EE 230: Optical Fiber Communication Lecture 6

From the movieWarriors of the Net

Nonlinear Processes in Optical Fibers

Polarization

• In molecules, P=μ+αE+βE2+γE3+…• In materials, P=X(o)+X(1)E+X(2)E2+X(3)E3+…

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.

Polarization

Molecules and their dipole moments

Distortion of an electron cloud in response to an E-field

Non-linear Polarization

Nonlinear Effects

• Stimulated Raman scattering

• Stimulated Brillouin scattering

• Four-wave Mixing

• Self-phase Modulation

• Cross-phase Modulation

Index of Refraction

Imaginary part of index: absorption

For a sample of absorbance A and thickness d, the imaginary part of the refractive index is equal to

d

A

4

10lnmax

Index of Refraction vs Wavelength

Refractive index vs Frequency for silica

Refractive Index for various materials

Wave slowing in a medium of higher Index

Nonlinear index of refraction

Real part of index is best described as a power series

n=n1+n2(P/Aeff)

Term in parentheses is the intensity. For silica fiber, n22.6x10-11 μm2/mW

Interaction Length

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

L

eff

eL

1

Nonlinear parameter

Propagation constant is power-dependent

effA

n

22

PNL

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

Effective Length and effective area

Single Mode Gaussian Approximation

Fundamentals of Photonics - Saleh and Teich

Fiber Optic Communiocation Systems - Agrawal

Gaussian Pulse Mode Field Diameter

Fiber Optics Communication Technology-Mynbaev & Scheiner

w0/a=0.65+1.619V-3/2+2.879V-6 forV between 1.2 and 2.4. Otherwise,use w0/a=(ln V)-1/2

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).

Phase modulation

• Self-modulation: φNL= γPLeff

• Cross-modulation: φNL= 2γPotherLeff

Effect of these phase changes is a frequency chirp (frequency changes during pulse), broadening pulse and reducing bit rate-length product

Self Phase Modulation

Pulse Spreading due to Self Phase Modulation

Gaussian Pulse in a Kerr Medium

Phase change of gaussian pulse

Instantaneous frequency shift

Instantaneous Frequency chirp

Solitons

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

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

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)

Stimulated Raman scattering

• Oscillations are Si-O bonds in the glass, frequency ≤3.3x1013 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

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

Four-wave Mixing

Taylor Series expansion of β(ω)

Through the cubic term:

where

...62

332210

i

i

i d

d

Importance of Taylor Series terms

Group velocity Vg, dispersion D, and dispersion slope S

1

1

gV

22

2 c

D

233

2

2

42

cc

d

dDS

Four-Wave Mixing Phase-Matching Requirement

Phase mismatch M needs to be small for FWM to occur significantly

2143 M

FWM in a WDM system

ω1=ω2=ω (power lost from one signal wavelength)

ω3=ω+Χ where Χ is the difference in frequency between adjacent channels

ω4=ω-Χ

Substitute in phase mismatch expression to get M=β2Χ2

Want β2 to be big to minimize FWM!

Four Wave Mixing