Spatial modeling of optical crosstalk in InGaAsP Geiger-mode APD focal plane arrays

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1 Spatial modeling of optical crosstalk in InGaAsP Geiger-mode APD focal plane arrays Brian Piccione, Xudong Jiang, and Mark A. Itzler * Princeton Lightwave, Inc., 2555 Route 130 South, Cranbury, New Jersey 08512, USA * mitzler@princetonlightwave.com Abstract: We report a spatial model of optical crosstalk in InGaAsP Geiger-mode APD focal plane arrays created via non-sequential ray tracing. Using twenty-four equivalent experimental data sets as a baseline, we show that experimental results can be reproduced to a high degree of accuracy by incorporating secondary crosstalk effects, with reasonable assumptions of material and emission source properties. We use this model to categorize crosstalk according to source and path, showing that the majority of crosstalk in the immediate neighborhood of avalanching pixels in the present devices can be attributed to direct line-of-sight emissions Optical Society of America OCIS codes: ( ) Arrays; ( ) Avalanche photodiodes (APDs); ( ) Noise in imaging systems. References and links 1. D. Renker, Geiger-mode avalanche photodiodes, history, properties and problems, Nucl. Instrum. Methods Phys. Res. A 567(1), (2006). 2. B. F. Aull, A. H. Loomis, D. J. Young, R. M. Heinrichs, B. J. Felton, P. J. Daniels, and D. J. Landers, Geigermode avalanche photodiodes for three-dimensional imaging, Linc. Lab. J. 13, (2002). 3. B. F. Aull, A. H. Loomis, D. J. Young, A. Stern, B. J. Felton, P. J. Daniels, D. J. Landers, L. Retherford, D. D. Rathman, and R. M. Heinrichs, Three-dimensional imaging with arrays of Geiger-mode avalanche photodiodes, in Integrated Optoelectronic Devices 2004 (International Society for Optics and Photonics2004), pp M. A. Itzler, M. Entwistle, M. Owens, K. Patel, X. Jiang, K. Slomkowski, S. Rangwala, P. F. Zalud, T. Senko, and J. Tower, Geiger-mode avalanche photodiode focal plane arrays for three-dimensional imaging ladar, in Proc. SPIE (2010), pp C. 5. M. A. Itzler, M. Entwistle, M. Owens, X. Jiang, K. Patel, K. Slomkowski, T. Koch, S. Rangwala, P. F. Zalud, and Y. Yu, InP-based Geiger-mode avalanche photodiode arrays for three-dimensional imaging at 1.06 μm, in SPIE Defense, Security, and Sensing (International Society for Optics and Photonics 2009), pp O A. L. Lacaita, F. Zappa, S. Bigliardi, and M. Manfredi, On the bremsstrahlung origin of hot-carrier-induced photons in silicon devices, IEEE Trans. Electron Dev. 40(3), (1993). 7. J. Jackson, P. Hurley, B. Lane, A. Mathewson, and A. Morrison, Comparing leakage currents and dark count rates in Geiger-mode avalanche photodiodes, Appl. Phys. Lett. 80(22), (2002). 8. Y. Kang, H. Lu, Y.-H. Lo, D. Bethune, and W. Risk, Dark count probability and quantum efficiency of avalanche photodiodes for single-photon detection, Appl. Phys. Lett. 83(14), (2003). 9. I. Rech, A. Ingargiola, R. Spinelli, I. Labanca, S. Marangoni, M. Ghioni, and S. Cova, Optical crosstalk in single photon avalanche diode arrays: A new complete model, Opt. Express 16(12), (2008). 10. R. D. Younger, K. A. McIntosh, J. W. Chludzinski, D. C. Oakley, L. J. Mahoney, J. E. Funk, J. P. Donnelly, and S. Verghese, Crosstalk analysis of integrated Geiger-mode avalanche photodiode focal plane arrays, in Proc. SPIE (2009), pp Q E. Sciacca, G. Condorelli, S. Aurite, S. Lombardo, M. Mazzillo, D. Sanfilippo, G. Fallica, and E. Rimini, Crosstalk characterization in Geiger-mode avalanche photodiode arrays, IEEE Electron. Dev. Lett. 29(3), (2008). 12. Q. Chau, X. Jiang, M. A. Itzler, M. Entwitsle, B. Piccione, M. Owens, and K. Slomkowski, Analysis and modeling of optical crosstalk in InP-based Geiger-mode avalanche photodiode FPAs, in SPIE Sensing Technology + Applications (International Society for Optics and Photonics 2015), pp O M. A. Itzler, U. Krishnamachari, Q. Chau, X. Jiang, M. Entwistle, M. Owens, and K. Slomkowski, Statistical analysis of dark count rate in Geiger-mode apd FPAs, in SPIE Security + Defence (International Society for Optics and Photonics 2014), pp OSA 16 May 2016 Vol. 24, No. 10 DOI: /OE OPTICS EXPRESS 10635

2 14. I. Rech, A. Ingargiola, R. Spinelli, I. Labanca, S. Marangoni, M. Ghioni, and S. Cova, A new approach to optical crosstalk modeling in single-photon avalanche diodes, IEEE Photonics Technol. Lett. 20(5), (2008). 15. E. D. Palik and G. Ghosh, Handbook of optical constants of solids (Academic Press Inc., 1997). 16. J. A. McCaulley, V. M. Donnelly, M. Vernon, and I. Taha, Temperature dependence of the near-infrared refractive index of silicon, gallium arsenide, and indium phosphide, Phys. Rev. B Condens. Matter 49(11), (1994). 17. J. S. Blakemore, Semiconducting and other major properties of gallium arsenide, J. Appl. Phys. 53(10), R123 R181 (1982). 18. Z. Hang, H. Shen, and F. H. Pollak, Temperature dependence of the E 0 and E gaps of InP up to 600 C, Solid State Commun. 73(1), (1990) B. F. Aull, D. R. Schuette, D. J. Young, D. M. Craig, B. J. Felton, and K. Warner, A Study of Crosstalk in a 256 x 256 Photon Counting Imager Based on Silicon Geiger-Mode Avalanche Photodiodes, IEEE Sens. J. 15, (2015). 21. M. A. Itzler, U. Krishnamachari, M. Entwistle, X. Jiang, M. Owens, and K. Slomkowski, Dark Count Statistics in Geiger-Mode Avalanche Photodiode Cameras for 3-D Imaging LADAR, IEEE J. Quantum Electron. 20, (2014). 1. Introduction As the sensitivity of imaging systems increases, so does the need to understand the origins of imaging noise. Nowhere is this more apparent than for cameras based upon Geiger-mode avalanche photodetector (GmAPD) technology, whereby arrays of avalanche photodetectors are biased above their breakdown voltage, producing avalanches of charge carriers following the creation of an initial electron-hole pair [1]. While such behavior ultimately enables cameras with single-photon sensitivity [2 5], it also carries the burden of extreme sensitivity to noise: single electron-hole pairs created via thermal excitation [6] are similarly multiplied to produce macroscopic currents, resulting in false positive dark counts at a much higher rate than that seen from lower-gain architectures [7,8]. In addition to increased susceptibility to dark carriers arising from thermal excitation and material defects, GmAPD arrays are much more likely to produce dark counts arising from the phenomenon known as optical crosstalk, in which electroluminescence generated in the active region of one pixel is detected by a neighboring pixel [9,10]. In addition to the broadband, blackbody emission common to all photodetector technologies, GmAPDs have been shown to emit strongly near the multiplication layer bandgap following detection events [10], as their high internal gain creates a large population of charge carriers shortly after the initiation of an avalanche. It is this emission (on the order of 10 5 photons per electron) [6], in conjunction with single-photon sensitivity, which raises the relative impact of optical crosstalk in GmAPD arrays. With optical crosstalk predicted to increase as array pitch is decreased [11], and dark count rate directly limiting both spatial and temporal resolution even in devices with existing crosstalk mitigation mechanisms [12], it is clear that a fundamental understanding of optical crosstalk is necessary to further advance photon-counting array technology. Here, we report a spatial model of optical crosstalk in InGaAsP GmAPD focal plane arrays, using twenty-four equivalent experimental data sets as a baseline for calibration. By incorporating secondary crosstalk effects, we obtain an accurate reproduction of the experimental spatial crosstalk map. We then use this model to explore the origin of each crosstalk event, providing insight into methods for future reduction. 2. Optical crosstalk in GmAPD arrays While the exact distribution of optical crosstalk in GmAPD arrays will of course depend explicitly on device structure, modern GmAPD array designs operate on the same principles and share many common design elements. Figure 1(a) shows a typical InP-based GmAPD layer structure, where illumination enters through an anti-reflection coating on the back side of the wafer, traveling through the bulk of the InP substrate and a filter layer before being 2016 OSA 16 May 2016 Vol. 24, No. 10 DOI: /OE OPTICS EXPRESS 10636

3 absorbed. The resultant photoexcited charge carriers are then multiplied in a separate multiplication region, before finally being collected by the p-contact metal below. Fig. 1. (a) Pixel-level overview of typical GmAPD device layer structure, including a microlens array used in increasing optical fill factor. (b) Array-level overview of optical crosstalk, illustrating an array of pixels optically isolated by etched trench structures. Light emitted from the multiplication region of pixel 0 (with spectral properties given in [10]) can reach neighboring active regions through three distinct vectors: back side metal reflection (A), back side SiN x reflection (B), and direct line-of-sight transmission (C). θ denotes the angle between back side reflections and the back side surface normal, used in defining the InP/SiN x /air interface critical angle of 17.7 beyond which all photons approaching the interface undergo total internal reflection. Though the total crosstalk seen by each pixel is frequently a combination of both back- and front-side contributions, in the interest of simplified categorization, all pixels will henceforth be referred to as Type A or Type B if their primary backside contribution involves a metal or SiN x reflection, respectively. With its high population of electron-hole pairs during avalanche breakdown, the multiplication layer is the physical origin for the majority of crosstalk in III-V GmAPD arrays, the bulk of which has been shown previously to be the result of radiative recombination [10]. As shown in Fig. 1(b), there are three possible paths for emission from the multiplication region of one pixel to reach the absorption region of another. Paths A and B represent photons which first travel towards the back side of the substrate, reflecting off of either the n-contact metal apertures (A) or anti-reflective SiN x windows (B). Path C illustrates what is known as line-of-sight crosstalk, which is transmitted from pixel to pixel without reflecting off of the back side. All three paths have been identified previously [13], and various mechanisms have already been implemented in an attempt to mitigate their impacts. Paths A and B are the impetus for the aforementioned crosstalk filter layer, which is designed with a bandgap that absorbs the majority of higher-energy photons from direct InP recombination while still allowing slightly lower-energy input photons to reach the absorption region from the wafer back side. Path A is additionally attenuated by ensuring the n-metal is absorptive, and path C is typically interrupted by trenches on the front side between each pixel, etched through the multiplication layer. Crosstalk remains a limiting factor to III-V GmAPD camera performance despite these mitigation strategies, however, and previous modeling of similar systems did not account for them in their analysis [9, 14]. For further improvement, then, we now turn towards physical modeling, with an eye towards understanding the major crosstalk vectors which still remain after accounting for the state-of-the-art in crosstalk mitigation mechanisms OSA 16 May 2016 Vol. 24, No. 10 DOI: /OE OPTICS EXPRESS 10637

4 3. Spatially-resolved optical crosstalk model In order to accurately model the effects of optical crosstalk on dark count rates, an appropriate physical model must first be chosen. While wave-optical techniques (e.g. finite-difference time-domain, finite-difference frequency-domain) were considered, their substantial computational overhead is unnecessary when the majority of physical dimensions are much longer than the modeled wavelengths. Instead, we began with the assumption that the crosstalk emission source is incoherent and exploited the speed of non-sequential ray tracing (Zemax 14) for our model. 3.1 Model setup As shown in Fig. 2, our model is a full-scale three-dimensional reproduction of a 5 5 pixel GmAPD array with 50 µm pitch, 16 µm-diameter active areas, anti-reflective-film coated GaP microlens array, and the epitaxial structure from Fig. 1(a). The emission source was modeled as a cylinder with a diameter of 16 µm and thickness equal to that of the multiplication layer, positioned in the center of the multiplication region, emitting ten million rays in all directions. All ray origins were distributed randomly throughout the emission volume, with no preference for direction of emission. As noted in Fig. 2(a), all side walls were modeled as perfectly absorbing. All surface reflections were fixed as perfectly specular, and bulk and surface scattering were disabled. In choosing material properties and the emission source spectrum, we first had to deal with a discrepancy between the available emission source spectrum (measured at room temperature in [10]) and the available experimental data set (obtained at 20 C, see Section 3.2). In evaluating how to proceed, we noted that the refractive indices and band gaps of the two binary materials systems bracketing the composition of our detectors (InP and GaAs) have very similar temperature dependences near room temperature ( K 1 at 1.1 µm and 0.25 nm/k for InP, K 1 at 1.1µm and 0.27 nm/k for GaAs) [15 17], meaning the crosstalk emission spectrum (largely determined by InP multiplier bandgap) and absorption band edge (determined by our InGaAsP absorber) do not shift substantially relative to each other over a temperature difference of 40. Left with the choice, then, of either approximating material properties and the emission source spectrum at 20 C, or using well known room-temperature material properties [18] and spectral data [10], we assumed that the latter would have the smaller impact on results and used room-temperature properties for all materials. For power measurement at each individual pixel, cylindrical detectors with 16 µm diameters and spatially uniform detection efficiency were placed in the center of each pixel s absorption layer. In addition to these discrete detectors, two inert monitor layers, flanking the device absorption layer, were inserted in order to observe a detailed spatial distribution of rays in a critical region (see insert of Fig. 2(a)). Detectors count all photons with energies above the absorption band edge of the modeled epitaxial structure, with no regard given to photon detection efficiency (PDE) or potential arm state of the modeled detector OSA 16 May 2016 Vol. 24, No. 10 DOI: /OE OPTICS EXPRESS 10638

5 Fig. 2. (a) Schematic of basic model geometry, shown in cross-section. The emission source is modeled as a cylinder in the center of the multiplication layer, with thickness equal to the multiplication layer thickness and diameter equal to the lithographically-defined active area size (16 µm in this case). Individual detectors at each pixel are modeled as cylinders in the center of the absorption layer, again with diameters equal to 16 µm. Inset: Detailed view of the epitaxial layer structure, as modeled. (b) Interior view of three-dimensional 5 5 pixel model geometry, showing emission from pixel in far corner, where rays are grouped by color according to number of surface interactions. Front side trenches outline all pixels in a square grid, with sidewalls etched normal to the front-side wafer surface. All epitaxial layers detailed in Fig. 1(a) have been modeled but are left transparent for illustrative purposes OSA 16 May 2016 Vol. 24, No. 10 DOI: /OE OPTICS EXPRESS 10639

6 3.2 Results Results from the simulation are presented in Fig. 3. Figure 3(a) shows experimental crosstalk results for a 5 5 pixel sub-array averaged from three Princeton Lightwave GmAPD LADAR cameras with lithographically defined active areas of 16 µm [19], pixel pitches of 50 µm, and epitaxial structures as described in Fig. 1, biased 3V above breakdown voltage and held at 20 C. Crosstalk was extracted from these cameras over a time gate of 2 µs using methods outlined in [13], conservatively assuming all dark counts with interarrival times less than 5 ns to the previous dark count event are considered crosstalk events. Experimental data is the total number of crosstalk counts including those associated with crosstalk chains, whereby the avalanche associated with one crosstalk event, hereafter referred to as a 1st order event, can itself emit photons and trigger 2nd order events, and so on [20]. For the data presented below, an excess bias of ~3V was used to obtain 30% PDE. Additional averaging was performed between the eight equivalent octants of each crosstalk data set as delimited by the eight axes of mirror symmetry in a square lattice. In this plot, the active pixel is shown in the bottom left of the array (0, 0) and the relative intensity of all data points is normalized to that of the first nearest neighbor (0, 1). Additionally, Type B pixels (see Fig. 1(b)) are outlined in thick borders. Figure 3(b) shows discrete detector results from the simulation described in Section 3.1, while Fig. 3(c) shows the simulation plotted against experimental results, with Type A and B pixels separated. Fig. 3. (a) Experimental and (b) simulated spatial maps of optical crosstalk magnitude with active pixels at (0,0) (denoted with an A ), normalized to first-nearest-neighbor crosstalk. Experimental results show cumulative crosstalk (including 2nd- and higher-order crosstalk events), while the simulation shows 1st order crosstalk events only. Type B pixels are shown with black borders, and only photons with energy above the absorber band gap are counted. Color map scale is common to both (a) and (b). (c) Data from (a) and (b) plotted as a function of distance on a logarithmic scale, with reflections further separated according to source of backside reflection. 3.3 Analysis As seen in Fig. 3, the simulation results generally reproduce the experimental results well, with the power-versus-distance plot of Type A reflections generally matching the trend of the experimental data, and a near-constant offset for Type B reflections. The reduction of 2016 OSA 16 May 2016 Vol. 24, No. 10 DOI: /OE OPTICS EXPRESS 10640

7 crosstalk with increasing distance has been reproduced, as has the locally non-monotonic behavior of both reflection lines. In the two-dimensional spatial plots of Figs. 3(a) and 3(b), Type B pixels stand out from their nearest neighbors, and the trend of generally higher Type B crosstalk as a simple function of distance (Fig. 3(c)) has been reproduced as well. Still, there is a noticeable difference in relative values beyond the normalization point, and we must look closer at the model assumptions in order to determine the source of the discrepancy Cumulative crosstalk contributions In early attempts at improving the fit, unknown emission source variables such as shape, size, and position were varied substantially, with the only physical bounds being the known outer diameter of the device active region and thickness of the multiplication layer (a random angular distribution of emission was assumed in every run, as was a uniform dispersion of ray origins within each simulated source). Cylinders, rings, and oblate spheroids of various dimensions, all centered about the active pixel radial axis of symmetry, were tested at various vertical positions within the multiplication region, all changes resulted in measured power shifts of less than one percent for points beyond the nearest neighbor pixels. With these changes occurring within the ~1% error seen between identical simulation runs with the same conditions, we returned the emission source shape to the simple cylinder used to generate Fig. 3(b) and turned elsewhere to explain the discrepancy. As described thus far, the simulation includes only one emission source, which is located at the center of the primary avalanche pixel at (0,0). While this level of simulation can be used effectively as a stand-in for 1st order crosstalk events, however, it does not include the 2ndand higher-order events discussed above which are included as part of the experimental data set. In order to approximate the effects of higher-order crosstalk, we used our 1st order results from Fig. 3 as a baseline for adding 2nd-order effects. As shown in Fig. 4(a), we begin with the 1st-order simulation results from Fig. 3(b), now illustrating a full quadrant of the simulated data, as opposed to a single octant, for the sake of clarity within this figure. From here, 2nd-order effects are simulated (Fig. 4(b)) by first re-centering the 1st-order map such that the original active pixel has been moved to one of its 12 nearest neighbors (delimited by the dark border in the 1st-order map). From here, the original map is scaled by a variable factor, from this point referred to as the nearest-neighbor crosstalk probability per avalanche. In Fig. 4, the one example map showing 2nd-order events illustrates a secondary avalanche centered at (0,1), with the original map scaled by a nearest-neighbor crosstalk probability per avalanche of 3.5%, and then multiplied by the number of secondary avalanches the new active pixel is expected to generate. The number of secondary avalanches each new active pixel generates is simply its normalized value from Fig. 4(a), meaning that it is expected the primary avalanche that generated crosstalk in pixel (0,1) would initiate one secondary avalanche, while 0.04 avalanches would be initiated in pixel (1,1), and so on. Under this scheme, the first nearest neighbors to (0,1) in the (0,1)-centered map (Fig. 4(b)) have a 2nd-order crosstalk probability of = 0.035, the first nearest neighbors to (1,1) in the (1,1)-centered map have a 2nd-order probability of = , and so on. After repeating this process for all 12 nearest neighbors, adding all 12 2nd-order maps to the 1st-order map, and renormalizing, we end up with a renormalized map of crosstalk (Fig. 4(c)) that includes both 1st-and 2nd-order effects OSA 16 May 2016 Vol. 24, No. 10 DOI: /OE OPTICS EXPRESS 10641

8 Fig. 4. Illustration of method for adding 2nd-order crosstalk events to 1st-order effects. a) The 1st-order model results from Fig. 3(b) are presented as a full, symmetric quadrant of data, with the 12 nearest neighbors to the active pixel delimited with a thick border. b) 2nd-order effects are simulated by first re-centering the map from a) so that the active pixel is now at (0,1). After this, all numbers in a given 2nd-order map are multiplied by the nearest-neighbor crosstalk probability per avalanche (3.5% in this illustration), and then multiplied by the number of secondary avalanches the new active pixel represents (1 for pixel (0,1), as determined by its normalized value in a)). c) The map from a) is added to 2nd-order maps which have been generated for all 12 pixels outlined in a), and then re-normalized. This produces a simulated cumulative crosstalk map. The 3.5% nearest-neighbor crosstalk probability per avalanche is not a measured or theoretical value, but rather was determined empirically by plotting the normalized crosstalk power versus distance plot extracted from Fig. 4(c), and monitoring the fit of the new cumulative model to that of the cumulative experimental data first shown in Fig. 3(c). As seen in Fig. 5, adding 2nd-order crosstalk with a 3.5% nearest-neighbor crosstalk probability per avalanche to 1st-order crosstalk does not alter the relative offset of Type B crosstalk substantially, but the fit of Type A crosstalk to experimental data has been markedly improved. In sum, the average error of Type B pixels was increased slightly from + 41% to + 48%, while the average error of Type A pixels was reduced from 28% to + 1.7%. The 3 increase in the normalized value of pixel (1,1) is a result of the (1,1) pixel being flanked on two sides by pixels with high probabilities of 2nd-order emission ((0,1) and (1,0)), while the relative increase of all other Type A pixels is attributed to their being surrounded by Type B pixels, which have a higher probability of 2nd-order emission due to their higher 1st-order crosstalk detection probabilities OSA 16 May 2016 Vol. 24, No. 10 DOI: /OE OPTICS EXPRESS 10642

9 Fig. 5. Normalized crosstalk power as a function of distance, showing the same cumulative experimental crosstalk and simulated 1st-order crosstalk from Fig. 3(c), now plotted alongside the results from the cumulative crosstalk simulation obtained via the methodology shown in Fig. 4. With a physically grounded path to obtaining Fig. 5 and the excellent fit of the cumulative Type A points in particular, we believe that the model up to this point replicates the experimental system quite well. However, the near-constant offset of the Type B pixels points to a remaining discrepancy. It is important to note that this does not necessarily mean that Type B pixels were over-estimated on an absolute scale: due to their normalization relative to the first Type A pixel, this discrepancy could conversely indicate that all Type A pixels were systematically under-estimated. In re-evaluating all assumptions that went into building the model, we noted that by plotting both pixel types on the same graph, we are assuming that they are indeed comparable, with every ray detected in the model equally likely to trigger an avalanche event regardless of pixel type. However, this more generally assumes that each pixel has an equivalent PDE with respect to the absorption of crosstalk photons, and we believe this is most likely not the case. The PDE of the detectors used in this study was measured at 30% via methods detailed elsewhere [5], but this number is calculated for light incident on the PDA through the backside aperture. This assumes a constant quantum efficiency of ~80% for light travelling roughly normal to the absorption layer. However, the path length of light traversing the detector regions is highly variable in the case of crosstalk: some light is reflected off of the PDA back side and approaches detectors from above (path A or B in Fig. 1(b)), but some light travels directly from the emitter source to a neighboring detector (path C in Fig. 1(b)). Light travelling along path A or B should thus be absorbed with a quantum efficiency on the order of ~80%, but light travelling along path C, approaching the absorption layer at a shallow angle, could experience a path length through the absorber as long as the active region diameter. In the case of the latter, this means an absorption length over an order of magnitude longer than the former, with a quantum efficiency of 100%. Without a quantification of the exact path length each ray traverses through the active cylinder of each absorption region (unavailable in the current simulation tools), and thus a ray path-specific PDE, it is difficult to quantify this effect and make final adjustments to the cumulative model results shown in Fig. 5. However, with the stark difference in the expected magnitude of metal- and InP/SiN x /airback side reflections (with the metal stack absorbing nearly all light traveling path A and InP/SiN x /air reflecting a high percentage of light traveling path B), we expect that as a trend, 2016 OSA 16 May 2016 Vol. 24, No. 10 DOI: /OE OPTICS EXPRESS 10643

10 type A pixels should receive proportionally fewer crosstalk photons traveling perpendicular to the absorber layer, and therefore should experience a higher PDE than the experimentally measured 30%. Though we believe this ray-dependent PDE to be the most likely explanation for our remaining discrepancy, it is important to note that regardless of its root cause (PDE difference, error in surface property assumptions, etc.), the issue at hand is a near-constant offset of one pixel type relative to another. Meanwhile, the 3.5% nearest-neighbor crosstalk probability per avalanche was used to correct only the shape of the simulated Type A curve in Fig. 5, i.e. the relative values of crosstalk magnitude within one type. Thus regardless of any final adjustment which would be applied uniformly to a given type, the 3.5% value, primarily affecting only relative values within a type, should hold as a realistic metric for this system. This is useful in further analysis, as it allows for further calibration. While the 3.5% nearest-neighbor crosstalk probability per avalanche was initially calculated from 2nd-order crosstalk effects, there is no difference in the physics describing the photon emission behavior of a primary avalanche versus a secondary avalanche. Because of this, the percentage must also describe the nearest-neighbor crosstalk probability per avalanche for the primary avalanche. If we begin with this value, and multiply it by the ratio of simulated rays generated by the primary avalanche per simulated rays detected at the nearest neighbor (NN), we can obtain a derived value for the real number of photons generated per avalanche in the physical device: photons collected at NN 10 photons generated 6.5photons generated = (1) 1 Avalanche 53,700photons collected at NN 1 Avalanche The resultant value of 6.5 photons generated per avalanche is of course valid only for the overbias in use during experimental crosstalk measurements, and without knowledge of avalanche size, in carriers, it is difficult to extract deeper meaning from the number such as radiative recombination rates of InP in general. Still, this number is useful in furthering the applicability of the simulated results: multiplying the number of rays generated in the model by 1 avalanche per 6.5 photons, we can see that by choosing to emit 10 7 rays, we have effectively simulated primary avalanches. Now that we know the number of primary avalanches, we can divide the absolute number of rays detected at each pixel in the simulation by that number to obtain the probability for a photon entering a detector active region per primary avalanche. Finally, to obtain a true crosstalk probability, we must multiply these values by the probability that a photon entering an active region will initiate an avalanche event and be detected. Nominally, this probability is simply the device PDE. With the PDE likely variable as a function of ray path we cannot yet claim ideally calculated crosstalk probabilities, but we have multiplied our results uniformly by our measured PDE 30% in order to obtain a rough comparison. As seen in Fig. 6, the absolute crosstalk probabilities obtained via this method are generally within a factor of ~2 from experimental results. At this point, the two largest remaining discrepancies are very likely the ray-dependent PDE unaccounted for in the model, as well as an imperfect experimental methodology for extracting pure crosstalk events from background, uncorrelated dark count events. While these results leave room for further improvement, it is noteworthy that they are at the point where a precise calculation of raydependent PDE would leave imperfections in experimental methodology as the sole remaining discrepancy. Upon improving the model further in its next iteration, the results can ultimately be used for refining experimental measurement techniques OSA 16 May 2016 Vol. 24, No. 10 DOI: /OE OPTICS EXPRESS 10644

11 Fig. 6. Absolute cumulative crosstalk probabilities per primary avalanche event (primary avalanche located at pixel (0,0), denoted with an A ). a) Experimental averages obtained from three cameras at 3V overbiases corresponding to a PDE of 30%, and b) simulated results obtained by scaling the cumulative model results shown in Fig. 5 by the factors obtained from Eq. (1) Generalized crosstalk paths Though the full cumulative crosstalk analysis is crucial in obtaining the fit in Fig. 5, the cumulative total is still dominated by 1st order crosstalk, and as such, it is useful to use the initial 1st-order ray trace results from Fig. 3 to identify on a more general level the most common crosstalk paths. To this end, we have plotted the cumulative power of all rays emitted by the simulated primary avalanche at (0,0) which cross the two transparent monitor planes surrounding the absorption layer (see Fig. 2(a)). The upper monitor in Fig. 7(a) is primarily the result of rays reflected off of the device back side, while the lower monitor in Fig. 7(b) largely shows rays which have approached the absorption region from below, giving an approximate spatial map of line-of-sight crosstalk. The upper monitor shows two features of note, namely, the halo of low power surrounding the source, and the smaller circles of high power at longer distances. The halo was first noted in [9] as a result of transmission through the backside SiN x for shallow angles and reflection past the InP/SiN x /air critical angle, while the smaller circles at longer distances are images of the back side metal apertures, formed by high reflectivity from SiN x and low reflectivity of the metal. Within the lower monitor, the most immediately striking feature is the aforementioned discrepancy between pixels which lie in the same column or row of the source pixel and those which do not: though pixel (1,1) is a distance of 1.41 pixels away from the source, the total power traversing the lower monitor in its location is on par with that of pixel (0,4) OSA 16 May 2016 Vol. 24, No. 10 DOI: /OE OPTICS EXPRESS 10645

12 Fig. 7. Detailed spatial maps of transmitted power through the upper (a) and lower (b) monitor layers first illustrated in Fig. 2(a), for light emitted from the primary avalanche in pixel (0,0). Only light with energy above the absorber bandgap is registered. Figure 7(b) begins to illustrate the uniqueness of pixels in the same column or row as the active pixel in a qualitative sense, but because it includes some rays which reflected off of the back side and passed through the absorption layer from above, it cannot be used to directly quantify the directional dependence of crosstalk traversing the present etched isolation trenches. In order to better understand the magnitude of this effect, we categorized all rays which reached the discrete detectors according to whether or not they reflected off of the device back side. We then scaled these results to include 2nd-order effects using the same methodology and probability from Fig. 4, effectively creating cumulative back-side-reflected and line-of-sight only maps. We then divided the line-of-sight portion by the total detected power in order to obtain a cumulative line-of-sight fraction for each pixel. In order to isolate the effect of directional dependence we focused our analysis on only one type of pixel, and chose Type B over Type A due to the significantly higher magnitude of crosstalk recorded for all Type B pixels. The results of this cumulative line-of-sight fraction analysis are shown in Fig. 8. In this figure, the five unique Type B pixels have been classified according to their angular deviation from the normal to the source pixel trench walls; i.e., the 0 line contains (0,4) and (0,2), the 45 line (2,2) and (4,4), and the 27 point is (2,4). As seen in this data, the line-of-sight fraction is reduced at minimum by a factor of five when comparing the 0 and 45 degree lines, with the 27 point lying in between. With the 27 and 45 pixels all lying beyond the 17.7 InP/air critical angle as defined from the trench etch normal, any line-of-sight fraction which remains is entirely contributed by multiple reflections off of front side trench walls, and as seen by the absolute magnitude of their crosstalk in Fig. 3, this removal of normal line-ofsight crosstalk could have a significant impact on reducing the cumulative total. With this analysis it is important to note that the data still says little of the total path length each individual ray takes through the detector cylinder, as rays can still approach a detector from below while clipping the side of a cylinder laterally, or otherwise pass through nearly perpendicular to the layer from below without reaching the device back side within the confines of our 5 5 modeled grid. Due to these uncertainties, this implementation is useful in highlighting the stark line-of-sight fraction dependence on pixel location only OSA 16 May 2016 Vol. 24, No. 10 DOI: /OE OPTICS EXPRESS 10646

13 Fig. 8. Simulated cumulative line-of-sight fraction versus distance plot for all Type B pixels, grouped by angular deviation from normal incidence to the trench walls (e.g. the 0 line includes pixels (4, 0) and (2,0)). This grouping illustrates the dominance of the line-of-sight crosstalk vector for pixels in the same column or row of the active pixel Relationship between simulation results and array-level cumulative crosstalk The results presented thus far are primarily to crosstalk probabilities of individual pixels, or types of pixels. Expanding upon their relationship to array-level crosstalk can help better contextualize their meaning, but to do this, we must first decide upon a reasonable correction factor for the Type B offset from Fig. 5. Again assuming that the near-constant offset is the manifestation of an unknown physical mechanism common to all pixels of one type, we reduce the value of all 1st-order Type B pixels by 48% and then add in 2nd-order contributions again using the method outlined in Fig. 4. After this, we convert our 5 5 adjusted line-of-sight and back side reflection crosstalk magnitudes to a 9 9 sub-array centered about the active pixel and then sum the contributions from all paths, obtaining the percentages shown in Table 1. This table provides insight into the primary physical causes behind crosstalk for the sub-array, which itself was measured to represent an average of 31% of all crosstalk across the three full cameras considered here (with array-level cumulative values obtained via the same spatial methods shown to correlate well with those obtained via temporal analysis methods elsewhere) [21]. Table 1. Percentage contribution of all physical crosstalk paths to cumulative crosstalk probability total for 9 9 sub-array. Physical cumulative crosstalk path Contribution to 9 9 cumulative total Total line-of-sight 82% Normal line-of-sight 58% Non-normal line-of-sight 24% Total back side reflections 18% Pure SiNx reflections 12% Pure metal reflections 6% Table 1 is useful in understanding the magnitude of crosstalk within the immediate neighborhood of an active pixel; however, the numbers shown are strongly influenced by a few dominant pixels. Breaking down the percentages further to look at the individual pixels reveals the origin of this line-of-sight dominance: as shown in Table 2, the four unique pixels closest to the active pixel combined constitute 56% of the 9 9 sub-array total, and the crosstalk in these pixels is almost entirely dominated by line-of-sight photons. Conversely, the contributions beyond these 16 nearest neighbors are largely due to back-side reflections, and as shown in Fig. 8, the relative influence of back side reflections grows quickly as a function of distance OSA 16 May 2016 Vol. 24, No. 10 DOI: /OE OPTICS EXPRESS 10647

14 Table 2. Line-of-sight contributions for the 16 nearest neighbors surrounding the active pixel, alongside their magnitude contribution to the 9 9 and cumulative totals. Percentages are sums of the total crosstalk percentages for all pixels equivalent by symmetry, i.e. the percentages for (0,1) represent the sum of crosstalk counts for (0,1), (1,0), (0,-1), and ( 1,0). Pixels denoted with an asterisk lie in the same column as the active pixel. Unique pixel position Cumulative line-of-sight percentage Contribution to 9 9 cumulative total Contribution to cumulative total (0,1)* 100% 37% 11% (0,2)* 95% 12% 4% (1,1) 100% 4% 1% (1,2) 84% 3% 1% Considering this analysis in light of the fact that the 9 9 sub-array represents only 31% of the cumulative device crosstalk, we can state that while the line-of-sight vector dominates crosstalk for nearest neighbors, back side reflections are likely responsible for the majority of cumulative device crosstalk. This can have important implications for future crosstalk mitigation strategies: imaging applications with stringent requirements on spatial precision may best be served by interrupting the line-of-sight vector, while applications which use the array as a single super-pixel may benefit most by interrupting the back side vector in order to reduce the cumulative total. In the case of the former, modifying trenches in order to eliminate the normal line-of-sight may be the single most effective strategy to begin with. 4. Conclusion We have successfully modeled crosstalk in an InGaAsP GmAPD array with crosstalk mitigation mechanisms in place. By accurately accounting for these mechanisms and material properties, and by incorporating 2nd-order crosstalk, we have improved upon previous modeling efforts, reproducing the previously unexplained non-monotonic behavior in the crosstalk versus distance relationship for a single type of back side reflection. The distinction between normal- and non-normal line-of-sight crosstalk has been identified as critical in describing spatial crosstalk patterns. Line-of-sight crosstalk has been shown to dominate over all other paths in contributing to cumulative crosstalk values for the immediate neighborhood of active pixels, while the influence of back side reflections grows as a function of distance and is likely responsible for the majority of total, cumulative crosstalk across the array. By prioritizing which crosstalk is most problematic to a given application, it may now be possible to improve performance by focusing device design efforts on interrupting the relevant path OSA 16 May 2016 Vol. 24, No. 10 DOI: /OE OPTICS EXPRESS 10648