What is Fraunhofer Diffraction

 

Fraunhofer diffraction at the slit

 

 

Phase difference:

 

Every point x of the opening is the starting point of an elementary wave (Huygens principle). The superposition of all partial beams diffracted at the angle F corresponds to a summation of the field strengths from x = 0 to x = b (with a fixed F) in the direction kF. outgoing partial waves and results in the field strength at the wave front taking into account the phase relationships

 

 

 

The angle element DF / p corresponds to the proportion of that emanating from the gap b Elementary waveswhich is scattered in the direction of F, F + DF. The ratio (Dx / b) corresponds to a portion of the beam of width Dx passing through the gap b.


 

Using the expression for the position-dependent phase differencewe get the field strength

 

 

This gives the amplitude of the field strength as a function of the angle F:

 

 

The intensity is proportional to the square of the amount of the field strength:

By means of relationships

 

we receive

 

 

 

 

 

 

 

Calculation of the intensity minima

 

With

follows

 

The condition for intensity minima results from this

 

 

(Note: For diffraction at the grating, this is the condition for intensity maxima!)

 

 

Calculation of the intensity maxima

 

It's the problem

 

to solve for the occurrence of relative maxima.

The intensity equation can be in the simplified form

 

 

write down. By differentiating for a and setting to zero it follows

 

 

 

So the transcendent equation must be

 

 

be solved.

 

This equation has the solutions

 

 

 

Where the coefficients km Solutions to the equation

 

are with