Even 100 years after its introduction by Louis de Broglie, the wave-nature of matter is often regarded as a mind-boggling phenomenon. To give an intuitive introduction to this field, we here discuss the diffraction of massive molecules through a single, a double, and a triple slit, as well as a nanomechanical grating. While the experiments are in good agreement with undergraduate textbook predictions, we also observe pronounced differences resulting from the molecules' mass and internal complexity. The molecules' polarizability causes an attractive van der Waals interaction with the slit walls, which can be modified by rotating the nanomechanical mask with respect to the molecular beam. The text is meant to introduce students and teachers to the concepts of molecule diffraction, supported by problems and solutions that can be discussed in class.

## I. INTRODUCTION

Due to its conceptual simplicity, diffraction at single and double slits is often used to illustrate the principles of wave optics. Extending this idea to matter waves then serves to illustrate the wave-particle duality and the superposition principle for massive particles in quantum mechanics. The present work summarizes the analogies and differences of light and particle optics in the context of recent molecule diffraction experiments. We show that a proper description needs to include internal particle properties such as the polarizability, even though the de Broglie wavelength contains only information about the center of mass motion.

Matter-wave diffraction through a few slits ( $ N = 1 \u2212 3$) has been demonstrated for
electrons,^{1–3} neutrons,^{4,5} atoms,^{6–9} and molecules,^{10,11} using mostly micro- or nanopatterned membranes. The possibility
to realize even complex structures with nanometer precision in these materials led to the
fabrication of slits^{6} and gratings,^{12} sieves,^{13,14} and holograms.^{15} They are used to focus beams of neutral helium^{16} and study weakly bound clusters^{17,18} as well as higher-order matter-wave
interference.^{11,19} Nanomechanical
gratings became essential for three-grating interferometers with electrons,^{20,21} neutrons,^{22} atoms,^{23} and molecules^{24–26} up to masses
beyond 25 000 u—and even for experiments with antimatter.^{27}

Here, we put molecule diffraction at nanomechanical masks into a pedagogical context. These
experiments can serve in high school and undergraduate teaching, since they can be related
directly to textbook knowledge of classical optics. Our work expands on an earlier
discussion of fullerene diffraction in this journal in 2003,^{28} explaining in more detail how to prepare and quantify transverse
and longitudinal coherence of the molecular matter wave. To visualize the wave-particle
duality, we use fluorescence imaging: Each molecule can be identified as a single particle
on the detector, and a large number of molecules then lead to the emergence of a diffraction
pattern, as shown before for single photons^{29} and electrons.^{30}

While diffraction of molecules and light share many fundamental features, we also see
pronounced differences resulting from the particles' mass and electronic structure.
Molecules fall visibly in the gravitational field, and they interact with the grating walls
because of van der Waals forces. These forces are studied by rotating the grating relative
to the molecular beam, which change with the average distance between the molecules and the
slit wall. The role of the internal molecular structure opens numerous interesting
questions, which are treated as problems and solutions for class work in the supplementary
material.^{31}

## II. THEORY

### A. Fraunhofer diffraction

Like many textbooks, we discuss diffraction through a thin mask in the far-field
approximation, that is, in the Fraunhofer regime. In this regime, the curvature of the
wavefronts can be neglected, and the diffracted intensity pattern is described by the
Fourier transform of the transmission function of the diffracting mask (see the
supplementary material^{31}). Assuming an
incident plane wave, the far-field assumption is valid when $ L 2 \u226b w 2 / \lambda dB$,
where *L*_{2} is the distance between the mask and the detector, *w* is the width of the coherently illuminated mask, and $ \lambda dB$ is the de Broglie wavelength (see Fig. 1). For *w* = 300 nm and $ \lambda dB \u2243 3 \xd7 10 \u2212 12$ m, the transition region is <50 mm and, thus, considerably smaller than $ L 2 = 590$ mm in our experiments.
However, even when this requirement is not met, the far-field may still be a reasonable
approximation (see the supplementary material^{31}).

*m*and velocity

*v*with Planck's constant

*h*. If we neglect all internal properties, the intensity distribution

*I*at an angle

*θ*depends on the grating period

*d*, the slit width

*s*, and the number

*N*of coherently illuminated slits

*N*-slit interference.

*s*. This induces a transverse momentum uncertainty with a full width at half maximum (FWHM) of

^{31}). By measuring the beam's width, that is, its position uncertainty at the detector, we can, thus, extract $ \Delta p$.

^{4,10}Within this envelope, a multipath interference pattern is formed when the wave is diffracted at two or more slits. Increasing the number

*N*of illuminated slits sharpens the principal interference fringes and causes the occurrence of

*N*− 2 secondary maxima. The expected diffraction patterns behind a single, a double, and a triple slit are shown in Fig. 2.

### B. Coherence

In order to observe multislit diffraction, the wavelets along different paths from the source to the detector need to have defined phase relations, i.e., they need to be coherent. Mathematically, coherence is defined as the normalized correlation function of the waves in space and time. There are several examples for coherent sources: lasers in optics or Bose–Einstein condensates in matter-wave science, where all photons or atoms are highly correlated over macroscopic distances in space and time. However, even thermal light and molecular beams can be prepared to be coherent.

It is often useful to distinguish between transverse and longitudinal coherence. As for laser radiation, they describe the distance along the propagation direction (longitudinal) and perpendicular to it (transverse) over which the phase of the matter wave is correlated.

Transverse coherence can be visualized as the uncertainty in the particle's transverse
position, which makes it impossible to predict through which grating slit the particle
will go. Since both the transverse coherence function and the diffraction pattern behind a
single slit are described by a Fourier transform, they share the same functional form: $ ( sin \u2009 ( x ) / x ) 2$.^{32} If we define the transverse coherence
width *X _{T}* by the distance between the first order minima of
this coherence function, we find $ X T = 2 L 1 \lambda dB / s$, which grows inversely
proportional with the source width

*s*(see the supplementary material

^{31}).

Longitudinal coherence is a measure of spectral purity of the beam and, thus, depends on
the distribution of molecular velocities. If the distribution is too broad, the
interference pattern vanishes, as soon as the constructive interference of one wavelength
overlaps with the destructive interference of another one. This definition leads to the
respective coherence length $ X L = \lambda 2 / \Delta \lambda $. It differs by a factor of $ 1 / 2 \pi $ from a definition based on
propagating Gaussian wave packets.^{33} However, it has proven surprisingly useful in real-world experiments, which are not
necessarily well described by Gaussian wave packets (see the supplementary material^{31}).

While the transverse coherence grows with the distance behind the source, this is not the
case for the longitudinal coherence *X _{L}*, because the spectrum
does not change in free flight. We can, however, increase

*X*by spectral filtering, that is, by selecting molecular velocities.

_{L}## III. EXPERIMENTAL SETUP

To record molecular diffraction patterns, we use the experimental setup sketched in Fig. 1.^{34} A thin film of the molecule phthalocyanine (PcH_{2}, mass *m* = 514.5 u) is coated onto a window, which is mounted onto a vacuum
chamber at a base pressure of $ P < 1 \xd7 10 \u2212 7$ mbar. To launch the matter wave, we focus a laser at *λ* = 420 nm with a 50×
objective onto the film. This is where the first quantum effect comes into play: The
position of the emitted molecules is defined by the spot size *s*_{1} of the focused laser beam, which is twice the laser waist $ s 1 = 2 w 0 = ( 1.7 \xb1 0.5 )$ *μ*m. This localized evaporation can be seen as a position measurement.
Approximating the evaporation spot as a rectangle, we can use Eq. (2) to estimate the associated transverse momentum
uncertainty of $ \Delta p \u2243 3.5 \xd7 10 \u2212 28$ kg m/s. When the molecules reach the grating after $ L 1 = 1.55$ m, this has evolved into
a position uncertainty. Heisenberg's principle, thus, helps us to prepare the transverse
coherence required to illuminate several slits by the same molecular wave function. For
PcH_{2} moving at *v* = 250 m/s, the transverse coherence width
amounts to $ X T \u2243 5.7$ *μ*m, i.e., 57 times the grating periods of 100 nm.

The grating was milled with a focused beam of gallium ions into an ultra-thin membrane of
amorphous carbon with a thickness of $ T = 21 \xb1 2$ nm. A four-axis manipulator
allows for 3D translations of the grating and a rotation around the *y*-axis.
In our ( $ N = 1 \u2212 3$)-slit gratings, the mean
geometrical slit width amounts to $ s geo = 80 \xb1 5$ nm, and the period is $ d geo = 100 \xb1 5$ nm.^{11} The bars between the slits have a cross section of
21 × 20 nm^{2}, only an order of magnitude wider than the diameter of the
diffracted molecule itself (1.5 nm). In the rotation experiments, we use a grating with $ d geo = 101 \xb1 2$ nm and $ s geo = 61 \xb1 1$ nm. The maximum width of the
patterned area amounts to 5 *μ*m and, thus, acts as a collimator in the *x*-direction. Additionally, the beam is loosely collimated by a
piezo-controlled slit (S_{x}) to prevent transmission through
membrane defects.

After passing the grating, the molecules are collected on a quartz plate $ L 2 = 0.59$ m further downstream. To
visualize the molecular density pattern, we excite PcH_{2} with 60 mW of laser light
at 661 nm focused onto a spot size of 400 × 400 *μ*m^{2} and collect
the laser-induced fluorescence with a 20× microscope objective. A band filter transmitting
in the wavelength region from 700 to 725 nm is used to separate the fluorescence from the
laser light, and the image is recorded with an electron multiplying (EM) CCD camera.

In the experiments, we use phthalocyanine (cf. Fig. 1)
due to its high thermal stability and fluorescence quantum yield. This allows detection of
single molecules with high contrast, which is required to visualize even weak signals.^{34} Furthermore, PcH_{2} is non-polar
and, therefore, essentially unperturbed by residual charges in the mechanical masks.^{35}

We use a thermal beam, containing a wide range of molecular velocities and, thus, de
Broglie wavelengths. If all these velocities were to overlap at the detection screen, it
would obscure many of the details present in the diffraction pattern. To prevent this, we
restrict different velocities to different regions of the quartz plate. For that purpose, we
use a horizontal delimiter S_{y} (see Fig. 3). In combination with the source, it defines the free-flight parabolas
of the molecules in the presence of gravitational acceleration *g*. As the
free-fall distance $ H = g ( L 1 + L 2 ) 2 / ( 2 v 2 )$ depends on
the molecular velocity *v*, slow molecules fall further than fast ones, and
thus, they are separated at the position of the detector. In our few-slit diffraction
experiment, the masks themselves serve as velocity selectors, for the case of the
diffraction grating, an additional slit is introduced just before it.

## IV. DIFFRACTION THROUGH A FEW SLITS

### A. Results

In Fig. 4, we show the results of molecular
matter-wave diffraction at a single, a double, and a triple slit. The electron micrographs
of the respective masks are shown in the upper trace, and the molecular diffraction
patterns are shown in the middle trace. They span molecular velocities *v* between 140 and 430 m/s, corresponding to de Broglie wavelengths $ \lambda dB$ between 5.5 and 1.8 pm. Each bright dot corresponds to a single molecule that scatters
thousands of fluorescence photons during detection. At first glance, we observe a
qualitative difference between the single-slit pattern and the other two: Diffraction at a
single slit leads to a structureless, broad signal while the double- and triple-slit
patterns exhibit a sub-structure. This is what we expect based on Eq. (1) and Fig. 2.

To illustrate the level of detail in the patterns, we vertically sum over the velocity
band between 140 and 150 m/s as shown in the lower trace of Fig. 4. Here, all patterns share the same envelope resulting from single-slit
diffraction. The triple slit pattern also reveals the expected secondary maximum in
between the principal diffraction orders. Furthermore, the width of the zeroth diffraction
order decreases with increasing *N* from 15.6 ± 0.1 *μ*m for
the double slit to 12.0 ± 0.1 *μ*m for the triple slit.

It is often stated in textbooks that the wavelength has to be comparable to the grating
period to observe interference. However, in the current experiments $ \lambda dB$ is five orders of magnitude smaller than *d*, and still we observe
high-contrast interference. To achieve this, we have to fulfill three conditions.

First, the transverse coherence has to be large enough to engulf all slits that shall
contribute to multislit interference. As discussed before, *X _{T}* covers dozens of grating periods and, thus, exceeds the minimum requirement by far.

Second, the collimation angle has to be smaller than the diffraction angle $ \theta = \lambda dB / d$ to prevent the diffraction
orders from overlapping. In our experiment, the width of the source ( $ s 1 = 1.7$ *μ*m) and the grating (*w* = 280 nm for the triple slit) at
a distance of $ L 1 = 1.55$ m define a collimation
angle of $ \theta = ( w + s 1 ) / L 1 = 1.3$ *μ*rad. This is well below the diffraction angle of 31 *μ*rad for PcH_{2} moving at 250 m/s.

Finally, the resolution of the detector has to be sufficient to resolve the pattern. This
is achieved using fluorescence microscopy: Each pixel in Fig. 4 images 400 × 400 nm^{2} on the fluorescence screen.^{34} Curve fitting allows to determine the
barycenter of the fluorescence curve with 10 nm accuracy and hence can easily resolve
features on the *μ*m-scale, as observed here.

### B. Differences between light and matter waves

The patterns in Fig. 4 display many features also
expected for light: the quantitative separation of the principal diffraction peaks, the
emergence of the intermediate peaks for *N* = 3, and the narrowing of the
principal diffraction orders with increasing *N*. However, there are also
some major differences. The most outstanding feature is that the diffraction orders are
not parallel but bent. Our source emits a distribution of molecular velocities, which is
reflected in the spread of de Broglie wavelengths $ \lambda dB$.
As the molecules follow free-flight parabolas in the gravitational field, all diffraction
orders except the zeroth should be curved, as can be seen in Fig. 4.

The narrowing of the diffraction orders with increasing *N* is limited in
our experiments. In the model of Eq. (1),
the diffraction orders get sharper as long as *N* increases. In the
experiments, however, the width of the diffraction orders has a lower bound defined by the
transverse collimation angle, i.e., deviations from the assumption of an incident plane
wave.

Also the envelope function hints at differences in the diffraction mechanism. From the
micrographs, we extract a geometrical slit width $ s geo = 80 \xb1 5$ nm. According to Eq. (2), this should lead to a diffraction envelope
with a FWHM of 35 ± 2 *μ*m at the detector for *v* = 145
m/s. However, we observe a width of 53 ± 1 *μ*m, corresponding to an
effective slit width $ s eff$ of 53 ± 1 nm. The reason for this reduction is the van der Waals, or more generally, the
Casimir–Polder interaction.

## V. VAN DER WAALS INTERACTIONS

^{36}As the maximum distance between molecule and the nearest grating bar is 40 nm during transmission, we are in the short range limit, known as the van der Waals interactions. Here, the induced dipole moments interact with their mirror images in the material. While the attractive potential scales with $ 1 / x 6$ between two isolated particles, we have to integrate over the half-sphere of the grating, resulting in a potential, which scales with $ 1 / x 3$.

^{37,38}Approximating the grating thickness

*T*as infinite, the potential $ V pot$ between the molecules and the grating can be written in the form $ V pot ( x ) = \u2212 C 3 / x 3$. The factor

*C*

_{3}includes the frequency-dependent polarizability of the particle and the dielectric function of the material grating, i.e., the response of both interaction partners to oscillating electric fields. This leads to a position-dependent phase $\varphi $, which is imprinted onto the molecular matter wave in each slit

^{4}times the size of the diffracted particles.

^{39}For ultra-thin gratings, however, the molecule can actually be thicker than the grating itself.

^{40}Thus, the approach to the grating and the departure from it also have to be considered.

^{41}

To fully describe the interaction, we would have to characterize the molecule, the grating,
and the interaction between them to a high level.^{41,42} Such an analysis is very demanding: Each molecule consisting of *n* atoms has $ 3 n \u2212 6$ vibrations and many (often
up to 500) rotational levels excited. Flexible molecules can adopt a number of different
conformations, which may interconvert within picoseconds. Furthermore, the polarizability is
often not isotropic, even for rigid molecules such as PcH_{2}. In consequence, the
force depends on the orientation of the molecule during the transit through the grating.
Moreover, charges implanted in the grating material may lead to an attractive force several
times stronger than expected.^{41}

Here, we resort to a phenomenological analysis: Close to the grating walls, the phase shift
becomes so large that small position changes cause large phase fluctuations and the
interference terms are averaged out. In a few nanometer distance, the molecule may even be
adsorbed by the grating.^{43} Hence, we
divide the slit into two regions: In the center $\varphi $ is small and multislit diffraction is
possible. Close to the grating walls, however, the molecule cannot contribute to the
diffraction pattern any more. In this picture, the influence of the van der Waals
interactions reduces the slit width from $ s geo$ to an effective slit width $ s eff$.^{39}

There are several options to modify the van der Waals interactions. First, altering the
grating material changes the coefficient *C*_{3}, which determines
the phase shift. Second, one can minimize the grating's thickness *T*—ultimately to just a single layer of atoms. This has recently been
demonstrated using patterned single-layer graphene, which was stable enough to withstand the
impact of fast molecules, and to yield high contrast molecule diffraction patterns.^{40} Finally, we can change the molecule-grating
distance and interaction length by rotating the grating.

## VI. DIFFRACTION THROUGH A ROTATED GRATING

Rotating the grating modifies not only the effective grating period $ d eff$ and slit width $ s eff$ but also the interaction time and distance between the molecule and the grating walls. The
molecules get close only to the edges of the grating, as shown in Fig. 5. This effect has been used to characterize nanomechanical gratings,^{44,45} and the ensuing reduction in slit
width was key to study weakly bound clusters.^{46} In these experiments, the maximum angle of 42° was limited by the
membrane thickness.^{45} Here we use an
ultra-thin grating with a thickness of $ T = 21 \xb1 2$ nm and a large opening
fraction to achieve rotation angles up to 60°. This reduces the effective period by a factor
of $ cos (60\xb0)=0.5$.

### A. Results

The diffraction patterns recorded at $ \theta grat=0\xb0,\u200440\xb0$, and 60° are shown in Fig. 6. They span molecular velocities *v* from
500 to 110 m/s, corresponding to de Broglie wavelengths $ \lambda dB$ between 1.6 and 7.1 pm. For $ \theta grat=0\xb0$, the pattern is dominated by the zeroth
and both first diffraction orders. Rotating the grating broadens the envelope function and
shifts the position of the diffraction orders. For $ \theta grat=40\xb0$, the effective grating period $ d eff = d geo \u2009 cos \u2009 ( \theta grat )$ is
reduced to 77 nm, resulting in larger diffraction angles. Due to the grating thickness,
the slit width is reduced from $ s geo = 61$ nm to $ s eff = 32$ nm. This confinement of
the matter wave in the effective slit widens the single-slit envelope and leads to a
stronger population of higher diffraction orders. In consequence, diffraction up to the $\xb16th$ orders can be observed. At 60°, the
effective period $ d eff$ is half of the geometrical one, and the slit width is reduced by a factor of 5. At this
angle, the diffraction envelope is about five times wider than under normal incidence.

We show typical traces for $ v \u223c 290$ m/s in the right column
of Fig. 6. To assess $ s eff$,
we fit a Gaussian to the maxima of the diffraction orders and convert its FWHM to the
corresponding slit width utilizing Eq. (2).
The width of the signal increases from $39\xb11$ (0°) to $184\xb12\u2009\mu $m (60°), corresponding to a decrease in $ s eff$ from 36 ± 1 to 8 ± 1 nm. Comparing the effective slit widths to the geometrical one (Table I) shows that the difference between them
decreases with larger rotation angle: While the difference amounts to 25 ± 2 nm at
perpendicular angle of incidence, it is reduced to only 4 ± 3 nm at 60°. However, such
values have to be treated with care. Prior experiments have shown that the width of the
envelope may be smaller than expected for slit widths of a few nanometer.^{45} In consequence, $ s eff$ extracted for 60° represents an upper bound.

. | 0° . | 40° . | 60° . |
---|---|---|---|

$ s geo$ (nm) | 61 ± 1 | 32 ± 2 | 12 ± 2 |

FWHM (μm) | 39 ± 1 | 69 ± 1 | 184 ± 2 |

$ s eff$ (nm) | 36 ± 1 | 20 ± 1 | $ 8 \xb1 1 $ |

$ \Delta ( s geo \u2212 s eff )$ (nm) | 25 ± 2 | 12 ± 3 | $ 4 \xb1 3 $ |

. | 0° . | 40° . | 60° . |
---|---|---|---|

$ s geo$ (nm) | 61 ± 1 | 32 ± 2 | 12 ± 2 |

FWHM (μm) | 39 ± 1 | 69 ± 1 | 184 ± 2 |

$ s eff$ (nm) | 36 ± 1 | 20 ± 1 | $ 8 \xb1 1 $ |

$ \Delta ( s geo \u2212 s eff )$ (nm) | 25 ± 2 | 12 ± 3 | $ 4 \xb1 3 $ |

## VII. SUMMARY AND OUTLOOK

We have demonstrated molecular diffraction at a single-, double-, and triple slit as well
as a rotated nanomechanical grating. Within the framework of the de Broglie hypothesis, the
patterns agree astonishingly well with predictions from general wave optics, as used for
light. However, we also observe pronounced differences, associated with the molecular mass
and complex internal dynamics: Molecules fall visibly in the gravitational field and they
are attracted by nearby walls. In the experiments presented here, the de Broglie wavelength
ranges between $2\u2009\u2009and\u20096$ pm. Even though it is smaller than each
molecule by about three orders of magnitude, we can see diffraction and a high-contrast
interference pattern: Grating diffraction, thus, translates a relative path length
difference of a few picometers into peak separations on the order of dozens of micrometers.
This is a magnification by more than 10^{6}. In this respect, our experiments
resemble small-angle X-ray scattering, aimed to reveal long range order in bio-systems. We
have shown how this magnification can be utilized to visualize the force due to the van der
Waals interaction, which is here on the atto-Newton level. It is interesting to see that
matter-wave-based quantum technologies—using full multigrating interferometers—have started
to generate impact in force and acceleration sensing applications.^{47–49}

Matter-wave diffraction requires delicate setups, making it challenging for students to
gain hands-on experience. A good alternative are online simulators, which provide a detailed
lab environment and offer the possibility to perform realistic experiments life in
class.^{50,51}

The similarities and differences in the diffraction of matter and light are a good starting
point to introduce matter-wave diffraction in introductory classes on quantum physics. To
facilitate this, we include in the supplementary material^{31} a number of problems and solutions related to our present
experiments.

## ACKNOWLEDGMENTS

The authors thank Thomas Juffmann and Joseph Cotter for work on that experiment as well as Yigal Lilach for writing the masks. This project has received funding from the Austrian Science Fund (FWF) within Project No. P-30176.

## REFERENCES

*Principles of Optics*

*Atom Interferometry: Proceedings of the International School of Physics “Enrico Fermi”*