Chapter 7 Interference of Light - Erbion

Chapter 7Interference of Light

Lecture Notes for Modern Optics based on Pedrotti & Pedrotti & Pedrotti

Instructor: Nayer EradatInstructor: Nayer EradatSpring 2009

4/12/2009 1Interference of Light

Interference of light

Interference phenomenon depends on superposition of twodepends on superposition of two or more individual waves under strict conditions.

Constructive interference leadsConstructive interference leads to enhancement of the resulting amplitude with respect to that of the constituents.

Destructive interference leads to diminution of the resulting amplitude with respect to that of p pthe constituents.

Interference produces an alternating spatial pattern of g p pfringes.

4/12/2009 2Interference of Light

Two‐beam interference: vector treatment

( )1 01

Interference of the plane waves of same frequency arriving at point P:

cos ks tω φ= − +E E ( )( )

1 01 1 1

2 02 2 2

1 2

cos

cos

The combined disturbance at point P is:

are rapidly verying function

ks t

ks t

s

ω φ

ω φ

= +

= − +

= +

E E

E E

E E E

E 15s in time with frequencies of 10 and average to zero very quicklyHz are rapidly verying functionsE

( ) 22

s in time with frequencies of ~10 and average to zero very quickly. Irradiance or radiant power density is a measure of time-average of the square of the field (detector reading)

/ et

Hz

I E W cm E= ∝ -151 in practice t averaging time for the eye >>T(period 10 s)30

s⎛ ⎞⎜ ⎟⎝ ⎠

( ) ( )( )

1 2 1 20 0

1 1 2 2 1 20 2

I c c

I c

ε ε

ε

= = + +

= + +

E E E E E E

E E E E E E

i i

i i i

1 2 12

1 21) if and are orthogonal no inte

I I I I= + +

E E

1 2

rference happens. cos 0

2) if and are parallel interference has i ff t 1

θ

θ

=

E E

( )

1 2

01 0212 0

maximum effect cos 1

3) if and have angle the interference term is:

I cos

h i

c

k

θ

δ

ε δ

δ φ φ

=

=

E E

E Ei

th h diff b t th

4/12/2009 Interference of Light 3

( )2 1 2 1where is k s sδ φ φ= − + − the phase difference between the waves. For a purely monochromatic wave is independent of time (a constant phase relationship between two waves). δ

Two‐beam interference

( )

1 2 1 2

2 2

Interference of the plane waves of same frequency arriving at point P: 2 cos

1 1 h

I I I I I

k

δ

δ δ δ φ φ

= + +

( )2 201 021 0 01 1 0 02 12 0 1 2 2 1 2 1

1 1; ; I cos 2 cos where 2 2

For

I cE I cE c I I k s sε ε ε δ δ δ φ φ= = = = = − + −E E

mutually incoherent field

i

and low quality laser sources the phases are random functions of time that

very on time scales much larger than the E field oscillations but shorter than the detector averaging time.

s

The interferenc

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

1 2

e term in this case will be:

2 cos 0 for all incoherent sources and low quality lasers and

Light from independent sources even if they are same kind of lasers do not interfere.

I I k s s t t

I I I

φ φ− + − =

= +1 2

( )( ) ( )2 1

For light from same lase split and recombined the phase difference at detector

0 if the light paths are equal and constant and the interference term is:

2 cos

t t

I I k s s

φ φ δ− = =

−

mutually coherent beams

( ) ( ) ( )( ) ( )( )2 cost t I I k s sφ φ+ − = −1 2 22 cosI I k s s( ) ( ) ( )( ) ( )( )( ) ( )

1 2 1 1 2 2 1

2 1

2 cos

Even if travel distance difference is nonzero 0 so long as The coherence time of a source is inversely proportional to the range of the frequency components

c

t t I I k s s

t t t t t

φ φ

φ φ δ δ

+ =

− + ≅ <<

that

10

0

1make up the electric field .

Coherence length is the distance that electric field travels in coherence time:

For mutually coherent sources we assume the length traveled from

c

c t c

t

L l c ct

τν

τ

= =Δ

= = =

the sources is much shorter than the


For mutually coherent sources we assume the length traveled from

1 2 1 2

the sources is much shorter than the

coherence length, then the irradiance of the combined fields is: 2 cos .I I I I I δ= + +

Two‐beam interference

1 2 1 22 cos

is the total phase difference between the beam from the point of separation It can be due to

I I I I I δ

δ

= + +

is the total phase difference between the beam from the point of separation. It can be due to a) path length differenceb) phase shift at reflecting beam splittersc) differing indeci

δ

es of refraction in the separate pathsc) differing indecies of refraction in the separate paths depanding on the sign of the cos the interference will be constructive (+) or destructive ( ). A pattern of

alternating maxima and minima (fringes) will

form at

δ −

the plane of observation due to varying δform at

( )( )

2max 1 2 1 2 01 02

2

the plane of observation due to varying at different locations of the observation screen.

2

2

I I I I I E E

I I I I I E E

δ

⎧ = + + = +⎪⎨⎪ ( )min 1 2 1 2 01 02

1 2 0 max 0 min

2

If 4 and 0Fring visibility i

I I I I I E E

I I I I I I

= + − = −⎪⎩= = → = =

s a measure of fringe contrast

max min

max min

and is defined as:

To ensure maximum visibility in experimental tti k th lit d

I IFringe visibilityI I

−=

+


settings, we make sure the amplitudes are samein both arms when they arrive at the screen.

Conservation of energy

( )2

Rewriting the irradiance for the interference pattern of two equal amplitude components, we get:

( )20 0 0 0

2

2 cos 2 1 cos

Using 1 cos 2cos we get2

I I I I Iδ δ

δδ

δ

= + + = +

+ =

204 cos

2The energy is not conserved at each poi

I I δ=

0nt of the interference pattern 2but it is conserved over one spatial period 2

I II I

≠

= 0but it is conserved over one spatial period 2 .So inteference causes redistribution of the energy of sources over the fringe pattern.

The frine analysis perform

avI I=

ed here was for plane harminc waves but the results areThe frine analysis performed here was for plane harminc waves but the results are general and hold for spherical and cylindrical waves. The only difference is that thesewaves do not have constant amplitudes and irradiances as they propagate.


Young’s (1802) double slit experiment

To demonstrate the interference pattern and fringes Young generated two coherent point sources by splitting the wavefront of a wave generated by a point source to two in-phase point sources. Today we uese a laser as the first source. We can assume that the amplitudes of the splitted waves are equal. Our goal is to develope an expression for the irradiance at an arbitrary point, on a P

screen at a

( )20

distance from the sources. is the separation of the sources. The irradiance at is:4 cos / 2 where is the phase difference of the waves arriving at

In this geometry phase difference is

L a PI I Pδ δ=

caus

2 1

ed by the optical path difference

sinThe phase difference associated

s s a θΔ = − ≅

( ) ( )2 1

2

pwith is

2 /k s sδ π λ

π

Δ

= − = Δ

Δ⎛ ⎞20

20

4 cos

sin4 cos

I I

aI I

πλ

π θλ

Δ⎛ ⎞= ⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠


λ⎝ ⎠

Young’s (1802) double slit experiment2

0sin4 cos aI I π θλ

⎛ ⎞= ⎜ ⎟⎝ ⎠

( )constructive interference sin , 0, 1,...destructive interference sin 1/ 2 , 0, 1,...

a m ma m m

y ay

θ λθ λ

π

→ = Δ = = ±⎧⎪⎨ → = Δ = + = ±⎪⎩

⎛ ⎞20

,ma

For sin tan then 4 cos

bright fringe m

y ayy L I IL L

ay m yL

πθ θλ

π πλ

⎛ ⎞<< → = ≅ ≅ ⎜ ⎟⎝ ⎠

= → x , 0, 1,...m L maλ⎧ = = ±⎪

⎪ Lλ( )

,min

1/ 21dark fringe ,2

0 1

m

am Lay m y

L am

λπ πλ

⎪⎪

+⎪ ⎛ ⎞= + → =⎨ ⎜ ⎟⎝ ⎠⎪

⎪ = ±0, 1,...

Fringe separtion (peak to peak):

m

Lλ

⎪ ±⎪⎩

maxima minima 1 is constant for all fringes.

The minima are situated mi

m mLy y y y

aλ

+Δ = Δ = − =

dway between the maxima.We can design experiments with this technique to measure the


We can design experiments with this technique to measure the wavelength or width of the very tiny slits.

Formation of the fringes with analysis of the crests and valleys of the spherical waves from two sources

maximaCrests of both sources

minimaOne valley and one crest


The bright fringe surfaces for two coherent point source generated by rotating the pattern around the x axisg y g p


Double‐slit experiments with virtual sources ILloyd’s mirrory

The interference pattern is generated by superposition of the light from the actual source S and the virtual source S’ that is image of the S on the mirror MM’


Double‐slit experiments with virtual sources IIFresnel’s mirrorFresnel s mirror

The interference pattern is generated by superposition of the light from the two virtual source S1 and S2 that are images of the S on the mirror M1 and M2


Double‐slit experiments with virtual sources IIFresnel’s biprismsFresnel s biprisms

The interference pattern is generated by superposition of the light from the two virtual source S1 and S2 that are formed by refraction in the two halves of the biprism. p


Interference in dielectric filmsDielectric here means index of refraction is real and positive. No considerable absorption hapens at the wavelength of interesthapens at the wavelength of interest.To create two coherent sources from one we can use two techniques:1) Amplitude division. For example by partial reflection at two different interfaces.

2) Wavefrot division. Fro example placing two slits in front of the waverfront. A transparent film bounded by parallel p

( )lanes divides a beam of light into two parts.

amplitude division some of the light is reflected from the first interface and some from the second interface. Two reflected portions from first and second planes can be brought together by a lens to interfer at a point on a screen.


Thin film interference ( ) ( )0The path difference is: fn AB BC n ADΔ = + −

⎡ ⎤( ) ( ) ( )0

0 it can be shown

0Snell's law: sin sin

f f

i f t

n AE FC n AD n EB BF

n nθ θ=

⎡ ⎤Δ = + − + +⎣ ⎦

=

( )0

sin / 2 sin

2 sin sin2sin sin

t t

t t

i i f

AE AG AC

AE AC nAE AD ADAD AC n

θ θ

θ θθ θ

= =

= ⎫= =⎬= ⎭

0 2f

fn AD AEn n⎭

= = ( ) ( ) ( )( )

0 0

2 2 cos this can be related to the angle of inceidence

f f

f f f t

AE FC n AE FC n AD

n EB BF n EB n t θ

⎡ ⎤+ → + − =⎣ ⎦

Δ = + = → Δ =

through the Snell's law. For normal incidence 2 .

The phase difference corresponding to the path fn tΔ =

difference is: 2 / .

The net phase difference has to take into account the possible phase change by the reflection.

kδ π λΔ = Δ =

P r

p p p g y optical path difference, equivalent path difference arrising from the phaΔ Δ

( )

se change.

Constructive interference: where 0,1, 2,...

D t ti i t f 1/ 2 h 0 1 2P r m mλ

λ

Δ + Δ = =

Δ + Δ +


( )r

Destructive interference: 1/ 2 where 0,1, 2,...

What is the value of for dirrerent types of interface?P r m mλΔ + Δ = + =

Δ

Reflected wave amplitudeReflected wave amplitude

Amplitude of the reflected electromagnetic wave at an interface

a bn nE E−na nb

Er

( ) i

Three cases are recognized:1) If n external reflection then E and E have different

a br i

a b

b

E En n

n

=+

< ( )a r i1) If n external reflection then E and E have different

signs so the reflected wave has bn<

( )

half-cycle phase difernce

with respect to the incident wave.2) If n internal reflection then E and E have the samen> ( )a r i

a

2) If n internal reflection then E and E have the same

signs so the reflected wave and incident wave are in-phase.

3) If n then

b

b r

n

n E

>

= 0 no reflection happens.= Ei

Amplitude of reflected EM wave from f a b

r ia b

n nE En n−

=+

an interface

Condition for constructive and destructive i f b hi filinterference by a thin film

Constructive: 0 2 cos 2 cosn t m n t mθ λ θ λΔ + Δ = + = → =No relative phase shift at interfaces :

r

r

Constructive: 0 2 cos 2 cos

1Destructive: 0 2 cos2

p f t f t

p f t

n t m n t m

n t m

θ λ θ λ

θ λ

Δ + Δ = + = → =

⎛ ⎞Δ + Δ = + = +⎜ ⎟⎝ ⎠

0, 1, 2, 3,...

i

m =Half - cycle relative phase shift at interface :

1λ θ λ θ λ⎛ ⎞Constructiv r1e: 2 cos 2 cos

2 2

1for normal incidence 22

p f t f t

f

n t m n t m

n t m

λ θ λ θ λ

λ

⎛ ⎞Δ + Δ = + = → = +⎜ ⎟⎝ ⎠

⎛ ⎞= +⎜ ⎟⎝ ⎠

r

21Destructive: 2 cos 2 cos

2 2

f

p f t f tn t m n t mλ θ λ θ λ

⎜ ⎟⎝ ⎠

⎛ ⎞Δ + Δ = + = + → =⎜ ⎟⎝ ⎠

for normal incidence 2

0, 1, 2, 3,...fn t m

m

λ=

=

Antireflecting coatingFor antireflecting coatings we want the reflected wave to vanish.

r

s

1if then the reflected wave is out of phase and desctructive interference will happen. 2

Usually n so both reflec

p

f a

m

n n

λ⎛ ⎞Δ + Δ = +⎜ ⎟⎝ ⎠> > rtons are external and no phase shift happens at the interfeces 0. Δ =

To minimize the film thickness and loss we take 1 and for one layer antireflection coating

2 this is the so-called qu2 4p

m

t tλ λ

=

Δ = = → = arter wavelength layer.

This works only for one wavelength. In order to make it for a brader spectrum designers use many layers.An antireflection coating is perfect if amplitudes of the waves coming from both surfaces are equal.

To achieve that we need a special relationship between the indexes of the layers. The reflection coefficient 1 /2 1

2 1

1 /of a surface is: the r at two interface have1 /

n nrn n

−=

+

0

to be equal.

1 /At air-film interface

1 /f

a f

n nr

n n n−

− ⎫= ⎪+ ⎪0 2

0 00

For maximum extinction

1 /for n 1

1 /At film-substrate interface

1 /

Example: for the

f f sa f f s f s f s

s f f airf s

s f

n n n nr r n n n n nn n n n

rn n

− −

−

+ ⎪→ = → = → = = → =⎬− ⎪= ⎪+ ⎭( )yellow green 550nm that is eye's most sensitive range 1 22 the clasest materialn =


Example: for the ( )2

yellow-green 550nm that is eye s most sensitive range 1.22 the clasest material

to this is index is MgF 1.38.fn

with n

=

=

Dielectric mirrors, filters, laser mirrorsWe can also design complete reflectors with alternating high-low /4 layers that λ

( )create constructive interference for the reflected light here we have phase shift . Advantage of these mirrors is that they don't heat up (no absorption) and are perfect reflectors for a specific wavelength Good for making laser resonatorsperfect reflectors for a specific wavelength. Good for making laser resonators.

We can also design low-pass, high-pass, band-pass filters with very high precision.Optical thin film design is a branch of optics that deals with these applications.


Localized and non‐localized finger

Non‐localized fringes form everywhere and we can place a screen on their path to see g y p pthem. For example fringes from a double slit are observable at every distance from the sources.

Localized fringes form at a specific location in space and the screen has to be placed there i h F l f i f hi fil l li d i fi i d dto view them. For example fringes from a thin film are localized at infinity and we need a

converging lens to project them on a screen. The fringes in the picture (on the right side) are fringes of equal inclination of Heidinger fringes formed by parallel rays not possible with point sources.p


Generating non‐localized real fringes with dielectric films using pointGenerating non localized real fringes with dielectric films using point sources


Interference at a wedgeInterference at a wedge• Path difference between two rays is

almost equal to 2talmost equal to 2t

• Wave 1 has zero phase shift due to reflection from the glass‐air interface.

• Wave 2 has 1800 phase shift due to reflection from air‐glass interface.

• That is why the left corner appears

12

• That is why the left corner appears dark. Light path difference is zero but phase difference due to reflection is 1800 or half a cycle.

a br i

n nE E−=

+a bn n+

Fringes of equal thickness

For a wedge with verying thickness the path difference 2 cos varies even if the angle of incidence is

constant. For a fixed direction of incident light bright or dark fringes will be associated w

f tn tΔ =

ith a particulat thickness (fringes of equal thickness).

Fringes can be viewd with the arrangement in the figure and are called Fizeu Fringes. At normal incidence cos 1 and the light path difference i

θ =s 2 p fn tΔ =g p

bright2 1 dark

2

p f

f r

mn t

m

λ

λ

⎧⎪+ Δ = ⎨⎛ ⎞+⎜ ⎟⎪⎝ ⎠⎩


2⎝ ⎠⎩

Newton rings


Film thickness measurements


Stokes relations

In reality many internal reflections happen in the films. The Stokes relations offer a formalism to track the y y pprflection and transmssion coefficients for the electric fields arriving at an interface. L

i t r

et E amplitude of the incident light, E amplitude of the tranmitted light, E amplitude of the reflected light

We define the reflection coefficient as: and the transmission coefficient t as:rEr r

= = =

= t tE=

iE

ri

t

EAt the interface E is devided th to parts

EAccording to the principle of ray reversibility both cases in fig a and b are valid When they happen

i

i

i

ErEtE

=⎧⎨ =⎩

According to the principle of ray reversibility both cases in fig a and b are valid. When they happen together (figure c) we mark the r

( )( )

2 2i

everse senario by primed parameters. But b and c are physically

E = r ' ' 1equivalent. So Stokes relations between the amplitude coefficients

0 ' '

itt E tt r

E

⎫⎧ + → = −⎪ ⎪⎨ ⎬⎪ ⎪( )0 ' '

for angles of incidenceir t tr E r r= + → = −⎪ ⎪⎩ ⎭

related through the Snell's law.There is a phase differenceπThere is a phase difference between the rays incident from each side.

π


Multiple beam interference


Multiple beam interference I

We now consider interference between a narrow beam of amplitude E and angle of incidence0

i

We now consider interference between a narrow beam of amplitude E and angle of incidence and the multiple reflections from inside of a film of thickness , index surrounded by air.

We want to finft nθ

d superposition of the reflected waves from the top of the plate.

h h diff b h i fl d b i 2k kδ θΔ

( )0

1 0

The phase difference between the successive reflected beams is: 2 cos

Incident beam: E

f t

i t

i t

k k n t

e

E rE e

ω

ω

δ θ= Δ =

= ( )( ) ( )

1 0

2 0

3

' '

' '

i tE tt r E e

E tt r

ω δ−=

= ( ) ( )230

i tE e ω δ−

( ) ( )

( )( ) ( )( )

354 0

12 10

' '

... ' ' for N=2,3,...

this form does not hold for E that never passes through the film

i t

i t NNN

E tt r E e

E tt r E e

ω δ

ω δ

−

− −+

=

=

( ) ( )

1

12 3 20 0 0

1 2

this form does not hold for E that never passes through the film.

' 'i t NNi t i t iR N

N NE E rE e tt E r e E e r tt e rω δω ω δ

∞ ∞⎡ − − ⎤− −⎣ ⎦

= =

= = + = +∑ ∑ ( ) ( )4 2

2

1

N i N

Ne δ

∞− −

=

∞

⎛ ⎞⎜ ⎟⎝ ⎠

∑


2 2 2

2

11 ... where ' and x 1 the series converges1

N i

Nx x x x r e

xδ

∞− −

=

= + + + = = <−∑

Multiple beam interference II' ' i th St k l ti hi

ii t tt r eE E

δω

−⎛ ⎞+⎜ ⎟

( ) ( )

0 2

2

0 02 2

using the Stokes relationships1 '

1 1

1 1

i tR i

i ii t i t

R i i

E E e rr e

r re r eE E e r E e

r e r e

ωδ

δ δω ω

δ δ

−

− −

− −

= +⎜ ⎟−⎝ ⎠⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟= − =⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠

( ) ( )2 2 20 2 2

1 1

1 1 us

1 1

i t i i t i

R R i i

r e r e

e e e eI E E r

r e r e

ω δ ω δ

δ δ

− −

−

⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠⎡ ⎤ ⎡ ⎤− −⎢ ⎥ ⎢ ⎥= =

− −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

2i

20

ing 2cos =e and Ri R

i

EIeI E

δ δδ −+ =⎣ ⎦ ⎣ ⎦

( ) ( )22 2 2

0 4 2 4 2

1 cos 2 1 cos2

1 2 cos 1 2 cosR R R i

rI E E r I I

r r r rδ δ

δ δ⎛ ⎞⎛ ⎞− −

= = → = ⎜ ⎟⎜ ⎟ ⎜ ⎟+ − + −⎝ ⎠ ⎝ ⎠

( )22

4 2

Using the conservation of energy equation we find

2 1 cos1 2 cos

i R T

T T i R i

I I I

rI E I I I

r rδ

= +

−= = − = −

+ −( )4 2 2

4 2

1 2 cos 2 1 cos1 2 cosi i

r r rI I

r rδ δ

δ δ⎛ ⎞ ⎛ ⎞+ − − −

=⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ −⎝ ⎠ ⎝ ⎠

( )22

4 2

1

1 2 cosT i

rI I

r r δ

⎛ ⎞−⎜ ⎟=⎜ ⎟+ −⎝ ⎠


Multiple beam interference III

( ) ( )222 12 1 cos rr δ ⎛ ⎞−⎛ ⎞− ⎜ ⎟( ) ( )

( )

4 2 4 2

12 1 cos and

1 2 cos 1 2 cos

A minimum for reflected irradiance occurs when 1 cos 0 cos 1 2

R i T i

rrI I I I

r r r r

m

δδ δ

δ δ δ π

⎛ ⎞− ⎜ ⎟= =⎜ ⎟⎜ ⎟ ⎜ ⎟+ − + −⎝ ⎠ ⎝ ⎠− = → = → =

2 cos condition for minimum reflectance

This also prov

f tm n tλ θΔ = =

( )22cos 1

4 2

1ides the condition for maximum transmission

1 2 cosT i T i

rI I I I

r rδ

δ=

⎛ ⎞−⎜ ⎟= ⎯⎯⎯→ =⎜ ⎟+ −1 2 cos

All the secondary and higher order reflections are in-phase with each other and they all have phase differenc

r r δ

π

⎜ ⎟+⎝ ⎠

e with the first reflection. So they all cancel out with the first reflection.

' ' EE 2 2021 2

1 0

' ' 1 for interface of air and glass 1, 1.5, 0.04 and 96% of the light is

cancelled in the first

tt r EE r n n rE rE

= = − = = =

reflection so ignoring the higher order reflections is perfectly justified.When cos 1 the reflection maximum occursδ = −When cos 1 the reflection maximum occurs.

1 1 2 cos condition for maximum reflectance2 2 f tm m n t

δ

δ π λ θ

= −

⎛ ⎞ ⎛ ⎞= + → Δ = + =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

2⎛ ⎞ ⎛ ⎞


and in tha( )

22 2

2 22

4 1t case and 11

R i T ir rI I I I

rr

⎛ ⎞ ⎛ ⎞−⎜ ⎟= = ⎜ ⎟⎜ ⎟ +⎝ ⎠+⎝ ⎠

Chapter 7 Interference of Light - Erbion

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