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A: Wave Phenomena A.5 Resolution

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Resolution Resolution refers to the ability to distinguish two objects that are close together. E.g. Two distant stars. If you can see them separately you can resolve them. If they look like just one dot, you can’t resolve them.

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Rayleigh Criterion If the light from two separate objects passes through a slit, two diffraction patterns will be observed on a screen:

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If the angular separation is small, the two diffraction patterns will lie on top of one another:

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This can lead to the two images being indistinguishable from one another:

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The Rayleigh criterion: Two objects can be just resolved if the central maximum of the diffraction pattern of one of the sources is formed at the position of the first minimum of the other.

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If we consider the resultant image intensity of the two diffraction patterns superimposed upon one another it can be seen how two images become indistinguishable (unresolved): Simulation

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As a result of the Rayleigh criterion, θ (the angle between the central maximum and first minimum) also gives the minimum possible angle formed by light rays passing from the two sources to the aperture of the detector (e.g. The eye) b x θ θ D Central maximum 2 (Minimum 1) Central maximum 1 Two sources: θ = 1.22 λ b θ = x D Note: The small angle approximation states that if D is large: θ ≈ tanθ ≈ sinθ So...

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E.g. A spy satellite travels at a distance of 50km above Earth's surface. How large must the aperture of the lens be so that it can resolve objects with a size of 2mm and thus read a newspaper? Assume the light has a wavelength of 400nm.

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Answer 1.Draw the situation: 2. Determine the minimum value of θ using the small angle approximation 3. Determine b from 4. b = 12.2m. θ = 1.22 λ b b 2mm θ θ r s 50 km Central maximum of r Central maximum of s

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Better resolution We can see that increasing the distance to the viewed objects brings the two diffraction patterns closer together, ultimately to the limit stated by the Rayleigh criterion. The only way to then improve resolution (to decrease the minimum possible angle between the central maxima) is then to increase b or decrease λ b x θ θ D Central maximum 2 Central maximum 1

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CDs, DVDs and Blu-ray To store more information on a BD than a DVD the pits are made smaller. This means that the aperture for the reflected laser light decreases, thus increasing θ and decreasing resolution of the detector. To compensate, a shorter wavelength of light (405nm compared to 650nm) is used.

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Radio Telescopes Radio waves emitted by bodies in space have very long wavelengths (typically 20cm). Clearly a small reflector aperture would result in large values of θ and low resolution. Hence reflector dishes with very large apertures are used. The Arecibo observatory in Puerto Rico has a reflector aperture of 305m. googlemap

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