A Robust Continuous-Time Multi-Dithering Technique forLaser Communications using Adaptive Optics

Dimitrios N. Loizos Paul P. Sotiriadis Gert CauwenberghsElectrical and Computer Engineering Electrical and Computer Engineering Division of Biological Sciences

The Johns Hopkins University The Johns Hopkins University University of California, San DiegoBaltimore, MD 21218 Baltimore, MD 21218 San Diego, CA 92093

dloizos@jhu.edu pps@jhu.edu gert@ucsd.edu

Abstract- A robust system architecture to achieve optical coherency free optimization. Several methods that had been proposed in thein multiple-beam free-space laser communication links with adaptive past, such as multi-dithering [4] and sequential perturbations [8],optics is introduced. It is based on deterministic multi-dithering and as well as the recently proposed stochastic parallel gradient descentgradient descent flows and accounts for phase delays in the dither signal,during propagation in the atmosphere, as well as saturation of the optical [5] and decoupled stochastic parallel gradient descent [2] methodsphase shifters. The architecture has been mathematically analyzed and seem very promising and can potentially enhance the quality of lasersimulation results of a VLSI implementation of the architecture are communications.presented and found in agreement with the theoretical model. The purpose of the present work is to propose a system architecture

that can implement the multi-dither algorithm for adaptive opticsrobustly and in continuous-time. Similar architectures have been

The inherent property of high carrier frequency characterizing laser proposed in the past ([4], [9], [10]), however the one proposedcommunications has always been one of their major advantages over here features several properties that make it attractive for use inconventional RF communications, as it provides a larger modulation modem laser communications. First, it is meant for implementationbandwidth and therefore higher data rates, and less interference in a VLSI circuit, combining low-power consumption, compact size,between optical systems. The reason that laser communications have faster computation, higher perturbation frequencies and control ofnot expanded as expected, so far, has been the difficulty to compen- multiple channels. Second, it compensates for possible phase shiftssate for atmospheric turbulence. Turbulence causes random optical of the dither signal in the case of long-distance communications.inhomogeneities in the atmosphere that result in beam spreading, Finally, it accounts for saturation of the phase controllers when thebeam wander and scintillation (intensity fluctuation), and therefore multi-dither algorithm dictates continuous increase or decrease in thedistort the transmitted data [1]. controlling phase.

Over the last few decades, significant work has been done in thefield of adaptive optics in order to alleviate or at least mitigate wave- II. THE MULTI-DITHER ALGORITHMfront distortion and therefore enhance the use of laser communication The purpose of the multi-dither algorithm is to maximize the valuesystems. The techniques that have been so far proposed can be divided of a performance metric of the optical system, usually the wave-into two main categories [2]: wave-front conjugation and model-free front's intensity. This is done by calculating the gradient of theoptimization. Wave-front conjugation requires the reconstruction of metric and then, using the gradient descent method, find its maximumthe wave-front at the receiver and provides information for both phase value. First, a brief overview of the gradient descent method is givendistortion and amplitude modulation [3]. Model-free optimization and then the mathematical approach to the multi-dither algorithm istechniques are computationally simpler and focus on optimizing presented.a system performance metric with respect to phase, without anyknowledge of the propagation path [2], [4], [5]. This is usually done A. The gradient descent methodby superimposing a perturbation to a control signal and calculating Consider the real vector u T(U, U2, U T and a scalar costthe gradient of the metric.

function J(u). The basic form of the gradient descent method isAlthough the wave-front conjugation technique has the advantage expressed by equation (1) where E is a positive constant and u evolves

of being able to derive the additional information of amplitude in time according to the differential equation (gradient flow)modulation, phase-only correction of the wave-front is theoreticallysufficient for a propagation distance in the near field, and has du = _EVJ(U) (1)been also demonstrated to provide significant improvement in the dttransmitted beam quality for large distance optical communications Taking the derivative of J(u(t)) with respect to time gives[6]. Most of the adaptive optics techniques nowadays use phase-onlycorrection for the wave-front control. Moreover, under conditions dJ(u(t)) = (VJ(u) d -=-e VJ(u) 2 (2)of strong intensity modulation used in several laser communication dt dtschemes, the wave-front can be difficultly reproduced, imposing indicating that J(u(t)) is a strictly decreasing function of time unlessconstraints in the use of the wave-front conjugation technique [5]. u has reached an equilibrium point of (1).The factors discussed above, as well as the emergence of high- Assuming that J does not have saddle points, (2) implies that the

bandwidth phase controllers (e.g. [7]) that make possible small high- gradient flow (1) reaches a (local) minimum of J. By flipping thefrequency phase perturbations of the transmitted beam, have shifted sign of E, we get the gradient flow du/dt =EVJ(U) that reaches athe research interest in adaptive optics techniques towards model- (local) maximum of J.

0-7803-9390-2/06/$20.00 ©C2006 IEEE 3626 ISCAS 2006

A variation of (1) having practical advantages is derived by keeping Note that there is only one constant term on the right hand sideonly the sign information of the partial derivatives in VJ(u), i.e. of (7). Assuming that the frequency differences (wi- wj), i :7 j are

not very small, passing signal (7) through a low pass filter givesdt

= esgn(VJ(u)) (3) J(ui) cos(wjt) = (8)

where the sgn function is applied element-wise to vector VJ(u). 2 o9 uFollowing similar steps as before we get that (where over-line means low-passing). This process is performed in

d (U (V j (U)) T dun 09i parallel for all channels, j = 1, 2, ..., n, and in this way the gradient

J = :Tdu (4) of J is reconstructed.dt dt i=1 1AiI The above analysis indicates that both the dither frequencies, wj,

which leads to the same qualitative behavior as (1) but requires much as well as the dither frequency differences wi-wj, i 7 j should

simpler, ±1, control signals. be larger than the bandwidth of the low-pass filter, in order to avoidcrosstalk between the dithering signals of the channels.

B. Sinusoidal multi-dithering Expression (8) was derived with the implicit assumption that thedetection is synchronous, i.e. zero phase difference between the

In order to converge to the local (global) minimum/maximum, cosines. In reality however, the phases of the perturbation signals arethe gradient descent/ascent method needs the information of the delayed by = wi T where T is a summation of several propagationgradient's value for metric J. In our approach, this is provided delays and processing times: the time the laser beam travels from thethrough the multi-dither algorithm. Let n be the size of vector u, transmitter to the receiver, the processing time it takes the receiverthat represents the control signals of the phase controllers. In the to estimate the instantaneous power and transmit it back to the phase(deterministic) sinusoidal multi-dithering method, sinusoidal dithers controller, the time the feedback signal travels from the receiver to theof different frequency for each element (channel) of vector u, are phase controller at the transmitter, plus any delay due to processingsuperimposed to the control u in the phase controller's module. By taking this into account we get

i = U + 6 = U + a [CoS(wlt), cos(w2t), * cos(wt)] (5) C nt) T-dela) cos(w t) = 2 cos(Oj). (9)

where a is a small scalar parameter. Evaluating J using Taylor series u

we get that Combining the above multi-dithering approach for estimating thegradient vector with the modified gradient descent, (3), we get

J(fi) J(u + 0) = J(u) + (VJ(u))TO + ... du =n sgn[() (10)____ dtfi T~delaCOS(WJt)Ij)= J(u) + aZXn | cos(wit) + . . dt

tu(6) The sign of e was flipped since we want to maximize J. Note that

To extract the entries of the gradient vector VJ(u) we first multiply cos(9oj) does not influence the result as long as the delay correspondsJ(ui) with cos(wjt), j = 1, 2, ... , n. This gives to less than 900 phase shift. The proposed system architecture

described in the next Section is able to compensate for larger phaseJ(i)cos(wjt) J(u) cos(wjt) shifts.

+ a +-J cos(2wt) Written in a more compact way, equation (10) can take the form

a 092&u3=d sg J(u - T)j) (I11)+2 i cos ((wi - wj) t) (7) dtVJ(u(t

2E 0J u III. SYSTEM ARCHITECTURE2 9ui cos ((wi + ) t) The configuration of the overall optical communication system

+ H.O.T. is shown in Fig. 1. A modulated high power laser beam is split

Trarsritter~~~~~~~~~~~~~~poe

Modulated \

Amplifier 0 Phase Shifter

|-Phase Controller -------------------------------------------------

Fig. 1. Optical system configuration

3627

3-phasemultiplier phase selection 3 oscillator buffer capacitor array buffer

received metric <,~ u'j to phase shifter

amplifier 5th order quantizer sign selection charge J 2rrChebyshev LPF pump controller

Fig. 2. Block diagram for one channel of the phase controller

into n different channels. The beams at each channel are amplified opposite to what we would expect. To compensate for this, a 3-and then phase shifted according to the controls provided by the phase oscillator (phases at 00, 1200 and 2400) is proposed to bephase controller. Each beam follows a different path through the used in the architecture, giving the flexibility to choose the phaseatmosphere, that is considered of unknown properties. The wave- of the oscillator closest to that of the perturbation after propagatingfront at the receiver is a combination of the wave-fronts from all through the atmospheric medium. An additional degree of freedomchannels and its intensity is detected through a photodiode. The for choosing the phase is given by the sign selection block (Fig. 2).desired performance metric is calculated and the information of its Graphically, this can be represented as a division of the phase domainvalue is transmitted back to the phase controller. The phase controller into 6 equally sized subdomains (Fig. 3). Depending on the phasecalculates the appropriate phase shifts that maximize the metric and delay of the perturbation, the appropriate phase of the oscillator andadjusts the control of the phase shifters, closing the feedback loop. the correct sign can be selected so that the phase subdomain where theThe purpose of this work, as mentioned above, is to present a phase delay lies on is chosen. At this point it is worth mentioning

system architecture for the phase controller. The phase controller that the sign selection block (Fig. 2) gives also the possibility ofcan control multiple channels. In Fig. 2, the system architecture for performing minimization instead of maximization of the performancechannel j, j = 1 ... n is shown. The control u', that adjusts the metric, if this is required.phase shifters, constitutes of two components: a slow-varying oneprovided by the charge pump, and a high-frequency small amplitude 1200 60Odither coming from the oscillator. The amplitude of the dither is set 15O/ 300by appropriately adjusting the ratio between C and the capacitance 180° \ 0°of the capacitor array. The signal u' controls the phase alteration ofthe transmitted laser beam from this channel. 210 3-3,,

After the performance metric J has been computed in the receiver, 240 300and the information of its value has been transmitted to the phasecontroller, synchronous detection is performed at every channel, asdictated by (7). The phase selection attribute will be discussed later. Fig. 3. Phase domain - the dashed lines represent the boundaries of the

in subdomains and the bold lines the selected phase for the synchronous detectorAfter the metric and the oscillator's signal have been multiplied in at these subdomainschannel j, the output is low-passed by a 5th order Chebyshev filter,that features sharp cutoff and low ripple in the bandpass. Accordingto the analysis of Section II, this lowpass filtering should provide us B. Quantizationwith the partial derivative of J with respect to uj. This information is Instead of using directly the gradient of J, it is proposed to useenough to perform gradient descent maximization. However, we have only the information of its sign, provided by the quantization of thechosen to quantize this partial derivative and retrieve only the sign lowpass filter's output. As mentioned in Section II, this information isinformation for reasons that will be explained later. A controllable sufficient to perform the gradient descent algorithm. Moreover, givencharge pump, then, scales the magnitude of the sign information (the that the correct phase subdomain has been selected, this informatione used in Section II) and updates the voltage on capacitor C. is independent of any phase offset oi between the perturbation of theThe 27r controller compares the value of control u; with an upper metric and the selected phase of the oscillator. However, the choice of

and lower threshold. These thresholds indicate the upper and lower keeping only the sign information of the gradient was mainly madelimits that the controls u of the phase shifters can take before the for issues of stability of the feedback system.phase shifters saturate. If u' exceeds or drops below these thresholds,the voltage on capacitor C is instantaneously set to a predefined Lowpass Filtervalue. The voltage drop or jump on the capacitor and therefore on H(f)

uj corresponds to a 27r phase shift on the phase shifter. Therefore, y x

the phase shifters are effectively driven away from saturation without sgn(x) or f(x)any impact on the instantaneous phase shift added to each channel.

A. Phase selection Fig. 4. Simplified closed loop diagram for the feedback of each channel

As was discussed in Section II, in the case where the atmosphericpath introduces a phase delay greater than 900 to the perturbation, Consider the feedback loop for each channel which can be reducedsynchronous detection might fail, giving a signal whose sign is to the system of Fig. 4. Assuming that the lowpass filter H(f) has

3628

2.30- 2 30I

2 0i 2. 02

,1-70i 70i-1f;20 >22

1 30.2 >320

1.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 01 1 O r(a) L1~(a)2.2 2,2j43LLiu 22u 52u 42 u132j

me ( ) tnle s )

2_2-5N J(u)X,

c I X2a a5 2J(tu)g (b) rX(b)

X,O ~ou 2-L Xu i2 You 6O X k 1 ou 2 6 237u 4.2u 6.2u 2XXt'men s t ime i:s)

Fig. 5. Simulation results for perturbation frequencies between 100-800MHz Fig. 6. Simulation results for perturbation frequencies between 100-800MHzand high gain in the charge pump and low gain in the charge pump

at least two poles, the closed system will potentially oscillate at the Similar conclusions have been derived by simulations of the systemfrequency at which the Barkhausen phase criterion for oscillation is at frequencies in the range between 100 and 800kHz.satisfied, i.e. when the phase response of the filter reaches 1800. For V. CONCLUSIONthe system to indeed oscillate, the Barkhausen amplitude criterion A system architecture implementing a multi-dithering algorithmfor oscillation needs to be also satisfied. In the case where a limiter with gradient descent for adaptive optics has been described and an-function is used in the closed loop, the criterion is always satisfied and alyzed. The architecture compensates for phase delays and saturationthe loop will always oscillate. However, the amplitude of oscillation of the controlled optical phase shifters. The system is capable ofis controllable by adjusting the bounds of the limiting function. If, on minimizing or maximizing the performance metric and the gradientthe other hand, a linear feedback is used, the system can potentially descent optimization is achieved using only the sign information ofbe made stable, if all the gains and phase responses of the system the gradient. Simulation results of a VLSI implementation in SiGeare known. Nonetheless, it may not be easy to predict these gains, technology have been presented and demonstrate the validity of theMoreover, if the system starts oscillating, the amplitude of oscillation architecture.will be limited only by the non-linearities of the components.

REFERENCESIV. SIMULATION RESULTS [1] T. Weyrauch and M.A. Vorontsov, "Free-space laser communications

The proposed architecture with 8 control channels has been de- with adaptive optics: Atmospheric compensation experiments," J. Opt.Fiber Commun. Rep., 2004, pp. 355-379.

signed in a 0.5,uim SiGe technology. The design of the channels has [2] M.A. Vorontsov, "Decoupled stochastic parallel gradient descent opti-been done in such a way so that the oscillation frequency and the mization for adaptive optics: integrated approach for wave-front sensorcutoff frequency of the lowpass filters are tunable over at least 3 information fusion," J. Opt. Soc. Am. A, February 2002, Vol. 19, No. 2,decades. Some preliminary simulation results are presented that prove pp. 356-368.the validity of the architecture. Simulations were performed for the [3] M.D. Levenson, "High-resolution imaging by wave-front conjugation,"

Optics Letters, May 1980, Vol. 5, No. 5, pp. 182-184.case of minimization of the performance metric, but as discussed [4] T.R. O'Meara, "The multidither principle in adaptive optics," J. Opt.earlier, the system can also perform maximization by appropriately Soc. Am., March 1977, Vol. 67, No. 3, pp. 306-315.controlling the sign selection signal (Fig. 2). [5] M.A. Vorontsov, G.W. Carhart, M. Cohen and G. Cauwenberghs, "Adap-

tive optics based on analog parallel stochastic optimization: analysis andAhe firstchannesimuatienw conductedzwith pterturbaIOnOfreencs fr experimental demonstration," J. Opt. Soc. Am. A, August 2000, Vol. 17,

the 8 channels between 100-800MHz and steps of 100MHz, and a No. 8, pp. 1440-1453.performance metric to be minimized J(u) = i(ui- 2)2. The [6] B.M. Levine, E.A. Martinsen, A. Wirth, A. Jankevics, M. Toledo-cutoff frequency of the lowpass filters was set to 10MHz. Fig. 5(b) Quinones, F. Landers and T.L. Bruno, "Horizontal line-of-sight tur-demonstrates how the metric is minimized, while Fig. 5(a) shows bulence over near-ground paths and implications for adaptive optics

the expected oscillation oftecotrolincorrections in laser communications," Applied Optics, 20 July 1998, Vol.the expected oscillation of the controlling signals ui, j= 1 . . . 8, 37, No. 21, pp. 4553-4560.around the expected value of 2V, after convergence. A limit cycle of [7] T. Bifano, J. Perreault, P. Bierden and C. Dimas, "Micromachinedamplitude around 100mV can be observed for the signals ui, due to Deformable Mirrors for Adaptive Optics," Proc. SPIE, 2002, Vol. 4825,the high gain (parameter e in Section II) that had been selected for pp. 10-13.

the charge pump. Convergence isach[8] J.W. Hardy, "Active Optics: A New Technology for the Control of Light,"the charge pump. Convergence iS achieved in almost lbuls. Proc. IEEE, June 1978, Vol. 66, No. 6, pp. 651-697.A second simulation with the exact same setup as the first one [9] W.B. Bridges, P.T. Brunner, ST Lazzara, T.A. Nussmeier, T.R. OMeara,

but with a considerably lower gain for the charge pump was run. J.A. Sanguinet and W.P. Brown, Jr., "Coherent Optical Adaptive Tech-Fig. 6(a) shows that the amplitude of the limit cycle dropped, to a niques," Applied Optics, February 1974, Vol. 13, No. 2, pp. 291-300.point~.thtircial elgbe hl i.6(). eosrtsta [10] J.E. Pearson, W.B. Bnidges, S. Hansen, T.A. Nussmeier and M.E.

' ' ~~~~~~~~~~Pedinoff,"Coherent optical adaptive techniques: design and performancethe convergence time increased, indicating a trade-off between the of an 18-element visible multidither COAT system," Applied Optics,convergence time of the system and the amplitude of the limit cycle. March 1976, Vol. 15, No. 3, pp. 611-621.

3629