# Interference and diffraction

### Young’s double slit

Often quoted as a demonstration of the wave nature of light.  It uses a single slit followed by a double slit to produce “fringes”.  These are areas of light and dark caused by constructive and destructive interference.

Formula:

l = ws / D

l = wavelength of light,

w = fringe width (centre to centre)

s = slit spacing

D = distance of screen from the double slits.

Note:  The use of the double slit is so that we have two sources that are coherent.  This means that the two sets of light waves are emitted with a constant phase difference.  As a result the fringes stay in fixed positions.

Questions are often set on the effect of altering the various variables in the equation.  E.g. a greater wavelength of light would result in wider fringes as l is proportional to w (for fixed s and D).

Important to remember that red light has a greater wavelength than violet, so passing white light through the double slit arrangement would result in a dispersion of the coloured spectrum – with the red fringes showing greatest spacing.

### Diffraction by a single slit

Fringes are also seen when a point source of light is viewed through a narrow single slit. In this case the fringes fade away from the central bright maximum.  The central fringe is twice the width of the other fringes.

The narrower the slit the greater the diffraction.

The longer the wavelength the greater the diffraction.

 Intensity Position on screen

Diffraction is when a wave passes through a narrow opening and is bent.  Maximum diffraction occurs when the wavelength of the wave most closely matches the width of the gap.

Note the brightness of Young’s double slit arrangement fades out at the edges two – this is due to the first single slit superimposing a single slit diffraction pattern on top of the double slit fringes.

In this example the dotted line represents the original single slit diffraction pattern.  The unbroken line represents the double slit pattern as modulated by the single slit pattern.

 Intensity Position on screen

### Diffraction Gratings

A plate with many closely spaced is called a diffraction grating.  If there are many closely spaced slits then fewer but sharper fringes are seen.  This is because each of the many slits transmit light but they cancel one another out in most directions.  Each slit produces a diffracted set of waves but they reinforce each other only in certain directions.

The closer the slits, the more widely spaced are the diffracted beams.

The longer the wavelength the more widely spaced are the diffracted beams

White light gives a mini-spectrum of colours in each diffracted beam.

Formula:

nl = d sin q

n = order number of the beam

l = wavelength

d = slit spacing

q = angle of diffraction of the nth order

This needs to be derived:

Consider two slits on the grating (Q and P) in close up and follow two paths of diffracted light from the two slits.

 Q   P

To understand this derivation you have to be confident of two facts.

1.  The angle of diffraction of the waves (q) is equal to the angle QPY.  Go double check this fact and make sure you are confident of it.

2.  The path difference between the two paths is equal to length QY.  Again battle with this one until you are confident.

From trigonometry sin q = opposite / hypotenuse = sin [angle QPY] = QY / PQ (using fact number 1)

For constructive interference, the path difference must be equal to nl (so the waves arrive in phase with each other)

Or put another way path difference = QY = nl - for constructive interference (using fact number 2)

Put these two ideas tighter and:

sin q = QY / PQ = nl / d   (where PQ = the slit spacing, d)

OR:  sin q = nl / d   rearranges to give: nl = d sin q

Grating diffraction pattern:

see diagram – for effect of single slit diffraction upon the widely spaced maxima.  The dotted line is the single slit diffraction pattern that moderates the intensity of the grating fringes.

 Position on screen