Obtain the equation fot resolving of optical instrument.

Explanation

The effect of diffraction has an adverse impact in the image formation by the optical instruments such as microscope and telescope. For a single rectangular slit, the half angle `theta` subtended by the spread of central maximum (or position of firs minimum) is given by the relation.
`a sin theta = lambda`
Similar to a rectangular slit, when a circular aperture or opening (like a lens or the iris of our eye) forms an image of a point object, the formed will not be a point but a diffraction pattern of concentric circles that becomes fainter while moving away from the center. These are known as Airy.s discs. The circle of central maximum has the half angular spread given by the equation,
`a sin theta = 1.22 lambda`
Here, the numerical value 1.22 comes for central maximum formed by circular aperutres. This involves higher level mathematics which is avoided in this discussion.
`a sin theta = .22 lambda`

For small angles, `sin theta approx theta`
`a theta = 1.22 lambda`
Rewriting further, `theta=(1.22lambda)/(a)and(r_(0))/(f)=(1.22lambda)/(a)`
`r_(0)=(1.22lambdaf)/(a)`
When two point sources close to each another form image on the screen, the diffraction pattern of one point source can overlap with another and blurred image. To obtain a good image of the two sources, the two point sources must be resolved i.e. the point sources must be imaged in such a way that their images are sufficiently for apart that their diffraction pattern do not overlap, According to Rayleigh.s criterison, for two point objects to be just resolved, the minimum of one coincides with the first minimum of the other and vice versa. Such an image is said to be just resolved image of the object. The Rayleigh.s criterion is said to be limit of resolution. According to Rayliegh.s criterion the two point sources are said to be just resolved when the distance between the two maxima is at least `r_(o)`. The angular resolution has a unit in radian (rad) and it is given by the equaiton,
`theta = (1.22 lambda)/(a)`
it shows that the frist order diffraction angle must be as small as possible for greater resolution. This further shows that for better resolution, the wavelenght of light used must be as small as possible and the size of the aperture of the instrument used must be as large as possible. The equation (4) is used to calculate spacial resolution.
The inverse of resolution is called resolving power. This implies, smaller the resolution, greater is the resolving power of the instrument. The ability of an optical instrument to separate or distinguish small or closely adjecent objects through the image formation is said to be resolving of the instrument.