Optoelectronics and optical communication
(FFFN25, FYST50)
Week 3: Guided-wave optics
Fiber optics
Cord Arnold
Summary Dielectric Waveguides
No fundamental mode cutoff !
Dielectric waveguides 2D waveguides Coupling in WG
Fiber optics introduction
• Light is guided in the core of optical
fibers by total internal reflection (TIR)
• TIR enables low loss propagation over
large distance
• Light propagates inside the fiber in
form of modes
• Optical fibers can be classified as
single mode and multimode fibers
• Information transfer rate of multimode
fibers is limited by modal dispersion
• The modal dispersion is greatly
reduced in graded index (GRIN)
fibers
Optical fiber
Typical dimensions 2a/2b [μm]/[μm]
8/125 ; 50/125 ; 62.5/125 85/125 100/140
Materials:
Core: fused silica SiO2
Cladding: fused silica SiO2 co-doped
with Ti, Ge, or B
Refractive index change:
n1 = 1.44 … 1.46
Δ = 0.001 … 0.02
Fractional refractive index
Guided rays in step index fibers
Meridional rays
The trajectory of meridional rays lie in
planes that pass through the axis of
the fiber. The ray is guided if θ:
Skewed rays
Skewed rays lie in planes shifted from the
fiber axes by a distance R. The rays are
identified by the angles θ and φ. The ray
trajectory is confined within a cylindrical
shell with an inner and outer radius R
and a, respectively. The propagation
condition is the same as for meridional
rays: Condition for guidance
Numerical aperture
Graded index fiber
p - Grade profile parameter
Lower order modes:
θ ↓ Path ↓ nav ↑ v↓
Higher order modes:
θ ↑ Path ↑ nav ↓ v ↑
Average
propagation speed
is the same for both
low and high order
modes
Guided waves
Each component of monochromatic EM
field (Er, Eφ ,Ez, Hr, Hφ ,Hz,) in a fiber obeys
Helmholtz equation:
In cylindrical coordinates:
Solution is product of three terms:
Radial variation
Azimuthal variation
Axial variation
Equation for radial profile u(r)
Axial propagation of the wave is accounted by
Azimuthal variation (i.e. with φ) is periodic:
β –propagation constantwhere
Helmholtz equitation in step-index fibers
1,core kar << β
2,cladding kar >> β
Fiber V parameter
Normalization for kT and γ :
Guidelines for solving Helmholtz equation:
Guiding X < V (i.e. kT<k0NA)
Fiber V parameter
•Maxwell equations:
Each of EM field components obey:
(5.3-12)
(5.3-13)
•Boundary conditions:
are continuous at r=a
• Scaling factors for the field components
- Field distributions
• Characteristic equation (i.e. dispersion
relation) for β
- For each index l several solutions m
are obtained
- βlm, kTlm, γlm, ulm(r)
l- azimuthal index (l = 0,1,2,…)
m-radial index (m= 1,2,3…)
Characteristic equation for weakly guiding fibers
Weakly guiding fibers:
�Paraxial rays :
�Propagating wave: TEM
Propagating modes: Linearly Polarized LPlmWith two orthogonal polarizations (x and y)
Characteristic equation for LPlm modes:
(9.2-11)
(9.2-14)
Intensity distribution of (a) LP01 and (b) LP34
LHS RHS
The graphical/numerical solution of the
characteristic equation yields Xl,m, ϒl,m,
βl,m, and ul,m(r,φ).
Finding the modes, example for V=10
( )
( )
( )
( )yK
yKy
xJ
xJx
l
l
l
l
0
1)0(
0
1)0(
=
+=
=
+==
l=0
Zeros of J1, J-1
LP01, LP02, LP03
V
Bessel functions of the first kind
( )
( )
( )
( )
( )
( )
( )
( )yK
yKy
xJ
xJx
yK
yKy
xJ
xJx
l
l
l
l
l
l
l
l
1
1)1(
1
1)1(
1
1)1(
1
1)1(or
=
+=
=
+=
−=
+−=
−=
+−===
l=1
Zeros of J0
LP11, LP12, LP13
V
( )
( )
( )
( )
( )
( )
( )
( )yK
yKy
xJ
xJx
yK
yKy
xJ
xJx
l
l
l
l
l
l
l
l
2
1)2(
2
1)2(
2
1)2(
2
1)2(or
=
+=
=
+=
−=
+−=
−=
+−===
l=2
Zeros of J1, J-1
LP21, LP22
V
( )
( )
( )
( )
( )
( )
( )
( )yK
yKy
xJ
xJx
yK
yKy
xJ
xJx
l
l
l
l
l
l
l
l
3
1)3(
3
1)3(
3
1)3(
3
1)3(or
=
+=
=
+=
−=
+−=
−=
+−===
l=3
Zeros of J2, J-2
LP31, LP32
V
( )
( )
( )
( )
( )
( )
( )
( )yK
yKy
xJ
xJx
yK
yKy
xJ
xJx
l
l
l
l
l
l
l
l
4
1)4(
4
1)4(
4
1)4(
4
1)4(or
=
+=
=
+=
−=
+−=
−=
+−===
l=4
Zeros of J3, J-3
LP41, LP42
V
6.38 9.76
2.405 5.52 8.65
3.832 7.016 10.17
5.14 8.42
Modes of Optical Fibers
LP01
Y (vertically)
polarizedX (horizontally )
polarized
+ two orthogonal polarizations:
l=0, m=1 LP11
l=1, m=1
Number of modes and mode cutoffFrom graphical solution of characteristic
equaWon: V↑ M ↑
Second mode cutoff ↔ single mode condition
l\m 1 2 3 4
0 0 3.832 7.016 10.17
1 2.405 5.52 8.65 11.79
2 3.832 7.016 10.17 13.32
3 5.14 8.42 11.62 14.8
4 6.38 9.76 13.02 16.22
For each l there are as many modes m as Jl+1(x) has
roots in the interval 0<x<V.
a
c
a
aV
c
61.2NA
1
NA405.2
2
405.2NA2
0c
0
<⇔
>⇔
<=
ν
πλ
λπ
cc v νλλ <>⇒ or
The fiber is single mode
Mode quiz
LPlm
Mode quiz
l=0, m=1 l=1, m=1 l=2, m=1 l=0, m=2
l=3, m=1 l=1, m=2 l=4, m=1 l=2, m=2
l=0, m=3 l=5, m=1 l=3, m=2 l=1, m=3
V>0 V>2.405 V>3.832 V>3.832
V>5.14 V>5.52 V>6.38 V>7.016
V>7.016 V>7.588 V>8.42 V>8.65
2 4 4
4 4 4 4
4 4 4
2
2
Degeneracy
http://www.rp-photonics.com/passive_fiber_optics.html
Propagation constants and group velocities
(large V)
Propagation constants: Group velocity
Δ↑ NA↑ � easy to couple light
Δ↑ Δν↑ � light pulses spread � difficult
transmit information at high rates
2
2
ml,2
0
2
1
2
T
2
0
2
1ml,a
Xknkkn −=−=β
Single mode fibers
Effective refractive index n(V)
for the fundamental mode
Dispersion relation ω(β01)
Supports only one, fundamental LP01
mode
☺☺☺☺Advantages:
• No multi-modal dispersion
• No modal noise
• Lower losses
High information
transmission rates
over long distances
���� Disadvantages:
Difficult to couple � Higher
tolerances � Higher price for
telecommunication components
λ1<λ2n1>n2a1>a2
Small core, small NA, or large
wavelength.
Close to
second
mode.
Far from
second
mode
Practical example:
Corning ® SMF 28 Single-Mode Optical Fiber
Core diameter: 8.2 μm
Cladding diameter: 125 μm
Coating diameter: 245 μm
Refractive index difference: 0.34%
Effective group refraction index:
@ 1310 nm 1.4677 ( SiO2 1.4468)
@ 1550 nm 1.4682 ( SiO2 1.4440)
Numerical aperture @1310nm: 0.14
→Acceptance angle 0.14 rad / 8о
Cutoff wavelength: 1260nm
Mode field diameter:
@ 1310 nm :9.2±0.4μm
@ 1550 nm :10.4±0.8μm
Wavelength Attenuation dB/km
850 nm 1,81
1300 nm 0,35
1310 nm 0,34
1383 nm 0,5
1550 nm 0,19
Polarization maintaining fibers
Types of polarization maintaining fiber
• PANDA Polarization-maintaining And
Absorption reducing Fibers
• Elliptical clad
• Bow-tie
y
x
• In conventional single-mode fiber (SMF) LP01
mode has two orthogonal polarizations (x,y),
i.e. SMF two orthogonal polarization modes.
• Since polarization mode dispersion (PMD) is
vanishingly small uncontrolled power transfer
between two linear polarizations may occur.
=> Elliptical polarization
• Breaking circular symmetry of conventional
SMF enables two polarization modes propagate
at different speeds and become uncoupled.
Fiber connectors• SMA – typical (in lab applications)
multimode fiber connector
• ST- One of the most commonly used
fiber optic connectors in networking
applications. For both short distances
applications and long line systems.
• FC/PC - Widely used precise
(single/multimode ) fiber optic
Connector
• SC -Used frequently for newer network
applications. Square, keyed connector
with push-pull mating, 2.5mm ferrule
and molded housing for protection.
Insertion loss
dB
Back
reflection dB
FC/PC < 0.3 < 40
FC/APC “angle
polished”< 0.25 < 60
SMA 0.5 … 1
Summary
SM:
λ1<λ2n1>n2a1>a2
LP11y
x
End of lecture