Diffraction: Propagation of wave based on Huygens s principle.

Transcription

1 Diffraction: In addition to interference, waves also exhibit another property diffraction, which is the bending of waves as they pass by some objects or through an aperture. The phenomenon of diffraction can be understood using Huygens s principle which states that Every unobstructed point on a wavefront will act a source of secondary spherical waves. The new wavefront is the surface tangent to all the secondary spherical waves. Propagation of wave based on Huygens s principle. According to Huygens s principle, light wave incident on two slits will spread out and exhibit an interference pattern in the region beyond. The pattern is called a diffraction pattern. On the other hand, if no bending occurs and the light wave continues to travel in straight lines, then no diffraction pattern would be observed (shown below). (a) (b) (a) Spreading of light leading to a diffraction pattern. (b) Absence of diffraction pattern if the paths of the light wave are straight lines. We shall restrict ourselves to a special case of diffraction called the Fraunhofer diffraction. In this case, all light rays that emerge from the slit are approximately parallel to each other. For a diffraction pattern to appear on the screen, a convex lens is placed between the slit and screen to provide convergence of the light rays

2 Fraunhofer Diffraction: In this case, source and screen are placed at infinite distances from the obstacle. In this case, for getting diffraction, we need two convex lenses, one to make the light from the source parallel and the other to focus the light after parallel rays. The incident wavefronts are plane and the secondary wavelets, which originate from the unblocked position of the wavefront, are in the same phase at every point in the plane of the obstacle. Fraunhofer diffraction can be easily observed in practice. Fraunhofer Diffraction of light by a slit of width a Consider a slit AB of width a perpendicular to the plane of the paper. Let a plane wavefront propagating normally to the slit is incident on it Let the diffracted light is focused by means of a convex lens on a screen placed in the focal plane of the lens. According to Huygens-Fresnel, every point on the wavefront in the plane of the slit is a source of secondary spherical wavelets, which spread out to the forward direction. The secondary wavelets are focused on the screen at point O and it is a bright central image. The secondary wavelets travelling at an angle θ with normal and focused at P on the screen For finding the intensity at point P, consider the path difference between secondary wavelets from A and B in direction θ From figure, path difference =BC= ABsinθ=a sinθ --- (1) Phase difference (δ)= (path difference)= a sinθ ---(2) Let us consider that the width of the slit is divided into n slits. The phase difference between any two consecutive waves from these would be ( total phase difference)= ( a sinθ)= d ---(3) But the resultant amplitude, R = ( ) ( ) Where x is the amplitude of each source ---(4)

3 R = ( a sinθ) ( a sinθ) R = ( a sinθ) ( a sinθ) R = ( ), where α = a sinθ is small, hence sin ( ) R = = R =, A= nx = amplitude due to n sources Intensity I = R 2 = A 2 ( ) 2 Case-1 Principal Maxima : Intensity depends on amplitude, therefore R =, where α = a sinθ The value of R will be maximum when α = 0, because i = 1 maximum value α = 0 = a sinθ, =, which is the principle maxima Case-II Minimum intensity position: The intensity will be minimum when R = 0, sin = 0, α = ± nπ = a sinθ a sinθ =± n -----(5)

4 Case III Secondary maxima R = maximum value for sin = = ( ) = a sinθ a sinθ = ( ) ----(6) Expression for intensity for secondary maxima Intensity distribution for a single slit diffraction In addition to principle maxima at θ =, there are secondary maxima in between equally spaced minima and the condition for these secondary maxima is = ( ) = 0 ( ) =0 Hence, either = or ( ) =, but = condition satisfies for minima So, condition for maxima is ( ) = = = = tan Solution for such equations can be obtained from graphical method plot graph for both the equation and identify the point where the above condition satisfies

5 The above figure shows both y = and y = tan graph and it is clear from the graph that the common points for both graphs are = ( ). But = is the condition for principal maxima hence condition for secondary maxima is = ( ) Intensity of Double-Slit Diffraction Patterns The intensity of double slit-diffraction pattern is the result of interference of diffraction pattern from each slit. Hence, the resultant pattern is an interference pattern with diffraction pattern as bracket. I = R 2 = A 2 ( ) 2 cos 2 (δ/2) Where cos(δ/2) is from interference component

6 From the above pattern it is clear that some of the interference maximas, which coincide with diffraction minima are missing, these maxima are also known as missing order. Diffraction due to n-slit or Grating An arrangement of large number of equally spaced slit is known as diffraction grating and the diffraction pattern obtained is known as diffraction spectrum. A diffraction grating consists of a large number N of slits each of width e and separated from the next by a distance d, as shown below Path difference between adjacent rays is given below (e+d)sinθ = ±nλ (7) λ = ( ) = where N is grating element i.e., number slit per unit length = ( )

7 1. Determination of λ of beam of light from unknown source using grating n =3 n =2 θ n =1 n =0 n =1 when a beam of monochromatic light falls normally on grating it splits into diffraction pattern as shown above λ =, ----(8) n= order of maxima, N= grating element, θ= angle at which n th maxima diffracted. 2. Finding the maximum order for a grating (n max ) Rearranging equation (8) n =2 n =3 n = n max =, for maximum value for sinθ = 1 3. Dispersive power of grating: Dispersive power of grating is defined as ration of change in angle with respect to change in wavelength Dispersive power = Differentiating equation (8) dλ = = Dispersive power is directly proportional to n and N and inversely to cosθ.

8 4. Resolving power of grating: Resolving power of grating is the quality of grating element in how clearly it can separate two close waves in the spectrum. Resolving power = Let s consider a beam of light consisting to two wavelength λ 1 and λ 2 which are very close to each other, falls normally on a grating λ 2 = λ 1 + d λ, where d λ is very small let n th maxima for λ 1 forms at an angle θ n, nth maxima for λ 1 + d λ forms at angle θ n + dθ. As shown in the figure below. Form equation (7), we get following condition (e+d)sinθ n = nλ 1 (e+d)sin(θ n +dθ) = n(λ 1 +dλ)=nλ 1 +ndλ (9) If we exam carefully λ 1 has maxima at θ n than at angle (θ n +dθ) it will have first minima only when the path difference between the rays at θ n and (θ n +dθ) is now substituting this in equation (9) we get = ndλ = nn Thus resolving power of a grating is directly proportional to the order of spectrum(n) and the grating element(n)