Interference by division of Wavefront, Physics tutorial

Introduction:

The interference in light is as real as an effect as the interference in sound or water waves, and there is one example of it well-known to everyone - the bright colors of a thin film of oil spread out on a water surface. There are mainly two reasons why the interference of light is noticed in certain cases and not in others. At first, the light waves encompass very short wavelengths - the visible portion of the spectrum expands only from 400 mm for violet light to 700 mm for red light. Secondly, each and every natural source of light emits light waves merely as short trains of random pulses, in such a way that any interference that takes place is averaged out throughout the period of observation by the eye, unless special methods are used.

Similar to standing waves and beats, the phenomenon of interference is mainly based on the superposition of two or more individual waves under instead strict conditions which will soon be clarified. If interest lies mainly in the effects of enhancement or diminution of light waves, such effects are generally stated to be due to the interference of light. If enhancement or constructive interference and diminution or destructive interference conditions alternate in a spatial display, that is, the interference is stated to produce a pattern of fringes as in the double slit interference pattern.

The similar condition might lead to the enhancement of one color at the expense of the other color, generating interference colors as in the case of soap film and oil slicks.

Wave motion:

The wave is a disturbance in a material or medium, where the individual portions of the material might merely represent periodic motion, whereas the waveform itself moves via the material.

Definition: Wave motion is stated as the movement of a deformation of a material or medium, where the individual portions or elements of the material just move backward and forward, up-and-down, or in the cyclical pattern.

This seems as if something is in reality moving all along the material, however in reality it is just the distortion moving, where one portion affects the next.

1) The waves are transmitted via a medium: The energy and the wave are both formed or made through the successive action of people standing up and down. When there were no people in the stadium, no wave could exist and no energy could be transmitted. We state the people at the stadium, the water at the beach, the air molecules transmitting sound and so on, the medium via which such waves are transmitted.

2) The medium itself is not propagated: For the wave to work, each and every person in the stadium just requires to stand up and sit back down. The wave travels around the stadium, however the people don't.

Transverse wave and longitudinal wave:

The Waves can be categorized into transverse waves and longitudinal waves according to the direction of the vibration.

1) The transverse wave is one in which the vibrations are at right angles or perpendicular to the direction of travel of the wave. For illustration: Light and water waves.

2) The longitudinal wave is one in which the vibrations are all along the direction of travel of the wave. For illustration: sound waves.

Characteristics of waves:

170_characteristics of waves.jpg

Amplitude (A): It is the maximum disturbance, or height of a wave, is termed its amplitude. It is noted in the diagram that this is the distance from the midline of a wave to the top of a crest or to the bottom of the trough.

Wavelength (λ): It is the distance from a specific point on one wave, to the similar point on the next is termed as the wavelength. This can be symbolized via the symbol 'λ'. Wavelength is evaluated or measured in meters.

Frequency (f): It is the number of waves passing each and every second is termed its frequency, 'f'. The frequency is evaluated in units termed as hertz (Hz).

Wave speed, wavelength and frequency are associated in the given equation:

Wave speed (m/s) = Frequency (Hz) x wavelength (m)

v = f λ

Principle of superposition:

The Principle of superposition defines that whenever two waves of the similar type meet up at a point in space, the resulting displacement at that point is the vector sum of the displacements that the two waves would individually produce at that point.

Interference signifies to the superposing of two or more coherent waves to produce areas of maxima and minima in space, according to the principle of superposition.

A) Constructive interference: The Constructive interference takes place if two or more waves arrive at the screen in phase with one other, in such a way that the amplitude of the resulting wave is the sum of the amplitude of the individual waves.

B) Destructive interference: The Destructive interference takes place whenever two or more waves arrives 'π' out of phase with one other.

Phase difference between the waves:

Constructive interference: n (2π)

Destructive interference: (n + 1/2) 2π

Path difference:

                                                         2 sources in phase              2 sources π out of phase

Constructive interference:                         nλ or 0                              (n + 1/2) λ

Destructive interference:                          (n + 1/2) λ                             nλ or 0 

Young's double slit experiment:

A simple experiment of the interference of light was illustrated through Thomas Young in the year 1801. It gives solid evidence that the light is a wave.

2018_Youngs double slit experiment.jpg

Two slit A and B are illuminated by the same monochromatic source of light S. This ensures that the wave leaving A and B are coherent.

It was Thomas Young who first observed that if a transparent screen is placed formed in the region where the beams overlap. At O, along the perpendicular bisector of AB, the waves due to A and B are in phase. Hence, a bright band or fringe is formed. This is because the two light wave coming from A and B would have traveled the same distance at the point O. Bright and dark bands are subsequently formed on either side of O. The bright bands occur where d = nl where n = 0, 1, 2, 3... i.e. constructive interference takes place.

On the other hand the dark bands are formed where d = λ (n + 1/2)

These alternate bright and dark bands are known as interference fringes. At point O, the path difference is zero.

Fringe Separations:

1461_Fringe separations.jpg

In this section you will see that the thickness Y between two adjacent bright or dark fringes and 'a' is the distance between the slits A and B. D is the distance of the slit from the screen and the wavelength of the monochromatic light is 'λ'.

In Figure given above P is the position of nth bright fringe, then path difference at that point P is provided by

BP - AP = BM = nλ.

Consider a Δ NPO, then

tan θ = Xn/D...........................................Eq.4

Similarly in Δ AMB,

sin θ = nλ/a...........................................Eq.5

In practice, D is large and Xn is very small, therefore Θ is small,

Hence tan θ ≈ sin θ

Therefore Xn/D = nλ/a

Position of the nth bright fringe from origin O is provided by the relation

Or Xn = nλD/a................Eq.6

The distance of the next bright fringe from O is given by Xn + 1

Xn + 1 = (n+1)λD/a................Eq.7

[i.e. replaced n with n+1]

Hence the spacing Y between the nth and (n+1)th fringes can be determined by subtracting Eq.6 from Eq.7

Y = Xn+1 - Xn

Therefore on substituting values

Y = (n+1)λD/a - nλD/a

Therefore the expression for fringe width is

Y = λD/a ................Eq.8

Thus it can be found from Eq.8) that fringe width differs directly proportional to D and l and inversely proportional to distance between slits a. Therefore using expression in Eq. 8, one can estimate wavelength of light easily.

Fresnel's Biprism:

1291_Fresnel Biprism.jpg

The alternative procedure to the classic Young's slits experiment for evaluating the wavelength of light is that due to Fresnel. The apparatus is representing in the figure shown above.

Monochromatic light from the narrow slit 'S' falls on the bi-prism, the axis of which should be in line by the slit. The refracting angles of the bi-prism are extremely small, generally around 0.25o. This prism makes two virtual images of the slit S1 and S2 in the plane of S, and such two virtual images act as the sources for two sets of waves that overlap and generate an interference prototype on the screen.

The fringes are much brighter than such generated through Young's slits, due to the reason of high amount of light which can pass via the prism as compared with that passing via the double slit arrangement.

The formula employed is similar as for Young's slits, the only problem being the measurement of the separation of the two virtual sources S1 and S2.

This can be completed through placing a convex lens among the bi-prism and the screen or eyepiece and evaluating the separation (s) of the images of s1 and s2 generated by the lens. When the object and image distances (u and v) are found, the value of 'd' can be computed from

d/s = u/v

By employing a two position process removes the requirement to compute u and v. When s1 and s2 are the separations of the two images slits in the two positions then:

d= [s1s2]1/2

Wavelength of light = x d/D = x [s1s2]1/2/D

Here 'x' is the fringe width.

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