Nonlinear Fiber Optics - 4 ed. Agrawal

11.3. Tapered Fibers with Air Cladding 437
the radial coordinate and w is the mode-field radius. If we assume for simplicity that
n 2 = 0 for air, we obtain [48]
γ = 2n 2
λw 2 [1 − exp(−4a2 /w 2 )] = 2n (
2
λa 2 lnV 1 − 1 )
V 4 , (11.3.2)
where we used the relation a/w ≈ √ lnV [49].
At this point, it is important to remember that the V parameter itself depends on the
core radius as V =(2πa/λ)(n 2 1 −1)1/2 , where we uses n c = 1 for the refractive index of
air cladding. As an example, for a tapered fiber with 2-μm core diameter (a = 1 μm),
V ≈ 7 at a wavelength near 1 μmifweusen 1 = 1.45 for the refractive indices of silica
core. For these values γ ≈ 100 W −1 /km from Eq. (11.3.2). Clearly, a tapered fiber
with a 2-μm diameter is a highly nonlinear fiber as γ is enhanced for it by a factor of
50 compared with standard fibers. Such large values of γ are a direct consequence of
the strong mode confinement resulting from air cladding. The effective mode area of a
2-μm diameter tapered fiber is only about 1.6 μm 2 if we use A eff = πw 2 = πa 2 /lnV .It
is also important to note that such a fiber supports multiple modes since its V parameter
does not satisfy the single-mode condition V < 2.405.
The nonlinear parameter γ can be enhanced even further for tapered fibers by reducing
their core diameter to below 2 μm. Moreover, if the fiber is designed such that
V < 2.405, it would support a single mode. It is evident from Eq. (11.3.2) that γ can
be maximized for a specific value of V . For this purpose, we first express a in terms of
V in Eq. (11.3.2) and obtain
γ(V )=(8π 2 n 2 /λ 3 )(n 2 1 − 1) lnV (
V 2 1 − 1 )
V 4 , (11.3.3)
where we used n 2 = 1 for the air cladding. By setting dγ/dV equals 0, γ is found to
become maximum for V ≈ 1.85 [48], and the maximum value is about 370 W −1 /km if
we use n 1 = 1.45 and λ = 1 μm.
Dispersive properties of tapered fibers are also quite different than those of conventional
fibers and depend strongly on their core diameter. They can be studied by
solving the eigenvalue equation (2.2.8) numerically for specific values of the radius a
and the refractive index n 1 (ω) of the tapered core with n 2 = 1 for the cladding. The
dispersion of the silica material can be included by employing the Sellmeier equation
(1.2.6). The solution of Eq. (2.2.8) for m = 1 provides the propagation constant β(ω),
or the effective refractive index of the fundamental fiber mode, as a function of ω. The
mth-order dispersion parameter can then be calculated using β m =(d m β/dω m ) ω=ω0 ,
where ω 0 is the carrier frequency of the optical pulse launched into the fiber.
Figure 11.8 shows the wavelength dependence of the second- and the third-order
dispersion parameters, β 2 and β 3 , calculated numerically with this approach for several
values of the core diameter d ≡ 2a of a tapered fiber. The main point to note is
that the ZDWL of the fiber shifts toward visible wavelengths as the core diameter is
reduced to below 3 μm. This feature is of practical significance because it allows the
dispersion to become anomalous at wavelengths near 800 nm, where Ti:sapphire lasers
provide intense ultrashort pulses routinely. As a result, optical solitons can form inside
tapered fibers at such wavelengths. Figure 11.8(b) shows that β 3 is also enhanced for