1

Chapter 10 DiffractionMarch 9, 11 Fraunhofer diffraction: The single slit

10.1 Preliminary considerationsDiffraction: The deviation of light from propagation in a straight line.There is no essential physical distinction between interference and diffraction.

Huygens-Fresnel Principle: Every unobstructed point of a wave front serves as a source of spherical wavelets. The amplitude of the optical field at any point beyond is the superposition of all these wavelets, taking into account their amplitudes and phases.

Fraunhofer (far field) diffraction: Both the incoming and outgoing waves approach being planar. a2/R<<1 where R is the smaller of the two distances from the source to the aperture and from the aperture to the observation point. a is the size of the aperture. The diffraction pattern does not change when moving the observation plane further away.Fresnel (near field) diffraction: Plane of observation is close to the aperture. General case of diffraction. The diffraction pattern changes when the observation plane moves.

S

P

aR1

R2

2

Coherent line source:

y

x

P (x,y)

dy'

r

-D/2

D/2

' )exp(

),(2/

2/dy

r

ikryxE

D

DLε

L is the source strength per unit length.This equation changes a diffraction problem into an integration (interference) problem.

Mathematical criteria for Fraunhofer diffraction:The phase for the rays meeting at the observation point is a linear function of the aperture variables. S y'

P

y' sinWaves from a point source:Harmonic spherical wave:

),(or ),cos(),( )( tkrier

AtrEtkr

r

AtrE

A is the source strength.

3

10.2 Fraunhofer diffraction10.2.1 The single slit

y

x

P (x,y)

y'r

-D/2

D/2

R

The slit is along the z-axis and has a width of D.

2sin

)0()(

II

sin)2/( ),exp(sin

sin)2/(

]sin)2/sin[()exp(

' )]sin'(exp[

' )exp(

),(

2/

2/

2/

2/

kDikRR

D

kD

kDikR

R

D

dyR

yRik

dyr

ikryxE

L

L

D

DL

D

DL

In the amplitude, r is approximated by R.

In the phase, r is approximated by R-y' sinif D2/R<<1.Fraunhofer diffraction condition

The overall phase is the same as a point source at the center of the slit.

Integrate over z gives the same function.

22

22

'2

cossin'

sin'2')'(

yR

yR

RyyRyr

4

maxima ,...47.3 ,46.2 ,43.1 ,0 ,tan

minima ...3 ,2 , ,0sin

0)sincos(sin2

)0(3

I

d

dI

2sin

)0()(

II sin

2

kD

y

x

P (x,y)

y'r

-D/2

D/2

R

2sin

)0()(

II

I/I(0)=0.047 0.016

(©WIU OptoLab)

DkD

DkD

2/

peaks side theof Widths

2

2/

22

peak central theof Width

5

Phasor model of single slit Fraunhofer diffraction: rolling paper

6

Read: Ch10: 1-2Homework: Ch10: 3,7,8,9Due: March 27

7

10.2.2 The double slit z

x

P (x,z)R-a sin

R

ab

)sinexp(1)exp(sin

')]sin'(exp[

')]sin'(exp[),(

2/

2/

2/

2/

ikaikRR

b

dzzRikR

dzzRikR

zxE

L

ba

ba

L

b

b

L

2

2

0 cossin

4)(

II

sin)2/(

sin)2/(

factor.) 1/2 omit the (We slit. oneonly is there whenaxis at theintensity ,Let 2

0

kb

ka

R

bI L

The result is a rapidly varying double-slit interference pattern (cos2) modulated by a slowly varying single-slit diffraction pattern (sin2/2).

March 23 Double slit and many slits

8

(©WIU OptoLab)

sin

cos/sin

)/sin sin()0()( 2

2a

b

bII

Single-slit diffraction Two-slit interference

Envelope Fringes

Question:Which interference maximum coincides with the first diffraction minimum?

b

am

ma

b

sin

sin

“Half-fringe” (split fringe) may occur there. Our author counts a half-fringe as 0.5 fringe.

2

2

0 cossin

4)(

II

half-fringe

9

10.2.3 Diffraction by many slits

z

x

P (x,z)

R-a sin

R

a

b

R-2a sin

)2exp(1

)2exp(1)exp(

sin

])1(2exp[...)4exp()2exp(1)exp(sin

')'(...')'(')'(')'(),(

)]sin'(exp[)'( ,/2/)1(

2/)1(

2/2

2/2

2/

2/

2/

2/

i

NiikRbC

NiiiikRbC

dzzFCdzzFCdzzFCdzzFCzxE

zRikzFRCbaN

baN

ba

ba

ba

ba

b

b

L

22

0 sin

sinsin)(

N

II

sin)2/(

sin)2/(

kb

ka

10

6

4

sin

sinsin22

N

ba

N

22

0 sin

sinsin)(

N

II

Principle maxima:Minima (totally N-1):

Subsidiary maxima (totally N-2):

... ,2 , ,0

)1(

... ,3

,2

, N

N

NNN

2

)32( ..., ,

2

5 ,

2

3

N

N

NN

sin)2/(

sin)2/(

kb

ka

11

Phasor model of three-slit interference: rotating sticks

12

Read: Ch10: 2Homework: Ch10: 10,11,12Due: April 3

13

10.2.4 The rectangular aperture

Coherent aperture:

Aperture

)exp(dS

r

ikrE A

2

2

2222

222

222

/)(1

/)(21

/)(2/)(1

)()(

RZzYyR

RZzYyR

RZzYyRzyR

zZyYXr

ZYXR

dS=dydz P(Y,Z)rR

x

y

z

Y

Z

Aperture

/)(exp)exp(

),( dSRZzYyikR

ikRZYE A

March 25 Rectangular aperture

Fraunhofer diffraction condition

X

14

).sin2

( 2

'

),sin2

( 2

' ,

'

'sin

'

'sin)exp(

/)(exp)exp(

/)(exp)exp(

),(

2/

2/

2/

2/

Aperture

b

aA

a

a

b

b

A

A

kb

R

Ykb

ka

R

Zka

R

ikRab

dydzRZzYyikR

ikR

dSRZzYyikR

ikRZYE

Rectangular aperture:

22

'

'sin

'

'sin)0(),(

IZYI

dS=dydz P(Y,Z)rR

x

y

z

Y

Za

b

15

22

'

'sin

'

'sin)0(),(

IZYI

kb

Rm YRkbY

2 ... ,2 ,2/' Y minimum:

ka

Rm ZRkaZ

2 ... ,2 ,2/' Z minimum:

.2/'

,2/'

RkbY

RkaZ

16

Read: Ch10: 2Homework: Ch10: 14,17Due: April 3

17

10.2.5 The circular aperture

Importance in optical instrumentation: The image of a distant point source is not a point, but a diffraction pattern because of the limited size of the lenses.

2

12

222

12

0 0

2

0 0

0

2

0 0

Aperture

/

)/(4

/

)/(2

)exp(

)/(2)exp(

cos/exp)exp(

)cos(/exp)exp(

/)(exp)exp(

),(

)cos(

coscossinsin

Rkaq

RkaqJ

R

AE

Rkaq

RkaqJa

R

ikR

dRkqJR

ikR

ddRqikR

ikR

ddRqikR

ikR

dSRZzYyikR

ikRZYE

q

qqZzYy

A

A

aA

aA

aA

A

2

1)(lim

)()]([

])cos(exp[2

)(

)cosexp(2

1)(

1

0

1

2

0

2

00

u

uJ

uJuuJudu

d

dvvumvii

uJ

dvviuuJ

u

mm

mm

m

m

Bessel functions:.

2

1

2

1

sin

)sin(2)0(

/

)/(2)0()(

ka

kaJI

Rkaq

RkaqJII

March 27 Circular aperture

P(Y,Z)

R

x

y

z

Y

Z

q

a

18

2

1

sin

)sin(2)0()(

ka

kaJII

J0(u)

J1(u)

u

)0(/)( II

sinka

3.83

0.018

q1

lensa for 22.1 ,2

22.11 D

f

a

Rq

Radius of Airy disk:

83.3/ 0)/( 11 RkaqRkaqJ

DP

f

19

Read: Ch10: 2Homework: Ch10: 25,28Due: April 3

20

March 30 Resolution of imaging systems

10.2.6.0 Equivalence between the far field and the focal plane diffraction pattern

1 1 2

2 1 2

22 22

1

( / ) exp( ) exp[ ( sin )]

( '/ ) exp( ) exp[ ( sin )]

( )exp ( ) constant

( )

(does not depend on )

' ',

E y R C ikR C ik R a

E y f C ikL C ik L a

Eik L R

E

y y y f

R f y R

Two coherent point sources:

P

a sin

Ra

y

• This applies to any number of arbitrarily distributed point sources in space.• Far field and focal plane produce the same diffraction pattern, but with different sizes.• R is replaced by f in the focal plane pattern.A lens pulls a far-field diffraction pattern to its focal plane, reduces the size by f/R.

P'

a sin

a

f

y'L

21

10.2.6 Resolution of imaging systems

Overlap of two incoherent point sources:

Rayleigh’s criterion for bare resolution:The center of one Airy disk falls on the first minimum of the other Airy disk.We can actually do a little better.

Angular limit of resolution:

D

22.1min

Df

qD

fq

22.1

22.1

1

1

D P

f

Image size of a circular aperture:

D

P2S1

S2

P1

far away f

Image size of a far point source:

lensDf

q 22.11 D

P

f

22

Question: Comparing the circular with the square aperture, why does the square aperture produce a smaller diffraction pattern? (/D vs. 1.22 /D)

Wavelength dependence: CD DVD

Angular limit of resolution: D

22.1min

Our eyes: '1mm 2

nm 550min About 1/3000 rad

Spot distance on the retina: 20 mm/3000=6.7mSpace between human photoreceptor cells on the retina: 5-7m.Pixel size of a CCD camera: ~7.5 m.

Pupil diameter Focal length Human cone photoreceptor cells150m

23

Read: Ch10: 2No homework

24

Diffraction grating: An optical device with regularly spaced array of diffracting elements.Transmission gratings and reflection gratings.

Grating equation:i

m

a

i

m

a

ma im )sin(sin

Blazed grating: Enhancing the energy of a certain order of diffraction.Blaze angle: Specular reflection: mir 2

April 1, 3 Gratings

m=0

12

-2-1

i r

0

a

specular

0th

25

Grating spectroscopy:

m

i

Na

kaka

N

NII

cos

2

cos

2)sin)(sin2/(

2

sin

sinsin)(

22

0

Angular width of a spectral line due to instrumental broadening. Inversely proportional to Na.

Angular dispersion:

m

mim a

m

d

dma

cos)sin(sin

dm

d

Angular width for a spectral line:N-slit interference

Between two minima, (N-1)/N to (N+1)/N .

26

Limit of resolution:

)sin(sin

2 :for criterion sRayleigh'

coscos dispersion from separationAngular

cos

2)( h wavelengteach of idthAngular w

min

separationwidthmin

separation

width

im

mm

m

m

NamN

a

m

a

m

d

d

Na

min

Resolving power:

Question: Why does the resolving power increase with increasing order number and with increasing number of illuminated slits?

Barely resolved two close wavelengths:

mN min

separation

width

27

Free spectral range:

m

mma fsrim

)1()()sin(sin

sinm

m =1

m =3

m =2

fsr

In higher order diffraction the spectrum is more spread in angle. This results in a higher resolving power but a narrower free spectral range.

mfsr

28

Read: Ch10: 1-2Homework: Ch10: 32,33,34,39,41Due: April 10