Fundamental mode solutions

Fig. 12-4 Numerical solution of the eigenvalue equations of Table 12-4 (a) for the first twelve modes. The values along the dashed line are the cutoff values of U, and the dashed curve is the fundamental mode solution for a fiber with and K fixed.

We begin with a brief review of the weak-guidance approximation for fundamental modes on circular fibers. In Sections 13-2 and 13-4, we showed that the two fundamental modes are virtually TEM waves, with transverse fields that are polarized parallel to one of a pair of orthogonal directions. The transverse field components for the x- and y-polarized HEj j modes are given by Eq. (13-9) relative to the axes of Fig. 14-1. The spatial variation Fq (r) is the fundamental-mode solution (/ = 0) of the scalar wave equation in Table 13-1, page 288. Hence... 

By analogy with the fundamental-mode solution for the elliptical fiber of infinite parabolic profile given in Table 16-1, page 356, we assume that the solution 4 (x, y) of the scalar wave equation of Eq. (16-3) can be approximated by setting [1, 2]... 

The propagation constants associated with the two fundamental-mode solutions of the scalar wave equation are generally distinct because of the... 

Clearly there must be a transition from the fundamental solutions of Eq. (18-54) to those of Eq. (18-33) as the fibers become similar. In the situation where the fibers are nearly identical, the fundamental-mode solutions of the composite waveguide may be expressed, for later comparison with Eq. (18-33), in the general form... 

The principle of operation of this sensor is based on the fact that, as the fundamental optical mode travels through the MNF, its shape is modified depending on the index contrast between the solution in the channel and the polymer. Consequently, the change of the fundamental mode results in variation of the MNF... 

In papers , unsteady-state regime arising upon propagation of the stationary fundamental mode from linear to nonlinear section of a single-mode step-index waveguide was studied via numerical modeling. It was shown that the stationary solution to the paraxial nonlinear wave equation (2.9) at some distance from the end of a nonlinear waveguide has the form of a transversely stable distribution ( nonlinear mode ) dependent on the field intensity, with a width smaller than that of the initial linear distribution. 

The nonlinear modes are usually referred to as solutions to the nonlinear Helmholtz equation in the waveguide cross-section. For their investigation, power-dispersion diagrams are commonly used that give values of critical powers and are helpful in stability analysis of the fundamental mode. ... 

Figure 1. Monotonic behaviour of the fundamental and second-harmonic modes. Solution of Eqs. (7) for the initial conditions =0.1 + 0.1 and < 20 — 0.

The solution of Orr- Sommerfeld equation has four fundamental modes as... 

Second-harmonic generation, which was observed in the early days of lasers [18] is probably the best known nonlinear optical process. Because of its simplicity and variety of practical applications, it is a starting point for presentation of nonlinear optical processes in the textbooks on nonlinear optics [1,2]. Classically, the second-harmonic generation means the appearance of the field at frequency 2co (second harmonic) when the optical field of frequency co (fundamental mode) propagates through a nonlinear crystal. In the quantum picture of the process, we deal with a nonlinear process in which two photons of the fundamental mode are annihilated and one photon of the second harmonic is created. The classical treatment of the problem allows for closed-form solutions with the possibility of energy being transferred completely into the second-harmonic mode. For quantum fields, the closed-form analytical solution of the... 

The solutions (58) are monotonic and eventually all the energy present initially in the fundamental mode is transferred to the second-harmonic mode. 

Generally, the second-harmonic generation is described by the quantum state (127) and we use this state in our further calculations. Classical solutions discussed earlier, ua x) = sech x and = tanh x, indicated that the amplitudes of the two modes are monotonic functions of time and that eventually all the energy from the fundamental mode will be transferred into the second-harmonic mode, assuming that there was no second-harmonic signal initially. It is well known [20,48], however, that the quantum solution has oscillatory character and does not allow for the complete power transfer. Using the state (127) we find that the mean photon numbers evolve in time according to the formulas... 

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