The single mode fiber coupling calculation can be significantly expanded by using Physical Optics Propagation. The coupling is still computed by an overlap integral, but the use of Physical Optics gives major benefits:

• Any complex mode can be defined, the calculation is not restricted to Gaussian modes
• The fiber coupling overlap integral can be computed on any surface where the receiver fiber mode is known. This includes, but is not restricted to, the surface that represents the fiber
• External programs, such as Beam Propagation and Finite-Difference-Time Domain codes, can be used to compute the mode structure of a fiber (or any integrated optic device) and can express this as a complex amplitude distribution suitable for use in this calculation using the .zbf file format or DLL interface. See this article for an example. 
• Diffraction effects due to the beam being truncated on apertures, or due to propagation over long distances, can be accurately modelled

To set up the POP calculation, click on Analysis > Physical Optics > Physical Optics Propagation and select the following options:

This sets up a Gaussian mode of radial waist 4.6 µ to start on surface 1, and to propagate through the system to the image surface where we compute its overlap integral with an identical mode. On the Beam Definition tab, we use a sampling of 256 x 256, and press the Auto button in order to set the size of the starting array.

The Physical Optics window reports the fiber coupling, see the region outlined in red below. The POPD optimization operand gives all the Physical Optics data via the Merit Function Editor, and is often  a more useful reference. See the description of the POPD operand in the Optimization chapter of the User's Guide for full details. The POPD operand uses the Saved Settings of the POP analysis window, so if you have not saved these settings please do so now. 

Here is the phase of the coupling beam at the image surface, and the important data for the coupler:

POPD Data Item (obtained via the Merit Function)  Description  Value  0  Total Fiber Coupling 0.994  1  System Efficiency 0.998  2  Receiver Efficiency 0.996  10  Pilot Beam Waist  4.57 µ  23  Effective Beam Width  4.84 µ  26  M2  1.082

The phase is the most useful property to look at, because the irradiance profile is almost perfectly Gaussian (M² is 1.082). The phase of the receiver mode is exactly zero everywhere, so the phase shows us the degree of mismatch directly. To show the phase, go to the 'Display' tab of the POP window settings, and configure it like so:



Note the shape of the phase profile, which shows parabolic and quartic terms: equivalent to focus and spherical aberration. Note also the truncation of the phase profile at the edge of the lens. From the system efficiency, we know that less than 1% of energy is being lost due to the size of the lens.

If we optimize the total fiber coupling (remember the fiber/lens spacing is the only variable) we get a small improvement:

POPD Data Item  Description  Value  0  Total Fiber Coupling 0.994  1  System Efficiency 0.999  2  Receiver Efficiency 0.996  10  Pilot Beam Waist  4.57 µ  23  Effective Beam Width  4.84 µ  26  M2  1.081

The fiber coupling has not improved significantly because the majority of the phase error occurs where there is little energy:

{This plot was produced by taking a POP window showing a cross-section of irradiance, cloning it (Window > Clone), setting the clone to show phase instead of intensity, and then using Window...Overlay to overlay the irradiance plot.} 

The file at this point is saved as after POP.zmx.

So far, the differences between the paraxial Gaussian, single mode and full POP calculations have been subtle, with the answers being similar but with more physical insight being available as the analysis has become more sophisticated. However, as the length of the coupler is increased, diffraction effects become more important, and the differences in the approaches, and the superiority of the POP method,  becomes more obvious.

If the lens-lens distance is set to 20 mm, the ray-based single mode coupling calculation hardly changes: FICL reports a coupling efficiency of 0.99. This is because the rays in this space are nearly collimated, and rays always travel in a straight line.

However, the POP calculation predicts a coupling efficiency of 0.57: almost half. This is because the Gaussian mode diffracts and changes size in the optical space between the two lenses. After 20 mm propagation, the Gaussian mode has increased in size to 0.15 mm 1/e² width, which is now comparable to the 0.12 mm lens size. As a result, a significant amount of energy is diffracted at the aperture of the second lens, as this overlay of the irradiance immediately before and after the aperture of the second lens, and the phase profile of the beam shows. The beam as it focusses onto the receiver fiber is significantly non-Gaussian and has an M² of 2.45.

This is different to what the rays see. Expanding the y-scale on the layout so that the ray-distribution can be seen shows:

In fact, because of the spherical aberration, the rays predict that the beam is smaller on the second lens than on the first. When propagating over long distances with nearly-collimated beams, Physical Optics calculations are more accurate than rays.

Furthermore, POP allows rigorous optimization of the coupler. Setting the fiber/lens distance fixed (as we have already optimized it) and making the 20 mm interlens separation variable, a few cycles of optimization yields an optimum lens separation of 2.072 mm. Experimentally, a figure of 2 mm has been found, and the difference between these numbers is insignificant. Using the Universal Plot, we can see the sensitivity of the fiber coupling efficiency to the variation of the lens-lens separation:

Similarly, as the source fiber mode propagates to the receiver fiber, changing the lens-lens distance changes the M² beam quality parameter.