Fig. 1. Refractive (or phase) index n (solid line curve), and group index n_g (dashed line curve) of silica, as a function of the wavelength λ in a vacuum. At 1.3 μm, there is no group velocity dispersion since n_g is constant, i.e., dn_g/dλ is null. This is the basic cause; the fact that d²n/dλ² = 0 at 1.3 μm is just a consequence of dn_g/dλ = −λ · d²n/dλ².

Fig. 2. Refractive (or phase) index n (solid line curve), and group index n_g (dashed line curve) of silica, as a function of the wavelength λ in a vacuum. At 1.55 μm, the slope of the tangent to the curve n(λ), i.e., dn/dλ or n′(λ), equates minus 0.012 μm⁻¹, and gives the chromatic, or first-order, dispersion; the one to the curve n_g(λ), i.e., dn_g/dλ or n′_g(λ), equates plus 0.007 μm⁻¹, and gives the group velocity dispersion, or second-order dispersion. Since 1/c = 1 ns/300 mm, the dispersion parameter D(1.55 μm) = (1/c) · n′_g(1.55 μm) equates + 23 ps/(nm km).

Fig. 3. Refractive (or phase) index n(λ) (solid line), and group index n_g(λ) (dashed line), in the theoretical case of a material without any group dispersion. The straight line representing n(λ) crosses the ordinate axis corresponding to λ = 0, at the constant value of n_g(λ) = n(0).

Fig. 4. Refractive (or phase) index n(λ) (solid line curve), with group dispersion. The tangent T₀(λ) to the curve n(λ) at λ₀ crosses the ordinate axis corresponding to λ = 0, at the value n_g(λ₀) of the group index for λ₀.

Fig. 5. Refractive (or phase) index n(λ) (solid line curve), and group index n_g(λ) (dashed line curve) of silica. The tangent to the curve n(λ), at λ₀, crosses the extended ordinate axis, where λ = 0 μm, at the value n_g(λ₀) of the group index for λ₀, as shown for λ₀ equal to 0.85 μm, or to 1.3 μm.

Fig. 6. Effective index n_eff of the fundamental mode, n₁ being the refractive index of the core, and n₂ being the one of the cladding: (a) classical representation, as a function of the normalized spatial frequency λ_c/λ, in a vacuum; (b) inverted representation as a function of the normalized wavelength λ/λ_c, in a vacuum.

Fig. 7. Geometrical construction to derive the effective group index n_g-eff(λ/λ_c) of the fundamental mode (solid line curve) from its effective phase index n_eff(λ/λ_c) (dashed line curve). The tangent to n_eff(λ/λ_c) crosses the ordinate axis at the value of the corresponding effective group index n_g-eff(λ/λ_c), as shown for λ = 1.25 λ_c. One sees easily that n_g-eff is about equal to the refractive index of the core n₁, in the practical domain of use of a single-mode fiber, i.e., λ_c < λ < 1.5 λ_c; above 1.5 λ_c, the mode starts to widen a lot and the curvature loss increases drastically.

Fig. 8. Two possible geometrical constructions for relating group index n_g to phase index n: (a) with the dependence in wavelength λ, i.e., the spatial period of the wave, the tangent T₀(λ) to the phase index curve crosses the ordinate axis at the value n_g(λ₀) of the group index; (b) with the dependence in angular frequency ω, a double-slope line DS₀(ω) is drawn from where the tangent T₀(ω) to the phase index curve crosses the ordinate axis, and the group index n_g(ω₀) equates DS₀(ω₀).

Fig. 9. Geometrical construction relating the effective group index n_g-eff to the effective index n_eff of the fundamental mode of a fiber, with the dependence in normalized spatial frequency λ_c/λ. A double-slope line is drawn from where the tangent to the effective index curve crosses the ordinate axis, and the group index n_g(λ_c/λ₀) equates DS₀(λ_c/λ₀), as shown for λ_c/λ₀ = 0.8, i.e., for λ₀ = 1.25 λ_c. As in Figure 7, one easily sees that n_g-eff is about equal to the refractive index of the core n₁, in the practical domain of use of a single-mode fiber, i.e., λ_c < λ < 1.5 λ_c, or 0.67 < λ_c/λ < 1.

Fig. 10. Geometrical construction relating group birefringence B_g to phase birefringence B(λ) (solid line curve); the tangent T₀(λ) to the curve B(λ) at λ₀, crosses the ordinate axis corresponding to λ = 0, at the value B_g(λ₀) of the group birefringence for λ₀. This figure is obviously derived from Figure 4.