## Fiber Modes

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.

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