Nonlinear Fiber Optics - 4 ed. Agrawal

2.2. Fiber Modes 29
across the core-cladding interface requires that Ẽ z , ˜H z , Ẽ φ , and ˜H φ be the same when
ρ = a is approached from inside or outside the core. The equality of these field components
at ρ = a leads to an eigenvalue equation whose solutions determine the propagation
constant β for the fiber modes. Since the whole procedure is well known [5]–[7],
we write the eigenvalue equation directly:
[ ][ J
′
m (pa)
pJ m (pa) + K′ m(qa) J
′
m (pa)
qK m (qa) pJ m (pa) + n2 c K ′ ] (
m(qa) mβk0 (n 2 1
n 2 =
− ) 2 n2 c)
1
qK m (qa) an 1 p 2 q 2 , (2.2.8)
where a prime denotes differentiation with respect to the argument and we used the
important relation
p 2 + q 2 =(n 2 1 − n 2 c)k 2 0. (2.2.9)
The eigenvalue equation (2.2.8) in general has several solutions for β for each
integer value of m. It is customary to express these solutions by β mn , where both
m and n take integer values. Each eigenvalue β mn corresponds to one specific mode
supported by the fiber. The corresponding modal field distribution is obtained from Eq.
(2.2.3). It turns out [5]–[7] that there are two types of fiber modes, designated as HE mn
and EH mn .Form = 0, these modes are analogous to the transverse-electric (TE) and
transverse-magnetic (TM) modes of a planar waveguide because the axial component
of the electric field, or the magnetic field, vanishes. However, for m > 0, fiber modes
become hybrid, i.e., all six components of the electromagnetic field are nonzero.
2.2.2 Single-Mode Condition
The number of modes supported by a specific fiber at a given wavelength depends on
its design parameters, namely the core radius a and the core-cladding index difference
n 1 −n c . An important parameter for each mode is its cut-off frequency. This frequency
is determined by the condition q = 0. The value of p when q = 0 for a given mode
determines the cut-off frequency from Eq. (2.2.9). It is useful to define a normalized
frequency V by the relation
V = p c a = k 0 a(n 2 1 − n 2 c) 1/2 , (2.2.10)
where p c is obtained from Eq. (2.2.9) by setting q = 0.
The eigenvalue equation (2.2.8) can be used to determine the values of V at which
different modes reach cut-off. The procedure is complicated, but has been described
in many texts [5]–[7]. Since we are interested mainly in single-mode fibers, we limit
the discussion to the cut-off condition that allows the fiber to support only one mode.
A single-mode fiber supports only the HE 11 mode, also referred to as the fundamental
mode. All other modes are beyond cut-off if the parameter V < V c , where V c is the
smallest solution of J 0 (V c )=0orV c ≈ 2.405. The actual value of V is a critical design
parameter. Typically, microbending losses increase as V /V c becomes small. In practice,
therefore, fibers are designed such that V is close to V c . The cut-off wavelength
λ c for single-mode fibers can be obtained by using k 0 = 2π/λ c and V = 2.405 in Eq.
(2.2.10). For a typical value n 1 − n c = 0.005 for the index difference, λ c = 1.2 μm for
a = 4 μm, indicating that such a fiber supports a single mode only for λ > 1.2 μm. In