4Physics logo  Diffuse Reflectance Measurements of
Standard Diffusers

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Michael R. Cohen
4Physics, Harrisburg, PA

Louis J. Small, III
Shippensburg University, Physics Department
Shippensburg, PA 17257

Abstract

Lambert's law states that the intensity of light scattered from a point on a reflecting surface follows a cosine relationship with the polar angle of the scattered light. No dependence on azimuthal angle of the incident or scattered light is assumed. It can be investigated by looking at diffuse reflectance as measured via the bidirectional reflectance distribution function (BRDF). Formally, BRDF can be defined as the surface radiance divided by the incident surface irradiance. In this work, we compare the BRDF of three materials engineered to be "lambertian" diffusers: Spectralon® from Labsphere, Inc., and Accuflect® B6 and G6 from Accuratus Corporation.

Introduction

Lambert's law states that the intensity of light scattered from a point on a reflecting surface follows a cosine relationship; I(θs) = I0 cos(θs ), where θs is the polar angle of the scattered light and I0 is the incident light intensity at the point. No dependence on azimuthal angle of the incident or scattered light is assumed, as indicated in figure 1. (Incident light intensity also follows a cosine relationship as the illuminated region of surface varies with the cosine of the incidence angle, θi, from the surface normal, I0 = Iinccos(θi).) Further, no allowance for polarization effects is included.

Lambertian scatter definition
Fig. 1. Lambertian, or diffuse, scatter of light is governed by a cosine intensity relationship

Since the effective area viewed by an observer looking at a surface from a polar angle θ increases with angle by 1/cos(θ), Lambert's law implies that the reflected power from an extended surface is constant with polar angle.

Lambert's law is a definition of an ideal. There is no physical principle requiring it to be followed, but it approximates what is observed in reflection from rough surfaces where many randomizing events can occur before the light leaves the surface. A number of materials are commonly referred to as lambertian diffusers, meaning that they follow Lambert's law. Depending on application, only general reflective properties may be significant rather than detailed angular reflectance behavior. Some materials are specified with regard to overall white light reflectivity (total reflected vs. incident power) and uniformity of spectral reflectance. This study begins to look at the detailed angular reflectance of some “lambertian” materials.

Method

Diffuse reflectance is often measured via the bidirectional reflectance distribution function (BRDF). Formal definition of BRDF is straightforward, following the original definition by Nicodemus, et al.1 In radiometric terms, it is the surface radiance (into some specified polar and azimuthal angles) divided by the incident surface irradiance (at specified source polar and azimuthal angles). The scattered surface radiance is the light flux scattered through solid angle Ωs per unit illuminated surface area per unit projected solid angle. (The projected solid angle is the solid angle multiplied by cos(θs) at scattering angle θs .) The incident surface irradiance is the incident light flux per unit illuminated surface area (assuming a uniform incident beam).

BRDF defined
(1)

For simplicity, the scatter function of Stover2 can be used as it allows for bulk scatter in addition to surface scatter and permits non-uniform incident beam profiles. It simply uses the cosine-corrected scattered radiance rather than solely the surface radiance (which has the effect of removing the factor of cos(θs) from the projected solid angle) to yield scatter per unit illuminated surface area per unit solid angle. Effectively, the bidirectional scatter distribution function (BSDF) is the scattered power per unit solid angle divided by the incident power:

Stover's BSDF
(2)

In these expressions, the angular dependence on incident and scattering polar and azimuthal angles has been suppressed. The geometry defined in measuring BSDF or BRDF is shown in figure 2 below.

A perfect lambertian reflector with reflection coefficient R would disperse the incident power Pi uniformly through a solid angle of 2π steradians. The reflection coefficient can be expressed as the integrated reflected power, divided by the incident power. We can use this fact to evaluate the BRDF of an ideal lambertian reflector:

Reflectance
(3)

With unit reflectivity, the BRDF would be

Lambertian diffuser BRDF
(4)

 

Scattering geometry
Fig. 2. Definition of geometry for BSDF. Pi is the incident power at polar angle θi and azimuthal angle φi, illuminating surface region A. Po is the specular reflection, dPs is the scattered power into solid angle s at polar angle θs and azimuthal angle φs .

Experiment

A scatterometer was designed and constructed for this preliminary study of the scatter function, independent of source polarization and scattered light polarization. Follow on work will address the question of depolarization in diffuse scatter. Also, this study was performed at a single wavelength. A tunable, monochromatized source is being developed for further investigations.

The scatterometer consists of a mounting stage for the diffuser sample with height and rotation adjustment, and a pair of two-axis goniometer arms for the light source and the scatter detector. The goniometer arms have set point adjustments every 10 degrees in polar angle and every 5 degrees in azimuthal angle. A laser diode (650 nm, 5 mW, unpolarized) is the incident light source. It's beam profile is nominally Gaussian, with an asymmetric 1 x 3 mm footprint. A beamsplitter cube divides the beam, with one portion going to an unbiased Si photodiode to monitor beam intensity while the other continues to the sample. The beam monitor is read by a UDT Instruments Model 351 transimpedance amplifier. The scatter detector is also an unbiased Si photodiode (6.8 mm sq.) located 204 mm from the sample plane, read by a Graseby Optronics S-3768 transimpedance amplifier. It subtends a solid angle of about 1.1x10-3 sr. The polar angle of the incident beam is limited to a maximum of 60° to maintain the full illuminated area within the scatter detector's line-of-sight.

In operation, the incident laser goniometer arm is aligned with the detector arm to establish a system sightline. The sample height is adjusted to align with the sightline, placing the sample surface at the rotation center of the goniometers. The two arms can then be adjusted independently to set the pair of polar and azimuth angles. Proper tracking can be verified by replacing the diffuse reflector sample with a highly specular reflector to check angles and view factors.

In this study, three diffuse reflectance samples were examined: Spectralon® from Labsphere, Inc. and Accuflect® B6 and G6 from Accuratus Corporation.

Spectralon® is a teflon-type composite solid, as shown in figure 3. It has favorable physical and optical characteristics. It has established use as a replacement diffuse reflectance standard for loose-packed PTFE powder (the NIST-defined and traceable refelctance standard) in applications where a rigid, chemically inert solid is preferable.

Spectralon micrpgraph
Fig. 3. Photomicrograph of Labsphere Spectralon® sample showing 50 – 100 µm clusters of particles. Sample had grazing illumination.

Accuflect® is an alumina based ceramic with the ability to withstand extreme chemical and thermal environments expected for aluminas. B6 (figure 4 below) is an unglazed version of the product, while G6 (figure 5 below) is the glazed version for use where water absorption, normal for the porous ceramic, can pose a difficulty. Visual comparison of the two samples shows G6 with a higher specular reflectivity than B6.

Accuflect B6
Fig. 4. Photomicrograph of Accuratus Accuflect® B6, lit with grazing illumination. The typical particle size appears to be in the range of 10 – 15 µm.

 

Accuflect G6
Fig. 5. Photomicrograph of Accuratus Accuflect® G6 (glazed), lit with grazing light incidence. Particle size estimate (10 – 15 µm) agrees with observations of the B6 sample, although larger particles, or clusters, are also evident.

None of the samples in this study were provided as technical-grade, traceable standards, but they are hoped to be relatively representative of the respective materials.

Data

Following instrument alignment and intensity calibration of the laser diode, azimuthal scans were taken of the scattered light from the central region of each sample. This data was acquired for defined scattering polar angle, θs, and polar and azimuthal incidence angles, θi and φi, respectively. Due to mechanical interference between the goniometer arms, the azimuthal range was restricted to 0° - 120°.

Figure 6 shows the BRDF for the Labsphere Spectralon® sample at a single light incidence geometry. The polar plot overlays the BRDF vs. azimuthal scattering angle, φs , for selected polar scattering angles, θs . The incident polar and azimuthal angles are θi = φi = 0°.

In fig. 6A, the scale of BRDF values ranges from an ideal specular reflector (BRDF = 0, except on specular) to an ideal lambertian scatterer (BRDF = 1/π). Fig. 6B shows the same data with a limited BRDF range to exhibit the variations in the measurements. Within experimental error, no significant variation in BRDF is observed with respect to scattering polar angle or scattering azimuthal angle. Overall, the Spectralon® sample exhibits a BRDF value indistinguishable from the ideal 1/π.

Spectralon azimuthal scatter data.
Fig. 6A. Azimuthal BRDF data for Labsphere Spectralon® sample at scattering polar angles of 20°, 40°, and 60°. Unpolarized light was incident along the sample normal.

 

Spectralon azimuthal scatter data.
Fig. 6B. Azimuthal BRDF data for Labsphere Spectralon® sample at scattering polar angles of 20°, 40°, and 60°. Unpolarized light was incident along the sample normal. The radial scale ranges from BRDF=0.26 sr-1 to BRDF=0.33 sr-1.

Figures 7A,B present equivalent views for Accuflect® B6. Light is again normally incident to the surface while azimuthal scans of the scattered light were acquired at selected scattering polar angles, as indicated.

Accuflect B6 azimuthal scatter data.
Fig. 7A. Azimuthal BRDF data for Accuratus Accuflect® B6 sample at scattering polar angles of 20°, 40°, and 60°. Unpolarized light was incident along the sample normal.

The unglazed B6 ceramic compares very favorably with Labsphere's Spectralon® under these conditions. Both materials exhibit essentially Lambertian behavior with BRDF independent of scattering polar and azimuthal angle, to within experiment accuracy. The nominal BRDF value found for Accuflect® B6 may differ from 1/π by up to 10%, indicating a slightly smaller diffuse reflectance than the ideal (0.9 ≤ R ≤ 0.95).

Accuflect B6 azimuthal scatter data.
Fig. 7B. Azimuthal BRDF data for Accuratus Accuflect® B6 sample at scattering polar angles, θs, of 20°, 40°, and 60°. Unpolarized light was incident along the sample normal. The radial scale ranges from BRDF=0.26 sr-1 to BRDF=0.33 sr-1.

Figures 8A,B present equivalent views for Accuflect® G6, the glazed material. Light is again normally incident to the surface while azimuthal scans of the scattered light were acquired at selected scattering polar angles, as indicated.

The glazed G6 ceramic compares well with Labsphere's Spectralon® under these conditions. Accuflect® G6 does not show any significant azimuthal scattering dependence. It does have a higher specular reflectivity, as would be expected for a glazed ceramic, though its general (polar-averaged) reflectivity is essentially the same as the unglazed B6 material (0.9 ≤ R ≤ 0.95). For light incident along the sample normal, the observed reflectance decreases with increasing polar angle. (This result is anticipated for light transmitted through the glaze layer, diffusely scattered from the base ceramic, and then undergoing a polar angle dependent transmission back through the specularly reflective glaze.) Further work will examine the specular reflectivity of this sample in greater detail.

Accuflect G6 azimuthal scatter data.
Fig. 8A. Azimuthal BRDF data for Accuratus Accuflect® G6 sample at scattering polar angles of 20°, 40°, and 60°. Unpolarized light was incident along the sample normal.

 

Accuflect G6 azimuthal scatter data.
Fig. 8B. Azimuthal BRDF data for Accuratus Accuflect® B6 sample at scattering polar angles, θs, of 20°, 40°, and 60°. Unpolarized light was incident along the sample normal. The radial scale ranges from BRDF=0.26 sr-1 to BRDF=0.33 sr-1.


Acknowledgments

The authors wish to gratefully acknowledge Labsphere, Inc. and the Accuratus Corporation for supplying the samples used in this effort.

References

1. Nicodemus, F. E., Richmond, J. C., Hsia, J. J., Ginsberg, I., and Limperis, T., Geometric Considerations and Nomenclature for Reflectance, U.S. Dept. of Commerce, NBS Monograph 160, 1977.

2. Stover, John C., Optical Scattering, Measurement and Analysis, 2nd ed., pp. 19-22, SPIE Optical Engineering Press, Bellingham, WA, 1995.


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