The waves spread out from

the opening!

5. Diffraction1) What it is? The bending of waves behind obstacles or apertures into the ”shadow region”, that can be considered as interference of many waves.

2) Haw to observe?Diffraction is most pronounced when the wavelength of the wave is similar to the size of the obstacle or aperture. For example, the diffraction of sound waves is commonly observed because the wavelength of sound is similar to the size of doors, etc.

•Light will diffract around a single slit or obstacle.•The resulting pattern of light and dark stripes on a screen is called a diffraction pattern (fringes).•This pattern arises because different points along a slit create wavelets that interfere with each other just as a double slit would.

1

3) Diffraction from a single slit

D/2

D/2

1r

2r

x

y

21

121 sin D

First minimum (dark):

1sinD

Minima (dark fringes):

,...3,2,1

sin

m

mD m tanxy

D

mm

sin tan smallFor

Dm

Dm

tot 2

D

mxm

y

2

Example: In order to obtain a good single slit diffraction pattern, the slit width could be:A. /100 ; B. /10; C. ; D. 10;

E.100

3a) Diffraction from a single slit (intensity)

Minima (dark fringes):

,...3,2,1

sin

m

mD m

Example: Light of wavelength 610 nm is incident on a slit 0.20 mm wide and the diffraction pattern is produced on a screen that is 1.5 m from the slit. What is the width of the central maximum?

mmm

m

mmy

D

mxym

2.91510.91020.0

106105.122 3

3

9

1

?2

5.1

20.0

610

1

y

mx

mmD

nm

Example: Light of wavelength 687 nm is incident on a single slit 0.75 mm wide. At what distance from the slit should a screen be placed if the second dark fringe in the diffraction pattern is to be 1.7 mm from the center of the screen?

mm

m

mmx

m

Dyx

D

mxy m

m

93.0687.02

75.07.1

106872

1075.0107.19

33

?

7.1

75.0

687

2

x

mmy

mmD

nm

4

4) Multiple slits (Diffraction grating)•A diffraction grating consists of a large number of equally spaced narrow slits or lines. •A transmission grating has slits, while a reflection grating has lines that reflect light.

•The more lines or slits there are, the narrower the peaks.

Two slits

Six slits

Parameters:

NdLdD

; ;

0 :for is This D5

The maxima of the diffraction pattern are defined by:

4a) Diffraction gratings and not monochromatic light

...2,1,0

sin

m

md m

4b) The Spectrometer and Spectroscopy

•A spectrometer makes accurate measurementsof wavelengths using a diffraction grating or prism.

•The wavelength can be determined to high accuracy by measuring the angle at which the light is diffracted.

•Atoms and molecules can be identified when they are in a thin gas through their characteristic emission lines.

Example: A diffraction grating has 5000 lines per cm. The angle between the central maximum and the fourth order maximum is 47.2°. What is the wavelength of the light?

?

4

2.47

1025000/1

//

4

4

1

m

cmcm

LNNLd

md m sin

md m /sin

nm

cm

367cm1067.3

4/2.47sin1025

4

Example: A diffraction grating has 6000 lines per centimeter ruled on it. What is the angular separation between the second and the third orders on the same side of the central order when the grating is illuminated with a beam of light of wavelength 550 nm?

?

550

106

6000/1

23

15

1

nm

m

cmd

dm

md

m

m

/sin

sin

6.40

66.0106105502sin

99.0106105503sin

23

1592

1593

mm

mm

7

5) Circular apertures. Limits of resolution

D 1sinThin slits:

Circular apertures: D 22.1sin 1

DR 22.1 1

Example: The Earth and Moon are separated by about 400x106 m. When Mars is 5x1010 m from Earth, could a person standing on Mars resolve the Earth and its Moon as two separate objects without a telescope? Assume a pupil diameter of 5 mm and λ = 550 nm.

?/

5

550

105

1040010

6

R

mmD

nm

mx

my

Dx

yR 22.1

yes!

1011.022.1105

1055022.122.1

108105

10400

33

9

310

6

R

R D

m

m

x

y

x

yRayleigh criterion: two images are just resolvable when the center of the diffraction disk of one image is directly over the first minimum in the diffraction pattern of the other

8