Optics

# Equation of lens manufacturers

Descartes is credited with discovering the lens makers equation

Daily we see people around us using glasses for visual rest (sunglasses) or for visual correction (prescription glasses). The first manufacturers of these accessories followed empirical recipes to melt the glass and thus mold and polish it. A little later, these techniques were improved thanks to an understanding of the principles governing refraction. Understanding the principles of refraction, it was possible to formulate an equation that related the geometric characteristics of the lens (the radii of curvature of the light incidence and emergence faces), the physical characteristics of the material (refraction index of the material in relation to the external environment) and its convergence power.

Historically, this discovery was attributed to René Descartes who, in addition to being a physicist, was a philosopher and mathematician. This equation, called “from the lens makers”, is also known as “Halley’s equation”. It relates the focal length f of a thin lens to the radii of curvature 1 and 2 of its faces, the absolute refractive index ( lens ) of the material the lens is made of and the absolute refractive index ( half ) of the medium in which the lens is immersed:

If the curvature face is convex – R > 0, that is, the sign of the radius of curvature R will be positive.
If the curvature face is concave – R < 0, that is, the sign of the radius of curvature R will be negative
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When the lens is in the air, lens = 1, a biconvex lens will be converging as the distance f is positive. However, if we place this lens in a liquid whose index of refraction is greater than that of the lens ( medium > n ), its focal length will be negative, indicating that the lens becomes divergent in this medium. Likewise, a diverging lens immersed in this liquid will become converging.