Figure 2 shows the variation of the overall CIE x values in dependency of the reflectivity of the substrate surface (which is assumed to be either of specular or of diffuse nature and which is varied in between 70% and 100%) and the globe-top sizes. The phosphor concentrations for the globe-tops with the radii of 1 mm, 2 mm and 3 mm were adjusted to achieve overall CIE x values of about 0.3300 and 0.3800 (for a diffuse reflectivity of the substrate surface of 85%).
In the case of a specular substrate surface reflectivity, the same phosphor concentrations as in case of a diffuse one were used.
An increase of the substrate surface reflectivity to values larger than 85% goes along with an increase of the overall CIE x values for all three globe-top radii. This can be attributed to the fact that with increasing reflectivity the backscattered light becomes less and less absorbed. Since the yellow light emitted from the phosphor particles is more prone to be backscattered than the blue LED light, the substrate reflectivity in particularly affects the yellowish fraction of the overall emission spectrum and therefore also the overall CIE x values increase with increasing substrate reflectivity. In principle, as suggested in [24], a higher reflectivity of the substrate surface could be therefore applied in order to lower the amount of phosphor which is necessary in order to achieve a desired CCT value. The lowest increase of the CIE x value (and therefore the lowest deviation from the initial value) with increasing substrate reflectivity shows the hemispherical globe-top with the smallest of the three radii since in this case also the coverage of the substrate surface with the globe-top material is the lowest.
This behaviour reverses in case that the reflectivity of the substrate surface is reduced to values lower than 85%. In this case, the overall CIE x values decrease for all the different globe-top radii since a comparably larger amount of the (mainly yellow) backscattered light is absorbed by the substrate surface. Therefore, again, the hemispherical globe-top with the smallest radius shows the lowest decrease of the CIE x values and the lowest deviation from the initial value. In practice, it is most likely that the reflectivity of the substrate surface decreases (and not increases) during the lifetime of the LED assembly because of materials degradation. One can therefore conclude that a smaller globe-top size has some advantages in regard of long-term maintenance of a given colour temperature.
In addition, a comparison of specular and diffuse substrate surface reflectivities shows that for the same value of the reflectivity, a specular reflectivity gives reason for somewhat lower CIE x values than a diffuse one. This behaviour is a bit more pronounced for the larger globe-top radii than for the smaller ones. Since for both types of reflectivity the amount of phosphor is the same (with respect to the individual globe-top radii), the use of diffuse substrate surfaces opens another possibility to reduce the required volume concentration and therefore amount of phosphor to achieve a desired CCT value.
In order to highlight this more in detail, the following approximation can be done, in which hemispheres with the different radii and the respective phosphor concentrations are considered, however the volume of the LED die is neglected. Assuming a density of the phosphor of 4.56 g/cm3 as it is typical for Ce-YAG based systems [34] and of 0.96 g/cm3, which is an order of range value for PDMS [35], the required amounts of phosphor for the three globe-top radii in case of the diffuse substrate are given by 0.00048 g (1 mm), 0.00171 g (2 mm) and 0.00378 g (3 mm) for a CIE x value of 0.33 as well as 0.00079 g (1 mm), 0.00272 g (2 mm) and 0.00592 g (3 mm) in the case of a CIE x value of 0.38. Considering the above mentioned differences for specular (for a CIE x value of 0.33 and a globe-top with a radius of 1 mm 0.00049 g, for 2 mm 0.00175 g and 3 mm 0.003841 g would be required) and diffuse substrate reflectivities, this means that in case of a CIE x value of 0.33 and a specular substrate surface reflectivity an about 1% higher amount of phosphor for the globe-top with the radius of 1 mm would be required and an about 2.5% higher amount for the globe-top with the radius of 3 mm.
These differences of diffuse and specular surface reflectivities are also reflected in a comparison of the mean of the CIE x values for two perpendicular directions (line-by-line and column-by-column averaging) of the data matrix as a function of the pixel number as shown in Figures 3 and 4. Figure 3 compares the mean of the CIE x values for a globe-top with a radius of 1 mm and a substrate surface of either diffuse (left column) or specular (middle column) reflectivity. For a diffuse reflectivity of the substrate surface of 85%, for which the phosphor concentration is adjusted in order to achieve a CIE x value of about 0.3300, the Figure shows that for normal viewing directions the CIE x values are a bit lower than for larger viewing angles. This is in accordance with a recent study by Sun et al. [14], in which it was concluded that the size of a hemispherical globe-top should be increased in normal direction in order to gain white light-emission with low variation of the colour temperature for different viewing angles. The same trend can be observed in case of a diffuse reflectivity of either 70% and 100%. For a specular reflectivity and the same conditions, this behaviour is even more pronounced. This means, the CIE x values become comparably lower in particular for normal viewing directions. This is also emphasized in the difference graphs for the mean of the CIE x values as a function of the pixel number for a specular and a diffuse reflectivity as shown in the right column of Figure 3.
The same behaviour can be also observed in case of the globe-top with a radius of 3 mm, for which the difference graphs for diffuse and specular reflectivities of the substrate surface are shown in the left column of Figure 4.
A detailed comparison of these images with the images shown in the right column of Figure 3 highlights that for a globe-top with 3 mm radius there are in particular more pronounced differences between diffuse and specular reflectivity for larger viewing angles than in the 1 mm case.
The middle column of Figure 4 shows the difference graphs for the mean of the CIE x values for globe-tops having radii of 1 mm and 3 mm for a diffuse reflectivity. While there is no significant difference for a diffuse reflectivity of 85% (in particular since also the phosphor concentrations were adjusted in order to achieve the same CIE x value of about 0.3300 for the different radii), there is a shift to the bluish with increasing globe-top size for decreasing surface reflectivities and a shift to the yellowish with increasing globe-top size for increasing surface reflectivities. However, for the same type of (diffuse) reflectivity, the globe-top radius has no obvious impact on angular dependent colour variations.
A similar behaviour as in case of phosphor concentrations which give reason for CIE x values of about 0.3300 can also be observed for phosphor concentrations giving reason for CIE x values of about 0.3800. This is emphasized in the images of the right column of Figure 4, which show the difference of the mean of the CIE x values as a function of the pixel number for specular and diffuse reflectivities for globe-top radii of 3 mm.
As a conclusion, these studies show that the colour maintenance is the better the smaller the globe-top radius is and that with a substrate surface having a diffuse reflectivity not only higher CIE x values but also lower angular variations of the CIE x values can be realized.
Figures 5 and 6 finally compare the radiant fluxes (blue radiant flux, yellow radiant flux and total radiant flux) for globe-top sizes of 1 mm and 3 mm. For this comparison, again a diffuse reflectivity of the substrate surface of 85% is taken as a reference system. For the same globe-top radii, an increasing substrate surface reflectivity of up to 100% gives reason for higher radiant fluxes, in particular for the yellow radiant flux, which again is due to the higher contribution of the yellow light on the overall backscattered light. Again, this enhancement is more pronounced for globe-tops with larger radii, since in this case the coverage of the substrate by the globe-top material is also larger.
In addition, a comparison of specular and diffuse reflectivity shows, that the observed lower CIE x values for a specular reflectivity as shown in Figure 2 are mainly caused by a reduction of the yellowish fraction of the light.
Contrarily, in case that the reflectivity of the substrate surface is reduced to 70%, the radiant fluxes decrease. Again, the yellow radiant flux is more affected than the blue one, since a larger portion of the backscattered (mainly yellow) light now becomes absorbed by the substrate surface.
Figure 7 compares the radiant flux ratios for the blue, the yellow and the total radiant fluxes of LED packages with a hemispherical globe-top in dependency of the globe-top size (all with respect to a globe-top radius of 1 mm). As obvious, the larger the globe-top radii the larger are the radiant fluxes. As evident from the right side image of Figure 7, the differences in the radiant fluxes of globe-tops with smaller and larger radii are the more pronounced the higher the phosphor concentration is, this means the lower the CCT values are. To conclude and contrary to colour maintenance, which is favoured by smaller globe-top sizes, in terms of the radiant fluxes a larger globe-top radius is advantageously.
In order to highlight the impact of the different surface reflectivities on the colour variations in detail, Figure 8 shows these variations with respect to MacAdam ellipses of step-size 2 (blue dotted), 4 (red solid) and 5 (gray dotted). The ellipses are centerd around a CIE x value of about 0.3300, which represents the reference configuration with a diffuse substrate surface reflectivity of 85%. An increase or a reduction of the surface reflectivity to 100% and 70%, respectively, causes colour variations, which match the outer limits of a MacAdam ellipse of step-size 2 for a globe-top radius of 1 mm and even of step-size 4 in case of a globe-top with a radius of 3 mm (for diffuse reflectivity, see left image of Figure 8). Using a surface with a specular instead of a diffuse reflectivity causes a shift of these deviations to the bluish (middle image of Figure 8). This means, the colour deviation is even more pronounced for a reduction of the surface reflectivity as compared to the case of a diffuse reflectivity of 85%, but is diminished for an increase of the surface reflectivity. However, the relative colour deviations are also in case of a specular reflectivity quite similar to the relative deviations in case of a diffuse reflectivity. This is shown in the right image of Figure 8, for which the phosphor concentrations were varied so that also for a specular reflectivity a CIE x value of about 0.3300 is achieved for a surface reflectivity of 85% (for 1 mm 5.17 vol.%, for 2 mm 2.29 vol.% and for 3 mm 1.49 vol.%). This behaviour is quite similar for globe-tops with phosphor concentrations giving reason for CIE x values of about 0.3800, despite the fact that the overall colour deviations are not that pronounced (see Figure 9).
Therefore, and as discussed above, from the viewpoint of CCT maintenance, a globe-top with smaller size is preferable, since such a globe-top configuration shows the lowest deviation from the initial CIE x value in case that the reflectivity of the substrate surface diminishes because of long term materials degradation.
However, this is contradictory to the request of higher radiant fluxes, which are favoured by larger globe-top sizes.
Anyhow, as mentioned above, besides the optical behaviour, also the thermal behaviour should be considered in a comprehensive discussion of optimized globe-top configurations. Due to the fact that the silicone matrix has a comparable low thermal conductivity (typically 0.2 W/mK) and due to the heat generated within the CCE upon colour conversion (Stokes shift, quantum efficiency lower than unity), there may be a notable thermal load of the CCEs upon device operation [25],[30],[36],[37]. In order to study the impact of the different globe-top configurations on the respective thermal load, Figure 10 shows the calculated temperature profiles along the vertical directions of the LED packages. In this figure, a height z of 0 μm in vertical direction coincides with the top of the PCB. As evident from the left image of Figure 10, which shows the temperature profiles for the three hemispherical globe-top configurations with radii of 1 mm, 2 mm, and 3 mm for a CIE x value of about 0.3300 under the assumption of a quantum efficiency of 100%, the by far highest maximum temperatures are reached for the globe-top with a radius of 1 mm. Such high maximum temperatures indeed may have some drawbacks upon device operation. For example, the luminescence intensity of most of the phosphor materials decreases with increasing temperature. Besides an overall reduction of the luminous efficacy, this may also cause some shifts of the chromaticity coordinates due to the lower fraction of yellowish emission. In addition, higher temperatures also enhance the risk for long term-materials degradation and deterioration of the optical properties. Therefore, from a thermal point of view, globe-tops with larger sizes are preferable in order to keep the thermal stress of the materials as low as possible. However, besides the overall temperature, also the variation of the temperature with increasing current is of importance. Generally, the higher the current the higher is the thermal load which is induced in the CCE. Considering the fact that, as mentioned above, for a Ce doped YAG phosphor a temperature variation of about 40 K may cause a colour deviation which matches the outer limits of a MacAdam ellipse of step 2 [25], larger temperature variations may also cause notable colour variations. As obvious, going from 350 mA to 1000 mA causes a much larger temperature variation in case of the smallest globe-top radius in comparison with those of larger radii. This means that in particular globe-tops with small radii are prone to suffer from temperature dependent colour variations upon device operation at different currents.
These thermal impacts (higher maximum temperatures and larger temperature variations for different driving currents in case of smaller globe-top radii) again also dimish the aforementioned advantages of smaller globe-top radii in regard of CCT maintenance and restricts their applicability in particular to applications for which the current is not varied upon device operation.