Free-Space Optical Communication Through Atmospheric Turbulence Channels Xiaoming Zhu and Joseph M

Home , Intensity modulation, Optical communication, Optical fiber

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 8, AUGUST 2002 1293 Free-Space Optical Communication Through Atmospheric Turbulence Channels Xiaoming Zhu and Joseph M. Kahn, Fellow, IEEE

Abstract—In free-space optical communication links, atmo- of communication systems. Aerosol scattering effects caused spheric turbulence causes fluctuations in both the intensity and by rain, snow and fog can also degrade the performance of the phase of the received light signal, impairing link performance. free-space optical communication systems [5], [6], but are not In this paper, we describe several communication techniques to mitigate turbulence-induced intensity fluctuations, i.e., signal treated in this paper. fading. These techniques are applicable in the regime in which the Atmospheric turbulence has been studied extensively and var- receiver aperture is smaller than the correlation length of fading ious theoretical models have been proposed to describe turbu- and the observation interval is shorter than the correlation time lence-induced image degradation and intensity fluctuations (i.e., of fading. We assume that the receiver has no knowledge of the signal fading) [7]–[13]. Two useful parameters describing tur- instantaneous fading state. When the receiver knows only the marginal statistics of the fading, a symbol-by-symbol ML detector bulence-induced fading are , the correlation length of inten- can be used to improve detection performance. If the receiver sity fluctuations and , the correlation time of intensity fluctu- has knowledge of the joint temporal statistics of the fading, max- ations. When the receiver aperture can be made larger than imum-likelihood sequence detection (MLSD) can be employed, the correlation length , then turbulence-induced fading can be yielding a further performance improvement, but at the cost of reduced substantially by aperture averaging [13]. very high complexity. Spatial diversity reception with multiple receivers can also be used to overcome turbulence-induced fading. Because it is not always possible to satisfy the condition We describe the use of ML detection in spatial diversity reception , in this paper, we propose alternative techniques to reduce the diversity gain penalty caused by correlation between for mitigating fading in the regime where . At the the fading at different receivers. In a companion paper, we bit rates of interest in most free-space optical systems, the describe two reduced-complexity implementations of the MLSD, receiver observation interval during each bit interval is which make use of a single-step Markov chain model for the fading correlation in conjunction with per-survivor processing. smaller than the turbulence correlation time . Throughout this paper, we will assume that and . The Index Terms—Atmospheric turbulence, free-space optical communication, MLSD, spatial diversity reception. techniques we consider are based on the statistical properties of turbulence-induced signal intensity fading, as functions of both temporal and spatial coordinates. Our approaches can be I. INTRODUCTION divided into two categories: temporal-domain techniques and REE-SPACE optical communication has attracted con- spatial-domain techniques. F siderable attention recently for a variety of applications In the temporal-domain techniques, one employs a single re- [1]–[4]. Because of the complexity associated with phase or ceiver. If the receiver has knowledge of the marginal fading frequency modulation, current free-space optical communi- distribution, but knows neither the temporal fading correlation cation systems typically use intensity modulation with direct nor the instantaneous fading state, a maximum-likelihood (ML) detection (IM/DD). Atmospheric turbulence can degrade the symbol-by-symbol detection technique can be used. If the re- performance of free-space optical links, particularly over ceiver further knows the joint temporal fading distribution, but ranges of the order of 1 km or longer. Inhomogeneities in the not the instantaneous fading state, the receiver can use a ML se- temperature and pressure of the atmosphere lead to variations quence detection (MLSD) technique to mitigate turbulence-in- of the refractive index along the transmission path. These index duced fading. inhomogeneities can deteriorate the quality of the received In the spatial-domain techniques, one must employ at least image and can cause fluctuations in both the intensity and the two receivers to collect the signal light at different positions or phase of the received signal. These fluctuations can lead to an from different spatial angles. To maximize the diversity recep- increase in the link error probability, limiting the performance tion gain, the multiple receivers should be placed as far apart as possible, so that the turbulence-induced fading is uncorrelated at the various receivers. In practice, however, it may not always Paper approved by W. C. Kwong, the Editor for Optical Communications of the IEEE Communications Society. Manuscript received May 24, 2001; re- be possible to place the receivers sufficiently far apart. Hence, in vised August 19, 2001. This work was supported by the DARPA STAB Program this paper, making use of the spatial correlation of turbulence-in- under Contract DAAH01-00-C-0089 and by the DARPA MTO MEMS Program duced fading, we derive the optimal ML detection scheme for under Contract DABT63-98-1-0018. This paper was presented in part at the IEEE Conference on Global Communications, San Francisco, CA, November correlated spatial diversity reception. 27–December 1, 2001. The remainder of this paper is organized as follows. In Sec- The authors are with the Department of Electrical Engineering and Computer tion II, we review the theories used to model atmospheric tur- Sciences, University of California at Berkeley, Berkeley, CA 94720-1772 USA (e-mail: [email protected]; [email protected]). bulence and use them to derive the spatial and temporal coher- Publisher Item Identifier 10.1109/TCOMM.2002.800829. ence properties of the optical field in weak turbulence chan-

0090-6778/02$17.00 © 2002 IEEE 1294 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 8, AUGUST 2002 nels. We then use the coherence properties introduced to de- Here, is the wavenumber spectrum structure parameter, rive the joint probability distribution of the light intensity re- which is altitude-dependent. Hufnagel and Stanley gave a ceived on turbulence channels, characterizing the correlation of simple model for [9]: signal fading over both time and space. In Section III we describe the temporal domain techniques. We first use the mar- (3) ginal distribution of fading to derive a ML symbol-by-symbol detector for systems using on-off keying (OOK). We then use where is parameter describing the strength of the turbulence the joint temporal distribution of fading to derive a MLSD for and is effective height of the turbulent atmosphere. For at- OOK. In Section IV,we introduce the spatial-diversity reception mospheric channels near the ground ( m), can vary scheme, where the instantaneous fading state at each receiver from 10 m for strong turbulence 10 m to for is unknown. With knowledge of the joint spatial distribution of weak turbulence, with 10 m often quoted as a typical fading, ML detection can help to mitigate the diversity reception “average” value [8]. Other models and recent measurements of gain penalty caused by correlation among multiple receivers. the vertical profile of can be found in [11], [12]. We also numerically demonstrate that in the dual-receiver case, this ML detection scheme has better performance than conven- B. Spatial and Temporal Coherence of Optical Signals tional equal-gain combining (EGC). We present our conclusions Through Turbulence in Section V. As we show in Section III, the high computational complexity To describe spatial coherence of optical waves, the so-called of the MLSD makes it impractical for most applications. In a mutual coherence function (MCF) is widely used [8]: companion paper [14], we describe two reduced-complexity im- (4) plementations of the MLSD, which make use of a single-step Markov chain model for the fading correlation in conjunction where is the complex optical field. Setting in in with per-survivor processing. (4), we obtain the spatial MCF . The Rytov method is frequently used to expand the optical field : II. OPTICAL COMMUNICATION THROUGH ATMOSPHERIC TURBULENCE (5) In this section, we first review the theories used to model atmospheric turbulence. We then use these to derive the spatial where is the field amplitude without air turbulence: and temporal coherence properties of the optical field in weak turbulence channels. Finally, we present the joint spatial and (6) temporal distribution of turbulence-induced fading. The exponent of the perturbation factor is: A. Modeling of Atmospheric Turbulence Atmospheric turbulence can be physically described by Kol- mogorov theory [7]–[10]. The energy of large eddies is redis- (7) tributed without loss to eddies of decreasing size until finally dissipated by viscosity. The size of turbulence eddies normally where is the log-amplitude fluctuation and is the phase ranges from a few millimeters to a few meters, denoted as the fluctuation. We assume and be homogeneous, isotropic inner scale and the outer scale , respectively. We can ex- and independent Gaussian random variables. This assumption press the refractive index as , where is valid for long propagation distances through turbulence. is the average index and is the fluctuation component in- In order to characterize turbulence-induced fluctuations of the duced by spatial variations of temperature and pressure in the log-amplitude , we use the log-amplitude covariance function: atmosphere. The correlation function of is defined as

(1) (8) Since the random disturbance is Gaussian-distributed under the Setting in (1), we obtain . which describes assumption of weak turbulence, we can use the Rytov method the spatial coherence of the refractive index. to derive the normalized log-amplitude covariance function for To study the spatial coherence of the refractive index, two positions in a receiving plane perpendicular to the direction many models have been proposed, which assume exponential, of propagation [13]: Gaussian or other solvable function forms for . We define the wavenumber spectrum to be the spatial (9) Fourier transform of . A widely used model with good accuracy was proposed by Kolmogorov, which assumes where is the distance between and . We define the cor- the wavenumber spectrum to be relation length of intensity fluctuations, , such that . When the propagation path length satisfies the condition (2) , where is the wavelength and and are ZHU AND KAHN: FREE-SPACE OPTICAL COMMUNICATION 1295 inner and outer length scales, respectively, can be approxi- mated by [13]

(10)

In most free-space optical communication systems with visible or infrared lasers and with propagation distance of a few hun- dred meters to a few kilometers, (10) is valid. We note that aerosol forward scattering can further degrade the coherence of the optical field and thus affect the correlation length. In this paper, however, we focus only on atmospheric turbulence effects. Atmosphere turbulence also varies with time and leads to intensity fluctuations that are temporally correlated. Modeling the movement of atmospheric eddies is extremely difficult and a simplified “frozen air” model is normally employed, which assumes that a collection of eddies will remain frozen in relation to one another, while the entire collection is translated along some Fig. 1. Standard deviation of the log-amplitude fluctuation versus propagation direction by the wind. Taylor’s frozen-in hypothesis can be ex- distance for a plane wave. pressed as [10]: for the covariance of the log-amplitude fluctuation of plane (11) and spherical waves [10]: where is the velocity of the wind, which has an average and a fluctuation .If is negligible and is transverse to the direction of light propagation, then temporal correlation (13) becomes analogous to spatial correlation; in particular, the correlation time is . Assuming a narrow beam prop- agating over a long distance, the refractive index fluctuations (14) along the direction of propagation will be well-averaged and will be weaker than those along the direction transverse to prop- Fig. 1 shows the standard deviation of the log-amplitude fluc- agation. Therefore we need only consider the component of the tuation for a plane wave, computed using (13), as a func- wind velocity vector perpendicular to the propagation direction tion of the propagation distance . In Fig. 1, we again assume a . The turbulence correlation time is therefore wavelength of 529 nm and assume to be constant. Fig. 1 shows that for propagation distances of a kilometer, varies (12) from 10 to 1 for different values of . Consider the propagation of light through a large number of elements of the atmosphere, each causing an independent, iden- C. Probability Distributions of Turbulence-Induced Intensity tically distributed phase delay and scattering. By the Central Fading Limit Theorem, the marginal distribution of the log-amplitude As discussed previously, when the propagation distance is is Gaussian: long, log-amplitude fluctuations can become significant. In this section, we will derive the statistical properties of the log-am- (15) plitude fluctuations, which we refer to as “intensity fading” or simply “fading.” The marginal distribution of fading is derived The light intensity is related to the log-amplitude by in Section I, while the joint spatial and temporal distribution of (16) fading are derived in Section II. Where is the ensemble average of log-amplitude . 1) Marginal Distribution of Fading: In this section, we de- From (15) and (16), the average light intensity is: rive the marginal distribution of fading at a single point in space at a single instant in time. The marginal distribution is used in (17) symbol-by-symbol ML detection, which is discussed in Sec- Hence, the marginal distribution of light intensity fading in- tion III-A. duced by turbulence is log-normal: For propagation distances less than a few kilometers, variations of the log-amplitude are typically much smaller than vari- (18) ations of the phase. Over longer propagation distances, where turbulence becomes more severe, the variation of the log-ampli- 2) Joint Spatial and Temporal Distributions of Fading: In tude can become comparable to that of the phase. Based on the this section, we derive the joint spatial and temporal distribu- atmosphere turbulence model adopted here and assuming weak tions of fading. The joint spatial distribution describes the fading turbulence, we can obtain the approximate analytic expression at multiple points in space at a single instant of time and is used 1296 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 8, AUGUST 2002

in Section IV in evaluating the performance of spatial diversity III. ML DETECTION OF ON-OFF KEYING IN reception. The joint temporal distribution describes the fading TURBULENCE CHANNELS at a single point in space at multiple instants of time. This dis- In this paper, we consider intensity modulation/direct detec- tribution is the basis for the MLSD introduced in Section III. tion (IM/DD) links using on-off keying (OOK). In most prac- We assume that the log-amplitude at receivers is described tical systems, the receiver signal-to-noise ratio (SNR) is limited by a joint Gaussian distribution. From (9), the auto-covariance by shot noise caused by ambient light much stronger than the de- receivers in a plane transverse matrix of the log-amplitude at sired signal and/or by thermal noise in the electronics following to the direction of propagation is given by the photodetector. In this case, the noise can usually be modeled to high accuracy as additive, white Gaussian noise that is statis- tically independent of the desired signal. Let denote the bit interval of the OOK system and assume that the receiver inte- grates the received photocurrent for an interval during (19) each bit interval. At the end of the integration interval, the re- sulting electrical signal can be expressed as: where is the distance between points and in the receiver plane. Based on “frozen-in” model and ignoring wind velocity (24) fluctuations, (19) can also be modified to describe temporal where is the received signal light intensity and is the am- fading correlation by making the substitution bient light intensity. Both of these quantities can be assumed to (20) be constant during the integration time. The optical-to-electrical conversion coefficient is given by where is the time interval between observations. In all that follows, we assume a communication system using OOK, in which (25) case, is the bit interval. We denote the covariance matrix of a string of bits as shown in (21) at the bottom of the page. From where is the quantum efficiency of the photodetector, is the the previous discussion about spatial and temporal correlation electron charge, is the signal wavelength, is Plank’s con- (see Section II-B), the correlation time equals . stant, is the speed of light. The additive noise is white and The joint distribution of intensity for a sequence of trans- Gaussian and has zero mean and covariance , independent mitted On bits is of whether the received bit is Off or On. In this section, we will describe symbol-by-symbol ML detection and compute its error probability in the absence of error- correction coding. We will then describe the MLSD.

A. Symbol-by-Symbol ML Detection 1) Description: We assume that the receiver has knowledge (22) of the marginal distribution of the turbulence-induced fading, but has no knowledge of the channel’s instantaneous fading. After subtraction of the ambient light bias , the signal Similarly, when an On bit is transmitted, the joint distribution is described by the following conditional densities when of intensity at receivers is given by the transmitted bit is Off or On, respectively

Off (26)

On On

(23) (27)

(21) ZHU AND KAHN: FREE-SPACE OPTICAL COMMUNICATION 1297

Fig. 3. Normalized threshold for maximum-likelihood symbol-by-symbol Fig. 2. Likelihood ratio versus normalized received signal amplitude for detection versus log-amplitude standard deviation for different values of the different values of the log-amplitude standard deviation. noise covariance.

The optimal maximum a posteriori (MAP) symbol-by- can be ignored when the bit rate is not high and multipath effects symbol detector decodes the bit as [16], [17] are not pronounced, we have (28) Bit Error Off Off (32) where is the probability that a On bit or Off bit is transmitted. is the conditional distribution that if a bit (On Bit Error On On (33) or Off) is transmitted, a signal level will be received. If On and Off bits are equiprobable, or if their a priori probabilities are unknown, the symbol-by-symbol ML detector decodes the bit as B. Maximum-Likelihood Sequence Detection (MLSD) (29) The MLSD exploits the temporal correlation of turbulence- induced fading and is thus expected to outperform the symbol- The likelihood function is by-symbol ML detector. For a sequence of transmitted bits, On the MLSD computes the likelihood ratio of each of the 2 Off possible bit sequences and chooses

(30) In Fig. 2, we see that the likelihood ratio increases monotoni- cally with for , so that ML detection can be imple- mented by simply thresholding the received signal. In Fig. 3, we plot the optimal ML threshold for versus the log-am- (34) plitude standard deviation . We see that as increases, the ML threshold decreases toward zero, because turbulence-in- Here, each can take the value Off or On, so that . duced fading increases the fluctuation of the On-state signal level, while leaving the fluctuation of the Off-state signal level unchanged. In Fig. 3, we also see that as the additive Gaussian noise covariance increases, the fluctuations of the Off and On states become more closely equal and the threshold increases toward 1/2. 2) Error Probability Calculation: The bit-error probability (35) of OOK can be computed as: Off Bit Error Off On Bit Error On The complexity of MLSD is proportional to , because it (31) requires computing an -dimensional integral for each of 2 bit where Bit Error Off and Bit Error On denote the bit- sequences. This complexity is excessive for most applications. error probabilities when the transmitted bit is Off and On, re- In a companion paper [14], we describe two reduced-complexity spectively. Without considering intersymbol interference, which implementations of the MLSD, which make use of a single-step 1298 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 8, AUGUST 2002

Markov chain model for the fading correlation in conjunction The ML detector employs the decision rule . Since with per-survivor processing. the log-amplitude follows a joint log-normal distribution, calculation of the likelihood function in (39) involves multi-dimen- IV. SPATIAL DIVERSITY RECEPTION sional integration. We emphasize that this decision rule has been derived under the assumption that the receiving party knows the Spatial diversity reception, which has been well-studied for fading correlation but not the instantaneous fading state. application at radio and microwave frequencies, has the poten- The bit-error probability of the ML receiver is given by tial to mitigate the degradation caused by atmospheric turbulence [16]–[20]. Spatial diversity reception in free-space optical Off Bit Error Off On Bit Error On communication has been proposed and studied in [18]–[20]. (40) Ibrahim [19] has studied the performance of spatial-diversity where Bit Error Off and Bit Error On denote the optical reception on turbulence channels, assuming that turbu- bit-error probabilities when the transmitted bit is Off and On, lence-induced fading is uncorrelated at each of the optical re- respectively. Without considering intersymbol interference, ceivers. In order for this assumption to hold true, the spacing which can be ignored when the bit rate is not high and multipath between receivers should exceed the fading correlation length effects are not pronounced, we have: in the plane of the receivers. It may be difficult to satisfy this assumption in practice, for various reasons. Available space may Bit Error Off Off (41) not permit sufficient receiver spacing. In power-limited links, and which often employ well-collimated beams, the receiver spacing required for uncorrelated fading may exceed the beam diameter. Bit Error On On (42)

A. Maximum-Likelihood Diversity Detection on Turbulence To evaluate the optimal ML diversity detection scheme, we Channels compare it with the conventional equal-gain combining (EGC) In the case of spatial diversity reception with receivers, scheme [16]. In the EGC scheme, we assume that the receiving part has knowledge of the marginal distribution of the channel the received signal is described by an -component vector . fades, in each receiver, but has no knowledge of the fading cor- Taking account of correlation between the receivers, (26) and relation or the instantaneous fading state. For each individual (27) are modified as follows: receiver output, we can find an optimum threshold . Then the Off (36) EGC detector adds together the receiver outputs with equal gains and compares the sum to the threshold

On (43)

The error probability of EGC is

Bit Error (44)

(37) where where is the noise covariance of the th receiver and Bit Error Off On

(45)

B. Numerical Simulation for Dual Receivers (38) In this section, we present numerical simulations of the Here, is the covariance matrix of the log-amplitudes in the performance of spatial-diversity detection for the dual-receiver receivers, as in (19). The likelihood function is case, which is illustrated in Fig. 4. As described above, the ML receiver [Fig. 4(a)] has full knowledge of the turbulence-in- On duced fading correlation matrix , while the EGC receiver [Fig. 4(b)] has knowledge only of the marginal distribution of Off fades at receivers. We assume that and and define the electrical signal-to-noise ratio . We have used the expressions given in the previous section to numerically compute the average bit-error probabilities. In Fig. 5(a) we plot the simulation results, assuming (39) and , varying the normalized correlation ZHU AND KAHN: FREE-SPACE OPTICAL COMMUNICATION 1299

from 0 to 0.9. In Fig. 5(b), we present corresponding results assuming . Comparing Fig. 5(a) and (b), we see that turbulence-induced fading causes a greater degradation of the bit-error probability when the standard deviation is larger. Diversity reception with two receivers can improve the performance as compared to a single receiver. With two receivers, ML detection achieves better performance than EGC for a given SNR. The advantage of ML over EGC is more pronounced when the correlation between the two receivers is high. It is also more pronounced when the SNR is high, so that errors are caused mainly by turbulence-induced fading, as op- posed to noise.

V. C ONCLUSIONS Fig. 4. Dual-branch reception on atmospheric turbulence channels with correlated turbulence-induced fading. (a) Maximum-likelihood detection. Free-space optical communication through atmosphere tur- (b) Equal-gain combining with threshold detection. bulence is now under active research and various methods have been proposed to mitigate turbulence-induced communication signal fading. In this paper we present detection techniques for free-space optical communication systems in which and . These techniques can help to mitigate the effects due to turbulence-induced log-amplitude fluctuations. The ML detection schemes over turbulence channels are studied based on the statistical distributions of turbulence-induced fading. If the instantaneous fading state is unknown but the marginal fading statistics are known, we can apply ML symbol-by-symbol detection to improve detection efficiency. If the temporal correlation of fading is known, i.e., the joint temporal distribution of turbulence-induced fading, we can apply MLSD, leading to a further improvement in detection performance. Spatial diversity reception can also help to mitigate turbulence-induced fading. When the spacing between receivers is not much greater than the fading correlation length, diversity (a) gain is reduced by correlation, but ML detection can be used to overcome some of this loss. We have shown that in the dual-receiver case, ML diversity reception outperforms the conventional EGC method.

REFERENCES [1] P. F. Szajowski, G. Nykolak, J. J. Auborn, H. M. Presby, and G. E. Tourgee, “High-power optical amplifier-enabled 1550-nm terrestrial free-space optical data-link operating at 10 Gb/s,” in Proc. IEEE Mil. Commun. Conf, Atlantic City, NJ, Oct.–Nov. 31–3, 1999, pp. 687–689. [2] V. Hsu, J. M. Kahn, and K. S. J. Pister, “Wireless communications for Smart dust,” Univ. of California, Berkeley, Electron. Res. Lab. Tech. Memo. M98/2, 1998. [3] J. M. Kahn, R. H. Katz, and K. S. J. Pister, “Mobile networking for smart dust,” in Proc. ACM/IEEE Intl. Conf. on Mobile Computing and Networking (MobiCom 99), Seattle, WA, Aug. 17–19, 1999, pp. ???–???. [4] P. B. Chu, N. R. Lo, E. Berg, and K. S. J. Pister, “Optical communication (b) using micro corner cuber reflectors,” in Proc. IEEE MEMS Workshop, Nagoya, Japan, Jan. 26–30, 1997, pp. 350–355. Fig. 5. Bit-error probability of dual-branch receiver versus average electrical [5] D. Sadot, A. Melamed, and N. S. Kopeika, “Effects of aerosol for- signal-to-noise ratio using maximum-likelihood detection (lines with circles) ward scatter on the long- and short-exposure atmospheric coherence and equal-gain combining (lines without symbols) for different values of & , diameter,” Waves in Random Media, vol. 4, no. 4, pp. 487–498, Oct. the normalized correlation between the two receivers. The line with squares 1994. represents the bit-error probability using a single receiver. The turbulence- [6] S. Arnon and N. S. Kopeika, “Effect of particulates on performance of induced fading has standard deviation ' aHXI in (a) and ' aHXPS in (b). optical communication in space and an adaptive method to minimize In both (a) and (b), this fading has mean iaH. such effects,” Appl. Opt., vol. 33, no. 21, pp. 4930–4937, July 1994. 1300 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 8, AUGUST 2002

[7] M. Born and E. Wolf, Principles of Optics, 6th ed. city??, Canada: Xiaoming Zhu was born in 1974 and received the Pergamon, 1980. B.S. degree from Tsinghua University, Beijing, [8] J. W. Goodman, Statistical Optics. New York: Wiley, 1985. China, in 1997. He is currently working toward [9] A. Ishimaru, Wave Propagation and Scattering in Random the Ph.D. degree in the Department of Electrical Media. New York: Academic, 1978, vol. 1–2. Engineering and Computer Sciences, University of [10] S. Karp, R. Gagliardi, S. E. Moran, and L. B. Stotts, Optical Chan- California, Berkeley. nels. New York: Plenum, 1988. His areas of interests includes wireless communi- [11] N. S. Kopeika, A. Zilberman, and Y. Sorani, “Measured profiles of cation, free-space optical communication, and optical aerosols and turbulence for elevations of 2–20 km and consequences fiber communication networks. on widening of laser beams,” in Proc. SPIE Optical Pulse and Beam Propagation III, vol. 4271, 2001. [12] A. Zilberman, N. S. Kopeida, and Y. Sorani, “Laser bean widening as a function of elevation in the atmosphere for horizontal propagation,” in Proc. SPIE Laser Weapons Technology II, vol. 4376, 2001. [13] L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Joseph M. Kahn (M’90–SM’98–F’00) received the Random Media: SPIE Optical Engineering Press, 1998. A.B., M.A. and Ph.D. degrees in physics from Uni- [14] X. Zhu and J. M. Kahn, “Markov chain model in maximum-likelihood versity of California at Berkeley in 1981, 1983, and sequence detection for free-space optical communication through atmo- 1986, respectively. spheric turbulence channels,” IEEE Trans. Commun., submitted for pub- He is currently a Professor in the Department lication. of Electrical Engineering and Computer Sciences [15] D. L. Fried, “Statistics of a geometric representation of wavefront dis- at University of California at Berkeley. In 2000, tortion,” J. Opt. Soc. Amer., vol. 55, no. 11, pp. 1427–1435, Nov. 1965. he co-founded StrataLight Communications, Inc., [16] G. L. Stuber, Principles of Mobile Communication. New York:Kluwer where he is currently Chief Scientist. From 1987 to 90 he was a Member of Technical Staff in the Academic , 1996. Lightwave Communications Research Department [17] J. G. Proakis, Digital Communication, 3rd ed. New York: McGraw- of AT&T Bell Laboratories, where he performed research on multi-gi- Hill, 1995. gabit-per-second coherent optical fiber transmission systems, setting several [18] M. Srinivasan and V. Vilnrotter, “Avalanche photodiode arrays for op- world records for receiver sensitivity. He joined the faculty of University of tical communication receivers,”, NASA TMO Progress Rep. 42-144, California at Berkeley in 1990. His current research interests include wireless 2001. communication using antenna arrays, free-space optical communication, [19] M. M. Ibrahim and A. M. Ibrahim, “Performance analysis of optical re- optical fiber communication, and wireless communication for networks of ceivers with space diversity reception,” Proc. IEE—Commun., vol. 143, sensors based on micro-electromechanical systems. no. 6, pp. 369–372, December 1996. Prof. Kahn received the National Science Foundation Presidential Young In- [20] X. Zhu and J. M. Kahn, “Maximum-likelihood spatial-diversity recep- vestigator Award in 1991. He is a Fellow of the IEEE, and is a member of the tion on correlated turbulent free-space optical channels,” presented at the IEEE Communications Society, the IEEE Information Theory Society, and the IEEE Conf. on Global Commun., San Francisco, CA, Nov.–Dec. 27–1, IEEE Lasers and Electro-Optics Society. From 1993 to 2000, he served as a 2000. technical editor of IEEE Personal Communications Magazine.