Fiber Fundamental Mode Solver

This program computes the fundamental mode (LP01) of a step-index optical fiber by solving the Bessel functions of the fiber using a dichotomy method.
It outputs the effective index (neff), the mode field diameters (MFDs), the effective area (Aeff), the V-number, and optionally the chromatic dispersion and group velocity.

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Mode Field Diameter (MFD)

The mode field diameter characterizes the transverse extent of the fundamental mode and is often more relevant than the core diameter.
Three definitions are implemented:

• Gaussian approximation (1/e² criterion)
  Approximates the fundamental mode by a Gaussian and defines the MFD at the 1/e² intensity points.

• Near-field rms definition

$$ d_N = 2 \sqrt{ \frac{2 \int_0^\infty r^3 , |\psi(r)|^2 , dr}{\int_0^\infty r , |\psi(r)|^2 , dr} } $$

where ψ(r) is the radial field profile.
This is also referred to as the 4σ definition.

• Petermann II definition

$$ w_P = \sqrt{ \frac{2 \int_0^\infty |\psi(r)|^2 r , dr}{\int_0^\infty \left(\frac{d\psi}{dr}\right)^2 r , dr} } $$

$$ d_P = 2 , w_P $$

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Effective Area (Aeff)

Two approaches are available:

• Gaussian approximation

$$ A_{\mathrm{eff}} \approx \pi \ w_0^2 $$

• Rigorous field integral

$$ A_{\mathrm{eff}} = \frac{\left( 2\pi \int_0^\infty |\psi(r)|^2 r , dr \right)^2} {2\pi \int_0^\infty |\psi(r)|^4 r , dr} $$

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Chromatic Dispersion and Group Velocity

The solver can also compute and plot the spectral dependence of:

• Effective index $n_\mathrm{eff}(\lambda)$
• Group velocity $v_g(\lambda)$
• Chromatic dispersion $D(\lambda)$ (in ps/nm/km)

The formulas used are:

• Group index:

$$ n_g(\lambda) = n_\mathrm{eff}(\lambda) - \lambda , \frac{dn_\mathrm{eff}}{d\lambda} $$

• Group velocity:

$$ v_g(\lambda) = \frac{c}{n_g(\lambda)} $$

• Chromatic dispersion:

$$ D(\lambda) = -\frac{\lambda}{c} , \frac{d^2 n_\mathrm{eff}}{d\lambda^2} $$

where $c$ is the speed of light in vacuum.

⚠️ To reduce numerical noise, the first and last points (for group velocity) and the two first/last points (for dispersion) are discarded in the plots.

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V-Number

The solver computes the V-number:

$$ V = \frac{2\pi a}{\lambda} , \mathrm{NA} $$

where a is the fiber core radius and NA the numerical aperture.

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Advantages

Compared to simple approximations (e.g. the Marcus approximation, which can be inaccurate for practical fibers),
this solver provides higher precision while remaining significantly simpler to use than a full vectorial electromagnetic mode solver.
It also allows direct visualization of the mode profile, MFDs, and the dispersion characteristics of the fiber.

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References

• Introduction to Fiber Optics, Ajoy Ghatak and K. Thyagarajan
• ITU-T G.650.1, Definitions and test methods for mode field parameters
• G. P. Agrawal, Nonlinear Fiber Optics (for dispersion definitions)