Math 5900 – Summer 2011

Lecture 4: Interference of Light

Gernot LaicherUniversity of Utah - Department of Physics & Astronomy

Light = travelling electromagnetic wave (em wave)

Different colors = different frequencies of em waves.

Mechanical waves need medium to travel (e.g., air molecules for sound waves).

Light can travel through vacuum (needs no medium)

Speed of light in vacuum approx. 3x108m/s (different in other media, e.g. 2x108m/s in glass).

Wave propagation:

Wave crests move with the speed of light.

Period T (of electric field vector oscillation) :

Time required for one crest to move forward by a distance equal to its wavelength.

Frequency f: f=1/T

In short:

Huygen’s Principle:

Every point on a wave-front may be considered a source of secondary spherical wavelets which spread out in the forward

direction at the speed of light. The new wave-front is the tangential surface to all of these secondary wavelets.

Diffraction: Occurs when wave encounters obstacle.

Example +Demo: Plane wave encounters a single slit.

Intensity distribution of diffraction pattern is mathematically quite involved. Location of the so-called minima fairly easy to understand.

a<<

a

d

y

Dark region on screen: Sum of all the waves coming from the opening interfere destructively (amplitudes add up to zero).

E.g, when

a

d

y

Use trigonometry to relate path length difference to position y on the screen:

a

d

y

Destructive interference occurs for

Relate min to distance y on the screen:

For small angles

Between these two minima is “central maximum” for y=0 (=0).

Between these two minima : “Central maximum” for y=0 (=0).

Determine wavelength of laser by measuring distance between the two minima (2 ymin) on both sides of the central maximum and d and a.

Additional minima from single slit diffraction:

Shape of intensity distribution on screen:

Diffraction pattern of a circular aperture

Without derivation: For circular aperture (a hole) of diameter c the diffraction pattern is an “airy disk”.

Angle of first minimum (dark circle) (measured from axis that goes through center of the central bright disc):

Small angle approximation ( in radians)

For small angles

ymin = radius of first dark ringc = diameter of holed = distance from hole to screen

Babinet’s Principle

The diffraction patterns from an opaque body is identical to that from a hole of the same size and shape except for the overall forward beam intensity.

Example:

Diffraction pattern produced by circular disc = same from circular hole of the same size.

We will use this principle to determine the thickness of a hair and the

size of small particles from their respective diffraction patterns.

Double Slit Diffraction

Wave fronts of light

Double slit

Screen

Wave going through this slit travels a bit further to get to this particular place on the screen.Waves from the two slits are out of

phase by half a wavelength in that directionWaves annihilate each other in that direction

(“destructive interference”).Darkness on that place on the screen.

Dark

The waves going through both slits travel the same distance to the screen.Waves from the two slits in phase.Waves add together to twice the amplitude (“constructive interference”). Bright spot in center.

Dark

Bright

Dark

Dark

Dark

Bright

Bright

Dark

Bright

The light exits the slits in all directionssimultaneously.

A pattern of bright and dark regions appears.

d

y

Condition for intensity maximum:

d

y

For small angles

Intensity distribution of diffraction pattern of double slit depends on each single slit width.

Single slit pattern superimposed on double slit pattern!

Basically, “pure” double slit intensity distribution is multiplied by the single slit intensity distribution.

Diffraction grating acts in many ways like a double slit. However, bright spots are much more narrow/less broad.

Makes it easier to separate peak location of two separate but close wavelengths.

Diffractiongrating

d sin n (= difference in path length)

n

Whenever d sin n = n: All waves are in phase (constructive interference); n = 0,1,2….Otherwise they cancel each other (destructive interference).

Different meansconstructive interferencefor different n !

d