Width of central maximum

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The behaviour of light, as a rule, depicts the conduct of apparent, bright, and infrared light. Young’s double-hole experiment demonstrates the experiment, or the modern, two-hole experiment, or the double-hole experiment, that both light and matter exhibit properties of both wave and particle. 

 The screen’s diffraction pattern will be at a distance L (which is very much greater than the width of the central maximum) from the slit. 

A single slit forms an interference pattern with a centre maximum. The width of the central maximum is wider than the other maxima, which are narrower and dimmer.

Diffraction

The spreading of waves as they move through or around a barrier is referred to as diffraction. When it comes to light, diffraction occurs when a light wave passes through a corner or through an opening or slit that is physically the same size or smaller than the light’s wavelength.

Diffraction is an optical process with a regular pattern where the light splits or refracts into several beams that travel in different directions. Their directions depend on the spacing of the grating and the wavelength of the light.  

Diffraction due to a single slit

For any screen point, we will measure a/2 lengths from its centre to determine the angle of the screen. We’ll start with the condition of the black fringes to understand the pattern. 

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, there is a change of the behaviour pattern of fringe, which is known as fringe shift.

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

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.

Young’s Single Slit Diffraction

In 1801, Thomas Young demonstrated the wave nature of light with his double-slit experiment. Monochromatic light is shone through two tiny slits in this experiment. After travelling through each slit, the waves superimpose on a distant screen, resulting in alternative brilliant and dark fringes. The intensity and fringe width of all the bright fringes are the same.

Monochromatic light is transmitted through a single slit of limited width in a single slit experiment, and an identical pattern appears on the screen. The width and intensity of the single-slit diffraction pattern decrease as we travel away from the central maximum, unlike the double-slit diffraction pattern.

The central maximum

On either side of the centre, there are two minima, which are separated from each other by a distance that is equal to the width of the central maxima. 

For very small value of  θ,

sin θ ≈ θ 

⇒ λ = a sin θ ≈ aθ

⇒ θ = y/D = λ/a

⇒ y = λD/a

⇒  θ = 2λD/a

⇒ 2θ = 2λ/a

The formula for Single Slit Diffraction

When a monochromatic light of frequency falls on a slit with a width of a, the force on a screen L away from the slit can be transmitted as a component of the point made with the first course of light here as a component of θ. Here, the point made with the first course of light is taken as θ. It is given by,

I(θ) = Io Sin^2α/α^2 (1)

Here, α= π*λ

Sin θ and I0, situated at θ=0, is the intensity of the central bright fringe.

Diffraction Maxima and Minima: Bright edges show up at points

θ= 0, θ=sin-1 (±3λ/2), θ=sin-1 (±5λ/2) ..

θ= 0 is the focal most extreme.

I.e θ=sin-1 ±(2n+1)λ/2 (2)

Dark fringes compared to the condition,

asinθ = m λ (3)

 with m = ± 1, ± 2, ± 3 ………..

Diffraction via a single slit appears as an envelope over the obstruction design between the two slits in a double-slit setup.

Fringe width

The angular width of the central maximum, given by 2θ, is the angular distance between the two first-order minima (on one or the other side of the centre).

In the diffraction formula, the slit width is inversely proportional to the width of the central maximum. As a result, as the slit width narrows, the centre maximum widens, and as the slit width widens, the central maximum narrows. As a result of this behaviour, it may be deduced that light bends more as the aperture size decreases.

width of central maxima formula

The distance between the first secondary minimums on either side of the core bright fringe determines the width of the central maximum.

for the width of central maximum equation,

We know that for the first secondary minimum, n=1

asinθ=nλ

asinθ=λ

sinθ=λ/a equation (i)

 for small angles,

sinθ=y/D equation (ii)

 from equation (i) and equation (ii),

y/D= λ/a

y= λD/a

 width of central maximum=2y=2λD/a

therefore, w=2λD/a

 As the slit width, a, increases, the width of the central maximum, w, decreases.

 The angular width of central maximum=2θ

asinθ=λ

sinθ=λ/a

 for small angles,

asinθ=λ

θ=λ/a

 Angular width= 2λ/a

 Conclusion

Spreading waves are more likely to occur when passed through a small opening. If, for example, a door is left open, sound waves that enter the room can be heard even if the geometry of ray propagation suggests that there should be no sound in the room. In the same way, ocean waves that pass through a breakwater’s aperture can reverberate across the bay. Diffraction, which is the bending of a wave around the margins of an opening or a barrier, is displayed by all sorts of waves. Sound and ocean waves are examples of diffraction. Wave diffraction allows waves entering a breakwater opening to propagate throughout the bay.