# Why is the formula for diffraction gratings not the same as for a double-slit diffraction formula?

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?

• If you show the 2 formulae, the question might be clearer.
– JEB
May 21 '20 at 2:18
• It's should be the same formula. What is $D$? What is $s$? What is $w$? Looks like the Airy pattern formula. May 21 '20 at 4:00
• Oops sorry I will make it clearer :)
– KSP
May 21 '20 at 4:03
• 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). May 21 '20 at 9:03

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 $$f$$.

• 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. May 21 '20 at 9:02
• 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. May 21 '20 at 19:45
• So are you saying the double-slit formula and diffraction grating is the same?
– KSP
May 23 '20 at 4:16
• 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. May 23 '20 at 5:03