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Optics Letters

Radial diffraction of light in the radial momentum state space

Tianlong Ma, Dongkai Zhang, Xiaodong Qiu, Yuanyuan Chen, Lixiang Chen

Published Aug 12, 2020• 5 citations
Publisher

Abstract

While linear or angular position and momentum can be linked by a continuous or discrete Fourier transform, there are some subtle problems in the analogous Fourier relationship between radial position and radial momentum in history. Here we exploit radial position and newly introduced radial momentum variables to report a radial version of light's diffraction. The mask with single or multiple radial slits confines the light to a radial transmission function. As a result, in the radial momentum state space, we can observe the diffraction sidebands generated on the transmitted light due to a transverse restriction of the radial range. Our experimental results clearly reveal the intriguing diffraction behaviors between radial position and radial momentum variables at the single-photon level, making them another candidate for fundamental tests of quantum mechanics and for a variety of quantum information applications.

DOI
10.1364/ol.400878

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While linear or angular position and momentum can be linked by a continuous or discrete Fourier transform, there are some subtle problems in the analogous Fourier relationship between radial position and radial momentum in history. Here we exploit radial position and newly introduced radial momentum variables to report a radial version of light's diffraction. The mask with single or multiple radial slits confines the light to a radial transmission function. As a result, in the radial momentum state space, we can observe the diffraction sidebands generated on the transmitted light due to a transverse restriction of the radial range. Our experimental results clearly reveal the intriguing diffraction behaviors between radial position and radial momentum variables at the single-photon level, making them another candidate for fundamental tests of quantum mechanics and for a variety of quantum information applications.
As is well known, angular position and orbital angular momentum (OAM) of photons are a conjugate pair of variables that have been extensively explored for quantum information science and technology. In contrast, the radial degrees of freedom remain relatively unexplored. Here we exploit the radial variables, i.e., radial position and radial momentum, to demonstrate Einstein-Podolsky-Rosen correlations between down-converted photons. In our experiment, we prepare various annular apertures to define the radial positions and use eigenmode projection to measure the radial momenta. The resulting correlations are found to violate the Heisenberg-like uncertainty principle for independent particles, thus manifesting the entangled feature in the radial structure of two-photon wave functions. Our work suggests that, in parallel with angular position and OAM, the radial position and radial momentum can offer a new platform for a fundamental test of quantum mechanics and for novel application of quantum information.
The angular profile and the orbital angular momentum of a light mode are related by Fourier transform. Any modification of the angular distribution, e. g. via diffraction off a suitably programmed spatial light modulator, influences the orbital angular momentum spectrum of the light. This holds true even at the single photon level. We observe the influence of various angular masks on the orbital angular momentum spectrum, both in the near and the far field, and describe the resulting patterns in terms of angular diffraction. If photons are entangled in their orbital angular momentum, diffraction of one photon affects the orbital angular momentum spectrum of its partner photon, and angular ghost diffraction can be measured in the coincidence counts. We highlight the role of the angular Fourier relationship for these effects.
While the uncertainty principle for linear position and linear momentum, and more recently for angular position and angular momentum, is well established, its radial equivalent has so far eluded researchers. Here we exploit the logarithmic radial position, ln r , and hyperbolic momentum, P H , to formulate a rigorous uncertainty principle for the radial degree of freedom of transverse light modes. We show that the product of their uncertainties is bounded by Planck’s constant, Δ ln r · Δ P H ≥ ℏ / 2 , and identify a set of radial intelligent states that satisfy the equality. We illustrate the radial uncertainty principle for a variety of intelligent states, by preparing transverse light modes with suitable radial profiles. We use eigenmode projection to measure the corresponding hyperbolic momenta, confirming the minimum uncertainty bound. Optical systems are most naturally described in terms of cylindrical coordinates, and our radial uncertainty relation provides the missing piece in characterizing optical quantum measurements, providing a new platform for the fundamental tests and applications of quantum optics.
This experiment investigates the Heisenberg Uncertainty Principle (HUP) through an optical analogue using Single-slit diffraction of a monochromatic He–Ne laser. The spatial confinement of light through slits of varying width is compared with the resulting diffraction patterns, providing a means to examine the relationship between position uncertainty ∆x and transverse momentum uncertainty ∆px. Diffraction patterns were recorded using a photodiode detector, and the angular positions of the first minima were determined and used to estimate the spread in momentum via the de-Broglie relation. The results demonstrates and are consistent with the expected inverse relationship. The measured diffraction patterns were also found to adhere to the Fraunhofer diffraction relation, supporting the validity of the experimental model and the goodness of fit of the data.The calculated uncertainty products were further found to be consistent with the expected magnitude of Planck's constant. and therefore the the findings support the interpretation of diffraction as a classifcal manifestation of the Foruier relation between position and momentum space. Thus providing a physically intuituve demonstration of Heisenberg's Uncertainty Principle.
Diffraction properties of light can be derived from Quantum Mechanics and Relativity. Using the fact that position and momentum are conjugate variables, we show that the momentum distribution of light coincides with the well-known angular spectrum distribution. The momentum distribution links quantum theory and relativity to classical diffraction theory. We also show that the Huygens Principle and the momentum distribution are conjugate expressions at the diffraction aperture. These considerations lead to the geometrical theory of diffraction.