Hecht; 11/3/2010; 9-1

Chapter 9. Interference Differential wave equation is linear. → Superposition principle applies. The electric fields are summed vectorially. Interference is the interaction of multiple waves in which the resultant irradiance deviates from the sum of each irradiance. 9.1 General Considerations The total electric field is given by a vector sum 1 2 3 ...= + + +b b b b But we can only measure its intensity. Interference of two plane waves

( )1 1 1 1, cosor t E k r tω ε⎡ ⎤= − +⎣ ⎦ib , ( )2 2 2 2, cosor t E k r tω ε⎡ ⎤= − +⎣ ⎦ib

Irradiance or intensity is defined as

( ) ( )21 2 1 2~

T TI ⇒ + + ⇒ib b b b b 2 2

1 2 1 22T T T+ + ib b b b 1 2 12 I I I≡ + +

The time-average is defined as ( )1( ) ' 't T

T tf t f t dt

T+

= ∫

The interference term

12 1 2 1 2 1 1 2 22 coso oTI E E k r k rε ε⎡ ⎤= = + − −⎣ ⎦i i i ib b

↑ ↑ ↑ δ , phase difference ↑

1 2 1 1 2 2cos coso oE E k r t k r tω ε ω ε⎡ ⎤ ⎡ ⎤− + − +⎣ ⎦ ⎣ ⎦i i i ( ) ( )ω ε ω ε− + − +⎡ ⎤ ⎡ ⎤⇒ + +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦1 1 2 2

1 21 1. .2 2

i k r t i k r to oE E e c c e c ci ii

If 1 2 and o oE E are parallel

2 21 2 1 22

T T TI = + + ib b b b

2221

1 2 cos2 2

ooo o

EEE E δ⇒ + +

↑ ↑ 1I 2I

1 2 1 22 cosI I I I I δ= + + For 0 cos 1δ< < : Constructive interference For 1 cos 0δ− < < : Destructive interference If 1 2 and o oE E are perpendicular

→ 1 2 0o oE E =i and 1 2I I I= + (No interference) • In the complex expoential form

( ) ( )1 1

1 1, i k roE r t E e ε+

=i , ( ) ( )2 2

2 2, i k roE r t E e ε+

=i

1 21~2

I E E•

Hecht; 11/3/2010; 9-2 9.2 Conditions for Interference A. Temporal and Spatial Coherence A quasi-monochromatic source produces a mix of photon wavetrains. ↑ Oscillation during <10ns The net wave resembles a sinusoid during the temporal coherence time and then it changes phase.

(a) Monochromatic point source: The disturbance at 1 'P will appear at 2 3 4', ', 'P P P in sequence at later times. By watching 4 'P one can predict what happens at 1 'P → Temporally coherent (Coherence time is infinity in this case). (b) The point source changes frequency from moment to moment: No correlation between 1 4' and 'P P . But close points 2 3' and 'P P are somewhat correlated. The longest distance over which the wave is sinusoidal is called coherence length. Points 1 2 3, , P P P in (a) and (b) are correlated since they are on the same spherical wavefront. → Spatially coherent (c) Many monochromatic point sources with the same frequency ν : Same wavefront pattern at every period 1/τ ν= . → 1 2 3', ', 'P P P are temporally coherent. Assume each point source changes phase randomly emitting 10ns long sinusoid. → Waves at 1 2 3', ', 'P P P are correlated only for t<10ns. 1 2 and P P are completely uncorrelated.

Hecht; 11/3/2010; 9-3 9.3 Wavefront-Splitting Interferometers A. Young’s Experiment Natural light through a pinhole → A spherical wave. Not temporally coherent but spatially coherent. The primary wave passes through two pinholes in phase. → Pinholes are two coherent secondary sources. Interference is observed at the screen. Mathematical description The path difference

1 1 2sinS B a r rθ= ≈ − → 1 2θ ≈ −a r r → ≈ −1 2ya r rs

,

↑ /y sθ ≈ . Constructive interference for 1 2r r mλ− =

→ msy ma

λ≈ : m-th bright region

→ /m m aθ λ≈ : Angular separation between fringes

↑

/y sθ ≈

Hecht; 11/3/2010; 9-4 Phase difference between two spherical waves

( )1 22πδλ

= − ≈yk r r as

The irradiance

1 2 1 22 cosI I I I I δ= + + → ( )24 cos /2oI I δ= → 24 cosoaI I ysπλ

⎛ ⎞= ⎜ ⎟⎝ ⎠

↑ Assuming 1 2 0I I I= =

• Incident white light → Bright spot near optic axis with m=0. Color bands for higher m ( my is a function on λ ). • Coherence length clΔ < The path difference 1S B → No interference Other Interferometers (a ) Fresnel’s Double Mirror (b) Fresnel Double Prism (c) Lloys’d Mirror The reflected beam at glancing angle undergoes 180o phase shift

Hecht; 11/3/2010; 9-5 9.4 Amplitude-Splitting Interferometers A. Dielectric Films - Double-Beam Interference

The path length different between two reflected waves ( ) 1fn AB BC n ADΛ = + − → 2 cosf tn d θΛ =

The phase difference

4

cosfo t

o

nk d

πδ π θ π

λ= Λ ± ⇒ ± : π± from int./ext. reflections (sign is immaterial)

Constructive interference for 2 mδ π=

( )cos 2 14

ftd m

λθ = +

Destructive interference for ( )2 1mδ π= +

( )cos 24

ftd m

λθ =

Fringes of equal inclination

Interference from extended source: Waves of the same incident angle meet at P1 or P2. Fringes at P1 and P2 are fringes of equal inclination.

Hecht; 11/3/2010; 9-6 Haidinger fringe Fizeau fringes Fringes created from a wedge shaped film

Interference maximum, ( )2

2 2 1fm

o

nd m

λ= +

Soap film

The black region at the top has thicknes less than /4fλ → Two waves are out of phase Newton’s ring

Fringes are created from nearly parallel plates with a point contact

Interference fringe of concentric circles

( )2

2 2 1fm

o

nd m

λ= +

The radius of the m-th bright ring ( )1

2m fx m Rλ= +

Hecht; 11/3/2010; 9-7 B. Mirrored Interferometers Michelson-Interferometer

Compensator C makes each beam traverses the beamsplitter three times. It negates the dispersion (No need for monochromatic waves). Mach-Zehnder Interferometer Two separate beam paths make it difficult to align. An object interposed in one beam alters OPL difference. → Change of fringe pattern Sagnac Interferometer

Hecht; 11/3/2010; 9-8 9.5 Types and Localization of Interference Fringes Real fringe : Two beams converge and make fringes on screen.

Virtual fringe : Lens is required to project fringes on screen. Localized fringes : Observable only over a particular surface. Extended sources always form localized fringes.

Nonlocalized fringes : Real and exist everywhere. Small sources form these fringes. Upper half : Real and nonlocal Lower half : Virtual and localized at infinity 9.6 Multiple Beam Interference OPL difference between adjacent rays as before. 2 cosf tn d θΛ = The electric field of the reflected beam.

For an incident wave i toE e ω−

1i t

r oE E re ω−= ,

2 ' ' i i tr oE E tr t e δ ω−= , : okδ = Λ

3 23 ' ' i i t

r oE E tr t e δ ω−= : 2' 1ir e δ < . . . ( )1(2 3)' ' i N i tN

Nr oE E tr t e δ ω− −−=

→ 2

' '1 '

ii t

r o i

r tt eE E e rr e

δω

δ− ⎡ ⎤

= +⎢ ⎥−⎣ ⎦

( )2

1

1

ii t

o i

r eE e

r e

δω

δ−

⎡ ⎤−⎢ ⎥⇒

−⎢ ⎥⎣ ⎦

↑ 'r r= − and 2' 1tt r= − The reflected intensity

( )

( )2

*4 2

2 1 cos/2

1 2 cosr r r i

rI E E I

r rδ

δ−

= =+ −

When mλΛ = → 2 mδ π= → 0E = , the 1st reflection is canceled by others

When ( )12m λΛ = + → 12

2mδ π ⎛ ⎞= +⎜ ⎟⎝ ⎠

→ ( )2

21 o

rE Er

=+

Hecht; 11/3/2010; 9-9

Coefficient of Finesse is defined as 2

2

21

rFr

⎛ ⎞≡ ⎜ ⎟−⎝ ⎠

Then

( )( )

2

2

sin /21 sin /2

r

i

FII F

δδ

=+

( )2

11 sin /2

t

i

II F δ

=+

Airy function is defined as

( ) ( )2

11 sin /2F

θδ

≡+

^

Airy function 1 - Airy function A. The Fabry-Perot Interferometer It consists of two highly reflecting surfaces separated by d. It is used as laser resonant cavity, etalon, interferometer. ↑ ↑ Fixed d Variable d

Intensity transmission ratio

( )2

11 sin /2

t

i

II F δ

=+

πδ θλ

=2 2 cosf t

o

n d

Hecht; 11/3/2010; 9-10 9.7 Applications of Single and Multilayer Films Anti-reflection coating, heat reflector, beamsplitter, dichroic mirror, broad and narrow band-pass filters. A. Mathematical Treatment The tangential field at (just below) boundary I. '

I tI rIIE E E= + , (1)

( )'1 coso

I tI rII iIIo

H E E nεθ

μ= −

At (just above) boundary II 'o oik h ik h

II tI rIIE E e E e −= + , (2)

( )'1 coso oik h ik h o

II tI rII iIIo

H E e E e nεθ

μ−= −

where 1 /cos iIIh n d θ= ' and tI rIIE E in terms of and II IIE H in (2). and I IE H in terms of and II IIE H in (1). ( ) ( ) 1cos sin /I II o II oE E k h H i k h Y= − ,

( ) ( )1 sin cosI II o II oH E Y i k h H k h= − +

where 1 1 cosoiII

o

Y nεθ

μ≡ , or 1 1 /coso

iIIo

Y nεθ

μ≡

↑ (For //E plane of incidence) In matrix form

( ) 1

1

cos sin /sin cos

I IIo o

I IIo o

E Ek h i k h YH HiY k h k h

−⎡ ⎤⎡ ⎤ ⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦

(3)

↑ 1M≡ , characteristic matrix For multilayer of number p

( 1)1 2

( 1)

.. pIp

pI

EEM M M

HH+

+

⎡ ⎤⎡ ⎤= ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦

Rewrite eq. (3) using the fields given outside the film

( ) 1iI rI tII

iI rI o tII s

E E EM

E E Y E Y+⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥− ⎣ ⎦⎣ ⎦

→ ( )

11 12

21 22

11

s

o s

r m t m Y tr Y m t m Y t

+ = +

− = + → (4)

where 0 cosoo iI

o

Y nεθ

μ= and coso

s s iIIo

Y nεθ

μ=

Hecht; 11/3/2010; 9-11 B. Antireflection Coatings At normal incidence with /2ok h π= or /4oh λ= ,

−⎡ ⎤

= ⎢ ⎥−⎣ ⎦1

1

0 /0

i YM

iY

→ From (4), the reflectance ⎡ ⎤−

= ⎢ ⎥+⎣ ⎦

22121

o s

o s

n n nRn n n

→ = 0R for 21 o sn n n=

A double-layer, quarter-wavelength antireflection coating

2 2 11

2 1 21

0 / / 00 /0 0 /0

i Y n ni YM

iY n niY− −− ⎡ ⎤ ⎡ ⎤⎡ ⎤

= ⇒⎢ ⎥ ⎢ ⎥⎢ ⎥ − −−⎣ ⎦ ⎣ ⎦ ⎣ ⎦

↑ Normal incidence

From (4), ⎡ ⎤−

= ⇒⎢ ⎥+⎣ ⎦

22 22 12 22 1

0o s

o s

n n n nRn n n n

for 2

2

1

s

o

n nn n

⎛ ⎞=⎜ ⎟

⎝ ⎠

C. Multilayer Periodic System(High reflectivity mirror) Increased /H Ln n → Widened high-reflectance region Increased number of stack → Higher reflectance peak Increased max. reflectance by an additional H-layer : ( )mg HL Ha

Decreased left-peak by an eighth-wave L-layer on both sides : ( ) ( ) ( )0.5 0.5mg L HL H L a

Decreased right-peak : ( ) ( ) ( )0.5 0.5mg H L HL H a

Hecht; 11/3/2010; 9-12