Fourier limit

The achievable pulse length is determined by the total number of modes that can contribute to the pulse. The broader the frequency comb the shorter the possible pulse length, ideally reaching the so-called Fourier limit. In fact, the spectral width is usually limited by the width over which the GVD and higher order terms can be compensated for by mode pulling [5,6]. Cavity modes that are outside this bandwidth are suppressed without the help of the Kerr-lens effect and do not oscillate. 

Fig. 6. Cyclotron signal of a single 12C5+ ion. The full width of the resonance is 20 mHz corresponding to a relative line width of 10-9. The measurement time is given by the Fourier limit to 80 s...

The advantage of pulsed lasers is the large photon flux during the pulse time AT, which allows the ionization of the excited molecules before they decay by relaxation into lower levels where they are lost for further ionization. Their disadvantages are their large spectral bandwidth, which is generally larger than the Fourier-limited bandwidth An > 1/AT, and their low duty cycle. At typical repetition rates of = 10 to 100 s and a pulse duration of AT = 10 s, the duty cycle is only lO- -lO- ... 

High-resolution CARS can be also performed with injection-seeded pulsed dye lasers [334, 351]. If the radiation of a single-mode cw dye laser with frequency o) is injected into the cavity of a pulsed dye laser that has been mode matched to the Gaussian beam of the cw laser (Vol. 1, Sect. 5.8), the amplification of the gain medium is enhanced considerably at the frequency co and the pulsed laser oscillates on a single cavity mode at the frequency co. Only some milliwatts of the cw laser are needed for injection, while the output of the single-mode pulsed laser reaches several kilowatts, which may be further amplified (Vol. 1, Sect. 5.5). Its bandwidth Av for pulses of duration At is only limited by the Fourier limit Av = 1 f(27tAt). 

The spectral resolution An of most time-resolved techniques is, in principle, confined by the Fourier limit Av = a/AT, where AT is the duration of the short light pulse and the factor a 1 depends on the profile / (i) of the pulse. The spectral bandwidth Av of such Fourier-limited pulses is still much narrower than that of light pulses from incoherent light sources, such as flashlamps or sparks. Some time-resolved coherent methods based on regular trains of short pulses even circumvent the Fourier limit Av of a single pulse and simultaneously reach extremely high spectral and time resolutions (Sect. 7.4). 

In principle, the lower limit ATniin of the pulse width is given by the Fourier limit ATmin = aI8v, where a 1 is a constant that depends on the time profile of the pulse (Sect. 6.2.2). The larger the spectral width of the gain profile is, the smaller ATmin becomes. In reality, however, the dispersion effects, which increase with Sv, become more and more important and prevent reaching the principal lower limit of AJniin- In Fig. 6.21 the achievable limit ATmin is plotted against the spec-... 

Similar to cw lasers, active media with a broad spectral gain profile can be used for pulsed lasers. With wavelength-selective optical elements inside the laser resonator, the laser wavelength can be tuned across the whole gain profile. However, the drawback is the widening of the pulse length AT with decreasing spectral width Av due to the principal Fourier limit AT > 27t/Av. 

Fig. 6.73 Autocorrelation signal SocG (r) for different pulse profiles without background suppression (upper part) and with background suppression (lower part) (a) Fourier-limited Gaussian pulse (b) rectangular pulse (c) single noise pulse and (d) continuous noise...

R. Seiler, T. Paul, M. Andrist, F. Merkt, Generation of programmable near Fourier-limited pulses of narrow band laser radiation from the near infrared to the vacuum ultraviolet. Rev. Sci. Instrum. 76, 103103 (2005)... 

With a short laser pulse of duration At, which has a Fourier-limited spectral bandwidth Amore than one energy level can be excited simultaneously if their energy separation AE < HAco. For simplicity, we restrict the discussion here to two atomic/molecular levels Ei and E2 (see Figure 2.10). The wave function of the excited species is now a linear combination of the wave functions t/, and j/2, the atom/molecule is said to be in a coherent superposition of the two states 11) and 2). 

In general, thermal radiation sources have the bandwidth hco, which is much larger than the Fourier limit Aco= l/T. Therefore, the finite interaction time imposes no extra limitation. This may change, however, when lasers are considered (Sects. 2.7.5 and 3.4). 

The spectral bandwidth of a single-mode pulsed laser with pulse duration AT is, in principle, limited by the Fourier limit, that is. 

When the output of a stable cw dye laser (Av 1 MHz) is amplified in three amplifier cells, pumped by a copper-vapor laser with a Gaussian time profile I t) with the halfwidth A/, Fourier-limited pulses with Av Cl 40 MHz and peak powers of 500kW can be generated. These pulses are wavelength tunable with the wavelength of the cw dye laser. 

The pulse width of most excimer lasers lies within 5—20 ns. Recently, long-pulse XeCl lasers have been developed, which have pulse widths of T > 300 ns [5.204]. They allow amplification of single-mode cw dye lasers with Fourier-limited bandwidths of Av < 2MHz at peak powers of R> 10 kW. 

A more reliable technique for achieving really Fourier-limited pulses is based on the amplification of a cw single-mode laser in several pulsed amplifier cells. The expenditure for this setup is, however, much larger because one needs a cw dye laser with a cw pump laser and a pulsed pump laser for the amplifier cells. Since... 

Its principle is illustrated in Fig. 6.20 The setup consists of two cw Tiisapphire near-infrared single-mode ring lasers with wavenumbers v and V2. The output radiation of these lasers is amplified by nanosecond pump laser pulses, resulting in amplified Fourier-limited pulses in the near IR. Tunable VUV radiation... 

We have already discussed quantum-beat spectroscopy (QBS) in connection with beam-foil excitation (Fig.6.6). There the case of abrupt excitation upon passage through a foil was discussed. Here we will consider the much more well-defined case of a pulsed optical excitation. If two close-lying levels are populated simultaneously by a short laser pulse, the time-resolved fluorescence intensity will decay exponentially with a superimposed modulation, as illustrated in Fig. 6.6. The modulation, or the quantum beat phenomenon, is due to interference between the transition amplitudes from these coherently excited states. Consider the simultaneous excitation, by a laser pulse, of two eigenstates, 1 and 2, from a common initial state i. In order to achieve coherent excitation of both states by a pulse of duration At, the Fourier-limited spectral bandwidth Au 1/At must be larger than the frequency separation ( - 2)/ = the pulsed excitation occurs at... 

The broad gain medium of Ti S Au — 10 Hz) allows the build-up of very short pulses with a Fourier limitation of roughly 1 / Ai/ = 10 fe. Recently, pulse lengths of about 5.5 fs have been attained directly from a Ti S laser, cor-respondmg to about two optical cycles [9.152]. Using subsequent compression of pulses which have first been spectrally broadened by self-pheise modulation as already discussed in comiection with Fig. 8.50, a pulse length of 4.6 s has been attained [8.153]. Techniques for ultra-short laser-pulse generation are... 

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