Wave superposition

• If two waves are in the same place at the same time they superpose.

• This means that their amplitudes add together vectorially

Positively when they are in phase

Wave superposition• If two waves are in the same place at the same time they

superpose.• This means that their amplitudes add together vectorially

Negatively when they are in antiphase

The conditions for two waves to interfere with each other

• The waves must be coherent.

This means there must be a constant phase difference between them ( which also implies that they have to be of the same frequency.)

Interference between water waves from coherent sources

Interference Is best understood with relatively long waves.

Both water waves or microwaves (3cm ) provide a reasonable model.

Coherence of the waves is ensured by obtaining the waves from a single frequency microwave source ( a monochromatic source)

The waves are passed through two slits and are diffracted in the process. (That is they begin to spread out as if the slit was at the centre of a circular wave front)

On a wave front diagram the waves positively superpose where they cross and negatively superpose in the centre of the gaps.

A microwave detector moved normal to the source detects positive and negative superposition called interference fringes.

Each peak is produced by positive interference. Each trough occurs because of negative interference

Central maximum phase difference 0

Here the path lengths from S1 and S2 to X are the same

S1

S2

X

Next maximum phase difference 2π

Note that when the phase difference is 2π the path length from one of the slits is longer by a single wavelength

S1

S2

2π

Path difference 0

Path difference 2π

Path difference 4π

0.670m

0.645m

In this case λ= 0.670-0.645= 0.025m.

Measuring the difference in path length to the first fringe

Zero or central fringe

1st fringe

Notes

• If you measure over 2 fringes you would have to divide your answer by 2 and so on. (the fringes are equally spaces).

• You may choose to use maxima OR minima. (they are equally spaced)

• Measurement of path difference is impractical for interference involving visible light! Why?

Interference with Light Sources

laser

double slit

The laser is a coherent light source which is divided into two by the fine double slit The screen is at a distance of 5-10m. The interference pattern below is produced. The fringes are equally spaced.

The geometry of the situation gives us the relationship

D

ws

D

W is the distance between adjacent fringes

S is the slit separation

In reality you would measure the total distance between the centre of several visible fringes and divide by the number of dark intervals between them to achieve a better value for w.

Diffraction from a Single Slit

Through a narrow single slit the wave front spreads out. If the slit is wide the spreading is slight. If the slit is comparable in width with the wavelegth of the wave the spreading is large.

Diffraction of water waves from above

Diffracted laser light from a single slit projected onto a screen

intensity

Zero order principle maximum

2nd order

Principle maximum

minimum

minimum

minimum

minimum

1st order

Principle maximum

1st order

Principle maximum

2nd order

Principle maximum

The diffraction grating

• A typical diffraction grating is an arrangement of identically spaced diffracting elements. Normally a large number of parallel lines are ruled on glass. The diffracting elements are the gaps between the ruled lines. typically there would be around 600 lines per mm.

Each line acts as a very narrow slit

θ

θ

The light through each slit is diffracted in all directions.

Consider the light through two slits diffracted at angle θ to the normal

A

B

C

If the light diffracted through angle θ at A is in phase with the light diffracted through θ at B it must be in phase with the light at this angle through every other slit.

θ

A

N

θ

Notice that:

to be in phase

the path difference (ie the distance A to N)

has to be a multiple of the wavelength λ

i.e (n λ )

X

Y

d

The path difference is the length AN

The distance AN = d sinθ So d sinθ = n λ

What you see

• When the waves interfere constructively through each slit they are at an angle given by the formula

• d sinθ = n λ

Question

• When a grating of 300 lines per millimetre is illuminated with parallel beam of monochromatic light normal to it a second order principle maximum is observed at 18.90 to the straight through direction. Calculate the wavelength of the light.