*As of now, only high-school level diffraction covered. Will go into more detail in future once prerequisite maths learned. I really struggled with this!* # Introduction - Wavelets, Wavefronts, Ever wondered why a *CD* reflects different colours? When a wave travels through a small gap (known as a 'slit'), it actually bends inwards (or outwards, depending on perspective!), creating an effect known as **diffraction.** Let's dive into it! ![[diffraction_wavefront.png]] *Fig 1: A visualisation of diffraction! Here, we're using ==wavefronts== to portray the movement of the wave.* We've got to explain why a wave diffracts. A common, textbook-y way to explain why is the use of *wavelets* - infinitely small point sources along the slit which the wave passes through. Although this explanation works, it's difficult for the human mind to grasp the concept of an 'infinity', which automatically disqualifies it from settling down in our minds. # The Diffraction Grating We've got a specific equation for single-slit diffraction... $d\sin \theta = n\lambda$ Where: - $d$ is the *average separation between slits* - $n$ is the *number of maxima* - $\lambda$ is the *wavelength of the light incident on the slits* - $\theta$ is the *angular width of the number of maxima* ![[double-slit diagram.png]] *Fig 2: The equation, visualised - to find the angular width of the second maximum.* When we approximate this equation for a small angle, we get: $\theta = \frac{n\lambda}{d}$ Which is what we usually do, since the wavelength of light is so small the width of the central maximum is also going to be small! For single slit equations, since the path difference is just the difference between the diffraction grating, we can just write it as: $\theta = \frac{n\lambda}{b}$ Where $b$ refers to the *WIDTH OF THE SLIT.* ![[double-slit diagram no label.png]] **Huygens' Principle** allows this to happen - according to it, ***there is an infinite number of sources (wavelengths) when a light passes through a slit.*** So we're just taking the last sources from the slit as our distance to find the path difference between waves. ## The Double-Slit Equation Alongside the single slit, the **double-slit model** is pretty commonly used as a special example for the single-slit equation. To set the scene, [Young's Double-Slit equation](https://pressbooks.online.ucf.edu/phy2053bc/chapter/youngs-double-slit-experiment), in which Thomas Young shone monochromatic light on two slits, allowed us to derive the trademark: $d\sin \theta = n\lambda$ Since he used a double-slit to prove this as true, we can just adapt the equation for $\theta$: $\theta = \frac{n\lambda}{d}$ When $\theta$ is small, as it often is. When we use $D$ as the *distance to the screen that the spectrum is being projected on*, we can write $\tan\theta$ and therefore $\theta$ as: $\theta = \frac{s}{D}$ Where $s$ is the *distance being fringes* - as the equation can be used to find the distance between successive light or dark fringes! So the formula in terms of $s$ would be: $s = \frac{n\lambda D}{d}$ Where: - $n$ is the *number of fringes we want to find the distance to* - $\lambda$ is the *wavelength of light incident on the slits* - $D$ is the *distance to the screen from the slit* - $d$ is the *slit separation* - $s$ is the *linear distance between each fringe.* Otherwise known as the **Double-Slit Equation.** Remember to keep the diffraction grating equation and this one every time you're asked a question about how the spectrum changes! For example, if the wavelength of light incident on the slit decreases, then the width of the maxima will decrease. If the number of slits increases *for the same size of slit*, **the distance between each slit will decrease** (if the distance between each slit stays the same, the distance to each maxima will stay the same - see next paragraph for universal changes!) and **so the distance to each maxima will increase.** However, **the intensity of each maximum will increase**; as per the wavelets from Huygens' Principle, the intensity of the beam from each of the slits will be the same as the progenitor beam and such we're getting more sources at the maxima. **When the number of slits is increased, the WIDTH OF EACH INDIVIDUAL MAXIMUM will decrease!** That's because there are more **secondary maxima** in the diffraction spectra - which we'll see later! %%second vs fourth order APPROXIMATE diagram%% More details about why this is the case below! ## Attributes of a Double-Slit Spectrum %%show what we'd expect%% In fact, the double-slit spectrum is actually a **MIXTURE** of the single-and-double slit interference patterns: ![[double-single overlap.png]] *Fig 2: What we actually see! Red: double-slit spectrum, Blue: single-slit spectrum* Same goes for the three, four-slit spectra - just that the maxima get taller and taller, whilst the secondary maxima get shorter and shorter. # Example Questions *TBA* # What's Next? Navigate back to the home page here: [[Home]] Navigate back to the physics home page here: [[The (Incomplete) Physics Almanac]] Navigate to any waves' articles: [[Waves Foundations]]