No index of refraction for SSFM?

[Vancuum]

I'm trying to use opticommpy's built-in SSFM function for the NLSE to generate data for an ML algorithm that will estimate the parameters of the NLSE. In order to do this, I need to label the data generated by SSFM with time and space coordinates. For that, I need to know the propagation speed of light in the fiber. I noticed that the SSFM function uses the speed of light in a vacuum to calculate $\beta_2$. Is there a reason for this? How would including the index of refraction of the fiber affect the simulation? Can I make a first order approximation and say that the propagation speed of light in the fiber would be $c\n$

[edsonportosilva]

Hello, @Vancuum,

The parameter $\beta_2$ in optical fibers is directly related to how the refractive index of the fiber material changes with wavelength. Specifically, $\beta_2$ quantifies how different frequency components of a pulse travel at different group velocities, causing pulse broadening. The $\beta_2$ parameter is defined as the second derivative of the propagation constant $\beta$ with respect to angular frequency $\omega$:

$$ \beta_2 = \frac{d^2\beta}{d\omega^2} $$

The propagation constant $\beta$ is given by:

$$ \beta(\omega) = \frac{n_{\text{eff}}(\omega)\omega}{c} $$

where:

• $n_{eff}(\omega)$ is the effective refractive index of the guided mode,
• $\omega$ is the angular frequency,
• $c$ is the speed of light in vacuum.

The dispersion parameter $D$ of the fiber is often used in practice and is related to $\beta_2$ by:

$$ D = -\frac{2\pi c}{\lambda^2} \beta_2 $$

Rewriting:

$$ \beta_2 = -\frac{\lambda^2D}{2\pi c} $$

where:

• $\lambda$ is the wavelength in vacuum,
• $c$ is the speed of light in vacuum.

The calculation of $\beta_2$ is necessary to define the dispersion operator in the SSFM, and it is done through $D$, which is provided as a fiber parameter. This parameter, therefore, includes the information on the refractive index of the fiber.

So, the final answer to your question is: the refractive index of the fiber is indirectly passed to the SSFM via the dispersion parameter $D$ of the fiber. That is the case for all SSFM applied to optical communications, as far as I know. I guess you probably could find other versions of the SSFM applied to problems in other fields where the information on the refractive index maybe used differently, but that is outside my knowledge.

You can find the full theory explanation for what I wrote above in textbooks such that: AGRAWAL, G. P. Fiber-Optic Communication System.