Chapter 11. Fraunhofer DiffractionChapter 11. Fraunhofer DiffractionLast lecture

• Numerical aperture of optical fiber• Allowed modes in fibers• Attenuation• Modal distortion, Material dispersion, Waveguide dispersion

This lecture • Diffraction from a single slit• Diffraction from apertures : rectangular, circular• Resolution : diffraction limit• Diffraction from multiple-slits

Diffraction regimesDiffraction regimes

Fraunhofer DiffractionFraunhofer Diffraction

Fraunhofer diffraction• Specific sort of diffraction

– far-field diffraction– plane wavefront– Simpler maths

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

11-1. Fraunhofer Diffraction from a Single Slit11-1. 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.

r0

Δ

( )

( )

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⎝( ){ }0exp i k r tω

⎞+ Δ −⎡ ⎤⎟ ⎣ ⎦⎜ ⎟

⎠

why?

r0

Δ

0

, ., , .

.sinL L

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

The path difference s

lit

Substituting we obtai

d

n

dE E s

θΔ =

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

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

−

−

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

⎡ ⎤⎛ ⎞= −⎡ ⎤⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠ ⎣ ⎦

∫

( ) ( ) ( )

( ) ( ) ( )

( )

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*

0 0 02 20

sin2 sin exp

1 1 sin sin2 2

L

LP P

E bi i kr tr

The irradiance at point P is given by

E bI = c E E c Ir

ββ ωβ

β βε εβ β

⎛ ⎞= −⎡ ⎤⎜ ⎟ ⎣ ⎦⎝ ⎠

⎛ ⎞= =⎜ ⎟

⎝ ⎠

2 10 2sin ( ), = sinI I c kbβ β θ=

0 0

sinsinc 1 0, lim sinc lim 1

sin 0, , 1, 2,1 sin2

The function is for

The zeroes of irradiance occur when or w khen mb m

β β

ββ ββ

β β θ π

→ →= =

= ±=

=

±== K

2 10 2sin ( ), = sinI I c kbβ β θ=

The angular width of the central maximum : ( )1 12sin sinm m bλθ θ θ=+ =−Δ ≈ − =

,, 2 / ,

1 22

sinIn terms of the length y on the observation screen

and in terms of wavelength kwe can write

y b ybf f

Zeroes in the irradiance pattern will occur when

b y mf

The maximum in the irradiance pattern is a

f

β=

y

t

m fyb

λ π

π πβλ λ

π π

θ

λλ

=

= =

≅

= ⇒ =

2 2

sin cos sin cos sin 0

sin tancos

0.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

y

f

Fraunhofer Diffraction pattern from a Single Slit

2 10 2sin ( sin )I I c kb θ=

2 LW Lb

θ λ ⎛ ⎞= Δ = ⎜ ⎟⎝ ⎠

11-2. Beam spreading due to diffraction11-2. Beam spreading due to diffraction

16-3. Rectangular Apertures16-3. Rectangular Apertures

( )20 sinc sin

When t of the rectangular aperture are comparable,a diffraction pattern is observed inboth the x - and y - dimensions, governed in each dimension by the formula we have already developed :

he length a and width b

I I α= ( )2

1 sin2

c

where

k a

Zeroes in the irradiance pattern are observed when

m f m fy or xb a

α θ

λ λ

β

=

= =

x

y

Square AperturesSquare Apertures

Circular AperturesCircular Apertures

xdsdA =∫∫=

Area

iskAp dAe

rEE θsin

02

22

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)

Fraunhofer Diffraction from Circular Apertures: Bessel Functions

Fraunhofer Diffraction from Circular Apertures: Bessel Functions

832.3sin21sin === θθγ kDkR (first zero)

( ) ( ) 212

( ) 0J

I Iγ

θγ

⎡ ⎤= ⎢ ⎥

⎣ ⎦

⎭⎬⎫

⎩⎨⎧

=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

( )I γ

Dλθθ 22.1

21min =Δ=

: Airy pattern

: Far-field angular radius

Airy pattern and Airy discAiry pattern and Airy disc( ) ( ) 212

( ) 0J

I Iγ

θγ

⎡ ⎤= ⎢ ⎥

⎣ ⎦

Airy disc

Comparison : Slit and Circular AperturesComparison : Slit and Circular Apertures

Circular aperture

(Airy function)

Single slit(sinc function)

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 a function of the Airy disc. » Two light sources must be separated by at least the

diameter of first dark band.» Called Rayleigh Criterion Image blurring

due todiffraction

Rayleigh Criterion : Two light sources must be separated by at least the diameter of first dark band.

separatedconfused

Rayleigh limit

Rayleigh LimitRayleigh Limit

Resolution limit of a lens:

(f = focal length)

min min 1.22

0.611.222

fx fD

NA NA

λθ

λ λ

⎛ ⎞= Δ = ⎜ ⎟⎝ ⎠

⎛ ⎞≈ =⎜ ⎟⎝ ⎠

minx λ≈The resolution of a microscope isroughly equal to the wavelength.

11-5. Fraunhofer Diffraction from Double Slits11-5. Fraunhofer Diffraction from Double SlitsNow 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

− − + − − −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

= =

( ) ( )( ) ( )

( ) ( )( )

( )

0

0

2 2* 2

0 0 20

exp exp 2 cos

exp exp 2 sin

exp2 cos 2 sin

2

1 1 4sin4cos2 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 cr

α α α

β β β

ωα β

β

βε ε αβ

+ − =

− − =

−⎡ ⎤⎛ ⎞ ⎣ ⎦= ⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

222

0 0 020

sin 14 cos , 2

LE bI = I where I cr

βα εβ

⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

Fraunhofer Diffraction from a Double SlitFraunhofer Diffraction from a Double Slit

22

0 2

sin4 cos

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

I I βαβ

⎛ ⎞= ⎜ ⎟

⎝ ⎠

Fraunhofer Diffraction from a Double SlitFraunhofer Diffraction from a Double Slit

Single Slit

Double Slit

11-6. Fraunhofer Diffraction from Many Slits (Grating)

11-6. 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 2sin Nαα

⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

2 2

0sin sin

sinPNI I β α

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

Diffraction grating equationDiffraction grating equation

λθ ma =sin m=1

m=2

m=0

m=1

Fraunhofer Diffraction from Multiple SlitsFraunhofer Diffraction from Multiple Slits

N = 2

N = 3

N = 4

N = 5