Some ideas about representations of aspheric optical surfaces Ding-Qiang Su and Ya-Nan Wang

Nanjing Astronomical Instrument Factory, Academia Sinica, Nanjing, China. Received 21 May 1984. 0003-6935/85/030323-04$02.00/0. © 1985 Optical Society of America. Having read the communication by Rodgers,1 we would

like to express our opinions and explain some facts. First we discuss the fundamental concepts and our points

of view: (a) The ideal shape of each surface in an optical system,

which is determined by the best image quality, exists objec­tively. The goal of testing and manufacturing an optical surface is to reach the ideal shape of the surface. Hence there is no need to find a new method for testing and manufacturing when a new representation is used to describe that ideal shape.

(b) According to the Weierstrass theorem it can be dem­onstrated that a power polynomial is accurate enough to represent the ideal shape of an optical surface only if its order is high enough. It is impossible to find another representation that can yield significantly better image qualities than that of a power polynomial because any function can be expanded into a power polynomial.

(c) Like a power polynomial, many other polynomials and functions can be accurate enough to represent the ideal shape of an optical surface. There does exist a speed of convergence problem in the approximation of a function. It is of signifi­cance to find a function that can represent the ideal shape of a surface as accurately as its power polynomial counterpart but using fewer coefficients. We have done some work on this.

The number of coefficients is not the only factor that should be considered. There are two other important factors: First, this kind of function must be of universal significance and thus suitable for a wide variety of aspheric surfaces. Second, the coefficients in the new function should be easier to determine and convenient for studying optical systems.

(d) In addition to polynomial and explicit functions, im­plicit functions F{x,y) = 0, parametric equations x = f1(t),y = f2(t), splines, and a combination of some primary functions can also be used to represent an optical aspheric surface. For example, by using parametric equations Chretien2 gave the strict solution to the shapes of surfaces in an aplanatic two-mirror system. Early in 1957 Su3 suggested using a segment function, like splines, to represent the shape of the correcting plate in a Schmidt system.

(e) The effect of using one function to fit another function depends not only on what kind of function is used but also on how the coefficients are determined. Special attention was given to this requirement in our methods.4 The coefficients are determined by minimizing the rms value of the difference

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of the slopes of the normals to the surface with respect to the ideal surface, i.e., the rms image diameter.

The process has been used to design many aspheric systems successfully.5-11 The basic function used in the process is the power polynomial; but the other function can be used in the process too.

There are a number of specific points to be addressed con­cerning the design for a submillimeter telescope as configured in Fig. 1.

Fig. 1. Four-mirror submillimeter telescope configuration as de­signed for the Arizona-Max Planck Institut 10-m SMT.

(a) The construction of the submillimeter telescope men­tioned by Rodgers1 is different from the original design of Meinel, Meinel, Su, and Wang (MMSW).5 The differences are as follows:

(1) The distance between primary and secondary mirrors is changed from 5515 to 5000 nm. This places the secondary twice as far from the caustic than in the original MMSW de­sign, easing the problem of crowding of the rays at the rim of the secondary. Also, the height (1399 mm) of the marginal ray at the secondary mirror is much higher than in the original MMSW design (997 mm).5

(2) The positions of mirrors 3 and 4 are different from the MMSW design.

(3) The radius of curvature of mirror 4 is also different. (4) The final ƒ ratio is changed from 6 to 10. These dif­

ferences have a large effect on the imagery. (b) Figure 2(a) in Rodgers1 is not our design result. (c) According to the spot diagrams and the transverse ab­

errations curves, Fig. 2(a) in Rodgers1 is only free from spherical aberration, but Figs. 2(c) and (d) are aplanatic. It is obvious that image qualities in a system that eliminates only spherical aberration are poorer than in one that is aplanatic. This fact does not bear on which function is better to express the surface because any function can be used to express the surface in a system that is free from spherical aberration and is also an aplanatic system.

(d) For comparison with Rodgers, we took a similar system wherein the positions of all mirrors and the shapes of mirrors 1 and 4 were exactly the same, and the ƒ ratio was nearly the same as his. We calculated this system using our method4 of

Fig. 2. Spot diagrams: (a) Rodgers' best design.1 (b) Our design whose secondary mirror and third mirror are described by a twenty-fourth-degree polynomial on a base sphere and a tenth-degree polynomial on a base sphere, respectively. (b-1) At a planar surface; (b-2) at an optimally curved surface having a -210.81-mm radius, (c) Our design whose secondary mirror is described by a twenty-fourth-degree polynomial on a base sphere; the third and fourth mirrors are described by a tenth-degree polynomial on base spheres, (c-1) At a planar surface; (c-2) at an optimally curved surface having a 160.87-mm radius. The Airy disk at 30 μm with a 0.74-mm (1.51-sec of arc) diameter is indicated

by three circles.

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eliminating spherical aberration and satisfying the sine con­dition. The secondary and tertiary mirrors are expressed by sphere plus power polynomials up to the twenty-fourth and tenth orders, respectively. The construction we obtained is shown in Table I, and the spot diagrams of this system in Fig. 2(b). For comparison the best spot diagrams Rodgers ob­tained [in his Fig. 2(c)] are shown in Fig. 2(a). The residual spherical aberration in our design is much less than that of Rodgers, and both size and symmetry of images in our design are also a little bit better than those of Rodgers. So the conic-plus-polynomial has the ability to express the shapes of aspheric surfaces in this telescopic design.

(e) Rodgers says that his Fig. 2(b) is the spot diagram of the system eliminating spherical aberration and satisfying the sine condition, mirrors 2 and 3 being described by twenty-sec­ond-degree polynomials. We calculated this system using our method in the same conditions. Mirrors 2 and 3 are also represented by twenty-second-degree polynomials, but the spot diagram is almost the same as Fig. 2(b) in this Letter.

Let us now discuss a general principle of optical design. Rodgers designs and appraises the optical system according to the principle of eliminating spherical aberration and sat­isfying the sine condition. For comparison with his system, we do the same as in the above. But in general it is better to define a merit function to express the quality of the optical

system. Furthermore, the number of aspheric surfaces is not necessarily limited to two. For example, it is better to design the submillimeter telescope based on the following conditions: (1) the merit function is defined by the rms value of image size in the 3-min of arc diameter of the field of view; (2) mirrors 2, 3, and 4 are sphere-plus-polynomial (the primary mirror is still taken as a sphere); (3) the power polynomial is up to twenty-fourth order for mirror 2 and to tenth order for mirrors 3 and 4; (4) all coefficients of polynomials (whole number is 19) are variables.

The construction we obtained is shown in Table II and the spot diagrams in Fig. 2(c) and Fig. 3. The image qualities are much better than those in Fig. 2(b). The maximum spread of images is only 0.09 sec of arc in the 3-min of arc diameter of the field of view in the optimally curved image surface. This diameter is not only much less than the Airy disk diam­eter at 30 μm but is also excellent for a ground-based optical telescope.

The ray fan diagrams shown in Fig. 4 indicate a significant difference that may explain why our polynomial program leads to a better design. Figure 4 [2(b-l)] is our approxima­tion to the Rodgers design [Fig. 2(c)]. Our ray fans show no oscillations until the scale is greatly magnified, as in Fig. 4 [2(c-2)]. These differences are probably due to subtleties within the optimization programs that are yet to be identified.

Table I. Four-Mirror System with Two Aspheric Surfaces

Table II. Four-Mirror System with Three Aspheric Surfaces

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Fig. 3. Spot diagrams of Fig. 2(c-2) amplified by 10. The diameter of the circle is 0.074 mm (0.151 sec of arc).

Fig. 4. Transverse ray aberrations at full field for several designs as optimized using our program. Note the relative absence of residual zonal oscillations. (a) Design shown in Fig. 2(b-l); (b) design shown

in Fig. 2(c-l); (c) design shown refocused in Fig. 2(c-2).

The result, however, is that an excellent design can be achieved for such a fast spherical primary; use of polynomials up to the twenty-fourth order is still superior to alternate functions thus far proposed.

Acknowledgements are due to Aden and Marjorie Meinel at the University of Arizona, John Stacey at JPL for their participation in preparation of this report, and to Yi Mei-liang, Cao Chang-xin, Shao Lian-zhen, and Liang Ming for helpful discussions and generous help.

References 1. J. M. Rodgers, "Nonstandard Representations of Aspheric Sur­

faces in a Telescope Design," Appl. Opt. 23, 520 (1984). 2. H. Chrétien, "Le télescope de Newton et Le télescope aplanéti-

que," Rev. Opt. 1, 11, 49 (1922). 3. Ding-Qiang Su, "The Research of Schmidt Correcting Lens," J.

Teaching Res. Nanjing Univ. Sci. 1, 5 (1958). 4. Ding-Qiang Su, Ya-Nan Wang, and Lan-Juan Wang, "The Nu­

merical Calculation of Aspherics in an Optical System," Acta Astron. Sin. 25, 86 (1984).

5. A. B. Meinel, M. P. Meinel, Ding-Qiang Su, and Ya-Nan Wang, "Four-Mirror Spherical-Primary Submillimeter Telescope De­sign," Appl. Opt. 23, 3020 (1984).

6. Ding-Qiang Su and Ya-Nan Wang, "Automation Correction of Aberration in Astro-optical Systems," Acta Astron. Sin. 15, 51 (1974) [Chin. Astron. 2, 171 (1978)].

7. Ding-Qiang Su and Lan-Juan Wang, "A Flat-field Reflecting Focal Reducer," Opt. Acta 29, 391 (1982).

8. Ding-Qiang Su, Lian-Zhen Shao, and Ming Liang, "A Configu­ration of the Optical System for a 5m Telescope," Opt. Acta 29, 1237 (1982).

9. Mei-Liang Yi, "The Design and Study of the Aspherical Plate Corrector of the View Field for Cassegrain System," Acta Astron. Sin. 23, 398 (1982).

10. Lan-Juan Wang and Ding-Qiang Su, "The Studies of Two Types of Catadioptric Telescope," Acta Opt. Sin. 3, 132 (1983).

11. Lian-Zhen Shao and Ding-Qiang Su, "Improvement of Chromatic Aberration of an Aspherical Plate Corrector for Prime Focus," Opt. Acta 30, 1267 (1983).

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