Lecture 6 Nonlinearities

1. Lecture 6 Nonlinearities By Halim

2. Light – Matter Interaction ´,k´,E´ ,k,E Normally, Induced Dipole Reradiation (electronic response) 1. Optical interactions depend on the Electric field in the light wave. 2. Valence/outer `bound’ electrons that respond to this field. But, 3. Does this idea work when you go to high light Intensities? NO!

3. Start of Nonlinear Optics Nonlinear optics started by the discovery of Second Harmonic generation shortly after demonstration of the first laser in 1960 by Ali Javan. (Peter Frankenet al 1961)

4. 2. The Essence of Nonlinear Optics When the intensity of the incident light to a material system increases the response of medium is no longer linear Output Input intensity

5. Response of an optical Medium The response of an optical medium to the incident electro magnetic field is the induced dipole moments inside the medium

6. Nonlinear Susceptibility Dipole moment per unit volume or polarization The general form of polarization

10. Nonlinear Polarization • Permanent Polarization • First order polarization: • Second order Polarization • Third Order Polarization

11. e a0 N How does optical nonlinearity appear The strength of the electric field of the light wave should be in the range of atomic fields

12. Nonlinear Optical Interactions • The E-field of a laser beam • 2nd order nonlinear polarization

13. Sum Frequency Generation Application: Tunable radiation in the UV Spectral region.

14. Application: The low frequency photon, amplifies in the presence of high frequency beam . This is known as parametric amplification. Difference Frequency Generation

15. Phase Matching • Since the optical (NLO) media are dispersive, • The fundamental and the harmonic signals have • different propagation speeds inside the media. • The harmonic signals generated at different points • interfere destructively with each other.

16. Third Order Nonlinearities • When the general form of the incident electric field is in the following form, The third order polarization will have 22 components which their frequency dependent are

17. The Intensity Dependent Refractive Index • The incident optical field • Third order nonlinear polarization

18. The total polarization can be written as One can define an effective susceptibility The refractive index can be defined as usual

19. By definition where

20. Third order nonlinear susceptibility of some material

21. Processes due to intensity dependent refractive index • Self focusing and self defocusing • Wave mixing • Degenerate four wave mixing and optical phase conjugation

22. Self focusing and self defocusing • The laser beam has Gaussian intensity profile. It can induce a Gaussian refractive index profile inside the NLO sample.

23. Wave mixing

24. PCM M PCM s M Optical Phase Conjugation • Phase conjugation mirror

25. What is the phase conjugation The signal wave The phase conjugated wave

26. A1 A2 A3 A4 Degenerate Four Wave Mixing • All of the three incoming beams A1, A2 and A3 should be originated • from a coherent source. • The fourth beam A4, will have the same Phase, Polarization, and • Path as A3. • It is possible that the intensity of A4 be morethan that of A3

27. General Overview of Nonlinearities- 2 categories • Nonlinear inelastic scattering processes include • Stimulated Raman scattering (SRS) • Stimulated Brillouin scattering (SBS) • Nonlinear effects from intensity-dependent variations in the refractive index include • Self-phase modulation (SPM) • Cross-phase modulation (XPM) • Four-wave mixing (FWM)

28. Basic Effects of Nonlinearities • SBS, SRS, and FWM result in gains or losses in a channel. • The power variations depend on the optical signal intensity. • These processes provide gains to some channels while depleting power from others • These effects produce crosstalk between the wavelength channels. • FWM can be suppressed through special arrangements of fibers having different dispersion characteristics. • SPM and XPM affect only the phase of signals, which causes chirping in digital pulses. This can worsen pulse broadening due to dispersion, particularly in very high-rate systems, such as 40 Gb/s. • When any of these nonlinear effects contribute to signal impairment, an additional amount of power will be needed at the receiver to maintain the same BER as in their absence. This additional power is the power penaltyfor that effect.

29. Power Penalty for Nonlinear Effects • When any nonlinear effect contributes to signal strength reduction, the amount of optical power reduction (in decibels) is the power penalty for that effect

30. Effective Length and Area • Nonlinear effects increase with distance, but are offset by the continuous decrease in signal power along the fiber due to attenuation • A simple model assumes the power is constant over an effective length Leffgiven by • Nonlinear effects increase with the light intensity. For a given optical power, this intensity is inversely proportional to the area of the fiber core. • In practice one can use an effective cross-sectional area Aeff, which assumes a uniform intensity distribution across most of the core.

31. Stimulated Raman Scattering • In stimulated Raman scattering a silica molecule absorbs energy from an incident photon giving it a lower energy and a longer wavelength • The modified photon is called a Stokes photon. • Because the optical signal wave that is injected into a fiber is the source of the interacting photons, it is called the pump wave because it supplies power for the generated wave. • The power transferred to a higher-wavelength channel increases approximately linearly with channel spacing up to about 16 THz (or 125 nm at 1550-nm), and then drops off sharply for larger spacing.

32. Stimulated Brillouin Scattering • Instimulated Brillouin scattering (SBS)a strong optical signal generates an acoustic wave that produces variations in the refractive index. • The index variations cause lightwaves to scatter in the backward direction. • The backscattered light experiences gain from the forward-propagating signals, which leads to depletion of the signal power. • Below a signal level called the SBS threshold, the transmitted power increases linearly with the input level and SBS is negligible. • Beyond the SBS threshold, the % increase in signal depletion grows with signal strength • Beyond the SBS limit any additional launched optical power is scattered backward in the fiber.

33. Kerr Effect • The refractive index n of many optical materials has a weak dependence on optical intensity I (power/Aeff) given by • Here n0 is the ordinary refractive index of the material and n2 is the nonlinear index coefficient. • n2 is about 2.6  10-8 μm2/W in silica, between 1.2 - 5.1  10-6 μm2/W in tellurite glasses, and 2.4  10-5 μm2/W in As40Se60 chalcogenide glass. • The refractive index nonlinearity is the Kerr nonlinearity.

34. Self-Phase Modulation (SPM) • The Kerr nonlinearity produces a carrier-induced phase modulation of the propagating signalcalled the Kerr effect. • In single-wavelength links, the Kerr effect gives rise to self-phase modulation (SPM). • This converts light power fluctuations in a wave to spurious phase fluctuations in the same wave. • In a medium having an intensity-dependent refractive index, a time-varying signal intensity will produce a time-varying refractive index. • The leading edge of a pulse will see a positive dn/dt, whereas the trailing edge will see a negative dn/dt. • This leads to frequency chirping, in that the rising edge of the pulse shifts toward lower frequencies, and the trailing edge toward higher frequencies.

35. Cross-Phase Modulation (XPM) • Cross-phase modulation (XPM) appears in WDM systems and has a similar origin as SPM. • The refractive index nonlinearity converts optical intensity fluctuations in a particular wavelength channel to phase fluctuations in another copropagating channel. • XPM only appears when the two interacting light beams or pulses overlap in space and time. • When multiple wavelengths propagate in a fiber, the total phase shift for an optical signal with frequency ωi is

36. Four-Wave Mixing (FWM) • Four-wave mixing (FWM) is a third-order nonlinearity in optical fibers that is analogous to intermodulation distortion in electrical systems. • When wavelength channels are located near the zero-dispersion point, 3 optical frequencies will mix to produce a 4th intermodulation product given by • If this frequency falls in the transmission window of the original frequencies, it can cause severe crosstalk.

37. FWM Mitigation • If the chromatic dispersion is low, or if there are regions of both positive and negative dispersion in the DWDM operating band, then a large number of FWM terms can be generated by the DWDM signals. • If G.653 dispersion-shifted fibers are used for DWDM in the C-band, the positive and negative dispersion regions around 1550 nm can generate a large number of interfering in-band signals. • The G.655 fiber has a chromatic dispersion value ranging from about 3 to 9 ps/(nm · km) in the entire C-band.

38. Wavelength Converters • One beneficial application of XPM and FWM techniques is for performing wavelength conversion in WDM networks. • An optical wavelength converter is a device that can translate information on an incoming wavelength directly to a new wavelength without entering the electrical domain.

39. Solitons • A pulse shape known as a solitontakes advantage of nonlinear effects in silica, particularly SPM resulting from the Kerr nonlinearity, to overcome the pulse-broadening effects of GVD. • Solitons are very narrow, high-intensity optical pulses that retain their shape through the interaction of balancing pulse dispersion with the nonlinear properties of an optical fiber. • If the relative effects of SPM and GVD are controlled just right, and the appropriate pulse shape is chosen, the pulse compression resulting from SPM can exactly offset the pulse broadening effect of GVD.

40. Dispersive Pulse Propagation • When a dispersive pulse traverses a medium with a positive GVD parameter, the leading part of the pulse is shifted toward a longer wavelength so that the speed in that portion increases. • In the trailing half, the frequency rises so the speed decreases. • Consequently, in addition to a spectral change with distance, the energy in the center of the pulse is dispersed to either side, and the pulse eventually takes on a rectangular-wave shape.

41. Soliton Pulse Propagation • When a narrow high-intensity pulse traverses a medium with a negative GVD parameter, GVD counteracts the chirp produced by SPM. • GVD retards the low frequencies in the front end of the pulse and advances the high frequencies at the back. • The high-intensity sharply peaked soliton pulse changes neither its shape nor its spectrum as it travels along the fiber.

42. Phase Shifts of a Soliton Pulse • The first-order effects of the dispersive and nonlinear terms are complementary phase shifts Phase shift for nonlinear processes: Phase shift for Dispersion effect: