1 MIT 2.71/2.710 Optics 11/02/05 wk9-b-1 Wave description of optical imaging systems MIT 2.71/2.710 Optics 11/02/05 wk9-b-2 Thin transparencies coherent illumination: plane wave <~50λ >~λ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = − λ π z i z y x a 2 exp ) , , ( Field after transparency: ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ = + λ π z i y x g z y x a 2 exp ) , ( ) , , ( in ( ) { } , exp ) , ( ) , ( in y x i y x t y x g φ = Field before transparency: Transmission function: =0 =0 assumptions: transparency at z=0 transparency thickness can be ignored
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1
MIT 2.71/2.710 Optics11/02/05 wk9-b-1
Wave description of optical imaging systems
MIT 2.71/2.710 Optics11/02/05 wk9-b-2
Thin transparenciescoherent
illumination:planewave
<~50λ
>~λ
⎭⎬⎫
⎩⎨⎧=− λ
π zizyxa 2 exp),,(
Field after transparency:
⎭⎬⎫
⎩⎨⎧=+ λ
π ziyxgzyxa 2 exp ),(),,( in
( ){ } , exp ),(),(in yxiyxtyxg φ=
Field before transparency:
Transmission function:
=0
=0
assumptions: transparency at z=0transparency thickness can be ignored
2
MIT 2.71/2.710 Optics11/02/05 wk9-b-3
Diffraction: Huygens principleincident
planewave
),(),,( in yxgzyxa =+
Field after transparency:
d
Field at distance d:contains contributions
from all spherical wavesemitted at the transparency,
summed coherently
MIT 2.71/2.710 Optics11/02/05 wk9-b-4
Huygens principle: one point source
incomingplane wave
opaquescreen
l
x´
x=x0
sphericalwave
3
MIT 2.71/2.710 Optics11/02/05 wk9-b-5
Simple interference: two point sources
incomingplane wave
opaquescreen
x´
d1
d2
l
a
intensity
alλ
=Λx=–a/2
x=a/2
( ) ( ).2cos12cos4),(),(:Intensity
.cos42exp2),(:Amplitude
22
22
22
2
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎭⎬⎫
⎩⎨⎧ ′⎟
⎠⎞
⎜⎝⎛+=⎟
⎠⎞
⎜⎝⎛ ′
=′′=′′
⎟⎠⎞
⎜⎝⎛ ′
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧′++′
+=′′
xl
all
xal
yxeyxI
lxa
l
yaxili
liyxe
λπ
λλπ
λ
λπ
λπ
λπ
λ
MIT 2.71/2.710 Optics11/02/05 wk9-b-6
Diffraction: many point sources
incomingplane wave
opaquescreen
x´
l
x
x=x1
x=x2
x=x3
x=…
many spherical waves,tightly packed
4
MIT 2.71/2.710 Optics11/02/05 wk9-b-7
Diffraction: many point sources,attenuated & phase-delayed
incomingplane wave
thintransparency
x´
l
x
x=x1
x=x2
x=x3
x=…
{ }11 exp φit
{ }22 exp φit
{ }33 exp φit
MIT 2.71/2.710 Optics11/02/05 wk9-b-8
Diffraction: many point sourcesattenuated & phase-delayed, math
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
5
MIT 2.71/2.710 Optics11/02/05 wk9-b-9
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 Optics11/02/05 wk9-b-10
Diffraction from an obscuration
6
MIT 2.71/2.710 Optics11/02/05 wk9-b-11
Diffraction from a small obscuration
MIT 2.71/2.710 Optics11/02/05 wk9-b-12
Example: circular aperture
2r0
input field gin(x,y)
5m 8m 10m
x
y y’
x’
y’
x’
y’
x’
gout(x,y;5m)
r0=10mm
gout(x,y;8m) gout(x,y;10m)
7
MIT 2.71/2.710 Optics11/02/05 wk9-b-13
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 Optics11/02/05 wk9-b-14
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)
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MIT 2.71/2.710 Optics11/02/05 wk9-b-15
The blinking spot (Poisson spot)
MIT 2.71/2.710 Optics11/02/05 wk9-b-16
Example: circular aperture
2r0
input field gin(x,y)
z’
x
y
x’
gout(x;z)r0=??
(from Hecht, Optics, 4th edition, page 494)
9
MIT 2.71/2.710 Optics11/02/05 wk9-b-17
Fraunhofer diffraction
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) 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
propagation distance l is “very large”
( )λ
λ max22
22 yxllyx +>>⇔<<+approximation valid if
MIT 2.71/2.710 Optics11/02/05 wk9-b-18
Fraunhofer diffractionx
y
l→∞
x´
y´
),(out yxg ′′( )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
′=
′=
10
MIT 2.71/2.710 Optics11/02/05 wk9-b-19
Fraunhofer diffractionx
y
l→∞
x´
y´
),(out yxg ′′( )yxg ,in
spherical waveoriginating at x
l→∞ plane wave propagatingat angle –x/l⇔ spatial frequency –x/(λl)
MIT 2.71/2.710 Optics11/02/05 wk9-b-20
Fraunhofer diffractionx
y
l→∞
x´
y´
),(out yxg ′′( )yxg ,in
spherical wavesoriginating atvarious pointsalong x