# Statistical Analysis of Silicon Photomultiplier Output Signals

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling of SiPM Signals

#### 2.1. Approximation of Output Pulse

#### 2.2. Stochastic Process of Photon Detection

#### 2.3. Statistical Analysis

## 3. Experimental Study

#### 3.1. Test Setup

#### 3.2. Event Rate and DC Current

#### 3.3. Pulse Analysis

## 4. Discussion

- The rise time of ${e}_{p}\left(t\right)$ is longer than the rise time of ${v}_{p2}\left(t\right)$ even for an ideal square optical pulse, which may affect the accuracy of pulse arrival time estimation. In fact, the waveform for $0<t<{T}_{w}$ is given by the first integral in (21). Therefore, the estimated ${e}_{p}\left(t\right)$ corresponds to the original waveform (1) since ${v}_{vp2}\left(t\right)$ is its lossy derivative.
- As a consequence of (6), ${e}_{p}\left(t\right)$ starts to decrease after reaching its maximum. Therefore, there exists some optimum pulse width ${T}_{w}$, above which the amplitude of ${e}_{p}\left(t\right)$ does not grow.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

APD | Avalanche photodiode |

SPAD | Single-photon avalanche detector |

SiPM | Silicon photomultiplier |

LIDAR | Light detection and ranging |

PDE | Photon detection efficiency |

DCR | Dark count rate |

FPGA | Field programmable gate array |

VCSEL | Vertical-cavity surface-emitting laser |

LED | Light-emitting diode |

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**Figure 3.**Approximation of the normalized output waveform for ON Semi SiPMs: (

**a**) 10 $\mathsf{\mu}$m-pixel device (MICRORB-10010) with the double-exponential response (1): ${V}_{0}=1.343$, $\alpha =8.076\times {10}^{7}$ s${}^{-1}$, $\beta =1.044\times {10}^{9}$ s${}^{-1}$; (

**b**) 35 $\mathsf{\mu}$m-pixel device (MICRORB-10035) with the fast component and multiple-exponential approximation (2): ${V}_{1}=0.5896$, ${\alpha}_{1}=1.233\times {10}^{7}$ s${}^{-1}$, ${\beta}_{1}=0.4610\times {10}^{9}$ s${}^{-1}$, ${V}_{2}=11.096$, ${\alpha}_{2}=0.5069\times {10}^{9}$ s${}^{-1}$, ${\beta}_{2}=0.5914\times {10}^{9}$ s${}^{-1}$.

**Figure 4.**Fitting the function (5) to the measured capacitive-output waveform ${v}_{p2}$ for MICRORB-10010: ${V}_{1}=2.315$, $\alpha =5.860\times {10}^{7}$ s${}^{-1}$, $\beta =1.045\times {10}^{9}$ s${}^{-1}$, $\gamma =0.7067\times {10}^{9}$ s${}^{-1}$.

**Figure 8.**Mean and standard deviation of $v\left(t\right)$ for ${T}_{w}=20$ ns, and ${\mathsf{\Lambda}}_{0}=1$: (

**a**) Normalized mean ${e}_{p}\left(t\right)$ computed using (21). The dashed line shows a single pulse of ${v}_{p2}$ for comparison. (

**b**) Normalized standard deviation ${s}_{p}\left(t\right)$ computed using (22).

**Figure 9.**Block diagram (

**a**) and photograph (

**b**) of the experimental setup. The measuring chamber is wrapped in aluminum foil to block the ambient light.

**Figure 11.**Measuring the event rate: (

**a**) Example record with multi-event pulses. (

**b**) Estimated event rate as a function of DC current.

**Figure 12.**Raw signals captured by oscilloscope (Channel 1—SiPM output ${v}_{p2}$; Channel 3—FPGA pulse to VCSEL, Channel 4—sync pulses). The red circle shows electrical crosstalk from sync pulses.

**Figure 13.**Statistical processing of recorded signals: (

**a**) Estimation of $e\left(t\right)$ and $s\left(t\right)$ for ${\lambda}_{0}$ = 5.54 MHz and ${\mathsf{\Lambda}}_{0}$ = 27.7 MHz computed from 9998 records. (

**b**) Estimation of $SN{R}_{0}$ for different pulse amplitudes.

Parameter | Typical |
---|---|

Breakdown Voltage $\left({V}_{BR}\right)$ | 25 V |

Overvoltage $\left({V}_{OV}\right)$ | 7 V (10 V max) |

Number of Microcells | 620 |

Responsivity (905 nm, typ. ${V}_{OV}$) | 240 kA/W |

Dark Count Rate | 2.6 MHz |

Dark Current | 1.5 $\mathsf{\mu}$A |

Afterpulsing | 1% |

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**MDPI and ACS Style**

Kolka, Z.; Barcik, P.; Biolkova, V. Statistical Analysis of Silicon Photomultiplier Output Signals. *Sensors* **2022**, *22*, 9134.
https://doi.org/10.3390/s22239134

**AMA Style**

Kolka Z, Barcik P, Biolkova V. Statistical Analysis of Silicon Photomultiplier Output Signals. *Sensors*. 2022; 22(23):9134.
https://doi.org/10.3390/s22239134

**Chicago/Turabian Style**

Kolka, Zdenek, Peter Barcik, and Viera Biolkova. 2022. "Statistical Analysis of Silicon Photomultiplier Output Signals" *Sensors* 22, no. 23: 9134.
https://doi.org/10.3390/s22239134