key: cord-0181716-b1ljlh2d
authors: Thakur, R. Basu; Steiger, A.; Shu, S.; Faramarzi, F.; Klimovich, N.; Day, P. K.; Shirokoff, E.; Mauskopf, P. D.; Barry, P. S.
title: Development of Superconducting On-chip Fourier Transform Spectrometers
date: 2021-11-12
journal: nan
DOI: nan
sha: 0bca76dba00d843b4ad77d3a3bab9d3b9127d1a1
doc_id: 181716
cord_uid: b1ljlh2d

Superconducting On-chip Fourier Transform Spectrometers (SOFTS) are broadband, compact and electronic interferometers. Being extremely compact, SOFTS can fit into standard antenna coupled detector architectures. SOFTS will enable kilo-pixel spectro-imaging focal planes enhancing sub-millimeter science; particularly cluster astrophysics / cosmology, CMB-science and line intensity mapping. This proceeding details the development, design and bench-marking of RF on-chip architecture of SOFTS for Ka and W bands.

In thin film superconductors like NbTiN and NbN, increasing supercurrent I modifies the density of states, increasing kinetic inductance 1 . This in turn alters the phase velocity, ultimately enabling current-controlled delays in a transmission line geometry. For a transmission line (length , width w, inductance-per-square L and impedance Z 0 ) with inductance and capacitance per unit length of L and C respectively, the current-controlled delay is given by Eq. 1. The characteristic currents I * , I * are determined by the material properties and device geometry. See our previous work in 1-10 GHz and 25-40 GHz range 2,3 . 

We employ this core idea to realize Superconducting On-chip Fourier Transform Spectrometers (SOFTS 2 ). A broadband input is split in two parallel transmission lines where a relative phase delay is introduced with current-biasing. The recombined signal is an interferogram like a classical FTS. In this paper we present a thorough superconducting circuit design, numerical characterization, device fabrication and calibration plans. Fig. 1 shows the schematic, Left-to-right: Two inputs are a broadband antenna with band-defining filter, and a precision calibrator / bolometric load. These are combined with phasing via a hybrid coupler (HC). Each HC feeds a Superconducting Transmission Line (STL) via a diplexer, to DC bias the STLs while transmitting RF. Each STL has independent current controlled phase-delays. The phase-delayed signals are combined with a HC (sum/difference modes) and read out with low noise detectors (MKIDs/TESs) which measure interferograms as function of DC sweeps. While initial tests will be done with DC bias, audio-band AC biasing is possible and will be explored for multiplexing advantage. A bolometric load (∼ O(100) µK CMB √ s) will be the calibration input. The other input will see the "sky" via a broadband polarized antenna. The STLs are equivalent to optical arms of a standard FTS. With current biasing one arm, relative phase delay is added such that following the final hybrid coupler, two detectors see the interfered power analogous to the symmetric and antisymmetric ports of a classical FTS. With detectors at both outputs, a precise calibration load and a completely symmetric interferometer design, we minimize systematics as well as maintain ability to perform transfer function calibration via RF cascade studies. Subsystems have been designed for Ka and W-bands, Fig. 3 , and they are adaptable to higher frequencies. The STL is largely band independent, and design optimization with respect to fractional-resolution vs. loss can be done for near-THz frequencies. We are testing two SOFTS architectures in parallel with the Ka and W band devices. The variations in superconducting material, geometry and optical coupling are intentional, see Tab 1 and Fig. 2 . There are a few reasons why the W-band chip has a larger footprint than the Ka band device, such as, (i) to design a 4-port split block we needed to put the probes in a row which takes up more space (ii) to lower δ ν we use a long STL for the W-band device. The Ka-band chip in contrast is optimized for 1 GHz resolution (δ ν). We will down-select based on measurements, i.e., take the best aspects of each design and develop a unified SOFTS architecture. For CMB-science we need δ ν ∼ O(1) GHz, and for line-intensity mapping it should be an order of magnitude lower. NB: SOFTS resolution is tunable, i.e., we can always have high δ ν by applying less current. Here we strictly quote the smallest δ ν which depends on device geometry and material. 

N-port S-matrices of each subsystem are linked following Fig. 1 . MathWorks' RF Blockset software * is used to model this cascade network, Fig. 4 shows a simplified flow diagram. We input single tones on the antenna port (port 1 of first hybrid-coupler HC (0) ) and monitor the power at the two outputs (ports 2 and 3 of HC (3) ) and the reflected power in the second input port (port 4 of HC (0) ). All powers are monitored as dissipation across 50 Ω impedances. In * https://www.mathworks.com/help/simrf/ the ideal case, i.e., for negligible reflections and loss from the transmission lines, and low cross-talk between output ports of each hybrid-coupler, we can model the power measured in the symmetric and anti symmetric ports (ports 2 and 3 of HC (3) ). For unit voltage input on the antenna port only, this is shown in Eq. 2, with ∆ τ = τ 2 − τ 1 .

For any 90-deg hybrid-coupler, S 34 ≈ S 21 ≈ i and S 31 ≈ S 24 ≈ 1. So one can reduce the two equations, up to constants as ≈ 1 ± cos(2πν∆ τ(I)) which are the classical FTS outputs. RF cascade simulations allow us to comprehensively model frequency dependencies as well as multi-path effects from reflections and cross-talk. Due to these non-idealities we can expect mild anharmonicities. The ability to model and understand these anharmonicities is indeed a major advantage for SOFTS, i.e., we can understand the spectrometer performance as a pure circuit model, as compared to optical FTS where multipath effects are challenging to accurately model and correct for. Here we are using simulated S-parameters, laboratory measurements will be used to construct A SOFTS as shown in our previous publication 2 .

For every single-tone input we scan 2 ns of delay, via L (I) modeling from Eq. 1, to produce interferograms. FFTs of these interferograms generate a transfer function (A SOFTS ), i.e., observed frequencies for single tone inputs. Fig. 5 shows relevant figures for Ka-band studies and W-band simulations are done identically. Digital signal processing with the transfer function enables accurate spectral recovery.

Error correction implies accounting for device non-ideality generated anharmonicities, so that we have accurate spectral recovery. Each input tone is a unit vector in frequency space, named U k , where only the k th element is 1, e.g. ν min = 1, 0, 0, ... T , ν min +δ ν = 0, 1, 0, ... T .

Each single tone input generates multi-tone output given by V k = A SOFTS · U k , see Fig. 5 . Since U k is essentially a delta function in frequency space, we pursue inversion following least-squares method, Eq. 3 (further explanation in acknowledgements). A SOFTS (transfer function matrix) for port-2 of the output hybrid coupler.

A T SOFTS A SOFTS −1 A T SOFTS is analogous to a Green's function for the observed SOFTS spectrum. Suppose that B true is the true multichroic sky-signal and B obs is the SOFTS spectrum that is readout. Then using Eq. 4 we can recover B true .

We demonstrate this spectral reconstruction by considering the CMB power as measured by the SOFTS devices. Our method has fractional errors |B obs − B true |/B true 10 −11 . This is a major achievement over classical optical FTSs (wavelengths ∼ 0.1 − 10mm), where reconstruction of multi-path and alignment issues are far less accurate. 4 Optical coupling and device hardware

Ka-band SOFTS mask was shown in Fig. 2 , in essence it is an inverted microstrip architecture where 35nm of NbTiN is the workhorse superconductor. Fabrication steps will be identical to our published work on the measurement of Ka-band phase-delay 3 . We have fabricated a PCB for mounting the SOFTS chip, and an OFHC copper housing to encase the chip and PCB for laboratory testing. These were designed focusing on reduction of cross-talk and radiative losses. The actual SOFTS chip is ∼ 6mm × 1mm, and the PCB is necessary to wirebond RF and DC lines, and to connectorize at the housing ports. RF measurements will be done with Ka-band VNA; signals from which will travel via V-connectors, then transmitted via CPW lines on this PCB, Fig. 7 . The PCB allows us to connect the CPW lines to on-chip microstrips. Over the Ka-band we have -20 dB cross-talk and <-10 dB reflections. Detailed measurements will be needed, as material properties and geometries play subtle roles. NB: Ultimately for antenna coupled SOFTS such a PCB is not needed. 

W-band SOFTS housing with wave-guide coupling has also been designed and fabricated. Waveguide coupling is done using a radial probe 4 . STLs were deposited with a thin film (40 nm) of Niobium Nitride (NbN) and etched using a reactive ion etching (RIE) process.

Other circuit elements including the probes and the hybrids were deposited with a thicker Niobium (Nb) film (150 nm) using a lift-off process. Silicon Nitride was used as the dielectric layer (500 nm) and it was deposited on top of the circuit using a Plasma-Enhanced Chemical Vapor Deposition (PECVD) method. The "skyplane" was then deposited on top of the dielectric. Lastly, the silicon below the probes are etched away for improved probe coupling. Fig.8 .a shows the layer stack-up view of the device. Chip-housing consists of three parts with the top parts being the split-block waveguide as shown in Fig.8 , and a chip holder which consists of the waveguide backshorts and acts as a heat sink for the SOFTS chip. We have two extrude cuts on the chip holder to place PCBs for DC biasing of the STLs.

In this proceeding we outlined detailed circuit-theoretic modeling of Superconducting Onchip Fourier Transform Spectrometers (SOFTS). We discussed devise design and hardware progress for Ka and W-band SOFTS, including fabrication layout of SOFTS chips and their necessary optical coupling technologies. These bands were chosen based on commercial availability of VNA and parts. SOFTS design is completely flexible. Furthermore we elucidated comprehensive RF cascade simulations of our complete devices, and demonstrated that such on-chip circuits have fractional errors in spectral recovery at levels of 10 −11 . Although due to COVID-19 device fabrication and testing has been delayed, our imminent work will involve measurements of both Ka and W-band devices. We will bench-mark these measurements based on the framework developed in this publication. assembly for SOFTS chip. 4 waveguide flanges fittings for each SOFTS port. Through holes to mount the top part of the housing to the chip holder and cryogenic stage. c) Split block is a transition from the waveguides to on-chip probes. d) Chip holder design has backshorts for probes and extrude cuts for DC bias PCBs.

Density of states in a superconductor carrying a supercurrent

Superconducting on-chip fourier transform spectrometer

Nonlinearity and wide-band parametric amplification in a nbtin microstrip transmission line

Initial design of a w-band superconducting kinetic inductance qubit

Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions

Acknowledgements We demonstrated spectral recovery without noise. Robust recovery with noise is also possible in this framework; we thank Robert Webber (Caltech) for pointing this out 5 . We thank Rick LeDuc at JPL-MDL for device fabrication support. The W-band housing is being made using micro-mill machining by Matt Underhill at ASU. This research is supported by NASA award NNH18ZDA001N-APRA, and by the University of Chicago College Summer Research Scholarship.