Wave Optics - Interference and Diffraction - Lecture Slides | PHYS 270, Study notes of Physics

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Wave Optics

Wave Optics

• Water wave passing though an opening
• Sharp shadow
• (^) The wave nature of light is needed to

explain various phenomena

• Interference
• Diffraction
• (^) The particle nature of light was the basis

for ray (geometric) optics

Interference

• Constructive interference occurs when the path-length difference is integer

multiple of the wavelength

• (^) In small-angle approximation, angle of bight fringes is
• (^) The position of bright fringes can be found
• (^) The fringe spacing is independent of m

 r  d sin   m 

, m 0 , 1 , 2 , 3 , ...

d

m L

y Ltan y m

   





,m 0 , 1 , 2 , 3 ...

d

m

m

 

d

L

y y y

m 1 m



Interference

• Destructive interference occurs when the path-length

difference is a half-integer number of wavelengths

• Dark fringes are located exactly halfway between the bright

fringes.

m 0 , 1 , 2 , ...

r m  

,m 0 , 1 , 2 , ...

d

L

y' m

m



Examples

• A double slit is illuminated simultaneously with orange light of wavelength

600 nm and light of an unknown wavelength. The m = 4 bright fringe of

the unknown wavelength overlaps the m = 3 bright orange fringe. What is

the unknown wavelength?

• (^) Light of 600 nm wavelength illuminates a double slit. The intensity pattern

shown in the figure below is seen on the screen 2.0 m behind the slits.

What is the spacing (in mm) between the slits?

• (^) A double-slit interference pattern has a fringe spacing of 4.0 mm. How far

from the central maximum is the first position at which the intensity is

equal to I 1

?

• (^) Two narrow slits 50 μm apart are illuminated with light of wavelength 500

nm. What is the angle of the m = 2 bright fringe in radians? In degrees?

Single Slit Diffraction

• Huygens’ principle - Each point on a wave front is the source of a

spherical wavelet that spreads out at the wave

speed

• At a later time, the shape of the wave front is

the line tangent to all the wavelets

Single Slit diffraction

• Continuing the process we find the general condition
• (^) Angles of dark fringes (small angle)
• The position of the dark fringes
• The width of the central maximum
• (^) The light intensity pattern looks

a sin p p 1 , 2 , 3 , ...

p

p 1 , 2 , 3 , ...

a

p

p





p 1 , 2 , 3 , ...

a

p L

y

p

a

2 L

w

Circular-Aperture Diffraction

Light of wavelength λ passes through a circular aperture of diameter D , and is

then incident on a viewing screen a distance L behind the aperture, L >> D.

The diffraction pattern has a circular central maximum, surrounded by a

series of secondary bright fringes shaped like rings.

The angle of the first minimum in the intensity is

The width of the central maximum on the screen is

Examples

• A diffraction grating produces a first-order maximum at an

angle of 20°. What is the angle of the second-order maximum?

• (^) Light of 600 nm wavelength illuminates a single slit. The

intensity pattern shown in the figure below is seen on a screen

2.0 m behind the slits. What is the width of the slit?

Interferometers

• A device that makes practical use of

interference is called an interferometer.

• (^) Based on division and recombination of a single

wave.

• (^) Acoustical interferometer
  • (^) Constructive: Δr = mλ => L = m λ/
  • (^) Destructive: Δr =(m + ½)λ => L = (m + 1/2) λ/
  • (^) If you increase/decrease the length L, the number

of maxima appearing changes

2

L

m





 

Measuring Indices of Refraction

A Michelson interferometer can be used to

measure indices of refraction of gases. A

cell of thickness d is inserted into one arm of

the cell. When the cell contains a vacuum,

the number of wavelengths inside the cell is

When the cell is filled with a specific gas, the

number of wavelengths spanning the distance d

is

Filling the cell has increased the lower path by

wavelengths. By counting fringe shifts as the cell is filled, one

can determine n.

Examples

• A Michelson interferometer uses red light with a wavelength 0f

656.45 nm from a hydrogen discharge lamp. How many bright-

dark-bright fringe shifts are observed if mirror M 2

is moved

exactly 1 cm?