## Abstract

In this paper, resonance energy transfer (RET) networks between chromophores are used to implement fluorescent taggants with temporally coded signatures. Because the temporal signature of such a fluorescent taggant is a phase-type distribution defined by the geometry of its RET network, the taggant design is not constrained by resolvable dyes and has a significantly larger coding capacity than spectrally or lifetime coded fluorescent taggants. Meanwhile, the detection process becomes highly efficient when the signatures are coded in the time domain. The taggant identification method is based on the multinomial distribution of detected photons and Maximum Likelihood Estimation, which guarantees high accuracy even with only a few hundred photons and also applies to a mixture of taggants in multiplex detection. Therefore, these temporally coded fluorescent taggants have great potential for both *in situ* and Lidar applications.

© 2016 Optical Society of America

## 1. Introduction

Fluorescent taggants have been widely used for labeling and identification applications, and they are usually made of organic dyes, quantum dots (QDs), metal complexes, etc [1–7]. Because fluorescent taggants with different compositions absorb or emit light in distinct wavelength regions, their spectral characteristics are often used as their optical signatures. While a few taggants can often be selected for reliable spectral discrimination, this approach suffers from poor scalability when target applications require a larger taggant library. The spectra of fluorescent materials are not easy to alter, which greatly constrains the number of resolvable taggants that can be made of finite available fluorescent materials.

Within this context, this paper explores using the time-resolved fluorescence signal of a taggant as an alternative way to encode its optical signature. While time-resolved fluorescence has been considered for labeling applications in previous studies, it is so far limited to the intrinsic exponential decays of fluorescence, and hence constrained by the resolvable lifetimes of available chromophores [8–10]. Based on the two observations that 1) resonance energy transfer (RET) networks can generate virtually arbitrary time-resolved fluorescence signals and 2) RET networks can be precisely built with chromophores through DNA self-assembly, we propose RET network based fluorescent taggants, which can potentially bring significantly larger coding capacity and flexibility to taggant design.

On the detection side, time-resolved photon detection with a single pair of interrogation and detection wavelengths facilitates the detection of all taggants when the signatures are encoded in the time domain. Meanwhile, because the detected photons follow a temporal multinomial distribution, statistical methods such as Maximum Likelihood Estimation (MLE) enable a reliable and convenient taggant identification even under low light conditions. Further, a mixture of taggants, in multiplex detection, can also be formulated in MLE and resolved by the Expectation Maximization (EM) algorithm. With these unique advantages, the fluorescent taggants with temporal signatures have great potential for both *in situ* and Lidar applications.

## 2. Resonance energy transfer network based fluorescent taggants

The fluorescent taggant we propose in this paper is based on molecular-scale RET networks that can be implemented by placing chromophores in various geometries by using hierarchical DNA self-assembly with sub-nanometer precision [11–14]. The exciton dynamics in a RET network behaves as an absorbing Continuous Time Markov Chain (CTMC), and the time-resolved fluorescence signal of a RET network is a phase-type distribution configured by its physical geometry [15]. Therefore, RET networks with distinct geometries can be designed and fabricated to achieve different temporal signatures beyond the intrincis characterisitics of constituent dyes. Because any positive-valued distribution can be theoretically approximated with a phase-type distribution to arbitrary precision [16], these temporally coded fluorescent taggants have a significantly larger coding capacity than spectrally or lifetime coded taggants.

#### 2.1 Resonance energy transfer networks

A RET network can be built by placing multiple chromophores in a physical geometry where each chromophore may interact with the others through resonance energy transfer (Fig. 1). A chromophore is a molecule that can absorb photons at a specific wavelength and reemit at a longer wavelength via fluorescence. RET is an energy transfer mechanism between two chromophores where the donor chromophore, initially in its excited state, transfers energy to the acceptor chromophore through non-radiative dipole-dipole coupling (Fig. 2) [17]. The fundamental parameters that govern the transfer rates between the chromophores in a RET network are defined by the molecules we choose to create the network and the separation between them. A fully specified chromophore network can be conveniently and economically fabricated with sub-nanometer precision through DNA self-assembly [11, 12]. This process makes it feasible to synthesize large scale chromophore networks and fine-tune their transfer rates within a wide range [13, 18, 19]. Compared with other molecular self-assembly methods [20, 21], DNA self-assembly has unique advantages for fabricating RET networks such as the capability of making irregular and asymmetrical geometries with sub-nanometer precision, low cost and high throughput.

The transfer rate of the RET process between a chromophore pair, first derived by Förster, is:

where ${\tau}_{D}^{0}$ is the intrinsic fluorescence lifetime of the donor, ${k}^{2}$ is the mutual orientation of the chromophore pair, $r$ is the distance between the chromophore pair, and ${R}_{0}$ is the Förster radius, i.e., the distance at which the transfer efficiency is 50%. The Förster radius of a chromophore pair mainly depends on the properties of the two chromophores, such as the quantum yield ${\varphi}_{D}^{0}$ and emission spectra ${I}_{D}\left(\lambda \right)$ of the donor and the molar absorption coefficient of the acceptor ${\epsilon}_{A}(\lambda )$.The intrinsic fluorescence lifetime ${\tau}^{0}$ of a chromophore is determined by the rates of all the intrinsic relaxation pathways including both radiative pathway (i.e., fluorescence) and nonradiative pathways. In the presence of RET to an acceptor chromophore, another relaxation pathway exists with the rate ${k}_{RET}$ in Eq. (1). The relaxation event through each pathway is an exponentially distributed random variable in the time domain, and, as a result, the de-excitation of the chromophore is also exponentially distributed. Between a RET pair, the excited state lifetime of the donor chromophore is shown in Eq. (2), and the transfer efficiency is shown in Eq. (3).

#### 2.2 CTMC model of the exciton dynamics in a RET network

After a chromophore is excited in a RET network, the exciton dynamics in the RET network is an absorbing Continuous Time Markov Chain [15]. CTMC is a continuous-time stochastic process with a finite or countable state space $S$, in which the time spent in each state is exponentially distributed. The Markov property of a CTMC means that the conditional probability distribution of future states of the process (conditional on both past and present states) depends only on the present state and not on the sequence of events that preceded it. A CTMC is defined by its discrete state space $S$, a transition matrix $Q$ that indicates the transition rate between each pair of states, and an initial probability distribution $\pi \left(0\right)$.

An absorbing CTMC has at least one absorbing state ${S}_{Ai}\text{}\left(i=1,\dots ,n\right)$, which only has incoming transition rates, and the probability of the system having transitioned into an absorbing state approximates 1 as time increases to infinity: $\underset{t\to \infty}{\mathrm{lim}}Prob\left(X\left(t\right)\in \left\{{S}_{Ai},i=1,\dots ,n\right\}\right)=1$. Additionally, the absorption probability of each absorbing state ${S}_{Ai}\left(i=1,\dots ,n\right)$, i.e., the probability of the system transitioning into each specific absorbing state, is ${P}_{Ai}=Prob\left(X\left(\infty \right)={S}_{Ai}\right)$, which depends on the initial probability distribution $\pi \left(0\right)$ and the transition matrix $Q$ [22].

Figure 3 shows the exciton dynamics between a chromophore pair and its corresponding CTMC model. It is assumed that an ultrashort pulse laser only excites the donor chromophore and at most one chromophore remains excited between the pair at any time, which is feasible from our experience with wavelength division multiplexing and fabricated RET networks. In the CTMC, each chromophore has a transient state ${S}_{T}$ to indicate whether it is in its excited state, and the exciton decay through each intrinsic relaxation pathway of each chromophore is represented as an absorbing state ${S}_{A}$, i.e., ${S}_{A1}$: ‘Donor Fluoresces’, ${S}_{A2}$: ‘Acceptor Fluoresces’, ${S}_{A3}$: ‘Donor Nonradiative Decay’, ${S}_{A4}$: ‘Acceptor Nonradiative Decay’. (Although not shown in Fig. 3(a), nonradiative decay exists as an exciton relaxation pathway and is included in Fig. 3(b).) The initial state vector $\pi \left(0\right)$ and the transition matrix $Q$ of the CTMC are shown in Eqs. (4) and (5).

For a larger RET network with more than two chromophores, its CTMC will simply enclose more states and the transition rates between them (see Fig. 4).

#### 2.3 Temporally coded signatures of RET networks

The time-resolved fluorescence of a RET network has a phase-type distribution that is defined by the absorbing CTMC of the exciton dynamics in the network. Because the time to RET transfer, after the donor is excited, follows an exponential distribution between each chromophore pair and the sequence of RET transfers between an exciton entering and leaving (i.e., decaying) a chromophore network is a random process, the time to exciton decay follows a phase-type distribution [15]. The RET transfer between a chromophore pair with a transfer rate of ${k}_{RET}$ physically implements a phase transition with a transition rate of $\lambda ={k}_{RET}$ in the phase-type distribution, and the geometry of the chromophore network controls how these phases are convolved and mixed to form the phase-type distribution. Specifically, the time to exciton decay in a RET network is the time to absorption in the absorbing CTMC of its exciton dynamics, and the CTMC specifies the distribution of this phase-type random variable through its transition matrix $Q$ which contains the RET transfer rate between each chromophore pair and the decay rate of each relaxation pathway.

Consider the RET pair in Fig. 3 for simplicity. An Alexa Fluor 488 dye and an Alexa Fluor 594 dye are chosen as the donor and acceptor respectively, which are placed 10nm apart. The state probabilities of the four absorbing states in the CTMC, ${\pi}_{SA}\left(t\right)=\left[{\pi}_{SA1}\left(t\right),{\pi}_{SA2}\left(t\right),{\pi}_{SA3}\left(t\right),{\pi}_{SA4}\left(t\right)\right]$ (${S}_{A1}$: ‘Donor Fluoresces’, ${S}_{A2}$: ‘Acceptor Fluoresces’, ${S}_{A3}$: ‘Donor Nonradiative Decay’, ${S}_{A4}$: ‘Acceptor Nonradiative Decay’), are shown in Fig. 5(a) and their sum monotonically approximates 1 as the input exciton is increasingly likely to have decayed as time passes. Specifically, the blue and red solid curves in Fig. 5(a) correspond to the fluorescence of the two chromophores. By taking the derivative of these two curves and normalizing each, the conditional probability density function (PDF) of the *time to fluorescence* (TTF) from each chromophore, ${f}_{T}(t|X\left(\infty \right)={S}_{A1})$ and ${f}_{T}(t|X\left(\infty \right)={S}_{A2})$, can be derived (see Fig. 5(b)), which are phase-type distributions. This method can be generalized to larger RET networks with more chromophores.

Because phase-type distributions form a dense set in the field of all positive-valued distributions and can approximate any positive-valued distribution [16], RET networks can achieve virtually arbitrary temporal signatures, which brings an unprecedented coding capacity to fluorescent taggants. The methods of approximating a general distribution with a phase-type distribution have been well investigated [23–25], which provide guidance on designing a RET network given a target signature. Further, because network geometry is leveraged to create signatures in this approach, it is not constrained by resolvable chromophores unlike spectrally or lifetime coded fluorescent taggants. To illustrate this point, six different RET networks were designed using the same set of three dye molecules, i.e., Alexa Fluor 430, Alexa Fluor 594 and Alexa Fluor 750, and their temporal signatures were simulated and compared. The excitation and emission spectra of the three dyes are plotted in Fig. 6, which shows that AF430 can be excited at a wavelength about 450nm with negligible interference with the other dyes and the fluorescence of AF750 can be measured at a wavelength about 780nm with negligible crosstalk from the other dyes. Each of the six RET networks adopts a wire geometry that contains one AF430 chromophore, one AF750 chromophore and a certain number (${N}_{i}=i,i=1,\dots ,6$) of AF594 chromophores in between (see Fig. 7). The distance between two adjacent chromophores equals their Förster radius ${R}_{0}$. The AF430 chromophore functions as an optical antenna and can be excited by a pulsed laser source, and the AF594 chromophores function as mediators that propagate excitons downwards, and the AF750 chromophore functions as an emitter and its time-resolved fluorescence is used as the temporal signature.

In each RET network, RET transfer can occur between both adjacent and non-adjacent chromophores. Because an exciton may decay from any chromophore in a RET network, the end-to-end transfer efficiency from AF430 to AF750 varies as a function of the length of the chromophore wire and its pairwise transfer efficiencies. By building the CTMC model for each RET network, its temporal signature can be simulated by deriving the conditional PDF of the TTF from its AF750 chromophore (see Fig. 8).

As shown in Fig. 8, the mean time to fluorescence from the AF750 chromophore increases with the number of mediators, which can be intuitively explained by a longer time an average exciton takes to reach the AF750 in a longer chromophore wire. Additionally, because each RET transfer incurs a convolution with an exponential distribution, the distribution of the time to reach the AF750 becomes less concentrated in a longer wire and resembles a hypoexponential distribution with more exponential stages. It should be noted that they are not exactly hypoexponential distributions due to the nonzero back transfers rates. The difference between two temporal signatures determines the difficulty of discriminating them, and can be quantified as the Kullback–Leibler (KL) divergence between two phase-type distributions. For the six temporal signatures in Fig. 8, their pairwise KL divergences are shown in Table 1. Given a detection resolution and signal strength, the difference between any two signatures should exceed a threshold to maintain a low misidentification probability, which will be further discussed in the next section.

Meanwhile, the conversion probability of a fluorescent taggant is another metric of interest in practical applications, which is the probability of the taggant emitting a fluorescence photon after excitation and proportionally affects its fluorescence intensity. For each of the six designed RET networks, the conversion probability is the probability of an exciton fluorescing from its emitter AF750 after entering the network from its antenna AF430, and, in CTMC, corresponds to the absorption probability of the absorbing state for the emitter fluorescence ${P}_{A\_ef}=Prob\left(X\left(\infty \right)={S}_{A\_ef}\right)$. The conversion probabilities of the six chromophore wires from short to long are respectively 0.0301, 0.0127, 0.0051, 0.0021, 0.0008, and 0.0003, the decrease of which indicates a higher probability of an exciton decaying in the middle of a longer chromophore wire. While chromophores can be placed closer in RET networks to increase their conversion probabilities, the divergence between their temporal signatures may decrease and make taggant identification more challenging, which poses a tradeoff in the design process. Nevertheless, if uniform fluorescence intensity is required from all taggants, parameters such as the concentration of each fabricated RET network ensemble can be increased to compensate for their different conversion probabilities.

It is noteworthy that organic dyes are relatively more suitable for making the temporally coded fluorescent taggants when compared with other fluorescent molecules such as fluorescent proteins and QDs. These fluorescent taggants rely on the RET transfers within different chromophore networks to generate distinct temporal signatures. Among common fluorescent molecules, organic dyes are the most suitable and widely used for making RET pairs [26]. There is a large and diverse collection of commercially available organic dyes. Their optical properties (e.g., absorption/emission spectra, extinction coefficient, quantum yield) and RET properties as donors and acceptors (i.e., Förster radius) are well characterized, and these synthetic dyes are continually improved in terms of photostability, solubility and conjugation strategies. Meanwhile, organic dyes have relatively narrower absorption bands, and individual dyes can be selectively excited in a chromophore network. The large selection of commercially available dyes cover a wide spectral range from UV to NIR, and RET pairs can be flexibly cascaded and organized to create different RET networks. Further, organic dyes typically exhibit mono-exponential decays while QDs often exhibit multi-exponential decays [26]. Multi-exponential distributions would make it more challenging to model the exciton dynamics and identify taggants based on temporal discrimination.

## 3. Detection method

To identify fluorescent taggants with spectrally coded signatures, their absorption or emission spectra need to be fully or partially characterized and analyzed using statistical methods such as principal component analysis (PCA) and cluster analysis [27–30]. Therefore, a spectrometer or spectral filters are required to select different wavelengths for each detection procedure. Moreover, a time-gated measurement is often necessary to minimize the effect of background emission and other noise sources [27, 31].

In contrast, a time-resolved photon detection system with a single pair of interrogation and detection wavelengths facilitates the detection of all taggants when their fluorescent signatures are encoded in the time domain. The process of single photon arrivals is captured in a multinomial distribution model and the taggant identification method is based on Maximum Likelihood Estimation which yields a misclassification probability that exponentially decreases with the number of detected photons. The number of required photons to guarantee a high identification accuracy can be as low as a few hundred, which is orders of magnitude lower than in spectrally-coded approaches. Further, the identification of a mixture of taggants, in multiplex detection, can also be formulated in MLE and reliably resolved by the Expectation Maximization (EM) algorithm [32].

#### 3.1 Detection system

A lab demo is performed to illustrate the detection of temporally coded fluorescent taggants (Fig. 9). Powered by a pulsed laser diode driver, a laser diode emits ultrashort pulses to interrogate the RET network based fluorescent taggant in a sample cuvette. A spectral filter can be inserted between the laser diode and the sample cuvette if the laser spectrum needs to be attenuated. The fluorescence of the taggants are focused onto the aperture of a single-photon avalanche diode (SPAD), and a spectral filter is placed before the SPAD to only pass the wavelength region where temporal signatures are encoded. With a timing resolution as high as ~40ps, the photon detection signal of the SPAD is fed into a Time-Correlated Single Photon Counting (TCSPC) module [33]. This time-resolved photon counting module records the detection times of individual photons relative to the SYNC signal with a timing resolution as high as several picoseconds and reconstructs the time-resolved histogram of photon counts. The TCSPC measurements of fluorescent taggants are usually performed in a laboratory where steps are taken to minimize the background noise. Portable TCSPC systems have recently been built and *in situ* detection has been demonstrated [34, 35]. When a lower timing resolution is sufficient for distinguishing temporal signatures, a high-speed gated ICCD camera may be used instead of a TCSPC.

The fluorescent taggant tested in the detection system is a chromophore wire fabricated using the tile based DNA self-assembly [11, 12], with a source ATTO 488, two mediators ATTO 565 and an emitter ATTO 610. The DNA sequences and the nucleobases chosen for dye conjugation are shown in Fig. 10, and the chromophore–DNA conjugation was achieved by using a primary amino modifier group on Thymidine to attach an NHS ester-modified dye molecule. Compared with the previous theoretical derivation, more practical aspects were taken into account when designing and fabricating this fluorescent taggant for experimental demonstration, and ATTO dyes were chosen due to their better photostability and optical properties such as strong absorbance and high quantum yields. Because the energy transfer from the source to the emitter in the chromophore wire relies on the energy migration between mediators, we chose ATTO 565 as the mediator chromophore because of its short Stokes shift to improve the homotransfer capability and the chromophore wire can be extended without significantly reducing the conversion probability.

Excited at the excitation maximum 501nm of ATTO 488, the time-resolved fluorescence of the chromophore wire was detected at three different wavelengths, 523nm, 592nm and 634nm, respectively corresponding to the emission spectral peaks of ATTO 488, ATTO 565 and ATTO 610. Figure 11 shows the TCSPC data and the results after smoothing with a Savitzky-Golay filter and normalization. The gradual change of the detected temporal signal from an exponential-like decay at 523nm to a hypoexponential-like distribution at 634nm is consistent with the shape change between the simulated temporal signals in Fig. 8. The exponential decay of the source mainly contributes to the fluorescence detected at 523nm, and the hypoexponential-like distributions of the mediators and the emitter contribute to the fluorescence detected at 592nm and 634nm. The number of temporally convolved exponential distributions increases with the distance from the source chromophore in the exciton flow, which results in a less concentrated distribution with a longer average TTF.

Despite the observed shape change that originates from the theoretical CTMC description of exciton dynamics, a precise curve fitting between theoretical and observed temporal signatures remains difficult to achieve and beyond the scope of this paper. Although the exciton dynamics in a RET network is a CTMC, an accurate curve fitting would first need to precisely quantify the discrepancy between the fixed RET transfer rates in theory and the dynamic RET transfer rates in reality. While the calculation of RET transfer rates are based on fixed parameters in theory, these rates are nondeterministic in reality due to the molecular-scale physical variation and conformational fluctuations and follow unknown distributions. Parameters such as inter-dye distances [36, 37] and their dipole orientations [38, 39] control the RET transfer rates and vary between individual structures within a fabricated ensemble and undergo incessant dynamic changes. The attempts to capture the effects of such parameter variations require complex and laborious simulation and experimental procedures such as molecular dynamics (MD) simulation and single-molecule FRET (smFRET). While these methods can partially explain the observed discrepancy, they deviate from the focus of this paper, which is to use RET networks of different geometries to create distinct temporal signatures based on the CTMC description of exciton dynamics. Additionally, other non-ideal aspects may also contribute to the discrepancy such as the wavelength crosstalk between chromophores and the yield of complete RET networks in the fabricated ensemble.

Because of the discrepancy between theoretical and fabricated temporal signatures, the temporal probability distributions of photon detection should be finely measured for fabricated fluorescent taggants as the reference for taggant identification in real-world applications. Meanwhile, a more thorough comparison with experimental results is an important aspect of our future work.

#### 3.2 Taggant identification

Because the temporal signatures can be exactly characterized as temporal probability distributions unlike emission spectra and the process of single photon detection can be captured by a multinomial distribution, MLE can be applied to reliably and conveniently identify a taggant. In the special case of single exponential decays, the methods of estimating fluorescence lifetime have been extensively investigated, and MLE is the most statistically efficient estimator and asymptotically achieves the Cramér–Rao lower bound (CRLB), which is the theoretical limit of the variance of any unbiased estimator [40, 41]. Only a few hundred photons are needed to yield an estimate of lifetime with 10% relative error when the background photon count is negligible, and the required number of photons only slightly increases in the presence of 20% background fluorescence [41]. Additionally, the Kullback-Leibler minimum discrimination information has been successfully used to classify a measured signal among a set of lifetimes in low light conditions [42, 43]. Equivalent to MLE in a finite regime, this method has the lowest misclassification probability.

Beyond single exponential decays, the Kullback-Leibler minimum discrimination information can also be used to locate the temporal taggant signature that best matches a measured signal because this classification method only requires the multinomial distribution model of photon detections. Assuming a taggant library contains $S$ taggants, the temporal signature of each taggant is represented by the probability density function (PDF) of a temporal probability distribution ${f}_{i}\left(t\right)\left(i=1,\dots ,S\right)$. The detection system has a finite measurement window *T*, which is equally divided into $k$ channels of width $\Delta T$. For taggant $i$, the probability of a photon being detected in channel $j$ is ${p}_{j}\left(i\right)$ in the absence of background photons (Eq. (6)).

If $N$ photons are detected from taggant $i$, their distribution over the $k$ channels follows a multinomial distribution (Eq. (7)), where $n=\left[{n}_{1},\dots ,{n}_{k}\right]$ is the number of photons in each channel and ${\sum}_{j=1}^{k}{n}_{j}}=N$. Given sufficient measurement window *T* and number of channels $k$, each type of taggant has a unique pattern of distributing photons, which is a multinomial distribution parameterized by $\left[{p}_{1}\left(i\right),\dots ,{p}_{k}\left(i\right)\right]\left(i=1,\dots ,S\right)$.

Within this context, the task of taggant identification becomes to classify the measured signal, i.e., $n=\left[{n}_{1},\dots ,{n}_{k}\right]$, among the $S$ patterns. The Kullback-Leibler minimum discrimination information is calculated between the measurement and each pattern (Eq. (8)). With the calculated minimum discrimination information for all $S$ patterns, the detected taggant is expected to be the one that yields the lowest value of ${I}^{*}$.

When classifying a measured fluorescence signal using the Kullback-Leibler minimum discrimination information, the probability of misclassification has been theoretically derived and experimentally verified for dyes with single exponential decays [43]. Because the results are based upon the multinomial distribution model, they are applicable to general temporal signatures. The probability of incorrectly classifying a detected taggant of type $i$ as type $i\text{'}$ ($P\left({i}^{\prime}\text{|}i\right)$) can be expressed using the Gaussian error function by approximating $\Delta {I}_{i\text{'}i}={I}^{*}\left({i}^{\prime}\right)-{I}^{*}\left(i\right)$ as a normal distribution (Eq. (9)). This misclassification probability decreases exponentially with the signal strength $N$.

When more than two taggants exist in the taggant library ($S>2$), the error probability of misclassifying taggant $i$ is

Consider the six temporal signatures in Fig. 8 and a measurement window $T=50ns$ with $k=256$ equally divided channels. The error probability of misclassifying each taggant predicted by Eq. (10) is plotted in Fig. 12. Taggants 5 and 6 have higher misclassification probabilities than the others due to the shorter KL divergence between their temporal signatures. Nevertheless, the misclassification probability of each taggant decreases exponentially with the number of detected photons, and only 500 photons are needed to reach the accuracy of at most 1 error in 10,000 classifications for all taggants.

If the taggant detection is carried out in a low-light environment and the background emission is filtered out by the spectral filter before the SPAD, it is reasonable to neglect background photons because of the extremely low dark count rate (~2Hz) of modern SPAD’s [33]. However, when they are not negligible, the background photons can be modeled as a uniform distribution over the $k$ channels in the measurement window [41]. For taggant $i$, the probability of detecting a photon in channel $j$ in the presence of background photons is now

where ${p}_{j}\left(i\right)$ is the probability of a fluorescence photon being detected in channel $j$ in a background-free environment (Eq. (6)) and $b$ is the portion of photons due to the background noise. When the modified patterns of detecting photons ($p{\text{'}}_{j}\left(i\right)$) of the six taggants are used for taggant identification in a background noise of $b$, the number of photons required for the same identification accuracy of 1 error in 10,000 classifications increases with $b$(see Fig. 13). The increase remains trivial when$b<30\%$, and fewer than 1,000 photons are needed in this region. Aside from this theoretical analysis of the effect of noise on the identification accuracy, a practical approach to incorporating noise and other non-ideal aspects (e.g., the instrument response function of the detection system) is to accurately measure the pattern ${p}_{j}\left(i\right)$of each taggant in a similar or identical environment prior to taggant detections and use the measured patterns as the reference for classification.

#### 3.3 Multiplex detection

Multiple taggants sometimes exist as a mixture, and it is often desired to recognize each constituent taggant. Least squares based methods are commonly used to resolve a mixture of fluorescent taggants with spectrally coded signatures in multiplex detection [44–46]. However, they often require an even finer spectral characterization due to the assumed Gaussian noise model and produce ambiguous results as the number of taggants or their spectral overlap increases.

In contrast, a mixure of temporally coded signatures can be more reliably and conveniently resolved using the statistical methods for mixture models such as the EM algorithm. Assume $N$ photons have been detected from a sample and their distribution over the $k$ channels is $n=\left[{n}_{1},\dots ,{n}_{k}\right]$. Given the prior knowledge that the sample is a mixture of two taggants from the six taggants in Fig. 7, their identities ($Ta{g}_{1}$ and $Ta{g}_{2}$) and fractions ($p$ and $1-p$) constitute a mixture model behind the measured time-resolved signal. The photon distribution given this mixture model follows a multinomial distribution:

where ${p}_{j}\left(Tag\right)\left(j=1,\dots ,k\right)$ is the probability of detecting a photon in each time bin given a specific taggant (Eq. (6)). The parameters of this mixture model can be estimated through maximizing the likelihood of observing the measured signal, i.e., ${\sum}_{j=1}^{k}{n}_{j}\cdot \mathrm{log}\left[p\cdot {p}_{j}\left(Ta{g}_{1}\right)+\left(1-p\right)\cdot {p}_{j}\left(Ta{g}_{2}\right)\right]$. However, this likelihood is not convenient to directly optimize due to the sum inside the logarithm. Instead, the EM algorithm dynamically calculates the probabilities of a detected photon being from the two taggants, i.e., ${T}_{j,1}$ and ${T}_{j,2}\left(j=1,\dots ,k\right)$, and optimizes the target likelihood through iteratively evaluating and maximizing the expected log-likelihood $\sum}_{j=1}^{k}{n}_{j}\left\{{T}_{j,1}\mathrm{log}\left[p\cdot {p}_{j}\left(Ta{g}_{1}\right)\right]+{T}_{j,2}\mathrm{log}\left[\left(1-p\right)\cdot {p}_{j}\left(Ta{g}_{2}\right)\right]\right\$. As a result, the parameters can be separately updated in each iteration until the target likelihood reaches its maximum, which is outlined as follows.

- 1.
*$Ta{g}_{1}=1;Ta{g}_{2}=6;p=\mathrm{0.5.}$**// starting point of the three variables*- 2.
- 2.1 E-step:
*With the current estimates of $Ta{g}_{1},Ta{g}_{2}$*,*and $p$*,*calculate ${T}_{j,1}$**and${T}_{j,2}\left(j=1,\dots ,k\right)$:*$${T}_{j,1}=\frac{p\cdot {p}_{j}\left(Ta{g}_{1}\right)}{p\cdot {p}_{j}\left(Ta{g}_{1}\right)+\left(1-p\right)\cdot {p}_{j}\left(Ta{g}_{2}\right)};\text{}{T}_{j,2}=\frac{\left(1-p\right)\cdot {p}_{j}\left(Ta{g}_{2}\right)}{p\cdot {p}_{j}\left(Ta{g}_{1}\right)+\left(1-p\right)\cdot {p}_{j}\left(Ta{g}_{2}\right)}.$$ - 2.2 M-step:
*With the latest values of ${T}_{j,1}$**and ${T}_{j,2}\left(j=1,\dots ,k\right)$*,*update $Ta{g}_{1}$*, $Ta{g}_{2}$*and p:* - 2.3 if termination condition is met:
break

else:

go back to 2.1

After the three parameters are initialized with a starting point, the algorithm enters the iterations of Expectation and Maximization steps. Each iteration consists of an Expectation step and a Maximization step. The Expectation step calculates the probabilities of a detected photon being from the two taggants, i.e., ${T}_{j,1}$ and ${T}_{j,2}\left(j=1,\dots ,k\right)$, based on the current estimates of the identities and fractions of the two taggants. With the current values of ${T}_{j,1}$ and ${T}_{j,2}$ as the weights of each detected photon related to the two taggants, the Maximization step updates the fractions of the two taggants and their identities by respectively maximizing the weighted likelihood of observing the photon detection times for each taggant. The iterative process reaches an end when a termination condition is met. For example, the target likelihood can be evaluated at the end of each iteration, and when the change of this value between adjacent iterations falls below a threshold, it can be concluded that the target likelihood has been maximized. Because, dependent on its starting point, the EM algorithm may converge to a local maximum, it may be necessary to run the EM algorithm with different starting points to improve the chance of locating the maximum likelihood and the correct values of the three parameters.

Because this approach to multiplex detection is still based on the multinomial distribution model and MLE, it is the most statistically efficient. Further, a model selection criterion such as Bayesian Information Criterion (BIC) can be used to estimate the number of existing taggants in a mixture if this information is absent prior to a detection. However, it is noteworthy that a higher number of photons is often necessary in multiplex detection because more parameters are to be estimated in the mixture model.

## 4. Lidar integration

Lidar is a remote sensing technology that identifies objects of interest and measures distance by illuminating a target with a laser and analyzes the reflected light, which has wide applications in archaeology, forestry, atmospheric physics, anti-poaching, etc. Because the fluorescence emitted by a target is often used to determine the presence of the target and its distance, natural or artificial fluorescent materials become common targets in Lidar applications [27–31, 47]. However, these fluorescence Lidar applications usually use the spectral characteristics of the fluorescent materials by measuring the intensity of the reflected light within one or multiple wavelength bands, which has limited the ability to tag different objects and resolve them in multiplex detection.

With the unique advantages of a RET network based fluorescent taggant with temporal signatures, a large number of different objects can be conveniently coded in the same wavelength region with a small set of chromophores. The single channel of interrogation and detection wavelengths will make the detection procedure highly efficient, and the superior reliability of target identification under low light conditions will potentially increase the detection range.

While the fluorescence being detected in fluorescence Lidar applications is often in the wavelength range between 350nm and 710nm which overlaps the solar spectrum, its spectral and temporal characterisitics can still be accurately measured by taking measures to reduce the background signal such as using optical filters and time-gating technqiues. Modern ICCD cameras are highly sensitive cameras that have single photon detection capability and high speed time-gating (min. gate width ~5ns) capability with a resolution of ~40ps. Many modern fluorescence Lidar systems use ICCD cameras and have high-resolution temporal measurement capability, and they can operate in full daylight [48–50]. These implementations are compatible with the temporally coded fluorescent taggants if their timing resolution is sufficient to resolve the temporal signatures. Combined with the high-speed time-gating technique, TCSPC can potentially further improve the timing resolution when it becomes necessary for taggant discrimination.

There are additional aspects to take into account when designing a fluorescent taggant for Lidar applications, which are partially considered in a previous work [31] that fabricated fluorescent taggants using nanocrystals and demonstrated their far-field (~3km) detection capability. For example, the laser source in common fluorescence Lidar applications often has a UV wavelength around 350nm due to the higher eye-safe power density in this region, and commercially available dyes suitable for this excitation wavelength should be considered as the source chromophore such as ATTO 390 and Alexa Fluor 350. In addition, the mediator and the emitter should have low absorptivity so that their direct excitation is reduced and their excitation mostly comes from the source chromophore through RET transfer.

## 5. Conclusion

This paper proposes using RET networks to implement fluorescent taggants with temporally coded signatures. Compared with spectrally or lifetime coded fluorescent taggants, these temporally coded fluorescent taggants have a significantly larger coding capacity and flexibility in taggant design. Meanwhile, with these fluorescent taggants, the taggant detection and identification process becomes highly efficient and reliable even with only a few hundred photons under low light conditions. Additionally, the MLE based taggant identification can also resolve a mixture of taggants in multiplex detection. These unique properties make the temporally coded fluorescent taggants a superior candidate for both *in situ* and Lidar applications.

## Acknowledgments

This project was supported by the National Security Science and Engineering Faculty Fellowship (NSSEFF) ONR (N00014-15-1-0032).

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