Lens grinding formula

f = focal length

The lens grinder formula , also known as the lens maker equation , indicates how the refractive power of a thin spherical lens is related to its shape. The shape of the lens is described by the radii of the spheres that form the surfaces of the lenses. Further parameters that have an influence on the refractive power are the thickness of the lens, the refractive index of its material and the refractive index of the surrounding medium.

Be there

• ${\ displaystyle R_ {1}, R_ {2}}$the spherical radii (here it should be noted that the two radii have the same sign if the center points are on the same side of the lens [convex-concave lens], but different signs if the lens is biconvex or biconcave; see also: spherical Lenses ),
• ${\ displaystyle d}$the thickness of the lens (measured at the height of the optical axis ),
• ${\ displaystyle n_ {0}}$ the refractive index of the medium outside the lens,
• ${\ displaystyle n}$ the refractive index of the lens material,
• ${\ displaystyle f}$ the focal length of the lens and
• ${\ displaystyle D}$ the refractive power.

For optical systems with the same media in object space (1) and image space (2) ( ), the following generally applies: ${\ displaystyle n_ {1} = n_ {2} = n_ {0}}$

${\ displaystyle D = {\ frac {1} {f}} = {\ frac {n-n_ {0}} {n_ {0}}} \ left ({\ frac {1} {R_ {1}}} - {\ frac {1} {R_ {2}}} + {\ frac {(n-n_ {0}) d} {nR_ {1} R_ {2}}} \ right)}$

If the external medium is air under the same conditions, the following applies approximately: ${\ displaystyle \ Rightarrow n_ {1} = n_ {2} = n_ {0} \ approx 1}$

${\ displaystyle D = {\ frac {1} {f}} = \ left (n-1 \ right) \ left ({\ frac {1} {R_ {1}}} - {\ frac {1} {R_ {2}}} + {\ frac {(n-1) d} {nR_ {1} R_ {2}}} \ right)}$

For thin lenses , the thickness of which is much smaller than the spherical radii, the equation for the so-called lens grinder formula is simplified:

${\ displaystyle D = {\ frac {1} {f}} = \ left (n-1 \ right) \ left ({\ frac {1} {R_ {1}}} - {\ frac {1} {R_ {2}}} \ right) \ ;.}$

Individual evidence

1. ^ Wilhelm Raith, Clemens Schaefer : Electromagnetism (=  textbook of experimental physics . Volume 2 ). 8th, completely revised edition. Walter de Gruyter, Berlin a. a. 1999, ISBN 3-11-016097-8 , pp. 386-387 .
2. Eugene Hecht: Optics . 4th, revised edition. Oldenbourg, Munich a. a. 2005, ISBN 3-486-27359-0 , pp. 267 .
3. Wolfgang Zinth , Ursula Zinth: Optics. Rays of light - waves - photons . 3rd, improved edition. Oldenbourg, 2011, ISBN 978-3-486-70534-8 , pp. 93 .