We have developed a technique to perform optical two-dimensional Fourier-transform (2DFT) spectroscopy in a reflection geometry. Various reflection 2DFT spectra are obtained for an atomic vapor. The technique is useful for the cases where optical 2DFT spectroscopy cannot be performed in the transmission geometry.
© 2013 Optical Society of America
Optical two-dimensional Fourier-transform (2DFT) spectroscopy is a powerful tool for studying structure and dynamics of matter. A few examples are structural information in proteins , ultrafast hydrogen-bond dynamics in water , energy transfer in photosynthesis , and many-body interactions in semiconductors  or atomic vapors .
The implementation of 2DFT spectroscopy in the optical regime is fundamentally different compared to nuclear magnetic resonance (NMR), where the concept of 2DFT spectroscopy originated. Advances in experimental techniques have played a crucial role in the development of optical 2DFT spectroscopy since the first demonstrations in 1998 [6,7]. Unlike the relatively standardized equipment in NMR, numerous approaches are used in optical 2DFT spectroscopy. These approaches include passive stabilization [8–10], active stabilization [11, 12], using pulse shapers  and phase modulators . An approach of using incoherent light  has been proposed recently.
Despite the differences in these methods, they are all performed in a geometry where the generated signal is transmitted through the sample. In contrast, reflection experiments can provide access to a signal when a transmitted signal cannot be used, for example, in the study of optically dense media, interface or surface properties, and stand-off detection for remote sensing. Transient four wave mixing (TFWM) signals have been observed in reflection from both semiconductor quantum wells  and atomic vapors [17, 18]. Optical 2DFT spectroscopy is based on TFWM, thus performing it in reflection should also be possible. Compared to transmission experiments, the implementation of reflection 2DFT spectroscopy has different requirements on signal detection and phase stabilization. For example, the sample position affects the phase of excitation pulses and signal in reflection experiments, while this is not the case in transmission experiments. In this work, we implement reflection optical 2DFT spectroscopy. The experimental setup and the data processing are discussed, and the technique is demonstrated for an atomic vapor.
Optical multi-dimensional Fourier-transform spectroscopy is based on nonlinear wave mixing of a sequence of excitation pulses in a sample. The phase of the induced nonlinear signal needs to be tracked for performing a numerical Fourier transform. Consequently, the experiment requires subwavelength stability and precision for stepping time delays. In our experiment, this is achieved by using the JILA-MONSTR , which consists of nested Michelson interferometers with active phase stabilization. It is fed with the output of a Ti:Sapphire oscillator and generates four phase-locked pulses propagating along four corners of a square. The time delay of the pulses can be controlled with subcycle precision.
Three pulses A*, B and C (with respective wave vectors kA, kB and kC) are used to perform TFWM in the box geometry, as shown in Fig. 1(a). The TFWM signal in the phase-matched direction kS = −kA + kB + kC is recorded as S(τ, T, t). Here τ is the time delay between the first and second pulses, T is the time delay between the second and third pulses, and t is the emission time, as shown in Fig. 1(b). A 2DFT spectrum S(ωτ, T, ωt) is constructed by Fourier-transforming the time domain TFWM signal with respect to τ and t. The spectrum S(ωτ, T, ωt) is a rephasing (non-rephasing) spectrum if the conjugated pulse A* arrives first (second). A variety of spectra, such as correlation, zero-quantum, double-quantum and three-dimensional  spectra, can be generated by using different time ordering and/or scanning different delay(s).
To measure the TFWM signal field, heterodyne detection is realized by spectral interferometry  between the signal and reference pulse D. The phase of the reference pulse has to remain locked to the excitation pulses and the signal. In transmission experiments, this requirement is achieved by common optical paths or locking the reference to pulse C if the reference is routed around the sample. In transmission, the position of the sample does not affect the phase of the signal relative to the excitation pulses. In reflection experiments, however, the phase does depend on the sample position, and thus mechanical vibration of the sample causes phase fluctuations. Routing the reference around the sample is no longer a good option since it is difficult, if possible, to combine the signal and the reference while locking the reference phase to the excitation pulses and the sample position. A solution is to reflect beam D off the sample and use it as the reference. For the phase-matching direction considered, the signal co-propagates with the reference so that they remain phase-locked even in the presence of sample vibrations. Spectral interferometry can be performed, as illustrated in Fig. 1(a). The sample is slightly tilted such that the reference and signal beams are offset from beam A. Prior to arriving at the sample, the reference pulse is attenuated and its delay is adjusted with respect to the last excitation pulse to achieve proper conditions for spectral interferometry. Compared to transmission experiments with the reference beam being routed around the sample, an additional complication in this scheme is that the reflected reference pulse is reshaped by the resonant optical response of the sample. The phase distortion of the reference pulse needs to be characterized by an auxiliary spectral interferometry experiment and to be corrected in data analysis. This issue will be discussed in the next section.
Reflection 2DFT spectroscopy is tested with a potassium (K) atomic vapor. A K atom can be considered as a three-level system, as shown in Fig. 1(c). The corresponding 2DFT spectra can be calculated and have been experimentally characterized in the transmission scheme [5, 21]. A strong TFWM signal in reflection has been observed [17, 18] at temperatures higher than 500 °C. To prepare the sample, a K vapor is loaded into a titanium cell  which can be heated up to 800 °C. The optical access is provided by a sapphire window bonded to the cell. Although the K vapor is an ideal test system, we note that our scheme is applicable to other samples in general and we have also obtained reflection 2DFT spectra of GaAs quantum wells.
3. Data processing
A 2DFT spectrum is constructed from the spectral interferograms that are acquired at incremental steps of one time delay, τ or T. The TFWM signal field is extracted from interferograms at different time delays and is Fourier-transformed along the direction of the scanned time. In this section, the procedure of extracting the TFWM signal from an interferogram is discussed. Particularly, the discussion includes distortion of the TFWM signal due to the reference phase and how the distortion can be corrected in the analysis.
The reference pulse and the signal field propagate co-linearly into a spectrometer where an intensity interferogram is recorded. The TFWM signal ÊS and the reference pulse ÊR can be written as ÊS = ẼS(ω)e−iωτS eiϕS(ω) and ÊR = ẼR(ω)e−iωτR eiϕR(ω), respectively. Here ẼS,R, τS,R, and ϕS,R are the electric field spectral magnitude, delay and spectral phase, respectively. The recorded interferogram ISI is14, 23, 24], which is used in our experiment. The sign of the interference terms is flipped if the phase of one excitation pulse is changed by π, while the background terms are not sensitive to the phase. In our experiment, the phases of pulses A* and B are toggled by liquid crystal phase modulators such that a proper combination of interferograms are acquired to eliminate the background terms and to suppress the scatter from the excitation pulses, leaving only the interference terms. The term , which can be isolated via Fourier filtering, is
The time delay τS − τR can be easily measured and the amplitude ẼR can be determined from the reference spectrum. To extract the signal field ẼSeiϕS(ω), the reference spectral phase ϕR(ω) must also be determined. If the reference spectral phase is a constant, it can be neglected in calculating the signal amplitude. However, if the reference spectral phase varies, which is the case in the present experiment, the variation has to be characterized in order to extract the signal (even the amplitude).
Phase of the reflected reference
The reference pulse is reflected off the sample and its spectrum is reshaped by the sample’s optical response. This reshaping is shown in Fig. 2. The black dashed line is the laser spectrum and the blue solid line is the spectrum of the reference pulse reflected off a K vapor at 550 °C. Two resonant features in the reflection spectrum correspond to the two transitions of the three-level system. The reflection also modifies the reference phase, which can be measured through an auxiliary spectral interferometry experiment. An interferogram between the reflected reference pulse and a pulse split from the reference before it is incident on the sample is recorded to extract ϕR(ω). The measured reference phase, shown in Fig. 2 as the red curve, can be used to correct the phase in the interferograms so that an accurate TFWM signal can be retrieved.
Correction of the reference phase
The TFWM signal can be distorted if the reference phase is not taken into account. This distortion is clearer in the time domain. A time-resolved TFWM signal is a Fourier transform of the interferogram described by Eq. (2). Without correcting the reference phase, the time-resolved TFWM signal is shown in Fig. 3(a), where the TFWM amplitude is plotted as a function of the emission time. Time zero is defined by the arrival of the last excitation pulse (pulse C) in these experiments. Based on causality, the TFWM signal should appear only after time zero. However, the retrieved TFWM signal in Fig. 3(a) appears both after and before time zero. The signal at negative times is an artifact due to the distortion of the reference phase upon reflection.
To correct this distortion, the interference term represented by Eq. (2) is multiplied by a phase term e+iϕR(ω), where ϕR(ω) is the measured reference phase shown in Fig. 2. Therefore, the phase term e−iϕR(ω) in Eq. (2) can be eliminated. The resulting interferogram is Fourier-transformed into the time domain, and the retrieved TFWM signal is shown in Fig. 3(b). The signal is no longer distorted by the reference phase. The signal is corrected such that it appears only at positive emission times. The spike at about −6.3 ps is due to the back reflection of the reference beam from the windows of the vapor cell. An accurate TFWM signal field can now be extracted from the time-resolved TFWM signal with the reference phase corrected.
To further reduce the noise such as back reflections and scattering of the excitation pulses, the time resolved TFWM signal can be truncated by setting the amplitude to zero at negative emission times, as shown in Fig. 3(c). This truncation does not alter the TFWM signal since the signal appears only at positive emission times. The truncated signal is then Fourier-transformed back into the frequency domain.
Following the same procedure, the TFWM signal field is retrieved for each step of the scanned time delay. A 2DFT spectrum can be generated by Fourier-transforming a series of signals with respect to the scanned time delay.
Reflection 2DFT spectroscopy is performed on a K vapor. The vapor cell is heated to 550 °C and the atomic density is about 1018 cm−3. The laser spectrum is tuned to cover both D1 and D2 lines (Fig. 2). The laser repetition rate is 76 MHz, the pulse duration is about 150 fs, the average power is 10 mW/beam, and the beam diameter at the sample is 50 μm. All excitation beams are co-linearly polarized. 2DFT spectra are obtained by scanning τ up to 13 ps with T fixed at 500 fs. The resolution along the emission frequency axis is about 5 GHz.
The rephasing 2DFT amplitude (absolute value) spectrum is shown in Fig. 4(a). Two peaks on the diagonal are associated with the D1 and D2 lines (Fig. 1(c)), and two cross peaks represent the coupling between the transitions. Fig. 4(b) shows the non-rephasing 2DFT amplitude spectrum. The pattern of the spectra are consistent with the spectra of a K vapor at 250 °C taken in transmission . However, it is not possible to obtain 2DFT spectra in transmission for a K vapor at 550 °C due to high optical density. The spectra in Fig. 4 have much broader linewidths than the spectra at 250 °C in , indicating that the dephasing rate is much faster at the temperature of 550 °C. The rephasing and non-rephasing spectra can be used to construct the correlation spectrum , as shown in Fig. 4(c). The peaks in the correlation spectrum are asymmetric. The frequency-frequency correlation function can be calculated  from the asymmetry to provide insight into the non-Markovian dynamics in the atomic vapor.
We implement reflection 2DFT spectroscopy in which the TFWM signal is reflected off the sample. The fourth beam co-propagates with the signal and serves as the reference for spectral interferometry. The phase of the reflected reference pulse is measured and corrected in the data processing to generate accurate 2DFT spectra. Rephasing, non-rephasing and correlation spectra are obtained for an optically dense K atomic vapor.
This work was supported by the National Science Foundation through the JILA Physics Frontier Center and the Chemical Sciences, Geosciences, and Biosciences Division Office of Basic Energy Sciences of the Department of Energy.
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