13. Fresnel diffraction

Remind! Diffraction regimes

( ) ( )02

exp,

ikrzEE x y d di r

ξ ηλ ∑

= ∫∫

( ) ( )2 22r z x yξ η= + − + −Screen (x,y)Aperture (ξ,η)

r

( ) ( )00

exp( )

ikrEE P F dAi r

θλ ∑

= ∫∫

Fresnel-Kirchhoff diffraction formula

( ) cos zFr

θ θ= =

( ) ( )2 22 2

2 2 2 2

1 112 2 2 2

2 2 2 2

x yx yz zz z z z

x y x yzz z z z z z

ξ ηξ η

ξ η ξ η

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

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

Obliquity factor :

( ) ( ) ( )

( ) ( )

2 20

2 2

, exp exp2

exp exp 2

E kE x y ikz i x yi z z

k ki i x y d dz z

λ

ξ η ξ η ξ η∑

⎡ ⎤= +⎢ ⎥⎣ ⎦⎡ ⎤ ⎡ ⎤× + − +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∫∫

Screen (x,y)Aperture (ξ,η)

r( ) ( ) ( )

( ) ( )

( ) ( )

2 20

2 2

2 2

, exp exp2

exp exp 2

exp exp 2

E kE x y ikz i x yi z z

k ki i x y d dz z

k kC i i x y d dz z

λ

ξ η ξ η ξ η

ξ η ξ η ξ η

∑

∑

⎡ ⎤= +⎢ ⎥⎣ ⎦⎡ ⎤ ⎡ ⎤× + − +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤= + − +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∫∫

∫∫

( ) ( ) ( )2 2, ( , ) exp exp 2k kE x y C U i i x y d dz z

ξ η ξ η ξ η ξ η⎡ ⎤ ⎡ ⎤= + − +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∫∫

( ) ( )2 2

2( , ) ,kjzE x y U e

ξ ηξ η

+⎧ ⎫∝ ⎨ ⎬

⎩ ⎭F

( ) ( )

( )

, ( , ) exp

( , ) exp sin sin

kE x y C U i x y d dz

C U ik d dξ η

ξ η ξ η ξ η

ξ η ξ θ η θ ξ η

⎡ ⎤= − +⎢ ⎥⎣ ⎦

⎡ ⎤= − +⎣ ⎦

∫∫

∫∫

( ){ }( , ) ,E x y U ξ η∝F

Fresnel diffraction

Fraunhofer diffraction

This is most general form of diffraction– No restrictions on optical layout

• near-field diffraction• curved wavefront

– Analysis somewhat difficult

Fresnel Diffraction

Screen

z

Curved wavefront(parabolic wavelets)

Fresnel (near-field) diffraction

( ) ( )2 2

2( , ) ,kjzU x y U e

ξ ηξ η

+⎧ ⎫≈ ⎨ ⎬

⎩ ⎭F

λ16

2Dz ≥

( ) ( )[ ]2max223

4ηξ

λπ

−+−⟩⟩ yxz

• Accuracy can be expected for much shorter distances

( , ) 2 4for U smooth & slow varing function; x D zξ η ξ λ− = ≤

Fresnel approximation

Accuracy of the Fresnel Approximation

In summary, Fresnel diffraction is …

13-7. Fresnel Diffraction by Square Aperture

(b) Diffraction pattern at four axial positions marked by the arrows in (a) and corresponding to the Fresnel numbers NF=10, 1, 0.5, and 0.1. The shaded area represents the geometrical shadow of the slit. The dashed lines at represent the width of the Fraunhofer pattern in the far field. Where the dashed lines coincide with the edges of the geometrical shadow, the Fresnel number NF=0.5.

( )dDx /λ=

Fresnel Diffraction from a slit of width D = 2a. (a) Shaded area is the geometrical shadow of the aperture. The dashed line is the width of the Fraunhofer diffracted beam.

2 / : Fresnel numberFN a zλ=

2w2a

( ) ( ) ( ) ( )[ ] ηξηξηξλ

ddyxz

kjUzj

eyxUjkz

2

exp,, 22

⎭⎬⎫

⎩⎨⎧ −+−= ∫ ∫

∞

∞−

2 / : Fresnel numberFv N a zλΔ = =

Fresnel diffraction from a wire

Fresnel diffraction from a straight edge

From Huygens’ principle to Fresnel-Kirchhoff diffraction

Huygens’ principle

Given wave-front at t

Allow wavelets to evolve for time ∆t

r = c ∆t ≈ λ

New wavefront

What about –r direction?(π-phase delay when the secondary wavelets, Hecht, 3.5.2, 3nd Ed)

Construct the wave front tangent to the wavelets

Every point on a wave front is a source of secondary wavelets.i.e. particles in a medium excited by electric field (E) re-radiate in all directionsi.e. in vacuum, E, B fields associated with wave act as sources of additional fields

secondary wavelets

Secondarywavelet

Huygens’ wave front construction

Incompleteness of Huygens’ principle

Fresnel’s modification Huygens-Fresnel principle

Huygens’ Secondary wavelets on the wavefront surface O

O

P

Spherical wave from the point source SObliquity factor:

unity where θ=0 zero where θ = π/2

Huygens-Fresnel principle

daFer

er

EdaFerr

EE ikr

A

ikrs

rrik

Asp

pp

)(1'

1)('

1 ')'( θθ ∫∫∫∫ ⎥⎦⎤

⎢⎣⎡== +

Kirchhoff modificationFresnel’s shortcomings :

He did not mention the existence of backward secondary wavelets, however, there also would be a reverse wave traveling back toward the source.He introduce a quantity of the obliquity factor, but he did little more than conjecture about this kind.

Gustav Kirchhoff : Fresnel-Kirchhoff diffraction theoryA more rigorous theory based directly on the solution of the differential wave equation.He, although a contemporary of Maxwell, employed the older elastic-solid theory of light.He found F(θ) = (1 + cosθ )/2. F(0) = 1 in the forward direction, F(π) = 0 with the back wave.

⎟⎠⎞

⎜⎝⎛ <<= ∫∫ 22

, )(1'

1 ' πθπ -daFer

er

EE ikr

A

ikrsp

p

θ

Fresnel-Kirchhoff diffraction formula

( )ππθπ

<<⎭⎬⎫

⎩⎨⎧ +−

= +∫∫ θ -daerr

ikEE rrik

A

sp

p

, '

12cos1

2)'(

Arnold Johannes Wilhelm Sommerfeld : Rayleigh-Sommerfeld diffraction theoryA very rigorous solution of partial differential wave equation.The first solution utilizing the electromagnetic theory of light.

dar

eEi

EpA

ikr

Op cos1 θλ ∫∫=

This final formula looks similar to the Fresnel-Kirchhoff formula,therefore, now we call this the revised Fresnel-Kirchhoff formula,or, just call the Fresnel-Kirchhoff diffraction integral.

Fresnel-Kirchhoff diffraction integral

λ/2

Z1

Z2 Z3

Spherical wave from source Po

Huygens’ Secondary wavelets on the wavefront surface S

Obliquity factor: unity where χ=0 at C zero where χ=π/2 at high enough zone index

: Fresnel Zones

: Fresnel Zones

The average distance of successive zones from P differs by λ/2 -> half-period zones.Thus, the contributions of the zones to the disturbance at P alternate in sign,

Z1

Z2 Z3

For an unobstructed wave, the last term ψn=0.

Whereas, a freely propagating spherical wave from the source Po to P is

Therefore, one can assume that the complex amplitude of

1 exp( )iksi sλ

⎛ ⎞= ⎜ ⎟⎝ ⎠

(1/2 means averaging of the possible values,more details are in 10-3, Optics, Hecht, 2nd Ed)

: Diffraction of light from circular apertures and disks

(a) The first two zones are uncovered,

(b) The first zone is uncovered if point P is placed father away,

(c) Only the first zone is covered by an opaque disk,

12

1

1

: Babinet principle

RVariation of on-axis irradiance Diffraction patterns from

circular apertures

RP

(consider the point P at the on-axis P)

112ψ≈

Fresnel diffraction from a circular aperture

Poisson spot

Babinet principle

At screen At complementaryscreen

{ }Amplitude of Sψ

{ }Phase of Sψ

{ }Amplitude of CSψ S CS UNψ ψ ψ+ =

{ }Phase of CSψ

Sψ CSψ

without screen

: Straight edge

Damped oscillatingAt the edge

Monotonically decreasing

13-6. The Fresnel zone plateThe average distance of successive zones from P differs by λ/2 -> half-period zones.Thus, the contributions of the zones to the disturbance at P alternate in sign,

Assumeplane wavefronts

22 22 2 2

0 0 00 02 4n

nR r n r r nr r

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

( )0 0 nR nr rλ λ≈ >>

R1O

R3

R4

RN

R2

P

0 2r N λ

+

0r

Z1

Z2 Z3

If the even zones(n=even) are blocked 1 3 5( )Pψ ψ ψ ψ= + + + Bright spot at P

It acts as a lens!

2

0nRr

nλ=

( )0 0 nR nr rλ λ≈ >>

R1O

R3

R4

RN

R2

P

0 2r N λ

+

0r2

11 0 ( 1) Rf r n

λ= = =

Fresnel zone-plate lens

0 11

2 2nnR r Rn n

λ= =nR

( )sin sin tan nn m m m

m n

R mR mf R

λθ λ θ θ= ⇒ = =∼

( )( ) ( ) 11

1 12m n nRf R R nR

m mnλ λ⎛ ⎞= = ⎜ ⎟⎝ ⎠

21

mRfmλ

=

1f2f3f

Fresnel zone-plate lens has multiple foci.

Sinusoidal zone plate: This type has a single focal point.

Binary zone plate: The areas of each ring, both light and dark, are equal.It has multiple focal points.

Fresnel zone-plate lens

For soft X-ray focusing

Fresnel lens: This type has a single focal point.Focusing efficiency approaches 100%.