Fundamentals of Diffraction and Image Formation

# The thin lens considered as a phase transformer

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).

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).

Image IV-1 Thin lens : front view and side view. [zoom...]Info

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.

The total phase delay accumulated by the wave can therefore be written:

Let UL(x,y) and U'L(x,y) be the complex fields located immediately in front and below the lens.

The effect introduced by the lens can therefore be described by a phase transform U'L(x,y)=UL(x,y)tL(x,y) , where :

IV-1 Info

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

IV-2 Info

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:

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:

then the phase transform can be written:

This relation neglects the finite size of the lens. We will take it into account later.

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:

Image IV-2- Diverse types of converging lenses [zoom...]Info
Image IV-3- Diverse types of diverging lenses [zoom...]Info