This annex presents the basics of slab
waveguides and optical fibers. It is based on the reference book of L.B.
Jeunhomme, "Single-mode fiber optics", Second edition, Chapter 1 "Basic
theory", Marcel Dekker, Inc (New York, Basel), 1990.
We consider a symmetric slab waveguide
of width 2a, core refractive index n2 and cladding refractive index
n1 (Fig. A-1). The propagation direction is z, the direction
orthogonal to the guide is x.
Fig. A-1 Slab waveguide geometry
The Maxwell's equations in dielectric
materials lead to two self-consistent types of solutions. The first involves
only Ey, Hx and Hz (transverse electric TE)
and the second Hy, Ex and Ey (transverse
magnetic TM), where E and H are the electric and magnetic fields. For the TE
case, the Maxwell's equations reduce to
|
(A-1) |
where b is the propagation
constant (b =w/c = 2p/l). A similar equation can be found for TM case (we limit further the
study to the TE case only). The field variation along the x-axis will exhibit
sinusoidal behavior where k2nj2 > b2 (oscillating field) and exponential behavior elsewhere (evanescent
field). Guided modes have propagation constant that fulfill the following
relation
|
(A-2) |
For b greater than kn1
the field is evanescent everywhere and thus, carries no energy. For b smaller than kn2,
the field is oscillating everywhere and radiates laterally the energy
(radiative modes).
We define a transverse propagation
constant u/a and a transverse decay constant v/a defined as
|
(A-3) |
where u and v are chosen positive. We
define also a dimensionless parameter V called the normalized frequency
|
(A-4) |
Two kind of solutions are found from
the field continuity condition at |x| = a
-
Even TE modes
-
Odd TE
modes
The continuity conditions
v = u×tan(u) or v = -u/tan(u) and the condition u2 + v2 = V2
imply that the structure can only support discrete modes. The fundamental mode
TE0 is always present and unique as long as the V < p/2. The first
even mode TE0 appears for V = p/2. The third mode (even
TE1) appear at V = p and so on. Each time the parameter V reaches a multiple of p/2, a new mode
reaches its cutoff (for which v = 0 and b = kn2).
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