A diffraction grating is an optical element that diffracts light into its constituent wavelength (colors). It is an excellent device used in laboratory to study the spectra.
Construction : A diffraction grating is consisting of a large number of equidistance parallel fine scratches (line) of equal width on an optically flat surface of transparent material. It is possible to put a large number of lines (scratches) per centimeter on the transparent material, e.g. the 6000 lines/cm on it using a diamond tipped tool. Since all the lines are drawn in same plane that's why it is called a plane transmission grating. The scratches are opaque ( non - transparent part ) but the areas between the scratches can transmit light. Thus, a diffraction grating becomes a multitude of parallel slit sources when light falls upon it.
There are typically two different types of diffraction grating used in general. A diffraction grating can be a reflection grating or a transmission grating. The most common type of diffraction grating is a plane grating. A transmission grating is produced in the same way as a reflection grating. In order to prepare a plane transmission grating the lines are drawn on plane transparent surface and in case of reflection grating these lines or scratches are drawn against a silver coating surface.
Transmission gratings are usually prepared with an anti-reflection coating whereas reflection gratings are normally coated with a reflective coating, usually aluminum. Transmission gratings offer high efficiency and are generally easier to align as compare the reflection grating.
The original grating so produced is quite expensive ( also known as master grating ), so their replicas are generally used in laboratory. A replica grating can be prepared in laboratory using the following procure. A thin layer of cellulose acetate solution is poured over the surface of an original grating and then it is allowed to dry. Now this thin layer of cellulose acetate is stripped - off from the original grating and then it is allowed to dry. Now this thin layer of cellulose acetate is stripped - off from the original grating surface. The impression of this grating retains this thin film. In order to prepare a plane transmission grating this thin film. In order to prepare a plane transmission grating this thin film is fixed in between two transparent glass plates and in case of reflection gratings this thin film is fixed against a silver polished surface.
Theory : Consider a parallel beam of light of wavelength 'l' incident normally on the grating surface. This case of plane transmission grating can be considered as a case of N - similar parallel slit. The width of each slit is 'a' and separated by an opaque spacing of width 'b'. The sum of this slit width 'a' and opaque spacing 'b' is known as grating element. [(a + b) = Grating element].
According Huygens's principle, each point of the incident wave front acts like a new wave front, so each slit becomes a new source and emits out secondary wavelets emerging from all the points in a single slit I a direction ?, (? = Angle of diffraction, we are considering those secondary wavelets which are diffracted in a direction ?) can be considered to be equivalent to a single wave of amplitude starting from the centre of the slit.
If the total number of slits in grating is assumed to be N, thus the waves diffracted from all the slits in direction are equivalent to N parallel waves each from the middle points of the slits S1, S2, S3 ....... SN-1, SN respectively.
In result, these N parallel waves interfere and gives interface pattern i.e. maxima and minima on screen.
Now from equation (3) and equation (4) it is clear that the phase difference between the successive wavelets is 2B. Now in order to calculate the resultant amplitude of N waves in a direction ?, having a common phase 2B and common amplitude. We use vector polygon method (As in case of single slit).
Let us draw equal distance AB1, B1 B2.......BN-1 BN representing equal amplitude R with a common phase difference of 2B among them.
Equation (11) gives the intensity of principle maxima, since the number of lines 'N' are very large hence these maxima are very intense and called principle maxima.
(2) Position of principle maxima : The position of principle maxima can be evaluated as
Equation (13) is known as grating equation. If we put n=0, we have ? = 0 i.e. at point P0. On screen all the waves arrive in same phase and a central bright fringe will be found there. This central maxima is also known as zero order principle maxima.
(3) Position of minima : In order to calculate the position of minima we have to consider only those value of B for which only the numerator of the term sin NB / Sin B is zero. i.e. for minima where m has all integer value except m ? 0, N, 2N ......... because for these values of m, sin B = 0 and gives the position of corresponding principle maxima.
From these values, now it is clear that there are (N-1) equally spaced minima exist between two consecutive principle maxima corresponding to 0 and N.
(4) Condition for secondary maxima : There are (N-1) equally spaced minima exist between two consecutive principle maxima, so there should be (N-2) other maxima known as secondary maxima between two adjacent principle maxima. These maxima are known as secondary maxima.
In order to calculate the position of these secondary maxima, on differentiating equation (10) with respect to B and equating it equal to zero, we have the root of equation (19), other than those, for which B = ± nπ (this value of B gives position of principle maxima) gives rise to position of secondary maxima.
(5) Intensity of secondary maxima : It is clear from equation (22) that with increase in N (number of slits), the intensity of secondary maxima decreases. Hence in general secondary maxima are not visible