Page 396 - Fiber Bragg Gratings
P. 396
8A Erbium-doped fiber lasers 373
where a and y are the gain and loss for the lasing mode n. With negligible
propagation losses, the loss is given as
where R O(X) and R HR(X) are the reflectivity of the output coupling and
high reflector gratings, respectively, at wavelength, A.
The gain in terms of the steady-state ion population densities may
be defined as
where the emission and absorption cross-sections are a e and a a, respec-
tively.
The gain for two-moded lasers may be simplified in terms of the
saturation intensity in the strong pump regime to
where the saturation intensity is a function of the emission and absorption
cross-sections, and a 0 is the small-signal unsaturated gain. The lasing
photon intensities J x and / 2 are periodic within the cavity as a result of
the formation of standing waves.
Using Eq. (8.4.4) in Eq. (8.4.1), the intensity of each steady-state
mode is
The slope efficiency at single-mode operation is then
where a p is the ground-state absorption cross-section, Aj is the lasing
mode area, and A p is the pump mode area. Substituting Eq. (8.4.5) into
Eq. (8.4.1) and solving for the loss for the second mode results in the
condition for stable single-mode operation,
