Quantum models for synchronously pumped type I optical parametric oscillators (SPOPO) are presented. The study of the dynamics of SPOPOs, which typically involves millions of coupled signal longitudinal modes, is significantly simplified when one considers the "supermodes", which are independent linear superpositions of all the signal modes diagonalizing the parametric interaction. In terms of these supermodes the SPOPO dynamics becomes that of about a hundred of independent, single mode degenerate OPOs, each of them being a squeezer. One derives a general expression for the squeezing spectrum measured in a balanced homodyne detection experiment, valid for any temporal shape of the local oscillator. Realistic cases are then studied using both analytical and numerical methods: the oscillation threshold is derived, and the spectral and temporal shapes of the squeezed supermodes are characterized.

We have developed a full multimode theory of a synchronously pumped type-I optical parametric oscillator. We calculate the output quantum fluctuations of the device and find that, in the degenerate case ͑coincident signal and idler set of frequencies͒, significant squeezing is obtained when one approaches threshold from below for a set of well-defined "supermodes," or frequency combs, consisting of a coherent linear superposition of signal modes of different frequencies which are resonant in the cavity.Optical parametric oscillators ͑OPO's͒ are among the best sources of squeezed ͓1͔, correlated ͓2͔, and entangled ͓3͔ light in the so-called continuous-variable regime. They have allowed physicists to successfully implement demonstration experiments for high-sensitivity optical measurements and quantum information protocols. In order to maximize the quantum effects, one needs to optimize the parametric downconversion process. This has been achieved so far by using either intense pump lasers or resonant cavities. Having in mind that the parametric process is an almost instantaneous one, femtosecond mode-locked lasers are the best pump sources in this respect, as they generate very high peak optical powers with high coherence properties. Furthermore, they minimize the thermal effects in the linear crystal which often hamper the normal operation of parametric devices. Mode-locked lasers have been already used extensively to generate nonclassical light, by either pumping a parametric crystal ͓4,5͔ or an optical fiber ͓6͔. However, in such singlepath configurations, perfect quantum properties are only obtained when the pump power goes to infinity. This is the reason why mode locking is often associated with Q switching and pulse amplification ͓7͔ in order to reach even higher peak powers, at the expense of a loss in the coherence properties between the successive pump pulses. In contrast, intracavity devices produce perfect quantum properties for a finite power: namely, the oscillation threshold of the device. It is therefore tempting to consider devices in which one takes advantage of the beneficial effects of both high peak powers and resonant cavity buildup. Such devices exist: they are the so-called synchronously pumped OPO's ͑SPOPO's͒. In a SPOPO the cavity round-trip time is equal to that of the pumping mode-locked laser, so that the effects of the successive intense pump pulses add coherently, thus reducing considerably its oscillation threshold. Such SPOPO's have already been implemented as efficient sources of tunable ultrashort pulses ͓8-13͔, and their temporal properties have been theoretically investigated ͓14-16͔. Let us mention that mode-locked OPO's have also been developed: in such devices, the cavity is resonant only for the signal modes and idler modes, and the pump pulses are not recirculating. Degenerate mode-locked OPO's have been used to generate pulsed squeezed light in the picosecond regime ͓5͔.In this Rapid Communication, we make a complete multimode quantum analysis of SPOPO's and show theoretic...