Some diffraction slits/gratings questions

[lol_nl]

Homework Statement

First I have one general question about single-slit diffraction patterns. I've never fully understood this phenomenon. In my book it is stated that there will be minima when the waves from the top and the bottom of the slit differ exactly a whole wavelength, because one can then divide the slit into pairs of gaps that will then interfere destructively. Then maxima should occur more or less in between these minima, when the top and bottom waves differ k+0.5 (k integer from 0 on) wavelengths. Is it correct that the intensity then decreases with I = I0 (1/(2k+1))? Each time one divides the slit into an odd number of gaps, from 3, 5, 7 etc. on, there will still be pairs of gaps interfering destructively, so only 1/3, 1/5, 1/7, etc. of the intensity remains.
Then I do not understand why one doesn't consider the situation when pairs of gaps differ k wavelength. E.g. you divide the slit into two gaps and the top of these two gaps differ one wavelength, so a/2 sin(theta) = k(lambda). But one can reduce that to a sin(theta) = (2k)(lambda), which overlaps with the formula for the maxima! Where am I getting stuck?

Then there is one problem I couldn't solve.
1) A plane wave of wavelength 590 nm is incident on a slit with a width of a = 0.40 mm. A thin converging lens of focal length +70 cm is placed between the slit and a viewing screen and focuses the light on the screen.
(a) How far is the screen from the lens?
(b) Calculate the angle (theta) of the first diffraction minimum

Homework Equations

Diffraction formulas for single silt, lens formula..

The Attempt at a Solution

For the problem: I can't see how you can derive the distance from this information and don't understand what the lens does to the interference pattern.

 

[hikaru1221]

I don't remember much about diffraction, so I'll try to give you my opinions as much as I can.

First, I don't believe the intensity decreases that slowly. I recall that the second maximum is a lot smaller the 1st one. I also remember that the equation sin(theta)=k*lambda/a (k is any integer but 0) is for minima only. Have a look at this:
http://en.wikipedia.org/wiki/Diffraction#Single-slit_diffraction

And about the problem, consider minima for convenience. Each minimum corresponds to only one angle, i.e. all the light coming from all points at the slit under that angle will give us the corresponding minimum. Because all the lights of a minimum comes out under only 1 angle, they are parallel. Now in order that all those lights can focus at one point of the screen to give us one-pointed minimum pattern, what should the distance between the lens and the screen be?

 

[lol_nl]

Ah. Because the light rays are perpendicular when they reach the lens and the lens has to focus all the light rays to one point, that point has to be the focal point of the lens.
So the distance is 70 cm. Then (b) can simply be solved using sin(theta)= lambda/a.

Thanks.

 

[hikaru1221]

Is it true that all of them are perpendicular to the lens?
If all the lights are focused at one point, how can we observe the diffraction pattern?

 

[lol_nl]

The three light rays on your drawing are parallel to each other (first order minimum, so two halves interfere destructively) and create a minimum on the lens that is then focused onto the screen. And the light rays between them... they should at least be perpendicular in pairs that produce destructive interference, right?

 

[hikaru1221]

I guess you meant "parallel" instead of "perpendicular", right?
The rays I drew don't have to correspond to the first minimum. They just come out at the same angle.