Beat length

What makes EIM susceptible to be mistaken are configurations with regions, where no guided mode can be determined. If more than an estimate of the propagation constant is required, e.g. in order to calculate the half-beat length of a buried channel waveguide directional coupler configuration from the mode indices of the symmetric and anti-symmetric supermodes, EIM should be superseded by a more accurate method. 

The nonlinear directional coupler is potentially a useful device because it has four ports, two input and two output, and because the outputs can be manipulated with either one or two inputs. Optimally the two channels are identical, and the coupling occurs through field overlap between the two channels. As a result, when only one of the channels is excited with low powers at the input, the power oscillates between the two channels with a beat length Lb, just like what occurs in a pair of weakly coupled identical pendulii. As the input power is increased, a mismatch is induced in the wavevectors of the two channels, which decreases the rate of the power transfer with propagation distance. This leads to an increase in the effective beat length. There is a critical power associated with this device, for which an infinitely long, lossless NLDC acts as a 50 50 splitter, that is, Lb oo. 

Figure 7. A number of all-optical guided wave devices and their responses to increasing power. (a) Half beat length directional coupler. (b) One beat length directional coupler. (c) Distributed feedback grating relector. (d) Nonlinear Mach-Zehnder interferometer. (e) Nonlinear mode mixer. (f) Nonlinear X-switch. For nonlinear media (n2 0), the input power determines the output state.

If Zb denotes the beat length, i.e. the distance along the waveguide in which there is total transfer of power from one fiber to the other fiber and back again, then Eqs. (18-35) and (18-39) give... 

As the separation increases, )3+ - )S and the beat length becomes exponentially large. The transfer of power is clearly a consequence of interference, or beating, between the fundamental mode fields in Eq. (18-37), and depends only on the difference between the scalar propagation constants. There is no need to consider polarization corrections to the propagation constants in order to study cross-talk on the composite waveguide. 

The polarization corrections cause interference effects between pairs of fundamental modes with the same propagation constant. For example, if the modes associated with are excited, the difference between and 6Py+ accounts for the apparent rotation of the total transverse fields as they propagate [9]. This was examined in Section 14-7, and can be characterized by a beat length 4n/(SP + —SPy ). We can compare the cross-talk beat length of Eq. (18-40) with the rotation beat length. For large separation, Eqs. (18-40), (18-42) and (37-88) give... 

Consequently the fraction of total power transferred between the two fibers is F, and the beat length z, = InFIC. Using the figures from the previous section a 1 % difference in core radii of the two fibers results in a maximum exchange of 10% of total power. We consider this problem again in Chapter 29. 

We recall from Eqs. (11-2) and (11-3) that the total field on a cylindrically symmetric fiber can be represented by a summation of modes. It is then clear that the total field can undergo a significant change in a distance equal to the beat length, z, between a pair of modes. We need the largest beat length, which is given by... 

The above results will be accurate provided the fiber radius does not change significantly over a distance equal to the appropriate beat length in Eq. (19-5), i.e. 5p/p(z) < 1 where dp = z dp(z)/dz. For a fiber supporting more than the fundamental mode z is given by Eq. U9-3), which leads to... 

The slowness criterion requires the twist be only slight over a distance equal to the beat length between the two fundamental modes in order for the above description to be accurate. By setting and 2 = j8, in Eq. (19-5), the criterion is expressible as... 

Thus the two local fundamental modes interfere, and all power is transferred from one mode to the other mode and back again over a distance equal to the local beat length z,... 

When both fibers have a step profile, a difference of 1 % in core radii is sufficient to reduce power transfer to 10% of total power, as shown in Section 18-20. Consequently 6pjp need be only a few per cent for the fibers to be effectively isolated, and as C(zc) is exponentially small, the taper angle, 2, will be minute. Since 21 > Zb, the critical length for total power transfer between tapered fibers is large compared with the beat length for total power transfer between... 

Thus a fraction F of the total power is exchanged between the two modes and the beat length is InFIx. At the resonance of Eq. (28-21), F = 1/2, and 50 per cent power transfer occurs. We emphasize that the solution, Eq. (28-23), is only accurate at, or very close to, resonance. This situation minimizes the power loss to radiation and to the backward-propagating fundamental mode. 

Hence a fraction of total power is transferred from one fiber to the other fiber and back in the beat length z, = 2nF/C along the waveguide. Only when the fibers are identical is complete power tranter achieved, and since C is very small compared with or 21 ihe fibers must be virtually identical for F to be... 

The solution given by Eq. (29-11) depends only on the coupling coefficient since F = 1 for identical fibers. We derived an expression for C in Section 18-14 assuming a step profile. The normalized coefficient pC/(2A) is plotted against the normalized separation d/p for various values of the common fiber parameter V in Fig. 29-2(a). For example, the beat length Zj, = InjC is approximately 2.8 x 10 p for K= 2.4, A = 0.005 and dip = 4. Given V, the value of C increases with decreasing separation as the fields... 

When V = 2.6, A = 0.005 and d/p = 4, the beat length 2 S 1.7 x 10 p. This is approximately six times larger than the result for step-profile fibers with the same separation at cutoff of the second mode, and is due to the exponentially small difference for the Gaussian profile compared with — ci profile. 

The values of D in Eq. (29-38) show that cross-talk is greater in an array than for two fibers with the same separation. Furthermore, the beat length varies inversely with D and therefore maximum power transfer occurs over a shorter distance in arrays. 

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