Page 449 - High Power Laser Handbook

416 Fi b er L a s er s Intr oduction to Optical Fiber Lasers 417

The relative refractive index difference of a step-index optical
fiber is defined as

n − n
∆= co cl (15.1)
n co

where ∆ is typically below 2 percent. In the weakly guiding regime,
reflected and transmitted fields around core and cladding interfaces
are well approximated without considering field orientations. Linearly
polarized modes can then be used to describe the modes in the optical
fiber. By ignoring the vector nature of the fields, the weakly guiding
approximation greatly simplifies the theoretical analysis of an optical
fiber. The numerical aperture (NA) of an optical fiber is defined as

2
NA = n − 2 n (15.2)
co cl
A very important parameter in an optical fiber is normalized fre-
quency, which is defined as

2πρ NA
V = (15.3)
λ

where λ is the vacuum wavelength. The guided modes of a wave-
guide can be obtained from the Helmholtz eigenvalue equation,
which is derived from Maxwell's equations and which ensures all rel-
evant field continuities at all boundaries. A guided mode can be seen
as a robust fundamental spatial distribution that propagates at the
propagation constant β and maintains a constant wavefront. It can be
expressed as
θ
Er(, ,) = E r(, ) e − iz (15.4)
β
θ
z
0
where β is propagation constant, E is transverse mode distribution,
0
and z is the propagation distance. Due to the Helmholtz equation’s
unique scaling characteristics, a waveguide’s mode properties are
entirely determined once the normalized frequency V is known. It is
worth noting that proportionally scaling both ρ and λ leads to modes
with the same relative field distribution and propagation constant.
The effective mode index n can be obtained from the relation
eff
2 πn
β = eff (15.5)
λ
In the weakly guiding regime, waveguide modes can be repre-
sented as LP , where LP stands for linearly polarized and l and m are
lm