Multi-Plane Spatially Resolved Phase Structuring Using Optical Communication Modes

Abstract: We present a deterministic framework for three-dimensional beam shaping that enables versatile control of intensity and phase, pixel-by-pixel, across multiple axial planes. Conventional multi-plane holographic techniques typically rely on iterative optimization and mitigate inter-plane crosstalk through phase randomization, introducing speckle noise and thereby limiting deterministic phase control. Here, target fields are synthesized as a linear superposition of free-space communication modes obtained from the singular value decomposition of a coupling operator connecting a source plane to multiple target planes. Because these modes form orthogonal and energy-efficient transmission channels between the source and receiving spaces, their superposition yields volumetric wavefields with enforced phase coherence and reduced inter-plane crosstalk, without iterative refinement. We experimentally demonstrate high-fidelity reconstruction of intensity and phase profiles across multiple planes using a single phase-only spatial light modulator, including arbitrary structured phase singularity patterns. The proposed approach establishes communication-mode optics as a practical and physically grounded framework for multi-plane beam shaping, particularly in applications where phase structure and coherence across depth are essential.

• The paper introduces a deterministic SVD-based method that synthesizes arbitrary complex fields across multiple planes using orthogonal communication modes, avoiding iterative optimization and random phase masks.
• Experiments reconstruct three-plane intensity targets with mean-square errors below 0.101, signal-to-background ratios above 5.5, and speckle contrast from 0.14 to 0.36.
• The paper demonstrates phase-singularity sheets and shows that extrinsic modes suppress inter-plane crosstalk, while the mode spectrum identifies diffraction-limited spacing and energy-feasibility constraints.

This paper introduces a deterministic framework for multi-plane beam shaping in which target complex-valued fields across several axial planes are synthesized as a linear superposition of free-space communication modes derived from the singular value decomposition (SVD) of the source–receiver coupling operator (2603.15222). The central claim is that because these modes form orthogonal, energy-efficient transmission channels between the source aperture and the set of receiving planes, their superposition enforces phase coherence and suppresses inter-plane crosstalk by construction—without iterative optimization or random-phase decorrelation—and therefore enables deterministic, pixel-by-pixel phase control that conventional multi-plane holography cannot provide.

Motivation and relation to prior work

Iterative multi-plane holographic techniques—global Gerchberg–Saxton variants and stochastic gradient descent schemes—reconstruct prescribed intensity patterns at multiple depths but suffer inter-plane crosstalk from overlapping defocused fields. Crosstalk is typically mitigated by imposing random phase distributions on the targets, which suppresses coherent leakage but precludes deterministic phase control and introduces speckle noise. Even crosstalk-penalized algorithms such as compensatory GGS and double-constraint SGD cannot eliminate all speckle, a limitation the authors attribute to vortex stagnation in iterative Fourier-transform methods. The consequence is fundamental: intensity-only multi-plane holography produces layered images rather than a single physically consistent wavefront, sacrificing wavefront curvature fidelity, depth-dependent aberration correction, volumetric topological field engineering (e.g., phase-singularity sheets), and phase-gradient force landscapes.

Communication mode theory, established via SVD of scalar Green's-function coupling operators, previously existed mainly as an analytical tool in simplified prolate-spheroidal geometries; a practical experimental demonstration for continuous-depth-of-field volume shaping appeared only recently in the authors' prior work. This paper extends that line to spatially resolved multi-plane structuring of arbitrary complex fields, which the authors state is the first general, experimentally validated application of SVD modal optics to this problem.

Mode taxonomy and design rules

The source plane is discretized as $251 \times 251$ points with spacing $4\lambda$; each receiving plane is sampled as a $21 \times 21$ array whose point spacing grows linearly with distance so that all planes subtend the same solid angle from the aperture and support the same number $N = 100$ of strongly coupled modes at $\lambda = 532$ nm. A key contribution is the classification of these modes into two sub-categories:

• Intrinsic modes: inherent to a given receiving plane and independent of the other planes; their coupling strengths decay plane-to-plane with the same inverse-square law as intensity from an on-axis point source.
• Extrinsic modes: arising from the receiving space as a whole due to the presence of the other planes.

The sum $N_{\text{int},n} + N_{\text{ext},n} = N$ is constant per plane, but the split varies: for three equally spaced planes ($L = L_0$), the extrinsic counts are 3, 15, and 3 for planes one through three, with the middle plane hosting the largest extrinsic population. Supplementary analysis shows that excluding extrinsic modes degrades reconstruction markedly, establishing them as the mechanism responsible for automatic inter-plane crosstalk suppression—an implication of direct practical value, since it identifies which modes must be retained rather than truncating purely by singular-value magnitude.

As the inter-plane separation shrinks, extrinsic strong modes increase in number, but below a threshold (here around $L = 0.125L_0$) the total number of strongly coupled modes falls sharply, leaving too few usable channels. This defines a minimum separation distance imposed by diffraction: pushing below it demands weakly coupled modes whose excitation requires exponentially large source amplitudes, and because the weak-mode fall-off follows the universal tunneling-escape behavior of waves, the authors argue no other wave-shaping technique can produce physically realizable solutions in that sub-diffraction regime—a strong claim that positions the method as also a diagnostic for feasibility.

Synthesis procedure

Target profiles are projected onto the receiving eigenfunctions, and the required source function is computed as $$\ket{\Psi_T} = \sum_j s_j^{-1} \braket{\Phi_{R,j}|\Phi_T}\ket{\Psi_{S,j}}$$, analogous to a Fourier series truncated at well-coupled modes. Simulations demonstrate intensity-only targets (binary digits '1', '2', '3', one per plane) reconstructed from the first 350 well-coupled modes, and structured phase targets in which each digit carries phase $\pi/2$ inside its contour and $4\lambda$0 outside with zero amplitude along the contour—2D phase-singularity sheets. Notably, singularity placement here is achieved simply by prescribing the target, rather than by inverse-design phase-gradient maximization. Reconstructed singular points reach intensities as low as −24 dB; approaching mathematically exact sheet singularities requires including weakly coupled modes, lowering the minimum to −41 dB at the cost of impractically high source amplitudes.

Experimental implementation

The computed source function cannot be encoded directly on a phase-only SLM: it contains reactive near-field components beyond the free-space propagation limit and large-amplitude diffraction orders incompatible with 10-bit phase quantization. Instead, the field propagated to $4\lambda$1—fully radiative, with higher diffraction orders spatially separated—is encoded using the Arrizón Type-3 complex-field CGH algorithm, whose main drawback is low diffraction efficiency owing to the limited modulation depth ($4\lambda$2). A 4f system filters the first diffraction order; intensities are recorded on a camera mounted on a translation stage, and phase is retrieved without a reference beam via single-beam multiple-intensity reconstruction (SBMIR).

For the intensity-only case, measured MSE remains below 0.101 across all three planes, signal-to-background ratio exceeds 5.5, and speckle contrast stays between 0.14 and 0.36:

Plane	MSE	SBR	Speckle contrast
1	0.045	11.68	0.139
2	0.073	9.31	0.231
3	0.101	5.70	0.357

Although no phase constraints were imposed in this case, SBMIR-retrieved phases are smooth and mutually consistent—the authors emphasize this as evidence that the reconstructions constitute a single global solution of the scalar wave equation rather than independent plane images. For the phase-singularity case, phase MSE ranges from 0.071 (plane 1) to 0.192 (plane 3), with local phase standard deviations within digit interiors below $4\lambda$3, confirming sharp $4\lambda$4 transitions co-located with dark contours persisting across all planes. Higher-resolution demonstrations using $4\lambda$5 receiving arrays (400 strong modes per plane, source synthesized from 912 well-coupled modes) show moderate error increases (e.g., speckle contrast up to ~0.56, phase MSE up to 0.25) but preserve structural features and coherent phase transitions. Additional measurements in a non-paraxial configuration confirm qualitative robustness, though the paraxial heuristic mode count applies only near paraxial conditions.

Limitations and open questions

Several limitations are acknowledged explicitly. First, the SVD computation is expensive: roughly 3.4 hours for 600 modes and nearly 7 hours for 1200 modes on unaccelerated hardware—acceptable only because it is a one-time offline calculation per fixed geometry, whereas iterative algorithms must re-run per target. Second, the achievable phase fidelity degrades toward deeper planes (phase MSE rising from 0.071 to 0.192), and exact sheet singularities remain inaccessible within the strongly coupled subspace. Third, tightly spaced receiving planes enter a regime where strong plus partially coupled modes are insufficient, forcing either degraded accuracy or high-energy weak-mode solutions confined outside the target region. Fourth, the encoding path imposes constraints of its own: the Arrizón algorithm's low diffraction efficiency, and the requirement to encode the Fresnel-propagated field rather than the source function itself. Whether the approach scales to many more planes, larger apertures, or dynamic reconfigurable geometries—and how the non-paraxial mode-counting heuristic should be formalized—remain open questions raised but not resolved by this work.

Conclusion

The paper establishes SVD communication-mode optics as a deterministic, experimentally validated route to multi-plane synthesis of arbitrary complex-valued fields with a single static phase-only hologram, achieving speckle-suppressed, phase-coherent reconstructions—including multi-plane phase-singularity sheets—that intensity-only iterative methods cannot deliver. Its dual value lies both in the synthesis capability itself and in the diagnostic power of the mode spectrum, which quantifies when a desired volumetric structure is physically constructible given a finite aperture.