1
4. Optical Fibers
2
Anatomy of an Optical Fiber
• Light confined to core with higher index of refraction
• Two analysis approaches
– Ray tracing
– Field propagation using Maxwell’s equations
3
Optical Fiber Analysis
• Calculation of modes supported by an optical fiber
– Intensity profile
– Phase propagation constant
• Effect of fiber on signal propagation
– Signal attenuation
– Pulse spreading through dispersion
4
Critical Angle
• Ray bends at boundary between materials
– Snell’s law
• Light confined to core if propagation angle is greater than the critical angle
– Total internal reflection (TIR)
11
212 sinsin n
n1122 sinsin nn
1
211 sin n
nc
5
Constructive Interference
• Propagation requires constructive interference
– Wave stays in phase after multiple reflections
– Only discrete angles greater than the critical angle are allowed to propagate
6
Numerical Aperture
• The acceptance angle for a fiber defines its numerical aperture (NA)
• The NA is related to the critical angle of the waveguide and is defined as:
• Telecommunications optical fiber n1~n2,
22
21sin nnNA i
1
21212121
22
21 2
n
nnnnnnnnnNA
21nNA1
21
n
nn
7
Modes
• The optical fiber support a set of discrete modes
• Qualitatively these modes can be thought of as different propagation angles
• A mode is characterized by its propagation constant in the z-direction z
• With geometrical optics this is given by
• The goal is to calculate the value of βz
• Remember that the range of βz is
izi kn sin01
ozo knkn 12
8
Optical Fiber Modes
• The optical fiber has a circular waveguide instead of planar
• The solutions to Maxwell’s equations
– Fields in core are non-decaying
• J, Y Bessel functions of first and second kind
– Fields in cladding are decaying
• K modified Bessel functions of second kind
• Solutions vary with radius r and angle • There are two mode number to specify the mode
– m is the radial mode number
– is the angular mode number
9
Bessel Functions
10
Transcendental Equation
• Under the weakly guiding approximation (n1-n2)<<1
– Valid for standard telecommunications fibers
• Substitute to eliminate the derivatives
aγK
aγKγa
akJ
akJak
l
'l
Tl
T'l
T
2221
2 oT knk 222
22okn
x
xJlxJxJ l
ll 1'
x
xKlxKxK l
ll 1'
aK
aKa
akJ
akJakor
aK
aKa
aκJ
akJak
l
l
Tl
TlT
l
l
l
TlT
1111
HE ModesEH Modes
11
Bessel Function Relationships• Bessel function recursive relationships
• Small angle approximations
xJxJ nn
n 1
xKxK nn
xJxJx
nxJ nnn 11
2
xJxJx
xJ 210
2
xKxKx
nxK nnn 11
2
xKxKx
xK 210
2
0
2
2
!1
00.57722
ln
nx
n
nx
xK nn
12
Lowest Order Modes• Look at the l=-1, 0, 1 modes
• Use bessel function properties to get positive order and highest order on top
• l=-1
• l=0
yK
yKy
xJ
xJx
1
2
1
2
yK
yKy
xJ
xJx
1
2
1
2
yK
yKy
xJ
xJx
1
0
1
0
yK
yKy
xJ
xJx
0
1
0
1
yK
yKy
xJ
xJx
0
1
0
1
yK
yKy
xJ
xJx
0
1
0
1
yK
yKy
xJ
xJx
1
2
1
2
akx T ay 22
21
22 2nnayxV
13
Lowest Order Modes cont.• l=+1
• So the 6 equations collapse down to 2 equations
yK
yKy
xJ
xJx
1
0
1
0
yK
yKy
xJ
xJx
1
2
1
2
yK
yKy
xJ
xJx
1
2
1
2
yK
yKy
xJ
xJx
1
2
1
2
yK
yKy
xJ
xJx
0
1
0
1
lowest modes
14
Modes
0 2 4 6 8 100
5
10
15
20
x (kT a)
LHS
, RH
S
15
Fiber Modes
16
Hybrid Fiber Modes
• The refractive index difference between the core and cladding is very small
• There is degeneracy between modes
– Groups of modes travel with the same velocity (z equal)
• These hybrid modes are approximated with nearly linearly polarized modes called LP modes
– LP01 from HE11
– LP0m from HE1m
– LP1m sum of TE0m, TM0m, and HE2m
– LPm sum of HE+1,m and EH-1,m
17
First Mode Cut-Off• First mode
– What is the smallest allowable V
– Let y 0 and the corresponding x V
– So V=0, no cut-off for lowest order mode
– Same as a symmetric slab waveguide
0
5772.02
ln
221
limlim0
0
1
00
1
y
yy
yK
yKy
VJ
VJV
yy
01 VJ
18
Second Mode Cut-Off• Second mode
VJV
VJ 12
2
2
221
221
limlim
2
01
2
01
2
y
yy
yK
yKy
VJ
VJV
yy
xJxJx
nxJ nnn 11
2
xJxJx
xJ 012
2
VJV
VJVJV o 11
22
0VJo
405.2V
19
Cut-off V-parameter for low-order LPlm modes
m=1 m=2 m=3
l=0 0 3.832 7.016
l=1 2.405 5.520 8.654
20
Number of Modes
• The number of modes can be characterized by the normalized frequency
• Most standard optical fibers are characterized by their numerical aperture
• Normalized frequency is related to numerical aperture
• The optical fiber is single mode if V<2.405
• For large normalized frequency the number of modes is approximately
22
21
2nnaV
22
21 nnNA
NAaV2
14
Modes# 22
VV
21
Intensity Profiles
22
Standard Single Mode Optical Fibers
• Most common single mode optical fiber: SMF28 from Corning
– Core diameter dcore=8.2 m
– Outer cladding diameter: dclad=125m
– Step index
– Numerical Aperture NA=0.14
• NA=sin()
• =8°
• cutoff = 1260nm (single mode for cutoff)
• Single mode for both =1300nm and =1550nm standard telecommunications wavelengths
23
Standard Multimode Optical Fibers
• Most common multimode optical fiber: 62.5/125 from Corning
– Core diameter dcore= 62.5 m
– Outer cladding diameter: dclad=125m
– Graded index
– Numerical Aperture NA=0.275
• NA=sin()
• =16°
• Many modes
24
5. Optical Fibers Attenuation
25
Coaxial Vs. Optical Fiber Attenuation
26
Fiber Attenuation
• Loss or attenuation is a limiting parameter in fiber optic systems
• Fiber optic transmission systems became competitive with electrical transmission lines only when losses were reduced to allow signal transmission over distances greater than 10 km
• Fiber attenuation can be described by the general relation:
where is the power attenuation coefficient per unit length
• If Pin power is launched into the fiber, the power remaining after propagating a length L within the fiber Pout is
PdzdP
LPP inout exp
27
Fiber Attenuation
• Attenuation is conveniently expressed in terms of dB/km
• Power is often expressed in dBm (dBm is dB from 1mW)
34.4
log10
log10
log10
10
10
10
eLL
P
eP
L
P
P
LkmdB
in
Lin
in
out
dBmmW
mWmWP 10
1
10log1010 10
mWmWdBmP 50110127 10
27
28
Fiber Attenuation
• Example: 10mW of power is launched into an optical fiber that has an attenuation of =0.6 dB/km. What is the received power after traveling a distance of 100 km?
– Initial power is: Pin = 10 dBm
– Received power is: Pout= Pin– L=10 dBm – (0.6)(100) = -50 dBm
• Example: 8mW of power is launched into an optical fiber that has an attenuation of =0.6 dB/km. The received power needs to be -22dBm. What is the maximum transmission distance?
– Initial power is: Pin = 10log10(8) = 9 dBm
– Received power is: Pout = 1mW 10-2.2 = 6.3 W
– Pout - Pin = 9dBm - (-22dBm) = 31dB = 0.6 L
– L=51.7 km
nWmWPout 10110 1050
29
Material Absorption
• Material absorption
– Intrinsic: caused by atomic resonance of the fiber material
• Ultra-violet
• Infra-red: primary intrinsic absorption for optical communications
– Extrinsic: caused by atomic absorptions of external particles in the fiber
• Primarily caused by the O-H bond in water that has absorption peaks at =2.8, 1.4, 0.93, 0.7 m
• Interaction between O-H bond and SiO2 glass at =1.24 m
• The most important absorption peaks are at =1.4 m and 1.24 m
30
Scattering Loss
• There are four primary kinds of scattering loss
– Rayleigh scattering is the most important
where cR is the Rayleigh scattering coefficient and is the range from 0.8 to 1.0 (dB/km)·(m)4
• Mie scattering is caused by inhomogeneity in the surface of the waveguide
– Mie scattering is typically very small in optical fibers
• Brillouin and Raman scattering depend on the intensity of the power in the optical fiber
– Insignificant unless the power is greater than 100mW
kmdBcRR /1
4
31
Absorption and Scattering Loss
32
Absorption and Scattering Loss
33
Loss on Standard Optical Fiber
Wavelength SMF28 62.5/125
850 nm 1.8 dB/km 2.72 dB/km
1300 nm 0.35 dB/km 0.52 dB/km
1380 nm 0.50 dB/km 0.92 dB/km
1550 nm 0.19 dB/km 0.29 dB/km
34
External Losses
• Bending loss
– Radiation loss at bends in the optical fiber
– Insignificant unless R<1mm
– Larger radius of curvature becomes more significant if there are accumulated bending losses over a long distance
• Coupling and splicing loss
– Misalignment of core centers
– Tilt
– Air gaps
– End face reflections
– Mode mismatches
35
6. Optical Fiber Dispersion
36
Dispersion
• Dispersive medium: velocity of propagation depends on frequency
• Dispersion causes temporal pulse spreading
– Pulse overlap results in indistinguishable data
– Inter symbol interference (ISI)
• Dispersion is related to the velocity of the pulse
37
Intermodal Dispersion
• Higher order modes have a longer path length
– Longer path length has a longer propagation time
– Temporal pulse separation
– vg is used as the propagation speed for the rays to take into account the material dispersion
gv
L
38
Group Velocity
• Remember that group velocity is defined as
• For a plane wave traveling in glass of index n1
• Resulting in
1
gv
cn 1
c
n
nn
c
n
cc
n
g1
11
11
1
gg n
cv
1
1
111
nnn g
39
Intermodal Dispersion
• Path length PL depends on the propagation angle
• The travel time for a longitudinal distance of L is
• Temporal pulse separation
• The dispersion is time delay per unit length or
1sin
gg v
L
v
PL
12 sin
1
sin
1
gg vvL
1sinL
PL
12 sin
1
sin
1
gg vvD
40
Step Index Multimode Fiber
• Step index multimode fiber has a large number of modes
• Intermodal dispersion is the maximum delay minus the minimum delay
• Highest order mode (~c) Lowest order mode (~90°)
• Dispersion becomes
• The modes are not equally excited
– The overall dispersed pulse has an rms pulse spread of approximately
2
11
2 sin
11
n
n
c
n
vvg
cgg
c
n
vvg
gg
1
1 90sin
11
c
n
n
nn
c
n
n
n
c
nD ggg 1
2
211
2
11 1
21
c
nD g
41
Graded Index Multimode Fiber
• Higher order modes
– Larger propagation length
– Travel farther into the cladding
– Speed increases with distance away from the core (decreasing index of refraction)
– Relative difference in propagation speed is less
42
Graded Index Multimode Fiber
• Refractive index profile
• The intermodal dispersion is smaller than for step index multimode fiber
arnn
ara
rn
rn
21
2
1
21
21
4
21
inter
c
nD g
43
Intramodal Dispersion
• Single mode optical fibers have zero intermodal dispersion (only one mode)
• Propagation velocity of the signal depends on the wavelength
• Expand the propagation delay as a Taylor series
• Dispersion is defined as
• Propagation delay becomes
• Keeping the first two terms, the pulse width increase for a laser linewidth of is
2
22
2
1
go
googg
z
g
g
vD
1intra
intra2intra 2
1 DD ooogg
intraDg
44
Intramodal Dispersion
• Intramodal dispersion is
• There are two components to intramodal dispersion
• Material dispersion is related to the dependence of index of refraction on wavelength
• Waveguide dispersion is related to dimensions of the waveguide
1
1intra
zzD
waveguidematerial1
1
1
1intra
1DD
c
nn
cD zgzg
45
Material Dispersion
• Material dispersion depends on the material
1
1material
1
zgn
cD
46
Waveguide Dispersion
• Waveguide dispersion depends on the dimensions of the waveguide
• Expanded to give
where V is the normalized frequency
• Practical optical fibers are weekly guiding (n1-n2 <<1) resulting in the simplification
1
1waveguide
zg
c
nD
11
2
22
1
11waveguide 2
zzgg
VV
VV
n
n
c
nD
NAakV
bVV
Vc
nnD gg
2
221
waveguide
47
Total Intramodal Dispersion
• Total dispersion can be designed to be zero at a specific wavelength
• Standard single mode telecommunications fiber has zero dispersion around =1.3 m
• Dispersion shift fiber has the zero dispersion shifted to around =1.55 m
48
Standard Optical Fiber Dispersion
• Standard optical fiber
– Step index ≈0.0036
– Graded index ≈0.02
• Dispersion
– Step index multi-mode optical fiber (Dtot~10ns/km)
– Graded index multi-mode optical fiber (Dtot~0.5ns/km)
– Single mode optical fiber (Dintra~18ps/km nm)
4
21
intertot
c
nDD g
intratot DD
21
inter
c
nDD g
tot
49
What is the laser linewidth?
• Wavelength linewidth is a combination of inherent laser linewidth and linewidth change caused by modulation
– Single mode FP laser laser~2nm
– Multimode FP laser or LED laser~30nm
– DFB laser laser~0.01nm
• Laser linewidth due to modulation
– f~2B
2mod
2laser
Bc
fc
cf
cf
22
2
2