### Physical layout and numerical simulation

Table of Contents

The layout of the electromagnetic structure of the coupler, tunable sub-delay lines, interconnecting transmission lines and ground–signal–ground–signal–ground waveguides was designed and simulated by the 2.5D EMX electromagnetic tool. S-parameters of the entire electromagnetic structure were used for co-design with the device models of the transistor switches. Parasitic resistances and capacitances were extracted from 45-nm CMOS NMOS SOI device models supplied by GlobalFoundries, using Siemens’s Calibre tool. The full circuit was designed and simulated in Cadence Virtuoso, using S-parameter analysis for small-signal analysis and harmonic balance for large-signal linearity analysis. For a description of digital logic circuits controlling the thermometer decoding used for activating switches in the coarse and medium tuning portions of the tunable transmission lines, see Supplementary Section 3.

### Measurement

The chip was probed on a microwave probe station, after wirebonding DC bias pads and pads of input bits to a printed circuit board (Supplementary Fig. 9). De-embedded S-parameter measurements shown in Figs. 3 and 5 were made using an Agilent 8722ES 40-GHz vector network analyser. The digital input bits that drive the on-chip thermometer decoding logic were supplied using a Digilent Analog Discovery 2 static I/O module. Keithley source meter units were used as power supplies for the logic and to bias the trimming varactors. Linearity measurements were made using an Agilent 8564EC spectrum analyser. An HP 83623A continuous-wave source was used for 1-dB compression-point measurements and power-combined Anritsu 68369B RF synthesizers were used in the third-order intermodulation product (IIP3) measurement in Supplementary Fig. 11.

### Calculation of channel capacity, incorporating effects of beam squint

Here we estimate the channel capacity of the delay element for a typical multicarrier modulation scheme in which data are transmitted as a combination of orthogonal narrowband subcarrier signals, that is, orthogonal frequency-division multiplexing. This is the most commonly used modulation scheme in cell-phone technology today. The maximum usable bandwidth of the signal processed by the delay element (either a pure phase shifter or a TTD element) is *B*. The absolute frequency of the subcarrier, *f*, can be related to the frequency of the carrier, *f*_{c}, by

$$f=\zeta {f}_{{\rm{c}}}$$

(2)

*ζ* is the ratio of the subcarrier frequency to the carrier frequency. Therefore, if \(f\in \left[{f}_{{\rm{c}}}-\frac{B}{2},{f}_{{\rm{c}}}+\frac{B}{2}\right]\) and fractional bandwidth *b* is *B*/*f*_{c}, \(\zeta \in \left[1-\frac{b}{2},1+\frac{b}{2}\right]\). On an integrated circuit or on a discrete printed circuit board, the inter-antenna spacing is typically designed to be *λ*_{c}/2. The array gain can then be calculated for each subcarrier as *g*(*ζφ* − *φ*_{F}), in which *φ* is the angle of arrival or angle of departure and *φ*_{F} is the angle of focus of the beam. Let us assume that the signal transmitted had no beam squint and only examine the effects on angle of arrival at the receiver. The antenna array gain is^{37,38}

$$g(x)=\frac{\sin \left(\frac{N\pi x}{2}\right)}{\sqrt{N}\sin \left(\frac{\pi x}{2}\right)}{{\rm{e}}}^{j\frac{(N-1)\pi x}{2}}$$

(3)

in which *x* = *ζφ* − *φ*_{F}. In other words, the gain for a subcarrier frequency *ζ* at an angle of arrival of *φ* is equivalent to the gain for a carrier frequency at an angle of arrival of *φ*′ = *ζφ*. Clearly, the effect of beam squint worsens when *φ* is substantially different from *φ*_{F}. Assume that a signal of total power *P* and bandwidth *B* and that consists of *N*_{f} evenly spaced subcarriers is received by an array of *N* delay elements. Here the Gaussian thermal noise is *σ*^{2}. If only pure phase shifters were used, the channel capacity with beam squint is^{41,42}

$${C}_{{\rm{squint}}}({\varphi }_{{\rm{F}}},\varphi ,b)=\frac{B}{{N}_{{\rm{f}}}}\mathop{\sum }\limits_{n=0}^{{N}_{{\rm{f}}}-1}{\log }_{2}\left(1+\frac{2P{| g({\zeta }_{n}\varphi -{\varphi }_{{\rm{F}}})| }^{2}}{B{\sigma }^{2}}\right)$$

(4)

in which \({\zeta }_{n}=1+\frac{(2n-{N}_{{\rm{f}}}+1)b}{2{N}_{{\rm{f}}}}.\)

If only TTD elements were used in the array, no beam squint is experienced. The channel capacity would be^{41,42}

$${C}_{{\rm{no-squint}}}({\varphi }_{{\rm{F}}},\varphi )=B{\log }_{2}\left(1+\frac{2P{| g(\varphi -{\varphi }_{{\rm{F}}})| }^{2}}{B{\sigma }^{2}}\right)$$

(5)

It should be noted that the number of delay elements, *N*, is constrained by the maximum area on chip that can be allocated for a phased array or an array of TTD elements. Projections of channel capacity in Fig. 4b,c are computed for a maximum area of 2.1 mm^{2} using equations (4) and (5). It is then clear that the improved channel capacity using TTD elements in arrays comes at great implementation cost and an arrangement of them in arrays on chip is unrealistically expensive. For simplicity, the performance of the Q-TTD element reported here is approximated to that of a TTD within a bandwidth that gives less than 3 ps of deviation from a TTD. In this computation, *N*_{f} = 2,048 and normalized angles, *φ* and *φ*_{F}, are both equal to 0.9 to maximize antenna gain. Further, with a typical receive power of around −80 dBm and Gaussian noise spectral density of *σ*^{2} (equal to *kT*, *k* being the Boltzmann constant), at 290 K, the power-to-noise-density ratio (*P*/*σ*^{2}) would be around 2 GHz. The representative comparison in Fig. 4c, made using these assumptions, shows the advantage of dense integration of these Q-TTD elements, both in doubling of available channel capacity in comparison with on-chip phase shifters and TTD elements and in the monotonic increase in channel capacity with increased operational bandwidth and power-to-noise-density ratio (Fig. 4b).

### Projected array factor for the three types of delay element

Designing for a boosted channel capacity necessitates the calculation of a critical metric in array theory, the array factor, for the three types of array. It determines the efficacy of the delay elements in directing the beam in a target direction with minimal squint, large array gain and small frequency dependence. For brevity, we will consider a set of N identical antennas (transducers) and their corresponding delay elements arranged in 1D (linear) array and oriented in the same direction, with each transducer element having the same radiation pattern. This should simplify the analysis to consider the effect of only the delay elements themselves.

The array factor is a function of the positions of the delay elements in the array and the weights used to quantify the net contribution to the field strength in a particular direction. Referring to Supplementary Fig. 12a, the inter-element separation, *d*_{x}, equals *λ*/2. *λ* is the midband frequency of the linear array. For this simple configuration, it can be shown that for a phased array, the array factor (denoted AF in the following equations) is^{38}

$${{\rm{A}}{\rm{F}}}_{{\rm{P}}{\rm{h}}{\rm{a}}{\rm{s}}{\rm{e}}{\rm{d}}-{\rm{A}}{\rm{r}}{\rm{r}}{\rm{a}}{\rm{y}}}=\mathop{\sum }\limits_{n=0}^{N-1}{a}_{n}\exp \left(j\omega n[{\delta }_{(\theta )}-\frac{{\omega }_{0}}{\omega }{\tau }_{0}]\right)$$

(6)

in which *a*_{n} is the weight assigned to the *n*th element. Here all *a*_{n} are equal to 1. *ω*_{0} is the centre frequency, *ω* is the frequency component of interest in the incoming beam and *τ*_{0} is an effective time increment to steer the beam to the desired angle, *θ*. Here *δ*_{(θ)} is the effective angle-dependent ‘projected delay’ that equals \(\frac{{d}_{x}}{c}\sin \theta \), with *c* being the speed of light. Similarly, the array factor of a TTD element is a simple variation of equation (6) that excludes frequency dependence. It is

$${{\rm{AF}}}_{{\rm{TTD}}}=\mathop{\sum }\limits_{n=0}^{N-1}{a}_{n}\exp \left(j\omega n\left[{\delta }_{(\theta )}-{\tau }_{0}\right]\right)$$

(7)

Now, as the time-delay variations across frequency of the Q-TTD in Fig. 4c show, there is a deviation of Δ*τ* = ±3 ps about the 50-ps reference state, within the band of interest. That is, the frequency-dependent projected-phase term has a slight variation from 0.94 to 1.06. Therefore, the array factor of a Q-TTD element is

$${{\rm{AF}}}_{{\rm{Q-TTD}}}=\mathop{\sum }\limits_{n=0}^{N-1}{a}_{n}\exp \left(j\omega n\left[{{{\epsilon }}_{{\rm{Q-TTD}}}\delta }_{(\theta )}-{\tau }_{0}\right]\right)$$

(8)

with *ε*_{Q-TTD} ∈ [0.94, 1.06] for the band from 14 GHz to 24 GHz.

These array factors can now be determined for any target angle to discover what degree of beam squint is imparted by each type of array. For ease, we will consider only a *θ* of 45° here. Also, as mentioned in the main text, we will impose a realistic silicon chip area of 2.1 mm^{2}, which would restrict the phased array comprising miniature phase shifters from ref. ^{16} to 15, the TTD elements from ref. ^{22} to 6 and the number of Q-TTD elements to 16. Figure 4d shows the projected array factors, normalized with respect to their maximum amplitudes for these three types of array and assuming that the arrays have infinite phase resolution, for simplicity. As expected, the phased array has poor fidelity with respect to frequency (Fig. 4d, left), owing to the narrow bandwidth over which a constant group delay is maintained. It smears out the signal across the spectrum and its gain in the target direction diminishes rapidly for a frequency offset. A TTD array, on the other hand, does have a directive response (Fig. 4d, centre) but its gain is rather weak owing to the limited number of on-chip elements. A Q-TTD element, however, has not only very directive gain owing to its high packing density (Fig. 4d, right) but also a high array factor that exceeds 0.8 across its band (Supplementary Fig. 12c.3). It produces group-delay variations similar to that of a TTD phase shifter^{43,44}. These key features combine to give a channel capacity that surpasses both on-chip phased and true-timed arrays greatly.