Search for the optimal parameters of relaxing nonlinearity to obtain self-trapping of an ultrashort pulse in a photonic crystal Denis V. Novitsky * B. I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, Nezavisimosti Avenue 68, BY-220072 Minsk, Belarus We theoretically study the conditions for realization of trapping of a femtosecond pulse inside a one-dimensional photonic crystal with the relaxing cubic nonlinearity. A number of variants is considered: focusing and defocusing nonlinearities of the layers, a half-linear system, structures with differing values of nonlinearity parameters of periodically alternating layers. The results seem to be useful to make the optimal choice of the system characteristics to obtain self-trapping. I. INTRODUCTION A one-dimensional photonic crystal is a set of period- ically alternating layers with certain thicknesses and re- fractive indices. If the layers of such a structure consist of materials possessing non-resonant cubic (Kerr) nonlin- earity, one can observe pulse compression and stationary solitonic pulses formation in such photonic crystals at submillimeter distances [1, 2]. However, even the fastest (electronic) mechanism of Kerr nonlinearity is character- ized by relaxation times of femtosecond order. Therefore, considering interaction of ultrashort (femtosecond) light pulses with nonlinear photonic crystals, one has to take into account non-instantaneousness of nonlinearity. The necessity to consider nonlinearity relaxation was realized soon after the birth of nonlinear optics. From the end of 1960s scientists studied influence of relaxation in the framework of the Debye model on such effects as laser beam self-focusing [3–5], filament formation [6], paramet- ric amplification [7], and pulse compression [8]. In recent years much attention is given to investigation of mod- ulation instability in non-instantaneous nonlinear media [9–12], resulting in pulse train generation [13, 14], insta- bility of speckle patterns [15], solitary pulse reshaping [16]. However, there are only a few works on nonlinear optics of photonic crystals which take into account relaxation of nonlinearity of media composing a structure. Per- haps, this is connected with necessity to adjust different time parameters of the structure – nonlinearity relaxation time, pulse duration and pulse transmission time through the system. As a result, relaxation can be ignored in many cases. Here we can refer to several papers which are important for the present investigation. In Ref. [17] the possibility of compression of femtosecond light pulse in a one-dimensional photonic crystal with relaxing cu- bic nonlinearity was studied, the existence of two com- ponents (focusing and defocusing) of nonlinearity being significant. Further, it was shown [18] that, for identical nonlinearity of all the layers of one-dimensional periodic structure, one can observe not only decreasing of com- pression efficiency, but even pulse self-trapping at high * dvnovitsky@tut.by enough intensities. Pulse energy can be stored inside the system for a long time (more than 100 times longer than the pulse transmission time through the linear crystal), the shape of intensity distribution along the structure corresponding to the shape of the pulse. The process of light redistribution and radiation emission from the crystal appears to be very slow. If the intensity of the incident pulse is further increased, the most part of radi- ation is rapidly reflected due to pulse trapping near the very entrance of the photonic crystal. In Ref. [19] the effects of spectral transformations con- nected with self-trapping were studied. These transfor- mations were observed both in transmission and reflec- tion and depend on the regime of light interaction with the nonlinear system. In the regime of strong interac- tion (pulse is trapped near the entrance), quasimonochro- matic and continuum-like radiation inside the band gap appears in reflection. In the weak interaction regime (pulse is trapped near the exit from the crystal), the narrow peak near the edge of the forbidden gap can be observed in the spectrum of transmitted light. This paper is a logic continuation of Refs. [18, 19] and purposes the aim to study the possibility of ultrashort pulse self-trapping for the different variants of nonlinear- ity of the photonic crystal layers and to determine the optimal values of nonlinearity parameters for observation of this localization effect. II. THE MAIN FEATURES OF THE SELF-TRAPPING EFFECT Light propagation in a one-dimensional nonlinear peri- odic structure is described by the Maxwell wave equation ∂ 2 E ∂z 2 - 1 c 2 ∂ 2 (n 2 E) ∂t 2 =0, (1) with the dependence of refractive index on light intensity I = |E| 2 as follows, n(z,t)= n 0 (z)+ δn(I,t). (2) Here E is the electric field strength, n 0 (z) is the linear part of refractive index varying along the z axis, δn is the nonlinear part of refractive index and is governed by