Modes are fields that maintain the same transverse distribution and polarization at all distances along the waveguide axis. They can be regarded as transverse resonances of the fields in the waveguide.
Fiber modes can be classified into the bound modes and the radiation modes. Then, the electric and magnetic field vectors can be written as(7)
where j=1,2,…M; and are the forward, backward, and radiation electric and magnetic fields, respectively, and and are the modal amplitudes.
If the waveguide has a refractive index profile that does not change along its length z, that is, , such waveguide is called translationally invariant. Then the electric and magnetic fields can be written in the form(8)
where is the propagation constant. At a given frequency , the value of is calculated from an eigenvalue equation. The eigenvalue equation, also called dispersion relation or characteristic equation, is a transverse resonance condition, resulting from the field solutions that are bounded everywhere, approach zero sufficiently fast at infinity, and satisfy all boundary conditions at the core/cladding interface.
In weakly guiding fiber, a small but nonzero index difference maintains total internal reflection, but the medium is nearly homogeneous as far as polarization effects are concerned. Then, the modes of weakly guiding fibers are nearly TEM waves, with the longitudinal components of the electric and magnetic fields much weaker than the transverse components. For weakly guiding fibers, the eigenvalue equation is given by(9)
where , and . The eigenvalue equation (9) is a transcendental equation and may be solved graphically by plotting its right- and left-hand sides versus X.
The range of allowed is bounded by(10)
where . The lower limit of allowed values of the bound mode propagation constant is called modal cutoff. Far from cutoff, a mode is well confined in the core, while close to cutoff the mode significantly extends into the cladding region. In contrast to planar waveguides, not all fiber modes escape from the core completely at the cutoff as shown in Fig. 5(a). Mode profiles of the first four modes are plotted in Fig. 5(b).
Figure 5. (a) The fraction of power in the core of the step-index weakly guiding optical fiber [Snyder and Love, Optical Waveguide Theory, 1983], (b) Mode profiles of the lower -order fiber modes.