Laser rate equations

The rate equations of the proposed model for the population inversion N and number of photons ϕ are almost identical to the conventional ones⁶⁰:

where N is the population inversion, ϕ is the number of photons, σ_st is the stimulated emission cross-section, R_p is the pumping rate, τ₂ is the effective lifetime of upper laser level, τ_cav is the cavity lifetime, V_N is the mode volume in the active medium, V_ϕ is the mode volume in the cavity, and c is the speed of light. The four-level laser scheme is assumed. Since NRLs do not have ‘mode’ volumes, the effective 3D volume of the photon distribution should be used instead for V_ϕ and V_N. From the stationary solutions of Eqs. (⁷) and (⁸), Eq. (¹) can be derived with the pumping threshold:

where η_p is the cavity pumping efficiency in Eq. (¹) and η0p is the practical efficiency of the pumping process. It is noteworthy that η_o = η0pη0em in Eq. (¹) is the total practical efficiency that combines the imperfections in both the pumping and emission processes, η0p and η0em, respectively.

According to Fig. 6a, we estimated V_ϕ as the sphere volume without the spherical cap with polar angle θ:

where θ=sin−1(1/β) and δD_cav is the effective penetration depth of the emission light for the used scattering medium. For τ_cav, the lifetime for the integrating spheres is used⁶¹:

where d¯ is the mean distance between wall reflections and δτ_cav is the effective dwell time of the emission light for the used scattering medium. We numerically calculated d¯ (see Supplementary Fig. ⁵). It is worth noting that d¯θ<1≈d¯θ=0=2/3Dcav. The effective dwell time and penetration depth of the used scattering medium were calculated based on the measured scattering and absorption coefficients⁶². Specular reflection between the air and scattering medium interface was also considered roughly using the effective refractive index and measured volume fraction of the employed scattering medium. The detailed equations can be found in the Supplementary Information. In this work, both δD_cav/D_cav and δτ_cav/τ_cav were less than 0.05. The known properties of 1% Nd:YAG were utilized, τ₂ = 230 μs and σ_st = 6.5 × 10⁻²³ m² from ref. ²⁹.