Interference in Thin Films
A thin film is the soap film or a film of oil spread over water surface which appears bright or dark when viewed in a monochromatic light and appears coloured when viewed in white light. (Interference in Thin Films)
As an example , let us consider that there is a soap film or a thin film of oil of refractive index (mu) is spread over water. When this film is seen through the reflected waves of the white light shows beautiful colours. The thin film is viewed due to the light waves reflected at the upper and lower surfaces of thin films. Both of these waves originate from the same source and hence the coherent waves have been generated by division of amplitude and so a beautiful colour pattern if observed due to interference. (Interference in Thin Films)
Consider that a parallel sided thin transparent film of thickness t ( XX* = YY*= t) and refractive index (mu) is taken under experimental consideration. Let a ray of light AB of monochromatic light is incident at point B on the upper surface of the film. A part of this light wave goes along BC following the laws of reflection and the other part goes along along BD following the laws of refraction. (Interference in Thin Films)
On reaching the point D on the lower surface X*Y* of the film, a part undergoes reflection along DF and the other part undergoes transmission and emerge out as DE and is called transmitted part of the incident wave BD. When the reflected part of the wave DF reaches the point F on the upper surface of the thin film, a part of it undergoes refraction as FG and the other part is reflected back into the film as FH. (Interference in Thin Films)
In this way , this process of multiple reflections and refractions will continue at the upper and lower surfaces of the thin film until the intensity of the primary incident wave AB die out due to the fact that there is a loss of energy of the wave each time, it undergoes reflection and refraction at both the surfaces of the film as shown in Figure 4.31. (Interference in Thin Films)
As an outcome of the multiple reflections and refractions, at the upper and lower surfaces of the thin film, the two sets of waves are obtained. The set of waves BC, FG, …..reflected from the upper surface XY of the film called Reflected Set of waves and the other set of waves DE, HT,… transmitted from the lower surface X*Y*of the film called Transmitted Set of waves. The interference pattern becomes visible when these sets of waves, Reflected Set or Transmitted Set are focused by our eye lens on the retina. (Interference in Thin Films)
Interference in Reflected set of waves:
The path difference between the successive reflected waves BC and FG is obviously given as follows. Path difference = (BD + DF)in oil film – BL path in rarer medium
When a wave of light is reflected at the surface of a denser medium after travelling in a rarer medium an additional phase difference of pi (180o) or a path difference of half of the wavelength is introduced. As the light beam BC undergoes reflection by the upper surface of the thin film backed by a denser medium, so an additional path difference of half of the wavelength or an additional phase difference of pi (180o) is introduced. Thus, the net path difference will be (Interference in Thin Films)
For Bright Fringe ( Maxima or Constructive Interference) There is a bright Fringe if the path difference between the interfering waves is integral multiple of their wavelength. (Interference in Thin Films)
Under the case when the condition given by eqn. (0.04) is satisfied, the film will appear bright. (Interference in Thin Films)
For Dark Fringe (Minima or Destructive Interference) There is a dark Fringe if the path difference between the interfering waves is an odd number multiple of the half of their wavelength. (Interference in Thin Films)
Interference in Transmitted set of waves:
When the thin film observed from the other side of it i.e. through the transmitted set of wave, then there is no additional path difference between the two waves DE and HT because of the reason that both of these rays emerge out of the film after suffering reflection backed by a rarer medium D and F. Therefore, the film will appear bright or dark when the eqns. (0.06) and (0.07) are satisfied respectively. (Interference in Thin Films)
It is here, observed and revealed that the conditions for maxima and minima when viewed in transmitted set of waves are just opposite to those for reflected set of waves. Therefore, the reflected and transmitted systems are said to be complementary to each other implying that thin film which appears bright in reflected set of waves will appear dark in transmitted set of waves and vice versa. (Interference in Thin Films)
Colours in thin film:
A thin film may be viewed as as a bright and a dark depending on the path difference between the light waves coming from the upper and the lower surface of the film after reflection or refraction when the thin film is seen with a monochromatic light. But when the thin film of a soap solution or of an oil are seen with a white light then bright anb brilliant colours are seen. (Interference in Thin Films)
This is because of the reason that the path difference between the light waves coming after the reflection from the upper and lower surfaces of the film depends upon the refractive index of the material of the thin film, thickness of the film (t) and the angle of reflection (r). When a specific colour is seen from a specific position of the eye, the other parameters like t and r are fixed . But refractive index of the material of the film varies with the wavelength of the light wave Therefore, only a particular wavelength will satisfy the condition for the film to appear bright. (Interference in Thin Films)
Since the white light is of composite nature consisting of different wavelengths ranging from 2700 Ao (Violet colour)to 7800 Ao (Red colour), therefore, the condition for maxima and minima (constructive interference and destructive interference) for the different constituent colours occurs at different positions (points) of thin film. As an example, if the path difference at certain place is equal to 1for red colour(7800 Ao), it will be 1.5 for blue colour (5000 Ao), so the blue colour will be reflected more pronounced at this place and the red colour will not be reflected at all, whereas other intermediate colours will have intermediate colours. (Interference in Thin Films)
Therefore, there is a mixture of of colours at each place and the composition of this mixture is different at different places. Various beautiful colour shades are observed on account of this reflected light waves. Thus the particular portion of the film appears of a particular colour, when seen from the particular position of the eye when the chromatic light (white light) is incident of the thin film. (Interference in Thin Films)
Interference and conservation law of energy
Let us suppose that a1 and a2 are the amplitudes of the two light waves emitted by two coherent sources respectively. The intensity of light at any point on the screen under the consideration that no interference is taking place between the light waves from the two sources, will be the sum of the intensities of the individual waves. Thus, I = I1 + I2 or
I = a12 + a22 – – – -(0.08)
In case, the light waves from the two coherent sources interfere we shall get maxima and minima. The intensities of these maxima and minima are given by
Imax = ( a1 + a2)2 and Imin = ( a1 – a2)2 – -(0.09)
As the maxima and minima are formed when the interference takes place, thus, the resultant intensity varies from Imin = ( a1 – a2)2 to Imax = ( a1 + a2)2 Therefore, the average intensity in the presence of interference Iav = (Imax + Imin)/2 . Therefore, Iav = [( a1 + a2)2 +(a1 + a2)2)]/2 => Iav = a12 + a22 -(0.10)
The conclusion can be drawn from eqns. (0.08) and (0.10) That when the interference takes place, the light energy which disappears at the region of destructive interference appears at the region of constructive interference so that the average intensity of the light remains the same. This can also be stated in other words that during the phenomenon of interference, the energy can neither be created at the points of constructive interference nor it can be destroyed at the points of destructive interference but it is merely transferred from the point of destructive interference to the points of constructive interference. (Interference in Thin Films)
If both of the coherent waves are of the same amplitude, a1 = a2 = a (say), then Imax = ( a + a)2 = 4a2 and Imin = ( a1 – a2)2 = 0 and Iav = [(4a2+ 0)]/2 = 2a2 The intensity distribution graph shown below in Figure 4.32 predict that the conservation law of energy is obeyed in the phenomenon of the interference of light,
Key Points
- It is very important to note that the conditions for the thin films to appear bright or dark in the transmitted set of waves are just the contrary to those in the reflected set of waves. Therefore, the film which appears bright in the reflected set of waves will appear dark in the transmitted set of light waves and vice versa.
- The soap or the oil films appear coloured due to the phenomenon of interference when a chromatic or white light falls on such films.
- The conservation law of energy is not violated by the phenomenon of the interference, rather this phenomenon is strictly in accordance with the conservation law of energy.