---
title: Heterogeneous Angular Spectrum Approach
url: https://www.emergentmind.com/topics/heterogeneous-angular-spectrum-approach
type: topic
---

# Heterogeneous Angular Spectrum Approach

Heterogeneous Angular Spectrum Approach denotes a family of angular-spectrum-based formulations in which the classical angular spectrum method is adapted to nonuniform propagation or control conditions. In its most direct form, it extends homogeneous-medium angular-spectrum propagation to heterogeneous media by introducing medium-dependent correction terms into the spectral propagator; in more recent optical work, it also denotes the use of deliberately heterogeneous illumination angular spectra as task encoders for diffractive networks. The shared mathematical core is the decomposition of a field into plane-wave components, for example \(A(f_x,f_y)=\mathcal{F}\{U(x,y,0)\}\), followed by propagation, rearrangement, or conditioning of those components in a way that reflects heterogeneity in sound speed, attenuation, interface geometry, or illumination structure [1906.08156] [2601.04825].

## 1. Classical basis and the meaning of heterogeneity

The homogeneous angular spectrum method represents a monochromatic field on a plane by its transverse spatial-frequency content and propagates each spectral component independently. In acoustics, if \(P(k_x,k_y;z)\) is the transverse Fourier transform of the pressure field, homogeneous propagation is given by
\[
P(k_x,k_y; z+\Delta z)=P(k_x,k_y; z)\,\exp(i k_z \Delta z),
\qquad
k_z=\sqrt{k_0^2-k_x^2-k_y^2},
\]
with evanescent components handled separately [1906.08156]. In optics, the same principle appears as
\[
U(x,y,z)=\mathcal{F}^{-1}\{A(f_x,f_y)\,H(f_x,f_y;z)\},
\]
with the exact scalar transfer function
\[
H(f_x,f_y;z)=\exp\!\Big(i k z \sqrt{1-(\lambda f_x)^2-(\lambda f_y)^2}\Big),
\]
and an evanescent cutoff for \(\sqrt{f_x^2+f_y^2}>1/\lambda\) [2601.04825].

In heterogeneous formulations, the independent phase progression of each plane-wave component is no longer sufficient. The departure from homogeneity may arise from a spatially varying sound speed \(c(\mathbf{r})\), from a depth-dependent stratification \(c(z)\), from non-planar and absorbing interfaces in thin-film optics, or from task-specific subsets of illumination angular components in diffractive optical computing [1906.08156] [2007.01133] [2407.06535] [2601.04825].

This suggests that “heterogeneous” has two technically distinct uses in the literature. In acoustic propagation, it refers to heterogeneity of the propagation medium itself. In diffractive-network control, it refers to heterogeneous angular codes imposed on an otherwise fixed optical system. The mathematical object is still the angular spectrum, but the role played by heterogeneity differs across these lines of work [1906.08156] [2601.04825].

## 2. Heterogeneous-medium ASA for trans-skull acoustics

The best-defined propagation-oriented Heterogeneous Angular Spectrum Approach was introduced for trans-skull imaging and focusing. The starting point is a heterogeneous Helmholtz form
\[
\big(\nabla^2+k_0^2\big)\tilde p(\mathbf{r},\omega)=\lambda(\mathbf{r})\,\tilde p(\mathbf{r},\omega),
\]
with
\[
\mu(\mathbf{r})=\frac{c_0^2}{c^2(\mathbf{r})},
\qquad
\lambda(\mathbf{r})=k_0^2\big(1-\mu(\mathbf{r})\big),
\]
where \(c_0\) is a reference sound speed and \(k_0=\omega/c_0\) [1906.08156].

Under weak heterogeneity and one-way propagation assumptions, the method uses a numerical marching scheme derived from an implicit solution. In discretized form,
\[
P^{n+1}\approx P^n e^{i k_z\Delta z}
+\frac{e^{i k_z\Delta z}}{2 i k_z}\,\big(\Lambda * P^n\big)\,\Delta z,
\]
with the convolution term evaluated efficiently as
\[
\big(\Lambda * P^n\big)(k_x,k_y;z_n)=
\mathcal{F}_k\big[\tilde p^n(x,y)\,\lambda(x,y;z_n)\big].
\]
Each axial step therefore requires an inverse FFT to recover \(\tilde p^n\), multiplication by \(\lambda\), a forward FFT, and application of the homogeneous phase factor plus the heterogeneity correction [1906.08156].

The formulation assumes linear acoustics, weak heterogeneity, a one-way forward-propagating field, and longitudinal waves only. Density variations and reflection terms are neglected in the derivation used in the paper, and shear-mode conversion is neglected because the incidence angles were largely below the skull’s critical angle of approximately \(26^\circ\) [1906.08156].

For trans-skull focusing, the method backpropagates a desired focal point to the aperture plane and extracts element-wise delays and amplitudes,
\[
\tau(x)=\frac{\arg\{\tilde p(x,0)\}}{\omega},
\qquad
A(x)=\frac{|\tilde p(x,0)|}{\max_x |\tilde p(x,0)|}.
\]
These corrections are then used to drive the array. For passive acoustic mapping, the same marching operator backpropagates measured aperture data and forms an intensity map
\[
I(x,y;z)=\|\tilde p(x,y;z)\|^2.
\]
The source location is taken as the maximum of \(I\) over the reconstruction region [1906.08156].

Quantitatively, the corrected trans-skull focusing reduced focal location error from \(2.2\pm0.7\) mm to \(1.0\pm0.4\) mm across \(0.25\)–\(1.5\) MHz and apertures of \(50\)–\(100\) mm. In passive acoustic mapping, the method produced an average of \(75\%\) error reduction and \(40\) to \(60\%\) increase in peak intensity relative to homogeneous ASA; one reported case decreased localization error from \(2.9\pm1.8\) mm to \(0.7\pm0.5\) mm. Reported runtimes were \(25.2\pm8.5\) ms per case for focusing delay and amplitude computation, \(166\pm37\) ms per frequency for heterogeneous-ASA passive acoustic mapping versus \(44\pm4\) ms for homogeneous ASA, and approximately \(540\pm18\) ms per frequency for a 3D passive acoustic mapping example [1906.08156].

A notable implementation feature is stabilization of backpropagation. The aperture spectrum is tapered with a Tukey window with cosine fraction \(R=0.25\) to suppress evanescent growth, and the transverse FFTs are zero-padded by \(4\times\). The study used \(\Delta x=\Delta y=200~\mu\text{m}\) and \(\Delta z=50~\mu\text{m}\) [1906.08156].

## 3. First-order analytical correction in continuously stratified media

A related development treats the special case \(c=c(z)\), where the medium is continuously stratified in depth. In this setting the heterogeneous term becomes multiplicative rather than convolutional, and the transformed equation reduces to
\[
\frac{d^2P}{dz^2}+k_z^2P=\lambda(z)\,P,
\]
with
\[
\mu(z)=\frac{c_0^2}{c^2(z)},
\qquad
\lambda(z)=\frac{\omega^2}{c_0^2}\big(1-\mu(z)\big).
\]
Using the ansatz
\[
P(k_x,k_y,z)=A(k_x,k_y,z)e^{i k_z z}
\]
and neglecting \(d^2A/dz^2\), the paper derives a first-order analytical solution [2007.01133].

The resulting stratified propagation operator is
\[
P(k_x,k_y,z)=
P_0(k_x,k_y)\,
\exp\!\left\{
i\left[
k_z z-\frac{k_0^2}{2k_z}\int_0^z (1-\mu(z'))\,dz'
\right]
\right\}.
\]
The phase correction can be written as
\[
d\phi=\frac{k_0^2}{2k_z}\big(1-\mu(z)\big)\,dz,
\]
so the first-order correction acts as an accumulated depth-dependent phase term rather than an amplitude correction [2007.01133].

The validity regime is explicitly first-order and WKB-like. The paper states short-wavelength and weak-heterogeneity conditions, including
\[
\left|
\frac{\partial\lambda/\partial z}{2 i k_z \lambda}
\right|\ll 1,
\qquad
\left|
\frac{k_0^2}{k_z^2}(1-\mu)
\right|\ll 1,
\]
together with the truncation condition
\[
\frac{1}{4}\left(\frac{k_0}{k_z}\right)^2
\left[
k_0\int_0^z (1-\mu(z'))\,dz'
\right]^2
\ll 1.
\]
The method is therefore intended for continuous stratification, slow depth variation, and forward propagation only [2007.01133].

For passive source localization, the corrected operator is used to backpropagate narrowband aperture data to candidate depths, inverse transform to real space, and accumulate
\[
I(x,y,z)=\sum_\omega
\left\|
\mathcal{F}_k^{-1}\!\left[P(k_x,k_y,z)\right]
\right\|^2.
\]
The reported mean localization error improved from \(1.2\pm0.3\) to \(0.49\pm0.3\) wavelengths at scalings relevant to biomedical (\(kL\sim 500\)), underwater (\(kL\sim 800\)), and atmospheric (\(kL\sim 10\)) applications. Total computation was on the order of milliseconds on standard hardware, specifically \(225\pm84\) ms for the stratified formulation compared with \(78\pm63\) ms for homogeneous ASA [2007.01133].

This stratified result is mathematically narrower than the trans-skull heterogeneous-medium formulation, but it clarifies the structure of the correction. It shows that when heterogeneity collapses to a depth-only profile, the general heterogeneous ASA admits an explicit first-order phase law rather than a full marching convolution [2007.01133].

## 4. Heterogeneous angular-spectrum encoding in diffractive networks

A distinct optical use of the concept appears in diffractive neural networks, where the illumination angular spectrum is used as a task encoder for a fixed diffractive stack. A plane wave is shaped by an amplitude mask \(M(x,y)\), sent through a \(2f\) system, and the field at the object plane is proportional to the Fourier transform of the mask,
\[
U_{\text{obj}}(x,y)\propto
\mathcal{F}\{M(\xi,\eta)\}\big|_{f_x=x/(\lambda f),\,f_y=y/(\lambda f)}.
\]
Bright regions in \(M\) therefore select allowed angular components, and different masks correspond to different functionalities [2601.04825].

The paper explicitly defines “heterogeneous” to mean that the angular-component sets used to encode tasks can differ in content and shape across tasks. Examples include disjoint concentric rings, quadrant squares, and learned nonoverlapping or overlapping patterns. This control mechanism operates at fixed wavelength and polarization and within a very narrow paraxial range, with \(\text{NA}\approx 7.5\times 10^{-3}\) and \(\theta_{\max}\approx 0.43^\circ\) [2601.04825].

The diffractive network used in the simulations had four phase-only layers, each \(300\times300\) pixels with pixel pitch \(10~\mu\text{m}\times10~\mu\text{m}\), separated by \(5\) cm of free space. Illumination was a plane wave at \(\lambda=550\) nm unless otherwise specified. The amplitude-only mask was followed by a \(2f\) system with a lens of \(300\times300\) pixels, pitch \(10~\mu\text{m}\), focal length \(f=20\) cm, and aperture radius \(r=1.5\) mm. Free-space propagation was computed by the angular spectrum method with the Rayleigh–Sommerfeld transfer function and evanescent cutoff [2601.04825].

The training problem was cast as multi-task image-to-image regression. For tasks \(t\in\{1,\dots,M\}\), each encoded by a distinct mask \(M_t(x,y)\), the loss was
\[
\mathcal{L}
=
\frac{1}{N}\sum_{i=1}^N \sum_{t=1}^M
\big\|
\hat I^{(i,t)}-I_{\mathrm{GT}}^{(c(i,t))}
\big\|_2^2.
\]
An energy-preserving variant added
\[
\exp\!\Big\{-\gamma\sum_{t=1}^M \|\hat I^{(i,t)}\|_1\Big\}
\]
to balance fidelity and throughput, because pure \(L_2\) regression could preserve only approximately \(1\%\)–\(8\%\) of the energy [2601.04825].

The paper reported several multi-task demonstrations. For MNIST digit translation with four tasks, learned masks outperformed ring and square masks by approximately \(0.7\) dB average PSNR, with SSIM also higher. For EMNIST letters mapped to digits and Greek letters, learned masks beat ring masks by approximately \(0.6\) dB PSNR on average. In an eight-task setting, learned masks improved average PSNR by approximately \(1.4\) dB versus rings and approximately \(0.9\) dB versus rectangles; predefined rings sometimes failed, including cases that misproduced lowercase instead of uppercase outputs [2601.04825].

Robustness experiments showed that spatially coherent broadband operation with \(32\) training wavelengths in \(400\)–\(700\) nm, evaluated over \(500\), achieved average PSNR of approximately \(14.3\) dB for the \(\ell-1\) task and approximately \(14.5\) dB for the \(\ell+1\) task, with visible speckle in backgrounds. A spatially incoherent monochromatic case using modal expansion with \(L=25\) during training and \(L=10{,}000\) at test achieved approximately \(16.1\) dB and \(16.7\) dB, respectively. Thresholding learned masks to pure binary after training changed performance negligibly [2601.04825].

The same framework was also evaluated on multi-dataset classification. Angular spectrum encoding achieved \(92.5\%\), \(81.6\%\), \(68.2\%\), and \(87.1\%\) on MNIST, FashionMNIST, KMNIST, and EMNIST, respectively, for an average of \(82.35\%\), outperforming wavelength encoding at \(79.95\%\) average and a no-encoding baseline at \(73.5\%\). Specialized single-dataset networks achieved \(93.2\%\), \(83.5\%\), \(77.8\%\), and \(88.2\%\) [2601.04825].

This optical line of work does not treat a heterogeneous propagation medium. Instead, it uses heterogeneous angular codes to control a fixed diffractive network. A plausible implication is that the angular spectrum becomes a multiplexing degree of freedom analogous to wavelength or polarization, but with the control implemented at fixed wavelength and polarization [2601.04825].

## 5. Related optical generalizations: thin-film inverse synthesis and arbitrary-plane diffraction

Related ASA developments broaden the computational role of the angular spectrum in optics. In inverse synthesis for semiconductor thin films, ASA is used as a forward model for reconstructing both optical and geometrical properties from transmission spectroscopy. The field is propagated componentwise with
\[
P_{\text{AS}}(f_x,\Delta z)=
\exp\!\big(i k\Delta z \sqrt{n^2-(\lambda f_x)^2}\big),
\]
combined with per-component absorption
\[
P_{\text{abs}}(f_x,\Delta z)=
\exp\!\left(
-\frac{2\pi}{\lambda}
\frac{\kappa n \Delta z}{\sqrt{n^2-(\lambda f_x)^2}}
\right),
\]
so that \(P_{\text{total}}=P_{\text{AS}}\cdot P_{\text{abs}}\). Interfaces are handled by normal-incidence Fresnel coefficients and shape-based modifiers, with a top-surface parameterization \(S(x)=c_1 x+c_2 x^2\) that encodes tilt and curvature [2407.06535].

The inverse problem minimizes
\[
J(\theta)=
\sqrt{\frac{1}{N}\sum_\lambda
[T_{\text{meas}}(\lambda)-T_{\text{model}}(\lambda;\theta)]^2},
\]
using a genetic algorithm followed by an interior-point local optimizer. The forward model includes multi-reflection, partial coherence, Gaussian beam illumination, and Savitzky–Golay smoothing of the simulated transmittance. On an amorphous silicon thin film on glass, the ASA-based inverse synthesis achieved RMSE of approximately \(0.559\%\), comparable to a transfer-matrix-based inverse synthesis at approximately \(0.555\%\). Geometry optimization reduced standard deviations of material parameters by factors between approximately \(2.85\times\) and \(8.00\times\) [2407.06535].

A different extension treats diffraction between arbitrary planes by angular spectrum rearrangement. Here the propagation medium remains homogeneous, but the geometry of source and observation planes is arbitrary. The rotated spectrum is formed by
\[
\mathbf{k}_{\mathrm{t}}=\mathbf{R}\,\mathbf{k}_{\mathrm{r}},
\qquad
\mathbf{R}=\mathbf{R}_y(\theta)\mathbf{R}_z(\phi),
\]
with explicit mappings to \((k_u,k_v,k_w)\), after which a sparse rearranged spectrum \(F(k_u,k_v)\) is synthesized by a matrix triple product
\[
E(u,v)=\Omega(v,k_v)\,F(k_u,k_v)\,\Omega(u,k_u).
\]
The paper states that this avoids interpolation and Jacobian-induced errors while retaining efficiency comparable to the control method [2312.06167].

Quantitatively, for scalar diffraction the control method reached a peak error of \(7.2\) at \(\theta=90^\circ\), whereas the proposed rearrangement method yielded “exact” results across angles. For vectorial diffraction, the control method reached errors of \(8.7\), and selective merging reduced time consumption to at most \(3\%\) of that before merging in the scalar case and to \(15\%\) in a vectorial case with an allowed error of \(1.4\times10^{-5}\) [2312.06167].

These two optical directions are not labeled “Heterogeneous Angular Spectrum Approach” in the same way as the trans-skull literature, but they show the same methodological tendency: the angular spectrum is treated as a flexible computational state variable that can absorb heterogeneity in geometry, absorption path length, boundary shape, or plane orientation rather than only free-space phase accumulation [2407.06535] [2312.06167].

## 6. Assumptions, limitations, and terminological variation

Across the propagation-oriented formulations, the main assumptions are weak or slowly varying heterogeneity, one-way forward propagation, and limited treatment of reflections or multiple scattering. In the trans-skull method, backward waves are not modeled, density variations and reflection terms are neglected in the main operator, and shear-mode conversion is omitted. In the stratified analytical correction, the approximation is explicitly first-order and assumes continuous depth variation. In optical thin-film inverse synthesis, the model is scalar and uses normal-incidence Fresnel coefficients. In diffractive-network control, the demonstrated encoding operates in a very small angular bandwidth, with capacity bounded by the available NA and SNR [1906.08156] [2007.01133] [2407.06535] [2601.04825].

The performance trade-offs also differ by application. In trans-skull acoustics, heterogeneous correction adds modest computational overhead relative to homogeneous ASA while yielding sub-millimeter-scale improvements. In stratified localization, the correction improves error from \(1.2\pm0.3\) to \(0.49\pm0.3\) wavelengths at a runtime increase from \(78\pm63\) ms to \(225\pm84\) ms. In diffractive networks, multiplexing one fixed diffractive stack over several tasks incurs a modest fidelity trade-off relative to specialized single-task networks, and pure fidelity training can yield low throughput of approximately \(1\%\)–\(8\%\) preserved energy [1906.08156] [2007.01133] [2601.04825].

A distinct use of the acronym HASA appears in work on ferromagnetic resonance in heterogeneous thin films. There, HASA is not a propagation algorithm but a sensitivity-based interpretation of the angular evolution of multi-peak FMR spectra. The key condition is
\[
\frac{dH_{\text{res}}}{dM_{\text{eff}}}=0
\]
in the magnetically dominating region, which defines a dynamic homogeneity angle at which multiple resonance lines can collapse into a single peak. The paper distinguishes this bulk effect from the surface critical angle in spin-wave resonance, where higher-mode intensities vanish without line-position coalescence [1208.6573].

This terminological variation suggests that the phrase is not fully standardized across disciplines. In acoustics and computational optics, it usually denotes an angular-spectrum propagation or synthesis method adapted to heterogeneity in medium or geometry. In diffractive optical computing, it denotes heterogeneous angular-spectrum control signals. In magnetic resonance, it denotes an angular-sensitivity framework centered on a zero-derivative resonance condition [1906.08156] [2601.04825] [1208.6573].

Source: https://www.emergentmind.com/topics/heterogeneous-angular-spectrum-approach