Lamb-dip

Pump and probe waves may also be generated by a single laser when the incident beam is reflected back into the absorption cell (Fig. 2.6). 

The saturated population difference in such a case of equal intensities I = h = t of the two counterpropagating waves with wavevectors = — 2 is then 

Inserting (2.33 and 2.23) into (2.34) yields in the weak-field approximation (5q 1) 

This represents the Doppler-broadened absorption profile a (o) with a dip at the line center co = coo (Fig. 2.6c), which is called a Lamb dip after W.E. Lamb, who first explained it theoretically [206]. For o) = cpq the saturated absorption coefficient drops to Q s( o) = (1 — o)- The depth of the Lamb dip is So = Bikl/(cys), 

I = II = I2 being the intensity of one of the counterpropagating waves that form the standing wave. For coo — co ys the saturated absorption coefficient becomes 0 s = Q o(l — 5 0/2), which corresponds to the saturation by one of the two waves. 

Inserting (7.27 and 7.17) into (7.28) yields in the weak-field approximation So 1) after some elaborate calculations [7.2] the saturated-absorption co- 

The Lamb dip can be understood in a simple, conspicuous way for (0 (00 the incident wave is absorbed by molecules with the velocity components Vz = +( n o=FKs/2)/, the reflected wave by other molecules with 

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Figure Bl.4.5. The Lamb dip spectrum of the CO 6-5 transition obtained with the Cologne THz BWO spectrometer. The dip is of order 30-40 kHz in width and the transition frequency is detennined to 0.5 kHz [M].

Figure 2.5 Typical (gaussian) absorption line showing a HWHM (half width at half maximum) of Av and a Lamb dip (dashed curve)...

Natural line broadening is usually very small compared with other causes of broadening. However, not only is it of considerable theoretical importance but also, in the ingenious technique of Lamb dip spectroscopy (see Section 2.3.5.2), observations may be made of spectra in which all other sources of broadening are removed. 

Figure 2.6 Three typical groups of molecules, with velocities (molecules 1 and 2), 0 (molecules 3-5), and —v (molecules 6 and 7) towards the source in a Lamb dip experiment...

Molecules such as 3,4 and 5 in Figure 2.6, which have a zero velocity component away from the source, behave uniquely in that they absorb radiation of the same frequency Vj-es whether the radiation is travelling towards or away from R, and this may result in saturation (see Section 2.3.4). If saturation occurs for the set of molecules 3, 4 and 5 while the radiation is travelling towards R, no further absorption takes place as it travels back from R. The result is that a dip in the absorbance curve is observed at Vj-es, as indicated in Figure 2.5. This is known as a Lamb dip, an effect which was predicted by Lamb in 1964. The width of the dip is the natural line width, and observation of the dip results in much greater accuracy of measurement of v es. 

Figure 9.24 Laser Stark spectrum of FNO showing Lamb dips in the components of the line of the ij vibrational transition. (Reproduced, with permission, from Allegrini, M., Johns, J. W. C. and McKellar, A. R. W., J. Molec. Spectrosc., 73, 168, 1978)...

A further feature of the spectmm in Figure 9.24 is the sharp spike at the centre of each P-shaped transition. The reason for this is that saturation of the transition has occurred. This was discussed in Section 2.3.5.2 in the context of Lamb dips in microwave and millimetre wave spectroscopy and referred to the situation in which the two energy levels involved, m(lower) and n(upper), are close together. Under these circumstances saturation occurs when... 

Figure 9.26 (a) Doppler line shape with a Lamb dip. (b) As in (a) but with modulation and phase-... 

This situation corresponds to the well-known saturation effect in the emission of most gas laser transitions, where, for the same reason, fewer upper-state molecules can contribute to the gain of the laser transition at the center of the doppler-broadened fluorescence line than nearby. When tuning the laser frequency across the doppler-line profile, the laser intensity therefore shows a dip at the centerfrequen-cy, called the Bennet hole or Lamb dip after W.R. Bennet who discovered and explained this phenomen, and W.E. Lamb 2) who predicted it in his general theory of a laser. 

In both cases (i. e. emission or absorption saturation) the halfwidth of this/Lamb dip is slightly dependent on laser power but mainly determined by the interaction time of the individual molecules with the standing light wave in the cavity. This time may be limited by the finite lifetimes rb of upper or lower states, by the average time l/aup between two disturbing collisions, or by the transit time Tt of the gas molecules across the laser beam. This last limitation becomes important at low pressures of the absorbing gas and for transitions between long-lived states (see Section IV.3). 

The important point is that the Lamb-dip widths for most visible and near infrared transitions at low pressures are several orders of magnitude smaller than the doppler widths and are therefore well suited for high resolution spectroscopy. When probing with a... 

Fig. 15. Lamb dip spectroscopy of Left side Stark spectrum (with...

When the probe is placed inside the resonator each line exhibits a Lamb dip with a halfwidth of 2.9 Mc/sec, which is mainly limited by inhomogeneities of the electric field (see Fig. 15 b). 

Line centers of a number of previously unresolved SF transitions in the fundamental Vj band have been observed by Rabinowitz et al. with this Lamb-dip technique. Measurements of the dip width at small saturation levels yielded a determination of the SFe-SFe cross-sections for phase interrupting collisions. 

With a monomode CO2 laser, delivering a constant power on any rotational line of either the 10.4 jtx or the 9.4 n band, Bordfe and Henry studied the shape of Lamb dip profiles in CO2 which became asymmetric when two rotational lines were allowed to oscillate simultaneously, since the gains are tightly coupled by rotational thermalization. 

The hyperfine structure of the R(127) rotational line in the 11-5 band of molecular iodine at X = 6328 A could be resolved by Lamb dip spectroscopy with a halfwidth of 4.5 MHz, and the influ-enie of isotopic substition and has been studied 38, 338a)k)... 

The idea of using the same medium as absorber and active material has been proposed and realized by several authors 340-343) Leg and Skolnick 40) used a neon gas discharge at low current and low pressure as saturable absorber inside the cavity of a He-Ne laser oscillating at X = 6328 A. The Lamb-dip halfwidth obtained was 30 Mc/sec compared to 1500 Mc/sec for the doppler line. The disadvantage of this arrangement is that the frequency of the neon transitions depends upon pressure and current 341) in the absorption cell, and this limits the stability and reproducibility of the Lamb dip center frequency. 

Lamb dip spectroscopy provides a very sensitive tool for studying small frequency shifts and broadening of spectral lines which normally would be undetectable because they may be small compared to the doppler width. These investigations yield information about collisions at low pressures, where the effect of far distant collisions is not suppressed by the more effective close collisions. This allows the potential between the collision partners at large intermolecular distances to be examined. 

The last two chapters discussed spectroscopic studies which used coincidences between laser lines and transitions in other atoms or molecules. These investigations have been performed either with lasers as external light sources, or inside the laser cavity. In the latter case coupling phenomena occur between the absorbing species and the laser emission, one example of which is the saturation effect employed in Lamb dip spectroscopy and laser frequency stabilization. This chapter will deal with spectroscopic investigations of the laser medium itself and some perceptions one may obtain from it. 

Fig. 6. Direct absorption, AM and FM spectra of the hyperfine components of the B-X R,(34)27-0 transition of I2 at 541 nm. Note the hyperfine Lamb-dips are observed in the absorption spectrum...

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