PDF 4. Fresnel and Fraunhofer Diffraction

4. 4. Fresnel Fresnel and and Fraunhofer Fraunhofer Diffraction

Diffraction

[ ]

3 2

max 2

2 ( )

)

4 ( ξ η

λ

π _{− + −}

>> x y

z 2

) (ξ²+η²

>>k

λ z

>>

z Full-wave

solution

Rayleigh- Sommerfeld

Fresnel (near field)

Fraunhofer (far field)

Vectoranalysis Scalar approximation

z (x,y) 1.6 mm 4 mm

(ξ,η)

D_ξ= 50μm

λ=

632.8nm

0.633mm

D_η= 25μm

O(x, y) = 2mm

O(x, y) = 1m

[ ]

3 2

max 2

2 ( )

)

4 ( ξ η

λ

π _{− + −}

>> x y

z 2

) (ξ²+η²

>>k

λ z

>>

z Full-wave

solution

Rayleigh- Sommerfeld

Fresnel (near field)

Fraunhofer (far field)

Vectoranalysis Scalar approximation

z (x,y) 1.6 mm 4 mm

(ξ,η)

D_ξ= 50μm

λ=

632.8nm

0.633mm

D_η= 25μm

O(x, y) = 2mm

O(x, y) = 1m

Apertures Near-field diffraction Far-field diffraction

Intensity Intensity

( ) P U ( ) P

²

I =

( ) P u ( ) P , t

²

I =

• Instantaneous Intensity

( ) ( ) P , t u P , t

²

I =

(Average over many oscillation cycles)

Huygens

Huygens - - Fresnel Fresnel Principle Principle

( ) ( ) ( )

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

Fresnel

Fresnel Approximation (I) Approximation (I)

⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝ + ⎛ −

⎟ ⎠

⎜ ⎞

⎝ + ⎛ −

≈

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}

⎭ ⎬

⎫

⎩ ⎨

⎧ − + −

= ∫ ∫

^∞

∞

−

Fresnel diffraction integral

( x y ) U ( ξ η ) ( h x ξ y η ) d ξ d η

U , = ∫ ∫

^∞

, − , −

∞

−

( ) = ⎢⎣ ⎡ (

²

+

²

) ⎥⎦ ⎤

exp 2

, x y

z jk z

j y e

x h

jkz

λ

LSI

Fresnel

Fresnel Approximation (II) Approximation (II)

Fresnel diffraction integral

( ) ( ) ( ) 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

η

ξ

λ ξ

/ ,

/ 2

2

2 2 2

2

, )

, (

=

= +

+

⎭ ⎬

⎫

⎩ ⎨

= F ⎧

Positive vs. Negative Phases 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

Accuracy of the

Accuracy of the Fresnel Fresnel Approximation Approximation

( ) ( )

[ ]

²max

2 3 2

4 ξ η

λ

π − + −

〉〉 x y

z

• Accuracy can be expected for much shorter distances

Fresnel

Fresnel Approximation and Angular Spectrum Approximation and Angular Spectrum

( ) ( ) ( )

0

1

2 exp ,

2 2

⎪ ⎪

⎩

⎪⎪ ⎨

⎧ ⎥⎦ ⎤

⎢⎣ ⎡ − −

=

^{X Y}

Y X

f z f

j f

f H

λ λ λ

π λ

2

1

2

+

_Y

〈

X

f

f

otherwise

( ) ( )

⎭ ⎬

⎫

⎩ ⎨

⎧ ⎥⎦ ⎤

⎢⎣ ⎡ +

=

^{2 2}

,

exp x y

j z z

j f e

f H

jkz Y

X

λ

π F λ

( )

[

^{2 2}

]

exp

_{X Y}

jkz

j z f f

e − +

= πλ

( ) ( ) ( ) ( )

2 1 2

1

2 2

2

2 X Y

Y X

f f f

f λ λ λ

λ − ≈ − −

−

Fresnel

Fresnel Diffraction between Diffraction between Confocal Confocal Spherical Surfaces 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

, ,

( )

{ }

e

jkz

Fraunhofer

Fraunhofer Diffraction Diffraction

( ) ( ) ( ξ η ) ξ η

λ η π

λ U ξ j z x y d d

z j e y e

x U

y z x

j k jkz

⎥⎦ ⎤

⎢⎣ ⎡ − +

= ∫ ∫

^∞

∞

−

+

2

exp ,

,

) 2 (

2 2

( )

2

max 2

2

η

ξ +

〉〉 k z

( )

{ }

_{f x z f y z}

y z x

j k jkz

Y X

z U j

e e

λ

η

λ

λ ξ

^{/ , /}

) 2 (

,

2 2

=

= +

= F

Examples of

Examples of Fraunhofer Fraunhofer Diffraction (I) Diffraction (I)

Fraunhofer diffraction from a

rectangular aperture. The central lobe of the pattern has half-angular widths

y y

x

x

λ / D and θ λ / D

θ = =

The Fraunhofer diffraction

pattern from a circular aperture

produces the Airy pattern with

the radius of the central disk

Examples of

Examples of Fraunhofer Fraunhofer Diffraction (II) Diffraction (II)

25 .

0

= 0

η

16

2

/

1

= m

η

+

16

2

/

1

= m

η

−

Examples of

Examples of Fraunhofer Fraunhofer Diffraction (III) Diffraction (III)

( ) ( ) ⎟

⎠

⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝

⎥⎦ ⎛

⎢⎣ ⎤

= ⎡

w f w

j m t

_A

rect 2 rect 2

2 2 sin

exp

, η π

₀

ξ ξ η

ξ

Fresnel

Fresnel Diffraction by Square Aperture 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=

( ^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.

Talbot Images Talbot Images

( ) [ m ( L ) ]

t

_A

1 cos 2 /

2

, η 1 πξ

ξ = +

( ) ⎥

⎦

⎢ ⎤

⎣

⎡ ⎟

⎠

⎜ ⎞

⎝ + ⎛

⎟ ⎠

⎜ ⎞

⎝

⎟ ⎛

⎠

⎜ ⎞

⎝ + ⎛

= L

m x L

x L

m z y

x

I πλ π 2 π

2 cos cos

cos 2

4 1

, 1

₂ ^{2 2}