optical fibres - dispersion part 2 [group velocity ... - ucy€¦ · ece 455 – lecture 06 6 •...

ECE 455 – Lecture 06 1

Optical Fibres

- Dispersion Part 2

[Group Velocity Dispersion]

• HMY 445

• Lecture 06

• Fall Semester 2016

Stavros IezekielDepartment of Electrical and

Computer Engineering

University of Cyprus

ECE 455 – Lecture 06

CHROMATIC DISPERSION – WHAT

WE KNOW SO FAR

2


© U

. of W

ashin

gto

nIf we have a pulse of light which is not monochromatic (it contains a group of

wavelengths), then we will have chromatic dispersion:

Optical fibre

z

Intramodal (chromatic) dispersion

Material dispersion Waveguide dispersion

Due to nonlinear

relationship between

n and λ

Due to refractive

index profile of the

fibre. Changes with

λ.

• Dispersion due to fact that group

velocity changes with wavelength


dispersion no⇒= pg vv

dispersion⇒≠ pg vv

kvp

ω= Phase velocity

dk

dvg

ω= Group velocity

k

ω

dispersion normal⇒

< pg vv

pg vv =

dispersion anomalous⇒

> pg vv


GROUP VELOCITY DISPERSION

5


• So far we have considered just two, very closely spaced frequencies within the

group emitted by an optical source such as a laser:

Wavepackets

• If we consider the entire spectrum emitted by the source, we still obtain a

modulated waveform, with a group velocity and phase velocity as before.

ω0

Intensity

(arbitrary units)

ω

2 δω

Wave packet –

A short pulse composed of

the sum of waves over a

finite bandwidth.

We now look at the whole

spectrum.


© U

CS

D


• We can prove the properties of the wavepacket by using the Fourier transform:

∫∫∞

∞−

∞

∞−

− =↔= dtetfFdeFtftjtj ωω

πωωω )(

2

1)()()(

Time domain Frequency domain

ω0

ωω0 - δω ω0 + δω

F(ω)This represents optical source

spectrum; has a gaussian

profile

peak frequency

(12)

Fourier


• We can think of F(ω) as being equal to some spectrum G(ω) which is identical in

shape, but centred at ω = 0 instead of ω0:

• By inspection, )()( 0ωωω −=GF

∫∞

∞−

−=− dtetgGtj )(

00)(

2

1)(

ωω

πωω

∫∞

∞−

−= dteetgtjtj ωω

π0)(

2

1∫∞

∞−

=

dtetf

F

tjω

π

ω

)(2

1

)(

ω0ωω0 - δω ω0 + δω

F(ω)G(ω)

0- δω δω

(13)

(14) (15)


tjetgtf 0)()(

ω−=• Hence:

Corresponds to sinusoid

at optical frequency ω0

Impulse response of:

G(ω)

0

-1,5

-1

-0,5

0

0,5

1

1,5

0 0,5 1 1,5 2

gives:g(t)

0N.B. Fourier transform of a gaussian

pulse is also gaussian in shape

(16)


• In other words, the impulse response associated with the optical source takes on the

form of a wavepacket:

g(t)

f (t)

t

• This wavepacket represents a pulse of light emitted by the optical source, and it contains

a range of frequencies (i.e. wavelengths).

• We now need to examine what will happen to the group velocity of this pulse as it

propagates along a fibre.


• Consider an optical pulse launched into a single mode fibre. Due to the spectral width

of the source, this pulse consists of a group of wavelengths which travel at the group

velocity:

optical power

wavelength λ

distance

dk

dvg

ω=

λ0


• So the time taken for the wavegroup to travel a distance L along the fibre is given by the

group delay τg:

ωτ

d

dkL

v

L

g

g ==

• The phase velocity of the peak wavelength λ0 is given by:

c

nk

n

c

kvp

ωω=⇒==

• Substituting (18) into (17):

+==ω

ωω

τ

d

dnn

cd

dk

L

g 1

(17)

(18)

(19)


• Eqn. (3) shows that the group delay per unit length depends on both n and dn/dω. It is

also dependent on the frequency ω. However, we prefer to work with wavelength λinstead:

λ

n

• From the inverse relationship between frequency and wavelength (c = fλ = ωλ/2π), we

find that:

−=

+=λ

λω

ωτ

d

dnn

cd

dnn

cL

g 11

ω

n

instead of....

(20)


−=

+=

+=∴

λλ

ωλ

λω

ωω

τ

d

dnn

c

d

d

d

dnn

c

d

dnn

cL

g

1

1

1

Derivation of equation (20):

ωλ

ωπ

ωλ

ωπ

λλλ −=−=⇒=⇒=⇒=2

22 c

d

dc

f

cfc


Group Refractive Index

n

c

kvp ==

ω

g

gn

c

dk

dv ==

ω

• Imagine we have a fibre with core refractive index n. In this case,

• If we transmit a spread of wavelengths, then we can regard the resulting group as

encountering a group refractive index, and this is defined via:

g

gv

cn =

(21)

(22)

(23)


λλ

τ

d

dnn

Lcn

g

g −==∴

• n varies with wavelength:

dispersionvvnnd

dngg ⇒≠⇒≠⇒≠ 0

λ

• In fact, ng will also be wavelength dependent:

2

2

λλ

λ d

nd

d

dng −=

(24)

(25)


λ

n

ng

1.31 µm

Point of inflection

2

2

λλ

λ d

nd

d

dng −=

02

2

=λd

nd

Minimum 0=λd

dng

For silica glass: At 1.31 µm, n has

a point of inflection, ng is

minimum, and the group velocity

is therefore maximum.


Group velocity dispersion (GVD)

• We know that:

• An optical source emits a spread of wavelengths centred on λ0.

• This can be represented by a wavepacket which travels at the group velocity

and therefore “sees” a group index ng.

• However, ng and thus the group velocity vg and delay τg are all wavelength

dependent.

• Each different spectral component emitted by the source will travel at

different group velocities, and this GVD is the cause of material dispersion.


λ

=

c

n

L

g

gτ

λ0 λ0 + δλ

)(1

0λτ gL

)(1

0 δλλτ +gL

• Consider the delay difference (per unit length) for a wavelength δλ away from the

central wavelength λ0:

• If the wavelength difference is sufficiently small, we can neglect second-order

terms in a Taylor series expansion to get:

0

)()( 00

λλ

τδλλτδλλτ

d

d

LL

ggg =−+

gτδ

δλ

(26)


• Consider the delay difference (per unit length) for a wavelength δλ away from the

central wavelength λ0:

0

11

λλ

τ

δλ

δτ

d

d

LL

gg =

−=λ

λτ

d

dnn

cL

g 1

• From (20):

2

21

λλ

δλ

δτ

d

nd

cL

g −=∴

Material

dispersion

Dmat

Units: ps/(nm.km)

(27)


2

2

λλ

d

nd

cDmat

−=

LDmatmat λσσ =

spread

in time

(ps)

spread in

wavelength (nm)

(11)

Group index, refractive index

and material dispersion for

silica glass (SiO2)

material dispersion

(ps/nm-km)length (km)


DISPERSION MANAGEMENT

23


Dispersion modified fibres

• For conventional single-mode optical fibre:

– minimum attenuation occurs at 1.55 µm

– minimum dispersion occurs at 1.3 µm

• Furthermore, optical amplifiers operate in the 1.55 µm

region

• In response to this, dispersion modified fibres have been

developed to provide minimal dispersion at 1.55 µm


• Structure dependent losses (waveguide losses) have little

effect on overall attenuation, so changing the refractive

index profile in single-mode fibre will have negligible impact

on attenuation.

– However, changing the refractive index will modify the

waveguide dispersion term, and this can be used to our

advantage.

• In fact, the refractive index profile can be tailored to shift

the dispersion zero to 1.55 µm or to flatten the dispersion

vs. wavelength profile so that dispersion is almost zero

between 1.3 µm and 1.55 µm


Changing the refractive index profile changes the waveguide

dispersion:

Dispersion

shifted

Dispersion

flattened


Dispersion

shifted

Dispersion

flattened

optical fibres - dispersion part 2 [group velocity ... - ucy€¦ · ece 455 – lecture 06 6 •...

Documents