Finite length effects on first hyperpolarizability tensor for chiral carbon nanotubes with application to second harmonic far‐field radiation pattern

Abstract

The general form of the first hyperpolarizability third-rank tensor β for chiral carbon nanotubes (CNTs) with symmetry Ln p 22 is calculated with group projector technique method. For this purpose, the group natural factorization has been adopted and the associated irreducible representations, parameterized in terms of suitable quantum numbers, then used to construct the group projector operator in the relevant vector space. The followed approach allows to model the finite length effects on the tensor and to study its convergence to the invariant form as a function of CNT topology. As an applicative example, to elucidate the role played by finite length effects, the obtained results are applied to the study of second harmonic (SH) far-field radiation pattern emitted by a chiral CNT within the linear dielectric homogeneity approximation. The particular case of a tube forming an angle of π/4 with the propagation direction of the linearly polarized exciting laser beam is treated. It is demonstrated that the radiation pattern consists of two main lobes with high directivity along the specular reflected and transmitted directions. According to the model developed and for the cases considered, finite length effects introduce asymmetries in the intensity distribution of the two lobes.

Advances in nonlinear optical spectroscopies: a historical perspective of developments and applications presented at ECONOS

“…The first hyper‐polarizability of CNTs was also studied theoretically. The effects of electron statistics on the frequency dependence of the hyper‐polarizability were analyzed, as well as finite length effects with respect to the SHG far‐field radiation pattern . The CNT‐based generation of the second‐ and third‐harmonic radiation was also demonstrated with femtosecond pulses .…”

“…The effects of electron statistics on the frequency dependence of the hyper-polarizability were analyzed, [164] as well as finite length effects with respect to the SHG far-field radiation pattern. [165] The CNT-based generation of the second-and third-harmonic radiation was also demonstrated with femtosecond pulses. [166] It was concluded that harmonicgeneration processes are ideally suited for the nonlinear spectroscopy and structure analysis of CNTs.…”

Recent advances in linear and nonlinear Raman spectroscopy. Part IV

“…The general form of the first hyperpolarizability third-rank tensor for chiral carbon nanotubes with symmetry Lnp22 was calculated by the group projector technique by De Dominics and Fantoni [102] to study the second-harmonic far-field radiation pattern. Bunkin and Pershin [103] studied modifications of the spectrum of water in the presence of protein and DNA and single-wall carbon nanotubes at low concentration by four-wave mixing spectroscopy.…”