Since we are skipping over all of the derivation of the wave equation in fibers (the derivation is covered in “Classical Optics”), I will just state the starting point we will be working from in this section, which is the Helmholtz form of the wave equation:

\begin{equation} \nabla^2E + n^2(\omega)\frac{\omega^2}{c^2}E = 0 \end{equation}

The solutions to this equation really depend on the form of the propagation constant $n^2(\omega)\frac{\omega^2}{c^2} = k^2$, since this will state whether or not there are guided modes or radiation modes. Although we gave some attention to the radiation modes over in the “Classical Optics” section, we will pay them no mind here since they do not play an important role in our discussion. My recommendation (unless, of course, you would like to become a professor) is to just know that the radiation modes exist, and look to references 1 and 2 cited in the first section if you have any misgivings about this whole radiation mode business.

**Table of Contents:**

**EIGENVALUE EQUATION**

From our starting point, we will again break the Helmholtz into cylindrical coordinates due to the geometry of the fiber:

\begin{equation} \frac{\partial^2E}{\partial{\phi^2}} + \frac{1}{\phi}\frac{\partial{E}}{\partial\phi} + \frac{1}{\phi^2}\frac{\partial^2E}{\partial\phi^2} + \frac{\partial^2{E}}{\partial{z^2}} + n^2k_0^2E = 0 \end{equation}

Like we did in the classical optics section, we can validly assume that this equation is separable because of the symmetry of the problem: the $E$-field propagation in the z-direction is independent of the propagation in the r- or $\phi$-directions. As such, we can solve the wave equation for z-direction propagation with the substitution:

\begin{equation} E_z(r,\omega) = A(\omega)F(\rho)e^{im\phi}e^{i\beta{z}} \end{equation}

where “A” is a constant to normalize the modes, “$\beta$” is the propagation constant, and “m” is an integer. Substitution of this equation into the cylindrical form of the Helmholtz yields:

\begin{equation} \frac{d^2F}{d\rho^2} + \frac{1}{\rho}\frac{dF}{d\rho} + [\kappa^2-\frac{m^2}{\rho^2}]F = 0 \end{equation}

where $\kappa^2 = n^2k_0^2-\beta^2$. This equation is a well-known equation for Bessel functions, and its general solution can be expressed as a combination of Bessel functions and Neumann functions.

This can then produce an eigenvalue equation for the z-direction propagation, which I won’t write here but has solutions denoted as $\beta_{mn}$ that corresponds to the modes that the fiber can support. Then the corresponding solution of the original cylindrical Helmholtz equation would yield the modal distribution. For m = 0, the modes are denoted as the TE or TM waves, same as the planar waveguides. However, for nonzero m, the modes are hybrid.

**SINGLE MODE CONDTION**

The single mode condition arises when we set $\gamma = 0$ in the equation:

\begin{equation} \kappa^2 + \gamma^2 = (n_1^2-n_2^2)k_0^2 \end{equation}

The value of $\kappa$ at this point determines the mode cut-off condition. The normalized frequency V again is a helpful parameter to determine the number of modes that a fiber can support:

\begin{equation} V = \kappa_ca = k_0a(n_1^2-n_2^2)^{1/2} \end{equation}

The process of determining what value of V will support how many modes comes from solving the eigenvalue equation, and is quite complicated. Since we are typically only interested in single mode propagation, though, it is useful just to know that a single mode fiber supports only the HE$_{11}$ mode, which is also referred to as the *fundamental mode*. For the fundamental mode, there is cutoff if $V > 2.405$. Having a V-number below this can ensure single mode propagation.

**CHARACTERISTICS OF THE FUNDAMENTAL MODE**

The field distribution of the fundamental mode has three potentially nonzero $r$, $\phi$, and $z$ components, or in Cartesian coodinates x, y, and z. Thinking back to how we defined polarization in either the x- or y-direction in the LPIC section and combining this with the fact that light coming into the fiber will likely have different distributions along these axes, we can then see that either the x- or y-polarized electric field component would dominate as the field propagates through a fiber. Thus, for all intents and purposes, the light is linearly polarized. For this reason, the LP_{mn}, or linearly polarized, notation is commonly used to describe the modal numbers in fibers. For example, the $HE_{11}$ mode would become the $LP_{01}$ mode. If we now assume that the light is polarized along the x-axis, the electric field for the fundamental mode would be given as:

\begin{equation} E(r,\omega) = \hat{x}A(\omega)F(x,y)e^{i\beta(\omega)z} \end{equation}

where F(x,y) gives the transverse distribution. For the core region, this is given as:

\begin{equation} F(x,y) = J_0(\kappa\rho),~~\rho\leq{a} \end{equation}

and for the cladding region:

\begin{equation} F(x,y) = (a/\rho)^{1/2}J_0(\kappa{a})e^{-\gamma(\rho-a)},~~\rho\geq{a} \end{equation}

The frequency dependence of the propagation constant $\beta(\omega)$ comes from both the frequency dependence of the refractive indices as well as $\kappa$, which is what gives us the value for the material dispersion!