# Why do reflection gratings work?

The law of reflection (that the angles of incidence and reflection are equal) can be derived directly from Maxwell's equations, or from Fermat's principle. However, reflection gratings completely defy this law, and from light incident at a fixed angle comes a whole diffraction pattern - light reflected at a range of different angles.

My teachers and textbooks tend to gloss over this point. Why do reflection gratings work, and why is there not a contradiction with the law of reflection?

• Do you understand how transmission gratings work? Commented Jun 24, 2020 at 16:24
• The grating structure breaks the translational symmetry in the plane of the grating just as a transmission grating (or even simpler a 1- or 2-slit pattern) breaks the symmetry. Both impose additional constraints on the solutions to Maxwell's equations leading to diffraction. Commented Jun 24, 2020 at 16:27
• Yep I get transmission gratings. I suppose one can write off this question by asking "why does diffraction happen in the first place?" but in the case of reflection I would find that answer unsatisfactory because we have a law derived from first principles that tells us that when light hits a mirror it should reflect at the angle of incidence, yet Maxwell's equations seem to be defied in this case. Commented Jun 24, 2020 at 16:27
• Could you explain your comment a bit more @JonCuster? Commented Jun 24, 2020 at 16:31
• What needs clarification? The fact that the grating has structure should impact solutions to Maxwell's equations, right? So, why insist that reflection from a not-smooth surface should be the same as reflection from a smooth surface? Commented Jun 24, 2020 at 16:35