[Bored chemist]

The properties concerned are energy and momentum.

Incoming photon transfers energy and momentum to an electron. Why does the electron then emit a photon with the same energy, same momentum vector parallel to the surface, but reversed momentum vector perpendicular to the surface? If the temperature is above absolute zero, the electrons are all jiggling about and quite capable of transferring energy and momentum to the mass of the mirror.

Balls
Balls also obey the same conservation rules in  spite of being able to transfer momentum + energy.
Yet the classical physics explanation says that i=r.

In effect you get a pair of simultaneous equations in velocity and angle and you have to satisfy the conditions that there's no breach of the conservation laws.
The only solution is  i=r.

[hamdani yusuf]

A simplified model could view it as a result of Lenz's Law.
- An electromagnetic wave approaching a metallic reflecting surface induces a current in the metal which opposes the external magnetic field. If it is a good reflector, the opposing field will be equal and opposite to the incoming wave.
- The incoming wave cancels the "mirror image" wave in the metal, so the incoming wave does not propagate into the metal surface.
- But the external wave and the wave in the metal surface does have constructive interference, producing an outgoing wave which is a mirror-image of the incoming wave, and the angle of incidence = the angle of reflection.

See: https://en.wikipedia.org/wiki/Lenz%27s_law

Of course, this is only part of the story, since non-conductive surfaces (eg glass) can also act as partial mirrors if they are in a medium with a very different speed of light.

I just want to add that the molecules of dielectric media can act as antenna array/lattice that partially scatter incoming wave to all directions similar to radiation pattern of dipole antennae. The remaining wave not completely canceled by first layer will be scattered by the next layers,  and so on. It is easier to demonstrate the mechanism using longer wavelength than visible light, such as microwave and radio wave,  since mechanical structure of the antenna array can be easily observed and manipulated at will.
In this video you can learn how an antenna array can control direction of transmitted em wave.

[alancalverd]

It's a very clear exposition of the propagation and interference of a monochromatic, coherent em wave from a structure whose periodicity matches that of the wave, but

(a) it isn't clear how this would apply to an incoherent, broad spectrum incident on a periodic structure (say polished metal) whose periodicity is a thousandth of the wavelength of the beam, or an aperiodic structure (glass, liquid metal)

(b) in describing microwave dishes, it assumes that we know that i = r without explaining why!   

[Bored chemist]

It's a very clear exposition of the propagation and interference of a monochromatic, coherent em wave from a structure whose periodicity matches that of the wave, but

(a) it isn't clear how this would apply to an incoherent, broad spectrum incident on a periodic structure (say polished metal) whose periodicity is a thousandth of the wavelength of the beam, or an aperiodic structure (glass, liquid metal)

(b) in describing microwave dishes, it assumes that we know that i = r without explaining why!   

Thank heavens it was an addition, rather than a grass roots explanation.

[hamdani yusuf]

It's a very clear exposition of the propagation and interference of a monochromatic, coherent em wave from a structure whose periodicity matches that of the wave, but

(a) it isn't clear how this would apply to an incoherent, broad spectrum incident on a periodic structure (say polished metal) whose periodicity is a thousandth of the wavelength of the beam, or an aperiodic structure (glass, liquid metal)

(b) in describing microwave dishes, it assumes that we know that i = r without explaining why!   

as long as the periodicity is much less than wavelengths, we'll get most of reflected light have same angle as incident light, and very few will be scattered in other directions. So it doesn't make much difference if the wavelength is 10 times or 100 times the periodicity of the structure.

As explained in the video,  the direction of the radio wave from antenna array is determined by the phase difference among individual antenna. It just happen that in case of reflection, with phase difference created by incoming wave, the direction of the reflected wave will have same angle as incoming wave.

[alancalverd]

Again, you have stated the experimental observation, but not the mechanism of reflection.

[hamdani yusuf]

The mechanism of reflection has already been answered by evan's post that I've quoted. The video has shown that phase difference among antenna array's elements determine direction of wave propagation. My previous answer said that in specular reflection, angle of incoming wave determines the phase difference among antenna array's elements, which in turn determines the angle of reflection. Which part of those isn't clear yet?

[alancalverd]

The fact that a mirror is not a static regular array of dipole elements with spacing of the order of λ/2, but a sea of delocalised electrons.

[hamdani yusuf]

The fact that a mirror is not a static regular array of dipole elements with spacing of the order of λ/2, but a sea of delocalised electrons.

The mechanism I described above is not limited for a static regular array of dipole elements with spacing of the order of λ/2.The spacing can be much smaller than that. You can still reflect 30 mm microwave using a grid with spacing in the order of microns.
Regular mirrors consist of a glass plate coated with metallic layer on the rear side. Most of reflected light is produced by the metal. The irregularities of the glass molecules as well as metallic layer do scatter the light, but as long as its size is much smaller than the wavelength and random, the effect is negligible since they tend to cancel each other.

[alancalverd]

Again you are stating the phenomenon but not the mechanism. The reflection of radio waves by re-radiation and interference from a dipole array was well explained, but it required and presumed the array to be of similar dimensions and periodicity to the incoming wavelength. The maths simply doesn't work if the array doesn't meet those conditions, but the phenomenon is obvious.

The question remains: what is the mechanism of reflection of em radiation from a nonperiodic surface? The intriguing answer seems to be that although the interaction is a quantum phenomenon, it can only be described by a continuous wave equation.

[hamdani yusuf]

The maths simply doesn't work if the array doesn't meet those conditions, but the phenomenon is obvious.

Can you show how it doesn't work? What does your math predict?

[alancalverd]

The interference pattern of a dipole array depends on the phase relationship of adjacent dipoles, as shown in your excellent RCAF clip.   If the array periodicity is λ/2 you get constructive interference and maximum lobes perpendicular to the plane of the array. If you place your reflector dipoles λ/4 behind the array, you get unidirectional transmission. But if the phase relationships are not exactly as stated you get wobbly lobes and partial reflection.

In an optical mirror you don't have dipoles or a regular array of anything commensurate with λ, so you need another mechanism to explain optical reflection.

[Bored chemist]

The interference pattern of a dipole array depends on the phase relationship of adjacent dipoles, as shown in your excellent RCAF clip.   If the array periodicity is λ/2 you get constructive interference and maximum lobes perpendicular to the plane of the array. If you place your reflector dipoles λ/4 behind the array, you get unidirectional transmission. But if the phase relationships are not exactly as stated you get wobbly lobes and partial reflection.

In an optical mirror you don't have dipoles or a regular array of anything commensurate with λ, so you need another mechanism to explain optical reflection.

Just out of curiosity, what happens as you move the elements closer together?
What pattern would you get from an extended array of conductors at lambda/10 or lambda/100?

[alancalverd]

Which part of those isn't clear yet?

The absence of dipoles.

[Bored chemist]

Which part of those isn't clear yet?

The absence of dipoles.

Well, while you wait, you could look at this

Just out of curiosity, what happens as you move the elements closer together?
What pattern would you get from an extended array of conductors at lambda/10 or lambda/100?

[evan_au]

lambda/10 or lambda/100?

Waves tend to ignore individual structures much less than λ/4, and go around them.
A series of structures much less than λ/4 apart, appears much like a continuous surface.

[Bored chemist]

lambda/10 or lambda/100?

Waves tend to ignore individual structures much less than λ/4, and go around them.
A series of structures much less than λ/4 apart, appears much like a continuous surface.

Yes.
I know that, and that's 'why I asked Alan to do the calculation for " extended array of conductors at lambda/10 or lambda/100".
It will show that the angle of incidence is equal to... of you know the rest.
At which point he has to accept that  hamdani yusuf's explanation is valid.

[alancalverd]

I refer the hon gent to the second paragraph of reply #19 above.