We introduce the two-point propagation field (TPPF)-a real-valued, phase-sensitive quantity defined as the functional derivative of the single-photon detection probability with respect to an infinitesimal opaque perturbation placed between the source and detection slits. The TPPF is analytically derived and shown to exhibit a stable, high-frequency sinusoidal structure with periods of 4 similar to 7 nm near the X-ray detection slit. This structure enables shot-noise-limited displacement detection with similar to 200 pm precision for 6 keV X-rays using total photon counts on the order of 1 x 10(7) and detector photon counting as low as 287. Beyond displacement detection, the TPPF physically performs a Fourier-Radon transformation of the projection data, providing a pathway to non-iterative frequency-domain tomography. Two conceptual strategies-a central blocker and off-axis multi-slit arrays-are estimated to lower the required incident photon budget by more than one order of magnitude each, yielding combined reductions of two to three orders of magnitude with near-term detector development. The TPPF concept, originally developed in a perturbative study of single-particle propagation, bridges quantum measurement questions with practical high-resolution X-ray physics. This work provides the foundational physics required for future discrete sampling and 3D numerical reconstruction algorithms.

Lasers emitting visible light based on high harmonic generation (HHG) have significantly enhanced measurement capabilities, enabling new applications across precision metrology, attosecond science, and ultrafast time-resolved spectroscopy. This paper discusses the theoretical framework of HHG with a focus on nonlinear effects, examining in depth second-harmonic generation (SHG) and third-harmonic generation (THG) mechanisms, as well as a thermal nonlinear model for pump stability analysis. The current state of HHG within integrated optical circuits is reviewed, with a particular emphasis on its implementation in high-index doped silica glass micro-ring resonators (HDSG MRRs). We conclude by addressing future directions for optimizing these systems to expand their applicability in advanced photonic technologies, highlighting their potential for innovation in both applied and fundamental sciences.

We investigate properties of optical elements in the optical cavity of an x-ray laser oscillator with emphasis on the power loss of a monochromatic Gaussian radiation beam upon passing through them using analytical and numerical approaches. Here we assume the optical cavity comprises of refractive lenses and Bragg crystals exploiting symmetric Bragg scattering. For the lens, we include focusing and curvature effects while we include angular filtering due to finite Darwin width of the Bragg crystal. Our results indicate the feasibility of x-ray power out-coupling in the range of 0.5%–28% with intracavity reflectivity of 67%–95% at 10.321 keV for nominal (≥65% reflectivity) cases. We also show the possibility of achieving high out-coupling of 35%–80% while restricting the intracavity reflectivity in the 19%–62% by adopting lossy cavities with 10–20 μm thin crystals. Published by the American Physical Society 2025

Routine beam-based characterization of the NSLS-II storage ring indicated probable seasonal variations of the ring circumference and energy. We arranged regular direct measurements of the ring circumference during 2023 and compared the results with the archived data of the magnet and undulator settings. A clear correlation between the ring circumference and energy was observed. These seasonal variations of the ring energy are found to be the main reason for regular lattice corrections required to keep the machine performance stable for user operations. The measurement techniques and results are discussed in this article.

We analyze the evolution of a particle wave function when it propagates through free space in the longitudinal z-direction from a thin entrance slit to a detector behind a thin exit slit parallel to the horizontal y-axis. We consider an extra aperture slit between the two slits to probe the evolution of the wave function and close the aperture slit starting from wide open until the detection counting rate in a repeated experiment drops to half. When all the slits are long and thin, the 1D Schroedinger equation gives the wave function evolution until the final detection. The width of the aperture slit in the vertical x-direction depends on the z-position of the slit providing an approximate description of the wave function evolution. The width of the function characterizing this dependence starts from the entrance slit. It grows wider until it reaches a maximum and then shrinks narrower and finally collapses into the exit slit where the particle is detected. Thus the envelope of this function has a spindle shape with its pointed ends at the two slits. Hence it is very different from the well-known wave function of the Schroedinger equation with the initial condition at the entrance slit, which is narrow only at the beginning, then grows wider until it reaches the exit slit, where it is much larger than the slit width. However, the phase information is lost because the aperture slit distorts the wave function. To keep the phase information, we replace the aperture slit with a thin pin (parallel to the y-axis) that blocks the wave function. We then study its perturbative effect on the counting rate of the detector. This analysis provides a function to probe the process of the wave function collapse right before the detection. We show this function is real-valued, with amplitude and phase information, and is closely related to the wave function.

We analyze the evolution of a particle wave function when it propagates through free space in the longitudinal z-direction from a thin entrance slit to a detector behind a thin exit slit parallel to the horizontal y-axis. We consider an extra aperture slit between the two slits to probe the evolution of the wave function and close the aperture slit starting from wide open until the detection counting rate in a repeated experiment drops to half. When all the slits are long and thin, the 1D Schroedinger equation gives the wave function evolution until the final detection. The width of the aperture slit in the vertical x-direction depends on the z-position of the slit providing an approximate description of the wave function evolution. The width of the function characterizing this dependence starts from the entrance slit. It grows wider until it reaches a maximum and then shrinks narrower and finally collapses into the exit slit where the particle is detected. Thus the envelope of this function has a spindle shape with its pointed ends at the two slits. Hence it is very different from the well-known wave function of the Schroedinger equation with the initial condition at the entrance slit, which is narrow only at the beginning, then grows wider until it reaches the exit slit, where it is much larger than the slit width. However, the phase information is lost because the aperture slit distorts the wave function. To keep the phase information, we replace the aperture slit with a thin pin (parallel to the y-axis) that blocks the wave function. We then study its perturbative effect on the counting rate of the detector. This analysis provides a function to probe the process of the wave function collapse right before the detection. We show this function is real-valued, with amplitude and phase information, and is closely related to the wave function.

We present a detailed derivation of a formula for the small-gain calculation for an x-ray free electron laser oscillator (XFELO) based on a medium-energy (3–4 GeV) storage ring. We found harmonic lasing and strong focusing are essential for this beam energy range. Taking the small-signal low-gain formula developed by Kim and his colleagues, we modified it in such a way that the gain can be calculated without the “no focusing approximation,” and a strong focusing can be applied, as well as harmonic lasing. In this formula, the gain is represented as a product of two factors with one of them depending only on the harmonic number, undulator period, and gap. Using this factor, we show that it is favorable to use harmonic lasing to achieve hard x-ray FEL working in the small-signal low-gain regime with the medium-energy electron beam. Our formula also allows FEL optimization by varying the vertical gradient of the undulator, the vertical dispersion, and the horizontal and vertical focusing, independently. As an example, we applied this formula to study the feasibility of an XFELO option for the National Synchrotron Light Source II (NSLS-II) upgrade. Since a quite high peak current is required for the FEL, collective effects of beam dynamics in medium-energy synchrotrons significantly affect the electron beam parameters. We carried out a multiparameter optimization taking collective effects into account. Note, even though our example is for a ring-based XFELO at 3 to 4 GeV, the formula and, in particular, the approach developed here may be applied to other types of FELs. Published by the American Physical Society 2024

To align with the global trend of integrating synchrotron light source (SLS) and free electron laser (FEL) facilities on one site, in line with examples such as SPring-8 and SACLA in Japan and ELETTRA and FERMI in Italy, we actively explore FEL options leveraging the ultralow-emittance electron beam of the NSLS-II upgrade. These options show promising potential for synergy with storage ring (SR) operations, thereby significantly enhancing our facility’s capabilities. Echo-enabled harmonic generation (EEHG) is well-suited to SR-based FELs, and has already been demonstrated with the capability of generating extremely narrow bandwidth as well as high brightness, realized using diffraction-limited short pulses in transverse planes and Fourier transform-limited bandwidth in the soft X-ray spectrum. However, regarding a conventional EEHG scheme, the combination of the shortest seed laser wavelength (256 nm) and highest harmonic (200) sets the short wavelength limit to λ = 1.28 nm. To further extend the short wavelength limit down to the tender and hard X-ray region, a vital option is to shorten the seed laser wavelength. Thanks to recent advances in high harmonic generation (HHG), packing 109 photons at one harmonic within a few-femtosecond pulse could turn such a novel HHG source into an ideal seeding for EEHG. Thus, compared to the cascaded EEHG, the HHG seeding option could not only lower the cost, but also free the SR space for accommodating more user beamlines. Moreover, to mitigate the SASE background noise on the sample and detector, we combine the HHG seeding EEHG with the crab cavity short pulse scheme for maximum benefit.

Free-electron-laser-based beamlines utilize fully coherent laser pulses with extremely narrow bandwidth allowing direct use of X-rays without monochromators. This could be very beneficial for all users of current and future fourth-generation diffraction-limited synchrotron light sources (DL-SLSs) who need narrowband full-coherence high-brightness X-ray pulses. Based on our previous finding, i.e. that separating the two stages of echo-enabled harmonic generation (EEHG) with a few extra bending-magnet sections provides an effective way to increase the momentum compaction of chicane 1, one can simultaneously achieve adequate prebunching at extremely high harmonics as well as keep the energy modulation to the ideal minimum. This could open the door for cascaded EEHG, toward fully coherent tender and hard X-ray wavelengths. Built on our compact design of a twin-pulse seeding electron beam with an adjustable delay and timing jitter at the level of a few femtoseconds, a cascaded EEHG can be implemented, which includes two EEHG beamlines, where the radiation pulse generated by the first beamline with harmonic h1 could be used as the input seed laser pulse to the second beamline with harmonic h2. Hence, the second radiator could potentially reach very high harmonics [h = h1(20)h2(25-100)] from 500 to 2000, corresponding to tender and hard X-ray wavelengths. It is demonstrated that the cascaded EEHG scheme is compatible with almost any current or planned fourth-generation DL-SLS, with significant benefits for space-limited storage rings in particular. The main advantage is that this scheme requires almost no change of the storage-ring lattice and is fully compatible with other beamlines. Current proposals for rings with much longer straight sections would add self-amplified spontaneous emission as another viable option for storage-ring-based free-electron lasers.

We studied the feasibility of a hard x-ray FEL oscillator (XFELO) based on a 3 to 4 GeV storage ring considered for the low-emittance upgrade of NSLS-II. We present a more detailed derivation of a formula for the small-gain gain calculation for 3 GeV XFELO published in the proceedings of IPAC'21 [1]. We modified the small-signal low-gain formula developed by K.J. Kim, et.al. [4{6] so that the gain can be derived without taking the \no focusing approximation" and a strong focusing can be applied. In this formula, the gain is cast in the form of a product of two factors with one of them depending only on the harmonic number, undulator period, and gap. Using this factor, we show that it is favorable to use harmonic lasing to achieve hard x-ray FEL working in the small-signal low-gain regime with the medium-energy electron beam (3-4 GeV). Our formula also allows FEL optimization by varying the vertical gradient of the undulator, the vertical dispersion, and the horizontal and vertical focusing, independently. Since a quite high peak current is required for the FEL, the collective effects of beam dynamics in medium-energy synchrotrons significantly affect the electron beam parameters. We carried out a multiple-parameter optimization taking collective effects into account and the result indicates the XFELO is feasible for storage ring energy as low as 3 GeV, with local correction of betatron coupling.

To analyze nonlinear dynamic systems, we developed a new technique based on the square matrix method. We propose this technique called the "convergence map" for generating particle stability diagrams similar to the frequency maps widely used in accelerator physics to estimate dynamic aperture. The convergence map provides similar information as the frequency map but in a much shorter computing time. The dynamic equation can be rewritten in terms of action-angle variables provided by the square matrix derived from the accelerator lattice. The convergence map is obtained by solving the exact nonlinear equation iteratively by the perturbation method using Fourier transform and studying convergence numerically. When the iteration is convergent, the solution is expressed as a quasiperiodic analytical function as a highly accurate approximation and hence the motion is stable. The border of stable motion determines the dynamical aperture. As an example, we applied the new method to the nonlinear optimization of the NSLS-II storage ring and demonstrated a dynamic aperture comparable to or larger than the nominal one obtained by particle tracking. The computation speed of the convergence map is 30 to 300 times faster than the speed of the particle tracking, depending on the size of the ring lattice (number of superperiods). The computation speed ratio is larger for complex lattices with low symmetry, such as particle colliders.

Having previously reported that separating the two stages of echo-enabled harmonic generation (EEHG) with one or more bending magnet (BM) sections allows the BMs to serve as the desired source of momentum compaction, here we demonstrate that this arrangement can greatly reduce the total energy modulation required by any 4th generation synchrotron light source, leading to higher repetition rates as well as stronger coherent radiation output power, with significant benefits. Since the EEHG beamline performance is mainly determined by the momentum compaction, beam emittances and beta functions of a storage ring lattice, allowing for different separations between the two stages is a straightforward way to increase the momentum compaction of chicane 1. This also enables pump-probe capabilities in a novel context, where twin-pulse seeding on the same electron bunch would allow two distinct radiation pulses with an adjustable delay in the range of 0.1 to 10 ps. In the twin-pulse seeding scheme, the same electron bunch could undergo modulation from two distinct laser pulses. Later stages would produce independent harmonics in subsequent straight sections. There are two variations of this twin-pulse seeding scheme, supporting different scientific applications. With a common modulation in stage 1, the first option allows simultaneously two independent radiation sources, with a full coverage of the EUV (2.5 to 50 nm) to soft X-ray (1.25 to 2.5 nm) spectrum; for the second option, the same stage 2 undulator could generate two coherent pulses both fitting within the FEL bandwidth, or at distinct harmonics. We present particle tracking simulation studies based on the APS-U lattice, including quantum excitation and radiation damping. These simulations indicate that there is no degradation of the modulated longitudinal phase space even when the two stages are separated by as many as 10 BM sections.

The goal of the NSLS-II online optimization project is to improve the beam quality for the user experiments. To increase the beam lifetime and injection efficiency, we have developed a model-independent online optimization of nonlinear beam dynamics using advanced algorithms, such as Robust Conjugate-Gradient Algorithm (RCDS). The optimization objective is the injection efficiency and optimization variables are the sextupole magnet strengths. Using the online optimization technique, we increased the NSLS-II dynamic aperture and reduced the amplitude-dependent tune shift. Recently, the sextupole optimization was successfully applied to double the injection efficiency up to above 90% for the high-chromaticity lattice being developed to improve the beam stability and to increase the single-bunch beam intensity. Minimizing the beam perturbation during injection is the second objective in this project, realized by online optimization of the injection kickers. To optimize the full set of kicker parameters, including the trigger timing, amplitude, and pulse width, we upgraded all kicker power supplies with the capability of tunable waveform width. As a result, we have reduced the injection transient by a factor of 29, down to the limit of 60 μm.

Prebunching via echo-enabled harmonic generation (EEHG) is an efficient way to reduce the radiator length and improve the longitudinal coherence as well as output stability in storage-ring-based free-electron lasers. We propose a conceptual design, which uses two straight sections to seed coherent extreme-ultraviolet (EUV) and soft X-ray emission with nearly MHz repetition rate. To take the large energy spread (10 −3 ) of a storage ring into account and utilize the existing bending magnets between the two straight sections as the first chicane, we implement a special modeling tool, named EEHG optimizer. This tool has been successfully applied to maximize the prebunching with a reasonably low energy modulation, thereby generating intense coherent X-ray pulses within a short undulator length (a few meters) limited by the available space of a storage ring. Numerical simulations confirm that the optimized EEHG parameters can be directly applied to generate a 10 MW scale peak power with fully coherent ultrafast EUV to soft X-ray pulses based on the NSLS-II parameters. This method can be easily extended to other types of diffraction-limited storage rings.

from the drift mode having wide measurement span covering energy jitter range from 10 −4 to 10 −1 . The key advantage of this method is that it allows us to extract the electron beam energy spread concurrently with the ongoing experiment and enables online optimization of the electron beam especially for future high charge single-shot ultrafast electron diffraction (UED) and ultrafast electron microscopy (UEM) experiments. Furthermore, real-time energy measurement enables the filtering process to remove off-energy shots, improving the resolution of time-resolved UED. As a result, this method can be applied to the entire UED user community, beyond the traditional electron beam diagnostics of accelerators used by accelerator physicists.

We report picosecond bunch length measurements using an interferometric method for a 3 MeV electron beam having bunch charge ranging from 1 to 14 pC. The method senses the single-cycle sub-terahertz (THz) pulse emitted by each electron bunch as coherent transition radiation which, in turn, is analyzed using a Michelson-type interferometer, forming an interferogram that is then processed to yield the nominal electron bunch length. This sub-THz coherent radiation intensity was measured using a quasi-optical detector (QOD) operated at room temperature. This experiment was quite challenging since the divergence angle of the sub-THz pulse emitted by the low-energy electron bunch exceeds ±10°, and its pulse energy at the entrance to the detector was as low as 100 pJ. When compared to a conventional helium-cooled silicon composite bolometer designed for frequencies above 0.5 THz, the QOD provided much better signal-to-noise ratio in the ∼80 GHz frequency range, which was critical for the successful measurement of the bunch length.

Having previously reported on bunching via echo-enabled harmonic generation (EEHG) as an effective way to improve the longitudinal coherence in the NSLS-II storage ring [X. Yang et al., Sci. Rep. 12, 9437 (2022)], we demonstrate that this EEHG scheme can be easily adopted to any fourth generation diffraction-limited synchrotron light source with significant benefits. The advantage of the scheme is that it requires no change of the lattice and is fully compatible with other beamlines. Since the EEHG performance is mainly determined by the momentum compaction, beam emittances, and beta functions of a SR lattice, we have identified these crucial parameters and successfully built a generalized model, which can predict the performance of nearly any SLS. Regarding the fourth generation SLSs, momentum compactions are often significantly smaller; thus, to cover the x rays with a photon energy of up to 1 keV, we utilize a specific design, including a 250 nm seed-laser wavelength. Our model predicts that for most of the current and future fourth generation SLSs, the EEHG scheme can produce significant prebunching up to harmonic 200 and, thus, generate a few MW scale peak power at 1.25 nm wavelength.

We discuss recent work on the construction of a numerical simulation of the fast orbit feedback system at NSLS-II. The simulation operates in the time domain and also includes the spatial domain, i.e. all beam position monitors (BPMs) and correctors. It can accept inputs of the real beam orbit and the measured orbit response matrix. It can also add errors to every stage of the calculation. We present methods to verify the simulation results by comparing the simulation results with experimental data collected at NSLS-II. The results are in very good agreement. The effects of errors from BPMs and correctors, as well as the amplitude of excitation, on the feedback performance, are also explored. This simulation can be used to predict the behavior and performance of the system for a future upgrade.

We develop a tool for calculation to study the conditions for a hard x-ray FEL oscillator at 3 GeV. We show that the approach developed by K.J. Kim, et.al. [1–3] for the small signal low gain formula can be modified so that the gain can be derived wihout taking the “no focusing approximation” adopted in the approach, so that a strong focusing can be applied. We also derive the formula to allow for gain calculation of harmonic lasing. We show that the relation of gain with harmonic number is not sufficient to determine whether harmonic lasing is favorable, also the relation between harmonic number, the undulator field strength, gap, period and energy spread must be analyzed together.