This article explores problems that can appear in the beam intensity profiles when using Physical Optics Propagation (POP) in OpticStudio. The beams can be undersampled and/or can lack an adequate guard band around the beam. We demonstrate how to identify these problems and how to fix them.

Authored By Erin Elliott

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Introduction

This article is Part 2 of a series that works through an example system to show the correct way to use Physical Optics Propagation (POP).

In Part 1, we discussed the example system and looked at the beam file viewer. In part 2, in this article, we’ll look at the beam intensities and problems that can arise with the intensities. In Part 3, we’ll look at the beam phases and problems that can arise in phase. Parts 1 and 3 of this series can be found in the Related Articles section.

Problems in the beam intensity data

As discussed in Part 1, the Beam File Viewer is used to inspect the beam at various surfaces in the example system (shown below). 

Because we saved the beam files during the POP run, we can inspect the beam at any surface by choosing the appropriate File in the Beam File Viewer.

Surface 1 is a dummy surface located at the object plane. Thus, the beam at Surface 1 shows the beam that was launched into the system. It appears as expected: a Gaussian beam with a 6.4 micron waist radius (shown below).

At Surface 2, however, the grid width has become very large (124.2 mm) and the beam looks poorly resolved. 

At Surface 4 at the front of the first lens, the beam is even more poorly sampled with a grid width of 253.8 mm. Zooming in on the beam will show distinct pixels. The pixel edges represent fast changes in intensity, which will cause high-frequency artifacts to appear in both X and Y further along in the beam propagation. Simply put, the beam is undersampled, which will cause issues later on.

     

Fixing the sampling problem

To fix this problem, we need to increase the grid width of the starting grid. The beam has traveled a long optical distance, from focus to the front of the first lens, which is essentially a Fourier transform of the beam. In Fourier transforms, the resolution at a surface is inversely proportional to the grid width at the previous surface. In other words, the grid width in one space determines the pixel size in the following space, so a wider grid size at the first surface results in a higher resolution at the second surface.

In this case, we'll adjust the starting grid width by changing the X-/Y-Width setting from 0.1 to 0.4 mm. This will result in a better resolution at Surfaces 2 through 4. 

After running POP, check that the beam at Surface 1 is still adequately resolved by zooming in on the beam at that surface. Here, we can see that the sampling across the beam is still okay.

     

At Surface 4, the grid width is now a more reasonable value of 126 mm; zooming in shows the sampling across the beam is now much improved.

     

Adjusting sampling in collimated space

Moving on, let’s take a look at the beam when it gets to Surface 9, where we’ve placed a circular obscuration. We can see from the image below that the grid width hasn’t changed, but we can see that the circular obscuration is pixellated. As with the previously undersampled beam at Surface 2, the sharp edges along the X and Y edges of the pixels will create nonphysical artifacts further along in the propagation.

Inspecting the layout plot shows that the beam is not changing much in size between Surfaces 3 (the front of the first lens) and Surface 9 (the obscuration), because the beam isn’t traveling a long optical distance. So in this case, if we change the sampling at Surface 3, we can automatically apply a similar change to all the remaining surface in the collimated space.

Backing up to the beam file on Surface 3, we see that the total grid width is 120 mm. We can resample this beam in the Surface 3 Properties...Physical Optics menu in the Lens Data Editor. Change the total grid width to 30 mm by setting X-Y-Width: 30, which will give us 4x more pixels across the beam.

After running POP again, we can see that the grid width at Surface 3 is 30 mm, as expected (see below). Moving forward to Surface 9 and zooming in, we can also see that the central obscuration is now much better resolved, which will minimize artifacts from the pixel edges.

     

An undersampled image

With the previous two changes, the sampling looks good up through Surface 13, the back of the second singlet. Unfortunately, inspecting the beam intensity at the image plane (Surface 15) shows that it is no longer well sampled.

Again, the beam is traveling a long optical distance from the back of the second lens to focus. So, the resolution in the image plane is inversely proportional to the total grid width at the back of the second lens. Increasing the grid width at the back of the second lens will fix the sampling problem in the image plane.

Here, we will do this by resampling at the dummy surface (Surface 14) after the second lens. The starting grid width is 30 mm. 

In the Surface Properties, change the grid width by setting X-/Y-Width: 60 mm. 

After running POP again, we can check that the beam is still well sampled at Surface 14. At the image plane, the resolution across the beam is improved by a factor of 2.

Comparison to geometric ray trace results    

For this example system, the geometric rays are a good representation of the Gaussian beam as long as we’re not near a focus. Thus, we can cross-check the POP results using the geometric ray results. The Footprint Diagram is particularly useful for this cross-check, because it plots the geometric ray intersections with a surface and the aperture of that surface.

An example of a cross-check using the Footprint Diagram is shown below. For the geometric rays, we’ve set the Gaussian Apodization to 1, so the edge of the rays represent the 1/e² points in intensity. In the Beam File Viewer, we can see that Surface 9 has a grid semi-diameter of 7.8 mm. For comparison, under Surface 9 Properties...Aperture, set Maximum Radius: 7.2 mm. This will create a footprint plot with the same scale as the beam file. Comparing the two show that the beams are in good agreement in size.

If we carry out the same comparison in the focal plane, though, the two diagrams disagree. That’s because the geometric rays come to a perfect focus, while the Gaussian Beam has a finite waist size. The Footprint Diagram cannot be used to cross-check the POP results near a focus.

The Poisson Spot

Surface 11 occurs before the second lens and is some distance away from the central obscuration. The beam file at that surface shows some interesting features. There are Fresnel rings rippling outward from the edge of the aperture, as expected. There’s also a bright spot in the center of the beam. That spot is known as Poisson’s spot.

As a quick history lesson, in 1818 Poisson was a judge in a competition held by the French Academy of Sciences. Fresnel entered the competition with a wave diffraction theory of light. Poisson, a supporter of the particle theory of light, said that it was absurd to expect a bright spot to appear after a central obscuration. The debate was resolved when the president of the organization was able to experimentally recreate the bright spot and Fresnel won the competition.

Sampling of the Fresnel rings

Incidentally, a cross-section of the beam at Surface 11 shows that the sampling of the ripples (also called Fresnel rings) and the bright spot could be improved. 

The sampling could be improved by decreasing the grid width at Surface 3 below 30 mm. However, doing so would leave just a small guard band around the beam. A better solution would be to increase the total sampling from 1024 to 2048. Note that the change must be made in both the POP window and in the Physical Optics properties of the individual surfaces that we’ve already adjusted. The figures below shows the improved result after increasing the sampling to 2048. The sampling could be increased still further if the computation time, limited by the hardware of your computer, doesn’t become too long.

     

Previous Article: Using Physical Optics Propagation (POP), Part 1: Inspecting the beams

Next Article: Using Physical Optics Propagation (POP), Part 3: Inspecting the beam phases.

 

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