to the Weierstrass Institute for Applied Analysis and Stochastics, Germany

In all existing mode-locked lasers coherence time of the medium $T_2$ is much smaller than the pulse duration, so the ultimate limit on the laser pulse duration is set by the bandwidth of the gain medium and medium polarization is adiabatically eliminated [1]. Another situation arises when the interaction between light and matter is coherent [2-3], i.e. pulse duration is smaller than the medium coherence time $T_2$. In this case it is possible to generate optical pulses from a laser with spectral bandwidth larger than active medium bandwidth. Stabilization mechanism of pulse formation in this case is related to $\pi$ pulse formation in a gain medium and $2\pi$ pulse of self-induced transparency in an absorbing medium [4-5]. In the previous study of coherent passive mode-locking absorber and gain medium were implemented in the same sample [4-5]. In the real laser systems gain and absorber medium are separated in space. In the present work perform a more detailed numerical analysis of passive mode-locking in a two-section (or single-section) laser including coherent effect when absorber and gain medium are placed separately in the cavity. We demonstrate numerically the possibility of mode-locked pulse formation in a two-section and in a single section. We perform numerical simulations for the ring and linear cavities and compare the results. Our numerical analysis is based on the system of Maxwell-Bloch equations for slow envelopes of the polarization of two-level gain and absorbing medium, placed in the ring optical cavity, the population difference of the medium and slow envelope of the electric field. This model allows to perform sufficiently complete modeling of evolution of an extended two-level medium in a cavity, taking into account multi-mode character of radiation and the nonlinear coherent effects accompanying light interaction with two-level medium [2-3]. It has been shown that in this case the coherent mode-locking can occur in a two-section laser with a ring cavity in the case when generating is unidirectional. The duration of the laser pulse can be significantly smaller than the relaxation time of the polarization of gain medium, i.e., pulse spectrum broader than the gain bandwidth of the laser medium. In these examples when the transition dipole moment of the absorber medium increases pulse duration can decreased until hundreds of fs. Pulse energy in this case is keeping approximately constant. Finally we considered the examples of the different scenarios of the pulse propagation inside of the laser with a linear cavity in a two-section laser when coherent mechanism of mode-locking is realized, and without coherent mechanism. These examples show that in the linear resonators mode-locking regimes in which coherent mechanisms play a significant role are possible. All examples mentioned above illustrate the role of coherent interaction in the amplifier and absorber. This is the formation of $\pi$ - like pulses in the amplifier, which are for the absorber $2\pi$-like pulses. 1. H. A. Haus, “Mode-locking of lasers”. IEEE Journal of Selected Topics in Quantum Electronics, vol.6, pp. 1173-1185, 2000. 2. P. G. Kryukov and V. S. Letokhov, “Propagation of a light pulse in a resonantly amplifying (absorbing) medium”, Physics-Uspekhi, vol. 12, pp. 641–672, 1970. L. Allen and J. H. Eberly. Optical resonance and two level atoms. Wiley, New York, 1975. 3. V. V. Kozlov, “Self-induced transparency soliton laser via coherent mode locking”, Physical Review A, vol. 56, pp. 1607–1612, 1997. 4. M. A. Talukder, C. R Menyuk, “Analytical and computational study of self-induced transparency mode locking in quantum cascade lasers”, Physical Review A, vol. 79, p. 063841, 2009.