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Abstract
Sub-wavelength metallic grooves behave as Fabry-Perot nanocavities able to resonantly enhance the absorption of light as well as the intensity of the electromagnetic field. Here, with a one-mode analytical model, we investigate the effect of a correlated disorder on 1D groove arrays i.e., randomly shaped and positioned grooves on a metallic layer. We show that a jitter-based disorder leads to a redistribution of energy compared to the periodic case. In an extreme case, a periodic diffracting array can be converted into a highly scattering array (98% at λ = 2.8 µm with a 1 µm full width at half maximum). Eventually, we show that the optical response of combinations of variously shaped grooves can be well described by the individual sub-set behaviors.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Resonant nanostructures are used to tailor the optical response of large surfaces, and in particular to selectively enhance absorption or thermal emission [1]. For this purpose, Fabry-Perot nanocavities are among the most commonly studied nanostructures as they exhibit appealing features: angular tolerance, strong electromagnetic field enhancement, subwavelength dimensions, and possibility to tailor the spectral response on a large bandwidth [2–4]. These nanocavities, which are typically made of a metal-insulator-metal stack, can be either horizontal (often referred to as MIM nanoantenna) or vertical (grooves) and still exhibit the same optical behavior [5]. The fabrication of such groove arrays is rather challenging in comparison with MIM arrays [4,6,7], but we can make use of scalable techniques [8]. Besides, grooves can be combined in the same subwavelength period to tune the absorption on a wider spectral range [9].
These studies on groove resonators have been done with periodic boundary conditions, while other studies have introduced shape or position disorder in the case of other plasmonic resonators [10–16]. Considering the influence of disorder on resonators is of interest for several reasons. First, fabrication processes relying on colloids are very useful for large surfaces but cannot produce periodic arrays in most cases. Second, periodicity induces diffracted orders, which can be undesired in some applications such as thermal emission [17], optical filters [18], augmented reality [15], optical stealth [19]. Finally, it has been shown that position disorder can enhance some optical properties such as the resonance bandwidth or the balance between radiative and non-radiative losses [12,15]. However, the introduction of disorder increases the computational cost of simulating the optical response with classical electromagnetic solvers (modal methods, FEM, FDTD) [20,21].
In this article (in section 2), we introduce an analytical one-mode model [22] that reduces the computational burden of simulating random groove arrays. This model is used to study the effect of a correlated disorder, and it is shown in section 3 that disorder converts absorption/diffraction into scattering. We are able to design a structure that is completely scattering over a given spectral band. These results are then extended to a combination of two groove arrays that can be independently disordered in section 4, leading to custom spectral templates. We finally show that the behavior of such disordered combinations of grooves can be well described by the behavior of the subsets.
2. Analytical model
The model described hereafter is able to compute the optical response of a 1D-array of grooves (permittivity $\epsilon _d$) carved in a metallic layer (permittivity $\epsilon _m$). This type of structure allows to introduce simplifying assumptions that significantly decrease the computation time. The main assumption is the use of a one-mode formulation for the field within each groove. This has been previously used for 1D lamellar metallic gratings [22], and here we extend these results to multi-groove arrays whose positions and shapes can be randomly chosen. An illustration of such a structure is shown on Fig. 1. It is made of $n_G$ grooves, and each groove is defined by its position $x^{(j)}_{c}$, width $w^{(j)}$ and height $h^{(j)}$. This complex pattern is repeated with a period $L_x$. The light is incident on the structure with an angle $\theta$, a wavevector $k^{(0)}$ and a transverse magnetic polarization. In the following, the $e^{-i\omega t}$ time evolution term is not explicitly written, as all calculations are done for monochromatic fields. The grooves have subwavelength dimensions and do not support any propagative TE modes, so only the TM case is described in the following.
Fig. 1. Scheme of a periodic multi-groove structure (period $L_x$). Each groove is defined by its position $x^{(j)}_{c}$, width $w^{(j)}$ and height $h^{(j)}$. The grooves are carved in a metallic layer. $\epsilon _d$ and $\epsilon _m$ are the permittivities inside the groove and of the metal layer.
The expression of the magnetic field inside each groove (zone B of Fig. 1) is simplified thanks to a one-mode formulation [22], which is based on three assumptions:
Then, the propagative mode used to describe the field within groove $(j)$ is written as:The magnetic field in the upper medium (zone A of Fig. 1) is described using a Rayleigh expansion:
In order to compute the optical response of the groove structures, the reflected amplitudes $R_n$ are linked to the groove amplitudes $A_0^{(j)}$ and $B_0^{(j)}$ thanks to the continuity relations of the electromagnetic field at each interface. Besides, the wavevectors $k_{z, d}^{(j)}$ and $k_{x, d}^{(j)}$ can be expressed as a function of the groove widths and the permittivities $\epsilon _d$ and $\epsilon _m$.
2.1 Metal-insulator-metal groove wavevector
The metal-insulator-metal groove acts as a waveguide, whose wavevector must satisfy the round-trip condition along the $x$ axis:
where $r_{d, m}$ is the TM reflexion coefficient between the dielectric inside the cavity and the metallic walls. With the small width assumption of Eq. (1), Eq. (6) is simplified to:2.2 Resolution
At the groove apertures, the tangential electromagnetic fields ($H_y$ and $E_x$) are continuous, and the derivative $\frac {\partial _z H_{y} }{\epsilon }$ is also continuous since the Maxwell-Ampere equation leads to $E_{x}= \frac {\partial _z H_{y} }{\omega \epsilon }$.
Then, at the interface between a dielectric and a metal, a surface impedance condition is used [24], to avoid computing the field within the metal layer. This condition writes as follows:
with $\eta _A = \frac {\epsilon _A k_0}{i \sqrt {\epsilon _m}}$.The first step of the resolution is to express each amplitude of the Rayleigh expansion $R_n$ in Eq. (5) as a function of the amplitudes of the propagative mode in each groove $A_0^{(j)}$ and $B_0^{(j)}$. It is done by projecting the $H_{y, A}$ field on one reflected mode $n$. For this, the following notations are introduced: $\beta _{\pm }^{(n)} = 1 \pm \frac {ik_z^{(n)}}{\eta _A}$, $\alpha _{\pm }^{(j)} = 1 \pm \frac {ik_{z,d}^{(j)}\epsilon _A}{\eta _A\epsilon _d}$, $\gamma ^{(j)} = \frac {i k_{z,d}^{(j)} - \eta _B}{ik_{z,d}^{(j)} + \eta _B}$.
The following integrals are defined over each groove aperture $(j)$ on their width $w^{(j)}$:
Then, the surface impedance condition at the bottom of groove $(j)$ links the amplitude of the down-propagating mode to the up-propagating mode inside the groove:
Thus, projecting the $H_{y, A}$ field on mode $e^{-ik_x^{(n_0)}x}$ and using the interface conditions gives:The second step of the resolution is to integrate the interface conditions over the aperture of groove $(j_0)$, to express a given groove mode amplitude $A_0^{(j_0)}$ as a function of the Rayleigh amplitude $R_n$:
Injecting Eq. (15) into Eq. (16), gives a linear system on the amplitudes $A_0^{(j)}$ which can be written in matrix form:
with the following expressions for the matrix and vector coefficients:To solve this system, the sum on all the reflected modes is truncated at the $\pm N$ modes ($N$ should be large enough so that all propagative modes, i.e., modes so that $\mathfrak {Im}(k_z^{(n)}) = 0$, are computed). The values of both the groove amplitudes $A_0^{(j)}$ and the Rayleigh amplitudes $R_n$ are then found, and the overall reflectivity of the surface can be found as the sum of the energy in each propagative mode:
2.3 Numerical example
This model is used to compute the optical response of a single groove array and is compared to an exact Maxwell’s equations solver, the B-spline Modal Method (BMM) [21]. The groove has subwavelength dimensions with width $w^{(1)}=0.25$ µm, depth $h^{(1)}=3$ µm and is repeated with a period $L_x=6$ µm. Gold is used for the metallic parts and its permittivity is modelled following a Drude model in agreement with experimental data in the infrared range [25]: $\epsilon _{Au}(\lambda )=1-1/(\lambda _p / \lambda (\lambda _p/\lambda + i\gamma ))$ with $\lambda _p=159$ nm and $\gamma =0.0075$. Thus, the metal skin depth $d_m$ is of the order of 20 nm in the mid-IR range.
Figure 2(b) shows the specular reflection spectra computed using the analytical model (blue dashed curve) and the BMM tool (red curve) for the structure shown in the left half of Fig. 2(a). Light is incoming at a normal incidence and with TM-polarization. As expected, this structure exhibits Fabry-Perot resonances at the fundamental wavelength $\lambda _1 = 14.3$ µm and at the first odd harmonic $\lambda _3 = 4.65$ µm. The position and depth of the resonances are computed with a satisfying precision, which confirms that the optical response of the structure is mainly determined by the fundamental groove mode. Figure 2(c) shows the specular reflection spectra computed using the analytical model (blue dashed curve) and the BMM tool (red curve) for the structure shown in the right half of Fig. 2(a). The structure is much more complex than the one for Fig. 2(b), and its optical response is therefore much less straightforward. However, Fabry-Perot resonances corresponding to the deeper slits are visible at $\lambda =10.1$ µm, $\lambda =11.8$ µm and $\lambda = 13.3$ µm. Again, despite the complexity of this structure, both the analytical model and the BMM tool are in perfect agreement, which confirms that the optical response of the structure is mainly determined by the fundamental groove mode.
Fig. 2. Study of the specular reflection of two different periodic structures, i.e., a one-groove array and a five-groove array, as shown in (a). (b) Comparison between the specular reflection given by the analytical model and a BMM tool used as reference : The unit cell is $6$ µm large and consists in a single groove of dimensions $w=0.25$ µm and $h=3$ µm (c) Same comparison for a unit cell which is 10 µm large and consists in 5 grooves of different depths and widths, such that $w^{(j+1)} > w^{(j)}$ and $h^{(j+1)} > h^{(j)}$, with $w^{(1)} = 100$ nm, $w^{(5)} = 500$ nm, $h^{(1)} = 1$ µm, $h^{(5)} = 2.6$ µm.
This analytical model therefore gives accurate results in terms of reflection spectrum for groove arrays. Its main benefit is a large gain in computation time for more complex structures. Thus, with an 8 core CPU (Intel Xeon 3.6 GHz) and 16 GB RAM, computing the response of one groove at a given wavelength needs 22 ms (resp. 40 ms) with analytical model (resp. BMM), but if the number of grooves is increased to 20, the computation time becomes 3.1 s (resp. 18.1 s) with analytical model (resp. BMM). When computing the optical response of large, disordered arrays, this speed gain gets even more significant.
3. Single groove disorder
3.1 Jitter disorder for subwavelength and diffracting groove arrays
In this section, we apply this model to the study of disordered arrays. A given groove array is disordered by introducing a jitter, i.e., by randomly shifting the position of each groove within a given area. The Random Factor (RF) is defined as the breadth of the applied jitter, as shown in Fig. 3(a). Noteworthily, this type of disorder does not modify the groove density when the position shifts are drawn within a uniform distribution. The case $\mathrm {RF} = 0$ corresponds to the perfectly ordered case, and $\mathrm {RF} = L_x$ corresponds to a highly disordered case. It is the highest random factor so that the groove density is conserved. Besides, the random draws where two grooves are closer than the metal skin depth are discarded.
Fig. 3. Definition of both single- and bigroove random factors (RF). (a) For a single groove array, each groove is displaced by drawing randomly a new position within a uniform distribution of breadth RF and centered on the original groove position. (b) The same uniform random draw principle is applied to bigroove arrays, where in addition a different random factor is used for both groove types: RF$^{(1)}$ and RF$^{(2)}$
The analytical model relies on periodic boundary conditions, and in order to apply it to disordered arrays, only $m$ grooves are considered and boundary conditions are applied with a super-period $m L_x$. The choice of the $m$ does not affect the resulting behavior significantly if it is large enough, i.e., the periodic repetition of $m$ disordered grooves has the same properties as a fully disordered array. In the following, we chose $m=20$ as a good trade-off between disorder and computation time. Also, this super-period gives access to more reflected angles $\theta$, following the Rayleigh decomposition described in Eq. (5). Thus, the computed $R_n$ represent all possible directions for the reflected field, namely the specular reflection ($n=0$), diffracted orders of the ordered array ($n = mk, k \in \mathbb {Z}$) and scattering ($n \in [mk+1; (m+1)k - 1], k \in \mathbb {Z}$). This is summarized in the following variables:
Figure 4 shows the effect of a jitter disorder on the groove array previously studied in Fig. 2(a) for two different values of the period $L_x$ : the first is subwavelength (Fig. 4(a)) and the second is diffractive (Fig. 4(b)).
Fig. 4. Specular Reflection, Diffracted Orders, Scattering, and Absorption as a function of the Random Factor $\mathrm {RF}$. The computations are done at $\lambda _3 = 4.65$ µm for a groove of dimensions $h=3$ µm, $w=0.25$ µm and of (a) period $L_x = 3$ µm and (b) period $L_x = 6$ µm. The grooves are disordered by a uniform jitter. Solid lines correspond to averages and error bars indicate maximum and minimum values over 20 repetitions. Error bars are shown only on S to improve readability.
Here the Random Factor (RF) is progressively increased, starting from a periodic array ($\mathrm {RF}=0$). For each RF value, 20 draws are made. The solid lines are the mean values, and error bars indicate minimum and maximum values obtained on these 20 random structures. These error bars are shown only on S to improve readability, because S is our quantity of interest.
As the RF is increased, both the subwavelength and diffractive case see an increase in scattering ($S$). However, while for the diffractive structure it is the energy sent into the diffracted orders ($D$) which is redirected into scattering, for the subwavelength structure it is the absorption ($A$) which is redirected. In both cases, the specular reflection ($R$) is relatively unchanged. To summarize, disordering a surface does not affect significantly its specular reflection, but increases its scattering. In the diffractive case, this scattered energy comes from a diminution of the diffracted orders, while absorption is unchanged; in the subwavelength case, it is the diminution of absorption which balances the increase of scattering.
Though expected, as scattering is the disordered equivalent of diffraction, this result was not obvious due to the complex coupling mechanism between the groove modes and the incident field. It must also be emphasized that scattering is only observed on the resonance spectral bandwidth. Far from the resonance wavelength, the grooves are no longer interacting with the incoming light and due to their subwavelength dimensions, the surface behaves as a mirror. This resonant behavior clearly distinguishes this metasurface from an ordinary rough surface.
These results show that disordering a groove array enables the redistribution of the reflected energy from diffracted orders to scattering. In the next section, we study a highly diffractive structure and disorder it, in order to obtain a structure which converts the maximum amount of energy into scattering.
3.2 Highly scattering metasurface
Figure 5(a) shows the optical response of a highly diffractive structure with $w=4$ µm, $h=2$ µm and $L_x=8$ µm with a low quality factor resonance ($Q \simeq 3$). There is a high quality factor resonance in the middle of this resonance, which is due to a Rayleigh-Wood anomaly. However, this structure geometry does not satisfy the condition $w \ll \lambda$ on the spectral range of interest. In order to check the results of the analytical model, the specular and diffracted reflection spectra computed with BMM are also plotted on Fig. 5(a). These results are in fair agreement with those obtained by the analytical model, proving that the constraint on the groove width can, in some cases, be relaxed. Here, only the fundamental mode in the groove contributes to the resonant behavior. The other propagative modes in the groove are not contributing and can thus be neglected.
Fig. 5. (a) Comparison of the spectra computed by the analytical model and the BMM tool for the structure presented in inset, where the assumption $w \ll \lambda$ is violated. (b) Optical intensity distribution as a function of RF at $\lambda = 2.8$ µm, computed by the analytical model. The super-period consists in 20 grooves and the results are averaged on 20 iterations. (inset) The structure for which these results are computed, with $w=4$ µm, $h=2$ µm and $L_x=8$ µm.
Figure 5(b) shows the optical behavior when this array is disordered. In this case too, disordering the grooves converts nearly all the diffracted energy into scattered energy. Meanwhile, the specular reflection and absorption are very weak (below $2\%$). Therefore, when the Random Factor is large enough, the structure sends all the incident optical energy into scattering.
The spectral bandwidth of this fully scattering behavior is investigated in Fig. 6. It shows the evolution of the spectra on the 2 to $4$ µm range as the Random Factor is increased from 0 to 8 µm. The results are averaged on 20 random draws, and the minimum and maximum values are given as error bars.
Fig. 6. Study of the linewidth of the resonance described in Fig. 5, computed by the analytical model. As in this previous figure, the super-period consists in 20 grooves and the results are averaged on 20 draws. (a) RF = 0 µm, (b) RF = 2 µm, (c) RF = 4 µm, (d) RF = 8 µm
These results show that the scattering has a resonant behavior, so that its spectral shape is similar to that of the specular reflection. The full width at half maximum is nearly 1 µm. The error bars when $\mathrm {RF}=4$ µm are very large, which is due to the dispersion of the random draws, as some are close to the ordered structure while others are strongly disordered. The high-quality factor resonance at $2.66$ µm disappears as the RF is increased. This is expected since it was due to coupling between the groove mode and diffracted orders that no longer exist.
4. Bigroove disorder
4.1 Optical behavior
The previous section showed the optical response of ordered and disordered arrays of grooves. In this section, the optical response of bi-groove arrays, i.e., ordered or disordered arrangements of two different types of grooves, is discussed. The study of these arrays is the first step towards designing more complex structures whose reflection and scattering spectra are tunable on wider spectral ranges.
For this, Fig. 3(b) presents the first bi-groove array studied. It consists in a first groove of dimensions $w^{(1)}$ and $h^{(1)}$ and a second one of dimensions $w^{(2)}$ and $h^{(2)}$, both are equally spaced and repeated with a period $L_x$. The presence of two different types of grooves makes it possible to disorder one type more than the other, so we define RF$^{(1)}$ (resp. RF$^{(2)}$) the random factor of the first groove array (resp. second groove array), as shown in Fig. 3(b). The resulting structure consists of two independently disordered sub-arrays written $\{ w^{(1)}, h^{(1)}, \textrm {RF}^{(1)}\}$ and $\{ w^{(2)}, h^{(2)}, \textrm {RF}^{(2)}\}$. The first periodic sub-array ($\{ w^{(1)}, h^{(1)}, \textrm {RF}^{(1)}=0\}$) is identical to the one studied in Fig. 2(b).
Figure 7 shows the specular reflection and scattering spectra of the ordered and disordered bi-groove arrays. These spectra are computed by the analytical model for one single random draw.
Fig. 7. Spectra of Specular Reflection, Diffracted Orders and Scattering computed by the analytical model for ordered and disordered bigroove arrays of period $6$ µm and dimensions {$w^{(1)} = 0.25$ µm, $h^{(1)} = 3$ µm} and {$w^{(2)}=0.25$ µm, $h^{(1)} = 2$ µm}. (a) RF$^{(1)}$ = RF$^{(2)}$ = 0 (b) all disordered (c) RF$^{(2)}$ = 0 (d) RF$^{(1)}$ = 0
In the ordered case (Fig. 7(a)), the bigroove structure exhibits three resonances between 4 and 16 µm. The fundamental resonance at $\lambda _1^{(1)}=14.3$ µm and its harmonic resonance $\lambda _3^{(1)}=4.65$ µm correspond to those found in Fig. 2 for the array made of groove $\{ w^{(1)}, h^{(1)}\}$. The second groove ($\{ w^{(2)}, h^{(2)}\}$) also exhibits a fundamental resonance at $\lambda _1^{(2)}=9.78$ µm. This indicates that the grooves are behaving independently, as was previously shown [5]. When both sub-arrays are disordered (Fig. 7(b)), scattering is enhanced within the three resonance bandwidths, partially substituting for either absorption at $\lambda _1^{(1)}$ and $\lambda _1^{(2)}$ or diffraction at $\lambda _3^{(1)}$. In Fig. 7(c) (resp. Figure 7(d)), only the first (resp. second) sub-array is disordered, which leads to the scattering being enhanced only at the first groove resonances $\lambda _1^{(1)}$ and $\lambda _3^{(1)}$ (resp. second groove resonance $\lambda _1^{(2)}$). Interestingly, the disorder of one subarray does not affect the response of the other one. One intuitive explanation is based on both the independence of the grooves and the resonant nature of these phenomena. Thus, it is possible to design a custom structure that diffracts on a given spectral band, scatters on a second one and absorbs on yet another.
Moreover, the behavior of these disordered arrays seems to be directly resulting from the behavior of the individual groove arrays. Indeed, it has previously been shown that it was possible to express the absorption of a combination of two nanoresonators as a function of their individual responses [5]. For instance, the probability $(1-A^{(1,2)})$ for a photon to escape from absorption by the bigroove can be expressed as the product of the independent probabilities $(1-A^{(1)})$ and $(1-A^{(2)})$ to escape absorption by the single groove resonators taken separately. This leads to the following expression for absorption $A^{(1,2)}=1-(1-A^{(1)})(1-A^{(2)})$.
This expression can be extended to scattering and diffraction:
where $R^{(1)}$, $D^{(1)}$, $S^{(1)}$ (resp. $R^{(2)}$, $D^{(2)}$, $S^{(2)}$) are the specular reflection, diffraction and scattering of the first sub-array (resp. second sub-array).The spectra corresponding to Eqs. (26–28) (crosses) are also shown on Figs. 7(a-d), showing a very good agreement between the analytical model and the bigroove model.
When the resonances are spectrally closer, couplings between the grooves may appear and the Eqs. (26–28) may not be valid anymore. To illustrate this, a bigroove structure with close resonances is introduced in Fig. 8: $\{w^{(1)}=0.3$ µm, $h^{(1)}=2.2$ µm, $\textrm {RF}^{(1)}\}$, $\{w^{(2)}=0.17$ µm, $h^{(2)}=1.9$ µm, $\textrm {RF}^{(2)} \}$, repeated with a period $L_x=6$ µm.
Fig. 8. Spectra of reflection and diffraction computed by the analytical model for two single groove arrays with close resonances and the corresponding bigroove array: {$w^{(1)}=0.3$ µm, $h^{(1)}=2.2$ µm} and {$w^{(2)}=0.17$ µm, $h^{(2)}=1.9$ µm}, with a period $L_x=6$ µm. (a) Individual spectra with RF$^{(1)}$ = RF$^{(2)}$ = 0 (b) Individual disordered spectra (c) Product of individual spectra and bigroove spectrum with RF$^{(1)}$ = RF$^{(2)}$ = 0 (d) Product of individual disordered spectra and bigroove disordered spectrum.
Figures 8(a-b) shows the reflective and scattering spectra of the individual sub-arrays in the ordered and disordered cases. The first (resp. second) groove exhibits a resonance at $\lambda =10.6$ µm (resp. $\lambda =9.4$ µm). Figure 8(c) shows the spectra of the periodic bigroove array, and there is a fair agreement with the bigroove model, Eqs. (26)–28. When the array is disordered, discrepancies between the curves appear, as shown in Fig. 8(d). This is due to coupling that takes place between the different types of grooves, and is not taken into account in the product of the individual spectra.
5. Conclusion
We have developed a one-mode analytical model able to accurately describe the optical response of 1D-groove assemblies. Such a simplified model could be adapted to the analysis of other resonant nanostructures (whose optical behavior depends on the propagation of a single mode), and could also be extended to multimode nanostructures. With this model, we showed the redistribution of energy that occurs in correlatively disordered groove arrays. The energy which is either absorbed or sent into diffracted orders in the periodic case is converted into scattering, with an increasing efficiency as the disorder grows. Hence, a patterned metallic surface can become highly diffusive on a given spectral range. Besides, these results stand for a combination of two differently shaped grooves, whose optical response behave independently, thus allowing the design of custom spectral responses. Finally, we have shown that the behavior of disordered groove combinations can be straightforwardly described by the behavior of the independent subsets.
Funding
Direction Générale de l’Armement (Scholarship).
Acknowledgments
This work was supported by two DGA-AID scholarships
Disclosures
The authors declare no conflicts of interest.
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