Calculation of the mode composition in the horn constriction at the incidence plane of a linearly polarized plane wave

The parameters of the back-to-back horn are defined in [3]. They consist of two openings and a constriction connecting them, see the details in [3] and a part of the horn in Figure 1. One of the openings defines the radiation pattern of the receiving system, and the other is facing the receiving antenna array. The constriction of the horn determines the modal composition of the electromagnetic field passing through the horn. For a given constriction diameter, up to 38 modes of a circular waveguide are propagating in the operating frequency range.

Figure 1: The front part and the constriction of the back-to-back horn of the LSPE-SWIPE receiving system [3] used for the calculation of the mode composition in the constriction.

For correct numerical modeling of the frequency response of the receiving system, it is necessary to know the distribution of the field of the incident wave on the matrix of receiving cells. The most consistent would be modeling a back-to-back horn irradiated by a plane electromagnetic wave with a receiving matrix installed in the opposite opening. However, this task requires a lot of computational resources. Therefore, the problem of irradiating the receiving matrix with an incoming radiation was divided into two parts. The first problem was to determine the mode composition of the radiation passing through the constriction of a back-to-back horn when a plane electromagnetic wave falls on it (Figure 1). The second problem was to calculate the power absorbed by the receiving matrix when this matrix is irradiated by the electromagnetic field of a round waveguide port with a calculated mode composition.

The result of the calculation of the mode composition in a back-to-back horn is shown in Figure 2. It can be seen that over the entire operating frequency range of the LSPE-SWIPE, modes with numbers 1 (main), 2, and 4 are present in the horn. Starting from the frequency of 150 GHz, mode 7 appears; amplitudes of modes 7, 9, and 13 are associated with the achievement of the corresponding cutoff frequencies of these modes. The amplitudes of the remaining modes are negligible compared to the amplitudes of the listed modes and can be ignored. The calculated mode amplitudes can be used to calculate the frequency response of antenna arrays with integrated bolometers for the main and auxiliary frequency channels of the LSPE-SWIPE.

Figure 2: The mode composition of the electromagnetic field in the constriction of a bidirectional horn as a function of frequency.

Calculation of the frequency response of a planar antenna matrix with integrated cold-electron bolometers

The frequency response of the receiving matrix was calculated as follows. The radiation incident on the matrix is formed by the waveguide port of a circular waveguide with a diameter corresponding to the wide part of the bidirectional horn facing the receiving system (on the right in Figure 1). An RC circuit was used as an equivalent circuit for CEB at high frequencies. In the process of numerical simulation, we calculated the dependence of the power, Pi, released on the active resistance of the i-th RC-chain of the array of receiving cells, on the frequency of the incident radiation, and the total power in all receiving cells:

(1)

The summation was performed over the cells included in one frequency channel. Thus, as a result of the calculation, the frequency response of the receiving system of one frequency channel was obtained. This calculation principle was used for all types of receiving matrices of the LSPE-SWIPE auxiliary frequency channels.

Main frequency channel of the LSPE-SWIPE

The LSPE-SWIPE 145 GHz main channel receiving system is based on an array of dipole antennas with integrated CEBs (Figure 3). The receiving cells are located within a circle under the opening of a bidirectional horn. Its characteristic feature is that the antenna dipoles and wires that provide the CEB DC bias are located in the same layer. This significantly simplifies and reduces the cost of manufacturing such structures.

Figure 3: Receiving system of the LSPE-SWIPE145 GHz main channel. a) A quarter of receiving cells matrix on the 14 mm chip. b) A single cell of the receiving system based on a dipole antenna.

The frequency response of the LSPE-SWIPE 145 GHz main frequency channel is shown in Figure 4. The received frequency band of the main frequency channel at the level of 0.5 is 45 GHz. At 220 GHz there is a spurious resonance that can be suppressed with a low-pass filter.

Figure 4: Frequency response of the LSPE-SWIPE 145 GHz main frequency channel.

Auxiliary frequency channels of the LSPE-SWIPE

The cells are located within a circle with a diameter of 4.5 mm; while one half of the circle is occupied by cells for 210 GHz, the second half is occupied by cells for 240 GHz. Receiving cells with integrated CEBs are located on a 260 µm thick silicon substrate with a silicon dioxide layer. The difficulty of the considered receiving system is the existing specific sample holder, which allows the receiving of the signal from the front side of the pixel only with the back-short placed at the back side of the pixel. Such configuration leads to tougher constraints and less number of free parameters than in [10], where the signal was coming to the antennas through the substrate.

In this work, the receiving system based on a bow-tie dipole antenna was calculated (Figure 5) with the DC bias lines connected to the central parts of the antennas.

Figure 5: a) Receiving cell array based on bow-tie antennas; half of the 7 mm plate on the left are 210 GHz and on the right 240 GHz frequency channels. b) A receiving system cell based on a bow-tie antenna.

The frequency response of the receiving matrix with bow-tie antennas is shown in Figure 6. Rather good band separation is visible in spite of a certain cross-talk of the 210 GHz channel. The frequency response width at 50% of the maximum level for the 210 GHz channel is 26 GHz and for the 240 GHz channel is 38 GHz. In addition, the radiofrequency (RF) tails of the 240 GHz channel above 270 GHz are supposed to be suppressed by a band-pass filter.

Figure 6: Frequency response of a matrix of receiving cells based on bow-tie antennas for 210 GHz and 240 GHz frequency channels.

Noise-equivalent power calculations for the LSPE-SWIPE

The noise-equivalent power (NEP) is a measure of a minimal signal that can be detected, and it qualifies the sensitivity of a detector. The NEP is the ratio of the total system noise (which includes both the internal noise of the detector and the photon noise of the received signal, depending on the signal power) with respect to the responsivity, which can be calculated using the heat balance equations [6].

The power load for the LSPE-SWIPE 145, 210, and 240 GHz frequency channels should be 11, 12.4, and 16 pW, respectively, as stated in Table 4 in [2]. The total NEP level should be approx. 7 × 10−17 W/Hz1/2 for the 145 GHz channel, 8.5 × 10−17 W/Hz1/2 for the 210 GHz channel, and 1.3 × 10−16 W/Hz1/2 for the 240 GHz channel at the working point for a background (photon-noise) limited operation. However, the readout system is based on SQUIDs, and this system has an input current noise of 4–10 pA/Hz1/2. To optimize the receiver for better noise characteristics, we consider optimized CEBs with a single superconductor–insulator–normal (SIN) tunnel junction and a single superconductor–normal (SN) contact [12]. Combined together, they form a SINS structure.

This solution can help reaching better noise characteristics than those of CEBs with two SIN tunnel junctions due to several reasons. First, the responsivity is increased by a factor of two due to hot electrons tunneling only through one SIN junction. Second, the bolometer resistance is decreased twice, which helps in array matching with a SQUID readout (the total resistance of this array should be 1 Ohm). Third, the electron cooling efficiency is increased by a factor of two due to the two-fold increase in the readout for the same power going through the system. Last but not least, the absence of a second SIN tunnel junction suppresses the Coulomb blockade, so the absorber volume can be decreased by a factor of four, leaving the capacitance unchanged.

Therefore, the current responsivity is increased from 40–45 nA/pW to 80–100 nA/pW. Thus, the total NEP for this CEB concept should also be two times better than the total NEP for CEBs with two SIN tunnel junctions and it should be close to the required photon NEP level for the LSPE-SWIPE frequency channels. This comparison for arrays of 66 SINS CEBs with a 6 pW power load is shown in Figure 2 in [12].

For better NEP estimation we should also account for the matching of the CEB array with the SQUID readout system. The mismatching can happen if there is dynamic resistance of the CEB array at the working point with a parasitic resistance of connecting wires higher than 1 Ohm. Then the noise of the SQUID readout system is multiplied by the square root of the ratio of the obtained dynamic resistance of the CEB array and the connecting wires to the required value of 1 Ohm. The NEP estimations for different numbers of CEBs in a parallel array for 145, 210, and 240 GHz frequency channels accounting for possible mismatching with the SQUID readout system are shown below with a SQUID noise of 10 pA/Hz1/2.

In Figure 7 and Figure 8 it is seen that the total NEP level (green curves), which includes the NEP of SINS junctions (blue curves) and the NEP of the SQUID readout system (purple curves), nearly approaches the photon NEP level (red line), so the increase in the number of CEBs improves the total NEP. For 200 SINS CEBs of the 240 GHz frequency channel (Figure 9, solid curves), the total NEP curve reaches the photon NEP line at the working point. The minimal value of total NEP in this case is 1.15 × 10−16 W/Hz1/2.

Figure 7: NEP estimations for the 145 GHz channel with 11 pW power load. Dashed curves are for 44 SINS CEBs; solid curves are for 200 SINS CEBs.

Figure 8: NEP estimations for the 210 GHz channel with 12.4 pW power load. Dashed curves are for 44 SINS CEBs; solid curves are for 200 SINS CEBs.

Figure 9: NEP estimations for the 240 GHz channel with 16 pW power load. Dashed curves are for 44 SINS CEBs; solid curves are for 200 SINS CEBs.

These estimations were calculated for an input current noise of the SQUID readout system in the worst case scenario of 10 pA/Hz1/2. If this noise is reduced twice (i.e., to 5 pA/Hz1/2) the minimal value of the total NEP for 200 SINS CEBs of 240 GHz frequency channel falls below 7 × 10−17 W/Hz1/2. In addition, the photon-noise limit can be easily reached for all frequency channels with the use of the Quasiparticle Cascade Amplifier (QCA), which can be added to the SINS CEB. The great benefits of using the QCA are described in detail in [12].