Interference and Diffraction

Interference and Diffraction

0 2

0 2

Constructive InterferenceDestructive Interference

path difference (x) phase difference (?)

0 2

0

x

2

φ

λ

xWhen x <

φλx ) MOD(2 When x

d

)( tiAe

tiAe

To point P

dsin

Phase difference when arriving at P is

λθdαφφ /sin212

Original phase difference phase from path difference

Two oscillators or two sources

n equally spaced oscillators; equal amplitude; differentin phase (different starting phase, different path length)

a

1

Huygen’swavelets

A B

2

CPlanewave

)])1(cos()2cos()cos([cos φntφtφttAR

E.g.

n slitsequally spaced

D EABDE 2

What is the result of the sum?

)])1(cos()2cos()cos([cos φntφtφttAR

)][ )1(2 φnitiiφtiiφtiti eeeeAR

Using complex amplitude to solve the problem

)]1[ )1(2 φniiφiφti eeeAeR

)exp(1

)exp(1

iφ

inφ

a

aaaa

nn

1

11 12

)2/sin(

)2/sin(

1

1 2/)1(2/2/

2/2/

2/

2/

φ

nφeAe

ee

ee

e

eAe

e

eAeR φniti

iφiφ

inφinφ

iφ

inφti

iφ

inφti

2

2

222/)1(222

)2/sin(

)2/sin(

)2/sin(

)2/sin(

φ

nφA

φ

nφeeAR φniti

1 1

1A

)])1(cos()2cos()cos([cos φntφtφttAR

φ

2A 3A

4A

5A

6A

RAφ

O

TQ

M

r )2/sin(2 rA

AA 1

6 OQT

)2/6sin(2 rAR

)2/sin(2 nrAR

nOQT

n = 6

)2/sin(

)2/sin()2/sin(2);2/sin(2

n

AAnrArA RR 2

0 )2/sin(

)2/sin(

n

II

2

22

)2/sin(

)2/sin(

φ

nφARI

0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

n/2 = 2

I

()

Max = n2

n/2 =

n = 6

0 2 4 6 8 100

5

10

15

20

25

30

35

40

I

()

2

22

)2/sin(

)2/sin(

φ

nφARI

n = 6

a

1

Huygen’swavelets

A B

2

C1sinaAB 2sinaAC

DE

1sin2 aAD 2sin2 aAE

….

….

Phase difference

/sin2 1a

….

2

Path difference

Phase difference

/sin2 2a

….

2

1st Constructive Interference: 1sina

2nd Constructive Interference: 2sin 2 a

In general, constructive Interference: na n sin

2

2

222/)1(222

)2/sin(

)2/sin(

)2/sin(

)2/sin(

φ

nφA

φ

nφeeARI φniti

/sin2 na

2

0

sinsin

sinsin)(

n

n

n

a

naII

I0

=

Width: W10 < W5 ; Height: n2; Integrated intensity?

d/2x

0

d

sinx

Fraunhofer Single Slit:tiAe

)][ )1(2 φnitiiφtiiφtiti eeeeAR

iφtiAe

x

…..

sin

2x

d

xitid

xiti dxeAedxeeAR0

sin2

0

sin2

d

xid

xi

dxiei

dxe0

sin2

0

sin2

sin2

sin2

1

ddi

ei 0

sin2

|sin2

)1(sin2

sin2

diti e

iAeR

)(sin2

sinsinsin

dididiti eee

iAeR

)sinsin(2sin2

sin

diei

Aediti

sin

)sinsin(sin

d

ddeAe

diti

2

0

sin

)sinsin(

d

dII 2

0 AdI

x = 0 and x = d/2; path difference BC = dsin/2. phase difference = (2/)*dsin/2 = dsin/ Destructive interference: dsin/ = , 2, 3, …= n. dsin = n.

d/2

BC

}Interference betweena pair of waveletsfrom the top andcenter of the slit

next pair

0

d

The Rayleigh Criterion

http://www.kshitij-pmt.com/resolution-of-single-slit-and-circular-apertures

Barely ResolvedWell Resolved

Resolution of single slit and circular aperture:

Single Slit:d

d

sin1

sin

Circular aperture:

Minimum 1

Airy rings

b

a

n sets of double slits

The red ones:

)][ 111 )1(2

1

φnitiiφtiiφtiti eeeeAR

The blue ones:

)]

[12

12122

)1(

2

2

φniiφti

iφiφtiiφiφtiiφti

e

eeeAR

sin2

1 aφ

sin2

2 bφ

)]1][1[ 1112 )1(2

21

φniiφiφiφti eeeeAeRRR

22

0

sinsin

sinsin

sinsin

sin2sin)(

a

na

b

bII

2/cos2)(1

11 2

2/2/2/2/2

2222

2

2

2 φeeeee

ee iφiφiφiφ

iφ

iφiφ

What about n slits each with a slit width of d?

a

d22

0

sinsin

sinsin

sin

sinsin)(

a

Na

d

dII

If d is very small, upper cap is more flat! a equals to a lot d!

Diffraction Geometry

Plane normal

X

X’

Y

Y’

11a2

32a

1’

2’

3’

1a’, 2a’

L

M N

2

dP K

S

Phase difference between different atoms interacted with X-ray.

Atoms in neighboring plane: 2dsin

0

a

Laue’s Equations:

acos acos0

Constructive interference: |acos-acos0| = h

Similarly in the y direction: |bcos - bcos0| = k

Similarly in the y direction: |ccos - ccos0| = l

Integer

Reciprocal lattice and diffraction:S0S

S-S0

A

O

-S

u

n

v

m

OA = pa1+qa2+ra3

p, q, r:integers

OAOASS

2

)-(2- difference Phase 0

)-(

0SS

321 bbb lkh b1, b2, b3: base vectors of G

)(2)()(2 321321 rlqkphrqplkh aaabbb

constructive interference

Path difference: uA + Av = Om+On= S0OA + (-S) OA = (S0-S)OA.

If

Ewald Sphere

k

k

k

k

|k| = |k| = |k| = 2/

12 B22 B

k

k

k

k1/

kk k = G

Diffraction codition

k

k

k

k

Reciprocal lattice

O