Interference due to division of amplitude

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SALONI SHARMA

Phase Changes Due To Reflection

An electromagnetic wave undergoes a phase change of 180° upon reflection from a medium of higher index of refraction than the one in which it was traveling (similar to a reflected pulse on a string)

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There is no phase change when the wave is reflected from a boundary leading to a medium of lower index of refraction (similar to a pulse in a string reflecting from a free support)

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INTERFERENCE IN THIN FILMS

When a film of oil spreads over the surface of water, or a thin glass plate is illuminated by light, interference occurs between the light waves reflected from the film, and also between the light waves transmitted through the film.

Interference in reflected System

Interference in reflected rays

Interference in transmitted rays

• Interference due to an infinitely thin film

When the thin of the film is small compared to the wave length of light (𝜆 >> t). The effective path difference between the interfering waves in reflected light for a film is 2ut cosr - 𝜆/2. When the film is excessively thin such that its thickness t is very small compared to the wavelength of light then 2ut cos r is almost zero. Hence the effective path difference becomes 𝜆/2. This is condition for minimum intensity. Hence every wave length will be absent and the film will appear black in reflected light and it will appear bright in transmitted light.

• Interference in a thick film:

When the thickness of the film is large compared to the wave length of light (t>> 𝜆), the path difference at any point of the film will be large. In this case the condition of constructive interference,

2µt cosr = (2n + 1)𝝀/2,at a given point is satisfied by large number of wave length with the value of n different for different colours. At the same point the condition of destructive interference 2µt cos r = n𝝀 is also satisfied for another set of large number of wavelength.

The thus in the case of a thick film illuminated by white light, the colors are not observed in the reflected light and its appears uniformly illuminated.

Colours of thin films

• When a thin film is illuminated by monochromatic light and seen in reflected light, it will appear bright if 2µt cosr = (2n + 1) 𝝀/2 and dark if 2µt cosr = n𝝀.
• However, if the film is illuminated by white light the film shows different colors. The eye looking at the film receives the waves of light reflected from the upper and lower surfaces of the film. For a thin film these rays are very close to each other. The path difference between interfering rays is (2µt cosr - 𝝀/2). The path difference depends upon t, the thickness of the film and upon r which depends upon the inclination of the incident rays. Since white light consist of a continuous range of wave lengths. For a particular value of t and r i.e. at a particular point of the film and for a particular position of the eye, the waves of only certain wavelength satisfy the constructive interference. Therefore only those color will be present in the reflected system with maximum intensity. The other neighbouring wavelengths will be present with less intensity. There will also be certain wavelengths which satisfy the condition of minima. Such wavelengths of colors will be absent from the reflected light, as a result the point of the film will appear colored.

Fringes of equal inclination or Haidinger fringes

• Let us consider that a thin film is illuminated by an extended monochromatic light source. When the film is of uniform thickness, the path difference 2µt cos r between the coherent beams is only due to the change in r. If the thickness of a film is large, the path difference will change appreciably even when r changes in a very small way. In this case each fringe represent the locus of all point on the film, ray from which are equally inclined to the normal. There fringes are called fringes of equal inclination. The fringes of equal inclination are known as Haidinger fringes. In this case all the pairs of interfering rays of equal inclination pass through the plate as a parallel beam and hence meet at infinity. The other pairs of different inclination meet at different points at infinity.

Necessity of an Extended Light Source

• An extended source is necessary to enable the eye to see whole of the film and observed the interference pattern due to whole film.
• If we have a point source whole of the film does not come into focus simultaneously, whereas with broad or extended source the whole of the film can be brought into focus at the same time and at same position.

Thin film with point source

Let us consider a thin film and a narrow source of light at S, then for each incident ray E1,E2 we get a pair of parallel interfering rays in different directions. The incident ray E1 produces interference fringes because E1r and E1r′ parallel to each other reach the eyeat position 1, where as the incident ray E2 meet the surface at point A1 at some different angle and is reflected along E2r and E2r'. Here E2r and E2r' do not reach the eye. Therefore, the position AB of the film is visible whereas the position A’B’ is not visible.

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Position 1

Position 2

Thin film with extended source

If extended source of light is used the incident ray E1 after reflection from the upper and the lower surface of the film emerges as E1r,E1r' which reach the eye. Also incident ray E2 from some other point of the source after reflection from the upper and lower surface of the film emerges as E2r,E2r' which also reach the eye. Therefore, in the case of such a source of light, the rays incident at different points on the film are well adjust by the eye and the field of view is large.

Due to this reason, to observe interference phenomenon in this film, a broad source of light is required. With a broad source of light, rays of light are incident at different angels and the reflected parallel beams reach the eyes or the microscope objective. Each such ray a light has its origin at a different point on the source

Interference due to wedge shaped film

Let us consider a thin wedge shaped film of refractive index µ, bounded by two plane surfaces inclined at an angle θ. Let the film be illuminated by a parallel beam of monochromatic light the interference occurs between the incident ray reflected at the upper and lower surfaces of the film so that equidistant alternate dark and bright fringes becomes visible. The interfering rays in reflected light are AR1 and CR2, both originating from the same incident ray SA. To evaluate the path difference between these two rays the perpendiculars CD and CG are drawn from C on AR1 and AB. If 𝜃 is an angle of wedge – shaped film is very small AR1 and CR2 will be almost parallel and after the perpendicular CD the paths will be equal.

• Subtracting we get,
• 2𝜇(tn+1−tn)cos𝜃 = 𝜆,
• Substituting the value in equation (9) 2𝜇𝛽tan𝜃 cos𝜃 = 𝜆 2𝜇𝛽𝑠𝑖𝑛𝜃 = 𝜆 So that 𝛽 = 𝜆 2𝜇𝑠𝑖𝑛𝜃 ≈ 𝜆 2𝜇𝜃 (if wedge angle is very small 𝑠𝑖𝑛𝜃 ≈ 𝜃) 𝛽 = 𝜆 2𝜇𝜃 From this equation it is clear that the wedge film is form an angle 𝜃 is very small then the fringe width of the reflected light is independent of n. Therefore for dark fringe and bright fringes are equal width and if the wedge angle 𝜃 is increase then fringe width is also decrease and if wavelength is increase then fringe width is also increase.

Fringes of equal thickness or fizeau fringes

• The path difference ∆ = 2𝜇t cos𝜃 between the rays reflected from the wedge shaped film of constant wedge angle depends on the thickness of the film at that place where light is incident normally on the film. Due to this reason the interference fringes will be the locus of all those points at which the thickness t of film has a constant value. So these fringes are straight fringes of equal thickness and are called Fizeau Fringes.
• The fringe width of these fringes 𝛽 = 𝜆/2𝜇𝜃 depends on the wedge angle while it does not depend on the thickness of film. These fringes are formed within film. So they are also called localized fringes. This types of fringes are observed in Michelson's interferometer

NEWTON’S RING

• When a plano-convex lens with its convex surface is placed on a plane glass plate, an air film of gradually increasing thickness is formed between the lens and the glass plate. The thickness of the air film is almost zero at the point of contact O and gradually increases as one proceeds towards the periphery of the lens. If monochromatic light is allowed to fall normally on the lens, and the film is viewed in reflected light, alternate bright and dark concentric rings are seen around the point of contact. These rings were first discovered by Sir Isaac Newton, hence named as Newton's Rings. If it is viewed with the white light then coloured fringes are obtained. The experimental arrangement of the Newton’s Ring apparatus is shown in figure

A parallel beam of monochromatic light is reflected towards the lens L. Consider a beam of monochromatic light strikes normally on the upper surface of the air film. The beam gets partly reflected and partly refracted. The refracted beam in the air film is also reflected partly at the lower surface of the film. The two reflected rays, i.e. produced at the upper and lower surface of the film, are coherent and interfere constructively or destructively. When the light reflected upwards is observed through microscope M which is focused on the glass plate, a pattern of dark and bright concentric rings are observed from the point of contact O. These concentric rings are known as Newton's Rings.

Diameter of Newton’s Rings

• Consider a ring of radius r due to thickness t of air film as shown in the figure

Condition for a bright ring �(constructive interference in thin film)

Condition for a dark ring� (Destructive interference in thin film)

Applications of Newton’s Ring

Determination of Refractive Index of the Liquid

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