Refraction, lens equation, optical power

A lens is most commonly used for refraction imaging. A lens is the simplest optical system. It consists of a transparent medium (most often glass, sometimes plastic) bounded by two refractive surfaces (usually spherical).

There are two types of lenses: A lens can be:
 * Converging  : parallel rays intersect at the image focus $$ \to $$ the focus is real (f > 0);
 * Diverging  : parallel rays diverge in the image focus; they intersect during retrograde extension at the subject focus$$ \to $$ the focus is virtual (f < 0).

A lens is called thin if its thickness is small compared to the radii of its spherical surfaces. In a homogeneous environment: $$ f=f' $$.
 * Thin : the thickness of the lens is negligible compared to its focal length;
 * Thick : the thickness of the lens is not negligible compared to its focal length.

Thick lens
$$ \left(\frac{n_2}{n_1}-1\right)\left(\frac{1}{r_1}+\frac{1}{r_2}\right) = \frac{1}{f} $$
 *  n2 : index of refraction of the lens material n1 : index of refraction of the surrounding medium;
 * r1 a r2 : radii of curvature.

Thin lens
$$ \frac{1}{a}+\frac{1}{a'} = \pm \frac{1}{f} $$ This equation is valid for both converging and diverging lenses. When substituting the values ​​a, a' we follow the sign convention:
 * a : object distance,
 * a‘ : image distance,
 * f : focal length.


 * a > 0 : the object is located in the object space (in front of the lens);
 * a < 0 : the object is located in the image space (behind the lens);
 * a‘ > 0 : the image is located in the image space; it is real;
 * a‘ < 0 : the image is located in the object space; it is virtual.

Optical power
Optical power is a quantity expressing the refractive power of a lens.

Calculation of optical power
$$ \varphi = \frac{1}{f} $$ Unit: diopter (D) = m -1 ( f must be given in meters! ). 1 D is the optical power of a lens with a focal length of 1 m. φ  > 0 applies to converging lenses and φ  < 0 to diverging lenses.
 * φ : optical power,
 * f : focal length.

Image formation on ray diagram
(Note: The distance of the object ( x ) from the lens: what is the image)

In case of a diverging lens, its distance from the lens does not matter. The image is always apparent, direct and reduced.
 * x > 2f : real, inverted and reduced;
 * x = 2f : real, inverted, same size as object;
 * x < 2f : real, inverted, magnified;
 * x = f : the image is at infinity;
 * x < f : apparent, direct, magnified.