Fully parallel, high-speed incoherent optical method for performing discrete Fourier transforms

TL;DR — In 1978 Goodman, Dias, and Woody showed that a shelf of LEDs, a photographic mask, and a row of photodiodes computes a full matrix–vector product in one pass of light: every multiply-accumulate happens simultaneously, in the transit time of photons, with the "multiplications" done by attenuation for free. They pitched it as a DFT engine, which was the wrong application at the wrong time — analog precision was poor, the mask was frozen film, and nobody's workload was dominated by dense matrix–vector products. Today, neural-network inference is exactly that workload, and the photonic tensor cores now being commercialized are, architecturally, this paper with a silicon-photonics costume on. The bibliographic details are verified; my account of the paper's contents is from knowledge of the literature.

The idea, as they had it

The construction is almost embarrassingly simple, which is why it's beautiful. You want to compute y = Ax, where x \in \mathbb{R}^N_{\ge 0} and A has nonnegative entries (we'll fix that restriction shortly).

Take a vertical column of N light sources — LEDs — and drive each one so its intensity encodes x_i. Anamorphic optics (cylindrical lenses) smear each source's light out so that source i uniformly illuminates column i of a two-dimensional transparency mask. The mask's transmittance at cell (j, i) is proportional to A_{ji}. A second set of cylindrical optics then collects all the light emerging from row j of the mask onto detector j. Detector j therefore integrates

which is the matrix–vector product. Every multiplication is an attenuation; every accumulation is photons landing on the same detector. All N^2 MACs occur in parallel, in the ~nanosecond it takes light to cross the bench. The energy cost of the arithmetic itself is essentially zero — you pay for generating the light and reading the detectors, and that cost is amortized across the whole fan-out.

The subtlety the paper had to solve is that incoherent light carries only nonnegative intensities, while the DFT matrix F_{jk} = e^{-2\pi i jk/N} is complex. Their fix — and this trick recurs throughout analog computing — is to decompose complex numbers into nonnegative components. My recollection is that they used a three-component representation along unit vectors at 120° in the complex plane (any complex number is a nonnegative combination of 1, e^{2\pi i/3}, e^{4\pi i/3}), so a complex N-point DFT becomes a modestly larger real nonnegative matrix–vector product; the electronics recombine the components afterward. The general lesson stands regardless of the exact decomposition: sign and phase cost you a small constant factor in an incoherent system, nothing more.

The framing worth noticing: everyone else in 1978 was racing to make the coherent Fourier optics of lenses (a lens literally computes a continuous 2D Fourier transform at its focal plane) useful, which demands lasers, interferometric stability, and phase control. Goodman's group went the other way — give up phase, use cheap incoherent LEDs, and get an arbitrary discrete linear transform, because the mask can encode any matrix at all. The DFT was the demo; the machine is a general matrix–vector engine. That generality is the whole reason the paper matters now.

Why it couldn't work then

Nothing at the edges was fast or precise. The optical transit is instantaneous, but you must drive N sources and read N detectors every cycle. In 1978, LEDs could be modulated at roughly MHz–tens-of-MHz rates, and detector/amplifier chains at similar speeds with maybe 1% analog accuracy — call it 6–7 effective bits on a good day. A 32-point transform at 10 MHz is ~10^{10} MAC/s, which genuinely beat contemporary digital hardware (a minicomputer managed roughly 10^6 FLOPS; even dedicated FFT boxes were orders of magnitude below the optical throughput). But that headline number couldn't be sustained into a system: the surrounding electronics to feed and drain it at that rate didn't exist at reasonable cost.

The matrix was frozen. The mask was photographic film. Changing A meant developing a new transparency. Programmable spatial light modulators were primitive — liquid-crystal devices with millisecond response and poor uniformity. So the machine could only ever accelerate one fixed transform.

And the killer: the DFT was the wrong target. The FFT computes an N-point DFT in O(N \log N) operations on hardware riding Moore's law, in stable digital arithmetic with as many bits as you like. An analog device offering O(1)-time transforms at 6-bit accuracy was competing against an algorithm and an exponential. There was no workload where you needed billions of dense, arbitrary matrix–vector products per second and could tolerate low precision. The paper was a solution waiting fifty years for its problem.

What changed

Three things, and they arrived together.

The workload. Transformer inference is repeated dense matrix–vector multiplication — that is nearly the entire FLOP budget. Better still, inference is famously tolerant of low precision: INT8 is standard, 4-bit weights are routine, and there is credible work at lower. The precision that killed analog optics for scientific FFTs is a non-issue, or at least a negotiable one, for neural nets. And the economics inverted: in a datacenter GPU, the energy of a MAC is dominated not by arithmetic but by moving weights and activations through wires — roughly 0.1–1 pJ/MAC at INT8 all-in. Optics does fan-out and accumulation with no charging of wires.

The devices. Silicon photonics gives modulators at tens of GHz, waveguides, on-chip germanium photodetectors, and wavelength-division multiplexing, all in CMOS-compatible processes. The frozen film mask has modern replacements: meshes of Mach–Zehnder interferometers whose phase shifters encode the matrix (Shen et al., Nature Photonics 2017 — "Deep learning with coherent nanophotonic circuits"), microring weight banks, and nonvolatile phase-change-material cells that hold a weight with zero static power (Feldmann et al., Nature 2021, who also used frequency combs to run many input vectors through the same matrix simultaneously — batching by color).

The theory of when optics wins. Hamerly and colleagues (PRX 2019) made the energy argument crisp: because one input photon stream fans out across an entire row, the optical energy per MAC scales like 1/N, and for large enough N you can approach sub-femtojoule MACs — below the Landauer-adjacent floor of digital switching — with the true cost concentrated in DACs, ADCs, and the laser wall plug. This is Goodman's amortization argument, made quantitative.

What a serious 2026 revival looks like

You would not build a DFT engine; you would build a transformer-inference tile, and the block diagram is Goodman's with each element upgraded:

• Sources → high-speed modulators. Activations arrive electronically, hit DACs, and modulate light at 10–50 GHz. Reuse Goodman's insight that the input is intensity, not phase, where possible — incoherent/photoelectric schemes are dramatically less sensitive to fabrication and thermal drift than interferometric meshes.
• Film mask → nonvolatile analog weights. PCM cells or trimmed microrings hold the weight matrix. Weights change rarely in inference, so slow writes are fine — exactly the regime where the "fixed mask" stops being a bug and becomes the feature (no weight movement energy at all).
• Detectors → integrated photodiodes + ADCs, followed by electronic nonlinearity, softmax, and normalization. Nobody sane does the nonlinear parts optically; the linear layers are >95% of the FLOPs, so Amdahl is on your side.
• Signs via differential pairs. Goodman's nonnegativity decomposition survives intact: represent signed weights as A^+ - A^- on paired detectors. Fifty years old, still the right answer.
• Hardware-in-the-loop training. The genuinely modern ingredient. Analog crossbars have crosstalk, shot noise, and ~4–6 effective bits. Rather than calibrating these away, train through the physical device — physics-aware backpropagation in the style of Wright et al. (Nature 2022, "Deep physical neural networks") — so the weights absorb the hardware's idiosyncrasies. Fine-tuning a pretrained model on-device for a few epochs plausibly recovers most of the accuracy gap.

The honest engineering question is whether the ADC/DAC energy at the boundary eats the optical savings. The answer depends on N: the conversion cost is O(N) per matrix–vector product while the computation is O(N^2), so big matrices favor optics. Transformer layers with d_{\text{model}} of 4k–16k are exactly the right size class.

Has it been vindicated? What's still open

Partially vindicated, honestly. The lineage is direct: Shen et al. 2017 reignited the field; Feldmann et al. 2021 demonstrated a photonic tensor core doing convolutions in parallel; Lightmatter built Envise, an inference accelerator on precisely this template (notably, their commercial traction shifted toward photonic interconnect — Passage — which tells you where the near-term money is); Lightelligence and several successors continue on compute. Free-space descendants revive the original bulk-optics geometry at scale, with megapixel SLMs as the "mask." Academic demos have run keyword spotting, MNIST-class vision, and small transformer layers optically with digital-comparable accuracy after hardware-aware training.

What remains genuinely open: no photonic system has yet beaten a same-node GPU on end-to-end energy per token for a real model, once lasers, thermal tuning, and converters are counted; analog precision drifts with temperature and aging, and continuous calibration is an unsolved systems problem; and on-chip training (not just inference) through an optical forward pass is still a research frontier. The gap between "the physics computes the MAC for free" and "the system wins" is exactly where the field sits.

The meta-lesson is the one worth carrying around: Goodman's machine was never wrong, it was unemployed. An architecture whose only trick is dense, low-precision, fixed-matrix multiplication at enormous throughput had to wait for a civilization whose dominant computational expense is dense, low-precision, mostly-fixed-matrix multiplication.

Where to read

The paper is at doi.org/10.1364/ol.2.000001 (Optics Letters 2, 1, 1978) — it is two pages and a pleasure. Read alongside: Shen et al., "Deep learning with coherent nanophotonic circuits" (Nature Photonics 2017) for the modern reboot; Hamerly et al. (PRX 2019) for the energy-scaling argument that makes the case quantitative; Feldmann et al. (Nature 2021) for the WDM tensor core; and Wright et al. (Nature 2022) for training through imperfect physical hardware. Goodman's textbook Introduction to Fourier Optics supplies the background the 1978 paper assumes in its first sentence.