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Spherical refracting surface modeling.
Description
with :
- arrow : indicates direction of light propagation.
- $
n_{ini}$ : refractive index of the initial medium. - $
n_{fin}$ : refractive index of the final medium. - $
\overline{SC}$ : algebraic distance between vertex S and center C of curvature on optical axis.
!!!! BE CAREFUL :
!!!! In the same way as we use in English the single word "mirror" to qualify a "reflecting surface", in French is use the single word "dioptre" to qualify a "refracting surface".
!!!! The term "dioptre" in English is a unit of mesure of the vergence of an optical system. In French, the same unit of measure is named "dioptrie".
!!!! So keep in mind the following scheme :
!!!!
!!!! refracting surface : EN : refracting surface , ES : superficie refractiva , FR : dioptre.
!!!! A crystal ball forms a spherical refracting surface : un "dioptre sphérique" in French.
!!!!
!!!! unit of measure : EN : dioptre , ES : dioptría , FR : dioptrie.
!!!! My corrective lens for both eyes are 4 dioptres : "4 dioptries" in French.
Spherical refracting surface.
Analytical study
A spherical refracting surface in analytical paraxial optics is defined by three quantities :
- $
n_{ini}$ : refractive index of the initial medium (the medium on the side on the incident light). - $
n_{fin}$ : refractive index of the final medium (the medium on the side on the emerging light, after crossing the refracting surface). - $
\overline{SC}$ : the algebraic distance between the vertex S (sometimes called "pole", is the centre of the aperture) and the center of curvature C of the refracting surface.
! USEFUL : The whole analytic study below also applies to a plane refracting surface. We just need to remark that a plane surface is a spherical surface whose radius of curvature tends towards infinity.
Consider a point object $B_{obj}$ whose orthogonal projection on the optical axis gives the point object $A_{obj}$. If the point object is located on the optical axis, then $B_{obj}=A_{obj}$ and we will use to named it point object $A_{obj}$. The point object $B_{obj}$ can be real as well as virtual.
The calculation of the position of the point image $B_{ima}$, conjugated point of the point object $B_{obj}$ by the refracting surface, is carried out in two steps :
- I use the spherical refracting surface equation (known too as the "conjuction equation" for a spherical refracting surface) to calculate the position of the point $
A_{ima}$, $A_{ima}$ being the orthogonal projection on the optical axis of the point image $B_{ima}$.
$\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=\dfrac{n_{fin}-n_{ini}}{\overline{SC}}$
To perform this I need to know the algebraic distance $\overline{SA_{obj}}$, and the calculation of the algebraic distance $\overline{SA_{ima}}$ along the optical axis gives me the position of $A_{ima}$.
- I use the "transverse magnification equation" for a spherical refracting surface, to calculate the algebraic value of the transverse magnification $
\overline{M_T}$, then to derive the algebraic length $\overline{A_{ima}B_{ima}}$ of the segment $[A_{ima}B_{ima}]$, that is the algebraic distance of the point image $B_{ima}$ from its orthogonal projection $A_{ima}$ on the optical axis.
By definition : $\overline{M_T}=\dfrac{\overline{A_{ima}B_{ima}}}{\overline{A_{obj}B_{obj}}}$.
Its expression for spherical refracting surface : $\overline{M_T}=\dfrac{n_{ini}\cdot\overline{SA_{ima}}}{n_{fin}\cdot\overline{SA_{obj}}}$.
I know $\overline{SA_{obj}}$, $n_{ini}$ and $n_{fin}$, I have previously calculated $\overline{SA_{ima}}$, so I can calculate $\overline{M_T}$ and deduced $\overline{A_{ima}B_{ima}}$
! USEFUL : The conjuction equation and the transverse magnification equation for a plane refracting surface are obtained by rewriting these equations for a spherical refracting surface in the limit when $|\overline{SC}|\longrightarrow\infty$.
Then we get for a plane refracting surface :
!
! * conjuction equation : $\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=0$.
!
! * transverse magnification equation : $\dfrac{n_{ini}\cdot\overline{SA_{ima}}}{n_{fin}\cdot\overline{SA_{obj}}}$ (unchanged).
!
! This generalizes and completes the knowledge you get about plane refracting surfaces seen in your pedagogical paths in plain and hills.