9.4.3 Resolvance of Diffraction Gratings

• In order to know if a diffraction grating is able to resolve two wavelengths, the resolving power of the diffraction grating must be found
  • This is based on the Rayleigh criterion as applied to diffraction gratings and their output
• The resolving power, R, of a diffraction grating is given by:

• Where:
  • R = the resolving power of the grating (no unit)
  • λ = the wavelength of incident light (m)
  • Δλ = the smallest difference in wavelength that the grating can resolve (m)

• The resolving power is also equal to 

R = N × m

• Where:
  • N = the total number of slits on the diffraction grating (or those illuminated by an incident beam of light)
  • m = the order of diffraction
• Therefore, combining the two equations gives: 

• • Order of diffraction, m = 3
  • Wavelength, λ₁ = 164.99 nm
  • Wavelength, λ₂ = 165.20 nm
  • Beam width = 0.25 mm

• • The value of the incident wavelength, λ, can be determined from the mean of the two wavelengths:

= 165.095 nm 

• • The difference between the two wavelengths, Δλ, is:

Δλ = λ₂ − λ₁ = 165.20 − 164.99 = 0.21 nm

• • The resolving power can be calculated using:

Step 4: Input the relevant values

= 786.2

• • The equation relating resolving power and number of slits is given by:

R = N × m

• • Rearranging for N and substituting the values for R and m:

= 262.1

• • The value 262.1 is the number of lines illuminated for the beam of 0.25 mm

• • Since 262.1 is the number of lines illuminated for the beam of 0.25 mm, then 4 times more lines must be illuminated to account for 1 mm:

262.1 × 4 = 1048.4 lines per mm

State 7: State the final answer

• • The minimum amount of lines needed per mm to resolve the relevant calcium lines is: 1049 lines per mm