Wave description of optical imaging systems -...

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

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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

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

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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


2r0


12m 15m 18m

x

y y’

x’

y’

x’

y’

x’

gout(x,y;12m)

r0=10mm


MIT 2.71/2.710 Optics11/02/05 wk9-b-14


2r0


20m 25m 30m

x

y y’

x’

y’

x’

y’

x’

gout(x,y;20m)

r0=10mm


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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


2r0


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

′=

′=

10

MIT 2.71/2.710 Optics11/02/05 wk9-b-19


y

l→∞

x´

y´


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


y

l→∞

x´

y´


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

11

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


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

12

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

13

MIT 2.71/2.710 Optics11/02/05 wk9-b-25


……

Λ

incidentplanewave


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


Λ

d

θ d = Λ sinθ


From the geometrywe find

Therefore, interference isconstructive iff

Λ=⇔

⇔=Λλθ

λθ

m

m

sin

sin

14

MIT 2.71/2.710 Optics11/02/05 wk9-b-27


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

15

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


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

≈

16

MIT 2.71/2.710 Optics11/02/05 wk9-b-31


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

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