The output peak power is measured with an integrating sphere coupled to a spectrometer (LCS-100 from Labsphere). As the output beam is Lambertian and strongly divergent, the entrance of the integrating sphere is positioned very close to the output surface for accurate measurements. The evolution of the output peak power versus the pump peak power is reported in Fig. 4, with and without the “back” mirror. The maximum peak power is 116 W for a pump peak power of 3405 W corresponding to a brightness of 167 W/cm2/sr and an optical efficiency of 3.4%. In comparison, a Ce:YAG material in a similar setup provides an optical efficiency of 7.7% [12].

In order to understand the origin of the limited performance, we modelized the output power as follows. Self-absorption losses are taken in account by separating the emission in two contributions P1 and P2., with respect to the propagation distance in the crystal.

P1 is the power related to "one-way" photons that are emitted in the output escape cone of the \(w.t\) output face.

P2, is the power related to photons emitted in the backward emission cone and that propagate towards the output face after a reflection on the back mirror.

The expression of P1 is found by integration between z = 0 and z = *l* of all the emission in the output cone.

$${{P}_{1}=T}_{LED}.{\eta }_{r}\frac{{\lambda }_{p}}{{\lambda }_{em}}{P}_{pump}\text{ }(1-{e}^{-t{\alpha }_{p}})\frac{1-cos{\theta }_{TIR}}{2}\frac{1}{{\alpha }_{1}Kl}\left(1-{e}^{-{\alpha }_{1}Kl}\right)$$

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where

\({T}_{LED}\) is the transmission of the \(w.l\) face for the LED: As the LED emission cone is 30°, this quantity can be approximated by Fresnel transmission at normal incidence (\({T}_{LED}=0.91)\),

\({\lambda }_{p}\) is the average pump wavelength (367 nm),

\({\lambda }_{em}\) is the average emitted wavelength (455 nm),

\({P}_{pump}\) is the peak pump power (from 0 to 3405 W),

\({\alpha }_{1}\) is the loss propagation coefficient for these photons propagating from 0 to \(l\) in the concentrator. The loss has 3 contributions:

$${\alpha }_{1}={\alpha }_{0}+{\alpha }_{l}+{\alpha }_{ESA}$$

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where

α0 corresponds to the losses of the concentrator including the crystal quality, including diffusion loss coefficient and the polishing quality (α0 = 4.10-3 cm1),

\({\alpha }_{l}\) corresponds to the self-absorption losses \(({\alpha }_{l}\)= 1.7 10− 2 cm-1) and is calculated using spectral absorption data presented in Fig. 2,

\({\alpha }_{ESA}\) corresponds to the losses induced by excited state absorption \({\alpha }_{ESA}={\sigma }_{ESA}{n}_{2}\), \({n}_{2}\) being the population density of the upper state level. As in [12], we neglect the decrease of \({n}_{2}\), caused by ESA because of the low pump power density of LEDs. Therefore, the relation between \({n}_{2}\) and the pump power \({P}_{pump}\) is:

$${n}_{2}=\tau \frac{{P}_{pump}{T}_{LED}}{lwt}\frac{{\lambda }_{p}}{hc}\left.{\left(1-e\right.}^{-t{\alpha }_{p}}\right)$$

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where τ is the lifetime of the upper level (τ = 41 ns).

The expression for P2 is similar to the expression for P1 but with an extra reflection on the opposite face and an additional propagation between *l* and *2l* in order to reach the output face.

In the case of a back mirror, the expression of P2 is:

$${P}_{2}={RP}_{1}{e}^{-{\alpha }_{2}Kl}$$

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where

$${\alpha }_{2}={\alpha }_{0}+{\alpha }_{2l}+{\alpha }_{ESA}$$

\({\alpha }_{2l}\) corresponds to the self-absorption losses for a propagation between *l* and *2l.* It is calculated similarly as \({\alpha }_{l}\) by considering the spectral density \({I}_{l}\left(\lambda \right)\) and \({I}_{2l}\left(\lambda \right)\) presented on Fig. 2 \(({\alpha }_{2l}\)= 9 10− 3 cm-1).

Without the back mirror, a Fresnel reflection on the opposite face must be considered, the expression of P2 is therefore:

$${P}_{2}={(1-{T}_{Fresnel})P}_{1}{e}^{-{\alpha }_{2}Kl}$$

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where \({T}_{Fresnel}\) is the Fresnel transmission of the \(w.t\) interfaces taking all directions in the output escape cone and all polarizations in account (\({T}_{Fresnel}\)=0.86).

For the calculation of the output power, we consider the light transmitted by the output \(w.t\) interface (related to \({T}_{Fresnel}\) ) and the light reflected by this interface coming back after a reflection on the opposite face. Multiple reflection light can be simulated as in [22]. It corresponds to propagation lengths in the concentrator larger than \(2l\).

Therefore, the output power with a back mirror having a reflectivity \(R\) is:

$${P}_{2Dmirror}=({P}_{1}+{P}_{2})\frac{{T}_{Fresnel}}{1-{e}^{-2{\alpha }_{3}Kl}R\left(1-{T}_{Fresnel}\right)}$$

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where \({\alpha }_{3}\) correspond to losses for photons propagating distances larger than \(2l\) in the concentrator\({\alpha }_{3}={\alpha }_{0}+{\alpha }_{3l}+{\alpha }_{ESA}\)

For those photons, self-absorption losses are estimated from our measurements presented in Fig. 2. Considering the spectral density \({I}_{2l}\left(\lambda \right)\) and \({I}_{3l}\left(\lambda \right)\) a loss coefficient namely \({\alpha }_{3l}\) can be deduced \(({\alpha }_{3l}\)= 4 10− 3 cm-1). As the spectral shape evolves only slightly between \(2l\) and \(3l\), we can suppose that this value can also be used for longer propagation distances than \(3l\).

Without the back mirror, the output power is found by replacing \(R\) by \(1-{T}_{Fresnel}\):

$${P}_{2D}=({P}_{1}+{P}_{2})\frac{{T}_{Fresnel}}{1-{e}^{-2{\alpha }_{3}Kl}{\left(1-{T}_{Fresnel}\right)}^{2}}$$

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The simulations are presented in Fig. 4. In order to separate the different limitations, we calculate also the output power with and without excited-state absorption (ESA) and self-absorption. Figure 4 shows that self-absorption has a stronger influence than ESA. ESA effect is more visible with the back mirror since longer propagation distances render this configuration more sensitive to losses.

The simulations are adjusted to the experimental data thanks to three loss parameters that have not been measured: the effective reflectivity R of the back mirror (taking the large divergence of the output beam in account), the cross section of the excited-state absorption and the effective loss coefficient \({\alpha }_{0}\) taking in account crystal imperfections (diffusion measured to be 10-3cm-1 and polishing quality). For a better accuracy, those parameters are adjusted by fitting the output power obtained in the "3D configuration" (see Section 5) which is much more sensitive to losses because of long propagation distances inside the concentrator. Despite the indirect adjustment, Fig. 4 shows that the 2D configuration simulations agree with experimental value\(s.\)