This angle is obviously related to the θ = λ/L angle but it’s not the same: in fact, it’s a slightly wider angle. The so-called Rayleigh criterionregards two point sources as just resolved when the principal diffraction maximum of one image coincides with the first minimum of the other, as shown below. If the distance is greater, the two points are (very) well resolved, and if it is smaller, they are regarded as not resolved. Hence, telescopes, microscopes (using light or electron beams or whatever) have a limited resolving power. That may make it difficult to see them, in a telescope, a microscope, or whatever device. If we have two point sources right next to each other, they will create two Airy disks, as shown above, which may overlap. Spatial resolution of a microscope: the Rayleigh criterion versus Dawes’ limit However, these secondary minima do not play any role in determining the resolving power of a lens, or a telescope, or an electron microscope, etc, and so you can just forget about them for the time being.įor small angles (expressed in radians), we can use the so-called small-angle approximation and equate sinθ with θ: the error of this approximation is less than one percent for angles smaller than 0.244 radians (14°), so we have the amazingly simply result that the first minimum occurs at an angle θ such that: The second, third, fourth etc minimum occurs at an angle θ such that Δ = 2λ, 3λ, 4λ, etc. It’s good to remind ourselves of the geometry of the situation: below we picture the array of oscillators, and so we know that the first minimum occurs at an angle such that Δ = λ. the first black ring) occurs at an angle θ such that sinθ = λ/L, with λ the wavelength of the light and L the lens diameter. The spatial resolution is also known as the angular resolution, which is quite appropriate, because it refers to an angle indeed: we know the first minimum (i.e. What’s actually going on is that the lens acts as a slit or, if it’s circular (which is usually the case), as an aperture indeed: the wavefront of the transmitted light is taken to be spherical or plane when it exits the lens and interferes with itself, thereby creating the ring-shaped diffraction pattern that we explained in the previous post. In practice, diffraction causes a blurring of the image, and may make it difficult to distinguish two separate points, as shown below (credit for the image must go to Wikipedia again, I am afraid). For light going through an aperture, this generates the so-called Airy pattern. In my previous post, I derived and explained the general formula for the pattern generated by a light beam going through a slit or a circular aperture: the diffraction pattern. Pre-script (dated 26 June 2020): This post did not suffer too much from the attack on this blog by the the dark force.
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