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Anisotropic fibres

In document Optical fibre couplers (Page 61-65)

2 4 THE PERTURBATION APPROACH 2.4.0 Perturbation results in the scalar regime

2.4.2 Anisotropic fibres

A simple model for a weakly-guiding, weakly-birefringent fibre consists of a circular core, with slightly different refractive indices for the two polarization states, embedded in an infinite isotropic cladding, as shown in Fig. 2.6. The difference may be caused by stress, say, inherent in the fibre structure. The propagation constants for the orthogonally polarized fundamental modes differ slightly, and this difference is approximated in (2-41), derived from the decoupled scalar wave equations for the two polarization states. (2-41) only applies if the difference in the core indices is small enough that the two modal field distributions themselves are virtually identical, i.e., if the fraction,

T[ , of power propagating in the core is the same for each mode.

Where larger anisotropy is present, the fundamental mode patterns and propagation constants differ significantly, and the difference does not vary linearly with the core anisotropy. Direct solution of the scalar wave equation for the circular fibre gives ß as a function of core index, or U as a function of V, equivalently [Snyder & Love, 1983, Fig. 14-4] and the resulting variation, shown in Fig. 2.12, is only linear with index locally.

The perturbation expression (2-41) predicts that dß/dn should be proportional for each index, to the core power fraction, rj, which monotonically approaches unity as core index increases, for a given fibre geometry. This is clearly not the case in general, since the slope in Fig. 2.12 peaks at a finite value of the index, and decreases thereafter as n increases. Only when the core-cladding index difference is small, (A « 1) is (2-41) valid.

51

FIG 2.12

Fundamental mode propagation constant varies non-linearly with core index in a circular fibre, as does birefringence. Weakly-anisotropic fibres possess birefringence is proportional to anisotropy.

From (2-41), the birefringent beat length is :

t _ 2x 2k _ k

Ißa-

ß b l (k T |ln a - n j ) i i n j s j

(249)

where the profile anisotropy param eter 5ab is defined in (2-28). The fibre acts as in the same m anner as a bulk optic retardance device, with retardance equal to 2kL/Lb , where L is the fibre length. Obviously, since the fibre can be any desired length, the retardance and its environm ental sensitivity may both be extrem ely large, making such fibres useful in sensing devices.

The orthogonally polarized modes o f a birefringent fibre are coupled physically by perturbations in the fibre along its length, such as bends, twists and externally imposed stress at an angle to the optical axes. External stress applied parallel to the optical axes m erely reinforces or reduces the birefringence. To couple the modes, such perturbations have to occur on length scales smaller than, or com parable to, the birefringent beat length. This is analogous to the geometric coupling o f alm ost-degenerate scalar modes o f a fibre, w hich gives rise to hybrid m odes - when the scalar degeneracy split is large enough, coupling between scalar modes does not occur. If the fibre birefringence is sufficient, the split in the degeneracy o f the orthogonally polarized m odes is too large for externally applied perturbations to couple them, (i.e. there is no resonance between the birefringent modes and the external perturbations) and the modes propagate undisturbed.

This is the reason why the designers o f polarization-m aintaining fibres try to m axim ize the birefringence, subject to other considerations, such as brittleness o f the stressed fibre p refo rm s. B irefrin g en t fibres are therefore ch aracterised by their polarization-holding ability, over a given length, w ith one figure o f m erit being the "h p aram eter". This is defin ed by m easuring the tim e-averaged am ount o f pow er cross-coupled to the unwanted polarization state from a given input polarized parallel to one optical axis, after distance L :

5 3

Clearly, depolarization is inevitable in the long run, but the better the fibre design, the longer the length scale for this to occur.

In coherent systems, and the like, polarization preservation is also required in all passive components, such as couplers, used in the system. It is intuitive that in birefringent couplers, if the two fibres are arranged with their optical axes misaligned, there is the opportunity for one to act as a physical perturbation on the other causing depolarization of the fields. The length scale of any perturbation is related to its physical cause, and in the case of a coupler, it is due to the evanescent beating between the modes. Therefore, we are interested in the resonance between the intrinsic fibre birefringence with its length scale LB

= 2tz/(ßa - ßh ) and the perturbation length scale Lc = n/C due to evanescent beating.

Clearly, the greatest depolarizing effect in birefringent couplers must occur when the optical axes are misaligned by 45° and the ratio of the evanescent and birefringent beat length scales is close to unity. Performance of a polarization-preserving coupler is optimal when the optical axes are aligned and the birefringent beat length is much shorter than that due to evanescent beating. This is examined in detail in Chapter 5.

CHAPTER 3

In document Optical fibre couplers (Page 61-65)