ИСТИНА

he process of spontaneous parametric down conversion (SPDC), in which the pumping wave of a frequency $\omega_p$ decays into the signal and idler modes of frequencies $\omega_s$ and $\omega_s$ obeying the frequency relations~\cite{Boyd}: % \begin{equation} \omega_p=\omega_s+\omega_i \label{eq:1} \end{equation} % is widely used to generate entanglement between a pair of optical modes. For quantum applications, several designs of the nonlinear optical elements supporting the SPDC process have been suggested. Among them are the domain structure design, used to achieve proper parametric wavelengths~\cite{Harris}, and the nonlinear waveguide design~\cite{NonlinearWaveguides} for confining the parametric modes in a small volume. In these two approaches the optical properties of the nonlinear medium are modulated in the longitudinal and the transverse dimensions, respectively, by means of which the required temporal and spatial spectra of the parametric fields are achieved. The optical communication point of view on the spatial modes of the quantum light requires spatial mode profile to match the fiber modes in order to communicate the quantum light in a lossless manner. However, the application of entangled optical states for local tasks may benefit from complex spatial spectrum structure, which is far from the ones suitable for optical communication. This is why the development of the SPDC sources with a broadband spatial spectrum is important~\cite{Kolobov}. In the present report I study the formation of two-mode entangled states in a micron scale thickness slab (Fig. 1) in which the SPDC process takes place. This geometry is capable to simultaneously support two kinds of modes, namely, the waveguide modes and the radiation modes~\cite{Waveguide}, whereby the spatial spectrum can be significantly enriched as compared with the traditional approaches. As a result, the two channels for entangled modes generation are accessible in the nonlinear slab in which the signal field and idler field profiles are given by~\cite{Leaky} % \begin{figure}[t] \includegraphics[width=0.8\columnwidth]{waveguide.eps}% This imports a EPS figure \caption{\label{fig:Figure} The SPDC process proceeding in a several micron scale thickness nonlinear slab can give rise to both the waveguide and the radiation kinds of modes (see expansion \eqref{eq:2}).} \end{figure} % % \begin{equation} \hat{E}^{\mu}(x,z)=\sum_{l=1}^N{}C_l^{\mu}(z)\hat{A}_l^{\mu}(x)+\int_{-\infty}^{+\infty}{}dq{}c^{\mu}(q,z){}\hat{a}^{\mu}(q,x) \label{eq:2} \end{equation} % where $\hat{A}^{\mu}_l(x)$, $\hat{a}^{\mu}(q,x)$ are the waveguide and radiation mode profile operators, respectively, $C_l^{\mu}(z)$, $c^{\mu}(q,z)$ are their expansion coefficients with $l$ and $q$ being the waveguide mode number and the transverse wave-vector of the radiation mode component, the superscript $\mu$ indicates the signal or idler modes, $N$ is the number of the waveguide modes supported by the slab; $x$ and $z$ are the transverse and longitudinal coordinate, respectively. Different scenarios of entangled mode formation in the slab is analysed. To quantify the amount of two-mode entanglement the criterion based on the quadrature components covariance matrices is applied. It has been shown that different types of modes carrying entanglement can be facilitated in the slab configuration, depending on the slab's material and thickness parameters. Possible applications of the two-mode entanglement carried by the radiation modes in the planar slab geometry are considered.