Hi Barry,

I think the point here is that in "classical" electrodynamics, the energy of a wave is neatly distributed to the halfwaves but quantum mechanics tells us that each energy quantum is spread over many wavelengths.

So, a photon can "try" several paths, and the "decision" about the final path can be delayed for the time t = length of photon / speed of light. During this time, each path which proves to be a dead end street is removed from the "list of possible paths". In the present phenomenon, only the path through the glass remains on the "list" and is chosen with probability p = 1 instead of, e. g., p = 0.7 without coating.

The mechanism behind the double slit experiment is the same (photon "tries" all possible paths) with the difference that the diffraction at the edges of the slits generates an uncountable set of paths (with more than one remaining possible), in contrast to the three paths in the coating case.

BTW, without quantum mechanics and the odd behavior of photons, the transparency should be unaffected by the coating but this would confront us with a serious problem because the energy of both reflected waves cannot just disappear during destructive interference.

Joerg Fricke Thanks for the discussion of this; it's a tough one where you have to invoke the probability function collapsing to an observed view to provide an explanation. Mind boggling.

Alternatively, we're living in a simulation! ;-)

Look into simple view: The Snell–Descartes law. Changing the index of refraction changes the reflectivity and the transmission through.

I really had fun doing my master thesis on Interference Polarizing Filters: Building quarter wavelength mirrors on each side of the the middle layer that was 1/2 wavelength layer. Building my own Michelson Interferometer to measure the layer thickness of every layer.

Yes, this due, presumably, to the change in velocity of light in different media, another very interesting phenomena. I had thought about this for a method to deliver information faster than light speed (in vacuo), by passing light through helium, it is slightly slowed (c/1.000036). But if you then move the helium, the light transmission speed increases, 'carried' by the speed of the flow. You'd have to get past 10,800 metres per second, though :-)

Dear Barry,

This is the multi-layer coatings which combine films of different refractive indexes and thicknesses with a quarter wave thickness, which is used in camera lenses and range finders to reduce their reflectance in some specific wavelength. The reflectance R of a coated surface is given by, R= (ns-nf/ns+nf)2

A single layer coating is effective and inexpensive to produce. This coating with a quarter wave thickness is used in camera lenses and range finders to reduce their reflectance in some specific wavelength. This coating is known as quarter wavelength anti-reflection coating or quarter wavelength impedance transducer.

Theory

The intensity of the net reflected light depends on the optical thickness of the film. Optical thickness t of a film is given by,

t=nd

Where n is the refractive index and d is the physical thickness of the film. If the optical thickness of the film is a quarter waves, the phase difference of the two reflected waves is pai. They become out of phase and interfere destructively. Then, the net intensity of the reflected wave is at minimum. If the optical thickness of the film is a half wave, the phase difference of the two reflected waves is 2 pai. They become in-phase and interfere constructively. Then, the net intensity of the reflected wave is at maximum. Here, they all obtain the uniform optical thickness of a specific quarter wavelength. At that specific wavelength, maximum reflectance is obtained. This maximum reflectance is bracketed by a region called .

Refer the site:

http://www.u.arizona.edu/~waisze/report.html

Ashish

Thanks Ashish. I appreciate the conventional explanation. The issue is, though, not the reflected light interfering destructively, but how does that increase the amount of light transmitted? That, I think, can only be addressed by using quantum mechanics.

Hello Barry Nicholson ,

You are not the first, nor will you be the last, person to ask this particular question. I think the first time I saw someone express such honest befuddlement in print was in the following reference book/buyer's guide,

[1] Anonymous; Coatings; in the Encyclopedia section of Teddi C. Laurin (editor); The Optical Industry & Systems Purchasing Directory, Book 2, 27th Edition; Optical Publishing Co., Inc.; 1981; pp. E-12 - E-13.

The author of the entry in the section entitled "Coatings" was, like yourself, bemused about how a MgF2 coating performed its antireflection function. He, of course, provided the reader with the conventionally unhelpful explanation. Let's see if we can do a little bit better.

In analyzing a system such as antireflection coatings most, if not all, the approaches start at the transfer function, which relates input to output. Optical, electrical, and communication engineers all use this approach because it usually gives the right answer with a minimum of work; it's alegebraically efficient. What many people forget, or sometime never knew in the first place, is that this approach is incomplete.

The systems we study are sometimes described by linear DE's (Differential Equations). It does not matter if they are ODE's (Ordinary DE's), PDE's (Partial DE's), or coupled (simultaneous) DE's. The general solution of a linear DE is the sum of the complementary and particular solutions. When you employ a transfer function, or a similar approach, you are only using the particular solution. Usually this is not a problem because the complementary solution is usually a transient solution, i.e., it vanishes at long times. The particular solution can be considered the steady state solution even if it oscillates.

But ignoring the complementary solution can be a big problem in, at least, two situations: 1) when the complementary solution isn't a transient solution, i.e., when it does not disappear at long times (a pole in the right-half of the real-imaginary plane); and 2) when you are trying to understand something like an antireflection coating, when the time sequence does not make any sense, usually, indicated when someone says "How did the wave know how to do that?"

If you are interested in the first case, see my RG URL: Method Pole-Zero Cancellation

But your real interest is in the second case. What is important to understand is that to make sense of the complementary/steady state solution, you need to understand what happened at t=0 when the incident light wave first impinged upon the coated lens. Initially, part of the incident wave was reflected backwards from the outer surface of the antiflection layer, but after the standing wave is established within the antireflecting layer, then the out of phase wave from the interface between the inner surface of the antireflecting layer and the outer surface of the glass cancels the reflected wave from the outer surface of the coating. In this case the complementary/transient solution disappears at long times, but without being aware of it you cannot understand how the waves "know" what to do because there is a time incongruity.

Remember, "Leap, and the net suddenly appears." This is a leap of faith, but when you decide to leap into the quantum abyss, a spider's web suddenly appears, which will ensnare you and ultimately exhausts you to death since no one really understands quantum mechanics. It is a leap of despair because you are trying to explain a mystery, antireflecting coatings, with an enigma, quantum mechanics.

Forget the philosophy, that is your choice, but never forget the backstory, i.e., what happens at t=0 and its immediate, subsequent neighborhood.

Regards,

Tom Cuff

Hi, again, Barry Nicholson ,

I'm not sure you have access to reference [1] mentioned in my previous post so let me just quote from the section where the writer expresses her or his genuine confusion about the antireflecting properties of MgF2. After giving the standard explanation in words, the writer goes on, in the next paragraph, to express their honest confusion: "This is all fine, but is it not logical to expect the transmission through the interface to be reduced by the 2.5% reflected at the first surface? It would seem quite reasonable to expect only 97.5% of the incident energy to be transmitted. However, theory as well as actual measurement tells us that the transmission will be 100% ... But - how do we manage to get that 2.5% of energy back into the lens? (p. E-12)" In this same paragraph, the author mentions that some books invoke the conservation of energy principle, though, the writer is still not convinced as the last statement of the paragraph makes abundantly clear. Rarely has there been such an honest author.

I know that my dig at quantum mechanics in my last post probably appears to be "over the top" as they say, but I am not saying you can't use quantum mechanics, if you want. For example, in optics there is the phenomenon called "total internal reflection", which can be defeated by bringing another glass surface to within a quarter of a wavelength of the first. Such an arrangement results in an evanescent wave conveying energy to the second piece of glass in what is referred to as "frustrated internal reflection". In many ways this phenomenon closely resembles quantum mechanical tunneling. However, it can be explained using Maxwell's equations for EM waves, see,

[2] Max Born, Emil Wolf; Principles of Optics, Sixth Edition; Cambridge University Press; 1980; pp. 47-51.

However, I remember a paper on RG, where the authors offered a quantum mechanical derivation of this same classical phenomenon. If I can find it, I will post its URL to this question. The point being that you can use whatever approach makes you feel comfortable.

For the antireflecting coating, I just happen to believe that, unlike Las Vegas, what happens at t=0 does not necessarily stay at t=0. There, I've said it, and I will bother you no more. Thanks for an interesting and honest question.

Regards,

Tom Cuff

Hi Barry Nicholson ,

Here is the URL I promised: Article Quantum (not frustrated) theory of the total internal reflec...

Regards,

Tom Cuff