## Introduction: diffraction of waves

Since diffraction arises from the interaction of matter with waves, we have to understand something about the behavior
of waves. We need to understand first how to describe waves, both the terminology and (at some level) the mathematics.
Diffraction results from the addition of waves scattered by different objects, so we also have to understand how waves add up and
how this reflects the relative positions of the objects scattering them.

Electromagnetic waves, such as X-rays, vary over time and space. Although there is also a magnetic component that varies
perpendicular to the electric field component, it is the variation in the electric field that interests us. That is because
X-rays interact with matter through their interaction with charged particles, particularly electrons.

We take the general case of monochromatic wave packet:Ψ_{1}=Ae^{iφ1}
e^{iωt} , Ψ_{2}=Ae^{iφ2}e^{iωt}
.
When two waves diffract we get their wave sum : Ψ=Ψ_{1}
+Ψ_{2}=A(e^{iφ1}+
e^{iφ2})e^{iωt}=A
e^{i(ωt+φ1)}(1+e^{i(φ2-φ1)
})

the image below describes sum of two waves with relative phases of Δφ=45^{o}.

sum of two wave with relative phase of Δφ which is different from 45^{o} can be plotted using
the function : **two_waves(angle)** ( see downloads section )

The intensity of such waves with different phase can be described as : I=ΨΨ^{*}=
A^{2}(2+2cos(Δφ)

the image below describes the intensity (I) as function of the relative phase Δφ .

In order to demonstrate the diffraction of such waves we make "phase map" as described in the image below:

By this map we can understand the diffraction better. The white circles indicate positive phase of the light wave and the black
circles indicate negative phase, when the size of the
circles indicate the amplitude of the positive phase. For example: for phase of π/2 the amplitude is 1 and the radius of the white
circle is 1/7. For phase of π/4, the amplitude is 0.707 and the radius of the circle is 0.707/7.
For zero amplitude a black dot is drawn