Image formation under monochromatic illumination

We consider an object located at a distance do in front of the lens; at a distance di (in the image plane) we have a field distribution Ui (xi,yi) (see figure IV-9).

We define the magnification Gs =di /do .We assume that :

(the Gaussian lens formula is satisfied).

We first consider the image predicted by geometric optics. It is simply a reversed and dilated version of the object: 

On the wave optics point of view, and taking into account the linear property of the propagation phenomenon, the expression of Ui is a superposition integral (Huygens-Fresnel principle): 

where h(Mi,Mo) is the amplitude of the field produced at the point Mi(xi,yi) by a point source of unit amplitude and located at the point Mo (xo,yo) in the object plane.

h(Mi,Mo) fully characterizes the optical properties of the system. It is called the point-spread function of the imaging system.

To determine h, we consider a point source in the object plane δ(Mo) (see figure IV-10).

This point source emits a spherical wavelet which propagates on a distance do until the plane tangent to the lens. At this plane, the field amplitude is:

This field meets the pupil function P(x,y) characterizing the finite dimensions of the lens. We apply the phase transform:

to this field to obtain the transmitted field U'L. Using Fresnel diffraction law on a finite distance di gives the expression of the point-spread function: 

h(Mi,Mo) represents the Fraunhofer diffracting figure of the lens aperture (the pupil) centered at point xi =-Gsxo ; yi =-Gsyo .