Time-resolved vs. frequency- resolved spectroscopy

Transcription

1 Time-resolved vs. frequency- resolved spectroscopy 1

2 Response to the impulsation Response to the impulsation Using the equation: Linear response to the δ function: It implies that the response function φ(t) is the response to the δ function where 2

3 Short δ-like stimulus 3

4 4

5 Wave packet A wave packet is a localized disturbance that results from the sum of many different wave forms: u(x,t)=asin (kx-ωt) f(x,t)= A i sin (k i x-ω i t) k=1/λ ---wavenumber ω frequency i If the packet is strongly localized, more frequencies are needed to allow the constructive superposition in the region of localization and destructive superposition outside the region. 5

6 f(x,t)= A i sin (kix-w(ki)t) i The amplitude A(k) contains the coefficients of the linear superposition of the plane wave solutions.these coefficients can in turn be expressed as a function of f(x,t) evaluated at t = 0 by inverting the Fourier transform relation above: Maxwell distribution of wavenumbers: 6

7 2-D wave packet 7

8 In physics, a wave packet is an envelope or packet containing an arbitrary number of wave forms. In quantum mechanics the wave packet is ascribed a special significance: it is interpreted to be a "probability wave" describing the probability that a particle or particles in a particular state will be measured to have a given position and momentum Time-eolving localized state concept Is essential in modern femto-dynamics Heller 1981, Zewail (1996) 8

9 Superposition of great many waves can lead to extremely well localized (temporally, spatially ) wave packet, but the waves are distributed over a large frequency range; recall Heisenberg uncertainty Delta-like functions can only be realized approximately by a short impulsion, e.g., an ultrashort laser pulse (today femtosecond (fs) pulses are easily generated and the first generation of laser systems producing attosecond pulses is becoming available, see F. Krausz and P. Corkum, Laser Focus World 38, 7 (2002) and A. Baltuska et al., Nature 42 (6923), 611 (2003); Nature 42 (6928), 189 (2003)). In a spectroscopic experiment using one such short impulsion as stimulus, one monitors, in real time, the response of the system to the impulsion and one uses the term time-resolved spectroscopy. 9

10 Techniques The response, called a free induction decay in NMR or EPR, is the response function which, by means of a Fourier transformation, can be converted in the power spectrum of the system. Several important spectroscopic techniques rely on this principle, such as, for instance, Fourier-transform microwave spectroscopy, Fourier-transform NMR and EPR spectroscopy, fs laser spectroscopy. 10

11 Long periodic stimulus 11

12 Following the development of the response equations one sees that are complex eigenvalues and are eigenfunctions of the response operator R Depending on the magnitude of the eigenvalue the system responds more or less strongly to the stimulus, i.e. it is more or less susceptible to it. The eigenvalue s therefore called the susceptibility of the system 12

13 susceptibility It is a function of the angular frequency of the stimulus and it represents the power spectrum of the system, as will be explained in Exercise. Equation shows that is the Fourier transform of the response function 13

14 14

15 By analogy, and with equation we can write the response function φ(t) as inverse Fourier transform of the susceptibility 15

16 If one knows the susceptibility at all frequencies, one can determine the response function φ(t) by inverse Fourier transformation. To obtain the complete response behaviour of a linear system using oscillatory stimuli, one must measure susceptibilty at each angular frequency. A determination can be made either by sweeping the frequency of a tunable monochromatic radiation source (lasers, microwaves,: : :) or, equivalently, by dispersing a white light source into its spectral components using a grating or a prism. Spectroscopic experiments carried out in the frequency domain are called frequency-resolved experiments. 16

17 Equivalence between time-resolved and frequency resolved experiments Whether one measures χ(ω) as a function of ω in a frequencyresolved experiment, or φ(t) as a function of t in a time-resolved experiment, the same information is obtained. The susceptibility χ(ω) can be converted into φ(t) by means of an inverse Fourier transformation, and vice versa. Time- and frequency-resolved spectroscopy are thus two sides of the same medal. Whether time-resolved or frequency-resolved spectroscopy is used to extract the spectrum of a sample depends on experimental convenience and on the availability of suitable sources of short pulses or monochromatic radiation. 17