The quantum ultraviolet catastrophe and the noise of an ultrashort pulse mode-locked laser

The Quantum \Ultraviolet Catastrophe" and the Noise of anUltrashort Pulse Mo de-Lo cked LaserY. Takushima, H. A. Haus andF. X. KartnerDepartment of Electrical Engineering and Computer Science andResearch Lab oratory of Electronics, Massachusetts Institute of Technology77 Massachusetts Avenue, Cambridge, Massacusetts 02139 USAABSTRACTThe ultraviolet quantum catastrophe of the zero-p oint uctuations in an L-C circuit of nite Q is revisited.The zero-p oint energy is shown to approach the value h!o=2 only in the limit of an in niteQ.AFabry-Perotresonator, on the other hand, has b ounded zero-p oint energies of its mo des that are equal to h!n=2. We analyzethe noise of an ultra-fast mo de-lo cked laser when the slowly varying envelope approximation (SVEA) is notvalid. It is found that quantum correlations exist that are not present in the SVEA. The correlations b ecomeevident in the sp ectrum of the zero-p oint uctuations.Keywords:ultraviolet catastrophe, zero-p oint uctuation, mo de-lo cked lasers, slowly-varying envelope ap-proximation1. INTRODUCTIONThe zero-p oint energy of a simple oscillator of niteQis in nite. A discussion of some of the physical dicultiesencountered in this context is in the b o ok of Gardiner and Zoller.1These authors compute the zero-p oint energyof a L-C circuit and nd that it diverges. This is a kind of \ultraviolet catastrophe" of an electromagneticresonator. The divergent term dep ends logarithmically up on the upp er limit of the integral over frequency.This divergence can b e avoided by simply cutting o the integral at some high cuto frequency. A consequenceis that the zero-p oint uctuation energy of the L-C circuit is not any more equal to h!0=2, where!=1pLC.In this pap er we take up the question of the ultraviolet catastrophe. We revisit the problem analyzed byGardiner and Zoller and show that the divergence disapp ears in the limit of in niteQ. Thus, at least in thislimit one may assign unequivocally a nite energy of h!0=2 to the resonator. However, the in nity still remainswhen theQis nite. In this case one may argue that no L-C circuit can b ehave as a lump ed circuit at allfrequencies. At frequencies at which the size of the circuit elements b ecomes comparable to a wavelength, theelements are not describ ed any more as an inductance and a capacitance. A natural cut-o frequency emerges.This is a p ersuasive argument. However, there is still a correction to the zero-p oint energy of a resonant circuitof niteQthat dep ends on the cut-o frequency.In order to obtain a b etter mo del of the zero-p oint uctuations of a resonator, we lo ok in detail at thezero-point energy in a Fabry-Perot resonator. Each resonance has assigned to it a bandwidth equal to the freespectral range. Weshow that the individual resonances of the Fabry-Perot have zero-p oint uctuations equalto h!n=2, where!is the resonance frequency, for any value ofQ.The resonator of a mo de-lo cked laser is of the Fabry-Perot type.The analyses of noise in mo de-lo ckedlasers used the slowly varying envelope approximation (SVEA), in which the same zero-p oint energy h!0=2was assigned to all axial mo des, where!0is the carrier frequency. In ultra-short-pulse mo de-lo cked lasers, thebandwidth approaches an o ctave. In this case, the di erence in the zero-p oint energies of the axial mo des hasto b e taken into account. We address this issue in this pap er. For the analysis of the quantum noise of anultra-short-pulse mo de-lo cked laser we use a lter with a rectangular transmission characteristic (such as anidealized Bragg mirror). This mo del avoids excitation at and near zero frequency, which would o ccur, if theusual (approximated) Lorentzian lter were used. The mo de-lo cking is assumed to b e active, as by injection ofan ultra-short pulse into a saturable absorb er inside the laser cavity.We nd that the phase uctuations andtiming jitter uctuations are correlated, as opp osed to the result of an analysis using the SVEA.