Diffraction of a Gaussian Beam with Limited cross Section by a Volume Phase Grating under Waveguide Mode Resonance

Abstract

In this work, the diffraction of a Gaussian beam on a volume phase grating was researched theoretically and numerically. The proposed method is based on rigorous coupled-wave analysis (RCWA) and Fourier transform. The Gaussian beam is decomposed into plane waves using the Fourier transform. The number of plane waves is determined using the sampling theorem. The complex reflected and transmitted amplitudes are calculated for each RCWA plane wave. The distribution of the fields along the grating for the reflected and transmitted waves is determined using inverse Fourier transform. The powers of the reflected and transmitted waves are determined based on these distributions. Our method shows that the energy conservation law is satisfied for the phase grating. That is, the power of the incident Gaussian beam is equal to the sum of the powers of the reflected and transmitted beams. It is demonstration of our approach correctness. The numerous studies have shown that the spatial shapes of the reflected and transmitted beams differ from the Gaussian beam under resonance. In additional, the waveguide mode appears also in the grating. The spatial forms of the reflected and transmitted beams are Gaussian in the absence of resonance. It was found that the width of the resonance curves is wider for the Gaussian beam than for the plane wave. However, the spectral and angular sensitivities are the same as for the plane wave. The resonant wavelengths are slightly different for the plane wave and the Gaussian beam. Numerical calculations for four refractive index modulation coefficients of the grating medium were carried out by the proposed method. The widths of the resonance curves decrease with the increasing in the refractive index modulation. Moreover, the reflection coefficient also increases.

Features of the Resonance in a Rectangular Dielectric Surace-Relief Gratings Illuminated with a Limited Cross Section Gaussian Beam

“…Let us consider the case when the beam propagates in air at the angle and it is incident on the infinite grating at the angle in the medium with the refractive index ( Figure 1 ). Grating parameters were selected based on the result of several our previous numerical experiments [ 10 , 22 , 23 ]. Moreover, structures with such parameters can be fabricated experimentally.…”

“…Particularly, it is a laser beam where the intensity distribution in the cross-section is described by the Gaussian function. The result of numerical analysis depends significantly on the beam cross-section size under resonance, as shown in [ 23 , 24 ]. Reflection coefficient of the resonant structure decreases and the FWHM increases when the beam width decreases.…”

“…Reflection coefficient of the resonant structure decreases and the FWHM increases when the beam width decreases. However, the angular and spectral sensitivities practically do not change [ 23 ]. The method described in [ 23 ] is based on the representation of limited beam in the form of the decomposition into the plane waves using the Fourier transform and RCWA [ 19 , 20 ] for calculations.…”

“…However, the angular and spectral sensitivities practically do not change [ 23 ]. The method described in [ 23 ] is based on the representation of limited beam in the form of the decomposition into the plane waves using the Fourier transform and RCWA [ 19 , 20 ] for calculations. The field distribution of the reflected and transmitted beams is determined by the inverse discrete Fourier transform.…”

“…Therefore, using the finite diameter light beam diffraction on the grating [ 23 ] based on the RCWA method and the sampling theorem [ 25 ] will provide additional information regarding the nature of the resonance phenomena, in the dielectric grating system given in the work [ 12 ]. The RCWA is asymptotically accurate [ 20 ] and converges faster than the other methods for dielectric periodic structures.…”