Implementation of Photonic Crystals into Davis LUT module for GATE simulation

Abstract

The performance of Positron Emission Tomography (PET) detectors has been constrained by the photodetector collection of optical photons emitted in the scintillator, which was limited to photons reaching the exit surface with an angle larger than the critical angle. Photonic Crystals (PhCs) are periodic nanostructures with sizes comparable to the optical photons’ wavelengths, which can break through the critical angle limit. Thorough experimental investigation of PhCs effect on optical harvest in scintillator detectors is complex and costly. Simulation can overcome these challenges. Mainstream software such as GATE does not support PhCs simulation. Here, we generalize the GATE optical model by incorporating the PhCs optical model into the LUT Davis model. We can model the performance of advanced scintillator detectors via the generalized LUT Davis model. The scintillator and PhCs materials tested in this work were lutetium oxyorthosilicate and titanium dioxide, respectively. Scintillators with a cross-section of 3×3 mm² or 10×10 mm² and a thickness varying from 9 mm to 18 mm with a step size of 3 mm were modeled with a PhCs interface to the photodetector. Among the 4 tested PhCs configurations, the best optical photon harvest was improved by 62.4% compared to traditional coupling with variable results between PhCs structures. The energy resolution only slightly improved. We thus investigated the angular distribution of collected optical photons, which can guide the optimization of photodetectors’ detection efficiency at specific angles.

I. INTRODUCTION

THE energy resolution of Positron Emission Tomography (PET) detectors is currently close to 10% [1]. Good energy resolution allows PET detectors to precisely distinguish photoelectric events from Compton events. The energy resolution of the PET detector is inversely proportional to the square root of the number of collected optical photons (LY_coll) [2] for most of scintillators. Unfortunately, if an optical photon impinges on the scintillator-photodetector interface (commonly using a coupling medium such as optical grease or glue) with an angle larger than the critical angle, it will not be transmitted through the exit surface of the crystal but reflected inside the crystal. The optical photon information is thus lost, which impairs energy resolution. Previous work has experimentally demonstrated that photonic crystals (PhCs) can improve scintillator detector performance, such as light collection, energy resolution, and timing resolution [3] [4] [5] [6] [7] [8] as they can break through critical angle limits. However, developing advanced scintillation detectors by experimental methods [9] engages high costs and slows down development. Simulation methods are powerful tools that could help design PhCs by sampling the large parameter space of geometric features and observe several important characteristics individually such as light yield, reflections patterns, transport time in the crystal, extraction yield at the interface, which is not feasible experimentally.

Here, we propose a model [10] to investigate the PhCs effect on the scintillator detector’s light yield and energy resolution with Monte Carlo simulations. Optical photon propagation inside PhCs follows the wave optics theory instead of ray optics theory and, therefore, cannot be described with existing models such as the LUT Davis model currently used in GATE [11]. The LUT Davis model maps the photon incidence angles with reflection/transmission values and with the distribution of reflected/transmitted photons and stores the information into look-up tables (LUT) to be read into GATE for the simulation of optical boundaries [12]. Our approach to build new ‘PhCs LUTs’ consists in solving the Maxwell equation for PhCs with the Finite Difference Time Domain (FDTD) method in commercial software (e.g., Ansys Lumerical), then use the Fourier optics theory to obtain the energy distribution of light passing through the PhCs structure [13] [14]. This energy distribution, called the far field, is representative of the optical photons’ behavior at the scintillator-photodetector interface and thus determines their collection and angular distribution. GATE models optical photon transportation within regions far larger than visible wavelengths, so the far field must be converted into a LUT that maps the reflection and transmission pattern of photons at a crystal-PhCs-photodetector interface and build a PhCs LUT.

In this work, we present the development of PhCs LUTs and their application to GATE simulations of lutetium oxyorthosilicate (LSO) crystals. The light collection, energy resolution, and angular distribution are compared for four different PhCs structures with different patterns and materials. PhCs were made of ~400 nm titanium dioxide (TiO₂) pillars which shape was either conical or cylindrical and in contact with LSO on one side and air or optical grease on the other side. LSO crystals had 3×3 mm² and 10×10 mm² cross-section areas, with varying thickness from 9 mm to 18 mm (Fig. 1 (a)) to study the performance of PhCs for thinner or thicker detectors, as we anticipate the PhCs to modify the light propagation inside the scintillator when photons are reflected off the exit face that is coated with the PhCs.

Fig. 1: Scintillator detector configuration in GATE. Photonic Crystals simulation configuration. Cross-section view of four photonic Crystals’ structures in unit cell. Their geometry size is labeled in plot.

(a) Coupling conditions have: ‘No PhCs’, ‘PhCs1’, ‘PhCs 2’, ‘PhCs 3’, and ‘PhCs 4’. (b) micro-scale view of Photonic Crystals structure in simulation. The blue block represents scintillator, Lutetium Oxyorthosilicate (LSO) and the red cylinders the PhCs made of titanium dioxide (TiO₂). The PhCs are patterned as square lattice. (c) Refractive index spectrum. Red and blue curve indicate LSO and TiO₂ refractive index respectively. (d) PhCs 1: Nano-cylinder deposited on scintillator and coupling environment is air. (e) PhCs 2: Nano-cylinder deposited on scintillator and coupling environment is optical grease. (f) PhCs 3: Truncate cone deposited on scintillator and coupling environment is air. (g) PhCs 4: Truncate cone deposited on scintillator and coupling environment is optical grease.

II. Materials and methods

A. Photonic Crystals modeling

1). Materials and Nanostructure Configuration

Photonic crystals are nanostructures designed to control the flow of electromagnetic waves, particularly light, in a manner analogous to how semiconductor crystals control the flow of electrons. Optical photon propagation does not follow the traditional geometrical optics rule in PhCs. The solution of Maxwell’s equations describing the behavior of electromagnetic fields can describe optical photon propagation inside PhCs modeling light as an electromagnetic wave. We solved Maxwell equations in PhCs using FDTD, a numerical method to solve partial differential equations [13] [14] implemented in commercial software (Ansys Lumerical). We considered LSO as the scintillator material (blue block in Fig. 1 (b)) and TiO₂ as PhCs material, (red cylinders in Fig. 1 (b)), in simulation for the following reasons: (1) LSO is commonly used as scintillator material in commercial PET detector [15].

(2) TiO₂ has an energy bandgap of 3.3 eV (corresponding to optical wavelength of 375 nm) and will not absorb most of the scintillation and Cherenkov photons emitted by the scintillator. (3) The deposition of TiO₂ upon LSO has been widely implemented [6] [5]. TiO₂ always has a larger refractive index than LSO within the emission spectrum (Fig. 1 (c)). The refractive index of photonic crystals does not necessarily need to exceed that of the scintillator.

We chose a square lattice of TiO₂ as the PhCs pattern (Fig. 1 (b)), in which the PhCs elements (which may be rods, spheres, or other shapes) are arranged in a regular grid pattern with square symmetry. The lattice spacing and the size of the individual elements are typically close to the wavelength of visible light. Four PhCs structures were modeled in this study; the cross-sectional views of the unit cell are shown in Fig. 1 (d):

1. cylinder-air coupling,

2. cylinder-optical grease coupling,

3. truncate cone-air coupling, and

4. truncate cone-optical grease coupling,

which are abbreviated as ‘PhCs 1’, ‘PhCs 2’, ‘PhCs 3’, and ‘PhCs 4’, respectively. In this paper, we refer ‘No PhCs’ to conventional coupling scenario: polished scintillator coupled with photodetector by optical grease. Optical grease has a refractive index of 1.47 and glue has a refractive index of 1.56. The glue is used to couple scintillator with enhanced specular reflector (ESR) reflector. The truncated cone has a cover radius of 100 nm and a cone angle of 20°. We chose the nano-cylinder dimension to match that of PhCs described in [6], which we used as a qualitative reference for our initial model. However, a perfect cylindrical shape is hard to manufacture, and a truncated cone geometry is closer to the structure of fabricated PhCs [16]. We simulated two materials within the PhCs structure (around the pillars): optical grease (conventional coupling in benchtop detectors and equivalent to glue coupling, yielding a lower index mismatch with the scintillator), and air (similar to manufactured PhCs) [16].

2). From PhCs wave simulations to Davis LUT model

Young’s double slit experiment stated that if multiple optical photons were shot through and crossed the double slit, these discrete optical photons would fall onto the screen and form dark-bright bands [17]. The bright regions indicate high energy where more optical photons fall, while the dark regions mean low energy where fewer optical photons fall. These dark and bright regions can predict incident optical photons destiny, like Davis LUT in GATE. Ansys Lumerical can calculate these dark and bright regions. Other software exist that can model PhCs, such as CAMFR [18] that can compute the near field from PhCs modelling. However, it cannot project the near field to far field, which is needed to build the LUTs.

GATE processes discrete optical photon transport in scintillators typically with a size much larger than the optical wavelength, which is handled by the Davis LUT model [19] or others large-scale models [20] [21]. Therefore, we must obtain the energy map for each PhCs structure to build a specific LUT and ultimately predict the optical photons’ fate at the scintillator-PhCs interface in GATE simulations.

Ansys Lumerical FDTD was used to calculate two types of energy maps: the near field, computing the energy distribution in a region with a size comparable to the visible wavelength, and the far field, computing the energy distribution in a region much larger than the visible wavelength (typically more than ten times). Both fields were calculated in (V/m)² and the near field was projected following Green’s function theorem to calculate the far field.

As shown in

Fig. 2 (a), a plane wave with λ= 420 nm was directed perpendicular to the scintillator (LSO)-PhCs (TiO₂) interface (green arrow). Five bright spots are displayed in the far field indicative of higher energy than that in the dark blue region (negligible energy deposition). The energy distribution is representative of the angular distribution of photons traversing the PhCs (e.g., bright spots show angles at which most photons would be transmitted) and the transmission coefficient. The energy distribution was calculated in the same way for 30 wavelengths between 370 nm and 800 nm, corresponding to the LSO scintillation emission spectrum. The plane wave’s polar angles (θ in Fig. 2 (b)) range from 0 to 89.9 degrees with a step size of 1 degree; we chose three azimuth angles (φ in Fig. 2 (b)): 0, 22.5, and 45 degrees and used the square lattice symmetry property to infer the distribution for other azimuthal angles. The following steps are applied to build PhCs LUT:

Fig. 2: The procedure to build Photonic crystals Look-Up-Table.

(a) a plane wave with wavelength of 420 nm was directed to the scintillator-PhCs interface with a 90° angle (green arrow). Firstly, FDTD was used to solve Maxwell equations and compute the transmitted energy distribution in the region very close to the interface. Then, the FDTD-based software decomposed the near field into multiple plane waves and propagated them to the far region to compute the far field. (b) Polar angle and azimuth settings for plane waves.

1. fix polar angle (e.g., θ₁=0°) and azimuth angle (e.g., φ₁=0°); sample 30 wavelengths from 370 nm to 800 nm with equal step size; calculate 30 energy maps and transmission coefficients.

2. normalize and sum these energy maps and transmission coefficients based on LSO emission spectrum probability density function.

3. repeat step 1 and step 2 for other two azimuth angles: φ₂=22.5° and φ₃=45°.

4. flip normalized energy map along y=x to obtain two new normalized energy maps and transmission coefficients corresponding to φ₄=67.5° and φ₅=90°.

5. overlap and average these five energy maps and calculate the mean value of five transmission coefficients.

6. find bright spots of these five overlapped energy maps and mark down their central positions.

7. calculate energy fraction for each bright spot among the overlapped energy maps.

8. build angular distribution for Davis LUT according to the energy fraction

9. repeat step 1-step 6 for all 90 polar angles.

The PhCs LUTs for the Davis model account for the incidence angle, similar to the previous approach, so no modifications were needed into GATE source code to read and use them. Far fields of different PhCs are displayed in Fig. 3. All of them depict unit cells of four PhCs. Although the magnitude of the optical intensity varied among different PhCs, only the relative magnitude of optical intensity within a given far field energy map matters because it will be normalized to determine the optical photons’ angular distribution. Transmissivity versus incident angles is presented in Fig. 4.

Fig. 3:

(a)-(d): Far Field and Transmissivity for Photonic Crystals Structures PhCs 1-4 described in Fig. 1 (d) (e) (f) (g).

Fig. 4: Transmission coefficient at different incident angles for different PhCs structures.

Blue curve represents traditional coupling scenario: pure optical grease couples scintillator with photodetector and indicates critical angle around 56°. The magenta curve represents an air coupling of a polished scintillator with a photodetector entrance window and shows a critical angle around 33°. This curve shows the effect of the coupling material’s refractive index on transmissivity value for ideal polished interface.

The critical angle of ~56° corresponded to the shoulder of the blue curve, which is a traditional coupling scenario (scintillator-optical grease with refractive index 1.47), and critical angle of ~ 34° corresponded to scintillator-air coupling (magenta curve), and no critical angles appear in the PhCs couplings, which all exhibit flatter curves with overall lower transmission when incident angle is small. The red curve has an acceptable match with the transmission curve in [6], providing a qualitative validation of our far field energy distributions. These results confirm previous works ( [3] [4]) showing that photons reflected back in the crystal without PhCs can now be transmitted and therefore collected with PhCs applied. The next step described in section B is to carry out detector simulations with the PhCs LUTs. Although these curves give an indication of the PhCs behavior, to calculate the fraction of transmitted photons for a given structure their angular distribution impinging on the PhCs surface is needed and can only be obtained through full Monte Carlo simulations.

B. Scintillator detector configuration in GATE simulations

We set the source as mono-energetic with 511 KeV gamma photons, an activity of 10000 Becquerel for all configurations and an isotropic emission. We simulated both photoelectrical process and Compton process. In this simulation work, two scintillator cross-section areas were taken into consideration: 3×3 mm² and 10×10 mm², among which both have four thicknesses: 9, 12, 15, and 18 mm, shown in Fig. 1 (a). We chose these geometries for the following reasons: (1) a crystal cross-section of 3×3 mm² is currently the most common in commercial systems [22], and there is a trend to decrease the scintillator thickness for cost and space constraints reasons, (2) a crystal size of 10×10 mm² was previously tested in monolithic scintillator detectors [16]. All scintillators’ surfaces were polished, and five sides were wrapped with an ESR reflector to inhibit optical photon transmission. The scintillator was coupled with one photodetector on one face and the coupling scenarios were: traditional coupling (scintillator-optical grease with refractive index of 1.47) and PhCs 1-4 couplings. The exit surface (coupling surface) was totally covered by photodetector. The photodetector was modeled with a photon detection efficiency of 1 in all configurations. Therefore, once an optical photon exits the scintillator from scintillator-photodetector interface, it is automatically detected.

C. Simulated physics characteristics

1). Number of collected optical photons and energy resolution

We measured the PhCs ability to harvest optical photons by estimating the average number of collected optical photons per gamma event, for all scintillator geometries and coupling conditions. Fig. 5 shows the simulated pulse height spectrum for a 12 mm-thick scintillator with a 10×10 mm² cross-section area. The number of collected optical photons for each gamma event was histogrammed and fitted with Gaussian distribution. The energy resolution was estimated by dividing the Full-Width-Half-Maximum (FWHM) by the Gaussian model mean value, corresponding to the peak position.

Fig. 5: Pulse height spectrum for scintillator with 10×10 mm² cross-section and 12 mm thickness.

Different face color represents different coupling scenarios. E_r refers to energy resolution and LY_coll is the average number of collected optical photons per Gamma events. ‘No PhCs’ refers to polished scintillator coupled with photodetector by optical grease.

2). Angular distribution of collected optical photons

We made histograms of collected optical photons momentum for each configuration. These angular distributions can guide the optimization of photodetector Quantum Efficiency (QE) at specific angles according to the optical photons’ angular distribution from the scintillators.

III. Results

A. Number of collected optical photons.

PhCs ability to harvest optical photons is greater in scintillators with large cross-section areas than in scintillators with small cross-section areas. As shown in Fig. 6 (a)(b) and Table 1, the average number of collected photons per event (LY_coll) was greater for 10×10 mm² crystals than 3×3 mm² crystals when considering the same coupling scenario and thickness.

Fig. 6: Average number of collected optical photons per Gamma events (LY_coll) and Energy resolution (E_r). Four marker colors represent four crystals’ thickness. Different marker shapes indicate different coupling conditions between scintillator and photodetector.

(a) and (c): 10×10 mm² cross-section area scintillator. (b) and (d): 3×3 mm² cross-section area scintillator.

Table 1:

Average number of collected optical photons per Gamma events (LY_coll). The second row represents crystal thickness in mm.

	10×10 mm2				3×3 mm2
	9	12	15	18	9	12	15	18
No PhCs	3483	3379	3297	3168	2537	2325	2192	2023
PhCs 1	4203	3821	3821	3244	2224	1893	1681	1481
Gain (%)	20.67%	13.08%	15.89%	2.39%	−12.34%	−18.58%	−23.31%	−26.79%
PhCs 2	5657	5396	4967	4743	3459	2968	2634	2416
Gain (%)	62.42%	59.69%	50.65%	49.72%	36.34%	27.66%	20.16%	19.43%
PhCs 3	4272	3881	3586	3299	2243	1934	1636	1472
Gain (%)	22.65%	14.86%	8.77%	4.14%	−11.59%	−16.82%	−25.37%	−27.24%
PhCs 4	5338	4869	4583	4300	3093	2648	2414	2161
Gain (%)	53.26%	44.1%	39.01%	35.73%	21.92%	13.89%	10.04%	6.82%

For example, 10×10 mm² crystal with 9 mm thickness for ‘PhCs 1’ has LY_coll of 4203 while 3×3 mm² crystal with 9 mm thickness for ‘PhCs 1’ has LY_coll of 2224. The difference between two cross-sections for conventional optical grease coupling (‘No PhCs’) is not as significant as other PhCs couplings. However, PhCs do not always have superiority to ‘No PhCs’. For instance, conventional optical grease coupling ‘No PhCs’ can harvest more optical photons than ‘PhCs 1’ and ‘PhCs 3’ but less than ‘PhCs 2’ and ‘PhCs 4’ for any thickness when crystals have a 3×3 mm² cross-section. As the crystal thickness increases, the LY_coll decreases. The number of collected photons decreases with increasing length, due to absorption of the light along the crystals and an increase of reflections on the surfaces. The longer the crystal, the longer the photon travels in length. Furthermore, these photons have a higher probability of being absorbed before reaching the exit face of the crystal. For example, in Fig. 6 (a) the LY_coll decreased from 5657 (magenta star, 9 mm thickness) to 4743 (cyan star, 18 mm thickness) for all ‘PhCs 2’. There are small peaks with a greater number of collected photons right of the photopeak for each histogram in Fig. 5, due to this “depth-of-interaction (DOI) walk” (Fig. S1). The maximum increase of the average number of collected optical photons LY_coll was 62.4% (from 3483 for ‘No PhCs’ to 5657 with ‘PhCs 2’ for a 9 mm thick crystal of 10×10 mm² cross-section).

B. Energy resolution estimation

Previous studies showed that the energy resolution is related to the light yield through the following equation in most cases:

$${\mathbf{E}}_{\mathbf{resolution}}\text{α}\sqrt{\frac{1}{{\mathbf{LY}}_{\mathbf{coll}}}}$$ Eq 1

where ${\text{E}}_{\text{resolution}}$ represented energy resolution [2]. However, Eq 1 is valid for most scintillators. For example, ‘No PhCs’ in the 10×10 mm² case harvested fewer optical photons than any PhCs, while the energy resolution was not always worse than any PhCs energy resolution for any crystal thickness (Fig. 6 (a), Table 1 and Fig. 6 (c), Table 2).

Table 2:

Energy resolution (Percentage value). The second row represents crystal thickness in mm.

	10×10 mm2				3×3 mm2
	9	12	15	18	9	12	15	18
No PhCs	12.3	12.1	12.7	11.6	12.1	13.6	13.3	13.7
PhCs 1	12.6	12.2	12.2	12.6	14.4	14.6	15.9	16.2
Gain (%)	−2.44%	−0.83%	3.94%	−8.62%	−19.01%	−7.35%	−19.55%	−18.25%
PhCs 2	11.5	12.5	12.4	12.6	14.1	15.4	15.6	15.3
Gain (%)	6.5%	−3.31%	2.36%	−8.62%	−16.53%	−13.24%	−17.29%	−11.68%
PhCs 3	11.5	12.4	13.1	12.8	14.2	15.2	15.3	15.9
Gain (%)	6.5%	−2.48%	−3.15%	−10.34%	−17.36%	−11.76%	−15.04%	−16.06%
PhCs 4	11.4	11.6	11.8	12.4	14.5	13.5	16.5	15.1
Gain (%)	7.32%	4.13%	7.09%	−6.9%	−19.83%	0.74%	−24.06%	−10.22%

As shown in Fig. 5, ‘No PhCs’ (in blue) appeared very narrow, but its peak position was very low. Meanwhile, PhCs 2 (green) had a greater peak position and light collection yield, but its FWHM was very large and degraded the energy resolution. PhCs 4 (cyan) exhibited the best E_resolution, although it did not collect the greatest number of optical photons, indicating that there is a combined effect of light collection spread (FWHM) and peak position (expected).

‘PhCs 2’ surpassed any other coupling scenario in terms of collected optical photons for both cross-section areas 3×3 mm² and 10×10 mm² but its energy resolution did not show superiority considering all coupling scenarios, all thicknesses, and cross-section areas. The energy resolution was thus not always inversely proportional to the square root of average number of collected optical photons.

C. Collected optical photons’ angular distribution.

To better understand differences in light collection, which are likely influenced by the light angular distribution at the PhCs-scintillator interface, we investigated the distribution of collected optical photons’ angles between the photon momentum and the surface normal vector pointing outwards of the scintillator at the exit surface.

Results are displayed in Fig. 7. The top and bottom rows show the histograms of collected optical photons’ angles for the 3×3 mm² and 10×10 mm² crystals, respectively. The magnitude of the histograms, related to the number of collected photons LY_coll shown in Fig. 6, is greater for shorter crystals (e.g. from the 9 mm-thick scintillators in magenta to the 18 mm-thick crystal in cyan).

Fig. 7: Collected optical photons’ angular distribution histogram before entering photodetector. 0 degree means normal incidence.

First row: scintillator with 3×3 mm² cross-section area. Second row: scintillator with 10×10 mm². Different face color histogram indicates different scintillator thickness. Five columns represent five coupling scenarios.

The first column (‘No PhCs’ coupling) in Fig. 7 revealed a critical angle around 56°, confirming the critical angle shown in the transmission curve of optical grease coupling (Fig. 4, blue curve). In contrast, there are no critical angles with PhCs, consistent with the transmission curves shown in Fig. 4. Fewer optical photons exit the interface between the scintillator and photodetector with a smaller angle in the ‘No PhCs’ case as shown in Fig. 7, left column. This phenomenon can be explained by several reasons: (1) Optical photons behavior at all scintillator surfaces (side surfaces and ‘No PhCs’) can be approximated as specular reflection, which does not change photon momentum’s tangential component along surface and only flip photon momentum’s normal component at surface, as shown in Fig. S3 (a) below. (2) Fig. S3 (b) below (magenta star represents Gamma-scintillator interaction point) indicates the relationship between number of emitted scintillation photons and these photons’ emission direction: the closer θ is to 0° or 180°, the fewer emitted photons in this direction will be (ds=r²sin(θ)d_θd_ϕ, where r is unit cycle radius) and larger photon momentum’s y-component magnitude will be. (3) Transmitted optical photons’ incident angles at photodetector interface are determined by their photon momentum’s y-component magnitude: the larger the y-component magnitude is, the smaller the incident angle will be. (4) All surfaces will not change emitted optical photons’ y-component magnitude no matter how optical photons bounce inside scintillator before being detected or transmitted outside at side surfaces. (5) fewer detected optical photons have smaller incident angles.

IV. Discussion

In this study, we developed a new simulation model for PhCs and applied it to four different geometries. We studied the effect of PhCs, scintillator thickness, and scintillator cross-section area on the number of detected optical photons and energy resolution and angular distribution. Photonic Crystals coupling can break through the critical angle limit. However, some PhCs structures may decrease the collection of optical photons with incident angles smaller than the critical angle for a traditional optical grease or glue coupling between the scintillator and the photodetector, such as ‘PhCs 1’ (Fig. 4 and Fig. 6 (a)(b)). The PhCs geometries and coupling materials can be optimized to enhance the transmission at all incident angles, as shown by ‘PhCs 2’ and ‘PhCs 4’ exhibiting higher transmission than ‘PhCs 1’ and ‘PhCs 3’. ‘PhCs 2’ and ‘PhCs 4’ contain optical grease around the TiO₂ pillars instead of air, so the larger refractive index results in greater light collection. We chose nano-cylinders with specific dimensions to compare our results with PhCs previously studied experimentally [6] and obtained good agreement (Fig. 4). Some discrepancies are due to our consideration of a more realistic LSO scintillation spectrum broader than that used in the simulation study where only one wavelength of 420 nm was used, and more azimuth angles included in our FDTD model to obtain a realistic illumination of the PhCs structure for more accurate computational results. We also considered truncated cones since PhCs fabrication processes cannot easily produce a perfect geometry like a nano-cylinder. Heterogenous structures, such as aperiodic mix of nano-cylinder and nano-cone, usually appear after fabrication yet are difficult to identify once the PhCs are encapsulated. Modeling these heterogeneous structures in Ansys Lumerical is challenging because we cannot set a specific structure in one unit. To validate our simulation results, we will measure the energy resolution of a scintillator detector with PhCs coupling with a benchtop setup. Light collection results in Fig. 6 (a)(b) reasonably match light collection in [16] and similarly show a slight improvement of optical photon collection over a conventional coupling, as expected. Some discrepancy arises from the different lattice (hexagonal in the experimental study in [16], square in this study), the difference in reflector (Teflon wrapping in [16] vs ESR in this study), and a 100% photodetection efficiency in this study. In contrast, the energy resolution in Fig. 6 (c)(d) is not consistent with [16]. Fig. 5 shows a photopeak broadening for all PhCs, which degrades the energy resolution even though the peak position increases. Other effects like scintillator thickness and smaller aspect ratio (smaller cross-section related to the thickness) increase the variance in the number of collected photons between events, causing the broadening of the energy spectrum.

Our light collection results showed an expected specular behavior for a traditional optical grease coupling, while the PhCs coupling scenarios showed a very different behavior. As such, a higher light yield did not necessarily result in better energy resolution (Fig. 6 (a)(b) and Fig. 6 (c)(d)). The overall energy resolution in the 10×10 mm² cross-section area is better than 3×3 mm² for all coupling scenarios and crystal thicknesses on account of a greater light collection. LY_coll increases when the scintillator becomes shorter. This is because emitted optical photons go through fewer reflections at shorter scintillators’ side surfaces and have shorter travel length in shorter crystal; therefore, fewer optical photons will transmit toward outside side surfaces and have less chance to be absorbed by scintillator.

The angular distribution of collected photons can assist in designing customized photodetectors because most commercial photodetectors’ QE is measured when the laser is placed in air, orthogonal to the photodetectors’ surface, and therefore does not provide a realistic measurement for scintillation detector assemblies. Optical photons do not exit the scintillator-photodetector interface perpendicularly but with a wide range of angles. Simulations can provide unique information on light angular distribution and help optimize the QE for some specific transmitted photons’ angles entering photodetector. Angles in Fig. 7 refer to transmitted photons’ angles inside the scintillator, before refraction toward the photodetector. We do not have transmitted photons’ angles outside the scintillator in this simulation work and will obtain them in the future work. As the photodetector QE might not be easily tunable by vendors, PhCs structures could be optimized to decrease the collected optical photons’ exit angles and bring them as close as possible to 0 degree.

DOI walk effect can explain the existence of a second peak on the right side of the main peak from the pulse height spectrum shown in Fig. 5. We decomposed the spectrum based on the DOI with 3 mm bins, and the results are presented in Fig. S1. The DOI was measured from the entrance face (i.e., low DOI was closest to the entrance face). As DOI increases, the pulse height spectra shifted towards higher values, corresponding to more photons detected. Therefore, these peaks accumulate on the right side and form the noticeable second peak from the pulse height spectrum in Fig. 5.

PhCs cannot always guarantee a better optical photon harvest because some optical photons may bounce many times and then be transmitted through the sides and lost or get absorbed by the bulk of the scintillator after interacting with PhCs and continuing moving within the scintillator. To investigate the outer transmission and the absorbed optical photons, new simulations with a gamma source intensity of 2 kBq and all other parameters unchanged were carried out. Previous simulations with stronger source intensity could generate the information of photons transmitted outside photons at scintillator side surfaces but the computer had memory limit to extract it. Therefore, we reduced the gamma source activity.

Results in Fig. S2 show that ‘No PhCs’ yield to fewer lost photons (transmitted outside and absorbed) than ‘PhCs 1’ and ‘PhCs 3’ for a 3×3 mm² cross-section. Therefore, ‘No PhCs’ can collect more optical photons than ‘PhCs 1’ and ‘PhCs 3’ at the same cross-section area. The opposite is true for ‘PhCs 2’ and ‘PhCs 4’ with provide the highest light collection (Fig. 6 (a)(b)). We attribute this light loss to the angular distribution of photons impinging on these five sides after they were reflected off the PhCs interface. Not shown here, a tally of the number of reflections revealed no difference between the PhCs, indicating that the transmission is likely due to other factors such as the incidence angle. Studying the angular distribution on each face for various configurations represents a volume of work beyond the scope of this paper.

V. Conclusion

In this work, we developed a model to simulate photonic crystals in GATE. We have used the simulation method to demonstrate that scintillator detectors’ energy resolution depends not only on light collection but also on other factors: scintillator thickness, cross-section area, and scintillator-photodetector coupling case. The modeling method, implemented in Monte Carlo simulation (GATE) and FDTD (Ansys Lumerical), fulfills a critical function in scintillator detector investigations, and it can reduce the cost of developing advanced scintillator detectors. In the future, we will conduct simulation investigations on the effect of PhCs on pixelated scintillator detectors [23]. To validate our simulation results, we will manufacture crystal LSO with configurations discussed in this paper (Coupling: ‘No PhCs’ and ‘PhCs 1-4’; Cross-section area: 3×3 mm² and 10×10 mm²; Thickness: 9, 12, 15 and 18 mm; single-ended readout), and then measure their light collection as well as energy resolution. Once the interface is validated against experimental PhCs characterization (in progress), we plan to release PhCs LUTs in GATE.

Timing resolution was characterized through detection time resolution but did not improve (results not shown). It could be improved by enhancing the extraction of the first optical photon, reaching the scintillator exit face for each gamma event [24] with improved geometries. The methods proposed in this paper will allow us to optimize the PhCs structure for timing, studying Cerenkov and scintillation photons separately.