Gaussian approximation fundamental modes

It can be shown [5.1,5.24] that in nonfocal resonators with large Fresnel numbers N the field distribution of the fundamental mode can also be described by the Gaussian profile (5.32). The confocal resonator with d = R can be replaced by other mirror configurations without changing the field configurations if the radius Rf of each mirror at the position zo equals the radius R of the wavefront in (5.37) at this position. This means that any two surfaces of constant phase can be replaced by reflectors, which have the same radius of curvature as the wave front - in the approximation outlined above. 

At the mirror surfaces is = 1 b = b, which means that the phase front is identical with the mirror surface. [Due to diffraction this is not quite true at the mirror edges at larger distances from the axis, where the approximation (5.32) is not correct]. At the center of the resonator is Zq = 0 b becomes infinite. At the beam waist the constant phase surface becomes a plane z = 0. This is illustrated in Fig.5.8 which shows the phase fronts and intensity profiles of the fundamental mode at different locations inside a confocal resonator. It can be shown [5.15] that also in nonconfocal resonators with large Fresnel numbers N the field distribution of the fundamental mode can be described by the Gaussian profile (5.27). 

In Section 14-10, we introduced the concept of profile volume. We showed in the case of clad power-law profiles of equal volume that some properties, such as the range of single-mode operation and the fundamental-mode intensity distribution are insensitive to profile shape, whereas other properties, such as waveguide dispersion, depend critically on profile shape. Within the Gaussian approximation, we can demonstrate directly the insensitivity of the intensity distribution to profile shape. 

Table 15-1 Gaussian approximation for circular fibers. Gaussian approximation for the fundamental-mode fields of weakly guiding, circular fibers. Parameters are defined inside the back cover and coordinates are illustrated in Fig. 14-1...

If we substitute the Gaussian approximation of Eq. (15-2) into Table 13-2, page 292, we obtain the expressions in Table 15-2 for fundamental-mode quantities on an arbitrary profile. We have generalized the function i/-the fraction of modal power within the core-and define tf(R) to be the fraction of modal power within normalized radius R = r/p. This is a more useful quantity for profiles with no well-defined core-cladding interface. The expressions for Vg and D follow from Eqs. (13-17) and (13-18). If for a particular profile the... 

Table 15-2 Gaussian approximation for fundamental-mode properties. Definitions are taken from Table 13-2, page 292, for modal quantities. The approximate...

The fundamental-mode properties of the weakly guiding, step-profile fiber were given in analytical form in the previous chapter, but, nevertheless, numerical solution of a transcendental eigenvalue equation is required. Within the Gaussian approximation the propagation constant is given explicitly, and all other modal properties have much simpler analytical forms, at the expense of only a slight loss of accuracy [4, 5]. 

The fundamental modes of the infinite parabolic profile fiber have a Gaussian spatial variation it is the exact solution of the scalar wave equation. Thus, the essence of the Gaussian approximation is the approximation of the fundamental-mode fields of an arbitrary profile fiber by the fundamentalmode fields of some parabolic profile fiber, the particular profile being determined from the stationary expression for the propagation constant in Table 15-1. Clearly this approach can be generalized to apply to higher-order modes, by fitting the appropriate solution for the infinite parabolic profile [9]. 

We discussed the Gaussian approximation for the fundamental modes of this profile in Section 15-2. For all m = 1 modes, Eq. (15-17) reduces to... 

Apart from the / = 0 fundamental modes, there are analytical solutions for / = 1 and / = 2, when Eq. (15-21) reduces to a cubic and a quadratic polynomial, respectively. The solution for Rq and are presented in Table lS-4 [9], together with the exact cutoff values [7]. If we compare the approximate and exact values of U, the error is less than 1.2 % for the 1 = m = 1 modes if I > 3, and is less than 0.9 % for the 1 = 2, w = 1 modes if K > 4.S. In both cases the error increases as V approaches V, because the field distribution F, is a poor approximation to the spread-out fields of the Gaussian profile near cutoff. 

If we use the step-profile approximation to determine the values of U for the fundamental modes of the Gaussian-profile fiber, then there is no limitation on the range of V, unlike the Gaussian approximation of Table 15-2. Furthermore, there is so little difference between the approximate and exact values of U, that a plot is indistinguishable from the dashed curve in Fig. 15-1 (b). Plots of the intensity distribution calculated from the step-profile approximation are virtually coincident with the exact solution away from the fiber axis, particularly for small values of V, whereas the Gaussian approximation leads to a significant error, as is evident in the V = 1.592 plot of Fig. 15-l(c). The Gaussian approximation has a far field which decreases too rapidly, as explained in Section 15-5, whereas the choice of field in Eq. (15-22) ensures the correct far field behavior for R > 1. 

The Gaussian approximation was introduced in Chapter 15 to provide simple, but accurate, analytical expressions for fundamental-mode quantities of interest on circular fibers of arbitrary profile. Here we show how to generalize this approximation and describe fundamental-mode propagation on weakly guiding fibers of arbitrary cross-section. 

The basis of the Gaussian approximation for circular fibers is the observation that the fundamental-mode field distribution on an arbitrary profile fiber is approximately Gaussian. Coupled with the fact that the same field on an ihfinite parabolic-profile fiber is exactly Gaussian, the approximation fits the field of the arbitrary profile fiber to the field of an infinite parabolic-profile fiber. The optimum fit is found by the variational procedure described in Section 15-1. Now in Chapter 16, we showed that the fundamental-mode field distribution on an elliptical fiber with an infinite parabolic profile has a Gaussian dependence on both spatial variables in the cross-section. Accordingly, we fit the field of such a profile to the unknown field of the noncircular fiber of arbitrary profile by a similar variational procedure, as we show below [1, 2],... 

Table 17-1 Gaussian approximation for fundamental modes of noncircnlar fibers.

It is readily verified, by repeating the derivation in Table lS-1, page 339, for planar waveguides, that the above expression is the spot-size equation for the Gaussian approximation to the fundamental modes of a step-profile, planar waveguide of core half-width Py. 

For the fundamental modes on a circular fiber, is independent of 0. Consequently —tiy and p = p in Eq. (18-21). In other words, the propagation constants for the circular and elliptical fibers are identical for slight eccentricity, provided the core areas are equal [6]. The latter condition is equivalent to requiring equal profile volumes, as is clear from Eq. (17-13). Hence the present result is consistent with the more general result of Section 17-3, which showed that, within the Gaussian approximation, P = on an arbitrary, elliptical-profile fiber of slight eccentricity, provided the profile volumes are equal. 

Gaussian approximation, Tq is the spot size of Eq. (20-24). Here p, is the radius or spot size for the beam. Pj is the total beam power and Pq is the power entering the fundamental mode. 

The efficiency with which beams excite the fundamental modes of circular fibers, with the fields of Eq. (13-9), is of particular interest when the fiber is single moded. In order to account for weakly guiding fibers of otherwise arbitrary profile, when analytical solutions of the scalar wave equation for Fo (r) are not available, we use the Gaussian approximation of Chapter 15. The radial dependence of the fundamental-mode transverse fields is approximated in Eq. (15-2) by setting... 

The spot size, r, depends on the particular profile shape. Examples, including the step and Gaussian profiles are given in Table 15-2, page 340. If we approximate Fo(r) by Eq. (20-24), it follows that all of the results for fundamental-mode excitation of the infinite parabolic-profile fiber, derived earlier in this chapter, apply equally to arbitrary profile fibers provided the appropriate expression for r is substituted into Table 20-1. 

A further consequence of the Gaussian approximation is a simple and accurate description of power loss at junctions between single-mode fibers. In practice, junctions are imperfect because of (a) mismatches between fibers, (b) tilts and (c) offsets, as illustrated in Fig. 20-3. If the incident, fundamental-mode fields of the fiber for z < 0 are described by the Gaussian approximation and have spot size Ps, i.e. Fo(r) = exp( — r /2pf), then each junction imperfection can be regarded as a Gaussian beam incident on the fiber in z > 0. If we also use the Gaussian approximation for the latter, as described above, the fractional... 

Fig. 20-4 (a) The fraction of total power in a uniform beam that excites modes of a step-profile fiber as a function of the tilt angle 0j, where P includes all modes with the same values of U in Fig. 14-4, and bm is the total excited power [2]. (b) Variation of the excitation efficiency with the fiber parameter for on-axis illumination, where solid curves denote the exact solution of Eq. (20-27c) and the dashed curve is the Gaussian approximation of Eq. (20-28a). (d) The corresponding curves for the fundamental mode for various ratios of beam to core radii calculated from Eqs. (20-27c) and (20-28b). (c) Plots of Pq/Pi for the fundamental mode and different ratios of beam to core radii. 

In Fig. 20-4(d) we show how an increase in beam radius affects the fraction of power exciting the fundamental mode, as calculated from Eq. (20-27b). For larger values of V, less power enters the HE, mode as p, increases. The dashed curves are the Gaussian approximation of Table 20-1 with r = p/(21n i.e. 

The expression in Eq. (20-30) for the fundamental-mode efficiency is also the result which we would obtain using the Gaussian approximation of Eq. (20-24) for an arbitrary profile. Thus we have a general expression for lens illumination. For example. Table 15-2, page 340, gives Tq = p/(21n for the step profile, and at K = 2.4 the error between Eq. (20-30) and an exact analysis is less than 1 % [10]. 

Fig. 20-7 (a) The fraction of power, calculated from Eq. (20-34X that enters the fundamental mode as a function of the angular spread 0 of a diffuse source for step (sX Gaussian (g) and infinite parabolic (p) profile fibers, (b) The percentage error in the geometric optics analysis of totally incoherent illumination of a multimode fiber as a function of the fiber parameter. The solid curve is the exact result calculated from Eq. (20-39) and the dashed curve is the approximation of Eq. (20-41) [11]. 

The transition loss for Fig. 23-4(a) is due to the mismatch over AA between the fields of the straight section and the offset fields of the curved section. We use the Gaussian approximation of Eq. (15-2) to describe the radial distribution of the fundamental-mode fields relative to the fiber axis and obtain... 

When each fiber in isolation has a Gaussian refractive-index profile, we can determine the coupling coefficient using the Gaussian approximation of Chapter 15 for the fundamental modes. The profile for the composite waveguide is given in terms of the radial coordinates of Fig. 27-1 (b) by... 

In Section 15-5, we showed ho w to derive the far field of a weakly guiding fiber from a knowledge of the Gaussian approximation to the fundamental mode. As explained below Eq. (15-14), this requires solution of... 

We can solve Eq. (36-51) for an arbitrary profile, n r), if we approximate the fundamental-mode fields with the Gaussian approximation of Chapter 15. The solution of the scalar wave equation for the straight fiber is given in terms of the spot size Tq for the particular profile by Eq. (15-2). To solve Eq. (36-51), we set... 

---