This article describes the use of vignetting factors in modeling a system with fixed apertures. Vignetting factors can be used to determine the size and shape of the beam which passes through the system unobscured. These factors also provide a mechanism for efficient optimization of such systems.

**Authored By Sanjay Gangadhara **

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## Introduction

Vignetting describes the effects by which the brightness of an image is reduced at its edge relative to its center.

Vignetting of the input beam is generally caused by surface apertures. It may be intentional on the part of the designer in order to limit aberrations, or it may be an unintentional consequence of overfilling a system composed of optical components with fixed sizes.

This effect can be modeled in OpticStudio using four scale factors and the tangential angle: VCX, VCY, VDX, VDY, and TAN.

In this article, examples are given on how to define vignetting factors both manually and automatically. An example is also given to show one of the main benefits of vignetting factors.

## Setting the Values for the Vignetting Factors: Manually

In principle, the user may specify any set of values for the vignetting factors. One use for this capability is to shape the input beam seen by the optical system.

Consider the singlet lens system provided in the file Vignetting example.ZMX (the file for this system is available for download from the top of this article). In this system, the lens is illuminated by an on-axis circular beam with a 10 mm diameter. The diameter of the beam is defined by the system aperture:

Imagine now that we wanted the system to be illuminated by an elliptical beam with a size of 8 x 6 mm. This can be done by modifying the size of the pupil seen by our on-axis field point. The appropriate vignetting factors VDX and VDY are determined from the following formulas:

and

where* P' _{x} *and

*P'*are scaled normalized pupil coordinates.

_{y}The vignetting factors can be specified in **Setup...Editors...Field Data Editor**:

The resultant shape of the beam may be observed in the Spot Diagram:

Setting the Values for the Vignetting Factors: Automatically

What if we didn’t want to specify the vignetting factors ourselves? We can let OpticStudio calculate the desired factors for us.

Re-open Vignetting example.ZMX. In this file, we will now place an elliptical aperture on the stop surface, with the desired size.

The marginal rays are now vignetted by the aperture, since our input beam (circular, 10 mm diameter) overfills the aperture. OpticStudio can determine how to modify the pupil size into which rays are launched to ensure no vignetting of the input beam, using the ‘Set Vignetting’ function in the System Explorer:

This function calculates the appropriate vignetting factors for each defined field point, to ensure that the top (*P _{x} *= 0,

*P*= 1), bottom (

_{y}*P*= 0,

_{x}*P*= -1), left (

_{y}*P*= -1,

_{x}*P*= 0) and right (

_{y}*P*= 1,

_{x}*P*= 0) marginal rays from each field point pass through all apertures in the system. For this case, OpticStudio finds the same values for the vignetting factors as we calculated manually:

_{y}However, do not underestimate the capability of the ‘Set Vignetting’ tool! In more complex systems which may be tilted and/or decentered, and/or contain asymmetric apertures, this tool can be very useful in helping the user determine the maximum beam size that can be passed through the system from each field point.

For example, open the file Cooke 40 degree field.zmx, located in the directory {Zemax}\Samples\Sequential\Objectives\. In this file, we will tilt and decenter the second element of the triplet using the Tilt/Decenter Elements tool (found on the Lens Data Editor toolbar):

For more details on tilting and decentering elements with this tool, see the article entitled “__How to tilt and decenter a sequential optical component__”. As a result of the tilt and decenter, portions of the input beam from each field point are vignetted:

The ‘Set Vignetting’ tool may then be used to determine the appropriate vignetting factors:

which ensure no vignetting of the beam:

Using Vignetting Factors for Efficient Optimization

One of the main benefits of using vignetting factors in OpticStudio is their aid in optimizing vignetted systems efficiently.

There are two different pupil sampling algorithms used by OpticStudio for optimization: Gaussian Quadrature (GQ) and Rectangular Array (RA). The GQ algorithm is much more efficient, but this algorithm does not account for vignetting; the algorithm assumes that all launched rays make it to the image plane. Thus, if rays are vignetted in the system (e.g. due to surface apertures), the GQ algorithm cannot be used, and the RA algorithm must be chosen instead.

However, if vignetting factors are used to modify the pupil seen by each field point in the system, then (in principle) all of the rays that OpticStudio launches from each field point will make it through the system - there will be no vignetting. In this case, the GQ algorithm can be used.

Let’s consider an example. Re-open the file Cooke 40 degree field.zmx. Then, change the Semi-Diameter of surfaces 5 and 6 to “5”:

Click on ‘Set Vignetting’ in the Field Data Editor to define the appropriate vignetting factors:

Now, we will evaluate the RMS spot radius in this system using the Merit Function. To do so, build a default merit function in the Optimization Wizard with the following inputs:

The merit function value is 9.93E-3, corresponding to an RMS spot radius of 9.93 microns:

If we increase the sampling (i.e. the number of rings and arms used in the GQ algorithm), the merit function value does not change significantly, indicating that our original result is well-sampled. The number of rays needed to generate this result corresponds to the number of TRAC operands in the merit function; there are 63 TRAC operands, i.e. 63 rays are needed.

Let’s now use the RA algorithm to sample the pupil. We systematically increase the number of rays in the sampling grid until a result that is similar to the GQ algorithm is found. We find that a grid of 10x10 is needed:

corresponding to the presence of 298 TRAC operands. Thus, with the RA algorithm we require 298 rays, or over four times as many as needed by the GQ algorithm to generate the same result. This demonstrates the power of the GQ algorithm. If we had not used vignetting factors to eliminate vignetting in this system, however, we would have been constrained to using the RA algorithm.

There are some situations in which the GQ algorithm cannot be used, because vignetting factors do not appropriately describe the vignetted pupil. Those include systems with extremely asymmetric or unusual apertures, or when vignetting is present in systems where aberrations are dominated by higher order terms. In those cases, the RA algorithm must be used during optimization, with the “Delete Vignetted” option selected. However, for optical systems with circular, elliptical, or rectangular apertures, vignetting factors can be safely used to describe the pupil, and the GQ algorithm may be chosen.

KA-01393

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