This article explains how to model an elliptical surface in OpticStudio using the Conjugate Surface in Sequential Mode. It presents an elliptical mirror example which perfectly focuses light from one focal point to the other.

**Authored By Nam-Hyong Kim, Yuan Chen, Kevin Scales**

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## Introduction

An elliptical mirror can image light from one focal point to the other without aberration. To model this in Sequential Mode, a Standard Surface or a Conjugate Surface can be used. If the foci locations of the ellipse are known, the Conjugate Surface is the superior method. Rather than having to compute the necessary radius of curvature and conic constant from these points, the Conjugate Surface allows them to be input directly as (`x`,`y`,`z`) coordinates. An example is available in the Article Attachments.

For more information on modeling an ellipse with the Standard Surface, refer to the OpticStudio Help Files at, "**The Setup Tab...Editors Group...Lens Data Editor...Sequential Surfaces (lens data editor)...Standard**."

## Before getting started

Before continuing with this article, ensure that you are familiar with the basic operations in Sequential Mode, as well as how to tilt and decenter optical components. If you are unfamiliar with these concepts, see the following articles:

If you are comfortable with both of these operations, proceed to the next section.

## Using the Conjugate Surface

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. Also note that like any other surface in OpticStudio, 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.

This elliptical mirror can be constructed using the Standard Surface but as we will see shortly, it is much easier to use the Conjugate Surface instead.

From the Article Attachments, open the file "**conjugate surface start.zar**." This file represents a simple fold mirror that reflects rays from a point source through 90 degrees, as shown below. The STOP surface is located on a dummy surface (Surface 1) positioned immediately before the fold mirror.

We will make Surface 2 a Conjugate Surface type with the appropriate parameters to perfectly image the object to the image surface. Navigate to **Surface 2 Properties...Type** and change **Surface Type: Conjugate**. Then, select **Make Surface Global Coordinate Reference** to define the local axis of Surface 2 as the Global Coordinate Reference point.

Update the 3D Layout.

The Conjugate Surface requires the XYZ coordinates of both foci; the OBJECT and IMAGE Surface locations in this case. The object and image coordinates, as referenced to the mirror vertex, can be obtained from the Prescription Data report. Open the Prescription Data under **Analyze...Reports...Prescription Data** and open the **Settings**. Click **Clear All** to remove all of the data from the report. Now, we can select only the setting we are interested in: **Global Vertex**.

With "Global Vertex" selected, the global object and image coordinate are reported in the Prescription Data report. 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!

When we have different X and Y values, extra caution shall be taken on inputting the tilt angle value. Please make sure that you have put enough digits into the editor, otherwise you will get the coma aberration as the system has a small tilt compared to the nominal position.

It’s the same case to enter the thickness if you calculate the digits manually and enter it in the lens data editor. If the thickness is not close to the nominal one, we will get defocus in this case. To skip the manual calculation, you can use the merit function in the sample file to calculate the distance, then copy and paste this value into the lens data editor. We can see this in the attached file ‘conjugate surface end_modify.zar’. The value of 500 we get here matches the magnitude of –500 that we get with a thickness solve on the Conjugate surface thickness. Of course, using this method, you must check for yourself that the thickness sign correctly matches the mirror space.

An alternative way is to set the thickness as variable and then optimize the system to get the spot size equal to 0.

## Aperture-decentered elliptical mirror system

Another sample file is provided for you, ‘Conjugate_General_Demo.zar’ so that you can modify the parameters directly to build your aperture-decentered elliptical mirror system.

When we form the image of F1 via the elliptical mirror to F2, the relative position of surface vertex shall be calculated. The example demonstrated above used the distance **a**, then set a tilt value of the elliptical surface. Another way is to add the distance **b** along the Z axis and then generate a decenter by the distance CF1, which is the **c **value shown above. The ray is tilted by angle if the chief ray is incident on the surface vertex. In the sample file, this is the angle that can be put into the Tilt About X in the first Coordinate Break surface to aim the chief ray at the vertex. Assuming the mirror is at the right side of the focal point, we have Y1 or Y2 = **c** and Z1=Z2= **-b** as it’s the relative position.

One thing that might be worth mentioning is that the order value of the first coordinate break is set to 1. This means that the tilts are done first, in the order around local z, then around the new y, and then around the new x, and then finally the decenters are done.

Knowing the decenter value to the surface vertex, we could set the tilt angle as variable and optimize it so that the chief ray could reach the desired aperture position.

We see initially that the tilt is set to –20 degrees, and the rays hit well away from the vertex. They focus perfectly, as expected for an ellipse that the Conjugate surface models, but if we want to strike the vertex without performing the arctan function separately, we can target the RAGY and RAGZ to be exactly zero. Change the weights to 1 and optimize, leaving the tilt as a variable. We see that the value of –36.86990 is quickly attained.

Be careful that you can only use this simplified setup with Conjugate surface. If you want to build an off-axis parabolic surface, refer to this article for more information: How to model an off-axis parabolic mirror at finite conjugates – Knowledgebase (zemax.com)

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