Multiplane Light Conversion

• Multiplane Light Conversion (MPLC) is a programmable optical system that deterministically implements arbitrary linear transformations on spatial modes using cascaded phase masks and free-space propagation.
• MPLC employs inverse design algorithms such as wavefront matching and gradient-based optimization to maximize mode overlap, reduce crosstalk, and ensure high fidelity in mode transformations.
• MPLC finds practical applications in telecommunications, quantum information processing, and photonic signal processing, offering scalable, low-loss, and reconfigurable performance for advanced optical systems.

Multi-plane Light Conversion (MPLC) is a class of programmable optical systems that deterministically implement arbitrary linear transformations between finite sets of optical spatial modes, with applications across optical communications, quantum information, and photonic signal processing. By cascading phase-only elements (phase masks) with linear propagation (typically free-space diffraction), MPLC realizes highly efficient, low-loss, and reconfigurable mode transformations, often with scaling advantages not achievable by integrated waveguide-meshes. Recent advances have focused on the optimization, fabrication, and practical deployment of MPLC architectures for classical and quantum photonic systems.

1. Mathematical Foundations and Physical Principle

MPLC implements a unitary (or in some cases general linear) transformation on transverse optical modes by interleaving thin, phase-only modulation planes and propagation sections. Mathematically, the field at the output is

$$E_{\rm out} = (P_M\,F_{M}\,P_{M-1}\,F_{M-1}\cdots P_1\,F_1)\,E_{\rm in}$$

where each $P_m$ is a diagonal operator multiplying the field by $\exp[i\phi_m(x,y)]$, and each $F_m$ represents propagation over a distance $z_m$, typically modeled by the Fresnel integral or angular spectrum method. If $N$ is the number of encoded modes, a sufficiently large number $M$ of planes allows MPLC to approximate any desired transformation on the $N$-dimensional design subspace (Zhang et al., 2023, Kupianskyi et al., 2022).

Given two bases of orthonormal input and output modes, $\{u_n\}$ and $\{v_n\}$, MPLC can synthesize any $P_m$0 unitary $P_m$1 where $P_m$2, and—via block-encoding and more—general (nonunitary) linear maps as well (Taguchi, 2024). The phase profiles $P_m$3 are found by inverse design algorithms maximizing the mode overlap, minimizing crosstalk, or optimizing other figures of merit, subject to physical and fabrication constraints.

2. Inverse Design Algorithms and Optimization Strategies

The most established MPLC design techniques are:

A. Wavefront Matching Algorithm (WMA)

Iteratively alternates forward-propagating the input modes and backward-propagating the target modes through the cascaded planes. At each plane, the phase mask is updated by maximizing the complex overlap between the local forward and backward fields:

$P_m$4

This maximizes the local phase match for all channels (Rothe et al., 2024, Kupianskyi et al., 2022, Zhang et al., 2023).

B. Gradient-Based and Direct-Search Algorithms

For scenarios with nonconvex objectives (e.g., explicit trade-offs between insertion loss, extinction ratio, modal uniformity), direct-search methods (random pixel-wise perturbations with explicit cost functions) or analytic gradient ascent can achieve higher mode extinction ratios and uniformity at controllable insertion loss penalty, particularly in high-dimensional, programmable multiplexers (Rothe et al., 2024). These approaches readily accommodate constraints such as device insertion loss $P_m$5, mode extinction ratio $P_m$6, and mode uniformity.

C. Physical Neural Network (PNN) Training

Treats the sequence of MPLC operations as a differentiable neural network (parameterized by mask profiles and plane distances), optimized end-to-end via stochastic gradient descent. Allows training on mini-batches for very high mode counts (hundreds), and direct integration with hyperparameter optimization frameworks (Zhu et al., 2023).

D. Block-Encoding and Redundancy

For general linear (possibly nonunitary) matrix conversion, block-encoding schemes embed the target matrix in a higher-dimensional unitary and configure the MPLC to realize this unitary, offering superior convergence and iterative tuning compared to decompositions based solely on SVD (Taguchi, 2024).

3. Scaling Laws, Complexity, and Entropy Engineering

A. Minimal Plane Count and Scalability

Exact universality for an $P_m$7-mode transformation requires $P_m$8 to $P_m$9 degrees of freedom. For unitary $\exp[i\phi_m(x,y)]$0 maps, minimal synthesis generally requires $\exp[i\phi_m(x,y)]$1 for arbitrary $\exp[i\phi_m(x,y)]$2, but practical applications (e.g., multiplexers, mode sorters) often achieve $\exp[i\phi_m(x,y)]$3 with high fidelity (Bade et al., 2018, Labroille et al., 2014, Sureka et al., 16 Mar 2026). Recent results have demonstrated that approximate synthesis—allowing bounded elementwise errors $\exp[i\phi_m(x,y)]$4—achieves sub-quadratic scaling of phase shifters:

$\exp[i\phi_m(x,y)]$5

with $\exp[i\phi_m(x,y)]$6 as low as 1.2–1.6 for moderate error tolerance ($\exp[i\phi_m(x,y)]$7) (Taguchi, 2024).

B. Shannon Matrix Entropy in Mixer Engineering

The normalized Shannon entropy $\exp[i\phi_m(x,y)]$8 of a mixer $\exp[i\phi_m(x,y)]$9 quantifies the "mixing strength." Low-entropy mixers (i.e., strong diagonals, few significant couplings) can substantially reduce the required number of phase-shifter layers without major performance degradation. Entropy-optimal designs typically achieve $F_m$0–$F_m$1 for the best trade-off (Taguchi, 2024).

C. Few-Layer Redundancy and Local Unimodality

Redundancy (i.e., several extra layers beyond the theoretical minimum) removes spurious local minima in the mask-parameter optimization landscape, allowing highly accurate iterative configuration even in the presence of quantization, crosstalk, and hardware nonidealities (Taguchi et al., 2023).

4. Fabrication Technologies and Device Architectures

A. Reflective and Transmissive Phase Planes

Traditional MPLC architectures deploy reflective phase masks on glass or silicon, fabricated by multi-step, multi-level lithography or, more recently, direct writing laser (DWL) grayscale lithography. This enables vertical depth quantization <10 nm, surface roughness <3 nm, and $F_m$2 16-bit phase control, yielding measured mode-conversion fidelities >90% (Gurung et al., 14 Jul 2025).

B. Integrated and Miniaturized MPLCs

Recent advances include monolithic MPLCs inscribed by femtosecond-laser nano-grating writing within a glass chip, producing fully encapsulated geometric-phase holograms. Demonstrations include compact 3-mode and 10-mode Hermite-Gaussian sorters in sub-mm$F_m$3 glass chips (Būtaitė et al., 6 Feb 2026). This approach could enable robust, alignment-free packaging and high-density photonic integration.

C. Reconfigurable and MEMS Platforms

Spatial light modulators (SLMs) and MEMS phase arrays allow electronically programmable phase masks. Switching rates up to kHz have been demonstrated on MEMS devices, enabling rapid (re)configuration and in-situ adaptation to system drifts and environmental variations (Rocha et al., 23 Jan 2025).

D. Multi-Pass and Folded Optical Layouts

Many demonstrators employ folded geometries, where the beam is reflected multiple times between an SLM and fixed mirror, each reflection corresponding to a new phase plane. Cascading multiple SLMs or using monolithic phase-plate stacks facilitates scaling to larger numbers of modes or planes with well-controlled alignment (Lib et al., 2024).

5. Performance Benchmarks and Applications

A. Insertion Loss, Crosstalk, and Fidelity

• State-of-the-art MPLCs achieve per-mode insertion loss (IL) <1 dB theoretically, 3–5 dB with scattering losses; mode-dependent loss (MDL) <0.5 dB; crosstalk <–25 dB per channel; bandwidth >100 nm (Zhang et al., 2023).
• High-dimensional spatial multiplexers (up to 45 modes) show IL ≈ 4 dB, average crosstalk ≈ –28 dB, and robust performance under strong fiber bending (Bade et al., 2018).
• Miniaturized glass-embedded MPLCs presently exhibit per-plane efficiency ~80% and simulated mode purities 80–85% (Būtaitė et al., 6 Feb 2026).

B. High-Speed, Adaptive, and Self-Configuring Operation

• Feedback-controlled MPLCs enable real-time mode-matching and adaptation, with demonstrated dynamic recovery times in the second-to-ms regime and proof-of-principle mode sorting across 7 spatial modes (Korichi et al., 2022, Rocha et al., 23 Jan 2025).
• In-situ optimization algorithms (hardware-in-the-loop wavefront matching) directly compensate all system aberrations and misalignments (Rocha et al., 23 Jan 2025).

C. Scalability in Mode Number and Application Breadth

• Linear scaling of required planes with mode number $F_m$4 for "pixel-mode" spatial interferometers (demonstrated up to $F_m$5), in contrast to $F_m$6 scaling for integrated MZI meshes (Sureka et al., 16 Mar 2026).
• Applications comprise telecommunication mode-multiplexers/demultiplexers, universal spatial mode sorters, tunable beamsplitters, high-dimensional quantum state processors, and programmable linear optical circuits (Lib et al., 2021, Martinez-Becerril et al., 2024, Zhang et al., 2020, Sureka et al., 16 Mar 2026).

D. Quantum Information Processing

• MPLCs are deployed for universal, high-dimensional entanglement processing, with randomized unitary transformations on entangled photon pairs, three-basis entanglement certification, and efficient mode-basis reformatting for distributed quantum channels (Lib et al., 2021).
• Programmable MPLCs enable optimal discrimination of non-orthogonal quantum states, with experiments demonstrating unambiguous sorting with error rates below minimum-error limits in up to 7 dimensions (Goel et al., 2022).

6. Design Guidelines and Practical Considerations

Parameter	Typical Value/Recommendation	Impact
Plane count $F_m$7	$F_m$8 for arbitrary unitaries	High fidelity, low crosstalk
Mixer entropy $F_m$9	$z_m$0–$z_m$1	Optimal mixing vs. scalability
Phase quantization	$z_m$2 8–16 bits per mask	Minimizes quantization error
Insertion loss (IL)	$z_m$33 dB typical (to $z_m$41 dB possible)	System efficiency, optical networks
Crosstalk	$z_m$5–20 dB	Channel isolation, especially in MDM
Fabrication technique	DWL grayscale, etched silica/Si, SLM, MEMS	Determines scalability, reconfigurability

Design should target the minimal number of planes and phase-depth/entropy respecting the trade-off between optical complexity and required performance (error tolerance, crosstalk). Block-encoding is generally superior for nonunitary matrix conversion, and feedback-driven optimization is crucial for practical deployments (Taguchi, 2024, Taguchi et al., 2023, Rothe et al., 2024).

7. Outlook and Emerging Trends

Ongoing research includes miniaturization via monolithic, laser-written geometric-phase elements (Būtaitė et al., 6 Feb 2026), further reduction in loss by employing dielectric metasurfaces or advanced multi-layer reflective coatings (Gurung et al., 14 Jul 2025), and the application of advanced optimization (machine learning, direct search) for higher mode counts, complex circuit topologies, and hybrid spatial-wavelength-polarization devices (Rothe et al., 2024).

Key directions are:

• Integration with photonic chips for hybrid MPLC/silicon photonics systems;
• Scaling to hundreds of modes for petabit-class communications (Zhang et al., 2023, Bade et al., 2018);
• Acceleration and robustness for field-deployed, rapidly varying environments (e.g., real-time mode unscrambling in turbulence or fiber networks) (Rocha et al., 23 Jan 2025, Korichi et al., 2022);
• Expanding the operational basis to include time-bin, frequency, or polarization degrees of freedom in the same physical MPLC stack (Zhang et al., 2020).

A significant feature is the sub-quadratic scaling of the necessary hardware for approximate linear maps, exemplified by the combined use of low-entropy mixing, plane-count reduction, and model-aware co-design (e.g., weight quantization in photonic neural networks) (Taguchi, 2024). Block-encoding architectures simplify configuration and, alongside quantization-aware training, position MPLC as a scalable route to compact, high-throughput, and energy-efficient optical matrix computing in both classical and quantum domains.