Contents I. Introduction
For range finding sensors based on the time-of-flight (TOF) measurement with single-photon avalanche diodes (SPAD) two basic working principles are known: direct and indirect [1]. In the direct technique a highly accurate electronic stopwatch is used to determine the time between the emission of a short laser pulse and the reception of the signal reflected by the target object. Since in most of the published sensors only the arrival time of the first detected photon is recorded [2], [3], the dead time of the SPADs has no effect on the distance measurement. Therefore, we focus on the indirect techniques in this paper. The indirect principle uses intensity modulated light and the reflected light signal is integrated in several time windows of equal width. If standard or avalanche photodiodes (APD) are used as detector, each incident photon generates an electron-hole pair which is separated by the electric field. The resulting charge is stored in a capacitance and converted into a voltage signal to gain information about the mean intensity during the respective time span [4]. Since SPADs are APDs biased above the breakdown voltage, a single photon entering the active area can cause the SPAD to avalanche break. This generates a macroscopic self-sustaining current flow through the SPAD preventing the detection of subsequent photons [5]. To detect another photon the current through the SPAD has to be quenched by decreasing the voltage below the breakdown level and recharge it again. Therefore, the integration of the incident light is performed by single photon counting in case of SPADs. The statistic of the counting process is the main source of uncertainty in this kind of measurement [6] and has been analyzed in [7] for different indirect techniques. Except for the quenching and reset circuit, the circuity for signal processing is fully digital avoiding additional signal noise. Anyway, noise sources in the time domain are present. Since the jitter of the SPADs and further electronics are in the range of 100 ps they are neglectable [6]. Other sources of noise are crosstalk, which can be reduces by proper design of SPADs, and afterpulsing [1], what describes a triggering of the SPAD after recharge without an incident photon. To reduce afterpulsing the SPADs are kept off for a certain time between current quenching and recharging, this is called hold-off time. Afterpulsing is caused by charge carriers released from traps in the pn-junction. Since the lifetime of these carriers is exponentially distributed, the afterpulsing probability decreases accordingly with increasing hold-of time [8]. During the time needed for quenching the current, keeping the SPAD off, and recharging, no photons can be detected. This time is called dead time and it affects the number of counted photons as well as the variance of the photon count [9]. Since the distance information is obtained from the number of counted photons in each time interval, the dead time affects the distance measurement as well. In this paper we present an in-depth analysis of the influence on common measurement techniques. The presented models help to understand the dead time effects, develop techniques for their compensation, and, hence, to improve the measurement precision and reliability. For the indirect TOF measurement basically two different kinds of modulation are known: pulsed light (PL) and continuous wave (CW) [6] modulation. Due to the possibility of adjusting the duty cycle to enable sufficient optical power and, hence, achieve a requested range when eye safety regulations have to be taken into account [10], the PL modulations is preferred in high range applications. However, we will analyze both modulation types in this paper.
