How can I calculate the effective refractive index of the grating in a fiber - either in FDTD or MODE Solutions?

For example on the simplest model - FBG from KB.

# Effective refractive index of the FBG

**mdudek**#1

**msaygin**#2

Hi.

Could you please clarify what do you mean by refractive index of the grating?

I ask you this, because effective refractive index is a parameter that appear naturally in waveguides for propagation of the corresponding eigen modes.

In my view, in case of a grating you can’t simply define the effective refractive index for the following reason.

The grating structure is a distributed entity, which operates by constructive/destructive interference originated from a multitude of interfaces. Therefore, in general, the field amplitude will be varied over the grating’s span, due to gradual energy redistribution between the forward and backward propagating waves.

**mdudek**#3

Thank you for your reply.

I had in mind the effective refractive index of the fiber as in the Bragg reflected wavelength relation:

```
λ_B = 2 * n_eff * Λ
```

where *λ_B* is the reflected wavelength, *Λ* is the grating period and *n_eff* is the effective refractive index.

**msaygin**#4

Then, the n_eff parameter can be calculated as n_eff = λ_B/(2Λ), where Λ is known from geometry and λ_B is the wavelength derived from the reflection spectrum (the maximum in the reflection spectrum).

I hope that this will help.

**mdudek**#5

Technically yes, but it is the opposite of what I would like to achieve. To calculate (more like “find”) n_eff your way, I need to do a sweep of possible wavelengths.

Is there maybe an other way?

**msaygin**#6

I thought that an alternative method would converge to what I said, however, I am not 100 % sure.

Probably, someone else can add extra on the question.

**fgomez**#7

The expression

is based on the Bragg condition for constructive interference of reflections. Typically, you know the desired center wavelength for reflection (λ_B) and you want to find the period for a given Bragg grating characterized by n_eff. This effective index is usually estimated from an average of the effective indices of the sections of the Bragg grating; for instance, in this Bragg grating example, n_eff is the effective index of a waveguide with a width that is the average of the two sections that form the unit cell of the periodic structure.

An alternative way to find the effective index is based on a band structure calculation, where we can find the wavevector component along the propagation direction for the mode supported at the desired wavelength and use the relation with effective index:

k = 2*pi*n_eff/lambda

This is illustrated in this example as well.

**mdudek**#8

Thank you, it was really helpful.

I have another question - regarding the Bragg grating example. For the trick to avoid re-calculating modes for all interesting wavelengths you use relation n_eff/n_g, with information that:

the ratio between effective and group index, n_eff/n_g, can be calculated with the FDE eigenmode solver in MODE.

Could you explain how to do it?

**fgomez**#9

Hi @mdudek,

FDE is the eigonmode solver in MODE Solutions. You can add a FDE simulation region to your simulation file as in the attached exampleBragg_grating_Si.lms (298.0 KB). In this example the grating is formed by waveguide sections w1 and w2. I also included a waveguide with the average width (“waveguide”), which you can use to estimate the effective and group indices required for the wavelength-sweep trick. For the FDE analysis disable w1 and w2, and enable waveguide; then, set the FDE region as active.

The cross section of the FDE simulation region should be set to be the same as the one for the EME simulation region. Using the Frequency analysis tab in the Eigensolver analysis window you can calculate the effective and group indices as shown below:

**umair.korai**#11

Hi @fgomez,

Using this example, nr_waveguide3d.fsp, I am changing the dimensions and refractive index of the core 500 nm x 220 nm and 3.47 (for example as silicon) and re-run the simulation but the effective index is not same as it has to be (say around 2.4). The refractive index is like below.

Here is the updated file, can you please tell me that why effective index appears to be like this?

nr_waveguide3d_Silicon_core.fsp (2.0 MB)

Thanks,

Umair