Chemin optics

The optical way is a tool of the geometrical Optique and undulatory.

It is defined as the distance which the Lumière in the vacuum would have traversed for the length of time that it puts to carry out a way in a given medium. It is thus a quantity which depends on the Speed of light in the medium.

The Principe of Fermat states that the ways borrowed by the light to go from a point to another have a stationary optical way.

Chemin optics and index of refraction

Into the mediums other than the vacuum, the properties Diélectrique S of materials introduce a modification of the Speed of light. Speed of light, noted v is related to the optical index N of the medium by the relation:

$v\; =\; \backslash \; frac\; \{C\}\; \{N\}$

with C speed of light in the vacuum.

In the case of a homogeneous medium, for which N is the same one in any point, the optical way to go from a point has towards a point B in straight line, which one notes $\backslash \; mathcal\; L\_\; \{AB\}$, is simply given by the geometrical distance between the point has and the point B multiplied by the index of refraction N . One has as follows:

$\backslash \; mathcal\; L\_\; \{AB\}\; =\; N\; \backslash ,\; AB$

where AB is the geometrical distance between the point has and the point B .

Example: A luminous ray traverses 5 cm in a layer of water. In parallel, another luminous ray (identical to preceding) crosses 5 cm of air. Water has as an index of refraction N = 1,33 and the air an index appreciably equal to that of the vacuum N = 1. In water, the optical way of the luminous ray will be worth D = 1,33 × 5 = 6,65 cm. In the air, it will be worth = 1×5 = 5 cm. The optical way will be longer in water than in the air.

If they had left at the same time, the luminous ray which crosses the air arrives " avant" the other. that can be checked with a laser and mirrors (one in water and one in the air) by systems of interferences. The difference in phase between the ray having crossed water and that having crossed air will be function length of crossed water and report/ratio of the indices water/air.

General case: curved way and inhomogenous medium

That is to say an unspecified curve C , in an inhomogenous medium ( N can vary in various points of space). One looks for the optical way of the light traversing this curve C . For that, one considers two so unspecified points them belonging to the curve C , infinitely close and distant of a distance D S .

Locally, the optical way is that of the simple case: a luminous ray in straight line. One can thus write:

$\backslash \; mathrm\; \{D\}\; \backslash \; mathcal\; L\; =\; N\; \backslash \; left\; (S\; \backslash \; right)\; \backslash ,\; \backslash \; mathrm\; \{D\}\; s$

with N (S) the index of the medium in a point S of the curve. To find the way optical separating two points has and B on this unspecified curve, it is enough to make the integral sum of all the elements D L on the curvilinear coordinates S delimited by the points has and B :

$\backslash \; mathcal\; L\_\; \{AB\}\; =\; \backslash \; int\_\; \{has\}\; ^\; \{B\}\; N\; \backslash \; left\; (S\; \backslash \; right)\; \backslash ,\; \backslash \; mathrm\; \{D\}\; s$

Equation iconale

The equation iconale (or eikonale) can be obtained starting from the optical way $\backslash \; mathcal\; L$.

$\backslash \; mathcal\; L\; =\; \backslash \; int\_\; \{A\_0\}\; ^\; \{has\}\; N\; \backslash ,\; \backslash \; mathrm\; \{D\}\; s$

By noting has ₀ the point of coordinates R ₀ and has a generic point of coordinates R located on another surface of wave.

$\backslash \; mathcal\; L\; (\backslash \; vec\; \{R\})\; =\; \backslash \; int\_\; \{\backslash \; vec\; \{r\_0\}\}\; ^\; \{\backslash \; vec\; \{R\}\}\; N\; \backslash ,\; ds$

This notation brings the following differential:

$\backslash \; mathrm\; \{D\}\; \backslash \; mathcal\; L\; =\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; mathcal\; L\; \backslash ,\; \backslash \; mathrm\; \{D\}\; \backslash \; vec\; \{R\}\; =\; N\; \backslash \; mathrm\; \{D\}\; S$

One can also write it with the unit vector U defining the direction of propagation of the light wave.

$n\; \backslash \; mathrm\; \{D\}\; S\; =\; N\; \backslash \; left\; (\backslash \; mathrm\; \{D\}\; \backslash \; vec\; \{R\}.\; \backslash \; vec\; \{U\}\; \backslash \; right)$

What implies the equation iconale geometrical optics:

$n\; \backslash \; vec\; \{U\}\; =\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; mathcal\; L$

with:

$n\; =\; \backslash \; left|\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; mathcal\; L\; \backslash \; right|$

and $\backslash \; nabla$ the formal operator Nabla.

Fundamental law of geometrical optics

The fundamental law of geometrical optics is the following one:

$\backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; \backslash \; left\; (N\; \backslash \; vec\; \{U\}\; \backslash \; right)\}\; \{\backslash \; mathrm\; \{D\}\; S\}\; =\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; n$

This expressed law in a very general way can refer to a surface separating two different indices. One poses the vector NR , normal on surface. The vector grad (N) is carried by NR .

$\backslash \; Delta\; \backslash \; left\; (N\; \backslash \; vec\; \{U\}\; \backslash \; right)\; =\; \backslash \; vec\; \{NR\}\; \backslash \; int\_\; \{-\; \backslash \; frac\; \{\backslash \; epsilon\}\; \{2\}\}\; ^\; \{+\; \backslash \; frac\; \{\backslash \; epsilon\}\; \{2\}\}\; \backslash ,\; ||\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; (N)\; ||\; \backslash ,\; \backslash \; mathrm\; \{D\}\; s$

And by posing I _s the value of the integral

$\backslash \; n\_2\; \backslash \; vec\; \{\backslash \; u\_2\}\; -\; \backslash \; n\_1\; \backslash \; vec\; \{\backslash \; u\_1\}\; =\; \backslash \; I\_s\; \backslash \; vec\; \{NR\}$

What points out the law of Snell-Descartes.

Analogy enters optics and mechanics

In mechanics, one writes

$\backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; \backslash \; vec\; \{p\}\}\; \{\backslash \; mathrm\; \{D\}\; T\}\; =\; F$

that is to say

$\backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; \backslash \; left\; (p\; \backslash \; vec\; \{U\}\; \backslash \; right)\}\; \{\backslash \; mathrm\; \{D\}\; S\}\; =\; \backslash \; frac\; \{\backslash \; vec\; \{F\}\}$

By noticing the analogy with the equation

$\backslash \; frac\; \{\backslash \; mathrm\; \{D\}\; \backslash \; left\; (N\; \backslash \; vec\; \{U\}\; \backslash \; right)\}\; \{\backslash \; mathrm\; \{D\}\; S\}\; =\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; (N)$

One can write that under the following particular conditions (in Physique of the particles, one uses the value C like unit speed):

• for a mass unit m = 1;
• for a speed unit || v || = || U || = 1.

F drift of a potential which is function only of the index of refraction:

$\backslash \; vec\; \{F\}\; =\; -\; \backslash \; overrightarrow\; \{\backslash \; nabla\}\; \backslash \; left\; (-\; \backslash \; frac\; \{n^2\}\; \{2\}\; \backslash \; right)$

One can push the analogy by recalling that geometrical optics is the approximation low wavelengths of undulatory optics.

The general idea of this analogy (had a presentiment of in the years 1830 by Hamilton, then reformulated by Louis de Broglie in 1923) is to associate momentum p of the particle and the Vecteur of wave K of the wave. The electron microscope is a concrete implementation.

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