An elliptical mirror can image light from one focal point to the other without aberration. The Standard Surface can model elliptical surfaces by using the conic constant k with a value between -1 and 0. When k = -1 a parabolic surface results, in which one focal point is at infinity. When k = 0, a spherical surface results, in which the two focal points are co-located. For more information about how to model an ellipse with the standard surface, please refer to chapter 11, section "Surface types > Standard > Modeling an ellipse with the standard surface" of the user manual.
  
The Conjugate Surface provides a convenient way of modeling an elliptical surface when the two foci locations are known. Rather than having to compute the necessary radius of curvature and conic constant, the conjugate surface allows you to enter the {x,y,z} coordinates of the two foci locations directly. Like any other surface in ZEMAX, the vertex of the conjugate surface defines the local coordinate origin, which means that the foci locations are specified with respect to the surface vertex.

The diagram below shows a finite-conjugate imaging system in which the object and the image surfaces are co-located with the foci of the elliptical mirror. The object and the image surfaces are centered on, and orthogonal to, the chief-ray. This elliptical mirror can be constructed using the standard surface but as will see shortly,  it is much easier to use the conjugate surface instead.



If any of the steps in this example is unclear, please refer to following articles first.
Designing A Singlet in ZEMAX
How to Tilt and Decenter a Sequential Optical Component

Open the attached "conjugate surface start.zmx" file. 






The stop is located at the dummy surface #1. We will make the surface #2 a conjugate surface type with the appropriate parameters to perfectly image the object to the image surface.

Make surface #2 type to "Conjugate" and set it as the global coordinate reference.



Update the 3D layout



The conjugate surface requires the XYZ coordinates of both foci; the object and the image surface location in this case.  The object and image coordinates, referenced to the mirror vertex ,can be obtained from the Prescription report by following these steps.

1) Open the Prescription Data under Reports >Prescription Data and apply the following settings



3) The global object and image coordinate are reported in the Prescription Data. Since the conjugate surface is the global coordinate reference, the reported XYZ coordinates are referenced to the vertex of the conjugate surface.

Enter the coordinates in the Lens Data Editor.



Update the 3D layout to see the result!