General concept

The main idea of the Physically Motivated Basis Spectra (PMBS) approach is that the plenoptic function R(X, Y, Z, φ, ϑ, λ) is described as the weighted sum of the n_max physical basis spectra S_n(λ), which exists in the LED light source:

Equation 1 assumes that although the plenoptic function R(X, Y, Z, φ, ϑ, λ) varies spectrally as a function of the angular and spatial dimension, its underlying relative spectral distributions are constant. R(X, Y, Z, φ, ϑ, λ) can therefore be described by the weighted sum of this constant basis spectra. Thereby each basis spectrum requires a physical origin in the system. Then all spatial and angular variations are described by the amplitude distributions A_n(X, Y, Z, φ, ϑ) which change as a function of angular and spatial dimension.

However, A_n(X, Y, Z, φ, ϑ) cannot be measured directly because there is no optical filter, which generally measures only one basis spectrum. The M_n(X, Y, Z, φ, ϑ) obtained with the ILMD-based measurements depend on the effective spectral sensitivities s_τ,eff,n(λ) of the ILMD-system. The sensitivity s_τ,eff,n(λ) is the product of the transmission function of the actual optical filter τ_n(λ) used to perform the measurement, the optional transmission function of a neutral density filter τ_ND(λ) and the spectral system sensitivity function s_Sys(λ). Thus, a measured ray file is defined as.

If it is assumed that besides the measured signal M_n(X, Y, Z, φ, ϑ) the relative basis spectra S_n(λ) as well as all spectral transmissions and sensitivities and therefore s_τ,eff,n(λ) are known, only the n_max A_n(X, Y, Z, φ, ϑ) are unknown variables. If m different optical filters are used, they produce m different effective spectral sensitivities s_τ,eff,m(λ) and thus m spectrally weighted measurement values M_n(X, Y, Z, φ, ϑ), which are given by

These equations are a system of linear equations, which has a unique solution if the number of different optical filter measurements equals the number of physical basis spectra m = n_max. Equation 3 can be rewritten in its matrix form as

The matrix M_Sτ summarizes all available information about the effective spectral sensitivity of the measurement system and the physical basis spectra of the DUT. Eq. 4 therefore relates the measurement values M_n(X, Y, Z, φ, ϑ) to the unknown amplitudes A_n(X, Y, Z, φ, ϑ) which will be assigned to the light source model’s n_max physical basis spectra. The required distributions A_n(X, Y, Z, φ, ϑ) can be reconstructed by inverting the matrix M_Sτ. The inverted matrix is called the spectral reconstruction matrix. The reconstructed distributions can be used to calculate the plenoptic function and thus to create the spectral ray file according to Eq. 1.

It is important to note that these equations are only valid if absolute values are used. If absolute values are not available directly, they can be reconstructed by a comparison of the measured ray file and a spectral measurement within the goniometric setup. The details are described in [1]. Further, there are some limitations, which shall be mentioned. The most limiting assumption is the constant basis spectra assumption, which implies that PMBS does not cover nonlinear effects such as quenching caused by phosphor saturation due to a high radiant flux of the LED [20] or phosphor self absorption. Chromatic effects resulting from a primary optic cannot be modeled with the PMBS approach as well. Although it is common in near field photometry, it shall be mentioned that all interrelated measurements require the steady state condition with respect to electrical and thermal operation conditions.

Workflow of PMBS

Based on these relations, a general workflow for PMBS can be concluded. The workflow is visualized in Fig. 2 and serves as an outline for the next subsections. The process starts with the determination of the individual semiconductor and phosphor basis spectra using a spectral measurement. Subsequently, the goniophotometric measurement with n_max different suitable filters are performed. The spectral sensitivities of the measurement system and the basis spectra are combined to form the spectral reconstruction matrix . Subsequently, the ray files are used to solve the system of linear equations by applying the reconstruction equation Eq. 4.

Figure 2:

Workflow to apply PMBS (published under CC BY-SA 4.0 from [1]).

Basis spectra estimation

Each LED basis spectrum can be described by a set of a model parameters x_n = {p₁ … p_j}, which parametrize a phenomenological model function such as a Gaussian distribution, a second order Lorentzian or an even more complex function. The number of parameters j depend on the specific phenomenological model used. Examples are provided in [21, 22]

In order to precisely estimate a set of basis spectra S_i(λ) out of a measured LED spectrum S_M(λ), we assume that all basis spectra occur in the measurement and that at least the number of fundamental different basis spectra (different colors, phosphor) is known prior to the process. Based on this knowledge a set of model parameters x = {x₁ … x_nmax} can be estimated using an optimization algorithm, which minimizes the residual sum of squares between the sum of all basis spectra models and the measured spectrum.

In order to also include a phosphor spectrum into the modeling process, a more complex procedure is necessary. It basically includes the determination of the model spectrum, which has the least probability to be an LED spectrum. Than it applies a smoothing spline on the measured spectrum minus all LED spectra (excluding the determined spectrum). The process can be repeated until convergence is reached. More information and examples are provided in [1]

The basis spectra can be optimized by incorporating an additional LED attribute, which is the spatial separation of colored LEDs. Most often, the different LED colors lie next to each other. A green LED is green, a red LED is red and a white LED is white (or rather blue and yellow). The green, red and white LEDs do not overlap in the 3-dimensional spatial dimension. Although there might be a remote phosphor, which overlaps with all LEDs, the single colored chips are both spatially separated and highly localized in the three dimensional spatial dimension.

A set of single measurement images M_n(X, Y) can be used to reconstruct the spectral images (or irradiances) A_n(X, Y) of the individual basis spectra using the mathematical relations described in Eq. 4. The reconstructed A_n(X, Y) can be used to quantify the “spatial separation” of the individual spectral sources and thus be used as additional merit or constraint for an additional optimization.

Determination of optical filters

In theory any arbitrary filter combination is sufficient for a spectral reconstruction with PMBS. An example is shown in Fig. 3. The RGBW LED system consists of four basis spectra and, therefore, PMBS requires four different ILMD-based filter measurements. According to Eq. 4, the four basis spectra can be reconstructed with, for instance, the filter combinations A-D. However, at a closer look no one would consider to choose filter combination B or C during the application of PMBS in this example because they do not provide an accurate measurement of the blue part of the spectrum. During the application of PMBS the ideal reconstruction Eq. 4 has to be extended to the real reconstruction Eq. 6. This equation considers that the reconstructed A_n(X, Y, Z, φ, ϑ) differ by ΔA_n(X, Y, Z, φ, ϑ) due to the deviations η_n(X, Y, Z, φ, ϑ) of the measurements M_n(X, Y, Z, φ, ϑ)

The uncertainty η_n(X, Y, Z, φ, ϑ) from Eq. 6 has two different physical origins

The first factor consists of uncertainties η_τ,n(X, Y, Z, φ, ϑ) regarding the effective spectral sensitivities s_τ,eff,n(λ) and their interactions with the basis spectra. The uncertainty η_τ,n(X, Y, Z, φ, ϑ) depends on the individual filter used in the measurement and is added to the true value M_t,n(X, Y, Z, φ, ϑ). The second term η_N describes uncertainties, which apply for all filter measurements independent of the specific filter function s_τ,eff,n(λ) at each step of the goniophotometric measurements. Examples are noise from the sensor chip or the quantification errors caused by the ray file resolutions.

Figure 3:

Different combinations of optical glass edge absorption filters to apply PMBS on a RGBW LED-system (published under CC BY-SA 4.0 from [1]).

A detailed estimation of the reconstruction performance of a filter combination needs to consider the individual filter based η_τ,n(X, Y, Z, φ, ϑ) as well as its error propagation. Thus, the performance will differ for each filter combination and selecting a robust filter combination becomes an essential part of applying PMBS.

In order to determine a robust filter combination, a Monte Carlo simulation, which incorporates precise filter models, uncertainty distributions as well as the different filter combinations, can be carried out. Suitable models, uncertainty distributions and a modeling and evaluation procedure can be found in [23].

Figure 4 shows a comparison of a Monte Carlo simulation and a real reconstruction performance of 15 possible filter combinations for the RGBW system from Fig. 3. Altogether six glass absorption filters were considered. The y-axis shows the mean of the measured reconstruction performance as chromaticity coordinate distance Δu’ν′. It bases on a comparison of the reconstructed spectral LED model to a angularly resolved spectral measurements in the goniometric setup. The x-axis shows the selection number/ranking of a specific filter combination. The lower the selection number, the better the simulated reconstruction performance.

Figure 4:

Validation of Monte Carlo based filter selection (published under CC BY-SA 4.0 from [1]).

The reconstruction performance can be divided into roughly three groups. The first group results in a Δu’ν′ ≈ 0.01, the second group in a Δu’ν′ ≈ 0.025 and the last group in a Δu’ν > 0.1. The last group consists of filter combinations, which cannot reconstruct the spectra. Impossible chromaticity distance values such as Δu’ν > 0.6 result from physically impossible chromaticity coordinates, which occur due to partly negative spectral distributions. The impossible area is marked in Fig. 4. The Monte Carlo based filter selection succeeds in separating these three groups. That way, the filter selection identifies a robust and suitable filter combination to apply PMBS.

Comparison to the Blue/Yellow approach

To quantitatively compare both methods, an LED with a strong angular color uniformity was chosen. Angularly resolved spectral measurements have been performed to assess the results. Figure 5 shows the LED as well as its color shadow on a photographic image. The angularly resolved spectral measurements are evaluated as chromaticity coordinates in the CIE u’ν′ diagram and shown in Fig. 5 as well.

Figure 5:

Tested white LED and measured angular color shift (published under CC BY-SA 4.0 from [1]).

Figure 6 shows the reconstructions of LED spectra as well as a comparison of the reconstructed chromaticity coordinates to the angularly resolved spectral measurements for both approaches. The comparison of reconstructed spectra to measured spectra shows that the deviations are smaller in the case of PMBS. The comparison of the reconstructed chromaticities to the measurements lead to the same conclusion. PMBS is not only able to describe the relative chromaticity shift more precisely, but also differs less in terms of the absolute chromaticity differences, which is Δu’ν′_mean ≈ 0.002 Note that the required measurement amount of both methods is identical in the case of a phosphor converted white LED.

Figure 6:

Comparison of the Blue/Yellow and PMBS approach (published under CC BY-SA 4.0 from [1]).

Spectral ray tracing

To validate the suitability of the obtained spectral ray files, a far field measurement of a wavelength dependent optical system including the measurement object was compared to ray tracing simulations in the commercial ray tracer OptisWorks from OPTIS. The experimental setup containing the temperature controlled (TEC controller) LED source as well as the dispersion prism on a rotating stage is shown in Fig 7. The LED source uses three separated RGBW LEDs. In the experiment the configuration used is red (upper left), red-white, (middle), and white (bottom right), which can also be noticed in the photographic image of the experimental setup. The optical system consists of one dispersion prism because it shows a strong wavelength dependence and because the simple geometry minimizes deviations between the optical system and the CAD geometry, which can disturb the assessment of the spectral ray files.

Figure 7:

Experimental setup (left) and simulation setup (right) (published under CC BY-SA 4.0 from [1]).

The simulation setup is shown in Fig. 7 on the right hand side. In both simulation and measurement the prism was aligned with the rotating plate. As in the measurement, the photometric center of the intensity sensor in the simulation was defined at the center of the prism. The ray file editor from OptisWorks was used to assign the basis spectra to the reconstructed ray files. As the prism was located in the near field of the source, the influence of the prism is different for each LED source. Figure 8 shows a photographic image of the obtained distribution on the left hand side. It consists of a solely white part in the upper right, a red/white mixture in the middle and a solely red part in the lower left. Furthermore, both parts with a white LED show a chromatic shift from blue to red caused by the prism. The characteristic shape arises due to the placement of the source and the aperture of the prism.

Figure 8:

Qualitative comparison: Photographic Image (left), Colorimetric far field measurement (middle) and spectral ray tracing result (right) (published under CC BY-SA 4.0 from [1]).

A colorimetric intensity distribution of the refracted part of the light was goniometrically measured with an automotive goniometer and the colorimeter C3300 from LMT. The measurement result is shown in Figure 8 in the middle. The image on the right hand side shows the filtered result of the spectral ray tracing simulation in OptisWorks. The filtering was performed with the OPTIS XMP-filter in order to decrease the noise in the simulation. Furthermore, only the three major sequences, which represent 92% of the radiometric power reaching the sensor, contribute to the simulation result. Both images aim to visualize the results in true colors. Therefore, results from the measurement and the simulation provided in the CIE spectral tristimulus values were converted in RGB values. It has to be noted that the results remain false colors if watched on a printed page or a display. The comparison of Fig. 8 middle and right shows a very good qualitative agreement between the simulation and the measurement. Both the general shape and color of each region as well as the overlapping regions obtained with the spectral ray files fit the measurement. This can also be verified by the red spot in the upper left corner. It is caused by a reflection of the red LED within the prism and occurs in both measurement and simulation