The Optical Fiber and Light Wave Propagation Xavier Fernando Ryerson Comm. Lab.
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The Optical Fiber and Light Wave Propagation
Xavier Fernando
Ryerson Comm. Lab
The Optical Fiber• Fiber optic cable functions as a ”light
guide,” guiding the light from one end to the other end.
• Fiber categories based on propagation:– Single Mode Fiber (SMF)– Multimode Fiber (MMF)
• Categories based on refractive index profile– Step Index Fiber (SIF)– Graded Index Fiber (GIF)
Step Index Fiber• Uniform ref. index of n1 (1.44 < n1 < 1.46) within
the core and a lower ref. index n2 in the cladding.
• The core and cladding radii are a and b. Typically 2a/2b are 8/125, 50/125, 62.5/125, 85/125, or 100/140 µm.
• SIF is generally made by doping high-purity fused silica glass (SiO2) with different concentrations of materials like titanium, germanium, or boron.
n1 n2
n1>n2
Different Light Wave Theories
• Different theories explain light behaviour • We will first use ray theory to understand
light propagation in multimode fibres• Then use electromagnetic wave theory to
understand propagation in single mode fibres
• Quantum theory is useful to learn photo detection and emission phenomena
Refraction and Reflection
Snell’s Law: n1 Sin Φ1 = n2 Sin Φ2
When Φ2 = 90,
Φ1 = Φc is the
Critical Angle
Φc=Sin-1(n2/n1 )
Step Index Multimode Fiber
1
221
22
21 12 n
n
n
nn
Fractional refractive-index
profile
Ray description of different fibers
Single Mode Step Index Fiber
Protective polymerinc coating
Buffer tube: d = 1mm
Cladding: d = 125 - 150 m
Core: d = 8 - 10 mn
r
The cross section of a typical single-mode fiber with a tight buffertube. (d = diameter)
n1
n2
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Only one propagation mode is allowed in a given wavelength. This is achieved by very small core diameter (8-10 µm)SMF offers highest bit rate, most widely used in telecom
Step Index Multimode Fiber
• Guided light propagation can be explained by ray optics
• When the incident angle is smaller the acceptance angle, light will propagate via TIR
• Large number of modes possible• Each mode travels at a different velocity
Modal Dispersion
• Used in short links, mostly with LED sources
Graded Index Multimode Fiber
• Core refractive index gradually changes towards the cladding
• The light ray gradually bends and the TIR happens at different points
• The rays that travel longer distance also travel faster
• Offer less modal dispersion compared to Step Index MMF
Refractive Index Profile of Step and Graded Index Fibers
a b
n1
n2n2
n1
a b
n = n =
Ste
p
Gra
ded
Step and Graded Index Fibers
n1
n2
21
3
nO
n1
21
3
n
n2
OO' O''
n2
(a) Multimode stepindex fiber. Ray pathsare different so thatrays arrive at differenttimes.
(b) Graded index fiber.Ray paths are differentbut so are the velocitiesalong the paths so thatall the rays arrive at thesame time.
23
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Total Internal Reflectionin Graded Index Fiber
n decreases step by step from one layerto next upper layer; very thin layers.
Continuous decrease in n gives a raypath changing continuously.
TIR TIR
(a) A ray in thinly stratifed medium becomes refracted as it passes from onelayer to the next upper layer with lower n and eventually its angle satisfies TIR.(b) In a medium where n decreases continuously the path of the ray bendscontinuously.
(a) (b)
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Total Internal Reflectionin Graded Index Fiber - II
nb
nc
O O'Ray 1
A
B'
B
AB
B' Ray 2
M
B' c/nb
c/na12
B''
na
a
b
c We can visualize a graded indexfiber by imagining a stratifiedmedium with the layers of refractiveindices na > nb > nc ... Consider twoclose rays 1 and 2 launched from Oat the same time but with slightlydifferent launching angles. Ray 1just suffers total internal reflection.Ray 2 becomes refracted at B andreflected at B'.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Skew Rays
Fiber axis
12
34
5
Skew ray1
3
2
4
5
Fiber axis
1
2
3Meridional ray
1, 3
2
(a) A meridionalray alwayscrosses the fiberaxis.
(b) A skew raydoes not haveto cross thefiber axis. Itzigzags aroundthe fiber axis.
Illustration of the difference between a meridional ray and a skew ray.Numbers represent reflections of the ray.
Along the fiber
Ray path projectedon to a plane normalto fiber axis
Ray path along the fiber
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Maxwell’s Equations
……..(1) (Faraday’s Law)
E: Electric Field
……….(2) (Maxwell’s Faraday equation)
H: Magnetic Field
……….(3) (Gauss Law)
……….(4) (Gauss Law for magnetism)
Taking the curl of (1) and using and
The parameter ε is permittivity and μ is permeability.
…….(5)
In a linear isotropic dielectric material with no currents and free of charges,
Maxwell’s Equations
But from the vector identity
……(6)
Using (5) and (3), …….(7)
Similarly taking the curl of (2), it can be shown
………(8)
(7) and (8) are standard wave equations. Note the Laplacian operation is,
2
2
2
2
22 11
z
EE
rr
Er
rrE
1
Note pv
Maxwell’s Equation
• Electrical and magnetic vectors in cylindrical coordinates are give by,
.…..(9)
……(10)
• Substituting (9) and (10) in Maxwell’s curl equations
….(11)
….(12)
….(13)
Maxwell’s Equation• Also
----------(14)
----------(15)
----------(16)
• By eliminating variables, above can be rewritten such that when Ez and Hz are known, the remaining transverse components Er , Eφ, Hr , Hφ, can be determined from (17) to (20).
Maxwell’s Equation …………..(17)
…………..(18)
…..........(19)
.………… (20)
Substituting (19) and (20) into (16) results in ….…(21)
…….(22)
Electric and Magnetic ModesNote (21) and (22) each contain either Ez or Hz only. This may
imply Ez and Hz are uncoupled. However. Coupling between Ez and Hz is required by the boundary conditions.
If boundary conditions do not lead to coupling between field components, mode solution will imply either Ez =0 or Hz =0. This is what happens in metallic waveguides.When Ez =0, modes are called transverse electric or TE modes
When Hz =0, modes are called transverse magnetic or TM modes
However, in optical fiber hybrid modes also will exist (both Ez and Hz are nonzero). These modes are designated as HE or EH modes, depending on either H or E component is larger.
Wave Equations for Step Index Fibers
• Using separation of variables
………..(23) • The time and z-dependent factors are given by
………..(24)
• Circular symmetry requires, each field component must not change when Ø is increased by 2п. Thus
…………(25)• Where υ is an integer.• Therefore, (21) becomes
….(26)
Wave Equations for Step Index Fibers• Solving (26). For the fiber core region, the solution must
remain finite as r0, whereas in cladding, the solution must decay to zero as r∞
• Hence, the solutions are – In the core, (r < a),
Where, Jv is the Bessel function of first kind of order v
– In the cladding, (r > a),
Where, Kv is the modified Bessel functions of second kind
Bessel Functions First Kind Bessel Functions Second kind
Modified Bessel first kind Modified Bessel Second kind
Propagation Constant β
• From definition of modified Bessel function
• Since Kv(wr) must go to zero as r∞, w>0. This implies that
• A second condition can be deduced from behavior of Jv(ur). Inside core u is real for F1 to be real, thus,
• Hence, permissible range of β for bound solutions is
Meaning of u and w
• Both u and w describes guided wave variation in radial direction– u is known as guided wave radial direction phase
constant (Jn resembles sine function)
– w is known as guided wave radial direction decay constant (recall Kn resemble exponential function)
Inside the core, we can write, 221
22 kuq
Outside the core, we can write, 22
22 kw
V-Number (Normalized Frequency)
All but HE11 mode will cut off when b = 0.
Hence, for single mode condition,
405.2)(2
cVNAa
V
22
21
222
22
12222 2
)( nna
akkawuV
Define the V-Number (Normalized Frequency) as,
2
221
22
2
2
22 )/(
nn
nk
V
awb
Define the normalized propagation const b as,
0 2 4 61 3 5V
b
1
0
0.8
0.6
0.4
0.2
LP01
LP11
LP21
LP02
2.405
Normalized propagation constant b vs. V-numberfor a step index fiber for various LP modes.© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
0
0.5
1
1.5
0 1 2 3
V - number
V[d2(Vb)/dV2]
[d2(Vb)/dV2] vs. V-number for a step index fiber (after W.A. Gambling etal., The Radio and Electronics Engineer , 51, 313, 1981)
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Field Distribution in the SMF
n2
Light
n2
n1
y
E(y)
E(y,z,t ) = E(y)cos(t – 0z)
m = 0
Field of evanescent wave(exponential decay)
Field of guided wave
The electric field pattern of the lowest mode traveling wave along theguide. This mode has m = 0 and the lowest . It is often referred to as theglazing incidence ray. It has the highest phase velocity along the guide.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Mode-field Diameter (2W0)
In a Single Mode Fiber,
)/exp()( 20
20 wrErE
At r = wo, E(Wo)=Eo/e
Typically Wo > a
Cladding Power Vs Normalized Frequency
Vc = 2.4
Modes
Power in the cladding
Lower order modes have higher power in the cladding larger MFD
Higher the Wavelength More the Evanescent Field
y
E(y)
Cladding
Cladding
Core
2 > 11 > c
2 < 11 < cut-off
vg1
y
vg2 > vg1
The electric field of TE0 mode extends more into thecladding as the wavelength increases. As more of the fieldis carried by the cladding, the group velocity increases.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Light Intensity
E
r
E01
Core
Cladding
The electric field distribution of the fundamental modein the transverse plane to the fiber axis z. The lightintensity is greatest at the center of the fiber. Intensitypatterns in LP01, LP11 and LP21 modes.
(a) The electric fieldof the fundamentalmode
(b) The intensity inthe fundamentalmode LP01
(c) The intensityin LP11
(d) The intensityin LP21
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Fiber Key Parameters
Fiber Key Parameters
Effects of Dispersion and Attenuation
Dispersion for Digital Signals
t0
Emitter
Very shortlight pulses
Input Output
Fiber
PhotodetectorDigital signal
Information Information
t0
~2² T
t
Output IntensityInput Intensity
²
An optical fiber link for transmitting digital information and the effect ofdispersion in the fiber on the output pulses.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Modal DispersionLow order modeHigh order mode
Cladding
Core
Light pulse
t0 t
Spread,
Broadenedlight pulse
IntensityIntensity
Axial
Schematic illustration of light propagation in a slab dielectric waveguide. Light pulseentering the waveguide breaks up into various modes which then propagate at differentgroup velocities down the guide. At the end of the guide, the modes combine toconstitute the output light pulse which is broader than the input light pulse.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Major Dispersions in Fiber
• Modal Dispersion: Different modes travel at different velocities, exist only in multimode fibers
• This was the major problem in first generation systems
• Modal dispersion was alleviated with single mode fiber– Still the problem was not fully solved
Dispersion in SMF• Material Dispersion: Since n is a function of
wavelength, different wavelengths travel at slightly different velocities. This exists in both multimode and single mode fibers.
• Waveguide Dispersion: Signal in the cladding travels with a different velocity than the signal in the core. This phenomenon is significant in single mode conditions.
Group Velocity (Chromatic) Dispersion = Material Disp. + Waveguide Disp.
t
Spread, ²
t0
Spectrum, ²
12o
Intensity Intensity Intensity
Cladding
CoreEmitter
Very shortlight pulse
vg(2)
vg(1)Input
Output
All excitation sources are inherently non-monochromatic and emit within aspectrum, ² , of wavelengths. Waves in the guide with different free spacewavelengths travel at different group velocities due to the wavelength dependenceof n1. The waves arrive at the end of the fiber at different times and hence result ina broadened output pulse.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Group Velocity Dispersion
Modifying Chromatic Dispersion
GVD = Material Disp. + Waveguide dispersion
• Material dispersion depends on the material properties and difficult to alter
• Waveguide dispersion depends on fiber dimensions and refractive index profile. These can be altered to get:– 1300 nm optimized fiber– Dispersion Shifted Fiber (DSF) – Dispersion Flattened Fiber (DFF)
Material and Waveguide Dispersions
0
1.2 1.3 1.4 1.5 1.61.1-30
20
30
10
-20
-10
(m)
Dm
Dm + Dw
Dw0
Dispersion coefficient (ps km-1 nm-1)
Material dispersion coefficient (Dm) for the core material (taken asSiO2), waveguide dispersion coefficient (Dw) (a = 4.2 m) and thetotal or chromatic dispersion coefficient Dch (= Dm + Dw) as afunction of free space wavelength,
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Material and waveguide dispersion coefficients in anoptical fiber with a core SiO2-13.5%GeO2 for a = 2.5to 4 m.
0
–10
10
20
1.2 1.3 1.4 1.5 1.6–20
(m)
Dm
Dw
SiO2-13.5%GeO2
2.5
3.03.54.0a (m)
Dispersion coefficient (ps km-1 nm-1)
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Different WG Dispersion Profiles
WGD is changed by adjusting fiber profile
Dispersion Shifting/Flattening
(Standard)
(Zero Disp. At 1550 nm)
(Low Dispersion throughout)
20
-10
-20
-30
10
1.1 1.2 1.3 1.4 1.5 1.6 1.7
0
30
(m)
Dm
Dw
Dch = D m + Dw
1
Dispersion coefficient (ps km -1 nm-1)
2
n
r
Thin layer of claddingwith a depressed index
Dispersion flattened fiber example. The material dispersion coefficient ( Dm) for thecore material and waveguide dispersion coefficient ( Dw) for the doubly clad fiberresult in a flattened small chromatic dispersion between 1 and 2.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Specialty Fibers with Different Index Profiles
1300 nm optimized
Dispersion Shifted
Specialty Fibers with Different Index Profiles
Dispersion Flattened
Large area dispersion shifted Large area dispersion flattened
Polarization Mode Dispersion• Since optical fiber has a single axis of
anisotropy, differently polarized light travels at slightly different velocity
• This results in Polarization Mode Dispersion• PMD is usually small, compared to GVD or
Modal dispersion • May become significant if all other dispersion
mechanisms are small
X and Y Polarizations
A Linear Polarized wave will always have two orthogonal components.
These can be called x and y polarization componentsEach component can be individually handled if
polarization sensitive components are used
Polarization Mode Dispersion (PMD)
Each polarization state has a different velocity PMD
Birefringence• Birefringence is the decomposition of a ray of light
into two rays types of (anisotropic) material• In optical fibers, birefringence can be understood
by assigning two different refractive indices nx and ny to the material for different polarizations.
• In optical fiber, birefringence happens due to the asymmetry in the fiber core and due to external stresses
• There are Hi-Bi, Low-Bi and polarization maintaining fibers.
Total Dispersion
For Multi Mode Fibers:
For Single Mode Fibers:
But Group Velocity Disp.
Hence,
(ΔTpol is usually negligible )
(Note for MMF ΔTGVD ~= ΔTmat
Permissible Bit Rate• As a rule of thumb the permissible total
dispersion can be up to 70% of the bit period. Therefore,
T
0.35 B :bits) Zero(Return to pulses width halfFor •
T
0.7 B :bits) ZeroReturn to(Non pulses width fullFor •
Total
Total
Disp. & Attenuation Summary
t0
Pi = Input light power
Emitter
OpticalInput
OpticalOutput
Fiber
PhotodetectorSinusoidal signal
Sinusoidal electrical signalt
t0
f1 kHz 1 MHz 1 GHz
Po / Pi
fop
0.1
0.05
f = Modulation frequency
An optical fiber link for transmitting analog signals and the effect of dispersion in thefiber on the bandwidth, fop.
Po = Output light power
Electrical signal (photocurrent)
fel
10.707
f1 kHz 1 MHz 1 GHz
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Fiber Optic Link is a Low Pass Filter for Analog Signals
Attenuation Vs Frequency
Fiber attenuation does not depend on modulation frequency
Attenuation in Fiber
Attenuation Coefficient
• Silica has lowest attenuation at 1550 nm• Water molecules resonate and give high
attenuation around 1400 nm in standard fibers• Attenuation happens because:
– Absorption (extrinsic and intrinsic)– Scattering losses (Rayleigh, Raman and Brillouin…)– Bending losses (macro and micro bending)
dB/km dB)(dB)0(
z
zPP
All Wave Fiber for DWDM
Lowest attenuation occurs at 1550 nm for Silica
Att
enu
atio
n
char
acte
rist
ics
Escaping wave
c
Microbending
R
Cladding
Core
Field distribution
Sharp bends change the local waveguide geometry that can lead to wavesescaping. The zigzagging ray suddenly finds itself with an incidenceangle that gives rise to either a transmitted wave, or to a greatercladding penetration; the field reaches the outside medium and some lightenergy is lost.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Bending Loss
Note:Higher MFD Higher Bending Loss
Micro-bending losses
Fiber Production
The Fiber Cable
Preform feed
Furnace 2000°C
Thicknessmonitoring gauge
Take-up drum
Polymer coater
Ultraviolet light or furnacefor curing
Capstan
Schematic illustration of a fiber drawing tower.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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