Proceedings of Topical Meeting on Optoinformatics (pdf-format, 1.21 ...

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10 OPTOINFORMATICS’05 UNCERTAINTY RELATIONSHIPS FOR ANGULAR POSITION AND ANGULAR MOMENTUM OF LIGHT Eric Yao and Miles Padgett, Department <strong>of</strong> Physics and Astronomy, University <strong>of</strong> Glasgow, Glasgow, Scotland Sonja Franke-Arnold and Stephen Barnett, Department <strong>of</strong> Physics, University <strong>of</strong> Strathclyde, Glasgow, Scotland E-mail: e.yao@physics.gla.ac.uk We confirm both the form <strong>of</strong> the angular uncertainty relationship and the Fourier-nature <strong>of</strong> the relationship for more complex aperture functions. The angular uncertainty relation suggests a new method for secure free space optical communications. Heisenberg’s uncertainty principle is commonly encountered as a relationship between the standard deviations <strong>of</strong> momentum and position, ∆x ∆p ≥ h 2 [1] . Although frequently associated with quantum mechanics, it applies to any pair <strong>of</strong> variables that are related by a Fourier-transform, whether they be classical or quantum observables. Momentum and position are both unbounded and continuous variables and the relationship between them is a continuous Fourier-transform. Angular momentum is more complicated since its conjugate variable, angle, is a repeating function within a 2π range. We show that, when applied to light beams, measurements in the distribution <strong>of</strong> angular momentum states are subject to a discrete Fourier-transform relationship with angular position. The 2π cyclic nature <strong>of</strong> angular measurement raises issues with the formulation <strong>of</strong> an angular uncertainty relationship which stem from the fact that angle is an unbounded variable and its associated standard deviation is ill-defined. However, in keeping with the work <strong>of</strong> Pegg and Barnett [2] , if one defines angle only over the 2π range, one discovers a self-consistent set <strong>of</strong> relationships that provide predictive formulae relating to the limiting accuracy <strong>of</strong> possible measurements. One area <strong>of</strong> quantum physics where angular momentum is becoming <strong>of</strong> extreme importance is that <strong>of</strong> optical beams. For a beam with helical phase fronts described by exp(il φ) the orbital angular momentum is lh per photon and can be measured in a variety <strong>of</strong> ways. Most simply, diffractive components can confirm whether or not a particular beam, or indeed photon <strong>of</strong> light, is described by a specific value <strong>of</strong> l . We have previously measured the angular momentum <strong>of</strong> an l =0 light beam after passage through an angular restriction [3] . We confirmed that the resulting spread in orbital angular momentum states has a Fourier relationship with the angular transmission function <strong>of</strong> the aperture. Furthermore, for an aperture centred at φ =0, with a width characterised by a standard deviation over a 2π range, the relative uncertainties obey the relationship ∆L∆φ ≥ h 1− 2πP( −π ) 2 - an expression compatible with the Pegg Barnett Phase.
SAINT-PETERSBURG, October 17 – 20, 2005 11 In this work we concentrate on single photons, initially with non-zero l , and measure the statistical distribution <strong>of</strong> l -states after transmission through generalised apertures. Experimentally we generate a low intensity laser beam in a precise l -state using a He-Ne laser whose beam is collimated and expanded to fill the aperture <strong>of</strong> an addressable spatial light modulator. The modulator is programmed with a diffraction grating containing a fork dislocation to produce a beam with a helical phase in the first diffraction order. A second spatial light modulator is used to analyse the l -state. If the index <strong>of</strong> the analysing hologram is opposite to that <strong>of</strong> the incoming beam, it transforms the beam back into a plane wave which can be focused to pass through a diffraction limited pinhole. Cycling through various indices whilst monitoring the power transmitted through the pinhole allows the l -state <strong>of</strong> the incoming beam to be deduced. The same analysing hologram can be combined with the angular aperture giving a single optical component that both sets the angular aperture and measures the resulting l -distribution. We count the individual photons transmitted through the pinhole using a photon counter. We confirm experimentally both the form <strong>of</strong> the angular uncertainty relationship and the Fourier-nature <strong>of</strong> the relationship for more complex aperture functions. 1. E. Merzbacher, Quantum Mechanics (John Wiley & Sons, Brisbane,1998). 2. S.M. Barnett and D.T. Pegg, “Quantum theory <strong>of</strong> rotation angles”, Phys.Rev.A 41, 3427 (1990). 3. S. Frank-Arnold, S.M. Barnett, E. Yao, J. Leach, J. Courtial and M.J. Padgett, “Uncertainty principle for angular position and angular momentum”, New J. Phys. 6, 103 (2004).
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