Interference of waves 12 – Learn in Fun
Interference
It is the phenomenon re-distribution of energy of the medium due to the superposition on two coherent waves.
The energy is uniformly distributed in the medium when a source of light sends the energy into it. But when the two sources emit the light of nearly same wavelength, same frequency and either in same phase or with constant phase difference, in a same medium, then the energy of the medium gets redistributed non-uniformly. This redistribution of the energy of the medium is called Interference of light.
The phenomenon of redistribution of light energy in a medium due to the superposition of two light waves coming from two coherent sources of light is called the Interference of light.
As a result of redistribution of light energy in the medium, it observed that the intensity of light at some points in the medium is maximum and the interference at such points is said to be constructive interference while at some other points the intensity is minimum and the interference at such points is said to be destructive interference. When the amplitude of the waves are equal, then at the points of destructive interference, the resultant intensity of light is zero implies that complete darkness. Alternate bright and dark bands of light are formed which are called interference fringes.
Conservation Law of Energy is strictly followed when the phenomenon of Interference of light takes place. In the interference of light, the energy is neither created nor destroyed but it is merely redistributed. The energy that disappears at the points of destructive interference appears at the points of constructive interference and vice-versa. Therefore, in interference, the energy is merely redistributed and is strictly, in accordance with the conservation law of energy.
Young’s Double Slit Experiment
Sir Thomas Young in 1801-1802 demonstrated the phenomenon of interference of light experimentally. In his experiment, he used sodium vapour lamp as a source of monochromatic light to illuminate a narrow rectangular slit S of about 1mm width. This narrow slit acts as a source to illuminate two parallel narrow slits S1 and S2 arranged symmetrically and parallel to the slit S at a distance about 10 cm from it.
The experimentally recommended values of separation between the two slits Sa and S2 is about 2mm and of their width not more than 0.3 mm. Two coherent sources S1 and S2 are thus, obtained as per Huygens’ Principle and therefore, the interference takes place. Interference Fringes consisting of alternately bright and dark fringes/bands will be obtained on the screen which will disappear if one of the slits is closed and so the intensity of light becomes uniform.
The interference pattern obtained on the screen in Young’s Double Slit Experiment consists of a bright fringe at the center of the fringe pattern surrounded by the alternate dark and bright fringes on both sides of the central bright fringe.
Bright Fringes due to Constructive Interference:
There are certain points in the medium where crest of one wave falls on the crest of the other or the trough of one wave falls on the trough of the other. This gives rise to maximum resultant amplitude and hence the resultant intensity of light will be maximum due to constructive interference. Such points on the screen will correspond to the position of bright fringes.
Dark Fringes due to Destructive Interference:
There are certain points in the medium where crest of one wave falls on the trough of the other or the trough of one wave falls on the crest of the other. This gives rise to minimum resultant amplitude and hence the resultant intensity of light will be minimum due to destructive interference. Such points on the screen will correspond to the position of dark fringes.
Mathematical Conditions For Constructive and destructive interference
Let us suppose that the two narrow slits S1 and S2 fixed very closed to each other and at are illuminated by a monochromatic source of light S. The two slits S1 and S2 are also at equal distances from the monochromatic source S as shown below in in Figure 4.27
the wavefronts from the two slit sources S1 and S2 which act as the two coherent sources, spread out in all possible directions and in the region they superpose on the screen, a pattern of fringes will be observed containing alternate bright and dark fringes. At the center point O on the screen the intensity is always maximum corresponding to central maxima or principal maxima.
Condition for Maxima and minima
Consider that the two coherent waves from the sources S1 and S2 reaching a point P on the screen at any time t are given by
This eqn. (04) shows that the resultant displacement due to the superposition at a point P on the screen is a simple harmonic wave with amplitude A and phase difference of theta in comparison to the harmonic wave from the slit source S1. The amplitude A of the resultant simple harmonic wave is determined by squaring and adding the eqns. (02) and (03). Thus,
As is known that the intensity of a light wave is proportional to its amplitude. For convenience and simplicity, it is assumed here that the intensity of light wave is equal to the square of its amplitude. Thus, the intensity of light at any point P on the screen is taken to be
Constructive Interference: Constructive interference at any point on the screen corresponds to the maximum intensity or formation of bright fringe at that point and so from eqn. (05), it is follows that the intensity of light will be maximum, when
Thus, the interference at some points on the screen will be constructive interference when the phase difference between the two interfering coherent waves reaching the screen possess a phase difference which is an integral multiple of 2(pi). The above condition for constructive interference can also be expressed in terms of path difference between the two waves. Knowing that the path difference between the two waves equal to the wavelength is equivalent to the phase difference of 2(pi). Thus, when the two waves from the the sources S1 and S2 reaching the point P on the screen have a path difference x, then the phase difference between the two waves will be
The eqn. (07) gives the condition for constructive interference in terms of path difference between the two interfering coherent waves. Therefore, the point P will be the point of maximum intensity, if the two light waves reaching the point P will have the path difference an integral multiple of the wavelength of light.
Condition for destructive Interference (minima):
It is evident that eqns. (08) and (09) provide the condition of destructive interference in terms of phase difference and path difference respectively. From these eqns. It follows that the point P on the screen will be a point of minimum intensity of light if the phase difference between the light waves in an odd number multiple of pi or the path difference is an odd number multiple of half of the wavelength of the light.
Some Important Results: The expressions of the intensity of light in the following few cases are of great importance and of immense use in the study of the relations of the intensity, maximum intensity, minimum intensity and the width of the slits while undergoing the phenomenon of interference of light.
Ratio of intensity of light at maxima and minima: The expressions for the intensity of maxima and minima using equation (05) can be expressed as
Ratio of intensity of light due to two sources: The intensity of light due to a slit source is proportional to the width of the slit. Therefore, for the two slits S1 and S2 having width w1 and w2 send the light wave if intensities I1 and I2 of the respective slits, then
coherent and Incoherent source of light
Two sources are regarded as coherent sources when they continuously emit the light waves of same wavelength or same frequency and they are either in same phase or have a constant phase difference.
Two independent sources hardly possess coherence and could not produce the interference of good quality. In practice, the coherent sources are produced from a single parent source.
In Figure 4.28, A single spherical wavefront emitted from the parent source S illuminates the two slits S1 and S2 which act as the coherent sources as the frequency or the wavelength of the waves emitted from them are the same and may be in same phase or having constant phase difference. The coherent sources so obtained are capable of emitting the waves of almost equal amplitude.
There are generally two convenient methods of producing the coherent waves. By Division of wave front and by Division of the amplitude.
By Division of wavefront: The wave is divided into two parts as shown in Figure 4.28which are made to superpose after travelling different optical paths and interference is produced. Interference produced in Young’s Double Slit Experiment and Fresnel’s Biprism are studied using the division of the wavefront. The parent source is a point source in this method of producing coherent sources.
By division of Amplitude: In the method of producing the coherent sources, by dividing the amplitude of the incoming wavefront into two or more parts by partial reflections and refraction of the wave as shown in Figure 4.29 R and R* are the two partially reflected waves acting as the coherent waves which superpose to produce interference of light
As an example, interference due to thin films, Formation of Newton’s Rings, Michelson’s interferometers are studied by producing the coherent waves by producing division of wavefront.
Incoherent Sources: When the two light sources are not capable of producing the waves having constant phase difference or in same phase are called Incoherent sources. Two independent sources of light may have same frequency or wavelength but they may not have constant phase difference or zero phase difference and so may not be coherent in nature.
The two independent sources can not be coherent: The two independent sources can not act as the coherent sources perfectly due to the following reasons.
*In case of two independent sources, the light is emitted by the individual atoms but not by the bulk of matter as whole.
** The emission of light takes place through the millions and millions of atoms independently, therefore, these millions of atoms of a source are impossible to to emit the waves in same phase. The light emitted by a monochromatic source maintains monochromaticity for about a time span of 10-8 sec and after this span the atoms responsible for coherent emission undergo change. Thus, the interference pattern and the phase undergo a change by 108 times in every second. however such rapid changes, our eyes failed to respond and a uniform illumination on the screen is observed.
Conditions for Obtaining Coherent Sources:
*The difference of the path between the waves from the two slit sources reaching the screen must not be larger than 30 cm otherwise, the phase difference produced on account of the path difference will not be constant.
**The two sources emit monochromatic light (single wavelength). However, in case the light emitted by the two sources is not monochromatic then the different colours will produce different interference pattern causing an overlapping of the fringes of different colours.
***The two light sources must not be independent but be obtained from a single source by using any methods like division of wavefront or division of amplitude.
Sustained Interference
The interference pattern is considered to be a Sustained Interference when the positions of the maximum and minimum intensity of light remain fixed all along the screen. It is also called as Permanent Interference. There are various steps to be undertaken for producing sustained interference.
Conditions For Sustained Interference:
- The two sources must emit the light waves continuously and uninterrupted.
- The amplitude of the two interfering waves emitted by the two sources preferably be equal.
- The two sources of light should be very narrow and their width not exceeding 0.33 mm.
- The two sources of light must be placed very close to each other. The distance of separation between the sources should not exceed 2 mm.
- The two sources must emit the light waves of same frequencies or same wavelength implying that the two sources must be coherent in nature.
- The light waves emitted by two sources must be necessarily in either same phase or a constant phase difference must be maintained between them.
Theory of interference Fringes
Let us consider that S1 and S2 are the two coherent sources of light obtained from the same parent source by employing the method of division of wavefront and are separated through a distance of d as shown in Figure 4.30.
The screen is placed at a distance of D from the plane of two slit sources S1 and S2. Let us further suppose that P is a point on on the screen at a distance y from the center of the screen where the wavefronts reach from S1 and S2 after travelling unequal distances x = [2yd/(D + D)]= [2yd/2D]=Yd/D. The type and nature of the interference depends upon the path difference between these waves (S2P -S1P). If the path difference between the waves reaching the point P on the screen from the two coherent sources is x, then
x = (S2P -S1P)—–(16)From the triangle S2PL, we can write, S2P2 = S2L2 + LP2 = D2 + (y + d/2)2, Also from triangle S1PK, S1P2 = S1K2 + KP2 = D2 + (y – d/2)2, On subtracting, we get, (S2P + S1P)(S2P -S1P)= [D2 + (y + d/2)2]-[D2 + (y – d/2)2 (S2P + S1P)(S2P -S1P)= (y + d/2)2 – (y – d/2)2 = 4y.d/2= 2yd (S2P + S1P) x = 2yd => x = [2yd/(S2P + S1P)]- – (17)
Since the point P lies very close to the point O, the center of the screen, this implies that S2P = S1P = D, so, eqn. (17)can be written as x = [2yd/(D + D)]= [2yd/2D]=Yd/D Thus, x =Yd/D – – – – – -(18)
Positions of maxima and Minima on the Screen: It has been shown that the point P will be a center of a bright fringe in the interference pattern when the path difference between the wave reaching the point P from S2 and S1 is integral multiple of the wavelength of light. Thus,
It has also been shown earlier that the point P will be a center of a dark fringe in the interference pattern when the path difference between the wave reaching the point P from S2 and S1 is odd number multiple of half wavelength of light. Thus,
Fringe width: The separation between two consecutive bright and dark fringes is called as fringe width. Thus, width of dark fringe = Separation between two consecutive bright fringes.
In a case when the two coherent waves producing interference have equal amplitude a (say), then the intensities of maxima and minima can be written as Imax = (a + a)2 = 4a2 and Imin = (a – a)2 = 0
The interference pattern will consist of alternately dark and bright fringe on both sides of the bright fringe called central maxima. All the fringes in this case will be of equal width such that the dark fringes are completely dark and bright fringes will have uniform intensity 4a2. B1, B2, – – – -are the positions of bright fringes while D1, D2 – – – – are the positions of dark fringes on both sides of the central bright fringe which is formed at the center of the screen.
Conclusions: *All the fringes are of equal width as the fringe width is independent of n, the number of fringe. *For a particular wavelength, d must be small and D should be fairly large to get the fringes of proper width. *Fringes produced by the light of shorter wavelength are narrower as compared to those produced by longer lemda. *The angular width of of a fringe is (Wavelength/d)
Key Points:
- The redistribution of energy takes place in the medium when the two light waves interfere in that medium.
- The sustained and good quality interference is observed only when two coherent sources are used which send the waves of same frequency/wavelength in same phase or with constant phase difference.
- The width of all fringes (central, dark, and bright) is same in the interference pattern produced by two narrow slits.
- The fringes obtained in Young’s double slit experiment are hyperbolic in shape but the fringes appear straight in small interference pattern.
- The intensity of all the bright fringes is same in the interference pattern produced by the two narrow slits.
- The fringe width in the interference pattern produced by two narrow slits , decreases (1) when using the light of smaller wavelength, (2) when the distance between slits and screen D decreases, (3) the distance between the two slits d, increases and (4) when the medium between the slits and the screen is replaced by a medium of higher refractive index.