Photon avalanche (PA) is a unique upconversion mechanism exhibiting a giant optical nonlinear response,1^–3 which promises applications in advanced technologies such as microlasers,4 integrated photonic circuits,5 super-resolution microscopy,6^,7 optical sensing,8^,9 and quantum optics.10 Unlike traditional upconversion processes11 and avalanche-like mechanisms,12 the crucial trick to realize the PA effect is to incorporate a non-resonant ground state absorption (GSA) process with a much stronger (usually ${10}^4$ times) excited state absorption (ESA) process and maintain an efficient cross relaxation. In other words, an effective looping cycle should be kept to ensure the accumulation of electrons in the intermediate excited states, eventually leading to a burst of emission intensity. Due to the harsh excitation and energy transfer requirements, along with unignorable surface quenching in nanoparticles, the PA effect has long been restricted only in bulk materials.13^–20

Recently, efficient PA phenomena have been reported separately in ${\mathrm{Tm}}^{3+}$-doped and ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}$-codoped nanoparticles by harnessing interionic energy transfer processes.2^,3 However, the PA effect achieved in the ${\mathrm{Tm}}^{3+}$ singly doped mechanism under 1064 nm excitation was only restricted to 800 nm emission.2 Importantly, PA emissions with giant optical nonlinearity (≥ 20th order) have not yet been realized in many other ion systems. For example, lanthanide ions such as ${\mathrm{Tb}}^{3+}$, ${\mathrm{Eu}}^{3+}$, ${\mathrm{Dy}}^{3+}$, and ${\mathrm{Sm}}^{3+}$, which are widely used in many fields such as quantum cutting and lighting, cannot efficiently generate PA.21 The unfavorable energy-looping-free characteristics of those energy levels are contrary to the prerequisites of triggering an efficient PA effect, making it challenging to find a proper PA excitation scheme (see Fig. S1 in the Supplementary Material). Although the PA effect can appear in some of these ions, the observed nonlinearities are relatively low, and single-nanoparticle brightness cannot be detected.22 Therefore, developing a bright, huge-nonlinearity, universal mechanism with relevance to practice for generating PA emissions in various emitters and realizing advanced photonics applications remains a formidable task.

In a three-level PA scheme based on a traditional single-type-ion strategy [Fig. 1(a)], the electrons at the ground state are first promoted to the metastable state through non-resonant GSA and followed by the higher-energy excited state to be populated via the resonant ESA. When the excitation intensity exceeds a critical threshold, the ESA and cross-relaxation process constitute a positive energy-looping cycle, boosting the population of the emitting levels and leading to a sharp emission burst. To achieve a high-efficiency PA effect in nanosized materials, a strategy of dual-type-ion doping was proposed to address issues such as large energy loss related to nonradiative relaxation, surface (-coupled concentration) quenching in nanomaterials [Fig. 1(a)]. This dual-type-ion (${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}$) PA mechanism proposed in our previous work can also obtain ${\mathrm{Yb}}^{3+}$ sensitizer sublattices avalanched and thereby propagate the PA energy to different sites.3 However, the challenge of these existing single/dual-type-ion PA mechanisms in extending the PA effect has been met with limited success, i.e., ${\mathrm{Ho}}^{3+}$ and ${\mathrm{Tm}}^{3+}$ ions only. Some important emitters with a large gap between the ground state and the emitting states such as ${\mathrm{Tb}}^{3+}$, ${\mathrm{Eu}}^{3+}$, ${\mathrm{Dy}}^{3+}$, and ${\mathrm{Sm}}^{3+}$ can hardly get avalanched likewise because the cooperative sensitization upconversion (CSU) process between the sensitizers ${\mathrm{Yb}}^{3+}$ and these ions is inefficient,11 which hinders multiplexing capability and extremely nonlinear applications. Here, we proposed a powerful strategy named cascade migration photon avalanche (cMPA) to enrich the library of PA-achievable emitters. In this cMPA mechanism, $A^{3+}$ collects the PA energy from the nano-engine in the first MPA network (MPA-I). Subsequently, it acts as the nano-engine of the secondary sublattice energy network (MPA-II), and the PA energy can further migrate to other emitters $X^{3+}$ [Fig. 1(b)]. In other words, $A^{3+}$ is a crucial accumulator in this cMPA mechanism to bridge the MPA-I and the MPA-II networks. For the proof of concept, we selected a ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}$-codoped core as the PA nano-engine of MPA-I to establish the cMPA system. ${\mathrm{Tm}}^{3+}$ ion was chosen as the accumulator because it is capable of accepting PA energy from the MPA-I network via the ${\mathrm{Yb}}^{3+}$ sublattice and further incorporation with ${\mathrm{Gd}}^{3+}$ sublattice,11 which can provide energy for a variety of ions that cannot be easily excited by ${\mathrm{Yb}}^{3+}$ sublattice. Therefore, the ${\mathrm{Tm}}^{3+}/{\mathrm{Gd}}^{3+}$ network was designed as an MPA-II network for the cMPA system.

To demonstrate the feasibility of the above-designed cMPA mechanism, two theoretical models of cMPA were established based on differential rate equations, from which simulations and analysis were conducted. We first studied the ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}/{\mathrm{Tm}}^{3+}$-codoped nano-engine for the cMPA system (see Method S1 and Table S1 in the Supplementary Material). According to the calculations, clear excitation thresholds ($420\mathrm{kW}{\mathrm{cm}}^{-2}$) and $\text{S}$-shaped curves of excitation-dependent population rates of the emitting states could be observed for the ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}/{\mathrm{Tm}}^{3+}$-codoped system [Fig. 1(c) and Fig. S2 in the Supplementary Material]. The obtained optical nonlinearities of ${\mathrm{Pr}}^{3+}$ emission at 484 nm and ${\mathrm{Tm}}^{3+}$ emissions at 475 nm and 452 nm were 19th order, 34th order, and 48th order, respectively. Under the excitation of the 852 nm laser, an efficient PA effect can be initiated in the ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}$-doped core, then the avalanching energy migrates to the ${\mathrm{Tm}}^{3+}$-doped shell through the ${\mathrm{Yb}}^{3+}$ sublattice network. Since the $475/452\mathrm{nm}$ emissions of ${\mathrm{Tm}}^{3+}$ correspond to three-/four-photon upconversion, the optical nonlinearities of the two emissions can be further amplified but showed a discrepancy in the amplification factor. Next, the whole cMPA system of ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}/{\mathrm{Tm}}^{3+}/{\mathrm{Gd}}^{+}/X^{3+}$-codoped nanoparticles was theoretically modeled and studied (see Appendix A and Table S1 in the Supplementary Material). It was found that the cascade MPA networks enable the energy transfer from the ${{}^{6}P}_{7/2}$ state of ${\mathrm{Gd}}^{3+}$ ions to other well-matched emitters and finally arouse PA from them. Intriguingly, a positive PA evolution with optical nonlinearity being enhanced can be observed during the energy transfer from ${\mathrm{Tm}}^{3+}$ ions to ${\mathrm{Gd}}^{3+}$ ions, allowing cMPA to further amplify the optical nonlinear response [Fig. 1(d) and Fig. S3 in the Supplementary Material]. In this context, the group of emitters ${\mathrm{Tb}}^{3+}$, ${\mathrm{Eu}}^{3+}$, ${\mathrm{Dy}}^{3+}$, and ${\mathrm{Sm}}^{3+}$ can all get avalanched under a fixed excitation scheme, driving extremely nonlinear applications and multiplexing capability.

Subsequently, experimental studies were carried out under the guidance of the aforementioned theory. To ensure the PA energy migrating to the ${\mathrm{Gd}}^{3+}$-mediated network for achieving effective cMPA, the PA effect of ${\mathrm{Tm}}^{3+}$ ions is very crucial and has to be optimized in terms of emission intensity and optical nonlinearity [Fig. 2(a)]. We synthesized multilayered core-shell nanoparticles by controlling the ${\mathrm{Yb}}^{3+}$ doping concentrations ${\mathrm{NaYF}}_4:\mathrm{Yb}/\mathrm{Pr}(x/0.5\%)@{\mathrm{NaYF}}_4:\mathrm{Yb}/\mathrm{Tm}(y/4\%)@{\mathrm{NaYF}}_4:\mathrm{Yb}/\mathrm{Pr}(x/0.5\%)@{\mathrm{NaYF}}_4$ ($x=15\%,25\%;y=3\%,10\%$). The nanoparticles were synthesized according to our previously reported protocol23 (see Method S2 in the Supplementary Material), which were characterized by transmission electron microscopy (see Fig. S4 in the Supplementary Material). A lab-built microspectroscope was utilized to characterize the PA effect (see Appendix B and Fig. S5 in the Supplementary Material). Under the 852 nm continuous-wave (CW) excitation, multicolor emissions from the ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}/{\mathrm{Tm}}^{3+}$ system were obtained, assigned to the transitions of ${{}^{3}P}_0\to {{}^{3}F}_2$ (642 nm), ${{}^{1}D}_2\to {{}^{3}H}_4$ (609 nm), ${{}^{3}P}_0\to {{}^{3}H}_6$ (605 nm), ${{}^{3}P}_1\to {{}^{3}H}_5$ (525 nm), ${{}^{3}P}_0\to {{}^{3}H}_4$ (484 nm), and ${{}^{1}D}_2\to {{}^{3}F}_4$ (452 nm) [Fig. 2(b)]. When appropriately increasing the ${\mathrm{Yb}}^{3+}$ concentration in the cascade shell (${\mathrm{Yb}}^{3+}/{\mathrm{Tm}}^{3+}$) from 3% to 10%, the emission intensity of ${\mathrm{Tm}}^{3+}$ emitters at 452 nm was significantly enhanced. However, the 10% doping concentration of ${\mathrm{Yb}}^{3+}$ ions in the shell was close to the 15% ${\mathrm{Yb}}^{3+}$ doped in the PA core, resulting in a significant energy leakage for the core and weakening its PA optical nonlinearity. Therefore, the ${\mathrm{Yb}}^{3+}$ doping concentration in the avalanched ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}$-codoped core was elevated from 15% to 25% accordingly. As the excitation-intensity-dependent emission curves shown in Fig. 2(c), a maximum optical nonlinearity order of 63.3 ($62.3\pm 0.8$ from three independent measurements, Fig. S6 in the Supplementary Material) was obtained when the ${\mathrm{Yb}}^{3+}$ concentrations in the core and shell layer were 25% and 10%, respectively. In our experiments, the excitation-emission power dependences were measured from nanoparticle ensembles. Therefore, optimizing the doping concentration can facilitate to establish highly efficient energy migrating pathways, enabling bright PA emission from ${\mathrm{Tm}}^{3+}$ ions with ultrahigh nonlinearity order [Fig. 2(d)].

To verify the possibility of energy transfer in the ${\mathrm{Gd}}^{3+}$-based MPA-II network, we designed and fabricated multilayered nanoparticles with composites of ${\mathrm{NaYF}}_4:\mathrm{Yb}/\mathrm{Pr}(25\%/0.5\%)$$@{\mathrm{NaGdF}}_4:\mathrm{Yb}/\mathrm{Tm}(10\%/4\%)@{\mathrm{NaGdF}}_4:\mathrm{Tb}(20\%)@{\mathrm{NaYF}}_4$, as shown in Fig. 3(a).24^–26 High-angle annular dark-field scanning transmission electron microscopy (HADDF-STEM) and energy dispersive spectroscopy (EDS) mapping images confirmed the formation of the heterogeneous multilayered core-shell structures with clear component boundaries (see Fig. S7 in the Supplementary Material). As shown in the mechanistic diagram [Fig. 3(b)], the entire cMPA process for ${\mathrm{Tb}}^{3+}$ emission can be explained as below. The ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}$-codoped core, as the role of the PA nano-engine, was first illuminated, and a huge optical nonlinear response was activated for both ${\mathrm{Pr}}^{3+}$ and ${\mathrm{Yb}}^{3+}$ ions. Next, the energy states of the ${\mathrm{Tm}}^{3+}$ ion, including the high-lying state ${{}^{1}I}_6$, were populated and thereby got avalanched by a five-step upconversion process from the ${\mathrm{Yb}}^{3+}$ sublattice. The avalanched behavior can rapidly propagate within and beyond the MPA-I network and then arrive in the MPA-II network via the energy transfer from the ${{}^{1}I}_6$ state (${\mathrm{Tm}}^{3+}$) to the ${{}^{6}P}_{7/2}$ state (${\mathrm{Gd}}^{3+}$). Finally, the avalanched ${\mathrm{Gd}}^{3+}$ network can easily pump the emitting states of ${\mathrm{Tb}}^{3+}$ ions and enable multi-color PA emissions including 544 nm (${{}^{5}D}_4\to {{}^{7}F}_5$), 585 nm (${{}^{5}D}_4\to {{}^{7}F}_4$), and 620 nm (${{}^{5}D}_4\to {{}^{7}F}_3$). In the experiment, three characteristic emissions of ${\mathrm{Tb}}^{3+}$ ions at 544 nm, 585 nm, and 620 nm were obviously observed from the ${\mathrm{Tb}}^{3+}$-doped PA nanoparticles under 852 nm excitation [Fig. 3(c)], which demonstrated the ability of the cMPA mechanism for propagating the PA effect to ${\mathrm{Tb}}^{3+}$ ions through the cascaded migration networks of ${\mathrm{Yb}}^{3+}$ and ${\mathrm{Gd}}^{3+}$ sublattice. The emission spectra of ${\mathrm{Tb}}^{3+}$ emitters under different excitation intensities were recorded, and the excitation-emission power dependence exhibited a typical $\text{S}$-shaped curve with a giant optical nonlinearity up to 42.6 ($43.6\pm 0.9$) order and a PA threshold at $370\mathrm{kW}{\mathrm{cm}}^{-2}$ [Fig. 3(d)]. The nonlinearity order for 484 nm emission of ${\mathrm{Pr}}^{3+}$ emitters was 25.7, while it was 36.6 for ${\mathrm{Tm}}^{3+}$ 452 nm emission. The multi-photon upconversion process within the MPA-I network further amplified the nonlinearity of the PA engine, resulting in the nonlinearity enabled by the cMPA mechanism far exceeding that by the ${\mathrm{Tm}}^{3+}$ singly doped PA mechanism.22 To further confirm that the PA energy of ${\mathrm{Tb}}^{3+}$ emitters originated from the ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}$-codoped nano-engine, we also switched to 980 nm excitation for the nanoparticles and investigated the upconversion luminescence likewise. No obvious characteristic emission of ${\mathrm{Pr}}^{3+}$ ions was observed under 980 nm excitation [Fig. 3(e)]. In contrast, ${\mathrm{Tb}}^{3+}$ ions can be excited through the energy migration-mediated upconversion (EMU) process, exhibiting intense emissions. As expected, the inactivation of the PA engine core made the ${\mathrm{Tb}}^{3+}$ emissions unavalanched [Fig. 3(f)].

Theoretically, the avalanched ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}$-codoped core can also alternatively arouse the PA effect on ${\mathrm{Tb}}^{3+}$ ions through the ${\mathrm{Yb}}^{3+}$-mediated CSU process. We also fabricated ${\mathrm{Tb}}^{3+}$-doped CSU nanoparticles and studied its PA properties [Fig. 3(g)]. It is well known that, in this case, the energy of a pair of excited ${\mathrm{Yb}}^{3+}$ has to be simultaneously transferred to one adjacent ${\mathrm{Tb}}^{3+}$ ion, meaning that the CSU process is a very inefficient, second-order cooperative sensitization process.27 Under 852 nm excitation, the luminescent spectra of ${\mathrm{Tb}}^{3+}$-doped CSU nanoparticles were measured, and they overlapped very well with that of ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}$-codoped PA nanoparticles as shown in Fig. 3(h), indicating that no emission can be observed from ${\mathrm{Tb}}^{3+}$ ions. This verified that the proposed cMPA mechanism is indispensable as well as powerful to arouse PA emissions from ${\mathrm{Tb}}^{3+}$ ions.

To further extend PA behavior to many other emitters through the proposed cMPA mechanism, we synthesized ${\mathrm{Eu}}^{3+}$-, ${\mathrm{Dy}}^{3+}$-, and ${\mathrm{Sm}}^{3+}$-activated four-layered core-shell nanoparticles, respectively, ${\mathrm{NaYF}}_4:\mathrm{Yb}/\mathrm{Pr}(25\%/0.5\%)@{\mathrm{NaGdF}}_4:\mathrm{Yb}/\mathrm{Tm}(10\%/4\%)@{\mathrm{NaGdF}}_4:X@{\mathrm{NaYF}}_4$ ($X=20\%$ Eu, 5% Dy, or 5% Sm). As the ions ${\mathrm{Eu}}^{3+}$ and ${\mathrm{Dy}}^{3+}$ have a similar structure of intrinsic electronic energy states to the ${\mathrm{Tb}}^{3+}$ ion, they can capture the PA energy from the ${\mathrm{Gd}}^{3+}$-based cMPA network [Figs. 4(a) and 4(d)]. Two characteristic emission bands from ${\mathrm{Eu}}^{3+}$ ions were observed with peaks at 592 nm (${{}^{5}D}_0\to {{}^{7}F}_1$) and 617 nm (${{}^{5}D}_0\to {{}^{7}F}_2$) [Fig. 4(b)], achieving a 45.3 ($44.9\pm 2.2$) order optical nonlinearity at the 592 nm emission when the laser power was approaching the PA threshold [$\sim 330\mathrm{kW}{\mathrm{cm}}^{-2}$, Fig. 4(c)]. Similarly, the characteristic emission peak of ${\mathrm{Dy}}^{3+}$ ions was observed at 574 nm (${{}^{4}F}_{9/2}\to {{}^{6}H}_{11/2}$) with a giant optical nonlinearity up to 45.4 ($44.7\pm 1.2$) order [Figs. 4(e) and 4(f)]. The PA threshold of ${\mathrm{Dy}}^{3+}$-doped cMPA nanoparticles was around $400\mathrm{kW}{\mathrm{cm}}^{-2}$. The variation in their PA thresholds may be attributed to the discrepancies of energy transfer efficiency among different ions, after all the ions with less pumping efficiency would require higher energy to get avalanched. Similar to the result of ${\mathrm{Tb}}^{3+}$ ions, here ${\mathrm{Eu}}^{3+}$ and ${\mathrm{Dy}}^{3+}$ ions did not exhibit any PA nature under 980 nm excitation (see Fig. S8 in the Supplementary Material). In the case of ${\mathrm{Sm}}^{3+}$-doped nanoparticles, the energy transfer mechanism is analogous to that of ${\mathrm{Tb}}^{3+}$, ${\mathrm{Eu}}^{3+}$, and ${\mathrm{Dy}}^{3+}$ ions [see Fig. S9(a) in the Supplementary Material]. With the same cMPA mechanism, a certain PA emission can also be observed from ${\mathrm{Sm}}^{3+}$ ions at 563 nm (${{}^{4}G}_{5/2}\to {{}^{6}H}_{5/2}$) [see Fig. S9(b) in the Supplementary Material], although its intensity was too weak to measure the excitation-intensity-dependent emission curve, which was mainly due to the largely enhanced non-radiative relaxation effect for ${\mathrm{Sm}}^{3+}$ ions with very dense energy states.11 Overall, the developed cMPA mechanism can promote the full-spectrum PA emission from these lanthanide emitters, including but not limited to the visible light at 400 to 700 nm [Fig. 4(g)]. The ${\mathrm{Tm}}^{3+}$ singly doped PA mechanism under 1064 nm excitation can also enable avalanching emission from a series of lanthanide ions.12 However, the energy looping locked the energy mainly to the ${{}^{3}H}_4$ state featuring an 800 nm PA emission, and the higher-lying energy states such as ${{}^{1}I}_6$ cannot be effectively populated, resulting in very weak emission and low optical nonlinearities from the acceptor ions (see Fig. S10 in the Supplementary Material).

With the capability of generating the high-efficiency PA effects from many lanthanide ions, our proposed cMPA mechanism provides new insights into fascinating applications of relevance, i.e., super-resolution microscopic imaging.28^,29 As a demonstration, we implemented low-power, single-NIR-CW-beam optical nanoscopy. We built a laser-scanning microscope coupled with an 852 nm CW beam for cMPA super-resolution microscopy [Fig. 5(a), see Appendix B]. First, the ${\mathrm{Tm}}^{3+}$-doped cMPA nanoparticles distributed on the slide were imaged. By adjusting the excitation intensity, a series of images at 605 nm of ${\mathrm{Pr}}^{3+}$ emitters [Fig. 5(b)] and at 452 nm of ${\mathrm{Tm}}^{3+}$ emitters [Fig. 5(c)] were obtained. In the 452 nm detection channel of ${\mathrm{Tm}}^{3+}$, the diffraction-unlimited full width at half-maximum (FWHM) of the effective point spread function (PSF) was measured to be 48 nm ($\lambda /17$) at an excitation intensity of $320\mathrm{kW}{\mathrm{cm}}^{-2}$ [Fig. 5(c-iv)]. In comparison, a diffraction-limited FWHM of 225 nm was observed at a high-power laser excitation of $1403\mathrm{kW}{\mathrm{cm}}^{-2}$ [Fig. 5(c-ii)], demonstrating a five-fold improvement in the lateral resolution. It is worth mentioning that it would take time for the PA energy looping cycles to build up an efficient PA effect. In the scanning imaging experiments, we took a $100\mu \mathrm{s}$ pixel dwell time, compatible with a standard confocal microscope, and achieved super-resolution imaging at high speed.30^,31 The improvement of the imaging resolution reflected the obtained non-steady-state PA nonlinearity orders during imaging were slightly lower than the steady-state test results. In the 605 nm detection channel of ${\mathrm{Pr}}^{3+}$, the lateral resolution was improved by three-fold [Figs. 5(b-ii) and 5(b-iv)]. These results confirmed that the lateral resolution was higher in the 452 nm channel, corroborating the higher nonlinearity order of PA in ${\mathrm{Tm}}^{3+}$ ions compared to ${\mathrm{Pr}}^{3+}$ ions. ${\mathrm{Tm}}^{3+}$ ions exhibited higher sensitivity to the variation of excitation intensity, thereby enabling clearer separation of two adjacent nanoparticles (separated by 200 nm) in the 452 nm channel [Figs. 5(b-v) and 5(c-v)]. To further verify the super-resolution imaging capability in $X^{3+}$ ($X=\mathrm{Tb}/\mathrm{Eu}/\mathrm{Dy}$), we took ${\mathrm{Tb}}^{3+}$ ions as an example and imaged ${\mathrm{Tb}}^{3+}$-doped cMPA nanoparticles. A series of images at 484 nm of ${\mathrm{Pr}}^{3+}$ ions [Fig. 5(d)] and at 545 nm of ${\mathrm{Tb}}^{3+}$ ions [Fig. 5(e)] emission peaks were captured. By varying the excitation intensity, the images showed a diffraction-unlimited FWHM of 80 nm ($\lambda /10$) in the 545 nm detection channel [Fig. 5(e-iv)]. Compared with the diffraction-limited FWHM of 238 nm [Fig. 5(e-ii)], the lateral resolution was improved by three-fold. Similarly, the resolution improvement of the ${\mathrm{Tb}}^{3+}$ channel is more significant than that of the ${\mathrm{Pr}}^{3+}$ channel [Fig. 5(d-iv)], which is consistent with the observed excitation power-dependent curves. These results further confirm that the PA energy can be effectively transferred to the activator ions in the shell layer through the proposed cMPA mechanism, enabling long-distance PA energy transfer pathways that hold promise for the applications of many other ions in super-resolution microscopy.

In addition, taking advantage of the full-spectrum and tunable threshold properties of cMPA emissions, we demonstrated multi-color cMPA super-resolution microscopy using a single excitation beam [Fig. 6(a)]. The ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}$-codoped PA nanoparticles and the ${\mathrm{Tb}}^{3+}$-doped cMPA nanoparticles were mixed and evenly dispersed on the coverslip for multi-color super-resolution imaging. Interestingly, cMPA nanoprobes with different PA thresholds provided a new dimension of excitation-power-controlled multiplexing imaging. The PA threshold of ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}$-codoped PA nanoparticles ($\sim 70\mathrm{kW}{\mathrm{cm}}^{-2}$) is lower than that of the ${\mathrm{Tb}}^{3+}$-doped cMPA nanoparticles ($\sim 370\mathrm{kW}{\mathrm{cm}}^{-2}$). By adjusting the excitation power, we could selectively detect emissions from different nanoparticles [Figs. 6(b)–6(d)]. We have experimentally demonstrated that our full-spectrum cMPA mechanism can be used for dual-color super-resolution microscopic imaging [Figs. 6(e) and 6(f)]. Furthermore, the cMPA nanoparticles do not exhibit photobleaching or photodarkening (see Fig. S11 in the Supplementary Material), providing high-efficiency emission and excellent photostability during long-term laser-scanning imaging and monitoring.

By cooperating with the ${\mathrm{Gd}}^{3+}$ and ${\mathrm{Yb}}^{3+}$ sublattice networks in a multilayered core-shell structure, we have successfully developed a universal mechanism that can generate abundant energy transfer paths, leading to full-spectrum PA with ultrahigh nonlinearities (45th and 63rd order). Through elaborate design and control of this cMPA mechanism, different emitters with mismatched energy states against the PA rules can be activated for an extremely optical nonlinear response. By optimizing the ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}$-codoped nano-engine, we observed a bright 452 nm emission of ${\mathrm{Tm}}^{3+}$ with a giant optical nonlinearity of 63rd order. Via the cMPA strategy, other spatially separated ions (${\mathrm{Tb}}^{3+}$, ${\mathrm{Eu}}^{3+}$, ${\mathrm{Dy}}^{3+}$, ${\mathrm{Sm}}^{3+}$) can be activated by this optimized interior PA engine, largely enriching the spectral range of PA emission not limited to the visible range. The observed optical nonlinearity up to 45th order and the achieved bright single-nanoparticle imaging verify that cMPA is a very efficient and powerful PA mechanism of practical significance. As an example, we realized sub-50-nm single-beam super-resolution imaging with the advantages of single-nanoparticle sensitivity and non-photobleaching. With the cMPA concept, we also proposed excitation power as a new dimension for multiplexing and demonstrated dual-color super-resolution imaging. The inherited full-spectrum nonlinear emissions are capable of multiphoton microscopy with single-beam shaping and imaging adaptive optics. In addition, the demonstrated cMPA might also be extended to other luminophores, such as inorganic-organic hybrid luminescence systems, to generate significant optical nonlinearities from them.32 We envision that the cMPA capabilities will provide new insights into designing next-generation nonlinear nanomaterials with significant optical nonlinearity, enabling cutting-edge applications such as low-power single-beam-driven diffraction-unlimited sensing,2 miniaturized lasers,4 optical data storage,33^,34 and lithography.35

The photon avalanches of the system doped with ${\mathrm{Yb}}^{3+}/{\mathrm{Pr}}^{3+}\text{-}{\mathrm{Yb}}^{3+}/{\mathrm{Tm}}^{3+}/{\mathrm{Gd}}^{3+}\text{-}{\mathrm{Gd}}^{3+}/X^{3+}$ can be described by a set of rate equations. The energy scheme and energy transfer pathway are shown in Fig. S3 of the Supplementary Material. The rate equations for each of the energy states can be established as follows: $${\mathrm{Pr}}^{3+}({}^3{H}_4):\frac{\text{d}{n}_0}{\mathrm{d}t}=-\frac{I_{852}}{\hslash \upsilon }{\sigma }_0{n}_0-w_1{n}_{Yb1}{n}_0+w_1^{\prime }{n}_{Yb0}{n}_5+{\beta }_1{n}_1+\frac{n_6}{{\tau }_6}+b_{70}\frac{n_7}{{\tau }_7}+b_{80}\frac{n_8}{{\tau }_8}-c_1{n}_0{n}_7-c_5{n}_0{n}_7-c_4{n}_0{n}_7-c_3{n}_0{n}_5-c_6{n}_0{n}_8-c_2{n}_0{n}_6-c_7{n}_0{n}_6,$$(1)$${\mathrm{Pr}}^{3+}({}^3{H}_5):\frac{\mathrm{d}{n}_1}{\mathrm{d}t}=-{\beta }_1{n}_1+{\beta }_2{n}_2+\frac{n_5}{{\tau }_5}+b_{71}\frac{n_7}{{\tau }_7}+b_{81}\frac{n_8}{{\tau }_8},$$(2)$${\mathrm{Pr}}^{3+}{(}^3{H}_6):\frac{\mathrm{d}{n}_2}{\mathrm{d}t}=-{\beta }_2{n}_2+{\beta }_3{n}_3+b_{82}\frac{n_8}{{\tau }_8}+c_5{n}_0{n}_7+c_4{n}_0{n}_7+c_3{n}_0{n}_5,$$(3)$${\mathrm{Pr}}^{3+}{(}^3{F}_2):\frac{\mathrm{d}{n}_3}{\mathrm{d}t}=-{\beta }_3{n}_3+{\beta }_4{n}_4+b_{73}\frac{n_7}{{\tau }_7}+c_3{n}_0{n}_5,$$(4)$${\mathrm{Pr}}^{3+}{(}^3{F}_4):\frac{\mathrm{d}{n}_4}{\mathrm{d}t}=-{\beta }_4{n}_4+{\beta }_5{n}_5+w_2{n}_{Yb0}{n}_6-w_2^{\prime }{n}_{Yb1}{n}_4+c_2{n}_0{n}_6+c_7{n}_0{n}_6,$$(5)$${\mathrm{Pr}}^{3+}({}^1{G}_4):\frac{\mathrm{d}{n}_5}{\mathrm{d}t}=-\frac{I_{852}}{\hslash \upsilon }{\sigma }_5{n}_5+\frac{I_{852}}{\hslash \upsilon }{\sigma }_0{n}_0-{\beta }_5{n}_5+w_1{n}_{Yb1}{n}_0-w_1^{\prime }{n}_{Yb0}{n}_5+w_3{n}_{Yb0}{n}_7-w_3^{\prime }{n}_{Yb1}{n}_5-2c_1{n}_0{n}_7-c_3{n}_0{n}_5+c_2{n}_0{n}_6+c_7{n}_0{n}_6-\frac{n_5}{{\tau }_5},$$(6)$${\mathrm{Pr}}^{3+}({{}^{1}D}_2):\frac{\text{d}{n}_6}{\text{d}t}=-w_2{n}_{Yb0}{n}_6+w_2^{\prime }{n}_{Yb1}{n}_4-\frac{n_6}{{\tau }_6}+{\beta }_7{n}_7+c_5{n}_0{n}_7+c_4{n}_0{n}_7+c_6{n}_0{n}_8-c_2{n}_0{n}_6-c_7{n}_0{n}_6,$$(7)$${\mathrm{Pr}}^{3+}({{}^{3}P}_0):\frac{\text{d}{n}_7}{\text{d}t}=-w_3{n}_{Yb0}{n}_7+w_3^{\prime }{n}_{Yb1}{n}_5-\frac{n_7}{{\tau }_7}-{\beta }_7{n}_7+{\beta }_8{n}_8-c_1{n}_0{n}_7-c_5{n}_0{n}_7-c_4{n}_0{n}_7,$$(8)$${\mathrm{Pr}}^{3+}({{}^{3}P}_1):\frac{\text{d}{n}_8}{\text{d}t}=-{\beta }_8{n}_8+\frac{I_{852}}{\hslash \upsilon }{\sigma }_5{n}_5-\frac{n_8}{{\tau }_8}-c_6{n}_0{n}_8,$$(9)$${{\mathrm{Yb}}_1}^{3+}({{}^{2}F}_{7/2}):\frac{\text{d}{n}_{Yb1}}{\text{d}t}=-\frac{I_{852}}{\hslash \upsilon }{\sigma }_0{n}_0+\frac{n_{Yb1}}{{\tau }_{Yb}}+w_1{n}_{Yb1}{n}_0-w_1^{\prime }{n}_{Yb0}{n}_5-w_2{n}_{Yb0}{n}_6+w_2^{\prime }{n}_{Yb1}{n}_4-w_3{n}_{Yb0}{n}_7+w_3^{\prime }{n}_{Yb1}{n}_5-w_4{n}_{Yb1}{n}_{Yb11},$$(10)$${\mathrm{Yb}}_1^{3+}({{}^{2}F}_{5/2}):\frac{\text{d}{n}_{Yb0}}{\text{d}t}=\frac{I_{852}}{\hslash \upsilon }{\sigma }_0{n}_0-\frac{n_{Yb1}}{{\tau }_{Yb}}-w_1{n}_{Yb1}{n}_0+w_1^{\prime }{n}_{Yb0}{n}_5+w_2{n}_{Yb0}{n}_6-w_2^{\prime }{n}_{Yb1}{n}_4+w_3{n}_{Yb0}{n}_7-w_3^{\prime }{n}_{Yb1}{n}_5+w_4{n}_{Yb1}{n}_{Yb11},$$(11)$$\begin{gathered} {\mathrm{Yb}}_2^{3+}({{}^{2}F}_{7/2}):\frac{\text{d}{n}_{Yb11}}{\text{d}t}=-\frac{I_{852}}{\hslash \upsilon }{\sigma }_0{n}_0+\frac{n_{Yb11}}{{\tau }_{Yb}}-w_5{n}_{Yb11}{n}_9-w_6{n}_{Yb11}{n}_{10}-w_7{n}_{Yb11}{n}_{12} \\ -w_8{n}_{Yb11}{n}_{14}-w_9{n}_{Yb11}{n}_{15}+w_4{n}_{Yb1}{n}_{Yb11}, \end{gathered}$$(12)$$\begin{gathered} {\mathrm{Yb}}_2^{3+}({{}^{2}F}_{5/2}):\frac{\text{d}{n}_{Yb10}}{\text{d}t}=\frac{I_{852}}{\hslash \upsilon }{\sigma }_0{n}_0{n}_0-\frac{n_{Yb11}}{{\tau }_{Yb}}+w_5{n}_{Yb11}{n}_9+w_6{n}_{Yb11}{n}_{10}+w_7{n}_{Yb11}{n}_{12} \\ +w_8{n}_{Yb11}{n}_{14}+w_9{n}_{Yb11}{n}_{15}-w_4{n}_{Yb1}{n}_{Yb11}, \end{gathered}$$(13)$${\mathrm{Tm}}^{3+}({{}^{3}H}_6):\frac{\text{d}{n}_9}{\text{d}t}=-w_5{n}_{Yb11}{n}_9+w_{10}{n}_{16}{n}_{17}+\frac{b_{912}{n}_{12}}{{\tau }_{12}}+\frac{b_{914}{n}_{14}}{{\tau }_{14}}+\frac{b_{915}{n}_{15}}{{\tau }_{15}}+\frac{n_{10}}{{\tau }_{10}}+\frac{n_{13}}{{\tau }_{13}}-c_{12}{n}_9{n}_{15}-c_8{n}_9{n}_{12},$$(14)$${\mathrm{Tm}}^{3+}({{}^{3}F}_4):\frac{\mathrm{d}{n}_{10}}{\mathrm{d}t}=-w_6{n}_{Yb11}{n}_{10}-\frac{n_{10}}{{\tau }_{10}}+\frac{b_{1012}{n}_{12}}{{\tau }_{12}}+\frac{b_{1014}{n}_{14}}{{\tau }_{14}}+\frac{b_{1015}{n}_{15}}{{\tau }_{15}}+\frac{b_{1016}{n}_{16}}{{\tau }_{16}}+{\beta }_{11}{\mathrm{n}}_{11}+2c_8{n}_9{n}_{12}+c_9{n}_{12}{n}_{14}+c_{10}{n}_{12}{n}_{14},$$(15)$${\mathrm{Tm}}^{3+}({{}^{3}H}_5):\frac{\text{d}{n}_{11}}{\text{d}t}=w_5{n}_{Yb11}{n}_9+\frac{b_{1114}{n}_{14}}{{\tau }_{14}}+\frac{b_{1116}{n}_{16}}{{\tau }_{16}}-{\beta }_{11}{n}_{11}+{\beta }_{12}{n}_{12},$$(16)$${\mathrm{Tm}}^{3+}({}^3{H}_4):\frac{\mathrm{d}{n}_{12}}{\mathrm{d}t}=-w_7{n}_{Yb11}{n}_{12}+\frac{b_{1214}{n}_{14}}{{\tau }_{14}}+\frac{b_{1215}{n}_{15}}{{\tau }_{15}}-\frac{n_{12}}{{\tau }_{12}}-{\beta }_{12}{\mathrm{n}}_{12}+{\beta }_{13}{n}_{13}+c_{12}{n}_9{n}_{15}-c_8{n}_9{n}_{12}-c_9{n}_{12}{n}_{14}-c_{10}{n}_{12}{n}_{14},$$(17)$${\mathrm{Tm}}^{3+}({{}^{3}F}_3/{{}^{3}F}_2):\frac{\mathrm{d}{n}_{13}}{\mathrm{d}t}=w_6{n}_{Yb11}{n}_{10}+\frac{b_{1315}{n}_{15}}{{\tau }_{15}}-\frac{n_{13}}{{\tau }_{13}}-{\beta }_{13}{n}_{13}+{\beta }_{14}{n}_{14}+c_{12}{n}_9{n}_{15},$$(18)$${\mathrm{Tm}}^{3+}({{}^{1}G}_4):\frac{\text{d}{n}_{14}}{\text{d}t}=w_7{n}_{Yb11}{n}_{12}-w_8{n}_{Yb11}{n}_{14}-\frac{n_{14}}{{\tau }_{14}}-{\beta }_{14}{n}_{14}+{\beta }_{15}{n}_{15}-c_9{n}_{12}{n}_{14}-c_{10}{n}_{12}{n}_{14},$$(19)$${\mathrm{Tm}}^{3+}({{}^{1}D}_2):\frac{\text{d}{n}_{15}}{\text{d}t}=w_8{n}_{Yb11}{n}_{14}-w_9{n}_{Yb11}{n}_{15}-\frac{n_{15}}{{\tau }_{15}}-{\beta }_{15}{n}_{15}+{\beta }_{16}{n}_{16}+c_9{n}_{12}{n}_{14}+c_{10}{n}_{12}{n}_{14}-c_{12}{n}_9{n}_{15},$$(20)$${\mathrm{Tm}}^{3+}({{}^{1}I}_6):\frac{\text{d}{n}_{16}}{\text{d}t}=w_9{n}_{Yb11}{n}_{15}-w_{10}{n}_{16}{n}_{17}-\frac{n_{16}}{{\tau }_{16}}-{\beta }_{16}{n}_{16},$$(21)$${{\mathrm{Gd}}_1}^{3+}({{}^{8}S}_{7/2}):\frac{\text{d}{n}_{17}}{\text{d}t}=-w_{10}{n}_{16}{n}_{17}+w_{11}{n}_{18}{n}_{19}+\frac{n_{18}}{{\tau }_{18}},$$(22)$${{\mathrm{Gd}}_1}^{3+}({}^6{P}_{7/2}):\frac{\mathrm{d}{n}_{18}}{\mathrm{d}t}=-w_{11}{n}_{18}{n}_{19}+w_{10}{n}_{16}{n}_{17}-\frac{n_{18}}{{\tau }_{18}},$$(23)$${{\mathrm{Gd}}_2}^{3+}({{}^{8}S}_{7/2}):\frac{\mathrm{d}{n}_{19}}{\mathrm{d}t}=-w_{11}{n}_{18}{n}_{19}+w_{12}{n}_{20}{n}_{21}+\frac{n_{20}}{{\tau }_{20}},$$(24)$${{\mathrm{Gd}}_2}^{3+}({{}^{6}P}_{7/2}):\frac{\mathrm{d}{n}_{20}}{\mathrm{d}t}=-w_{12}{n}_{20}{n}_{21}+w_{11}{n}_{18}{n}_{19}-\frac{n_{20}}{{\tau }_{20}},$$(25)$$X^{3+}(\text{ground}):\frac{\mathrm{d}{n}_{21}}{\mathrm{d}t}=-w_{12}{n}_{20}{n}_{21}+\frac{n_{22}}{{\tau }_{22}},$$(26)$$X^{3+}(\text{excited}):\frac{\mathrm{d}{n}_{22}}{\mathrm{d}t}=w_{12}{n}_{20}{n}_{21}-\frac{n_{22}}{{\tau }_{22}}.$$(27)In Eqs. (1)–(27), $n_i$ ($i=0-8$) denotes the population densities of the ${{}^{3}H}_4$, ${{}^{3}H}_5$, ${{}^{3}H}_6$, ${{}^{3}F}_2$, ${{}^{3}F}_4$, ${{}^{1}G}_4$, ${{}^1\mathrm{D}}_2$, ${{}^{3}P}_0$, and ${{}^{3}P}_1$ states of ${\mathrm{Pr}}^{3+}$; $n_i$ ($i=9-16$) denotes the population densities of the ${{}^{3}H}_6$, ${{}^{3}F}_4$, ${{}^{3}H}_5$, ${{}^{3}H}_4$, ${{}^{3}F}_2/{{}^{3}F}_3$, ${{}^{1}G}_4$, ${{}^{1}D}_2$, and ${{}^{1}I}_6$ states of ${\mathrm{Tm}}^{3+}$; and $n_i$ ($i=17-22$) denotes the population densities of the ${{}^{8}S}_{7/2}$, ${{}^6\mathrm{P}}_{7/2}$, ${{}^{8}S}_{7/2}$, and ${{}^{6}P}_{7/2}$ states of ${\mathrm{Gd}}^{3+}$, the ground and excited states of $X^{3+}$, whereas $n_{Yb1}$ and $n_{\mathit{Yb}11}$ denote the population densities of the ${{}^{2}F}_{7/2}$ and ${{}^{2}F}_{5/2}$ states of ${\mathrm{Yb}}^{3+}$, respectively. ${\beta }_i$ ($i=1$, 2, 3, 4, 5, 7, 8) is the nonradiative decay ratio of the ${{}^{3}H}_5$, ${{}^{3}H}_6$, ${{}^{3}F}_2$, ${{}^{3}F}_4$, ${{}^{1}G}_4$, ${{}^{3}P}_0$, and ${{}^{3}P}_1$ states of ${\mathrm{Pr}}^{3+}$, respectively. ${\beta }_i$ ($i=11-16$) is the nonradiative decay ratio of the ${{}^{3}H}_5$, ${{}^{3}H}_4$, ${{}^{3}F}_3$, ${{}^{1}G}_4$, ${{}^{1}D}_2$, and ${{}^{1}I}_6$ states of ${\mathrm{Tm}}^{3+}$, respectively. ${\tau }_i$ ($i=5$, 6, 7, 8) refers to the radiative lifetime of the ${{}^{1}G}_4$, ${{}^{1}D}_2$, ${{}^{3}P}_0$, and ${{}^{3}P}_1$ states of ${\mathrm{Pr}}^{3+}$. ${\tau }_i$ ($i=10$, 12, 13, 14, 15, 16) refers to the radiative lifetime of the ${}^3{F}_4$, ${{}^{3}H}_4$, ${}^3{F}_3$, ${{}^{1}G}_4$, ${}^1{D}_2$, and ${}^1{I}_6$ states of ${\mathrm{Tm}}^{3+}$. ${\tau }_i$ ($i=18$, 22) refers to the radiative lifetime of the ${{}^{6}P}_{7/2}$ of ${\mathrm{Gd}}^{3+}$ and the excited state of $X^{3+}$, respectively. ${\tau }_{\mathit{Yb}}$ is the lifetime of the ${{}^{2}F}_{5/2}$ state of ${\mathrm{Yb}}^{3+}$. $b_{ij}$ ($i>j$) is the branching ratio for the radiative transition from the initial state $i$ to the terminal state $j$ of ${\mathrm{Pr}}^{3+}$ and ${\mathrm{Tm}}^{3+}$. $c_i$ ($i=1$, 2, 3, 4, 5, 6, 7) denotes the ${\mathrm{Pr}}^{3+}-{\mathrm{Pr}}^{3+}$ cross-relaxation processes ${{}^{3}P}_0+{{}^{3}H}_4\to {{}^{1}G}_4+{{}^{1}G}_4$, ${{}^{1}D}_2+{}^3{H}_4\to {{}^{3}F}_4+{{}^{1}G}_4$, ${{}^{1}G}_4+{{}^{3}H}_4\to {{}^{3}H}_6+{{}^{3}F}_2$, ${{}^{3}P}_0+{}^3{H}_4\to {{}^{1}D}_2+{{}^{3}H}_6$, ${{}^{3}P}_0+{{}^{3}H}_4\to {{}^{3}H}_6+{{}^{1}D}_2$, ${{}^{3}P}_1+{{}^{3}H}_4\to {{}^{1}D}_2+{{}^{3}F}_2$, ${{}^{1}D}_2+{{}^{3}H}_4\to {{}^{1}G}_4+{}^3{F}_4$, respectively. $c_i$ ($i=8$, 9, 10, 12) denotes the ${\mathrm{Tm}}^{3+}-{\mathrm{Tm}}^{3+}$ cross-relaxation processes ${{}^{3}H}_4+{{}^{3}H}_6\to {{}^{3}F}_4+{{}^{3}F}_4$, ${{}^{1}D}_2+{{}^{3}F}_4\to {{}^{3}H}_4+{{}^{1}G}_4$, ${{}^{1}D}_2+{{}^{3}F}_4\to {{}^{1}G}_4+{{}^{3}H}_4$, ${{}^{1}D}_2+{{}^{3}H}_6\to {{}^{3}F}_2+{{}^{3}H}_4$, respectively. $w_i$ ($i=1$, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12) is the rate of energy transfer processes ${{}^{2}F}_{5/2}({\mathrm{Yb}}^{3+})+{{}^{3}H}_4({\mathrm{Pr}}^{3+})\to {{}^{2}F}_{7/2}({\mathrm{Yb}}^{3+})+{{}^{1}G}_4({\mathrm{Pr}}^{3+})$, ${{}^{1}D}_2({\mathrm{Pr}}^{3+})+{{}^{2}F}_{7/2}({\mathrm{Yb}}^{3+})\to {{}^{3}F}_4({\mathrm{Pr}}^{3+})+{{}^{2}F}_{5/2}({\mathrm{Yb}}^{3+})$, ${{}^{3}P}_0({\mathrm{Pr}}^{3+})+{{}^{2}F}_{7/2}({\mathrm{Yb}}^{3+})\to {{}^{1}G}_4({\mathrm{Pr}}^{3+})+{{}^{2}F}_{5/2}({\mathrm{Yb}}^{3+})$, ${{}^{2}F}_{5/2}({\mathrm{Yb}}^{3+})+{{}^{2}F}_{7/2}({\mathrm{Yb}}^{3+})\to {{}^{2}F}_{5/2}({\mathrm{Yb}}^{3+})+{{}^{2}F}_{7/2}({\mathrm{Yb}}^{3+})$, ${{}^{2}F}_{5/2}({\mathrm{Yb}}^{3+})+{{}^{2}F}_{7/2}({\mathrm{Yb}}^{3+})\to {{}^{3}H}_6({\mathrm{Tm}}^{3+})+{{}^{3}H}_5({\mathrm{Tm}}^{3+})$, ${{}^{2}F}_{5/2}({\mathrm{Yb}}^{3+})+{{}^{2}F}_{7/2}({\mathrm{Yb}}^{3+})\to {}^3{H}_5({\mathrm{Tm}}^{3+})+{{}^{3}F}_2({\mathrm{Tm}}^{3+})$, ${{}^{2}F}_{5/2}({\mathrm{Yb}}^{3+})+{{}^{2}F}_{7/2}({\mathrm{Yb}}^{3+})\to {{}^{3}H}_4({\mathrm{Tm}}^{3+})+{{}^{1}G}_4({\mathrm{Tm}}^{3+})$, ${{}^{2}F}_{5/2}({\mathrm{Yb}}^{3+})+{{}^{2}F}_{7/2}({\mathrm{Yb}}^{3+})\to {{}^{1}G}_4({\mathrm{Tm}}^{3+})+{{}^{1}D}_2({\mathrm{Tm}}^{3+})$, ${{}^{2}F}_{5/2}({\mathrm{Yb}}^{3+})+{{}^{2}F}_{7/2}({\mathrm{Yb}}^{3+})\to {{}^{1}D}_2({\mathrm{Tm}}^{3+})+{{}^{1}I}_6({\mathrm{Tm}}^{3+})$, ${{}^{3}H}_6({\mathrm{Tm}}^{3+})+{{}^{1}I}_6({\mathrm{Tm}}^{3+})\to {{}^{8}S}_{7/2}({\mathrm{Gd}}^{3+})+{{}^{6}P}_{7/2}({\mathrm{Gd}}^{3+})$, ${{}^{8}S}_{7/2}({\mathrm{Gd}}^{3+})+{{}^{6}P}_{7/2}({\mathrm{Gd}}^{3+})\to \text{excited}(X^{3+})+\text{ground}(X^{3+})$, respectively. ${w^{\prime }}_i$ ($i=1$, 2, 3) is the rate of back-energy transfer processes ${{}^{1}G}_4({\mathrm{Pr}}^{3+})+{{}^{2}F}_{7/2}({\mathrm{Yb}}^{3+})\to {{}^{3}H}_4({\mathrm{Pr}}^{3+})+{{}^{2}F}_{5/2}({\mathrm{Yb}}^{3+})$, ${{}^{2}F}_{5/2}({\mathrm{Yb}}^{3+})+{{}^{3}F}_4({\mathrm{Pr}}^{3+})\to {{}^{2}F}_{7/2}({\mathrm{Yb}}^{3+})+{{}^{1}D}_2({\mathrm{Pr}}^{3+})$, ${{}^{2}F}_{7/2}({\mathrm{Yb}}^{3+})+{{}^{1}G}_4({\mathrm{Pr}}^{3+})\to {{}^{3}P}_0({\mathrm{Pr}}^{3+})+{{}^{2}F}_{5/2}({\mathrm{Yb}}^{3+})$, respectively. $I_{852}$ is the excitation intensity of the 852 nm laser beam. $\hslash$ is the Planck constant, and $\upsilon$ is the frequency of the 852 nm laser light. The difference in the wavenumber between 852 nm and ${{}^{2}F}_{5/2}$ (${\mathrm{Yb}}^{3+}$) is $1489{\mathrm{cm}}^{-1}$ and between 852 nm and ${{}^{1}G}_4$ (${\mathrm{Pr}}^{3+}$) is $1884{\mathrm{cm}}^{-1}$, indicating very weak GSA. ${\sigma }_0$, ${\sigma }_5$, and ${\sigma }_{\mathit{Yb}0}$ are the cross-sections of absorption processes ${{}^{3}H}_4\to {{}^{1}G}_4({\mathrm{Pr}}^{3+})$, ${{}^{1}G}_4\to {{}^{3}P}_1({\mathrm{Pr}}^{3+})$, and ${{}^{2}F}_{7/2}\to {{}^{2}F}_{5/2}({\mathrm{Yb}}^{3+})$ under excitation of the 852 nm laser. A numerical solution can be obtained by solving these rate equations with a computer. The parameters (Table S1 in the Supplementary Material) were varied within a reasonable range to investigate the photon avalanche process.