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Abstract
Light emission by fluorophores can be computed from the knowledge of the absorption spectrum. However, at long wavelengths, the calculated emission may diverge if the decay of the imaginary part of the permittivity is not modelled with precision. We report a technique to obtain the permittivity of fluorophores such as dye molecules from fluorescence measurements. We find that the Brendel-Bormann model enables to fit the emission spectra accurately.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
Corrections
Elise Bailly, Kevin Chevrier, Camilo Perez de la Vega, Jean-Paul Hugonin, Yannick De Wilde, Valentina Krachmalnicoff, Benjamin Vest, and Jean-Jacques Greffet, "Method to measure the refractive index for photoluminescence modelling: erratum," Opt. Mater. Express 13, 2165-2165 (2023)https://opg.optica.org/ome/abstract.cfm?uri=ome-13-8-2165
12 August 2022: A minor correction was made to an author name.
1. Introduction
Fluorophores such as rare earth ions, dye molecules or quantum dots are widely used for lighting applications. They can be coupled to resonant metasurfaces in order to control the emission properties. Metasurfaces can be used to improve light extraction [1–21], to control the emission spectrum [22–27] or the directivity [28–34]. Novel light-matter interaction regimes have been explored in these platforms including strong coupling [35–42] and Bose-Einstein condensation [43,44]. A summary can be found in recent reviews [13,45]. As discussed in the reviews, the interplay between the emitters and the metasurface is accounted for by modelling the emitters as individual electric dipoles randomly oriented inside the structure whose contributions can be added incoherently. The dipole emission spectrum is taken to be the emission spectrum of the emitters which accounts for the Stokes shift compared to the absorption spectrum. Such an approach yields the power emitted by a metasurface up to a proportionality factor related to the excitation. It is therefore not quantitative and cannot be used to model the efficiency of the source. A quantitative model should include a description of the temperature dependent population of the electronic states.
In related systems, such as light emitting diodes (LEDs), a statistical approach is routinely used to establish a model of light emission. Indeed, the emitted power can be given quantitatively by a generalized Kirchhoff law [46,47]. This relation states that the emitted flux is given by the product of the absorptivity of the device and the blackbody radiance. The latter depends on the temperature $T$ and on a so-called photon chemical potential denoted by $\mu$ which is the difference of the quasi-Fermi levels in the conduction band and the valence band. For LEDs, $\mu$ is equal to $eV$ where $e$ is the electron charge and $V$ is the voltage applied to the diode by the driving electrical source. Therefore, maximizing the light emission by a given system can be achieved by maximizing its absorptivity in the active region. Besides, this model yields an upper bound of the emitted power. This figure of merit is useful when considering metasurface design and efficiency optimization. A record-efficiency light emitting diode [48] has been designed following this approach. In principle, the same approach could also be used for fluorophores coupled to resonant metasurfaces. However, this method has not been reported so far. As will be shown below, a key difficulty to be addressed is the introduction of a suitable permittivity model for the fluorophores. The purpose of this article is to solve this outstanding issue.
For many applications such as reflectivity measurements in the strong coupling regime [35,36,49], the permittivity of the fluorophores is modelled using a Lorentzian model. These fits are often in great agreement with experimental data from reflectivity measurements. However this Lorentzian model fails to retrieve the experimental photoluminescence using Kirchhoff law, as it can be seen in Fig. 1. More sophisticated models of the permittivity include a sum of Lorentzian lines, or a sum of poles [50–53]. These models suffer from the same drawback and do not enable to retrieve the experimental photoluminescent emission. Hence, there is a need to identify an alternative model of the permittivity to enable the use of Kirchhoff law to compute photoluminescence.
Fig. 1. Maps of photoluminescence as a function of the energy and the parallel wave-vector, normalized by their maximum values: (a) Experimental photoluminescent emission of TDBC (b) Simulated emission using Kirchoff law and a Lorentzian model. This model is defined as follows: $\epsilon _{\mathrm {L}}(\omega ) = n_{\mathrm {0}}^2+\dfrac {f_{\mathrm {L}}}{E_{\mathrm {a}}^2-\omega ^2-i\omega \gamma _{\mathrm {L}}}$ with $f_{\mathrm {L}}=0.55$ $\mathrm {eV}^2$, $n_{\mathrm {0}}=1.65$, $E_{\mathrm {a}}=2.09$ eV and $\gamma _{\mathrm {L}}=0.027$ eV. These parameters were fitted from the experimental reflectivity map of Fig. 3(b). All data are normalized.
In this article, we show that the Brendel-Bormann model [54–56] can be used successfully. It corresponds to an infinite sum of Lorentz oscillators whose amplitudes are given by a Gaussian function centered on the transition frequency. If necessary, the model can be improved by adding other infinite sums of oscillators, centered at different frequencies.
To quantitatively assess the fundamental connection between the absorption and emission, we use the relation $\sigma _{em}(\omega )=\sigma _{abs}(\omega )\exp [\hbar (\mu -\omega )/k_B T]$ where $\sigma _{em}$ is the emission cross section, $\sigma _{abs}$ is the absorption cross section, and $\mu$ is the photon chemical potential which accounts for the pumping. The presence of the exponential factor $\exp (-\hbar \omega /k_B T)$ is responsible for the red shift of the emission spectrum known as Stokes shift. This relation has been derived by several authors for systems in equilibrium ($\mu =0$) and later extended to account for pumping (see e.g. [57,58] and references therein for a discussion of earlier contributions by Kennard, Stepanov, Neporent and McCumber). A related relation is the fluctuation-dissipation theorem which connects the power spectral density of the current fluctuations (i.e. the source of emission) to the imaginary part of the permittivity of the material. A generalized form of this theorem valid for a non-zero chemical potential has been reported in Ref. [59]. Knowing the absorption cross section, it is possible in principle to use this relation to compute the emission cross section and retrieve the experimental Stokes shift. This relation can be used as a simple test of a permittivity model. To proceed, we write the absorptivity coefficient using either the refractive index $n$, $\kappa _{abs} = 2\mathrm {Im}(k)=2\mathrm {Im}(n)\omega /c$ or using the absorption cross section: $\kappa _{abs} = N\sigma _{abs}$ where $N$ is the number of fluorophores per unit volume. Therefore, we obtain that $\sigma _{abs} \propto \mathrm {Im}(n)$. The spectra of Fig. 2 were obtained in this way, for a Brendel-Bormann model (Fig. 2(a)) or for a Lorentzian model (Fig. 2(b)).
Fig. 2. Emission and absorption cross sections for TDBC, computed with (a) the Brendel-Bormann or (b) the Lorentzian model of permittivity. The Brendel-Bormann model is defined in Eq. (5) and the associated parameters are gathered in Table 1. The Lorentzian model is defined in Fig. 1. We observe that the Brendel-Bormann model reproduces properly the Stokes shift between absorption and emission whereas the Lorentzian model fails to properly reproduce the spectral behavior of the emission cross section. All data are normalized.
This figure shows that the Brendel-Bormann model reproduces the spectral dependance and Stokes shift that was already observed experimentally by several groups [40,60]. It shows that the Lorentzian model cannot be used to compute the photoluminescent emission. We will show in this article the procedure to obtain the parameters of the Brendel-Bormann model in order to compute with precision the photoluminescent emission. We will also discuss why this model is well-suited for photoluminescence calculations.
2. Experimental measurements
The fluorophore considered in this article is a J-aggregated dye (TDBC) deposited on a thin ($50\,$nm) continuous silver film thermally evaporated on a glass substrate. A highly concentrated ($8.75\,{10}^{-3}\,$mol.L$^{-1}$) aqueous solution of TDBC is spin coated on a silver film to form a 17 nm-thick continuous layer. The sample is investigated through photoluminescence and reflectometry measurements from the glass side as shown in Fig. 3(a). For this purpose, the sample sits on an inverted microscope. The objective’s back-focal plane is imaged onto the slit of a spectrometer coupled to an EMCCD camera. The resulting images are resolved both in energy and in-plane wavevector [61]. For reflectometry measurements, white thermal light is focused on the sample through the objective. For photoluminescence measurements, a 445 nm laser is used. A polarizer is inserted on the detection path to select the Transverse Magnetic (TM) polarization of the signal.
Fig. 3. Reflectivity maps of a 17-nm-thick layer of TDBC on a 50-nm-thick layer of silver: (a) Structure of the sample illuminated from glass in Kretschmann geometry, leading to a surface plasmon excitation (b) Experimental data, (c) Simulated reflectivity map for an anisotropic model of TDBC permittivity and (d) Simulated reflectivity map for an isotropic model of TDBC permittivity. Simulation realized with a Brendel-Bormann model, after optimization on photoluminescence. All data gathered in Table 1.
An example of reflectivity map is given in Fig. 3(b). It is seen that there is a dip due to the resonant excitation of a surface plasmon at the interface metal/TDBC. The presence of a gap is the signature of strong coupling between the surface plasmon and the TDBC so that polaritons are formed. An example of photoluminescence map is given in Fig. 1(a) where the map displays emission peaks at the same frequency and angular dependence than the reflectivity dip. This points to a mechanism of emission assisted by the polaritons. First, TDBC molecules relax by emitting a polariton. The leakage of the polariton into glass generates the observed photoluminescence. We note that the upper polariton is clearly seen in reflection but hardly seen in photoluminescence. We will explain this feature in the next section.
3. Data analysis
3.1 Methods
3.1.1 Cost function
The purpose of the paper is to obtain an accurate model of the permittivity $\epsilon _{\mathrm {model}}(p_{1},p_{2},\ldots )$ of TDBC by fitting its parameters $p_{1},p_{2},\ldots$ of the permittivity model. We aim at fitting theoretically the experimental emitted power $P_{\mathrm {em}}^{\mathrm {exp}}(\theta,\lambda )$ such that:
3.1.2 Kirchhoff law
The theoretical emitted power $P_{\mathrm {em}}^{\mathrm {th}}$ is computed using the local Kirchhoff law [62]. It states that the emitted power $dP_{e}$ at a wavelength $\lambda$, emitted by a layer of thermalized emitters in the direction $\mathbf {u}$ ( i.e. for a given angle θ) and a given polarization state $l$ can be cast in the form:
The model accounts for the observed correlation between the minimum of the reflectivity and the maximum of emission. Indeed, for an opaque body, an increased absorptivity due to surface waves excitation leads to a reduced reflectivity. Hence, the model accounts for enhanced emission due to the excitation of surface plasmons through the absorptivity term. The model also explains why the upper polariton is observed in reflectivity but hardly seen in photoluminescence. The key difference between both polaritons in emission is due to the Wien term $\exp (-\hbar \omega /k_B T)$ so that emission at high frequencies is reduced. Just like the Stokes shift, this difference can be seen as a consequence of the exponentially decaying occupation of excited states for a thermalized system.
3.1.3 Brendel-Bormann model: a suitable model to account for the photoluminescence
It is necessary to be able to compute the absorption in the emitting layer in order to obtain the emitted power, and consequently to have an appropriate dielectric permittivity model. As shown in Fig. 1 or in Fig. 2(b), a Lorentzian model, commonly used for dye molecules such as TDBC, is not able to reproduce the experimental photoemission. In particular, it is clearly seen in Fig. 2(b) that the imaginary part of the permittivity should decay much faster than for a Lorentzian at large wavelengths. In this article, we use the Brendel-Bormann model [54–56] for the dielectric permittivity:
with $X_{\mathrm {k}}$ an infinite sum of Lorentz oscillators whose amplitudes are given by a Gaussian function:In order to compute it numerically, we use the formalism and the formula defined in [55].
This model provides a function of the complex variable $\omega$ which is analytical in a complex half-space so that it satisfies Kramers-Kronig relations by construction. The Gaussian distribution of oscillators with frequency $x$ and width $\nu _{\tau k}$ is a model for the distribution of transitions between the rovibrational states of two electronic levels. As a consequence of the distribution of oscillators accounted by the Gaussian function, any incident frequency $\omega$ is resonant with only a small fraction $\nu _{\tau \,k}/\sigma _k$ of the transitions, so that the imaginary part can be drastically reduced as compared with a simple oscillator. In other words, a Lorentzian model accounts for the homogeneous broadening of a single transition between two states, whereas the Brendel-Bormann model accounts for the inhomogeneous broadening due to a large number of narrow transitions between two bands.
3.1.4 Data processing
Before performing the optimization procedure, we first filter some noise on the experimental photoluminescence map data. First of all, the signal of the emission map is convoluted with a 3-by-3 matrix of ones. Secondly, the background contribution is evaluated by averaging the value of the signal far from the emission area in a square between -30$^{\circ }$ and +30$^{\circ }$ and between 520 nm and 560 nm. This average value is then subtracted from the photoluminescence signal.
In the optimization procedure, the following assumptions are made. Firstly, the medium, the chemical potential $\mu$ and the temperature $T$ are all considered to be homogeneous within the entire volume of the layer. Secondly, Wien’s approximation is valid at $300$K: $\exp (\frac {\hbar \omega -\mu }{k_{B}T}) >> 1$. With these assumptions, and after normalization by the maximum emitted power at $\lambda _{0}$, Eq. (3) can be simplified:
To model our system, we can first choose the permittivity to be isotropic. The photoluminescence map therefore displays an additional mode in the gap [63] in contrast with the experimental data (Fig. 3(b,d)). However, the TDBC displays some anisotropy, due to the formation of oriented aggregates when packing TDBC molecules together. To account for the anisotropy, we introduce an additional term of $\epsilon$ in the perpendicular direction of the layer (denoted $\epsilon _{\perp }$) as an additional parameter of the cost function $f_{\mathrm {cost}}(\epsilon )$. With this feature included in our model, we retrieve the two branches of the experimental data (Fig. 3(b,c)).
The silver layer is described by a Drude model:
4. Results
4.1 Comparison of experimental and theoretical photoluminescence map
The experimental emission was correctly reproduced by the simulation as can be seen in Fig. 4 using $m=3$ (see Eq. (4)). Both the experimental data and the simulations are obtained in the TM configuration. Note that the upper polariton branch in the experimental photoluminescence map fades into the detector noise. As a consequence, the optimization procedure is not completely able to assess its quantitative contribution to the permittivity. This explains the slight discrepancies between experimental and simulated data regarding the upper polariton (Fig. 3(b,c)).
The parameters of the refractive index model are shown in Table 1 and the associated permittivity and refractive index are plotted as a function of $\lambda$ in Fig. 5. The results reported in Table 1 show that Lorentzian lines of the Brendel-Bormann model are much narrower than Gaussian lines ($\nu _{\mathrm {\tau k}} \ll \sigma _{\mathrm {k}}$) as expected for inhomogeneous broadening due to a large number of narrow transitions between rovibrational levels.
Fig. 4. Maps of photoluminescence as a function of the energy and the parallel wave-vector, normalized by their maximum values: (a) Experimental photoluminescent emission of TDBC and (b) simulated emission after optimization of the Brendel-Bormann model. All data are normalized by their maximum values.
Table 1. Brendel-Bormann model parameter values
Fig. 5. (a) The complex refractive index n of TDBC obtained with a Brendel-Bormann model, (b) the complex dielectric permittivity $\epsilon$ of TDBC and (c) the imaginary part of the complex dielectric permittivity of TDBC obtained with a Brendel-Bormann model (red dashed line) or with a Lorentzian model (brown plain line) compared to the Planck function $\mathrm {I_{BB}}$ (blue dotted line).
When comparing with the literature on J-aggregated TDBC in solution [60,64], we observe a blue shift and a broadening of the emission and absorption spectra (Fig. 2(a)). As shown in [60], the monomer TDBC spectra are broader and strongly blue-shifted. An intermediate situation, similar to our observation, arises when the J-aggregates interact with their environment [64].
4.2 Discussion
As said in the introduction, it is possible in principle to compute the emission cross section from the absorption cross section using $\sigma _{em}(\omega )=\sigma _{abs}(\omega )\exp [\hbar (\mu -\omega )/k_B T]$ and then retrieve the experimental Stokes shift. Yet, the model of permittivity has to be extremely accurate at the low energy tail of the absorption spectrum to capture correctly the decay of the absorption spectrum. While this tiny absorption does not affect reflectivity measurements, it becomes critical when computing photoluminescence as it is multiplied by an exponentially varying term $\exp (-\hbar \omega /k_B T)$. If we consider an emitting system at temperature $300\,$K ($25\,$meV) with an emission region whose spectral width is on the order of $0.1$ eV, the factor $\exp (-\hbar \omega /k_B T)$ varies from 1 to $e^4\approx 55$ in this band. The Brendel-Bormann model precisely enables to account for a fast decay of the imaginary part of the permittivity at long wavelengths which can compensate the exponential increase of the Planck function. To illustrate quantitatively this behavior, we use a logarithmic scale to plot in Fig. 5(c) the imaginary part of the permittivity when using either the Lorentzian model or the Brendel-Bormann model. We also plot the Planck function. It is clearly seen that the slow decay of the Lorentzian model cannot compensate the increase of Planck function resulting in the unphysical divergence that is shown in Fig. 1 and Fig. 2.
5. Conclusion
We have shown that it is possible to model quantitatively the photoluminescence of TDBC mediated by surface plasmons. This is a very good example of light emission by an ensemble of emitters interacting with a resonant surface. The interplay between the two systems is accounted for by the local Kirchhoff law [62] which introduces the absorptivity in the TDBC layer. We have shown that simple permittivity models fail to enable the photoluminescence calculation using Kirchhoff. Using the Brendel-Bormann model, the parameters of the model were found by fitting the experimental emission. This model captures accurately the decay of the absorption spectrum at long wavelengths so that it can be used to compute light emission by fluorophores. These results pave the way to the design of complex metasurfaces to control light emission, creating novel light sources with new properties.
Funding
H2020 European Research Council (754387, Marie Sklodowska-Curie 754387); Agence Nationale de la Recherche (ANR-17-CE24-0046, ANR-18-CE30-0014); Labex (ANR-10-IDEX-0001-02 PSL*, ANR-10-LABX-24).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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