Fundamental modes propagation constant

The scalar propagation constants P+ and for the fundamental modes of the composite waveguide are given by Eq. (18-35) in terms of the fundamental mode propagation constant for either fiber in isolation and C of Eq. (18-42). We explained in Section 13-5 that polarization corrections are required to correctly distinguish between the propagation constants of each pair of fundamental modes associated with P+ or P-. To determine each correction, we substitute the approximate transverse electric field of Eq. (18-36) into Eq. (13-12), where I now denotes the interface of both fibers. Thus, in the notation of Section 18-12, and with the help of Eqs. (18-36) and (18-33), we obtain 5 by setting... 

Consider a composite waveguide consisting of two step-profile fibers of core radii p and p + 5p, and common core and cladding indices and n, . Using the notation of the previous section, the difference — 2 in fundamental-mode propagation constants is given by — of Eq. (18-13a), and C follows from Eq. (18-42), assuming the fibers are well separated. Hence... 

Thus if Hq and n (po) are sufficiently dissimilar, the perturbed and unperturbed fields can differ greatly within the perturbation region. Even if n s n(po) we need to retain Eq. (18-64) without further approximation in order to describe correctly polarization effects due to the nonuniformity, e.g. the difference in fundamental-mode propagation constants. 

When the two fibers of Section 29-4 have slight differences 611 0 and Sp in their core indices and radii, respectively, the fundamental-mode propagation constants are no longer equal and differ by a small amount <5)S = - 2- The corresponding miniscule... 

When the fibers in Fig. 29-1 are absorbing, the refractive-index profile of each fiber and of the composite waveguide has the complex form given in Table 11-2, page 232. The fundamental-mode propagation constant for each fiber in isolation is also complex, so we set... 

Fig. 13.5 Calculated propagation constants (ft) for the fundamental modes of glass MNFs with refractive indices of 1.46 (silica), 1.48 (fluoride), 1.54 (phosphate), 1.89 (germinate), and 2.02 (tellurite), respectively. A circle marked on each curve corresponds to the maximum radius of the single mode MF. Radiation wavelength is X 633 nm. Reprinted from Ref. 62 with permission. 2008 Optical Society of America...

In an adiabatically tapered axially symmetric MNF, the propagation constant of the fundamental mode, p(z), is a slow function of the coordinate z along the axis of the MNF and (13.4) is modified as follows ... 

Effective refractive index of the w-th HOCM of an LPG jSc Propagation constant of the fundamental core mode... 

From this expression we obtain a dispersion curve that relates V and b, as shown in Fig. 6. When the waveguide parameters ri2, and a and wavelength X are given, the propagation constant of any mode can be obtained from (Eq. 30). The mode number is labeled in the order of increasing N, and TEq is the fundamental mode. This mode number N corresponds to the number of nodes in the field distribution. 

In the case of lateral compression, the propagated fundamental mode perfectly degenerates into x- and y-polarized modes with different propagation constants. According to the photoelastic effect and the stress strain relationship, the effective index changes of the two polarized modes ean be expressed as ... 

Fundamental modes of waveguides of arbitrary cross-section 285 13-6 Polarization corrections to the scalar propagation constant 286 13-7 Higher-order modes of circular fibers 287... 

First we consider fundamental modes and then higher-order modes. Like the exact propagation constant P, the scalar propagation constant p is largest for fundamental modes. It is convenient to distinguish between fibers of circular cross-section and waveguides of arbitrary cross-section. 

The fundamental and HEi , modes of a fiber of circular cross-section are formed from the scalar wave equation solution with no azimuthal variation. Hence in Eq. (13-8) depends only on the radial position r. There is no perferred axis of symmetry in the circular cross-section. Thus, in this exceptional case, the transverse electric field can be directed so that it is everywhere parallel to one of an arbitrary pair of orthogonal directions. If we denote this pair of directions by x- and y-axes, as in Fig. 12-3, then there are two fundamental or HEi , modes, one with its transverse electric field parallel to the x-direction, and the other parallel to the y-direction. The symmetry also requires that the scalar propagation constants of each pair of modes are equal. 

The solution of Eq. (13-8) for the fundamental modes by definition has the largest value of Then, as for the circular fiber there are two modes associated with this solution, one polarized along the x-direction and the other polarized along the y-direction. Both modes have the same scalar propagation constant P, but, because the cross-section is not circular, we know the exact... 

Table 13-1. Accordingly, the LP designation of fundamental modes in Section 13-4 is not applicable. However, if we ignore all polarization properties of the fiber, then all four modes have the same propagation constant given by i.e. SPi = 0. Then there is no constraint on the field directions and they can be expressed in a plane polarized form, e.g. = F, (r) cos x or = F,(r) x...

The higher-order modes of waveguides with noncircular cross-sections are constructed from each pair of solutions Pj (x, y) and Pg (x, y) of Eq. (13-8) and their corresponding scalar propagation constants and p. The transverse electric fields of these modes are polarized along the same optical axes as the fundamental modes of Section 13-5. There are two pairs of higher-order modes. Each pair has fields given by Eq. (13-10), with p and P(X) y) replaced by Pg and Pj(x,y) for one pair, and by p and Pj,(x,y) for the other pair. The polarization corrections Sp, Sp, SPy and SPy are obtained from Eq. (13-11) with the appropriate field substituted for e,. 

The small polarization correction to the scalar propagation constant due to structural anisotropy is given by Eq. (13-11). For an isotropic fiber of circular cross-section, the corrections for the two fundamental modes are identical, i.e. SPx = Py This is not the case for the anisotropic fiber, since the parameters in Eq. (13-19) depend on polarization. However, SP — SPy is small compared to the difference in propagation constants in Eq. (13-20), Px — Py, since the fiber is weakly guiding, and can be ignored. 

If V is below the cutoff value 2.405 of the second mode in iFig. 14-4, the fiber is single moded and only the even and odd fundamental modes can propagate. Both modes have the same propagation constant j8. Consequently the group velocity and the transit time of Eq. (11-36) are independent of polarization. In the weak-guidance approximation, the expression for in Table 14—3 follows from Eq. (13-17), and is plotted against V in Fig. 14—3(d) as the dimensionless quantity (n — c)/cA. 

If we substitute the spot size back into Table 15-1, we deduce the expressions for the propagation constant and modal parameter in Table 15-2, which in turn lead to the remaining expressions for fundamental-mode quantities. 

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]. 

Consider a pulse within which only the two fundamental modes are excited. Waveguide dispersion describes the spread in each mode, but because of elUpticity the spread for each polarization is different. In addition, the sUght difference dpj —SPy between corrected propagation constants implies the respective group velocities are unequal and consequently there will be intermodal dispersion between the two modes. Intermodal dispersion which relies on polarization difference is often referred to as a birefringence effect. 

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