This article explains the geometry behind the OpticStudio curvature cross-section analysis. Two important aspects to understand are the conventions used for the tangential and sagittal curvature directions, and the placement of the cross-section in situations where the aperture is decentered.

**Authored By Shawn Gay**

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## Introduction

In this article, we walk through how surface curvature is defined in OpticStudio and how that relates to the geometry in a Curvature Cross-Section analysis. A point of possible confusion is clarified for situations when the surface is not rotationally symmetric and the tangential and sagittal curvature becomes multi-valued at the surface vertex due to the arbitrary choice of the tangential direction at that point.

## System setup

Our discussion begins with a simple Standard Surface with a strong hyperbolic component (see the attachment "CurvatureExampleZernikeStandardSag.ZAR"). The surface has azimuthal symmetry centered on the surface vertex. The surface entries in the Lens Data Editor, as well as the 3D layout cross-section for surfaces 1 and 2, are shown in the figure below. The lens material is not relevant as we are only discussing the geometry of the surface.

A visualization of the surface sag is shown in the next figure. While it may be difficult to see, the curvature is decreasing as the distance from the surface vertex increases.

## Curvature directions in the Surface Curvature analysis

It is important to keep in mind the directions that OpticStudio uses in the curvature calculations. OpticStudio allows you to calculate curvature in the x, y, tangential, and sagittal directions. While the x- and y-directions are invariant across the entire surface, the tangential and sagittal directions vary from point to point on the surface. The convention used by OpticStudio is that the tangential direction points radially away from the surface vertex and the sagittal direction is perpendicular to this and the local surface normal at the point. For a square cross-section of the surface, the x- and y-direction vectors are shown at every point in the following figure:

The corresponding tangential and sagittal direction vectors are shown below. The central position highlighted in orange corresponds to the surface vertex.

In the OpticStudio Surface Curvature analysis, a visualization is produced of the curvature values over the entire surface visible through the aperture. Note that at the surface vertex, an arbitrary choice is made as to the tangential direction. In the Surface Curvature analysis, the convention that OpticStudio adopts is to align the tangential direction with the x-direction at the surface vertex.

## Curvature directions in the Surface Curvature Cross-Section analysis

In the OpticStudio Surface Curvature Cross Section analysis, the user specifies an orientation angle for the desired cross-section. OpticStudio will sweep through a curve on the surface in the plane corresponding to the cross-section orientation and report back the curvature values along this curve. As an example, consider a cross-section oriented at 0-degrees (i.e., the slice generated in the x-z plane). The following two figures show how the x- and y-directions (top) compare to the tangential and sagittal directions for a 0-degree cross section (bottom).

For a rotated cross section, the convention used by OpticStudio is for the tangential direction at the surface vertex to line up with the tangential direction vectors along the rest of the cross-section. The following figure shows an example of a 45-degree cross-section with the corresponding tangential and sagittal directions. For surfaces that are not azimuthally symmetric, this can cause some confusion when comparing Surface Curvature Cross Section results with the Surface Curvature results for tangential or sagittal curvature at the surface vertex.

## Cross-sections for decentered apertures

When an aperture is decentered, the cross-section in the OpticStudio analysis always goes through the center of the aperture, but does not in general pass through the surface vertex. The decentering results in tangential and sagittal curvature directions that do not in general correspond to the x- or y-directions, even along the 0-degree cross-section. The top figure shows the x- and y-directions while the bottom figure shows the tangential and sagittal curvature directions.

## Cross-section calculation examples

We now show some calculation results for the Surface Curvature and related Surface Curvature Cross-Section analyses.

### Case 1: No aperture decentering

To demonstrate an intuitive case, consider the 0-degree cross sections of the X- and Y-curvatures for a case with no aperture decentering. The first figure below shows the X- and Y-curvature maps with dashed lines representing the cross-section paths and black arrows pointing from the surface vertex point to the corresponding point in the cross-section plot. Along this cross-section, the X- and Y-curvatures have the exact same values at the surface vertex, but the X-curvature falls off much faster than the Y-curvature as the test point moves to either side along the cross-section path. The second figure below shows a similar presentation of cross-sections as the first, but this time for the tangential and sagittal curvature directions. It’s interesting to note that although the 2D maps look quite different, along this particular cross-section, the X- and Y-curvature cross sections have exactly the same values as the tangential and sagittal curvature cross sections. The reason why is because this happens to be a cross-section where the locally defined tangential and sagittal directions line up with the X- and Y-direction vectors (within a 180-degree flip to the left of the surface vertex, but due to the symmetry of the surface, this has no effect on the curvature values).

### Case 2: Decentered apertures

Example cross-section curvature calculations for our sample system with a decentered aperture applied are shown below. The aperture on surface 1 has been decentered by (-10,-10). The aperture is square with side length of 25 mm.

The results shown in the first two figures are x- and y-curvature values for 0- and 30-degree cross section orientations. The results shown in the bottom two figures are tangential and sagittal curvature values for 0- and 30-degree cross section orientations.

There are several differences to note between the Surface Curvature and the Surface Curvature Cross Section output. First, the false color plots in the Surface Curvature do not display values in the vignetted areas whereas the cross-section plots do. Second, note that the x- and y-axis values in the Surface Curvature analysis are plotted with respect to the surface vertex location whereas the cross-section x-axis corresponds to position along the cross-section projection, zeroed at the aperture center.

## A case without azimuthal symmetry

If a surface is not rotationally symmetric, the arbitrary definition of the tangential direction at the surface vertex means that the tangential and sagittal curvatures are multi-valued at that point. To illustrate how this looks in the Curvature Cross Section analysis, we show an example using a Zernike Standard Sag Surface (see below and refer to the "CurvatureExampleZernikeStandardSag.ZAR" sample file.)

The tangential and sagittal curvature results as calculated by the Surface Curvature analysis are shown below. Remember that the curvature value reported at the surface vertex in this analysis uses the convention that the tangential direction aligns with the +x direction. As such, it appears that there is a discontinuity in the curvature as you sweep through any cross-section except for the 0-degree case through the center. This is not really a discontinuity though, just a calculation artifact due to the fact that the tangential direction vector is really undefined at the surface vertex.

Tangential (top) and sagittal (bottom) curvature reported in the Surface Curvature analysis.

The first figure below shows tangential and sagittal curvature plots for surface without azimuthal symmetry, 0-deg cross-section with no decentering. The second figure shows tangential and sagittal curvature plots for surface without azimuthal symmetry, 30-deg cross-section with no decentering. Due to the difference in the tangential directions in the two different cross-sections, the curvature values reported are significantly different at the surface vertex. Once again, if you happen upon a situation like this, don’t let it worry you—it is just an artifact of the arbitrary choice of tangential direction used in the calculations and not a physical discontinuity.

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