Coherence & Optical FibresUnit-III (B.Tech-II sem)

BYDr Vishal Jain

COHERENCE

Two wave sources are perfectly coherent if they have a constant phase difference and the same frequency. Coherence is an ideal property of waves that enables interference. 

Coherent sources are those which emits light waves of same wave length or frequency and have a constant phase difference. 

Types of Coherence

Temporal Coherence• It is measure of ability of a beam to interfere

of another portion of it self

Spatial Coherence• It refer to ability of two separate portion of

wave to produce interference

Temporal Coherence Spatial Coherence

.

Temporal Coherence is concerned with the phase correlation of waves at a given point in space at two different instant of time.

Spatial coherence is concerned with the phase correlation of two different points across a wave front at a given instant of time

The type of coherence related with time The type of coherence related with position

It is known as longitudinal coherence. It is known as transverse coherence

The temporal coherence of light is related to frequency bandwidth of the source. monocromaticity

Spatial coherence is related to size of light source

. Temporal coherence measure in interferometer such as Michelson Interferometer

Spatial coherence measure in interferometer such as Young’s Double Slit Interferometer

.

Coherence time, Coherence Length and Spectral Purity Factor

The average time interval for which definite phase relationship exists in knowledge is known as coherence time

The distance L for which the wave field remains sinusoidal is given by coherence Length L

cL

Coherence Length In Terms of Frequency

1t

cL

c

----------------Eq 1

----------------Eq 2

----------------Eq 3

By Differentiate equation 3 we got

22

cc

& Spectral Purity Factor is given by

QSo

QccL

2

QL

cQ

cL ---------Eq 4

---------Eq 5

---------Eq 6---------Eq 7

---------Eq 8

Problems & SolutionsQ.1. Light of wavelength 4800Å has a length of 25 waves. What is the coherence length and coherence time.Sol:- Given Wavelength 4800Å so the coherence length L=25 X 4800 X 10-10 m

= 12 X 10-6m

So coherence Time ssm

mCL 14

8

6

100.4/103

1012

Q.2. Imagine that we crop a continuous laser beam ( to be perfectly mochromatic wavelength 6000Å) into 0.5ns pulse using some sort of shutter. Compute the coherence length, band width and the line width.Sol:- Given Wavelength 6000Å, time = 0.5 X 10 -9s,

m s m s c L15 . 0 / 10 3 10 5. 08 9

z Hs

9910 2

10 5. 01 1

ms m

H mc

z138

9 2 7 2

10 16 . 28/ 10 3

10 2 ) 10 5. 6(

Interference Visibility

A quantitative measurement of the coherence of a light source is equal to the visibility V of the fringes.The fringe visibility is defined as

minmax

minmax

IIIIV

---------- eq (1)

A Value V ranging between 0 to 1 if V = 1 implies very high contrast fringes, the fringes are washed away when V=0

A value greater than 0.88 indicates that light is highly coherent.

Interference Visibility as a measure of Coherence

Let us consider two interfering waves, each of intensity I0 . Both the waves consists of coherent part Ic and Incoherent part Iinc .

If C degree of coherence then.

oc CII

oinc ICI )1(

------------- eq . 1

------------- eq . 2

When superimposed. The coherence part shall interfere adding there by their amplitudes where as for the incoherent parts the intensities are simply added. Thus

incc III 24max

incII 20min

------------- eq . 3

------------- eq . 4

Putting the values from eq 1 and 2 to eq 3 and 4

oo ICCII )1(24max

oICI )1(20min

------------- eq . 5

------------- eq . 6

As we know interference visibility given as

CICICCIICICCI

IIIIV

oo

ooo

)1(2)1(24)1(2)1(24

minmax

minmax

Therefore it is seen that degree of contrast is a measure of degree of coherence between waves of equal intensities000

Optical Fibre (Introduction)

• An optical fiber is essentially a waveguide for light

• It consists of a core and cladding that surrounds the core

• The index of refraction of the cladding is less than that of the core, causing

rays of light leaving the core to be refracted back into the core

• A light-emitting diode (LED) or laser diode (LD) can be used for the source

• Advantages of optical fiber include:

– Greater bandwidth than copper

– Lower loss

– Immunity to crosstalk

– No electrical hazard

Optical Fiber Construction

Daniel Colladon first described this “light fountain” or “light pipe” in an 1842 article titled On the reflections of a ray of light inside a parabolic liquid stream. This particular illustration comes from a later article by Colladon, in 1884.

A wall-mount cabinet containing optical fiber interconnects. The yellow cables are single mode fibers; the orange and aqua cables are multi-mode fibers: 50/125 µm OM2 and 50/125 µm OM3 fibers respectively.

Optical Fiber• Optical fiber is made from thin strands of either glass or plastic

• It has little mechanical strength, so it must be enclosed in a

protective jacket

• Often, two or more fibers are enclosed in the same cable for

increased bandwidth and redundancy in case one of the fibers

breaks

• It is also easier to build a full-duplex system using two fibers,

one for transmission in each direction

Optical Fibre Working Principle• Optical fibers work on the principle of total internal reflection• With light, the refractive index is listed• The angle of refraction at the interface between two media is governed by

Snell’s law:

n1 sin1 n2 sin2

Types of Fiber on the basis of modes of propagationIn digital multimode fiber systems, a light pulse separates into multiple spatial paths or modes. Each component reaches the receiver at a slightly different time, broadening the received pulse. Single-mode fiber solves the differential mode delay problem, allowing data rates to be increased until chromatic dispersion.

Single mode fiber has a lower power loss characteristic than multimode fiber, which means light can travel longer distances through it than it can through multimode fiber. Within a data center, it's typical to use multimode which can get you 300-400 meters. If you have very long runs or are connecting over longer distance, single mode can get you 10km, 40km, 80km, and even farther.

The multi-mode fiber has has much larger core diameter than single mode fiber.The core diameter of multimode fiber is typically 50–100 micrometers, while that of single mode fiber is between 8 and 10.5 micrometers.

Modes and Materials

• Since optical fiber is a waveguide, light can propagate in a number of modes

• If a fiber is of large diameter, light entering at different angles will excite

different modes while narrow fiber may only excite one mode

• Multimode propagation will cause dispersion, which results in the

spreading of pulses and limits the usable bandwidth

• Single-mode fiber has much less dispersion but is more expensive to

produce. Its small size, together with the fact that its numerical aperture is

smaller than that of multimode fiber, makes it more difficult to couple to

light sources

Types of Fiber on the basis on Index • In step-index fibers the index of refraction changes radically between the core

and the cladding. • Graded-index fiber is a compromise multimode fiber, but the index of

refraction gradually decreases away from the center of the core• Graded-index fiber has less dispersion than a multimode step-index fiber

Numerical Aperture and Angle of Acceptance

• The numerical aperture of the fiber is

closely related to the critical angle and

is often used in the specification for

optical fiber and the components that

work with it

• The numerical aperture is given by the

formula:

• The angle of acceptance is twice that

given by the numerical aperture

22sin.. cladcoreoiAN

i

r

P R

Q

Incident ray

Launching end

Total Internally reflected ray Axis of

Fibreµ core

Cladding µclad

Normal

Reflected ray

µ o

θ

Numerical Aperture and Angle of AcceptanceConsider a cylindrical fibre wire which consists of an inner core of refractive index µcore and an outer cladding of refractive index µclad.

Let µo. be the refractive index of the medium from which the light ray enters the fibre. This end is known as launching end. Let a ray of light enters the fibre at an incidence angle of i. to the axis of fibre as shown in figure. This ray refracted at an angle r and strikes the core-cladding interface at an angle θ. Let θ is greater than critical angle θc. As long as the angle θ is greater than critical angle θc, the light will stay within the optical fibre.

Now we shall calculate the angle of incidence i for which θ is greater than and equal to θc So that the light remains within the core.Applying the snell’s law of refraction at the point of entry P.

ri coreo sinsin ----------- eq 1

From triangle PQR it is seen thator 90 , or 90

Or cos)90sin(sin or ----------- eq 2

Substituting the value of sin r from equation 2 to eq 1

cossin coreo i ----------- eq 3

cossino

corei

If Incidence angle i is increased beyond a limit , θ will drop below the critical value θc and the ray will escape from the side walls of the fibre. The largest value of i (imax) occures when θ = θc.

----------- eq 4

So the eq 4 can be written as

co

corei

cossin max ----------- eq 5

Applying the snell’s law of refraction at the core cladding interface

ocladccore 90sinsin

core

cladc

sin

2

2

1coscore

cladc

----------- eq 6

So the eq 5 can be written as

2

22

2

2

max 1sino

cladcore

core

clad

o

corei

NAi cladcore 22maxsin

221max sin cladcorei

Here imax is the angle of acceptance

So the angle of acceptance is defined as the maximum angle that a light ray can have relative to the axis of the fibre and propagate down the fibre.

Numerical Aperture:- It is also known as figure of merit for optical fibre. It is defined as sine of acceptance angle. NA= io = imax

Propagation Condition

NA in terms of Fractional Refractive index Δ

If i is the angle of incidence of an incident ray, then the ray will be able to propagate only if i < io or sin i < sin io

or 22sin cladcorei

core

cladcore

The fractional refractive index change Δ is defined as the ratio of refractive index difference between core and cladding to the refractive index of core. It is expressed as

corecladcore

core

cladcorecladcorecladcorecladcoreNA

2

2)()())((22

corecladcoreNA 2

2)(

Or if

corecladcore

2

)(

22 2corecoreNA

than

V- NumberThis is an important parameter of optical fibre given by the relation

222cladcore

aV

Where a is the radius of the core and λ is free space wave length.

The maximum number of modes (Nm) supported by a single mode step index fibre is determined by.

2

21 VNm

If V< 2.405, the fibre will support only one mode and known as single mode optical fibre If V>2.405, the fibre will support many modes simultaneously. This is known as multi-mode fibre The wavelength corresponding to the value V=2.0405 known as cutoff wavelength this is expressed as

405.2V

c

Problems and Solutions

Q.1. Calculate the refractive indices of the core and cladding material of a fibre from the following data NA=0.22 and Δ=0.012Sol:- Given NA=0.22 and Δ=0.012

We know the relation

22 2corecoreNA

42.1012.02

22.02

NA

core

core

cladcore

42.1

42.1012.0 cladso

40.1clad

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