Metrological Evaluation of Optical Surfaces Using Goniometric and Spectrophotometric Techniques

BRDF is a function describing the transformation of radiance of a surface element in the given observation angle at a given wavelength λ, L_r(θ_r,ϕ_r,x_r,y_r,λ), depending on the illumination direction (θ_i,ϕ_i) and the observation direction (θ_r,ϕ_r) in relation to the irradiance incident on the environment from a given direction at a given wavelength E_i(θ_i,ϕ_i,λ). [3] (1) ρθi,ϕi,θr,ϕr,xr,yr,λ=dLrθr,ϕr,xr,yr,λdEiθi,ϕi,λ \rho \left( {{\theta _i},{\phi _i},{\theta _r},{\phi _r},{x_r},{y_r},\lambda } \right) = {{{\rm{d}}{L_r}\left( {{\theta _r},{\phi _r},{x_r},{y_r},\lambda } \right)} \over {{\rm{d}}{E_i}\left( {{\theta _i},{\phi _i},\lambda } \right)}} where: ρ is BRDF, i is the direction of illumination described by its zenithal angle θ_i and its azimuthal angle ϕ_i, r is the direction of observation described by its zenithal angle θ_r and its azimuthal angle ϕ_r, E_i is incident irradiance, L_r is reflected radiance, and λ is wavelength.

The BRDF function is a more general fundamental physical property, as it can define the reflection characterization for any angle of incident optical beam and any detection direction across all possible wavelengths. The spectral radiance factor is a more specific quantity derived from the spectral BRDF for a particular wavelength and a reference Lambertian surface under specific geometries, used when a full BRDF measurement is impractical.

The spectral radiance factor is the spectrally defined physical quantity that indicates the ratio of the radiance L_e,n of a surface element in a given direction and the radiance L_e,d of a reflection perfect or a transmission perfect scatterer irradiated and observed in the same way: (2) βeλ=Le,nλLe,dλ {\beta _e}\left( \lambda \right) = {{{L_{e,n}}\left( \lambda \right)} \over {{L_{e,d}}\left( \lambda \right)}}

The definition applies in a given direction under given irradiation conditions for a surface element of the environment that does not itself emit radiation [3].

The spectral radiance factor is thus an essentially relative quantity.

The robot-based goniospectrophotometer RoboCapp (see Fig. 2) consists of a motorized circular ring with an external diameter of 1.3 m, which allows the rotation of the detection system around the sample.

The sample positioning is achieved by the 6-axis Mitsubishi robotic arm located in the center of the circular ring. A sample holder is placed on the top of the robotic arm. Due to the 6 degrees of freedom provided by the robot, it is possible to measure in any arbitrary geometric configuration (see Fig. 1), which allows for any desired orientation of the sample with respect to the incident optical beam.

Fig. 1.

Description of notation and angles [4].

Fig. 2.

CMI robo-goniospectrophotometer facility (a photo documentation from the sources of the Department of Radiometry and Photometry of CMI).

To produce monochromatic optical radiation suitable for precise spectrophotometric and colorimetric applications, the experimental setup integrates a high-intensity laser-driven light source (LDLS) with a diffraction grating monochromator. This configuration allows wavelength tuning across a broad spectral range, extending from the visible into the infrared light spectrum, with a spectral bandwidth of approximately 5 nm. The output optical beam from the monochromator is subsequently collimated using a reflective optical assembly. The polarization plane of the illumination beam is regulated using a high-extinction-ratio film polarizer, which is integrated into a computer-controlled motorized rotation stage.

The RoboCapp detection system (see Fig. 3) was developed through a joint effort between the CMI and the Measurement Standards Laboratory of New Zealand (MSL). The unit has been specifically engineered to achieve an exceptionally wide dynamic range of 10 orders of magnitude within the visible spectral range, accommodating optical powers from the fW level up to several tens of μW [5].

Fig. 3.

Diagram of the RoboCapp detection system [6].

The system incorporates an aperture, a mirror, and sensors integrated into a small, compact integrating sphere. To suppress inter-reflections, a mirror with a focal length of 150 mm is employed in place of a lens. The mirror is tilted by 5.5° along both horizontal and vertical axes to minimize stray light and improve measurement fidelity.

The integrating sphere, manufactured from sintered halon with an internal diameter of 25 mm, contains three ports. A 10 mm entrance port directs the focused optical beam from the mirror. Two additional recessed ports house the detectors: a Hamamatsu photomultiplier and a Si low-noise Si photodiode. This dual-sensor configuration ensures continuous coverage across the entire dynamic range of the instrument.

A key feature of the design is its insensitivity to the polarization of the detected optical light beam. As BRDF characterization exhibits dependence on both vertical and horizontal polarization of the incident optical radiation, the detection system was constructed to ensure equal detection efficiency for both polarization orientations. Therefore, two measurements are performed using vertically and horizontally polarized incident light.

In measurement configurations typical for the RoboCapp system, where the sample surface over a relatively small spot area is illuminated, the spectral radiance factor is determined by [1] (3) βeλ=ϕrλϕiλRr2+rr2rr2cosθr {\beta _e}\left( \lambda \right) = {{{\phi _r}\left( \lambda \right)} \over {{\phi _i}\left( \lambda \right)}}{{R_r^2 + r_r^2} \over {r_r^2\cos \left( {{\theta _r}} \right)}} where: ϕ_r(λ) is the spectral reflected radiant flux, ϕ_i(λ) is the spectral incident radiant flux, R_r is the distance from the detector aperture to the sample surface, and r_r is the detector aperture radius. The quantities ϕ_r(λ) and ϕ_i(λ) generate proportional photocurrent signals, I_r(λ) and I_r(λ), respectively, in the RoboCapp Si photodiode that are given by the equations (4) ϕrλ=kIrλsλ {\phi _r}\left( \lambda \right) = k{{{I_r}\left( \lambda \right)} \over {s\left( \lambda \right)}} (5) ϕiλ=kIiλsλ {\phi _i}\left( \lambda \right) = k{{{I_i}\left( \lambda \right)} \over {s\left( \lambda \right)}} where: k is a constant value derived by the geometrical factor of the RoboCapp detector integrating sphere, and s(λ) is the Si photodiode spectral responsivity.

Applying (4) and (5) to (3), the measurement equation for the spectral radiance factor as determined by RoboCapp is obtained: (6) βeλ=IrλIiλRr2+rr2rr2cosθr {\beta _e}\left( \lambda \right) = {{{I_r}\left( \lambda \right)} \over {{I_i}\left( \lambda \right)}}{{R_r^2 + r_r^2} \over {r_r^2\cos \left( {{\theta _r}} \right)}} where: R_r value is measured with a non-contact calibrated laser meter, r_r value is calibrated by the CMI length department, I_r(λ) and I_i(λ) are measured by custom-made switched integrator amplifier electronics [7], [8].

For the characterization of the quasi-Lambertian diffuse standard material sample Spectralon 99, the 0°/45° measurement geometry was selected. The measurement was conducted within the visible light spectrum, covering wavelengths between 400 nm and 780 nm.