Deep-space and near-earth optical communications by coded orbital angular momentum (OAM) modulation

Transcription

1 Deep-space and near-earth optcal communcatons by coded orbtal angular momentum (OAM) modulaton Ivan B Djordjevc Unversty of Arzona, Depart Electrcal & Computer Eng, 130 E Speedway Blvd, Tucson, AZ 8571, USA van@ecearzonaedu Abstract: In order to acheve mult-ggabt transmsson (projected for 00) for the use n nterplanetary communcatons, the usage of large number of tme slots n pulse-poston modulaton (PPM), typcally used n deep-space applcatons, s needed, whch mposes strngent requrements on system desgn and mplementaton As an alternatve satsfyng hghbandwdth demands of future nterplanetary communcatons, whle keepng the system cost and power consumpton reasonably low, n ths paper, we descrbe the use of orbtal angular momentum (OAM) as an addtonal degree of freedom The OAM s assocated wth azmuthal phase of the complex electrc feld Because OAM egenstates are orthogonal the can be used as bass functons for N-dmensonal sgnalng The OAM modulaton and multplexng can, therefore, be used, n combnaton wth other degrees of freedom, to solve the hgh-bandwdth requrements of future deep-space and near-earth optcal communcatons The man challenge for OAM deepspace communcaton represents the lnk between a spacecraft probe and the Earth staton because n the presence of atmospherc turbulence the orthogonalty between OAM states s no longer preserved We wll show that n combnaton wth LDPC codes, the OAM-based modulaton schemes can operate even under strong atmospherc turbulence regme In addton, the spectral effcency of proposed scheme s N /log N tmes better than that of PPM 011 Optcal Socety of Amerca OCIS codes: (060605) Free-space optcal communcaton; ( ) Modulaton; ( ) Atmospherc turbulence; (1880) Detecton; (106085) Space nstrumentaton References and lnks 1 B Moson and J Hamkns, Deep-space optcal communcatons downlnk budget: modulaton and codng, The Interplanetary Network Progress Report 4-154, Aprl-June 003 (Jet Propulson Laboratory, Pasadena, Calforna, 15 August 003), pp 1-8, B Moson and J Hamkns, Coded modulaton for the deep-space optcal channel: serally concatenated pulseposton modulaton, The Interplanetary Network Progress Report (Jet Propulson Laboratory, Pasadena, Calforna, 15 May 005), pp 1-5, 3 F Xu, M-A Khalgh, and S Bourennane, Coded PPM and multpulse PPM and teratve detecton for freespace optcal lnks, J Opt Commun Netw 1(5), (009) 4 S J Dolnar, J Hamkns, B E Moson, and V A Vlnrotter, Optcal modulaton and codng, n Deep Space Optcal Communcatons, H Hemmat, ed (Wle, 006), pp H Hemmat, Interplanetary laser communcatons, Optcs & Photoncs News 18(11), 7 (007) 6 L C Andrews and R L Phlps, Laser Beam Propagaton through Random Meda (SPIE Press, 005) 7 I B Djordjevc and M Arabac, LDPC-coded orbtal angular momentum (OAM) modulaton for free-space optcal communcaton, Opt Express 18(4), (010) 8 I B Djordjevc, Orbtal angular momentum (OAM) based LDPC-coded deep-space optcal communcaton, Proc SPIE 793, 79306, (011) (nvted paper) 9 C Paterson, Atmospherc turbulence and orbtal angular momentum of sngle photons for optcal communcaton, Phys Rev Lett 94(15), (005) 10 J A Anguta, M A Nefeld, and B V Vasc, Turbulence-nduced channel crosstalk n an orbtal angular momentum-multplexed free-space optcal lnk, Appl Opt 47(13), (008) (C) 011 OSA 18 July 011 / Vol 19, No 15 / OPTICS EXPRESS 1477

2 11 J Leach, J Courtal, K Skeldon, S M Barnett, S Franke-Arnold, and M J Padgett, Interferometrc methods to measure orbtal and spn, or the total angular momentum of a sngle photon, Phys Rev Lett 9(1), (004) 1 G Gbson, J Courtal, M Padgett, M Vasnetsov, V Pas ko, S Barnett, and S Franke-Arnold, Free-space nformaton transfer usng lght beams carryng orbtal angular momentum, Opt Express 1(), (004) 13 M T Grunesen, W A Mller, R C Dymale, and A M Swet, Holographc generaton of complex felds wth spatal lght modulators: applcaton to quantum key dstrbuton, Appl Opt 47(4), A3 A4 (008) 14 M P C Fossorer, Quas-cyclc low-densty party-check codes from crculant permutaton matrces, IEEE Trans Inf Theory 50(8), (004) 15 S Ln and D J Costello, Error Control Codng: Fundamentals and Applcatons (Prentce Hall, 004) 16 I B Djordjevc, M Arabac, and L Mnkov, Next generaton FEC for hgh-capacty communcaton n optcal transport networks, J Lghtwave Technol 7(16), (009) 17 J Chen, A Dholaka, E Eleftherou, M Fossorer, and X-Y Hu, Reduced-complexty decodng of LDPC codes, IEEE Trans Commun 53(8), (005) 18 I B Djordjevc and M Arabac, LDPC-coded orbtal angular momentum (OAM) modulaton for free-space optcal communcaton, Opt Express 18(4), (010) 19 J D Jackson, Classcal Electrodynamcs (Wley, 1975) 0 I B Djordjevc, M Arabac, L Xu, and T Wang, Generalzed OFDM (GOFDM) for ultra-hgh-speed optcal transmsson, Opt Express 19(7), (011) 1 H G Batshon and I B Djordjevc, Beyond 40 Gb/s per wavelength optcal transmsson usng coded hybrd subcarrer/ampltude/ phase/polarzaton modulaton, IEEE Photon Technol Lett (5), (010) H G Batshon, I B Djordjevc, L Xu, and T Wang, Multdmensonal LDPC-coded modulaton for beyond 400 Gb/s per wavelength transmsson, IEEE Photon Technol Lett 1(16), (009) 3 I B Djordjevc and H G Batshon, Generalzed hybrd subcarrer/ampltude/phase/polarzaton LDPC-coded modulaton based FSO Networkng, n Proceedngs of IEEE 1th Internatonal Conference on Transparent Optcal Networks (ICTON 010) (IEEE, 010), paper ThB34 4 W Sheh and I B Djordjevc, OFDM for Optcal Communcatons (Elsever/Academc Press, 009) 5 S M Alamout, A smple transmt dversty technque for wreless communcatons, IEEE J Sel Areas Comm 16(8), (1998) 6 A Goldsmth, Wreless Communcatons (Cambrdge Unversty Press, 005) 7 D Tse and P Vswanath, Fundamentals of Wreless Communcaton (Cambrdge Unversty Press, 005) 8 T M Duman and A Ghrayeb, Codng for MIMO Communcaton Systems (Wley, 007) 9 I B Djordjevc, L Xu, and T Wang, PMD compensaton n multlevel coded-modulaton schemes wth coherent detecton usng BLAST algorthm and teratve polarzaton cancellaton, Opt Express 16(19), (008) 30 J G Proaks, Dgtal Communcaton (McGraw Hll, 001) 31 I B Djordjevc and G T Djordjevc, On the communcaton over strong atmospherc turbulence channels by adaptve modulaton and codng, Opt Express 17(0), (009) 3 I B Djordjevc, Adaptve modulaton and codng for communcaton over the atmospherc turbulence channels, n Proceedngs of IEEE Photoncs Socety Summer Topcals (IEEE, 009), paper TuD33 33 M Arabac, I B Djordjevc, R Saunders, and R M Marcocca, Polarzaton-multplexed rate-adaptve nonbnary-quas-cyclc-ldpc-coded multlevel modulaton wth coherent detecton for optcal transport networks, Opt Express 18(3), (010) 1 Introducton Power-effcent modulaton schemes, such as pulse-poston modulaton (PPM), are wdely adopted n current deep-space optcal communcatons [1 6] The very large bandwdth of these lnks (compared to RF lnks) has made the low spectral effcency of PPM less of a concern However, n order to acheve mult-ggabt transmsson (projected for 00) for the use n nterplanetary communcatons [5], the use of large number of tme slots n PPM s needed, whch mposes strngent requrements on system desgn and mplementaton On the other hand, we have shown recently [7,8] that through the use of orbtal angular momentum (OAM) modulaton we can satsfy hgh-bandwdth demands of future nterplanetary communcatons, whle keepng the system cost and power consumpton reasonably low The OAM s assocated wth azmuthal phase of the complex electrc feld Because OAM egenstates are orthogonal, n prncple, arbtrary number of bts per sngle photon can be transmtted The ablty to generate/analyze states wth dfferent OAMs, by usng nterferometrc or holographc methods [9 13] allows the realzaton of deep-space/near-earth optcal communcaton systems wth ultra-hgh photon effcences expressed n terms of number of bts per photon The OAM modulaton and multplexng can, therefore, be used, n combnaton wth other degrees of freedom, to solve the hgh-bandwdth requrements of future deep-space and near-earth optcal communcatons By ncreasng the number of OAM (C) 011 OSA 18 July 011 / Vol 19, No 15 / OPTICS EXPRESS 1478

3 states we can ncrease the aggregate data rate of the system, whle enablng relable transmsson at these hgher speeds by usng low-densty party-check (LDPC) codes [14 17] Dramatc ncrease n photon effcency through OAM modulaton and multplexng wll provde revolutonary capabltes for future deep-space/near-earth applcatons The man challenge for OAM deep-space communcaton represents the lnk between a spacecraft probe and the Earth staton because n the presence of atmospherc turbulence the orthogonalty between OAM states s no longer preserved In ths paper, we wll show that n combnaton wth LDPC codes, OAM-based modulaton schemes can operate even under strong atmospherc turbulence regme The followng mult-ggabt enablng approaches for deep-space and near-earth optcal communcatons based on OAM are descrbed: () OAM multplexng (Sec ); () N- dmensonal OAM modulaton (Sec ); and () orthogonal OAM dvson multplexng (Sec 3), smlar to OFDM [4] We wll also descrbe dfferent approaches to deal wth atmospherc turbulence nduced OAM crosstalk ncludng: (1) equalzaton; () OAM-tme codng, whch s analogous to the space-tme codng; and (3) adaptve modulaton and codng (AMC) Notce that our prevous paper on OAM modulaton [18] deals wth terrestral FSO communcatons, and t employs volume holograms to perform OAM modulaton/demodulaton, whle n ths paper nstead we use multmode fber (MMF) to perform OAM modulaton/demodulaton Moreover, Ref [18] does not dscuss at all nether orthogonal OAM dvson multplexng nor compensaton of OAM crosstalk effects ntroduced by atmospherc turbulence Further, we provde detaled descrpton of OAM demodulator and teratve demapper/decoder Fnally, we descrbe both drect detecton and coherent detecton OAM demodulators The paper s organzed as follows In Secton, we descrbe the concepts of OAM multplexng and modulaton We also descrbe an N-dmensonal OAM-based coded modulaton scheme sutable for deep-space and near-earth applcatons The OAM dvson multplexng s descrbed n Secton 3 Dfferent methods to re-establsh the orthogonalty of OAM modes, after transmsson over atmospherc turbulence channels, are descrbed n Secton 4 We evaluate the sutablty of OAM-based coded-modulaton schemes for deepspace/near-earth applcatons n Secton 5 Secton 6 s devoted to concludng remarks Deep-space and near-earth optcal Communcatons based on LDPC-coded OAM modulaton In ths secton, we descrbe how to smultaneously ncrease the photon and spectral effcences based on N-dmensonal OAM coded modulaton The angular momentum, L, of the classcal electromagnetc feld can be wrtten as [19] L 1 1, k x, y, z k k 4c E AdV V 4c E r A dv (1) V where E s the electrc feld ntensty, A s the vector potental assocated wth magnetc feld 1 ntensty H by H A and electrc feld ntensty by E = -c A / t, and c s the speed of lght The second term n (1) s dentfed wth the OAM because of the presence of the angular momentum operator r term Among varous optcal beams that can carry OAM, Laguerre-Gauss (LG) vortex beams can easly be mplemented For example, a feld dstrbuton of LG beam travelng along z-drecton can be expressed n cylndrcal coordnates (r,,z) (r denotes the radal dstance from propagaton axs, denotes the azmuthal angle and z denotes the propagaton dstance) as follows [10]: l r jkr z z 1 1 tan p! 1 r j p l l r w z z z R zr jl ul, p r,, z Lp e e e e, p l! wz wz w z () (C) 011 OSA 18 July 011 / Vol 19, No 15 / OPTICS EXPRESS 1479

4 where w(z) = w 0 [1 + (z/z R )] 1/ wth w 0 beng the zero-order Gaussan radus at the wast, z w s the Raylegh range (wth λ beng the wavelength), k = π/λ s the propagaton R 0 / constant, and L l p() s the assocated Laguerre polynomal, wth p and l representng the radal and azmuthal mode numbers, respectvely It can be seen from () that lth mode of LG beam has the azmuthal angular dependence of the form exp(-jl), and consequently, l s also called the azmuthal mode number (ndex) For l = 0, feld u(r,,z) becomes a zero-order Gaussan beam, that s the TEM 00 mode For p = 0, L l p() = 1 for all ls, so that the ntensty of a LG mode s a rng of radus proportonal to ( l ) 1/, as llustrated n Fg 1(a) It can be shown [10] that for fxed p, the followng prncple of othogonalty s satsfed: * ump, rdrd, n m (3) m, p, n, p m, p,, n, p,, u u u r z u r z rdrd 0, n m Clearly, dfferent OAM states for fxed p are mutually orthogonal and as such they can be used as bass functons for OAM modulaton The number of OAM states to be used depends on atmospherc turbulence strength For N = L + 1, the correspondng OAM states can be ndexed by l where l takes values n the set {-L,-L + 1,,-1,0,1,,L} By ncreasng the number of dmensons (e, the number of OAM egenstates), we can ncrease the aggregate data rate of the system The transmtter and recever block dagrams for OAM modulaton are shown n Fg 1 Prevously, we studed the use of volume holograms based multdmensonal terrestral FSO communcaton [18] The terrestral FSO system descrbed n [18] employs the volume holograms to perform OAM multplexng/de-multplexng Instead of usng the set of holograms to mpose dfferent OAM states, here we use mode multplexer/demultplexer based on multmode fbers as shown n Fg 1(b)/(c) Durng mode multplexng, a contnuous wave laser dode sgnal s splt nto N branches by usng a power spltter (such as 1:N star coupler) to feed N Mach-Zehnder modulators (s), each correspondng to one of the N OAM modes The mode demultplexer can be mplemented by propagatng the multmode sgnal n opposte drecton, as shown n Fg 1(c) The jth nput to the jth, 1 j N, corresponds to the jth coordnate of the sgnal to be modulated (see Fg 1b) The output of mode multplexer s expanded by telescope and transmtted towards remote destnaton The OAM multplexng s obtaned by summng up N ndependent streams per each OAM state nto a sngle optcal beam Clearly, when OAM transmtter and recever are mplemented as shown n Fg 1(b,c) we are not able to dstngush between OAM modes wth azmuthal mode numbers of the same magntude but of opposte sgn Nevertheless, dstngushable modes l = 0,1,,L (for fxed p) are stll orthogonal The mode detector shown n Fg 1(e), nstead of beng mplemented based on Fg 1(c) t can be mplemented based on ntegrated rng-shaped detector wth N-dfferent pn regons to capture dfferent l-modes, as llustrated n Fg 1(a) for p = 0 Such a mode detector wll have N outputs that correspond to N-projectons along OAM modes After ths generc descrpton of OAM based deep-space/near-earth optcal communcaton systems, we provde more detals of OAM transmtter and recever The K dfferent bt streams comng from dfferent nformaton sources are encoded usng (n,k m ) (m = 1,,K) LDPC codes, as shown n the Fg 1(d) The outputs of the encoders are nterleaved by the K n block nterleaver The block nterleaver accepts bts from the encoders row-wse and outputs bts column-wse to the mapper, whch accepts K bts at each tme nstance The mapper determnes the correspondng M N -ary sgnal constellaton pont by s N N C N 1, j j; 1,,, M (4) j where M s the number of ampltude levels per OAM state and C N s the normalzaton constant The set {Φ 1, Φ,,Φ N } represents a set of N orthogonal OAM bass functons, whle wth φ,j we denoted OAM modulaton coordnates of th constellaton pont The sgnals are (C) 011 OSA 18 July 011 / Vol 19, No 15 / OPTICS EXPRESS 1480

5 then modulated, mode multplexed and sent over the deep-space or near-earth optcal channel (see Fg 1) The number OAM modes N to be used s determned by the desred fnal rate and current channel condtons The smplest OAM-based coded modulaton scheme wth drect detecton can be descrbed by the followng set of constellaton ponts for N = 3 and M = {(0,0,0), p=0 l=0 l=1 l=l (a) N-dmensonal OAM transmtter OAM coord 1 Mode-multplexer Laser dode Power spltter Channel 1 Channel N Taper-core MMF Claddng n 1 >n Core n Expandng telescope OAM coordnate N (b) N-dmensonal OAM recever Compressng telescope Source channels 1 K LDPC encoder r 1 =k 1 / n LDPC encoder r K =k K / n MMF FSO compressng telescope Interleaver Kxn Modes detector Mode-demultplexer Taper-core K (c) Mapper & s (d) APP Demapper N-dm symbols Bt LLRs Calculaton PD 1 PD PD N Modes multplexer Extrnsc LLRs Projecton along OAM mode 1 Projecton along OAM mode N to FSO expandng telescope LDPC Decoder 1 1 LDPC Decoder K K Fg 1 Multdmensonal LDPC-coded OAM modulaton scheme: (a) ntensty spatal dstrbuton for p = 0, (b) MMF-based mode-multplexer and N-dmensonal transmtter confguratons, (c) MMF-based N-dmensonal OAM recever, (d) overall transmtter archtecture and (e) overall recever archtecture : Mach-Zehnder modulator, MMF: multmode fber, PD: photodetector, APP: a posteror probablty, LLRs: log-lkelhood ratos (e) (C) 011 OSA 18 July 011 / Vol 19, No 15 / OPTICS EXPRESS 1481

6 (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1)} The N-dmensonal sgnal constellaton can be obtaned as the N-dmensonal Cartesan product of a one-dmensonal sgnal constellaton orgnatng from non-negatve pulse-ampltude modulaton (PAM) At the recever sde (see Fg 1(d)), after OAM demodulaton (or mode demultplexng) and photodetecton, the outputs of the N branches of the demodulator are sampled at the symbol rate, and the correspondng samples, after analog-to-dgtal converson, are forwarded to the a posteror probablty (APP) demapper The orthogonalty among OAM modes n realstc deep-space and near-earth optcal communcaton systems can be re-establshed by varous MIMO and equalzaton technques, as dscussed n Secton 4 The APP demapper provdes the bt LLRs requred for teratve LDPC decodng (see Fg 1e) The lth branch n Fg 1(f) represents the projecton along lth OAM state (coordnate) Let T L L L S S S S S S S denote the transmtted sgnal constellaton pont for system shown n Fg 1(a), R denote the receved constellaton pont, and S 0 denote the reference constellaton pont The superscrpt (l) (l = -L,,-1,0,1,L) s used to denote the lth OAM coordnate When the deepspace/near-earth optcal communcaton system s based on N-dmensonal transmtter and recever shown n Fg 1(c,d), then the correspondng sgnal constellaton pont s defned as T 0 1 N 1 S S S S Let P(R S ) denote the condtonal probabltes that are estmated by collecton of OAM crosstalk hstograms, obtaned by propagatng suffcently long tranng sequence over the deep-space/near-earth optcal channel, whle P(S) denotes a pror probablty of symbol S The symbol log-lkelhood ratos (LLRs) can be calculated by where S R P S R S log, P S0 R P s determned by Bayes rule as S R S PS R S PS P P S R (6) P By substtutng Eq (6) nto Eq (5) we obtan P R S P S P R S P S P R S S log log log log a S, P R S S 0 P S0 P R S P 0 0 P R S0 (7) where P P a S log S / S 0 Let us denote by c j the jth bt n the observed symbol S bnary representaton c = (c 0,c 1,,c b-1 ) (b s the number of bts needed to represent the symbol S) The pror symbol LLRs for the next teraton are determned by Because a S b1 Pc 0 P S j Pc 0 j b1 0 j0 log log log b1 P S j0 Pc Pcj j j0 (5) (8) (C) 011 OSA 18 July 011 / Vol 19, No 15 / OPTICS EXPRESS 148

7 j 0 c Pc j Lc j Pc 0, j 0 log c jlc j,, c j 1 where L(c j ) = log[p(c j = 0)/P(c j = 1)], the pror symbol LLRs become where b c Lc a j j j1 Fnally, the pror symbol estmate can be obtaned from (9) s (10) b c L c s ˆ, (11) a j D,e j j1 out n L ˆ D, e cj L cj L c j (1) In Eq (1), we use L(c (n) j) [L(c (out) j)] to denote the LDPC decoder nput (output) The bt LLRs L(c j ) are determned from symbol LLRs by Sexp Lac c k k Sexp Lac c k exp c: c j0 : c 0, k j Lc ˆ j log exp c: c j1 : ck 0, k j Therefore, the jth bt relablty s calculated as the logarthm of the rato of a probablty that c j = 0 and probablty that c j = 1 In the nomnator, we perform the summaton over all symbols S havng 0 at the observed poston j, whle n the denomnator over all symbols S havng 1 at the same poston j Wth L a (c k ) we denoted the pror (extrnsc) nformaton determned from the APP demapper The nner summaton n (13) s performed over all bts of symbol S, selected n the outer summaton, for whch c k = 0, kj The bt LLRs are forwarded to LDPC decoders, whch provde extrnsc bt LLRs for demapper accordng to (1), and are used as nputs to (11) as the pror nformaton Notce that deep-space/near-earth optcal communcaton systems shown n Fg 1 are based on drect detecton When coherent detecton s used, the correspondng modulator can be: () one-dmensonal based at null pont (see Fg 1(b)), () two-dmensonal modulator based on I/Q modulator (see Fg (a)) and () four-dmensonal (see Fg (b)) When I/Q modulator s used, n addton to N OAM orthogonal modes, we employ the nphase (I) and quadrature (Q) components so that the correspondng space s N-dmensonal On the other hand, when four-dmensonal modulator s used, wth two coordnates beng I- and Q-coordnates n x-polarzaton and two coordnates beng I- and Q-coordnates n y polarzaton, the correspondng space s 4N-dmensonal The 4N-dmensonal scheme allows even 4N-bts to be transmtted per sngle symbol, whch represents an energy-effcent communcaton scheme However, the complexty of such recever mght be too hgh even for near-earth applcatons The one-dmensonal coherent detecton scheme seems to be a good compromse between optcal lnk energy-effcency and recever complexty In Fg (c), we show the recever confguraton wth coherent detecton that assumes that one-dmensonal s are used One laser s used to detect all coordnates, whose output s used as an nput to 1:N power spltter The output of balanced detector n (n = 0,1,,N-1) provdes the projecton along OAM state n Ths coherent detecton recever scheme; composed of modedemultplexer, balanced coherent detectors, power spltter and local laser; can be ntegrated on a sngle chp usng hybrd opto-electronc ntegrated crcut (OEIC) technology Notce that dfferent orthogonal bass functons can be used nstead of orthogonal OAM egenstates (eg, [0], [1 3]) For example, the orthogonal subcarrers can be used as the (13) (C) 011 OSA 18 July 011 / Vol 19, No 15 / OPTICS EXPRESS 1483

8 Nth complex coordnate Re{} Im{} DAC DAC LPF LPF From powerspltter / I Q to modes multplexer (a) Q x I x I/Q-MOD n x-pol From powerspltter PBS 3 db / / PBC to modes multplexer from compressng telescope Modedemultplexer Q y (b) I y Balanced coh det 1 Local laser Balanced coh det N (c) I/Q-MOD n y-pol BCJR Equalzer Extrnsc LLRs Bt LLRs Calculaton LDPC Decoder 1 1 LDPC Decoder K Fg Coherent detecton based OAM modulaton: (a) two-dmensonal modulator, (b) fourdmensonal modulator and (c) coherent recever archtecture (correspondng to onedmensonal modulators) bass functons for multdmensonal sgnalng, whch was the subject of Ref [0] However, the subcarrer-based multdmensonal sgnalng cannot be used to smultaneously mprove SNR senstvty and spectral effcency For example, f we are concerned wth SNR senstvty we can use the orthogonal subcarrers as bass functons n multdmensonal sgnalng to ncrease the Eucldean dstance between sgnal constellaton ponts, whle keepng the total sgnal bandwdth the same In proposed scheme, however, we can smultaneously mprove the SNR senstvty and spectral effcency Namely, by ncreasng the number of OAM egenstates, used for bass functons, we can dramatcally ncrease the aggregated data rate, whle keepng the symbol rate fxed, as the ncrease of the number of OAM modes does not lead to bandwdth expanson In order to acheve mult-ggabt transmsson, whch s projected for 00, for the use n nterplanetary communcatons [5], the use of better spectral effcent schemes, such as proposed one, s needed The current deep-space optcal communcatons are based on PPM, and n order to acheve the mult-gb/s data rate, predcted by Hemmat n [5], a huge number of tme-slots wll be needed n PPM, whch ntroduces too strngent requrements from mplementaton pont of vew 3 Orthogonal OAM dvson multplexng The transmtter and recever confguratons for orthogonal OAM dvson multplexng are shown n Fg 3(a), and (b), respectvely It has certan smlartes wth OFDM [4] The transmtter confguraton up to the mapper s very smlar to that shown n Fg 1(e) The mapper accepts K bts at tme nstance from the (K n) nterleaver column-wse and determnes the correspondng M-ary (M = K ) sgnal constellaton pont ( I,, Q, ) assumng a K (C) 011 OSA 18 July 011 / Vol 19, No 15 / OPTICS EXPRESS 1484

9 two-dmensonal (D) constellaton such as M-ary PSK or M-ary QAM The coordnates correspond to n-phase (I) and quadrature (Q) components of M-ary D constellaton The D constellaton ponts, after seral-to-parallel (S/P) converson, are used as the nputs to I/Q modulators of OAM transmtters shown n Fg 3(a) On the recever sde, upon OAM demultplexng and detecton as shown n Fg 3(b), the soft estmates of symbols s k carred by kth OAM state are forwarded to the APP demapper, whch determnes the symbol LLRs The soft symbol estmates n the kth OAM state s k are obtaned by usng one of the OAMcrosstalk compensaton technques descrbed n next secton The bt LLRs needed for LDPC decoders are calculated n the bt LLR calculaton block n smlar fashon to that descrbed n Secton 1 Source channels K LDPC encoder r 1 =k 1 / n LDPC encoder r K =k K / n Interleaver Kxn K QAM symbols Buffer Mapper + S/P converter (a) s & N-dmensonal OAM multplexer to FSO expandng telescope from FSO compressng telescope N-dmensonal OAM demux, photodetectors & P/S converter LPF LPF (b) ADC ADC APP Demapper Bt LLRs Calculaton LDPC Decoder 1 1 LDPC Decoder K K Fg 3 Deep-space optcal communcaton by orthogonal OAM dvson multplexng: (a) transmtter and (b) recever confguratons 4 Compensaton of OAM crosstalk ntroduced by atmospherc turbulence In ths secton, we descrbe several approaches to compensate for atmospherc turbulence effects OAM Channel Model Atmospherc turbulence s caused by varatons n the refractve ndex of the transmsson medum due to nhomogenetes n temperature and pressure caused by solar heatng and wnd The optcal turbulent channel has been ntensvely studed, and varous models have been proposed to descrbe turbulence-nduced performance degradaton [6] The OAM-based FSO channel model wth coherent detecton can be descrbed as follows: h11 h1 h1, N h1 h h, N, y Hx w, H, (14) h N,1 hn, hn, N where x = [x 1 x x N ] T s transmtted vector wth th complex number component beng the n-phase/quadrature (I/Q) sgnal constellaton pont transmtted usng th OAM state; y = [y 1 y y N ] T s the receved vector; w = [w 1 w w N ] T s the vector of nose samples; and H represents the channel matrx wth rank(h) = N that corresponds to the number of OAM states not experencng fadng (due to atmospherc turbulence) The channel matrx coeffcents can be determned by usng a tranng sequence The OAM FSO channel model (14) s smlar to a wreless communcaton MIMO model [5 9] and, therefore, dfferent technques used n spatal nterference cancellaton are drectly applcable here Below we descrbe equalzaton, OAM-tme codng, and adaptve modulaton and codng Other methods sutable to deal wth atmospherc turbulence nduced crosstalk among OAM states nclude: OAM dversty; lnear (C) 011 OSA 18 July 011 / Vol 19, No 15 / OPTICS EXPRESS 1485

10 nterfaces: zero-forcng nterface and mnmum mean-square error (MMSE) nterface; nonlnear nterfaces: vertcal and dagonal BLAST [9], teratve atmospherc turbulence cancellaton; parallel decomposton of OAM-MIMO by sngular value decomposton (SVD) of the channel matrx; and OAM-beam formng OAM crosstalk compensaton by equalzaton The scalar representaton of Eq (14), N y h x w ; 1,,, N (15), j j j1 ndcates that th OAM detector output can be represented as a fnte mpulse response (FIR) flter That s, the channel model (15) s smlar to an ntersymbol nterference (ISI) model so that varous channel equalzaton technques [6 30] can be used to compensate for OAM crosstalk ncludng a feed-forward equalzer (FFE), decson-feedback equalzer (DFE), maxmum-lkelhood sequence detector (MLSD), and a turbo equalzer OAM-tme codng If we allow the sgnal desgns to be extended over multple OAM states and over multple symbol tmes, we can refer to these desgns as the OAM-tme codes, n analogy to the spacetme codes used n wreless communcatons [5 9] Most space-tme codes desgned for wreless communcatons are straghtforwardly applcable here, but they are most probably not optmal Under the assumpton that the FSO channel s quas-statc for the duraton of N s symbols, whch s true for mult-ggabt transmsson, the OAM FSO channel nputs and outputs are matrces, wth dmensons correspondng to space-coordnate (the number of OAM modes) and tme-coordnate (the number of symbol ntervals) Ths can be modeled by where Y HX W, (16) Y y, y,, y ( Y ), X x, x,, x ( X ) 1 Ns j NNs 1 Ns j NNs W w1, w,, w ( W ) s N j NNs and where N s the number of OAM modes Let us observe an OAM-tme code n whch the recever has the knowledge of the channel matrx H Under maxmum-lkelhood (ML) detecton, the optmum transmt matrx s obtaned by the followng mnmzaton: Ns ˆ arg mn, NN s XX 1 X y Hx where the mnmzaton s performed over all possble OAM-tme nput matrces Adaptve modulaton and codng (AMC) There are many parameters that can be vared at the transmtter sde to accommodate FSO channel condtons, ncludng data rate, power, codng rate, and combnatons of dfferent adaptaton parameters In several of recent papers (eg, [31 33]), we have studed varous adaptve modulaton and codng schemes sutable for dealng wth atmospherc turbulence for dfferent multlevel modulaton schemes Ths study can be extended to coded OAM modulaton schemes 5 Performance analyss In Fg 4, we show the BER performance of (grth-1, column-weght-3) LDPC(1510,7560)- coded N-dmensonal (ND) OAM modulaton versus PPM On horzontal axs we use average (17) (18) (C) 011 OSA 18 July 011 / Vol 19, No 15 / OPTICS EXPRESS 1486

11 Bt-error rate, BER Bt-error rate, BER number of photons per dmenson (n db-scale) so that ND-OAM and PPM can be compared The results of smulatons shown n Fgs 4-6 are obtaned for Posson deep-space optcal channel model for the average number of background photons of 01 We assume that resdual crosstalk on a gven OAM mode s gven by n x = 01 From Fg 4 t s clear that as we ncrease the number of dmensons, the mprovement of ND-OAM over PPM s larger The mprovement (measured at BER of 10 6 ) ranges from 067 db for N = to 81 db for N = 8 Moreover, the spectral effcency S E of ND-OAM s N SE,ND-OAM log / 1/ Ts N (19) S log M / M / T log M / M E,PPM s tmes better than that of PPM In (19), wth T s we denoted the sgnalng nterval (recprocal of a symbol rate) For the same number of dmensons (M = N), the spectral effcency of ND- OAM s N /log N tmes better For example, for M = N = 8, the spectral effcency of ND- OAM s 1334 tmes better In Fgs 5-6, we evaluate performance of LDPC-coded ND-OAM n the presence of atmospherc turbulence, under medum (Fg 5) and strong (Fg 6) turbulence regmes (For defnton of Rytov standard devaton σ R see [6]) For N = 8, the degradaton n performance ranges from 0 db n medum turbulence regme to 87 db n strong turbulence regme (at BER of 10 6 ) ND-OAM, n x =01: PPM: N= N=4 N=8 M= M=4 M= Average number of photons per bt, n b Fg 4 LDPC(1510,7560)-coded ND-OAM modulaton versus PPM ND-OAM: N= N=4 N= R =1 n x = Average number of photons per bt, n b [db] Fg 5 LDPC(1510,7560)-coded ND-OAM modulaton n medum turbulence regme (σ R = 1) (C) 011 OSA 18 July 011 / Vol 19, No 15 / OPTICS EXPRESS 1487

12 Bt-error rate, BER Bt-error rate, BER ND-OAM: N= N=4 N= R =4 n x = Average number of photons per bt, n b [db] Fg 6 LDPC(1510,7560)-coded ND-OAM modulaton n strong turbulence regme (σ R = ) In Fg 7, we evaluate (grth-8, column-weght-4) LDPC(430,34)-coded ND-OAM scheme for use n near-earth applcatons Both drect and coherent detecton schemes are evaluated In smulatons, we assume that OAM crosstalk dstrbuton follows the Gaussan dstrbuton (the same assumpton was used n [10]), whch motves the use of sgnal-to-noseand-crosstalk rato (SCNR) nstead of sgnal-to-nose rato (SNR) Drect detecton schemes can operate properly n weak turbulence regme, whle n strong turbulence regme they exhbt error floor phenomena (not shown n fgure) and as such must be combned wth repetton MIMO schemes On the other hand, the coherent detecton schemes wth M = and N can operate even n strong turbulence regme for reasonable hgh SCNRs, as shown n Fg 7 R = LDPC-coded OAM (dr det): M=, N= M=, N=4 Back-to-back LDPC-coded OAM (dr det): M=, N= M=, N=4 M=, N=8 M=4, N= M=4, N= LDPC-coded OAM (coh det): R =10: M=, N= M=, N=4 R =0: M=, N= Back-to-back: M=, N= Sgnal-to-nose-and-crosstalk-rato, SNCR [db] Fg 7 BER performance of LDPC(430,34)-coded OAM modulaton based FSO systems n weak (σ R = 03), medum (σ R = 1) and strong (σ R = ) turbulence regmes 6 Concludng remarks In ths paper, we studed the use of OAM as an addtonal degree of freedom for modulaton and multplexng n deep-space and near-earth applcatons The followng mult-ggabt enablng approaches for deep-space communcaton based on OAM have been descrbed: OAM multplexng, N-dmensonal OAM modulaton, and orthogonal OAM dvson multplexng We demonstrate by Monte Carlo smulatons, over the atmospherc turbulence channels, that proposed schemes can operate under strong atmospherc turbulence and provde dramatc mprovement n throughput We have shown that the mprovement of ND-OAM (C) 011 OSA 18 July 011 / Vol 19, No 15 / OPTICS EXPRESS 1488

13 over PPM, for the same number of dmensons, (measured at BER of 10 6 ) ranges from 067dB for N = to 81 db for N = 8 Moreover, the spectral effcency, for the same number of dmensons, of ND-OAM s N /log N tmes better The proposed OAM-based schemes, therefore, represent the excellent canddates for next generaton deep-space and near-earth optcal communcatons In addton to next generaton deep-space/near-earth optcal communcaton enablng technologes, we have descrbed dfferent approaches to deal wth atmospherc turbulence nduced OAM crosstalk ncludng equalzaton, OAM-tme codng, and adaptve modulaton and codng Notce there s a sgnfcant effort to mprove the result from [] n terms comng closer to the channel capacty lmts Some researchers are tryng to mprove ths result even for a fracton of db Wth the proposed scheme, however, we are able to outperform LDPC coded PPM for even 81 db for N = 8, and at the same tme mprove spectral effcency 1 tmes (also for N = 8), ndcatng that the proposed scheme s a promsng canddate for next generaton deep space and near-earth optcal communcatons Some other applcatons of nterest of OAM-based systems descrbed n ths paper nclude: provdng ultra-hgh-speed Internet connecton, ncreasng data rate and reducng system cost and deployment tme n access networks, enablng ground-to-satellte/satellte-to-ground FSO communcatons at hgh-speed, enablng ultrahgh-speed ntersatellte communcatons, and possbly enablng arcraft-to-satellte/ satellte-to-arcraft hgh-data-rate communcatons Acknowledgments Ths work was supported n part by the Natonal Scence Foundaton (NSF) under Grants CCF and ECCS (C) 011 OSA 18 July 011 / Vol 19, No 15 / OPTICS EXPRESS 1489