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Since the Zernike polynomials are fit to a single vector quantity (surface normal or sag data), they do not represent the full rigid-body motion of an optical surface in six DOF. For instance, fitting Zernike terms to the surface displacements of a flat optical surface yields no information about whether the surface was laterally displaced or rotated about the optical axis. When computing optical element errors, it is common practice to remove the rigid-body errors and represent the higher-order surface deformations in the optical model using techniques discussed in Section 4.5.
4.5 Representing Elastic Shape Changes in the Optical
Model
There are several commonly utilized methods to represent finite-element-derived optical surface displacements within commercial optical design software (such as Code V4 and Zemax5). This process is depicted in Fig. 4.14. These methods
include polynomial surface definitions, surface interferogram files, and uniform arrays of data that use either sag or surface normal displacements. General discussion and application of representing finite-element surface displacements using the above optical modeling techniques is discussed by Doyle et al.6
Engineering judgment determines the “best” modeling approach for a specific application and is dependent on the optical system, optical model, and the desired accuracy. Uncertainties such as material properties, boundary conditions, and load conditions along with understanding limitations and approximations in the accuracy of the models should also enter into the decision as to the most applicable approach.
4.5.1 Polynomial surface definition
Polynomial surface definitions use a base surface definition plus the addition of polynomial terms to describe the shape of an optical surface. This definition allows finite-element displacement data to be fit to polynomials and added as perturbations to the base surface for ray tracing in the optical model. Polynomial options include Zernike polynomials, X-Y polynomials, aspheric polynomials, and others. The user can select the polynomial set that best represents the FEA displacements. The shape of an optical surface is defined by the sag displacement from the tangent plane. Thus, the polynomials must be fit to optical surface sag displacements. Finite-element-derived sag deformations, for example, can be
represented by Zernike coefficients as perturbations to the base surface shown below:
2 2 2 sag . 1 1 1 i i cr a Z k c r
¦
(4.11) The first term is the nominal conic surface definition, and the second termrepresents the perturbations to the base surface represented by the Zernike coefficients ai and the Zernike polynomials Zi.
The accuracy of this approach is dependent on the accuracy of the polynomial fit to the surface displacements. Fitting to a larger number of terms provides the potential of an improved fit and hence accuracy. The maximum number of terms allowed in the fit is dependent upon the optical design software.
4.5.2 Interferogram files
Surface interferogram files are 2D data sets that represent surface normal deviations that are assigned to optical surfaces in the optical model. This file format is also used to represent interferometrically measured topographical fringe maps created during optical testing. Use of an interferogram file is an approximate technique to represent a deformed surface shape, as compared to ray tracing off a deformed optical surface represented using a polynomial surface definition. The approximation lies in the computation of the optical errors for a given ray. A ray is traced to the undeformed surface, and the intersection coordinates are used to determine the surface error as defined by the interferogram file from which ray deviations and OPD are computed. The error associated with this approximation is a function of the ray angle and the spatial variation and magnitude of the displacement field. The error in this approximation in representing FEA optical-surface deformations consistent with mechanical perturbations is typically negligible for most applications.
Interferogram file data can be represented in two formats: Zernike polynomials (Standard or Fringe) or as a uniform rectangular array (or grid array) and require finite element displacements to be converted into surface-normal displacements.
The Zernike polynomial format provides a more accurate representation relative to a grid array if an accurate fit is achieved. Code V places no limit on the number of Zernike polynomial terms that may be used to represent the surface normal displacements. Surface deformations and slope data is computed directly from the polynomial representation. The grid format is useful when an accurate Zernike fit cannot be achieved.
The interferogram file data may be scaled in the optical model, which is useful in performing design trades by scaling surface errors due to unit g-loads, thermal soaks, or thermal gradients. As with assigning rigid-body perturbations to an optical surface, understanding and relating the finite element coordinate system to the optical surface coordinate system is necessary for a successful surface-error representation. For instance, in Code V, a positive surface
deformation represents a “bump” on the optical surface, as shown in Fig. 4.15. This is consistent with measuring the surface from the “air” side of the element.
In addition, it is necessary to align and place the interferogram file at the correct location and with the proper orientation on the optical surface. Commands are available to scale, mirror (reverse or flip), rotate, and decenter the interferogram file to the correct position. Test cases should always be run to verify that the position and orientation of the interferogram files are correct.
4.5.3 Uniform grid arrays of data
Uniform grid arrays of data are useful in representing optical surface displacements when an accurate polynomial fit cannot be achieved. Grid arrays are able to represent high-frequency spatial variations seen in edge roll-off, localized mounting effects, or quilting of a lightweight optic. For example, two residual surface-displacement maps after adaptive correction (gravity loading on the left and thermal loading on the right) are shown in Fig. 4.16. The percent of the RMS surface error represented by a 66-term and 231-term Standard Zernike polynomial is shown in Table 4.3. A large fraction of the surface displacements is not included in the Zernike fit for each of these two cases. A uniform array provides a much more accurate representation. For example, a 51 u 51 array represents over 98% and 99% of the RMS surface error for the two cases, respectively.
Positive Surface Deformation Direction of Light
Surface normal
Surface normal
Figure 4.15 Sign convention for Code V surface interferogram files.
Figure 4.16 Surface displacements due to gravity (left) and thermal soak (right) after
Table 4.3 Percent of RMS surface error represented by 66- and 231-term Standard
Zernike polynomial and a 51 x 51 uniform grid array.
66-Term Fit 231-Term Fit Grid 51x51
Gravity 5% 32% 98%
Thermal Soak 4% 40% 99%
The loss in accuracy in representing surface displacements with uniform arrays of data is two-fold; first, interpolation is required to create a uniform rectangular array from a non-uniform FEA mesh; and second, errors result from ray tracing in the optical model for incident rays that do not coincide with a data point. In this case, a second interpolation step is used within the optical model to compute the surface errors. Two common uniform-array formats are Code V’s surface interferogram files and the Zemax Grid Sag surface.
4.5.3.1 Grid Sag surface
The Grid Sag surface is a Zemax surface definition that uses a uniform array of sag displacements and/or slope data to define perturbations to a base surface. The base surface has a shape defined by a base plane, sphere, conic asphere, or polynomial plus additional sag terms defined by a rectangular array of sag values, defined as
2 2 2 sag ( , ) 1 1 1 i i cr z x y k c r . (4.12)
Zemax offers two interpolation routines, linear and bicubic, to determine the surface errors during optical ray tracing. If only sag displacements are provided, the linear interpolation routine is used to compute the slope terms using finite differences. If, in addition to the sag displacements, the first derivatives in the x and y directions w(ds)/wx, w(ds)/wy, and the cross-derivative terms w(ds)2/wxwy are
supplied by the user, then Zemax’s bicubic interpolation may be used. The rotation values help ensure a smooth fit over the boundary points.
4.5.3.2 Interpolation
In general, creating a grid interferogram file or Grid Sag surface requires the surface displacements computed at the finite-element grid points to be interpolated to a uniform grid, as shown in Fig. 4.17. The accuracy of the interpolation method is critical for high-performance optical systems, such as near mounting locations where regions of rapidly varying displacements commonly exist.
One method to interpolate surface data to a uniform grid uses Delaunay triangulation techniques including nearest neighbor, linear, and cubic. Another method to interpolate data to a uniform grid is to use the finite element shape
FEA Computed Surface Deformations
Non-Uniform Grid Uniform Grid
Figure 4.17 Interpolating FEA displacements to a uniform grid.
Delaunay Triangulation: Nearest Neighbor Delaunay Triangulation: Cubic Interpolation Shape Function Interpolation
Figure 4.18 Interpolation using Delaunay triangulation and FE shape functions. functions.7,8 In this approach, values are interpolated to the grid points using the
shape functions from the surface element in which the grid point falls.
For 3D models, interpolation may be performed by creating a set of “dummy” plate elements to be modeled on the optical surface. The structural thickness of the plate elements can be made arbitrarily small. Alternatively, the 2D shape functions on the solid element face can be used to perform the interpolation.
An example of interpolating a finite-element mesh to a uniform grid is shown for Delaunay triangulation techniques (nearest neighbor and cubic) and cubic finite element shape functions in Fig. 4.18. The interpolation from a FEA mesh to a rectangular array is more accurate using cubic interpolation (as compared to linear interpolation). In this case, a surface of “dummy” plate elements is required to provide the nodal rotations. Note for accurate edge effects, a surface coat of “dummy” plate elements should wrap around the optic to avoid erroneous edge effects (see Section 10.2.3.3 for more detail).