Matrix Fourier Optics Overview

• Matrix Fourier Optics is a framework that describes linear optical propagation through matrix operators acting on spatial-frequency bases, extending traditional plane-wave and lens-based methods.
• It unifies several approaches—including ABCD matrix formalism, Jones matrices for polarization, and discrete Fourier transforms in finite-dimensional systems—to provide a cohesive operator perspective.
• Applications range from beam propagation and multilayer media analysis to the design of optical matrix multipliers in advanced imaging and computational photonic systems.

Matrix Fourier Optics denotes a broad operator-centered view of optical propagation in which fields are represented in a basis—most commonly transverse spatial frequency, but also modal, polarization, or discrete channel bases—and optical elements act through matrices, transfer functions, or integral kernels on the corresponding coefficients. In this sense, it includes classical Fourier optics based on plane-wave decomposition and lens-mediated Fourier transformation, paraxial canonical-transform formalisms, matrix-valued spectral propagators for anisotropic media, and finite-dimensional optical realizations of discrete Fourier transforms and general matrix-vector multiplication (Mansuripur, 2020, Fan et al., 2010, Tanuwijaya et al., 12 Feb 2025). The term is therefore broader than thin-lens imaging alone, but narrower than “all matrix optics”: different state spaces are propagated in different subfields, and the choice of basis determines both the mathematical form and the physical interpretation of the optical operator (Acquaroli, 2018).

1. Scope, terminology, and relation to adjacent formalisms

A central organizing idea is that Fourier optics is not merely a computational trick; it is embedded in how waves propagate, how lenses form fields, and how optical systems reorganize information into spatial-frequency variables (Mansuripur, 2020). In the standard scalar setting, a field in a transverse plane is expanded into plane-wave components indexed by spatial frequencies $(s_x,s_y)$, and propagation acts diagonally on that spectrum. Matrix Fourier Optics extends this viewpoint by emphasizing the operator that acts on the chosen representation: a scalar transfer function in isotropic free space, a Jones matrix in polarization-sensitive propagation, or a finite $N\times N$ transform in discrete optical computing (Filipovich et al., 2024, Tanuwijaya et al., 12 Feb 2025).

This usage overlaps with, but is not identical to, paraxial $ABCD$ optics. $ABCD$ matrices propagate ray vectors or Gaussian-beam $q$-parameters through first-order systems, whereas Fourier-optical operators act on wave amplitudes or spectra. A plausible implication is that “matrix Fourier optics” is best reserved for wave-level transform operators, even when those operators are generated by the same paraxial systems that admit an $ABCD$ description (Fan et al., 2010, Ding et al., 2021).

It also overlaps only partially with transfer-matrix methods for layered media. Acquaroli’s thin-film formalism propagates forward and backward plane-wave amplitudes through a stratified medium by multiplying $2\times2$ matrices, and the review explicitly notes that this transfer-matrix method can be viewed as the exact 1D-per-spatial-frequency propagator for a stratified medium (Acquaroli, 2018). That connection is structural rather than literal: the method is not about diffraction integrals, finite pupils, or image formation, but it becomes a building block of Fourier optics when a finite beam is decomposed into angular components and each component is propagated separately (Acquaroli, 2018).

A further distinction concerns universality. Some devices implement arbitrary or at least dense finite-dimensional matrices, whereas others implement only diagonal operators in the Fourier basis. The 2025 “Metapinhole” work is explicit on this point: its metagrating should be read as an angle-dependent transfer function $H(k_x,k_y)$, hence as a diagonal operator in the plane-wave basis, not as a general mode-mixing transform (Abouelatta et al., 6 Sep 2025).

2. Continuous Fourier-optical operator foundations

The continuous foundation is the Fourier-transform pair. For a transverse field $a_0(x,y)$, Mansuripur writes

$$A_0(s_x,s_y)=\iint a_0(x,y)\exp[-i2\pi(s_x x+s_y y)]\,dx\,dy,$$

$N\times N$0

In this representation, the Fourier spectrum is the angular spectrum of plane waves, with direction cosines $N\times N$1; propagating components satisfy $N\times N$2, while components outside that unit disk are evanescent (Mansuripur, 2020).

Free-space propagation acts diagonally in this basis. In exact angular-spectrum form,

$N\times N$3

and in the paraxial regime the transfer function becomes

$N\times N$4

Equivalently,

$N\times N$5

This is the canonical Fourier-optical statement that propagation is multiplication in the spatial-frequency domain and convolution in real space (Mansuripur, 2020).

The lens enters as a quadratic-phase element that converts angular-spectrum content into focal-plane position. In standard thin-lens notation,

$N\times N$6

and propagation by $N\times N$7 to the back focal plane yields, up to quadratic phase and normalization,

$N\times N$8

This is the physical basis of coherent filtering and $N\times N$9 processing (Mansuripur, 2020).

The same operator logic extends beyond ordinary field propagation. The Wigner distribution propagates under paraxial free space as a shear,

$ABCD$0

and the van Cittert–Zernike theorem states that the complex degree of mutual coherence is the Fourier transform of the normalized source intensity (Mansuripur, 2020). A plausible implication is that Matrix Fourier Optics is not confined to amplitudes; it naturally includes phase-space and coherence operators whenever the underlying transformation remains linear.

A broader review of Fourier methods in optics makes the same systems-theoretic point in a different language, emphasizing transfer functions, impulse responses, cross-correlations, and photonic neural couplings as instances of a common transform-based formalism (Froehly et al., 2019).

3. Canonical transforms, $ABCD$1 systems, and finite Fourier groups

In paraxial first-order systems, the relevant matrix is the symplectic ray-transfer matrix

$ABCD$2

A standard kernel for an $ABCD$3 system with $ABCD$4 is

$ABCD$5

This shows that every first-order paraxial optical system is a quadratic-phase-modulated Fourier-like transform, and that the Fourier transform appears as the special case $ABCD$6 (Mansuripur, 2020).

A more formal operator treatment identifies every lossless paraxial system with a unitary operator whose coordinate-space matrix element is exactly the Collins/Fresnel kernel. In the notation of Fan and coauthors,

$ABCD$7

while system concatenation obeys

$ABCD$8

This is an exact matrix-optics statement at the operator level: multiplying $ABCD$9 matrices is equivalent to composing Fresnel-type transforms (Fan et al., 2010).

Fractional Fourier structure appears whenever the underlying phase-space action is a rotation. For propagation on a constant Gaussian curvature surface, the curved-surface $ABCD$0 matrix is

$ABCD$1

which is formally identical to a fractional Fourier-transform system (Ding et al., 2021). A later operator-theoretic development goes further, stating that circular and hyperbolic fractional-order Fourier transformations are Weyl pseudo-differential operators and that their products can be composed through Weyl calculus (Pellat-Finet, 1 Oct 2025).

The finite-dimensional analogue is developed by Atakishiyev and collaborators, who replace the continuous paraxial-optical phase space by a finite oscillator model ruled by $ABCD$2. In that model, the continuous Fourier group $ABCD$3 acquires an exact finite unitary representation on $ABCD$4-pixel images, with isotropic and anisotropic Fourier transforms, rotations, and gyrations realized as explicit $ABCD$5 matrices (Wolf et al., 2011). This is one of the clearest strict meanings of “matrix Fourier optics”: a discrete image space on which the optical Fourier group acts exactly and unitarily.

4. Matrix-valued propagation in layered, anisotropic, and crystal media

The simplest matrix-valued propagators arise in stratified isotropic media. Acquaroli represents the field in each layer by forward and backward plane-wave amplitudes,

$ABCD$6

and propagates them through interfaces and homogeneous layers with $ABCD$7 matrices. The total multilayer operator is

$ABCD$8

The same formalism yields reflectance, transmittance, absorptance, internal field profiles, and Bloch dispersion of distributed Bragg reflectors (Acquaroli, 2018). The review explicitly remarks that, for a finite beam, one would decompose the beam into angular components and apply this transfer-matrix method to each angle and polarization separately before inverse transforming, so the thin-film method becomes a per-spatial-frequency building block of Fourier optics (Acquaroli, 2018).

For genuinely vector anisotropic media, scalar Fourier optics is insufficient. The 2025 GTMM paper targets transverse-homogeneous, longitudinal-inhomogeneous bianisotropic media using the four-component transverse state

$ABCD$9

obeying

$q$0

with $q$1 a $q$2 state matrix determined by $q$3, $q$4, $q$5, $q$6, and $q$7. For each transverse spatial frequency, the slab yields $q$8 Jones reflection and transmission matrices $q$9 and $ABCD$0, and the reflected or transmitted beam is reconstructed by inverse Fourier synthesis (Qin, 11 Sep 2025). This is matrix Fourier optics in a literal reciprocal-space sense: the transfer function is matrix-valued rather than scalar.

The 2024 non-uniform Fourier crystal-optics paper makes this explicit for arbitrary anisotropic dielectrics. Its central result is a $ABCD$1 transition matrix

$ABCD$2

which maps the two transverse Fourier components on an input plane to the full three-component electric-field spectrum after propagation in the crystal (Xie et al., 2024). The paper’s explicit aim is to bridge the analytically convenient $ABCD$3-based crystal-optics eigenanalysis and the $ABCD$4-based Fourier-optical propagation formalism (Xie et al., 2024).

A common misconception is that any polarization-sensitive optical element can be handled by a single Jones matrix. These papers show the limitation of that view: once the medium is spatial-frequency dependent, longitudinally structured, or anisotropic in a nontrivial way, the correct operator generally depends on $ABCD$5 and may act between spaces of different dimension, as in the $ABCD$6 crystal propagator (Qin, 11 Sep 2025, Xie et al., 2024).

5. Discrete operators, metasurfaces, and lens-free Fourier-domain devices

A major contemporary direction is the direct fabrication of finite-dimensional optical operators. The meta-DFT metasurface is designed to implement an $ABCD$7 complex DFT with a ninth reference channel,

$ABCD$8

so that

$ABCD$9

Each input component is encoded as the illumination amplitude and phase on one metalens, each output component is read at a distinct focal point, and complex output recovery is achieved by four phase-shifted measurements with the integrated reference metalens (Tanuwijaya et al., 12 Feb 2025). The paper emphasizes that this is not a continuous Fourier-transforming lens sampled after the fact; it is a discrete matrix operator synthesized directly in a chosen basis and extensible to arbitrary complex-valued matrix-vector multiplication (Tanuwijaya et al., 12 Feb 2025).

A complementary line aims to preserve continuous Fourier transformation while extending the accessible angular spectrum. The 2017 dielectric Fourier metasurface is designed around the exact nonparaxial mapping

$2\times2$0

rather than the paraxial approximation $2\times2$1, and uses a metasurface phase that is approximately angle-dispersion-free: $2\times2$2 The reported device operates from $2\times2$3 to $2\times2$4 and from $2\times2$5 to $2\times2$6, making it a nonparaxial Fourier-transform element rather than an ordinary thin lens (Liu et al., 2017).

An even more specialized case is the metapinhole, which does not create a physical Fourier plane but directly implements an angular transfer function $2\times2$7. In discrete notation, its action is naturally written as

$2\times2$8

with $2\times2$9 diagonal in the Fourier basis. The paper is explicit that this is the spectral-filtering part of a $H(k_x,k_y)$0 system, not a universal dense matrix processor (Abouelatta et al., 6 Sep 2025).

Free-space optical matrix multiplication can also be realized without metasurfaces by using programmable spatial masks and Fourier summation. In the 2024 optical matrix-multiplication work, a coherent field on a Gaussian lattice carries a tensor-product structure $H(k_x,k_y)$1; one SLM encodes the input vector, a second SLM encodes matrix coefficients, and a cylindrical lens performs the contraction over one coordinate because the zero spatial-frequency component equals the integral along that axis (Koni et al., 2024). This again illustrates the central algebra of Matrix Fourier Optics: local multiplication in one plane combined with Fourier-domain summation yields finite-dimensional linear algebra.

6. Computational frameworks, applications, and limitations

Modern software frameworks increasingly treat Fourier-optical systems as differentiable operator graphs. TorchOptics models scalar modulation by

$H(k_x,k_y)$2

and free-space propagation either as a Rayleigh–Sommerfeld convolution or as an angular-spectrum transfer-function operator,

$H(k_x,k_y)$3

For polarization, it explicitly uses matrix Fourier optics in the Jones sense,

$H(k_x,k_y)$4

and for partial coherence it propagates the mutual coherence tensor $H(k_x,k_y)$5, which is naturally an operator kernel (Filipovich et al., 2024). A plausible implication is that current computational practice has normalized the operator viewpoint even when full dense matrices are never assembled explicitly.

Application domains are correspondingly broad. The optical-CNN accelerator combines silicon photonics and free-space optics so that convolutional layers are realized by optical Fourier transforms and pointwise Fourier-domain multiplication, functionally approximating operators of the form

$H(k_x,k_y)$6

or, in the hardware pathway emphasized in the paper, repeated forward transforms plus a spatial reordering (Cottle et al., 2020). Fourier-conjugate adaptive optics relocates the adaptive element to a Fourier-conjugate image plane, thereby enlarging usable field of view in deep imaging and exploiting angular memory effect more effectively than standard conjugate AO (Amitonova, 2018). Thin-film transfer matrices remain indispensable for multilayer spectra, cavities, local field enhancement, and DBR dispersion (Acquaroli, 2018).

Several limitations recur across the literature. First, basis choice matters: diagonal operators in the plane-wave basis are dense in real space, and vice versa (Abouelatta et al., 6 Sep 2025). Second, scalar models fail when vector polarization structure, anisotropy, or bianisotropy are essential (Qin, 11 Sep 2025, Xie et al., 2024). Third, discrete optical matrix processors are not automatically universal; many are fixed or only block-diagonal/diagonal in a chosen basis (Abouelatta et al., 6 Sep 2025). Fourth, physical readout is often intensity-only, so complex-field recovery may require auxiliary reference channels or multiple measurements, as in meta-DFT and hybrid optical Fourier accelerators (Tanuwijaya et al., 12 Feb 2025, Cottle et al., 2020).

The most defensible general interpretation is therefore plural rather than singular. Matrix Fourier Optics is not one formalism but a family of closely related operator frameworks in which optical propagation is expressed as basis-dependent matrix action: scalar transfer functions in spatial-frequency space, canonical-transform kernels generated by symplectic matrices, Jones and higher-rank spectral propagators in anisotropic media, and fabricated finite-dimensional transforms in metasurfaces and optical linear-algebra hardware (Mansuripur, 2020, Fan et al., 2010, Xie et al., 2024). This suggests that the field’s unifying principle is not any specific device or equation, but the recognition that optical systems are linear transforms whose most revealing representation is often a matrix in an appropriately chosen Fourier-related basis.