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What is the refractive index of a common mirror (mercury coated)? As a mirror completely reflects the light ray, do it have infinite refractive index?

Rahul R
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The refractive index of glass varies with the type of glass, but is usually about 1.3 to 1.5. The metallic coating on the glass is typically silver or aluminum. (I think your reference to mercury may be due to a misunderstanding of material about an old process for depositing tin onto the glass. Mercury is liquid at room temperature, so it can't be the actual backing.)

A real-valued index of refraction is used to describe a material that is an insulator, typically a dielectric. Reflection from a conductor, such as a metal, is not the same as reflection from a dielectric. The intensity and polarization of the reflection are different. A conductor cannot be described by a real-valued index of refraction.

Suppose we consider a perfect conductor for simplicity. An electromagnetic wave can't exist as a solution to Maxwell's equations inside such a conductor, and energy can't be dissipated in it through ohmic heating because it's a perfect conductor. Therefore by conservation of energy, the light is 100% reflected.

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@BenCrowell is right that the refractive index of a metal is not a real number. Good conductors tend to have small real part of refractive index and large imaginary part. As an example, I happen to remember that gold at around 800 nm wavelength (where it’s a decent mirror) has refractive index of around $n=0.3+5i$. This means that the wave inside the metal, incident at some angle to the dielectric/metal interface, is refracted closely toward the surface and is highly attenuated.

Gilbert
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A perfect mirror can be defined as having a reflection R=1 and a phase change upon reflection of 180 degrees. The Fresnel formula for reflection at an interface of air with material at perpendicular incidence is $R=\frac{1-n}{1+n}$. R is the amplitude reflection, the intensity reflection (energy) is $|R|^2$. So a perfect mirror would have to have $n \rightarrow \infty$.

The refractive index (RI) $n$ is in general complex valued. Dielectrics do not have very large values of n at optical wavelengths so they are not very suitable as mirrors. Metals do have large RIs. For aluminium the RI varies about $n \simeq 1-6i$ at optical wavelengths. This gives $|R|^2 \simeq 36/(4+36) = 0.9 = 90%$, the other 10% is absorbed and turned into heat, and a phase change of about 170 degrees. Good enough for a mirror.

my2cts
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Of course Ben Crowell is right, but let me add a few things.

Mirrors are usually coated with Aluminum. The refractive index of Aluminum is 1.373.

The refractive index of Mercury is 1.7442. But Mercury is a liquid, and therefore it is not used in mirrors.

There are no perfect mirrors. When a photon hits a mirror, three things can happen, elastic scattering (reflection), inelastic scattering (refraction), or absorption and re-emission (that is how visible light is re-emitted from non-conductors).

Mirrors, like with Aluminum:

  1. reflect (elastically scatter) most visible light

  2. refract (inelastically scatter) most non-visible light

  3. absorb and re-emit on the surface very little ratio of photons

Ao even with Aluminum, some of the photons will get absorbed or inelastically scattered. The index of refraction would not be infinite. To do that, the speed of light in that medium would be 0 (if the refraction would be infinite). That would mean it somehow traps light. We do not have a material that can do that.

Árpád Szendrei
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    There are a large number of errors in this answer. Mirrors do not refract. Refraction is in general not inelastic.The refractive index values here do not pertain to optical wavelengths. Absorption and reemission generally does not occur in a mirror. – my2cts Oct 14 '18 at 08:14
  • @my2cts Yes, mirrors do refract at a certain (but very low ratio of the photons) ratio. What you are talking about is a perfect reflector, and does not exist. Refraction is inelastic scattering. Absorption and re-emission happens too with a mirror (again at a very low ratio). – Árpád Szendrei Oct 14 '18 at 17:04
  • Your statements are inaccurate. I don't believe that you can produce references to support these. – my2cts Oct 14 '18 at 17:28
  • @my2cts your statements are inaccurate and lack any kind of knowledge in QM level reflection/refraction. Can you support your statements with references? Please give me a reference that states that refraction is not elastic scattering. – Árpád Szendrei Oct 14 '18 at 17:33
  • I don't need references for this. Photons retain their energy when passing through glass. Btw I am not going to discuss with you. I asked you for references, several times. I claim that you have no real understanding of the subject. – my2cts Oct 14 '18 at 18:09
  • @my2cts ok so you have no references for your statements, of QM reflection/refraction, or refraction being not inelastic scattering. – Árpád Szendrei Oct 14 '18 at 18:14
  • @my2cts now I think I understand where we disagree. You are talking about refraction through glass (transparent media) for example. You are correct, that is elastic scattering. That will keep the mirror image, phase and energy of the photons. – Árpád Szendrei Oct 14 '18 at 18:28
  • @my2cts just for curiosity. Please tell me whether you say that refraction through glass is a) elastic scattering b) absorption-reemission c) something else? – Árpád Szendrei Oct 14 '18 at 18:30
  • The passing of light through glass is what the term refraction refers to. You are right that refraction is elastic, but it is not scattering. But we are discussing mirrors here, so metals I to which light penetrates only of a distance equal to a fraction of a wavelength. @ArpadSzendrei – my2cts Oct 14 '18 at 18:36
  • Refraction through glass is not scattering and it is not absorption/reemission. So that leaves option c. I have explained what it is on this site before. – my2cts Oct 14 '18 at 18:39