Single-Slit Diffraction (DP IB Physics): Revision Note

Single Slit Intensity Pattern

• When monochromatic light passes through a single rectangular slit, a diffraction pattern can be observed on a faraway screen

• This pattern, similar to the double slit interference pattern, contains:

  • bright fringes of maximum intensity, produced by constructive interference

  • dark fringes of zero or minimum intensity, produced by destructive interference

The diffraction pattern produced by a laser beam diffracted through a single slit onto a screen is different to the diffraction pattern produced through a double slit

• However, the single-slit diffraction and double-slit interference patterns are slightly different

• The central maximum of the single-slit diffraction pattern is:

  • wider and brighter than the other bright fringes

  • wider than that of the double-slit interference pattern

• On either side of the wide central maxima for the single slit diffraction pattern are much narrower and less bright maxima

  • These get dimmer as the order increases

Features of the single-slit intensity pattern

• If a laser emitting blue light is directed at a single slit, where the slit width is similar in size to the wavelength of the light, its intensity pattern will be as follows:

The intensity pattern of blue laser light diffracted through a single slit

• The features of the single slit intensity pattern are: 

  • the central bright fringe has the greatest intensity of any fringe and is called the central maximum

  • the dark fringes are regions with zero intensity

  • the intensity of each bright fringe gradually decreases on either side of the central maxima

Effect of changing wavelength

• If the wavelength passing through the gap increases, the wave diffracts more

• This means the angle of diffraction of the waves increases as they pass through the slit

  • As a result, the width of the bright maxima also increases

• Red light

  • has the longest wavelength on the visible light spectrum

  • produces a diffraction pattern with wider fringes (due to a larger angle of diffraction)

• Blue light

  • has a shorter wavelength on the visible light spectrum

  • produces a diffraction pattern with narrower fringes (due to a smaller angle of diffraction)

Fringe width depends on the wavelength of the light 

• If the blue laser is replaced with a red laser:

  • the light diffracts more as the waves pass through the single slit

  • the fringes in the intensity pattern appear wider

The intensity pattern of red laser light shows that longer wavelengths diffract more than shorter wavelengths

Effect of changing slit width

• If the slit is made narrower:

  • the angle of diffraction is greater

  • the waves spread out more beyond the slit

• For a narrower slit, the intensity graph shows that: 

  • the intensity of the maxima decreases

  • the width of the central maxima increases

  • the spacing between fringes is wider

Single slit diffraction equation

• These properties of wavelength and slit width for single slit diffraction for the first minimum can be explained using the equation:

• Where:

  • = the angle of diffraction of the first minimum (°)

  • = wavelength of incident light (m)

  • = slit width (m)

• This equation tells us:

  • the longer the wavelength of light, the larger the angle of diffraction

  • the narrower the slit width then the larger the angle of diffraction

Slit width and angle of diffraction are inversely proportional. Increasing the slit width leads to a decrease in the angle of diffraction, hence the maxima appear narrower

Single slit geometry

• The diffraction pattern made by waves passing through a slit of width can be explained by considering Huygen's principle, in which

  • each point on a wavefront acts as a source of secondary waves, or wavelets

  • the wavelets are coherent and interfere with each other

• The central maximum is produced by:

  • wavelets travelling straight forward

  • constructive interference due to all wavelets travelling the same distance

Position of the first minimum

• For wavelets travelling to the first minimum at an angle to the centre line (which splits the slit in half):

  • the wavelets on one half of the slit will travel a greater distance than the wavelets on the other half

  • if all the wavelets from one half cancel out all the wavelets from the other half, this will result in a dark region (destructive interference)

• Every point on the wavefront can be paired with another point a distance away

  • For all the wavelet pairs to cancel (or interfere destructively), the path difference must be

The geometry of single-slit diffraction

• If the distance between the slit and the screen is considerably larger than the slit width, :

  • the waves can be considered to travel nearly parallel to each other

Determining the path difference using two parallel waves

• For two rays, and , travelling parallel to each other at an angle between the normal and the slit, where:

  • emerges from the top edge of the slit

  • emerges from the centre of the slit, exactly halfway down

• The path difference is the extra distance travelled by , which is equal to:

path difference = 

• For the two rays, and , to interfere destructively:

path difference =

• Therefore, the angle of the first minimum occurs when:

• Since the distance between the slit and the screen is much larger than the slit width , the angle is very small

• Therefore, the small-angle approximation can be applied:

• This leads to the equation for the angle of diffraction of the first minimum:

• Using similar reasoning, additional minima occur at:

• Where = 1, 2, 3... etc.

• Note: In general, the condition for destructive interference for single slit diffraction can be described by . This expression should be used for angles at which the small-angle approximation is not valid

Width of the central maximum

• The central maximum extends from the centre to the first minimum on each side

• The distance to the first minimum can be found using trigonometry

• Where

  • = angle of the first minimum (rad)

  • = distance from the centre to the first minimum (m)

  • = distance from the slit to the screen (m)

Determining the width of the central maximum

• Using the small-angle approximation :

• The total width of the central maximum is equal to , which gives:

• Substituting the expression for the angle of the first minimum:

Double Slit Modulation

• When light passes through a double slit, two types of interference occur:

  • The diffracted rays passing through one slit interfere with the rays passing through the other

  • Rays passing through the same slit interfere with each other

• This produces a double-slit intensity pattern where the single-slit intensity pattern modulates (adjusts) the intensity of the light on the screen

  • The single-slit intensity pattern has a distinctive central maximum and subsequent maxima at lower intensity

  • The double-slit interference pattern has equally spaced intensity peaks with maxima of equal intensity

  • Together, the combined double slit intensity pattern has equally spaced bright fringes, but now within a single slit 'envelope' 

The double slit interference pattern is modulated inside the single slit intensity pattern

• This is assuming that:

  • The slit width is not negligible

  • The distance between the slits is much greater than their width