The off-axis parabolic mirror is an important design form in the optics industry. This article demonstrates how to model an off-axis parabolic mirror according to some manufacturer specifications. It also demonstrates use of the Chief Ray Solve for centering the resulting image surface on chief ray path.
Authored By Nam-Hyong Kim, Yuan Chen, reviewed by Kevin Scales
Introduction
An off-axis parabolic mirror has the advantage of not having to obscure the input beam to access the image plane. OpticStudio can easily model any off-axis portion of a surface, parabolic or not. This tutorial will show you how to model an off-axis parabolic mirror. The concept shown here applies to any decentered surface and is not limited to off-axis parabolic mirrors.
Parabolic mirror design specifications
We will model a commercially-available off-axis parabolic mirror. The goal of this exercise is to be able to tilt the mirror about the X axis at any point along the optical axis (Z axis). The specifications for the mirror are as follows.
| Off-axis Distance | 150mm |
| Focal Length | 1000mm |
| Component Physical Diameter | 203mm |
Back surface of the substrate is perpendicular to the optical axis.
If you are unsure about any of the procedures used in this tutorial, please refer to the either "How to design a singlet lens" or "How to tilt and decenter a sequential optical component" before proceeding further.
Entering the basic geometry
To start the parabolic miror design, we will first define the system settings. Make the following adjustments in the System Explorer.
- Set Aperture...Aperture Type: Entrance Pupil Diameter and Aperture Value: 100
- Set Units...Lens Units: Millimeters
- Set Wavelengths...Wavelength 1: 0.550 um
Next, we can begin defining the system geometry. Add one surface to the Lens Data Editor after the STOP surface. Then enter the following parameter values on Surfaces 1-3. Note that the Image surface has a user-defined Semi-Diameter of 30 mm, as indicated in the solve box.
The "sag" or z-coordinate of the Standard surface is given by:
where c is the curvature (the reciprocal of the radius), r is the radial coordinate in lens units and k is the conic constant. The conic constant is less than -1 for hyperbolas, -1 for parabolas, between -1 and 0 for ellipses, 0 for spheres, and greater than 0 for oblate ellipsoids. To make the mirror surface parabolic, enter Conic: -1.
Because the focal length of a mirror is half the radius of curvature, enter Radius: -2000 mm. The sign of the radius of curvature is negative since the center of curvature is to the left (toward -Z-axis) of the mirror. Additionally, because Surface 1 and the image surface are co-located, we will choose not to draw Surface 1 in the layout so that we can see only the image surface at that location. Set the following property in the Surface Properties dialog.
To make the mirror substrate flat and orthogonal to the optical axis, choose the following options in the Surface Properties dialog. We will set Thickness: 40 mm since the manufacturer does not specify the substrate thickness on their website.
Open the 3D Layout with the following settings.
Add the off-axis distance
In the Tilt/Decenter tab of Surface Properties of Surface 2, specify Decenter Y: -150 mm.
From the manufacturer's specification, the off-axis distance is 150 mm and the physical diameter of the mirror is 203 mm. Specify the correct aperture size and location in the Aperture Tab of the Surface Properties menu.
Open the 3D Layout.
Note that the rays are moving away from the coordinate system. In order to center the image surface and make it orthogonal to the chief ray, insert a Coordinate Break surface before the image surface and place a Chief Ray solve on the Decenter Y and the Tilt About X parameters. OpticStudio will automatically calculate the amount of decenter and tilt needed to make the chief ray hit at the center of this surface at normal incidence.
Update the 3D Layout.
Perfect!
Using Off-Axis Conic Freeform
Since OpticStudio 20.2’s release, there has been available the Off-Axis Conic Freeform surface, which requires less setups to model the decentered aperture. More details are included in: The Setup Tab » Editors Group (Setup Tab) » Lens Data Editor » Sequential Surfaces (lens data editor) » Off-Axis Conic Freeform.
This surface will generate the sag distribution at the decenter value Y_0. The coordinate system for the surface is decentered and tilted to the center of the off-axis conic, as shown by the (y, z) coordinate system sketched below. 
The surface can do much more than build off-axis parabolic mirrors but here we only focus on how to model the same system as the one built above.
Apart from entering the surface radius of curvature, the conic coefficient and the decenter distance, we need to insert a coordinate break surface to get the surface tilted at the angle that it’s supposed to be. The value is Θ=atan(Y_0/(Radius of Curvature)).
Be careful that the system performance is extremely sensitive to data accuracy of the tilt angle. Especially for the case where aperture is not far away from the surface vertex, the manual input of tilt angle results in coma aberration in the image space. While building the same system, even setting the angle as the optimization target, we can have some residual coma if the input value is close to the correct value. As it’s too close to the correct value, the optimization might not be able to find the desired value. A correctly-aligned OAP will focus a collimated beam to a spot size of exactly zero to within the limits of floating-point arithmetic, so the recommended method is to set the tilt about X as variable with value 0 and then optimize the spot size to zero. If this cannot be achieved, then the system is otherwise misaligned.
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