Prism Interactionnel: Metasurface & Choreography

• Prism Interactionnel is an interdisciplinary concept bridging programmable metasurfaces for optical beam steering and formal choreography in probabilistic concurrent systems.
• It enables precise, digital control of wave steering and nonreciprocal amplification, allowing real-time spatial decomposition of frequency components.
• It also provides a formal method for modeling and verifying interactive system behaviors, ensuring synchronized global interactions in probabilistic frameworks.

A prism interactionnel, depending on context, refers to either (i) a fully programmable, nonreciprocal metasurface that generalizes the spatial decomposition function of classical optical prisms, or (ii) a formal choreography specification centering system interactions as first-class entities for modeling and verifying probabilistic concurrent systems, particularly in the context of the PRISM probabilistic model checker. Both usages share the emphasis on structured, interaction-centered decomposition—whether of electromagnetic waves into spatially separated beams or of system executions into global, probability-aware interaction traces.

1. Programmable Nonreciprocal Metasurface Prism: Physical Principles

A prism interactionnel in electromagnetics is a deeply subwavelength, planar metasurface that acts analogously to a classical glass prism by decomposing an incident polychromatic wave into spatially separated beams, but with three major advances: (1) complete electronic programmability of the refraction law for each frequency component, (2) intrinsic nonreciprocity and directionality via amplification, and (3) sub-wavelength, integrated form factor (Taravati et al., 2020).

The fundamental operating equation stems from the generalized Snell’s law for metasurfaces. For incident frequency $\omega$ and wavelength $\lambda(\omega)$, the angle of refraction is given by

$$\theta(\omega) = \sin^{-1} \left[ \frac{\lambda(\omega)}{2\pi} \frac{\partial \varphi(x,\omega)}{\partial x} \right]$$

with $\varphi(x,\omega)$ the transmission phase profile imparted along the metasurface. Programmable variation of $\varphi(x,\omega)$ for each $\omega$ enables arbitrary steering of frequency components.

2. Metasurface Supercell Architecture and Nonreciprocity

Each metasurface supercell implements a five-stage chain: receive microstrip, unilateral transistor amplifier, tunable RF phase shifter, secondary amplifier, and transmit microstrip. The use of transistor amplifiers (e.g., Mini-Circuits Gali-2+) imparts strong forward gain ($|S_{21}| > 20\,{\rm dB}$) and high reverse isolation ($|S_{12}| < -20\,{\rm dB}$). The expressions for $S_{21}(\omega)$ and $S_{12}(\omega)$, which properly account for multiple reflections and nonreciprocal transmission, ensure programmable gain and phase at each cell while strongly suppressing back-propagating waves.

By digital bias control (via varactors and transistor biasing), each cell’s amplifiers and phase shifters can be independently tuned for each frequency. The metasurface thereby achieves spatially and spectrally variable beamforming and gain.

3. Digital Programming and Real-Time Operation

An external FPGA orchestrates the metasurface dynamics. The desired phase and amplitude (for each tone and cell) are encoded as digital words in a memory array, periodically output through DACs to the bias networks. Typical configurations use 6 bits for phase and 6 bits for gain per amplifier, yielding sub-degree and sub-dB control and aggregate beam-steering latencies $\lambda(\omega)$0. This supports real-time, frequency-dependent spatial mapping and gain control across all cells, enabling dynamic adaptation to evolving electromagnetic requirements (Taravati et al., 2020).

4. Performance Characterization

A $\lambda(\omega)$1 prototype fabricated on RO4350 at $\lambda(\omega)$2 demonstrated:

• Forward gain $\lambda(\omega)$3 across $\lambda(\omega)$4–$\lambda(\omega)$5
• Backward gain $\lambda(\omega)$6 ($\lambda(\omega)$7 $\lambda(\omega)$8 isolation)
• Far-field spatial separation of tones at $\lambda(\omega)$9, $$\theta(\omega) = \sin^{-1} \left[ \frac{\lambda(\omega)}{2\pi} \frac{\partial \varphi(x,\omega)}{\partial x} \right]$$0, $$\theta(\omega) = \sin^{-1} \left[ \frac{\lambda(\omega)}{2\pi} \frac{\partial \varphi(x,\omega)}{\partial x} \right]$$1 with individual beam angles ($$\theta(\omega) = \sin^{-1} \left[ \frac{\lambda(\omega)}{2\pi} \frac{\partial \varphi(x,\omega)}{\partial x} \right]$$2, $$\theta(\omega) = \sin^{-1} \left[ \frac{\lambda(\omega)}{2\pi} \frac{\partial \varphi(x,\omega)}{\partial x} \right]$$3, $$\theta(\omega) = \sin^{-1} \left[ \frac{\lambda(\omega)}{2\pi} \frac{\partial \varphi(x,\omega)}{\partial x} \right]$$4) and gain $$\theta(\omega) = \sin^{-1} \left[ \frac{\lambda(\omega)}{2\pi} \frac{\partial \varphi(x,\omega)}{\partial x} \right]$$5–$$\theta(\omega) = \sin^{-1} \left[ \frac{\lambda(\omega)}{2\pi} \frac{\partial \varphi(x,\omega)}{\partial x} \right]$$6, with isolation $$\theta(\omega) = \sin^{-1} \left[ \frac{\lambda(\omega)}{2\pi} \frac{\partial \varphi(x,\omega)}{\partial x} \right]$$7

Full-wave electromagnetic simulations concurred with empirical data. The device thus combines the functionalities of a beam-steering phased array, frequency demultiplexer, and nonreciprocal amplifier in a planar form factor.

Metric	Forward	Backward	Bandwidth
Gain ($$\theta(\omega) = \sin^{-1} \left[ \frac{\lambda(\omega)}{2\pi} \frac{\partial \varphi(x,\omega)}{\partial x} \right]$$8)	$$\theta(\omega) = \sin^{-1} \left[ \frac{\lambda(\omega)}{2\pi} \frac{\partial \varphi(x,\omega)}{\partial x} \right]$$9 dB	$\varphi(x,\omega)$0 dB	4.3% CF BW
Angle resolution	$\varphi(x,\omega)$1 degree	$\varphi(x,\omega)$2 degree	--
FPGA update rate	$\varphi(x,\omega)$3 ns	$\varphi(x,\omega)$4 ns	--

5. Applications and Outlook in Wave Engineering

The metasurface-based prism interactionnel supports applications—including real-time dynamic holography, adaptive or multi-beam radar, and agile multi-band wireless links—that demand both spatial dispersion (the prism effect) and nonreciprocal gain/isolation. The device’s digitally programmable nature suggests straightforward integration into 5G/6G and advanced radio/photonic architectures, offering “all-active,” reconfigurable, frequency- and direction-dependent spatial mapping heretofore inaccessible to passive or reciprocal metamaterials (Taravati et al., 2020).

6. Interactionnel Choreography in Formal Systems: The PRISM Perspective

In the domain of formal methods, “interactionnel” refers to the design of specification languages and models that foreground global, interaction-centered execution. The “probabilistic choreography language for PRISM” allows for end-to-end specification of concurrent systems as global interaction terms, supporting constructs such as probabilistic branching, parallel execution, and recursion (Carbone et al., 11 Mar 2025).

Syntax and Semantics

The core language syntax is:

• $\varphi(x,\omega)$5 where probabilistic branches $\varphi(x,\omega)$6 encode $\varphi(x,\omega)$7 sending one of the labels $\varphi(x,\omega)$8 to $\varphi(x,\omega)$9 with probability $\varphi(x,\omega)$0. System executions are modeled by a probabilistic labeled transition system (pLTS).

Compilation to PRISM

Compilation produces a set of PRISM modules—one per participant—with local state variables mirroring each participant’s control-point in the choreography. Each synchronous interaction is mapped to synchronized labels across modules, while probabilistic branching becomes localized, unsynchronized (internal $\varphi(x,\omega)$1-transitions) with proper probability weights. The translation is formally shown to be probability-preserving via a bisimulation between the source choreography and the PRISM Markov Decision Process, ensuring equivalence of reachability and PCTL properties (Carbone et al., 11 Mar 2025).

Concrete Example

A single sender Alice flips a biased coin (0.6/0.4), sending “H” or “T” to Bob:

• Global choreography: $\varphi(x,\omega)$2
• The PRISM model encodes the probabilistic choice as an internal (unsynchronized) command in Alice’s module, and the subsequent sends as synchronized labels between Alice and Bob.
• The model-checking query $\varphi(x,\omega)$3 correctly yields $\varphi(x,\omega)$4, matching the intended global probability semantics.

A distinguishing feature is the preservation of global semantics: individual modules rendezvous on the prescribed labels, with no hidden orchestration, and the execution traces’ probability distributions are invariant under the translation (Carbone et al., 11 Mar 2025).

7. Comparative Analysis and Significance

Across both domains, the interactionnel prism framework reflects a paradigm in which global, compositional decomposition—whether of signals or system behaviors—is rendered programmable, precisely controllable, and modularized around explicit interactions. In electromagnetics, this supports unprecedented functional density and adaptability in frequency–space mapping and signal routing. In formal verification and system modeling, it yields clear, analyzable, and faithful representations of concurrent, probabilistic system behaviors centered around explicit communications. Both approaches have demonstrated significant advances in their respective performance benchmarks and formal guarantees (Taravati et al., 2020, Carbone et al., 11 Mar 2025).

A plausible implication is that advances in one domain—for example, digital programmability and supervised optimization in metasurface hardware—could inform methodologies for compositional synthesis or optimization in the other, such as automated construction of choreography modules or reward-guided interactive dialog systems. The interactionnel perspective thus bridges physical and computational instantiations of modular, precisely orchestrated interaction.