Modes of a photonic crystal fiber

In this example, we show how to use MODE to study a photonic crystal fiber (PCF).  Using the built-in analysis tools, we will calculate the effective index and dispersion of the PCF, as well as estimate how efficiently light can be coupled in to the PCF, and how much loss can result from bending the fiber.

Problem definition: More details

Consider a micro-structured silica fiber that uses a PCF of air holes with pitch = 23.2 microns and hole radius = 5.8 microns, with the central air hole missing.  Here, the user will learn to:

• Create a PCF using the Object Library
• Find the effective index of the fundamental mode at 1550 nm using Modal Analysis
• Measure both the waveguide and the total (including material) dispersion of the PCF using Frequency Analysis
• Analyze the total loss, including both the propagation loss and the macro-bending loss, in 90 degree bend using Bend Analysis

Modeling Instructions

This page is divided into 4 sections. The first section describes how to set up the model (PCF structure + simulation parameters). Sections 2 and 3 describe how to perform Modal analysis and Frequency analysis, and section 4 demonstrates how to use the built-in parameter sweeping tool to study the loss as function of bending.

Set up model

The physical structures to be modeled are created using the STRUCTURES tab in the Layout Editor. The first step is to create the fiber with the air holes.

• Begin by starting MODE.  You can save the MODE Simulation Project file (extension *.lms) at any point in this process. To do so, choose SAVE in the FILE menu.
• Press the MATERIAL DATABASE button

  

  and add a new sellmeier material with the following properties:

• Press on arrow on the STRUCTURES button

  

  and select a CIRCLE from the pull-down menu. Set the properties of the circle according to the following table.

• Press on arrow on the COMPONENTS button

  

  and select PHOTONIC CRYSTALS from the pull-down menu. This will open the object library window.
• Select HEXAGONAL LATTICE PC H-CAVITY from the list and press the INSERT button.
• Set the properties of the PC Cavity according to the following table.

• Press on the SIMULATION button

  

  to add an EIGENMODE SOLVER simulation region. Note that if your button does not look like the button to the left, you will need to press on the arrow to get the simulation region. Set the properties according to the following table.

• Press the Zoom EXTENT

  

  button to resize the view in the Layout Editor.

Modal Analysis

• Select the ANALYSIS tab with the RUN ACTIVE SIMULATION button

  

• Under the Modal analysis tab, enter the following parameters:

• Click the MESH STRUCTURE button to see the meshed PCF.
• Press the CALCULATE MODES button. Once you see a few modes appear in the index table, press the STOP button on the progress window.
• Once we determine the effective index of the fundamental mode to be near 1.4436, return to the layout editor by LAYOUT button

  

  Select the MODE object and set the number of mesh cells in x and y to be 120. This will make the simulations slower, but more accurate.
• Back to the MODAL ANALYSIS tab, now select the SEARCH NEAR N option, uncheck use max index, and enter the effective index of 1.4436.  Then press the CALCULATE MODES button.
• In addition to the Modal Analysis tab, one can also view the calculated modes using the Visualizer. In the Object tree as shown below, under the Eigensolver simulation region there is a EigensolverDataGroup called "data", which contains a material monitor as well as all the modes in the current mode list. One can then right click on the object and choose to Visualize the different datasets corresponding to each object.

For example, one can visualize the Electromagnetic fields of mode1:

Frequency Analysis

• Under the Frequency analysis tab, select the mode that you want to track (by clicking on it in the mode table), and enter the following parameters:

• Click on the FREQUENCY SWEEP button to begin the scan. The scan will take about a minute.
• To plot the calculated dispersion as a function of wavelength, select the FREQUENCY PLOT tab in the bottom righthand corner of the frequency analysis window. Then select "Dispersion" in the plot pull down menu. The plot can be seen above the frequency plot tab. If you press the PLOT IN NEW WINDOW you will get a new window.
• To determine the fraction of the total measured dispersion results from the waveguide geometry, as opposed to bulk material dispersion, we need to return to the layout editor and change the material properties of the fiber.  Return to the layout editor by clicking the LAYOUT button

  

  MODE will automatically close the Analysis Tab.  
• Select the fiber structure, set the material from "Corning 7980 Silica" to  "

Bend Analysis

In the following step of analyzing the photonic crystal fiber, we examine how the total loss in a 90 degree bend varies as a function of the radius of curvature. Before running the sweep, switch back to LAYOUT and set the material for the fiber back to "Corning 7980 Silica" and recalculate the modes. To perform a series of simulations to investigate the effect of systematic changes on cavity performance, it is convenient to take advantage of the built-in parameter sweep tool in MODE. There are 3 steps to setting up the bend analysis parameter sweep:

Step 1: Make data available to parameter sweep

We are interested in obtaining the loss from the fundamental mode, but the mode solver generally finds multiple modes. In order to keep track of the desired mode, we need to store a reference copy of the fundamental mode as a D-CARD (the D-CARD is a convenient data storage system which allows you to define, analyze, and transport data between analysis routines and applications).  Within the MODE LIST, highlight the fundamental mode, right click and "Add selected modes to global deck", a D-CARD with the default name "global_mode1" will be created.  Double click on this D-CARD and change the name to "FUND" (note that the name should be entered in capitals).

Next, we want to save the loss of the fundamental mode at each bend radius setting.

• Switch back to LAYOUT and edit the MODE data group by right clicking on the object with your mouse. Select the Edit object option from the list as shown in the figure above.
• In the Analysis->Variables tab click the bottom ADD button. This will add a result which the data group can give to the parameter sweep. Rename the result to "loss" as shown in the bottom left window.
• Switch to the Analysis_>Script tab shown in the right part of the image below. Then add the following lines of script commands which will add data to the "loss" result once a simulation has run. Press OK to save your changes.

fund = bestoverlap("FUND"); # find the mode that best overlaps with "FUND"

loss = getdata(fund,"loss"); # save the loss result for this mode

Step 2: Create a sweep

• Open the Optimization and Sweeps window using the VIEW menu at the top of the graphical user interface, or by right clicking on the top title bar of the MODE GUI.  
• Press on the CREATE NEW PARAMETER SWEEP button

  

  to add a new sweep to the simulation. Right click on the parameter sweep and choose to edit the parameter sweep. Set the properties according to the following screenshot. Notice that the only results you can chose are the results which are seen in the screenshot of the Analysis->Variables tab above.
• Here, the values specified for "roc" (radius of curvature) follow the equation roc = 2/linspace(1,5.5,10)^2, so we select "Values" as shown below and enter each value one by one. If we wish to sweep the parameter in evenly spaced intervals, it would be easier to select "Range" and enter the Start/Stop values instead.

Step 3: Run the sweep and plot the data

• In the Modal analysis window, select the "bent waveguide" option under modal analysis, and click "Calculate Modes". This will enable the sweep to change the value for the bend radius.
• Press the RUN SWEEP button

  

  in the Optimizations and Sweeps Window to run the sweep.
• Just like in Modal Analysis, we can use the Visualizer to plot the results.

However, here it is more useful to plot \( text{loss} \cdot \text{roc} \cdot \pi / 2 \) (instead of just loss), since we want to determine the loss in a 90 degree bend. Note that this is the actual bend angle, not the bend orientation.

To do this, open the script prompt window using the VIEW menu at the top of the graphical user interface, or by right clicking on the top title bar of the MODE GUI. Copy and paste the following commands into the script prompt and press ENTER on the keyboard to execute them.  

result = getsweepresult('sweep','loss');
loss = result.loss;
roc = result.roc;
plot(roc*1e3,loss*roc*pi/2,"radius (mm)","loss in 90 degree bend (dB)","","logplot");

Results and Discussion

The PCF is constructed by perforating a circular fiber of radius 300 um with air holes forming a hexagonal lattice.  The lattice constant is 23.2 um, and the air hole radius within the photonic crystal lattice is 5.8 um.  The cavity itself is formed by removing a central hole. Since the material of the air hole is specified as "etch" (with mesh order 1), whereas the mesh order of the fiber is 2, MODE will use the refractive index data from the air holes in regions where the two objects overlap. For more details about mesh order, refer to the mesh order page in the Reference Guide.

It is always a good idea to start with a relatively coarse mesh (a good rule of thumb is 50 grid points in each of the x and y direction).  This way the simulation will run quickly and still provide reasonable results. When high accuracy is required, increase the number of grid points. Note that the more grid points you use, the more memory required.

It is important to make sure that periodic structures are properly discretized in order to get accurate results. In general, it is always good to ensure that you can fit an integer number of mesh cells within each period of the device, and within the span of the simulation region. In this example, the x and y span are both 12 times the period of the structure along the respective axis, and there are 60/12=5 mesh cells per period.

In the screen shot below, or if you view the mesh in MODE by clicking on VIEW SIMULATION MESH

it is possible to see that there is a mesh cell at the same point at each  period of the hole array. If the mesh lines fall at different locations, each hole will have a slightly different size and shape, reducing the accuracy of the simulation.

Note that by setting the y min boundary condition to symmetric the lowest order mode with electric field polarized along the x axis has been chosen.

Modal Analysis Results

Before beginning Modal Analysis, one should always mesh the structure to see that the material properties that will be used in the calculation are correct.  Next, we provide an estimate of the mode effective index to the solver.   In this case, we use the SEARCH IN RANGE option because we do not yet know the effective index of the fundamental mode. With this setting, MODE will iteratively move through the effective index range (n1 to n2) specified, and locate any modes where the majority of the optical energy is located in the interior (and not along the boundaries) of the computation region. Once you see a number (say, more than 5) modes appear in the index table, press the STOP button on the progress window.

You should now have in front of you the following results:

By selecting various modes within the table, the spatial intensity profile of the mode will be plotted in the window.  Note that mode #1, with the highest effective refractive index, consists of a central intensity lobe - this is the fundamental mode that we are looking for (with effective index of approximately 1.4436). We can now use this value to speed up the calculation process when using a finer mesh with the SEARCH NEAR N option. Note that the fundamental mode is found in the first attempt, but now with a higher spatial resolution.  The effective index may shift slightly due to the higher resolution meshing.

The modes of interest are typically those that have energy near the center of the photonic crystal fiber (and away from the PML boundary layers).  Modes found near the PML boundary conditions tend to be artificial, and are automatically hidden.  The ADVANCED OPTIONS tab can be set so that MODE returns all of the modes found, should you be interested in doing so.

For each mode listed in the mode table, the effective index, propagation loss and polarization properties (see Mode List and Deck for a more precise definition) are shown.

Frequency Analysis Results

The frequency dependence of the effective index and propagation loss of a particular mode can be calculated with the frequency analysis tool. From this the associated modal group velocity, group effective index, modal delay and dispersion can also be determined. This type of analysis is done in the Frequency Analysis Tab. Here we determine the waveguide dispersion of the lowest order mode over a broad range in frequency (or wavelength).  

We can get the dispersion plot below by following the instructions in the "Modeling instructions" section.  Note that this is the plot of the total dispersion (i.e. material dispersion + waveguide dispersion), which at 1.55 microns is equal to 25.8 ps/(nm×km). To calculate the dispersion to greater precision, the number of grid points could be increased.  It is good practice to double the number of grid points and see if the results change significantly.

Now, to determine the fraction of the total measured dispersion results from the waveguide itself, we need to remove the material dispersion from the Corning material. To do this, we need to determine the refractive index of the Corning material at the wavelength of interest (1550nm) using the MATERIAL EXPLORER (press the MATERIAL EXPLORER button in the Material Database):

Note that the material is set to "Corning 7980 Silica". To determine the refractive index, set the min and max wavelengths to 1.55 microns and press "Fit and plot", and read off the real part of the effective index (1.4440) from the Re(index) plot. Once we set the index of the fiber to be a non-dispersive material of index 1.4440 and re-perform the sweep, the frequency sweep plot will show that the (waveguide only) dispersion is equal to about 1.3 ps/(nm×km).  Thus, the material dispersion is the dominant component of the total dispersion for this micro-structured fiber design.

Bend Analysis Results

The plot below shows that at around .3 meters the loss arising from the bend begins to increase dramatically. For smaller radius of curvature bends, the fundamental mode of interest begins to couple significantly to cladding modes within the photonic crystal fiber, leading to a complicated loss versus bend radius relationship.