2.2. WAVE PROPAGATION 29
2.2.1 Maxwell’s Equations
Like all electromagnetic phenomena, propagation of optical fields in fibers is governed
byMaxwell’s equations. For a nonconducting medium without free charges, these
equations take the form [30] (in SI units; see Appendix A)
∇×E=−∂B/∂t, (2.2.1)
∇×H=∂D/∂t, (2.2.2)
∇·D= 0 , (2.2.3)
∇·B= 0 , (2.2.4)whereEandHare the electric and magnetic field vectors, respectively, andDandB
are the corresponding flux densities. The flux densities are related to the field vectors
by the constitutive relations [30]
D=ε 0 E+P, (2.2.5)
B=μ 0 H+M, (2.2.6)whereε 0 is the vacuum permittivity,μ 0 is the vacuum permeability, andPandMare
the induced electric and magnetic polarizations, respectively. For optical fibersM= 0
because of the nonmagnetic nature of silica glass.
Evaluation of the electric polarizationPrequires a microscopic quantum-mechanical
approach. Although such an approach is essential when the optical frequency is near
a medium resonance, a phenomenological relation betweenPandEcan be used far
from medium resonances. This is the case for optical fibers in the wavelength region
0.5–2μm, a range that covers the low-loss region of optical fibers that is of interest
for fiber-optic communication systems. In general, the relation betweenPandEcan
be nonlinear. Although the nonlinear effects in optical fibers are of considerable in-
terest [31] and are covered in Section 2.6, they can be ignored in a discussion of fiber
modes.Pis then related toEby the relation
P(r,t)=ε 0∫∞−∞χ(r,t−t′)E(r,t′)dt′. (2.2.7)Linear susceptibilityχis, in general, a second-rank tensor but reduces to a scalar for
an isotropic medium such as silica glass. Optical fibers become slightly birefringent
because of unintentional variations in the core shape or in local strain; such birefrin-
gent effects are considered in Section 2.2.3. Equation (2.2.7) assumes a spatially local
response. However, it includes the delayed nature of the temporal response, a feature
that has important implications for optical fiber communications through chromatic
dispersion.
Equations (2.2.1)–(2.2.7) provide a general formalism for studying wave propaga-
tion in optical fibers. In practice, it is convenient to use a single field variableE.By
taking the curl of Eq. (2.2.1) and using Eqs. (2.2.2), (2.2.5), and (2.2.6), we obtain the
wave equation
∇×∇×E=−1
c^2∂^2 E
∂t^2−μ 0∂^2 P
∂t^2