# Erroneous results when solving fiber mode eigen equation

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• p_rob
In summary, the fiber mode eigen equation is a mathematical equation used to determine the modes and corresponding eigenvalues of a fiber optic system. It helps to understand the behavior of light propagation in optical fibers, which is crucial for designing efficient and reliable communication systems. Erroneous results can occur when solving this equation due to incorrect assumptions, numerical errors, or physical limitations. To avoid this, careful consideration of assumptions and input parameters is necessary, along with using multiple methods for verification and experimental measurements. Common sources of error include numerical errors, approximations in modeling, and inaccurate measurement data. These erroneous results can have a significant impact on the design and performance of fiber optic systems, leading to inefficient use of resources, reduced performance, and potential system failure.
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?

Rob

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.

## 1. What is the fiber mode eigen equation and why is it important?

The fiber mode eigen equation is a mathematical equation used to determine the modes and corresponding eigenvalues of a fiber optic system. It is important because it helps to understand the behavior of light propagation in optical fibers, which is crucial for designing efficient and reliable communication systems.

## 2. What are erroneous results in the context of solving the fiber mode eigen equation?

Erroneous results refer to incorrect or inaccurate solutions obtained when solving the fiber mode eigen equation. These can be caused by various factors, such as incorrect assumptions, numerical errors, or physical limitations of the system.

## 3. How can erroneous results be avoided when solving the fiber mode eigen equation?

To avoid erroneous results, it is important to carefully consider all assumptions and input parameters in the equation. It is also helpful to use multiple methods or software tools for verification and cross-checking of results. Additionally, conducting experimental measurements can help to validate the accuracy of the solutions.

## 4. What are some common sources of error when solving the fiber mode eigen equation?

Some common sources of error when solving the fiber mode eigen equation include numerical errors due to discretization of the equation, approximations made in the modeling of the fiber system, and simplifications made in the calculation process. Additionally, inaccurate measurement data or assumptions about the system can also contribute to erroneous results.

## 5. How can erroneous results impact the design and performance of fiber optic systems?

Erroneous results can have a significant impact on the design and performance of fiber optic systems. They can lead to inefficient use of resources, reduced system performance, and even system failure. Therefore, it is critical to carefully validate and verify solutions obtained from the fiber mode eigen equation to ensure the accuracy and reliability of the system.

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