If you have a an incoherent light source, why does passing it through a single slit make it coherent? Surely it makes a wavefront, which will be a coherent wave front, but there are still multiple wavelengths coming through still. Lastly, passing the now coherent wave front through a double slit will merely result in an interference pattern, correct? It wouldn't result in any other useful characteristics, would it?
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$\begingroup$ What evidence do YOU have, or know of, that passing incoherent EM radiation through a single slit, makes it coherent ?? Over what path length or time interval, would such coherence last, if the original source were say a hot lamp filament, for example. So why do we use lasers, instead of slits, to make coherent EM radiation ?? $\endgroup$– user26165Commented Nov 6, 2013 at 2:24
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3 Answers
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Let's recall the easy way to make a diffraction pattern from a double slit. You just shine a laser on it. Why does this create a diffraction pattern? Well we know that the intensity of the light is determined by the electric field, so to figure out why the intensity of the light is the way it is, we just need to figure out why the electric field is the way it is.
To determine what the electric field is on the far side of the double slit (away from the laser) we only need to know what the electric field is at the two slits. This is because the only way information about the electric field can get to the other side is through the slits. Since the laser produces an electric field which oscillates sinusoidally at a specific frequency, we know that the field at each slit must be oscillating sinusoidally with the same frequency as the laser and hence each other. Also since the distance from the laser is the same, the electric field oscillations at the two slits are in phase (even if the distances to the laser were different, they would still be out of phase by a constant phase, and you would still see a diffraction pattern). So two things are necessary: the light needs to be oscillating sinusoidally at a single frequency (I will call this the monochromicity condition), and the oscillation at both slits needs to be offset by a constant phase (I will call this the phase coherence condition).
I will not explain why these two conditions are sufficient for diffraction (I assume you know that already), but let's see if they are necessary.
Suppose first that the monochromicity condition is met, but the phase coherence condition is not met, so that the phase difference between the two silts varies as a function of time. This could be accomplished for example by holding your double slit up to the sun but using a band-pass filter to filter the sunlight. In this case, I claim you would see no diffraction pattern because as the phase difference between the two slits varies, the location of the diffraction maxima and minima will quickly vary, and the diffraction pattern gets washed out. Thus the phase coherence condition is met.
Is the monochromicity condition necessary. Let's suppose you shine a green laser and a red laser on a diffraction grating at the same time. The electric field at the slits will be a sum of two sine functions. What pattern do you see? You can decompose the electric field on the screen into a piece from the red laser and a piece from the green laser. Each piece individually would give you a nice diffraction pattern, but they interfere with each other creating a complicated pattern of maxima and minima. Moreover, they oscillate at different frequencies, so these maxima and minima will move around very quickly and the pattern will wash out to something that doesn't look much like a diffraction pattern. As the number of lasers increases, this effect would get worse and worse. So the monochromicity condition is necessary.
Now let's take our favorite incoherent light source, an incandescent light bulb, and shine it on a double slit. What happens? Well the light bulb is emitting essentially as a black body so it is emitting radiation at all frequencies and so does not satisfy the monochromicity condition. Also since the light bulb is a large object composed of many independent sources with different path lengths to the slit, the phases at the slits will be uncorrelated and we will not satisfy the coherence condition.
After all this reading we are ready to see the magic. By first passing the light from the light bulb through a single slit, it is possible to arrange things so that the electric field in the double slits satisfies both conditions.
Let's start with the monochromicity condition. We know that a single slit produces a single slit diffraction pattern. Actually the electric field is a superposition of electric fields oscillating at all frequencies, and so the electric field will be a superposition of many interference patterns. But we can put the slits of the double slit at the maxima of the interference pattern for the wavelength we are interested in. Then the electric field from most the other frequencies will be filtered out because they will be closer to their diffraction minima, and we will be left with a sufficiently monochrome electric field to produce a diffraction pattern.
What about phase coherence? Well that is easy. The light takes the same amount of time to propogate from the single slit to either double slit, so the electric field at either of the double slits will always be the same, so they must be in phase.
Since the electric field at the double slit satisfies both the monochromicity and phase coherence conditions, you will see the double slit diffraction pattern.
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$\begingroup$ Diffraction by a single slit, and double slit interference experiments were performed before laser or other coherent light sources existed. The slits do not render incoherent radiation sources coherent, nor is source coherence necessary to get either diffraction or double slit interference. The input single slit isolates radiation from a small region of an incoherent source, which ensures that the radiation arrives at the double slits phase related. The coherence length is short, but that doesn't change the phase difference at the slits. Small frequency range reduces color fringing. $\endgroup$– user26165Commented Nov 10, 2013 at 8:15
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$\begingroup$ You raise a good point that with the setup involving an incoherent source and a single slit, the light reaching the double slits will have a short coherence time, but since the light at either slit is in phase with each other (that is the phase drifts the same way for both slits), they will still produces the desired diffraction pattern. What is important is that the waveform from the single slit is spatially coherent, which guaranteed because the single slit acts like a point. $\endgroup$ Commented Nov 10, 2013 at 15:51
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$\begingroup$ The only thing you said that I take issue with is that the single slit isolates radiation from a small region of an incoherent source. All the light emitted from the surface of the source facing the single slit makes it in, but because the single slit is so narrow, if you are downstream you can't tell where the light originally came from, so it looks like the light originally came from a point source. $\endgroup$ Commented Nov 10, 2013 at 15:55
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If the slit is thin enough and you are only going to use light from one plane, then the new wavefront is dominated by contributions from only one part of the original wave and all parts of it (the wavefront) share the same phase.
answered Nov 1, 2013 at 22:51
dmckee --- ex-moderator kittendmckee --- ex-moderator kitten
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$\begingroup$ What is the drop in intensity? $\endgroup$– user24082Commented Nov 1, 2013 at 22:55
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$\begingroup$ @Anonymous in general many orders of magnitude. Actually dmckee's wonderfully succinct answer even tells you (in a subtle way) how you would calculate it: you find the positions of the "dominating radiators" on the original wavefront by Fermat's principle (same as the method of stationary phase) and then apply the stationary phase formula to calculate what gets through. You get better "yield" by focusing on the slit, but the slit works by throwing out "incoherent bits" so there will always be a hefty loss unless the light is coherent and unaberrated to begin with. $\endgroup$ Commented Nov 2, 2013 at 5:35
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Ok I will keep is simple. For interference, we need waves coming from two sources, in constent phase diffrence and same frequency. One way to achieve this would be to have two sources, say two 60W light bulbs to illuminate each slit. But light arriving from these two bulbs will be slightly diffrent in freequency and will not have constent phase diffrence and will not produce interference pattern. To solve this problem, one light bulb is used through one slit, then light passing through the double slits is coming from the same one source, in constent phase diffrence, hence acting as two coherent source of light to achieve better interference patterns.
