Programmable 2-bit STCM: EM Analog Operations

• Programmable 2-bit STCM is a metasurface platform where each meta-atom, controlled by two bits, toggles among four discrete states to enable dynamic EM wave manipulation.
• It employs space-time coding sequences—modulated via rapid switching of PIN diodes—to perform analog operations such as first-order differentiation and integration on spatial energy profiles.
• Experimental validations demonstrate low amplitude and phase errors, showcasing its potential for applications in beam steering, edge detection, and adaptive wireless communications.

A programmable 2-bit STCM (Space-Time Coding Metasurface), as realized in programmable electromagnetic platforms, denotes a metasurface in which each meta-atom can be programmed into one of four discrete states via two digital control bits, enabling active, time-varying manipulation of electromagnetic (EM) waves at subwavelength scales. Recent research demonstrates that such architectures can implement not just conventional EM manipulations (beam steering, focusing), but also perform direct analog mathematical operations—most notably first-order differentiation and integration—on spatial energy distributions of incident waves, achieved by careful design of space-time coding sequences that modulate the meta-atom states in both space and time. The 2-bit resolution (four quantization levels per meta-atom) provides sufficient granularity for harmonic synthesis in real-world metasurface hardware, as validated experimentally at microwave frequencies (Shi et al., 4 Jan 2026).

1. Architecture and Space-Time Coding Principle

Each meta-atom in a 2-bit STCM is a reflection-type unit cell, typically implemented on a printed circuit board (PCB) and embedding two electronically addressable PIN diodes. By setting these two diodes independently to ON (logic 1) or OFF (logic 0), the system generates four discrete reflection states denoted “00”, “01”, “10”, and “11”. At a representative frequency (e.g., 10.3 GHz), these four states yield phase steps of approximately 90° and amplitude response spanning from –0.06 dB to –3.25 dB. The phase and amplitude values for each state are as follows:

Coding State (PIN1, PIN2)	Amplitude (dB)	Phase (°)
00 (OFF,OFF)	–0.06	12
01 (ON,OFF)	–2.86	102
10 (OFF,ON)	–3.25	186
11 (ON,ON)	–0.98	289

Time-varying coding is achieved by modulating each meta-atom’s digital control inputs with periodic sequences of length $L$ (typically $L=16$ time slots per period $T_m$), producing a temporally modulated reflection coefficient for each unit cell. The instantaneous response $r_k(t)$ of the $k$-th meta-atom is synthesized by sequencing through $L$ different states $r_k^{(C_n)}$ according to a precomputed codeword $C_n$. Fourier analysis of $r_k(t)$ yields spatial and spectral harmonic components at multiples of the modulation frequency, forming the basis for harmonic-specific functionality (Shi et al., 4 Jan 2026).

2. Mathematical Framework for Calculus Operations

The space-time-coding metasurface implements analog calculus operations through spatial Fourier synthesis at selected harmonics. For a linear array of $N$ meta-atoms, the reflected field at angle $\theta$ for the $m$-th harmonic is

$$F_{\text{out},m}(\theta) = \sum_{i=1}^{N} f(x_i)\, r_{k,m}\, e^{-j(2\pi/\lambda) x_i \sin\theta}$$

where $f(x_i)$ is the near-field amplitude profile, $r_{k,m}$ is the $m$-th harmonic coefficient, and $\lambda$ is wavelength. This construction enables convolution between the incident spectrum and the metasurface’s transfer function in spatial-frequency ($k_x$) space.

Target transfer functions for first-order differentiation and integration in the Fourier domain are

• Differentiator: $T_{\text{diff}}(k_x) = j k_x$
• Integrator: $T_{\text{int}}(k_x) = \frac{1}{j k_x}$

By optimizing the space-time coding sequence ($\{C_n\}$) for each meta-atom, the metasurface’s transfer function $R_m(\theta)$ at harmonic $m$ is matched to these operator functions, emulating analog mathematical manipulation of the incoming waveform (Shi et al., 4 Jan 2026).

3. Space-Time Coding Sequence Optimization

To realize desired operator profiles, each column of the metasurface is assigned an independent time-varying digital codeword (length $L=16$; elements in $\{00,01,10,11\}$). A genetic algorithm minimizes a cost function that quantifies the deviation between the achieved and target harmonic transfer for each column. For example, for first-order differentiation on the $+1$st harmonic,

$$\text{err}_k^{\text{diff}} = \left|(k-N+1)jA - \frac{1}{L} \sum_{n=1}^{L} r_k^{(C_n)} e^{-j2\pi n k / L}\right|$$

with $A$ a scaling parameter. Simultaneous implementation of multiple operators (e.g., differentiation at $+1$st harmonic and integration at $+2$nd harmonic) is achieved by expanding the cost function to include all relevant harmonics with appropriate amplitude constraints. The optimized $16 \times 16$ codeword matrix is loaded onto the metasurface controller (Shi et al., 4 Jan 2026).

4. Hardware System and Experimentation

The hardware system comprises:

• A metasurface array of $N \times M$ meta-atoms, each with 2-bit programmable states.
• An FPGA board to sequence space-time codewords and to drive the PIN-diode bias lines via a digital-to-analog interface.
• Level-shifting and switching circuitry connecting the FPGA outputs to the metasurface.
• Synchronization of the space-time modulation clock (typ. 1 MHz) via phase-locked loops.

In experimental setups, a microwave horn antenna illuminates the metasurface inside an anechoic chamber. The metasurface’s output at specific harmonics (e.g., $+1$st or $+2$nd) is captured via harmonic-selective down-conversion, allowing direct comparison of theoretical, simulated, and measured far-field patterns for a variety of operator implementations (Shi et al., 4 Jan 2026).

5. Experimental Results and Performance Assessment

Extensive measurements confirm the programmable 2-bit STCM’s capability to perform analog calculus operations on spatial energy distributions:

• For a programmed single-beam incident waveform, the differentiator sequence produces two split lobes with a central null, as expected from the theoretical derivative.
• For two-beam incidence (phase opposition), the integrator sequence yields a single broad lobe with nearly flat amplitude over a $30^\circ$ span.
• Main-lobe amplitude and phase errors are $<0.5$ dB and $<10^\circ$ within central angular spans. Sidelobe deviations ($2$–$4$ dB) arise from PIN-diode nonlinearities and quantization error.
• The meta-atom’s phase step stability ($90^\circ\pm5^\circ$) and amplitude uniformity ($>-3.5$ dB) are maintained over $10.0$–$10.6$ GHz.

Limitations include the intrinsic quantization granularity of 2-bit coding and non-idealities in switching elements, which set bounds on achievable operator fidelity and angular resolution (Shi et al., 4 Jan 2026).

6. Scalability, Programmability, and Applications

The programmable 2-bit STCM supports rapid reconfiguration: any of the four states per meta-atom is set digitally; the entire metasurface can be reprogrammed in $\lesssim1$ ms per codeword matrix update (by FPGA I/O speed and diode switching time). Scalability to higher numbers of meta-atoms and to 3-bit or higher coding (increasing the number of discrete states to 8 or 16 per element) would allow narrower amplitude/phase steps, broader bandwidth, and multi-operator parallelism. Applications include:

• Real-time analog EM preprocessing (e.g., edge detection, space-domain equalization).
• Multifunctional beam-forming for adaptive wireless communications and advanced 5G/6G scenarios.
• Wave-based computational kernels for analog signal processing and microwave AI (Shi et al., 4 Jan 2026).

7. Comparative Context and Future Directions

The programmable 2-bit STCM advances the paradigm of direct, in situ electromagnetic analog computing by moving beyond static, spatial-only programmable metasurfaces. By leveraging time-varying coding sequences mapped to harmonic spectral space, such metasurfaces achieve dynamic, operator-specific control over incident wavefronts without additional post-processing. The integration of space-time coding with digitally addressable meta-atoms is a critical step toward multifunctional, software-defined EM analog processors. Increasing the bit depth per meta-atom and deploying more advanced control electronics will further broaden the functional scope and quantitative accuracy of such systems for real-time, physics-native information manipulation (Shi et al., 4 Jan 2026).