2.2. WAVE PROPAGATION 33
where a prime indicates differentiation with respect to the argument.
For a given set of the parametersk 0 ,a,n 1 , andn 2 , the eigenvalue equation (2.2.33)
can be solved numerically to determine the propagation constantβ. In general, it
may have multiple solutions for each integer value ofm. It is customary to enumerate
these solutions in descending numerical order and denote them byβmnfor a givenm
(n= 1 , 2 ,....). Each valueβmncorresponds to one possible mode of propagation of the
optical field whose spatial distribution is obtained from Eqs. (2.2.27)–(2.2.32). Since
the field distribution does not change with propagation except for a phase factor and sat-
isfies all boundary conditions, it is an optical mode of the fiber. In general, bothEzand
Hzare nonzero (except form=0), in contrast with the planar waveguides, for which
one of them can be taken to be zero. Fiber modes are therefore referred to ashybrid
modesand are denoted by HEmnor EHmn, depending on whetherHzorEzdominates.
In the special casem=0, HE 0 nand EH 0 nare also denoted by TE 0 nand TM 0 n, respec-
tively, since they correspond to transverse-electric (Ez=0) and transverse-magnetic
(Hz=0) modes of propagation. A different notation LPmnis sometimes used for
weakly guiding fibers [33] for which bothEzandHzare nearly zero (LP stands for
linearly polarized modes).
A mode is uniquely determined by its propagation constantβ. It is useful to in-
troduce a quantity ̄n=β/k 0 , called themode indexoreffective indexand having the
physical significance that each fiber mode propagates with an effective refractive in-
dex ̄nwhose value lies in the rangen 1 >n ̄>n 2. A mode ceases to be guided when
n ̄≤n 2. This can be understood by noting that the optical field of guided modes decays
exponentially inside the cladding layer since [32]
Km(qρ)=(π/ 2 qρ)^1 /^2 exp(−qρ) for qρ
1. (2.2.34)When ̄n≤n 2 ,q^2 ≤0 from Eq. (2.2.26) and the exponential decay does not occur. The
mode is said to reach cutoff whenqbecomes zero or when ̄n=n 2. From Eq. (2.2.25),
p=k 0 (n^21 −n^22 )^1 /^2 whenq=0. A parameter that plays an important role in determining
thecutoff conditionis defined as
V=k 0 a(n^21 −n^22 )^1 /^2 ≈( 2 π/λ)an 1√
2 ∆. (2.2.35)
It is called thenormalized frequency(V∝ω) or simply theVparameter. It is also
useful to introduce a normalized propagation constantbas
b=β/k 0 −n 2
n 1 −n 2=
n ̄−n 2
n 1 −n 2. (2.2.36)
Figure 2.5 shows a plot ofbas a function ofVfor a few low-order fiber modes obtained
by solving the eigenvalue equation (2.2.33). A fiber with a large value ofVsupports
many modes. A rough estimate of the number of modes for such a multimode fiber
is given byV^2 /2 [23]. For example, a typical multimode fiber witha= 25 μm and
∆= 5 × 10 −^3 hasV18 atλ= 1. 3 μm and would support about 162 modes. However,
the number of modes decreases rapidly asVis reduced. As seen in Fig. 2.5, a fiber with
V=5 supports seven modes. Below a certain value ofVall modes except the HE 11
mode reach cutoff. Such fibers support a single mode and are called single-mode fibers.
The properties of single-mode fibers are described next.