The following example illustrates how MODE can calculate the modes of a commercially available photonic crystal fiber, used for non-linear applications, by importing an SEM cross section. In order to couple light into this fiber, an optical lens system with a focused spot is used. While smaller spot sizes allow for a maximum coupling efficiency to the fiber, a larger spot size may be used to reduce the alignment sensitivity of the coupling efficiency. The initial study of coupling efficiency and alignment sensitivity is performed with an idealized Gaussian beam. Finally, the optical lens system that produces the desired beam is modeled in ASAP and the sensitivity of the input coupling efficiency to beam misalignment is calculated in MODE, demonstrating the interoperability features of these two products.
Problem Definition
A schematic diagram of the system to be modeled is shown above. The photonic crystal fiber is NL-15-670, manufactured by NKT Photonics. We would like to solve for the fundamental modes of this fiber, then determine the coupling efficiency to an ideal Gaussian beam. We will assume that we have a passive alignment system to couple light from the macroscopic optical system into the fiber, and that the alignment tolerance is ±1 mm. We want to find the spot-size that gives less than 3 dB reduction in coupling efficiency over this range of possible misalignment. We then use a simple lens design to produce this spot-size, and calculate the real spot using ASAP. We import this beam from the lens system into MODE and calculate the coupling efficiency and alignment sensitivity, making sure that we have achieved our goal of having less than 3 dB reduction in coupling efficiency over ±1 mm misalignment of the spot and the fiber mode.
Part 1. Calculate the modes of a small core photonic crystal fiber (NL-15-670) for use in non-linear applications
The fundamental mode is overlapped with a Gaussian beam to find the approximate spot-size that should be used for input coupling and meets our alignment tolerance of ±1 um.
For the first step of the problem, open MODE. Open the example file pcf_ASAP.lms included with this example. This file has imported a png image of NL-15-670 fiber manufactured by Crystal Fibre. The image, shown below, was taken from the specification sheet for NL-15-670 and is included with this example in the file pc_fiber_NL-15-670.png. The png image was imported using the image import wizard in MODE. A ring object and a series of circles were placed around the imported image to fill in gaps in the image import near the edges. The material was chosen to be Corning’s 7980 Silica and its dispersive properties are defined by Sellmeier coefficients available from Corning. In the following steps, we’ll find the modes of this fiber and calculate the loss.
Cross sectional image of fiber NL-15-670
Step 1. Find the effective index of the fundamental mode with a quick calculation
A screenshot of the structure as laid out in the MODE Layout Editor is shown below. We anticipate finding two, nearly degenerate modes contained near the fiber core. We’ll do a quick simulation with metal boundary conditions to find the effective index of the modes.
MODE layout editor
- If you are in analysis mode (the Analysis window is open), press LAYOUT to go back to the layout editor.
- Edit the MODE EigenmodeSolver to have the following properties:
Property |
Value |
---|---|
x position |
0 um |
x span |
10 um |
y position |
0 um |
y span |
10 um |
grid points x |
50 |
grid points y |
50 |
boundary conditions |
metal |
- Press the run button to start the analysis and make sure the wavelength is set to 0.8 microns. Then, press "mesh structure". You should see the meshed structure in the plot area on the right side of the analysis window.
Eigensolver Anaylsis
- Since we don’t know the effective index of the structure, choose to "search in range". Define the range to be from 1.45332 (the maximum index) to 1.4 and click on the button to Calculate Modes. You will quickly find two nearly degenerate modes corresponding to two possible polarization states, as shown below. The modes have effective indices of 1.409994 and 1.409347.
One of the two calculated modes.
Step 2. Calculate the modes of the full structure, including the loss
Now that we know where to look for the modes, we can include the full structure and perform a more detailed calculation. We'll increase the size of the Eigenmode solver area, modify the boundary conditions to be able to calculate losses (PML) and recalculate the modes:
- If you are in analysis mode (the Analysis window is open), press LAYOUT to go back to the layout editor.
- Edit the MODE EigenmodeSolver to have the following properties:
Property |
Value |
---|---|
x position |
0 um |
x span |
24 um |
y position |
0 um |
y span |
21 um |
grid points x |
100 |
grid points y |
100 |
boundary conditions |
PML |
- Go back to the Analysis tab and select "search near n". Uncheck the "use max index" checkbox and enter a value of n=1.409 for the region of effective index to search for modes. We can search for 10 modes, as before. Click on the button to Calculate Modes (it will automatically mesh the structure for you before calculating the modes). We find two modes that are well confined to the core of the structure. Click on the "log scale" radio button below the main plot window. You will see the screenshot shown in the following figure. The losses are calculated to be less than 5e10-8 dB/km – well below the limit where intrinsic material loss will dominate. Furthermore, at a distance of only 4 um from the center of the fiber, the electric field intensity has fallen by approximately 17 orders of magnitude. This indicates that we can save time by restricting our analysis to a region of 10x10 um^{2} around the fiber center and return to the original metal boundary conditions. At the same time, we’ll be able to increase the resolution near the fiber core, and obtain a more accurate calculation of the effective index.
One of the two fiber modes, on a log scale. Note how rapidly the electric field intensity decays away from the fiber core.
Step 3. A detailed calculation of the modes, restricted to a region near the fiber core
Now we know where to look for the modes, and we know that the losses are negligible and that we can restrict ourselves to a region of 10x10 um^{2} near the fiber core:
- If you are in analysis mode (the Analysis window is open), press LAYOUT to go back to the layout editor.
- Edit the MODE EigenmodeSolver to have the following properties:
Property |
Value |
---|---|
x position |
0 um |
x span |
10 um |
y position |
0 um |
y span |
10 um |
grid points x |
200 |
grid points y |
200 |
boundary conditions |
metal |
- Click run to go into analysis mode. Again, make sure that the “search near n” checkbox is checked and that the value of n=1.409 is entered. We can search for 10 modes, as before. Click on the button to Calculate Modes. You will find the 2 fundamental modes, as well as 8 higher order modes that are supported. One of the fundamental modes is shown below.
Linear scale |
Log scale |
Step 4. Overlap with a Gaussian beam
We will now calculate the overlap with an ideal Gaussian beam to determine the best spot-size we should use to couple with the fundamental modes of this fiber.
- Switch to the OVERLAP ANALYSIS tab and select the first mode, as shown below. Note that the effective modal area of the fundamental mode is approximately 1.2 um^{2}.
Overlap analysis
- In the Beam tab, we want to set the Gaussian parameters so that we can match the effective areas. The effective area of a Gaussian beam is πw_{0}^{2}. To match the modal area of 1.2 um^{2}, we need a waist radius, w_{0} = sqrt(1.2/π) um = 0.6 um, which can be set in the GUI. Click "Create beam" to place the Gaussian beam in the DECK.
- The overlap between the Gaussian beam and the two fundamental modes can be found. For the first mode, it is approximately 98% while for the second mode it is less than 1%. The coupling to the higher order modes is very small. The power coupling is lower than the overlap because the Gaussian beam is in a material of index 1 which does not match perfectly with the effective indices of the fundamental modes which are approximately 1.4.
- Set the "x shift" to an offset of 1 um and recalculate the overlap. You’ll see that it drops to approximately 5.7% - clearly lower than our 3 dB limit for this misalignment.
- Set the Gaussian beam to a radius of 2 um. And recalculate the overlap at an “x shift” of 0 and 1 um. You’ll see that the overlap is approximately 26.7% and 16.8% respectively. This variation is within the 3dB requirement of our system.
As a result of our preliminary analysis, we know that we need an optical system that can deliver an approximately Gaussian spot with a waist radius of approximately 2 um. The waist radius is the radius by which the field amplitude has fallen 1/e from the maximum.
Part 2: An optical system to deliver the approximately Gaussian spot
The lens we choose is an R-Biotar from Warren Smith's "Modern Lens Design", page 237.
Using ASAP, open and run the file BIOTAR.inr. This file constructs the lens and uses a 2 milliradian Gaussian beam that focuses to a spot that is approximately 4.5 um in diameter (FWHM). The incident Gaussian profile, ray trace through the lens system, and focused Gaussian profile are shown in figures below. The field data is exported to the file pcfinput.fld and will be imported into MODE in the next step.
View of the lens system.
The profile of the incident Gaussian beam in ASAP.
The ray trace through the lens system in ASAP.
The field profile of the focused spot that will be coupled to the photonic crystal fiber.
Part 3: Calculate the real coupling efficiency and the alignment tolerance
Now that we have our beam from a real optical system, we can calculate the coupling efficiency, as well as perform an alignment sensitivity analysis.
Step 1. Import the fld file
Move the mouse pointer to the DECK table and right-click to LOAD->From ASAP Interface File (*.fld). Browse the file pcfinput.fld that you have created in ASAP. The data is imported as a d-card called "fld_data". Select this d-card and you see the imported data plotted in the right-hand window, as shown below.
A screenshot of the OVERLAP tab with the imported data from ASAP shown in the right-hand image. Note that the data is not displayed on the same scale because it was not calculated on the same size window.
Step 2. Calculate the overlap
Click "Calculate" to see that the overlap is approximately 8% and 0.05% with the first and second modes, respectively. Again, the power coupling is reduced due the mismatch between the effective indices of the modes with air.
Step 3. Perform an alignment tolerance calculation
Redock the analysis window and load the alignment.lsf script file. Running the script will take a few minutes, depending on the speed of your computer. At the end, you'll have the two images shown below that display the alignment sensitivity of the power coupling to the two fundamental modes. The power coupling drops to approximately half of the peak value with a misalignment of approximately 2 um.
The region where the reduction is power coupling is less than 3dB is seen in the following plot. It clearly satisfies our alignment specification of ±1 um.