Optical-Surface Rigid-Body Errors

Mechanical and thermal loads that act on an optical system can significantly degrade optical performance by changing the position of optical elements and creating optical element misalignments. Positional or rigid-body errors include translations and rotations of a surface in six DOF. Translation of the optic along the optical axis is called despace, changes in lateral position are called decenter, and tip and tilt refer to rotations about the lateral axes as shown in Fig. 4.1. For non-rotationally symmetric optics, rotation about the optical axis must also be considered. These rigid-body errors result in optical system pointing errors and wavefront aberrations.

Despace Decenter Tip / Tilt

4.1.1 Computing rigid-body motions

Computing the rigid-body motions of an optical element or surface using FEA depends upon the application and the desired model fidelity. For small optical elements where elastic deformations are considered insignificant, the rigid-body motions can be determined using a single node coupled with a lumped mass representation. This approach is illustrated in Fig. 4.2.

Using shell or solid elements to model an optical element where multiple nodes represent the optical surface requires post-processing of the FEA data to extract the rigid-body motions. One approach internal to FEA codes is the use of an interpolation element that is tied to the optical surface nodes to compute the average rigid-body motions. The method may be employed for static and dynamic mechanical loads but is not recommended for thermal loading. Use of the interpolation element for thermal loads does not account for the radial motion of the node when computing axial or despace rigid-body displacements (this is described in more detail in Section 4.2.1).

An alternative approach for computing rigid-body motions of an optical surface that is represented by multiple nodes is to perform a least-squares best-fit. This requires exporting the FEA surface displacements into an auxiliary software algorithm for post-processing. The equations for performing the least-squares fit are presented below.

For an optical surface that is represented by a grid of nodes, the rigid-body motion of the surface (three translations, Tx, Ty, Tz, and three rotations, Rx, Ry, Rz)

may be computed as the area-weighted average motion. The rigid-body nodal displacements dx dy_i, _i, anddz at a given node position x_i, i, yi, and zi, due to

optical element rigid-body motions in six DOF are expressed as

. i x i y i z i y i x i z i z i x i y dx T z R y R dy T z R x R dz T y R x R          (4.1)

The squared error E between the actual optical-surface nodal displacements

dxi, dyi, and dsi and the rigid-body nodal displacements dx dyi, i, anddz is i

defined as 2 2 2 ( ) ( ) ( ) . i i i i i i i i E

¦

Note that the sag displacement ds is used in these calculations. This calculation is discussed in more detail in Section 4.2.1. The best-fit motions are found by taking partial derivatives with respect to each term and setting the result to zero. For example, the resulting equation for translation in the x direction is

.

i x i i y i i z i i

i i i i

w T  w z R  w y R w dx

¦

¦

¦

¦

Repeating this for each of the six rigid-body equations results in six simultaneous equations to solve for the average rigid-body motions.

4.1.2 Representing rigid-body motions in the optical model

The rigid-body errors computed from the FEA model may be represented in the optical model by using standard tilt and decenter commands that are commonly used to develop folded optical systems. These commands may be applied to perturb individual or groups of surfaces. Rigid-body errors applied to a double Gauss lens assembly are shown in Fig. 4.3. Adding FEA-derived surface errors to an optical model requires consistency between the mechanical and optical models in regards to units, geometry, and coordinate systems. In the FEA model, the displacement of nodes may be defined using either local or global coordinate systems. In an optical model, the coordinate system of an optical surface is nominally defined by a local coordinate system at the vertex. For on-axis optics, where the vertex is at the geometric center of the optic, maintaining consistency between the mechanical and optical coordinate systems is straightforward. For off-axis optics where the vertex is off-center or not physically on the optical substrate where typically the mechanical coordinate system is located, it is more challenging. In this instance, coordinate systems may be defined within the optical model using dummy surfaces and coordinate breaks that are located at the physical center of the substrate consistent with the mechanical model. Alternatively, within the FEA model, the vertex motions of an off-axis surface may be determined by adding a rigid link that relates the average rigid-body motions of the optical surface to the vertex location.

Single Element Decenter Doublet Tilt Single Element Despace

image 'Y s2 x0 y0 x1 y1 x2 y2x3 y3xi yi 'D s3 object

Figure 4.4 In the optical model, decenters and tilts are nominally applied to the local

coordinate system defining the surface.

Nominally applying rigid-body errors to optical surfaces in the optical model is done by tilting or decentering the local coordinate system that defines the surface, as illustrated in Fig. 4.4. This results in cumulative errors since each local coordinate system is defined relative to the local coordinate system of the preceding surface. A common method to uncouple the perturbations is to specify a decenter and return, which, as the name implies, returns the local coordinate system of the surface following the tilted and decentered surface to the original coordinate frame. Repeating this command for each of the surfaces allows rigid- body errors to be defined independently. Other methods to uncouple the rigid- body errors include the use of global coordinates as well as coordinate breaks and/or dummy surfaces. In general, applying rotations to an optical surface in the optical model is order dependent. However, for small rotations such as those typically computed by a linear finite element analysis, the order of the rotations can generally be neglected.

It is recommended that the rigid-body surface errors be separated from the higher-order elastic optical surface deformations and represented in the optical model using tilts and decenters. The residual surface deformations can be represented through polynomial fits or interpolated arrays. This approach affords the greatest accuracy and in addition provides greater insight into the behavior of the optical system.