Dead time effects in the indirect time-of-flight measurement with SPADs

Indirect time-of-flight measurement with SPADs is performed by counting incident photons in several time windows. Since SPADs exhibit dead time not all incident photons can be counted within a given time window. This affects the expected values and, hence, the variance of the distance measurement. For photon detection rates close to the inverse of the dead time, which defines the maximum count rate of a SPAD, the probability of photon detection cannot be assumed constant within the window anymore. In this paper the effects of dead time on the photon counts are analyzed by employing statistical calculations. Based on these a model to correct such effects can be derived.

I. Introduction

For range finding sensors based on the time-of-flight (TOF) measurement with SPADs 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. The indirect principle uses intensity modulated light and the reflected light signal is integrated in several time windows of equal width by counting the incident photons. The number of counted photons in each time window provides information about the mean intensity of the received light. Basically two different kinds of modulation are known: pulsed and continuous wave modulation [2]. In this paper we focus on pulsed light modulation, due to possibility of adjusting the duty cycle to enable sufficient optical power and hence achieve a requested range when taking eye safety regulation into account [3]. In the pulsed light modulation the reflected light is integrated in three windows with a width equal to the width of emitted pulse as shown in Fig. 1. Since the maximum TOF is limited to the pulse width, it is received within the first two windows, while the third receives ambient illumination only. The TOF is given by the pulse being split between first two windows as indicated by the hatched area in Fig. 1, though it is superimposed by the ambient light. From the counts of the three windows the TOF is calculated by $$\begin{equation*} T_{\mathrm{T}\mathrm{O}\mathrm{F}}= \frac{x_{2}-x_{3}}{x_{1}+x_{2}-2x_{3}}T_{\mathrm{P}} \tag{1} \end{equation*}$$ where the ambient light suppression is performed by the subtraction of . (1) is determined from the expected value of counted photons in each window. By integration of the photon detection rate these values are determined as $$\begin{align*} & x_{1}=(T_{\mathrm{P}}-T_{\mathrm{T}\mathrm{O}\mathrm{F}})\lambda_{\mathrm{A}}+T_{\mathrm{P}}\lambda_{\mathrm{B}},\\ & \qquad x_{2}=T_{\mathrm{T}\mathrm{O}\mathrm{F}}\lambda_{\mathrm{A}}+T_{\mathrm{P}}\lambda_{\mathrm{B}} \tag{2}\\ & \qquad\quad \text{and}\ x_{3}=T_{\mathrm{P}}\lambda_{\mathrm{B}} \end{align*}$$