiC material is listed in Table 2. From the Table 2, we may note that the amplitude of both ΔT and δT depend crucially on the thermal conductivity of the material and the convective heat transfer coefficients used in simulations. A more efficient cooling mechanism is necessary, especially in Zerodur and Pyrex, to remove the excessive heat to minimize the mirror seeing effects.

Table 2. The temperature difference for different glass material computed for heat transfer coefficient h1 = 5 W/m2 and h2 = 15 W/m2.

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A diurnal variations of δT computed between two reference points located at the rib wall and the center of hex-cell is shown in Fig. 4. Even though the surface temperature is always above the ambient, the impact of ΔT and δT are manifestly more pronounced during the noon hours.

Fig. 4 Temperature difference between two reference points located at hexagonal cell-center (C) and cell-edge (E).

4.2. Thermally induced surface deformation

The non-uniform temperature field such as shown in Fig. 3(a), causes the mirror substrate to deviate away from its nominal defined shape. Thermally induced shape change can be highly irregular in lightweighted structures. The surface distortions for each temperature input field were computed by solving the FEM structural model. The reference temperature Tref for the undeformed mirror geometry is assumed to be 20°C. The impact of lightweighted hex-cell geometry is clearly noticeable in the simulated image showing the surface deformation in Fig. 5(a) for Zerodur mirror. The corresponding axial displacements w along the radial line AA′ of the mirror faceplate for different substrate materials are also plotted in Figs. 5(b)–5(d). The surface displacement w is positive in Pyrex and SiC substrates and gradually increases in radially outward direction. However, for Zerodur material, w can be both positive or negative because of zero crossing of the CTE around 25°C. For a given temperature field, the surface undulations, represented by δw, are not uniform over the entire mirror surface but exhibit same periodicity as the underlying hex-cell structure. The displacement close to the mirror edges is expressed by ξw. The gradient across the mirror surface is approximated by the slope w′ = dw/dr. The numerical values computed for δw, ξw and w′ during the peak mirror temperature at t = 14 : 00 are summarized in Table 3.

Fig. 5 Thermally induced structural deformation: (a) for the Zerodur mirror and the corresponding z-displacements w (nm) along the radial line AA′ of the mirror faceplate for (b) Zerodur, (c) for Pyrex and (d) for SiC at different times of the day. The ambient temperature range was 20–30°C and h = 5 W/m2.

Table 3. Parameters describing the mechanical deformation of the surface.

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The FEM studies were also performed at other temperature range of interest, i.e., 0°C–10°C and 10°C–20°C. The surface deformation data was found almost similar to that given in Table 3.

5. Optical analysis

5.1. Surface sag and Zernike polynomials

The sag data for the axisymmetric mirror segment was imported into MATLAB and converted into full aperture using co-ordinate transformation of the form:
[xyzΔs]=[cosθsinθ00sinθcosθ0000100001][xyzΔs]
(11)
where θ = /3 and N = 2, 3, 4, 5 and 6. Equation(11) maps the surface sag data from the original mirror segment (unprimed coordinate) to other (primed coordinate) sectors.

The MATLAB function TriScatteredInterp which makes use of Delaunay triangulation algorithm, was used for interpolating irregularly spaced sag data over an uniformly sampled array of points. A typical example of surface sag for full aperture Pyrex mirror is shown in Fig. 6(a). The sag values vary between 2.8μm–6.13μm and the root mean square (RMS) sag over the full aperture is 5.3μm. The RMS wavefront error is twice the RMS sag. The diurnal variation of RMS sag for Pyrex, SiC and Zerodur, shown in Figs. 6(b) and 6(c), is consistent with variation of their CTE under specified temperature range. The optical aberrations of the thermally deformed surface were further analyzed by fitting Zernike annular polynomials to 2D sag data. The main advantages of Zernike polynomials arise due to their orthogonality over the unit circle, rotational invariance, completeness property and direct relationship of expansion coefficients with known aberrations of the optical system [27

27. R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976) [CrossRef] .

,28

28. V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Appl. Opt. 33, 8125–8127 (1994) [CrossRef] [PubMed] .

]. Each Zernike term is a product of three components, namely, a normalization factor, a radial part and an azimuthal part of type cos or sin. The surface sag Δs(ρ, ϕ; ε) for an annular pupil with obscuration ratio ε, radial variable ρ and azimuth angle ϕ can be written in terms of Zernike annular polynomial as:
Δs(ρ,ϕ;ε)=j=1ajZj(ρ,ϕ;ε)
(12)
where j is a single index polynomial-ordering term and aj are expansion coefficients [29

29. We have chsoen the normalization factor and the indexing scheme similar to Noll’s notaion which is also followed in Zemax ray-tracing software.

].

Fig. 6 (a) Surface sag for thermally deformed Pyrex mirror at t = 14 hrs and h = 5 W/m2. The variation of RMS surface sag with daytime (b) for Pyrex and SiC and (c) for Zerodur mirror.

The j-th order Zernike polynomials Zj(ρ, ϕ; ε) in Eq. (12) is defines as [28

28. V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Appl. Opt. 33, 8125–8127 (1994) [CrossRef] [PubMed] .

]
Zj(ρ,ϕ;ε)={2(n+1)Rnm(ρ;ε)cosmϕm0,jeven2(n+1)Rnm(ρ;ε)sinmϕm0,joddn+1Rn0(ρ;ε)m=0
(13)
where n is radial degree and m is azimuthal frequency such that mn and n − |m| is even. The procedure to obtain expressions for radial component Rnm(ρ;ε) in Eq. (13) is discussed in Ref [28

28. V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Appl. Opt. 33, 8125–8127 (1994) [CrossRef] [PubMed] .

].

Fig. 7 Zernike expansion coefficients corresponding to (a) defocus, primary, secondary and other higher order spherical aberrations which contribute to overall all figure change and (b) the remaining low-amplitude terms which contribute to mid-frequency surface errors.

Table 4. Low-order radial Zernike aberration terms.

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Table 5. Predominant low-amplitude Zernike terms consisting of azimuth and radial parts.

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Fig. 8 Residual map obtained by subtracting the reconstructed Zernike surface from the original sag data.

5.2. Image quality

The surface aberration data of thermally deformed mirror was analyzed using Zernike polynomials and the optical performance was evaluated in a commercial ray tracing software Zemax. A simple telescope design was created in Zemax as shown in Fig. 9. The design conforms to optical prescription of M1 given in Section 3. The aberration data can be incorporated into the optical model of the telescope by assigning a suitable surface type –the Zernike Standard Sag, to the M1 in Zemax. This surface type is commonly used for describing mechanical deformations. Selecting Zernike Standard Sag surface allows user to define the standard surface (e.g. plane, spheres, conics etc) plus additional Zernike terms that can be imposed on the standard surface to represent surface aberrations [32

32. Zemax, Optical Design Program (User’s Manual, 2012).

]. The aberration data for M1 along with number of Zernike terms, normalized radius of the aperture and a list of externally computed Zernike coefficients was directly imported into Zemax using Extra Data Editor. The M2 was assumed to be free from any surface irregularity.

Fig. 9 Optical model of a Gregorian-type reflecting telescope in Zemax. M1: primary mirror; M2: secondary mirror; F1: primary focus; F2: secondary focus or image plane.

In Zemax, there are many ways to assess the optical performance of the system. We used spot diagrams and modulation transfer function to evaluate the performance of the telescope system. The thermal distortions are assumed to become worse when the mirror temperature reaches its maximum at t = 14 : 00 hrs. For the worst case scenario, the effect of thermally induced aberrations on the telescope image quality can be readily seen in the spot diagram shown in Fig. 10. Except for Zerodur material, the RMS spot radius (Figs. 10(a)–10(c)) for Pyrex and SiC is significantly larger than the size of the airy disc, implying the diffraction limited performance of M1 made of high CTE materials, is severely compromised by thermal heating. The contribution of low amplitude Zernike terms (Fig. 7(b)) is estimated by setting the radial terms (Fig. 7(a)) to zero in Zemax. Clearly, the low amplitude terms add significant scatter to the energy distribution as shown in Figs. 10(d)–10(f). The envelop of the spot scatter is also indicative of the periodicity and shape of the ‘surface bumps’ caused by uneven temperature distribution resulting from underlying hex-cell structure of the M1. Another way to examine the optical performance is to compute the contrast variations of the imaging system at different spatial frequencies. The variation of contrast, i.e., MTF for Pyrex and SiC is plotted in Figs. 11(a) and 11(b), respectively. Considering the severity of the thermal distortions, a steep fall in contrast at relatively low-frequencies is not completely unexpected.

Fig. 10 The spot diagram showing the telescope performance for a on-axis 0° field point. The blue traces represent the ray distribution in the image plane due to temperature induced surface imperfections in M1. The pattern has an envelop of six-fold symmetry due to hexagonal nature of the elements, as expected. The diffraction limited Airy disc is indicated by the black circle. The specified wavelength is 0.55 μm and the radius of the airy disc is close to 15 μm. Top row (a)–(c), when all the Zernike terms were included. Bottom row (d)–(f), without piston, focus and spherical terms.
Fig. 11 The system contrast in terms of transverse modulation transfer function at various spatial frequencies for a 0° field angle (a) for Pyrex and (b) for the SiC mirror. For Zerodur (not shown), the ray aberrations are well within the diffraction limits and three MTF curves would overlap.

The preceding studies were also repeated for convective heat transfer coefficient h = 15 W/m2. The corresponding RMS spot radii for the Zerodur, Pyrex and SiC were found to be 3.3μm, 1305μm and 1798μm. This is only a moderate improvement. This shows that forced circulation of ambient air alone in high CTE materials (Pyrex and SiC) is not enough to keep the thermal distortions within tolerable limits. For effective thermal control, the temperature of the circulated air has to regulated few degrees below the ambient temperature. A detailed CFD analysis is however needed to arrive at the optimal coolant air temperature that would minimize the mirror seeing and thermal distortions of the M1. The numerical studies presented in this paper would complement those efforts. Finally, we draw the attention of the scientific community to consider the present scheme of opto-thermal analysis for optimization of the solar telescope design.

6. Conclusions

Acknowledgments

We would like to thank Dr. Ping Zhou at Large Optics Fabrication and Testing Group, Arizona University, for suggesting to use a faster and more versatile MATLAB code ‘SAGUARO’ for the Zernike analysis.

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W. Cao, N. Gorceix, R. Coulter, K. Ahn, T. R. Rimmele, and P. R. Goode, “Scientific instrumentation for the 1.6 m New Solar Telescope in Big Bear,” Astron. Nachr. 333, 636–639 (2010) [CrossRef] .

9.

P. Emde, J. Kühn, U. Weis, and T. Bornkessel, “Thermal design features of the solar telescope GREGOR,” Proc. SPIE 5495, 238–246 (2004) [CrossRef] .

10.

A. Miyashita, R. Ogasawara, G. Macaraya, and N. Itoh, “Temperature control for the primary mirror of Subaru Telescope using the data from ‘Forecast for Mauna Kea observatories’,” Publ. Nat. Astron. Observ. Jpn 7, 25–31 (2003).

11.

C. M. Lowne, “An investigation of the effects of mirror temperature upon telescope seeing,” Mon. Not. R. Astron. Soc. 188, 249–259 (1979).

12.

B. Bohannan, E. T. Pearson, and D. Hagelbarger, “Thermal control of classical astronomical primary mirrors,” Proc. SPIE 4003, 406–416 (2000) [CrossRef] .

13.

R. J. S. Greenhalgh, L. M. Stepp, and E. R. Hansen, “Gemini primary mirror thermal management system,” Proc. SPIE 2199, 911–921 (1994) [CrossRef] .

14.

A. Ahmad, Handbook of Optomechanical Engineering (CRC Press, 1997), Chap. 5.

15.

T. L. Gray, M. W. Smith, L. E. Cohan, and D. W. Miller, “Minimizing high spatial frequency residual in active space telescope mirrors,” Proc. SPIE 7436, 74360M (2009.) [CrossRef]

16.

J. M. Tamkin, “A study of image artifacts caused by structured mid-spatial frequnecy fabrication errors on optica surfaces,” Ph. D. dissertion (Univesity of Arizona, 2010).

17.

R. N. Youngworth and B. D. Stone, “Simple estimates for the effects of mid-spatial-frequency errors on image quality,” Appl. Opt. 39, 2198–2209 (2000) [CrossRef] .

18.

E. Segato, V. D. Deppo, S. Debei, G. Naletto, G. Cremonese, and E. Flamini, “Method for studying the effects of thermal deformations on optical systems for space application,” Appl. Opt. 50, 2836–2845 (2011) [CrossRef] [PubMed] .

19.

Y. Ding, Z. You, E. Lu, and H. Cheng, “A thermo-optical analysis method for a space optical remote sensor optostructural system,” Opt. Eng. 43, 2730–2735 (2004) [CrossRef] .

20.

F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer (Wiley, 2001).

21.

R. K. Banyal and B. Ravindra, “Thermal characteristics of a classical solar telescope primary mirror,” New Astron. 16, 328–336 (2011) [CrossRef] .

22.

F. M. Gottsche and F. S. Olesen, “Modelling of diurnal cycles of brightness temperature extracted from METEOSAT data,” Remote Sens. Environ. 76, 337–348 (2001) [CrossRef] .

23.

M. H. Sadd, Elasticity: Theory, Applications, and Numerics (Academic Press, 2009), Chap. 2.

24.

M. Süß, T. Volkmer, and P. Eisenträger, “GREGOR M1 mirror and cell design: Effect of different mirror substrate on the telescope design,” Proc. SPIE 7736, 77391l (2010) [CrossRef] .

25.

T. Westerhoff, M. Schäfer, A. Thomas, M. Weissenburger, T. Werner, and A. Werz, “Manufacturing of the ZERODUR 1.5-m primary mirror for the solar telescope GREGOR as preparation of light weighting of blanks up to 4-m diameter,” Proc. SPIE 7739, 77390M–77390M-9 (2010) [CrossRef] .

26.

K. B. Doyle, V. L. Genberg, and G. J. Michels, Integrated Optomechanical Analysis (SPIE Press, 2002), Chap. 4 [CrossRef] .

27.

R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976) [CrossRef] .

28.

V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Appl. Opt. 33, 8125–8127 (1994) [CrossRef] [PubMed] .

29.

We have chsoen the normalization factor and the indexing scheme similar to Noll’s notaion which is also followed in Zemax ray-tracing software.

30.

D. W. Kim, B. J. Lewisa, and J. H. Burgea, “Open-source data analysis and visualization software platform: SAGUARO,” Proc. SPIE 8126, 81260B–81260B-10 (2011) [CrossRef] .

31.

R.A. Applegate, E.J. Sarver, and V. Kemsara, “Are all aberrations equal?,” J. Refract. Surg. 18, S556–S562, (2002) [PubMed] .

32.

Zemax, Optical Design Program (User’s Manual, 2012).

OCIS Codes
(110.6770) Imaging systems : Telescopes
(120.6810) Instrumentation, measurement, and metrology : Thermal effects
(160.4670) Materials : Optical materials
(220.0220) Optical design and fabrication : Optical design and fabrication
(230.4040) Optical devices : Mirrors
(080.1005) Geometric optics : Aberration expansions

ToC Category:
Imaging Systems

History
Original Manuscript: December 3, 2012
Revised Manuscript: February 19, 2013
Manuscript Accepted: February 25, 2013
Published: March 13, 2013

Citation
Ravinder K. Banyal, B. Ravindra, and S. Chatterjee, "Opto-thermal analysis of a lightweighted mirror for solar telescope," Opt. Express 21, 7065-7081 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-6-7065


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References

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  2. R. Gilmozzi, “Science and technology drivers for future giant telescopes,” Proc. SPIE5489, 1–10 (2004). [CrossRef]
  3. O. von der Lühe, “History of solar telescopes,” Exp. Astron.25, 193–207, (2009). [CrossRef]
  4. K. L. Stephen, R. R. Thomas, and W. Jeremy, “Advanced technology solar telescope,” Earth Moon Planets104,77–82, (2009). [CrossRef]
  5. C. J. Sánchez, M. Collados, D. Soltau, R. López, J. L. Rasilla, and B. Gelly, “Current concept for the 4m European Solar Telescope (EST) optical design,” Proc. SPIE7652, 76520S (2010). [CrossRef]
  6. S. S. Hasan, D. Soltau, H. Kärcher, M. Säß, and T. Berkefeld, “NLST: India’s National Large Solar Telescope,” Astron. Nachr.331, 628–635 (2010). [CrossRef]
  7. W. Schmidt, O. von der Lühe, R. Volkmer, C. Denker, S. K. Solanki, H. Balthasar, N. B. Gonzalez, Th. Berkefeld, M. Collados, A. Fischer, C. Halbgewachs, F. Heidecke, A. Hofmann, F. Kneer, A. Lagg, H. Nicklas, E. Popow, K.G. Puschmann, D. Schmidt, M. Sigwarth, M. Sobotka, D. Soltau, J. Staude, K.G. Strassmeier, and T.A. Waldmann, “The 1.5 meter solar telescope GREGOR,” Astron. Nachr.331, 796–809 (2012).
  8. W. Cao, N. Gorceix, R. Coulter, K. Ahn, T. R. Rimmele, and P. R. Goode, “Scientific instrumentation for the 1.6 m New Solar Telescope in Big Bear,” Astron. Nachr.333, 636–639 (2010). [CrossRef]
  9. P. Emde, J. Kühn, U. Weis, and T. Bornkessel, “Thermal design features of the solar telescope GREGOR,” Proc. SPIE5495, 238–246 (2004). [CrossRef]
  10. A. Miyashita, R. Ogasawara, G. Macaraya, and N. Itoh, “Temperature control for the primary mirror of Subaru Telescope using the data from ‘Forecast for Mauna Kea observatories’,” Publ. Nat. Astron. Observ. Jpn7, 25–31 (2003).
  11. C. M. Lowne, “An investigation of the effects of mirror temperature upon telescope seeing,” Mon. Not. R. Astron. Soc.188, 249–259 (1979).
  12. B. Bohannan, E. T. Pearson, and D. Hagelbarger, “Thermal control of classical astronomical primary mirrors,” Proc. SPIE4003, 406–416 (2000). [CrossRef]
  13. R. J. S. Greenhalgh, L. M. Stepp, and E. R. Hansen, “Gemini primary mirror thermal management system,” Proc. SPIE2199, 911–921 (1994). [CrossRef]
  14. A. Ahmad, Handbook of Optomechanical Engineering (CRC Press, 1997), Chap. 5.
  15. T. L. Gray, M. W. Smith, L. E. Cohan, and D. W. Miller, “Minimizing high spatial frequency residual in active space telescope mirrors,” Proc. SPIE7436, 74360M (2009.) [CrossRef]
  16. J. M. Tamkin, “A study of image artifacts caused by structured mid-spatial frequnecy fabrication errors on optica surfaces,” Ph. D. dissertion (Univesity of Arizona, 2010).
  17. R. N. Youngworth and B. D. Stone, “Simple estimates for the effects of mid-spatial-frequency errors on image quality,” Appl. Opt.39, 2198–2209 (2000). [CrossRef]
  18. E. Segato, V. D. Deppo, S. Debei, G. Naletto, G. Cremonese, and E. Flamini, “Method for studying the effects of thermal deformations on optical systems for space application,” Appl. Opt.50, 2836–2845 (2011). [CrossRef] [PubMed]
  19. Y. Ding, Z. You, E. Lu, and H. Cheng, “A thermo-optical analysis method for a space optical remote sensor optostructural system,” Opt. Eng.43, 2730–2735 (2004). [CrossRef]
  20. F. P. Incropera and D. P. DeWitt, Fundamentals of Heat and Mass Transfer (Wiley, 2001).
  21. R. K. Banyal and B. Ravindra, “Thermal characteristics of a classical solar telescope primary mirror,” New Astron.16, 328–336 (2011). [CrossRef]
  22. F. M. Gottsche and F. S. Olesen, “Modelling of diurnal cycles of brightness temperature extracted from METEOSAT data,” Remote Sens. Environ.76, 337–348 (2001). [CrossRef]
  23. M. H. Sadd, Elasticity: Theory, Applications, and Numerics (Academic Press, 2009), Chap. 2.
  24. M. Süß, T. Volkmer, and P. Eisenträger, “GREGOR M1 mirror and cell design: Effect of different mirror substrate on the telescope design,” Proc. SPIE7736, 77391l (2010). [CrossRef]
  25. T. Westerhoff, M. Schäfer, A. Thomas, M. Weissenburger, T. Werner, and A. Werz, “Manufacturing of the ZERODUR 1.5-m primary mirror for the solar telescope GREGOR as preparation of light weighting of blanks up to 4-m diameter,” Proc. SPIE7739, 77390M–77390M-9 (2010). [CrossRef]
  26. K. B. Doyle, V. L. Genberg, and G. J. Michels, Integrated Optomechanical Analysis (SPIE Press, 2002), Chap. 4. [CrossRef]
  27. R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am.66, 207–211 (1976). [CrossRef]
  28. V. N. Mahajan, “Zernike annular polynomials and optical aberrations of systems with annular pupils,” Appl. Opt.33, 8125–8127 (1994). [CrossRef] [PubMed]
  29. We have chsoen the normalization factor and the indexing scheme similar to Noll’s notaion which is also followed in Zemax ray-tracing software.
  30. D. W. Kim, B. J. Lewisa, and J. H. Burgea, “Open-source data analysis and visualization software platform: SAGUARO,” Proc. SPIE8126, 81260B–81260B-10 (2011). [CrossRef]
  31. R.A. Applegate, E.J. Sarver, and V. Kemsara, “Are all aberrations equal?,” J. Refract. Surg.18, S556–S562, (2002). [PubMed]
  32. Zemax, Optical Design Program (User’s Manual, 2012).

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