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Optical vortex knots – one photon at a time Sebastien J. Tempone-Wiltshire, Shaun P. Johnstone & Kristian Helmerson
received: 06 December 2015 accepted: 24 March 2016 Published: 18 April 2016
Feynman described the double slit experiment as “a phenomenon which is impossible, absolutely impossible, to explain in any classical way and which has in it the heart of quantum mechanics”. The double-slit experiment, performed one photon at a time, dramatically demonstrates the particlewave duality of quantum objects by generating a fringe pattern corresponding to the interference of light (a wave phenomenon) from two slits, even when there is only one photon (a particle) at a time passing through the apparatus. The particle-wave duality of light should also apply to complex three dimensional optical fields formed by multi-path interference, however, this has not been demonstrated. Here we observe particle-wave duality of a three dimensional field by generating a trefoil optical vortex knot – one photon at a time. This result demonstrates a fundamental physical principle, that particlewave duality implies interference in both space (between spatially distinct modes) and time (through the complex evolution of the superposition of modes), and has implications for topologically entangled single photon states, orbital angular momentum multiplexing and topological quantum computing. Since Einstein’s description of the photon in 19051, used to explain the photoelectric effect, the quantized nature of light has been accepted. Numerous experiments since have shown the particle-wave duality of photons and other quantum objects. These include two slit interference experiments with single photons2–4, electrons5–7, atoms8 and even much larger macroscopic objects such as C60 buckyballs9. In the realm of single photons, more complex intensity and phase structures have been investigated including both analog10 and digital11,12 holograms at ultralow light intensities. It has also been shown that single photons can carry the orbital angular momentum associated with optical vortices13. In all of these experiments, the intensity (and phase) profiles resulting from interference were essentially two-dimensional (2D). Further propagation in the far-field produced a self-similar 2D pattern. Here we report the first measurements of a three dimensional (3D) complex optical field - one photon at a time - by demonstrating that the distribution of single photons evolves to the full complex field. These results are unique in that not only do we measure a complex three dimensional vortex structure within the optical field, produced one photon at a time, but we also show that the effects of diffraction on a single photon are the same as for the entire optical field, even though diffraction is a ‘non-local’ effect. The three dimensional knotted structure investigated in this paper is made from optical vortices. An optical vortex14 is a point in an optical field about which the field undergoes some integer multiple of 2π phase winding, leading to an intensity zero at the vortex core as at this point the phase of the field is undefined. These vortices have been shown to carry the orbital angular momentum of an optical field and thus can be both ‘positive’ and ‘negative’, depending upon the sense of rotation of the angular momentum. They therefore arise from the global structure of the field, not the local structure. These vortices trace out lines along the direction of propagation of the optical field that can only be removed by a transfer of angular momentum to some object they interact with, or through annihilation with a vortex of opposite sign. An optical vortex knot is a vortex line in an optical field that follows the multiply-connected topology of a knot. As such, it is an inherently three-dimensional optical field that exhibits both complex, but measurable, phase and intensity profiles. Optical vortex knots can be constructed in the paraxial approximation by a superposition of Laguerre-Gaussian (LG) beams15. Isolated optical vortex knots, including the trefoil (Fig. 1b), cinquefoil and a Hopf link were demonstrated in the laboratory15 by generating the appropriate superposition of LG modes using a spatial light modulator (SLM). We generate our optical field employing methods similar to those of refs 15 and 16, using a complex valued, phase only hologram17 displayed upon a SLM. A reference Gaussian beam is also generated from the SLM and made to interfere with the knotted optical vortex field. The vortices leave a characteristic forked structure in the School of Physics and Astronomy, Monash University, Victoria 3800, Australia. Correspondence and requests for materials should be addressed to S.J.T.-W. (email: [email protected])
Scientific Reports | 6:24463 | DOI: 10.1038/srep24463
1
www.nature.com/scientificreports/
Figure 1. Measuring the optical fields knotted topology. Different transverse planes of the optical field along the beam propagation direction (indicated by the black arrow) are imaged onto the camera by moving the position of the final imaging lens before the camera (see optical set up in left of Fig. 2). The length scales in both the propagation direction and transverse plane are shown. (a) Vortices within each plane (indicated by red dots) are located by the characteristic forked structure they create in each interferogram, due to the 2π phase winding about each vortex. (b) The vortices located in 25 separate planes are then ‘stitched’ together to form the three dimensional trefoil knot nodal structure.
resulting interferogram, as shown in Fig. 1a, which facilitates identifying their locations. A schematic of the optical set up is shown on the left in Fig. 2, with the inset showing the hologram used to generate the knotted vortex structure and the reference Gaussian beam. A number of different transverse planes of the knotted optical field (Fig. 1) are imaged onto the camera by translating a final imaging lens (which also provides magnification of the field) in the direction of propagation. The vortices within each image plane are located and then ‘stitched’ together to visualise the three dimensional trefoil knot vortex structure (Fig. 1b). To demonstrate that the optical vortex knot can be generated with single photons, we perform the measurement in a regime where on average there is less than a single photon in the apparatus at any given time. The SLM is illuminated with a laser intensity of
OPEN
Optical vortex knots – one photon at a time Sebastien J. Tempone-Wiltshire, Shaun P. Johnstone & Kristian Helmerson
received: 06 December 2015 accepted: 24 March 2016 Published: 18 April 2016
Feynman described the double slit experiment as “a phenomenon which is impossible, absolutely impossible, to explain in any classical way and which has in it the heart of quantum mechanics”. The double-slit experiment, performed one photon at a time, dramatically demonstrates the particlewave duality of quantum objects by generating a fringe pattern corresponding to the interference of light (a wave phenomenon) from two slits, even when there is only one photon (a particle) at a time passing through the apparatus. The particle-wave duality of light should also apply to complex three dimensional optical fields formed by multi-path interference, however, this has not been demonstrated. Here we observe particle-wave duality of a three dimensional field by generating a trefoil optical vortex knot – one photon at a time. This result demonstrates a fundamental physical principle, that particlewave duality implies interference in both space (between spatially distinct modes) and time (through the complex evolution of the superposition of modes), and has implications for topologically entangled single photon states, orbital angular momentum multiplexing and topological quantum computing. Since Einstein’s description of the photon in 19051, used to explain the photoelectric effect, the quantized nature of light has been accepted. Numerous experiments since have shown the particle-wave duality of photons and other quantum objects. These include two slit interference experiments with single photons2–4, electrons5–7, atoms8 and even much larger macroscopic objects such as C60 buckyballs9. In the realm of single photons, more complex intensity and phase structures have been investigated including both analog10 and digital11,12 holograms at ultralow light intensities. It has also been shown that single photons can carry the orbital angular momentum associated with optical vortices13. In all of these experiments, the intensity (and phase) profiles resulting from interference were essentially two-dimensional (2D). Further propagation in the far-field produced a self-similar 2D pattern. Here we report the first measurements of a three dimensional (3D) complex optical field - one photon at a time - by demonstrating that the distribution of single photons evolves to the full complex field. These results are unique in that not only do we measure a complex three dimensional vortex structure within the optical field, produced one photon at a time, but we also show that the effects of diffraction on a single photon are the same as for the entire optical field, even though diffraction is a ‘non-local’ effect. The three dimensional knotted structure investigated in this paper is made from optical vortices. An optical vortex14 is a point in an optical field about which the field undergoes some integer multiple of 2π phase winding, leading to an intensity zero at the vortex core as at this point the phase of the field is undefined. These vortices have been shown to carry the orbital angular momentum of an optical field and thus can be both ‘positive’ and ‘negative’, depending upon the sense of rotation of the angular momentum. They therefore arise from the global structure of the field, not the local structure. These vortices trace out lines along the direction of propagation of the optical field that can only be removed by a transfer of angular momentum to some object they interact with, or through annihilation with a vortex of opposite sign. An optical vortex knot is a vortex line in an optical field that follows the multiply-connected topology of a knot. As such, it is an inherently three-dimensional optical field that exhibits both complex, but measurable, phase and intensity profiles. Optical vortex knots can be constructed in the paraxial approximation by a superposition of Laguerre-Gaussian (LG) beams15. Isolated optical vortex knots, including the trefoil (Fig. 1b), cinquefoil and a Hopf link were demonstrated in the laboratory15 by generating the appropriate superposition of LG modes using a spatial light modulator (SLM). We generate our optical field employing methods similar to those of refs 15 and 16, using a complex valued, phase only hologram17 displayed upon a SLM. A reference Gaussian beam is also generated from the SLM and made to interfere with the knotted optical vortex field. The vortices leave a characteristic forked structure in the School of Physics and Astronomy, Monash University, Victoria 3800, Australia. Correspondence and requests for materials should be addressed to S.J.T.-W. (email: [email protected])
Scientific Reports | 6:24463 | DOI: 10.1038/srep24463
1
www.nature.com/scientificreports/
Figure 1. Measuring the optical fields knotted topology. Different transverse planes of the optical field along the beam propagation direction (indicated by the black arrow) are imaged onto the camera by moving the position of the final imaging lens before the camera (see optical set up in left of Fig. 2). The length scales in both the propagation direction and transverse plane are shown. (a) Vortices within each plane (indicated by red dots) are located by the characteristic forked structure they create in each interferogram, due to the 2π phase winding about each vortex. (b) The vortices located in 25 separate planes are then ‘stitched’ together to form the three dimensional trefoil knot nodal structure.
resulting interferogram, as shown in Fig. 1a, which facilitates identifying their locations. A schematic of the optical set up is shown on the left in Fig. 2, with the inset showing the hologram used to generate the knotted vortex structure and the reference Gaussian beam. A number of different transverse planes of the knotted optical field (Fig. 1) are imaged onto the camera by translating a final imaging lens (which also provides magnification of the field) in the direction of propagation. The vortices within each image plane are located and then ‘stitched’ together to visualise the three dimensional trefoil knot vortex structure (Fig. 1b). To demonstrate that the optical vortex knot can be generated with single photons, we perform the measurement in a regime where on average there is less than a single photon in the apparatus at any given time. The SLM is illuminated with a laser intensity of
