Amplified stimulated emission in upconversion nanoparticles for super-resolution nanoscopy
上转换纳米颗粒中增强的受激辐射在超分辨率纳米成像中的应用

Yujia Liu $^{1, 2, 3, 4 *}$ , Yiqing Lu $^{1, 2 *}$ , Xusan Yang $^{3 *}$ , Xianlin Zheng $^{1, 2}$ , Shihui Wen $^{1, 5}$ , Fan Wang $^{1, 2, 5}$ , Xavier Vida1 $^{1, 2}$ , Jiangbo Zhao $^{1, 2}$ , Deming Liu $^{1, 2}$ , Zhiguang Zhou $^{1, 5}$ , Chenshuo Ma $^{1}$ , Jiajia Zhou $^{5}$ , James A. Piper $^{1, 2}$ , Peng Xi $^{1, 3}$ & Dayong Jin $^{1, 2, 5}$
刘宇佳 $^{1, 2, 3, 4 *}$ ，卢一青 $^{1, 2 *}$ ，杨旭山 $^{3 *}$ ，郑先林 $^{1, 2}$ ，文世辉 $^{1, 5}$ ，王凡 $^{1, 2, 5}$ ，Xavier Vida1 $^{1, 2}$ ，赵江波 $^{1, 2}$ , 刘德明 $^{1, 2}$ , 周志光 $^{1, 5}$ , 马晨硕 $^{1}$ ,周佳佳 $^{5}$ , 詹姆斯·A·派珀 $^{1, 2}$ , 席鹏 $^{1, 3}$ & 金大勇 $^{1, 2, 5}$

Lanthanide-doped glasses and crystals are attractive for laser applications because the metastable energy levels of the trivalent lanthanide ions facilitate the establishment of population inversion and amplified stimulated emission at relatively low pump power

^{1 - 3}

. At the nanometre scale, lanthanide-doped upconversion nanoparticles (UCNPs) can now be made with precisely controlled phase, dimension and doping level

^{4, 5}

. When excited in the near-infrared, these UCNPs emit stable, bright visible luminescence at a variety of selectable wavelengths

^{6 - 9}

, with single-nanoparticle sensitivity

^{10 - 13}

, which makes them suitable for advanced luminescence microscopy applications. Here we show that UCNPs doped with high concentrations of thulium ions (

{Tm}^{3 +}

), excited at a wavelength of 980 nanometres, can readily establish a population inversion on their intermediate metastable

^{3} H_{4}

level: the reduced inter-emitter distance at high

{Tm}^{3 +}

doping concentration leads to intense cross-relaxation, inducing a photon-avalanche-like effect that rapidly populates the metastable

^{3} H_{4}

level, resulting in population inversion relative to the

^{3} H_{6}

ground level within a single nanoparticle. As a result, illumination by a laser at 808 nanometres, matching the upconversion band of the

^{3} H_{4} \to^{3} H_{6}

transition, can trigger amplified stimulated emission to discharge the

^{3} H_{4}

intermediate level, so that the upconversion pathway to generate blue luminescence can be optically inhibited. We harness these properties to realize low-power super-resolution stimulated emission depletion (STED) microscopy and achieve nanometrescale optical resolution (nanoscopy), imaging single UCNPs; the resolution is 28 nanometres, that is,

1 / 36

th of the wavelength. These engineered nanocrystals offer saturation intensity two orders of magnitude lower than those of fluorescent probes currently employed in stimulated emission depletion microscopy, suggesting a new way of alleviating the square-root law that typically limits the resolution that can be practically achieved by such techniques.
镧系元素掺杂玻璃和晶体在激光应用中具有吸引力，因为三价镧系离子具有亚稳态能级，这有利于在相对较低的泵浦功率下建立人口反转和增强的受激辐射

^{1 - 3}

。在纳米尺度上，镧系元素掺杂上转换纳米颗粒（UCNPs）现在可以制备出具有精确控制的相、尺寸和掺杂水平的材料

^{4, 5}

。当在近红外波段激发时，这些 UCNPs 可发出稳定、明亮的可见光发光，且发光波长可选范围广泛

^{6 - 9}

，同时具备单纳米颗粒灵敏度

^{10 - 13}

，使其适用于先进发光显微镜技术。本文展示了高浓度钬离子掺杂的 UCNPs（

{Tm}^{3 +}

），在 980 纳米波长激发下，可在中间亚稳态

^{3} H_{4}

能级上迅速建立粒子数反转：高

{Tm}^{3 +}

掺杂浓度下发光体间距缩短，导致强烈的交叉弛豫，引发类似光子雪崩效应，迅速填充亚稳态

^{3} H_{4}

能级，从而在单个纳米颗粒内相对于

^{3} H_{6}

基态实现人口倒置。因此，当激光以 808 纳米波长（与

^{3} H_{4} \to^{3} H_{6}

跃迁的上转换带匹配）照射时，可触发增强的受激辐射，使

^{3} H_{4}

中间能级放空，从而光致抑制上转换路径以产生蓝光发光。我们利用这些特性实现了低功耗超分辨率受激辐射耗尽（STED）显微镜，并达到了纳米级光学分辨率（纳米显微镜），成功成像单个单克隆纳米颗粒（UCNPs）；分辨率为 28 纳米，即波长的

1 / 36

分之一。这些工程化纳米晶体提供的饱和强度比目前在受激发射淬灭显微镜中使用的荧光探针低两个数量级，这为缓解通常限制此类技术实际可实现分辨率的平方根定律提供了一种新方法。

To investigate conditions for amplified stimulated emission from single

Yb /

Tm co-doped

{NaYF}_{4}

UCNPs, we built a dual-laser confocal microscope (Extended Data Fig. 1). As illustrated in Fig. 1a, the photon upconversion process comprises absorption of 980 nm excitation by the

{Yb}^{3 +}

sensitizers, stepwise transfer of that energy onto the scaffold energy levels of the

{Tm}^{3 +}

emitters, and eventually upconverted emission from the two-photon

^{3} H_{4}

, three-photon

^{1} G_{4}

or four-photon

^{1} D_{2}

levels of

{Tm}^{3 +}

. In the presence of population inversion between the

^{3} H_{4}

intermediate level and the

^{3} H_{6}

ground level, a probe laser beam with wavelength corresponding to the energy gap (that is, 808 nm for

^{3} H_{4} \to^{3} H_{6}

) will trigger stimulated emission to discharge the

^{3} H_{4}

level, consequently inhibiting upconverted emission from higher excited levels (for example,

^{1} G_{4}

and

^{1} D_{2}

).
为了研究单个

Yb /

钬共掺杂

{NaYF}_{4}

超微纳米颗粒（UCNPs）中受激辐射增强的条件，我们构建了一台双激光共聚焦显微镜（扩展数据图 1）。如图 1a 所示，光子上转换过程包括

{Yb}^{3 +}

敏化剂吸收 980 nm 激发光，将该能量分步转移至

{Tm}^{3 +}

发射体的支架能级，最终从两光子

^{3} H_{4}

三光子

^{1} G_{4}

或四光子

^{1} D_{2}

能级发射。当

^{3} H_{4}

中间能级与

^{3} H_{6}

基能级之间存在反转分布时，波长与能隙对应（即

^{3} H_{4} \to^{3} H_{6}

为 808 nm）的探针激光束将触发受激辐射，使

^{3} H_{4}

能级放空，从而抑制来自更高激发能级的上转换发射（例如

^{1} G_{4}

和

^{1} D_{2}

）。

Figure 1 b shows confocal images of single UCNPs highly doped with

8 % {Tm}^{3 +}

ions (and

20 % {Yb}^{3 +}

). The upconversion emission under
图 1b 显示了高度掺杂

8 % {Tm}^{3 +}

离子（和

20 % {Yb}^{3 +}

）的单个 UCNPs 的共聚焦图像。在

Figure 1 | Probing upconversion luminescence using dual-laser illumination. a, Energy level diagrams of

Yb / Tm

co-doped UCNPs under 980 nm illumination (left), and under both 980 nm and 808 nm illumination (right). See text for details.

b

, Confocal images in 455 nm upconversion emission of the

8 % Tm

-doped UCNPs under continuouswave 980 nm laser (left) and under both 980 nm and 808 nm dual laser (right) illumination. c, As

b

but for

1 % Tm

-doped UCNPs. For

b

and

c

, the 980 nm and 808 nm laser powers measured at the objective back aperture were 1 mW and 5 mW , respectively. Each inset shows the luminescence signal profile along the diagonal white line in that panel across a typical nanocrystal. Pixel dwell time, 4 ms ; scale bar, 500 nm .
图 1 | 采用双激光照明探测上转换发光。a，

Yb / Tm

共掺杂 UCNPs 在 980 nm 照明下的能级图（左），以及在 980 nm 和 808 nm 联合照明下的能级图（右）。详见正文。

b

，

8 % Tm

掺杂的 UCNPs 在 980 nm 连续波激光（左）和 980 nm 与 808 nm 双激光（右）照射下的 455 nm 上转换发光共聚焦图像。c，与

b

相同，但针对

1 % Tm

掺杂的 UCNPs。对于

b

和

c

，980 nm 和 808 nm 激光在物镜后孔径处的功率分别为 1 mW 和 5 mW。每个插图显示了该面板中沿对角白线跨典型纳米晶体的发光信号轮廓。像素驻留时间，4 ms；刻度尺，500 nm。

Figure 2 | Competition between absorption and stimulated emission.
图 2 | 吸收与受激辐射之间的竞争。
a, Transient response of the 455 nm emission from

8 % Tm

-doped UCNPs under synchronous 980 nm and 808 nm pulses ( 1 ms duration). The 980 nm laser power was fixed at 1 mW , while the 808 nm laser power was varied from 0 to 40 mW (both at the objective back aperture). b, Diagrams illustrating net absorption (left) and net stimulated emission (right) between

^{3} H_{4}

and

^{3} H_{6}

levels when the UCNPs under 980 nm excitation were probed with the 808 nm laser, which then led to either inhibition or enhancement of the further upconverted emission. See text for details. c, As a but for

1 % Tm

-doped UCNPs.
a, 455 nm 发射的瞬态响应，来自

8 % Tm

掺杂的 UCNPs 在同步 980 nm 和 808 nm 脉冲（1 ms 持续时间）下的响应。980 nm 激光功率固定为 1 mW，而 808 nm 激光功率在 0 到 40 mW 之间变化（均在物镜后孔径处）。b, 示意图展示了当 UCNPs 在 980 nm 激发下被 808 nm 激光探测时，

^{3} H_{4}

与

^{3} H_{6}

能级之间的净吸收（左）和净受激辐射（右），这导致了进一步上转换发光的抑制或增强。详见正文。c，对于

1 % Tm

掺杂的 UCNPs，情况类似。
continuous-wave (CW) 980 nm excitation was clearly inhibited once a CW 808 nm probe beam was applied. By contrast, under the same experiment conditions, the UCNPs with low doping concentration (conventionally

1 % {Tm}^{3 +}

and

20 % {Yb}^{3 +}

) showed negligible optical switching effects (Fig. 1c).
连续波（CW）980 nm 激发在施加 CW 808 nm 探测光束后被明显抑制。相比之下，在相同的实验条件下，低掺杂浓度（常规上

1 % {Tm}^{3 +}

和

20 % {Yb}^{3 +}

）的 UCNPs 显示出微乎其微的光开关效应（图 1c）。

Since the 808 nm probe matches only the

^{3} H_{4} \to^{3} H_{6}

transition (no upconversion emission is observed under 808 nm excitation alone, Extended Data Fig. 2), inhibition of visible upconversion emission by the 808 nm probe is concrete evidence for population inversion and net stimulated emission on

^{3} H_{4} \to^{3} H_{6}

; if this were not so, visible upconversion emission would be enhanced because of the increased optical pumping to

^{3} H_{4}

. To confirm this, we measured the transient response of the upconversion emission at 455 nm (Fig. 2) and at 650 nm (Extended Data Fig. 3) using synchronized 980 nm and 808 nm pulses. Initially, the emission from 8% Tm-doped UCNPs is indeed enhanced (Fig. 2a), indicating that the absorption of 808 nm dominates (cartoon shown in Fig. 2b left). The more intense the 808 nm beam, the more obvious the initial enhancement. However increase of the upconversion emission is rapidly truncated once sufficient 980 nm energy is transferred to

{Tm}^{3 +}

(

^{3} H_{4}

) acceptors to establish a population inversion, whereupon stimulated emission on

^{3} H_{4} \to^{3} H_{6}

dominates and upconversion emissions
由于 808 nm 探针仅与

^{3} H_{4} \to^{3} H_{6}

跃迁匹配（在仅 808 nm 激发下未观察到上转换发射，见扩展数据图 2），808 nm 探针对可见上转换发射的抑制是

^{3} H_{4} \to^{3} H_{6}

态中人口倒转和净受激发射的直接证据；若非如此，由于对

^{3} H_{4}

（

^{3} H_{4}

）受体中足够建立人口倒置时，上转换发光的增强迅速被截断，此时

^{3} H_{4} \to^{3} H_{6}

上的受激发光占主导，上转换发光

Figure 3 | The photon-avalanche-like process underlying enhanced population inversion. a, Energy level diagram of

Yb / Tm

co-doped UCNPs including typical cross-relaxation pathways among

{Tm}^{3 +}

emitters. Solid arrows, excitation and emission; curved arrows, non-radiative relaxation; dashed arrows connected by dotted lines, energy transfer processes. b, Diagrams of the photon-avalanche-like process, which stems from the intense cross-relaxation among emitters in a highly doped nanocrystal only (sensitizers are not shown). UC, upconversion; CR, cross-relaxation. c, Transient response of the upconverted emission measured at 800 nm from

8 %

and

1 %

Tm-doped UCNPs after the 980 nm excitation was switched on at time

= 0 ms

(in the absence of the 808 nm probing beam). The 980 nm laser power was 1 mW at the objective back aperture.
图 3 | 增强人口反转背后的光子雪崩式过程。a，

Yb / Tm

共掺杂 UCNPs 的能级图，包括

{Tm}^{3 +}

发射体之间的典型交叉弛豫路径。实心箭头表示激发和发射；曲线箭头表示非辐射弛豫；虚线连接的箭头表示能量转移过程。b，光子雪崩样过程的示意图，该过程源于高掺杂纳米晶体中发射体之间的强烈交叉弛豫（敏化剂未显示）。UC，上转换；CR，交叉弛豫。c，在时间

= 0 ms

（无 808 nm 探测光束）开启 980 nm 激发后，

8 %

和

1 %

钆掺杂 UCNPs 在 800 nm 处测得的上转换发射瞬态响应。980 nm 激光功率为 1 mW，位于物镜后孔径处。
from the three- and four-photon levels of

^{1} G_{4}

and

^{1} D_{2}

are shortcircuited (cartoon shown in Fig. 2b right). By contrast, for 1% Tm-doped nanocrystals, the build-up of upconversion emission is accelerated by the probe 808 nm beam (that is, enhancement; corresponding to Fig. 2b left), showing no sign of population inversion over the same period of time (Fig. 2c).
从

^{1} G_{4}

和

^{1} D_{2}

的三个和四个光子能级中，光子被短路（如图 2b 右侧所示的示意图）。相比之下，对于 1%钬掺杂的纳米晶体，探测光 808 nm 光束加速了上转换发光的积累（即增强效应；对应于图 2b 左侧），在同一时间段内未观察到人口反转的迹象（图 2c）。

The differences shown above between the two UCNPs of different

{Tm}^{3 +}

concentrations indicate that the critical population inversion is driven by a key mechanism related to the doping concentration, which we ascribe to a photon-avalanche-like process. This refers to the intense cross-relaxation between emitters such that those in a higher excited energy level have high probability for transfer of energy to a nearby emitter in the ground or lower energy level instead of undergoing other relaxation pathways, forming a positive feedback process accelerating the accumulation of emitters in intermediate excited energy levels. Unlike ordinary photon avalanche, however, the condition of non-resonant ground-level absorption is not met with co-doped UCNPs containing sensitizers and emitters, so that the threshold behaviour of photon avalanche is absent

^{14 - 17}

(see Methods and Extended Data Fig. 4 for discussion). For the Tm-doped UCNPs, the similarly spaced energy levels of

{Tm}^{3 +}

provide abundant crossrelaxation pathways (Fig. 3a). The cross-relaxation coefficients are
上述两种不同

{Tm}^{3 +}

浓度 UCNP 之间的差异表明，临界反转人口是由与掺杂浓度相关的关键机制驱动的，我们将其归因于一种类似光子雪崩的过程。这指的是发射体之间强烈的交叉弛豫，使得处于较高激发能级中的发射体更可能将能量转移给附近处于基态或较低能级的发射体，而非通过其他弛豫途径，从而形成正反馈过程，加速发射体在中间激发能级中的积累。然而，与普通光子雪崩不同，共掺杂敏化剂和发光体的 UCNPs 不满足非共振基态吸收条件，因此光子雪崩的阈值行为缺失

^{14 - 17}

（详见方法部分及扩展数据图 4 讨论）。对于钬掺杂的 UCNPs，相似的能级间隔

{Tm}^{3 +}

提供了丰富的交叉放松路径（图 3a）。交叉放松系数为

Figure 4 | Optical switching efficiency with respect to Tm doping concentration. a, Depletion ratio of the upconversion emission at 455 nm , as a result of intermediate-level stimulated emission, under continuouswave 980 nm excitation at a fixed intensity of

0.66 MW {cm}^{- 2}

and 808 nm intensities up to

39 MW {cm}^{- 2}

, measured for a series of UCNPs with incremental Tm doping concentrations from 0.5 to

8 mol %

. Dashed lines are fitted curves using the form

η = {(1 + I_{808} / I_{sat})}^{- 1}

, where

I_{sat}

denotes the saturation intensity.

b

, The depletion ratio data in

a

are inverted to
图 4 | Tm 掺杂浓度对光开关效率的影响。a, 在连续波 980 nm 激发下，激发强度为

0.66 MW {cm}^{- 2}

，808 nm 激发强度为

39 MW {cm}^{- 2}

，测量了一系列钕掺杂浓度从 0.5 到

8 mol %

的 UCNPs 的 455 nm 上转换发射的耗尽比，这是由于中间能级受激辐射引起的。虚线为拟合曲线，采用公式

η = {(1 + I_{808} / I_{sat})}^{- 1}

，其中

I_{sat}

表示饱和强度。

b

，图

a

中的耗尽比数据经倒置处理后得到。
illustrate the linear relation

η^{- 1} = 1 + I_{808} / I_{sat}

. In the inset, the dashed lines are fitted using all data points; in the main panel, linear curve fitting for Tm concentrations

\geq 3.5 %

uses only data points at low 808 nm intensity. At high 808 nm intensity, the reciprocals of depletion ratio start to deviate from linearity towards plateaus. Error bars in

a

and

b, \pm 1

s.d. with

n = 5

each. c, Inverse of the saturation intensity values obtained by linear curve fitting in

b

in relation to the Tm doping concentration, plotted on a

\log - \log

scale in the main panel and on a linear scale in the inset.
展示线性关系

η^{- 1} = 1 + I_{808} / I_{sat}

。在插图中，虚线是使用所有数据点拟合得到的；在主图中，对 Tm 浓度

\geq 3.5 %

的线性拟合仅使用了 808 nm 强度较低时的数据点。在 808 nm 强度较高时，耗尽比的倒数开始偏离线性趋势并趋向于平台期。

a

和

b, \pm 1

中的误差条为标准差，每个误差条对应

n = 5

。c，主图中以

\log - \log

刻度绘制，插图中以线性刻度绘制，图中为通过

b

中线性拟合得到的饱和强度值的倒数与 Tm 掺杂浓度的关系。
enhanced quadratically as the doping concentration increases (assuming that energy transfer between two ions is inversely proportional to the sixth power of their separation

^{18, 19}

, and that the average separation distance is the cube root of the reciprocal of the doping concentration), so that cross-relaxation dominates when the emitters are close enough to each other (Fig. 3b).
随着掺杂浓度的增加，效应呈二次增强（假设两个离子之间的能量传递与它们之间距离的六次方成反比，且平均距离为掺杂浓度的倒数立方根），因此当发射体彼此足够接近时，交叉弛豫效应占据主导地位（图 3b）。

The presence or absence of the photon-avalanche-like process was experimentally verified by measuring the transient response of the upconversion emission from the

^{3} H_{4}

level under excitation by a single wavelength of 980 nm . The 800 nm upconversion emission of the

8 % Tm

-doped nanocrystals exhibited an ‘S’ shape over the build-up period (Fig. 3c), which is a signature of photon avalanche

^{14, 15, 17, 20}

. In contrast, for the

1 % Tm

-doped nanocrystals, the increase of 800 nm emission took place immediately upon 980 nm excitation. Theoretically, the precondition for photon avalanche is when the cross-relaxation coefficient surpasses the intrinsic decay rate from the higher excited level to the ground level, thereby producing positive feedback

^{14, 16}

. By fitting the transient responses of the

8 % Tm

-doped nanocrystals to rate equations, we show that the rate parameters obtained satisfy this requirement (Extended Data Fig. 5); in contrast, for

1 %

Tm-doped UCNPs where the cross-relaxation coefficients are reduced quadratically with the doping concentration (that is, dividing the coefficients by 64), the photon avalanche condition is not satisfied.
通过测量在 980 nm 单波长激发下

^{3} H_{4}

能级上转换发光的瞬态响应，实验上验证了光子雪崩样过程的存在或缺失。

8 % Tm

掺杂纳米晶体的 800 nm 上转换发射在建立过程中呈现出“S”形（图 3c），这是光子雪崩的特征

^{14, 15, 17, 20}

。相比之下，

1 % Tm

掺杂纳米晶体的 800 nm 发射强度在 980 nm 激发后立即开始增加。理论上，光子雪崩的先决条件是当交叉弛豫系数超过从高激发态到基态的固有衰减率时，从而产生正反馈

^{14, 16}

。通过将

8 % Tm

掺杂纳米晶体的瞬态响应拟合到速率方程，我们表明所得速率参数满足此要求（扩展数据图 5）；而对于

1 %

钬掺杂的 UCNPs，其交叉弛豫系数随掺杂浓度呈二次衰减（即系数除以 64），光子雪崩条件不成立。

To determine the critical

{Tm}^{3 +}

doping concentration for enhanced population inversion, a series of ten batches of

40 - nm

UCNPs were synthesized with incremental Tm concentrations from

0.5 mol %

8 mol %

(Extended Data Fig. 6). Under dual-laser CW illumination, the optical depletion ratio (that is, the ratio of the emission strength in the presence of the 808 nm depletion probe to that in the absence of that probe) of the 455 nm emission was measured as a function of the 808 nm intensity. As shown in Fig. 4a, within the available power range of the 808 nm laser, only UCNPs with Tm doping concentration

> 2 %

can have more than

50 %

of their emission switched off. For Tm doping concentrations above

4 %

, optimum depletion ratios above

90 %

can be achieved.
为了确定增强人口反转的临界

{Tm}^{3 +}

掺杂浓度，合成了 10 批

40 - nm

UCNPs，其 Tm 浓度从

0.5 mol %

逐渐增加到

8 mol %

（扩展数据图 6）。在双激光连续波（CW）照射下，测量了 455 nm 发射的光耗比（即在 808 nm 耗散探针存在时与不存在时的发射强度比）随 808 nm 强度变化的关系。如图 4a 所示，在 808 nm 激光的可用功率范围内，仅当钬掺杂浓度

> 2 %

时，UCNPs 的发射强度可有超过

50 %

被关闭。当钬掺杂浓度高于

4 %

时，可实现最佳耗尽比超过

90 %

。

The saturation intensities, denoting the particular values of the probe intensity that halve the upconverted emission, were calculated from a plot of the inverse of the depletion ratio against the 808 nm intensity via linear curve fitting (Fig. 4b)

^{21, 22}

. The linearity found for UCNPs with Tm doping

\geq 3.5 %

holds only at low 808 nm intensity; at greater intensities, the fit becomes dual- or multi-segment, which suggests that the depletion efficiency for the highly-doped UCNPs is power dependent,
饱和强度，即探针强度使上转换发射强度减半的特定值，通过对 808 nm 强度与耗尽比倒数作图并进行线性拟合计算得到（图 4b）

^{21, 22}

。对于掺钬的 UCNPs，所观察到的线性关系仅在 808 nm 强度较低时成立；当强度增大时，拟合曲线变为双峰或多峰，这表明高掺钬 UCNPs 的耗尽效率与功率相关。
and can be substantially amplified by the photon-avalanche-like effect. Those data points appearing with a linear fit in the low 808 nm intensity range show reduction of saturation intensities by more than two orders of magnitude (from

71.4 MW {cm}^{- 2}

0.19 MW {cm}^{- 2}

) as the Tm doping increases from

0.5 %

8 %

. Once the stimulated emission depletion at low intensity levels of the 808 nm illumination consumes the established population inversion, a balance is re-established between absorption and stimulated emission, and further inhibition of upconversion emission is diminished at higher 808 nm intensities (Extended Data Fig. 7).
并且可以通过类似光子雪崩效应显著增强。在 808 nm 强度较低范围内呈现线性拟合的数据点显示，随着 Tm 掺杂量从

0.5 %

增加到

8 %

，饱和强度减少了两个数量级（从

71.4 MW {cm}^{- 2}

到

0.19 MW {cm}^{- 2}

）。当 808 nm 光照在低强度水平下引起的受激辐射耗尽消耗掉已建立的反转人口时，吸收与受激辐射之间的平衡重新建立，且在更高 808 nm 强度下上转换发光的抑制作用进一步减弱（扩展数据图 7）。

The small saturation intensity of highly doped UCNPs suggests that the photon-avalanche-like effect in single UCNPs alleviates the square root law that all current super-resolution nanoscopy methods obey. Plotting the inverse of saturation intensity against the Tm doping concentration on a log-log scale shows a slope of 3.13 (Fig. 4c), suggesting that the saturation intensity is proportional to the average

Tm - Tm

distance to the ninth power. Both Fig. 4 c and b confirm that the saturation intensities in upconversion materials are highly dependent on the doping concentration of emission centres, showing that the photon-avalanche-like effect plays the key role in bringing down the intensity requirement for stimulated emission depletion.
高掺杂 UCNPs 的较低饱和强度表明，单个 UCNPs 中的光子雪崩效应缓解了当前所有超分辨率纳米成像方法所遵循的平方根定律。将饱和强度倒数与钕掺杂浓度在对数-对数坐标系中作图，得到斜率为 3.13（图 4c），表明饱和强度与平均

Tm - Tm

距离的九次方成正比。图 4c 和 b 均证实，上转换材料中的饱和强度对发光中心掺杂浓度高度依赖，表明光子雪崩效应在降低受激辐射耗尽所需强度方面发挥了关键作用。

We further explored the use of highly doped UCNPs for optical super-resolution imaging; the 808 nm beam was spatially modulated to produce a doughnut-shaped point spread function (PSF) that overlapped with the Gaussian PSF of the 980 nm excitation beam at the focal plane (illustrated in Fig. 5a). The nanocrystals on the periphery of the 980 nm PSF would therefore be expected to be optically switched off by the 808 nm beam, leading to an effective excitation spot smaller than the optical diffraction limit, similar to the situation in STED microscopy

^{21}

. To prove the concept, two samples of monodispersed UCNPs, both doped with

8 % Tm

and

20 % Yb

, were carefully characterized by transmission electron microscopy (TEM), showing average sizes of 39.8 and 12.9 nm , respectively (Fig. 5b and c). A region with a size comparable to the optical diffraction limit but containing three

40 - nm

UCNPs was selected, and a sequence of far-field optical super-resolution images that clearly resolve the adjacent UCNPs were recorded (Fig. 5d). Spot sizes of 48.3 nm were obtained by our upconversion-STED microscopy at an 808 nm intensity of

9.75 MW {cm}^{- 2}

(Fig. 5e and f). As the measured full-width at half-maximum (FWHM) is a convolution between the theoretical resolution and the physical size of the nanoparticle, this only gives the upper bound of the actual PSF size. Deconvolution based on the simple Pythagorean equation

^{23}

shows this result corresponds
我们进一步探讨了高掺杂 UCNPs 在光学超分辨成像中的应用；通过空间调制 808 nm 光束，生成与 980 nm 激发光束在焦平面上的高斯点扩散函数（PSF）重叠的环形 PSF（如图 5a 所示）。因此，位于 980 nm PSF 周边的纳米晶体预计会被 808 nm 光束光学关闭，导致有效激发斑点尺寸小于光学衍射极限，与 STED 显微镜中的情况类似

^{21}

。为了验证这一概念，对两种单分散 UCNPs 样品进行了仔细表征，这两种样品均掺杂了

8 % Tm

和

20 % Yb

，通过透射电子显微镜（TEM）测得平均尺寸分别为 39.8 和 12.9 nm（图 5b 和 c）。选取一个尺寸与光学衍射极限相当但包含三个

40 - nm

UCNPs 的区域，记录了一系列远场光学超分辨率图像，清晰分辨出相邻的 UCNPs（图 5d）。通过我们的上转换-STED 显微镜在

9.75 MW {cm}^{- 2}

808 nm 强度下获得了 48.3 nm 的斑点尺寸（图 5e 和 f）。由于测得的全宽半高（FWHM）是理论分辨率与纳米颗粒物理尺寸的卷积，这仅给出了实际点扩散函数（PSF）尺寸的上限。基于简单毕达哥拉斯方程

^{23}

的去卷积分析表明，该结果与实验结果一致。

Figure 5 | Super-resolution imaging of the highly-doped UCNPs. a, Diagrams of the upconversion-STED super-resolution imaging, in which a Gaussian excitation profile ( 980 nm ) and a Gauss-Laguerre mode ‘doughnut’ depletion profile ( 808 nm ) at far field are employed. b, c, Inset, TEM images of two samples of

8 % Tm

-doped UCNPs revealing an average size of 39.8 and 12.9 nm ; main panel, size dispersion. Scale bar,

100 nm . d

, The resolution enhancement obtained with an increase of depletion intensity. The lines are fitted to

d_{STED} = d_{C} / {(1 + a I_{STED})}^{1 / 2}

, in which

d_{C}

is the FWHM of the confocal spot (

I_{STED} = 0

), and

a

is a constant proportional to

I_{sat}

. The CW 980 nm excitation was kept at

0.66 MW {cm}^{- 2}

. Pixel dwell time, 4 ms ; error bars,

\pm 1 s

.d. with

n = 5

each. Insets, upconversion-STED images of

40 - nm

UCNPs producing the data shown. e, Confocal (left) and super-resolution (right) images of the

40 - nm

8 % Tm

-doped UCNPs. The 980 nm and 808 nm intensities were 0.66 and
图 5 | 高掺杂 UCNPs 的超分辨成像。a，上转换-STED 超分辨成像示意图，其中采用远场处的高斯激发光谱（980 nm）与高斯-拉盖尔模式“甜甜圈”耗散光谱（808 nm）。b, c, 插图，两种

8 % Tm

掺杂 UCNPs 的 TEM 图像，显示平均尺寸为 39.8 和 12.9 nm；主图，尺寸分散度。刻度尺，

100 nm . d

，随着耗尽强度增加获得的分辨率提升。曲线拟合至

d_{STED} = d_{C} / {(1 + a I_{STED})}^{1 / 2}

，其中

d_{C}

为共焦光斑的全宽半高（

I_{STED} = 0

），

a

为与

I_{sat}

成正比的常数。CW 980 nm 激发保持在

0.66 MW {cm}^{- 2}

。像素驻留时间，4 ms；误差条，

\pm 1 s

。d.与

n = 5

Published online 22 February 2017.
在线发表于 2017 年 2 月 22 日。

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Acknowledgements This project was primarily supported by the Australian Research Council (ARC) Future Fellowship Scheme (D.J., FT 130100517), the ARC Centre of Excellence for Nanoscale BioPhotonics (CE140100003), the Natural Science Foundation of China (61428501, 31327901, 61475010), and the National Instrumentation Project of China (2013YQ03065102). Y. Lu acknowledges support from a Macquarie University Research Fellowship.
致谢本研究主要得到澳大利亚研究理事会（ARC）未来研究员计划（D.J.，FT 130100517）、澳大利亚研究理事会纳米生物光子学卓越研究中心（CE140100003）、国家自然科学基金委员会（61428501，31327901，61475010）以及中国国家仪器项目（2013YQ03065102）。Y. Lu 感谢麦考瑞大学研究基金会的资助。

Author Contributions D.J. and P.X. conceived the project. D.J., P.X. and J.A.P. supervised the research. Y. Liu, X.Y., F.W., X.Z. and Z.Z. conducted the optical experiments. S.W., J. Zhao, D.L., J. Zhou, and C.M. synthesized the upconversion nanoparticles. Y. Lu carried out the modelling. Y. Lu, Y. Liu, X.Y., X.Z., X.V., P.X. and D.J. analysed the results, prepared the figures and wrote the manuscript. All authors participated in discussion and editing of the manuscript.
作者贡献 D.J. 和 P.X. 提出了该项目。D.J.、P.X. 和 J.A.P. 监督了研究工作。Y. Liu、X.Y.、F.W.、X.Z. 和 Z.Z. 进行了光学实验。S.W.、J. Zhao、D.L.、J. Zhou 和 C.M. 合成了上转换纳米颗粒。Y. Lu 进行了建模。Y. Lu、Y. Liu、X.Y.、X.Z.、X.V.、P.X. 和 D.J. 分析了结果、制作了图表并撰写了论文。所有作者均参与了论文的讨论和修改。

Author Information Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests. Readers are welcome to comment on the online version of the paper. Correspondence and requests for materials should be addressed to P.X. (xipeng@pku.edu.cn), Y. Lu (yiqing.lu@mq.edu.au) or D.J. (dayong.jin@uts.edu.au).
作者信息重新印刷和版权信息请访问 www.nature.com/reprints。作者声明无任何利益冲突。读者欢迎对论文的在线版本发表评论。来信及资料请求请发送至 P.X.（xipeng@pku.edu.cn）、Y. Lu（yiqing.lu@mq.edu.au）或 D.J.（dayong.jin@uts.edu.au）。

Reviewer Information Nature thanks Y. D. Suh and the other anonymous reviewer(s) for their contribution to the peer review of this work.
审稿人信息《自然》杂志感谢 Y. D. Suh 及匿名审稿人对本文同行评审的贡献。

METHODS 方法

Dual-laser confocal/super-resolution microscope. The optical system was built on a sample scanning configuration employing a 3-axis closed-loop piezo stage (stage body MAX311D/M, piezo controller BPC303; Thorlabs). Illustrated in Extended Data Fig. 1, a single-mode fibre-coupled 980 nm diode laser (LE-LS-980-300-FCS, LEO Photonics; maximum output power 300 mW ) was used as the excitation source. After collimation, the excitation beam was transmitted through a long-pass dichroic mirror (ZT860lpxr, Chroma), then reflected by a second short-pass dichroic mirror (T750spxrxt, Chroma), and focused through an oil-immersion objective (UPlanAPO, Olympus;

100 \times, NA = 1.4

) onto the sample slide. The first dichroic mirror also allowed the 808 nm probing beam from a polarization-maintaining fibre-coupled diode laser (LU0808M250-1C16F30A, Lumics; single mode, linear polarization, maximum output power 250 mW ) to merge with the 980 nm beam. The luminescence signal from the sample was collected by the same objective, split from the excitation and probing beams by the second dichroic mirror, before being coupled into a multi-mode fibre (M24L01, Thorlabs). The other end of the fibre was connected to a single-photon avalanche diode (SPAD; SPCM-AQRH-13-FC, PerkinElmer) capable of being time-gated electronically. To select upconversion emission bands, different band-pass filters (FF01-448/20-25 and FF01-660/13-25, Semrock) were inserted in the detection path for both transient response measurement and confocal imaging. The distal end of the multi-mode fibre could also be switched to a miniature monochromator (MicroHR Auto, Horiba) equipped with a second SPAD for measuring upconversion emission spectra (Extended Data Fig. 2).
双激光共聚焦/超分辨率显微镜。光学系统基于样品扫描配置构建，采用三轴闭环压电舞台（舞台主体 MAX311D/M，压电控制器 BPC303；Thorlabs）。如扩展数据图 1 所示，采用单模光纤耦合的 980 nm 半导体激光器（LE-LS-980-300-FCS，LEO Photonics；最大输出功率 300 mW）作为激发光源。经过准直后，激发光束通过长波长分光镜（ZT860lpxr，Chroma），再由第二个短波长分光镜（T750spxrxt，Chroma）反射，并通过油浸物镜（UPlanAPO，Olympus；

100 \times, NA = 1.4

）聚焦到样品载玻片上。第一个分光镜还允许来自偏振保持光纤耦合二极管激光器（LU0808M250-1C16F30A，Lumics；单模，线性偏振，最大输出功率 250 mW）的 808 nm 探测光束与 980 nm 光束合并。样品的发光信号由同一物镜收集，经第二个二色镜与激发光和探测光分离后，耦合入多模光纤（M24L01，Thorlabs）。光纤的另一端连接至单光子雪崩二极管（SPAD；SPCM-AQRH-13-FC，PerkinElmer），该器件可实现电子时间门控。为选择上转换发射带，在检测路径中插入不同带通滤光片（FF01-448/20-25 和 FF01-660/13-25，Semrock），用于瞬态响应测量和共聚焦成像。多模光纤的远端还可以切换到一个配备第二个 SPAD 的微型单色仪（MicroHR Auto，Horiba），用于测量上转换发射光谱（扩展数据图 2）。

For measuring the 800 nm transient response under 980 nm excitation only (Fig. 3c), the distal end of the multi-mode fibre connected to the SPAD was substituted for the distal end of the polarization-maintaining fibre of the 808 nm laser, with a band-pass filter (FB800-10, Thorlabs) also inserted in the beam path.
为了测量在 980 nm 激发下 800 nm 的瞬态响应（图 3c），将连接到 SPAD 的多模光纤的远端替换为 808 nm 激光的偏振保持光纤的远端，并在光路中插入一个带通滤光片（FB800-10，Thorlabs）。

For acquiring optical super-resolution images, a quarter-wave plate (WPQ10M-808, Thorlabs) was used to first transform the 808 nm beam from linear polarization to circular polarization. In practice a half-wave plate (WPH10M-808, Thorlabs) was also used to facilitate the adjustment towards quality circular polarization. Then, a vortex phase plate (VPP-1a, RPC Photonics) was inserted in the 808 nm beam path, so that a doughnut-shaped PSF was generated at the focal plane. In both cases of dual-laser confocal and super-resolution imaging, the two beams of 980 and 808 nm were carefully aligned to ensure precise overlapping of their PSFs in both

X - Y

and

Z

directions.
为了获取光学超分辨率图像，首先使用四分之一波片（WPQ10M-808，Thorlabs）将 808 nm 光束从线性偏振转换为圆偏振。实际操作中，还使用了一个半波片（WPH10M-808，Thorlabs）以方便调整至高质量圆偏振状态。随后，在 808 nm 光路中插入一个涡旋相位板（VPP-1a，RPC Photonics），使焦平面上生成环形点扩散函数（PSF）。在双激光共聚焦成像和超分辨率成像两种情况下，980 nm 和 808 nm 的两束光均经过精确对准，以确保其 PSF 在

X - Y

和

Z

方向上实现精确重叠。
Synthesis of UCNPs. To produce

40 - nm NaYF F_{4} : Yb, Tm

UCNPs with precisely controlled size, we first synthesized core nanocrystals using the organometallic method, followed by epitaxial growth via a hot-injection protocol. To obtain the core nanocrystals, 5 ml of a methanol solution of

{LnCl}_{3} (Ln = Y / Yb / Tm, 1.0 mmol

in total) together with 6 ml oleic acid (OA) and 15 ml 1-octadecene (ODE) were added to a 50 ml three-neck round-bottom flask. The resulting mixture was heated at

170^{\circ} C

under argon flow for 30 min to form a transparent solution of light yellow colour. The solution was cooled down to

50^{\circ} C

, and 5 ml of methanol solution containing

4.0 mmol {NH}_{4} F

and 2.5 mmol NaOH was added with vigorous stirring for 30 min . Then, the mixture was slowly heated to

150^{\circ} C

and kept for 20 min under argon flow to remove methanol and residual water. Next, the solution was quickly heated to

300^{\circ} C

, and maintained at that temperature for 1.5 h . The slurry was then cooled down and the products were isolated by addition of ethanol and centrifugation. After washing with cyclohexane/ethanol several times, the core nanocrystals were re-dispersed in cyclohexane at a concentration of

20 mg {ml}^{- 1}

.
UCNPs 的合成。为了制备具有精确控制尺寸的

40 - nm NaYF F_{4} : Yb, Tm

UCNPs，我们首先采用有机金属法合成核心纳米晶体，随后通过热注射法实现外延生长。为了获得核心纳米晶体，将 5 ml 甲醇溶液（总共

{LnCl}_{3} (Ln = Y / Yb / Tm, 1.0 mmol

）与 6 ml 油酸（OA）和 15 ml 1-十八烯（ODE）加入到 50 ml 三口圆底烧瓶中。将混合物在氩气流下于

170^{\circ} C

下加热 30 分钟，形成浅黄色透明溶液。溶液冷却至

50^{\circ} C

，加入 5 毫升含

4.0 mmol {NH}_{4} F

和 2.5 毫摩尔氢氧化钠的甲醇溶液，剧烈搅拌 30 分钟。随后，混合物缓慢加热至

150^{\circ} C

，并在氩气流下保持 20 分钟以除去甲醇和残留水分。随后，溶液迅速加热至

300^{\circ} C

，并保持该温度 1.5 小时。将浆液冷却后，通过加入乙醇并离心分离产物。经环己烷/乙醇多次洗涤后，将核心纳米晶体重新分散于环己烷中，浓度为

20 mg {ml}^{- 1}

。

The same procedures were repeated one more time, until the step where the reaction solution was slowly heated to

150^{\circ} C

and kept for 20 min . Instead of further heating to

300^{\circ} C

to trigger nanocrystal growth, the solution was cooled down to room temperature to yield the shell precursors.
相同的操作步骤重复了一次，直到将反应溶液缓慢加热至

150^{\circ} C

并保持 20 分钟。与之前不同的是，此次并未进一步加热至

300^{\circ} C

以触发纳米晶体生长，而是将溶液冷却至室温，从而获得壳层前驱体。

For epitaxial growth, 1.5 ml as-prepared core nanocrystals were added to a 50 ml flask containing 6 ml OA and 6 ml ODE . The mixture was heated to

170^{\circ} C

under argon for 30 min , and then further heated to

300^{\circ} C

. Next, 0.25 ml asprepared shell precursors were injected into the reaction mixture and ripened at

300^{\circ} C

for 4 min , followed by the same injection and ripening cycles for approximately 16 times. Finally, the slurry was cooled down to room temperature and the formed nanocrystals were purified according to the same procedure used for the core nanocrystals.
对于外延生长，将 1.5 ml 新鲜制备的核心纳米晶体加入到含有 6 ml OA 和 6 ml ODE 的 50 ml 烧瓶中。混合物在氩气氛围下加热至

170^{\circ} C

，保持 30 分钟，随后进一步加热至

300^{\circ} C

。随后，将 0.25 ml 预制壳前驱体注入反应混合物中，在

300^{\circ} C

下成熟 4 分钟，然后重复相同的注入和成熟循环约 16 次。最后，将浆液冷却至室温，并按照与核心纳米晶体相同的程序对形成的纳米晶体进行纯化。

The protocol for synthesizing

13 - nm {NaYF}_{4}

nanocrystals co-doped with

20 % Yb

and

8 % Tm

is similar to the one above for producing the core nanocrystals, except for the temperature control profile. 5 ml of methanol solution containing 0.72 mmol

{YCl}_{3}, 0.2 mmol {YbCl}_{3}

and

0.08 mmol {TmCl}_{3}

were mixed with 6 ml OA and 15 ml ODE in a 50 ml flask. The mixture was slowly heated up to

150^{\circ} C

under argon flow, and kept isothermally for 30 min to form a clear solution. The solution was cooled down to room temperature, and 5 ml of methanol solution containing

4.0 mmol {NH}_{4} F

and 2.5 mmol NaOH was added with vigorous stirring for 30 min .
合成共掺杂

13 - nm {NaYF}_{4}

和

20 % Yb

的

8 % Tm

纳米晶体的协议与上述制备核心纳米晶体的协议类似，仅温度控制曲线不同。将含 0.72 mmol

{YCl}_{3}, 0.2 mmol {YbCl}_{3}

和

0.08 mmol {TmCl}_{3}

的 5 ml 甲醇溶液与 6 ml OA 和 15 ml ODE 在 50 ml 烧瓶中混合。混合物在氩气流下缓慢加热至

150^{\circ} C

，并保持恒温 30 分钟以形成清晰溶液。溶液冷却至室温，加入 5 ml 含

4.0 mmol {NH}_{4} F

和 2.5 mmol NaOH 的甲醇溶液，剧烈搅拌 30 分钟。

Remaining in the argon atmosphere, the mixture was slowly heated up to

110^{\circ} C

to evaporate methanol and then to

150^{\circ} C

to evaporate residual water. After that, the solution was heated up to

300^{\circ} C

in 23 min , and kept at that temperature for another 1.5 h . The reaction was then stopped, and the slurry was cooled down to room temperature. The nanocrystals were washed with cyclohexane/ethanol for four times, and finally re-dispersed in cyclohexane.
在氩气气氛中，混合物缓慢加热至

110^{\circ} C

以蒸发甲醇，随后加热至

150^{\circ} C

以蒸发残留水分。之后，溶液在 23 分钟内加热至

300^{\circ} C

，并在此温度下保持 1.5 小时。反应停止后，浆液冷却至室温。纳米晶体用环己烷/乙醇洗涤四次，最后重新分散在环己烷中。
Preparation of sample slides. Sample slides carrying individual-distributed UCNPs were carefully prepared. To achieve the best imaging quality, an embedding medium with refractive index matching that of the immersion oil was made first. 2.4 g Mowiol 4-88 (Sigma-Aldrich) was mixed with 6 g glycerol in a 50 ml centrifuge tube. After stirring on a magnetic stirrer for

1 h, 6 ml

Milli-Q water was added, and the stirring continued for another 2 h . Then, 12 ml tris(hydroxymethyl)aminomethane (Tris)- HCl buffer (

0.2 M, pH 8.5

) was added, and the solution was water-bathed at

50^{\circ} C

under constant agitation until the Mowiol was largely dissolved. Any remaining solids were removed by centrifugation at

7, 500 g

for 30 min .
样品载玻片的制备。携带单个分布的 UCNPs 的样品载玻片被小心制备。为了获得最佳成像质量，首先制备了折射率与浸没油匹配的包埋介质。将 2.4 g Mowiol 4-88（Sigma-Aldrich）与 6 g 甘油在 50 ml 离心管中混合。在磁搅拌器上搅拌后，加入

1 h, 6 ml

Milli-Q 水，继续搅拌 2 小时。随后加入 12 ml Tris-HCl 缓冲液（

0.2 M, pH 8.5

），将溶液置于

50^{\circ} C

水浴中恒温搅拌直至 Mowiol 基本溶解。最后通过

7, 500 g

离心 30 分钟去除残留固体。

To prepare a sample slide, a cover slip was washed with pure ethanol and then Milli-Q water under ultrasonication, and then treated with

50 μ l

polylysine solution (

0.1 %

H_{2} O w / v

). After 30 min , the polylysine was washed off with Milli-Q water, and the cover slip was air-dried.

20 μ l

of the UCNPs (diluted to 0.01 mg

{ml}^{- 1}

in cyclohexane) were dropped onto the treated surface, which was immediately washed with

500 μ l

cyclohexane twice. After being air-dried, the cover slip was put over a clean glass slide spread with

10 μ l

as-prepared embedding medium, and any air bubbles were squeezed out by gentle force. The sample was kept at room temperature for another 24 h to ensure complete dryness before measurement.
制备样品载玻片时，先用纯乙醇清洗盖玻片，再在超声波处理下用 Milli-Q 水清洗，随后用

50 μ l

聚赖氨酸溶液（

0.1 %

在

H_{2} O w / v

中）处理。30 分钟后，用 Milli-Q 水洗去聚赖氨酸，并将盖玻片自然风干。将 UCNPs（用环己烷稀释至 0.01 mg/mL）滴加至处理过的表面，随即用环己烷洗涤两次。空气干燥后，盖玻片置于涂有

10 μ l

制备好的包埋介质的清洁玻璃载片上，并用轻柔力量挤出气泡。样品在室温下静置 24 小时，确保完全干燥后进行测量。
Analytical interpretation of the photon-avalanche-like process. The photon avalanche (PA) phenomenon was first discovered in 1979 in Pr-doped

{LaCl}_{3}

and

{LaBr}_{3}

crystals

^{34}

, followed by reconfirmations in crystals and glasses doped with other lanthanides including

{Sm}^{3 +}, {Nd}^{3 +}, {Tm}^{3 +}, {Er}^{3 +}

and

{Ho}^{3 +}

(refs 17, 20, 35). Later in the 90s various models were proposed to interpret the PA process. Two aspects of PA, namely (1) the intrinsic requirement of upconversion material to enable PA, and (2) the pumping condition in order to observe a PA threshold, have been shown with clear mathematical derivation

^{14 - 16, 36}

. These approaches were adopted here to explain the results obtained in this work.
光子雪崩效应的分析解释。光子雪崩（PA）现象首次于 1979 年在钪掺杂的

{LaCl}_{3}

和

{LaBr}_{3}

晶体中被发现，随后在掺杂其他镧系元素（包括

{Sm}^{3 +}, {Nd}^{3 +}, {Tm}^{3 +}, {Er}^{3 +}

和

{Ho}^{3 +}

）的晶体和玻璃中得到重新确认（参考文献 17、20、35）。20 世纪 90 年代，提出了多种模型试图解释 PA 过程。PA 的两个关键方面，即（1）上转换材料实现 PA 的内在要求，以及（2）观察 PA 阈值所需的泵浦条件，已通过清晰的数学推导得到证实

^{14 - 16, 36}

。本文采用这些方法来解释本研究中获得的结果。

To investigate the underlying cause of enhanced population inversion in highlydoped

{NaYF}_{4} : Yb, Tm

upconversion nanocrystals, consider the simplified energy level diagram illustrated in Extended Data Fig. 4a, which involves two energy levels associated with the sensitizer

{Yb}^{3 +}

ions (

^{2} F_{7 / 2}

and

^{2} F_{5 / 2}

, represented as S 1 and S 2 , respectively) and three associated with the emitter

{Tm}^{3 +}

ions

(^{3} H_{6},^{3} H_{5} /^{3} F_{4}

and

^{3} F_{2, 3} /^{3} H_{4}

, represented as 1,2 and 3 , respectively). Assuming: (a) the excitation photons are absorbed by

{Yb}^{3 +}

only; (b) energy transfer happens between

{Yb}^{3 +}

on the excited level and

{Tm}^{3 +}

on both the ground and the intermediate levels; © cross-relaxation happens between

{Tm}^{3 +}

on the ground and the upper-excited levels; and (d) other non-radiative relaxation pathways are negligible, the following independent rate equations can be obtained
为了探究高掺杂

{NaYF}_{4} : Yb, Tm

上转换纳米晶体中人口倒置增强的根本原因，考虑扩展数据图 4a 中简化的能级图，该图涉及与敏化剂

{Yb}^{3 +}

离子相关的两个能级（

^{2} F_{7 / 2}

和

^{2} F_{5 / 2}

，分别表示为 S 1 和 S 2）以及与发射体

{Tm}^{3 +}

离子

(^{3} H_{6},^{3} H_{5} /^{3} F_{4}

和

^{3} F_{2, 3} /^{3} H_{4}

相关的三个能级（分别表示为 1、2 和 3）。假设：(a) 激发光子仅被

{Yb}^{3 +}

吸收；(b) 能量传递发生在激发态的

{Yb}^{3 +}

与基态和中间态的

{Tm}^{3 +}

之间；(c) 交叉弛豫发生在基态的

{Tm}^{3 +}

与上激发态之间；以及(d) 其他非辐射弛豫途径可忽略，则可得到以下独立速率方程：

\begin{matrix} \frac{d n_{S 2}}{d t} = P n_{S 1} - W_{S} n_{S 2} - c_{1} n_{S 2} n_{1} - c_{2} n_{S 2} n_{2} \\ \frac{d n_{1}}{d t} = - c_{1} n_{S 2} n_{1} + W_{2} n_{2} + b W_{3} n_{3} - k n_{1} n_{3} \\ \frac{d n_{3}}{d t} = c_{2} n_{S 2} n_{2} - W_{3} n_{3} - k n_{1} n_{3} \end{matrix}

where

P

is the absorption rate of

{Yb}^{3 +} (P = σ λ I / h c

, in which

λ

is the excitation wavelength,

I

is the excitation intensity at

λ, σ

is the absorption cross-section of

{Yb}^{3 +}

λ, h

is the Planck constant, and

c

is the speed of light);

W_{S}

is the intrinsic decay rate of excited

{Yb}^{3 +}

;

c_{i}

is the upconversion coefficient between excited

{Yb}^{3 +}

and

{Tm}^{3 +}

on level

i

;

W_{i}

is the intrinsic decay rate of

{Tm}^{3 +}

on level

i

;

b

is the branching ratio for

{Tm}^{3 +}

decaying from level 3;

k

is the cross-relaxation coefficient; and

n

is the population of ions on an energy level satisfying
其中，

P

为

{Yb}^{3 +} (P = σ λ I / h c

的吸收率，其中

λ

为激发波长，

I

为激发强度，

λ, σ

为

{Yb}^{3 +}

在

λ, h

处的吸收截面，

c

为光速；

W_{S}

是激发态

{Yb}^{3 +}

的固有衰减率；

c_{i}

是激发态

{Yb}^{3 +}

与

{Tm}^{3 +}

在能级

i

上的上转换系数；

W_{i}

是

{Tm}^{3 +}

在能级

i

上的固有衰减率；

b

是

{Tm}^{3 +}

从能级 3 衰变的分支比；

k

是交叉弛豫系数；以及

n

是满足特定能级条件下的离子数密度。

\begin{matrix} n_{S 1} + n_{S 2} = 1 \\ n_{1} + n_{2} + n_{3} = 1 \end{matrix}

Let

R_{1} = c_{1} n_{S 2}, R_{2} = c_{2} n_{S 2}

, and selecting equations (2), (3) and (5), we have
令

R_{1} = c_{1} n_{S 2}, R_{2} = c_{2} n_{S 2}

，并选取方程（2）、（3）和（5），我们得到

\begin{aligned} \frac{d n_{1}}{d t} = - R_{1} n_{1} + W_{2} n_{2} + b W_{3} n_{3} - k n_{1} n_{3} \\ \frac{d n_{3}}{d t} = R_{2} n_{2} - W_{3} n_{3} - k n_{1} n_{3} \\ n_{1} + n_{2} + n_{3} = 1 \end{aligned}

To solve equation (6) at the equilibrium state (that is,

d n_{1} / d t = d n_{3} / d t = 0

), eliminating

n_{1}

and

n_{2}

leads to
要解方程（6）在平衡状态（即

d n_{1} / d t = d n_{3} / d t = 0

）下，消去

n_{1}

和

n_{2}

可得

\begin{aligned} A n_{3}^{2} + B n_{3} - R_{1} R_{2} = 0 \\ A = k (R_{2} + W_{3} + b W_{3} - W_{2}) \\ B = k W_{2} - k R_{2} + b W_{3} R_{2} + R_{1} R_{2} + R_{1} W_{3} + W_{2} W_{3} \end{aligned}

Therefore 因此

n_{3} = \frac{B}{2 A} [sgn (B) {(1 + \frac{4 A R_{1} R_{2}}{B^{2}})}^{1 / 2} - 1]

Note that the other root of the quadratic equation, which is negative, is omitted.
请注意，二次方程的另一个根（即负数根）已被省略。
The analytical solution of equations (7) and (8) appears identical to that derived by Joubert et al.

^{14}

when they modelled PA in

{LiYF}_{4}

{Nd}^{3 +}

, except in their case

R_{1}

and

R_{2}

are constant (due to the sole dopant of

{Nd}^{3 +}

) whereas here they are proportional to

n_{S 2}

. However, the number of sensitizers in a single nanocrystal is limited; therefore under sufficient excitation intensity, it is reasonable to assume that the population of

n_{S 2}

approaches its asymptotic limit, so that

R_{1}

and

R_{2}

can be treated as constants in equations (7) and (8). This allows the previous analyses to be adopted for the

Yb - Tm

upconversion system here.
方程（7）和（8）的解析解与 Joubert 等人推导的结果完全一致。

^{14}

当他们对 PA 在

{LiYF}_{4}

中建模时：

{Nd}^{3 +}

，除了在他们的情况下

R_{1}

和

R_{2}

是常数（由于

{Nd}^{3 +}

中仅有单一掺杂剂）而在此处它们与

n_{S 2}

成正比。然而，单个纳米晶体中的敏化剂数量是有限的；因此，在足够的激发强度下，假设

n_{S 2}

的浓度趋近于其渐近极限，从而在方程(7)和(8)中将

R_{1}

和

R_{2}

视为常数。这使得先前对

Yb - Tm

上转换系统的分析可直接应用于本研究。

To understand whether the upconversion material is capable of PA (the first aspect of PA mentioned above), assuming

A R_{1} R_{2} / B^{2} ≪ 1

, from equation (8) we have
要判断上转换材料是否具备 PA 能力（即上述 PA 的第一个方面），假设

A R_{1} R_{2} / B^{2} ≪ 1

，根据式（8）可得

n_{3} \approx {\begin{cases} R_{1} R_{2} / B & when B > 0 \\ - B / A & when B < 0 \end{cases}

It can be seen that, when

B > 0, n_{3}

is inversely proportional to

B

; therefore any change in

B

has negligible impact on

n_{3}

. When

B < 0, n_{3}

becomes proportional to

B

, so that the change in

B

will lead to an obvious difference in

n_{3}

. This

B < 0

regime is where PA happens, allowing

n_{3}

to increase dramatically towards population inversion.
可以看出，当

B > 0, n_{3}

与

B

成反比时，因此

B

的任何变化对

n_{3}

的影响可以忽略不计。当

B < 0, n_{3}

与

B

成正比时，

B

的变化将导致

n_{3}

出现明显差异。这种

B < 0

状态是 PA 发生的情形，使

n_{3}

能够剧烈增加直至达到种群反转。

According to the expression of

B

in equation (7), if an upconversion material offers

k < b W_{3}

, it always gives

B > 0

, therefore PA can never happen. On the other hand, only if

k > b W_{3}

, both regimes may exist for the material, and PA starts when

B

transits from positive to negative. The requirement of

k > b W_{3}

is thus the precondition for PA.
根据方程（7）中

B

的表达式，如果一种上转换材料具有

k < b W_{3}

，它总是产生

B > 0

，因此相变（PA）永远不会发生。另一方面，只有当

k > b W_{3}

时，该材料可能同时存在两种工作模式，且当

B

从正值过渡到负值时，PA 开始发生。因此，

k > b W_{3}

的条件是 PA 发生的先决条件。

This intrinsic requirement can be viewed in another way, illustrated first by Auzel and Chen

^{16}

. For equation (6), at the beginning of the photon upconversion process, one can assume

n_{1} \approx 1

. Therefore at the equilibrium state,
这一内在要求还可以从另一个角度来理解，首先由 Auzel 和 Chen

^{16}

进行阐述。对于方程（6），在光子上转换过程的初始阶段，可以假设

n_{1} \approx 1

。因此，在平衡状态下，

\begin{aligned} - R_{1} + W_{2} n_{2} + b W_{3} n_{3} - k n_{3} = 0 \\ R_{2} n_{2} - W_{3} n_{3} - k n_{3} = 0 \end{aligned}

Equation (10) can be rewritten as
方程（10）可以重新写为

\begin{aligned} n_{2} = \frac{R_{1}}{W_{2}} + \frac{k - b W_{3}}{W_{2}} n_{3} \\ n_{3} = \frac{R_{2}}{W_{3} + k} n_{2} \end{aligned}

Equation (11) can be translated as the block diagram shown in Extended Data Fig. 4b, which corresponds to a typical feedback system. It is explicit that

k > b W_{3}

represents positive feedback, while

k < b W_{3}

gives negative feedback. Thus, the precondition of an upconversion material for PA essentially means the crossrelaxation intensifies to a point that it begins to enhance rather than quench the populations of the excited levels.
方程（11）可转化为扩展数据图 4b 所示的方框图，该图对应于典型的反馈系统。其中，

k > b W_{3}

表示正反馈，而

k < b W_{3}

表示负反馈。因此，功率放大器（PA）用上转换材料的先决条件本质上意味着交叉弛豫效应增强到一定程度，开始增强而非抑制激发态的粒子数。

Now we come to the second aspect of PA mentioned above, namely, the PA threshold. First, let

R_{1} = 0

in equation (7), then we have
现在我们来讨论上述 PA 的第二个方面，即 PA 阈值。首先，令方程（7）中的

R_{1} = 0

，则有

\begin{aligned} A n_{3}^{2} + B^{'} n_{3} = 0 \\ A = k (R_{2} + W_{3} + b W_{3} - W_{2}) \\ B^{'} = k W_{2} - k R_{2} + b W_{3} R_{2} + W_{2} W_{3} \end{aligned}

Therefore, 因此，

n_{3} = max [0, - B^{'} / A]

Hence, the condition for

n_{3}

to be a positive value is
因此，

n_{3}

为正值的条件是

- \frac{B^{'}}{A} = \frac{(k - b W_{3}) R_{2} - k W_{2} - W_{2} W_{3}}{k (R_{2} + W_{3} + b W_{3} - W_{2})} > 0

Since

k > b W_{3}

, this leads to the pumping threshold for PA
自

k > b W_{3}

以来，这导致了 PA 的泵送阈值。

R_{2, th} = \frac{(k + W_{3}) W_{2}}{k - b W_{3}}

So that for

R_{2} \leq R_{2, th}, n_{3} = 0

. Since the value of zero cannot be plotted on a

\log - \log

scale when we plot

n_{3}

against

R_{2}

, the PA threshold is therefore distinct for the case of zero ground-level absorption

^{36}

.
因此，对于

R_{2} \leq R_{2, th}, n_{3} = 0

。由于在绘制

n_{3}

与

R_{2}

时，零值无法在

\log - \log

刻度上绘制，因此在地面吸收为零

^{36}

的情况下，PA 阈值是不同的。

When

R_{1} > 0

, clear observation of the PA threshold depends on the deviation of

n_{3}

from that in the case of zero ground-level absorption. This can be semi-quantified by the values of

n_{3}

at the threshold

R_{2, th}

. For

R_{1} = 0, n_{3} (R_{2, th}) = 0

; whereas for

R_{1} > 0

, note that at

R_{2, th}

we have

B = R_{1} R_{2} + R_{1} W_{3}

, and equation (8) becomes
当

R_{1} > 0

时，PA 阈值的清晰观察取决于

n_{3}

与零地面吸收情况下的偏差。这可以通过阈值

R_{2, th}

处的

n_{3}

值进行半定量分析。对于

R_{1} = 0, n_{3} (R_{2, th}) = 0

时；而对于

R_{1} > 0

时，需注意在

R_{2, th}

处有

B = R_{1} R_{2} + R_{1} W_{3}

，此时方程 (8) 变为

\begin{aligned} n_{3} (R_{2, th}) = \frac{\sqrt{B^{2} + 4 A R_{1} R_{2}} - B}{2 A} \\ = \frac{\sqrt{{(R_{1} R_{2} + R_{1} W_{3})}^{2} + 4 R_{1} R_{2} k (R_{2} + W_{3} + b W_{3} - W_{2})} - (R_{1} R_{2} + R_{1} W_{3})}{2 k (R_{2} + W_{3} + b W_{3} - W_{2})} \end{aligned}

Combining equations (15) and (16),

n_{3} (R_{2, th})

can be solved as
将方程（15）和（16）结合，

n_{3} (R_{2, th})

可解为

n_{3} (R_{2, th}) = \frac{(k W_{2} + k W_{3} + W_{2} W_{3} - b W_{3}^{2})}{[\sqrt{1 + \frac{4 k W_{2} W_{3} (1 + b) (k - b W_{3} + W_{2}) (k + W_{3})}{{(k W_{2} + k W_{3} + W_{2} W_{3} - b W_{3}^{2})}^{2}} R_{1}} - 1]}

Taking a first-order approximation, we have
进行一阶近似，我们得到

\begin{aligned} n_{3} (R_{2, th}) & \approx \frac{(k + W_{3}) W_{2}}{k W_{2} + k W_{3} + W_{2} W_{3} - b W_{3}^{2}} R_{1} \\ = \frac{(k + W_{3}) W_{2}}{(k + W_{3}) W_{2} + (k - b W_{3}) W_{3}} R_{1} \end{aligned}

Therefore,

n_{3}

will be close to zero at the threshold only if

R_{1}

is very small, so that the threshold will distinctly appear. This aspect of PA, first explained by Goldner and Pellé in the context of laser action

^{36}

, indicates that non-resonant ground-level absorption is necessary for the threshold to be observed. In the same paper, they further revealed that resonant pumping for the ground level can give a higher population in the upper level and thus favour laser action.
因此，

n_{3}

仅在

R_{1}

非常小时才会接近零，从而使阈值明显出现。PA 的这一特性首次由 Goldner 和 Pellé 在激光作用的背景下提出

^{36}

，表明非共振基态吸收是观察到阈值的必要条件。在同一论文中，他们进一步揭示了对基态的共振泵浦可使上能级人口增加，从而有利于激光作用。

Combining the two aspects of PA mentioned above, we find that, while observation of threshold behaviour provides sufficient experimental evidence for the positive feedback in PA, an upconversion material capable of such PA-like positive feedback does not necessarily show the threshold, unless under the condition of non-resonant ground-level absorption. In particular, for co-doped upconversion materials containing sensitizers and emitters, the ground-level absorption typically has a similar magnitude to the excited-level absorption (as both are via energy transfer from the sensitizers), so that a threshold is rarely observed.
将上述 PA 的两个方面结合起来，我们发现，虽然观察到阈值行为为 PA 中的正反馈提供了充分的实验证据，但能够产生这种 PA 样正反馈的上转换材料并不一定表现出阈值行为，除非在非共振基态吸收的条件下。特别是对于同时掺杂敏化剂和发光体的上转换材料，基态吸收通常与激发态吸收具有相似的量级（因为两者均通过敏化剂的能量转移实现），因此阈值现象很少被观察到。

Using the rate equations and parameters reported for

{LiYF}_{4} : {Nd}^{14}

, the numerical simulation results drawn as Extended Data Fig. 4c show similar behaviour of the excited-level populations when the pumping rate exceeds the threshold value, regardless of the presence/absence of the actual threshold (the latter case is simulated by fixing

β = R_{1} / R_{2} = 1

to represent resonant ground-level absorption). Similarly, using the model and parameters we find for our

{NaYF}_{4} : Yb, Tm

UCNPs, the threshold can be determined from the simulation by setting the ground-level upconversion coefficient far smaller than the excited-level upconversion coefficients (we chose

c_{1} = 1

), as shown in Extended Data Fig. 5d. The behaviour of the excited-level populations again appears very similar when the pumping rate exceeds the threshold value, regardless of the presence/absence of the virtual threshold.
使用

{LiYF}_{4} : {Nd}^{14}

中报告的速率方程和参数，数值模拟结果如扩展数据图 4c 所示，当泵浦速率超过阈值时，激发能级人口的行为相似，无论实际阈值是否存在（后一种情况通过将

β = R_{1} / R_{2} = 1

固定为共振基态吸收来模拟）。同样地，使用我们为

{NaYF}_{4} : Yb, Tm

UCNPs 得到的模型和参数，可以通过将基态上转换系数设置为远小于激发态上转换系数（我们选择

c_{1} = 1

）来从模拟中确定阈值，如扩展数据图 5d 所示。当泵浦率超过阈值时，激发能级人口的行为再次表现出高度相似性，无论虚拟阈值是否存在。
Numerical simulation of the

Y b - T m

upconversion system. Assuming rapid non-radiative decays for

^{3} H_{5} \to^{3} F_{4}

and

^{3} F_{2, 3} \to^{3} H_{4}

, and thus each pair being combined into one level, the energy level diagram of the

Yb - Tm

upconversion system in Fig. 3a is redrawn as Extended Data Fig. 5a, with rate parameters labelled on each pathway. The following model of rate equations is established

Y b - T m

上转换系统的数值模拟。假设

^{3} H_{5} \to^{3} F_{4}

和

^{3} F_{2, 3} \to^{3} H_{4}

的衰减为快速非辐射衰减，因此每对能级被合并为一个能级。图 3a 中

Yb - Tm

上转换系统的能级图被重新绘制为扩展数据图 5a，并在每个路径上标注了速率参数。建立了以下速率方程模型

\begin{aligned} \frac{d n_{S 2}}{d t} = & P_{980} n_{S 1} - W_{S} n_{S 2} - (c_{1} n_{1} + c_{2} n_{2} + c_{3} n_{3} + c_{4} n_{4}) n_{S 2} \\ \frac{d n_{2}}{d t} = & c_{1} n_{S 2} n_{1} - c_{2} n_{S 2} n_{2} - W_{2} n_{2} + b_{32} W_{3} n_{3} + b_{42} W_{4} n_{4} \\ + b_{52} W_{5} n_{5} + 2 k_{31} n_{1} n_{3} + k_{41} n_{1} n_{4} \end{aligned}

\begin{aligned} \frac{d n_{3}}{d t} = & P_{808} (n_{1} - n_{3}) + c_{2} n_{S 2} n_{2} - c_{3} n_{S 2} n_{3} - W_{3} n_{3} + b_{43} W_{4} n_{4} \\ + b_{53} W_{5} n_{5} - k_{31} n_{1} n_{3} + k_{41} n_{1} n_{4} + 2 k_{51} n_{1} n_{5} \\ \frac{d n_{4}}{d t} = & c_{3} n_{S 2} n_{3} - c_{4} n_{S 2} n_{4} - W_{4} n_{4} + b_{54} W_{5} n_{5} - k_{41} n_{1} n_{4} \\ \frac{d n_{5}}{d t} = & c_{4} n_{S 2} n_{4} - W_{5} n_{5} - k_{51} n_{1} n_{5} \end{aligned}

where

P_{980}

is the absorption rate of

{Yb}^{3 +}; P_{808}

is the absorption/stimulated emission rate of

{Tm}^{3 +}

(so that the term of

P_{808} (n_{1} - n_{3})

introduced into

d n_{3} / d t

represents the net effect of absorption and stimulated emission);

W_{S}

is the intrinsic decay rate of excited

{Yb}^{3 +}

;

c_{i}

is the upconversion coefficient between excited

{Yb}^{3 +}

and

{Tm}^{3 +}

on level

i

;

W_{i}

is the intrinsic decay rate of

{Tm}^{3 +}

on level

i

;

b_{i j}

is the branching ratio for

{Tm}^{3 +}

decaying from level

i

to level

j

satisfying

\sum_{j = 1}^{i - 1} b_{i j} = 1; k_{i j}

is the crossrelaxation coefficient between

{Tm}^{3 +}

on level

i

and level

j

; and

n

is the population of ions on an energy level satisfying
其中

P_{980}

是

{Yb}^{3 +}; P_{808}

的吸收率，

{Yb}^{3 +}; P_{808}

是

{Tm}^{3 +}

的吸收/受激辐射率（因此，引入到

d n_{3} / d t

中的

P_{808} (n_{1} - n_{3})

项代表吸收和受激辐射的净效应）；

W_{S}

是激发态

{Yb}^{3 +}

的固有衰减率；

c_{i}

是激发态

{Yb}^{3 +}

与

{Tm}^{3 +}

在能级

i

之间的上转换系数；

W_{i}

是

{Tm}^{3 +}

在能级

i

上的固有衰减率；

b_{i j}

是

{Tm}^{3 +}

从能级

i

衰变到能级

j

满足

\sum_{j = 1}^{i - 1} b_{i j} = 1; k_{i j}

的分支比，其中

\sum_{j = 1}^{i - 1} b_{i j} = 1; k_{i j}

是能级

{Tm}^{3 +}

与能级

i

之间的交叉弛豫系数；且

n

是满足上述条件的能级上的离子数。

\begin{matrix} n_{S 1} + n_{S 2} = 1 \\ n_{1} + n_{2} + n_{3} + n_{4} + n_{5} = 1 \end{matrix}

We use this rate-equations model of equations (19) and (20) to fit the transient response obtained from the

8 % Tm

-doped UCNPs under dual-laser excitation. The rate parameters yielded are summarized in Extended Data Fig. 5b, and the simulation results are plotted in Extended Data Fig. 5c, showing good consistency with the measurements.
我们采用方程组(19)和(20)构成的速率方程模型，对双激光激发下

8 % Tm

掺杂 UCNPs 的暂态响应进行拟合。所得速率参数汇总于扩展数据图 5b，模拟结果绘制于扩展数据图 5c，与测量结果呈现良好一致性。

The cross-relaxation coefficients are found to be orders of magnitude larger than the decay rates as well as the upconversion coefficients. In particular, it is seen
交叉弛豫系数发现比衰减率以及上转换系数大几个数量级。特别地，可以观察到
that the precondition for PA for the lowest three energy levels (that is,

k_{31} > b_{31} W_{3}

) has been satisfied; whereas if

k_{31}

is reduced quadratically with the doping concentration, for

1 % Tm

-doped nanocrystals we have

k_{31}^{'} = 2.3 \times 10^{3}

, so that

k_{31}^{'} < b_{31} W_{3}

. These support the assertion that intense cross-relaxation in the PA-like process plays the key role in enhancing population inversion.
PA 条件在最低三个能级（即

k_{31} > b_{31} W_{3}

）已满足；而当

k_{31}

随掺杂浓度呈二次方关系减小时，对于

1 % Tm

掺杂的纳米晶体，我们得到

k_{31}^{'} = 2.3 \times 10^{3}

，从而满足

k_{31}^{'} < b_{31} W_{3}

。这些结果支持了这样一种观点：即在 PA 样过程中，强烈的交叉弛豫在增强粒子反转中起着关键作用。

It should be noted that the modelling conducted here does not address all of the processes in the

Yb - Tm

upconversion system. On the one hand, Extended Data Fig. 2 indicates that both the blue peak at 480 nm and the red peak at 650 nm become negligible for high

{Tm}^{3 +}

doping concentrations, suggesting substantial cross-relaxation involving the

^{1} G_{4}

level. This may well include (

n_{4}, n_{2}

)

\to (n_{3}, n_{3})

(n_{4}, n_{3}) / (n_{3}, n_{4}) \to (n_{2}, n_{5})

and

(n_{4}, n_{4}) \to (n_{3}, n_{5})

, in addition to the sole

(n_{4}, n_{1}) \to (n_{3}, n_{2}) / (n_{2}, n_{3})

cross-relaxation process involving

n_{4}

that has been considered in the rate equations of equation (19). On the other hand, our current system is not able to characterize

{Tm}^{3 +}

emission bands outside the visible range, such as the one-photon emission at

1.8 μ m

and the five-photon emission at 350 nm . Nevertheless, the current modelling provides results reasonably close to the experimental data, which we believe is suited for illustrating the physical picture and forming a stepping stone for more precise investigation in the future.
需要指出的是，本文中进行的建模并未涵盖

Yb - Tm

上转换系统中的所有过程。一方面，扩展数据图 2 表明，当

{Tm}^{3 +}

掺杂浓度较高时，480 nm 处的蓝色峰和 650 nm 处的红色峰均变得可忽略，这表明涉及

^{1} G_{4}

能级的显著交叉弛豫现象。这可能包括（

n_{4}, n_{2}

）

\to (n_{3}, n_{3})

、

(n_{4}, n_{3}) / (n_{3}, n_{4}) \to (n_{2}, n_{5})

和

(n_{4}, n_{4}) \to (n_{3}, n_{5})

，以及在方程（19）的速率方程中考虑的唯一涉及

n_{4}

的

(n_{4}, n_{1}) \to (n_{3}, n_{2}) / (n_{2}, n_{3})

交叉弛豫过程。另一方面，当前系统无法表征可见光范围外的

{Tm}^{3 +}

发射带，例如

1.8 μ m

处的单光子发射和 350 nm 处的五光子发射。然而，当前建模结果与实验数据较为吻合，我们认为这足以阐明物理机制并为未来更精确的研究奠定基础。
Sample size. No statistical methods were used to predetermine sample size.
样本量。未采用统计方法预先确定样本量。
Data availability. The data that support the findings of this study are available from the corresponding authors upon reasonable request.
数据可用性。本研究中支持研究结果的数据可应合理请求从通讯作者处获取。
34. Chivian, J. S., Case, W. E. & Eden, D. D. The photon avalanche: a new phenomenon in

\Pr^{3 +}

-based infrared quantum counters. Appl. Phys. Lett. 35, 124-125 (1979).
34. 芝维安，J. S.，凯斯，W. E. & 埃登，D. D. 光子雪崩：基于

\Pr^{3 +}

的红外量子计数器中的新现象。应用物理快报 35, 124-125 (1979).
35. Joubert, M. F. Photon avalanche upconversion in rare earth laser materials. Opt. Mater. 11, 181-203 (1999).
35. 朱伯特，M. F. 稀土激光材料中的光子雪崩上转换。光学材料，11，181-203（1999）。
36. Goldner, P. & Pelle, F. Photon avalanche fluorescence and lasers. Opt. Mater. 5, 239-249 (1996).
36. 金德纳，P. & 佩勒，F. 光子雪崩荧光与激光. 光学材料 5, 239-249 (1996).

Extended Data Figure 1 | Optical layout of the dual-laser confocal/super-resolution microscope for probing

{NaYF}_{4}

: Yb , Tm nanocrystals.

{SMF}^{2}

single-mode fibre; PMF, polarization-maintaining fibre; MMF, multi-mode fibre; DC1 and DC2, dichroic filters; L1, L2 and L3, collimation/collection lenses; HWP, half-wave plate; QWP, quarter-wave plate; VPP, vortex phase plate; SPAD, single-photon avalanche diode; M, mirror.
扩展数据图 1 | 双激光共聚焦/超分辨显微镜的光学布局，用于探测

{NaYF}_{4}

：Yb、Tm 纳米晶体。

{SMF}^{2}

单模光纤；PMF，偏振保持光纤；MMF，多模光纤；DC1 和 DC2，二色滤光片；L1、L2 和 L3，准直/收集透镜；HWP，半波片；QWP，四分之一波片；VPP，涡旋相位片；SPAD，单光子雪崩二极管；M，反射镜。

Extended Data Figure

2

| Upconversion emission spectra of

8 %

(left) and

1 %

(right) Tm-doped UCNPs. The 980 nm and 808 nm laser powers measured at the objective back aperture were 1 and 10 mW , respectively.
扩展数据图

2

| Tm 掺杂 UCNPs 的向上转换发射光谱（左：

8 %

，右：

1 %

）。在物镜后孔处测得的 980 nm 和 808 nm 激光功率分别为 1 mW 和 10 mW。

Extended Data Figure 3 | Transient responses of

650 n m

upconversion emission under synchronous

980 n m

and

808 n m

pulses. Left,

8 %

Tm-doped UCNPs; right,

1 % Tm

-doped UCNPs. The 980 nm laser power was fixed at 1 mW , while the 808 nm laser power was varied from 0 to 40 mW (both measured at the objective back aperture).
扩展数据图 3 | 同步

980 n m

和

808 n m

脉冲下

650 n m

上转换发光的瞬态响应。左，

8 %

Tm 掺杂的 UCNPs；右，

1 % Tm

掺杂的 UCNPs。980 nm 激光功率固定为 1 mW，而 808 nm 激光功率在 0 至 40 mW 之间变化（均在物镜后孔径处测量）。

Extended Data Figure 5 | Simulation of the

Y b

-Tm upconversion system. a, Energy level diagram used for rate equation modelling. b, Rate parameters simulated for the

8 % Tm

-doped UCNPs. c, Comparison between the simulation and the experimental results for the

8 %

Tm-doped UCNPs under dual-laser excitation with different
扩展数据图 5 |

Y b

-Tm 上转换系统的模拟。a，用于速率方程建模的能级图。b，

8 % Tm

掺杂 UCNPs 的模拟速率参数。c，在双激光激发下，

8 %

Tm 掺杂 UCNPs 的模拟结果与实验结果的比较，采用不同的
b

absorption rates ( $s^{- 1}$ ) 吸收率（ $s^{- 1}$ ）		decay rates ( $s^{- 1}$ ) 衰减率 ( $s^{- 1}$ )
$P_{980}$	$P_{808}$	$W_{s}$	$W_{2}$	$W_{3}$	$W_{4}$	$W_{5}$
$2.75 \times 10^{4}$	$7.4 \times 10^{5}$	$7.6 \times 10^{3}$	$6.4 \times 10^{3}$	$1.8 \times 10^{4}$	$1.4 \times 10^{4}$	$3.2 \times 10^{3}$
upconversion coefficients 上转换系数				cross-relaxation coefficients 交叉弛豫系数
$c_{1}$	$c_{2}$	$c_{3}$	$c_{4}$	$k_{31}$	$k_{41}$	$k_{51}$
$6.23 \times 10^{4}$	$5.7 \times 10^{4}$	$7.4 \times 10^{4}$	$5.2 \times 10^{3}$	$1.5 \times 10^{5}$	$1.75 \times 10^{5}$	$4.8 \times 10^{5}$
branching ratios 分支比
$b_{31}$	$b_{32}$	$b_{41}$	$b_{43}$	$b_{51}$	$b_{52}$	$b_{53}$
0.27	0.73	0.18	0.58	0.24	0.23	0.20

pulse sequences (left and right). d, Numerical simulation of the emitter populations under 980 nm pumping only, using the obtained

c_{1}

value (left), and by artificially setting

c_{1} = 1

to reveal the threshold (right). Details of rate equation modelling are given in Methods.
脉冲序列（左和右）。d，在仅 980 nm 泵浦条件下，使用获得的

c_{1}

值（左）对发射体浓度进行数值模拟，以及通过人工设置

c_{1} = 1

以揭示阈值（右）。速率方程建模的详细信息见方法部分。

Extended Data Figure

6

| TEM images of the

{N a Y F}_{4} : Y b

, Tm nanocrystals. The Yb doping concentration was fixed at

20 mol %

, and

a - j

, the Tm doping concentrations are respectively

0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 6

and

8 mol %

. All nanocrystals have average sizes around 40 nm . Scale bar, 100 nm .
扩展数据图

6

| 透射电子显微镜（TEM）图像显示的

{N a Y F}_{4} : Y b

和 Tm 纳米晶体。钇（Yb）掺杂浓度固定为

20 mol %

，钬（Tm）掺杂浓度分别为

a - j

、

0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 6

和

8 mol %

。所有纳米晶体的平均尺寸约为 40 nm。刻度尺，100 nm。

Extended Data Figure 7 | Simulation of the optical switching at 455 nm for

8 %

Tm-doped UCNPs. a, The 455 nm depletion ratio as a function of the 808 nm probing rate, which is simulated using the rate parameters listed in Extended Data Fig. 5b. The curve fits well to

1 / (1 + I / I_{S})

except for
扩展数据图 7 | 455 nm 波长下

8 %

钕掺杂 UCNPs 的光开关模拟。a，455 nm 波长下的耗尽比与 808 nm 探测速率的关系，模拟结果基于扩展数据图 5b 中列出的速率参数。曲线与

1 / (1 + I / I_{S})

拟合良好，仅在
simulated using the rate parameters listed in Extended Data Fig. 5b. The 980 nm pumping is turned on at time

= 0 s

, while the 808 nm probing is turned on at time

= 0.5 ms

. Note the overlap between

n_{1} (^{3} H_{6}

ground level) and

n_{3} (^{3} H_{4}

intermediate level) once the 808 nm probing is turned on. the high power range.

b

, Emitter populations as a function of time, again
使用扩展数据图 5b 中列出的速率参数进行模拟。980 nm 泵浦在时间

= 0 s

开启，而 808 nm 探测在时间

= 0.5 ms

开启。注意，当 808 nm 探测开启后，

n_{1} (^{3} H_{6}

（基态水平）与

n_{3} (^{3} H_{4}

（中间能级）之间存在重叠。高功率范围。

b

，发射体浓度随时间的变化，再次

Extended Data Figure 8 | Time-series upconversion-STED images recorded for the same sample area under continuous laser excitation and scanning. The sample slide contains

40 - nm

UCNPs with

8 % Tm

and

20 % Yb

. The 455 nm upconversion emission photon count is colour coded.
扩展数据图 8 | 在连续激光激发和扫描条件下，对同一样品区域记录的时序上转换-STED 图像。样品载玻片中含有

40 - nm

UCNPs，其尺寸分别为

8 % Tm

和

20 % Yb

。455 nm 上转换发射光子计数以颜色编码显示。

The 980 and 808 nm laser powers at the objective back aperture were 1 and 30 mW , respectively. The scan step is 20 nm and the pixel dwell time is 4 ms . Scale bar, 500 nm .
980 nm 和 808 nm 激光功率在物镜后孔径处的值分别为 1 mW 和 30 mW。扫描步长为 20 nm，像素驻留时间为 4 ms。刻度尺，500 nm。

$^{1}$ Advanced Cytometry Laboratories, ARC Centre of Excellence for Nanoscale BioPhotonics (CNBP), Macquarie University, Sydney, New South Wales 2109, Australia. $^{2}$ Department of Physics and Astronomy, Macquarie University, Sydney, New South Wales 2109, Australia. $^{3}$ Department of Biomedical Engineering, College of Engineering, Peking University, Beijing 100871, China. $^{4}$ School of Life Sciences and Biotechnology, Shanghai Jiao Tong University, Shanghai 200241, China. $^{5}$ Institute for Biomedical Materials and Devices (IBMD), Faculty of Science, University of Technology Sydney, Sydney, New South Wales 2007, Australia.
$^{1}$ 先进细胞计量学实验室，纳米生物光子学卓越研究中心（CNBP），麦考瑞大学，悉尼，新南威尔士州 2109，澳大利亚。 $^{2}$ 物理与天文学系，麦考瑞大学，悉尼，新南威尔士州 2109，澳大利亚。 $^{3}$ 北京大学工学院生物医学工程系，中国北京市，邮编 100871。 $^{4}$ 上海交通大学生命科学与生物技术学院，中国上海市 200241。 $^{5}$ 悉尼科技大学科学学院生物医学材料与器件研究所（IBMD），澳大利亚新南威尔士州悉尼市 2007。
*These authors contributed equally to this work.
*这些作者对本研究贡献相同。

Amplified stimulated emission in upconversion nanoparticles for super-resolution nanoscopy 上转换纳米颗粒中增强的受激辐射在超分辨率纳米成像中的应用

Published online 22 February 2017.在线发表于 2017 年 2 月 22 日。

METHODS 方法

Amplified stimulated emission in upconversion nanoparticles for super-resolution nanoscopy
上转换纳米颗粒中增强的受激辐射在超分辨率纳米成像中的应用

Published online 22 February 2017.
在线发表于 2017 年 2 月 22 日。