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Pulse-by-pulse programmable synthesis of ultrafast optical waveforms

This paper presents a programmable pulse-by-pulse shaper that achieves deterministic spectral-temporal control of ultrafast pulses at ~20 MHz by synchronizing FPGA-driven electro-optic modulation, enabling the generation of complex waveforms and the mapping of pulse-index-dependent phase profiles into nonlinear spectral breathing dynamics.

Original authors: Shilong Liu, Gabriel Demontigny, Émile Dessureault, Stéphane Virally, Denis V. Seletskiy

Published 2026-09-09

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CC BY 4.0

Original authors: Shilong Liu, Gabriel Demontigny, Émile Dessureault, Stéphane Virally, Denis V. Seletskiy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of modern light science, researchers have long sought the ability to sculpt the shape of light itself. Imagine a beam of light not as a steady stream, but as a rapid-fire series of incredibly short bursts, each lasting only a fraction of a trillionth of a second. For decades, scientists could shape these bursts, but only by applying the same design to every single one in a row, like stamping a pattern onto a long ribbon of paper. The goal has always been to break this limitation: to treat every individual burst as a unique canvas, changing its color, timing, and shape from one moment to the next, even when they arrive millions of times per second. This level of control is crucial because it would allow scientists to probe the fastest chemical reactions and quantum events with a precision that matches the speed of the events themselves, rather than just watching a blurred average of them.

A team of researchers has now built a machine that achieves this elusive goal, turning the sequence number of a light pulse into a programmable dial. They created a system that can rewrite the properties of an ultrafast light pulse every time it arrives, operating at a rate of roughly 20 million pulses per second. Instead of using a static filter that affects all pulses equally, their device acts like a high-speed editor that assigns a unique instruction to each individual pulse as it passes through. By synchronizing a fast electronic signal with the arrival of each pulse, they can stretch the light out in time, paint a specific pattern onto it, and then compress it back. This process allows them to write a different "code" onto every single pulse in the train, effectively turning the pulse's position in the sequence into a new way to control the light.

The researchers demonstrated this capability through three distinct levels of control, each revealing a new way to manipulate the light. First, they showed they could simply flip the phase of the light, a property related to the timing of the wave's peaks and valleys, in a binary pattern. This created a spectral interference effect, similar to how light waves overlap to create patterns, but with the pattern shifting from one pulse to the next. Second, they programmed the pulses to move back and forth in time with extreme precision. By applying a specific type of phase shift, they could slide the position of each pulse forward or backward by tiny fractions of a second, creating a smooth, deterministic trajectory that the light followed pulse by pulse. This allowed them to create complex timing patterns, such as a pulse train that oscillates in a damped rhythm, all without moving any physical parts of the machine.

The most striking demonstration involved a phenomenon known as "breathing." The researchers programmed the pulses to periodically change their duration, stretching and compressing them in a rhythmic cycle. When these shaped pulses were sent through a special fiber optic cable, the nonlinear properties of the glass converted this time-based breathing into a change in the light's color spectrum. Instead of just getting wider or narrower in time, the pulses began to expand and contract in their range of colors. The team mapped out exactly how this breathing changed as they increased the strength of their programming. They discovered that at low levels, the light simply breathed gently as a single block of color. As they increased the control, the spectrum would split into multiple distinct peaks, like a single note breaking into a chord. At even higher levels, these peaks would merge back together into a single, broad, and intensely breathing envelope.

This work establishes a new way to think about light control, where the index of the pulse—simply its number in the sequence—becomes a programmable degree of freedom. In traditional systems, complex behaviors like this breathing or splitting often arise spontaneously from the internal dynamics of a laser, making them difficult to predict or control on demand. Here, the researchers proved that they could externally dictate these behaviors, writing a specific sequence of instructions that forces the light to evolve exactly as designed. By combining this pulse-by-pulse control with nonlinear optics, they created a platform where the evolution of light can be resolved and synthesized for every single pulse. This opens a path toward real-time, pulse-resolved synthesis of optical waveforms, offering a powerful new tool for exploring the fastest dynamics in nature.

Technical Summary: Pulse-by-pulse programmable synthesis of ultrafast optical waveforms

Problem Statement
While ultrafast optical pulse shaping is a central tool in modern photonics for controlling light-matter interactions and synthesizing arbitrary waveforms, established platforms face a fundamental limitation: they typically apply a single, static, or slowly refreshed spectral mask to a high-repetition-rate pulse train. Conventional spatial light modulators (SLMs) and liquid crystal on silicon (LCOS) devices offer high control dimensionality but suffer from refresh rates (typically ~50 Hz) far below the repetition rates of mode-locked lasers (MHz range). Faster approaches using digital micromirror devices (DMDs), acousto-optic modulators (AOMs), or electro-optic modulation (EOM) often operate in restricted modes, such as amplitude-only modulation or burst-level shaping, rather than enabling fully programmable, pulse-by-pulse control. Consequently, realizing fully programmable optical waveforms that can be updated at the rate of individual pulses in a high-repetition-rate train remains a long-standing goal in optical arbitrary waveform generation.

Methodology
The authors introduce a Programmable Pulse-by-Pulse Shaper (PPS) that treats the pulse index ( $n$ ) as an additional programmable dimension. The system operates at a repetition rate of approximately 20 MHz and combines a temporal 4f imaging system with FPGA-synchronized electro-optic modulation (EOM).

Temporal Stretching and Modulation: A picosecond-scale pulse train from a mode-locked fiber laser is temporally stretched using a chirped fiber Bragg grating (CFBG) with large group delay dispersion (GDD). This maps the spectral components of the pulse into a nanosecond-scale time window accessible to fast EOM.
FPGA-Driven Modulation: An FPGA generates RF waveforms synchronized with the pulse train. These waveforms drive the EOM, imprinting a programmed temporal modulation onto each stretched pulse.
Recompression and Spectral Mapping: After passing through the EOM, the pulses are recompressed. The temporal modulation is converted back into an effective spectral phase applied to the incident ultrashort pulse. By updating the RF waveform synchronously with the pulse index, the system implements a pulse-index-dependent spectral transfer function:
$\tilde{E}_{out}(\Omega,n) = \tilde{E}_{in}(\Omega,n)R(\Omega)\exp[i\phi_{EOM}(\Omega,n)]$
where $\phi_{EOM}$ can be expanded into zero-order (constant phase), first-order (group delay), and fractional-order terms.
Nonlinear Propagation: To demonstrate nonlinear dynamics, the shaped pulse train is launched into a nonlinear fiber stage dominated by self-phase modulation (SPM). The programmed temporal waveform evolution is converted into pulse-index-dependent spectral reshaping.
Characterization: The system utilizes Fourier-transform spectral interferometry with a reference pulse to reconstruct the temporal profile of individual pulses, allowing for pulse-resolved evaluation rather than averaged observables.

Key Contributions and Results
The paper demonstrates four distinct levels of programmable pulse synthesis:

Zero-Order Phase Coding: The system switches and scans the zero-order phase ( $\phi_{0,n}$ ) of individual pulses. By creating a temporal double-slit interference between a signal pulse and a reference pulse, the authors show that binary or multi-level phase sequences ( $-\pi/2 \to \pi/2$ ) can be mapped into double-slit-like spectral-temporal interference patterns. This results in subharmonic components in the RF spectrum (e.g., at $f_0/2$ ), confirming pulse-by-pulse phase control and offering a route to relative carrier-envelope phase (CEP) control.
First-Order Phase Programming: By programming the first-order spectral phase term ( $\tau_{1,n}\Omega$ ), the system controls the group delay of each pulse. The authors demonstrate deterministic temporal trajectories, including periodic triangular ramps (800 fs excursion over 80 pulses) and damped oscillatory movements. The system achieves a programmable delay resolution of approximately 170 attoseconds, limited by the timing jitter of the source rather than the shaper itself.
Fractional-Order Phase Engineering: The authors program fractional-order spectral phases (specifically with a Lévy index $\alpha=1$ ) to synthesize "temporal breathers." This creates a pulse-index-dependent variation in pulse duration. When these pulses propagate through a nonlinear fiber, the programmed temporal breathing is converted into a corresponding "spectral breathing" (periodic expansion and contraction of the spectral bandwidth) via SPM.
Phase-Level-Dependent Regime Mapping: By scanning the maximum programmed fractional-phase level ( $\Phi_m$ $Φ_{m}$ ), the authors construct a regime map of nonlinear spectral breathing. They identify three distinct morphological regimes:
- Regime I: Weakly modulated single-envelope breathing.
- Regime II: Multi-peak spectral splitting.
- Regime III: Strongly breathing merged-spectrum dynamics (where multi-peak structures merge into a dominant envelope while maintaining large breathing contrast).
  Experimental results align closely with numerical simulations based on a nonlinear Schrödinger-type model.

Significance
The paper claims to establish a significant step toward the vision of on-demand, pulse-by-pulse optical arbitrary waveform generation. By treating the pulse index as a programmable degree of freedom, the PPS moves beyond the limitations of static masks and slow refresh rates.

The work is significant for two primary reasons:

Programmable Nonlinear Dynamics: Unlike conventional nonlinear systems where dynamics like breathing or spectral splitting arise from self-organized processes governed by intrinsic laser parameters (gain, loss, dispersion), the PPS externally writes a designed phase sequence onto the pulse train. This allows for the deterministic synthesis, measurement, and classification of nonlinear optical states (such as breathing and spectral splitting) at the individual-pulse level.
Real-Time Resolution: The platform provides a route toward real-time, pulse-resolved optical arbitrary waveform synthesis, enabling the study of nonlinear pulse-resolved dynamics that were previously difficult to access or control on a pulse-by-pulse basis.

The authors note that the current implementation is limited by the electronic sampling resolution (1.25 GSa/s) of the FPGA used to write the phase patterns, which constrains the complexity of arbitrary waveform synthesis. However, they suggest that future improvements in FPGA/DAC architectures and modulator bandwidth could further expand the controllable degrees of freedom.

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