## Abstract

Superradiance is the anomalous radiance describing coherent photon emission from a gas. It plays a crucial role in atomic physics, quantum mechanics and astrophysics. Because the intensity of superradiant light beams is proportional to the number of particles squared, superradiance is also at the core of today’s most powerful light sources. Superradiant emission is intuitively expected when the distance between light-emitting particles is much smaller than the photon wavelength. Here, we break this assumption by predicting a never considered superradiance effect that holds even when the particle number per wavelength vanishes. We discover that a bunch of relativistic charged particles arranged in certain ways can generate an optical shock along the Vavilov–Cherenkov angle in vacuum, thereby concentrating broadband radiation in any spectral region into single light bullets. The process leaves clear experimental signatures, and we illustrate it in the form of a previously unrecognized nonlinear superradiant Thomson scattering. This concept may enable new forms of superradiant emission in advanced light sources, atomic physics systems, and unlock coherent emission in plasma accelerators.

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## Data availability

All data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. Source Data are provided with this paper.

## Code availability

All computer codes supporting the findings of this study are fully documented within the paper and its references. Reasonable requests for access to the codes should be directed to J.V. on behalf of the Osiris consortium.

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## Acknowledgements

We acknowledge stimulating discussions with A. Gover, who provided a significant contribution towards the clarification of the concept. We also acknowledge use of Marenostrum at the Barcelona Supercomputing Center, Piz Daint in Switzerland and SuperMUC at the Leibniz Research Center in Munich, Germany, under PRACE awards. This work was partially supported by the EU Accelerator Research and Innovation for European Science and Society (EU ARIES) under grant agreement no. 730871 (H2020-INFRAIA-2016-1). J.V. and M.P. acknowledge the support of FCT (Portugal) grants no. SFRH/IF/01635/2015 and PD/BD/150411/2019.

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## Additional information

**Peer review information** *Nature Physics* thanks Avraham Gover and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Extended data

### Extended Data Fig. 1 The role of the bunch transverse size on generalised superradiance.

(**a-c**), show the time-integrated radiation profile from co-propagating nonlinear Thomson scattering in a spherical detector in the far field. The angular coordinates in the spherical detectors are *θ* and *φ*. The colour scale is normalised to the maximum intensity in the left and right sides of each frame. The left side of each frame represents the radiation for a single electron, where most of the radiation is directed into an angle *φ*_{w} = *a*_{0}/*γ* ≃ 0.0447rad. Here, *a*_{0} = *e**A*/*m*_{e}*c*^{2} ≃ 4.47 is the normalised laser vector potential at the radial position of the particle and *γ* = 100 is the particle relativistic factor. The right side of each frame represents the radiation from the modulated electron bunch. The Vavilov–Cherenkov angle corresponds, in this case, to the angle of maximum emission for a single electron, *φ*_{w}. In (**a**) the bunch transverse size is *δ**r*_{β} = 3.1*c*/*ω*_{0} and *ω*_{0} is the laser frequency. In (**b**), *δ**r*_{β} = 1.0*c*/*ω*_{0}, and in (**c**) *δ**r*_{β} = 0.31*c*/*ω*_{0}. The right panels show that the integrated angular radiation profile becomes sharper as the bunch transverse size decreases. This is a result of superradiant emission in generalised superradiance. **d**, shows the temporal intensity profile of the superradiant light bursts. The different lines were obtained by integrating the spatiotemporal radiation profiles along any of the angular coordinates. Each line was normalised to its corresponding peak intensity. Each curve corresponds to the right half of panels (**a-c**), where black is for *δ**r*_{β} = 3.1*c*/*ω*_{0}, yellow corresponds to *δ**r*_{β} = 1*c*/*ω*_{0} and red to *δ**r*_{β} = 0.3*c*/*ω*_{0}.
Source data

### Extended Data Fig. 2 The role of the bunch divergence (or emittance) on generalised superradiance.

The figure shows the time-integrated radiation profile from co-propagating nonlinear Thomson scattering in a spherical detector in the far field. The angular coordinates in the spherical detectors are *θ* and *φ*. The colour scale is normalised to the maximum intensity in the left and right sides of each frame. The left side of each frame represents the radiation for a single particle, where most of the radiation is directed into an angle *φ*_{w} = *a*_{0}/*γ* ≃ 0.047, where *a*_{0} = *e**A*/*m*_{e}*c*^{2} ≃ 4.47 is the normalised laser vector potential at the position of the particle and *γ* = 100 is the particle relativistic factor. The right side of each frame represents the radiation from the modulated electron bunch. The Vavilov–Cherenkov angle *φ*_{VC} ≃ 0.06rad is higher than *φ*_{w}. Color scales were normalised to the peak values in all plots. In (**a**) *δ**p*_{⊥} = 1.0*m*_{e}*c* and in (**b**) *δ**p*_{⊥} = 0.1*m*_{e}*c*, where *δp*_{⊥} corresponds to the bunch transverse momentum spread. As a result of generalised superradiance, both frames (**a**) and (**b**) show that the maximum emission angle for the bunch does not coincide with the single-particle maximum emission angle, even though individual bunch particles in the right side of each frame execute the same trajectory as the single particle (shown in the left side). The time-integrated radiation profile becomes sharper for lower *δ**p*_{⊥}, approaching the ideal case of a zero emittance bunch with *δ**p*_{⊥} = 0.

### Extended Data Fig. 3 The role of the bunch energy spread on generalised superradiance.

The role of the bunch energy spread on generalised superradiance. **a-b**, show the time-integrated radiation profile from co-propagating nonlinear Thomson scattering in a spherical detector in the far field. The angular coordinates in the spherical detectors are *θ* and *φ*. The colour scale is normalised to the maximum intensity in the left and right sides of each frame. The left side of each frame represents the radiation for a single particle, where most of the radiation is directed into an angle *φ*_{w} = *a*_{0}/*γ* ≃ 0.047, where *a*_{0} = *e**A*/*m*_{e}*c*^{2} ≃ 4.47 is the normalised laser vector potential at the position of the particle and *γ* = 100 is the particle relativistic factor. The right side of each frame represents the radiation from the modulated electron bunch. The Vavilov–Cherenkov angle *φ*_{VC} ≃ 0.06rad is higher than *φ*_{w}. Color scales were normalised to the peak values in all plots. In (**a**), *δ**p*_{∥} = 10.0*m*_{e}*c*, and in (**b**) *δ**p*_{∥} = 1.0*m*_{e}*c*, where *δ**p*_{∥} is the bunch longitudinal momentum spread. As a result of generalised superradiance, both frames (**a**) and (**b**) show that the maximum emission angle for the bunch does not coincide with the single particle maximum emission angle, even though individual bunch particles in the right side of each frame execute the same trajectory as the single particle (shown in the left side). The time-integrated radiation profile becomes sharper for lower *δ**p*_{∥}, approaching the ideal case of a zero energy-spread bunch with *δ**p*_{∥} = 0. **c**, shows the temporal intensity profile of the superradiant light bursts. The different lines were obtained by integrating the spatiotemporal radiation profiles along any of the angular coordinates. Each line was normalised to its corresponding peak intensity. Each curve then corresponds to the right half of frames (**a-b**), where black corresponds to *δ**p*_{∥} = 10*m*_{e}*c*, and red to *δ**p*_{∥} = 1*m*_{e}*c*. The dashed black line is a scaled version of the solid black curve. Although the peak intensity is higher for *δ**p*_{∥} = 1*m*_{e}*c* than for *δ**p*_{∥} = 10*m*_{e}*c*, the duration of the superradiant light pulse remains identical in both cases. This indicates robustness of generalised superradiance to large (10% in this case) bunch energy spreads.
Source data

### Extended Data Fig. 4 Generalised superradiance in the context of spontaneous undulator radiation theory.

Generalised superradiance in the context of spontaneous undulator radiation theory. The figure shows the spectral linewidth functions of a single electron [*F*(*ω*), in red] and of the electron bunch [\({F}^{\prime}(\omega )\), in black] as a function of the emitted frequency *ω*. (**a-b**), illustrate an example of generalised superradiance with unmatched linewidth functions. The red line shows the spectral linewidth function of a long electron bunch with duration *σ*_{t}. The spectral width of the function \({F}^{\prime}(\omega )\) is *Δ**ω* = 2/*σ*_{t}. The bunch is transversely modulated at a frequency *ω*_{m}, and the bunch spectral function is maximum at frequencies *ω* = *n**ω*_{m}. The black line represents the spectral linewidth function of a single electron bunch, *F*(*ω*). The corresponding spectral width of *F*(*ω*) is *Δ**ω* = *ω*_{n}/(*n**N*_{w}), where *N*_{w} is the number of electron oscillations during the interaction length. The single electron resonant emission frequency is *ω* = *ω*_{n}. Here, the bunch and single electron linewidth functions are not matched, *ω*_{n} ≠ *n**ω*_{m}. As a result, emission will not be in a superradiant regime. **c**, Example of regular superradiance. The red line illustrates the spectral linewidth of a short electron bunch where *σ*_{t} ≲ 1/*ω*_{n}. The bunch spectral linewidth functionthus overlaps with the single electron spectral linewidth. **d**, Illustration of generalised superradiance with matched linewidth functions. Here the bunch modulation respects *n**ω*_{m} = *ω*_{n}. Emission will be in a superradiant regime even though the bunch is long such that *σ*_{t} ≫ 1/*ω*_{n}.

## Supplementary information

### Supplementary Information

Supplementary Figs. 1–5, Discussion and Tables 1 and 2.

## Source data

### Source Data Fig. 2e

Numerical data used to generate Fig. 2e.

### Source Data Fig. 3d-f

Numerical data used to generate Fig. 3d–f.

### Source Data Fig. 4

Numerical data used to generate Fig. 4a and the yellow squares in Fig. 4b.

### Source Data Extended Data Fig. 1d

Numerical data used to generate Extended Data Fig. 1d.

### Source Data Extended Data Fig. 3c

Numerical data used to generate Extended Data Fig. 3c.

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Vieira, J., Pardal, M., Mendonça, J.T. *et al.* Generalized superradiance for producing broadband coherent radiation with transversely modulated arbitrarily diluted bunches.
*Nat. Phys.* (2020). https://doi.org/10.1038/s41567-020-0995-5

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