Integrated thermal disturbance analysis of optical system of astronomical telescope

Transcription

1 Integrated thermal disturbance analysis of optical system of astronomical telescope * Dehua Yang, Zibo Jiang, Xinnan Li National Astronomical Observatories / Nanjing Institute of Astronomical Optics & Technology, Chinese Academy of Sciences, Nanjing, China ABSTRACT During operation, astronomical telescope will undergo thermal disturbance, especially more serious in solar telescope, which may cause degradation of image quality. As drives careful thermal load investigation and measure applied to assess its effect on final image quality during design phase. Integrated modeling analysis is boosting the process to find comprehensive optimum design scheme by software simulation. In this paper, we focus on the Finite Element Analysis (FEA) software-ansys-for thermal disturbance analysis and the optical design software-zemax-for optical system design. The integrated model based on ANSYS and ZEMAX is briefed in the first from an overview of point. Afterwards, we discuss the establishment of thermal model. Complete power series polynomial with spatial coordinates is introduced to present temperature field analytically. We also borrow linear interpolation technique derived from shape function in finite element theory to interface the thermal model and structural model and further to apply the temperatures onto structural model nodes. Thereby, the thermal loads are transferred with as high fidelity as possible. Data interface and communication between the two softwares are discussed mainly on mirror surfaces and hence on the optical figure representation and transformation. We compare and comment the two different methods, Zernike polynomials and power series expansion, for representing and transforming deformed optical surface to ZEMAX. Additionally, these methods applied to surface with non-circular aperture are discussed. At the end, an optical telescope with parabolic primary mirror of 900 mm in diameter is analyzed to illustrate the above discussion. Finite Element Model with most interested parts of the telescope is generated in ANSYS with necessary structural simplification and equivalence. Thermal analysis is performed and the resulted positions and figures of the optics are to be retrieved and transferred to ZEMAX, and thus final image quality is evaluated with thermal disturbance. Keywords: Integrated modeling, Thermal disturbance, Astronomical telescope, Power series, Zernike polynomials, Figure representation, Finite element method, Temperature field 1. INTRODUCTION An astronomical telescope is a multi-disciplinary integration of optics, mechanics and control sub-systems. With the increment of complicity of novel telescopes and with the development of information technology, system simulation method has been introduced into the field of astronomical telescope design to assess telescope performance and cost and further accelerates optimization process of the design. [1] To simultaneously achieve cost and performance goals, all components of the telescope must be developed, normally by commercial softwares, as blocks/models of an integrated system. [2] From the system-engineering point of view, for a complex telescope, an integrated telescope model plays a key role in verifying the interactions and interfaces between the blocks -various areas of the design. The models are actually virtual telescopes which are, respectively, encapsulated algorithm simulating independent aspect of the telescope, say, the telescope is represented by FEM software in the structural domain and a cope of the same telescope in the optical domain by any other optical software. [3] In this sense, the central work of integrated modeling with commercial software packages is the need for the data management and transformation between different models, further to synchronize them. We confine the thermal disturbance to be thermal effect on structure and optical components of the telescope, regardless with any seeing issues invoked, therefore, only thermal deformation is to be considered. In this narrowed sense, this paper is to discuss about basic issues related to structural-thermal-optical integrated performance analysis for * Correspondence: dhyang@niaot.ac.cn; Telephone: ; Fax: ; Nanjing Institute of Astronomical Optics & Technology (NIAOT), Bancang Str. 188, Nanjing, P. R. China Modeling, Systems Engineering, and Project Management for Astronomy III, edited by George Z. Angeli, Martin J. Cullum, Proc. of SPIE Vol. 7017, 70171N, (2008) X/08/$18 doi: / Proc. of SPIE Vol N-1

2 general optical telescopes. We review interface between thermal model and structural model. Complete polynomial with spatial coordinates is introduced to present temperature field analytically. Alternately, we borrow linear interpolation technique, derived from shape function in finite element theory, for temperature transformation to structural nodes, thereby the temperature from thermal model can be kept with as high fidelity as possible. Focus casts on the optics figures transformation between thermal disturbance analysis with Finite Element Analysis (FEA) software - ANSYS and the optical system design with optical design software-zemax. First, based on schematic data flow and analysis procedure, we elaborate the integrated modeling with thermal effect and optical system from an overview of point. Afterwards, we discuss about the establishment considerations of thermal model based assumed temperature field. Complete power series is introduced to express thermal field, if discrete temperatures are available/measured. In the next sections, the algorithms for optical figure transformation are discussed and commented on the widely used method Zernike polynomials and the power-series-expansion based complete polynomials. In addition, we discuss that these methods are essentially applicable to surfaces of non-circular geometry. At the end, we present an illustration with an optical telescope, which has a parabolic primary mirror of 900 mm in diameter. With necessary structural simplification and equivalence in ANSYS, finite element model is established with most interested parts of the telescope associated with the topic of this paper. To be specific, we are interested in the optical components and related structure which may affect the most thermally, therefore, we include the elevation assembly but omit the azimuth assembly of the telescope. Thus, thermal analysis is then performed, with assumed thermal field, to retrieve the resulted positions and figures of the optics to transfer into ZEMAX for the final image quality evaluation. The degradation of image quality is given in comparison of spot diagrams and interferograms in cases of being with or without thermal disturbance. 2. INTEGRATED THERMAL ANSYSIS OF OPTICAL TELESCOPE Integrated modeling method involves multi-discipline coupled analysis and communication between various engineering softwares. Fig. 1 shows the data flow and analysis process of the thermal disturbance affecting the optical system of a telescope. Beforehand, the Finite Element Model of telescope has been generated, say, in ANSYS and the optical system designed in ZEMAX base on opto-mechanical configuration of the telescope. Thermal disturbance as exterior load is to be applied as boundary condition on the FE model so that thermal deformation of structure as well as of optical components can be computed and hence the interested deflection (including figure distortion and rigid displacement) of optical components can be retrieved. Normally, the deflection data is given in the coordinate system defined in FE model for convenient establishment the model. It is necessary to transform the deflection to the required format used by ZEMAX or to be represented by any algorithm acceptable to ZEMAX. Therefore, Star Ray ZEMAX Image Quality Fig. 1 Thermal-optical analysis interface Proc. of SPIE Vol N-2

3 the first work is to define coordinate transformation relationship between the coordinate systems of the FE analysis domain and the optical disign domain; the next is to transfer the deflection information of optical components, as additional distortion of ideal figures, to the optical system design in ZEMAX. Thus, by ray tracing with light from star, image quality degradation of the telescope can be estimated. 3. THERMAL MODEL-TEMPERATURE FIELD REPRESENTATION In structural sense, thermal disturbance will cause such effect as thermal strain-induced deflection of mechanical parts and optical components in telescope system. Thermal strains can cause both pointing errors and surface errors. It is easy to evaluate the thermal effect for a telescope exposed to a uniform temperature field. However, it is normally out of the question to have such a uniform thermal environment for telescopes in operation, needless to talk about solar telescopes when the varying and strong solar radiation collected during observation. Even so thermal strains are repeatable and predictable if temperature is known, there are too many possibilities for temperature distribution over the structure to calculate all of them in advance and store the results in look-up tables for close-loop compensation. Some telescopes take advantage of the fact that thermal effects are slowly varying phenomenon to operate an FE analysis of the telescope with the temperature field actually measured. Afterwards, the calculation result is decomposed into the thermal induced axis misalignments, the actual pointing error and the optical figure error. The axis misalignment and pointing error will be expressed by transformation matrix for azimuth and elevation coordinate system, which may be compensated for by control system of the telescope, while the figure error of optical components will bring about degradation of image quality if no deformable mirror active optics available. [4] Since the temperatures are obtained/measured by sensors discretely located in the space, the next step is to reconstruction the temperature field that contains the whole telescope, or, in other words, to apply the measured temperature, with corresponding coordinates, to the structural model of the telescope established by FEM. There are several approaches possible for mapping the temperature to the structural model. One approach is to fit a function based on the discrete known temperature for representation of the temperature filed, thus, the structural model is to be submerged in the analytical temperature field and its mesh can be applied with specific temperatures on node level. Another approach is to fit a resultant function of the thermal effect of the structure model, then interpolate all the nodal effect of the structural model to retrieve the misalignment and figure errors of interest. We assume temperature is function of spatial coordinates (x, y, z) and introduce the complete polynomial of the three spatial variables to express the temperature field. The expression is as follows: n n k n j k i j k = = ijk (1) k= 0 j= 0 i= 0 t t( x, y, z) a x y z Where, a ijk is to-be-determined coefficient, which is determined by fitting, normally with least square algorithm, the polynomial with measured temperatures and corresponding positions (x, y, z); n is the highest order of the term in the polynomial, which is determined basically by the number of the measured temperatures. Reference 2 introduced another approach based on the concept of temperature seeds. The approach matches corresponding thermal and structural model nodes, within a specified dimension tolerance, and apply the same temperatures to those nodes. We resort that it is better/more accurate to the nodal temperatures in structural model by interpolation technique. [5] We superpose the two sets of nodes in the thermal model and structural model, and figure out up to eight closest thermal model nodes around a structural node within a given dimension range. Thus, borrowing the idea of shape function in finite element theory, we may use tri-linear, or the degenerated bi-linear and linear form, interpolation function to calculate the temperature and assign it to the encircled structural node with the temperatures of the surrounding thermal nodes. Thereby, the thermal load data can be transferred to the structure with as high fidelity as possible. A steady-state heat transfer analysis is firstly performed with thermal loads applied in Finite Element Analysis software, then, a complete set of nodal temperatures, the temperature field, is obtained for the next structural analysis. In some simple cases, the temperature field is detected and assumed to be with specific distribution or to be with uniform gradient in some direction. The nodal temperatures are easy to be calculated by simple interpolation. Then, by applying the nodal temperatures to the structural FE model, the overall thermal effect is calculated. [6] Proc. of SPIE Vol N-3

4 This paper uses equ. (1) to accomplish thermal mapping for the integrated analysis, which is to be detailed in the application section.. 4. INTERFACE WITH ZEMAX FIGURE REPRESENTATION FOR ZEMAX ZEMAX has predefined various model types of optical components, ranging from conventional standard surfaces to diffraction gratings, binary optics and others. To make the user interface as clear and clean as possible for definition of specific type of surface, ZEMAX uses different surface types to indicate what kinds of data are needed to define that type of surface. In ZEMAX, a standard surface can be a plane, spherical, or conic aspheric surface, whose sag or z-coordinate of the standard surface is given by [7] z standard 2 cr = (2) (1 + kcr ) where c is the curvature (the reciprocal of the radius), r is the radial coordinate in lens units and k is the conic constant. The conic constant is less than -1 for hyperbolas, -1 for parabolas, between -1 and 0 for ellipses, 0 for spheres, and greater than 0 for oblate ellipsoids. The coordinate system is defined with the sphere centered on the current optical axis, with the vertex located at the current axis position. ZEMAX treats planes as a special case of the sphere with infinite radius of curvature and conics as a special case as well. [7] ZEMAX uses parameter data to indicate the other types of surface. That is to say, the surface is expressed basically with the standard surface, z standard, in equ. (2) plus an additional term, z addition. Polynomial associated with spatial coordinates (x, y) is commonly used to represent z addition, whose coefficients are called parameter data. The surface expression is as follows, z = z + z = z + polynomial( x, y) (3) standard addition standard where, the polynomial normally has no constant term. ZEMAX has also predefined a great diversity of kinds of polynomial, such as Zernike polynomials and simple complete polynomials based on power series as following, z z AE ( x, y) N = + (4) standard i i i= 1 where, the additional summing term is a complete polynomial with two variables, x and y. There is no constant term, either. Besides, ZEMAX supports user-defined surfaces, which are created by writing software that defines the properties about the surface, and then dynamically linking the software into ZEMAX. [7] The concept of parameter data and the forms of equ. (3) and equ. (4) imply a convenient exterior interface to ZEMAX. The thermally induced deflection of the optical surfaces may be firstly expressed/fitted with polynomials like that in equ. (3), then, the original surfaces, regarded as the standard, can be updated with the extra thermal deformation of them. Thus, the degradation of image quality can be assessed by ZEMAX. 5. ZERNIKE POLYNOMIAL SURFACE FIT Assume the original surface figure is denoted with z origin, which may have been expressed in the form of equ. (3) in ZEMAX. With finite element software, we model the whole telescope including optical components and structure, and then compute thermal disturbance with thermal load properly applied. The thermal effect on the telescope is normally depicted as deformation of the structure and optical components. From the optical system point of view, the deformation is regarded as the disturbance of the distances between optical components and the surface figure distortion of them. Proc. of SPIE Vol N-4

5 Thus, we just need to focus on the distortion and displacement of the optical surfaces. Let us remind equ. (3), we borrow the idea of standard plus additional to deal with the thermally affected surface figure, and write the resultant deformed surface figure, z deform, like original plus thermally affected, thus, we have the following form zdeform = zorgin + zthermal (5) where, z thermal can be retrieved in discrete format nodal displacement from FEA calculation results. Basically, z thermal consists of rigid motion and local deformation of the optical surface. The Zernike polynomials are naturally one set of the candidates of fitting basis functions for the reconstruction of z thermal, which is defined as the following form (polar) z ( ρ, θ) = AZ ( ρ, θ) (6) thermal i i j= 0 where, n is the number of Zernike coefficients in the series, A i is the coefficient on the i th Zernike polynomial, ρ is the normalized radial ray coordinate, and θ is the angular ray coordinate. n (a) Original surface to be fitted (b) Fitted with 8-term Zernikes (c) Fitted with 15-term Zernikes Fig. 2 Surface figure fitted with power series polynomial (e) Fitted with 36-term Zernikes (d) Fitted with 24-term Zernikes Zernike polynomials are essentially one of an infinite number of complete sets of polynomials in two real variables. One of the interesting characteristics of Zernike polynomials is that the Zernikes are orthogonal in a continuous fashion over the interior of a unit circle, and in general, they will not be orthogonal over a discrete set of data points within a unit circle. They must be adapted in dependent form so as to keep orthogonality over otherwise shaped domain, for example, an annular shaped aperture (circular aperture with central obscuration). [7] However, this does not mean that the Zernikes can not be used over non-standard (filled circular) aperture. On the contrary, if we do not care the orthogonality of the components of the fitted surface/aperture with arbitrary shape, whatever basis function sets are applicable for fitting, including Zernike polynomials and the to-be-discussed complete power series polynomials in next section. Proc. of SPIE Vol N-5

6 BUD We try to apply Zernike polynomials to fit an example surface containing much information of high spatial frequency for the sake of deliberate comparison with the discussion in the latter section. As shown in Fig. 2, for convenience, we process the fitting work by Veeco software bundled together with its interferometer. Fig. 2 (a) shows the original surface to be fitted, which has a central hole in it. (b) shows the smooth surface fitted with 8-term Zernikes, (c) is fitted with 15-term Zernikes, (d) is fitted with 24-term Zernikes and (e) is fitted with 36-term Zernikes. It is seen that the more terms used in the Zernike polynomials the more high frequency details are represented. Nevertheless, even the figure fitted with 36-term Zernike polynomials is still a quite rough approximation of the original, for most information of high spatial frequency is filtered out/lost. In the next section, we will compare the residual fitted errors of the uses of Zernikes and complete power series. On the other hand, optical aberrations and figures are normally composed of information of low spatial frequency, and the first 36 terms of Zernikes are hence sufficient for their reconstruction. 6. SURFACE FIT WITH POWER SERIES COMPLETE POLYNOMIAL We now resort complete polynomial - power series with two spatial coordinates (x, y) to represent Z thermal, thus, we have i z ( xy, ) axy thermal n n j = ij j (6) j= 0 i= 0 where, n is the highest order in the series, a ij is the coefficient. This equation includes rigid motion, in other words, the U, BUD -BUD U (a) Fitted with 6-order polynomial (28 terms) (b) Fitted with 10-order polynomial (66 terms) (c) Fitted with 16-order polynomial (153 terms) r) Fig. 3 Surface figure fitted with power series polynomial (e) Fitted with 25-order polynomial (351 terms) (d) Fitted with 20-order polynomial (231 terms) Proc. of SPIE Vol N-6

7 constant term, which must be excluded before imported into ZEMAX, while it will be added to the distance between optical surfaces. Fig. 3 arrays a series of the fitted figures of the same surface fitted with Zernike polynomials above, where the central hole are also filled/interpolated with the fitted complete polynomial. In Fig. 3, (a) is the fitted figure by the complete power polynomial with highest order of 6, which is of the same order that the 24-term Zernike polynomials may possess. However, it is clear to see that only spatial information with comparatively low frequency is represented, even so, it is better than its Zernike counterpart is. As we see, with the increase of the order, more and more details of the figure are elaborated by the fitting polynomial. Nevertheless, this does not mean there is no upper limit of the order, the fact is that the higher is the order, the least square is more computational, and the machine truncation error is more serious. Empirically, is an appropriate order appointment. The residual errors of those fitted figures in Fig. 2 and Fig. 3 are plotted in Fig. 4 for usage comparison of Zernike polynomials and complete power series. It is clear to see the residual error of power series polynomial fitting is much less than that of Zernikes fitting at the same highest order of the polynomial. Moreover, it is predictable that the term incensement of Zernike polynomials produces much slower fading of the residual error. In addition, we see the fact that the 25-order complete power series does not derive expected least error but jumps to much greater number, though its fitted figure looks closest like the original surface. As confirms the argument mentioned above that there is an appropriate upper limit for the order of the complete polynomial generated with power series = 0) ) a - Power series. Zernike polynomials 4) = -U I I I I Highest order ofthe polynomial. Fig. 4 Residual fitted error comparison Compared with the forgoing Zernike polynomial in previous section, the common power series is prevailing and straightforward to use and easier to handle the fitting precision. This merit enables it especially suitable for surface with error/component of high spatial frequency. To be detailed, the handy Zernike polynomials are normally available up to 36 terms, whose highest order is 12. Although Zernike polynomials are characterized in orthogonality and it is possible to extend Zernike polynomials to more terms, the extension work is obviously troublesome in typing and programming. In fact, its orthogonality is dependent on the specific shape of the domain on which the Zernike polynomials are to apply, in other words, to keep orthogonality, domain of different shape requires different form of Zernike polynomials, as is actually a hard work from the mathematical and programming point of view. From the fitting point of view, aiming for the least residual error, it is not necessary to require orthogonality of basis functions/polynomials, the least square procedure will produce coefficients/amplifiers for each component of the fitted function/data. Actually, if we unite the like terms in the equation fitted by Zernike polynomials, we will eventually get a power series. Based on this Proc. of SPIE Vol N-7

8 understanding, complete polynomial type power series is dominant in usage for surface figure reconstruction and for programming, where only the highest required order, n, is needed to be defined beforehand and the formula is merely two nested loops of summation. The highest power order ZEMAX accepted is 20, which makes a maximum of 230 polynomial aspheric coefficients, excluding constant term. [7] Further, from the spatial information filtering point of view, that the higher is the order of the fitting basis function means the more figure error with higher spatial frequency can be filtered out/depicted. 7. APPLICATION TO AN OPTICAL TELESCOPE A telescope is taken for instance of the integrated thermal disturbance analysis. As its optical system shown Fig. 5, part 1 is the primary mirror which is a paraboloidal with a diameter of 900 mm, part 2 is a fold flat, part 3 is the secondary mirror which is also a paraboloidal sharing the same focus of the primary mirror. Part 4 is the imager at whose entrance where a converging lens exists. The distance between the primary and the secondary is 1640 mm from vertex to vertex. The parallel incident lights are converged by the primary mirror and reflected by the secondary to be a thinner parallel beam, which is shunted by the fold flat into the imager. Before imaged, the beam is converged into a point on the image plane by the entry lens. All the optical components use the same material of K4 glass, and the structure uses steel ,0 Fig. 5 Layout of the optical path Obviously, if the optical surfaces are ideally good, the image quality through this optical system is ideally good. Fig. 6 shows spot diagram with all of the 4 field lights at wavelength of nm by ZEMAX. The finite element model of the telescope is established with ANSYS. Shown in Fig.7, the finite element model is established only with most interested parts of the telescope, and necessary structural simplification and equivalence are applied, too. In the application of this paper s case, the deformation of the optical components and the deflection of the related structure parts are meaningful, which are the most thermally influential factors of the degradation of the image quality. The details of azimuth assembly, driving systems, bearings and so on are not of importance and hence excluded in the model. Additionally, the surfaces of the optical components are not necessarily precisely modeled as same as that defined with its figure equation. Because we are concentrating on the relative difference of the surfaces before and after the thermal load applied, the slight runout of the modeled surface from its accurate surface can only bring about tiny thermal and structural difference of the FEA results. Complete polynomial expressed by the power series in equ. (1) is used to generate temperature field of the example Proc. of SPIE Vol N-8

9 telescope. At first, a set of randomly selected points are scattered within the space occupied by the telescope, then they are assigned with random temperature of uniform distribution (0, 1) degrees Celsius, respectively. Then,the temperature field is to be fitted with the temperature and coordinate information of these points by equ. (1). Afterwards, the nodal temperatures in the finite element model of the telescope are simply interpolated in the fitted analytical temperature field, and applied as thermal body loads on the structural model to start a consequent static analysis, thereby, the thermally induced distortion and deflection of the structure and optical components are computed. Dt e.eee. e.eee it n e.eee. ieee it D e.eee. 2.eee it D e.eee. 3.eee it Fig. 6 Spot diagram at wavelength nm Fig. 7 Finite element model with parts of interest Fig. 8 Deflection contour plot of the telescope The deflection contour of the telescope is plotted in Fig. 8, it is seen that the telescope tube exhibits a conspicuous declination, which causes rigid motions, namely, piston, tip and tilt, of the mirrors. The tube-declination-induced rigid motions should be excluded when doing the analysis of figure reconstruction and image quality degradation. The contours of the distorted mirrors are shown in Fig. 9. It is seen that thermal distortion leads to figure error of low spatial Proc. of SPIE Vol N-9

10 frequency. For the smaller mirrors like the secondary and the tertiary fold mirror, the most distortion is rigid motion. In this case, Zernike polynomials are used for fitting all the three distorted surfaces. Table1 lists the rigid motions, the first three Zernike coefficients, of the three mirrors, where piston of both the primary and the secondary, and tip/tilt of the primary are much greater comparatively. The large piston induced by thermal expansion of tube is understandable because the steel tube, which mainly determines the distance between the mirrors, is the longest part in the interested assembly analyzed. On the other hand, because the primary mirror is much larger, in size, than the other two mirrors, which consequently makes it more spatially sensitive to thermal disturbance, it is predictable that the primary will contribute most to the final image degradation. The thermally affected image quality is tabulated in Table 2, where, comparison is listed for three cases of the full thermally affected case, the thermally affected case with all mirrors piston eliminated and the thermally affected case with all mirrors rigid motion eliminated. Corresponding spot diagrams of central field are shown in Fig. 10 and Fig. 11; and corresponding interferograms of the thermally distorted wavefront are shown in Fig. 12 and Fig. 13. We see that the rigid motion of the mirrors contributes ~60% degradation of the image quality. Fig. 9 Distortion contours of the mirrors Table 1 Zernike rigid motion of the mirrors Piston Tilt Tip Primary mirror Secondary mirror Fold mirror Table 2 Image quality of central field Image spot radius (um) Full thermally affected without piston without piston and tip/tilt RMS Max Proc. of SPIE Vol N-10

11 D: l PM D: l PM IMA: DDI3, -MMMM MM 1MM: DDI3, IMMM MM DM I PM DM I.WI PM 1MM: M.MI3 I.AAA MM 1MM: M.MI3, 2.MMM MM Fig. 10 Thermally affected spot diagram (micron) Fig. 11 Thermally affected spot diagram with piston, tip/tilt eliminated (micron) Fig. 12 Thermally affected interferogram Fig. 13 Thermally affected interferogram with piston, tip/tilt eliminated 8. CONCLUSION The integrated thermal disturbance analysis has been generally discussed with examples and applications in this paper. The discussion mainly covered temperature field representation, thermal load application, interface between optical model and structural model, surface figure reconstruction methods which has been compared and commented in more details. Based on these studies, we draw conclusions classified in the following points. Integrated modeling analysis involves multi-discipline coupled analysis and communication between various engineering softwares, where coordinate transformation and surface reconstruction are fundamental work. Borrowing the idea of shape function in finite element theory, tri-linear, or the degenerated bi-linear and linear form, interpolation function is introduced to calculate the nodal temperature of the structural model with the temperatures of the surrounding thermal nodes. Thereby, the thermal load data can be transferred to the structure Proc. of SPIE Vol N-11

12 with as high fidelity as possible. Complete power series polynomials are dominant in usage for surface reconstruction and for programming, where power series formula is simply two nested loops of summation. Moreover, complete polynomial with the three spatial variables is introduced to represent the temperature field. Compared with Zernike polynomials, the common complete power series is prevailing and straightforward to use and easier to handle the fitting precision. Especially, it is quite suitable for surface with error/component of high spatial frequency. From the fitting point of view, it is not necessary to require orthogonality of basis functions/polynomials and the fitted Zernike polynomials are eventually a set of power series. If we do not care the orthogonality of the fitting components, whatever basis function sets are applicable for fitting surface/aperture with whatever arbitrary shape. The thermally affected surface figure is written in the form of original plus thermally affected, borrowed from the concept of optical surface presentation in ZEMAX. Basically, the higher is the order of the fitting power series, the more details of the figure are elaborated, but the higher is the order, the more computational is the least square procedure, and the more is the fitting error. Order of is empirically recommended. ACKNOWLEDGEMENT The research is sponsored by the National Natural Science Foundation of China under Grant No The authors would like to thank their colleagues Dr. Yong Zhang and Mr. Heng Zu for their fruitful discussion about the work in this paper and for their help with the preparation of this paper. REFERENCES 1. Dehua Yang, Lingzhe Xu, Application Review of System Simulation Technique in Astronomical Telescope Design, Journal of System Simulation, to be published, John D. Johnston, Joseph M. Howard, Gary E. Mosier, et al., Integrated modeling activities for the James Webb Space Telescope: Structural-Thermal-Optical Analysis, Proc. of SPIE, 2003, 5487: George Angeli, Anna Seguson, Robert Upton, et al., Integrated modeling tool for large ground based optical telescopes, Proc. of SPIE, 2003, 5178: Peter Eisentraeger, Martin Suess, Verification of the Active Deformation Compensation System of the LMT/GMT by End-to-End Simulation, Proc. of SPIE, 2000, 4015: Wenyuan Weng, Finite Element Method in Computation of Mechanical Structure. Press of South East University, Dehua Yang, Kunxin Chen, Yongjun Liang, et al., Design evolution and evaluation of the segmented reflecting schmidt mirror cell of the lamost telescope, Proc. of SPIE, Vol. 5877, 2005: Focus Software, Inc., ZEMAX Optical Design Program: User s Guide, Version 7.0, March, Proc. of SPIE Vol N-12