Two-dimensional polarization imaging (2D POLIM) is an experimental method where correlations between fluorescence excitation- and fluorescence emission-polarization properties are measured. One way to analyze 2D POLIM data is to apply a so-called single funnel approximation (SFA). The SFA allows for quantitative assessment of energy transfer between chromophores with identical spectra [homo-FRET (Förster resonance energy transfer)]. In this paper, we run a series of computer experiments to investigate the applicability of the analysis based on the SFA to various systems ranging from single multichromophoric systems to isotropic ensembles. By setting various scenarios of energy transfer between individual chromophores within a single object, we were able to define the borders of the practical application of SFA. It allowed us to reach a more comprehensive interpretation of the experimental data in terms of uncovering the internal arrangement of chromophores in the system and energy transfer between them. We also found that the SFA can always formally explain the data for isotropic ensembles and derived a formula connecting the energy funneling efficiency parameter and traditional fluorescence anisotropy.

## I. INTRODUCTION

Excitation energy transfer [EET, i.e., energy transfer from a donor (D) in the excited state to an acceptor (A) in the ground state]^{1,2} is a common process between closely located chromophores in concentrated solutions of dyes, in molecular crystals, and within multichromophoric systems (fluorescent macromolecules), such as pi-conjugated polymers,^{3,4} natural light-harvesting complexes,^{5} and molecular aggregates.^{6} EET is also widely used in biological science as an indicator of the distance between fluorescent labels.^{7,8} Förster resonance energy transfer (FRET) is EET via a weak dipole–dipole coupling.^{2}

When the donor and the acceptor have distinguishable absorption and fluorescence spectra, EET is called hetero-FRET. Hetero-FRET can be studied in great detail using steady-state and time-resolved fluorescence spectroscopy because of substantial energetic/spectroscopic differences between the donor and the acceptor.^{9–11} For example, the efficiency of hetero-FRET can be assessed by comparing the fluorescence intensities of the spectrally resolved donor and acceptor.^{12}

However, these methods become inefficient when donors and acceptors are spectroscopically similar. Energy transfer in such systems is called homo-FRET. For example, homo-FRET occurs between closely located green fluorescent proteins used as labels in biological structures and between spectroscopic units inside a conjugated polymer chain.^{13} Homo-FRET can be assessed by fluorescence anisotropy (FA) measurements.^{11,14–17} The idea of this method is that polarized fluorescence can be induced by exciting an isotropic sample by linearly polarized light. In this case, fluorescence becomes anisotropic because the linearly polarized excitation light interacts with the absorbing molecules not equally, according to the Malus law. This effect is called photoselection. Subsequently, energy transfer “kills” the memory of the initial polarization induced by the polarized excitation, thus the absence of fluorescence polarization means an efficient homo-FRET.

However, the classical FA methodology can only be applied to samples with isotropic absorption.^{5,18–21} As soon as a sample or a studied object exhibits linear dichroism (or, in other words, anisotropic absorption), interpretation of FA becomes difficult and ambiguous. This is because the direction of the excitation polarization and the probed emission [parallel and perpendicular to the excitation electric vector, r = (I_{∥} − I_{⊥})/(I_{∥} − 2I_{⊥})] are fixed to the laboratory frame, and r then depends on the orientation of the sample relative to the laboratory frame. Furthermore, if the sample is anisotropic in terms of the orientation of the chromophores’ transition dipole moment, then FA depends not only on FRET efficiency but also on the degree of alignment and the principal orientation axis of the ensemble. Thus, the value of FA cannot directly represent the FRET efficiency in a system with linear dichroism. This means that FA measurements are not directly applicable at the single-molecule level because individual molecules and particles usually exhibit anisotropic absorption.^{5,21–23}

To overcome this problem and extract maximum quantitative information from fluorescence polarization measurements, an advanced polarization technique called two-dimensional polarization imaging (2D POLIM) was introduced several years ago.^{23,24} 2D here indicates that fluorescence intensity (I) is measured as a function of two variables: the orientation angle of the linearly polarized excitation light (*φ*_{ex}) and the orientation of the transmission axis of the polarization analyzer (*φ*_{em}) placed in front of the detector. The function I (*φ*_{ex}, *φ*_{em}) is called the 2D polarization portrait.

The ability to measure 2D polarization portraits brought a new theoretical framework for assessing energy transfer via a parameter called energy funneling efficiency (ε).^{24} This parameter is calculated within the so-called single funnel approximation (SFA).^{24} 2D POLIM and the analysis in the SFA framework have been successfully used to quantitatively assess energy transfer efficiency in various systems, including single molecules,^{21–23} conjugated polymer films,^{26,27} and natural light-harvesting system,^{5,25} and to detect protein aggregation in brain tissue.^{8}

Estimation of the energy transfer efficiency using the SFA has the following advantages:^{23} (i) It is insensitive to the preferential alignment of the sample, so it is applicable to multichromophoric systems with linear dichroism. (ii) It does not require any prior knowledge of the chromophores’ organization. Together with other polarization parameters extracted from the 2D portrait (modulation depth, phase, and luminescence shift), energy funneling efficiency ε gives a comprehensive picture of the internal organization and EET processes in the sample.

From its inception, it was clear that since the SFA is a simplification and assumes some conditions to the energy transfer processes, it must fail in some cases to explain fluorescence polarization properties. However, so far, the borders of the practical application of the SFA were not exploited theoretically. The SFA was applied without in-depth understanding of the conditions when mathematically the approximation can explain the experimental results. In addition, it was not considered that the fact that the SFA cannot fit the data might be used to extract additional information about the system. Based on these ideas, we started our simulation project regarding the validity of the SFA method.

In this paper, we present a series of computer experiments where we apply the SFA approach to diversely organized multichromophoric systems with different scenarios of internal EET. Surprisingly, we find that the SFA model can also be applied to multi-funnel systems under a few restrictions. We define the cases when the SFA fails, and we show that this, in fact, provides important information about the internal organization of the chromophores and EET between them. With an increase in the number of subsystems (from single molecule to ensembles), we found that the SFA methodology always works well for isotropic ensembled samples and that the energy transfer efficiency parameter has strict mathematical relations to classical fluorescence anisotropy.

## II. THE METHOD: TWO-DIMENSIONAL POLARIZATION IMAGING (2D POLIM)

Absorption and emission polarization properties can give information about the EET between chromophores or their orientation in space.^{2} Figure 1 illustrates the common experimental configurations to measure polarization properties, including the traditional one-dimensional [Figs. 1(b) and 1(c)] and the advanced two-dimensional [Fig. 1(a)] measurement schemes. Here, we say that a method is 1D when only one polarization angle is scanned and 2D when both polarization angles (the orientation of the excitation light polarization *φ*_{ex} and the angle of the analyzer *φ*_{em}) are scanned.

Figure 1(b) shows the experimental scheme used to measure fluorescence detected linear dichroism (FDLD). A sample consisted of many chromophores is excited by linearly polarized light with the polarization plane at angle *φ*_{ex}, and the fluorescence intensity I(*φ*_{ex}) is measured. Assuming that the fluorescence quantum yield and absorption coefficient are the same for all chromophores, the fluorescence intensity is determined by the excitation probability given by the Malus law,^{28} $Ii\phi ex\u223ccos2\phi ex\u2212\theta i0$, where *θ*_{i}^{0} is the orientation angle (phase) of *i*th chromophore. Thus, the total fluorescence intensity of the sample is

where $I\u0304$ is the fluorescence intensity averaged over *φ*_{ex}, M_{ex} is the modulation depth of fluorescence excitation (or FDLD), and *θ*_{ex} is the fluorescence excitation phase. M_{ex} ranges from 0 to 1 for an isotropic to fully anisotropic system, respectively. *θ*_{ex} gives the main orientation axis of the sample absorption and it ranges from 0 to π. Equation (1) acquires this form because a linear combination of cosines having the same angular frequency [the first sum in Eq. (1)] can always be expressed as a constant plus a cosine function with the same frequency and a certain phase.

To measure emission polarization [Fig. 1(c)], the sample is excited isotropically by a circular or randomly polarized light and a rotating analyzer is placed in front of the detector. Similar to Eq. (1), fluorescence intensity I(*φ*_{em}) can be expressed as

where *φ*_{em} and *θ*_{i}^{0} are the angles of the analyzer’s transmission axis and the orientation angle (phase) of *i*th chromophore, respectively, and M_{em} (also called polarization degree P) and *θ*_{em} are the emission modulation depth and phase, respectively.

The two methods described above can be combined to measure excitation and emission polarization simultaneously [Fig. 1(a)]. The sample then is excited with linearly polarized light at different angles *φ*_{ex}, while at the same time, the fluorescence from the sample is detected for several different orientations of the analyzer *φ*_{em}. Finally, a 2D function is obtained I(*φ*_{ex}, *φ*_{em}) by scanning both angles. This 2D function is called the two-dimensional polarization portrait (2D portrait).^{24}

Naturally, a 2D portrait contains all the information that can be obtained from traditional one-dimensional measurements when only one polarization angle is scanned. The advantage of the 2D polarization measurement, however, is that it allows for measuring correlations between the excitation and emission polarization properties. Analysis of such a polarization portrait brought a conceptually new idea in quantitative assessment of EET based on the so-called single funnel approximation (SFA), which will be discussed in detail below.^{24}

## III. COMPUTER EXPERIMENTS

### A. Replacing the real 3D system by its projection to the XY sample plane

In the experiments described in Fig. 1, it is important to use a low NA (numerical aperture) objective lens (in practice, <0.6 is sufficient) to exclude detection of the radiation coming from Z-oriented dipoles. It is necessary because a partial detection of Z component makes interpretation of the signal practically impossible. In the limit of zero NA, we are sure that we detect only the projection of the system to the XY plane.

In Fig. 2, there are two chromophores represented by dipoles A and B of an arbitrary length and orientation in 3D space. Because the electric field vector of the excitation light is in the XY plane, it can interact only with the projection of the dipoles to the XY plane. So, the apparent absorption cross sections visible for us are ∼*A*^{2} cos^{2} *ψ*_{A} and ∼*B*^{2} cos^{2} *ψ*_{B}, where ψ_{A} and ψ_{B} are the angles between the dipoles and the XY plane. The same concerns the emission. We can only detect the component of the wave with the electric field vector in the XY plane. It is equivalent to decreasing the emission intensity by the factors cos^{2} *ψ*_{A} and cos^{2} *ψ*_{B} for each dipole. To summarize, the response of the 3D system in the experiment shown in Fig. 1(a) is equivalent to that of a projection of the system to the XY plane.

The oscillator strength values of the chromophore visible in our experiment are ∼*A*^{2} cos^{2} *ψ*_{A} and ∼*B*^{2} cos^{2} *ψ*_{B}, and these values can be different, which is not convenient for calculations. However, we can always formally represent the response of the XY projection of each real dipole A and B by responses of sums of N_{A} and N_{B}, respectively, of smaller dipoles **e** of the same length. The value e^{2} is then the greatest common divisor of *A*^{2} cos^{2}*ψ*_{A} and *B*^{2} cos^{2} *ψ*_{B}. In other words, *A*^{2} cos^{2}*ψ*_{A} = *N*_{A}*e*^{2} and *B*^{2} cos^{2}*ψ*_{B} = *N*_{B}*e*^{2}, where N_{A} and N_{B} are the integer numbers. Thus, without losing the generality, for the modeling, it is sufficient to consider systems consisting of dipoles of identical length oriented in the XY plane. This is what we are going to use in the whole paper below, unless it is stated otherwise.

### B. Generation of 2D portraits

Let us describe how a polarization portrait I(*φ*_{ex}, *φ*_{em}) of a multichromophoric system with known organization and energy transfer can be calculated. Without losing generality (see Sec. III A), we assume that the transition dipole moments of absorption and emission of each chromophore are collinear, all have the same magnitude and lay in the XY sample plane. We also assume for simplicity that there is no nonradiative decay of the excited state (fluorescence quantum yield of the whole systems is unity). For a system consisting of N chromophores oriented in a 2D sample plane without EET between them, the function I(*φ*_{ex}, *φ*_{em}) is given by

where *φ*_{ex} and *φ*_{em} are the excitation and emission polarization angles, respectively, N is the number of chromophores, and *θ*_{i}^{0} is the orientation of the transition dipole moment of the chromophore *i*. The first term $cos2\phi ex\u2212\theta i0$ is the excitation probability of the chromophore *i*. The second term $cos2\phi em\u2212\theta i0$ is the fluorescence intensity of the chromophore *i* observed through a polarization analyzer polarizer oriented at the angle *φ*_{em}.

If there is EET between chromophores, then we need to include redistribution of the initial excitations between chromophores,

where T is the energy transfer matrix that describes the energy transfer/redistribution

The element *t*_{ij} gives the fraction of energy originally generated at chromophore *i*, which is transferred to chromophore *j* and emitted from there. Each column represents one chromophore. Because of energy conservation (fluorescence yield is unity), $\u2211j=1NTij=1$. The sum of elements in column *j* represents the total energy emitted from chromophore *j*. The matrix T represents the final (equilibrated) distribution of the fluorescence probability between chromophores and depends on the mutual FRET rates between individual chromophores and the competition between energy transfer with fluorescence. Using these equations, we can generate a 2D portrait of a multichromophoric system if we know the orientation of all chromophores (*θ*_{i}^{0}) and the matrix T.

Figure 3 illustrates how to generate the 2D portrait for a three-chromophore system. Equation (4) can be written as a multiplication of three matrices. In the first matrix, each column represents the excitation intensity of one chromophore and each row corresponds to *φ*_{ex} changing from 0° to 180° in 10° steps. The value $Ii\phi $ is the excitation probability of the *i*th chromophore. The second 3 × 3 matrix is the T matrix describing the redistribution of the energy between the three dipoles. The last matrix is the emission intensity of each chromophore projected on the transmission axis of a polarizer oriented at *φ*_{em}, which is scanned with 10° steps. The multiplication of the three yellow areas (marked in each matrix) generates one data point I(*φ*_{ex}, *φ*_{em}) on the 2D portrait according to Eq. (4).

### C. Single funnel approximation (SFA)

To use 2D portraits for a quantitative assessment of EET efficiency in multichromophoric systems, Camacho *et al.* developed the single funnel approximation (SFA) method.^{24} The main idea of the SFA is to decompose the 2D portrait I(*φ*_{ex}, *φ*_{em}) to a linear combination of two portraits: the NoET (no energy transfer) part and the ET (energy transfer) part with the weight coefficients (1 − ε) and ε, respectively [Fig. 4(a)]. The coefficient ε is called energy funneling efficiency,^{24}

Here, the ET part presents the fluorescence signal that does not show any photo-selection property. In other words, this emission comes from an effective emitter (energy funnel or energy acceptor) with fixed emission polarization properties regardless of the polarization orientation of the excitation light. The emitter is excited via energy transfer from the whole system. Due to these properties, $ET\phi ex,\phi em$ must be symmetric around a horizontal line *φ*_{ex} = θ_{f}—the absorption phase of the system [see the ET plot in Fig. 4(a)]. The coefficient ε is equal to the fraction of total energy emitted by the energy acceptor.

The NoET portrait describes the part of the fluorescence signal that exhibits photo-selection. For photo-selection, $NoET\phi ex,\phi em$ plot must be symmetric over its diagonal *φ*_{ex} = *φ*_{em} [see the NoET plot in Fig. 4(a)]. This signal is the sum of emission coming from each chromophore of the system with the intensity multiplied by the factor (1 − ε) because the fraction ε is given to the acceptor due to energy funneling. Basically, NoET presents the 2D portrait of the system as it would not have any energy transfer between chromophores at all. In this case, $NoET\phi ex,\phi em$ plot must be symmetric over its diagonal where *φ*_{ex} = *φ*_{em}. [See Fig. 4(a) where these axes of symmetry are marked on the corresponding 2D plots.]

Without losing generality, we can write the equation for the ET and NoET portraits more explicitly by replacing the absorption and photo-selected emission of individual chromophores with the absorption and emission of an equivalent system of chromophores of the same length oriented in the XY sample plane (see Sec. III A),^{24}

We also can simplify Eq. (7) by expressing the summation (which gives the polarization properties of absorption/fluorescence excitation) via the excitation modulation depth and phase [Eq. (1)],

Thus, the 2D portrait in the framework of the SFA is determined by seven polarization parameters: M_{ex}, M_{em}, *θ*_{ex}, *θ*_{em}, M_{f}, *θ*_{f}, and *ε*. The parameters M_{f} and *θ*_{f} are the modulation depth and the phase of the emission coming from the energy funnel (acceptor).

The funnel (or energy acceptor) can consist of more than one dipole being a pool of emitters. This is why its emission polarization degree M_{f} can be anything from zero to unity. The energy funneling efficiency ε is the coefficient in the linear combination of ET and NoET contributions. It ranges from zero (no energy transfer in the system) to unity when the system is a single funnel. The SFA assumes that there is only one acceptor for all chromophores of the system and each chromophore gives the same fraction of its excitation (*ε*) to the acceptor and nowhere else.

All these obviously imply some limitations on the applicability of SFA to real systems. Therefore, it is not always possible to decompose the function I(*φ*_{ex}, *φ*_{em}) according to Eq. (6). Figure 4(b) shows a simple example of a multi-funnel system and its computer-generated 2D portrait. The portrait was fitted using the SFA by the software previously developed for the analysis of real experimental data.^{8} As one can see, the fit is not successful because the residuals are very large. Although we formally obtained ET and NoET portraits and all corresponding polarization parameters, however, the residuals are negative for some angle combinations making the ET and NoET lose their physical meaning. It may seem natural that the single funnel approximation fails for a multi-funnel system; however, this is not always the case as will be discussed in detail later.

### D. The validity of SFA

The conditions described above bring us to the scientific motivation of this work, which is illustrated in Fig. 5. The main idea is to test the applicability of the SFA method by doing computer experiments. We set the dipole orientations and the T matrix to calculate a 2D portrait, as described in Sec. III B. These portraits are the “experimental data” that we analyze with the software previously developed for fitting real experimental data using the SFA [Fig. 5(i)].^{8,28} To quantitatively describe the difference between the “experimental data” and the fitting result, we calculated the value of fitting residual from the residual 2D portrait, $Residual=\u2211Residual(\phi ex,\phi em)2$.

If the experimental 2D plot and the best fit obtained in the framework of SFA match well, it means that the SFA approach mathematically works. In this case, we want to see whether the parameter ε can interpret the energy transfer in the system known from the matrix T. If the 2D portraits have obvious differences (the fitting residual is large), it means that the SFA fails. In this case, we will see the reasons for the discrepancies, consider how to further improve the SFA approach, and see if any useful information can be learned from the unsuccessful fitting. To simplify the simulations, in this section, we always use objects consisting of dipoles/molecules distributed in a 2D plane. However, the obtained results and conclusions are also valid for the dipole distributed in 3D (see Secs. III A–III C).

## IV. RESULTS AND DISCUSSIONS

### A. Examples of 2D portraits for systems with and without energy transfer

First, we would like to clarify some terminologies frequently used in this paper to avoid any confusion or misunderstanding. There are several levels considering the size of the object (the studied system). Chromophores (or dipoles) are the smallest spectroscopic units. Then, we have the system consisting of a small number of chromophores (or dipoles). Such a system can also be considered as a single molecule or a single multichromophoric system (Sec. IV B). The largest system consists of a large number of chromophores (or dipoles), the so-called “ensemble of molecules” (Sec. IV C). A system described above, no matter small or large, can also be split into many subsystems according to certain rules.

In this section, we give a preliminary introduction about what information we can learn from the 2D portraits by showing several computer-generated 2D portraits as examples (Fig. 6). In the calculations, we set the absorption strength of one dipole to unity as well as the excitation power of the linearly polarized light. In this case, according to Eq. (3), for a single dipole, the maximum value is one and the averaged value over the 2D portrait $I\phi ex,\phi em=1/4$. The averaged intensity in Fig. 6 scales with the number of dipoles.

For one chromophore oriented at 120° [Fig. 6(a)], the intensity maximum (the center of the yellow spot) locates on the diagonal of the portrait (120°, 120°). By changing the orientation of the chromophore, the yellow spot can be moved along the diagonal.

If there are two dipoles without EET between them, one more yellow spot appears on the diagonal [Fig. 6(b)]. Interestingly, the 2D portrait allows us to see that the system consists of two dipoles. However, this information is invisible from the one-dimensional measurements, such as fluorescence detected linear dichroism [I(*φ*_{ex}), blue cross section in Fig. 6(b)] or emission polarization degree [I(*φ*_{em}), red cross section in Fig. 6(b)]. Note that EET always blurs the features related to the orientation of the light-absorbing chromophores in the portrait, compare Fig. 6(b) with Fig. 6(e).

If we add more dipoles with random orientations and without EET, the intensity along the diagonal becomes constant. By this, we obtain the 2D portrait of an isotropic solution with fluorescence anisotropy r = 0.4. This portrait will look the same as the one shown in Fig. 6(d), which was calculated for three dipoles with orientations equally distributed over 180°.

Now let us compare geometrically identical systems with and without EET. In Fig. 6, the 2D portraits in the first row were calculated without EET and in the second row, with some EET. In the absence of EET, the polarization properties of the excitation characterized by I(*φ*_{ex}) (blue curve) and of the emission characterized by I(*φ*_{em}) (red curve) are exactly the same. EET can make the functions I(*φ*_{ex}) and I(*φ*_{em}) different [Figs. 6(e)–6(g)]. The difference between I(*φ*_{ex}) and I(*φ*_{em}) can be small [Fig. 6(f)] or large [Fig. 6(g)], even though the energy transfer efficiency is similar in the systems. Thus, the difference between excitation and emission polarization properties is sufficient but not a necessary signature of EET.

For reaching unambiguous conclusions about the presence of EET and its quantitative characterization, one needs to analyze a full 2D polarization portrait I(*φ*_{ex}, *φ*_{em}).^{23,29} The most universal analysis so far is based on the single funnel approximation proposed in 2012 by Camacho *et al.*^{24} (Sec. III C).

### B. Single-molecule level

#### 1. Extension of the SFA approach and its application to multi-funnel systems

By doing computer experiments, we found a situation when the SFA can exactly explain a multi-funnel system. In the following, we give an example in Fig. 7 and mathematically explain this extension of the SFA method to multi-funnel systems.

To be explained by the SFA, a multi-funnel system must meet the following requirements:

The system can be split into several subsystems, each of them obeying the criteria required for the SFA. This means that all chromophores in each subsystem transfer the same amount of energy to the corresponding funnel. In other words, each subsystem is a perfect single funnel system described by the SFA.

The absorption polarization properties (M

_{ex}, θ_{ex}) of all the subsystems must be the same.

Let us see how we can mathematically obtain this result. In the following, the index k is used to index individual subsystems. In a multi-funnel system, the funnels in each subsystem can have different funneling efficiency ε_{k} and polarization parameters M_{f,k} and θ_{f,k}. We also consider that each subsystem can have a different fluorescence excitation cross section *σ*_{k} due to a different number of chromophores. $\sigma \u0302k$ is the fluorescence excitation cross section of each subsystem normalized by the total fluorescence excitation cross section of the whole multi-funnel system,

where N is the number of subsystems. If we assume that all subsystems have the same absorption polarization properties defined by M_{ex} and θ_{ex}, the shape of the polarization portrait of the light possessing photo-selection from each system will be the same, let us call it NoET_{0}. The polarization plot of the emission coming from each funnel is different, let us call it ET_{k}.

Having all these, we can express the polarization plot of the whole multi-funnel system as

The signal coming from all independent chromophores is

where

Now let us look at the ET part. Because the polarization properties of absorption that are the same for each system are defined by M_{ex} and θ_{ex}, using Eq. (9), we can write

where *Abs*(*φ*_{ex}) and *F*_{k}(*φ*_{em}) are the corresponding modulation functions introduced to shorten the expression. The weighted sum of the modulation functions *F*_{k}(*φ*_{em}) is just another modulation function *F*_{m}(*φ*_{em}) of the same mathematical form and frequency with a certain modulation depth M_{m}, phase θ_{m}, and unknown pre-factor γ,

To find γ, let us recall that $\u222b0\pi F(\phi )d\phi =\u222b0\pi 12[1+M\u2061cos(2[\phi \u2212\theta ])]d\phi =\pi 2$. Thus, by integrating Eq. (15) and using Eq. (13), we obtain

and

where $ETm\phi ex,\phi em$ is the polarization portrait of the effective funnel and *ɛ*_{m} defined by Eq. (13) is the effective energy funneling efficiency.

Figure 7 shows an example of a multi-funnel system that consists of two single funnel systems with the same absorption polarization property (two blue ellipses have the same shape and orientation, M_{ex1} = M_{ex2} = 0.43 and θ_{ex1} = θ_{ex2} = 45°). Energy funneling efficiencies are different (*ε*_{1} = 0.2 and *ε*_{2} = 0.4), so are the overall absorption cross sections. The latter means different integrated emission intensities of the two systems. The two funnels are dipoles (M_{f1} = M_{f2} = 1) oriented at 90° (d1) and 60° (d2), respectively. By fitting the 2D portrait with the SFA, we obtain an effective energy transfer efficiency *ε*_{m}, which is a sum of *ε*_{i} of each subsystem weighted by their relative absorption strength. In this example, *ε*_{m} = 0.6*ε*_{1} + 0.4*ε*_{2} = 0.28. From SFA fitting, the M_{fm} and θ_{fm} of the complete system are 0.87° and 17.4°, respectively. The residuals are zero, indicating that the SFA works perfectly for such a system.

#### 2. When SFA fails

From the results presented in Sec. IV B 1, we see that the necessary condition for the SFA to fail is the different absorption polarization properties of the subsystems. This is enough for the SFA to fail if the polarization properties of the funnels in the subsystems are not the same. If the funnels are the same, the energy funneling efficiency must be different (otherwise the whole system is just one single funnel system). Figure 8 illustrates an example of a two-funnel system that cannot be described by the SFA.

The multi-funnel system in Fig. 8(a) consists of two single funnel systems with different absorption polarization properties (two blue ellipses have the same orientation θ_{ex1} = θ_{ex2} = 30° but different shape M_{ex1} = 0.34 and M_{ex2} = 0.67). Energy funneling efficiencies are different (*ε*_{1} = 0.2 and *ε*_{2} = 0.4), so are the overall absorption cross sections. The two funnels are dipoles (M_{f1} = M_{f2} = 1) oriented at 90° (d1) and 60° (d2), respectively. By fitting the 2D portrait with the SFA, the fitting residual is 0.02. The effective energy transfer efficiency *ε*_{m} ≠ 0.6*ε*_{1} + 0.4*ε*_{2}.

Considering the real experiments, the failure of SFA (high residuals of the fit) indicates that the sample is not a single funnel system, and subsystems have different absorption polarization properties. A single multichromophoric system (e.g., a single conjugated polymer chain) can be considered as a multi-funnel system described in Fig. 8(a). Each dipole represents one localized chromophore. Depending on the conformation of the polymer chain and the conjugation length of chromophores, the intra-chain energy transfer can happen efficiently between closely located chromophores (local EET),^{30,31} and there will be many subsystems (one subsystem is a group of closely located chromophores where intra-chain energy transfer can happen) with different absorption polarization properties. If we apply our SFA method to such multichromophoric systems possessing efficient local EET, we expect to obtain low ε but high residual. Thus, even though SFA fails, the fitting results can still provide us with important information about the arrangements of chromophores and energy transfer in the sample.

In our previous study,^{5} we tested the SFA method experimentally on two well-known multichromophoric systems: single LH2 complexes—the famous natural antenna of purple bacteria—and single molecules of conjugated polymers (CPs, polyfluorene bis-vinylphenylene derivatives). Most of those single molecules/complexes could be successfully fitted by the SFA method: 94% of the 181 isolated PFBV molecules, 85% of the 141 isolated PFBV-Rtx molecules, 91% of the 545 isolated LH2 complexes excited through the B800 ring, and 92% of the 410 isolated LH2 complexes excited through the B850 ring.

Now, we understand that those single molecules/complexes, which could not be fitted with the SFA in the paper cited above, must have been multi-funnel systems possessing efficient local EET, however, with each local antenna having different absorption polarization properties. In future experimental work, much more attention should be paid on the developing a practical criteria of SFA validity taking into account the presence of noise in the real data. This will allow us to use the full potential of SFA based analysis of energy transfer and organization of multichromophoric systems.

#### 3. Applicability of SFA is affected by the number of subsystems

It is interesting to check how the number of subsystems influences the applicability of SFA (Fig. 9). A large number of identical subsystems randomly oriented in space are an ensemble equivalent to, e.g., a frozen solution. As we will show later in Sec. IV C, the SFA explains polarization properties of ensembles of systems regardless of the internal organization of each system. It means that even if the SFA cannot explain the properties of one system (because it consists of several funnels), it should do so when these systems are collected in a large random ensemble.

Figure 9 shows an example of this effect. Each subsystem [Fig. 9(i)] contains two chromophores with angle β = 50° between them, and there is energy funneling from one chromophore to the other (ε = 0.8). We increase the number of subsystems N from 1 to 1000 and we fit the 2D portrait of each system by the SFA obtaining the fitting residuals and the apparent energy funneling efficiencies.

We can see that the SFA fits when N = 1 (trivial case), but it does not fit the data at all for N = 2 [Fig. 9(a)]. This is what we expected (see Sec. IV B 2). If we increase N further, the residual monotonically goes down reaching very small values again for large ensembles (N > 900), and ε becomes constant independent on N [Fig. 9(b)]. These results demonstrate that ensemble averaging makes the SFA method to be able to fit the data even though SFA cannot explain the properties of a small part of this ensemble containing several absorbers and funnels.

To summarize, when the SFA does not work, it means that the object can be seen as an ensemble of several light-absorbing antennas with different absorption polarization properties, and each antenna has its own funnel different by its polarization properties. Increasing the number of subsystems increases ensemble averaging and makes the fitting based on the SFA more and more accurate.

### C. Isotropic ensemble of molecules

#### 1. The relation between energy funneling efficiency **ε** and fluorescence anisotropy r

As we have shown above, the SFA works for large ensembles regardless of the internal organization of the systems. It means that it should also formally work for fluorescent molecules in solution, where fluorescence anisotropy r (or polarization degree P) is traditionally used to obtain information about the mutual orientation of the absorbing and emitting dipole transitions in the molecule, energy transfer within one molecule and/or between them, and rotational diffusion.^{11,14–17} In this section, we will show that there is a mathematical relation between *ε* and r (and P). It is worth to mention that neither fluorescence anisotropy nor energy funneling efficiency can distinguish the rotation and EET.

Fluorescence anisotropy r and polarization degree P are defined as

where the emission intensities S_{∥} and S_{⊥} are measured through a polarizer set parallel and perpendicular to the electric field vector of the linearly polarized excitation light, respectively. The reason why r and P are always equal to zero is photo-selection. While molecules in the ground state possess random orientational distribution, the initial distribution of molecules excited by linearly polarized light is not random due to Malus’ law. Under excitation by a short light pulse, the initial fluorescence possesses the highest absolute value of anisotropy, which later can decrease due to rotational diffusion and/or energy transfer.^{2} This highest anisotropy is called fundamental anisotropy, it is reached when the molecules do not rotate and there is no energy transfer between them. r depends on the angle β between the absorption and emission transition dipole moments of the molecule and the dimension of the space where the molecules are distributed,

Fluorescence anisotropy defined by Eq. (18) can be calculated from the 2D portrait I(*φ*_{ex,}*φ*_{em}) by taking just two values from the whole 2D function: S_{∥} = I(*φ*_{ex,}*φ*_{ex}) and S_{⊥} = I(*φ*_{ex,}*φ*_{ex} + π/2), where *φ*_{ex} is an arbitrary chosen orientation of the light laser polarization plane. Obviously, the value of fluorescence anisotropy of a solution should not depend on chosen *φ*_{ex}, which means that the 2D portrait is symmetric against its diagonal and the value on the diagonal is constant independently on *φ*_{ex}, M_{ex} = M_{em} = 0.

Examples of 2D portraits of isotropic solution with positive and negative values of fluorescence anisotropy are shown in Fig. 10. For positive r [Fig. 10(a)], we call the portrait as normal. For the normal portrait, the maximum intensity is reached at the diagonal *φ*_{ex} = *φ*_{em} (S_{∥} = I_{max}). For the negative r [Fig. 10(b)], we call the portrait as inverted because the conditions for the maximum intensity become the conditions for the minimum and vice versa. For the inverted portrait, the intensity reaches its minimum at the diagonal and maximum for *φ*_{em} = *φ*_{ex} ± π/2 (S_{⊥} = I_{max}). It is evident that an inverted portrait can be obtained from a normal portrait multiplied by a negative factor (to invert the function) with a constant offset added. In general, for any value of r, we can write

Here, we chose as the basis the portrait of a solution with r = 0.4 (β = 0°) and A and B are the constants. Now, let us compare the equation above with the definition of SFA (see Sec. III C),

$Isolutionr=0.4\phi ex,\phi em$ is the portrait a solution of chromophores with β = 0°, and without energy transfer, it possesses perfect photo-selection and can be called $NoET\phi ex,\phi em$ in terms of SFA. The constant intensity B obviously does not depend on the excitation polarization angle, thus it satisfies the definition of emission from a funnel, and so, the ET component for a random ensemble is completely unpolarized $ET\phi ex,\phi em$ = const.

So, we can see that any 2D portrait of a random ensemble can be presented as a weighted sum of an unpolarized funnel and a random ensemble with β = 0° with the coefficients (1 − ε) and ε, respectively,

Now, we need to find the relation between fluorescence anisotropy r and funneling efficiency ε.

Let us normalize $Isolutionr=0.4\phi ex,\phi em$ in such a way that the average value is equal to unity, then I_{max} = 1.5 and I_{min} = 0.5. Then, we obtain

And we obtain the following equations connecting ε with r and P:

#### 2. Energy funneling efficiency ε can be larger than one

From the equations above, we can see that for molecules randomly oriented in 3D space, the full range of r from +0.4 to −0.2 corresponds to the range of ε from 0 to 5/3 [Fig. 11(a)]; for molecules randomly oriented in the 2D plane, the full range of r from +0.4 to −2/7 corresponds to the range of ε from 0 to 2 [Fig. 11(b)]. In addition, we can see that as soon as the angle β between absorption and emission dipole moments becomes larger than the magic angle, anisotropy and polarization degrees become negative and the energy funneling efficiency becomes larger than 1.

The reason why we have never encountered ε > 1 before is that this situation is impossible for a single system with one energy funnel. For example, if we take one molecule with β = 90° between the absorbing and emitting transition dipoles, we will obtain ε = 1 [Fig. 12(a)]. When we combine several such systems to an ensemble, we get the multi-funnel situation where the SFA does not work in a general case (see Sec. IV B). Only when the ensemble becomes very large, the SFA method starts to be able to fit the data. For the large ensemble case where all the molecules have β = 90°, ε is equal to 2 [Fig. 12(b)]. This simulation result also fits our analytical solution derived in Sec. IV C 2. Thus, the situation with ε > 1 arises only for ensembles of objects (molecules), it is a consequence of photo-selection at the condition of very large reorientation of the transition dipole moment inside each object (molecule). So, in a way, the very large ε here is signaling a very large intra-molecular energy transfer without inter-molecular energy transfer in the ensemble.

## V. CONCLUSIONS

We presented a detailed study of the applicability of the single funnel approximation (SFA) to fit 2D polarization portraits of multichromophoric systems with different numbers, organization of chromophores, and energy transfer between them. Our work provides a better understanding of the parameters of the model and allows us to extract more information about the organization and energy transfer; see the summary in Fig. 13.

We found that the SFA can formally explain polarization properties of multi-funnel systems in some cases. The conditions are that the system can be split into several subsystems, which meet the SFA’s requirements, and the absorption polarization of all the subsystems is the same. The necessary condition for SFA not being able to fit the experimental data is the different absorption polarization properties of the subsystems. Thus, the failure of SFA gives additional information about the system.

We found that the SFA gives an exact formal description of polarization properties of solutions of fluorophores, and energy funneling efficiency ε can be unambiguously related to fluorescence anisotropy, $r=2\xd71\u2212\epsilon 5+\epsilon $. The funnel in this case is isotropic. Interpretation of ε for isotropic solutions is then the same as fluorescence anisotropy that ranges from +0.4 to −0.2 corresponding to the range of ε from 0 to 1.67 (3D) and ranges from +0.4 to −2/7 corresponding to the range of ε from 0 to 2 (2D).

## ACKNOWLEDGMENTS

This work was supported by the Swedish Research Council (Grant No. 2020-03530), the Knut and Alice Wallenberg Foundation (Grant No. 2016.0059), and the Lund Laser Center. J.S. acknowledges the China Scholarship Council (Grant No. 201608110147) for the Ph.D. scholarship.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors declare no conflicts of interest.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX: ANISOTROPY FOR DIPOLES DISTRIBUTED IN 2D PLANE

Calculation of anisotropy r for molecules randomly oriented in the 2D plane.

To measure the anisotropy of an isotropic sample, we use linearly polarized light to excite the sample and measure the fluorescence intensity parallel I_{∥} and perpendicular I_{⊥} to the direction of excitation electric field E. First, we consider that the electric field interacts with one dipole oriented on angle *φ*, as shown in Fig. 14. For each angle φ, we have two configurations: the emission transition dipole moment is rotated either to the left (+β) or to the right (−β). Hence, we have the emission intensities measured via vertical and horizontally oriented analyzer as

To calculate the anisotropy of a random ensemble, we need to integrate the angle *φ* from 0 to 2π and calculate the average value, so we get

The polarization P can be calculated by

and anisotropy r = 2P/(3-P), so

The maximum value of r is 0.4 when the absorption and emission transition dipole moment are collinear (β = 0°), r is equal to 0 for β = 45° (magic-angle in the case of 2D), and the minimum value of r = −2/7 when β = 90°.