Chapter 38: Diffraction (interference part 2)

Chapter 38: Diffraction
(interference part 2)

Diffraction is an interference effect like in
Ch 37, but usually refers more specifically to
bending of waves around obstacles (similar to
refraction). Diffraction also manifests itself
when waves from a large number (or even
continuous set) of sources interfere. This
happens when light illuminates diffraction
gratings, many slit screens, apertures, or even
crystals.

1

Diffraction from a single slit

What if slit width a >> λ?

2

Interference of light and bending of light
around obstacles:

NOT part of our everyday (pre-technological)
experience

Why?

•Interference effects require coherent light
sources

•Diffraction most significant when obstacle
sizes are comparable to wavelengths

Hard to satisfy conditions for
interference / diffraction in
laboratories before the 19th century!

3

Diffraction of light around a razor blade

4

Diffraction geometry
d1

d2
si

Fresnel diffraction (near-field): s ≈ dn

i ≈ dn

Fraunhofer diffraction (far-field): s >> dn

i >> dn

Fraunhofer is simpler since rays going to same point

on screen ≈ parallel 5

Fraunhofer diffraction

6

Fresnel diffraction

Poisson spot
of a penny

7

Diffraction Gratings

Spectroscope (using a diffraction grating to
separate wavelength components of light)

Spectrum of a a sodium-type street lamp

8

Multi-slit diffraction

Diffraction grating: Generalization from double-slit
experiment to multi-slit experiment.

principal maxima: d sinθ = mλ where m = 0, ±1, ±2…

m is the order of the principal maxima

note θ increases (pattern spreads) as λ increases 9

d

Relationship between height and width
of diffraction peak

If there was no interference: illumination by N slits
(each has intensity I0) => average over entire screen

NI0

⇒Due to conservation of energy, average over entire

screen is still NI0 even with interference

Intensity of a maximum (with maximum constructive
interference from N slits)

Emax = NE0 Imax = N 2 I0

=> 2-slit interference: Imax = 4I0

Angular separation between successive maxima:

θ m +1 −θm ≈ sinθm+1 − sinθm = λ

d

Assume each maximum is ∆θ wide

∆θ Imax Iavg

10

λ/d

∆θ Iavg

λ/d

Assume now that the intensity in a single maximum is

equal to the average intensity in λ / d :

I max ∆θ ≈ NI0 λ

d

With Imax = N 2I0 we get:

∆θ = NI0λ = 1 λ
N
dN 2I0 d

The bigger N (the number of slits), the taller,
and narrower the peaks.

11

Intensity Pattern

For multiple slits:

I = I0 sin ( Nβ )2

 sin β 


2β = 2π d sinθ β is HALF the slit-to-slit
λ
phase difference

note that as θ → 0, β → 0, sinβ → β

⇒ I = I0  Nβ 2 = I0N 2
 β 

Check for two-slit system (N = 2):

I = I0 sin (2β ) 2 = I0 ( 2sinβ cos β )2
 (sin
 sin β  β )2


= 4I0 cos2 β

12

Heightening and narrowing of diffraction
peaks as N increases

Compare with Michelson (single interference)
and Fabry-Perot (multiple interference) 13
interferometers

Resolution of Diffraction Gratings

Angular Dispersion: change in angular separation
due to different wavelength

d sinθ = mλ

∆[d sinθ ] = ∆[mλ]

d (cosθ ) ∆θ = m∆λ

∆θ = m (not same∆θ as in previous slide!)
∆λ
d cosθ

notaentghuatlaarndguislapredrsisiopenrsiniocnreinacsreesas(eims proves)

as o(irmdperomvesin) carseoarsdeesr m increases

Resolution: → important is the width of the

maxima and the separation

Definition: R ≡ λ ∆λ : smallest observable
∆λ
wavelength difference

For N-slit system: R = mN

Resolution increases (improves) as:

• order m increases 14

• number of slits/lines N increases

Example 38-1: sodium doublet at 589.0 nm
and 589.6 nm
a) how many slits required to resolve doublet?
b) screen is 4 m away, grating has 2000
slits/cm, what are positions of of two priniple
max. of first order.

15

Single-slit diffraction

Destructive interference: when ends of

slit differ by integer number of λ

sinθ = mλ where m = ±1, ±2, ±3…

a

what about m = 0? note θ increases (pattern

spreads) as λ increases 16

a

Single-slit intensity pattern

Minima at sinθ=(mλ)/a

Maxima approximately halfway between

minima

17

Example 38-3: a=0.10 mm, λ=633nm,
screen is 3 m away from slit
What is distance between minima on
either side of central maximum?

18

Intensity Pattern for Single-Slit
Diffraction

I = I max sin2 α
α2

α = πa sin θ
λ

for θ=0 recall:

lim sin α = α = 1
α→0 α α

for minima:

α = nπ = πa sin θ
λ

→ sin θ = n λ ,
a

where n = ±1, ±2, ±3…

19

Example 38-4: intensity ratios of 1st and
2nd maxima to the intensity of central
maximum for single slit?

20

How to get single-slit intensity pattern

Nd = a

I = lim I0 sin ( Nβ) 2
 
N →∞  sin β 

πd sin θ π  a  sin θ α
 N 
β = = = →0 as N →∞
λ λN

 sin α 2 sin2 α sin2 α
  α2 α2
=  = I0 = N 2I0
( ) ( )Ilim I0  α
 sin N
N →∞ 

N 21

Diffraction and Resolution

S D I
point Aperture
Aperture of Demo 39-1
optical system
Airy disk

22

θmin = 1.22 λ ≈ λ
D D

Rayleigh Criterion: Two point sources are just

resolved if the peak of the diffraction image of the

first source overlies (and is no closer than ) the

first minimum of the second source 23

L

Smin = L ⋅θ min ≅ Lλ with θ min ≅ λ

D D

24

Example 38-5: min separation between two
objects so that human eye can distinguish
them at
a) near point (25 cm)?
b) 5 m
Pupil diameter=2.5 mm

25

Example 38-6: Hubble telescope
D=2.4 m, 600 km above earth
a) θmin for visible (λ=550 nm) light?
b) ideal Smin for two objects on earth’s
surface

26

Slit width and grating patterns

Imult

Isingle

Imult Isingle

Demo
38-2

I = I Imult single = sin ( Nβ) 2  sin α 2
   α 
 sin β 

β = πd sin θ α = πa sin θ
λ λ

27

Diffraction pattern for multiple slits
where d=10a

note missing orders

28

X-ray diffraction

Recall that θ→0 when λ /d →0
therefore to see an effect on light due to an

intermediate object where d is small, λ must
be small

=> For diffraction on crystals use X-rays

θ3 29

NaCl (table salt) crystal
d≈0.1 nm

Bragg condition for constructive
Bragg
interference spot on

screen

Bragg's Law:

2d sin θ = nλ where n = 1, 2,3,…

Constructive interference

→ Bragg peak 30

Bragg Planes

31

X-ray diffraction pattern from
crystallized DNA

32

von Laue method for x-ray diffraction

33

Example 38-7: rock salt d=0.282 nm, what
wavelengths will appear in first and second
orders at 25o?

34