Optoelectronics and optical communication

(FFFN25, FYST50)

Week 3: Guided-wave optics

Fiber optics

Cord Arnold

Cord.Arnold@fysik.LTH.se

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