Propagation of Signals in Optical Fiber

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Chapter 2
Introduction to Optical Networks

• Propagation of Signals in Optical Fiber

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2.Propagation of Signals in Optical Fiber

• Advantages
• Low loss 0.2dB/km at 1550nm
• Enormous bandwidth at least 25THz
• Light weight
• Flexible
• Immunity to interferences
• Low cost
• Disadvantages and Impairments
• Difficult to handle
• Chromatic dispersion
• Nonlinear Effects

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2.1 Light Propagation in Optical Fiber

• Cladding refractive index 1.45
• core 810µm, 50µm, 62.5µm doped

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2.1.1Geometrical Optical Approach (Ray Theory)

• This approach is only applicable to multimode
  fibers.
• incident angle (???)
• refraction angle (???)
• reflection angle (???)
• Snells Law

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• gtCritical angle
• When total internal reflection occurs.
• let air refractive index
• acceptance angle (total reflection
  will occur at core/cladding interface)

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• 
  (2.2)
  

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• If ? is small (less than 0.01)
• For (multimode)
• Numerical Aperture NA
• Because different modes have different lengths of
  paths, intermodal dispersion occurs.

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• Infermode dispersion will cause digital pulse
  spreading
• Let L be the length of the fiber
• The ray travels along the center of the core
• The ray is incident at (slow ray)

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• Assume that the bit rate Bb/s
• Bit duration
• The capacity is measured by BL (ignore loss)
• Foe example, if

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• For optimum graded-index fibers, dT is shorter
  than that in the step-index fibers, because the
  ray travels along the center slows down (n is
  larger) and the ray traveling longer paths
  travels faster (n is small)

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• The time difference is given by (For Optical
  graded-index profile)
• and
• (single
  mode )
• If
• Long haul systems use single-mode fibers

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2.1.2 Wave Theory Approach

• Maxwells equations
• 
  D.1
• 
  D.2
• 
  D.3
• 
  
• 
  D.4
• the charge density, the current
  density
• the electric flux density, the
  magnetic flux density
• the electric field, the magnetic
  field

?
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• Because the field are function of time and
  location in the space, we denote them by
• and , where and t are
  position vector and time.
• Assume the space is linear and time-invariant the
  Fourier transform of is
• 
  2.4
• let be the induced electric polarization
• 
  2.5
• the permittivity of vacuum
• 
  2.6
• the magnetic polarization
• the permeability of vacuum
• ?????Fourier transform???

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• Locality of Response and related to
  dispersion and nonlinearities
• If the response to the applied electric field is
  local
• depends only on
• not on other values of
• This property holds in the 0.52µm wavelength
• Isotropy The electromagnetic properties are the
  same for all directions in the medium
• Birefringence The refraction indexes along two
  different directions are different (lithium
  niobate, LiNbO , modulator, isolator, tunable
  filter)

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• Linearity
• 
  (Convolution Integral) 2.7
• linear susceptibility
• The Fourier transform of is
• 
  2.8
• Where is the Fourier transform of
• ( is similar to the impulse
  response)
• is function of frequency
• gt Chromatic dispersion

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• Homogeneity A homogeneous medium has the same
  electromagnetic properties at all points
• The core of a graded-index fiber is inhomogeneous
• Losslessness No loss in the medium
• At first we will only consider the core and
  cladding regions of the fiber are locally
  responsive, isotropic, linear, homogeneous, and
  lossless.
• The refractive index is defined as
• 
  2.9
• For silica fibers

def
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• From Appendix D
• For (zero charge)
• (zero conductivity, dielectric
  material)
• For nonmagnetic material

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• Assume linear and homogenence

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• Take Fourier transform
• Recall
• 
  2.8
• Denote
• c speed of light
• (Locally response,
  isotropic, linear,
• 
  homogeneous, lossless) 2.9
• 
  

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• palacian operation
• 
  2.10
• (free space
  wave number)

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• For Cartesian coordinates
• For Cylindrical coordinates?. f and z
• n
• a radius of the core
• Similarly
  2.11
• Boundary conditions is finite
• and continuity of field at
  ?a
• References
• G.P. Agrawal Fiber-Optical Communication System
  Chapter 2
• John Senior Optical Fiber Communications,
  Principles and practice
• John Gowar Optical Communication Systems

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Fiber Modes
cladding
core
x
z
y
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• 
  must satisfy 2.10, 2.11 and the boundary
  conditions.
• let
• Where are unit vectors
• For the fundamental mode, the longitudinal
  component is
• the propagation constant

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• Bessel functions
• The transverse components
• For cylindrical symmetry of the fiber
• In general, we can write
• (Appendix E)

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• Where
• The multimode fiber can support many modes. A
  single mode fiber only supports the fundamental
  mode.
• Different modes have different ß,
• such that they propagate at different
  speeds.gtmode dispersion
• (We can think of a mode as one possible path
  that a guided ray can take)

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• For a fiber with core and cladding , if
  a wave propagating purely in the core, then the
  propagation constant is
• ? free space wavelength
• The wave number
• Similarly if the wave propagating purely in the
  cladding, then
• The fiber modes propagate partly in the cladding
  and partly in the core,
• so
• Define the effective index
• The speed of the wave in the fiber

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• For a fiber with core radius a , the cutoff
  condition is
• normalized wave number
• Recall
• V? when a? and ? ?
• For a single mode fiber, the typical values are
  a4µm and ?0.003

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• The light energy is distributed in the core and
  the cladding.

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• Since ? is small, a significant portion of the
  light energy can propagate in the cladding, the
  modes are weakly guided.
• The energy distribution of the core and the
  cladding depends on wavelength.
• It causes waveguide dispersion (different from
  material dispersion)
• ( Appendix E )
• For longer wave, it has more energy in the
  cladding and vice versa.

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• A multimode fiber has a large value of V
• The number of modes
• For example a25µm, ?0.005
• V28 at 0.8µm
• Define the normalized propagation constant (or
  normalized effective index)

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Polarization

• Two fundamental modes exist for all ?. Others
  only exist for ?lt ?cutoff,
• Linearly polarized field Its direction is
  constant.
• For the fundamental mode in a single-mode fiber

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• Fibers are not perfectly circularly symmetric.
  The two orthogonally polarized fundamental modes
  have different ß
• gtPolarization-mode dispersion (PMD)
• Differential group delay (DGD)
• ?t?ß/w typical value ?t0.5
• 100 km gt 50 ps
• Practically PMD varies randomly along the fiber
  and may be cancelled from an segment to another
  segment.
• Empirically, ?t 0.1-1
• Some elements such as isolators, circulators,
  filters may have polarization-dependent loss
  (PDL).

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2.2 Loss and Bandwidth
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• Recall
• Take the bandwidth over which the loss in dB/km
  is with a factor of 2 of its minimum.
• 80nm at 1.3µm, 180nm at 1.55µm
• gtBW35 THz

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• Erbium-Doped Fiber Amplifiers (EDFA) operate in
  the c and L bands, Fiber Raman Amplifiers (FRA)
  operate in the S band.
• All Wave fiber eliminates the absorption peaks
  due to water.

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2.2.1 Bending loss

• A bend with r 4cm, loss lt 0.01dB
• r? loss?

2.2.3 Chromatic Dispersion

• Different spectral components travel at different
  velocities.
• Material dispersion n(w)
• Waveguide dispersion, different wavelengths have
  different energy distributions in core and
  cladding gtdifferent ß, kn2lt ß lt kn1

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2.3.1 Chirped Gaussian Pulses

• Chirped frequency of the pulse changes with
  time.
• Cause of chirp direct modulation, nonlinear
  effects, generated on purpose. (soliton)

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Appendix E, or Govind P. Agrawal Fiber- Optic
Communication Systems 2nd Edition, John Wiley
Sons. Inc. PP4751

• A chirped Gaussian pulse at z0 is given by
• The instantaneous angular frequency

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• k The chirp factor
• Define The linearly chirped pulse the
  instantaneous angular frequency increases or
  decreases with time, (kconstant)
• Note
• Solve with the
  initial
• condition
  (E.7)
• We get
• 
  (E.8)
• A(z,t) is also Gaussian pulse

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• Broadening of chirped Gaussian pulses
• They have the same of broadening length.
• Note

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• In Fig 2.9, , it is true for standard
  fibers at 1.55µm
• Let be the dispersion length
• If , dispersion can be
  neglected

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• For kß2 lt 0
• For ß2 gt 0, high frequency travels faster
• gt the tail travels fasters gt
  compression
• gt make ßk lt 0, LD increases

?decreases
increases ? for certain z
(Fig 2.10)
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• Note ß2gt 0 , klt 0 ß2klt0

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2.3.2 Controlling the Dispersion Profile

• Def
• Chromatic dispersion parameter
• D
• D DM Dw
• The standard single mode fiber has small
  chromatic dispersion at 1.3 µm but large at 1.55
  µm

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• At 1.55µm loss is low, and EDFA is well
  developed.
• Dispersion becomes an issue
• We have not much control over DM, but Dw can be
  controlled by carefully designed refractive index
  profile.
• Dispersion shifted fibers, which have zero
  dispersion in 1.55µm band

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2.4 Nonlinear Effects

• For bit rate ?2.5 Gb/s, power a few mw
• Linear Assumption is valid
• Nonlinear effect appears for high power or high
  bit rate ? 10 Gb/s and WDM systems
• The first category relates to the interaction of
  lightwave with phonons (molecular vibrations)
• - Rayleigh Scattering
• Stimulated Brillouin Scattering (SBS)
• Stimulated Raman Scattering (SRS)

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• The second category is due to the dependence of
  the refractive index on the intensity
• - self-phase modulation (SPM)
• four-wave mixing (FWM)
• SBS and SRS transfer energy from short ? (pump)
  to long ?(stokes wave)
• Scattering gain coefficient, g, is measured in
  meter/watt and ?f.
• ? SPM induces chirping
• In a WDM system, variation of n depending on the
  intensity of all channels.
• gtYields Cross-phase modulation (CPM)
• gtinterchannel crosstalk

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• FWM,

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2.4.1 Effective Length and Area

• The nonlinear effect depends on fiber length and
  cross-section.

(for long link)
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• In addition nonlinear effect intensity

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2.4.1 Stimulated Brillouin Scattering (SBS)

• The scattering interaction occurs with acoustic
  phonons over ?f 15 MHz, at 1.55µm, stokes and
  pump waves propagate in opposite directions.
• If spacing gt 20 MHz gt no effects on different
  channels

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2.4.3 Stimulated Roman Scattering (SRS)

• SRS will deplete short wave power and amplifier
  long wave.

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2.4.4 Propagation in a Nonlinear Medium

• In a nonlinear medium, Fourier Transfer is not
  applicable.
• When the electrical field has only one component,
  we can write and as the scalar
  functions , and .
• Appendix F, contains higher order terms

nonlinear polarization
linear polarization
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• Because of symmetry , and
• The nonlinear response occurs less than
  100x10-15sec.
• If the bit rate is less than 100 Gb/s, then
• the third-order nonlinear susceptibility
  independent of time
• For simplicity, assume that the signals are
  monochromatic plane waves
• is constant in the plane perpendicular to
  the dispersion of propagation
• In WDM systems with n wavelengths at the angular
  frequencies

0
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2.4.5 Self-phase Modulation (SPM)

• Because n is intensity dependent
• gtinduces phase shift proportional to the
  intensity
• gtcreates chirping gt pulse broadening
• It is significant for high power systems.
• Consider a single channel case

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• Recall for linear medium
• Now, we have to modify as
• We get
• gt Phase changes with intensity

propagation constant changes with
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• 
  ,whose phase changes as , this phenomenon
  is referal as self- phase modulation (SPM)
• The intensity of the electrical field
• The intensity-dependent refractive index is
• The nonlinear index coefficient
• in
  silica fiber
• We take
  for example
• Because a pulse has its finite temporal extent
• gtThe phase shift is different in different parts
  of the pulse
• The leading edges have positive frequency shift
• The tailing edges have negative frequency shift
• gt SPM causes positive chirping

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2.4.6 SPM-induced chirp for Gaussian Pulses

• Consider an unchirped pulse with envelope
• which has unit peak
  amplitude and
• -width T01, and the peak power P01
• Define the nonlinear length as
• If link length ? LNL gt nonlinear effect is
  severe

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• From Appendix E, (E.18)
• After propagation L distance,
• The SPM-induced phase change is
• The instantaneous frequency is given by
• References Appendix E, and (Arg97)

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2.4.7 Cross-Phase Modulation

• In WDM systems, the intensity-dependent nonlinear
  effects (phase shift) are enhanced by other
  signals, this effect is referred to as
  cross-phase modulation (CPM)
• Consider two channels
• Recall

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• 2w1w2, 2w2w1, 3w1and 3w2 can be neglected
• 2w1-w2, 2w2-w1, are part of FWM.
• Consider the w1 channel, the CPM term is
• If E1E2
• Apparently CPM effect is twice of SPM.
• In practice, ß1 and ß2 are different
• gt The pulses corresponding to individual
  channel walk away from each other.
• gt can not interact further
• gt CPM is negligible for standard fibers
• Note for DSF, they travel at same velocity, CPM
  is significant

SPM
CPM
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2.4.8 Four-Wave Mixing (FWM)
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• wi, wj ,wk (three waves) generate
• wi wj wk (fourth wave)
• For example, channel spacing ?w
• w2 w1 ?w, w3 w1 2?w
• w1- w2 w3 w2, 2w2-w1 w3

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• Define
• The degeneracy factor
• The normalized Pijk(z,t) is given by
• If we assume that the optical signals propagate
  as plane waves over Ae and distance L, then the
  power is

(using Fig 2.15 and 2.36)
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• For example
• If another channel at
• Then FWM will interfere the channel.
• Practical FWM lacks of phase matching
• gt No significant influence (in normal fibers)

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2.4.9 New Optical Fiber Types

• A. DSF is not suitable for WDM due to nonlinear
  effect.
• To reduce nonlinear effect (different group
  
• velocities lack phase matching)
• gtto develop nonzero-dispersion fibers
• (NZ-DSF)
• a chromatic dispersion 16 ps/nm-km
• or -1 -6 ps/nm-km
• NZ-DSF has most advantage of DSF (in c-band)

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• Large Effective Area Fiber (LEAF)
• nonlinear effect for fix power

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Positive and Negative Dispersion Fibers For
Chromatic dispersion compensation
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