The signal to noise in two-dimensional spectra recorded in the pump-probe geometry can be significantly improved with a quasi-crossed polarizer configuration, often employed in linear dichroism measurements. Here we examine this method in detail and demonstrate how to analyse and interpret the amplified signals. The angle between transition dipole moments can be determined with better accuracy than in conventional anisotropy measurements, and the method can be used to selectively suppress individual peaks and to efficiently remove scattering contributions. We present spectra of the coupled CO-stretch modes of a Ruthenium-carbonyl complex in DMSO for experimental illustration.
© 2012 OSA
Two dimensional infrared spectroscopy (2D-IR) correlates the instantaneous frequency of a vibrational mode with the frequency of the same or a different mode after a certain waiting time. This makes it a powerful tool for studying inter- and intramolecular couplings of vibrational transitions. Equilibrium chemical exchange processes can be followed in real time on a picosecond time scale and the ultrafast fluctuations of the local environment of IR chromophores can be probed very sensitively by analysing their 2D lineshapes [1–9]. 2D experimental methods are under continuous development. The two frequency axes are accessed in the time domain or in the frequency domain and both collinear ”pump-probe” and non-collinear ”photon-echo” geometries are employed [10–18]. In addition, polarization control of the incident laser beams has permitted to simplify 2D-IR spectra [19, 20] and was used to gain structural information by correlating transition dipole moment orientations [21–25]. A 2D-experiment with only two independent beams is easier to build and operate than a 2D-IR set-up in the so-called boxcar arrangement with four independent beams [12, 26]. However, in the pump-probe geometry the signal is usually weaker, because it is proportional to the probe beam intensity, which must be kept small to avoid detector saturation. More recently, however, it was shown by Xiong et al.  that a polarizer placed behind the sample can significantly enhance signal to noise of 2D-spectra measured in the pump-probe geometry. The same principle has also been used in 2D electronic spectroscopy . With a proper choice of beam polarizations it is possible to generate a third order signal field ES (the equivalent of the echo signal in a boxcar geometry) that is polarized perpendicular to the probe beam. It is transmitted by the additional polarizer, which can selectively attenuate an intense probe pulse.
In this paper we briefly outline the close relationship between this form of 2D-IR signal amplification and linear dichroism experiments. For the latter, the explicit dependence of the signal on experimental parameters (polarizer alignment and quality, signal size, etc.) has already been derived in detail [29, 30]. We build on those results to demonstrate the full potential and some limitations of the method in 2D-IR spectroscopy. In particular, we discuss the complete suppression of scattering contributions and the precise determination of angles between transition dipole moment.
2. Measurement principle
In third order 2D spectroscopy, a signal field ES, generated by three matter-field interactions between sample and the incident laser beams, is detected phase-sensitively by interference with a local oscillator (LO) field ELO. Typically, the superposition of signal and LO fields are dispersed in a monochromator and detected as a function of ”probe” frequency ω3 by a square-law detector:
Fourier transform along t1, the time between first and second field interaction, yields the ”pump” frequency axis ω1 of the 2D spectrum, while t2 is the ”pump-probe” delay, the time between second and third interaction. The sensitivity of the set-up is determined by its capability of detecting small intensity changes due to ES on the background of the LO, hence ES should be made as large as possible and ELO should be kept small. In the perturbative limit, ES is proportional to the product of the three fields interacting with the sample ES ∝ E1E2E3, so these must ideally be equally strong. However, in a conventional pump-probe setup E3 = ELO, because the probe beam is responsible for the third field interaction and, at the same time, provides the LO field. A way around this, proposed by Zanni and co-workers , is to create a signal field that is polarized perpendicular to the probe beam. The probe beam can then be selectively blocked or attenuated by a polarizer placed behind the sample (see Fig. 1).27] and Myers et al.  used a pair of pump pulses at 45° with respect to a nearly X-polarized probe pulse:
When the LO intensity is adjusted for best noise on the detector and then kept constant, E3 = ELO/sinβ, as indicated in Fig. 1. The detected signal in Eq.(1) thus has a β-independent component Ā, proportional to the average of the signals recorded with parallel and perpendicular polarizations, and a linear dichroism component LD, enhanced by a factor ±cotβ. This is the result that has been derived with the help of Müller matrices in the context of pump-probe spectroscopy [29, 30]:
The pair of collinear pump pulses was generated by a fast-scanning Mach Zehnder interferometer with automatic delay and phase calibration, described in detail in . In all measurements the delay between the two pump pulses (coherence time t1) was scanned up to 4000 fs. The pump-pulse pair passed a computer-controlled λ/2 plate, changing the polarization between ±45° with respect to the laboratory Y-Axis every 20 scans of the interferometer (≈ 1 second/scan). For maximum signal amplification, an intense probe beam, centered at the same frequency, was produced by a second optical parametric amplifier (OPA, ≈ 2μJ per pulse), which doubled the available laser power. Before being focused onto the sample, the probe beam passed a wiregrid polarizer (LP1) with extinction ratio ≈ 10−4 (InfraSpecs, P03 model), oriented at an angle β from the X-axis. An identical wire grid polarizer (LP2), oriented in the Y-direction, attenuated the probe beam before it was dispersed in a monochromator and imaged onto 32 pixel MCT double array detector. We used a waveplate in front of LP1 to maintain constant light intensity at the sample for all angles β. Varying β thus only changed the amplitude of ELO, while E3 and the signal field intensity remained constant. Only for angles β close to 45°, when the LO became too large and saturated the detector, we used neutral density filter to lower its intensity, affecting the signal field at the same time. A small fraction of the probe beam was split off before LP1, passed the sample and LP2 and served as a reference to correct for shot to shot intensity fluctuations.
The 2D-signals were obtained by Fourier transformation of the light intensity (Eq. 1) with respect to t1. This isolates the oscillating term Re[ESELO], making the signal proportional to ES. In conventional pump-probe spectroscopy, the light intensity is measured with and without pump beam (chopper) and the log-ratio of the two measurements is taken (Eq. 9). The signal is then independent of the spectrum of the probe beam. The same is achieved in our 2D-scans by dividing the signal by the t1-averaged intensity.
To illustrate our method, we recorded 2D spectra of the Rhenium-carbonyl complex [Re(CO3) (dmpby)Br], dissolved in DMSO in the C=O stretching region (1860–2060 cm−1). The ground state bands at 1889, 1910, and 2018 cm−1 have been assigned to a’(2) antisymmetric stretching of axial CO and equatorial COs, a” antisymmetric stretching of equatorial COs and a’(1) symmetric stretching of all COs, respectively .
4. Signal Amplification and Peak elimination
In Fig. 2 we present a selection of 2D-IR spectra, showing the signals ΔA± and their difference S, at four different polarizer angles β = 2°, 5°, 10° and 45°. The 2D spectra measured at 45° correspond to the well-known parallel A|| and perpendicular A⊥ signals. Since the strongly coupled carbonyl ligands give rise to normal modes with nearly orthogonal transition dipole moments, the cross peaks are more pronounced in the perpendicular polarization measurements. An additional signature of the positive and negative anisotropy of the diagonal- and off-diagonal peaks is their opposite signs in the difference signals S in the right hand column.
As reported previously [27,28], the 2D spectra grow significantly more intense when β is decreased. However, only the difference spectra S are simply amplified and do not change shape. The amplification factors match well the predictions of Eq.(9), except for β = 2°, where it is slightly lower than expected (25 instead of 28). This is attributed to saturation effects, caused by the already large unamplified signals of the complex .
In the ΔA+ spectra (left column) the cross peaks reverse sign between β = 45° (negative bleach, blue) and β = 5° (positive bleach, red), while the diagonal peaks flip sign in the ΔA− spectra (central column). There are thus intermediate polarizer angles β =β0, for which Ā = LD cotβ in Eq. 9, where these peaks can be selectively eliminated from the 2D-spectra in a single measurement. The polarizer orientation β0 for peak elimination depends on the intramolecular angle ω between excited and probed transition dipole moments. At short pump-probe delays, before rotational diffusion takes place, β0 and ω are related by the relation  :
This relation is plotted in Fig. 3a. Peaks with positive anisotropy are eliminated in the ΔA−signal (red), those with negative anisotropy in the ΔA+ signal (black). The absolute value in equation 11 is due to our distinction between the two spectra recorded with pump-pulse polarizations ±45°, which requires that β > 0. Alternatively we may think of the ΔA− signal as the ΔA+ signal with the opposite tilt of the probe beam polarizer (β < 0). In this case β0 becomes a smooth function of ω, shown by the dotted line in Fig. 3.
It is in principle possible to scan β in order to determine β0 by peak elimination, similar to polarization angle scanning 2D-IR spectroscopy in the boxcar geometry . However, this is not the most practical or precise procedure, as it would involve measurements with low enhancement factors. In fact, peaks are best eliminated after the determination of β0 using the technique described in the next section.
5. Angle Determination
The determination of the angle ω between two transition dipole moments is one of the the main reasons why polarization is used in pump-probe and 2D-IR spectroscopy. Usually, signals are recorded with the pump beam polarized parallel and perpendicular to the probe beam, and ω can be found from the anisotropy
Advantages and limitations of intramolecular angle determination in polarization enhanced measurements are best illustrated with the help of a graphical evaluation: according to equation 9, the signals ΔA± recorded with different polarizer angles all lie on the same straight line when plotted against cotβ. In order to symmetrize the problem we again use positive and negative values of β instead of distinguishing between ΔA− (β < 0) and ΔA+ (β > 0). The conventional signals ΔA⊥ and ΔA|| then correspond to cotβ = −1 and cotβ = +1, respectively. The smaller |β| the further the signals ΔA± lie apart, and the more accurately we can determine the intercept of the line connecting them with the horizontal axis (see Fig. 3b). This intersection defines the angle β0 at which either the ΔA+ or the ΔA− signal vanishes . With determined graphically, we can calculate the intramolecular angle usingEq. 11(without absolute value).
There are two sources of error in the determination of β0: one is the noise on the measured signals ΔA±, represented by vertical error bars, the second is the accuracy in setting the pump and probe polarization angles, which gives rise to horizontal error bars in Fig. 3. The vertical error bars are approximately constant at all angles β, and therefore become less and less important as the signals are enhanced and move further apart in the plot. For a very small polarizer angle, however, the uncertainty in cotβ is so large that the line joining the signals becomes again ill-defined. With the parameters used in Fig. 3b, the intramolecular angle ω can hardly be defined from the conventional measurements (β = 45°). For β = 10° boundaries can be found (blue dashed lines), which are still limited by the vertical error bars, i. e. the noise of the signals. For β = 3°, on the other hand, the uncertainty of β is determining the range of possible values of cotβ0 (red dashed lines). Measurements at smaller polarizer angles with even larger horizontal error bars will not lead to further improvement. For nearly parallel or perpendicular transition dipoles the line joining the signals is much steeper and the horizontal uncertainty becomes dominant already for much smaller values of cotβ. The best choice of β for intramolecular angle determination thus strongly depends on the specific experimental conditions (noise, waveplate quality and alignment) as well as the transitions investigated. The largest improvement over conventional measurements can be expected for relative transition dipole orientations near magic angle, i. e. when the signals with parallel and perpendicular pump and probe polarizations are almost identical.
In our set-up β can be set very accurately relative to LP2 with the help of a motorized rotation stage controlling polarizer LP1. The largest error thus arises from an imperfect alignment of the pump-polarization, which we switch between ±45° using a half wave plate. A small angle between the propagation directions of pump and probe beams induces a further error, which can, however, be corrected for (see Appendix in Ref. ).
In Fig. 4, we plotted the intensity of two peaks in the 2D-IR spectra of the Rhenium complex as a function of cotβ (peak positions marked in Fig.2). This is done at a delay of 100 fs, to avoid problems due to rotational diffusion. The signal to noise ratio improves as β is made smaller, and the values of β0 and thus ω become more precise, as the line joining two distant points is more accurate than a line fitting two points that are close to each other and affected by the same noise. The angle found for the diagonal peak 1 is about 15° (cotβ = −2.2). For the second peak cotβ ≈ 3, which would be consistent with an angle ω = 90°
With increasing population time t2, rotational diffusion lowers the anisotropy, until the LD signal vanishes in an isotropic sample. In that case, signal enhancement is no longer possible and ΔA+ = ΔA− are reduced to the unamplified 2D-IR signals. Rotational diffusion of small molecules is usually well described by the spherical diffusor approximation, which predicts an exponential decay of the anisotropy α of Eq.12 . The metrics α′ and α″ in Eqs. 14 and 16 do not have the the same exponential behavior, and anisotropy decay data recorded with the crossed polarizer method should in principle be analyzed taking into account the time-dependent denominator in Eqs. 14. However, fitting α″ with a single exponential function can still be a good approximation, as the rotational diffusion decay time is then overestimated by less that 10% if anisotropy is initially positive, or underestimated by less than 5% if it is initially negative.
On the other hand, unlike the anisotropy α the normalized LD signal is also sensitive to other processes like vibrational relaxation or chemical dynamics. To isolate their time-dependence independent of rotational diffusion, measurements can be performed at mystic angle (tanβ = 1/3) .
6. Scattering suppression
When scattered light from different laser pulses reaches the detector, delay-dependent interference patterns arise, which can be much larger than the desired signal (that is due to interference between local oscillator and signal field). The scattering signal is usually dominated by interference between ELO and the pump fields E1 and E2, but interference of the two pump beams may also contribute significantly.
The quasi-crossed polarizer scheme offers an elegant way to suppress these scattering contributions. The idea is similar to a scheme recently proposed for the complete scatter suppression in 2D-IR measurements in the photon-echo geometry , but much easier to implement: the different laser fields must be manipulated in a way to reverse the sign of the signal field ES, without changing the delays between the other laser fields that may interfere on the detector. In this case, two independent measurements can be carried out with identical scattering contributions but inverted 2D-IR signals. Taking the difference between these two measurements yields scatter-free data.
Here we exploit the fact that only the Y -polarization component of light is transmitted by the polarizer LP2 behind the sample (Fig. 1). Hence we can manipulate the X-polarization components of the laser fields without affecting the scattering contributions. According to equation 7 inversion of the LD-component of the signal field ES can be achieved by switching the pump-polarization between X ±Y, or, alternatively, between Y ± X. In practice, the subtle but important difference between these two possibilities is the direction in which we rotate the waveplate in the pump-beam, which changes the delays of X and Y polarization components in different ways, as illustrated in Fig. 5a. If the waveplate is rotated between π/8 and 3π/8 with respect to the initial pump polarization direction (X), it induces a λ/2 delay of the X-components of the pump fields, without affecting their Y-components. The LD component of the signal is inverted while the scattering signal is unchanged.As a result, the difference between the two measurements ΔA+ − ΔA− not only isolates (and amplifies) the well-defined LD-component of the signal, but also fully eliminates background contributions. The LD signal may of course also be measured by rotating the waveplate in the opposite direction (between π/8 and −π/8), thus selectively delaying the Y-components of the pump fields (dotted arrow in Fig. 5). In this case, however, both LD signal and scattering background change sign and the latter becomes stronger in the difference signal.
Fig. 5b–d show 2D-IR spectra of the Rhenium complex at a damaged spot on the sample cell, causing large scattering contributions in the individual signals ΔA±, which hide the diagonal signals. Subtracting the two signals results in the LD spectrum shown in Fig. 5d, with very little scattering despite the poor sample quality.
It is interesting to compare the polarization method of scattering suppression to other schemes used in the pump-probe geometry. The most common technique is delay dithering, or in a more defined way, the variation of the waiting time t2 between the pump pulse(s) and the probe pulse in units of π/ω0, the center frequency of the pulses. The result is a pseudo inversion of the relative phase between pump and probe , which inverts the scattering contribution at ω0, without modifying the 2D-IR or pump-probe signal significantly . By summing over data recorded at three different delays, scattering can thus be suppressed by more than two orders of magnitude over a bandwidth of 10% . In interferometer-based 2D-IR spectroscopy it is also possible to chop the pump-beam (E2) in the static arm of the interferometer . This allows one to fully eliminate interference terms between E1 and the local oscillator, while the Fourier transform along t1 suppresses the static scattering contribution . Neither technique can suppress interference terms between E1 and E2. They can nevertheless be employed in combination with the crossed polarizer method to compensate for imperfect polarizers or alignment of the waveplate in the pump beam. When the pump-pulse pair is generated by a pulse shaper , the relative sign of the two pump beams may be inverted without changing their delay (true inversion), provoking an inversion of the signal. For full scattering suppression, including interference between E1 and E2, four measurements are then required .
In the crossed-polarizer scheme, the two measurements that are needed to deduce the scatter-free LD signal or the intramolecular angles should be carried out within the shortest possible time in order to reduce noise and minimize the effect of laser intensity drifts. The pump-pulse polarization can in principle be changed on a shot-to-shot basis using a photoelastic modulator (PEM)  instead of rotating a waveplate on a second to minute timescale. However, commercially available mid-IR modulators simultaneously change the refractive index for the X-and Y-components of the incident light, compromising the possibility to maintain a constant delay between the Y components of the scattered fields. A possible future improvement of our method could thus be the combination of fast polarization modulation (by a PEM) and compensating delay modulation (by a wobbling Brewster window ), which should lead to an effective delay change of only the X-components of the pump beams at half the repetition rate of the laser.
Several factors can limit the amplification of the 2D-IR signals. For imperfect polarizers and beyond the small signal approximation equations 9 and 11 must be modified as as discussed in detail in refs. [29, 30]. polarizers with an extinction ratio of at least 10−4 are necessary Note that mirrors between the two polarizers like flat steering mirror and parabolic mirror can reduce the effective extinction ratio by almost one order of magnitude in order to measure undistorted signals for β angles as small as 1° (60-fold amplification). In this case, however less than 1/1000 of the probe beam intensity is available as local oscillator, which means that there is usually not enough light on the detector array to achieve the best signal to noise ratio (S/N). We found that the optimum angle for best S/N is often close to β ≈ 3° (20-fold amplification). Smaller polarizer angles are usually also not useful for anisotropy measurements as the relative beam polarizations are then not known with sufficient accuracy. Most importantly, however, it must be remembered that only the linear dichroism signal is amplified. The two measurements of ΔA± should always be made together in order to be able to deduce the enhancement factors of the different bands and interpret the 2D spectra correctly.
The better determination of angles between transitions dipoles is an interesting feature of 2D-IR spectroscopy with crossed polarizers. As shown in section 5, the gain in precision is greatest for dipole orientations near magic angle, although bands with small anisotropy are only enhanced very little. Here we have applied the method to a molecule with almost perpendicular transition dipoles, which gives a maximum of signal amplification, but allows for only a moderate improvement of the angle determination. In fact, just like the usual anisotropy signal α, β0 is a very flat function of ω near 0° and 90°, and a small error on β0 induces a large error on the determination of ω. This illustrates that the new technique does not eliminate some drawbacks of the traditional measurements with parallel and perpendicular pump and probe pulses. In particular, band overlap and isotropic background signals (that vertically shift the points in Fig. 4 and change the value of β0) can still perturb the measurements. However, we believe that our method will make it possible to determine angles between transition dipoles of weak absorbers in conditions where this was not possible before because of noise limitations.
A very common problem in 2D-IR spectroscopy is the distinction between signals that arise from the direct excitation of a molecule and band-shifts induced by the overall heating of the sample as a result of energy dissipation. Since the latter signals are isotropic, they are completely suppressed in the LD difference spectra, whereas the bands of interest are amplified. The selective suppression of peaks in single spectra by adjusting β, on the other hand, may be well suited to distinguish bands with different anisotropy in congested spectral regions. This could equally highlight inhomogeneous anisotropy distributions and decay within a broad absorption band, without the often ambiguous subtractions of two separately measured spectra. One example is line broadening due to hydrogen bonding, which can cause both a spectral shift and alter the orientational relaxation times [35, 36].
Using polarizers, the different fields of a third order experiment in the pump probe geometry can be controlled independently, which makes it possible to boost anisotropic signals to levels that can otherwise only be reached in non-collinear experimental geometries. In a certain sense, polarization takes on the role of directional phase matching. At first sight, this seems to ’consume’ the polarization degree of freedom of the experiment, which is usually used for the determination of intramolecular angles. Here we have shown how this measurement can be carried out with even better precision when recording two amplified spectra with opposite polarizer orientations. This method also offers an elegant way of suppressing scattering signals and selected peaks in the 2D-IR spectrum, making it a very simple, versatile and powerful implementation of 2D-IR spectroscopy. polarization enhancement is also promising for fifth order experiments like transient 2D-IR spectroscopy, where dichroism can be even larger . Even when the delay between the first (electronic) excitation and the subsequent IR excitations becomes longer than the rotational diffusion time, amplification would still possible because the IR excitation restores the linear dichroism, which can be exploited in the way presented in this article.
This work was in part supported by the Swiss National Science Foundation Grant No. 200020-119814/1.
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37. Note that in our discussion of polarization-enhanced UV-IR pump-probe spectroscopy in ref. , we neglected the symmetry of the problem. Considering ΔA± signals separately, we wrongly stated that measurements with only little amplification were always necessary in order to reliably determine β0 and therefore concluded that our approach constituted only a modest improvement when compared to the conventional method of using parallel and perpendicular probing. This is actually not true and this method can be in fact much more precise for the determination of angles, especially near magic angle, or when the usual anisotropy measurements fail because of low signal to noise.