To answer your question as to how a half-silvered mirror works, first it might be a good idea to discuss how a full mirror works.
Classically speaking, the silver in the mirror has electrons that can freely move around. The electromagnetic fields of the incoming light accelerate the charged electrons in the silver, inducing electric current.
The currents flowing in the silver create their own electric fields, which by Lenz’s law, cancel out the electric field inside the silver, and in doing so, send an oppositely-shaped wave back out into the void (the reflected wave).
Because silver is not a perfect conductor of electricity, the topmost layer of silver does not completely cancel these fields, and so the light can actually penetrate a small distance into the metal (typically nanometers) before it’s finally converted into electric current.
If the silver coating is very thin, thinner than the penetration depth, then the component of the light wave that has penetrated through the metal will escape out the other side and keep going. That is, the resistance of the thin silver is high enough that the induced current doesn’t completely cancel out the electric fields of the photon.
This classical explanation is also the same as the quantum one.
They also make beamsplitters that are like you describe—I think they call them “Polka dot beamsplitters”. I don’t remember what they’re used for. They would work the same way, but if you have a focused laser beam, the beam spot would be so small that it would either hit a full-mirrored section or a transparent section, and not both. You would need to use a lens on both sides of the beam splitter to spread the beam out to encompass the whole beamsplitter, and then gather it back. I think as long as the polka dots are not on the same scale as the wavelength, it wouldn’t cause a problem.
To answer your question as to how a half-silvered mirror works, first it might be a good idea to discuss how a full mirror works.
Classically speaking, the silver in the mirror has electrons that can freely move around. The electromagnetic fields of the incoming light accelerate the charged electrons in the silver, inducing electric current.
The currents flowing in the silver create their own electric fields, which by Lenz’s law, cancel out the electric field inside the silver, and in doing so, send an oppositely-shaped wave back out into the void (the reflected wave).
Because silver is not a perfect conductor of electricity, the topmost layer of silver does not completely cancel these fields, and so the light can actually penetrate a small distance into the metal (typically nanometers) before it’s finally converted into electric current.
If the silver coating is very thin, thinner than the penetration depth, then the component of the light wave that has penetrated through the metal will escape out the other side and keep going. That is, the resistance of the thin silver is high enough that the induced current doesn’t completely cancel out the electric fields of the photon.
This classical explanation is also the same as the quantum one.
They also make beamsplitters that are like you describe—I think they call them “Polka dot beamsplitters”. I don’t remember what they’re used for. They would work the same way, but if you have a focused laser beam, the beam spot would be so small that it would either hit a full-mirrored section or a transparent section, and not both. You would need to use a lens on both sides of the beam splitter to spread the beam out to encompass the whole beamsplitter, and then gather it back. I think as long as the polka dots are not on the same scale as the wavelength, it wouldn’t cause a problem.