# Optical Scattering and Surface Roughness

For optical systems, the main reason to be concerned about surface roughness is that rough surfaces scatter light.  If a surface is intended to reflect or refract light, it is unusual for optical scattering to be desirable, so the scattering needs to be controlled by limiting the surface roughness.  This leads to an investigation of optical scattering versus surface roughness.  Half a century ago, Bennett and Porteus came up with the concept of Total Integrated Scatter (TIS, the total amount of light scattered by a surface) and found a functional relationship between TIS and surface roughness.  They published this in Bennett & Porteus, “Relation Between Surface Roughness and Specular Reflection at Normal Incidence,”JOSA 51, 123 (1961), which was later extended to refraction and non-normal incidence.

The equation that describes the functional relationship between optical scattering and surface roughness is

$TIS_{BP} \left(R_q \right) = R_0 \left[1-e^{- \left(\frac{4\pi R_q \cos \theta_i}{\lambda} \right)^2}\right]$

In this equation R0 is the  theoretical reflectance of the surface, Rq is the RMS roughness of the surface, θi is the angle of incidence on the surface and λ is the wavelength of light.

There are several rules of thumb we can learn from this equation.  The first is that optical scattering is proportional to reflectance; this means that surfaces intended to reflect light will inherently scatter more light than transmissive surfaces.  Second, scatter is related to Rq and not one of the other measures of scatter.  Third, shorter wavelengths will scatter more than longer ones.  And finally, more light scatters at normal incidence than grazing incidence.

Most of these rules of thumb make sense and correlate well with experience and observation, with Rq being the only one that is hard to relate to.  To help it make more sense, remember that for a typical surface Rq is about 1.4X Ra because it weights the larger deviations more heavily.  Bigger scratches scatter more light, so they should be weighted more heavily.  Using Rq simply reflects this weighting.

Equations are nice, but pictures convey concepts better.  To that end, here is a plot of the above equation:

Optical Scattering in % versus Surface Roughness in nm

For this plot, Ro = 1, λ = 500 nm (blue-green) and θ = 60°.

Now we can see that a surface with Rq = 25nm scatters about 10% of the light, which is pretty significant.   Dropping Rq down to about 17nm cuts the scatter in half.  Five percent scatter is still a lot of scatter, so this is the loosest tolerance we would think of using for surface roughness.  It is also important to note that this is for highly reflecting surfaces; transmitting surfaces will have scatter that is only 4% of this.  At the other end of the roughness spectrum, commercially polished glass has an Rq of 1.2nm, so scatter is well under 1%, and superpolished surfaces can achieve Rq = 0.1 – 0.2nm.  That’s about 1-2 atoms, so it’s a physical limit on scatter.

Single Point Diamond Turned surfaces can achieve Rq = 5nm and sometimes down to 3nm.  This is about as tight a spec as one could place on a molded plastic optic.  A loose spec would be SPI A-1.  The Society for the Plastics Industry has a range of specifications for the finish of molds for plastic, and A-1 is the smoothest.  It calls for Ra between 0.5 and 1.0 microinch (12.5-25nm).  Given that the measure is Ra (about 0.7 * Rq) and the surface could be as rough as 25nm (35nm Rq), this is a very loose tolerance for optical surfaces.

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