How Do Diffraction Gratings Affect Light Wavelengths?

[wolfson_1123]

Would appreciate anyhelp to solving any of these questions ty.

1 A diffraction grating of pitch 3 .27mm is illuminated at normal incidence with
light comprised of various wavelengths. If components of wavelength
495nm, 625nm and 990nm are present find:-
a) the angular positions of the first order maxima for these 3 wavelengths,
b) the angular positions of the second order maxima ( for all 3 ),and
c) the number of possible orders for each wavelength.

2 A diffraction grating has 600 lines per mm. If white light contains visible
wavelengths in the range 400nm to 700 nm, calculate the total angular
dispersion of the visible spectrum in 1st and 2nd order.

3 A transmission grating of pitch 1.75mm is illuminated by a collimated beam
of red light of wavelength 625nm. The beam is incident at at angle of 30o
with respect to the normal. At what angles would you expect to see the
zeroth order and the two first order maxima. Find by trial and error the
number of orders possible both sides of the zeroth order.

There were 8 questions and these are the ones i am stuck on tyty.

 

[wolfson_1123]

Managed to do questions 1 and 2, just need some help on q 3.

Thank you in advance

Wolfson.

 

[wais]

1. To find the angular position of the first order maxima, we can use the formula for diffraction grating: dsinθ = mλ, where d is the pitch of the grating, θ is the angle of diffraction, m is the order of the maxima, and λ is the wavelength of light.

a) For 495nm: θ = sin^-1(495x10^-9/3.27x10^-3) = 0.09 degrees
For 625nm: θ = sin^-1(625x10^-9/3.27x10^-3) = 0.11 degrees
For 990nm: θ = sin^-1(990x10^-9/3.27x10^-3) = 0.18 degrees

b) For 495nm: θ = sin^-1(495x10^-9/3.27x10^-3) = 0.18 degrees
For 625nm: θ = sin^-1(625x10^-9/3.27x10^-3) = 0.22 degrees
For 990nm: θ = sin^-1(990x10^-9/3.27x10^-3) = 0.36 degrees

c) The number of possible orders for each wavelength can be found by rearranging the formula: m = dsinθ/λ.
For 495nm: m = (3.27x10^-3)(sin0.09 degrees)/495x10^-9 = 1. For this wavelength, there is only one possible order.
For 625nm: m = (3.27x10^-3)(sin0.11 degrees)/625x10^-9 = 1. For this wavelength, there is only one possible order.
For 990nm: m = (3.27x10^-3)(sin0.18 degrees)/990x10^-9 = 1. For this wavelength, there is only one possible order.

2. The total angular dispersion can be found by subtracting the angles of the first and last order maxima.
For the first order: θ1 = sin^-1(λ/d) = sin^-1(400x10^-9/600x10^-3) = 0.38 degrees
θ2 = sin^-1(700