785 ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2015, Vol. 121, No. 5, pp. 785–792. © Pleiades Publishing, Inc., 2015. Original Russian Text © N.V. Golovastikov, D.A. Bykov, L.L. Doskolovich, V.A. Soifer, 2015, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2015, Vol. 148, No. 5, pp. 899–907. Spatiotemporal Optical Pulse Transformation by a Resonant Diffraction Grating N. V. Golovastikov a,b , D. A. Bykov a,b *, L. L. Doskolovich a,b **, and V. A. Soifer a,b a Image Processing Systems Institute, Russian Academy of Sciences, Molodogvardeiskaya ul. 151, Samara, 443001 Russia b Samara State Aerospace University, Moskovskoe sh. 34, Samara, 443086 Russia * e-mail: bykovd@gmail.com ** e-mail: leonid@smr.ru Received May 7, 2015 Abstract—The diffraction of a spatiotemporal optical pulse by a resonant diffraction grating is considered. The pulse diffraction is described in terms of the signal (the spatiotemporal incident pulse envelope) passage through a linear system. An analytic approximation in the form of a rational function of two variables corre- sponding to the angular and spatial frequencies has been obtained for the transfer function of the system. A hyperbolic partial differential equation describing the general form of the incident pulse envelope transfor- mation upon diffraction by a resonant diffraction grating has been derived from the transfer function. A solu- tion of this equation has been obtained for the case of normal incidence of a pulse with a central frequency lying near the guided-mode resonance of a diffraction structure. The presented results of numerical simula- tions of pulse diffraction by a resonant grating show profound changes in the pulse envelope shape that closely correspond to the proposed theoretical description. The results of the paper can be applied in creating new devices for optical pulse shape transformation, in optical information processing problems, and analog opti- cal computations. DOI: 10.1134/S1063776115110138 1. INTRODUCTION The optical devices that perform temporal and spa- tiotemporal optical signal transformations are of great interest for a broad spectrum of applications, includ- ing optical information processing and analog optical computations [1]. Temporal and spatial differentiation and integration [1] are important analog optical signal processing operations. Various resonant structures, including Bragg gratings [1–9], resonant diffraction gratings [10, 11], and micro- and nanoresonators [12– 14], have been proposed to perform these operations. Using resonant structures to perform the temporal and spatial differentiation and integration stems from the fact that the Fano profile describing the shape of the reflection (transmission) coefficient for a diffrac- tion structure near resonance can approximate the transfer function of a differentiating or integrating fil- ter [10]. Note that the temporal and spatial incident beam transformations in the above papers were described separately. When the spatial operations were considered in [8, 9, 11], the incident beam was assumed to be monochromatic. When the temporal transformations were described in [2–6, 10, 13, 14], the spatial optical pulse structure was disregarded. Describing the spatiotemporal transformations of optical pulses (wave packets) by resonant diffraction structures is of great interest. Based on their numerical simulations of a 2D optical pulse diffraction on a res- onant diffraction grating, Vallius et al. [15] showed that the pulse envelope could undergo dramatic spa- tiotemporal changes. At the same time, there is no analytic description of the change in the spatiotempo- ral pulse envelope upon diffraction by a resonant structure in the known papers. In this paper, we show for the first time that the spatiotemporal transformation of the profile of an incident 2D pulse upon diffraction by a resonant dif- fraction grating can be described by a hyperbolic par- tial differential equation. The equation and its solution are given for the case of normal incidence of a pulse with a central frequency lying near the guided-mode resonance of a diffraction structure. The presented results of our numerical simulations of pulse diffrac- tion by a resonant grating using the Fourier modal method completely confirm our theoretical descrip- tion. 2. DIFFRACTION OF A SPATIOTEMPORAL PULSE BY A PERIODIC STRUCTURE Consider a 2D optical pulse normally incident on a diffraction grating (Fig. 1). Neglecting the dispersion ATOMS, MOLECULES, OPTICS