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Light
Light
n2
Light
Light
n2
n1 > n 2
A planar dielectric waveguide has a central rectangular region of
higher refractive index n 1 than the surrounding region which has
a refractive index n2 . It is assumed that the waveguide is
infinitely wide and the central region is of thickness 2 a. It is
illuminated at one end by a monochromatic light source.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
n2
B

 
E
Light
A
k1

n 1 d = 2a
 
C
y
x
z
n2
A light ray travelling in the guide must interfere constructively with itself to
propagate successfully. Otherwise destructive interference will destroy the
wave.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
n2
A
B
1

E

k1
2
A
C

y


n1
n2
2a
x
1
z
B
Two arbitrary waves 1 and 2 that are initially in phase must remain in phase
after reflections. Otherwise the two will interfere destructively and cancel each
other.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
n2
A
1
E
Guide center
C a y


k

A
2
a
y
y
z
x
Interference of waves such as 1 and 2 leads to a standing wave pattern along the ydirection which propagates along z.
© 1999 S.O. Kasap,Optoelectronics (Prentice Hall)
Fie ld of evanescent wave
(exponential decay)
y
n2
Field of guided wave
E(y,z,t) = E(y)cos(t – 0z)
E(y)
Light
m=0
n1
n2
The electric field pattern of the lowest mode traveling wave along the
guide. This mode has m = 0 and the lowest . It is often referred to as the
glazing incidence ray. It has the highest phase velocity along the guide.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
y
n2
E(y)
Cl addi ng
m= 1
m= 0
Core
n1
n2
m= 2
2a
Cl addi ng
The electric field patterns of the first three modes (m = 0, 1, 2)
traveling wave along the guide. Notice different extents of field
penetration into the cladding.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
High order mode
Light pulse
Low order mode
Broadened
light pulse
Cladding
Core
Intensity
Intensity
Axial
0
Spread, 
t
Schematic illustration of light propagation in a slab dielectric waveguide. Light pulse
entering the waveguide breaks up into various modes which then propagate at different
group velocities down the guide. At the end of the guide, the modes combine to
constitute the output light pulse which is broader than the input light pulse.
© 1999 S.O. Kasap,Optoelectronics (Prentice Hall)
t
(a) TE mode
B //
By
 
(b) TM mode
E //
Ey
 
y
Bz
O
E
Ez
B
z
x (into paper)
Possible modes can be classified in terms of (a) transelectric field (TE)
and (b) transmagnetic field (TM). Plane of incidence is the paper.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
tan(ak1cosm  m/2)
m = 1, odd
m = 0, even
f(m)
10
89.17
5
88.34
c
87.52
86.68
m
0
82
84
86
88
90
Modes in a planar dielectric waveguide can be determined by
plotting the LHS and the RHS of eq. (11).
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Slope = c/n 2
Slope = c/n 1
TE2
TE1
cut-off
TE0
m
Schematic dispersion diagram,  vs.  for the slab waveguide for various TEm. modes.
cut–off corresponds to V = /2. The group velocity vg at any  is the slope of the  vs. 
curve at that frequency.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
y
y
Cladding
2 > 1
1 > c
v g1
1 < cut-off
E(y)
Core
v g2 > v g1
2 < 1
Cladding
The electric field of TE 0 mode extends more into the
cladding as the wavelength increases. As more of the field
is carried by the cladding, the group velocity increases.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
y
y
Cl addi ng

Core
n2 n1
r
z Fiber axis
n
The step index optical fiber. The central region, the core, has greater refractive
index than the outer region, the cladding. The fiber has cylindrical symmetry. We
use the coordinates r, , z to represent any point in the fiber. Cladding is
normally much thicker than shown.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Along the fiber
1
1, 3
3
(a) A meridional
ray always
crosses the fiber
axis.
Meridional ray
Fiber axis
2
2
1
2
1
Skew ray
Fiber axis
5
3
5
4
Ray path along the fiber
4
2
3
(b) A skew ray
does not have
to cross the
fiber axis. It
zigzags around
the fiber axis.
Ray path projected
on to a plane normal
to fiber axis
Illustration of the difference between a meridional ray and a skew ray.
Numbers represent reflections of the ray.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
(a) The electric field (b) The intensity in
of the fundamental
the fundamental
mode
mode LP01
(c) The intensity (d) The intensity
in LP 11
in LP 21
Core
E
Cladding
E01
r
The electric field distribution of the fundamental mode
in the transverse plane to the fiber axis z. The light
intensity is greatest at the center of the fiber. Intensity
patterns in LP 01, LP11 and LP21 modes.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
b
1
LP 01
0.8
LP 11
0.6
LP 21
0.4
LP 02
0.2
0
V
0
1
2
3
2.405
4
5
6
Normalized propagation constant b vs. V-number
for a step index fiber for various LP modes.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Lost
n2
n0
n1
 < c
B
Cladding
Propagates
 > c
A
Fiber axis
  max
Core
A
B
  max
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Maximum acceptance angle
max is that which just gives
total internal reflection at the
core-cladding interface, i.e.
when  = max then  = c.
Rays with  > max (e.g. ray
B) become refracted and
penetrate the cladding and are
eventually lost.
Cladding
Input
v g ( 1 )
Emitter
Very short
light pulse
v g ( 2 )
Intensity
Intensity
Core
Output
Intensity
Spectrum, ² 
Spread, ² 
1
o
2

0
t

All excitation sources are inherently non-monochromatic and emit within a
spectrum, ² , of wavelengths. Waves in the guide with different free space
wavelengths travel at different group velocities due to the wavelength dependence
of n1. The waves arrive at the end of the fiber at different times and hence result in
a broadened output pulse.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
t
Dispersion coefficient (ps km -1 nm -1 )
30
Dm
20
10
Dm + Dw
0
0
-10
Dw
-20
-30
1.1
1.2
1.3
1.4
 (m)
1.5
1.6
Material dispersion coefficient (Dm) for the core material (taken as
SiO2 ), waveguide dispersion coefficient (Dw ) (a = 4.2 m) and the
total or chromatic dispersion coefficient Dch (= Dm + Dw ) as a
function of free space wavelength,  
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Intensity
t
Output light pulse
n1 y // y
n1 x // x
Ey

Ex
Core
Ex
z
Ey
 = Pulse spread
t
E
Input light pulse
Suppose that the core refractive index has different values along two orthogonal
directions corresponding to electric field oscillation direction (polarizations). We can
take x and y axes along these directions. An input light will travel along the fiber with Ex
and Ey polarizations having different group velocities and hence arrive at the output at
different times
© 1999 S.O. Kasap,Optoelectronics (Prentice Hall)
-1
Dispe rsion coefficient (ps km
nm -1)
n
30
20
r
Dm
10
0
2
1
Dch = Dm + Dw
-10
Dw
-20
Thin laye r of c ladding
with a de pressed index
-30
1.1
1.2
1.3
1.4
(m)
1.5
1.6
1.7
Dispersion flattened fiber example. The material dispersion coefficient ( Dm) for the
core material and waveguide dispersion coefficient ( Dw) for the doubly clad fiber
result in a flattened small chromatic dispersion between 1 and 2.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Dispersion coefficient (ps km-1 nm-1)
20
Dm
10
SiO2-13.5%GeO2
0
a (m)
Dw
4.0
3.5
3.0
–10
2.5
–20
1.2
1.3
1.4
1.5
1.6
 (m)
Material and waveguide dispersion coefficients in an
optical fiber with a core SiO 2-13.5%GeO2 for a = 2.5
to 4 m.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Fiber
Digital signal
Information
Emitter
t
Photodetector
Information
Input
Output
Input Intensity
Output Intensity
² 
Very short
light pulses
0
T
t
t
0
~2² 
An optical fiber link for transmitting digital information and the effect of
dispersion in the fiber on the output pulses.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Output optical power
T = 4
1
0.61
0.5


t
A Gaussian output light pulse and some tolerable intersymbol
interference between two consecutive output light pulses (y-axis in
relative units). At time t =  from the pulse center, the relative
magnitude is e -1/2 = 0.607 and full width root mean square (rms)
spread isrms = 2.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Electrical signal (photocurrent)
1
0.707
Fiber
Sinusoidal signal
Photodetector
Emitter
t
Optical
Input
f = Modulation frequency
Pi = Input light power
Optical
Output
Po = Output light power
1 kHz
1 MHz
1 GHz
1 MHz
1 GHz
f
fel
Sinusoidal electrical signal
Po / Pi
0.1
0.05
0
t
0
t
1 kHz
fop
f
An optical fiber link for transmitting analog signals and the effect of dispersion in the
fiber on the bandwidth, fop.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
n2
n1
3
2
1
O
n
(a) Multimode step
index fiber. Ray paths
are different so that
rays arrive at different
times.
n2
O
O'
O''
3
2
1
2
3
n1
n2
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
n
(b) Graded index fiber.
Ray paths are different
but so are the velocities
along the paths so that
all the rays arrive at the
same time.
(a)
TIR
n decreases step by step from one layer
to next upper layer; very thin layers.
(b)
TIR
Continuous decrease in n gives a ray
path changing continuously.
(a) A ray in thinly stratifed medium becomes refracted as it passes from one
layer 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 bends
continuously.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
nc
B'
B
2
O
1
B
B'
c/nb
c/na
B'
Ray 2
A
B''
nb
na
A
Ray 1
M
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
We can visualize a graded index
fiber by imagining a stratified
medium with the layers of refractive
b indices na > nb > nc ... Consider two
close rays 1 and 2 launched from O
at the same time but with slightly
a different launching angles. Ray 1
just suffers total internal reflection.
O'
Ray 2 becomes refracted at B and
reflected at B'.
c
A solid with ions
Ex
Light direction
k
z
Lattice absorption through a crystal. The field in the wave
oscillates the ions which consequently generate "mechanical"
waves in the crystal; energy is thereby transferred from the wave
to lattice vibrations.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
A dielectric particle smaller than wavelength
Incident wave
Through wave
Scattered waves
Rayleigh scattering involves the polarization of a small dielectric
particle or a region that is much smaller than the light wavelength.
The field forces dipole oscillations in the particle (by polarizing it)
which leads to the emission of EM waves in "many" directions so
that a portion of the light energy is directed away from the incident
beam.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Field distribution
Microbending
Escaping wave
Cladding

Core
 

c

R
Sharp bends change the local waveguide geometry that can lead to waves
escaping. The zigzagging ray suddenly finds itself with an incidence
angle  that gives rise to either a transmitted wave, or to a greater
cladding penetration; the field reaches the outside medium and some light
energy is lost.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
B (m-1) for 10 cm of bend




= 633 nm
V 2.08

= 790 nm
V 1.67

0
2 4
6
8
10 12 14
16 18
Radius of curvature (mm)
Measured microbending loss for a 10 cm fiber bent by different amounts of radius of
curvature R. Single mode fiber with a core diameter of 3.9 m, cladding radius 48 m,
= 0.00275, NA = 0.10, V  1.67 and 2.08 (Data extracted and replotted from A.J.
Harris and P.F. Castle, IEEE J. Light Wave Technology, Vol. LT14, pp. 34-40, 1986;
see original article for discussion of peaks in B vs. R at 790 nm).
From S.O. Kasap, Optoelectronics and Photonics: Principles and Practices (Prentice Hall)
r
Buffer tube: d = 1mm
Protective polymerinc coating
Cladding: d = 125 - 150 m
n
Core: d = 8 - 10 m
n1
n2
The cross section of a typical single-mode fiber with a tight buffer
tube. (d = diameter)
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Drying gas es
Porous soot
preform wit h hole
Vapors: SiCl 4 + GeCl 4 + O 2
Fuel: H 2
Furnace
Preform
Burner
Furnace
Depos it ed s oot
Target rod
Depos it ed Ge doped SiO 2
Rotate mandrel
(a)
(b)
Clear s olid
glas s preform
(c)
Drawn fiber
Schematic illustration of OVD and the preform preparation for fiber drawing. (a)
Reaction of gases in the burner flame produces glass soot that deposits on to the outside
surface of the mandrel. (b) The mandrel is removed and the hollow porous soot preform
is consolidated; the soot particles are sintered, fused, together to form a clear glass rod.
(c) T he consolidated glass rod is used as a preform in fiber drawing.
© 1999 S.O . Kasap,Optoelectronics (Prentice Hall)
vg (m/s)
c/n2

TE 0

TE 4
TE 1

c/n1
 (1/s)


 

cut-off
 


Group velocity vs. angular frequency for three modes for a planar dielectric waveguide
which has n1 = 1.455, n2 = 1.44, a = 10 m (Results from Mathview, Waterloo Maple
math-software application). T E0 is for m = 0 etc.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
1.5
1
V[d2(Vb)/dV2]
0.5
0
0
1
2
3
V - number
[d2(Vb)/dV2] vs. V-number for a step index fiber (after W.A. Gambling et
al., The Radio and Electronics Engineer, 51, 313, 1981)
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
y = 5/2
n3

Medium 3
B'
B' B'
n2
B
Ray B
O
n1
y = 3/2
B'
Medium 2
A
B
Ray A
A A
Medium 1
B '

B''
B
O'
y = /2
/2
y=0
Step-graded-index dielectric waveguide. Two rays are launched from
the center of the waveguide at O at angles A and B such that ray A
suffers TIR at A and ray B suffers TIR at B'. Both TIRs are at critical
angles.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
0.25P
0.5P
O'
O
O
O
(a)
0.23P
(b)
(c)
Graded index (GRIN) rod lenses of different pitches. (a) Point O is on the rod face
center and the lens focuses the rays onto O' on to the center of the opposite face. (b)
The rays from O on the rod face center are collimated out. (c) O is slightly away from
the rod face and the rays are collimated out.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)