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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.


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.


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.


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.


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.


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


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.


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.


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


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


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.


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


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.


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,


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


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)


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.


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.


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


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.


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.


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.


Attenuation in Optical Fiber

)(

)log(101

L

inout

out

indB

ePP

P

P

L

G. Keiser (Ref. 1)

34.4dB

waveguides and optical fibers - concordia universityusers.encs.concordia.ca/~mojtaba/elec 425-6261...

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