This post answers the question: “What is mode propagation in optical fiber?”.  In order to consider concept of light propagation we will use Maxwell equations for electromagnetic waves. 

The Maxwell equations for electric and magnetic fields E and H  and electric and magnetic flux densities D and B are:


Another relationship between electromagnetic fields and flux densities with permittivity ε0 and permeability μ0, induced electric and magnetic polarizations P and M are:


Relation between P and E is P(r,t)=ε0χ(r, tt1)E(r, t1)dt1, here χ(r,t) is a linear susceptibility.

From the first Maxwell equation we can get ××E=1c22Et2μ02Pt2, here c=1μ0ε0 is a speed of light.

The permittivity ε(r,ω) is a complex quantity and can be represented with the refractive index n and the absorption coefficient α.  Both these quantities are frequency and linear susceptibility dependent. n(ω) is called a chromatic dispersionε=(n+iαc2ω)2, α=ωncImχ, n=(1+Reχ).

making general simplification we can obtain that 2E+n2(ω)k02E=0 this equation is called wave equation, here k0=2πλ=ωc is a free-space wave number and λ is a vacuum wavelength of optical field, ω it’s frequency.

In optical fibers induced magnetic polarization M=0, because optical fibers are non-magnetic.

Optical modes and mode propagation

Optical mode describes specific solution of the wave equation in accordance to the appropriate boundary conditions and whose spatial distribution does not change with propagation.

Types of fiber modes are:

  • guided modes;
  • leaky modes;
  • radiation modes.

It is very comfortable to use cylindrical coordinates (ρ, φ, z) for our equations, where wave equation will be 2Ezρ2+1ρEzz+1ρ22Ezφ2+2Ezz2+n2k02Ez=0, here n=n1, ρan2, ρ>ais a refractive index.

Electric field in cylindrical coordinates Ez(ρ, φ, z)=F(ρ)Φ(φ)Z(z). Using these two equations we can divert to three equations by correspondent coordinates ρ, φ, z. These equations can be resolved using Bessel functions with general solution F(ρ)=AJm(pρ)+AYm(pρ), ρaCKm(qρ)+CIm(qρ), ρ>a, here Jm, Ym, Km, Im are Bessel functions, and  A, A, C, C are constants. Meantime p, q are depending on refractive index and free-space wave number p2=n12k02β2, q2=β2n22k02, β is a propagation constant.

For certain values of parameters k0, a, n we can find propagation constant βnm for a given n and m. Every βnm corresponds to one possible propagation mode. Here we can introduce mode index  βk0, it is usually n1>βk0>n2 and means that each fiber propagates with an effective refractive index.

The propagation mode is reaching cutoff when βk0=n2. Cutoff condition is characterised with  normalised frequency  V=k0a(n12n22) and normalised propagation constant b=βk0n2n1n2.

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