MSc in Photonics Universitt Politècnic de Ctluny (UPC) Universitt Autònom de Brcelon (UAB) Universitt de Brcelon (UB) Institut de Ciències Fotòniques (ICFO) PHOTONICSBCN http://www.photonicsbcn.eu Mster in Photonics MASTER THESIS WORK Comprison of bem wnder correction by qudrnt nd idel detectors for eril quntum communiction links Jorge Gómez-Grcí Supervised by Dr. Verónic Fernández Mármol, (CSIC) Presented on dte 15 th October 015 Registered t
Comprison of bem wnder correction by qudrnt nd idel detectors for eril quntum communiction links Jorge Gómez Grcí Instituto de Tecnologís Físics y de l Informción, Consejo Superior de Investigciones Científics, (CSIC), C/Serrno 144, Mdrid, 8006, Spin E-mil: jorgegom@ucm.es Abstrct. We report n nlysis of bem wnder correction by both qudrnt nd n idel detector in close loop configurtion. The nlysis is performed with the im of finding the optimum detector for bem trcking of free spce quntum communiction links. The effect of bem wnder will be modelled s n ngle-of-rrivl fluctution t the receiver. A simultion using ry trcing softwre ws performed to simulte the correction for different vlues of the ngle of rrivl, rnging from those typicl from tmospheric turbulent regimes, with vlues of few mili, to lrger vlues up to 1, from lrger possible mislignments from the system. The simulting correcting system indictes tht qudrnt detector is only successful for lrge vlues of the ngle of rrivl (> 0. ), wheres for smller rnge no correction is shown due to the limiting fctor imposed by berrtions of the focusing lens. We lso show tht the optimum spot dimeter depends on the ngle of rrivl: if this is smller thn 0.5⁰, spot dimeter of t lest D spot /D gp =.5 should be used, wheres for lrger vlues of the ngle of rrivl, spot dimeter of t lest D spot /D gp =3.5 is required. Keywords: Bem wnder correction, free spce Quntum Key Distribution, turbulence trcking, quntum cryptogrphy. 1. Introduction Unlike clssicl cryptogrphy, bsed on integer fctoristion nd computer complexity, quntum cryptogrphy [1] uses fundmentl principles of quntum physics to ensure the secrecy of communiction systems. In prticulr, Quntum Key Distribution (QKD) enbles the informtionlly-secure distribution of common cryptogrphic key to two remote prties. This key cn be trnsmitted by guided or free spce mens. The first uses opticl fibre s the trnsmitting chnnel with world record trnsmissions reching 50 km []. Fibre-bsed systems re limited by bsorption in the fibre; nd birefringence mkes compenstion techniques necessry. On the contrry, these effects in free spce re lower, nd in scenrio where quntum repeters re still fr in the horizon, globl quntum communictions cn only be chieved using the tmosphere s the trnsmitting chnnel. Free spce communictions re useful both in lnd by linking points with poor connectivity in metropolitn networks nd s connections from ground to stellite or vice vers. In prticulr, for in-lnd links, free-spce QKD links re very useful due to their flexibility regrding instlltion nd portbility, since they cn be relocted ccording to the network needs [3]. However, free spce links re not exempt from certin tmospheric effects, such s bem wnder, which limits the efficiency of quntum communictions. Bem trcking techniques cn reduce drmticlly this effect nd improve the speed nd security of free spce QKD links. Although these techniques re commonly used in lser communiction systems they need to be dpted to quntum communiction links. Specificlly, the type of detector nd configurtion hve strong influence on the performnce of the correction system. We will present
comprtive study of bem wnder correction performed by n idel detector nd qudrnt photodiode, in vriety of conditions simulting bem wnder, in order to evlute the most suitble to our ppliction; being integrted in free-spce QKD system.. Principle of opertion of position sensitive detectors. Bem wnder is cused by rndom deflections of the bem s it propgtes through the tmosphere, which is in turn consequence of rndom vritions of the refrctive index due to stochstic chnges in the ir temperture. These vritions provoke rndom modifiction of the ngle of rrivl (the direction of propgtion of light bem) t the receiving sttion, ffecting directly to the pointing nd receiving stbility of the link. Bem wnder is hevily ffected by vritions in the ir density, time of the dy, wether, etc. Actively trcking nd linking the bem to fixed, optimum position not only mintins the lignment of the link, but lso llows reduction of the field of view (FOV) of the system, which usully is set to wider ngles to ccommodte ll the vritions of the signl. A smller FOV results in direct decrese of the error rte, since bckground noise is reduced, nd thus n increse of the secure key rte. A Fst Steering Mirror (FSM) fed by the signl given by Position Sensitive Detector (PSD) with Proportionl Integrl Derivtive (PID) controller cn be used to correct bem wnder. The FSM selected for the experiment is bovin-type mirror. This consists of bovin surrounded by some mgnets, fixed to the mirror; n electric current circultes inside the bovin, controlling the mgnets which move the mirror. They re known for their high precision, durbility nd short time response, nd more importntly, their ngulr resolution nd bndwidth re high enough for our purposes. It includes n internl position sensor tht controls the instntneous rottion of the mirror nd lso provides the rel position of the mirror. Using this vlue of the rel position together with the desired position (given by the PSD), the PID controller is cpble of rotting the FSM to the desired position (ctully, the position supported by the PSD is the position of the centroid of the spot). A PSD clcultes the position of light spot on its surfce by compring the signls of the four electrodiodes in which is divided. It bsiclly subtrcts the opposite signl in ech xis nd then normlises it with the totl detected intensity. Nming A, B, C nd D to the position detected in ech electrodiode (in consonnce with figure 1 ), the position detected in the x nd y xis will be: ( ) ( ) ( ) ( ) x B D A C, y A B C D (1) A B C D A B C D According to the number of ctive res in which the detector is divided, there re two sorts of PSDs: lterl effect (just one ctive re) nd qudrnt photodiodes (QD) (two to four ctive res). As we cn pprecite in figure 1, QD hs blind zone, i.e., gp mong the electrodiodes where the detector is unble to identify the position if the incoming bem is inside the gp. Hence, these detectors re used with unfocused bems. In contrst, since lterl effect detectors only hve one ctive re, they do not hve this gp nd my be used with focused bems. () Figure 1. Conceptul digrm of the two-type sensitive detectors: () qudrnt detector nd lterl effect detector.
The wvelengths tht lterl effect detectors nd QDs re sensitive to, depend on the quntum efficiency of the mterils which they re mde of. In generl, to detect wvelengths from visible light to 900 nm, the most efficient mteril is Si, wheres for the rnge from 1000 nm to 1600 nm InGAs is used insted. The QKD system previously developed in [4] employs wvelength of 850 nm for the quntum chnnel nd wvelength of 1550 nm for the synchronistion of the clocks of trnsmitter nd receiver. Nevertheless, s studied in [6], the synchronistion signl cn lso be employed for bem trcking. The trcking detectors must be then optimised for the wvelength of 1550 nm, i.e., they should be mde of InGAs. Fortuntely, most QDs in the mrket re mde of this mteril. In contrst, the mjority of lterl effect detectors re mde of Si nd therefore in order to compre the performnce of QDs nd lterl effect detectors, prtilly custom-mde lterl effect detector of InGAs ws chrcterised. However, previous work found its detected signl to be insufficient for proper correction, due to problems in the mplifiction phse [5]. 3. Description of close-loop correction strtegy. Previous work sttes tht compenstion of bem wnder in the receiver with the so-clled closed loop strtegy is, in most cses, the optimum strtegy for low distnce free spce link (up to km), [6]. figure shows digrm of the close loop strtegy integrted in the studied QKD system [4]. In the trnsmitter, two lsers of the sme wvelength encrypt the 0 s nd 1 s binry dt (which we will cll dt chnnel) nd n extr lser of different wvelength is used for synchronising the clocks of both trnsmitter nd receiver ( sync chnnel). The trcking of the bem is performed by the sme synchronistion lser, which simplifies the experimentl system. At the receiver, dichroic mirror seprtes the dt nd trcking/sync signls into two pths (trcking nd dt chnnels); n vlnche photodiode (APD) detects the synchronistion signl; nd QD performs the bem wnder correction. Two single photon vlnche diodes (SPADs) detect the quntum signl. Figure. Conceptul digrm of the receiver correction: closed loop strtegy. The close-loop correction strtegy is performed simultneously in both chnnels nd for it to be implemented eqully, the detectors in both chnnels must be plced in the sme equivlent plne. For our ppliction this should be the focl plne, since the bem spot tkes its smllest size. It should be stressed tht the purpose of correcting bem wnder is to reduce the FOV, nd thus decrese the bckground noise coupled into the system. However, this is in contrdiction with the need of using defocused bems with QDs. One solution for this problem my be incresing the spot size only in the trcking chnnel, where the QD is plced, while mintining the sme size in the dt chnnel. Previous work [5] suggests tht this could be done, for exmple, by incresing the berrtions in the trcking chnnel by plcing the lens (n chromtic doublet) before the detector in the opposite direction to how it is supposed to be plced (the direction where berrtions re minimised). Although incresing the berrtions seemed to be vlid wy to solve the problem, we will show tht, lthough correction is successful in the trcking chnnel, this will not be chieved in the quntum chnnel due to symmetries in both chnnels.
4. Modelling bem wnder Bem wnder cn be modelled s n ngle of rrivl (AOA),, in the receiver. The AOA,, in turbulence medium is described by turbulence theory s [3]: 1/ 3 h.91c n d(w G ), C n, () 3 1/ 3.4d 0 where d is the link distnce, h is the height over the opticl xis t which the bem impinges on the lens, 0 the rdius of the bem in the trnsmitter (we will use bem with 3.5 mm rdius), WG the short term bem rdius t the receiver perture (it tkes into ccount the divergence of 0 with the distnce) nd C n the refrctive index structure prmeter. The AOA,, ws modelled s n ngulr rottion of the source in the trnsmitter,, locted t distnce d from the focusing lens (see figure 3b ). () Figure 3. Setup of the close loop correction system: () simulted with the softwre OpticsLb, conceptul digrm before performing correction. For simplicity, the FSM nd the BS re not shown in. The setup used for the simultion, shown in Figure 3, tries to replicte the experimentl setup tested in [5][6], which includes source tht produces 7 mm of dimeter bem t wvelength of 650 nm, directed towrds FSM nd split by 50/50 bem splitter (BS) into the dt nd trcking chnnels. Both bems re focused by two 30 mm-focl-length lenses. In the trcking chnnel, the QD detects the position of the bem centroid nd sends this signl to the FSM through PID control, which tries to stbilise the bem. In the dt chnnel the PSD only observes the bem, nd n perture is plced to test the mount of opticl power tht psses through, which will ccount for how much the FOV is reduced. Approximting the AOA,, to which is given simply by, h rctn (3) d cn only be done when they coincide. Figure 4 is plot of nd s function of the distnce, for n extremely high turbulence medium C n 10 m (which is the worst cse s the difference increses with C n ). It shows tht cn be pproximted by for link shorter thn 30 m, under turbulence of 11 / 3 C n 10 m. 11 / 3 Figure 4. Comprison between nd (for C n 10 11 m / 3 ) s function of the distnce.
4.1. Dependence of the spot dimeter t the focl plne on h The ngulr rottion of the source mkes the bem enter the lens t n offset distnce from its centre, h, which will generte some berrtion in the bem. Essentilly, the min berrtion tht ppers is sphericl; rys striking the lens t greter distnce bove the xis h re focused nerer the lens. Therefore, the dimeter of the spot in the focl plne will be lrger for rys fr from the opticl xis (lrge h) thn the dimeter of the spot in the focl plne for prxil bem (low h). This mens the dimeter of the spot depends on h. For convenience, the spot dimeter in the focl plne will be simply nmed spot dimeter. In order to study the dependence of the spot dimeter with h, the distnce of the link d, nd the ngle of rrivl, simultion in the softwre OpticsLb ws performed using the setup shown in Figure 3. The source is moved using two types of movements: lterl nd ngulr trnsltions. Firstly, the source is trnslted lterlly for vlues of h close to the opticl xis (h = 1 mm) to vlues close to the edge (h = 7mm). Then, the source is centred in the opticl xis nd ngulrly rotted for ech distnce to keep h constnt. This is done for the sme vlues of h chosen for the lterl trnsltion. In figure 5, we hve plotted the spot dimeter s function of the distnce of the link, d, for severl vlues of h. Figure 5. Spot size in the focl plne of the lens s function of the link distnce d, between the source nd the lens, for different heights h. Continuous lines represent ngulr rottion nd the dsh lines correspond to the lterl displcement of the source. As it is depicted in figure 5, from distnce tht we will cll d 0, the spot dimeter becomes constnt with the distnce, which mens tht only depends on h nd not on d. Therefore, there is univocl reltion between the spot size nd h, which will be useful for our study. Therefore, if the source is fr enough from the lens; t certin distnce d 0, the rys cn be considered lmost prllel to the opticl xis (which mens tht 1 ). For the lrgest vlue of d 0 (h = 7 mm), distnce of 5 cm gives n error between the spot size for lterl displcement nd n ngulr rottion of the source of less thn 1%. Therefore this ws the miniml distnce between the source nd the focusing lens used for the simultions. If the source is closer (thn 5 cm) to the lens nd is ngulrly rotted, the behviour of the spot dimeter does completely chnge nd do not tend to lterl displcement of the source. The reson is tht the rys tht compound the bem cnnot be considered s prllel to the opticl xis, nd some other berrtions (prt from the sphericl) must be considered [6]. As consequence the spot dimeter becomes considerbly lrger. In summry, d 0 is the miniml distnce t which the source must be plced in order to void n incresed mount of berrtions, nd especilly to hve nd univocl reltionship between the spot dimeter nd h.
4.. Results of the simultion for bem wnder correction. Once we hve clrified tht we will work in region where the spot size only depends on h, we re going to nlyse the closed loop strtegy studied in [6]. In this previous work, experimentl dt tken t 30 m showed the movement of the centroid position in the focl re ws reduced lmost n order of mgnitude, for closed-loop correction-in-the-receiver configurtion. However, for quntum communiction system it is very importnt the size of the bem spot t the focl re, since it will delimit the field of view nd thus the improvement or worsening of the quntum bit error rte. For this purpose, the previous close-loop correction-t-the-receiver ws simulted with OpticsLb (see figure 3 ) for both n idel detector nd qudrnt detector. We hve modelled the turbulence s ngulr rottion of the source in the trnsmitter. It ws previously shown tht this ws good pproximtion s long s the distnce between the source nd the focusing lens ws kept below certin distnce (see figure 4 ). In order to evlute the correction, we will find the miniml dimeter, D 99%, tht n perture plced t the focl plne should hve to void losing more thn 1% of power. We will observe whether this dimeter hs decresed fter the correction hs been performed. Moreover, the detector in the trcking chnnel will be plced in the focl plne nd will be firstly, n idel detector nd fterwrds, QD with 50 μm gp, which ws the closest to tht used in [6]. The precision of the FSM controlled by QD will be 0.0005. The minimum distnce between the FSM nd the lenses will be 5 cm. As it ws previously commented in section, the PID controller rottes the FSM in such wy tht the position of the centroid of the spot, r, on the detector tends to 0 (See Figure 3b ). Since the spot is berrted, the centroid of the spot does not necessrily coincide with the geometricl centre of the spot. Hence, cncelling the position of the centroid does not generlly men tht the spot is completely centred. Moreover, lthough r 0 the bem will not pss through the centre of the lens. Since the scope of this study is evluting the performnce of both n idel nd qudrnt detectors for closed loop configurtion, the spot dimeter is n importnt fctor to consider, especilly for QD. Therefore, the correction of the QD will be evluted for different vlues of D spot / D gp. These vlues will be chieved by simulting the bem entering t the corresponding heights h of the focusing lens, using the univocl reltionship between the spot size nd h, discussed in the previous section. Tble 1 shows the four chosen vlues of D spot / D gp nd their corresponding vlues of h. Specificlly, this is chieved by simulting lterl displcement of the source tht produces those vlues of h t distnce of d 0. Note tht we used D / 1. 6 insted of 1.5; the reson is tht the QD is insensitive to smller spot D gp vlues thn 80 μm. In order to nlyse the bem wnder correction tht the detectors perform for certin D will fix the height h nd plot the dimeter of the perture tht collects the 99% of power, which we will cll D 99%, s function of the ngle of rrivl. For ech vlue of we clculte d (from eqution (3)) keeping h constnt. The reson of keeping h constnt is to hve fixed spot dimeter nd therefore seprte the influence of the ngle of rrivl from the spot size on the correction. Ech clculted distnce d will be the origin of the ngulr rottion (note tht there will be different distnce d for ech ). In the following, we will present the correction first for n idel detector nd then for QD. Tble 1. Vlues of D spot / D gp chosen for the study, nd their equivlent vlues of D spot nd h. D spot / D gp D spot (μm) h (mm) 0.5 5 1.1 1.6.5 3.5 80 15 175 3.1 4. 5.5 spot, we
4..1. Idel correction Firstly, one must tke into ccount tht under no correction D 99% does not depend on h. This is due to the wy the spot increses its size (s h increses), which is towrds the opticl xis. This mens tht the position of the furthest point of the spot remins unvried, so the top of the perture will lso remin unchnged. Figure 6 shows D99% s function of, under no correction nd fter correcting with n idel detector, i.e., detector tht cn exctly centre the position of the centroid, fulfilling r 0. This correction hs been mde for those heights shown in tble 1. Figure 6. Dimeter of the pertures necessry to collect the 99% of power in the focl plne of the lens, using n idel detector, s function of the ngle of rrivl. On one hnd it is esy to understnd tht if h increses (tht is incresing D D spot ), 99% will lso become lrger; this mens tht one must try to obtin the smllest vlues of h s possible. In order to understnd why D99% decreses with bigger vlues of, let rndom ngle 1 be smller thn nother ngle, 1. By eq. (3) for fixed vlue of h, d1 d. Knowing tht for n idel lens r F tn, it is cler tht under these conditions r1 r nd consequently the FSM hs to rotte more in order to mke r 0. Thus, nming h the height h fter performing the correction, for r the FSM hs rotte more nd hence h ' will be smller nd h' 1 h'. Since h is linked to the spot size ( Dspot' ) 1 ( Dspot' ) nd finlly, ( D99 % ) 1 ( D99% ). In summry, the smller the ngle of rrivl is, the bigger the dimeter of the perture will be. Figure 7 depicts this process.. () Figure 7. Digrm of the incidence upon lens of two bems plced t different distnces: () before correction, fter correction. To distinguish vribles before nd fter performing the correction, primed vribles re used in the ltter. For simplicity, we hve dropped out the FSM nd the BS from the digrm. For big vlues of, both r nd the geometricl centre of the spot re bove the opticl xis before correction nd fter correction, both r nd the centre of the spot re closer to the opticl xis; D 99% decreses (see Figure 8b ). However, smller vlues of produce tht the centroid of the spot is bove the opticl xis while its geometricl centre is underneth (see Figure 8 ). After performing the correction r is closer to the opticl xis, while the centre of the spot is further. This explins why D 99% fter correction grows for smll vlues of. Looking t
figure 6, it is cler tht if 0. then the method works: D 99% becomes smller fter correction. From figure 4, the ngle of rrivl due to tmospheric turbulence lone is considerbly smller thn 0. (regrdless of the distnce d). However, there might be lrger contributions to the ngle of rrivl different from this, such s, mislignments due to not optimised initil lignment, vibrtions of the building, etc. In conclusion, the correction using closed-loop strtegy is evident for 0. with considerble improvement s increses nd h (or D spot ) decreses. For 0. no improvement ws observed due to the deformtion of the spot cused by berrtions. () Figure 8. Digrm of the D 99%, nd r before nd fter correcting for: () smll vlues of lrge vlues of. 4... Correction with QD. Here we will study the performnce of QD for closed loop configurtion. As it ws previously mentioned, if the spot is too smll it flls into the gp, where the QD is not sensitive. The QD begins to detect the spot when t lest 0.1% of its power flls outside of the gp, which occurs for 80 m. However, the QD will not be ble to correct properly for spot of this D spot dimeter. This hppens becuse s the FSM brely rottes to perform the correction, the spot flls into the gp nd is no longer detectble. In figure 9, the spot size fter correction hs been plotted s function of, for different initil spot dimeters. In this figure, one cn observe tht regrdless of the initil vlue of D spot, the dimeter of the spot fter correction lwys tends to 80 μm, which is the minimum detectble spot dimeter. () Figure 9. Correction performed with QD: () spot dimeter fter correcting s function of the ngle of rrivl, for different initil spot dimeters, dimeter of the pertures necessry to collect 99% of power in the focl plne of the lens using QD nd no correction t ll, s function of. Similrly to figure 7, figure 9b shows D99% s function of, under no correction nd fter correcting with QD for different vlues of h. In this figure we cn observe tht if the initil spot dimeter is lrge enough, such s 175 μm, the QD behves more similrly to n idel detector ( D99% decreses with bigger vlues of ). However, t certin, D99% strts to grow gin. The reson is tht the spot dimeter fter correction hs reched the limit vlue of
80 μm nd the QD cnnot correct it ny more (s it is too smll to be detected). Therefore, s increses, r increses ( r F tn ), nd with it, lso D 99%. As generl conclusion, we cn extrct tht for lower vlues of [0,0.5 ], D spot 15 m is the optimum dimeter nd for higher ngles of rrivl [0.5,1 ] lrger spot is preferble. () Figure 10. Dimeter of the pertures necessry to collect the 99% of power in the focl plne using QD, n idel detector nd no correction t ll, s function of : () for h = 4 mm, for h = 5.5 mm. From figure 10 for h = 4 mm, which corresponds to D spot 15 m, the QD only behves similrly to n idel detector for [0,0.5 ]. However, in figure 10b, where h= 5.5 mm; both behviours re similr in the whole rnge. Therefore the lrger the spot the more idel the behviour of the correction becomes. However, how cn we increse the spot dimeter in the trcking chnnel (where the QD is plced) while keeping it smll in the dt chnnel? Previous work [5] suggested tht solution to improve the correction when using QD is to turn the lens round in the trcking chnnel, nd thus increse the spot dimeter in this chnnel through berrtions, while keeping it smller t the dt chnnel, which is desired to reduce the FOV. The correction for this configurtion ws experimentlly tested in the trcking chnnel but not in the dt chnnel. We hve simulted the sme conditions (with the setup shown in Figure 3 ) turning the doublet of the trcking chnnel in the opposite direction nd found there is no correction in the dt chnnel. The reson is tht the plnes where the detectors re plced (the focl plnes) in both chnnels re no longer equivlent. Due to the spot of the trcking chnnel being strongly berrted, its displcement from the centre of the focl plne r (trcking) will be different from tht of the dt chnnel r (dt). Since the correction is progrmmed to tke r (trcking) to zero, (s it is the one connected to the FSM vi the PID), the dt chnnel will not undergo the sme correction. On top of tht, this difference in the position of the centroid for both chnnels cnnot be voided, since it is found to chnge with the initil spot dimeter; since the turbulence produces rndom vrition of, the different vlues of will produce different D. Figure 11 shows the position of the centroid fter correction r for both chnnels. From the plot, it is cler tht ' regrdless of the detector used, r ' in the dt chnnel is different from tht of the trcking chnnel, which mens tht this strtegy will not work. spot Figure 11. r ' in the trcking nd dt chnnels for QD, s function of. The spot dimeter ws D spot 185 m, different from the previous vlues, since the lens ws plced in different position.
4. Conclusion On one hnd, we hve proved tht the closed loop correction strtegy, using n idel detector, is evident for high ngles of rrivl (lrger thn 0.⁰ pproximtely). However for lower vlues of the ngle of rrivl the correction is not present. This difference in performnce is due to berrtions. For low ngles of rrivl the berrtion limits the collecting perture D 99% ; however, for high vlues of the ngle of rrivl the devition of the position of the centroid, r, is much lrger thn the effect of berrtions themselves. Therefore, correcting (mking r = 0) improves the collecting perture nd thus, correction. Regrding the effect of the spot size in the correction, if the initil spot dimeter in the focl plne of the focusing lens is sufficiently lrge ( D / 3. 5 ), the qudrnt detector behves s n idel detector. The spot dimeter tht spot D gp should be used in ech ppliction depends on the rnge of the ngle of rrivl: for low vlues, [0,0.5 ], spot dimeter of D spot = 15 μm gurntees fvourble correction, wheres for higher vlues, [0.5,1 ], spot dimeter of t lest D spot = 175 μm is required. Finlly, we demonstrte tht turning round the lens in the chnnel tht performs the correction in order to increse the spot dimeter vi berrtions does not work for the dt chnnel, becuse the plnes where detectors re set re not equivlent. Acknowledgments I would like to thnk first to my supervisor Verónic Fernández nd her collegue Ntli Denisenko, who gve me this opportunity in which I hve been involved. I do not find the proper words to express ll the kindness nd comprehension with which they hve lwys treted me. To them I would like to dedicte this whole project. Moreover, I would like to thnk Emili Bgn for his support nd Crin Cojocru for her help with the documents to mke this reserch t CSIC. I would lso like to thnk CSIC for letting me be one more for severl months. We would lso like to thnk the project with reference TEC01-35673 from the Ministerio de Economí y Competitividd. References [1] C. H. Bennett nd G. Brssrd, Quntum cryptogrphy: public key distribution nd coin tossing, in Proc. of IEEE Int. Conf. on Computers, Systems nd Signl Processing, Bnglore, Indi, pp. 175 179 (1984). [] D. Stucki et l., High rte, long-distnce quntum key distribution over 50 km of ultr low loss fibres, New J. Phys., vol. 11, no.7 p. 075003, 009. [3] Willebrnd, H. nd Ghumn, B. S. (00). Free-Spce Optics: Enbling Opticl Connectivity in Tody s Networks. Sms Publishing..3., 4.1. [4] M. J. Grcí-Mrtínez, Ntli Denisenko, D. Soto, D. Arroyo, A.B. Orue, V. Fernndez, High-speed free-spce quntum key distribution system for urbn dylight pplictions, Applied Optics., vol. 5, no. 14, pp. 3311-3317, 013. [5] A. Crrsco-Csdo, Contribuciones ls comunicciones óptics en espcio libre: Utilizción de telescopios Cherenkov como receptores y corrección de bem wnder en comunicciones cuántics, (Ph. D. thesis, Universidd Crlos III de Mdrid, 015). [6] A. Crrsco-Csdo, Ntli Denisenko, V. Fernndez, Correction of bem wnder for free-spce quntum key distribution system operting in urbn enviroment, Opticl Engineering 53(8), 08411 (014). [7] E. Hetch, Optics, (Addison Wesley, 4th ed. 00).