Propagation constant modes

Figure 3. Normalized propagation constant as a function of normalized frequency for three low-order TE and TM modes n, 1.45, n2=l. 8, n3=l. 32.

Figure 7. Concept of optical sensor based on spectroscopy of guided modes. A change in the refractive index alters propagation constant of a guided mode.

Biolayer refractive index, n Mode propagation constant, ft... 

The phase-matching for three wave mixing process can be achieved in an optical fibre for modes which satisfy the condition 3 = i where 3, 3g and 3, are "the propagation constants for the three modes. For opticaX second harmonic generation this condition becomes 3 = 23, where 3 and 3... 

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 ... 

The field distribution of the -th Bloch mode is determined by the field amplitudes p and q , its propagation constant is / A, where... 

The transfer matrix method is known to be often unstable. If this is the case for the matrix A, other methods can be alternatively used to calculate the Bloch modes and their propagation constants " . ... 

This formally simple procedure is very difficult to perform, however. Because of radiation from the bend, the azimuthal propagation constant v to be found is complex. Since the bend radius of the waveguide is typically larger than the wavelength, the real part of v can be large, too. Moreover, a number of modes of each slice with very different values of their effective indexes are to be considered simultaneously. It causes very serious... 

Note that this procedure is suitable also for the calculation of eigenmodes in circular ring and disk microresonators. The cross-sections of disk microresonators are usually simpler, without any mid-slices, which leads to shorter calculation times. However, good estimate of initial values of the propagation constants of the disk modes for their search in the complex plane is more difficult, and their mode spectrum is denser than for ring microresonators with similar radii of curvature. 

Surface plasmon-polaritons (SPP), also referred as to surface plasma waves, are special modes of electromagnetic field which can exist at the interface between a dielectric and a metal that behaves like a nearly-iree electron plasma. A surface plasmon is a transverse-magnetic mode (magnetic vector is perpendicular to the direction of propagation of the wave and parallel to the plane of interface) and is characterized by its propagation constant and field distribution. The propagation constant, P can be expressed as follows ... 

Numerics has been used intensively in the field of integrated optics (10) since its early days, simply due to the fact that even the basic example of the slab waveguide requires the solution of a transcendental equation in order to calculate the propagation constants of the slab- guided modes. Of course, the focus was directed to analytical methods, primarily, as long as the power of a desktop computer did allow for a few coupled equations and special functions only e.g. to describe the nonlinear directional coupler in a coupled mode theory (CMT) picture. During the years, lots of analytic and semi-analytic approaches to solve the wave equation have been developed in order... 

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. 

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