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reflectance & reflectivity

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Reflectance and reflectivity: how are they different?

We know from common experience if not from physics that materials reflect light. We know that lighter materials reflect more light than darker materials; white reflecting the most and black the least. How should we describe this ability to reflect light? Should we refer to this ability as reflectance or reflectivity?

Science aims to avoid the ambiguous; its concepts and definitions precise, its methods exact. In optics reflectance and reflectivity are two entirely different but related concepts and the distinction between them is simple:

• reflectivity is the property of a material
• reflectance is the property of a particular sample of that material or a particular surface

The terms are not interchangeable even though both are defined by the proportion of the incident light that is reflected – so exactly what does the distinction between reflectivity and reflectance amount to?

example 1: silver

Take the following example: A piece of silver is polished until its surface is shiny. It reflects a certain proportion of the incident light. But what happens if our sample of silver is made increasingly thin? Light is reflected not just at the surface, the interface between the silver and the air but also from the atoms just below the surface. If the sample of silver is made extremely thin we find that some of the light that falls on it will start to pass through. Very thin layers of silver are transmissive as well as reflective. Thin silver films of the order 25nm{footnote}the nanometre (nm) is 10^-9 metre or 1 thousand-millionth of a metre{/footnote} will reflect only 10% of the incident light. By the time the thicknes has increased to 100nm the reflectance will have increased to about 90%{footnote}Transmission of light through thin silver films via surface plasmon-polaritons; Armando Giannattasio, Ian R. Hooper, and William L. Barnes, 2004{/footnote}.

Nothing about the silver has changed apart from its thickness. To the material itself, the silver we attribute the term “reflectivity”; to our specific sample of silver of a specific thickness we attribute the term “reflectance”.

It will become evident that our sample of silver, as soon as it becomes more than a few hundred atoms thick{footnote}a silver atom is between 100pm to 300pm depending on the way it is measured or defined. A picometre (pm) is 10^-12m; if we take the mean distance between silver atoms to be in the region of 200pm (or 0.2nm) we can see that the thickness of the film we are considering is in the region of 50 atoms at its thinnest to about 500 atoms at its greatest.{/footnote}, will no longer transmit any light. When this happens and for all thickness greater than this value the reflectance will equal the reflectivity. The reflectivity of silver is generally taken to be between 89-93% and is therefore the limiting value of reflectance{footnote}The problem with silver is that it soon tarnishes in air and the reflectance of any given sample gradually falls over time.{/footnote}.

Silver was chosen in this example because we normally think of it in the context of a mirror, a reflector of light. But we know that half-silvered mirrors used for beam splitting are designed specifically for transmitting some of the incident light at the same time as reflecting some of that incident light. The limiting value of reflectance at about 100nm and greater is also of importance to manufacturers of silvered mirrors. There is absolutely no gain in having silver any thicker than this; it won't reflect any more light and will only serve to increase manufacturing costs.

example 2: glass

This time we will consider a material we normally take to transmit light: glass. The most useful property of glass is that it is transparent to light. But it is never 100% transparent; some light is lost at the surface (it is reflected), some light is lost as it passes through the glass (it is absorbed) and some light is lost at the rear surface as the light exits the glass and re-enters air (internal reflection). What does this mean for the reflectance and reflectivity of glass?

First there are different types of glass and each one will have its own value of reflectivity depending on the refractive index n of the material. The reflectivity of crown glass is about 4%; flint glass somewhat greater at about 6%. The equations for the Reflectance R and Transmittance T of light at an interface between two media whose refractive indices are n₁ and n₂ respectivley are given below:

Figure 1: Reflectance at an interface between two media

Figure 2: Transmittance at an interface between two media

If we substitute typical values into these equations we can calculate both the Reflectance and Transmittance (i.e., the proportion or fraction of light reflected and transmitted) at the interface. The first boundary will be air-to-glass and the typical values of refractive index will be n₁=1.0 for air and n₂=1.5 for glass:

Figures 3a and 3b: Reflectance at an air-glass interface

We can undertake a similar substitution to find the values of Transmittance at the air-glass interface:

Figures 4a and 4b: Transmittance at an air-glass interface

The value of 0.96 for the Transmittance is exactly what we would expect. After all, if the incident light is 100% and the fraction reflected is 0.04 as shown above (0.04 = 4 hundredths = 4%) then the fraction transmitted must be 0.96 (0.96 = 96%). Figure 5 shows these values for the incident, reflected and transmitted (refracted) beam of light at the air-glass boundary:

Figure 5: the incident, reflected and transmitted beams at an air-glass boundary

The transmitted beam will at some point strike the rear surface of our sample of glass and once again part of the light will be reflected and some transmitted. Most of the light will pass straight through.The proportions reflected and transmitted will once again be 0.04 (4%) and 0.96 (96%). This time, however the values will be not be 4% and 96% of the incident light but 4% and 96% of the 96% transmitted through the front surface of the glass. The calculation is easy: the amount transmitted will be 0.96 x 0.96 = 0.9216 (92.16%). This is shown diagramatically in figure 6:

Figure 6: transmission and reflection at the rear glass-air boundary

The second reason we should not ignore the rear surface of our sample of glass is because light is reflected internally at this boundary. The light internally reflected light returns once again to the front surface where a proportion will be transmitted through the glass-air interface. This emergent light is in addition to the 4% of the light already reflected at the front surface. Once again the values are easy to calculate: the proportions reflected and transmitted will always be 4% and 96% at this and every subsequent internal reflection.

At the rear surface 4% of the 96% transmitted light will be reflected; the reflected amount is 0.04 x 0.96 =  0.0384 (3.84%). It is this 3.84% of the original incident light that returns to the front surface. Here 96% of it will emerge at the front glass-air interface; the exact amount transmitted is 0.96 x 0.0384 = 0.036864 or approximately 3.68%. Figure 7 shows the consequences of this second reflection - the internal reflection at the rear boundary:

Figure 7: transmission and reflection for the first internal reflection

So long as the light is not completely absorbed, internal reflections will continue alternately between the front and rear boundaries. However, the proportions reflected decrease geometrically. The proportions of light become almost insignificant after the next internal reflection as shown in Figure 8:

Figure 8: transmission and reflection for the second and third internal reflections

The example we have explored is quite instructive because, after two reflections, the total amount of light leaving the front surface is 4% (the proportion initally reflected) plus 3.68% transmitted after a single internal reflection. This yields 7.68%. Here we have the situation where the total light apparently reflected is almost double that of a single reflection. In other words, the surface reflectance is almost twice the reflectivity.

If the glass is very thin the contribution from the rear surface will be greatest. As the thickness of glass increases the contribution of the rear surface diminishes because of absorption of the light by imperfections in the glass. The important lesson is that when we consider reflectivity the thickness of the material must be great enough for there to be no contribution from the rear surface. In other words the glass must be sufficiently thick for all light to have been absorbed before reaching the rear boundary. The further pursuit of this line of thought is beyond the scope of this article but it leads to the Airy formula and the concept ot transmittivity.

exaple 3: porcelain

In order to appreciate the part played by the medium itself we will explore a third example: porcelain. Porcelain, specifically that made with the addition of bone ash, is translucent. It transmits light only diffusely.

As light passes through the porcelain the inherent opacity of the material will reduce the transmission of light. As the porcelain gets thicker and thicker, more and more light will be absorbed. At some point a thickness will be reached where no light gets through.

With increasing thickness of porcelain there is also the increasing opportunity for light to be reflected from within. This type of reflection is a diffuse reflection. The term scattering must not be used in this context - in science the term scattering has quite a different meaning. Some of the diffusely reflected light will be returned in the direction from which it originated; some will eventually be absorbed even after multiple diffuse reflections. The light that is returned to the front boundary will add to the light previously reflected at that interface and will be at a maximum when the porcelain is thick enough for no light to be transmitted. A typical diffuse reflection model is shown in Figure 9:

Figure 9: diffuse reflection model for a translucent material

The critical thickness will be different for different types of porcelain but at this thickness and all greater thicknesses the reflectance and the reflectivity will be the same. Up to that point the reflectance will always have been less than the reflectivity. The practical application of this is, of course, in dentistry. The porcelain of a crown has to be sufficiently thick for the metal substrate to have no effect.

the implications of diffuse reflection for reflectivity

The light reflected at the surface will conform to the simple law of reflection but the diffuse reflection from within the body of the material means that there will be a lesser but significant proportion of light reflected in almost every direction. When it comes to measuring reflectivity the conditions must be standardised and this happens in two important ways:

1. the incident light must be perpendicular to the surface of the material
2. the reflected light must be collected at all possible angles

The first condition requires us to measure reflectivity for normally incident light. The second condition requires us to integrate all the reflected light about a hemisphere.

Summary

White reflects more light than black because its reflectivity is greater, but the reflectance of a specific sample of white will depend on both the material and its thickness. That’s why two coats of “brilliant white” paint will almost invariably give a better result than one coat, despite what the advertising says.

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