wave description of optical imaging systems -...
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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
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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
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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
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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
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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
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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)
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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)
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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
′=
′=
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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
l→∞ plane waves propagatingat corresponding angles –x/l⇔ spatial frequencies –x/(λl)
superposition of superposition of
( ) yxl
yyl
xxiyxglyxg dd2- exp, );,( inout⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ ′
+⎟⎠⎞
⎜⎝⎛ ′
∝′′ ∫ λλπ
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MIT 2.71/2.710 Optics11/02/05 wk9-b-21
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 Optics11/02/05 wk9-b-22
Example: circular aperture
2r0 0
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
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MIT 2.71/2.710 Optics11/02/05 wk9-b-23
Gratings
MIT 2.71/2.710 Optics11/02/05 wk9-b-24
Diffraction from periodic array of holes
……
Λ
incidentplanewave
Period: ΛSpatial frequency: 1/Λ
A spherical wave is generatedat each hole;
we need to figure out how theperiodically-spaced spherical waves
interfere
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MIT 2.71/2.710 Optics11/02/05 wk9-b-25
Diffraction from periodic array of holes
……
Λ
incidentplanewave
Period: ΛSpatial frequency: 1/Λ
Interference is constructive in thedirection pointed by the parallel rays
if the optical path difference between successive rays
equals an integral multiple of λ(equivalently, the phase delay
equals an integral multiple of 2π)
Optical path differences
MIT 2.71/2.710 Optics11/02/05 wk9-b-26
Diffraction from periodic array of holes
Λ
d
θ d = Λ sinθ
Period: ΛSpatial frequency: 1/Λ
From the geometrywe find
Therefore, interference isconstructive iff
Λ=⇔
⇔=Λλθ
λθ
m
m
sin
sin
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MIT 2.71/2.710 Optics11/02/05 wk9-b-27
Diffraction from periodic array of holes
m=1
……
Λ
incidentplanewave
m=3
m=2
m=–1
m=–2m=–3
m=0
“straight-through” order (aka DC term)
Grating spatial frequency: 1/ΛAngular separation between diffracted orders: ∆θ ≈λ/Λ
several diffracted plane waves“diffraction orders”
1st diffracted order
2nd diffracted order
–1st diffracted order
MIT 2.71/2.710 Optics11/02/05 wk9-b-28
Fraunhofer diffraction from periodic array of holes
…
Λ
…
z →∞
Λ≈≈
λθθ msin
zmxΛ
≈′λ
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MIT 2.71/2.710 Optics11/02/05 wk9-b-29
Sinusoidal amplitude grating
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )⎥⎦⎤
⎢⎣⎡ +′+′+−′−′+′′×
⎭⎬⎫
⎩⎨⎧ ′+′
=
0000
222
out
41
41
21
expe,
zvyzuxzvyzuxyx
zyx
ziyxg
zi
λδλδλδλδδδ
λλ
λπ
x
y
u ≡ x’
v ≡ y’
Λ ψ
( ) ( )[ ] )(2cos1 21, 00in yvxuyxg ++= π
0
020
20
tan ,1vu
vu=
+=Λ ψ
MIT 2.71/2.710 Optics11/02/05 wk9-b-30
Sinusoidal amplitude grating
θ
……
incidentplanewave
0
1u
≡Λ
Only the 0th and ±1st
orders are visible
( ) ( )[ ]
xuixui
xuyxg
00 2 2
0in
e41
21e
41
2cos1 21,
ππ
π
−+ ++=
=+=
–θ
0uλλθ =Λ
≈
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MIT 2.71/2.710 Optics11/02/05 wk9-b-31
Sinusoidal amplitude grating
l→∞
oneplanewave
threeplanewaves
far field threeconverging
spherical waves
0th order
+1st order
–1st order
%25.6161 , %25
41
110 ===== −+ ηηη
1+η
1−η
0η
diffraction efficiencies
MIT 2.71/2.710 Optics11/02/05 wk9-b-32
Dispersion
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MIT 2.71/2.710 Optics11/02/05 wk9-b-33
Diffraction from a grating
…
Λ
…
z →∞
Λ≈≈
λθθ msin
MIT 2.71/2.710 Optics11/02/05 wk9-b-34
Dispersion from a grating
…
Λ
…
z →∞
white
Λ≈
(green)(green) λθ m
Λ≈
(blue)(blue) λθ m
Λ≈
(red)(red) λθ m
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MIT 2.71/2.710 Optics11/02/05 wk9-b-35
Prism dispersion vs grating dispersion
glassair
Blue light is refracted atlargerlarger angle than red:
normal dispersion
Blue light is diffracted atsmallersmaller angle than red:
anomalous dispersion