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由于双光子吸收过程属于非线性光学效应,吸收的光子数与光强并非线性关系,而是与光强I的平方成正比,与吸收截面δ成正比。而单光子吸收过程则不属于非线性光学效应,荧光光强与激发光强成线性关系。通过推导双光子荧光的空间分布公式来研究TPF显微成像的成像深度与分辨率。定义光斑半径为ω,束腰处的光斑半径为ω0,瑞利距离为$z_0 $,激发光的波长为λ。以激光的传播方向为轴,以束腰为中心建立柱坐标系,任意一点与光轴的距离为r,与中心的轴向距离$z $。高斯光束复振幅的空间分布可以表示为:
$$ E\left(r,z\right) = {E}_{0}\frac{{\omega }_{0}}{\omega \left(z\right)}\times \mathrm{exp}\left[ -\frac{{r}^{2}}{{\omega }^{2}\left(z\right)} \right]\times \mathrm{e}\mathrm{x}\mathrm{p}\left[ -ik\left(z+ \frac{{r}^{2}}{2R(z)}-\phi (z)\right) \right] $$ (1) 式中:E0为光束中心的复振幅;${\omega }\left(z\right) $表示沿轴方向各位置光斑大小。
$$ {\omega }^{2}\left(z\right)={\omega }_{0}^{2}\left[1+{\left(\frac{z}{{z}_{0}}\right)}^{2}\right] $$ (2) $$ {z}_{0}=\frac{\pi }{2}{\omega }_{0}^{2} $$ (3) 从公式(1)中可以看出,公式分为三部分,第一项与第二项为振幅衰减因子,最后一项为相位衰减因子。任意位置的振幅大小可以通过将该位置的深度以及与光轴的距离代入公式(1)来确定。
光强的大小为复振幅的平方,因此荧光光强的公式表示为:
$$ I={E}_{0}^{2}\frac{{\omega }_{0}^{2}}{{\omega }^{2}\left(z\right)}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{2{r}^{2}}{{\omega }^{2}\left(z\right)}\right) $$ (4) 将E02替换为I0,得到:
$$ I={I}_{0}\frac{{\omega }_{0}^{2}}{{\omega }^{2}\left(z\right)}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{2{r}^{2}}{{\omega }^{2}\left(z\right)}\right) $$ (5) 由于TPF的荧光强度则与激发光的平方成正比。因此TPF的荧光强度分布为:
$$ {I}_{\mathrm{T}\mathrm{P}\mathrm{M}}={A}{I}_{0}^{2}\frac{{\omega }_{0}^{4}}{{\omega }^{4}\left(z\right)}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{4{r}^{2}}{{\omega }^{2}\left(z\right)}\right) $$ (6) 式中:A为TPF激发光强度的平方的比例系数。若要得到TPF信号强度的轴向分布,应忽略位置与光轴的距离r的影响,仅考虑位置与中心的轴向距离$z $,这里令r=0,得到:
$$ {I}_{\mathrm{T}\mathrm{P}\mathrm{M}-{z}}={A}{I}_{0}^{2}\frac{{\omega }_{0}^{4}}{{\omega }^{4}\left(z\right)} $$ (7) 通过公式(7)可以看出,TPF信号强度轴向分布的特点。在轴向任意深度处的TPF信号强度与总光强的平方成正比,与激光的束腰半径的四次方成正比,而与在该深度下的光斑大小的四次方成反比。在成像深度为0时,TPF信号强度均为最大值。随着位置沿$z $轴向两边移动,荧光强度逐渐减小。
如果仅考虑位置与光轴的距离r的影响,可以得到单光子荧光和TPF信号的径向分布。令$z $=0,因此${\omega }\left(z\right) $=ω0,得到:
$$ {I}_{\mathrm{T}\mathrm{P}\mathrm{M}-{r}}={A}{I}_{0}^{2}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{4{r}^{2}}{{\omega }^{2}\left(z\right)}\right) $$ (8) 通过公式(8)可以看出TPF信号强度的径向分布不仅与距离r有关,还受到此处的光斑大小${\omega }\left(z\right) $的影响。在确定深度下,平面上某点的TPF信号强度与总光强的平方成正比,并且随着距离r的增加,该点的信号强度随r2呈e指数衰减。而在不同深度下,光斑越大,则信号强度随距离r的增加衰减得更快。
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公式(5)~(8)为理想情况下高斯光束的单光子荧光与TPF信号的强度分布,然而对较厚的生物样品进行显微成像时,组织对光会有吸收与散射效应。对于近红外光源,组织对光的吸收效应相较于散射效应十分微弱,可以忽略。只有非散射光子才能到达物镜,被探测仪器采集,称为弹道光子,弹道光子数量随着成像深度的增加呈指数衰减。入射光强与深度$z $的关系为:
$$ {I}_{\mathrm{T}\mathrm{P}\mathrm{M}-{r}}=\mathrm{A}{I}_{0}^{2}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{4{r}^{2}}{{\omega }^{2}\left(z\right)}\right) $$ (9) 定义lex为激发光在组织中的散射长度,并且TPF的荧光强度与激发光的平方成正比。则:
$$ {I}_{\mathrm{T}\mathrm{P}\mathrm{M}}={\mathrm{A}I}_{0}^{2}\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{2z}{{l}_{\mathrm{e}\mathrm{x}}}\right) $$ (10) $z $的方向以深入样品为正,散射效应只发生在样品内部,因此公式(10)在$z $>0时成立。根据公式(9)可以得到的最大成像深度${z}_{\mathrm{m}\mathrm{a}\mathrm{x}} $为:
$$ {z}_{\mathrm{m}\mathrm{a}\mathrm{x}}={l}_{\mathrm{e}\mathrm{x}}\mathrm{l}\mathrm{n}\left[{I}_{0}\sqrt{\eta \varphi }\sqrt{1/f\tau }\right] $$ (11) 式中:η为荧光量子效率;φ为荧光收集效率;f与τ为激发光脉冲的重复频率与脉冲宽度。从公式(11)中可以看出,增加激光器的输出功率、以及减小脉冲的重复频率与脉冲宽度都可以增加TPF的成像深度。
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在1.2节中已经推导出TPF信号强度的径向分布公式(8),得到了光强在样品的X~Y平面上的分布情况。激光器输出的光束为高斯光束,束腰位置的光斑最小。光斑大小为光强衰减为中心光强的e−2时的宽度,成像时两像素点之间的距离应不小于2ω0。因此TPF的横向分辨率为:
$$ {r}_{\mathrm{T}\mathrm{P}\mathrm{M}-{x}{y}}=2{\omega }_{0}=2\frac{\lambda {f}_{0}}{\pi \omega } $$ (12) 式中:λ为激发光的波长;ω为光束在物镜前的光斑大小。又因为:
$$ \frac{{f}_{0}}{\omega }=\mathrm{c}\mathrm{o}\mathrm{t}\alpha $$ (13) 式中:α为物镜的汇聚角。将公式(12)改写为:
$$ {r}_{\mathrm{T}\mathrm{P}\mathrm{M}-{x}{y}}=2\frac{\lambda }{\pi }\frac{\sqrt{1-{\mathrm{s}\mathrm{i}\mathrm{n}}^{2}\alpha }}{\mathrm{s}\mathrm{i}\mathrm{n}\alpha } $$ (14) 在实验中,很难去测量光束的汇聚角,但是往往物镜的数值孔径NA是已知的,NA=n·sinα,其中n为物镜与样品之间介质的折射率。对于普通物镜,介质为空气,此时n=1。对于油浸物镜,此时n=1.5。将数值孔径NA代入公式(14)中,得到:
$$ {r}_{\mathrm{T}\mathrm{P}\mathrm{M}-{x}{y}}=2\frac{\lambda }{\mathrm{\pi }}\frac{\sqrt{{n}^{2}-{NA}^{2}}}{NA} $$ (15) Zipfel等人根据高数值孔径衍射理论拟合出TPF显微成像的分辨率公式[11],根据纵向分辨率rTPM-z为光照点扩散函数的平方(IPSF2)纵向极限的1/e,横向分辨率rTPM-xy为IPSF2纵向极限的1/e,得到TPF显微成像的纵向与横向分辨率为:
$$ {r}_{\mathrm{T}\mathrm{P}\mathrm{M}-{z}}=\frac{0.532\sqrt{2\mathrm{l}\mathrm{n}2}\lambda }{n-\sqrt{{n}^{2}-{NA}^{2}}} $$ (16) $$ {r}_{\mathrm{T}\mathrm{P}\mathrm{M}-{x}{y}}=\frac{0.320\sqrt{2\mathrm{l}\mathrm{n}2}\lambda }{NA} $$ (17) 式中:NA为物镜的数值孔径。公式(17)为NA大于0.7时的横向分辨率,而NA小于0.7时的横向分辨率公式为:
$$ {r}_{\mathrm{T}\mathrm{P}\mathrm{M}-{x}{y}}=\frac{0.325\sqrt{2\mathrm{l}\mathrm{n}2}\lambda }{{NA}^{0.91}} $$ (18) 该实验使用物镜的数值孔径NA为0.65,可以计算出显微成像系统的横向分辨率理论上为453 nm,纵向分辨率为2.087 μm。
Two-photon fluorescence 3D microscopic imaging of mouse brain based on femtosecond pulses
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摘要: 双光子荧光(two-photon fluorescence,TPF)显微成像技术借助荧光探针实现样品中被标记成分的特异性成像,具有天然的三维层析能力、高成像深度与空间分辨率、以及更小的光漂白与光损伤,已经发展成为化学、医药学和生命科学领域的一项重要研究工具。文中通过分析高斯光束复振幅在空间中的分布,推导出TPF信号的纵向与径向分布公式,以此估算出文中的TPF显微成像系统的横向分辨率为453 nm,纵向分辨率为2.087 μm。使用飞秒激光器作为激发光源,搭建了TPF显微成像系统。在800 nm波长的飞秒脉冲激发下,测量了罗丹明B溶液的TPF光谱,从而选择636~703 nm作为显微成像的荧光探测窗口。随后开展了对罗丹明B染色的小鼠大脑切片的TPF显微成像实验研究,利用断层扫描成像的方式获得了小鼠大脑切片在0~14 μm深度内的荧光强度分布。通过三维重构完成了对生物样品的三维立体成像,获得了小鼠大脑中灰质与白质在不同深度的分布情况,实验结果证明了搭建的显微成像系统具有优异的成像深度与空间分辨能力。Abstract:
Objective Optical microscopy technology is to observe and record images of the microstructure of objects at a scale indistinguishable from the human eye, and has become an important tool for human observation of the microscopic world. Among them, electron microscopy has a resolution that breaks the limit of optical diffraction and can reach the nanometer level. However, it needs to provide a vacuum environment for electron acceleration, so it is not conducive to the observation of living samples. Optical microscopes are easy to operate and inexpensive, and are widely used in scientific research, industry, medicine and other fields. Conventional light microscopes rely on the contrast produced by differences in the optical properties of the sample for imaging, and do not require labeling or staining, but do not have sufficient specificity. In contrast, nonlinear optical microscopy not only realizes specificity imaging, but also has higher imaging depth and resolution. Among them, fluorescence microscopy technology with the help of fluorescent probes to label different components within the biological sample, through the detection of fluorescence signal to achieve imaging of the labeled components in the sample, to obtain its distribution within the sample. TPF (two-photon fluorescence) microscopic imaging technology based on the nonlinear effect of two-photon absorption, the fluorescence signal will not be excited outside the focal plane, and therefore has a high spatial resolution. TPF microscopy mostly uses infrared wavelength light source, which has lower phototoxicity and photobleaching to biological samples and higher imaging depth. In summary, this paper builds a TPF microscopy system based on femtosecond pulsed light source to study the imaging performance of the system on biological samples. Methods We build a TPF microscopic imaging system (Fig.1), using a Ti: sapphire femtosecond laser as the excitation laser source, with a central wavelength of 800 nm, a repetition frequency of 80 MHz, and a pulse width of 100 fs. The scanning optical path of the system was formed by lens and a scanning oscillator to complete the collimation, beam reduction, and two-dimensional deflection of the excitation light. The fluorescence signal is converted into an electrical signal and processed by a computer to obtain the imaging results. Results and Discussions The output spectrum of the femtosecond laser, and the fluorescence spectrum of rhodamine B were obtained using a TPF spectroscopic measurement system (Fig.2). The central wavelength of the femtosecond laser was 1 030 nm, and the half-height width of the spectrum was 14.47 nm. While the spectral range of the fluorescence covered from 620 nm to 710 nm, the intensity increased steeply from 620 nm, with a peak at 630 nm, and then the intensity decreased slowly with increasing wavelength, so the laser could effectively excite the TPF signal of the sample. The relationship between the two-photon fluorescence intensity and the excitation pulse power was analyzed by adjusting the power of the excitation pulse (Fig.4). The fluorescence intensity was linearly related to the square of the excitation power in the region of different concentrations of the samples. The ratio coefficient of fluorescence intensity to the square of the excitation power was larger in the region with higher concentration for the same excitation power. The fluorescence intensity distribution of the samples within 0-14 μm depth was obtained by 3D TPF microscopic imaging experiments of mouse brain sections (Fig.5). It was obtained that the gray matter portion within the mouse brain sample was located in the superficial layer within 6 μm of the sample, while the white matter portion was more widely distributed longitudinally. The depth distribution curves of the fluorescence intensity of different tissues were obtained by curve fitting, which led to an imaging depth of 12.9 μm for the system. The intensity distribution curves of the narrow slits of multiple samples were plotted, and by analyzing the minimum distance that the imaging system could resolve, a lateral resolution of at least 2.25 μm was derived. Conclusions A femtosecond laser was used as the excitation laser source. The fluorescence spectra of the rhodamine B solution samples were measured under excitation at 800 nm. Thus a detection window of 636-703 nm was selected for subsequent microscopic imaging experiments. TPF microscopic imaging experiments of mouse brain sections stained by rhodamine B were carried out to obtain the fluorescence intensity distribution of biological samples in the depth of 0-14 μm by tomography imaging. After three-dimensional reconstruction of the images, it was concluded that the gray matter portion within the mouse brain sample was located in a shallow layer within 6 μm of the sample, while the white matter portion was more widely distributed longitudinally, while the gray matter of the mouse brain had higher fluorescence intensity and had a higher density than the white matter portion. The experimental results demonstrate that the constructed microscopic imaging system has excellent spatial resolution and imaging depth, with an imaging depth of 12.9 μm and a lateral resolution of at least 2.25 μm. -
图 1 TPF显微成像以及TPF光谱系统。BS:分束片;L3、L4、SL、TL:透镜;GM:扫描振镜;OBJ:物镜;M:反射镜;F:滤波片;PMT:光电倍增管;PC:计算机
Figure 1. TPF microscopic imaging as well as TPF spectroscopy system. BS: beam splitter; L3, L4, SL, TL: lens; GM: scanning oscillator; OBJ: objective; M: reflector; F: filter; PMT: photomultiplier tube; PC: computer
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