Green light of 540 nm is diffracted by a grating of 200lines/mm. calculate the angular deviation of 3rd order image. is 10th order possible?
To calculate the angular deviation of the 3rd order image, we can use the equation for angular deviation caused by a diffraction grating:
θ = (m * λ) / d
Where:
θ is the angular deviation,
m is the order of the image,
λ is the wavelength of light,
and d is the spacing between the grating lines.
Given:
Wavelength λ = 540 nm (or 540 x 10^(-9) meters),
Grating spacing d = 1 / 200 lines/mm (or 1 / (200 x 10^6) meters).
Now we can substitute the values into the equation:
θ = (3 * 540 x 10^(-9)) / (1 / (200 x 10^6))
θ = (3 * 540 x 10^(-9)) * (200 x 10^6)
θ = 3 * 540 x (200 x 10^(-9)) * 10^6
θ = 3 * 540 x 200 x 10^(-3)
θ = 3 * 108000 x 10^(-3)
θ = 324 x 10^(-3)
θ = 0.324 radians
So, the angular deviation of the 3rd order image is approximately 0.324 radians.
Now, to determine if the 10th order is possible, we need to check if the angular deviation exceeds the limit of the grating.
The maximum order that can be observed is given by the equation:
m_max = d * sin(θ_max) / λ
Where:
m_max is the maximum observed order,
d is the spacing between grating lines,
θ_max is the maximum allowable angular deviation (for a specific diffraction grating),
and λ is the wavelength of light.
Since we know the grating spacing (d) and the wavelength (λ), we need to find the maximum allowable angular deviation (θ_max).
For a grating of N lines per mm, the formula for the maximum allowable angular deviation is:
sin(θ_max) ≈ m_max * λ / (N * 1000)
Given that N = 200 lines/mm:
sin(θ_max) ≈ 10 * λ / (200 * 1000)
sin(θ_max) ≈ λ / (200 * 10000) [simplifying]
sin(θ_max) ≈ 540 x 10^(-9) / (200 x 10^4) [substituting λ = 540 nm]
sin(θ_max) ≈ 27 x 10^(-9) / (10 x 10^3)
sin(θ_max) ≈ 2.7 x 10^(-12)
Now, we can plug the value of sin(θ_max) into the equation for m_max:
m_max = d * sin(θ_max) / λ
m_max = (1 / (200 x 10^6)) * (2.7 x 10^(-12)) / (540 x 10^(-9))
m_max = 2.7 x 10^(-12) / (200 x 540)
m_max = 2.7 x 10^(-12) / 108000
m_max ≈ 2.5 x 10^(-17)
Since m_max is very small (close to zero), it means that the 10th order is not possible, as it is beyond the maximum order that can be observed for this grating.
Therefore, the angular deviation of the 3rd order image is approximately 0.324 radians, and the 10th order is not possible.