ZEMAX Development Corporation thanks Dr. Reinhard Voelkel of Suss MicroOptics SA for the experimental data used in this article.

This article is also available in Japanese.

This article describes a commercial fiber coupler, which couples two pieces of Corning SMF-28e Fiber using SUSS MicroOptics FC-Q-250 microlens arrays:

The manufacturers' data is as follows:

Single Mode Fiber, Corning SMF-28e 1
Numerical Aperture	0.14
Core Diameter	8.3 µm
Mode Field Diameter @1.31 µ	9.2 ± 0.4 µm

Microlens Array, SUSS MicroOptics SMO399920 2
Substrate material	Fused Silica
Substrate thickness	0.9 mm
Internal Transmission	>0.99
Lens Diameter	240 µm
Lens Pitch	250 µm
Radius of Curvature	330 µm
Conic Constant	0
Numerical Aperture	0.17

The file single mode coupler.zmx in the attached zip file (which can be downloaded from the last page of this article) shows how to implement this system. Please note the following:

• The object/lens and lens/image distance has been set by hand to 0.1 mm as this is approximately the right value. This number is to be computed by the optimization routine later
• A pick-up solve is used to make the final lens-image thickness the same as the initial object-lens image. Since the lenses and fibers are identical (within manufacturing tolerances), the optical system should work either way round, and should therefore by symmmetric
• The separation of the two lenses is set to 2 mm, as this is the experimental distance used. Again, this distance will be computed by rigorous optimization later
• The system aperture is set using float by stop size on the rear face of the first lens. This means that the system aperture is set by the physical aperture of the lens. The fiber mode we propagate through this system can be clipped by this physical aperture. In this case, the fiber mode is significantly smaller than the physical aperture
• Be wary of the multiple definitions of the term "numerical aperture". It may use the sine of the marginal ray angle, the sine of the angle at which the intensity has fallen to 1/e² (both definitions are used in different calculations in ZEMAX, as we shall see) or the sine of the angle at which the intensity has fallen to 1% of peak, as used by Corning. Definitions matter!
• A Gaussian apodization has been applied to the aperture definition to highlight the Gaussian distribution of light. This is currently only approximate. The calculations we shall use later will be precise
• The lens is diffraction limited across most of its aperture, and is diffraction limited across the region illuminated by the fiber mode

The paraxial Gaussian beam calculation is the simplest of the analytic tools we will use to characterize the fiber coupler. Its use is recommended to get a "feel" for the performance of the system.

The mode field diameter of the fiber at 1.31 µ is 9.2 ± 0.4 µm according the the Corning datasheet. Therefore, we set up the paraxial Gaussian beam calculation (Analysis > Physical Optics > Paraxial Gaussian Beam) as follows:

The beam waist is always positioned relative to surface 1, which in this case is positioned at the same place as the object surface. Therefore, a Gaussian waist of 4.6 µ is positioned at the source fiber location. It then propagates through the optical system:

It can be seen from this that the 1/e² beam size is 65µ at surface 3 and  70µ at surface 4. The physical semi-diameter of these surfaces is 120µ. This means that energy outside approximately two beam-widths will be truncated. Note also that the beam is not in best focus on the image surface: it has a size of 5.3 µ whereas it should be 4.6 µ on the assumption of symmetry. We will optimize the thickness of surface 1 (which also controls the thickness of surface 5 via a pick-up solve) to improve this. Note that the thickness of surface 5 is on a pick-up solve because the system should give the same coupling when used either way round: we are using identical fibers and identical lenses (within manufacturing tolerances) and so we expect the best system to be symmetric.

ZEMAX has an optimization operand GBPS, Gaussian Beam Paraxial Size, which can be used to optimize the fiber/coupler lens distance. Because we know the system will work best if symmetric, we know that the desired Gaussian Beam Size is 4.6 µ, and so the merit function is a simple one-liner:

Optimizing the fiber/lens distance gives a value of  0.117 mm for the fiber/lens distance, and the following Gaussian beam data:

This as as much as a simple paraxial Gaussian analysis can tell us. The file at this point is saved as after Gaussian optimization.zmx.

The single-mode fiber coupling calculation (available under Analysis > Calculations > Fiber Coupling Efficiency) provides a more powerful capability for fibers with Gaussian-shaped modes. It performs two calculations: an energy-transport calculation and a mode-matching calculation. The system efficiency (S) is the sum of the energy collected by the entrance pupil which passes through the optical system, accounting for both the vignetting and transmission of the optics (if polarization is used), divided by the sum of all the energy which radiates from the source fiber:

where F_s(x,y) is the source fiber amplitude function and the integral in the numerator is only done over the entrance pupil of the optical system, and t(x,y) is the amplitude transmission function of the optics. The transmission is affected by bulk absorption and optical coatings if use polarization is checked on.

Aberrations in the optical system introduce phase errors which will affect the coupling into the fiber. Maximum coupling efficiency is achieved when the mode of the wavefront converging towards the receiving fiber perfectly matches the mode of the fiber in both amplitude and phase at all points in the wavefront. This is defined mathematically as a normalized overlap integral between the fiber and wavefront amplitude:

where F_r(x,y) is the function describing the receiving fiber complex amplitude, W(x,y) is the function describing the complex amplitude of the wavefront from the exit pupil of the optical system, and the ' symbol represents complex conjugate. Note that these functions are complex valued, so this is a coherent overlap integral.

T has a maximum possible value of 1.0, and will decrease if there is any mismatch between the fiber amplitude and phase and the wavefront amplitude and phase.

ZEMAX computes the values S and T. The total power coupling efficiency is the product of these numbers. A theoretical maximum coupling efficiency is also computed; this value is based upon ignoring the aberrations but accounting for all vignetting, transmission, and other amplitude mismatches between the modes.

Now in this calculation, the source and receiver modes are defined by their Gaussian NA, which is defined as the refractive index n of the object or image space surfaces times the sine of the half-angle to the 1/e² power point. This angle can be computed in one of two ways:

• From the divergence angle of the Gaussian beam calculation, using the mode field diameter to define the beam waist (as calculated on the previous page)
• From the 1% power NA given in the Corning datasheet, and computing the 1/e² power point from that

The appropriate value for NA is 0.09 for both receiver and source fibers, and so the calculation is set up as follows:

which gives the results:

We may use the FICL operand to optimize the coupling efficiency with the following one-line merit function:

And running a few cycles of optimization gives a fiber/lens thickness of 0.110 mm (was 0.117 with the simple Gaussian calculation) and the detailed results:

Note the following:

• The system efficiency has not changed significantly, as this is set by the apertures of the surface and the size of the modes, which do not change much for this slight refocus
• The receiver efficiency has improved as the refocus makes the source fiber mode, after transmission through the optical system, a better match to the receiver fiber mode
• The maximum efficiency indicates the improvement that can be made by e.g. adding aspherics, extra surfaces etc. In this case, the efficiency is about as high as it can be

The file at this point is saved in the attached archive as after FICL optimization.zmx.

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.

The preceding calculations have all ignored the effects of surface reflections and bulk absorption in the optical materials, both of which ZEMAX has detailed models of. In both the POP and Single Mode Fiber calculations, the switch Use Polarization in the Settings turns on the Polarization calculation so that surface transmission effects and volume absorption can be accounted for.

Re-open the after POP.zmx sample file and in the settings of both the Fiber Coupling calculation and POP calculation select "Use Polarization". Save the settings. Then click on General > Polarization and define the incident polarization to be linear in y:

As a result, the fiber coupling calculation from POP and FICL drop to around 86%. Note the change is in the system efficiency (energy transport) rather than in the mode coupling: the polarization effects are too slow as a function of angle for the mode shape to be changed, although a more extreme system may show changes because of this.

Under Tools > Coatings > Add Coatings to All Surfaces, add a single-layer MgF₂ coating to all glass surfaces:

With this coating in place, note that the coupling efficiencies improve

	No Polarization	Polarization, No coatings	Polarization and AR coatings	Polarization and HEAR1 coatings
POPD	99%	86%	92%	98%

ZEMAX has comprehensive fiber modelling capabilities.

• The simplest method, a paraxial Gaussian beam is useful to get a feel for the system and to understand its first-order properties.
• The ray-based Fiber Coupling calculation is useful where the fiber modes are Gaussian, and diffraction effects are negligible.
• Physical Optics provides a comprehensive solution to modelling fiber coupling, allowing any complex mode to be used as source or receiver fiber, with full treatment of diffraction effects.
• The effects of thin-film coatings and material absorption can also be included.
• Optimization of these systems is straightforward with the FICL and POPD operands
• Tolerancing can be achieved using the same merit function operands as optimization.

External References

1. Corning Datasheet PI1446, April 2005
2. SUSS Micro-Optics FC-Q-250 Microlens array