18. Fresnel Diffraction

(1)

18. Fresnel Diffraction

• This is most general form of diffraction

– No restrictions on optical layout

• near-field diffraction

• curved wavefront

– Analysis difficult

Fresnel Diffraction Fresnel Diffraction

Screen

Obstruction

(2)

Holmholtz Equation

( ) ^P ^t ^A ( ) ^P [ ^t ( ) ^P ]

u , = cos 2 πν + φ

( ) ^P ^t { ^U ( ) ^P ( ^j ^t ) }

u , = Re exp − 2 πν

( ) ^P ^A ( ) ^P [ ^j ( ) ^P ]

U = exp − φ

2

0

2 2

2

=

∂

− ∂

∇ t

u c

u n

( ^∇ ² ⁺ ^k ² ) ^U ⁼ ⁰ Holmholtz Equation

λ π π ν ²

2 =

= n c

k

(3)

Green’s Theorem

Let 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

( ) ^ds

n G U n

U G d

U G

G U

s

∫∫

∫∫∫ ^∇ ⁻ ^∇ ^υ ⁼ ^ _ ^ ^∂ _∂ ⁻ ^∂ _∂ ^ _ ^

ν

2 2

Where signifies a partial derivative in the outward normal direction at each point on S.

∂n

∂

(4)

Integral Theorem of Holmholtz and Kirchhoff

( ) ( )

01 01 1

exp r P jkr

G =

Kirchhoff’s G

( ) ( ) ( )

r ds jkr U n

r jkr n

P U U

S

∫∫   

 

 



 





∂

− ∂

 

 





∂

= ∂

01 01 01

01 0

exp exp

4 1

π

Free-space Green’s function

(5)

Kirchhoff’s Formulation of Diffraction

( ) ^ds

n U G

n G P U

U ∫∫

∑

 

 





∂

− ∂

∂

= ∂ π 4

1

0

Kirchhoff boundary conditions

are and

On

- ∑ , U ∂ U/ ∂ n

screen.

no were

there if

as

0. and 0

, On

- S

₁

U = ∂ U/ ∂ n =

(6)

Fresnel-Kirchhoff’s Diffraction Formula (I)

( ) ( ) ( )

01 01 01

01

1

1 exp

,

cos r

jkr jk r

r n n

P

G 



 



 −

∂ =

∂ G G

( ) ( ) ( ^>> ^λ )

≈

₀₁

01 01

01

exp for

,

cos r

r r jkr

n jk G G

( ) ( )

21 21 1

exp r

jkr P A

U =

( ) [ ( ) ] ( ) ( )

r ds n r

n r

r

r r

jk j

P A

U  

  −

= ∫∫ +

∑

2 , cos ,

cos

exp

₀₁ ₂₁

01 21

0

G G G

G λ

* Reciprocity Theorem of Helmholtz

(7)

Fresnel-Kirchhoff’s Diffraction Formula

( ) ( ) ( (II) )

r ds P jkr

U P

U

01 01 1

0

∫∫ exp

∑

= ′

( ) ( )

 

 



= 

′

21

21 1

exp 1

r

jkr A

P j

U λ

( ) ( )

 

  −

2 , cos ,

cos n G r G

₀₁

n G r G

₂₁

• 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.

(8)

First Rayleigh-Sommerfeld Solution

( ) ^ds

n U G n G

P U U

S

∫∫ ^ _ ^ ^∂ _∂ ⁻ ^∂ _∂ ^ _ ^

= 4

1

0

π

( ) ( ) ( )

01 01 01

01

1

~

exp ~ exp

r r jk r

P jkr

G

₋

= −

( ) ^ds

n U G P

U

_I

∫∫

∑

−

∂

= − π 4

1

0

( ) ( )

n P G n

P G

∂

= ∂

∂

₋ ₁ ₁

2 ( ) ^ds

n U G P

U

_I

∫∫

∑

∂

= − π 2

1

0

(9)

Second Rayleigh-Sommerfeld Solution

( ) ( ) ( )

01 01 01

01

1

~

exp ~ exp

r r jk r

P jkr

G

₊

= +

( ) ^G ^ds

n P U

U

_II ₊

∫∫

∑

^∂ _∂

= 4 π 1

0

G G

₊

= 2

( ) ^Gds

n P U

U

_II

∫∫

∑

∂

= ∂

π 2

1

0

(10)

Rayleigh-Sommerfeld Diffraction Formula

( ) ( ) ( ) ( ) ⁿ ^r ^ds

r P jkr

j U P

U

_I ₀₁

01 01 1

0

1 exp cos G , G

∫∫

∑

= λ

( ) ( ) ( )

r ds jkr n

P P U

U

_II

01 01 1

0

exp 2

1 ∫∫

∑

∂

= ∂ π

( ) ( )

21 21 1

exp r A jkr

P

U =

( ) [ ( ) ] ₍ ₎

s d r r n

r

r r

jk j

P A

U

_I

∫∫

∑

= +

₀₁

01 21

0

exp cos G , G

λ

( ) [ ( ) ] ₍ ₎

s d r r n

r

r r

jk j

P A

U

_II

∫∫

∑

− +

=

₂₁

01 21

0

exp cos G , G

λ

For the case of a spherical wave illumination,

(11)

Comparison (I)

( ) ^ds

n U G

n G P U

U ∫∫

_K ^K

∑

 

 





∂

− ∂

∂

= ∂ π 4

1

0

( ) ∫∫

∑

∂

− ∂

= ds

n U G

P

U

^K

π 2

1

0 1

( ) ^G ^ds

n P U

U

_II

∫∫

_K

∑

∂

= ∂ π 2

1

0

(12)

Comparison (II)

( ) [ ( ) ]

r ds r

r r

jk j

P A

U ψ

λ

exp

01 21

0

∫∫

∑

= +

( ) ( )

[ ]

( )















−

=

21 01

, cos

, 2 cos

1

r n

r n r

n

G G

G G G

G

ψ

Fresnel-Kirchhoff theory

First Rayleigh-Sommerfeld solution Second Rayleigh-Sommerfeld solution

* For a normal plane wave incidence,

[ ]













 +

= 1

cos

cos 2 1

1 θ

θ ψ

Fresnel-Kirchhoff theory

First Rayleigh-Sommerfeld solution

Second Rayleigh-Sommerfeld solution

(13)

Huygens-Fresnel Principle

( ) ( ) ( )

r ds P jkr

j U P

U θ

λ ^cos

exp 1

01 01 1

0

∫∫

∑

=

( ) ^P ^h ( ^P ^P ) ( ) ^U ^P ^ds

U

₀

∫∫

₀

,

₁ ₁

∑

=

( ) ( ) _θ

λ ^cos

exp , 1

01 01 1

0

r

jkr P j

P

h =

(14)

( ) ( ) ( )

r ds P jkr

j U P

U 1 exp cos

01 01 1

0

θ

λ ∫∫

∑

=

01

cos r

= z θ

( ) ( _ξ _η ) ( ) _ξ _η

λ r ^d ^d

U jkr j

y z x

U exp

,

₂

01

∫∫

01

∑

=

( ) (

²

)

²

2

01

= z + x − ξ + y − η

r

Only two assumptions : scalar theory + r

₀₁

>> λ

Huygens-Fresnel Principle

(15)

Fresnel Approximation

 





 



 



 

 +  −

 

 

 +  −

≈

2 2

01

2 1 2

1 1

z y z

z x

r ξ η

( ) _λ ( ) ^ξ ^η [ ( ^x ^ξ ) ( ^y ^η ) ] ^d ^ξ ^d ^η

z j k z U

j y e

x U

jkz

exp 2 ,

,

² ²

 



 

 − + −

= ∫ ∫

^∞

∞

−

( ) ⁽ ⁾ ( ) ^ξ ^η ⁽ ⁾ ⁽ ⁾ ^ξ ^η

λ

η λ ξ

η π

ξ

e d d

e U

z e j y e

x

U

^z ^j ^z ^x ^y

j k y

z x j k

jkz

∫ ∫

^∞

∞

−

+ + −

+

 



 

=

² ² ²

 ,

² ² ² ²

,

( )

( ) ⁽ ⁾

z y f z x f z

j k y

z x j k jkz

Y X

e U

z e j y e

x U

λ λ

η

ξ

λ ξ

/ ,

/ 2

2

2 2 2

2

, )

, (

=

= +

+

 



 

= F 

(16)

Positive vs. Negative Phases

y

Wavefront emitted

earlier

z Wavefront emitted

later

θ

z y

k G

Wavefront emitted

later

Wavefront emitted

earlier

(

₀₁

)

exp jkr

( ) _ ^

 

²

+

²

exp 2 x y

z j k

(

₀₁

)

exp − jkr

( ) _ ^

  −

²

+

²

exp 2 x y

z j k

( ^j ² ^πα ^y )

exp

( ^j ² ^y )

exp − πα

z=0

z=0 diverging

converging

(17)

Accuracy of the Fresnel Approximation

max

2 2

21 21 21

2 2

21 21 2 21 21

21 21

2 2

01 01 01

2 2

01 01 2 01 01

01 01 2

21 01

D= x-

(1 ) , since 2 2

1 1 1

2 Let

z r D

D D

z r z r

r r

z r D

D D

z r z r

r r r r D ξ

λ

∆ = − −

≈ − − ≈ ≅

∆ = − −

≈ − − ≈ ≅

 

∆ + ∆ =  +  >

 

( ) ( )

[ ]

²max

2 3 2

4 ξ η

λ

π ₋ ₊ ₋

〉〉 x y

z

• Accuracy can be expected for much shorter distances

Fresnel (near-field) Regime

(18)

Fresnel Diffraction between Confocal Spherical Surfaces

( ) ( ) ^ξ ^η ⁽ ⁾ ^ξ ^η

λ

η λ ξ

π

d d e

z U j

y e x

U

^j ^z ^x ^y

jkz ∞ − +

∞

∫ ∫

−

=

2

, ,

( )

{ }

_f _x _z _f _y _z

jkz

Y X

z U j

e

λ

η

λ

λ ξ ^,

⁼ ^/ ^, ⁼ ^/

= F

(19)

(20)

R

N

r

^N

r

o

N o

r _{= +} r N _{ } λ

    2

, , ,

N o o o

o o

o

N o o N

o

R r N r r N N

r r

R R r

Since R Nr f r f

r N N

λ λ λ

λ λ

   

   

=   +   − =   +      

= = = =

2 2 2

2 2

1 1

1

2 4

1

(21)

(22)

(23)

Fresnel Diffraction by Square Aperture

(b) Diffraction pattern at four axial positions marked by the arrows in (a) and

corresponding to the Fresnel numbers N_F=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 N_F=0.5.

( ^D)^d

x = 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.

(24)

Rectangular symmetric aperture

(25)

{ } {

²

} {

²

} {

²

}

²

*

2 1 2 1 2 1 2 1

1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

I = φφ = 4 ^  C ξ − C ξ + S ξ − S ξ ^{ }   C η − C η + S η − S η ^ 

18. Fresnel Diffraction