Magnetic-free non-reciprocity based on staggered commutation

Lorentz reciprocity is a fundamental characteristic of the vast majority of electronic and photonic structures. However, non-reciprocal components such as isolators, circulators and gyrators enable new applications ranging from radio frequencies to optical frequencies, including full-duplex wireless communication and on-chip all-optical information processing. Such components today dominantly rely on the phenomenon of Faraday rotation in magneto-optic materials. However, they are typically bulky, expensive and not suitable for insertion in a conventional integrated circuit. Here we demonstrate magnetic-free linear passive non-reciprocity based on the concept of staggered commutation. Commutation is a form of parametric modulation with very high modulation ratio. We observe that staggered commutation enables time-reversal symmetry breaking within very small dimensions (λ/1,250 × λ/1,250 in our device), resulting in a miniature radio-frequency circulator that exhibits reduced implementation complexity, very low loss, strong non-reciprocity, significantly enhanced linearity and real-time reconfigurability, and is integrated in a conventional complementary metal–oxide–semiconductor integrated circuit for the first time.


Supplementary Tables
Supplementary Table 1: Values of the components used in the integrated circulator: the inductor is an off-the-shelf surface-mount component, while the capacitors are implemented on the integrated circuit.

Component
Value Supplementary In this supplementary note, we provide an analysis of the staggered commutated network based on linear periodically-time-varying (LPTV) network theory 1, 2 .
As mentioned in the main text, LPTV analysis of commutation across an array of transmission lines is analytically challenging. Here, we consider an array of capacitors with commutating switches on both sides (Supplementary Figure 1). Since we are interested in electrically-small transmission lines enabling effectively a point parametric modulator, the approximation of the transmission lines with a capacitance is expected to be accurate. An array of N capacitors is considered, each of value C, and the switches are controlled by the signals denoted by p i (t) and q i (t) in Supplementary Figure 1. A high level of the control signals denotes that the corresponding switch is in the ON state, while a low level denotes the OFF state. While the analysis considered would generally apply to any commutated network (electronic, optical or otherwise), these control signals are highly representative of the clock signals that are generated on our complementary metal-oxide-semiconductor (CMOS) IC to control the transistor-based switches. Each switch is ON for a 1 N fraction of the commutation period T s . The switches are assumed to be ideal with infinite modulation ratio -in other words, in the ON state, the switches have zero resistance or perfect transmission, and in the OFF state, the switches have infinite resistance or perfect (reciprocal) isolation. Capacitive parasitics in the switches are ignored under the assumption that switching speed is significantly higher than the commutation frequency. The q i (t) control signals are delayed by T s /4 compared to the p i (t) signals representing staggered commutation by -90 • . An input voltage source V in and source resistance of value Z 0 models the feed transmission line, and a load resistance of Z 0 models the load transmission line.
In linear periodically-time-varying (LPTV) systems, the input and output spectra are related as 1, 2 : where X(f ) and Y (f ) are the input and output spectra, respectively, and f s = 1 Ts is the periodicity HTFs are then combined across the kernels to find the overall HTFs to V 2 or V 1 . The reader is directed to prior literature 2 for additional details on this analytical approach.
The single-state kernel is shown in Supplementary Figure 2. Furthermore, based on the timing of the control signals, a single period can be divided into the sub-periods τ 1 , τ 2 , τ 3 and τ 4 . Within each sub-period, the kernel has a valid LTI state-space description. The n-th period is shown in Supplementary Figure 2. The durations of the sub-periods are given by: It should be noted that we are implicitly assuming N ≥ 4 in the timing diagram of Supplementary Figure 2 so that p i (t) and q i (t) don't overlap with each other. As mentioned earlier, in each sub-period marked in Supplementary Figure 2, the kernel can be analyzed using LTI theory.
For example, in the first sub-period, we have: It should be noted that this equation only holds for nT s < t < nT s + Ts N . A similar LTI differential equation can be written for other sub-periods as well with A 2 = 0, A 3 = −1 Z 0 C , A 4 = 0 and B 2 = 0, B 3 = 0, B 4 = 0. We now define v c,k (t) as a signal that is equal to v c (t) within the k-th sub-period and zero otherwise. In other words, v c, is a windowing function given by w k (t) = 1, nT s + σ k−1 ≤ t < nT s + σ k and 0 elsewhere. Accounting for the boundary conditions at the beginning and end of each sub-period, we get: It should be noted that v in,k (t) = v in (t) × w k (t). Since (5) is valid over all time, a Fourier transform may be evaluated as below. where The superscript adopted in the notation clarifies that we are attempting to determine the harmonic transfer function to V c in the single-state kernel. In this equation, the G k functions are the switching-moment transfer functions of the system at the switching moments t = nT s + σ k .
Additional details related to the derivation of (7) may be found in prior literature 2  . The G k (f )s in (7) can be evaluated by following the steps delineated in prior literature 2 to obtain the generic form: where f C = 1 2πZ 0 C and Note that only β 2 and β 3 are of interest since H V 2,kernel n = H V c,kernel n,3 . Substituting for A k , B k and G k in (7), we derive: The overall transfer function H V 2 n (f ) from V in to V 2 in Supplementary Figure 1 can be calculated by performing a summation across all paths 2 as follows: This results in: Based on the definition of the scattering parameters, S 21 can be computed as: A similar procedure may be followed to calculate S 11 , requiring the computation of the transfer function from V in to V 1 instead. In this case, H V 1,kernel Subsequently, S 11 can be calculated as Similarly, by exciting the circuit with a voltage source at port 2 and performing a similar analysis, we can derive To summarize, the S-parameter matrix of the network shown in Supplementary Figure 1 at f s under the assumption of f C /f s << 1 or C >> 1 2πfsZ 0 can be written as: It can clearly be seen that S 21 and S 12 are reciprocal in magnitude response but are phase nonreciprocal. With -90 • staggering, the phase of S 21 is -90 • while the phase of S 12 is +90 • . Furthermore, it can be easily shown that the magnitudes of S 21 and S 12 approach unity as N → ∞ while the magnitudes of S 11 and S 22 approach 0. In other words, when f C /f s << 1 or C >> 1 2πfsZ 0 , at f s , strong signal transmission is seen across the staggered commutated network, particularly for large N , along with phase non-reciprocity. It should be mentioned that N = 4 and N = 8 are sufficient to ensure low loss transmission, as the magnitudes of S 21 and S 12 are -1.8dB and -0.45dB, respectively.