Question:
A laser beam, with a wavelength of 532 nm, is incident on a diffraction grating with a slit spacing of 1 µm. The laser beam enters the grating at an angle of 30° with respect to the normal. The diffraction pattern is observed on a screen located 2 m away from the grating.
a) Calculate the angular separation between the first-order and second-order maxima in the resulting diffraction pattern.
b) Determine the distance between the second-order maxima on the screen.
c) Explain why increasing the number of slits in the grating would affect the width of the diffraction pattern.
Answer:
a) The angular separation between the first-order and second-order maxima can be calculated using the formula:
θ = λ / d
where θ is the angular separation, λ is the wavelength of the laser beam, and d is the slit spacing of the grating.
Substituting the given values:
θ = (532 × 10⁻⁹ m) / (1 × 10⁻⁶ m) = 0.532°
Therefore, the angular separation between the first-order and second-order maxima is 0.532°.
b) The distance between two consecutive maxima can be calculated using the formula:
y = L tan(θ)
where y is the distance between two consecutive maxima, L is the distance between the grating and the screen, and θ is the angular separation between the maxima.
Substituting the given values:
y = (2 m) × tan(0.532 × (π/180) radians) ≈ 0.036 m
Therefore, the distance between the second-order maxima on the screen is approximately 0.036 m.
c) Increasing the number of slits in the grating would lead to a narrower diffraction pattern. This is because as the number of slits increase, the constructive interference becomes more pronounced and the destructive interference is reduced. As a result, the maxima become sharper and narrower, resulting in a narrower diffraction pattern.
Furthermore, according to the formula for the angular separation,
θ = λ / d
the larger the value of d (slit spacing), the smaller the angle θ. Hence, increasing the number of slits (decreasing d) would result in a larger θ and a narrower diffraction pattern.