Time-of-Flight Broadening

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 time of flight T = d/v of molecules with a mean thermal velocity v through a laser beam of diameter d may be smaller than the spon- 

In such cases, the linewidth of a Doppler-free molecular transition is no longer limited by the spontaneous transition probabilities (see 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 cosw t which oscillates with constant amplitude during the time interval T and which then suddenly stops oscillating. Its frequency spectrum is obtained from the Fourier transform 

This example can be applied to an atom which traverses a laser beam with a rectangular intensity profile (see Fig.3.13a). The oscillator amplitude x(t) is proportional to the field amplitude E = EQ(r)coso)t. If the interaction time T = d/v is small compared to the damping time T = 1/y the oscillation amplitude can be regarded as constant during the time T. The halfwidth of the line in frequency units is then 6v v/d. 

The field distribution across a laser beam which oscillates in the fundamental mode is given by (see Sect.5.3) 

There are two possible ways of reducing the time-of-flight broadening one may either enlarge the laser beam diameter w, or one may decrease the temperature, thus reducing the molecular velocity v. Both methods have been verified experimentally and will be discussed in Chap.10. 

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Excitation of the gas by an evanescent wave introduces a pure imaginary z-component of the wave vector kf and, hence, an additional time-of-flight broadening of the fluorescence lines. The fluorescence line intensity is determined by the gas volume where the polarization corresponding to the contributions given by either Eq. (7.40) or Eq. (7.41) is essentially nonzero. Therefore, the intensity of emission at the frequency w is proportional to the EW penetration depth, 5, whereas that at the frequency mo is proportional to the polarization memory length, lx = ut/t (see Section 2.4.3). The latter line is thus dominant in the spectrum if <5 [Pg.189]

Fig.3.13a,b. Time-of-flight broadening Frequency distribution of the transition probability of an atom traversing a laser beam, (a) With rectangular intensity profile (b) with a Gaussian intensity profile... 

Calculate the minimum beam diameter which is necessary to bring time of flight broadening in problem 3.2c below the natural linewidth. 

F1q. 12,2, Halfwidth of the "Lamb peak" in the output of an He-Ne laser at X = 3.39 ym with an internal methane cell as a function of the CH4 pressure (lower curve). The upper curve shows pressure broadening in an external CH4 cell. The two different intersets are mainly due to the different laser beam diameters in the two cells, causing different time of flight broadening [12.4]... 

In this chapter we discuss several techniques which can reduce or even completely avoid time-of-flight broadening. Some of these methods have already been realized experimentally while others are only theoretical proposals which could not be proved up to now. These techniques allow ultrahigh resolution, in some cases even within the natural linewidth. This raises the interesting question about the ultimate resolution limit and the experimental or fundamental factors that determine such a limit. 

The problem of time-of-flight broadening was recognized many years ago in electric or magnetic resonance spectroscopy in molecular beams [13.1], In these Rabi-type experiments the natural linewidth of the radio-frequency or microwave transitions is extremely small [because the spontaneous transition... 

A considerable reduction of time-of-flight broadening could be achieved by realization of Ramsay s ingenious idea of separated fields [13.2]. The molecules in the beam pass two phase-coherent fie lds which are spatially separated by a distance x = L Ax = d large compared with the extension AX = d of each field (Fig.13.1). The interaction of the molecules with the first field creates a dipole moment of each molecular oscillator with a phase depending on the interaction time t = d/v and the detuning of the... 

Note that the requirement 60 < A/2d for molecules which experience the same phase in the first zone is equivalent to the condition that the residual Doppler width 6o)p in the absorption profile of molecules moving within the angular slice 59 does not exceed the time-of-flight broadening 6o)jp = irv /d. This can immediately be seen from the relations... 

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