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Physical Review A

Poisson’s spot with molecules

Thomas J.G. Reisinger, Amil A. Patel, Herbert Reingruber, Katrin Fladischer, Wolfgang Ernst, G. Bracco, Henry I. Smith, Bodil Holst

Published May 14, 2009• 48 citations
Publisher

Abstract

In the Poisson-spot experiment, waves emanating from a source are blocked by a circular obstacle. Due to their positive on-axis interference an image of the source (the Poisson spot) is observed within the geometrical shadow of the obstacle. In this paper we report the observation of Poisson's spot using a beam of neutral deuterium molecules. The wavelength independence and the weak constraints on angular alignment and position of the circular obstacle make Poisson's spot a promising candidate for applications ranging from the study of large molecule diffraction to patterning with molecules.

DOI
10.1103/physreva.79.053823

Related papers

Free-standing, axially-symmetric diffraction gratings for neutral matter-waves: experiments and fabrication

Thomas J.G. Reisinger

Since de Broglie’s famous postulation of the wave-nature of material particles and its subsequent experimental verification, it is known that atoms and molecules behave like waves on a small scale. The investigation of matter waves is a fundamental topic in quantum mechanics. Furthermore, matter waves have many applications in a number of scientific fields. For example they are used to investigate materials by studying the diffraction of atoms (Helium Atom Scattering) or electrons (Low-energy electron-diffraction) from the material’s surface. The topic of this thesis has been to make nano-structured optical elements for matter-waves and use them in a number of experiments. A major result of this thesis is the performance of the classical Poisson-spot experiment with neutral matter-waves. The Poisson-spot refers to the bright interference spot observed in the shadow of a circular obstacle. The experiment gave convincing evidence of the wave nature of light at the beginning of the 19th century. The first two articles (A&B) in this thesis are concerned with the Poisson-spot experiment for matter waves and its applications. The experiment was performed using low-energy deuterium molecules (normal D2, 23.5 meV), which is described in the first article. In the second article the feasibility of using the Poisson-spot experiment to demonstrate the wave-nature of large molecules is studied. The wave properties of large molecules are an important topic in current research. The remaining three articles are concerned with Fresnel zone-plates. Fresnel zoneplates are axially symmetric diffraction gratings that can be used to focus waves. Loosely formulated the relationship between Fresnel zone-plates and the circular object in the Poissonspot experiment is the same as the relationship between multiple-slit gratings and a single straight edge. Paper C studies the application of microscopy with neutral atoms and molecules and the limitations of focusing matter waves with zone plates. The fabrication of the nanostructured free-standing zone-plates using electron-beam lithography is the topic of paper D. A second, new application of Fresnel zone-plates is introduced in paper E: There a zone plate is used for a direct size measurement of the so-called virtual source in a supersonic expansion. The main-result from the Poisson-spot experiment was the observation of the Poisson spot with a molecular-deuterium beam and the successful comparison of the collected experimental data with Fresnel-diffraction theory. The wave-length independence of the on-axis interferencecondition in the Poisson-spot experiment and the weak constraints on angular alignment and position of the circular object led us to conclude that the Poisson spot is a good candidate for demonstrating the wave nature of larger molecules. This idea is studied further in paper B, where among other things the feasibility of a Poissonspot experiment with the fullerene molecule C70 is examined. The main conclusion from this article was that the wave-nature of a molecular-beam can be demonstrated in diffraction experiments with circular discs that have varying amount of intended edge-corrugation. This is because the dependence of the bright spot on the edge-corrugation in the particle-model is different than in the wave-model. The result for the C70 Poisson-spot was that the experiment would be very challenging with count rates as low as 10−4 s−1. However, one possibility would be, due to the simplicity of the setup, to parallelize the experiment and measure the diffraction pattern behind many circular discs at once. This could be realized using the fabrication techniques discussed in chapter 4. Other applications of the Poisson spot could include the study of the Casmir-Polder potential, molecule-lithography and diffraction experiments with atom-lasers from Bose-Einstein condensates. While Fresnel zone-plates can be used to focus any type of wave they suffer from strong chromatic aberration since their focal length is inversely proportional to wavelength. In the case of neutral atom and molecule focusing this is what limits resolution. Paper C presents the highest resolution helium transmission-images hitherto and results from the first focusing of molecular deuterium with a zone plate. In addition the paper discusses the resolution-limit and finds that with presently available techniques a minimal spot size of 300 nm full-width-athalf- maximum is feasible using a zone plate of 200 μm diameter and a beam with speed ratio of about 500. The Fresnel zone-plates used to focus the supersonic-expansion beams need to be freestanding, since the low-energy atoms and molecules do not penetrate any material. In paper D an electron-beam lithography fabrication-process for free-standing zone-plates is presented. The process uses a 200-nm-thin layer of low-stress silicon-nitride for the material of the zone plates. The fabricated zone plates were tested in the supersonic-expansion beam apparatus. The transmission and first-order diffraction-efficiency are close to the theoretical prediction for the smaller 190 μm-diameter zone plates patterned with a single electron-beam write-field. A reduction to 70 % for the 388-μm-diameter zone-plate was observed which was attributed to stitching errors since it was stitched together from four write-fields. In chapter 4 additional unpublished results are presented on spatial-phase-locked electron-beam lithography, which aims at the reduction of stitching errors and other pattern-placement errors in electron-beam lithography. Finally in paper E one of the fabricated zone plates in addition to a previously existing one is used to measure the virtual source-width of a molecular deuterium supersonic-expansion beam. The virtual source-width is a measure of the beam’s temperature perpendicular to the beam axis. The beam’s energy distribution along the beam axis is measured using the timeof- flight method. Data sets for source temperatures T0=310 K and T0=106 K were collected in the stagnation pressure ranges p0=3-171 bar and p0=3-131 bar, respectively. The measured parameters were compared to a simple model of the expansion which explicitly includes the coupling between translational and rotational degrees of freedom. The data generally corresponded very well to the model, except for the virtual source size which was systematically about 2/3 of the model’s prediction. For the cold source-conditions the beam is increasingly heated due to condensation effects at increasing pressures, which results in a clear deviation from the model. This could be used to estimate the fraction of condensed molecules in the beam.

•Oct 4, 2011•1 citation

On the relative intensity of Poisson’s spot

Thomas J.G. Reisinger, Philipp M. Leufke, H. Gleiter +1 more

The Fresnel diffraction phenomenon referred to as Poisson's spot or spot of Arago has, beside its historical significance, become relevant in a number of fields. Among them are for example fundamental tests of the super-position principle in the transition from quantum to classical physics and the search for extra-solar planets using star shades. Poisson's spot refers to the positive on-axis wave interference in the shadow of any spherical or circular obstacle. While the spot's intensity is equal to the undisturbed field in the plane wave picture, its intensity in general depends on a number of factors, namely the size and wavelength of the source, the size and surface corrugation of the diffraction obstacle, and the distances between source, obstacle and detector. The intensity can be calculated by solving the Fresnel–Kirchhoff diffraction integral numerically, which however tends to be computationally expensive. We have therefore devised an analytical model for the on-axis intensity of Poisson's spot relative to the intensity of the undisturbed wave field and successfully validated it both using a simple light diffraction setup and numerical methods. The model will be useful for optimizing future Poisson-spot matter-wave diffraction experiments and determining under what experimental conditions the spot can be observed.

New Journal of Physics•Feb 6, 2017•12 citations

Poisson's spot and Gouy phase

I. G. da Paz, Rodolfo Reis Soldati, Luís A. Cabral +2 more

Recently there have been experimental results on Poisson spot matter-wave interferometry followed by theoretical models describing the relative importance of the wave and particle behaviors for the phenomenon. We propose an analytical theoretical model for Poisson's spot with matter waves based on the Babinet principle, in which we use the results for free propagation and single-slit diffraction. We take into account effects of loss of coherence and finite detection area using the propagator for a quantum particle interacting with an environment. We observe that the matter-wave Gouy phase plays a role in the existence of the central peak and thus corroborates the predominantly wavelike character of the Poisson's spot. Our model shows remarkable agreement with the experimental data for deuterium (${\text{D}}_{2}$) molecules.

arXiv:1609.09023•Physical Review A•Dec 12, 2016•6 citations

Impact of Casimir-Polder interaction on Poisson-spot diffraction at a dielectric sphere

Joshua Leo Hemmerich, Robert G. Bennett, Thomas J.G. Reisinger +5 more

Diffraction of matter waves is an important demonstration of the fact that objects in nature possess a mixture of particlelike and wavelike properties. Unlike in the case of light diffraction, matter waves are subject to a vacuum-mediated interaction with diffraction obstacles. Here we present a detailed account of this effect through the calculation of the attractive Casimir-Polder potential between a dielectric sphere and an atomic beam. Furthermore, we use our calculated potential to make predictions about the diffraction patterns to be observed in an ongoing experiment where a beam of indium atoms is diffracted around a silicon-dioxide sphere. The result is an amplification of the on-axis bright feature, which is the matter-wave analog of the well-known ``Poisson spot'' from optics. Our treatment confirms that the diffraction patterns resulting from our complete account of the sphere Casimir-Polder potential are indistinguishable from those found via a large-sphere nonretarded approximation in the discussed experiments, establishing the latter as an adequate model.

arXiv:1606.09472•Physical Review A•Aug 17, 2016•18 citations

Particle–wave discrimination in Poisson spot experiments

Thomas J.G. Reisinger, G. Bracco, Bodil Holst

Matter–wave interferometry has been used extensively over the last few years to demonstrate the quantum-mechanical wave nature of increasingly larger and more massive particles. We have recently suggested the use of the historical Poisson spot setup to test the diffraction properties of larger objects. In this paper, we present the results of a classical particle van der Waals (vdW) force model for a Poisson spot experimental setup and compare these to Fresnel diffraction calculations with a vdW phase term. We include the effect of disc-edge roughness in both models. Calculations are performed with D 2 and with C 70 using realistic parameters. We find that the sensitivity of the on-axis interference/focus spot to disc-edge roughness is very different in the two cases. We conclude that by measuring the intensity on the optical axis as a function of disc-edge roughness, it can be determined whether the objects behave as de Broglie waves or classical particles. The scaling of the Poisson spot experiment to larger molecular masses is, however, not as favorable as in the case of near-field light-grating-based interferometers. Instead, we discuss the possibility of studying the Casimir–Polder potential using the Poisson spot setup.

New Journal of Physics•Jun 28, 2011•26 citations

Electron wavelike behavior: A historical and experimental introduction

G. Matteucci

Following the Fresnel theory of light and the consequent scientific debate that his formulation generated, two key experiments with electrons, diffraction by means of a circular hole and a circular obstruction, have been realized to show the existence of the Fresnel zones and of the so-called ‘‘Poisson spot.’’ The basic arguments concerning the quantum mechanical nature of electrons can be introduced by taking advantage of the vivid impression stimulated by the experimental images.

American Journal of Physics•Dec 1, 1990•21 citations
In the Poisson-spot experiment, waves emanating from a source are blocked by a circular obstacle. Due to their positive on-axis interference an image of the source (the Poisson spot) is observed within the geometrical shadow of the obstacle. In this paper we report the observation of Poisson's spot using a beam of neutral deuterium molecules. The wavelength independence and the weak constraints on angular alignment and position of the circular obstacle make Poisson's spot a promising candidate for applications ranging from the study of large molecule diffraction to patterning with molecules.
Since de Broglie’s famous postulation of the wave-nature of material particles and its subsequent experimental verification, it is known that atoms and molecules behave like waves on a small scale. The investigation of matter waves is a fundamental topic in quantum mechanics. Furthermore, matter waves have many applications in a number of scientific fields. For example they are used to investigate materials by studying the diffraction of atoms (Helium Atom Scattering) or electrons (Low-energy electron-diffraction) from the material’s surface. The topic of this thesis has been to make nano-structured optical elements for matter-waves and use them in a number of experiments. A major result of this thesis is the performance of the classical Poisson-spot experiment with neutral matter-waves. The Poisson-spot refers to the bright interference spot observed in the shadow of a circular obstacle. The experiment gave convincing evidence of the wave nature of light at the beginning of the 19th century. The first two articles (A&B) in this thesis are concerned with the Poisson-spot experiment for matter waves and its applications. The experiment was performed using low-energy deuterium molecules (normal D2, 23.5 meV), which is described in the first article. In the second article the feasibility of using the Poisson-spot experiment to demonstrate the wave-nature of large molecules is studied. The wave properties of large molecules are an important topic in current research. The remaining three articles are concerned with Fresnel zone-plates. Fresnel zoneplates are axially symmetric diffraction gratings that can be used to focus waves. Loosely formulated the relationship between Fresnel zone-plates and the circular object in the Poissonspot experiment is the same as the relationship between multiple-slit gratings and a single straight edge. Paper C studies the application of microscopy with neutral atoms and molecules and the limitations of focusing matter waves with zone plates. The fabrication of the nanostructured free-standing zone-plates using electron-beam lithography is the topic of paper D. A second, new application of Fresnel zone-plates is introduced in paper E: There a zone plate is used for a direct size measurement of the so-called virtual source in a supersonic expansion. The main-result from the Poisson-spot experiment was the observation of the Poisson spot with a molecular-deuterium beam and the successful comparison of the collected experimental data with Fresnel-diffraction theory. The wave-length independence of the on-axis interferencecondition in the Poisson-spot experiment and the weak constraints on angular alignment and position of the circular object led us to conclude that the Poisson spot is a good candidate for demonstrating the wave nature of larger molecules. This idea is studied further in paper B, where among other things the feasibility of a Poissonspot experiment with the fullerene molecule C70 is examined. The main conclusion from this article was that the wave-nature of a molecular-beam can be demonstrated in diffraction experiments with circular discs that have varying amount of intended edge-corrugation. This is because the dependence of the bright spot on the edge-corrugation in the particle-model is different than in the wave-model. The result for the C70 Poisson-spot was that the experiment would be very challenging with count rates as low as 10−4 s−1. However, one possibility would be, due to the simplicity of the setup, to parallelize the experiment and measure the diffraction pattern behind many circular discs at once. This could be realized using the fabrication techniques discussed in chapter 4. Other applications of the Poisson spot could include the study of the Casmir-Polder potential, molecule-lithography and diffraction experiments with atom-lasers from Bose-Einstein condensates. While Fresnel zone-plates can be used to focus any type of wave they suffer from strong chromatic aberration since their focal length is inversely proportional to wavelength. In the case of neutral atom and molecule focusing this is what limits resolution. Paper C presents the highest resolution helium transmission-images hitherto and results from the first focusing of molecular deuterium with a zone plate. In addition the paper discusses the resolution-limit and finds that with presently available techniques a minimal spot size of 300 nm full-width-athalf- maximum is feasible using a zone plate of 200 μm diameter and a beam with speed ratio of about 500. The Fresnel zone-plates used to focus the supersonic-expansion beams need to be freestanding, since the low-energy atoms and molecules do not penetrate any material. In paper D an electron-beam lithography fabrication-process for free-standing zone-plates is presented. The process uses a 200-nm-thin layer of low-stress silicon-nitride for the material of the zone plates. The fabricated zone plates were tested in the supersonic-expansion beam apparatus. The transmission and first-order diffraction-efficiency are close to the theoretical prediction for the smaller 190 μm-diameter zone plates patterned with a single electron-beam write-field. A reduction to 70 % for the 388-μm-diameter zone-plate was observed which was attributed to stitching errors since it was stitched together from four write-fields. In chapter 4 additional unpublished results are presented on spatial-phase-locked electron-beam lithography, which aims at the reduction of stitching errors and other pattern-placement errors in electron-beam lithography. Finally in paper E one of the fabricated zone plates in addition to a previously existing one is used to measure the virtual source-width of a molecular deuterium supersonic-expansion beam. The virtual source-width is a measure of the beam’s temperature perpendicular to the beam axis. The beam’s energy distribution along the beam axis is measured using the timeof- flight method. Data sets for source temperatures T0=310 K and T0=106 K were collected in the stagnation pressure ranges p0=3-171 bar and p0=3-131 bar, respectively. The measured parameters were compared to a simple model of the expansion which explicitly includes the coupling between translational and rotational degrees of freedom. The data generally corresponded very well to the model, except for the virtual source size which was systematically about 2/3 of the model’s prediction. For the cold source-conditions the beam is increasingly heated due to condensation effects at increasing pressures, which results in a clear deviation from the model. This could be used to estimate the fraction of condensed molecules in the beam.
The Fresnel diffraction phenomenon referred to as Poisson's spot or spot of Arago has, beside its historical significance, become relevant in a number of fields. Among them are for example fundamental tests of the super-position principle in the transition from quantum to classical physics and the search for extra-solar planets using star shades. Poisson's spot refers to the positive on-axis wave interference in the shadow of any spherical or circular obstacle. While the spot's intensity is equal to the undisturbed field in the plane wave picture, its intensity in general depends on a number of factors, namely the size and wavelength of the source, the size and surface corrugation of the diffraction obstacle, and the distances between source, obstacle and detector. The intensity can be calculated by solving the Fresnel–Kirchhoff diffraction integral numerically, which however tends to be computationally expensive. We have therefore devised an analytical model for the on-axis intensity of Poisson's spot relative to the intensity of the undisturbed wave field and successfully validated it both using a simple light diffraction setup and numerical methods. The model will be useful for optimizing future Poisson-spot matter-wave diffraction experiments and determining under what experimental conditions the spot can be observed.
Recently there have been experimental results on Poisson spot matter-wave interferometry followed by theoretical models describing the relative importance of the wave and particle behaviors for the phenomenon. We propose an analytical theoretical model for Poisson's spot with matter waves based on the Babinet principle, in which we use the results for free propagation and single-slit diffraction. We take into account effects of loss of coherence and finite detection area using the propagator for a quantum particle interacting with an environment. We observe that the matter-wave Gouy phase plays a role in the existence of the central peak and thus corroborates the predominantly wavelike character of the Poisson's spot. Our model shows remarkable agreement with the experimental data for deuterium (D2{\text{D}}_{2}D2​) molecules.
Diffraction of matter waves is an important demonstration of the fact that objects in nature possess a mixture of particlelike and wavelike properties. Unlike in the case of light diffraction, matter waves are subject to a vacuum-mediated interaction with diffraction obstacles. Here we present a detailed account of this effect through the calculation of the attractive Casimir-Polder potential between a dielectric sphere and an atomic beam. Furthermore, we use our calculated potential to make predictions about the diffraction patterns to be observed in an ongoing experiment where a beam of indium atoms is diffracted around a silicon-dioxide sphere. The result is an amplification of the on-axis bright feature, which is the matter-wave analog of the well-known ``Poisson spot'' from optics. Our treatment confirms that the diffraction patterns resulting from our complete account of the sphere Casimir-Polder potential are indistinguishable from those found via a large-sphere nonretarded approximation in the discussed experiments, establishing the latter as an adequate model.
Matter–wave interferometry has been used extensively over the last few years to demonstrate the quantum-mechanical wave nature of increasingly larger and more massive particles. We have recently suggested the use of the historical Poisson spot setup to test the diffraction properties of larger objects. In this paper, we present the results of a classical particle van der Waals (vdW) force model for a Poisson spot experimental setup and compare these to Fresnel diffraction calculations with a vdW phase term. We include the effect of disc-edge roughness in both models. Calculations are performed with D 2 and with C 70 using realistic parameters. We find that the sensitivity of the on-axis interference/focus spot to disc-edge roughness is very different in the two cases. We conclude that by measuring the intensity on the optical axis as a function of disc-edge roughness, it can be determined whether the objects behave as de Broglie waves or classical particles. The scaling of the Poisson spot experiment to larger molecular masses is, however, not as favorable as in the case of near-field light-grating-based interferometers. Instead, we discuss the possibility of studying the Casimir–Polder potential using the Poisson spot setup.