Velocity Dispersion Analysis Research Articles

Multispacecraft Observations of Protons and Helium Nuclei in Some Solar Energetic Particle Events toward the Maximum of Cycle 25

The intricate behavior of particle acceleration and transport mechanisms complicates the overall efforts in formulating a comprehensive understanding of solar energetic particle (SEP) events; these efforts include observations of low-energy particles (from tens of keV to hundreds of MeV) by space-borne instruments and measurements by the ground-based neutron monitors of the secondary particles generated in the Earth atmosphere by SEPs in the GeV range. Numerous space-borne missions provided good data on the nature/characteristics of these solar particles in past solar cycles, but more recently—concurrently with the rise toward the maximum of solar cycle 25—the High-Energy Particle Detector (HEPD-01) proved to be well suited for the study of solar physics and space weather. Its nominal 30–300 MeV energy range for protons can enlarge the detection capabilities of solar particles at low Earth orbit, closer to the injection limit of many SEP events. In this work, we characterize three SEP events within the first six months of 2022 through spectral and velocity dispersion analysis, assessing the response of HEPD-01 to >M1 events.

Dynamic acceleration of energetic protons by an interplanetary collisionless shock

Context. Interplanetary collisionless shocks are known to be capable of accelerating charged particles up to hundreds of MeV. However, the underlying acceleration mechanisms are still under debate. Aims. We present the dynamic behaviors of energetic protons that are accelerated by an interplanetary shock that was observed with unprecedented high-resolution measurements by the Electron-Proton Telescope sensor of the Energetic Particle Detector suite on board the Solar Orbiter spacecraft on 2021 November 3. We constrain the potential acceleration mechanisms and processes. Methods. We first reconstructed the proton pitch-angle distributions (PADs) in the solar wind frame. Then, we examined the temporal flux profile, PAD, and the velocity distribution function of energetic protons close to the shock, and we qualitatively compared the observations with theoretical predictions. Moreover, we applied a velocity dispersion analysis (VDA) to an observed velocity dispersion event and derived the proton path length and release time at the shock. Then, we tested this derivation by comparing it with the shock motion and the magnetic field configuration. Results. We find that ∼1000–4000 keV protons exhibit a rapid-rise, rapid-decay temporal flux profile with a clear velocity dispersion ∼2 min before the shock, similar to impulsive solar energetic particle events. The proton path length based on the VDA of this event is consistent with the length derived from the shock motion and magnetic field configuration. The peak spectrum in this event appears to be steeper than the spectrum at the shock. Furthermore, we find that ∼50–200 keV proton fluxes peak between ∼10 and ∼20 s before the shock, with an inverse velocity dispersion. The velocity dispersion event and the inverse velocity dispersion event are both accompanied by magnetic kinks or switchbacks. In addition, two distinct proton populations appear near the shock. The first population at energies below ∼300 keV is characterized by a power-law spectrum with an index of ∼6–7 and a flux profile that increases before and decreases after the shock. The other population at energies above ∼300 keV shows a long-lasting, anti-sunward-beamed PAD across the shock and a flux profile that remains relatively constant before and increases slightly after the shock. Conclusions. These results suggest that the shock acceleration of energetic protons is highly dynamic due to temporal and/or spatial variations at the shock front. The observation of the velocity dispersion event further suggests that shock acceleration can be impulsive and efficient, which may be due to the interaction between the shock and magnetic kinks or switchbacks. Moreover, these results may support shock-drift acceleration and diffusive shock acceleration as candidate acceleration mechanisms at interplanetary shocks.

Relativistic electron beams accelerated by an interplanetary shock

Context. Collisionless shock waves have long been considered to be among the most prolific particle accelerators in the universe. Shocks alter the plasma they propagate through, and often exhibit complex evolution across multiple scales. Interplanetary (IP) traveling shocks have been recorded in situ for over half a century and act as a natural laboratory for experimentally verifying various aspects of large-scale collisionless shocks. A fundamentally interesting problem in both heliophysics and astrophysics is the acceleration of electrons to relativistic energies (> 300 keV) by traveling shocks. Aims. The reason for an incomplete understanding of electron acceleration at IP shocks is due to scale-related challenges and a lack of instrumental capabilities. This Letter presents the first observations of field-aligned beams of relativistic electrons upstream of an IP shock, observed thanks to the instrumental capabilities of Solar Orbiter. This study presents the characteristics of the electron beams close to the source and contributes to the understanding of their acceleration mechanism. Methods. On 25 July 2022, Solar Orbiter encountered an IP shock at 0.98 AU. The shock was associated with an energetic storm particle event, which also featured upstream field-aligned relativistic electron beams observed 14 min prior to the actual shock crossing. The distance of the beam’s origin was investigated using a velocity dispersion analysis (VDA). Peak-intensity energy spectra were anaylzed and compared with those obtained from a semi-analytical fast-Fermi acceleration model. Results. By leveraging Solar Orbiter’s high temporal resolution Energetic Particle Detector (EPD), we successfully showcase an IP shock’s ability to accelerate relativistic electron beams. Our proposed acceleration mechanism offers an explanation for the observed electron beam and its characteristics, while we also explore the potential contributions of more complex mechanisms.

Directional sources and distributed acoustic sensing deployments for near-surface characterization

Active near-surface surveys using distributed acoustic sensing (DAS) have been practically limited to 2D inline acquisitions followed by Rayleigh-wave dispersion analysis. In this study, we analyze different acquisition setups and the subsurface properties that can be estimated from them. First, we find a surface survey dedicated to Love-wave acquisition. We use a horizontal source and common-receiver sorting of the DAS data for a pure Love-wave velocity dispersion analysis yielding an S-wave velocity profile. Common-receiver sorting has the added benefit of practically eliminating the effect of the gauge length. Next, downhole DAS is used in a vertical seismic profile-type survey to recover a 1D P-wave velocity through checkshot analysis. Finally, we find that the horizontal portion of a fiber deployed in a deviated borehole records high-frequency data from surface sources. We use first arrivals for diving-wave tomography, yielding a 2D P-wave velocity model above and below the fiber. We validate the inversion results of all acquisitions through a comparison with geophone-derived velocity models. We conclude that source orientation plays a crucial role and should be considered during survey planning in conjunction with DAS directivity. As anticipated, fiber coupling can strongly influence the signal-to-noise ratio and needs to be adequately planned. In vertical boreholes, infilling offers a viable solution for deployment in existing wells. For deviated boreholes, installation inside the casing yields suboptimal coupling, but more adequate deployment protocols may yield high-frequency data useful for velocity model building and imaging. From a practical point of view, the acquisitions that we describe are difficult to justify due to their operational costs and limitations. Nonetheless, they highlight the potential of DAS in going beyond traditional analysis.

Unusually long path length for a nearly scatter-free solar particle event observed by Solar Orbiter at 0.43 au

Context. After their acceleration and release at the Sun, solar energetic particles (SEPs) are injected into the interplanetary medium and are bound to the interplanetary magnetic field (IMF) by the Lorentz force. The expansion of the IMF close to the Sun focuses the particle pitch-angle distribution, and scattering counteracts this focusing. Solar Orbiter observed an unusual solar particle event on 9 April 2022 when it was at 0.43 astronomical units (au) from the Sun. Aims. We show that the inferred IMF along which the SEPs traveled was about three times longer than the nominal length of the Parker spiral and provide an explanation for this apparently long path. Methods. We used velocity dispersion analysis (VDA) information to infer the spiral length along which the electrons and ions traveled and infer their solar release times and arrival direction. Results. The path length inferred from VDA is approximately three times longer than the nominal Parker spiral. Nevertheless, the pitch-angle distribution of the particles of this event is highly anisotropic, and the electrons and ions appear to be streaming along the same IMF structures. The angular width of the streaming population is estimated to be approximately 30 degrees. The highly anisotropic ion beam was observed for more than 12 h. This may be due to the low level of fluctuations in the IMF, which in turn is very probably due to this event being inside an interplanetary coronal mass ejection The slow and small rotation in the IMF suggests a flux-rope structure. Small flux dropouts are associated with very small changes in pitch angle, which may be explained by different flux tubes connecting to different locations in the flare region. Conclusions. The unusually long path length along which the electrons and ions have propagated virtually scatter-free together with the short-term flux dropouts offer excellent opportunities to study the transport of SEPs within interplanetary structures. The 9 April 2022 solar particle event offers an especially rich number of unique observations that can be used to limit SEP transport models.

Release Episodes of Electrons and Protons in Solar Energetic Particle Events

We analyzed a sample of 21 solar energetic particle (SEP) events with clear signatures in both near-relativistic electrons and high-energy protons spanning over ∼2.5 solar cycles from 1997 to 2016. We employed velocity dispersion analysis (VDA) for protons and fractional VDA (FVDA) for electrons, as well as time shifting analysis (TSA) in order to identify the solar release times (SRTs) of the electrons. We found that, for the majority of the events (62%), a simultaneous release was observed, while, for 14% of the events, electrons were released later than protons (i.e., delayed electrons); for 24% of the events, the opposite result was found (i.e., delayed protons). We found that the path length (L) traveled by the protons and electrons was not related to the aforementioned categorization. Moreover, we show that, in the case of simultaneous SEP events, protons and electrons are being released in close connection to type III and type II bursts, while the opposite is the case for delayed events. In addition, we demonstrate that, for the simultaneous events, both the proton and the electron release are established in heights < 5RS and that, especially for the well-connected simultaneous events, there is a co-occurrence of the type II burst with the release time of the particles.

Statistical Study of Release Time and Its Energy Dependence of In Situ Energetic Electrons in Impulsive Solar Flares

AbstractUsing the fraction velocity dispersion analysis method, it has been shown recently that in two impulsive solar energetic electron (SEE) events, the release times of near‐relativistic electrons at the Sun for outward‐propagating electrons are energy dependent and are delayed compared to those of the downward‐propagating electrons. In this work, we perform a statistical study of the release time and its energy dependence of near‐relativistic electrons in impulsive SEE events. We use in situ observations from the WIND spacecraft and remote hard X‐ray observations from the RHESSI and/or Fermi spacecraft. The difference in the release times between outward electrons and downward electrons for 29 events is obtained. In all events, the release of the outward‐propagating electrons is delayed from those precipitating downward. In 26 of the 29 events, the release times of outward‐propagating electrons also show clear energy dependence. In 15 of these 26 events, in situ electron data from more than five energy channels were available. The delay time as a function of energy for nine of these can be fitted by a form proposed by G. Li et al. (2021, https://doi.org/10.1029/2021GL095138). The implication of this energy‐dependent release on the Magnetohydrodynamics turbulence property at the electron acceleration site is discussed.

An analysis for unipole acoustic logging while drilling wavefields with varying eccentered collar positions

Abstract The unipole logging while drilling (LWD) tool is a recently developed acoustic logging tool that uses an azimuthal emitting source to generate acoustic waves in the borehole to measure the longitudinal and transverse wave speeds of the surrounding stratum. In this study, the unipole transmitter transducer is modeled as a surface force source, and the analytical solution for unipole LWD wavefield is derived. The received waveforms are composed of infinite-order contributions owing to the directivity of the excitation of the unipole source. Using up to eight lowest-order modes, the acoustic waveforms excited by the unipole are calculated in both soft and hard formations, and then slowness-time coherence and velocity dispersion analysis are performed. The pure mode waves are obtained using the mode wave decomposition method. The influence of tool eccentricity (off-center positioning) on the wavefields excited by the unipole transmitter is also discussed. Based on the addition theorem of the Bessel function, the analytical solution of the received waveforms when the tool is eccentered can be obtained. In soft and hard formations, the received waveforms with tool eccentricities of 0 to 90% are compared. Tool eccentricity has a smaller effect for the soft formation than the hard formation. The received waveforms on the side closest to the borehole wall are also affected most when the tool is off center, with a sharp increase in amplitude in the fast formation and a late arrival wave group in the soft formation. These conclusions are important for the design and data interpretation of a unipole LWD tool.

Measuring the Depth of Subsurface Defects in Additive Manufacturing Components by Laser-Generated Ultrasound

A new method to measure the depth of subsurface defects in additive manufacturing components is proposed based on the velocity dispersion analysis of Lamb waves by the wavelet-transform of laser ultrasound. Firstly, the mode-conversion from laser-generated surface waves to Lamb waves caused by subsurface defects at different depths is studied systematically. Secondly, an additive manufactured 316L stainless steel sample with six subsurface defects has been fabricated to validate the efficiency of the proposed method. The measured result of the defect depth is very close to the real designed value, with a fitting coefficient of 0.98. The defect depth range for high accuracy measurement is suggested to be lower than 0.8 mm, which is enough to meet the inspection of layer thickness during additive manufacturing. The result indicates that the proposed method based on laser-generated ultrasound (LGU) velocity dispersion analysis is robust and reliable for defect depth measurement and meaningful to improve the processing quality and processing efficiency of additive/subtractive hybrid manufacturing.

A study of the characteristic properties of SEP events observed by SOHO ERNE during solar cycle 24

Properties and association of Solar Energetic Particle (SEP) events with their solar sources during the 24th solar cycle were examined. SEPs observed by SOHO ERNE were used. SEP properties include solar release time, peak proton flux and total fluence in different energy channels. Solar release times of SEP events were found out using Velocity Dispersion Analysis (VDA). It is observed that for low Coronal Mass Ejection (CME) velocities, the difference between SEP and CME release times is larger compared to that for high velocities. For SEPs associated with C class Solar Flares (SF), the average time difference in the release times is around 78 min, whereas in M and X class flares the time differences are 38 and 31 min respectively. Correlation analysis of proton peak flux with the CME speed and SF peak flux reveals that the coefficients decrease with increase in particle energy in both the cases. The correlation coefficient between total proton fluence of SEP and the peak flux of SF is moderate and decrease with increase in energy and tends to increase beyond 60 MeV. The correlation between total proton fluence and CME speed is high and it increases initially and then decrease with increase in energy. The analysis of H/He ratio of SEP events shows that the average value of the ratio extends from 16 to 837.

Constraints on the Electron Acceleration Process in Solar Flare: A Case Study

AbstractCombining in situ measurements of energetic electrons and remote sensing observations of hard X‐rays and type III radio bursts, we examine the release times of energetic electrons in the July 23, 2016 event. We find that the releases of in situ energetic electrons from the Sun are delayed from those electrons that are responsible for the hard X‐rays. We further find that the release time of in situ electrons is a function of electron energy. Under the assumption that the acceleration mechanism for the upward propagating electrons is of Fermi‐type and is controlled by an energy‐dependent diffusion coefficient, we fit these release times by a simple functional form, related to the turbulence spectral index. Implications of our study on the underlying electron acceleration mechanisms and the magnetic reconnection process in solar flares are discussed. Our results demonstrate the power of the recently developed fractional velocity dispersion analysis (FVDA) method in solar flare studies.

CWT-Based Method for Extracting Seismic Velocity Dispersion

Time–frequency (T–F) analysis has powerful applications in various fields such as geoscience and engineering. Seismograms are distorted by dispersion and attenuation in such a way that they alter their amplitude and phase spectra. Velocity dispersion is usually neglected in the conventional seismic data processing. However, it has a severe effect on seismic data processing if dispersion is intense in high-attenuation media. Most of the studies on dispersion are theoretical models and laboratory measurements, whereas velocity dispersion analysis directly from field data is rare. In this letter, a method is proposed to estimate the dispersion of vertical seismic profile (VSP) data based on continuous wavelet transform (CWT) in time–frequency domain. The Morlet wavelet, three-parameter wavelet (TPW), and Cauchy wavelet are chosen and compared as mother wavelets in CWT to determine the variations of velocity with frequency. Then, the detailed process of extracting velocity dispersion of propagating waves in a dissipative medium is proposed by using their T–F spectra. Finally, the synthetic and field VSP data are used to demonstrate the validity of the proposed method. The results indicate that the velocity dispersion in synthetic data estimated by TPW and Cauchy wavelet generally match well with the theoretical values while it is not the case for the Morlet wavelet. The velocity dispersion of field data is weak and the estimated velocity basically matches well with the logging data.

Characterizing blood clots using acoustic radiation force optical coherence elastography and ultrasound shear wave elastography

Thromboembolism in a cerebral blood vessel is associated with high morbidity and mortality. Mechanical thrombectomy (MT) is one of the emergenc proceduresperformed to remove emboli. However, the interventional approaches such as aspiration catheters or stent retriever are empirically selected. An inappropriate selection of surgical devices can influence the success rate during embolectomy, which can lead to an increase in brain damage. There has been growing interest in the study of clot composition and using a priori knowledge of clot composition to provide guidance for an appropriate treatment strategy for interventional physicians. Developing imaging tools which can allow interventionalists to understand clot composition could affect management and device strategy. In this study, we investigated how clots of different compositions can be characterized by using acoustic radiation force optical coherence elastography (ARF–OCE) and compared with ultrasound shear wave elastography (SWE). Five different clots compositions using human blood were fabricated into cylindrical forms from fibrin-rich (21% red blood cells, RBCs) to RBC-rich (95% RBCs). Using the ARF–OCE and SWE, we characterized the wave velocities measured in the time-domain. In addition, the semi-analytical finite element model was used to explore the relationship between the phase velocities with various frequency ranges and diameters of the clots. The study demonstrated that the wave group velocities generally decrease as RBC content increases in ARF–OCE and SWE. The correlation of the group velocities from the OCE and SWE methods represented a good agreement as RBC composition is larger than 39%. Using the phase velocity dispersion analysis applied to ARF–OCE data, we estimated the shear wave velocities decoupling the effects of the geometry and material properties of the clots. The study demonstrated that the composition of the clots can be characterized by elastographic methods using ARF–OCE and SWE, and OCE demonstrated better ability to discriminate between clots of different RBC compositions, compared to the ultrasound-based approach, especially in clots with low RBC compositions.

Observations of Outward-propagating and Mirroring of the Same Energetic Electrons by Wind

Abstract We report an energetic electron event observed by the Wind spacecraft in which electrons were first detected to propagate away from the Sun, and subsequently detected to propagate back toward the Sun. Using the recently developed fractional velocity dispersion analysis (FVDA), the path lengths corresponding to these two sequential electron signals were found to be ∼1.5 and ∼3.1 au. The inferred release times for both populations were found to be the same, within uncertainty. No bidirectional halo electrons were identified during the event, indicating that the electrons were not propagating within a closed magnetic field configuration (e.g., a magnetic flux rope). These observations suggest that the same energetic electrons were observed twice at 1 au, with the second time being due to reflection occurring beyond 1 au, by perhaps magnetic kinks caused by a coronal mass ejection–driven shock.

First Solar Energetic Particles Measured on the Lunar Far-side

Abstract On 2019 May 6 the Lunar Lander Neutron & Dosimetry (LND) Experiment on board the Chang’E-4 lander on the far-side of the Moon detected its first solar energetic particle (SEP) event with proton energies up to 21 MeV. Combined proton energy spectra are studied based on the LND, SOHO/EPHIN, and ACE/EPAM measurements, which show that LND could provide a complementary data set from a special location on the Moon, contributing to our existing observations and understanding of space environment. We applied velocity dispersion analysis to the impulsive electron event and weak proton enhancement and show that electrons are released only 22 minutes after the flare onset and ∼15 minutes after the type II radio burst, while protons are released more than one hour after the electron release. The beam-like in situ electrons and clear velocity dispersion indicate a good magnetic connection between the source and Earth. This is remarkable because stereoscopic remote-sensing observations from Earth and STEREO-A suggest that the SEPs are associated with an active region nearly 113° away from the magnetic footpoint of Earth. This suggests that these SEPs did not propagate along the nominal Parker spiral normally assumed for ballistic mapping and that the release and propagation mechanism of electrons and protons are likely to differ significantly for this event.

The Role of Successive and Interacting CMEs in the Acceleration and Release of Solar Energetic Particles: Multi-viewpoint Observations

Abstract Gradual and large solar energetic particle (SEP) events (flux of ions with energy >10 MeV above 10 pfu) are primarily produced in shocks driven by fast and wide coronal mass ejections (CMEs). Past research, both in theory and statistics, has found that the situation where a fast primary CME (priCME) is preceded by previous CMEs (preCMEs) is favorable to a more efficient particle acceleration. However, the physical causes of this association is still a matter of debate, including the association of the acceleration and release of SEPs with the interaction of successive CMEs. Taking advantage of the twin Solar TErrestrial RElations Observatory spacecraft, we study 41 large SEP events in solar cycle 24 by multi-viewpoint observations. Although 21 events (∼51%) have a preCME identifiable in the Large Angle and Spectrometric Coronagraph, we determine that the priCMEs overlap the preCMEs in three dimension (3D) for only 11 events (∼27%). We further investigate the acceleration (using type II radio bursts) and release (using velocity dispersion analysis) of the particles for all potential instances of CME–CME interaction in 3D. We find that, for six of 11 events, the priCME is far away from catching up with the preCME when the particles are released. However, for the limited samples, the SEP peak intensity is significantly higher in the events in which the priCME is closest to impacting the preCME, indicating the potential for the increased seed population or more enhanced turbulence levels occurring closer to the preCME.

Identification of Two Distinct Electron Populations in an Impulsive Solar Energetic Electron Event

Abstract We examine the release times of energetic electrons in the 2001 April 25 event. An M2.7 flare occurred on 2001 April 25, from AR 09433, located at N18W09. The flare was observed in X-rays by GOES and the Yohkoh spacecraft. The Yohkoh observation also included hard X-ray (HXR) images for all four energy channels: L, M1, M2, and H. We use Yohkoh observation times as a proxy for the release time of energetic electrons that propagated downward, which are responsible for the HXRs. In situ >∼25 keV electrons were observed by the Wind spacecraft. For these electrons, we obtain the release time at the Sun using the recently developed Fractional Velocity Dispersion Analysis method. We find that the release times of outward-propagating energetic electrons are clearly delayed from those that propagated downward. Furthermore, these delayed releases are energy dependent. The implication of this delay on the underlying acceleration and trapping process of the >∼25 keV electrons at solar flares is discussed.

Properties of High-Energy Solar Particle Events Associated with Solar Radio Emissions

We have analysed 58 high-energy proton events and 36 temporally related near-relativistic electron events from the years 1997 – 2015 for which the velocity dispersion analysis of the first-arriving particles gave the apparent path lengths between 1 and 3 AU. We investigated the dependence of the characteristics of the proton events on the associations of type II, III, and IV radio bursts. We also examined the properties of the soft X-ray flares and coronal mass ejections associated with these events. All proton events were associated with decametric type III radio bursts, while type IV emission was observed only in the meter wavelengths in some of the events (32/58). Almost all proton events (56/58) were associated with radio type II bursts: 11 with metric (m) type II only, 11 with decametric–hectometric (DH) only, and 34 with type II radio bursts at both wavelength ranges. By examining several characteristics of the proton events, we discovered that the proton events can be divided into two categories. The characteristics of events belonging to the same category were similar, while they significantly differed between events in different categories. The distinctive factors between the categories were the wavelength range of the associated type II radio emission and the temporal relation of the proton release with respect to the type II onset. In Category 1 are the events which were associated with only metric type II emission or both m and DH type II and the release time of protons was before the DH type II onset (18/56 events). Category 2 consists of the events which were associated with only DH type II emission or both m and DH type II and the protons were released at or after the DH type II onset (31/56 events). For seven of the 56 events we were not able to determine a definite category due to timing uncertainties. The events in Category 1 had significantly higher intensity rise rates, shorter rise times, lower release heights, and harder energy spectra than Category 2 events. Category 1 events also originated from magnetically well-connected regions and had only small time differences between the proton release times and the type III onsets. The soft X-ray flares for these events had significantly shorter rise times and durations than for Category 2 events. We found 36 electron events temporally related to the proton events, which fulfilled the same path length criterion as the proton events. We compared the release times of protons and electrons at the Sun, and discovered that in 19 of the 36 events protons were released almost simultaneously (within {pm},7 minutes) with the electrons, in 16 events protons were released later than the electrons, and in one event electrons were released after the protons. The simultaneous proton and electron events and the delayed proton events did not unambiguously fall in the two categories of proton events, although most of the events in which the protons were released after the electrons belonged to Category 2. We conclude that acceleration of protons in Category 1 events occurred low in the corona, either by CME-driven shocks or below the CMEs in solar flares or in CME initiation related processes. It seems plausible that protons in Category 2 events were accelerated by CME-driven shocks high in the solar corona. Large delays of protons with respect to type III onsets in the events where protons were released after the electrons suggest late acceleration or release of protons close to the Sun, but the exact mechanism causing the delay remained unclear.

Statistical Analysis of Interplanetary Magnetic Field Path Lengths from Solar Energetic Electron Events Observed by WIND

Abstract We calculate the interplanetary magnetic field path lengths traveled by electrons in solar electron events detected by the WIND 3DP instrument from 1994 to 2016. The velocity dispersion analysis method is applied for electrons at energies of ∼27–310 keV. Previous velocity dispersion analyses employ the onset times, which are often affected by instrumental effects and the pre-existing background flux, leading to large uncertainties. We propose a new method here. Instead of using the peak or onset time, we apply the velocity dispersion analysis to the times that correspond to the rising phase of the fluxes that are a fraction, η, of the peak flux. We perform statistical analysis on selected events whose calculated path lengths have uncertainties smaller than 0.1 au. The mean and standard deviation, (μ, σ), of the calculated path lengths corresponding to η = 3/4, 1/2, and 1/3 of the peak flux is (1.17 au, 0.17 au), (1.11 au, 0.14 au), and (1.06 au, 0.15 au). The distribution of the calculated path lengths is also well fitted by a Gaussian distribution for the η = 3/4 and 1/3 cases. These results suggest that in these electron events the interplanetary magnetic field topology is close to the nominal Parker spiral, with little field line meandering. Our results have important implications for particles’ perpendicular diffusion.

Group velocity dispersion analysis of terahertz quantum cascade laser frequency comb

The frequency comb which is characterized by equally-spaced frequency lines with high mode coherence has received much attention since its first demonstration in near-infrared and optical frequency range. In the terahertz frequency range, the electrically-pumped terahertz quantum cascade laser (THz QCL) based on semiconductors is an ideal candidate for achieving frequency comb operation in a frequency range between 1 THz and 5 THz. The group velocity dispersion (GVD) is a key factor for the frequency comb. A higher GVD can pull the frequencies from their equidistant values and limit the comb bandwidth. Therefore the laser dispersion needs to be compensated for in order to make the total GVD sufficiently low and flat, such as using a Gires-Tournois interferometer (GTI) or the double chirped mirror (DCM). However, a successful design still depends on the knowledge of the total GVD in the laser. In this paper, we show how to calculate the GVD in metal-metal waveguide THz QCLs by taking into account the dispersions from the GaAs material, the waveguide, and the laser gain, which conduces to the understanding of the frequency comb behavior. The waveguide loss is modelled by the finite element method. The loss due to intersubband absorption is calculated by Fermi's gold rule. All the losses, i.e., waveguide loss, mirror loss, and intersubband absorption loss, are summed up to calculate the clamped gain. The material loss can be calculated by using the reststrahlen band model. Because of these losses and gain, the refractive index needs to be replaced by a complex refractive index. The real part of the complex refractive index is the refractive index, which can be calculated from the Kramers-Kronig relationship that connects the loss or gain with the refractive index. Then the GVD introduced by the material loss, waveguide loss, and clamped gain can be finally calculated. The results show that the total GVD of THz QCL is approximately –8 × 10<sup>5</sup>~8 × 10<sup>5</sup> fs<sup>2</sup>/mm which is strongly determined by the clamped gain. Finally, the developed numerical model is employed to study the dispersion compensation effect of a GTI mirror which is coupled into a QCL gain cavity. The design of the THz QCL based on GTI structure is more flexible and feasible than that of the DCM. The result shows that by carefully designing the geometry of GTI, the dispersion of a THz QCL can be compensated for, thus achieving the broadband terahertz frequency combs.