Editing Lens (section)

=== For a spherical surface ===
[[File:Refraction at spherical surface.svg|thumb|Simulation of refraction at spherical surface at [https://www.desmos.com/calculator/ax4rsqdot0 Desmos]]]
For a single refraction for a circular boundary, the relation between object and its image in the [[paraxial approximation]] is given by<ref>{{Cite web |date=2019-07-02 |title=4.4: Spherical Refractors |url=https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9B__Waves_Sound_Optics_Thermodynamics_and_Fluids/04%3A_Geometrical_Optics/4.04%3A_Spherical_Refractors |access-date=2023-07-02 |website=Physics LibreTexts |language=en |archive-date=26 November 2022 |archive-url=https://web.archive.org/web/20221126132929/https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9B__Waves_Sound_Optics_Thermodynamics_and_Fluids/04%3A_Geometrical_Optics/4.04%3A_Spherical_Refractors |url-status=live }}</ref><ref>{{Cite web |title=Refraction at Spherical Surfaces |url=https://personal.math.ubc.ca/~cass/courses/m309-01a/chu/MirrorsLenses/refraction-curved.htm |access-date=2023-07-02 |website=personal.math.ubc.ca |archive-date=26 October 2021 |archive-url=https://web.archive.org/web/20211026211612/https://personal.math.ubc.ca/~cass/courses/m309-01a/chu/MirrorsLenses/refraction-curved.htm |url-status=live }}</ref>

<math display="block">\frac {n_1}u + \frac {n_2}v = \frac {n_2-n_1}R</math>

where {{mvar|R}} is the radius of the spherical surface, {{math|''n''{{sub|2}}}} is the refractive index of the material of the surface, {{math|''n''{{sub|1}}}} is the refractive index of medium (the medium other than the spherical surface material), <math display="inline">u</math> is the on-axis (on the optical axis) object distance from the line perpendicular to the axis toward the refraction point on the surface (which height is ''h''), and <math display="inline">v</math> is the on-axis image distance from the line. Due to paraxial approximation where the line of ''h'' is close to the vertex of the spherical surface meeting the optical axis on the left, <math display="inline">u</math> and <math display="inline">v</math> are also considered distances with respect to the vertex.

Moving <math display="inline">v</math> toward the right infinity leads to the first or object focal length <math display="inline">f_0</math> for the spherical surface. Similarly, <math display="inline">u</math> toward the left infinity leads to the second or image focal length <math>f_i</math>.<ref>{{Cite book |last=Hecht |first=Eugene |title=Optics |publisher=Pearson |year=2017 |isbn=978-1-292-09693-3 |edition=5th |pages=164 |language=English |chapter=5.2.2 Refraction at Spherical Surfaces}}</ref>

<math display="block">\begin{align}
 f_0 &= \frac{n_1}{n_2 - n_1} R,\\
 f_i &= \frac{n_2}{n_2 - n_1} R
\end{align}</math>

Applying this equation on the two spherical surfaces of a lens and approximating the lens thickness to zero (so a thin lens) leads to the [[#Lensmaker's equation|lensmaker's formula]].

==== Derivation ====
[[File:Refraction in spherical surface.svg|thumb]]
[[File:Four spherical refractions.png|thumb|The four cases of spherical refraction]]
Applying [[Snell's law]] on the spherical surface, <math>n_1 \sin i = n_2 \sin r\,.</math>

Also in the diagram,<math display="block">\begin{align}
\tan (i - \theta) &= \frac hu \\
\tan (\theta - r) &= \frac hv \\
\sin \theta &= \frac hR
\end{align}</math>, and using [[Small-angle approximation|small angle approximation]] (paraxial approximation) and eliminating {{mvar|i}}, {{mvar|r}}, and {{mvar|θ}},

<math display="block">\frac {n_2}v + \frac {n_1}u = \frac {n_2-n_1}R\,.</math>