Monte Carlo Simulation of Optical Photon Detection Efficiency: A Geant4 Study with Surface Roughness, Incidence Angles, and Wavelength Dependence

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Elamrawy, Adel M. Ismail This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7464946/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This paper presents a comprehensive Monte Carlo simulation of optical photon detection using the Geant4 toolkit. Multiple scenarios were designed to analyze the influence of detector surface roughness, incidence angles, and photon wavelengths on detection efficiency. Simulated configurations included smooth, rough, and nanostructured surfaces, with incidence angles of 0°, 30°, and 60°, and wavelengths ranging from 450 nm to 630 nm. The outputs were processed using Python-based analysis pipelines to generate key performance indicators (KPIs) and statistical summaries. Results indicate that smooth and nanostructured surfaces significantly improve detection efficiency, while rough surfaces reduce photon transmission. Angle-dependent behavior shows strong degradation at oblique incidences, and wavelength-dependent performance aligns with published quantum efficiency curves of photodetectors. These findings validate the methodology and suggest potential applications in optical communication systems, medical imaging, and detector optimization. The paper concludes with recommendations for integrating machine learning approaches to further enhance predictive modeling and design automation. Physical sciences/Engineering Physical sciences/Materials science Physical sciences/Optics and photonics Physical sciences/Physics Geant4 KPI ALR MT EM GPS CSV VLC Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 1. Introduction Photon detection is a cornerstone of modern science and technology, underpinning applications as diverse as optical communication systems, medical imaging, high-energy physics experiments, and astrophysical observations . The ability of a detector to capture incoming photons with high efficiency directly impacts the performance and sensitivity of these systems. However, detection efficiency is not constant; it varies significantly with parameters such as surface quality of the detector, angle of photon incidence, and photon wavelength . Understanding and optimizing these dependencies is therefore essential for the design of high-performance photodetectors. Traditional experimental approaches to characterize these effects require costly, time-intensive laboratory setups and are often limited in the range of conditions that can be tested. To overcome these challenges, Monte Carlo–based simulation frameworks have emerged as a powerful complement to experimental studies. Among them, the Geant4 toolkit has become a standard in the physics and engineering communities due to its ability to model the transport of particles—including optical photons—through matter with high precision. Its flexibility allows researchers to study the influence of complex geometries, boundary conditions, and material properties on photon behavior without the prohibitive costs of large-scale experiments. This paper focuses on exploring three key parameters that critically determine detection efficiency: Surface roughness , which controls scattering, reflection, and absorption at material boundaries. Incidence angle , which governs photon entry paths and optical losses at oblique geometries. Photon wavelength , which interacts with the detector’s absorption spectrum and intrinsic quantum efficiency. To systematically investigate these effects, we employed Geant4-based simulations that replicate photon–detector interactions under controlled, reproducible conditions. The study considered smooth, rough, and nanostructured surfaces , with incidence angles at 0°, 30°, and 60°, and photon wavelengths of 450 nm (blue), 520 nm (green), and 630 nm (red). By combining multiple scenarios, the simulation generated a structured dataset exceeding 180,000 photon events , which was subsequently analyzed using Python-based statistical and visualization tools. The contribution of this work lies in bridging physics-driven Monte Carlo simulations with modern data analysis pipelines . Beyond replicating known experimental phenomena, the study provides structured evidence of how surface engineering and geometrical factors affect detection efficiency. Furthermore, the discussion outlines potential extensions using machine learning and deep learning approaches to accelerate detector optimization, thereby linking classical simulation with emerging data-driven methodologies. 2. Related Work Recent studies have systematically investigated several aspects highly relevant to our work: Surface Roughness and Scattering Losses. [ 1 ]and [ 2 ] established that rough optical boundaries in Geant4 induce diffuse scattering, which significantly reduces detection efficiency by diverting photons away from the acceptance cone. These findings underscore the importance of precise modeling of surface micro-structure in simulations . Nanostructuring and Enhanced Light Capture. Nanostructured surfaces such as perovskite nanowires or graded-index “black silicon” textures have been shown to improve photodetector performance via enhanced absorption and reduced reflection. [ 3 ] reported that nanostructured perovskite photodetectors deliver markedly higher photoresponsivity and external quantum efficiency [ 3 ]. Similarly, black silicon—featuring sub-micron needle-like structures—creates an effective refractive-index gradient that lowers Fresnel reflection to below 5% across visible wavelengths [ 4 ]. Incidence Angle Effects on Efficiency. [ 5 ] provided an analytical expression for the angular dependence of quantum efficiency, emphasizing that photonic collection varies with both angle and polarization. Additionally, [ 6 ] demonstrated in micropatterned Ge/Si quantum-dot photodiodes that resonant peaks in responsivity shift or diminish at oblique angles due to changes in field coupling and trapping structures [ 6 ]. Machine Learning for Detector Simulation and Optimization. More recently, learning-based approaches have been applied to detector system modeling and optimization. For example, neural networks have been trained to classify photon-hit patterns and predict detection efficiency under varied conditions, leading to accelerated design cycles and reduced simulation load. [ 7 ] used convolutional neural networks to classify scintillation events in pixelated detectors, while [ 8 ] employed gradient boosting models to predict detector response across hardware parameter sweeps. Synthesis. Our study extends these foundational works by implementing a controlled, multi-scenario Geant4 simulation that varies surface topology, incidence angle, and wavelength. We integrate rigorous data processing pipelines, including KPI computation, statistical testing, and signature visualizations for full transparency and reproducibility. Moreover, our introduction of a polar response map offers a novel, compact representation merging angular and spectral dependencies a format not seen in earlier literature. 3. Methodology 3.1 Simulation Framework The simulation framework was implemented using the Geant4 toolkit (version 11.2 patch-2) , which is widely employed in high-energy physics, medical physics, and optical detector studies. The simulation was executed in multi-threaded (MT) mode , enabling parallel event generation and thus significantly improving throughput compared to single-thread execution. This choice allowed us to scale up the number of simulated photon events while maintaining reproducibility across independent runs. For the physics configuration, we adopted the standard electromagnetic (EM) physics lists available in Geant4, which provide validated models for the interactions of optical photons and relevant charged particles. These include: Optical Processes : reflection, refraction, absorption, Rayleigh scattering, boundary processes (including polished and ground surfaces), and wavelength-dependent quantum efficiency. Electromagnetic Interactions : Compton scattering, photoelectric effect, bremsstrahlung, and ionization processes for charged particles (to ensure consistency in multi-particle environments). Surface Interaction Models : Unified model of optical boundary processes, enabling explicit handling of smooth, rough, and nanostructured surfaces by assigning surface finish parameters and scattering probabilities. The framework also incorporated the General Particle Source (GPS) in Geant4 to generate optical photons. This ensured flexible control over photon energy, angular distributions, and spatial emission characteristics. Configurable parameters (such as incidence angle and wavelength) were systematically varied across simulation scenarios. All simulation runs were conducted in batch mode , allowing reproducible execution through macro command files. Outputs were recorded in structured CSV format , containing event-by-event photon hit data, which was later processed using Python for KPI extraction, statistical analysis, and visualization. 3.2 Detector Geometry and Configurations The detector was modeled as a planar photodetector with well-defined boundaries, a configuration frequently adopted in Geant4 optical photon studies to allow systematic evaluation of surface and geometrical effects. The geometry was kept deliberately simple (rectangular slab) to isolate the influence of surface properties, incidence angles, and photon wavelengths without introducing confounding factors from complex shapes. Three distinct surface configurations were implemented to capture realistic variations in detector response: Smooth Surfaces : Representing ideally polished detector interfaces, characterized by specular reflection and minimal surface scattering. Rough Surfaces : Modeled using Geant4’s ground surface option, where micro-irregularities introduce diffuse scattering, leading to angular broadening and reduced photon transmission. Nanostructured Surfaces : Implemented as an effective medium approximation, mimicking sub-wavelength structures (e.g., gratings or nanopillars). These surfaces enhance photon trapping and absorption probabilities by modifying the boundary reflection profile. Optical boundary processes were used to control how photons interact with the detector interface. Specifically, Geant4’s unified optical model was applied to describe: Specular reflection at smooth surfaces. Diffuse reflection and Lambertian scattering at rough interfaces. Wavelength-dependent transmission and absorption across all surfaces, adjusted to represent realistic refractive index and absorption spectra of photodetector materials. 3.3 Photon Source and Parameters Photon generation was implemented using the General Particle Source (GPS) module of Geant4, which provides flexible control over the spatial, angular, and spectral properties of emitted photons. The source was configured to replicate realistic optical excitation conditions while enabling systematic variation across experimental parameters. Photon Type and Energy The source emitted optical photons explicitly defined by Geant4’s electromagnetic physics lists. Photon energy was set according to the desired wavelength using the relation E = hc / \lambda, where \lambda ranged from 450 nm (blue) to 630 nm (red) , covering the visible spectrum relevant for photodetectors. A central reference wavelength of 520 nm (green) was included as a benchmark, since many silicon-based detectors exhibit peak quantum efficiency in this region. Spatial Configuration Photons were emitted from a point-like source placed at a controlled distance from the detector surface. The emission center was aligned at (-2.0, -2.0, 1.4 \, \text{m}), ensuring consistent photon travel paths into the detector region. The geometry was fixed across all scenarios, eliminating positional bias. Angular Distribution To mimic diffuse light input, photons were generated with an isotropic angular distribution . Incidence angles relative to the detector plane were varied systematically at 0°, 30°, and 60° . This design allowed investigation of angular dependence in photon detection efficiency, including the effects of oblique incidence and enhanced scattering. Polarization Control A fixed polarization vector (1, 0, 0) was applied, enabling consistent tracking of polarization-dependent effects across simulations. Number of Events Each scenario was executed with 20,000 photon events to achieve statistically robust results. The number of photons was chosen to balance computational efficiency with precision, particularly in multi-threaded execution mode. 3.4 Data Collection The Geant4 simulations were configured to record event-by-event photon interactions in structured CSV logs. For each photon event, the following parameters were stored: hit_flag : Binary indicator of whether the photon successfully reached the detector’s sensitive surface. detector coordinates (det_x_mm, det_y_mm) : Spatial location of the photon hit, measured in millimeters relative to the detector reference frame. photon properties : Including wavelength (450, 520, 630 nm) and incidence angle (0°, 30°, 60°). surface condition : Smooth, rough, or nanostructured, defining the optical boundary properties. A total of 18 unique simulation scenarios were executed, corresponding to all combinations of surface type, incidence angle, and photon wavelength. Each scenario contained approximately 10,000 photon events , yielding a combined dataset of over 180,000 simulated optical photon interactions . The output logs were systematically organized in the project’s geant4_raw directory, with filenames encoding the scenario parameters (e.g., events_rough_angle30_wl520nm.csv ). This naming convention ensured reproducibility and facilitated automated analysis pipelines. All datasets were subsequently processed and validated to confirm event integrity. This included checking for corrupted entries, verifying coordinate ranges, and ensuring scenario consistency. The final combined dataset, sim_results_combined.csv , served as the primary input for downstream statistical analysis and visualization. 3.5 Data Analysis Pipeline To ensure reproducible and rigorous processing of the simulation outputs, a structured Python-based analysis pipeline was developed. The pipeline integrated widely used scientific computing libraries— Pandas for data handling, NumPy for numerical operations, and Matplotlib for visualization. The analysis workflow consisted of four dedicated scripts, each performing a specific stage: process_data.py : Aggregated the raw Geant4 CSV outputs from multiple scenarios, performed quality control (QC), removed corrupted or incomplete entries, and produced the combined dataset ( sim_results_combined.csv ). exploratory_stats.py : Computed descriptive statistics and key performance indicators (KPIs) including detection efficiency, angular loss ratios, wavelength-dependent transmission, and measures of spatial distribution. make_plots.py : Generated standard and publication-ready visualizations such as bar charts, line plots, heatmaps, scatter plots, and boxplots. These figures enabled direct comparison across surfaces, incidence angles, and wavelengths. compare_scenarios.py : Performed statistical comparisons between different surface types and conditions, highlighting performance trade-offs. The KPIs extracted from the processed datasets included: Detection Efficiency (η) : Fraction of generated photons that successfully hit the detector. Angular Loss Ratio (ALR) : Quantification of efficiency reduction as a function of incidence angle. Wavelength-Dependent Transmission (Tλ) : Variation of efficiency across the tested spectrum (450–630 nm). Spatial Distribution Uniformity (σx, σy) : Statistical spread of hit coordinates, used to evaluate diffuse scattering effects. Together, this automated analysis pipeline ensured that all results were systematically processed, reproducible, and suitable for integration into downstream statistical analysis and visualization tasks. 4. Results This section presents the outcomes of the Geant4-based simulations across all 18 experimental scenarios (three surface conditions × three incidence angles × three wavelengths). Results are structured into subsections focusing on detection efficiency, surface roughness effects, angle dependence, wavelength dependence, and spatial distributions. Quantitative results are summarized in tables, while figures illustrate trends and comparative analyses. 4.1 Detection Efficiency Table 1 summarizes the mean detection efficiency across all simulated scenarios. As shown in Fig. 1 (Surface Comparison by Wavelength) , smooth surfaces consistently demonstrated the highest baseline efficiency, while rough surfaces exhibited a reduction of approximately 15–25%. Nanostructured surfaces not only outperformed rough surfaces by ~ 27% on average, but in several cases exceeded smooth surfaces, particularly at 520 nm . Smooth Surfaces : Maximum efficiency across all tested wavelengths and angles, providing a reliable baseline. Rough Surfaces : Consistent efficiency losses attributed to diffuse scattering and back-reflection. Nanostructured Surfaces : Demonstrated improved photon capture efficiency, validating literature findings on nano-patterning. Table 1 Mean detection efficiency across all scenarios Surface Mean Efficiency (%) Std Dev (%) Smooth 82.5 3.1 Rough 62.3 4.5 Nanostructured 88.4 2.8 4.2 Effect of Surface Roughness Surface morphology strongly influenced photon transport. For 450 nm photons at normal incidence (0°) , efficiency on rough surfaces decreased by ~ 20% compared to smooth surfaces (see Fig. 2 ). At high angles ( 60° incidence ), losses exceeded 35%, confirming enhanced scattering losses with rough boundaries. These results align with previous experimental studies on rough silicon detectors [ 1 ] [ 2 ], which reported similar diffuse reflection phenomena. Figure 2 . Detection efficiency by incidence angle for different surfaces 4.3 Effect of Incidence Angle Figure 3 presents the angular dependence of efficiency for all surface types. 0° incidence : Maximum efficiency across all cases. 30° incidence : Moderate losses (~ 10%) observed. 60° incidence : Severe efficiency reduction (> 30%), most pronounced for rough surfaces. These angular losses are quantified in Table 2 , which reports the angular loss ratio (ALR) as a percentage relative to normal incidence. Table 2 Angular loss ratio (ALR) across surfaces and wavelengths Surface Incidence Angle (°) ALR (%) Smooth 0 0 Smooth 30 8 Smooth 60 28 Rough 0 0 Rough 30 15 Rough 60 38 Nanostructured 0 0 Nanostructured 30 6 Nanostructured 60 25 4.4 Effect of Wavelength The impact of photon wavelength on detection efficiency is summarized in Fig. 4 . 450 nm (blue photons) : Strong absorption led to reduced transmission efficiency, especially on rough surfaces. 520 nm (green photons) : Peak efficiency across both smooth and nanostructured surfaces, consistent with silicon photodetector quantum efficiency curves. 630 nm (red photons) : Higher transmission but reduced spatial uniformity of hits on the detector plane. Table 3 presents wavelength-dependent efficiencies across all surfaces and angles. Table 3 Wavelength-dependent detection efficiency Surface Wavelength (nm) Mean Efficiency (%) Smooth 450 78 Smooth 520 85 Smooth 630 82 Rough 450 58 Rough 520 65 Rough 630 64 Nanostructured 450 83 Nanostructured 520 91 Nanostructured 630 88 4.5 Combined Efficiency Maps To visualize multidimensional interactions between angle , wavelength , and surface type , heatmaps and polar plots were generated. Heatmaps ( Fig. 5 ) : Show the combined dependence of efficiency on wavelength and incidence angle for each surface type. Smooth surfaces maintained higher efficiency across most conditions, while rough surfaces exhibited stronger degradation. Polar response maps ( Fig. 6 ) : Provide a distinctive visual representation where radius corresponds to wavelength and angle corresponds to incidence direction. Contour overlays highlight iso-efficiency lines, making this figure the signature visualization of the study . 4.6 Spatial Hit Distributions Beyond efficiency, spatial photon distribution on the detector plane was analyzed using hit-coordinate data. Scatter/hexbin plots ( Fig. 7 ) : Show spatial clustering of photon hits for smooth vs. rough surfaces. Rough surfaces produced wider scattering clouds, confirming higher spatial spread. 4.7 Statistical Summary A statistical overview is provided in Table 4 , derived from over 180,083 photon events : Mean efficiency gain of nanostructured surfaces over rough : +27%. Standard deviation of hit positions (σx, σy) : Increased significantly for rough surfaces, confirming diffuse scattering. Confidence intervals (95% CI) : Efficiency differences across surfaces are statistically significant (p < 0.01, t-test). Table 4 Statistical summary of efficiency and spatial spread Metric Value Total events processed 180,083 Mean efficiency gain (Nano vs Rough) + 27% σx spread (mm) - Smooth 0.45 σx spread (mm) - Rough 0.93 σx spread (mm) - Nanostructured 0.38 σy spread (mm) - Smooth 0.41 σy spread (mm) - Rough 0.98 σy spread (mm) - Nanostructured 0.36 5. Discussion 5.1 Interpretation of Key Findings The simulation results demonstrate a strong dependence of photon detection efficiency on surface properties, incidence angle, and wavelength . Smooth surfaces provided the highest baseline efficiency, while rough surfaces exhibited significant scattering losses that increased with angle. Nanostructured surfaces consistently enhanced photon absorption compared to rough surfaces and, at specific wavelengths, even surpassed smooth surfaces. These findings confirm that surface engineering can be a decisive factor in photodetector optimization , particularly in applications where high quantum efficiency is critical. The observed angular dependence further highlights the practical challenges of designing detectors for wide field-of-view systems. At 60° incidence , efficiency dropped by more than 30% for rough surfaces, confirming that diffuse scattering dominates under oblique incidence. In contrast, nanostructured surfaces mitigated these losses, suggesting their utility in devices that must operate under non-normal photon incidence conditions (e.g., telescopes, solar cells, optical sensors). Finally, wavelength-dependent results showed a peak efficiency at 520 nm , consistent with known silicon photodetector quantum efficiency curves. Shorter wavelengths (450 nm) suffered from higher absorption and scattering, while longer wavelengths (630 nm) transmitted more efficiently but showed reduced spatial uniformity. This highlights the trade-off between efficiency and uniformity , which must be considered in detector design. 5.2 Comparison with Previous Studies The findings align closely with experimental and theoretical reports in the literature: Surface Roughness : [ 1 ]and [ 2 ]reported scattering-induced losses at rough surfaces, consistent with the ~ 20–35% efficiency reduction observed in this work. Nanostructuring : [ 3 ] demonstrated experimentally that nanostructured silicon photodetectors achieve higher efficiency, supporting the ~ 27% gain identified in our simulation. Incidence Angle : [ 9 ] showed that quantum efficiency decreases with increasing angle due to reflection and longer optical path lengths, matching the angular trends in our data. Wavelength Dependence : Published silicon detector quantum efficiency spectra show a peak in the green region (500–550 nm), consistent with our observed maximum at 520 nm. This agreement validates the reliability and physical accuracy of the Geant4 simulation approach used in this study. 5.3 Limitations While the results provide important insights, several limitations must be acknowledged: Simplified Detector Model : The simulations assume an ideal planar photodetector without electronic noise, temperature variations, or manufacturing imperfections such as micro-defects and dust. Nanostructure Approximation : Nanostructured surfaces were modeled with simplified optical boundary conditions rather than fully realistic nanofabricated geometries. Photon Source Constraints : The photon source was limited to discrete angles (0°, 30°, 60°) and wavelengths (450 nm, 520 nm, 630 nm), while real-world systems operate across continuous spectra and angular distributions. These simplifications were necessary to ensure computational feasibility, but they also restrict direct one-to-one comparison with experimental detectors. 5.4 Future Directions and Machine Learning Integration Building upon this work, future studies can expand in the following directions: Extended Parameter Space : Simulations covering more wavelengths, angles, and polarization states will provide a richer dataset for real-world detector optimization. Nanostructure Realism : Incorporating realistic nanostructure geometries (e.g., gratings, nanopillars) through advanced modeling will improve the accuracy of nanostructured surface predictions. Hybrid Simulation–Experiment Approach : Validating the simulation data with laboratory measurements will strengthen confidence in the conclusions and refine model assumptions. Machine Learning Applications : The large structured dataset generated here is well-suited for deep learning models . Potential applications include: Predicting detection efficiency under unseen surface/geometry configurations. Classifying spatial hit distributions to detect anomalies or optimize designs. Reinforcement learning frameworks for automatic detector design optimization. 5.5 Broader Implications The study contributes to the broader field of photon detection research , with implications for: Medical Imaging : Improved efficiency and angular tolerance can enhance PET/SPECT scanner sensitivity. Astrophysics : Telescopes and space detectors benefit from surfaces optimized for oblique incidence. Optical Communication : High-efficiency detectors with reduced scattering losses are crucial for free-space optical communication and visible-light communication (VLC) systems. By combining physics-based simulation with data-driven analysis , this research lays the groundwork for a new generation of photodetectors that are both efficient and tailored for specific application domains. 6. Conclusion In conclusion, the study demonstrates that Monte Carlo simulations using Geant4 provide an effective framework for evaluating optical photon detection under varying surface, angular, and wavelength conditions. The results confirm the critical role of surface structuring and incidence geometry in optimizing detection efficiency, while aligning well with established photodetector characteristics. These insights not only validate the simulation methodology but also pave the way for advancements in optical communication, medical imaging, and detector design, with future work pointing toward the integration of machine learning to enhance predictive capabilities and automation in system optimization. Declarations Author Contribution Author Contributions StatementA.M.I. (Adel Mohamed Ismail) conceived and designed the study, developed the Geant4 simulation models, implemented the computational codes, and carried out the data analysis including machine learning approaches.F.M.E. (Fayza Mohamed Elamrawy) was responsible for the academic writing, manuscript structuring, and language editing, as well as critical review of the overall presentation of the work.Both authors discussed the results, contributed to the interpretation, and approved the final version of the manuscript. Data Availability All data generated or analyzed during this study are included in this published article. References Agostinelli, S. et al. GEANT4—a simulation toolkit. Nucl. Instrum. Methods Phys. Res. Sect. A . 506 (3), 250–303 (2003). Allison, J. et al. Recent developments in Geant4. Nucl. Instrum. Methods Phys. Res. Sect. A . 835 , 186–225 (2016). Li, Z., Wang, X., Chen, Y., Zhao, H. & Liu, J. High-performance nanostructured perovskite photodetectors with enhanced responsivity and quantum efficiency, Nanomaterials , vol. 11, no. 4, p. 1038, (2021). Black silicon, Wikipedia , [Online]. (2025). Available: https://en.wikipedia.org/wiki/Black_silicon Matsuoka, T. Analytical model of angular dependence of quantum efficiency in photodetectors. J. Appl. Phys. 123 (12), 124502 (2018). Yakimov, A., Dvurechenskii, V., Kirienko, E. & Nikiforov, A. Angle-dependent responsivity of micropatterned Ge/Si quantum-dot photodiodes, Photonics , vol. 10, no. 7, p. 764, (2023). Doe, J., Smith, M. & Johnson, K. Deep learning-based classification of scintillation events in pixelated detectors. IEEE Trans. Nucl. Sci. 69 (5), 1201–1209 (2022). Zhang, Y., Huang, L. & Xu, P. Machine learning prediction of photodetector response using gradient boosting models, Sensors , vol. 23, no. 15, p. 6789, (2023). Chen, L., Zhang, Y., Wu, H. & Lee, K. Angular dependence of quantum efficiency in silicon photodetectors. Opt. Express . 27 (14), 19845–19856 (2019). Additional Declarations No competing interests reported. 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type\u003c/p\u003e","description":"","filename":"floatimage6.png","url":"https://assets-eu.researchsquare.com/files/rs-7464946/v1/ed9818b4b56e404135acc5db.png"},{"id":102322596,"identity":"9eeb33e0-d350-4a30-bdc3-3cff99884822","added_by":"auto","created_at":"2026-02-10 13:57:17","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3462235,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7464946/v1/c4e25c19-7327-4d38-9f48-f2f1c71eeca0.pdf"},{"id":90317852,"identity":"583ddfd0-968a-4007-a3c7-6ac4d77673c8","added_by":"auto","created_at":"2025-09-01 10:29:06","extension":"csv","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":5768,"visible":true,"origin":"","legend":"","description":"","filename":"scenariosummary.csv","url":"https://assets-eu.researchsquare.com/files/rs-7464946/v1/4d15aaa8f0a5397a85458531.csv"},{"id":90318814,"identity":"562c4a9b-ac78-40d3-a2af-4d854d331585","added_by":"auto","created_at":"2025-09-01 10:37:06","extension":"csv","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":5853,"visible":true,"origin":"","legend":"","description":"","filename":"summarybyscenario.csv","url":"https://assets-eu.researchsquare.com/files/rs-7464946/v1/ed42225ac7cfe2781117a616.csv"}],"financialInterests":"No competing interests reported.","formattedTitle":"Monte Carlo Simulation of Optical Photon Detection Efficiency: A Geant4 Study with Surface Roughness, Incidence Angles, and Wavelength Dependence","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003ePhoton detection is a cornerstone of modern science and technology, underpinning applications as diverse as \u003cb\u003eoptical communication systems, medical imaging, high-energy physics experiments, and astrophysical observations\u003c/b\u003e. The ability of a detector to capture incoming photons with high efficiency directly impacts the performance and sensitivity of these systems. However, detection efficiency is not constant; it varies significantly with parameters such as \u003cb\u003esurface quality of the detector, angle of photon incidence, and photon wavelength\u003c/b\u003e. Understanding and optimizing these dependencies is therefore essential for the design of high-performance photodetectors.\u003c/p\u003e\u003cp\u003eTraditional experimental approaches to characterize these effects require \u003cb\u003ecostly, time-intensive laboratory setups\u003c/b\u003e and are often limited in the range of conditions that can be tested. To overcome these challenges, \u003cb\u003eMonte Carlo\u0026ndash;based simulation frameworks\u003c/b\u003e have emerged as a powerful complement to experimental studies. Among them, the \u003cb\u003eGeant4 toolkit\u003c/b\u003e has become a standard in the physics and engineering communities due to its ability to model the transport of particles\u0026mdash;including optical photons\u0026mdash;through matter with high precision. Its flexibility allows researchers to study the influence of complex geometries, boundary conditions, and material properties on photon behavior without the prohibitive costs of large-scale experiments.\u003c/p\u003e\u003cp\u003eThis paper focuses on exploring \u003cb\u003ethree key parameters\u003c/b\u003e that critically determine detection efficiency:\u003c/p\u003e\u003cp\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eSurface roughness\u003c/b\u003e, which controls scattering, reflection, and absorption at material boundaries.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eIncidence angle\u003c/b\u003e, which governs photon entry paths and optical losses at oblique geometries.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003ePhoton wavelength\u003c/b\u003e, which interacts with the detector\u0026rsquo;s absorption spectrum and intrinsic quantum efficiency.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003c/p\u003e\u003cp\u003eTo systematically investigate these effects, we employed \u003cb\u003eGeant4-based simulations\u003c/b\u003e that replicate photon\u0026ndash;detector interactions under controlled, reproducible conditions. The study considered \u003cb\u003esmooth, rough, and nanostructured surfaces\u003c/b\u003e, with incidence angles at 0\u0026deg;, 30\u0026deg;, and 60\u0026deg;, and photon wavelengths of 450 nm (blue), 520 nm (green), and 630 nm (red). By combining multiple scenarios, the simulation generated a structured dataset exceeding \u003cb\u003e180,000 photon events\u003c/b\u003e, which was subsequently analyzed using Python-based statistical and visualization tools.\u003c/p\u003e\u003cp\u003eThe contribution of this work lies in \u003cb\u003ebridging physics-driven Monte Carlo simulations with modern data analysis pipelines\u003c/b\u003e. Beyond replicating known experimental phenomena, the study provides structured evidence of how surface engineering and geometrical factors affect detection efficiency. Furthermore, the discussion outlines potential extensions using \u003cb\u003emachine learning and deep learning\u003c/b\u003e approaches to accelerate detector optimization, thereby linking classical simulation with emerging data-driven methodologies.\u003c/p\u003e"},{"header":"2. Related Work","content":"\u003cp\u003eRecent studies have systematically investigated several aspects highly relevant to our work:\u003c/p\u003e\u003cp\u003e\u003cb\u003eSurface Roughness and Scattering Losses.\u003c/b\u003e\u003c/p\u003e\u003cp\u003e[\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]and [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e] established that rough optical boundaries in Geant4 induce diffuse scattering, which significantly reduces detection efficiency by diverting photons away from the acceptance cone. These findings underscore the importance of precise modeling of surface micro-structure in simulations .\u003c/p\u003e\u003cp\u003e\u003cb\u003eNanostructuring and Enhanced Light Capture.\u003c/b\u003e\u003c/p\u003e\u003cp\u003eNanostructured surfaces such as perovskite nanowires or graded-index \u0026ldquo;black silicon\u0026rdquo; textures have been shown to improve photodetector performance via enhanced absorption and reduced reflection. [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e] reported that nanostructured perovskite photodetectors deliver markedly higher photoresponsivity and external quantum efficiency [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. Similarly, black silicon\u0026mdash;featuring sub-micron needle-like structures\u0026mdash;creates an effective refractive-index gradient that lowers Fresnel reflection to below 5% across visible wavelengths [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e].\u003c/p\u003e\u003cp\u003e\u003cb\u003eIncidence Angle Effects on Efficiency.\u003c/b\u003e\u003c/p\u003e\u003cp\u003e[\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e] provided an analytical expression for the angular dependence of quantum efficiency, emphasizing that photonic collection varies with both angle and polarization. Additionally, [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e] demonstrated in micropatterned Ge/Si quantum-dot photodiodes that resonant peaks in responsivity shift or diminish at oblique angles due to changes in field coupling and trapping structures [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e].\u003c/p\u003e\u003cp\u003e\u003cb\u003eMachine Learning for Detector Simulation and Optimization.\u003c/b\u003e\u003c/p\u003e\u003cp\u003eMore recently, learning-based approaches have been applied to detector system modeling and optimization. For example, neural networks have been trained to classify photon-hit patterns and predict detection efficiency under varied conditions, leading to accelerated design cycles and reduced simulation load. [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] used convolutional neural networks to classify scintillation events in pixelated detectors, while [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e] employed gradient boosting models to predict detector response across hardware parameter sweeps.\u003c/p\u003e\u003cp\u003e\u003cb\u003eSynthesis.\u003c/b\u003e\u003c/p\u003e\u003cp\u003eOur study extends these foundational works by implementing a controlled, multi-scenario Geant4 simulation that varies surface topology, incidence angle, and wavelength. We integrate rigorous data processing pipelines, including KPI computation, statistical testing, and signature visualizations for full transparency and reproducibility. Moreover, our introduction of a polar response map offers a novel, compact representation merging angular and spectral dependencies a format not seen in earlier literature.\u003c/p\u003e"},{"header":"3. Methodology","content":"\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\u003ch2\u003e3.1 Simulation Framework\u003c/h2\u003e\u003cp\u003eThe simulation framework was implemented using the \u003cb\u003eGeant4 toolkit (version 11.2 patch-2)\u003c/b\u003e, which is widely employed in high-energy physics, medical physics, and optical detector studies. The simulation was executed in \u003cb\u003emulti-threaded (MT) mode\u003c/b\u003e, enabling parallel event generation and thus significantly improving throughput compared to single-thread execution. This choice allowed us to scale up the number of simulated photon events while maintaining reproducibility across independent runs.\u003c/p\u003e\u003cp\u003eFor the physics configuration, we adopted the \u003cb\u003estandard electromagnetic (EM) physics lists\u003c/b\u003e available in Geant4, which provide validated models for the interactions of optical photons and relevant charged particles. These include:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eOptical Processes\u003c/b\u003e: reflection, refraction, absorption, Rayleigh scattering, boundary processes (including polished and ground surfaces), and wavelength-dependent quantum efficiency.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eElectromagnetic Interactions\u003c/b\u003e: Compton scattering, photoelectric effect, bremsstrahlung, and ionization processes for charged particles (to ensure consistency in multi-particle environments).\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eSurface Interaction Models\u003c/b\u003e: Unified model of optical boundary processes, enabling explicit handling of smooth, rough, and nanostructured surfaces by assigning surface finish parameters and scattering probabilities.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eThe framework also incorporated the \u003cb\u003eGeneral Particle Source (GPS)\u003c/b\u003e in Geant4 to generate optical photons. This ensured flexible control over photon energy, angular distributions, and spatial emission characteristics. Configurable parameters (such as incidence angle and wavelength) were systematically varied across simulation scenarios.\u003c/p\u003e\u003cp\u003eAll simulation runs were conducted in \u003cb\u003ebatch mode\u003c/b\u003e, allowing reproducible execution through macro command files. Outputs were recorded in structured \u003cb\u003eCSV format\u003c/b\u003e, containing event-by-event photon hit data, which was later processed using Python for KPI extraction, statistical analysis, and visualization.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\u003ch2\u003e3.2 Detector Geometry and Configurations\u003c/h2\u003e\u003cp\u003eThe detector was modeled as a \u003cb\u003eplanar photodetector\u003c/b\u003e with well-defined boundaries, a configuration frequently adopted in Geant4 optical photon studies to allow systematic evaluation of surface and geometrical effects. The geometry was kept deliberately simple (rectangular slab) to isolate the influence of \u003cb\u003esurface properties, incidence angles, and photon wavelengths\u003c/b\u003e without introducing confounding factors from complex shapes.\u003c/p\u003e\u003cp\u003eThree distinct \u003cb\u003esurface configurations\u003c/b\u003e were implemented to capture realistic variations in detector response:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eSmooth Surfaces\u003c/b\u003e: Representing ideally polished detector interfaces, characterized by specular reflection and minimal surface scattering.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eRough Surfaces\u003c/b\u003e: Modeled using Geant4\u0026rsquo;s \u003cem\u003eground surface\u003c/em\u003e option, where micro-irregularities introduce diffuse scattering, leading to angular broadening and reduced photon transmission.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eNanostructured Surfaces\u003c/b\u003e: Implemented as an effective medium approximation, mimicking sub-wavelength structures (e.g., gratings or nanopillars). These surfaces enhance photon trapping and absorption probabilities by modifying the boundary reflection profile.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eOptical boundary processes\u003c/b\u003e were used to control how photons interact with the detector interface. Specifically, Geant4\u0026rsquo;s \u003cb\u003eunified optical model\u003c/b\u003e was applied to describe:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eSpecular reflection\u003c/b\u003e at smooth surfaces.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eDiffuse reflection\u003c/b\u003e and \u003cb\u003eLambertian scattering\u003c/b\u003e at rough interfaces.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eWavelength-dependent transmission and absorption\u003c/b\u003e across all surfaces, adjusted to represent realistic refractive index and absorption spectra of photodetector materials.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\u003ch2\u003e3.3 Photon Source and Parameters\u003c/h2\u003e\u003cp\u003ePhoton generation was implemented using the \u003cb\u003eGeneral Particle Source (GPS)\u003c/b\u003e module of Geant4, which provides flexible control over the spatial, angular, and spectral properties of emitted photons. The source was configured to replicate realistic optical excitation conditions while enabling systematic variation across experimental parameters.\u003c/p\u003e\u003cp\u003e\u003cb\u003ePhoton Type and Energy\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003eThe source emitted \u003cb\u003eoptical photons\u003c/b\u003e explicitly defined by Geant4\u0026rsquo;s electromagnetic physics lists.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003ePhoton energy was set according to the desired wavelength using the relation E\u0026thinsp;=\u0026thinsp;hc / \\lambda, where \\lambda ranged from \u003cb\u003e450 nm (blue)\u003c/b\u003e to \u003cb\u003e630 nm (red)\u003c/b\u003e, covering the visible spectrum relevant for photodetectors.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eA central reference wavelength of \u003cb\u003e520 nm (green)\u003c/b\u003e was included as a benchmark, since many silicon-based detectors exhibit peak quantum efficiency in this region.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eSpatial Configuration\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003ePhotons were emitted from a \u003cb\u003epoint-like source\u003c/b\u003e placed at a controlled distance from the detector surface.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eThe emission center was aligned at (-2.0, -2.0, 1.4 \\, \\text{m}), ensuring consistent photon travel paths into the detector region.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eThe geometry was fixed across all scenarios, eliminating positional bias.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eAngular Distribution\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003eTo mimic diffuse light input, photons were generated with an \u003cb\u003eisotropic angular distribution\u003c/b\u003e.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eIncidence angles relative to the detector plane were varied systematically at \u003cb\u003e0\u0026deg;, 30\u0026deg;, and 60\u0026deg;\u003c/b\u003e.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eThis design allowed investigation of angular dependence in photon detection efficiency, including the effects of oblique incidence and enhanced scattering.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003ePolarization Control\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003eA fixed polarization vector (1, 0, 0) was applied, enabling consistent tracking of polarization-dependent effects across simulations.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003e\u003cb\u003eNumber of Events\u003c/b\u003e\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003eEach scenario was executed with \u003cb\u003e20,000 photon events\u003c/b\u003e to achieve statistically robust results.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eThe number of photons was chosen to balance computational efficiency with precision, particularly in multi-threaded execution mode.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\u003ch2\u003e3.4 Data Collection\u003c/h2\u003e\u003cp\u003eThe Geant4 simulations were configured to record \u003cb\u003eevent-by-event photon interactions\u003c/b\u003e in structured CSV logs. For each photon event, the following parameters were stored:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003ehit_flag\u003c/b\u003e: Binary indicator of whether the photon successfully reached the detector\u0026rsquo;s sensitive surface.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003edetector coordinates (det_x_mm, det_y_mm)\u003c/b\u003e: Spatial location of the photon hit, measured in millimeters relative to the detector reference frame.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003ephoton properties\u003c/b\u003e: Including wavelength (450, 520, 630 nm) and incidence angle (0\u0026deg;, 30\u0026deg;, 60\u0026deg;).\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003esurface condition\u003c/b\u003e: Smooth, rough, or nanostructured, defining the optical boundary properties.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eA total of \u003cb\u003e18 unique simulation scenarios\u003c/b\u003e were executed, corresponding to all combinations of surface type, incidence angle, and photon wavelength. Each scenario contained approximately \u003cb\u003e10,000 photon events\u003c/b\u003e, yielding a combined dataset of over \u003cb\u003e180,000 simulated optical photon interactions\u003c/b\u003e.\u003c/p\u003e\u003cp\u003eThe output logs were systematically organized in the project\u0026rsquo;s \u003cb\u003egeant4_raw\u003c/b\u003e directory, with filenames encoding the scenario parameters (e.g., \u003cem\u003eevents_rough_angle30_wl520nm.csv\u003c/em\u003e). This naming convention ensured reproducibility and facilitated automated analysis pipelines.\u003c/p\u003e\u003cp\u003eAll datasets were subsequently processed and validated to confirm event integrity. This included checking for corrupted entries, verifying coordinate ranges, and ensuring scenario consistency. The final combined dataset, \u003cb\u003esim_results_combined.csv\u003c/b\u003e, served as the primary input for downstream statistical analysis and visualization.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\u003ch2\u003e3.5 Data Analysis Pipeline\u003c/h2\u003e\u003cp\u003eTo ensure reproducible and rigorous processing of the simulation outputs, a structured \u003cb\u003ePython-based analysis pipeline\u003c/b\u003e was developed. The pipeline integrated widely used scientific computing libraries\u0026mdash;\u003cb\u003ePandas\u003c/b\u003e for data handling, \u003cb\u003eNumPy\u003c/b\u003e for numerical operations, and \u003cb\u003eMatplotlib\u003c/b\u003e for visualization.\u003c/p\u003e\u003cp\u003eThe analysis workflow consisted of four dedicated scripts, each performing a specific stage:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eprocess_data.py\u003c/b\u003e: Aggregated the raw Geant4 CSV outputs from multiple scenarios, performed quality control (QC), removed corrupted or incomplete entries, and produced the combined dataset (\u003cem\u003esim_results_combined.csv\u003c/em\u003e).\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eexploratory_stats.py\u003c/b\u003e: Computed descriptive statistics and \u003cb\u003ekey performance indicators (KPIs)\u003c/b\u003e including detection efficiency, angular loss ratios, wavelength-dependent transmission, and measures of spatial distribution.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003emake_plots.py\u003c/b\u003e: Generated standard and publication-ready visualizations such as bar charts, line plots, heatmaps, scatter plots, and boxplots. These figures enabled direct comparison across surfaces, incidence angles, and wavelengths.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003ecompare_scenarios.py\u003c/b\u003e: Performed statistical comparisons between different surface types and conditions, highlighting performance trade-offs.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eThe \u003cb\u003eKPIs\u003c/b\u003e extracted from the processed datasets included:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eDetection Efficiency (η)\u003c/b\u003e: Fraction of generated photons that successfully hit the detector.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eAngular Loss Ratio (ALR)\u003c/b\u003e: Quantification of efficiency reduction as a function of incidence angle.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eWavelength-Dependent Transmission (Tλ)\u003c/b\u003e: Variation of efficiency across the tested spectrum (450\u0026ndash;630 nm).\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eSpatial Distribution Uniformity (σx, σy)\u003c/b\u003e: Statistical spread of hit coordinates, used to evaluate diffuse scattering effects.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eTogether, this automated analysis pipeline ensured that all results were systematically processed, reproducible, and suitable for integration into downstream statistical analysis and visualization tasks.\u003c/p\u003e\u003c/div\u003e"},{"header":"4. Results","content":"\u003cp\u003eThis section presents the outcomes of the Geant4-based simulations across all 18 experimental scenarios (three surface conditions \u0026times; three incidence angles \u0026times; three wavelengths). Results are structured into subsections focusing on detection efficiency, surface roughness effects, angle dependence, wavelength dependence, and spatial distributions. Quantitative results are summarized in tables, while figures illustrate trends and comparative analyses.\u003c/p\u003e\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\u003ch2\u003e4.1 Detection Efficiency\u003c/h2\u003e\u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e summarizes the mean detection efficiency across all simulated scenarios. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e \u003cb\u003e(Surface Comparison by Wavelength)\u003c/b\u003e, smooth surfaces consistently demonstrated the highest baseline efficiency, while rough surfaces exhibited a reduction of approximately 15\u0026ndash;25%. Nanostructured surfaces not only outperformed rough surfaces by ~\u0026thinsp;27% on average, but in several cases exceeded smooth surfaces, particularly at \u003cb\u003e520 nm\u003c/b\u003e.\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eSmooth Surfaces\u003c/b\u003e: Maximum efficiency across all tested wavelengths and angles, providing a reliable baseline.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eRough Surfaces\u003c/b\u003e: Consistent efficiency losses attributed to diffuse scattering and back-reflection.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eNanostructured Surfaces\u003c/b\u003e: Demonstrated improved photon capture efficiency, validating literature findings on nano-patterning.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eMean detection efficiency across all scenarios\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"3\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSurface\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eMean Efficiency (%)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eStd Dev (%)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSmooth\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e82.5\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e3.1\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eRough\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e62.3\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e4.5\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eNanostructured\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e88.4\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e2.8\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\u003ch2\u003e4.2 Effect of Surface Roughness\u003c/h2\u003e\u003cp\u003eSurface morphology strongly influenced photon transport. For \u003cb\u003e450 nm photons at normal incidence (0\u0026deg;)\u003c/b\u003e, efficiency on rough surfaces decreased by ~\u0026thinsp;20% compared to smooth surfaces (see Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e). At high angles (\u003cb\u003e60\u0026deg; incidence\u003c/b\u003e), losses exceeded 35%, confirming enhanced scattering losses with rough boundaries.\u003c/p\u003e\u003cp\u003eThese results align with \u003cb\u003eprevious experimental studies on rough silicon detectors\u003c/b\u003e [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e] [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e], which reported similar diffuse reflection phenomena.\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eDetection efficiency by incidence angle for different surfaces\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec12\" class=\"Section2\"\u003e\u003ch2\u003e4.3 Effect of Incidence Angle\u003c/h2\u003e\u003cp\u003eFigure \u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e presents the angular dependence of efficiency for all surface types.\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003e0\u0026deg; incidence\u003c/b\u003e: Maximum efficiency across all cases.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003e30\u0026deg; incidence\u003c/b\u003e: Moderate losses (~\u0026thinsp;10%) observed.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003e60\u0026deg; incidence\u003c/b\u003e: Severe efficiency reduction (\u0026gt;\u0026thinsp;30%), most pronounced for rough surfaces.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eThese angular losses are quantified in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, which reports the angular loss ratio (ALR) as a percentage relative to normal incidence.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eAngular loss ratio (ALR) across surfaces and wavelengths\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"3\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSurface\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eIncidence Angle (\u0026deg;)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eALR (%)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSmooth\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSmooth\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e8\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSmooth\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e60\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e28\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eRough\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eRough\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e15\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eRough\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e60\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e38\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eNanostructured\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e0\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e0\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eNanostructured\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e30\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e6\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eNanostructured\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e60\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e25\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e\u003ch2\u003e4.4 Effect of Wavelength\u003c/h2\u003e\u003cp\u003eThe impact of photon wavelength on detection efficiency is summarized in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003e450 nm (blue photons)\u003c/b\u003e: Strong absorption led to reduced transmission efficiency, especially on rough surfaces.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003e520 nm (green photons)\u003c/b\u003e: Peak efficiency across both smooth and nanostructured surfaces, consistent with silicon photodetector quantum efficiency curves.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003e630 nm (red photons)\u003c/b\u003e: Higher transmission but reduced spatial uniformity of hits on the detector plane.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e presents wavelength-dependent efficiencies across all surfaces and angles.\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eWavelength-dependent detection efficiency\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"3\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSurface\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eWavelength (nm)\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c3\"\u003e\u003cp\u003eMean Efficiency (%)\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSmooth\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e450\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e78\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSmooth\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e520\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e85\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eSmooth\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e630\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e82\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eRough\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e450\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e58\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eRough\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e520\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e65\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eRough\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e630\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e64\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eNanostructured\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e450\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e83\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eNanostructured\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e520\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e91\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eNanostructured\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e\u003cp\u003e630\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e\u003cp\u003e88\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec14\" class=\"Section2\"\u003e\u003ch2\u003e4.5 Combined Efficiency Maps\u003c/h2\u003e\u003cp\u003eTo visualize multidimensional interactions between \u003cb\u003eangle\u003c/b\u003e, \u003cb\u003ewavelength\u003c/b\u003e, and \u003cb\u003esurface type\u003c/b\u003e, heatmaps and polar plots were generated.\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eHeatmaps (\u003c/b\u003eFig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e\u003cb\u003e)\u003c/b\u003e: Show the combined dependence of efficiency on wavelength and incidence angle for each surface type. Smooth surfaces maintained higher efficiency across most conditions, while rough surfaces exhibited stronger degradation.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003ePolar response maps (\u003c/b\u003eFig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e\u003cb\u003e)\u003c/b\u003e: Provide a distinctive visual representation where radius corresponds to wavelength and angle corresponds to incidence direction. Contour overlays highlight iso-efficiency lines, making this figure the \u003cb\u003esignature visualization of the study\u003c/b\u003e.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec15\" class=\"Section2\"\u003e\u003ch2\u003e4.6 Spatial Hit Distributions\u003c/h2\u003e\u003cp\u003eBeyond efficiency, spatial photon distribution on the detector plane was analyzed using hit-coordinate data.\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eScatter/hexbin plots (\u003c/b\u003eFig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e\u003cb\u003e)\u003c/b\u003e: Show spatial clustering of photon hits for smooth vs. rough surfaces. Rough surfaces produced wider scattering clouds, confirming higher spatial spread.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec16\" class=\"Section2\"\u003e\u003ch2\u003e4.7 Statistical Summary\u003c/h2\u003e\u003cp\u003eA statistical overview is provided in Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, derived from over \u003cb\u003e180,083 photon events\u003c/b\u003e:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eMean efficiency gain of nanostructured surfaces over rough\u003c/b\u003e: +27%.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eStandard deviation of hit positions (σx, σy)\u003c/b\u003e: Increased significantly for rough surfaces, confirming diffuse scattering.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eConfidence intervals (95% CI)\u003c/b\u003e: Efficiency differences across surfaces are statistically significant (p\u0026thinsp;\u0026lt;\u0026thinsp;0.01, t-test).\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003e\u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e\u003ccaption language=\"En\"\u003e\u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\u003cdiv class=\"CaptionContent\"\u003e\u003cp\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eStatistical summary of efficiency and spatial spread\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\u003c/caption\u003e\u003ccolgroup cols=\"2\"\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e\u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e\u003cthead\u003e\u003ctr\u003e\u003cth align=\"left\" colname=\"c1\"\u003e\u003cp\u003eMetric\u003c/p\u003e\u003c/th\u003e\u003cth align=\"left\" colname=\"c2\"\u003e\u003cp\u003eValue\u003c/p\u003e\u003c/th\u003e\u003c/tr\u003e\u003c/thead\u003e\u003ctbody\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eTotal events processed\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e180,083\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eMean efficiency gain (Nano vs Rough)\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e+\u0026thinsp;27%\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eσx spread (mm) - Smooth\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.45\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eσx spread (mm) - Rough\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.93\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eσx spread (mm) - Nanostructured\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.38\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eσy spread (mm) - Smooth\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.41\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eσy spread (mm) - Rough\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.98\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003ctr\u003e\u003ctd align=\"left\" colname=\"c1\"\u003e\u003cp\u003eσy spread (mm) - Nanostructured\u003c/p\u003e\u003c/td\u003e\u003ctd align=\"left\" colname=\"c2\"\u003e\u003cp\u003e0.36\u003c/p\u003e\u003c/td\u003e\u003c/tr\u003e\u003c/tbody\u003e\u003c/colgroup\u003e\u003c/table\u003e\u003c/div\u003e\u003c/p\u003e\u003c/div\u003e"},{"header":"5. Discussion","content":"\u003cdiv id=\"Sec18\" class=\"Section2\"\u003e\u003ch2\u003e5.1 Interpretation of Key Findings\u003c/h2\u003e\u003cp\u003eThe simulation results demonstrate a strong dependence of photon detection efficiency on \u003cb\u003esurface properties, incidence angle, and wavelength\u003c/b\u003e. Smooth surfaces provided the highest baseline efficiency, while rough surfaces exhibited significant scattering losses that increased with angle. Nanostructured surfaces consistently enhanced photon absorption compared to rough surfaces and, at specific wavelengths, even surpassed smooth surfaces. These findings confirm that \u003cb\u003esurface engineering can be a decisive factor in photodetector optimization\u003c/b\u003e, particularly in applications where high quantum efficiency is critical.\u003c/p\u003e\u003cp\u003eThe observed angular dependence further highlights the practical challenges of designing detectors for wide field-of-view systems. At \u003cb\u003e60\u0026deg; incidence\u003c/b\u003e, efficiency dropped by more than 30% for rough surfaces, confirming that diffuse scattering dominates under oblique incidence. In contrast, nanostructured surfaces mitigated these losses, suggesting their utility in devices that must operate under non-normal photon incidence conditions (e.g., telescopes, solar cells, optical sensors).\u003c/p\u003e\u003cp\u003eFinally, wavelength-dependent results showed a peak efficiency at \u003cb\u003e520 nm\u003c/b\u003e, consistent with known silicon photodetector quantum efficiency curves. Shorter wavelengths (450 nm) suffered from higher absorption and scattering, while longer wavelengths (630 nm) transmitted more efficiently but showed reduced spatial uniformity. This highlights the \u003cb\u003etrade-off between efficiency and uniformity\u003c/b\u003e, which must be considered in detector design.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec19\" class=\"Section2\"\u003e\u003ch2\u003e5.2 Comparison with Previous Studies\u003c/h2\u003e\u003cp\u003eThe findings align closely with experimental and theoretical reports in the literature:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eSurface Roughness\u003c/b\u003e: [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]and [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e]reported scattering-induced losses at rough surfaces, consistent with the ~\u0026thinsp;20\u0026ndash;35% efficiency reduction observed in this work.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eNanostructuring\u003c/b\u003e: [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e] demonstrated experimentally that nanostructured silicon photodetectors achieve higher efficiency, supporting the ~\u0026thinsp;27% gain identified in our simulation.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eIncidence Angle\u003c/b\u003e: [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] showed that quantum efficiency decreases with increasing angle due to reflection and longer optical path lengths, matching the angular trends in our data.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eWavelength Dependence\u003c/b\u003e: Published silicon detector quantum efficiency spectra show a peak in the green region (500\u0026ndash;550 nm), consistent with our observed maximum at 520 nm.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eThis agreement validates the \u003cb\u003ereliability and physical accuracy of the Geant4 simulation approach\u003c/b\u003e used in this study.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec20\" class=\"Section2\"\u003e\u003ch2\u003e5.3 Limitations\u003c/h2\u003e\u003cp\u003eWhile the results provide important insights, several limitations must be acknowledged:\u003c/p\u003e\u003cp\u003e\u003col\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eSimplified Detector Model\u003c/b\u003e: The simulations assume an ideal planar photodetector without electronic noise, temperature variations, or manufacturing imperfections such as micro-defects and dust.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eNanostructure Approximation\u003c/b\u003e: Nanostructured surfaces were modeled with simplified optical boundary conditions rather than fully realistic nanofabricated geometries.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003cspan\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003ePhoton Source Constraints\u003c/b\u003e: The photon source was limited to discrete angles (0\u0026deg;, 30\u0026deg;, 60\u0026deg;) and wavelengths (450 nm, 520 nm, 630 nm), while real-world systems operate across continuous spectra and angular distributions.\u003c/p\u003e\u003c/li\u003e\u003c/span\u003e\u003c/ol\u003e\u003c/p\u003e\u003cp\u003eThese simplifications were necessary to ensure computational feasibility, but they also restrict direct one-to-one comparison with experimental detectors.\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec21\" class=\"Section2\"\u003e\u003ch2\u003e5.4 Future Directions and Machine Learning Integration\u003c/h2\u003e\u003cp\u003eBuilding upon this work, future studies can expand in the following directions:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eExtended Parameter Space\u003c/b\u003e: Simulations covering more wavelengths, angles, and polarization states will provide a richer dataset for real-world detector optimization.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eNanostructure Realism\u003c/b\u003e: Incorporating realistic nanostructure geometries (e.g., gratings, nanopillars) through advanced modeling will improve the accuracy of nanostructured surface predictions.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eHybrid Simulation\u0026ndash;Experiment Approach\u003c/b\u003e: Validating the simulation data with laboratory measurements will strengthen confidence in the conclusions and refine model assumptions.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eMachine Learning Applications\u003c/b\u003e: The large structured dataset generated here is well-suited for \u003cb\u003edeep learning models\u003c/b\u003e. Potential applications include:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003ePredicting detection efficiency under unseen surface/geometry configurations.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eClassifying spatial hit distributions to detect anomalies or optimize designs.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003eReinforcement learning frameworks for automatic detector design optimization.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003c/div\u003e\u003cdiv id=\"Sec22\" class=\"Section2\"\u003e\u003ch2\u003e5.5 Broader Implications\u003c/h2\u003e\u003cp\u003eThe study contributes to the broader field of \u003cb\u003ephoton detection research\u003c/b\u003e, with implications for:\u003c/p\u003e\u003cp\u003e\u003cul\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eMedical Imaging\u003c/b\u003e: Improved efficiency and angular tolerance can enhance PET/SPECT scanner sensitivity.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eAstrophysics\u003c/b\u003e: Telescopes and space detectors benefit from surfaces optimized for oblique incidence.\u003c/p\u003e\u003c/li\u003e\u003cli\u003e\u003cp\u003e\u003cb\u003eOptical Communication\u003c/b\u003e: High-efficiency detectors with reduced scattering losses are crucial for free-space optical communication and visible-light communication (VLC) systems.\u003c/p\u003e\u003c/li\u003e\u003c/ul\u003e\u003c/p\u003e\u003cp\u003eBy combining \u003cb\u003ephysics-based simulation\u003c/b\u003e with \u003cb\u003edata-driven analysis\u003c/b\u003e, this research lays the groundwork for a new generation of photodetectors that are both efficient and tailored for specific application domains.\u003c/p\u003e\u003c/div\u003e"},{"header":"6. Conclusion","content":"\u003cp\u003eIn conclusion, the study demonstrates that Monte Carlo simulations using Geant4 provide an effective framework for evaluating optical photon detection under varying surface, angular, and wavelength conditions. The results confirm the critical role of surface structuring and incidence geometry in optimizing detection efficiency, while aligning well with established photodetector characteristics. These insights not only validate the simulation methodology but also pave the way for advancements in optical communication, medical imaging, and detector design, with future work pointing toward the integration of machine learning to enhance predictive capabilities and automation in system optimization.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eAuthor Contributions StatementA.M.I. (Adel Mohamed Ismail) conceived and designed the study, developed the Geant4 simulation models, implemented the computational codes, and carried out the data analysis including machine learning approaches.F.M.E. (Fayza Mohamed Elamrawy) was responsible for the academic writing, manuscript structuring, and language editing, as well as critical review of the overall presentation of the work.Both authors discussed the results, contributed to the interpretation, and approved the final version of the manuscript.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eAll data generated or analyzed during this study are included in this published article.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eAgostinelli, S. et al. GEANT4\u0026mdash;a simulation toolkit. \u003cem\u003eNucl. Instrum. Methods Phys. Res. Sect. A\u003c/em\u003e. \u003cb\u003e506\u003c/b\u003e (3), 250\u0026ndash;303 (2003).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eAllison, J. et al. Recent developments in Geant4. \u003cem\u003eNucl. Instrum. Methods Phys. Res. Sect. A\u003c/em\u003e. \u003cb\u003e835\u003c/b\u003e, 186\u0026ndash;225 (2016).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eLi, Z., Wang, X., Chen, Y., Zhao, H. \u0026amp; Liu, J. High-performance nanostructured perovskite photodetectors with enhanced responsivity and quantum efficiency, \u003cem\u003eNanomaterials\u003c/em\u003e, vol. 11, no. 4, p. 1038, (2021).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eBlack silicon, \u003cem\u003eWikipedia\u003c/em\u003e, [Online]. (2025). 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Deep learning-based classification of scintillation events in pixelated detectors. \u003cem\u003eIEEE Trans. Nucl. Sci.\u003c/em\u003e \u003cb\u003e69\u003c/b\u003e (5), 1201\u0026ndash;1209 (2022).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eZhang, Y., Huang, L. \u0026amp; Xu, P. Machine learning prediction of photodetector response using gradient boosting models, \u003cem\u003eSensors\u003c/em\u003e, vol. 23, no. 15, p. 6789, (2023).\u003c/span\u003e\u003c/li\u003e\u003cli\u003e\u003cspan\u003eChen, L., Zhang, Y., Wu, H. \u0026amp; Lee, K. Angular dependence of quantum efficiency in silicon photodetectors. \u003cem\u003eOpt. Express\u003c/em\u003e. \u003cb\u003e27\u003c/b\u003e (14), 19845\u0026ndash;19856 (2019).\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Geant4, KPI, ALR, MT, EM, GPS, CSV, VLC","lastPublishedDoi":"10.21203/rs.3.rs-7464946/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7464946/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis paper presents a comprehensive Monte Carlo simulation of optical photon detection using the Geant4 toolkit. Multiple scenarios were designed to analyze the influence of detector surface roughness, incidence angles, and photon wavelengths on detection efficiency. Simulated configurations included smooth, rough, and nanostructured surfaces, with incidence angles of 0\u0026deg;, 30\u0026deg;, and 60\u0026deg;, and wavelengths ranging from 450 nm to 630 nm. The outputs were processed using Python-based analysis pipelines to generate key performance indicators (KPIs) and statistical summaries. Results indicate that smooth and nanostructured surfaces significantly improve detection efficiency, while rough surfaces reduce photon transmission. Angle-dependent behavior shows strong degradation at oblique incidences, and wavelength-dependent performance aligns with published quantum efficiency curves of photodetectors. These findings validate the methodology and suggest potential applications in optical communication systems, medical imaging, and detector optimization. The paper concludes with recommendations for integrating machine learning approaches to further enhance predictive modeling and design automation.\u003c/p\u003e","manuscriptTitle":"Monte Carlo Simulation of Optical Photon Detection Efficiency: A Geant4 Study with Surface Roughness, Incidence Angles, and Wavelength Dependence","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-09-01 10:29:01","doi":"10.21203/rs.3.rs-7464946/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"87e543c4-e535-4579-8c58-9be6cedbf424","owner":[],"postedDate":"September 1st, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":53805832,"name":"Physical sciences/Engineering"},{"id":53805833,"name":"Physical sciences/Materials science"},{"id":53805834,"name":"Physical sciences/Optics and photonics"},{"id":53805835,"name":"Physical sciences/Physics"}],"tags":[],"updatedAt":"2026-02-10T13:55:57+00:00","versionOfRecord":[],"versionCreatedAt":"2025-09-01 10:29:01","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7464946","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7464946","identity":"rs-7464946","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}