arXiv:1205.6518v2 [physics.optics] 31 May 2012 oriae epciey h variable The respectively. coordinates r aureGusa L)mdswihhv helical with a have structure, which phase modes (LG) Laguerre-Gaussian are of tum hwdta em ihatases mltd profile amplitude transverse a car- of Allen with beams (OAM). beams are that momentum showed modes angular these orbital of rying example commu- One free-space the nication. for increase to bandwidth freedom information of available degree additional an as modes a phase as thin as considered to be referred [3]. can commonly turbulence is type [2]. and this beam screen, of phase transmitted a distortion in across phase lead- result distortion A index phase atmosphere refractive a the the to of of ing change density dependent result- in spatial atmosphere change a the a of in pressure ing and variations temperature dependent time im- as- in randomly an the natural in The by aberrations [1]. length to age Atmospheric relation great in at channels. community studied tronomy between been cross-talk turbulence has atmospheric turbulence the that on effect communica- the has free-space is any channel for tions concern fundamental A ffcso topei ublneo communication a on turbulence the atmospheric study experimentally of we effects letter, this In [8–12]. communi- based cation OAM such affects [7]. turbulence mospheric space Hilbert large a of improved use be to the quantum shown with a been have with system transmitted distribution secu- key keys the alphabet, cryptographic large In of a 6]. vari- rity [5, of a advantages links as the communication OAM to addition optical of use free-space the in suggests able such as and integer, rno Rodenburg, Brandon nuneo topei ublneo ttso ih carry light of states on turbulence atmospheric of Influence hr a enrcn neeti h s fspatial of use the in interest recent been has There eety hr aebe eea tde nhwat- how on studies several been have there Recently, ψ 2 1 ℓ colo hsc n srnm,Uiest fGagw SUP Glasgow. of University Astronomy, and Physics of School h nttt fOtc,Uiest fRcetr 2 Wilmo 320 Rochester. of University Optics, of Institute The = 3 ℓ etefrAvne ntuetto,Dprmn fPhysic of Department Instrumentation, Advanced for Centre A ¯ h ( oetm(A)poaaigtruhsmltdatmospheric simulated through propagating (OAM) momentum hscostl nOMfreee oe,soigta turbul that number. mode showing of modes, irrespective p eleven range, a this for within r of OAM quality modes use in mode in cross-talk the degradation statistic this through the obeys introduced, turbulence is which simulated turbulence this aberration, introduce phase We varying randomly a r e htn[] neapeo uhbeams such of example An [4]. photon per ehv xeietlysuidtedgaaino oepuri mode of degradation the studied experimentally have We exp( ) CScodes: OCIS Mirhosseini, iℓθ ar nobtlaglrmomen- angular orbital an carry ) 4 r eateto hsc,Uiest fOtw,Otw,O K1N ON Ottawa, Ottawa, of University Physics, of Department and 1 1.30 7.55 270.5565 270.5585, 010.1330, ∗ atnP .Lavery, J. P. Martin θ 1 ai .Robertson J. David sterda n angular and radial the as ℓ sa unbounded an is nua momentum angular ∗ [email protected] † opldSpebr1,2018 18, September Compiled [email protected] tal. et 2 † 3 ie Padgett, Miles , eu Malik, Mehul 1

c h A uepsto esrdb h system. the by measured superposition the OAM phase to the Thin su- (c) added OAM sorter. is the mode of turbulence the measure on a incident in gives perposition measured locations spe- power these The to of CCD. a each focused These on elements. then locations spatial optical are cific refractive states two momentum which of transverse states (MS) use momentum sorter the transverse mode with into OAM states a onto OAM imaged of converts is ex- aperture beam an front order by first the The illuminated laser. (SLM) HeNe modulator panded light use the spatial by a prepared is OAM a carrying of beam A (a) 1. Fig. LG 7 ucio d ohse Y167 USA 14627, NY Rochester Rd, Hutchison 275 BLDG, t slsi rs-akbtenOMmds estudy We modes. OAM between cross-talk in esults 08OtclSceyo America of Society Optical 2018 ,Kli ulig lso 1 Q,Soln,UK Scotland, 8QQ, G12 Glasgow Building, Kelvin A, otltdb omgrvtruec theory. turbulence Kolmogorov by postulated s ,Dra nvriy uhm H L,UK 3LE, DH1 Durham, University, Durham s, ℓ neuiomydgae h uiyo l the all of purity the degrades uniformly ence fre oorm eni b.Ti sraie on realized is This (b). in seen hologram, -forked (a) b (c) (b) aeol pta ih ouao.Oc the Once modulator. light spatial hase-only yfrlgtbascryn ria angular orbital carrying beams light for ty Mirror ublne h ublnei oee as modeled is turbulence The turbulence. 1 HeNe Laser acl .O’Sullivan, N. Malcolm 2 n oetW Boyd W. Robert and Aperture f 1 Transformer N,Canada 6N5, Mode f 2 ℓ fre oormchanging hologram -forked f 1 n orbital ing 1 1 , 4 Mohammad CCD SLM 0 system utilizing OAM modes as the information carrier. 10 ∆=0 We generate a single OAM mode using a spatial light ∆=1 modulator (SLM). Atmospheric turbulence is then sim- ulated by the addition of a turbulent phase screen to the phase hologram displayed on the SLM shown in Fig. 1. ∆=2 Once the turbulence is applied, the phase aberrations −2 ∆ 10 result in a spread of the input mode power over neigh- s ∆=3 boring OAM modes, resulting in cross-talk between the channels. This spread in power is then measured for dif- ∆=4 ferent turbulence strengths. We generate turbulence phase screens according to ∆=5 Kolmogorov turbulence theory [3]. The aberrations in- troduced by atmospheric turbulence can be considered 10−4 −2 0 2 as normal random variables, where the ensemble average 10 10 10 2 D/r0 can be written as [φ(r1) − φ(r2)] , which is known D E as the phase structure function [8, 9]. Here, φ(r1) and Fig. 2. The average power (s∆) in detected mode ψ∆ is φ(r2) are two randomly generated phase fluctuations. plotted as a function of turbulence strength (D/r0) for From Kolmogorov statistics it can be shown that this an input mode with ℓ = 0 (see Eqn. 2). Experimental ensemble average must meet the requirement that data (crosses) is co-plotted with the theoretical predic- 5/3 tion given by Eqn. 2 taking into account the inherent 2 r1 − r2 [φ(r1) − φ(r2)] =6.88 . (1) cross-talk of the sorter (solid lines). The original theory

D E r0 from Ref. [9] is also plotted for comparison (dotted lines).

The value r0 is the Fried parameter, and is a measure of the traverse distance scale over which the refractive index is correlated [3]. To characterize the effect of tur- tum states [13, 14]. These elements transform a beam bulence on the optical system, the ratio D/r0 is consid- ered, where D is the aperture of the system. There are of the form exp(iℓθ), to exp(iℓx/a) at the output, where a is a scaling parameter. A lens is used to focus these two limiting cases for this ratio: when D/r0 < 1, the res- olution of the system is limited by its aperture, and when transverse momentum modes to discrete spots at a CCD placed in its focal plane. Adjacent, equally sized regions D/r0 > 1, the atmosphere limits the system’s ability to resolve an object [3]. are selected on the CCD image, with each region corre- Our theoretical analysis closely follows that of refer- sponding to a specific ℓ-value. The total counts over all the pixels in each region is summed to give the relative ence [9]. Consider a single OAM mode, ψℓ, transmitted through an ensemble average of many turbulent phase power in each OAM mode. For each input ℓ mode, the power is measured across all 11 regions, and normalized screens. The average detected power, s∆, in the mode, with respect to the power measured for ℓ = 0 with no ψ +∆, is given by ℓ turbulence applied. 1 2π 5/3 D θ − 1 −3.44 (ρ sin 2 ) A mode range of ℓ = 5 to ℓ = +5 was investigated, s∆ = ρ dρ dθe h r0  i cos∆θ, π Z0 Z0 and for 100 randomly generated phase screens the aver- (2) age power in each OAM mode was measured (Fig. 2). where ∆ is an integer step in the mode index of ℓ, and A range of turbulence levels characterised by D/r0 were ρ =2r/D [9]. tested. As predicted by Eqn. 2, the cross-talk between As shown in Fig. 1, we generate OAM modes by use OAM modes increases with turbulence. In the mid/high of a simple forked diffraction grating created using an turbulence regime we see good agreement between our SLM that is illuminated by an expanded gaussian beam measurements and the theory proposed in [9]. In the low produced by a HeNe laser. Rather than producing a turbulence regime, the cross-talk between modes arises pure Laguerre-Gaussian mode, this results in a helically from residual cross-talk in our mode sorter, which can be phased beam, which has a near-Gaussian intensity dis- attributed to the diffraction limit [13,14]. The weightings tribution in the image plane of the SLM. This approach of the known input states described by an N = 11 ele- maintains the ratio D/r0 independent of the mode index. ment column vector [I] are mapped by an N × N cross- A particular turbulent phase screen can then be added talk matrix onto the measured N element output vector to this hologram to simulate the presence of atmospheric [O] (Eqn. 3). For the case of zero residual cross-talk, this turbulence. The SLM is then imaged to the 8 mm diam- matrix would correspond to an identity matrix. For fi- eter input pupil of the OAM mode sorter (MS) to de- nite cross-talk, the coefficients a−j etc. are measured at compose the resulting beam into its constituent OAM zero turbulence. Consequently, this matrix is used to pre- modes. dict the measured OAM output spectrum for an input The mode sorter uses two refractive optical elements OAM state subject to the atmospheric cross-talk from which transform OAM states into transverse momen- our theoretical model (Eqn. 2).

2 ℓ = −1 ℓ = −3 distance point-to-point communications on earth, tur- 1 bulence is characterized more accurately by multi-plane turbulence. In such cases one can expect intensity fluctu- ations and scintillation effects, and the thin phase model 10−2 is insufficient. Knowledge of the limits atmospheric turbulence im- 10−4 poses on a free-space communication channel is very im- portant for designing an optical system operating in such ℓ = −4 ℓ =4 an environment. In this letter, we have experimentally 1 characterized the effects of thin-phase turbulence over a range of ℓ = −5 to ℓ = +5, and verified that turbulence

−2 degrades the mode quality independent of input mode 10 number. This result indicates that a system implement- ing adaptive optics to reduce the effects of turbulence 10−4 can operate independently of the communications chan- nel. The experimental data presented also indicates the ℓ =3 ℓ =5 potential working range of a free-space OAM channel 1 and the expected cross-talk for such a system. We ex- pect that our study provides useful information for the

−2 construction of practical quantum key distribution sys- 10 tems using OAM modes [15]. We acknowledge Dan Gauthier and Jonathan Leach 10−4 for helpful discussions. Our work was primarily sup- 2 −2 2 ported by the DARPA InPho program through the US 10− 1 102 10 1 10 Army Research Office award W911NF-10-1-0395. MPJL Fig. 3. The spread in power resulting from atmospheric was further supported by European collaboration EC turbulence was measured for a range of different propa- FP7 255914, PHORBITECH, and MJP is supported by the Royal Society. gating OAM modes ψℓ.

References

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