Seoul Nat’l Univ. NRL HoloTech 4. 4. Fresnel Fresnel and and Fraunhofer Fraunhofer Diffraction Diffraction
Seoul Nat’l Univ. NRL HoloTech
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
Seoul Nat’l Univ. NRL HoloTech
Apertures Near-field diffraction Far-field diffraction
Seoul Nat’l Univ. NRL HoloTech
IntensityIntensity
( ) ( ) 2PUPI =
( ) ( ) 2, tPuPI =
• Instantaneous Intensity
( ) ( ) 2,, tPutPI =
(Average over many oscillation cycles)
Seoul Nat’l Univ. NRL HoloTech
HuygensHuygens--FresnelFresnel PrinciplePrinciple
( ) ( ) ( ) dsr
jkrPUj
PU cosexp1
01
0110 θ
λ ∫∫∑=
01
cosrz
=θ
( ) ( ) ( ) ηξηξλ
ddr
jkrUjzyxU exp,, 2
01
01∫∫∑
=
( ) ( )22201 ηξ −+−+= yxzr
Seoul Nat’l Univ. NRL HoloTech
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
Seoul Nat’l Univ. NRL HoloTech
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
Seoul Nat’l Univ. NRL HoloTech
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 =
Seoul Nat’l Univ. NRL HoloTech
Examples of Examples of FraunhoferFraunhofer Diffraction (II)Diffraction (II)
25.00 =η16/2
1 m=+η16/21 m=−η
Seoul Nat’l Univ. NRL HoloTech
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
Seoul Nat’l Univ. NRL HoloTech
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