8. Diffraction gratings - condition for diffraction fringes (derivation), dispersion of a grating and resolving power of a grating. (3p.)

Question 1

Diffraction

Answer 1

Diffraction is the bending of a wave around an obstacle or through and opening.

Question 2

Diffraction grating

Answer 2

Diffraction grating is an optical component, which splits and diffracts light into several beams traveling in different directions. A large number of parallel, closely spaced slits constitutes a diffraction grating.

Question 3

Dark fringes

Answer 3

The first dark fringe occurs at 𝑑sin 𝜃 = 𝜆. As we move up on the screen, the next dark fringe occurs when the first ray interferes destructively with the one from ¼ the way down the slit.

d/4 * sin 𝜃 = 𝜆
d * sin 𝜃 = 2 𝜆

We can continue to 1/6, 1/8, etc., to get all the dark fringes at:

𝒅𝐬𝐢𝐧 𝜽 = 𝒎𝝀

Question 4

Dispersion

Answer 4

The angular separation ∆𝜃 per unit wavelength interval ∆𝜆 is called the dispersion D of the gratings:

𝑫 = m / (𝒅*𝐜𝐨𝐬 𝜽)

The largest possible dispersion means that we can separate lines of very close wavelengths.

Question 5

Resolving power

Answer 5

Resolving power is the measure of the smallest wavelength difference that can be resolved in a spectrum. It is defined by the following equation:

𝑹 = 𝜆/∆𝜆 = 𝑵𝒎

where ∆𝜆 is the smallest separation between two lines in a spectrum which can be distinguished. The resolution power, like the dispersion, increases with the order number. But unlike the dispersion, R is proportional to the number of lines N whereas is independent of their separation d.