Analysis and Performance of Lumped-Element Kinetic Inductance Detectors for W-Band

Lumped-element superconducting resonators are a promising technology for their use in millimeter-wave observations and quantum computing applications that require large arrays of extremely sensitive detectors. Among them, lumped-element kinetic inductance detectors (LEKIDs) have shown good performance in the submillimeter band in several earth-based telescopes. In this work, LEKIDs for their use as millimeter-wave receivers of astronomical applications are presented. LEKID arrays using a thin bilayer of superconducting titanium/aluminum (Ti/Al), deposited on the silicon substrate, have been designed and fabricated. The design of a dual-polarization LEKID with the goal of detection at the $W$ -band for two orthogonal polarizations is described and a fabricated array has demonstrated absorption at ambient temperature. Also, an approximate design methodology of the coupling parameter for LEKIDs’ readout, essential for dynamic range optimization of the detector under millimeter-wave radiation, is proposed. In addition, the resonance characteristics and coupling factor of the fabricated superconducting resonators using high-quality internal factor $Q_{i}$ under cryogenic temperatures have been analyzed. The design guidelines in this work are applicable to other LEKID arrays, and the presented superconducting Ti/Al thin-film LEKIDs can be used in future receiver arrays in the millimeter bands.


I. INTRODUCTION
K INETIC inductance detectors (KIDs) are used in high-sensitivity radio astronomy receivers. These detectors operate under the property of inductance change in a superconducting strip, when they are cooled down well below their superconducting critical temperature, usually near absolute zero, when they absorb a microwave signal taken by a telescope receiver system. KID-based receivers are built containing thousands of individual detectors, or instrument pixels, as their intrinsic multiplexing is a relatively simple way of receiver integration. Frequency-domain multiplex operation consists of a high amount of resonant circuits coupled to a single readout transmission line, making feasible such a high number of instrument pixels relatively easy [1]. Each individual KID is embedded in a resonant circuit having a resonant frequency different from the remainder KIDs. In this way, the readout system is in fact a frequency division multiplex communication system [2]. Former KID receivers operated in the submillimeter bands, but, nowadays, the use of new superconducting materials allows detection in the optical near-IR band [3] or even in the millimeter band [4], [5] showing good performance in all cases.
KIDs are based on superconducting microresonators with a characteristic resonant frequency and quality factor. The absorption of an incoming radiation, with energy higher than the superconducting gap, changes the kinetic inductance (L k ), leading to a change in the resonant frequency. Moreover, this absorption also produces dissipative losses in the resonator, decreasing the quality factor in comparison to the darkness condition. The reduction in the resonant frequency and quality factor, given proper calibration, enables the detection of radio signals from astronomical sources, using complex processing algorithms.
Several KID radio astronomy instruments for millimeterand submillimeter-wave observations are in operation, achieving even thousands of multiplexed detectors [6]- [10]. There are two main types of detectors: Antenna-coupled microwave kinetic inductance detectors (MKIDs) [2] and lumped-element kinetic inductance detectors (LEKIDs) [11]. The former are based on distributed resonators with coplanar waveguide (CPW) geometries coupled to an antenna that acts as an active receiver. The later, LEKIDs, which is the focus of this study, are based on lumped inductor-capacitor resonators, where the inductor acts directly as the effective optical absorber, and its design is shown in Fig. 1.
Great efforts have been made to develop dual-polarization LEKIDs and to improve the sensitivity at the W -band for future cosmic microwave background observations. For instance, Catalano et al. [5], [12], [13] showed that the superconducting proximity effect in a titanium/aluminum (Ti/Al) bilayer causes a decrease in the superconducting critical temperature that pushes the LEKID detection to the W -band. Regarding polarization sensitivity, earth-based instruments such as NIKA2 [9] use an external polarizer that separates the two linear polarizations into two independent arrays based on a Hilbert fractal structure. On the other hand, some additional works have focused on the development of dual-polarization LEKIDs demonstrating simultaneous orthogonal polarization sensitivity in the millimeter-and submillimeter-wave ranges [8], [14]. However, very little has been done in developing dual-polarization LEKIDs in the W -band. Therefore, in this work, we focus on both the requirements and present an optimized version of the BiKID approach [15] for the W -band, based on an optimized optical coupling design.
The response of LEKIDs is maximized when critical coupling is achieved, that is, the external quality factor (Q c ) equals the internal quality factor (Q i ) [16]. Whereas Q i is set by fixed parameters such as the optical background or operating base temperature, Q c can be modified by tuning the geometrical parameters as separation s. Therefore, a design methodology for microwave coupling to the LEKID has been developed and applied to a new prototype, with several external factors Q c .
This article presents the analysis, design, fabrication, and experimental tests of the bilayer LEKID prototypes working in the W -band (75-110 GHz). The remainder of this article is organized as follows. Section II deals with the KID design, concerning millimeter-wave signals to be detected, and low-frequency resonators' design, with special focus on external coupling. Section III describes fabrication and prototypes' assembly. Section IV presents experimental systems to perform ambient temperature and cryogenic tests. The tests  results are detailed in Section V. Finally, Section VI presents conclusions summarizing this article.

II. LEKID DESIGN
The presented LEKIDs are based on series superconducting capacitor-inductor resonators coupled to a single transmission line, as it was first proposed in [11]. The inductor, which acts as the effective absorber, is designed to absorb the incident radiation by matching optically to free space. In this case, the strip grating, deposited on the top of a high-resistive silicon substrate, was designed to be matched at the W -band to the free-space impedance, when a backshort is placed at the rear part of the silicon wafer. The linearly polarized wave absorption of this design is characterized elsewhere and peaks around 78 GHz [17].
A new dual-polarization LEKID is designed to absorb and distinguish millimeter-wave radiation of two orthogonal linearly polarized waves simultaneously. For this purpose, two LEKIDs, based on the design explained before, can be stacked one on the top of the other with perpendicular orientation as shown in Fig. 2. In this case, each of them will be dedicated to the detection of one of the two perpendicular polarizations of the incident waves.

A. MM-Wave Coupling Design
An equivalent circuit model of the dual-polarization LEKID is shown in Fig. 3.
The input admittance of the dual-polarization LEKID is where the admittance Y 1 is given by and γ = α Si + jβ Si is the complex propagation constant, where the real part is the attenuation constant α Si , due to dielectric losses, and the imaginary part β Si is the phase constant in silicon. The shunt admittances Y strips1 and Y strips2 are the equivalent circuits of the strips that compose the inductors of the LEKIDs. Silicon substrates thicknesses are l 1 and l 2 . Silicon wave impedance is δ Si = δ 0 / √ ε rSi with ε rSi = 11.9 and δ 0 ≈ 120π.
The strips that compose the LEKID inductors have width 2a, distance between strips g, and a sheet resistance R s (/sq). It is worth to note that even though the detector is operating at temperatures well below its critical temperature (Tc), an incident radiation with frequency (ν) that provides energy higher than the superconducting gap (), i.e., hν > 2, where h is the Planck constant, leads to a sheet resistance comparable to that in the normal conducting state just above the superconducting critical temperature [18].
The metallic strips present a resistive and inductive impedance for an incident electric field parallel to them [18]- [20], the admittance of which is given by where the reactance X L is If the incident electric field is orthogonal to the strips, the equivalent circuit is a capacitive admittance, according to [21,Sec. 5.18, eq.(1a)].
A dual-polarization LEKID with two stacked and orthogonal strips is shown in Fig. 4. The admittance Y in in (1) depends on incident wave polarization shown in Fig. 2. The admittance Y strips1 is inductive as (3) and Y strips2 is capacitive in case of a wave TEM V . On the other hand, admittances Y strips1 and Y strips2 are capacitive and inductive as (3), respectively, for TEM H .
The strip is designed to achieve efficient absorption, and therefore, for an inductive strip admittance in (3), the geometry (a, g) is calculated to fulfill real Y stripsL = 1/δ 0 with metal resistance R s (/sq) at cryogenic temperature. Then, the grounded silicon substrate presents the imaginary admittance required to match the imaginary part of the admittance in (3) at the center frequency for a thickness l 1 given by The second LEKID is formed by rotating the same strips' geometry 90 • with respect to the previous one on the silicon substrate, with thickness a half wavelength at the center frequency, which allows matching simultaneously in both the strips. This type of structure has narrow bandwidth behavior, since impedance matching is achieved with one single silicon dielectric layer l 1 .
The initial geometry of the strips for the LEKID design has been obtained for a 35-nm-thick Ti/Al film with Rs = 1.27 /sq measured just above Tc (see Fig. 5). A strip width 2a = 3 μm has been chosen to achieve the broadest bandwidth around 90 GHz, which together with a spacing g = 375 μm provides real(Y StripsL ) = 1/δ 0 in (3). Using those values, the inductive and capacitive admittances are calculated. Their reflection coefficients on air, from 65 to 110 GHz, are depicted in Fig. 6(a).
The thickness l 1 ≈ 292 μm of the grounded silicon substrate is calculated using (5), and the silicon substrate thickness between LEKIDs is l 2 = λ Si /2 ≈ 483 μm with λ Si = λ 0 / √ ε rSi , both calculated at 90 GHz. The obtained reflection coefficients, of the dual-polarization LEKID model in Fig. 2, for the two linear polarized incident waves (TEM V and TEM H ) and lossless silicon substrate are plotted in Fig. 6(b) from 65 to 110 GHz, showing a perfect matching and maximum absorption at 90 GHz. The loss tangent of high-resistive silicon in the millimeter bands at 4.2 K is around 1.12 × 10 −6 [22], and therefore, it has been considered negligible for the design. Moreover, the obtained impedance matching, for both incident waves, is not affected by the strip with capacitive performance.
With this initial design, the next step is a 3-D electromagnetic (EM) simulation at the W -band of the dual-polarized LEKID using the HFSS 3-D EM simulator from ANSYS, as a single unit cell with Floquet ports with master and slave boundaries, to simulate an array as a planar-periodic structure. The final millimeter-wave absorption, of the dual-polarization LEKID, for the two linearly polarized incident waves, is also  simulated with the complete structure composed of the meandered strips and interdigital capacitors (see Fig. 4), and it is shown in [17]. The absorption efficiency of each component, using the Field Calculator available in the HFSS 3-D EM simulator, is shown in Fig. 7 for both polarizations. This tool makes use of the Surface Loss function to estimate the

B. Low-Frequency Resonator Design
A typical LEKID superconducting resonator is coupled to the main microstrip readout line as it is shown in Fig. 1. The interdigital capacitor fixes its resonant frequency and allows frequency multiplexing. For readout, the resonator is coupled to a transmission line, selecting the coupling coefficient with a separation s (see Fig. 1). Absorption of photons at the millimeter wave results in a change in the resonator frequency and quality factor, and therefore, an accurate design of the coupling is crucial for dynamic range optimization of the detector under millimeter-wave radiation.
The connection of the inductive and capacitive parts forms a series RLC resonant circuit. The resistive part of this equivalent circuit comes from the conductor losses of both the inductive and capacitive parts. As we are dealing with superconducting resonators, this resistive part can be extracted from the two-fluid model and tends to zero when the operating temperature is much lower than its critical temperature [11].
LEKID resonators are absorption mode coupled [24], and the effect of an individual resonator on the readout line transmission is only present in a narrowband around its resonant frequency. Fig. 8 shows an equivalent circuit with two coupled resonators in a transmission test schematic [25].
The mutual inductance M is the main coupling effect, which transforms each series resonant circuit to a shunt resonant circuit. In a narrow frequency band, taking into account only one resonator and considering other resonances well-detuned, the equivalent circuit shown in Fig. 9 is useful to analyze the LEKID resonator quality factors and coupling factor and to obtain their relation to the S 21 scattering parameter. This last transmission parameter is in fact the tested one in an LEKID instrument.
The coupling factor k is the relation between the delivered power to external loads P E and the internally delivered power P 0 inside the resonator, at the resonant frequency, given by where Q i is the internal quality factor and Q c is the external quality factor [26]. For critical coupling, k is 1 and the power delivered to the resonator is maximum.  LEKID coupling is analyzed as a pair of asymmetric coupled microstrip lines, having a total coupling length equal to the contribution of meander line sections close to the readout line: = 4·(g + 2a) (see Fig. 1). In a typical LEKID, this coupling length is very short in terms of wavelength at the resonant frequency ( f 0 ), and the equivalent circuit can be analyzed as a differential length (dz) of a coupled lines pair, as it is shown in Fig. 10(a). All elements (C 1 , C 2 , C M , L 1 , L 2 , and M) have dimensions per unit of length. The readout microstrip linewidth (W 1 ) is typically chosen to have 50characteristic impedance, whereas the inductor strip width W 2 = 2a is narrow to optimize the LEKID millimeter-wave absorption, as well as to achieve an inductive value adequate for the resonant frequency, inside the selected radio frequency band for the readout.
Performing a circuit analysis of the schematic shown in Fig. 10(b), using typical values for the microstrip structure, it is straightforward to realize that the most relevant element in the coupling is the mutual inductance M dz. The equivalent circuit can be greatly simplified for design purposes, since all the capacitive elements have a negligible effect. Using the equivalent circuits shown in Figs. 8 and 9, the mutual inductance required to obtain a specific coupling for an LEKID resonator is where ω 0 is the resonant angular frequency 2π f 0 . Coupling design can be simplified using the symmetry properties of a two-port network [27]. The symmetry axis of an LEKID-coupled resonator is depicted in Fig. 11. Applying the odd mode condition, a short circuit (SC) in all points of the symmetry axis, the two-port network is converted into a one-port network, which is applied to the coupled lines' network. Moreover, due to the symmetry properties, the LEKID impedance Z KID , the inductance elements in the mutual inductance equivalent network and the coupling length are halved. The resultant equivalent networks are shown in Fig. 12(a) and (b). These two networks have identical  Y-parameters, their calculation being the first step for the proposed design process. By analysis of the mutual inductance circuit in Fig. 12(b), its Y 21 parameter is For a specific LEKID coupling design, the input data are dielectric substrate parameters, the width of coupled lines, W 1 and W 2 = 2a in Fig. 10(a), the coupling length , and the resonant frequency f 0 . Given the input data, and after selecting the desired coupling level: overcoupling, undercoupling, or critical coupling, the mutual inductance M is calculated according to (7). Inductances L 1 and L 2 in (8) are calculated from inductances per unit length L 1 and L 2 of the isolated microstrip lines, for widths W 1 and W 2 . The assumption of isolation of the microstrip lines, for L 1 and L 2 calculation purposes, is a good approach, because coupling of the LEKID resonators is very weak in general, and both the lines are separated by a sufficiently large distance s. The next step is to obtain the right separation s between lines using an asymmetrical coupled microstrip lines' electrical model [28]. This model is available in microwave circuit simulators, and through an optimization routine, or by manual tuning, the separation s value must be varied to achieve the desired Y 21 parameter, of the two-port network in Fig. 12(c), with the value calculated in (8) at the resonant frequency ω 0 . For an improved accuracy of the separation s, a 2-D EM simulator is used, taking as an initial value the obtained s value by electrical model simulation, to avoid the uncertainties of the coupled lines' electrical models for very weak couplings and short coupling lengths.
For the readout design of an LEKID, applying the present design method, Table I shows the obtained separation s for   Table I. three different coupling factors regarding an LEKID with an expected internal quality factor Q i = 200 000 and resonant frequency f 0 = 500 MHz. It has been considered a high-resistive silicon substrate with a thickness 285 μm, coupling length = 4 · (g + 2a) = 3 mm, and line widths W 1 = 220 μm for the 50-readout line and W 2 = 2a =3 μm for the inductor strip. The S 21 responses of an LEKID for the three coupling factors (k) with the parameters obtained in Table I are simulated and the results are shown in Fig. 13.

III. KIDS FABRICATION AND ASSEMBLY
The devices were fabricated following the technological process detailed in [17]. In summary, by means of confocal sputtering, a Ti/Al bilayer was deposited on a high-resistive silicon wafer 275-μm thick, with a 200-nm-thick Al layer on the rear part of the wafer, to be used as the ground plane and optical backshort. The physical dimensions for the inductor were calculated considering an initial approach of R s (g/2a) ∼ = 377 in (3). The LEKIDs were designed for single polarization, and two prototypes were fabricated and measured at cryogenic and ambient temperatures with 2a = 3 μm and g = 440 μm for optical coupling. The bilayer was characterized at cryogenic temperatures, obtaining R s = 1.27 /sq, T c = 782 mK, and an estimation of the kinetic inductance L k = 2.24 pH/sq. The preliminary  results for low-frequency cryogenic characterization, and optical absorption for single polarization, were presented in [17].
To verify the performance of a dual-polarization prototype, several stacked wafers were measured at ambient temperature as a proof of concept. An array of 11 × 11 single-polarization LEKIDs was fabricated on a 275-μm-thick silicon wafer, while an identical second one was fabricated without a ground plane. Both the wafers were stacked orthogonally with a 275-μmthick silicon wafer in between, to distinguish the polarization between two linear polarized incident waves. Fig. 14(a) shows a photograph of the stacked wafers placed in a test fixture, to characterize them at ambient temperature, whereas a photograph of the LEKID inductor is depicted in Fig. 14(b).
On the other hand, a new prototype with seven pixels was designed and fabricated modifying the coupling factor Q c between pixels. Separation s and orientation of each LEKID, with respect to the single 50-microstrip transmission line, were modified from pixel to pixel to tune external coupling Q c . This device has been designed following the methodology explained in Section II-B, which is crucial for future designs, enabling the optimized critical coupling, that is, Q i to be close to Q c under the desired optical load [16]. Fig. 15(a) and (b) shows the design of two pixels, where their coupling has been modified by rotating 90 • the original pixel orientation, increasing the total coupling length . In the remaining pixels, coupling has been modified by tuning the separation s between lines. This prototype was mounted inside a light-tight package made from bulk Al, where Al wirebonds were used to connect the microstrip lines to the readout chain as shown in Fig. 15(c).

IV. EXPERIMENTAL SYSTEMS
This section describes two experimental test setups used for KIDs' characterization: A W -band ambient temperature quasi-optical system used for millimeter-wave absorption test and a cryogenic test system to measure LEKIDs' resonant frequencies and quality factors.

A. Ambient Temperature Test System
The ambient temperature test system operates at the W -band. It has been set up to characterize the LEKIDs' absorption through a free-space measurement. The system consists of two horn antennas, two dielectric lenses that collimate the beam at the measurement plane, and a vector network analyzer (VNA) [17], [29], [30]. The quasi-optical system has a 4 f topology, with f the focal length 75 mm of polytetrafluoroethylene (PTFE) plano-convex lenses LAT075 (Thorlabs Inc., Newton, NJ, USA), as shown in Fig. 16. The horn antennas (QSH-SL-75-110-F-20) provide around 20-dB gain. They are rectangular horns with dimensions a × b = 14.8 mm × 11 mm at the aperture, where the beam radii that maximize the coupling to the fundamental Gaussian mode are ωx = 0.35a and ωy = 0.5b [30]. The beam waist radius at z = 0 is similar for both the coordinates bωr 0 ≈ 4.72 mm. The lenses' diameter is D = 50 mm and the beam radius at the antenna aperture is ω ≈ 5.37 mm, and since D > 4ω, the fractional power lost is lower than 3 × 10 −4 for the fundamental Gaussian mode [30]. The calculated output beam waist radius, at the middle of the system, is bωr 0,out = 16.87 mm, which defines the size of the beam spot for the absorption measurement of LEKID arrays at ambient temperature.

B. Cryogenic Test System
The experimental cryogenic system is the BlueFors dilution refrigerator (DR) LD-250 shown in Fig. 17. This cryostat consists of a DR backed by a two-stage pulse tube cooler that provides 60-and 4-K temperature stages, while the DR provides a 0.7-K stage and a variable 12 mK-1-K stage, where the detectors are mounted [see Fig. 17(a)]. Electrical cryogenic characterization has been performed using the following read-out system, which connects the external warm electronics with the LEKID array at cryogenic temperatures.
1) Cryogenic Harness: The cryogenic harness scheme is shown in Fig. 17, which has been carefully chosen to minimize thermal loading between stages. A stainless steel inner and outer conductor coaxial cable is used from room temperature to the 4-K stage, being thermalized in the 60-K stage by a dc block. A 20-dB attenuator reduces the 300-K radiation and dissipates the power in the 4-K stage. A cupronickel (CuNi) coaxial is used from the 4-K stage to the 12-mK stage. Again, two dc blocks and an extra 10-dB attenuator reduce the noise contribution from the warmer stages. Finally, a semirigid copper coaxial cable connects the last dc block to the LEKID package. Al wirebonds are used to connect a microstrip board to the LEKID chip. On the return path, a copper coaxial cable connects the package with a dc block at the 12-mK stage. Then, a superconducting NbTi coaxial cable carries the signal from this stage up to the 4-K stage where a SiGe low-noise amplifier (LNA) (Caltech-CITLF3) amplifies the signal (15-dB gain). Finally, the CuNi coaxial cable carries the signal to the output port of the cryostat.
2) Readout System: To measure the transmission characteristic of the LEKID array, a readout system has been assembled in a chassis. It is composed of several coaxial modules such as single-pole double-thru (SPDT), LNA, attenuator, power splitter, and quadrature modulator. Four SPDT RF switches enable switching between VNA connection and the I /Q demodulator, through a measurement mode option and it also switches between two different device under tests (DUTs) through a channel selection option. The VNA measurements work from 40 MHz to 2.6 GHz and the I /Q demodulation measurements from 500 MHz to 2.6 GHz.
For I /Q measurements, the local oscillator (LO) signal is generated by a commercial PSG analog signal generator and is split using a power splitter. A step attenuator is used as a variable attenuator to tune the readout signal power level to the required level for the LEKIDs. The signal coming from the LEKID cryostat output is amplified with a gain block consisting of three amplifiers (∼43-dB gain), to reach the RF power required by an I /Q demodulator. The block diagram of the readout is given in Fig. 18. The "IN" and OUT" ports are connected to the cryogenic setup shown in Fig. 17.

V. RESULTS
The measurements described in this section refer to the fabricated prototypes that have been detailed in Section IV. The measurements at ambient and cryogenic temperatures have been carried out, and their results have been compared with the simulations.

A. Measurements at Ambient Temperature
The dual-polarization prototype, made up of 11 × 11 LEKIDs, was characterized at ambient temperature. This prototype was measured for two orthogonal and linearly polarized waves in the 65-110-GHz frequency band. The results are depicted in Fig. 19. These measurements were made using the quasi-optical test-bench described in Section IV. A photograph of the sample under test is shown in Fig. 20. Simulations using the obtained parameters for the Ti/Al bilayers at ambient temperature (R s = 4 /sq) are also included. The actual tested silicon substrate height (h = 295 μm) confirms a good fitting between the simulated and experimental results for both polarizations, showing a maximum absorption around 75 GHz. The tolerance of the silicon wafer thickness is responsible for the frequency shift, and its real tested value has been updated in the simulations.
Moreover, these measurement results have been compared with the simulated absorption of all LEKID components, to obtain the absorption efficiency. Fig. 19(a) and (b) shows a comparison between the test results and simulations of each component, which confirms that the incident power is mainly absorbed in the inductor, as expected. The inductor absorption efficiency is reduced due to the dielectric loss, which has a negligible effect at cryogenic temperature.

B. Cryogenic Measurements
Cryogenic characterization of the seven pixels array was performed using the experimental setup explained in  Section IV-B. A VNA is used for measuring the S 21 parameter across the array. Fig. 21 shows the S 21 transmission amplitude through the array, where each minimum corresponds to a pixel. Due to the low pixel packing density in this prototype, negligible crosstalk between resonators is expected [31]. The resonant frequency, loaded quality factor (Q), external quality factor (Q c ), and internal quality factor (Q i ) were obtained following the procedure detailed in [32].
The EM simulation software Sonnet was used to confirm the estimated kinetic inductance (L k ) from electrical  characterization [17]. Fig. 22 shows the simulated resonant frequency of the lowest resonance as a function of the kinetic inductance. As can be seen, the experimental resonant frequency corresponds to L k = 2.2 pH/sq, very close to the estimated value. These simulations estimate the kinetic fraction by comparing the experimental resonance and the simulated one, obtaining α = 0.38.
The quality factors obtained for all the pixels are shown in Fig. 23. The internal quality factor, around 10 6 for all the pixels, indicates the excellent quality of the deposited film [33]. The external quality factor, as can be seen, has been tuned within an order of magnitude, following the procedure explained in Section II-B. The lowest value, 3.5 × 10 3 corresponds to the pixel shown in Fig. 15(a), whereas the highest value, 5 × 10 4 , corresponds to the pixel shown in Fig. 15(b). The obtained external quality factors agree reasonably well with the values from the design methodology. The differences  between measurement and simulation are related to small tolerances in the nanofabrication process.
Under these dark conditions, the external quality factor limits the loaded quality factor (overcoupling, Q i > Q c ).
However, under the optical load, the internal quality factor is expected to diminish, leading the LEKIDs to a critical optical coupling (Q c = Q i ) which is desirable for maximizing the response [16]. For simulating this effect, a temperature sweep has been performed as shown in Fig. 24(a). Fig. 24(b) shows the loaded, internal, and external quality factors as a function of bath temperature. As can be seen, Q i is reduced as the temperature is increased due to thermally excited quasiparticles. This effect can be compared with the response of the LEKIDs upon optical illumination [34]. As can be seen, the LEKID goes from an overcoupled to undercoupled regime, showing the importance of a proper microwave design for reaching the critical coupling (Q i = Q c ). Future experiments will be performed under optical illumination to choose the optimum external coupling.

VI. CONCLUSION
This work has demonstrated dual-polarization LEKID array absorption at the W -band at ambient temperature. A superconducting Ti/Al bilayer is used to push down the critical temperature (from 1.2 K of pure Al to 782 mK), and therefore push down the low-frequency limit imposed by the superconducting gap to this frequency band. The used topology allows us to detect simultaneously two orthogonal linearly polarized waves in one single pixel. The described design methodology is applicable to other millimeter-and submillimeter-wave bands.
On the other hand, a prototype with seven pixels was fabricated adjusting the external coupling factor, following a proposed method, to achieve critical coupling under the desired optical load. The results obtained exhibit a high internal quality factor. The range of the external quality factors applied has experimentally demonstrated a critical coupling when the bath temperature is increased to emulate an optical load. The dual-polarization LEKID design presented and its initial test results show a very promising technology for future polarimetry experiments. She joined the Department of Transport and Quantum Phenomena, Complutense University of Madrid, as a Research Assistant, in 2018, where she was involved in superconductor transport measurements. Then, she joined CSIC-INTA and IMDEA-Nanociencia. Her research interests lie in the area of superconducting resonators, including modeling, nanofabrication, and low-temperature microwave characterization of microwave devices with applications in astrophysics and quantum computing. She is also involved in the study of the interaction of superconductors with 2-D and magnetic materials. She collaborates actively with other researchers in the same field and contributes to several scientific conferences. In 1992, she joined the Department of Communications Engineering, University of Cantabria, where she is currently an Associate Professor. In the past years, she has worked in projects focused on the development of radiometers for space applications, like the Planck Mission, in particular in low-noise amplifiers at room and cryogenic temperatures. She is currently involved in several projects focused on the development of very low-noise receivers in the 30-and 40-GHz frequency bands for the QUIJOTE experiment and on the development of polarimetry receivers at the W -band. Her main research interests include the design and testing of microwave circuits in both hybrid and monolithic technologies.
Eduardo Artal (Life Member, IEEE) received the Engineer and Dr. Engineer in Telecommunication degrees from the Technical University of Catalonia, Barcelona, Spain, in 1976 and1982, respectively. From 1976 to 1990, he was an Assistant Professor with the Technical University of Catalonia. From 1979 to 1981, in a partial leave from the university, he was with Mier Allende S.A., Barcelona, where he was involved with the TV and FM radio re-emitters development. Since 1990, he has been a Professor with the University of Cantabria, Santander, Spain, where he was the Manager of the Telecommunication Engineering Course, from 1990 to 1994. From 1994 to 1998, he was the Manager of the National Program for Information and Communications Technologies with the "Plan Nacional de I + D," National Research and Development Plan in the Spanish Ministry of Education and Science, Madrid, Spain. His main areas of activities and contributions have been microwave circuits and systems, including monolithic microwave integrated circuits up to 50 GHz. His current research interests are low-noise millimeter-wave amplifiers and receivers for radio astronomy applications.
Juan Pablo Pascual (Senior Member, IEEE) was born in Santander, Spain, in 1968. He received the M.Sc. degree in physics (specialized in electronics) and the Ph.D. degree in electronic engineering from the University of Cantabria, Santander, in 1990 and 1996, respectively.
He currently works as an Associate Professor with the Communications Engineering Department, University of Cantabria. He has been involved in modeling and design projects with institutions like Alcatel Espacio, Tres Cantos, Madrid, Spain, OMMIC, Limeil Brevannes, France, ESA ESTEC Noordwijk, The Netherlands, Technical University of Darmstadt, Darmstadt, Germany (where he stayed during 1999), the PLANCK, the QUIJOTE Mission, and the "Terahertz Technology for Electromagnetic Sensing Applications" consortiums. He has participated in the Spanish Network of Excellence in Terahertz and has been responsible for a training program of Doctors for the industry (University of Cantabria-Erzia Tech., Santander, Spain). He has coauthored more than 60 contributions in international journals and congress. His research interests are active device modeling, monolithic microwave integrated circuit (MMIC) design methodology of nonlinear functions and subsystems, from microwaves to millimeter waves and terahertz, and system simulation.