1. Introduction

Two dimensional nuclear magnetic resonance (NMR) spectra are retrieved by propagating radio-frequency pulse sequences with carefully selected relative carrier phases and time delays perpendicularly to a static magnetic field. The signal phase from the induced nuclear magnetization in a sample is dependent on the phase of subsequent pulses. Careful selection of pulse sequences can allow an NMR spectroscopist to distinguish absolute dipole decay and inter-dipole dephasing times, and to isolate homogeneous from inhomogeneous dipole oscillation decay rates [1, 2]. Decay time information is used to provide image contrast for magnetic resonance imaging. In nonlinear optical spectroscopy, phase matching can accomplish analogous tasks, although the few hundred terahertz frequencies of ultrafast lasers move the study of molecular dynamics to a much faster timescale [3, 4, 5, 6]. Modern optical pulse shaping processes make a stricter analogy of NMR, collinear optical phase-cycled electronic spectroscopy, possible [7, 8, 9, 10]. Here, we show how phase and delay controlled collinear femtosecond pulse sequences generated by an acousto-optic modulator are used to retrieve fully phase-cycled two-dimensional spectra of rubidium vapor in both fluorescence and transmission modes with unprecedented speed.

2. Collinear time-resolved spectroscopy

Until recently, non-collinear phase matching techniques have been used to collect nonlinear spectra. Multiple pulses with distinct time delays and different propagation vectors are crossed in a sample. Provided that the sample is large with respect to λ ³, the cube volume of the excitation wavelength, two pulses with sufficiently different propagation vectors will assume all possible relative carrier phases over their crossed paths in a sample region. Third-order polarization associated with three pulse interactions emerges from the sample as a coherent field in a unique phase-matched direction. Pump pulses with variable time separation can be used in ‘transient grating’ experiments, provided they coexist within the dipole relaxation time frame. The amplitude and phase of the resulting nonlinear field is heterodyne detected using a local oscillator to yield the complete time resolved nonlinear polarization.

Recently, it has been shown [7, 8, 14] that a collinear geometry can be used to retrieve similar information through pulse carrier phase modulation. This phase cycling arrangement for the retrieval of third-order nonlinear polarization is shown next to its phase-matching geometry analog in Fig. 1. Although phase matching yields a desirable background-free signal, a few features of collinear spectroscopy make it irreplaceable in certain circumstances. Most obviously, it is possible to use this technology to retrieve nonlinear polarization from samples smaller than λ³, as no ‘spectral grating’ of dipole oscillators is needed to emit a coherent, phase-matched signal.

Collinear nonlinear spectroscopy also allows for transmission detected experiments. Two-dimensional spectroscopy on molecules that do not fluoresce would not excite an off beam-axis detector. As with two-photon microscopy, only photons which have traversed the focal volume are likely to have been involved in four-wave mixing. The absence of these interacted (and therefore absorbed) photons in the transmitted light can then be accounted for by nonlinear interactions in the focal volume. The promise of diffuse transmissive biological imaging with scatter-immune coherent radiation is exciting.

 

Fig. 1. Two-dimensional spectroscopy in phase matching and collinear arrangements. In the phase matching arrangement shown on the top plot, multiple pulses are spatially crossed in a time delayed sequence through an ensemble of dipole oscillators. The resulting nonlinear polarization coherently constructs in a phase matched direction for background free signal detection. In the collinear arrangement, shown on the bottom plot with a blue reference wave used to emphasize the relevance of the relative pulse carrier phases, the first pulse causes all dipole oscillators within the laser bandwidth to oscillate at their natural frequencies. The second pulse can intensify, suppress, or map the oscillations into other coherences depending on its phase. The third pulse serves as a local oscillator. Here, the first and last pulses are 1800 out of carrier phase with respect to the center pulse. τ_1,2 are the time delays, and k⃗ is the propagation vector.

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The third benefit of acousto-optically controlled nonlinear spectroscopy comes from the rapid update rate of acousto-optic modulators. Whereas phase matching techniques use mechanical translation stages, acousto-optic modulators are driven by software controlled radio-frequency (RF) waveforms. Consecutive laser pulses can be formed into any number of separate pulses with any delay and/or phase combinations provided they fit within the allowable time delay window of the shaper. This allows for extremely rapid nonlinear spectrum retrieval rates.

3. Two-dimensional spectroscopy

The four wave mixing nonlinear response function of one or more atomic or molecular transitions can be mapped on a two-dimensional (2D) spectrum. Peak locations and shapes on such a plot can elucidate diverse information such as transition frequencies, relative transition dipole strengths, dynamics, coupling strength, and transition correlation [5, 11, 12, 13, 3].

A useful example of such a nonlinear signal is the 2D photon echo peak. Here, a first interaction creates an oscillating dipole coherence from a ground state ensemble. Localized in-homogeneities within the ensemble promote dipole dephasing, and thus a deterioration of the macroscopically detectable polarization from the sample. The dipoles are allowed to evolve until a second pulse reverses their phases. Because individual dipole inhomogeneities are retained over the duration of the experiment, individual dipoles rephase at their original dephasing rates. When the ensemble has completely rephased, the re-appearing macroscopic polarization emits a photon echo signal whose phase and amplitude are detected with a third pulse. The homogeneous linewidth can be isolated from a Doppler-broadened ensemble through the photon echo signal. The information retrievable from the photon echo peaks in a two-dimensional spectrum is summarized in table 1.

Table 1. Information found in photon echo peaks. The presence, shape, and location of photon echo peaks exhibit useful information about the source molecules, as shown in this table.

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A theoretical 2D spectrum of rubidium vapor in 200 Torr of helium is shown in Fig. 2(b). The axes represent the difference frequencies between the 375 Terahertz (THz) center laser frequency and the transition frequencies of rubidium accessible by the laser bandwidth, shown in Fig. 2(a). This is down-sampled data, where pulses with a 375THz are mixed with atomic transitions which occur at 377.4 and 384.6THz. The difference frequencies (up to 384.6THz -375THz=9.6THz) are sampled by pulses separated by increments of less than 0.5/9.6THz=52 femtoseconds (accounting for the Nyquist sampling criterion).

The vertical axis of the spectrum Fig. 2(b) can be considered a Fourier-transformed free induction decay (FID) fluorescence signal from the first and second pulse-system interactions. The horizontal axis then represents the transform of the FID created by the third and fourth interactions. Cross-peaks represent coherence transfer processes corresponding to higher order polarization. The second and third interactions in our 2D sequence are combined into a single pulse, as the marginal information recovered from temporally separating these interactions is not crucial to demonstrating rapid phase-cycling. The fourth interaction can be regarded as a heterodyned phase and amplitude detection of all present linear and nonlinear polarization.

4. Phase cycling theory

Optical phase cycling theory has been adequately described in other references [8, 7]. A short overview, however, will be sufficient to understanding the work for this paper.

If the propagation vectors of the pulses and induced polarization in a collinear experiment can be ignored, we can define the pulse fields by their frequency ω, amplitude A(t), and phase Φ:

where c.c. is the complex conjugate of the directly preceding term. The laboratory observable collected after each three pulse experiment, whether in form of fluorescence or of absorption, is the total population in each level of Fig. 2(a). The total final population in both excited levels in rubidium, as a sum of all possible relative pulse phases and delays, is:

where ρ_x is the final population in level ‘x’, a_i are the amplitudes of each of the nonlinearities, α, β, and γ are the phase coefficients, Φ_x are the pulse phases, ω_x are the evolution frequencies of the nonlinear signals between interactions, and t_x are the times between the interactions. This equation represents the population created by all orders of nonlinearities for all 36 possible phase/frequency combinations by the three pulse sequence. To isolate particular nonlinearities (such as the photon echo signal), we need only the components of the summation containing the relevant coherence transfer processes. In the phase-cycling process, all signal associated with unwanted coherence transfer processes are subtracted from the total, 36 component signal.

 

Fig. 2. Level structure and linear/nonlinear spectra of rubidium.

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Because the marginal information gained by probing the system with four separate optical pulses is not useful for the demonstration of our technique, we use three pulses and combine the second and third interactions into the second pulse. The last pulse and interaction serves to heterodyne detect the sum of linear and nonlinear polarizations. So, in total, we are talking of three separate pulses containing four laser-system interactions. Though the three interactions shown in Eq. (2) are sufficient to create third-order polarization, we combine the second two interactions in subsequent equations, and treat the third pulse as the fourth interaction.

The constraint for producing observable fluorescence is that the sum total of coherence transfer processes result in the total absorption of one or more photons. In our case, this requires that the sum of the phase coefficients, Φ(and frequencies ω) for the entire coherence transfer pathway, is zero. This means that α+β+γ=0 for our four interactions. For the photon echo rephasing signal, α=γ=1, and, consequently, β=-2. The echo signal varies according to the corresponding factors of the phase exponential in Eq. (2). The goal of phase cycling is to isolate the signal which changes by this factor over the raster scan of the t ₁ and t ₂ dimensions.

If ΩL is the center laser frequency, and ω_a-Ω_L is the offset frequency between Ω_L and the frequency sampled by the time steps t ₁ and t ₂ on the horizontal and vertical axes, then all DC components in the fluorescence show up on the ω ₂=Ω_L and ω₁=Ω_L lines. In this case, the inter-pulse coherences have no effect on the signal. To remove the ω₂=Ω_L line, we could subtract two experiments where the last pulses differ by 180 degrees. Borrowing from NMR notation, where 00 and 1800 pulses are represented by X and X̄, we would use XXX-XXX̄. This means that we would perform two separate experiments: one with all three pulses in phase for a complete scan of t ₁ and t ₂, and one with the third pulse 1800 out of phase for a similar scan. To remove the ω ₁=Ω_L line, we would use XXX-X̄XX. We can combine four experiments to remove both DC lines: XXX-XXX̄-X̄XX+X̄XX̄. This would leave us with a free induction decay signal along the ω ₁=Ω ₂ line of our 2D frequency plot, as well as all nonlinear polarization peaks. In effect, we have subtracted all signals which remain constant for the phase changes of the first and third pulses.

If Y=900 and Ȳ=2700 in carrier phase, the linear peaks are removed and the photon echo signal isolated by the following combination of 16 experiments:

This subtracts all signals which do not follow the coherence transfer pathway of the photon echo signal. This 16-phase cycled time-domain data is then two-dimensionally Fourier transformed to yield a 2D frequency plot.

5. Experimental setup

For our rubidium sample, 64 time separations from 0.13 to about 3 picoseconds in 45 femtosecond steps between the first and second and the second and third pulses are sufficient to map the two 5P transitions. A total of 64×64=4096 pulses are needed to complete a single phase 2D spectrum. For photon echo signal isolation, the first and third pulses are separately cycled through 0, 90, 180, and 270 degrees for each delay step for a total of 16 time-scan experiments.

An acousto-optic pulse shaper is the ideal device for such experiments; it can be used to transform a single input pulse into a sequence of pulses with arbitrary relative phase and delay with each updated acoustic waveform, and therefore permit the retrieval of unique data points from consecutive laser pulses. In our case, this update rate is 10 microseconds. With no limitations in electronics, this means that unique data points can be collected at a rate of 100 Kilohertz. In an acousto-optic pulse shaper, a broadband, ultrashort optical pulse is introduced into a 4f (four focal point) configuration through a diffraction grating (Fig. 3). The dispersed light from the grating is focused onto an acousto-optic modulator. The RF pulse driven modulator diffracts the separated wavelengths through phonon/photon interactions. The diffraction angle is given by:

where λ is the center laser wavelength, ν_RF is the RF diffraction frequency, and υ_ac is the velocity of the RF generated acoustic wave in the modulator. The diffracted pulse is then refocused onto a second grating by a second spherical mirror.

Equation 4 shows a proportionality between the RF modulation wave and the light diffraction angle. The displacement of the diffracted beam on the second grating, x, is proportional to the change in diffraction angle by $\mathit{sin}\left(\theta \right)=\frac{x}{2f}$. From this, where t is time and dt is time change,

and

where δν_RF is the change in RF frequency. In our setup, f=37.5cm, λ=0.8µm, and ${\upsilon }_{\mathit{ac}}=4.2E3\frac{m}{s}$. c is the speed of light in air. This gives a delay of approximately a picosecond for every 4.3MHz change in the RF diffraction frequency. If three different RF frequencies are mixed, amplified, and used to drive the modulator, a single optical pulse entering the shaper will branch into three separate collinear pulses at the foot of the shaper. From the properties of Bragg phase matching in the modulator, the phase of the RF control pulse is directly transferred onto the phase of its resulting optical pulse. This is less useful for single pulse diffraction, but is very helpful in creating multiple pulses: the relative phases of multiple RF frequencies map directly onto the relative phases of the three resulting optical pulses.

 

Fig. 3. Experimental setup. 50fs, broadband pulses enter the shaper at 20KHz. They are spectrally dispersed by an 1800 line/millimeter diffraction grating and focussed to a zero-dispersion line in an acousto-optic modulator by a 37.5 centimeter focal length spherical mirror. The modulator deflects the pulses as discussed in the text. The pulses are recombined through a second identical lens and grating pair. The three pulses created by the three RF waveforms in the acousto-optic modulator are focussed on a rubidium cell by 20cm lenses. Isotropic fluorescence is collected by a large f-number lens and focussed on a photomultiplier tube. The transmission-echo signal is retrieved by isolating light just from the two transitions shown in Fig. 2(a). The transmitted light is filtered by a 4F notch-pass filter which allows only light within 0.3nm of the 5S _1/2 →5P _1/2 and 5S _1/2 →5P _3/2 transitions through. The back-reflected light is picked off and sent into a photomultiplier.

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It is worth mentioning that the diffraction equations shown above assume monochromatic light. This is, of course, not true. The larger the bandwidth of light propagated through the shaper, the more the pulses from the shaper are chirped in time and space. A first-order compensation of this non-ideality can be made by chirping the RF driving pulses in the AOM to instigate similar diffraction angles for all frequency components of the pulse. For the demonstration covered in this paper, such compensation was not necessary.

The pulses are created by two data channels and a marker channel from a Lecroy LW-420A arbitrary waveform generator with 1 mega-sample per channel extended memory. The entire 64×64×16 pulse sequence is loaded into the high speed memory of the waveform generator before beginning data capture. During data taking, channel 1 of the AWG is cycled through all 64×4 time/phase RF waveforms continuously, in synchronicity with incoming laser pulses. Through the sequencing capabilities of the waveform generator, channel 2 steps through 1 of its 64×4 time/phase waveforms at each completion of the channel 1 cycle. The channel 2 marker outputs a constant 80MHz. All three outputs are mixed up by 120MHz to create RF waveforms capable of driving the AOM. The 80MHz from the laser oscillator is divided by 4000 in the laser RF unit to provide the 20KHz trigger for the regenerative amplifier. We separately divide the 80MHz by 8 to provide a 10MHz synchronization clock for our arbitrary waveform generator and our RF frequency upconverting circuit. This synchronization is vital to the success of the experiment: during each laser pulse incidence in the modulator, the phase of the diffracting RF pulses must be reliably governed by the driving electronics. The laser pulses from the amplifier are 10 microjoule and 50fs FWHM. The delayed, shaped pulses used in the experiment are about 100 nanojoules each.

 

Fig. 4. Rapid phase cycling setup electronics. For proper synchronization of the RF drive pulse in the AOM, all RF frequencies must be derived from the 80MHz repetition rate of our oscillator laser. The RF unit for the laser amplifier provides a 20KHz synchronization signal for triggering the boxcar. A frequency divider provides a 10MHz signal to externally oscillate both the 120MHz RF source and the AWG. The Ch2 marker from the AWG is mixed up by the 120MHz to create a 200MHz RF pulse for the AOM. This creates the central of the three sampling pulses for the experiment. The Ch1 and Ch2 outputs are mixed with the 120MHz to create RF pulses which vary from 200MHz. Ch1 derived pulses increase in frequency to create more delayed pulses from the shaper. Ch2 RF pulses create pulses which are less delayed then the center 200MHz pulse. When all three RF pulses are mixed, bandpass (BP) filtered, amplified (AMP), and sent to the AOM, three optical pulses with arbitrary relative delays and phases are created. The 20KHz signal is shown with yellow lines, the 10MHz with green, and the sampling pulse from the Ch1 marker, necessary to allow the computer oscilloscope to sample the data from the boxcar with red. The red arrow from indicates data flow from the boxcar to the scope.

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The rubidium cell has a 0.5cm path length, and is heated to 100 degrees Celcius for this experiment. It contains 200 Torr of helium as a buffer gas. The fluorescence signal is detected using a Hamamatsu R3896 photomultiplier tube, amplified, and collected by a Stanford Research SRS-250 boxcar integrator. The boxcar data is sampled by a Gage CS1610 PCI computer based oscilloscope. The scope’s external clock function accepts a 20KHz trigger constructed by the remaining marker output of the arbitrary waveform generator. A short DC pulse from channel 2 of the arbitrary waveform generator prompts the computer based oscilloscope to begin collecting data at the start of every 65536-pulse sequence. National Instruments Labview and Mathworks Matlab were used to program the AWG and collect and display data.

In published work done in our group [7], a 2D rubidium spectrum with 16 phase cycling took about 20 hours to record. Here we demonstrate full 16 phase-cycled 2D spectrum retrieval in just over 3 seconds. With out the 20KHz limit imposed by our boxcar integrator, we could instantly decrease the 65536-pulse phase-cycled data retrieval time to about 0.6 seconds. Also, it is possible to isolate the photon echo signal from 10 relative phase combinations [18], which would reduce the time to retrieve a photon echo signal from an equal number of sampling points to under 0.4 seconds. This is one great benefit over phase-matched experiments, which rely on mechanical translation stages. It is also worth noting that not all 2D spectra require 64 steps in each dimension. A 32×32 step spectrum could be retrieved, for instance, at an update rate of 10Hz.

6. Results and analysis

Figure 5 shows photon echo data collected in rubidium in both fluorescence and transmission mode. The slightly different peak locations in the two sets of data is derived from a difference in the absolute delay of the RF AOM driving pulses relative to the center of the AOM. Essentially, the systems were calibrated to different center laser frequencies in the two cases. For a better comparison, the RF pulses would need to be calibrated for the same peak locations for both fluorescence and transmission modes. The transmission mode data has the appearance of higher signal to noise because of the narrow-pass filter which limits detected light to the two transmissions. Though true transmission-mode experiments should be done without filtering, this data serves as proof that the echo signal is present in the transmitted beam.

Both data sets are un-averaged. It was found that averaging did not greatly help the rubidium signal-to noise ratio. This indicates that the noise present in the fluorescence mode experiment is related to RF artifacts rather than to the rubidium cell. As this paper seeks to demonstrate rapid 2D data set recovery, averaging would also have increased the time to collect a spectrum and thus defeated the fast-spectroscopy benefit of our setup.

A few conclusions can be inferred from comparing the data with table 1. The most prominent is the presence of cross-peaks in the rubidium data. Indeed, from Fig. 2(a), we see that the 5S _1/2→5P _1/2 and 5S _1/2→5P _3/2 transitions share a ground state. A coherence transfer from one transition, through the ground state, and to the other transition instigated by the second pump pulse results in such peaks. As the excited state lifetimes of the 5P levels in rubidium far exceed the available delay window of the modulator, we were not able to extract meaning from the photon echo peak shapes.

7. conclusion

We have demonstrated rapid two-dimensional phase-cycled data retrieval on rubidium vapor. The three second data rate is impressive for a system capable of retrieving homogeneous linewidth, coupling information, and other four wave mixing signals. The collinear nature of the setup permits transmission mode detection, where nonlinear absorption, rather than fluorescence, is recorded.

 

Fig. 5. Rubidium photon echo data taken in fluorescence and transmission mode. This un-averaged data is retrieved from 16 time-scan experiments combined according to Eq. (3).

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