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. But we can put the slits of the double slit at the maxima of the interference pattern for the wavelength we are interested in.
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.
We know that a single slit produces a single slit diffraction pattern. Let's start with the monochromicity condition. 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. 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.Īfter all this reading we are ready to see the magic. 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. Now let's take our favorite incoherent light source, an incandescent light bulb, and shine it on a double slit. So the monochromicity condition is necessary. As the number of lasers increases, this effect would get worse and worse. 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. Each piece individually would give you a nice diffraction pattern, but they interfere with each other creating a complicated pattern of maxima and minima. 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. The electric field at the slits will be a sum of two sine functions. Let's suppose you shine a green laser and a red laser on a diffraction grating at the same time. Is the monochromicity condition necessary. Thus the phase coherence condition is met. 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. 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. 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. 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. 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). 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). 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. This is because the only way information about the electric field can get to the other side is through the slits. 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. 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. Let's recall the easy way to make a diffraction pattern from a double slit.
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