Authored By: Mark Nicholson 

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Introduction

Modeling wavefront aberration and PSF when imaging through a lenslet array is tricky, because multiple images are formed. In the attached file (which can be downloaded from the link at the end of this article) a 7 x 7 array of lenses forms a grid of spots on the image plane:



While all the geometric functions work well, the FFT and Huygens PSF struggle with systems like this, because wavefront error is measured relative to a single reference sphere, located in the exit pupil of the system. But what is 'the' exit pupil pupil of a system that forms 49 spatially separate images for each object point? Clearly there is no single reference sphere that can be applied, as the wavefront does not converge towards a single image location. OPD plots, wavefront error, PSF (and even the concept of a 'point-spread' function) have no meaning in such a system.

Physical Optics treats the wavefront as a single complex amplitude array, and can therefore treat the case where the beam interacts with a lenslet array more naturally. There are some setup considerations to be made, but otherwise analysis with Physical Optics is straightforward.

Setting Up POP
In a manner similar to the ray-based wavefront calculations, POP uses a reference so that the sampling requirements can be reduced. Using a reference means that Zemax needs only carry the 'difference compared to the reference' phase rather than the absolute phase of the beam. In POP, this reference is computed surface by surface, by using a pilot beam, which is the best-fit Gaussian beam that matches the beam being launched.

The pilot beam is then propagated from surface to surface using the Skew Gaussian beam calculation. At each surface, new beam parameters, such as the new waist, phase radius, or position are computed. The properties of the pilot beam are then used to determine if the actual distribution is inside or outside the Rayleigh range, and what propagation algorithms are appropriate.

However, the pilot beam approach suffers the exact same problem as the reference sphere when the beam is split into multiple beamlets. The phase reference for one beamlet is not a good reference for its neighbor. So in this case, we turn off the use of the pilot beam reference, and use a plane (flat) wavefront reference instead (note if the beam has an existing phase radius of curvature we could choose to use that instead).

Just double-click on the Lens Array surface, go to the Physical Optics tab, select Output Pilot Radius, and set it equal to Plane (this is already done in the sample file provided):



The phase of the input beam is now measured relative to a plane, instead of the best-fit Gaussian. The sampling of the beam needs to be set adequately high, of course, and this can be tested by looking at the phase of the beam after refraction through the lenslet array surface:



The beam can then be propagated to the image surface, and an array of spots with the correct relative intensity and diffraction structure can be seen:



although the diffraction structure is easier to see when logarithmic scaling is applied:







Array Size
There is one other trick that can be used with a file like this. Don't use the Auto button to set the size of the beam array!

This also arises from the one-beam-in, multiple-beams-out nature of the problem. When Zemax uses the pilot beam to do a first-pass analysis of the likely propagation of the beam through this system, it will use this data to compute the beam array size. This calculation will conclude that the incoming beam will form a small spot, and attempt to set the input beam size such that we get optimum sampling in both Fourier domains (the input and image domains).

However, the spot diagram on the first page of this article shows that the beam will actually retain its original size. It will be broken up into multiple beamlets, but the overall array size will not change. Therefore, just set the X and Y widths of the array to be some multiple of the input beam width, so that there is no significant energy truncated at the edge of the array:



and the setup of this calculation is complete.



Summary
Using POP to propagate a wavefront through a lens array and onto the focal plane is reasonably straightforward, as long as the single-beam-in, multiple-beams-out nature of the problem is understood. You need to:

1. Not use the pilot beam as a reference at the lens array surface, and instead use either a plane or user-defined (x, y) spheres as the phase reference,

2. Test the phase on the lenslet array surface directly to ensure it is adequately sampled

3. Do not use the Auto algorithm to compute the width of the array. Instead, just make the array sufficiently large that no significant energy is diffracted at the edge of the array.