Reduction of Doppler Width

If the source diameter is small compared with the slit width b and if b d (which means 1), the flux density behind the slit B is approximately constant across 

The density n(v)dv of molecules with velocities v = v inside the interval v to p + du in a molecular beam at thermal equilibrium, which effuses with the most probable velocity Pp = (2kr/m) / into the z-direction, can be described at the distance r = (z + from the source A as 

If the collimated molecular beam is crossed perpendicularly with a monochromatic laser beam with frequency ty propagating into the x-direction, the absorption probability for each molecule depends on its velocity component Vx. In Vol. 1, Sect. 3.2 it was shown that the center frequency of a molecular transition, which is coo in the rest frame of the moving molecule, is Doppler shifted to a frequency q according to 

Only those molecules with velocity components Vx in the interval Avx = hco jk around Vx = (co - coo)/k essentially contribute to the absorption of the monochromatic laser wave, because these molecules are shifted into resonance with the laser frequency co within the natural linewidth Scon of the absorbing transition. 

When the laser beam in the x-z-plane (y = 0) travels along the x-direction through the molecular beam, its power decreases as 

Let us assume molecules effusing into a vacuum tank from a small hole A in an oven which is filled with a gas or vapor at pressure p (Fig.9.1). The molecular density behind A and the background pressure in the vacuum tank shall be sufficiently low to assure a large mean-free path of the effusing molecules, such that collisions can be neglected. The number N( ) of 

The absorption cross section a((jj,v ) describes the absorption of a monochromatic wave of frequency w by a molecule with a velocity component v. Its spectral profile is represented by a Lorentzian (see Sect.3.6) 

The density n(v, x)dv of molecules with velocity components v in the interval dv at a point (x,z) in the molecular beam can be derived from (10.2). 

---

The exact determination of such a small splitting requires molecular beam techniques. The molecules are cooled in a supersonic expansion which leads to a population distribution among the rotational states corresponding to a Boltzmann distribution with 10-12 K. This leads to a considerable reduction of populated states and thereby to a simplification of the spectrum. In a supersonic expansion we find a considerable decrease in the line width due to the decrease in translational motion, especially when the laser is perpendicular to the expansion direction. The corresponding reduction in Doppler width allows an improved frequency resolution, which will later be shown to be crucial for the success of the experiment. 

A reduction of the Doppler width by a factor ItP is attainable. As an example, the barium resonance at 535 nm would, with an ion source temperature of 2000K and an acceleration of 60keV, have a residual Doppler width of about 1 MHz, compared with the natural linewidth of 19 MHz. In practice the experimental linewidths are 10-50 MHz depending on the natural linewidth, ion source, acceleration voltage stability and beam optics. 

The practice of cooling discharge tubes in liquid air allowed some reduction in the Doppler width, but this technique was not always fully exploited since it was common to use high current densities in order to obtain a bright atomic spectrum. A considerable advance wras made after the discovery of deuterium in 1932, since the Doppler width of these lines compared with those of the light isotope is smaller by the factor (see Fig. 3). 

Of a number of studies of this line since the war, the best resolution was obtained by Series [122]. Reduction of therf Doppler width was again achieved by liquid hydrogen cooling of a discharge as gentle as possible. Spectroscopic analysis was by means of multiple Fabry-Perot interferometers, which allow an extension of spectral range without sacrifice of resolving power. 

Particularly for polyatomic molecules with their complex visible absorption spectra, the reduction of the Doppler width is essential for the resolution of single lines [392]. This is illustrated by a section from the excitation spectrum of the SO2 molecule, excited with a single-mode frequency-doubled dye laser tunable around X = 304 nm (Fig. 4.4b). For comparison the same section of the spectrum as obtained with Doppler-limited laser spectroscopy in an SO2 cell is shown in Fig. 4.4a [391]. 

Ezekiel and coworkers [923] used the reduction of the Doppler width in a collimated molecular beam (Sect. 4.1) for accurate heterodyne spectroscopy. The beams of two argon lasers intersect the collimated beam of I2 molecules perpendicularly. The laser-induced fluorescence is utilized to stabilize the laser onto the centers of two hfs components of a visible rotational transition. The difference frequency of the two lasers then yields the hfs splittings. 

Doppler broadening of the emission lines will be much reduced. The same principle for line width reduction is used in absorption measurements, in which absorption from a continuous wavelength distribution by atoms is recorded at right angles to the collimated beam. The resulting Doppler width when atoms with a preferred direction of motion are used can be calculated in a similar way as for the case with evenly distributed directions of motion. In practical cases, linewidths of tens of MHz are obtained in optical absorption measurements on atomic beams at thermal velocities (a few hundred m/s). 

In the examples considered so far, the laser beam was crossed perpendicularly with the molecular beam, and the reduction of the Doppler width was achieved through the limitation of the maximum-velocity components v by geometrical apertures. One therefore often calls this reduction of the... 

For the example of Na atoms given above, the natural linewidth (10 MHz) is about 100 times smaller than the Doppler width a t 500 K. If optical cooling reduces the Doppler width by a factor of 100 this implies a reduction of... 

A third method, which comes more and more into use, is based on various optical double resonance techniques and requires generally at least two tunable lasers. The essential point of this method is the labelling of a selected molecular level by optical pumping with a "pump laser". A second "probe laser" is tuned through the spectral range of interest, but only those molecular transitions are monitored, which start from the "labelled" level, selected by the pump transition. The method works with pulsed and with cw lasers. Combined with one of the sub-Doppler techniques a drastic reduction of the line density as well as a reduction of the line width can be achieved. This makes double resonance techniques very attractive. 

---