Resonances

Waves are periodic both in time and space. The oscillations of the E-M field propagate repeating themselves. Unexpected, interesting effects arise when waves meet and interact with periodic structures. Of course, in order for two periodicities to “notice” / “feel” each other, they must be proportionate.

Let's first consider the ideal case: a plane wave (one constant wavelength) is incident perpendicularly on a flat grating with an infinitely large number of infinitely long transparent/opaque slits/bars. Schematically it looks like this:

Let's consider some direction (the angle α from the normal to the grating). Let's denote the pitch of the grating (the width of one slit + the width of one bar) as D, then the difference in the path of two rays (waves) from adjacent slits in this direction will be equal to

If the path difference is equal to an integer number of wavelengths, then all waves (from all slits) will arrive at point N in phase, amplifying each other.

If the path difference is equal to an odd number of half wavelengths, then the waves from neighboring slits will "cancel" each other. And it will be dark at the corresponding point in the focal plane of the lens.

But what happens if the path difference is not equal to an integer number of wavelengths? Since we are considering an infinite grating and an infinitely wide incident wave, then for each chosen slit in the grating there always will be another slit (not necessarily adjacent), such that the path difference will be an odd number of half wavelengths. And - again - at the corresponding point in the focal plane of the lens it will be dark.

Thus, under such ideal conditions (infinite grating, infinitely wide incident wave) in the focal plane of the lens we would get a series of infinitely bright, infinitely thin line-peaks of intensity separated by completely dark areas (due to mutual canceling of secondary waves). These peaks correspond to directions with path differences of 0, 1, 2, etc. wavelengths (exactly).

Mathematically, this looks as the sum of the delta-functions of the light intensity.

Under real conditions (finite grating dimensions and incident wave widths), the peaks of intensity will be finite. A small deviation of the angle (and path difference) will lead to only a partial decrease in illumination at the corresponding points ... That is, the picture is “smoothed out”, the light intensity maximums become wider. However, the picture remains amazing (for us, used to see geometric optics in common life 😉).

Let's take a more realistic case. If white light (a set of different wavelengths) falls on a diffraction grating, then (after the lens) the peaks for the different wavelengths will be shifted relative to each other. And rainbows will appear in the focal plane of the lens. Yeah, the peaks for different wavelengths are at different angles α (except for direct transmission at α=0 ). A small difference in wavelength will result in a small displacement of the focal plane, a large difference will result in a large displacement.

And again: this effect can be explained only from the standpoint of wave optics.