Phase Diffuser at the Transmitter for Laserom Link: Effet of Partially Coherent Beam on the Bit-Error Rate. O. Korotkova* a, L. C. Andrews** a, R. L. Phillips*** b a Dept. of Mathematis, Univ. of Central Florida; b Florida Spae nstitute ABSTRACT By using a omplex phase sreen model for a diffuser loated at the transmitter, analyti expressions are developed for the sintillation index of a lowest order aussian-beam wave in the pupil plane of the reeiver in weak and strong atmospheri onditions. The effet of partial oherene on the sintillation index is analyzed as a funtion of the propagation distane and the orrelation length of the diffuser. The redution in the sintillation level (due to the transmitter aperture averaging effet) is shown under all atmospheri onditions. The signal to noise ratio and the bit error rates (for OOK signal modulation) are disussed.. NTRODUCTON The interest in the (spatially) partially oherent beam as a tool improving the performane of the laser ommuniation systems was indiated reently in a number of publiations [-4]. Although the theoretial foundations of partial oherene in free spae were established in early 6s [5, 6] there is still a lak of knowledge about the behavior of partially oherent beams in the atmospheri turbulene. During the 7s and early 8s the expressions for the seond and the fourth order statistis (sintillation index, in partiular) of partially oherent aussian beam field in the atmosphere were established by several authors in weak and saturation regimes [7-]. However, only reently, based on the theory of optial sintillation for the oherent beams [], the authors suggested the model for partially oherent beam valid for all atmospheri onditions (inluding the fousing regime) [3,4]. Besides the fat that the model does not have restritions on the atmospheri spetrum, strength of turbulene and the degree of oherene, it also proves to be simple enough (in notieable differene with other models) to be used for the analysis of optial system performane. Sine [] was published, it was well understood that a partially oherent soure produes transmitter aperture averaging whih an be used as an additional effet to the (ommonly used) reeiver aperture averaging. Therefore, hanging the degree of spatial oherene (that is usually done by plaing different diffusers over the transmitter aperture) there is a way to ontrol (to some extent) the size of the olleting aperture, whih is ritial for the ost and flexibility of the system. However, only a few attempts were reently made to alulate the bit-error rates [, 5] of a partially oherent aussian beam propagating through the atmosphere. These studies were based on the quadrati approximation for the atmospheri spetrum, whih ould lead to the overestimation of atual performane level. We believe that with our model [3,4] one an predit the bit-error rate (as well as the probability of fades) of ertain ommuniation link with better auray. One of the major drawbaks of partially oherent beams is the additional broadening as ompared with a perfetly oherent beam, resulting in power loss at the reeiver. This leads to the optimization problem for the signal to noise ratio with onstraint, defining the optimal regime (the relation between the degree of oherene and transmitted power) for the operating system.
Diffuser Atmosphere aussian lens. MODEL A shemati diagram for the propagation of a partially oherent beam is shown in Figure [3]. We assume the transmitted beam wave is a TEM aussian-beam wave haraterized by parameters Detetor L L Laser Θ =, =, (.) F kw L L f π where k = (m - ) is the laser wave number (λ is hosen to be λ µm for all alulations), L(m) is the propagation distane to the Figure. Propagation of partially oherent beam olleting lens, F (m) is the phase front radius of urvature, and W (m) is the laser exit aperture radius. For illustrative purposes in all alulations for this paper we use the ollimated beam (F = ) with W =.5m. The diffuser is modeled as a thin omplex phase sreen with a aussian spetrum model [6] 3 n l ( ) = Φs κ exp l κ (.) 8π π 4 where κ (m - ) is the atmospheri wave number, l (m) is the lateral orrelation radius diretly related to the variane the aussian phase sreen also used in the literature [], by g g of l =, (.3) n model (.), <n > is the flutuation in the index of refration indued by the sreen. We also introdue the nondimensional quantity L q =, (.4) kl relating the strength of the diffuser (sreen) to first Fresnel zone. The effet of the diffuser is to reate a new aussian beam that an be haraterized by an effetive set of beam parameters. Following [4], we introdue parameters e, Θ e for the partially oherent beam (of radius W e and phase front radius of urvature F e ) inident on the olleting lens: e N s =, + 4q Θ e Θ = + 4q (.5) where and Θ are the orresponding pupil plane parameters of a perfetly oherent beam given by and = Θ +, Θ Θ = (.6) Θ + 4q N s = + (.7)
is the number of spekle ells in the transverse plane of the transmitted beam (the measure of the diffuser strength relative to the initial beam size). The olleting aussian lens is desribed by the soft aperture radius W and foal length F. The nondimensional L lens parameter Ω = will be used as well. kw 3. SCNTLLATON NDEX. FLUX VARANCE. n the weak flutuation regime in the pupil plane of the reeiver, the on-axis sintillation index of partially oherent aussian-beam was alulated in [4]: 5 / 6 5 7 5 / 6 B ( L, ) = 3.86 Re i F, ; ; Θe + i e e, (3.) 6 6 6 6 where is the Rytov variane, i is the imaginary unit, F is the Hypergeometri funtion, Re stands for the real part of the argument. Following [] the flux variane of the irradiane flutuations at the detetor plane (valid for all atmospheri onditions) is where [ ] ( L L, Ω ) = exp ( L + L, Ω ) + ( L + L, Ω ), (3.) + f ln x f ln y f ln x is the flux variane assoiated with large-sale flutuations given by Ω e η x ln x ( L + L f, Ω ) =.49 R.4 ( + Θ ) Ω + e η x e + e + Ω, (3.3) R = ( Θ ) ( ) e + Θe. 3 5 7 / 6 The quantity η x in (3.3) is the normalized large-sale utoff frequeny determined by the asymptoti behavior of ln x in weak turbulene and saturation regime []: η x R = 6/ 7 ( / ) B +. 56 / 7 / 5 B. (3.4) The small-sale flux variane defined by ln y ( L + L,Ω ) f 5/ 6 y 7. η =. 4η y + + Ω ln y in (3.3) is, (3.5) Fig. Flux variane ( L + L f, Ω ) as a funtion of the radius of the olleting lens W for oherent and partially oherent beams in weak turbulene. where the orresponding utoff frequeny is
/ 5 / 5 ( / ) (. ) η y = 3 B + 69 B (3.6) Fig. 3 Flux variane ( L + L f, Ω ) as a funtion of the radius of the olleting lens W for oherent and partially oherent beams in fousing regime. Fig. 4 Flux variane ( L + L f, Ω ) as a funtion of the radius of the olleting lens W for oherent and partially oherent beams in strong turbulene. Fig. shows the flux variane (3.) vs. W for a perfetly oherent beam (solid urve) and partially oherent beams: with l =. m (dashed urve) and with l =. m (dotted urve). The propagation distane L = m, 4 / 3 C n = m, leading to the Rytov variane =. 33. We see that in weak flutuation regime there is a signifiant redution of the flux, espeially for a point aperture W =. With the inrease of the olleting aperture size this effet dereases. n the analysis of the signal-to-noise ratio and the BER below we hoose the olleting aperture radius W =. m to onentrate primarily on the averaging effet due to the transmitter. Similar urves are generated in moderate (. 5 ) in Fig. 3 and in strong ( 6. 98 ) turbulene in Fig. 4, but as the strength of turbulene grows = = the absolute redution of flux beomes less. The aperture averaging fator of a partially oherent beam (AP) an be defined similarly to that of a perfetly oherent beam (AC), namely [6] ( L + L f,ω,q ) ( L + L,,q ) AP = The dependene of AP on q is inluded in the formula above expliitly to show the ontrast between AP and AC. There are two phenomena ontributing to AP: aperture averaging by the olleting lens and transmitter aperture averaging due to partial oherene. f (3.7) 4. SNAL TO NOSE RATO. n free spae the signal to noise ratio SNRP of a partially oherent beam an be defined similarly to that of perfetly oherent beam []: i s SNRP =, (4.) N
where i s is the mean value of the reeived signal urrent (proportional to the transmitted power) and N is the variane of the detetor noise. Sine partially oherent beams have greater divergene the reeived power depends upon the degree of oherene of the wave as well, i.e. more power is required for less oherent beams to sustain the same signal to noise ratio as the perfetly oherent beam (of the same size and phase front radius of urvature) produes. Therefore the relation between SNRP and the orresponding free spae signal to noise ratio of the oherent beam SNRC is dedued from the beam size of partially oherent beam at the reeiver, derived in [3]: Fig. 5 <SNRP> (db) as a funtion of SNRC (db) in weak turbulene for several values of l.. Fig. 6 <SNRP> (db) as a funtion of SNRC (db) in fousing regime for several values of l. SNRP SNRC SNRC = =, (4.) PP /PC + 4 q where PP is the reeived power of the partially oherent beam and PC is the power of oherent beam. n the atmosphere the standard definition of SNR is adapted similarly: SNRP SNRC SNRC = = (4.3) PP /PP + SNRC + 4 q +.63 + SNRC / 5 Fig. 7 <SNRP> (db) as a funtion of SNRC (db) in strong turbulene for several values of l. where brakets are used for averaging. n Fig. 5-7 we show the signal to noise ratio of the partially oherent beam <SNRP> (db) as a funtion of SNRC (db) in different atmospheri onditions. We note that in all regimes the atmospheri SNR of a perfetly oherent beam (dashed-and-dotted urves everywhere) is always below -db, whih orresponds to onventionally exepted BER of the operating system (on the order of -9 ). n weak flutuation regime (Fig. 5) and in fousing regime (Fig. 6) the use of the partial oherene provides a notieable improvement of the atmospheri SNR, namely in these regimes with SNRC in the range 4-5 (db) it is possible to obtain <SNRP> (db) exeeding -db. We note that the saturation effet of <SNRP> is taking plae for any strength of the diffuser and that of the turbulene (after some ertain level of <SNRP> is attained the inrease of
power does not produe any additional effet). Therefore, for partiular ommuniation link and fixed atmospheri regime there exists the optimal diffuser. For example, in Fig. 5 for SNRC =3dB the best hoie is the diffuser with l =. m (dotted urve) whih orresponds to <SNRP> = 5(dB). n Fig. 6 (fousing regime) the stronger diffuser together with greater amount of power are required to reah the BER= -9 ; here, for the fixed SNRC =3dB the diffuser with l =. m (dotted urve) would not provide with suffiient level of BER; for SNRC =5dB the diffuser with l =. m (solid urve) should be used. Based on Fig. 7 (strong turbulene) we note that no matter how strong the diffuser, the required level of BER annot be attained only with use of the partial oherene. n this regime only the ombination of different tools (multiaperture reeivers, adaptive optis, partial oherene) would be effiient for the mitigation the atmospheri effets. 5. BT-ERROR RATES. Following [] the probability of error (BER) for the aussian beam (OOK modulation sheme) in the atmospheri turbulene is based on the amma-amma probability density funtion of the intensity flutuations: ( ) ( α+ β ) / α+ β αβ SNRP x BER = erf x Kα β ( xaβ )dx ( α) ( β) Γ Γ (5.) where <SNRP> is defined in (4.3), K(x) is K-Bessel funtion, erf(x) is the omplementary error funtion, α and β are two parameters of the distribution, related orrespondingly to the large-sale x and the small-sale y intensity flutuations: α =, exp( ) ln x β = (5.) exp( ) ln y where ln x and ln y are given by (3.3) and (3.5). Fig. 8 The BER of partially oherent beams (logarithmi sale) as a funtion of <SNRP>(dB) in weak turbulene for oherent and partially oherent beams. Fig. 9 The BER of partially oherent beams (logarithmi sale) as a funtion of <SNRP>(dB) in fousing regime for oherent and partially oherent beams
Although the probability (5.) is valid in all atmospheri onditions we onentrate primarily on weak and moderate regimes where the partial oherene is proving to be very effiient. Using the onventional way to display the system performane, we plot BER (5.) versus <SNRP>dB (see Fig. 8) for two beams: perfetly oherent (dashed urve) and partially oherent, with l =. m (solid urve) in weak turbulene ( =. 33 ). The later diffuser is atually on the order of the optimal diffuser for this partiular senario (refer to Fig. 5). f the transmitted power of the operating system is fixed (an not be adjusted) the partially oherent beam an improve the BER up to the several orders of magnitude over the distanes under m and values of Cn on the order of -4 m -/3, whih orresponds to Rytov variane (for example, in Fig. 8). This inrease of Fig. The BER of partially oherent beams (logarithmi sale) as a funtion of <SNRP>(dB) in strong turbulene for oherent and partially oherent beams < BER is shown here to be ritial (BER attains the value -9 at <SNRP>=5dB while perfetly oherent beam an not produe this BER with any transmitted power). See Fig. 5 to find the orresponding values of the free-spae signal to noise ratio SNRC for eah beam. On the other hand if a ertain system performane is needed (BER should be set up permanently to some level) then the diffuser an be also used as a tool for the redution of the transmitting power. n Fig. 9 the BER of perfetly oherent beam (dashed urve) and partially oherent beam with l =. m (solid urve) are plotted in fousing regime (. 5 ). This hoie of the diffuser was disussed in Fig. 6. For <SNRP> = 5(dB) the BER = -9, but it requires very high level of input power (SNRC should be at least 5dB). n Fig. the BER of the same two beams as in Fig. 9 are displayed vs. <SNRP> in strong turbulene ( = 6. 98 ). n this ase partial oherene still redues the BER but by the amount whih is not suffiient for a good system performane. With the derease of orrelation distane l parameter α given by (5.) of the amma-amma distribution grows rapidly whih might ause some diffiulties in numerial integration of (5.). n suh ases single amma distribution might be used instead; hene the BER (5.) an be replaed by = γ γ BER = erf () Γ γ SNRP x x γ exp( γx)dx (5.3) where γ is the parameter of the distribution, related to the flux-variane (3.) by γ = ( L +, Ω ) L f (5.4) The integral (5.3) predits slightly higher levels of the BER ompared with that of (5.). The solid urve in Fig. 9 was generated with the help of this simplifiation.
6. CONCLUDN REMARKS n this paper we made the attempt to alulate the main harateristis of performane of the ommuniation system, (inluding the bit-error rates) using partially oherent beam in various atmospheri onditions. enerally partial oherene of the beam ontrols the part of the flux variane, related to large-sale flutuations, namely, ln x ( L + L f, Ω ) as l. n weak flutuation regime the ondition ln x ( L + L f, Ω ).5 ( L + L f, Ω ) holds, leading to the signifiant effet of partial oherene (strength of the diffuser). On the other hand, in strong regime we have ln x ( L + L f, Ω ) << ( L + L f, Ω ), therefore the effet of the diffuser on the total flux variane is less. This leads to the orresponding effets on the signal to noise ratio and the BER. As was noted in Setion 3 of this paper, the aperture averaging fator AP of the beam depends on the degree of oherene of the transmitted beam as well as the olleting aperture size. Hene, partial oherene of the soure (realizable by plaing a relatively light and heap diffuser over the laser exit aperture) an be used instead of more expensive and umbersome olleting lens, espeially in ases when the transmitter an be stationary but the reeiver system has to be mobile. 7. REFERENCES. J. C. Riklin and F. M. Davidson, Atmospheri turbulene effets on a partially oherent aussian Beam: impliations for free-spae laser ommuniation, J. Opt. So. Am. A 9, 794-8 ()... bur, E. Wolf Spreading of partially oherent beams in random media (JOSA A, Vol.9, 8, ). 3. Sidorovih et al. Mitigation of aberration in a beam-shaping telesope and optial inhomogeinity in a free-spae optial path using an extended light soure oupled to the telesope, Pro. SPE Vol.4635, 79-9 (). 4. T. Shirai, A. Dogariu, E. Wolf, Diretionality of some model beams propagating in atmospheri turbulene (Optis Letters, submitted). 5. L. Mandel, E.Wolf, Optial Coherene and Quantum Optis (Cambridge University Press, Cambridge, 995). 6. A. C. Shell, The Multiple Plate Antenna (Dotoral Dissertation, Massahusetts nstitute of Tehnology, Cambridge, Massahusetts, 96). 7. A.. Kon and V.. Tatarskii, On the theory of the propagation of partially oherent light beams in a turbulent atmosphere, Radiophys. Quantum Eletron. 5, 87-9 (97). 8. J. Carl Leader, ntensity flutuations resulting from a spatially partially oherent light propagating through atmospheri turbulene, J. Opt. So. Am. 69, 73 84 (979). 9. S. C. H. Wang and M. A. Plonus, Optial beam propagation for a partially oherent soure in the turbulent atmosphere, J. Opt. So. Am. 69, 97-34 (979).. L. Fante ntensity flutuations of an optial wave in a turbulent medium, effet of soure oherene, Opt. Ata 8, 3-7 (98).. V. A. Banakh, V. M. Buldakov, and V. L. Mironov, ntensity flutuations of a partially oherent light beam in a turbulent atmosphere, Opt. Spektrosk. 54, 54-59 (983).. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Sintillation with Appliations (SPE Press, Bellingham, ). 3. O. Korotkova, L. C. Andrews, R. L. Phillips Spekle propagation through atmosphere: effets of a random phase sreen at the soure, Pro. SPE Vol. 48 (). 4. O. Korotkova, L. C. Andrews, R. L. Phillips Seond and fourth order statistis of a partially oherent beam in atmospheri turbulene, JOSA A (submitted). 5. J. C. Riklin, Free-Spae Laser Communiation Using a Partially oherent Soure Beam (Dotoral Dissertation, The John Hopkins University, Baltimore, Maryland, ). 6. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPE Press, Bellingham, 998). * olga_korotkova@hotmail.om; ** landrews@pegasus..uf.edu; *** phillips@mail.uf.edu