Autocorrelation width

TABLE II Autocorrelation Widths and Spectral Bandwidths for Several Transform-Limited Pulse Shapes... 

Functional form of pulse shape Ratio of pulse to autocorrelation width, td/tc Fourier transform product, tdAi/... 

The physical concept described by Eq. (1-19) reveals that the path to achieve the narrowest PSF is through the reference modulation with the narrowest autocorrelation width. Since the autocorrelation of a point source is the sharpest, it looks like the best choice for reference modulation. However, it is not a good policy in the view point of energy. So, a lens array was proposed to approach the effect of point source. The phase modulation of a lens can be described as... 

Table 2.2. Autocorrelation width and spectral bandwidth for different transform-limited pulses taken from [183]. At and Ata are the FWHMs of I(t) and G (t) respectively. Au is the FWHM of the measurable frequency spectrum.

We now proceed to some examples of this Fourier transfonn view of optical spectroscopy. Consider, for example, the UV absorption spectnun of CO2, shown in figure Al.6.11. The spectnuu is seen to have a long progression of vibrational features, each with fairly unifonu shape and width. Wliat is the physical interpretation of tliis vibrational progression and what is the origin of the width of the features The goal is to come up with a dynamical model that leads to a wavepacket autocorrelation fiinction whose Fourier transfonn... 

The frill width at half maximum of the autocorrelation signal, 21 fs, corresponds to a pulse width of 13.5 fs if a sech shape for the l(t) fiinction is assumed. The corresponding output spectrum shown in fignre B2.1.3(T)) exhibits a width at half maximum of approximately 700 cm The time-bandwidth product A i A v is close to 0.3. This result implies that the pulse was compressed nearly to the Heisenberg indetenninacy (or Fourier transfonn) limit [53] by the double-passed prism pair placed in the beam path prior to the autocorrelator. 

The characterization of the laser pulse widths can be done with commercial autocorrelators or by a variety of other methods that can be found in the ultrafast laser literature. " For example, we have found it convenient to find time zero delay between the pump and probe laser beams in picosecond TR experiments by using fluorescence depletion of trans-stilbene. In this method, the time zero was ascertained by varying the optical delay between the pump and probe beams to a position where the depletion of the stilbene fluorescence was halfway to the maximum fluorescence depletion by the probe laser. The accuracy of the time zero measurement was estimated to be +0.5ps for 1.5ps laser pulses. A typical cross correlation time between the pump and probe pulses can also be measured by the fluorescence depletion method. 

Positions and Widths Using Doubled Chebyshev Autocorrelation Functions. 

Using an experimentally-optimized focusing lens (/ = 15 mm spot radius, w = 4.3 pm) and a 3 mm KNbOj crystal (cut for NCPM at 22°C and 858 nm AR-coated), up to 11.8 mW of blue average power with a spectral width up to AIsh 1-4 nm at 429 nm was generated with only 44.6 mW of incident fundamental. The maximum observed SHG conversion efficiency was as high as 30 %. The overall efficiency of the electrical-to-blue process was over 1 %, and the blue pulses were measured by autocorrelation to be -500 fs in duration. ... 

This effective Q,t-range overlaps with that of DLS. DLS measures the dynamics of density or concentration fluctuations by autocorrelation of the scattered laser light intensity in time. The intensity fluctuations result from a change of the random interference pattern (speckle) from a small observation volume. The size of the observation volume and the width of the detector opening determine the contrast factor C of the fluctuations (coherence factor). The normalized intensity autocorrelation function g Q,t) relates to the field amplitude correlation function g (Q,t) in a simple way g t)=l+C g t) if Gaussian statistics holds [30]. g Q,t) represents the correlation function of the fluctuat-... 

Exercise. Delta functions do not occur in nature. In any physical application L(t) has an autocorrelation time tc > 0 for a Brownian particle tc is at least as large as the duration of an individual collision. It is therefore more physical to write instead of the delta function in (1.3) some sharply peaked function (j>(t — t ) of width tc. Show that this leads to the same results provided that 1. 

Here <( t ) f(t")> is the autocorrelation function of the electromagnetic field. For the case of excitation by a conventional light source, where the amplitudes and the phases of the field are subject to random fluctuations, the field autocorrelation function differs from zero for time intervals shorter than the reciprocal width of the exciting source. In the limit 8v A, that is when the spectral width, 8v, of the source exceeds the inhomogenously broadened line width, the field autocorrelation function can be represented as a delta function... 

A time response function of the apparatus can be measured by upconversion of the excitation beam. The width of such measured instrument response function is 280fs (FWHM). Comparing this result with the width of the autocorrelation function of the dye laser 110fs we observe 170fs broadening of the instrument response function due to group velocity... 

The features discussed above can be readily interpreted in terms of the time-dependence of the autocorrelation function < t) >. The most important factor governing the changes in the widths of the spectra for the positive versus negative displacement is the initial decrease in < (t)>. When transforming from the time domain to the frequency domain, a slow decrease in corresponds to a narrow progression and a fast decrease in < m> corresponds to a broader progression. 

The width At of the autocorrelation function is inversely proportional to the steepness of the potential. 

The steeper the potential the faster the autocorrelation function diminishes and the broader is the spectrum. The width of S(t) is related to the breadth of the spectrum by... 

The Fourier transformation of this autocorrelation function, eiEt/h x( ) x(t))dt, gives an energy spectrum with lines centered at E°, and line shapes of a Lorentzian form. The widths of these lines are Tn = h/Tn and the rate constants can then be obtained as kn = l/r . 

The measured autocorrelation function must be normalized before useful parameters can be extracted from it. The normalization consists of establishing a baseline. Normally this is done experimentally by measuring a delayed and/or infinite-time baseline. Occasionally, this is done by fitting a baseline as one of the parameters. In any of these cases the baseline must be correctly established within at least 0.1 % or surprisingly large errors occur in any parameter which describes the width of the distribution. 

Figure 7 Schematic of the laser system used in the Raman FID and echo experiments. PC = Pulse compressor AOM = acousto-optic modulator PD = photodiode FB = feedback electronics PBS = polarizing beamsplitter 3PBF = 3-plate birefringent filter SDL/LDL = Stokes/Laser dye laser P = pellicle AC = autocorrelator OC = output coupler LBO/KDP = doubling crystals. Final pulses have widths of 0.5-1 ps and energies of 0.3-1 mJ (From Ref. 6.)... 

Figure 5.3 shows a schematic illustration of our HRS measurement system [26-28]. A mode-locked Ti sapphire laser (Spectra-Physics, Tsunami) was used to induce HRS. Pulses of 70 fs at 790 nm with a repetition rate of 82 MHz were spectrally narrowed by a custom-made laser line filter (Optical Coatings Japan). The obtained pulse width was 14 cm FWHM with a pulse duration of 1 ps (measured by autocorrelation). These pulses were introduced into an inverted microscope system (Nikon, TE-2000) with a 36x, 0.52 N.A. reflective microscope objective or a 100 X, 1.49 N.A. oil-immersion microscope objective. Backscattered photons were collected by the same objective and filtered by dichroic mirrors (Optical Coatings Japan). Finally, HRS signals were detected by a charge-coupled camera (Princeton Instruments, PIXIS 400B) with a polychromator (ACTON, SP2500i). 

The arrangement we used for interfacing the picosecond laser to the molecular beam (or free jet) is shown schematically in fig. 1. The laser is a synchronously pumped dye-laser system whose coherence width, time and pulse duration were characterized by the SHG autocorrelation technique. The pulse widths of these lasers are typically 1-2 ps, or 15 ps when a cavity dumper is used. For detection one of three techniques... 

To answer this question, we eomputed the resonance positions and widths of HCO 41] and HN2,[42] using both doubled and undoubled autocorrelation funetions obtained from the damped Chebyshev propagation. The results indieated tiiat the enforeed doubling of the autocorrelation function yields no appreeiable differenees in both positions and widths of the narrow resonances when compared with those obtained from a directly calculated autocorrelation function. The differences are plotted in Fig. 1 for the low-lying resonances of HN2. The largest differences are for resonanees with widths on the order of a few hundred wave numbers.[42]... 

Due to the finite propagation time T of the wavepackets, the Fourier transformation causes artifacts known as the Gibbs phenomenon [122]. In order to reduce this effect, the autocorrelation function is first multiplied by a damping function cos jtt/IT) [81,123]. Furthermore, to simulate the experimental line broadening, the autocorrelation functions will be damped by an additional multiplication with a Gaussian function exp — t/xd)% where zj is the damping parameter. This multiplication is equivalent to a convolution of the spectrum with a Gaussian with a full width at half maximum (FWHM) of /xd- The convolution thus simulates... 

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