Chapter 9

# Observation of double stars

Take 10 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.

If you have more questions, feel free to ask them on the forum.

Consider a source consisting of two points and , separated by a distance .

This source illuminates two Young's holes. We note the common wavelength issued by the two sources and it is assumed that the waves from the two sources have the same amplitude.

This case may be, for example, the two components of a double star as seen from Earth.

This exercise highlights the concept of spatial coherence : the spatial coherence of a light source is better as long as the source is limited in space.

Observation of double stars

The observation is done on a remote screen shot of the two slits in the distance .

We note the distance from the plane of the two slits with two point sources.

## Question

Determine the intensities and on the screen, an abscissa point , due to each of the stars.

### Hint

The stars are incoherent : the total intensity on the screen will be the sum of the intensities created by each star individually.

### Solution

The stars are incoherent : the total intensity on the screen will be the sum of the intensities created by each star individually.

We are interested in the illumination due to the source . The total path difference is :

The intensity is then :

In the same way, the intensity due to the second source is :

## Question

Deduce the total intensity on the screen and put in the form :

What is the meaning of the term  ?

### Hint

Using the trigonometric relationship :

### Solution

We obtain :

It is recognized in the second cosine term interference of Young's holes for a single point source. The fringes are rectilinear and the interfringe is equal to :

The first cosine is independent of the observation point. It is called visibility and noted :

Fringes contrast is defined by :

Contrast and visibility are therefore equal to the sign.

Fringes contrast

The previous figure shows the shape of the fringes on the screen, in terms of the value of contrast .

Zero contrast is obtained for a distance between sources such as :

Take, for example :

One can find this particular value of using a qualitative reasoning.

Indeed, when there is interference fringes, the light fringes due to the first star superimposed on the dark fringes due to the second star.

Therefore, unlike the orders of interference at a point of the screen should be half-integer.

In the case where this difference is equal to  :

We thus find :

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Experience of Young's slits
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Temporal coherence and Young slits ; displacement and interference fringes