Hierarchical Phased-array Antenna

• Hierarchical phased-array antennas are engineered multi-level systems that group antenna elements to perform frequency-dependent coherent summing, ensuring near-constant beamwidth across broad bands.
• They integrate precise phase-matching, multi-chroic filter banks, and optimized signal routing to reduce detector count and backend complexity in imaging and communication applications.
• Demonstrated in mm/sub-mm astronomical instruments and wireless beamforming, these arrays offer improved mapping speeds, sidelobe suppression, and scalable architectures for advanced applications.

A hierarchical phased-array antenna is an engineered, multi-level grouping and signal-summation architecture for antenna arrays, in which the effective pixel size varies with frequency by coherently summing neighboring elements in a frequency-dependent fashion. This approach efficiently matches pixel area to the optimal wavelength-dependent sampling criteria in imaging, communications, and sensing arrays, enabling near-constant beamwidth and sensitivity across extended bandwidths while often reducing detector count and backend complexity. Recent implementations span mm/sub-mm astronomical detector arrays, wireless beamforming, and scalable large-N architectures.

1. Architectural Principles and Mathematical Foundations

The hierarchical phased-array concept organizes antenna elements into a multi-stage structure where each stage aggregates signals from groups of elements into actively beamformed outputs. At each hierarchy level $\ell$, an array factor describes the far-field pattern as a function of phase-aligned summation over spatial coordinates:

$$AF_\ell(\theta,\phi) = \sum_{n=1}^{N_\ell} w_n^{(\ell)} \exp\left[j k\,\hat{\mathbf{u}} \cdot \mathbf{r}_n^{(\ell)}\right]$$

where $N_\ell$ is the number of subelements summed at level $\ell$, $w_n^{(\ell)}$ are (possibly tapered) weights, $k=2\pi/\lambda$, and $\mathbf{r}_n^{(\ell)}$ are geometric positions. By making the number of elements $N_\ell$ and array aperture $D_\ell$ frequency-dependent (typically $D_\ell \propto \lambda$), the main-lobe beamwidth

$$\theta_{\rm FWHM}^{(\ell)} \approx 1.02\,\frac{\lambda}{D_\ell}$$

is held approximately constant across the array’s designed operating bands (Cukierman et al., 2018, Cukierman et al., 2017).

In digital and RF communication contexts, similar hierarchical signal flow is formalized:

• First level (elements grouped as “tiles”):

$y_{1,b}(t) = \sum_{n=1}^{L} w_{1,b,n} x_n(t)$

• These outputs feed second-level “stations,” producing final beams:

$$y_{2,b}(t) = \sum_{i=1}^{T M_1} w_{2,b,i} y_{1,i}(t),\quad b=1,...,B$$

where $L$ is elements per tile, $T$ the number of tiles, $M_1$ intermediate beams per tile, and $B$ final beams (Faulkner et al., 2010).

2. Hierarchy Levels and Physical Implementation

Antenna elements—slot-dipoles, sinuous structures, dipoles—are lithographically patterned with precise geometric parameters and physical groupings. Typical hierarchy levels in mm/sub-mm focal planes are:

Hierarchy Level	Band(s) (GHz)	# Elements per Pixel	Pixel Pitch	Implementation
Level 1	170–365	1	2.5 mm	Single slot, BPF+KID
Level 2	125–170	4	5.0 mm	2×2 array, summed
Level N	variable	variable	$N^{1/2} d$	Multi-scale subarray

Physical layouts (e.g., a square or triangular grid of slots or sinuous antennas) prioritize equal path lengths to summing nodes, implementing phase-matched microstrip networks using superconducting materials (Nb, AlMn) and low-loss dielectrics (amorphous Si, SiN) for mm/sub-mm band operation (Shu et al., 2021, Martin et al., 31 Jan 2024, Huang et al., 12 Nov 2025, Cukierman et al., 2018, Cukierman et al., 2017).

In communication arrays and large-N radio astronomy, grouping is realized by grouping elements into tiles and stations, managed either in the RF domain (analog vector-modulators, phase-shifters) or after digitization (FPGA/ASIC beamforming).

3. Frequency-Band Splitting and Multi-Chroic Filter Banks

Hierarchical phased arrays for multi-band detection integrate on-chip lumped-element filter banks (Chebyshev, Butterworth, etc.) at each summing level. Each microstrip branch feeds a bandpass (BPF) or lowpass filter tuned to atmospheric windows or comms bands:

• Slot-dipole arrays for NEW-MUSIC (Huang et al., 12 Nov 2025) use 3rd/5th order Chebyshev BPFs with center frequencies covering 77–411 GHz.
• Sinuous and slot arrays for KIDs or bolometers often adopt three-pole Chebyshev BPFs (series capacitors, shunt inductors in Nb microstrip, $Z_0$ matched to 37–50 Ω), with band edges tracked to within a few percent via EM simulations (Shu et al., 2021, Martin et al., 31 Jan 2024, Cukierman et al., 2018).

Each band’s signal is routed post-filter either to direct readout (high frequencies) or summed for larger aperture formation (low frequencies). Isolation between bands is determined by $|S_{11}|$ and $|S_{21}|$ values; reported typical return/insertion loss at center is $|S_{11}| \leq -15$ dB, $|S_{21}| \leq -0.5$ dB.

4. Coherent Summing Networks and Beamforming

Hierarchical summing in physical arrays is achieved with quarter-wave transformers (Wilkinson dividers, hybrids), binary tree combiners, or multi-way trivider circuits. Path lengths are precisely equalized (within $< \lambda/20$ or $\lambda/10$) to ensure phase coherence.

Beamforming theory:

• The resulting far-field pattern is the product of the elemental beam and the hierarchy’s array factor.
• Directivity increases and main-lobe narrows with increased group size at low frequencies; e.g., summing four or more subelements gives a $\sim$6 dB gain and HPBW halving (Shu et al., 2021, Huang et al., 12 Nov 2025, Cukierman et al., 2018).
• Grating lobe suppression is maintained by designing element spacings below $0.5\,\lambda$ for all bands operated; e.g., slot spacings $d=2.5$ mm for $f_{\mathrm{min}} = 125$ GHz ($d \approx 0.5\,\lambda_{\min}$).

Sinuous hierarchies implement triangular and hexagonal subarrays, while slot-dipole arrays use square subarrays. Dividers and hybrids (planar, slotline) are used for microstrip recombination across grouped elements (Cukierman et al., 2017, Cukierman et al., 2018).

5. Signal Routing, Detector Coupling, and Readout Efficiency

Hierarchical phased arrays efficiently route band-separated signals to detectors (TES bolometers, KIDs) while minimizing readout and backend complexity:

• KID integration employs capacitive coupling pads (longitudinal or transverse) for selected bands, with coupling efficiency $\eta_c \geq 0.9$ by matching network impedance ($Z_{\rm KID}$, $Z_0$).
• TES integration uses matched resistive terminations on microstrip islands; hybrid/lumped-sum architectures allow replacement with KIDs for multiplexed readout (Cukierman et al., 2018).
• Detector count scales only logarithmically with the number of bands ($N_{\rm det} \propto N_{\rm elem} \cdot H_B$, $H_B$ harmonic number), as opposed to linear growth in conventional multichroic focal planes (Cukierman et al., 2017).
• In large-N comms arrays, hierarchical beamforming reduces active controller hardware—up to 87% reduction for $N=128$ elements—by employing dimensionality reduction via SVD and R basis vectors and a bank of $R \ll N$ amplitude/phase controllers (Xia et al., 2022).

6. Performance Metrics, Validation, and Scalability

Hierarchical phased arrays are characterized by their measured and simulated beam patterns, spectral response, optical/radiation efficiency, mapping speed enhancements, and fabrication tolerances.

Performance highlights:

• Beamwidths (HPBW) held near-constant across 3:1 or larger bandwidth (e.g., $7.6^\circ$ in 90–220 GHz, (Cukierman et al., 2018); $17.3^\circ$ at 157 GHz in slot-dipole arrays (Martin et al., 31 Jan 2024))
• Sidelobe suppression at $-10$ to $-15$ dB; polarization purity $\lesssim -20$ dB.
• Optical efficiencies for detector integration measured at $20\text{–}40\%$ (KIDs) (Shu et al., 2021, Martin et al., 31 Jan 2024) and simulated at $>80\%$ for integrated slot-dipole/filterbank arrays (Huang et al., 12 Nov 2025).
• Mapping speed gains up to $\sim2\times$ for broader bandwidths (Cukierman et al., 2017).
• Fabrication tolerances for microstrip width and capacitor area are controlled to yield beam pointing errors well below main-lobe HPBW, e.g., $<1^\circ$ at $400$ GHz (Huang et al., 12 Nov 2025).

Scalability is demonstrated in multi-scale extensions (three or more hierarchy levels, $8\times8$ superpixels for $75$–$125$ GHz bands) and in hardware-verified beamforming comms arrays (16, 8, 4 elements; 4, 3, 2 controllers) (Xia et al., 2022).

7. Applications, Limitations, and Future Directions

Applications include:

• Large-format mm/sub-mm focal planes for telescopes (e.g., CMB, Sunyaev–Zel’dovich surveys, dusty galaxy detection); compatibility with $\sim$10–50 m-class instruments (Martin et al., 31 Jan 2024, Huang et al., 12 Nov 2025, Shu et al., 2021, Cukierman et al., 2018).
• Wideband multi-beam/trans-millimeter polarimetry (e.g., 80–420 GHz, 2.4-octave bandwidth) (Huang et al., 12 Nov 2025).
• Satellite comms arrays, deep-space apertures, and large-N radio arrays with substantially reduced control hardware (Xia et al., 2022, Faulkner et al., 2010).

Limitations:

• Increased microwave/RF complexity in summing networks, susceptibility to loss and stray coupling (Cukierman et al., 2017).
• Fabrication tolerance criticality, especially for path-length matching and impedance/coupling element size.
• Precise modeling and calibration required for array-factor systematics, polarization control, and sidelobe suppression.

Future directions include adopting new materials (SiN$_x$ for lower-loss dielectric lines), further hierarchy levels for enhanced bandwidth and detector-count reduction, and integrating digital beamforming per-element as semiconductor and DSP technologies advance (Faulkner et al., 2010). This suggests continued convergence between RF and digital hierarchical beamforming, underpinning both astronomy and comms applications with scalable architectures.