Erroneous results when solving fiber mode eigen equation

[p_rob]

Hi,

I'm new to this forum and I couldn't find any specific sub-forum for fiber optics/waveguide theory, which my problem is regarding. Please do let me know if I should post this question some where else (and if so, where) on this forum. Anyways, here's my problem:

I want to find the effective index for the fundamental mode in a circularly symmetric fiber for different wavelengths. To do this, I used the standard eigenvalue equation for the propagation constant, see for instance Nonlinear fiberoptics by Agrawal:
$$\left(\frac{J'_0(pa)}{pJ_0(pa)}+\frac{K'_0(qa)}{qK_0(qa)}\right)\left(\frac{J'_0(pa)}{pJ_0(pa)}+\frac{n^2_2}{n^2_1}\frac{K'_0(qa)}{qK_0(qa)}\right)=0$$
where $p=\sqrt{n^2_1k^2_0-\beta^2}, q=\sqrt{\beta^2-n^2_2k^2_0}$, $J_0$ denotes the zeroth order Bessel function of the first kind, $K_0$ denotes the the zeroth order modified Bessel function and the primed versions denote their derivatives. $a$ is the fiber's core radius, $n_1$ is the refractive index in the core and $n_2$ is the refractive index in the cladding, $k_0=\frac{2\pi}{\lambda}$ where $\lambda$ is the wavelength and $\beta$ is the propagation constant.

My problem is that using this equation, I don't get the same solutions (and sometimes I don't get any solutions) in cases where I can see that there are solutions by simulating the electric field distribution in COMSOL for a straight circularly symmetric fiber (using the application provided in the wave optics library). So for instance, if I'm using the following parameters:
$$n_1=1.4632, n_2=1.4322, \lambda=2.3865 \mu m, a=2.75 \mu m$$
COMSOL gives me an effective refractive index, $n_{eff}$, (which is related to the propagation constant by $\beta=n_{eff}k_0$) of 1.4466. But if I plot the absolute value of the left hand side of the eigenvalue equation ranging from 1.4322 to 1.4632, there is no dip in the curve around 1.4466 as one would expect. It doesn't seem to be a discretization problem either as I've tried to use more points within the interval without being able to find any dip around the the value of 1.4466.

So my question is why I can't find this dip when I'm using the eigenvalue equation?

Thanks in advance
Rob

 

[AndyW]

Hi Rob,

Welcome to the forum! Fiber optics and waveguide theory can be a complex topic, so it's great that you're seeking help and clarification. You've definitely come to the right place.

From what you've described, it seems like the issue may be related to the assumptions and limitations of the eigenvalue equation you are using. While it is a commonly used equation for finding the propagation constant and effective index of a fundamental mode in a circularly symmetric fiber, it does have its limitations.

One potential issue could be the assumption of a perfectly circularly symmetric fiber. In reality, fiber optics can have various levels of imperfections and asymmetries, which can affect the accuracy of the eigenvalue equation. This could explain why you are not getting the expected dip in the curve.

Another factor to consider is the accuracy of the refractive index values you are using. Small variations in refractive index can greatly impact the effective index and propagation constant, so it's important to ensure that the values you are using are accurate and consistent.

Additionally, the eigenvalue equation assumes that the fiber is infinitely long, which may not be the case in your simulations. This could also contribute to the discrepancies between the results from the eigenvalue equation and the COMSOL simulation.

In summary, while the eigenvalue equation is a useful tool for calculating the effective index and propagation constant of a fundamental mode in a circularly symmetric fiber, it does have its limitations. I would recommend double-checking your assumptions, refractive index values, and the length of the fiber in your simulations to see if that helps resolve the discrepancies you are seeing.

I hope this helps! Good luck with your research.