-
根据光的标量衍射理论,光学显微成像系统的理论分辨率极限计算公式可以统一表示为:
$$ \delta = \gamma \cdot \frac{\lambda }{{\rm NA}} $$ (1) 式中:λ表示照明或发光波长;NA表示系统的数值孔径;系数γ的取值与照明光的相干性、待测目标是双点或双缝,以及所采用的分辨率判据等因素有关。通过模拟不同情况下目标经过光学系统形成的衍射斑,利用不同判据对像面衍射斑强度分布的不同要求得出不同情况下的γ取值。
-
点源经过衍射受限光学系统所成的像不是理想的几何点像,而是有一定大小的艾里斑。艾里斑沿径向的强度分布为[13]:
$$ I(\rho ) = {I_0}{\left[ {\frac{{2{J_1}({{ka\rho } \mathord{\left/ {\vphantom {{ka\rho } z}} \right. } z})}}{{{{ka\rho } \mathord{\left/ {\vphantom {{ka\rho } z}} \right. } z}}}} \right]^2} $$ (2) 式中:I0为艾里斑中心强度;k = 2π/λ为波矢;a表示衍射孔半径(透镜成像时为透镜孔径的一半);z为衍射孔径到像面距离;ρ为径向坐标;J1是第一类一阶贝塞尔函数。为了简化公式得到统一的分辨率判据,笔者定义无量纲量x与实际距离ρ的关系:x = kaρ/z,用来表示两点或两缝的相对间距。公式(2)可简化为:
$$ I(x) = {I_0}{\left[ {\frac{{2{J_1}(x)}}{x}} \right]^2} $$ (3) (1)非相干光照明下不同判据的两点分辨率
在非相干光照明或荧光显微成像条件下,两个点源经过衍射受限光学系统之后,所成图像为两个点源各自艾里斑强度的非相干叠加,从公式(3)可以得到像面坐标中的总强度分布为:
$$ I(x) = {I_0}{\left[ {\dfrac{{2{J_1}\left(x - \dfrac{{{\rm{d}}x}}{2}\right)}}{{x - \dfrac{{{\rm{d}}x}}{2}}}} \right]^2} + {I_0}{\left[ {\dfrac{{2{J_1}\left(x + \dfrac{{{\rm{d}}x}}{2}\right)}}{{x + \dfrac{{{\rm{d}}x}}{2}}}} \right]^2} $$ (4) 式中:dx表示两个艾里斑中心之间的距离。模拟中通过改变dx的值使两个艾里斑逐渐靠近(远离)达到极限判据情况,然后测量不同条件情况下dx的值,从而求得公式(1)中对应的系数γ值。由x = kaρ/z,可得像面上两点中心的实际距离δ′为:
$$ \delta '{{ = }}{\rm{d}}\rho {{ = }}{\rm{d}}x\dfrac{z}{{ka}} = \dfrac{{{\rm{d}}x}}{{2\pi }}\dfrac{\lambda }{{n'\sin \theta '}} $$ (5) 根据物像空间满足的正弦条件:
$$ \delta n\sin \theta = \delta 'n'\sin \theta ' $$ (6) 式中:n和n′分别表示物空间和像空间的折射率,可得到物面上两点的分辨率距离δ与dx的关系为:
$$ \delta = \dfrac{{{\rm{d}}x}}{{2\pi }}\dfrac{\lambda }{{n\sin \theta }} = \dfrac{{{\rm{d}}x}}{{2\pi }}\dfrac{\lambda }{{\rm NA}} $$ (7) 对比公式(7)和公式(1)可以计算得到不同成像条件下系数γ的取值为γ = dx/2π。
瑞利判据定义为第一个点光源经过衍射受限光学系统之后形成的艾里斑中心与第二个点源产生的艾里斑的第一零点重合。根据公式(2)~(4),计算模拟了两个艾里斑二维强度分布图及沿图中虚线的轮廓线,如图1所示。图中的蓝线(I1)和红线(I2)分别代表两点源在像面上各自形成的强度分布轮廓,绿线(Isum)代表两点源叠加强度分布轮廓。通过不断减小
dx的值模拟两个点源不断靠近,当dx = 3.833时(如图1(b)所示),蓝色线的极大值与红色线的第一极小值重合,满足瑞利判据条件,根据公式(7)可计算得到瑞利判据的系数γ = 0.61。此时,两点所产生的艾里斑强度叠加后的曲线,中心凹陷大小为峰值的27%。 图 1 两非相干点源衍射叠加形成的光斑强度分布随点源间距变化情况。蓝线(I1)和红线(I2)分别表示两点源在像面上各自强度分布轮廓,绿线(Isum)表示总的强度分布轮廓
Figure 1. Superposition intensity distributions formed by two diffracted incoherent point sources vs their separated distance. The blue (I1) and red (I2) lines represent the intensity distribution profiles of the two-point sources on the image plane, while the green one (Isum) represents total intensity distribution profile
对于Saprrow判据,其定义是当两个艾里斑叠加总强度中心主极大消失,在中心区域形成平顶时,所对应两点源之间的距离为分辨率极限,如图1(c)所示,在此情况下dx = 2.990。同理,根据公式(7)可计算得到Sparrow判据的系数γ = 0.47。
(2)相干光照明下不同判据的两点分辨率
在相干光照明条件下,两个点源经过衍射受限光学系统之后,产生的光场为两个艾里斑的复振幅叠加,此时像面总强度分布为:
$$ I(x) = {I_0}{\left| {\dfrac{{2{J_1}\left(x - \dfrac{{{\rm{d}}x}}{2}\right)}}{{x - \dfrac{{{\rm{d}}x}}{2}}} + {{\rm e}^{j\varphi }}\dfrac{{2{J_1}\left(x + \dfrac{{{\rm{d}}x}}{2}\right)}}{{x + \dfrac{{{\rm{d}}x}}{2}}}} \right|^2} $$ (8) 式中:φ为两个点源的相对相位差。当两点的距离dx仍取瑞利判据间隔时(dx = 3.833),两点光源叠加的艾里斑强度随二者相位差及其相对间距的变化关系如图2(a)所示,当φ分别为0、π/2和π三种情况时,对应的强度分布曲线如图2(b)所示。
图 2 相距为瑞利判据间隔的两个不同相位差的衍射相干点源叠加形成的光斑总强度分布。(a)两点光源叠加的艾里斑强度随二者相位差及相对间距的变化关系;(b) φ分别为0、π
/2和π时对应的强度分布曲线 Figure 2. Superposition total intensity distribution profiles formed by two diffracted coherent point sources with different phase difference at the separated distance of Rayleigh criterion. (a) Total intensity distribution profiles of the superimposed two Airy disk against their phase difference and the distance; (b) Intensity distribution profiles at φ = 0, π/2 and π, respectively
当φ = π/2时,I(x)分布与非相干情况完全相同,分辨率相同。当φ = 0时,两个点光源相位相同,I(x)不出现中心凹陷,因而两点在瑞利判据间隔下不能被分辨。若要能分辨,需进一步增大dx的值,以瑞利判据下中心下凹27%进行判断,如图3(c)所示,此时dx = 5.136,相应的γ = 0.82。同样地,在Saprrow判据下(如图3(b)所示),dx = 4.607,相应的γ = 0.73。
图 3 相干光源照明条件下两个相位相同(φ=0)点源的分辨率极限判据。(a)~(d)不同间距情况下两点源所形成艾里斑叠加强度分布
Figure 3. Criterion of resolution limit for two point sources with the same phase (φ=0) under coherent illumination. (a)-(d) Superposition total intensity distributin profiles under different separated distances of the two point sources
当φ=π时,两个点源相位相反。当两个点源不断靠近时,相干叠加后的总强度中心值始终为零,其像面总强度分布如图4中绿色曲线所示,两个点源一直可以分辨,成像分辨率比非相干照明下高。但是此种情况下不能根据合成强度轮廓线峰值位置准确确定两个点源的位置间距,例如当两个点源之间的距离dx=5.136时(如图4(a)所示),第一个点源中心刚好处于第二个点源艾里斑的第一级亮纹位置,此时为相长干涉,合成强度轮廓线峰值位置与两个点源艾里斑中心重合。但是当dx<5.136时,如图4(b)~(d)所示,两个点源的实际位置间距小于探测图像强度两峰值之间的距离,并且随着两个点源的不断靠近,其在像面的合成强度峰值越来越低。为了更清楚地看明这个趋势,图4(e)将4条曲线画在同一坐标系中。可以看到,合成强度两峰值随着dx减小,其强度值越来越小。图4(f)是其相应的每条曲线各自归一化强度分布,可以看到,合成强度两峰之间的间距变化很小。这说明在这种情况下,两点在瑞利判据间隔以下还是不能被有效分辨。
图 4 相干光源照明条件下两个相位相反(φ = π)点源的分辨率极限判断问题。(a)~(d) 两点源的间距dx不断减小时所形成艾里斑叠加强度分布;(e) dx取不同值的两点源叠加强度分布;(f) 图(e)中叠加总强度各自归一化强度分布
Figure 4. Criterion of resolution limit for two point sources with the opposite phase (φ = π) under coherent light illumination. (a)-(d) Superposition total intensity distribution profiles with the reducing separated distance dx of the two point sources; (e) Superposition intensity distributions profiles for different values of dx; (f) Respectively normalized intensity distributions profiles corresponding to (e)
-
与点源衍射形成艾里斑类似,单缝经过光学系统之后由于衍射形成明暗相间的条纹。垂直于缝扩展方向相应强度分布为:
$I\left(\rho \right)={{[\rm sin}\left(ka\rho /z\right)}/{({ka\rho /z)}]}^{2}$ ,同样令x = kaρ/z,则强度分布为:$ I\left(x\right)={({{\rm sin}x}/{x})}^{2} $ ,其像面上所形成的衍射光斑强度分布与点光源艾里斑强度分布对比结果如图5所示。可以看到,单缝衍射光斑的一级暗线和主峰半高宽比艾里斑更小一些,因此用双缝测量系统的分辨率更高一些。(1)非相干光照明或荧光显微成像时,像面上的总强度分布为两个单缝衍射强度分布的叠加:
$I\left(x\right)= {\left(\dfrac{{\rm sin}{x}}{x}\right)}^{2}+{\left[\dfrac{{\rm sin}\left(x+{\rm{d}}x\right)}{x+{\rm{d}}x}\right]}^{2}$ 。此时瑞利判据间距dx = 3.14,γ为0.5;Saprrow判据间距dx = 2.62,γ为0.42。(2)相干光照明时,像面上总强度分布为两个单衍射的复振幅叠加的模平方。当两个狭缝的相位差等于零时,总强度分布表达式为:
$I\left(x\right)={\left[\dfrac{{\rm sin}x}{x}+\left(\dfrac{{\rm sin}\left(x+{\rm{d}}x\right)}{x+{\rm{d}}x}\right)\right]}^{2}$ 。此时瑞利判据间距dx = 4.55,γ为0.71;Saprrow判据间距dx = 4.17,γ为0.66。 -
以上分别对非相干光和相干光照明情况下,光学显微成像系统对双点目标和双缝目标成像时,分辨率极限计算公式中的系数γ进行了计算,表1对上述结果进行了对比。
表 1 不同情形下分辨率极限公式中系数γ的取值
Table 1. Values of coefficients γ in resolution equation at different cases
Resolution criterion Target Coherent imaging Incoherent imaging Abbe Double slit - 0.5 Rayleigh Double point 0.82 0.61 Double slit 0.71 0.5 Sparrow Double point 0.73 0.47 Double slit 0.66 0.42 从表1可以看出,在相同条件下,当分别选用瑞利判据和Sparrow判据计算显微成像系统分辨率极限时,Sparrow判据比瑞利判据得到的分辨率极限更优一些。在同一判据下,在非相干光照明情况下,系数γ的取值更小,相应的成像系统分辨能力更强。相比于双点目标,在用双缝目标标定系统成像分辨率时,系数γ的取值也更小,测量得到的系统分辨能力也更优一些。
值得注意的是,在上述讨论相干光照明条件下,对于相位相反的两个点源,系统总能识别出来两个光斑,但是光斑成像位置与点源实际位置难以确定。当两点源距离小于艾里斑第一级亮环的半径时,经过成像系统后所拍摄图像始终显示总强度的第一个峰值刚好处于第二个点源艾里斑的一级亮环位置。同时,当相位相反的这两点相互靠近时,将造成点源衍射艾里斑在该方向上的压缩,如图4(e)所示。该现象可以用来产生小于衍射极限艾里斑的超振荡聚焦,笔者对此进行了模拟。
为了方便,这里以一维方向压缩进行说明,该结果可以扩展到二维压缩。如图6(a)所示,单位振幅的点光源P0衍射聚焦为艾里斑结构。通过在P0点左右各加入一个振幅相同、相位相反、间距为dx = 5.136的P−1和P+1两个点源,可以实现对初始点源P0的艾里斑在x方向的压缩,如图6(b)所示。通过增加±1级点源的强度,可进一步提高对中心点光源所形成艾里斑的压缩程度。定义±1级点源与中心点源强度比:β = I±1/I0,则衍射光斑随β变化过程如图6(b)~(d)所示。以中心点强度为参考,相应的沿虚线归一化强度分布如图6(e)所示。与原艾里斑相比,图6(d)压缩后中心点衍射光斑峰值半高宽减小36%。类似地,在±1级点源与中心点源强度相同时,通过减小±1级点源与中心点源的距离,同样可以提高压缩程度,如图7所示。图7(c)所示为压缩后中心点衍射光斑半高宽减小46%,但±1级点源距离中心点源不能太近,否则会导致中心点被淹没在±1级艾里斑中,如图7(d)所示。
图 6 超衍射极限聚焦。(a) 单点衍射聚焦艾里斑分布;(b) 3个振幅相同、两边两点(P−1和P+1)相位与中心点(P0)相位相反、相邻两点间距dx = 5.136的3个相干点源衍射聚焦艾里斑的分布;(c)~(d) 逐渐增加+1和−1级两个点源与中心点源的强度比β = I±1/I0,衍射艾里斑的强度分布;(e) 图(a)~(d)中沿虚线以中心点为参考归一化强度分布曲线
Figure 6. Focusing beyond the diffraction limit. (a) Airy disk distribution of single point diffraction focusing; (b) Diffraction focusing Airy pattern of three coherent point source when placing two points of P−1 and P+1 at dx=5.136 with the same intensity and π phase shift to the central point of P0; (c)-(d) Diffraction Airy disk patterns when increasing the ratio of intensities of the P−1 and P+1 points to the central point P0, β = I±1/I0; (e) The intensity distribution profiles along the dashed lines in (a)-(d)
Discussion on spatial resolution of microscopic imaging system (invited)
-
摘要: 空间分辨率是光学显微成像系统的核心指标,根据光学衍射理论,成像系统的空间分辨率由照明光波长与显微物镜的数值孔径共同决定。而在实际成像过程中,根据不同判据得出的显微成像系统分辨率略有差异,需要根据光源的相干性和被观测目标的结构等特征选择合适的判据来准确计算成像系统分辨率。通过理论分析和数值模拟,给出了不同情况下成像分辨率的计算方法,并对比了在相干光源和非相干光源照明下,对双缝目标和双点目标成像时成像分辨率的差异。
-
关键词:
- 光学显微成像 /
- 相干光和非相干光照明 /
- 分辨率判据
Abstract: Spatial resolution is a key specific parameter of the optical microscopic imaging system. According to the optical diffraction theory, the spatial resolution of imaging system is determined by the wavelength of illumination light and the numerical aperture of microscope objective. In the practical imaging process, the resolutions of microscopic imaging system obtained from different criteria are slightly different. It is necessary to select an appropriate criterion according to the coherence of light sources and the structural characteristics of obsversed targets to accurately calculate the resolutions of imaging system. In this paper, the calculation methods for spatial resolution under different conditions are provided via theoretical analysis and numerical simulation. Furthermore, we compare and discusse the difference of imaging resolutions under the illuminations of coherent and incoherent light sources for double points and double slits targets, respectively. -
图 1 两非相干点源衍射叠加形成的光斑强度分布随点源间距变化情况。蓝线(I1)和红线(I2)分别表示两点源在像面上各自强度分布轮廓,绿线(Isum)表示总的强度分布轮廓
Figure 1. Superposition intensity distributions formed by two diffracted incoherent point sources vs their separated distance. The blue (I1) and red (I2) lines represent the intensity distribution profiles of the two-point sources on the image plane, while the green one (Isum) represents total intensity distribution profile
图 2 相距为瑞利判据间隔的两个不同相位差的衍射相干点源叠加形成的光斑总强度分布。(a)两点光源叠加的艾里斑强度随二者相位差及相对间距的变化关系;(b) φ分别为0、π
/2和π时对应的强度分布曲线 Figure 2. Superposition total intensity distribution profiles formed by two diffracted coherent point sources with different phase difference at the separated distance of Rayleigh criterion. (a) Total intensity distribution profiles of the superimposed two Airy disk against their phase difference and the distance; (b) Intensity distribution profiles at φ = 0, π/2 and π, respectively
图 3 相干光源照明条件下两个相位相同(φ=0)点源的分辨率极限判据。(a)~(d)不同间距情况下两点源所形成艾里斑叠加强度分布
Figure 3. Criterion of resolution limit for two point sources with the same phase (φ=0) under coherent illumination. (a)-(d) Superposition total intensity distributin profiles under different separated distances of the two point sources
图 4 相干光源照明条件下两个相位相反(φ = π)点源的分辨率极限判断问题。(a)~(d) 两点源的间距dx不断减小时所形成艾里斑叠加强度分布;(e) dx取不同值的两点源叠加强度分布;(f) 图(e)中叠加总强度各自归一化强度分布
Figure 4. Criterion of resolution limit for two point sources with the opposite phase (φ = π) under coherent light illumination. (a)-(d) Superposition total intensity distribution profiles with the reducing separated distance dx of the two point sources; (e) Superposition intensity distributions profiles for different values of dx; (f) Respectively normalized intensity distributions profiles corresponding to (e)
图 6 超衍射极限聚焦。(a) 单点衍射聚焦艾里斑分布;(b) 3个振幅相同、两边两点(P−1和P+1)相位与中心点(P0)相位相反、相邻两点间距dx = 5.136的3个相干点源衍射聚焦艾里斑的分布;(c)~(d) 逐渐增加+1和−1级两个点源与中心点源的强度比β = I±1/I0,衍射艾里斑的强度分布;(e) 图(a)~(d)中沿虚线以中心点为参考归一化强度分布曲线
Figure 6. Focusing beyond the diffraction limit. (a) Airy disk distribution of single point diffraction focusing; (b) Diffraction focusing Airy pattern of three coherent point source when placing two points of P−1 and P+1 at dx=5.136 with the same intensity and π phase shift to the central point of P0; (c)-(d) Diffraction Airy disk patterns when increasing the ratio of intensities of the P−1 and P+1 points to the central point P0, β = I±1/I0; (e) The intensity distribution profiles along the dashed lines in (a)-(d)
表 1 不同情形下分辨率极限公式中系数γ的取值
Table 1. Values of coefficients γ in resolution equation at different cases
Resolution criterion Target Coherent imaging Incoherent imaging Abbe Double slit - 0.5 Rayleigh Double point 0.82 0.61 Double slit 0.71 0.5 Sparrow Double point 0.73 0.47 Double slit 0.66 0.42 -
[1] Yamanaka M, Smith N I, Fujita K. Introduction to super-resolution microscopy [J]. Microscopy, 2014, 63(3): 177-192. doi: 10.1093/jmicro/dfu007 [2] Ma Y, Wen K, Liu M, et al. Recent advances in structured illumination microscopy [J]. Journal of Physics: Photonics, 2021, 3(2): 024009. doi: 10.1088/2515-7647/abdb04 [3] Gustafsson M G L. Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy [J]. Journal of Microscopy, 2000, 198(2): 82-87. doi: 10.1046/j.1365-2818.2000.00710.x [4] Airy G B. On the diffraction of an object-glass with circular aperture [J]. Transactions of the Cambridge Philosophical Society, 1835, 5(3): 283-291. [5] Abbe E. A contribution to the theory of the microscope and the nature of microscopic vision [J]. Proceedings of Bristol Naturalists Society, 1876, 1(2): 200-261. [6] Abbe E. The relation of aperture and power in the microscope [J]. Journal of the Royal Microscopical Society, 1882, 2: 300-309. doi: 10.1111/j.1365-2818.1882.tb00190.x [7] Rayleigh L. Investigations in optics, with special reference to the spectroscope [J]. London, Edinburgh, and Dublin Philosophi-cal Magazine and Journal of Science, 1879, 8(49): 261-274. doi: 10.1080/14786447908639684 [8] Sparrow C M. On spectroscopic resolving power [J]. Astrophy-sical Journal, 1916, 44(2): 76-86. [9] Sayanna K R, Sagar D K, Goud S L. Effects of defocusing on the Sparrow limits for apodized optical systems [J]. Optics Communications, 2003, 217(1): 59-67. [10] You S T, Kuang C F, Zhang B L. Resolution criteria in double-slit microscopic imaging experiments [J]. Scientific Reports, 2016, 6(1): 33764. doi: 10.1038/srep33764 [11] Yamamoto K. Two-point resolution in bright background under partially coherent light [J]. Optical Review, 2020, 27(6): 486. [12] Reddy A N K, Khonina S N. Apodization for improving the two-point resolution of coherent optical systems with defect of focus [J]. Applied Physics B, 2018, 124(12): 229. doi: 10.1007/s00340-018-7101-z [13] Latychevskaia T. Lateral and axial resolution criteria in incoherent and coherent optics and holography near- and far-field regimes [J]. Applied Optics, 2019, 58(13): 3597-3603. doi: 10.1364/AO.58.003597 [14] Gao P, Yuan C. Resolution enhancement of digital holographic microscopy via synthetic aperture: a review [J]. Light: Advanced Manufacturing, 2022, 3(1): 105-120. doi: 10.37188/lam.2022.006