# How does Huygens's Principle Work?

I just learnt about Huygens's principle that each point of a wavefront acts as a point source producing wavelets. However, I tried drawing it in a diffraction situation, but it doesn't come out right. In theory, if the gap size and the wavelength is same, maximum diffraction occurs, but when I try to draw this on paper it doesn't produce semi-circular wavefronts. It makes the edges rounded while the middle is flat.

Something like this when I draw, but isn't it supposed to be semi-circular if wavelength is same as gap size? • No. It will be semi-circular if the slit size is much smaller than a wavelength, so that the source resembles a point source. A point source would produce only one wavelet, and ... a semi-circle. Anything else is elongated. – garyp Nov 8 '16 at 3:14
• Oh so even though the gap size is same as the wavelength, it isn't actually semicircular? Because this is what I have been taught by my teacher. – Hilkjh Nov 8 '16 at 3:25
• elongated semi-circular would make it descriptively right. – Mozibur Ullah Nov 8 '16 at 7:21

Using Huygens to draw pictures is good for illustrating diffraction but you have to be cautious when interpreting the diagrams.

The diagram below shows the Huygens construction for wavelets separated by one wavelength and there being 11 secondary sources in the slit. Lines $XX'$ and $YY'$ demarcate the region where one would expect to have light if one was considering light to travel in straight lines - geometrical optics.
What you must note is the envelope of the Huygens wavelets between those two lines is a straight line and its length stays constant as the waves travel further and further from the slit.
So by the time you get to 40 wavelength from the slit you almost get what you expected - a semicircular wavefront centred on the slit. For clarity in the second diagram I have just shown three 3 secondary sources in the slit.

This drawing of the wavelets can be avoided using Mathematics and refinements to the secondary wavelets.

One refinement is to have the amplitude of a secondary wavelet dependent on the direction of its travel being a maximum for the part of the wavelet which is travelling in the same direction as the incident light and getting less an less as the direction of travel moves away from the direction of the incident light.
Also as the wavelets move further from the slit their amplitude is reduced because a certain amount of energy now has to be spread over a longer wavelet.

So the diagrams which you draw give you an idea as to what happens but for the detail you must resort to methods which do not rely on drawing semicircles on paper.

• ahh i get it, so its just the first few wavefronts appear to be more flat, but when there is more wavefronts it becomes more semi-circular? – Hilkjh Nov 9 '16 at 3:42

The picture you posted from wikipedia is rather misleading. In fact, Huygens's Principle and what your teacher have taught you is correct: light passing through any aperture is diffracted as if it were coming from a number of aligned point sources (like in the image you provided). However, what this diagram mis-represents is that the sum of the circular wavefronts generated becomes somehow "flat". If this were indeed the case then light passed through a aperture much larger than its wavelength would create an image on a screen placed some distance away from the aperture that is a perfect point, with decaying intensity at the fringes (this seems to be what @garyp is suggesting in his comment).

However, in reality what we find is that the "image" - this just being the light collected at a plane located some distance much larger than the size of the aperture -- follows the pointspread distribution function.

This function can be rigorously derived fromt the assumption of Huygen's Principle, and is derived in Chapter 29 of the Feynman lectures (Lectures on Mechanics).

• So, how would a diagram look like if drawn correctly? – Hilkjh Nov 8 '16 at 6:45
• This answer is a little confused; it's not clear to me what point you are making. A plane wave incident on an aperture much larger than the wavelength would produce a flat top intensity profile, like the image on the left. A shadow. Not a point, as this answer suggests. The point spread function shown on the right is for an aperture about the size of a wavelength, not for a large aperture. Also, I think the drawing of Huygens' wavelets in the OP is correct. – garyp Nov 8 '16 at 12:48
• Wikipedia provides a good graphic of the pointspread function arising from huygen's principle. upload.wikimedia.org/wikipedia/commons/0/0f/… – zephyrus Nov 8 '16 at 18:43