Interference and Diffraction

Certain physical phenomena are most easily explained by

invoking the wave nature of light, rather than the particle

nature. These include: interference, diffraction, polarization.

Coherence

Oscillates in a simple pattern with time and varies in a smooth

way in space at any instant, then the light is said to be

coherent.

If, on the other hand, the phase of a light wave varies

randomly from point to point, or from moment to moment (on

scales coarser than the wavelength or period of the light) then

the light is said to be incoherent.

For example, a laser produces highly coherent light. In a laser,

all of the atoms radiate in phase. An incandescent or

fluorescent light bulb produces incoherent light. All of the

atoms in the phosphor of the bulb radiate with random phase.

Each atom oscillates for about 1ns, and produces a wave about

1 million wavelengths long.

Interference

Just like sound waves, light waves also display constructive and

destructive interference.

For incoherent light, the interference is hard to observe

because it is “washed out” by the very rapid phase jumps of

the light. Soap films are one example where we can see

Interference effects even with incoherent light.

Conditions for Interference

The following four conditions must be true in order for an

interference pattern to be observed.

-The source must be coherent – has a constant

Phase relationship

- Wavelengths must be the same monochromatic

- The Principle of Superposition must apply.

- The waves have the same polarization state.

Constructive and Destructive Interference

(The same for all waves!)

Two waves (top

and middle) arrive

at the same point in

space.

The total wave

amplitude is the

waves.

The waves can add

constructively or

destructively.

Constructive interference occurs when two waves are in

phase. To be in phase, the points on the wave must have

Δφ=(2π)m, where m is an integer.

When coherent waves are in phase, the resulting amplitude

is just the sum of the individual amplitudes. The energy

content of a wave depends on A2. Thus, I∝A2.

The resulting amplitude and intensity are:

Destructive interference occurs when two waves are a half

cycle out of phase. To be out of phase the points on the

wave must have Δφ=(2π)(m+½), where m is an integer.

The resulting amplitude and intensity are:

Coherent waves can become out of phase if they travel

different distances to the point of observation.

When both waves travel in the same medium the Interference

conditions are:

For constructive interference

Where m = an integer.

For destructive interference

Where m = an integer

Fringes

If at point P the path difference yields a phase difference of

180 degrees between the two beams a “dark fringe” will

appear. If the two waves are in phase, a “bright

fringe” will appear.

Example A 60.0 kHz transmitter sends an EM wave to a

receiver 21 km away. The signal also travels to the

receiver by another path where it reflects from a helicopter.

Assume that there is a 180° phase shift when the wave is

reflected.

(a) What is the

wavelength of

this EM wave?

(b) Will this situation give constructive interference,

destructive inference, or something in between?

The path length difference is Δl = 10 km = 2λ, a whole

number of wavelengths. Since there is also a 180° phase

Shift there will be destructive interference.

Michelson Interferometer

In the Michelson interferometer, a beam of coherent light is

incident on a beam splitter. Half of the light is transmitted

to mirror M1 and half is reflected to mirror M2.

The beams of light are reflected by the mirrors, combined

Together, and observed on the screen.

If the arms are of different lengths, a phase difference

between the beams can be introduced.

Example A Michelson interferometer is adjusted so that a

bright fringe appears on the screen. As one of the mirrors is

moved 25.8 μm, 92 bright fringes are counted on the screen.

What is the wavelength of the light used in the

interferometer?

Moving the mirror a distance d introduces a path length

difference of 2d. The number of bright fringes (N)

corresponds to the number of wavelengths in the extra

path length.

Thin Films

When an incident light ray reflects from a boundary with a

higher index of refraction, the reflected wave is inverted (a

180° phase shift is introduced).

A light ray can be reflected many times within a medium.

Interference in Thin Films

We have all seen the colorful patterns which appear in

soap bubbles. The patterns result from interference

of light reflected from both surfaces of the film. Some

colors undergo constructive some destructive

interference:

2t = mλn = mλ / n (m = 0, 1, 2…) Destructive (invisible)

2t = (m+1/2)λn = (m + ½)λ / n (m = 0, 1, 2…) Constructive (bright color)

Newton’s Rings

Example A thin film of oil (n=1.50) of thickness 0.40 μm is

spread over a puddle of water (n=1.33). For which

wavelength in the visible spectrum do you expect constructive interference for reflection at normal incidence?

Consider the first two

reflected rays. r1 is

from the air-oil

boundary and r2 is from

the oil-water boundary.

r1 has a 180° phase shift (n oil >n air),

but r2 does not (n oil

Example :

White light is incident on a soap film (n = 1.30) in air. The reflected light looks bluish because the red light (λ = 670 nm)

is absent in the reflection. What is the minimum thickness of

the soap film?

2t = mλ / n (m = 1, 2…) Destructive interference

t = 1λ / 2n = 258nm

Young’s Double-Slit Experiment

Place a source of coherent light

behind a mask that has two vertical

slits cut into it. The slits are L tall, their centers are separated by d, and their widths are a.

The slits become sources of waves that, as they travel

outward, can interfere with each other.

The pattern seen on the screen

There are

alternating

bright/dark

spots

An

intensity

trace

Interference Conditions

For constructive interference,

the path difference must be

zero or an integral multiple of

the wavelength:

For destructive interference, the

path difference must be an odd

multiple of half wavelengths:

d sinθ =(n+1/2)λ (n =0,±1,±2...)

n is called the order number

Gratings

A grating has a large

number of evenly spaced,

parallel slits cut into it.

Example Red light with λ=650 nm can be seen in three orders in a particular grating. About how many rulings per cm

does this grating have?

Example continued:

Since the m = 4 case is not observed, it must be that sinθ4>1.

We can then assume that θ3≈90°. This gives

Diffraction

Using Huygens’s principle: every

point on a wave front is a source

of wavelets; light will spread out

when it passes through a narrow

slit.

Diffraction is appreciable only when

the slit width is nearly the same size

or smaller than the wavelength.

Definition and Types of Diffraction

• Diffraction is the bending of a wave around an object

accompanied by an interference pattern

• Fresnel Diffraction - curved (spherical) wave front is

diffracted

• Fraunhofer Diffraction - plane wave is diffracted

The intensity pattern

On the screen

Example: Light from a red laser passes through a single slit

to form a diffraction pattern on a distant screen. If the width

of the slit is increased by a factor of two, what happens to the

width of the central maximum on the screen?

The central maximum occurs between θ=0 and θ as determined

by the location of the 1st minimum in the diffraction pattern:

From the previous picture, θ only determines the half-width

of the maximum. If a is doubled, the width of the maximum is halved.

Resolution of Optical Instruments

The effect of diffraction is to spread light out. When viewing

two distant objects, it is possible that their light is spread out

to where the images of each object overlap. The objects

become indistinguishable.

Resolvability

Rayleigh’s Criterion: two point sources are barely resolvable if

their angular separation θR results in the central maximum of the diffraction pattern of one source’s image is centered on

the first minimum of the diffraction pattern of the

other source’s image

For a circular aperture, the

Rayleigh criterion is:

where a is the aperture size of your instrument, λ is the wavelength of light used to make the observation,

and Δθ is the angular separation between the two

observed bodies.

To resolve a pair of objects, the angular separation between

them must be greater than the value of Δθ.

Example: The radio telescope at Arecibo, Puerto Rico, has a

reflecting spherical bowl of 305 m diameter. Radio signals

can be received and emitted at various frequencies at the

focal point of the reflecting bowl. At a frequency of 300 MHz,

what is the angle between two stars that can barely be

resolved?

X-Ray Diffraction

X-rays are electromagnetic radiation with wavelength ~1 Å =

10-10 m (visible light ~5.5x10-7 m)

X-ray generation

X-ray wavelengths to short to be resolved

by a standard optical grating

X-Ray Diffraction, cont’d

Diffraction of x-rays by crystal: spacing d of adjacent crystal planes on the order of 0.1 nm

→ three-dimensional diffraction grating with

diffraction maxima along angles where reflections

from different planes interfere constructive