Chaotic Synchronization Of Single-mode And Multi-mode Laser Systems

This paper mainly discusses the chaotic synchronization of a linear array including N coupled single-mode and multi-mode lasers. The chaotic synchronization of intensities between N lasers is investigated. By using the analytic signal method and the Gaussian filter method, the relation between the phase synchronization of the laser field and the corresponding intensity synchronization is analyzed. It is shown that the chaotic synchronization of laser intensities is due to the stable message transmission of the phase synchronization. It causes the chaotic synchronization of laser intensities between lasers without direct coupling. The relations of the chaotic synchronization between the intensities and phases of N coupled single-mode and multi-mode lasers are obtained.Since the mid and late of 1990's, many scientists discovered the phenomenon of chaotic intensity synchronization in a system of two or three single-mode lasers both theoretically and experimentally. To utilize the chaotic synchronization of lasers to the communications and messages transmission, it is necessary to investigate the chaotic synchronization of N lasers. So the investigation of the chaotic synchronization of anarray of many lasers, such as laser system of 6, 7, 8, 9, ...... , and N lasers becomes themain content of this paper. In an array of N lasers, as long as the parameters of the lasers can be suitably adjusted, the chaotic synchronization of laser intensities can occur. The chaotic synchronization of laser intensities can occur between the N - th and the first lasers, the (N -1)- th and the second lasers, ...... , and so on. When the number Nis odd, the chaotic synchronization appears between pairs of lasers. There is nochaotic synchronization between two nearest lasers and between the middle and the other lasers. When the number N is even, the chaotic synchronization appears between -pairs of lasers. There is also no chaotic synchronization between two nearest lasers. The real phases of lasers in the array can show some hints of phase synchronization, but they cannot provide clear picture of phase synchronization between these lasers. By virtue of the analytic signal and the Gaussian filter methods in the Hilbert space, we analyze the phase synchronization of the nearest lasers. The numerical simulations of dynamical equations of N lasers indicate that there exists chaotic synchronization of intensitiesbetween pairs of lasers though there are no directly couplings between these lasers except for the center pair when N is even. The messages of the chaotic synchronization of intensities between these pairs of lasers can be transmitted by the stable phase synchronization between nearest lasers in the array.We extend the theoretical model of spatially coupled single-mode laser array to the system of two and three multimode lasers. In the multimode lasers system, mode competition exists in intensities between the modes of each multimode laser. The chaotic synchronization of the total intensities is based on the chaotic synchronization of the intensities in each corresponding mode. The messages of the intensity synchronization are achieved through the phase synchronization between lasers without direct coupling.The calculation of the Lyapunov exponents in the array of N lasers shows that the laser system is in the chaotic states when the largest Lyapunov exponents are greater than zero. The parameters in the laser system are in the chaotic states. From the power spectrum of laser intensity in the array, it is also seen that there is no peak in the power spectrum that corresponds to a particular frequency when chaotic synchronization appears in the laser system. There is broadband of the power spectra with flat shape in laser intensities appears. This means that the laser system is indeed in the chaotic state.