A Novel Analysis for Light Patterns in Nano Structures

Nano antennas have a significant role in photonic nanodevices due to the ability of concentrating radiation in a small space region. The confined fields present a high sensitivity to the used materials in that regions. The electromagnetic wave models of nanoantennas usually operate in the frequency domain. However, electromagnetic wave models in the time domain are just as important and more advantageous, for example, in light pulses propagation. In this article, a new stochastic method based on the ray model of light in absorbing media is presented and light patterns on a target near a nanoantenna are obtained. The optical properties of the materials are described by their complex refractive index. When dealing with photons in the time domain, this formulation allows the calculation of the probability of a given photon movement. Different patterns, obtained with the new method, are compared with a Finite Element Tool ones, leading to model validation. The simulated structure included an aperture nanoantenna. With this method, the results show the light confinement and the extraordinary optical transmission phenomenon, meaning some photons on the target are refracted and re-transmitted by the metal. Thus, the output beam is more intense than the transmitted directly by the apertures.

phenomenon at a nano scale, which result from the coupling between light and surface waves, such as surface plasmon [3]. Several models, namely on frequency domain, that explain this behaviour have been developed with applications in sensor or filter design [4], [5].
For this reason, time-domain models have been a bit neglected. However, time-domain analysis offers some advantages in comparison with the frequency/spectral analysis. First, with device miniaturisation, the emitted power is also an important specification: one must consider pulse response instead of a continuous emission as applied in frequency analysis. If the emitted power is sufficiently low a particle-like model can be applied. Another interesting aspect is the pattern analysis in comparison with a spectral analysis. Light patterns on a certain target present some important properties that may allow us to design better sensors, detectors or even filters [1], [2], [4], [5], [6].
Then, modelling these devices' behaviour on the time-domain and for low intensities is an alternative approach which might be promising. In this article, a method is proposed to obtain light patterns on a target. It is based on a single-particle methodology, allowing us to obtain patterns on targets even near the nano structures, in this case, nano arrays or antennas. Based on this methodology, it is intended to quantify phenomena as EOT, namely, to understand the radiation that comes from the metal. It is a semi-classical approach associated with some wave optics and geometric optics concepts. This methodology was already validated on [1] for a single interface and on [2] for multiple interfaces. The aforementioned patterns, generated by multiple interfaces and complex structures, are compared and validated with electromagnetic profiles.

II. METHODOLOGY
The stochastic model presented in this article is based on inhomogeneous plane waves on absorbing media. A wave is define as inhomogeneous when there is no coincidence between the normal to the plane of constant phase,ê, and the normal to the plane of constant amplitude,ĝ [6], [7], [8], [9]. It is also known that the incident planes ofê andĝ may be different and distanced by an angle φ [1], [2]. There is an angle θ representative of real component of the wavevector (normal to the equi phase planes) and a different angle ψ responsible for its imaginary part (normal to the equi amplitude planes). Both angles are referenced with the surface normal vector.
For media characterised by a complex refractive index, the wavevector is also complex and then these media may support This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ inhomogeneous waves, as presented on expressions 1 and 2, beingŝ a vector perpendicular to the interface. The apparent refractive index components, N and K, can be defined using expressions 3, where n and k are respectively the real and imaginary parts of the complex refractive index. Based on that, expression 4 is obtained, considering cos(α) =ê ·ĝ. The value of K may be determined using one of the expressions of 3, based on the already determined N [1], [2], [6], [7].
On the interface between two media, a photon will interact with the interface and its directions may change.
That interaction leads to the reflection or refraction of that photon. On both cases the complex components of k must be conserved, due to boundary conditions, leading to expression 5 [6], [7]. From this relation it is also determined the relation for the angles φ (angle between the incident planes ofê andĝ), such that φ i = φ r = φ t [1], [2], [6], [7].
Thus, if the incident wave is a homogeneous wave, where the planes of constant phase and amplitude coincide, φ i = 0 − → α = θ − ψ. Then, this relation will be valid along all the simulation, since φ will be null in every interface [1], [6], [7]. s ∧k i =ŝ ∧k r =ŝ ∧k t (5) This relation leads to θ r = θ i and ψ r = ψ i , for the reflection case.
For the refraction scenario, the complex Snell's law should be used, as presented on expression 6. However, N t and K t are unknown parameters that must be determined in order to compute θ t and ψ t . Expression 7 may be obtained to compute N t from the incident wave parameters [7], where N s = N i sin(θ i ), K s = K i sin(ψ i ) and φ i will be null. After obtaining N t , K t must be calculated from expression 3 [1], [2], [6], [7].
Fresnel reflection and transmission coefficients are obtained from the boundary conditions.
In this approach, photons are used. Since photons represent a quantum of light, they can not be divided and consequently on the interface they will be either reflected or refracted. Using Fresnel coefficients it is possible to calculate the probabilities of being reflected or refracted. In expression 8 are presented the electromagnetic field coefficient for PM and PE waves (parallel magnetic and parallel electric waves) [1], [2], [6], [7], [8]. Furthermore, on expression 9 the reflection power coefficients are presented.
The transmission field coefficient may be determined by expression 10, as well as the transmission power coefficients by expression 11.
Usually, the determination of R and T is enough to understand what happens at an interface. However, in absorbing media (namely high absorbing media) other coefficients might be necessary to account for the phenomena. The correct expression is R + T = 1 + M , where M is the interference coefficient defined by expression 12. Surface conductivity is neglected in this analysis, and consequently there is no surface Joule-heat-loss (surface absorption) [1], [2], [6].
Then, the probability of being reflected at the interface is P ref lected = R and the probability of being refracted is P ref racted = T − M . The value R + T must be higher than 1, depending on the M coefficient values. However, they are complementary phenomena, since P ref lected + P ref racted = 1, meaning that M acts as a correction term [1], [2], [10].
The absorption (media) ratio might be obtained using Lambert-Beer law, however it is known that the absorption coefficient is dependent on the wave parameters, namely on the transmitted angles and on the incident angles [1], [2], [7]. This statement reveals that Lambert-Beer's law is only valid for homogeneous waves, i.e., when the propagated wave has its plane of constant phase coincident with its plane of constant amplitude.
Thus, the absorption ratio is better computed according to expression 13, where d is the propagated distance. This expression is similar to Lambert-Beer law and expression 13 tends to that law for α = 0 − → K = κ [1], [2].
In the case of TE waves the propagation direction, η T E coincides with the direction given by θ. On the other hand, the TM waves propagation direction will be dependent on the electrical permittivity phase, δ , and on the phase of the wavevector component perpendicular to the surface, δ k . The relation is presented on expression 14 [2], [11].
Although the propagation direction might be different, the angles of the wavevector remain the same. The boundary conditions are always dependent on θ and ψ, even that η = θ [2], [11]. For non absorbing media (δ = 0) η T M = θ.

A. Method Validation
To implement this method, a Python script is developed using the above methodology and expressions for a 2D analysis. On a metal film of thickness t, there are two equal slits of width d, spaced by s, as illustrated in Fig. 1. In order to have a more complex and real model, N emitters are placed at a distance d e to the metal surface. They emit a particle every δ degrees, each of them characterised as an inhomogeneous plane wave for a given direction and in a time interval Δt e . Then, each emitter generates a cylindrical wave (all particles are emitted at the same time). A target is placed at a distance D from the metal.
The method validation includes a set of experiments designed to validate the methodology proposed.
The first experiment starts with a double slit set-up, with slits widths of the order of the incident wavelength. The second experiment refers also to the same two-slit configuration, but, the slits widths are ten times smaller than the incident wavelength. On both tests, the used wavelength is 1550 nm quite far from the plasma resonances, as presented on [1]. The third test is identical to the second one, except on the operating wavelength, which is 550 nm (near the resonance). A huge increase on the number of photons on the target, is expected due to the EOT phenomenon, although the pattern might be identical. For the fourth set-up, the number of slits is increased from two to four in order to assess the differences in results. The fourth test is also identical to the second one, but with four slits instead of two. The fifth and last test allows us to confirm EOT phenomena, since a four-slit structure is considered, for a wavelength of 550nm (near the resonance), and with a thickness of 30nm, expecting to have even more photons on the pattern [1]. Table I sums up the considerations for these five tests.
A space resolution of 1 nm is defined, which leads to a time resolution of 3.3 fs, considering propagation in freespace. When considering the propagation in other materials, the time interval between photon movements is equal to the time resolution, 3.3 fs, being the propagation distance adjusted based on the materials' group velocity. For all the configurations, the emission angle δ is 0.1 • and in order to introduce more complexity on the structure to test this new approach, a metal extension of s a = 1000 nm is added in each side.
An application to simulate the same structure, is developed in a Finite Element Tool, COMSOL Multiphysics. In this case, the results are obtained applying a Finite Element method on Maxwell's equations in the time domain. A TM pulse emulates a plane wave with the same duration as on the developed tool. All the other external boundary conditions are defined as scattering boundary conditions in order to absorb wave components propagating perpendicular to them, being transparent only to its parallel components.
In the developed tool, a total of 250 cylindrical emitters are considered, each one emitting 720 particles (one every δ = 0.25 • ). The convergence of both methods must improve when the number of emitters and the number of emitted particles per emitter increases, since the emission conditions will tend to the emission of a plane wave, under the Huygens-Fresnel principle.
The Rakic's Drude-Lorentz model for the gold electrical relative permittivity is assumed, whose parameters are obtained from [12]. At λ = 1550 nm the gold electrical relative permittivity is characterised by gold = −96.957 + j11.504 while air = 1 + j0 is assumed for air. Other metals could be used to replace gold, for instance, aluminium, silver or copper. Although copper and aluminium could bring cost advantages, gold chemical stability is the highest. Another relevant aspect is the differences in the SPP resonances in each of these five materials, which will influence the nanoantennas transmission and, consequently, the obtained pattern. [1].
In Fig. 2 the results from the developed tool on the first configuration are presented, in blue. In the same figure, the red curve represents the results taken from the Finite Element Tool analysis. Both results consider all the particles or waves that pass on that position in the same time interval. In the developed tool that time is defined so that there is no active particle, i.e. all photons were absorbed or are already out of boundaries. The   Fig. 3. Then, it is possible to conclude that small slits transmit less photons. However, the obtained results differ from those predicted by classical theories, i.e., it does not follow the rule (d/λ) 4 . On the other hand, it is also verified that some photons are transmitted and pass from the metal, 1 on the pattern of Fig. 2 and 2 in Fig. 3. In these conditions, metal might reflect almost all the radiation.
The differences between these methods are expected since our model is stochastic and presents discrete results. However, intensity profiles are identical and running the model it is possible to verify that by increasing the photon number (decreasing δ, or increasing the number of emitters) the particle profile will tend to the electromagnetic result. Spatial and time resolution of the developed model are also important to approximate both results and obtain more precise results. It is possible to recognise the electromagnetic profile on the obtained results, following the more stressed areas (as verified in Figs. 2 and 3).
Another configuration of interest is the same as the second one. However, the emission takes place at λ = 550 nm, where gold is characterised by gold = −5.371 + j2.360. Using this relative permittivity, the gold reflectivity is lower than the previous one. As a consequence, the pattern illustrated in Fig. 4 has  less photons (12807 on total) and the maximum corresponds only to 27 photons. Nevertheless, as expected, the number of photons refracted by the metal increases, since the photons' wavelength is close to a resonance region. In this case a total of 13 from 12807 incident are transmitted by the metal. This configuration is equal to the second one, however the photons wavelength changes, leading to EOT photons on the target.
In order to better assess the validity of the developed method the next (fourth) configuration will be presented, which is identical to the second one. The pattern on the target is presented in Fig. 5, both for the developed tool (blue) and using the Finite Element Tool (red). Once again, the convergence of results from both tools is verified. The maximum number corresponds to 23 photons and the sum of photons on the pattern is 14986. Moreover, it is also observed that some photons pass through the metal, 14 of the previously mentioned 14986.
For λ = 550 nm the gold thickness is reduced to 30nm, since according to the previous results published on [1] it is an excellent thickness to increase the number of photons on the target. The propagation length computed on [1] for these conditions allows radiation/photons' propagation until other interfaces are reached, meaning that there will be a certain outcome probability. In Fig. 6 are the obtained results. In this case, there are 32744 photons on the target, of which 5929 (18.11%) come through the metal. The maximum of this pattern represents 32 photons, and consequently the comparison with previous figures should be done carefully. Comparing both figures, it is possible to verify that the pattern has more peaks than the previous one and that the intensity will be higher almost everywhere, including on the edges (on the metal extension s a ).

B. Discussion
Geometry or ray optics assumes that rays propagate along a straight-line optical path in a homogeneous medium [7], [8].
An ordinary ray optics model cannot predict interference and diffraction scenarios. However, by analysing not the rays but particles, and including phase variation by using a complex refractive index in the calculations, it is possible to overcome the interference and diffraction problems. Different photons can be in the same spatial coordinate since they are bosons. When the number of involved particles is high enough, the particle and the wave approaches tend to have the same result. A single photon cannot produce a diffraction pattern or interfere with others and, for that reason, it is impossible to extrapolate a pattern using a small set of photons. Thus, the pattern will be improved when the number of photons increases, i.e., decreasing the angle δ, leading to a more continuous cylindrical wavefront or increasing the number of emitters.
Another aspect to note is that this method allows us to obtain excellent results even for sub-wavelength structures. Since it is a single-particle methodology, the results are discrete, and the resolution of the patterns is an important parameter. For this reason, the results are not as smooth as when obtained using the classical electromagnetic theory framework. However, increasing the number of photons, the patterns tend to become identical. On the other hand, Maxwell's equations are verified for the average results. By analysing the most stressed areas of the obtained patterns, it is possible to identify the pattern from the FEM Tool.
Moreover, this method does not neglect the role of the metal in the observed pattern. First, the observed pattern and respective time response vary with the incident wavelength, influenced by variations in the metal electrical permittivity. Then, for specific wavelengths and incident angles, there is some probability of a given photon being transmitted by the metal, leading to an extraordinary transmission in comparison with what is expected by classical theories (Kirchhoff's, Bethe's or Bouwkamp's theories). They assume that metals reflect all incident light, predicting a decrease proportional to (d/λ) 4 [3], [4], [5]. It is verified that even at large wavelengths there is a certain non-null probability to have a transmitted photon from the metal.
The last test is the most complex one, since the incident wavelength is near the gold plasma resonance and the metal thickness is of the order of the propagation length of the air-metal interface, as previously presented on [1]. The complexity of this test relies on the fact that the light pattern will be more influenced by photons passing through the metal. Using the proposed methodology and approach it is possible to quantify phenomena such as extraordinary optical transmission (EOT). In this case, it is possible to verify an increase in the number of photons passing through the metal from tens to thousands or tens of thousands, by adjusting the structure dimensions and by changing the incident wavelength.

IV. CONCLUSION
In this article, a model of light propagation in media with absorption based on the ray model of light is presented and applied not only to waves but also to photons. When considering photons, the developed tool is not deterministic but rather based on stochastic processes. The validation of the proposed model results from a time-domain analysis of structures with a light pulse excitation. The obtained light patterns can then be compared with other published results.
The considered structures in this work include an emitter of photons or waves, a metal film with slits and a target. The obtained results suggest that some photons in the target pattern come from the metal and not from the aperture. This reveals the phenomenon of extraordinary optical transmission (EOT) in the time domain, i.e., on the pattern, there are more photons than the photons transmitted by the aperture. This effect is due to the surface plasmon resonances in the illuminated metal, resulting from the variation of the metal's permittivity with the light wavelength and its incident angles.
It is also verified that different structural configurations result in patterns of various shapes and intensities. Thus, a methodology to design structures like optical sensors, photodetectors, and optical filters might be developed. The reduced computational effort is the a significant advantage of employing this tool instead of a finite element one. The proposed methodology and developed tool represent an excellent alternative, returning quite interesting and suitable results for small light intensities. This novel approach is useful to quantify phenomena as EOT.