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Resolving Power of Plane Diffraction Grating

We know that the diffraction grating has ability to produce spectrum i.e., to separate the lines of nearly equal wavelengths and therefore it has resolving capability. The resolving power of a grating may be defined as its ability to form separate diffraction maxima of two wavelengths which are very close to each other. If A. is the mean value of the two wavelengths and dλ is the

difference between two then resolving power may be defined as

resolving power = λ/ dλ

 

Expression for resolving power. Let a beam of light having two wavelengths λ1 and λ2 is falling normally on a grating AB which has (a + b) grating element and N number of slits as shown in Figure·5.ll. After passing through grating rays forms the diffraction patterns which can be seen through telescope.

 

Now, if these patterns are very close together they overlap and cannot be seen separately.

 

However, if they satisfy the Rayleigh criterion, that is the wavelengths can be just resolved when central maxima due to one falls on the first minima of the other.

FIGURE 5.11 Resolving power

 

Let the direction of nth principal maxima for wavelength A.1 is given by

(a + b) sin θn = nλ l

or

N (a + b) sin θn = Nnλ1

and the first minima will be in the direction given by

 

N (a+b)sin (θn +dθn) =mλ1

 

where m is an integer except 0, N, 2N … ,because at these values condition of maxima will be

satisfied. The first minima adjacent to the nth maxima will be in the direction (en+ den) only

when m = (nN + 1). Thus

 

N (a + b) sin (θn + dθn) = (nN + 1) λ1

 

For just resolution, the principal maxima for the wavelength A.2 must be formed in the

direction (e11 + de11).

 

Therefore        (a +b) sin (θn + dθn) = nλ2

or                     N (a + b) sin (θn + dθn) = Nnλ2

 

 

Now equating the two equations

(nN + 1) λ.1 = Nn. λ2

(nN + 1)  λ =Nn (λ. + dλ.)

λ = Nn dλ.

λ1 = λ, λ21 = dλ, λ2 = λ. + dλ.

 

Thus resolving power of grating is found as

R.P. = λ/dλ = nN

 

Resolving power = order of spectrum x total number of lines on grating which can also be written as

N (a+ b) sin θ/λ = w sin θn

 

where, m= N(a +b) is the total width of lined space in grating.

 

R .P. MAX =N (a+b)/λ w/λ

θn= 90°