How can visible light penetrate dozens if not hundreds of meters deep when it's skin depth is about a few centimeters? The title says the question. I am wondering how it is possible for electromagnetic radiation with a frequency of 6.66*10^14 Hz, which has a skin depth of less than a millimeter, to penetrate that deep in water. Thank you for your time
 A: the penetration depth argument only holds for EM waves traveling through electrically conductive media like metals. Water is not electrically conductive and behaves instead as a dielectric medium.
A: Adding a little to the Doty answer.
666 THz (450 nm) is near the minimum optical absorption of water. In the ultraviolet at shorter wavelengths there is absorption from the tail of the first electronic transition of water. At longer wavelengths the harmonics of the O-H bonding are causing absorption.
The absorption spectrum of seawater is similar to pure water except there are also ions - and dissolved organic matter, suspended particles etc. At RF frequencies the absorption is significant as these ions can respond to the electric field. So one way to consider how the conductivity behaves is to think of the total conductivity being dependent on the concentration of each ions. Then the plasma frequency of the lightest ion will give the highest frequency where the electromagnetic waves will strongly absorb. In the case of sodium that works out to be somewhere around 250 GHZ so your light at 666 THz (450 nm) is above the cutoff frequency and doesn't induce motion of the ions and so the sea water is better treated as a dielectric.
A: The skin depth model doesn't apply when the conduction is limited by the inertia of the conducting charges rather than by the resistivity of the medium. At optical frequencies, the ions that give seawater its conductivity hardly move in response to the electromagnetic field.
