Thin lenses

1- Generalities

A lens is a system formed by the association of two diopters, at least one of which is spherical. It is a centered system whose axis is the line joining the two centers of these two diopters

The approximate stigmatism of lenses is related to that of the diopters forming these lenses; the conditions for stigmatism are satisfied in the Gaussian approximation: 

• Lens of small aperture 
• Object that is planar and perpendicular to the optical axis 
• Object of small extent (small angles).

3- Thin lenses

A lens is said to be thin if its thickness is small compared with the radii of curvature; for a thin lens one can show that 

$e\ll R_1$ , $e\ll R_2$, and $e\ll \mathrm{\mid }{R}_1-R_2\mathrm{\mid }$.

Since the lens is a centered dioptric system, it has principal foci: an object focus $F$ and an image focus $F^{\mathrm{\prime }}$, which are equal in magnitude and opposite in position; according to the expression for the focal distance of a thick lens:

 

$

5- Conjugation relation

Let us consider a thin lens as the combination of two diopters $D_1$ and $D_2$, and a real point $A$ on the principal axis.

Thus, using the relations for spherical diopters.

Since the lens is thin, that is, $S_1\equiv S_2$, then:

If the thin lens is now considered as a centered system immersed in a medium of refractive index $n$n (so $n^{\mathrm{\prime }}=n$), then one can write: 

and since $S_1\equiv S_2$, the conjugation relation of a centered system (SC) gives:

The convergence of a thin lens is defined as the ratio $C=1\mathrm{/}f$ in diopters; 

• if $C>0$ then $f^{\mathrm{\prime }}>\mathrm{0 }$ and the lens is converging​
• if $C<0$ then $f^{\mathrm{\prime }}<0$ and the lens is diverging
  

For a thin lens, Newton’s formula is written as:​

Geometric construction

6- Magnification

Magnification is given by: