Characterising the optical response of ultra-low-noise far-infrared 60-110 μ m transition edge sensors
Emily A. Williams, Stafford Withington, David J. Goldie, Christopher N. Thomas, Peter A. R. Ade, Rashmi Sudiwala
CCharacterising the optical response of ultra-low-noise far-infrared 60-110 µ m transition edge sensors E. A. Williams, S. Withington, D. J. Goldie, C. N. Thomas, P. A. R. Ade, and R. Sudiwala Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 OHE,United Kingdom School of Physics and Astronomy, Cardiff University, Cardiff CF24 3YB, United Kingdom (Dated: 20 August 2020)
Far-infrared Transition Edge Sensors (TESs) are being developed for the SAFARI gratingspectrometer on the cooled-aperture space telescope SPICA. In support of this work, we havedevised a cryogenic (90 mK) test facility for carrying out precision optical measurementson ultra-low-noise TESs. Although our facility is suitable for the whole of the SAFARIwavelength range, 34-230 µ m, we focus on a representative set of measurements at 60-110 µ musing a device having a Noise Equivalent Power (NEP) of 0.32 aW / √ Hz. The system isable to perform a range of measurements: (i) Dark electrical characterisation. (ii) Opticalefficiency with respect to a partially coherent beam having a modal composition identicalto that of an ideal imaging telescope. (iii) Optical saturation and dynamic range. (iv) Fastoptical transient response to a modulated thermal source. (v) Optical transient response inthe presence of high-level background loading. We describe dark measurements to determinethe operating characteristics of a TES, and then compare predicted optical behaviour withmeasured optical behaviour. By comparing electrical and optical transient response, we wereable to observe thermalisation in the device. We comment on the challenge of eliminatingstray light.
I. INTRODUCTION
Transition Edge Sensors (TESs) use the stronglytemperature-dependent resistance of a superconduct-ing film biased on its transition to detect energyand power with exceptional sensitivity . Dominant insubmillimetre-wave ground-based astronomy, TESs areunder further development for the next generation ofspace observatories, including LiteBIRD , SPICA ,and ATHENA , at submillimetre, infrared and X-raywavelengths respectively.The SAFARI instrument on SPICA is a grating spec-trometer covering the wavelength range 34-230 µ m inthree wavebands, each with a dedicated TES array. Thecomplete instrument will require of order 2250 TESsread out using superconducting frequency domain mul-tiplexers (FDM) . Because SAFARI will use a post-dispersed Martin-Puplett interferometer to provide bothhigh and low resolution observing modes, the TESs willbe arranged in three grating arrays of 5 ×
150 pixelseach. In addition, because the primary mirror of SPICAis cooled to 4 K, a Noise Equivalent Power (NEP) of 0.2aW / √ Hz is required to ensure that SAFARI is detec-tor noise limited. This NEP is two orders of magnitudesmaller than that needed for ground-based observatories,and can only be obtained by cooling the TES arrays,FDM filters and SQUID readout electronics to 50 mK.Here we discuss the design and construction of a cryo-genic test facility for characterising the dark and opticalbehaviour of ultra-low-noise TESs. The optical config-uration enables optical efficiency to be determined withrespect to a partially coherent beam having a modal com-position identical to that of an ideal imaging telescope. Inaddition to steady-state illumination, a fast infrared ther-mal source allows TES transient response to tiny changesin optical power to be examined over timescales of tens of milliseconds. It is also possible to measure transientresponse in the presence of thermal background loading,increasing to the point of optical saturation. This infor-mation is important because the dynamical behaviour ofthe TES arrays must be matched to other time constantsin the SAFARI instrument, such as the scan speed of theFTS, the slewing rate of the telescope, and readout anddata rates.Given the extreme sensitivity of these sensors, it wasimperative to address stray light elimination, and mag-netic and RF shielding, when configuring the appara-tus. Ultimately, these considerations have direct impli-cations for realising optimal TES performance in space-flight hardware. Here, we draw particular attention tosome of the subtle problems associated with eliminatingsteady-state and fluctuating stray light.We demonstrate TES characterisation using the opti-cal test facility on a representative device in an array forthe SAFARI M-band, 60-110 µ m. This is the first pro-totype SAFARI M-band array that we have fabricated,building on earlier demonstrations covering the 34-60 µ mand 110-210 µ m wavebands . We report dark charac-terisation, TES optical efficiency, and electrical and opti-cal transient response for parametric changes in steady-state background optical loading, the latter constitutingour first such measurements on any TES device. II. CRYOGENIC OPTICAL TEST FACILITY
The test facility was based on a two-stage adiabaticdemagnetisation refrigerator (ADR), backed by a pulsetube cooler (PTC). The PTC provided a 3.3 K temper-ature stage, with the ADR giving an intermediate 1 Kstage and final 50 mK stage. The ADR magnet currentwas controlled in a servo loop to hold the physical tem- a r X i v : . [ phy s i c s . i n s - d e t ] A ug FIG. 1. Simplified representation of the test facility, showingthe optical path from the blackbody hot load to the detectorarray. perature of the TES array constant, at 90 mK unlessotherwise specified in Section V, with an RMS deviationof 140 µ K from target over a typical measurement set.A simplified representation of the optical configura-tion is shown in Fig. 1: those components at the basetemperature of 50 mK are shown in blue, whereas thoseat 3.3 K are in red. A cryogenic blackbody infrared load(shown in purple), with a slow response time, illuminatedthe detector array through a thermal radiation filter, aband-defining filter stack, and an optical aperture. Thephysical, and therefore radiometric, temperature of thehot load could be varied from 3.3 K to 50 K to enablethe steady state optical response of the TES array to bemeasured.The optical configuration has the following key fea-tures: (i) The far-field aperture of the test system andthe input aperture of each detector ensure that the detec-tor is illuminated by a specific number of optical modes,having the same spatial forms, and present in the samerelative proportions, as those generated by an ideal imag-ing telescope. This feature is important because thedetectors are partially coherent, and therefore their op-tical behaviour depends on the coherence properties ofthe illuminating field. (ii) The filters that determine thespectral band are placed ahead of the far-field apertureto ensure their presence does not significantly alter thespatial forms of the illuminating modes at the detector.In this way, the efficiency of a detector could be mea-sured with respect to a well-defined partially coherentfew-mode field.The detector array was mounted in a light-tight en-closure, such that the detectors could be coupled to theincoming radiation through an array of micro-machinedinfrared lightpipes, or opened up to the beam to allowdirect illumination. Direct illumination enables the per-formance of the lightpipes to be separated from the in-trinsic performance of the detectors. An additional recesswas machined into the back of the light-tight enclosureto house the bias electronics and SQUID readout chips.A detailed schematic of the test facility is shown in Fig.2(a), and rendered in three dimensions in Fig. 2(b). Thehot load, filters, aperture, and detector and electronicsenclosure, indicated in Fig. 1, may be identified. Theoptical components at 3 K were mounted on two rods toenable rapid assembly and disassembly.The blackbody hot load consisted of a copper cone,coated on its inner face with a high emissivity SiC and C loaded epoxy. Kevlar threads were used to suspendthe cone from the 3.3 K level, and heating resistors and acryogenic thermometer were used to produce a thermalradiation field having a controllable temperature. Cru-cially, the response time of the hot load was determinedby connecting a wire between the Cu cone and one of the3.3 K rods, enabling the temperature of the hot load tobe swept quickly without incurring an excessive thermalload. A thermal blocking filter was mounted on the 3.3 Kstage to reduce long wavelength heating of the band-defining filters, 60-110 µ m, which were held at 50 mK.These custom-designed infrared high-pass and low-passfilters, made of patterned metallic films on polypropy-lene substrates , were angled and mounted in a black-ened tube to avoid multiply scattered radiation reachingthe detectors. As seen in Fig. 1, a thermal break wasnecessary between the hot load and thermal filter, andthe filter stack and detector. Simply leaving a gap riskedallowing stray light to enter the light path from the envi-ronment of the cryostat, whose radiometric temperatureis poorly defined and could be high. A non-touching opti-cal baffle was designed, and tested by placing a hot loadoutside of the path but inside of the cryostat. Radia-tion from this external load could not be detected by theTESs, despite their sensitivity, demonstrating that theoptical baffle worked exceedingly well.The main blackbody load allowed the TES array tobe radiated with a variable temperature, 3.3 K to 50 K,few-mode radiation field. Because of its mass, and theneed to limit the heat load on the ADR, the thermal timeconstant of the load was adjusted such that recovery tobase temperature following a temperature increase of 8K, of the order required for TES saturation in this config-uration, could be achieved in approximately 20 s. Thisarrangement provided a steady state optical source, inthe sense that it was significantly slower than the timeconstants of the TESs. A key requirement, however, wasto be able to measure the transient optical response fordifferent values of background optical loading. This wasachieved by placing a tiny ceramic resistor, 1 mm , onthe periphery of the optical beam between the hot loadand the thermal filter. The cross section of the resistorwas sufficiently small that the blockage of power fromthe main hot load was negligible. At these wavelengths,the resistor acts as a point source on the edge of thefield of view of the detectors. The chip resistor was sus-pended by two copper wires attached to the 3 K housingusing cryogenic epoxy. Given the small mass of the resis-tor, approximately 3 mg, and the diameter of the wire,0.127 mm, an input electrical power of less than 25 mWwas needed to achieve a calculated temperature increaseto within 1 mK of 10 K in under 2 ms. Various cur-rent waveforms could be applied to the resistor, allowingdifferent forms of modulation. This modulation schemeworked exceptionally well, enabling direct measurementof TES optical response to small changes in optical signalover a range of optical background loadings.Considerable attention was dedicated to the design ofthe detector and electronics enclosures. Two recesseswere machined into the front and back of a piece ofOxygen-free high thermal conductivity Cu rod. The frontrecess housed the TES chip, whereas the back recess FIG. 2. Schematics of the optical test facility design: (a) cross-section; (b) 3-D rendering without enclosing shields. housed the bias resistors and SQUIDs. By separating thedetectors and readout electronics by a machined wall, wewere able to prevent stray light from the readout elec-tronics reaching the TESs, an effect that we had seenin previous design iterations. Additionally, the front andback lids had baffled edges to ensure that stray light couldnot enter from the environment of the cryostat. Previ-ous experience had indicated that it is exceedingly diffi-cult to make removable lids having extremely low lightleakage. All of the inner surfaces of the assembly werecoated with infrared absorber, and great care was takenwith feedthrough wiring.Precise metrology was needed to ensure that the detec-tor and backing plate wafers were aligned laterally withrespect to each other and the exit apertures of the light-pipes. The challenge was made even more demandingby the fact that the distance between the TES absorbersand the exit apertures of the lightpipes was just 20 µ m,and that between the absorbers and backshorts 21 . µ m,along the whole length of the chip. The array was alignedlaterally by four dowel pins, and constrained by a fur-ther four dowel pins, positioned to account for differentialthermal expansion between the copper base plate and thesilicon detector assembly. The array was secured by G10fibreglass clamps fastened to raised bosses machined di-rectly into the enclosure base plate. Precision metrologywas carried out using a coordinate measuring machineand surface profiler to verify the positions of the dowelpins, the flatness of the base plate and the alignmentand orientation of the detector array with respect to thebackshort array. Crucially, the positions were measuredafter multiple cooling cycles, indicating that there wasno lateral creep. The bias resistors and SQUIDs weremounted in the back of the detector module. To improveheat sinking and reduce the emission of thermal radia-tion, the bias resistors and SQUID chips were clampeddirectly onto the copper base plate through apertures inthe fibreglass circuit board. Given the extreme sensitivity of the detectors, theymust be shielded from stray light, stray magnetic fields,electromagnetic interference, and microphonic pick-up.Of these, it proved particularly challenging to preventstray light entering the test module, and indeed pene-trating the TES enclosure. In a previous iteration ofthe test facility , where the hot load was thermally con-nected to a cylindrical radiation shield surrounding the50 mK components, we detected significant stray light,and proved by extensive measurements that this radia-tion was thermal in origin, had a spectrum longward of2 mm, and entered the TES enclosure through a paththat did not involve the lightpipes. Despite these inves-tigations, the exact source of this radiation and its routeto the TES array was never conclusively identified.To design the magnetic shielding, detailed finite el-ement modelling was carried out. The final designcomprised a nested arrangement of an e-beam weldedniobium inner can , and a high-permeability alloyCryophy R (cid:13) outer can . Re-entrant flanges were usedon the edges of the cans to prevent stray fields enteringthrough joints; these are detailed in Fig. 2(a). Finally,to minimise flux leakage, openings in the shields wererestricted to those in the face adjoining the 3 K copperbase ring on the left of Fig. 2(a). Overall, the flux at-tenuation factor with respect to the inner volume of themain cryostat was calculated to be of order 10 for DCand 10 for AC external fields of amplitude 40 nT.To reduce microphonic noise, great care was taken toprevent loose wiring sweeping through stray magneticfields in response to mechanical vibrations. Wiring pass-ing from the 3 K rod to the 1 K rod was therefore securedto Kevlar straps running between the 50 mK, 1 K and3 K stages. The chosen wiring scheme used twisted pairNb/Ti in a CuNi matrix for optimal thermal isolation,with all non-twisted-pair wire loops minimised.Figure 3 shows images of the test facility at variousstages of assembly. Figure 3(a) shows the TES array FIG. 3. Images of the test module through various stagesof assembly. (a) Detector enclosure with mounted TES ar-ray. The TES array is seen in blue, and the Si backing platewith micromachined Au plated backshorts is seen in grey. (b)Installed 50 mK sections including the electronics and detec-tor enclosures, enclosed filters and optical labyrinth. (c) Hotload prior to installation. The optical modulator can just beseen as a white chip suspended on fine Cu wires. (d) Assem-bled module viewed from the exterior of the magnetic shields,above the 3 K base plate. clamped within the detector enclosure, and supercon-ducting fan-out wiring. Installed 50 mK sections areshown in Fig. 3(b) prior to the hot load being mounted,including the electronics and detector enclosures, opticalfilter housing, and the 50 mK section of the main op-tical labyrinth. The 3 K rods can be seen, with wiringsecured to a Kevlar cradle between the different temper-ature stages. Figure 3(c) shows the hot load prior toinstallation, suspended from its outer housing by Kevlarthreads. The tiny optical modulator is also visible. Fi-nally, the module is viewed from the exterior of the mag-netic shielding in Fig. 3(d).
III. M-BAND TES ARRAY
TESs were fabricated on a 200 nm thick, amorphous,low-stress SiN x membrane, in a 3 ×
37 linear geometrysuitable for grating spectrometer readout . The arrayis shown installed in the test facility in Fig. 3(a). Theresults described in this paper were based on the repre-sentative TES shown in Fig. 4.Each TES consisted of a 50 × µ m Mo/Au super-conducting bilayer, Fig. 4(i), thermally coupled to a170 × µ m β -phase Ta FIR absorber (ii), forming anisland suspended from the surrounding wafer by fourSiN x support legs (iii) of length 640 µ m and cross-section0 . × . µ m. Nb wiring was deposited on two of the legsfor biasing and readout. Varying numbers of interdigi-tated Au bars were deposited on the upper surfaces ofthe bilayers, giving transition temperatures T C = 139 ± µ m side-length apertures penetrating both the Ta filmand the underlying SiN x membrane, with a 47% filling FIG. 4. Optical microscopy image of the tested ultra-low-noise FIR TES, with superconducting Mo/Au bilayer (i),meshed β -Ta absorber with Au thermalisation ring (ii) andfour SiN x support legs (iii). factor. Since the response time of a TES is fundamen-tally related to the heat capacities of its components, themeshed design was developed to reduce the heat capacityassociated with the absorber. The Ta film thickness wasincreased to 17 nm from the 8 nm of a solid absorber,to obtain an effective surface impedance matching freespace.To facilitate maximum power absorption by the ab-sorber, an optically flat reflective backshort was placedat a distance of λ C /4 behind each absorber, where λ C is the band-centre wavelength taken to be 85 µ m for theSAFARI M-band. The backshorts consisted of a high-conductivity sputtered Au film on an array of Si pillars,etched from the upper layer of a silicon-on-insulator (SoI)backing wafer, whilst also defining a recess into which theTES array chip was mounted. Nb breakout wiring wasdeposited on the backing plate.Under normal operation, the TESs would be illumi-nated by an array of micromachined lightpipes, mounteddirectly above the detector array. Tapered pyrami-dal lightpipes were manufactured having walls of thick-ness 150 µ m, 1350 µ m × µ m entrance apertures and120 µ m × µ m square exit apertures. The lightpipearray was then mounted such that the exit apertureswere aligned axially with the corresponding 170 µ m TESabsorbers. We have previously installed and tested thelightpipe array with the M-band TES array . The light-pipes were removed for the measurements described inthis paper however, so that the planar TES absorbercould be illuminated directly, allowing the TES opticalefficiency to be determined independently of the modaltransfer properties of the lightpipes.The device under test was voltage-biased with alow impedance source defined by a 1.45 mΩ bias re-sistor and read out using a two-stage SQUID ampli-fier as a low-noise current-to-voltage converter. TheSQUIDs were manufactured by Physikalisch-TechnischeBundesanstalt . A key requirement for maintaining sta-ble voltage bias of the low-impedance TESs is to keepthe total stray resistance below 0 . IV. TES ELECTROTHERMAL MODELLING
One aspect of TES characterisation using the opticaltest facility is the measurement of optical and electricalresponse times. Here we briefly describe the theoreticalmodel that was used for interpreting the experimentaldata described in Section V D.A voltage-biased TES self-regulates its temperature towithin a narrow range around the transition tempera-ture. Absorbed optical power causes the temperature ofthe bilayer to increase, increasing its resistance. How-ever, this in turn causes the dissipated Joule power todecrease, maintaining the operating point of the device.This negative feedback enhances many aspects of per-formance, including speeding up the device with respectto its intrinsic thermal response. Figure 5(a) shows aTh´evenin representation of the voltage-bias and read-out circuits. The TES is represented as a variable resis-tor with temperature- and current-dependent resistance R ( T , I ), where T is the bilayer temperature and I isthe current. V is the Th´evenin-equivalent bias voltageand V TES is the voltage across the TES. The current isread out using an inductively coupled SQUID. R L is thesum of the bias and stray resistances, and L representsthe sum of the input inductance to the SQUID and anyadditional stray inductance.To represent a TES bolometer with a large optical ab-sorber adjacent to the superconducting bilayer, a simplethermal model has been adopted featuring two heat ca-pacities, C and C , thermally linked with conductance G . Each heat capacity is also connected to the sur-rounding heat bath with conductance G/
2, where G isthe combined conductance of the TES legs, as illustratedin Fig. 5(b). The first heat capacity is taken to containthe bilayer, where Joule power is dissipated. The secondheat capacity may be interpreted as being associated withthe absorber and underlying dielectric. However, subjectto the constraints implied by the choice of conductanceto the heat bath, it is not necessary to define a physicallocation for this heat capacity in order to construct themodel.TES electrothermal behaviour is described by the fol-lowing electrical and thermal differential equations: L d I d t = V − IR L − IR ( T , I ) , (1) C d T d t = − P B1 + P + P J + P , (2) C d T d t = − P B2 − P + P , (3)where T is the temperature of the first heat capacity,equal to the bilayer temperature, and T is the temper-ature of the second heat capacity, equal to the absorbertemperature. P J = I R is the Joule power dissipatedin the bilayer. P B1 and P B2 are thermal powers to theheat bath from each heat capacity; P is the net power 𝑅 L 𝐿𝑉 𝑅(𝑇 , 𝐼) 𝐶 𝑃 B1 𝑃 J (a) (b) + - V TES
SQUID 𝐺 𝐶 𝐺𝑃 𝑃 B2 𝐺 𝑃 𝑃 FIG. 5. (a) Th´evenin equivalent representation of the TESbias circuit, where the TES is shown as a variable resistor R ( T , I ). R L is the internal resistance of the voltage source, V , and is the sum of a 1.45 mΩ bias resistor and an ≈ . L corresponds to the input inductanceof the SQUID and any stray wiring inductance. V TES is thevoltage across the TES. (b): Thermal circuit representing theTES as two heat capacities, C and C , where C containsthe superconducting bilayer, connected by a weak thermallink with conductance G . Each capacity is also coupled tothe heat bath via thermal conductance G/ G is thecombined thermal conductance of the TES legs. P B1 and P B2 represent net thermal power to the heat bath from C and C respectively, P is net thermal power between C and C , P J is Joule power input to C , and P and P are external powerinputs to C and C . flow between the heat capacities; and P and P are ad-ditional power inputs, for example optical absorption, tothe two heat capacities respectively. R L is the internalresistance of the voltage source, V , and is the sum of a1.45 mΩ bias resistor and an ≈ . L corresponds to the input inductance of the SQUID andany stray wiring inductance.It is standard practice in TES physics to expand non-linear terms , in this case P B1 , P B2 , P , P J and R ( T , I ),to first order in the small signal limit around the initialsteady state values I , R , T and T , givingdd t δIδT δT = − M δIδT δT + δVLδP C δP C , (4)where M = R L + R (1+ β ) L I R αT L − I R (2+ β ) C − α I R T − G − G C − G C − G C G + G C , (5)and δI = I − I , δT = T − T , δT = T − T .The resistance-temperature and resistance-current sen-sitivities are given by α = ( ∂ ln R/∂ ln T ) I and β =( ∂ ln R/∂ ln I ) T respectively. δV , δP and δP representsmall changes in the applied bias voltage and externalpower input to the first and second heat capacities. Thisresults in a solution for δI ( t ), δT ( t ) and δT ( t ) of theform: δIδT δT = (cid:88) i =1 ( a i − a i e − t/τ i ) (6)= (cid:88) i =1 ( b i v i − b i v i e − t/τ i ) , (7)where τ i = λ − , λ i being the eigenvalues and v i theeigenvectors of the matrix M , and b b b = − ( λ v λ v λ v ) − δVLδP C δP C . (8) V. MEASUREMENTSA. Preliminary thermal and electrical characterisation
On cooling without temperature regulation, the tem-perature of the electronics and detector enclosuresreached T B = 59 . R stray , and inductance, L ,were extracted from the measured circuit impedance withthe TES in its superconducting state , Z = R L + 2 πf L .The value R stray = 0 .
45 mΩ was significantly lower thanthe 1-2.5 mΩ measured by us using alternative config-urations. Low stray resistance is crucial to achievinga near-ideal TES voltage bias. A total inductance of L = ( L in + L stray ) = 108 nH was measured. Using themanufacturer’s value for the SQUID input impedance, L in = 80 nH, implies a stray inductance L stray = 28 nH.Knowledge of the total inductance is required for calcu-lating the TES response times, as described in SectionIV. B. TES characterisation: thermal conductance andoptical absorption
Figure 6(a) shows the TES I − V TES curve, for a se-lection of bath temperatures T B , where T B is taken tobe the temperature of the detector housing controlledthrough the residual current in the ADR magnet. Thecorresponding dissipated Joule power, P J = IV TES , isshown in Fig. 6(b), from T B = 60 mK (blue, top) to T B = 120 mK (green, bottom).Figure 6(c) compares a representative P J − V TES plateau measured using the optical test facility with aprevious measurement on a different TES, in a previoustest system without custom magnetic shielding. Powerhas been normalised to its mean value across the transi-tion and voltage to its value at the start of the transitionto allow a comparison between the two TESs, which weredesigned with different thermal conductances and tran-sition temperatures. The earlier measurement has been displaced on the power axis for clarity in Fig. 6(c). Themeasurement from the earlier test system exhibits sig-nificant transient current drops, caused by low-frequencyelectromagnetic interference. These artefacts are com-pletely absent in the new measurements despite the fivetimes greater sensitivity of the TES, suggesting that themagnetic shielding and anchored wiring of the test facil-ity were effective in eliminating pick up.In the steady state, the net power flow from the TESisland to the heat bath, P B , is equal to the Joule powerdissipated in the bilayer, P J , which is approximately con-stant across the transition as evident in Fig. 6(b). Figure6(d) shows P B calculated as mean P J over the transitionfor each T B . The power flow to the bath is described bythe expression P B = K ( T n C − T n B ) , (9)where K is a parameter that scales the overall heat flux,and n ∼ − n = 1 . K = 400 fW / K n and T C = 131 . G to the heat bath is thengiven by G = nKT ( n − = 108 . (cid:112) k B GT = 0 .
32 aW / √ Hz, where k B is theBoltzmann constant. This exceptionally low NEP com-bined with the saturation power corresponding to P B ata given temperature, for instance 3.8 fW at 90 mK, em-phasise the utility of the device.A crucial aspect of detector characterisation is deter-mining the optical absorption efficiency, for which it isimperative that out-of-band stray radiation is minimised.As described in Section II, the far-infrared absorbers wereilluminated with a blackbody hot load of temperature T BB , through band-defining filters. As T BB is increasedabove its base temperature T BB0 ≈ . P = P J ( T BB0 ) − P J ( T BB ) . (10)Figure 6(e) shows ∆ P against change in T BB , ∆ T BB = T BB − T BB0 , with the bath temperature T B maintained at90 mK. When the absorbed power is equal to the Joulepower at T BB0 , the TES reaches its normal state. Thisis the differential saturation power of the TES relativeto the incident power at T BB = T BB0 , indicated on Fig.6(e) (dashed cyan line and final added point). Whilst therapid increase in ∆ P above ∆ T BB ≈ T BB , linearwith ∆ T BB . This is attributed to long-wavelength straylight, as was observed to a far greater extent in our earliertest apparatus . The absorbed power ∆ P may thereforeby modelled as∆ P mod = α ∆ T BB + η opt ∆ P th , (11)where α is a stray light coefficient and η opt is the de-tector optical efficiency. The theoretical overall opticalthroughput to the detectors, P th ( T BB ) may be calculatedas V TES (nV)0123 I ( µ V ) (a) 0 . . V TES /V TES0 . . P J / P J (c)0 1 2 3 4 5 6 7 8 9∆ T BB (K)012345 ∆ P ( f W ) (e) 0 5 10 15 20 V TES (nV)0246810 P J ( f W ) (b)60 80 100 120 140 T B (mK)0123456 P B ( f W ) (d) FIG. 6. (a) TES current, I , against voltage across the TES, V TES for a subset of bath temperatures, T B : 60 mK (blue),100 mK (orange) and 120 mK (green). (b) CorrespondingJoule power P J against V TES for these bath temperatures.(c) Net thermal power to the heat bath, P B , equal to P J aver-aged over the transition, against T B . (d) P J normalised to itsmean value over the transition P J0 against V TES normalisedto its value at the normal end of the transition, for the testedM-band TES in the test facility (blue) and a longer wave-length device in a previous module design (orange). Datapoints for the longer wavelength device have been displacedto P J /P J0 = 0 . P against change in hot load temperature, ∆ T BB . Saturationpower at T B = 90 mK is indicated in cyan and model opticalpower ∆ P mod = αT BB + η opt ∆ P th in red, where α is a lin-ear stray light coefficient, η opt is the TES optical efficiencyand ∆ P th is the theoretical total throughput. Contributions αT BB and η opt ∆ P th are shown in orange, dashed, and green,dashed, respectively. P th = A Ω (cid:90) λ max λ min η filters hc λ hcλk B T BB − λ, (12)where A is the TES absorber area, Ω is the solid an-gle subtended by the optical aperture, λ min = 50 µ mand λ max = 160 µ m, η filters is the product of the ther-mal and band-pass filter transmission coeffcients, h isPlanck’s constant, k B is the Boltzmann constant and c is the speed of light. Equation 11 was fitted to the mea-sured data, giving ∆ P mod shown in Fig. 6(e) (red) with α = 0 .
035 and η opt = 1 .
08. This near-unity optical effi-ciency indicates close to ideal optical performance of theTES absorber-backshort assembly. The slight overesti-mation may be due, for example, to some small system-atic error in bias and readout circuit parameters, or inthe dimensions featuring in Eq. 12. The stray light coeffi-cient α is approximately three times lower than that mea-sured for this TES in our early test facility designs, wherethe hot load was thermally connected to a cylindricalradiation shield surrounding the 50 mK components .A significant reduction in stray light has therefore beenachieved in this test facility compared to earlier configu-rations, especially given that the thermal path from thehot load must now pass within the closed shielding inclose proximity to the detector enclosure before reachingthe 3 K plate of the cryostat. The origin of the remainingcontribution is the subject of further investigation, butnevertheless the quality of the data is pleasing given theexceedingly low powers involved. C. Calibration of optical modulator
To investigate the optical coupling between the resis-tive chip modulator and the detectors, the optical powerabsorbed by a TES was measured whilst the electricalpower dissipated in the chip resistor was increased, withthe primary hot load remaining at base temperature.Figure 7(a) shows P J − V TES for a subset of electricalpowers from 0 −
40 mW, demonstrating progressive ab-sorption across the full dynamic range up to TES satu-ration. Absorbed power ∆ P is shown against electricalpower P in in Fig. 7 for the modulator (blue) comparedto that of the hot load (orange). The functional form ofthe increase in ∆ P with P in for the optical modulatorreflects that of the hot load, indicating in-band illumi-nation of the detectors, and showing sensitivity to ≈ D. TES current response to modulation in bias voltageand optical power
The test system allowed us to compare the response ofa TES to a small step in bias voltage, δV , with that of anoptical pulse, δP . The TES was biased within its transi-tion, at 33% of its normal state resistance without opticalloading, corresponding to a bias voltage V ≈ . P in (mW)0123 ∆ P ( f W ) (b) 0 5 10 15 V TES (nV)0246 P J ( f W ) (a) FIG. 7. (a) Joule power P J against voltage across the TES, V TES , for increasing power input to the chip resistor from 0W (blue) to 38.6 mW (red). (b) Absorbed optical power, ∆ P ,against input P in to the chip resistor (blue) and to the hotload (orange). amplitude square wave was superimposed on the bias in-put, giving δV ≈ .
13 nV. The change in TES current asa function of time, δI ( t ), was averaged over 40 periods ofthe square wave, separately for the leading and trailingedges, corresponding to an increase in V and a return toinitial V respectively.Figure 8(a) shows the change in TES current, δI ( t ),on the leading edge of the voltage pulse, with the black-body hot load at base temperature, T BB = T BB0 = 3 . t = 0 is due to the electrical response ofthe TES and bias circuit. This is followed by a slowrelaxation towards the new, lower, steady state currentunder electrothermal feedback, corresponding to a nega-tive final current change δI f for positive δV . The changein Joule power, δP J , corresponding to δI f ≈ . δI ( t ) on absorbed opti-cal power, each measured response may be decomposedinto a sum of exponential contributions, to be comparedwith those predicted by small-signal electrothermal the-ory. A function having the form of Eq. 6 was thereforefitted to the measured data, but with the amplitudes, a i1 for i = 1 , , τ i , as free parame-ters rather than being calculated through Eqs. 5 and 8.These parameters were therefore not constrained to anyparticular inter-relation or progression with T BB .Figure 8(a) shows a least-squares fit to the mea-sured response (solid, red), with additive components(a i1 − a i1 e − t/τ i ) (lilac, orange, green). The shortest timeconstant, corresponding to the electrical contribution, was fixed at τ = 7 µ s ≈ L/ ( R L + R ). The precise valueassumed for τ has negligible effect on the fitting as itis 3-4 orders of magnitude smaller than the next largesttime constant, and only affects a small number of pointsdue to the sampling time. Fitted parameters a i1 and τ i are listed in Table I, indicating contributions of compa-rable amplitude from terms in two thermal constants τ and τ , where τ ≈ τ , rather than one as would beexpected from a single heat capacity model.The heat capacities, C and C , and linking thermalconductance G , were estimated by fitting the small sig-nal electrothermal model described by Eqs. 6-8 to δI ( t )at T BB = T BB0 . Steady state values I , R , T , T were simulated based on a parametric R ( T , I ) surface to capture coarse large signal behaviour. The resistance-current sensitivity β was derived from measurements ofthe circuit impedance in the high frequency limit , andthe relationship α = 100 β assumed . It was not the aimof this work to extract highly accurate electrothermalparameters or heat capacities for this particular device,but rather to interpret the forms of measured responseprofiles in the context of TES electrothermal behaviour.The values C = 34 . ± . C = 70 . ± . G = 1 . ± .
02 pW/K were obtained. That C ,containing the bilayer, is a factor of two smaller than C may offer some support to the notion that C is as-sociated with the larger volume of SiN x underlying theabsorber. A comparable ratio C /C = 1 . a i1 and timeconstants τ i are listed in Table I, and show strong sim-ilarity with those derived from the free amplitude andtime constant fit shown in Fig. 8(a).In the limit of small changes in I , R , T and T , andheat capacities and thermal conductances independent oftemperature, it is expected that δI ( t ) for positive δV isequal to the negative of that for negative δV . Figure 8(b)shows the sum of δI for leading and trailing δV , indicat-ing that the absolute responses are essentially identicaland therefore validating the use of the small signal limit.To measure the response δI ( t ) for a small step in op-tical power δP , the optical modulator was used to pro-vide a small modulation, superimposed on a steady statebackground from the hot load. A voltage square wavewas applied to the modulator, and the TES current av-eraged over 40 leading edges, δP positive, and 40 trailingedges, as the optical perturbation returns to zero. Anunexpected contribution to δI ( t ) was observed, with atime constant around 3 s, considerably longer than anyexpected either from the TES or optical modulator. Itis suspected that this contribution originates from straylight: for example, from some element of the modulewarming slowly and radiating as electrical power is ap-plied to the modulator chip resistor. The source of thisstray light has not yet been identified; however, possiblecandidates include the outer hot load housing to whichthe chip resistor is anchored, or the optical filters them-selves, which rely on their metallic patterning to heatsink the polypropylene substrate.Figure 8(c) shows δI ( t ) for positive δP , for T BB = T BB0 , with the stray light contribution subtracted. The
FIG. 8. (a) Change in TES current, δI , with time, t , in response to a small increase in bias voltage, δV at t = 0, with theblackbody hot load at base temperature, T BB = T BB0 . A decomposition into exponential contributions according to Eq. 6,with fitted amplitudes and time constants, is shown with additive terms (lilac, orange, green) and sum (red). (b) Sum of δI for leading and trailing δV , δI L and δI T respectively. (c) δI ( t ) for a positive step in optical power input, δP , at t = 0, for T BB = T BB0 . Simulated δI for power input δP to the second heat capacity is shown (red), scaled to match measured final δI .Right-hand axis shows δP calculated from this scaling factor. (d) δI L + δI T for the leading and trailing optical responses.TABLE I. First elements a i1 of the amplitude vectors a i , and time constants τ i , where i = 1 , ,
3, parameters of Equation 6forming a functional representation of δI . M/S indicates values (M) acquired from free fitting to measured δI at T BB = T BB0 ,and values (S) simulated from the small signal electrothermal model, with amplitudes given by Equations 7 and 8, and timeconstants from Equation 4. Step type is indicated as being either in bias voltage, δV , or optical power, δP , with δP indicatingpower input to the second heat capacity in simulations. Leading edge (L) describes positive δV or δP , and trailing edge Tnegative δV or δP returning to initial bias voltage or optical power.M/S Step L/T a (nA) a (nA) a (nA) τ ( µ s) τ (ms) τ (ms)M δV L 3.03 -7.11 -9.48 7 7.8 79.9M δV T -3.46 7.73 9.38 7 9.1 88.3S δV L 2.94 - 8.09 - 8.63 2.33 10.1 94.7M δP L - - -7.41 - - 84.5M δP T - 3.02 5.54 - 7.4 57.9S δP L 0 0.86 -8.13 2.33 10.1 94.7 rapid drop in current over the first few hundred millisec-onds is characteristic of the TES electrothermal responseto a step in incident power, demonstrating that the opti-cal modulator is capable of delivering a modulated signalwith steps that are abrupt on the timescale of the TESresponse.Table I lists freely varying amplitudes and time con-stants obtained through fitting Eq. 6 to δI ( t ). In thecase of the leading edge optical data, a convincing fit wasobtained with a single thermal time constant τ . This be-haviour is consistent with power input to the second heatcapacity, δP , for which the longest time constant domi-nates, since thermal flux must traverse the weak thermallink G before a response can be seen in the TES. Themodelled response for power δP into C is shown in Fig.8(c) scaled to match measured δI f and using C , C and G derived from the voltage step measurement at T BB0 without further fitting. The value of δP derived fromthe global scaling factor is 43.5 aW, half the 87.4 aW pre-dicted from the I − V TES measurement of Fig. 7 for thesame input power to the optical modulator. However,since the fitted stray light amplitude, 8.93 nA, is almostequal to the amplitude | a | = 7 .
41 nA, it is likely that M-band power accounts for roughly 50% of the 87.4 aWtotal, with the remainder arising from stray light absorp-tion at long timescales, rendering δP = 43 . δP = 43 . δI with δP in the small signal limit. Thisemphasises that not only is the TES eminently capableof detecting the extremely small signal applied in themeasurement shown, but would register absorbed powerdown to around 10 aW whilst remaining above the noiseassociated with the present experimental system.That the modelled response for power applied to thesecond heat capacitance is able to closely reproduce themeasured δI ( t ) suggests that the superconducting bilayeris indeed weakly thermally coupled to the site of opticalabsorption in this device, resulting in a slower current re-sponse to a change in optical signal than to a change inbias voltage. Since optical absorption by the impedance-matched absorber is expected to dominate over directabsorption by the bilayer, this also supports the sugges-tion that C represents the absorber within this simplemodel.0Contrary to expectation, a faster response was ob-served for optical δI ( t ) on the trailing edge than theleading edge. This is reflected in the sum of the re-sponses shown in Fig. 8(d), and in the amplitudes a i1 and time constants τ i listed in Table I. In this case, δI ( t )is well described by two thermal time constants of com-parable amplitude, in greater similarity to the electricalresponse than the leading edge optical. This asymme-try, as yet unexplained, would not have been revealedthrough measurements of TES response to bias voltagemodulation alone.A key functionality of the test facility is the ability tomeasure both the electrical and optical TES responsesunder background optical illumination from the black-body hot load. Figure 9(a) shows the leading edge elec-trical response to a small step in bias voltage, as in Fig.8(a), for a selection of blackbody hot load temperatures T BB increasing from base temperature, T BB0 ≈ . T BB ≈
12 K (lilac). As T BB increases, the mag-nitude of the final current change, δI f , decreases as TESresistance increases and approaches the normal state. Fi-nally, at the highest T BB , the absorbed optical power,∆ P = 3 . T B = 90 mK, and the TES returns to itsnormal state. In the absence of electrothermal feedback,the electrical time constant reverts to L/ ( R L + R n ) andthe TES behaves as a normal metal resistor with positive δI f .It is evident from Fig. 9(a) that the time taken forthe TES current to reach its new equilibrium increaseswith increasing optical power loading, as the resistance-temperature sensitivity α decreases, up until saturationand the vanishing of the electrothermal contributions tothe response time. This effect is likewise observed incorresponding optical response data for increasing T BB .Figure 9(b) shows freely fitted τ , the longest elec-trothermal time constant, against P for leading edge δV and δP . As expected, good agreement is observed be-tween τ for the electrical and optical responses. It isalso possible to calculate the expected value of τ withabsorbed power P without further fitting, using the smallsignal thermal model, measured β with T BB , and the val-ues obtained for C , C and G . This is also shown inFig. 9(b), where the small kinks arise from random errorin β . Measured τ is seen to approximately follow thesimulated trend, demonstrating that the observed slow-ing of the response with optical power is indeed an ex-pected consequence of TES electrothermal behaviour. ByTES saturation, τ has increased to five times its initialvalue, considerably slowing δI ( t ). This corresponds toan increase in the time taken for the current to relaxto within 10% of its final steady state value of approxi-mately seven times for δV and five times for δP . This as-pect of TES behaviour has important implications for thedesign of TES bolometers for optical instruments suchas SAFARI, dictating, for example, optical power as aproportion of available saturation power that may be ab-sorbed before the response time exceeds the lower limitset by the scanning of the Martin-Puplett interferometer.Figure 9(c) shows the fitted and simulated magnitude ofthe amplitude, a , associated with τ , which decreases with increasing absorbed power as expected. VI. CONCLUSIONS
We have devised and implemented a cryogenic test fa-cility for ultra-low-noise far-infrared transition edge sen-sors. These sensors are being developed for the SAFARIgrating spectrometer on the cooled-aperture space tele-scope SPICA. Although the experimental arrangement issuitable for the whole of the SAFARI wavelength range,34-230 µ m, we have focused on a representative set ofmeasurements at 60-110 µ m.A key feature of the optical configuration is its abilityto measure optical efficiencies with respect to a few-modebeam having modal characteristics identical to those ofan ideal imaging telescope. Moreover, the addition ofa fast infrared thermal source allows the direct measure-ment of the temporal response of TESs to tiny changes inoptical power. We have shown that it is possible to mea-sure transient optical response in the presence of steady-state background loading, all the way up to detector sat-uration.A crucial consideration in the design was the minimisa-tion of stray light, and the maximisation of magnetic andelectrical shielding. In the context of the detector mod-ule, considerable care went into enhancing sensitivity andoptical efficiency, achieving a high performance thermaland mechanical design, ensuring repeatability throughgood metrology, and eliminating stray light.Overall, the test facility performed well. The detectormodule had a base temperature of T B = 59 . µ K. Even at thiselevated temperature, the devices tested had an excep-tionally low NEP, 0.32 aW / √ Hz, making them suitablefor ultra-low-noise space applications. The optical effi-ciency was measured to be 108%, near-ideal with slightoverestimation possibly due to calibration, with an opti-cal saturation power of 4 fW. This indicates that theAu micromachined Si backshorts and meshed β -phaseTa absorber functioned according to design. This waspleasing as the thickness of the Ta film had been de-creased by 53% to compensate for the increased sheet re-sistance caused by meshing, thereby obtaining a effectiveimpedance closely matched to free space. Despite consid-erable effort to eliminate stray light, a long-wavelength < FIG. 9. (a) Change in TES current, δI , with time, t , in response to a small increase in bias voltage, δV at t = 0, for absorbedbackground optical power P from 0 fW (blue) to 3.7 fW (lilac), as indicated. (b) Fitted values for the longest time constant, τ , for δV (blue) and δP (orange) against P , with simulated progression (red). (c) Absolute values of amplitudes a associatedwith τ for δV (blue) and δP (orange), with simulated trends (red, black). of some optical component within the field of view. Themeasured functional forms imply the existence of a weakthermal link between the TES bilayer and the site of opti-cal absorption, despite our attempt to ensure fast thermalresponse by connecting the TES bilayer directly to a Authermalising bar around the periphery of the absorber.As predicted by electrothermal modelling, we found thatthe electrical and optical responses were slowed to severaltimes their dark values when background optical loadingwas applied, up to TES saturation. These observationsdemonstrate the importance of being able to measure op-tical and electrical transient response when refining TESdesign.Considerable care is needed to eliminate stray light,and indeed to avoid the scenario where a calibration loadheats a nearby surface, which subsequently re-radiatesover a relatively long time period. In the context of SA-FARI, the associated impact on TES responsivity and re-sponse speed, intimately related to other time constantsand scan speeds in the instrument, would be highly detri-mental to performance. Stray light control, and excellentthermal design, both of the detector chips and instru-ment, are central to the operation of ultra-low-noise as-tronomy instruments. ACKNOWLEDGMENTS
The authors are grateful to the European SpaceAgency CTP programme, 4000107657/13/NL/HB, andthe UK Space Agency NSTP programme, for fundingthis work. We would like to thank our colleagues at theEuropean Space Agency, in particular Astrid Heske, Pe-ter Verhoeve, and Kate Isaak, for their continued sup-port. Magnetic modelling of the shielding configurationwas carried out under the NSTP programme at the EarthObservation Navigation and Science Group of Airbus De-fence and Space, and we would like to thank ChristianTrenkel and Maike Lieser for their work throughout this project. Emily Williams is grateful for a PhD studentshipfrom the NanoDTC, Cambridge, EP/L015978/1.The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest. K. D. Irwin and G. C. Hilton, in
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