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We demonstrate a new scheme for measuring different tensor elements of the optical Kerr effect response. A dual-ring, polarization-dependent Sagnac interferometer is used to create two copropagating probe pulses that arrive at the sample at different times but that reach the detector simultaneously and collinearly. The tensor element of the response that is measured is determined by the polarization of the pump pulse. By controlling the relative timing of the probe pulses it is also possible to perform optical subtraction of two different tensor elements of the response at two different times, a strategy that can be used to enhance or suppress particular contributions to the OKE response.
©2007 Optical Society of America
1. Introduction
Ultrafast optical Kerr effect (OKE) spectroscopy [1–4] has become a broadly-used tool for studying dynamics in bulk liquids [5–7], liquid mixtures [8–11], polymer solutions [12, 13], confined liquids [14, 15], and other fluid systems [13, 16, 17]. In this technique, a pump laser pulse passes through a transparent sample, inducing a change in refractive index that is measured with a time-delayed probe pulse. In its most common implementations, OKE spectroscopy is used to monitor the xyxy tensor element of the third-order response as a function the delay time between the pump and probe pulses. This so-called depolarized response is sensitive to effects that include collective molecular reorientation, scattering from depolarized Raman-active intramolecular vibrations, and librational and collective scattering from depolarized intermolecular modes. The depolarized response corresponds to the transient birefringence induced by the pump pulse.
In many instances it is of interest to measure tensor elements of the OKE response other than xyxy. The most important additional tensor element of the OKE response is xxmm, where m is the “magic” angle (54.7°). This tensor element is known as the isotropic response [18], and it arises from isotropic changes that the pump pulse induces in the refractive index. Any tensor element of the OKE response in an isotropic system can be described using a linear combination of the depolarized response and the isotropic response [18].
While the isotropic response is insensitive to reorientation, it is affected by intramolecular and intermolecular modes with the appropriate symmetry. In some samples, such as proteins [19], the isotropic response from intramolecular modes is considerably more pronounced than the depolarized response. As a result, there is great interest in developing spectroscopic techniques that can cleanly separate the depolarized and isotropic OKE responses. A number of techniques for measuring the isotropic response have been reported previously, based on effects such as lensing [20] or spatial shifting of the probe pulse [21, 22].
Here we present a new strategy for determining different OKE tensor elements. The inspiration for this approach comes from the work of Trebino and Hayden on antiresonantring transient electronic spectroscopy [23]. In their setup, a Sagnac interferometer was used to create counterpropagating probe pulses for pump/probe spectroscopy. The sample was positioned away from the center of the ring, such that the probe pulse that propagated in the opposite direction from the pump pulse arrived at the sample considerably earlier than the pump pulse, whereas the other probe pulse arrived in proximity to or after the pump pulse. All of the light that enters such an interferometer exits in the same direction unless the probe pulses experience different absorption. This difference in absorption is the signal of interest in an electronic pump/probe experiment. Since the probe beams travel exactly the same path the interferometer is highly stable, and each beam experiences identical distortions due to any imperfect optics. Any deviation of the input beam splitter from perfect 50% reflectivity serves to create a local oscillator for the signal electric field [23].
Here we present an antiresonant ring scheme for measuring phase changes rather than absorption. We call our technique antiresonant-ring Kerr spectroscopy (ARKS). Our setup employs a double Sagnac interferometer, such that both probe beams propagate through the sample in the same direction. As a result, ARKS can be used to measure different tensor elements of the OKE response, and can even be used to combine responses at two different delay times in order to enhance or suppress particular contributions to the OKE decay.
2. Experimental setup
A diagram of our experimental setup is shown in Fig. 1. A KMLabs Ti:sapphire laser generates 50 fs pulses with a center wavelength near 800 nm. After a prism dispersion compensator and spatial filtering, the laser output is split into a strong pump beam and a weak probe beam. The pump beam traverses a computer-controlled optical delay line. The probe beam passes through a half-wave plate, a polarizer set at 45°, and then a quarter-wave plate (QWP) before entering the dual-ring interferometer via a polarizing beam cube (PBC). Orthogonal polarizations travel down each leg of the interferometer. The lengths of the two legs can be changed to adjust the timing of the probe pulses, and are generally set so that one probe pulse arrives at the sample several picoseconds or more before the other. The two probe beams are recombined in a second PBC, after which they are focused into the sample at the same spot as the pump beam. The probe beams are recollimated after the sample.
Fig. 1. Experimental setup for ARKS. HWP = half-wave plate, Pol = polarizer, QWP = quarter-wave plate, PBC = polarizing beam cube, 
For collecting the depolarized OKE signal, the input optics for the interferometer are set such that there is a single, horizontally-polarized probe beam. The pump polarization is set to 45°. An analyzer polarizer is placed immediately after the recollimating lens and is set to pass light that is vertically polarized. The input polarizer and QWP are then adjusted to implement a local oscillator for optical heterodyne detection (OHD) of the OKE signal [6, 24].
To measure other tensor elements of the OKE response, one axis of the QWP is set such that the polarization state of the input light is unaffected. After the sample the recollimated probe beams are fed back into the second PBC such that they are collinear with the incoming probe beams (see Fig. 1). The horizontally- and vertically-polarized components of the probe beam once again travel separate paths and are recombined at the original PBC. The two different polarization components ultimately travel along the exact same path, albeit in different directions, regardless of the delay between the probe pulses at the sample. Because the beams travel along identical paths, the dual-ring interferometer is relatively insensitive to vibration, as no significant motion of optics can occur during a single round trip of the interferometer. However, the laser must have good pointing stability so that the optical path through the interferometer is similar from pulse to pulse.
In the limit of weak probe beams, the output polarization of the dual-ring interferometer in the absence of a pump pulse should be identical to the input polarization (barring a 180° flip of the horizontal component of the polarization for each reflection). An analyzer polarizer is therefore placed after the output face of the first PBC and adjusted to pass light with a polarization that is orthogonal to the input polarization. In the presence of the pump beam, the two probe beams can experience different phase shifts in the sample, due to their different arrival times and different polarizations. Any differential phase shift introduces a polarization component that is orthogonal to the input polarization, and therefore leaks through the analyzer polarizer.
To optimize the signal-to-noise ratio of the data, we use a scheme that is similar to one we have reported previously [14]. The pump and probe beams are chopped at different frequencies with different rings of the same chopper wheel (Fig. 1). A small portion of the chopped probe beam is picked off to form a reference beam (not shown in Fig. 1) and is sent to a separate detector. The output of the reference detector is subtracted from that of the signal detector using an analog preamplifier. The output of the preamplifier is sent to a lockin amplifier that is referenced to the sum of the chopping frequencies. Care is taken to match the shape of the chopped signal and reference waveforms as well as possible.
The extinction ratio of polarizing beam cubes is not as high as that of other polarizing optics. Typical cubes we have employed have an extinction ratio for horizontal:vertical polarization on the order of 200 in the transmitted direction; in the reflected direction the extinction ratio for vertical:horizontal polarization is considerably lower. As a result, leakage of vertically-polarized light can be ignored, but there are significant reflections of the horizontal polarization component that propagate through the interferometer in undesired directions. It is important to understand the effects of these reflections in evaluating the performance of the spectrometer.
In Fig. 2 we consider all pathways that will allow horizontally-polarized light that enters the interferometer to exit in the direction in which the signal is detected. The desired path is labeled (a) in Fig. 2. The influence of the other pathways is minimized by a number of factors. First, there must be an even number of reflections for the beam to exit the dual-ring interferometer in the same direction as the signal, which limits the number of potential interfering pathways to seven, paths (b) through (h). Second, many of the paths involve reflections also change the path length of the probe beam such that it arrives at the detector at a significantly different time from the signal; this is the case for paths (b), (c), (e) and (h). These non-time-coincident reflections can reduce the dynamic range of the detection to some extent, but they do not interfere with the signal optically. Paths (d) and (g) do not contribute significantly to the signal because the probe pulse passes through the sample in the wrong direction. Path (h) passes through the sample in the proper direction, but involves four reflections and so is very weak.
Fig. 2. Pathways through which horizontally-polarized light that enters the ARKS apparatus can exit in the direction in which the signal is detected. Reflections of the predominantly transmitted horizontal polarization are indicated with red arrows.
Based on the above reasoning, we do not expect imperfect extinction of the PBCs to interfere with the desired signal. However, we must also consider a second effect of such leakage, namely that it can act as a local oscillator. The PBCs are composed of dielectric materials, and the leakage of horizontally-polarized light due to its reflection from the diagonal interfaces in these optics leads to a phase shift. The laser wavelength is far from any absorption in the PBS material, and so the phase of this reflection is not expected to have any significant dependence on wavelength over the bandwidth of the laser pulses. We have found that this phase shift is consistent for a given PBC, but varies among PBCs. The phase of the local oscillator therefore cannot be controlled, but the quadrature component of the non-resonant OKE signal is so much greater than the in-phase component that only the quadrature componenent of the local oscillator acts to amplify the signal. Introducing a variable wave plate in place of the input quarter-wave plate to control the phase of the local oscillator did not change the shape of OKE decays, and led only to a modest increase in signal intensity.
3. Results
To measure the depolarized OKE response we use the procedure described above. To measure OKE response functions of the form xxkk, where k refers to any direction in the xy plane, the dual-ring interferometer is employed. One probe beam (generally the one that is reflected from the first PBC, whose polarization we will call y) is set to arrive at the sample before the other (whose polarization we will call x). The time between the probe beams is set to be greater than the longest delay of interest. The pump beam, which is polarized in the k direction, is scanned from times slightly after the later probe beam out to the greatest delay desired, and data are collected as a function of delay time. In this scheme, the pump pulse does not affect the earlier probe pulse, and so only the later probe pulse experiences a phase change due to the pump. The xxkk tensor element of the OKE response is therefore probed.
Shown in Fig. 3 are typical data for carbon disulfide (CS2) at room temperature. The intensities of the decays have been normalized. The depolarized response is in good agreement with previous OHD-OKE data for this liquid [5, 25]. The xxxx (polarized) and xxyy responses are similar to the depolarized response, although the magnitude (and sign) of the electronic signal near zero time changes with respect to that of the nuclear signal at positive delay times for the different tensor elements. The fact that the electronic response is negative for some tensor elements of the response is expected, and indicates that the signal is indeed heterodyned. The isotropic response is quite small, in agreement with previous results [20, 26]. The dominance of the depolarized response in this liquid accounts for the strong similarities among the xyxy, xxxx and xxyy tensor elements. The lack of any component of the isotropic response that decays with the time scale of collective reorientation indicates that we can discriminate strongly against any contamination from the depolarized response.
Fig. 3. Normalized ARKS data for CS2 under difference polarization conditions. The data have been offset vertically for clarity.
To further assess the quality of the isotropic response measured with ARKS, in Fig. 4 we present data for sulfur monochloride (S2Cl2). Once again the decays have been normalized to the same maximum intensity. For this liquid, the contribution of Raman-active intramolecular vibrational modes to the OKE response is considerably stronger than that of collective orientational relaxation or intermolecular modes. The depolarized response exhibits oscillations due to a number of different vibrational modes [27–29]. The polarized response is similar to the depolarized response. The yyxx and isotropic responses are also similar to one another, but in contrast to the other two responses are dominated largely by the contribution of a single vibrational mode that is highly polarized. The fact that the isotropic decay oscillates around the baseline of the scan is additional evidence that the signal is heterodyned.
Fig. 4. Normalized ARKS data for S2Cl2 under difference polarization conditions. The data have been offset vertically for clarity.
To examine the spectral content the OKE decays for S2Cl2, we computed the power spectra corresponding to the depolarized and isotropic response functions. These power spectra, which are shown in Fig. 5, confirm the presence of a number of vibrational modes. The peaks at 102 cm-1, 205 cm-1, and 446 cm-1 are assigned to the v 4 (torsion), v 3 (S-S-Cl symmetric angle deformation) and v 2 (S-Cl symmetric stretch) modes, respectively, all of which have A symmetry in the C 2 point group of this molecule [29]. These modes are all polarized and, as is clear from Fig. 5, v 2 and v 3 are strongly so. The peaks at 238 cm-1 and 434 cm-1 belong to the v 6 (S-S-Cl asymmetric angle deformation) and v 5 (S-Cl asymmetric stretch) modes, which have B symmetry and therefore should be completely depolarized [29]. The power spectra in Fig. 5 confirm the depolarized nature of these vibrations. It is important to note that peaks do not appear in the power spectrum at the second harmonic of the fundamental frequencies, indicating again that our signal is proportional to the response rather than to its magnitude squared.
While the data in Fig. 5 are power spectra, we should also note that it is possible to perform Fourier-transform deconvolution on ARKS data to obtain the Bose-Einstein-corrected, low-frequency spectral density [6, 24]. To do so, however, requires that the pump and probe pulses be transform-limited at the sample. Due to the dispersion introduced by the PBCs, attaining transorm-limited pulses requires separate dispersion compensation of the pump and probe beams. While compensation of each beam is readily possible, we have not attempted it here.
We have previously demonstrated that by having two pump pulses of controllable polarization and intensity, it is possible to perform optical addition of two depolarized OKE response functions with a time shift between them [27, 30]. This strategy can be employed for enhancing or suppressing particular contributions to the depolarized response, which may be useful, for instance, in OKE microscopy. In the technique presented here, we instead have used two probe pulses. While for all of the measurements discussed above one probe pulse always arrives before the pump pulse, it is also possible to have both probe pulses arrive after the pump. In this manner, we can perform optical subtraction of two different tensor elements of the OKE response function with a time shift between them.
ARKS data for S2Cl2 illustrating the implementation of optical subtraction are shown in Fig. 6. For reference we show the xxxx and yyxx response functions obtained with one probe pulse arriving significantly before the pump pulse. In the lower two traces both probe pulses arrive after the pump pulse. By adjusting the relative timing of the two probe pulses, we can either enhance the response from the v 2 vibration (red trace) or suppress it virtually completely (green trace). By combining this technique with the previously demonstrated use of pump pulse pairs [27, 30], it will be possible to control the contribution of vibrational modes of any symmetry to the OKE signal.
Fig. 6. Normalized S2Cl2 ARKS data for the xxxx and yyxx tensor elements, as well as scans in which these tesnor elements are optically subtracted with different time shifts to enhance or suppress the contribution of the v 3 mode.
3. Conclusions
We have presented a new technique for obtaining different tensor elements of the OKE response function. Employing a dual-ring Sagnac interferometer makes possible the optical subtraction of the response functions from two different probe beams. ARKS is a versatile method of collecting OKE data, and also makes possible the enhancement or suppression of different contributions to the response when both probe pulses arrive after the pump pulse. We are currently working on implementing ARKS in an OKE microscope.
Acknowledgments
This work was supported by the National Science Foundation, Grant CHE-0608045.
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= vertical polarization, ↔ = horizontal polarization. Blue denotes polarizations before the sample and red polarizations after the sample. The mirror in the green block can be removed to measure the depolarized OKE response.