Journal of Russian Laser Research, Volume 38, Number 4, July, 2017 PHASE EFFECTS OF THE PARAMETRIC INTERACTION IN METAMATERIALS R. J. Kasumova, 1* Z. H. Tagiyev, 2 Sh. Sh. Amirov, 2,3 Sh. A. Shamilova, 1 and G. A. Safarova 1 1 Physics Department, Baku State University Baku, Azerbaijan 2 Azerbaijan Medical University Baku, Azerbaijan 3 Department of Electronics, Telecommunications, and Radio-Engineering Khazar University, Baku, Azerbaijan * Corresponding author e-mail: renajkasumova @ gmail.com Abstract We study the phase effects of the three-wave parametric interaction in metamaterials considering the negative refraction at the frequency of a signal wave. We analyze the efficiency of energy conversions between two direct waves with respect to the energy of the backward signal wave along with the dyna- mics of the signal-wave amplification in metamaterials. We show that there exist optimum values of the fundamental-wave intensity and the phase mismatch at which the efficiency of conversion is maximum. We obtain an analytic expression for the optimum value of the relative length of the metamaterial and present a numerical evaluation of the expected value for efficiency of the frequency parametric conversion in dielectric waveguides. A sufficient enhancement in the signal-wave amplification, which is possible at optimum values of the pump intensity and the metamaterial total length, leads to the parametric generation of the signal wave. Changing the frequency and power of the pump wave, one can realize a regular tuning of the frequencies of parametric converters. Keywords: metamaterial, parametric interaction, constant-intensity approximation. 1. Introduction Recently, intense researches were carried out with artificially created structures – metamaterials, which are attractive due to their unusual properties [1–3]. There are various experimental approaches to create these structures. At the initial stage of investigations, the metamaterials were considered to be periodic structures incorporated into the original material [4, 5]. However, higher losses in fabricated structures hinder progress in applications of the metamaterials. The next stage was to use the latest achievements in nanoplasmonics, namely, the results of studies of the electromagnetic field around the localized plasmons [6]. Since nonlinearities of plasmonic metals like silver [7] or gold [8] sharply in- crease near the resonances, their inserting into dielectric structures leads to a significant increase in the electromagnetic-field concentration in the vicinity of nanoparticles. So in the case of gold nanoparticles, because of a low work function of electrons from gold to silicon, the latter is used as a dielectric structure. Manuscript submitted by the authors in English first on April 3, 2017 and in final form on June 5, 2017. 1071-2836/17/3804-00211 c  2017 Springer Science+Business Media New York 211