For systems with a complex dispersion profile, such as microring resonators, instead of using a Taylor expansion of the dispersion that results in a set of different [beta] coefficients (i.e., dispersion to different orders), it is often more convenient and accurate to work with the integrated dispersion [D.sub.int] = [[omega].sub.[mu]] - ([[omega].sub.0] + [D.sub.1][mu]), with [mu] an integer representing the mode number
relative to the pump resonance (for which [mu] = 0), [[omega].sub.[mu]] is the angular frequency of the [mu]th mode, and [D.sub.1] is the angular free-spectral range evaluated at the pumped resonance (i.e., [D.sub.1] /2[pi] is the free-spectral range in units of Hz).
In contrast, the eighth structural resonance, that is, a mode with an even circumferential mode number
, was not visible in the Z-direction response.
The difference in natural frequency for each mode number
is very small.
Where in this equation, [m]: is the substituted matrix of mass, [[[phi].sub.i]]: is the mode vector of cost for ith mode, i is the mode number
, [M.sub.ei]: is the effective mass for mode number
i, providing [M.sub.ei] for each mode the mass participation factor in addition to the effective substituted factor of stiffness for each mode could be gained through sets of relation in Equations (8)-(10).
The angular momentum of the mode is calculated as L[??], where [??] is the Planck constant and L is called the mode number
The mode number
of the radiated OAM carrying beam is exactly equal to l.
Selected numerical results are presented to indicate the influence of the values of nonlocal parameter, Winkler modulus parameter, Pasternak modulus parameter, mode number
, aspect ratio, and the type of nonlocal plate theory, in detail.
This is emphasized by the mode number
which is almost the same per position and per mode order, according to the general data in Table 1.
Then, their perturbations can be Fourier-analyzed putting them proportional to exp[i(-[omega]t + m[phi] [k.sub.z]z)], where [omega] is the angular wave frequency (that, in general, can be a complex quantity), m is the mode number
(a positive or negative integer), and [k.sub.z] is the axial wave number.
where [w.sub.0] is the waist radius of Gauss beam with the propagating distance of Z = 0 m, w > (z) is the waist radius with propagation distance of Z > 0 m, [z.sub.R] = [pi][w.sub.0.sup.2]/[lambda] is Rayleigh distance, k = 2[pi]/[lambda] is wave vector, l is azimuth pattern (also called optical vortex topological charges), and p is radial mode number
(2) Obtain the frequency estimates and damping ratio estimates from each mode number
from the eigenvalues of the system matrix.
The misdiagnosis rate is related to mode number
. In order to exclude the influence of the mode number
, we define a relative misdiagnosis rate (RMR), and the aforementioned misdiagnosis rate will be referred to as absolute misdiagnosis rate (AMR).