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Overview from last week
• Optical systems act as linear shift-invariant (LSI) filters(we have not yet seen why)
• Analysis tool for LSI filters: Fourier transform– decompose arbitrary 2D functions into superpositions
of 2D sinusoids (Fourier transform)– use the transfer function to determine what happens to
each 2D sinusoid as it is transmitted through the system (filtering)
– recompose the filtered 2D sinusoids to determine the output 2D function (Fourier integral, aka inverse Fourier transform)
MIT 2.71/2.710 Optics11/01/04 wk9-a-1
Today
• Wave description of optical systems• Diffraction
– very short distances: near field, we skip– intermediate distances: Fresnel diffractionexpressed as a convolution– long distances (∞): Fraunhofer diffractionexpressed as a Fourier transform
MIT 2.71/2.710 Optics11/01/04 wk9-a-2
Space and spatial frequency representations
g(x,y)SPACE DOMAIN
G(u,v)
( ) ( ) ( ) yxyxgvuG vyuxi dd e , , 2 +−∫= π
2D2D Fourier transformFourier transform 2D2D Fourier integralFourier integralaka
inverse inverse 2D2D Fourier transformFourier transformSPATIAL FREQUENCY
DOMAIN
( ) ( ) ( ) vuvuGyxg vyuxi dd e , , 2 ++∫= π
MIT 2.71/2.710 Optics11/01/04 wk9-a-3
2D2D linear shift invariant systems
input outputconvolution with convolution with impulse responseimpulse response
gi(x,y) go(x’, y’)( ) ( ) ( ) yxyyxxhyxgyxg dd , ,, io −′−′=′′ ∫ inverse Fourier
inverse Fourier
transformtransformFo
urie
rFo
urie
r
trans
form
trans
form
LSILSI
multiplication with multiplication with transfer functiontransfer functionGi(u,v) Go(u,v)
( ) ( ) ( )vuHvuGvuG , ,, io =
MIT 2.71/2.710 Optics11/01/04 wk9-a-4
Wave description of optical imaging systems
MIT 2.71/2.710 Optics11/01/04 wk9-a-5
Thin transparenciescoherent
illumination:planewave
Transmission function:
( ){ } , exp ),(),(in yxiyxtyxg φ=>~λ
Field before transparency:
⎭⎬⎫
⎩⎨⎧=− λ
π zizyxa 2 exp),,(=0
Field after transparency:
⎭⎬⎫
⎩⎨⎧=+ λ
π ziyxgzyxa 2 exp ),(),,( in
=0
assumptions: transparency at z=0transparency thickness can be ignored<~50λ
MIT 2.71/2.710 Optics11/01/04 wk9-a-6
Diffraction: Huygens principle
MIT 2.71/2.710 Optics11/01/04 wk9-a-7
),(),,( in yxgzyxa =+
Field after transparency:
Field at distance d:contains contributions
from all spherical wavesemitted at the transparency,
summed coherently
incidentplanewave
d
Huygens principle: one point sourcex´
incomingplane wave
opaquescreen
sphericalwave
x=x0
l
MIT 2.71/2.710 Optics11/01/04 wk9-a-8
Simple interference: two point sourcesx´
incomingplane wave
opaquescreen
MIT 2.71/2.710 Optics11/01/04 wk9-a-9
d1
d2
intensity
a
alλ
=Λx=–a/2
x=a/2
l
MIT 2.71/2.710 Optics11/01/04 wk9-a-10
Two point sources interfering: math…
x´
intensity
alλ
=Λ
( ) ( )
( ) ( ).2cos12cos4
cos42exp2),(),(:Intensity
.cos42exp2
22exp
22exp42exp1
2/exp2/exp2exp1),(
:Amplitude
22
2
2
22
2
2
22
2
22
2
2222
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎭⎬⎫
⎩⎨⎧ ′⎟
⎠⎞
⎜⎝⎛+=⎟
⎠⎞
⎜⎝⎛ ′
=
=⎟⎠⎞
⎜⎝⎛ ′
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧ ′++′+=′′=′′
⎟⎠⎞
⎜⎝⎛ ′
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧′++′
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎭⎬⎫
⎩⎨⎧ ′
+⎭⎬⎫
⎩⎨⎧ ′−
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧′++′
+=
=⎥⎥⎦
⎤
⎢⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧ ′++′
+⎭⎬⎫
⎩⎨⎧ ′+−′
⎭⎬⎫
⎩⎨⎧−=′′
xl
all
xal
lxa
l
yaxili
liyxeyxI
lxa
l
yaxili
li
lxai
lxai
l
yaxili
li
lyaxi
lyaxili
liyxe
λπ
λλπ
λ
λπ
λπ
λπ
λ
λπ
λπ
λπ
λ
λπ
λπ
λπ
λπ
λ
λπ
λπ
λπ
λ
(paraxial approximation)
Diffraction: many point sourcesx´
incomingplane wave
opaquescreen
x
many spherical waves,tightly packedx=x1
x=x2
x=x3
x=…
l
MIT 2.71/2.710 Optics11/01/04 wk9-a-11
Diffraction: many point sources,attenuated & phase-delayed
x´
thintransparency
l
x
x=x1
x=x2
x=x3
x=…
{ }11 exp φit
{ }22 exp φit
{ }33 exp φit
incomingplane wave
MIT 2.71/2.710 Optics11/01/04 wk9-a-12
Diffraction: many point sourcesattenuated & phase-delayed, math
MIT 2.71/2.710 Optics11/01/04 wk9-a-13
incomingplane wave
Thin transparency x´
l
( )
( )
( )
( ) ...2exp
2exp
2exp
... from
wavespherical
fromwave
spherical
fromwave
sphericalfield
223
33
222
22
221
11
321
⎭⎬⎫
⎩⎨⎧ ′+−′
++
+⎭⎬⎫
⎩⎨⎧ ′+−′
++
+⎭⎬⎫
⎩⎨⎧ ′+−′
++=
=+⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛+⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛+⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=′
lyxxilii
lit
lyxxilii
lit
lyxxilii
lit
xxxx
λπ
λπφ
λ
λπ
λπφ
λ
λπ
λπφ
λ
x
( ) { } ( ) ( ) .ddexp),(exp),(2exp1,field22
yxl
yyxxiyxiyxtlili
yx⎭⎬⎫
⎩⎨⎧ −′+−′
⎭⎬⎫
⎩⎨⎧=′′ ∫∫ λ
πφλ
πλ
continuouslimit
( )yxg ,in
{ }),(exp),(),( in
yxiyxtyxgφ==
transmission function
Fresnel diffraction
( ) ( ) ( ) .ddexp),(2exp1,22
inout yxl
yyxxiyxglili
yxg⎭⎬⎫
⎩⎨⎧ −′+−′
⎭⎬⎫
⎩⎨⎧=′′ ∫∫ λ
πλ
πλ
( ) ⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧ +
⎭⎬⎫
⎩⎨⎧∗=′′
lyxili
liyxgyxg
λπ
λπ
λ
22
inout exp2exp1),(,
amplitude distributionat output plane
transparencytransmission
function(complex teiφ)
spherical wave@z=l
(aka Green’s function)
CONSTANT:NOT interesting
FUNCTION OF LATERAL COORDINATES:Interesting!!!
The diffracted field is the convolution convolution of thetransparency with a spherical wave
MIT 2.71/2.710 Optics11/01/04 wk9-a-14
Example: circular aperture
2r0
input field gin(x,y)
5m 8m 10m
x
y y’
x’
y’
x’
y’
x’
r0=10mm
gout(x,y;5m) gout(x,y;8m) gout(x,y;10m)
MIT 2.71/2.710 Optics11/01/04 wk9-a-15
Example: circular aperture
2r0
input field gin(x,y)
12m 15m 18m
x
y y’
x’
y’
x’
y’
x’
gout(x,y;12m)
r0=10mm
gout(x,y;15m) gout(x,y;18m)
MIT 2.71/2.710 Optics11/01/04 wk9-a-16
Example: circular aperture
2r0
input field gin(x,y)
20m 25m 30m
x
y y’
x’
y’
x’
y’
x’
gout(x,y;20m)
r0=10mm
gout(x,y;25m) gout(x,y;30m)
MIT 2.71/2.710 Optics11/01/04 wk9-a-17
Example: circular aperture
2r0
input field gin(x,y)
x’x
y
r0=??
z’
gout(x;z)
Image removed due to copyright concerns
MIT 2.71/2.710 Optics11/01/04 wk9-a-18
(from Hecht, Optics, 4th edition, page 494)
Fraunhofer diffractionpropagation distance l is “very large”
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) yxl
yyxxiyxgl
yxilili
yxg
yxl
yyxxilyxiyxg
lyxili
liyxg
yxl
yyyyxxxxiyxglili
yxg
yxl
yyxxiyxglili
yxg
dd2exp),(2exp1,
,dd2expexp),(2exp1,
,dd22exp),(2exp1,
,ddexp),(2exp1,
in
22
out
22
in
22
out
2222
inout
22
inout
⎭⎬⎫
⎩⎨⎧ ′+′−
⎭⎬⎫
⎩⎨⎧ ′+′
+≈′′
⎭⎬⎫
⎩⎨⎧ ′+′−
⎭⎬⎫
⎩⎨⎧ +
⎭⎬⎫
⎩⎨⎧ ′+′
+=′′
⎭⎬⎫
⎩⎨⎧ ′−+′+′−+′
⎭⎬⎫
⎩⎨⎧=′′
⎭⎬⎫
⎩⎨⎧ −′+−′
⎭⎬⎫
⎩⎨⎧=′′
∫∫
∫∫
∫∫
∫∫
λπ
λπ
λπ
λ
λπ
λπ
λπ
λπ
λ
λπ
λπ
λ
λπ
λπ
λ
( )λ
λ max22
22 yxllyx +>>⇔<<+approximation valid if
MIT 2.71/2.710 Optics11/01/04 wk9-a-19
Fraunhofer diffractionx
y
x´
),(out yxg
y´
l→∞ ′′( )yxg ,in
( ) yxl
yyl
xxiyxglyxg dd2- exp, );,( inout⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ ′
+⎟⎠⎞
⎜⎝⎛ ′
∝′′ ∫ λλπ
The “far-field” (i.e. the diffraction pattern at a largelongitudinal distance l equals the Fourier transform
of the original transparencycalculated at spatial frequencies
lyf
lxf yx λλ
′=
′=
MIT 2.71/2.710 Optics11/01/04 wk9-a-20
Fraunhofer diffraction
MIT 2.71/2.710 Optics11/01/04 wk9-a-21
x
y
x´
y´
l→∞
( )yxg ,in ),(out yxg ′′
l→∞ plane wave propagatingat angle –x/l⇔ spatial frequency –x/(λl)
spherical waveoriginating at x
Fraunhofer diffraction
MIT 2.71/2.710 Optics11/01/04 wk9-a-22
x
y
x´
y´
l→∞
( )yxg ,in ),(out yxg ′′
spherical wavesoriginating atvarious pointsalong x
l→∞ plane waves propagatingat corresponding angles –x/l⇔ spatial frequencies –x/(λl)
superposition of superposition of
( ) yxl
yyl
xxiyxglyxg dd2- exp, );,( inout⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ ′
+⎟⎠⎞
⎜⎝⎛ ′
∝′′ ∫ λλπ
Example: rectangular aperture
input field far field
x0
( ) ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=
00in rectrect,
yy
xxyxg ( ) ( ) ( )
⎟⎠⎞
⎜⎝⎛ ′
⎟⎠⎞
⎜⎝⎛ ′
×
⎭⎬⎫
⎩⎨⎧ ′+′
=
lyy
lxxyx
lyx
liyxg
li
λλ
λλ
λπ
0000
222
out
sincsinc
expe,
l→∞
free spacepropagation by
0xlλ
MIT 2.71/2.710 Optics11/01/04 wk9-a-23
Example: circular aperture
MIT 2.71/2.710 Optics11/01/04 wk9-a-24
2r00
22.1rlλ
l→∞
free spacepropagation by
input field far field
( )⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ +=
0
22
in circ,r
yxyxg ( ) ( ) ( )
( ) ( )
( ) ( )⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
′+′
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ′+′
×
⎭⎬⎫
⎩⎨⎧ ′+′
=
lyxr
lyxr
r
lyx
liyxg
li
λπ
λπ
π
λλ
λπ
220
220
1
20
222
out
2
2J
2
expe,
( ) ( )⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ ′+′l
yxrλ
π 2202
jinc
also known asAiry pattern, or