today - massachusetts institute of technology · mit 2.71/2.710 optics 11/03/04 wk9-b-1 today •...

$: Today - Massachusetts Institute of Technology · MIT 2.71/2.710 Optics 11/03/04 wk9-b-1 Today • Diffraction from periodic transparencies: gratings • Grating dispersion • Wave$
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


……

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 Optics11/03/04 wk9-b-4


Λ

d

θ d = Λ sinθ


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


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


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


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´


( ) 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


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


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


MIT 2.71/2.710 Optics11/03/04 wk9-b-28


fz

x x’ x”

( ) , 1exp , 22

f ⎟⎟⎠

⎞⎜⎜⎝

⎛ ′′′′

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−

′′+′′=′′′′∴

fy

fxG

fz

fyxiyxg

λλλπ

( ) ( ) d d2exp, 1exp ,22

f yxf

yyxxiyxgfz

fyxiyxg ∫∫

⎭⎬⎫

⎩⎨⎧ ′′+′′−

⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−

′′+′′=′′′′

λπ

λπ

sphericalwave-front


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´


( )∫∫⎭⎬⎫

⎩⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ′

+⎟⎠⎞

⎜⎝⎛ ′

∝′′ yxl

yyl

xx-iyxglyxg dd2 exp, );,( inout λλπ

x

yf

x´

y´


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´


( )∫∫ ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡′⎟⎟

⎠

⎞⎜⎜⎝

⎛−+′⎟⎟

⎠

⎞⎜⎜⎝

⎛−∝′′ 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´


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.

today - massachusetts institute of technology · mit 2.71/2.710 optics 11/03/04 wk9-b-1 today •...

Documents