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----
-title: 'Spherical refracting surface : overview'
-media_order: dioptre-1.gif
-published: false
-visible: false
----
-
-### What is a refracting interface ?
-
-#### Refracting surface : physical description
-
-* **Interface separating two transparent media of different refractive indices**.
-
-* can be **found in nature** :
-Examples :
-\- a **plane refracting surface** : the *flat and quiet surface of a lake*.
-\- a **spherical refracting surface** : a *fish ball aquarium*.
-
-
-Fig. 1. The spherical refracting interface of a fish ball aquarium.
-
-* **appears in the design and modeling of other optical elements** :
-Examples :
-\- a **glass window pane** is the combinaison of *two parallel plane refracting interfaces* (air/glass, then glass/air), separated by the thickness of the pane.
-\- a **lens** is composed of *two consecutive curved (often spherical) refracting interfaces* (air/glass, then glass/air) that are rotational symmetrical around a same axis.
-
-#### Refracting interface versus refracting surface
-
-!!!! *DIFFICULT POINT* : One plane or spherical refracting interface has two different optical behaviors for image formation,
-is characterized by two different sets of parameters, depending of the direction of the light propagation.
-!!!!
-!!!!Consider a plane interface (a thick window whose thickness and effect can be neglected) separating air and water,
-and two twins (Thompson and Thomson) at equal distances on both sides of the interfaces (Fig. 2a).
-!!!!
-!!!! 
-!!!! Fig. 2a : The situation is not symmetrical.
-!!!!
-!!!! * When Thompson (in air) looks at Thomson (in water), the light propagates from Thomson to Thompson’s eyes.
-The fact is that Thompson sees the image of his brother closer than the real position of his brother (Fig. 2b)
-!!!!
-!!!! 
-!!!! Fig. 2b. Thompson sees his brother closer than his real position in water.
-!!!!
-!!!! * In the opposite situation, when Thomson (in water) looks at Thompson (in air),
-the light propagates from Thompson to Thomson’s eyes.
-And the fact is that Thomson sees the image of his brother farther away than his real position (Fig. 2c)
-!!!! (Strictly speaking, the eye of a fish should be considered in this situation, eyes well adapted to the vision in water,
-and in direct contact with water. If not, we should consider that the Thompson’s dive mask is filled with water,
-to have Thomson’s eyes in contact with water and not add another water/air refracting interface
-(that of the dive mask) on the path of the light).
-!!!!
-!!!! 
-!!!! Fig. 2c. Thomson sees in brother farther than his real position in air.
-!!!!
-!!!! All this can be predicted and calculated, but this example shows that this air/water plane
-refracting interface corresponds to two different plane refracting surfaces :
-!!!!
-!!!! * First case , refracting surface such as :
-!!!! \- refracting index of the medium of the incident light : $n_{inc} = n_{water} = 1.33$
-!!!! \- refracting index of the medium of the emergent light : $n_{eme} = n_{air} = 1$
-!!!!
-!!!! *¨ Second case , refracting surface such as :
-!!!! \- refracting index of the medium of the incident light : $n_{inc} = n_{air} = 1$
-!!!! \- refracting index of the medium of the emergent light : $n_{eme} = n_{water} = 1.33$.
-!!!!
-
-#### Difference in terminology between Spanish, French and English
-
-!!!! *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._
-
-
-#### Non stigmatism of spherical refracting surfaces
-
-Ray tracing study of a **spherical refracting surface** :
-
-
-* **At each impact point** of the rays upon the spherical refracting surface, the **Snell-Descartes relation applies**.
-
-
-
-* A spherical refracting surface is **not stigmatic** : The *rays (or their extensions)* originating *from a same object point* and that emerge from the surfac egenerally *do not converge towards an image point*.
-
-
-
-* **If we limit the aperture** of the spherical refracting surface so that only the rays
-meeting the surface near the vertex are refracted through the surface.
-
-
-
-* **and if** the object points remain close enough to the optical axis, so that the **angles of
-incidence and refraction remain small**, then for each object point an image point can be almost
-defined, and therefore the spherical refracting surface becomes *quasi-stigmatic*.
-
-
-
-
-#### Gauss conditions / paraxial approximation and quasi-stigmatism
-
-When spherical refracting surfaces are used under the following conditions, named **Gauss conditions** :
-\- The *angles of incidence and refraction are small*
-(the rays are slightly inclined on the optical axis, and intercept the spherical surface in the vicinity of its vertex),
-then *the spherical refracting surfaces* can be considered *quasi-stigmatic*, and therefore they can be used to build optical images.
-
-Mathematically, when an angle $`\alpha`$ is small $`\alpha < or \approx 10 ^\circ`$, the following approximations can be made :
-$`sin(\alpha) \approx tan (\alpha) \approx \alpha`$, and $`cos(\alpha) \approx 1`$.
-
-*Geometrical optics limited to Gaussian conditions* is called *Gaussian optics* or *paraxial optics*.
-
-
-#### Thin spherical refracting surface
-
-We call **thin spherical refracting surface** a spherical refracting surface *used in the Gauss conditions*.
-
-
-### How is modeled a spherical refracting surface in paraxial optics ?
-
-
-#### Characterization of a spherical refracting surface
-
-* 2 distincts points : **vextex S** and **center of curvature C** on the optical axis,
-which defines $`\overline{SC}`$ : algebraic distance between vertex S and center C of curvature on optical axis.
-
-* 2 refractive index values :
-\- **$`n_{inc}`$ : refractive index of the medium of the incident light**.
-\- **$`n_{eme}`$ : refractive index of the medium of the emergent light**.
-
-* 1 arrow : indicates the *direction of light propagation*
-
-
-
-
-#### Analytical study
-
-
-* **Thin spherical refracting surface equation** = **conjuction equation** for a spherical refracting surface
-**$`\dfrac{n_{eme}}{\overline{SA_{ima}}}-\dfrac{n_{inc}}{\overline{SA_{obj}}}=\dfrac{n_{eme}-n_{inc}}{\overline{SC}}`$** (equ.1)
-
-* **Transverse magnification expression**
- **$`\overline{M_T}=\dfrac{n_{inc}\cdot\overline{SA_{ima}}}{n_{eme}\cdot\overline{SA_{obj}}}`$**
- (equ.2)
-You know $`\overline{SA_{obj}}`$, $`n_{inc}`$ and $`n_{eme}`$, you have previously calculated $`\overline{SA_{ima}}`$, so you 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 :* $`\dfrac{n_{fin}}{\overline{SA_{ima}}}-\dfrac{n_{ini}}{\overline{SA_{obj}}}=0`$ (equ.3)
-!
-! * *transverse magnification equation :* $`\dfrac{n_{ini}\cdot\overline{SA_{ima}}}{n_{fin}\cdot\overline{SA_{obj}}}`$
- (equ.2, unchanged)
-! but (equ.3) gives $`\dfrac{\overline{SA_{ima}}}{\overline{SA_{obj}}}=\dfrac{n_{inc}}{n_{eme}}`$.
-! Copy this result into (equ.2) leads to $`\overline{M_T}=+1`$.
-
-
-#### Graphical study
-
-##### 1 - Determining object and image focal points
-
-Positions of object focal point F and image focal point F’ are easily obtained from the conjunction equation (equ. 1).
-
-* Image focal length $`\overline{OF'}`$ : $`\left(|\overline{OA_{obj}}|\rightarrow\infty\Rightarrow A_{ima}=F'\right)`$
- (equ.1)$`\Longrightarrow\dfrac{n_{eme}}{\overline{SF'}}=\dfrac{n_{eme}-n_{inc}}{\overline{SC}}`$
-$`\Longrightarrow\overline{SF'}=\dfrac{n_{eme}\cdot\overline{SC}}{n_{eme}-n_{inc}}`$ (equ.4)
-
-* Object focal length $`\overline{OF}`$ : $`\left(|\overline{OA_{ima}}|\rightarrow\infty\Rightarrow A_{obj}=F\right)`$
- (equ.1) $`\Longrightarrow-\dfrac{n_{inc}}{\overline{SF}}=\dfrac{n_{eme}-n_{inc}}{\overline{SC}}`$
-$`\Longrightarrow\overline{SF}=-\dfrac{n_{inc}\cdot\overline{SC}}{n_{eme}-n_{inc}}`$ (equ.5)
-
-!!!! *ADVISE* :
-!!!! Memory does not replace understanding. Do not memorise (equ.4) and (equ.5), but understand
-!!!! the definitions of the object and image focal points, and know how to find these two equations
-!!! from the conjuction equation for a spherical refracting surface.
-!!!!
-
-! *NOTE 1* :
-! An optical element being convergent when the image focal point is real,
-! so when $`\overline{OF}>0`$ (with optically axis positively oriented in the direction of the light propagation),
-! you can deduce from (equ.4) that is spherical refracting surface is convergent if and only if its center
-! of curvature C is in the mmedium of highest refractive index.
-!
-
-##### 2 - Thin spherical refracting surface representation
-
-* **Optical axis = revolution axis** of the refracting surface, positively **oriented** in the direction of
-propagation of the light.
-
-* Thin spherical refracting surface representation :
-\- **line segment**, perpendicular to the optical axis, centered on the axis with symbolic
-**indication of the direction of curvature** of the surface at its extremities.
-\- **vertex S**, that locates the refracting surface on the optical axis.
-\- **nodal point C = center of curvature**.
-\- **object focal point F and image focal point F’**.
-
-! *NOTE 2*
-! The direction of the curvature does not presume the convergent or divergent character
-! of the diopter. It also depends on the refractive index values on each side of the spherical
-! refracting surface. look at what happens to the incident ray parallel to the optical axis
-in Figures 3 and 4, and 5 and 6 below, and review NOTE 1.
-!
-
-#### Examples of graphical situations, with analytical results to train
-
-!!!! *IMPORTANT* :
-!!!! Even for only one of the following figures, the real or virtual character of the
-!!!! image may depend on the position of the object. This paragraph is only for you
-!!!! to understand how to determine the 3 rays that determine the image. It is
-!!!! important not to memorize these figures, which would be limiting, misleading
-!!!! and without interest.
-!!!!
-!!!! All the useful numerical values are given for each figure, making it possible
-!!!! also to check that you master the analytical study of each presented case.
-!!!!
-
-
-
-* with **real objects**
-
-
-Fig. 4.
-
-
-Fig. 5.
-
-
-Fig. 6.
-
-
-Fig. 7.
-
-* with **virtual objects**
-
-
-Fig. 8.
-
-
-Fig. 9.
-
-
-Fig. 10.
-
-
-Fig. 11.
-