Transcript Lecture 2A: Ray Optics of Fibers

EE 230: Optical Fiber Communication Lecture 2
Fibers from the view of Geometrical
Optics
From the movie
Warriors of the Net
Total Internal Reflection
Snells Law :
n1 sin1  n2 sin2
Re flection Condition
1  3
When n1  n2 and as 1 increases eventually 2
goes to 90 deg rees and
n
n1 sinc  n2 or sinc  2
n1
c is called the Critical angle
For 1  c there is no propagating refracted ray
Reflection as a function of angle
The reflectivities of waves polarized
parallel and perpendicular to the plane of
incidence as given by the Fresnel equations
Fiber Optics Communication Technology-Mynbaev & Scheiner
This additional Phase
Shift is not accounted for
in geometrical wave
approach
Principal Types of Optical Fiber
Types of Fibers
•Single mode/Multi-mode
•Step Index/Graded Index
•Dispersion Shifted/Non-dispersion shifted
•Silica/fluoride/Other materials
•Major Performance Concerns for Fibers
•Wavelength range
•Maximum Propagation Distance
•Maximum bitrate
•Crosstalk
Understanding Fiber Optics-Hecht
Fabrication of Optical Fiber
• Fabrication of fiber preform:
macroscopic version with correct index
profile
• Drawing of preform down into thin fiber
• Jacketing and cabling
Step-Index Fiber
• Cladding typically pure silica
• Core doped with germanium to increase
index
• Index difference referred to as “delta” in
units of percent (typically 0.3-1.0%)
• Tradeoff between coupling and bending
losses
• Index discontinuity at core-clad
boundary
Basic Step index Fiber Structure
Fiber Optics Communication Technology-Mynbaev & Scheiner
Ray Trajectories in Step Index
fiber
Meridional Rays
Skew Rays
Coupling Light into an Optical
Fiber
Fiber Optics Communication Technology-Mynbaev & Scheiner
Acceptance Angle
The acceptance angle (i) is the largest incident
angle ray that can be coupled into a guided ray
within the fiber
The Numerical Aperature (NA) is the sin(i) this
is defined analagously to that for a lens
NA =
(n12 -
Where D º
n2
1
2 2
)
=
1
2 2
(2D n )
=
1
2
n(2D )
n1 - n2
n + n2
and n º 1
n
2
f# º
f
f
=
D FullAccep tan ceAngle
=
1
2 ×NA
Optics-Hecht & Zajac
n0
θ2
θ1
φ2
φ1
nCO
nCL
Numerical Aperture
From Snell’s Law,
nCO sin 1  nCL sin  2
no sin 1  nCO sin  2
 nCL
 nCO
1   c  sin 1 
For total internal reflection, θ2=90º



What value of φ1 corresponds to θc?
That is the maximum acceptance angle for the fiber.
φ2 = 90º-θc
sinφ2 = cos θc
n
sin  c  CL , so
nCO
2
2
nCO
 nCL
cos  c 
nCO
 n2  n2
CL
nCO sin  2  nCO  CO

nCO


  n 2  n 2  NA
CO
CL


Again from Snell’s Law,
n0 sin 1  nCO sin 2
 NA 

 n0 
1c  sin 1 
(= NA), so
For Corning SMF-28 optical fiber
nco=1.4504, nCL=1.4447 at 1550 nm
NA = 0.13
Acceptance angle = 7.35 degrees
Geometrical View of Modes
•Ray approximation valid in the
limit that l goes to zero
•Geometrical Optics does not
predict the existance of discrete
modes
•Maxwells Equations and
dielectric boundary conditions
give rise to the requirement that
the fields and phase reproduce
themselves each “cycle”
Fiber Optics Communication Technology-Mynbaev & Scheiner
Rays and Their E-field Distribution
Origin of Modal Dispersion
•
•
Straight path along fiber axis has distance L and velocity c/nCO for
transit time of LnCO/c
Path at maximum acceptance angle φc has distance L/cosφ2 where
φ2=90º-θc and thus a longer transit time.
 NA 

cos 2  1  sin 2 2  1  
 nCO 
2
LnCO
t 2  t1 
c
 nCO


 1
 nCL

•
Transit time difference equal to
•
•
Dispersion limits rate of signals that fiber can handle
If spread can be up to 70% of bit period, then maximum bit rate is
1.4cnCO/L(NA)2
Intermodal Dispersion
n1
L
D tSI = c n nD
2
D tSI @L
c nD for n1 @n2
(NA)2
D tSI @L
c 2n
Fiber Optics Communication Technology-Mynbaev & Scheiner
Bandwidth for Various Fiber Types
Bit Rate BR < 1
4D t
BRSI =
1 = c
4D tSI 4LnD
BRGI =
2c
n1LD 2
BRGI
BRSI
=
2c
n1LD 2
c
4LnD
=
8
D
No intermodal time shift for single
Mode Fiber
Fiber Optics Communication Technology-Mynbaev & Scheiner
Graded Index Fiber


 
n(  )  n1 1      for  <a
 a  

n(  )  n1 1    =n 2 for  >a
Fiber Optic Communication Systems-Agarwal
for   2 a "parabolic profile"
1

NA=n1  2  2 1    which varies with 
a
2
tGI 
Ln1 2
8c
Fiber Optic Communications-Palais
Ray Propagation in Graded-Index
Fiber
Graded Index Slab Uniform in X and Z
Fundamentals of Photonics - Saleh and Teich
Ray spreading comparison
L NA
t SI 
2cnCO
2
L NA

3
8cnCO
4
tGI
Comparison, continued
If NA=0.13 and nCO=1.45,
∆tSI/L=19 ps/m
∆tGI/L=0.039 ps/m
Graded-index fiber has substantially less
modal dispersion