Perforance Analysis of Atospheric Field Conjugation Adaptive Arrays Aniceto Belonte* a, Joseph M. Kahn b a Technical Univ. of Catalonia, Dept. of Signal Theory and Coun., 08034 Barcelona, Spain; b Stanford University, Departent of Electrical Engineering, Stanford, CA 94305, USA * belonte@tsc.upc.edu ABSTRACT Syste configurations based on single onolithic-apertures that are iune to atospheric fluctuations are being developed. Main goal is the iproveent of the perforance achievable in coherent, free-space optical counication systes using atospheric copensation techniques such as adaptive optics. As an alternative to a single onolithicaperture coherent receiver with a full-size collecting area, a large effective aperture can be achieved by cobining the output signal fro an array of saller receivers. We study the counication perforance of field conjugation adaptive arrays applied in synchronous laser counication through the turbulent atosphere. We assue that a single inforation-bearing signal is transitted over the atospheric fading channel, and that the adaptive array coherent receiver cobines ultiple dependent replicas to iprove detection efficiency. We consider the effects of log-noral aplitude fluctuations and Gaussian phase fluctuations, in addition to local oscillator shot noise. We study the effect of various atospheric paraeters and the nuber of branches cobined at the receiver. Keywords: Atospheric optical counications; Coherent receivers; Diversity cobining 1. INTRODUCTION With coherent counications, the coplex field of the received signal encodes the transitted inforation. To fully easure the coplex electrical field, i.e. aplitude and phase, of the light wave, in a coherent detection syste the incoing signal interferes with a local oscillator (LO). In a free-space laser receiver the presence of atospheric turbulence affects the coherence of the received signal that is to be ixed with the local oscillator. Light propagated through a turbulent atosphere contains speckle which will be present at the detector surface. Therefore, illuinating a single-eleent detector with a unifor LO bea will produce isatch of the aplitudes and phases of the two fields resulting in a loss in downconverted power. The downconverted coherent power is axiized when the spatial field of the received signal atches that of the local oscillator. [1],[] Syste configurations based on single onolithic-apertures that are iune to atospheric fluctuations are being developed. [3] Main goal is the iproveent of the perforance achievable in coherent, free-space optical counication systes using atospheric copensation techniques. As an alternative to this approach, the perforance of diversity cobining techniques, where two or ore statistically independent fading signals are cobined at the receiver, have been considered to iprove detection efficiency in coherent laser counication through the turbulent atosphere. [4][5] Diversity cobining consists of receiving redundantly the sae inforation signal over ultiple fading channels and to exploit the low probability of concurrence of deep fades in all the diversity channels. In this analysis, we consider a coherent fiber array consisting of densely packet ultiple subapertures, with each subaperture interfaced to a single-ode fiber, as a receiver structure for increasing the perforance of the atospheric laser syste. Instead of using a single onolithic-aperture coherent receiver with a full-size collecting area, a large effective aperture can be equally achieved by cobining the output signal fro a fiber array of saller subapertures in a close-packed arrangeent (see Fig. 1). A coherent fiber array offers an advantage in ters of the coupling efficiency as that the nuber of turbulence speckles over each subaperture in the array is uch saller than it would be over a single large aperture. Now, each receiver aperture can be saller than the scale on which the signal wavefront varies and the local oscillator phase can be atched to the signal to attain successful coherent reception. Output signals fro these receivers can then be cobined electronically to enhance the detection statistics. In general, the perforance of such a field conjugation adaptive should iprove with an increasing nuber of subapertures and, given a fixed collecting area, Free-Space Laser Counication Technologies XXIII, edited by Haid Heati, Proc. of SPIE Vol. 793, 7930Q 011 SPIE CCC code: 077-786X/11/$18 doi: 10.1117/1.873056 Proc. of SPIE Vol. 793 7930Q-1 Downloaded fro SPIE Digital Library on Feb 011 to 83.53.8.114. Ters of Use: http://spiedl.org/ters
the fiber array syste can offer superior perforance. Note that, due to the close spatial arrangeent of the subapertures in a coherent fiber array, atospheric fading on the array coponents is correlated or dependent. We consider a general statistical odel to describe the signal collected by the fiber array receiver after propagation through the atosphere. We consider the perforance of such receivers under the effects of log-noral aplitude fluctuations and Gaussian phase fluctuations, in addition to local oscillator shot noise.. A STATISTICAL MODEL FOR THE RECEIVER In a coherent counication receiver, the SNR γ 0 per unit bandwidth B for a quantu or shot-noise liited signal can be interpreted as the detected nuber of photons (photocounts) per sybol when 1/B is the sybol period. Coherently detected signals are odeled as narrowband RF signals with additive white Gaussian noise (AWGN). For a coherent receiver syste, in the presence of target speckle and atospheric turbulence, we ust consider fading signals, which are signals also affected by ultiplicative noise. In the fading AWGN channel, we let α denote the atospheric channel power fading and γ 0 α denote the instantaneous received SNR per pulse. For a shot-noise-liited coherent optical receiver, the SNR of the envelope detector can be taken as the nuber of signal photons detected on the receiver aperture γ 0 ultiplied by a heterodyne power ixing efficiency α. For systes with perfect spatial ode atching, the ixing efficiency is equal to 1. When the spatial odes are not properly atched, the contribution to the current signal fro different parts of the receiver aperture can interfere destructively and result in the reduced instantaneous heterodyne ixing and consequent fading. Note that, conditional on a realization of the atospheric channel described by α, this is an AWGN channel with instantaneous received SNR γ=γ 0 α. This quantity is a function of the rando channel power fading α, and is therefore rando. The statistical properties of the atospheric rando channel fade α, with probability density function (PDF) p α (α), provide a statistical characterization of the SNR γ=γ 0 α. In this study we define a statistical odel for the fading aplitude α (i.e., SNR γ) of the received signal after propagation through the atosphere. In a single-aperture, fiber-based coherent receiver, when the spatial field of the received signal E i (r) does not atch that of the local oscillator E (r), as described by the fiber-ode profile referred to the receiving aperture, the rando fading 4 α = d W ( ) Ei( ) E( ) π D r r r r (1) depends in aplitude and phase isatches of the two fields incident on the receiving aperture. Phase and aplitude isatches represent the aplitude fluctuations and phase distortions introduced by atospheric turbulence in the received signal. The circular receiving aperture of diaeter D is defined by the aperture function W(r), which equals unity for r D/, and equals zero for r >D/. In general, in Eq. (1) fading is a coplex agnitude α = αr + jαiwhere α r and α i represent integrals over the collecting aperture of the real and iaginary parts, respectively, of the optical fields reaching the receiver. These real and iaginary parts can be considered as the coponents of a coplex rando phasor. We need to study how aplitude and phase turbulence-induced fluctuations of the optical field define the statistics of the fading intensity α = αr + jαi. Fro Eq. (1), we note that the two rando agnitudesα r and α i can be expressed as integrals over the aperture and, hence, are the sus of contributions fro each point in the aperture. In order to proceed with the analysis, we could consider a statistical odel in which these continuous integrals are expressed as finite sus over statistically independent cells in the aperture. Under the assuption that the nuber of independent coherent regions is large enough, we can consider that α r and α i asyptotically approach jointly noral rando variables. Then, the probability density function of the length fading aplitude α can be well approxiate by a Rayleigh distribution. Just as in a speckle pattern, the Rayleigh distribution for the turbulence aplitude fading length is a consequence of the central-liit theore. However, under conditions of weak-turbulence in which the nuber of coherent ters is sall, the fading ay actually be the result of suing a sall nuber of ters. In this case, the fading α is not likely to be Rayleigh. Rather than assuing that α is always Rayleigh distributed for all conditions of turbulence, it is ore realistic to assue that α satisfied a generalized Rayleigh distribution that becoes Rayleigh only when the nuber of coherent ters N becoes large enough. Such a distribution is the Nakagai- distribution, [6] in essence a central chi-square distribution described by: Proc. of SPIE Vol. 793 7930Q- Downloaded fro SPIE Digital Library on Feb 011 to 83.53.8.114. Ters of Use: http://spiedl.org/ters
1 α pα ( α ) = ( N) exp N Γ ( ) ( α ). () where Γ is the coplete gaa function. The Nakagai- paraeter and fading paraeter N are a easure of turbulence effects. Here, N is the inverse of the fading ean-square value N = 1 α. (3) The paraeter characterizes the aount of turbulent fading. When 1, the nuber of contribution coherent areas N is large and the -distribution reduces to Rayleigh. Note that the Nakagai- distribution closely approxiates the Rice distribution [6] we have previously used to odel the ipact of atospheric turbulence-induced fading on free-space optical counication links using coherent detection. [3] Applying the Jacobian of the transforation α = γ γ0, the corresponding SNR γ distribution can be described according to a gaa distribution with a shape paraeter and a ean value γ = γ α γ given by 0 0 N p γ ( γ) 1 N γ N = exp γ. (4) γ Γ γ ( ) 0 0 When 1, the gaa distribution reduces to exponential distribution. Note that the PDF in Eq. (4) can equivalently be expressed in ters of its oent generating function (MGF), which it is closely related to the Laplace transfor of the distribution p γ ( γ ) and defined as the expected value of exp( sγ ) : ( ) = γ exp( γ) γ ( γ) M s d s p γ 0 γ0 γ0 1 s 1 s. = = α N The MGF is a useful tool for analyzing the average error probability in counication systes with fading. Also, it can be shown that the distribution oents are given by k γ ( k) k Γ( ) Γ + = which yields, by using the average SNR, an expression for (5) k γ, (6) 4 1 σ γ α = = 1. (7) γ α Since α r and α i can be considered jointly noral rando variables, it is possible to relate high-order oents with the 4 4 lower-order oents and replaces the fourth-order oent in Eq. (7) by α = 3α α. It results: 1 α =. (8) α The paraeter, by characterizing the aount of fading through the noralized SNR γ variance, gives ore control over the extent of the turbulence fading. When 1 and the nuber of contribution coherent areas is large, the noralized variance is one, as expected for Rayleigh distributions. When grows, and a very sall nuber of contribution ters add together, the noralized variance decreases. Now, the density function becoes highly peaked around the ean value γ = γ 0 N and there is just a sall fading to be considered. To have a easure of turbulence effects, it has been necessary to develop procedures to estiate and N. Equivalently, as Eqs. (3) and (8) describe fading paraeters and N in ters of fading first and second oents, we have needed to Proc. of SPIE Vol. 793 7930Q-3 Downloaded fro SPIE Digital Library on Feb 011 to 83.53.8.114. Ters of Use: http://spiedl.org/ters
establish closed expressions for α and α. In order to assess the ipact of turbulence on the heterodyne ixing and fading, the field aplitude without the effect of turbulence in the pupil plane ust be odified by a ultiplicative factor exp χ( r) jφ( r) where χ(r) and φ(r) represent the log-aplitude fluctuations (scintillation) and phase variations (aberrations), respectively, introduced by atospheric turbulence. Consequently, both γ and 1/r are described in ters of log-noral aplitude fluctuations and Gaussian phase fluctuations as characterized by their respective statistical variances, σ χ and σ φ. Also, the propagating ode of a single-ode fiber is well approxiated by an untruncated Gaussian function and the fiber-ode profile referred to the receiving aperture describing the local oscillator E (r) can be characterized by its fiber-ode field radius at the front surface of the receiving lens ω. We define a general odel for the output SNR of diversity systes over correlated fading channels. For field conjugation adaptive arrays, where the atospheric fading on the branches is correlated or dependent, we can solve the proble by transforing it into an independent proble using the technique of spatial whitening. At this way, the results for independent fading channels are easily extended to the ore general proble of correlated channels. 3. PERFORMANCE OF A CLOSELY PACKED HEXAGONAL COHERENT ARRAY In a field conjugation fiber array, the interediate-frequency signals at the output of the single-ode fibers needs to be co-phased and their aplitude independently adapted before they are sued to lessen signal fading associated with atospheric turbulence and reduce fiber coupling shortcoings. Note that this is equivalent to consider axiu ratio cobining (MRC) of the received signals by the array subapertures as they can be considered as branches of a diversity cobiner. [7] MRC diversity schees assue perfect knowledge of the branch aplitudes and phases, require independent processing of each branch, and need that the individual signals fro each branch be weighted by their signal to noise power ratios then sued coherently. A receiver with MRC will coherently cobine the diversity branches by weighting the by the coplex conjugate of their respective fading gains and adding the. Clearly, like in a MRC cobiner, the instantaneous SNR γ T for a suing coherent array is the power ratio of the phase-coherent addition of the signal aplitudes fro each eleent of the array to the incoherent addition of the noise. If an optiu voltage gain proportional to the aplitude of the signal itself is assued for each eleent in the array, and if equal noise powers are assued, the resultant coposite SNR γ T for an L-eleent coherent fiber array is the su of the array eleents SNR γ l. For independent subaperture signals and equal average branch SNR, i.e. γl = γ for all l {1,,,L}, the PDF of the received SNR γ T at the output of a perfect L-branch coherent array in the atosphere would be described a su of L independent and identically distributed gaa rando variables. This rando variable it is also described by the gaa distribution Eq. (4) with a shape paraeter (L). However, for close coherent fiber arrays receivers, with insufficient collecting apertures spacing, it is not realistic to assue that the cobined signals are independent of one another. In this scenario, the degree of correlation aong the different fadingsα l describing γ T will depend on several factors, including atospheric conditions and the exact geoetry of the coherent array receiver. It is worthy of entioning that the evaluation of the ost coon perforance easures of an optical coherent counication syste in the presence of atospheric fading can be accoplished based entirely on the knowledge of the MGF of the output SNR without ever having to copute its PDF. We will use MT ( s ) to estiate the perforance of the MRC coherent fiber array. This MGF-based approach is quite useful in siplifying our analysis. Figures 3 and 4 consider the ean and standard deviation of the SNR at the output of field conjugation adaptive arrays. We conteplate an L-eleent coherent fiber array and assue that the subapertures are arranged in a hexagonal closepacked array as shown in Fig.. The larger circle represents a single receiver aperture of diaeter D. The sall circles represent packed subapertures and each subaperture contains a lens that couples the received light into a single-ode fiber. For coparison of the receiver perforance between an L-eleent coherent array and a single large aperture, we force the hexagonal array to be packed within the liits of the single aperture area. The hexagonal distribution is the densest way to arrange circles in a plane. Still, note that each subaperture will have a pupil area slightly saller than 1/L ties the pupil area of the single receiver syste and an array fill factor needs to be considered in our analysis. For instances, for the array with L=7 eleents in Fig. 1, the array fill factor is 7/9. Also, the truncation paraeter of the pupil τ D ω describes the ratio of the receiver aperture diaeter to the diaeter of the backpropagated fiber ode. A large Proc. of SPIE Vol. 793 7930Q-4 Downloaded fro SPIE Digital Library on Feb 011 to 83.53.8.114. Ters of Use: http://spiedl.org/ters
value of τ or D ω represents a narrow Gaussian ode or a weakly truncated pupil. A unifor illuinated pupil is obtained by letting τ 0. The truncation paraeter ust be chosen to optiize the receiving syste perforance. Although the optiu paraeter ay depend on the level of atospheric turbulence considered, this dependency is very weak and we can chose the optiu value of 1.1 obtained in absence of turbulence, when the incident plane wave is fully coherent. [8] We study the ean SNR in Fig. 3 and the SNR noralized standard deviation in Fig. 4 as a function of several paraeters: the average turbulence-free SNR γ 0, the receiver aperture diaeter D, the nuber of subapertures L of the hexagonal distributed fiber array coherently sued, and the strength of atospheric turbulence. Turbulence is quantified by two paraeters: the phase coherence length r 0, which describes the spatial correlation of phase fluctuations in the receiver plane, [1] and the scintillation index σ β. [9] The value of the scintillation index σ β = 1 corresponds to strong scintillation, but still below the saturation regie. When we assue no scintillation, σ β =0, the effect of turbulence is siply to reduce the coherence length r 0. For a fixed coherent diaeter r 0, as aperture diaeter D is increased, the noralized aperture diaeter D/r 0 increases, and turbulence reduces the heterodyne downconversion efficiency. In Fig. 3, the ean SNR is plotted against the noralized aperture diaeter D/r 0 for different nuber of subapertures L on the hexagonal array. The SNR is expressed in db, referenced to the turbulence free SNR γ 0. This corresponds to the ean intensity fading α = 1 N according to the transforation γ = γα 0. Note that the ean SNR is just a representation of the fiber-coupling efficiency. The received signal bea ust be coupled into a single-ode fiber but atospheric turbulence degrades the spatial coherence of a laser bea and liits the fiber-coupling efficiency and, consequently, the available ean SNR at the output of each fiber in the array. As expected for one single onolithic aperture L=1, if D is less than r 0, the noralized ean SNR γ γ 0 reains constant. A truncation paraeter τ = 1.1 reduces the ean SNR by 5 db. When diaeter D is larger than r 0, atospheric turbulence liits the effective receiving aperture to the diensions of the coherence diaeter r 0 and the noralized ean SNR goes down very quickly. When the noralized aperture diaeter D/r 0 is large, an increase in the nuber of the hexagonal array subapertures iproves the situation significantly. For instances, when a large noralized aperture D/r 0 =10 is considered, increasing the nuber of subapertures L fro 1 to 19 coends the ean SNR by ore than 13 db. The ean SNR is just a representation of the fiber-coupling efficiency: the advantage of a fiber array in ters of the ean SNR and the fiber coupling efficiency is that the nuber of field coherence areas N over each subaperture is saller than it would be over a single large aperture. In Fig. 4, we plot the noralized SNR standard deviation (SNR uncertainty or relative error) σ γ γ against the noralized aperture diaeter D/r 0 for different values of the nuber of subapertures L on the hexagonal array. In the liit of weak turbulence (sall noralized aperture diaeter D/r 0 ), the noralized variance trends asyptotically to 0. When aperture diaeter is increased, SNR uncertainty grows steadily with the nuber of field coherence areas N over the receiving pupil until reaching a well define axi. In the liit of strong turbulence (large noralized aperture diaeter D/r 0 ), the SNR uncertainty becoes flat and very intense. When L=1, the SNR standard deviation reach a axiu value of alost 1.6 ( db). Once again, an increase in the nuber of the array subapertures will iprove the situation and decrease SNR uncertainty: As an exaple, when the noralized aperture D/r 0 =10 is considered, increasing the nuber of subapertures L fro 1 to 19 decrease the ean SNR by roughly 6 db, fro 1dB to -5 db. As we observe in Fig. 4, the effects of scintillation are noticeable for the sall aperture diaeters and ust be properly considered. For relatively sall apertures, aplitude scintillation is doinant, and noralized variance is virtually unaffected by wavefront phase distortions. When the aperture is larger, phase distortion becoes doinant and the scintillation index σ β is of little relevance in the SNR uncertainty. Also, it results of interest to consider the case of independent subapertures. These independent signals could be obtained by spacing the ultiple subapertures in the array. If the output of a perfect L-eleents cobiner in the atosphere would be described as a su of L independently fading signals, the output SNR uncertainty would be apparently better. That is the case in the liit of weak turbulence or sall noralized apertures, where, for instances, when 7-eleents arrays are considered, the uncertainty is alost 4 db saller for ideal independent subapertures. This 4-dB difference is the penalty for using closely packed arrays that cobine highly correlated signals. However, when strong turbulence or large noralized apertures are considered, field coherence areas are saller than any array subaperture used in this analysis and the signals collected by then are ostly uncorrelated. In this regie, the ultiple replicas cobined at the array receiver are statistically independent and to distance the eleents of the array would not iprove the SNR uncertainty. Proc. of SPIE Vol. 793 7930Q-5 Downloaded fro SPIE Digital Library on Feb 011 to 83.53.8.114. 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Figures 5 and 6 presents the SEP [10][11] for an L-eleents coherent fiber array receiver. Figure. 5 shows the perforance SEP vs. noralized aperture diaeter D/r 0, while Fig. 6 shows the perforance vs. turbulence-free SNR γ 0. In Fig. 5, for the sallest aperture diaeter considered, the turbulence-free SNR has a value γ 0 = 0 db. For any other aperture diaeter, the value of γ 0 is proportional to D.When we assue no scintillation, σ β =0, the effect of turbulence is siply to reduce the coherence length r 0. For a single large aperture, even using a relatively sall noralized aperture diaeter D/r 0 =1, turbulence introduces a strong perforance penalty at SEP. When ultiaperture array receivers are considered, in ost situations it yields a substantial perforance iproveent. An array with just L=19 subapertures yields significant iproveent for even the largest noralized apertures considered. The perforance of such array receiver is very close to the perforance expected in an AWGN syste. In Fig. 5, when large noralized apertures D/r 0 are considered, the SEP becoes independent of the scintillation index σ β, and tends toward an asyptotic value that is independent of noralized aperture diaeter D/r 0. Figure 6 shows the perforance for different values of L, the nuber of array eleents. Even using a relatively sall noralized aperture diaeter D/r 0 =, when a single large aperture receiver is used, turbulence introduces ore than a 30-dB perforance penalty at 10 3 SEP. When ultieleent arrays are used, perforance iproves arkedly. For instances, considering a sall array receiver with L=7 subapertures, at a SEP = 10-3 the SNR penalty is just below 10 db. This value should be contrasted with the 6-dB penalty observed in Fig. 6 when L=7 independently fading signals are ideally cobined. Also, although this results assue no scintillation, when we ipose a strong scintillation index of σ β =1 the penalty increase is less than 1 db at SEP = 10-3. 4. CONCLUSIONS We have nuerically evaluated the perforance of adaptive field conjugation array receivers in coherent laser counications through the turbulent atosphere. We analyze coherent fiber arrays consisting of densely packet ultiple subapertures in a hexagonal arrangeent and consider the effects of log-noral aplitude fluctuations and Gaussian phase fluctuations, in addition to local oscillator shot noise. By noting that the ipact of atospheric turbulence on coherent receivers can be statistically described by a Nakagai- probability density function, our odel uses fundaental principles of atospheric propagation and circuvents the need for a detailed description of the turbulence proble. For fiber adaptive arrays, where the atospheric fading on the subapertures is correlated or dependent, we can solve the proble by transforing it into an independent proble. A MGF-based approach used in our analysis has provided easily evaluable analytical expressions for the signal statistical oents and the sybol error probabilities. We have used it to study the effect of various paraeters on perforance, including turbulence level, signal strength, receive aperture size, and the nuber of subapertures in the coherent fiber array. We have separately quantified the effects of aplitude fluctuations and wavefront phase distortion on syste perforance, and have identified different regies of turbulence. For ost typical free-space laser counication situations, using coherent arrays with a reasonably sall nuber of subapertures such as L=19 increases the counication perforance by several decibels. The research of Aniceto Belonte was partially funded by the Spanish Departent of Science and Innovation MICINN Grant No. TEC 009-1005. REFERENCES [1] D. L. Fried, "Optical heterodyne detection of an atospherically distorted signal wave front," Proc. IEEE 55, 57-67 (1967). [] R. M. Gagliardi and S. Karp, Optical Counications (John Wiley & Sons, 1995). [3] A. Belonte and J. M. Kahn, "Perforance of synchronous optical receivers using atospheric copensation techniques, Opt. Express 16, 14151-1416 (008). [4] E. J. Lee and V. W. Chan, "Diversity Coherent and Incoherent Receivers for Free-Space Optical Counication in the Presence and Absence of Interference," J. Opt. Coun. Netw. 1, 463-483 (009). [5] A. Belonte and J. M. Kahn, "Capacity of coherent free-space optical links using diversity-cobining techniques," Opt. Express 17, 1601-1611 (009). Proc. of SPIE Vol. 793 7930Q-6 Downloaded fro SPIE Digital Library on Feb 011 to 83.53.8.114. Ters of Use: http://spiedl.org/ters
[6] M. Nakagai, The -distribution. A general forula of intensity distribution of rapid fading, in Statistical Methods in Radio Wave Propagation, W. C. Hoffan, ed. (Pergaon Press, 1960). [7] J. D. Parsons, Diversity techniques in counications receivers, in Advanced Signal Processing, D. A. Creasey, ed. (Peregrinus, 1985), Chap. 6. [8] P. J. Winzer and W. R. Leeb, Fiber coupling efficiency for rando light and its applications to lidar, Opt. Letters 3, 986-988 (1998). [9] J. W. Strohbehn, T. Wang, and J. P. Speck, On the probability distribution of line-of-sight fluctuations of optical signals, Radio Science 10, 59-70 (1975). [10] J. G. Proakis and M. Salehi, Digital Counications, (Mc Graw-Hill, 007). [11] M. K. Sion and M.-S. Alouini, A unified approach to the perforance analysis of digital counications over generalized fading channels, IEEE Proc. 86, 1860-1877 (1998). Figure 1. A coherent free-space optical counication syste is affected by the presence of atospheric turbulence in any ways. Aplitude scintillation and phase distortion in the receiver plane act as intense sources of noise distorting the quality of the optical signal available for processing and add together to deteriorate the overall counication perforance of the optical systes. In a field conjugation fiber array, the interediate-frequency signals at the output of the single-ode fibers are adaptively co-phased, have their aplitude separately adjusted, and then sued to itigate signal fading associated with atospheric turbulence. Figure. Adaptive arrays could be considered to alleviate the deteriorating effects of atospheric turbulence. We consider L- eleent coherent fiber arrays and assue that the subapertures are arranged in a hexagonal close-packed array. Proc. of SPIE Vol. 793 7930Q-7 Downloaded fro SPIE Digital Library on Feb 011 to 83.53.8.114. Ters of Use: http://spiedl.org/ters
Figure 3. Noralized ean coherent SNR vs. the noralized receiver aperture diaeter D/r 0. Perforance is shown for different values of the nuber of of supapertures L in the closely packed hexagonal coherent array. The case L=1 corresponds to a onolitic aperture (black line). When a single aperture is considered, D describes the receiver aperture diaeter. For coparison of the receiver perforance between an L-eleent coherent array and a single large aperture, we force the hexagonal array to be packed within the liits of the single aperture diaeter D. The analysis conteplates the array fill factor and considers a coupling-geoetry paraeterτ for each subaperture lens equal to 1.1. Figure 4. SNR noralized standard deviation vs. the noralized receiver aperture diaeter D/r 0. Perforance is shown for different values of the nuber of of supapertures L in the closely packed hexagonal coherent array. The case L=1 corresponds to a onolitic aperture (black line). When a single aperture is considered, D describes the receiver aperture diaeter. For coparison of the receiver perforance between an L-eleent coherent array and a single large aperture, we force the hexagonal array to be packed within the liits of the single aperture diaeter D. The analysis conteplates the array fill factor and considers a coupling-geoetry paraeterτ for each subaperture lens equal to 1.1.. Aplitude fluctuations are neglected (solid line) by assuing σ β =0. When scintillation is considered (dashed line), the scintillation index is fixed at σ β =1. The red, dotted line corresponds to the uncertainty associated with L=7 independent-subapertures liit where fading correlation is neglected. Proc. of SPIE Vol. 793 7930Q-8 Downloaded fro SPIE Digital Library on Feb 011 to 83.53.8.114. Ters of Use: http://spiedl.org/ters
Figure 5. SEP vs. noralized receiver aperture diaeter D/r 0 for QPSK with coherent detection and AWGN. Perforance is shown for different values of the nuber of of supapertures L in the closedly packed hexagonal coherent array. The case L=1 corresponds to a onolitic aperture (black line). When a single aperture is considered, D describes the receiver aperture diaeter. For coparison of the receiver perforance between an L-eleent coherent array and a single large aperture, we force the hexagonal array to be packed within the liits of the single aperture diaeter D. The analysis conteplates the array fill factor and considers a coupling-geoetry paraeterτ for each subaperture lens equal to 1.1. The turbulence-free SNR per sybol γ 0 is proportional to the square of the aperture diaeter D. For the sallest aperture considered, we assue γ 0 = 0 db. In all cases, aplitude fluctuations are neglected (solid line) by assuing σ β =0. When scintillation is considered for L=7 (red, dashed line), the scintillation index is fixed at σ β =1. Also for L=7, the red, dotted line corresponds to the SEP associated with the independent-subapertures liit where fading correlation is neglected.the no-turbulence, AWGN case is indicated by black, dashed lines. The best perforance shown inthis plots (dot black line) considers the AWGN liit withτ = 0 to obtain a unifor illuinated pupil. When copared with the τ = 1.1 case, the ean SNR is now 5-dB higher. Figure 6. SEP vs. nuber of photons per sybol for QPSK with coherent detection and AWGN. Perforance is shown for different values of the nuber of of supapertures L in the closedly packed hexagonal coherent array. The noralized aperture diaeter D/r 0 is set to. Other paraeters are siilar to those in Fig. 5. Once again, aplitude fluctuations are neglected (solid line) by assuing σ β =0. When scintillation is considered for L=7 (red, dashed line), the scintillation index is fixed at σ β =1. Also for L=7, the red, dotted line corresponds to the SEP associated with the independent-subapertures liit where fading correlation is neglected. Proc. of SPIE Vol. 793 7930Q-9 Downloaded fro SPIE Digital Library on Feb 011 to 83.53.8.114. Ters of Use: http://spiedl.org/ters