A.2      Optical Fiber Waveguide

A.2.3      Fundamental mode HE₁₁

For V < 2.405 the fiber only support the fundamental mode HE₁₁ (for slab waveguides the condition value was p/2). There are two HE₁₁ modes, one polarized along the x-axis and the second one along y-axis, but they are degenerated due to the circular symmetry of the fiber. In this case, the fiber is called "single-mode". The cutoff wavelength is defined as

The field distribution of the HE₁₁ mode is given by (either E_x of E_y can be taken as 0)

where the first of (sin(j),cos(j)) for E_z and of (cos(j),sin(j)) for H_z holds if E_x = 0, and the second holds if E_y = 0. Z₀ is the vacuum impedance, J_0,1 are the Bessel functions of order 0 and 1, K_0,1 the modified Bessel functions. The continuity equation is given by

The longitudinal components of the fields are on the order of u/kan with respect to the transverse components. Using (A-4) and (A-10) and the fact that D is smaller than 1 %, we can consider the mode as transversely polarized with a linear polarization. This leads to the denomination of the LP₀₁ mode.

A useful approximation for v(V) is given by

The corresponding u value is obtained from (A-4). The relative error in u compared to the exact solution is less than 0.1 % for 1.5 < V < 2.5 and increase to 1 % for 1 < V < 3.

The propagation constant b is also defined as in (A-3) and we define an effective refractive index of the mode, n_eff, as

We define the normalized propagation constant b, varying between 0 and 1 (only dependent of V)

Using the fact that D is small, we can write

The error is less than 0.2 % for V between 1.5 and 2.5 (less than 2 % for V between 1 and 3).

The group delay t, characterizes the propagation delay time per unit length of a modulated signal transmitted by the optical wave. It is obtained as

where c is the vacuum light speed. Neglecting the difference in the dispersive properties between the core and the cladding, the time delay can be expressed as

where N₂ = d(kn₂)/dk is the group index of refraction of the material with refractive index n₂, and (Vb)' = d(Vb)/dV well approximated by

The error is less than 1 % for 1.6 < V < 2.4 (less than 4 % for 1 < V < 3).

The dispersion description in fibers is more difficult as there is a mixing between the material dispersion (variation of N₂ and D with l) and the waveguide dispersion (variations of b and (Vb)' with l).