Something to Reflect Upon: Diffuse Reflectance Materials & Coatings A Technical Briefing by Robert Yeo, Director, Pro-Lite Technology Ltd October 2019 reflectance. Implicit in this definition is that the Key Words intensity of light reflected is equal to that incident DIFFUSE, DIFFUSE REFLECTANCE, SCATTER, SPECULAR, on to the specular surface, reduced only by the MIRROR, LAMBERTIAN, REFLECTOR, HAZE, BRDF, BTDF, reflectance factor of the material at that GLOSS, MATTE, COSINE RULE, LIGHT METROLOGY, wavelength. REMOTE SENSING, LIDAR Introduction Here’s something to reflect upon. Imagine a material which reflects light equally in all directions. It appears equally bright to an observer, regardless of the angle from which they view it. What I am describing is known to optical physicists as a Lambertian reflector. A material that exhibits Lambertian reflectance is one which is a perfectly matte, or diffusely reflecting. In physics, we say that the material obeys Lambert’s Cosine Law, named after Johann Lambert, who described the concept of a perfect diffuse reflector in his 1760 book “Photometria”. Figure 1: A Mirror Exhibits Specular Reflectance (i = r) In this article, I will detail the important Diffuse Reflectance characteristics of diffusely reflecting materials, Very few materials are perfect mirrors (or specular describe some commercially available Lambertian reflectors), with the exception of high quality laser materials and highlight some of the more optics. In practice, most materials display both interesting applications in science and industry. part-specular and part diffuse reflectance (or in lay Specular Reflectance terms, they are part glossy and part matte). So what is diffuse reflectance? The 18th century Let’s first consider a surface that is the opposite of Swiss polymath Johann Heinrich Lambert (1728- a diffusely reflecting material. A mirror is a specular 1777), described the concept of a perfect diffuse reflector, from which a ray of light incident at an reflector in his 1760 book “Photometria”. The angle i to the surface normal is reflected at the adjective “Lambertian” has become synonymous exact opposite angle r (see Figure 1). As with materials that are matte or diffuse. A diffusely commonly expressed for a mirror (or a mirror-like reflecting material behaves very differently to a surface), the angle of incidence equals the angle of www.pro-lite.co.uk Tech Brief: Light Field Imaging Page 1 perfect mirror. Instead of all of the light reflecting radiant flux reflected from the surface (in Watts) in the specular direction, the light reflects in all and 훺 (in steradians) the solid angle subtended. directions. For a theoretical perfect Lambertian (or The definition of intensity assumes that the light is diffuse) reflector, the intensity of light reflected emitted (reflected or transmitted) from a obeys Lambert’s Cosine Law (see Figure 2). theoretical point source of light. Figure 2: Lamberts Cosine Law for a Perfect Diffuse Reflector Figure 3: Intensity is the Flux per Unit Solid Angle (Steradian) Reflected from a Point Source of Light Lambert’s law states that the intensity of light reflected from a perfect diffuse reflector at an Intensity Versus Radiance angle subtended to the surface normal I is equal Lambert’s law describes how the light reflected to intensity reflected in the direction of the surface from a perfect diffuse reflector varies as a function normal I0 multiplied by the cosine of the angle of the cosine of the angle subtended to the surface subtended. This is expressed as follows normal. However, this cosine relationship appears (Equation 1): to contradict the observation that a Lambertian surface reflects with constant brightness (radiance 퐼휃 = 퐼0cos휃 or luminance) as the angle of view varies. Thus at normal incidence, the intensity is at a maximum, while at 90° (in the plane of the This apparent dichotomy is resolved when you surface), the intensity has dropped to zero. It is consider that the human vision system observes worth noting that “intensity” has an important light reflected as the intensity per unit area, which geometric definition, distinct from its use in the is termed radiance (or luminance). The radiance is vernacular (see Figure 3). It is the quantity of the radiant flux reflected per unit solid angle per radiant or luminous flux (“optical power”) reflected unit area, expressed as follows (Equation 3): in a particular direction per steradian, the unit of 훷 solid angle. In radiometric terms, radiant intensity 퐿 = in a defined direction is expressed as follows 훺퐴 (Equation 2): Where L is the radiance (Watts per steradian per 훷 sq. meter), 훷 is the radiant flux (Watts), 훺 the solid 퐼 = angle (steradians) and A is the area viewed (sq. 훺 meters). Where I is the radiant intensity (Watts per As shown previously (Equation 2), flux per unit steradian) in a specified direction, 훷 is the total solid angle is intensity, so radiance can therefore www.pro-lite.co.uk Tech Brief: Light Field Imaging Page 2 be expressed as intensity per unit area pane. Haze is a phenomenon which adversely (Equation 4): affects our perception of the quality of a glossy surface. 퐿 = 퐼/퐴 We know that intensity varies with the cosine of Quantifying Scatter the angle subtended to the surface normal (see The science of quantifying scattered light (either Equation 1). So the radiance observed at angle upon reflection or transmission) is referred to as theta (퐿휃) is equal to the product of the intensity “scatterometry” (see Figure 4). A scatter at normal incidence (퐼0) and the cosine of theta, measurement will reveal to what extent a material divided by the projected area at angle theta (퐴휃) as conforms to the theoretical ideal of a perfect shown below (Equation 5): specular or perfect diffuse surface. Scatter measurements are reported as the Bi-Directional 퐿휃 = 퐼0cos휃/퐴휃 Reflective Distribution Function (BRDF) or Bi- The observed surface at normal incidence is a circle Directional Transmissive Distribution Function of area 퐴. When viewed from a higher angle theta, (BTDF). These scatter functions express the ratio of the circle transforms into an ellipse, and the area incident irradiance to reflected (or transmitted) reduces in proportion to the cosine of theta radiance, for each angle of illumination, and for (Equation 6): each angle of reflectance (or transmittance). The BRDF for a Lambertian material is 1/휋 at all angles. 퐴휃 = 퐴0cos휃 So we can now express the radiance at angle theta as shown below (Equation 7): 퐿휃 = 퐼0cos휃/퐴0cos휃 The cos휃 factor appears in both the nominator and denominator, so abrogate. So we can finally show that the radiance is constant with angle (Equation 8): 퐿휃 = 퐼0/퐴0 To put it in lay terms, the brightness of a Lambertian (or perfect diffuse reflector) remains constant as you view it from different angles. This Figure 4: Principle of BRDF Scatterometry is because, the change in intensity with angle (the cosine relationship) is countered by an equal but Diffusely Reflecting Materials opposite change in the projected surface area that you view (also a cosine relationship). Thus the So far, so theoretical. What of commercially Lambertian surface will possess the same available diffusely reflecting optical-grade brightness (luminance or radiance) regardless of materials? Two companies lead in this field the angle that you view it from. worldwide. One is US firm Labsphere (www.labsphere.com) and the other is Near-Specular Diffuse Reflectance SphereOptics (www.sphereoptics.de), a German The proportion of light scattered at angles close to company. Both companies developed materials the specular direction is termed “haze”. Haze is that possess near-perfect Lambertian reflectance. what you observe from a dusty or frosted window Labsphere’s Spectralon® and SphereOptics’ Zenith www.pro-lite.co.uk Tech Brief: Light Field Imaging Page 3 Polymer® are forms of solid, sintered PTFE and Spheres coated with Spectralon/Zenith (PTFE) or possess the following attributes: Spectraflect (BaSO4) are commonly used in the radiometry and photometry of light sources • Near-perfect Lambertian reflectance over (lamps, LEDs, lasers etc). an almost 2휋 steradian field. • >99% total hemispherical reflectance in the 400-1500nm range. • >95% total hemispherical reflectance in the 250-2500nm range. • Available as undoped (white) as well as doped (grey-scale) reflectance material. • Non-wavelength selective, non- fluorescent. Whereas Spectralon and Zenith-Polymer are bulk scatterers and are deployed as solid, 3D reflectors, other coatings exist that can be applied by spray Figure 5: Labsphere Integrating Sphere Radiometer painting or electro-coating. For the UV-VIS-NIR band, Labsphere produces a spray coating called An internally illuminated integrating sphere Spectraflect® which is based on barium sulphate. provides a source of uniform radiance or irradiance This coating offers a slightly lower level of optical (see Figure 6). These Lambertian light sources are performance than PTFE albeit at lower cost. For used to test and calibrate image sensors and applications requiring a more durable, yet cost camera equipment. The uniform light source effective solution, Labsphere developed a coating corrects for aberrations inherent with lens called Permaflect®. For applications that require systems, imparts an absolute radiometric Lambertian transmittance, thin sections of Zenith calibration onto the camera, can be used to Polymer are offered as diffuser sheets. For determine the quantum efficiency of a sensor applications in the mid to far-infrared band, array, performs pixel gain normalisation of a sensor Labsphere offers an electroplated gold coating and corrects for photometric non-linearities. Large called InfraGold that offers ~94% diffuse integrating spheres are used to calibrate spectral reflectance from 800nm to 20μm. radiometers used on board satellites in earth observation science and remote sensing. Who Uses These Materials? The commercial products mentioned previously are specialist optical materials and their application is mostly limited to demanding optical applications in science and industry, rather than mass-market consumer applications, for which lower cost, lower performance solutions exist.