Pulse Fourier-limited

Suppose that a pulse Fourier transform proton NMR experiment is carried out on a sample containing acetone and ethanol. If the instrument is correctly operated and the Bq field perfectly uniform, then the result will he a spectrum in which each of the lines has a Lorentzian shape, with a width given hy the natural limit 1/(7tT2). Unfortunately such a result is an unattainable ideal the most that any experimenter can hope for is to shim the field sufficiently well that the sample experiences only a narrow distribution of Bq fields. The effect of the Bq inhomogeneity is to superimpose an instrumental lineshape on the natural lineshapes of the different resonances the true spectrum is convoluted by the instrumental lineshape. 

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. 

Nuclear Magnetic Resonance Spectroscopy. Like IR spectroscopy, NMR spectroscopy requires little sample preparation, and provides extremely detailed information on the composition of many resins. The only limitation is that the sample must be soluble in a deuterated solvent (e.g., deuterated chloroform, tetrahydro-furan, dimethylformamide). Commercial pulse Fourier transform NMR spectrometers with superconducting magnets (field strength 4-14 Tesla) allow routine measurement of high-resolution H- and C-NMR spectra. Two-dimensional NMR techniques and other multipulse techniques (e.g., distortionless enhancement of polarization transfer, DEPT) can also be used [10.16]. These methods are employed to analyze complicated structures. C-NMR spectroscopy is particularly suitable for the qualitative analysis of individual resins in binders, quantiative evaluations are more readily obtained by H-NMR spectroscopy. Comprehensive information on NMR measurements and the assignment of the resonance lines are given in the literature, e.g., for branched polyesters [10.17], alkyd resins [10.18], polyacrylates [10.19], polyurethane elastomers [10.20], fatty acids [10.21], cycloaliphatic diisocyanates [10.22], and epoxy resins [10.23]. 

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). 

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. 

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