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The project file used in this example is provided in the top-level page. Alternatively, the file can be created from scratch by following the steps described in the Modeling Instructions page. The figure below shows the step-index fiber geometry and index profile, together with a screenshot of the top XY plane view in Layout mode in FEEM after the simulation has been set up.

 

step_index_profile

step_index_xy_screenshot_zoom50

(Left) Index profile of the step-index fiber.  (Right) XY top plane view of the simulation setup in FEEM.

 

Note that the cladding radius used in the simulation setup is much smaller than the actual physical value. This is justified by the fact that the modes of interest are mostly confined in the core so it is not necessary to simulate the entire cladding. As explained in the Modeling Instructions page, we assign PEC boundary conditions to the outer boundary of the cladding circle so this becomes the boundary of the simulated region.

Run Simulation

Open the step_index_fiber.ldev project file and run the simulation by clicking the "Run" button icon_runpar2 in the "FEEM" tab of the tabbed toolbar.

 

Results

The settings specified under FEEM Solver Region in the Modeling Instructions page have been chosen so that the solver will find the first twenty modes supported by the fiber for a wavelength of \(\lambda=1.55\mu\text{m}\). Here we will analyze those results and validate them with a semianalytical calculation for the TM modes.

 

Semianalytical Solutions

It is possible to find the effective indices of the TE, TM and mixed modes (EH and HE) supported by the step-index fiber by solving transcendental equations numerically (see Related references at the top-level page). Here we only consider the semianalytical solutions for TM modes, which are calculated in the MATLAB script step_index_fiber.m.  You can download the text file step_index_fiber.txt, which contains the calculated results for the first couple of TM modes (this file and the MATLAB script were also used in this example). The MATLAB script uses a root finder to solve the characteristic equation for the effective index \( n \):

 

$$ \frac{J_1(h \; a)}{h \; a \; J_0(h \; a)} + \bigg( \frac{n_2}{n_1}\bigg)^2\frac{K_1(q \; a)}{q \; a \; K_0(q \; a)}=0 $$

 

where \( n_1 = 1.44 \) and \( n_2 = 1.4 \) are the material index values for the core and cladding, respectively. The core radius is denoted by \( a = 10 \mu \text{m} \). \( J_{\nu}(x) \) and \( K_{\nu}(x) \) are the Bessel function of first and second kind, with

 

$$ h \equiv k_0 \sqrt{n_1^2 - n ^2}, \quad q \equiv k_0 \sqrt{n^2 - n_2 ^2}, \quad  k_o = \frac{2 \pi}{\lambda}. $$

 

FEEM results

The effective index can be visualized by selecting the FEEM in the Objects Tree, right clicking on the "modeproperties" result in "Result View - FEEM"  and selecting Visualize>New Visualizer. The plot of effective index versus mode number will look like the one below. Note that the loss is exactly zero since we only used simple dielectric materials with real-valued refractive index. The staircase shape of the effective index plot is due to mode degeneracy in this fiber.

 

stepindex_fiber_neff_vs_modenumber_zoom70

Effective index of the first twenty modes supported by the step-index fiber calculated with FEEM.

 

According to the semianalytical solution described above, the first TM mode, TM01, has effective index 1.43729 (up to five decimal figures). We can use this values to search for this mode and compare the FEEM and semianalytical results. To modify the mode search:

Switch to Layout mode and go to the Modal Analysis tab in the Edit properties window of the FEEM.

Disable the option "use max index" and provide the new target value for the effective index, n.

It is also convenient to reduce the number of trial modes to 1 so that only the mode with the closest effective index will be reported in the results.

 

After running the new simulation, we find that the FEEM and semianalytical results are very close. A careful comparison reveals that the relative error is less than 0.001%, which confirms the validity of the FEEM results. For a more detailed discussion of convergence testing in FEEM, see the Graded-index fiber example here.

 

The modal fields can be visualized by selecting the FEEM in the Objects Tree, right clicking on the "fields" result in "Result View - FEEM"  and selecting Visualize>New Visualizer. The results for the magnetic field of the TM01 mode are shown below. We confirm that the polarization of this mode is indeed TM by comparing the amplitude of Hz with those of Hx and Hy. We find that the longitudinal component of the magnetic field is negligible compared to the transverse components.

 

Magnetic field intensity

Magnetic field intensity

|Hx|

|Hx|

|Hy|

|Hy|

Modal field profile for TM01 mode. The Hz field component (not shown) is negligible for this mode.

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