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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.

## Author information

Authors

### Contributions

All authors contributed to all aspects of this work.

### Corresponding author

Correspondence to J. Vieira.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended 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 φw = a0/γ 0.047, where a0 = eA/mec2 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 φVC 0.06rad is higher than φw. Color scales were normalised to the peak values in all plots. In (a) δp = 1.0mec and in (b) δp = 0.1mec, 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 δp, approaching the ideal case of a zero emittance bunch with δp = 0.

### 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 σt. The spectral width of the function $${F}^{\prime}(\omega )$$ is Δω = 2/σt. The bunch is transversely modulated at a frequency ωm, and the bunch spectral function is maximum at frequencies ω = nωm. The black line represents the spectral linewidth function of a single electron bunch, F(ω). The corresponding spectral width of F(ω) is Δω = ωn/(nNw), where Nw is the number of electron oscillations during the interaction length. The single electron resonant emission frequency is ω = ωn. Here, the bunch and single electron linewidth functions are not matched, ωnnωm. 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 σt 1/ωn. 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 nωm = ωn. Emission will be in a superradiant regime even though the bunch is long such that σt 1/ωn.

## 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