Huygens and Single Slit Diffraction

Huygens’ Constructions are geometric diagrams which are a fast way of getting to grips with several progressive wave properties like reflection, refraction, diffraction and interference, particularly with EM waves, but modelled in a ripple tank. Here’s an overview.

A plane wavefront consists of a large number of secondary circular wavelets whose speed is constant, if the medium is isotropic (uniform density, like glass or water). These superpose algebraically to form a new wavefront elsewhere in time and space.

Let’s look at diffraction at a single slit. When we see diffraction effects in a ripple tank, we see something like this…the wave smears out horizontally.

The most circular diffraction occurs when the gap size is of the order of 1 wavelength.

If we try the same trick with monochromatic laser light, however, we don’t just see the wave energy ‘smearing out’, as we can see here. Instead, we see a series of light and dark bands.  This applet shows the simplest single slit diffraction very well, but you need the Wolfram CDF player to run it. You can change the colour (wavelength in air) of the light and also change the width of the slit. If we use red light, the pattern is broader than with blue light. Of course, we can’t physically do this, since we usually only have a laser with one colour available. Wider slits allow more light in but produce a narrower set of light and dark bands. Think about what it would look like if we were able to use light of any color, and be able to vary the slit width. Notice the central bright maximum is twice as wide as the others.

Screen Shot 6
Single slit diffraction from a He/Ne laser wavelength 589nm

Now, how does it work? Imagine dividing the slit into two and the slit to consist of lots of secondary wavelets, as in the diagram. The diffracted rays can be thought of as being parallel because the screen is a very, very long way away.

Screen Shot 4

Now think about the first dark fringe. The light from a wavelet near the top end of the slit has a partner exactly halfway down the slit with a path difference of exactly half a wavelength.

The next wavelet down is similarly partnered, and so on all the way down the slit. This means that  every pair of wavelets is superposing destructively, or cancelling out, causing the first dark fringe.

Screen Shot 3.png

in some syllabuses written as:

Screen Shot 2.png

Screen Shot 3.png

Similarly, the light bands are formed because each wavelet is partnered with another with a path difference of  a whole wavelength.so, this time,

 

 

Screen Shot 5

or:

Screen Shot 1

 

We notice that the bright bands are much less bright, which this little construction explains. Imagine this time that the slit is divided into three. At a particular angle, two-thirds of the slit has a wavelet partner half a wavelength out of phase, leaving just one third of unpartnered contributions to form the first or subsequent maxima, so it is much less bright than the central bright maximum.

Screen Shot 1

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s