Criteria for stable guidance

As discussed above, the modes of an optical fiber are orthogonal solutions, therefore, in the absence of a perturbation to the fiber’s refractive index profile, we expect that the power coupled into a particular mode in a fiber will reside in that mode throughout the entirety of propagation. However, in Sec. 5.1.1, we will see that periodic index perturbations can lead to resonant transfer of energy from one mode to another. These perturbations do not necessarily need to be periodic in order for mode coupling to occur; a single perturbation (generally a bend or twist in the fiber) can cause power to unintentionally leak from one mode to another. If the fiber is subject to many perturbations, power will couple to neighboring modes which can in turn couple to other modes, leading to a random assortment of modes and relative group delays at the output of the fiber. This process is referred to as distributed mode coupling, and it is problematic for four-wave mixing (FWM) because it drastically reduces the effective length and thus conversion efficiency of a given frequency conversion process. The likelihood of undesired coupling goes as exp[−12Lc∆β], where Lc is the correla-

tion length of the perturbation, and ∆β is the difference in the propagation constants of the initial and final modes, proportional to the effective index difference ∆nef f

(Marcuse, 1984; Bjarklev, 1986; Ramachandran et al., 2015). The symmetry of the perturbation determines the difference in azimuthal index (l in LPl,m) between the

coupled modes. Given that the most prevalent perturbation expected is a microbend, which can be modeled as a perturbation with cos(φ) dependence (Marcuse, 1982), we expect maximum coupling between modes for which ∆l = 1. With these considera-

tions in mind, it follows that mode coupling for LP0,m modes will primarily result in

power transfer to the nearest neighbor LP1,m and LP1,m−1modes, and maximizing the

effective index difference, ∆nef f, between these modes will improve mode stability.

Determining a threshold ∆nef f for stable propagation is not straightforward – how-

ever, it is well known that polarization-maintaining (PM) fibers have birefringence on the order of ∼ 1 × 10−4 _{and can resist coupling between the polarization states over}

significant lengths (∼100 m). Therefore, a minimum criterion for mode stability can be approximated by ∆nef f > 1 × 10−4, though ideally ∆nef f is as large as possible.

Figure 2·3: (a) Several refractive index profiles as a function of q, where n(r,q) is given by Eq. (2.14). The fiber constants correspond to a typical multi-mode fiber: n1 = 1.48, N = 500, and ∆ = 0.03. (b)

Index splitting parameter δ (given by Eq. (2.16)) as a function of q. Dashed black lines delineate q for each of the profiles in (a).

Effective index splitting in a fiber is related to the functional form of the refractive index profile. For example, consider a multi-mode fiber with an arbitrary refractive index profile given by

n2(r, q) =        n2 1 1 − 2∆ _r a q
r < a n2 1(1 − 2∆) = n22 r ≥ a (2.14)

normalized index step given by (n1 − n2)/n1, and n2 is the refractive index of the

cladding. The factor q describes the shape of the profile such that q = 2 corresponds to parabolic index profile (purple line, Fig. 2·3(a)), and q → ∞ describes a step-index fiber (red line, Fig. 2·3(a)). Refractive index profiles for several q values are shown in Fig. 2·3(a). In such a fiber, the propagation constant for each mode, β, can be approximated by (Ghatak and Thyagarajan, 1998)

β_l,m2 _{≈ k}2₀n2₁  1 − 2∆_Nv  q q+2 (2.15)

where k0 = 2π/λ, λ is wavelength, N is the number of guided modes in the fiber, and v

depends on the radial and angular eigenvalues of the mode such that v ∼ (m+l +1)2_.

Given that β = k0nef f and ∆nef f ≈ [(n0,mef f)2 − (n 1,m

ef f)2]/2n1, the effective index

splitting between LP0,m and LP1,m in such a fiber is proportional to a value δ defined

by ∆nef f = ∆n1  1 N _q+2q δ, _{δ ≡}h(m + 2)q+22q − (m + 1) 2q q+2 i (2.16)

Fig. 2·3(b) shows δ as a function of q for several different modes, assuming n1, N ,

and ∆ are constant with changing q. The dashed vertical lines correspond to the index profiles in Fig. 2·3(a). Effective index splitting increases with q meaning that a step-index fiber is the best design for maximizing index splitting to achieve stable mode propagation. Accordingly, all of the fibers used in our experiments are as close to step-index as possible.

Fig. 2·3(b) also illustrates the fact that effective index splitting increases as a function of mode order. The increased stability of the highest order modes in the fiber relative to lower modes seems counter intuitive: increased mode volume should cause the effective index splitting between adjacent modes to decrease. Indeed this is the case, however, the symmetry of likely perturbations to the fiber (i.e. bends) acts as a selection rule which determines that only the splitting between LP0,m modes and

their nearest neighbor LP1,m modes is relevant. At higher mode orders, other LPν,n

modes are stacked between LP0,m and LP1,m in effective index; therefore the overall

adjacent mode splitting decreases, but the separation between the modes that are likely to couple actually increases. Effective index splitting in the context of mode stability is discussed further in (Ramachandran et al., 2006b), and the increased stability of higher radial order modes is experimentally demonstrated.

Another key aspect of mode stability is also discussed in (Ramachandran et al., 2006b) – as effective area (Aeff) is increased, the nearest neighbor effective index

splitting for a given LP0,m mode decreases. This behavior is a relevant consideration

for designing high power fiber amplifiers. For fundamental mode operation, increasing Aeff in order to facilitate higher power operation leads to smaller effective index

splitting, and thus unstable propagation. This issue can be mitigated to a certain extent by creating mechanisms for HOM loss (Wong et al., 2005; Fini, 2005; Dong et al., 2007; Ma et al., 2011; Jain et al., 2014), but lack of mode stability is still a fundamental problem which limits Aeff scaling in the fundamental mode. If instead

an intentionally multi-moded fiber is used, and the signal is guided in a HOM, then the decrease in effective index splitting with increased Aeff can be offset by simply

increasing the radial mode order. For this reason, HOMs are ideal candidates for Aeff scaling in fiber lasers (Ramachandran et al., 2008), and amplifiers with stable

propagation in modes with Aeff>6000 µm2 have been demonstrated (Nicholson et al.,

2012).