Diffraction and phase difference

[CINA]

Hi,

I'm having some trouble understanding the path difference between two parallel rays leaving a single slit, and was hoping you guys could help me. I'm having a hard time understand what path differences exist for each subsection of a slit and for which minimum. For example:

-What is the phase difference of two rays leaving the top and bottom of a slit ( width 'a' ) at the first minimum.

-What about two rays leaving from the top and a/2 at the first minimum? At the top and a/4?

-What about two rays leaving the top and bottom at the second minimum?

-What about two rays leaving the top and a/2 at the second minimum?

Does this just come from the a*sin θ = mλ?

 

[Ken G]

You need to know that a wave cycles through 2pi phase each time it propagates a distance equal to its wavelength. So if you know the wavelength, to get the phase at the screen you just need to calculate the distance and divide by the wavelength, and look at the fractional non-integer remainder. Multiply that fractional remainder by 2pi and that's the phase it arrives at the screen with. Do it again for another place in the slit, you get a different answer because the distance is different (you'll need nothing but the Pythagorean theorem). Take the difference between those answers, and you get the phase difference. After you do this a few times, you'll note some patterns that make it easier to extract the general rule there. (Working to lowest order in a/d simplifies these expressions drastically, and is normally done.)

 

[sophiecentaur]

The two slits (Young's) calculation tells you where you will get cancellation between the two elements, treating them as point sources. The textbook treatment for a single slit divides the (vertical) slit into a left half and a right half and works out the resultant for pairs of elemental slits which are separated by half the slit width and adds them up (progressing from simple interference to diffraction, that is). The angles of the nulls from all the pairs will coincide to give you the nulls on the single slit diffraction pattern. However, you need to do more than that to find the actual values away from the nulls.