Széchenyi István University Győr Hungary Solitons in optical fibers Szilvia Nagy Department of Telecommunications

2ESM Zilina 2008 Nonlinear effects in fibers History of solitons Korteweg—deVries equations Envelop solitons Solitons in optical fibers Amplification of solitons – optical soliton transmission systems Outline – General Properties

3ESM Zilina 2008 Brillouin scattering: acoustic vibrations caused by electro- magnetic field (e.g. the light itself, if P>3mW) acoustic vibrations caused by electro- magnetic field (e.g. the light itself, if P>3mW) acoustic waves generate refractive index fluctuations acoustic waves generate refractive index fluctuations scattering on the refraction index waves scattering on the refraction index waves the frequency of the light is shifted slightly direction dependently (~11 GHz backw.) the frequency of the light is shifted slightly direction dependently (~11 GHz backw.) longer pulses – stronger effect longer pulses – stronger effect Nonlinear effects in fibers

4ESM Zilina 2008 Raman scattering: optical phonons (vibrations) caused by electromagnetic field and the light can exchange energy (similar to Brillouin but not acoustical phonons) optical phonons (vibrations) caused by electromagnetic field and the light can exchange energy (similar to Brillouin but not acoustical phonons) Stimulated Raman and Brillouin scattering can be used for amplification Nonlinear effects in fibers

5ESM Zilina 2008 (Pockels effect: refractive index change due to ecternal electronic field refractive index change due to ecternal electronic field  n ~ | E | - a linear effect)  n ~ | E | - a linear effect) Nonlinear effects in fibers

6ESM Zilina 2008 Kerr effect: the refractive index changes in response to an electromagnetic field the refractive index changes in response to an electromagnetic field  n = K | E | 2  n = K | E | 2 light modulators up to 10 GHz light modulators up to 10 GHz can cause self-phase modulation, self- induced phase and frequency shift, self- focusing, mode locking can cause self-phase modulation, self- induced phase and frequency shift, self- focusing, mode locking can produce solitons with the dispersion can produce solitons with the dispersion Nonlinear effects in fibers

7ESM Zilina 2008 Kerr effect: the polarization vector the polarization vector if E = E  cos(  t), the polarization in first order is if E = E  cos(  t), the polarization in first order is Nonlinear effects in fibers Pockels Kerr

8ESM Zilina 2008 Kerr effect: the susceptibility the susceptibility the refractive index the refractive index n 2 is mostly small, large intensity is needed (silica: n 2 ≈10 −20 m 2 /W, I ≈10 9 W/cm 2 ) n 2 is mostly small, large intensity is needed (silica: n 2 ≈10 −20 m 2 /W, I ≈10 9 W/cm 2 ) Nonlinear effects in fibers

9ESM Zilina 2008 Gordon-Haus jitter: a timing jitter originating from fluctuations of the center frequency of the (soliton) pulse a timing jitter originating from fluctuations of the center frequency of the (soliton) pulse noise in fiber optic links caused by periodically spaced amplifiers noise in fiber optic links caused by periodically spaced amplifiers the amplifiers introduce quantum noise, this shifts the center frequency of the pulse the amplifiers introduce quantum noise, this shifts the center frequency of the pulse the behavior of the center frequency modeled as random walk the behavior of the center frequency modeled as random walk Nonlinear effects in fibers

10ESM Zilina 2008 Gordon-Haus jitter: dominant in long-haul data transmission dominant in long-haul data transmission ~L 3, ~L 3, can be suppressed by can be suppressed by regularly applied optical filters amplifiers with limited gain bandwidth can also take place in mode-locked lasers can also take place in mode-locked lasers Nonlinear effects in fibers

11ESM Zilina 2008 History of solitons John Scott Russel ( ) 1834, Union Canal, Hermiston near Edinbourgh, a boat was pulled after the stop of the boat a „wave of translation” arised 8-9miles/hour wave velocity traveled 1-2 miles long

12ESM Zilina 2008 History of solitons J. S. Russel, Report on Waves, 1844

13ESM Zilina 2008 History of solitons Snibston Discovery Park

14ESM Zilina 2008 History of solitons Scott Russel Aqueduct, 1995 Heriot-Watt University Edinbourgh

15ESM Zilina 2008 History of solitons 1870s J. Boussinesq, Rayleigh both deduced the secret of Russel’s waves: the dispersion and the nonlinearity cancels each other 1964 Zabusky and Kruskal solves the KdV equation numerically, solitary wave solutions: soliton 1960s: nonlinear wave propagation studied with computers: many fields were found where solitons appear

16ESM Zilina 2008 History of solitons 1970s A. Hasegawa proposed solitons in optical fibers 1980 Mollenauer demonstrated soliton transmission in optical fiber (10 ps, 1.5  m, 700 m fiber) 1988 Mollenauer and Smith sent soliton light pulses in fiber for 6000 km without electronic amplifier

17ESM Zilina 2008 In 1895 Korteweg and deVries modeled the wave motion on the surface of shallow water by the equation where hwave height  time in coordinates  space coordinate moving with the wave Korteweg—deVries equations

18ESM Zilina 2008 Korteweg—deVries equations Derivation of the KdV equation a wave h propagating in x direction can be described in the coordinate system ( ,  ) traveling with the wave as Using the original (x,t) coordinates:

19ESM Zilina 2008 Korteweg—deVries equations Stationary solution of the KdV equation Dispersive and nonlinear effects can balance to make a stationary solution

20ESM Zilina 2008 Korteweg—deVries equations Stationary solution of the KdV equation Dispersive and nonlinear effects can balance to make a stationary solution where  is the velocity of the solitary wave in the ( ,  ) space

21ESM Zilina 2008 Korteweg—deVries equations Stationary solution of the KdV equation

22ESM Zilina 2008 Korteweg—deVries equations The KdV equation and the inverse scattering problems the Schrödinger equation: if „potential” u(x,t) satisfies a KdV equation, is independent of time is independent of time u(x,0) → 0 as |x|→ ∞ u(x,0) → 0 as |x|→ ∞ the Schrödinger equation can be solved for t=0 for a given initial u(x,0) the Schrödinger equation can be solved for t=0 for a given initial u(x,0)

23ESM Zilina 2008 Korteweg—deVries equations The KdV equation and the inverse scattering problems t=0 scattering data can be derived from the t=0 solution t=0 scattering data can be derived from the t=0 solution the time evolution of  and thus the scattering data is known the time evolution of  and thus the scattering data is known u(x,t) can be found for each  (x,t) by inverse scattering methods. u(x,t) can be found for each  (x,t) by inverse scattering methods.

24ESM Zilina 2008 Korteweg—deVries equations Solutions of KdV equations with various boundary conditions in various dimensions soliton propagating and scattering

25ESM Zilina 2008 Korteweg—deVries equations Solutions of KdV equations with various boundary conditions in various dimensions soliton wave in the sea (Molokai)

26ESM Zilina 2008 Korteweg—deVries equations Solutions of KdV equations with various boundary conditions in various dimensions soliton wave in the sky

27ESM Zilina 2008 Korteweg—deVries equations Solutions of KdV equations with various boundary conditions in various dimensions two solitons 1D

28ESM Zilina 2008 Korteweg—deVries equations Solutions of KdV equations with various boundary conditions in various dimensions two solitons 2D

29ESM Zilina 2008 Korteweg—deVries equations Solutions of KdV equations with various boundary conditions in various dimensions crossing solitons

30ESM Zilina 2008 Korteweg—deVries equations Solutions of KdV equations with various boundary conditions in various dimensions crossing solitons

31ESM Zilina 2008 Korteweg—deVries equations Solutions of KdV equations with various boundary conditions in various dimensions airball soliton scattering

32ESM Zilina 2008 Korteweg—deVries equations Solutions of KdV equations with various boundary conditions in various dimensions airball soliton scattering – a pinch

33ESM Zilina 2008 Korteweg—deVries equations Solutions of KdV equations with various boundary conditions in various dimensions higher order soliton

34ESM Zilina 2008 Envelop solitons Envelop of a wave if the amplitude of a wave varies (slowly) envelop of the wave complex amplitude

35ESM Zilina 2008 Envelop solitons If the wave can be described by the wave equation for the envelop with, and reduction factor, ~1/2

36ESM Zilina 2008 Envelop solitons Normalization

37ESM Zilina 2008 Envelop solitons Solving the non-linear Schrödinger equation test function test function the new equation the new equation

38ESM Zilina 2008 Envelop solitons looking for solitary wave solution of the new equation looking for solitary wave solution of the new equation ifis a stationary solution ifis a stationary solution it can be shown, that C is independent of X

39ESM Zilina 2008 Envelop solitons the solutions the solutions which give which give 0 and  0 are phase constants  =  1/2 : amplitude and pulse width  transmission speed

40ESM Zilina 2008 Solitons in optical fibers envelop equation of a light wave in a fiber fiber loss rate per unit length:  with

41ESM Zilina 2008 Solitons in optical fibers Solitons can arise as solution of if the real part of the nonlinear term is dominant,

42ESM Zilina 2008 Solitons in optical fibers the condition for existence of a soliton: example:   ≈ 1500 nm |Ê| ≈ 10 6 V/m  < 2 ×10 −4 m −1 n 2 ≈ 1.2×10 −22 m 2 /V 2 n 2 ≈ 1.2×10 −22 m 2 /V dB/km

43ESM Zilina 2008 Solitons in optical fibers the normalized equation, with if  is small enough, perturbation techniques can be used

44ESM Zilina 2008 Solitons in optical fibers The solution of the normalized soliton equation in fibers with loss predicts the amplitude  of the soliton decreases as it propagates: the amplitude  of the soliton decreases as it propagates: the width  of the soliton increases the width  of the soliton increases their product remains constant their product remains constant

45ESM Zilina 2008 Solitons in optical fibers Effects of the waveguide manifest as higher order linear dispersion nonlinear dispersion of the Kerr coefficient nonlinear dissipation due to Raman processes (imaginary!!!)

46ESM Zilina 2008 Solitons in optical fibers Necessary condition for existence of a soliton  0 : pulse length [ps]  0 : pulse length [ps] P 0 :required pulse power [W] P 0 :required pulse power [W]  : wavelength [  m]  : wavelength [  m] D : dispersion [ps/(nm km)] D : dispersion [ps/(nm km)] S : cross-sectional area [  m 2 ] S : cross-sectional area [  m 2 ] e.g., S=60  m 2, =1.5  m, |D|=10 ps/(nm km)  0 =10 ps, P 0 =180 mW

47ESM Zilina 2008 Solitons in optical fibers Soliton generation needs low loss fiber (<1 dB/km) low loss fiber (<1 dB/km) spectral width of the laser pulse be narrower than the inverse of the pulse length spectral width of the laser pulse be narrower than the inverse of the pulse length Mollenauer & al. 1980, AT&T Bell Lab. Mollenauer & al. 1980, AT&T Bell Lab. 700 m fiber, 10 −6 cm 2 cross section 7 ps pulse, F 2+ color center laser with Nd:YAG pump 1.2 W soliton threshold

48ESM Zilina 2008 Amplification of solitons For small loss the soliton propagates with the product of its pulse length and height being constant reshaping is needed for long-distance communication application reshaping methods: induced Raman amplification – the loss compensated along the fiber induced Raman amplification – the loss compensated along the fiber repeated Raman Amplifiers repeated Raman Amplifiers Er doped amplifiers Er doped amplifiers

49ESM Zilina 2008 Amplification of solitons Experiment on the long distance transmission of a soliton by repeated Raman Amplification (Mollenauer & Smith, 1988) 41.7 km 3 dB coupler dependent coupler all fiber MZ interferometer signal in signal out pump in 1600 nm 1500 nm filter, 9 ps diode, spectrum analyzer

50ESM Zilina 2008 Amplification of solitons Erbium doped fiber amplifiers, periodically placed in the transmission line distance of the amplifiers should be less then the soliton dispersion length distance of the amplifiers should be less then the soliton dispersion length dispersion shifted fibers or filters for reshaping dispersion shifted fibers or filters for reshaping quantum noise arise spontaneous emission noise spontaneous emission noise Gordon—House jitter Gordon—House jitter

51ESM Zilina 2008 Optical soliton transmission systems The soliton based communication systems mostly use on/off or DPSK keying In soliton communication systems the timing jitters which originate from frequency fluctuation are held under control by narrow band optical filters frequency guiding filter frequency guiding filter e.g., a shallow Fabry-Perot etalon filter e.g., a shallow Fabry-Perot etalon filter (in non-soliton systems, these guiding filters destroy the signal, they are not used)

52ESM Zilina 2008 Optical soliton transmission systems It is possible to make the soliton “slide” in frequency sliding frequency guiding filters sliding frequency guiding filters each consecutive narrow-band filter has slightly different center frequency each consecutive narrow-band filter has slightly different center frequency center frequency sliding rate: f’= df/dz center frequency sliding rate: f’= df/dz the solitons can follow the frequency shift the solitons can follow the frequency shift the noise can not follow the frequency sliding, it drops out the noise can not follow the frequency sliding, it drops out

53ESM Zilina 2008 Optical soliton transmission systems Wavelength division multiplexing in soliton communication systems solitons with different center frequency propagate with different group velocity solitons with different center frequency propagate with different group velocity in collision of two solitons, they propagate together for a while in collision of two solitons, they propagate together for a while collision length: collision length:

54ESM Zilina 2008 Optical soliton transmission systems during the collision both solitons shifts in frequency (same magnitude, opposite sign) during the collision both solitons shifts in frequency (same magnitude, opposite sign) first part of the collision: the fast soliton’s velocity increases, while the slow one becomes slower first part of the collision: the fast soliton’s velocity increases, while the slow one becomes slower at the second part of the collision, the opposite effect takes place, symmetrically at the second part of the collision, the opposite effect takes place, symmetrically

55ESM Zilina 2008 Optical soliton transmission systems if during the collision the solitons reach an amplifier or a reshaper, the symmetry brakes if during the collision the solitons reach an amplifier or a reshaper, the symmetry brakes the result is non-zero residual frequency shift can arise, unless the result is non-zero residual frequency shift can arise, unless

56ESM Zilina 2008 Optical soliton transmission systems if a collision of two solitons take place at the input of the transmission if a collision of two solitons take place at the input of the transmission half collision half collision it can be avoided by staggering the pulse positions of the WDM channels at the input. it can be avoided by staggering the pulse positions of the WDM channels at the input.

57ESM Zilina 2008 J. C. Russel, Report of the fourteenth meeting of the British Association for the Advancement of Science, York, September 1844, p. 311 London, Boussinesq J. Math. Pures Appl., vol. 7, p. 55, Lord Rayleigh Philosophical Magazine, s5, vol. 1, p. 257, 1876, Proc. London Math. Soc. s1, vol. 17, p. 4, N.J. Zabusky, M.D. Kruskal, Phys. Rev. Lett., vol. 15, p. 240, A. Hasegawa, F.D. Tappert, Appl. Phys. Lett., vol. 23, p. 142, 1973.

58ESM Zilina 2008 L.F. Mollenauer, R.H. Stolen, J.P. Gorden, Phys. Rev. Lett., vol. 45, p. 1095, J.P. Gordon, H.A. Haus, Opt. Lett., vol. 11, p. 665, D.J. Korteweg, G, deVries, Phil. Mag. Ser. 5, vol. 39, p. 422, 1895.

59ESM Zilina 2008 J. Hecht, Understanding fiber Optics (fifth edition), Pearson Prentice Hall, Upper Saddle River, New Jersey, Columbus, Ohio, J. Gowar, Optical Communication Systems (second edition) Prentice-Hall of India, New Delhi, A. Hasegawa, Optical Solitons in Fibers Springer-Verlag, Berlin, Fiber Optic Handbook, Fiber, Devices, and Systems for Optical Communications, editor: M. Bass, (associate editor: E. W. Van Stryland) McGraw-Hill, New York, 2002.

60ESM Zilina 2008 J. Hietarinta, J. Ruokolainen, Dromions – The Movie E. Frenkel, Five lectures on soliton equations arXiv:q-alg/ v Contribution to Survays in Differential Geometry, vol. 3, International Press. Encyclopedia of Laser Physics Light Bullet Home Page,