Broadening transit-time

In many experiments in laser spectroscopy, the interaction time of molecules with the radiation field is small compared with the spontaneous lifetimes of excited levels. Particularly for transitions between rotational-vibrational levels of molecules with spontaneous lifetimes in the millisecond range, the transit time T = df v of molecules with a mean thermal velocity v passing through a laser beam of diameter d may be smaller than the spontaneous lifetime by several orders of magnitude. 

In such cases, the linewidth of a Doppler-free molecular transition is no longer limited by the spontaneous transition probabilities (Sect. 3.1), but by the time of flight through the laser beam, which determines the interaction time of the molecule with the radiation field. This can be seen as follows consider an undamped oscillator x = xq cos co )t that oscillates with constant amplitude during the time interval T and then suddenly stops oscillating. Its frequency spectrum is obtained from the Fourier transform 

This example can be applied to an atom that traverses a laser beam with a rectangular intensity profile (Fig. 3.19a). The oscillator amplitude x(t) is proportional to the field amplitude E = Eo r) cos cot. If the interaction time T = d/v s small compared to the damping time 7 = l/y, the oscillation amplitude can be regarded as constant during the time T. The full halfwidth of the absorption line is then = S.bvfd Su v/d. 

In reality, the field distribution across a laser beam that oscillates in the fundamental mode is given by (Sect. 5.3) 

There are two possible ways of reducing the transit-time broadening one may either enlarge the laser beam diameter 2w, or one may decrease the molecular velocity v. Both methods have been verified experimentally and will be discussed in Sects. 7.3 and 14.2. The most efficient way is to directly reduce the atomic velocity by optical cooling (Chap. 14). 

So far we have assumed that the wave fronts of the laser radiation field are planes and that the molecules move parallel to these planes. However, the phase surfaces of a focussed Gaussian beam are curved except at the focus. As Fig.3.20 illustrates, the spatial phase shift A(f = x27t/A experienced by an atom moving along the r direction perpendicular to the laser beam axis z is with r = R2-(R-x)2 — x v /2R for x R 

A somce of line broadening which is fairly common in molecular beam studies, and often dominant in ion beam studies is transit time broadening. This arises when the interaction time between the electromagnetic radiation field and the molecule is limited by the time the molecule spends in the radiation field. The transit time t is equal to dN, where v is the molecular velocity and d is the length of the radiation field. A typical 

There is a second effect that causes a collisional narrowing of spectral lines. In the case of very long lifetimes of levels connected by an EM transition, the linewidth is not determined by the lifetimes but by the diffusion time of the atoms out of the laser beam (Sect. 3.4). Inserting a noble gas into the sample cell decreases the diffusion rate and therefore increases the interaction time of the sample atoms with the laser field, which results in a decrease of the linewidth with pressure [104] until the pressure broadening overcompensates the narrowing effect. 

Molecules in a molecular beam with thermal velocities = 5 x 10 cm/s passing through a laser beam of 0.1-cm diameter have the mean transit time T = 2 xs. 

For a beam of fast ions with velodlies w = 3 x 10 cm/s, the time required to traverse a laser beam with d = 0.1 cm is already below 10 s, which is shorter than the spontaneous lifetimes of most atomic levels. 

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The transit time broadening has been further reduced by installing a small aperture in front of the photomultiplier. Thereby, only atoms that travel close to the axis of the enhancement cavity contribute to the signal. 

The 328 nm light is then coupled into a linear enhancement cavity placed inside the vacuum system. The metastable ions will ultimately be focused through the centre of the resonant mode of the cavity where they will interact with the light. The waist size of the fundamental mode of the cavity is around 100 /rm, chosen to make the transit-time broadening roughly equal to the natural width of the transition. A cavity is a convenient way of providing the counter-propagating beams required for the Doppler-free excitation of the two-photon transition,... 

Despite such successes, it was obvious that only a continuous wave laser source can do justice to the extremely sharp 1S-2S transition. Production of intense cw radiation near 243 nm remained long an elusive goal. Satisfactory power levels of several mW were first achieved by B. COUILLAUD et al. [19] by summing the frequency of a 351 nm argon laser and a 790 nm dye laser in a crystal of KDP. In the first cw experiment with this source [11], the power in the observation cell was further enhanced with a standing wave build-up cavity. Fig. 3 shows two-photon spectra recorded in this way. Although the resolution is much superior to the earlier pulsed spectra, it remains limited to a few MHz by laser frequency jitter, collision effects, and transit-time broadening. A further at least millionfold improvement in resolution should ultimately be achievable. 

In ion spectroscopy, however, the ions are under acceleration between collisions and, depending on the condition, the resonance may not last for the whole collision interval. This fact introduces a new source of uncertainty broadening which I call here transit" broadening (the quotation mark is to discriminate this effect from the usual transit time broadening which is much smaller). 

For Lamb-dip spectroscopy with ultrahigh resolution, the output beam of the powerful laser is expanded before it is sent through the sample cell in order to minimize transit-time broadening (Vol. 1, Sect. 3.4). A retroreflector provides the coun-terpropagating probe wave for Lamb-dip spectroscopy. The real experimental setup is somewhat more complicated. A third laser is used to eliminate the troublesome region near the zero-offset frequency. Furthermore, optical decoupling elements have to be inserted to avoid optical feedback between the three lasers. A detailed description of the whole system can be found in [222]. 

Vtr 0.4p/L 2 MHz, while for the perpendicular intersection of a laser beam with diameter 2m = 1 mm, the transit time broadening Ptr = 400 MHz gives a non-negligible contribution. 

Fig. 8.2 Linewidth of the Lamb peak in the output power of a HeNe laser at X = 339 pm with an intracavity CH4 absorption cell and different beam waists of the expanded laser beam, causing a different transit-time broadening [976]...

Lamb peaks (inverse Lamb dips) at the line centers of the absorbing transitions (Sect. 2.3). The line profiles of these peaks are determined by the pressure in the absorption cell, by saturation broadening, and by transit-time broadening (Vol. 1, Sect. 3.4). Center frequency coq, linewidth Aco, and line profile Pl(co) are measured as a function of the pressure p (Fig. 8.2). The slope of the straight line Aco p) yields the line-broadening coefficient [977], while the measurement of coo p) gives the collision-induced line shift. 

Since such small splittings can only be observed if the width of the Lamb peaks is smaller than the recoil shift, all possible broadening effects, such as pressure broadening and transit-time broadening, must be carefully minimized. This can be achieved in experiments at low pressures and with expanded laser beam diameters [1117], An experimental example is displayed in Fig. 9.3. 

The transit-time broadening can greatly be reduced by the optical Ramsey method of separated fields. The best resolution of the recoil splittings has indeed... 

Example 9.14 Assume the atoms start with vqz = 5 m/s. Their upward flight time is then t = vo /g = 0.5 s, their path length is z = vot - gt /2 = 1.25 m, and their total flight time is 1 s. Their transit time through a laser beam with the diameter d= cra close to the culmination point is Ttr = 90 ms, and the maximum transverse velocity is v < 0.45 m/s. The transit-time broadening is then less than 10 Hz. 

The problem of transit-time broadening was recognized many years ago in electric or magnetic resonance spectroscopy in molecular beams [1253]. In these Rabi experiments [1254], the natural linewidth of the radio frequency or microwave transitions is extremely small because the spontaneous transition probability is, according to Vol. 1, (2.22), proportional to co. The spectral widths of the microwave or RF lines are therefore determined mainly by the transit time AT = d/v of molecules with the mean velocity v through the interaction zone in the C field (Fig. 5.10a) with length d. 

Although transit-time broadening is greatly reduced by the Ramsey technique, the quadratic Doppler effect is still present and may prevent the complete resolution of the recoil components. This may cause asymmetric line profiles where the central frequency cannot be determined with the desired accuracy. As was shown by Helmcke et al. [1269, 1277], one of the recoil components can be eliminated if the upper level Pi of the Ca transition is depopulated by optical pumping with a second laser. In Fig. 9.63 the relevant level scheme, the experimental setup, and the measured central Ramsey maximum of the remaining recoil component are shown. 

The transit time broadening for molecules passing with velocity v through a Gaussian beam with waist v is 50% = 2(v/w) V21n2 = 2.4 v/w. The transit time is ftr = w/v. 

Ultrashort pulses may be also used for vibrational spectroscopy with high frequency resolution. As a first example we have demonstrated FT-CARS of a supersonic expansion. Several advantages of the technique should be noted. The effect of transit time broadening can be eliminated. Artifacts via the nonresonant part of the third order susceptibility are negligible. A possible dynamic Stark effect during the excitation process does not influence the ns signal transient. Precise spectroscopic information is provided without narrow-band laser sources. 

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