Chapter 16. Fraunhofer DiffractionChapter 16. Fraunhofer Diffraction

Fraunhofer ApproximationFraunhofer Approximation

( ) ( )22201 ηξ −+−+= yxzr

( ) ( ) ( ) ηξηξλ

ddr

jkrUjzyxU exp,, 2

01

01∫∫∑

=

Huygens-Fresnel Principle

Fraunhofer Approximation :

( ) ( ) ( ) ηξηξλπηξ

λddyx

zjU

zjeeyxU

yxz

kjjkz

⎥⎦⎤

⎢⎣⎡ +−= ∫ ∫

∞

∞−

+2exp,,

)(2

22

( )2

max22 ηξ +

⟩⟩kz

)(1)(21

)(21)(1)(

21

21

211

22

2222

22

01

ηξ

ηξηξ

ηξ

yxz

yxz

z

zyx

zyx

zz

zy

zxzr

+−++≈

+++−++=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

+⎟⎠⎞

⎜⎝⎛ −

+≈

FT

Fraunhofer DiffractionFraunhofer Diffraction

Fraunhofer diffraction• Specific sort of diffraction

– far-field diffraction– plane wavefront– Simpler maths

(참고) Fresnel Approximation

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

+⎟⎠⎞

⎜⎝⎛ −

+≈22

01 21

211

zy

zxzr ηξ

( ) ( ) ( ) ( )[ ] ηξηξηξλ

ddyxz

kjUzj

eyxUjkz

2

exp,, 22

⎭⎬⎫

⎩⎨⎧ −+−= ∫ ∫

∞

∞−

( ) ( ) ( ) ( ) ( )ηξηξ

ληξ

λπηξ

ddeeUezj

eyxUyx

zj

zkjyx

zkjjkz

∫ ∫∞

∞−

+−++

⎭⎬⎫

⎩⎨⎧

=2

222222

,,

( ) ( ) ( )

zyfzxf

zkjyx

zkjjkz

YX

eUezj

eyxUλλ

ηξηξ

λ/,/

222222

,),(==

++

⎭⎬⎫

⎩⎨⎧

= F

( ) ( )22201 ηξ −+−+= yxzr

Fresnel Diffraction• This is most general form of diffraction

– No restrictions on optical layout • near-field diffraction• curved wavefront

– Analysis difficult

Fresnel DiffractionFresnel Diffraction

Screen

Obstruction

16-1. Fraunhofer Diffraction from a Single Slit16-1. Fraunhofer Diffraction from a Single Slit• Consider the geometry shown below. Assume that the slit is very long in

the direction perpendicular to the page so that we can neglect diffraction effects in the perpendicular direction.

Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit

( )

( )

0

0

0

0

exp

0.

P

P

The contribution to the electric field amplitudeat point P due to the wavelet emanating fromthe element ds in the slit is given by

dEdE i kr tr

Let r r for the source element ds at sThen for any element

dEdEr

ω⎛ ⎞= −⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠

= =

⎛= ⎜

+ Δ⎝( ){ }0

0

exp

, ., , .

sinL L

i k r t

We can neglect the path difference in the amplitude term but not in the phase termWe let dE E ds where E is the electric field amplitude assumed uniform over the width of the slit

The path difference s

ω

θ

⎞+ Δ −⎡ ⎤⎟ ⎣ ⎦⎜ ⎟

⎠

Δ=

Δ =

( ){ } ( ) ( )

( ) ( )

/ 2

0 0 / 20 0

/ 2

00 / 2

.

exp sin exp exp sin

exp sinexp

sin

bL L

P P b

b

LP

b

Substituting we obtain

E ds EdE i k r s t E i kr t i k s dsr r

i k sEIntegrating we obtain E i kr tr i k

θ ω ω θ

θω

θ

−

−

⎛ ⎞ ⎛ ⎞= + − = −⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠

⎡ ⎤⎛ ⎞= −⎡ ⎤⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠ ⎣ ⎦

∫

Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit

( ) ( ) ( )

( ) ( ) ( )

( )

00

00

00

exp expexp

sin

1 sin2

exp exp exp2

exp2

LP

LP

L

Evaluating with the integral limits we obtain

i iEE i kr tr i k

where

k b

Rearranging we obtain

E bE i kr t i ir i

E bi kr tr i

β βω

θ

β θ

ω β ββ

ωβ

− −⎡ ⎤⎛ ⎞= −⎡ ⎤⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠ ⎣ ⎦

≡

⎛ ⎞= − − −⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎣ ⎦ ⎣ ⎦⎝ ⎠⎛ ⎞

= −⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠

( ) ( )00

2 2 2* 2

0 0 0 02 20

sin2 sin exp

1 1 sin sin sinc2 2

L

LP P

E bi i kr tr

The irradiance at point P is given by

E bI = c E E c I Ir

ββ ωβ

β βε ε ββ β

⎛ ⎞= −⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠

⎛ ⎞= = =⎜ ⎟

⎝ ⎠ ( )θβ sinsinsin 212

02

0 kbcIcII ==

Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit

2 2 2* 2

0 0 0 02 20

0 0

1 1 sin sin sinc2 2

sinsinc 1 0, lim sinc lim 1

1sin 0, sin 1, 2,2

LP P

The irradiance at point P is given by

E bI = c E E c I Ir

The function is for

The zeroes of irradiance occur when or when k b m m

β β

β βε ε ββ β

ββ ββ

β β θ π

→ →

⎛ ⎞= = =⎜ ⎟

⎝ ⎠

= = =

= = = = ± ± K

( )θβ sinsinsin 212

02

0 kbcIcII ==

Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit,

sin , 2 / ,

1 22

In terms of the length y on the observation screeny f and in terms of wavelength kwe can write

y b ybf f

Zeroes in the irradiance pattern will occur when

b y m fm yf b

The maximum in the irradiance pattern is at β= 0

θ λ π

π πβλ λ

π λπλ

≅ =

= =

= =

2 2

sin cos sin cos sin 0

sin tancos

.Secondary maxima are found from

dd

β β β β β ββ β β β β

ββ ββ

⎛ ⎞ −= − = =⎜ ⎟

⎝ ⎠

⇒ = =

1.43π

2.46π

3.47π

0

θβ sin21 kb=

β20 sin cII =

Fraunhofer Diffraction from a Single SlitFraunhofer Diffraction from a Single Slit

Note: x- and y-axes switched in book, Figs. 16-5a (here) and Fig. 16-1 do not match.

β20 sin cII =

16-2. Beam Spreading16-2. Beam Spreading

( )

sin

min

1 2

The angular width of the centralmaximum is defined as the angularseparation Δθ between the first minima oneither side of the central maximum,

yf

The first ima in the irradiance patternwill occur when

fm fy Δθb b b

The

θ θ

λλ λ

= ≅

±= = ⇒ =

2

width W of the diffraction pattern thusincreases linearly with distance from the slit,in the regions far from the slit where Fraunhoferdiffraction applies

LW = L Δθbλ

=

L

W

16-3. Rectangular Apertures16-3. Rectangular Apertures

When the length a and width b of therectangular aperture are comparable,a diffraction pattern is observed inboth the x - and y - dimensions, governedin each dimension by the formula wehave already developed. The irradiancepat

( )( )2 20 sinc sinc

1, sin2

tern is

I I

where k a

Zeroes in the irradiance pattern are observed when

m f m fy or xb a

α β

α θ

λ λ

=

=

= =

y

Circular AperturesCircular Apertures

xdsdA =

∫∫=Area

iskAp dAe

rEE θsin

0

222

2⎟⎠⎞

⎜⎝⎛+=

xsR 222 sRx −=

dssRerEE

R

R

iskAp

22sin

0

2−= ∫−

θ

θγ sin ,/ kRRsv ==

{ }⎭⎬⎫

⎩⎨⎧

=−= ∫− γγπγ )(212 1

0

221

10

2 Jr

REdvver

REE AviAp

(the first order Bessel function of the first kind)

Bessel FunctionsBessel Functions

832.3sin21sin === θθγ kDkR (first zero)

θγ sin21 kD=

( ) ( )( )

21

1 212

2 sin( ) 0

sinJ k D

I Ik D

θθ

θ⎡ ⎤

= ⎢ ⎥⎢ ⎥⎣ ⎦

⎭⎬⎫

⎩⎨⎧

=γγπ )(2 1

0

2 Jr

REE Ap

0)at (or, 0 when 21)(1 =→⎭⎬⎫

⎩⎨⎧

→ θγγγJ

1 1min2 2

2sin 3.832

First minimum in the Airy pattern is atDk D k D πθ θ θ

λ⎛ ⎞⎛ ⎞≅ = = ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

Fraunhofer Diffraction from Circular Apertures: The Airy Pattern

Fraunhofer Diffraction from Circular Apertures: The Airy Pattern

)0(/ II

Dλθθ 22.1

21min =Δ=

Airy patternsAiry patterns

Airy Disc

Slit and Circular AperturesSlit and Circular Apertures

Circular aperture

Single slit(sinc ftn)

Sin θ0 λ/D 2λ/D 3λ/D−λ/D−2λ/D−3λ/D

Intensity

16-4. Resolution16-4. Resolution

• Ability to discern fine details of object– Lord Rayleigh in 1896

» resolution is function of the Airy disc.

– Rayleigh: Limit of resolution» Two light sources must be separated by at least the

diameter of first dark band.» Called Rayleigh Criterion

Rayleigh CriterionRayleigh Criterion

Rayleigh Limit

Rayleigh LimitRayleigh Limit

Resolution limit of a lens:

(f = focal length)

NANA

Dfx

λλ

λ

61.02

22.1

22.1min

=⎟⎠⎞

⎜⎝⎛≈

⎟⎠⎞

⎜⎝⎛=

Fraunhofer Diffraction from a Double SlitFraunhofer Diffraction from a Double Slit

Now for the double slit we can imagine that we placean obstruction in the middle of the single slit. Then all that we have to do to calculate the field from the double slit is to change the limits of

( ) ( )( )

( )

( ) ( )( )

( )

( ) ( )( )

( )

/ 2

0 / 20

/ 2

0 / 20

/ 2

00 / 2

exp exp sin

exp exp sin

exp sin expexp

sin

a bL

P a b

a bL

a b

a b

LP

a b

integration.

EE i kr t i k s dsr

E i kr t i k s dsr

Integrating we obtain

i k s iEE i kr tr i k

ω θ

ω θ

θω

θ

+

−

− −

− +

+

−

⎛ ⎞= − +⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠

⎛ ⎞−⎡ ⎤⎜ ⎟ ⎣ ⎦

⎝ ⎠

⎛ ⎞ ⎡ ⎤= − +⎡ ⎤⎜ ⎟ ⎢ ⎥⎣ ⎦

⎣ ⎦⎝ ⎠

∫

∫

( )( )

( )

( ) ( ) ( )

( ) ( )

( ) ( )

/ 2

/ 2

0

0

0

0

sinsin

exp sin sinexp exp

sin 2 2

sin sinexp exp

2 2

expexp ex

2

a b

a b

L

LP

k si k

i kr t i k a b i k a bEr i k

i k a b i k a b

b i kr tEE ir i

θθ

ω θ θθ

θ θ

ωα

β

− −

− +

⎧ ⎫⎡ ⎤⎪ ⎪⎨ ⎬⎢ ⎥

⎣ ⎦⎪ ⎪⎩ ⎭− ⎧⎡ ⎤ + −⎛ ⎞ ⎡ ⎤ ⎡ ⎤⎪⎣ ⎦= −⎨⎜ ⎟ ⎢ ⎥ ⎢ ⎥

⎪ ⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎩⎫− − − +⎡ ⎤ ⎡ ⎤⎪+ − ⎬⎢ ⎥ ⎢ ⎥⎪⎣ ⎦ ⎣ ⎦⎭

−⎡ ⎤⎛ ⎞ ⎣ ⎦= ⎜ ⎟⎝ ⎠

( ) ( ) ( ) ( ) ( ){ }p exp exp exp exp

sin sin

i i i i i

where k a and k b

β β α β β

α θ β θ

− − + − − −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

= =

16-5. Fraunhofer Diffraction from a Double Slit16-5. Fraunhofer Diffraction from a Double Slit

( ) ( )( ) ( )

( ) ( )( )

( )

0

0

2 2* 2

0 0 020

exp exp 2 cos

exp exp 2 sin

exp2 cos 2 sin

2

1 1 4sin4cos 4 c2 2 4

LP

LP P

But we know thati i

i i i

Substituting we obtain

b i kr tEE ir i

The irradiance at point P is given by

E bI = c E E c Ir

α α α

β β β

ωα β

β

βε ε αβ

+ − =

− − =

−⎡ ⎤⎛ ⎞ ⎣ ⎦= ⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

22

2

2

0 00

sinos

12

LE bwhere I cr

βαβ

ε

⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟

⎝ ⎠

Fraunhofer Diffraction from a Double SlitFraunhofer Diffraction from a Double Slit

The irradiance at point P from a doubleslit is given by the product of thediffraction pattern from single slit and the interference pattern from a double slit.

22

0 2

sin4 cosI I βαβ

⎛ ⎞= ⎜ ⎟

⎝ ⎠

Fraunhofer Diffraction from a Double SlitFraunhofer Diffraction from a Double Slit

Single Slit

Double Slit

16-6. Fraunhofer Diffraction from Many Slits (Grating)

16-6. Fraunhofer Diffraction from Many Slits (Grating)

Now for the multiple slits we just need to againchange the limits of integration. For N even slits with width b evenly spaced a distance a apart, wecan place the origin of the coordinate system at t

( ) ( )( )

( ){/ 2 2 1 / 2

0 2 1 / 210

exp exp sinj N j a b

LP j a b

j

he center obstruction and label the slits with the index j (Note that the diagram does not exactly correspond with this).

EE i kr t i k s dsr

ω θ= − +⎡ ⎤⎣ ⎦

− −⎡ ⎤⎣ ⎦=

⎛ ⎞= −⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠

∑ ∫

( )( )

( ) }

( ) ( )( )

( )

( )( )

( )

2 1 / 2

2 1 / 2

2 1 / 2/ 2

010 2 1 / 2

2 1 /

2 1 / 2

exp sin

exp sinexp

sin

exp sinsin

j a b

j a b

j a bj NL

Pj j a b

j a b

j a b

i k s ds

Integrating we obtain

i k sEE i kr tr i k

i k si k

θ

θω

θ

θθ

− − +⎡ ⎤⎣ ⎦

− − −⎡ ⎤⎣ ⎦

− +⎡ ⎤⎣ ⎦=

= − −⎡ ⎤⎣ ⎦

− − +⎡ ⎤⎣ ⎦

− − −⎡ ⎤⎣ ⎦

+

⎧⎡ ⎤⎛ ⎞ ⎪= −⎡ ⎤ ⎨⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠ ⎣ ⎦⎪⎩

⎡ ⎤+ ⎢ ⎥⎣ ⎦

∫

∑

( ) ( ) ( )

( ) ( )

2

/ 20

10

exp 2 1 sin 2 1 sinexp exp

sin 2 2

2 1 sin 2 1 sinexp exp

2 2

j NL

j

i kr t i k j a b i k j a bEr i k

i k j a b i k j a b

ω θ θθ

θ θ

=

=

⎫⎪⎬⎪⎭⎧ ⎡ ⎤ ⎡ ⎤− − + − −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎪⎣ ⎦ ⎣ ⎦ ⎣ ⎦= −⎢ ⎥ ⎢ ⎥⎨⎜ ⎟

⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎪ ⎣ ⎦ ⎣ ⎦⎩⎫⎡ ⎤ ⎡ ⎤− − − − − −⎡ ⎤ ⎡ ⎤ ⎪⎣ ⎦ ⎣ ⎦+ −⎢ ⎥ ⎢ ⎥⎬

⎢ ⎥ ⎢ ⎥⎪⎣ ⎦ ⎣ ⎦⎭

∑

( ) ( ) ( ) ( ) ( )( ) ( ) ( ){ }

( )

/ 20

10

0

0

sin sin

expexp 2 1 exp exp exp 2 1 exp exp

2

exp

j NL

Pj

LP

Substuting using k a and k b and rearranging we obtain

b i kr tEE i j i i i j i ir i

We can rewrite this as

b i kr tEEr

α θ β θ

ωα β β α β β

β

ω

=

=

= =

−⎡ ⎤⎛ ⎞ ⎣ ⎦= − − − + − − − −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎝ ⎠

−⎡⎛ ⎞ ⎣= ⎜ ⎟⎝ ⎠

∑

( ) ( ) ( ){ }

( ) ( ){ }

( ) ( ) ( ) ( ) ( ){ }

/ 2

1

/ 2

010

/ 2

010

2 sin exp 2 1 exp 2 12

sinexp Re exp 2 1

sinexp Re exp exp 3 exp 5 exp 1

j N

j

j NL

j

j NL

j

i i j i ji

E b i kr t i jr

E b i kr t i i i i Nr

The last te

β α αβ

βω αβ

βω α α α αβ

=

=

=

=

=

=

⎤⎦ − + − −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

⎛ ⎞ ⎛ ⎞= − −⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎣ ⎦

⎝ ⎠⎝ ⎠⎛ ⎞ ⎛ ⎞

= − + + + + −⎡ ⎤ ⎡ ⎤⎜ ⎟ ⎜ ⎟⎣ ⎦ ⎣ ⎦⎝ ⎠⎝ ⎠

∑

∑

∑ L

( ) ( ) ( ) ( ){ }/ 2

1

2*

0 00

sinRe exp exp 3 exp 5 exp 1sin

. int

1 1 sin2 2

j N

j

LP P P

rm is a geometric series that converges to

Ni i i i N

The details of the last step are outlined in the book The irradiance at po P is given by

E bI c E E cr

αα α α αα

βε εβ

=

=

+ + + + − =⎡ ⎤⎣ ⎦

⎛ ⎞= = ⎜ ⎟

⎝ ⎠

∑ L

2 22 2

0sin sin sin

sinN NIα β αα β α

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

Fraunhofer Diffraction from Multiple SlitsFraunhofer Diffraction from Multiple Slits

2 2 22 2*

0 0 00

1 1 sin sin sin sin2 2 sin sin

sin, . , ' 'sin

lim

LP P P

m

The irradiance at point P is given by

E b N NI c E E c Ir

NWhen m the term is a maximum For this condition from L Hospital s rule

α

β α β αε εβ α β α

αα πα

→

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

=

( )

( )

sinsin coslim lim

sinsin cos

1 sin sin 0, 1, 2,2

, .

m m

d NN N Nd N

dd

The principal maxima the irradiance pattern occur forp pka a m mN N

For large N the principal maxima are bright and well separated This ana

π α π α π

αα αα

αα αα

π πα θ θ πλ

→ →= = = ±

= = = = = = ± ± K

,

sin

lysis givesus the grating equation

a mθ λ=

2sin

⎟⎟⎠

⎞⎜⎜⎝

⎛ββ

2

sinsin

⎟⎠⎞

⎜⎝⎛

ααN

N2

λθ ma =sin

m=1

m=2

m=0

2 2

0sin sin

sinNI I β α

β α⎛ ⎞ ⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

Diffraction grating equationDiffraction grating equation

λθ ma =sin m=1

m=2

m=0

m=1

m=0

Fraunhofer Diffraction from Multiple SlitsFraunhofer Diffraction from Multiple Slits

N = 2

N = 3

N = 4

N = 5