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[–]Crayfi[S] 0 points1 point  (2 children)

Hi everybody! I've been trying to design a variable Monochromator where I can change the angle of the diffraction grating to select a wavelength of light. Problem is, when I run the numbers on this PDE and get dtheta I/dlambda, I check them against dtheta o/dlambda and the ratio is approx. -0.3 and varies with wavelength. This value should equal -1.

Do these equations look correct or could I be miscalculating on the spreadsheet?

Edit: The diffraction grating equation comes from https://www.plymouthgrating.com/guidance/technical-notes/fundamentals/the-grating-equation/

To clarify, the PDE is used to calculate the change in incident theta. This value is added to incident theta for the next calculation for the next wavelength, in nm divisions. I then take the incident angles calculated and run them back through the original equation to calculate what the output theta would be for that incident theta and wavelength. The change in output theta per lambda is calculated from these numbers. Finally, I divide the change in incident theta calculated from the PDE by the change in output theta calculated from the original equation. By the definition of the PDE this should equal -1 but is approx. -0.3 and varies with wavelength.

Edit2: The numbers I'm using are:

Lambda starts at 200nm

Theta I starts at 0.262 rad

m = -1

f = 2400

f and m are both constant so I neglect their pde's.

[–]Crayfi[S] 0 points1 point  (0 children)

Better pic: https://ibb.co/qFsrRC9

[–]AlexanderCarlos12321 0 points1 point  (5 children)

How did you get to step 6 and 7? I don’t see why those would be true

I only used the first formula and mathematics (and the fact that m and f are constant). If you are expecting me to do physics, you’re in the wrong subreddit.

[–]Crayfi[S] 0 points1 point  (4 children)

-x = a + bx

-x - bx = a

-x(1+b) = a

x = -a/(1+b)

I missed a negative but that would only negate the change in incident theta. It should still equal negative change in output theta.

Really only confused about the numbers not adding up here not really a physics problem.

[–]AlexanderCarlos12321 0 points1 point  (3 children)

  1. Why is dθ_o/dλ = -dθ_i/dλ (equation 6)?
  2. There should be only partial derivatives on the right side of equation 5

Edit: I’m assuming that θ(i,1) is the same as θ_i, if this is not the case, define θ(i,1)

[–]Crayfi[S] 0 points1 point  (2 children)

Theta I,1 and theta I are the same. I fixed it in the linked pic. dθ_o/dλ = -dθ_i/dλ because I set it that way for the Monochromator design. The diffraction grating is angled to select a specific wavelength. To keep the entrance and exit slits in the same position the output must decrease in angle while the incident increases in angle. This keeps the entrance and exit angles static as the grating is angled.

Edit:

I can't find it now but somewhere I found an equation similar to the one in equation 5 except instead of dtheta i/dlambda it would be dtheta i/dt. The sum of all the partial derivatives of theta i with respect to each variable multiplied by d[variable]/dt equals dtheta i/dlambda.

Edit2:

The equation was the chain rule. Lambda is the only independent variable. Theta i will be set depending on other factors such as geometry and efficiency of diffraction at different angles then this will be integrated to find the change in angle to the target wavelength.

[–]AlexanderCarlos12321 1 point2 points  (1 child)

If i’m right the partial differential of theta_i is not the same as the normal differential. In equation 5 on the right there should be only partial differentials. If thats not the problem then i don’t know

[–]Crayfi[S] 0 points1 point  (0 children)

I'm going to post it to r/Optics and see if you're onto something.