2.1.

Lynx X-Ray Microcalorimeter

The LXM focal plane has been proposed to include a main array and two subarrays (Fig. 1).¹ The main array covers a 5 deg field-of-view with 1 in. pixels, with 3-eV energy resolution at 7 keV. In the central 1 deg of the main array, an enhanced main array provides finer imaging capability and improves energy resolution to 1.5 eV using 0.5 in. pixels. Adjacent to the main array, an ultra-hi-res array provides 0.3-eV energy resolution up to 0.75 keV, accepting a much greater x-ray flux than the main array. All told, fully populating the LXM focal plane requires 102,600 pixels. The burden to read out all of these pixels will be met using a combination of thermal and electrical multiplexing.

Fig. 1

Illustration of proposed LXM focal plane,²⁰ including detector and pixel requirements for the main array (green), the enhanced main array at its center (red), and the ultra-hi-res array next to it (black).

The thermal multiplexing is based on the hydra pixels described in detail elsewhere in these proceedings,²¹ each consisting of a microcalorimeter that is coupled to multiple discrete absorbers by thermal links with different thermal conductances. Hydra detectors have been under development for over a decade by the NASA Goddard Space Flight Center (GSFC).⁶ The links are varied so as to give a different characteristic pulse shape for x-ray photons absorbed in each of the absorbers. The rising edge of each pulse is used to determine in which pixel the photon was absorbed, and the energy of the pulse is determined using an optimal filter algorithm that is calibrated for each specific absorber element. NASA GSFC has demonstrated nine-absorber hydra TESs with $<3\mathrm{eV}$ resolution at 6 keV and 20-absorber hydra TESs with 3.4 eV at 6 keV.²¹

Readout in a photon starved environment can be considerably simplified with hydra designs in which multiple spatial pixels are connected to a single detector and a single readout channel. A multiplexing circuit reading out hydra detectors must then be able to track the maximum slew rate of the fastest hydra pixel. The reduction in detector count and resonator count significantly eases both power dissipation and complexity, in both the cryogenic components and in the room-temperature electronics, and could be critically enabling for a satellite instrument like the LXM.

Even with 25-fold thermal multiplexing the main array will require $\sim 3500$ high-speed TES detectors. We expect each of these detectors to receive x-ray pulses at $\sim 10$ counts per second (cps) for the highest resolution x-ray events (40 cps for the medium resolution case where the record length for determining the energy is four times shorter and 150 cps for the lowest energy resolution x-ray events),¹ with slew rates as high as 1.5 A/s, which the readout must be able to track. The total count rates in the other subarrays are similarly demanding and will require large multiplexer bandwidth to read out. Electrical multiplexing factors of hundreds are therefore required, for a range of bandwidths per sensor from 0.86 to 6 MHz. Table 1 in Sec. 3.2 summarizes the requirements for the different subarrays, along with their microwave SQUID multiplexer solutions.

Table 1

Requirements for the various subarrays of the LXM. The ultra-hi-res array will not be read out at the same time as the main array and enhanced main array so at most 16 HEMTs will be operated at once.

2.2.

Microcalorimeter Multiplexing Technologies

In a multiplexer, the signals from different input channels are multiplied by a basis set of modulation functions and summed into a common output channel. The signals are then separated and decoded by inversion of this basis. The three most important sets of orthogonal basis functions used to date to multiplex TES detectors are low duty-cycle boxcar functions used in time-domain multiplexing (TDM), sinusoidal functions used in frequency-domain multiplexing (FDM), and Walsh functions used in code-division multiplexing (CDM).

The most technically mature multiplexing scheme for the readout of TES microcalorimeters is time-division multiplexing (TDM).^22,23 In TDM, the sensors are dc-biased and each is inductively coupled to its own first-stage SQUID that is modulated in time by a low duty-cycle boxcar function. Rows of SQUIDs are switched on sequentially in time, and columns of SQUIDs are read out in parallel. TDM readout enabled rapid scaling from single-pixel demonstrations in the early 1990s to multiple eight-column by 30-row (240-pixel) microcalorimeter arrays for x-ray spectroscopy today. However, in a TDM column with $N$ sensors, each sensor’s SQUID signal is sampled for only $1/N$ of the time, so high-frequency SQUID noise is aliased into the signal band, a problem that grows as $\sqrt{N}$.²² Up to $N\approx 30$, this “multiplexing disadvantage” is almost negligible but it becomes significant for higher multiplexing factors. The readout noise referenced to the input of the multiplexer channel can be improved by increasing the coupling between the input coil and SQUID, but this decreases the maximum slew rate that can be achieved by the same factor.⁷ Therefore, there is a trade-off between multiplexing factor, input referred readout noise, and maximum measurable slew rate. Furthermore, the state-of-the-art TDM has only achieved a total bandwidth of $\sim 10\mathrm{MHz}$ per column, preventing the readout of large numbers of truly fast sensors.²³

An alternative multiplexing scheme is MHz-band frequency-division multiplexing (FDM).²⁴ In FDM, the sensors are placed in series LC resonant circuits and ac-biased at different frequencies in a 1- to 5-MHz band. So far, FDM of microcalorimeters has produced a 6-pixel demonstration with an average resolution of $\sim 3\mathrm{eV}$ at 6 keV.²⁵ FDM of bolometric sensors has achieved multiplexing factors of up to 176 detectors per line,^26,27 but the $\sim 10\mathrm{MHz}$ total bandwidth again limits its usefulness for fast sensors.

CDM uses a different set of orthogonal modulation functions: Walsh codes, in which during each time step the signals from all dc-biased TESs are summed with equal weight but different polarity patterns.²⁸ In $\mathrm{\varphi }$-CDM,²⁹ the current signals from $N$ microcalorimeters are passively summed in $N$ different SQUIDs with different coupling polarities. A 32-channel $\mathrm{\varphi }$-CDM circuit with an array of TES x-ray microcalorimeters has been demonstrated giving single pixel energy resolutions from 2.4 to 3.0 eV FWHM, and obtained an energy resolution of the coadded spectrum of 2.77 eV at $\mathrm{Mn}\mathrm{K}\alpha$ (5.9 keV).³⁰ While CDM eliminates the aliasing penalty of TDM, it does not ease the 10 MHz bandwidth limit.

For non-hydra x-ray microcalorimeters, all three techniques can provide a path to arrays of ${10}^3\mathrm{pixels}$, and all three techniques are under consideration for the ATHENA x-ray integral field unit, an $\sim 3000\text{-}\mathrm{pixel}$ array scheduled to launch in 2031. However, due to the scale of the LXM array ($\sim \mathrm{100,000}\mathrm{pixels}$) and the slew rates of the hydra microcalorimeters, more readout bandwidth is needed than existing TDM, CDM, or FDM techniques can reasonably provide. To perform the electrical multiplexing required for the LXM, a higher bandwidth multiplexing technique is required. As will be shown in the following sections, microwave SQUID multiplexing is capable of providing the necessary bandwidth.

2.3.

Principles of Microwave SQUID Multiplexing

Microwave multiplexing dramatically increases the available bandwidth by moving the FDM carrier signals into the gigahertz regime. The microwave SQUID multiplexer ($\mu \mathrm{MUX}$)^10,11 divides the bandwidth available in a high electron-mobility transistor (HEMT) amplifier among input channels of superconducting microwave resonators coupled to a common feedline, in a similar approach to that of microwave kinetic inductance detectors (MKIDs).³¹

A microwave SQUID multiplexer uses rf-SQUIDs inductively coupled to cryogenic sensors to modulate the frequency of microwave resonators (Fig. 2). By coupling these resonators to a common microwave feedline with each resonator designed to resonate at a different frequency, we can read out all the sensors simultaneously. A microwave tone placed on one resonance measures its frequency shift and thus its detector signal; a superposition of microwave tones, one for each resonator, measures all detectors at once.

Fig. 2

(a) A schematic representation showing just three channels of a microwave SQUID multiplexing circuit with TESs. (b) A photograph of a 33-channel $\mu \mathrm{MUX}$ chip. (c) A close-up photograph showing a quarter-wave microwave resonator capacitively coupled to a feedline (right) and terminated with an rf-SQUID (left).

This technique allows a dramatic increase in multiplexing factor. Balancing noise, cost, and availability of components, we have chosen to work with commercial HEMTs in the 4- to 8-GHz band. With high-$Q$ (narrow-band) superconducting resonators and accurate frequency placement, we can fill this HEMT bandwidth with thousands of nonoverlapping resonances, each with its own SQUID. By digitally synthesizing the superposition of probe tones, we can read out all the SQUIDs simultaneously.

Flux-ramp modulation is applied to the rf-SQUIDs, in order to linearize the SQUID response.³² By ramping the flux in the rf-SQUID at a fixed frequency ($f_r$) by a integer number of flux quanta ($N$), the resonant frequency of the resonator is modulated at a frequency of $N{f}_r$. The flux in the SQUID can be determined by the relative phase shift of the modulated signal averaged over the period of the flux-ramp. The flux-ramp is applied to the rf-SQUIDs in the multiplexer using a common set of wires to keep the wire count low. The limits that flux-ramp modulation place on the maximum measurable slew rate of the sensor and how this affects the bandwidth of the resonators will be described in Sec. 3.1. In the standard algorithm for flux-ramp demodulation, a single phase shift is determined for each period of the flux ramp signal. Therefore, the current through the sensor is measured at a sampling rate equal to the flux-ramp frequency.

2.4.

Microwave SQUID Multiplexing Demonstrations

To evaluate the suitability of microwave SQUID multiplexing to readout the LXM, we should review the relevant demonstrations. The first demonstration of microwave SQUID multiplexing of TES microcalorimeters on scales larger than a few pixels is from the pathfinder project, SLEDGEHAMMER. This instrument uses 300-kHz resonator-bandwidth microwave SQUID multiplexer designs. SLEDGEHAMMER is a prototype high-resolution energy-dispersive gamma-ray spectrometer consisting of an array of 512 TES microcalorimeters divided into two arrays.^15,33 As the SLEDGEHAMMER array is similar to the proposed LXM x-ray microcalorimeter array, that is, the one we will review.

The SLEDGEHAMMER instrument has demonstrated readout of 128 microwave SQUID multiplexed channels on a single set of coaxial cables.¹⁵ The readout noise was sufficiently low ($\sqrt{S_I}\approx 35\mathrm{pA}/\sqrt{\mathrm{Hz}}$) to not significantly degrade the energy resolution of the TESs (55 eV at 100 keV). A photograph of the fully assembled 256-TES sample box is shown in Fig. 3(a), with microwave signals entering and exiting the box through the subminiature version A (SMA) connectors in the four corners, passing through four 33-resonator multiplexing chips on either side, allowing for readout of the entire box on two pairs of coax. The resonator bandwidth of 300 kHz and spacing of 6 MHz for this demonstration were matched to the requirements of the slower gamma-rays sensors designed to operate at photon energies of up to 210 keV.

Fig. 3

(a) Photograph of the sample box containing $8\times 32$-TES microcalorimeter detector chips (center), eight microwave multiplexer chips (outer vertical columns) and chips for detector bias, Nyquist filtering, and signal routing. Coax to microstrip to coplanar waveguide (CPW) adapters connect to the feedline on the multiplexing chips. (b) The microwave transmission through four microwave SQUID multiplexer chips on one side of the box showing resonance dips from 132 resonators.

Although the SLEDGEHAMMER instrument is currently the largest array of TES microcalorimeters using microwave SQUID multiplexing, a similar multiplexing factor has been demonstrated on an array of 104 soft x-ray TESs. These sensors, with detector time constants of ${\tau }_{\mathrm{fall}}\approx 400\mu \mathrm{s}$ were integrated with microwave SQUID multiplexers as shown in Fig. 4(a), with the four microwave SQUID multiplexer chips at the outside of the box and the x-ray sensor array in the middle. The combined x-ray spectrum of 82 of the channels is shown in Fig. 4 for x-rays from the magnesium $\mathrm{K}\alpha$ line at 1.25 keV achieving an energy resolution of 2.04 eV, consistent with the resolution achieved using TDM with the same array.

Fig. 4

(a) Photograph of the sample box for the 100 channel demonstration of microwave SQUID multiplexed readout of soft x-ray TESs. (b) Combined spectra from 82 TESs showing 2.04 eV resolution of the magnesium $\mathrm{K}\alpha$ line at 1.25 keV.

The same multiplexer designs were also used at NASA GSFC to read out an array of hard x-ray TESs.³⁴ The array had previously been measured with a conventional SQUID readout, nonmultiplexed, achieving a resolution of 2.8 eV FWHM at 5.9 keV. Five pixels from this array were multiplexed simultaneously achieving spectral resolutions between 2.8 and 3.1 eV FWHM.

The multiplexers used for these demonstrations do not have sufficient bandwidth to track the rising edge of pulses from the faster hydra microcalorimeters planned for the the LXM. However, microwave SQUID multiplexers with 2 MHz of bandwidth per resonator spaced by 14 MHz have also been demonstrated, see Fig. 5(a), although they are at an earlier state of development. These multiplexers have demonstrated appropriate modulation ($\sim 1.5\mathrm{MHz}$), shown in Fig. 5(a), and readout noise of $20\mathrm{pA}/\sqrt{\mathrm{Hz}}$, shown in Fig. 5(b) and been measured at flux ramp rates as high as 800 kHz with $2{\mathrm{\varphi }}_0$ ramp amplitude. These higher bandwidth designs have been used to measure high-speed x-ray TESs (${\tau }_{\mathrm{fall}}\approx 30\mu \mathrm{s}$). Figure 6(b) shows a 1.35-keV pulse from magnesium $\mathrm{K}\alpha$ line at 1.25 keV. Figure 6(a) shows measured spectrum with a 1.37-eV FWHM resolution at 1.25 keV. Although these high-speed multiplexers have not yet been used in large scale demonstrations, they will soon be demonstrated in the HOLMES³⁵ and the TES x-ray spectrometer at second-generation Linac coherent light source (LCLS-II)³⁶ currently under development.

Fig. 5

(a) Measured ${|S_{21}|}^2$ as a function of frequency for seven different values of input current. The shifting of the resonance demonstrates the modulation range of the 2-MHz design. (b) Measured current noise spectral density from one channel of the microwave SQUID multiplexer with bandwidth of 2 MHz and the TES unbiased (i.e., in the superconducting state). The readout noise ($\sqrt{S_I}\approx 20\mathrm{pA}/\sqrt{\mathrm{Hz}}$) can be seen above the roll-off of the TES noise.

Fig. 6

(a) Measured spectrum showing 1.37 eV FWHM resolution of the magnesium $\mathrm{K}\alpha$ line at 1.25 keV measured at flux-ramp modulation at 800 kHz. (b) A fast x-ray pulse read out using microwave SQUID multiplexing and flux-ramp modulation at 800 kHz. Each sample represents the demodulation of one flux-ramp.

3.1.

Fundamental Performance of Microwave SQUID Multiplexing

The microwave SQUID multiplexer has two primary noise sources: HEMT amplifier noise and two-level-system (TLS) noise.³⁷ The TLS noise arises from the switching of two-level systems with electric dipole moments, coupled to the resonator. As these TLSs change state, they change the effective electrical length of the resonator, producing resonance-frequency noise. The power spectral density of this noise has characteristic $1/\sqrt{f}$ dependence on frequency. This dependence makes it the dominate signals close to the carrier. The HEMT is a broadband noise source and can be described by a noise temperature, typically $T_N\approx 3\mathrm{K}$. The HEMT noise dominates farther from the microwave carrier frequency because the TLS noise falls off with frequency. The frequency of modulation used is chosen to ensure the signal appears at a frequency where the TLS noise is subdominant.

The SQUIDs provide the gain that boosts both the signal and the noise of the detectors above the noise of the cryogenic HEMT amplifier. The effective gain is determined by two factors: the fractional change in the microwave transmission function and the power of the microwave tone probing the resonance. The optimal power per tone can be shown to be independent of the resonator bandwidth, so long as the frequency shift has been optimized to match, and is approximately $-71\mathrm{dBm}$. At this power level, we observe a readout noise level of 2 to $3\mu {\mathrm{\varphi }}_0/\sqrt{\mathrm{Hz}}$. With a typical input coupling of $M_{\mathrm{in}}\approx 230\mathrm{pH}$,^15,33 this adds an input-referred current noise of 18 to $27\mathrm{pA}/\sqrt{\mathrm{Hz}}$. This noise is low enough not to degrade the TESs proposed for the LXM. If necessary, the current noise can be improved by increasing the input coupling, but the optimization of the noise needs to be balanced against other important sensor parameters such as dynamic range and crosstalk as will be discussed later in this section.

The SQUID response is linearized through the flux-ramp modulation³² mentioned earlier. This is necessary because, while the response of an rf-SQUID to applied flux is periodic with a magnetic flux quantum ${\mathrm{\varphi }}_0$, this use of flux-ramp modulation provides a dynamic range that is effectively infinite, subject to a slew rate requirement. The flux-ramp demodulation algorithm, which must identify phase shifts of the SQUID response between one ramp period and the next, requires that the rapidly changing input signal during the rising edge of a pulse induce a change of less than half a flux quantum during a single flux ramp. Otherwise, it can for example confuse a phase shift slightly greater than $\pi$ with a phase shift slightly less than $-\pi$. Close to this limit, it can also degrade signal-to-noise and introduce small systematic errors. For accurate flux-ramp demodulation, we therefore require the maximum input signal slew rate SR to obey

A number of instruments under development plan to track the resonance frequency of each resonator using a feedback loop instead of using fixed frequency probe tones.^36,38 Since the probe tone spends more time deeper in the resonance less power per tone is transmitted to the amplifiers and mixers in the readout chain. This has the advantage of lowering the linearity requirements for the amplifiers and mixers. There is also potential to use this technique to ramp the resonator faster than specified in Eq. (1) and to get more than one demodulated phase data point per flux ramp. In this paper, we take a conservative approach and only consider techniques that have been previously demonstrated in TES microcalorimeters. Therefore, Eq. (1) will be used to define the maximum slew rate when considering configurations to meet the requirements of the LXM. However, any continuing advancements could be used to provide further margin.

Crosstalk requirements ultimately determine the frequency spacing between adjacent multiplexer channels. Due to the Lorentzian tails of each resonator’s microwave transmission (Fig. 7), a change in frequency of one resonator leads to a change in measured transmission at its neighbors. It can be shown¹¹ that a change in one resonance frequency of $\mathrm{\Delta }\omega$ leads to crosstalk into its neighboring resonators that in the worst case is $\frac{\mathrm{\Delta }\omega }{16n^2}$, where $n$ is number of bandwidths between resonators. Spacing resonators by $\sim 10\mathrm{times}$ their bandwidth thus leads to a maximum crosstalk below one part per thousand. Using the bandwidth set by input signal slew rate, this crosstalk constraint determines the number of resonances that can be packed in a block of readout bandwidth.

Fig. 7

Lorentzian transmission of a pair of resonators spaced at twice their bandwidth (left) and eight times their bandwidth.

For a given application, there is a chain of optimizations: (1) a target current noise determines an input coupling, (2) a target slew rate determines a flux-ramp repetition rate, (3) the flux-ramp rate determines resonator bandwidth, and (4) a crosstalk requirement determines the spacing between resonances and therefore the number of channels that can be multiplexed on a single line.

For example, a resonator bandwidth of 2 MHz allows for a $2{\mathrm{\varphi }}_0$ flux-ramp frequency of up to 500 kHz, satisfying the requirements [Eq. (1)] in which signals of interest have a peak slew rate of $\sim 2.3\mathrm{A}/\mathrm{s}$ for an $M_{\mathrm{in}}=230\mathrm{pH}$. In general, the required bandwidth (BW) of each microwave SQUID resonator is

3.2.

Microwave Multiplexing for the LXM Requirements

Table 1 summarizes the proposed readout resulting from optimization to meet the LXM system requirements. This table estimates the number of HEMT amplifiers necessary to read out the pixel count of the various LXM subarrays, with a $\sim 50\%$ safety margin of resonator bandwidth relative to expected detector slew rate, and with the resonators spaced by seven times the bandwidth for low crosstalk.

The optimization of thermal multiplexing relative to electrical multiplexing is described elsewhere in this proceedings,¹ but results in a requirement of at most 16 HEMTs to be read out at once. As shown in Table 1, this requirement is satisfied, since the ultra-hi-res array will not be read out at the same time as the others.

Design parameters for the main array predict that the maximum slew rate during a pulse can be as high as 1.5 A/s for the fastest pixel. For an input coupling of $M_{\mathrm{in}}\approx 170\mathrm{pH}$, this becomes $0.12{\mathrm{\varphi }}_0/\mu \mathrm{s}$, requiring a flux-ramp repetition rate of at least 250 kHz. To provide for this with a 50% margin, the proposed resonator bandwidth is 1.4 MHz spaced at 10 MHz, enabling readout of $\sim 400$ detectors (10,000 spatial pixels) on a single pair of coaxial cables and one HEMT. As described in Sec. 2.2, the National Institute of Standards and Technology (NIST) has designed and fabricated multiplexers that have 300 kHz and 2 MHz bandwidth resonators. Scaling parameters to make a 1.4-MHz bandwidth design for the main array is very low risk.

The LXM focal plane has been proposed to include two other subarrays. The enhanced main array covers a narrower field-of-view, but is intended to read out higher x-ray flux signals. This requires wider bandwidth per microwave multiplexer readout channel. The ultra-hi-res array uses non-hydra detectors to allow for 0.3-eV energy resolution at high count rates. This requires one microwave multiplexer channel per pixel, but at moderate bandwidth. The process of scaling microwave SQUID multiplexer designs to other bandwidths is described in the following section.

3.3.

Scaling the Bandwidth of Microwave SQUID Multiplexers

The resonance width can be defined in terms of a dimensionless parameter known as the quality factor ($Q={\omega }_0/\mathrm{BW}$). The total $Q$ of the resonator is determined by $Q_i$ due to internal losses and $Q_c$ due to the coupling between the resonator and the feedline as

An equation for $Q_c$ is derived in Ref. 11 considering for a circulating energy in the CPW resonator of characteristic impedance $Z_1$, how much power leaks out through the coupling capacitor $C_c$ onto the feedline $Z_0$. This yields

Across the 4 to 8 GHz bandwidth proposed for the LXM, the ideal coupling capacitor varies significantly. Therefore, the coupling is separately optimized for each sub-band. In practice, microwave simulation tools are used to simulate the coupling $S_{13}$ through the coupling capacitor, with port 1 and port 2 being the two sides of the feedline and port 3 the resonator.

Once the bandwidth of the resonator is optimized, the frequency shift of the resonator must be matched to the bandwidth. If the peak-to-peak SQUID response is less than the bandwidth, then the multiplexer does not achieve optimum gain. If the peak-to-peak SQUID response is greater than the bandwidth, then there are sections of the flux ramp where the SQUID response is insensitive to input flux. Furthermore, if the peak-to-peak response is greater than the bandwidth, it becomes possible for the resonator to bifurcate, toggling between two self-consistent states of resonance frequency and internal energy.

The frequency of a quarter-wave resonator is shifted due to the screening of the termination inductance by the rf-SQUID, shown schematically in Fig. 8. An equation for this shift is also derived in Ref. 11. The change in frequency for small changes in load inductance is

Fig. 8

Schematic of the inductor at the termination of the resonator $L_c$ coupled to an rf-SQUID with inductance $L_S$ with mutual inductance $M_C$. The rf-SQUID screens $L_c$ and gives and effective inductance is a function of the input flux of the SQUID.

Using this strategy, we have optimized the microwave SQUID multiplexer for a variety of maximum input slew rates. Since the newest hydra prototypes have faster slew rates than the final versions, we designed for wider bandwidth than optimum for the enhanced main array. With both a 2-MHz design and an 8-MHz design in hand, we will be able to measure all prototype hydra microcalorimeters and demonstrate resonator bandwidth over the needed range.

The 8-MHz bandwidth design and fabrication are complete and the devices are currently being tested. The capacitive coupling to the resonator, shown in Fig. 9(a), was increased to $C_c\approx 24$ fF to achieve 8-MHz resonator bandwidth at 6 GHz. The coupling between the resonator and the SQUID was been increased to $M_c\approx 10\mathrm{pH}$, to match the 8-MHz bandwidth. The coupling was calculated using a three-dimensional inductance extraction program and the device is shown in Fig. 9(b).

Fig. 9

(a) Example coupling capacitor between the resonator and the microwave feedline optimized for 8-MHz bandwidth. (b) Photograph of the gradiometric coupler between rf SQUID, resonator, input coupling coil, and flux-ramp coupling coil to optimize noise and slew rate for the LXM enhanced main array. The self inductances are on the main diagonal and the mutual inductances are on the off diagonal of the table.

3.4.

Advanced Spread Spectrum Multiplexers

Microwave SQUID multiplexers have the potential for more efficient bandwidth utilization under low count rate conditions by the implementation of a spread-spectrum multiplexing (SSMux) design.³⁹ SSMux operates by combining circuit elements developed for microwave SQUID multiplexers and Walsh code-division SQUID multiplexers (CDM).³⁰ The SSMux takes the signal from each detector pixel and deliberately spreads it in the frequency domain across multiple microwave SQUID resonators to increase the multiplexing factor, see Fig. 10.

As previously described, the bandwidth required by each pixel in a microwave SQUID multiplexer is set by the maximum slew rate on the leading edge of the pulse. However, at any given time, few pixels are in the steep part of the leading edge of a pulse, where the slew rate is highest. The multiplexing factor and slew-rate budget can therefore be increased by spreading the flux signal from each pixel over $N_{\mathrm{ss}}$ resonators in a Walsh code. In SSMux, the signal from each TES is coupled to multiple resonators and each resonator is coupled to multiple TESs in a Walsh code. In this scheme, the total number of microwave SQUIDs is still equal to the number of pixels—which may be hydras—but the high slew rate of a pixel on the steep rising edge of a pulse is divided between $N_{\mathrm{ss}}$ different resonators, reducing the slew rate in each, and reducing the bandwidth each resonator requires. The reduction in required bandwidth makes it possible to pack the resonators closer, allowing a higher multiplexing factor in each HEMT, either reducing the number of HEMT channels, increasing the number of pixels, or expanding engineering margins (Fig. 10).

Fig. 10

The electrical schematic of a four-pixel implementation of spread-spectrum SQUID multiplexer (SSMux). The current from each TES couple to all four SQUIDs shown, with coupling polarities modulating in a Walsh code. There are an equal number of SQUIDs and TESs, but the flux-slew burden from the leading edge of an x-ray pulse is shared among all SQUIDs.

Walsh CDM of TES detectors into time-division multiplexed SQUIDs is well established.³⁰ With the $N_{\mathrm{ss}}$ detectors coupling to the $N_{\mathrm{ss}}$ SQUIDs according to an orthogonal Walsh code, the code can be inverted, extracting independent measurements of each TES. When the Walsh code is inverted, combining the $N_{\mathrm{ss}}$ measurements reduces the effective SQUID noise amplitude by $\sqrt{N_{\mathrm{ss}}}$.

The mutual inductance $M$ can therefore be reduced by as much as $\sqrt{N_{\mathrm{ss}}}$. As long as only one of the TESs in the Walsh set is in a high-slew-rate condition at a given time, the maximum flux slew rate applied to this resonator is reduced by $\sqrt{N_{\mathrm{ss}}}$, and the resonators can be spaced closer in frequency space, increasing the multiplexing factor by up to $\sqrt{N_{\mathrm{ss}}}$. The analysis of this multiplex factor advantage is studied in detail in Ref. 39.

SSMux is both similar to and complementary to hydras. Hydras reduce the number of wires that must be extracted from the focal plane by implementing thermal multiplexing, but the scale of implemented hydras is constrained by pulse pileup. If an x-ray pulse arrives in one hydra subpixel during a pulse in another hydra subpixel, the energy resolution of both counts is significantly degraded. In contrast, SSMux does not reduce the number of leads extracted from the focal plane, but it scales more favorably with pulse pileup. An optimal implementation of an x-ray calorimeter system may incorporate both hydras and SSMux in a complementary configuration.