Adding a dimension to the infrared spectra of interfaces using heterodyne detected 2D sum-frequency generation (HD 2D SFG) spectroscopy

Abstract

In the last ten years, two-dimensional infrared spectroscopy has become an important technique for studying molecular structures and dynamics. We report the implementation of heterodyne detected two-dimensional sum-frequency generation (HD 2D SFG) spectroscopy, which is the analog of 2D infrared (2D IR) spectroscopy, but is selective to noncentrosymmetric systems such as interfaces. We implement the technique using mid-IR pulse shaping, which enables rapid scanning, phase cycling, and automatic phasing. Absorptive spectra are obtained, that have the highest frequency resolution possible, from which we extract the rephasing and nonrephasing signals that are sometimes preferred. Using this technique, we measure the vibrational mode of CO adsorbed on a polycrystalline Pt surface. The 2D spectrum reveals a significant inhomogenous contribution to the spectral line shape, which is quantified by simulations. This observation indicates that the surface conformation and environment of CO molecules is more complicated than the simple “atop” configuration assumed in previous work. Our method can be straightforwardly incorporated into many existing SFG spectrometers. The technique enables one to quantify inhomogeneity, vibrational couplings, spectral diffusion, chemical exchange, and many other properties analogous to 2D IR spectroscopy, but specifically for interfaces.

Molecular spectroscopies are some of the best tools for studying structures and dynamics. Two particularly useful variants are sum-frequency generation (SFG) and two-dimensional infrared (2D IR) spectroscopy. SFG spectroscopy provides a vibrational spectrum of molecular systems that lack an inversion center (1), and so has become a valuable tool for probing interfaces because no signal arises from the bulk. SFG spectroscopy has helped reveal the surface structure of liquids, characterize the surfaces of materials, and probe membrane proteins, to name only a few applications (2–5). 2D IR spectroscopy is also a vibrational spectroscopy, although not interface specific. 2D IR spectroscopy spreads the infrared spectrum into a second coordinate so that coupled vibrational modes are correlated by cross peaks, vibrational dynamics quantified by 2D line shapes, and energy transfer or chemical exchange revealed from peak intensities (6–8), in addition to many other capabilities not possible with linear one-dimensional (1D) vibrational spectroscopies like SFG spectroscopy. 2D IR spectroscopy is now being used to study protein structure and dynamics, solvent dynamics, charge transfer in semiconductors, and many other processes (9–12). These two techniques might be considered the core technologies of modern infrared spectroscopy.

In this article, we combine the surface sensitivity of SFG spectroscopy with the multidimensional capabilities of 2D IR spectroscopy in a technique that we call heterodyne detected (HD) 2D SFG spectroscopy. With this technique, one obtains 2D vibrational spectra of noncentrosymmetric systems. We demonstrate our approach by measuring the HD 2D SFG spectrum of carbon monoxide on polycrystalline platinum. Pt-CO is studied extensively to better understand electrochemical catalysis (13–16). The linear SFG spectrum of Pt-CO has been measured many times previously. The spectrum is almost always fit to a Lorentzian line shape (14–16), from which one concludes that Pt-CO is vibrationally narrowed, meaning that the structural dynamics of the CO and its surrounding environment are so fast that the system is homogeneous. By resolving the 2D line shape of Pt-CO, we find that there is a significant inhomogeneous contribution, which indicates that there is a slow component to the vibrational dynamics. Heterogeneity could be caused by some combination of surface crystallinity and weak hydrogen bonding of water to CO. Either way, the HD 2D SFG spectrum allows one to quantify the heterogeneity, which leads to a fundamentally different conclusion about the nature of the Pt-CO interface.

To implement HD 2D SFG spectroscopy, we have used many tricks-of-the-trade from the fields of 2D IR and 2D visible spectroscopy, including mid-IR pulse shaping, passive phase stabilization, and phase cycling. We collect absorptive spectra, which have the highest frequency resolution (17), while also showing how one can extract out the rephasing and nonrephasing signals that are sometimes preferable (18, 19). Another important piece of the methodology is heterodyne detection (20–24). Recently, landmark 2D SFG spectra were reported (25, 26). Those experiments, along with related pump-probe SFG experiments (27–29), proved that the signal strengths are sufficient for performing higher-order nonlinear spectroscopy at interfaces. However, one cannot interpret those spectra in the same manner as a 2D IR spectrum because the spectra are distorted by a complicated convolution of signals, as we discuss below. In contrast, HD 2D SFG spectra are mathematically analogous to 2D IR spectra, but with surface sensitivity and a few other interesting and informative differences that we outline below.

Qualitative Description of Pulse Sequences and Spectra

To provide a basis for comparison, we present a simplified description of the pulse sequences and spectra for each of the three techniques that we discuss in this article. Consider a system that has molecules in both the bulk and attached to a surface, such as CO in contact with a platinum plate illustrated in Fig. 1A. If one measures a SFG spectrum, one obtains a vibrational spectrum of the CO absorbed to the Pt because it is oriented. CO in the bulk electrolyte generates no signal because it is isotropic on length scales comparable to the visible pulse wavelength (30). There are many ways to generate an SFG spectrum, but they all use an infrared pulse that is on-resonance with the vibrational mode and a visible pulse to generate an emitted field with a frequency that is the sum of the two (2–4). Shown in Fig. 1B is a pulse sequence that uses a fs infrared pulse and a ps visible pulse (31). In the experiments reported here, we use a dichroic filter to frequency narrow the visible pulse, which is why it is asymmetric and broad in the time-domain pulse sequence. The goal is to measure the vibrational free induction decay that is created by the infrared pulse (dashed). The ps pulse has a narrow spectrum so that when the emitted field is sent through a spectrograph, one obtains a spectrum of the free induction decay. This pulse sequence also contains a local oscillator pulse at the same frequency as the emitted field, which heterodynes the free induction decay by interfering with it on the detector, allowing both the real and imaginary parts of the emitted field to be measured (20–23). The imaginary part produces a vibrational spectrum like that shown in Fig. 1C, which only contains signal from noncentrosymmetric CO at or near the surface.

Fig. 1.

Schematics of the molecular system, pulse sequences and spectra of linear SFG, 2D IR and HD 2D SFG spectroscopies. (A) The regions that signal is generated from for each technique. The pulse sequence and schematic spectrum, respectively, for (B, C) linear (1D) SFG, (D, E) 2D IR and (F, G) HD 2D SFG spectroscopy. E_n are mid-IR pulses, E_vis is the visible upconversion pulse, and E_LO is the local oscillator pulse. The coherences used to generate the spectra are shown dashed. In the 2D spectra, the phases of the peaks are labeled “+” and “−“.

Shown in Fig. 1 D and E is a 2D IR pulse sequence and schematic spectrum. There are many ways to collect a 2D IR spectrum, but the most common and most accurate method uses four femtosecond infrared pulses, where the first three pulses excite the vibrational modes of the molecule and the fourth pulse is the local oscillator that heterodynes the emitted field (dashed). The emitted field is analogous to the vibrational free induction decay in the SFG pulse sequence, but the 2D IR emitted field may include a vibrational echo (7). The additional pulses in the 2D IR pulse sequence generate another vibrational coherence during t₁, which modulates the emitted field. Thus, by measuring the heterodyned spectrum as a function of t₁, one obtains a correlation between t₁ and t₃, which is often visualized in frequency space (ω₁, ω₃) by Fourier transforming the data. A schematic 2D IR spectrum is given in Fig. 1E, which consists of two peaks that are out-of-phase by 180°. The negative peak on the diagonal is created by transitions at the fundamental frequency ω₀₁(ν = 0 → 1) while the positive peak is created by sequence transitions at frequency ω₁₂(ν = 1 → 2). Thus, the difference in frequencies between the positive and negative peaks along the probe axis gives the anharmonic shift of the vibrational mode (Δ = ω₁₂ - ω₀₁). The width of the peaks along the diagonal gives the total vibrational line width while the antidiagonal width gives the homogeneous line width. As a result, peaks that are inhomogeneously broadened are elongated along the diagonal (7, 8).

Shown in Fig. 1F is the pulse sequence that we use to collect a HD 2D SFG spectrum. Notice that it is the SFG pulse sequence with an additional two excitation pulses. That way, one correlates the t₁ and t₃ vibrational coherences by measuring the interferometric SFG spectrum as a function of t₁ and then Fourier transforming the data into a 2D spectrum. As we show mathematically below, the t₁ and t₃ vibrational coherences for HD 2D SFG spectroscopy are identical to those in 2D IR spectroscopy, barring a difference in polarizability vs. transition dipoles. As a result, one will obtain two peaks that are 180° out-of-phase and separated by Δ, just like in a 2D IR spectrum. However, the signs of the peaks may be reversed from those in a 2D IR spectrum, because the signs in heterodyned SFG spectroscopy also depend on the absolute orientation of the molecule (which is why centrosymmetric signals average to zero). As we show below, the HD 2D SFG spectrum can be interpreted in the same way as a 2D IR spectrum, but will only report molecules near the interface that are preferentially oriented.

Results

Shown in Fig. 2A is the linear SFG spectrum of a CO monolayer on a polycrystalline Pt surface using the pulse sequence shown in Fig. 1B. Because heterodyne detection is phase sensitive, both the imaginary and real parts of the spectra are measured (20–23). We focus on the imaginary spectrum (solid) because it has an absorptive line shape. In Fig. 2B, the spectrum is fit to both a Lorentzian and a Gaussian function, convoluted with the electric field of the visible pulse. Fits give full-width-at-half-maxima (fwhm) of 25 and 36 cm^-1 for the two line shapes, respectively, which are similar to previous reports (15, 16). Neither fit is ideal, but most previous reports used Lorentzian fits (14–16), thereby indicating homogeneous dynamics. A Gaussian line shape would indicate an inhomogeneous environment or structural distribution. The spectrum can also be fit to a Voigt, but the fit is not unique. Thus, one cannot conclude with certainty the nature of the vibrational dynamics from the SFG spectrum, which is a typical difficulty with standard linear spectroscopies.

Fig. 2.

Linear and HD 2D SFG scans of CO on polycrystalline Pt. (A) The real, imaginary and absolute SFG spectra. (B) Fittings to the imaginary 1D SFG spectrum. (C) Pump-probe spectrum. (D) The time evolution of the fundamental peak at 2,090 cm^-1 with 20 fs and (inset) 10 fs time steps

Shown in Fig. 2C is one particular time delay from the dataset using the HD 2D SFG pulse sequence in Fig. 1F. The spectrum shown has t₁ = 0, which makes it equivalent to a pump-probe spectrum (27, 28). This spectrum contains both negative and positive peaks, which are due to the ν = 0 → 1 and ν = 1 → 2 transitions, respectively, as discussed above. Shown in Fig. 2D is the amplitude of the spectrum at ω₃ = 2,090 cm^-1 as a function of t₁. The amplitude oscillates with a period of 80 fs, rather than the 16 fs period that corresponds to the natural vibrational frequency of 2,090 cm^-1, because the data was collected in the rotating frame (19, 32). The inset shows a few periods measured in smaller time steps to better illustrate the vibrational coherences.

Fourier transforming the spectrum along t₁ produces the HD 2D SFG spectrum shown in Fig. 3. The 2D spectrum resembles the schematic spectrum in Fig. 1G. There is a negative peak on the diagonal and a positive peak shifted off the diagonal along the ω₃ axis. It is also apparent that the spectra are elongated along the diagonal. Thus, by simple inspection, we get an estimate of the anharmonic shift of the CO stretch and learn that the line shape is inhomogeneously broadened.

Fig. 3.

Experimental and simulated HD 2D SFG spectra of a CO monolayer on a polycrystalline Pt surface. (A–C) Experimental absorptive, rephasing, and nonrephasing spectra with (D–F) their corresponding simulations.

The 2D spectrum in Fig. 3A has absorptive line shapes in both dimensions and thus the optimal frequency resolution. To obtain absorptive 2D spectra, one must sum together spectra collected with rephasing and nonrephasing pathways in order to cancel their individual phase twists (17). Because the first two excitation pulses in our experimental setup are collinear, both signals are generated in the same phase matching direction, detected simultaneously, and automatically summed, in an analogous manner to collecting 2D IR spectra in the pump-probe beam geometry (33, 34). However, it is sometimes advantageous to inspect the rephasing and nonrephasing spectra separately because correlated frequency fluctuations and cross peak patterns are different in the two types of spectra. Cycling the relative phase of the first two infrared pulses between 0 and and taking their linear combinations is one way of producing the two types of spectra (18, 19). Instead, we use spectral interferometry (7, 20–22) to obtain the complex HD 2D SFG signal and then Fourier transform it along t₁ to retrieve the rephasing and nonrephasing spectra in different quadrants, which are shown in Fig. 3 B and C. The rephasing spectra are oriented along the diagonal while the nonrephasing spectra are oriented perpendicular to the diagonal. Both spectra are phase twisted, because both spectra contain absorptive and dispersive components, which is why they are much broader than the absorptive spectrum. The ratio between the rephasing and nonrephasing spectra is 1.7, which is another indicator that the system is inhomogeneously broadened. For homogeneous line shapes, the intensities of the two spectra are equal.

Discussion

In this section we present the mathematical formalism that we use to interpret the HD 2D SFG spectrum and extract the homogeneous and inhomogeneous line widths. We contrast the signal to that measured by 2D IR spectroscopy.

Formalism of HD 2D SFG Spectroscopy

Each laser pulse interacts with the system either through the vibrational transition dipole μ or the electronic polarizability α to create a fourth-order macroscopic polarization, P⁽⁴⁾(t) (7, 8). Neglecting the polarization of the laser pulses, which will be a topic for a future article, P⁽⁴⁾(t) is written

[1]

where the integrals convolute over the pulse envelopes and are the Feynman paths associated with the fourth-order molecular response functions. In this equation, we have written the infrared transition dipoles, μ, and the Raman polarizability, α, in front of the integrals rather than in the response functions themselves, for reasons that become apparent below. Because the electronic molecular response associated with the nonresonant visible pulse is short lived, we approximate it as a Dirac delta function δ(t₄) so that becomes

[2]

where are the third-order responses that are measured in 2D IR spectroscopy. As a result, [1] simplifies to

[3]

Because the mid-IR pulses are fs in duration compared to the ps molecular response, it is valid to use a semiimpulsive approximation for the pump pulse durations, E_n(t) ∼ δ(t), which eliminates the convolutions in [3], illustrating that the macroscopic polarization closely resembles the desired molecular responses

[4]

In [4], we have included the experimentally controlled pulse delays as dependent variables as a reminder of which time variables are manipulated experimentally. The polarization creates an emitted field, E⁽⁴⁾(t₃) ∝ i ∗ P⁽⁴⁾(t₃), that is spatially overlapped with the local oscillator pulse E_LO. Both the signal and local oscillator are sent through a spectrograph that Fourier transforms the fields, which are then measured on a square-law detector to generate the actual signal

[5]

where represents a convolution. The first term is just the spectrum of the local oscillator, which is subtracted off. The second term is the homodyne detected fourth-order response, which is very weak and carries no phase information. The last term is the desired one, because it provides the actual fourth-order electric field. Because this field is scaled by E_LO, it is much larger than the homodyne response, which we ignore. A semiimpulsive approximation is also appropriate for the last term (20), E_LO = δ(t), so that the final signal is

[6]

To shift the spectra back to the IR region, one applies the relation ω₃ = ω_SFG - ω_vis to the spectra. From this equation it is apparent that HD 2D SFG spectroscopy measures the third-order molecular responses, , just like 2D IR spectroscopy. The differences are that the last interaction in HD 2D SFG spectroscopy depends on α instead of μ, and the HD 2D SFG signal is convoluted with the visible pulse that is absent in a 2D IR experiment. We show below that this convolution leads to a distorted signal. If a fs instead of a ps visible pulse is used, then the convolution drops out, the spectrum is undistorted, and the HD 2D SFG signal is identical to the 2D IR signal except for the scaling factors. We have not taken that route here because then one must actively scan the t₃ delay, rather than use a spectrograph to measure the entire t₃ response at once (20). The 2D spectrum is generated by collecting for a series of t₁ delays, which is then Fourier transformed to give a correlation between ω₁ and ω₃.

Following the notation above, we can simulate the third-order vibrational response just as we do for 2D IR spectroscopy. We start by drawing the Feynman diagrams from which the response functions can be written (see SI Text) (7). Because the experimental spectra are close to symmetric along the diagonal, we use a second-order Cumulant expansion and Bloch dynamics, which approximate the vibrational dynamics with a homogeneous lifetime (T₂) that includes time constants for both pure dephasing and vibrational lifetime and an inhomogeneous broadening constant, δΩ. With these approximations, the third-order vibrational response for the rephasing and nonrephasing pathways are

[7a]

[7b]

In the equations above we also used the harmonic approximation, which makes the ratio of the response functions of the fundamental and sequence transitions in HD 2D SFG spectroscopy equal, just as in 2D IR spectroscopy, because α depends linearly on the vibrational coordinate just as μ does (35). Because the experimental spectra have fundamental and sequence peaks of equal (but opposite) intensities, we conclude that the harmonic approximation is valid and that the metal surface does not appreciably weaken the Born-Oppenheimer approximation at these energies (36).

Vibrational Dynamics of Pt-CO and Its Structural Implications

Using the formalism above, we have fit the experimental 2D and linear SFG spectra to obtain Δ = -22 cm^-1, T₂ = 610 fs, and 1/δΩ = 588 fs. The simulated spectra are shown in Fig. 3 D–F. The fits reproduce the diagonal and antidiagonal line widths and predict a ratio of 1.4∶1 for rephasing:nonrephasing spectra. Thus, the simulations quantify the homogeneous and inhomogeneous contributions to the line width.

Shown in Fig. 2B is the simulated linear SFG spectrum. Because the simulations include both T₂ and δΩ, the line shape is a Voigt profile, rather than purely a Lorentzian or Gaussian. The inhomogeneous contribution accounts for about 40% of the total line width. Thus, inhomogeneity is a major contribution to the vibrational dynamics. The inhomogeneous distribution could be caused by the crystallinity of Pt or water hydrogen bonding to the CO or both. Previously, features in the OH stretch region have been attributed to H₂O molecules coadsorbed with the CO monolayer (37). Moreover, recent DFT calculations have shown that there are strong/weak hydrogen-bonding interactions between the coadsorbed H₂O and the bridge/atop-adsorbed CO molecules (38). Our results might provide further evidence of these hydrogen-bonding interactions between CO and H₂O around it.

While the line shape now better reflects the dynamics, it is still not a perfect fit to the linear SFG spectrum. Bloch dynamics is usually a poor approximation of vibrational dynamics, because the dynamics of most vibrational modes tend to evolve on a few ps time scale due to solvent and structural motions. Forcing fits into rigorous homogeneous and inhomogeneous contributions can be quite inaccurate (39). A better way is to actually measure the evolution of the vibrational dynamics by collecting a series of HD 2D SFG spectra as a function of the t₂ time delay and measuring the change in the ellipticity and nodal slope of the peaks (40). These parameters are related to the vibrational frequency fluctuation correlation function, which captures the spectral evolution of the line shape. Waiting time experiments should also be able to distinguish between the two origins of the inhomogeneity because crystallinity effects would not be time dependent, but hydrogen bonding would cause spectral diffusion. Waiting time experiments may also provide evidence for vibron hopping (41). While potentially very interesting, these dynamical “waiting time” experiments lie outside the scope of this initial report on HD 2D SFG spectroscopy. It would also be interesting to compare the inhomogeneous distributions of CO in the atop configuration on Pt(111) which may be more ordered.

Heterodyne Detection and Spectral Distortions

2D IR spectroscopy was originally implemented using a ps pump pulse (6), but most researchers now only use fs pulses because there is less distortion to the spectrum (33, 42). To understand the effects of the ps E_vis pulse on the HD 2D SFG spectrum, we present simulations in Fig. 4 A and B. Fig. 4A is a reproduction of Fig. 3B, but plotted with expanded axes. This spectrum was generated using E_vis = e^iωt-t/T with T = 600 fs, so that the visible pulse has a fwhm of 18 cm^-1, which matches the Raman filter that we use in our experimental setup. In Fig. 4B, we set E_vis = e^iωt(T = ∞) so that the spectrum is an exact measurement of the response. Notice that the 2D spectrum is unaltered along ω₁ but broadened along ω₃. In essence, E_vis is acting as a window function on the vibrational coherence during t₃ (20), by artificially decreasing the homogeneous lifetime. Clearly, one wants E_vis to perturb the spectrum as little as possible, which is why we used the narrowest Raman filter that can be commercially purchased. Smaller bandwidths can be obtained using gratings, although gratings will act as a Gaussian window function and thus produce a different 2D line shape (20), which may or may not be preferable. An alternative would be to use a fs E_vis pulse, but then one needs to scan t₃ in the time-domain (20).

Fig. 4.

Simulated HD 2D SFG spectra. Simulated result using visible upconversion pulse approximated as E_vis = e^iωt-t/T for (A) T = 600 fs (spectrum is identical to Fig. 3B) and (B) T = ∞ so that the spectrum is an exact Fourier transform of the molecular response. (C) Simulated experimental spectrum when the local oscillator is a linear SFG spectrum rather than an independent local oscillator.

Previous experiments that produced 2D plots of SFG spectra did not add an independent local oscillator pulse, but instead heterodyned the fourth-order 2D SFG signal with the second-order SFG signal (25, 26). Using the second-order SFG signal as a local oscillator leads to nonintuitive features, as has been previously reported (43). To describe those experiments mathematically, we replace E_LO by E⁽²⁾ in Eq. 5., but now the semiimpulsive approximation cannot be made. As a result, the interference term becomes

[8]

where the second bracketed term is the linear SFG signal. Thus, the spectrum is highly modulated along the ω₃ axis, because the “local oscillator” is now a very complicated function. In fact, the linear response will not necessarily contain all the frequencies needed to heterodyne the entire fourth-order signal. This effect is illustrated in Fig. 4C, which uses the same Bloch parameters that are obtained from simulating the experimental spectrum (Fig. 3). Notice that the sequence peak is now much weaker than the fundamental and the line shapes are distorted. The sequence peak is weak because R⁽¹⁾(ω₃) does not contain overtone frequencies. The only reason that the sequence peak is apparent at all is because the anharmonic shift is comparable to the line width so that the sequence band spectrally overlaps a little with the fundamental. In fact, one can get the wrong signs of peaks, or as noted previously, fake cross-peaks (43). In comparison, an independent LO pulse allows the 2D spectrum to be cleanly interpreted and simulated.

Conclusions

We have demonstrated the direct analog of 2D IR spectroscopy with surface-selective sensitivity. It is straightforward to implement HD 2D SFG spectroscopy, because one can either add a mid-IR pulse shaper to an existing HD SFG spectrometer or add an upconversion pulse and visible local oscillator to a 2D IR spectrometer. Instead of a pulse shaper, one could use an etalon or two individual pump pulses, although etalons will distort the spectra along ω₁ (33, 42) while individual pumps pulses will require an additional phasing process (34), and both techniques lack the many benefits of pulse shaping. To estimate the difficulty in performing a HD 2D SFG experiment on a particular system, one can approximate the expected nonlinear SFG response from the ΔOD of a 2D IR signal for a similar system, because the difference for both is μ², excluding the orientational contribution. Considering that our experiments were performed with only a few μJ of mid-IR pulse energy, whereas most modern SFG spectrometers generate 15–20 μJ, there are many systems that will have easily measureable HD 2D SFG signals. Of course, this paper has only probed an isolated vibrational mode, but HD 2D SFG spectra will also exhibit cross peaks in coupled systems. Along with theoretical developments (44, 45), we believe that HD 2D SFG spectroscopy will become another useful tool in the fields of surface science and multidimensional spectroscopy.

Methods

Detailed information about the experiment and a schematic of the instrument is provided in SI Text. In brief, E₁ and E₂ are generated by a mid-IR pulse shaper that together are 0.5 μJ. E₃ is 0.1 μJ. E_vis is at 800 nm and 2 μJ. E_LO is created using a 5%Mg:LiNbO₃ crystal in a phase stable design with a beam geometry for heterodyne detection. All pulses are P-polarized. The heterodyned signal is dispersed on a 400 × 1,340 CCD camera. t_LO = 2.5 ps, which is controlled by a pair of ZnSe wedges while t₂ = t_vis = 0. Data are collected with φ₁ = 0 and π, which are subtracted to remove the linear SFG spectrum and background scatter. Phase cycling is also used to shift the apparent response frequency to 326 cm^-1, which has a 80 fs period. We collect t₁ = 0 to 2,000 fs in steps of 20 fs (7). E₃ is not passively phase stabilized, causing φ₃ to drift at about 2°/ min (20), which we monitor with reference scans of the linear SFG spectrum every 80 s that are used to phase correct the data. The local oscillator and homodyned signal are removed by Fourier transforming the measured spectrum along ω₃, filtering out the zero frequency component, and inverse Fourier transform back to frequency space.