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4. Optical Fibers
Stephen Schultz
Fiber Optics
Fall 2005
1
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
Stephen Schultz
Fiber Optics
Fall 2005
2
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
Stephen Schultz
Fiber Optics
Fall 2005
3
Critical Angle
•
Ray bends at boundary between materials
n2 sin 2  n1 sin 1
– Snell’s law
 2  sin 1  n2 n sin 1 

•

1
Light confined to core if propagation angle is greater than the critical angle
– Total internal reflection (TIR)
1   c  sin 1  n2 n 

Stephen Schultz
Fiber Optics
Fall 2005
1

4
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
Stephen Schultz
Fiber Optics
Fall 2005
5
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:
NA  sin  i   n12  n22
•
Telecommunications optical fiber n1~n2,
NA  n12  n22 
n1  n2  n1  n2  
NA  n1 2 
Stephen Schultz

Fiber Optics
Fall 2005
2 n12
n1  n2
n1
n1  n2
n1
6
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 bz
•
With geometrical optics this is given by
•
The goal is to calculate the value of βz
•
Remember that the range of βz is
Stephen Schultz
b zi  n1 k0 sin i 
n2 ko  b z  n1 ko
Fiber Optics
Fall 2005
7
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
– n is the angular mode number
Stephen Schultz
Fiber Optics
Fall 2005
8
Bessel Functions
Stephen Schultz
Fiber Optics
Fall 2005
9
Transcendental Equation
•
Under the weakly guiding approximation (n1-n2)<<1
– Valid for standard telecommunications fibers
J l' kT a 
K l' γ a 
kT a
 γa
J l kT a 
K l γ a 
kT2  n12 ko2  b 2
•
 2  b 2  n22 ko2
Substitute to eliminate the derivatives
J l x 




J x   J l 1 x  l
'
l
kT a
K l x 




K x   K l 1 x  l
'
l
x
J l 1 kT a 
K  a 
  a l 1
or
J l κ a 
K l  a 
kT a
HE Modes
Stephen Schultz
x
J l 1 kT a 
K  a 
  a l 1
J l kT a 
K l  a 
EH Modes
Fiber Optics
Fall 2005
10
Bessel Function Relationships
•
Bessel function recursive relationships
K n x   K n x 
J n x   1 J n x 
n
2n
J n 1  x  
J n x   J n 1  x 
x
2n
K n 1  x    K n  x   K n 1  x 
x
2


K 0 x   K1 x   K 2 x 
x
2
J 0  x   J1  x   J 2  x 
x
•
Small angle approximations

 x

ln
   0.5772 n  0


2
K n x   
n


n

1
!
2



n0
 

2  x
Stephen Schultz
Fiber Optics
Fall 2005
11
Lowest Order Modes
•
Look at the l=-1, 0, 1 modes
x  kT a
•
•
y  a
V  x2  y2 

a n12  n22
Use bessel function properties to get positive order and highest order on top
l=-1
J x 
K y
x 2
  y 2
J 1  x 
K 1  y 
x
l=0
x
J 1  x 
K y
  y 1
J 0 x 
K0  y
x
x
J1 x 
K y
y 1
J 0 x 
K0  y
Stephen Schultz
J 0 x 
K y
 y 0
J1  x 
K1  y 
J 2 x 
K2 y
x
y
J1  x 
K1  y 
J x 
K y
x 2
y 2
J1  x 
K1  y 
•
2
Fiber Optics
Fall 2005
J1 x 
K y
y 1
J 0 x 
K0  y
12
Lowest Order Modes cont.
•
l=+1
x
J 0 x 
K y
 y 0
J1  x 
K1  y 
x
•
x
J 2 x 
K y
y 2
J1  x 
K1  y 
J 2 x 
K y
y 2
J1  x 
K1  y 
So the 6 equations collapse down to 2 equations
x
J1 x 
K1  y 
x
y
J 0 x 
K0  y
J 2 x 
K y
y 2
J1  x 
K1  y 
lowest modes
Stephen Schultz
Fiber Optics
Fall 2005
13
Modes
20
LHS, RHS
15
10
5
0
0
2
4
6
8
10
x (kT a)
Stephen Schultz
Fiber Optics
Fall 2005
14
Fiber Modes
Stephen Schultz
Fiber Optics
Fall 2005
15
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 (bz 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
– LPnm sum of HEn+1,m and EHn-1,m
Stephen Schultz
Fiber Optics
Fall 2005
16
First Mode Cut-Off
•
First mode
– What is the smallest allowable V
– Let y  0 and the corresponding x V
1 2
y   
J V 
K y
2 y
V 1
 lim y 1
 lim
0
y

0
y

0
y
J 0 V 
K0  y 
 
 ln    0.5772
2
J1 V   0
– So V=0, no cut-off for lowest order mode
– Same as a symmetric slab waveguide
Stephen Schultz
Fiber Optics
Fall 2005
17
Second Mode Cut-Off
•
Second mode
2
1 2
y   
J V 
K y
2 y
V 2
 lim y 2
 lim
2
y

0
y

0
J1 V 
K1  y 
1 2
   
2 y
J 2 V  
2
J1 V 
V
2n
J n 1  x  
J n x   J n 1  x 
x
2
J 2  x   J1  x   J 0  x 
x
2
2
J1 V   J o V   J1 V 
V
V
J o V   0
V  2.405
Stephen Schultz
Fiber Optics
Fall 2005
18
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
Stephen Schultz
Fiber Optics
Fall 2005
19
Number of Modes
•
The number of modes can be characterized by the normalized frequency
V
•
2

a n12  n22
Most standard optical fibers are characterized by their numerical aperture
NA  n12  n22
•
Normalized frequency is related to numerical aperture
V
•
•
2
a NA

The optical fiber is single mode if V<2.405
For large normalized frequency the number of modes is approximately
# Modes 
Stephen Schultz
4

2
V
2
Fiber Optics
Fall 2005
V  1
20
Intensity Profiles
Stephen Schultz
Fiber Optics
Fall 2005
21
Standard Single Mode Optical Fibers
•
Most common single mode optical fiber: SMF28 from Corning
– Core diameter dcore=8.2 mm
– Outer cladding diameter: dclad=125mm
– 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
Stephen Schultz
Fiber Optics
Fall 2005
22
Standard Multimode Optical Fibers
•
Most common multimode optical fiber: 62.5/125 from Corning
– Core diameter dcore= 62.5 mm
– Outer cladding diameter: dclad=125mm
– Graded index
– Numerical Aperture NA=0.275
• NA=sin()
• =16°
• Many modes
Stephen Schultz
Fiber Optics
Fall 2005
23
5. Optical Fibers Attenuation
Stephen Schultz
Fiber Optics
Fall 2005
24
Coaxial Vs. Optical Fiber Attenuation
Stephen Schultz
Fiber Optics
Fall 2005
25
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:
 a P
dz
where a is the power attenuation coefficient per unit length
dP
•
If Pin power is launched into the fiber, the power remaining after propagating a
length L within the fiber Pout is
Pout  Pin exp  a L
Stephen Schultz
Fiber Optics
Fall 2005
26
Fiber Attenuation
•
Attenuation is conveniently expressed in terms of dB/km
a dB km  
P 
10
log 10  out 
L
 Pin 
 Pine aL 
10

  log 10 
L
P
 in 
10
 aL log 10 e
L
 4.34a

•
Power is often expressed in dBm (dBm is dB from 1mW)
 10 mW 
  10 dBm
P  10 mW  10 log 10 
1
mW


27
 10

P  27 dBm  1 mW 10   501mW


Stephen Schultz
Fiber Optics
Fall 2005
27
Fiber Attenuation
•
Example: 10mW of power is launched into an optical fiber that has an
attenuation of a=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– a L=10 dBm – (0.6)(100) = -50 dBm
Pout  1050 10  1 mW  10 nW
•
Example: 8mW of power is launched into an optical fiber that has an
attenuation of a=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 mW
– Pout - Pin = 9dBm - (-22dBm) = 31dB = 0.6 L
– L=51.7 km
Stephen Schultz
Fiber Optics
Fall 2005
28
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 mm
• Interaction between O-H bond and SiO2 glass at =1.24 mm
• The most important absorption peaks are at =1.4 mm and 1.24 mm
Stephen Schultz
Fiber Optics
Fall 2005
29
Scattering Loss
•
There are four primary kinds of scattering loss
– Rayleigh scattering is the most important
a R  cR
•
•
1
dB / km
4
where cR is the Rayleigh scattering coefficient and is the range from 0.8 to
1.0 (dB/km)·(mm)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
Stephen Schultz
Fiber Optics
Fall 2005
30
Absorption and Scattering Loss
Stephen Schultz
Fiber Optics
Fall 2005
31
Absorption and Scattering Loss
Stephen Schultz
Fiber Optics
Fall 2005
32
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
Stephen Schultz
Fiber Optics
Fall 2005
33
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
Stephen Schultz
Fiber Optics
Fall 2005
34
6. Optical Fiber Dispersion
Stephen Schultz
Fiber Optics
Fall 2005
35
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
Stephen Schultz
Fiber Optics
Fall 2005
36
Intermodal Dispersion
•
Higher order modes have a longer path length
– Longer path length has a longer propagation time
– Temporal pulse separation
 
L
vg
– vg is used as the propagation speed for the rays to take into account the
material dispersion
Stephen Schultz
Fiber Optics
Fall 2005
37
Group Velocity
•
Remember that group velocity is defined as
•
For a plane wave traveling in glass of index n1
•
Resulting in
b


n1  n1

c c 

1  n1

 n1 

c  

 b 
vg  





b  n1
1

c
n1g

c
1
c
 b 
vg  


n1g
  
Stephen Schultz
Fiber Optics
Fall 2005
n1g  n1  
n1

38
Intermodal Dispersion
•
•
•
•
Path length PL depends on the propagation angle
L
PL 
sin 1
The travel time for a longitudinal distance of L is
Temporal pulse separation
PL
L

vg vg sin 1

1
1 
  L 

 v sin  v sin  
2
g
1
 g
The dispersion is time delay per unit length or
Stephen Schultz

Fiber Optics
Fall 2005

1
1 

D

 v sin  v sin  
2
g
1
 g
39
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°)
n1g
1
1


v g1 v g sin 90
c
n1g n1
1
1


v g 2 vg sin  c
c n2
•
Dispersion becomes
n1g  n1  n1g
  1 
D
c  n2  c
•
 n1  n2  n1g

 

 n2  c
The modes are not equally excited
– The overall dispersed pulse has an rms pulse spread of approximately
n1g 
D
c 2
Stephen Schultz
Fiber Optics
Fall 2005
40
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
Stephen Schultz
Fiber Optics
Fall 2005
41
Graded Index Multimode Fiber
•
Refractive index profile
2

r

n1 1  2   

a

nr   
 n 1 2   n
2
 1

•
ra
ra
The intermodal dispersion is smaller than for step index multimode fiber
n1g 2
Dinter 
c 4
Stephen Schultz
Fiber Optics
Fall 2005
42
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
2
 g 1
2  g
 g   g o   o   
 o   

 2
2
•
Dispersion is defined as
Dintra 
•
 g

Propagation delay becomes

  1    b z 





  v g     
 g   g o   o    Dintra 
•
1
o   2 Dintra
2

Keeping the first two terms, the pulse width increase for a laser linewidth of
 is
 g  Dintra 
Stephen Schultz
Fiber Optics
Fall 2005
43
Intramodal Dispersion
•
Intramodal dispersion is
Dintra 
•
There are two components to intramodal dispersion
Dintra 
•
•
  b z    b z b1 




      b1  
1 n1g b z n1g   b z 

  Dmaterial  Dwaveguide

c  b1
c   b1 
Material dispersion is related to the dependence of index of refraction on
wavelength
Waveguide dispersion is related to dimensions of the waveguide
Stephen Schultz
Fiber Optics
Fall 2005
44
Material Dispersion
•
Material dispersion depends on the material
Dmaterial 
Stephen Schultz
1 n1g b z
c  b1
Fiber Optics
Fall 2005
45
Waveguide Dispersion
•
Waveguide dispersion depends on the dimensions of the waveguide
Dwaveguide 
•
Expanded to give
n1g   b z 


c   b1 
n1g n1g  2  2  b z 
  b z 


 
Dwaveguide  
V

2
V

2 

c n1   V  b1 
V  b1 
where V is the normalized frequency
V  k a NA
•
Practical optical fibers are weekly guiding (n1-n2 <<1) resulting in the
simplification
Dwaveguide
Stephen Schultz
n

1g
 n2 g 
2
V b
V
2
c
V
Fiber Optics
Fall 2005
46
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 mm
Dispersion shift fiber has the zero dispersion shifted to around =1.55 mm
Stephen Schultz
Fiber Optics
Fall 2005
47
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)
n1g 
Dtot  Dinter 
c 2
– Graded index multi-mode optical fiber (Dtot~0.5ns/km)
n1g 2
Dtot  Dinter 
c 4
– Single mode optical fiber (Dintra~18ps/km nm)
Dtot  Dintra
Stephen Schultz
Fiber Optics
Fall 2005
48
What is the laser linewidth?
•
Wavelength linewidth is a combination of inherent laser linewidth and
linewidth change caused by modulation
  2laser  2mod
•
– 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
c
f 

f
c
 2


 
 
Stephen Schultz
2
c
2
c
f
2B
Fiber Optics
Fall 2005
49