Bennet hole

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 practice this condition may be fulfilled not only in excitation, e.g. by means of a pulsed laser or a continuous dye laser with insufficient frequency selectivity, but also by means of fines from a continuous gas laser working in simultaneous axial mode u>i (multimode) generation regime see Fig. 3.10(a). Let Au>i = u>i+1 — uii = itc/L denote the mode separation in a laser, L being the resonator length. Then, as pointed out in [110, 127, 231], broad line approximation works if Awj is smaller than the width of the Bennet holes r en [268, 320] in the absorption contour see Fig. 3.10(6). The positions of the Bennet holes are determined by the condition ujq — w/ + kv = 0, where luq is the central transition frequency, k is the wave vector and v is the velocity of the absorbing particle. The broad fine approximation is valid if the following conditions are fulfilled (see Fig. 3.10) ... 

Fig. 2.4 Velocity-selective saturation of a Doppler-broadened transition (a) the Bennet hole in the lower- and the Bennet peak in the upper-state population distribution Ndv ). (b) Saturated absorption profile when the saturating laser is tuned across the Doppler-profile of a molecular transition dashed curve)...

Eq. (2.29) illustrates a remarkable result Although at each frequency co the monochromatic laser burns a Bennet hole into the velocity distribution N vf)y this hole cannot be detected just by tuning the laser through the absorption profile. The absorption coefficient... 

The Bennet hole can, however, be detected if two lasers are used ... 

Fig. 2.6 Saturation of an inhomogeneous line profile (a) Bennet hole and dip produced by a monochromatic running wave with co coq (b) Bennet holes caused by the two counterpropagat-ing waves for co coq and for o) = (joo dashed curve) (c) Lamb dip in the absorption profile o s( )...

Note The width of the Lamb dip in (2.35a) is Scouo = This corresponds, however, to the velocity interval Ai = 2y /k because the opposite Doppler shifts Act) = (coo — (o) = of the two Bennet holes add when the laser frequency co is tuned. 

Fig. 9.2 Generation of the recoil doublet of Lamb dips (a) population peaks in the upper-state population for coo (b) Bennet holes in the lower-state population (c) recoil doublet in the output power Pl(co) of the laser...

When the absorbing molecules with the resonant absorption frequency coq are placed inside the laser resonator, the standing laser wave of frequency co coq burns two Bennet holes into the population distribution Ni (Vz) (Fig. 9.2b and Sect. 2.2), which, according to (9.3), appear at the velocity components... 

The velocity-selective minimum in the velocity distribution AN(vz) at i , = co — coo)/k, which is often called a Bennet hole [7.1], has the homogeneous width (according to Sect. 3.6)... 

Because of saturation the population density n (v2,)dv of the absorbing subgroup in the interval dv = 6ui/k around v = (co - 0)22)/k decreases while that of the upper state E2 increases correspondingly. This causes a dip in the population distribution n2(v ) (Bennet hole) and a corresponding peak in the distribution n2(v ) of the upper state (see Figs.3.16,17)... 

Fig.3.17. Saturation of an inhomogeneously broadened transition. Generation of a Bennet hole in the lower and a population peak in the upper level...

This is drastically changed when besides the saturating light wave, a second wave is used which can probe the Bennet hole burned by the saturating wave. We shall assume the probe wave to be sufficiently weak for saturation by the probe to be negligible, and that the saturating wave has a given frequency u)g. The absorption coefficient a(u)) for the probe wave is then... 

This is a Voigt profile with a Bennet hole centered at w - (see Fig.3.18). The halfwidth 6o) of this hole is determined by the saturated homogeneous linewidth = yA + Sq and the depth 4Sq/y is also determined by the intensity of the saturating field and the Einstein coefficient B 2 transition [3.19]. 

Due to saturation, both waves burn a Bennet hole into the population distribution nj(v ) which are located symmetrical to v = 0 at V2= (o)j2 " because of the opposite Doppler shift for both waves (Fig.3.19a). If the laser frequency u is tuned across the Doppler width of the absorption line, both holes coincide at u = 03 2 - v = 0. At w = 03 2 intensity I = 1 + 2 of the saturating wave is twice the intensity 1 or I2 and the Bennet hole... 

Fig,10,17. (a) "Hole burning" in the lower level population distribution ni(V2) of an absorbing transition and generation of a corresponding population peak in the upper level, (b) Increase of Bennet hole width with increasing saturating intensity... 

The spectral width y of this "Bennet hole" [10.28] represents the homo-g.eneous linewidth of the molecular transition and is, as mentioned above. 

Fig.10.18. (a) Bennet hole burned symmetrically into the velocity distribution n(vz) by two counterpropagating waves of frequency co + coq. 

In order to detect this Bennet hole, which has been burned into the population distribution n (v ) by the so-called pump wave, a second light wave, called the probe wave has to be sent through the sample to probe the population depletion in the hole. This probe wave may be either split from the same laser which provides the pump wave, or it may also come from another laser. 

This causes a Bennet hole in the population distribution n (v ) (see Fig. 10.17). When the second laser is tuned over the absorption profile of the transition a c, the Bennet hole causes a decrease in absorption of this laser. This can be monitored for instance by the corresponding decrease of the fluorescence intensity on the transition c m (Fig.10.31). 

Since the two transitions are coupled only by those molecules within the velocity range Av which have been pumped by one of the lasers, the doubleresonance signals show similarly small homogeneous linewidths as in saturation spectroscopy with a single laser. However, for precise spectroscopy the common Bennet hole should be exactly at the oentev of the population distribution and not anywhere around v = 0. This can be achieved for instance by using the Lamb dip, produced in the standing wave of the pump field, to stabilize the pump laser frequency w to the center frequency... 

Fig, 13,8. Recoil shift of Bennet holes in the lower state population (b) and of Bennet peaks in the upper state population (a). Recoil doublet in the output of a laser with internal absorption cell, generated by the shift of the Lamb dip against the Lamb peak... 

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