Cascaded-Forward (CaFo) in Photonics & Neural Networks

Updated 3 July 2026

CaFo is a multifaceted framework that unites cascaded forward Brillouin scattering in photonic devices with novel deep learning algorithms that avoid backpropagation.
It employs sequential, modular cascades to generate multiple Stokes orders in optics and to train neural network blocks independently via forward-only signal propagation.
Experimental implementations in microspheres, fiber systems, and multimodal few-shot learning pipelines highlight CaFo’s practical impact on sensor technology and energy-efficient computation.

Cascaded-Forward (CaFo) refers to a diverse set of technical constructs that leverage sequential, multi-stage information transfer—across nonlinear optics, optomechanics, and modern machine learning. The term most commonly denotes (1) cascaded forward Brillouin scattering and lasing (CaFo) in photonic waveguides and microresonators, and (2) the Cascaded Forward (CaFo) algorithm(s) for block-wise deep neural network training without backpropagation. In recent literature, “CaFo” additionally designates a pipeline that fuses multiple large pre-trained models for few-shot learning. All CaFo paradigms are unified by generative or signal-processing feedback chains in a forward-only, typically modular configuration.

1. CaFo in Nonlinear Optics: Cascaded Forward Brillouin Scattering and Lasing

Cascaded forward Brillouin scattering (CaFo) is a nonlinear optoacoustic process in which multiple Stokes orders are generated via forward-stimulated Brillouin interactions. Energy- and momentum-conservation lead to $\omega_p - \omega_s = \Omega_B$ and $k_p - k_s \approx q \approx 0$ , where $\omega_p,\,\omega_s$ are optical pump and Stokes frequencies, $\Omega_B$ is the Brillouin shift (acoustic frequency), and $q$ is the acoustic phonon wavevector. Intra-modal CaFo typically occurs in either fiber waveguides or high-Q whispering-gallery-mode (WGM) microresonators—such as As $_2$ S $_3$ chalcogenide microspheres—with strong electrostrictive optomechanical coupling and negligible radiation pressure (Shanavas et al., 2022, Hayashi et al., 2018).

The process can be initiated and further amplified through seeding (e.g., via backward stimulated Brillouin scattering, BSBS), producing a comb of discrete optical lines separated by the characteristic frequency $\Omega_B$ , with Stokes beams generated up to 25th order at threshold pump powers $\lesssim$ 1 mW in $\sim$ 100 μm diameter WGM spheres (Shanavas et al., 2022).

2. Theoretical Description and Modeling of CaFo

The optical–acoustic interaction underlying CaFo is formulated via coupled-mode equations and Hamiltonian treatments encompassing all Stokes and anti-Stokes orders (Wolff et al., 2016). For a lossless waveguide, the Hamiltonian

$k_p - k_s \approx q \approx 0$ 0

describes photon and phonon dynamics with an electrostrictive interaction $k_p - k_s \approx q \approx 0$ 1 (photoelastic and boundary contributions). The relevant envelope operators yield coupled classical equations in the slowly-varying envelope approximation:

$k_p - k_s \approx q \approx 0$ 2

$k_p - k_s \approx q \approx 0$ 3

Without optical dispersion, the intensity profile remains unchanged and only the phase is modulated—a signature of pure phase grating, with the optical field's harmonics given by Bessel function amplitudes for each Stokes order (Wolff et al., 2016). In real materials, finite dispersion induces spatial exponential growth/suppression of acoustic amplitude. The intrinsic Brillouin gain $k_p - k_s \approx q \approx 0$ 4 and modal overlap parameters define experimental thresholds (Shanavas et al., 2022).

3. Implementation and Experimental Realizations

In microresonator platforms, CaFo is demonstrated using high-index As $k_p - k_s \approx q \approx 0$ 5S $k_p - k_s \approx q \approx 0$ 6 glass spheres of diameter 100–125 μm and optical quality factors $k_p - k_s \approx q \approx 0$ 7. These are fabricated by fiber taper-pull and thermal reflow, and overcoupled to tapered silica fibers ( $k_p - k_s \approx q \approx 0$ 8500 nm waist) to enable a broadened resonance supporting multiple Stokes lines (Shanavas et al., 2022). Pumping is performed at 1550 nm via a single-frequency diode laser swept into resonance; Stokes orders are detected via heterodyne beating on InGaAs photodiodes and analyzed with RF spectrum analyzers. Cascaded Stokes orders ( $k_p - k_s \approx q \approx 0$ 9) are observed, with comb spacing accurately matching finite-element predictions of $\omega_p,\,\omega_s$ 0 and thresholds corroborating model estimates.

In fiber systems, CaFo can be seeded by BSBS: counter-propagating pump and probe beams in highly nonlinear fiber create backward Stokes waves that induce forward cascades spanning radial acoustic modes up to ∼1 GHz, at signal-to-noise ratios $\omega_p,\,\omega_s$ 14 dB (Hayashi et al., 2018). The resulting multi-line spectrum is sensitive to strain and temperature, enabling distributed sensing.

4. Applications of CaFo in Photonics

Cascaded forward Brillouin lasing platforms provide sensitive, comb-resolved sensors and on-chip photonic microwave sources (Shanavas et al., 2022). Gas sensing exploits the $\omega_p,\,\omega_s$ 2-fold frequency shift of higher Stokes orders for improved refractive index or acoustic parameter detection. For RF and microwave photonics, the equally spaced Stokes comb serves as a frequency synthesizer; unlike backward Brillouin systems, strict free spectral range matching is unnecessary, allowing relaxed fabrication tolerances and wide tunability via geometry. Chip-scale integration is facilitated by translating sphere geometries to planar microdisks/wedges and optimizing the optical–acoustic overlap through waveguide engineering.

5. CaFo Algorithms in Neural Network Training

The Cascaded Forward (CaFo) algorithm represents a non-backpropagation paradigm for deep neural network optimization (Zhao et al., 2023, Spyra et al., 2 Nov 2025, Spyra, 23 Sep 2025). A CaFo-trained model decomposes into $\omega_p,\,\omega_s$ 3 sequential “blocks” (e.g., Conv→ReLU→Pool→BN) with per-block, independently trained classifiers (“predictors”) that map block outputs directly to label distributions. Each block–predictor pair is trained via supervised loss (cross-entropy, MSE, or sparsemax), omitting the need for negative examples as required by Forward-Forward (FF). At inference, predicted class probabilities are aggregated block-wise, typically by summing predictor outputs.

Training variants include:

CaFo-Rand: blocks are randomly initialized and frozen; only predictors are learned.
CaFo-DFA: blocks are pre-trained using Direct Feedback Alignment (DFA) with fixed random feedback, then predictors are fitted.

No global backward pass is needed; block-wise parallelism is straightforward, and block predictors are compatible with energy- and memory-constrained hardware.

6. Performance and Efficiency of CaFo Learning Methods

Empirical evaluations demonstrate that CaFo closes much of the gap to backpropagation on CNNs if DFA-trained blocks are used (test accuracy within 1–2 percentage points) but at a considerable computational/energy penalty—CaFo-DFA incurs 4× more training time and energy than BP, due to synthetic gradient computations and fragmented compute kernels (Spyra, 23 Sep 2025, Spyra et al., 2 Nov 2025). CaFo-Rand, while moderately more energy-efficient (up to 19% less energy, 9% less memory on CIFAR-10), suffers from degraded accuracy ( $\omega_p,\,\omega_s$ 413 percentage points drop on CIFAR-10). In contrast, Mono-Forward (MF) closes both efficiency and accuracy gaps to BP in MLPs. The elimination of negative sampling (required by FF) and removal of global backward locking are key conceptual advantages of CaFo; its limitations chiefly stem from either insufficient feature separability (Rand-CE) or DFA-induced overhead.

7. CaFo in Multimodal Few-Shot Learning via Foundation Model Cascades

A recent CaFo instantiation in few-shot vision learns by cascading outputs from four foundation models: GPT-3 (“prompt” for textual context), DALL·E (“generate” synthetic images), CLIP and DINO (contrastive visual encoders), unified through a learnable, adaptive cache model (Zhang et al., 2023). The sequential pipeline expands a $\omega_p,\,\omega_s$ 5-shot dataset with synthetic data, enriches textual prompts, and adaptively fuses predictions using nonlinear similarity-based weighting. The resulting system achieves state-of-the-art accuracy on multiple few-shot benchmarks, demonstrating that CaFo-style multimodal cascades can leverage complementary pre-training paradigms. Each foundation model adds unique gains; combining all four yields a top-1 ImageNet accuracy increase of 8.46% over zero-shot CLIP with negligible computational overhead.

In summary, Cascaded-Forward (CaFo) encapsulates a generative and architectural philosophy: signal flow and learning are arranged as modular, parallelizable cascades, either in photonic nonlinear media or in deep models eschewing global backpropagation. In both optics and machine learning, CaFo structures facilitate robust, scalable, and, in specific forms, hardware-conscious information processing (Shanavas et al., 2022, Wolff et al., 2016, Zhao et al., 2023, Spyra et al., 2 Nov 2025, Spyra, 23 Sep 2025, Zhang et al., 2023).