# Fiber optic components; developing models from a datasheet

Optical fibers are extremely important components in nano-photonics research, where they are often used to couple to and from waveguides, quantum dots, lasers etc. Although studying these components may not be central to your scientific goals; accurate fiber simulation with Lumericals optical solvers can be crucial for optimizing your experimental set-up, understanding alignment errors and controlling for tolerances in manufacturing. Based on user feedback we have recently developed a set of cleaved face optical fiber components, available in our object model library. These include:

1. A cleaved face single mode step index fiber.
1. A cleaved face multimode fiber with a power law core

It is difficult to anticipate the needs of all users, but these fiber optic components can be edited and should be flexible enough to suite most applications. Nonetheless, translating a particular fiber from a datasheet into a component model may present some difficulty due to a lack of glass models, unknown doping profiles and/or understanding what parameters in the datasheets are of importance. Although we cannot provide accurate glass chemistry and doping models; we can demonstrate how to identify important parameters in the datasheet and use this information to build realistic fiber components for simulation. In this example I will present a simple workflow, demonstrating how to translate a datasheet into a component model, and discuss methods of taking the it further.

### Objective: Develop an accurate fiber model in Lumerical based on a datasheet.

To begin with let’s look at the cleaved face fiber optic components. Rather than simply meshing two cylinders, these objects are defined as planar solids with a top and bottom cleaving angle. Fiber cleaving is a controlled break that typically results in a perfectly flat end face; however, this process, is not exact. The addition of angled end faces allow the user to more realistically model the experimental set-up. One can find the end face angle through microscopic measurements, or as a means of exploring tolerances. Furthermore, a flat angle is not always preferred. A predefined angle can be useful for coupling at non-normal incidence such as with a grating coupler.

For the average user working at an optics bench a fiber datasheet will provide an abundance of information; much of which will be inapplicable. Some information may of interest but will ultimately not enter the model, and a few given parameters are crucial. Given a Corning® SMF-28 Ultra Optical Fiber datasheet we can attempt to sort these into three categories.

#### Table 1. Fiber Datasheet

Important Interesting Inapplicable
Mode-field diameter Dispersion Environmental specs
Glass geometry Max attenuation Mechanical spec
Performance characteristics Attenuation vs. wavelength Macrobend loss
Polarization mode dispersion

Obviously the zeroth step is to ensure you are using a fiber that is appropriate to your frequency bands here we see that 1260nm cut-off is given, and upper wavelength values of 1625nm are given. If your research is being done in a lab, we can most likely ignore things like environmental and mechanical specifications. In the second column we have placed information that would be crucial for telecom engineers looking into the long haul signal characteristics but in most applications, one would not be concerned with the minimal dispersion and loss present in these fibers. A possible exception may be looking at the pulse shape evolution. It may be appropriate to perform sanity checks on these numbers through analytic methods or numerically in INTERCONNECT. Modelling these components in INTERCONNECT is well documented and not the subject of this post. See optical fiber primitive element and/or AppGallery example.

The important parameters are glass geometry, mode-field diameter, and performance characteristics. Glass geometry, and performance characteristics can be immediately plugged into the cf_fiber structure group. Mode field diameter (MFD) is defined as the radial distance from the optical axis such that $$P( r) = 1 - e^{-2} \approx 0.865$$ This parameter describes the mode spatial profile and so is likely the most relevant performance metric for your simulation. The question is how to reproduce this result in the component model?

### Method: Adjust the cladding index, assuming the core is pure silica, to find the MFD.

In order to find the core cladding index variation that produces the desired MFD we iterate the following steps.

1. Choose $\Delta$n, and find the modes using FDE
2. Calculate the intensity distribution
3. Minimizes $|P( r) - e^{-2}|$ to find r
4. Compare this result with given MFD, change $\Delta$n, repeat.

Here is a project file and a script that performs the first 3 steps.
cf_fiber.lms (318.3 KB)
MFD_Solve2D.lsf (3.0 KB)

First we set up the simulation with the datasheet parameters. The core diameter and the MFD diameter for 1310nm are given as 8.2 $\mu m$, and 9.2 $\mu m$ respectively. Since the MFD is approximately ~ $7\lambda_0$ across for this single mode device; we should suspect that the index variation is quite small. An initial guess of 0.005 seems sufficient, but some experimentation may be required to find a good initial guess. Also defined here are the limits and resolution of the radial distances to sweep over.

The modes returned by FDE are given below with the core structure drawn onto a linear scale color plot.

Due to cylindrical symmetry the modes are degenerate in polarization. Next notice that the an effective area of 63.4516 $\mu m$ is found, corresponding to a radius of D $= 8.9883 \mu m$. If we use the power and impedance integration option under the modal analysis tab we find the power contained in this region is 87%.

Running the radial sweep defined above we minimize $|P( r) - e^{-2}|$ to find D $= 8.901 \mu m$ within 0.1%.

The FDE analysis gives the effective area based on the definition developed by G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

$$\text{Effective Area} = \frac{(\int_{}^{}|E|^{2}dA)^{2} }{\int_{}^{}|E|^{4}dA}$$

This has the advantage of not assuming a normal distribution; however, it has been shown through experiment that the effective area and MFA may differ, by up to 10%. See this Corning® paper for a discussion on the topic. Care should be taken when using these definitions interchangeably. Manufacturing tolerances are given as $\pm0.4\mu m$; thus, we have found an reasonable approximation of this device.

To improve upon the model we could attempt to manually bisect upon the desired MFD, by choosing an updated $\Delta$n. You may want to adjust the resolution of your radial sweep by increasing the number of points or by reducing the limits since we are now is close to our solution.

After a few a iterations of adjusting $\Delta$n

For dn = 0.004585, MFD was found to be 9.2 micron to within plus or minus 0.0013743% This differs from the desired value 9.2micron by 0nm : increase dn

A solution where the integrated power differs from the expected value in the 5th decimal place seems like good result.

At this time let us plot the power integrated vs radius and consider the limits of numerical integration. We see here a plot with a discontinuities in it’s slope due to granular nature of sweeping over a rectilinear grid in cylindrical coordinates. The mesh size here is about 50nm, while our step size is $dr \approx 3.7nm$. Although the step size is well below the grid size we continue to add mesh cells to the integration region at each concentric circle step. Keep in mind that it will not be possible to continue refining our step size; unless we also reduce our mesh size.

In the data sheet under Performance Characteristics we are given that the Numerical Aperture for this fiber is “0.14 as calculated from the 1% power level in the far field scan”. Plotting the energy density in the far field (see below) we find the $ln(0.01) \approx -4.61$, point at a 11$^{\circ}$ half angle. This corresponds to an NA = 0.19 which given the inexact measurement technique and error associated with cleaving seems reasonable.

### Taking It Further

• #### Different Methods

Here we have used a filtering function to perform the integration which I believe is the most general and easiest code to understand. This method is especially versatile when dealing with non-uniform grids, which is often the case in FDE where using mesh override regions cover the guiding structure is recommended. We could speed things up if we knew the limits of integration beforehand which would be possible on a uniform mesh. If speed is critical we could make use of the fact that the definition depends on the mode being cylindrical symmetric, and collapse the geometry into 1D radial vector before minimizing. Alternatively one could use a MATLAB, or SciPy fitting toolbox to find the Gaussian.

• #### Other Application

Users interested in the polarization mode dispersion PMD could measure the power coupling between the degenerate modes. Obtaining models for polarization maintaining fibers (PM-fibers) could immediately be accomplished by breaking this degeneracy using a Panda or Bow-Tie configuration. Other PM-fiber component designs such as elliptic cladding or stress-induced birefringence could be modeled by adjusting the cladding vertices of the cf_fiber structure group, or using a grid attribute. Alternatively, one may be looking at performing these calculations a number of times. Then it may be worthwhile to automate the bisection method. Finally, although the spatial profile fitting, that we have performed here will be the most important result for the vast majority of users, some advanced models may require accounting for the dispersive properties of the fiber. In this case one could build custom materials by fitting the MFD at 1310nm and 1550nm. Then given the effective group index (or Dispersion) fit the local material slope to obtain accurate models within broadband simulations.

If you have any examples of fiber models that you developed, using this or other methods please share them here. It would be very interesting to see your results. Are there other applications that you are interested in? Comment below with questions and/or suggestions. Based on user feedback we may look at different fiber application in the future.

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