Hecht; 11/28/2010; 10-1

Chapter 10 Diffraction 10.1 Preliminary Considerations Diffraction is a deviation of light from rectilinear propagation. It occurs whenever a portion of a wavefront is obstructed. No significant distinction between interference and diffraction. ↑ ↑ Superposition of a few waves, many waves Huygens-Fresnel Principle Every point on a wavefront serves as a source of spherical secondary wavelets with the same frequency. The amplitude of the optical field at a later time is the superposition of all the secondary wavelets. If the wavelength is larger than the aperture, the wave will spread out at a large angle. [Picture P445] A. Opaque Obstructions There is no optical field beyond it. The incident wave and the electron-oscillator fields superimpose to yield zero field beyond the screen. A screen = Apertured screen + Small disk ↓ ↓ Diffraction field + Electron-oscillator field ⇒ Zero field Unpaired electron-oscillator at the rim → Not negligible near the aperture. B. Fraunhofer-Fresnel Diffraction A point source at infinity → Aperture → Screen ↑ ↑ ↑

Distance Ol . Distance Sl . Diffraction pattern. Plane wave. Sl The diffraction pattern

Short dist. Same shape as the aperture. Geometrical optics applies. Intermediate Recognizable aperture shape with fringes. Fresnel or Near-field diffraction. Long dist. Smooth pattern with no resemblance to the aperture. Fraunhofer or Far-field diffraction.

Reducing the wavelength. Fraunhofer diffraction → Fresnel diffraction → Geometrical optics The point source closer to the aperture. → Not plane wave at the aperture. → Fresnel diffraction even if Sl → ∞ Fraunhofer diffraction when 2 /R a λ> R: Smaller distance of Ol and Sl . a: aperture size.

Hecht; 11/28/2010; 10-2 Fraunhofer diffraction can be obtained by lenses.

(S and P are equivalently at infinity)

P and S are on the focal points of lenses L1 and L2 , respectively. → P is an image of S In general, an image is a Fraunhofer diffraction pattern C. Several Coherent Oscillators A linear array of N identical coherent point sources

Electric field at an infinite distance, P ( ) ( ) ( ) ( ) ( ) ( )21

1 2 ... Ni kr ti kr t i kr to o o NE E r e E r e E r eωω ω−− −= + + +

Assuming same wave amplitudes at P

→ ( ) ( ) ( ) ( )12 1

1 ...Ni kr t i i i

oE E r e e e eω δ δ δ −− ⎡ ⎤= + + + +⎢ ⎥⎣ ⎦ : sinkdδ θ=

→ ( ) ( ) ( ) ( )1 1 /2 sin /2 sin /2sin /2 sin /2

i kr N i kR ti to o

N NE E r e e E r eδ ωω δ δδ δ

+ −⎡ ⎤ −− ⎣ ⎦ ⎛ ⎞ ⎛ ⎞= ⇒⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

↑ ( )1

12 1 sinR N d rθ= − + , the distance from the center to P

The intensity at P : [ ][ ]

2

2

sin ( /2)sinsin ( /2)sino

N kdI I

kdθθ

=

The principal maxima : 2

oI I N= for ( /2)sin sin mkd m d mθ π θ λ= → = • If there is a phase shift between adjacent sources of sinkdδ θ ε= + , the principal maxima occur at sin /md m kθ λ ε= −

Hecht; 11/28/2010; 10-3 An idealized line source Each point emits a spherical wavelet

( )cos ω⎛ ⎞= −⎜ ⎟⎝ ⎠

oEE kr t

r

The electric field from an infinitesimal segment iyΔ

( )cos ω⎛ ⎞ ⎛ ⎞= − Δ⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠o

i i ii

E NE kr t yr D

: D is the array length

: N is total number of sources The total field at P

( )1

cos ω=

= − Δ∑N

Li i i

i i

EE kr t yr

where ( )1 limL oNE E N

D →∞≡

→ ( )/2

/2

cos ω

−

−= ∫

D

LD

kr tE E dy

r (1)

From the Problem 9.15

2

2sin cos ...2yr R yR

θ θ⎛ ⎞

= − + +⎜ ⎟⎝ ⎠

(2)

The third term can be ignored if the phase ( )2 2/2

cos2

θ⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

Dk

R is very small

→ Fraunhofer condition Insert (2) into (1)

( )/2

/2

cos sinθ ω−

= − −⎡ ⎤⎣ ⎦∫D

L

D

EE k R y t dyR

→ ( )cossin '

'L

kR tE E D

Rωβ

β−⎛ ⎞

= ⎜ ⎟⎝ ⎠

: ( )' /2 sinkDβ θ≡

For D λ>> , E becomes a circular wave in xz-plane (No diffraction and no wave in vertical direction) ↑ It looks like a spherical wave coming from the center that is cut by the horizontal plane. The irradiance

( ) ( ) ( )2

2sin '0 0 sinc ''

I I Iβθ ββ

⎛ ⎞= ⇒⎜ ⎟

⎝ ⎠ (3)

Hecht; 11/28/2010; 10-4 10.2 Fraunhofer Diffraction Goto Section D first then come back here A. The Single Slit

From Eq. (4) of Sec D.

( )

( )/2 /2 /

/2 /2

i kR ty b z a ik Yy Zz RAy b z a

E eE e dzdyR

ω−= = − +

=− =−= ∫ ∫

→ ( ) sin sini kR t

AlbE eER

ω α βα β

− ⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

where , sin2 2 2kbZ klY kl

R Rα β θ= = =

Irradiance

( ) ( )22sin sinI I o α βθ

α β⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

• Very narrow slit, 0b → . The irradiance

( ) ( ) ( )2

2sin '0 0 sinc'

I I Iβθ ββ

⎛ ⎞= ⇒⎜ ⎟

⎝ ⎠

Compare this with the previous result Eq. (3).

( ) 0I θ = for , 2 ...β π π= ± ±

↑ Destructive interference whenever the path difference between the top and the bottom rays is mλ , or sinb mθ λ= .

( , , )X Y Z

( , , )x y z

Hecht; 11/28/2010; 10-5 B. The Double Slits Waves from each slit will interfere

The total electric field at P using Eq. (4)

( ) ( )/2 /2 /2 /2

/ /

/2 /2 /2 /2

l b l a bik Yy Zz R ik Yy Zz R

l b l a b

E C e dzdy C e dzdy+

− + − +

− − − −

= +∫ ∫ ∫ ∫

→ ( ) ( )/2 /2

/ /

/2 /2

sin b a bik Zz R ik Zz R

b a b

E lC e dz e dzββ

+− −

− −

⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟

⎝ ⎠ ⎢ ⎥⎣ ⎦∫ ∫ :

2klY

Rβ =

→ sin sin2 cosiE lbCe γ β α γβ α

− ⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

: sin2 2kbZ kb

Rα θ= = , sin

2 2kaZ ka

Rγ θ= =

The irradiance

( )2 2

2sin sin4 cosoI I β αθ γβ α

⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

: oI is the irradiance from each slit

↑ ↑ ↑ Diffraction. Interference. An interference maximum and a diffraction minimum correspond to the same θ → Missing order For a mb= , there are m bright fringes within the central diffraction peak

( , , )X Y Z

( , , )x y z

Hecht; 11/28/2010; 10-6 C. Diffraction by Many Slits The total electric field at P

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

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

...N a bl b l a b l a b l

l b l a b l a b l N a b

E C F z dzdy C F z dzdy C F z dzdy C F z dzdy− ++ +

− − − − − − − − −

= + + + +∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫

→ ( ) ( )( )2 1/ / /sin sin 1 ...NikZa R ikZa R ikZa RE lbC e e eβ α

β α−− − −⎛ ⎞ ⎛ ⎞ ⎡ ⎤= + + + +⎜ ⎟ ⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠⎝ ⎠

→ sin sin sinsin

i N NE lbCe γ β α γβ α γ

− ⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

where 2klY

Rβ = , sin

2 2kbZ kb

Rα θ= = , sin

2 2kaZ ka

Rγ θ= =

The irradiance

( ) ( ) 2 22

2

0 sin sin sinsin

I NIN

β α γθβ α γ

⎛ ⎞ ⎡ ⎤⎛ ⎞= ⎜ ⎟ ⎜ ⎟ ⎢ ⎥⎝ ⎠⎝ ⎠ ⎣ ⎦

(0)I , peak of the total I at the screen center oI , peak irradiance by one slit The principal maxima for 0sinγ = , 0, , 2 ...γ π π= ± ±

(N-1) minima for 0sinNγ = , 2 3, , ...N N Nπ π πγ = ± ± ±

A subsidiary maxima at 3 5, ...2 2N Nπ πγ = ± ±

The irradiance of the first subsidiary maximum

22 1(0)

3 22I I

π⎛ ⎞≈ ≈⎜ ⎟⎝ ⎠

Hecht; 11/28/2010; 10-7 D. The Rectangular Aperture

a

bz

x

y

dSP(X,Y,Z)

R

r

The differential electric field at P from a differential source AE dS

( )( )i kr t

AedE E dS

r

ω−

=

Radial distance

( ) ( ) ( ) ( ) ( )2 22 2 2 2 2 21 / 2 / 1 /r X Y y Z z R y z R Yy Zz R R Yy Zz R⎡ ⎤= + − + − = + + − + ≈ − +⎣ ⎦

↑ In the Fraunhofer region R → ∞ The total electric field at P

( )

( )/2 /2 /

/2 /2

i kR ty b z a ik Yy Zz RAy b z a

E eE e dydzR

ω−= = − +

=− =−= ∫ ∫ (4)

→ ( ) sin ' sin '

'

i kR tAabE eER

ω α βα β

− ⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

: ' /2 , ' /2kaZ R kbY Rα β= =

The irradiance

22sin ' sin '( , ) (0)

' 'I Y Z I α β

α β⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

The maxima along 'β axis : 2

(0)'m

IIβ

= for 3 5' , ,...2 2mπ πβ = ± ± and ' 0α =

The maxima along 'α axis : 2

(0)'m

IIα

= for 3 5' , ,...2 2mπ πα = ± ± and ' 0β =

Hecht; 11/28/2010; 10-8 E. The Circular Aperture

The electric field at P

( )

( )( )

( ) ( )2

/ / cos

0 0

i kR t i kR t aik Yy Zz R i k q RA A

Aperture

E e E eE e dS e d dR R

ω ω πρ φ

ρ φ

ρ ρ φ− −

− + − −Φ

= =

= ⇒∫∫ ∫ ∫

↑ ↑ ↑ 0Φ = due to axial symmetry

cos , sincos , sin

z yZ q Y q

ρ φ ρ φ= == Φ = Φ

= ( )02 /J k q Rπ ρ , Bessel function of zero order

→ ( )

212

i kR tAE e R kaqE a J

R kaq R

ω

π− ⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

The irradiance

( ) ( )2 22 2

1 12

/ 2 /2(0)

/ /A J kaq R J kaq RE AI I

kaq R kaq RR⎡ ⎤ ⎡ ⎤

= ⇒⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

↑ 2 2 2(0) /2AI E A R=

since ( )1 0/ 1/2

uJ u u

==

Using / sinq R θ=

( ) 2

12 sin(0)

sinJ ka

I Ika

θθ

⎡ ⎤= ⎢ ⎥

⎣ ⎦ : Airy pattern

The radius of the first dark ring : 1 1.222Rr

aλ

= → half Dλθ ≈ , D=2a, aperture dia.

Diffraction at the focal point of a lens : 1 1.22 fqDλ

≈

Hecht; 11/28/2010; 10-9 F. Resolution of Imaging Systems

Rayleigh criterion : Two spots can be resolved when one Airy disk falls on the first minimum of the second Airy disk. → ( )min

1.22 /Dϕ λΔ =

→ ( )min1.22 /l f DλΔ = : Limit of resolution

Resolving power is defined as ( )min

1/ ϕΔ or ( )min1/ lΔ

G. The Zeroth-Order Bessel Beam One solution of the differential wave equation is ( ) ( ) ( ), , , i k z t

oE r z t J k r e ωθ −⊥∝ : Bessel beam

The irradiance ( ) ( )2, , , oI r z t J k rθ ⊥∝ : Independent of z → No diffraction

Not possible to create the perfect Bessel beam due to its infinite extent.

Hecht; 11/28/2010; 10-10 H. The Diffraction Grating A diffraction grating produces periodic alterations in the phase and/or amplitude of the incident wave.

Transmission amplitude grating Reflection phase grating A multiple-slits modulates the amplitude. Ruling on a reflective surface Transmission phase grating Ruling of parallel notches on a flat glass plate. (Periodic change of the refractive index)

Grating equation Constructive interference condition sin ma mθ λ= : Normal incidence ( )sin sinm ia mθ θ λ− = : Oblique incidence The main disadvantage of these gratings → Most of the incident energy is wasted in the specular reflection Blazed grating can shift energy from the zeroth order

to a higher order

and i mθ θ are measured from the normal to the grating plane not to the groove surface. Grating Spectroscopy Two and Three Dimensional Gratings