Q. Explain Rayleigh's criterion of just resolution of two spectral lines of equal intensities giving suitable intensity distribution carves. Show how the resolving power of a plane transmission grating depends on:
(i) Number of ruled lines
(ii) Width of ruled surfaces (space)
Deduce suitable formula is support of your answer.
What do you understand by 'resolution'? Explain. What is meant by resolving power of a grating? Deduce an expression for the same, and discuss its dependence on various constant of grating.
Explain Rayleigh criterion for resolution and apply it to distinguish between the resolving power and dispersive power of a diffraction grating.
Sol. Resolution means to see two close objects just as separate. Rayleigh purposed a criterion which can manifest when two closed object can be seen as just as separate. This criterion is known as Lord Rayleigh's criterion of resolution.
According to this criterion two images are said to be just resolved if the position of central maxima of the intensity curves of one coincides with the first minima of the other and vice-versa. This is known as Rayleigh's criterion of limiting resolution.
If the distance between the objects is greater than the resolution limit, they appear as completely resolved (separate) and in case if the distance between the objects is less than the resolution limit, they appeared as un-resolved (mixed).
Let the intensity distribution pattern of two spectral lines having wavelength l1, l2 are shown resolved (separate) and in case if the distance between the objects is Fig. (a), (b), (c). In case both the spectral lines are completely resolved.
Now if the difference in wavelength is reduced, two spectral lines get closer. A condition comes when the central maxima of one spectral line l1 is coincide with the 1st minima of other spectral line l2 of vice versa. In this case the resultant intensity has a dip in between the central maxima of both the wavelength. This condition of spectral lines is called just resolution condition.
If the difference in wavelength is reduced after the condition of just resolution the diffraction pattern of both the spectral lines overlapped to each other. This is the condition of un-resolved.
Resolving Power of a Grating: Refer of Prob.12 (a) (iii).
Formula for the Resolving Power of Grating:
The resolving power of a grating is defined as its ability to show two spectral lines of very close wavelengths as separate.
It is measured in terms of the ration l/dl, l is wavelength of a spectral line and d is the difference in wavelength between this line and a neighboring line which is just resolved.
Consider a beam of light consisting of wavelengths l and l + dl, is incident normally on a grating surface.
Let ?n and ?n + d?n is the direction of principle maxima of wavelength l and l + dl, then
Let the nth principle maxima of wavelength l is formed at P1, if 1st minima of this wavelength is formed in the direction (?n + d?n)