1. This teaching material is a part of e-Photon/ONe Master study in Optical Communications and Networks Course and module: Author(s): No part of this presentation can be reused without the permission of author(s). Users are requested to ask for permission by specifying the purpose of the usage. http://www.e-photon-one.org Introduction to optical networks – Light propagation Chapter 8 : Non-linear effects in optical fibres Laurent Dupont, Michel Morvan : GET-ENST Bretagne, firstname.name@enst-bretagne.fr Roberto Gaudino : POLITO, roberto.gaudino@polito.it

2. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (2) Outline Foreword The Kerr Effect  Non-linear polarisation  Self-phase modulation  Cross-phase modulation  Four wave mixing Stimulated Raman and Brillouin scattering  Elastic and inelastic scattering  Spontaneous Raman scattering  Stimulated Raman Scattering and Raman amplification  Stimulated Brillouin Scattering

3. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (3) Foreword This chapter describes synthetically the non-linear phenomena that can be encountered in optical fibres. It is mainly devoted to the major effects like the Kerr effect and the Stimulated Raman and Brillouin Scattering. The basic physical phenomena and the origins are described. They require some notions in solid state physics and quantum mechanics. Their complexity should deserve a dedicated course for a complete description but it is outside the scope of this course. The consequence of these non-linear effects on transmission will be detailed in the transmission course.

4. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (4) Molecular structure of fused silica Silica is the material of glass optical fibres. Fused silica is an amorphous (i.e. non crystalline) silicon dioxide. Unlike cristobalite which is a crystalline form of silica with an ordered assembly of SiO 4 tetrahedral basic brick, fused silica present a disordered assembly of such basic bricks. Crystalline structure : cristobaliteAmorphous structure

5. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (5) Outline Foreword The Kerr Effect  Non-linear polarisation  Self-phase modulation  Cross-phase modulation  Four wave mixing Stimulated Raman and Brillouin scattering  Elastic and inelastic scattering  Spontaneous Raman scattering  Stimulated Raman Scattering and Raman amplification  Stimulated Brillouin Scattering

6. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (6) Non-linear polarisation and non-linear effects in optical fibres Non-linear material polarisation : with If Kerr Effect

7. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (7) The Kerr effect As presented previously, the Kerr effect is due to the non-linear response of the material. It means that the index of the silica is now depending on the optical field propagation through it : where n 2 =3,2.10 -20 m 2 /W Even if silica is a weakly non-linear medium, the Kerr has significant influence due to the field intensity within the fibre core combined with the propagation distance. The Effect Kerr can not be neglected at high optical powers.

8. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (8) Effective area of an optical fibre The effective area depends on the fibre type    A eff Radius  MFD : Mode Field Diameter Approximated Gaussian profile

9. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (9) Effective area of various fibre types (typical values) The DSF fibre present the smallest effective area among the line fibres. Among special fibres, the effective area is particularly small in Dispersion Compensating Fibres. This implies that one has to be extremely cautious when fixing the DCM input power levels in dispersion compensated links. ITU-T fibre typeA eff @ 1550 nm (µm²) G.652 SMF85 G.653 (DSF)46 G.654 (CSF)88 G.655 (NZDSF)52 (D>0), 56 (D<0) and 73 DCF23 SMF : Single Mode Fiber DSF : Dispersion Shifted Fiber CSF : Cut-off Shifted Fiber NZDSF : Non-Zero DSF DCF : Dispersion Compensating Fiber

10. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (10) - Self Phase Modulation (SPM) : If an intensity modulated signal propagates in the fibre, the intensity modulation induces an index modulation of the fibre and in return a phase modulation to the signal.  The signal modulates itself  The SPM induced phase modulation broadens the signal spectrum. - Cross Phase Modulation (XPM) : In the case of a multi–channel propagation, the index modulation induced by the Kerr–effect modulates the other channels and vice-versa. - Four Wave Mixing (FWM) : In the case of a multi–channel propagation and under phase matching conditions, new frequencies are generated in the fibre causing crosstalk and power depletion. The different consequences of the Kerr effect

11. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (11) SPM or Self-Phase Modulation P(t) t + ++ + + - + - - The signal phase modulation speed is proportional to its amplitude modulation speed : the higher the bit rate, the higher the SPM. We have with hence

12. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (12) Self Phase Modulation : the equations Without dispersion, SPM alone induces a phase shift over a propagation distance z equal to : Where  is the non-linear coefficient of the fibre, defined as : The phase modulation generates a frequency modulation or « chirp » :

13. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (13) Effective fibre length The previous equation can be written in a simpler way : L eff is called the effective length of the fibre. As non- linear effects depend on optical intensity, L eff indicates how far along the fibre the non-linearities occur. with L P For large L L eff P in P in /e

14. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (14) Self-phase modulation Spectral broadening of the pulse As shown by the equations, the non-linear phase follows exactly the power shape of the optical pulses. The frequency chirp is then proportional to the derivative of the optical power. If pulses propagate under the non-linear regime :  the optical frequency will decrease on the pulse leading edge (red shift).  the optical frequency will increase on the pulse trailing edge (blue shift).

15. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (15) Non-linear length of a fibre In order to easily estimate the influence of the Kerr effect and the SPM in particular, a non-linear length has been defined as the propagation length for which a Gaussian envelope pulse has been phase shifted by  due to SPM. We find that : Where  is the non-linear parameter and P 0 is the pulse peak power. This length has to be compared with the dispersion length T 0 is the initial pulse width and  2 the second derivative of propagation constant of the fibre

16. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (16) The different propagation regimes of an optical pulse Depending on the relative values of fibre length L, non-linear length L NL and dispersion length L D, we can distinguish four different propagation regimes : 1°) L NL >> L and L D >> L : the optical pulse will keep its shape during propagation. 2°) L NL >> L and L D  < L : the pulse will be broadened and distorted by chromatic dispersion. 3°) L NL  > L : the pulse will be distorted by SPM and its spectrum will be broadened. 4°) L NL  0, these two effects can balance each other to maintain the pulse shape unchanged : this is the soliton effect.

17. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (17) Cross Phase Modulation (XPM) In the case of multi-channel propagation at various wavelength, the different channels modulate themselves via SPM but also each other via the fibre index modulation. The efficiency of Cross Phase Modulation (XPM) depends on :  The fibre chromatic dispersion  The SOP of the different channels  Channel spacing  Channel power XPM induces non-linear crosstalk.

18. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (18) Four Wave Mixing (FWM) Under specific phase and wave vectors matching conditions, four different waves will interact in the fibre in a non-linear way. The easiest way to obtain FWM in a fibre is to propagate two waves at angular frequencies  1 and  2 that will create new waves at frequencies  3 and  4 such as: The phase matching condition is : This phenomenon is strongly dependent on channel spacing and chromatic dispersion. The generated waves may cause crosstalk if they are at the same wavelength as incident channels.

19. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (19) Four wave mixing (FWM) The total power at new frequencies generated via four wave mixing is proportional to : Non-linear Fibre Where P is the power per channel P FWM PsPs Hence :

20. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (20) Some solutions to mitigate the Kerr effect in fibres Decrease the field intensity by increasing the effective area. e.g. by using G.652 or G.655 LEAF (Large Effective Area Fibre) fibres. In the case of single channel transmission, the increase of the chromatic dispersion will automatically lower the SPM. But the problem is reported to the chromatic dispersion compensation if DCF is used as SPM may be high in such fibres. In the case of multi-channel transmission, the increase of the channel spacing and /or chromatic dispersion will decrease XPM effects. Use orthogonal SOPs for adjacent channels to decrease interaction

21. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (21) Outline Foreword The Kerr Effect  Non-linear polarisation  Self-phase modulation  Cross-phase modulation  Four wave mixing Stimulated Raman and Brillouin scattering  Elastic and inelastic scattering  Spontaneous Raman scattering  Stimulated Raman Scattering and Raman amplification  Stimulated Brillouin Scattering

22. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (22) The Raman scattering phenomenon Light is usually scattered by matter in an elastic way. It means that no energy is exchanged between light and matter : the incident and the scattered photons have the same wavelength hence the same energy. This is the Rayleigh scattering. But in some cases, energy can be transferred from the photon to the atom or molecule or vice-versa during the scattering process : this is the Raman inelastic scattering. The photon can loose or gain energy and hence change its frequency.  If the photons lose energy, the scattered wave has a longer wavelength and is called the Stokes wave.  If the photons gain energy, the scattered wave has a shorter wavelength and is called the anti-Stokes wave.

23. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (23) Phonons in solids Phonons are quantum modes of vibrations in solids. Just like for the photon, the energy of a phonon is proportional to its frequency :  Low frequency phonons corresponding to the medium vibration modes are called acoustic phonons.  Much higher frequency phonons are encountered in crystals where there are more than one atom per cell. These phonon are called optical phonons because they can be excited using photons (usually IR). Just like photons, phonons can be seen both as waves and particles. Phonons are play a major role in solids properties like thermal conductivity and electrical conduction.

24. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (24) The Raman scattering equations The Raman scattering is observed in crystalline solids but also in amorphous solids like fused silica. It is a quantum process that can be described by the two following equations : Equation (1) describes the conservation of energy and equation (2) the conservation of impulsion.  St, K St p,kpp,kp  St, k St Stokes  St, K St p,kpp,kp  St, k St Anti-Stokes

25. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (25) Spontaneous Raman scattering (1928 - C. V. Raman) The excited vibrational state of the molecule or atom will be transmitted to the material and propagates as a phonon with energy : s > p s < p Pump photon Stokes photon Anti-Stokes photon Pump photon Relaxed state Excited vibrational state

26. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (26) Raman gain can be obtained in any fibre. The combination of several pumps at different wavelengths allow to expand the gain bandwidth. The maximum Raman gain is obtained for a frequency shift of 13 THz in fused silica. Raman amplification depends on the relative SOP of the pump and the signal. Raman gain is obtained both in co and counter-propagative regime. Raman gain profile (nm)  = 100 nm (at 1550 nm) Pump wave Gain curve 1450 1550

27. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (27) Stimulated Raman Scattering As for the lasing process, Stimulated Raman Scattering (SRS) occurs when the pump power has reached a certain threshold. In the absence of fibre losses, the total number of photons is constant, Hence we have : s < p Pump photon Signal photon

28. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (28) Raman amplification Raman amplification uses the Stimulated Raman Scattering. It allows to amplify a signal within the fibre itself. The pump wave will transfer its energy to the signal wave. Hence, in the continuous wave case, the process is governed by the set of following equations : These equations can be solved analytically if we neglect the pump depletion, i.e. the loss of pump power due to the Raman process.

29. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (29) Raman effective length Under the previously defined conditions, we find that : L eff is called the effective length. It represents the fibre length that can be effectively considered due to the pump absorption. In order to find the full signal power, we have to integrate the intensity over the full Raman gain band.

30. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (30) Raman threshold The Raman threshold is defined as the input pump power for which the signal (Stokes) and pump powers are equal at the fibre output. Under the assumption of a Lorentzian Raman gain spectrum, the Raman pump threshold for a co-propagating pump wave has been found [R.G. Smith in Applied Optics 1972] to be well approximated by : For typical line fibres used at 1550 nm, we find pump powers about 600 to 800 mW, which is quite high. These powers have to be coupled inside the line fibre using specific pump muxes.

31. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (31) Raman amplification in counter-propagating mode Raman pump Several 100’s of mW Pump MUX Line fibre In-line EDFA amplification site

32. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (32) Power versus distance for Raman and non-Raman fibre hops -20 -16 -12 -8 -4 0 Signal Power (dBm) 100806040200 Distance (km) Signal power with Raman pumping Signal power without Raman pumping Counter directional pumping

33. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (33) Effect of Stimulated Brillouin Scattering Stimulated Brillouin Scattering (SBS) is a particular case of SRS where the involved phonons are acoustic ones. SBS gain is narrow band and the Stokes wave is always propagating backwards. The frequency shift is about 11 GHz in fused silica. The SBS threshold is much lower than for SRS : several mW are enough in certain conditions. Usually, a low frequency dithering of the signal broadens the spectrm and hence raises the threshold s - 11 GHz TX s

34. Michel Morvan & Marc Wuilpart Introduction to Optical Networks, VIII – Non-linear Effects in Optical Fibres (34) Bibliography Charles Kittel : « Introduction to solid state physics », fifth edition, John Wiley and Sons, Inc Govind Agrawal : « Non-linear fiber optics », Academic Press, Inc Richard Feynman, « The Feynman lectures on physics », Addison Wesley Publishing Company, Inc « L’optique guidée monomode et ses applications », in French, by a group of Thomson-CSF engineers, Masson