Quasi-Planar Array: Design Principles

• Quasi-planar arrays are non-uniform planar configurations that depart from dense layouts while retaining two-dimensional organization for improved degrees of freedom.
• They enable virtual aperture formation and optimized beam synthesis, enhancing resolution and reducing hardware complexity in various signal and imaging applications.
• These designs extend to superconducting receiver architectures and topological frameworks, balancing aperture density, sidelobe control, and fabrication constraints.

A quasi-planar array is an array configuration that departs from a dense, conventional planar aperture while retaining a fundamentally two-dimensional spatial organization. In the antenna and signal-processing literature, the term covers arrays whose elements lie on a plane but are non-uniformly distributed, arrays that are planar in a rotated coordinate frame determined by a target direction, and apertures that sample a plane only sparsely in one in-plane dimension (Goel et al., 2022, Costa et al., 2023, Chen et al., 23 Sep 2025). In superconducting heterodyne instrumentation, a related usage denotes a layered architecture with a quasi-two-dimensional local-oscillator distribution network (Shan et al., 2018). In discrete geometry, by contrast, “quasi-planar” denotes a drawing in the plane with no $k$ pairwise crossing edges, which is terminologically related but conceptually distinct from sensor or radiator arrays (Fox et al., 2011).

1. Terminology and geometric scope

The literature does not use a single universal formal definition of a quasi-planar array. Instead, the designation is tied to how a nominally two-dimensional aperture departs from a fully dense planar realization while preserving planar or near-planar geometry.

Context	Quasi-planar meaning	Source
Sparse array processing	Sensors lie on one plane, but the grid is non-uniform	(Goel et al., 2022)
Directivity optimization	Array lies on a plane dependent on $(\theta_0,\phi_0)$	(Costa et al., 2023)
Photoacoustic imaging	Two parallel linear arrays form a $2\times16$ sparse planar aperture	(Chen et al., 23 Sep 2025)
SIS mixer receivers	Quasi-two-dimensional LO waveguide network in a layered planar architecture	(Shan et al., 2018)
Topological graph theory	Plane drawing with no $k$ pairwise crossing edges	(Fox et al., 2011)

In sparse array processing, the rectangular coprime planar array (RCPA) is explicitly described as “a 2D sparse array whose elements lie on a plane,” with all sensors lying in one plane, such as $z=0$, and with non-uniform sensor coordinates along each axis following a coprime pattern rather than uniform spacing (Goel et al., 2022). In directivity optimization, the relevant geometry is planar in a rotated frame but may be tilted relative to the device axes, so the resulting structure is planar in the mathematical formulation and quasi-planar relative to the physical axes (Costa et al., 2023). In photoacoustic imaging, the term is defined operationally: two 16-element linear arrays placed side-by-side in parallel sample both $x$ and $y$, but very sparsely in $y$, so the aperture is not a full 2D matrix array (Chen et al., 23 Sep 2025).

This suggests that quasi-planarity is best understood as a geometric and architectural class rather than a single topology. The common feature is partial or transformed access to a two-dimensional aperture: non-uniform sampling, tilted-planar realization, sparse in-plane sampling, or layered quasi-2D interconnect.

2. Sparse planar realizations and virtual apertures

A canonical quasi-planar construction in array processing is the RCPA, built from the conventional one-dimensional coprime array with coprime integers $M,N$ and inter-element spacing $d\leq \lambda/2$. One convenient form of the 1D sensor locations is

$(\theta_0,\phi_0)$0

with total number of distinct sensors

$(\theta_0,\phi_0)$1

The 2D array is then formed by taking this full 1D coprime set on both axes,

$(\theta_0,\phi_0)$2

so that the total number of physical sensors is

$(\theta_0,\phi_0)$3

For the example $(\theta_0,\phi_0)$4, $(\theta_0,\phi_0)$5, $(\theta_0,\phi_0)$6, and the array is a $(\theta_0,\phi_0)$7 rectangular planar array with 36 sensors located at $(\theta_0,\phi_0)$8 (Goel et al., 2022).

The central construct is the two-dimensional difference co-array,

$(\theta_0,\phi_0)$9

which acts as a virtual planar array. For far-field narrowband sources, the covariance between sensors depends only on the spatial lag $2\times16$0. In the model

$2\times16$1

with

$2\times16$2

the steering-vector element at sensor $2\times16$3 is

$2\times16$4

Vectorizing and reshaping the covariance produces an autocorrelation vector defined on $2\times16$5, and MUSIC, after constructing a Hermitian Toeplitz matrix on $2\times16$6, treats $2\times16$7 as the sampling grid of a virtual planar array (Goel et al., 2022).

The significance of quasi-planarity in this setting is not merely physical placement but co-array quality. The RCPA lag set is contiguous over a large region,

$2\times16$8

and the paper reports that the difference co-array has “very few holes,” with hole percentage improved from $2\times16$9 for conventional CPA to $k$0 for the proposed RCPA (Goel et al., 2022). The resulting virtual aperture is therefore much closer to a hole-free dense grid than traditional coprime planar constructions.

This co-array structure directly controls degrees of freedom (DOF) in under-determined DOA estimation. The paper states that for 2D co-array processing, $k$1, and that the virtual planar aperture has effectively $k$2 virtual sensors for one parametric form, while the physical sensor count is only $k$3 (Goel et al., 2022). In simulation, with $k$4, the geometry estimates up to 49 uncorrelated sources in 2D using the virtual array, with RMSE $k$5 for reasonable SNR and snapshots (Goel et al., 2022).

The same work also ties quasi-planar sparsity to beam synthesis. For a planar array with sensors at $k$6, the array factor is

$k$7

The optimization objective is written as

$k$8

subject to a main lobe at $k$9, interference suppression $z=0$0 dB or more below the main lobe, and sidelobe levels at least 17 dB below the main lobe in $z=0$1. This is formulated as a second-order cone program. The optimized RCPA beam pattern reports measured directivity $z=0$2 dBi, interference suppression $z=0$3 dB and $z=0$4 dB for the two tested interferences, and sidelobe over-requirement percentage $z=0$5 (Goel et al., 2022).

3. Distinctive planes, finite apertures, and geometry diversity

A second meaning of quasi-planarity arises when a nominally volumetric array is analytically reduced to a plane that depends on the desired angle of departure or arrival. In the directivity formulation

$z=0$6

with

$z=0$7

the positions are expressed in rectangular coordinates $z=0$8 and the observation unit vector is

$z=0$9

For the omnidirectional case $x$0, maximizing the numerator of the directivity leads, after taking $x$1 and $x$2, to the plane relation

$x$3

and therefore to

$x$4

All element positions then lie in a plane perpendicular to the desired direction, so the 3D optimization reduces to a 2D optimization in $x$5, with $x$6 determined by the plane constraint (Costa et al., 2023).

When $x$7 is not aligned with the global $x$8-axis, this plane is tilted: $x$9 The array is therefore planar in the rotated coordinate frame but quasi-planar relative to the device’s physical axes (Costa et al., 2023). OUPA, the optimal uniform planar array method, restricts the geometry to a UPA $y$0 on this plane and reduces the non-convex design to a one-dimensional optimization over $y$1,

$y$2

solved by the successive evaluation and validation (SEV) method. For small $y$3, the achievable directivity by GA optimization demonstrates gains of $y$4 dBi compared with traditional beamforming using steering vectors for ULA and UCA, and gains of $y$5 dBi compared with an improved UCA method; for $y$6, OUPA is reported at $y$7 dBi and GA-stall at $y$8 dBi, while for extensive quasi-squared configurations such as $y$9, OUPA surpasses $y$0 dBi (Costa et al., 2023).

A finite-aperture perspective makes the geometry problem more explicit. For a rectangular planar fluid antenna array, the aperture is

$y$1

with port positions $y$2 and a minimum inter-port distance constraint $y$3. The analysis of uniform random placement shows that the minimum pairwise distance $y$4 follows a Rayleigh law under the Chen–Stein Poisson approximation, with

$y$5

so that

$y$6

and the mean scales as $y$7, in contrast to the $y$8 behavior in the linear case (Zhang et al., 21 May 2026). The same source introduces a universal CRB for joint elevation–azimuth estimation governed by a $y$9 geometric inertia matrix

$M,N$0

with

$M,N$1

and proves that both its trace and determinant are invariant to the azimuth look direction (Zhang et al., 21 May 2026).

The associated CRB expressions,

$M,N$2

$M,N$3

show why boundary-focused geometries improve estimation precision: maximizing $M,N$4 reduces the CRB. The same paper proves an intrinsic precision–ambiguity trade-off: maximizing the geometric determinant drives ports toward the aperture boundary but simultaneously increases sidelobe-induced spatial ambiguity (Zhang et al., 21 May 2026). This suggests that quasi-planar array design is not exhausted by maximizing aperture spread; interior coverage and spatial regularization remain central when beam ambiguity matters.

4. Sparse quasi-planar apertures in photoacoustic imaging

In photoacoustic imaging (PAI), quasi-planarity is defined operationally rather than by co-array algebra or rotated-plane optimization. The reported probe consists of two 16-element linear ultrasound arrays placed side-by-side in parallel, giving a $M,N$5 aperture: 16 elements in the $M,N$6-direction and 2 in the $M,N$7-direction (Chen et al., 23 Sep 2025). Because the aperture is sampled in both $M,N$8 and $M,N$9, but very sparsely in $d\leq \lambda/2$0, it is not a full 2D matrix array, yet it is more than a single line.

The geometry can be written as two rows of elements on a surface $d\leq \lambda/2$1,

$d\leq \lambda/2$2

The aperture length in $d\leq \lambda/2$3 is approximately $d\leq \lambda/2$4, and the quasi-planar extent in $d\leq \lambda/2$5 is approximately $d\leq \lambda/2$6 (Chen et al., 23 Sep 2025). A plastic optical fiber bundle is routed through the gap between the two transducers to deliver laser pulses, and the paper references an acoustic receiving angle of 14 degrees for the underlying linear probe, producing a fan-shaped imaging plane parallel to the illumination plane (Chen et al., 23 Sep 2025).

Reconstruction uses a 3D delay-and-sum beamformer. For a voxel $d\leq \lambda/2$7, the time-of-flight to element $d\leq \lambda/2$8 is

$d\leq \lambda/2$9

and the reconstructed initial pressure is approximated by

$(\theta_0,\phi_0)$00

with $(\theta_0,\phi_0)$01 channels (Chen et al., 23 Sep 2025). Because DAS with sparse views produces arc-shaped bright artifacts and relatively large bright spots, the reported post-processing retains voxels with signal strength above 50% of the maximum and filters out voxels below 50% of the maximum (Chen et al., 23 Sep 2025).

The system-level significance is that a sparse quasi-planar aperture provides real-time 3D localization without the channel count of a dense planar array. The hardware uses a Nd:YAG-based PhotoSonus series tunable laser with repetition rate 20 Hz and pulse width 10 ns, a Flash DAQ32 with 32 analog channels, and a custom printed circuit board with on-board signal amplifiers (Chen et al., 23 Sep 2025). Simulation studies using k-Wave show five distinct point positions reconstructed as distinct bright spots in 3D, while agarose phantom experiments with pencil leads and chicken-breast experiments with three black iron wires demonstrate 3D localization at different depths and different longitudinal planes (Chen et al., 23 Sep 2025).

The trade-off is explicit. Linear arrays “cannot provide reconstruction of 3D images, which makes it impossible to locate chromophores in 3D space,” whereas true planar arrays require hundreds to thousands of elements and correspondingly thousands of ADC channels (Chen et al., 23 Sep 2025). The quasi-planar probe therefore occupies an intermediate regime: only 32 channels, no mechanical scanning, and real-time or near-real-time 3D positioning, but poor elevational resolution, strong limited-angle behavior, and reconstruction artifacts from sparse $(\theta_0,\phi_0)$02-sampling (Chen et al., 23 Sep 2025).

5. Quasi-two-dimensional receiver architectures in radio astronomy

In SIS mixer array receivers, quasi-planarity refers less to the radiator distribution than to the integration architecture. The proposed planar SIS mixer array receiver replaces the traditional 3D assembly of single pixels with a layered architecture in which each functional layer is monolithic and serves all pixels in the array (Shan et al., 2018). The concept features membrane-based on-chip waveguide probes and a quasi-two-dimensional local-oscillator distribution waveguide network, allowing dual-polarization, balanced mixing, and sideband separation to be implemented in the same planar circuit (Shan et al., 2018).

The architecture is organized into layers. The top layer is a horn antenna array; the middle layer contains a tree-structure LO waveguide network spreading in a horizontal plane; and the bottom layer is a single mixer chip carrying all pixels, with membrane-based planar antennas, OMTs, hybrids, LO/signal diplexers, SIS mixers, and IF outputs (Shan et al., 2018). The LO network is quasi-2D because it is laid out predominantly in a horizontal plane, with local vertical 90° bends at the ends of the branches to feed the mixer chip (Shan et al., 2018). In the demonstration, the chip size is 13 mm $(\theta_0,\phi_0)$03 10 mm, far larger than a conventional SIS chip (Shan et al., 2018).

The electromagnetic implementation combines localized 3D-to-planar coupling with planar superconducting circuitry. The membrane is a 6 $(\theta_0,\phi_0)$04m Si device layer in an SOI wafer, with the handler wafer removed where membranes are needed and with approximately 30 $(\theta_0,\phi_0)$05m clearance above and below the membrane in the waveguide assembly (Shan et al., 2018). The signal probe and LO probe both exhibit impedance near 100 $(\theta_0,\phi_0)$06 over the targeted band, and the chip uses both microstrip and coplanar waveguide lines, with reported values around $(\theta_0,\phi_0)$07 for microstrip and $(\theta_0,\phi_0)$08 for CPW at 145 GHz, together with synthesized lines for intermediate impedances (Shan et al., 2018).

The prototype performance is representative of the quasi-planar integration concept rather than of large-array beamforming. The paper reports a minimum DSB noise temperature of approximately 70 K at LO $(\theta_0,\phi_0)$09 GHz, cross-polarization level better than 1% in rotation tests, and cross-polarization below $(\theta_0,\phi_0)$10 dB across the beam in raster scans (Shan et al., 2018). At the same time, it identifies the principal challenges for scaling: LO distribution isolation, sensitivity of quadrature hybrids to linewidth and dielectric thickness, membrane robustness and alignment, large-chip fabrication yield, and IF LNA power consumption (Shan et al., 2018).

This usage broadens the quasi-planar concept from aperture geometry to system architecture. The array is “planar” at the circuit level and “quasi-two-dimensional” at the LO-distribution level, while retaining only localized 3D machining for waveguide interfaces. A plausible implication is that quasi-planarity, in instrumentation, often denotes the replacement of bulky per-pixel 3D assemblies by layered, monolithic, shared structures.

6. Quasi-planarity as a combinatorial condition

A distinct usage appears in topological graph theory, where quasi-planarity has no direct connection to antenna or imaging apertures. A topological graph is drawn in the plane with vertices as points and edges as non-self-intersecting arcs; a graph is $(\theta_0,\phi_0)$11-quasi-planar if it does not contain $(\theta_0,\phi_0)$12 pairwise crossing edges (Fox et al., 2011). The case $(\theta_0,\phi_0)$13 is exactly planarity.

For simple topological graphs, where every pair of edges meets at most once, one result states that every $(\theta_0,\phi_0)$14-vertex simple $(\theta_0,\phi_0)$15-quasi-planar graph contains at most

$(\theta_0,\phi_0)$16

edges, where $(\theta_0,\phi_0)$17 denotes the inverse Ackermann function and $(\theta_0,\phi_0)$18 depends only on $(\theta_0,\phi_0)$19 (Suk, 2011). A closely related result improves the bound for simple topological $(\theta_0,\phi_0)$20-quasi-planar graphs to

$(\theta_0,\phi_0)$21

and for graphs whose edges are drawn as $(\theta_0,\phi_0)$22-monotone curves proves the upper bound

$(\theta_0,\phi_0)$23

(Fox et al., 2011).

This literature is relevant to the present topic only as a terminological boundary. It shows that “quasi-planar” can denote a forbidden crossing pattern in arrangements of curves rather than a physical array on a plane. The juxtaposition is useful because it dispels a common misconception: quasi-planarity does not always mean “almost planar hardware.” In one major research tradition, it means “no large clique in the crossing graph of edges” (Fox et al., 2011).

7. Cross-domain principles and recurrent trade-offs

Taken together, the literature suggests that quasi-planar arrays are usually introduced to occupy a technically useful middle ground between a one-dimensional structure and a dense planar or volumetric one. In sparse coprime arrays, the objective is to increase DOF and aperture through a virtual planar co-array while keeping physical sensor count limited (Goel et al., 2022). In directivity optimization, the objective is to capture most of the volumetric directivity benefit by restricting the solution to a plane dictated by the target direction (Costa et al., 2023). In PAI, the objective is to fill the gap between 2D linear-array imaging and high-channel-count 3D planar imaging (Chen et al., 23 Sep 2025). In SIS receiver design, the objective is to replace complicated three-dimensional LO distribution and packaging by a quasi-2D layered network and a monolithic planar chip (Shan et al., 2018).

The dominant trade-offs are likewise recurrent. Sparse and virtual-aperture designs improve DOF, beamwidth, and resolution, but they depend on hole structure, co-array processing, or pattern optimization, and they remain sensitive to sidelobes and interference constraints (Goel et al., 2022). Distinctive-plane and finite-aperture designs improve directivity or CRB by pushing elements toward geometrically extreme positions, but this may worsen sidelobes and spatial ambiguity (Costa et al., 2023, Zhang et al., 21 May 2026). In biomedical imaging, channel count and cost are reduced dramatically, but elevational resolution and volumetric fidelity are limited by sparse in-plane sampling (Chen et al., 23 Sep 2025). In superconducting receivers, monolithic quasi-planar integration improves packaging density and functional integration, but LO isolation, cross-talk, membrane tolerance, and fabrication yield become limiting constraints (Shan et al., 2018).

A consistent technical theme is that quasi-planarity is rarely a purely geometric descriptor. It usually encodes a design methodology: exploit a plane, a near-plane, or a sparsely sampled plane to obtain a favorable compromise among aperture, hardware complexity, channel count, fabrication difficulty, and algorithmic recoverability. In that sense, quasi-planar array design is best understood as a family of constrained two-dimensional architectures whose performance depends as much on virtualization, optimization, and integration strategy as on the physical element coordinates themselves.