Photon counting in the Infrared with e-APD devices

J. L. Gach*a,b, aAix Marseille Université, CNRS, CNES, LAM (Laboratoire d'Astrophysique de Marseille) UMR

7326, 13388, Marseille, France; bFirst Light Imaging Group, Europarc Ste Victoire – Bât. 6, Route de Valbrillant – Le Canet 13590

Meyreuil France

ABSTRACT

Since the first photon counting systems in the visible, developped by Boksenberg and his collabortors in 1972, many groups around the world improved photon counting techniques. In the 2000's in the visible, EMCCDSs (electron multiplying charge coupled devices) allowed to replace the classical image intensifier photon counting systems by solid state devices and improved a lot the QE. But EMCCDs suffer from several issues, and the most important of them is the excess noise factor which prevents to know what is the exact incoming number pf photons in the case of multiple photons per pixel. In the infrared there was no equivalent to EMCCDs up to the development of e-APD sensors and cameras made with HgCdTe material (electron initiated avalanche photo diode). We will present the concepts of photon counting in the infrared with such devices and the main properties and advantages of infrared e-APDs compared to their visible counterpart, EMCCDs and the possibility to detect the photon energy (color) with such devices.

Keywords: Photon counting, e-APD, EMCCD, infrared, imaging

1. INTRODUCTION

Photon counting devices appeared a long time ago, as soon as it was possible to amplify the photonic signal sufficiently to overcome the readout noise of the electronics. This introduction shows the evolution of photon counting techniques through the ages up to modern photon counting in the visible.

1.1 The photomultiplier tube, first photon counting device

The fluxes involved especially for sources of low emission in astronomy (but not only) are of the order of a few photons per hour, which would correspond to photocurrents of some zepto (1.10-21) ampere, impossible to measure in the current state of technology. It was therefore necessary to amplify this extremely weak signal to be able to detect it. The first idea was that of the photomultiplier tube combining photoelectric effect and secondary electronic emission.

Figure 1 : Left, An image intensifier tube, right its principle using electron multiplication over dynodes in a vacuum tube

*jean-luc.gach@lam.fr; phone +33 495044119; www.lam.fr; www.first-light.fr

The first tube was invented by NJ. Harley Iams and Bernard Salzberg of RCA in 1934 [1]. Figure 1 illustrates a photomultiplier tube embodiment and illustrates the principle of operation. An incident photon is transformed into a primary electron by a photocathode immersed in vacuum, this electron is accelerated and focused on a series of dynodes by an electrostatic field. During the impact of the electron on the dynode, one or more secondary electrons are emitted by ionization, these secondary electrons are in turn accelerated and focused by an electrostatic field on the next dynode. The anode collects the final jet of electrons and the measurement of the current coming from the anode makes it possible to measure the initial photonic signal amplified by the gain factor of the system which is essentially a function of the number of dynodes, their geometry and the field electric acceleration. This device is interesting since it shows the basic principles of photon counting that can be sliced in 3 independent processes, the photo-conversion done by the photocathode in this example, the amplification itself combining two different effects, electron acceleration in an electric field and secondary emission by impact ionization and finally current collection and detection on the anode of the tube.

1.2 Image intensified photon counting devices

Lallemand was the first to evoke the concept of bidimensional photon counting since he wrote in his 1936 historical publication [2] about the electronic camera (translated from French language): "This camera is a real photon counter, the photographic plate playing the role of totalizer of photon pulses at the same time as faithfully reproducing the point of arrival of each photon'. However, it will be necessary to wait until 1971 and Gezari [3][4] to see appearing the first experiments of bidimensional photon counting by image intensifier tubes. In this case the image is no longer formed directly on the photo plate as in the Lallemand Camera but on a phosphor screen. An optical fiber bundle coupling or lens system allows cascading intensifier tubes and thus obtain much higher amplification rates than those obtained by Lallemand and its electronic camera. Figure 2 shows a sectional view of a 2nd generation microchannel plate (MCP) intensifier tube. Figure 3 illustrates the operation of a microchannel plate. Each channel is a tube of 5 to 40 m diameter covered with resistive material. A potential difference is applied on each side of the MCP. An electron striking the wall will generate a number of secondary electrons accelerated by the electric field internal to the channel in the manner of dynodes. Each channel behaves like a micro photomultiplier tube.

Figure 2: Structure of a microchannel plate (MCP) amplified image intensifier tube

Figure 3: The MCP plate is a set of microtubes (SEM view left) coated with a resistive material acting as elementary micro photomultipliers and paving the plane to form a bidimensional image.

The Gezari device was described in more detail by Labeyrie et al. in 1974 [5], and mentioned the possibility of autocorrelations with a few photons per image. Figure 4 shows the detector used by Labeyrie and Gezzari. At the same time, Boksenberg understood from 1972 [6] the ability to detect single photons with this type of detector. He was also the first to couple an intensifier to an electronic reading system and not a photo plate (this was a plumbicon tube).

Figure 4: Detector employed by Labeyrie et al. to perform speckle interferometry experiments. The detector is a Kodak TriX film associated with a cascade of image intensifiers visible in the foreground.

Figure 5 : Left: Boksenberg's photon counting camera, Right: a photon image produced by this camera.

In 1975 photon counting techniques were then established and described in detail by Boksenberg [7]. The process differs slightly from standard imaging since the photon accumulation is not done on the imager itself but on a computer’s memory. Each elementary image is sliced to a threshold level (usually 5RON), and the photons are detected by computing the center of gravity of each blob (see figure 6). Each image is a short time exposure and the accumulation is

done by combining the positions detected in thousands of elementary images taken during the whole integration. The system has a big advantage since the slicing process is nonlinear it removes completely the noise in the image and therefore photon counting imagers have absolutely no readout noise. The price to pay is nonlinearity at high fluxes, since when several photons hit the detector at the same place on the same elementary image it is counted as one photon. This can be mitigated by avoiding the case using a fast readout and very short exposure times. This has been developed in Gach et al. [8].

Figure 6 : Top : photon counting principle. the signal produced by an amplified photon in the image is sliced and a gravity computation is made to determine the initial photon impact localization. Bottom : an integration example, with an elementary 1/50th second integration image on the left, 10s accumulation reconstruction in the center and 5 min on the right (source Hamamatsu).

Subsequently the systems inspired by the one of Boksenberg were developed with refinements mainly concerning the acquisition which benefited from the advent of the computers. Thus Blazit [9][10] describes an autocorellation electronic system for speckle applications. In 1980 Boulesteix and Marcelin [11] describe a system with real-time acquisition, and showed that intensified photon counting systems (IPCS) were more efficient than very low flux coupled charge coupled devices (CCDs). In 1995 & 1999 respectively EBCCDs (Williams et al. [12]) and EBAPS (Aebi et al. [13]) systems were introduced placing the detector directly in the vacuum tube and the avoiding the phosphor electron-to-photon conversion.

1.3 EmCCD photon counting devices

In 2002 the high quantum efficiency AsGa photon counting system was introduced (Gach et al. [8]). It was a real improvement of QE with such systems which were suffering from very poor quantum efficiency in the reddest part of the visible spectrum with classical bialkali or multialkali (Na2KSb)Cs photocathodes. But it has been showed (for example by Boutboul [14]) that it is impossible to reach very high quantum efficiencies with such systems because on one side the photocathode must be thin to avoid phonon scattering and on the other side thick to maximize the absorption (Beer-Lambert law). Solid state systems were the only way to improve photon counting. In this sense the EMCCDs were a very promising track. The first avalanche multiplication experiments in CCDs (Charge Coupled Device) are reported very early in the literature as early as 1983 by Madan et al. [15]. Various other authors have conducted subsequent experiments on the same theme, among which Gadjar & Burke in 1988 [16] or Hynecek in 1992 [17] and 1994 [18]. However, it will be necessary to wait for the work of Jerram et al. to see the first commercially available EMCCDs [19] from Marconi Applied Technologies or Hynecek at the same time [20] from Texas Instruments. EMCCDs were coupling then the advantages of solid state photoconversion with QE near 100%, an electron multiplication system and a detection circuitry on the same integrated device. As the multiplication process is done on a pixel per pixel basis, the gravity detection step of photon counting is no more necessary in this case. The first photon counting image with an EMCCD reported was in 2002 by Gach et al. [21] with a cryogenically cooled Marconi CCD65 (576x288 pixels) originally designed for TV night vision applications.

Figure 7 : The first EMCCD photon counting system reported in the literature in 2002 during the scientific detector workshop [21], using a cryogenically cooled Marconi CCD65 (left) and a photon counting regime image at G=3000 (right).

As for image-tube-based photon counting, EMCCD photon counting does not permit to know the number of output electrons knowing the output amplified level. This is due to the stochastic nature of electron multiplication in the amplification register and therefore given an amplification level and input number of electrons, there is a probability to have an output level of electrons, and these probabilities overlap as plotted in figure 8.

Figure 8 : Output probability intensity of an EMCCD at a gain of 200 for various numbers of input electrons.

Basden and his collaborators showed in [22] that this probability can be expressed as:

�(�) =��−1�

−��

�� (� − 1)!

Where g is the mean gain and n the number of input electrons. This is the major limitation of photon counting EMCCDs, since one still needs to maintain a very high frame rate to overcome the nonlinarity at high fluxes due to photon coincidence or is obliged to use the sensor for very faint fluxes in the case of low framerates.

2. EXCESS NOISE FACTOR

2.1 McIntyre’s theory of ionization

McIntyre's theory [23] formalizes the stochastic effect of the process in various materials for avalanche triggered multiplication. His formalism uses the parameter k which is the ratio of the ionization probabilities for holes and electrons per unit length, respectively. The theory makes possible to calculate the average gain <G> for a junction of length W as well as excess noise factor F2

<G> or ENF as a function of the average gain <G> that quantifies the stochastic effect of multiplication :

⟨�⟩ =(1 − �)

����−��(1 − �)� − �

�<�>2 = �⟨�⟩ + (1 − �) �2 −

1

⟨�⟩�

It will be noted that in the literature the excess noise factor is noted F or F2 according to the authors. Figure 9 shows the excess noise factor for APD diodes in various materials.

Figure 9 : Excess noise factor (ENF) versus gain for APD diodes in various semiconductor materials according to McIntyre's theory.

It is lovely to see that the results of Robbins et al. [24] who have measured the ENF in EMCCDs follow perfectly this theory with k=0, meaning a ionization exclusively done by electrons which is the case in an electron populated channel of an EMCCD. It is important to maintain F2 as low as possible to minimize the noise introduced by the ENF in the final total noise that scales as :

���� = ��2(�� × � + ����) +����

2

⟨�⟩2

Where QE is the quantum efficiency, S is the signal, Dark is the dark signal, lec is the amplifier noise and <G> the gain.

2.2 Excess Noise Factor in Mercury Cadmium Telluride avalanche photodiodes

Leveque et al. described a strong dissymmetry of k for Mercury Cadmium Telluride (MCT) in 1993 [25] and had already predicted a favorable situation for making avalanche photodiodes with a small ENF. Figure 9 shows the evolution of the ionization ratio between holes and electrons (factor k) as a function of the composition of the material. For materials with a low concentration of mercury, the electrons are the main contributors of the multiplication process, whereas for those with a high concentration (> 53%) the holes are the majority. It should be noted that in the case where k> 1 (majority of holes), the McIntyre theory applies, simply replace k by 1 / k. The strong asymmetry of the factor k in MCT made possible to envisage the realization of infrared diodes whose ENF would be equivalent to the one of silicon, and this was a big issue for high speed telecommunication applications where no low-ENF infrared photodiodes were existing.

Figure 9: variation of the ionization ratio between holes and electrons (k factor) in the MCT material as a function of the cadmium concentration on the abscissa, extracted from Leveque et al. [25].

Surprisingly, Beck et al. reported in 2001 an MCT MWIR avalanche photodiode structure whose excess noise does not follow McIntyre's theory. Indeed they observed F2 ~ 1 for large gains [26] as illustrated in Figure 10 followed by a more detailed publication in 2004 [27].

Figure 10: Measurements of F2 excess noise by Beck et al. [26] for their MCT MWIR diodes showing that the McIntyre theory (dashed curve) is not respected in the case of this avalanche diode.

2.3 Dead space theories

The failure of McIntyre’s theory can be explained in a highly dissymmetric material by the fact that an electron needs to acquire kinetic energy to be able to generate a new impact ionization, and this was not taken into account in McIntyre’s theory where there is a non-null probability that a newly generated electron generates in turn immediately a new impact ionization. As early as 1974, Okuto and Crowell [28] hypothesized non-localized ionization coefficients, meaning that an electron that had recently passed through the conduction band could not in turn immediately generate a new impact ionization and had to acquire sufficient kinetic energy, which is obtained only after having traveled a given distance in the material, accelerated by the electric field. This model will then be identified in the literature as "hard dead space model". The impact of the introduction of this "dead space" is a self-organization of the multiplication process which then becomes localized in space at specific points and thus decreases the stochastic aspect of the process. Other authors will then develop this theory including Saleh in 1990 [29], a theory that will correctly predict the ENF of certain diodes (Hayat 2000 [30]). Other researchers worked in parallel to the refinement of this theory, this is the case of Marsland in 1990 [31]. The same author made a good summary of the state of these different variants in 2011 [32] and showed that all these works eventually lead to close results. Nevertheless this theory, although relatively close to the observations concerning the MCT, was not entirely satisfactory. In particular, the appearance of resonances for gains of 2, 4, 8, and 16 predicted by Marsland (Figure 11) with F = 1 was not really observed.

Figure 11: Predictions of ENF over gain according to Marsland's work showing a resonance for M gains of 2.4, 8 and 16.

Derelle and her collaborators have built a model involving not only the notion of "dead space" but also phonon and alloy scattering computed by Monte Carlo simulation in good agreement with the observations [33] made on diodes developed by Rothman and his collaborators [34]. Today the question does not seem completely closed, and teams continue to work on the subject with interesting results like those of Bertazzi et al. in 2010 [35] or Belotti et al. in 2011 [36].

3. PHOTON COUNTING IN THE INFRARED

3.1 C-RED One e-APD camera

As early as 2008 the prospects offered by e-APDs, and their simplicity of manufacture which was compatible with the standard processes opened the way to build imagers using this technology. In collaboration with the companies SOFRADIR, BIOSPACE and the CEA-LETI laboratories, ONERA and IPAG, a project was launched to realize a 320x256 sensor using this technology [37]. At the same time, Finger and his collaborators did a fairly similar job [38] in collaboration with SELEX. The manufacturer of the sensor SOFRADIR has however decided not to pursue this sector so that subsequently it is the component developed by SELEX (now Leonardo) and G. Finger that we will use in particular at First Light Imaging for the C-RED One camera [39]. Figure 12 shows a picture of the camera.

Figure 12: First Light Imaging’s C-RED One camera using Leonardo’s SAPHIRA e-APD sensor.

.

3.2 Noise & speed performance of C-RED One enables photon counting

Compared to other short wave infrared cameras (SWIR), the C-RED One camera developed by First Light Imaging offers a higher refresh rate, lower RMS noise, low background noise, excellent cosmetics and wide spectral response in the J, H and K bands (0.8 to 2.5 μm). The results obtained with this sensor and the C-RED One camera are particularly impressive since we hold the record of infrared imaging speed (3500 FPS) and simultaneously the readout noise record (0.3 electron), which is particularly interesting for adaptive optics and interferometry applications. This project, funded by the European Horizon 2020 "SME Instrument" research program (grant agreement 673244), has demonstrated that the use of e-ADP detectors provides unprecedented results for high frequency imaging of natural guide stars (NGS) and near-infrared wavefront sensors on large telescopes. These and other results were developed in [39][40]. The other application of such a camera is interferometry and several results obtained with C-RED One were presented in this conference by the MIRC-X team [41][42] at the Chara interferometer. Figure 13 shows the typical readout noise vs gain of the C-RED One camera, and one could see that above gains of 20 to 30, the camera enters in sub-electron readout noise regime enabling photon counting capability.

Figure 13: Typical readout noise of a C-RED One camera as a function of the APD gain.

Furthermore, the high readout rate of up to 3500 FPS in single read prevents from saturation even at relatively high fluxes when used like EMCCD photon counting cameras, with a simple thresholding of the image. This is really a new feature in the infrared since most of the cameras at these wavelength are much noisier and very slow scan, clearly discarding any possibility of counting single events.

3.3 Proportional photon counting

An excess noise factor close to 1 makes it possible finally to consider a count of photons in proportional mode and no longer by thresholding, in other words the possibility to go back to the initial number of electrons knowing the number of electrons at the output of the amplified system. The advantage is obviously to overcome the problem of coincidence and maintain good linearity even at very high flux. However, the limit from which this differentiation is possible has never been formalized. The central limit theorem makes it possible to estimate that the distribution of the charges at the output of an amplifier system, resulting from the joint probability of a large number of elementary processes, follows a normal distribution. The definition of the excess noise factor makes it possible to deduce the parameters, since the variance of a photonic source S electron signal is S, the amplified signal variance is the combination of S Gaussian amplification processes (one per electron) of variance σ2

A. The total variance of the signal will therefore be:

�(�) = � + � ��2

�

1

�(�) = � + ���2

�(�) = �(1 + ��2)

But by definition

�(�) = �2�

Therefore

��2 = �2 − 1

This makes it possible to characterize the amplification spectrum of a photodiode. For a number of electrons n greater than one, the output distribution will be the convolution of the n elementary distributions, for normal distributions, the resultant is a normal distribution of mean E = n and variance V = nσ2

A. The spectrum f (x) is obtained by combining the distributions for n = 1 to ∞. Figure 14 shows the output spectrum for an evenly distributed source of photons, and for different ENF, while figure 15 shows the spectrum on a real source, convolved with the poissonian distribution.

Figure 14: Standard output spectrum of an APD photodiode for a uniform distribution of the number of injected electrons and for various excess noise factors.

Figure 15: Normalized (input referred) spectrum for a 2-photon (left) and 4-photon (right) source by elementary integration for an excess noise factor of 1.02 and 1.05 and for an electronic noise = M / 5 (M being the multiplication gain)

The conclusion is that F2 needs to be as low as 1.05 to start to see the 1 vs 2 photon intensity peaks, and 1.02 is preferable to be able to do proportional photon counting up to 5 simultaneous photons. The current ENF of saphira devices (~1.2) prevents from using now proportional photon counting, but standard photon counting at high speed is possible. Note also that in this model a simple constant gain vs wavelength model is used, but we know that in heterojunction e-APDs like the saphira device this is not the case. But this would permit color detection in the infrared.

4. PHOTON COUNTING AND COLOR DETECTION

4.1 Basic principle

The application of photon counting techniques would make it possible to produce color-sensitive detectors if the absorption layer is merged with the multiplication layer. A photon absorbed more or less deeply will be amplified differently as shown in Figure 16. But the average penetration length depends on the wavelength of the photons and the absorbing material as shown in Figure 17. Combining these two effects is therefore possible to detect the wavelength of the photons. It is even possible to tune the wavelength response of the detector with multiple thresholding of the output diode amplitude (patented [43]).

Figure 16: A photon penetrating deeper into an APD photodiode (right) will be less amplified than a less penetrating photon (left) because the amplitude of the output pulse is proportional to this amplification. By exploiting this property it is possible to go back to the wavelength.

Figure 17: Depth of photon absorption in a semiconductor versus wavelength for silicon (left) and MCT with c = 2.4m (right).

4.2 evidences

This effect has been shown by Kirn et al. with a Hamamatsu avalanche photodiode with MESA structure. However, Kirn and his collaborators saw this as a parasitic effect and did not exploit the property [44]. Moreover using silicon is not very favorable because of a relatively high ENF. The use of a low-excess-noise material is more favorable than silicon, so there would be a benefit in using e-APD photodiodes instead. Because diodes in the saphira devices have also a MESA structure, this effect has also been raised but misinterpreted as a QE variation vs vavelength (Figure 18).

Figure 18: QE vs wavelength measured by G. Finger (private communication). It is spotted here a QE variation vs wavelength for different gains (orange circle), but this is due to the absorption length of redder photons that penetrate deeper to the gain region. In fact this is not a QE variation vs wavelength but a gain variation vs wavelength that is falsely seen as a QE variation (in the reddest part, of the plot at G=221, the actual gain is smaller), demonstrating the ability of Saphira devices to detect colors.

4.3 Dark current reduction

Being able to threshold the pixel level permits also to discriminate bulk dark current generated in the 3.5m material cutoff multiplication region of the e-APD. Indeed this is where the majority of the bulk dark is generated due to the long cutoff wavelength of the material. These thermally generated electrons will be less amplified than those generated in the 2.5m cutoff photon absorber. By thresholding properly the photoevents it is then possible to reject most of the thermal events generated in the bulk material.

5. CONCLUSION

We showed in this paper that photon counting in the infrared is made possible by using e-APD imagers like the C-RED One Camera, even at quite high flux taking the advantage of the high frame rate of the camera and then avoiding

coincidence non-linear effects. But the Excess noise factor measured with the saphira devices is still too high and needs to be decreased to a value of 1.05 to 1.02 to do proportional photon counting. One track forward is to cool down deeper the e-APD FPA, down to 40K for example, to have a more favorable excess noise factor. This will be done in the future, as well as real color detection experiments, made possible by the low excess noise factor behavior of e-APDs.

REFERENCES

[1] H. Iams and B. Salzberg, "The Secondary Emission Phototube," Proc.of the IRE 23, 1935, 55 (1934). [2] A. Lallemand, "Application de l'optique électronique à la photographie," Comptes rendus de l'Académie des

Sciences de Paris 203, 990 (1936). [3] D. T. Gezari, A. Labeyrie, and R. V. Stachnik, "Diffraction Limited Resolution through Holographic

Techniques.," Bulletin of the American Astronomical Society 3, 244 (1971). [4] D. Y. Gezari, A. Labeyrie, and R. V. Stachnik, "Speckle Interferometry: Diffraction-Limited Measurements of

Nine Stars with the 200-INCH Telescope," Astrophysical Journal 173, L1 (1972). [5] A. Labeyrie, "Speckle interferometry observations at Mount Palomar," Nouvelle Revue d'Optique 5(3), 141-151

(1974). [6] A. Boksenberg, "Performance of the UCL image photon counting system.," Proceedings of ESO/CERN

conference on auxiliary instrumentation for large telescopes, 295-316 (1972). [7] A. Boksenberg, "Television Detector Techniques," Image Processing Techniques in Astronomy, 59-79 (1975). [8] J.‐L. Gach et al., "Fabry‐Pérot Observations Using a New GaAs Photon‐counting System," Publications of the

Astronomical Society of the Pacific 114(799), 1043-1050 (2002). [9] A. Blazit, L. Koechlin, and J. L. Oneto, "On Line Digital Correlation of Photon Counting TV Images for Stellar

Interferometry," Image Processing Techniques in Astronomy, 54-79 (1976). [10] A. Blazit, D. Bonneau, L. Koechlin, and A. Labeyrie, "The digital speckle interferometer: preliminary results on

59 stars and 3C 273.," Astrophysical Journal Letters 214, L79 - L84 (1977). [11] J. Boulesteix and M. Marcelin, "Marseille Observatory I. P. C. S. (Image Photon Counting System)," Two

Dimensional Photometry: proceedings of the ESO workshop, 119 (1980). [12] G. M. Williams, A. L. Rheinheimer, and V. W. Aebi, "Electron-bombarded back-illuminated CCD sensors for

low-light-level imaging applications," Proc. SPIE 2415, 211–235 (1995). [13] V. W. Aebi and J. J. Boyle, "Electron bombarded active pixel sensor," patent US 6285018 B1 (1999). [14] T. Boutboul, A. Akkerman, A. Gibrekhterman, A. Breskin, and R. Chechik, "An improved model for

ultraviolet- and x-ray-induced electron emission from CsI," Journal of Applied Physics 86(10), 5841-5849 (1999).

[15] S.K. Madan, B. Bhaumik, and J.M. Vasi, "Experimental observation of avalanche multiplication in charge-coupled devices," IEEE Transactions on Electron Devices 30(6), 694-699 (1983).

[16] S.A. Gajar and B.E. Burke, "Charge amplification by impact ionization in charge-coupled devices," IEEE Transactions on Electron Devices 35(12), 2435-2436 (1988).

[17] J. Hynecek, "CCM-a new low-noise charge carrier multiplier suitable for detection of charge in small pixel CCD image sensors," IEEE Transactions on Electron Devices 39(8), 1972-1975 (1992).

[18] J. Hynecek, "Charge multiplying detector (CMD) suitable for small pixel CCD image sensors," patent US5337340 A (1994).

[19] P. Jerram et al., "The LLLCCD: low light imaging without the need for an intensifier," Proc. SPIE 4306, 178–186 (2001).

[20] J. Hynecek, "Impactron – a new solid state image intensifier," IEEE Transactions on Electron Devices 48(10), 2238-2241 (2001).

[21] J.L. Gach, C. Guillaume, O. Boissin, and C. Cavadore, "First Results of an L3CCD in Photon Counting Mode," Scientific Detectors for Astronomy. Astronomy. Astrophysics and Space Science Library 300, 611-614 (2004).

[22] A. G. Basden, C. A. Haniff, and C. D. Mackay, "Photon counting strategies with low light level CCDs," Monthly Notices of the Royal Astronomical Society 345(3), 985-991 (2003).

[23] R.J. McIntyre, "Multiplication noise in uniform avalanche diodes," IEEE Transactions on Electron Devices 13(1), 164-168 (1966).

[24] M. S. Robbins and B. J. Hadwen, "The Noise Performance of Electron Multiplying charge-coupled devices," IEEE transactions on electron devices 50(5), 1227-1232 (2003).

[25] G. Leveque, M. Nasser, D. Bertho, B. Orsal, and R. Alabedra, "Ionization energies in CdxHg1-xTe avalanche photodiodes," Semiconductor Science and Technology 8, 1317-1329 (1993).

[26] J.D. Beck, C.F. Wan, M. A. Kinch, and J. E. Robinson, "MWIR HgCdTe avalanche photodiodes," Proc. SPIE 4454, 188-197 (2001).

[27] J. D. Beck et al., "The HgCdTe Electron Avalanche Photodiode," Proc. SPIE 5564, 44-53 (2004). [28] Y. Okuto and C.R. Crowell, "Ionization coefficients: a nolocalized property," Physical review B 10(10), 4284-

4296 (1974). [29] B.E. Saleh, M.M. Hayat, and M.C. Teich, "Effect of dead space on the excess noise factor and time response of

avalanche photodiodes," IEEE transactions on electron devices 37(9), 1976-1984 (1990). [30] M.M. Hayat M.A. Saleh, B.E. saleh, and M.C. Teich, "Noise Factor for Thin GaAs and AlGaAs Avalanche

photodiodes," IEEE transactions on electron devices 47(3), 625-633 (2000). [31] J.S. Marsland, "On the effect of ionization dead spaces on avalanche multiplication and noise for uniform

electric fields," Journal of applied physics 67, 1929 (1990). [32] J. S. Marsland, "Comparison of different models of non-local impact ionization for low noise avalanche

photodiodes," physica status solidi 8(9), 2577–2581(2011). [33] S. Derelle et al., "A Monte Carlo Study of Hg0.7Cd0.3Te e-APD," IEEE transactions on electron devices 56(4),

569-577 (2009). [34] J. Rothman et al., "High performance characteristics in pin MW HgCdTe e-APDs," Proc. SPIE 6542, 654219

(2007). [35] F. Bertazzi, M. Moresco, M. Penna, M. Goano, and E. Belotti, "Full-Band Monte Carlo Simulation of HgCdTe

APDs," Journal of Electronic Materials 39(7), 912-917 (2010). [36] E. Bellotti, M. Moresco, and F. Bertazzi, "A 2D Full-Band Monte Carlo Study of HgCdTe-Based Avalanche

Photodiodes," Journal of Electronic Materials 40(8), 1651-1656 (2011). [37] P. Feautrier, et al., “Advances in detector technologies for visible and infrared wavefront sensing,” Proc. SPIE

8447, 84470Q (2012). [38] G. Finger et al., "SAPHIRA detector for infrared wavefront sensing," Proc. SPIE 8453, 84530T (2012). [39] P. Feautrier, J.L. Gach, and P. Wizinowich, “State of the art IR cameras for wavefront sensing using e-APD

MCT arrays,” AO4ELT4 conference, (2015). [40] J.L. Gach et al., “C-RED One: ultra-high speed wavefront sensing in the infrared made possible,” Proc. SPIE

9909, 990913 (2016). [41] S. Kraus et al., “The MIRC-X 6-telescope imager at CHARA: key science drivers, instrument specifications and

operation,” Proc. SPIE 10701, 1070156 (2018). [42] N. Anugu et al., “CHARA/MIRC-X: sensitivity improvements with an ultra low noise detector,” Proc. SPIE

10701, 1070157 (2018). [43] J.L. Gach, “Spectral sensitive solid-state photodetector”, Patent US2013335742, (2013). [44] Th. Kirn et al., "Wavelength dependance of avalanche photodiode (APD) parameters," Nuclear instrument and

methods in physics research A 387, 202-204 (1997).