Thin lens approximationexample

A lens is said to be thin if a ray entering through one side at point (x,y) emerges on the opposite side at the same point (x,y) (negligible ray translation).

Let e0 be the maximum thickness of the lens, φL=kne(x,y) he phase delay introduced by the lens and φA=ke0-ke(x,y) the phase delay introduced by the remaining part of the free space that lies between the planes tangent to the entrance and exit sides of the lens.

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Thin lens approximationequation

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Thin lensmagnification equation

A relatively simple mathematic calculation shows that the thickness function can be written (within the paraxial approximation)[]:

Sign convention: R>0 for a center of curvature to the right of the surface, and R<0 in the opposite case. This sign convention, adopted for R and consequently for f, allows applying tL to all types of converging (figure IV-2) and diverging (figure IV-3) lenses:

Focal length ofthin lensformula

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Types ofthinlenses

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Therefore, a thin lens simply delays the incident wave phase with a quantity proportional to its thickness at any point (x,y) (see figure IV-1).

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If we neglect the constant phase term and we group the characteristic dimensions of the lens (n, R1, R2 ) into only one number f, called focal distance, such that:

Thin lens approximationformula

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where R1 and R2 are the radius of curvature of the entrance and exit sides of the lens. By replacing Eq. (IV-2) in (IV-1) we obtain: