fore, the performance of eye diagrams and that of BER is consis-tent with each other very well. The square in Figure 2(b) corre-sponds to the best measured BER with regard to different residualdispersions, which agree with the simulation results very well.

The measured mean PMD of the 300 km transmission line is0.8 ps. As the number of recirculations increases, the mean PMDincreases linearly with the number of recirculations.

Figure 3 shows the BER evolution versus transmission distancefor dispersion compensating using TDC and short fibers, respec-tively. The BER with TDC is similar to that with short fiberswithin 600 km transmission, but the clock signal cannot be recov-ered any more when the distance increases further. However, theclock signal could be recovered until 1500 km transmission for thecase of short fibers. The reason is that the combined effect of PMDand group delay ripple in FBG result in more serious samplingtiming shift. The sampling time may no longer be at the center ofthe eye diagram, which lead to the BER degradation.

5. CONCLUSIONS

Both PMD and group delay ripple in FBG can induce the phaseshift of recovered clock signal, and consequently affect the sam-pling time shift. Therefore the sampling time may be not at thecenter of eye diagrams, and this will result in significant BERdegradation in high speed optical transmission, which is experi-mentally demonstrated in 42.8 Gbit/s recirculating transmission.

ACKNOWLEDGMENTS

This work is supported by the National Hi-tech Research andDevelopment Program of China (863 Program) (No.2006AA01Z253 and No. 2006AA01Z261). The authors also ac-knowledge the donation of VPI software suite from Alexander vonHumboldt foundation.

REFERENCES

1. E. Iannone, F.Matera, A. Galtarossa, G. Gianello, and M. Schiano,Effect of polarization dispersion on the performance of IM-DD com-munication systems, IEEE Photon Technol Lett 5 (1993), 1247–1249.

2. C.R. Menyuk and B.S. Marks, Interaction of polarization mode disper-sion and nonlinearity in optical fiber transmission systems, J LightwaveTechnol 24 (2006), 2806–2826.

3. L.E. Nelson, T.N. Nielsen, and H. Kogelnik, Observation of PMD-induced coherent crosstalk in polarization-multiplexed transmission,Photon Technol Lett 13 (2001), 738–740.

4. Y. Benlachtar, R.I. Killey, and P. Bayvel, The effects of polarization-mode dispersion on the phase of the recovered clock, J LightwaveTechnol 24 (2006), 3944–3952.

5. N. Cheng and John C. Cartledge, Power penalty due to the amplitudeand phase response ripple of a dispersion compensating fiber bragggrating for chirped optical signals, J Lightwave Technol 24 (2006),3363–3369.

6. R.L. Lachance, M. Morin, and Y. Painchaud, Group delay ripple in fibreBragg grating tunable dispersion compensators, Electron Lett 38 (2002),1505–1507.

7. N. Litchinitser, Y. Li, M. Sumetsky, P. Westbrook, and B. Eggleton,Tunable dispersion compensation devices: Group delay ripple and sys-tem performance, OFC TuD 2 (2005), 163–164.

8. D.C. Zhang, X.L. Li, X.R. Zhang, J.H. Li, A.S. Xu, Z.Y. Wang, Z.Y.Chen, et al., 42.8 Gbit/s electro-absorption modulated NRZ transmis-sion over 1200 km standard singlemode fibre, Electron Lett 43, (2007),237–239.

© 2007 Wiley Periodicals, Inc.

A HYBRID APPROACH FOR MODELINGSTOCHASTIC RAY PROPAGATION INSTRATIFIED RANDOM LATTICES

Anna Martini,1 Massimo Franceschetti,1,2 and Andrea Massa1

1 Department of Information and Communication Technology,University of Trento, Via Sommarive 14, I-38050 Trento, Italy;Corresponding author: andrea.massa@ing.unitn.it2 Department of Electrical and Computer Engineering, University ofCalifornia at San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0407

Received 27 April 2007

ABSTRACT: The present contribution deals with ray propagation insemi-infinite percolation lattices consisting of a succession of uniformdensity layers. The problem of analytically evaluating the probabilitythat a single ray penetrates up to a prescribed level before being re-flected back into the above empty half-plane is addressed. A hybrid ap-proach, exploiting the complementarity of two mathematical models indealing with uniform configurations, is presented and assessed throughnumerical ray-tracing-based experiments to show improvements uponprevious predictions techniques. © 2007 Wiley Periodicals, Inc.Microwave Opt Technol Lett 49: 3068–3073, 2007; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22934

Key words: stochastic ray tracing; stratified random media; percolationtheory

1. INTRODUCTION

This article deals with wave propagation in stratified randomlattices where the electromagnetic source is external to the half-plane filled by the obstacles and it radiates a plane monochromaticwave impinging on the lattice with a known incidence angle �. Byassuming that the dimension of each site is large with respect to thewavelength, the wave is modeled in terms of a collection ofparallel rays that undergo specular reflections on occupied cells.The aim is analytically estimating the probability Pr{0 3 k} thata single ray reaches a prescribed level k inside the lattice beforebeing reflected back in the above empty half-plane.

The problem concerned with a uniform random grid where sitesare occupied with a known probability q � 1 � p was addressedin [1]. Ray propagation was modeled in terms of a one-dimen-sional stochastic process and Pr{03 k} was evaluated by applyingthe theory of the Martingale random processes [2] and the so-

Figure 3 BER versus transmission distance

3068 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 12, December 2007 DOI 10.1002/mop

called Wald’s approximation. The extension to the two-dimen-sional case, where an isotropic source is located inside the randomlattice, has been proposed in [3]. Moreover, the dual case of smallobstacles has been dealt with in [4 and 5]. As far as the one-dimensional percolation model is concerned, the approach pro-posed in [1] and referred to as Martingale approach (MTGA), hasbeen successively extended to the nonuniform case, where theoccupancy probability changes according to a known obstacles’density q(j), j being the row index [6]. To apply the theory of theMartingale random processes and the Wald’s approximation, theray jumps following the first one are assumed independent, iden-tically-distributed, with mean and standard deviation approachingzero. Both mathematical considerations and numerical experi-ments have shown that these conditions are verified provided that(a) the incidence angle is not far from 45 ° or a large number ofreflections takes place, (b) the percolation lattice in hand is dense,and (c) the density profile does not present abrupt changes in valuebetween adjacent levels and a significant variation along the lat-tice. With reference to the last condition, it is evident that theMTGA fails when dealing with ray propagation in stratified ran-dom lattices, since such configurations are characterized by step-like variations in the density profile. To overcome such a draw-back, an ad hoc formulation, referred to in the following asMultilayer Martingale approach (MMTGA), has been described in[7]. Starting from the observation that a stratified random grid ismade up by a succession of uniform density layers, the propagationinside each single layer is faithfully described by the MTGA [butstill under conditions (a) and (b)]. Thus, mathematically bindingthe terms predicting propagation in each single uniform layer leadsto a formal description of the ray propagation in the whole strat-ified lattice.

Another approach for to estimating the probability Pr{0 3 k}is the so-called Markov approach (MKVA) [8], where the raypropagation is modeled by means of a Markov chain [9]. Byobserving that whenever a ray hits an occupied vertical face it doesnot change its vertical direction of propagation, only reflections onoccupied horizontal faces, whose number is independent from theincidence angle �, play a relevant role in evaluating Pr{0 3 k}.Likewise the MTGA, the MKVA properly works provided thatsome conditions are verified. In particular, it looses accuracy,when (i) the incidence angle deviates from 45 ° and (ii) theobstacles’ density increases. However, there are no requirementson the density profile and thus, when stratified random lattices aretaken into account, the MKVA allows reliable predictions [underconditions (i) and (ii)] and a customized formulation is not needed.

A comparison between the MMTGA and the MKVA whendealing with ray propagation in stratified random lattices has beenpresented in [10]. Numerical experiments have pointed out that theMMTGA outperforms the MKVA when dense lattices are consid-ered, whereas the MKVA offers more reliable predictions whenthe stratified grid is constituted by sparse layers. Such resultssuggest that it could be profitable to consider a hybrid procedureexploiting in a complementary fashion the MTGA [1] and theMKVA [8].

The article is organized as follows. In Section 2, the problem isstated and the mathematical formulation is briefly resumed. Sec-tion 3 deals with the numerical validation, showing improvementsof the proposed approach upon previous results. Final commentsand conclusions are drawn in Section 4.

2. PROBLEM STATEMENT AND MATHEMATICALFORMULATION

Let us consider a stratified random grid (see Fig. 1) made up by asuccession of uniform layers denoted by the index n. The n-th layer

is made up by Ln levels characterized by the same occupancyprobability qn and identified by the relative index i � 1, . . . , Ln.Accordingly, each single level inside the grid is identified by ln,i

� j � i � �t�1

n�1Lt and the density profile is mathematically

described in terms of

q�ln,i� � qn; n � 1, 2, . . . ; i � 1, . . . , Ln. (1)

Ray propagation along the whole lattice is described by means ofthe Markov chain depicted in Figure 2, where states j� and j�

denote a ray traveling inside the level j with positive and negativedirection, respectively. Such a schema allows one to mathemati-cally bind state-of-the-art building blocks denoting the probabilitythat a ray freely crosses layer n (i.e., the probability that a ray,traveling with positive direction in the level ln,1, reaches level ln,Ln

before being reflected back in level ln,1,Pn � Pr�ln,1� �ln,Ln

� ≺ln,1� �).

The mutual exclusive event is referred to as Qn � 1� Pn � Pr�ln,1

� �ln,1� ≺ln,Ln

� �. Moreover, it is worth noting that, dueto symmetry, the probability Pr�ln,Ln

� �ln,1� ≺ln,Ln

� � that a ray freelycrosses layer n traveling with negative direction is equal to Pn, andPr�ln,Ln

� �ln,Ln

� ≺ln,1� � � Qn. Now, let us focus on the probabilities of

transition between adjacent layers, i.e., Pr�ln,Ln

� �ln�1,1� ≺ln,Ln

� � and Pr

Figure 1 Example of a propagating ray in a stratified lattice. The grid isa realization of the obstacles’ density distribution reported on the left-handside. [Color figure can be viewed in the online issue, which is available atwww.interscience.wiley.com]

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 12, December 2007 3069

�ln�1,1� �ln,Ln

� ≺ln�1,1� �. As far as the first term is concerned, a ray

traveling with positive direction through level ln,Ln reaches nextlayer (n � 1), keeping its direction of propagation, with probability

pn�1 (i.e., the probability that the horizontal face between layer nand layer n � 1 is free). Such an event holds true whatever thenumber of reflections on vertical faces occurring at level ln,Ln, sincethey do not change the vertical direction of propagation of the ray.Similar reasoning leads to Pr�ln�1,1

� �ln,Ln

� ≺ln�1,1� � � pn.

Now, with reference to the Markov chain model and by assum-ing that level k belongs to the K-th layer, i.e., lK,1 � k � lK,LK, thefollowing closed form relation is obtained

Pr�0�k� �p1

1

P1� p1�

n�2

K �1 � Pn

pnPn�

qn

pnpn�1�

. (2)

The building blocks Pn, n � 1, . . . K, can be evaluated either bymeans of the MKVA [8],

Pn � Pn�MKVA� �

pn

1 � �Ln � 2�qn, (3)

or through the MTGA [1],

Pn � Pn�MTGA� � �1, i � 1,

pn

qen�1 � pen

�Ln�1�

Ln � 1 � , i � 1 , (4)

where pen � 1 � qen � pntan ��1 is the effective probability that

a ray crosses a level with occupancy probability qn without anyreflections and LK � (k � lK,1). The core idea of this article is tofully exploit the complementarity of the MKVA and of the MTGAin describing ray propagation in uniform random lattices. For eachuniform layer of the grid, a choice between (3) and (4) is per-formed on the basis of the incidence condition and of the obsta-cles’ density of the layer at hand. To rigorously define the choicecriterion (i.e., the range of qn and � values such that one approachrather than the other is more suitable to be applied) numericalexperiments, reported in the next section, have been carried out.

In passing, we observe that a priori applying either (3) or (4)along the whole lattice (i.e., � n, n � 1, . . . , K), independentlyfrom qn and �, leads to the approaches compared in [10], i.e., theMKVA and the MMTGA, respectively.

3. NUMERICAL VALIDATION

In this section, selected numerical results, assessing the effective-ness of the hybrid solution compared with the MMTGA and theMKVA, are reported. As a reference solution, the penetrationprobability Pr{03 k} has been numerically estimated by means ofcomputer-based ray tracing experiments performed according tothe procedure described in [1]. To quantify the prediction accu-racy, let us define the prediction error

�k–�PrS�0�k� � Pr�0�k��

maxk

�PrS�0�k� 100, (5)

where the subscript S indicates numerically-computed values, andthe mean error

��–

�k�1

kMAX

�k

kMAX, (6)

Figure 2 Markov chain modeling the ray propagation inside a stratifiedrandom lattice. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com]

3070 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 12, December 2007 DOI 10.1002/mop

kMAX being the total number of levels in the numerical experimentat hand. Moreover, to analyze the mean behavior when N densityprofiles are considered, let us define the global mean error

�–

�i�1

N

��i

N, (7)

where ��i is the mean error relative to the i-th profile.

3.1. CalibrationThe aim of this section is to figure out a choice criterion accordingto that either the MKVA (3) or the MTGA (4) is selected asbuilding block Pn, n � 1, . . . , K, in (2). Towards this end, theuniform density profiles obtained by varying q between 0.05 and0.4 (Higher q values have been not taken into account, since forvalues higher than the so-called percolation threshold qc (qc �0.40725 for the two-dimensional case) the propagation is inhibited[11]) with a step of 0.05, have been considered. Moreover, differ-ent incidence conditions have been taken into account, namely � �{15°, 30°, 45°, 60°, 75°} and Pr{03 k} has been evaluated in thefirst kMAX � 10 levels.

With reference to the obtained mean error values (see Fig. 3),

the following rule has been stated:

● if qn 0.2 [Figs. 3(a)–3(c)], then Pn � Pn(MKVA) whatever

�;● if 0.2 � qn � 0.3 [Figs. 3(d)–3(f)], then Pn � Pn

(MTGA) if �� 30 ° and Pn � Pn

(MKVA) elsewhere;● if qn � 0.3 [Figs. 3(g) and 3(h)], then Pn � Pn

(MKVA) if � �15 ° and Pn � Pn

(MTGA) otherwise.

3.2. Numerical Assessment and ComparisonsThis section is aimed at assessing the proposed approach, referredto in the following as hybrid approach (HYBA), by means of anexhaustive numerical validation. In particular, an analysis on therole of the problem parameters in affecting the estimation accu-racy, along with a comparison with the MKVA and the MMTGA,will be presented by considering different test cases. Three- andfour-layers profiles having Ln � 8, n � 1, . . . , K � 1 and LK �kMAX � 8(K � 1), obtained by taking into account all the possiblecombinations of the occupancy probability values qn � {0.05,0.15, 0.25, 0.35}, n � 1, . . . , K, have been considered. Moreover,different impinging directions, � � {15°, 30°, 45°, 60°, 75°}, havebeen assumed and the penetration probability has been estimatedin the first kMAX � 32 levels.

Figure 3 Uniform profiles � Mean error � � versus � when (a) q � 0.05, (b) q � 0.10, (c) q � 0.15, (d) q � 0.20, (e) q � 0.25, (f) q � 0.30, (g)q � 0.35, and (h) q � 0.40. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com]

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 12, December 2007 3071

Firstly, let us analyze the behavior of the global mean error �for different � values. With reference to Figure 4, by comparingthe plots concerned with three- and four-layers profiles, it can beobserved that the number of layers K does not affect the predictionaccuracy of the considered approaches. In particular, concerningthe HYBA, ��4L � �3L� � 0.04 whatever �, where subscripts“4L” and “3L” refer to three- and the four-layers profiles, respec-tively. As far as the dependence on the incidence angle is con-

cerned, it is evident that the accuracy of the HYBA (as well as thatof the other approaches) increases as � tends to 45°. This is fullypredictable, since both the building blocks (3) and (4) ensure thebest performances when � � 45 ° [8]. Finally, it turns out that onaverage the HYBA outperforms both the MKVA and the MMTGAwhatever � and in a more significant fashion when � � 75 °

���MMTGA

�HYBA� � 2.5 and ��MKVA

�HYBA� � 1.8�.

The second test case deals with two four-layer profiles. Theformer, indicated as “4LS,” is made up by sparse layers (i.e., q1 �q3 � 0.05, q2 � q4 � 0.15), whereas the latter, “4LD,” is dense(i.e., q1 � q3 � 0.25, q2 � q4 � 0.35). By analyzing Table 1,where the mean error values when � � 45 ° are reported, it can beobserved that, as expected, the HYBA satisfactorily performs in

both cases ����4LD

��4LS�

HYBA

� 1�, while the MKVA and the

MMTGA are sensitive to the obstacles’ density, allowing morereliable predictions in correspondence with profile “4LS”

����4LD

��4LS�

MKVA

� 2� and “4LD” ����4LS

��4LD�

MMTGA

� 1.7�, respec-

tively.The last test case deals with a four-layers profile consisting of

very sparse and very dense layers in alternated succession (i.e.,q1 � q3 � 0.05, q2 � q4 � 0.35). The plots in Figure 5, as wellas the �� values reported in Table 2, point out that the HYBAoutperforms both the MKVA and the MMTGA whatever �. Theeffectiveness of the HYBA is more evident when � � 75 ° [Fig.5(e) and last row of Table 2] as confirmed by the following indexes��MKVA

��HYBA� 4.11 and

��MMTGA

��HYBA� 4.22. A final observation is

concerned with the role of the variation size in the occupationprobability value between adjacent layers, i.e., �n,n�1 � � qn�1

� qn�. By considering the case � � 45 ° and comparing the ��values of the test case in hand (�n,n�1 � 0.3) with those of theprevious test case (�n,n�1 � 0.1, Table 1), it can be observed thatperformances are not affected by �n,n�1. Such an event points outthe reliability of the Markov chain model in correctly predictingthe change in the slope of Pr{03 k} that occurs in correspondencewith the border between adjacent layers.

4. CONCLUSIONS

In this article, a hybrid formulation for predicting the ray propa-gation in stratified random lattices has been presented. The pro-posed solution exploits the positive features of the MTGA and theMKVA in dealing with uniform random grids, compensating at thesame time their drawbacks. Numerical experiments have demon-strated the effectiveness and the reliability of the proposed solutionas well as the improvements with respect to other stochasticapproaches. Summarizing, the following considerations can bedrawn:

● Both the number of layers and the variation in the occupationprobability values between adjacent layers do not affect

Figure 3 Continued

0

1

2

3

4

5

6

15 30 45 60 75

<φ>

θ

HYBAMKVA

MMTGA

(a)

0

1

2

3

4

5

6

15 30 45 60 75

<φ>

θ

HYBAMKVA

MMTGA

(b)

Figure 4 Multilayers profiles � Global mean error � versus � for(a) the three-layers profiles and (b) the four-layers profiles. [Color figurecan be viewed in the online issue, which is available at www.interscience.wiley.com]

TABLE 1 Mean Error �� for Four-Layers Profiles “4LS” (q1 �

q2 � 0.05 and q2 � q4 � 0.15 and “4LD” (q1 � q3 � 0.25 andq2 � q4 � 0.35) when � � 45°

HYBA MKVA MMTGA

4LS 0.71 0.71 1.554LD 0.74 1.42 0.92

3072 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 12, December 2007 DOI 10.1002/mop

performances. Such a behavior assesses the effectiveness ofthe Markov chain model (see Fig. 2) and the relative solution(2) in modeling propagation in the whole stratified lattice.

● Unlike the MMTGA and the MKVA, the performances of thehybrid method are not affected by the obstacles’ density.

● The hybrid technique outperforms the other methods, what-ever the incidence angle and the obstacles’ density.

● Whatever the approach, the most reliable predictions areobtained when � � 45°.

ACKNOWLEDGMENT

This work has been supported in Italy by the “Progettazione di unLivello Fisico ‘Intelligente’ per Reti Mobili ad Elevata Riconfigu-rabilita,” Progetto di Ricerca di Interesse Nazionale - MIURProject COFIN 2005099984.

REFERENCES

1. G. Franceschetti, S. Marano, and F. Palmieri, Propagation withoutwave equation toward an urban area model, IEEE Trans AntennasPropag 47 (1999), 1393–1404.

2. R.M. Ross, Stochastic processes, Wiley, New York, 1983.3. S. Marano and M. Franceschetti, Ray propagation in a random lattice:

A maximum entropy, anomalous diffusion process, IEEE Trans An-tennas Propag 53 (2005), 1888–1896.

4. M. Franceschetti, J. Bruck, and L. Schulman, A random walk model ofwave propagation, IEEE Trans Antennas Propag 52 (2004), 1304–1317.

5. M. Franceschetti, Stochastic rays pulse propagation, IEEE Trans An-tennas Propag 52 (2004), 2742–2752.

6. A. Martini, R. Azaro, M. Franceschetti, and A. Massa, Ray propaga-tion in nonuniform random lattices, Part 2, J Opt Soc Am A 24 (2007),2363–2371.

7. A. Martini, M. Franceschetti, and A. Massa, Stochastic ray propaga-tion in stratified random lattices, IEEE Antennas Wireless Propag Lett,in press.

8. A. Martini, M. Franceschetti, and A. Massa, Ray propagation in anon-uniform random lattice, J Opt Soc Am A 23 (2006), 2251–2261.

9. J.R. Norris, Markov chains, Cambridge University Press, Cambridge,1998.

10. A. Martini, L. Marchi, M. Franceschetti, and A. Massa, Stochastic raypropagation in stratified random lattices—Comparative assessment oftwo mathematical approaches, Prog Electromagn Res 71 (2007), 159–171.

11. G. Grimmett, Percolation. Springer-Verlag, New York, 1989.

© 2007 Wiley Periodicals, Inc.

0

0.5

1

0 4 8 12 16 20 24 28 32

Pr{0

->

k}

k

HYBAMKVA

MMTGAExperiments

(a)

0

0.5

1

0 4 8 12 16 20 24 28 32

Pr{0

->

k}

k

HYBAMKVA

MMTGAExperiments

(b)

0

0.5

1

0 4 8 12 16 20 24 28 32

Pr{0

->

k}

k

HYBAMKVA

MMTGAExperiments

(c)

0

0.5

1

0 4 8 12 16 20 24 28 32

Pr{0

->

k}

k

HYBAMKVA

MMTGAExperiments

(d)

0

0.5

1

0 4 8 12 16 20 24 28 32

Pr{0

->

k}

k

HYBAMKVA

MMTGAExperiments

(e)

Figure 5 Multilayers profiles � Pr{0 3 k} versus k for a four-layersprofile with q1 � q3 � 0.05 and q2 � q4 � 0.35 when (a) � � 15°, (b)� � 30°, (c) � � 45°, (d) � � 60°, and (e) � � 75°. [Color figure can beviewed in the online issue, which is available at www.interscience.wiley.com]

TABLE 2 Mean Error �� for Four-Layers ProfileCharacterized by q1 � q3 � 0.05 and q2 � q4 � 0.35

� HYBA MKVA MMTGA

15° 3.95 3.95 5.3430° 2.59 3.20 3.0445° 0.55 1.28 0.6260° 0.94 3.43 1.7275° 0.89 3.66 3.76

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 12, December 2007 3073