This article discusses the conventions used by OpticStudio in defining the tangential and sagittal planes.

**Authored By Misato Hayashida**

## Introduction

Understanding tangential and sagittal directions is crucial when analyzing optical performance through tangential and sagittal responses in wavefront, optical path difference, MTF and other important analysis features in OpticStudio. This article explains the convention in OpticStudio and how to rotate the orientation of the sagittal and tangential planes.

## Tangential and sagittal planes for off-axis object fields

The tangential plane is a plane that contains the local object z axis and an object field point. The sagittal plane is the plane that is orthogonal to the tangential plane with which it intersects along the chief ray. For a rotationally symmetric system, field points along the y axis alone define the system’s imaging properties, and these points should be used. In this case, the tangential plane is the YZ plane, and sagittal plane is the plane that contain the chief ray and the x pupil axis.

For a non-rotationally symmetric system, however, the above generalization is not applicable. In OpticStudio, the convention is that the tangential plane is the plane that always contains the y-axis in the pupil local space. Note that this is different from ordinary text-book convention for off-axis field point which is shown in the figure below (top), where the tangential plane is the plane that contains the field point and the z-axis in the object local space.

Some algorithms in OpticStudio such as POWF and Fast Semi-Diameter computation adapt the textbook convention of the tangential and sagittal directions, rather than the OpticStudio convention. See Help Files for more detail. The tangential/sagittal discussed in this article are all based on the OpticStudio convention given above, and not the tangential/sagittal described in conventional textbook.

## Optical path difference and tangential angle

As explained in the previous section, the tangential and sagittal plane in OpticStuio always contain the y and x axis in pupil space coordinates, respectively, no matter where the field point lies in object space. The tangential and sagittal planes can be rotated along the z axis in pupil space, using a field definition called Tangential Angle (TAN) (or Vignetting Angle (VAN), in OpticStudio versions 16.5 or earlier).

Tangential rotation using the tangential angle (TAN) is essentially the same as rotating the pupil coordinates. This can be easily observed in the Wavefront Map.

The wavefront on the left is for TAN = 0 degree. On the right, tangential/sagittal direction is rotated for 45 degrees by TAN = 45 degrees. TAN can provide interesting information in a variety of analyses done in pupil coordinates, such as Optical Path Differences and Ray Fan diagrams.

It is important to note that the tangential and sagittal planes are rotated in the pupil space, not in object or image spaces:

Understanding the above distinction becomes important as we discuss MTF responses to the TAN in the next section.

## FFT MTF and field rotation

One analysis feature in which the tangential angle, or TAN, may become useful is the FFT MTF analysis. FFT MTF computes the tangential/sagittal responses in the pupil space coordinates, and TAN is necessary to fully explore the FFT MTF responses in non-rotationally symmetric system.

Interesting properties of FFT MTF with TAN are:

- For rotationally symmetric systems, simply defining TAN in a field provides the same effect as rotating the field in the object space. It is much easier to define TAN than to compute field values that precisely rotate the field in object space.
- For non-rotationally symmetric systems, tangential angle and field rotation will not yield the same MTF. Instead, for a given field rotation, the tangential response is computed along the tangential direction defined by TAN, and both field rotation and TAN are independent components necessary to fully analyze system.

Let us demonstrate this using the Double Gauss 28 degree sample file: Zemax/Samples/Sequential/ Objective/Double Gauss 28 degree.zmx.**Open** the file and adjust the field values such that **Field 1...X Angle (°): 14.0** and **...Y Angle (°): 0.0**, and **Field 2...X Angle (°): 9.998585** and **...Y Angle (°): 9.998585**. The resulting three fields all have the same “magnitude” of 14 degrees.

Open a FFT MTF analysis, to plot of the tangential responses of the three fields.

The MTF response is different for each field, which may seem counter-intuitive at first, because the Double Gauss 28 degree system is rotationally symmetric and the fields are all at 14 degrees. However, the tangential/sagittal MTFs are always computed in y/x direction, no matter what angle the field is rotated, resulting in the difference in MTF responses. This can be seen more clearly in the FFT Surface MTFs.

Now, let us rotate the tangential plane. Defining TAN = 45 degrees for field 1 will yield the following MTF surface:

Here, tangential/sagittal planes are rotated along pupil z-axis, and the result is identical to the MTF response of Field 3, which is Field 1 rotated by 45 degrees.

For non-rotationally symmetric system, however, these results will not always agree; rather, the combination of field rotation and TAN needs be employed to fully analyze the MTF properties of an imaging system.

## Huygens/geometric MTF and field rotation

Note that the tangential angle (TAN) will not have any effect on Huygens MTF or Geometric MTF for fields without vignetting. This is because the computation of Huygens/Geometric MTF is carried out in image space coordinates. For Huygens/Geometric MTF, the tangential response corresponds to the image of a periodic target oriented with lines along the image space X axis, and the sagittal response corresponds to the image of a periodic target oriented with lines along the image space Y axis. Huygens/Geometric MTF is ignorant of any changes in pupil space coordinates, so the tangential/sagittal direction is always in y/x in the image space no matter what the tangential angle is. This is different from FFT MTF, where tangential and sagittal responses are computed in pupil space.

In Huygens/Geometric MTF, rotating tangential/sagittal planes is achieved by simply rotating the image plane in the Lens Data Editor. This is equivalent to employing TAN in FFT MTF.

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