**Scott Lerner and Jose Sasian**

*FIGURE 1. Standard explicit polynomial representation of an aspheric certifying mirror using 10 aspheric terms results in a large residual error (vertical axis), shown as a function of the aperture (horizontal axis). The peak-to-valley residual error is 2000 waves at 633 nm. In contrast, a parametrically defined representation of the same surface has virtually no residual error.*

Modern trends in fields such as conformal optics, optical testing, and optical lithography have compelled optical designers to take a fresh look at how an optical surface is defined. Aided by lens-design software, the optical designer tries to find the ideal surface or surfaces that will meet the requirements of the application at hand. Traditionally, spherical surfaces, conicoids (paraboloids, hyperboloids, and ellipsoids), splines, and polynomial aspherics are used to meet design specifications. In some current applications, however, these traditional surfaces no longer permit the optical designer to satisfactorily complete a design or take it to the limits of what is possible. Thus, there is an ongoing effort in lens design to come up with new surface descriptions that can represent the ideal surfaces a given optical system needs. A very interesting surface description is that of parametrically defined aspheric surfaces.^{1, 2}

**Parametric versus explicit**

Surfaces that are currently used in lens-design software fall into the class of explicitly defined surfaces. The sag *z* of these explicitly defined surfaces is represented as a function of the transverse coordinates *x* and *y* (sag *z* is the difference in *z* coordinates between a surface point lying on the optical axis and one at the edge of the aperture). For example, a paraboloid is defined by *z* = (*x ^{2}* +

*y*)/2

^{2}*R,*where

*R*is the vertex radius of curvature. This surface representation is convenient because the sag z is explicitly given and there is no need to solve for it. However, the class of surfaces that can be explicitly defined does not contain all the surfaces that are of interest and useful in lens design.

A powerful approach to enlarging the class of surfaces that can be programmed in lens design software is the use of parametrically defined surfaces. In a parametrically defined surface, the surface sag z is defined by two or more equations and at least an auxiliary variable. For example, an ellipsoidal surface can be represented by two parametric equations:

*r = A* sin(*θ*)

*z = B* (cos(*θ*) - 1)

where *r = (x ^{2} + y^{2})^{1/2},* A is the semi-minor axis of the ellipse,

B the semi-major axis, and q the auxiliary variable. The advantage of a parametric representation is that it can be used to represent a much wider class of surfaces than what is possible with an explicitly defined approach.

For example, a precise small mirror (certifying mirror) can be used to simulate the normal rays reflected from a large mirror under test and certify that a null corrector is properly constructed. The certifying mirror effectively has the shape of the geometrical wavefront associated to the ray normals to the paraboloid. For the case of testing a large paraboloid mirror, the small certifying mirror is parametrically defined by

and

null

where R is the vertex radius of curvature and P is the distance between mirrors. This surface representation is simple and perfect. In comparison and contrast, the standard explicit surface representation cannot perfectly simulate such a surface even if dozens of aspheric terms are used. When 10 aspheric terms are used, the standard representation of an aspheric surface has a very large peak-to-valley residual error of 2000 waves at a wavelength of 633 nm (see Fig. 1).

The problem of the certifying mirror is one example of a larger class of design problems that involve surfaces near a ray caustic. These surfaces in particular benefit from the use of parametrically defined surfaces. Near a caustic, the ray heights no longer are linear as a function of the aperture and therefore standard surfaces fall short in matching an aberrated wavefront. By using parametrically defined surfaces, the wavefront correction can be improved by several orders of magnitude simply because such surfaces are capable of representing the ideal surface that is needed or to approach it more closely.

**The Taylor expansion**

The parametric Taylor surface is an example of a surface formulation that accounts for the nonlinear behavior of ray heights. For the certifying mirror, a finite Taylor expansion of each of the equations that define the surface has been used to approximate the exact parametric equations. The wavefront residual using such a surface with 10 terms is less than 1/100 of a wave peak-to-valley. This compares to the wavefront residual of 2000 waves with the polynomial representation. As the exact equation for the surface radius is defined by an odd function, the Taylor expansion of the equation will contain only odd polynomial terms of the auxiliary parameter. Likewise, the exact equation for the surface sag is an even function and will contain only even polynomial terms of the auxiliary parameter.

*FIGURE 2. Beam expander/contractor contains a parametrically defined aspheric plano-convex lens. Notice that the aspheric lens is placed near the ray caustic.*

Its general formulation allows the parametric Taylor surface to be used to solve the larger problem of representing surfaces near caustics. The problem of a 13-to-1 beam expander/contractor illustrates the generality of the parametric Taylor description. The example beam expander/contractor consists of two concentric spherical mirrors and a plano-convex aspheric lens (see Fig. 2). The parametric Taylor representation properly describes the aspheric lens; the resulting system has a root-mean-square (rms) wavefront residual of less than 0.001 waves at a design wavelength of 633 nm. In contrast, the polynomial surface converges to a solution for which the rms wavefront residual is 6000 waves. In this example, the truncated parametric Taylor representation performs more than seven orders of magnitude better.

Parametrically defined surfaces solve the problem of defining surfaces where the normal to the surface becomes perpendicular to the optical axis and therefore can be used to represent highly aspheric surfaces and to properly simulate fabrication errors in steep surfaces (see Fig. 3).

Because optimization time is proportional to the square of the number of variables, optimization of complex systems using conventional polynomial or spline surfaces can be very slow. An important advantage of parametrically defined surfaces is that the number of variables needed to represent a given surface can be small, greatly reducing optimization time.

**Putting theory into practice**

In the field of extreme-ultraviolet (EUV) lithography, the projection-camera optics that produce the aerial images to print electronic circuits on silicon wafers consist of aspheric mirrors. The lens designer has no means to determine whether a given design done with standard polynomial surfaces has reached its limits of performance. Adding a more flexible and general surface definition to lens design software might allow designers to improve the performance of EUV lithographic lens systems, where an improvement of even a few percent is significant.

**FIGURE 3. Exaggerated representation of a periodic error in a conformal dome using a parametrically defined surface shows that the periodic error conforms to the surface in a realistic manner. In contrast, conventional polynomial representations improperly represent periodic errors on steeply curved surfaces.**

Like any other aspheric surfaces, the fabrication and testing of parametrically defined surfaces is an issue. Single-point diamond turning, computer-controlled polishing, magnetorheological finishing, and holographic testing are techniques that can be used to deal with the fabrication of parametrically defined surfaces. From the fabrication point of view, parametrically defined surfaces are no different from other standard aspheric surfaces.

Some existing optical-design software can do ray-tracing calculations on parametrically defined surfaces, as long as the designer resorts to user-defined surfaces. However, these surfaces are not a standard option that the user can readily select. It is likely that in the near future some class of parametric surfaces will be a standard option in lens-design software.

**REFERENCES**

- S. A. Lerner, Optical design using novel aspheric surfaces, Ph.D. dissertation, University of Arizona (2000).
- S. A. Lerner, J. M. Sasian, Appl. Opt. 39, 28 (2000).

**SCOTT LERNER** is an optical engineer at Lawrence Livermore National Laboratory, 7000 East Avenue L-495, Livermore, CA 94550; e-mail: [email protected]. JOSE SASIAN is an associate professor at the Optical Sciences Center, the University of Arizona, 1630 East University Blvd., Tucson, AZ 85721; e-mail: [email protected].