Chapter 9

# Malus' theorem, superposition of a plane wave and of a spherical wave

Take 20 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

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A converging lens is used , pierced at its center, as two-wave interference system.

A point source , monochromatic wavelength , is placed at the focal object of the lens.

It follows that the wave emerging from the lens is plane and that directly transmitted through the hole is spherical.

The hole has a diameter on the output face and a depth along the axis.

## Question

Give the analytical expressions of waves that overlap ; be adopted as the origin of phase waves in and it is assumed that these two waves have the same amplitude.

### Solution

The source is the focus of the lens : it emits a spherical wave.

The lens transforms into plane wave.

The optical path to go from to through the lens is equal to that to go from to (use the theorem of Malus) through . Therefore : Expression of the plane wave in is : For the spherical wave that goes directly from to , the optical path is (with ) : The amplitude of the spherical wave is : Whether, assuming that this wave has the same amplitude as the plane wave : The resulting amplitude in is : ## Question

What is the intensity in the plane located at the same distance of the output face of as , as a function of the cylindrical coordinate ?

Deduce the nature of the interference fringes.

### Solution

The light intensity is : The fringes are here rings (corresponding to ) centered on . A bright ray is given by : (with an integer)

So : ## Question

Calculate the radius extreme bright fringes knowing that : ; ; (index of the glass of )

### Solution

The point is in the interference field if . is calculated by taking then taking .

We find and .

In the latter case, take and recalculate . We find .

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Twyman's interferometer