The intensity distributions near the focal point for tight focusing of azimuthally polarized Laguerre–Gaussian beam including Higher Order Modes are calculated based on vector diffraction theory. For Higher Order as well as the fundamental lower Order Mode A-TEM01*, a small focal hole at the centre is obtained for the focusing under high NA lens. In a limit of NA=0.95, the Higher Order A-TEMp1* Mode beams of p=1, 2, 3, 4, 5 and 6 are established numerically in detail. It is observed that one of the Higher Order Mode A-TEM51* generates some interesting results. And it is also observed that the intensity distribution of the different Mode has little variation among the degree of truncation (b) of the input beam beside the pupil. This work is important for optical manipulation, optical data storage, micro-machines and optical trapping applications for low refractive index particles.

We present a new theoretical Model for analyzing the impact of Higher Order Modes on transverse Mode instability in bent Yb-doped fiber amplifiers. The Model indicates a comparative analysis of Mode instability using the normalized propagation constant in both straight and bent fibers amplifiers. The impact of different bending radii, on bent fibers is investigated in the normalized propagation constant. The results are then compared with the straight fibers. Simulation analysis showed that the interference between the fundamental Mode (LP01) and the first Higher Order Mode (LP11) has the greatest effect on the Mode instability, comparing the other Modes. Considering all the existed Modes, the behavior of Mode instability in the fibers with different V-numbers are simulated and explained physically.

Based on the geometric background of Sombor index and motivating by the Higher Order connectivity index and the Sombor index, we introduce the pathcoordinate of a path in a graph and a degree-point in a Higher dimensional coordinate system, and define the Higher Order Sombor index of a graph. We first consider mathematical properties of the Higher Order Sombor index, show that the Higher Order connectivity index of a starlike tree is completely determined by its branches and that starlike trees with a given maximum degree which have the same Higher Order Sombor indices are isomorphic. Then, we determine the extremal values of the second Order Sombor index for all trees with n vertices and characterize the corresponding extremal trees. Finally, the chemical importance of the second Order Sombor index is investigated and it is shown that the new index is useful in predicting physicochemical properties with high accuracy compared to some well-established.

In Order to achieve a low phase noise oscillator, designing a high Q resonator is inevitable. Exciting Higher Order Modes is an appropriate method to increase the quality factor. Existence of unwanted Modes is a major issue that appears in Higher Order Mode excitation. The oscillator circuit cannot oscillate in the desirable Mode due to low Mode separation. This paper presents a design methodology for DR which leads to spurious performance improvement in Higher Order Modes in addition to the rejection of unwanted frequencies. The goal is controlling the frequency separation between resonance and unwanted frequencies in Order to implement in low noise oscillators. In an eight-GHz dielectric resonator excited by TE22δ Mode, the quality factor equals to 12450 which is Higher than a DR excited by TE01δ Mode with a quality factor of 8100. Also, unwanted frequencies at 8. 05 GHz and 8. 3 GHz are suppressed by the proposed technique with minimum impact on the main resonance Mode.

Purpose: To evaluate the correlation between refractive error and Higher-Order aberrations (HOAs) in patients with myopic astigmatism.Methods: HOAs were measured using the Zywave II aberrometer over a 6 mm pupil. Correlations between HOAs and myopia, astigmatism, and age were analyzed.Results: One hundred and twenty-six eyes of 63 subjects with mean age of 26.4±5.9 years were studied. Mean spherical equivalent refractive error and refractive astigmatism were -4.94±1.63 D and 0.96±1.06 D, respectively. The most common Higher-Order aberration was primary horizontal trefoil with mean value of 0.069±0.152 mm followed by spherical aberration (-0.064±0.130 mm) and primary vertical coma (-0.038±0.148 mm). As the Order of aberration increased from third to fifth, its contribution to total HOA decreased: 53.9% for third Order, 31.9% for fourth Order, and 14.2% for fifth Order aberrations. Significant correlations were observed between spherical equivalent refractive error and primary horizontal coma (R=0.231, P=0.022), and root mean square (RMS) of spherical aberration (R=0.213, P=0.031); between astigmatism and RMS of total HOA (R=0.251, P=0.032), RMS of fourth Order aberration (R=0.35, P<0.001), and primary horizontal coma (R=0.314, P=0.004). Spherical aberration (R=0.214, P=0.034) and secondary vertical coma (R=0.203, P=0.031) significantly increased with age. Conclusion: Primary horizontal trefoil, spherical aberration and primary vertical coma are the predominant Higher-Order aberrations in eyes with myopic astigmatism.