Performance of synchronous optical receivers using atmospheric compensation techniques

Transcription

1 Pefomance of synchonous optical eceives using atmospheic compensation techniques Aniceto Belmonte, and Joseph. Kahn. Technical Univesity of Catalonia, Depatment of ignal Theoy and Communications, Bacelona, pain. tanfod Univesity, Depatment of Electical Engineeing, tanfod, CA 94305, UA Abstact: We model the impact of atmospheic tubulence-induced phase and amplitude fluctuations on fee-space optical links using synchonous detection. We deive exact expessions fo the pobability density function of the signal-to-noise atio in the pesence of tubulence. We conside the effects of log-nomal amplitude fluctuations and Gaussian phase fluctuations, in addition to local oscillato shot noise, fo both passive eceives and those employing active modal compensation of wave-font phase distotion. We compute eo pobabilities fo -ay phase-shift keying, and evaluate the impact of vaious paametes, including the atio of eceive apetue diamete to the wave-font coheence diamete, and the numbe of modes compensated. 008 Optical ociety of Ameica OCI codes: (00.330) Atmospheic tubulence; ( ) tatistical optics; (00.080) Adaptive optics; ( ) Optical communications; ( ) Lida. Refeences and links. D. L. Fied, "Optical heteodyne detection of an atmospheically distoted signal wave font," Poc. IEEE 55, (967).. D. L. Fied, Atmospheic modulation noise in an optical heteodyne eceive, IEEE J. Quantum Electon. QE-3, 3- (967). 3. J. H. Chunside and C.. cintye, "ignal cuent pobability distibution fo optical heteodyne eceives in the tubulent atmosphee. : Theoy," Appl. Opt. 7, 4-47 (978). 4. J. H. Chunside and C.. cintye, "Heteodyne eceives fo atmospheic optical communications," Appl. Opt. 9, (980). 5. K. A. Winick, "Atmospheic tubulence-induced signal fades on optical heteodyne communication links," Appl. Opt. 5, (986). 6.. Pelot, "Tubulence-induced fading pobability in coheent optical communication though the atmosphee," Appl. Opt. 46, (007). 7. A. Belmonte, "Influence of atmospheic phase compensation on optical heteodyne powe measuements," Opt. Expess 6, (008). 8. E. Ip, A. P. T. Lau, D. J. F. Baos, and J.. Kahn, "Coheent detection in optical fibe systems," Opt. Expess 6, (008). 9.. P. Cagigal and V. F. Canales, "peckle statistics in patially coected wave fonts," Opt. Lett. 3, (998). 0. J. W. Goodman, peckle Phenomena in Optics. Theoy and Applications (Ben Robets & Company, 007).. R. J. oll, Zenike polynomials and atmospheic tubulence, J. Opt. oc. Am. 66, 07- (976)... Bon and E. Wolf, Pinciples of Optics (Cambidge Univesity Pess, 999) 3. J. W. tohbehn, T. Wang, and J. P. peck, On the pobability distibution of line-of-sight fluctuations of optical signals, Radio cience 0, (975). 4. R. F. Pawula,. O. Rice, and J. H. Robets, Distibution of the phase angle between two vectos petubed by Gaussian noise, IEEE Tans. Commun. CO-30, (98). 5. L. C. Andews, R. L. Phillips, and C. Y. Hopen, Lase Beam cintillation with Applications (PIE Pess, 00).. Intoduction Evaluating the pefomance of a heteodyne o homodyne eceive in the pesence of atmospheic tubulence is geneally difficult because of the complex ways tubulence affects the coheence of the eceived signal that is to be mixed with the local oscillato. The # $5.00 UD Received 7 Jun 008; evised 3 Jul 008; accepted 0 Aug 008; published 6 Aug 008 (C) 008 OA eptembe 008 / Vol. 6, o. 8 / OPTIC EXPRE 45

2 downconveted heteodyne o homodyne powe is maximized when the spatial field of the eceived signal matches that of the local oscillato. Any mismatch of the amplitudes and phases of the two fields will esult in a loss in downconveted powe. Adaptive compensation of atmospheic wave-font phase distotion to impove the pefomance of atmospheic systems has been an impotant field of study fo many yeas. In paticula, modal compensation method involves coection of seveal modes of an expansion of the total phase distotion in a set of basis functions. Hee, we study in a unified famewok the effects of both wavefont distotion and amplitude scintillation on the pefomance of synchonous (coheent) eceives utilizing wavefont compensation. The effects ascibed to tubulence ae andom and subsequently must be descibed in a statistical sense. Ealy woks quantified tubulence-induced fading though its mean and vaiance [,], although these ae not adequate to fully chaacteize system pefomance. Late analyses have attempted to ovecome these limitations and fully chaacteize the statistics of heteodyne optical systems by assuming a highly simplified model of atmospheic effects [3,4]. An altenate appoach, aimed at ovecoming the limitations of pevious wok, is based on numeical simulation of heteodyne optical systems [5,6]. Unfotunately, none of these pio woks have esulted in an accuate statistical desciption of the pefomance of phase-compensated homodyne o heteodyne systems. Recently, a full-wave simulation of beam popagation has been used to examine the uncetainty inheent to the pocess of optical powe measuement with a pactical heteodyne eceive because of the pesence of efactive tubulence [7]. The emainde of this pape is oganized as follows. In ection, we define a mathematical model fo the eceived signal afte popagation though the atmosphee. By noting that the downconveted signal cuent can be chaacteized as the sum of many contibutions fom diffeent coheent egions within the apetue, we show that the pobability density function (PDF) of this cuent can be well-appoximated by a modified Rice distibution. In ou model, the paametes descibing the PDF depend on the tubulence conditions and the degee of modal compensation applied in the eceive. In ection 3, we compute the eo pobability fo QPK modulation in the pesence of multiplicative noise fom atmospheic tubulence and additive white Gaussian noise (AWG). We povide analytical expessions fo the eo pobability of synchonous communication systems, and use them to study the effect of vaious paametes on pefomance, including tubulence level, signal stength, eceive apetue size, and the extent of wavefont compensation.. Fist-ode statistics in optical homodyne o heteodyne detection When a eceived signal expeiences atmospheic tubulence duing tansmission, both its envelope and its phase fluctuate ove time. In the case of coheent modulation, phase fluctuations can seveely degade pefomance unless measues ae taken to compensate fo them at the eceive. Hee, we assume that afte homodyne o heteodyne downconvesion is used to obtain an electical signal, the eceive is able to tack any phase fluctuations caused by tubulence (as well as those caused by lase phase noise), pefoming ideal coheent (synchonous) demodulation. Unde this assumption, analyzing the eceive pefomance equies knowledge of only the envelope statistics of the downconveted electical signal. In a coheent eceive, downconvesion fom the optical domain to the electical domain can be achieved using eithe heteodyne o homodyne methods [8]. The pefomance compaison between homodyne and heteodyne depends on the modulation scheme being consideed. This pape analyzes QPK modulation with synchonous homodyne o heteodyne detection, assuming the dominant noise is local oscillato shot noise. Fo conceteness, equations ae given fo the heteodyne case, but the signal-to-noise atio (R) and eo-ate pefomance ae the same fo the homodyne case. We assume the heteodyne downconvete uses a beamsplitte and a pai of photodetectos, compising a balanced eceive. (Using BPK modulation, it is possible to employ a single-quadatue downconvete fo homodyne but not heteodyne. Hence, fo BPK, homodyne can achieve a # $5.00 UD Received 7 Jun 008; evised 3 Jul 008; accepted 0 Aug 008; published 6 Aug 008 (C) 008 OA eptembe 008 / Vol. 6, o. 8 / OPTIC EXPRE 45

3 given eo ate at a 3-dB lowe signal powe than heteodyne in the shot-noise-limited egime.) In ode to assess the impact of tubulence, both log-amplitude and phase fluctuations should be consideed. As a consequence, the field in the pupil plane is expessed as A= A exp χ( ) jφ, () whee A s is the amplitude without the effect of tubulence, and χ() and φ() epesent the logamplitude fluctuations (scintillation) and phase vaiations (abeations), espectively, intoduced by atmospheic tubulence. In the heteodyne downconvete, the infomationcaying photocuent i s at the output of the balanced eceive is: ( ) ( ) [ ] ( ) x, () i = ηa0 A d W exp χ( ) cos Δ f t+δφ φ whee η is the quantum efficiency of the photodetecto, A 0 is the amplitude of the local oscillato, and f and φ ae, espectively, the diffeences between the fequencies and phases of the signal and local oscillato. The cicula eceiving apetue of diamete D is defined by the apetue function W(), which equals unity fo D/, and equals zeo fo >D/. We cos u v = cos ucosv+ sin usin v, obtaining can ewite the cosine using ( ) { [ ] ( ) ( ) ( ) [ ft φ] dw ( ) χ( ) φ ( ) } i = ηa0 A cos Δ f t+δ φ d W exp χ cos φ + + sin Δ +Δ exp sin. Taking the time aveage of the beat fequency f oscillations of Eq. (3), the aveage detected (3) signal powe i may be witten as i = η D A0 A + i 4, (4) whee and i ae nomalized vesions of the integals in Eq. (3): = D ( ) exp ( ) cos ( ) 4 dw χ φ i = D d W ( ) exp χ( ) sin φ( ). 4 In this last step, the tigonometic and exponential tems have been consideed timeindependent, as the time aveage is taken ove a peiod that is shot compaed with the coheence time associated with atmospheic tubulence. It is impotant to note that and i epesent integals ove the collecting apetue of the eal and imaginay pats, espectively, of the nomalized optical field eaching the eceive. These eal and imaginay pats can be consideed as the components of a complex andom phaso. If the noise is dominated by local oscillato shot noise, the aveage noise powe pe unit bandwidth is i 4 = eη D A0, whee e is the electonic chage. The R pe unit bandwidth γ = i i can be expessed using Eq. (4) as η γ = D A. (6) e 4 ote that the R in the pesence of tubulence, γ, is popotional to the R in the absence of tubulence, γ0 = ( η e) 4 D A : (5) # $5.00 UD Received 7 Jun 008; evised 3 Jul 008; accepted 0 Aug 008; published 6 Aug 008 (C) 008 OA eptembe 008 / Vol. 6, o. 8 / OPTIC EXPRE 453

4 γ γ = 0. (7) The constant of popotionality is = + i, a andom scale facto epesenting the effect of both the amplitude and phase fluctuations of the optical field. The statistical popeties of the andom vaiable, with mean p, povide a statistical and PDF ( ) chaacteization of the R γ. Using the aveage R γ = γ0 and the Jacobian of the tansfomation = γ γ = γ γ, we obtain the PDF of the R: 0 p γ = γ. (8) γ γ ( γ ) p Tuning attention to, we study how amplitude and phase fluctuations of the optical field define the statistics of the fading intensity = + i. We note that the two andom magnitudes and i ae expessed in Eq. (5) as integals ove the apetue and, hence, ae the sums of contibutions fom each point in the apetue. In ode to poceed with the analysis, we conside a statistical model in which these continuous integals ae expessed as finite sums ove statistically independent cells in the apetue: exp χ cosφ k k k = exp χ sin φ, i k k k = whee χ k is the log-amplitude and φ k is the phase of the kth statistically independent cell. A simila appoach has been used to analyze the statistics of the tehl atio [9]. The mean logamplitude can be extacted out of the sums, yielding exp χ exp χ χ cos φ ( ) k k k = exp χ exp χ χ sin φ. ( ) i k k k = Unde the assumption that, the numbe of independent cells, is lage enough, we can conside that and i asymptotically appoach jointly nomal andom vaiables: ( ) ( ) i i, (, ) exp exp i i = σ σ i σ σ i p (9) (0), () whee, i and σ, σ i ae the means and vaiances of, i, which ae equied to evaluate the joint PDF. To estimate these means and vaiances, we ecall that and i can be consideed as the eal and imaginay pats of a andom phaso. Hence, it is possible to evaluate the means and vaiances of and i by using the classical statistical model fo speckle with a non-unifom distibution of phases [0]. Afte some algebaic manipulation, mean values can be obtained as = exp χ exp ( χk χ) φ () + φ ( ) j i = exp χ exp( χk χ) φ () φ ( ), and the vaiances ae given by () # $5.00 UD Received 7 Jun 008; evised 3 Jul 008; accepted 0 Aug 008; published 6 Aug 008 (C) 008 OA eptembe 008 / Vol. 6, o. 8 / OPTIC EXPRE 454

5 ( ) expχ exp χk χ σ = + φ + φ 4 ( ) ( ) ( ) exp( k ) φ () φ ( ) φ () φ ( ) expχ χ χ + +, 4 expχ exp χk χ σ i = φ φ 4 ( ) ( ) exp( k ) φ () φ ( ) φ () φ ( ) expχ χ χ. 4 (3) φ (ω) is the chaacteistic function of the phase, i.e., the Fouie tansfom of its PDF. We point out that because, i esult fom atmospheic tubulence, we can conside phases φ k that obey zeo-mean Gaussian statistics: pφ ( φ ) = σφ exp( φ σφ ). In this case, the chaacteistic function of the phase is: φ ( ω) σ ω φ = exp. In [], the statistics of phase abeations caused by atmospheic tubulence wee chaacteized, consideing a Kolmogoov spectum of tubulence. In that analysis, classical esults fo the phase vaiance σ φ wee extended to conside modal compensation of atmospheic phase distotion. In such modal compensation, Zenike polynomials ae widely used as basis functions because of thei simple analytical expessions and thei coespondence to classical abeations []. It is known that the esidual phase vaiance afte modal compensation of J Zenike tems is given by σ φ D = CJ (4), (5) whee the apetue diamete D is nomalized by the wavefont coheence diamete 0, which descibes the spatial coelation of phase fluctuations in the eceive plane []. In (5), the coefficient C J depends on J []. Fo example, abeations up to tilt, astigmatism, coma and fifth-ode coespond to J = 3, 6, 0 and 0, espectively. Ideally, it is desiable to choose J lage enough that the esidual vaiance Eq. (5) becomes negligible. If it is also assumed that the log-amplitudes χ k ae nomal andom vaiables [3], we can use enegy consevation, and the expessions fo the mean of exponential functions of Gaussian vaiables, to obtain classical esults fo the log-amplitude and amplitude means: The iadiance β exp ( χ χ) χ = σ χ χ χ = σ ( ) exp k exp χ. k has a mean given by exp( σ χ ) vaiance σ χ is often expessed as a scintillation index σ β =exp(4σ χ ). ubstitution of Eq. (4) and Eq. (6) into Eq. () and Eq. (3) yields (6). The log-amplitude # $5.00 UD Received 7 Jun 008; evised 3 Jul 008; accepted 0 Aug 008; published 6 Aug 008 (C) 008 OA eptembe 008 / Vol. 6, o. 8 / OPTIC EXPRE 455

6 = exp σ χ exp σφ i = 0 σ = exp σ exp σ exp σ + σi = exp( σφ ). ( φ ) ( χ) ( φ ) At this point, we still need to detemine the numbe of statistically independent patches o cells pesent in the apetue. An analytical expession to estimate can be defined by = ( ) ( ) (7) d W C, (8) whee W() again chaacteizes the collecting apetue with aea = (/4)D. Hee, C() is the coheence function descibing the wavefont distotion intoduced by atmospheic tubulence [] 5/3 C ( ) = exp (9) 0 Physical insight into Eq. (8) may be obtained by consideing the limiting case in which the eceive apetue is much geate than the coheence diamete 0, i.e., D» 0. In this case, 5/3 8 = d exp 6.88 D. (0) 0 0 The integal can be solved in a closed fom to obtain = [.007 ( 0 /D) ]. To a good appoximation, the apetue can be consideed to consist of (D/ 0 ) independent cells, each of diamete 0. In the opposite exteme of an apetue much smalle than the coheence diamete, D«0, we have C() and =. This esult indicates that as the apetue gets smalle, the numbe of cells appoaches unity. Values of < ae not possible. An analytical expession fo valid fo all apetue diametes is given by which can easily be integated to yield D 5/3 8 = d exp 6.88 D, () 0 0 5/3 0 6 D D 5 0 =.09 Γ,.08. () Hee, Γ(a,x) is the lowe incomplete gamma function. Equation () will be used to estimate the value of needed to evaluate Eq. (7). Tuning again to the distibution function of inteest, a joint PDF of intensity and phase θ can be found by substituting a = cosθ, a i = sinθ into Eq. (), and multiplying the esulting expession by the Jacobian of the tansfomation, /. To obtain the maginal PDF of the fading intensity, we integate with espect to θ: # $5.00 UD Received 7 Jun 008; evised 3 Jul 008; accepted 0 Aug 008; published 6 Aug 008 (C) 008 OA eptembe 008 / Vol. 6, o. 8 / OPTIC EXPRE 456

7 p ( ) ( cosθ ) ( sinθ) dθ exp exp i i = 4 σ σ σ σ. (3) Unfotunately, the integal in Eq. (3) cannot be pefomed in closed fom. ince and i ae jointly nomal andom vaiables, it is possible to obtain the mean and vaiance of the intensity 4 4 fading as σ σ σ = σ + σ + 4σ. It is instuctive to conside the = + i + and ( i ) weak-tubulence, nea-field egime, whee values of σ φ and σ χ ae small but the logamplitude vaiance σ χ can be neglected in compaison to the effect of phase abeations. Fom Eq. (7), we obtain σ = 0, = σφ, and σi = σ φ. In this case, the fading intensity is defined as the sum of a constant (coheent) tem with amplitude and a andom (incoheent) tem i with zeo mean and vaiance σ i. uch a andom vaiable, defined as the sum of a known dominant phaso plus a andom phaso sum, is chaacteized by a Rice PDF [0]: p ( ) + a a exp σ σ, (4) σ = I 0 whee a = and σ = σ i. The paamete a epesents the coheent intensity that dominates ove the fluctuating esidual halo, whose intensity is epesented by σ. It is convenient to expess the basic Rice PDF (4) in tems of the mean = σ + a and the contast paamete σ / a, a measue of the stength of the esidual halo to the coheent component: ( + ) ( + ) + p ( ) = exp( ) exp I 0, (5) which is often efeed to as the modified Rician PDF. The vaiance of the intensity fading σ can be expessed in tems of mean value and the paamete as σ ( ) = + ( + ). It is possible to extend this convenient Rice distibution to stonge tubulence egimes, in which both the phase abeations σ φ and the log-amplitude vaiance σ χ need to be consideed. We can expect the geneal maginal PDF Eq. (3) to behave like the Rice PDF Eq. (5) povided that we can find a set of equivalent Rice paametes ( and ) that makes Eq. (7) and Eq. (9) have identical mean and vaiance. Compaing the means and vaiances of the two distibutions, the equied values of and can be computed as functions of, σ, and σ i though the elations = σ + σi + (6) 4 4 ( + ) ( + ) = ( σ + σi ) + 4 σ. Afte some algeba, the contast paamete / can be obtained fom Eq. (6) as σ + σi + a 4 + a ( σi σ ) ( σi σ ) =, (7) a # $5.00 UD Received 7 Jun 008; evised 3 Jul 008; accepted 0 Aug 008; published 6 Aug 008 (C) 008 OA eptembe 008 / Vol. 6, o. 8 / OPTIC EXPRE 457

8 whee the mean and vaiances σ, σ ae obtained with the help of Eq. (7). We have just edefined the Rician model so the dominant component can be a phaso sum of two o moe dominant waves, and can be subject to amplitude fluctuations (scintillation). The paamete anges between 0 and. It can be shown that when the dominant tem is vey weak ( 0), intensity fading becomes negative-exponential-distibuted, just as in a speckle patten. Likewise, when the dominant tem is vey stong ( ), the density function becomes highly peaked aound the mean value, and thee is no fading to be consideed. When is lage, it can be shown that the PDF of is, except fo a skewness facto, appoximately Gaussian with mean. Consequently, the Rice distibution can descibe the entie ange of fading that needs to be consideed in ou analysis. Using Eq. (5) to expess the PDF of the fading intensity, and applying Eq. (8), we find that the R γ is descibed by a noncental chi-squae distibution with two degees of feedom: whee the mean R ( + ) γ ( + ) + γ pγ ( γ ) = exp( ) exp I, (8) γ γ γ γ 0 = γ0 depends on tubulence-fee R 0 γ and the paamete descibing tubulence effects. While we have a plausible model leading to the Rice and noncental chi-squae distibutions, the choice of the beta distibution in [6] is based on a esemblance of the obseved cuves in some numeical simulations, and not on any statistical easoning. It is not clea how the paametes of the beta distibution must be (heuistically) fitted. Also, it esult difficult to extent those esults to coheent optical eceives using atmospheic compensation techniques. The famewok of ou appoach is mathematically moe obust and leads to esults easily extendable to systems employing active modal compensation. 3. Pefomance of coheent eceives In the pesence of tubulence, the eceived powe is scaled by the fading intensity, a andom vaiable with PDF p ( ), which depends on atmospheic popagation. At the eceive, the signal is petubed by AWG, which is statistically independent of the fading intensity. Hence, the instantaneous R γ is popotional to. The symbol-eo pobability (EP) p s (E) of an ideal coheent eceive is obtained by aveaging the EP conditioned on the R γ ove the PDF of the instantaneous R, p γ (γ): ( ) γ ( γ ) ( γ ) p E = d p E p γ. (9) 0 Although ou model can accommodate vaious modulation/detection schemes, in this pape, we conside -ay phase-shift keying (-PK) with ideal coheent detection based on maximum-likelihood pinciples. In this case, the EP conditioned on the instantaneous R is given by [4] sin / / p ( E γ ) dθ exp γ = /. (30) sin θ Using Eqs. (30) and (8) in Eq. (9), afte some algeba, we obtain # $5.00 UD Received 7 Jun 008; evised 3 Jul 008; accepted 0 Aug 008; published 6 Aug 008 (C) 008 OA eptembe 008 / Vol. 6, o. 8 / OPTIC EXPRE 458

9 Fig.. ymbol-eo pobability (EP) vs. tubulence-fee R pe symbol γ 0 fo QPK with coheent detection and additive white Gaussian noise (AWG). Pefomance is shown fo diffeent values of: (a) the nomalized eceive apetue diamete D/ 0, and (b) the numbe of modes J emoved by adaptive optics. Amplitude fluctuations ae neglected by assuming σβ =0. Tubulence is chaacteized by the phase coheence length 0. In (a), D/ 0 anges fom 0. (weak tubulence) to 0 (stong tubulence). In (b), the compensating phases ae expansions up to tilt (J=3), astigmatism (J=6), and 5th-ode abeations (J=0). The no-coection case (J=0) is also consideed. The no-tubulence case is indicated by black lines. ( ) sin / / sin γ + θ exp θ /. (3) ( + ) sin θ γ sin ( + ) sin θ γ sin p ( E) = d Although this integal cannot be put in a closed fom, we ae able to cay out the integation in Eq. (3) using a simple 30-point Gaussian-Legende quadatue fomula, which yields high accuacy. It is useful to have a simple closed-fom uppe bound on the eo pobability, which can be obtained by simply noting that sin θ. The integal in Eq. (30) is educed to p exp sin. (3) ( E γ ) γ By using this uppe bound along with the PDF of the instantaneous R Eq. (8) in the symbol eo pobability Eq. (9), afte some algeba, we obtain an uppe bound fo the EP in an -PK eceive: p ( E) + sin + + γ sin + γ sin ( ) exp ( ) γ ( ). (33) Compaison with the exact esult Eq. (3) shows that the uppe bound Eq. (33) estimates the EP with sufficient accuacy to be useful in many pactical situations. Figues -3 show the effect of atmospheic tubulence on QPK with coheent detection using modal-compensated heteodyne o homodyne eceives. We study the EP Eq. (3) as a function of seveal paametes: the aveage tubulence-fee R γ 0 pe symbol, the eceive apetue diamete D, the numbe of spatial modes J emoved by the compensation system, and the stength of atmospheic tubulence. Tubulence is quantified by two paametes: the phase # $5.00 UD Received 7 Jun 008; evised 3 Jul 008; accepted 0 Aug 008; published 6 Aug 008 (C) 008 OA eptembe 008 / Vol. 6, o. 8 / OPTIC EXPRE 459

10 Fig.. EP vs. coheence diamete 0 fo QPK with coheent detection and AWG. In (a), no phase compensation is employed, and pefomance is shown fo diffeent values of the scintillation index σ β. In (b), the scintillation index is fixed at σ β = 0.3, and pefomance is shown fo diffeent values of J, the numbe of modes coected by adaptive optics. The seveity of atmospheic tubulence inceases when 0 deceases. In all cases, we assume the tubulencefee R pe symbol is γ0 = 0 db, and the eceive apetue diamete is D = 0 cm. In (a), the σ β anges fom 0.3 (weake tubulence) to (stonge tubulence). In (b), the compensating phases ae expansions though tilt (J=3), astigmatism (J=6), and 5th-ode abeations (J=0). The no-tubulence case is indicated by black lines. coheence length 0 and the scintillation index σ β. We conside two nonzeo values of the scintillation index. The value σβ = 0.3 coesponds to elatively low scintillation levels, while σβ = coesponds to stong scintillation, but still below the satuation egime. When the tubulence eaches the satuation egime, wavefont distotion becomes so sevee that it would be unealistic to conside phase compensation. In most pactical fee-space links, amplitude fluctuations ae not satuated [5]. Figue pesents the EP vs. tubulence-fee R γ 0. Figue (a) shows the pefomance fo diffeent values of the nomalized apetue diamete D/ 0, while Fig. (b) shows the pefomance fo diffeent values of J, the numbe of modes compensated. We assume no scintillation, σβ =0, so the effect of tubulence is simply to educe the coheence length 0. Fo a fixed apetue diamete D, as 0 is educed, the nomalized apetue diamete D/ 0 inceases, and tubulence educes the heteodyne o homodyne downconvesion efficiency. Even using a elatively small nomalized apetue diamete D/ 0 =, tubulence intoduces moe than a 5- db pefomance penalty at 0 3 EP. When phase coection is used, as in Fig. (b), in most situations, compensation of just a few modes yields a substantial pefomance impovement. Compensation of J = 0 modes yields significant impovement fo even the lagest nomalized apetues consideed. Fo example, fo a nomalized apetue D/ 0 =0, at a EP = 0 3, the R penalty is just ove 5 db. This value should be contasted with the 5-dB penalty obseved in Fig. (a) fo D/ 0 = when J = 0, i.e., no modes ae compensated. Figue shows the EP vs. the phase coheence diamete 0. Figue (a) shows the pefomance fo diffeent values of the scintillation index σ β, while Fig. (b) shows the pefomance fo diffeent values of J, the numbe of modes compensated. In all cases pesented, the tubulence-fee R has a value γ0 = 0 db. The apetue diamete is fixed in evey plot to D = 0 cm. As we obseve in Fig. (a), fo stong tubulence, coesponding to small values of 0, the EP is substantially independent of the scintillation index σ β. In this egime, phase distotions have a lage impact, and high-ode phase coections may be equied. We note that in this egime of stong tubulence, the coheent pat of is vey # $5.00 UD Received 7 Jun 008; evised 3 Jul 008; accepted 0 Aug 008; published 6 Aug 008 (C) 008 OA eptembe 008 / Vol. 6, o. 8 / OPTIC EXPRE 460

11 Fig. 3. EP vs. nomalized eceive apetue diamete D/ 0 fo QPK with coheent detection and AWG. In (a), no phase compensation is employed, and pefomance is shown fo diffeent values of the scintillation index σ β. In (b), the scintillation index is fixed at σ β = 0.3, and pefomance is shown fo diffeent values of J, the numbe of modes coected by adaptive optics. In all cases, the tubulence-fee R pe symbol γ0 is popotional to the squae of the apetue diamete D. Fo the smallest apetue consideed, we assume γ0 = 0 db. In (a), σ β anges fom 0.3 (weake tubulence) to (stonge tubulence). In (b), the compensating phases ae expansions up to tilt (J=3), astigmatism (J=6), and 5th-ode abeations (J=0). The notubulence case is indicated by black lines. weak, 0, and intensity fading becomes negative-exponential distibuted, i.e. p γ γ = γ exp γ γ. In this case, EP Eq. (3) educes to p ( ) ( ) ( E) γ sin γ sin tan cot ( ) sin γ γ sin + + = +. (34) Fo the stongest tubulence (i.e., smallest 0 ) consideed, γ 0 and the EP Eq. (34) asymptotes to a maximum value ( )/. In the plots shown, whee =4, we have p s (E) 3/4. Figue 3 consides the effect of apetue diamete on pefomance. It pesents the EP as a function of the nomalized apetue diamete D/ 0 fo a constant phase coheence length 0 and constant scintillation index σ β. Fo the smallest apetue diamete consideed, the tubulencefee R has a value γ0 = 0 db. Fo any othe apetue diamete, the value of γ0 is popotional to D. Figue 3 illustates the concept of an optimal apetue diamete in coheent fee-space links. This optimal apetue diamete, which minimizes the EP, exhibits two diffeent egimes in ou studies. Fo elatively small apetues, amplitude scintillation is dominant, and pefomance is vitually unaffected by wavefont phase coection. When the apetue is lage, phase distotion becomes dominant, and high-ode phase coection may be needed to impove pefomance to acceptable levels. In [7], the simulation of beam popagation was used to examine the uncetainty inheent to the pocess of optical powe measuement with a pactical heteodyne eceive because of the pesence of efactive tubulence. Phase-compensated heteodyne eceives wee also consideed fo ovecoming the limitations imposed by the atmosphee by the patial coection of tubulence-induced wavefont phase abeations. As in the cuent analysis, simulations indicated that the optimal apetue diametes, those minimizing the elative eo, sepaate two diffeent egimes in ou simulations. The egime dominated by amplitude scintillation was defined fo elatively small apetues and it was vitually unaffected by phase-font coections. When lage apetues wee consideed, phase distotion was the elevant effect of tubulence, amplitude fluctuations # $5.00 UD Received 7 Jun 008; evised 3 Jul 008; accepted 0 Aug 008; published 6 Aug 008 (C) 008 OA eptembe 008 / Vol. 6, o. 8 / OPTIC EXPRE 46

12 wee of little influence, and we needed high-ode phase coections to decease the uncetainty to acceptable levels. In Fig. 3(a), no phase compensation is employed (J=0), and the pefomance is shown fo diffeent values of the scintillation index σ β. Hee, the optimal nomalized apetue diamete is close to D/ 0 = 0.3, and inceases slightly with inceasing scintillation index. In any case, when σ β becomes too lage, the optimization is of little pactical significance. In Fig. 3(b), we conside stong scintillation, σ β =, and show the pefomance fo diffeent values of J, the numbe of modes compensated. As we incease J, the optimized value of D/ 0 inceases, and the optimized EP impoves substantially. Even fo such stong scintillation, with compensation of J = 0 modes and optimized D/ 0, excellent EP pefomance is obtained. In this case, the optimized D/ 0 is athe lage (close to 4). Fo the lage values of D/ 0 consideed in these plots, the coheent pat of is vey weak ( 0) and fading intensity becomes negative-exponential-distibuted, such that the EP is descibed by Eq. (34). In Fig. 3(a), when lage nomalized apetues D/ 0 ae consideed, the EP becomes independent of the scintillation index σ β, and tends towad an asymptotic value that is independent of nomalized apetue diamete D/ 0. In Fig. 3(b), at lage values of D/ 0, the EPs also tend towad asymptotic values, independent of nomalized apetue diamete D/ 0, which depend only weakly on the scintillation index σ β. 4. Conclusions We have studied the impact of atmospheic tubulence-induced scintillation and phase abeations on the pefomance of fee-space optical links in which the eceive uses modal wavefont compensation and synchonous homodyne o heteodyne detection. We have defined a mathematical model fo the signal eceived afte popagation though the atmosphee and afte modal compensation. By noting that the down conveted electical signal cuent can be chaacteized as the sum of many contibutions fom diffeent coheent egions within the apetue, we showed that the PDF of this signal can be descibed by a modified Rice distibution. The paametes descibing the PDF depend on the tubulence conditions and the numbe of modes compensated at the eceive. We have povided analytical expessions fo the symbol eo pobability (EP) fo synchonous detection of -PK with additive white Gaussian noise, and have used them to study the effect of vaious paametes on pefomance, including signal level, apetue diamete, tubulence stength, and the numbe of modes compensated. We have sepaately quantified the effects of amplitude fluctuations and wavefont phase distotion on system pefomance, and have identified two diffeent egimes of tubulence, depending on the eceive apetue diamete nomalized to the coheence diamete of the wavefont phase. When the nomalized apetue diamete is elatively small, amplitude scintillation dominates and, as phase fluctuations have little impact, pefomance is vitually independent of the numbe of modes compensated. When the nomalized apetue is lage, amplitude fluctuations become negligible, and phase fluctuations become dominant, so that high-ode phase compensation may be needed to impove pefomance to acceptable levels. We have found that fo most typical link designs, wavefont phase fluctuations ae the dominant impaiment, and compensation of a modest numbe of modes can educe pefomance penalties by seveal decibels. Acknowledgment The eseach of Aniceto Belmonte was suppoted by a pain EC ecetay of tate fo Univesities and Reseach Gant Fellowship. He is on leave fom the Technical Univesity of Catalonia. The eseach of Joseph. Kahn was suppoted, in pat, by aval Reseach Laboatoy awad G035. # $5.00 UD Received 7 Jun 008; evised 3 Jul 008; accepted 0 Aug 008; published 6 Aug 008 (C) 008 OA eptembe 008 / Vol. 6, o. 8 / OPTIC EXPRE 46