Accurate analysis of coupling efficiency is critical in the design of fiber coupling systems. This article demonstrates the use of several fiber coupling efficiency analyses in OpticStudio.

Authored By Mark Nicholson, Kristen Norton

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Introduction

Simulation of single-mode fiber coupling efficiency is handled well by OpticStudio Sequential Mode. This article demonstrates how to set up a coupling system and examines the multiple tools available in Sequential Mode for beam and fiber coupling analysis, including Paraxial Gaussian Beam Propagation, Single-Mode Fiber Coupling, and Physical Optics Propagation. Accounting for losses due to partial reflection and material absorption is also discussed.
 
Zemax, LLC thanks Dr. Reinhard Voelkel of Suss MicroOptics SA for the experimental data used in this article.

Setting up the initial design

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	8.2 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 Article Attachments 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 symmetric
• 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 OpticStudio, 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.

Using the Paraxial Gaussian Beam calculation

The Paraxial Gaussian Beam Data analysis 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 wavelength 1.31 µm is 9.2 ± 0.4 µm according to the Corning datasheet. Navigate to Analyze...Gaussian Beams...Paraxial Gaussian Beam and set up the analysis 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 radius of 4.6 µm 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 65.6 µm at Surface 3 and 70.0 µm at Surface 4. The physical semi-diameter of these surfaces is 120 µm. 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 5.6 µm, whereas it should be 4.6 µm 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 symmetry. Note that the thickness of Surface 5 has a Pick-Up Solve, because the system should give the same coupling when used in either direction: we are using identical fibers and identical lenses (within manufacturing tolerances) and so we expect the best system to be symmetric.

OpticStudio has an optimization operand GBPS, Gaussian Beam Paraxial Size, which can be used to optimize the distance between the fiber and coupling lens. Because we know the system will work best if symmetric, we know that the desired Gaussian Beam Size is 4.6 µm at Surface 6, and so the Merit Function is a simple one-line operand.

Go to Optimize...Optimize! to run a Local Optimization.

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

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

Using the Single-Mode Fiber Coupling calculation

The Single-Mode Fiber Coupling calculation (available under Analyze...Fiber Coupling...Single Mode Coupling) 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 Fs(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, T:

where Fr(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.

OpticStudio computes the total power coupling efficiency as the product of S and T.  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.

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 in the previous example).
• 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.

This produces the following results.

We may use the FICL operand to optimize the coupling efficiency with the following one-line Merit Function.

And running 10 cycles of optimization, the fiber/lens thickness has changed to 0.107 mm (was 0.117 mm after the simple Gaussian calculation) with the following fiber coupling 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 total coupling efficiency is the product of system and receiver efficiency

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

Using the Physical Optics calculation

The single mode fiber coupling calculation can be significantly expanded by using Physical Optics Propagation (POP). 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, go to the Analyze Ribbon...Physical Optics, and use the following settings.

In the Beam Definition tab, start by entering the X- & Y-Sampling, and Waist X & Waist Y. Then, click the Automatic button to calculate the initial widths between data points.

This sets up a Gaussian mode of radial waist 4.6 µm 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.

The Physical Optics Propagation window displays the fiber coupling results; see the text highlighted below the plot in the following screenshot. The POPD optimization operand reports 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 Help Files for more information. The POPD operand uses the saved settings of the POP analysis window, so if you have not saved these settings please do so now. The Save button is boxed in red in the screenshot, below.

Here is a cross-section of the phase of the coupling beam at the image surface.

The phase is the most useful property to look at, because the irradiance profile is almost perfectly Gaussian (M² = 1.086). The phase of the receiver mode is exactly zero everywhere, so the phase shows us the degree of mismatch directly.

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.

The POPD operand can be used in the merit function to calculate the total fiber coupling efficiency, the system efficiency, the receiver efficiency, the ideal beam waist size, the actual beam size, the resulting M² value, and many more diagnostic quantities.  Here is a screenshot of these POPD results in the Merit Function Editor.

Set the Target and Weight to 1 for the total fiber coupling efficiency operand (POPD Data = 0). If we run an optimization (remember the fiber/lens spacing is the only variable) we get a small improvement.

The file at this point is saved as “after POP.zmx”. The fiber coupling improved slightly, but the majority of the phase error occurs where there is little energy.

To create an overlay plot like the one shown above, go to the toolbar of the Physical Optics Propagation analysis, and click the Clone button to create a second copy of the window.  In the second window, expand the settings, and go to the Display tab. Change the Data setting to Irradiance. Then, go back to the first window, and click the Active Overlay button in the analysis toolbar. These steps are shown in the screenshot below.

Clicking the Active Overlay button will open the Overlay Series window.  Use the following settings in the Available Series and Series Settings tabs, then click OK.

Try changing the lens to lens spacing to 20mm. The POP calculation now predicts a coupling efficiency of 0.57. 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 0.14 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. We can see this in an overlay of the irradiance immediately before and after the aperture of the second lens. The beam as it focusses onto the receiver fiber is significantly non-Gaussian and has an M² > 2.

POP also 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 2.15 mm.  This file is saved as “after interlens optimization.zmx”.  Using the Universal Plot, we can see the sensitivity of the fiber coupling efficiency to the variation of the lens-lens separation. Navigate to Analyze...Universal Plot...1-D...New and define the settings below. 

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

Accounting for surface transmission and bulk absorption

The preceding calculations have all ignored the effects of surface reflections and bulk absorption in the optical materials, both of which OpticStudio can accurately model. In both the POP and Single Mode Fiber calculations, the switch Use Polarization in the analysis Settings turns on the Polarization calculation, so that losses due to Fresnel reflections and volume absorption can be accounted for.

Re-open the “after POP.zmx” sample file, and in the settings of both the Fiber Coupling analysis and the Physical Optics Propagation...General, check on Use Polarization. Save the settings. Then go to the System Explorer...Polarization and define the incident polarization to be linear in the Y-direction. 

As a result, the fiber coupling calculation from POPD and FICL drop to around 86%. If you’re looking in the merit function editor, note that the FICL operand also requires settings the “Pol?” flag to 1. You'll notice that 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.

In the toolbar of the Lens Data Editor, click the Add Coatings to All Surfaces button, and add an AR coating (single layer of MgF2) to all glass surfaces.

With this coating in place, the POPD coupling efficiency increases to about 93%.  Similarly, adding the HEAR1 coating further increases the efficiency to 99%.

References

1. Corning. 2005. Corning SMF-28e Optical Fiber Product Information. January. http://www.tlc.unipr.it/cucinotta/cfa/datasheet_SMF28e.pdf.

2. SUSS MicroOptics. n.d. Products. https://www.suss-microoptics.com/en/products.

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