1. EE 230: Optical Fiber Communication Lecture 6 From the movie Warriors of the Net Nonlinear Processes in Optical Fibers

2. Polarization In molecules, P=μ+αE+βE 2 +γE 3 +… In materials, P=X (o) +X (1) E+X (2) E 2 +X (3) E 3 +… If multiple electric fields are applied, every possible cross term is generated. At sufficiently high values of E, quadratic or higher terms become important and nonlinear effects are induced in the fiber.

3. Polarization Molecules and their dipole moments Distortion of an electron cloud in response to an E-field

4. Non-linear Polarization

5. Nonlinear Effects Stimulated Raman scattering Stimulated Brillouin scattering Four-wave Mixing Self-phase Modulation Cross-phase Modulation

6. Index of Refraction

7. Imaginary part of index: absorption For a sample of absorbance A and thickness d, the imaginary part of the refractive index is equal to

8. Index of Refraction vs Wavelength Refractive index vs Frequency for silica Refractive Index for various materials Wave slowing in a medium of higher Index

9. Nonlinear index of refraction Real part of index is best described as a power series n=n 1 +n 2 (P/A eff ) Term in parentheses is the intensity. For silica fiber, n 2  2.6x10 -11 μm 2 /mW

10. Interaction Length where α (in cm -1 ) is the loss coefficient of the fiber. 0.1 dB/km=2.3x10 -7 cm -1.

11. Nonlinear parameter Propagation constant is power-dependent

12. Propagation in Single Mode Fiber Understanding Fiber Optics-Hecht Geometrical optics is not useful for single mode fiber, must be handled by full E & M treatment Think of guiding as diffraction constrained by refraction Fields are evanescently damped in the cladding

13. Effective Length and effective area

14. Single Mode Gaussian Approximation Fundamentals of Photonics - Saleh and Teich Fiber Optic Communiocation Systems - Agrawal

15. Gaussian Pulse Mode Field Diameter Fiber Optics Communication Technology-Mynbaev & Scheiner w 0 /a=0.65+1.619V -3/2 +2.879V -6 for V between 1.2 and 2.4. Otherwise, use w 0 /a=(ln V) -1/2

16. Mitigation If P is high in a fiber application, the nonlinear component of the index is minimized by increasing the effective area of the fiber. Fiber designed for this purpose is called LEAF fiber (Large Effective Area).

17. Phase modulation Self-modulation: φ NL = γPL eff Cross-modulation: φ NL = 2γP other L eff Effect of these phase changes is a frequency chirp (frequency changes during pulse), broadening pulse and reducing bit rate-length product

18. Self Phase Modulation

19. Pulse Spreading due to Self Phase Modulation

20. Gaussian Pulse in a Kerr Medium Phase change of gaussian pulse Instantaneous frequency shift Instantaneous Frequency chirp

21. Solitons

22. Nonlinear scattering Signal photon scatters off oscillation that is present in the material, gains or loses frequency equivalent to that of the material oscillation At high powers, beating of signal frequency and scattered frequency generates frequency component at the difference that drives the material oscillations

23. Stimulated Brillouin Scattering Sound waves represent alternating regions of compressed material and expanded material Index of refraction increases with density of polarizable electrons and thus with compression Scattering is induced by index discontinuities

24. SBS, continued Transfer of energy into acoustic wave results in backwards scattering in fiber Brillouin frequency shift equal to 2nv/λ, where n is the mode index and v is the speed of sound in the material For fiber, scattered light is 11 GHz lower in frequency than signal wavelength (speed of sound is 5.96 km/s)

25. Stimulated Raman scattering Oscillations are Si-O bonds in the glass, frequency ≤3.3x10 13 Hz Scattered photon can come off decreased by that amount (Stokes) or increased by that amount (anti-Stokes) Stokes shift scatters 1550 nm light up to 1870 nm light

26. Raman shift in silica Spectrum shows major peaks at 1100, 800, and 450 cm -1 Those vibrational oscillations occur at 33, 24, and 13.5 THz Raman gain spectrum shows maximum at 12-14, 18, 24, and 33 THz

27. Four-wave Mixing

28. Taylor Series expansion of β(ω) Through the cubic term: where

29. Importance of Taylor Series terms Group velocity V g, dispersion D, and dispersion slope S

30. Four-Wave Mixing Phase-Matching Requirement Phase mismatch M needs to be small for FWM to occur significantly

31. FWM in a WDM system ω 1 =ω 2 =ω (power lost from one signal wavelength) ω 3 =ω+Χ where Χ is the difference in frequency between adjacent channels ω 4 =ω-Χ Substitute in phase mismatch expression to get M=β 2 Χ 2 Want β 2 to be big to minimize FWM!

32. Four Wave Mixing