Fiber-Optic Communication via the Nonlinear Fourier Transform

Transcription

1 Fiber-Optic Communication via the Nonlinear Fourier Transform Frank R. Kschischang Department of Electrical & Computer Engineering University of Toronto IC Research Day SwissTech Convention Center, EPFL 3 June 6

2 Thank You To: Mansoor Yousefi (University of Toronto, Technical University of Munich, Télécom ParisTech) Gerhard Kramer (Technical University of Munich) Siddarth Hari (University of Toronto) Alan P.-T. Lau (Hong Kong Polytechnical University) René-Jean Essiambre (Nokia Bell Labs) The organizers: Michael Gastpar Rüdiger Urbanke

3 Part I: Fiber-Optic Communication 3

4 Illustrating the principle of total internal reflection.

5 Source: Nicolas Rapp, Fortune Magazine, July 3, 5

6 Source: 6

7 eunetworks Optical Fiber Network Map Source: 7

8 Fiber-Optic Communication Systems Hybrid Amplifier SSMF Hybrid Amplifier ROADM SSMF Hybrid Amplifier ROADM SSMF Hybrid Amplifier TX RX Forward pump Backward pump Forward pump Backward pump Forward pump Backward pump span span span N Physics: Enabling Technologies Low-loss (. db/km) optical fiber, huge bandwidth ( 5 THz bandwidth) Optical amplifiers RX TX ROADM λ λ 3 Laser transmitters and Mach-Zehnder modulators TX RX 8

9 Historical Capacity Evolution System capacity (Tb/s) Pbit/s Tbit/s Tbit/s Tbit/s Gbit/s Gbit/s Gbit/s TDM Research WDM Research SDM Research Year Space-division multiplexing has already exceeded the nonlinear Shannon limit Nokia 6 < COPYRIGHT 6 Nokia. ALL RIGHTS RESERVED. > Nonlinear Shannon capacity limit estimate (8 Tb/s) Courtesy of R.-J. Essiambre (Nokia Bell Labs) [IEEE Photonics J., Apr. 3] TDM: Time-division multiplexing WDM: Wavelength-division multiplexing SDM: Space-division multiplexing 9

10 Nonlinear Shannon Limit Estimate (Single Polarization) 5 km of standard single-mode fiber (SSMF) Spectral eficiency (bits/s/hz) Shannon AWGN Record laboratory experiment s Nonlinear Shannon limit of spectral efficiency Courtesy of R.-J. Essiambre (Nokia Bell Labs) [J. Lightwave Techn., Feb. ] Signal-to-noise ratio, SNR* (db) *SNR is the signal average power average divided by the AWGN power per symbol Experimentally demonstrated fiber capacity have leveled off to ~6% of maximum spectral efficiency since Nokia 6 < COPYRIGHT 6 Nokia. ALL RIGHTS RESERVED. >

11 Fiber-Optic Communication Systems: Challenges Reliability P e < 5

12 Fiber-Optic Communication Systems: Challenges Reliability P e < 5 Speed Gb/s per-channel data rates (or even more)

13 Fiber-Optic Communication Systems: Challenges Reliability P e < 5 Speed Gb/s per-channel data rates (or even more) Non-Linearity The fiber-optic channel is non-linear in the input power!

14 The Kerr Effect Kerr electro-optic effect (DC Kerr effect) An effect discovered by John Kerr in 875 It produces a change of refractive index in the direction parallel to the externally applied electric field The change of index is proportional to the square of the magnitude of the external field

15 The Kerr Effect in Optical Fibers Optical Kerr Effect (or AC Kerr effect) No externally applied electric field is necessary The signal light itself produces the electric field that changes the index of refraction of the material (fused silica) The change in index in turn changes the signal field The change in index of refraction is proportional to the square of the field magnitude changes index of refraction optical fiber P WDM channel generates distortion channel of interest f 3

16 Nonlinear Effects in Fibers intra-channel inter-channel signal noise signal signal signal noise signal signal NL phase noise parametric amplification self-phase modulation NL phase noise WDM nonlinearities SPM-induced NL phase noise modulation instability isolated pulse SPM IXPM IFWM XPM-induced NL phase noise XPM FWM Key NL = nonlinear; SPM = self-phase modulation; (I)XPM = (intra-channel) cross-phase modulation; (I)FWM = (intra-channel) four-wave mixing; WDM = wavelength-division multiplexing Courtesy of R. J. Essiambre

17 System Model Hybrid Amplifier coherent fiber-optic communication system standard-single-mode fiber ideal distributed Raman amplification SSMF Hybrid Amplifier ROADM SSMF Hybrid Amplifier ROADM SSMF Hybrid Amplifier TX RX Forward pump Backward pump Forward pump Backward pump Forward pump Backward pump span span span N But, could also consider systems with inline dispersion-compensating fiber lumped amplification Q(t, ) Q(t, L) TX AMP AMP RX SSMF }{{} N spans 5

18 Generalized Nonlinear Schrödinger Equation Q(t, z) is the complex baseband representation of the signal (the full field, representing co-propagating DWDM signals) Transmitter sends Q(t, ) Receiver gets Q(t, L), where L is the total system length Evolution of Q(t, z) The generalized non-linear Schrödinger (GNLS) equation expresses the evolution of Q(t, z): Q(t, z) z + jβ Q(t, z) t }{{} dispersion jγ Q(t, z) Q(t, z) } {{ } nonlinearity = n(t, z) }{{} noise No loss term (since ideal distributed Raman amplification is assumed). 6

19 System Parameters n(t, z) is a circularly symmetric complex Gaussian noise process with autocorrelation E [ n(t, z)n (t, z ) ] = αhν s K T δ(t t, z z ), where h is Planck s constant, ν s is the optical frequency, and K T is the phonon occupancy factor. Second-order dispersion β ps /km Loss α.65 5 m Nonlinear coefficient γ.7 W km Center carrier frequency ν s 93. THz Phonon occupancy factor K T.3 7

20 Solving the GNLS Equation Throughout propagation over an optical fiber, stochastic effects (noise), linear effects (dispersion) and nonlinear effects (Kerr nonlinearity) interact. noise nonlinearity dispersion Even in the absence of noise, solving the GNLS equation requires numerical techniques. 8

21 Split-Step Fourier Method h divide fiber length into short segments consider each segment as the concatenation of (separable) nonlinear and linear transforms for distributed amplification, an additive noise is added after the linear step. Q(t, z ) Q(t, z + h) step size h 9

22 Nonlinear Step In the absence of linear effects, the GNLS equation has the form Q(t, z) z = jγ Q(t, z) Q(t, z), with solution Q(t, z + h) = Q(t, z ) exp(jγ Q(t, z ) h). Nonlinear Step is best taken in the time domain.

23 Linear Step Now use the previous solution as input to the linear step, i.e., let ˆQ(t, z ) = Q(t, z ) exp(jγ Q(t, z ) h) be the input to the linear step. The linear form of the GNLS equation is Q(t, z) = jβ Q(t, z) z t. Defining Q(t, z) = π it can be shown that Linear Step... Q(ω, z) exp(jωt)dω, i.e., Q(t, z) Q(ω, z + h) = Q(ω, z ) exp... is best taken in the frequency domain. ( j β ) ω h. F Q(ω, z),

24 Split-Step Propagator Putting this together, we have Q(t, z + h) = F F Q(t, z ) exp(jγ Q(t, z ) h) }{{} exp nonlinear step ( j β ) ω h } {{ } linear step where F is the Fourier transform operator. In practice: extensive use is made of the FFT step size can be adapted 3 linear and nonlinear steps can be offset by a half-step See: Sinkin, Holzlohner, Zweck, Menyuk, J. Lightwave Tech., 3.

25 Launching a Pulse z = km 8 q(t) t (soliton units) 3

26 Launching a Pulse z = km 8 q(t) t (soliton units) 3

27 Launching a Pulse z = 5 km 8 q(t) t (soliton units) 3

28 Launching a Pulse z = km 8 q(t) t (soliton units) 3

29 Current Approaches Digital Backpropagation q(t, ) q(t, L ) = K NLS (q(t, )) q(t, L ) = K NLS (q(t, L q(t, L) )) Wavelength-division multiplexing (WDM) input output noise W guard band }{{} out of band }{{} COI f

30 Digital Backpropagation Digital backpropagation = split-step Fourier method, using a negative step-size h, performed at the receiver Full compensation (involving multiple WDM channels) generally impossible, even in absence of noise (due to wavelength routing) Noise is neglected (cf. zero-forcing equalizer) Single-channel backpropagation typically performed, after extraction of desired channel using a filter 5

31 Current Achievable Rates pulse shapes constellation I(x) R(x) Amplitude time 6

32 Capacity Estimation See: R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, B. Goebel, Capacity limits of optical fiber networks, J. Lightw. Technol., vol. 8, pp. 66 7, Sept./Oct.. System Model ( ) T s Q(t, ) Q(t, L) t = kt s TX CHANNEL LPF DSP 7

33 Achievable Rates from Memoryless Capacity Estimate 9 Spectral Efficiency (bits/s/hz) L= km, Eq. only L= km, BP L= km, Eq. only L= km, BP L=5 km, Eq. only L=5 km, BP Shannon Limit (AWGN) (BP adds.55 to.75 bits/s/hz relative to EQ) SNR (db) B. P. Smith and F. R. Kschischang, J. Lightwave Techn., vol. 3, pp. 7 53,. 8

34 Schematic Achievable Rate Curve 5 Rate [bits/s/hz] 5 log( + SNR) Nonlinearity kicks in SNR [db] noise-limited nonlinearity-limited 9

35 Optical Fiber Capacity Bounds C [bit/symbol] upper bound lower bounds power bound Upper bound: Transmit power G. Kramer, M. I. Yousefi, and F. R. Kschischang, Upper Bound on the Capacity of a Cascade of Nonlinear and Noisy Channels, Proc. IEEE Info. Theory Workshop, Jerusalem, Israel, Apr. 5. M. I. Yousefi, G. Kramer, and F. R. Kschischang, Upper Bound on the Capacity of the Nonlinear Schrödinger Channel, Proc. th Canadian Workshop on Info. Theory, St. John s, NL, Jul. 5. 3

36 Summary Pulse propagation in optical fibers is well-modeled by the stochastic nonlinear Schrödinger equation: Q(t, z) z + jβ Q(t, z) t }{{} dispersion jγ Q(t, z) Q(t, z) } {{ } nonlinearity = n(t, z) }{{} noise z is distance along the fiber; TX at z =, RX at z = L t is retarded time, i.e., t = τ β z where τ is ordinary time and β is a constant Q(t, z) is the complex envelope of the propagating signal β is the chromatic dispersion coefficient γ is the nonlinearity parameter V (t, z) is space-time white Gaussian noise noise dispersion nonlinearity 3

37 Part II: Nonlinear Fourier Transform b a b a NFT(q)(λ) 5 spectrum of L Rλ 5 Iλ 3

38 Central Question: Does fiber nonlinearity really place an upper limit on achievable spectral efficiency? To try to answer this question, we have to understand the nonlinearity more deeply. Spoiler: We will not give a satisfactory answer in this talk. 33

39 Evolution Equations An evolution equation (in + dimensions) is a partial differential equation for an unknown function q(t, z) of the form q z = K(q), where K(q) is an expression involving only q and its derivatives with respect to t. Heat: K(q) = c q tt NLSE: K(q) = j ( q tt (t, z) + q(t, z) q(t, z) ) KdV: K(q) = q ttt (t, z) + q(t, z)q t (t, z) (Here subscripts denote partial derivatives, thus q tt = t q(t, z).) 3

40 Normalized NLS Equation Changing variables (to so-called soliton units ): q = Q P, z = z L, t = t T, with T = we get the: β L and P = Normalized NLS equation γl (and then dropping the primes ) jq z (t, z) = q tt + q(t, z) q(t, z) + v(t, z). 35

41 Isospectral Flow A Key Idea We seek an invariant under evolution (in the absence of noise), e.g., let L be a linear differential operator (depending on q(t, z)). It may be possible to find an L whose (eigenvalue) spectrum remains constant, even as q evolves (in z) according to some evolution equation. z L q(t, ) q(t, z) q(t, L) L(q(t, )) L(q(t, z)) L(q(t, L)) Constant Spectrum 36

42 Example: Isospectral Families of Matrices Let L(z) be a family of diagonalizable square matrices whose entries are functions of z. Clearly, the eigenvalues of these matrices in general depend on z. For some families (isospectral families), it might be the case that while the entries of the matrix change with z, the eigenvalues remain constant. Each member of an isospectral family is similar to a constant diagonal matrix Λ, i.e., L(z) = G(z)ΛG (z) for some similarity transformation G(z). 37

43 Isospectral Families of Operators Let H be a Hilbert space, and let L(z) be a family of diagonalizable bounded linear operators operating on H, e.g., L(q(t, z)) = + q(t, z). t If the eigenvalues of L(z) do not depend on z, then we refer to L(z) as an isospectral family of operators. For each z, L(z) is similar to a multiplication operator Λ (the operator equivalent of a diagonal matrix), i.e., L(z) = G(z)ΛG (z), for some operator G(z). The spectrum of an operator is defined as σ(l) = {λ L λi is not invertible}. Spectrum can be discrete (like matrices), continuous, residual, etc. 38

44 The Lax Equation We have L(z) = G(z)ΛG (z), where Λ does not depend on z. Assuming that L(z) varies smoothly with z, we can form dl(z) dz = G ΛG + GΛ ( G G G ) = G } {{ G ( } GΛG ) ( GΛG ) G }{{}}{{}} {{ G } M(z) L(z) L(z) M(z) = M(z)L(z) L(z)M(z) = [M, L] () where [M, L] = ML LM is the commutator bracket. In other words, every diagonalizable isospectral operator L(z) satisfies the differential equation (). The converse is also true. 39

45 Lax Pairs Lemma Let L(z) be a diagonalizable family of operators. Then L(z) is an isospectral family if and only if it satisfies dl dz for some operator M, where [M, L] = ML LM. = [M, L], () Definition The operators L and M satisfying () are called a Lax Pair (after Peter D. Lax, who introduced the concept [968]).

46 Lax Pairs Recall that L may depend on a function q(t, z) and so can M. The commutator bracket [M, L] can create nonlinear evolution equations for q(t, z) in the form L z = [L, M] q z = K(q), where K(q) is some, in general nonlinear, function of q(t, z) and its time derivatives.

47 KdV Equation An example of this is the Korteweg-de Vries (KdV) equation (which arises in the evolution of waves on shallow water surfaces). Let q(t, z) be a real-valued function and choose L = t + 3 q, M = 3 t 3 + q t + q t. The Lax equation L z = [M, L] is easily simplified to 3 q z 3 (q ttt + qq t ) + (some terms) t, where is the zero operator. The zero-order term of this equation, which must be zero, produces the KdV equation q z = q ttt + qq t.

48 NLS equation If q(t, z) varies based on the NLS equation jq z = q tt + q q then the spectrum of L = j ) t q(t, z) q (t, z) t ( (3) is preserved during the evolution (Zakharov-Shabat 97). In addition, the operator M is given by ( jλ M = j q(t, z) ) λq(t, z) jq t (t, z) λq (t, z) jqt (t, z) jλ + j q(t, z) Thus the NLS equation is indeed generated by a Lax pair! 3

49 Eigenvalues and Eigenvectors of L The eigenvalues of the operator L, which are constant in an isospectral flow, are defined via Lv = λv () Taking the z derivative of () and using the Lax equation L z = [M, L], we can show that an eigenvector of L evolves according to the linear equation v z = Mv. Furthermore, we can re-write () as v t = Pv for some operator P. Thus we get Evolution of Eigenvectors of L The eigenvectors of L evolve according to the linear system v z = Mv (5) v t = Pv (6)

50 Zero-Curvature Condition Combining equations (5) and (6) by using the equality of mixed derivatives, i.e., v tz = v zt, the Lax equation () is reduced to the Zero-curvature Condition P z M t + [P, M] =. (7) Note that the nonlinear equation derived from (7) results as a compatibility condition between the two linear equations (5) and (6). Thus certain nonlinear evolution equations possess a certain hidden linearity. We can work with the Lax pair (L, M) or equivalently with the pair (P, M). 5

51 Integrable Systems We refer to a system described by a Lax pair M, L, measured at distance L, as the integrable system S = (L, M; L). x(t) = q(t, ) y(t) = q(t, L) L z (q) = [M(q), L(q)] input waveform output waveform Definition (Lax Convolution) The action of an integrable system S = (L, M; L) on the input q(t, ) is called the Lax convolution of q with S. We write the system output as q(t, L) = q(t, ) (L, M; L). Definition (Integrable communication channels) A waveform communication channel C : x(t) v(t, z) y(t) with inputs x(t) L (R) and space-time noise v(t, z) L (R, R + ), and output y(t) L (R), is said to be integrable if the noise-free channel is an integrable system. 6

52 Spectrum of L The Zakharov-Shabat operator for the NLS equation has two types of spectra: the discrete (or point) spectrum, which occurs in C + and corresponds to solitons. The continuous spectrum, which in general includes the whole real line, corresponds to the non-solitonic (or radiation) component of the signal. I(λ) R(λ) 7

53 Eigenspace Associated with λ Associated with each point λ in the spectrum of L is a two-dimensional eigenspace in which eigenvectors v evolve in time according to ( ) jλ q(t, z) v t = Pv = q v. (t, z) jλ Assuming q(t, z) is pulse-like, decaying to zero as t, the P operator approaches a diagonal matrix, and the eigenvectors approach v(t, λ) (αe jλt, βe jλt ) T, α, β C. 8

54 Canonical Eigenvectors For example, the canonical eigenvectors at t ( ) ( ) e jλt, e jλ t are bounded for λ in the upper (lower) half-complex plane respectively. We can propagate these vectors forward in time according to v t = Pv, let them interact with q(t, z), and measure what is scattered at t = + in the basis ( ) e jλt, ( ) e jλ t The corresponding coefficients a(λ) and b(λ) are called the nonlinear Fourier coefficients. 9

55 Continuous and Discrete Spectra It can be shown that these scattering projections are well defined at all points λ R, the real line continuous spectrum at those points λ C +, the upper-half complex plane, where a(λ) = (necessarily isolated points due to analyticity of a(λ)) discrete spectrum 5

56 Computing the Fourier coefficients ( v (, λ) e ) jλt ( ) jλ q(t) v t = q v (t) jλ v (, λ) v (+, λ) ( ) e jλt ṽ (, λ ) ( ) e jλt Since the nonlinear Fourier coefficients are time independent, simply compute them at t = +, by projecting v (+, λ) and ṽ (+, λ) onto the basis v (+, λ) and ṽ (+, λ) to obtain [ v (+, λ), ṽ (+, λ) ] = [ ṽ (+, λ), v (+, λ) ] S, where S = ( a(λ) b (λ ) ) b(λ) a (λ. ) 5

57 Nonlinear Fourier Transform For the purpose of developing the inverse transform, it is sufficient to work with the ratios ˆq(λ) = b(λ) a(λ) and q(λ j) = b(λ j) a (λ j ). 5

58 Nonlinear Fourier Transform Definition (Nonlinear Fourier transform) Let q(t) be a sufficiently smooth function in L (R). The nonlinear Fourier transform of q(t) with respect to a given Lax operator L consists of the continuous and discrete spectral functions ˆq(λ) : R C and q(λ j ) : C + C where ˆq(λ) = b(λ) a(λ), q(λ j) = b(λ j) a, j =,,, N, (λ j ) in which λ j are the zeros of a(λ). Here, the spectral coefficients a(λ) and b(λ) are given by a(λ) = lim v t e jλt, b(λ) = lim v t e jλt 53

59 The (ordinary) Fourier Transform The nonlinear Fourier transform measures, for each fixed eigenvalue λ, the interaction of a propagating eigenvector (initialized as a complex exponential) with a waveform q(t). The ordinary Fourier transform measures, for each fixed frequency ω, the interaction of the complex exponential e jωt with a waveform q(t) via the inner product Q(ω) = q(t), e jωt = q(t)e jωt dt. An eigenvector evolution version of the ordinary Fourier transform is recovered by defining ( ) q(t) v t = v, jω giving v = α ( ) + β ( t ) q(τ)e jωτ dτ e jωt 5

60 Example: NFT of a Rectangular Pulse Consider the rectangular pulse { A, t [t, t ]; q(t) =, otherwise. Let T = t t and T = t + t, and let = λ + A. After some work, we find that the zeros of a(λ) in C +, satisfy j tan(t A + λ ) = + A λ, giving rise to the discrete spectrum. The continuous spectrum is given by ( ˆq(λ) = A jλ e jλt ) jλ cot( T ) A Te t +t sinc(tf ). 55

61 Discrete & Continuous Spectra of Rectangular Pulse ˆq(λ) R(λ) I(λ) ˆq(λ) R(λ) I(λ) λ λ A = A = ˆq(λ) R(λ) I(λ) λ A = 6 56

62 NFT Properties Let q(t) ( q(λ), q(λ k )) be a nonlinear Fourier transform pair. (The ordinary Fourier transform as limit of the nonlinear Fourier transform): If q L, there is no discrete spectrum and q(λ) Q(λ), where Q(λ) is the ordinary (linear) Fourier transform of q (t) Q(λ) = q (t)e jλt dt. (Weak nonlinearity): If a, then âq(λ) a q(λ) and ãq(λ k ) a q(λ k ). In general, however, âq(λ) a q(λ) and ãq(λ k ) a q(λ k ). (Constant phase change): ejφ q(t)(λ) = e jφ q(t)(λ) and e jφ q(t)(λ k ) = e jφ q(t)(λ k ). 57

63 Fourier Transform Properties (cont d) (Time dilation): q( t a ) = a q(aλ) and q( t a ) = a q(aλ k ); (Time shift): q(t t ) e jλt ( q(λ, ) q(λ k )); (Frequency shift): q(t)e jωt ( q(λ ω), q(λ k ω)); (Parseval identity): q(t) dt = Ê + Ẽ, where Ê = π log ( + ˆq(λ) ) dλ, Ẽ = N I (λ j ). The quantities Ê and Ẽ represent the energy contained in the continuous and discrete spectra, respectively. j= 58

64 The Key NFT Property (Lax convolution): If q (t) = q (t) (L, M; L), then q (λ) = c(λ, L) q (λ) and q (λ k ) = c(λ, L) q (λ k ). For the NLS equation, c(λ, L) = exp( jλ L). 59

65 Signals and Systems LTI systems: y(t) = h(t) x(t) + z(t) Y (ω) = H(ω)X (ω) + Z(ω) Nonlinear Integrable systems: y(t) = x(t) (L, M; L)+z(t) {Ŷ (λ) = H(λ) ˆX (λ) + Ẑ(λ), Ỹ (λ j ) = H(λ j ) X (λ j ) + Z(λ j ) where the channel filter is H(λ) = e jλz. We see that the operation of the Lax convolution in the nonlinear Fourier domain is described by a simple multiplicative (diagonal) operator (i.e., filters), much in the same way that the ordinary Fourier transform maps x(t) y(t) to X (ω) Y (ω). 6

66 Launching a Pulse (revisited) z = km 8 q(t) t (soliton units) 6

67 Launching a Pulse (revisited) z = km 8 q(t) t (soliton units) 6

68 Launching a Pulse (revisited) z = 5 km 8 q(t) t (soliton units) 6

69 Launching a Pulse (revisited) z = km 8 q(t) t (soliton units) 6

70 Launching a Pulse (revisited) 6 5 z = km Im(λ) Re(λ) q(λ) λ 6

71 Launching a Pulse (revisited) 6 5 z = km Im(λ) Re(λ) q(λ) λ 6

72 Launching a Pulse (revisited) 6 5 z = 5 km Im(λ) Re(λ) q(λ) λ 6

73 Launching a Pulse (revisited) 6 5 z = km Im(λ) Re(λ) q(λ) λ 6

74 Launching a Pulse (revisited) q(t) z = km Im(λ) Re(λ) 3 3 t (soliton units) 63

75 Launching a Pulse (revisited) q(t) z = km Im(λ) Re(λ) 3 3 t (soliton units) 63

76 Launching a Pulse (revisited) q(t) z = 5 km Im(λ) Re(λ) 3 3 t (soliton units) 63

77 Launching a Pulse (revisited) q(t) z = km Im(λ) Re(λ) 3 3 t (soliton units) 63

78 Launching a Pulse (revisited) q(t) z = km Im(λ) Re(λ) 3 3 t (soliton units) 63

79 Nonlinear FDM Immediate application: encode data in the nonlinear spectrum! {λ j, q(λ j ), ˆq(λ)} TX S/P. INFT. P/S D/A LPF channel {λ j, q(λ j ), ˆq(λ)} RX. P/S. NFT. S/P A/D LPF See also: A. Hasegawa and T. Nyu, Eigenvalue Communication, J. of Lightwave Technology, vol., pp , March

80 Multisoliton Communication Multisolitons are signals whose NFT spectrum only contains the discrete component, i.e., the corresponding L operator has only discrete eigenvalues Im(λ) 3 Im(λ).5 q Re(λ) q Re(λ) t 5 5 t sech(t) Satsuma-Yajima -soliton 65

81 Generation of Multisoliton Signals Inverse NFT problem: Given a set of points in C +, compute a signal q(t) such that the eigenvalues of its corresponding L operator are exactly these points Several approaches: Riemann Hilbert system of linear equations, Hirota bilinearization scheme, Darboux transformation Darboux method (recursive construction) Starting with a multisoliton signal q(t, λ, λ,..., λ k ), and a solution, φ(t, λ k+ ), of ( ) jλ q(t, z) v t (t, λ) = q v(t, λ), (t, z) jλ compute q = q j(λ φ k+ λ k+ ) φ φ + φ. q has eigenvalues of q and new eigenvalue from φ 66

82 On-Off Eigenvalue Encoder (OOK) 5 6 Im(λ) 3 b = 3 b = b = b = q(t) Re(λ) 5 5 t 5 6 Im(λ) 3 b = 3 b = b = b = q(t) Re(λ) 5 5 t OOK signals for input bits [ ] and [ ] 67

83 On-Off Eigenvalue Encoder (OOK) 5 6 Im(λ) 3 b = 3 b = b = b = q(t) Re(λ) 5 5 t 5 6 Im(λ) 3 b = 3 b = b = b = q(t) Re(λ) 5 5 t OOK signals for input bits [ ] and [ ] Problem: Spectral efficiency drops as the number of points is increased 67

84 Multi-eigenvalue position encoding 68 Multi-eigenvalue position encoding Im(λ) q(t) t 6 q(t) t.5.5 Re(λ)

85 Multi-eigenvalue position encoding Generate ( N ) + ( N ) + + ( N k ) signals for a given grid of N points, and at most k eigenvalues in a transmit signal Select cut-off bandwidth and pulsewidth SE (in bits/s/hz) 3 3 B c (in GHz) T c (in ns) 3 Spectral efficiency as a function of the cutoff parameters (N=5, =.j, k = 5) 69

86 Multi-eigenvalue position encoding Achievable spectral efficiencies k M M T c B c SE (in bits/s/hz) 3, ns 6.5 GHz.3 5, ns 5.8 GHz.7 5,369,936,569 ns 3.6 GHz 3. Table : Spectral efficiencies of the multi-eigenvalue position encoder (N = 5 and =.) 7

87 Experimental Demonstration ECL laser AWG I Q I/Q modulator EDFA AOM LO Coherent Receiver I x Q x I y Q y Real-time Oscilloscope & offline DSP PC OBPF 3dB AOM Coupler Fiber recirculating loop EDFA 5 km SSMF Raman pump Z. Dong, S. Hari, T. Gui, K. Zhong, M. I. Yousefi, C. Lu, P.-K. A. Wai, F. R. Kschischang, and A. P.-T. Lau, Nonlinear Frequency Division Multiplexed Transmissions based on NFT, IEEE Photonics Techn. Letters, vol. 7, no. 5, pp. 6 63, Aug. 5. 7

88 eigenvalues (3km) (,,,) (,,,) (,,,) (,,,) (,,,) 5 3 (,,,) 5 3 (,,,) (,,,) (,,,) (,,,) (,,,) (,,,) (,,,) (,,,) (,,,) 7

89 3 eigenvalues (8km) (,,) (,,) (,,) (,,) (,,) (,,) (,,) 73

90 Received Signal (8km) 7

91 Recent Result (Yousefi and Yangzhang, 6) Achievable rate [bits/d] 8 6 AWGN NFDM WDM P [dbm] ( km, normal-dispersion, NFDM vs 5 channel WDM in 6 GHz bandwidth) M. I. Yousefi and X. Yangzhang, Linear and Nonlinear Frequency-Division Multiplexing, arxiv:63.389v, May 6. 75

92 Conclusion Pulse train transmission and WDM may not be the best transmission strategy for the nonlinear optical channel. All deterministic distortions are zero for all users under NFDM: NFDM removes inter-channel interference (cross-talk) between users of a network sharing the same fiber channel; NFDM removes inter-symbol interference (ISI) (intra-channel interactions) for each user; with NFDM, information in each channel of interest can be conveniently read anywhere in a network without knowledge of the distance or any information about other users; spectral invariants are remarkably stable features of the NLS flow. Exploiting the integrability of the NLS equation, NFDM modulates non-interacting degrees of freedom and thereby does not suffer from cross-talk, a major limiting factor for prior work. 76

93 Conclusions FPU Zabusky-Kruskal Gardner et al. Lax RM Miura Hirota Zakharov-Shabat Hasegawa Ablowitz et al. WDM, optical solitons Hasegawa s 993 s The NFT can be used for transmission of information NFT solves at least two major problems it removes ISI spectral data appear to be more robust to noise compared with time-domain processing (e.g., digital backpropagation) Is it better than existing methods? We do not know yet. So far we can match existing methods. However, our insights suggest that the achievable rates for NFDM will trend upwards for increasing SNRs. disadvantages: critically relies on integrability it is hard to implement or sometimes to simulate. WDM disp/nonlin man NFT 77

94 Open Problems Plenty of open problems: Entropy (rate) evolution (production) in NLS equation with additive noise Rigorous bounds on capacity Vector models (for polarization, multimode, multicore) Discrete models Development of fast algorithms 78

95 References A. Hasegawa and T. Nyu, Eigenvalue Communication, J. of Lightwave Technology, vol., pp , March 993. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, B. Goebel, Capacity limits of optical fiber networks, J. Lightw. Technol., vol. 8, pp. 66 7, Sept./Oct.. 3 M. I. Yousefi and F. R. Kschischang, Information Transmission using the Nonlinear Fourier Transform, Part I: Mathematical Tools, Part II: Numerical Methods, Part III: Discrete Spectrum Modulation, IEEE Trans. on Inform. Theory, July. E. Meron, M. Feder and M. Shtaif, On the achievable communication rates of generalized soliton transmission systems, arxiv:7.,. 5 P. Kazakopoulos and A. L. Moustakas, Transmission of information via the non-linear Schrödinger equation: The random Gaussian input case, ArXiv:.79, 3. 6 S. Wahls and H. V. Poor, Fast numerical nonlinear Fourier transforms, to appear in IT Transactions, 5. 79

96 References (cont d) 7 H. Bülow, Experimental assessment of nonlinear Fourier transform based detection under fiber nonlinearity, ECOC. 8 J. E. Prilepsky, et al., Nonlinear inverse synthesis and eigenvalue division multiplexing in optical fiber channels, PRL, Jul.. 9 H. Terauchi and A. Manuta, Eigenvalue modulated optical transmission system based on digital coherent technology, OECC, 3. H. Terauchi, Y. Matsuda, A. Toyota, and A. Maruta, Noise tolerance of eigenvalue modulated optical transmission system based on digital coherent technology, OECC/ACOFT,. S. Wahls, S. T. Le, J. E. Prilepsky, H. V. Poor, S. K. Turitsyn, Digital backpropagation in the nonlinear Fourier domain, SPAWC, Stockholm, Sweden, Jun. 5. V. Aref, H. Bülow, K. Schuh, W. Idler, Experimental demonstration of nonlinear frequency division multiplexed transmission, ECOC 5, Sept. 5. 8

97 6 Steele Prize For their discovery in the 97s of the mathematical framework underlying the nonlinear Fourier transform, Clifford S. Gardner, John M. Greene, Martin D. Kruskal and Robert M. Miura received the prestigious 6 Leroy P. Steele Prize for a Seminal Contribution to Research, awarded by the American Mathematical Society. From the Steele Prize Announcement: Nonlinearity has undergone a revolution: from a nuisance to be eliminated, to a new tool to be exploited. 8

98 Acknowledgment This work was supported: by the Natural Sciences and Engineering Research Council of Canada through the Discovery Grants Program, and by the Institute for Advanced Study, Technische Universität München, through a Hans Fischer Senior Fellowship. I am profoundly grateful for all support. 8

99 Acknowledgment This work was supported: by the Natural Sciences and Engineering Research Council of Canada through the Discovery Grants Program, and by the Institute for Advanced Study, Technische Universität München, through a Hans Fischer Senior Fellowship. I am profoundly grateful for all support. Thank You! 8