Experimental alert: Someone may be able to answer this question experimentally simply by going to a shopping mall and finding the right piece of holographic jewelry.
My question is whether the type of front-view "thick film" hologram often seen in jewelry can be used to create a true mirror, one in which you can see an image of yourself reflected in the hologram.
I'm afraid I messed up the terminology quite badly in my first try at this question! The type of hologram I was trying to ask about is correctly called a reflection (or thick film or volume) hologram because of its ability to reflect light back in the direction from which it came. It does that using only photographic emulsion, via a wave exclusion effect similar to the one that gives peacock feathers and opals their bright colors. (For the record, I was originally thinking, quite incorrectly, that "reflection holograms" meant the ones that use holes in a smooth metallic surface, such as those seen on almost any credit card. Now I'm unsure what those ones are called.)
At least one non-SE site correctly points out that you cannot create a holographic mirror by using a transmission hologram, which is the kind where the light source passes through a film and you view it from the other side. (Again, I was using this term incorrectly in my first version of this question.) That's not a very deep answer, however, since light from the viewer plays no role at all in what is being seen in a transmission hologram. You can't reflect something whose light plays no role in the image being shown!
So, my question is this: Can the wave exclusion effect that is used in volume holograms, which lack any true metallic surfaces despite their shiny appearance, be configured in a way that makes it possible to reflect true images of objects in front of the hologram?
I suspect it's possible for two reasons: (1) Somewhere I still have a thick-film hologram of the insides of a watch. I clearly recall watching a bright spot move left and right on the image of a curved metal part within the watch as I moved a background light move left and right in front of the hologram. While a moving bright spot is hardly a complete image, it does indicate that a hologram is capable of a visible response to an object not in the original hologram. (2) I'm not aware of anything from diffraction theory that says you cannot use multilayer diffraction patterns to create simple, mirror-like reflecting surfaces. (Nor am I aware of anything that says you can for sure, either.)
So, experimenters: Does anyone out there have a hologram in hand that seems to reflect non-trivial light patterns?
And theorists: Regardless of how one would create it in the lab, is it mathematically possible to create multilayer diffraction patterns that, like metallic mirrors, would reflect light in a way that depends on the incident angle of the light?
Best Answer
After appending my 2019-04-30 update to my much older answer, the above excellent, to-the-point and from-the-trenches expert answer came in. I immediately changed the designated answer from mine to the new one. There's probably still some fun reading below in my old reply and update, though... :)
Alas, I must answer my own question: I found a very explicit example online description of someone who created a thick-film transmission hologram of a convex mirror. She (or he) describes seeing her own face clearly, even if only in monochrome. So, if I accept this description at face value, it clearly is possible to create a realistic mirror using only wave-exclusion diffraction effects. Cool!
Also, I am amused (or is it chagrined?) that this reminded me of the importance of reading long articles all the way to the end, even if you feel you already got the point. This description of an actual holographic mirror was hidden at the very end of the long posting on I mentioned in my question about how transmission holograms cannot form mirrors.
2019-04-30 Update
As noted in the comments below, the above link to an explicit description of a holographic mirror unfortunately is no longer available, not even in Internet archives.
However, this draft book chapter PDF on reflection using Denisyuk transmission holograms seems to provide pretty good coverage of the issues.
Still, as I get older I find I like finding the simplest possible explanations of things. The simplest proof that true holograhic mirrors can exist is this: You can see your own face in a pool of calm water.
Why? Well, the reason why thick film holograms can reflect light at all is because any change in refractive index in a transparent medium creates an amplitude -- a probability -- for light to be reflected back in the direction in which it came. Metal mirrors are just extreme examples of this effect, since the Fermi surface electrons in metals create a nearly 100% probability that photons will be reflected.
The quantum mechanical details of reflections works in transparent materials are covered delightfully in my favorite Richard Feynman book, QED: The Strange Theory of Light and Matter. In addition to its relevance here for understanding what is possible with holograms, I recommend QED strongly to anyone interested in understanding just how utterly and completely weird quantum mechanics really is.
Feynman discusses how properly space layers of changes in refractive index can create a surface that, at least for certain frequencies, has a nearly 100% probability of reflecting light. A holographic mirror!
Finally, take a contemplative look at this image (or a real example from your kitchen) of a roll of very layers of Mylar film:
Nearly everyone has at sometime noticed at some level of consciousness how remarkably metallic such rolls look, almost like aluminum foil. That is because even though the distances between the film layers are not wave-coherent as they would in a photographic hologram, they do collectively reflect more and more light, until the surface looks remarkably metallic... which is to say, remarkably like a mirror.
Such a roll of Mylar film thus can plausibly be construed as a crude mechanically constructed hologram, and thus a proof that at least at some level of quality, transparent materials can indeed be configured to create plausibly effective, metallic-looking reflective mirrors.