The inverse problem for Surface Plasmon Resonance measurements  on a thin layer of aluminium in the Kretschmann configuration, is solved with a Particle Swarm Optimization method. The optical indexes as well as the geometrical parameters are found for the best fit of the experimental reflection coefficient in s and p polarization, for four samples, under three theoretical hypothesis on materials: the metal layer is pure, melted with its oxyde, or coated with oxyde. The influence of the thickness of the metal layer on its optical properties is then investigated.
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In 1959, the optical absorption in the Kretschmann configuration  due to the excitation of the surface plasma wave between the metal and air has first been seen and discussed by Turbadar , however, without relating this effect to surface plasma waves . Turbadar  published measurements of the reflectance by four aluminium layers deposited on a glass prism, in this setup. The aluminium films were deposited on glass by evaporation from tungsten loops after pre-fusing in vacuum. The pressure during evaporation was about 10−5 mm Hg. The evaporation time was not strictly controlled but was usually less than 30 seconds. The substrate, polish optical glass, was chemically cleaned and then cleaned by ionic bombardment immediately before evaporation. The substrate was made of a right angle prism, with the Al film deposited on its hypotenuse . The experimental results were compared to theoretical curves, but some discrepancies were found and never explained, especially for small thicknesses of aluminium. The sample was supposed to be a pure Aluminium layer above a glass prism, with bulk optical index.
We propose an explanation of these differences, by comparing the best fits of the experimental data, under the hypothesis of the possible oxydation of aluminium before optical measurements. The possible dependence of the optical index of aluminium with the thickness of the layer is also discussed. This inverse problem method is similar to that proposed in Ref. , but uses the Particle Swarm Method (PSO)  instead of the evolutionary method. For plasmonic applications, a recent benchmark has proved that the PSO could be more efficient than an evolutionary method, based on selection procedure [6, 7]. The PSO is used to retrieve the experimental parameters that were unknown in the original paper by Turbadar . The PSO and the models are described in the section 2, the results of the inverse problem procedure are proposed in the section 3 before a physical discussion in Sec. 4 and concluding remarks.
2. The inverse problem procedure
The inverse problem procedure consists in searching the best set of parameters for the models to fit the experimental data, with the Particle Swarm Optimization method. Multilayer models of the experiments are used in this inverse search procedure, with various hypothesis on the nature of the Aluminium deposition. We first describe the PSO algorithm and then we introduce the models.
2.1. The Particle Swarm Optimization method
In 1995, Kennedy and Eberhart proposed a new method for optimization problems . The PSO mimics the behavior of a swarm of bees in search of pollen, in a domain of search. The position of the pollen corresponds to the minimum of a fitness function, corresponding to the problem: the distance between the model and the experimental data from Turbadar  in the mean square sense. The fitness function F is the root of the sum of square of the sum of differences between the experimental R and the computed data Rt. In Ref.  were reported the measurement of the reflectance R of a thin Aluminium layer deposited on a glass prism. Measurements of R were recorded in both s and p polarizations, as functions of the angle of incidence. The fitness function is therefore
Any input of the model (or possible parameters) forms a vector x(t) which size depends on the degree of freedom of the model. This vector mimics a bee, which flies in the domain of search and therefore depends on a virtual time t. Each bee communicates good positions to each other and adjust its own position x(t) and velocity V(t) based on these best positions followingEquation 2 is used to calculate the particle new velocity using its previous velocity and the distances between its current position and its own best found position p(t) and the swarm global best g(t). Then the particle moves toward a new position following equation 3.
The success of PSO depends on the exogenous parameters c1, c2 and ω. Some general recommandations for these parameters and for the initialization of the algorithm was proposed recently . In this study, the acceleration coefficients are c1 = 0.738 and c2 = 1.51 and the inertial weight ω linearly decreases from 0.9 to 0.4. As mentioned in a previous study , the results are hardly dependent on these three last parameters. The convergence is reached if the relative distance between the bees is less than 1% in the hyper-espace of search. Hundred realizations of the same algorithm help to check the stability of the method and reveals that the number of iterations t is less than 300.
All numerical results have been verified by using evolutionary method based on evolution strategies described in [9–11]. This method involves random variations and selection in their respective search processes for the optimum. The evolutionary loop is repeated until the distance between the particles in the final population is lower than 1%. In each loop, pairs of elements of the initial population are randomly recombinated and then randomly mutated. Then the μ best elements of this population replaces the initial population. The size of the initial population is μ = 14 and of the secondary population is λ = 100 (after mutation). In the investigated cases, the performances of both method is about the same. Each realization takes less than a few seconds on a personal computer. This property is directly linked to the simplicity of the models.
2.2. The models
Two multilayer models are proposed to describe the experiments. Both are based on the computation of the reflectance of the metal layer in the Kretschmann configuration . All experimental data were obtained with an illumination wavelength λ0 = 550 nm and the medium above the metal layer was air (relative permittivity εAir=1) . Figure 1 illustrates the experimental setup, where the aluminium layer is deposited on a glass prism.
In the first model, only one metal layer is considered. In the second, one a two layers system is introduced, to be able to introduce aluminium oxyde above the surface of the metal.
2.2.1. Single layer
Figure 1(a) shows the multilayer model of the experiments. The fitness function F (Eq. 1) evaluation requires the computation of the reflectance in both polarizations (s) and (p). These generalized Fresnel coefficients are easy to calculate, assuming monochromatic incoming plane wave :Fig. 1). The coefficients depends on the polarization and on the permittivites of two successive media (Fig. 1): 13].
The following material hypothesis are successively investigated:
- H1a The layer is made of pure aluminium with bulk complex relative permittivity ε2 = εAl. The vector of unknown parameters is x(t) = (e, εV). This model was used in the reference paper .
- H1b The layer is made of pure aluminium with unknown complex relative permittivity εAl. The vector of unknown parameters is x(t) = (e, εV, εAl).
- H1c The layer is made of an homogeneous composite medium (HCM) of aluminium and aluminium oxyde Al2O3, with bulk relative permittivities εAl and εAl2O3. In the Bruggeman formalism, oxyde and metal are both supposed to be dispersed in the HCM . This formulation is also known in optics as the Geometrical Effective Medium Approximation (GEMA) which was developed for absorbing multilayer structures  The vector of unknown parameters is x(t) = (e,εV, fo), where fo is the fraction of aluminium oxyde, therefore fo ∈ [0; 1]. :
- H1d The layer is made of a mixing of pure aluminium and aluminium oxyde Al2O3, with respectively bulk relative permittivities εAl2O3 and εAl. The Maxwell-Garnet model  of inclusion is used:
The aluminium oxyde is considered as inclusions in the metal. The vector of unknown parameters is x(t) = (e,εV, fo).
Models H1c and H1d, and the two followings (H2a and H2b) are not directly related to the mode of elaboration, but to the properties of aluminium itself: its speed of corrosion is greater than 0.19 nm per day and can reach 4.6 nm per day in air . Therefore, if the reflectance measurement has been made in the few days following the deposition, oxyde could appear in the layer (hypothesis H1c and H1d), or on the aluminium layer (following hypothesis H2a and H2b).
2.2.2. Two layers
Figure 1(b) shows the multilayer model of the experiments. The theoretical reflectance can be written18] and extended in Ref. .
In this case, the aluminium oxyde is supposed to be an homogeneous layer of thickness e3, above the aluminium of thickness e2 with e2 + e3 = e. The following material hypothesis are successively investigated:
- H2a The bulk complex relative permittivities ε2 = εAl and ε3 = εAl2O3 can be used. The vector of unknown parameters is x(t) = (e2, e3, εV).
- H2b The complex relative permittivity ε2 = εAl is unknown as it depends on the mode of deposition and on the thickness of the layer. The vector of unknown parameters is x(t) = (e2, e3, εV, εAl = ε2).
Six models of the problem are proposed. The next section is devoted to their comparison in their capability in fitting the experimental data of Ref. . Their performance in the fitting is an indicator of their availability to describe the reality.
3. Numerical fit of the experimental data
The velocity of the particles in PSO (Eq. 2) is computed from uniform laws in a bounded domain of acceptable parameters. The domain of search is limited to the following wide intervals: , fo ∈ [0; 1], ei ∈ [0; 50] nm, e ∈ [0; 50] nm, ℜ(ε2) ∈ [−50; 50], and ℑ(ε2) ∈ [0; 50]. To ensure numerical stability, the reduction of Eq. 4 and 9 to common denominator is preferred before computation.
In computations, the imaginary part of the permittivity ε2 is only due to the contribution of aluminium. In models H1a, H1c, H1d, and H2a, the bulk value of the relative permittivities of Aluminium and Aluminium oxyde are used. The value given in Ref.  is εAl = (0.79 + 5.3ι)2 = −27.6 + 8.39ι, differs strongly from Palik’s one: εAl = (0.958 + 6.690ι)2 = −43.8 +12.8ι . The relative permittivity of aluminium oxide is also subject to variations: εAl2O3 = 1.772 = 3.13 , εAl2O3 = 1.72 = 2.89 . Therefore, computations using all four combinations of these results help to chose between these values. The best fits are obtained with εAl = (0.958 + 6.690ι)2 = −43.8 + 12.8ι and εAl2O3 = 1.772 = 3.13 .
Figures 2–5 are the superimposition of the experimental data retrieved from Ref.  (circles) and the computed reflectance from Eq. 4 and 9 (plain curves), for the best parameter set obtained with the models mentioned above.
The values obtained from the PSO least square fit of the experimental data seem to be in agreement but a more careful study is required to select the best model for each sample, by benching the fitness, the optical index of the prism and the thicknesses in Tab. 1.
Table 1 gives the best parameters for the four thicknesses h1 in the experiments : h1 ≈ 2.5 nm, 8 nm, 12.5 nm and 19 nm. The best fit is usually obtained for the minimum of the fitness function, but the corresponding thickness should also be realistic, and therefore close to hi. Our numerical study shows that the best geometrical and material parameters are highly sensitive to the model, and therefore, the agreement between the best and the indicative thickness hi is a criterion of quality of the model.
The value of the optical characteristic of the prism εV is critical to get the correct position of the slope change in the curves (between 40° and 50°) and therefore, the dispersion of the optical index of the prism is slightly dependent on the model. For example, Tab. 1 reveals that both the Geometrical Effective Medium Approximation (H1c) and the Maxwell-Garnett (H1d) models give comparable parameters but the best thickness from these two models are far from the indicative thickness hi excepted for the thinner sample. Therefore, a specific discussion seems to be necessary for each sample.
h1 ≈ 2.5 nm. The best optical index of the prism is the same for all models. The best fit is obtained apparently for the two layers models (H2a and H2b). However, for these models, the best thickness includes a thick layer of aluminium oxyde. Surprisingly the worse fitness function (H1c and H1d) corresponds to a thickness closest to the experimental estimation h1. For this model, the Bruggeman and the Maxwell-Garnett formalisms are used and therefore, the inclusion medium and the host medium are both supposed to be dispersed in an homogenized composite medium . Both models H1c and H1d give an effective relative permittivity of the medium which differs from the bulk one. This deviation can be explained by the presence of oxyde mixed in the metal layer. The fraction of aluminium oxyde is about 34% and therefore the real part of the effective index is greater than this of bulk. Moreover, the imaginary part of the optical index is half of this of bulk. The inclusion of oxyde modifies the absorption of the material and therefore decreases its electric conductivity. the behavior of the relative permittivity (H2b) supports this conclusion. For this very small thickness of aluminium, these results show that the hypothesis of oxydation is relevant.
h1 ≈ 8 nm. The variations of the fitness function are important. The best fit is obtained with the two layers model, leaving free the relative permittivity of gold. In this case, the thickness of aluminium is 5.2 nm and the thickness of Al2O3 is 3.2 nm. The best relative permittivity of Aluminium is close to the bulk one, but the best results are obtained for a free relative permittivity of aluminium (H1b and H2a). The corresponding relative permittivities could be compared to the bulk values also indicated in the Table . The real part of the result of the fit is close the Palik’s one, whereas the imaginary part is close to Turbadar’s one. The increase of the real part of the optical index can only be explained by a transition between the cases of the very small thickness (h1 ≈ 2.5 nm), where the effective medium is less metallic than pure aluminium.
h1 ≈ 12.5 nm. The best fits are obtained with the models H1b and H2b where the relative permittivity of aluminium is also determined by the computation. This case is close to the previous one. For this thickness, the oxydation consists therefore in a layer above the aluminium plate and again, the electric conductivity of the medium is more influenced than the dielectric part. The optical behavior of the medium tends to the bulk one.
h1 ≈ 19 nm. Whatever the model is, the fits of the experimental results have about the same quality: the values of F are close to each other. Therefore the quality of the fit is not sufficient to determine the best model and therefore to conclude on the material nature of the layer, even if the performance of H1a could suggest that the layer can be considered as pure aluminium. On the other hand, the computed thickness can be compared to the indicative hi and therefore, three models could be selected: H1a (the pure metal layer), H2a and H2b (the thickness of oxyde being negligible). The Maxwell-Garnett and the Bruggeman models are not acceptable for this thickness. Under H1d, the best fraction of oxyde is lower than 2%. Therefore, for the sample of thickness h1 ≈ 19 nm, the layer can be considered as pure aluminium. This result is confirmed by the performance of the simplest model H1a, where a single layer of Aluminium is considered, with the relative permittivity of bulk.
The oxydation of Aluminium seems to be a good explanation of the discrepancies between the experimental data and the original computed curves . The real part of the effective index of the layer is much greater than that of pure aluminum, for small thicknesses. If the thickness is greater than 8 nm, its behavior tends to that of the bulk metal. The imaginary part of the relative permittivity decreases drastically for small thickness. This last behavior is coherent with the hypothesis of inclusions of the oxyde in the metal which decrease its electric conductivity. The increase of the real part in the present case confirms that the possible cristalline structure of aluminum in thin layer cannot be invoked. This material property of nanostructures tends actually to decrease the real part of the optical index . Therefore, the hypothesis of aluminium oxydation is relevant.
4. Physical discussion
A perturbation approach and a modal approach are proposed to explain the results. The first one illustrates two different optical regimes as a function of the aluminium thickness. The second approach explains the shape of the reflectance, in the light of surface plasmon resonance.
4.1. Perturbation approach
The explanation of the dependence of the reflectance curves with the thickness can now be made. For simplicity, the single layer only is under consideration. The exponential fe = exp(2ιew2) in Eq. 4, in the range of incidence angles: Φg ∈ [25°,65°] governs the physical discussion. Table 2 gives the minima and the maxima of the real and imaginary parts of the function fe.
- The very small thickness (e = 2 nm). The first column in Tab. 2 shows that fe is close to 1 and the series of the reflectance can be written:
The zero order of the series is predominant and the shape of the curves are hardly dependent on the thickness e. The reflectance is characterized by a drastic change when the angle of illumination overpass the critical angle . In this case, the plasmon resonance effect is not dominant, even if the effective permittivity of the layer is that of a metal. DL0 is independent on the thickness but involves the relative permittivities of the substrate, the layer and the above medium.
- The small thickness (e > 20 nm). The last column in Tab. 2 shows that fe is close to 0. This case has been considered in a previous paper . The discussion presented here, makes the link between the resonance, and the series deduced from polynomial division of the numerator by the denominator in the reflectance formula. The reflectance can be approximated:
The zero order term does not able to describe the rapid change of slope of the reflectance in p polarization (the sign of the plasmon resonance), on the contrary of the higher order term of the series. This factor DL1 is the product of the Fresnel reflection coefficient of the interface 2 – 3 ( ) multiplied by . This last term can be expressed by the product of the Fresnel transmission coefficient of the interfaces 1 – 2 and 2 – 1. DL1 has a resonance behavior as shown in Fig. 6.
Figures 6 and 7 illustrate the closeness of DL0 (resp DL1) and the reflectance curves in (p) and (s) polarization, for very small (resp. small) thickness. The chosen parameters are typical of the results in Tab. 1 for a single layer. Even if the permittivity of the layer is characteristic of a metal in the case of very small thickness, the plasmon resonance cannot be observed. On the contrary, for a thickness around 20 nm, the plasmon can be observed. Consequently, the thickness plays a key role in the plasmon resonance.
Therefore, the thickness is a critical parameters for the surface plasmon resonance. In the case of nanometric thickness, the plasmon cannot be excited and the ultra thin layer modifies the reflectance in a quasi on-off manner. To generate a plasmon resonance, the thickness of aluminium should reach a given threshold, which depends on the deposition mode and on the material. In this case, the shape of the reflectance is strongly linked to the thickness of metal. A S matrix formulation can help to characterize the plasmon resonance.
4.2. The S matrix and the plasmon resonance
The plasmon resonance has been fully described by Otto in Ref.  and in . The theoretical dispersion relation of plasmons is commonly obtained without illumination, a linear system relating the amplitudes of the fields in all materials, being deduced. Therefore, this homogenous system admits non trivial solution if the determinant of the associated matrix is equal to zero [25–28]. This approach is the general application of a general law on resonances, which claims that it corresponds to obtain something from nothing. However, the properties of the S matrix in the plasmon resonance conditions have nevertheless rarely been specified even if this formalism is broadly used to describe resonances in quantum mechanics, acoustics. . . .
In electromagnetism, the scattering matrix characterizes the optical response of a system to the illumination. It relates the output fields to the input field. It has been introduced as the generalized Fresnel’s coefficient of the settlement . This coefficient can be calculated directly by using, for example, the polynomial formulation of reflection by stratified planar structures , or the recurrence relations proposed in Ref. . Il the case of a single metal layer between two dielectrics media, it can be easily deduced from the Fresnel’s coefficient, in (p) polarization25]. The solution of this equation is complex  and the resonance is not exactly excited. Consequently, the quality of the resonance can be illustrated by the vanishing of the real part and of the imaginary part of D(S−1) (Fig. 10–11). If D(S−1) is close to zero, the S matrix admits a pole, which is also known to be the mathematical expression of the resonance . In this case at least one of the eigenvalues of the matrix S−1 is close to zero, the determinant being the product of the eigenvalues.
Figure 10 shows that the real part of the determinant does not vanish. The change of slope of the reflectance curve is due to the vanishing of the imaginary part of D(S−1). This behavior explains that the shape of the curves is not characteristic of the plasmon resonance, on the contrary of the curves in Fig. 11, for which the real part and the imaginary part of D(S−1) vanish for about the same angle Φg.
In this paper, the retrieval of unknown experimental parameters in  has been done with the Particle Swarm Optimization method. The oxydation of the aluminium layer has helped to explain the obtained results, especially in the case of very thin layers. The shape of the reflectance curves in p polarization has been explained in the cases of very thin nanometric layers and of thin layers with thickness greater than about 10 nm. For very thin layer, the shape of the reflectance curve is hardly dependent on the thickness. In the case of thicker layer, the resonance behavior of the first order term of the series, assuming exp(2ιew2) close to 0, can explain the reflectance change of slope and the plasmon resonance. The determinant of the inverse of the S matrix helps to characterize the plasmon resonance. The oxydation of very thin layer could be an explanation of the shape of the reflectance in Kretschmann configuration. The proposed particle swarm method could be applied to thin layers made of other metals and to other inverse problem of experimental studies as phase recovering , sensors  or theoretical studies where the best parameters are searched [35, 36].
The first proposal to put to the front the results by Turbadar, and to explain them is due to Andreas Otto. The author wishes to thank Andreas for having provided the idea of this study, and for discussions on this stimulating subject. The ethics about the inclusion of authors according to their participation in the writing of the paper, led him to not accept to be co-author.
This work was supported by the “ Conseil Régional de Champagne Ardenne”, the “ Conseil Général de l’Aube” and the Nanoantenna European Project (FP7 Health-F5-2009-241818).
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