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How to Model an Off-Axis Ellipsoidal Mirror at Finite Conjugates

This article explains how to model an off-axis ellipsoidal mirror when the light source is a one focal point of an ellipsoidal mirror and image (detector) in another focal point of an ellipsoidal mirror.
Mike Sluch (msluch@yahoo.com)
01/12/2018

Introduction

Off-axis ellipsoidal mirrors (OAEs) are often used to bring a diverging beam from one focus point to another focus point at a 90 degree angle for the Fourier Transform Infrared (FTIR) spectrometer applications. However, OAEs may also be used in visual spectrometers and interferometers. Setting up such a system in Zemax OpticStudio is fairly simple and is similar for the setting of an off-axis parabolic mirror (OAPs) [1]. However, it is a little different from an OAP case because the OAE has two focal points. Moreover, surface parameters used in SolidWorks drafting software are different from the standard surface parameters used in OpticStudio (Fig.1).

Fig.1 Ellipsoidal Mirror, 50.8 mm diameter, F1=62.81 mm, F2=158.23 mm

A simple OpticStudio example made in sequential and non-sequential ray-tracing modes are provided to illustrate OAEs setting. This article shows how to model OAEs in sequential and non-sequential modes starting from SolidWorks drawing.

An OAE mirror consists of a small section cut off from a large ellipsoidal mirror shown in Fig.1.  Standard drafting software, like SolidWorks, as well as the mirror manufacturer (Aero Research Associates, http://aerorese.com/) which make OAEs, uses semi major (a) axis and semi minor (b) axis and eccentricity (e) to define the elliptical surface. These parameters are different from OpticStudio standard surface parameters (radius of curvature and conic constant). To convert SolidWorks parameters (a, b, c, e)  to the OpticStudio standard surface parameters (elliptical: radius of curvature R and conic constant k).  The ellipse parameters used in SolidWorks are: a=110.52 mm, b=70.49mm, c=85.12 mm where c2+b2=a2, and e=c/a  e is eccentricity, e= 0.770177343. The converting formulas are below:

The conic constant k=-e2 and R=a*(k+1),                  (1)

Thus OpticStudio standard surface parameters are k=-0.59, R=44.96 mm according to equations (1).

The elliptical mirror orientation in SolidWorks defined by F1=158. 23 mm, F2 =62.81 mm and angle=21.568 degrees. Please note that the angle between F1 and F2 is 90 degrees (Fig.1)

The sample file for our setup, EllipticalOffAxisV1.zar, is provided in the archive (.ZAR) file attached to this article. The input beam starts at the point source located at the first focus of the ellipsoidal mirror. The mirror has a first focal length of 62.81 mm (F1) and the second focal length of F2=158.23 mm. A coordinate break surface is used for steering the beam at the 21.567 degree angle in respect to the optical axis of the system (Fig.2), radius of curvature and conic constant are calculated from (1).

Fig.2 Off-axis ellipsoidal mirror in OpticStudio’s Lens Data Editor

The OAE mirror also has an aperture placed on it, so that the portion of the OAP used in the model corresponds to that section of the mirror which the beam will interact with (Fig.3). Aperture Y decenter and maximum radius taken from the drawing (Fig.1)

Fig.3 Aperture setting.

It is interesting to see what the elliptical mirror looks like without the aperture (Fig.4) and with the aperture as shown in Fig.5. The incoming ray from first focus (F1) and reflected ray to second focus (F2) make a 90 degree angle (Fig.6)

Fig.4 OAEs without aperture

Fig.5 OAEs with aperture.

Fig.6. Incoming ray from F1 and reflected ray to F2 makes a 90 degree angle.

Results from the Spot Diagram confirm that the beam is brought to focus without aberration.

Modeling OAEs in Non-Sequential Mode

There is no elliptical object in the OpticStudio object library. In order to model OAEs in non-sequential mode, a user has to create an elliptical model in SolidWorks (Fig.1) and then save the model as a step file and import the step file into OpticStudio (Fig.7).An example OpticStudio file and SolidWorks Step file are provided as the attachment NS-EllipticalOffAxisV1.zar.

Fig.7 Ellipsoidal mirror in OpticStudio non-sequential mode

Summary

Modeling an off-axis elliptical mirror is fairly straight-forward in Zemax OpticStudio. It is important, however, that the steps be performed in the proper order to ensure a correct setup of the system.
References.

1. How to Model an Off-Axis Parabolic Mirror at Finite Conjugates, Zemax Knowledge Base