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Orbital angular momentum of photons

BACKGROUND

 ORBITAL ANGULAR MOMENTUM (OAM) The phase fronts of light beams in orbital angular momentum (OAM) eigenstates rotate, clockwise for positive OAM values, anti-clockwise for negative values. The phase front with 0 OAM doesn't rotate at all. OAM = -1 hbar per photon (right-hand helical phase front) OAM = 0 (plane wave) OAM = +1 hbar per photon (left-hand helix) OAM = +2 hbar per photon (two-fold helix) OAM = +3 hbar per photon (three-fold helix: fusilli)
 SPIN Spin eigenstates rotate: at each point, the polarisation vector rotates in the transverse plane (in the movies below the beam is travelling from left to right). They possess angular momentum, in much the same way in which rotating masses do. spin = -1 hbar per photon (right-hand circular polarisation) spin = +1 hbar per photon (left-hand circular polarisation) There is no spin eigenstate for spin 0. However, the two spin eigenstates can be superposed to give a spin-0 state. As one would expect of a state with no angular momentum, it doesn't rotate: spin = 0 (linear polarisation)

 CONSERVATION OF ANGULAR MOMENTUM Angular momentum can never be 'created', only exchanged (just like energy and momentum). Take, for example, the absorption of a photon by a microscopic particle: the photon's angular momentum is transferred to the particle, which starts spinning (provided friction can be overcome). This conservation of angular momentum is a direct consequence of the isotropy of space, i.e. the fact that empty space 'looks the same' in all directions.
ORBITAL ANGULAR MOMENTUM OF LIGHT. In 1992 it was found theoretically by L. Allen et al. that Laguerre-Gaussian light beams possess an orbital angular momentum (OAM) of l hbar (i.e. l h/ (2 pi)) per photon, where l is the so-called azimuthal index of the beam (L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Orbital angular momentum of light and the transformation of Laguerre-Gaussian modes, Phys. Rev. A 45, 8185-8189 (1992)). These beams are examples of light beams with an intensity structure that is symmetric about the beam's axis and a phase structure of l intertwined phase fronts (see figure to the right) - the quantum-mechanical eigenfunctions of the OAM operator Lz = i hbar d/d phi. Since then, several groups, including our own, have demonstrated the mechanical effects of the OAM in that such beams: when microscopic particles absorb such light, they begin to rotate (conservation of angular momentum). Specifically, this has been demonstrated very elegantly in Optical Tweezers, where the same light beam traps ('tweezes') the particle, resulting in a microscopic tool that can not only hold microscopic particles still and translate them in the x, y, and z direction (this is what Optical Tweezers do), but also rotate them about the beam's axis.

 ANGULAR MOMENTUM AND ROTATIONS Mechanical effect of angular momentum: makes things rotate Conservation of angular momentum due to isotropy of space, i.e. its invariance under rotations Quantum-mechanical eigenstates of angular momentum operators change phase under rotations but look the same otherwise
QUANTUM NATURE. Although the OAM was always measured 'per photon', there was always some doubt whether it really is a property associated with individual photons. (In the paper by Allen et al. it originally 'popped' out of a semi-classical calculation.) An experiment performed in 2001 by Alois Mair and colleagues in Anton Zeilinger's group (A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Entanglement of the orbital angular momentum states of photons, Nature 412, 313-316 (2001)) answered this question by performing a fundamentally quantum-mechanical experiment with photons with OAM (Mair et al. were able to show that the orbital angular momenta of a pair of photons generated in parametric down-conversion are entangled; it doesn't get much more quantum than that!). For their experiment Mair et al. had to be able to detect photons with one particular value of the OAM. This was done with holograms which 'flattened' the phase fronts of photons with one chosen value of the OAM, allowing these photons to be focussed through a pin hole and detected. This procedure is a beautifully simple way to detect photons in one particular OAM stateā albeit with limited efficiency: quite a few photons are not detected although they should be.

POTENTIAL FOR QUANTUM COMMUNICATIONS. Shortly afterwards it was proposed by Gabriel Molina-Terriza and co-workers at the University of Barcelona that OAM could be used to encode information in much the same way in which spin angular momentum can be used in quantum cryptography to encode information in the form of polarisation (left- and right-hand circular polarisation are the quantum-mechanical spin eigenstates with respective spins of -hbar and +hbar per photon) (Gabriel Molina-Terriza, Juan P. Torres, and Lluis Torner, Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector State of Angular Momentum, Phys. Rev. Lett. 88, 013601 (2002)). But OAM has a big advantage: whereas a single photon has only 2 distinct spin states, it has in principle infinitely many distinct OAM states. Information can be encoded either way (or as different colours), or in both ways simultaneously, thereby multiplying the number of distinguishable states. In principle, a single photon can in this way carry an arbitrarily large amount of information. The only trouble was that no method had been found that could distinguish all these OAM states with good efficiency.

This is where our work comes in.

ORBITAL ANGULAR MOMENTUM SORTER

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HISTORY. Together with theoreticians Steve Barnett and Sonja Franke-Arnold from the University of Strathclyde, Jonathan Leach, Johannes Courtial, and Miles Padgett, were investigating quantum-mechanical entanglement of the OAM of down-converted photons (independently from Zeilinger's group). The project was lead by Miles. It became apparent that an efficient way to measure the orbital angular momentum of individual photons with good efficiency would be immensely useful for this work. Amongst other things it would also be of potentially great importance for communications (see above). Miles convinced Johannes that this is an interesting and important problem, who - a few days later - came up with the design of the OAM sorter.

 Interferometer that sorts input beams into two 'ports', A1 and B1, according to their orbital angular momentum (OAM) state. The beams are rotated with respect to each other through an angle alpha. If alpha = 180°, photons with an OAM that is an even multiple of hbar (i.e. Lz = ..., -4 hbar, -2 hbar, 0, 2 hbar, 4 hbar, ...) come out of one port, while those whose OAM is an odd multiple of hbar come out of the other port.
CONFIGURATION The mode sorter consists of a number of interferometers that route light beams - and in fact individual photons - into different 'output ports' according to their OAM. Each interferometer divides photons passing through it into two different classes; the first interferometer, for example, is designed so that photons with even values of l come out of one port, while those with odd values emerge from the other. This is done simply by rotating the light beam in one arm of the interferometer through 180° about the beam axis and re-combining it with the non-rotated beam in a beam splitter cube. In the absence of any other phase shift, a beam with an even l value will interfere constructively with itself rotated through 180° and emerge from one side of the beam splitter cube, while a beam with an odd l value will interfere destructively and come out of another side of the beam splitter cube. (Try this out with the interactive demonstration on the right: select the same value of l for the top 2 beams (the input beams) and vary their rotation angles, either by typing a number into the corresponding text field or by dragging one of the red or blue 'node lines'. The brightness of the superposition of the two beams (bottom) will change with their relative rotation angle.) The photons within each class can be subdivided in subsequent interferometers: photons with even values of l, for example, are sorted into those whose l values are/aren't multiples of 4 (by rotation through 90° of the beam in one arm). In principle, the number of output ports can be arbitrarily large and the device 'sorts' with perfect efficiency. With single-photon detectors in each output port it can therefore measure the OAM of individual photons.

EXPERIMENT. An experimental demonstration was set up which could sort 4 different OAM states (to distinguish more states, the setup would have had to be extended by more interferometers, making the experiment much more difficult - see below). A possible way was identified to demonstrate that our setup can indeed sort single photons: the intensity of light passing through the mode sorter can be lowered to levels corresponding to less than one photon being in each interferometer at any one time (on average). This is sufficient to demonstrate sorting of individual photons - a well-rehearsed argument.

WEI AND XUE'S IMPROVEMENTS Haiqing Wei and Xin Xue (from Gazillion Bits Incorporated, CA, and McGill University, Canada) proposed a simplification of our original OAM sorter. The original OAM sorter treats states with odd values of l just like those with even values, after making their l values even (i.e. make them an integer multiple of 2; similarly, l values are turned into integer multiples of 4, 8, 16, etc. later in the sorting process). Making the odd l values even can be done simply by adding +1 to the l value with a 'spiral phase plate' or equivalent component (like, for example, a hologram of a spiral phase plate). However, this is problematic because in practice such components work with only limited efficiency. Wei and Xue realised that the phase shift due to a rotation of the beam for odd l values differs from that for even l values only by a constant phase. This constant phase can be subtracted simply by introducing an additional phase shift in the other arm of the interferometer, so that an interferometer that sorts even l values can, just by changing the path length in the other arm, be used to sort odd l values.

Wei and Xue also brought to our attention that together with Andrew Kirk from McGill University, Canada, they have also described a method for sorting Hermite-Gaussian (HG) modes which predates our OAM sorter (X. Xue, H. Wei, and A. G. Kirk, Beam analysis by fractional Fourier transform, Opt. Lett. 26, 1746-1748 (2001)). This HG mode sorter and our OAM sorter, although developed independently, are strikingly similar: both setups use a 'tree' of Mach-Zehnder interferometers to sort the incoming light in successive stages, but in one arm of each Mach-Zehnder interferometer the HG mode sorter performs a fractional Fourier transform operation whereas our OAM sorter rotates the beam. In conjunction with a cylindrical-lens mode converter, the HG mode sorter could be used as a Laguerre-Gaussian (LG) mode sorter; as LG modes are in OAM eigenstates, such a device could therefore be used as a single-photon OAM sorter.

CHALLENGES

There are a number of problems associated with this technology that need to be overcomebefore it becomes practical. One obvious problem is the interferometric stability required in each of the numerous interferometers - in order to distinguish 2N states with our mode sorter, 2N-1 interferometers are required. (Just keeping the 3 interferometers required for distinction of 4 states aligned keeps Jonathan quite busy.) It might be possible to modify the scheme so that 2N states can be distinguished with only N interferometers (all the interferometers in any one of the N stages are identical). A reduction in the size of the setup - which at the moment takes up about 1 square meter of optical table - might help make individual interferometers more stable.

Even if these problems can be solved, orbital angular momentum encoding is unfortunately not compatible with today's optical fibres. The trouble is that optical fibres - at least most fibres in use today and to the best of our knowledge - alter the light's OAM state. For example, it was shown in our group a couple of years ago that a multi-mode fibre, which was 'stressed' in the middle (a piece of weight was resting on it), converted light with no OAM into light with an OAM of 1hbar per photon (D. McGloin, N. B. Simpson, and M. J. Padgett, The transfer of orbital angular momentum from a stressed fibre-optic waveguide to a light beam, Appl. Opt. 37, 469-472 (1998)). Bending of fibres also tends to cause difficulties. All this would suggest that, at the very least, one would have to be very careful when sending OAM-encoded photons over optical fibres. Future work might change this, of course.

KEY PUBLICATIONS

• J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Measuring the orbital angular momentum of a single photon, Phys. Rev. Lett. 88, 257901 (2002)
• H. Wei, X. Xue, J. Leach, M. J. Padgett, S. M.Barnett, S. Franke-Arnold, E. Yao, and J. Courtial, Simplified measurement of the orbital angular momentum of photons, Opt. Commun. 223, 117-122 (2003)