Download EE 230: Optical Fiber Communication Lecture 6

EE 230: Optical Fiber Communication
Lecture 6
Nonlinear Processes in Optical Fibers
From the movie
Warriors of the Net
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
Distortion of an electron cloud in response to an E-field
Molecules and their dipole moments
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
Amax ln 10
4d
Index of Refraction vs Wavelength
Refractive Index for various materials
Refractive index vs Frequency for silica
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, n22.6x10-11 μm2/mW
Interaction Length
Leff 
1 e
L

where α (in cm-1) is the loss coefficient of the
fiber. 0.1 dB/km=2.3x10-7 cm-1.
Nonlinear parameter
2n2

Aeff
Propagation constant is power-dependent
 NL    P
Propagation in Single Mode
Fiber
Geometrical optics is not useful for
single mode fiber, must be handled by full E & M
treatment
Understanding Fiber Optics-Hecht
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
w0/a=0.65+1.619V-3/2+2.879V-6 for
V between 1.2 and 2.4. Otherwise,
use w0/a=(ln V)-1/2
Fiber Optics Communication Technology-Mynbaev & Scheiner
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:
   0  1   
2
where
d 
i 
i
d
i
2
 
2

3
6
 
3
 ...
Importance of Taylor Series terms
Group velocity Vg, dispersion D, and
dispersion slope S
Vg 
D
1
1
 2c

2
2
dD  2c 
 4c 
S
  2   3   3  2
d   
  
2
Four-Wave Mixing Phase-Matching
Requirement
Phase mismatch M needs to be small for
FWM to occur significantly
M   3     4    1     2 
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