Diffraction of Light waves

Light waves superimpose each other and the redistribution of energy due to this can be observed in terms of well defined patterns of maxima and minima. Wherein, maxima refers to more energy and minima refers to less energy. Diffraction can also be called as interference in secondary wavelets.

1. Diffraction

2. Introduction The phenomenon of diffraction was first discovered by Grimaldi in 1665. Bending of light around the edge of an obstacle is called diffraction. Sound waves bend round objects of a similar size to their wavelength. The wall has a similar size to the sound's wavelength. The effect is called diffraction. Sound Light has a very small wavelength. So only very small objects or gaps can affect its direction. The wall blocks the light and the person can't see round the corner.

3. Huygen’s Principle Huygen postulated a model for light waves, similar to water waves. On the basis of this model, he proposed a hypothesis for geometrically constructing the position of a wavefront . Postulate 1: Each point on a wavefront (primary wavefront) acts as a source of new disturbances, called secondary wavelets, which travel in all directions with the velocity of light in that medium. Postulate 2: The surface touching these secondary wavelets tangentially in the forward direction at any instant gives a new wavefront, called a secondary wavefront at that instant.

4. Construction of Secondary Wavefronts Huygen postulated that the action of the secondary wavelets was confined only to the points at which they touched the forward envelope, and thus no backward wavefront exists. According to Stoke’s law, the intensity of the spherical wavelets varies as where θ is the angle between the direction of propagation of the wavelet and the normal at that point. For backward direction, θ =180°; therefore, intensity = 0. Construction of secondary wavefront

5. Phenomenon of Diffraction Diffraction of light is the phenomenon of bending of light waves around the corners of an obstacle or an aperture placed in its path, and their spreading into the region of the geometrical shadow. Diffraction becomes significant when the dimensions of the aperture or obstacle are comparable to the wavelength of the light. It occurs due to mutual interference of the secondary wavelets starting from portions of the primary wavefront , which are allowed to pass through the aperture.

6. Difference between Interference and Diffraction Fringes

7. Types of Diffraction Fresnel diffraction occurs when the source and the screen are at a finite distance from the diffracting aperture, such that the incident and diffracted wavefronts are spherical or cylindrical. Fraunhofer diffraction occurs when the source and the screen are at an infinitely large distance from the diffracting aperture, such that incident and diffracted wavefronts are plane.

8. Resultant of Multiple Simple Harmonic Motions Let us assume that a particle is simultaneously acted upon by n SHM vibrations. All the vibrations have the same amplitude A, and δ represents the phase difference between successive vibrations. A phase difference δ exists between bc and ab , bc and cd, etc. The phase difference between cd and ab is 2 δ , between de and ab is 3 δ , etc. Here, Resultant R can be given by Resultant of multiple SHMs

9. Fraunhofer Diffraction at Single Slit If the source and the screen are at infinitely large distances from the diffracting aperture, both the incident and the diffracted wavefronts can then be assumed to be plane wavefronts . The resulting diffraction is called Fraunhofer diffraction. The phase difference δ between waves originating from any two consecutive parts is expressed as The expression for intensity can be written as Schematic of single-slit diffraction

10. Fraunhofer Diffraction at Single Slit Principal Maxima The resultant R can be written as R will maximize to A T for =0, that is, Thus,  = 0 or the secondary wavelets that travel normal to the slit result in maxima on the screen. These are known as the principal maxima . Position for Minimum Intensity The intensity on the screen would be minimum if sin=0.

11. Fraunhofer Diffraction at Single Slit Secondary Maxima In addition to the principal maxima for  = 0, weak secondary maxima are also observed between equally spaced minima. Intensity of the first and second secondary maximum can be given by I 1 and I 2 respectively, ; Intensity of the principal maxima can be given by Thus, the intensity of secondary maxima decreases progressively and most of the incident light energy is concentrated in the principal maxima.

12. Fraunhofer Diffraction at Single Slit The principal maximum is at ɸ = 0 and the secondary maxima occur at and so on. Minima positions that lie between secondary maxima are given as ɸ =± π , ±2 π , ±3 π , and so on. Thus, the secondary maxima are not exactly midway between two minima but are displaced towards the centre of the diffraction pattern. Schematic representation of variation of intensity versus ɸ

13. Fraunhofer Diffraction at Double Slit Now consider the diffraction pattern obtained from an arrangement having two slits, each having a width ‘ d ’ and separated by a distance ‘ a ’. The resulting fringe pattern will then consist of the interference pattern due to the two slits. Each individual slit will also produce a diffraction pattern. PD between the two interfering waves is Path difference between interfering waves

14. Fraunhofer Diffraction at Double Slit The total intensity of the fringe pattern can be given by The first term of above equation is called the interference factor and the second term is called the diffraction factor . If the zero of diffraction envelope coincides with a maxima of interference pattern, we see no intensity on the screen. This order of interference pattern is, therefore, missing and is called the missing order . Double-slit diffraction pattern with interference

15. N- slit Diffraction or Plane Diffraction Grating Plane Transmission Grating In this gratings are constructed using ruling of equidistant parallel lines on a transparent material like glass using a fine diamond point. The ruled lines are opaque to light, whereas the space between any two lines is transparent to light and act as slits. Plane or Concave Reflection Gratings Another method of producing gratings is by drawing lines on a plane or a concave silvered surface. Light then gets reflected from a point situated between two lines. Diffraction due to plane grating

16. Resolving Power The ability of an optical instrument to form two separate diffraction patterns of two objects is called its resolving power. A criterion for resolution of two point sources by an optical instrument was first proposed by Rayleigh. According to the Rayleigh’s criterion, two point sources are resolvable by an optical instrument if the central maximum of the diffraction pattern of one of them falls over the first minimum of the diffraction pattern of the other. Similarly, two spectral lines are resolvable if the central maximum of the diffraction pattern of one of the wavelengths falls over the first minimum due to the other wavelength, or vice versa.

17. Resolving Power Schematic representation of Rayleigh’s criterion (a) Widely separated wavelengths (b) Resultant showing dip (c) Very close wavelengths

18. Resolving Power of Grating The ability of a grating to form separate diffraction maxima for two very close wavelengths and thereby to resolve the two wavelengths is called the resolving power of the grating. Resolving Power of a grating is given as

19. Applications of Concepts Diffraction gratings are capable of breaking white light into its constituent colours . Prisms also carry out the same task. Unlike in prisms, however, in a diffraction grating, deflection of any specific colour is proportional to its wavelength. The spectra produced by gratings are, therefore, easier to calibrate compared to that produced by prisms. Diffractive optics is also used in holography for reconstructing three-dimensional images of objects using laser light.