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What is the Rayleighs criteria of resolution? Define resolving power and derive the expression for the resolving power of grating.
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An optical instrument is said to be able to resolve two-point objects if their corresponding diffraction pattern is distinguishable.

The resolving power of an instrument is ability of the instrument to produce separate image of object which are very close to each other.

Rayleigh's criterion of resolution: - According to Rayleighs criterion, two-point sources are resolvable by a optical instrument when the central maxima in the diffraction pattern of one falls over the first minimum due to other and vice versa.

Resolving power of grating: -

One of the important property of grating is ability to separate spectral lines which have nearly same wavelength. Resolving power of grating is defined as the capacity to form separate maxima of two wavelength which are very close to each other.

This is measured by λ/dλ where dλ is the smallest difference between two wavelengths an λ is mean wavelength. Let AB be the plane diffraction grating of grating constant (a+b) .let the beam of two wavelengths λ and λ+dλ is incident normally on the grating

P1 is the nth principal maximum of wavelength λ at an angle θ_n and P2 is nth principal maxima due to wavelength λ + dλ at an angle $θ_n$+ $dθ_n$

According to Rayleigh`s criterion, two wavelengths will be just resolved if principal maxima of λ + dλ in the direction $θ_n$+$dθ_n$ falls on the first minima of λ in the direction $θ_n$ +$dθ_n$

We know, in grating equation for maxima is (a+b) sinθ = nλ

And minima is N (a+b) sinθ = mλ (m = nN ±1 )

At $p_2$

Maximum due to wavelength λ+dλ in the direction $θ_n$+ $dθ_n$ is given by

(a+b)sin($θ_n$ + $dθ_n$ ) = n(λ+dλ) ………………….(1)

Minima due to wavelength λ in the direction $θ_n$ + $dθ_n$ is given by

N(a+b)sin($θ_n$+ $dθ_n$ )=(nN+1)λ …………………(2)

Multiply equation. (1) by N

N(a+b)sin($θ_n$+$dθ_n$ )= Nn(λ+dλ) ……………….(3)

From equation. (2) and (3)

(nN+1)λ = Nn(λ+dλ)

nNλ +λ = Nnλ + Nndλ

λ/dλ=nN Resolving power of grating

i,e resolving power of grating is proportional to number. of orders and number of lines /cm on grating

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Rayleigh’s criterion of resolution:

• Rayleigh’s proposed a criterion for the resolution of two close objects or spectral lines on the basis of the resultant intensity distribution curves of two close point’s objects or two close spectral lines.

• This criterion is called the Rayleigh’s criterion of resolution

Resolving power of a grating:

• It is defined as the ability of a grating to form two well separated diffraction patterns corresponding to two close wavelengths.

• If a grating can just resolve the spectral lines of wavelength of λ and λ+dλ, then its resolving power is expressed as

$$RP = \frac{λ}{dλ}$$