double-slit-experimentopticsrefraction
I recently learnt the young's double slit experiment and its basic formulas, but i wondered what would (mathematically) happen if i put one of the 2 slits into a refractive medium such as water or glass. If a formula exists please show the derivation.
Note : The medium is only in between the slit and the screen , everything else is the same .
All I got was two parallel bright fringes instead, like the ones you would get by shining a torch through two very thick slits.
It means the separation between slit is not close and the slits size is not small enough! Those two light beam must overlap to have interference. Small slit size is required to have large diffraction, the optimal slit size is certainly small than wavelength $\lesssim\lambda\approx0.5\mu m$ which gives you large diffraction. However, larger slits size is ok, but you have to (a) Make two slits as close as possible (b) move the setup far from the screen. You will know that it is enough when the light beam can overlap.
For the slit, you need better tools than a knife as well as a better material. First, you should use a shape cutter. Second, you need a material that can have a sharper edge such as film. I believe that film was used in the first few experiments of this kind. You have mentioned a hair is enough so $10\mu m$ should probably be ok, you just need to move the screen further away.
For the light source, you should always use a laser, since a high coherent light is required. Any laser out there is ok, it just cost 1 dollar and I can sure you can borrow a laser pointer near you. As I remember when I was doing Michelson Morley experiment, a tungsten light only gives interference pattern for $<0.1m$ with short coherent length, but a laser can have coherent length $>2m$. It means your life can be easier as you can use a 20 times larger slit with a laser!
Edit: Additional info on the methods Young used for this experiment.
The wiki about Young' interference experiment has quoted his paper on "On the nature of light and colours" (Also around page p.140 in the book Method and Appraisal in the Physical Sciences). The relevant excerpt is:
In order that the effects of two portions of light may be thus combined, it is necessary that they be derived from the same origin, and that they arrive at the same point by different paths, in directions not much deviating from each other. This deviation may be produced in one or both of the portions by diffraction, by reflection, by refraction, or by any of these effects combined; but the simplest case appears to be, when a beam of homogeneous light falls on a screen in which there are two very small holes or slits, which may be considered as centres of divergence, from whence the light is diffracted in every direction.
So, I guess the experiments were carried out as follow:
Since his results cover all color, so it is very likely that he used sunlight rather than other light source such as candle (There was no light bulk at that time). Also, there is no diffraction grating, so it is likely that he was just using a simple prism.
For home experiments carried out these day, we can use LED as a monochromatic light source so that step 1 and 2 can be skipped. If you use a torch, you still need the step 2.
Your question touches upon the characteristic features and controversies of quantum mechanics. You want to know whether any theory can predict or explain which slit a photon passed through in a double-slit experiment.
With a few caveats, the answer is that there is no such theory. Relativity, quantum field theory, string theory etc say nothing about the puzzles in quantum mechanics. In quantum mechanics, it makes no sense to speak of the behaviour of a system in between your observations. In that time, it won't have definite values for observable quantities and photons etc won't follow definite paths, but superposition of all possible paths.
In other words, when you write
In Young's double-slit experiment, we know that a photon goes through either one of the slits but we don't know which one, and it ends up on a screen.
you need to be careful. If we didn't perform the measurement, all we know is that the photon was in a superposition of all possible paths, some going through the first slit and some going through the second slit. The classical intuition that the photon must have gone through one of the two slits and not the other is incorrect.
Now, of course, many over the years objected to this situation, and attempted to construct so-called "hidden-variable" theories, in which a system had predictable behaviour, including which path in a double-slit experiment. As it turned out, though, there are strong constraints on such a theory (e.g. Bell's inequalities ) - the fact is that experiments demonstrate quantum mechanical rather than classical behaviour.
It seems quite unlikely that any theory in the future could be constructed that agrees with our observations and predicts/explains which path a photon travelled through in a double-slit experiment. The interference fringes on the screen result from the fact that the particles don't travel through a definite slit.
Best Answer
The main effect of placing a refractive medium in one of the beam paths is that the phase of that beam will be delayed by an amount depending on the refractive index of the medium and the thickness of the medium. If the phase of one of the beams is delayed slightly, the interference fringes will shift laterally. Delay the phase by 1/2 cycle, and the fringes will shift by 1/2 of their spacing.
Note that the whole fringe pattern will not be shifted laterally. The fringe spacing (which is greatest in the middle and gets smaller toward the outside) will not be changed in the pattern; only the locations of the bright and dark fringes will change. In other words, for small shifts the amount of lateral shift is proportional to the fringe spacing. If the amount of shift is an integral number of full cycles, then the fringe pattern will return to its original form. That's not exactly true. If the total phase shift is greater than the coherence length (in cycles) of the light source, then the light from the two slits will not interfere to form fringes. Fringe contrast is a function of the mutual coherence of the two beams.
So, a two-slit interferometer with an adjustable-thickness refractive medium in one path can be used to measure the coherence function of a light source.