We describe an experimental implementation of a free-space 11-dimensional
communication system using orbital angular momentum (OAM) modes. This system
has a maximum measured OAM channel capacity of 2.12 bits/photon. The effects of
Kolmogorov thin-phase turbulence on the OAM channel capacity are quantified. We
find that increasing the turbulence leads to a degradation of the channel
capacity. We are able to mitigate the effects of turbulence by increasing the
spacing between detected OAM modes. This study has implications for
high-dimensional quantum key distribution (QKD) systems. We describe the sort
of QKD system that could be built using our current technology.Comment: 6 pages, 5 figure

Boyd, Robert W.

Lavery, Martin P. J.

Leach, Jonathan

Malik, Mehul

Mirhosseini, Mohammad

O'Sullivan, Malcolm

Padgett, Miles J.

Rodenburg, Brandon

Optics Express

English

arXiv

We describe an experimental implementation of a free-space 11-dimensional communication system using orbital angular momentum (OAM) modes. This system has a maximum measured OAM channel capacity of 2.12 bits/photon. The effects of Kolmogorov thin-phase turbulence on the OAM channel capacity are quantified. We find that increasing the turbulence leads to a degradation of the channel capacity. We are able to mitigate the effects of turbulence by increasing the spacing between detected OAM modes. This study has implications for high-dimensional quantum key distribution (QKD) systems. We describe the sort of QKD system that could be built using our current technology. (C) 2012 Optical Society of America</p

Heriot Watt Pure

    Heriot-Watt University Research Gateway                                                          Heriot-Watt UniversityInfluence of atmospheric turbulence on optical communications using orbital angularmomentum for encodingMalik, Mehul; O'Sullivan, Malcolm; Rodenburg, Brandon; Mirhosseini, Mohammad; Leach,Jonathan; Lavery, Martin P. J.; Padgett, Miles J.; Boyd, Robert W.Published in:Optics ExpressDOI:10.1364/OE.20.013195Publication date:2012Link to publication in Heriot-Watt Research GatewayCitation for published version (APA):Malik, M., O'Sullivan, M., Rodenburg, B., Mirhosseini, M., Leach, J., Lavery, M. P. J., ... Boyd, R. W. (2012).Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding.Optics Express, 20(12), 13195-13200. 10.1364/OE.20.013195Inﬂuence of atmospheric turbulence onoptical communications using orbitalangular momentum for encodingMehul Malik,1,∗ Malcolm O’Sullivan,1 Brandon Rodenburg,1Mohammad Mirhosseini,1 Jonathan Leach,2 Martin P. J. Lavery,3Miles J. Padgett,3 and Robert W. Boyd1,21The Institute of Optics, University of Rochester, Rochester, New York 14627 USA2Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5 Canada3School of Physics and Astronomy, University of Glasgow, Glasgow, UK∗memalik@optics.rochester.eduAbstract: We describe an experimental implementation of a free-space11-dimensional communication system using orbital angular momentum(OAM) modes. This system has a maximum measured OAM channel ca-pacity of 2.12 bits/photon. The effects of Kolmogorov thin-phase turbulenceon the OAM channel capacity are quantiﬁed. We ﬁnd that increasing theturbulence leads to a degradation of the channel capacity. We are ableto mitigate the effects of turbulence by increasing the spacing betweendetected OAM modes. This study has implications for high-dimensionalquantum key distribution (QKD) systems. We describe the sort of QKDsystem that could be built using our current technology.© 2012 Optical Society of AmericaOCIS codes: (270.5568) Quantum cryptography; (200.2605) Free-space optical communica-tion; (010.1330) Atmospheric turbulence.References and links1. C. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int.Conf., 175–179 (Bangalore, 1984).2. P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt,“Long-distance quantum key distribution in optical ﬁbre,” New J. Phys. 8, 193–193 (2006).3. R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jen-newein, J. Perdigues, P. Trojek, B. ¨ Omer, M. F¨ urst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Wein-furter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486(2007).4. M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel en-coding: security analysis,” J. Phys. A-Math. Gen. 35, 10065–10076 (2002).5. S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography withqutrits,” New J. Phys. 8, 75 (2006).6. N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-Level sys-tems,” Phys. Rev. Lett. 88, 127902 (2002).7. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communica-tion,” Phys. Rev. Lett. 94, 153901 (2005).8. G. Tyler and R. W. Boyd, “Inﬂuence of atmospheric turbulence on the propagation of quantum states of lightcarrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009).9. B. Smith and M. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).10. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological chargeconservation,” J. Opt. Soc. Am. A 25, 225–230 (2008).#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  1319511. F. Roux, “Inﬁnitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled bipho-ton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).12. W.K.WoottersandB.D.Fields,“Optimalstate-determinationbymutuallyunbiasedmeasurements,”Ann.Phys.-New York 191, 363–381 (1989).13. V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, “Pixelated phase computer holograms for the accurateencoding of scalar complex ﬁelds,” J. Opt. Soc. Am. A 24, 3500–3507 (2007).14. M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex ﬁelds withspatial light modulators: application to quantum key distribution,” Appl. Opt. 47, A32–A42 (2008).15. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efﬁcient sorting oforbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).16. M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractiveelements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–2115 (2012).17. A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).18. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1431(1965).19. C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38,2161–2170 (1999).1. IntroductionSince Bennett and Brassard introduced the ﬁrst quantum key distribution (QKD) protocol in1984 [1], the ﬁeld of QKD has rapidly developed to the extent that QKD systems are commer-cially available today. Secure transmission of a quantum key has been performed over 148.7km of ﬁber [2] as well as over 144 km of free space [3]. One of the limiting aspects of thesekey distribution systems is that they use the polarization degree of freedom of the photon toencode information [2, 3]. The use of polarization encoding limits the maximum amount ofinformation that can be encoded on each photon to one bit. In addition, it places a low boundon the amount of error an eavesdropper can introduce without compromising the security of thetransmission [4]. Because of these limitations, there has been great interest in exploring otherways to encode information on a photon that would allow for higher data transmission rates andincreased security [5,6].Here we report results on the use of orbital angular momentum (OAM) modes of a photonfor optical communication. The motivation for doing so is that OAM modes span an inﬁnite-dimensional basis. Hence, there is no limit to how much information one can send per photonin such a system. The large dimensionality of this protocol also provides a much higher level ofsecurity than the two-state approach [4]. However, in a practical communication system usingOAM modes, the maximum number of modes that can be used is limited by the size of thelimiting aperture in the system. This occurs because the radius of an OAM mode increaseswith the mode number. In addition, since these are spatial modes, they are highly susceptible toturbulence. Recently, there have been several theoretical studies on how atmospheric turbulenceaffects OAM modes [7–11]. In this paper, we experimentally study the effects of atmosphericturbulence on the channel capacity of an OAM communication channel at high light levels.Also, we describe a high-dimensional QKD system that could be built using OAM modes forencoding.2. Quantum key distribution systemIn a conventional QKD system, the sender (Alice) and receiver (Bob) must have at least twomutually unbiased bases (MUBs) in which to send and receive their key. The maximum numberof MUBs in an N-dimensional QKD system is equal to N +1, for when N is a prime number[12]. For the polarization-based implementation of the BB84 protocol [1], N is equal to 2 andthere are three available MUBs. Here we describe a proposed OAM-based QKD system witha dimensionality d = 11, and 12 possible mutually unbiased bases. Out of these, we choose to#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13196HeNe SLM AOMPinhole ALICESorterEMCCDBOBFT Lens (OAM)Imaging Lens (ANG)R1 R2 BSFig. 1. Proposed QKD Setup. Our source is a HeNe laser operating at 633 nm that is mod-ulated by an acousto-optic modulator (AOM) to carve out single-photon pulses. The OAMand ANG states, as well as the turbulence phase screens are generated by the spatial lightmodulator (SLM) and 4f system. R1 and R2 are custom refractive elements used to sortthe OAM and ANG modes, and the beam splitter (BS) acts as the passive basis selector.The modes are detected on an electron-multiplying camera (EMCCD).use only two in order to maximize our key transmission rate, which is inversely proportional tothe number of MUBs that are used. The ﬁrst basis is made up of 11 OAM modes and is calledthe OAM basis. The OAM modes have the formΨl,0OAM = A0W(r/R)exp(ilθ) (1)where A0 is the spatially uniform ﬁeld amplitude,W(x) is an aperture function such thatW(x)=1 for |x| ≤ 1 and zero otherwise, r and θ are the radial and azimuthal coordinates, and l is theOAM quantum number. The second basis is called the ANG basis and is made up of angularmodes, which are a linear combination of the 11 OAM modes and have the formΨnANG =1√115∑l=−5Ψl,0OAMexp(i2πnl/11). (2)These modes are so named because their intensity proﬁle looks like an angular slice thatmoves around the center of the beam as one changes the relative phases of the componentOAM modes. In order to characterize the effects of turbulence on our system, we choose touse pure vortex modes with a circular aperture function instead of Laguerre-Gauss modes inour system. We do this because our theoretical approach closely follows that of reference [8],which is also based on pure vortex modes.A schematic of our proposed QKD system is shown in Fig. 1. We use a HeNe laser modu-latedbyanacousto-optical modulator(AOM)asoursource.TheAOMcarves outsingle-photonpulses for the QKD protocol. The OAM and ANG modes are generated by Alice using a holo-gram generated on a Holoeye PLUTO phase-only spatial light modulator (SLM) in conjunctionwith a 4f system [13,14]. Figure 2 shows some of the holograms we used and the OAM andANG modes generated by them.Thegeneratedmodesarethenimagedacrossafree-spacechannelontoBob’sdetectionplane.One of the key hurdles to using OAM modes to perform QKD has been the need of a method ofefﬁciently sorting single photons carrying OAM modes. This problem was recently overcomeby a method that uses two phase screens to conformally map an input OAM state to a shiftedspot in the fourier plane of a lens [15]. In our setup, we are using custom refractive elementsdesigned according to this method in order to carry out efﬁcient sorting of OAM modes [16].The OAM sorter maps the input OAM modes to a row of spots on our detector.Since the complex ANG modes are mostly non-overlapping in intensity, they can be sorted#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13197Fig. 2. Upper row: Holograms written onto the SLM to generate OAM and ANG modes.Lower row: CCD images of the corresponding modes generated (OAM mode number l =−1, 3, and 5, and ANG mode number n = 1, 5, and 9).based on their angular position. For convenience, we use the ﬁrst element (R1) of the OAMsorter to map the angular intensities to a partially overlapping row of spots, which are imagedonto the detector. The beam splitter (BS) acts a passive selector of Bob’s measurement basis,randomly measuring photons in either the OAM or ANG basis. We use an electron-multiplyingCCD (EMCCD) as our single-photon detector. While we intend on using this setup to performQKD in the future, currently we are using it at high light levels for the purposes of characteriz-ing the effects of turbulence on the OAM channel.3. Effects of turbulence on channel capacityAtmospheric turbulence plays a large role in determining how practical a free-space commu-nication system may be. Here, we study the effects of weak turbulence on the OAM commu-nication channel. In our system, the aperture of the sender is imaged onto the aperture of thereceiver. For such a system, turbulence can be modeled as a thin-phase screen in the sendingaperture [17]. The effects of turbulence are characterized in terms of crosstalk in the OAM andANG channels. However, the effects of thin phase turbulence on the ANG channel cannot bestudied in our system because of two reasons. First, the sorter is limited in that it only maps thenon-overlapping intensities of the complex ANG modes to a row of spots. Any phase informa-tion in the ANG modes is not conveyed. Second, the thin-phase turbulence is in an image planeof the ANG channel. Since the ANG modes are imaged from the SLM to the sorter elementR1, and then intensity-mapped to the detector, the effects of turbulence added at the SLM arenot apparent in the sorted ANG modes.However, we can study the effects of turbulence on the OAM communication channel, asthe sorter sorts these modes based on their orthogonal phase distributions. The weak turbulenceregime can be approximated by the Kolmogorov thin-phase model of turbulence. The phasestructure function in this model obeys the relationship [φ(r1)−φ(r2)]2  = 6.88r1−r2r05/3(3)where r0 is Fried’s coherence diameter [18], a measure of the transverse distance scale overwhich refractive index correlations remain correlated. We generate turbulence phase screensbased on this model [19] and add them to the pattern written on the SLM. An example ofthree Kolmogorov phase screens with different values of r0 is shown in Fig. 3. The crosstalkintroduced by this type of turbulence into an OAM channel can be written in the form [8]#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13198Fig. 3. Sample Kolmogorov turbulence phase screens that are added onto the SLM forthree different values of turbulence strength, D/r0 =(a) 5.12, (b) 10.25, and (c) 102.5.Each screen has a resolution of 512×512 pixels and the phase wraps from 0 to 2π.Pds =1π 10ρdρ 2π0dθe−3.44(D/r0)5/3(ρsinθ/2)5/3cos[(d−s)θ] (4)where ρ = r/R, and Pds is the conditional probability that Bob detects an OAM mode withquantum number d given that Alice sent an OAM mode with quantum number s. Here, D isthe diameter of the receiving aperture. From these probabilities, we can calculate the channelcapacity of our OAM channel asC = max[H(x)−H(x|y)]= max{Ps}−∑sPslog2(Ps)+∑sPs∑dPdslog2(Pds). (5)Here, N is the total number of modes in our system and Ps is the probability that Alice sentmode s. Experimental data for the channel capacity of the OAM communication channel withN = 11,9,7,5, and 3 are shown in Fig. 4. The channel capacity, C, is plotted as a functionof the turbulence strength, D/r0, where D is the diameter of the photon ﬁeld at the SLMand r0 is Fried’s coherence diameter. For each data point, we average over 20 different tur-bulence screens. The theoretical crosstalk is calculated by Eq. (4) and corrected for the inherent  00.52Channel Capacity (bits per photon)11.5D/r0101 100 10-1N=11N= 9N= 7N= 5N= 3Pol.Fig. 4. Inﬂuence of Kolmogorov turbulence on the channel capacity of an OAM communi-cation channel for a system dimensionality of N = 3,5,7,9 and 11. The channel capacityof a system using polarization modes is plotted as a dotted line for comparison.#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13199  00.20.40.60.811.21.41.6Channel Capacity (bits per photon)D/r0101 100 10-1MS = 4MS = 2MS = 1Fig. 5. Effects of changing the spacing between detected modes (MS) on the channel ca-pacity of a 3-dimensional OAM communication system in the presence of atmosphericturbulence. For a mode-spacing of 4, the maximum channel capacity of a 3-dimensionalsystem is seen to approach the theoretical maximum of log2(3)=1.585.crosstalk introduced by the sorter [15]. The theoretical channel capacity as calculated by Eq.(5) is plotted in Fig. 4, as well as the channel capacity of a polarization-based two-dimensionalsystem, which is essentially immune to the effects of turbulence.As can be seen, the maximum channel capacity for N =11 is much lower than the theoreticalmaximum of log2(11)=3.46 bits/photon. This is because the OAM sorter is not perfect andintroduces some inherent crosstalk [15]. Nonetheless, our higher dimensional systems performmuch better than the polarization-based system below a certain turbulence strength. This isindicatedinFig.4bythepointatwhichthechannelcapacitycurveforanN-dimensionalsystemcrosses the dotted line for a polarization-based system. At a turbulence strength of D/r0 = 10,the channel capacity vanishes for all systems.Since OAM modes constitute an inﬁnite-dimensional basis, one can reduce the effects ofthe inherent sorter crosstalk and turbulence by increasing the spacing (MS) between detectedmodes to greater than one. We study this effect on a subset of our system with N = 3, forthree different mode-spacings (MS = 1, 2 and 4). It is clear from Fig. 5 that increasing themode spacing between detected modes boosts the initial channel capacity and makes our systemslightly more resistant to turbulence. The maximum channel capacity for a mode-spacing offour is close to the theoretical maximum of log2(3)=1.585. In addition, the channel capacityfor this mode-spacing starts decreasing at a value of D/r0 that is almost an order of magnitudegreater than that for a mode-spacing of one.4. ConclusionsWe characterized the effects of Kolmogorov thin-phase turbulence on the channel capacity ofan OAM-based communication system at high light levels. We found that turbulence affects thechannel capacity drastically, and the effects can be mitigated by increasing the mode-spacing.We were not able to study the effects of thin-phase turbulence on the ANG channel due to theinherentlimitationsofourmodesorter.Wearecurrentlyworkingonimprovingtheperformanceof our mode sorter and extending our study to thick-phase turbulence. We would like to thankDr. D. J. Gauthier for useful discussions. This work was supported by the DARPA/DSO InPhoprogram and the Canadian Excellence Research Chair (CERC) program.#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13200

Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding

Influence of atmospheric turbulence onoptical communications using orbitalangular momentum for encodingMehul Malik,1,∗ Malcolm O’Sullivan,1 Brandon Rodenburg,1Mohammad Mirhosseini,1 Jonathan Leach,2 Martin P. J. Lavery,3Miles J. Padgett,3 and Robert W. Boyd1,21The Institute of Optics, University of Rochester, Rochester, New York 14627 USA2Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5 Canada3School of Physics and Astronomy, University of Glasgow, Glasgow, UK∗memalik@optics.rochester.eduAbstract: We describe an experimental implementation of a free-space11-dimensional communication system using orbital angular momentum(OAM) modes. This system has a maximum measured OAM channel ca-pacity of 2.12 bits/photon. The effects of Kolmogorov thin-phase turbulenceon the OAM channel capacity are quantified. We find that increasing theturbulence leads to a degradation of the channel capacity. We are ableto mitigate the effects of turbulence by increasing the spacing betweendetected OAM modes. This study has implications for high-dimensionalquantum key distribution (QKD) systems. We describe the sort of QKDsystem that could be built using our current technology.© 2012 Optical Society of AmericaOCIS codes: (270.5568) Quantum cryptography; (200.2605) Free-space optical communica-tion; (010.1330) Atmospheric turbulence.References and links1. C. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int.Conf., 175–179 (Bangalore, 1984).2. P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt,“Long-distance quantum key distribution in optical fibre,” New J. Phys. 8, 193–193 (2006).3. R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jen-newein, J. Perdigues, P. Trojek, B. ¨Omer, M. Fu¨rst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Wein-furter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486(2007).4. M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel en-coding: security analysis,” J. Phys. A-Math. Gen. 35, 10065–10076 (2002).5. S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography withqutrits,” New J. Phys. 8, 75 (2006).6. N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-Level sys-tems,” Phys. Rev. Lett. 88, 127902 (2002).7. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communica-tion,” Phys. Rev. Lett. 94, 153901 (2005).8. G. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of lightcarrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009).9. B. Smith and M. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).10. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological chargeconservation,” J. Opt. Soc. Am. A 25, 225–230 (2008).#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  1319511. F. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled bipho-ton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).12. W. K. Wootters and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys.-New York 191, 363–381 (1989).13. V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, “Pixelated phase computer holograms for the accurateencoding of scalar complex fields,” J. Opt. Soc. Am. A 24, 3500–3507 (2007).14. M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields withspatial light modulators: application to quantum key distribution,” Appl. Opt. 47, A32–A42 (2008).15. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting oforbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).16. M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractiveelements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–2115 (2012).17. A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).18. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1431(1965).19. C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38,2161–2170 (1999).1. IntroductionSince Bennett and Brassard introduced the first quantum key distribution (QKD) protocol in1984 [1], the field of QKD has rapidly developed to the extent that QKD systems are commer-cially available today. Secure transmission of a quantum key has been performed over 148.7km of fiber [2] as well as over 144 km of free space [3]. One of the limiting aspects of thesekey distribution systems is that they use the polarization degree of freedom of the photon toencode information [2, 3]. The use of polarization encoding limits the maximum amount ofinformation that can be encoded on each photon to one bit. In addition, it places a low boundon the amount of error an eavesdropper can introduce without compromising the security of thetransmission [4]. Because of these limitations, there has been great interest in exploring otherways to encode information on a photon that would allow for higher data transmission rates andincreased security [5, 6].Here we report results on the use of orbital angular momentum (OAM) modes of a photonfor optical communication. The motivation for doing so is that OAM modes span an infinite-dimensional basis. Hence, there is no limit to how much information one can send per photonin such a system. The large dimensionality of this protocol also provides a much higher level ofsecurity than the two-state approach [4]. However, in a practical communication system usingOAM modes, the maximum number of modes that can be used is limited by the size of thelimiting aperture in the system. This occurs because the radius of an OAM mode increaseswith the mode number. In addition, since these are spatial modes, they are highly susceptible toturbulence. Recently, there have been several theoretical studies on how atmospheric turbulenceaffects OAM modes [7–11]. In this paper, we experimentally study the effects of atmosphericturbulence on the channel capacity of an OAM communication channel at high light levels.Also, we describe a high-dimensional QKD system that could be built using OAM modes forencoding.2. Quantum key distribution systemIn a conventional QKD system, the sender (Alice) and receiver (Bob) must have at least twomutually unbiased bases (MUBs) in which to send and receive their key. The maximum numberof MUBs in an N-dimensional QKD system is equal to N + 1, for when N is a prime number[12]. For the polarization-based implementation of the BB84 protocol [1], N is equal to 2 andthere are three available MUBs. Here we describe a proposed OAM-based QKD system witha dimensionality d = 11, and 12 possible mutually unbiased bases. Out of these, we choose to#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13196HeNe SLMAOMPinholeALICESorterEMCCDBOBFT Lens (OAM)Imaging Lens (ANG)R1 R2 BSFig. 1. Proposed QKD Setup. Our source is a HeNe laser operating at 633 nm that is mod-ulated by an acousto-optic modulator (AOM) to carve out single-photon pulses. The OAMand ANG states, as well as the turbulence phase screens are generated by the spatial lightmodulator (SLM) and 4 f system. R1 and R2 are custom refractive elements used to sortthe OAM and ANG modes, and the beam splitter (BS) acts as the passive basis selector.The modes are detected on an electron-multiplying camera (EMCCD).use only two in order to maximize our key transmission rate, which is inversely proportional tothe number of MUBs that are used. The first basis is made up of 11 OAM modes and is calledthe OAM basis. The OAM modes have the formΨl,0OAM = A0W (r/R)exp(ilθ) (1)where A0 is the spatially uniform field amplitude, W (x) is an aperture function such that W (x) =1 for |x| ≤ 1 and zero otherwise, r and θ are the radial and azimuthal coordinates, and l is theOAM quantum number. The second basis is called the ANG basis and is made up of angularmodes, which are a linear combination of the 11 OAM modes and have the formΨnANG =1√115∑l=−5Ψl,0OAM exp(i2πnl/11). (2)These modes are so named because their intensity profile looks like an angular slice thatmoves around the center of the beam as one changes the relative phases of the componentOAM modes. In order to characterize the effects of turbulence on our system, we choose touse pure vortex modes with a circular aperture function instead of Laguerre-Gauss modes inour system. We do this because our theoretical approach closely follows that of reference [8],which is also based on pure vortex modes.A schematic of our proposed QKD system is shown in Fig. 1. We use a HeNe laser modu-lated by an acousto-optical modulator (AOM) as our source. The AOM carves out single-photonpulses for the QKD protocol. The OAM and ANG modes are generated by Alice using a holo-gram generated on a Holoeye PLUTO phase-only spatial light modulator (SLM) in conjunctionwith a 4 f system [13, 14]. Figure 2 shows some of the holograms we used and the OAM andANG modes generated by them.The generated modes are then imaged across a free-space channel onto Bob’s detection plane.One of the key hurdles to using OAM modes to perform QKD has been the need of a method ofefficiently sorting single photons carrying OAM modes. This problem was recently overcomeby a method that uses two phase screens to conformally map an input OAM state to a shiftedspot in the fourier plane of a lens [15]. In our setup, we are using custom refractive elementsdesigned according to this method in order to carry out efficient sorting of OAM modes [16].The OAM sorter maps the input OAM modes to a row of spots on our detector.Since the complex ANG modes are mostly non-overlapping in intensity, they can be sorted#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13197Fig. 2. Upper row: Holograms written onto the SLM to generate OAM and ANG modes.Lower row: CCD images of the corresponding modes generated (OAM mode number l =−1, 3, and 5, and ANG mode number n = 1, 5, and 9).based on their angular position. For convenience, we use the first element (R1) of the OAMsorter to map the angular intensities to a partially overlapping row of spots, which are imagedonto the detector. The beam splitter (BS) acts a passive selector of Bob’s measurement basis,randomly measuring photons in either the OAM or ANG basis. We use an electron-multiplyingCCD (EMCCD) as our single-photon detector. While we intend on using this setup to performQKD in the future, currently we are using it at high light levels for the purposes of characteriz-ing the effects of turbulence on the OAM channel.3. Effects of turbulence on channel capacityAtmospheric turbulence plays a large role in determining how practical a free-space commu-nication system may be. Here, we study the effects of weak turbulence on the OAM commu-nication channel. In our system, the aperture of the sender is imaged onto the aperture of thereceiver. For such a system, turbulence can be modeled as a thin-phase screen in the sendingaperture [17]. The effects of turbulence are characterized in terms of crosstalk in the OAM andANG channels. However, the effects of thin phase turbulence on the ANG channel cannot bestudied in our system because of two reasons. First, the sorter is limited in that it only maps thenon-overlapping intensities of the complex ANG modes to a row of spots. Any phase informa-tion in the ANG modes is not conveyed. Second, the thin-phase turbulence is in an image planeof the ANG channel. Since the ANG modes are imaged from the SLM to the sorter elementR1, and then intensity-mapped to the detector, the effects of turbulence added at the SLM arenot apparent in the sorted ANG modes.However, we can study the effects of turbulence on the OAM communication channel, asthe sorter sorts these modes based on their orthogonal phase distributions. The weak turbulenceregime can be approximated by the Kolmogorov thin-phase model of turbulence. The phasestructure function in this model obeys the relationship〈[φ(r1)−φ(r2)]2〉= 6.88∣∣∣∣r1 − r2r0∣∣∣∣5/3(3)where r0 is Fried’s coherence diameter [18], a measure of the transverse distance scale overwhich refractive index correlations remain correlated. We generate turbulence phase screensbased on this model [19] and add them to the pattern written on the SLM. An example ofthree Kolmogorov phase screens with different values of r0 is shown in Fig. 3. The crosstalkintroduced by this type of turbulence into an OAM channel can be written in the form [8]#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13198Fig. 3. Sample Kolmogorov turbulence phase screens that are added onto the SLM forthree different values of turbulence strength, D/r0 =(a) 5.12, (b) 10.25, and (c) 102.5.Each screen has a resolution of 512×512 pixels and the phase wraps from 0 to 2π .Pds =1π∫ 10ρdρ∫ 2π0dθe−3.44(D/r0)5/3(ρ sinθ/2)5/3 cos [(d − s)θ ] (4)where ρ = r/R, and Pds is the conditional probability that Bob detects an OAM mode withquantum number d given that Alice sent an OAM mode with quantum number s. Here, D isthe diameter of the receiving aperture. From these probabilities, we can calculate the channelcapacity of our OAM channel asC = max[H(x)−H(x|y)]= max{Ps}[−∑sPs log2 (Ps)+∑sPs ∑dPds log2(Pds)]. (5)Here, N is the total number of modes in our system and Ps is the probability that Alice sentmode s. Experimental data for the channel capacity of the OAM communication channel withN = 11,9,7,5, and 3 are shown in Fig. 4. The channel capacity, C, is plotted as a functionof the turbulence strength, D/r0, where D is the diameter of the photon field at the SLMand r0 is Fried’s coherence diameter. For each data point, we average over 20 different tur-bulence screens. The theoretical crosstalk is calculated by Eq. (4) and corrected for the inherent  00.52Channel Capacity (bits per photon)11.5D/r010110010-1N=11N= 9N= 7N= 5N= 3Pol.Fig. 4. Influence of Kolmogorov turbulence on the channel capacity of an OAM communi-cation channel for a system dimensionality of N = 3,5,7,9 and 11. The channel capacityof a system using polarization modes is plotted as a dotted line for comparison.#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13199  00.20.40.60.811.21.41.6Channel Capacity (bits per photon)D/r010110010-1MS = 4MS = 2MS = 1Fig. 5. Effects of changing the spacing between detected modes (MS) on the channel ca-pacity of a 3-dimensional OAM communication system in the presence of atmosphericturbulence. For a mode-spacing of 4, the maximum channel capacity of a 3-dimensionalsystem is seen to approach the theoretical maximum of log2(3) = 1.585.crosstalk introduced by the sorter [15]. The theoretical channel capacity as calculated by Eq.(5) is plotted in Fig. 4, as well as the channel capacity of a polarization-based two-dimensionalsystem, which is essentially immune to the effects of turbulence.As can be seen, the maximum channel capacity for N = 11 is much lower than the theoreticalmaximum of log2(11) = 3.46 bits/photon. This is because the OAM sorter is not perfect andintroduces some inherent crosstalk [15]. Nonetheless, our higher dimensional systems performmuch better than the polarization-based system below a certain turbulence strength. This isindicated in Fig. 4 by the point at which the channel capacity curve for an N-dimensional systemcrosses the dotted line for a polarization-based system. At a turbulence strength of D/r0 = 10,the channel capacity vanishes for all systems.Since OAM modes constitute an infinite-dimensional basis, one can reduce the effects ofthe inherent sorter crosstalk and turbulence by increasing the spacing (MS) between detectedmodes to greater than one. We study this effect on a subset of our system with N = 3, forthree different mode-spacings (MS = 1, 2 and 4). It is clear from Fig. 5 that increasing themode spacing between detected modes boosts the initial channel capacity and makes our systemslightly more resistant to turbulence. The maximum channel capacity for a mode-spacing offour is close to the theoretical maximum of log2(3) = 1.585. In addition, the channel capacityfor this mode-spacing starts decreasing at a value of D/r0 that is almost an order of magnitudegreater than that for a mode-spacing of one.4. ConclusionsWe characterized the effects of Kolmogorov thin-phase turbulence on the channel capacity ofan OAM-based communication system at high light levels. We found that turbulence affects thechannel capacity drastically, and the effects can be mitigated by increasing the mode-spacing.We were not able to study the effects of thin-phase turbulence on the ANG channel due to theinherent limitations of our mode sorter. We are currently working on improving the performanceof our mode sorter and extending our study to thick-phase turbulence. We would like to thankDr. D. J. Gauthier for useful discussions. This work was supported by the DARPA/DSO InPhoprogram and the Canadian Excellence Research Chair (CERC) program.#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13200

Mehul Malik

Malcolm O’Sullivan

Brandon Rodenburg

Mohammad Mirhosseini

Jonathan Leach

Martin P. J. Lavery

Miles J. Padgett

Robert W. Boyd

Crossref

We describe an experimental implementation of a free-space 11-dimensional communication system using orbital angular momentum (OAM) modes. This system has a maximum measured OAM channel capacity of 2.12 bits/photon. The effects of Kolmogorov thin-phase turbulence on the OAM channel capacity are quantified. We find that increasing the turbulence leads to a degradation of the channel capacity. We are able to mitigate the effects of turbulence by increasing the spacing between detected OAM modes. This study has implications for high-dimensional quantum key distribution (QKD) systems. We describe the sort of QKD system that could be built using our current technology

Malik, M.

O’Sullivan, M.

Rodenburg, B.

Mirhosseini, M.

Leach, J.

Lavery, M.P.J.

Padgett, M.J.

Boyd, R.W.

Enlighten

Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk       Malik, M., O’Sullivan, M., Rodenburg, B., Mirhosseini, M., Leach, J., Lavery, M.P.J., Padgett, M.J., and Boyd, R.W. (2012) Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding. Optics Express, 20 (12). p. 13195. ISSN 1094-4087   Copyright © 2012 OSA  This paper was published in Optics Express and is made available as an electronic reprint with the permission of OSA. The paper can be found at the following URL on the OSA website. Systematic or multiple reproduction or distribution to multiple locations via electronic or other means is prohibited and is subject to penalties under law.    http://eprints.gla.ac.uk/74531/     Deposited on: 23 January 2013   Influence of atmospheric turbulence onoptical communications using orbitalangular momentum for encodingMehul Malik,1,∗ Malcolm O’Sullivan,1 Brandon Rodenburg,1Mohammad Mirhosseini,1 Jonathan Leach,2 Martin P. J. Lavery,3Miles J. Padgett,3 and Robert W. Boyd1,21The Institute of Optics, University of Rochester, Rochester, New York 14627 USA2Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5 Canada3School of Physics and Astronomy, University of Glasgow, Glasgow, UK∗memalik@optics.rochester.eduAbstract: We describe an experimental implementation of a free-space11-dimensional communication system using orbital angular momentum(OAM) modes. This system has a maximum measured OAM channel ca-pacity of 2.12 bits/photon. The effects of Kolmogorov thin-phase turbulenceon the OAM channel capacity are quantified. We find that increasing theturbulence leads to a degradation of the channel capacity. We are ableto mitigate the effects of turbulence by increasing the spacing betweendetected OAM modes. This study has implications for high-dimensionalquantum key distribution (QKD) systems. We describe the sort of QKDsystem that could be built using our current technology.© 2012 Optical Society of AmericaOCIS codes: (270.5568) Quantum cryptography; (200.2605) Free-space optical communica-tion; (010.1330) Atmospheric turbulence.References and links1. C. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int.Conf., 175–179 (Bangalore, 1984).2. P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt,“Long-distance quantum key distribution in optical fibre,” New J. Phys. 8, 193–193 (2006).3. R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jen-newein, J. Perdigues, P. Trojek, B. ¨Omer, M. Fu¨rst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Wein-furter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486(2007).4. M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel en-coding: security analysis,” J. Phys. A-Math. Gen. 35, 10065–10076 (2002).5. S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography withqutrits,” New J. Phys. 8, 75 (2006).6. N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-Level sys-tems,” Phys. Rev. Lett. 88, 127902 (2002).7. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communica-tion,” Phys. Rev. Lett. 94, 153901 (2005).8. G. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of lightcarrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009).9. B. Smith and M. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).10. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological chargeconservation,” J. Opt. Soc. Am. A 25, 225–230 (2008).#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  1319511. F. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled bipho-ton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).12. W. K. Wootters and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys.-New York 191, 363–381 (1989).13. V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, “Pixelated phase computer holograms for the accurateencoding of scalar complex fields,” J. Opt. Soc. Am. A 24, 3500–3507 (2007).14. M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields withspatial light modulators: application to quantum key distribution,” Appl. Opt. 47, A32–A42 (2008).15. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting oforbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).16. M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractiveelements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–2115 (2012).17. A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).18. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1431(1965).19. C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38,2161–2170 (1999).1. IntroductionSince Bennett and Brassard introduced the first quantum key distribution (QKD) protocol in1984 [1], the field of QKD has rapidly developed to the extent that QKD systems are commer-cially available today. Secure transmission of a quantum key has been performed over 148.7km of fiber [2] as well as over 144 km of free space [3]. One of the limiting aspects of thesekey distribution systems is that they use the polarization degree of freedom of the photon toencode information [2, 3]. The use of polarization encoding limits the maximum amount ofinformation that can be encoded on each photon to one bit. In addition, it places a low boundon the amount of error an eavesdropper can introduce without compromising the security of thetransmission [4]. Because of these limitations, there has been great interest in exploring otherways to encode information on a photon that would allow for higher data transmission rates andincreased security [5, 6].Here we report results on the use of orbital angular momentum (OAM) modes of a photonfor optical communication. The motivation for doing so is that OAM modes span an infinite-dimensional basis. Hence, there is no limit to how much information one can send per photonin such a system. The large dimensionality of this protocol also provides a much higher level ofsecurity than the two-state approach [4]. However, in a practical communication system usingOAM modes, the maximum number of modes that can be used is limited by the size of thelimiting aperture in the system. This occurs because the radius of an OAM mode increaseswith the mode number. In addition, since these are spatial modes, they are highly susceptible toturbulence. Recently, there have been several theoretical studies on how atmospheric turbulenceaffects OAM modes [7–11]. In this paper, we experimentally study the effects of atmosphericturbulence on the channel capacity of an OAM communication channel at high light levels.Also, we describe a high-dimensional QKD system that could be built using OAM modes forencoding.2. Quantum key distribution systemIn a conventional QKD system, the sender (Alice) and receiver (Bob) must have at least twomutually unbiased bases (MUBs) in which to send and receive their key. The maximum numberof MUBs in an N-dimensional QKD system is equal to N + 1, for when N is a prime number[12]. For the polarization-based implementation of the BB84 protocol [1], N is equal to 2 andthere are three available MUBs. Here we describe a proposed OAM-based QKD system witha dimensionality d = 11, and 12 possible mutually unbiased bases. Out of these, we choose to#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13196HeNe SLMAOMPinholeALICESorterEMCCDBOBFT Lens (OAM)Imaging Lens (ANG)R1 R2 BSFig. 1. Proposed QKD Setup. Our source is a HeNe laser operating at 633 nm that is mod-ulated by an acousto-optic modulator (AOM) to carve out single-photon pulses. The OAMand ANG states, as well as the turbulence phase screens are generated by the spatial lightmodulator (SLM) and 4 f system. R1 and R2 are custom refractive elements used to sortthe OAM and ANG modes, and the beam splitter (BS) acts as the passive basis selector.The modes are detected on an electron-multiplying camera (EMCCD).use only two in order to maximize our key transmission rate, which is inversely proportional tothe number of MUBs that are used. The first basis is made up of 11 OAM modes and is calledthe OAM basis. The OAM modes have the formΨl,0OAM = A0W (r/R)exp(ilθ) (1)where A0 is the spatially uniform field amplitude, W (x) is an aperture function such that W (x) =1 for |x| ≤ 1 and zero otherwise, r and θ are the radial and azimuthal coordinates, and l is theOAM quantum number. The second basis is called the ANG basis and is made up of angularmodes, which are a linear combination of the 11 OAM modes and have the formΨnANG =1√115∑l=−5Ψl,0OAM exp(i2πnl/11). (2)These modes are so named because their intensity profile looks like an angular slice thatmoves around the center of the beam as one changes the relative phases of the componentOAM modes. In order to characterize the effects of turbulence on our system, we choose touse pure vortex modes with a circular aperture function instead of Laguerre-Gauss modes inour system. We do this because our theoretical approach closely follows that of reference [8],which is also based on pure vortex modes.A schematic of our proposed QKD system is shown in Fig. 1. We use a HeNe laser modu-lated by an acousto-optical modulator (AOM) as our source. The AOM carves out single-photonpulses for the QKD protocol. The OAM and ANG modes are generated by Alice using a holo-gram generated on a Holoeye PLUTO phase-only spatial light modulator (SLM) in conjunctionwith a 4 f system [13, 14]. Figure 2 shows some of the holograms we used and the OAM andANG modes generated by them.The generated modes are then imaged across a free-space channel onto Bob’s detection plane.One of the key hurdles to using OAM modes to perform QKD has been the need of a method ofefficiently sorting single photons carrying OAM modes. This problem was recently overcomeby a method that uses two phase screens to conformally map an input OAM state to a shiftedspot in the fourier plane of a lens [15]. In our setup, we are using custom refractive elementsdesigned according to this method in order to carry out efficient sorting of OAM modes [16].The OAM sorter maps the input OAM modes to a row of spots on our detector.Since the complex ANG modes are mostly non-overlapping in intensity, they can be sorted#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13197Fig. 2. Upper row: Holograms written onto the SLM to generate OAM and ANG modes.Lower row: CCD images of the corresponding modes generated (OAM mode number l =−1, 3, and 5, and ANG mode number n = 1, 5, and 9).based on their angular position. For convenience, we use the first element (R1) of the OAMsorter to map the angular intensities to a partially overlapping row of spots, which are imagedonto the detector. The beam splitter (BS) acts a passive selector of Bob’s measurement basis,randomly measuring photons in either the OAM or ANG basis. We use an electron-multiplyingCCD (EMCCD) as our single-photon detector. While we intend on using this setup to performQKD in the future, currently we are using it at high light levels for the purposes of characteriz-ing the effects of turbulence on the OAM channel.3. Effects of turbulence on channel capacityAtmospheric turbulence plays a large role in determining how practical a free-space commu-nication system may be. Here, we study the effects of weak turbulence on the OAM commu-nication channel. In our system, the aperture of the sender is imaged onto the aperture of thereceiver. For such a system, turbulence can be modeled as a thin-phase screen in the sendingaperture [17]. The effects of turbulence are characterized in terms of crosstalk in the OAM andANG channels. However, the effects of thin phase turbulence on the ANG channel cannot bestudied in our system because of two reasons. First, the sorter is limited in that it only maps thenon-overlapping intensities of the complex ANG modes to a row of spots. Any phase informa-tion in the ANG modes is not conveyed. Second, the thin-phase turbulence is in an image planeof the ANG channel. Since the ANG modes are imaged from the SLM to the sorter elementR1, and then intensity-mapped to the detector, the effects of turbulence added at the SLM arenot apparent in the sorted ANG modes.However, we can study the effects of turbulence on the OAM communication channel, asthe sorter sorts these modes based on their orthogonal phase distributions. The weak turbulenceregime can be approximated by the Kolmogorov thin-phase model of turbulence. The phasestructure function in this model obeys the relationship〈[φ(r1)−φ(r2)]2〉= 6.88∣∣∣∣r1 − r2r0∣∣∣∣5/3(3)where r0 is Fried’s coherence diameter [18], a measure of the transverse distance scale overwhich refractive index correlations remain correlated. We generate turbulence phase screensbased on this model [19] and add them to the pattern written on the SLM. An example ofthree Kolmogorov phase screens with different values of r0 is shown in Fig. 3. The crosstalkintroduced by this type of turbulence into an OAM channel can be written in the form [8]#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13198Fig. 3. Sample Kolmogorov turbulence phase screens that are added onto the SLM forthree different values of turbulence strength, D/r0 =(a) 5.12, (b) 10.25, and (c) 102.5.Each screen has a resolution of 512×512 pixels and the phase wraps from 0 to 2π .Pds =1π∫ 10ρdρ∫ 2π0dθe−3.44(D/r0)5/3(ρ sinθ/2)5/3 cos [(d − s)θ ] (4)where ρ = r/R, and Pds is the conditional probability that Bob detects an OAM mode withquantum number d given that Alice sent an OAM mode with quantum number s. Here, D isthe diameter of the receiving aperture. From these probabilities, we can calculate the channelcapacity of our OAM channel asC = max[H(x)−H(x|y)]= max{Ps}[−∑sPs log2 (Ps)+∑sPs ∑dPds log2(Pds)]. (5)Here, N is the total number of modes in our system and Ps is the probability that Alice sentmode s. Experimental data for the channel capacity of the OAM communication channel withN = 11,9,7,5, and 3 are shown in Fig. 4. The channel capacity, C, is plotted as a functionof the turbulence strength, D/r0, where D is the diameter of the photon field at the SLMand r0 is Fried’s coherence diameter. For each data point, we average over 20 different tur-bulence screens. The theoretical crosstalk is calculated by Eq. (4) and corrected for the inherent  00.52Channel Capacity (bits per photon)11.5D/r010110010-1N=11N= 9N= 7N= 5N= 3Pol.Fig. 4. Influence of Kolmogorov turbulence on the channel capacity of an OAM communi-cation channel for a system dimensionality of N = 3,5,7,9 and 11. The channel capacityof a system using polarization modes is plotted as a dotted line for comparison.#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13199  00.20.40.60.811.21.41.6Channel Capacity (bits per photon)D/r010110010-1MS = 4MS = 2MS = 1Fig. 5. Effects of changing the spacing between detected modes (MS) on the channel ca-pacity of a 3-dimensional OAM communication system in the presence of atmosphericturbulence. For a mode-spacing of 4, the maximum channel capacity of a 3-dimensionalsystem is seen to approach the theoretical maximum of log2(3) = 1.585.crosstalk introduced by the sorter [15]. The theoretical channel capacity as calculated by Eq.(5) is plotted in Fig. 4, as well as the channel capacity of a polarization-based two-dimensionalsystem, which is essentially immune to the effects of turbulence.As can be seen, the maximum channel capacity for N = 11 is much lower than the theoreticalmaximum of log2(11) = 3.46 bits/photon. This is because the OAM sorter is not perfect andintroduces some inherent crosstalk [15]. Nonetheless, our higher dimensional systems performmuch better than the polarization-based system below a certain turbulence strength. This isindicated in Fig. 4 by the point at which the channel capacity curve for an N-dimensional systemcrosses the dotted line for a polarization-based system. At a turbulence strength of D/r0 = 10,the channel capacity vanishes for all systems.Since OAM modes constitute an infinite-dimensional basis, one can reduce the effects ofthe inherent sorter crosstalk and turbulence by increasing the spacing (MS) between detectedmodes to greater than one. We study this effect on a subset of our system with N = 3, forthree different mode-spacings (MS = 1, 2 and 4). It is clear from Fig. 5 that increasing themode spacing between detected modes boosts the initial channel capacity and makes our systemslightly more resistant to turbulence. The maximum channel capacity for a mode-spacing offour is close to the theoretical maximum of log2(3) = 1.585. In addition, the channel capacityfor this mode-spacing starts decreasing at a value of D/r0 that is almost an order of magnitudegreater than that for a mode-spacing of one.4. ConclusionsWe characterized the effects of Kolmogorov thin-phase turbulence on the channel capacity ofan OAM-based communication system at high light levels. We found that turbulence affects thechannel capacity drastically, and the effects can be mitigated by increasing the mode-spacing.We were not able to study the effects of thin-phase turbulence on the ANG channel due to theinherent limitations of our mode sorter. We are currently working on improving the performanceof our mode sorter and extending our study to thick-phase turbulence. We would like to thankDr. D. J. Gauthier for useful discussions. This work was supported by the DARPA/DSO InPhoprogram and the Canadian Excellence Research Chair (CERC) program.#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13200

Enlighten: Publications

Enlighten – Research publications by members of the University of Glasgow 
http://eprints.gla.ac.uk 
 
 
 
 
 
 
Malik, M., O’Sullivan, M., Rodenburg, B., Mirhosseini, M., Leach, 
J., Lavery, M.P.J., Padgett, M.J., and Boyd, R.W. (2012) Influence of 
atmospheric turbulence on optical communications using orbital angular 
momentum for encoding. Optics Express, 20 (12). p. 13195. ISSN 1094-
4087 
 
 
Copyright © 2012 OSA 
 
This paper was published in Optics Express and is made available as an 
electronic reprint with the permission of OSA. The paper can be found at 
the following URL on the OSA website. Systematic or multiple 
reproduction or distribution to multiple locations via electronic or other 
means is prohibited and is subject to penalties under law. 
 
 
 
http://eprints.gla.ac.uk/74531/ 
 
 
 
 
Deposited on: 23 January 2013 
 
 
Influence of atmospheric turbulence on
optical communications using orbital
angular momentum for encoding
Mehul Malik,1,∗ Malcolm O’Sullivan,1 Brandon Rodenburg,1
Mohammad Mirhosseini,1 Jonathan Leach,2 Martin P. J. Lavery,3
Miles J. Padgett,3 and Robert W. Boyd1,2
1The Institute of Optics, University of Rochester, Rochester, New York 14627 USA
2Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5 Canada
3School of Physics and Astronomy, University of Glasgow, Glasgow, UK
∗memalik@optics.rochester.edu
Abstract: We describe an experimental implementation of a free-space
11-dimensional communication system using orbital angular momentum
(OAM) modes. This system has a maximum measured OAM channel ca-
pacity of 2.12 bits/photon. The effects of Kolmogorov thin-phase turbulence
on the OAM channel capacity are quantified. We find that increasing the
turbulence leads to a degradation of the channel capacity. We are able
to mitigate the effects of turbulence by increasing the spacing between
detected OAM modes. This study has implications for high-dimensional
quantum key distribution (QKD) systems. We describe the sort of QKD
system that could be built using our current technology.
© 2012 Optical Society of America
OCIS codes: (270.5568) Quantum cryptography; (200.2605) Free-space optical communica-
tion; (010.1330) Atmospheric turbulence.
References and links
1. C. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int.
Conf., 175–179 (Bangalore, 1984).
2. P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt,
“Long-distance quantum key distribution in optical fibre,” New J. Phys. 8, 193–193 (2006).
3. R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jen-
newein, J. Perdigues, P. Trojek, B. ¨Omer, M. Fu¨rst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Wein-
furter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486
(2007).
4. M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel en-
coding: security analysis,” J. Phys. A-Math. Gen. 35, 10065–10076 (2002).
5. S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with
qutrits,” New J. Phys. 8, 75 (2006).
6. N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-Level sys-
tems,” Phys. Rev. Lett. 88, 127902 (2002).
7. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communica-
tion,” Phys. Rev. Lett. 94, 153901 (2005).
8. G. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light
carrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009).
9. B. Smith and M. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
10. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge
conservation,” J. Opt. Soc. Am. A 25, 225–230 (2008).
#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012
(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13195
11. F. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled bipho-
ton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
12. W. K. Wootters and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys.-
New York 191, 363–381 (1989).
13. V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, “Pixelated phase computer holograms for the accurate
encoding of scalar complex fields,” J. Opt. Soc. Am. A 24, 3500–3507 (2007).
14. M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with
spatial light modulators: application to quantum key distribution,” Appl. Opt. 47, A32–A42 (2008).
15. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of
orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
16. M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive
elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–
2115 (2012).
17. A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).
18. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1431
(1965).
19. C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38,
2161–2170 (1999).
1. Introduction
Since Bennett and Brassard introduced the first quantum key distribution (QKD) protocol in
1984 [1], the field of QKD has rapidly developed to the extent that QKD systems are commer-
cially available today. Secure transmission of a quantum key has been performed over 148.7
km of fiber [2] as well as over 144 km of free space [3]. One of the limiting aspects of these
key distribution systems is that they use the polarization degree of freedom of the photon to
encode information [2, 3]. The use of polarization encoding limits the maximum amount of
information that can be encoded on each photon to one bit. In addition, it places a low bound
on the amount of error an eavesdropper can introduce without compromising the security of the
transmission [4]. Because of these limitations, there has been great interest in exploring other
ways to encode information on a photon that would allow for higher data transmission rates and
increased security [5, 6].
Here we report results on the use of orbital angular momentum (OAM) modes of a photon
for optical communication. The motivation for doing so is that OAM modes span an infinite-
dimensional basis. Hence, there is no limit to how much information one can send per photon
in such a system. The large dimensionality of this protocol also provides a much higher level of
security than the two-state approach [4]. However, in a practical communication system using
OAM modes, the maximum number of modes that can be used is limited by the size of the
limiting aperture in the system. This occurs because the radius of an OAM mode increases
with the mode number. In addition, since these are spatial modes, they are highly susceptible to
turbulence. Recently, there have been several theoretical studies on how atmospheric turbulence
affects OAM modes [7–11]. In this paper, we experimentally study the effects of atmospheric
turbulence on the channel capacity of an OAM communication channel at high light levels.
Also, we describe a high-dimensional QKD system that could be built using OAM modes for
encoding.
2. Quantum key distribution system
In a conventional QKD system, the sender (Alice) and receiver (Bob) must have at least two
mutually unbiased bases (MUBs) in which to send and receive their key. The maximum number
of MUBs in an N-dimensional QKD system is equal to N + 1, for when N is a prime number
[12]. For the polarization-based implementation of the BB84 protocol [1], N is equal to 2 and
there are three available MUBs. Here we describe a proposed OAM-based QKD system with
a dimensionality d = 11, and 12 possible mutually unbiased bases. Out of these, we choose to
#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012
(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13196
HeNe SLMAOM
PinholeALICE
Sorter
EMCCD
BOB
FT Lens (OAM)
Imaging Lens (ANG)
R1 R2 BS
Fig. 1. Proposed QKD Setup. Our source is a HeNe laser operating at 633 nm that is mod-
ulated by an acousto-optic modulator (AOM) to carve out single-photon pulses. The OAM
and ANG states, as well as the turbulence phase screens are generated by the spatial light
modulator (SLM) and 4 f system. R1 and R2 are custom refractive elements used to sort
the OAM and ANG modes, and the beam splitter (BS) acts as the passive basis selector.
The modes are detected on an electron-multiplying camera (EMCCD).
use only two in order to maximize our key transmission rate, which is inversely proportional to
the number of MUBs that are used. The first basis is made up of 11 OAM modes and is called
the OAM basis. The OAM modes have the form
Ψl,0OAM = A0W (r/R)exp(ilθ) (1)
where A0 is the spatially uniform field amplitude, W (x) is an aperture function such that W (x) =
1 for |x| ≤ 1 and zero otherwise, r and θ are the radial and azimuthal coordinates, and l is the
OAM quantum number. The second basis is called the ANG basis and is made up of angular
modes, which are a linear combination of the 11 OAM modes and have the form
ΨnANG =
1√
11
5
∑
l=−5
Ψl,0OAM exp(i2πnl/11). (2)
These modes are so named because their intensity profile looks like an angular slice that
moves around the center of the beam as one changes the relative phases of the component
OAM modes. In order to characterize the effects of turbulence on our system, we choose to
use pure vortex modes with a circular aperture function instead of Laguerre-Gauss modes in
our system. We do this because our theoretical approach closely follows that of reference [8],
which is also based on pure vortex modes.
A schematic of our proposed QKD system is shown in Fig. 1. We use a HeNe laser modu-
lated by an acousto-optical modulator (AOM) as our source. The AOM carves out single-photon
pulses for the QKD protocol. The OAM and ANG modes are generated by Alice using a holo-
gram generated on a Holoeye PLUTO phase-only spatial light modulator (SLM) in conjunction
with a 4 f system [13, 14]. Figure 2 shows some of the holograms we used and the OAM and
ANG modes generated by them.
The generated modes are then imaged across a free-space channel onto Bob’s detection plane.
One of the key hurdles to using OAM modes to perform QKD has been the need of a method of
efficiently sorting single photons carrying OAM modes. This problem was recently overcome
by a method that uses two phase screens to conformally map an input OAM state to a shifted
spot in the fourier plane of a lens [15]. In our setup, we are using custom refractive elements
designed according to this method in order to carry out efficient sorting of OAM modes [16].
The OAM sorter maps the input OAM modes to a row of spots on our detector.
Since the complex ANG modes are mostly non-overlapping in intensity, they can be sorted
#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012
(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13197
Fig. 2. Upper row: Holograms written onto the SLM to generate OAM and ANG modes.
Lower row: CCD images of the corresponding modes generated (OAM mode number l =
−1, 3, and 5, and ANG mode number n = 1, 5, and 9).
based on their angular position. For convenience, we use the first element (R1) of the OAM
sorter to map the angular intensities to a partially overlapping row of spots, which are imaged
onto the detector. The beam splitter (BS) acts a passive selector of Bob’s measurement basis,
randomly measuring photons in either the OAM or ANG basis. We use an electron-multiplying
CCD (EMCCD) as our single-photon detector. While we intend on using this setup to perform
QKD in the future, currently we are using it at high light levels for the purposes of characteriz-
ing the effects of turbulence on the OAM channel.
3. Effects of turbulence on channel capacity
Atmospheric turbulence plays a large role in determining how practical a free-space commu-
nication system may be. Here, we study the effects of weak turbulence on the OAM commu-
nication channel. In our system, the aperture of the sender is imaged onto the aperture of the
receiver. For such a system, turbulence can be modeled as a thin-phase screen in the sending
aperture [17]. The effects of turbulence are characterized in terms of crosstalk in the OAM and
ANG channels. However, the effects of thin phase turbulence on the ANG channel cannot be
studied in our system because of two reasons. First, the sorter is limited in that it only maps the
non-overlapping intensities of the complex ANG modes to a row of spots. Any phase informa-
tion in the ANG modes is not conveyed. Second, the thin-phase turbulence is in an image plane
of the ANG channel. Since the ANG modes are imaged from the SLM to the sorter element
R1, and then intensity-mapped to the detector, the effects of turbulence added at the SLM are
not apparent in the sorted ANG modes.
However, we can study the effects of turbulence on the OAM communication channel, as
the sorter sorts these modes based on their orthogonal phase distributions. The weak turbulence
regime can be approximated by the Kolmogorov thin-phase model of turbulence. The phase
structure function in this model obeys the relationship
〈[φ(r1)−φ(r2)]2〉= 6.88
∣
∣
∣
∣
r1 − r2
r0
∣
∣
∣
∣
5/3
(3)
where r0 is Fried’s coherence diameter [18], a measure of the transverse distance scale over
which refractive index correlations remain correlated. We generate turbulence phase screens
based on this model [19] and add them to the pattern written on the SLM. An example of
three Kolmogorov phase screens with different values of r0 is shown in Fig. 3. The crosstalk
introduced by this type of turbulence into an OAM channel can be written in the form [8]
#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012
(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13198
Fig. 3. Sample Kolmogorov turbulence phase screens that are added onto the SLM for
three different values of turbulence strength, D/r0 =(a) 5.12, (b) 10.25, and (c) 102.5.
Each screen has a resolution of 512×512 pixels and the phase wraps from 0 to 2π .
Pds =
1
π
∫ 1
0
ρdρ
∫ 2π
0
dθe−3.44(D/r0)5/3(ρ sinθ/2)5/3 cos [(d − s)θ ] (4)
where ρ = r/R, and Pds is the conditional probability that Bob detects an OAM mode with
quantum number d given that Alice sent an OAM mode with quantum number s. Here, D is
the diameter of the receiving aperture. From these probabilities, we can calculate the channel
capacity of our OAM channel as
C = max[H(x)−H(x|y)]
= max
{Ps}
[
−∑
s
Ps log2 (Ps)+∑
s
Ps ∑
d
Pds log2(Pds)
]
. (5)
Here, N is the total number of modes in our system and Ps is the probability that Alice sent
mode s. Experimental data for the channel capacity of the OAM communication channel with
N = 11,9,7,5, and 3 are shown in Fig. 4. The channel capacity, C, is plotted as a function
of the turbulence strength, D/r0, where D is the diameter of the photon field at the SLM
and r0 is Fried’s coherence diameter. For each data point, we average over 20 different tur-
bulence screens. The theoretical crosstalk is calculated by Eq. (4) and corrected for the inherent
 
 
0
0.5
2
Ch
an
ne
l C
ap
ac
ity
 (b
its
 pe
r p
ho
ton
)
1
1.5
D/r0
10110010-1
N=11
N= 9
N= 7
N= 5
N= 3
Pol.
Fig. 4. Influence of Kolmogorov turbulence on the channel capacity of an OAM communi-
cation channel for a system dimensionality of N = 3,5,7,9 and 11. The channel capacity
of a system using polarization modes is plotted as a dotted line for comparison.
#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012
(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13199
  
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Ch
an
ne
l C
ap
ac
ity
 (b
its
 pe
r p
ho
ton
)
D/r0
10110010-1
MS = 4
MS = 2
MS = 1
Fig. 5. Effects of changing the spacing between detected modes (MS) on the channel ca-
pacity of a 3-dimensional OAM communication system in the presence of atmospheric
turbulence. For a mode-spacing of 4, the maximum channel capacity of a 3-dimensional
system is seen to approach the theoretical maximum of log2(3) = 1.585.
crosstalk introduced by the sorter [15]. The theoretical channel capacity as calculated by Eq.
(5) is plotted in Fig. 4, as well as the channel capacity of a polarization-based two-dimensional
system, which is essentially immune to the effects of turbulence.
As can be seen, the maximum channel capacity for N = 11 is much lower than the theoretical
maximum of log2(11) = 3.46 bits/photon. This is because the OAM sorter is not perfect and
introduces some inherent crosstalk [15]. Nonetheless, our higher dimensional systems perform
much better than the polarization-based system below a certain turbulence strength. This is
indicated in Fig. 4 by the point at which the channel capacity curve for an N-dimensional system
crosses the dotted line for a polarization-based system. At a turbulence strength of D/r0 = 10,
the channel capacity vanishes for all systems.
Since OAM modes constitute an infinite-dimensional basis, one can reduce the effects of
the inherent sorter crosstalk and turbulence by increasing the spacing (MS) between detected
modes to greater than one. We study this effect on a subset of our system with N = 3, for
three different mode-spacings (MS = 1, 2 and 4). It is clear from Fig. 5 that increasing the
mode spacing between detected modes boosts the initial channel capacity and makes our system
slightly more resistant to turbulence. The maximum channel capacity for a mode-spacing of
four is close to the theoretical maximum of log2(3) = 1.585. In addition, the channel capacity
for this mode-spacing starts decreasing at a value of D/r0 that is almost an order of magnitude
greater than that for a mode-spacing of one.
4. Conclusions
We characterized the effects of Kolmogorov thin-phase turbulence on the channel capacity of
an OAM-based communication system at high light levels. We found that turbulence affects the
channel capacity drastically, and the effects can be mitigated by increasing the mode-spacing.
We were not able to study the effects of thin-phase turbulence on the ANG channel due to the
inherent limitations of our mode sorter. We are currently working on improving the performance
of our mode sorter and extending our study to thick-phase turbulence. We would like to thank
Dr. D. J. Gauthier for useful discussions. This work was supported by the DARPA/DSO InPho
program and the Canadian Excellence Research Chair (CERC) program.
#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012
(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13200


O’Sullivan, Malcolm

Caltech Authors - Main

Influence of atmospheric turbulence on
optical communications using orbital
angular momentum for encoding
Mehul Malik,1,∗ Malcolm O’Sullivan,1 Brandon Rodenburg,1
Mohammad Mirhosseini,1 Jonathan Leach,2 Martin P. J. Lavery,3
Miles J. Padgett,3 and Robert W. Boyd1,2
1The Institute of Optics, University of Rochester, Rochester, New York 14627 USA
2Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5 Canada
3School of Physics and Astronomy, University of Glasgow, Glasgow, UK
∗memalik@optics.rochester.edu
Abstract: We describe an experimental implementation of a free-space
11-dimensional communication system using orbital angular momentum
(OAM) modes. This system has a maximum measured OAM channel ca-
pacity of 2.12 bits/photon. The effects of Kolmogorov thin-phase turbulence
on the OAM channel capacity are quantified. We find that increasing the
turbulence leads to a degradation of the channel capacity. We are able
to mitigate the effects of turbulence by increasing the spacing between
detected OAM modes. This study has implications for high-dimensional
quantum key distribution (QKD) systems. We describe the sort of QKD
system that could be built using our current technology.
© 2012 Optical Society of America
OCIS codes: (270.5568) Quantum cryptography; (200.2605) Free-space optical communica-
tion; (010.1330) Atmospheric turbulence.
References and links
1. C. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int.
Conf., 175–179 (Bangalore, 1984).
2. P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt,
“Long-distance quantum key distribution in optical fibre,” New J. Phys. 8, 193–193 (2006).
3. R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jen-
newein, J. Perdigues, P. Trojek, B. Ömer, M. Fürst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Wein-
furter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486
(2007).
4. M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel en-
coding: security analysis,” J. Phys. A-Math. Gen. 35, 10065–10076 (2002).
5. S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with
qutrits,” New J. Phys. 8, 75 (2006).
6. N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-Level sys-
tems,” Phys. Rev. Lett. 88, 127902 (2002).
7. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communica-
tion,” Phys. Rev. Lett. 94, 153901 (2005).
8. G. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light
carrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009).
9. B. Smith and M. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
10. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge
conservation,” J. Opt. Soc. Am. A 25, 225–230 (2008).
#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012
(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13195
11. F. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled bipho-
ton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
12. W. K. Wootters and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys.-
New York 191, 363–381 (1989).
13. V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, “Pixelated phase computer holograms for the accurate
encoding of scalar complex fields,” J. Opt. Soc. Am. A 24, 3500–3507 (2007).
14. M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with
spatial light modulators: application to quantum key distribution,” Appl. Opt. 47, A32–A42 (2008).
15. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of
orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
16. M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive
elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–
2115 (2012).
17. A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).
18. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1431
(1965).
19. C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38,
2161–2170 (1999).
1. Introduction
Since Bennett and Brassard introduced the first quantum key distribution (QKD) protocol in
1984 [1], the field of QKD has rapidly developed to the extent that QKD systems are commer-
cially available today. Secure transmission of a quantum key has been performed over 148.7
km of fiber [2] as well as over 144 km of free space [3]. One of the limiting aspects of these
key distribution systems is that they use the polarization degree of freedom of the photon to
encode information [2, 3]. The use of polarization encoding limits the maximum amount of
information that can be encoded on each photon to one bit. In addition, it places a low bound
on the amount of error an eavesdropper can introduce without compromising the security of the
transmission [4]. Because of these limitations, there has been great interest in exploring other
ways to encode information on a photon that would allow for higher data transmission rates and
increased security [5, 6].
Here we report results on the use of orbital angular momentum (OAM) modes of a photon
for optical communication. The motivation for doing so is that OAM modes span an infinite-
dimensional basis. Hence, there is no limit to how much information one can send per photon
in such a system. The large dimensionality of this protocol also provides a much higher level of
security than the two-state approach [4]. However, in a practical communication system using
OAM modes, the maximum number of modes that can be used is limited by the size of the
limiting aperture in the system. This occurs because the radius of an OAM mode increases
with the mode number. In addition, since these are spatial modes, they are highly susceptible to
turbulence. Recently, there have been several theoretical studies on how atmospheric turbulence
affects OAM modes [7–11]. In this paper, we experimentally study the effects of atmospheric
turbulence on the channel capacity of an OAM communication channel at high light levels.
Also, we describe a high-dimensional QKD system that could be built using OAM modes for
encoding.
2. Quantum key distribution system
In a conventional QKD system, the sender (Alice) and receiver (Bob) must have at least two
mutually unbiased bases (MUBs) in which to send and receive their key. The maximum number
of MUBs in an N-dimensional QKD system is equal to N + 1, for when N is a prime number
[12]. For the polarization-based implementation of the BB84 protocol [1], N is equal to 2 and
there are three available MUBs. Here we describe a proposed OAM-based QKD system with
a dimensionality d = 11, and 12 possible mutually unbiased bases. Out of these, we choose to
#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012
(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13196
HeNe SLMAOM
PinholeALICE
Sorter
EMCCD
BOB
FT Lens (OAM)
Imaging Lens (ANG)
R1 R2 BS
Fig. 1. Proposed QKD Setup. Our source is a HeNe laser operating at 633 nm that is mod-
ulated by an acousto-optic modulator (AOM) to carve out single-photon pulses. The OAM
and ANG states, as well as the turbulence phase screens are generated by the spatial light
modulator (SLM) and 4 f system. R1 and R2 are custom refractive elements used to sort
the OAM and ANG modes, and the beam splitter (BS) acts as the passive basis selector.
The modes are detected on an electron-multiplying camera (EMCCD).
use only two in order to maximize our key transmission rate, which is inversely proportional to
the number of MUBs that are used. The first basis is made up of 11 OAM modes and is called
the OAM basis. The OAM modes have the form
Ψl,0OAM = A0W (r/R)exp(ilθ) (1)
where A0 is the spatially uniform field amplitude, W (x) is an aperture function such that W (x) =
1 for |x| ≤ 1 and zero otherwise, r and θ are the radial and azimuthal coordinates, and l is the
OAM quantum number. The second basis is called the ANG basis and is made up of angular
modes, which are a linear combination of the 11 OAM modes and have the form
ΨnANG =
1√
11
5
∑
l=−5
Ψl,0OAM exp(i2πnl/11). (2)
These modes are so named because their intensity profile looks like an angular slice that
moves around the center of the beam as one changes the relative phases of the component
OAM modes. In order to characterize the effects of turbulence on our system, we choose to
use pure vortex modes with a circular aperture function instead of Laguerre-Gauss modes in
our system. We do this because our theoretical approach closely follows that of reference [8],
which is also based on pure vortex modes.
A schematic of our proposed QKD system is shown in Fig. 1. We use a HeNe laser modu-
lated by an acousto-optical modulator (AOM) as our source. The AOM carves out single-photon
pulses for the QKD protocol. The OAM and ANG modes are generated by Alice using a holo-
gram generated on a Holoeye PLUTO phase-only spatial light modulator (SLM) in conjunction
with a 4 f system [13, 14]. Figure 2 shows some of the holograms we used and the OAM and
ANG modes generated by them.
The generated modes are then imaged across a free-space channel onto Bob’s detection plane.
One of the key hurdles to using OAM modes to perform QKD has been the need of a method of
efficiently sorting single photons carrying OAM modes. This problem was recently overcome
by a method that uses two phase screens to conformally map an input OAM state to a shifted
spot in the fourier plane of a lens [15]. In our setup, we are using custom refractive elements
designed according to this method in order to carry out efficient sorting of OAM modes [16].
The OAM sorter maps the input OAM modes to a row of spots on our detector.
Since the complex ANG modes are mostly non-overlapping in intensity, they can be sorted
#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012
(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13197
Fig. 2. Upper row: Holograms written onto the SLM to generate OAM and ANG modes.
Lower row: CCD images of the corresponding modes generated (OAM mode number l =
−1, 3, and 5, and ANG mode number n = 1, 5, and 9).
based on their angular position. For convenience, we use the first element (R1) of the OAM
sorter to map the angular intensities to a partially overlapping row of spots, which are imaged
onto the detector. The beam splitter (BS) acts a passive selector of Bob’s measurement basis,
randomly measuring photons in either the OAM or ANG basis. We use an electron-multiplying
CCD (EMCCD) as our single-photon detector. While we intend on using this setup to perform
QKD in the future, currently we are using it at high light levels for the purposes of characteriz-
ing the effects of turbulence on the OAM channel.
3. Effects of turbulence on channel capacity
Atmospheric turbulence plays a large role in determining how practical a free-space commu-
nication system may be. Here, we study the effects of weak turbulence on the OAM commu-
nication channel. In our system, the aperture of the sender is imaged onto the aperture of the
receiver. For such a system, turbulence can be modeled as a thin-phase screen in the sending
aperture [17]. The effects of turbulence are characterized in terms of crosstalk in the OAM and
ANG channels. However, the effects of thin phase turbulence on the ANG channel cannot be
studied in our system because of two reasons. First, the sorter is limited in that it only maps the
non-overlapping intensities of the complex ANG modes to a row of spots. Any phase informa-
tion in the ANG modes is not conveyed. Second, the thin-phase turbulence is in an image plane
of the ANG channel. Since the ANG modes are imaged from the SLM to the sorter element
R1, and then intensity-mapped to the detector, the effects of turbulence added at the SLM are
not apparent in the sorted ANG modes.
However, we can study the effects of turbulence on the OAM communication channel, as
the sorter sorts these modes based on their orthogonal phase distributions. The weak turbulence
regime can be approximated by the Kolmogorov thin-phase model of turbulence. The phase
structure function in this model obeys the relationship
〈[φ(r1)−φ(r2)]2〉= 6.88
∣
∣
∣
∣
r1 − r2
r0
∣
∣
∣
∣
5/3
(3)
where r0 is Fried’s coherence diameter [18], a measure of the transverse distance scale over
which refractive index correlations remain correlated. We generate turbulence phase screens
based on this model [19] and add them to the pattern written on the SLM. An example of
three Kolmogorov phase screens with different values of r0 is shown in Fig. 3. The crosstalk
introduced by this type of turbulence into an OAM channel can be written in the form [8]
#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012
(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13198
Fig. 3. Sample Kolmogorov turbulence phase screens that are added onto the SLM for
three different values of turbulence strength, D/r0 =(a) 5.12, (b) 10.25, and (c) 102.5.
Each screen has a resolution of 512×512 pixels and the phase wraps from 0 to 2π .
Pds =
1
π
∫ 1
0
ρdρ
∫ 2π
0
dθe−3.44(D/r0)
5/3(ρ sinθ/2)5/3 cos [(d− s)θ ] (4)
where ρ = r/R, and Pds is the conditional probability that Bob detects an OAM mode with
quantum number d given that Alice sent an OAM mode with quantum number s. Here, D is
the diameter of the receiving aperture. From these probabilities, we can calculate the channel
capacity of our OAM channel as
C = max[H(x)−H(x|y)]
= max
{Ps}
[
−∑
s
Ps log2 (Ps)+∑
s
Ps ∑
d
Pds log2(Pds)
]
. (5)
Here, N is the total number of modes in our system and Ps is the probability that Alice sent
mode s. Experimental data for the channel capacity of the OAM communication channel with
N = 11,9,7,5, and 3 are shown in Fig. 4. The channel capacity, C, is plotted as a function
of the turbulence strength, D/r0, where D is the diameter of the photon field at the SLM
and r0 is Fried’s coherence diameter. For each data point, we average over 20 different tur-
bulence screens. The theoretical crosstalk is calculated by Eq. (4) and corrected for the inherent
 
 
0
0.5
2
C
ha
nn
el
 C
ap
ac
ity
 (
bi
ts
 p
er
 p
ho
to
n)
1
1.5
D/r0
10
1
10
0
10
-1
N=11
N= 9
N= 7
N= 5
N= 3
Pol.
Fig. 4. Influence of Kolmogorov turbulence on the channel capacity of an OAM communi-
cation channel for a system dimensionality of N = 3,5,7,9 and 11. The channel capacity
of a system using polarization modes is plotted as a dotted line for comparison.
#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012
(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13199
 
 
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
C
ha
nn
el
 C
ap
ac
ity
 (
bi
ts
 p
er
 p
ho
to
n)
D/r0
10
1
10
0
10
-1
MS = 4
MS = 2
MS = 1
Fig. 5. Effects of changing the spacing between detected modes (MS) on the channel ca-
pacity of a 3-dimensional OAM communication system in the presence of atmospheric
turbulence. For a mode-spacing of 4, the maximum channel capacity of a 3-dimensional
system is seen to approach the theoretical maximum of log2(3) = 1.585.
crosstalk introduced by the sorter [15]. The theoretical channel capacity as calculated by Eq.
(5) is plotted in Fig. 4, as well as the channel capacity of a polarization-based two-dimensional
system, which is essentially immune to the effects of turbulence.
As can be seen, the maximum channel capacity for N = 11 is much lower than the theoretical
maximum of log2(11) = 3.46 bits/photon. This is because the OAM sorter is not perfect and
introduces some inherent crosstalk [15]. Nonetheless, our higher dimensional systems perform
much better than the polarization-based system below a certain turbulence strength. This is
indicated in Fig. 4 by the point at which the channel capacity curve for an N-dimensional system
crosses the dotted line for a polarization-based system. At a turbulence strength of D/r0 = 10,
the channel capacity vanishes for all systems.
Since OAM modes constitute an infinite-dimensional basis, one can reduce the effects of
the inherent sorter crosstalk and turbulence by increasing the spacing (MS) between detected
modes to greater than one. We study this effect on a subset of our system with N = 3, for
three different mode-spacings (MS = 1, 2 and 4). It is clear from Fig. 5 that increasing the
mode spacing between detected modes boosts the initial channel capacity and makes our system
slightly more resistant to turbulence. The maximum channel capacity for a mode-spacing of
four is close to the theoretical maximum of log2(3) = 1.585. In addition, the channel capacity
for this mode-spacing starts decreasing at a value of D/r0 that is almost an order of magnitude
greater than that for a mode-spacing of one.
4. Conclusions
We characterized the effects of Kolmogorov thin-phase turbulence on the channel capacity of
an OAM-based communication system at high light levels. We found that turbulence affects the
channel capacity drastically, and the effects can be mitigated by increasing the mode-spacing.
We were not able to study the effects of thin-phase turbulence on the ANG channel due to the
inherent limitations of our mode sorter. We are currently working on improving the performance
of our mode sorter and extending our study to thick-phase turbulence. We would like to thank
Dr. D. J. Gauthier for useful discussions. This work was supported by the DARPA/DSO InPho
program and the Canadian Excellence Research Chair (CERC) program.
#167358 - $15.00 USD Received 25 Apr 2012; revised 17 May 2012; accepted 21 May 2012; published 25 May 2012
(C) 2012 OSA 4 June 2012 / Vol. 20,  No. 12 / OPTICS EXPRESS  13200


ar
X
iv
:1
20
4.
57
81
v2
  [
qu
an
t-p
h]
  4
 Ju
n 2
01
2
Influence of atmospheric turbulence on
optical communications using orbital
angular momentum for encoding
Mehul Malik,1,∗ Malcolm O’Sullivan,1 Brandon Rodenburg,1
Mohammad Mirhosseini,1 Jonathan Leach,2 Martin P. J. Lavery,3
Miles J. Padgett,3 and Robert W. Boyd1,2
1The Institute of Optics, University of Rochester, Rochester, New York 14627 USA
2Department of Physics, University of Ottawa, Ottawa, ON K1N 6N5 Canada
3School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom
*memalik@optics.rochester.edu
Abstract: We describe an experimental implementation of a free-space
11-dimensional communication system using orbital angular momentum
(OAM) modes. This system has a maximum measured OAM channel ca-
pacity of 2.12 bits/photon. The effects of Kolmogorov thin-phase turbulence
on the OAM channel capacity are quantified. We find that increasing the
turbulence leads to a degradation of the channel capacity. We are able
to mitigate the effects of turbulence by increasing the spacing between
detected OAM modes. This study has implications for high-dimensional
quantum key distribution (QKD) systems. We describe the sort of QKD
system that could be built using our current technology.
© 2018 Optical Society of America
OCIS codes: (270.5568) Quantum cryptography; (200.2605) Free-space optical communica-
tion; (010.1330) Atmospheric turbulence.
References and links
1. C. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” Proc. IEEE Int.
Conf., 175–179 (Bangalore, 1984).
2. P. A. Hiskett, D. Rosenberg, C. G. Peterson, R. J. Hughes, S. Nam, A. E. Lita, A. J. Miller, and J. E. Nordholt,
“Long-distance quantum key distribution in optical fibre,” New J. Phys. 8, 193–193 (2006).
3. R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jen-
newein, J. Perdigues, P. Trojek, B. ¨Omer, M. Fu¨rst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Wein-
furter, and A. Zeilinger, “Entanglement-based quantum communication over 144 km,” Nat. Phys. 3, 481–486
(2007).
4. M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. Cerf, “Quantum key distribution using multilevel en-
coding: security analysis,” J. Phys. A-Math. Gen. 35, 10065–10076 (2002).
5. S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental quantum cryptography with
qutrits,” New J. Phys. 8, 75 (2006).
6. N. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-Level sys-
tems,” Phys. Rev. Lett. 88, 127902 (2002).
7. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communica-
tion,” Phys. Rev. Lett. 94, 153901 (2005).
8. G. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light
carrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009).
9. B. Smith and M. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
10. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge
conservation,” J. Opt. Soc. Am. A 25, 225–230 (2008).
11. F. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled bipho-
ton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
12. W. K. Wootters and B. D. Fields, “Optimal state-determination by mutually unbiased measurements,” Ann. Phys.-
New York 191, 363–381 (1989).
13. V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, “Pixelated phase computer holograms for the accurate
encoding of scalar complex fields,” J. Opt. Soc. Am. A 24, 3500–3507 (2007).
14. M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with
spatial light modulators: application to quantum key distribution,” Appl. Opt. 47, A32–A42 (2008).
15. G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, “Efficient sorting of
orbital angular momentum states of light,” Phys. Rev. Lett. 105, 153601 (2010).
16. M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, G. D. Love, M. J. Padgett, and J. Courtial, “Refractive
elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–
2115 (2012).
17. A. T. Young, “Seeing: its cause and cure,” Astrophys. J. 189, 587–604 (1974).
18. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427–1431
(1965).
19. C. M. Harding, R. A. Johnston, and R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38,
2161–2170 (1999).
1. Introduction
Since Bennett and Brassard introduced the first quantum key distribution (QKD) protocol in
1984 [1], the field of QKD has rapidly developed to the extent that QKD systems are commer-
cially available today. Secure transmission of a quantum key has been performed over 148.7
km of fiber [2] as well as over 144 km of free space [3]. One of the limiting aspects of these
key distribution systems is that they use the polarization degree of freedom of the photon to
encode information [2, 3]. The use of polarization encoding limits the maximum amount of
information that can be encoded on each photon to one bit. In addition, it places a low bound
on the amount of error an eavesdropper can introduce without compromising the security of the
transmission [4]. Because of these limitations, there has been great interest in exploring other
ways to encode information on a photon that would allow for higher data transmission rates and
increased security [5, 6].
Here we report results on the use of orbital angular momentum (OAM) modes of a photon
for optical communication. The motivation for doing so is that OAM modes span an infinite-
dimensional basis. Hence, there is no limit to how much information one can send per photon
in such a system. The large dimensionality of this protocol also provides a much higher level of
security than the two-state approach [4]. However, in a practical communication system using
OAM modes, the maximum number of modes that can be used is limited by the size of the
limiting aperture in the system. This occurs because the radius of an OAM mode increases
with the mode number. In addition, since these are spatial modes, they are highly susceptible to
turbulence. Recently, there have been several theoretical studies on how atmospheric turbulence
affects OAM modes [7–11]. In this paper, we experimentally study the effects of atmospheric
turbulence on the channel capacity of an OAM communication channel at high light levels.
Also, we describe a high-dimensional QKD system that could be built using OAM modes for
encoding.
2. Quantum key distribution system
In a conventional QKD system, the sender (Alice) and receiver (Bob) must have at least two
mutually unbiased bases (MUBs) in which to send and receive their key. The maximum number
of MUBs in an N-dimensional QKD system is equal to N + 1, for when N is a prime number
[12]. For the polarization-based implementation of the BB84 protocol [1], N is equal to 2 and
there are three available MUBs. Here we describe a proposed OAM-based QKD system with
a dimensionality d = 11, and 12 possible mutually unbiased bases. Out of these, we choose to
HeNe SLMAOM
Pinhole
ALICE
Sorter
EMCCD
BOB
FT Lens (OAM)
Imaging Lens (ANG)
R1 R2
BS
Fig. 1. Proposed QKD Setup. Our source is a HeNe laser operating at 633 nm that is mod-
ulated by an acousto-optic modulator (AOM) to carve out single-photon pulses. The OAM
and ANG states, as well as the turbulence phase screens are generated by the spatial light
modulator (SLM) and 4 f system. R1 and R2 are custom refractive elements used to sort
the OAM and ANG modes, and the beam splitter (BS) acts as the passive basis selector.
The modes are detected on an electron-multiplying camera (EMCCD).
use only two in order to maximize our key transmission rate, which is inversely proportional to
the number of MUBs that are used. The first basis is made up of 11 OAM modes and is called
the OAM basis. The OAM modes have the form
Ψl,0OAM = A0W (r/R)exp(ilθ ) (1)
where A0 is the spatially uniform field amplitude, W (x) is an aperture function such that W (x) =
1 for |x| ≤ 1 and zero otherwise, r and θ are the radial and azimuthal coordinates, and l is the
OAM quantum number. The second basis is called the ANG basis and is made up of angular
modes, which are a linear combination of the 11 OAM modes and have the form
ΨnANG =
1√
11
5
∑
l=−5
Ψl,0OAM exp(i2pinl/11). (2)
These modes are so named because their intensity profile looks like an angular slice that
moves around the center of the beam as one changes the relative phases of the component
OAM modes. In order to characterize the effects of turbulence on our system, we choose to
use pure vortex modes with a circular aperture function instead of Laguerre-Gauss modes in
our system. We do this because our theoretical approach closely follows that of reference [8],
which is also based on pure vortex modes.
A schematic of our proposed QKD system is shown in Fig. 1. We use a HeNe laser modu-
lated by an acousto-optical modulator (AOM) as our source. The AOM carves out single-photon
pulses for the QKD protocol. The OAM and ANG modes are generated by Alice using a holo-
gram generated on a Holoeye PLUTO phase-only spatial light modulator (SLM) in conjunction
with a 4 f system [13, 14]. Figure 2 shows some of the holograms we used and the OAM and
ANG modes generated by them.
The generated modes are then imaged across a free-space channel onto Bob’s detection plane.
One of the key hurdles to using OAM modes to perform QKD has been the need of a method of
efficiently sorting single photons carrying OAM modes. This problem was recently overcome
by a method that uses two phase screens to conformally map an input OAM state to a shifted
spot in the fourier plane of a lens [15]. In our setup, we are using custom refractive elements
designed according to this method in order to carry out efficient sorting of OAM modes [16].
The OAM sorter maps the input OAM modes to a row of spots on our detector.
Since the complex ANG modes are mostly non-overlapping in intensity, they can be sorted
Fig. 2. Upper row: Holograms written onto the SLM to generate OAM and ANG modes.
Lower row: CCD images of the corresponding modes generated (OAM mode number l =
−1, 3, and 5, and ANG mode number n = 1, 5, and 9).
based on their angular position. For convenience, we use the first element (R1) of the OAM
sorter to map the angular intensities to a partially overlapping row of spots, which are imaged
onto the detector. The beam splitter (BS) acts a passive selector of Bob’s measurement basis,
randomly measuring photons in either the OAM or ANG basis. We use an electron-multiplying
CCD (EMCCD) as our single-photon detector. While we intend on using this setup to perform
QKD in the future, currently we are using it at high light levels for the purposes of characteriz-
ing the effects of turbulence on the OAM channel.
3. Effects of turbulence on channel capacity
Atmospheric turbulence plays a large role in determining how practical a free-space commu-
nication system may be. Here, we study the effects of weak turbulence on the OAM commu-
nication channel. In our system, the aperture of the sender is imaged onto the aperture of the
receiver. For such a system, turbulence can be modeled as a thin-phase screen in the sending
aperture [17]. The effects of turbulence are characterized in terms of crosstalk in the OAM and
ANG channels. However, the effects of thin phase turbulence on the ANG channel cannot be
studied in our system because of two reasons. First, the sorter is limited in that it only maps the
non-overlapping intensities of the complex ANG modes to a row of spots. Any phase informa-
tion in the ANG modes is not conveyed. Second, the thin-phase turbulence is in an image plane
of the ANG channel. Since the ANG modes are imaged from the SLM to the sorter element
R1, and then intensity-mapped to the detector, the effects of turbulence added at the SLM are
not apparent in the sorted ANG modes.
However, we can study the effects of turbulence on the OAM communication channel, as
the sorter sorts these modes based on their orthogonal phase distributions. The weak turbulence
regime can be approximated by the Kolmogorov thin-phase model of turbulence. The phase
structure function in this model obeys the relationship
〈[φ(r1)−φ(r2)]2〉= 6.88
∣∣∣∣r1− r2r0
∣∣∣∣
5/3
(3)
where r0 is Fried’s coherence diameter [18], a measure of the transverse distance scale over
which refractive index correlations remain correlated. We generate turbulence phase screens
based on this model [19] and add them to the pattern written on the SLM. An example of
three Kolmogorov phase screens with different values of r0 is shown in Fig. 3. The crosstalk
introduced by this type of turbulence into an OAM channel can be written in the form [8]
Fig. 3. Sample Kolmogorov turbulence phase screens that are added onto the SLM for
three different values of turbulence strength, D/r0 =(a) 5.12, (b) 10.25, and (c) 102.5.
Each screen has a resolution of 512×512 pixels and the phase wraps from 0 to 2pi .
Pds =
1
pi
∫ 1
0
ρdρ
∫ 2pi
0
dθe−3.44(D/r0)5/3(ρ sinθ/2)5/3 cos [(d− s)θ ] (4)
where ρ = r/R, and Pds is the conditional probability that Bob detects an OAM mode with
quantum number d given that Alice sent an OAM mode with quantum number s. Here, D is
the diameter of the receiving aperture. From these probabilities, we can calculate the channel
capacity of our OAM channel as
C = max[H(x)−H(x|y)]
= max
{Ps}
[
−∑
s
Ps log2 (Ps)+∑
s
Ps ∑
d
Pds log2(Pds)
]
. (5)
Here, N is the total number of modes in our system and Ps is the probability that Alice sent
mode s. Experimental data for the channel capacity of the OAM communication channel with
N = 11,9,7,5, and 3 are shown in Fig. 4. The channel capacity, C, is plotted as a function
of the turbulence strength, D/r0, where D is the diameter of the photon field at the SLM and
r0 is Fried’s coherence diameter. For each data point, we average over 20 different turbulence
 
 
0
0.5
2
Ch
a
n
n
e
l C
a
pa
ci
ty
 (b
its
 p
e
r 
ph
o
to
n
)
1
1.5
D/r0
10110010-1
N=11
N= 9
N= 7
N= 5
N= 3
Pol.
Fig. 4. Influence of Kolmogorov turbulence on the channel capacity of an OAM communi-
cation channel for a system dimensionality of N = 3,5,7,9 and 11. The channel capacity
of a system using polarization modes is plotted as a dotted line for comparison.
  
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Ch
a
n
n
e
l C
a
pa
ci
ty
 (b
its
 p
e
r 
ph
o
to
n
)
D/r0
10110010-1
MS = 4
MS = 2
MS = 1
Fig. 5. Effects of changing the spacing between detected modes (MS) on the channel ca-
pacity of a 3-dimensional OAM communication system in the presence of atmospheric
turbulence. For a mode-spacing of 4, the maximum channel capacity of a 3-dimensional
system is seen to approach the theoretical maximum of log2(3) = 1.585.
screens. The theoretical crosstalk is calculated by eq. (4) and corrected for the inherent crosstalk
introduced by the sorter [15]. The theoretical channel capacity as calculated by eq. (5) is plotted
in Fig. 4, as well as the channel capacity of a polarization-based two-dimensional system, which
is essentially immune to the effects of turbulence.
As can be seen, the maximum channel capacity for N = 11 is much lower than the theoretical
maximum of log2(11) = 3.46 bits/photon. This is because the OAM sorter is not perfect and
introduces some inherent crosstalk [15]. Nonetheless, our higher dimensional systems perform
much better than the polarization-based system below a certain turbulence strength. This is
indicated in Fig. 4 by the point at which the channel capacity curve for an N-dimensional system
crosses the dotted line for a polarization-based system. At a turbulence strength of D/r0 = 10,
the channel capacity vanishes for all systems.
Since OAM modes constitute an infinite-dimensional basis, one can reduce the effects of
the inherent sorter crosstalk and turbulence by increasing the spacing (MS) between detected
modes to greater than one. We study this effect on a subset of our system with N = 3, for
three different mode-spacings (MS = 1, 2 and 4). It is clear from Fig. 5 that increasing the
mode spacing between detected modes boosts the initial channel capacity and makes our system
slightly more resistant to turbulence. The maximum channel capacity for a mode-spacing of
four is close to the theoretical maximum of log2(3) = 1.585. In addition, the channel capacity
for this mode-spacing starts decreasing at a value of D/r0 that is almost an order of magnitude
greater than that for a mode-spacing of one.
4. Conclusions
We characterized the effects of Kolmogorov thin-phase turbulence on the channel capacity of
an OAM-based communication system at high light levels. We found that turbulence affects the
channel capacity drastically, and the effects can be mitigated by increasing the mode-spacing.
We were not able to study the effects of thin-phase turbulence on the ANG channel due to the
inherent limitations of our mode sorter. We are currently working on improving the performance
of our mode sorter and extending our study to thick-phase turbulence. We would like to thank
Dr. D. J. Gauthier for useful discussions. This work was supported by the DARPA/DSO InPho
program and the Canadian Excellence Research Chair (CERC) program.


Influence of Atmospheric Turbulence on Optical Communications using Orbital Angular Momentum for Encoding

Abstract

Similar works

Full text

Available Versions

Heriot Watt Pure

Heriot Watt Pure

Crossref

Enlighten

Enlighten: Publications

Caltech Authors - Main

Caltech Authors - Main