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MIT 2.71/2.710 Optics12/07/05 wk14-b-1
3D Optics
MIT 2.71/2.710 Optics12/07/05 wk14-b-2
2D Optics 3D Optics
lenses gratings
Light interacts with medium on a sequence of discrete surfaces
GRadient INdex optics(GRIN)
VolumeHolograms
Light interacts with medium throughout a volumethroughout a volume
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MIT 2.71/2.710 Optics12/07/05 wk14-b-3
MIT 2.71/2.710 Optics12/07/05 wk14-b-4
Imaging with traditional lenses
depthof
field
lens
A A imaged
B
B
blurred
image plane
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MIT 2.71/2.710 Optics12/07/05 wk14-b-5
Imaging with 3D lenses
depthof
field
3D lens
B
B
blocked
image plane
A
A imaged
Contrast Defocus ⇔
MIT 2.71/2.710 Optics12/07/05 wk14-b-7
z=0 µm z=50 µm
z=200 µm z=250 µm
originalobject
Microturbine provided by Chee Wei Wong, Alan Epstein, MIT
Object Distance = 46 cmObject Distance = 46 cm
Resolution accomplished Resolution accomplished ≤≤ 100 100 µµmmraw imagesraw images
profileprofilereconstructionreconstruction
Arnab Sinha, George BarbastathisMIT 3D Optical Systems groupOptics Express 11:3202, 2003
Shape recovery (“profilometry”) with 3D lenses
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MIT 2.71/2.710 Optics12/07/05 wk14-b-8
index ofrefraction
n0+n1n0– n1
3D lens: plane-to-plane wave volume hologram
MIT 2.71/2.710 Optics12/07/05 wk14-b-9
Making the (simplest) 3D lens
photosensitivematerial
referencepoint source
of
sθ
plane wavesignal beam
referenceplane
( ) ( )[ ]rkkr ⋅−+= rs10 cosnnnafter exposure
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MIT 2.71/2.710 Optics12/07/05 wk14-b-10
Operating the (simplest) 3D lens
3D objectobjective
lens 3D lens
collectorlens
digitalcamera
visiblecolumn
x∆ y∆
z∆ L
of
sθ
xy
z
mm1m100~ ÷µL
MIT 2.71/2.710 Optics12/07/05 wk14-b-11
Operating a multiplex 3D lens
3D objectobjective
lens
collectorlens
digitalcamera
visiblecolumns
3D lens(multiplex)
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MIT 2.71/2.710 Optics12/07/05 wk14-b-12
Operating a hyperspectral multiplex 3D lens
3D objectobjective
lens
collectorlens
digitalcamera
visiblerainbow slices
3D lens(multiplex)
MIT 2.71/2.710 Optics12/07/05 wk14-b-13
Rainbow volume holographic imaging
slice #1slice #1 slice #2slice #2
2½D shape2½D shape
Wenyang Sun, George BarbastathisMIT 3D Optical Systems group
Optics Letters 30:976, 2005
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MIT 2.71/2.710 Optics12/07/05 wk14-b-14
Multiplex imaging of 3D fluorescent object
3D object:fluorescent beads
in water
three three ““slicesslices”” throughthroughfluorescent (3D) object, stacked fluorescent (3D) object, stacked
along along longitudinallongitudinal direction direction ……
…… are viewed are viewed simultaneouslysimultaneously and and sideside--byby--sidesideon the digital cameraon the digital camera
Wenhai Liu,* Demetri Psaltis,* George Barbastathis***Caltech Optical Info. Proc. group **MIT 3D Optical Systems group
Optics Letters 27:854, 2002
MIT 2.71/2.710 Optics12/07/05 wk14-b-15
Camera pixel layout for 4D (3D spatial + spectral) imaging#1
λ1 λ2 λw
slice index #2 #N
M pixels
W pixels
3D object
#1#2 #Nslice index
WNMWzyx
×××=××× λ
#1
#2
#W
slitindex
camera die
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MIT 2.71/2.710 Optics12/07/05 wk14-b-16
3D opticalimage
traditionalcamera
0 100 200 300 400 500 600 70040
60
80
100
120
140
160
180
200
220
240
irradiance acrossimage cross-section
Sun Light (completely passive) illumination
~10cm
~5m
objective
0 100 200 300 400 500 600 7060
80
100
120
140
160
180
MIT 2.71/2.710 Optics12/07/05 wk14-b-17
Temporal Heterodyning
input signal
local oscillator
low pass filter
transducer
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MIT 2.71/2.710 Optics12/07/05 wk14-b-18
Temporal Heterodyning
input signal
local oscillator
low pass filter
transducer
( )φω +tA ii cos
( )tA LL cos ω
( )( )φωω −− tAA cos iLiL
MIT 2.71/2.710 Optics12/07/05 wk14-b-19
1D Spatial Heterodyning
input signal
local oscillator
( ){ }φ+xkiA ii cos
( ){ }xkiA LL cos
?
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MIT 2.71/2.710 Optics12/07/05 wk14-b-20
1D Spatial Heterodyning
input signal
local oscillator
( ){ }φ+xkiA ii exp
( ) ( ) 2SSSRR expexp φ++ xikAxikA
thintransparency
MIT 2.71/2.710 Optics12/07/05 wk14-b-21
1D Spatial Heterodyning
SRi kkk ++
SRi kkk −+
SRi kkk +−
SRi kkk −−
input signal
local oscillator
( ){ }φ+xkiA ii exp
( ) ( ) 2SSSRR expexp φ++ xikAxikA
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MIT 2.71/2.710 Optics12/07/05 wk14-b-22
1D Spatial Heterodyning
input signal
local oscillator
( ){ }φ+xkiA ii exp
( ) ( ) 2SSSRR expexp φ++ xikAxikA
MIT 2.71/2.710 Optics12/07/05 wk14-b-23
1D Spatial Heterodyning
input signal
local oscillator
( ){ }φ+xkiA ii exp
( ) ( ) 2SSSRR expexp φ++ xikAxikA
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MIT 2.71/2.710 Optics12/07/05 wk14-b-24
1D Spatial Heterodyning
SRi kkk +−
input signal
local oscillator
( ){ }φ+xkiA ii exp
( ) ( ) 2SSSRR expexp φ++ xikAxikA
MIT 2.71/2.710 Optics12/07/05 wk14-b-25
1D Spatial Heterodyning
SRi kkk +−
low pass filter
input signal
local oscillator
( ){ }φ+xkiA ii exp
( ) ( ) 2SSSRR expexp φ++ xikAxikA
( )( ) ⎭
⎬⎫
⎩⎨⎧
⎥⎦
⎤⎢⎣
⎡++
′+−
S
SRiSRi exp
φφxkkk
iAAA
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MIT 2.71/2.710 Optics12/07/05 wk14-b-26
1D Spatial Heterodyning in the Fourier domain
input signal
local oscillator
( ){ }φ+′′xki i exp
( ) ( ) 2SSSRR expexp φ+′′+′′ xikAxikA
ℑ⎟⎠⎞
⎜⎝⎛ −
πλδ2
ifkx
MIT 2.71/2.710 Optics12/07/05 wk14-b-27
1D Spatial Heterodyning in the Fourier domain
input signal
local oscillator
( ) ( ) 2SSSRR expexp φ+′′+′′ xikAxikA
ℑ ℑ
spatial filter
( )⎥⎦⎤
⎢⎣⎡ +−
−′π
λδ2
SRi kkkfx
⎟⎠⎞
⎜⎝⎛ −
πλδ2
ifkx ( ){ }φ+′′xki i exp
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MIT 2.71/2.710 Optics12/07/05 wk14-b-28
33D Spatial Heterodyning in the Fourier domain
input signal
local oscillator
( ){ }φ+′′⋅rk i exp i
( ) ( ) 2SSSRR expexp φ+′′⋅+′′⋅ rkrk iAiA
⎟⎠⎞
⎜⎝⎛ −
πλδ2
ifkx
ℑ ?
MIT 2.71/2.710 Optics12/07/05 wk14-b-29
33D Spatial Heterodyning in the Fourier domain
input signal
local oscillator
( ) ( ) 2SSSRR expexp φ+′′⋅+′′⋅ rkrk iAiA
⎟⎠⎞
⎜⎝⎛ −
πλδ2
ifkx
ℑ ?
( )⎥⎦⎤
⎢⎣⎡ +−
−′π
λδ2
SRi kkkfx
( ){ }φ+′′⋅rk i exp i
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MIT 2.71/2.710 Optics12/07/05 wk14-b-30
33D Spatial Heterodyning in the Fourier domain
input signal
local oscillator
( ) ( ) 2SSSRR expexp φ+′′⋅+′′⋅ rkrk iAiA
⎟⎠⎞
⎜⎝⎛ −
πλδ2
ifkx
ℑ ?
( )⎥⎦⎤
⎢⎣⎡ +−
−′π
λδ2
SRi kkkfx
( ){ }φ+′′⋅rk i exp i
MIT 2.71/2.710 Optics12/07/05 wk14-b-31
33D Spatial Heterodyning in the Fourier domain
input signal
local oscillator
( ) ( ) 2SSSRR expexp φ+′′⋅+′′⋅ rkrk iAiA
⎟⎠⎞
⎜⎝⎛ −
πλδ2
ifkx
ℑ ?
( )⎥⎦⎤
⎢⎣⎡ +−
−′π
λδ2
SRi kkkfx
( ){ }φ+′′⋅rk i exp i
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MIT 2.71/2.710 Optics12/07/05 wk14-b-32
Phase matching as a filter for 3D spatial heterodyning
Recording Bragg-matched readout
Diffracted beams from elemental thin slicesare all in phase in phase → strong diffraction
MIT 2.71/2.710 Optics12/07/05 wk14-b-33
Recording Bragg-mismatched readout
Diffracted beams from elemental thin slicesare out of phase out of phase → no diffraction
Phase matching as a filter for 3D spatial heterodyning
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MIT 2.71/2.710 Optics12/07/05 wk14-b-34
Recording Bragg-mismatched readout
Rθ∆
θλθsin2R LL
=Λ
=∆Angle Bragg selectivity
θ
Λ
L
θ
Phase matching as a filter for 3D spatial heterodyning
MIT 2.71/2.710 Optics12/07/05 wk14-b-35
33D Spatial Heterodyning in the Fourier domain
input signal
local oscillator
( ) ( ) 2SSSRR expexp φ+′′⋅+′′⋅ rkrk iAiA
⎟⎠⎞
⎜⎝⎛ −
πλδ2
ifkx
( )⎥⎦⎤
⎢⎣⎡ +−
−′π
λδ2
SRi kkkfx
propagation along z
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MIT 2.71/2.710 Optics12/07/05 wk14-b-36
33D Spatial Heterodyning in the Fourier domain
input signal
local oscillator
( ) ( ) 2SSSRR expexp φ+′′⋅+′′⋅ rkrk iAiA
⎟⎠⎞
⎜⎝⎛ −
πλδ2
ifkx
( )⎥⎦⎤
⎢⎣⎡ +−
−′π
λδ2
SRi kkkfx
propagation along z
MIT 2.71/2.710 Optics12/07/05 wk14-b-37
3D Spatial Heterodyning in k-space
Recording Bragg-matched readout
Rk
Sk
Kλπ2
radius
=k
RS kkK −=
pk
K
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MIT 2.71/2.710 Optics12/07/05 wk14-b-38
Recording Bragg-matched readout
Rk
Sk
Kλπ2
radius
=k
RS kkK −=
pk
K
dk
Kkk += Rd
3D Spatial Heterodyning in k-space
MIT 2.71/2.710 Optics12/07/05 wk14-b-39
Recording Bragg-mismatched readout
Rk
Sk
Kλπ2
radius
=k
RS kkK −=
pk
K
dk
( )[ ] zxxKkk ˆˆ ˆ dRd zk+⋅+=k=d k
⎟⎠⎞
⎜⎝⎛ ∆= LkII zd
2matched-Braggdiffracted 2
1sinc
zkd∆
x
z
3D Spatial Heterodyning in k-space
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MIT 2.71/2.710 Optics12/07/05 wk14-b-40
Applications: 3D holographic data storage
⎟⎟⎠
⎞⎜⎜⎝
⎛−
πλ
δ2
i,mfkx ( )yxgm ′′,
MIT 2.71/2.710 Optics12/07/05 wk14-b-41
Applications: 3D holographic data storage
⎟⎟⎠
⎞⎜⎜⎝
⎛−
πλ
δ2
i,nfkx ( )yxgn ′′,
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MIT 2.71/2.710 Optics12/07/05 wk14-b-42
Applications: 3D holographic optical correlators
( )yxgm , ⎟⎟⎠
⎞⎜⎜⎝
⎛−′
πλ
δ2
i,mfkx
strongest
MIT 2.71/2.710 Optics12/07/05 wk14-b-43
Applications: 3D holographic optical correlators
( )yxgn , ⎟⎟⎠
⎞⎜⎜⎝
⎛−′
πλ
δ2
i,nfkx
strongest
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MIT 2.71/2.710 Optics12/07/05 wk14-b-44
Applications: 3D holographic optical imaging
( )zyxf ,,
⎟⎠⎞
⎜⎝⎛∆
×⎟⎠⎞
⎜⎝⎛
′∆′
×
×⎟⎠⎞
⎜⎝⎛ ′−′
zz
xx
zyfkxf
sliceslit
,,2
0
πλ
MIT 2.71/2.710 Optics12/07/05 wk14-b-45
Applications: 3D holographic optical imaging
( )zyxf ,,
⎟⎠⎞
⎜⎝⎛∆
×⎟⎠⎞
⎜⎝⎛
′∆′
×
×⎟⎠⎞
⎜⎝⎛ ′−′
zz
xx
zyfkxf
sliceslit
,,2
0
πλ
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MIT 2.71/2.710 Optics12/07/05 wk14-b-46
Applications: 3D holographic hyperspectral imaging
( )λ,,, zyxf spectral components ofselected voxel are spread
out on detector plane
MIT 2.71/2.710 Optics12/07/05 wk14-b-47
Applications: 3D multiplex hyperspectral imaging
( )λ,,, zyxf spectral components ofmultiplemultiple voxelss areare spread
out on nonnon--overlapping locationsoverlapping locationsinin detector plane
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MIT 2.71/2.710 Optics12/07/05 wk14-b-48
Applications: probing unknown 3D holograms
⎟⎠⎞
⎜⎝⎛ −
πλδ2
0fkx ( )yxI ′′,
?
( )zyx ′′′′′′ ,,ε
MIT 2.71/2.710 Optics12/07/05 wk14-b-49
Optical imaging with 3D pupil
( )00 , yyxx −−δ ( )00 ,;, yxyxh ′′( )zyx ′′′′′′ ,,ε
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MIT 2.71/2.710 Optics12/07/05 wk14-b-50
Optical imaging with 3D pupil
( )00 , yyxx −−δ ( )00 ,;, yxyxh ′′
( ) { } ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ ′+′−
+⎟⎟⎠
⎞⎜⎜⎝
⎛ ′+⎟⎟
⎠
⎞⎜⎜⎝
⎛ ′+ℑ=′′
22
22
21
20
20
21
0
21
000 22
1,1,1,;,f
yxf
yxfy
fy
fx
fxyxyxh
λλλε
( )zyx ′′′′′′ ,,ε
MIT 2.71/2.710 Optics12/07/05 wk14-b-51
Optical imaging with 2D pupil
( )00 , yyxx −−δ ( )00 ,;, yxyxh ′′
( ) { } ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ ′+′−
+⎟⎟⎠
⎞⎜⎜⎝
⎛ ′+⎟⎟
⎠
⎞⎜⎜⎝
⎛ ′+ℑ=′′
22
22
21
20
20
21
0
21
000 22
1,1,1,;,f
yxf
yxfy
fy
fx
fxyxyxh
λλλε
( )zyx ′′′′′′ ,,ε
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MIT 2.71/2.710 Optics12/07/05 wk14-b-52
Optical imaging with 3D pupil
( )0xx −δ ( )0; xxh ′
( ) { } ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ ′−⎟⎟
⎠
⎞⎜⎜⎝
⎛ ′+ℑ=′
22
2
21
20
21
00 22
1,1;f
xf
xfx
fxxxh
λλε
( )zx ′′′′ ,ε
simplify to 1D lateral dimension + longitudinal dimension
MIT 2.71/2.710 Optics12/07/05 wk14-b-53
Optical imaging with 3D pupil: building the Hopkins matrix
( )xδ ( )0;xh ′( )zx ′′′′ ,ε
L
θs
( ) ( ) [ ]( ) 2ss cossinexpexp, θθε zxikikzzx ++=′′′′
Plane-to-plane wave hologramOn-axis reference
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MIT 2.71/2.710 Optics12/07/05 wk14-b-54
Optical imaging with 3D pupil: building the Hopkins matrix
( )xδ ( )0;xh ′( )zx ′′′′ ,ε
L
θs
MIT 2.71/2.710 Optics12/07/05 wk14-b-55
Optical imaging with 3D pupil: building the Hopkins matrix
( )xδ ( )0;xh ′
L
θs
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MIT 2.71/2.710 Optics12/07/05 wk14-b-56
Optical imaging with 3D pupil: building the Hopkins matrix
( )xx d−δ ( )xxh d;′
L
θs
MIT 2.71/2.710 Optics12/07/05 wk14-b-57
Optical imaging with 3D pupil: building the Hopkins matrix
( )xx d2−δ ( )xxh d2;′
L
θs
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MIT 2.71/2.710 Optics12/07/05 wk14-b-58
Optical imaging with 3D pupil: building the Hopkins matrix
( )0; xxh ′
L
θs
( )0xx −δ
x′
0x
MIT 2.71/2.710 Optics12/07/05 wk14-b-59
Optical imaging with 3D pupil: building the Hopkins matrix
( )0; xxh ′
L
θs
( )0xx −δ
x′
0x
Hopkinsmatrix
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MIT 2.71/2.710 Optics12/07/05 wk14-b-60
Incoherent imaging with 3D pupil
L
θs
object:incoherent
superpositionof point sources
( )( )( )( )( )
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−M
M
xfxf
fxfxf
d2d-0dd2 ( )
( )( )
( )( )
?
d2d0
dd2
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
′−
′−
′′
M
M
xgxg
gxgxg
image
MIT 2.71/2.710 Optics12/07/05 wk14-b-61
Incoherent imaging with 3D pupil
L
θs
( )( )( )( )( )
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−M
M
xfxf
fxfxf
d2d-0dd2
object:incoherent
superpositionof point sources
object
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MIT 2.71/2.710 Optics12/07/05 wk14-b-62
Incoherent imaging with 3D pupil
L
θs
object
( )( )( )( )( )
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−M
M
xfxf
fxfxf
d2d-0dd2
Hopkins matrix
MIT 2.71/2.710 Optics12/07/05 wk14-b-63
Incoherent imaging with 3D pupil
L
θs
object
( )( )( )
( )( )
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
′−
′−
′′
M
M
xgxg
gxgxg
d2d0
dd2 ( )
( )( )( )( )
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−M
M
xfxf
fxfxf
d2d-0dd2
image Hopkins matrix
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MIT 2.71/2.710 Optics12/07/05 wk14-b-64
Incoherent imaging with clear (2D) pupil
object
( )( )( )
( )( )
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
′−
′−
′′
M
M
xgxg
gxgxg
d2d0
dd2 ( )
( )( )( )( )
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−M
M
xfxf
fxfxf
d2d-0dd2
image Hopkins matrix
MIT 2.71/2.710 Optics12/07/05 wk14-b-65
Defocused response of 3D pupils
( )00 ,, zzyxx −−δ ( )00 ,;, zxyxq ′′( )zyx ′′′′′′ ,,ε
?
33
MIT 2.71/2.710 Optics12/07/05 wk14-b-66
Defocused response of 3D pupils
?
( )00 ,, zzyxx −−δ ( )00 ,;, zxyxq ′′( )zyx ′′′′′′ ,,ε
( ) ( ) ( )⎭⎬⎫
⎩⎨⎧ ′′+−×
⎭⎬⎫
⎩⎨⎧ ′′+′′
⎭⎬⎫
⎩⎨⎧=′′′′′′ ∫∫ 2
1
22
1
exp2exp,dd2exp,,f
zyxif
yyxxiyxypxzizyxPλ
πλ
πλ
π
( ) ( ) ( )zyxPzyxzyxg ′′′′′′×′′′′′′=′′′′′′ ,,,,,, ε
( ) ⎥⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛ ′+′−
′′=′′
22
22
22 211,,,
fyx
fy
fxGyxq
λλλ
( ) ( )0Fresnel, zyxp =
( )yxp ,
MIT 2.71/2.710 Optics12/07/05 wk14-b-67
Blinking spot of 3D pupil
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MIT 2.71/2.710 Optics12/07/05 wk14-b-68
Applications: 3D holographic optical imaging
( )zyxf ,,
⎟⎠⎞
⎜⎝⎛∆
×⎟⎠⎞
⎜⎝⎛
′∆′
×
×⎟⎠⎞
⎜⎝⎛ ′−′
zz
xx
zyfkxf
sliceslit
,,2
0
πλ
3D pupil
MIT 2.71/2.710 Optics12/07/05 wk14-b-69
3D optics for imaging: summary
1m1m 1cm1cm 100100μμmm 11μμmm 10nm10nm100m100mObject sizeObject size
Working distanceWorking distance
1mm1mm
10cm10cm
1m1m
1km1km
Microscopy
……
1cm1cm
Ladar/Lidar
35
MIT 2.71/2.710 Optics12/07/05 wk14-b-70
Probing unknown 3D holograms
⎟⎠⎞
⎜⎝⎛ −
πλδ2
0fkx ( )yxI ′′,
?
( )zyx ′′′′′′ ,,ε
MIT 2.71/2.710 Optics12/07/05 wk14-b-71
Diffraction from deformed 3D holograms
36
MIT 2.71/2.710 Optics12/07/05 wk14-b-72
Experimental arrangement: transmission geometry
3D
MIT 2.71/2.710 Optics12/07/05 wk14-b-73
Experiment: diffraction from indented 3D hologram
37
MIT 2.71/2.710 Optics12/07/05 wk14-b-74
Fringe deformation due to indenter (transmission)indenter tip
MIT 2.71/2.710 Optics12/07/05 wk14-b-75
Deformed transmission 3D hologram in k-space
38
MIT 2.71/2.710 Optics12/07/05 wk14-b-76
Deformed transmission 3D hologram in k-space
MIT 2.71/2.710 Optics12/07/05 wk14-b-77
Deformed transmission 3D hologram in k-space
39
MIT 2.71/2.710 Optics12/07/05 wk14-b-78
Experimental arrangement: reflection geometry
3D
MIT 2.71/2.710 Optics12/07/05 wk14-b-79
Experiment: diffraction from indented 3D hologram