Page 520 - Engineering Electromagnetics, 8th Edition

502 ENGINEERING ELECTROMAGNETICS

with TE and TM waves in the slab. In our fiber, we do not apply transverse resonance
directly,butratherimplicitly,byrequiringthatallfieldssatisfytheboundaryconditions
7
at the core/cladding interface, ρ = a. We have already applied conditions on the
transverse fields to obtain Eq. (151). The remaining condition is continuity of the
z components of E and H.In the weak-guidance approximation, we have neglected
all z components, but we will consider them now for this last exercise. Using Faraday’s
law in point form, continuity of H zs at ρ = a is the same as the continuity of the
z component of ∇× E s , provided that µ = µ 0 (or is the same value) in both regions.
Specifically

(∇× E s1 ) z = (∇× E s2 ) z (154)
ρ=a ρ=a
The procedure begins by expressing the electric field in (151) in terms of ρ and φ
components and then applying (154). This is a lengthy procedure and is left as an
exercise (or may be found in Reference 5). The result is the eigenvalue equation for
LP modes in the weakly guiding step index fiber:
J −1 (u) w K −1 (w) (155)
J (u) =− u K (w)

This equation, like (139) and (140), is transcendental, and it must be solved for u and
w numerically or graphically. This exercise in all of its aspects is beyond the scope of
our treatment. Instead, we will obtain from (155) the conditions for cutoff for a given
mode and some properties of the most important mode—that which has no cutoff,
and which is therefore the mode that is present in single-mode fiber.
The solution of (155) is facilitated by noting that u and w can be combined to give
anew parameter that is independent of β and depends only on the fiber construction
and on the operating frequency. This new parameter, called the normalized frequency,
or V number, is found using (150a) and (150b):

2
2
2
V ≡ u + w = ak 0 n − n 2 2 (156)
1
We note that an increase in V is accomplished through an increase in core radius,
frequency, or index difference.
The cutoff condition for a given mode can now be found from (155) in conjunction
with (156). To do this, we note that cutoff in a dielectric guide means that total
reflectionatthecore/claddingboundaryjustceases,andpowerjustbeginstopropagate
radially, away from the core. The effect on the electric field of Eq. (151) is to produce
a cladding field that no longer diminishes with increasing radius. This occurs in the
modifiedBesselfunction, K(wρ/a),whenw = 0.Thisisourgeneralcutoffcondition,

7 Recall that the equations for reflection coefficient (119) and (120), from which the phase shift on
reflection used in transverse resonance is determined, originally came from the application of the
field boundary conditions.