Laguerre-Gaussian Laser Beam Second Harmonic Generation in Arrays of Vertically Aligned Carbon Nanotube

In this present work, we have studied second harmonic generation by nonlinear interaction of Laguerre-Gaussian laser in arrays of vertically aligned carbon nanotubes. These arrays of carbon nanotube are organized in a dielectric surface and the interaction of Laguerre-Gaussian laser beam is taken perpendicularly to the length of carbon nanotube. As the Laguerre-Gaussian laser beam field interacts with carbon nanotube arrays, the nonlinearity in the medium is arisen by electrostatic restoration force and causes the excursion in electron cylinder with respect to ion cylinder. The nonlinearity results the second and third harmonic electron excursion in carbon nanotube. We have derived the analytic expressions of second harmonic nonlinear current density and second harmonic electric field in carbon nanotubes. The results and discussions of this theory is explained by various graphical profiles. Owing to presence of surface plasmons resonance, enhanced second harmonic electric field amplitude is observed for laser beam frequency approaches near the 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{2}$$\end{document} times of the electron plasma frequency. The peak amplitude of second harmonic electric field is seen at Laguerre-Gaussian laser beam-normalized propagation distancer/w0L similar to 0.46\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r/{w}_{0L }\sim 0.46$$\end{document}. One can tune and control the second harmonic electric field amplitude by varying the Laguerre-Gaussian laser beam index (l, p), laser beam initial width(w0L)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({w}_{0L })$$\end{document}, laser beam frequency, carbon nanotube radii, inter carbon nanotube separation, equilibrium electron density of carbon nanotube and electron-ion collisional frequency. The large amplitude second harmonic field is generated by using the Laguerre-Gaussian laser beam compared with only Gaussian laser beam.