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Optics Express

Young’s double-slit experiment: noise-resolution duality

Yakov I. Nesterets, Timur E. Gureyev

Published Feb 4, 2015• 6 citations
Publisher

Abstract

Statistical aspects of Young's double-slit diffraction experiment are analysed quantitatively. It is shown that the signal-to-noise ratio and the spatial resolution in the detected diffraction pattern satisfy a duality relationship which implies that both of them cannot be improved simultaneously beyond a certain limit if the total number of particles forming the image is fixed. As a consequence of this duality, it is possible to estimate the minimal number of particles that have to be detected in order for two slits separated by a given distance to be resolved with a confidence level corresponding to a pre-defined signal-to-noise ratio, e.g. according to the Rose criterion. These results are related to the recently introduced imaging system quality characteristic which combines the spatial resolution and the noise sensitivity, and allows one to estimate the efficiency with which imaging quanta are utilised in a system to deliver maximal amount of information about the imaged object. The presented results can be useful for applications where the imaging quanta are at a premium or where minimization of the radiation dose is important.

DOI
10.1364/oe.23.003373

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Statistical aspects of Young's double-slit diffraction experiment are analysed quantitatively. It is shown that the signal-to-noise ratio and the spatial resolution in the detected diffraction pattern satisfy a duality relationship which implies that both of them cannot be improved simultaneously beyond a certain limit if the total number of particles forming the image is fixed. As a consequence of this duality, it is possible to estimate the minimal number of particles that have to be detected in order for two slits separated by a given distance to be resolved with a confidence level corresponding to a pre-defined signal-to-noise ratio, e.g. according to the Rose criterion. These results are related to the recently introduced imaging system quality characteristic which combines the spatial resolution and the noise sensitivity, and allows one to estimate the efficiency with which imaging quanta are utilised in a system to deliver maximal amount of information about the imaged object. The presented results can be useful for applications where the imaging quanta are at a premium or where minimization of the radiation dose is important.
Several variations of the Heisenberg uncertainty inequality are derived on the basis of `noise-resolution duality' recently proposed by us. The same approach leads to a related inequality that provides an upper limit for the information capacity of imaging systems in terms of the number of imaging quanta (particles) used in the experiment. These results are useful in the context of biomedical imaging constrained by the radiation dose delivered to the sample, or in imaging (e.g., astronomical) problems under low light conditions. References R. Bach, D. Pope, S.-H. Liou and H. Batelaan, Controlled double-slit electron diffraction. New J. Phys. 15:033018, 2013. doi:10.1088/1367-2630/15/3/033018 H. H. Barrett and K. J. Myers, Foundations of image science. Wiley, Hoboken, 2004. http://au.wiley.com/WileyCDA/WileyTitle/productCd-0471153001.html H. Bateman and A. Erdelyi, Tables of integral transforms. McGraw–Hill, New York, 1954. http://resolver.caltech.edu/CaltechAUTHORS:20140123-101456353 M. Born and E. Wolf, Principles of optics. Cambridge University Press, Cambridge, 1999. doi:10.1017/CBO9781139644181 I. J. Cox and C. J. R. Sheppard, Information capacity and resolution in an optical system. J. Opt. Soc. Am. A 3:1152–1158, 1986. doi:10.1364/JOSAA.3.001152 M. G. Cowling and J. F. Price, Bandwidth versus time concentration: the Heisenberg–Pauli–Weyl inequality. SIAM J. Math. Anal. 15(1):151–165, 1984. doi:10.1137/0515012 F. de Hoog, G. Schmalz and T. E. Gureyev, An uncertainty inequality. Appl. Math. Lett. 38:84–86, 2014. doi:10.1016/j.aml.2014.07.009 I. Dreier, W. Ehm, T. Gneiting and D. Richards, Improved bounds for Laue's constant and multivariate extensions. Math. Nachr. 228:109–122, 2001. doi:10.1002/1522-2616(200108)228:1<109::AID-MANA109>3.0.CO;2-V V. A. Epanechnikov, Non-parametric estimation of a multivariate probability density. Theor. Probab. Appl. 14:153–158, 1969. doi:10.1137/1114019 A. C. Fabian, K. A. Pounds and R. D. Blandford, Frontiers of x-ray astronomy. Cambridge University Press, Cambridge, 2004. P. B. Felgett and E. H. Linfoot, On the assessment of optical images. Phil. T. R. Soc. A 247:369–407, 1955. doi:10.1098/rsta.1955.0001 G. B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3:207–238, 1997. doi:10.1007/BF02649110 T. E. Gureyev, Ya. I. Nesterets, F. de Hoog, G. Schmalz, S. C. Mayo, S. Mohammadi and G. Tromba, Duality between noise and spatial resolution in linear systems. Opt. Express 22:9087–9094, 2014. doi:10.1364/OE.22.009087 T. E. Gureyev, S. C. Mayo, Ya. I. Nesterets, S. Mohammadi, D. Lockie, R. H. Menk, F. Arfelli, K. M. Pavlov, M. J. Kitchen, F. Zanconati, C. Dullin and G. Tromba, Investigation of imaging quality of synchrotron-based phase-contrast mammographic tomography. J. Phys. D: Appl. Phys. 47:365401, 2014. doi:10.1088/0022-3727/47/36/365401 M. R. Howells, T. Beetz, H. N. Chapman, C. Cui, J. M. Holton, C. J. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. A. Shapiro, J. C. H. Spence and D. Starodub, An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy. J. Electron Spectrosc. 170:4–12, 2009. doi:10.1016/j.elspec.2008.10.008 E. Laeng and C. Morpurgo, An uncertainty inequality involving L1L^1L1 norms. P. Am. Math. Soc. 127:3565–3572, 1999. http://www.jstor.org/stable/119352 Ya. I. Nesterets and T. E. Gureyev, Young's double-slit experiment: noise-resolution duality. Opt. Express 23:3373–3381, 2015. doi:10.1364/OE.23.003373 C. Shannon: Communication in the presence of noise. P. IRE 37:10–21, 1949. doi:10.1109/JRPROC.1949.232969
Double-slit interference experiments using monochromatic hard X-rays with the energy of 25 keV are presented. The experiments were performed at a synchrotron source with a distance of 110 m between the interferometer and the detector to produce an interference pattern with a sufficiently broad period that could be adequately sampled by a photon-counting detector with 75 micrometre pixels. In the single-particle version of the experiment, over one million image frames with a single registered photon in each one were collected. The sum of these frames showed a clear presence of the interference pattern with the expected period. Subsequent analysis provided an objective estimation of the minimal number of detected photons required to determine, in accordance with the Rose criterion, the presence of the photon interference. Apart from a general theoretical interest, these investigations were aimed at exploring the possibility of medical X-ray phase-contrast imaging in photon-counting regime at minimal radiation doses.
We present time-resolved diffraction and two-slit interference experiments using a streak camera as a detector for femtosecond pulses of photons. These experiments show how the diffraction pattern is built by adding frames of a few photons to each frame. It is estimated that after 300 photons the diffraction pattern emerges. With time resolution we can check the speed of light and put an upper limit of 2 ps at our resolution to the time for wave function collapse in the quantum measurement process. We then produce interference experiments with photons of different energies impinging on the slits, i.e. we know which photon impinges on each slit. We show that for poor time resolution, no interference is observed, but for high time resolution, we have interference that is revealed as beats of 100 GHz frequency. The condition for interference is that the two pulses should overlap spatially at the detector, even if the pulses have different energies but are generated from the same pulse of the laser. The interference seems to be in agreement with classical theory at first sight. However, closer study and analysis of the data show deviations in the visibility of the interference fringes and of their phase. These experiments are discussed in connection with quantum mechanics and it may be concluded that the time resolution provides new data for understanding the longstanding and continuing arguments on wave-particle duality initiated by Newton, Young, Fresnel, Planck and others. A thought experiment is presented in the appendix to try to distinguish the photons at the detector by making it sensitive to colour.
In this paper, we analyze the statistics of detection data in a general double-double-slit experiment. The two particles are detected at random times which are not equal in general and because we do not have any constraint on the distances of left and right screens from their slits and the ratio they can be detected in completely different timescales. As the detection of the first particle leads to collapse of the wave function, there is no a straightforward and agreed-upon method to study this problem in the orthodox formalism which lacks a clear prediction of these random events and therefore the quantum state afterward. This is not the case in the Bohmian framework which we implement in this paper, and we can predict the system up to the end of experiment. The main result is the joint distribution of detection data including the arrival time and position of the particles on left and right screens. As one of the main consequences, we see that although the joint spatial distribution can be affected by a change to the relative location of screens, the marginals on each side remain intact, compatible with signal locality. At the end, we see how this result is very sensitive to quantum equilibrium conditions.
Summary form only given. The essence of wave-particle duality is that particles passing through slits form a pattern that can only be explained by resorting to wave mechanics. Here we present results that provide insight into the buildup of the multiple-slit interference pattern. The perfect time resolved measurements would ideally involve a single-particle source and an array of single-particle noiseless detectors featuring perfect quantum efficiency and time resolution. In previous experiments for electrons [1], photons [2] and molecules [3], detection of single particles was limited by minimum acquisition times, thus denying access to precise timing information. We make a significant step forward, approaching the perfect measurement very closely and showing real-time interference pattern buildup of single-photon detections as a result of passing through a system of multiple slits. These measurements accessed precise timing information, allowing us to analyse the interference pattern's formation process and effects of entanglement.
Imaging is the estimation of the spatial distribution of a physical object by measuring the optical radiation it emits, or by making use of an optical wave that interacts with the object, via reflection or transmission, for example, before being measured by a detector. The resolution of an imaging system is limited by the inability to localize the optical field at points of the object. Under otherwise ideal conditions, resolution is limited by diffraction. The sensitivity of an imaging system is limited by the uncertainty in the measurement. Under ideal conditions, this is determined by photon noise, which depends on the statistical fluctuations of the light. In conventional imaging systems, an extended detector, such as a CCD camera or an array detector, measures the spatial distribution of the optical intensity, which is proportional to the photon flux density. In interferometric systems, the spatial distribution of the optical field is inferred from measurements of the optical intensity. With the emergence of coherence theory, imaging systems based on measurements of the second-order coherence function at pairs of points in the detection plane were developed. An example is the imaging of an incoherent object based on the van Cittert-Zernike theorem. Imaging systems based on measurement of intensity correlation, or the photon coincidence rate, at pairs of points, were developed in the 1960s. A classic example of the photon-correlation imaging of an object emitting thermal light is stellar imaging using a Hanbury-Brown-Twiss intensity-correlation interferometer. More recently, two-photon light, which may be generated via spontaneous parametric downconversion in a second-order nonlinear optical crystal, has been used for imaging. This type of two-photon (or biphoton) imaging, which has come to be called quantum imaging, is also based on the measurement of photon coincidence by the use of photon-counting array detectors or by scanning two photon-counting detectors at pairs of points. To compare the resolution and sensitivity of imaging systems based on the aforementioned types of measurements, it is necessary to derive expressions for the measured quantities in terms of the object distribution. The point-spread functions based on such expressions can be used to assess the resolution. One measure of the sensitivity of the imaging process is the signal-to-noise ratio (SNR) of the measured variables. The statistical nature of the light source must be known in order to determine the SNR. The purpose of this chapter is to compare the resolution and sensitivity of photon-correlation imaging systems that make use of thermal light and two-photon light. We will henceforth refer to these two imaging modalities as classical and quantum photon-correlation imaging, or simply classical and quantum imaging, respectively. Clearly, light in other quantum states can also be used for imaging.