Superposition of two plane waves

Let us consider two plane waves propagating towards z positives, and wave vectors that are not collinear to z but belong to the {x,z} plane. Figure 3 illustrates this scenario.

In this case, we simply have:

for the first wave and:

for the second one. Keeping in mind that:

and that :

Figure 4 shows the interference field.

The distance that separates two consecutive zones of the same nature is the interfringe distance. For z constant, the abscissa x of the bright fringe at a given k order is such that:

In the direction x,for the bright fringe that is consecutive to the preceding one, the optical phase varies from and we have:

following the direction x, the distance that separates the two bright fringes (the distance between fringes in x) is thus defined by:

where .

Equivalently, in the direction z, it follows that the interfringe in z is:

where .

In the case that it follows that:

Therefore, the fringes are parallel to the z axis.

To give some numerical values, let us consider microns and , we obtain . The interfringe is of the same order as the length of the waves. Therefore, moving from one bright fringe to another offers a sub-micrometric sensibility.