Accordance of energy conservation law of differential equations which describe light propagation in nonabsorbent mediums of periodical structures

摘要

Systems of linear differential equations with constant coefficients which describe light diffraction on transparent periodical mediums, obtained basing on Maxwell equations are presented in this work. Respective equations for H-mode-polarization and E-mode-polarization of electromagnetic waves, in which electric field strength of a wave is the main variable, were found. Equations for E-mode-polarization are more complicated than for H-mode-polarization by their structure. All equations were examined for accordance to energy conservation law, which means that divergence of Poynting-vector in a grating should equal zero. Solutions obtained from systems of equations for H-mode-polarization by parabolic approximation and by taking into account the second derivative were compared. We discovered that for thick gratings, if variable component of refraction index is less than 0.01, error at parabolic approximation is not considerable and it can be neglected.