Prostiranje svetlosti u kompleksnim fotonskim rešetkama sa zasićujućom nelinearnošću

Abstract

U ovoj disertaciji teorijski i eksperimentalno su analizirani linearni i nelinearni efekti koji prate prostiranje vidljive laserske svetlosti kroz različite jednodimenzionalne (1D) fotonske rešetke napravljene od materijala sa zasićujućom (saturacionom) nelinearnošću, kao što je, na primer, litijum niobat (LiNbO3). Zbog nelinearnog odziva kakvim se odlikuju ovakvi materijali, moguće je formiranje stabilnih prostornih lokalizovanih struktura na granicama između dve uniformne rešetke istih, odnosno različitih perioda, kao i unutar binarnih superrešetki, kako u nelinearnom režimu, tako i u linearnom kada do lokalizacije dolazi u okolini defekta. U doktorskoj tezi razvijeni su odgovarajući matematički modeli i primenjene različite numeričke metode za dobijanje odgovarajućih rezultata, dok su neki od rezultata potvrđeni i eksperimentalno u nelinearnim 1D fotonskim rešetkama proizvedenim u LiNbO3. Korišćeni matematički modeli zasnovani su na sistemima spregnutih diferencno-diferencijalnih jednačina, tačnije 1D diskretnim nelinearnim Šredingerovim (Schrödinger) jednačinama sa nelinearnostima Kerovog (Kerr) i zasićujućeg tipa, dok je stabilnost dobijenih rešenja ispitivana primenom metoda linearne analize stabilnosti. Za rešavanje stacionarnih i dinamičkih jednačina izloženih u tezi, korišćene su numeričke metode pod nazivom Gaus-Njutnova metoda (Gauss-Newton) i Runge-Kuta (Runge-Kutta) metoda, respektivno. Dobijeni numerički i eksperimentalni rezultati pokazuju da prisustvo defekta utiče na formiranje lokalizovanih modova kako u nelinearnom, tako i u linearnom režimu, tj. pri malim upadnim snagama nedovoljnim za ispoljavanje nelinearnog odziva sredine. Rezultati pokazuju da pored pojave linearnih lokalizovanih stanja, u nelinearnom režimu dolazi do narušavanja simetrije lokalizovanih rešenja kada su ispunjeni odgovarajući uslovi. U slučaju prostiranja svetlosti kroz binarne superrešetke, rezultati ukazuju na postojanje novih tipova lokalizovanih rešenja, kao i otvaranje dodatnog procepa u zonskoj strukturi posmatrane rešetke. Pored ovoga, prikazani su i numerički rezultati dobijeni analizom međusobnih interakcija solitonskih rešenja, kao i ispitivanjem osobina modova lokalizovanih na površini rešetke.

In this dissertation the linear and nonlinear effects accompanying the propagation of visible laser light through a different one-dimensional (1D) photonic lattices made of material with a saturable nonlinearity, such as, for example, lithium niobate (LiNbO3) are both investigated, theoretically and experimentally. Due to the nonlinear response of these materials to the intensity of incident radiation, the light passing through the photonic lattice causes local change in the refractive index, providing necessary conditions for the formation of stable localized spatial structures at the interfaces between two uniform lattices of the same or different periods, and also within the binary superlattices, in the nonlinear regime, as well as in the linear regime when the localization occurs in the vicinity of the defect. In this dissertation, appropriate mathematical models are developed and various numerical methods for obtaining the relevant results are applied. Some of the results have been confirmed experimentally in 1D nonlinear photonic lattices fabricated in LiNbO3. Mathematical models are based on systems of coupled difference- differential equations, namely the 1D discrete nonlinear Schrödinger equation with the nonlinear term of Kerr and saturable type, while the stability of the obtained solutions was investigated using the method of linear stability analysis. To solve stationary and dynamic equations presented in this dissertation, numerical methods such as the Gauss-Newton and Runge-Kutta methods are performed, respectively. The obtained numerical and experimental results show that the presence of the defect affects the formation of localized modes in the nonlinear, as well as in the linear regime, i.e. for intensities insufficient for the manifestation of the nonlinear response of the medium. Furthermore, the results indicate to the possible appearance of the symmetry breaking of the mode profiles of certain nonlinear localized solutions when appropriate conditions are met. In the case of light propagation through binary superlattices, results show the emergence of new types of localized solutions and the opening of an additional gap in the observed energy spectrum of the lattice. Additionally, numerical results obtained by analyzing the mutual interaction of soliton solutions, as well as examining the properties of localized modes at the surface of the lattice, are presented as well.