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What we often refer to as Snell's law:

$$n_1 \sin(\theta_1) - n_2 \sin(\theta_2) = 0$$

has quite a bit of history behind it. It can be demonstrated in several ways, one of which is by asserting that along the boundary there is no discontinuity in phase from one side to the other, or if there is, that it's constant.

What I usually call "the grating equation"

$$n_1 \sin(\theta_1) - n_2 \sin(\theta_2) = \frac{m \lambda}{d}$$

where $m$ is the integer order and $\lambda$ and $d$ are the wavelength and periodic grating spacing can be demonstrated by asserting that at a periodic array of points spaced $d$ apart there is no discontinuity in phase, but what happens between those points is now unconstrained. That's how multiple non-zero orders are now possible.

Question: Proper, periodic diffraction gratings are much more recent items than glass lenses. Where does the grating equations come from? Who first wrote it, did it it precede actual diffraction grating experiments, and does it have another name?

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Your two equations are very different in fact: Snell's law can be derived using geometric optics, while the grating equation requires wave theory of light. The grating (as an optical instrument) was invented by Joseph von Fraunhofer in 1823, when the wave theory was already available but not universally recognized. Apparently the equation is due to him as well.

Source: S. Sternberg, A history of 19th century spectroscopy, Appendix F to his book Group Theory and Physics, Cambridge UP, 1994.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – HDE 226868
    Jan 18, 2022 at 13:59
  • $\begingroup$ I see all comments have been moved to chat. I've added a bounty as I'm still looking for something that cites the actual first instance of grating equation as we currently recognize it. If you can locate that it will be great! I disagree that the two equations are different; Snell's law is just m=0 transmission diffraction which gets all the light when the strength of the grating goes to zero. It's certainly true that they originated in different contexts - their histories are different. $\endgroup$
    – uhoh
    Mar 27, 2022 at 19:37

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