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 \(\vec k\). These are the paths of energy.

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) :

\(s(M,t) = A(M)\cos (\omega t - \vec k.\vec r + {\phi _0})\)

The period \(T\), the frequency \(f\), the pulse \(\omega\) and the wave vector \(\vec k\) whose modulus is :

\(k=\frac {2\pi}{\lambda_0}\)

Where \(\lambda_0\) 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 :

    \({\lambda _0} = cT = \frac{c}{f}\;\;\;\;\;and\;\;\;\;\;{k_0} = \frac{\omega }{c} = \frac{{2\pi }}{{{\lambda _0}}}\)

  • In a material medium where the wave propagates at the speed \(v = c / n\) : (\(n\) is the index of the medium assumed to be homogeneous)

    \(\lambda = vT = \frac{v}{f} = \frac{{{\lambda _0}}}{n}\;\;;\;\;k = \frac{\omega }{v} = n{k_0}\)

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

In a homogeneous medium of index \(n\), a rectilinear light ray is determined by an arbitrary point \(O\) and its unit vector \(\vec u\).

\(M\) is any point of this radius, \(r=\vec u.\overrightarrow {OM}\) is the length traveled by the light between \(O\) and \(M\), counted positively in the direction of propagation.

The phase of the wave in \(M\) can be written :

\(\Phi (M,t) = \omega t - \vec k.\vec r + {\phi _0} = \omega t - k.r + {\phi _0} = \omega t - n{k_0}r + {\phi _0} = \omega t - 2\pi \frac{{nr}}{{{\lambda _0}}} + {\phi _0}\)

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

\({\Phi _{M/O}} = \Phi (M,t) - \Phi (O,t) = - \vec k.\overrightarrow {OM} = - \frac{{2\pi }}{{{\lambda _0}}}\;n\;r\;\;\;\;\;(with\;:\;\vec k = n\;\frac{{2\pi }}{{{\lambda _0}}}\;\vec u)\)

Here one sees the interest of the concept of optical path, \(\delta = nr\) and :

\({\Phi _{M/O}} = - \frac{{2\pi }}{{{\lambda _0}}}\;n\;r = - \frac{{2\pi }}{{{\lambda _0}}}\delta\)

AttentionOptical path concept

Is a medium defined at any point \(M(x,y,z)\) by an index \(n(x,y,z)\) ; we define the optical path \(L_{AB}\) between two points \(A\) and \(B\), along a curve \((C)\) by :

\({L_{AB}} = (AB) = \int_{\;A,(C)}^{\;B} {n(x,y,z)\;d\ell }\)

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

\({L_{AB}} = (AB) = \int_{\;A,(C)}^{\;B} {\frac{c}{v}(vdt)} = \int_{\;A,(C)}^{\;B} {cdt} = c\Delta t\)

The optical path is equal to \(c\) times the time taken for light to get from \(A\) to \(B\) in the medium of index \(n\).

Moreover, \((AB) = (BA)\) and, for a homogeneous medium, \((AB) = n\;AB\), \(AB\) is the distance between points \(A\) and \(B\).

And the phase of the wave in \(M\) can be written :

\({\Phi _{M/O}} = {\phi _M} - {\phi _O} = - \frac{{2\pi }}{{{\lambda _0}}}\;(OM) = - \frac{{2\pi }}{{{\lambda _0}}}{\delta _{OM}}\)

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.

We consider the case of the following figure :

The light source \((S)\) is placed in the object focal plane of a lens \((L)\).

The rays emerge parallel ; \((\Pi)\) is a wave plane and the optical paths \((SM_1)\) and \((SM_2)\) are equal :

\((SM_1)=(SM_2)\)

FondamentalStigmatism

Two points \(A\) and \(B\) will be stigmatic with respect to an optical system \((S)\) if the optical path \((AB)\) is independent of the beam passed through the system.

An object point \(A\) and image \(B\) by an optical system consisting of mirrors and lenses.

Considering any two light rays connecting \(A\) to \(B\) and intersect \(P\) and \(Q\) with a wave surface \((\Sigma')\).

According to the theorem of Malus :

\((AP)=(AQ)\)

According to the principle of reversibility of light, \(BP\) and \(BQ\) are light rays and :

\((BP)=(BQ)\)

Is :

\((PB)=(QB)\)

Therefore :

\((AP)+(PB)=(AQ)+(QB)=(AB)\)

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 \(A\) to \(B\) does not depend on the rays, since there is stigmatism.

Therefore, the optical paths \((APB)\) and \((AQB)\) are equal.