2.1         Optical fiber

2.1.3        Fiber Parameters

a)   Effective refractive index

We consider a single-mode, step index optical fiber and we define a relative index difference D :

In practice, for step index fibers, the relative index difference D is smaller than 1 % and weak guidance is admitted. We define a dimensionless parameter V called the normalized frequency

where a is the core radius, l the wavelength and k = 2p/l. The V parameter indicates how far away from the cutoff (condition where the mode is no more guided) a given mode is. The closer to the cutoff the mode is, the deeper the evanescent field extends in the cladding. A single-mode-operating condition is possible when V < 2.405. The propagation constant b in the fiber corresponds to the solutions of the mode equations and then an effective refractive index of the guided mode n_eff can be defined as

A good approximation of the effective refractive index n_eff is found in Appendix A and it can be expressed in terms of the V parameter, the wavelength and the fiber parameters (n₁, n₂ and a)

As the relative refractive index difference is small, the longitudinal components can be almost neglected and the mode is considered transversely linearly polarized (LP₀₁ mode). We also define a cutoff wavelength under which another mode appear

b)   Group refractive index

The group delay t that characterize the propagation time per unit length is well approximated by (Appendix A)

where N₂ = d(kn₂)/dk is the group index of refraction of the cladding. For silica in the wavelength range 1300-1500 nm, the relative difference between N₂ and n₂ is less than 1.5 % [2-1]. The group delay is the inversely proportional to the group velocity and the group refractive index is defined as

c)   Dispersion

There are two differences between the light traveling in vacuum and along an optical fiber :

-         Time delay : expresses the propagation delay per unit length

-         Dispersion : expresses the time delay variation with the wavelength

Dispersion effects are a problem in telecommunication transmission since they broaden the pulse width. Four kind of dispersion can be identified :

-         Material dispersion : the refractive index of the cladding and core are frequency dependent; for fused silica, the dispersion is negative at wavelength below 1300 nm and positive above 1300 nm

-         Waveguide dispersion : equation (2-6) takes into account that the propagation constant for a given mode is wavelength dependent in a non linear manner, leading to another dispersion component

-         Modal dispersion : each propagation mode has its own propagation parameters and then different modes travel at different velocities for the same wavelength

-         Polarization mode dispersion : the birefringence in fibers modifies the propagation constant

For the single-mode fibers used in our experiments, the modal dispersion was negligible. Due to the sign change of the material dispersion around 1300 nm a zero-dispersion propagation is possible when the waveguide and material dispersion compensate each other. For step-index fibers, the zero-dispersion wavelength is close to 1300 nm. The tailoring of the waveguide structure (core profiling or segmenting) modifies the waveguide dispersion and then the zero-dispersion condition can be shifted (for example at 1550 nm) [2-2].