Theoretical Analysis Of Injection-Locked Dye Lasers: An Application To The Generation Of Optical Pulse Sequences With Phase Control

Theoretical analysis of injection - locked dye lasers:An application to the generation of optical pulse sequences with phase controlFrank Spano and Warren S. WarrenDepartment of Chemistry, Princeton UniversityWashington Road, Princeton, New Jersey 08544AbstractA theoretical model based on the laser rate equations is used to describe injectionlocking in pulsed dye lasers. It is shown that, in general, the frequency of the injectionlocked pulse is not equal to the frequency of the cw light -an unexpected result since thetwo are related through stimulated emission. We discuss how to accommodate this feature inthe design of a specialized delay line used to create optical pulse sequences with variablephase shifts between the pulses.In a previous publicatonl we described an injection locked laser system used to produceoptical pulse sequences containing high -power (100 kw), subnanosecond (150 psec), transformlimited pulses, with adjustable phase shifts between then. These sequences are ideal forstudying fast relaxation processes in the condensed phase. The system operates by injectingacousto -optically modulated radiation from a ring dye laser into a simple cavity, which istransversely pumped by an atmospheric nitrogen laser. The resulting pulse has essentiallyall of its energy concentrated into the narrow injected frequency band of the ring laser,and its phase replicates the phase of the injecting field. A two -pulse sequence is createdby double pumping the dye cavity with two N2 lasers while simultaneously injecting two AOmodulated pulses which have a prescribed phase relationship between them.One major limitation of this system is that the time delay between pulses cannot besmaller than the risetime of the AOM. We have therefore devised a second technique wherebya single injection locked pulse is directed into an optical delay line provided by an un-equal arm Michaelson interferometer. The path length difference is actively stabilized towithin a small fraction of a wavelength by observing the fringe patterns formed by the in-jected ring laser light (which is essentially cw in this application). If the injected ringlight and the injection locked pulse have the same frequency, (intuitively, stimulatedemission must take place at the injected frequency) then a path length difference whichyields (for example) a dark fringe for the ring leser light would result in the formation oftwo pulses which are 180 degrees out of phase.To obtain the desired phase relation betweenthe pulses, stabilization of the fringe pattern of the injecting radiation is a very attrac-tive approach; interferometric stabilization by a single mode He -Ne lasers is commonplace,but this approach would only be useful here if it were possible to align the He -Ne beam andthe injection locked pulse to be collinear to within a microradian. This difficulty is notpresent in our configuration since the injection locking imposes collinearity between the cwinjected radiation and the final injection locked pulse (see Fig. 1).In fact, however, spectroscopic confirmations that the pulses do indeed retain the de-sired phase relation (such as looking for no fluorescence after two pulses with a 180°phase shift) seemed to give erratic results. One possible explanation of this occurrence isthat perhaps the frequency relationship between the injected cw ring laser field and thefinal injection locked pulse is not so simple. This relationship might depend on an uncon-trolled system parameter which can be different for different experimental runs. Previoustheories2,3,4 used to decribe injection locking in the pulsed regime are unable to accountfor a possible frequency difference. We have therefore developed a more realistic theorythat, unlike earlier theories, treates the injected electric field coherently.Figure 1 illustrates the main features of the problem. A coherent electromagnetic fieldat a frequency win is injected through the partially reflecting mirror M1 from the left.Mirror M2 is 100% reflecting. Eventually, a steady -state standing wave is established with-in a cavity of length L. The pump pulse is applied transversely, and its amplitude isassumed to have a Gaussian time dependence. The resulting evolution of the coherent elec-tric field is calculated using the following transport equations for the right and lefttravelling complex electric fields Er and Es respectively, and the rate equation for theexcited state population N1: