It is clear from Fig. 2a–d &Fig. 3 that the 2D and 1D spatial distributions of the PDErel inside the G-APD cells of the MPPC were flat and uniform, whereas the 2D and 1D distribution plots of the 1.5 p.e. PCR and the prompt optical crosstalk probability (Pct(X,Y)) exhibited non-uniformity within the G-APD cells. The PCR at a 1.5 p.e. threshold in the central area of the G-APD cell was significantly lower than that in the edge area. To investigate this phenomenon, an optical microscope photo of the tested MPPC was recorded, as shown in Fig. 4, which indicated that the gap spacings between the G-APD cells were not all equal because of the required surface metal electrode wiring. Comparing Fig. 2c, d with Fig. 4 reveals that smaller gaps in the G-APD cells lead to less-uniform POCT and Pct(X,Y) inside the G-APD cell and stronger localized POCT in the G-APD edge area. We then needed to determine why this interesting phenomenon occurs.

Based on our analysis, we propose two potential reasons for the non-uniform POCT phenomenon in this MPPC. The first possible reason for the non-uniform POCT phenomenon is that the horizontal orientation electric field in the gap between two G-APD cells was larger because of the narrower depletion region, thus resulting in greater Pct. However, even if the electric field in the gap region is large, it cannot exceed the electric field in the photosensitive region. Otherwise, the PDE in the gap region would be even higher than in the photosensitive region, which would contradict the experimental results. The second possible reason is that the crosstalk photons emitted from different locations in the source G-APD cell traveled different distances to reach the nearest neighbor G-APD cell along a straight line. The probability that the photons emitted from the central area of the source G-APD cell will reach the nearest neighboring G-APD cell is less than the probability that the photons emitted from the edge area of the source G-APD cell will arrive to the nearest neighboring G-APD cell. Therefore, three hypotheses are needed to comprehensively explain these phenomena: First, the emission of the photons is localized in space (as shown in Fig. 6); Second, the non-uniformity is mainly a result of the emitted photons (EPs) propagating along a straight line to the nearest neighboring G-APD cell (analyzed further vide infra); Third, the penetration depth of the EPs that cause the non-uniformity is very limited in the silicon substrate material (i.e., far less than the G-APD cell size of the MPPC; 100 µm). If the penetration depth of the EPs causing the non-uniformity is > 100 µm, it is impossible to show the clear unevenness of Pct (X, Y) inside the G-APD cell.

The first hypothesis mentioned above is well-founded. The avalanche multiplication process has been studied by theory [27], simulations [28] as well as by experiment [21]. The results have shown that a narrow tube or “filament” with high carrier density is locally created in the p-n junction of a G-APD cell very fast, and the avalanche lateral begins to diffuses over the other part of the junction at the same time. If the avalanche is quenched by a high value resistor (usually called quenching resistor), the lateral diffusion is usually limited to about 10 micrometers around the starting point [27], while the voltage drop across the p-n junction of an G-APD cell decreased rapidly. It’s natural that during an avalanche process in an G-APD cell, the impact ionization rate and the emission rate of photons are the highest when the high density carrier filament is created, and is exponentially decayed over time [26, 27, 29]. A representative plot of the photon emission rate versus time is shown in Fig. 7, which indicates that photon emission mainly occurs during the early stage of the Geiger avalanche discharge of an G-APD cell, i.e., when the avalanche carriers are still limited in the “seed area” (a small area around the starting point where the photo-electron or dark initial carrier generated) [26, 29]. Although the emitted photons created from the filament can be absorbed in another location of the high-field region and cause a second filament [11, 29], the photon emission probability from the second filament reduced a lot because of the exponentially decreasing voltage drop across the p-n junction of the G-APD cell. Consequently, the photon emission during a certain Geiger avalanche in a G-APD cell is basically space-localized.

The second hypothesis can also be understood intuitively. If the EPs emitted from the seed area are absorbed by the very G-APD cell itself, they then participate in the avalanche diffusion process [28, 29]. However, if an EP propagates to the adjacent G-APD cell and is absorbed, an optical crosstalk event is triggered. The closer the seed area is to the adjacent G-APD cell, the higher the probability that the EP will propagate directly to the adjacent G-APD cell and be absorbed, thus leading to a higher probability of an optical crosstalk event. The non-neighboring G-APD cells are further from the seed point, so the probability of direct crosstalk events decays exponentially with distance. A schematic diagram illustrating the emission and absorption of direct EPs is presented in Fig. 6b. Let us imagine that following laser excitation, avalanche seed areas are generated at point 1 and 2, respectively. Point 1 represents the central area of the G-APD cell, and point 2 is located at the edge; therefore, owing to the different propagation distances, the probability that EPs generated at point 1 reach the adjacent G-APD cell is clearly lower than the probability that EPs generated at point 2 reach the adjacent G-APD cell (i.e., the *P*ct at point 2 is greater than the *P*ct at point 1). If there are optical isolation trenches between the G-APD cells to prevent direct EP propagation to the adjacent G-APD cells, the non-uniformity of the spatial distribution of *P*ct (X, Y) within the G-APD cell should be greatly reduced. To verify this concept, we measured the spatial distribution of R1.5p.e. (X, Y) and R0.5p.e. (X, Y) in an SiPM with optical isolation trenches, as shown in Fig. 7&Fig. 8.

It is clear from Fig. 8 that the distribution of *R*1.5p.e. and *P*ct (X, Y) inside the G-APD cells in the FBK SiPM was much more uniform and flatter than that of the MPPC comprising 100-µm G-APD cells (shown in Fig. 2). Nevertheless, since the G-APD cell pitch in the FBK SiPM was smaller than that in the MPPC, the non-uniformity of the *P*ct distribution may be not appeared due to fluctuations in the experimental data. To further verify the influence of the optical isolation trenches on the non-uniformity of the *P*ct (X, Y), we measured the *R*1.5p.e. (X, Y) and *P*ct (X, Y) of an MPPC with smaller G-APD cells (25 µm, square) without isolation trenches (i.e., the same structure as the MPPC shown in Fig. 5). The spatial distribution characteristics, *R*1.5p.e. (X, Y) and *P*ct (X, Y) are presented in Fig. 10 & Fig. 11.

It is clear from Fig. 10 & Fig. 11 that inside the G-APD cells of the MPPC that did not contain optical isolation trenches, the *P*ct (X, Y) was still non-uniform, although the G-APD cells were only 25 µm, and the Pct (X, Y) at the edge area was still higher than that in the central area. This confirmed that the optical isolation trenches in the FBK SiPM (see Fig. 8) obviously improved the uniformity of the *P*ct distribution. These results validated the aforementioned hypothesis 2, which posited that the non-uniformity was mainly contributed by the EPs that travel along a straight line to the nearest neighboring G-APD cell; these results also supported the rationality of hypothesis 1. Imagining that after the laser spot at a given position irradiating on the photosensitive surface of a G-APD cell, if the emission position of the generated photons is random inside the G-APD cell, it would be impossible to obtain non-uniform spatial distribution of *P*ct.

As for the third hypothesis, it has already been observed by Ref [30] and our group’s previous work[23]. The results in Ref[23] indicated that the penetration depth of this type of EPs was only about 22 µm; thus, the corresponding wavelength of the propagating photons in an MPPC substrate (i.e., silicon) was approximately 850 nm[12, 21]. Correspondingly, the equivalent effective dominant wavelength of the EPs that contributed to the non-uniform distribution of Pct (X, Y) is 850 nm. Although the wavelength of the EPs ranged from 450 to 1600 nm[31], the absorption depth of the EPs with wavelengths shorter than 850 nm, was too shallow, and these photons were mainly absorbed by the source G-APD cells themselves. In contrast, EPs with wavelengths around 850 nm were easily absorbed by the adjacent G-APD cells, owing to their suitable penetration depth, and this led to the non-uniformity of Pct (X, Y) in the source G-APD cell. The penetration depth of EPs with wavelengths longer than 850 nm, was too large to contribute to the non-uniformity of Pct (X, Y) in the source G-APD cell.