1. Introduction

In the last years, great efforts have been addressed to develop infrared (IR) devices capable of capturing the dynamics of objects that heat and cool at high speed, due to its impact on medical, commercial, and military applications [1,2]. Today's thermal cameras are possible thanks to un-cooled thermo-resistive detectors known as micro-bolometers [3,4], which are the most sensitive devices able to sense the long (LWIR) and mid-infrared bands (MIR), and are based on quite simple materials with a high TCR coefficient [5]. In this way, amorphous silicon is the most commercialized material due to its low-cost [6], nevertheless a wide variety of new material are currently being investigated as novel candidates [7,8].

Despite the great advantages of micro-bolometers, limitations arise when their use as thermal detectors is extended to fast-frame-rate applications. These limitations are inherent to their large thermal mass that reduces their time of response to a few milliseconds [9,10]. At the light of these concerns, the idea of using micrometer-sized antennas emerged as a strategy to develop high-speed infrared devices, allowing this way, the integration or “loading” of nano-meter-scale bolometers with smaller thermal mass [11–13]. Analogous to the microwave antennas, micro-antennas capture the infrared radiation through resonant modes of current induced along their volume [14,18]. The use of a nano-bolometer as an antenna's load element allows the detection of this resonant current mode due to the ohmic losses that increase its temperature, thus defining the way for detecting infrared waves. This type of architecture will allow bringing the size of a bolometer to the nanometer-scale without reducing the collecting area of the final-device (related to the antennas’ effective area), resulting in shorter detection times (microseconds [15,16]).

However, should be underlined that the correct design of infrared systems based on micro-antenna arrays requires proper strategies to assess the arrays’ fundamental parameters, commonly obtained with the scattering properties under the illumination of an external source. These fundamental parameters concerns, first, the shortening of the wavelength in metals at optical frequencies, conducting to the scaling of the effective resonance-wavelength λ_eff of the individual antennas [17]. Relevant contributions have addressed this issue for a wide variety of antennas’ designs and materials [18–21]. Second, the significant changes of the micro-antennas’ scattering properties due to the strong dipolar interactions between them, developed when incorporated into an array [22,23]. Relevant contributions have unveiled well-defined short (evanescent) and long-range interaction regimes inducing, in either scenarios, the shift of resonances towards longer wavelengths [24–26]. Finally yet importantly, feeding the micro-antennas’ gap with nano-bolometer elements introduce an effective gap-capacitance, which changes the scattering properties of the array [27], conducting to an effective frequency of operation f_OP. The frequency of operation entirely depends on the impedance matching between the nano-load element and the antennas arrays; the better the impedance coupling between both elements, the better the match between the frequency of operation and frequencies of resonances of the micro-antennas (in the short-circuit configuration). Since nano-bolometers are necessary high-resistive materials (because its high TCR coefficient), we conduct efforts to address in a proper way this last issue. We underline that those valuable findings usually arise from reflectance (and absorbance) measurements and simulations performed on diluted systems of micro-antennas, from which the fundamental properties of the antennas are inferred in an indirect way.

In this work, we analyze (for the first time) the collective behavior of micro-antennas arrays from a fundamental point of view, which leverages the concepts of “collective input impedance” and “matching of nano-loading elements,” in order to obtain the collective resonances of the array in a proper manner [27–29]. The used perspective entirely differs from the optical approach, and introduces an accurate way to unveil the contrasting nature of the even and odd resonance modes of micro-antennas [30,31], and further, to understand their dependence on the dipolar interaction. At the light of these results, we subsequently address efforts to optimize a rectangular array of micro-dipole antennas for the efficient sensing of the thermal wavelengths (around λ = 10.6 µm), with special attention to the capacitive (and resistive) effect that bolometers introduce at the gap of the antenna’s array. The analysis is conducted at the aid of finite-element-simulations (COMSOL Multiphysics), software package that enable us to get, as underlined, one the most fundamental parameters of the micro-antennas by the use of port analysis. It is important to stress that despite the relevant contributions reporting on the impedance of nano-antennas, most of the times, the reports only concern the case of individual or single metal nano-antennas, or even more, the study is restricted to the fundamental mode (the odd mode) despite its low-resistance. The collective behavior of the odd and even mode is stills missing. It will be shown that tuning the behavior of the collective even mode becomes a relevant issue in the performance of nano-bolometers coupled to micro-antennas.

Following this introduction, we present in section (2) the numerical strategy used to obtain the input impedance of micro-antennas (individual and in array form), and hereupon evaluate their odd and even resonances modes in a proper manner. For comparison purposes, we also introduce a numerical methodology to obtain the absorbance spectra of the micro-antennas, and so, to indirectly assess to the resonances of devices from an optical point of view; all those numerical models built in COMSOL Multiphysics. In section (3), we discuss first the numerical impedance of single micro-antennas with stress aimed to understand the nature of the odd and even modes, entirely resistive and capacitive, respectively. We subsequently analyze the IR absorbance spectra to obtain the resonances by an optical approach (being there the frequencies where the absorbance is maximum), and thanks to their physical interpretation, we will draw a connection between electrical resonant modes and the optical resonances in the short and open-circuit configurations. In section (4), we embed the antennas into an infinite two-dimensional square array of antennas, described by the lattice parameters “a_x” and “a_y,” and evaluate (for the first time) the emerging collective-input impedance. Then, we discuss the contrasting shift behavior of the collective and even modes in terms of the longitudinal and transverse dipolar interaction; the analogy with the optical resonances in the short and open-circuit configurations is discussed. Finally, in section (5), we evaluate the effect on the array's performance that results from introducing nano-bolometers with a load resistance of Z₀= 500 Ω and we will find the optimal operating frequencies, entirely determined by the impedance coupling between the bolometer and the micro-antenna.

2. Methodology

To address the above issues, we consider the case of titanium micro-dipoles, individual and in array form, on the top of a silicon dioxide substrate (Fig. 1). We choose titanium because of its relatively short skin-depth and consider dipole-type micro-antennas (with length L, gap G, thickness T and width W) due to the intensity of its resonant modes [32,33]. However, this study can easily be extended to other relevant antenna designs, as well as other materials.

 

Fig. 1. (a) Schematic of a single titanium dipole, and (b) rectangular array of titanium dipoles, with lattice parameters a_x and a_y.

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The antennas systems here considered lie on the surface of a semi-infinite silicon dioxide wafer. The substrate is used to reduce the heat dissipation from the bolometer, as well as for electrical insulation [34]. Since the systems are on a dielectric-metal interface, their resonances appear at longer wavelengths than in the case of vacuum-suspended systems. The relationship ${\lambda _{res}} = \lambda _{res}^0/\sqrt {{\varepsilon _{eff}}} $, based an effective-medium model (${\varepsilon _{eff}} = 1 + {\varepsilon _{substrate}}/2$), provides the relationship between the resonances of both media [35,36]. For this study, we shall only be concerned with the dielectric-metal interface resonances.

The optical behavior of micro-dipoles in the infrared region (individual and in array form) is analyzed by numerically solving Maxwell's equations, in which we consider as source of illumination a normal-incident plane wave. The numerical solutions of the differential partial equations are obtained by using COMSOL Multiphysics, software package based on the finite-element-method. By using this method, the simulation space is split into a mesh with a large number of elements; while Maxwell's equations are solved exactly at the nodes, the solution is found by interpolation at all other points. To build in a proper manner the micro-dipole models, it is required the refractive index of materials at infrared wavelengths (titanium and silicon dioxide) reported in the literature [37,38]. It must be underlined that this study is conducted by using two separate optical models (described below) that make use of different boundary conditions for both, the individual and the array form of micro-antennas.

In the simulation of an individual antenna, we treat the sidewalls and the bottom surface of the simulation space as regions with ideal absorbance or perfect-matched layers surfaces (PML surfaces). The upper surface is used to generate the monochromatic plane-wave (of arbitrary amplitude E = 1 V/m) whose frequency is swept from 10 to 70 THz (∼30 to 4.3 µm). Once the model is ran, the infrared absorption of the individual antenna is obtained from numerical data by integrating the electromagnetic energy absorbed within the antenna's volume, as a function of the frequency.

In contrast, the absorbance A of the antennas within a periodic array is analyzed by making use of the Floquet Boundary Conditions (FBC), which allows the simulation of the large number of antennas contained in the array [39–41]. Under these conditions, the array can be modeled as a single rectangular cell containing a single antenna. The periodic Floquet conditions are defined on the side faces of the unit cell, the unit cell's top face is used to set a periodic port with monochromatic excitation of variable frequency (of arbitrary amplitude E = 1 V/m), and the unit cell's bottom face is used as a periodic reception port.

The reflectance R and transmittance T of the array (so, the absorbance) are found by performing an eigen-mode analysis of the two-port system [42,43]. Through the analysis of the two ports, one used for the input and the other for the output of the light, the numerical solver can generate the scattering parameters or S parameters [42,44]. The transmittance and reflectance of light are defined in terms of the S parameters throughout the relationships [44]:

Finally yet importantly, the input impedance Z_in is one of the most fundamental parameters of micro-antennas, accurately unveiling the nature of its resonant modes. This parameter is obtained here by driving the terminals of the antenna with an alternate voltage V_AC, and thereafter measuring the I_AC current flowing through the dipole terminals. In this way, the Z_in impedance is found through the V_AC/I_AC ratio. We use a lumped port element at the gap of the dipole to drive their arms with a sinusoidal voltage of amplitude V_AC = 1V; the port impedance is arbitrarily set to Z₀ = 50 Ω during the first part of the study. By sensing both, the amplitude and the phase of the current through the port, the software package performs a one-port analysis obtaining the micro-dipole impedance Z_in. Further, we take advantage of the port impedance as a manner to analyze the mismatch effects introduced by nano-load elements feeding the micro-dipole, such as nano-bolometers do. In this way, changing the impedance of the port allows us to determine how the resistance of nano-bolometers tunes the operating frequency of the devices. With this strategy, the return losses parameter S₁₁ indicates us directly the operating frequency of the system.

3. Results and discussion

3.1 Optical and electrical resonances of individual antennas

The power absorbed by an individual dipole (2 microns long, 100 nm width and 250 nm gap’s length) in the short circuit and open circuit configurations are shown as a function of the light’s frequency (Fig. 2(a)). The short-circuit configuration is easily obtained by covering the gap with the same antenna material, resulting in a perfectly matched device or continuous dipole. In the open-circuit configuration, the gap is simply deemed to be covered with air.

 

Fig. 2. (a) Optical absorption of an individual micro-dipole in both, short and open circuit configuration; the frequencies of maximum absorption indicate the dipole optical resonances. (b) Input impedance of an individual micro-dipole; the frequencies where reactance X vanishes indicate the even (low resistance) and odd (high resistance) electrical resonant modes. The even and odd resonant modes are equivalent to the short and open-circuit optical resonances.

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The optical resonances of the individual dipoles are indirectly obtained from the absorption plot, by identifying the frequencies that maximize the light’s absorption. In the short circuit configuration, the titanium dipole shows its optical resonance at a frequency of ∼28 THz (∼10.6 µm), while in the open-circuit configuration the dipole resonates at a frequency of 48 THz (∼6.2 µm). In contrast to the short-circuit dipole, the resonance of the open-circuit dipole can be tuned by means of the gap size (as discussed below), which in turn modulates the dipole interaction between the antenna arms. In both cases, harmonic resonances of these modes will appear at higher frequencies if the frequency range is increased.

To support the optical results, it is required the evaluation of the input impedance Z_in of the individual micro-dipole. As mentioned, this fundamental parameter is evaluated by considering the micro-dipole as an emitting antenna, whose terminals are drove by an alternating V_AC voltage. By making use of this strategy, the numerical impedance of the dipole has been evaluated as a function of the voltage-source frequency (Fig. 2(b)). For sake of clarity, we have separated the real part of the input impedance referred to as the R_in resistance, and the imaginary part, referred to as the Χ_in reactance (which contains both, capacitive and inductive contributions).

The resonant modes of the micro-dipole are defined as the frequencies where the reactance vanishes to zero; in this condition, the current I_AC along the antenna is in phase with the source’s voltage V_AC, and so, all the electromagnetic energy flows within the system as an electric current. Results show that the resonant modes, indicated on the reactance plot with the aid of open and filled circles, appear at frequencies around ∼28 THz and 46 THz (10.6 µm and ∼6.5 µm), respectively. The first resonance is known as the odd mode or series resonance (described by an equivalent series RCL circuit), its derivative is positive and exhibits a low resistance (56 Ω). The second resonance is known as the even mode or parallel resonance (described by an equivalent parallel RCL circuit), its derivative is negative and features a high resistance (330 Ω). In contrast to the odd mode, the even mode is strongly dependent on the gap capacitance.

The odd mode is the first natural resonance of the micro-dipole. Since the intrinsic (or natural) impedance of the dipole for this resonant mode is completely real (this means, that the intrinsic reactance of dipole vanish to zero [28]), then the mode it is not affected by the gap capacitance. Hence, their nature is entirely resistive and can be understood as a continuous or short-circuited dipole. As we will discuss in detail below, the wavelength of this mode scales with the length L of the dipole following the geometrical relationship 2L =λ_eff [17]. In contrast to the odd mode, the intrinsic reactance of the even mode vanishes thanks to the mutual cancellation of the dipole's natural inductance and the capacitance introduced by the gap (the dipole's intrinsic inductance is non-zero since the excitation frequency is above the system's natural resonance). This second resonant mode can be understood as an open circuit dipole and therefore strongly depends on the gap. These concepts allow a direct equivalence to be established between the odd and even resonant modes of the emitting antennas (port-excited), with the optical resonances of the receiving antennas (light-illuminated), in the short-circuit and open-circuit configurations, respectively: the odd (even) mode is electrical counterpart of the short (open) circuit optical resonances. This equivalence is discussed for all the results presented along this analysis.

We briefly show the behavior of the odd and even resonant modes of the individual open-circuit dipoles, first, by varying the antenna's length (with a fixed gap size), and second, yet importantly, by changing the gap size that is the space where the nano-bolometer of an infrared detector is placed. This additional information will allow us to perform a better assessment of the antennas embedded in arrays, and of the effect of placing a bolometer in the gap.

The input impedance for some individual dipoles with different lengths (but same gap G = 250 nm) is evaluated as a function of the light’s frequency (Fig. 3(a)). With the help of open and closed circles, we identify the odd and even resonant modes, respectively. As the length of the antenna increases, both modes move towards lower frequencies or longer wavelengths (Fig. 3(b)). Both modes’ resonant length λ_res increases linearly with the antenna's length, however, the even resonant mode exhibits smaller variations with the length and the physical mechanism beyond its behavior differs from the odd mode. From an optical point of view, relevant contributions concerning receiving dipoles (in short-circuit configurations), establish a linear dependence of the resonance length λ_res with the length of the antenna [17], given by the relationship:

 

Fig. 3. Input impedance of a titanium dipole as a function of: (a) the antenna length and, (c) the gap size. Odd and even resonant modes (X_in = 0) are indicated using open and closed circles, respectively. For sake of clarity, the resonant wavelength of odd and even modes are shown as a function of (c) the dipole length and (d) the gap size.

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On the other hand, the impact of the gap's capacitive on the micro-dipole input impedance can be modulated through the gap size G: the smaller the gap, the greater the capacitance, shifting the input impedance to lower frequencies. To demonstrate this behavior, the input impedance of various antennas with different gap sizes are shown (Fig. 3(c)), while fixing the antenna length (L = 2 µm). As expected, the even mode is remarkably affected by the reduction of the gap’s size because of its capacitive nature, shifting its wavelength from ∼6 to 9 microns, while the odd mode remains relatively constant at ∼ 10.2 microns (Fig. 3(d)). It must be noted that both modes will overlap (or degenerate) when the gap’s size vanishes to zero.

3.2 Collective impedance in micro-dipole arrays: longitudinal and transverse interaction

The input impedance and resonance frequencies of an individual micro-dipole change when the antenna is embedded in a periodic array with a large number of elements. This is due to the longitudinal and transverse dipole interaction between the micro-dipole and its closest neighbors. In this section, we are concerned with showing how the resonances of the titanium micro-dipoles change with the lattice parameters of a rectangular array a_x and a_y.

The input impedance of some dipole antenna arrays with the same longitudinal spacing a_y = 4 µm, but with different transverse spacing a_x, is shown as a function of frequency (Fig. 4(a)). With the help of open and closed circles, we identify the frequencies in which the odd and even modes (X_in = 0) of each array occur. The findings indicate that decreasing the transverse distance between antennas causes the shift of impedance plots. In the one hand, the impedance amplitude strongly increases from ∼320 to ∼400 Ω, reaching the maxima of resistance near the lattice parameter a_x∼2.9 µm. The curve becomes narrower as it reaches its greatest amplitude, and broadens again for smaller lattice parameter values. On the other hand, the wavelengths of the resonant modes shift from those of an individual antenna (Fig. 4(b)): the even mode linearly shits towards shorter wavelengths (or higher frequencies), while the odd mode wavelength exhibits slight oscillations without significantly changing its value.

 

Fig. 4. Input impedance of titanium dipoles integrated in an infinite rectangular antenna array, with a_x and a_y grid parameters. Impedance is shown as a function of: (a) the transverse distance between dipoles, defined by the grid parameter a_x, while keeping grid parameter a_y = 4 µm, and (c) the longitudinal distance between dipoles, defined by the grid parameter a_y, while keeping grid parameter a_x = 4 µm. The odd and even modes are shown using open and closed circles, respectively. The wavelength dependence of the resonant modes is shown as a function of both network parameters, in (b) and (d), respectively.

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This behavior can be understood in terms of the transverse dipolar interaction between a micro-dipole and its two closest neighbors, the one to the left and the one to the right of the dipole’s axes. The electric field of the closest neighbors overlap in a constructive manner along the micro-dipole (because of the axial symmetry) and reduce its effective width W, confining in this way the paths of the resonant currents. As consequence, the capacitance and inductance decrease, inducing the particular shift of the even resonant mode towards shorter wavelengths due to its capacitive nature (${f_{res}} = 1/2\pi \sqrt {LC} $). In contrast, due to their resistive nature the odd mode does not change with the capacitance, this mode shifts since the micro-dipole’s resistance changes with the dipolar interaction too, which in turn increase the effective resistance of the dipole, however, this change is not so strong. From an optical point of view, relevant contribution are analyzed the optical counterpart of this odd mode, by analyzing optical dipoles in the short-circuit configuration [25].

The effect of the longitudinal interaction between the antennas is evaluated similarly. The input impedance of some arrays with the same transverse separation a_x = 4 µm, but with different longitudinal spacing a_y, is shown as a function of the frequency (Fig. 4(c)). Results show the impedance and the resonant modes are unaffected for lattice parameters greater a_y > 2.5µm (tip-to-tip dipole distances greater than 500 nm), indicating that the longitudinal dipolar interaction is weak in the long regime. For smaller distances, however, the dipole interaction becomes stronger inducing the shift of the resonant modes and the impedance curve width (Fig. 4(d)). The dipolar interaction causes both “odd and even” resonant modes to shift towards higher wavelengths (lower frequencies).

This effect can be understood by considering the tip-to-tip dipolar interaction between the micro-dipole and its two closest neighbors. In this case, the sign of the effective charge at the tips of the micro-dipole is opposite to that of its neighbors, what lengthens the paths of the resonant modes of the micro-dipole. As consequence, the effective length of the dipole L increases, shifting both modes towards longer wavelengths (L = 2λ_res).

On the other hand, as we have mentioned, there exists an equivalence between the optical properties of receiving antennas (illuminated by a linear planar wave) and the electrical properties of transmitting antennas (fed with an alternating voltage across their terminals). To unveil this equivalence, we first evaluate and show the optical absorbance spectra (through color mapping) for dipole arrays in the “short-circuit” configuration (Fig. 5(a) and Fig. 5(c)).

 

Fig. 5. Optical absorbance spectra of a rectangular titanium dipole array. The absorbance for the case of short-circuited dipoles is shown as a function of (a) the transverse distance, described by the lattice parameter a_x, and (c) the longitudinal distance, described by the lattice parameter a_y. The absorbance for open circuit dipoles is shown as a function of (b) the transverse distance and (d) the longitudinal distance. The frequencies where the optical absorbance is higher, properly coincide with the odd and even resonant modes (indicated by dotted lines) obtained using emitting antennas.

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The color map in Fig. 5(a) (Fig. 5(c)) show the absorbance spectra for short-circuit dipole arrays with the same longitudinal (transverse) lattice parameter a_y = 4µm (a_x= 4µm), but with different transverse (longitudinal) distance, showing the impact of dipolar interaction on the arrays’ absorbance. The optical resonances of short-circuit dipoles are indirectly determined by identifying the frequencies at which the absorbance is highest. These resonant frequencies form a small red band on the color mapping about 28 THz (26 THz), which increases (decreases) slightly as the antennas are brought together transversely (longitudinally). As results show, these types of arrays absorb about ∼32% of the incident light power regardless of the a_x and a_y parameter values; the remainder light is mostly transmitted towards the substrate. In the case of transverse interaction, we note that a considerable increase in the width of the absorption spectrum, and a second maximum in the closest spaced arrays occurs.

Once the short-circuit dipoles’ absorbance is evaluated, we use color mapping to compare the optical resonances (the small red band) with odd resonant modes obtained from the electrical impedance approach. Odd resonant modes are shown by using a bold dotted line on the map. This allows us to compare and underlined the agreement between both concepts.

To unveil the equivalence between the even mode and the open circuit resonance we evaluate the optical absorbance for arrays in this configuration. Similarly, we present two color maps, Fig. 5(b) and Fig. 5(d), with which we clearly show the effect of transverse and longitudinal dipole interactions, respectively. The absorbance of the open-circuit dipoles increases from 10% to 20% and from 15% to 8%, when the lattice parameter ax and ay decrease, respectively. Similarly, we also include on the maps the even resonant modes obtained by using the electrical approach. The numerical results allow us to observe a difference between optical and electrical resonances. The absorption maxima form a small band around 52 THz and 48 THz (for the ax and ay parameters, respectively), while the even modes are presented at 48 THz and 44 THz.

Finally yet importantly, we present the complete study of the even and odd resonant frequencies for titanium dipole arrays when both types of dipole interaction, transverse and longitudinal, are simultaneously tuned (Fig. 6(a) and (b)). The intensity of the color maps show us the frequency at which appear the odd and even resonant modes for each lattice parameter point (a_x, a_y). With the help of equidistant lines, we identified some resonance frequencies for odd (28, 30, 32, and 34 THz) and even (44, 46, and 48 THz) modes. This allows us to emphasize that there are many arrangements with different parameters (a_x, a_y) with the same resonant frequency for the odd (even) mode, but with a different odd (even) mode. As we will show in the following section, the difference between the two comes from the resistive part R of the input impedance.

 

Fig. 6. Complete analysis of odd (a) and even (b) resonant modes for any pair of lattice parameters (a_x, a_y); the equipotential lines on the mapping show the pairs (a_x, a_y) that exhibit the same resonant frequency.

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3.3 Micro-antenna arrays with a nano-bolometer as a load resistance

Incorporating a resistive nano-bolometer in the gap of the antenna shift the frequency at which the array's absorption is highest; we will arbitrarily call this frequency “frequency of operation” f_OP, which is different from the even and odd resonance of the antennas f_res. The operating frequency of arrays f_OP tunes according to the nano-bolometer resistance. For highly resistive nano-bolometers, the operating frequency will be very close to the resonance of a dipole array in an “open-circuit” configuration. In the case of a good coupling between the impedance of the nano-bolometers and dipoles, the operating frequency will be the same as the dipole resonance in “short-circuit” configuration.

Designing an infrared detector based on nano-bolometers coupled to micro-antennas requires therefore to correctly knowing the resistance of the nano-bolometer, as well as the input impedance of the antennas within the array, which depends on the network parameters a_x, a_y. As we will show in the following, each geometrical configuration of micro-dipoles has its own input impedance and therefore will have a different operating frequency f_OP despite having the same resistive nano-bolometer in its gap.

To get insight into how the operating frequency f_OP depends on the impedance matching between the nano-bolometers and micro-dipoles, we make use of the port properties into simulations. The port has the dual purpose of feeding the dipole terminals with a voltage of V_AC = 1 V, as well as introducing a resistance Z₀, which accounts for the effects of the nano-bolometers. For this study, we consider 500 Ω nano-bolometers feeding the arrays already analyzed (L = 2 µm and G = 250 nm). By performing the one-port analysis of the array’s, the scattering parameter S₁₁ [dB], also known as the return loss parameter, is obtained. This parameter measures the power loss in the returned/reflected signal between the dipole and the port (or nano-bolometer). The operating frequency of the arrays f_OP can be determined by identifying the frequency at which the return losses are the lowest f_min(S11). It is must be carefully underlined that replacing nano-bolometers by lumped-port elements is a numerical strategy that has been successfully employed to describe, in a first approximation, emitting micro-antennas. This strategy can be improved by introducing ports with complex impedances, and by removing the air-gap capacitance; however, due to the gap’s size, those contribution can be discarded, in a first time.

The return loss of a series of arrays with different lattice parameter a_x and same parameter a_y= 4 µm, are shown as a function of frequency (Fig. 7(a)). Results show that arrays reach their minimum value at different frequencies f_OP and with different amplitudes S₁₁, despite having the same load resistance. It should be underlined, that the better the impedance match between the nano-bolometer and the micro-dipoles, the better the energy transfer between them, and so, the better the performance of infrared devices. This condition is achieved by the array with lattice parameter a_x = 3 µm, exhibiting return losses around of 18 dB, at the frequency of ∼46 THz. Based on the resonance figures in the previous section, it is verified that the frequency is that of the even mode. This behavior can be understood by considering the resistance R of the arrays as a function of the lattice parameter a_x, in which the bold asterisks indicates the resistance value of the best array (a_x = 3 µm), around ∼ 400 Ω, unveiling the match with the nano-bolometer (Fig. 7(b)). In a similar way, it has been determined the array that best matches the resistance of the nano-bolometer, for arrays with different lattice parameter a_y (a_x fixed to 4 µm), resulting in a_x of four microns (Fig. 7(c) and (d)). We remark that despite the lattice parameter optimization there are still losses of power. By comparing the impedance of both elements, we can appreciate that matching is not perfect. Despite these limits, it must be stressed that the lack of this optimization procedure could lead to much more substantial losses, as the results show (e.g. ∼8 dB instead of ∼ 18 dB).

 

Fig. 7. (a) Return losses of a series of micro-dipole arrays with different lattice parameter a_x, (b) real part of the input impedance of the micro-dipoles arrays as a function of a_x, (c) return losses of a series of micro-dipoles arrays with different lattice parameter a_y, and (d) real part of the input impedance of the micro-dipoles as a function of a_y.

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However, thanks to these results, it can be realized that the dipolar interactions, and so, the match between the elements, could be enhanced by increasing the coordination number, this means, by increasing the quantity of closest neighborhoods. Using hexagonal arrays could represent a strategy to reach higher impedance values (this, by using a single lattice parameter), and to better match the nano-bolometer to the antennas. On the other hand, the use of tapered micro-dipoles, which are slightly variations of half-wave dipoles, could improve the impedance match, too. In contrast to individual half-wave dipoles, the individual tapered dipoles reach higher input impedances [45].

Through the results presented in this study, we have unveiled the effect of the lattice parameters on the collective input impedance of arrays, and so, discussed the overall performance of antenna-based infrared devices.

4. Conclusions

In summary, the collective odd and even resonant modes arising from rectangular arrays of titanium micro-dipoles are analyzed in a proper manner, by introducing the concepts of input impedance and “nanoload” elements, with stress aimed at the efficient sensing of thermal wavelengths, around 10.6 µm. First, the entirely resistive nature of the odd mode and the entirely capacitive nature of the even mode are unveiled through the analysis of its input impedance. At the light of these results, the opposite shift of the resonance wavelength of both modes, as the size of the dipole’s gap decreases, is stablished and understood. Subsequently, the collective even and odd modes of micro-dipoles embedded in a square array are evaluated in terms of the lattice parameters (a_x, a_y). The shift of both modes towards longer wavelengths, as the lattice parameter a_y decreases, is explained in terms of the tip-to-tip dipolar interaction and its effect of the effective length of dipoles. In contrast, the opposite shift of the modes, as the lattice parameter a_x decreases, is understood in terms of the longitudinal dipolar interaction and its effect on the effective capacitance and width of micro-dipoles. By analyzing the absorbance spectra of the micro-antenna arrays, the equivalence of optical resonances counterpart, in the short and open-circuit configurations, with the odd and even modes is presented. Finally, the effect on the array's performance that results from introducing highly resistive nano-bolometers is optimized by exploiting the natural high-resistance of the collective even modes.