# Adding a phase surface to an off-axis mirror

Phase surfaces in OpticStudio are infinitely thin surfaces that add phase delays to the rays in an optical system. Though not physical, the surfaces are useful tools for tolerancing effects such as surface irregularity and index inhomogeneity by adding various perturbations to the rays. They can also be used to add measured data to surfaces. In this article, you’ll learn how to position a Zernike Fringe Phase surface at the surface of an off-axis parabola (OAP). We will also add a power error to the OAP, as an example of adding a perturbation that is centered on the off-axis part. The technique can be applied to any kind of surface.

Authored By Erin Elliot

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## Introduction

In OpticStudio, there are several surfaces that enable phase delays: Zernike Standard/Fringe Phase surfaces use Zernike polynomials, while the Grid Phase surface uses a grid of defined points. Each is unique in its applicability, as the Zernike surfaces allow us to simulate irregularity, while the Grid Phase allows us to add measured interferometric data to a surface.

In this article, we will use a Zernike Fringe Phase surface to simulate an error in the radius of curvature of an off-axis parabola (OAP) model. Several considerations in the positioning of the surface need to be taken to ensure that the phase surface does not interfere with the model beyond its intended purpose. Following that, we will use the Z4 term to add a power perturbation to the OAP.

## OAP system layout

The figures below show a starting OAP design that we’ll use to demonstrate. The OAP has a diameter of 50.8 mm, a focal distance of -187.5 mm, and a beam divergence angle of -36.9°. (See 'OAPWithPhaseSurface_v01_StartingSystem.zar' in the article attachments.) The stop is positioned at the front edge of the mirror. There are two coordinate breaks: one to shift the parent mirror downward, and one to follow the chief ray from the OAP vertex to the image plane. Figure 1: A layout plot of the example off-axis parabola design, with a back focal distance of -187.5 mm.  Figure 2: The Lens Data Editor for the OAP design.

## Insert and position the phase surface

In the OAPWithPhaseSurface_v02_OptimizePosition.zar sample file, a coordinate break and a Zernike Fringe Phase surface have been added at lines 5 and 6 in the Lens Data Editor, as shown in the figure below. The pickup in line 6 ensures that the Zernike Fringe Phase surface has the same base radius of curvature as the OAP. The radius of curvature of the phase surface does not introduce phase errors to the rays, but it does give a physical curvature to the phase surface, which affects where in space the rays encounter the surface. Figure 3:  Inserting a Zernike Fringe Phase surface in rows 5 and 6 of the Lens Data Editor.

We’ll use an optimization to correctly position the phase surface. The variables are shown in Figure 3. The table below describes the purpose of each variable.

 Variable Function Thickness of surface 2 ("stop") Positions the stop at the front edge of the OAP Thickness of surface 4 ("OAP") Positions the phase surface (surface 6) at the Z location of the OAP vertex Decenter Y of surface 5 ("go to OAP vertex") Positions the phase surface (surface 6) at the Y location of the OAP vertex Tilt About X of surface 5 ("go to OAP vertex") Tilts the phase surface (surface 6) to be tangent to the OAP vertex Thickness of surface 7 ("follow chief ray to image") Places the image plance at the best focus

Table 1: A listing of the variables seen in Figure 3, and the purpose of each variable.

## Position the stop

The thickness of the stop surface was set as a variable in the starting design, which enables us to position it at the front edge of the OAP. In the Merit Function shown in Figure 4, the operand in line 7 is set up to control that thickness. It functions by calculating the global Z coordinate (RAGZ) of the +Y ray at the stop, and the global Z coordinate (RAGZ) of the +Y ray at the OAP. These two values must be equal using the DIFF operand. The DIFF operand is the only row that contains a weight, and is the only row in the figure below that impacts the Merit Function value. Figure 4: In the Merit Function Editor, lines 1 through 7 ensure that the stop is positioned at the front edge of the OAP.

## Position the phase surface

The phase surface should be positioned at the vertex of the OAP, and should have a tilt tangent to the vertex of the OAP. To accomplish this, we’ll shift the phase surface back along the Z axis, starting from the vertex of the parent mirror. This distance is controlled by the thickness of line 4 in the Lens Data Editor (see Figure 3), which is set to a variable. The Decenter Y and Tilt About X terms in line 5 are also set to variables. This permits the phase surface to be tilted and positioned as needed.

In the Merit Function shown below, RAGY, RAGZ, and DIFF have been used to require that the Y and Z coordinates of the chief ray hitting the OAP and the chief ray hitting the phase surface are equal. Lines 11, 15, and 19 control the Y coordinate, and lines 12, 16, and 20 control the Z coordinate. The RAID and DIFF operands (lines 13, 17, and 21) require that the angle of incidence on the OAP and phase surface are equal, so that the phase surface and the OAP have the same tilt at the center of the OAP. Figure 5: Lines 9 through 21 in the Merit Function Editor are used to control the Y decenter, Z decenter, and tilt of the phase surface.

## Position the image plane

The final piece of the Merit Function is simply a default Merit Function for optimizing the RMS spot size. This controls the distance to the image plane so that the image stays in focus when the phase surface moves. The thickness of line 7 in the Lens Data Editor varies for this reason (see Figure 3).

After optimizing, the phase surface will have the correct position and tilt with respect to the OAP. See the OAPWithPhaseSurface_v02_OptimizePosition.zar file to inspect the system after optimization.

## Move the phase surface away from the mirror

In the file OAPWithPhaseSurface_v03_OffsetPhaseSurface.zar, the phase surface has been backed up slightly using the thickness of line 5, as shown below, so that it physically occurs after the OAP surface. This offset is not strictly required; the phase surface would work correctly even if the rays travel slightly in the reverse direction to reach the phase surface.  But the offset does make the layout plot easier to understand.

The distance to the focal point (the thickness of line 7) was optimized to keep the system in focus after the added distance in line 5. For that optimization, a Merit Function containing only RMS spot size was used. Figure 6: In line 5 of the LDE, the phase surface is offset slightly in Z. This ensures that all points on the phase surface are physically located to the left of the OAP.

## Final system

In the final system, the phase surface is positioned properly, as shown below. It is decentered in Y and Z to the vertex of the off-axis part. It is tilted to match the tilt at the vertex of the off-axis part, and it has a curvature similar to that of the off-axis part. This ensures that rays that arrive at a given XYZ coordinate on the phase surface encounter a similar XYZ intercept on the surface of the off-axis parabola itself, preventing registration errors between the phase surface and the OAP. Figure 7: The phase surface is positioned correctly, so that it is very near the OAP surface.

## Add phase to the OAP

Now that the phase surface is correctly positioned, the phase surface can be used for tolerancing surface irregularity or adding measured surface data. As a simple example, suppose that the OAP has an error in its radius of curvature. Changing the Radius of Curvature of the parent parabola would not result in a power deformation centered on the part center. The correct way to add a power error to the OAP, then, is to use the Zernike Phase Surface that is centered on the off-axis part.

A complete list of the Zernike Fringe Phase terms are available in the OpticStudio Help files. The first few terms are shown below. Figure 8:  The first five Zernike Fringe Phase polynomials.

Power is held in the Zernike fringe term, Z4. The total phase delay added by the term is: The radius ρ is normalized by the Norm Radius column on the Zernike surface and should be set to just slightly larger than the clear aperture on the part, as shown below. The Z4 coefficient has units of waves. Setting Z4 to 0.5, you can calculate that at ρ = 1, the peak-to-valley wavefront added should be 1 wave. Figure 9: Adding power to the Zernike Fringe Phase surface.

The wavefront map confirms the result, as shown below. Figure 10: A wavefront map shows that we’ve added 1 wave of P-V power to the off-axis part.

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