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Welcome to SFG-PF

If you are not familiar with the SFG-PF routine, please take a few minutes to read the following introduction. Otherwise, you may skip this step by clicking on the links below.


Sum-Frequency Generation (SFG) is a nonlinear optical process mixing two input light beams (frequencies ω1 and ω2) to produce a new one at frequency ω3 = ω1 + ω2. It has become popular within the past decades when used as infrared-visible SFG spectroscopy. Using a tuneable or broadband infrared (IR) laser as one of the input beams leads to the resonant excitation of vibration modes, the symmetry properties of SFG making it an efficient interface-specific vibrational spectroscopy able to probe submonolayer molecular adsorbates or the topmost layer of a bulk.

You may have a look at the homepage of the Physico-chimie des interfaces group to discover examples of applications of SFG spectroscopy to various kinds of interfaces. Our team is also one of the few groups to routinely use the twin spectroscopy DFG (generation of a beam at the difference frequency ω3 = ω1 - ω2).

Modelling the SFG response

As for more classical (linear) vibrational spectroscopies, vibration modes essentially contribute as lorentzian resonances in a SFG spectrum recorded as a function of the IR wavenumber (ω). However, contrary to the linear case, the resonances interfere with each other, and with any additional constant contribution to the SFG signal. This is why data analysis is not as straightforward as in the linear case and requires caution. In order to extract the relevant parameters hidden in their SFG spectra, many SFG spectroscopy users fit their data according to equation :


The first term in this equation represents the constant contribution to SFG, often attributed to the substrate. The sum runs over all the vibration modes excited in the probed wavenumber range. Each vibration j is modeled by a complex lorentzian (amplitude A, phase φ, resonance wavenumber ω and width Γ in cm-1). For DFG, the imaginary part of the resonant lorentzian denominator becomes negative.

This equation is at the heart of the SFG-PF problem.

How to find the correct fit ?

Fitting an SFG spectrum boils down to finding a value for each parameter which accounts best for the experimental data.

Once a set of parameter is obtained and the corresponding fitting function determined, a question remains : is this set of parameters unique ? In other words, are there other sets describing the same fitting function ? The question may seem odd, but Le Rille and Tadjeddine [1] have shown in 1999 that, in the simple case of one resonance interfering with a constant background, there are two equivalents sets which build up the same fitting function.

We have generalized their method and proved in 2009 [2] that there are in fact 2N equivalent sets of parameters for a single fitting function with N vibration modes and a nonvanishing constant term. In addition, this amount reduces to 2N-1 when the constant term χ0 vanishes.

The SFG-PF routine calculates all the equivalent sets of parameters up to 5 resonances.

Preparing your input

In a first step, please provide the number of vibration modes (N) in the form.

A second form opens, in which you have to

  • choose between SFG and DFG ;
  • enter the value of the constant contribution (χ0), which must be real and positive ;
  • enter a value for each parameter, four per resonance : A, ω, Γ and φ (in degrees).

A table displays the 2N (resp. 2N-1) sets of parameters. It may be opened as a CSV file to be transferred to a scientific software for plotting and analyzing. Depending on the routine you use, a GIF file of the spectrum can be provided.

Important notes

- The SFG-PF routines run using the open source software Scilab 5.4. Two versions using a web interface are available on this site : with or without GIF display (the routine runs faster without graphical output).
- If you want to study ghost peaks (for a definition, see [2]), simply add a resonance with a zero amplitude and run the routine. Half of the sets of parameters should have a nonvanishing amplitude for this resonance.
- Beyond infrared-visible SFG and DFG, the routines apply generally to all resonant spectroscopies described by the above equations. The units for ω and Γ have been chosen here as IR wavenumbers, but the algorithms apply to any energy or frequency resonant denominator (for example in eV or Hz).
- If you intend to use the SFG-PF routines and their outputs for any kind of publication, we kindly ask you to acknowledge the authors and cite reference [2].

Author : B. Busson. Many thanks to J.-M. Teuler and A. Tadjeddine

[1] Le Rille, A. and Tadjeddine, A. J. Electroanal. Chem. (1999) 467, 238-248. DOI : 10.1016/S0022-0728(99)00047-9

[2] Busson, B. and Tadjeddine, A. J. Phys. Chem. C (2009) 113, 21895-21902. DOI : 10.1021/jp908240d