MIT 2.71/2.710 Optics11/03/04 wk9-b-1
Today
• Diffraction from periodic transparencies: gratings• Grating dispersion
• Wave optics description of a lens: quadratic phase delay• Lens as Fourier transform engine
MIT 2.71/2.710 Optics11/03/04 wk9-b-2
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
MIT 2.71/2.710 Optics11/03/04 wk9-b-3
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 Optics11/03/04 wk9-b-4
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
MIT 2.71/2.710 Optics11/03/04 wk9-b-5
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 Optics11/03/04 wk9-b-6
Fraunhofer diffraction from periodic array of holes
…
Λ
…
z →∞
Λ≈≈
λθθ msin
zmxΛ
≈′λ
MIT 2.71/2.710 Optics11/03/04 wk9-b-7
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 Optics11/03/04 wk9-b-8
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
≈
MIT 2.71/2.710 Optics11/03/04 wk9-b-9
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 Optics11/03/04 wk9-b-10
Dispersion
MIT 2.71/2.710 Optics11/03/04 wk9-b-11
Diffraction from a grating
…
Λ
…
z →∞
Λ≈≈
λθθ msin
MIT 2.71/2.710 Optics11/03/04 wk9-b-12
Dispersion from a grating
…
Λ
…
z →∞
white
Λ≈
(green)(green) λθ m
Λ≈
(blue)(blue) λθ m
Λ≈
(red)(red) λθ m
MIT 2.71/2.710 Optics11/03/04 wk9-b-13
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
MIT 2.71/2.710 Optics11/03/04 wk9-b-14
The ideal thin lens as a Fourier transform engine
MIT 2.71/2.710 Optics11/03/04 wk9-b-15
Fresnel diffraction
( ) ( ) ( ) .ddexp),(2exp1;,22
inout yxl
yyxxiyxglili
lyxg⎭⎬⎫
⎩⎨⎧ −′+−′
⎭⎬⎫
⎩⎨⎧=′′ ∫∫ λ
πλ
πλ
The diffracted field is the convolution convolution of the transparency with a spherical waveQ: how can we “undo” the convolution optically?
ReminderReminder
x
y
arbitrary l
x´
y´
),(out yxg ′′( )yxg ,in
coherentplane-waveillumination
MIT 2.71/2.710 Optics11/03/04 wk9-b-16
Fraunhofer diffraction
x
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
′=
′=
Q: is there another optical element who
can perform a Fourier
transformation without having to go
too far (to ∞ ) ?
ReminderReminder
MIT 2.71/2.710 Optics11/03/04 wk9-b-17
The thin lens (geometrical optics)
Ray bending is proportionalproportionalto the distanceto the distance from the axis
object at ∞(plane wave)
point object at finite distance(spherical wave)
f (focal length)
MIT 2.71/2.710 Optics11/03/04 wk9-b-18
The thin lens (wave optics)
incoming wavefronta(x,y)
outgoing wavefronta(x,y) t(x,y)eiφ(x,y)
(thin transparency approximation)
MIT 2.71/2.710 Optics11/03/04 wk9-b-19
The thin lens transmission function
∆0
∆(x,y)
( )
( )
( )
( ) ( ) ( )
( ) ( )⎭⎬⎫
⎩⎨⎧ +
⎟⎟⎠
⎞⎜⎜⎝
⎛−−−
⎭⎬⎫
⎩⎨⎧ ∆≈
⎭⎬⎫
⎩⎨⎧ ∆−+∆=
⎟⎟⎠
⎞⎜⎜⎝
⎛−
+−∆≈∆
⎟⎟⎠
⎞⎜⎜⎝
⎛⎥⎦
⎤⎢⎣
⎡ +−−+⎟
⎟⎠
⎞⎜⎜⎝
⎛⎥⎦
⎤⎢⎣
⎡ +−−−∆≈∆
⎟⎟⎠
⎞⎜⎜⎝
⎛ +−−+⎟
⎟⎠
⎞⎜⎜⎝
⎛ +−−−∆=∆
21112exp2exp,
,122exp,
112
,
211
211,
1111,
22
210lens
0lens
21
22
0
2
22
21
22
10
2
22
21
22
10
yxRR
niniyxa
yxniyxa
RRyxyx
RyxR
RyxRyx
RyxR
RyxRyx
λπ
λπ
λπ
λπ
MIT 2.71/2.710 Optics11/03/04 wk9-b-20
The thin lens transmission function
∆0
∆(x,y)
( )
( ) length focal theis 1111 where
exp2exp,
21
22
0lens
⎟⎟⎠
⎞⎜⎜⎝
⎛−−≡
⎭⎬⎫
⎩⎨⎧ +−
⎭⎬⎫
⎩⎨⎧ ∆≈
RRn
f
fyxiniyxa
λπ
λπ
this constant-phase term can be omitted
MIT 2.71/2.710 Optics11/03/04 wk9-b-21
Example: plane wave through lens
plane wave: exp{i2πu0x}angle θ0, sp. freq. u0≈ θ0 /λ
( )⎭⎬⎫
⎩⎨⎧ +−=
fyxiyxa
λπ
22
lens exp, lens,
MIT 2.71/2.710 Optics11/03/04 wk9-b-22
( ) { }
( ) { } ( )⎭⎬⎫
⎩⎨⎧ +−−=
⎭⎬⎫
⎩⎨⎧ +−=
+
+
fyfuxifuiyxa
fyxixuiyxa
λλπλπ
λππ
2202
0
22
0
expexp,
exp2exp,
:lensafter wavefront
Example: plane wave through lens
fu λ0
ignore
spherical wave,converging
off–axis
back focal plane
f
MIT 2.71/2.710 Optics11/03/04 wk9-b-23
Example: spherical wave through lens
( ) ( )
( ) { }⎭⎬⎫
⎩⎨⎧ +−∆=
⎭⎬⎫
⎩⎨⎧ ++
⎭⎬⎫
⎩⎨⎧=−
fyxiniyxa
fyxxifiyxa
f
λππ
λπ
λπ
22
0lens
220
exp2exp,
:functionion transmisslens
exp2exp,
:) distance propagated (has wavespherical
0x
spherical wave,diverging off–axis
front focal plane f
MIT 2.71/2.710 Optics11/03/04 wk9-b-24
Example: spherical wave through lens
( ) ( ) ( )⎭⎬⎫
⎩⎨⎧
++⎟⎠⎞
⎜⎝⎛ +∆=×= −+ f
xxif
xifniyxayxayxaλ
πλ
πλ
π 020
0lens 22exp,,,
lensafter wavefront
ignore
spherical wave,diverging off–axis
front focal plane
fx0 angleat ≈θ
plane wave
0x
MIT 2.71/2.710 Optics11/03/04 wk9-b-25
Diffraction at the back focal plane
fz
thintransparency
g(x,y)
thinlens
back focal planediffraction pattern
gf(x”,y”)
x x’ x”
MIT 2.71/2.710 Optics11/03/04 wk9-b-26
Diffraction at the back focal plane
fz
x x’ x”
1D calculation1D calculation
( ) ( ) ( )
( ) ( )
( ) ( ) ( ) xfxxixgxg
fxixgxg
xzxxixgxg
′⎭⎬⎫
⎩⎨⎧ ′−′′′=′′
⎭⎬⎫
⎩⎨⎧ ′−′=′
⎭⎬⎫
⎩⎨⎧ −′
=′
∫
∫
+
−+
−
d exp
exp
d exp
2
lensf
2
lenslens
2
lens
λπ
λπ
λπField before lens
Field after lens
Field at back f.p.
g(x,y) gf(x”,y”)
MIT 2.71/2.710 Optics11/03/04 wk9-b-27
Diffraction at the back focal plane
fz
x x’ x”
1D calculation1D calculation
( ) ( ) xfxxixg
fz
fxixg d 2exp 1exp
2
f ∫⎭⎬⎫
⎩⎨⎧ ′′−
⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−
′′=′′
λπ
λπ
( ) ( ) d d2exp, 1exp ,22
f yxf
yyxxiyxgfz
fyxiyxg ∫∫
⎭⎬⎫
⎩⎨⎧ ′′+′′−
⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−
′′+′′=′′′′
λπ
λπ
2D version2D version
g(x,y) gf(x”,y”)
MIT 2.71/2.710 Optics11/03/04 wk9-b-28
Diffraction at the back focal plane
fz
x x’ x”
( ) , 1exp , 22
f ⎟⎟⎠
⎞⎜⎜⎝
⎛ ′′′′
⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−
′′+′′=′′′′∴
fy
fxG
fz
fyxiyxg
λλλπ
( ) ( ) d d2exp, 1exp ,22
f yxf
yyxxiyxgfz
fyxiyxg ∫∫
⎭⎬⎫
⎩⎨⎧ ′′+′′−
⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛−
′′+′′=′′′′
λπ
λπ
sphericalwave-front
g(x,y) gf(x”,y”)
Fourier transformof g(x,y)
MIT 2.71/2.710 Optics11/03/04 wk9-b-29
Fraunhofer diffraction vis-á-vis a lensx
y
l→∞
x´
y´
),(out yxg ′′( )yxg ,in
( )∫∫⎭⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ ′
+⎟⎠⎞
⎜⎝⎛ ′
∝′′ yxl
yyl
xx-iyxglyxg dd2 exp, );,( inout λλπ
x
yf
x´
y´
),(out yxg ′′( )yxg ,in
f
( )∫∫ ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ ′+⎟⎟⎠
⎞⎜⎜⎝
⎛ ′∝′′ yx
fyy
fxx-iyxgfyxg dd2 exp, );,( inout λλ
π
MIT 2.71/2.710 Optics11/03/04 wk9-b-30
Spherical – plane wave dualityx
yf
x´
y´
),(out yxg ′′( )yxg ,in
( )∫∫ ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎦
⎤⎢⎣
⎡′⎟⎟
⎠
⎞⎜⎜⎝
⎛−+′⎟⎟
⎠
⎞⎜⎜⎝
⎛−∝′′ yxy
fyx
fxiyxgyxg dd2 exp, ),( inout λλ
π
point source at (x,y)amplitude gin(x,y)
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
fy
fx
λλ, towards
plane wave oriented
each output coordinate(x’,y’) receives ... ... a superposition ...
... of plane wavescorresponding to
point sources in the object
MIT 2.71/2.710 Optics11/03/04 wk9-b-31
Spherical – plane wave dualityx
yf
x´
y´
),(out yxg ′′( )yxg ,in
a plane wave departingfrom the transparency
at angle (θx, θy) has amplitudeequal to the Fourier coefficient
at frequency (θx/λ, θy /λ) of gin(x,y)
( ) ( ) ( )ffff yxyx θθλλθ
λλθ ,, towards =⎟⎟
⎠
⎞⎜⎜⎝
⎛××
produces a spherical wave converging
( )∫∫ ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ ′+⎟⎟⎠
⎞⎜⎜⎝
⎛ ′∝′′ yx
fyy
fxx-iyxgyxg dd2 exp, ),( inout λλ
π
each output coordinate(x’,y’) receives amplitude equal
to that of the correspondingFourier component
MIT 2.71/2.710 Optics11/03/04 wk9-b-32
Conclusions
• When a thin transparency is illuminated coherently by a monochromatic plane wave and the light passes through a lens, the field at the focal plane is the Fourier transform of the transparency times a spherical wavefront
• The lens produces at its focal plane the Fraunhoferdiffraction pattern of the transparency
• When the transparency is placed exactly one focal distance behind the lens (i.e., z=f ), the Fourier transform relationship is exact.