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7. GUIDED WAVES
[email protected]
1396; office Δ013 ΙΤΕ
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Στέλιος Τζωρτζάκης
METY-490 Photonic Materials
Fiber Optics
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Introduction
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Optical fibers are long, thin
strands of very pure glass
few 10s µm in diameter.
They are arranged in
bundles called optical
cables and used to transmit
light signals over long
distances.
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Historical Perspective
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Optical Era
Electrical Era
• Telegraph;
1836
• Optical Fibers;
• Telephone;
1876
• Optical Amplifiers; 1990
• Coaxial Cables; 1840
• WDM Technology; 1996
• Microwaves;
• Multiple bands;
1948
1978
2002
• Microwaves and coaxial cables limited to B ∼ 100 Mb/s.
• Optical systems can operate at bit rates >10 Tb/s.
• Improvement in system capacity is related to the high frequency of
optical waves (∼200 THz at 1.5 µm).
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History (cont.)
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1930 – Patents on tubing
1950 – Patent for two-layer glass wave-guide
1960 – Laser first used as light source
1965 – Light fiber discovered (Charles K. Kao, Nobel Prize 2009)
1970s – Refining of manufacturing process
1980s – Optical Fiber technology becomes backbone of long
distance telephone networks in NA.
Traffic segmentation by client
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Parts of Optical Fiber
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Core – thin glass center of the fiber where light
travels.
 Cladding – outer optical material surrounding
the core.
 Buffer Coating – plastic
coating that protects
the fiber.

How are fibers made
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Making of the pre-form
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How are fibers made cont.
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Principle of Drawing a Fibre
Drawing Tower
Types of Fibers
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nc
step-index
multimode
nf
nc
step-index
singlemode
nc
nf
nc
nc
nf
GRIN
nc
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Five Generations of Fiber Technologies
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Fiber waveguiding: Total Internal Reflection
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Total Internal Reflection in a Step Index Fiber
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escapes core
cladding
nt
core
ni
escapes core
i
stuck in core
i
i
sin  c  
nt
 nti
ni
critical angle
i  c for TIR
Graded Index Fiber
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nc
n varies
quadratically
nf
nc
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The V Parameter
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V  2
a
NA
o
a = fiber radius
o = incident wavelength
• known as the “V-parameter” or the fiber parameter
• an important parameter that governs the number of modes
• for V<2.405 the fiber is single-mode; the corresponding minimum
wavelengh is called cut-off wavelength
Fibers carry modes of light
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number of modes ~ V 2
a mode is :
•
a solution to the wave equation
•
a given path/distribution of light
higher # modes gives more light, which is not always desirable
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Modes (cont.)
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Example of # of Modes @ 850nm
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Silica step-index fiber has nf = 1.452, nc = 1.442 (NA = 0.205)
SELFOC graded index fiber with same NA
diameter
(microns)
2.5
50
200
400
1000
# step-index
modes
2
1.4 E3
22 E3
92 E3
2.4 E6
# GRIN
modes
1
716
11 E3
46 E3
1.2 E6
high # modes implies classical optics
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Controlling the # of Modes
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From the V parameter, we see that we can reduce the number of
modes in a fiber by reducing:
(1) NA
(2) diameter (wrt )
This is exactly the case in single mode fibers.
Fiber modes math
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The wave equation in cylindrical symmetry:
This is solved by using the method of separation of variables:
and
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Fiber modes: Bessel functions
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NA of a Fiber
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max
NA  noutside sinmax 
The NA defines a cone of acceptance for light
that will be guided by the fiber
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NA of a Step Index Fiber
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n2
n1
n0
90-t
t
max
must be > critical angle
NA  n0 sin  max 
NAstep 
n12  n2 2
n0
NAstep  n12  n2 2
NA changes with n
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1.4
1.4
1.2
1
air
NA n f  1.457 1.00
0.8
NA n f  1.457 1.33
0.6
water
0.4
0.2
0
0
1.4
1.4
1.5
1.6
nf
1.7
1.8
1.8
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NA is sensitive to n
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1.4
1.4
1.2
1%
change
1
NA n f  1.457 1.00
0.8
NA n f  1.472 1.00
5%
change
0.6
NA n f  1.530 1.00
0.4
0.2
0
0
1.4
1.4
1.5
1.6
nf
1.7
1.8
1.8
NA and Acceptance angle
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1.4
1.4
1.2
water
1
NA  i  1.00
NA  i  1.33
0.8
air
0.6
0.4
0.2
0
0
0
0
15
30
45
i
60
75
90
90
i
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NA of a GRIN Fiber
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The condition below assures a ray will have enough fiber
to bend back towards the center axis:
NA  n1
D
2
NA in air
 is a parameter describing how n changes
in the GRIN fiber
NA and Light Flux
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light gathering power ~ NA2
max
Example: Fiber with NA of 0.66 has 43% flux-carrying
capacity of a fiber with an NA of 1.0
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NA and # of Modes
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killed ray
propagated ray
large
NA
small
NA
Angle Preservation
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2




 n0 
 1 
 n1 
 
2

2
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Angle Preservation (2)
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In an ideal fiber, the angle of incidence
will equal the exit angle.
Rough surfaces, bending, and other real-world imperfections will
case a change in the exit cone.
example: critical bend radius
Critical Bend radius
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Polarization and Fibers
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entrance
exit
cladding
core
meridional ray: stays in the same plane
skewed ray: rotates in many planes about center
Polarization
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If the ray rotates during propagation, then the
polarization state will change

E

S

E

S

B

B
linear polarized beam translates into elliptical beam
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What to Do?
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•In some applications, polarization is not needed:
example: diffusive spectroscopy
•In others it is critical:
example: OCT (Optical Coherence Tomography)
•must remove the circular symmetry of the fiber
change n profile so that polarizations are not coupled
sort of like birefringence
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Losses & Dispersion
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Attenuation
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A (dB / Km)  
P 
10
log10  out 
L
 Pin 
Fibers are made of “glass”
- commonly high-quality fused silica (SiO2)
- some trace impurities (usually controlled)
Losses due to:
- Rayleigh scattering (~ -4)
- absorption (silica, impurities, dopants)
- mechanical stress
- coatings
Attenuation Profiles
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IR absorption
Rayleigh
Scattering
𝑂𝐻 − absorption
89% transmission
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Losses Management
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Optical Amplifiers
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Dispersion: The Basics
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Light propagates at a finite speed
fastest ray
slowest ray
fastest ray: one traveling down middle (“axial mode”)
slowest ray: one entering at highest angle (“high order” mode)
there will be a difference in time for these two rays
Modal Dispersion
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t 
L  n1 
  1
c  n2 
modal dispersion increases with:
L ~ NA2
usually the biggest dispersion problem in
step index multi-mode fibers
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Types of Dispersion in Fibers
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modal
- time delay from path length differences
- usually the biggest culprit in step-index
material
- n() : different times to cross fiber
-(note: smallest effect ~ 1.3 m)
waveguide
- changes in field distribution
-(important for SM)
nonlinear
- n can become intensity-dependent
NOTE: GRIN fibers tend to have less modal dispersion
because the ray paths are shorter
Dispersion in Fibers Math
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In guided optics we use β (instead of k) for the particular mode under consideration,
thus defining the effective refractive index nβ, the effective group index Nβ, and
the effective group-velocity dispersion Dβ for the mode
The exact frequency dependence of these parameters depends on the parameters
of the fiber, which are the V number and the normalized index difference Δ.
In the case of a step-index fiber, it is also convenient to use the normalized guide
index b
n1, and n2, the core and
cladding index respectively
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Dispersion in Fibers Math cont.
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In the weakly guiding approximation, where Δ is small, the effective refractive index
nβ, the effective group index Nβ, and the effective group-velocity dispersion Dβ for
the mode are:
where N2 is the group index and D2 is the group-velocity dispersion of the fiber
cladding
Dispersion Effect
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Dispersion-Compensating Fibers
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Dispersion Management Schemes
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Optical Phase Conjugation
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Phase conjugation
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When a nonlinear-optical effect produces a light wave proportional to E*, the
process is called a phase-conjugation process. Phase conjugators can cancel out
aberrations.
exp ikz 
Distorting
medium
  x, y 
exp i  k  z  2i  x, y 
exp ikz  i  x, y 
A normal mirror leaves
the sign of the phase
unchanged
exp i  k  z  i  x, y 
A phase-conjugate
mirror reverses the
sign of the phase
exp[i  k  z  i  x, y   i  x, y ]
exp i  k  z  i  x, y 
The second traversal through the medium cancels out
the phase distortion caused by the first pass!
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Signal Modulation
Modulation Formats
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Wavelength Division Multiplexing (WDM)
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Multiplexing - WDM
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WDM
Multiplexed signal
Signal 1
Signal 1
Signal 2
Signal 2
DEMUX
MUX
Signal 3
Signal 3
Single-mode Fiber
Signal 4
Signal 4
Wavelengths travel independently
Data rate and signal format on each wavelength is
completely independent
Designed for SM fiber
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Multiplexing - WDM
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WDM – Wave Division Multiplexing
Earliest technology
Mux/Demux of two optical wavelengths
(1310nm/1550nm)
Wide wavelength spacing means
Low cost, uncooled lasers can be used
Low cost, filters can be used
Limited usefulness due to low mux count
Multiplexing - DWDM
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DWDM – Dense Wave Division Multiplexing
Mux/Demux of narrowly spaced wavelengths
400 / 200 / 100 / 50 GHz Channel spacing
3.2 / 1.6 / 0.8 / 0.4 nm wavelength spacing
Up to 160 wavelengths per fiber
Narrow spacing = higher cost implementation
More expensive lasers and filters to separate ’s
Primarily for Telco backbone – Distance
Means to add uncompressed Video signals to existing fiber
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Multiplexing - CWDM
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CWDM – Coarse Wave Division Multiplexing
Newest technology (ITU Std G.694.2)
Based on DWDM but simpler and more robust
Wider wavelength spacing (20 nm)
Up to 18 wavelengths per fiber
Uses un-cooled lasers and simpler filters
Significant system cost savings over DWDM
DWDM can be used with CWDM to increase channel count or link
budget
CWDM Optical Spectrum
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20nm spaced wavelengths
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DWDM vs. CWDM Spectrum
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1.6nm Spacing
dB
1470
1490
1510
1530
1550
1570
1590
1610
Wavelength
Summary
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
Optical fibers carry modes of light

Step-index, GRIN, single mode & multimode

NA is related to acceptance cone and n’s.

How Step-index and GRIN fibers propagate light.

Factors that change light propagation in fibers:
 mechanical aspects (bending,
tapers, etc)
 attenuation
 dispersion
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Assignments
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
A7.1 Fiber Loop Polarization Controller: how it works.

A7.2 Erbium Doped Fiber Amplifiers (EDFAs): physics and how
are they used in fiber networks.

A7.3 Stimulated Raman Scattering (SRS) in fibers.

A7.4 Polarization Maintaining Fibers: explain the principle and
give examples.

A7.5 In-Fiber Bragg Gratings (FBGs) : what are they and what
are they used for.

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A7.6 Eye Diagrams and Bit Error Rates (BER) : explain.
Problems
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Problems
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1. A step-index silica fiber has a core index of 1.452, a cladding index of 1.449, and a
core diameter of 8 μm. What are its numerical aperture and acceptance angle? What
is the value of its V number at 850 nm wavelength? Is it single moded at this
wavelength? What is the cutoff wavelength for its single-mode operation?
V  2
a
o
NA
NAstep  n12  n2 2
Problems
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2. A step-index single-mode fiber for transmitting a signal at λ = 1.35 μm
is 100 km long. At this wavelength, the fiber has the following parameters for its silica
cladding: n2 = 1.446, N2 = 1.466, and D2 = −0.0027. Its core has a radius of a = 4 μm
and an index of n1 = 1.450. What are the propagation constant, the group velocity,
and the group-velocity dispersion of the signal propagating as the guided mode of
the fiber? If a 10-ps pulse that has a spectral width of Δλ = 2 nm is sent through the
fiber, what is its transmission time through the fiber? What is the pulse duration
when it arrives at the other end of the fiber?
For V=2:
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