3D Opticsweb.mit.edu/course/2/2.710/OldFiles/Fall06/2.710-wk14-b-ho.pdf · 3 MIT 2.71/2.710 Optics 12/07/05 wk14-b-5 Imaging with 3D lenses depth of field 3D lens B B blocked image

1

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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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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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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input signal

local oscillator

( ){ }φ+xkiA ii exp

( ) ( ) 2SSSRR expexp φ++ xikAxikA

thintransparency

MIT 2.71/2.710 Optics12/07/05 wk14-b-21


SRi kkk ++

SRi kkk −+

SRi kkk +−

SRi kkk −−

input signal

local oscillator



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input signal

local oscillator



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input signal

local oscillator



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SRi kkk +−

input signal

local oscillator



MIT 2.71/2.710 Optics12/07/05 wk14-b-25


SRi kkk +−

low pass filter

input signal

local oscillator



( )( ) ⎭

⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡++

′+−

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


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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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


input signal

local oscillator


⎟⎠⎞

⎜⎝⎛ −

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


input signal

local oscillator


⎟⎠⎞

⎜⎝⎛ −

πλδ2

ifkx

ℑ ?

( )⎥⎦⎤

⎢⎣⎡ +−

−′π

λδ2

SRi kkkfx

( ){ }φ+′′⋅rk i exp i

MIT 2.71/2.710 Optics12/07/05 wk14-b-31


input signal

local oscillator


⎟⎠⎞

⎜⎝⎛ −

πλδ2

ifkx

ℑ ?

( )⎥⎦⎤

⎢⎣⎡ +−

−′π

λδ2

SRi kkkfx

( ){ }φ+′′⋅rk i exp i

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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

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Recording Bragg-mismatched readout

Diffracted beams from elemental thin slicesare out of phase out of phase → no diffraction


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Rθ∆

θλθsin2R LL

=Λ

=∆Angle Bragg selectivity

θ

Λ

L

θ


MIT 2.71/2.710 Optics12/07/05 wk14-b-35


input signal

local oscillator


⎟⎠⎞

⎜⎝⎛ −

πλδ2

ifkx

( )⎥⎦⎤

⎢⎣⎡ +−

−′π

λδ2

SRi kkkfx

propagation along z

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MIT 2.71/2.710 Optics12/07/05 wk14-b-36


input signal

local oscillator


⎟⎠⎞

⎜⎝⎛ −

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


Rk

Sk

Kλπ2

radius

=k

RS kkK −=

pk

K

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Rk

Sk

Kλπ2

radius

=k

RS kkK −=

pk

K

dk

Kkk += Rd


MIT 2.71/2.710 Optics12/07/05 wk14-b-39


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


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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


( )zyxf ,,

⎟⎠⎞

⎜⎝⎛∆

×⎟⎠⎞

⎜⎝⎛

′∆′

×

×⎟⎠⎞

⎜⎝⎛ ′−′

zz

xx

zyfkxf

sliceslit

,,2

0

πλ

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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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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

25

MIT 2.71/2.710 Optics12/07/05 wk14-b-50


( )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


( )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

26

MIT 2.71/2.710 Optics12/07/05 wk14-b-52


( )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

27

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( )xδ ( )0;xh ′( )zx ′′′′ ,ε

L

θs

MIT 2.71/2.710 Optics12/07/05 wk14-b-55


( )xδ ( )0;xh ′

L

θs

28

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( )xx d−δ ( )xxh d;′

L

θs

MIT 2.71/2.710 Optics12/07/05 wk14-b-57


( )xx d2−δ ( )xxh d2;′

L

θs

29

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( )0; xxh ′

L

θs

( )0xx −δ

x′

0x

MIT 2.71/2.710 Optics12/07/05 wk14-b-59


( )0; xxh ′

L

θs

( )0xx −δ

x′

0x

Hopkinsmatrix

30

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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


L

θs

( )( )( )( )( )

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−M

M

xfxf

fxfxf

d2d-0dd2

object:incoherent

superpositionof point sources

object

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L

θs

object

( )( )( )( )( )

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−M

M

xfxf

fxfxf

d2d-0dd2

Hopkins matrix

MIT 2.71/2.710 Optics12/07/05 wk14-b-63


L

θs

object

( )( )( )

( )( )

=

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

′−

′−

′′

M

M

xgxg

gxgxg

d2d0

dd2 ( )

( )( )( )( )

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−M

M

xfxf

fxfxf

d2d-0dd2

image Hopkins matrix

32

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

34

MIT 2.71/2.710 Optics12/07/05 wk14-b-68


( )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

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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

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MIT 2.71/2.710 Optics12/07/05 wk14-b-77


39

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Experimental arrangement: reflection geometry

3D

MIT 2.71/2.710 Optics12/07/05 wk14-b-79

Experiment: diffraction from indented 3D hologram

40

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Fringe deformation due to indenter (reflection)

z

indenter tip

MIT 2.71/2.710 Optics12/07/05 wk14-b-81

Deformed reflection 3D hologram in k-space

Documents

3D Opticsweb.mit.edu/course/2/2.710/OldFiles/Fall06/2.710-wk14-b-ho.pdf · 3 MIT 2.71/2.710 Optics 12/07/05 wk14-b-5 Imaging with 3D lenses depth of field 3D lens B B blocked image