Study of Modulation in Complex Refractive Indices Induced by Ultrafast Relativistic Electrons Using Infrared and THz Probe Pulses

Abstract

Objective

Achieving ultra-precise temporal resolution in ionizing radiation detection is essential, particularly in positron emission tomography, where precise timing enhances signal-to-noise ratios and may enable reconstruction-less imaging. A promising approach involves utilizing ultrafast modulation of the complex refractive index, where sending probe pulses to the detection crystals will result in changes in picoseconds (ps), and thus a sub - 10 ps coincidence time resolution can be realized. Towards this goal, here, we aim to first measure the ps changes in probe pulses using an ionizing radiation source with high time resolution.

Approach

We used relativistic, ultrafast electrons to induce complex refractive index and use probe pulses in the near-infrared (800 nm) and terahertz (THz, 300 μm) regimes to test the hypothesized wavelength-squared increase in absorption coefficient in the Drude free-carrier absorption model. We measured BGO, ZnSe, BaF₂, ZnS, PBG, and PWO with 1 mm thickness to control the deposited energy of the 3 MeV electrons, simulating ionization energy of the 511 keV photons.

Main results

Both with the 800 nm and THz probe pulses, transmission decreased across most samples, indicating the free carrier absorption, with an induced signal change of 11% in BaF₂, but without the predicted Drude modulation increase. To understand this discrepancy, we simulated ionization tracks and examined the geometry of the free carrier distribution, attributing the mismatch in THz modulations to the sub-wavelength diameter of trajectories, despite the lengths reaching 500 μm to 1 mm. Additionally, thin samples truncated the final segments of the ionization tracks, and the measured initial segments have larger inter-inelastic collision distances due to lower stopping power (dE/dx) for high-energy electrons, exacerbating diffraction-limited resolution.

Significance

Our work offers insights into ultrafast radiation detection using complex refractive index modulation and highlights critical considerations in sample preparation, probe wavelength, and probe-charge carrier coupling scenarios.

1. Introduction

The standard method of detecting ionizing radiation is via the scintillation process [1], [2], where the energy of the ionizing radiation is converted within a scintillation crystal to optical photons that are detected with photodetectors [2], [3]. The scintillation-based method has sufficient sensitivity to detect a single ionizing radiation particle interaction. However, the temporal characteristics of the scintillation process are limited by the time it takes for the free charge carriers to migrate through the lattice, drop into an excited energy state of a nearby impurity site, and deexcite, and, as a result, it is difficult to precisely obtain the arrival time of ionizing radiation to within less than a few hundred picoseconds [4]-[6]. Applications of high temporal precision ionizing radiation detection include time-of-flight positron emission tomography (TOF-PET), where better annihilation photon pair coincidence time resolution enhances reconstructed image signal-to-noise ratio [7]-[9]. Therefore, alternative ways to detect ionizing radiation with higher temporal resolution are of particular interest to investigate.

One possible alternative detection method to achieve higher temporal resolution is to utilize material responses that happen much faster than scintillation photon production, such as modulation of optical properties (a.k.a. optical modulation), including the free carrier absorption effect [10]-[12]. When the ionizing radiation enters a detection crystal, it releases energy through inelastic collisions which then create charge carriers. The instantaneous creation of the charge carriers, which occurs in the sub-femtosecond (fs) realm, leads to an increase in local charge density, and therefore a change in the complex refractive index of the material [10]. This transient modulation takes place significantly earlier than the charge carrier recombination that is required for scintillation, offering the potential to dramatically reduce timescales and increase temporal precision of radiation detection [5], [10], [13].

A recent demonstrations of radiation detection via optical modulation showed promising results [14]-[16] with a potential to utilize the ultrafast timescales of optical property modulation. In a Cadmium Telluride crystal, annihilation photons induced changes in the refractive index which were detected by shifts in the fringes of an interference pattern formed between a reference beam and one that penetrated the crystal [14]. As the modulation signal measured was a result of accumulation of multiple ionizing radiation particle interactions, the temporal precision was not elucidated. More recent efforts have been focused toward single-shot measurements with ultrafast time resolution [16]. In that recent work, synchrotron X-ray laser pulses with femtosecond temporal resolution [17] were used to excite the detection samples and we demonstrated that the resulting optical modulation is indeed transient [16]. Furthermore a, state-of-the-art technique we refer to as interferometric spectral encoding [18] - [21] was tested to circumvent the need for fast electronics.

These preliminary efforts, however, offered limited applicability to PET due to the characteristics of the X-ray pulses. For instance, the energy of the X-ray photons was much lower than that of the annihilation photons, and the number of incident X-ray photons was orders of magnitude greater than that of the 511 keV photons. The incident X-ray pulses generated a sea of charge carriers with a homogeneous density across the laser diameter and thin layer, which differs in geometry from the tortuous, stochastic ionization trajectories formed by single annihilation photons. Using an ionizing radiation source with higher energy can mitigate these limitations, as longer ionization tracks are formed, and the total deposited energy can be controlled by reducing the sample thickness to produce energy deposition similar to that of PET. In addition, further material response investigation is required for improved sensitivity [16].

To achieve the highest sensitivity and temporal precision for ionizing radiation detection using the optical modulation mechanism, an effective test bed is necessary to study and characterize the resulting ultrafast and non-equilibrium changes in the optical properties. There are qualities to look for in the ideal test bed: i) higher radiation particle energy with high temporal precision, and ii) availability of a variety of optical probe wavelengths. Compared to X-ray photons of 10 keV or below, the annihilation photons and the resulting photo-electrons have energy above 400 keV. This higher energy produces more inelastic collisions during the ionization process and longer trajectories, resulting in a different spatial profile for the induced complex refractive index modulation, which is generated along the ionization tracks. Focusing on studying the individual ionization trajectories, it is advantageous to have a lower number of particles compared to the X-ray case, which led to a homogeneous volume of charge carriers [18]. In addition, the availability of varied ranges of optical and electromagnetic probe wavelengths is critical to survey wavelength dependent signal generation as the modulation is hypothesized to be a function of probe wavelength. For example, one of the possible mechanisms inducing the modulation is the Drude model [22], where free charge carriers induce increased absorption (complex component of the refractive index) with square of the probe wavelength, favoring longer-wavelength sources in the infrared or terahertz ranges as described in the equation below:

$$\Delta \alpha =\left(\frac{e^3{\lambda }^2}{4{\pi }^2{c}^3{ϵ}_{0}n}\right)[\frac{\Delta N_e}{{m_{ce}^{\ast }}^2{\mu }_e}+\frac{\Delta N_h}{{m_{ch}^{\ast }}^2{\mu }_h}]$$ (1)

[23] ($\Delta \alpha$: change in absorption coefficient, $\mathsf{e}$: electronic charge, ε₀: permittivity of free space, $\mathsf{c}$: speed of light, $\lambda$: probe wavelength, $n$: refractive index of the unperturbed material, $m_{ce∕h}^{\ast }$: effective mass of electrons/holes, ${\mu }_{e∕h}$: electron/hole mobility, and $N_{e∕h}$: change in electron/hole carrier density).

A promising candidate for the desired test bed is an Ultrafast Electron Diffraction (UED) facility [24] - [26] that utilizes the de Broglie wavelength of ultrafast electron pulses to probe atomic structures of optically excited materials. The relativistic electrons possess energies in 3-4 MeV with temporal precision, or bunch lengths, in the hundreds of femtoseconds realm, which stem from the synchronized ultrafast laser pulses. For the purpose of probing radiation induced optical modulation, the master laser pulses can be further delayed in order to be used as optical probes, which is the reverse configuration of typical UED experiments, i.e., electron-pump and laser-probe setups [27], [28].

Here, we propose to use the ultrafast electron pulses as the excitation source in order to investigate critical information such as material response, optical probe wavelength dependency, and refinement of the optics setup to boost detection sensitivity. Multiple parameters of materials properties such as Z, density and bandgap must be considered to understand the exact mechanisms involved in the optical modulation process. Also of interest is the role of increased ionization energy that can lead to different spatiotemporal characteristics of the ionization charge track. For understanding the role of optical probes, based on previous reports [14]- [16], we hypothesize that free carrier absorption described by the Drude model [29], [30] will be the dominant mechanism. With an 800 nm laser probe, both standard transmission measurements and more sensitive interferometric measurements will be performed. Finally, a terahertz (THz) probe will be tested which is predicted to incur a much stronger absorption per the Drude model, which describes the nonlinear increase in the absorption coefficient with longer wavelengths [31].

2. Materials and Methods

2.1. Materials

In this work, we used bulk crystals for detection materials to ensure that a sufficient fraction of the high electron energy was absorbed. There were two major selection requirements of good stopping power for the ionizing particles (relatively high effective Z and density) and high bandgap, for increased energy absorption and good transmission of the infrared (~800 nm) and THz (30 μm to 3 mm) probe emissions, respectively. Conventional scintillation crystals such as barium fluoride (BaF₂), lutetium–yttrium oxyorthosilicate (LYSO), bismuth germanate (BGO) and lead tungstate (PWO) already meet both requirements with strong stopping power due to high effective Z and density. Semiconductor crystals such as zinc sulfide (ZnS), zinc telluride (ZnTe), and lead-bismuth gallate (PBG) tend to have lower stopping power but more appealing bandgap values. For the terahertz (THz) experiment, among the above crystals, we selected BGO, ZnTe, PWO, LYSO since they were transparent to the THz probe pulses. The detailed detection material properties are summarized in Table 1. Most of the samples had thickness of 500 μm, while BaF₂ and ZnTe crystals were thicker (1 mm) and PBG thinner (200 μm). The thicknesses were chosen to reduce the deposited energy to match closer to the 511 keV energy following a simulation method from previous report [37]. The samples were all mounted on translational stages inside the vacuum chamber.

Table 1.

Detection crystals studied in this paper and their material properties.

Sample	Effective Z Number	Density (g/cm3)	Bandgap (eV)	Scintillator	Thickness	Used in 800 nm	Used in THz
BaF2	52.2	4.89	6.86 [32]	Yes	1 mm	Y	N
Bi4Ge3O12 (BGO)	74.2	7.13	3.20 [33]	Yes	500 μm	Y	Y
Lu1.8Y0.2SiO5 (LYSO)	64.6	7.1	6.8-7.4 [34]	Yes	500 μm	N	Y
PbBiGa (PBG)	75.7	7.91	1.98 (measured)	No	200 μm	Y	N
PbWO4	75.2	8.28	3.5 [35]	Yes	500 μm	N	Y
ZnS	27.1	4.09	3.54 [32]	No	500 μm	Y	N
ZnTe	47.2	6.34	2.26 [36]	No	1 mm	N	Y

2.2. Methods

2.2.1. Ultrafast Electron Beam

To generate ionization tracks, we used relativistic electron beams at the Korea Atomic Energy Research Institute’s Ultrafast Electron Diffraction facility, (KAERI-UED). The electron pulse generation was initiated from the same master laser which later on was used as an optical probe (Fig. 1 (a)), and the experimental setup and conditions to generate electrons were the same as described in the earlier work from KAERI [24]. The energy of the electron pulse was set at 3.1 MeV, with a repetition rate of 50 Hz, and charge quantity of 2 pC per pulse, with an exception of PBG crystal measurement which was operated at 2.5 MeV to compensate for the reduced crystal thickness; and pulse duration or bunch length of the ultrafast electron beam was 30fs (r.m.s.). The electron beam condition mentioned above was consistently maintained for measurements across all crystals using both types of probes: 800 nm and THz. The diameter of the electron pulse was measured to be 500 μm, which corresponds to 130 nm between any incident electrons, sufficiently separating ionization trajectories as the electrons progress and disperse through the sample.

Fig. 1.

(a) Schematic of the Ultrafast Electron Diffraction (UED) facility and transmission measurement setup with optical laser probe pulses (800 nm). The laser pulses from the master laser are split and sent to generate electrons (blue) and also used as probe pulses sent to the sample (orange). For the interferometric measurements, a pair of polarizers (POL-_{1, 2}) and birefringent plate (a-BBO_{1, 2}) were used. The boxcar integrator accepted trigger output from the master laser and integrated signal from the balanced photodetector. (b) Samples loaded onto the holder. (c) Inside of the vacuum chamber where the electron pulses and laser overlapped on the sample. (d) Closer view of the mounted samples.

2.2.2. Optical Probe Measurement Setup

For optical measurements, the master laser pulse was split and routed to a longer, variable optical path which was controlled by the mechanical stages. The laser pulse had repetition rate of 50 Hz, center wavelength of 800 nm, beam diameter of 500 μm and pulse duration of 45 femtoseconds (FWHM) [24]. The laser beam was aligned such that it overlapped with the electron beam onto the sample which was mounted inside the vacuum chamber, and the laser path and electron path were approximately colinear to each other. For time-resolved measurements, the delay stage was scanned from the shortest to the longest distances, in step sizes of 50 – 200 μm, which correspond to time intervals of 168 – 670 femtoseconds, based on the time it takes for the probe pulse to travel (c = 3 x 10⁸ m/s). To measure the transmission changes, the laser pulse was collected after it passed through the sample and exited the vacuum chamber (Fig 1 (a)). The setup measures transmission rather than reflection, which is crucial for capturing all signals induced by electrons across the entire thickness of the sample. This approach accounts for the longer ranges and the generation of secondary particles arising from the higher particle energy of the electron pulses.

Additionally, for increased sensitivity by removing the baseline transmission, an interferometric measurement setup was installed, by adding a matching pair of polarizers (Princeton Scientific) and birefringent plates (Newlight Photonics), indicated by asterisks (Fig 1 (a)). The birefringent plate introduced a 4 ps delay in the vertical versus horizontal polarization components of the probe pulse, similar to the earlier work [16]. The first pair of polarizer and birefringent plate placed before the vacuum chamber and the second pair installed after the chamber. The polarizers were set 90 degrees to each other, while the birefringent plates were set close to 90 degrees with finer adjustment in rotation in the direction orthogonal to the optical axis, to reach a most sensitive detection point. We noted that the time-zero point (t=0) was shifted due to the additional optics and resulting increase in the path length.

A silicon photodetector with area of 0.8 mm² and response time of 1 ns (Thorlabs, DET10A2) was used to measure the changes in the optical probe pulse intensity after passing through the sample, with a lens used to focus the probe onto the active area. As the arrival timing of both the laser pulses and electron pulses are known a priori, the pulses were averaged in order to increase the signal-to-noise ratio. A custom-made boxcar integrator was used to collect data within a designated time window following the arrival of the laser pulses, utilizing the trigger output signal from the master laser that generates the electrons.

2.2.3. THz Generation and Detection Setup

The terahertz probe pulses were generated from KAERI as described in the prior work, with the output of 1uJ, and center frequency of 0.35 THz [38]. Due to the strong power output of the THz probe, the transmitted intensity was directly measured by using a high electron mobility transistor (HEMT) THz sensor (TeraSense), which was then read out by an oscilloscope (Fig. 2).

Fig. 2.

Schematic of the THz probe setup. (a) THz beam path within the vacuum chamber. (b) Samples were mounted on translational stages to align with the THz beam path. (c) The transmitted THz pulses were focused by a lens to a HEMT detector, which was then read out by an oscilloscope.

To align the THz path to measure the intensity change due to the electron irradiation, the sample mount was rotated by 45 degrees and an additional monitoring camera was installed. In particular, scintillating samples were utilized to visualize the position of the electron beam, with its reflectance recorded by the camera. The signal generated from the HEMT detector was further recorded through the boxcar integrator, in a manner similar to the optical probe measurement to integrate over the region-of-interest and also to average signals to increase the signal-to-noise ratio. We obtained data from an extended scan range of the delay stage, which corresponds to the time between the electron pulses and the laser probe pulse, as the time-zero point, or onset of electron beam, in the THz setup was expected to be shifted due to the added optical component and modified beam path.

2.2.4. Data Analysis

To compare the signal intensities with and without electron irradiation, we subtracted the data with electron pulses from the data without them to eliminate systematic error or drift. We assumed that the onset of the electron beam was instantaneous relative to the total scan time, and that the charge carrier lifetime was significantly longer than the electron pulse width. In MATLAB, a moving average with a 13-ps window was applied to the datasets to remove high-frequency noise and increase the signal-to-noise ratio. The onset of the electron beam was extracted by identifying the maximum derivative of the signal. We then obtained median transmission values before and after the electron beam onset, and the reported values are median and standard deviation in the time trace before and after the electron beam onset. Notably, the BGO measurement in transmission mode with the probe laser beam exhibited a continual decrease in the signal, so we directly show the transmitted laser beam intensity.

2.2.5. Simulation of Ionization Trajectories

We used a Monte Carlo method (pyPENELOPE) to simulate the ionization trajectory induced by the ultrafast electron pulses [37], [39]. The electron energies of 3 MeV and 500 keV were selected to represent ultrafast electrons and photoelectrons from annihilation photons, respectively, in a PET scenario. The cutoff energy was set to 200 eV and BGO was selected as a material with thickness unlimited as well as with thickness of 500 μm. From the full trajectory, inelastic collisions where nonzero energy was deposited were selected, and the distances between each point were calculated. All of the analysis was done in MATLAB.

3. Results

3.1. Transmission Measurement with 800 nm Optical Probe Beam

To investigate the hypothesized modulation in complex refractive index induced by the ultrafast electrons, we first measured transmission changes induced by the ultrafast electrons with the 800 nm probe beam in samples with highest densities, 200 μm thick PBG crystal and 500 μm thick BGO. The baseline transmission through the PBG crystal was measured by comparing laser intensities without the sample and observed to be 72.6 % (data not shown), indicating that the PBG crystal was transmissive for the 800 nm probe. In Fig. 3 (a), the transmission changes measured in PBG as a result of electron beam irradiation (electron energy of 2.5 MeV) are shown. With the onset of electron beam (indicated by the gray line in Fig. 3 (a)) the transmission increased from 1.1 x 10⁻⁴ to 2.4 x 10⁻³. Next, we measured transmission of probe laser pulse in BGO, with increased thickness (500 μm) and increased electron beam energy (3.1 MeV). The transmission from BGO exhibited more gradual decrease over scanned time range as shown in Fig. 3 (b). With the electron beam, the average value was 2.1, and the mean of the first 3 ps of the measurement was 2.4 whereas the last 3 ps decreased to 1.7.

Fig. 3.

Transmission changes in high density samples of PBG and BGO. (a) Transmission increases as the electron beam (vertical gray line) turns on in PBG (dotted lines indicate median values of before and after electron beam). (b) In BGO, we observed more gradual, continued decrease in transmission of laser probe pulses, with electron beams on (red) versus without electron pulses (blue).

3.2. Interferometric Measurement with 800 nm Optical Probe Beam

To further investigate the ways to improve the transient modulation intensity, we tested the interferometric spectral encoding measurement techniques in different samples. In Fig. 4, the optical modulation obtained from BGO, BaF₂, and ZnS is shown, with the onset of electron beam indicated by the gray vertical line. In BaF₂, we saw a decrease in the signal intensity of 8.9 x 10⁻³ to 7.9 x 10⁻³ for e-beam off and on, respectively. On the contrary, in BGO, the signal increased with the electron beam, from 6.4 x 10⁻³ to 7.3 x 10⁻². Lastly, in ZnS, the changes in signal intensity were minimal, not exceeding the noise level.

Fig. 4.

Interferometric measurement of electron pulse-induced optical modulation in BaF₂, BGO, and ZnS crystals. (The onset of the electron beam is highlighted by the vertical gray line, and for each sample, median value before and after the electron beam are presented as dotted horizontal lines.)

3.3. Transmission Measurement with THz Probe

Next, we tested our hypothesis using the THz probe, predicting that free-carrier absorption would increase with longer probe wavelengths. In Fig. 5, signal obtained from the four materials PWO, LYSO, ZnTe, and BGO are shown, after median filtering to reduce high frequency noise. In PWO and LYSO (Fig. 5 (a)), decrease in transmission was observed from 4.6 x 10⁻⁴ to 2.6 x 10⁻⁴, and 4.0 x 10⁻⁴ to 2.7 x 10⁻⁴, respectively. We measured a transmission decrease in ZnTe from 4.8 x 10⁻⁴ to 1.3 x 10⁻⁴, and in BGO, the median signal intensity decreased from 5.3 x 10⁻⁴ to 3.8 x 10⁻⁴ (Fig. 5 (b)).

Fig. 5.

Absorption induced by electron beam, measured using a THz probe in (a) PWO and LYSO, and (b) ZnTe and BGO. The onset of electron beam is indicated by the vertical line, with transmission before and after onset. Across the samples, transmission decreased after the arrival after electron pulses. For visualization, LYSO and ZnTe were plotted with an offset of 5 x 10⁻⁴. The median value of signal intensity before and after the electron beam for each sample is shown as horizontal dotted

A summary of signal intensity changes is provided in Fig. 6, while direct comparison may not be applicable, the signal-to-noise ratio in intensity changes were higher with signals from the 800 nm probe pulses and using interferometric measurement setup, compared to the transmission measurement with THz probes. Most measurements exhibited a decrease in signal, except for PBG and BGO. In PBG, the signal was a direct measurement of transmission and may indicate the involvement of mechanisms other than free carrier absorption. In BGO, a signal increase was measured with the interferometric setup, while direct transmission decreased, suggesting that the interferometric setup may detect contributions beyond transmission, as it is sensitive to total changes in the complex refractive index. For PET applications, arrival timing can be extracted from both increases and decreases in signal, and further optimization focused on maximizing signal intensity will be an important area of focus.

Fig. 6.

(a) Signal intensity changes measured with an 800 nm probe laser pulse. Note that PBG was measured using the transmission setup, while other samples were measured using the interferometric setup. (b) Transmission changes measured by THz probe pulses.

3.4. Simulation of Ionization Trajectories and Charge Carrier Distribution

In Fig. 7, the simulated ionization tracks induced from 3 MeV electrons in BGO are shown for the unlimited thickness (Fig. 7(a)) and thickness of 500μm (Fig. 7(b)), in the Z direction (the direction in which the electrons enter the crystal). Each dot in the trajectory represents a single inelastic collision. Considering the low charge carrier mobilities in scintillator materials, we examined the intrinsic geometry of ionization tracks without considering the carrier diffusion [10], [40]. In the sample of unlimited thickness (Fig. 7(a)), we observe the effect of higher stopping power toward the end of the trajectory due to lower electron energy, forming a conglomerated cluster of charge carriers. In contrast, in samples with limited thickness (Fig. 7(b)), the full energy is not deposited, and the trajectories are more linear form without the clustered structure. We visualize the charge carrier distribution along the ionization trajectory, illustrating that the diameters of the charge carrier distribution are minute compared to the length of the entire track (Fig. 7(c)).

Fig. 7.

Simulation of ionization tracks induced by the ultrafast 3 MeV electrons. (a) Ionization tracks in BGO, with unlimited thickness. (b) Tracks simulated in BGO, 500 μm thick crystal. (c) Visualization of ionization trajectory and charge carrier distribution surrounding inelastic collision.

We further quantify the differences in geometries in terms of the distance between inelastic collisions and confirm that the reduced thickness case had greater distances (Fig. 8(a)). To pinpoint the source of the discrepancy in the distances, we then compared the inelastic collisions from 3 MeV down to 2.5 MeV with those at 500 keV or less, as shown in Fig. 8(b). The electrons with higher energy underwent inelastic collisions with mean distance of 0.96 μm (median 0.56 μm) between collisions whereas the lower energy electrons had shorter distances between collisions, with mean distance of 0.37 μm (median 0.17).

Fig. 8.

Comparison of inter-inelastic collision distances in different scenarios with (a) unlimited thickness case (blue) versus 500 μm sample (red) and (b) electron energy of 3 MeV - 2.5 MeV (blue) versus 500 keV (red).

4. Discussion

In this study, we employed ultrafast, high-energy electrons (2-3 MeV) as the ionization source to probe a broad spectrum of materials, focusing particularly on the infrared and terahertz ranges. Previous X-ray experiments utilized lower-energy photons (1.8 – 9.5 keV) as the ionization source, with particle numbers of 1.6×10¹² and 8.7×10¹², for 1.8 keV and 9.5 keV, respectively. In contrast, our approach used 1.3 × 10⁷ electrons, significantly reducing the particle count and allowing us to investigate individual ionization trajectories and the resulting charge carrier densities, deviating from the homogeneous layer of charge carriers studied in previous X-ray experiments. These earlier studies utilized detection materials like diamond, YAG, and BGO, along with a probe laser pulse containing spectra in the visible wavelength. In our research, we selected a wider array of materials, ranging from scintillators to semiconductors, and expanded the pulse wavelengths into the near-infrared and terahertz regimes to explore the hypothesized Drude response.

In Fig. 3, we studied optical transmission changes induced by ionization produced from a coherent electron beam in PBG and BGO crystals. PBG had intrinsic transmission of 72% (data not shown) which showed that the crystal was transmissive with a 800 nm probe which is relatively close to PBG’s bandgap energy at 625 nm. It is noteworthy that the transmission increased with the onset of the electron beam, i.e., contrary to the hypothesized free carrier absorption, suggesting that another mechanism, in addition to the proposed Drude model, contributes to the observed signal. One possible explanation could be a bandgap modification [23] in the material that occurs as a result of the electron irradiation, which can bring either a bandgap increase or decrease, depending on the material properties and probe wavelength [41]. In BGO, the intrinsic transmission was 76% even with a higher thickness of 500 μm, compared to the PBG crystal, confirming the higher bandgap of BGO crystal. The transmission decrease with electron beam in BGO is in agreement with the Drude model, suggesting that in the probe wavelength regimes that are further away from the bandgap, free carrier absorption is indeed the dominating mechanism. We also observed continual decrease in the signal, which may suggest mechanisms that take place over longer timescales, such as thermal effects, are occurring.

In addition, we obtained interferometric measurements (Fig. 4) with 800 nm probe laser pulses in BaF₂, BGO, and ZnS. In BaF₂ and BGO, the intensity changes were greater than those measured from ZnS, potentially due to the lower density of ZnS. The signal decreased in BaF₂, while in BGO, the intensity decreased with the onset of the electron beam. BGO is photorefractive and has strong nonlinear optical properties, which BaF₂ lacks. This difference affects the signal behaviors, as the interferometric setup is sensitive to phase changes in addition to the transmission of probe pulses. In BGO, the signal exhibited transient changes, contrary to the transmission changes, indicating the temporal sensitivity of the interferometric setup. For low-signal applications, such as positron emission tomography, this increase in the signal-to-noise ratio and the ability to capture transient signals can be advantageous for improved sensitivity in detecting the effects of individual 511 keV photon interactions.

Next, we observed the effects of longer wavelength using the THz probe via simulation of the ionization tracks. As seen in Fig. 5, the transmission decreased in the THz wavelength range in all the tested samples, confirming the free carrier absorption. However, when we compiled the data taken from both probe types, as shown in Fig. 6, the modulation intensity measured by THz probe was less than expected from the Drude model. Since the Drude free carrier absorption is expected to increase proportionally to the square of the wavelength, the THz probe with a wavelength of 300 μm should yield a signal increase 10⁵ times stronger than that of the 800 nm probe. (In this study, the increased absorption would lead to a decrease in transmission). To find possible explanations, we examined the associated electron ionization trajectory and considered the charge carrier distribution induced by the ultrafast electrons and how the interaction between the THz probe and ionization tracks can be dependent on the wavelength.

In Fig. 7, we simulated ionization trajectories with unlimited and reduced sample thicknesses to obtain geometries that explain the THz signal intensity being lower than predicted by the Drude model. The key difference is that the "glomeruli-like" clusters exist only at the ends of the trajectories in the unlimited thickness case, forming conglomerate charge carrier volumes detectable by the THz probe. On the contrary, with reduced sample thickness, ionization trajectories are truncated, and only the beginning part of the trajectory, which is more linear and has a small diameter of micrometers, is present in the crystal. Due to the small diameters [42] (Fig. 7(c)), charge carriers along the truncated trajectories are difficult to detect by the THz probe wavelength, which has several micrometers (300 μm in this study), due to diffraction-limited resolution [43], [44], despite the length of trajectories reaching several hundreds of micrometers. Furthermore, when we take a closer look at the distances between the inelastic collisions (Fig. 8(a)), the 500 μm thickness sample had longer mean distance. This indicates that in the limited sample thickness case, the beginning part of the trajectory has a lower charge carrier density than at the end due to lower dE/dx for higher electron energy, further lowering the probability of detection when using the THz probe. In Fig. 8(b), more direct comparison of beginning of trajectory represented by inelastic collisions from 3 MeV down to 2.5 MeV versus 500 keV case, confirming the importance of the conglomerated tail ends of the ionization trajectories. We expect that lowering the comparison threshold energy i.e., from 500 keV to 200 keV (3 MeV to 2.8 MeV versus 200 keV) will increase this discrepancy even further.

Therefore, we hypothesize that for THz detection, it is important to reconcile the diffraction limited resolution issue to utilize the full power of wavelength-squared increase from the longer wavelength. One approach is to use thicker samples or lower electron energy such that the higher density charge carrier volume at the ends of the ionization trajectories as shown in Fig. 7(a) can be probed as well. An alternative approach would involve using a near-field THz probe, which can achieve spatial resolutions down to sub-wavelength [45] - [47] scale for detecting smaller charge carrier distributions. The exact requirement for spatial resolution needs investigation, considering the tradeoff between the higher resolution and detection volume. For applications in PET, the detection settings are considered to be more favorable to THz probes, as the ample crystal thicknesses and lower particle energy nearly guarantee the entire ionization trajectory is completely absorbed and the probing region covers the clusters of charge carriers that are formed at the end of tracks.

On the other hand, free carrier absorption and the Drude model for the optical probe in the near IR regime of 800 nm may be a limited option for radiation detection due to the low signal-to-noise ratio and it may be advantageous to consider alternative mechanisms such as bandgap modification which can bring orders or magnitude changes in the complex refractive index [23], [29], [41].

One of the mechanisms we envision as suitable is exploiting the local charge carrier densities along the ionization trajectories, as shown by calculations reaching up to 10¹⁸/cm³ [10], depending on the material properties. This density is high enough to induce nonlinear modulations in the complex refractive index, even in the visible wavelengths, and mechanisms such as the band-filling effect and induced transparency can arise immediately [30]. This gating-like transparency is advantageous, as the signal intensity or number of transmitted photons is a function of the number of input photons, which we can control. Since we have learned that the volume occupied by the charge carriers along the ionization trajectories is sub-wavelength with visible probe pulses, the probe pulse parameters can be optimized for high coupling efficiency. This optimization can effectively address the diffraction-limited resolution by leveraging photonics techniques commonly used to investigate nanostructures. Key parameters to be optimized include both the detection crystal's material properties and the probe pulses. For example, factors such as the bandgap properties of the crystals, as well as the pulse duration, wavelength, and spatial distribution, should be considered.

As next steps, we will investigate means to increase the sensitivity. For the optical probe approach, we will experimentally investigate various combinations of detection crystals and probe wavelengths to identify viable bandgap modification mechanisms. Installation of nearfield probe and utilizing thicker samples will be investigated for THz probe approach.

5. Conclusion

In this study, we report the first-time measurement of optical modulation induced by an ultrafast 3 MeV electron beam, using both an 800 nm probe and a THz probe beam. Initially, we employed the 800 nm probe on samples of PBG, ZnS, BGO, and BaF₂, observing both increases and decreases in probe pulse transmission with and without electron pulses. The signal is the combination of transient modulations and signal trends over a longer duration. Additionally, we utilized the interferometric spectral encoding measurement technique to enhance detection sensitivity and focus specifically on transient optical modulations, thereby excluding longer-duration modulations. Subsequently, we tested the long-wavelength response using the THz probe, based on the Drude model, which predicts a wavelength-squared increase in absorption relative to the probe wavelength. However, in experiments we observed a decrease in transmission across all tested samples without the anticipated signal increase proportional to the square of the wavelength. Simulation of ionization tracks led us to hypothesize that low charge carrier density at the start of the ionization trajectory, which is the part of the trajectory absorbed by a thin crystal, might account for the discrepancy in expected signal enhancement.

Based on our experimental and simulation findings, we propose two distinct strategies for future research with optical and THz probes. For the optical probe, considering the limited signal-to-noise ratio, exploring alternative mechanisms such as bandgap modification [41] may prove beneficial. For the THz probe, strategies to overcome diffraction-limited resolution—such as using thicker samples, employing lower energy ionizing radiation, and utilizing near-field probes—are recommended to fully capture the ionization-induced signal modulation predicted by the Drude model.