1. Introduction

Optical multi-dimensional Fourier transform spectroscopy (FTS) originates conceptually from the development of nuclear magnetic resonance (NMR) in the 1970s [1]. The most common implementation is two-dimensional (2D) spectroscopy, which correlates the oscillation frequencies of a nonlinear signal during two different time periods. A 2D Fourier transform spectrum has several clear advantages over traditional one dimensional spectroscopy or non-Fourier transform 2D spectroscopy results. First, congested spectra can be disentangled by spreading them out over two dimensions, and coupling between resonances can be identified by the presence of cross-peaks in the 2D spectra. Second, 2D FTS can separate real and imaginary parts of the nonlinear signal, resulting in higher frequency resolution compared to measuring the magnitude only. Furthermore, the frequency resolution of 2D spectra is only limited by the maximum separation between excitation pulses, therefore, overcoming the trade-off between time and frequency resolutions imposed by the time-bandwidth product [2]. This powerful combination make 2D Fourier transform spectroscopy the ideal tool for unraveling coupling among resonances and dynamics in complex systems.

2D FTS in the optical regime was pioneered by I. Noda and coworkers [3,4]. Femtosecond 2D Fourier transform spectroscopy has been implemented in the infrared/ near-infrared regime for the studies of molecular vibrational and electronic states. Tanimura and Mukamel’s proposal for 2D Raman spectroscopy [5] was followed by experimental implementation [6–8], however it was found that cascade processes dominated when Raman excitation was used, motivating a shift to IR excitation. The anharmonic nuclear potential and correlated fluctuations in the transition energies of two coupled molecular vibrations were studied by 2D IR spectroscopy [9,10]. The absorptive correlation spectrum was obtained by summing complementary spectra of two vibrational coherences that oscillate with conjugate frequencies in the initial evolution time period [11]. At shorter wavelengths, 2D electronic spectra of the dye IR144 in methanol were measured. Separate real absorptive and imaginary dispersive spectra were obtained and compared to theoretical predictions [12 ,13]. Recently, the electronic couplings in the molecular complex of photosynthetic protein were studied by 2D FTS in the visible range [14,15].

We have implemented 2D Fourier transform spectroscopy at near visible wavelengths (~800 nm) to study optical excitations in semiconductors, particularly excitons [16,17]. The requirement of stabilizing the phase between excitation pulses and measuring the signal coherently is more demanding for shorter wavelengths compared to IR, since the fluctuations of the path length and beam direction cause larger phase errors. One approach to 2D experiments in the visible wavelength is to maximize the stability of the apparatus passively and simply measure the phase delays [12,13], however this requires resampling the data, which must be done with great care. Another approach relies on either a pulse shaper and/or diffractive optics to generate inherently phase-stable excitation pulses [14,15,18]. There are certain limitations for each implementation. For example, spurious pulses often overlap with main peaks from an acousto-optic pulse shaper in time domain and cause harmful amplitude modulation. Furthermore, the range of delay between pulse pairs is constrained by the frequency resolution to about tens of picoseconds, making studies, for example, of the population relaxation process in semiconductors infeasible. In the case of diffractive optics approach, the reference pulse also passes through the sample, possibly introducing undesirable dynamical modifications in systems with strong excitation induced effects, such as semiconductors. In our experiments, active interferometric stabilization based on feedback electronics is used to maintain the phase stability of the excitation pulse pairs and the reference pulse [19,20].

2. Experiment

Two-dimensional Fourier transform spectroscopy is an extension to the technique of transient four-wave mixing (TFWM) where the signal field is completely characterized in amplitude and phase by heterodyne detection. The signal is generated by a sequence of three excitation pulses. The time delays are denoted as the evolution time τ (between the first and the second pulse), the mixing time T (between the second and the third pulse) and the detection time t (after the third pulse). The first pulse excites all the transitions within its spectrum, causing rapid polarization oscillations during the evolution period. The second pulse arrives after a delay τ, enhancing or suppressing such oscillations depending on the transition frequencies and the delay. After a mixing period, T, to allow population relaxation, the third pulse induces the nonlinear signal field:

where μ_ij and μ_kl are the dipole moments of the participating transitions and we have lumped all dynamics into D(τ, T, t). A Fourier transform with respect to τ and t yields a 2D spectrum S2D(ω_τ, ω_t) linking the absorption frequencies ω_τ during evolution period and the emission frequency ω_t during detection period. The sign in the exponential of Eq. (1) is determined by the coherence pathway, which can be controlled by the time ordering of the excitation pulses. The case where the system evolves with conjugate frequencies during the evolution and detection periods is called a rephasing measurement and corresponds to the time ordering that generates a photon echo in the presence of inhomogeneous broadening. The sign of frequencies is usually defined by ω_t thus for a rephasing measurement ω_τ is negative and the data are plotted in the lower right quadrant of the plane of the entire (ω_τ,ω_t) plane, i.e., ω_τ<0, ω_t>0. The non-rephasing data corresponds to the two frequencies having the same sign in the two periods and appears in the upper right quadrant of the plane, i.e., ω_τ>0, ω_t>0 [21].

The biggest challenge in implementing 2D spectroscopy is the requirement of preserving and extracting the optical phase information during both the evolution and the detection periods. The measurement of the optical phase is accomplished by heterodyne detection of the nonlinear signal with a reference pulse (or a local oscillator). The phase evolution of the signal due to change in delay τ has to be recorded, and the reference phase must be kept constant relative to that of the third pulse during the entire data-taking period. Therefore beam path stability has to be maintained both between the first two excitation pulses and between the third pulse and reference. Achieving interferometric stability between non-collinear beams is difficult and requires careful designs as explained in the remaining part of this section.

The output of a Kerr-lens mode-locked Ti:sapphire laser emitting 100 fs pulses at a repetition rate of 76 MHz is split into three excitation beams with equal power. They are aligned in the standard “box” geometry, i.e., on the three corners of a square with beam directions labeled as k _a, k _b and k _c as shown in Fig. 1(a). These three excitation pulses are focused onto a 20 μm diameter spot on the sample by a 10 cm focal length lens. For this arrangement of the excitation pulses, the emitted transient four-wave mixing signal is isolated in the phase-matched direction k _s = - k _a + k _b + k _c. The non-collinear three-pulse geometry is used so that the electronic coherence order can be selected by choosing the phase-matched direction, temporal pulse order and Fourier transform variables. When k _a arrives first, a rephasing measurement is obtained, whereas in a non-rephasing measurement it arrives second. If a collinear geometry is used, phase cycling schemes have to be utilized to isolate the desired nonlinear signal as is routinely performed in NMR and a recent optical FTS measurement based on an acousto-optic pulse shaper performed on Rubidium vapor [18,22].

The excitation delay τ is stabilized with a separate Helium-Neon CW laser, which follows the same optical path of the Ti:sapphire beam between two dichroic beam splitters to form a folded Michelson interferometer with the beam paths of pulse 1 and 2 as two arms, as shown in the right box in Fig. 1(b). The second dichroic beam splitter is common to both arms of the interferometer. This design relies on the fact that rotational motion of the dichroic is minimal. The recombined He-Ne beam exiting the interferometer is detected with a silicon photodiode. The error signal monitoring the relative arm lengths is sent to a loop filter and fed back to a piezoelectric actuator mounted on the back of a mirror in one arm, which corrects for any fluctuation. Figure 2(a) presents the error signal while the delay is locked along with the unlocked error signal from the He-Ne interferometer. Fluctuations in the error signal drops to below 1% when the servo loop is enabled, which means the phase change between pulse 1 and 2 is reduced to less than 0.5% of one optical cycle, taking into account the fact that the He-Ne beam double passes the interferometer. The fluctuation of delay τ is suppressed to below 0.01π (peak-to-peak deviation) with active stabilization.

 

Fig. 1. (a) Excitation scheme and (b) experimental setup for 2D Fourier transform spectroscopy. The delay between first two excitation pulses is stabilized and scanned by a stabilization interferometer (enclosed in the right box). The reference phase is locked by another stabilization interferometer (enclosed in the left box). CMP: chirped mirror pair; D: photodiode; BS: beam splitter; DBS: dichroic beam splitter; PZT: piezoelectric actuator.

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Fig. 2. (a) Error signal from the excitation phase stabilization interferometer while delay τ is unlocked and locked; (b) Error signal from the reference stabilization interferometer while the reference phase is unlocked and locked. Each locked error signal is shown in smaller scale in inset.

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After acquisition of the nonlinear signal at a particular delay position, the feedback loop is disabled and delay τ is stepped to the next position by a picomotor actuator with step size of 30 nm. The He-Ne interferometer fringe goes through one period if the Ti:sapphire beam path length changes by half a He-Ne laser wavelength, 316.4 nm. The He-Ne interference signal is monitored while the picomotor steps, and an overall offset is removed to make the fringes oscillate symmetrically around zero. Once the signal crosses the zero level, the servo loop is enabled to lock delay τ at the steepest slope of the fringe and allow the acquisition of the nonlinear signal, as shown in Fig. 3(a). The change of path length from one locking point to another is exactly 316.4 nm, as the small error due to overshooting zero by a partial step is corrected by the piezoelectric actuator once locking is established. The same process of stepping and locking is repeated for every delay t until all the measurements are finished, as shown schematically in Fig. 3(b). Delay τ is scanned 2048 steps from 0 to 2.17 ps with step size of 1.06 fs. Then the experimental data are padded with zero to 8196 steps since the nonlinear signal has decayed to background level with delay τ larger than 2 ps. This gives the spectral range of 474 THz and resolution of 0.12 THz along the absorption frequency axis after the Fourier transform.

 

Fig. 3. (a) The error signal from the He-Ne interferometer during locking and stepping (the circled dots are where delay τ is locked). (b) The flowchart of locking and stepping delay τ by monitoring the error signal. N is the total number of measurements.

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The reference beam for heterodyne detection is derived from the third excitation pulse before it is focused. The reference pulse is routed around the cryostat and combined with the signal beam at a beam splitter with 10% reflectivity. Glass wedges are inserted into the reference path to compensate for delay and chirp induced by the cryostat windows. The portion of the reference transmitted through the beam splitter is used to generate an error signal to correct any relative fluctuations between the path length of the reference and third pulse. A spectral interferogram is obtained from the combined reflected reference and transmitted signal.

Fluctuations between reference and the third pulse are significant due to the turbulent air flow around the cold cryostat used to hold and cool the sample. Therefore it is necessary to actively stabilize the reference phase relative to the third beam. As shown in the left box in Fig. 1(b), an interferometer is formed with part of the reference and the transmitted third beam. They are focused with a 20 cm singlet focal lens onto to one end of a single mode fiber (mode field diameter 5.6 μm), where a spatial fringe pattern is formed. The fiber core is much smaller than a single spatial fringe. The resulting interferometric error signal is detected with a high gain photodetector. The error signal is fed back to a piezoelectric actuator, which moves a mirror in the reference beam to compensate the path length fluctuations. Figure 2(b) shows the error signal from the stabilization interferometer, with fluctuation of relative path length reduced to less than 5% of one wavelength. The reference phase is locked to a constant value with peak-to-peak deviation below 0.1π during the whole 2D experiment. Long term phase drift, which is exacerbated by the presence of the cryostat, is monitored by the PZT voltage and compensated by moving a glass wedge in the reference used to set the relative delay to the signal.

Spectral interferograms are obtained using a spectrograph and CCD camera. To optimally use the dynamic range of the CCD camera, the reference is set to be approximately 10 times stronger than the signal. The combined signal and reference beam is focused with a 10× microscope objective into a single mode fiber (NA 0.12 and mode field diameter 5.6 μm) to ensure complete spatial mode matching for interference, while a linear polarizer in front of the fiber ensures polarization overlap. The light exiting the fiber is coupled into a 0.25 m imaging spectrograph and imaged onto a thermo-electric cooled CCD camera (16 bits dynamic range and 1024×256 array of 26 μm pixels). The fiber is mounted at the entrance of the spectrometer and matches the numerical aperture of the spectrograph (f= 3.9, NA = 1/(2f)). Therefore, the light fills the grating completely and the optimal frequency resolution of 0.1 nm is achieved. The reference beam arrives 5 to 8 ps earlier than the signal to obtain an interferogram with the maximum number of fringes allowed by the spectrograph/CCD resolution.

During data acquisition, the laser beam direction is also monitored in two dimensions by a pair of quadrant detectors and actively stabilized by feedback and piezoelectric actuators on a turning mirror. The experimental setup is enclosed in a sturdy acrylic box to minimize the influences of air flow, environmental noise and temperature variations.

3. Data collection and processing

To test our apparatus, we use a sample consisting of ten periods of a 10 nm GaAs quantum well separated by 10 nm Al_0.3Ga_0.7As barriers held at ~10 K in an optical cryostat. Due to quantum confinement, the heavy-hole and light-hole valence bands are energetically split by approximately 6 meV, resulting in two excitonic resonances. The heavy-hole and light-hole excitons can be excited simultaneously with excitation pulses tuned to about 802 nm. The two exciton resonances are coupled quantum mechanically through a common ground state for linearly polarized excitation.

The complex field of the nonlinear signal is obtained by a standard Fourier transform spectral interferometry approach [23,24]. An example of a spectral interferogram is shown in Fig. 4(a) and the corresponding retrieved spectral magnitude and phase are shown in Fig. 4(b). Strictly speaking, the phase measured is the phase difference between the signal and reference. Material dispersion to all three excitation pulses and the reference has been compensated with a pair of negative dispersion chirped mirrors and verified to be near transform-limited by power spectrum, autocorrelation measurements and time-bandwidth product calculation. The interferometry measurement and analysis are repeated at each excitation pulse delay to obtain a 2D map as a function of emission frequency ω_t and delay τ.

 

Fig. 4. (a) A typical spectral interferogram between the signal and reference, where the reference arrives 7.66 ps earlier; (b) The magnitude (solid line) and phase (dotted line) of the signal retrieved by Fourier transform spectral interferometry.

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Before generating a 2D spectrum, we first use this data set to verify the overall performance of the system by analyzing the phase evolution with excitation delay τ at the frequency of the heavy-hole resonance. In addition to testing the entire apparatus, it also serves as an out-of-loop characterization, whereas the data in Fig. 2 are in-loop. According to Eq. (1), the change of signal phase at a certain emission frequency ω_t is ω_tΔτ, where the change of delay τ between two acquisitions of spectral interferograms is Δτ= 1.06 fs. At the resonance frequency of heavy-hole exciton, 372.12 THz, the phase change is 0.79π. The constant phase shift is subtracted, leaving the residual fluctuations shown in Fig. 5. Only the initial data for which the signal is strong are shown. The experimental result of phase change agrees with the expected value quite well, with a mean of 0.02π and a standard deviation of 0.06π These results are consistent with stability of the excitation and reference of 0.01π and 0.1π (peak-to-peak deviation), respectively. Note that this analysis is only valid if the signal is dominated by a diagonal peak, which we show below. The presence of the cross-peak from the light-hole exciton adds a systematic variation to the phase, worsening the apparent fluctuations.

 

Fig. 5. The fluctuations of signal phase change as a function of delay τ at the frequency of 372.12 THz. The nominal value of 0.79π has been subtracted, leaving the a mean value of 0.02π and a standard deviation of 0.06π.

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A direct Fourier transform with respect to t converts the two-dimensional map into the final 2D spectrum with dimensions of emission frequency ω_t and absorption frequency ω_τ. The complex field of the nonlinear signal is shown in two dimensions and can be presented in real and imaginary parts. The correct separation of a 2D spectrum in the real and imaginary parts requires the determination of an overall constant phase due to the arbitrary phase shift between the reference pulse and the third pulse. This is done by applying a constant phase rotation to the complex four-wave mixing signal retrieved from the interferogram at τ= 0, and fitting the real part to an independent spectrally resolved pump-probe measurement [25]. The third excitation pulse with a power equal to the total of first two pulses is used as the pump beam. The pump power is chosen to reproduce excitation conditions in the 2D experiments since excitation induced effects dominate the optical nonlinear response of semiconductors. A tracer pulse that travels along the signal path with half of the pump power is used as the probe. The delay between the pump and probe equals the mixing time T used in the 2D experiments. The differential transmission signal along the probe is spectrally resolved. A typical pump-probe measurement and the phase corrected FWM spectrum are shown in Fig. 6.

 

Fig. 6. A typical pump-probe measurement (dotted) and the least-squared match of phase corrected FWM spectrum (solid). The maximal mismatch is below 10%.

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Both the rephasing and non-rephasing measurements of the 2D spectra are shown in Fig. 7. The amplitude plot of the 2D spectrum presents two diagonal and two off-diagonal peaks clearly. Diagonal peaks lying close to diagonal lines result from oscillations at the same frequency during both absorption and emission time periods. These peaks resemble the linear absorption spectrum. The spectral line width along the diagonal direction in the rephasing measurement shown in Fig. 7(d) indicates the inhomogeneous width while the homogeneous linewidth is measured in the direction perpendicular to the diagonal. In the case of the heavy hole exciton, the homogeneous linewidth is on the order of 0.8 meV, which depends on both the spectral width of the excitation pulses and the excitation power (~ 1 mW). Off-diagonal peaks indicate that the oscillation frequency changes during or between the two periods as a result of coupling between the two resonances. In the simplest model, heavy hole and light hole excitons are coupled through common conduction band states while different spin valence bands are involved. Therefore, the appearance of the off-diagonal peaks is expected. There is an asymmetry between the intensity of the off-diagonal peaks, most evident in the rephasing measurements shown in Fig. 7(d)–(f). For T > 0, such asymmetry could result from population relaxation from a higher energy state (light hole) into a lower state (heavy hole) and is observed in 2D FTS performed on molecular systems [14]. Here, an asymmetry is observed even when T = 0 and population relaxation does not have time to take place. Many-body interactions between excitons can explain such effects [26]. The imaginary part of a 2D spectrum measures the transient change in the refractive index, while the real part approximates the resonant absorption of a probe field at frequency ω_t, induced by the excitation frequency ω_τ. Distinctive real and imaginary lineshapes reveal dominant microscopic many-body interactions responsible for the nonlinear response [27].

“Nonrephasing” measurements, shown in Fig. 7(a)–(c), correspond to time ordering of the excitation pulses such that the conjugated field - k _a arrive after k _b. Nonrephasing measurements are also referred to as “virtual echo” since these measurements sample quantum mechanical pathways with the same phase evolution direction during τ and t, i.e., exp{i(ω_ττ2002B;ω_tt)} not allowing cancellation of dephasing within an ensemble with inhomongeneously distributed oscillation frequencies. In the case of a simple two-level system, nonrephasing measurements carry the same information as those of rephasing measurements. In a more complicated system, the combination of rephasing and nonrephasing measurements can be used to emphasize certain spectroscopic features and reveal the degree of correlation of inhomogeneity between the coupled modes [21]. Here, the strength of the off-diagonal peaks weakens in the nonrephasing measurements compared to the rephasing measurements as expected from a third order perturbation calculation.

 

Fig. 7. The 2D spectra of a GaAs multi-quantum well sample. Both the non-rephasing (upper row) and rephasing (lower row) data are shown in amplitude (A, D), real part (B, E) and imaginary part (C, F). Contour spacing is 5%. Note that absorption frequencies (vertical axis) in rephasing measurements are negative. Solid while lines are drawn to help identify the diagonal peaks corresponding to heavy hole (HH) and light hole (LH) excitons. The background level in the 2D spectrum is typically about 5% of the maximum peak strength.

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We note that there seem to be tails along the absorption frequency direction on each peak in the 2D spectrum. These tails most likely arise from the fact that delay τ is only scanned in one direction and single sided Fourier transform is performed. We have observed similar tails in numerical simulations based on modified optical Bloch equations. These tails may be suppressed by either taking symmetric scans of τ or applying some kind of window functions as routinely used in NMR. However, we did not employ these procedures due to complications of the nonlinear response in semiconductors compared to those in molecules. For example, it has been previously demonstrated that both free induction decay and photon echo occurs simultaneously in semiconductors in the presence of disorder [28]. These effects produce unique features in 2D FTS and are still under investigation.

4. Summary

Optical two-dimensional Fourier transform spectroscopy in the near-IR wavelength regime is demonstrated. With active interferometric stabilization, relative phase fluctuations between excitation pulses are suppressed to below 0.01π and the reference phase error is reduced to less than 0.1π during whole data-taking time. The measured signal phase change per excitation delay step has a mean error of 0.02π from the expected value and a standard deviation of 0.06π. Semiconductor quantum wells have been investigated with this technique and the coupling between localized optical excitations is distinctly presented in 2D spectra.

The authors gratefully acknowledge the helpful discussions with D. M. Jonas and M. DeCamp. This work is supported by DOE/BES. S. T. C. is a staff member of the NIST Quantum Physics Division.