Chapter 9

Geometrical and waves optics

Optical path and Malus' theorem

FondamentalPrinciple of inverse return of light

Light rays of geometrical optics are tangent, at any point, to the direction of propagation of the light wave.

The light rays are parallel to the wave vector . These are the paths of energy.

Light rays

Principle of reversibility of light :

"The laws of reflection and refraction are independent of the direction of travel of light".

If we reverse the direction of light propagation, the light rays are unchanged.

FondamentalPhase of a light wave

For a monochromatic light (within the scalar theory of light) :

The period , the frequency , the pulse and the wave vector whose modulus is :

Where is the wavelength in vacuum.

The frequency of a visible electromagnetic wave determines its color, again characterized by its wavelength, provided the propagation medium is specified.

The link between the temporal and spatial variations is given by the speed of propagation and depends on medium.

  • In a vacuum :

  • In a material medium where the wave propagates at the speed : ( is the index of the medium assumed to be homogeneous)

Phase difference between two points on the same ray of light :

In a homogeneous medium of index , a rectilinear light ray is determined by an arbitrary point and its unit vector .

is any point of this radius, is the length traveled by the light between and , counted positively in the direction of propagation.

Optical path

The phase of the wave in can be written :

Thus, the phase difference between the points and is (actually, the relative phase of the wave in compared to ) :

Here one sees the interest of the concept of optical path, and :

AttentionOptical path concept

Is a medium defined at any point by an index ; we define the optical path between two points and , along a curve by :

The optical path is equal to the distance that would cross the light in vacuum during the same time it takes to cover the curve in the considered medium. Indeed :

The optical path is equal to times the time taken for light to get from to in the medium of index .

Moreover, and, for a homogeneous medium, , is the distance between points and .

And the phase of the wave in can be written :

FondamentalMalus' theorem

"In an isotropic medium, after any number of reflections and refractions, rays from the same point source remain perpendicular to the wave surfaces".

For a plane wave, the rays are parallel to each other and perpendicular to the planes of waves.

For a spherical wave, the light rays are precisely the wave spheres rays.

This theorem will be admitted and justified by some examples.

Malus' theorem

We consider the case of the following figure :

Applying the theorem of Malus

The light source is placed in the object focal plane of a lens .

The rays emerge parallel ; is a wave plane and the optical paths and are equal :


Two points and will be stigmatic with respect to an optical system if the optical path is independent of the beam passed through the system.

An object point and image by an optical system consisting of mirrors and lenses.

Considering any two light rays connecting to and intersect and with a wave surface .


According to the theorem of Malus :

According to the principle of reversibility of light, and are light rays and :

Is :

Therefore :

The optical path between two points combined with a stigmatic optical system is independent of the radius between them.

It may also indicate that the propagation time of the rays emitted simultaneously from point to does not depend on the rays, since there is stigmatism.

Therefore, the optical paths and are equal.

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Fermat's principle