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I understand how to derive the formula for diffraction gratings as you just have to compare the light rays approaching at a maxima point. The formula for diffraction grating formula is shown below : $$d \sin{\theta} = mλ$$ where $d =$ distance between slits/slit separation

From my understanding, the double-slit formula is derived from small-angle approximations but we can't assume small angles for a diffraction grating as there is a lot more interference and diffraction. Formula for double slits is shown below : $$w = \frac{mλD}s$$ Where $w =$ distance between fringes, $m =$ fringe order. $D =$ distance between slits, and screen $s =$ distance between slits/slit separation.

What I can't seem to do is visualize this scenario as I do not fully understand why the angle, $\theta$ between the centre and a maxima point cannot be small for a diffraction grating as well? And so small angle approximations can't be used?

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  • $\begingroup$ If you show the 2 formulae, the question might be clearer. $\endgroup$
    – JEB
    May 21, 2020 at 2:18
  • $\begingroup$ It's should be the same formula. What is $D$? What is $s$? What is $w$? Looks like the Airy pattern formula. $\endgroup$
    – SuperCiocia ♦
    May 21, 2020 at 4:00
  • $\begingroup$ Oops sorry I will make it clearer :) $\endgroup$
    – KSP
    May 21, 2020 at 4:03
  • $\begingroup$ The principle maxima from the grating fall precisely at the directions where the two-slit pattern gives a maximum. Both are given by $d \sin \theta = n \lambda$ where $\theta$ is the angle from the normal (and I assumed normal illumination). $\endgroup$
    – Andrew Steane
    May 21, 2020 at 9:03

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Writing your second equation using the same symbols of the first equation $(s\rightarrow d)$: $$ w = \frac{m \lambda D}{d}.$$

Then, from geometry, $w = D \tan\Theta$, so you end up with: $$ d\tan\Theta = m\lambda $$ for the double slit diffraction pattern. Which, as you are saying, is different from the general diffraction grating formula: $$ d\sin\Theta = m\lambda.$$

You can see that these two are the same in the small angle approximation, because for small $\Theta$ you have $\sin\Theta \approx \Theta \approx \tan\Theta$.

The reason for the small angle is that the diffraction pattern satisfying the grating equation is the far-field (Fraunhofer) diffraction pattern. For which you would need the screen to be very far away from the slits/grating. Alternatively, you can have a lens, that brings the far-field pattern at its focal length $f$.

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  • $\begingroup$ Careful: the angle theta here is defined differently in your two cases. In fact the two-slit and the grating cases are identical. That is, the principle maxima from the grating fall precisely at the directions where the two-slit pattern gives a maximum. $\endgroup$
    – Andrew Steane
    May 21, 2020 at 9:02
  • $\begingroup$ Yes both gratings give you the maxima at the same angle $\Theta$. My point here was to reconcile the spatial separation $w$ on the screen. $\endgroup$
    – SuperCiocia ♦
    May 21, 2020 at 19:45
  • $\begingroup$ So are you saying the double-slit formula and diffraction grating is the same? $\endgroup$
    – KSP
    May 23, 2020 at 4:16
  • $\begingroup$ Yes the double slit is just a special case of the grating. The equations for the angular separation are the same. The difference here is because the second equation is for $w$ which is the spatial separation on the screen. $\endgroup$
    – SuperCiocia ♦
    May 23, 2020 at 5:03

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