• Two-Beam Interference

(1)

This Lecture

• Two-Beam Interference

• Young’s Double Slit Experiment

• Virtual Sources

• Newton’s Rings

• Film Thickness Measurement by Interference

• Stokes Relations

• Multiple-Beam Interference in a Parallel Plate Last Lecture

• Superposition of waves

• Laser

Chapter 7. Interference of Light

(2)

Two-Beam Interference

( ) ( ) ⁽ ⁾

1 2

1 01 1 1 01 1 1

2 02 2 2 02 2 2

:

, cos cos

Consider two waves E and E that have the same frequency

E r t E k r t E ks t

E r t E k r t E ks t

ω

ω φ ω φ

= ⋅ − + = − +

r r

r

r r r r r

r r r r r r

k r

1

k r

2

S₂

S₁

(3)

Two-Beam Interference

Interference term

(4)

Two-Beam Interference

The interference term is given by

The irradiances for beams 1 and 2 are given by :

1 2 01 02

cos(

1 1

) cos(

2 2

) E E r r ⋅ = E r ⋅ E r ks − ω φ t + ks − ω φ t +

{

α ≡ ks1 +φ1 , β ≡ ks2 +φ2

}

1 2 01 02

01 02

2 2 cos( ) cos( )

cos( 2 ) cos( )

cos( ) cos

E E E E t t

E E t

E E E E

α ω β ω

α β ω β α

β α δ

⋅ = ⋅ − −

= ⋅ ⎡⎣ + − + − ⎤⎦

= ⋅ −

= ⋅

r r r r

r r r r r r

{2 cos( ) cos( )A B =cos(A+B) cos(+ A B− )}

(

1 2

) (

2 1

)

k s s

δ ≡ − + φ φ −

: phase difference

1 2

2

1 2

cos

I = + + I I I I δ

when Er₀₁ Er₀₂ (same polarization)

1 2

2

1 2

cos

I = + + I I I I δ

for purely monochromatic light

(5)

Interference of mutually incoherent beams

1 2

2

1 2

cos

I = + + I I I I δ

( )

{

1 2 2 1

}

cos δ = cos k s − s + φ ( ) t − φ ( ) t

“mutual incoherence” means φ₁(t) and φ₂(t) are random in time

cos δ = 0

1 2

I = + I I

Mutually incoherent beams do not interference with each other

(6)

Interference of mutually coherent beams

1 2

2

1 2

cos

I = + + I I I I δ

( )

{

1 2 2 1

}

cos δ = cos k s − s + φ ( ) t − φ ( ) t

“mutual coherence” means [ φ₁(t) - φ₂(t) ] is constant in time

1 2

2

1 2

cos

I = + + I I I I δ

Mutually coherent beams do interference with each other

( )

{

¹ ²

}

cos δ = cos δ = cos k s − s

(7)

The total irradiance is given by

There is a maximum in the interference pattern when

This is referred to as constructive interference.

There is a minimum in the interference pattern when

This is referred to as destructive interference

Interference of mutually coherent beams

(8)

Visibility

Visibility = fringe contrast

{ ⁰ ¹ }

min max

min

max

≤ ≤

+

≡ − V

I I

I V I

When

Therefore, V = 1

max

4

0

I = I I

_min

= 0

(9)

Conditions for good visibility

Sources must be:

same in phase evolution

in terms of

time (source frequency) Æ temporal coherence space (source size) Æ spatial coherence

Same in amplitude Same in polarization

Very good Normal

Very bad

(10)

Young’s Double Slit Experiment

Hecht, Optics, Chapter 9.

(11)

Assume that y << s and a << s.

The condition for an interference maximum is

The condition for an interference minimum is Relation between

geometric path difference and phase difference :

a

s

y

Δ

θ

Young’s Double Slit Interference

k 2 π

δ λ

⎛ ⎞

= Δ = ⎜ ⎝ ⎟ ⎠ Δ

(12)

On the screen the irradiance pattern is given by

Assuming that y << s :

Bright fringes:

Dark fringes:

Young’s Double Slit Interference

max min

y y s

a

⎛ λ ⎞

Δ = Δ = ⎜ ⎟ ⎝ ⎠

(13)

Interference Fringes From 2 Point Sources

Figure 7-5

(14)

Interference Fringes From 2 Point Sources

Two coherent point sources : P₁ and P₂

(15)

Interference With Virtual Sources:

Fresnel’s Double Mirror

(16)

Interference With Virtual Sources:

Lloyd’s Mirror

mirror

Rotation stage Light

source

Figure 7-7

(17)

Interference With Virtual Sources:

Fresnel’s Biprism

(18)

7-4. Interference in Dielectric Films

(19)

Analysis of Interference in Dielectric Films

Figure 7-12

(20)

Analysis of Interference in Dielectric Films

The phase difference due to optical path length differences for the front and back

reflections is given by

(21)

Analysis of Interference in Dielectric Films

Also need to account for phase differences Δ_r due to differences in the reflection process at the front and back surfaces

Constructive interference

Destructive interference

Δ_r Δ = Δ_p + Δ_r

Î Δ_r

(22)

Fringes of Equal Inclination

Fringes arise as ∆ varies due to

changes in the incident angle: θ_i Æ θ_t

Constructive interference

Destructive interference

Figure 7-13

(23)

7-5. Fringes of Equal Thickness

When the direction of the incoming light is fixed, fringes arise as ∆ varies due to changes

in the dielectric film thickness :

Constructive interference Æ Bright fringe

Destructive interference Æ Dark fringe

(24)

7-6. Fringes of Equal Thickness: Newton’s Rings

Figure 7-17

(25)

Newton’s rings

θ

R-t_m R

t_m r_m

Maxima (bright rings) when,

n

t_m << R

2 2

2 2 2

( )

2

m m

m

r t

R r R t R

t

= + − → = +

2 2

m m

r ≈ Rt

2 ( 1, )

2 2

m f r

t λ mλ n λ

Δ = + = = Δ =

2 1

2 2

tm λ ^⎛m ^⎞λ Δ = + =⎜⎝ + ⎟⎠

2

rm m Rλ

→ =

Minima (dark rings) when,

reflection transmission

2 1

m 2

r ^⎛m ^⎞λR

→ = ⎜ − ⎟

⎝ ⎠

(26)

7-7. Film-thickness measurement by interference

(27)

7-8. Stokes Relations in reflection and transmission

E_i is the amplitude of the incident light.

The amplitudes of the reflected and transmitted beams are given by

From the principle of reversibility

Stokes relations

r = eⁱ^π r’

(28)

7-9. Multiple-Beam Interference in a Parallel Plate

I

0

I

R

I

T

0

I

R

R ≡ I

0

I

T

T ≡ I

Reflectance Transmittance

(29)

Find the superposition of the reflected beams from the top of the plate.

The phase difference between neighboring beams is

Given that the incident wave is

Δ

δ = k Δ = 2 n d

_f

cos θ

_t

Multiple-Beam Interference in a Parallel Plate

r

r’

t

2

0 0

I = E

2

T T

I = E

2

R R

I = E

d

t’

(30)

The reflected amplitude resulting from the superposition of the reflected beams from the top of the plate is given by

Define

Therefore

Multiple-Beam Interference in a Parallel Plate

(31)

The Stokes relations can now be used to simplify the expression

Multiple-Beam Interference in a Parallel Plate

(32)

The reflection irradiance is given by

Multiple-Beam Interference in a Parallel Plate

(33)

Multiple-Beam Interference in a Parallel Plate

The transmitted irradiance is given by

(34)

Multiple-Beam Interference in a Parallel Plate

Minima in transmitted irradiance and maxima in reflected irradiance occur when Minima in reflected irradiance and maxima in transmitted irradiance occur when

• Two-Beam Interference

This Lecture

• Two-Beam Interference

• Young’s Double Slit Experiment

• Virtual Sources

• Newton’s Rings

• Film Thickness Measurement by Interference

• Stokes Relations

• Multiple-Beam Interference in a Parallel Plate Last Lecture

• Superposition of waves

• Laser

Chapter 7. Interference of Light

Two-Beam Interference

( ) ( ) ( )

( ) ( ) ( )

k r

k r

Two-Beam Interference

Two-Beam Interference

cos(

) cos(

) E E r r ⋅ = E r ⋅ E r ks − ω φ t + ks − ω φ t +

{

}

α ω β ω

α β ω β α

β α δ

(

) (

)

k s s

δ ≡ − + φ φ −

2

cos

I = + + I I I I δ

2

cos

I = + + I I I I δ

Interference of mutually incoherent beams

2

cos

I = + + I I I I δ

( )

{

}

cos δ = cos k s − s + φ ( ) t − φ ( ) t

cos δ = 0

I = + I I

Interference of mutually coherent beams

2

cos

I = + + I I I I δ

( )

{

}

cos δ = cos k s − s + φ ( ) t − φ ( ) t

2

cos

I = + + I I I I δ

( )

{

}

cos δ = cos δ = cos k s − s

Interference of mutually coherent beams

Visibility

Visibility = fringe contrast

{ 0 1 }

≤ ≤

+

≡ − V

I I

I V I

4

I = I I

= 0

Conditions for good visibility

same in phase evolution

time (source frequency) Æ temporal coherence space (source size) Æ spatial coherence

Same in amplitude Same in polarization

Young’s Double Slit Experiment

( ) ( ) ⁽ ⁾

( ) ( ) ⁽ ⁾

{ ⁰ ¹ }

n d θ ^⎛ m ^⎞ λ