Page 450 - High Power Laser Handbook

418 Fi b er L a s er s Intr oduction to Optical Fiber Lasers 419

azimuthal and radial mode numbers, respectively. The effective mode
index falls into the following range:

n > n > n (15.6)
co eff cl
Modes with n < n have oscillating fields in the cladding and
eff
cl
are no longer guided. They are sometimes referred to as radiation
modes or, if discrete, leaky modes. A useful interpretation for leaky
modes is that the total internal reflection condition is no longer met,
resulting in power loss at each reflection at the core-cladding inter-
face. The normalized propagation constant b, defined in Eq. (15.7), is
a useful parameter for measuring how strongly a mode is guided.
For guided modes, b falls between 0 and 1. A mode is no longer
guided when b ≤ 0.

n − n 2
2
b = eff cl (15.7)
2
2
n − n cl
co
The effective mode index of a single-mode optical fiber and an
optical fiber that supports few modes is illustrated in Fig. 15.2a and
15.2b, respectively. The fundamental mode LP always has the high-
01
est effective mode index, followed by the second order mode LP . It
11
is also worth noting that the effective mode index difference between
modes can serve as a rough measure of how easily two modes can be
phase matched for intermodal coupling, though a more rigorous
analysis of intermodal coupling will also have to involve the spatial
overlap integral between the modes and the external perturbation.
Modes cannot couple to each other in the absence of external pertur-
bations due to their orthogonal nature. As the number of guided
modes increases in a waveguide, the effective mode index difference
gets smaller, which means it is generally easier for intermodal cou-
pling to occur.

n n
n co n co
LP 01
LP 01
LP 11
n cl n cl

r r
(a) (b)
Figure 15.2 Effective mode index in (a) a single-mode fiber and (b) a fiber
that supports few modes.