Dark Current and Single Photon Detection by 1550 nm Avalanche Photodiodes Dead Time Corrected Probability Distributions and Entropy

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Dark Current and Single Photon Detection by

1550 nm Avalanche Photodiodes: Dead Time

Corrected Probability Distributions and Entropy

Rates

NICOLE MENKART,1,2,* JOSEPH D. HART,3THOMAS E. MURPHY,1,2,

& RAJARSHI ROY 2,4

1Department of Electrical & Computer Engineering, University of Maryland, College Park, College Park,

MD 20742, USA

Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, College

Park, MD 20742, USA

3Optical Sciences Division, U.S. Naval Research Laboratory, Washington, DC, 20375, USA

4Department of Physics, University of Maryland, College Park, College Park, MD 20742, USA

*nmenkart@umd.edu

Abstract:

Single photon detectors have dark count rates that depend strongly on the bias level

for detector operation. In the case of weak light sources such as novel lasers or single-photon

emitters, the rate of counts due to the light source can be comparable to that of the detector

dark counts. In such cases, a characterization of the statistical properties of the dark counts is

necessary. The dark counts are often assumed to follow a Poisson process that is statistically

independent of the incident photon counts. This assumption must be validated for speciﬁc types

of photodetectors. In this work, we focus on single-photon avalanche photodiodes (SPADs)

made for 1550nm. For the InGaAs detectors used, we ﬁnd the measured distributions often diﬀer

signiﬁcantly from Poisson due to the presence of dead time and afterpulsing with the diﬀerence

increasing with the bias level used for obtaining higher quantum eﬃciencies. We ﬁnd that when

the dead time is increased to remove the eﬀects of afterpulsing, it is necessary to correct the

measured distributions for the eﬀects of the dead time. To this end, we apply an iterative algorithm

to remove dead time eﬀects from the probability distribution for dark counts as well as for the

case where light from an external weak laser source (known to be Poisson) is detected together

with the dark counts. We believe this to be the ﬁrst instance of the comprehensive application

of this algorithm to real data and ﬁnd that the dead time corrected probability distributions are

Poisson distributions in both cases. We additionally use the Grassberger-Procaccia algorithm

to estimate the entropy production rates of the dark count processes, which provides a single

metric that characterizes the temporal correlations between dark counts as well as the shape of

the distribution. We have thus developed a systematic procedure for taking data with 1550nm

SPADs and obtaining accurate photocount statistics to examine novel light sources.

1. Introduction

Single photon counting has been used for many decades to study the quantum properties of

light and its interactions with atoms and molecules. Single photon detection was used to

lay the foundations of coherent states emitted by laser sources and non-classical states of

light ﬁelds [1, 2]. More recently, it is an indispensable technique for the characterization of

single photon emitters (SPEs) which are used widely in quantum information applications [3].

Traditionally, photomultiplier tubes were used to count photons [4] in the visible and ultraviolet

spectrum, but with the development of ﬁber optic telecommunications, semiconductor-based

detectors in the 1550nm region came into widespread use for weak signals at the single photon

arXiv:2210.01921v1 [physics.optics] 4 Oct 2022

level. Geiger-mode single-photon avalanche diodes (SPADs) have found widespread use over the

past two decades [5–8].

All single-photon detectors exhibit dark counts in the absence of light: in avalanche photodiodes,

dark counts occur when avalanches are triggered by electrical carriers that are thermally generated

or emitted by trapping levels in the semiconductor [9, 10]. The rate at which these avalanches

are triggered, called the dark count rate, is heavily dependent on detector settings such as the

quantum eﬃciency and dead time [6, 7, 11]. Dark counts are a source of noise in any application

and increase with the bias voltage needed to enhance the quantum eﬃciency of the detector.

There have been several studies of dark current in avalanche photodiodes [5,12] and numerical

models have been developed to compare with experimental measurements [13].

However, a complete statistical characterization of the dark counts in 1550 nm SPADs has

not been reported. The purpose of our research is to explore the dark counts of SPADs by

using time-tagged photodetector measurements that can be used to generate histograms of

their interarrival time distributions, probability distributions, and entropy rates. A simplifying

assumption often made about the statistics of dark counts is that they follow a Poisson distribution.

We examine this assumption and show that it is not generally valid without proper processing of

the data. This processing is essential when one wants to accurately characterize the statistics of

single photon emitters or novel light sources.

It is long understood that Poisson statistics are not always an appropriate model for counts

from photomultiplier tubes [4, 14], image sensors such as CCD or CMOS sensors [15], general

purpose digital pulse processing systems [16], and SPAD photocounts [13,17]. However, it is still

often assumed that their dark counts follow Poisson statistics [9] whose probability distribution

is given by eqn. 1 [18, 19]

𝑃(𝑛)=(𝑟𝑇)𝑛𝑒(𝑟𝑇 )

𝑛!,(1)

where

𝑃(𝑛)

is the probability of

𝑛

counts being detected in the time interval

𝑇

, assuming

𝑟

is the

average rate of detected counts. We will deﬁne

𝜆=𝑟𝑇

as the average number of counts in the

interval

𝑇

. The detector quantum eﬃciency,

𝜂

, relates the optical power to the observed photon

count rate, 𝑟, as:

𝑟=𝜂(𝑃/ℎ𝜈),(2)

where 𝑃is the average optical power incident on the detector, and ℎ𝜈 is the energy per photon.

In many experiments the dark count rates are relatively small in comparison with the photon

count rates, therefore their eﬀect on the distribution is negligible. However, there are other

practical cases where the dark count rate is comparable to or only slightly less than the photon

count rate. In these instances, we aim to quantitatively explore the dark count distributions and

their inﬂuence on a weak attenuated coherent light source which is known to be Poisson [1, 2].

We study the deviations of dark current statistics from Poisson statistics due to dead time and

afterpulsing [4, 20] and propose a method for correcting the histograms. First, we recommend

extending the detector dead time to eliminate the eﬀects of afterpulsing as observed from the

interarrival distributions. Second, we suggest calculating the adjusted occurrence rate of dark

counts from the measured rate by using a standard formula. This enables us to determine the

correct slope for the interarrival time distribution of a Poisson process. Lastly, we will show that

a further step is necessary to correct the probability distributions for dead time which involves

implementing an iterative algorithm [21, 22].

2. Instrumentation

Single-photon avalanche diodes are named as such because when the reverse bias of its p-n

junction is raised above the breakdown voltage, just a single carrier can trigger an electrical

avalanche process, leading to a measurable current [9, 10]. To detect a subsequent photon, the

Electron

Electrical response

of detector

ττ τ

SPAD

Response

(a)

Discriminator

Output

(b)

Dead Time

Fig. 1. A schematic depicting the detector response to an electron carrier, which may

be triggered by a photon, dark current, etc., in the presence of dead time. (a) The

avalanche photodiode’s electrical response (black curve) to a triggered electron (blue

circle) including afterpulsing, the additional ﬂuctuations seen after the initial response.

After each response, there is a dead time (shaded blue boxes) when the detector cannot

detect any incoming electrons. The dead time,

𝜏

, is adjustable by the user. (b) Electrical

signal sent from the output of the detector after the SPAD response passes through a

discriminator.

bias voltage must be reduced to near or below the breakdown value. Restoring the SPAD to its

operative level, a process called quenching, is achieved by a quenching circuit that introduces a

ﬁnite recovery time, known as dead time, during which the device cannot respond to another

incident photon [23]. There are two main quenching modes: passive quenching (PQ) and

active quenching (AQ). PQ SPADs are paralyzable detectors where photons arriving during

the dead time are not counted and the dead time is extended [13]. Alternatively, AQ SPADs,

a type of nonparalyzable detector used in our experiments, will not count photons arriving

during the dead time nor will the dead time be extended by the quenching circuit. If a carrier is

triggered by photon absorption, the generated current will precisely mark the photon arrival time.

However, avalanches can also be triggered by dark current, thereby marking the time of avalanche

generation. Another phenomenon observed in SPADs is afterpulsing, a type of correlated noise

found in real, non-ideal detectors where more than one electric pulse is generated per event due

to traps holding extra charge carriers [12, 18, 24]. Figure 1 depicts the detection behavior of AQ

SPADs in the presence of dead time. Fig. 1(a) shows the SPADs electrical response (black curve)

to an electron (blue circle) which may be triggered by a photon or dark current. The additional

ﬂuctuations of the curve after the initial response are the afterpulsing. After each response, there

is a pre-determined dead time, represented by the shaded blue boxes, during which the detector

cannot detect any incoming electrons. In some detectors, the dead time

𝜏

is a parameter that

can be controlled by the user, hence the shaded boxes are of diﬀerent lengths. Fig. 1(b) shows

the electrical signal that is sent from the output of the detector after the SPAD response passes

through a discriminator.

All our experimental data is taken using InGaAs Geiger-mode avalanche photodetectors from

Aurea Technologies [25] operated in continuous mode and designed for 1550 nm wavelengths.

Our SPADs have a range of selectable dead times from 1

𝜇

s to 999

𝜇

s and three quantum

eﬃciency settings, 10%, 20%, and 30%, which correspond to the percentage of incident photons

that are detected and depend on the bias voltage applied to the APD. We observed dark current

using two chosen dead time values: 20

𝜇

s and 500

𝜇

s. The ﬁrst is the shortest dead time that

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DarkCurrentandSinglePhotonDetectionby1550nmAvalanchePhotodiodes:DeadTimeCorrectedProbabilityDistributionsandEntropyRatesNICOLEMENKART,1,2,*JOSEPHD.HART,3THOMASE.MURPHY,1,2,&RAJARSHIROY2,41DepartmentofElectrical&ComputerEngineering,UniversityofMaryland,CollegePark,CollegePark,MD20742,USA2Institutefor...

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Dark Current and Single Photon Detection by 1550 nm Avalanche Photodiodes Dead Time Corrected Probability Distributions and Entropy

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