Question

Derive the relation a sin theta = lambda for the first minimum of the diffraction pattern produced due to a single slit of width a using light of wavelength lambda

Answer 1

The angle of the first order minima can be explain by the Image given below.

let, the single slit width as a.

Now let's imagine it separated into two  equal parts.

According to the Huygen''s construction, we think about a point on the very upper part of the single slit, and the one more point which is below it with the distance of a/2,

Think about parallel rays (as the screen is distant ) from both these points present on slit, at angle θ to the axis of symmetry. ( In diffraction demonstration, the width of the given single slit is approx 10 µm , while the distance to the visual display unit (screen) is approx 1 m.)

The ray has to move an additional distance which is coming from the point at the distance of a/2 below  (a sin θ/2). suppose, this distance is 1/2 λ (where,  λ = wavelength)

, i.e. if

a sin θ  =  λ

then they mess with in destructive manner as they are π/2 phase offset.

Now, for all the point, a point on the very upper part of the single slit, one more point which is below it with the distance of a/2 and, at the angle that satisfies the condition a sin θ  =  λ, they all mess up in destructive manner.

therefore, the first minimum possess sin θ  =  λ/a, while sin θ  = –λ/a is also a minimum on the other side of axisymmetry .

These two minima restrict or limit the broad central diffraction maximum.