This article explains how to model an off-axis parabolic mirror when the source is a finite distance away from the mirror.

**Authored By Sanjay Gangadhara**

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## Introduction

Off-axis parabolic mirrors (OAPs) are often used to bring a collimated beam to focus without aberration. However, an OAP may also be used to form a perfectly collimated beam from a point source (i.e. a source at finite conjugates). Setting up such a system in OpticStudio is fairly simple, but it is important to perform the setup steps in the correct order. A simple example is provided to illustrate this point.

## Modeling an OAP at infinite conjugates: a review

Before modeling the finite conjugate case, let’s review the model of an OAP at infinite conjugates, i.e. a system in which a collimated beam is brought to focus. Such a system is also described in the article entitled “How to model an off-axis parabolic mirror.”

The sample file is provided to represent this system, "Infinite conjugates.ZMX," is provided in the archive (ZAR) file attached to this article. In this system, the input beam has a diameter of 20 mm and the mirror has a focal length of 100 mm. A Coordinate Break surface is used to decenter the mirror by 10 mm with respect to the input beam:

The 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:

The Coordinate Break on Surface 4 is used to restore the coordinate system, after which we propagate by 100 mm, i.e. the focal length of the mirror. The coordinate system is then adjusted to align the image plane with the Chief Ray by using a Chief Ray solve on the "Decenter Y" and "Tilt About X" parameters of Surface 5:

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

## Modeling an OAP at finite conjugates

To model an OAP at finite conjugates, we simply need to perform in reverse the steps used to model the mirror at infinite conjugates. At infinite conjugates, the steps were:

- Propagate to the location of the mirror.
- Decenter the coordinate system.
- Place the mirror in the system with a decentered aperture.
- Restore the coordinate system relative to step 2 above.
- Propagate to the focal plane of the mirror.
- Decenter and tilt the coordinate system to align the image plane with the Chief Ray.

Thus, to model the OAP at finite conjugates, we simply:

- Tilt and decenter the coordinate system, so that the Chief Ray is tilted and decentered.
- Propagate from the focal plane of the mirror to the mirror itself.
- Decenter the coordinate system, so that the Chief Ray hits the center of the OAP.
- Place the mirror in the system with a decentered aperture.
- Restore the coordinate system relative to step 3 above.
- Propagate to the location of the image plane.

An example which has been set up following these steps ("Finite conjugates.zmx") is provided in an archive file also attached to this article. In this system, we are modeling the same OAP as we modeled in the infinite conjugate example; that is, an OAP with the same focal length and the same decentered aperture. Based on our results from the infinite conjugate model, we have also “hard-coded” in the value of the tilt and decenter to use in step 1 above:

The "Order" flag for this Coordinate Break surface is set to 1, since the "Order" flag used in the infinite conjugate model was 0. For more information on why we flip the "Order" flag, see the OpticStudio Help File "The Setup Tab > EditorsGroup (Setup Tab) > Lens Data Editor > Sequential Surfaces (lens data editor) > Coordinate Break."

To evaluate the performance of this model, we let OpticStudio know that the system is afocal in image space by navigating to **System Explorer...Aperture **and activating the "Afocal Image Space" setting:

The results from the Spot Diagram again indicate perfect imaging, this time to a collimated beam (angular radius = 0):

The system looks identical to the case for infinite conjugates:

We would have to fletch the rays to tell which direction energy is propagating!

KA-01695

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