Harmonic Beam Steering

• Harmonic beam steering is a technique that uses time modulation in arrays and metasurfaces to independently control beam directions across multiple harmonics.
• Advanced digital preprocessing and optimization enable precise beam control with high efficiency and reduced side-lobe levels in various electromagnetic platforms.
• Applications span from THz communications and radar to nonlinear photonics and RIS-enabled sensing, offering versatile multi-frequency and multi-target capabilities.

Harmonic beam steering is the class of techniques that achieve directional control of radiation at harmonic (integer-multiple) or combinatorial frequencies generated by temporal, spatial, or spatiotemporal modulation of array elements or meta-atoms. Unlike conventional (static) phased arrays or metasurfaces, which steer a single-frequency beam by imposing a progressive spatial phase gradient, harmonic beam steering exploits temporal coding to generate and independently direct beams at multiple harmonic frequencies—substantially expanding the controllable degrees of freedom for wave manipulation in both the space and frequency domains. Applications span from THz communications and radar to nonlinear photonics, real-time multi-target illumination, and reconfigurable intelligent surface (RIS)-aided sensing (Shabanpour, 2020).

1. Physical Principles of Harmonic Beam Steering

Standard phased-array beam steering synthesizes a main lobe at angle θ₀ by applying a spatially progressive phase shift across the aperture, such that the array factor constructively interferes at the desired direction for the input frequency. In harmonic beam steering, the modulation of meta-atom or antenna parameters in time (e.g., phase, amplitude, load impedance, or transmission/reflection coefficient) introduces additional spectral components (harmonics or intermodulation products) in the reradiated field.

For a metasurface or array element modulated periodically with period T₀, the reflected (or transmitted) field contains frequency components at $f_c + m f_0$, $m \in \mathbb{Z}$, where $f_c$ is the carrier and $f_0=1/T_0$ is the modulation frequency. Each harmonic $m$ experiences a distinct effective spatial phase gradient:

$$\Delta\psi_m = \frac{2\pi}{\lambda_m} d \sin \theta_m$$

which yields a steering law:

$$\theta_m = \arcsin\left(\frac{m}{L\,d/\lambda}\right)$$

where $L$ is the sequence length (for digital coding), $d$ the element spacing, and $\lambda$ the wavelength at the carrier (Shabanpour, 2020). This allows simultaneous, independent steering of beams at multiple harmonic frequencies using a common hardware platform and control sequence.

2. Analytical Frameworks and Closed-Form Results

2.1. Space–Time Digital Metasurfaces

For large-aperture digital metasurfaces with $m \in \mathbb{Z}$0, the radiated power in the m-th harmonic, under uniformly coded elements and linear time-gradient, admits the closed-form:

$m \in \mathbb{Z}$1

with $m \in \mathbb{Z}$2 the fundamental power. End-fire harmonics (where $m \in \mathbb{Z}$3) require King–Thomas correction, yielding:

$m \in \mathbb{Z}$4

These formulas rely on the main-lobe dominance (Elliott–King) and the assumption of a single “1” in each $m \in \mathbb{Z}$5-interval coding period (Shabanpour, 2020). By tailoring the digital sequence, the metasurface can dynamically program the number, angle, and intensity of simultaneously radiated harmonic beams.

2.2. Time-Modulated Arrays

Time-modulated arrays (TMAs) excite each element with periodic ON–OFF or multilevel sequences, producing spectral lines at $m \in \mathbb{Z}$6. Advanced approaches (e.g., digitally preprocessed sequences or periodic stair-step pulses) use optimization to minimize power in unwanted harmonics and steer each selected harmonic $m \in \mathbb{Z}$7 to a desired angle $m \in \mathbb{Z}$8 (Maneiro-Catoira et al., 2024, Maneiro-Catoira et al., 2024). The array factor at each harmonic is determined by the DFT of the digital sequence per element and can be analytically described as:

$m \in \mathbb{Z}$9

Selection of the sequence $f_c$0 is usually by optimization to match $f_c$1 to the intended direction and amplitude taper.

3. Implementation Modalities: Architectures and Materials

Harmonic beam steering has been realized in a variety of electromagnetic platforms and at multiple frequency ranges:

• Space–Time Digital Metasurfaces: Large-area arrays of phase-only digital meta-elements controlled by bias voltages with programmed time gradients (binary or multilevel) (Shabanpour, 2020). Quantization level $f_c$2 and physical aperture set side-lobe suppression and beamwidth.
• Reconfigurable Intelligent Surfaces (RIS): Thin-film 1×4 patch arrays on flexible substrates with PIN diode loading and programmable baseband control (MCU/FPGAs) can steer harmonics on both reflection and transmission sides (Xie et al., 2024).
• Space–Time-Coded Metamaterial Antennas: Leaky-wave antennas with varactor-loaded CRLH unit cells modulated by FPGA-generated digital sequences achieve independently steered spatial beams for each harmonic, with programmable nonreciprocity (Vosoughitabar et al., 2023).
• Nonlinear Optical Phased Arrays: Second-harmonic (SH) generation in transition-metal dichalcogenide monolayers (e.g., MoS₂) coupled to plasmonic rods, where the phase of nonlinear polarization is encoded via the optical pump to steer the SH beam (Busschaert et al., 2019).
• Dielectric Metalattices: High-index 1D lattices exploit interference between lattice-coupled multipoles to route energy into specific diffraction (harmonic) orders for beam steering and splitting, subject to generalized Kerker conditions (Liu et al., 2017).

A comparison of selected implementations is shown below.

Architecture	Control Mechanism	Harmonic Orders	Reference
Digital Metasurface (RF/microwave)	Binary/L-step sequence	Multi	(Shabanpour, 2020, Xie et al., 2024)
TMA with Optimized Digital Sequence	Discrete-time switch	Multi	(Maneiro-Catoira et al., 2024, Maneiro-Catoira et al., 2024)
Leaky-wave ST-coding MTM Antenna	Varactor, FPGA	Multi	(Vosoughitabar et al., 2023)
Nonlinear Optical Array (SHG)	Pump phase	SH only	(Busschaert et al., 2019)
Dielectric Photonic Metalattice	Geometry, multipole	Multi (diffract.)	(Liu et al., 2017)

4. Design Constraints, Performance Metrics, and Trade-offs

4.1. Efficiency and Side-Lobe Level

Radiation efficiency into the selected harmonics is a primary concern. Digital preprocessing of TMA sequences yields up to 82% efficiency for $f_c$3 harmonics, compared to 32% for unprocessed rectangular TMA (Maneiro-Catoira et al., 2024). Side-lobe levels as low as –23 dB have been demonstrated by numerical optimization. Using stair-step pulses, first harmonic beamforming reaches $f_c$4, with the total system efficiency (including switching losses) $f_c$5 for a 30-element array at SLL = –17 dB (Maneiro-Catoira et al., 2024).

4.2. Scan Angle Range and Angular Resolution

The maximum steerable angle for a given harmonic is dictated by the spatial sampling (element spacing), modulation rate, and quantization:

$f_c$6

For an optical SH array with $f_c$7 and $f_c$8, the achievable scan range is $f_c$9 (Busschaert et al., 2019). RIS arrays exhibit front/back steering spans of up to $f_0=1/T_0$0 for the first harmonics (Xie et al., 2024).

4.3. Hardware Simplicity and Scalability

Time-modulated arrays with optimized binary sequences require only high-speed SPDT switches per element, digital memory, and a clock, omitting the need for VGAs or analog phase shifters (Maneiro-Catoira et al., 2024). Stair-step pulse architectures use SP4T switches for phase control and SPST switches for amplitude (Maneiro-Catoira et al., 2024). Inkjet-printed RIS approaches offer cost-effective, scalable implementations without a ground shield (Xie et al., 2024).

4.4. Limitations

• Harmonic amplitude decreases as $f_0=1/T_0$1 for a binary square wave; energy quickly drops for higher orders (Xie et al., 2024).
• Mutual coupling between elements can induce pattern non-uniformity and degrade angular precision.
• Speed of beam adaptation is limited by switching/control frequency (313 Hz in RIS prototypes) (Xie et al., 2024).
• Quantization and meta-atom size limit side-lobe suppression and power concentration.

5. Methodological Variants: Nonlinear and Multipolar Approaches

Harmonic beam steering has distinct manifestations in nonlinear and diffractive regimes.

• Nonlinear Harmonic Steering: The phase and amplitude of the nonlinear polarization (e.g., SHG in TMDCs) establish the beam direction for the generated harmonic (Busschaert et al., 2019). In a phased array, the main lobe at 2ω is swept by adjusting the phase increment δ between optical pumps. Uniform efficiency over the steering range ($f_0=1/T_0$2 peak variation) is attainable by matching nonlinear emission from coupled sources.
• Multipolar Interference in Photonic Lattices: Beam steering in metalattices relies on the engineered interplay of electric and magnetic multipoles (Mie resonances) modulated by lattice coupling. Suppression of undesired diffraction orders is achieved by satisfying generalized Kerker-type amplitude/phase relations among multipolar coefficients (Liu et al., 2017). The steering efficiency, reflection, and transmission at each order are given in explicit analytic forms in terms of geometry and material parameters.

6. Applications and Impact

Harmonic beam steering enables a broad set of functions not achievable with conventional analog or digital beamformers:

• Multibeam and Multifrequency Operation: Simultaneous beams at carrier and multiple harmonics without additional feeds or RF chains (Shabanpour, 2020).
• Simultaneous Transmission and Reception, Full Duplex, and Nonreciprocity: Space–time coding can break time-reversal symmetry in ST-MTM antennas for unidirectional or duplex operation (Vosoughitabar et al., 2023).
• Programmable Multiuser/Multi-Target Systems: RIS- and ST-metasurface-based implementations allow dynamic allocation of beams and frequencies to different spatial users or targets (Xie et al., 2024).
• Nonlinear Optoelectronics: Programmable optical SH beam steering for on-chip LIDAR, optical interconnects, and quantum photonics (Busschaert et al., 2019).
• Device-Free Sensing and In-Band Sensing: Harmonic beams for side-channel sensing without disturbing the primary communication link (Xie et al., 2024).

7. Outlook and Future Directions

Principal development trends in harmonic beam steering include:

• Scaling thin-film and RIS arrays to larger $f_0=1/T_0$3 dimensions for higher angular resolution and narrower beamwidths (Xie et al., 2024).
• Integration of higher-performance switching devices (e.g., RF MEMS, varactors) to enable faster adaptation, broader bandwidth, and lower insertion loss (Vosoughitabar et al., 2023).
• Adaptive digital preprocessing to mitigate inter-element coupling and nonuniformity, potentially using in-situ calibration and machine learning for real-time compensation (Xie et al., 2024).
• Extension to higher harmonics by advanced coding/multilevel modulation, electromagnetically engineered element designs and superoscillatory synthesis (Maneiro-Catoira et al., 2024).

Fundamental limitations remain set by the rapid decay of harmonic content with order, mutual coupling, and digital control resolution; however, continued co-development of algorithmic, material, and hardware innovations is actively expanding the feasible regime of agile, efficient, harmonic-domain programmable beam steering across electromagnetic and optical platforms.