waveguides and optical fibers - concordia universityusers.encs.concordia.ca/~mojtaba/elec 425-6261...
Post on 27-Apr-2018
239 Views
Preview:
TRANSCRIPT
Waveguides and Optical Fibers
Light
n2
A planar dielectric waveguide has a central rectangular region ofhigher refractive index n1 than the surrounding region which hasa refractive index n2. It is assumed that the waveguide isinfinitely wide and the central region is of thickness 2 a. It isilluminated at one end by a monochromatic light source.
n2
n1 > n2
Light
LightLight
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Dielectric Waveguides
n2
n2
d = 2a
k1
Light
A
B
C
E
n
1
A light ray travelling in the guide must interfere constructively with itself topropagate successfully. Otherwise destructive interference will destroy thewave.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
z
y
x
m
anmm
cos
22 1
mmm
nk
sin
2sin 1
1
mmm
nkk
cos
2cos 1
1
ΔΦ (AC) = k1(AB + BC) - 2Φ = m(2π), k1 = kn1 = 2πn1/λ
m=0, 1, 2, …
Waveguide condition
A plane wave propagate a waveguide with a n1/n2 core-cladding, along Z-direction.
BC = d/cosθ and AB = BC cos(2θ)
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)
ztyEtzyE mm cos)(2,,1
mmmmm ykztykEtzyE
2
1cos
2
1cos2,, 01
y
E(y)m = 0 m = 1 m = 2
Cladding
Cladding
Core 2an
1
n2
n2
The electric field patterns of the first three modes (m = 0, 1, 2)traveling wave along the guide. Notice different extents of fieldpenetration into the cladding.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Low order modeHigh order mode
Cladding
Core
Ligh t pulse
t0 t
Spread,
Broadened
light pulse
IntensityIntensity
Axial
Schematic illustration of light propagation in a slab dielectric waveguide. Light pulseentering the waveguide breaks up into various modes whic h 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)
Ey (m) is the field distribution along y axis and
constitute a mode of propagation.
m is called mode number. Defines the number of
modes traveling along the waveguide. For every
value of m we have an angle θm satisfying the
waveguide condition provided to satisfy the TIR as
well. Considering these condition one can show that
the number of modes should satisfy:
m ≤ (2V – Φ)/π
V is called V-number and it is a characteristic parameter of the waveguide
21
2
2
2
1
2nn
aV
For V ≤ π/2, m= 0, it is the lowest mode of propagation referred to single
mode waveguides.
The cut-off wavelength (frequency) is a free space wavelength for v = π/2
For V=π/2, The wavelength is the cut-off wavelength, above this wavelength only one-mode (m=0) will propagate
E
By
Bz
z
y
O
B
E// Ey
Ez
(b) TM mode(a) TE mode
B//
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)
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)
i
n2
n1 > n
2
Incident
light
Reflected
light
r
z
Virtual reflecting plane
Penetration depth,
z
y
The reflected light beam in total internal reflection appears to have been laterally shifted byan amount z at the interface.
A
B
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Goos-Hanchen shift
i
n2
n1 > n
2
Incident
light
Reflected
light
r
When medium B is thin (thickness d is small), the field penetrates tothe BC interface and gives rise to an attenuated wave in medium C.The effect is the tunnelling of the incident beam in A through B to C.
z
y
d
n1
A
B
C
© 1999 S.O. Kasap, Optoelectronics (Prent ice Hall)
Optical Tunneling
Waveguide Dispersion
Intermodal dispersion: In multimode waveguides the lowest mode has the slowest group velocity, the highest mode has the highest group velocity
Intramodal Dispersion; In single mode waveguide:• Waveguide Dispersion: as there is no prefect
monochromatic light• Material Dispersion: due to the n(λ)
c
nn
VVL gg
21
maxmin
11
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)
n
y
n2 n
1
Cladding
Core z
y
r
Fiber axis
The step index optical fiber. The central region, the core, has greater refractiveindex than the outer region, the cladding. The fiber has cylindrical symmetry. Weuse the coordinates r, , z to represent any point in the fiber. Cladding isnormally much thicker than shown.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
For the step index optical fiber Δ = (n1 – n2)/n1
is called normalized index difference
Fiber axis
12
34
5
Skew ray1
3
2
4
5
Fiber axis
1
2
3
Meridional 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)
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)
ztjrEE lmlmLP exp.
LPs (linearly polarized waves) propagating along the fiber have either TE or TM type represented by the propagation of an electric field distribution Elm(r,ɸ) along z.
ELP is the field of the LP mode and βlm is its propagation
constant along z.
12 2
1
2
2
2
1
1
21
n
nn
n
nn
For weakly guided fibers, i.e.:
The guide modes are visualized by traveling waves that are almost polarized, called linearly polarized (LP)
V-number
2
2VM
405.22
2
12
2
2
1 nna
Vc
offcut
2
1
2
2
2
1121 2// nnnnnn
Normalized index difference
For V = 2.405, the fiber is called single
mode (only the fundamental mode
propagate along the fiber). For V > 2.405
the number of mode increases according
to approximately Most SM fibers designed
with 1.5<V<2.4
For weakly guided
fiber Δ=0.01, 0.005,..
0 2 4 61 3 5
V
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)
2
2
2
1
2
2
2/
nn
nkb
kn2 <β<kn1 Propagation condition
b changes between 0 and 1
Since βlm of an LP mode depends on the waveguide properties and the source wavelength, the lightPropagation is described in terms of a “normalized propagation” constant that depends only on the V-number.
21
12
12
2
2
1 222
nna
nna
V
Cladding
Core max
A
B
< c
A
B
> c
max
n0
n1
n2
Lost
Propagates
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.
Fiber axis
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
0
2
12
2
2
1maxsin
n
nn
2/12
2
2
1 )( nnNA
0
maxsinn
NA NA
aV
2
Example values for n1=1.48, n2=1.47;
very close numbers
Typical values of NA = 0.07…,0.25
Numerical Aperture---Maximum Acceptance Angle
0
1max
)90sin(
sin
n
n
c
o
1
2sinn
nc
Numerical aperture
Optical waveguides display 3 types of
dispersion:
These are the main sources of dispersion in
the fibers.
• Material dispersion: different
wavelength of light travel at
different velocities within a given
medium.
Due to the variation of n1 of the core wrt
wavelength of the light.
•Waveguide dispersion: β depends
on the wavelength, so even within
a single mode different
wavelengths will propagate at
slightly different speeds.
Due to the variation of group velocity wrt V-
number
t
Spread, ²
t0
Spectrum, ²
12o
Intensity Intensity Intensity
Cladding
CoreEmitter
Very short
light 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 in
a broadened output pulse.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
DL
• Modal dispersion, in waveguides with more than one propagating mode. Modes travel with different group velocities.
Due to the number of modes traveling along the fiber with different group velocity and different path.
Low order modeHigh order mode
Cladding
Core
Ligh t pulse
t0 t
Spread,
Broadened
light pulse
IntensityIntensity
Axial
Schematic illustration of light propagation in a slab dielectric waveguide. Light pulseentering the waveguide breaks up into various modes whic h 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)
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 a
function of free space wavelength,
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
mDL
2
2
d
nd
cDm
DL
2
2
2
2
22
984.1
cna
ND
g
PDL
pm DDDL
Material
Dispersion
Coefficient
Waveguide
Dispersion
Coefficient
Dp is called profile
dispersion; group velocity
depends on Δ
Core
z
n1x
// x
n1y
// y
Ey
Ex
Ex
Ey
E
= Pulse spread
Input light p ulse
Out put light pulse
t
t
Intensity
Suppose that the core refractive index has different values along two orthogonaldirections corresponding to electric f ield oscillation direc tion (polarizations). We cantake x and y axes along these directions. An input light w ill travel along the fiber w ith Ex
and Ey polarizations having different group velocities and hence arrive at the output at
different times
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Polarization Dispersion
Material and waveguide dispersion coefficients in anoptical fiber with a core SiO2-13.5%GeO2 for a = 2.5
to 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)
e.g.
For λ =1.5, and 2a = 8μm
Dm=10 ps/km.nm and
Dw=-6 ps/km.nm
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 = Dm + Dw
1
Dispersion coefficient (ps km-1 nm-1)
2
n
r
Thin layer of cladding
with a depressed index
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)
t0
Emitter
Very short
light pulses
Input Output
Fiber
Photodetector
Digital 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)
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)
0.5P
O'O
(a)
0.25P
O
(b)
0.23P
O
(c)
Graded index (GRIN) rod lenses of different pitches. (a) Point O is on the rod facecenter 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 fromthe rod face and the rays are collimated out.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
z
A solid with ions
Light direction
k
Ex
Lattice absorption through a crystal. The field in the waveoscillates the ions which consequently generate "mechanical"waves in the crystal; energy is thereby transferred from the waveto lattice vibrations.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Sources of Loss and Attenuation in Fibers
Absorption depends on
materials, amount of
materials, wavelength, and
the impurities in the
substances.
It is cumulative and depends
on the amount of materials,
e.g. length of the fiber optics.
d)1(
α is the absorption per unit length and d
is the distance that light travels
Scattered waves
Incident waveThrough wave
A dielectric particle smaller than wavelength
Rayleigh scattering involves the polarization of a small dielectricparticle 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 sothat a portion of the light energy is directed away from the incidentbeam.© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
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)
Attenuation in Optical Fiber
)(
)log(101
L
inout
out
indB
ePP
P
P
L
G. Keiser (Ref. 1)
34.4dB
top related