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It is known that the beam splitter in the form of coupled waveguides (BS) is one of the main devices used in quantum optics and quantum technologies. A BS has two independent parameters: one is the reflection coefficient  R  or the transmission coefficient  T , where    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$R+T=1$$\end{document}    R  +  T  =  1; the second is the phase shift    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\phi $$\end{document}   ϕ. In various applications of quantum optics, these coefficients are considered constant. This is due to the fact that the frequency dependence of these coefficients is usually not taken into account, or this dependence is such that it cannot affect the constancy of these coefficients. It is shown that the coefficients  R ,  T  and phase shift    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\phi $$\end{document}   ϕare generally values that depend on the frequencies of incoming photons, the interaction time of photons in the BS, and the type of BS. It is established that in general,  R ,  T  and    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\phi $$\end{document}   ϕcannot be considered constant coefficients, and the criteria for when they can be considered constant are defined. The results obtained must be taken into account when analyzing and planning experiments where the beam splitter is presented in the form of coupled waveguides.

Theory of a frequency-dependent beam splitter in the form of coupled waveguides