30 CHAPTER 2. OPTICAL FIBERS
where the speed of light in vacuum is defined as usual byc=(μ 0 ε 0 )−^1 /^2. By introduc-
ing the Fourier transform ofE(r,t)through the relation
E ̃(r,ω)=∫∞−∞E(r,t)exp(iωt)dt, (2.2.9)as well as a similar relation forP(r,t), and by using Eq. (2.2.7), Eq. (2.2.8) can be
written in the frequency domain as
∇×∇×E ̃=−ε(r,ω)(ω^2 /c^2 )E ̃, (2.2.10)where the frequency-dependentdielectric constantis defined as
ε(r,ω)= 1 +χ ̃(r,ω), (2.2.11)andχ ̃(r,ω)is the Fourier transform ofχ(r,t). In general,ε(r,ω)is complex. Its real
and imaginary parts are related to therefractive index nand theabsorption coefficient
αby the definition
ε=(n+iαc/ 2 ω)^2. (2.2.12)
By using Eqs. (2.2.11) and (2.2.12),nandαare related toχ ̃as
n=( 1 +Reχ ̃)^1 /^2 , (2.2.13)
α=(ω/nc)Imχ ̃, (2.2.14)where Re and Im stand for the real and imaginary parts, respectively. Bothnandα
are frequency dependent. The frequency dependence ofnis referred to aschromatic
dispersionor simply as material dispersion. In Section 2.3, fiber dispersion is shown
to limit the performance of fiber-optic communication systems in a fundamental way.
Two further simplifications can be made before solving Eq. (2.2.10). First,εcan
be taken to be real and replaced byn^2 because of low optical losses in silica fibers.
Second, sincen(r,ω)is independent of the spatial coordinaterin both the core and the
cladding of a step-index fiber, one can use the identity
∇×∇×E ̃≡∇(∇·E ̃)−∇^2 E ̃=−∇^2 E ̃, (2.2.15)where we used Eq. (2.2.3) and the relationD ̃=εE ̃to set∇·E ̃=0. This simplification
is made even for graded-index fibers. Equation (2.2.15) then holds approximately as
long as the index changes occur over a length scale much longer than the wavelength.
By using Eq. (2.2.15) in Eq. (2.2.10), we obtain
∇^2 E ̃+n^2 (ω)k^20 E ̃= 0 , (2.2.16)where the free-space wave numberk 0 is defined as
k 0 =ω/c= 2 π/λ, (2.2.17)andλis the vacuum wavelength of the optical field oscillating at the frequencyω.
Equation (2.2.16) is solved next to obtain the optical modes of step-index fibers.