Abstract

Temporal photon correlation measurement, instrumental to probing the quantum properties of light, typically requires multiple single photon detectors. Progress in single photon avalanche diode (SPAD) array technology highlights their potential as high-performance detector arrays for quantum imaging and photon number–resolving (PNR) experiments. Here, we demonstrate this potential by incorporating a novel on-chip SPAD array with 42% peak photon detection efficiency, low dark count rate and crosstalk probability of 0.14% per detection in a confocal microscope. This enables reliable measurements of second and third order photon correlations from a single quantum dot emitter. Our analysis overcomes the inter-detector optical crosstalk background even though it is over an order of magnitude larger than our faint signal. To showcase the vast application space of such an approach, we implement a recently introduced super-resolution imaging method, quantum image scanning microscopy (Q-ISM).

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (6)

M. Castello, G. Tortarolo, M. Buttafava, T. Deguchi, F. Villa, S. Koho, L. Pesce, M. Oneto, S. Pelicci, L. Lanzanó, P. Bianchini, C. J. R. Sheppard, A. Diaspro, A. Tosi, and G. Vicidomini, “A robust and versatile platform for image scanning microscopy enabling super-resolution FLIM,” Nat. Methods 16(2), 175–178 (2019).
[Crossref]

C. Bruschini, H. Homulle, I. M. Antolovic, S. Burri, and E. Charbon, “Single-photon SPAD imagers in biophotonics: Review and Outlook,” Light: Sci. Appl. 8(1), 87 (2019).
[Crossref]

R. Tenne, U. Rossman, B. Rephael, Y. Israel, A. Krupinski-Ptaszek, R. Lapkiewicz, Y. Silberberg, and D. Oron, “Super-resolution enhancement by quantum image scanning microscopy,” Nat. Photonics 13(2), 116–122 (2019).
[Crossref]

A. C. Ulku, C. Bruschini, I. M. Antolovic, Y. Kuo, R. Ankri, S. Weiss, X. Michalet, and E. Charbon, “A 512 $\times$× 512 SPAD image sensor with integrated gating for widefield FLIM,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–12 (2019).
[Crossref]

C. Zhang, S. Lindner, I. M. Antolovic, J. Mata Pavia, M. Wolf, and E. Charbon, “A 30-frames/s, 252 $\times$× 144 SPAD Flash LiDAR with 1728 Dual-Clock 48.8-ps TDCs, and Pixel-Wise Integrated Histogramming,” IEEE J. Solid-State Circuits 54(4), 1137–1151 (2019).
[Crossref]

E. Toninelli, P.-A. Moreau, T. Gregory, A. Mihalyi, M. Edgar, N. Radwell, and M. Padgett, “Resolution-enhanced quantum imaging by centroid estimation of biphotons,” Optica 6(3), 347 (2019).
[Crossref]

2018 (7)

2017 (7)

Y. Israel, R. Tenne, D. Oron, and Y. Silberberg, “Quantum correlation enhanced super-resolution localization microscopy enabled by a fibre bundle camera,” Nat. Commun. 8(1), 14786 (2017).
[Crossref]

I. M. Antolovic, S. Burri, C. Bruschini, R. A. Hoebe, and E. Charbon, “SPAD imagers for super resolution localization microscopy enable analysis of fast fluorophore blinking,” Sci. Rep. 7(1), 44108 (2017).
[Crossref]

A. N. Otte, D. Garcia, T. Nguyen, and D. Purushotham, “Characterization of three high efficiency and blue sensitive silicon photomultipliers,” Nucl. Instrum. Methods Phys. Res., Sect. A 846, 106–125 (2017).
[Crossref]

J. Kröger, T. Ahrens, J. Sperling, W. Vogel, H. Stolz, and B. Hage, “High intensity click statistics from a 10$\times$×10 avalanche photodiode array,” J. Phys. B: At., Mol. Opt. Phys. 50(21), 214003 (2017).
[Crossref]

R. Kruse, J. Tiedau, T. J. Bartley, S. Barkhofen, and C. Silberhorn, “Limits of the time-multiplexed photon-counting method,” Phys. Rev. A 95(2), 023815 (2017).
[Crossref]

S. Burri, C. Bruschini, and E. Charbon, “LinoSPAD: A Compact Linear SPAD Camera System with 64 FPGA-Based TDC Modules for Versatile 50 ps Resolution Time-Resolved Imaging,” Instruments 1(1), 6 (2017).
[Crossref]

A. Classen, J. von Zanthier, M. O. Scully, and G. S. Agarwal, “Superresolution via Structured Illumination Quantum Correlation Microscopy (SIQCM),” Optica 4(6), 580 (2017).
[Crossref]

2016 (3)

M. Unternährer, B. Bessire, L. Gasparini, D. Stoppa, and A. Stefanov, “Coincidence detection of spatially correlated photon pairs with a monolithic time-resolving detector array,” Opt. Express 24(25), 28829 (2016).
[Crossref]

S. Burri, H. Homulle, C. Bruschini, and E. Charbon, “LinoSPAD: a time-resolved 256$\times$×1 CMOS SPAD line sensor system featuring 64 FPGA-based TDC channels running at up to 8.5 giga-events per second,” Opt. Sens. Detect. IV 9899, 98990D (2016).
[Crossref]

D. Bronzi, S. Tisa, F. Villa, F. Zappa, and A. Tosi, “SPAD Figures of Merit for Photon-Counting, Photon-Timing, and Imaging Applications: A Review,” IEEE Sens. J. 16(1), 3–12 (2016).
[Crossref]

2015 (4)

2014 (4)

C. Veerappan and E. Charbon, “A Substrate Isolated CMOS SPAD Enabling Wide Spectral Response and Low Electrical Crosstalk,” IEEE J. Sel. Top. Quantum Electron. 20(6), 299–305 (2014).
[Crossref]

A. Rundquist, M. Bajcsy, A. Majumdar, T. Sarmiento, K. Fischer, K. G. Lagoudakis, S. Buckley, A. Y. Piggott, and J. Vučković, “Nonclassical higher-order photon correlations with a quantum dot strongly coupled to a photonic-crystal nanocavity,” Phys. Rev. A 90(2), 023846 (2014).
[Crossref]

Y. Israel, S. Rosen, and Y. Silberberg, “Supersensitive Polarization Microscopy Using NOON States of Light,” Phys. Rev. Lett. 112(10), 103604 (2014).
[Crossref]

M. J. Stevens, S. Glancy, S. W. Nam, and R. P. Mirin, “Third-order antibunching from an imperfect single-photon source,” Opt. Express 22(3), 3244 (2014).
[Crossref]

2013 (2)

T. Ono, R. Okamoto, and S. Takeuchi, “An entanglement-enhanced microscope,” Nat. Commun. 4(1), 2426 (2013).
[Crossref]

O. Schwartz, J. M. Levitt, R. Tenne, S. Itzhakov, Z. Deutsch, and D. Oron, “Superresolution microscopy with quantum emitters,” Nano Lett. 13(12), 5832–5836 (2013).
[Crossref]

2012 (2)

O. Schwartz, R. Tenne, J. M. Levitt, Z. Deutsch, S. Itzhakov, and D. Oron, “Colloidal quantum dots as saturable fluorophores,” ACS Nano 6(10), 8778–8782 (2012).
[Crossref]

F. Villa, B. Markovic, S. Bellisai, D. Bronzi, A. Tosi, F. Zappa, S. Tisa, D. Durini, S. Weyers, U. Paschen, and W. Brockherde, “SPAD smart pixel for time-of-flight and time-correlated single-photon counting measurements,” IEEE Photonics J. 4(3), 795–804 (2012).
[Crossref]

2011 (1)

M. D. Eisaman, J. Fan, A. Migdall, and S. V. Polyakov, “Invited Review Article: Single-photon sources and detectors,” Rev. Sci. Instrum. 82(7), 071101 (2011).
[Crossref]

2010 (3)

G. Brida, M. Genovese, and I. R. Berchera, “Experimental realization of sub-shot-noise quantum imaging,” Nat. Photonics 4(4), 227–230 (2010).
[Crossref]

C. B. Müller and J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104(19), 1–4 (2010).
[Crossref]

H. Ta, A. Kiel, M. Wahl, and D. P. Herten, “Experimental approach to extend the range for counting fluorescent molecules based on photon-antibunching,” Phys. Chem. Chem. Phys. 12(35), 10295–10300 (2010).
[Crossref]

2009 (2)

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3(12), 696–705 (2009).
[Crossref]

G. Brida, I. P. Degiovanni, F. Piacentini, V. Schettini, S. V. Polyakov, and A. Migdall, “Scalable multiplexed detector system for high-rate telecom-band single-photon detection,” Rev. Sci. Instrum. 80(11), 116103 (2009).
[Crossref]

2008 (5)

A. E. Lita, A. J. Miller, and S. W. Nam, “Counting near-infrared single-photons with 95% efficiency,” Opt. Express 16(5), 3032–3040 (2008).
[Crossref]

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), 8381 (2008).
[Crossref]

M. Mičuda, O. Haderka, and M. Ježek, “High-efficiency photon-number-resolving multichannel detector,” Phys. Rev. A: At., Mol., Opt. Phys. 78(2), 1–4 (2008).
[Crossref]

A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’Tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, “Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths,” Nat. Photonics 2(5), 302–306 (2008).
[Crossref]

V. I. Klimov, A. A. Mikhailovsky, D. W. McBranch, C. A. Leatherdale, and M. G. Bawendi, “Quantization of Multiparticle Auger Rates in Semiconductor Quantum Dots,” Science 1011, 1011–1014 (2008).

2007 (3)

L. A. Jiang, E. A. Dauler, and J. T. Chang, “Photon-number-resolving detector with 10 bits of resolution,” Phys. Rev. A: At., Mol., Opt. Phys. 75(6), 062325 (2007).
[Crossref]

E. J. Gansen, M. A. Rowe, M. B. Greene, D. Rosenberg, T. E. Harvey, M. Y. Su, R. H. Hadfield, S. W. Nam, and R. P. Mirin, “Photon-number-discriminating detection using a quantum-dot, optically gated, field-effect transistor,” Nat. Photonics 1(10), 585–588 (2007).
[Crossref]

P. Eraerds, M. Legré, A. Rochas, H. Zbinden, and N. Gisin, “SiPM for fast Photon-Counting and Multiphoton Detection,” Opt. Express 15(22), 14539–14549 (2007).
[Crossref]

2003 (4)

D. Achilles, C. Silberhorn, C. Sliwa, K. Banaszek, and I. A. Walmsley, “Fiber-assisted detection with photon number resolution,” Opt. Lett. 28(23), 2387–2389 (2003).
[Crossref]

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C. Bruschini, H. Homulle, I. M. Antolovic, S. Burri, and E. Charbon, “Single-photon SPAD imagers in biophotonics: Review and Outlook,” Light: Sci. Appl. 8(1), 87 (2019).
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I. M. Antolovic, C. Bruschini, and E. Charbon, “Dynamic range extension for photon counting arrays,” Opt. Express 26(17), 22234 (2018).
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I. M. Antolovic, S. Burri, C. Bruschini, R. A. Hoebe, and E. Charbon, “SPAD imagers for super resolution localization microscopy enable analysis of fast fluorophore blinking,” Sci. Rep. 7(1), 44108 (2017).
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S. Burri, C. Bruschini, and E. Charbon, “LinoSPAD: A Compact Linear SPAD Camera System with 64 FPGA-Based TDC Modules for Versatile 50 ps Resolution Time-Resolved Imaging,” Instruments 1(1), 6 (2017).
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S. Burri, H. Homulle, C. Bruschini, and E. Charbon, “LinoSPAD: a time-resolved 256$\times$×1 CMOS SPAD line sensor system featuring 64 FPGA-based TDC channels running at up to 8.5 giga-events per second,” Opt. Sens. Detect. IV 9899, 98990D (2016).
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C. Bruschini, H. Homulle, I. M. Antolovic, S. Burri, and E. Charbon, “Single-photon SPAD imagers in biophotonics: Review and Outlook,” Light: Sci. Appl. 8(1), 87 (2019).
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I. M. Antolovic, S. Burri, C. Bruschini, R. A. Hoebe, and E. Charbon, “SPAD imagers for super resolution localization microscopy enable analysis of fast fluorophore blinking,” Sci. Rep. 7(1), 44108 (2017).
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S. Burri, C. Bruschini, and E. Charbon, “LinoSPAD: A Compact Linear SPAD Camera System with 64 FPGA-Based TDC Modules for Versatile 50 ps Resolution Time-Resolved Imaging,” Instruments 1(1), 6 (2017).
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S. Burri, H. Homulle, C. Bruschini, and E. Charbon, “LinoSPAD: a time-resolved 256$\times$×1 CMOS SPAD line sensor system featuring 64 FPGA-based TDC channels running at up to 8.5 giga-events per second,” Opt. Sens. Detect. IV 9899, 98990D (2016).
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C. Bruschini, H. Homulle, I. M. Antolovic, S. Burri, and E. Charbon, “Single-photon SPAD imagers in biophotonics: Review and Outlook,” Light: Sci. Appl. 8(1), 87 (2019).
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C. Zhang, S. Lindner, I. M. Antolovic, J. Mata Pavia, M. Wolf, and E. Charbon, “A 30-frames/s, 252 $\times$× 144 SPAD Flash LiDAR with 1728 Dual-Clock 48.8-ps TDCs, and Pixel-Wise Integrated Histogramming,” IEEE J. Solid-State Circuits 54(4), 1137–1151 (2019).
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I. M. Antolovic, C. Bruschini, and E. Charbon, “Dynamic range extension for photon counting arrays,” Opt. Express 26(17), 22234 (2018).
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S. Burri, C. Bruschini, and E. Charbon, “LinoSPAD: A Compact Linear SPAD Camera System with 64 FPGA-Based TDC Modules for Versatile 50 ps Resolution Time-Resolved Imaging,” Instruments 1(1), 6 (2017).
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I. M. Antolovic, S. Burri, C. Bruschini, R. A. Hoebe, and E. Charbon, “SPAD imagers for super resolution localization microscopy enable analysis of fast fluorophore blinking,” Sci. Rep. 7(1), 44108 (2017).
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S. Burri, H. Homulle, C. Bruschini, and E. Charbon, “LinoSPAD: a time-resolved 256$\times$×1 CMOS SPAD line sensor system featuring 64 FPGA-based TDC channels running at up to 8.5 giga-events per second,” Opt. Sens. Detect. IV 9899, 98990D (2016).
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W. Luo, Q. Weng, M. Long, P. Wang, F. Gong, H. Fang, M. Luo, W. Wang, Z. Wang, D. Zheng, W. Hu, X. Chen, and W. Lu, “Room-Temperature Single-Photon Detector Based on Single Nanowire,” Nano Lett. 18(9), 5439–5445 (2018).
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B. Cabrera, R. M. Clarke, P. Colling, A. J. Miller, S. Nam, and R. W. Romani, “Detection of single infrared, optical, and ultraviolet photons using superconducting transition edge sensors,” Appl. Phys. Lett. 73(6), 735–737 (1998).
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B. Cabrera, R. M. Clarke, P. Colling, A. J. Miller, S. Nam, and R. W. Romani, “Detection of single infrared, optical, and ultraviolet photons using superconducting transition edge sensors,” Appl. Phys. Lett. 73(6), 735–737 (1998).
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L. A. Jiang, E. A. Dauler, and J. T. Chang, “Photon-number-resolving detector with 10 bits of resolution,” Phys. Rev. A: At., Mol., Opt. Phys. 75(6), 062325 (2007).
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G. Brida, I. P. Degiovanni, F. Piacentini, V. Schettini, S. V. Polyakov, and A. Migdall, “Scalable multiplexed detector system for high-rate telecom-band single-photon detection,” Rev. Sci. Instrum. 80(11), 116103 (2009).
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M. Castello, G. Tortarolo, M. Buttafava, T. Deguchi, F. Villa, S. Koho, L. Pesce, M. Oneto, S. Pelicci, L. Lanzanó, P. Bianchini, C. J. R. Sheppard, A. Diaspro, A. Tosi, and G. Vicidomini, “A robust and versatile platform for image scanning microscopy enabling super-resolution FLIM,” Nat. Methods 16(2), 175–178 (2019).
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A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’Tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, “Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths,” Nat. Photonics 2(5), 302–306 (2008).
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I. Gyongy, N. Calder, A. Davies, N. A. W. Dutton, R. R. Duncan, C. Rickman, P. Dalgarno, and R. K. Henderson, “A 256$\times$×256 , 100-kfps, 61% Fill-Factor SPAD Image Sensor for Time-Resolved Microscopy Applications,” IEEE Trans. Electron Devices 65(2), 547–554 (2018).
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F. Villa, B. Markovic, S. Bellisai, D. Bronzi, A. Tosi, F. Zappa, S. Tisa, D. Durini, S. Weyers, U. Paschen, and W. Brockherde, “SPAD smart pixel for time-of-flight and time-correlated single-photon counting measurements,” IEEE Photonics J. 4(3), 795–804 (2012).
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M. Agnew, R. Warburton, I. Gyongy, N. Dutton, R. Henderson, D. Faccio, and J. Leach, “Imaging quantum correlations with a single-photon detector array,” in Imaging and Applied Optics 2016 (OSA, Washington, D.C., 2016), p. IT4E.4.

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I. Gyongy, N. Calder, A. Davies, N. A. W. Dutton, R. R. Duncan, C. Rickman, P. Dalgarno, and R. K. Henderson, “A 256$\times$×256 , 100-kfps, 61% Fill-Factor SPAD Image Sensor for Time-Resolved Microscopy Applications,” IEEE Trans. Electron Devices 65(2), 547–554 (2018).
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W. Luo, Q. Weng, M. Long, P. Wang, F. Gong, H. Fang, M. Luo, W. Wang, Z. Wang, D. Zheng, W. Hu, X. Chen, and W. Lu, “Room-Temperature Single-Photon Detector Based on Single Nanowire,” Nano Lett. 18(9), 5439–5445 (2018).
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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 $\times$× 256 photon counting imager based on silicon Geiger-mode avalanche photodiodes,” IEEE Sens. J. 15(4), 2123–2132 (2015).
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A. Ficorella, L. Pancheri, G. F. Betta, P. Brogi, G. Collazuol, P. S. Marrocchesi, F. Morsani, L. Ratti, and A. Savoy-Navarro, “Crosstalk mapping in CMOS SPAD arrays,” European Solid-State Device Research Conference (2016), pp. 101–104.

Fiore, A.

A. Divochiy, F. Marsili, D. Bitauld, A. Gaggero, R. Leoni, F. Mattioli, A. Korneev, V. Seleznev, N. Kaurova, O. Minaeva, G. Gol’Tsman, K. G. Lagoudakis, M. Benkhaoul, F. Lévy, and A. Fiore, “Superconducting nanowire photon-number-resolving detector at telecommunication wavelengths,” Nat. Photonics 2(5), 302–306 (2008).
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Figures (8)

Fig. 1.
Fig. 1. Photon correlation setups. a. Hanbury Brown and Twiss intensity interferometer. A beamsplitter splits the incident photons to two correlated detectors. A single photon emitter source will be characterized by a dip at the zero time delay of the second order photon correlation. Higher order correlations demand more beamsplitters and detectors. b. The SPAD array photon correlation setup. A SPAD array is positioned at the image plane of a scanning confocal microscope, resulting in the splitting of the beam by diffraction onto 23 detectors. Inset. An optical image of the SPAD array.
Fig. 2.
Fig. 2. SPAD array crosstalk characterization. a. Typical second order correlation of photon arrival times for two nearest-neighbour detectors in the array, in response to homogeneous illumination by a thermal source (analyzed from $10^7$ detections over ∼43 s). Note the sharp peak at zero time delay attributed to crosstalk. b. Crosstalk linearity in detection rate. Each colored set of markers represent a different nearest neighbour detector pair. Lines of corresponding color represent a linear fit for each pair. c. Characterization of crosstalk dependence on detector distance. Each bar shows the crosstalk probability averaged over all detector pairs at a certain distance. Error bars represent one standard deviation of the distribution over these pairs. The values suggest that crosstalk is significant mostly for nearest neighbour detectors. Inset. Visualization of neighbour rank. A photon (green arrow) is absorbed in the upper left detector. The neighbouring detectors are ranked by distance from the excited detector. Nearest neighbours are rank 1.
Fig. 3.
Fig. 3. Second order photon correlations. Second order photon arrival time correlations ($G^{(2)}(\tau )$) for: a. Classical light (large ensemble of QDs), b. same as a after crosstalk correction, c. single QD and d. single QD after crosstalk correction. All values are normalized by the total number of measured photons ($N_p$). The correlation peaks are centered at integer multiples of the laser inter-pulse separation (400ns) broadened by the QDs emission lifetime (∼26 ns). In panel a and c, the high crosstalk peaks at zero time delay can be clearly seen. In panel b the zero delay peak is featureless with respect to non-zero peaks, as expected for classical light sources following Poissonian statistics. The zero delay peak in panel d is significantly lower than the non-zero delay peaks, as expected from an antibunched light source. Classical and single QD correlation curves were analyzed from $10^7$ detections over ∼103 s and 8×106 detections over ∼105 s respectively. e. Histograms of the normalized second order correlation function at zero time delay ($g^{(2)}(0)$) after crosstalk correction for 19 single QD measurements, utilizing the SPAD array (top panel) and on a similar sample with a standard HBT setup using two separate SPADs and a split optical fiber (bottom panel). Note the very good agreement of the distributions’ mean value and width. The mean value is much smaller than $0.5$, indicative of single photon emitters.
Fig. 4.
Fig. 4. Third order photon correlations. Crosstalk corrected third order photon correlations for: a. Classical light (large ensemble of QDs), b. single QD. Photon triplets are histogramed according to the difference in their arrival times, $\tau _1$ and $\tau _2$ being the delays between the arrival of one (randomly selected) photon of the triplet and the arrival times of the other two photons respectively. The colorbar represents number of triplets in histogram bin ($t_{clk}= {10}\, {\rm ns}$ binning in both axes). Negative values are a result of crosstalk over-correction due to noise. The observed grid of peaks corresponds to the 2.5 MHz frequency of the pulsed excitation. The peaks’ profile match the fluorescence lifetime of the QDs. Note the decimation of peaks along the two axis and one of the diagonals in panel b, indicating photon antibunching (low $g^{(2)}(0)$). Insets are the second order correlation estimations attained by full vertical binning of the $G^{(3)}$ values. c. The three possible pathways for crosstalk to form false $G^{(3)}(0,0)$ triplets from incident photons (green arrows) and crosstalk events (blue arrows). d. Histogram of crosstalk corrected $g^{(3)}(0,0)$ values from 19 different QDs.
Fig. 5.
Fig. 5. Narrowing the point spread function with Q-ISM. A 1 µm × 1 µm confocal scan of a single CdSe/CdS/ZnS QD. a. CLSM image - summing counts over all detectors for each scan position. b. ISM image - the intensity image generated by each detector is shifted before summation. c. Q-ISM image - $\Delta G^{(2)}$ for each detector pair is shifted and then summed. d. Cross-sections for the different analyses: CLSM (blue circles, dashed line), ISM (red triangles, dotted line) and Q-ISM (yellow diamonds, dash-dot line). The values for the CLSM cross section were radially averaged to reduce blinking artifacts. These artifacts do not affect ISM and Q-ISM images. Scale bar: 0.25 µm.
Fig. 6.
Fig. 6. Crosstalk temporal stability. Standard deviation of crosstalk probability versus the averaging window used to to estimate it (Allan Deviation) over a measurement of ∼47 s (blue circles), and the expected Allan Deviation from shot noise (black line). The close agreement indicates stability of the crosstalk probability.
Fig. 7.
Fig. 7. Single quantum dot (QD) blinking. The fluorescence intensity measured from a static single QD with constant illumination presents three distinct fluorescent states: A bright ’on’ state, a dark ’off’ state and an intermediate intensity ’grey’ state.
Fig. 8.
Fig. 8. Estimating the resolution enhancement in Q-ISM. Scale bars are 250 nm

Equations (25)

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p i , j C T = G i , j ( 2 ) ( 0 ) G i , j ( 2 ) ( ) n i + n j ,
G c o r r ( 2 ) ( τ ) { G m e a s ( 2 ) ( 0 ) i j n i p i , j C T τ = 0 G m e a s ( 2 ) ( τ ) τ 0 ,
G ~ ( 2 ) ( T ) T p u l s e 2 + T p u l s e 2 d τ G c o r r ( 2 ) ( T + τ ) | T = k T p u l s e   ,   k Z
G ~ D C ( 2 ) = N p D C R T p u l s e N d 1 N d
g ( 2 ) ( 0 ) = G ~ ( 2 ) ( 0 ) G ~ D C ( 2 ) G ~ ( 2 ) ( T ) T   0 G ~ D C ( 2 )   ,
G ~ ( 3 ) ( T 1 , T 2 ) T p u l s e 2 + T p u l s e 2 d τ 1 d τ 2 G c o r r ( 3 ) ( T 1 + τ 1 , T 2 + τ 2 ) | T 1 , 2 = k 1 , 2 T p u l s e   ,   k 1 , 2 Z
G ~ D C ( 3 ) ( T ) = G ~ ( 2 ) ( T ) D C R T p u l s e N d 2 N d
g ( 3 ) ( 0 ) = G ~ ( 3 ) ( 0 , 0 ) G ~ D C ( 3 ) ( 0 ) G ~ ( 3 ) ( T 1 , T 2 ) T 1 , 2   0     T 1 T 2 G ~ D C ( 3 ) ( T ) T   0   ,
G c o r r ( 2 ) ( τ ) = G m e a s ( 2 ) ( τ ) | τ 0   ,
j n i p i , j C T | j i   ,
G c o r r ( 2 ) ( 0 ) = G m e a s ( 2 ) ( 0 ) i j n i p i , j C T .
G i , j ( c o r r ) ( 2 ) ( x , y , 0 ) = G i , j ( m e a s ) ( 2 ) ( x , y , 0 ) ( n i + n j ) p i , j C T ,
G ~ i , j ( 2 ) ( x , y , T ) T p u l s e 2 + T p u l s e 2 d τ G i , j ( c o r r ) ( 2 ) ( x , y , T + τ ) | T = k T p u l s e   ,   k Z
Δ G i , j ( 2 ) ( x , y ) = G ~ i , j ( 2 ) ( x , y , T ) T 0 G ~ i , j ( 2 ) ( x , y , 0 )
G c o r r ( 3 ) ( τ 1 , τ 2 ) = G m e a s ( 3 ) ( τ 1 , τ 2 ) | τ 1 0 τ 2 0 τ 1 τ 2   ,
G c o r r ( 3 ) ( τ 1 , τ 2 ) = G m e a s ( 3 ) ( τ 1 , τ 2 ) i j k n i p i , j C T | τ 1 = 0 τ 2 = 0 τ 1 = τ 2   ,
G c o r r ( 3 ) ( 0 , 0 ) = G m e a s ( 3 ) ( 0 , 0 ) i j k [ G ( i , j ) c o r r ( 2 ) ( 0 ) p j , k C T + n i p i , j C T ( p j , k C T + 1 2 p i , k C T ) ] ,
I ( x , y ) = A e ( x x 0 ) 2 2 σ x 2 e ( y y 0 ) 2 2 σ y 2 + B   ,
V [ G i , j ( c o r r ) ( 2 ) ( 0 ) ] = V [ G ~ i , j ( m e a s ) ( 2 ) ( 0 ) ] + V [ ( n i + n j ) p i , j C T ]   ,
G ~ i , j ( m e a s ) ( 2 ) ( 0 ) = n i n j 2 N p u l g ( 2 ) ( 0 ) + ( n i + n j ) p i , j C T ,
V [ G i , j ( c o r r ) ( 2 ) ( 0 ) ] = n i n j 2 N p u l g ( 2 ) ( 0 ) + ( n i + n j ) p i , j C T + ( n i + n j ) [ p i , j C T ] 2   .
G ( c o r r ) ( 2 ) ( 0 ) = 1 2 i j G i , j ( c o r r ) ( 2 ) ( 0 )
V [ G ( c o r r ) ( 2 ) ( 0 ) ] g ( 2 ) ( 0 ) N p 2 2 N p u l + 6 p C T N p ,
g ( e s t ) ( 2 ) ( 0 ) = G ( c o r r ) ( 2 ) ( 0 ) G ( 2 ) ( )
δ [ g ( e s t ) ( 2 ) ( 0 ) ] = 1 G ( 2 ) ( ) g ( 2 ) ( 0 ) + 12 p C T p p h ,