Overcoming experimental obstacles in two-dimensional spectroscopy of a single molecule Open Access

Two-dimensional electronic spectroscopy (2DES) is a well-established ultrafast technique for studying energy transfer dynamics.^1–3 However, most of these experiments are performed on ensembles of molecules. While ensemble studies reveal the collective behavior, single molecule spectroscopy provides insight into the heterogeneity of the system by transforming the ensemble average into a distribution of spectroscopic properties.^4–6 

The most common form of 2D spectroscopy uses three laser pulses to excite the sample. The emitted coherent radiation is then detected after interference with a fourth pulse acting as a local oscillator. Different nonlinear signals are separated by spatial phase matching. This technique requires a sample volume larger than λ³, so it cannot be used for a single-molecule experiment.⁷ Action-based 2D spectroscopy, on the other hand, uses a fourth pulse to bring the system into a population state from which an incoherent action signal is detected, e.g., a fluorescence photon,⁸ a photocurrent,⁹ or a photoelectron.¹⁰ Although these two techniques are conceptually similar, they are not identical since the interaction of the fourth pulse with the sample results in additional excitation pathways.^11–14 

Fluorescence-detected two-dimensional electronic spectroscopy (F-2DES) uses four collinear pulses to excite the sample and detects the incoherent fluorescence signal.^8,15,16 Different phase-cycling schemes can be used to extract the phase information from the fluorescence signal.^17–20 Because an incoherent fluorescence signal is detected, spatial phase matching and sample volume do not play a role. Thus, this approach can be combined with single-molecule fluorescence spectroscopy to measure the 2D spectra of individual molecules. However, a significant challenge for such an experiment is the limited photostability of the dye molecules at room temperature. In general, most dye molecules lead to about 10⁶ detected photons before permanently photobleaching.²¹ Performing a nonlinear experiment with this limited number of photons is challenging.

Recently, we demonstrated 2D spectroscopy of single dibenzoterrylene (DBT) molecules in a PMMA matrix using fluorescence as a reporter.²² We have shown that about 10⁵ photons are sufficient to measure nonlinear 2D spectra of such a molecule. Here, we discuss the key experimental decisions that led to this result.

A single molecule emits at most one photon per laser pulse. Due to the limited collection and detection efficiency of about 1%, a typical detection rate is about three orders of magnitude lower than the laser pulse rate. The first design decision is, therefore, to use a laser oscillator with a megahertz repetition rate (80 MHz in our case). With a kilohertz system, it simply takes too long to acquire 10⁶ detection events. As a consequence of the low photon rate at the detector, it is convenient to store the arrival time of each photon on a picosecond time scale using a time tagger. Otherwise, one would have to store one intensity value per time interval, which would be zero most of the time. The detection scheme must be photon-efficient. Rapid phase cycling combined with lock-in detection allows simultaneous measurement of all linear and nonlinear processes. Each photon is used to determine linear absorption, rephasing and non-rephasing spectra, and all other mixing products. This requires acousto-optic modulation of broadband laser pulses and lock-in demodulation of a stream of discrete photodetection events. We will discuss how to compensate for the spectral dispersion of the modulators, how to generate the reference frequency, and how to remove the nonlinearity due to detector dead time. The measurement sequence must be fast because a fluorescent molecule bleaches within a few tens of seconds. To this end, we have chosen fast mechanical delay stages to generate the pulse sequence. We will describe how the stage motion can be linearized to interferometric accuracy and how other phase effects can be corrected. Finally, the acquired photon stream contains much more information than just the 2D spectrum. We show how the stream can be sliced by the time the molecules spend in the excited state before photoemission.

Our setup follows the design proposed by Tekavec et al.,⁸ with some modifications. The laser system is a Ti:Sa oscillator (Laser Quantum, Venteon Power) with a repetition rate of 80 MHz and a pulse length of about 20 fs. After pre-compensation of the group delay dispersion in a pulse shaper in double pass configuration,²³ we use two Michelson interferometers cascaded in a Mach–Zehnder interferometer to generate the sequence of four pulses (Fig. 1). Three mechanical delay stages (SmarAct, 2× SLC-2430, 1× SLC-24150) control the delays between the pulses t₂₁, t₃₂, and t₄₃.

However, placing the AOM at the center of curvature of a spherical folding mirror avoids this effect.^24,25 In a ray optics image, all spectral components are reflected back into themselves. In a wave optics image, the wavefront of the Gaussian beam must match the spherical shape of the mirror for perfect reflection, otherwise the focus waist size ω₀ will be different in the second pass [Fig. 2(a)]. In addition, the AOM efficiency depends on the waist size. In our case, the optimal waist size for the AOM is about 150 μm, resulting in a single-pass efficiency of 78%. To keep the overall size of the interferometer small, we have chosen R = 200 mm as the radius of curvature of the spherical mirror. This results in a slightly smaller waist size (⁠$≈100$ μm), which is optimized by the two lenses (L1 and L2) in front of the interferometer. Our double-pass efficiency is 39%, a little lower than the square of the single-pass efficiency. The opening angle of the beam is about 5 mrad, still small enough compared to the deflection angle of 11 mrad. We have chosen quartz as the material for the AOM crystal because it results in comparatively low group delay dispersion (see Table I).

We designed the interferometer in a 3D modeling software (Autodesk Inventor) and built it over a custom-designed plate [Figs. 1(b) and 1(c)], inspired by Ref. 26. The interferometer is relatively compact (baseplate dimensions 35 × 40 cm², 1.5 cm thick). All four arms of the interferometer are perfectly identical. Thermal expansion will thus, to a first approximation, only change the less sensitive “population time” t₃₂ of the Mach–Zehnder interferometer and not the two Michelson interferometers. To drive the quartz AOMs, we need high RF power (≃6 W), which heats up the AOMs and thus the baseplate. We use cast aluminum for the baseplate and copper pedestals to mount the baseplate on the optical table for better heat dissipation. We place the interferometer in an acoustic foam box to reduce the effect of ambient vibration on the interferometer and to dampen thermal fluctuations of the environment. The baseplate temperature is ∼10 K above room temperature.

The interferometer has two outputs. One output is directed to a home-built fluorescence microscope. The light reflected via a dichroic beam splitter (DC) is used to excite the sample using a high NA (1.3) objective (Olympus UPlanFL 100×). The fluorescence signal from the sample is collected by the same objective and detected by two single-photon counting avalanche photodiodes (APD, Excelitas SPCM-AQRH). A short pass (SP) filter and a long pass (LP) filter are used to suppress laser back reflection. The APD photon pulses are registered using a time tagger (Swabian Instruments Time Tagger 20).

The second output of the interferometer is used to generate a reference signal for lock-in detection that effectively stabilizes the interferometer. A grating in the focus of a single-mode fiber coupler selects a narrow spectral range that is detected by using a photodiode. This stretches the pulse in the time domain and allows interference over a large delay range. The accessible range depends on the light intensity on the diode and the tolerable noise level. In our configuration, it is about 200 μm travel corresponding to 2.6 ps delay in t₂₁ and t₄₃. The time interval t₃₂ is spent in many processes in a population state that is phase-independent. In these cases, we can use the full travel of 100 mm, corresponding to 600 ps of delay. Since the spectral resolution is determined by the length of the delay range, the spectral width of the reference signal essentially determines the ultimate spectral resolution of our instrument, which is currently around 2.5 nm. Our choice of AOM double pass, in contrast to other experimental designs,¹⁷ results in only a single reference beam containing the beating between all 4 fields. The reference diode signal will contain six beat frequencies centered at Ω_ij, i ≠ j = 1, …, 4, as shown in Fig. 2(b). Our decision to record photodetection events rather than analog intensity values requires that we also generate reference events from the diode signal. We use three phase-locked loops (details given in the following). Each loop is locked to one of the beat signals Ω₂₁, Ω₃₂, and Ω₄₃, recovering the phase fluctuations across one delay stage. The remaining three beat frequencies can be calculated as a linear combination of the locked ones.

A second type of reference signal is generated by electronically mixing and low-pass filtering the AOM drive signals. This produces beat signals similar to the interference across the delay stages. Another three PLLs are used to detect these electronically mixed phases and compare them to the optical reference phases. This provides information about the drift and vibration of the interferometer and allows us to interferometrically determine the position of the delay stages.

Our experimental design thus differs from Tekavec et al.⁸ in three aspects: double-pass configuration of the AOM, event-based photon counting, and phase-locked loops for phase recovery. The double-pass configuration allows for broadband spectral operation using diffractive acousto-optic devices. The use of photon counters to measure the fluorescence signal provides the high sensitivity required for single emitter detection. As a result of our interferometer design, one reference signal contains all the beating frequencies, and we use three PLLs to individually track the phase variation between the pulse pairs over three delay stages.

The interferometer is only passively stabilized. Fluctuations and drifts in the path length difference are not compensated, but are followed synchronously by a phase-locked loop (PLL). The purpose of a phase-locked loop is to recover frequency and phase of a harmonic signal in the presence of other signals and noise. A feedback controller optimizes the frequency of an electronic oscillator until the phase difference to the signal is zero. Tuning the PLL to a signal frequency requires only tuning the oscillator, leaving all filters unchanged, which simplifies the circuit. Phase-locked loops are used in telecommunications and at the reference input of a lock-in amplifier.

We implemented a Costas loop²⁷ in a field-programmable gate array (FPGA) using the Labview programming environment (National Instruments) on a USB-7856R data acquisition board.²⁸  Figure 3(a) shows the schematic of the loop, which is repeated at a sampling rate of 1 MHz. First, the phase difference between the input signal and a local oscillator is determined by multiplication and low-pass filtering. This step determines which of the three beat frequencies will be locked. The filter has a cutoff frequency of 4 kHz and a roll-off of 24 dB/oct. A notch filter (4 kHz width) helps suppress the closest of the other beat signals. The phase difference is computed by a numerical atan2 function that takes into account the signs of the arguments and thus maps to a full 2π interval. The second component is the loop filter, which we implement as a proportional–integral (PI) feedback controller. It determines the response of the PLL to varying input signals. The third component is the numerical analog of a voltage controlled oscillator (VCO). We increment the phase of the local oscillator at each iteration of the 1 MHz acquisition loop by a fixed value Δϕ₀ equal to the nominal frequency of the reference signal, rolling over at 2π. The output of the PI controller additionally modifies this phase increment within limits corresponding to ±1 kHz frequency. In this way, the loop can only lock to its assigned frequency component. We close the loop by calculating the sine and cosine of the phase. This implementation has the advantage that the loop locks at zero phase difference regardless of the start condition.

With optimized PI parameters (see in the following), we recover the phase of the beat signals once per microsecond. For the time tagger, we generate a trigger pulse when the phase crosses zero. To reduce the jitter to less than 1 μs, we linearly interpolate successive phase values in a faster digital loop to a 5 ns time base. The time delay between the zero crossing of the phase at the analog input and the generated trigger adds to the overall phase response of all other electronics and is zeroed during alignment (see in the following).

The response of the PLL depends on the P and I parameters. To find the optimum values, we use a function generator as PLL input. We generate a sinusoidal signal with a frequency of 20 kHz, frequency modulated by a shift of 100 Hz in a square wave. We set the nominal frequency of the PLL to f₀ = 20 kHz and measure the phase response of the PLL for different I parameters, keeping P = 2000 constant, and for different P parameters, keeping I = 3.3 constant. (The values given have arbitrary units.) The response time τ_R of the PLL is the time it takes for the PLL to lock to the input signal, or in other words, the time it takes for the phase difference Δφ to relax to zero. As expected for a PI controller, we observe that as the I value increases, the response time of the PLL decreases, but at higher I values, the phase response becomes oscillatory and the response time increases again [Fig. 3(d)]. A similar type of phase response is observed for different P values [Fig. 3(e)]. From these measurements, we find the optimal value for our PLL as P = 2000 and I = 3.3, which corresponds to the smallest response time of about 1.35 ms. This means that our PLL can follow all acoustic vibrations of a few 100 Hz frequency, independent of the nominal PLL frequency in the range between 10 and 130 kHz [Fig. 3(f)]. We chose the three beat frequencies from this range as Ω₂₁ = 20 kHz, Ω₃₂ = 68 kHz, and Ω₄₃ = 32 kHz. The corresponding RF frequencies of the AOMs are 78.000, 78.010, 78.044, and 78.060 MHz, as the double-pass configuration doubles the frequency modulation.²⁵ 

Demodulation of the fluorescence signal at Ω₂₁, Ω₃₂, and Ω₄₃ yields the linear signal contribution due to the interaction of fields 1 and 2, 2 and 3, and 3 and 4, respectively. To recover any nonlinear signal, demodulation must be performed at the appropriate mixing frequencies. For example, demodulation at Ω₂₁ + Ω₄₃ yields the non-rephasing contribution, while demodulation at Ω₂₁ − Ω₄₃ yields the rephasing contribution. A 2D Fourier transform of the demodulated interferogram along the t₂₁ and t₄₃ delay gives the 2D spectra in the spectral domain. In case of the non-rephasing and rephasing signal, the PLL operating at Ω₃₂ is not needed, as the system stays in a phase insensitive population state during delay t₃₂. The “double quantum” process,³⁰ for example, is in contrast obtained by demodulating at Ω₂₁ + 2Ω₃₂ + Ω₄₃.

We use closed-loop slip-stick stages (SmarAct) for the three delays. The resolution of the position sensor is specified as 1 nm and the unidirectional repeatability as ±40 nm. Important for the experiment is the deviation between real and measured positions, i.e., the linearity of the stage. The interferometer allows us to measure the stage position independently. For this purpose, we compare the phase of the optical reference signal with that of the electronic reference (taking into account the factor of two due to the AOM double-pass) and convert the phases into spatial distances using the central wavelength of the reference diode. We find periodic and very reproducible deviations of about ±500 nm between the expected and realized stage position [Fig. 4(a)]. One source is the periodic error of the position sensor. Another source is the pitch and yaw rotation of the stage in combination with the ∼9.5 mm beam height above the stage. We use a look-up table to pre-compensate for these deviations, which drastically reduces the stage non-linearity.

Once the stages are linearized, we measure three 1D interferograms along the three delay stages, calculate the spectral phases ϕ_ij(ω_ij), and fit a third-order polynomial.¹⁹ The constant term gives the phase offset at the reference wavelength. It is set to zero to make the linear spectrum real and positive.¹⁹ The linear term corresponds to a time shift. We use it to fine-tune the zero position of the delay stages. Thus, the constant and linear terms are set to zero by the phase correction terms obtained during alignment. The quadratic term in the spectral phase corresponds to the dispersion (GDD) difference between the transmitted and reflected beam due to the finite thickness of the beam splitter. To compensate for this GDD mismatch, we place three compensator plates of appropriate thickness in the three arms of the interferometer. As shown in Fig. 4(b), the GDD difference between the two arms becomes negligible after the addition of the compensator plate.

Care must be taken when dealing with a nonlinear signal because the detector may operate in a nonlinear regime and thus “pollute” the measured nonlinear response of the sample. Single photon detectors have an intrinsic nonlinearity due to their dead time: the time it takes to get ready to detect a second photon after a first photon has been detected. If a photon arrives within this dead time, the detector will not respond, which contributes to a pseudo-nonlinear signal. To test this concept, we measure 2D spectra of continuous-wave laser reflection at 8 × 10⁵ cps (counts per second) on an otherwise blank sample. Ideally, one would detect no nonlinear signal at the demodulation frequencies Ω₂₁ ± Ω₄₃. However, as shown in Fig. 5 (top panel, “default”), we observe some nonlinear signal in the rephasing spectra and a similar observation was also made for the non-rephasing case.

Our detector has a dead time of 35 ns, as shown in the autocorrelation histogram in Fig. 5 (bottom panel left). To test the influence of the detector dead time on the nonlinear signal, we artificially increase the dead time by removing all photons that come earlier than 100 ns after a preceding photon in our photon stream data. We recompute the rephasing spectra with the artificially increased dead time (Fig. 5 top panel, “artificial”). As expected, the nonlinear signal amplitude increases to almost 3 times that of the “default” scenario.

We have assumed Poisson statistics above, which is strictly valid only for a continuous-wave laser as used in the demonstration. In the real experiment we are dealing with single molecule fluorescence emission after pulsed laser excitation. Figure 5 (bottom right) shows the corresponding histogram together with an antibunching model.³¹ It is measured as a cross correlation between two detectors, so that also photon pair separations below the detector dead time are registered. The detector dead time removes photons that follow within less than 35 ns after a first photon, marked in red. For dead time correction, a model needs to estimate the number of these red photons. At low photon numbers and low signal-to-noise ratio of the resulting spectra, the assumption of a constant autocorrelation is certainly sufficient. If the high signal-to-noise ratio of an ensemble experiment requires a very precise estimation of the influence of the detector dead time, one could improve the model by taking into account the details of the correlation. One could artificially extend the dead time to an integer multiple of the laser repetition rate to average over integer multiples of the exponential decays.

Detector nonlinearity becomes relevant when the probability of missing a photon due to dead time approaches the ratio of nonlinear to linear signal amplitude. The latter is a few percent, so photon count rates above about 10⁵ counts per second become critical for our detector (Excelitas SPCM-AQRH). These intensities are high for single molecule fluorescence, but we apply the dead time correction anyway.

As a test system, we used dibenzoterrylene (DBT) molecules in a polymethyl methacrylate (PMMA) matrix to demonstrate the general applicability of our technique. Other dye molecules and biological samples, e.g., light harvesting complexes can be studied in a similar manner. Even though DBT is known to be relatively photostable compared to the other dyes, we are optimistic that the 2D spectra of other single emitters will also be possible to measure as we needed only around 10⁵ photons to measure nonlinear 2D spectra.

One advantage of our technique is that we capture the raw photon stream, which contains a wealth of information.³² One example is the time the molecule spends in the excited state between excitation and emission. A more complex molecule than DBT may have two or more emitting states with different excited state lifetimes.⁶ Thus, photons detected early after excitation would be predominantly from one state and late photons would be predominantly from the other state. Sorting the photon stream data by arrival time after excitation could distinguish these states and give better insight into the energy transfer mechanisms.

To demonstrate this concept, we plot a histogram of the delay τ between photon detection and the previous laser pulse, the so-called time-correlated single photon counting (TCSPC) trace in Fig. 6(b). We observe a single exponential decay corresponding to an excited state lifetime of 5 ns. We split the photon stream into two parts: “fast” and “slow,” corresponding to delays τ smaller and larger than 3 ns, respectively. We choose this cutoff to obtain an almost equal number of photons in both parts. For each part, we compute the 2D spectra separately [Fig. 6(c)]. These two spectra would reflect the difference in the emitting state. For our system, we do not observe any difference in the 2D line shape between the “fast” and “slow” emission times. Only the noise level changes proportionally to $1/Nphotons$⁠, since in our definition the nonlinear signal amplitude is independent of the total number of photons detected. The difference in signal amplitudes between the two spectra lies within this increased noise level.

In time-resolved spectroscopy, decay-associated spectra (DAS) are calculated for each state or species from a time- and spectrally-resolved dataset. The above-mentioned analysis can be seen as a first step in using the photon arrival time as the relevant time axis and the 2D spectrum as the spectral axis. A similar approach has recently been demonstrated for (linear) Fourier spectroscopy and near-infrared photons.³³ 

While coherently detected 2D spectroscopy is a standard technique, its signal is too weak for single emitters. The closest is four-wave mixing spectroscopy of a single epitaxial quantum dot at cryogenic temperatures.³⁴ For soft matter at room temperature, even detecting the absorption of a single molecule is challenging.^35,36 Coherent 2D spectroscopy of single molecules is equivalent to detecting pump-induced variations in these weak signals and thus seems out of reach. The fluorescence of a single emitter is much easier to detect than its absorption. Fluorescence-detected 2D spectroscopy of a single molecule is possible,²² but still challenging.

In this article, we have discussed the key experimental problems that need to be solved to measure 2D spectra of single molecules. First of all, no fluorescence photon should be wasted. Rapid phase cycling together with lock-in detection allows measuring all mixing products simultaneously. To this end, we presented the double-pass configuration for spectrally broadband acousto-optic modulation and an accurate three-channel phase-locked loop for phase recovery. Second, the three slip-stick stages allow almost random access to the three temporal delays, but care must be taken to achieve the required linearity of the position sensor. Finally, artifacts in spectral phase and detector nonlinearity must be removed. Both can be done by appropriate correction schemes. Since the experiment is based on photon counting, all statistical methods of single molecule data analysis³² can be applied. We have demonstrated the slicing of the stream by photon arrival time to potentially distinguish different emitting states. We are confident that such data analysis in combination with an optimized sampling scheme^37,38 and excitation of the molecule by surface plasmons³⁹ will open the door to a wide range of systems, from photosynthesis to plasmonic strong coupling.

We thank M. Theisen, M. Heindl, and C. Schnupfhagn for their work on the pulse shaper, F. Paul for his help in the early stages of this experiment, and R. Weiner for the electronics. We acknowledge the financial support from the German Science Foundation (DFG) via IRTG OPTEXC and Project No. 524294906.

The authors have no conflicts to disclose.

Sanchayeeta Jana: Investigation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). Simon Durst: Investigation (equal); Software (equal); Visualization (equal); Writing – review & editing (equal). Lucas Ludwig: Software (equal). Markus Lippitz: Conceptualization (lead); Software (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.