Ultra-sensitive THz microwave kinetic inductance detectors for future space telescopes

Aims. Future actively cooled space-borne observatories for the far-infrared, loosely deﬁned as a 1–10 THz band, can potentially reach a sensitivity limited only by background radiation from the Universe. This will result in an increase in observing speed of many orders of magnitude. A spectroscopic instrument on such an observatory requires large arrays of detectors with a sensitivity expressed as a noise equivalent power NEP = 3 × 10 − 20 W / √ Hz. Methods. We present the design, fabrication, and characterisation of microwave kinetic inductance detectors (MKIDs) for this frequency range reaching the required sensitivity. The devices are based on thin-ﬁlm NbTiN resonators which use lens-antenna coupling to a submicron-width aluminium transmission line at the shorted end of the resonator where the radiation is absorbed. We optimised the MKID geometry for a low NEP by using a small aluminium volume of ≈ 1 µ m 3 and fabricating the aluminium section on a very thin (100 nm) SiN membrane. Both methods of optimisation also reduce the effect of excess noise by increasing the responsivity of the device, which is further increased by reducing the parasitic geometrical inductance of the resonator. Results. We measure the sensitivity of eight MKIDs with respect to the power absorbed in the detector using a thermal calibration source ﬁltered in a narrow band around 1.5 THz. We obtain a NEP exp ( P abs ) = 3 . 1 ± 0 . 9 × 10 − 20 W / √ Hz at a modulation frequency of 200 Hz averaged over all measured MKIDs. The NEP is limited by quasiparticle trapping. Conclusions. The measured sensitivity is sufﬁcient for spectroscopic observations from future, actively cooled space-based observato-ries. Moreover, the presented device design and assembly can be adapted for frequencies up to ≈ 10 THz and can be readily implemented in kilopixel arrays.


Introduction
Radiation in the far-infrared (FIR) part of the electromagnetic spectrum, loosely defined as the 0.03-1 mm wavelength range, represents about half the energy generated in the Universe since the Big Bang and includes information from processes mostly invisible at other wavelengths (Dole et al. 2006). Unfortunately, observations are notoriously difficult: The Earth's atmosphere is at best partially transparent. Even at the altitudes of stratospheric balloons, line emission from residual water vapour can contaminate astrophysical emission lines. Additionally, selfemission from the telescope creates significant radiation loading, limiting instrument sensitivity. Only an actively cooled space telescope with a temperature of about 4 K in combination with background-limited detectors allows an instrument that is limited only by the Universe background. A spectroscopic instrument on such an observatory requires large-format detector arrays with a pixel count of ≈10 5 with a sensitivity expressed in a noise equivalent power (NEP) of ∼3 ×10 −20 W/

√
Hz (Farrah et al. 2019;Hailey-Dunsheath et al. 2021). The combination of sensitivity, high radiation frequency, and pixel count presents a major challenge for future detector systems. Several detector systems exist that are progressing towards this goal.
Transition edge sensors (TESs; Irwin & Hilton 2005) are used in many ground-based and balloon-borne observatories. Recent work (Audley et al. 2016;Khosropanah et al. 2016;Williams et al. 2020) demonstrates the progress in horn-coupled devices operating in a 60-110 µm wavelength band reaching a NEP ∼ 3 × 10 −19 W/ √ Hz using a thermal calibration source (Williams et al. 2020). Nagler et al. (2020) suggest a possible route towards even better sensitivities, reaching an electrical NEP < 10 −20 W/ √ Hz by means of a proximity-effect TES. Quantum capacitance detectors (QCDs; Echternach et al. 2013) are a less mature technology, but are the most sensitive: they have shown single-photon counting at 1.5 THz and background limited performance at power levels as low as 10 −20 W, corresponding to a NEP < 10 −20 W/ √ Hz using a 451 pixel array of antenna-coupled detectors operating at 1.5 THz (Echternach et al. 2021).
Microwave kinetic inductance detectors (MKIDs) are microwave resonators whose resonance frequency depends on the amount of radiation absorbed (Day et al. 2003). Antenna-coupled MKIDs have reached sensitivities of NEP = 3 × 10 −19 W/ A&A 665, A17 (2022) (d) Cross-sectional diagram of the detector assembly (not to scale). The detector chip, as depicted in panels a-c, is coupled to a Si lens array using a spacer wafer with a Ta absorbing mesh, with an opening aligned to the antenna to enable radiation coupling. The vacuum gap t is created using spin-on PermiNex ® glue pillars as indicated. (Baselmans et al. 2017). Operation at radiation frequencies above 1 THz with a similar sensitivity and array size was demonstrated by Bueno et al. (2017Bueno et al. ( , 2018. Lumped-element KIDs have almost reached similar sensitivities for devices coupled to an on-chip filterbank (McGeehan et al. 2018). MKIDs are especially attractive because of their ease of multiplexing, allowing kilopixel arrays with only a single readout line. They are also the only detectors whose sensitivity to ionising radiation has been tested extensively for large arrays with relevant sensitivities (Karatsu et al. 2019), with an extrapolated dead time in L2 of less than 4% depending on array geometry. Additionally, Karatsu et al. (2016) found negligible changes in device performance after irradiating the devices with a proton beam simulating the worst-case scenario for radiation absorption of a five-year observation in L2.
In this paper, we present a MKID design that is scalable for a 0.1-10 THz frequency band and for which we measure a NEP exp (P abs ) = 3.1 ± 0.9 × 10 −20 W/

√
Hz at a modulation frequency of 200 Hz averaged over all devices, using a thermal calibration source band-pass filtered around 1.5 THz. Here, P abs is the THz power absorbed in the detector. The design is optimised for low NEP and high responsivity, and will be easily scalable to kilopixel arrays because the readout requirements and fabrication are similar to earlier large system demonstrators (Baselmans et al. 2017).

Device design
We explain the device design referring to Fig. 1: the detector is an antenna-coupled hybrid MKID made from a wide niobium-titanium-nitride (NbTiN) coplanar waveguide (CPW) section loaded with an interdigitated capacitor (IDC) fabricated on a silicon wafer. At the shorted end of the CPW, the MKID consists of an aluminium section on a sillicon-nitride (SiN) membrane of 100 nm in thickness. The latter has a leaky-slot antenna and aluminium CPW line (hereafter referred to as the THz line) to couple to and absorb the THz radiation. The entire structure is a microwave resonator with a resonance frequency F 0 ≈ 3 GHz. Radiation absorption in the aluminium THz line modifies the Cooper pair-and quasiparticle density which results in a frequency shift of the resonator read-out using homodyne detection at F 0 (Day et al. 2003). Aluminium is chosen as the active material for its superior performance in terms of NEP with respect to other materials.
The first step in optimising the MKID design is to minimise NEP GR , which is the fundamental limit in MKID sensitivity, and is given by (Day et al. 2003;de Visser et al. 2014): where ∆ is the energy gap of the superconductor, n qp the density of thermally excited quasiparticles, τ * R the experimentally observed quasiparticle recombination time (Flanigan et al. 2016), and V the volume of the central strip of the THz line. To minimise NEP GR , we aim to reduce the volume (Hailey-Dunsheath et al. 2021), which implies that the THz line must be short while still absorbing all the power coupled in from the antenna. This requires a thin and narrow central line. We use a central conductor width w c = 0.4 µm (Fig. 1c) fabricated from a sputter-deposited 16 nm thick aluminium film with a sheet resistance of R s = 4.1 Ω/ , corresponding to a resistivity ρ = 6.5 × 10 −8 Ωm, and critical temperature T c = 1.54 K. To match the line to the input impedance of the antenna, Z antenna = 141 + 36i Ω, obtained from simulations in CST ® , we use a gap width w g = 1.2 µm, resulting in a characteristic impedance of the THz line of Z THz line = 194 − 103i Ω. The high imaginary component is caused by the high aluminium sheet resistance. Despite this, we find a good match between the antenna and the THz line, with the reflection coëfficient |S 11 | = −14.5 dB. Subsequently we use simulations in SONNET ® based upon the method described by Endo et al. (2020) to calculate the fraction of the power absorbed in the THz central line with respect to the total power absorbed. We find a high value of 91.5% caused by the high line impedance. Most of the remaining power, 8.2%, is absorbed in the ground plane, and the remaining losses are due to SiN absorption and re-radiation. We note that quasiparticles created in the ground plane do not contribute to the detector response because they diffuse away from the THz A17, page 2 of 10 line. A THz line length of 50 µm is sufficient to absorb 90% of the radiation, the minimum THz line length in our devices is 113 µm.
We further reduce the aluminium volume by using a single THz line coupled to the antenna, in contrast to the dual THz line design from Bueno et al. (2017). To prevent quasiparticle diffusion from the central line to the ground plane, we use a small NbTiN plug creating an Andreev mirror close to the antenna feed point (Andreev 1964), as shown in Fig. 1c. The distance d = 2.5 µm in Fig. 1c is motivated by two factors. First, the THz current density, and with that the ohmic losses in the NbTiN plug, strongly depend on the distance d from the start of the THz line, which we simulate in CST ® . For d > 2.5 µm, we find that the losses become independent of d and are limited only by losses in the aluminium film. Second, quasiparticle creation is most intense at the start of the narrow section of the THz line where the current density peaks. Initial quasiparticle excitations will relax to the gap edge of aluminium in about t II = 1.3 ns (Kozorezov et al. 2000), which corresponds to a diffusion distance d = t II 2ρN 0 e 2 ≈ 1.9 µm, using N 0 the single particle density of states in aluminium N 0 = 1.08 × 10 47 m −3 J −1 and e the electron charge. Hence, d must be larger than this value to prevent quasiparticles created at the start of the narrow section, which have not yet relaxed to the gap energy of the aluminium to escape over the NbTiN Andreev barrier.
A low NEP GR alone is not sufficient for the detector to actually reach this sensitivity experimentally. It is also crucial that the measured noise, and thereby the NEP, is dominated by photon fluctuations and not by excess noise sources. We therefore maximise the photon noise power spectral density as measured with an MKID, which is given by ) (2) when using the MKID phase θ as observable, as is the case in this work. Here, we omit the photon bunching term which is negligible for the power and frequency used in this work. The first term represents the photon number fluctuations, with F the photon frequency, h Planck's constant, f the modulation frequency, and dθ/dP abs the MKID phase responsivity to the absorbed power. It is useful to reformulate this equation in terms of quasiparticle excitations, as an MKID measures photon fluctuations indirectly by measuring the quasiparticles created. Using dN qp /dP abs = τ * R η pb /∆, P abs = N qp ∆/(2τ * R η pb ) (Flanigan et al. 2016) and the Mattis-Bardeen result for dθ/dn qp (Mattis & Bardeen 1958), we get where the first two terms express the fluctuations in quasiparticle number due to photon fluctuations and the last term represents the MKID responsivity to quasiparticle number changes, dθ/dN qp where N qp = n qp V is the total number of quasiparticles. Furthermore, β = 1 + 2t/λ sin h2t/λ ∼ 2 for thin films, with λ the bulk magnetic penetration depth and t the film thickness, |σ| the absolute value of the aluminium surface conductance σ = σ 1 − iσ 2 , which depends on the energy gap, quasiparticle density, and microwave frequency (Mattis & Bardeen 1958). Q is the resonator quality factor, α = L k,c /L tot , which is the ratio of the kinetic inductance of the THz central line with respect to the total inductance. We maximise S θ in three ways: (i) We increase both τ * R and η pb by placing the THz line on a very thin membrane (Fyhrie et al. 2016), in contrast with the 1 µm thick membrane used by Bueno et al. (2017). This enhances the re-trapping of 2∆ phonons emitted by quasiparticle recombination. The result is an increase in the phonon escape time τ esc which increases the experimental quasiparticle recombination Here τ R is the single particle recombination time as defined by Kaplan et al. (1976) and τ B = 0.26 ns is the pair-breaking time in aluminium (Guruswamy et al. 2014). The membrane thickness is approximately 100 nm, which is limited by its mechanical strength. We find τ esc /τ B = 0.65 using a simple phonon-trapping model , which is a factor four increase with respect to a solid substrate. A larger ratio of τ esc /τ B also increases η pb ; we estimate η pb = 0.37 based on the work of Guruswamy et al. (2014), which will be smaller for the same THz line on a solid substrate. (ii) We minimise the THz central line volume V as discussed before, and (iii) we maximise the kinetic inductance fraction α. For the kinetic inductance of the THz line itself, this is automatically achieved by using a narrow and thin THz line. However, α in Eq. (2) is defined with respect to the entire resonator and a λ/4 distributed resonator has a significant geometric inductance. To reduce the geometric inductance, we include a large IDC to ground in the MKID as shown in Fig. 1a, following Noroozian et al. (2009). The IDC-section has a much larger capacitanceto-inductance ratio than a CPW line, and therefore the same resonance frequency is obtained with less parasitic inductance in the NbTiN section. The effect is greatest for the shortest THz lines: α ≈ 0.6 for a 113 µm-long THz line, while a device without the IDC would have α ≈ 0.42. This is a relatively small difference because the THz line itself has a very high (kinetic) inductance due to its small dimensions and high sheet resistance.
The last step in our effort to maximise the photon noise (and GR noise) visibility is to reduce the TLS noise, which we achieve by widening the capacitive section of the resonator, including the IDC (Gao et al. 2008). We use a central line width w = 40 µm and a gap width of g = 8 µm. Wider structures become impractically large because of the onset of self-resonances within the IDC itself. Using a narrower gap increases the IDC capacitance, but will increase the TLS noise (Wenner et al. 2011). We therefore limit the ratio w/(w + 2g) to about 0.70. We note that an increase in Q does not provide a useful optimisation, because both the photon noise level and the TLS noise level scale with Q 2 .
For the radiation coupling, we use a leaky wave antenna (Neto 2010) because this antenna allows for relatively large feature sizes, enabling high radiation frequencies (Yurduseven et al. 2016), and because it must be fabricated on a membrane, compatible with our design optimisation discussed above. The antenna feed is placed in the second focus of a synthesised elliptical Si lens with a diameter of 1.550 mm. The antenna uses a small vacuum gap t λ to launch a leaky wave into the Si (Neto 2010). The antenna design is optimised for 1.5 THz resulting in a slot opening angle of θ = 45 • , a slot length of H = 200 µm, and a vacuum gap of t = 4 µm, as shown in Figs. 1b,d. The entire design of the KID can be optimised for frequencies up to ∼10 THz by changing the leaky slot dimensions and vacuum gap height.

Fabrication and assembly
The device fabrication is largely identical to that of Bueno et al. (2017)  define most of the structures. We start with a ρ = 10 kΩcm Si wafer on which we deposit 150 nm SiN on both sides using low-pressure chemical vapour deposition (LPCVD). The SiN is patterned to define the membrane areas on the front side of the wafer and etch masks on the back side, which are used in the last fabrication step to remove the Si, creating a free-standing membrane. Subsequently, we sputter deposit 207 nm NbTiN (T c = 15 K , ρ = 228 × 10 −8 Ωm) to define the resonators. The NbTiN etch requires a short over-etch which removes approximately 50 nm from the SiN. To define the THz line and antenna we sputter deposit a 16 nm aluminium film and pattern it in two steps: First we perform UV contact lithography and wet etching to define the readout line bridges and a large patch of aluminium covering the membrane area. We then use electron beam lithography and wet etching to define the final patterns, similarly to Mirzaei et al. (2020), resulting in a high-resolution definition of the structures as shown in Fig. 1c. The small black structures on the aluminium are small voids created during the 15-min 175 • C bake of the PMMA e-beam resist, likely caused by thinfilm dewetting (Thompson 2012). In the last step, we open the membranes by potassium-hydroxide (KOH) etching (8 h in 75 • C KOH:H 2 O 1:4), followed directly with a RCA2 clean (10 min in 70 • C HCl:H 2 O 2 :H 2 O 1:1:5) to remove potassium ions on the back side of the membrane.
The assembled detector array consists of the device chip, a 250 µm Si spacer wafer, and a laser-machined Si lens array commercially obtained from Veldlaser. A cross-sectional sketch of the detector assembly is given in Fig. 1d, and a photograph is shown in Fig. 2b. The spacer wafer top has a 40 nm β-Ta absorbing mesh (R s = 51 Ω/ ) with a design similar to that used by Baselmans et al. (2017) that acts as a low-pass filter at the MKID readout frequencies and as THz radiation absorber to eliminate stray radiation. Above each antenna, a 600 µm diameter hole allows radiation coupling to the antenna. On the other side, the spacer wafer has 4 µm thick lithographically defined pillars made from PermiNex ® , a spin-on photo-sensitive glue used to define the vacuum gap t as indicated in Fig. 1d and to provide a permanent bond between the chip and the spacer wafer. To assemble the final detector, we align the spacer wafer with the lens array and glue them together with two spots of 2216 epoxy (see Fig. 2b), which is cured for 30 minutes at 90 • C. At this temperature the PermiNex ® is unaffected. In the next step, we align the lens-spacer assembly to the chip and press the assembly together using a spring-loaded tool. The PermiNex ® is cured for 15 minutes at 180 • C to create a permanent bond between spacer and chip, which is found to be resilient against more than 20 thermal cycles down to 120 mK. A photograph of the entire detector chip without the lens array is given in Fig. 2a. The chip has 27 detectors, with four different device designs that differ in THz line length, which can be 913, 663, 238, or 113 µm. This corresponds to V = 6.0, 4.4, 1.6, and 0.85 µm 3 respectively, including the small volume near the NbTiN plug. All devices are designed with a coupling Q factor Q c = 120 × 10 3 .

Experiment
In the experiment, we measure the sensitivity, NEP exp (P s , f ), at several powers from a thermal radiator source P s using the setup depicted in Fig. 2. We denote with f the post-detection modulation frequency of the spectral shape of the NEP. We mount the detector assembly in a holder which itself is mounted inside a light-tight box mechanically anchored to the cold stage of an adiabatic demagnetisation refrigerator (ADR). All interfaces and venting holes of the holder and light-tight box are equipped with labyrinths coated with a radiation-absorbing layer from carbonloaded epoxy and SiC grains. A thermal radiation source weakly coupled to the 3 K stage of the cooler enclosed in a 3 K box is placed directly above, and can be heated to 30 K. In the first experiment, we used a 24 mm diameter radiator with a radiation absorbing layer from carbon-loaded epoxy and 0.2 mm SiC grains, as shown in Fig. 2c. Radiation coupling between the source and the detector is achieved through a 1.85 mm aperture at 42 mm from the lens surface. The use of the aperture reduces the throughput to the detectors, and therefore higher temperatures are needed to reach the same amount of source power at the detectors. This reduces the sensitivity of the radiation power to errors in the measured temperature. Furthermore, it allows a check as to whether the coupled power matches the beam pattern of the detector. We use eight filters in total -mounted on the radiator box, the light-tight box, and the sample holder -to define a pass-band of around 1.5 THz. These are the exact same filters as discussed by de Visser et al. (2014). The second experiment was identical, except that we used a 40 mm diameter radiator with 1 mm SiC grains and no aperture, to cross-check our results: In this case, the antenna pattern does not play a role because the opening angle to the radiator is much larger than the beam opening angle. Also, the total power coupled to the detector chip largely exceeds the power coupled to the single-mode antenna beams. Stray coupling to this excess radiation will result in A17, page 4 of 10 Fig. 3. Noise, response and NEP for a single detector. (a) Measured phase noise power spectral density for several source powers, increasing from bottom to top. There is a clear transition from detector-limited noise (lower curves) to photon-noise-limited noise, characterised by a white spectrum rolled-off with τ * R (highest curves). The insert shows the τ * R for the same source powers. (b) KID phase response around the lowest source power used to obtain the NEP, P s = 0.145 aW (P abs = 5.63 zW), which corresponds to a radiator temperature of 4 K. The data represent a measurement obtained by sweeping the radiator temperature from 3.95 to 6.21 K and back to 3.95 K. The data is plotted on a semi-logarithmic scale to show the absence of any response at low powers. The fits and fitted range are indicated as well. (c) MKID response and fits for increasing and decreasing radiator temperature around a higher source power, P s = 12.2 fW. We indicate the source power as well as the absorbed power in panels b and c for convenience. (d) Measured NEP for the same set of powers, now expressed in power absorbed in the detector. The legend is also valid for panel a.
overestimation of the coupling efficiency. In all cases, a polariser is used on the holder to allow only radiation in co-polarisation with respect to the antenna. We note that, because of the increased sensitivity of the detectors, we needed to upgrade the light-tight box with respect to the setup in de Visser et al. (2014) -as indicated in the insert of Fig. 2c -with better stray-light control around the filter mount. We use a CRYOPHY ® outer shield and niobium inner shield at 3 K surrounding the light-tight box and radiator to limit time-dependent magnetic fields. The theoretical coupling between the absorbing aluminium volume in the MKID aligned with the aperture is calculated using a CST simulation of the antenna beam and the tool provided by Zhang et al. (2019) to be η opt,calc. = 4.3%. The reason for the low coupling is the (intended) large spill-over loss between the small pin-hole aperture and the lens-antenna beam pattern. This includes the losses in the ground plane of the THz line and the antenna to THz line mismatch, which are discussed above. For the 40 mm diameter radiator without aperture, we find η opt,calc. = 76%.
We began experiments with a frequency sweep with the chip at 120 mK and the radiator at 3 K, which represents fully dark conditions. We find 23 out of 27 MKIDs with an average Q i = 0.23 × 10 6 ± 0.13 × 10 6 and an average Q c = 6 × 10 4 ± 2 × 10 4 at a readout power of −118 dBm. We selected a subset of the detectors that couple well to the aperture. For each detector we obtained NEP exp (P s , f ) using where P is the radiation power, given now by P s , which is the power from the source that can couple to a single mode. P s is calculated from the radiator temperature and the measured filter transmission using the formalism explained in detail in Appendix B of Ferrari et al. (2018). Figure 3a shows the phase noise power spectral density S θ ( f ) for one representative device, which has a resonance frequency of 2.74 GHz and an aluminium length of 913 µm. The spectrum is obtained from 64 s of time domain data sampled at 50 kilosample/sec with the readout tuned to the MKID resonance frequency while maintaining a constant black body temperature. At low powers, shown by the lower (blue) lines, we observe a noise spectrum with a 1/f slope. At increasing power, the noise level at frequencies f < 1 kHz increases sharply until the noise becomes white for P s > 1 fW.
The insert in the figure shows τ * R for the same powers obtained from a fit to the cross power spectral density of the resonator amplitude and phase noise (de Visser et al. 2012), which is given in Appendix A. We observe a saturation of the recombination time at around 0.3 ms and a reduction for increasing powers as expected. Subsequently, we obtain the responsivity dθ/dP s by a linear fit to the MKID phase response upon a small change of the black-body temperature. Care was taken to sweep the radiator temperature slowly to prevent hysteresis caused by a temperature difference between the thermometer and the radiating surface. In Fig. 3b, we show the response measurement and fit around the lowest P s of 0.145 aW (corresponding to a 4 K radiator temperature). We fit the response curve over a range of 1 aW < P s < 0.1 fW (which corresponds to a range in absorbed power 0.039 aW < P abs < 39 aW, as we show later): the high power limit is used so as to reach a statistically significant response which is still linear; the low end was chosen so as to reduce the sensitivity to drifts. It is clear from the figure that there is negligible hysteresis and that the fits are excellent over the whole power range spanning a factor 1000, with no response below P s = 1 aW. This proves that the setup is light tight; radiation leaks would allow unfiltered radiation to couple to the detector. The total integrated power per mode for a 4 K radiator is 10 6 times larger than the power in the narrow band around 1.5 THz, and it is strongly temperature dependent. Even a weak coupling to broad-band stray radiation would therefore result in a significant response of the detector at 4 K (P s = 0.145 aW) in Fig. 3b. For higher powers, we fit a symmetric power range around the power at which we measure the noise, as shown in Fig. 3c.  Fig. 3. The black line shows the theoretical curve. The absorbed power is evaluated from the source power using the measured efficiency as indicated in the insert, and the mean value of the points, η opt = 0.0389 ± 0.004, is given by the line. The statistical error bars are smaller than the dots. (b) The averaged coupling efficiencies for the two experiments. The red dots represent the results for the large radiator, where all detectors couple almost identically to the source as expected; the calculated coupling is given by the line. The black dots give the coupling for the pinhole setup, where only the best aligned detector is expected to reach the theoretical coupling value given by the grey square. We note that, in both cases, the measured coupling exceeds the theoretical value by the same factor of ∼1.65. (c) The measured NEP exp (P abs , f = 200 Hz) for the pinhole experiment, as displayed in Fig. 2c, together with the dark NEP obtained at the same modulation frequency using a change in bath temperature to obtain the response. (d) The measured NEP exp (P abs , f ) for the pinhole experiment at the lowest radiator temperature for all measured MKIDs. We observe no dependence of the NEP on aluminium volume. We note that panel a and Fig. 3 are given for the detector with F 0 = 2.74 GHz.
A possible source of systematic error is the radiator temperature readout: the detector response starts at P s = 1 aW, corresponding to a radiator temperature of 6.3 K, where a 50 mK temperature change would result in a 10% change in P s . For the full aperture setup, the response starts at a temperature of 4.5 K, which is due to the higher coupling. At this temperature, a 50 mK temperature change will result in a 20% change in P s . We therefore performed a cross-calibration between the Cernox ® radiator thermometer and another similar thermometer. The temperature difference between the two is less than 15 mK in the temperature range of the experiment, contributing less than 3% uncertainty in radiator power.
We can now obtain NEP exp using Eq. (4), and use it to determine the detector coupling efficiency η opt using the photon noise from the radiator as an absolute calibration source (Ferrari et al. 2018). This procedure is only valid if the MKID is photon-noise limited, which is the case for source powers exceeding 5 fW; here the spectrum is white with a single Lorentzian roll-off (Janssen et al. 2013). The optical efficiency is given by (Ferrari et al. 2018) where F ν is the filter transmission, O ν is the photon occupation number, and P s,ν is P s per unit frequency ν. The result is given in the insert of Fig. 4a. We get η opt = 0.039 ± 0.004 with very little scatter between the individual datapoints. We can now obtain the experimental NEP as a function of the power absorbed, NEP exp (P abs , f ), using Eq. (4) with P = P abs = η opt P s . The result is shown in Fig. 3d. At the lowest powers, the NEP at f = 200 Hz reaches a value of NEP exp (P abs f = 200 Hz) = 3.3 ± 0.3 × 10 −20 W/ √ Hz. At lower frequencies, the NEP is higher because of the 1/f noise. For higher powers, the NEP spectral shape becomes flatter, and for P abs > 53 aW the NEP is frequency independent down to 1 Hz. In Fig. 4a, we give NEP exp (P abs , f = 200 Hz) together with the theoretical value of the photon noise background from the radiation source and obtain a result very similar to those of de Visser et al. (2014), Baselmans et al. (2017) and Ferrari et al. (2018), but with ten times greater sensitivity. For a comparison of the NEP as a function of the absorbed power as shown in Fig. 4a for all measured MKIDs, we refer to Appendix C. For all MKIDs, the microwave readout power absorbed by the quasiparticle system, P abs,qp , is significant, especially at low absorbed powers: For example, at the lowest value of P abs = 5.63 zW, we estimate that P abs,qp 20 aW. At values of P abs ≥ 1 fW, P abs,qp and P abs become similar.
In Fig. 4b, we give the coupling efficiency for all MKIDs for the aperture-coupled radiator by the black dots, and the theoretical coupling for a well-aligned detector by the grey square. We observe that the MKID with the highest resonance frequency has the maximum coupling, as expected as this is the detector in the chip centre as can be seen in Fig. 2a. The measured η opt = 7%, exceeding the expected value of 4.3% by a factor 1.63. We observe six MKIDs with similar coupling but some scatter, which can be explained by a small misalignment of the aperture, causing unequal coupling to the six detectors at equal distance from the centre device. The red dots represent the results with the large radiator; we now see that all detectors couple identically with a mean value of η opt = 1.27 ± 0.04, exceeding the expected value by a factor 1.68. The fact that both experiments yield an identical coupling with respect to the theoretical value implies that both setups are equivalent and systematic errors such as an error in the beam pattern calculation, in-band stray radiation coupling with the large radiator, or a difference in temperature between the radiator surface and thermometer can be excluded. The higher measured transmission is therefore most likely caused by a higher transmission of our eight quasi-optical filters compared to the data provided by their supplier QMCI ® , which gives a peak transmission of 13.5%. A A17, page 6 of 10 potential reason is that QMCI ® performs the filter measurements at room temperature.
In Fig. 4c we give NEP exp (P abs , f = 200 Hz) for eight KIDs in the pinhole experiment and find a mean value of NEP exp (P abs , f = 200 Hz) = 3.1 ± 0.9 × 10 −20 W/ √ Hz. We also give the dark NEP (Baselmans et al. 2008;Janssen et al. 2014). To obtain the dark NEP, we measure the response of the MKIDs to a change in chip temperature while keeping the radiator in dark conditions, that is, T BB = 3 K. Under these conditions, the number of quasiparticles in the aluminium N qp can be calculated from the chip temperature, the volume of the aluminium section of the resonator, and the energy gap (Janssen et al. 2014). Rewriting N qp in terms of FIR power allows the dark NEP to be calculated using Eq. (4) by replacing the responsivity term dθ/δP by the dark responsivity given by Baselmans et al. (2008): where we obtain τ * R using a fit to the cross power spectral density of the phase-and amplitude noise as discussed above. For the operating temperature of 120 mK, a good fit cannot always be obtained. We find a mean value of NEP dark , ( f = 200 Hz) = 3.8 ± 1.3 × 10 −20 W/ √ Hz, in good agreement with NEP exp (P abs f = 200Hz). A similar agreement between dark and optical NEP was found by Janssen et al. (2014). Finally, panel d shows the NEP exp (P abs , f ) at the lowest radiator power for the same set of MKIDs. All have a similar sensitivity.

Discussion
The observed reduction in photon noise level for P abs < 228 aW (see Fig. 3a) is inconsistent with a NEP limited by a thermal reservoir of residual quasiparticles, which was the conclusion in previous work . However, it is consistent with quasiparticle trapping as discussed by de . The optimisation of the MKID responsivity, as discussed in the context of Eq. (3), is now especially relevant. A high responsivity allows for the device to remain background limited even when the photon-noise level drops. The use of a thin membrane, increasing both τ * R and η pb , is therefore an important optimisation. The downside is a strong reduction in instantaneous dynamic range when compared to the results of Baselmans et al. (2017). We find an instantaneous dynamic range of a few hundred aW, with readout re-tuning the devices operate up to 30 fW absorbed power.
Interestingly, we observe in Fig. 4 that the NEP exp (P abs , f = 200 Hz) is independent of the aluminium volume. More precisely, we observe that both the noise S θ and responsivity dθ/dP are independent of volume. For the noise, this is not surprising, because we are limited in this regime by amplifier noise and device noise, but the responsivity is expected to scale with the inverse of the volume, as can be seen from Eqs. (2) and (3). We find from the dark NEP measurement that dθ/dN qp does scale inversely with volume as expected. We also find that the low temperature quasiparticle lifetime τ * R scales inversely with volume. A smaller effect is that α is slightly smaller for the smallest volume devices due to parasitic inductance in the IDC, and the net result is the observed volume independence of the responsivity to power. Interestingly, the observed volume (or line length length) dependence in τ * R is not seen in the Appendix of de Rooij et al. (2021).
A better understanding of the trapping mechanism and the transition between trapping-dominated hybrid MKIDs and devices limited by quasiparticle creation due to the readout power will be important for further optimisation, which is especially relevant for the 1/f noise. If trapping is caused by surface states, as argued by de , device optimisation is possible using thicker aluminium films as this will reduce trapping and increase τ * R . Interestingly, thicker films will also increase τ * R as well as η pb due to enhanced 2∆ phonon recycling caused by the longer dwell time of the phonons in the film. This might partially compensate the responsivity reduction caused by an increase in the volume and reduction in α.
It is interesting to consider the intrinsic energy resolution for the possibility of single photon counting in the FIR. This resolution is given by Moseley et al. (1984): dE NEP exp (P abs , f = 200 Hz) · τ * R . We find an energy resolution averaged over all MKIDs of dE/h = 0.5 ± 0.2 THz, with a best value of dE/h = 0.24 THz for one of the devices with the smallest volume. Reliable single photon detection should therefore be possible for frequencies above a few THz. For more details, we refer to Appendix B.

Conclusions
In conclusion, we present a new MKID design, where radiation coupling and absorption take place in a small aluminium volume fabricated on a 100 nm SiN membrane embedded in a NbTiN resonator. The device is optimised for high quasiparticle responsivity and low NEP. We obtain a mean NEP exp (P abs , f = 200 Hz) = 3.1 ± 0.9 × 10 −20 W/ √ Hz with a time constant varying from 0.33 msec to 0.03 msec. The NEP is limited by 1/f noise and quasiparticle trapping, limiting the performance in dark conditions at low modulation frequencies. At absorbed powers of 53 aW and higher the device reaches background-limited operation with a flat noise level down to 1 Hz. The antenna design, device geometry, and assembly using PermiNex ® spin-on glue allows scaling to frequencies up to ≈10 THz. Importantly, creating kilopixel arrays from the presented device design is straightforward; the fabrication and readout are largely identical to those in the work presented by Bueno et al. (2018). All presented data, measurement data, and analysis scripts will be made available in Baselmans et al. (2022).