Tilted and/or decentered surfaces

Tilted and/or decentered surfaces

Tilted surfaces are described by tilting the coordinate system in which the surface is described. It is useful to think of each surface having two (right-handed) coordinate systems: a base coordinate system, and a local coordinate system that is tilted and possibly decentered with respect to the base coordinate system. The base coordinate system normally has its origin on the z-axis of the previous system and is separated from it by the thickness of the previous surface. The local origin may be decentered by amounts dcx, dcy, dcz from the vertex of the base coordinate system. In addition, the local coordinate system may be tilted with respect to the base system by angles tla, tlb, tlc about the x, y, and z-axes.

An optical surface is defined by z' = f(x',y'). The sag of the surface near the origin is given approximately by (x'²+y'²)/2R in the local coordinate system, as shown in the figure. Note that the radius of curvature R has the same sign as the sag of the surface. This provides the basis of the sign convention: the sign of the radius of curvature is the same as the sag of the surface near the origin of the local coordinate system. This rule is valid regardless of the tilt.

The tilt angles tla, tlb, tlc are Euler angles (meaning that each rotation is made in the coordinate system resulting from the previous rotations) given in degrees. The figure below shows the sign conventions that are used. The angles shown are positive rotations. The sign conventions used in OSLO are the same as those used for most other optical design programs. Note that these conventions are not the same as are used in other disciplines (e.g. computer graphics), and are not self-consistent. (The sign convention for tlc is different from that for tla and tlb).

Since tilting and decentering are non-commutative, the program must be told which comes first. OSLO uses a special datum (called dt) to accomplish this. If tilting is carried out after decentering, tilt operations are carried out in the order tla, tlb, tlc. If tilting is carried out before decentering, tilt operations are carried out in the order tlc, tlb, tla. This method of specifying tilted and decentered surfaces allows one to restore the original position and orientation of a surface that has been tilted and decentered by picking up all the tilt and decentering data with negative signs.

In a normal system, the next surface is expressed in the local coordinate system of the current surface, that is, the thickness is measured along the z-axis of the local coordinate system. However, OSLO also has a return_coordinates command (rco) that causes the thickness to be measured along the z-axis of the base coordinate system of the current surface. An option to the command allows you to measure the thickness along the z-axis of any previous local coordinate system. Thus you can, for example, refer the coordinates of any surface to some base surface (e.g. surface 1). This provides a way to specify coordinates globally. Return_coordinates are not the same as the global coordinates used in OSLO SIX; you cannot transform back and forth between local and global coordinates when using rco.

A common error in using return_coordinates is using a tilt angle that is wrong by 180 degrees. The reason for the error is that there are two ways to express the same surface. Suppose you have a system where a beam, initially traveling from left to right, reflects from a mirror that is tilted at 45 degrees. Then the beam is traveling perpendicular to the original beam, as shown below.

The question then arises about whether the next surface should be tilted by +90 degrees or -90 degrees with respect to the original system, since it doesn’t seem to make any difference. If the surface is tilted by -90 degrees, the curvature as shown is positive, whereas if the surface is tilted by +90 degrees, the curvature is negative.

Actually, it does make a difference, and the correct answer is +90 degrees. While the two cases are geometrically identical, for the purpose of ray tracing they are not. A line intersects a sphere in two places, so in solving the ray trace equations, there must be a convention as to which intersection point to choose.

In OSLO, the ray trace equations are set up to use the normal intersection points predicted by the standard sign convention. This convention, when generalized to handle tilted surfaces, states that the z-axis of each local coordinate system should point in the same direction as the beam propagation after an even number of reflections, and in the opposite direction after an odd number of reflections. In the present case, the beam has undergone one reflection by the 45 degree mirror, so the z-axis of the next local coordinate system should point against the direction of beam propagation, as shown in the figure to the right.

In OSLO, there is a special flag on each surface called the ASI (alternate surface intersection) flag. If you turn it on, the ray trace intersection points will be opposite from the normal, so you could handle a -90 degree tilt by using the ASI flag. But then you would have to deal with the next surface in a different way from normal, and the path of least resistance is to set your system up according to standard conventions.

Actually, the easiest and best way to work with systems that have tilted mirrors is to use the bend (ben) command in OSLO. The bend command automatically sets the optical axis after reflection to be coincident with the ray that connects the previous vertex with the current vertex, i.e. it in effect propagates the optical axis through the system. The new coordinate system is adjusted, if necessary, so that the meridional plane remains the meridional plane after reflection.

Next Page | Previous Page | Chapter Summary | Table of Contents