1. INTRODUCTION

Talbot–Lau interferometry has attracted significant attention in the optics community, since it allows one to coherently generate self-images of periodic structures using spatially incoherent light sources. This is of practical relevance in classical light optics [1,2], and it opens new avenues in fields of research in which coherent sources or refractive optical elements are not readily available. This holds for x-ray physics [3,4], plasmonics [5], and ultrasound imaging [6]. For the same reason Talbot–Lau interferometry was proposed to be a promising tool for matter waves, particularly for massive objects [7]. The idea has been successfully demonstrated in experiments with electrons [8], atoms [9 –12], and hot molecules [13,14].

Many of these earlier Talbot–Lau experiments were built using microfabricated diffraction gratings. However, complex particles of high polarizability are best manipulated using masks of light. This avoids the perturbing influence of van der Waals interactions or static charges that occur when particles transit material masks with slits as narrow as 100 nm [15,16,13]. In response to the need, standing lightwave phase gratings [17,18] and absorptive photoionization gratings [19] as well as photofragmentation gratings [20] have been introduced to quantum optics experiments with macromolecules and molecular clusters. Especially, pulsed optical diffraction elements offer high versatility since the light–matter interaction can be timed with nanosecond precision and tuned via the laser wavelength, energy, or pulse length.

Here, we revisit the concept of optical time-domain matter-wave interferometry (OTIMA), which uses the periodic selection, diffraction, and detection of nanoparticles in a series of three pulsed standing lightwave gratings [19,21]. We focus on the practical limits imposed by the specifications of available laser systems and optical components at a wavelength of 157 nm. Particular attention is paid to the influence of a running wave that is superimposed to each standing lightwave grating. This offset is caused by the finite laser coherence and the mirror reflectivity. We further study the relevance of the local mirror roughness and demonstrate how to cope with global mirror surface deformations (flatness) on the level of 10 nm. The improved characterization of the OTIMA interferometer allows us to gain new information about optical cross sections and polarizabilities of particles such as molecular clusters of anthracene and ferrocene.

2. EXPERIMENTAL LAYOUT

The experimental sequence is as follows: molecules are evaporated inside an Even–Lavie valve [22] in the presence of high-pressure seed gas (1–7 bar of neon). A nozzle releases short (30 μs) dense pulses of molecules that subsequently cool and cluster in the adiabatic expansion. With a mean velocity of around $900{\mathrm{ms}}^{-1}$, the particle beam is skimmed, collimated, and transmitted into the interferometer chamber via two pumping stages. In our present work, we focus on clusters of anthracene (${\mathrm{Ac}}_N=({\mathrm{C}}_{14}{\mathrm{H}}_{10}{)}_N$) and ferrocene (${\mathrm{Fc}}_N={({\mathrm{C}}_{10}{\mathrm{H}}_{10}\mathrm{Fe})}_N$), since their volatilization and ionization properties are well suited for our analysis. For the measurements presented here, the molecular samples were heated to 430 and 500 K, respectively. A typical mass distribution of ferrocene clusters is shown in Fig. 1(b).

 

Fig. 1. Setup for cluster interferometry using three pulsed, absorptive standing lightwave gratings. (a) Mirror deformations shift the nodes of the standing waves within the laser spot. This reduces the fringe visibility due to averaging over the phase-shifted interference patterns. This is indicated by the solid and dashed semiclassical paths for two particles that start with the same velocity and direction but at different positions. A mirror reflectivity $\mathcal{R}<0.96$ and limited laser coherence are the reasons why a running wave overlays the periodic gratings. The figure is drawn not to scale to illustrate the effects of minuscule mirror deformations. (b) Mass spectrum of ferrocene clusters that are detected after the third grating pulse using VUV photoionization in a time-of-flight mass spectrometer. We compare the interference signal $S_{\mathrm{Int}}$ (blue dashed line) with a reference signal $S_{\mathrm{Ref}}$ (red solid line), as shown in the inset for the model system ${\mathrm{Fc}}_7$. (c) The normalized signal contrast $S_N$ (see text) is shown for the clusters ${\mathrm{Fc}}_3$ to ${\mathrm{Fc}}_9$.

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OTIMA interferometry, as depicted in Fig. 1(a), combines three VUV photodepletion gratings (${\mathrm{G}}_1–{\mathrm{G}}_3$). These standing lightwave gratings are formed by retroreflecting three fluorine (${\mathrm{F}}_2$) excimer laser beams ($\lambda =157.6\mathrm{nm}$, $E\approx 3\mathrm{mJ}$, $\tau =8\mathrm{ns}$) from the surface of a single 5 cm diameter ${\mathrm{CaF}}_2$ dielectric mirror. Each beam has a flat-top profile focused to an area of $A=10\mathrm{mm}\times 1\mathrm{mm}$, and neighboring beams are separated by 20 mm. The direction of the elongated waist is aligned with the cluster beam. At a photon energy of 7.9 eV (157 nm), the clusters efficiently ionize or fragment after absorption of a single photon. Since both processes deplete the particle beam, the gratings represent absorptive (beam depletion) masks with a period of $d=\lambda /2=78.8\mathrm{nm}$. Clusters that survive the interferometer sequence are post-ionized with a fourth ${\mathrm{F}}_2$ laser pulse and detected by time-of-flight mass spectrometry. This allows us to compare particles with different masses and optical properties in one and the same experimental run.

The absorptive character of grating ${\mathrm{G}}_1$ confines the molecules in a comb of starting positions, so that their wave functions expand over several nodes and antinodes of the standing lightwave by the time the second grating is triggered. Rephasing of the wavelets behind ${\mathrm{G}}_2$ leads to multipath interference and to the formation of a cluster density pattern, which can be resolved with the third grating pulse ${\mathrm{G}}_3$. This final pulse samples the interference pattern and transmits only those clusters whose wave function is aligned with the grating nodes. As in other realizations of Talbot–Lau interferometry, the coherent self-imaging of the diffraction grating occurs on a characteristic time scale, the Talbot time $T_T=m{d}^2/h$, where $m$ is the particle’s mass, $d$ the grating period, and $h$ Planck’s constant.

We measure the fringe visibility by recording the transmitted number of particles behind ${\mathrm{G}}_3$ as a function of the cluster mass. For this, we choose a fixed pulse separation time $T$ between two grating pulses and use two complementary measurement settings to visualize interference. In the interference setting, the cluster signal $S_{\mathrm{Int}}$ is recorded, while the two pulse separation times $T=T_3-T_2=T_2-T_1$ are kept equal to within $\mathrm{\Delta }T<1\mathrm{ns}$. Interference then enhances or reduces (depending on the phase) the cluster signal for masses close to the Talbot criterion. In the reference mode ($S_{\mathrm{Ref}}$), we set an imbalance in the pulse separation times by $\mathrm{\Delta }T>50\mathrm{ns}$. For the given divergence and mass range in the molecular beam, this is sufficient to wash out the interference pattern [23].

We define the cluster interference contrast as the normalized signal difference $S_N=(S_{\mathrm{Int}}-S_{\mathrm{Ref}})/S_{\mathrm{Ref}}$, with $S_N\in$ $[-1,1]$ [Fig. 1(c)]. In the hypothetical case of a perfectly flat mirror surface and in the absence of external accelerations, $S_N$ is positive and equal to the theoretical visibility $\mathcal{V}$ of the interference pattern (see Section 4). For a deformed mirror [see Fig. 1(a)], however, $S_N$ is given by the visibility of the cluster density pattern at ${\mathrm{G}}_3$ and the relative displacement $\mathrm{\Delta }D=\mathrm{\Delta }{x}_1-2\mathrm{\Delta }{x}_2+\mathrm{\Delta }{x}_3$ of this Talbot image with respect to ${\mathrm{G}}_3$. Maximal constructive interference occurs for $\mathrm{\Delta }D=zd$ and maximal destructive interference for $\mathrm{\Delta }D=(z+1/2)d$, with $z\in \mathbb{Z}$. The grating shifts $\mathrm{\Delta }{x}_i$ ($\mathrm{i}=\mathrm{1,2},3$) refer to a common zero line. In the present experiments, the corrugated mirror surface leads to negative $S_N$. Inertial or electromagnetic accelerations $a$ may introduce additional fringe displacements, which are, however, negligible for the parameters of our present experiments. If the mirror surface deformation exceeds 5–10 nm in the small effective region where the light pulse and the cluster cloud interact, different particles will experience differently shifted interferometers, as indicated by the two semiclassical paths in Fig. 1(a). The effective interaction region is set by the width of the detecting laser beam, i.e., 1–2 mm. Averaging will, thus, lead to reduced visibility of the density pattern at the time of ${\mathrm{G}}_3$. We account for the shift and reduction of the fringes by multiplying the theoretical visibility $\mathcal{V}$ [Eq. (7)] by the fringe factor $\mathrm{\Omega }\in$ $[-1,1]$, so that $S_N=\mathrm{\Omega }·\mathcal{V}$. On the one hand, this factor contains the interferogram shift for a single particle relative to ${\mathrm{G}}_3$. On the other hand, $\mathrm{\Omega }$ takes into account the reduction of the visibility due to averaging overshifted interferograms. It is determined by the mirror geometry in combination with the chosen pulse timings $T_1$, $T_2$, $T_3$ and the length of the detected cluster cloud.

3. MODELING FINITE MIRROR REFLECTIVITY AND FINITE LASER COHERENCE

The 1D intensity profile of a standing lightwave along $x$ is ideally described by $I(x)=4I_0{\cos }^2(2\pi x/\lambda )$, with $I_0=E/(A\tau )$ the incident laser intensity. In reality, however, we need to take into account imperfections of the interferometer mirror and the grating lasers. At VUV wavelengths around 157 nm, state-of-the-art ${\mathrm{CaF}}_2$ mirrors are limited to a reflectivity of $\mathcal{R}=96\%$. This may further degrade in the presence of hydrocarbon contaminations [24]. In addition, our free-running ${\mathrm{F}}_2$ lasers (EX 50, Gam Laser Inc.) have a limited longitudinal and transverse coherence. They are expected to emit dominantly at 157.63 nm (ca. 90% of the total intensity) and 157.52 nm (ca. 10%) with a linewidth of $\mathrm{\Delta }\lambda \approx 1\mathrm{pm}$ [25]. This corresponds to a longitudinal coherence length of about 1 cm. The transverse coherence is determined by the width of the emitting aperture and amounts to 50–100 μm at the location of our interferometer mirror in 2 m distance from the laser exit.

These imperfections add an incoherent sum of running waves to the standing wave light field, which reduces visibility $V_{\lambda }$ of the laser grating. This reduces, in turn, the expected interference contrast of the diffracted matter-wave in dependence of the particle’s VUV absorption cross section. Inhomogeneities within each grating, due to local roughness of the mirror surface, cause each individual particle to average over different standing wave phases during the laser pulse. Similarly, a divergent or tilted (with respect to the mirror surface) particle beam as well as mirror vibrations can lead to averaging over grating phases during the 8 ns pulse. For example, particles with a velocity of $1000{\mathrm{ms}}^{-1}$ orthogonal to the grating vector will accumulate over 8 μm of the standing light field. Thus, a particle beam with a divergence of 1 mrad will average the field over 8 nm in the grating direction. These effects strongly depend on the absorption cross section of each particle at the laser wavelength $\lambda$.

To account for this running wave contribution in our model, we distribute the incident laser intensity among a one-directional nonreflected contribution, $I(x)=(1-\mathcal{R})I_0$, two counterpropagating running waves $I(x)=2\mathcal{R}(1-\mathcal{C})I_0$, and the standing lightwave $4\mathcal{R}\mathcal{C}{I}_0{\cos }^2(2\pi x/\lambda )$. We quantify the coherent contribution by $\mathcal{C}\in$ $[0,1]$:

The transmission function $t$ is given by the cluster survival probability, which is the probability of not absorbing any photon during the grating pulse. The mean number of absorbed photons per pulse depends on the optical absorption cross section ${\sigma }_{\lambda }$ at $\lambda =157\mathrm{nm}$. It is given by $n(x)=I(x)\lambda {\sigma }_{\lambda }\tau /(hc)$. Assuming Poissonian statistics and averaging the transmission function $t$ over one grating period, we obtain

4. INFLUENCE OF THE SECOND GRATING

The established interferometer theory [26] describes the fringe visibility as a function of the molecular beam depletion in all three gratings and includes the dipole interaction in ${\mathrm{G}}_2$ between the particle’s optical polarizability ${\alpha }_{\lambda }$ and the electric laser light field. This allows us, in principle, to extract the particle’s optical properties from the interference contrast as a function of the laser power in ${\mathrm{G}}_2$.

In a purely absorptive grating, the difference between the mean number of absorbed photons in the antinodes and the nodes of the standing wave $n_0$ determines the effective open fraction of the grating and the interference contrast. The action of the first and third grating in a Talbot–Lau interferometer is fully characterized by $n_0^{(1)}$ and $n_0^{(3)}$.

In the second grating, the electric field additionally imprints a relevant spatially dependent phase shift onto the traversing matter waves, which is given by

It is convenient to introduce the dimensionless parameter $\beta$ as the ratio of the absorption cross section and the optical polarizability ${\alpha }_{\lambda }$:

The signal behind the third grating in a symmetric Talbot–Lau configuration, as shown in Fig. 1, can be Fourier expanded and expressed as a function of the relative grating shift $\mathrm{\Delta }D$ (see [21] for details):

The visibility of the fringe pattern is then defined as

If the pulse separation time fulfills the Talbot criterion $T=l·T_T$ (with $l\in \mathbb{N}$), the interference pattern is unaffected by the phase contribution, since all terms containing $\beta$ in Eq. (6) drop out. In this case, the visibility $\mathcal{V}$ is only determined by $n_0^{(i)}$ of the three gratings, as seen in Fig. 2. For symmetric pulse delays deviating from the Talbot time, $T_2-T_1=T_3-T_2\ne l·T_T$, the optical polarizability ${\alpha }_{\lambda }$ influences the interference contrast. The time dependence of the contrast would theoretically allow us to extract $\beta$ from such curves. Experimentally, however, changes of the time delay cause changes of the grating phases because of the mirror curvature and corrugation. In practice, measuring the contrast as a function of the laser energy at a fixed delay is more robust and allows us to retrieve $\beta$ from a fit to the data.

 

Fig. 2. Theoretical visibility $\mathcal{V}$ for different values of the optical polarizability. (a) Contrast as a function of the pulse separation time $T$, in multiples of the Talbot time $T_T$. It is plotted for varying phase shifts (polarizabilities) in the antinodes of the second standing wave (${\mathrm{\Phi }}_0^{(2)}=7.5$, dotted; ${\mathrm{\Phi }}_0^{(2)}=3$, dashed; ${\mathrm{\Phi }}_0^{(2)}=1.5$, solid). (b) Contrast as a function of ${\mathrm{\Phi }}_0^{(2)}$, i.e., for varying optical polarizability ${\alpha }_{\lambda }$, plotted for three different pulse separation times ($T=0.75T_T$, dashed; $T=0.9T_T$, dotted, $T=T_T$, solid). For $T=T_T$, the particle polarizability does not affect the interference contrast at all, while at $0.75T_T$ the contrast oscillates with ${\mathrm{\Phi }}_0^{(2)}$. The arrows and colors link plot (a) with plot (b). For both panels, the transmission through all three gratings was fixed by setting $n_0^{(i)}=6.$

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5. EXPERIMENTAL RESULTS

In order to extract the optical properties for all clusters, several parameters need to be measured in the experiment: An overall reduction of the interference contrast due to mirror deformations, common to all cluster numbers $N$, is included in the factor $\mathrm{\Omega }$. The visibility $V_{\lambda }$ is assumed to be the same for all gratings, since the laser beams are equally well aligned to the same mirror surface, and the laser properties are known to be insensitive to the detailed laser conditions, such as its gas pressure or temperature [25]. The cluster interference contrast will, however, depend on $V_{\lambda }$ since the grating quality influences the open fraction of the gratings $n_0^{(i)}$. Since the cluster size and structure are relevant, the parameter $\beta$ must be fitted for every cluster number, individually.

In order to extract these parameters from our measurements, we proceed as follows: We record the normalized contrast $S_N$ for all clusters in the accessible mass spectrum simultaneously as a function of the laser power in ${\mathrm{G}}_2$. To attenuate the laser energy, we use the VUV absorption of oxygen [27] in a pressure cell filled with air. This cell is inserted into the evacuated beam guiding system, separated by two ${\mathrm{CaF}}_2$ windows. It allows adjusting the transmitted laser power from a few percent up to 90% of the power at the cell entrance window, without compromising the timing performance or the beam profile. The laser pulse separation time $T$ is set to the Talbot time of the hexamer ($T=17.38\mathrm{\mu }\mathrm{s}$ for ferrocene ${\mathrm{Fc}}_6$, $T=16.65\mathrm{\mu s}$ for anthracene ${\mathrm{Ac}}_6$) such that high contrast is expected around these masses (see Fig. 1). In addition, we record the transmission $t^{(i)}$ through every grating, individually, and feed this information into the theoretical model (Sections 3 and 4; see also [21]). The free parameters are adjusted until the theoretical predictions fit the experimental data. We start with one value of $\mathrm{\Omega }$ and fit $V_{\lambda }$ for the cluster that fulfills the Talbot criterion (the hexamer). This contrast depends only on the absorption in the standing wave ($n_0^{(i)}$), not on the optical polarizability ${\alpha }_{\lambda }$, as shown in Fig. 2. The resulting visibility $V_{\lambda }$ and the measured grating transmissions are used to calculate $n_0^{(i)}$ with Eq. (2) for all cluster numbers $N$. With these values of $n_0^{(i)}$, the individual $\beta$ is fitted for those clusters whose Talbot time does not match the chosen pulse separation time (here, ${\mathrm{Fc}}_7$, ${\mathrm{Fc}}_8$, ${\mathrm{Fc}}_9$ and ${\mathrm{Ac}}_7$, ${\mathrm{Ac}}_8$, ${\mathrm{Ac}}_9$). The minimization of the least-squares fit in $V_{\lambda }$ and all $\beta$ values is repeated for varying $\mathrm{\Omega }$ to find the best match for all parameters. The results of this fitting procedure are shown in Fig. 3 for anthracene and Fig. 4 for ferrocene clusters. The routine converges very well, and we find good agreement between the model and the experimental data.

 

Fig. 3. Measured interference contrast as a function of the mean number $n_0^{(2)}$ of standing wave photons absorbed in the antinode of the second grating. (a) Interference measurements for three different laser energies reveal the shift of the maximal interference contrast toward higher masses for lower laser pulse energy. The triangles link the plots (a) and (b). (b) Normalized contrast for anthracene clusters of size $N=6$–9: ${\mathrm{Ac}}_6$ (gray, dashed–dotted), ${\mathrm{Ac}}_7$ (red, dotted), ${\mathrm{Ac}}_8$ (light red, solid), ${\mathrm{Ac}}_9$ (brown, dashed). The lines are fits based on the model discussed in the text. Error bars represent one standard deviation of statistical error.

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Fig. 4. Quantum interference of ferrocene clusters from $N=6–9$. It is shown as a function of number of absorbed photons in ${\mathrm{G}}_2$, which is also a measure for the open fraction of the grating. ${\mathrm{Fc}}_6$ (gray, dashed–dotted fit), ${\mathrm{Fc}}_7$ (dark blue, dotted fit), ${\mathrm{Fc}}_8$ (light blue, solid fit), ${\mathrm{Fc}}_9$ (blue, dashed fit). Error bars represent one standard deviation of statistical error.

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Figure 3(a) displays the normalized contrast $S_N$ for clusters between $N=3–9$, where the laser pulse energy in ${\mathrm{G}}_2$ is varied to yield an average $n_0^{(2)}$ in the octamer ${\mathrm{Ac}}_8$ of 3.5 (left panel), 2.3 (middle panel), and 1.6 (right panel). Even though the pulse delay time was set to match the Talbot time of ${\mathrm{Ac}}_6$, we find the highest contrast at higher masses. This agrees with the expected shift of the resonance as a function of the particle polarizability, and it includes the contribution of the running wave background. Figure 3(b) traces $S_N$ as a function of the absorbed photon number in ${\mathrm{G}}_2$, for clusters composed of $N=\mathrm{6,7},\mathrm{8,9}$ anthracene molecules.

The results of the model fit are collected in Table 1. We find a value $V_{\lambda }=0.75$, indicating high quality of the standing lightwaves at a distance of 1 mm to the mirror surface. Repeating the same experimental sequence with identical source settings, and the mirror retracted to 2 mm to the cluster beam, we find a decrease in the light grating visibility to $V_{\lambda }=0.70$. This is consistent with a coherence length of the fluorine lasers of about 1 cm. The overall reduction of the fringe visibility by almost a factor of two compared with the ideal theory value shows there is still room for development of high-quality VUV optics. The reduced contrast is partially due to the finite mirror reflectivity ($\mathcal{R}<0.96$). Even larger influence is attributed to the finite mirror curvature. Independent measurements on different mirror species revealed a global mirror curvature of about 6 km, bending the surface by up to 100 nm over 25 mm. Even this tiny displacement across the laser-cluster interaction zone (Fig. 1) may cause measurable phase averaging. To the first order, one might expect $\beta$ to be independent of the cluster number $N$. For localized and noninteracting electrons, the absorption cross section and the polarizability should grow with the number of molecular constituents and their participating electrons. This assumption is essentially confirmed by Table 1. One may see a small decrease in $\beta$ from $N=7$ to $N=9$, both in the case of anthracene and ferrocene. But the data are also compatible with the observation that the $\beta$ parameters vary by more than the one sigma systematic error of the values in Table 1 when the experiment is repeated daily with changing source conditions. Depending on the source properties, such as its opening time, temperature, and the conditions of the adiabatic expansion, the average cluster conformation, absorption cross section, and polarizability may vary [28]. For similar source and interferometer settings, the $\beta$ parameters of anthracene were, however, found to be robust on the level of 10%–20%, in all recent experiments. There is good agreement between the model and many dozens of data points, as well as the reproducibility of the $\beta$ parameters.

Table 1. Optical Parameters of Laser Grating and Molecular Clusters^a

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Based on this successful concept, we can also analyze the behavior of ferrocene clusters in OTIMA interferometry, as shown in Fig. 4. The data quality is comparable with the case of anthracene: the model fits the data well and allows us to determine the optical cluster parameters.

6. CONCLUSION

We have presented new analysis of matter-wave interferometry with pulsed photodepletion gratings in the vacuum ultraviolet regime. Our description takes into account real-world experimental limitations of the mirror flatness and roughness as well as the limited reflectivity of the VUV optics. This is essential for better understanding of the OTIMA concept, which is a generic idea for matter-wave interferometry with complex nanoparticles. Our method allows us to extract for the first time quantitatively the $\beta$ parameter of organic clusters, i.e., the ratio of their absorption cross section and polarizability at 157 nm. This is remarkable, as it is generally demanding to assess optical properties of weakly bound van der Waals clusters in the vacuum ultraviolet.

Matter-wave experiments have been proposed to test the linearity of quantum mechanics with massive particles [26,29 –32]. Such studies require the best possible understanding of all mechanisms, which may cause an experimental deviation from theoretical predictions. This includes quantum decoherence in the presence of molecular collisions, scattering, or emission of photons [14]. The good quantitative description that could be achieved here underlines that it is promising to further proceed with explorations of high-mass quantum interference.