Introduction

Super-resolution fluorescence microscopy encompasses a variety of techniques that can break the diffraction limit, and are of utmost importance for studies in life science and beyond1,2,3,4,5,6,7. As a typical super-resolution technique, stimulated emission depletion (STED) microscopy provides diffraction-unlimited imaging resolution while preserving the merits of confocal fluorescence microscopy, such as optical sectioning, molecular specificity, and fast imaging without the need for mathematical data postprocessing1,2,3,8,9,10. However, further dissemination of STED microscopy has been constrained by some intrinsic issues. To compete with the ultra-fast spontaneous fluorescence, a high-power ultra-fast laser with probe-defined depletion wavelength is required, which generally leads to serious photobleaching, phototoxicity effects, and re-excitation background in the samples11,12. In addition, high-precision overlapping of multiple pairs of lasers is typically required for multi-chromatic probes and for multicolor super-resolution imaging to match their different energy states, which results in high complexity, high cost, and low usability (Fig. 1a)13,14. Meanwhile, the spectral overlap among the utilized dyes (typically with broad emission spectra) would also lead to cross-talk between channels15, and thereby compromises its utility and impact.

Fig. 1: Schematics of two distinct emission inhibition mechanisms.
figure 1

a In traditional STED, the wavelengths of the excitation and depletion laser beams typically have to be carefully adapted with the varied spectroscopic properties of multi-chromatic probes. Multiple pairs of laser beams have to be temporally and spatially overlapped with high-precision for multicolor imaging. For multi-chromatic imaging, a unified excitation or depletion light wavelength would lead to low excitation or depletion efficiency and possible channel cross-talk. b The proposed stimulated-emission induced excitation depletion (STExD) strategy can generate depopulation simultaneously for multiple emitting states of multi-chromatic probes by utilizing a single depletion wavelength to de-excite their common sensitizer. In STExD, the depletion laser is defined by the common sensitizer, and thereby the strong dependence of the depletion wavelength on the emitting states and emission colors will be abnormally broken.

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Lanthanide-doped upconversion nanoparticles have become an important group of luminophores for diverse applications thanks to their unique excitation and emission properties, and have been employed for STED nanoscopy featuring low-power, near-infrared (NIR) laser excitation and ultrahigh photostability16,17,18,19,20,21. However, photo-switching in lanthanide nanoparticles is unfortunately rather limited and difficult to tune, and the depletion laser needs to directly act on the emitting activators for emissive-state depopulation, which is essentially the same as in traditional STED implemented with organic dyes and quantum dots17,18. Since upconversion nanoparticles comprise two types of optical centers, forming a typical binary sensitizer-activator system, suppression of the energy flow from the sensitizer to the activator would be a viable method to inhibit the emission of these multi-chromatic fluorophores. Here, we establish a powerful emission inhibition mechanism of cutting off the excitation chain realized in lanthanide-doped nanoparticles, named stimulated-emission induced excitation depletion (STExD) manifesting cascade amplified depletion, low depletion saturation intensity, and depletion light in the second near-infrared (NIR-II) tissue optical window. Different from the traditional depletion approach in STED of directly depleting the emitting state, the proposed STExD approach occurs in the sensitizer and thus allows emission inhibition to be achieved for many different emitters using a single depletion wavelength (Fig. 1b).

Results

Cascade amplified depletion reduces the saturation intensity

To implement this, we employ Nd3+ as the sensitizer22 as it has a typical quasi-four-level configuration suitable for lasing23. As a long-lived energy level, the 4F3/2 state of Nd3+ is in stark contrast with the ultra-fast decaying 4I11/2 state. Together with a large emission cross-section, a population inversion can be readily established between the 4F3/2 and 4I11/2 states, and the depletion saturation intensity (defined as the applied depletion intensity at which depletion efficiency (DE) reaches 50%) can be one or two orders of magnitude lower than those of organic dyes (Fig. 2a, Supplementary Fig. 1a, and Supplementary Table 1). Our numerical simulations on the effect of population inversion show that the depletion saturation intensity decreases with an increasing inverted population in this quasi-four-level system (Methods and Supplementary Fig. 1).

Fig. 2: Cascade amplified depletion of STExD in NaYF4:Nd nanoparticles.
figure 2

a Energy level diagram of Nd-upconversion system illustrating the emission inhibition process with the 1064-nm beam. b Full-spectrum emission inhibition, and the depletion efficiencies of different emissions of NaYF4:Nd (3%) nanoparticles under 740 and 1064 nm co-irradiation. c Emission intensity of one-photon (867 nm), two-photon (588 nm), and three-photon (450 nm) emissions from NaYF4:Nd (3%) nanoparticles versus depletion laser intensity (I740 = 78.3 kW cm−2, I1064 = 0–2.5 MW cm−2). Inset: the measured values of Isat were 116.2 ± 1.3, 42.4 ± 0.7, and 23.8 ± 0.4 kW cm−2 of 867, 588, and 450 nm emissions, respectively. d Comparison between calculated depletion efficiencies (DE, emission intensity ratio (I740-I740&1064)/I740) of one-, two-, and three-photon emissions. e The measured depletion efficiency and saturation intensity (Isat, 50% emission off) of 588 nm emission versus the Nd3+ concentration. NaYF4:Nd (x%) nanoparticles with x = 1, 2, 3, 4, 5, 8, 10. Data were presented as mean values ± standard deviation (SD). Error bars are defined as the SD of n = 3 independent measurements.

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Guided by the results from numerical simulations, a lab-made microscopy system coupled with a NIR continuous-wave (CW) excitation beam at 740 nm24 and a NIR-II CW depletion beam at 1064 nm was built to perform spectroscopic and imaging studies (Supplementary Fig. 2). Next, we synthesized singly-doped NaYF4:Nd (3%) nanoparticles (Supplementary Figs. 3a, 4a, c and Supplementary Table 2) and experimentally investigated the optical emission inhibition characteristics of Nd3+ ions. Upon excitation with a 740-nm laser (Supplementary Fig. 5a), ground-state absorption (GSA, 4I9/2 → 4F7/2) was initiated for Nd3+ ions. This is followed by three-step sequential processes of energy transfer upconversion (ETU) to the other Nd3+ ions to generate luminescence from higher levels, including 4F3/2 + 4I9/2 → 4I9/2 + 4F3/2 (ETU1), 4F3/2 + 4F3/2 → 4I9/2 + 2P1/2 (ETU2), and 4F3/2 + 4G5/2 → 4I9/2 + 4D3/2 (ETU3). The intensities of one-photon emissions at 867 and 900 nm (4F3/2 → 4I9/2), two-photon emissions at 523 nm (4G9/2 → 4I9/2), 533 nm (4G7/2 → 4I9/2), 573 nm (4G5/2 → 4I9/2), 588 nm (2P1/2 → 4I15/2), 640 nm (4G11/2 → 4I15/2), and 660 nm (2G9/2 → 4I15/2), and three-photon emissions at 415 nm (2P3/2 → 4I11/2) and 450 nm (4D3/2 → 4I15/2) were simultaneously collected (Fig. 2b). Their nonlinearity was confirmed by the measured power dependence of the emission intensity (Supplementary Fig. 6). When the 1064-nm laser was applied, all emission bands were sharply inhibited, which can be attributed to the stimulated emission process (4F3/2 → 4I11/2) induced by the 1064-nm laser as discussed above. Importantly, the re-excitation background by the depletion beam was not observed here (Fig. 2b and Supplementary Fig. 5a, b). It is worth noting that the depletion efficiencies were found to be different for different emission bands. With the co-irradiation of the 1064-nm beam (2.5 MW cm−2), relatively low depletion efficiency for the one-photon emission at 867 and 900 nm was observed, approximately 87%. In contrast, two-photon emission at 588 nm and three-photon emission at 450 nm exhibited much higher depletion efficiencies of up to 97.8 ± 0.5% and 99.3 ± 0.3%, respectively (Fig. 2b and Supplementary Table 3). The slight discrepancy in depletion efficiency between 415 and 450 nm emissions may be attributed to the excited state absorption (ESA) process of 4G5/2 → 2P3/2 driven by the depletion beam. The depletion saturation intensity of the three-photon emission at 450 nm was determined to be 23.8 ± 0.4 kW cm−2, nearly five times lower than that of the one-photon emission at 867 nm, and three orders of magnitude lower than that of traditional STED fluorophores25,26 (Fig. 2c).

We ascribed this difference to a cascade amplified depletion effect, originating from the stepwise ETU process. As the stimulated emission process directly depleted the 4F3/2 state, the depletion efficiency of one-photon emissions would be equivalent to the percentage of the electrons depleted from the 4F3/2 state. However, since pumping an ion to the two-photon emitting state i.e., 2P1/2, needs to consume two electrons at the excited state 4F3/2, the population in 2P1/2 would exhibit a squared relationship to that of the 4F3/2 state, which leads to a much higher depletion efficiency for the two-photon emission than for the one-photon emission, and similarly also for the three-photon emission. Thus, higher-order emissions relying on multiple energy transfer processes will be depleted more efficiently with the effect of cascade amplified depletion. For example, assuming that one-photon emission has a depletion efficiency of 50%, the depletion efficiencies for two-photon and three-photon emissions rapidly increase to 75 and 87.5%, respectively (Supplementary Fig. 7a). Theoretically, the depletion efficiency of an n-photon emission process can be calculated as DEn = 1-ηn, where η denotes the remaining ratio of one-photon emission intensity in the presence of a depletion beam, leading to a higher depletion rate (η−1) of multiphoton emissions (Methods, Fig. 2d and Supplementary Fig. 7). The alleviation in saturation intensity enabled by the cascade amplified depletion was consistent with our theoretical expectations and verified by numerical simulations (Methods and Supplementary Figs. 8, 9). When increasing the doping concentration (1–10%), the high depletion efficiency was hardly affected and the saturation intensities were slightly elevated (Fig. 2e), which could be ascribed to the fact that the enhanced cross-relaxation (CR) process of 4G9/2 + 4I9/2 → 4G5/2 + 4I11/2 would compete with the stimulated emission process (Fig. 2e and Supplementary Fig. 10a, b). In addition, the brightness of particles first increases and then saturates with increasing doping concentration, governed by the laser absorption and the CR process of 4F3/2 + 4I9/2 → 4I15/2 + 4I15/227,28 (Supplementary Fig. 10c, d). To investigate the optical depletion kinetics, the time-resolved intensity curves were acquired through modulating excitation beam or depletion beam. With the co-irradiation of the pulsed 740-nm beam and CW 1064-nm beam, the measured decay lifetime of the 2P1/2 state decreased from 30.8 ± 0.5 to 12.7 ± 0.2 μs when increasing the depletion intensity (Supplementary Fig. 11a), suggesting the effect of the sensitizing 4F3/2 states, greatly quenched by stimulated emission, on the decay time of this higher-lying level18,29,30. At the exaction of the CW 740-nm beam, the emission can be obviously inhibited by the pulsed 1064-nm beam (Supplementary Fig. 11b). The time required to reach a new steady-state decreased from 51.7 ± 1.9 to 13.8 ± 0.1 μs with the increasing depletion intensities.

A single NIR-II CW laser enables STExD for multi-chromatic emitters

The verified emission inhibition in the Nd3+ self-sensitized system lays the groundwork for extending the STExD strategy to other Nd3+-sensitized luminescence systems22. As proof-of-concept experiments, a series of Nd3+-sensitized nanoparticles were prepared and studied. We first characterized the emission inhibition performance of NaYF4: Nd/Yb/Er (3/1/5%) nanoparticles (Supplementary Figs. 3b, 4b, c). As shown in Fig. 3a, upon excitation at 740 nm, the activator Er3+ obtained excitation energy from Yb3+ and Nd3+ via multistep energy transfer, displaying characteristic emissions centered at 655 nm (4F9/2 → 4I15/2), 540 nm (4S3/2 → 4I15/2), 525 nm (2H11/2 → 4I15/2), 506 nm (4G11/2 → 4I13/2), 472 nm (2K15/2 → 4I13/2), and 410 nm (2H9/2 → 4I15/2)31 (Fig. 3b and Supplementary Fig. 12a), and the 588 nm emission of Nd3+ is also present32. When the 1064-nm depletion beam was applied, the de-excitation effect imposed on Nd3+ sensitizers indirectly depleted the emitting states of the activators. We unmixed the emission spectrum of Er3+ from Nd3+ by normalizing the 588 nm emission band. When Er3+-activated nanoparticles were excited by the 740-nm laser (non-resonant with Er3+) instead of the 800-nm laser (resonant with Er3+, Supplementary Fig. 5c), the depletion efficiency of 655 nm emission was measured to be 90.3 ± 0.4% (Fig. 3b, c and Supplementary Table 3), which by far exceeds the previously reported depletion efficiency of 27% in Er3+-activated nanoparticles16, with a saturation intensity of 160.9 ± 1.7 kW cm−2. The higher depletion efficiency of the 655 nm emission than that of the emissions from higher states was attributed to the 1064-nm laser-induced ESA processes of 4F9/2 → 2H9/2 and 4S3/2 → 2K15/2. The STExD effect in Nd/Yb/Er system was also supported by theoretical calculations (Supplementary Note 1 and Supplementary Figs. 13a, 14a).

Fig. 3: STExD strategy extended to various Nd-sensitized nanoparticles.
figure 3

a Optical emission inhibition mechanisms in Nd/Yb/Er and Nd/Yb/Ho systems. b Full-spectrum emission inhibition of NaYF4:Nd/Yb/Er (3/1/5%) nanoparticles under 740 and 1064 nm co-irradiation (I740 = 78.3 kW cm−2; I1064 = 2.5 MW cm−2). c Emission intensity at the 655 nm of Er3+ and the 645 nm of Ho3+ versus the depletion intensity (I740 = 78.3 kW cm−2; I1064 = 0–2.5 MW cm−2). The measured Isat were 160.9 ± 1.7 (Er-655 nm) and 85.8 ± 0.7 (Ho-645 nm) kW cm−2. Data were presented as mean values ± SD. Error bars are defined as the SD of n = 3 independent measurements. d–h Full-spectrum emission inhibition of other nanoparticles NaYF4:Nd/Yb/Ho (3/0.5/2%), Nd/Pr (3/0.5%), Nd/Eu (3/15%), Nd/Yb/Tm (3/1/0.2%), and Gd/Tb@Gd/Nd/Yb/Tm (50/15@50/3/5/1%) under 740 and 1064 nm co-irradiation.

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We next verified STExD in NaYF4:Nd/Yb/Ho (3/0.5/2%) nanoparticles (Supplementary Figs. 3c, 4c). Co-irradiation with a 1064-nm laser beam efficiently quenched the 645 nm (5F5 → 5I8), the 540 nm (5F4 → 5I8), and the 488 nm (5F3 → 5I8) emission bands of Ho3+ originally generated by the 740-nm laser (Fig. 3a, d, Supplementary Fig. 5d, and Supplementary Table 3). To the best of our knowledge, this is the first report of optically controlled emission inhibition in Ho3+-activated nanoparticles. These results were also in accordance with theoretical calculations (Supplementary Note 2 and Supplementary Figs. 13b, 14b). The depletion efficiency of the 645 nm emission was ~82.2 ± 0.3%, with a saturation intensity of 85.8 ± 0.7 kW cm−2 (Fig. 3c). The ESA processes of 5I7 → 5F4, 5I6 → 5F4 (phonon-assisted), and 5I5 → 5F3 compromised the depletion efficiencies of Ho3+ emissions, i.e., 650, 540, and 488 nm emissions, respectively. As an extensible mechanism, the STExD was also found to be applicable to other Nd-sensitized systems, including NaYF4:Nd/Pr, Nd/Eu, Nd/Yb/Tm, and NaYF4:Gd/Tb@ Gd/Nd/Yb/Tm nanoparticles (Supplementary Fig. 15). The effect of emission inhibition was achieved for Pr3+ (642, 610, and 485 nm emissions), Eu3+ (615 nm emission), Tm3+ (475 nm emission), and Tb3+ (620 and 545 nm emissions) (Fig. 3e–h and Supplementary Table 3). The observed depletion efficiencies of the emissions from Pr3+ and Tm3+ were relatively low due to the ESA processes of 740 nm (3F2 → 3P0 of Pr3+) and 1064 nm (3F2 → 1D2 of Pr3+ and 3F4 → 3F2 of Tm3+) (Supplementary Fig. 15).

Two-color STExD imaging implemented with two low-power NIR CW lasers

The versatile STExD mechanism described here allows applications in super-resolution fluorescence microscopy. A STExD microscope equipped with a 740-nm Gaussian excitation beam and a 1064-nm donut-shaped depletion beam was built to perform super-resolution imaging using different Nd3+-sensitized probes (Fig. 4a and Supplementary Fig. 2). Using such a pair of fixed NIR wavelength, low-power CW lasers, we achieved super-resolution imaging of multi-chromatic fluorophores by simply switching detection filters. As a demonstration, NaYF4:Nd nanoparticles were firstly imaged with visible emission bands from 500 to 700 nm. The sub-diffraction PSF FWHM of 34 nm was achieved, and a sequence of super-resolution images with low depletion intensities were captured (Fig. 4b, c). In addition, the red emission bands of NaYF4:Nd/Yb/Er and NaYF4:Nd/Yb/Ho nanoparticles were also imaged with obtained effective PSFs FWHM of 44 nm and 76 nm, respectively (Fig. 4d, e). In addition to the ability to image different probes with a single pair of simple, fixed-wavelength, and low-power NIR lasers, another capability is to perform imaging of multiple targets by multicolor super-resolution imaging. To demonstrate this, we performed two-color STExD imaging using these Nd3+-sensitized nanoparticles. The 645 nm red emission from the Ho3+ emitters was chosen as the channel 1 signal and the 588 nm emission from Nd3+ emitters as channel 2. To this end, Ce3+ ions were added as co-dopants to quench the 645 nm emission of Nd3+ (Supplementary Fig. 16). Due to the narrow emission band feature of lanthanide ions, the emission bands of interest can be easily separated with a dichroic mirror. The top row of Fig. 4f displays the diffraction-limited microscopic images of two channels and their superimposed image. It is evident that the two types of nanoparticles can be clearly distinguished in different channels, when subject to one and the same 740-nm excitation beam. Under co-irradiation with a 1064-nm beam, the effective PSFs in both channels were significantly narrowed.

Fig. 4: STExD imaging for multi-chromatic probes and two-color STExD.
figure 4

a Simplified schematic illustrating the STExD super-resolution microscopy system with two-channel detection. b Super-resolution imaging and line profile analyses of NaYF4:Nd (3%) nanoparticles (16.7 ± 0.9 nm). c Measured effective PSF FWHM plotted versus the depletion intensity (I740 = 78.3 kW cm−2; I1064 = 0–4.5 MW cm−2). Data were presented as mean values ± SD. Error bars are defined as the SD of n = 3 independent measurements. d Super-resolution imaging and line profile analyses of NaYF4:Nd/Yb/Er (3/1/5%) nanoparticles (20.5 ± 1.0 nm). e Super-resolution imaging and line profile analyses of NaYF4: Nd/Yb/Ho (3/0.5/2%) nanoparticles (21.3 ± 0.9 nm). f Two-color super-resolution imaging of NaYF4:Nd/Ce (3/20%, 17.9 ± 1.2 nm) and Nd/Yb/Ho (3/0.5/2%, 21.3 ± 0.9 nm) nanoparticles. The 645 nm emission from NaYF4:Nd/Yb/Ho was captured by channel 1 (in green). The 588 nm emissions from NaYF4:Nd/Ce were captured by channel 2 (in red). I740 = 78.3 kW cm−2; I1064 = 4.5 MW cm−2.

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STExD imaging of immunolabelled subcellular filaments

These nanoprobes naturally allow super-resolution imaging based on cellular immunolabeling. As a proof of concept, we imaged subcellular actin protein structures of HeLa cells, where NaYF4:Nd (3%) nanoparticles were surface modified with polyacrylic acid (PAA) to acquire hydrophilicity and biocompatibility, and then conjugated with amino-phalloidin molecules (Supplementary Fig. 17). Fixed HeLa cells were incubated with these phalloidin modified nanoparticles and imaged under the STExD microscope. The actin fibers of the HeLa cells were clearly observed under the excitation at 740 nm (Fig. 5a, b). The spatial resolution was significantly improved with co-irradiation at 1064 nm. Two regions of interest from the imaged sample were magnified, and line profiles were fitted with Gaussian functions (Fig. 5c–h). The FWHMs of the line profiles were below 100 nm, revealing fine structures of the subcellular actin filaments. Furthermore, the brightness of these nanoprobes did not show any decrease over 30-min laser scanning imaging, exhibiting excellent photostability (Fig. 5i).

Fig. 5: STExD imaging of immunolabelled subcellular filaments.
figure 5

a Fluorescence image of actin protein in HeLa cells immunolabeled with phalloidin conjugated NaYF4:Nd (3%, 10.3 ± 1.0 nm) nanoparticles under 740 nm Gaussian beam excitation. Pixel dwell time: 100 μs. I740 = 78.3 kW cm−2. b Differential Interference Contrast (DIC) image of the same sample. Three independent samples were used for this experiment with similar results obtained. c, f Magnified areas from white-dotted squares in (a). d, g Super-resolution imaging of the same regions as in (c, f) under co-irradiation with the 740 nm Gaussian beam and the 1064 nm donut-shaped beam. I740 = 78.3 kW cm−2; I1064 = 3.0 MW cm−2. e, h Analyses of line profiles, indicated with white arrows in (c, d), as well as in (e, f), respectively. i Photon counts were recorded with and without the co-irradiation of the 1064 nm donut-shaped beam. Insets show the corresponding images with the measured positions marked with white crosses.

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Discussion

In this study, we proposed and demonstrated the STExD strategy with cascade amplified depletion for effective emission inhibition of a variety of lanthanide emitters and for multicolor super-resolution imaging with one fixed pair of CW NIR lasers. The concept of STExD properly combines the physical processes of sensitization and stimulated emission, and makes full use of two upconversion characteristics: the binary sensitizer-activator system and the cascade multiphoton process. The continuous development of new design and synthesis methods is expected to produce even smaller and brighter nanoparticles, which will further enhance the detection signal-to-noise ratio, imaging resolution, and immunolabeling effect for diverse applications. The imaging speed can also be further increased by employing multi-focus or spinning disk scanning modes33.

This STExD strategy will provide a rational method to overcome some inherent limitations associated with traditional STED techniques, such as high-intensity and photobleaching, inflexibility of lasers scheme, and complicated optical design for multicolor imaging. The employed low-intensity-level CW lasers of NIR excitation and NIR-II depletion, combining the merits of the multiphoton excitation process, hold a potential for low-phototoxicity, deep-tissue super-resolution imaging in vivo34,35. This STExD mechanism can also circumvent the constraints of very limited photo-switching upconversion activators for the emerging upconversion STED super-resolution techniques17,18. One can also extend the STExD strategy to down-conversion lanthanide-doped nanoparticles and other sensitizer-activator (donor-acceptor) nanosystems36,37 (e.g., Föster resonance energy transfer and organic-inorganic hybrid luminescence systems), expanding the bandwidth for multichannel super-resolution imaging. We expect that our study will lend new and exciting fundamental insights into multiplexed STED mechanism with low-power NIR-II depletion, expanding broad applications in diffraction-unlimited optical sensing, optogenetics manipulation19, optical data storage, and lithography38,39.

Methods

Synthesis of NaYF4:Nd (3%) nanoparticles

NaYF4:Nd (3%) nanoparticles (10.6 ± 1.0 nm) were synthesized using previously reported methods40. In a typical procedure, 5 mL Ln(CH3CO2)3 aqueous solution with 0.03 mmol Nd3+ and 0.97 mmol Y3+ was added into a 100-mL round bottom flask containing 7.5 mL OA and 17.5 mL ODE. The mixture under stirring was heated to 120 °C for 10 min to remove water, then further heated to 150 °C for 40 min to form the lanthanide-oleate precursor. After cooling down to 40 °C, 0.148 g NH4F and 0.761 g sodium oleate were added to the flask. Subsequently, the mixture was stirred for at least 2 h at 40 °C, then heated to 100 °C under a vacuum for 30 min, and then filled with argon. Under anhydrous and oxygen-free conditions, the mixture was heated to 270 °C and incubated for 1.5 h, and then cooled down to room temperature. The nanoparticles were precipitated by the addition of 15 mL ethanol and collected by centrifugation at 7500 r.p.m (relative centrifuge force (RCF) = 4788×g) for 5 min. The obtained nanoparticles were washed several times using ethanol and cyclohexane and were finally re-dispersed into 8 mL cyclohexane for subsequent use. By varying the concentration of Nd3+ and Y3+, a series of NaYF4:Nd (x%) nanoparticles with x = 1, 2, 4, 5, 8, and 10 were obtained by the same procedure.

NaYF4:Nd (3%) nanoparticles (16.7 ± 0.9 nm) were synthesized using a similar method. After cooling the solution down to 40 °C, 10 mL of NH4F-methanol solution (0.4 M) and 2.5 mL of NaOH-methanol solution (1.0 M) were mixed and quickly injected into the flask. The same following steps as in the synthesis of NaYF4:Nd (3%) nanoparticles (10.6 ± 1.0 nm) were repeated. NaYF4:Nd/Ce (3/20%) nanoparticles (17.9 ± 1.2 nm) were synthesized using similar methods as NaYF4:Nd (3%) nanoparticles (16.7 ± 0.9 nm).

Synthesis of NaYF4:Nd/Yb/Er (3/1/5%) nanoparticles

A similar method was used to synthesize the NaYF4:Nd/Yb/Er (3/1/5%) nanoparticles (8.4 ± 1.0 nm). In a typical procedure, 5 mL Ln(CH3CO2)3 aqueous solution with 0.03 mmol Nd3+, 0.01 mmol Yb3+, 0.05 mmol Er3+, and 0.91 mmol Y3+ was added, and the same following steps as in the synthesis of NaYF4:Nd (3%) nanoparticles (10.6 ± 1.0 nm) were repeated. NaYF4:Nd/Yb/Er (3/1/5%) nanoparticles (20.5 ± 1.0 nm) were synthesized using similar methods as NaYF4:Nd (3%) nanoparticles (16.7 ± 0.9 nm) in addition to changing the final reaction temperature to 300 °C.

Synthesis of NaYF4:Nd/Yb/Ho (3/0.5/2%) nanoparticles

For the synthesis of NaYF4:Nd/Yb/Ho (3/0.5/2%) nanoparticles (9.1 ± 1.0 nm), 5 mL Ln(CH3CO2)3 aqueous solution with 0.03 mmol Nd3+, 0.005 mmol Yb3+, 0.02 mmol Ho3+, and 0.945 mmol Y3+ was added, and the same following steps as in the synthesis of NaYF4:Nd (3%) nanoparticles (10.6 ± 1.0 nm) were repeated. NaYF4:Nd/Yb/Ho (3/0.5/2%) nanoparticles (21.3 ± 0.9 nm) were synthesized using similar methods as NaYF4:Nd/Yb/Er (3/1/5%) nanoparticles (20.5 ± 1.0 nm).

Synthesis of NaYF4:Nd/Pr (3/0.5%) nanoparticles

For the synthesis of NaYF4:Nd/Pr (3/0.5%) nanoparticles (9.6 ± 0.9 nm), 5 mL Ln(CH3CO2)3 aqueous solution with 0.03 mmol Nd3+, 0.005 mmol Pr3+, and 0.965 mmol Y3+ was added, and the same following steps as in the synthesis of NaYF4:Nd (3%) nanoparticles (10.6 ± 1.0 nm) were repeated.

Synthesis of NaYF4:Nd/Eu (3/10%) nanoparticles

For the synthesis of NaYF4:Nd/Eu (3/10%) nanoparticles (10.4 ± 1.1 nm), 5 mL Ln(CH3CO2)3 aqueous solution with 0.03 mmol Nd3+, 0.1 mmol Eu3+, and 0.87 mmol Y3+ was added, and the same following steps as in the synthesis of NaYF4:Nd (3%) nanoparticles (10.6 ± 1.0 nm) were repeated.

Synthesis of NaYF4:Nd/Yb/Tm (3/1/0.2%) nanoparticles

For the synthesis of NaYF4:Nd/Yb/Tm (3/1/0.2%) nanoparticles (8.9 ± 0.9 nm), 5 mL Ln(CH3CO2)3 aqueous solution with 0.03 mmol Nd3+, 0.01 mmol Yb3+, 0.002 mmol Tm3+, and 0.958 mmol Y3+ was added, and the same following steps as in the synthesis of NaYF4:Nd (3%) nanoparticles (10.6 ± 1.0 nm) were repeated.

Synthesis of NaYF4:Gd/Tb (50/15%)@NaYF4:Gd/Nd/Yb/Tm (50/3/5/1%) nanoparticles

The NaYF4:Gd/Tb (50/15%)@NaYF4:Gd/Nd/Yb/Tm (50/3/5/1%) core-shell nanoparticles (18.4 ± 1.4 nm) were synthesized using a previously reported protocol with some modification41. Firstly, NaYF4:Gd/Tb (50/15%) core nanoparticles (12.8 ± 1.1 nm) were synthesized following steps in the synthesis of NaYF4:Nd (3%) nanoparticles (16.7 ± 0.9 nm), except with changing the dopants to 0.5 mmol Gd3+, 0.15 mmol Tb3+, and 0.35 mmol Y3+. Another 5 mL Ln(CH3CO2)3 aqueous solution with 0.5 mmol Gd3+, 0.03 mmol Nd3+, 0.05 mmol Yb3+, 0.01 mmol Tm3+, and 0.41 mmol Y3+, was added into a 100-mL round bottom flask containing 7.5 mL OA and 17.5 mL ODE. The solution was heated to 120 °C for 10 min to remove water, then further heated to 150 °C for 40 min to form the lanthanide-oleate precursor. Subsequently, the solution was cooled to 80 °C. The 8-mL suspension of bare core NaYF4:Gd/Tb (50/15%) nanoparticles (12.8 ± 1.1 nm) was pipetted into the flask and kept at this temperature for 30 min to remove cyclohexane. After the solution cooled down to 40 °C, the following steps were kept the same as the synthesis of core nanoparticles.

Ligand-exchange for PAA-nanoparticles

The ligand-exchange process of nanoparticles was prepared using previously reported methods42. Typically, 5 mL DMF and 50 mg NOBF4 were added into 5 mL cyclohexane containing 0.1 mmol as-prepared nanoparticles and vigorously stirred for 5 min. The nanoparticles were precipitated by adding an excess of toluene and collected by centrifugation at 14,000 r.p.m. (RCF = 16,684×g). for 20 min. The product was re-dispersed into 1 mL water and added into PAA solution (1.5 wt%) drop by drop. The mixture was stirred over 2 h and centrifugation was performed at 15000 r.p.m. (RCF = 19,152×g) for 20 min. The nanoparticles were re-dispersed in water to obtain a colorless and transparent colloidal solution.

Amino-phalloidin linking to PAA-nanoparticles

Amino-phalloidin linking with PAA-nanoparticles was carried out according to the commonly used methods43. In a typical experiment, 20 μL EDC (0.2 mg μL−1, MES) and 20 μL NHS (0.3 mg μL−1, MES) solution were prepared and added into 1 mL PAA-nanoparticles solution (1 mg mL−1, MES). The mixture was stirred for 30 min at room temperature, and then centrifugation at 15,000 r.p.m. (RCF = 19,152×g) for 20 min to acquire the NHS-nanoparticles. the precipitate was immediately re-dispersed in 1 mL HEPES solution. About 10 μL amino-phalloidin DMSO solution (1 mg mL−1, DMSO) was added to 200 μL NHS-nanoparticles solution, and the mixture was stored at 4 °C overnight for subsequent use.

HeLa cells culture and labeling

HeLa cells were cultured and labeled according to commonly-used culture and immunostaining techniques26. The HeLa cells were cultured in 96-well confocal plates with about 15,000 cells in each unit overnight. On the second day, the cells were rinsed with PBS buffer and fixed using immunostaining fixative for 20 min at room temperature. The fixed cells were washed and permeabilized with immunostaining permeate for 20 min. Washing followed by the addition of immunostaining blocking solution for 20 min. Then, the as-prepared phalloidin-nanoparticles were dissolved in HEPES buffer after centrifugation. And a volume of 200 µL solution (200 µg mL−1) was added. The staining reaction of actin protein was kept at 4 °C overnight. The cells were then rinsed with HEPES buffer three times for subsequent imaging.

Characterization

The transmission electron microscopic (TEM) imaging and the analysis of energy dispersive spectroscopy (EDS) for the as-prepared nanoparticles were characterized by a transmission electron microscope (JEOL-1400 PLUS) and analyzed with Fiji/ImageJ. The analysis of inductively coupled plasma (ICP) emission spectrometer was characterized using a plasma emission spectrometer (SPECTRO ARCOS MV). The as-prepared nanoparticles were dissolved in nitric acid and diluted to about 10 ppm for ICP analysis.

Optical system for spectroscopic and microscopic studies

As shown in Supplementary Fig. 2, a lab-made optical system for spectroscopic measurement and super-resolution imaging was set up. The non-descanned detectors (NDD) do not possess the function of confocal imaging. The 740 nm CW laser beam was generated by a Ti: Sapphire Oscillator (Mira HP, Coherent) and filtered with a band-pass filter (FF01-725/40-25, Semrock). A pinhole in combination with two lenses was used as a spatial filter to optimize the beam profile. The 1064 nm CW laser beam was generated by a single-mode diode laser (MAX-SPS500D, Max Photonics) and filtered with a band-pass filter (FF01-1064/5-25, Semrock). A single-mode fiber was used to optimize the beam profile. The laser beam power was controlled by two pairs of half-wave plates and polarization beam splitters placed in the optical path. These two NIR laser beams were then spatially overlapped using a dichroic mirror (T820dcspxrxt-UF2, Chroma), and then directed into a multiphoton laser scanning microscope (FV1000MPE-S with a motorized inverted stand, IX81, Olympus). In the super-resolution imaging, the 1064-nm beam was modulated by a 1064-nm vortex phase plate (VPP-m1064, RPC Photonics) to perform a 2π helix phase, generating a donut-shaped beam, and was made circularly polarized by another quarter-wave plate attached to the objective lens. The coaxial laser beams were reflected by a short-pass dichroic mirror (T690spxxf-UF1, Chroma), and focused on the sample through an oil-immersed objective (OL1, UPLXPO100XO, ×100/1.45, Olympus). A spectrometer (QE65000, Ocean Optics) was used for spectroscopic studies, where the visible emission (400–700 nm) was collected through the same objective and filtered by a short-pass filter (FF02-694/sp-25, Semrock). Photomultiplier tubes (PMT) were employed for imaging studies. Another objective (OL2, ×20/0.5, Olympus) was set on the condenser holder above the sample to collect the NIR emission, which was filtered by an 800 nm long-pass filter, and then detected by another spectrometer (QE65000, Ocean Optics). The two spectrometers can collect the spectra signal from the same sample region of interest simultaneously. A pair of polarization beam splitters and half-wave plates were placed in the optical path to adjust the laser powers, which were recorded by a power meter by using a pellicle beam splitter to reflect 5% of laser power. The size of the laser focus spot was defined as the full width at 1/e2 of maximum, and the laser intensities were calculated by dividing the powers by the area of laser spots.

Luminescence rise/decay times measurement

For the measurement of luminescence rise time and decay time, before coupling in the two beams, the 740 nm laser beam was modulated with a chopper (model SR540, Stanford), and the time-resolved luminescence intensity was recorded to perform the emission decay time measurements of the nanoparticles. The emission photons were selected by different filters and were detected by a PMT. The trigger signal from the chopper was synchronized with a time-correlated single-photon counter (NanoHarp, Picoquant). The chopper modulation frequency was 500 Hz. The chopper was placed in the focal plane where the beam was tightly focused to shorten the falling edge of the laser pulse.

Multicolor super-resolution imaging

In the two-color super-resolution microscopic imaging, a short-pass dichroic mirror (610spxxr-UF1, Chroma) was used to separate the luminescence to fall onto two PMT detectors. A band-pass filter (ET645/30x, Chroma) was placed in front of PMT1 to collect the 645 nm emission signal of NaYF4:Nd/Yb/Ho (3/0.5/2%) nanoparticles. A band-pass filter (ET590/33 m, Chroma) was placed in front of PMT2 to collect the 588 nm emission signal from NaYF4:Nd/Ce (3/20%) nanoparticles.

Numerical simulation on cascade amplified depletion of STExD

The electron transition processes in lanthanide ions can be described by a set of rate equations coupling the populations of each of the energy states. Here, we numerically simulated the cascade amplified depletion in the NaYF4:Nd (3%) system. The rate-equation-based theoretical analysis and simulation were performed using MATLAB (R2019a). As shown in Supplementary Fig. 8, the rate equations of the proposed energy transfer processes were derived as follows:

$${{{{{{\rm{Nd}}}}}}}^{3+}({\,\!}^{4}{{{{{\rm{I}}}}}}_{9/2}):\frac{d{n}_{0}}{dt} = {-\rho}_{1}{\sigma }_{0}{n}_{0}+{w}_{1}{n}_{3}{n}_{3}+{w}_{2}{n}_{3}{n}_{5}-{c}_{1}{n}_{0}{n}_{3}-{c}_{2}{n}_{0}{n}_{6}\\ \quad +{\beta }_{1}{n}_{1}+{b}_{30}\frac{{n}_{3}}{{\tau }_{3}}+\frac{{n}_{5}}{{\tau }_{5}}+\frac{{n}_{6}}{{\tau }_{6}}$$
(1)
$${{{{{{\rm{Nd}}}}}}}^{3+}({\,\!}^{4}{{{{{\rm{I}}}}}}_{11/2}):\frac{d{n}_{1}}{dt}=-{\rho }_{2}{\sigma }_{13}{n}_{1}+{\rho }_{2}{\sigma }_{31}{n}_{3}+{c}_{2}{n}_{0}{n}_{6}-{\beta }_{1}{n}_{1}+{\beta }_{2}{n}_{2}+{b}_{31}\frac{{n}_{3}}{{\tau }_{3}}+\frac{{n}_{9}}{{\tau }_{9}}$$
(2)
$${{{{{{\rm{Nd}}}}}}}^{3+}({\,\!}^{4}{{{{{\rm{I}}}}}}_{15/2}):\frac{d{n}_{2}}{dt}=2{c}_{1}{n}_{0}{n}_{3}-{\beta }_{2}{n}_{2}+{\beta }_{3}{n}_{3}+\frac{{n}_{7}}{{\tau }_{7}}+\frac{{n}_{8}}{{\tau }_{8}}+\frac{{n}_{10}}{{\tau }_{10}}$$
(3)
$${{{{{{\rm{Nd}}}}}}}^{3+}({\,\!}^{4}{{{{{\rm{F}}}}}}_{3/2}):\frac{d{n}_{3}}{dt}={\rho }_{2}{\sigma }_{13}{n}_{1}-{\rho }_{2}{\sigma }_{31}{n}_{3}-2{w}_{1}{n}_{3}{n}_{3}-{w}_{2}{n}_{3}{n}_{5}-{c}_{1}{n}_{0}{n}_{3}-{\beta }_{3}{n}_{3}+{\beta }_{4}{n}_{4}-\frac{{n}_{3}}{{\tau }_{3}}$$
(4)
$${{{{{{\rm{Nd}}}}}}}^{3+}({\,\!}^{4}{{{{{\rm{F}}}}}}_{7/2}):\frac{d{n}_{4}}{dt}={\rho }_{1}{\sigma }_{0}{n}_{0}-{\beta }_{4}{n}_{4}+{\beta }_{5}{n}_{5}$$
(5)
$${{{{{{\rm{Nd}}}}}}}^{3+}({\,\!}^{4}{{{{{\rm{G}}}}}}_{5/2}):\frac{d{n}_{5}}{dt}=-{\rho }_{2}{\sigma }_{59}{n}_{5}-{w}_{2}{n}_{3}{n}_{5}+{c}_{2}{n}_{0}{n}_{6}-{\beta }_{5}{n}_{5}+{\beta }_{6}{n}_{6}-\frac{{n}_{5}}{{\tau }_{5}}$$
(6)
$${{{{{{\rm{Nd}}}}}}}^{3+}({\,\!}^{4}{{{{{\rm{G}}}}}}_{9/2}):\frac{d{n}_{6}}{dt}=-{c}_{2}{n}_{0}{n}_{6}-{\beta }_{6}{n}_{6}+{\beta }_{7}{n}_{7}-\frac{{n}_{6}}{{\tau }_{6}}$$
(7)
$${{{{{{\rm{Nd}}}}}}}^{3+}({\,\!}^{4}{{{{{\rm{G}}}}}}_{11/2}):\frac{d{n}_{7}}{dt}=-{\beta }_{7}{n}_{7}+{\beta }_{8}{n}_{8}-\frac{{n}_{7}}{{\tau }_{7}}$$
(8)
$${{{{{{\rm{Nd}}}}}}}^{3+}({\,\!}^{2}{{{{{\rm{P}}}}}}_{1/2}):\frac{d{n}_{8}}{dt}={w}_{1}{n}_{3}{n}_{3}-{\beta }_{8}{n}_{8}+{\beta }_{9}{n}_{9}-\frac{{n}_{8}}{{\tau }_{8}}$$
(9)
$${{{{{{\rm{Nd}}}}}}}^{3+}({\,\!}^{2}{{{{{\rm{P}}}}}}_{3/2}):\frac{d{n}_{9}}{dt}={\rho }_{2}{\sigma }_{59}{n}_{5}-{\beta }_{9}{n}_{9}+{\beta }_{10}{n}_{10}-\frac{{n}_{9}}{{\tau }_{9}}$$
(10)
$${{{{{{\rm{Nd}}}}}}}^{3+}({\,\!}^{4}{{{{{\rm{D}}}}}}_{3/2}):\frac{d{n}_{10}}{dt}={w}_{2}{n}_{3}{n}_{5}-{\beta }_{10}{n}_{10}-\frac{{n}_{10}}{{\tau }_{10}}$$
(11)

Here ni (i = 0 to 10) represents the population of Nd3+ ions on the 4I9/2, 4I11/2, 4I15/2, 4F3/2, 4F7/2, 4G5/2, 4G9/2, 4G11/2, 2P1/2, 2P3/2, and 4D3/2 states, respectively. τi (i = 3, 5, 6, 7, 8, 9, 10) represents the radiative lifetimes of the 4F3/2, 4G5/2, 4G9/2, 4G11/2, 2P1/2, 2P3/2, and 4D3/2 states, respectively. βi (i = 1 to 10) represents the non-radiative decay rates of the 4I11/2, 4I15/2, 4F3/2, 4F7/2, 4G5/2, 4G9/2, 4G11/2, 2P1/2, 2P3/2, and 4D3/2 states, respectively. w1 and w2 denote the coefficients of the interionic energy transfer processes: 4F3/2 + 4F3/2 → 4I9/2 + 2P1/2 and 4F3/2 + 4G5/2 → 4I9/2 + 4D3/2. c1 and c2 denotes the coefficient of cross-relaxation processes: 4I9/2 + 4F3/2 → 4I15/2 + 4I15/2 and 4I9/2 + 4G9/2 → 4I11/2 + 4G5/2. σ0 denotes the absorption cross-section for the ground-state absorption process: 4I9/2 → 4F7/2. σ13 and σ31 denote the absorption cross-section for the excited state absorption process: 4I11/2 → 4F3/2 and the stimulated emission cross-section for the stimulated emission process: 4F3/2 → 4I11/2. σ59 denotes the absorption cross-section for the excited state absorption process: 4G5/2 → 2P3/2. ρ1 and ρ2 are the laser intensities of the 740 and 1064 nm beams. The values used for the main parameters are tabulated in Supplementary Table 4. To be consistent with the experimental condition, the intensity of 740 nm beams is set to 78 kW cm−2, while the intensity of the 1064 nm beam is varied from 0 to 2.0 MW cm−2. The change in population of the different electronic energy states with increasing depletion intensities reflects the emission inhibition behavior of each of the emission bands. As shown in Supplementary Fig. 9, the populations of 4F3/2, 2P2/1, 4D3/2 states, corresponding to 867, 588, and 450 nm emissions, show a sharp downtrend with increased depletion intensities, confirming the effective emission inhibition processes arising in NaYF4:Nd (3%) system. The higher-order emissions have a more efficient depletion, which is well in accordance with the proposed mechanism of cascade amplified depletion. In addition, according to the simulated population of 4F3/2 and 4I11/2, we can easily derive the value of the population inversion in the classic four-level Nd3+ system. By varying the decay lifetimes of the 4F3/2 and 4I11/2 states (the radiative lifetime of 4F3/2 and the non-radiative lifetime of 4I11/2), the relationship between the population inversion and the depletion saturation intensity was analyzed, and the results are shown in Supplementary Fig. 1.

Analytic description of the cascade amplified depletion effect

Based on the proposed STExD mechanism, the excitation depletion phenomenon could occur in any sensitizer-activator luminescence system when the stimulated emission process is inhibiting the energy of the sensitizer. Here, to better understand the STExD mechanism and the cascade amplified depletion process, a simplified sensitizer-activator system with a four-level sensitizer and four-level activator (Supplementary Fig. 7b) is used to analyze the relationship between the depletion of the activator and the sensitizer. This system could be expressed by the steady rate equations:

$$\frac{d{n}_{{{{{{\rm{S}}}}}}0}}{{dt}}=-{\alpha }_{{{{{{\rm{p}}}}}}}{n}_{{{{{{\rm{S}}}}}}0}{+{w}_{1}{n}_{{{{{{\rm{S}}}}}}2}{n}_{0}+{w}_{2}{n}_{{{{{{\rm{S}}}}}}2}{n}_{1}+{w}_{3}{n}_{{{{{{\rm{S}}}}}}2}{n}_{2}+\beta }_{{{{{{\rm{S}}}}}}1}{n}_{{{{{{\rm{S}}}}}}1}$$
(12)
$$\frac{d{n}_{{{{{{\rm{S}}}}}}1}}{{dt}}={{\alpha }_{{{{{{\rm{d}}}}}}}{n}_{{{{{{\rm{S}}}}}}2}-\beta }_{{{{{{\rm{S}}}}}}1}{n}_{{{{{{\rm{S}}}}}}1}$$
(13)
$$\frac{d{n}_{{{{{{\rm{S}}}}}}2}}{{dt}}=-{\alpha }_{{{{{{\rm{d}}}}}}}{n}_{{{{{{\rm{S}}}}}}2}-{w}_{1}{n}_{{{{{{\rm{S}}}}}}2}{n}_{0}-{w}_{2}{n}_{{{{{{\rm{S}}}}}}2}{n}_{1}-{w}_{3}{n}_{{{{{{\rm{S}}}}}}2}{n}_{2}{+\beta }_{{{{{{\rm{S}}}}}}3}{n}_{{{{{{\rm{S}}}}}}3}$$
(14)
$$\frac{d{n}_{{{{{{\rm{S}}}}}}3}}{{dt}}={\alpha }_{{{{{{\rm{p}}}}}}}{n}_{{{{{{\rm{S}}}}}}0}-{\beta }_{{{{{{\rm{S}}}}}}3}{n}_{{{{{{\rm{S}}}}}}3}$$
(15)
$$\frac{d{n}_{0}}{{dt}}=-\bigg(\frac{d{n}_{1}}{{dt}}+\frac{d{n}_{2}}{{dt}}+\frac{d{n}_{3}}{{dt}}\bigg)$$
(16)
$$\frac{d{n}_{1}}{{dt}}={w}_{1}{n}_{{{{{{\rm{S}}}}}}2}{n}_{0}-{w}_{2}{n}_{{{{{{\rm{S}}}}}}2}{n}_{1}-{\beta }_{1}{n}_{1}+{\beta }_{2}{n}_{2}-\frac{{n}_{1}}{{\tau }_{1}}$$
(17)
$$\frac{d{n}_{2}}{{dt}}={w}_{2}{n}_{{{{{{\rm{S}}}}}}2}{n}_{1}-{w}_{3}{n}_{{{{{{\rm{S}}}}}}2}{n}_{2}-{\beta }_{2}{n}_{2}+{\beta }_{3}{n}_{3}-\frac{{n}_{2}}{{\tau }_{2}}$$
(18)
$$\frac{d{n}_{3}}{{dt}}={w}_{3}{n}_{{{{{{\rm{S}}}}}}2}{n}_{2}-{\beta }_{3}{n}_{3}-\frac{{n}_{3}}{{\tau }_{3}}$$
(19)

In these equations, nSi (i = 0 to 3) represents the population of sensitizer ions at different energy states. ni (i = 0 to 3) represents the population of activator ions at different energy states. βi and τi (i = 1 to 3) represent the non-radiative decay rates and the radiative lifetimes of different energy states in activator. βS1 and βS3 denote the non-radiative decay rate of the nS1 and nS3 state of the sensitizer. wi (i = 1 to 3) denotes the energy transfer upconversion coefficients from sensitizer to activator. αp and αd represent the absorption rates for the excitation beam and depletion beam, respectively.

In a typical STExD system, the population of higher states would always be much smaller than that of the lower states, therefore, the non-radiative decay from higher states to lower states can be neglected. The Eqs. (17) to (19) describing the emitting states of the activator can then be rewritten as:

$${n}_{1}=\frac{{w}_{1}{n}_{{{{{{\rm{S}}}}}}2}{n}_{0}}{{w}_{2}{n}_{{{{{{\rm{S}}}}}}2}+{\beta }_{1}+\frac{1}{{\tau }_{1}}}$$
(20)
$${n}_{2}=\frac{{w}_{2}{n}_{{{{{{\rm{S}}}}}}2}{n}_{1}}{{w}_{3}{n}_{{{{{{\rm{S}}}}}}2}+{\beta }_{2}+\frac{1}{{\tau }_{2}}}$$
(21)
$${n}_{3}=\frac{{w}_{3}{n}_{{{{{{\rm{S}}}}}}2}{n}_{2}}{{\beta }_{3}+\frac{1}{{\tau }_{3}}}$$
(22)

Equation (20) expresses the relation between n1 and nS2. When the value of nS2 is decreased by the stimulated emission process in the sensitizer, two parts of the equation, w1nS2n0 and w2nS2 can affect the value of n1, representing the first-step and second-step ETU processes, respectively. In a typical upconversion system, the energy depletion effect caused by the second-step ETU process is always far less than the energy excitation effect caused by the first-step ETU process. Here, for clarity and simplicity, we assume that the multiphoton upconversion system is a standard multiphoton excitation system. The term w2nS2 is neglected.

Therefore, we have:

$${n}_{1}=\frac{{w}_{1}{n}_{{{{{{\rm{S}}}}}}2}{n}_{0}}{{\beta }_{1}+\frac{1}{{\tau }_{1}}}$$
(23)

with the combination of Eqs. (21) and (23), we could obtain:

$${n}_{2}=\frac{{w}_{1}{w}_{2}{{n}_{{{{{{\rm{S}}}}}}2}}^{2}{n}_{0}}{({\beta }_{1}+\frac{1}{{\tau }_{1}})({\beta }_{2}+\frac{1}{{\tau }_{2}})}$$
(24)

Similarly, by combining Eqs. (22) and (24), we could also get:

$${n}_{3}=\frac{{w}_{1}{w}_{2}{{w}_{3}{n}_{{{{{{\rm{S}}}}}}2}}^{3}{n}_{0}}{({\beta }_{1}+\frac{1}{{\tau }_{1}})({\beta }_{2}+\frac{1}{{\tau }_{2}})({\beta }_{3}+\frac{1}{{\tau }_{3}})}$$
(25)

Equations (23) to (25) show an apparent relationship between each order of emission and the population of the sensitizer. The one-photon emission from n1 state depends linearly on the population of nS2 state of the sensitizer, while the two-photon and three-photon emissions from the n1 and n2 states appear to have a quadratic and cubic relation to the population of sensitizer. When the excited state population of the sensitizer was decreased by the stimulated emission process, it can be determined that the higher-order emissions show higher depletion than the lower-order emissions, which is in accordance with the proposed mechanism of cascade amplified depletion.

In addition, from these equations, we can approximately assume that the depletion efficiency of the one-photon emission scales with the excited state population of the sensitizer, which means that their saturation intensities are also the same. As for the two-photon emission, achieving 50% DE only requires the sensitizer to be depleted by about 30%, indicating a much lower saturation intensity for this emission than for the one-photon emission will be expected. Similarly, when the DE of the sensitizer reaches 20%, the three-photon emission already reaches a DE of 50%. According to the formula:

$$\eta =\frac{1}{1+\frac{I}{{I}_{{{{{{\rm{sat}}}}}}}}}$$
(26)

where η denotes emission depletion ratio (1-DE). Isat denotes depletion saturation intensity. I denotes the applied depletion intensity. By determining the depletion intensity corresponding to depletion efficiencies of 20, 30, and 50% on the sensitizer emission, we found that the depletion saturation intensity of two-photon emission is effectively reduced to 40% of Isat of the one-photon emission, while this saturation intensity of three-photon emission is only a quarter. In conclusion, the cascade amplified depletion process in the STExD system can provide an effective way to further decrease the depletion intensity requirement. With the amplified depletion, the higher-order emissions can perform more efficient depletions than the traditional stimulated emission depletion (Fig. 2d).

Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.