Applied Physics Research; Vol. 11, No. 4; 2019 ISSN 1916-9639 E-ISSN 1916-9647 Published by Canadian Center of Science and Education 1 A Novel Theory for the Scattering of P-Polarized Hermite-Gaussian Electromagnetic Beams by a Double Metallic Nano-Slit I. Guzman-Montaño 1 , G. Montiel-Gonzalez 1 , J. Sumaya-Martinez 1 , P. Rosendo-Francisco 1 , & J. Reyes-Gomez 2 1 Facultad de Ciencias, Universidad Autónoma del Estado de México. Instituto Literario #100 Col. Centro. Toluca, México. C.P. 50000. Correo 2 Facultad de Ciencias de la Universidad de Colima, Bernal Diaz del Castillo #340, Col. Villas San Sebastian, Colima, Colima, Mexico. C.P. 28045 Correspondence: J. Sumaya-Martinez, Facultad de Ciencias, Universidad Autónoma del Estado de México. Instituto Literario #100 Col. Centro. Toluca, México. C.P. 50000. Correo. E-mail: jsm@uaemex.mx Received: October 29, 2018 Accepted: November 30, 2018 Online Published: July 15, 2019 doi:10.5539/apr.v11n4p1 URL: http://dx.doi.org/10.5539/apr.v11n4p1 Abstract We present a rigorous theory for oblique incident Hermite-Gaussian beams, diffracted by two optical nano-slits of width  and separation d, in a thick metallic screen for the case of polarization TM (P). The far field spectra as a function of several opto-geometrical parameters, wavelength λ, slit width , separation d, incidence angle θ ୧ and Hermite order m is analyzed. In the vectorial diffraction region given when λ/ >0.2, where  is the incident wavelength and as a function of the separation between slits d; we have numerically analyzed: the far field spectra, the energy diffracted along the incident beam direction (E ୧ ), and the validity of an approximate diffraction (scalar) property, namely E ୧ = Nτ/λ. Keywords: Difracction, Scattering, Nano-Slits 1. Introduction Currently there are several rigorous theories of diffraction by plane electromagnetic waves (Enriquez et al., 2011) and Gaussian beams (Mata et al., 1993, 1994) by two slits in metallic screens of zero thickness. However, these theories do not consider Hermite-Gauss beams or oblique incidence, nor thick screens with nonzero thickness (Mata et al., 2008). In this paper we present a novel rigorous theory of diffraction that allows to consider the illumination by Hermite-Gaussian beams at oblique incidence onto two nano-slits of width  and separation  in screens with infinite conductivity and thickness ℎ. In particular, we analyze the coupling between slits through the numerical study of the diffracted energy along the direction of the incident ( ௜ ) beam energy as a function of the parameter of separation  between the slits. It is revealed the existence of oscillations in the energy  ௜ . We also show that in the case of TM(S) polarization, the energy  ௜ is special because when compared to other diffraction patterns. Finally, we show that the scalar property valid at the scalar region (λ/ <0.2)  ௜ = / (Alvarez-Cabanillas, 1995) is not longer valid. 2. A Novel Vector Theory of Diffraction In Figure 1 we have two slits on a screen of infinite conductivity, and non-zero thickness denoted by ℎ. In this screen you have two parallel to the Oz axis,  wide and spaced slits . The display is in the gap and impinges perpendicularly on it a Hermite-Gaussian beam with wavelength  = 2/ and order m. We will use the complex representation for the fields and omit the time factor going forward  ௜ఠ௧ .  is the magnetic field when you have the TM (magnetic field parallel to the axis Oz) polarization. The  field satisfies the Helmholtz equation (Mata et al., 1994) ∂ ଶ H/ ∂x ଶ +∂ ଶ H/ ∂y ଶ +k ଶ H=0 (1) Denote by H I the solution of Eq (1) in the region I ( > ℎ / 2), expressed by a plane wave expansion: H  (x, y) = ଵ √ଶ஠ ׬ A(α) ୩ ୩ e ୧(஑୶ஒ୷) dα + ଵ √ଶ஠ ׬ B(α) ୩ ୩ e ୧(஑୶ାஒ୷) dα (2)