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4. 4. FresnelFresnel and and FraunhoferFraunhoferDiffractionDiffraction

Seoul Nat’l Univ. NRL HoloTech

[ ]3 2max

22 )()(4

ηξλπ

−+−>> yxz2

)( 22 ηξ +>>

kzλ>>z

Full-wave solution

Rayleigh-Sommerfeld

Fresnel(near field)

Fraunhofer(far field)

Vectoranalysis Scalar approximation

z(x,y)4 mm1.6 mm

(ξ,η)

Dξ= 50μm

λ=632.8nm

0.633mmDη= 25μm

O(x, y) = 2mm O(x, y) = 1m

[ ]3 2max

22 )()(4

ηξλπ

−+−>> yxz2

)( 22 ηξ +>>

kzλ>>z

Full-wave solution

Rayleigh-Sommerfeld

Fresnel(near field)

Fraunhofer(far field)

Vectoranalysis Scalar approximation

z(x,y)4 mm1.6 mm

(ξ,η)

Dξ= 50μm

λ=632.8nm

0.633mmDη= 25μm

O(x, y) = 2mm O(x, y) = 1m

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Apertures Near-field diffraction Far-field diffraction

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IntensityIntensity

( ) ( ) 2PUPI =

( ) ( ) 2, tPuPI =

• Instantaneous Intensity

( ) ( ) 2,, tPutPI =

(Average over many oscillation cycles)

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HuygensHuygens--FresnelFresnel PrinciplePrinciple

( ) ( ) ( ) dsr

jkrPUj

PU cosexp1

01

0110 θ

λ ∫∫∑=

01

cosrz

=θ

( ) ( ) ( ) ηξηξλ

ddr

jkrUjzyxU exp,, 2

01

01∫∫∑

=

( ) ( )22201 ηξ −+−+= yxzr

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FresnelFresnel Approximation (I)Approximation (I)

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

+⎟⎠⎞

⎜⎝⎛ −

+≈22

01 21

211

zy

zxzr ηξ

( ) ( ) ( ) ( )[ ] ηξηξηξλ

ddyxz

kjUzj

eyxUjkz

2

exp,, 22

⎭⎬⎫

⎩⎨⎧ −+−= ∫ ∫

∞

∞−

Fresnel diffraction integral

( ) ( ) ( ) ηξηξηξ ddyxhUyxU , ,, −−= ∫ ∫∞

∞−

( ) ( )⎥⎦⎤

⎢⎣⎡ += 22

2exp, yx

zjk

zjeyxh

jkz

λ

LSI

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FresnelFresnel Approximation (II)Approximation (II)

Fresnel diffraction integral

( ) ( ) ( ) ( ) ( )ηξηξ

ληξ

λπηξ

ddeeUezj

eyxUyx

zj

zkjyx

zkjjkz

∫ ∫∞

∞−

+−++

⎭⎬⎫

⎩⎨⎧

=2

222222

,,

( ) ( ) ( )

zyfzxf

zkjyx

zkjjkz

YX

eUezj

eyxUλλ

ηξηξ

λ/,/

222222

,),(==

++

⎭⎬⎫

⎩⎨⎧

= F

Seoul Nat’l Univ. NRL HoloTech

Positive vs. Negative PhasesPositive vs. Negative Phases

y

Wavefrontemitted earlier

z Wavefrontemitted

later

θ

z

y k

Wavefrontemitted

later

Wavefrontemitted earlier

( )01exp jkr

( )⎥⎦⎤

⎢⎣⎡ + 22

2exp yx

zkj

( )01exp jkr−

( )⎥⎦⎤

⎢⎣⎡ +− 22

2exp yx

zkj

( )yj πα2exp

( )yj 2exp πα−

z=0

z=0

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Accuracy of the Accuracy of the FresnelFresnel ApproximationApproximation

( ) ( )[ ]2max223

4ηξ

λπ

−+−⟩⟩ yxz

• Accuracy can be expected for much shorter distances

Seoul Nat’l Univ. NRL HoloTech

FresnelFresnel Approximation and Angular SpectrumApproximation and Angular Spectrum

( )( ) ( )

0

12exp,

22

⎪⎪⎩

⎪⎪⎨

⎧⎥⎦⎤

⎢⎣⎡ −−

=YX

YX

ffzjffH

λλλ

πλ122 ⟨+ YX ff

otherwise

( ) ( )⎭⎬⎫

⎩⎨⎧

⎥⎦⎤

⎢⎣⎡ += 22

, exp yxz

jzj

effHjkz

YX λπ

λF

( )[ ]22exp YXjkz ffzje +−= πλ

( ) ( ) ( ) ( )22

1122

22 YXYX

ffff λλλλ −−≈−−

Seoul Nat’l Univ. NRL HoloTech

FresnelFresnel Diffraction between Diffraction between ConfocalConfocal Spherical SurfacesSpherical Surfaces

( ) ( ) ( )ηξηξ

ληξ

λπ

ddeUzj

eyxUyx

zjjkz +−

∞

∞−∫ ∫=

2

,,

( ){ }zyfzxf

jkz

YXU

zje

λληξ

λ /,/,

=== F

Seoul Nat’l Univ. NRL HoloTech

FraunhoferFraunhofer DiffractionDiffraction

( ) ( ) ( ) ηξηξλπηξ

λddyx

zjU

zjeeyxU

yxz

kjjkz

⎥⎦⎤

⎢⎣⎡ +−= ∫ ∫

∞

∞−

+2exp,,

)(2

22

( )2

max22 ηξ +

⟩⟩kz

( ){ }zyfzxf

yxz

kjjkz

YXU

zjee

λληξ

λ /,/

)(2

,

22

==

+

= F

Seoul Nat’l Univ. NRL HoloTech

Examples of Examples of FraunhoferFraunhofer Diffraction (I)Diffraction (I)

Fraunhofer diffraction from a rectangular aperture. The central lobe of the pattern has half-angular widths

yyxx DD / and / λθλθ ==

The Fraunhofer diffraction pattern from a circular aperture produces the Airy pattern with the radius of the central disk subtending an angle D/22.1 λθ =

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Examples of Examples of FraunhoferFraunhofer Diffraction (II)Diffraction (II)

25.00 =η16/2

1 m=+η16/21 m=−η

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Examples of Examples of FraunhoferFraunhofer Diffraction (III)Diffraction (III)

( ) ( ) ⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛

⎥⎦⎤

⎢⎣⎡=

wwfmjt A 2

rect2

rect2sin2

exp, 0ηξξπηξ

( ) ( ) ⎟⎠⎞

⎜⎝⎛

⎥⎦⎤

⎢⎣⎡ −⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛≈ ∑

∞

−∞= zwyzqfx

zwmJ

zAyxI

qq λ

λλλ

2sinc2sinc2

, 20

222

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FresnelFresnel Diffraction by Square ApertureDiffraction 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=

( )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.

Seoul Nat’l Univ. NRL HoloTech

Talbot ImagesTalbot Images

( ) ( )[ ]LmtA /2cos121, πξηξ +=

( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛+=

Lxm

Lx

LzmyxI πππλ 2cos2coscos21

41, 22

2