# Why do higher modes propagate more in the cladding of an optical fiber than lower modes?

I am trying to understand the theory of inter-modal dispersion in optical fibers. It seems quite obvious that if higher modes have a greater angle of incidence in the fiber than lower modes, the path length of higher modes through the fiber is larger. This is because higher modes undergo more reflections, but they also have a greater part of the light wave traveling is the cladding. Here the speed of light is a little higher than in the core and therefore the higher modes are moving faster than lower modes. The theory of a part of the light wave traveling in the cladding has to do with the evanescent field I think, but why do higher modes have a greater part of the wave traveling in the cladding in comparison with lower modes?

For optical fibers with cylinder geometry there are an orbital angular momentum (OAM) mode number $$\ell\in\mathbb{Z}$$, and a radial mode number $$m\in\mathbb{N}$$. The average radial position grows with $$m$$. A fixed OAM $$\ell$$ leads to a centrifugal term in the radial equation, which diminishes the number of radial modes. See e.g. the RP photonic encyclopedia.
At a smaller angle of incidence on the boundary (higher mode), the field of the evanescent wave penetrates more deeply into the optically rarer medium (cladding). See the derivation here, in which the characteristic depth of penetration of the evanescent electric field is given by $$\frac{c}{\omega}\left((n_1\sin(\theta_I))^2-n_{2}^{2}\right)^{-1/2}.$$
• Crucially, there is a magnetic field associated with the evanescent wave. Neither the electric nor the magnetic field are allowed to be discontinuous at the boundary, and this applies no matter how many boundaries you have. So to address your proposed situation, an evanescent wave could extend into a third, adjacent medium. If the refractive index is greater than $n_2$, this evanescent wave could even give rise to a propagating wave in the third medium (the 'resonant tunneling' process). Jun 2, 2015 at 19:36