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We analyze theoretically a case of antiphase dynamics in the self-pulsing regime involving two orthogonal polarizations in intracavity second-harmonic generation. We show that, for this model, antiphase dynamics may lead to a nonreciprocal independence of the two polarizations as a result of partial overlap between the pulses. In the case in which two modes oscillate with one polarization and a single mode oscillates with orthogonal polarization, we find that the two modes can display chaos while the orthogonal mode remains periodic, despite coupling among all the modes.

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In a multimode laser operating near steady state, we determine analytically relations which connect the power spectrum density of each modal intensity and of the total intensity at the same frequency. We prove that, if the laser is in an antiphase regime, these relations become independent of the initial condition. This property rests on the existence of widely different time scales for the oscillation frequencies and their damping. Numerical simulations indicate that these relations remain true when a small amplitude modulation is applied to the control parameter.PACS numbers: 42.50. Ne, 42.55.Rz Recently, multimode lasers have been intensively studied as examples of spontaneous self-organized timeperiodic systems. This regime has been called antiphase dynamics (AD) in laser physics. It is a manifestation of the coherence property of nonsteady modal intensities that can be displayed by multimode lasers. It should not be confused with the electric field coherence of the single mode laser. AD has been reported in lasers in the case of spontaneous self-pulsing [1][2][3], in the presence of an external modulation [4,5], in the noise spectrum at steady state [6], in the transient relaxation to steady state [7,8], in the chaotic regime [9,10], and in the routes to chaos [11].A laser oscillating on N modes is characterized by N modal intensities I n ͑t͒, n 1, 2, . . . , N. The rate equation limit, where only model intensities and population inversion are coupled, applies to all the lasers in which AD has been reported up to now. For such lasers, the sum of the modal intensities SI͑t͒ P N n1 I n ͑t͒ is the total intensity. In the case of self-pulsing, AD means that each modal intensity is periodic, though with different phases and/or frequencies, but the total intensity is also periodic. When the dynamics is characterized by the relaxation frequencies, AD means that each modal intensity is driven by a number of frequencies (smaller than or equal to the mode number) while the total intensity is driven by only one frequency, the one which is related to the single mode frequency. The purpose of this Letter is to put forward yet another signature of AD by deriving universal relations between the power spectra of I n ͑t͒ and SI͑t͒. Universality in this context means that the relations are independent of the initial condition, i.e., of the preparation of the system. We shall first show that, under rather weakly constraining conditions a general relation can be found between the power spectrum of the total intensity, the modal intensities, and the intensity phases. We shall then use the known phase properties of the AD regime in two specific examples to reduce these relations to closed relations between the power spectra of the modal and the total intensities.

We investigate the properties of power spectra of a two-mode laser and predict that universal relations among the peaks of the power and noise spectra for different modes but the same frequency hold for deterministic and stochastic perturbations. These results are confirmed experimentally.PACS numbers: 42.50. Ne, 42.55.Rz In recent studies on multimode solid state free-running lasers (i.e., in the absence of either mode or phase locking), it has appeared that the peaks of the power spectra verify remarkable relations [1]. The purpose of this Letter is to present a study of the two-mode case for which additional and more explicit results can be obtained. We also present experimental results which confirm the theoretical analysis.Multimode free-running solid-state lasers can be described by the Tang, Statz, and deMars rate equations [2]e 2 dI n dtwhich couple the intensity of the N modes I n to the space average of the population inversion z 0 and the population gratings z n . The incoherent pumping is represented by w $ 1, the gain of mode n relative to the gain of the first mode is g n # 1, and e 2 is the photon lifetime divided by the atomic inversion lifetime. In agreement with experimental data [3], we have assumed that e is mode independent.The essential feature which will be exploited in this Letter is that e is a small parameter with typical values in the range 10 22 to 10 23 . In a previous paper [1], we have shown that the smallness of e can be used to derive universal relations for the power spectrum of such lasers. What makes this derivation possible is that the linear stability analysis around the steady state is governed by complex eigenvalues with different scaling for the real and imaginary parts. The imaginary parts, which determine the oscillation frequencies V j , are O͑1͞e͒ while their real parts, which determine the damping rates, are O͑1͒. The main result obtained in [1] is a connection between the peak of the power spectrum P͑I n , V j ͒ for mode I n at frequency V j and the peak of the power spectrum P͑SI, V j ͒ for the total intensity SI ϵ P N n 1 I n at the same frequencyThis result holds in the limit e ! 0. It involves the parameter w nmj u nj 2 u mj , where u nj is the phase of the nth component of the eigenvector associated with the eigenvalue l j whose imaginary part is V j as shown in [1]. In a number of situations, the phase difference may be independent of the preparation of the system. In this case the resulting power spectrum relation becomes universal.In the case of two modes, there are two frequencies V L , V R and g 2 ϵ g. It can be shown analytically that w 12L p 1 O͑e͒ and w 12R O͑e͒. This result indicates that the low frequency V L is associated with antiphase dynamics while the relaxation oscillation frequency V R is associated with inphase dynamics. Therefore the power spectra equalities becomeThe universality of these relations stems from the fact that they relate peaks of different modal intensities at the same frequency. Hence they express a relation between differe...