Theoretical Analysis of Coherent Passive and Self Mode-Locking in Lasers R.M. Arkhipov Weierstrass Institute for Applied Analysis and Stochastics Leibniz Institute in Forschungsverbund Berlin e. V. Berlin Germany arkhipov@wias-berlin.de M.V. Arkhipov Faculty of Physics, Saint-Petersburg State University Saint-Petersburg Russia arkhipm@mail.wplus.net Abstract—In the present work we perform numerical analysis of coherent mode-locking in lasers containing either coherent gain and absorbing medium or only one gain medium. Our numerical simulations indicate the possibility of mode-locked pulse formation when coherence time of the medium is larger than the pulse duration Keywords— coherent effects, passive mode-locking, self- induced transperency Passive mode-locking is a technique for generation of short optical pulses with high repetition rates [1]. Commonly in passive mode-locked lasers coherence time of the medium T 2 is much smaller than the pulse duration, so medium polarization is adiabatically eliminated [1]. Another situation arises when the interaction between light and matter is coherent, i.e. pulse duration is smaller than the medium coherence time T 2 . In this case coherent effects of pulse propagation in a resonant absorbing and amplifying medium can take place [2]. A well-known coherent phenomenon which was discovered by Mcall and Hahn is the self-induced transparency (SIT) [3]. This phenomenon arises when pulse duration is smaller than polarization relaxation time T 2 and leads to a soliton-like pulse propagation in a resonant medium. In the Ref. [4-5] theory of coherent mode-locking based on SIT in a 2 section laser was developed. It has been demonstrated that when the interaction between light and matter in the cavity is coherent mode-locked pulse in the absorbing medium is 2π pulse of SIT and in the gain medium it propagates as a π pulse. In [5] the authors proposed a quantum cascade lasers (QCLs) structure for experimental observation of the coherent mode-locking in QCL. In the [4,5] it was assumed that the electric field interacts with absorber and amplifier simultaneously, i.e. both absorber and amplifying “mixed” medium are homogeneously distributed in the laser cavity. This situation is however not entirely realistic but it can be applied to the designed structure of QCL proposed in [5]. In the real laser systems gain and absorber medium are separated in space. In the present work we perform a more detailed numerical analysis of passive mode-locking in a 2 and 1 section laser including coherent effects 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 laser (self- synchronization). 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 2- level gain and absorbing medium, 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]. Our numerical simulations indicate that when medium coherence time is too large, coherent mode-locked pulse arises. But the area of this pulse in absorbing medium is smaller than 2π, so the medium does not become completely inverted and then returns to the ground state during the pulse propagation as in the case of classical SIT. Moreover we have found that the pulse area in the gain medium is differs from π but is quite close to it. Finally our numerical simulations demonstrated that presence of the absorbing medium is unnecessary – stable coherent self mode-locking regime is preserved when absorbing medium is absent in the cavity. R.M. Arkhipov would like to acknowledge the support of EU FP7 ITN PROPHET, Grant No. 264687. R.M. Arkhipov would like to acknowledge Dr. I.A. Chekhonin and Dr. A.G. Vladimirov for helpful discussions. REFERENCES [1] H. A. Haus, “Mode-locking of lasers”. IEEE Journal of Selected Topics in Quantum Electronics, vol.6, pp. 1173-1185, 2000. [2] L. Allen and J. H. Eberly. Optical resonance and two level atoms. Wiley, New York, 1975. [3] S. L. McCall, E.L. Hahn, “Self-induced transparency”, Physical Review, vol. 183, pp .457, 1969. [4] V. V. Kozlov, “Self-induced transparency soliton laser via coherent mode locking”, Physical Review A, vol. 56, pp. 1607–1612, 1997. [5] 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.