Tutorials in vibrational sum frequency generation spectroscopy. I. The foundations

APPENDIX: MATHEMATICAL FORMULATION OF BB-VSFG

The response of a material subject to an electric field E (such as the electric field of light from a laser beam) is given by the polarization density, P, of the material. This can be expressed as P=ϵ0χ(1)E+ϵ0χ(2)E2+ϵ0χ(3)E3+⋯,(A1)where ϵ0 is the vacuum permittivity and χ(n) is the nth order electric susceptibility. χ(n) can be thought of as the degree to which the material is affected by the field—how much P changes for a given E—and mathematically takes the form of a tensor. In linear optics and linear spectroscopy, all terms except the first term on the RHS of Eq. (A1) are negligible. However, if the electric fields are intense enough, higher order susceptibilities become important.

In nonlinear spectroscopy, such as VSFG, intense electric fields from lasers are used specifically to cause a nonlinear material response, as this allows the spectroscopist access to the higher order susceptibilities, which contain different molecular information than can be accessed via linear spectroscopy. Specifically in SFG, the polarization induced in the sample by the IR and the VIS beams through χ(2) leads to emission of light at the sum frequency. Molecular information encoded in χ(2) is in turn encoded in the generated sum frequency light.

As the χ(2) (a 3×3×3) tensor is (under the electric dipole approximation) only nonzero in the absence of inversion symmetry (at the interface), SFG is inherently a surface specific spectroscopy. Furthermore, the polarization dependence of SFG arises because different polarization combinations probe different elements of the χ(2) tensor, so different molecular information can be obtained by looking at spectra recorded with different polarizations, as has been discussed in the main text.

It is now instructive to consider more mathematically the SFG process, as this provides a deeper insight into many of the phenomena discussed in the main text. The formulation below draws heavily on the excellent presentations given in Refs. 1414. I. V. Stiopkin, H. D. Jayathilake, C. Weeraman, and A. V. Benderskii, J. Chem. Phys. 132, 234503 (2010). https://doi.org/10.1063/1.3432776 and 1515. C. Weeraman, S. A. Mitchell, R. Lausten, L. J. Johnston, and A. Stolow, Opt. Express 18, 11483 (2010). https://doi.org/10.1364/OE.18.011483 but is presented more pedagogically. Overall, we consider the process as as two-step process consisting of initial excitation by a broadband IR pulse, followed by upconversion by a narrowband VIS pulse (even if these steps happen simultaneously in practice).Excitation by the broadband (and temporally short), IR pulse EIR induces a response in the molecule S(t). This response can be expressed mathematically in the time domain as: S(t)=Aexp⁡(iϕ)δ(t)−iΘ(t)∑jBjΓjexp⁡(−Γjt−iωjt).(A2)The first term in Eq. (A2) represents the nonresonant response of the system. A and ϕ represent the magnitude and phase of the nonresonant response, respectively, and i is the imaginary unit. δ(t) represents a delta function, as the nonresonant response is only nonzero when the IR laser pulse EIR is present. This first term can be conceptualized as simply a quantity that determines the strength of the nonresonant response (through A) and also the phase of the nonresonant response (through ϕ). The value of the phase determines how the nonresonant background interferes with the resonant signal, whether constructively or destructively. This interference between complex contributions to the signal is a hallmark of coherent spectroscopies such as VSFG.The second term in Eq. (A2) represents the resonant response of the system to the IR pulse. This term is expressed as a sum over different vibrational resonances j, which are generally modeled as Lorentzian line shapes in VSFG for simplicity.xx xxIn spectral fitting, it is sometimes seen that a Gaussian distribution of Lorentzian lines (a Voigt profile) is used, in certain circumstances.1010. A. G. Lambert, P. B. Davies, and D. J. Neivandt, Appl. Spectrosc. Rev. 40, 103 (2005). https://doi.org/10.1081/ASR-200038326 Each of these line shapes has an amplitude Bj, a linewidth (HWHM) Γj, and a central frequency ωj. Written in the time domain, this can be conceptualized as a wave with an amplitude Bj, frequency ωj, and that decays with a decay constant Γj. Θ(t) represents a step function because the resonant response is necessarily zero before the IR pulse hits and is only nonzero at later times. This reflects the principle of causality—the molecule cannot respond to the laser pulse before the laser pulse has arrived! Plotting Eq. (A2) for a single resonance would lead to a function that looks like the emitted IR (orange line) in Fig. 7.The molecular response S(t) caused by the IR laser pulse EIR results in a first-order polarization P(1)(t) in the sample, which can be written as: P(1)(t)=∫−∞∞EIR(t−t′)S(t)dt′,(A3)Equation (A4) is a convolution of the molecular response S(t) with the electric field of the short IR pulse EIR. If you are unfamiliar with the idea of a convolution, it can be thought of as expressing the result we would get if we blended two functions together, and this gives an idea of how one is modified by the presence of the other.xxi xxiThe integral in Eq. (A4) does this, essentially scanning t′ so that EIR is scanned through S(t), and the function that results from this is the convolution. If the IR pulse were infinitely short (a delta function in time), then P(1)(t)=S(t), but in reality the finite temporal width of the IR pulse broadens the molecular response slightly in time, with the result that the first-order polarization turns on with the rising edge of the IR laser pulse and slowly decays with the decay time of the excited vibrational resonance(s).The visible laser pulse EVIS then interacts with P(1)(t) to induce a second-order polarization, P(2)(t), written in the time domain as P(2)(t)=P(1)(t)∫−∞∞Q(t)EVIS(t−t′)dt′.(A5)Equation (A5) is another convolution, here between the visible electric field EVIS and Q(t)—the response of the molecule to the field EVIS. However, in the case of VSFG, the visible laser pulse is chosen not to be resonant with any electronic transition present in the molecule. As a result, any excitation by the visible pulse dephases (decays) essentially instantaneously, in contrast to the resonant IR excitation that dephased over a longer timescale dictated by the relaxation time of the excited oscillator. This means that the molecular response to the visible pulse can be written asThis is simply a delta function in time. As a result, the convolution integral at the end of Eq. (A5) simplifies to give P(2)(t)=P(1)(t)EVIS(t),(A6) =[EIR(t)⊛S(t)]EVIS(t).(A7)Thus, we see that the overall second-order polarization induced by both laser pulses can be conceptualized in the time domain as a convolution of the intrinsic molecular response S(t) with the short IR pulse EIR(t), which is multiplied by the temporally broad visible pulse EVIS(t). We know from fundamental electromagnetism that the SF light we measure in the experiment, ESF, is proportional to this second-order polarization. Thus, ESF(t)∝P(2)(t)=[EIR(t)⊛S(t)]EVIS(t).(A8)However, this is expressing our SF light in the time domain. Working in the time domain facilitates the previous discussion and is easier to conceptualize initially, but we need to analyze our SF light in the frequency domain to get useful spectroscopic information from it. We can do this by taking the Fourier transform of Eq. (A8), and in so doing, we obtain an expression for the SF light in the frequency domain, ESF(ω)∝[E~IR(ω)S~(ω)]⊛E~VIS(ω),(A9)where E~IR(ω) represents the Fourier transform of EIR(t) and so on, and multiplication in the time domain becomes convolution in the frequency domain and vice versa. From Eq. (A9), we can gain some useful insights into the nature of our BB-SFG measurement. First, we now see mathematically that the spectral width covered by the measurement is dictated by the spectral width of our IR pulse, E~IR—if E~IR=0, then we have no output SF light. Second, we see that in measuring our SF light on a spectrometer, we convolve our desired signal (the emission from the excited oscillators) with the visible pulse in frequency space, E~VIS(ω). As the emission from the excited oscillators is our “ideal” narrowband signal, it is broadened by the finite spectral width of E~VIS(ω), and this defines our spectral resolution—it is, therefore, clear that having as narrow a visible pulse (in frequency space) as possible is beneficial (recalling that convolution of a function with something infinitely narrow simply returns the original function). Practically speaking, this Fourier transform is performed physically in the VSFG experiment by the spectrometer.While the mathematical formulation of BB-VSFG illuminates some aspects of the process that are not clear from a more simplistic picture, a computer simulation of the process is even more insightful and can allow the effect of different input pulse durations, shapes, and wavelengths on the final VSFG spectrum to be analyzed. A simple homemade Python program for doing this based on the description given here and inspired by the work presented in Refs. 1414. I. V. Stiopkin, H. D. Jayathilake, C. Weeraman, and A. V. Benderskii, J. Chem. Phys. 132, 234503 (2010). https://doi.org/10.1063/1.3432776 and 1515. C. Weeraman, S. A. Mitchell, R. Lausten, L. J. Johnston, and A. Stolow, Opt. Express 18, 11483 (2010). https://doi.org/10.1364/OE.18.011483 can be found online—see Ref. 3737. J. D. Pickering, see https://github.com/james-d-pickering/SFG_upconversion_simulator for “SFG upconversion simulator” (2021)..

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