Expression for fringe width

Introduction

Fringe width is obtained from Young’s double-slit experiment. This experiment was first performed by Physicist Thomas Young in 1801. So, to perform this experiment a double slit is required and monochromatic light is required. When this experiment was performed, the pattern of bright lines and dark lines showed up. From there, the concept of fringe width came into the light. Then the expression for fringe width was derived. So, let us discuss more to understand Young’s double-slit experiment, Fringe width, Expression for fringe width, and Position of bright fringes.

Body

What is Young’s double-slit experiment;

Young’s double-slit experiment is a quantum mechanics experiment. In 1801, Thomas Young performed an experiment in which he made a small hole in cardboard and passed the light from that small hole. As the light passes through the hole,  it falls upon the cardboard with two small holes of dark and bright spots. The bright (constructive fringe) and dark (destructive fringe) come from here. Fringe width is the interference pattern of dark and bright fringe which are parallel. The experiment was first performed by using light. And it also displays the nature of quantum mechanical phenomena. The change in path length of both waves results in path shift and creates an interference pattern.

This experiment can also be performed with entities like electrons and photons. When this experiment was performed with electrons, it exhibited the same behavior as fired towards the double slit. It is possible to do this experiment with larger entities like electrons and photons. But it is difficult because of the increased size. Molecules were the largest entities on which Young’s double-slit was performed and each comprised 2000 atoms and whose mass was 25,000 atomic mass units.

What is Fringe width;

The distance between two consecutive bright(constructive fringe) and dark(destructive fringes) is called fringe width. The fringe formed at the centre is known as the central bright fringe. All the fringes are of equal length and hence, the distance between them is also equal and so is the position of bright fringes. The fringe width depends on the wavelength of light, the distance between the slits, and slits separation. The positions of the bright(constructive interference )fringes are at maximum intensity and the dark(destructive interference) fringes are at minimum intensity. When the distance between the dark and the bright fringes remains constant throughout, then it is known as linear fringe width.

 And therefore, the ratio of fringe width for bright to dark fringe is 1. It is denoted by β.

Expression for fringe width

Path difference between the lights from two slits S1 and S2 respectively and reaching to the point P on the screen is yD/d

yD/d = nλ

 y = nλd/D

Thus, the fringe width =   β = λD/d.

Displacement in fringes

In the path of one of the sources the thickness ‘t’ and refractive index ‘r’ are introduced, then optical path difference is changed and a fringe shift occurs.

What is a fringe shift?

When the phase relationship between the source component is changed, then there is a change of behaviour pattern of fringe, and that behaviour change is known as fringe shift.

On a viewing surface, the interaction alternates between constructive interference and destructive interference causing alternate lines of dark and light.

If fringe pattern will perform in the water

If the fringes are measured in the water, the fringe width will be narrower, because in water the wavelength of light is less. Thus, the fringe will decrease. This is the reason why this experiment is performed in air.

  Fringe width is given by,  β = λD/d.

The angular width, ϴ = λd =  βD.

Fringe shape in Young’s experiment with two slits

The shape of the fringe varies on the position and orientation of the screen, according to a careful investigation of the interference of fringes in Young’s double-slit experiment. The shape of fringes is hyperbolic on a screen which is placed parallel in position to the plane having the coherent sources. By considering the path difference between the interfering waves of visible light and understanding that fringes are actually of straight lines.

What are the path differences and phase differences?

The distance travelled by the two waves is known as path difference(λ). The difference in the phase angle of two waves is known as phase difference. For a complete wave, the wavelength varies is λ and the phase is changed through 2p. The path difference is directly proportional to the phase difference.  

Let the path difference of two waves be λ

The phase difference between them is 2p

If the path difference is x, then 

Path difference= 2πλ × x

Phase difference = 2πλ × path difference

Position of Bright Fringes

For maximum intensity or bright fringe to be formed at P

Path difference, Δz = nλ (n = 0, ±1, ±2, . . . .)

i.e., xd/D = nλ

or

x = nλD/d

The nth bright fringe’s distance from the centre is

xn = nλD/d

Similarly, the (n-1)th bright fringe’s distance from the centre is

xn-1  = (n -1)λD/d

Fringe width,  β = xn  – xn-1  = nλD/d – (n -1)λD/d = λD/d

(n = 0, ±1, ±2, . . . .)

Position of bright fringe if the wavelength is changed

The position of the bright fringe depends on the wavelength. The position of the central bright fringe does not change. Because it depends on the path difference between them. So, the position will remain the same but the screen will keep closer in case of a lower wavelength. But, because of the wavelength, the other bright points will change.

x  = n λD/d

Where, 

  λ is the wavelength

D is the  separation between screen and slit

D is the distance between the slits.

The distance between central bright and another bright will be small for lower wavelengths. And the distance between them will be bigger for higher wavelengths.

Conclusion

The purpose of this experiment was to know the results of the screen behind the slits. Young’s double-slit experiment proves that whatever passes through the two slits is a wave, exhibiting the interference. This experiment displays quantum mechanics very clearly. The experiment represents that the entities behave like waves and particles. But, we don’t know the exact position of quantum entities, we know the probability where they can exist. The intensity of fringes decreases as they get away from central maxima. The experiment can also be performed with three slits. Young’s double-slit experiment produces alternate bright and dark fringes if three slits are employed.