Modeling the Variance of Passive SiPMs in the Nonlinear Regime

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript presents an analytical and phenomenological framework for modeling the variance of the charge response of silicon photomultipliers (SiPMs) across the full dynamic range, including nonlinear regimes. The authors derive exact solutions for two limiting cases (instantaneous and very long light pulses), propose an empirical interpolation model for intermediate pulse durations, and validate it against Monte Carlo simulations and experimental measurements. The manuscript is well written, and the validation is thorough, but the incremental novelty and the lack of mechanistic explanation for the phenomenological model require minor revision.

1. The core idea of modeling SiPM nonlinearity phenomenologically has been extensively studied (e.g., Vinogradov, Jeans, Van Dam et al.). The authors should explicitly compare their approach with existing variance models and clearly articulate what new physical insight or predictive capability this work provides beyond previous models.
2. The proposed f(x) function (Eq. 33) is purely phenomenological. The authors must provide a physical interpretation for each term and parameter (α, β, ε, γ, δ) and justify the functional forms chosen. Why is the interpolation between two limits valid for arbitrary pulse durations? What is the physical meaning of α, β, and ε? How do they relate to recovery dynamics?
3. Table 1 shows that for very short pulses (Td/τ = 0.01), α = 0 and β and ε are not well-determined (e.g., ε = 1.142). The authors should discuss the identifiability and uncertainty of these parameters, especially when fitting experimental data with multiple free parameters.
4. The model assumes passive-quenching SiPMs and Poissonian photon statistics (except Appendix A). The authors should explicitly discuss the limitations and applicability of their model to other SiPM types (e.g., digital SiPMs, different pixel geometries) and non-Poissonian light sources in practical scenarios.
5. The model treats correlated noise via the terms γ·e^{-δx} and c·e^{-dx}. The authors should clarify why the suppression of correlated noise is modeled as a simple exponential decay with x, and whether this is physically motivated or just a convenient approximation. Also, the crosstalk suppression effect observed in simulations (f(x) below unity) is not captured by the model—this should be acknowledged and discussed.
6. The experimental data show good agreement with fits, but the number of free parameters is large relative to the data range (especially for LYSO scintillator, where β and ε were fixed). The authors should provide a sensitivity analysis or discuss how robust the parameter extraction is under practical measurement conditions.
7. The resolution definition in Eq. (39) includes a factor √N that is not standard in all SiPM literature. The authors should justify this normalization and clarify how it facilitates comparison across SiPMs with different N.
8. Minor Issues: Some citations are incomplete (e.g., Ref. [31] appears to be a student thesis—please verify).

 

Author Response

Dear Editor,

We would like to express our gratitude to Reviewer 1 for the detailed reading of our manuscript, the constructive suggestions, and the valuable feedback provided. We have addressed all the reviewer’s points step-by-step and revised the manuscript accordingly. Below is the point-by-point response to the reviewer’s comments.

1. The core idea of modeling SiPM nonlinearity phenomenologically has been extensively studied (e.g., Vinogradov, Jeans, Van Dam et al.). The authors should explicitly compare their approach with existing variance models and clearly articulate what new physical insight or predictive capability this work provides beyond previous models.

We thank the reviewer for pointing out the need to position our work relative to the literature. We would like to clarify that previous studies such as Van Dam et al. and Jeans focus on modeling the mean response rather than variance fluctuations. To the best of our knowledge, charge variance was only modeled by Stoykov for short pulses assuming zero intrinsic variance and no correlated noise (which reduces to equation (15) in our framework when ). Regarding the work of Vinogradov et al. [23], their framework models excess noise due to nonlinearity by adopting Stoykov's model for short pulses and a non-paralyzable dead-time model for long pulses, incorporating pixel recovery corrections in [33].

To address the reviewer’s suggestion and articulate the predictive capabilities of our framework:

• We have added two new references [23,33] replacing a previous citation (Ref. [24] in the original manuscript).
• A dedicated comparative discussion has been incorporated into Section 4 ("Comparison with previous model") and Figure 6, specifically analyzing the differences in photon-counting resolution predictions between our model and the models by Vinogradov et al.
• We show that while Vinogradov's models provide good agreement for long rectangular pulses, they break down for short pulses, non-rectangular pulse shapes, and high saturation levels, where our proposed model maintains high accuracy.

2. The proposed f(x) function (Eq. 33) is purely phenomenological. The authors must provide a physical interpretation for each term and parameter (α, β, ε, γ, δ) and justify the functional forms chosen. Why is the interpolation between two limits valid for arbitrary pulse durations? What is the physical meaning of α, β, and ε? How do they relate to recovery dynamics

We have revised the description of equation (33) and introduced a dedicated new figure (Figure 2) to provide a clearer physical and mathematical justification:

• The proposed  replaces the traditional Excess Noise Factor to capture relative charge fluctuations per triggered pixel across arbitrary illumination durations.
• The parameter  serves as the main interpolation weight:  corresponds to the short-pulse limit, whereas  retrieves the long-pulse limit.
• The parameters  and  were introduced to provide functional flexibility for arbitrary intermediate pulse shapes. Mathematically,  controls the location of the peak excess fluctuations in , while  fine-tunes the asymptotic saturation behavior as .
• The term  models the attenuation of correlated noise fluctuations as the pool of un-triggered pixels decreases exponentially.

3. Table 1 shows that for very short pulses (Td/τ = 0.01), α = 0 and β and ε are not well-determined (e.g., ε = 1.142). The authors should discuss the identifiability and uncertainty of these parameters, especially when fitting experimental data with multiple free parameters.

We appreciate the reviewer noticing this nuance. We have clarified the parameter sensitivity and uncertainties in Section 3.2:

• For very short pulses (), , which mathematically eliminates the contribution of the terms containing  and  in . Thus, these parameters play no role in the short-pulse limit.
• For short but finite pulses (), the value  reflects a lower sensitivity of the objective function to  in this regime.
• We have updated the text in Section 3.2 to discuss parameter identifiability explicitly and added representative numerical uncertainties (standard errors) for the fitted parameters to illustrate stability across different pulse ratios.

4. The model assumes passive-quenching SiPMs and Poissonian photon statistics (except Appendix A). The authors should explicitly discuss the limitations and applicability of their model to other SiPM types (e.g., digital SiPMs, different pixel geometries) and non-Poissonian light sources in practical scenarios.

We have updated the title, abstract, introduction, Section 5.1, the conclusions and Appendix A to clearly define the domain of applicability of our model:

• SiPM Architecture: The restriction to passive-quenching SiPMs is now explicitly highlighted in the title, abstract, introduction and conclusions. The fundamental differences in recovery dynamics mean the model is not directly applicable to active-quenching or digital SiPMs.
• Pixel Geometry: We have noted in Section 5.1 that microcell geometry primarily affects the probability and spatial cascade of correlated noise. The empirical parameters  and  effectively absorb these geometry-dependent net crosstalk and afterpulsing effects.
• Photon Statistics: While primary derivations in the main text assume Poissonian statistics, we explicitly clarify in Appendix A that the theoretical framework incorporates super-Poissonian light (such as negative binomial distributions) and can be adapted to arbitrary photon statistics provided the mean and variance of the incoming photon distribution are known.

5. The model treats correlated noise via the terms γ·e^{-δx} and c·e^{-dx}. The authors should clarify why the suppression of correlated noise is modeled as a simple exponential decay with x, and whether this is physically motivated or just a convenient approximation. Also, the crosstalk suppression effect observed in simulations (f(x) below unity) is not captured by the model—this should be acknowledged and discussed.

We have expanded the discussion on correlated noise suppression in Section 2.3 and Section 3.2:

• Exponential Suppression: As established in our previous work, modeling suppression via exponential decay terms ( and ) is a convenient approximation that aligns very well with Monte Carlo data. Physically, this form is justified because the fraction of available, untriggered microcells capable of sustaining secondary crosstalk avalanches decreases exponentially with increasing photon occupancy .
• Sub-unity  Effect: We have explicitly acknowledged that for certain parameters, simulated  curves exhibit slight sub-unity values (). The phenomenological model does not capture this minor feature; however, we clarify in the text that this deviation is extremely small and has a negligible impact on the overall integrated charge and photon-counting resolution predictions.

6. The experimental data show good agreement with fits, but the number of free parameters is large relative to the data range (especially for LYSO scintillator, where β and ε were fixed). The authors should provide a sensitivity analysis or discuss how robust the parameter extraction is under practical measurement conditions.

We have expanded Section 5.4 to address fit parameter robustness under practical experimental conditions. We emphasize that when fitting experimental data covering a restricted dynamic range (such as the LYSO measurements, where ), attempting to fit all parameters simultaneously leads to overparameterization. Setting  and  is a necessary and practical constraint that stabilizes the extraction of primary parameters without compromising accuracy over the measured illumination range.

7. The resolution definition in Eq. (39) includes a factor √N that is not standard in all SiPM literature. The authors should justify this normalization and clarify how it facilitates comparison across SiPMs with different N.

We have revised the text in Section 2.1 and Section 2.4 to better justify the  scaling factor:

• Standard relative resolution metrics inherently depend on the total number of microcells  of the tested device.
• Scaling the relative charge resolution  and the photon-counting resolution by  yields a normalized, dimensionless resolution metric that is independent of . This normalization allows direct, fair comparisons across SiPM devices with different microcell counts and pixel densities.

8. Minor Issues: Some citations are incomplete (e.g., Ref. [31] appears to be a student thesis—please verify).

We have reviewed and updated the bibliography. We have clarified that Ref. [31] corresponds to an undergraduate research project conducted within the DACIU academic program. Additionally, DOIs and publication details for several references have been checked and corrected.

 

Reviewer 2 Report

Comments and Suggestions for Authors

The authors proposed a phenomenological model for the variance of the passive SiPM charge response, which can be used for a wide range of light pulse durations. The model considered pixel recovery and correlated noise as well. Besides, the model was verified by simulations and experimental results. It provides a meaningful research account of SiPM in this work, though SiPM is not a very new topic. I think it can be accepted for publication after some issues are clarified. The detailed comments and suggestions are shown below.

1. Since the paper is constricted to passive SiPM, I recommend that the authors mention passive SiPM in the title.
2. The equation (33) and f(x) is the key part in the paper, however, I don’t think the authors make a clear and thorough explanation to this function. What is the physical meaning of this function?
3. Every symbol appearing in the equation should be indicated and explained in the subsequent text next to the equation. I find a lot of symbols in the equation which are not indicated in the following text. I need to take a guess on their meanings.
4. Speaking of Fig. 1, the authors provided the discussion of Fig.1 far behind the figure. It impairs the readability of the paper. The reader wants to know the physical meaning of the variables and implication of the curves when they see them in the figure. Since it is based on the phenomenological model, the parameters used in the model are very important. The authors need to explain why the values of the parameters were chosen when they calculated the data for Fig. 1.
5. In Fig.2 and Fig.3, why there seems to be great discrepancies between fitting and simulation results of f(x)? Especially, when compared with the upper two figures in Fig.2 and Fig.3.
6. I wonder the fitted values listed in Table 1 and Table 2 are the optimal. The authors need to explain why/how the values were determined.
7. When the correlated noise was included, it seemed that the discrepancy between fitting and simulation is larger than that for the case when the noise was neglected. The authors should explain the reason.
8. What are the exact waveform of the laser and LED pulse? Could the authors provide the waveform by oscilloscope? Why the authors didn’t provide the results measured by the control photodiode. I wonder how they convert the light intensity into x.
9. I guess the word “ansatz” in line 246 in Page 8 is German, please change it into English.

Author Response

Dear Editor,

We would like to express our gratitude to Reviewer 2 for the positive evaluation of our work, recognizing the meaningful contribution of our research, and for providing constructive comments and suggestions. We have carefully revised the manuscript to clarify every point raised by the reviewer. Below is our point-by-point response to the reviewer’s comments.

1. Since the paper is constricted to passive SiPM, I recommend that the authors mention passive SiPM in the title.

We agree with the reviewer’s recommendation. The title of the manuscript has been updated to "Modeling the variance of passive SiPMs in the nonlinear regime". In addition, we have explicitly highlighted the scope restriction to passive-quenching SiPMs in the abstract, introduction, Section 2.3, and the conclusions.

2. The equation (33) and f(x) is the key part in the paper, however, I don’t think the authors make a clear and thorough explanation to this function. What is the physical meaning of this function?

We have expanded the explanation surrounding equation (33) and introduced a dedicated new figure (Figure 2) in Section 2.3 to illustrate its behavior. Specifically, we clarify that  replaces the traditional Excess Noise Factor () to capture the relative charge variance per triggered pixel across arbitrary illumination durations. The function parameterizes how pixel recovery dynamics and correlated noise suppression continuously modify charge fluctuations as the device transitions into the saturation regime. Furthermore, Figure 2 visually demonstrates the physical effect and influence of each model parameter across intermediate pulse durations.

3. Every symbol appearing in the equation should be indicated and explained in the subsequent text next to the equation. I find a lot of symbols in the equation which are not indicated in the following text. I need to take a guess on their meanings.

We apologize for any lack of clarity in the original text. Although parameters such as , , and  were defined in preceding equations, and  and  were mentioned prior to equation (33), we have restructured the surrounding text in Section 2.3. All parameters are now explicitly enumerated and described immediately following equation (33) to ensure complete clarity and self-containment.

4. Speaking of Fig. 1, the authors provided the discussion of Fig.1 far behind the figure. It impairs the readability of the paper. The reader wants to know the physical meaning of the variables and implication of the curves when they see them in the figure. Since it is based on the phenomenological model, the parameters used in the model are very important. The authors need to explain why the values of the parameters were chosen when they calculated the data for Fig. 1.

To improve readability and structural flow, we have reorganized the figures:

• Figure 1 now displays only the exact analytical limiting cases (instantaneous and very long light pulses), allowing readers to grasp the fundamental boundaries immediately upon encountering the figure in Section 2.1.
• A new figure (Figure 2) has been introduced in Section 2.3 to show intermediate pulse durations and detailed model curves, accompanied by explicit explanations of the chosen parameter values and their physical implications.

5. In Fig.2 and Fig.3, why there seems to be great discrepancies between fitting and simulation results of f(x)? Especially, when compared with the upper two figures in Fig.2 and Fig.3.

We have added an explanatory remark in Section 2.3 regarding the representation of . The function  isolates the relative variance after stripping away the primary trend of mean charge scaling with . Consequently, this representation inherently amplifies small residual differences and statistical fluctuations in Monte Carlo data. While these detailed nuances appear visually magnified in  plots, their actual impact on the integrated charge resolution  is negligible, as demonstrated by the excellent agreement in the corresponding upper panels.

6. I wonder the fitted values listed in Table 1 and Table 2 are the optimal. The authors need to explain why/how the values were determined.

We have updated Section 3.2 to describe the optimization method explicitly. The parameters reported in Tables 1 and 2 were determined via least-squares fitting performed simultaneously or sequentially on the simulated datasets for  and , ensuring robust convergence to optimal global parameters.

7. When the correlated noise was included, it seemed that the discrepancy between fitting and simulation is larger than that for the case when the noise was neglected. The authors should explain the reason.

We have expanded the discussion in Section 3.2 to explain this behavior. When correlated noise (crosstalk and afterpulsing) is included, secondary avalanches induce additional local pixel occupancy, which subtly suppresses subsequent pixel recovery. Because this secondary recovery suppression mechanism is not explicitly incorporated into the empirical formulation of , slight deviations between the simulation and fit emerge in the presence of noise. This omission keeps the model manageable while having only a marginal effect on the resulting charge resolution.

8. What are the exact waveform of the laser and LED pulse? Could the authors provide the waveform by oscilloscope? Why the authors didn’t provide the results measured by the control photodiode. I wonder how they convert the light intensity into x.

We have expanded Section 5.3 to clarify these experimental details:

• Pulse Waveforms: The input light pulse profiles were reconstructed by processing the output waveforms measured with the SiPM alongside its single-photon response function. We have included a new figure (Figure 10) to illustrate these reconstructed light pulse profiles (exponential decay for the laser/scintillator and a charge-discharge model for the LED).
• Control Photodiode & Calibration to : The control photodiode provides a linear relative measurement of the light intensity introduced into the integrating sphere. Because its raw photocurrent is arbitrary, it must be calibrated to the mean number of impinging photons per pixel . This conversion is performed in the linear regime (), where , allowing us to establish a direct proportionality constant between the photodiode reading and  across the full dynamic range. The wording in Sections 5.2 and 5.3 has been thoroughly revised to make this calibration procedure explicit.

9. I guess the word “ansatz” in line 246 in Page 8 is German, please change it into English.

We have revised the text and replaced the term "ansatz" with “expression”.