Photon Orbital Angular Momentum: generation ...

Invited Paper

Photon Orbital Angular Momentum: generation, measurement and application to QKD M. J. Padgett1, D. Giovannini1, M. Lavery1, J. Romero1,2 , S. M. Barnett2, F. Miatto2, R. W. Boyd3, and J. Leach3 1

SUPA, School of Physics & Astronomy, University of Glasgow, Glasgow. UK. 2

SUPA, Department of Physics, Strathclyde University, Glasgow. UK. 3

Department of Physics, University of Ottawa, Ottawa. Canada. [email protected]

Abstract: The information carried by a photon can be encoded in one or more of many different degrees of freedom. Beyond the two-dimensional space of polarisation (spin angular momentum) our interest lies in the unbounded yet discrete state space of Orbital Angular Momentum (OAM). We examine how photon pairs can be generated and measured over a large range of OAM states. Secure Quantum Communication requires the ability to encode information onto a single photon in two or more complimentary basis. For this purpose, a basis set can be understood as a variable whose value can be known, without revealing any information on the value for the complimentary basis. Commonly, a variable of choice is polarization. Using linear polarization a single bit of information can be encoded as vertically or horizontally polarized light, representing a 1 or a 0. Alternatively the bit of information can be encoded as right circular or left circular, again corresponding to a 1 or a 0. However, at the level of single photons, if the recipient chooses to measure in the linear basis, then irrespective of the outcome of that measurement (i.e. 1 or 0) no similar measurement can be made of the circular states and visa versa. But polarization is not the only basis in which the information in the photon can be encoded. In principle, any basis set along with its complementary set can be used to encode the information, the unifying feature is that the two corresponding variables are subject to an uncertainty relationship. It is immediately apparent that arrival time and measured frequency or transverse position and transverse momentum are also possible basis sets in which to encode information. The advantage of these basis sets is that whereas polarization is restricted to one bit per photon, the continuous nature of time/frequency or position/momentum potentially gives an unlimited information capacity even on a single photon.

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11=3

=4

Figure 1: Helical Phase fronts corresponding to different OAM states. The surfaces of constant phase map out  intertwined helical surfaces.

Central to our work has been another basis set, namely, orbital angular momentum (OAM). As identified by Allen et al. in 1992, OAM arises from the spatial phase structure of an optical beam [1]. Specifically helical phase fronts € carry an OAM corresponding to  per photon (Figure 1), and light beams described by an exp(iφ ) cross-section carrying OAM have been the subject of intense study for the last 20 years [2]. When described in terms of Laguerre Gaussian a mode, these OAM states form a discrete, complete, orthonormal basis set with angular position (continuous but periodic) as a complimentary variable. This discrete/periodic nature raises fundamental questions as € € the corresponding uncertainty relationship [3] ensures to whether the inherent security that one associates with quantum techniques. Electro-Optical Remote Sensing, Photonic Technologies, and Applications VI, edited by G. W. Kamerman, O. Steinvall, K. L. Lewis, R. C. Hollins, T. J. Merlet, M. T. Gruneisen, M. Dusek, J. G. Rarity, G. J. Bishop, J. Gonglewski, Proc. of SPIE Vol. 8542, 85421P · © 2012 SPIE · CCC code: 0277-786/12/$18 · doi: 10.1117/12.975993 Proc. of SPIE Vol. 8542 85421P-1 Downloaded From: http://spiedigitallibrary.org/ on 03/31/2014 Terms of Use: http://spiedl.org/terms

Within a classical context the potential to use OAM as a variable in an optical communication system was demonstrated using diffractive optics to generate and measure the OAM value of an optical beam within a demonstrator free-space optical system [4], (Figure 2). 1 of 8 hologram patterns (transmitter)

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CCD camera Hologram pattern (receiver) Figure 2: Spatial light modulators can be used to transform a light beam into one of many OAM states that can then be used as the information carrying variable in a free-space optical link.

Within a quantum context most research has revolved around studies of OAM produced by parametric downconversion [5,6]. Typically a UV pump beam incident on a non-linear crystal, such as BBO, produces photon pairs that are entangled in their spatial states. Although the classical examination of the output shows it to be spatially incoherent and hence possess a range of OAM values [7] at the level of individual photon pairs, the OAM of the signal and pump beams always sum to give that of the pump beam [5]. In virtually all work to date, this OAM has been measured using forked diffraction gratings [8] to couple specific OAM states to single mode fibre giving a modal selectivity 99% or more. This selective coupling is greatly facilitated by the use of spatial light modulators acting as the diffractive components, allowing the detected state to be switched at video frame rates [9]. Our demonstration of an angular form of the Einstein Podolsky Rosen (EPR) paradox [10] and other tests [11] showed that orbital angular momentum has the potential to give a secure basis for a communication system. One point to note is that to ensure the unbiased nature of the OAM and angles states, each of the angle states must be obtained as a coherent addition of the range of OAM states [12]. For a finite number of OAM states this means the resulting angle states are complicated azimuthal functions with slightly overlapping intensity profiles (Figure 3). By comparison set of hard-edged, and hence none overlapping, angle states would require an infinite range of OAM states.

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*1)  =-1,0,1

 =-2,-1….2

 =-3,-2....3

 =-4,-3….4

Figure 3: Intensity cross sections of unbiased angle states plotted on a log scale over three orders of magnitude obtained from the equally weighted superposition of 3, 5, 7 and 9 adjacent OAM states.

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The requirement for precise intensity shaping of the detected photons can also be implemented using the spatial light modulator approach allowing the complicated correlations between the down converted photons to be studied, see figure 4 [12].

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Angle

BBO crystal f =300mm

lens

f =600mm lens

or

idler SLM 1

f=00mm lens

SMF

signal SPADs

SMF

Figure 4: Spatial light modulators (SLM) enable the measurement of the complex spatial states of individual photons, allowing the nature of the correlations between signal and idler modes to be explored.

Although OAM is potentially unbounded, in practice one only generates and measures states over a finite range of values and one aspect of our work has been to both characterize and extend this range. For generating OAM states by parametric down-conversion, making small changes to the phase matching conditions can generate orbital angular momentum states over 50 values [13,14] (c.f. 2 for polarization), see Figure 5. Various aspects come into play in the generation, transmission and detection of OAM. Whereas a property such as polarisation or frequency is associated with a single spatial mode, e.g. the input or output from a single mode fibre, OAM is by definition a multi mode parameter space. OAM modes are no different from any other modal set, their generation, lossless transmission or measurement requires optical systems with a suitable Fresnel number. In the case of generation by parametric down-conversion this leads to consideration of both the aperture of the non linear crystal (and pump beam) and the range of angles over which the phase-matching gives emission [15]. For transmission along optical fibre, minimizing coupling between spatial modes is essential [16], and for free space transmission atmospheric turbulence introduces cross-talk between states [17].

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20

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Figure 5: The range of OAM states produced by parametric down conversion can be increased by making small adjustments to the phase matching conditions of the nonlinear process. We take φ as the measure of our collinearity, with φ =0 corresponding to the collinear case.


In principle, different variables (polarization, time/frequency, position/momentum, angular momentum/angular position) can be combined to give massive amounts of information encoded on a single photon. Of course each of these basis sets creates technical challenges both in its generation and detection. Futhermore, loss of entanglement, cross talk between states and a degradation both of information capacity and security, can be introduced to different variables by different mechanisms. For spatial states, like orbital angular momentum, then atmospheric turbulence can have a detrimental effect on the fidelity of OAM channels [18]. As an alternative to using OAM as a quantum variable it can simply be employed for the multiplexing of classical channels. Each classical channel can then itself utilize a further variable to carry quantum secure information. Rather than encoding information directly, different values orbital angular momentum can therefore be used to give multiple classical or quantum channels [19]. Indeed we have show the operation of orbital angular momentum separator which routes different angular momentum states to different lateral positions [20]. Most previous techniques to measure or collect OAM states worked on a single OAM value at a time, hence although light could be encoded into one of many different OAM states, this high data capacity was not accessible to the measurement system. Our new approach is based upon beam reformatting where the helically phased wave associated with OAM is reshaped to give a linear phase ramp that when focus gives a spot, the lateral position of which is proportional to the OAM value of the incident beam [21], see figure 5.

Figure 5: The reformatter converts the helically phased waved associated with OAM to a linear phase ramp, which when focused gives a laterally displaced spot.

Each of these classical channels could then be encoded with secure information. Despite being a subject of study for 20 years it is only our recent work that has shown a practical approach to the multistate rather than sequential measurement or separation of orbital angular momentum. The orbital angular momentum of light was only recognized as a macroscopic description of the light beam 20 years ago, and as a resource for quantum studies/applications within the last 10 [22]. In this work we have discussed both the generation and measurement of orbital angular momentum both in terms of understanding the nature of the quantum variable and its applications to communication and other systems. [1] [2] [3] [4] [5] [6] [7] [8] [9]

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