Chapter 9

Geometrical and waves optics

To test the understanding of the lesson

Question

Define the conditions of Gauss.

Hint

Solution

The Gaussian approximation (after the German physicist Carl Friedrich Gauss) is the linear approximation of geometrical optics obtained when the rays of the angles of incidence is low (that is to say in a direction close to the normal to the surface of the optical instrument) and the point of incidence is close to the optical axis.

We then say that working with paraxial rays.

Question

Define the optical path . What is its signification ?

Hint

Solution

  • The optical path is defined by :

    where is the optical index of the medium at the point .

  • The optical path is equal to the distance that would cross the light in vacuum during the same time

    takes to cover the curve in the medium considered.

    Indeed :

Note :

For a homogeneous medium, , where is the index of the medium and the distance between and .

Question

Definition of rigorous stigmatism and aplanatism.

Hint

Solution

  • Stigmatism :

    An optical system has the stigmatism properties (it is stigmatic) if the image of a point through this system is a point.

  • Aplanatism :

    An optical system has the aplanatism property (it is aplanatic) if the image of an extended object perpendicular to the optical axis through this system is also perpendicular to the optical axis.

    A photo camera is aplanatic : photographing a person, feet and head are very clear in the plan of film.

Question

State the laws of Snell -Descartes.

Hint

Solution

  • Refracted and reflected rays are in the plane of incidence (defined by the normal to the optical diopter and the incident ray).

  • is the angle of incidence, the angle of réfraction and the angle of refraction (defined with respect to the normal to the diopter) :

    Where and are the indices of medium from both sides of the crossed diopter.

Question

  • Give the conjugate and magnification formulas of Newton (with double origins at focus ) for a thin lens.

  • Give the conjugate and magnification formulas of Descartes for a thin lens with origin in the center.

Hint

Solution

  • Newton formulas :

  • Descartes formulas :

Question

Definition of two mutually synchronous sources.

Hint

Solution

Two synchronous waves have same frequency and a phase shift which is independent of time.

Interference usually refers to the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves or matter waves.

Question

A quasi-monochromatic source has a spectral width in frequency . What is the relationship between and the coherence time  of this source ?

Hint

Solution

Question

  • Definition of an interference fringe.

  • Definition of an order of interference.

Hint

Solution

  • Constructive interference occurs when the phase difference between the waves is a multiple of (we have a light fringe), whereas destructive interference occurs when the difference is an odd multiple of (dark fringe).

  • The interference order is, in point  :

    Where is the wavelenght of the used source in vacuum.

Question

Give the formula of Fresnel, giving the illumination (also called light intensity) at two-waves interference.

Hint

Solution

  • If and are the intensities of only two sources :

    Where is the optical path difference.

  • In the case where the intensities are the same ( ) :

Question

Define the contrast of the fringes (or visibility factor).

Hint

Solution

The definition of fringe contrast is :

Question

In the case of the Michelson interferometer :

  • What is the expression of the optical path difference in the configuration "air blade of parallel faces" ?

  • What is the expression of the optical path difference in the configuration "air corner" ?

Hint

Solution

  • Configuration "air blade of parallel faces" :

  • Configuration "air corner" :

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