摘要： Coexistence of multiple dynamical states of operation is known in multimode lasers. In this work, we examine the symmetry properties of various laser models based on the delay differential equation (DDE) approach [1], and explain a vast class of experimental observations in mode-locked semiconductor lasers. In these models, the complex electric field amplitude ( )E t is presented by the equation having the form: 1 (cid:16) (cid:74) (cid:5) E t ( ) exp( (1) where dot means differentiation with respect to time, (cid:74) is the width of a Lorentzian bandwidth limiting element T(cid:77)(cid:32) (cid:58) is the term accounting the frequency detuning (cid:58) between one of the (such as a filter or gain profile); cavity modes and the transmittance peak of the filter; T is the cold cavity round trip time; is a function describing gain, saturable losses or/and nonlinearity. ) i R t T E t T (cid:77) R t T(cid:16) E t ( ) (cid:32) (cid:16) (cid:16) (cid:16) (cid:14) ) ( ( ) ( ) The phase parameter (cid:77) is 2(cid:652)-periodic translationally symmetrical. Continuous variation of the phase parameter (cid:77) allows for ‘unwrapping’ and overcoming this periodicity, unveiling translational symmetry in the system. The translational symmetry explains the appearance of multiple coexisting branches in the system, and multistability of the invariant solutions. We study this effect for several different systems, namely, a mode- locked single-section semiconductor laser, a mode-locked semiconductor laser with optical feedback, and a figure-eight semiconductor laser with a nonlinear amplifying loop mirror. By means of numerical continuation technique, we show that variation of the frequency detuning is the key to unveiling translationally symmetrical branches of laser mode-locked regimes, and their multistability. The multistable operations can demonstrate either slightly different repetition rates or/and very different temporal pulse profiles varying from Gaussian pulses to square waves. In order to corroborate our theoretical results experimentally, we explore single-section semiconductor quantum dash (Qdash) laser, and demonstrate switching and coexistence of two mode-locked operation states. The mode-locked regime repetition rate decreases with the pump current as a result of a thermally induced increase of the cavity length (see Fig. 1(a)). At 150 mA, the rf spectrum jumps discretely by approximately 2 MHz. This is accompanied by a jump in the optical spectrum of approximately 2 nm. After the jump, the repetition rate starts to decrease again until the next jump. This change of the cavity length induces the variation of the longitudinal mode spacing and, therefore, detuning between the dominant cavity mode and the peak of the gain profile, physically corresponding to the parameter (cid:77) in the model. The numerical bifurcation diagram plotted for the repetition rate evolution of the mode-locked operation is shown in Fig. 1(b), and the corresponding time traces are given in Fig. 1(c). The translational symmetry is ubiquitous in DDE laser models, and possibly, more effects of multistability related to multimode laser operation can be explained in this way.