^Resolving power of optical instruments

Suppose a convex lens is used to form the image of an object. Consider a parallel beam of light falling on it. If the lens is well corrected for aberrations, then geometrical optics tells us that the beam will get focused to a point, producing a sharp image point. However, because of diffraction, the beam instead of getting focused to a point gets focused to a spot of finite area in the form of alternate bright & dark concentric circles around a central bright disc as shown in figure. This spot is called the diffraction pattern.

A detailed analysis shows that the radius of the central bright region is approximately given by,

Thus the conclusion is a parallel beam of light incident on a convex lens gets focused to a spot of radius,   because of diffraction effects.  Where f is the focal length of the lens and 2 a (= d) is the diameter of the circular aperture or the diameter of the lens.

^Diffraction spread

From the above discussion it is clear that a parallel beam of light of wavelength λ on passing through an aperture of size d gets diffracted into a beam of angular width,

If a screen is placed at distance D, this beam spreads over a linear width,

If the diffraction spread y is small, only then the concept of ray optics will be valid.

^Position of maxima

Also it is found that at P for n^th maxima

In a similar way the angular position of nth bright fringe is

i.e. angular positions of 1^st, 2^nd, 3^rd maxima are

From above relation we can say that each secondary maxima occupies an angular width of , thus linear width of each secondary maxima is . As the central maxima is a spread between the angular positions to , thus its angular width is i.e. central maxima is twice wider than secondary maxima. Linear width of central maxi. is 

^Diffraction & Huygens’ theory

Diffraction can be explained using Huygens’ theory. According to this theory all parts of the slit AB will become source of secondary wavelets, which all start in the same phase at that position. The wavefronts from any two corresponding points such as (1, 13), (2, 12), (3, 11) etc. from the two halves of the slit travel identical distances to reach O thus have zero path difference, hence they add constructively to produce a bright fringe at point O, centre called central maxima or central bright fringe.

^Young’s double slit experiment

A common experiment to study interference of two light waves is YDSE. In this experiment overlapping of wave fronts of light waves coming from two slits S₁ & S₂ is studied by placing a screen at some distance from the slits. Let slits are separated by a distance ‘d’ & screen is situated ‘D >>d’ distance away from slits. Wavefronts reaching O from S₁ & S₂ are of equal path length produce no phase difference & thus we get maximum intensity at O (called central maxima, CM).

If the overlapping of waves is studied at a point P situated ‘y’ distance above or below the central maxima, then the intensity of the resultant wave depends on the phase difference between the waves S₂P & S₁P. If point P is situated ‘y’ distance above point ‘O’, then the path S₂P is longer than S₁P by an amount d sinθ. As here d << D, thus

thus path difference (p or Δx or simply x) can be expressed as

Phase & path difference for a sinusoidal wave are related as

Using above relation for conditions of maximum intensity we can say that maximum intensity is achieved at following positions from  central maxima

Using above relation for conditions of minimum intensity we can say that minimum intensity is achieved at following positions from  central maxima

For two waves of equal intensities the intensity of the resultant wave varies as square of the cos of Φ/2 i.e.

Here 4a² is the maximum intensity of the resultant wave at central maxima.

Facts

1. Fringe width of any dark or bright fringe is same & is

2. When interference is studied with white light, each of the seven colours produces its own fringe pattern, having different fringe width & due to overlapping blurred fringes are observed. However central fringe is white, on either side the nearest fringe is blue and farthest fringe is red & then uniform illumination.

3. If a transparent sheet of thickness ‘t’ & refractive index ‘m’ is introduced in one of paths of interfering waves, the entire fringe pattern displaces towards the side in which the sheet is inserted by a distance   without any change in the fringe.

Also no. of fringes shifted is