Wavelets and Multiple Slits (Diffraction and Interference Patterns) – AS Physics Revision

Huygens and his Wavelets (plus a little about the photoelectric effect)

English: Refrection in the manner of Huygens. ...

English: Refrection in the manner of Huygens. Deutsch: Wellenbrechung nach dem Huygens-Fresnel-Prinzip. (Photo credit: Wikipedia)

  • Before we had quantum theory, the best way to explain light was as a wave
  • A person called Huygens came up with a theory that each point on a wavefront became a new source
  • That way, there were infinite point sources, but they interfered, leaving a normal wave shape, which would behave correctly
  • This theory worked for explaining reflection, refraction and diffraction
  • Huygens managed to show that light always takes the path which is the quickest, even if that path is actually a longer distance
  • His theory also predicted things like Snell’s Law, and the fact that light is slower in more dense materials
  • The photoelectric effect eventually came along and proved that light had to be a particle… sort of… leading to quantum theory
  • I’ve got another post on the photoelectric effect, but there’s something I forgot to mention in it (read other post first)
  • Wave theories predicted that even low-frequency waves would make electrons go flying, if you left them firing at the metal plate for long enough. This just wasn’t the case, though
  • Quantum theory was the best way to explain the threshold frequency – where higher-frequency waves caused electrons to be emitted, but waves below that frequency didn’t, even if they were really intense

Young’s Slits

  • This bit isn’t actually about the weird quantum stuff, yet. That’s later on.
  • A good way to get two coherent sources (which will interfere, producing a pattern for you to study) is to have one source, but send it through two slits
  • The two slits (if they’re small enough, compared with the wavelength, and close together) become identical sources of their own, as light diffracts through them and interferes
  • Diffraction is when a wave spreads out after encountering an edge
  • As well as having narrow slits, close together, it’s also important to use monochromatic light, if you want to make measuring the interference pattern easy (a blurry pattern is difficult to measure precisely)

Diffraction Gratings (more than two slits)

  • When you add more slits, you get a sharper interference pattern
  • Diffraction gratings with loads of really tiny slits really close together can be created
  • If the grating has “100 slits per metre”, the slit separation is 1/100 (and that doesn’t just work for the number 100)
  • You’re more likely to be told how many slits there are per millimetre, though, so multiply by 1000 before dividing 1 by it (“32 slits per mm”… 1/(32*1000) = 31.25 *10^(-6) metre slot separation)
  • The closer together the slits are, the more spread out (and sharp) the pattern will be
  • The longer the wavelength, the more spread out the pattern
  • Diffraction gratings are great for using spectrometers to accurately measure the wavelength of light

How The Pattern Forms

  • Say you’ve got a screen, L metres away from your slits, which the interference pattern shows up on
  • Light leaves each slit at various angles (thanks to diffraction)
  • A place on the screen will have light from one slit reaching it, and light from another slit reaching it (or more, if there are more slits)
  • In some of these places, the light will constructively interfere, and in other places, it’ll destructively interfere – which is why you get the light and dark pattern
  • At certain angles, the light leaving one slit will travel exactly one wavelength further than the light leaving the other slit(s), reach the screen, and produce a bright fringe
  • This angle depends on the slit separation and the wavelength
  • There will probably be several angles (or orders) where this happens
  • Right in the middle is the zero order, the next one out (on both sides, as it’s symmetrical) is the first order, the second one out is the second order and so on…
  • There are equations to help you find the angles which will produce bright fringes, the distance to the screen the bright fringe separation, the wavelength or the slit separation

The Equations

  • n λ = d Sin(θ)
  • Where n is the order you’re looking for (a positive integer), lambda is the wavelength, d is the slit separation and theta is the angle at which the order appears
  • That one’s straightforward – just substitute things in
  • You can also show that Sin(θ) = x/L, where L is the distance from slits to screen, and x is the distance between bright fringes on the pattern on the screen (from the middle of one bright patch to the middle of another bright patch)
  • This is proven by drawing a line straight from the middle of a slit to a bright patch, then drawing another line from the same starting place, to the next bright patch, and joining them up to make a right angle triangle
  • Now for the trigonometry: You want the angle over by the slits, so x, the fringe spacing, is Opposite. You need it as a Sine, so you’re using Sin = Opposite / Hypotenuse. The Hypotenuse is unknown, but you can assume that the angle is so small that the Hypotenuse is basically L, the distance from slits to screen
  • You can use that strange approximation to work out θ, x or l
  • FOR SINGLE SLIT DIFFRACTION, IT’S DIFFERENT (it’s very important not to get mixed up. See below…)

About Matt

I like writing, filmmaking, programming and gaming, and prefer creating media to consuming it. On the topic of consumption, I'm also a big fan of eating.
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