Multi-Plane Ray Tracing Overview

• Multi-plane ray tracing is a technique that models light propagation by partitioning space into successive 2D layers to capture discrete deflections, diffraction, and interference.
• In gravitational lensing, it applies the thin-lens approximation by discretizing mass distributions into 2D planes, yielding measurable outcomes like magnification and shear statistics.
• Wave-optical and volumetric extensions overcome geometric limitations, enabling efficient computation of interference patterns and accurate modeling in rendering and radiative transfer.

Searching arXiv for the cited topic and papers. Multi-plane ray tracing is a set of propagation formulations in which transport is decomposed into successive planes or layers. In gravitational lensing, the standard form divides the mass distribution between observer and source into a sequence of 2D planes perpendicular to the line of sight and propagates rays with discrete deflections. In wave optics, the same geometry is expressed through a multi-plane Kirchhoff-Fresnel integral, so that interference and diffraction remain explicit and caustic singularities are regularized. Related layered constructions also appear in deep point-cloud rendering, broadband diffractive optics, and multi-group radiative transfer, while several recent works contrast plane-based schemes with fully 3D or volumetric integration (Killedar et al., 2011, Feldbrugge, 2020, Dai et al., 2019, Meem et al., 2019, Frostholm et al., 2018, Au et al., 2023).

1. Terminology and problem classes

Across these works, “multi-plane” denotes several related but non-identical discretizations of propagation. In lensing, it refers to multiple thin lens planes or scattering screens; in neural rendering, to a layered volume of camera frustum; in diffractive optics, to distinct image planes at different propagation distances; and in Lampray, each “plane” can be understood as one frequency bin per direction in a multi-group long-characteristics solve (Killedar et al., 2011, Dai et al., 2019, Meem et al., 2019, Frostholm et al., 2018).

Context	Plane or layer representation	Distinctive property
Gravitational multi-plane lensing	Sequence of 2D projected mass planes	Deflections occur at planes under the thin-lens approximation
Wave-optical multi-plane lensing	Successive integrals over lens planes	Interference fringes remain finite near caustics
Volumetric plasma or 3D gravitational tracing	Full 3D refractive index or potential	Continuous deflection, no projection onto planes
Neural point-cloud rendering	$P \times H \times W$ layered frustum volume	Visibility is learned rather than fixed by z-buffering
Broadband diffractive optics	Multiple propagation distances $z_1,z_2,\ldots$	Distinct images reconstructed in different planes
Adaptive-mesh radiative transfer	Multi-group long characteristics	Captures shadows and colliding beams correctly

A common source of ambiguity is that the same phrase can describe either a geometric discretization of space or a wave-propagation formalism defined over multiple planes. The literature considered here treats both usages explicitly. This suggests that “multi-plane ray tracing” is best understood as a family of plane-wise propagation strategies rather than a single algorithm.

2. Thin-lens multi-plane tracing in gravitational lensing

In the conventional multiple lens-plane method, the intervening mass distribution from a cosmological $N$-body simulation is divided into a sequence of 2D planes perpendicular to the line of sight, with all matter in a box placed onto a single slice. Light rays are propagated from the observer to the source and are deflected only at these planes according to the projected gravitational potential, so the method is an explicit realization of the thin-lens approximation (Killedar et al., 2011).

The 2D deflection angle is written as

$\boldsymbol{\alpha}_{2D} = \nabla \psi_{2D},$

with lensing potential

$$\psi_{2D}(\mathbf{x}) = \frac{1}{\pi} \int d^2 x' \, \kappa(\mathbf{x}') \ln |\mathbf{x} - \mathbf{x}'|.$$

Magnification is then expressed as

$\mu = \frac{1}{(1 - \kappa)^2 - |\gamma|^2},$

where convergence $\kappa$ and shear $\gamma$ are computed via derivatives of $\psi_{2D}$ (Killedar et al., 2011).

This formulation is computationally efficient, but the data emphasize two limitations. First, lensing only changes the direction of rays at the planes, with no deflection between planes. Second, projection artifacts can arise because extended 3D structures, such as filaments, are artificially compactified onto individual planes. In the cited comparison at source redshift $z_s = 0.5$, the multi-plane method yielded a broader magnification probability distribution with a larger high-magnification tail, a standard deviation $z_1,z_2,\ldots$0, mean $z_1,z_2,\ldots$1, and more high-shear events (Killedar et al., 2011).

In this setting, caustics are singularities of the ray map, and the geometric-optics intensity is obtained from the Jacobian determinant. The same literature notes that geometric multi-plane ray tracing is insensitive to certain parameters, including the redshifts of the lens planes, because only deflection angles and geometric coefficients matter (Feldbrugge, 2020). This limitation motivates wave-optical generalizations.

3. Wave-optical multi-plane propagation

For coherent electromagnetic waves lensed by plasmas or gravitational lenses, the multi-plane problem is formulated through the multi-plane Kirchhoff-Fresnel integral. With $z_1,z_2,\ldots$2 lens planes at redshifts $z_1,z_2,\ldots$3, the total delay function is a sum over segment delays,

$z_1,z_2,\ldots$4

and the observed amplitude is

$z_1,z_2,\ldots$5

The difficulty is that these are high-dimensional, oscillatory integrals that are only conditionally convergent (Feldbrugge, 2020).

The method introduced in “Multi-plane lensing in wave optics” replaces direct treatment of the full $z_1,z_2,\ldots$6-dimensional integral by a recursive sequence of lower-dimensional integrals,

$z_1,z_2,\ldots$7

with $z_1,z_2,\ldots$8 and $z_1,z_2,\ldots$9. At each stage, Picard-Lefschetz theory is used to deform the contour to steepest-descent manifolds, transforming conditionally convergent oscillatory integrals into absolutely convergent ones; FFTs can then be used for the convolution-type Gaussian kernels arising between adjacent planes (Feldbrugge, 2020).

The principal significance of this construction is twofold. First, it reduces the computational cost from exponential in $N$0 to polynomial in $N$1 and in the dimensionality $N$2 of each lens plane. Second, it retains diffraction and interference near caustics, where geometric optics diverges. The data further state that, for multi-plane lensing, the interference pattern depends on the redshifts of the lens planes, and that this redshift sensitivity is invisible to geometric ray tracing (Feldbrugge, 2020).

The practical implication is that wave patterns can probe physical parameters not inferred from geometric optics. The cited applications include pulsars and fast radio bursts in radio astronomy, as well as gravitational wave sources, for which interference fringes are sensitive to lens-plane redshifts and to the distribution of compact objects along the line of sight (Feldbrugge, 2020).

4. Volumetric and fully 3D alternatives

Several works in the dataset treat multi-plane methods as approximations to more general volumetric propagation. In cosmological gravitational lensing, a 3D ray-tracing code was developed that avoids projections and the thin-lens approximation by integrating the full null geodesic equations through the 3D potential over the entire path from observer to source (Killedar et al., 2011). The weak-field spacetime is written as a perturbed FLRW metric,

$N$3

and the rays satisfy

$N$4

The potential and its derivatives are computed on a 3D grid using FFTs (Killedar et al., 2011).

In the cited tests, the 3D method reproduced the expected magnification profile for a Schwarzschild lens, matched analytic weak-lensing predictions for an NFW lens to $N$5 error for most impact parameters, and at $N$6 produced a narrower magnification distribution with suppressed high-magnification tail, $N$7, and mean $N$8 (Killedar et al., 2011). The discrepancy relative to the multi-plane method was attributed to projection effects and shot noise from treating discrete particles in a plane as point masses.

A closely related contrast appears in plasma lensing. “Investigating Extreme Scattering Events by Volumetric Ray-tracing” models extreme scattering events by numerically ray-tracing through analytic, volumetric plasma lens models via the eikonal equation,

$N$9

with the re-parameterized system

$\boldsymbol{\alpha}_{2D} = \nabla \psi_{2D},$0

Delaunay triangulation connecting the rays approximates the evolving wavefront, and barycentric mapping from observer-plane pixels to source-plane triangles accounts naturally for multiple imaging and caustics (Au et al., 2023).

This volumetric method is explicitly contrasted with standard multi-plane algorithms that project scattering effects onto discrete 2D screens. The volumetric approach is essential for thick lenses, arbitrary refractive-index distributions, and sheet-like or turbulent structures; the trade-off is higher computational cost and sensitivity to ray sampling density (Au et al., 2023). In this literature, the main controversy is therefore not whether plane-based tracing is useful, but when the planar approximation becomes physically misleading.

5. Layered image formation in rendering and diffractive optics

Outside lensing, multi-plane constructions appear in image synthesis and wavefront engineering. In neural point-cloud rendering, the input point features are not projected directly to the 2D image plane. Instead, they are projected into a layered volume of camera frustum, subdivided uniformly between scene-dependent minimum and maximum depth into $\boldsymbol{\alpha}_{2D} = \nabla \psi_{2D},$1 planes, producing a $\boldsymbol{\alpha}_{2D} = \nabla \psi_{2D},$2 voxel grid (Dai et al., 2019). Feature aggregation in each voxel is weighted by image-plane proximity and depth closeness, and the resulting 3D feature volume is processed by a U-Net style 3D CNN with dilated convolutions in the last encoder layer.

The network predicts, for each depth plane, an RGB image $\boldsymbol{\alpha}_{2D} = \nabla \psi_{2D},$3 and a blending weight $\boldsymbol{\alpha}_{2D} = \nabla \psi_{2D},$4, and the final image is

$\boldsymbol{\alpha}_{2D} = \nabla \psi_{2D},$5

The paper’s stated advantage over z-buffering and first-hit ray tracing is that visibility is learned rather than imposed geometrically, which avoids ghosting effects due to false visibility check and occlusions caused by noise interferences. The reported experiments show better or comparable PSNR, SSIM, and LPIPS versus Neural Point-based Graphics and Pix2Pix, more stable renderings near object boundaries, and stronger robustness to noisy and relatively sparse point clouds; in a temporal-consistency user study, this approach received 61 out of 80 votes, versus 10 for NPG (Dai et al., 2019).

Broadband diffractive-optical elements provide a different multi-plane formulation. Here the device topography is optimized so that distinct intensity patterns are reconstructed at different propagation distances $\boldsymbol{\alpha}_{2D} = \nabla \psi_{2D},$6 under broadband illumination. The field is propagated by the Fresnel diffraction integral,

$\boldsymbol{\alpha}_{2D} = \nabla \psi_{2D},$7

with phase determined by pixel height. The data state explicitly that direct ray tracing is inadequate in this regime and that wavefront-based numerical propagation is required (Meem et al., 2019).

The experiments included a two-plane device projecting a “+” at $\boldsymbol{\alpha}_{2D} = \nabla \psi_{2D},$8 mm and “×” at $\boldsymbol{\alpha}_{2D} = \nabla \psi_{2D},$9 mm, and a three-plane device projecting three different digits at $$\psi_{2D}(\mathbf{x}) = \frac{1}{\pi} \int d^2 x' \, \kappa(\mathbf{x}') \ln |\mathbf{x} - \mathbf{x}'|.$$0 mm, $$\psi_{2D}(\mathbf{x}) = \frac{1}{\pi} \int d^2 x' \, \kappa(\mathbf{x}') \ln |\mathbf{x} - \mathbf{x}'|.$$1 mm, and $$\psi_{2D}(\mathbf{x}) = \frac{1}{\pi} \int d^2 x' \, \kappa(\mathbf{x}') \ln |\mathbf{x} - \mathbf{x}'|.$$2 mm. Crosstalk was quantified by global SSIM, and imaging efficiency in two-plane devices reached up to $$\psi_{2D}(\mathbf{x}) = \frac{1}{\pi} \int d^2 x' \, \kappa(\mathbf{x}') \ln |\mathbf{x} - \mathbf{x}'|.$$3. The same work also reported visible/NIR multiplexing, flat BDOEs using index-contrast top-coating, reflective BDOEs, and magnification controlled by source distance through

$$\psi_{2D}(\mathbf{x}) = \frac{1}{\pi} \int d^2 x' \, \kappa(\mathbf{x}') \ln |\mathbf{x} - \mathbf{x}'|.$$4

These results underscore a general point: once the optical system is fundamentally diffractive, “multi-plane” must be understood as a wave-propagation problem rather than as geometrical ray tracing (Meem et al., 2019).

6. Multi-group long characteristics and methodological boundaries

Lampray extends the multi-plane idea into radiation hydrodynamics by solving the time-independent radiative transfer equation on adaptive meshes with long-characteristics rays, multiple frequency groups, and HEALPix angular discretization. The transport equation is

$$\psi_{2D}(\mathbf{x}) = \frac{1}{\pi} \int d^2 x' \, \kappa(\mathbf{x}') \ln |\mathbf{x} - \mathbf{x}'|.$$5

and along a ray in a frequency bin,

$$\psi_{2D}(\mathbf{x}) = \frac{1}{\pi} \int d^2 x' \, \kappa(\mathbf{x}') \ln |\mathbf{x} - \mathbf{x}'|.$$6

Mean intensities are accumulated per cell for chemistry, and the code couples to Krome with an H-He-C-O network containing 36 species and 240 reactions (Frostholm et al., 2018).

The algorithm employs interpolation between cell and ray domains, including Triangular-Shaped Cloud interpolation in the transverse plane, virtual fine cells at AMR boundaries, and a separate ray-domain decomposition in which all points along a ray reside in the same memory space. Data movement from cell- to ray-domain is performed with a direct hash-table lookup algorithm. The implementation was reported to achieve robust weak scaling up to $$\psi_{2D}(\mathbf{x}) = \frac{1}{\pi} \int d^2 x' \, \kappa(\mathbf{x}') \ln |\mathbf{x} - \mathbf{x}'|.$$7 MPI ranks, with communication during the decomposition switch becoming the dominant bottleneck at large core counts (Frostholm et al., 2018).

Lampray is relevant to the broader topic because it makes explicit what plane-wise tracing does well. It avoids diffusion, captures shadows, and treats colliding beams correctly, complementing moment-based methods that cannot model sharply anisotropic fields and that suffer from artificial diffusion of shadows (Frostholm et al., 2018). This suggests a broader methodological boundary across the literature: multi-plane or long-characteristics tracing is preferred when pathwise transport, anisotropy, and multiple imaging or visibility ordering are central, whereas pure moment closures, first-hit heuristics, or simple thin-lens projections can become inadequate.

Taken together, these works define multi-plane ray tracing not as a single canonical procedure but as a spectrum of plane-wise propagation methods whose validity depends on the governing physics. In geometric lensing it is a thin-lens approximation; in wave optics it becomes a recursive integral over lens planes; in rendering it is a layered visibility representation; in diffractive optics it is an explicitly Fresnel-domain design problem; and in radiation hydrodynamics it is a multi-group long-characteristics transport scheme. The principal technical question in every case is the same: whether the relevant physics can be represented as interactions localized on planes, or whether continuous 3D propagation and full wavefront evolution must be retained (Feldbrugge, 2020, Killedar et al., 2011, Au et al., 2023, Meem et al., 2019)