# A Theory Of Metallic Reflexion

##### 6 minutes • 1226 words

The most striking properties of metals are:

- The power of brilliantly reflecting light at all angles of incidence

This is well shown by the mirrors of reflecting telescopes.

- The opacity, which causes a train of waves to be extinguished before it has proceeded many wave-lengths into a metallic medium.

These two attributes are probably connected because certain non-metallic bodies-e.g., aniline dyes, strongly absorb the rays in certain parts of the spectrum and reflect those rays with metallic brilliance.

Malus first noticed a third quality in which metals differ from transparent bodies.

- The polarization of the light reflected from them

This is closely related to the other two. In 1830, Sir David Brewster[70] showed that:

- plane-polarized light incident on a metallic surface remains polarized in the same plane after reflexion if its polarization is either parallel or perpendicular to the plane of reflexion.
- But in other cases the reflected light is polarized elliptically.

This discovery suggested to the mathematicians a theory of metallic reflexion.

Elliptic polarization is obtained when plane-polarized light is totally reflected at the surface of a transparent body. This analogy between the effects of total reflexion and metallic reflexion led to the surmise that the latter phenomenon might be treated in the same way as Fresnel had treated the former, namely, by introducing imaginary quantitics into the formulae of ordinary reflexion.

On these principles mathematical formulae were devised by MacCullagh[71] and Cauchy.[72]

To explain their method, we shall suppose the incident light to be polarized in the plane of incidence. According to Fresnel’s sinc-law, the amplitude of the light (polarized in this way) reflected from a transparent body is to the amplitude of the incident light in the ratio

where i denotes the angle of incidence and r is determined from the equation

MacCullagh and Cauchy assumed that these equations hold good also for reflexion at a metallic surface, provided the refractive index μ is replaced by a complex quantity

where ν and κ are to be regarded as two constants characteristic of the metal. We have therefore

If then we write

so that equations defining U and ν are obtained by equating separately the real and the imaginary parts of this equation, we have

and this may be written in the form

where

The quantities

mean the ratio of the intensities of the reflected and incident light, while δ measures the change of phase experienced by the light in reflexion.

The case of light polarized at right angles to the plane of incidence may be treated in the same way.

When the incidence is perpendicular, U evidently reduces to ν(1 + κ2)

, and ν reduces to -tan-1κ. For silver at perpendicular incidence almost all the light is reflected, so

…

is nearly unity: this requires cos ν to be small, and κ to be very large. The extreme case in which κ is indefinitely great but ν indefinitely small, so that the quasi-index of refraction is a pure imaginary, is generally known as the case of ideal silver.

The physical significance of the two constants ν and κ was more or less distinctly indicated by Cauchy; in fact, as the difference between metals and transparent bodies depends on the constant κ, it is evident that κ must in some way measure the opacity of the substance. This will be more clearly seen if we inquire how the elastic-solid theory of light can be extended 80 as to provide a physical basis for the formulae of MacCullagh and Cauchy.[73]

Fresnel’s sine-formula was the starting-point of our investigation of metallic reflexion. It is a consequence of Green’s elastic-solid theory.

The differences between Green’s results and those which we have derived arise solely from the complex value which we have assumed for `μ`

.

We have therefore to modify Green’s theory in such a way as to obtain a complex value for the index of refraction.

Take the plane of incidence as plane of xy, and the metallic surface as plane of yz. If the light is polarized in the plane of incidence, so that the light-vector is parallel to the axis of z, the incident light may be taken to be a function of the argument

Χ

where

here i denotes the angle of incidence, ρ the inertia of the aether, and n its rigidity.

Let the reflected light be a function of the argument

where, in order to secure continuity at the boundary, b and c must have the same values as before. Since Green’s formulae are to be still applicable, we must have

where

{\displaystyle \sin i=\mu \sin r}, but μ has now a complex value. This equation may be written in the form

Let the complex value of μ2 be written

the real part being written ρ1/ρ in order to exhibit the analogy with Green’s theory of transparent media: then we have

But an equation of this kind must (as in Green’s theory) represent the condition to be satisfied in order that the quantity

may satisfy the differential equation of motion of the aether; from which we see that the equation of motion of the aether in the metallic medium is probably of the form

This equation of motion differs from that of a Greenian elastic solid by reason of the occurrence of the term in

…

But this is evidently a “viscous” term, representing something like a frictional dissipation of the energy of luminous vibrations: a dissipation which, in fact, occasions the opacity of the metal.

Thus, the term which expresses opacity in the equation of motion of the luminiferous medium appears as the origin of the peculiarities of metallic reflexion.[74] It is curious to notice how closely this accords with the idea of Huygens, that metals are characterized by the presence of soft particles which camp the vibrations of light.

There is, however, one great difficulty attending this explanation of metallic reflexion, which was first pointed out by Lord Rayleigh.[75] We have seen that for ideal silver μ2 is real and negative: and therefore A must be zero and ρ1 negative; that is to say, the inertia of the luminiferous medium in the metal must be negative. This seems to destroy entirely the physical intelligibility of the theory as applied to the case of ideal silver.

The difficulty is a deep-seated one, and was not overcome for many years.

The direction in which the true solution lies will suggest itself when we consider the resemblance which has already been noticed between metals and those substances which show “surface colour”—e.g. the aniline dyes. In the case of the latter substances, the light which is so copiously reflected from them lies within a restricted part of the spectrum.

It therefore seems probable that the phenomenon is not to be attributed to the existence of dissipative terms, but that it belongs rather to the same class of effects as dispersion, and is to be referred to the same causes. In fact, dispersion means that the value of the refractive index of a substance with respect to any kind of light depends on the period of the light.

We have only to suppose that the physical causes which operate in dispersion cause the refractive index to become imaginary for certain kinds of light, in order to explain satisfactorily both the surface colours of the aniline dyes and the strong reflecting powers of the metals.