18. Fresnel Diffraction

• This is most general form of diffraction

– No restrictions on optical layout • near-field diffraction

• curved wavefront

– Analysis difficult

Fresnel DiffractionFresnel Diffraction

Screen

Obstruction

Holmholtz Equation( ) ( ) ( )[ ]PtPAtPu φπν += 2cos,

( ) ( ) ( ){ }tjPUtPu πν2expRe, −=

( ) ( ) ( )[ ]PjPAPU φ−= exp

02

2

2

22 =

∂∂

−∇tu

cnu

( ) 022 =+∇ Uk Holmholtz Equation

λπνπ 22 ==

cnk

Green’s TheoremLet U(P) and G(P) be any two complex-valued functions of

position, and let S be a closed surface surrounding a volume V. If U, G, and their first and second partial derivatives are single-valued and continuous within and on S, then we have

( ) dsnUG

nGUdUGGU

s∫∫∫∫∫

∂∂

−∂∂

=∇−∇ υν

22

Where signifies a partial derivative in the outward normal

direction at each point on S.n∂

∂

Integral Theorem of Holmholtz and Kirchhoff

( ) ( )01

011

exprjkrPG =

Kirchhoff’s G

( ) ( ) ( ) dsr

jkrn

Ur

jkrnUPU

S∫∫

∂∂

−

∂∂

=01

01

01

010

expexp41π

Free-space Green’s function

Kirchhoff’s Formulation of Diffraction

( ) dsnGUG

nUPU ∫∫

∑

∂∂

−∂∂

=π41

0

Kirchhoff boundary conditions

are and On - nU/, U ∂∂∑screen. no were thereif as

0. and 0 ,On - 1 =∂∂= nU/US

Fresnel-Kirchhoff’s Diffraction Formula (I)

( ) ( ) ( )01

01

0101

1 exp1,cosr

jkrr

jkrnnPG

−=

∂∂

( ) ( ) ( )λ>>≈ 0101

0101 forexp,cos r

rjkrrnjk

( ) ( )21

211

expr

jkrAPU =

( ) ( )[ ] ( ) ( ) dsrnrnrr

rrjkjAPU

−+

= ∫∫∑ 2

,cos,cosexp 2101

0121

01210 λ

* Reciprocity Theorem of Helmholtz

Fresnel-Kirchhoff’s Diffraction Formula (II)

( ) ( ) ( )dsrjkrPUPU

01

0110

exp∫∫∑

′=

( ) ( )

=′

21

211

exp1r

jkrAj

PUλ

( ) ( )

−

2,cos,cos 2101 rnrn

• restricted to the case of an aperture illumination consisting of a single expanding spherical wave.

• Kirhhoff’s boundary conditions are inconsistent! : Potential theory says that

“If 2-D potential function and it normal derivative vanish together along any

finite curve segment, then the potential function must vanish over the entire plane”.

Rayleigh-Sommerfeld theory

• the scalar theory holds.

• Both U and G satisfy the homogeneous scalar wave equation.

•The Sommerfeld radiation condition is satisfied.

First Rayleigh-Sommerfeld Solution

( ) dsnGUG

nUPU

S∫∫

∂∂

−∂∂

=1

41

0 π

( ) ( ) ( )01

01

01

011 ~

~expexpr

rjkrjkrPG −=−

( ) dsn

GUPU I ∫∫∑

−

∂∂−

=π41

0

( ) ( )nPG

nPG

∂∂

=∂

∂ − 11 2

( ) dsnGUPU I ∫∫

∑ ∂∂−

=π21

0

Second Rayleigh-Sommerfeld Solution

( ) ( ) ( )01

01

01

011 ~

~expexpr

rjkr

jkrPG +=+

( ) dsGnUPU II +

∑∫∫ ∂

∂=

π41

0

GG 2=+

( ) GdsnUPU II ∫∫

∑ ∂∂

=π21

0

Rayleigh-Sommerfeld Diffraction Formula

( ) ( ) ( ) ( )dsrnrjkrPU

jPU I 01

01

0110 ,cosexp1

∫∫∑

=λ

( ) ( ) ( )dsrjkr

nPUPU II

01

0110

exp21∫∫∑ ∂∂

=π

( ) ( )21

211

exprjkrAPU =

( ) ( )[ ] ( ) sdrnrr

rrjkjAPU I ∫∫

∑

+= 01

0121

01210 ,cosexp

λ

( ) ( )[ ] ( ) sdrnrr

rrjkjAPU II ∫∫

∑

+−= 21

0121

01210 ,cosexp

λ

For the case of a spherical wave illumination,

Comparison (I)

( ) dsn

GUGnUPU K

K∫∫∑

∂∂

−∂∂

=π41

0

( ) ∫∫∑ ∂

∂−= ds

nGUPU K

π21

01

( ) dsGnUPU KII ∫∫

∑ ∂∂

=π21

0

Comparison (II)( ) ( )[ ] ds

rrrrjk

jAPU ψλ

exp

0121

01210 ∫∫

∑

+=

( ) ( )[ ]

( )

( )

−

−

=

21

01

2101

,cos

,cos

,cos,cos21

rn

rn

rnrn

ψ

Fresnel-Kirchhoff theory

First Rayleigh-Sommerfeld solution

Second Rayleigh-Sommerfeld solution

* For a normal plane wave incidence,

[ ]

+

=

1

cos

cos121

θ

θ

ψ

Fresnel-Kirchhoff theory

First Rayleigh-Sommerfeld solution

Second Rayleigh-Sommerfeld solution

Huygens-Fresnel Principle

( ) ( ) ( ) dsrjkrPU

jPU θ

λcosexp1

01

0110 ∫∫

∑

=

( ) ( ) ( )dsPUPPhPU 1100 ,∫∫∑

=

( ) ( ) θλ

cosexp1,01

0110 r

jkrj

PPh =

( ) ( ) ( ) dsr

jkrPUj

PU cosexp1

01

0110 θ

λ ∫∫∑=

01

cosrz

=θ

( ) ( ) ( ) ηξηξλ

ddr

jkrUjzyxU exp,, 2

01

01∫∫∑

=

( ) ( )22201 ηξ −+−+= yxzr

Only two assumptions : scalar theory + λ>>01r

Huygens-Fresnel Principle

Fresnel Approximation

−

+

−

+≈22

01 21

211

zy

zxzr ηξ

( ) ( ) ( ) ( )[ ] ηξηξηξλ

ddyxz

kjUzj

eyxUjkz

2

exp,, 22

−+−= ∫ ∫

∞

∞−

( ) ( ) ( ) ( ) ( )ηξηξ

ληξ

λπηξ

ddeeUezj

eyxUyx

zj

zkjyx

zkjjkz

∫ ∫∞

∞−

+−++

=2

222222

,,

( ) ( ) ( )

zyfzxf

zkjyx

zkjjkz

YX

eUezj

eyxUλλ

ηξηξ

λ/,/

222222

,),(==

++

= F

Positive vs. Negative Phasesy

Wavefront emitted earlier

z Wavefront emitted

later

θ

z

y k

Wavefront emitted

later

Wavefront emitted earlier

( )01exp jkr

( )

+ 22

2exp yx

zkj

( )01exp jkr−

( )

+− 22

2exp yx

zkj

( )yj πα2exp

( )yj 2exp πα−

z=0

z=0

diverging

converging

Accuracy of the Fresnel Approximation

max

2 221 21 21

2 2

21 21 21 2122121

2 201 01 01

2 2

01 01 01 0120101

221 01

21 01

D= x-

(1 ) , since 22

(1 ) , since 22

1 1 12

Let

z r D

D Dz r z rrr

z r D

D Dz r z rrr

Dr r

ξ

λ

∆ = − −

≈ − − ≈ ≅

∆ = − −

≈ − − ≈ ≅

∆ + ∆ = + >

( ) ( )[ ]2max223

4ηξ

λπ

−+−⟩⟩ yxz

• Accuracy can be expected for much shorter distances

Fresnel (near-field) Regime

Fresnel Diffraction between Confocal Spherical Surfaces

( ) ( ) ( )ηξηξ

ληξ

λπ

ddeUzj

eyxUyx

zjjkz +−

∞

∞−∫ ∫=

2

,,

( ){ }zyfzxf

jkz

YXU

zje

λληξ

λ /,/,

=== F

NR Nr

or

N or r N λ = + 2

, , ,

N o o oo o

oN o o N

o

NR r N r r Nr r

rR RSince R Nr f r fr N N

λ λ λ

λ λλ λ

= + − = +

= = = =

22 22 2 2

2 21 1

1

2 4

1

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.

Rectangular symmetric aperture

{ } { } { } { }2 2 2 2*2 1 2 1 2 1 2 1

1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )4

I C C S S C C S Sφφ ξ ξ ξ ξ η η η η = = − + − − + −

Fresnel integrals2 2

0 0( ) cos , ( ) sin2 2t tC dt S dtα απ πα α

= =

∫ ∫

Cornu spiral2 2

0 0( ) cos , ( ) sin2 2t tC dt S dtα απ πα α

= =

∫ ∫

Straight edge

Homework:

Plot the Fresnel diffraction patterns

of “your full name” object

at several distances from the object.