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Optical systems frequently use aspheric surfaces to improve performance. Typically the designer uses a combination of experience and trail and error to decide which surfaces to make aspheric, then the shape of the aspheric surface is optimized as part of the optical system. In this paper, a method of optimally choosing which surfaces are to be made aspheric is developed. This is implemented by allowing the surface number of the asphere to become a variable in the optimization. Imaginary surfaces are used as an intermediary to make the problem continuous for the optimization. The method of implementation and design examples are given.
©2005 Optical Society of America
1. Introduction
The usage of aspherics is becoming more and more common today, with the improvement of the manufacturing ability of aspherics. For example aspherics is effective for the reduction of size and cost of data projector lenses or the achievement of higher NA of microlithography lenses [1–4]. The position of aspherics affects the performance of the optical systems very much. Even if the same number of aspherics is used, the effect of aspherics differs depending on the their position. When some aspherics are used, the combination of possible surface numbers of aspherics is very large. To find the most effective position among all the combination is one of important works of optical designers. For the decision of the appropriate position of aspherics, the deep insight into the function of aspherics and the rich experience are indispensable. If the best surface numbers of aspherics can be automatically determined as parameters of optimization just like surface curvatures or surface separations, the burden of the selection of the position of aspherics would be remarkably reduced and the reliability of the selection would be improved. In this paper I propose the method to automatically determine the best surface numbers of aspherics.
2. Formalization
For the optimization of optical systems nonlinear optimization methods such as the damped least square method are used. In these methods the independent variables are real numbers and the derivatives as to the independent variables are used for the search of the optimum. In order to include the aspheric surface number as a parameter of the optimization, it is necessary to define the real surface number of aspherics, that is, the aspherics between 2 consecutive optical surfaces. If the value of the aspheric surface number is integer, the surface indicated by the integer is aspherics. This method is akin to the glass parameters Nd and Vd. Although the real glasses have the discrete and specific Nd and Vd, the fictitious glass is defined as a continuous function of Nd and Vd and used as the parameters of the optimization.
2.1 Expression of aspherics with 2 imaginary surfaces
The aspherics can be viewed as a thin layer on a sphere. In Fig. 1 a thin glass layer is after the sphere on the rear side of the glass. The numbers on the section drawing show the surface numbers. The red line shows the aspherics. Table 1 shows the lens data.
Table 1. Lens data of Fig. 1
In Fig. 2 a thin glass layer is before the sphere on the front side of the glass. Table 2 shows the lens data.
Table 2. Lens data of Fig. 2
The aspherics in the air between 2 surfaces can be defined as a thin glass layer in the air. The lens in Fig. 3 is equivalent to Fig. 1, but the aspherics is expressed as a thin air layer in a glass. Table 3 shows the lens data.
Table 3. Lens data of Fig. 3
In Fig. 4 a thin air layer is after the sphere on the front side of the glass. Table 4 shows the lens data.
Table 4. Lens data of Fig. 4
The aspherics in the glass between 2 surfaces can be defined as a thin air layer in the glass. In this way the aspherics between 2 surfaces can be expressed by inserting 2 imaginary surfaces. 2 imaginary surfaces have the same paraxial curvature and the distance between 2 imaginary surfaces is 0. Then the imaginary surfaces have no paraxial effect.
2.2 Continuity of imaginary surfaces between optical surfaces
When the 2 imaginary surfaces move from a surface k to the next surface k+1, the structure of imaginary surfaces changes as follows.
- The curvature of imaginary surfaces changes from the curvature of the surface k C k to the curvature of the surface k+1 C k+1
- When the imaginary surfaces are in a glass, the media between the imaginary surfaces is the air.
- When the imaginary surfaces are in the air, the refractive index between the imaginary surfaces changes from the front side index of surface k N k to the rear side index of surface k+1 N′k+1.
- The aspheric deviation from the sphere is given to the rear side imaginary surface just after the surface k.
- The aspheric deviation from the sphere is given to the front side imaginary surface just before the surface k+1.
- When the imaginary surfaces move from a surface k to the next surface k+1, the aspheric deviation from the sphere is gradually transferred from the rear side imaginary surface to the front side imaginary surface.
- The attention on the sign of the aspheric deviation is necessary. The aspheric deviation on the front side imaginary surface at surface k+1 has the opposite sign to the aspheric deviation on the rear side imaginary surface at surface k. By this definition the aspherics expressed with the imaginary surfaces has a consistent role between the surfaces k and k+1.
2.3 Implementation
The condition of the continuity is simply fulfilled with the linear interpolation. The character of the asherics is expressed by the aspheric deviation z(x,y). Here x and y are coordinates on a plane perpendicular to the optical axis. Suppose that the distance between the optical surfaces k and k+1 is d, the distance from the optical surface k to the imaginary surfaces is ad, and the distance from the imaginary surfaces to the optical surface k is bd. Here a+b=1. The curvature C of imaginary surfaces and the refractive index N between the imaginary surfaces are determined as follows.
When the refractive index between optical surfaces k and k+1 is denoted as N′k, the aspheric deviation z 1(x,y) of the front side imaginary surface and the aspheric deviation z 2(x,y) of the rear side imaginary surface are determined as follows.
When N > N′k,
When N < N′k,
By this definition, When N > N′k,
When N < N′k,
The coefficient a which shows the position of the imaginary surfaces is a function of the real surface number. When the real surface number is an integer k, the position of imaginary surfaces coincides with the optical surface k. In order that the construction of the optical system is continuous to the real surface number, the coefficient a need to be a continuous function of the fraction r of the real surface number such that,
The simplest among these functions is,
But with this function the position of imaginary surfaces is not a smooth function of the real surface number at integer numbers, because the surface separations before and after optical surfaces are not generally equal. One method to eliminate this defect is to use a smooth function such that,
where a′(r) is the differential of a(r). One example of these functions is,
The aspherics with real surface number is implemented in the conventional lens design software as follows.
- The real surface number and the aspherical coefficients are kept independent of the surface data.
- The position of the imaginary surfaces is determined by the real surface number.
- The structure of the imaginary surfaces is determined by the position of the imaginary surfaces and the aspherical coefficients.
- The imaginary surfaces are inserted to the surface data.
- The optical system with the imaginary surfaces is evaluated.
In the following the aspherics with the real surface number is called the floating aspherics.
3.Examples
I implemented this method in my personal design program. Figure 5 shows the starting point of the first 2 examples. The numbers on the section drawing show the surface numbers. The design condition of this lens is:
FNO=2.0, f=50mm, tangent of the field of view =0.25, no vignetting, distortion < 5%, overall length <50mm
3.1 Example 1
With 1 aspherics the lens is modified to FNO=1.8 and the tangent of the field of view =0.28. The aspherical coefficients up to 10th degree are used. The merit function consists of the polychromatic rms spot size at 4 fields. The wavelengths are 486. 1nm, 587.6nm, and 656.3nm. The number of sample rays is 20 at each field. Figure 6 shows the merit function values for the different position of aspherics. The scale of the merit function value is arbitrary. The merit function value with the aspherics on the surface 3 and 6 is the best.
In the optimization with the floating aspherics, the initial value of the surface number was set to 13. The global optimization with escape function was applied to get 20 solutions [5,6]. Figure 7 shows the real surface number for each solution. Figure 8 shows the merit function value of each solution. The surface numbers in Fig. 7 are real numbers, but the merit function values in Fig. 8 are the values after choosing the nearest integer for the aspheric surface number. The CPU time of the global optimization is 732 seconds on 2.5GHz CPU. The solution 3 is the best with the aspherics on the surface 6. Figure 9 shows this solution. The red line shows the aspherics.
3.2 Example 2
With 4 aspherics the lens is modified to FNO=1.5 and the tangent of the field of view =0.33. The initial values of the surface numbers was set to (3 6 9 12). Figure 10 shows the real-valued surface numbers for each solution. Figure 11 shows the merit function value of each solution. The surface numbers in Fig. 10 are real numbers, but the merit function values in Fig.11 are the values after choosing the nearest integers for the aspheric surface numbers. The CPU time of the global optimization is 1137 seconds on 2.5GHz CPU. The solution 19 is the best with the aspherics on the surfaces (3 5 6 12). Figure 12 shows this solution. In this way the plural floating aspherics can be used simultaneously.
In the case of 4 aspherics among 12 surfaces the combination of aspheric surface numbers is 495 and it is not practical to try all the combinations. The quality of the lens depends on the combination of aspheric surface numbers in a complicated way. The result of the local optimization is restricted to the neighborhood of starting point. The global optimization with escape function finds many local minima automatically and it is expected that the set of local minima covers the promising region of the combination of aspheric surface numbers.
The genetic algorithm has been proposed as another way of global optimization of optical systems [7,8]. The genetic algorithm can treat both continuous variables and discrete variables and it seems natural to include the aspheric surface numbers as discrete variables. But as far as I know, the efficiency and the reliability of the genetic algorithm is still uncertain for the optical design. The global optimization with escape function is a simple extension of conventional optimization method, with which we have a lot of experience and know-how. The method of floating aspherics matches the conventional optimization method and offers the practical solutions.
3.3 Example 3: ArF microlithography lens
In this example NA=0.85 and the rms opd is controlled below 0.01 wavelength over the whole field. The sensitivity to manufacturing errors is very large for microlithography lenses. I designed 3 lenses with different sensitivity control. The first lens is without sensitivity control, the second lens is with sensitivity control on tilt, and the third lens is with sensitivity control on curvature error. The starting lens data and the design conditions are derived from a patent data [9]. 7 aspherics are used and aspherical coefficients up to 14th degree are used. In this practical design the floating aspherics is very useful to get good solutions under the tight design conditions. Figure 13 shows the solutions. The red line shows the aspherics.
4.Concluding remarks
The optical design with real surface numbers was proposed. The fundamental idea and the implementation of this method are a simple extension of the conventional optical design method. This method was implemented and the expected performance was confirmed. This method can be extended to diffractive optical elements (DOEs), and decentered surfaces as well as aspherics. This method will contribute to the improvement of the quality and the reduction of the period of the optical design with special surfaces.
References and links
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7. I. Ono, S. Kobayashi, and K. Yoshida, “Global and multi-objective optimization for lens design by real-coded genetic algorithms” in Proceedings of International Optical Design Conference 1998, L. R. Gardner and K. P. Thompson, eds., Proc. SPIE 3482, 110–121 (1998). [CrossRef]
8. K. E. Moore, “Algorithm for global optimization of optical systems based on genetic competition”, in Proceedings of Optical Design and Analysis Software, R. C. Juergens, ed., Proc. SPIE 3780, 40–47 (1999). [CrossRef]
9. Japan Patent, P2004-252119A.