Multi-Plane Light Conversion Techniques

• Multi-Plane Light Conversion (MPLC) is a phase-only beam shaping technique that cascades multiple phase masks with free-space propagation to approximate unitary modal transforms.
• Design methods like wavefront matching, gradient ascent, and differentiable network training enable high-fidelity conversion with reduced crosstalk and optimized performance.
• MPLC finds applications in mode-division multiplexing, quantum state manipulation, and optical computing, with ongoing research focused on low-loss, self-configuring architectures.

Multi-Plane Light Conversion (MPLC) is a linear, phase-only beam-shaping architecture that implements prescribed spatial transformations by cascading multiple phase-modulating planes separated by free-space propagation. In its standard free-space form, each plane applies a transverse phase profile and each propagation segment diffractively mixes the field, so that the full cascade approximates a unitary mapping on a chosen modal subspace. Across the arXiv literature, MPLC appears both as a programmable optical processor and as a fabrication paradigm for static diffractive devices, with applications spanning mode-division multiplexing, quantum state manipulation, spatial mode sorting, optical computing, imaging, and linear-optical interferometry (Zhang et al., 2023, Labroille et al., 2014).

1. Physical principle and operator description

At its most basic level, an MPLC alternates a phase operator with a propagation operator. If the $n$th plane imparts a phase $M_n(x,y)=\exp\!\big(i\phi_n(x,y)\big)$ and the inter-plane propagation over distance $z$ is denoted by $\mathcal{P}(z)$, the cascade can be written as

$$U^{(N)}=\mathcal{P}(z_N)\,M_N\,\cdots\,\mathcal{P}(z_1)\,M_1\,U^{(0)}.$$

Equivalently, several papers write the overall transform as an ordered product of phase masks and Fresnel propagators, emphasizing that phase multiplication and free-space diffraction are both unitary in the ideal paraxial, lossless, infinite-aperture limit (Zhang et al., 2023, Labroille et al., 2014).

Free-space propagation is usually modeled by the Fresnel integral,

$$U(x,y; z)=\frac{e^{ikz}}{i\lambda z}\iint U_0(x',y')\exp\!\left(\frac{ik}{2z}\big[(x-x')^2+(y-y')^2\big]\right)\,dx' dy',$$

or, in angular-spectrum form, by multiplication with a propagation transfer function in the spatial-frequency domain (Zhang et al., 2023, Lib et al., 2024). The essential physical point is that a single phase mask can only multiply the local field, whereas inter-plane diffraction spreads energy and creates overlap between previously separated spatial regions; repeated phase–propagation cycles thereby synthesize transformations that a single plane cannot realize (Lib et al., 2024).

The literature consistently treats MPLC as a near-unitary transform on a designed finite-dimensional modal subspace rather than as a globally controlled optical system. This distinction is important. The intended map is defined by a set of input modes $\{u_i\}$ and target output modes $\{v_j\}$, with transmission matrix elements

$T_{ji}=\langle v_j,\,U^{(N)}u_i\rangle.$

Within that subspace, MPLC can approximate arbitrary unitary mode changes with high fidelity; outside it, the device transmits a much larger set of uncontrolled channels (Zhang et al., 2023, Boucher et al., 2020). A common misconception is that “arbitrary unitary” means uniformly controlled behavior over all transverse degrees of freedom. The full-characterization study instead shows that the design subspace is preferentially transmitted, while the uncontrolled sector can become speckle-like and, in the large-mode regime, statistically resembles a filtered random scattering medium (Boucher et al., 2020).

2. Inverse design, optimization, and configuration

The canonical design method is wavefront matching, also described as the wavefront matching method or WMM. It iteratively propagates the input modes forward and the desired output modes backward, then updates each phase plane so that the local phase best aligns the two field sets. This method is prominent across telecom, quantum, and general spatial-sorting papers, including high-dimensional sorters, quantum processors, and practical tutorials (Kupianskyi et al., 2022, Lib et al., 2024).

Wavefront matching is not the only design route. In the low-plane limit, a gradient-ascent method with an objective that explicitly penalizes crosstalk and can steer residual light into a background region was shown to outperform WMM for high-dimensional spatial sorters. In that setting, for random “speckle mode” sorting with $M=5$ planes and $M_n(x,y)=\exp\!\big(i\phi_n(x,y)\big)$0 pixels per plane, WMM produced average crosstalk for $M_n(x,y)=\exp\!\big(i\phi_n(x,y)\big)$1 modes of $M_n(x,y)=\exp\!\big(i\phi_n(x,y)\big)$2, whereas the tailored gradient-ascent objective reduced it to $M_n(x,y)=\exp\!\big(i\phi_n(x,y)\big)$3 at broadly similar efficiencies (Kupianskyi et al., 2022). A related computer-generated-hologram optimization based on direct search, initialized by WMM and constrained by insertion-loss and mode-extinction-ratio criteria, improved average mode extinction ratio by as much as $M_n(x,y)=\exp\!\big(i\phi_n(x,y)\big)$4 dB at the expense of insertion loss deterioration of $M_n(x,y)=\exp\!\big(i\phi_n(x,y)\big)$5 dB, and was applied both to conventional LP modes and to Schmidt modes derived from a measured transmission matrix (Rothe et al., 2024).

A further extension treats MPLC explicitly as a differentiable physical neural network. In that framework, each layer is a phase modulation followed by propagation, and all mask phases and even inter-mask distances can be trained jointly with automatic differentiation. The reported 45-mode design used 8 phase plates of $M_n(x,y)=\exp\!\big(i\phi_n(x,y)\big)$6 pixels, $M_n(x,y)=\exp\!\big(i\phi_n(x,y)\big)$7, $M_n(x,y)=\exp\!\big(i\phi_n(x,y)\big)$8, and $M_n(x,y)=\exp\!\big(i\phi_n(x,y)\big)$9 fixed at $z$0 mm, and about $z$1 million trainable phase parameters; across batch sizes $z$2, all trained models achieved $z$3 average coupling efficiency and $z$4 dB insertion loss (Zhu et al., 2023). This suggests that MPLC optimization can be framed not only as adjoint optics but also as large-scale physical-network training.

Another branch of the literature abstracts MPLC into fixed unitary mode mixers interleaved with programmable phase arrays. In that representation,

$z$5

with $z$6 fixed $z$7-port unitaries and $z$8 diagonal phase-shifter arrays. For this model, few-layer redundancy with $z$9 or $\mathcal{P}(z)$0 was shown to regularize the optimization landscape, enabling L-BFGS-based iterative configuration with orders-of-magnitude better accuracy and a 20-fold speed-up compared to previous MPLC heuristics (Taguchi et al., 2023). This line of work also underpins approximate optical matrix realizations with low-entropy mode mixers and sub-quadratic phase-shifter scaling under bounded max-norm error thresholds (Taguchi, 2024).

More recently, configuration has moved from purely offline inverse design to in situ self-configuration. A forward-only optimization procedure measured the real transmission matrices of individual planes and updated each phase profile on a MEMS phase-only light modulator, so that the physical device itself absorbed aberrations, misalignments, and unknown component response. The reported proof-of-principle system optimized up to $\mathcal{P}(z)$1 parameters, converged in minutes, and operated with rapid MPLC switching at up to kiloHertz rates (Rocha et al., 23 Jan 2025). This suggests a shift from “designing an MPLC model” to “training the realized MPLC.”

3. Device architectures and implementation platforms

The dominant experimental architecture has been the reflective multi-pass free-space layout. In early telecom work, a multipass reflective cavity between a patterned reflective phase plate and a spherical mirror realized seven reflections for three-mode conversion; the device was implemented both with an LCOS SLM prototype and with a fabricated silica phase plate (Labroille et al., 2014). The same basic idea persists in later programmable systems: several nominal “planes” are realized as successive reflections from distinct regions of a single SLM, with mirrors or prisms routing the beam between those regions.

A particularly explicit implementation is the 10-plane programmable converter built from a single Hamamatsu X13138-02 SLM, a dielectric mirror, and a right-angle prism. In that system, the SLM–mirror distance was $\mathcal{P}(z)$2 mm, the SLM–prism distance was $\mathcal{P}(z)$3 mm, each plane occupied $\mathcal{P}(z)$4 SLM pixels, and the geometry was aligned to single-pixel accuracy by imaging successive planes and using $\mathcal{P}(z)$5-phase steps as fiducials (Lib et al., 2024). The related high-dimensional QKD experiment used a reflective multi-pass arrangement with the same 10-plane structure, the same Hamamatsu X13138-02 SLM, and effective 10-plane sequences for both photons in the apparatus (Lib et al., 2024).

Static reflective plates remain important because SLM-based MPLCs incur substantial loss. In the silica phase-plate realization of the three-mode telecom multiplexer, the fabricated reflective plate used $\mathcal{P}(z)$6 phase levels with $\mathcal{P}(z)$7 nm step depth and $\mathcal{P}(z)$8m lateral resolution, and the cavity mirrors had $\mathcal{P}(z)$9 reflectivity (Labroille et al., 2014). A newer fabrication route based on direct writing laser grayscale lithography transfers continuous-relief profiles into silicon by reactive ion etching and adds a reflective Al coating; in that study, a four-plane system reached $$U^{(N)}=\mathcal{P}(z_N)\,M_N\,\cdots\,\mathcal{P}(z_1)\,M_1\,U^{(0)}.$$0 fidelity for Gaussian-to-LG conversion, with sub-10 nm vertical resolution, surface roughness below 3 nm, and profile fidelity $$U^{(N)}=\mathcal{P}(z_N)\,M_N\,\cdots\,\mathcal{P}(z_1)\,M_1\,U^{(0)}.$$1 (Gurung et al., 14 Jul 2025). This suggests a practical pathway from programmable prototypes to lower-loss static masks.

A distinct direction is monolithic glass integration. One approach embeds geometric-phase holograms inside fused silica by femtosecond direct laser writing of birefringent nanogratings, yielding fully encapsulated transmissive MPLCs. Proof-of-concept glass-embedded devices included a 3-mode and a 10-mode Hermite–Gaussian sorter in chip volumes of roughly $$U^{(N)}=\mathcal{P}(z_N)\,M_N\,\cdots\,\mathcal{P}(z_1)\,M_1\,U^{(0)}.$$2 and $$U^{(N)}=\mathcal{P}(z_N)\,M_N\,\cdots\,\mathcal{P}(z_1)\,M_1\,U^{(0)}.$$3 mm$$U^{(N)}=\mathcal{P}(z_N)\,M_N\,\cdots\,\mathcal{P}(z_1)\,M_1\,U^{(0)}.$$4, respectively, with chip format $$U^{(N)}=\mathcal{P}(z_N)\,M_N\,\cdots\,\mathcal{P}(z_1)\,M_1\,U^{(0)}.$$5 cited for the larger device (Būtaitė et al., 6 Feb 2026). A more advanced volumetric platform distributed $$U^{(N)}=\mathcal{P}(z_N)\,M_N\,\cdots\,\mathcal{P}(z_1)\,M_1\,U^{(0)}.$$6–$$U^{(N)}=\mathcal{P}(z_N)\,M_N\,\cdots\,\mathcal{P}(z_1)\,M_1\,U^{(0)}.$$7 birefringent planes over $$U^{(N)}=\mathcal{P}(z_N)\,M_N\,\cdots\,\mathcal{P}(z_1)\,M_1\,U^{(0)}.$$8–$$U^{(N)}=\mathcal{P}(z_N)\,M_N\,\cdots\,\mathcal{P}(z_1)\,M_1\,U^{(0)}.$$9 mm inside fused silica and used geometric phase to implement not only scalar MPLC functions but also vectorial, polarization-dependent transforms (Korichi et al., 20 Apr 2026). These works replace external mirror alignment with intrinsic monolithic registration, at the cost of current efficiency limitations and polarization selectivity.

MPLC has also been reinterpreted as a programmable interferometer on free-space pixel modes. In one beam-array study, an SLM-based MPLC implemented arbitrary $$U(x,y; z)=\frac{e^{ikz}}{i\lambda z}\iint U_0(x',y')\exp\!\left(\frac{ik}{2z}\big[(x-x')^2+(y-y')^2\big]\right)\,dx' dy',$$0 transformations on two Gaussian beams and mapped the full parameter space of the target beamsplitter-like unitary (Martinez-Becerril et al., 2024). In a later pixel-mode interferometer architecture, spatial modes arranged on a 2D lattice underwent programmable unitaries on up to $$U(x,y; z)=\frac{e^{ikz}}{i\lambda z}\iint U_0(x',y')\exp\!\left(\frac{ik}{2z}\big[(x-x')^2+(y-y')^2\big]\right)\,dx' dy',$$1 pixel modes, including Hadamard transformations, permutations, and partial unitaries (Sureka et al., 16 Mar 2026). In these formulations, MPLC is not merely a mode sorter but a direct free-space alternative to integrated interferometer meshes.

4. Performance metrics and observed operating regimes

MPLC performance is usually reported through insertion loss, mode-dependent loss, crosstalk, sorting fidelity, overlap fidelity, singular-value spectra, or task-specific figures such as secure key rate. In telecom-oriented mode multiplexing, the first MPLC demonstration reported a typical 3-mode multiplexer with mode selectivity better than $$U(x,y; z)=\frac{e^{ikz}}{i\lambda z}\iint U_0(x',y')\exp\!\left(\frac{ik}{2z}\big[(x-x')^2+(y-y')^2\big]\right)\,dx' dy',$$2 dB and a total insertion efficiency of $$U(x,y; z)=\frac{e^{ikz}}{i\lambda z}\iint U_0(x',y')\exp\!\left(\frac{ik}{2z}\big[(x-x')^2+(y-y')^2\big]\right)\,dx' dy',$$3 dB across the full C-band; with coating improvements, the projected efficiency was $$U(x,y; z)=\frac{e^{ikz}}{i\lambda z}\iint U_0(x',y')\exp\!\left(\frac{ik}{2z}\big[(x-x')^2+(y-y')^2\big]\right)\,dx' dy',$$4 dB (Labroille et al., 2014). Scaling to a 45-mode graded-index-fiber multiplexer based on a separable HG design, the measured devices typically showed an average 4 dB insertion loss and $$U(x,y; z)=\frac{e^{ikz}}{i\lambda z}\iint U_0(x',y')\exp\!\left(\frac{ik}{2z}\big[(x-x')^2+(y-y')^2\big]\right)\,dx' dy',$$5 dB cross-talk across the C band (Bade et al., 2018).

High-dimensional mode sorters exhibit the familiar plane-count versus crosstalk trade-off. In prototype few-plane sorters, random orthogonal speckle modes were sorted with five planes up to $$U(x,y; z)=\frac{e^{ikz}}{i\lambda z}\iint U_0(x',y')\exp\!\left(\frac{ik}{2z}\big[(x-x')^2+(y-y')^2\big]\right)\,dx' dy',$$6 modes, OAM modes were sorted with three planes for $$U(x,y; z)=\frac{e^{ikz}}{i\lambda z}\iint U_0(x',y')\exp\!\left(\frac{ik}{2z}\big[(x-x')^2+(y-y')^2\big]\right)\,dx' dy',$$7 Laguerre–Gaussian modes, and Zernike modes were sorted with five-plane designs up to $$U(x,y; z)=\frac{e^{ikz}}{i\lambda z}\iint U_0(x',y')\exp\!\left(\frac{ik}{2z}\big[(x-x')^2+(y-y')^2\big]\right)\,dx' dy',$$8 modes (Kupianskyi et al., 2022). The corresponding measured average crosstalk rose with mode count, reflecting the low-plane regime and the finite resolution of single-SLM multi-pass implementations. The same paper emphasized that tailored objectives can strongly reduce crosstalk without requiring more planes (Kupianskyi et al., 2022).

The most systematic subspace-level characterization comes from full transmission-matrix reconstruction and singular-value decomposition. For MPLCs designed to shape $$U(x,y; z)=\frac{e^{ikz}}{i\lambda z}\iint U_0(x',y')\exp\!\left(\frac{ik}{2z}\big[(x-x')^2+(y-y')^2\big]\right)\,dx' dy',$$9, $\{u_i\}$0, or $\{u_i\}$1 modes, the first $\{u_i\}$2 singular values stand out from the rest, forming a spectral gap that quantifies preferential transmission of the design subspace. The overlap metric

$\{u_i\}$3

was reported as high for Gaussian-to-fiber transforms—for example, $\{u_i\}$4 for $\{u_i\}$5—but markedly lower for Gaussian-to-LG conversion, reflecting the larger resource demand of widening higher-order modes (Boucher et al., 2020). The same study argued that, in the large-mode regime, the uncontrolled sector of the device behaves like a random scattering medium with a limited number of controlled channels (Boucher et al., 2020).

Quantum and interferometric demonstrations add task-specific benchmarks. In high-dimensional QKD, a 10-plane MPLC programmed as a spatial-mode sorter achieved five-dimensional QKD with all six MUBs and 25-dimensional QKD with two tailored MUBs. For $\{u_i\}$6, the mean symbol error rate was $\{u_i\}$7, the depolarizing-noise bound gave $\{u_i\}$8 bits per sifted photon, and full SDP certification on tomographically complete data gave $\{u_i\}$9 bits per sifted photon; for $\{v_j\}$0, the reported values were $\{v_j\}$1, $\{v_j\}$2, $\{v_j\}$3, and a block-biased lower bound $\{v_j\}$4 bits per sifted photon (Lib et al., 2024). In reconfigurable two-beam interferometry, experimental sampling of the full $\{v_j\}$5 parameter space yielded an average transformation fidelity of $\{v_j\}$6 (Martinez-Becerril et al., 2024). In pixel-mode interferometers, experimental Hadamard unitaries on $\{v_j\}$7 modes showed average transmissivities $\{v_j\}$8 and average crosstalks $\{v_j\}$9 (Sureka et al., 16 Mar 2026).

5. Application domains

The original and still most mature application area is space-division and mode-division multiplexing. MPLC maps arrays of single-mode inputs to selected eigenmodes of few-mode or multimode fibers, acts as a demultiplexer when reversed, and is especially effective when the target basis is separable, as in Hermite–Gaussian approximations to graded-index-fiber modes (Labroille et al., 2014, Bade et al., 2018). The same architecture has been extended beyond pure modal multiplexing: a five-plane design simultaneously sorted four wavelengths and three spatial modes into a 2D array of Gaussian spots, with IL $T_{ji}=\langle v_j,\,U^{(N)}u_i\rangle.$0 dB, MDL $T_{ji}=\langle v_j,\,U^{(N)}u_i\rangle.$1 dB, and average crosstalk $T_{ji}=\langle v_j,\,U^{(N)}u_i\rangle.$2 dB (Zhang et al., 2020).

Quantum optics constitutes a second major domain. A five-plane programmable MPLC processed entangled photons in high dimensions, certified three-dimensional entanglement in two mutually unbiased bases, implemented 400 Haar-random four-mode transformations, and converted one photon from a pixel basis to fiber LP modes for entanglement distribution (Lib et al., 2021). Another MPLC performed simultaneous sorting of non-orthogonal, overlapping states of light for unambiguous state discrimination in dimensions $T_{ji}=\langle v_j,\,U^{(N)}u_i\rangle.$3 to $T_{ji}=\langle v_j,\,U^{(N)}u_i\rangle.$4, using an auxiliary output mode to implement the Naimark extension and the basis change in a single optical element (Goel et al., 2022). In the QKD setting already noted, MPLC supplied the multi-basis spatial-mode sorters required for deterministic high-dimensional measurements (Lib et al., 2024).

Optical computing and programmable linear optics form a third application class. Approximate optical matrix realization with MPLC-like architectures has been investigated through low-entropy mode mixers, Shannon matrix entropy, and block-encoding of general linear maps, with reported sub-quadratic scaling of phase-shifter count under tolerated max-norm error thresholds (Taguchi, 2024). Beam-array unitaries on reconfigurable MPLCs were explicitly proposed as building blocks for quantum and classical information processing (Martinez-Becerril et al., 2024), while pixel-mode interferometers demonstrated tunable beamsplitters, Hadamard unitaries, boosted-Bell-measurement unitaries, and partial unitaries on select subsets of modes (Sureka et al., 16 Mar 2026).

A further emerging domain is compact integrated structured-light processing. The volumetric glass platform demonstrated multimode unitary transformations, LG–HG conversion, complex beam splitting, polarization-controlled spatial operations, optical Skyrmion-topology conversion, and telecom-wavelength spatial-mode and polarization multiplexers in a device with a compact form factor of only a few cubic millimeters (Korichi et al., 20 Apr 2026). This suggests that MPLC is increasingly functioning as a general-purpose structured-light processor rather than only as a telecom mode sorter.

6. Practical limits, common misconceptions, and research directions

A recurring misconception is that MPLC is inherently low-loss because it is “phase-only.” In the ideal mathematical model, phase multiplication and free-space propagation are unitary; experimentally, however, insertion loss is often dominated by SLM reflectivity, diffraction efficiency, fill factor, metallic coatings, finite apertures, and scattering from phase discontinuities or nanostructures. The 2024 QKD experiment explicitly reported average MPLC insertion losses per photon of $T_{ji}=\langle v_j,\,U^{(N)}u_i\rangle.$5 dB for the $T_{ji}=\langle v_j,\,U^{(N)}u_i\rangle.$6 configuration and $T_{ji}=\langle v_j,\,U^{(N)}u_i\rangle.$7 dB for the $T_{ji}=\langle v_j,\,U^{(N)}u_i\rangle.$8 configuration on the SLM-based device, while noting that static phase plates would reduce loss substantially (Lib et al., 2024). The self-configuring MEMS MPLC similarly estimated an experimental efficiency of about $T_{ji}=\langle v_j,\,U^{(N)}u_i\rangle.$9 for a four-plane prototype because of modulator reflectivity and diffraction efficiency (Rocha et al., 23 Jan 2025). “Phase-only” therefore does not imply “low-loss” in realized hardware.

A second misconception is that universality automatically follows with a small number of planes. The literature is more cautious. Tutorials explicitly state that there is no rigorous theory for the minimum number of planes in the general case (Zhang et al., 2023), and several studies show that fidelity depends strongly on plane count, pixel resolution, spacing, mode family, and objective function (Kupianskyi et al., 2022, Martinez-Becerril et al., 2024). In pixel-mode interferometers, the required number of planes scales approximately linearly with mode number for the studied targets, but diffraction-induced field bulge penalizes naive attempts to stack many independent two-mode circuits in parallel (Sureka et al., 16 Mar 2026). A plausible implication is that MPLC universality is best understood as a high-dimensional design freedom with nontrivial geometry-dependent resource costs, rather than as a generic guarantee for shallow devices.

The main experimental constraints are alignment, aberrations, finite resolution, and speed. Free-space multi-pass systems demand precise plane-to-plane registration; the 10-plane build tutorial was motivated precisely by the need for single-pixel alignment with standard components (Lib et al., 2024). SLM platforms are flexible but slow and lossy; phase quantization and refresh-rate limits are repeatedly cited (Lib et al., 2024). These issues motivate three current research directions. The first is in situ self-configuration, in which the physical device is optimized directly and unknown imperfections are absorbed into the learned masks (Rocha et al., 23 Jan 2025). The second is static low-loss fabrication, exemplified by grayscale lithography for reflective masks and by monolithic glass-embedded or volumetric nanograting MPLCs (Gurung et al., 14 Jul 2025, Būtaitė et al., 6 Feb 2026, Korichi et al., 20 Apr 2026). The third is algorithmic refinement, including low-plane gradient objectives, direct-search CGH optimization, differentiable physical-network training, and entropy-aware approximate synthesis (Kupianskyi et al., 2022, Rothe et al., 2024, Zhu et al., 2023, Taguchi, 2024).

Taken together, these developments show MPLC evolving from a specialized free-space mode multiplexer into a broad optical platform. Its defining features remain the same—phase-only planes, diffraction-mediated coupling, and programmable or fabricated unitary transforms—but the surrounding ecosystem now includes self-configuring hardware, monolithic glass implementations, grayscale-lithography masks, vectorial geometric-phase devices, and free-space pixel-mode interferometers. This suggests that future MPLC research will be shaped less by the question of whether the architecture is universal in principle, and more by how efficiently specific transforms can be compiled, aligned, fabricated, and stabilized in physically realistic systems (Boucher et al., 2020, Rocha et al., 23 Jan 2025, Korichi et al., 20 Apr 2026).