Theory of diffraction

Theory of diffraction

Diffraction describes a variety of processes which obtain when waves, such as light, approaches an obstacle of dimension of the order of their wavelength, and is characterized by the apparent bending of the waves around the object, such as is demonstrated across.
What is transmitted in the one case is a sharp image of the aperture, and in the other a diffracted image of the aperture (seen , whereby most of the light is transmitted on axis, but at wider angles, because of interference effects, a diffraction pattern obtains.

▲Figure 1-6:-Single slit diffraction

Whilst this example demonstrates the principle in transmission, the same applies were the aperture replaced by a reflecting surface; would be reflected in the case of a narrow mirror light would be reflected in all directions. Note that in the case of reflection, specular reflection obtains, where the peak in intensity is transmitted in the same angle as the angle of incidence.
This is termed single slit diffraction

Reflection Diffraction Grating

A reflection diffraction grating is a surface which, on the microscopic scale, is made up of a large number of rectangular grooves of width comparable to the wavelength of light to be considered. 
On shining light upon this surface, diffraction at each of the grooves obtains; each groove acting as a (coherent) source of light, emitting a cylindrical wave.
The coherence of these cylindrical waves is an important aspect since any phase difference between adjacent grooves is due solely to geometry and not from the source.
It is the interference of the light from these numerous sources that is of interest. 
Diffraction can be visualised by the following, whereby light of wavelength   is incident at an angle   to the normal of a diffraction grating of groove spacing, d. Light is diffracted along angles   into a number, m, of diffraction orders.

▲Figure 1-7:- Plane reflection grating diffraction

It is the interference between the waves diffracted from each groove the provides wavelength discrimination as a function of angle.
A sign convention exists for the definition of angles and orders. In general angles are measured from the grating normal to the incident wavefront. Should diffraction occur on the opposite side of the grating normal, then negative angles are used.This view can be simplified to the following, where one can consider two adjacent grooves separated by a distance, d.

Figure 1-8:- Path difference between neighbouring rays

The geometrical path difference, △, between the path of the incident wavefront between A and B and the diffracted wavefront between C and D is

Now, for constructive interference to obtain, adjacent rays must differ by integer number of wavelengths. This leads to the grating equation.
where G is the groove density, G= 1/d.
For a given incidence angle  , there shall therefore be a set of discrete angles for which constructive interference shall be observed. At all other angles, there will be some measure of destructive interference.
Here m is the diffraction order and is an integer.
Since the absolute value of the sine function cannot exceed unity, then:
| m / d | < 2
For a particular wavelength the above gives the possible diffraction orders present.
Specular reflection (α=β) always exists, this is the m=0, zero order position, where the grating simply acts as a mirror and the component wavelengths of the incident wavefront are not separated.
In this case, one refers to the angular deviation, 2K, between the incidence and diffraction directions, defined as:
Further defined is the scan angle,  , measured from grating normal to the dissector of the beams
Now, substituting, the grating equation becomes:
For a given monochromator K is a constant, therefore one can determine select a wavelength by determining the required grating angle.