Interference Focusing for Mitigating Cross-Phase Modulation in a Simplified Optical Fiber Model

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1 Interference Focusing for Mitigating Cross-Phase Modulation in a Simplified Optical Fiber Model Hassan Ghozlan Department of Electrical Engineering University of Southern California Los Angeles, CA USA ghozlan@usc.edu Gerhard Kramer Department of Electrical Engineering University of Southern California Los Angeles, CA USA gkramer@usc.edu arxiv:003.6v cs.it] 30 Apr 00 Abstract A memoryless interference network model is introduced that is based on non-linear phenomena observed when transmitting information over optical fiber using wavelengthdivision multiplexing. The main characteristic of the model is that amplitude variations on one carrier wave are converted to phase variations on another carrier wave, i.e., the carriers interfere with each other through amplitude-to-phase conversion. For the case of two carriers, a new technique called interference focusing is proposed where each carrier achieves the capacity pre-log, thereby doubling the pre-log of / achieved by using conventional methods. The technique requires neither channel time variations nor global channel state information. Generalizations to more than two carriers are outlined. I. ITRODUCTIO The additive white Gaussian noise AWG channel, suitably modified, is a good model for many problems encountered in practice. For example, a parallel AWG channel is accurate for communication over copper cables with orthogonal frequency-division multiplexing OFDM. An AWG channel with multiplicative noise models multi-path fading for wireless communication. The capacities ] of these channels have been studied in great detail. In contrast, the capacities of fiber-optic channels have attracted less interest in the information theory community see, e.g., the tutorial paper ]. Perhaps this is because it was not until recently that it became necessary to communicate efficiently over fiber; optical fiber has long been viewed as having bandwidth to burn. However, the relentless increase in traffic demand and advances in optical technology have made determining fiber capacity of great interest. II. FIBER CHAEL MODELS The fiber channel suffers impairments such as propagation loss, dispersion, and Kerr non-linearity. Optical amplifiers such as Erbium-doped fiber amplifiers EDFAs compensate the attenuation in fiber links without electronic regeneration, and as a result amplified spontaneous emission ASE noise becomes a significant problem. Dispersion arises because the propagating medium absorbs energy through the oscillations of bound electrons, causing a frequency dependence of the material refractive index 3, p. 7]. The Kerr effect is caused by anharmonic motion of bound electrons in the presence of an intense electromagnetic field, causing an intensity dependence of the material refractive index 3, p. 7, 65]. Let Az, t be a complex number representing the slowly varying component or envelope of a single mode, linearly polarized, electric field at position z and time t. Suppose we use a retarded-time reference frame with T = t β z where β is the reciprocal of the group velocity. Suppose further that the ASE noise is negligible. The evolution of Az,T is then governed by the generalized non-linear Schrödinger LS equation 3, p., 50]: i A z + iα A β A T +γ A A = 0 where i =, α is the attenuation constant, β is the group velocity dispersion GVD coefficient, γ = n ω 0 /ca eff, n is the non-linear refractive index, ω 0 is the carrier frequency, c is the speed of light, and A eff is the effective cross-section area of the fiber. One usually normalizesaz,t usinge αz/ which effectively lets one set α = 0 see 3, p. 50, 6]. We are interested in studying the impact of non-linearities, so we consider the simplified model whereβ = 0, i.e., there is no dispersion or completely-compensated dispersion. Eq. with α = 0 and β = 0 has the exact solution 3, p. 98] AL,T = A0,Te iγl A0,T. where L is the fiber length. In other words, Kerr non-linearity leaves the pulse shape unchanged but causes an intensitydependent phase shift. The phase shift phenomenon is called self-phase modulation SPM. Suppose now that two optical fields at different carrier frequencies ω and ω are launched at the same location and propagate simultaneously inside the fiber. The fields interact with each other through the Kerr effect 3, Ch. 7]. Specifically, neglecting fiber losses by setting α = 0, the propagation is governed by the coupled LS equations 3, p. 6, 7]: i A z β A i A z β T +γ A + A A = 0 3 A T +γ A + A A +id A T = 0 where A j z,t is the time-retarded, slowly varying component of field j, j =,, the β j are GVD coefficients, the γ j are nonlinear parameters, and d = β β where the β j

2 are reciprocals of group velocities. We will assume that ω and ω are sufficiently close so that we can set d = 0. We further simplify and choose β = β = 0. The coupled LS equations 3- have the exact solutions 3, p. 75] A L,T = A 0,Te iγl A0,T + A 0,T A L,T = A 0,Te iγl A0,T + A 0,T where z = 0 is the point at which both fields are launched. Kerr non-linearity again leaves the pulse shapes unchanged but causes interference through intensity-dependent phase shifts. The interference phenomenon is called cross-phase modulation XPM. XPM is an important impairment in optical networks using wavelength-division multiplexing WDM, see ]. Equations 3-6 generalize naturally to launching and receiving fields at different locations, and to using K fields with K >. However, it seems prudent to emphasize that ignoring dispersion, or memory, is considered unrealistic for optical networks. On the other hand, our results do show that a new method called interference focusing is needed to approach capacity without dispersion. It seems natural to ect that this method will be useful with dispersion also, and this is the subject of ongoing work. The reason we study a memoryless model is to take a first step in gaining understanding. To strengthen the link to realistic channel models, we point out that a -parameter phenomenological model that captures the effects of XPM in optically-routed WDM networks was recently proposed in ]. The model is memoryless in that each received symbol Y is related to the transmitted signal only through the current transmitted symbol X as follows: 5 6 Y = Xe iφp +Z 7 where Φ P is a Gaussian random variable with variance c Var X and c is a parameter that accounts for system specifications, e.g., the number of WDM channels. Z is AWG with variance σ Z = + c Var X 3 where is the noise variance in the absence of non-linearities and c is another system-specific parameter. The authors of ] use 7 to accurately predict the channel capacities obtained from fullfield numerical simulations reported in 5], 6]. We make two observations. First, the WDM channels in 5], 6] are made approximately memoryless by using reverse propagation to compensate dispersion. Second, the Φ P in 7 is approximately Gaussian if Φ P is a sum of many weighted terms of the form A k 0,T, k =,,...,K, similar to 5-6. III. ITERFERECE ETWORK MODEL Equations 3-6 and their generalizations to K frequencies motivate the following memoryless interference network model based on sampling the fieldsa k z,t,k =,,...,K, at z = 0 and z = L. Transmitter k sends a string of symbols Xk n = X k,,x k,,,x k,n while receiver k sees Yk n = Y k,,y k,,,y k,n. We model the input-output relationship at each time instant j as K Y k,j = X k,j i h kl X l,j +Z k,j 8 l= for k =,,...,K where Z k,j is circularly-symmetric complex Gaussian noise with variance. All noise random variables at different receivers and different times are taken to be independent. The terms ih kk X k,j model SPM and the terms ih kl X l,j, k l, model XPM. We regard the h kl as channel coefficients that are time invariant. These coefficients are known at the transmitters as well as the receivers, although we shall later see that we need local channel state information only. We use the power constraints n n E X k,j ] P k, k =,,...,K. 9 j= Definition : The pre-log r k achieved by user k, whose information rate is R k P,...,P K,, is R k P,...,P K, r k = 0 P /,...,P K / logp k / for k =,...,K. Thus, a K-user transmission scheme may be studied by computing the pre-log K-tuple r,r,...,r K. One achieves the pre-log K-tuple /,/,...,/ if all users use only amplitude modulation or only phase modulation. The main point of our work is to show that one can, in fact, achieve the ultimate pre-log K-tuple,,...,. We again emphasize that we have ignored dispersion. Furthermore, the validity of 8 depends on the amplification, the network topology, the type of fiber, and so forth. For instance, when performing distributed amplification with stimulated Raman scattering, then additional phase noise should be included at high transmit powers. Also, our analysis assumes P k / for all k, and it assumes perfect channel knowledge. Determining the capacity for finite P k / and partial channel knowledge are the subjects of ongoing work. IV. TWO-USER ITERFERECE CHAEL Consider the -user interference channel for which 8 without the time indexes becomes Y = X ih X +ih X +Z Y = X ih X +ih X +Z. We propose an interference focusing scheme in which the transmitters focus their phase interference on one point by constraining their transmitted signals to satisfy h X = mπ, m =,,3,... 3 h X = mπ, m =,,3,... In other words, the transmitters use multi-ring modulation with specified spacings between the rings. We thereby remove XPM interference and - reduce to Y k = X k e ih kk X k +Z k, k =,. 5 This channel is effectively an AWG channel since h kk is known by receiver k and the SPM phase shift is determined by the desired signal X k. Multi-ring modulation was used in ], 5], 6] for symmetry and computational reasons only. We here find that it is useful for improving rate.

3 It remains to show that the pre-log pair r,r =, is achieved under the constraints 3-. We show this in two steps: we first determine the information rate for one ring and then extend the analysis to many rings. A. One Ring Consider the AWG channely = X+Z with X = Pe iφ where Φ is uniformly distributed on the interval 0, π. The achievable rate R is given by R = IX;Y = hy hy X = E logp Y Y] logπe 6 The probability density of Y can be shown to be, p. 688] p Y y = π e y A +P/ y A P I 0 7 where I 0 is the modified Bessel function of the first kind of order zero and y A = y. Therefore, we have ] hy = E log π e Y A +P/ Y A P I 0. 8 ext, we derive an upper bound on I 0 z that we will use in the process of lower bounding hy. Lemma : We have π I 0 z e z z, z 0. 9 Proof: We have cosx x /π for 0 x π/ by using the infinite product form for cosine x ] cosx = π n. 0 We thus have π n= I 0 z = e zcosθ dθ π 0 π/ ] π e z θ /π dθ + e 0 dθ π 0 π/ π e z = Q z + z where Qz = z π e x / dx. Finally, we observe that Qz 0 and πe z / z / for z 0. Using Lemma in 8, we have hy E log π E log π e YA P / Y A P/ Y A P/ ] = log 3π P 3 + E log Y A ]. 3 The last ectation in 3 is y A y A=0 e y A +P/ y A P I 0 log y A dy A. Setting z = y A / and ν = P/, ression is loge z z +ν I 0 zνlnzdz z=0 = loge Γ 0, P ] P +ln 5 where Γa,x is the upper incomplete Gamma function 7, p. 60] and where the second step follows by 8]. Inserting 5 into 3, and then 3 into 6, gives IX;Y 8 log P πe 8 log P πe + loge Γ 0, P 6 7 where we have made use of Γ0,x 0 for x 0. The desired pre-log therefore satisfies B. Multiple Rings RP/ r = P/ logp/. 8 Consider multiple rings with X = P j e iφ, j =,...,, where is the number of rings. The power levels P j allowed under interference focusing take the form mp 0 where m is a positive integer and p 0 is the minimum non-zero power level that depends on the channel coefficients. For example, for the -user interference channel p 0 = π/h for transmitter and p 0 = π/h for transmitter. The power levels must further satisfy E X ] P where P is the power constraint of the user being considered our pre-log analysis is based on a pointto-point AWG channel because interference is removed by interference focusing. The achievable rate R is given by R = IX;Y = IX A,Φ;Y = IX A ;Y+IΦ;Y X A 9 where X = X A e iφ. The term IX A ;Y can be viewed as the amplitude contribution while the term IΦ;Y X A is the phase contribution. Suppose that, for simplicity, we choose the rings to be spaced uniformly in amplitude as P j = aj p 0 30 where a is a positive integer. We further use a uniform frequency of occupation of rings with P XA P j = /, j =,,...,. The power constraint is therefore aj p 0 P. 3 j= ote that x Γ0,x = 0.

4 Phase Contribution: Using 7, we have IΦ A ;Y X A = j= j= IΦ A;Y X A = P j 3 8 log P j πe. 33 We show in the Appendix that by choosing a in 30 to scale as logp/, and choosing to satisfy 3, then we have P/ j= logp j/ logp/ 3 The pre-log of the phase contribution is therefore at least /. Amplitude Contribution: We have IX A ;Y = HX A HX A Y 35 where HX A = log. We show in the Appendix that scales as P//logP/ if a scales as logp/. We bound HX A Y using Fano s inequality as HX A Y HX A ˆX A HP e +P e log 36 where ˆX A is any estimate of X A given Y, P e = Pr ˆX A X A ] and HP e is the binary entropy function with a general logarithm base. Suppose we use the minimum distance estimator ˆX A = arg min x A X A Y A x A 37 where Y A = Y and X A = { P j : j =,...,}. We show in the Appendix that P e j 38 j= where j = P j P j /. For the power levels 30, we have j = ap 0 / for all j, and hence P e ap We see from 39 that P/ P e = 0 if a scales as logp/ recall that scales as P//logP/. We thus have P/ HX A Y = 0 by using 36. Consequently, we have IX A ;Y P/ logp/ = log P/ logp/ =. 0 We conclude from 9, 3, and 0 that interference focusing achieves the largest-possible pre-log of. Each user can therefore loit all the phase and amplitude degrees of freedom simultaneously. V. K-USER ITERFERECE ETWORK We outline how to apply interference focusing to problems with K >. Define the interference phase vector Ψ = Ψ,Ψ,...,Ψ K ] T where Ψ k = K l= h kl X l and the instantaneous power vector Π = X,..., X K ] T. The relation between the Ψ and Π in matrix form is Ψ = H SP Π+H XP Π 3 where H SP is a diagonal matrix that accounts for SPM and H XP is a zero-diagonal matrix that accounts for XPM. Example 3: Suppose the XPM matrix for a 3-user interference network is H XP = 0 / 3/5 3/ 0 /3 5/6 /5 0 Suppose that each transmitter knows the channel coefficients between itself and all the receiving nodes. The transmitters can thus use power levels of the form Π = π lcm,6m, lcm,5m, lcm5,3m 3 ] = π m,0m,5m 3 ] 5 where lcma,b is the least common multiple of a and b, and m,m,m 3 are positive integers. We thus have H XP Π = π m m m 3 which implies that the phase interference has been einated. Example 3 combined with an analysis similar to Section IV shows that interference focusing will give each user a prelog of even for K-user interference networks. However, the XPM coefficients h kl must be rationals. Modifying interference focusing for real-valued XPM coefficients is an interesting problem. It is clear from Example 3 that interference focusing does not require global channel state information. VI. COCLUSIO We introduced an interference network model based on a simplified optical fiber model. We assumed that there was no dispersion, or that dispersion was compensated. The non-linear nature of the fiber-optic medium causes the users to suffer from amplitude-dependent phase interference. We introduced a new technique called interference focusing that lets the users take full advantage of all the available amplitude and phase degrees of freedom. Several generalizations are interesting to study further, e.g., introduce group velocity, focus interference on multiple points, study low and intermediate signal-to-noise ratio, investigate partial channel knowledge, and so on.

5 ACKOWLEDGMET H. Ghozlan was supported by a USC Annenberg Fellowship. G. Kramer was supported by SF Grant CCF We are grateful to the reviewers for providing constructive criticisms that helped to improve the paper. APPEDIX PHASE MODULATIO WITH MULTIPLE RIGS We derive the key relations to prove that uniform phase modulation contributes / to the pre-log for specially-chosen multi-ring modulations. For 3 we compute j= ap 0 j = ap so to satisfy the power constraint we choose 3 j= 7 = 3+ +8P/ap 0. 8 We choose a to scale as logp/ so scales as P//logP/. ext, consider the sum in 3. The logarithm is an increasing function so we have aj p 0 ax p 0 log log dx 9 We can therefore write P x=0 j= logp j/ logp/ = lna p 0 / loge 50 loga p 0 / = P logp/ 5 where 5 follows becausea scales as logp/, scales as P//logP/, and p 0 is independent of P and. MIIMUM DISTACE ESTIMATOR We derive the bound 38 for the estimator 37. Let P e,j be the error probability when X A = P j. We have P e = j= P e,j and Pr Y A P + P, j = PK + P K Pr Y A, j = P e,j = Pj + P j Pr Y A +Pr Y A Pj+ P j+, otherwise. 5 Conditioned on X A = P j, Y A is a Ricean random variable, and hence we compute 9, p. 50] Pj + P j+ Pj Pj + P j+ Pr Y A = Q, / / 53 3 The solution for should be positive and rounded down to the nearest integer but we ignore these issues for notational simplicity. where Qa, b is the Marcum Q-function 0]. Consider the following bounds. Upper bound for b > a 0, UBMG] Qa,b b a. 5 Lower bound for b < a 0, LBaS] Qa, b a b a+b ]. 55 The bound 55 implies Qa,b a b. 56 We use 53 and 5 to write Pj + P j+ Pr Y A j+. 57 where j = P j P j /. Similarly, we use inequality 56 to write Pj + P j Pr Y A j. 58 Collecting our results, we have P e + j = + j= j= j= j+ j REFERECES ] C. E. Shannon. A mathematical theory of communications. Bell System Tech.., vol. 7no. :pp , , uly and Oct. 98. ] R.-. Essiambre, G. Kramer, P.. Winzer, G.. Foschini, and B. Goebel. Capacity its of optical fiber networks. ournal of Lightwave Technology, 8:66 70, Feb.5, 00. 3] G. P. Agrawal. onlinear Fiber Optics. Academic Press, 3rd edition, 00. ] B. Goebel, R.-. Essiambre, P.. Winzer, and. Hanik. Phenomenological Fiber-optic Channel Model for Rapid Capacity Limit Estimation. in preparation, ] R.-. Essiambre, G. Foschini, G. Kramer, and P. Winzer. Capacity Limits of Information Transport in Fiber-Optic etworks. Physical Review Letters, vol. 0:paper 6390, October ] R.-. Essiambre, G. Foschini, P. Winzer, and G. Kramer. Capacity its of fiber-optic communication systems. in Proc. OFC, page paper OThL, March ] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. ew York, 97. 8] Wolfram Research Inc. Mathematica Edition: Version ] ohn G. Proakis and Masoud Salehi. Digital Communications. McGraw- Hill, 5th edition, ] G. E. Corazza and G. Ferrari. ew bounds for the Marcum Q-function. IEEE Transactions on Information Theory, vol. 8no. :pp , ov 00.