Single-pass Free-electron Laser Research Articles

The Influence of Increased Electron Energy Spread on the Radiation of the Second Harmonic in Free Electron Lasers

Free electron lasers (FELs) are becoming more and more popular as sources of radiation for research purposes in many fields of science. They are frequently employed in second harmonic generation (SHG) studies. SHG is an important effect of nonlinear responses of matter to irradiation. It should be separated from the second harmonic of the radiation source, FELs, for correct analysis of the response. Using an analytical model for harmonic powers evolution in a single-pass FEL, we demonstrate the dependence of FEL harmonics on the key parameters of beams and undulators and show that it is possible to reduce the second FEL harmonic content if the energy spread of an electron beam is increased. For LEUTL FEL radiation in the visible range, we analytically demonstrate a reduction in the second harmonic power by an order of magnitude if electron energy spread is increased twice, which still allows efficient bunching at the fundamental wavelength. The method is valid regardless of the photon energy and, therefore, the absorption edge of the target material, and as such, it is valid also in the X-ray band.

On the physics of generation of even and odd harmonics in the undulators of free electron lasers

We theoretically explore the physics of even and odd harmonics generation and radiation in undulators of single pass free electron lasers (FELs). Planar undulators are studied as a most common design. The results are obtained analytically and some of them also numerically. In the analytical form the contributions of various factors to the harmonic generation are distinguished and their effect is analyzed. The results are verified with numerical simulations and compared with reported experimental data; the data for best documented FELs in terms of harmonic generation, LEUTL and LCLS, is analyzed. The reasons for even harmonic generation are identified; we show that they differ for the spontaneous undulator radiation and for the stimulated FEL radiation. Comparative analysis is performed with various theoretical models for the FEL harmonic powers. The advantages and possibilities of amplification and radiation the second FEL harmonic in the LCLS-II FEL with variable deflection parameter is demonstrated for the harmonic lasing self-seed (HLSS) FEL design. The possibility to suppress the radiated second harmonic in a FEL by a guiding field of a weak second harmonic of the main undulator field with negative phase is shown. This can be used to reduce the reflected signal of the second harmonic generation (SHG) when the nonlinear harmonic response of the mater is investigated.

Analytical study of higher harmonic bunching and matrix formalism in linear high-gain free-electron laser model

Short-wavelength single-pass free electron lasers (FEL) are capable of generating transverse-coherent and high-power radiations. Compared with the self-amplified spontaneous emission (SASE) FEL scheme, seeded FEL schemes can, in principle, improve the longitudinal coherence of radiation pulses. In recent years, the research community has pursued the realization of high-repetition-rate (MHz) and high-average-power (kW) seeded FELs. To alleviate the requirements of external seeding lasers, many novel seeding schemes have been proposed, which are currently limited by the state-of-the-art laser systems. An analytical study of a recently proposed seeding scheme based on direct-amplification-enabled harmonic generation (DEHG) is presented herein. In contrast to the complete numerical simulations and optimization design, this study starts from one-dimensional FEL equations of motion and derives an analytical formula for the harmonic bunching factor at the chicane exit after the electron beam traverses the modulator undulator. Additionally, to obtain a quick estimate of downstream high-gain FEL performance, we constructed a linear high-gain FEL transport matrix, including the three-dimensional effects incorporated by the Ming-Xie formula. As an application of the analytical formulas, we discuss three different cases for designing and optimizing the seeding scheme. We expect this analytical approach to shed light on seeding designs that aim to produce the desired high-brightness mirobunched electron beam.

Theoretical analysis of the influence of electron beam parameters on the harmonic powers in free electron lasers

We theoretically analyse the influence of the electron beam parameters on the radiation in free electron lasers (FELs). We consider three FEL installations, SPARC, LEUTL and SwissFEL, where several experiments with different values of the beam emittance, energy spread and current were conducted and documented in details. Based on the available data and on our analytical approach to the FEL harmonic power computation, we distinguish the key factors, affecting the harmonic generation, and compare our theoretical results with the available experimental data. Alternative theoretical approaches were also employed for the harmonic power estimation. At the reviewed installations and experiments the variation of the emittance, energy spread and electron current occurred; we model these experiments with account for all variations, and analyze and compare the results with each other for different sets of parameters; we compare the results with the available experimental data and measurements. We demonstrate as the emittance, the energy spread and the beam current density values interplay in their influence on the radiation characteristics of a single pass FEL and we show as individual variations of these parameters can mutually compensate each others influence to some degree.

Spontaneous and Stimulated Undulator Radiation in Symmetric and Asymmetric Multi-Periodic Magnetic Fields

In this work, the radiation from electrons in multi-periodic undulator fields with symmetric and asymmetric harmonics was analyzed using generalized Bessel functions formalism. The asymmetric, symmetric, and anti-symmetric periodic magnetic fields with harmonics were studied in order to get the enhanced radiation of the high harmonics of undulator radiation (UR). The effect on the spontaneous and stimulated UR was explored. The exact integral forms for the Bessel coefficients were obtained for undulators with general symmetric and asymmetric field harmonics. Spectral properties of the radiation from several configurations of the undulator fields with harmonics were compared with each other. The resulting spontaneous UR spectrum and harmonic intensities were obtained analytically in the form of integrals and compared with the respective results that were obtained numerically with SPECTRA program. The dimensionless scaling parameter of a free electron laser (FEL)—the Pierce parameter (ρ)—was computed and compared for the different considered undulators. We studied the differences in the behavior of the high-gain single pass FEL harmonics and the spontaneous UR harmonics in the same undulators. The undulators with variable deflection parameter (k) were considered. The effect of the k parameter (deflection parameter for a common planar undulator) on the spontaneous UR and on the high-gain FEL radiation was explored. In this context, an experiment with variable strength undulators at FLASH 2 FEL was analyzed; the shorter saturated length in high harmonic self-seeding (HHSS) regime vs. self-amplified spontaneous emission (SASE) is explained.

Generation of Coherent X-ray Harmonic Radiation in a Single-Pass Free-Electron Laser with Phase Shift of Electrons Relative to Photons

Evolution of bunching and harmonic power is theoretically studied for a single-pass free-electron laser (FEL) in which electrons are periodically phase-shifted relative to photons in intervals between undulators with harmonics of magnetic field. Analytical calculations are used to reveal the configuration of magnetic field of an asymmetric elliptical and planar undulators that make it possible to suppress the fundamental and amplify several higher harmonics. The evolution of power and bunching is analyzed using a phenomenological model of FEL that describes violation of bunching between the undulator stages and a relatively high sensitivity of electron–photon interaction at wavelengths of higher harmonics to energy spread, size, deflection of the beam from the axis, etc. Expressions for the Bessel coefficients of the planar and elliptical undulators with field harmonics are analytically studied. Such undulators can be used in phase-shifted FELs for an increase in the power of higher (X-ray) harmonics by a factor of up to 100 in comparison with conventional FELs.

Undulator and free-electron laser radiation with field harmonics and off-axis effects taken into account analytically

We present an analytic description of the generation of undulator radiation (UR) harmonics in a generalized elliptic undulator with field harmonics. The obtained analytic expressions for generalized Bessel and Airy functions describe the UR spectral lines and radiation intensity in the general and particular cases of two-frequency planar and helical undulators and also for other elliptic and planar undulators. We analytically describe the effect of finite electron beam size, beam emittance, off-axis deflection of electrons, electron energy spread, and beam-bending permanent magnetic components. The obtained analytic expressions distinguish the contributions made by each field component and by the beam and undulator parameters to the generation of radiation harmonics. Using an analytic model of single-pass free-electron lasers (FELs), we analyze the power evolution of harmonics in the LCLS, LEUTL, and SACLA FEL experiments. We discuss the effect that the beam and undulator parameters have on the generation of harmonics. It is shown that the strong second harmonics of radiation observed in experiments are caused by the detected beam deviation from the axis by 15μm over one gain length of 1.5 m in the X-ray LCLS FEL and by the large beam cross section of ∼ 200;μm in the LEUTL FEL. The results of modeling are fully consistent with experiments, which confirms the validity of the presented theoretical description of FEL power and radiation spectral density.

Генерация когерентного рентгеновского излучения гармоник в однопроходном лазере на свободных электронах со сдвигом фаз электронов относительно фотонов

Theoretical study of the radiated FEL power and bunching is conducted in single-pass free electron laser (FEL) with repetitive shift of the photon pulse with respect to the electron bunch in between the sections of the undulator with field harmonics. Using the analytical calculations, the asymmetric elliptic and planar undulator fields are identified for the undulators to be able to reduce the fundamental tone and enhance the harmonic radiation. This effect is amplified by the phase shift kπ/n, k=2,4,6… of the electrons and photons between the undulator sections. This allows n-th FEL harmonic reach its peak power beyond the fundamental saturation FEL length. It is not possible to achieve in a common FEL due to fast increase of the energy spread, induced by the fundamental tone along FEL and its early saturation. The phenomenological model of the FEL is used for the analysis of the bunching coefficients and harmonic powers evolution in the FEL. The model describes the debunching between the undulator sections, where the phase shift is imposed, and accounts for the higher sensitivity to losses of the electron-photon interaction at high harmonic wavelengths. The losses are due to the beam energy spread and emittance, deviation off the axis, diffraction etc. Explicit expressions for the Bessel coefficients of the planar and elliptic undulators with field harmonics are obtained and analyzed. The combination of the undulator field harmonics is identified, which reduces the fundamental tone and enhances the 3rd and 5th UR harmonics as compared with the radiation from a planar undulator. We propose the use of these undulators in a FEL with the electron-photon phase shift, which increases high harmonic powers in X-ray band by ~100 times with respect to common FELs.

Analytical Description of Nonlinear Harmonic Generation Close to the Saturation Region in Free Electron Lasers

The phenomenological modeling of nonlinear harmonic generation in single-pass free electron laser (FEL) experiments was performed. The gradual saturation of the harmonic power and its evolution around the saturated regime was modeled. The model includes all major losses for each harmonic with account for the fact that the undulator radiation (UR) and the electron-wave interaction at higher harmonic wavelengths are more sensitive to the beam energy spread, divergence, and other factors. The results of the phenomenological modeling are compared with the experimental data of the corresponding FELs and with the numerical simulation. The proposed description of nonlinear harmonic generation appears to be in good agreement with the FEL experiments SALCA, LEUTL, LCLS, and others under various conditions. The results also match their numerical simulations; in contrast to the latter, our phenomenological model allows rapid, still, and accurate modeling of the FEL harmonic power and bunching evolution with a PC or even an engineering calculator. The model permits studying FEL radiation from undulators with arbitrary magnetic field configuration. Using the analytical FEL model, we demonstrate that, in contrast to the helical undulator, in whose spectrum the fundamental and the second harmonics prevail, the FEL radiation from the elliptical undulator with the third field harmonic has a distinct fifth UR harmonic as well as the fundamental one, whereas other harmonics can be ignored even close to the saturation length.

Effect of the third undulator field harmonic on spontaneous and stimulated undulator radiation.

The effect of undulator field harmonics on spontaneous and stimulated undulator radiation, both on and off the undulator axis, is studied. Bessel factors for the undulators with field harmonics have been analytically calculated and numerically verified. The influence of the third undulator field harmonic on single-pass free-electron laser radiation is explored. Harmonic generation at the LCLS and SPring-8 free-electron lasers is modeled and analyzed.

Comparative Analysis of Phenomenological and Numerical Modeling of Experimentswith Single-Pass Free-Electron Lasers

We model the evolution of radiation harmonics in a single-pass free-electron laser using a phenomenological model, in which the main losses are taken into account separately for each of the harmonics. The modeling results are compared with the corresponding FEL experiments and the simulations, which we performed using the PERSEO code. A wide wavelength range (0.15–500 nm) is considered. The phenomenological description based on the use of several basic FEL parameters, such as the electron beam current, average energy, energy spread, and emittance of electrons, yields the results that correspond well to the LEUTL, SPARC, and LCLS experiments in various conditions. The evolution of the radiation power, bunching, and electron energy spread in the Spring 8 FEL at the follow-up stage is also considered. The necessity to improve electron beam parameters for generation of harmonics in this FEL is shown. The phenomenological model allows assessing the operation of the available and future FELs fast. Using it, one can model FELs with nearly any undulator with allowance for higher harmonics of their magnetic fields. Moreover, this model makes it possible to describe easily various effects in designed FELs, e.g., those with dephasing of electrons and photons, filtering of harmonics, and other features.

Harmonic Generation in Planar Undulators in Single-Pass Free Electron Lasers

The paper presents a theoretical study of the harmonic power evolution in a single-pass free-electron laser (FEL) and a comparison with some of the experimental results. The phenomenological approach which considers the main FEL parameters (current density, the Lorentz factor, electron energy spread, and the beam geometry) provides a description of any undulator. With due regard to the real laser beam parameters, the effect from high-order harmonic radiation and the beam deviation from the undulator axis on the FEL irradiation is investigated. The Bessel coefficients are calculated for the undulator in the presence of the second periodic field. The phenomenological approach provides the free-electron laser simulation experiments at Sorgente Pulsata Auto-amplificata di Radiazione Coerente (SPARC), SPring-8 Angstrom Compact Free-Electron Laser (SACLA) and Linac Coherent Light Source (LCLS). These studies include the high-harmonic generation, a comparison of the emission power evolution with the experimentally obtained data on the harmonic generation including even and odd high-order harmonics, gain and saturation lengths and the respective power. A possible effect from the third harmonic of the planar undulator magnetic field is studied in relation to the FEL irradiation. It is shown that this effect is insignificant even at the magnetic-field harmonic amplitude equaling ~1/10 of the amplitude of the ideal magnetic field in the periodic undulator. It is found that the second-harmonic generation in FEL lasing is rather weak, that is in good agreement with the measurement results as well as the behavior of the high-order harmonics.

Slippage boosted spectral cleaning in a seeded free-electron laser

The realization of fully coherent light sources at extreme ultraviolet to x-ray region has been a long-standing challenge for laser technologies. While modern single pass free-electron lasers (FELs) hold the ability to produce very intense x-ray radiation on few-femtosecond timescale, the output radiation pulses usually have noisy spectra and limited temporal coherence since the amplification starts from electron noise. A promising way for producing stable transform-limited pulses is based on the harmonic up-conversion techniques with a conventional laser as the seed. However, it is found that the insignificant phase error in the seed laser may be eventually multiplied by the harmonic number, leading to a degradation of the output temporal coherence at x-ray wavelength. Here, we report for the first time on the demonstration of a slippage boosted spectral cleaning technique to mitigate the impact of seed laser induced phase errors and to significantly improve the temporal coherence of a seeded FEL with large phase errors in the seed laser. Experimental results indicate the possibility of generating fully coherent x-ray radiation pulses with this technique.

Generation of Soft X-Ray Radiation in a Compact Free-Electron Laser with Harmonic Multiplication

Single-pass free-electron lasers (FELs) with harmonic multiplication in the X-ray range are studied theoretically with the objective of producing intense radiation with the minimum possible FEL length, beam energy, and seed laser frequency. A cascade FEL with a two-frequency undulator for electron bunching at wavelengths of higher harmonics is proposed. The dynamics of radiation power in a FEL with harmonic multiplication cascades is investigated based on a well-proven phenomenological FEL model that covers all major losses for each harmonic in each FEL cascade. The effect of the beam quality on X-ray generation in a FEL is examined. It is demonstrated that electron beams with energy spread σe ≤ 0.0002 are required for efficient use of harmonics in a cascade X-ray FEL. The evolution of power of harmonics in several cascade FELs at wavelengths of ~2–3 nm with multiplication of seed harmonics at a wavelength of 13.51 nm (Mo/Si reflection peak) and 11.43 nm (MoRu/Be reflection peak) is studied. It is demonstrated that the power increases to ~1 GW at a FEL length up to 40 m. In addition, the operation of multicascade FELs with a seed F2 excimer 157-nm UV laser and harmonics of lasers with CO2 and C2H2 at 30 nm with a beam of electrons with an energy of ~0.6 GeV and a current of ~1 kА is studied. The power of the modeled X-ray radiation of these FELs at λ = 2.5 and 3.3 nm increases to ~0.5 GW at ~30 m.

Phenomenological and numerical analysis of power evolution and bunching in single-pass X-ray FELs.

The harmonic power and bunching evolution in X-ray single-pass free-electron lasers (FELs) is modelled and the harmonic generation in a phase-shifted two-frequency FEL is explored. The advanced phenomenological FEL model, which is validated numerically and experimentally, is employed. The model accounts for major losses for each harmonic individually; it is compared with reported experimental data and with PERSEO numerical simulations, which are performed here for a variety of experiments. The latter cover the radiation wavelength range 0.15-300 nm. The phenomenological description is based on a few key FEL parameters: electron beam section, current, energy and its spread and divergence. The model is employed for modelling harmonic bunching and power evolution in a phase-shifted X-ray FEL with a two-frequency undulator, where lower harmonics with numbers less than nth are suppressed by the electron-photon phase shift of kπ/n, k = 2, 4, …, between the undulator sections. The benefits of the two-frequency phase-shifted FEL are highlighted. FEL-induced energy spread is shown to be three times lower than in a FEL without the phase-shift. The high-power harmonic and sub-harmonic radiation in such a FEL is demonstrated. In particular, powerful ∼14 GW X-ray radiation at λ5 = 0.15 nm from electrons with energy of 5.47 GeV and beam current ∼3.66 kA is possible in a two-frequency phase-shifted FEL at 30 m; this constitutes half of a FEL length where a common planar undulator radiates the same wavelength and power at the fundamental harmonic. Moreover, about a three times lower energy spread is induced by the dominant fifth harmonic, and the harmonic power can be thousands of times higher than in a common planar undulator FEL.

Two-frequency undulator in a short SASE FEL for angstrom wavelengths

We perform a theoretical study and modeling of harmonic evolution in an x-ray single-pass free-electron laser (FEL) with a two-frequency undulator and phase shifters. For this purpose a phenomenological FEL model is developed and validated; it accounts for all losses individually for each harmonic. The model was tested with FEL experiments, covering radiation wavelengths ranging from 0.15–500 nm. An example of the LCLS experiment modeling with 1.5 angstrom radiation is presented. The phenomenological description is based on a few main FEL parameters: electron beam current, energy and its spread, and emittance. The results match well with measurements and numerical simulations in various conditions. We model the harmonic bunching and power evolution in an x-ray FEL with a two-frequency undulator, in which the first and third harmonics are suppressed between the undulator sections by phase shifts of kπ/5, k = 2, 4,… of the electrons with respect to the photons. The advantages of the two-frequency undulator in such phase-shifted FELs are highlighted. The possibility to produce intense x-ray radiation by the dominant fifth harmonic in the linear regime is demonstrated. The radiation at subnanometric wavelength λ5 = 0.15 nm from electrons with E = 5.47 GeV energy and I ∼ 3.66 kA current already reaches ∼14 GW power at 30 m. This is shorter than the saturation length of the common undulator FEL with its fundamental harmonic at the same wavelength. The power boost and the reduction of the saturation length are complemented by 2.5 times lower electron energy and 3 times lower FEL-induced energy spread. The harmonic power increase can be up to 104 times greater than that in a common planar undulator FEL.

Two-Frequency Undulators for Generation of X-Ray Radiation in Free-Electron Lasers

We present a theoretical study and a computer simulation of characteristics of the undulator radiation in single-pass free-electron lasers (FELs). Using a phenomenological model describing the dynamics of the radiated power in FELs with allowance for the basic loss, we study generation of harmonics in the X-ray range in a FEL with a two-frequency undulator. We study the possibility to achieve a hundredfold increase in the radiation intensity of the nth harmonic in a FEL, in which the electron-phase shift by kπ/n with respect to photons occurs between undulator sections, where k = 2, 4, . . . . The advantages of using a two-frequency undulator in a single-pass FEL and the possibility of generating the high-power X-ray radiation by the FEL at the harmonic wavelengths 2.3–3.3 nm in the linear regime are demonstrated. The FEL is compared with the two-frequency undulator and the conventional plane undulator. Additionally, generation of radiation having a power of tens of megawatts is studied at the wavelength λ ≈ 3.27 nm in a multistage FEL with a length of 40 m, an off-the-shelf excimer ultraviolet seed laser, which operates at a wavelength of 157 nm, and an electron beam having an energy of about 0.6 GeV.

Single-pass high-gain tapered free-electron laser with transverse diffraction in the postsaturation regime

It has been well known that the resonant interaction of an ultrarelativistic electron beam and the radiation field in the single-pass high-gain free electron laser (FEL) amplifier leads to the optical gain guiding. The transverse Laplacian term of the slowly varying wave equation in the linear regime can be approximated as a constant detuning parameter, i.e., |∇⊥2|∼kR/zR where kR is the resonant wave number and zR is the Rayleigh range of the laser. In the post-saturation regime, the radiation power begins to oscillate about an equilibrium for the untapered case while continues to grow by undulator tapering. Moreover, in this regime the gain guiding decreases and the simple constant detune is no longer valid. In this paper we study the single-pass high-gain FEL performance in the post-saturation regime with inclusion of diffraction effect and undulator tapering. Our analysis relies upon two constants of motion, one from the energy conservation and the other from the adiabatic invariant of the action variable. By constructing a two-dimensional axisymmetric wave equation and the coupled one-dimensional electron dynamical equations, the performance of a tapered FEL in the postsaturation regime can be analyzed, including the fundamental mode profile, the power efficiency and the scaled energy spread. We begin the analytical investigation with two different axisymmetric electron beam profiles, the uniform and bounded parabolic ones. It is found that the tapered FEL power efficiency can be smaller but close to the taper ratio provided the resonant phase remains constant and the beam-wave is properly matched. Such a tapered efficiency is nearly independent of transverse electron beam size before significant electron detrapping occurs. This is essentially different from the untapered case, where the power extraction efficiency is around the essential FEL gain bandwidth (or ρ, the Pierce or FEL parameter) and depends on the beam size. It is also found that the power enhancement due to undulator tapering is attributed more by the field increase outside the transverse electron beam than that inside the transverse electron beam. Several scaling properties on the taper ratio and the transverse electron beam size are also discussed in this paper.3 MoreReceived 29 January 2018DOI:https://doi.org/10.1103/PhysRevAccelBeams.21.060702Published 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 SocietyPhysics Subject Headings (PhySH)Physical SystemsSynchrotron radiation & free-electron lasersAccelerators & Beams

Multicascade X-Ray Free-Electron Laser with Harmonic Multiplier and Two-Frequency Undulator

The feasibility of generation of powerful x-ray radiation by a cascade free-electron laser (FEL) with amplification of higher harmonics using a two-frequency undulator is studied. To analyze the FEL operation, a complex phenomenological single-pass FEL model is developed and used. It describes linear and nonlinear generation of harmonics in the FEL with seed laser that takes into account initial electron beam noise and describes all main losses of each harmonic in each FEL cascade. The model is also calibrated against and approved by the experimental FEL data and available results of three-dimensional numerical simulation. The electron beam in the undulator is assumed to be matched and focused, and the dynamics of power in the singlepass FEL with cascade harmonic multipliers is investigated to obtain x-ray laser radiation in the FEL having the shortest length, beam energy, and frequency of the seed laser as low as possible. In this context, the advantages of the two-frequency undulator used for generation of harmonics are demonstrated. The evolution of harmonics in a multicascade FEL with multiplication of harmonics is investigated. The operation of the cascade FEL at the wavelength λ = 1.14 nm, generating ~30 MW already on 38 m with the seed laser operating at a wavelength of 11.43 nm corresponding to the maximal reflectivity of the multilayered mirror MoRu/Be coating is investigated. In addition, the operation of the multicascade FEL with accessible seed UVlaser operating at a wavelength of 157 nm (F2 excimer UV-laser) and electron beam with energy of ~0.5 GeV is investigated. X-ray radiation simulated in it at the wavelength λ ~ 3.9 nm reaches power of ~50 MW already at ~27 m, which is by two orders of magnitude shorter than 3.4 km of the x-ray FEL recently put into operation in Europe.

Sub-picosecond electron bunch length measurement using coherent transition radiation at SXFEL

Longitudinal electron bunch length plays a significant role in single-pass free-electron lasers (FEL), as the high-gain FEL process depends strongly on the high peak current of electron bunches. Longitudinal electron bunch length was measured by detecting the interferogram of coherent transition radiation generated by electron bunches using a THz interferometer and a Golay cell (spectral range 0.02–20 THz) at Shanghai X-ray free-electron laser. The detailed process of measurement and data analysis are discussed herein. Furthermore, the electron bunch length was estimated based on the dispersive strength $$ R_{56} $$ of the bunch compressor and the energy spread $$ \delta $$ of electron bunches, which were obtained via experiments. The comparison showed that the measured bunch length was consistent with the estimated bunch length.