Fluorescence lifetime is an important contrast modality of selectively detecting different types of dyes and their interactions, where possible contaminating effects of hardly controllable factors such as exciting light flux and local exciting dye concentration are absent (1, 2). Time‐ and frequency‐domain detections of fluorescence lifetime have been elaborated both in microscopy and flow cytometry (2-5).

Phasor plot is a graphical replacement of the fluorescence decay curve of a fluorescent system. Although excellent accounts of phasor plots are existing in the literature (6, 7), we recast it now from the viewpoint of the general systems and communication theory. The approach of systems theory is that the examined object is treated as a black box, and the structure of the black box is inferred from the distortion of shapes of signals introduced into the black box (8) (Fig. 1). The shape of the input signal is distorted upon interactions with the elements of black box. The types and strengths of interactions can be found via a careful analysis of the shape distortions of the output signal in a noninvasive manner, not necessitating structural changes of the black box. The black box can also be conceived as an information channel influencing the transmitted information meant by the shape of input signal and the received information represented by the shape of the output signal. For the sake of concreteness, let us first mention an LR‐circuit (9) (first row of Fig. 1). Upon excitation of the circuit with a square pulse, the elicited current will show a degree of smoothing of both the front and rear of the input signal, dictated by the relaxation time constant, τ. The relaxation time expresses the degree of inertia of the system against the perturbation and here is dictated by the inductivity to resistance ratio (L/R) (9). As a second example, let us take a microscope objective (second row of Fig. 1). The image of a rectangular brightness distribution of an object placed in the focal plane will no more be rectangular in the image plane, but more‐or‐less blurred at the edges. The degree of blurring is characterized by the critical distance (d)—or reciprocal resolving power—of microscope resolution, which is dictated by the ratio of the detected wavelength and the objective's numerical aperture (10, 11). As the third example, let us take a fluorescent dye solution in a cuvette, excited by a square light pulse at a suitable wavelength for eliciting fluorescence (third row of Fig. 1). Then, the shape of the evoked fluorescence signal will also show distortions characteristic of the dye solution. At the front of the signal, there is no distortion because of the instantaneous absorption of photons (10⁻¹⁵ s). However, at the rear, some lengthening of emission is seen governed by the storage time of excitation, the fluorescence lifetime, τ, which in turn is defined as the reciprocal of the sum of radiative and nonradiative rate constants, k_f and k_nr, respectively.

A key to understanding of phasor‐plots is the observation that the output signal of a black box can be forecasted for any input signal shape if the systems response to an infinitely narrow excitation pulse (Dirac‐ or δ‐pulse) is known (Fig. 2A). This function is called impulse response function or weight function h(t), and considered as a fingerprint of the system (8-11). Its significance lies in that, in the knowledge of it, the response of the system mathematically can be found out for any input signal shape via the operation called convolution. During convolution, each particular g(t₀) value of a g(t) function is replaced by a weighted average of the whole g(t) function with a weighting function h(t) centered on the particular t₀ time point. This operation is analogous to smoothing clothes with an iron: where g(t) is the cloth with the creases to be removed, and the weight function h(t) is the iron. An important mathematical theorem is that of convolutions. It states that any convolution can be transformed into the product of two complex numbers, the so‐called Fourier‐transforms of the convolved quantities, with the consequence that any time function of the black box can be rendered visible by appropriate geometric features in the two‐dimensional plane (8-11). A commonplace example for Fourier‐transform is splitting up into different color beamlet components of a white light beam by a prism, or a drop of water.

We saw above that the impulse response or weight function represents the system's structure as a finger print. Its Fourier‐transform is called the transfer function. Because this is a complex number, it can be written as a product of a modulation m and complex exponential term containing a phase ϕ (Fig. 2B). The traditional way of representing lifetime measurement data is separately representing the modulation and phase in a common frame as monotonously decreasing and increasing functions of the circular frequency (12). However, by expanding the complex exponential of the phase via Euler's rule into the sum of a cosine (real) and a sine (imaginary) term, and by multiplying them with the modulation term, the transfer function can be resolved into the sum of a real and an imaginary term, called B and A, respectively (Fig. 2C). This second alternative representation of the transfer function is called a phasor‐plot (or AB‐, SG‐, ND‐plot) (6, 13).

In the case of image formation in a microscope, the δ‐pulse corresponds to a point in the object plane. The impulse response of the microscope is the point‐spread function (PSF) whose width is proportional to the critical distance given by the Abbe's formula. The H(ω) transfer function for the microscope is the Fourier‐transform of the PSF, called now optical transfer function (OTF). Imaging quality of microscopes can be improved by purposeful alteration of the OTF, called spatial filtering (9-11).

We introduced phasor plots above as a graphical representation of the transfer function of the system, which in turn has been obtained by Fourier‐transformation of the system's impulse response function. However, it can be obtained without any assumption on the functional form of the impulse response by a harmonic excitation of the system, that is, carrying out the Fourier‐transform at the hardware level, in a phase‐modulation measurement (Fig. 3A). Here, both the demodulation and phase shift can be determined from either the fluorescence response curve or the associated demodulation‐phase curve pair (Fig. 3B) and AB‐plot (Fig. 3C) at the circular frequency of excitation (ω₀), from which the apparent phase and demodulation lifetimes can also be determined.

In contrast, the single δ‐pulse excitation above only fixed the shapes of these curves and did not fix the actual values of phase, demodulation, and associated lifetimes. However, this can be made with a trick. If the single δ‐pulse is repeated many times, resulting in a “δ‐comb,” then the pulse series acts similarly to a harmonic excitation, with a circular frequency dictated by the pulse repetition rate. At this circular frequency, the phase and modulation lifetimes can be read off analogously to the harmonic case from the phase‐demodulation curve pair or from the AB‐plot (14).

We saw above that the A and B quantities are quadrature components of the complex transfer function, which is the Fourier‐transform of the impulse response or weight function of the dye solution. If A is plotted against B, then the complex plain can be divided into three regions (Fig. 4A). For a single component solution having a single fluorescence lifetime there is a strict relationship between the demodulation and the phase: m = cos(ϕ). Consequently, the corresponding (m, ϕ) pairs constitute a circle, the “universal circle” of radius 0.5, and origin of (0.5, 0), on which fluorescence lifetime increases by proceeding with ϕ anti‐clockwise from horizontal direction (ϕ = 0). The m = cos(ϕ) relationship for the “universal circle” conveniently understandable, for example, because the horizontal diameter of the circle is seen at 90° from all points of the circle, according to Thales' theorem. Points inside the “universal circle” obey the m < cos(ϕ) relationship, referring to ground‐state heterogeneity or the presence of Förster resonance energy transfer (FRET) for the donor (measured at the donor side). In these cases, the impulse response function of the system is composed only of positive amplitude components. However for excited state reactions such as solvent relaxation and FRET toward an acceptor (measured at the acceptor side), phasor points can also get outside of the circle, obeying the m > cos(ϕ) relationship (6, 7, 15). In the case of excited state reactions, the impulse response function contains also negative amplitude (pre‐exponential factor) component(s), representing indirect excitation by the molecular environment. Physical rotation can also be conceived as a special energy transfer process between the different orientations of the oscillating molecular dipole. The parallel polarized component of fluorescence decays via both the natural decay of the excited state, and via the molecule rotating out from the orientation of parallel emission. The corresponding phasor point falls inside the circle mimicking ground state heterogeneity. In contrast, the perpendicularly polarized component of fluorescence decays via the natural decay of the excited state, and via the molecule rotating into the orientation of perpendicular emission, the latter described by a negative amplitude, rendering the corresponding phasor point to fall outside of the circle, like for FRET toward the acceptor (16).

Ground state heterogeneity and FRET from the donor can be distinguished via the geometrical shapes defined by the phasor points. For ground state heterogeneity, they constitute straight line intervals with the cutting points of their elongations with the circle giving the lifetimes of the mixed components (12, 13) (Fig. 4B). For FRET from the donor, phasor points constitute banana‐like curved geometrical shapes (18) (Fig. 4C). For an explanation of the latter, we should think of that, in the absence of FRET the phasor point is the mixture of pure donor and pure background phasors locating on the circle to left and to the right, dividing the connecting bisector according to the lever rule: the distance of the mixture point from one end point of the bisector is proportional to the fluorescence intensity for the opposite point. Because of this, the starting point is located to the left inside the circle, but close to the perimeter. The endpoint of the FRET trajectory is the mixture of the close to 100%‐quenched donor and background. The phasor point representing this mixture locates to the right, inside the circle, close to the perimeter. Besides, the starting and the end points are located such a way that their connecting intersection cuts the circle at the pure donor and background points. At intermediate FRET efficiencies the phasor is a mixture of the quenched donor and background points.

In their work, Nichani et al. (in this issue, page 1265) developed further the concept of FRET trajectory by sensitizing FRET for indicating the time evolution of a biological process called apoptosis. The development of apoptosis is indicated by the level of caspase‐3 enzyme, which in turn is indicated by a FRET probe via the degree of FRET relief achieved by scissoring by caspase‐3 the linker domain fusing together the donor and acceptor, the three forming an apoptosis‐indicating FRET bioprobe. In the bioprobe, engineered green fluorescent protein serves as FRET donor and Alexa Fluor‐546 as the acceptor. For truly indicating caspase‐3 levels, prerequisites are the high enough dynamic range ensured by the large starting FRET efficiency, the low enough bioprobe level for avoiding cross‐FRET contamination between the nearby intact probes as well as between the intact probes and the liberated acceptors and donors. There are also technical requirements such as the spectral purity of the donor channel. As they demonstrate, phasor plots can be advantageously exploited also in flow conditions at a single excitation frequency for real time tracking a signaling event. However, the true interpretation of the phasor plots requires also the precise knowledge of all components of the donor fluorescence.

The multiplexing capability of flow cytometry is well illustrated by the authors via correlating donor fluorescence lifetime with FRET‐channel fluorescence, as an indicator for the presence of acceptor in the bioprobes. In steady state flow cytometry and fluorescence microscopy, FRET efficiency is determined generally in a simple donor quenching experiment or in the ratiometric FRET scheme involving two or three signal channels. In the latter case, however, absolute FRET efficiency can only be determined in the knowledge of a calibrating factor called α (or G), responsible for balancing quantum efficiencies and collection efficiencies of fluorescence for the donor and acceptor (19). Direct measurement of FRET efficiency via fluorescence lifetime obviates the need for α. Alternatively, fluorescence lifetime might also aid in determining α, and consequently, instrumental spectroscopic parameters such as fluorescence transmission efficiencies, or absorption coefficient of the donor and acceptor prevailing in the cellular milieu. The α‐factor is also an important parameter for determining the donor‐acceptor molar ratio of the actual FRET sample in the ratiometric FRET scheme. Besides the role in FRET measurements, fluorescence lifetime has also an important role in converting fluorescence anisotropy values to rotational mobilities.

In fluorescence spectroscopy, phasor plots have first been introduced by D. Jameson and his colleagues, but before that, they were also applied in relaxation kinetics, chemical and dielectric, under the name of “Cole‐and‐Cole plots” (6, 7). Recently these plots have been revitalized by the fact that their usage greatly widens the applicability of single‐frequency fluorescence lifetime imaging microscopy (FLIM) and the realization that phasor plots can be applied not only in the frequency domain FLIM, but also in time domain FLIM (14, 18). Theoretically, patch‐clamping in electrophysiology might also be a field for a fruitful exploitation of the phasors. Based on the close correspondence between open ionic channels and excited dyes, the decay kinetics of open channels is described by the same formalism as those for fluorescing dyes, by taking over the role of fluorescence lifetime by the mean open time of channels. In addition to the model‐independent, that is, free of assumed functional form, characterization of channel kinetics, for example, collective behavior of channels (cooperativity), synchronization of channel openings—the counterparts of excited state dye interactions—as the function of their cluster size might be revealed by the graphical counterparts of the decay curves. Besides, the direct application of the phase‐modulation detection of currents, as an analogue of phase‐modulation FLIM, might also convey some extra information.

Homo‐FRET is a FRET process taking place between dyes of identical type having small Stokes' shift. In the limiting case of high excitation intensities close to saturation, homo‐FRET might be exchanged for interactions between excited dyes in close proximity, called super/sub‐radiance or super/sub‐fluorescence. In this case, radiative rate might be either increased or decreased due to emission dipole synchronization by the net effect of the dye local fields, the exciting light, the emitted fluorescence, and the morphology of the dye cluster (20). Even if the mean excited state lifetime of the dyes stays constant, the lifetime heterogeneity might increase manifesting in a broadening of lifetime distributions. Phasor plots might represent these events by an analogue broadening of distributions and falling points outside of the universal circle.

Taking together, flow cytometry made a leap‐forward with the introduction of spectacular representation of FRET‐processes in the complex plane of phasor plots.

荧光寿命是选择性检测不同类型的染料及其相互作用的重要对比方式，其中不存在难以控制的因素（如激发光通量和局部激发染料浓度）的污染效应（1、2）。荧光寿命的时域和频域检测已在显微镜和流式细胞仪中进行了详细描述（2-5）。

相量图是荧光系统的荧光衰减曲线的图形替换。尽管文献（6、7）中已经很好地描述了相量图，但现在我们从一般系统和传播理论的角度对它进行了重铸。系统理论的方法是将被检查对象视为黑匣子，并根据引入黑匣子的信号形状的失真来推断黑匣子的结构（8） （图。1）。输入信号的形状在与黑匣子的元素相互作用时会失真。可以通过以无创方式仔细分析输出信号的形状失真来发现交互作用的类型和强度，而不必改变黑匣子的结构。黑匣子也可以被认为是一个信息信道，它影响由输入信号的形状表示的发送信息和由输出信号的形状表示的接收信息。为了具体起见，让我们首先提到一个LR电路（9）（图1的第一行）。在用方波脉冲激励电路时，所产生的电流将显示输入信号前后的平滑度，具体取决于松弛时间常数τ。弛豫时间表示系统抵抗扰动的惯性程度，在此由电感率与电阻比（L / R）（9）决定。作为第二个示例，让我们以显微镜物镜（图1的第二行）为例。放置在焦平面中的对象的矩形亮度分布的图像在图像平面中将不再是矩形，而是在边缘处或多或少地变得模糊。模糊程度由临界距离（d）或显微镜分辨率的倒数分辨能力，它由检测到的波长与物镜的数值孔径（10、11）之比决定。作为第三个例子，让我们在比色杯中取一个荧光染料溶液，该溶液由一个正方形波长的脉冲在合适的波长激发以激发荧光（图1第三行）。然后，诱发的荧光信号的形状也将显示染料溶液的变形特性。在信号的前面，由于光子的瞬时吸收而没有失真（10 ^-15s）。然而，在后，发射的一些延长被看作由激发的存储时间的约束，荧光寿命，τ，其又被定义为辐射和非辐射速率常数之和的倒数，ķ _˚F和ķ _NR，分别。

理解相量曲线的关键是观察到，如果已知系统对无限窄的激发脉冲（狄拉克或δ脉冲）的响应，则可以预测任何输入信号形状的黑盒输出信号（图。 2A）。该函数称为脉冲响​​应函数或权重函数h（t），被视为系统的指纹（8-11）。它的意义在于，通过已知的知识，可以通过称为卷积的操作从数学上找出系统对于任何输入信号形状的响应。在卷积，每个特定克（吨₀一的）值克（吨）函数由整个g（t）函数的加权平均值代替，其中加权函数h（t）以特定的t ₀时间点为中心。此操作类似于用熨斗将衣服弄平：其中g（t）是要去除折痕的布料，权重函数h（t）是铁。一个重要的数学定理是卷积。它指出，任何卷积都可以转换为两个复数的乘积，即卷积量的所谓傅立叶变换，其结果是，黑盒的任何时间函数都可以通过两个中适当的几何特征呈现出来三维平面（8-11）。傅里叶变换的一个常见示例是通过棱镜或水滴将白光束分成不同颜色的子束分量。

上面我们看到，脉冲响应或权重函数将系统的结构表示为指纹。它的傅里叶变换称为传递函数。因为这是一个复数，所以它可以写成调制m和包含相位ϕ的复指数项的乘积（图2B）。表示寿命测量数据的传统方式是在一个普通帧中分别表示调制和相位，作为圆形频率的单调递减函数（12）。但是，通过根据欧拉规则将相位的复数指数扩展为余弦（实数）和正弦（虚数）项的和，并将它们与调制项相乘，可以将传递函数解析为a的和。实项和虚项，分别称为B和A（图2C）。传递函数的第二种替代表示形式称为相量图（或AB-，SG-，ND-图）（6，13）。

在显微镜下成像时，δ脉冲对应于物平面中的一个点。显微镜的脉冲响应是点扩散函数（PSF），其宽度与阿贝公式给出的临界距离成正比。显微镜的H（ω）传递函数是PSF的傅立叶变换，现在称为光学传递函数（OTF）。显微镜的成像质量可以通过有目的地更改OTF（称为空间滤波）来提高（9-11）。

我们在上面介绍了相量图，作为系统传递函数的图形表示，而该传递函数又是通过系统冲激响应函数的傅立叶变换获得的。但是，可以在不对系统谐波激励的脉冲响应的功能形式进行任何假设的情况下，即在相位调制测量中在硬件级别执行傅里叶变换的情况下获得该信号（图3A） 。这里，无论是解调和相移可以从任一荧光响应曲线或在激发的圆频率相关联的解调相位曲线对（图3B）和AB-图（图3C）（ω确定₀），由此也可以确定视在相位和解调寿命。

相反，以上的单个δ脉冲激励仅固定了这些曲线的形状，而没有固定相位，解调和相关寿命的实际值。但是，这可以通过技巧来实现。如果单个δ脉冲重复多次，导致“δ梳”，则脉冲序列的作用类似于谐波激励，其圆形频率由脉冲重复频率决定。在此圆形频率下，可以从相位解调曲线对或从AB图（14）类似于谐波情况读取相位和调制寿命。

我们在上面看到，A和B数量是复数传递函数的正交分量，复数传递函数是染料溶液的脉冲响应或重量函数的傅立叶变换。如果将A与B作图，那么复杂的平原可以分为三个区域（图4A）。对于具有单个荧光寿命的单一成分溶液有解调和相位之间的关系严格：米= COS（φ）。因此，相应的（米，φ）对构成一个圆，半径0.5的“通用圈”和原点的（0.5,0），在其上的荧光寿命增加通过程序与φ从水平方向逆时针（φ= 0）。所述米= COS（φ）为“通用圈”方便地理解的，例如关系，因为圆的水平直径被认为是在90℃从圆的所有点，根据泰勒斯定理。“通用圆”内的点服从m  <cos（ϕ）关系，是指供体的基态异质性或Förster共振能量转移（FRET）的存在（在供体侧测量）。在这些情况下，系统的脉冲响应函数仅由正振幅分量组成。但是，对于激发态反应（例如溶剂弛豫和朝向受体的FRET（在受体侧测量）），相量点也可以到达圆之外，服从m  > cos（ϕ）关系（6，7，15）。对于激发态反应，脉冲响应函数还包含负振幅（指数前因子）分量，表示分子环境的间接激发。物理旋转也可以被认为是在振荡分子偶极子的不同方向之间的一种特殊的能量转移过程。荧光的平行极化分量既通过激发态的自然衰减，也通过分子从平行发射方向旋转出来而衰减。相应的相量点位于模仿基态异质性的圆内。相比之下，荧光的垂直极化分量会通过激发态的自然衰减以及通过旋转为垂直发射方向的分子而衰减，16）。

可以通过相量点定义的几何形状区分供体的基态异质性和FRET。对于基态异质性，它们以其伸长的切点与圆构成直线间隔，从而给出了混合组分（12、13）的寿命（图4B）。对于来自供体的FRET，相量点构成香蕉状的弯曲几何形状（18）（图4C）。为了对后者进行解释，我们应该想到，在没有FRET的情况下，相量点是纯施主相量和纯背景相量的混合物，其位于左右圆上，并根据杠杆将连接等分线分开规则：混合点到等分线束的一个端点的距离与相对点的荧光强度成正比。因此，起点位于圆内的左侧，但靠近周长。FRET轨迹的终点是接近100％淬灭的供体和背景的混合物。表示该混合的相量点位于圆内的右侧，靠近周长。除了，起点和终点的位置应使其相互连接的交点在纯供体和背景点处切圆。在中等FRET效率下，相量是淬灭供体和背景点的混合。

在他们的工作中，Nichani等人。（此问题，第XXX页）通过使FRET敏感，以指示称为凋亡的生物过程的时间演变，进一步发展了FRET轨迹的概念。凋亡的发展由caspase-3酶的水平指示，而FRET探针则通过caspase-3剪切连接体域将供体和受体融合在一起而实现的FRET缓解程度来指示FRET缓解指示凋亡的FRET生物探针。在生物探针中，工程绿色荧光蛋白充当FRET供体，Alexa Fluor‐546作为受体。为了真正指示caspase-3的水平，先决条件是足够大的动态FRET效率确保足够高的动态范围，足够低的生物探针水平，以避免附近完整探针之间以及完整探针与释放的受体和供体之间的交叉FRET污染。还存在技术要求，例如供体通道的光谱纯度。正如他们所证明的，相量图也可以在流动条件下以单个激发频率被有利地利用，以实时跟踪信号事件。但是，对相量图的真实解释还需要对供体荧光的所有成分有确切的了解。相量图还可以有利地在流动条件下以单个激励频率利用，以实时跟踪信令事件。但是，对相量图的真实解释还需要对供体荧光的所有成分有确切的了解。相量图还可以有利地在流动条件下以单个激励频率利用，以实时跟踪信令事件。但是，对相量图的真实解释还需要对供体荧光的所有成分有确切的了解。

作者通过将供体荧光寿命与FRET通道荧光相关联，很好地说明了流式细胞仪的多重能力，以此作为生物探针中受体存在的指标。在稳态流式细胞仪和荧光显微镜下，通常在简单的供体淬灭实验中或在涉及两个或三个信号通道的比例式FRET方案中确定FRET效率。但是，在后一种情况下，绝对FRET效率只能在称为α（或G）的校准因子的知识下确定，该校准因子负责平衡供体和受体的荧光的量子效率和收集效率（19）。通过荧光寿命直接测量FRET效率，消除了对α的需求。备选地，荧光寿命也可能有助于确定α，并因此确定仪器的光谱参数，例如荧光透射效率或细胞环境中普遍存在的供体和受体的吸收系数。在比例FRET方案中，α因子也是确定实际FRET样品的供体-受体摩尔比的重要参数。除了在FRET测量中发挥作用外，荧光寿命还具有将荧光各向异性值转换为旋转运动的重要作用。

In fluorescence spectroscopy, phasor plots have first been introduced by D. Jameson and his colleagues, but before that, they were also applied in relaxation kinetics, chemical and dielectric, under the name of “Cole‐and‐Cole plots” (6, 7). Recently these plots have been revitalized by the fact that their usage greatly widens the applicability of single‐frequency fluorescence lifetime imaging microscopy (FLIM) and the realization that phasor plots can be applied not only in the frequency domain FLIM, but also in time domain FLIM (14, 18). Theoretically, patch‐clamping in electrophysiology might also be a field for a fruitful exploitation of the phasors. Based on the close correspondence between open ionic channels and excited dyes, the decay kinetics of open channels is described by the same formalism as those for fluorescing dyes, by taking over the role of fluorescence lifetime by the mean open time of channels. In addition to the model‐independent, that is, free of assumed functional form, characterization of channel kinetics, for example, collective behavior of channels (cooperativity), synchronization of channel openings—the counterparts of excited state dye interactions—as the function of their cluster size might be revealed by the graphical counterparts of the decay curves. Besides, the direct application of the phase‐modulation detection of currents, as an analogue of phase‐modulation FLIM, might also convey some extra information.

均相FRET是在具有较小斯托克斯位移的相同类型染料之间进行的FRET过程。在接近饱和的高激发强度的极限情况下，可将均FRET交换为紧密接近的激发染料之间的相互作用，称为超/亚辐射或超/亚荧光。在这种情况下，由于染料局部场，激发光，发射的荧光和染料簇的形态的净效应，由于发射偶极同步，辐射速率可能会增加或减少（20）。即使染料的平均激发态寿命保持恒定，寿命异质性也可能增加，表现为寿命分布变宽。相量图可能通过分布的扩展和普遍圈外的下降点的模拟来表示这些事件。

综上所述，流式细胞术在相量图的复杂平面中引入了引人注目的FRET过程表示方法，从而实现了飞跃。