Imaging with One Lens

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Imaging with One Lens is the fifth lecture within the Waves and Optics section of PH1011. It covers focal lengths, the thin lens equation, linear magnification and angular magnification.

Focal Lengths
The definition of the focal length is the distance from the lens at which incoming parallel rays (eg from a source at infinite distance) converge at one point. The paraxial approximation allows tanθ = sinθ = θ, and these calculations work for thin lenses (small distance in comparison to the object and image distances). Images are real if they form on the positive side of the lens and virtual if they form on the same side that they came from. real images are always inverted.

As the angle alpha is conserved, both tangents can be set to equal and rearranged to give the linear magnification; h'/h = -v/u. The thin lens formula 1/u + 1/v = 1/f can be found by setting the two angles on the right focal distance equal and deriving from there.

Angular Size
Angular size α = h'/d (α measured in radians) as α = sinα for thin lenses); this proves that image height is dependent on the angle is is seen at.

Summary
The focal length is the distance at which incoming parallel rays converge. Real images form for positive v (image distance) and virtual images for positive u; the inverse of the focal length is equal to the sum of the inverses of the image and object distances. Linear magnification is given by negative image distance divided by object distance, and angular magnification is given by image height divided by distance.