How Does Fraunhofer Diffraction Prove the Equation y = R * m * (wavelength) / a?

[relinquished™]

I am to prove that in

y = R * m * (wavelength) / a

where y is the distance between two minima, a is the width of the slit, R is the length between the screen and the slit grating, and m is an integer which is the order of the minima.

I know I have to use the paraxial approximation where tan x is approximately sin x which is approx. x, but I can't seem to apply it. When I refer to my textbooks, they state that y is the difference between the central maximum and the first minimum when m=1. Is this applicable bet. two minima as well?

 

[Claude Bile]

Start with the condition for constructive interferance;

dsin(theta) = m(lambda)

Then apply the small angle approximation. It is quite simple.

Your textbook is referring to y as the difference between the first maximum and the centre of the pattern, which is equivalent (within the small angle approximation) to the distance between successive minima.

Claude.

 

[M98Ranger]

To prove the equation y = R * m * (wavelength) / a, we first need to understand the concept of Fraunhofer diffraction. Fraunhofer diffraction refers to the diffraction pattern produced when a plane wave of light passes through a narrow slit and is then observed on a screen placed at a distance from the slit. This pattern consists of a series of bright and dark fringes, with the central maximum being the brightest and the fringes becoming increasingly dimmer as they move away from the central maximum.

Now, let's consider the equation y = R * m * (wavelength) / a. Here, y represents the distance between two minima, a is the width of the slit, R is the distance between the slit and the screen, m is an integer which represents the order of the minima, and wavelength is the wavelength of the incident light.

To understand how this equation is derived, we need to apply the paraxial approximation. This approximation states that for small angles, the tangent of an angle is approximately equal to the sine of that angle. In other words, tan x ≈ sin x ≈ x. This approximation is used in Fraunhofer diffraction because the angles involved are very small.

Now, let's look at the diffraction pattern produced by a narrow slit. The central maximum is located at the center of the pattern and is the brightest point. The first minimum is located on either side of the central maximum, and it is the point where the intensity of the light is zero. The distance between the central maximum and the first minimum is y.

According to the paraxial approximation, the angle of diffraction for the first minimum is given by sin θ ≈ θ. This means that the angle of diffraction for the first minimum is approximately equal to the distance between the central maximum and the first minimum, which is y, divided by the distance between the slit and the screen, which is R. Mathematically, this can be written as θ ≈ y/R.

Now, we also know that the angle of diffraction is related to the wavelength of light and the width of the slit by the equation sin θ = m * (wavelength) / a, where m is the order of the minima. Combining this equation with the paraxial approximation, we get θ ≈ y/R = m * (wavelength) / a. Rearr