# Using Physical Optics Propagation (POP), Part 3: Inspecting the beam phases

This article is Part 3 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, we looked 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 the phase profiles of the beam.

Authored By Erin Elliott

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## Introduction

This article is Part 3 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, we looked at the beam intensities and problems that can arise with the intensities.

In this article, we will look at the beam phases and problems that can arise in phase.

## The example system

As discussed in Part 1, we’re using the Beam File Viewer to inspect the beam at various surfaces in the example system.  Figure 1: The layout plot and the Lens Data Editor for the example system

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

## Inspecting the phase data

Now that the amplitude sampling is good, let’s inspect the sampling of the phase.  In general, phase profiles change more quickly and are more challenging to sample than the beam intensity profiles.

A lens generally imparts a quadratic phase across an incident beam.  In OpticStudio, the phase is plotted from -pi to pi only.  If the lens imparts a phase change of more than pi, a plot of the phase will show “phase wrapping.”  For example, a phase of 3pi/2 will be displayed as pi/2.  The phase wrapping is just a plotting convention, and does not indicate actual discontinuities in the phase introduced to the beam by the lens!

In the Beam File Viewer, let’s look at the front of the lens, surface 4.  The intensity profile looks fine. Figure 2: Intensity profile at surface 4, the front of the first lens

Now, we can ask the Beam File Viewer to display phase data instead of the intensity. Figure 3: Setting the Beam File Viewer to display phase data

The resulting false color plot looks good (Figure 4). A rind appears in the data because of phase wrapping. But all portions of the phase appear to be well sampled.

We can double-check that the phase is well sampled by looking at a cross-section plot. (Under “Show As” in the Beam File Viewer settings, choose “Cross X” or “Cross Y.”)  Figure 5 shows the cross-section.  You can see that without phase wrapping, the phase introduced by the lens would have a smooth shape and that it is well sampled. (In this case, the shape of the phase goes as r4, because the lens is an asphere with an r4 term present.  A spherical lens will go as r2.)

It’s important to check every surface in a system to make sure that the phase is well sampled in all cases! Figure 4: The phase at surface 4, the front of the first lens. The rings in the data are caused by plotting convention which produces phase wrapping. Figure 5: A cross-section of the phase at surface 4, the front of the first lens, shows that the phase introduced by the lens is well sampled.

## A faster system

Let’s also look at the phase for a faster system. The system below has an NA of 0.2 and a focal distance of roughly 40 mm. The lenses have an F/# of about 2.4. Again, both lenses carry aspheric terms on their curved surfaces to correct the spherical aberration. Figure 6: A faster system to analyze.

An NA of 0.2 corresponds to a waist radius of about 1.56 microns. I’ll run POP with the settings shown in Figure 7. I’ve also adjusted the sampling before and after the lenses in the method described in Part 1. Figure 7: POP settings for the faster system.

At the front of the first lens, the beam intensity looks correct and well sampled. But if I plot the phase, I get the pattern shown in Figures 8, 9, and 10. The first few rings in the phase are sampled correctly. Moving toward the edge of the aperture, the slope of the phase is steeper, and the rings are not sampled well. This causes aliasing: the phase is changing quickly, but the function is sampled more slowly, resulting in strange geometric patterns that don’t represent the true phase in the lens. Figure 8: The phase profile at the front of lens 1 is undersampled, which produces strange patterns due to aliasing. Figure 9: Aliasing appears toward the edge of the lens's phase function. Figure 10: A cross-section of the phase at the front of the lens also shows the undersampled regions clearly.

## Predicting the sampling needed

So, how much would we have to increase the grid size to adequately sample the phase profile of this lens? The sampling can be calculated using the Merit Function Editor in OpticStudio (A ZPL macro could also be used.) The Merit Function calculation is shown below in Figure 11.

The strategy used is:

1. Measure the optical path difference for two rays near the edge of the beam, that are separated by a small distance in the pupil (This is likely where the steepest phase change occurs).
2. Calculate the number of pixels that would be needed to sample the phase change between those two rays.
3. Multiply by the full aperture diameter to calculate the total number of pixels needed across the full aperture.

The calculation assumes that we need 4 pixels per 1 wave of optical path difference. The calculation across the full pupil shows that a grid of 38,000x38,000 pixels is needed. For 2 pixels per 1 wave of optical path difference, lower than is really needed, a grid size of 16,000 x 16,000 pixels would be needed. Storing one array of this size will require 4.3 GB of RAM. For more information, search “Memory Requirements” in the OpticStudio Help Files.

Calculations of this size may not be possible, depending on your computer’s hardware, or may take a prohibitively long computation time.

For many fast systems, though, the ray-based Fiber Coupling algorithm is appropriate and POP is not needed. For the majority of fiber coupling systems, the diffraction effects from the edges of the lenses aren’t significant. In that case, use ray-based coupling calculations. Figure 11: Using the Merit Function editor to calculate the sampling required to adequately sample the phase change due to a fast lens.

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