
1.IntroductionMany rangefinding photoreceivers designed for the eyesafe spectral region near 1550 nm are assembled from InGaAs avalanche photodiodes (APDs) and resistivefeedback transimpedance amplifiers (RTIAs), followed by threshold pulsedetection and timestamping circuits. This general photoreceiver configuration—whether deployed as a singleelement sensor or as multiple parallel channels in a largeformat focal plane array—is applicable to military rangefinding and targeting, as well as to civilian applications, such as autonomous vehicle navigation and hazard avoidance. InGaAs APDs are attractive for these applications because they are sensitive beyond a $1.4\text{}\mu \mathrm{m}$ wavelength, where higher laser pulse energies can be used without creating an ocular hazard, and because the APD avalanche gain makes better use of weaker optical signals. Together, these qualities enable faster collection of threedimensional scene data from longer range or using smalleraperture optics. However, accurate modeling of electrooptic systems based on InGaAs APD photoreceivers requires an accurate model of APD photoreceiver false alarm rate (FAR), and the standard FAR model by Rice^{1}—which is widely applied to photodiodebased photoreceivers—requires modification to accurately model false alarms from APD photoreceivers. A simplified block diagram of the signal chain of a rangefinding APD photoreceiver is shown in Fig. 1. The APD converts incident optical power (Watts) to an output photocurrent (amps); the transimpedance amplifier (TIA)—characterized by a conversion gain in Ohms if its feedback is primarily resistive—then converts the photocurrent to a potential (volts). It is often convenient to work in units of quanta per signal pulse, such as photons for the optical signal and electrons for the output of the APD, in which case a conversion gain in units of reciprocal capacitance (e.g., $\mathrm{V}/{\mathrm{e}}^{}$) can also be defined, based on the peak deflection of the output voltage of the TIA in response to a current pulse containing a given electron count. In general, though, whether the conversion gain of the TIA is expressed in Ohms or volts per electron, it is a function of photocurrent signal pulse shape because TIA bandwidth is finite and TIA gain spectra are not necessarily white. In the following discussion, a fixed photocurrent signal pulse shape that results in a fixed conversion gain is assumed. The receiver diagrammed in Fig. 1 is a leadingedge detector—one of the most common methods of timeofflight rangefinding. The potential at the TIA output (${V}_{\text{out}}$; see Fig. 1) is fed into a threshold comparator that discriminates signal pulses from noise based on amplitude. When ${V}_{\text{out}}$ passes through the detection threshold (${V}_{\mathrm{th}}$) with positive slope, a digital pulse is generated that is timestamped by a timetodigital converter. If the comparator transitions upon reception of an optical signal, that event is a true positive; if the comparator transitions in the absence of a signal, that event is a false alarm. The pulse detection efficiency (${P}_{d}$) of the photoreceiver is the ratio of true positives to transmitted pulses. Its FAR is the probability that, in the absence of a signal return, within an infinitesimal time interval ($t$, $t+\mathrm{d}t$), the potential at the TIA output transitions through the detection threshold with positive slope. A receiver operating characteristic (ROC) is a plot of the truepositive rate (TPR) compared with the falsepositive rate (FPR) of a photoreceiver. If a rangefinding photoreceiver is operated using a range gate (${t}_{\text{gate}}$) during which reception of a single pulse return is possible, then Poisson statistics are used to calculate the FPR from the FAR, applying the definition that one or more false alarms during ${t}_{\text{gate}}$ constitute a false positive. The probability of false positive (${P}_{\mathrm{FP}}$) is unity minus the probability of zero false alarms occurring during the range gate. The FPR is Eq. (2)$$\mathrm{FPR}=\frac{{P}_{\mathrm{FP}}}{{t}_{\text{gate}}}=\frac{1\mathrm{exp}(\mathrm{FAR}\times {t}_{\text{gate}})}{{t}_{\text{gate}}}\text{\hspace{0.17em}}(\mathrm{Hz}).$$Due to the term $\mathrm{exp}(\mathrm{FAR}\times {t}_{\text{gate}})$ in Eq. (2), the ROC for a rangefinding photoreceiver depends on the range gate to which the probability of false positives applies. This prevents preparation of a general ROC for a rangefinding photoreceiver. However, a general plot of ${P}_{d}$ against FAR can be computed that characterizes a receiver, which may informally be termed an ROC, as it permits easy computation of an ROC once the range gate has been specified. This latter type of ROC is analyzed in this paper. The statistics of true positives and false alarms, which determine the ROC of a photoreceiver, depends on the pulseheight distributions of ${V}_{\text{out}}$ when a signal is present (true positives) and when a signal is not present (false alarms). These distributions are illustrated graphically in Fig. 2, where the solid curve is the distribution of ${V}_{\text{out}}$ in the absence of a signal return, and the dashed curve is the distribution of ${V}_{\text{out}}$ in the presence of a signal return. When a signal is received, the noise sources that cause ${V}_{\text{out}}$ to vary under dark conditions—such as the amplifier circuit noise and the shot noise on dark current—are also present, such that the distribution of ${V}_{\text{out}}$ when a signal is present is the convolution of its distribution under dark conditions with a separate distribution that characterizes the signal shot noise. The ${P}_{d}$ is the fraction of an ensemble of identically prepared signal pulses that will result in ${V}_{\text{out}}\ge {V}_{\mathrm{th}}$. Assuming that the photoreceiver is in an armed state where it is capable of responding to the reception of a signal pulse, ${P}_{d}$ is the complementary cumulative distribution function (CCDF) of ${V}_{\text{out}}$, evaluated at ${V}_{\mathrm{th}}$, in the presence of a signal return; graphically, the CCDF is the shaded area under the dashed curve in Fig. 2. Only part of the information required to compute the FAR is contained in Fig. 2—the probability that, in the absence of signal, ${V}_{\text{out}}$ is passing through ${V}_{\mathrm{th}}$. It is also necessary to determine the joint probability that ${V}_{\text{out}}$ has positive slope as it passes through ${V}_{\mathrm{th}}$. Rice^{1} published equations for FAR based on the assumption that both ${V}_{\text{out}}$ and its first time derivative are Gaussian distributed, using the bivariate normal distribution in his foundational 1944/1945 paper “Mathematical analysis of random noise.” To obtain accurate results for APDbased photoreceivers, Rice’s equation must be modified to account for the amplitude distribution of the APD output, published by McIntyre^{2} in 1972 Eq. (3)$${P}_{\text{McIntyre}}(n)=\frac{p\mathrm{\Gamma}(\frac{n}{1k}+1)}{n(np)!\times \mathrm{\Gamma}(\frac{nk}{1k}+1+p)}\times {\left[\frac{1+k(M1)}{M}\right]}^{p+\frac{nk}{1k}}\times {\left[\frac{(1k)(M1)}{M}\right]}^{np},$$The deviation of the McIntyre distribution from Gaussian—and its positive skew in particular—is more pronounced when a small number of primary electrons is multiplied (small $p$), when the mean avalanche gain is large (large $M$), and when the ionization rate ratio is closer to unity ($k\to 1$). This can be observed in the McIntyre distributions in Fig. 3, comparing different values of $p$, $M$, and $k$ but the same average output of $\u27e8n\u27e9=600\text{\hspace{0.17em}}{\mathrm{e}}^{}$. It is more important to use the McIntyre distribution when calculating FAR than ${P}_{d}$ because the number of primary electrons in the zerosignal condition is much smaller than during signal reception and because the detection threshold of the photoreceiver must be set many standard deviations into the tail of the noise distribution to achieve technologically useful FAR, whereas differences in ${P}_{d}$ smaller than a few percent are usually considered negligible. As long as the signal level is outside the photoncounting regime (tens of photons or stronger), the Gaussian approximation is sufficiently accurate to calculate ${P}_{d}$. 2.Avalanche Photodiode Photoreceiver Output StatisticsThe output of an analog APD photoreceiver is the superposition of the outputvoltage noise of the TIA with the voltage response of the TIA to the charge or current from the APD. The output of the APD is statistically independent from the noise of the TIA, so the random variable representing the output of the photoreceiver is the sum of two independent random variables, and its distribution is the convolution of their individual distributions. The McIntyre distribution is a discrete electroncount distribution, so it is convenient to refer all quantities to the node between APD output and TIA input, and to work in units of electrons. Assuming a TIA conversion gain ($G$), then ${V}_{\text{out}}$ and ${V}_{\mathrm{th}}$ are represented by equivalent charges at the TIA input (${n}_{\text{out}}={V}_{\text{out}}/G$ and ${n}_{\mathrm{th}}={V}_{\mathrm{th}}/G$, respectively). Moreover, although ${V}_{\text{out}}$ is a continuous variable that can take on any value as a result of circuit noise, the fluctuations of ${V}_{\text{out}}$ due to the circuit noise of the TIA can be discretized and referred to the TIA input in units of charge. Writing the discrete probability distributions of the TIA inputreferred noise and the APD output symbolically as ${P}_{\mathrm{TIA}}$ and ${P}_{\mathrm{APD}}$, the probability that the output of the APD and the inputreferred noise of the TIA will sum to a particular quantity of charge, ${n}_{\text{out}}$, is given by the discrete convolution Eq. (4)$${P}_{\mathrm{RX}}({n}_{\text{out}})=({P}_{\mathrm{TIA}}*{P}_{\mathrm{APD}})({n}_{\text{out}})\equiv \sum _{i}{P}_{\mathrm{TIA}}(i){P}_{\mathrm{APD}}({n}_{\text{out}}i).$$This model presents some difficulties of interpretation, since the noise of the TIA is an analog value characterized by the continuous Gaussian distribution of its output voltage, whereas the charge output of the APD is quantized and obeys the discrete McIntyre distribution. Furthermore, the McIntyre distribution does not address temporal statistics—it gives the probability that a certain number of electrons will eventually be output by an APD but not whether all those output electrons will simultaneously contribute to the instantaneous current. Although the number of photons arriving in a laser pulse and the number of photoelectrons generated by its reception are both discrete quantities, whether or not all of them contribute to ${n}_{\text{out}}$ depends on the laser pulse shape and the frequency response of the TIA. A related issue is that, to apply the McIntyre distribution to FAR calculations, chargeintegration times must be defined so that discrete electron counts can be computed from dark current and background photocurrent. In practice, the lack of rigor inherent in using the Gaussian distribution as though it was a discrete distribution is not a serious difficulty for the noise levels and conversion gains that are characteristic of the TIAs used in rangefinding. As long as the voltage noise of the TIA is equivalent to hundreds of electrons or more at its input, little accuracy is lost if the random variable representing the inputreferred noise of the TIA (${n}_{\mathrm{TIA}}$, in units of electrons) is restricted to integer values so that the Gaussian distribution function ${P}_{\mathrm{TIA}}(n)$ can be interpreted as the probability of the TIA noise taking on a value within a band of unit width centered on ${n}_{\mathrm{TIA}}$. For the purpose of convolving ${P}_{\mathrm{TIA}}({n}_{\mathrm{TIA}})$ with the output distribution of the APD Eq. (5)$${P}_{\mathrm{TIA}}({n}_{\mathrm{TIA}})=\frac{1}{\sqrt{2\pi \mathrm{var}({n}_{\mathrm{TIA}})}}\text{\hspace{0.17em}}\mathrm{exp}[\frac{{({n}_{\mathrm{TIA}}{\overline{n}}_{\mathrm{TIA}})}^{2}}{2\mathrm{var}({n}_{\mathrm{TIA}})}],$$Primary (unmultiplied) dark current and photocurrent are generated by Poisson processes, so the APD output distribution in the convolution of Eq. (4) must account for the distribution of the primary electron count ($p$) in Eq. (3). A Poissonweighted sum of McIntyre distributions is used Eq. (6)$${P}_{\mathrm{APD}}(n)=\sum _{p}{P}_{\text{Poisson}}(p)\times {P}_{\text{McIntyre}}(n)=\sum _{p}\mathrm{exp}(\u27e8p\u27e9)\frac{{\u27e8p\u27e9}^{p}}{p!}\times {P}_{\text{McIntyre}}(n).$$In the dark condition, the primary directcurrent (DC) dark current and background photocurrent integrate to an average electron count $\u27e8{p}_{\mathrm{DC}}\u27e9$. Reception of an optical pulse generates $\u27e8{p}_{\text{signal}}\u27e9$ primary carriers, and since the DC current is also present, the average primary electron count in the illuminated condition is In Fig. 2, assuming the TIA contributes no offset, the mean voltage when no signal is present (${V}_{\text{dark}}$) is expressed as and the mean output voltage when a signal is present (${V}_{\text{signal}}$) is expressed asEq. (9)$${V}_{\text{signal}}=G\times M\times (\u27e8{p}_{\mathrm{DC}}\u27e9+\u27e8{p}_{\text{signal}}\u27e9).$$To clarify the effective integration times that relate APD currents to the primary electron counts ${p}_{\mathrm{DC}}$ and ${p}_{\text{signal}}$—as well as to the multiplied output electron count in Eq. (3)—it is helpful to consider two limiting amplifier cases: (1) an ideal RTIA that generates an instantaneous output voltage proportional to the instantaneous output current of the APD and (2) a switched capacitivefeedback transimpedance amplifier (CTIA), similar to those used in many imaging readout integrated circuits, where an output voltage is generated that is proportional to the total charge delivered by the APD during some fixed exposure time. In the case of an ideal RTIA, the response of the photoreceiver is not determined by the total number of electrons generated from the photons received in a signal pulse, but rather by the maximum photocurrent that flows as a result. The Shockley–Ramo theorem allows the instantaneous current at the terminals of an APD, $i(t)$, to be calculated from the instantaneous count of electrons and holes within its junction, ${n}_{e}(t)$ and ${n}_{h}(t)$, and their respective saturation velocities in units of cm/s, ${v}_{se}$ and ${v}_{sh}$, as^{3}^{–}^{5} Eq. (10)$$i(t)\approx \frac{q}{w}[{v}_{se}{n}_{e}(t)+{v}_{sh}{n}_{h}(t)]\text{\hspace{0.17em}}(\mathrm{s}),$$Equation (10) can be recast in terms of junction transit times for electrons (${t}_{e}=w/{v}_{se}$) and holes (${t}_{h}=w/{v}_{sh}$) as If the laser pulse is much shorter than both junction transit times, then all of the carriers generated from the pulse will be present inside the junction simultaneously, and the APD output pulseheight distribution can be calculated using the average photon number of the laser pulse (${N}_{\text{signal}}$) and the primary quantum efficiency (QE) of the APD to find the mean primary photoelectron count and the mean multiplied signal electron countIf, however, the laser pulse duration (${t}_{\text{pulse}}$) is longer than the junction transit time, only a portion of the pulse energy will contribute to the response of the ideal RTIA photoreceiver. For a rectangular pulse of duration ${t}_{\text{pulse}}$ in seconds, the average primary electron count resulting from a signal pulse is approximately Eq. (14)$$\u27e8{p}_{\text{signal}}\u27e9\approx \mathrm{QE}\times {N}_{\text{signal}}\frac{{t}_{e}}{{t}_{\text{pulse}}}.$$A calculation similar to Eq. (14) applies to the combined dark current (${I}_{\text{dark}}$) and background photocurrent (${I}_{\text{background}}$), regardless of whether the signal pulse is longer or shorter than the junction transit time. ${I}_{\text{dark}}$ and ${I}_{\text{background}}$ are both generated by Poisson processes, and for most SWIR APD designs, the majority of the dark current originates in the InGaAs lightabsorption layer because that alloy has the narrowest bandgap among those from which the device is fabricated. Consequently, dark current and background photocurrent experience the same avalanche gain statistics and can be grouped into a single quantity, ${I}_{\mathrm{DC}}$. The associated average primary electron count from this combined DC current is then Eq. (15)$$\u27e8{p}_{\mathrm{DC}}\u27e9\approx \frac{{I}_{\text{dark}}+{I}_{\text{background}}}{qM}{t}_{e}=\frac{{I}_{\mathrm{DC}}}{qM}{t}_{e},$$The average multiplied electron count from dark current and background photocurrent is When the TIA does not have a separate outputvoltage offset, $\u27e8{n}_{\mathrm{DC}}\u27e9$ is the inputreferred form of ${V}_{\text{dark}}$, and Eq. (16) is a restatement of Eq. (8). In the case of a switched CTIA, charge from the APD may be accumulated over a current integration time (${t}_{\mathrm{int}}$) that is longer than ${t}_{e}$. In that case, ${t}_{\mathrm{int}}$ replaces ${t}_{e}$ in Eqs. (14) and (15). However—because dark current from the detector integrates too quickly, a ramped detection threshold that exactly tracks the charge integrated since the last reset is difficult to implement, and the settling time following a switched reset is too long—it is impractical to use switched CTIAs for laser rangefinding. Instead, CTIAs that are continuously reset through a lowpass filter or RTIAs that have some integrating character are commonly employed. The simplest example of the latter is an RTIA with too little bandwidth to match the rise time of the photocurrent pulse from the APD. When the bandwidth of an RTIA is too low for ${V}_{\text{out}}$ to track the input photocurrent waveform, the photocurrent charge deposited on its input shifts the input potential from virtual ground. Current flows in the feedback resistor of the RTIA until the potential at the input has been restored to its normal operating point, effectively giving the RTIA some chargeintegrating character. This is not helpful in a telecommunications application, where rapid settling is required to resolve “0” symbols following “1” symbols. The canonical “eye diagram” closes when the receiver circuit cannot keep pace with the optical modulation. However, rangefinding is different because the optical pulses are very sparse—typically once per 100 ms, and no faster than once per $1\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{s}$—so rise times in the order of tens of nanoseconds do not hamper reception of consecutive pulses. For either the case of a continuously reset CTIA or a realworld RTIA, an effective DC current integration period (${t}_{\mathrm{DC}}$) can be extracted from circuit simulations for use in place of ${t}_{e}$ in Eq. (15). The same circuit simulation produces a pulseshapespecific value of the conversion gain; this value is used instead of Eq. (14) to determine the signal response. Unfortunately, because the details of the transfer function of the TIA determine the quantitative relationship between the voltage noise at the TIA output and fluctuations of ${I}_{\text{dark}}$ and ${I}_{\text{background}}$ at the TIA input, it is usually not possible to apply analytic methods to estimate ${t}_{\mathrm{DC}}$ with useful accuracy. Instead, a simulation program with integrated circuit emphasis (SPICE) model of the TIA can be used to arrive at ${t}_{\mathrm{DC}}$. In such an SPICE model, the APD is represented by a DC current source equal to ${I}_{\mathrm{DC}}$, a transientcurrent source waveform, ${I}_{\mathrm{AC}}(t)$, derived from the laser pulse shape, a capacitor corresponding to the junction and interconnect capacitance of the APD, and a current noise source of spectral intensity (${S}_{I}$). InGaAs APDs typically operate with subnanosecond rise time so, for most nanosecondscale pulses used in rangefinding, the transient part of the current source can be approximated as the product of the APD spectral responsivity ($R$) and the opticalpower waveform of the signal pulse, $P(t)$ where $P(t)$ is in units of Watts, and the spectral responsivity of the APD isEq. (18)$$R=M\times \mathrm{QE}\frac{\lambda}{1.23984}\text{\hspace{0.17em}}(\mathrm{A}/\mathrm{W}),$$The spectral intensity of the current noise source that models multiplied shot noise on the dark current and background photocurrent is Eq. (19)$${S}_{{I}_{\mathrm{DC}}}=2qMF{I}_{\mathrm{DC}}\text{\hspace{0.17em}}({\mathrm{A}}^{2}/\mathrm{Hz}),$$SPICE models cannot simulate the full amplitude distribution of noise modeled by Eq. (4), but the standard deviation of ${V}_{\text{out}}$ is accessible. The procedure for extracting ${t}_{\mathrm{DC}}$ from an SPICE model of an APD photoreceiver is to simulate the RMS noise on ${V}_{\text{out}}$, in the absence of an optical signal, both with the noise source representing the APD (${V}_{\text{noise}}$) and without the noise source representing the APD (${V}_{\text{noise}\text{\hspace{0.17em}}\mathrm{TIA}}$). The noise contributed by the TIA is uncorrelated with the noise contributed by the APD, so the voltage noise attributable to the combined dark current and background photocurrent is Eq. (21)$${V}_{\text{noise}\text{\hspace{0.17em}}\mathrm{APD}}=\sqrt{{V}_{\text{noise}}^{2}{V}_{\text{noise}\text{\hspace{0.17em}}\mathrm{TIA}}^{2}}\text{\hspace{0.17em}}(\mathrm{V}\text{\hspace{0.17em}}\mathrm{RMS}).$$Conversion gain for the specific pulse shape modeled by ${I}_{\mathrm{AC}}(t)$ is extracted from the SPICE simulation by dividing the swing in ${V}_{\text{out}}$ in response to ${I}_{\mathrm{AC}}(t)$ by the total integrated charge delivered by ${I}_{\mathrm{AC}}(t)$ Eq. (22)$$G=\frac{{V}_{\text{signal}}{V}_{\text{dark}}}{\frac{1}{q}\int {I}_{\mathrm{AC}}(t)\mathrm{d}t}\text{\hspace{0.17em}}({\mathrm{V}/\mathrm{e}}^{}),$$The conversion gain is used to express the RMS voltage noise associated with the APD dark current and background photocurrent in units of electrons Eq. (23)$${n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{APD}}=\frac{{V}_{\text{noise}\text{\hspace{0.17em}}\mathrm{APD}}}{G}\text{\hspace{0.17em}}({\mathrm{e}}^{}).$$The same relationship defines the TIA inputreferred charge noise (${n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{TIA}}$), the square of which is equal to the variance appearing in Eq. (5) Eq. (24)$$\mathrm{var}({n}_{\mathrm{TIA}})={n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{TIA}}^{2}={\left(\frac{{V}_{\text{noise}\text{\hspace{0.17em}}\mathrm{TIA}}}{G}\right)}^{2}\text{\hspace{0.17em}}({\mathrm{e}}^{2}).$$The noiseequivalent input (NEI) of the photoreceiver is found using $G$ to refer the outputvoltage noise in the zerosignal condition (${V}_{\text{noise}}$) to an equivalent amount of input charge (${n}_{\text{noise}}$) and using the product of the mean gain and QE of the APD to refer ${n}_{\text{noise}}$ to an equivalent input level in photons Eq. (25)$$\mathrm{NEI}=\frac{{V}_{\text{noise}}}{G\times \mathrm{QE}\times M}\text{\hspace{0.17em}}(\text{photons}).$$The effective DC current integration time for use in Eq. (15) is found by equating the value for ${n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{APD}}$ given by the SPICE model to the value given by the Burgess variance theorem,^{6}^{,}^{7} which underlies the noisecurrent spectralintensity theorem of Eq. (19) Eq. (26)$${n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{APD}}^{2}=\u27e8{p}_{\mathrm{DC}}\u27e9{M}^{2}F=\left(\frac{{I}_{\mathrm{DC}}}{qM}{t}_{\mathrm{DC}}\right){M}^{2}F\Rightarrow {t}_{\mathrm{DC}}=\frac{q{n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{APD}}^{2}}{{I}_{\mathrm{DC}}MF}\text{\hspace{0.17em}}\mathrm{.}$$This calculation arrives at a value for ${t}_{\mathrm{DC}}$ that is calibrated such that an analytic calculation of the variance of ${V}_{\text{out}}$ matches an SPICE simulation, properly accounting for the transfer function of the TIA acting on the APD noise spectrum, which Eq. (19) models as white within the TIA bandwidth. With ${t}_{\mathrm{DC}}$, the full noise distribution of the APD photoreceiver can be computed using Eq. (4). It should be emphasized that, because the conversion gain is used to relate outputvoltage levels to input electron count, the inputreferred charge noise as well as ${t}_{\mathrm{DC}}$ is the function of the laser pulse shape. Conversion gain for pulses with a greater fraction of their energy outside the gain spectrum of the TIA will be lower, resulting in larger values of inputreferred charge noise such as ${n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{APD}}$ and, therefore, larger values of ${t}_{\mathrm{DC}}$. These methods were applied to compute ${P}_{\mathrm{RX}}({n}_{\text{out}})$ for a rangefinding photoreceiver assembled from a $75\text{}\mu \mathrm{m}$diameter InGaAs APD characterized by 80% QE at 1550 nm, $k=0.2$, and ${I}_{\text{dark}}=2.2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nA}$ when operating at $M=10$. The APD was paired with a TIA characterized by a 3dB bandwidth of 31 MHz, and—when responding to 4ns full width at half maximum Gaussianshaped laser pulses—${t}_{\mathrm{DC}}=10.2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{ns}$ and ${n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{TIA}}=244\text{\hspace{0.17em}}{\mathrm{e}}^{}$. With these parameters, $\u27e8{p}_{\mathrm{DC}}\u27e9=14\text{\hspace{0.17em}}{\mathrm{e}}^{}$ at $M=10$. The photoreceiver output distribution was calculated for dark conditions (no signal or background photocurrent), with the APD operating at mean avalanche gains of $M=5$, 10, and 20. The distributions computed by the convolution of Eq. (4) are compared in Fig. 4 to Gaussian distributions having the same means and variances. In this case, at avalanche gains greater than about $M=10$, the divergence of the highoutput tails of the photoreceiver distributions from their Gaussian approximations causes an FAR model based on the Gaussian approximation to underpredict the value of ${V}_{\mathrm{th}}$ required to extinguish false alarms below a given rate. 3.${P}_{d}$ and False Alarm Rate for Avalanche Photodiode PhotoreceiversThe probability of detecting a signal return pulse is the conditional probability that: (1) the photoreceiver is ready to register the pulse at the time it arrives and (2) the pulse into the decision circuit exceeds the detection threshold. Since the decision circuit only fires when its input voltage rises through its detection threshold, assuming that the laser pulse repetition period is many multiples of the settling time (${t}_{\text{settle}}$) of the amplifier, the probability that the receiver is active at the time a signal pulse arrives is the probability that zero false alarms have occurred within the preceding ${t}_{\text{settle}}$. We can estimate ${t}_{\text{settle}}\approx 2{t}_{\text{rise}}\approx 0.7/\mathrm{BW}$ and compute Eq. (27)$${P}_{d}\approx \mathrm{exp}(\frac{0.7}{\mathrm{BW}}\mathrm{FAR})\times [1\sum _{{n}_{\text{out}}\le {n}_{\mathrm{th}}}{P}_{\mathrm{RX}}({n}_{\text{out}})],$$Assuming typical design and operation—with $\mathrm{FAR}<1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{kHz}$ and $\mathrm{BW}>10\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{MHz}$—the exponential prefactor is essentially unity, and the detection efficiency is given by the second quantity—the CCDF of ${P}_{\mathrm{RX}}$ evaluated at the detection threshold. The exponential prefactor is primarily relevant in the photoncounting regime, when receivers may operate with detection threshold closer to the noise floor to sense weak signals, resulting in high FAR. When the average signal level is in the order of 10 photons or fewer, the divergence of the McIntyre distribution from its Gaussian approximation is large enough (Fig. 3) that it may be advisable to compute ${P}_{d}$ using Eq. (27). However, outside the photoncounting regime, the Gaussian approximation may be used, resulting in Eq. (28)$${P}_{d}\approx \frac{1}{2}\{1\mathrm{erf}[\frac{{n}_{\mathrm{th}}(\u27e8{n}_{\mathrm{DC}}\u27e9+\u27e8{n}_{\text{signal}}\u27e9)}{\sqrt{2({n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{TIA}}^{2}+(\u27e8{n}_{\mathrm{DC}}\u27e9+\u27e8{n}_{\text{signal}}\u27e9)MF)}}\left]\right\},$$In the case of an RTIA characterized by transimpedance (${G}_{\mathrm{\Omega}}$ in Ohms) and BW in Hz, responding to a laser pulse of peak power ${P}_{\text{signal}}$ in Watts, ${P}_{d}$ can also be written as Eq. (29)$${P}_{d}\approx \frac{1}{2}[1\mathrm{erf}(\frac{{V}_{\mathrm{th}}[R({P}_{\text{signal}}+{P}_{\text{background}})+{I}_{\text{dark}}]{G}_{\mathrm{\Omega}}}{{G}_{\mathrm{\Omega}}\sqrt{2\{2qMF[R({P}_{\text{signal}}+{P}_{\text{background}})+{I}_{\text{dark}}]+{S}_{I\text{\hspace{0.17em}}\mathrm{TIA}}\}\mathrm{BW}}}\left)\right],$$In developing an expression for FAR that accounts for the nonGaussian distribution of APD output, we follow Rice’s^{1} calculation of the Gaussian case. Rice analyzes a noisy current waveform defined in terms of uncorrelated random variables for its current ($I$) and the slope of its current ($\eta $) at every point in time, $t$. A false alarm occurs when the current transitions through a threshold value (${I}_{\mathrm{th}}$) with a positive slope. Rice shows that the probability of this occurring during the infinitesimal time interval ($t$, $t+\mathrm{d}t$) is Eq. (30)$${\mathrm{PDF}}_{\mathrm{FA}}=\mathrm{d}t{\int}_{0}^{\infty}\eta P(I={I}_{\mathrm{th}},\eta ;t)\mathrm{d}\eta \text{\hspace{0.17em}}(\mathrm{Hz}),$$Eq. (31)$$P{(I,\eta ;t)}_{\text{Gaussian}}=\frac{1}{2\pi \sqrt{\mathrm{var}(I)\mathrm{var}(\eta )}}\text{\hspace{0.17em}}\mathrm{exp}[\frac{1}{2}(\frac{{[I\overline{I}]}^{2}}{\mathrm{var}(I)}+\frac{{[\eta \overline{\eta}]}^{2}}{\mathrm{var}(\eta )}\left)\right]\text{\hspace{0.17em}}({\mathrm{A}}^{2}\text{\hspace{0.17em}}{\mathrm{Hz}}^{1}).$$Noting that—for $I(t)$ not to diverge—the average slope ($\overline{\eta}$) has to be zero, and substitution of Eq. (31) in Eq. (30) gives Eq. (32)$${\mathrm{PDF}}_{\mathrm{FA}\text{\hspace{0.17em}}\text{Gaussian}}=\frac{\mathrm{d}t}{2\pi}\sqrt{\frac{\mathrm{var}(\eta )}{\mathrm{var}(I)}\text{\hspace{0.17em}}}\text{\hspace{0.17em}}\mathrm{exp}[\frac{1}{2}(\frac{{[{I}_{\mathrm{th}}\overline{I}]}^{2}}{\mathrm{var}(I)}\left)\right]\text{\hspace{0.17em}}(\mathrm{Hz}).$$The FAR is just Eq. (32) without the differential $\mathrm{d}t$. Rice relates the variances of the current and its slope to its autocorrelation function ($\psi $) at zero time lag ($\tau $) as (respectively) Eq. (33)$$\mathrm{var}(I)={\psi}_{0}\equiv {\underset{t\to \infty}{\mathrm{lim}}\frac{1}{t}{\int}_{0}^{t}I(t)I(t+\tau )\mathrm{d}t}_{\tau =0}\text{\hspace{0.17em}}({\mathrm{A}}^{2})$$Eq. (34)$$\mathrm{var}(\eta )={\psi}_{0}^{\u2033}\equiv {\frac{{\partial}^{2}}{\partial {\tau}^{2}}\psi }_{\tau =0}\text{\hspace{0.17em}}({\mathrm{A}}^{2}{/\mathrm{s}}^{2}).$$The autocorrelation function is itself related to the spectral intensity of the noisy current, by inversion of the Wiener–Khintchine theorem^{8}^{–}^{10} Eq. (35)$$\psi (\tau )={\int}_{0}^{\infty}{S}_{I}(f)\mathrm{cos}(2\pi f\tau )\mathrm{d}f\text{\hspace{0.17em}}({\mathrm{A}}^{2}).$$Therefore Eq. (36)$$\mathrm{var}(I)={\psi}_{0}={\int}_{0}^{\infty}{S}_{I}(f)\mathrm{d}f\text{\hspace{0.17em}}({\mathrm{A}}^{2})$$Eq. (37)$$\mathrm{var}(\eta )=4{\pi}^{2}{\int}_{0}^{\infty}{f}^{2}{S}_{I}(f)\mathrm{d}f\text{\hspace{0.17em}}({\mathrm{A}}^{2}{/\mathrm{s}}^{2}).$$Substituting Eqs. (36) and (37) into Eq. (32), the FAR for Gaussiandistributed noise is Eq. (38)$${\mathrm{FAR}}_{\text{Gaussian}}=\frac{1}{2\pi}\sqrt{\frac{4{\pi}^{2}{\int}_{0}^{\infty}{f}^{2}{S}_{I}(f)\mathrm{d}f}{{\int}_{0}^{\infty}{S}_{I}(f)\mathrm{d}f}}\text{\hspace{0.17em}}\mathrm{exp}\{\frac{1}{2}[\frac{({I}_{\mathrm{th}}\overline{I}{)}^{2}}{\mathrm{var}(I)}\left]\right\}\text{\hspace{0.17em}}(\mathrm{Hz}),$$When the noise spectrum is white (constant ${S}_{I}$) over a finite bandwidth, ${S}_{I}$ cancels out in the radical and Eq. (38) becomes Eq. (39)$${\mathrm{FAR}}_{\text{Gaussian}}=\sqrt{\frac{1}{3}}\mathrm{BW}\text{\hspace{0.17em}}\mathrm{exp}[\frac{{({I}_{\mathrm{th}}\overline{I})}^{2}}{2{I}_{\text{noise}}^{2}}]=\sqrt{\frac{1}{3}}\mathrm{BW}\text{\hspace{0.17em}}\mathrm{exp}[\frac{{({V}_{\mathrm{th}}{V}_{\text{dark}})}^{2}}{2{V}_{\text{noise}}^{2}}]\text{\hspace{0.17em}}(\mathrm{Hz}),$$Eq. (40)$${I}_{\text{noise}}^{2}=\mathrm{var}(I)=\mathrm{BW}\times ({S}_{I\text{\hspace{0.17em}}\mathrm{TIA}}+{S}_{I\text{\hspace{0.17em}}\mathrm{DC}})=\mathrm{BW}\times [{S}_{I\text{\hspace{0.17em}}\mathrm{TIA}}+2qMF({I}_{\mathrm{DC}})]\text{\hspace{0.17em}}({\mathrm{A}}^{2}).$$Within the Gaussian approximation, the threshold that must be set to achieve a specified FAR is found from Eq. (39) as Eq. (41)$$\frac{{V}_{\mathrm{th}}{V}_{\text{dark}}}{{V}_{\text{noise}}}=\frac{{I}_{\mathrm{th}}\overline{I}}{{I}_{\text{noise}}}=\sqrt{2\text{\hspace{0.17em}}\mathrm{ln}\left(\frac{\sqrt{3}\mathrm{FAR}}{\mathrm{BW}}\right)}.$$Calculating FAR with better accuracy at threshold levels set high in the tail of the output distribution of an APD photoreceiver requires using the convolution of the McIntyredistributed output of the APD with the Gaussiandistributed TIA noise, ${P}_{\mathrm{RX}}({n}_{\text{out}})$, given by Eq. (4), in place of the Gaussian distribution used by Rice. ${P}_{\mathrm{RX}}({n}_{\text{out}})$ is an electroncount distribution (referred to the node between the APD and the TIA), but it can be used for the current distribution through a changeofvariable. Assuming the charge associated with electron count (${n}_{\text{out}}$) is transported in time (${t}_{\mathrm{ref}}$), the current can be rewritten Following the rule for changeofvariable of a probability density function, the current distribution is Eq. (43)$${P}_{\mathrm{RX}}(I)=\frac{d}{dI}{n}_{\text{out}}(I){P}_{\mathrm{RX}}[{n}_{\text{out}}(I)]=\frac{{t}_{\mathrm{ref}}}{q}{P}_{\mathrm{RX}}({n}_{\text{out}})\text{\hspace{0.17em}}({\mathrm{A}}^{1}).$$The joint probability distribution of the current and its slope, equivalent to Eq. (31), is Eq. (44)$$P{(I,\eta ;t)}_{\text{McIntyre}}=\frac{1}{\sqrt{2\pi \mathrm{var}(\eta )}}\frac{{t}_{\mathrm{ref}}}{q}{P}_{\mathrm{RX}}({n}_{\text{out}})\mathrm{exp}[\frac{{\eta}^{2}}{2\mathrm{var}(\eta )}]\text{\hspace{0.17em}}({\mathrm{A}}^{2}\text{\hspace{0.17em}}{\mathrm{Hz}}^{1}).$$Substitution of the modified joint probability distribution into Eq. (30) gives Eq. (45)$${\mathrm{PDF}}_{\mathrm{FA}\text{\hspace{0.17em}}\text{McIntyre}}=\frac{\mathrm{d}t}{2\pi}\sqrt{\frac{\mathrm{var}(\eta )}{\mathrm{var}(I)}}\frac{{t}_{\mathrm{ref}}}{q}{P}_{\mathrm{RX}}({n}_{\mathrm{th}})\sqrt{2\pi \mathrm{var}(I)}\text{\hspace{0.17em}}(\mathrm{Hz})$$Eq. (46)$${\mathrm{FAR}}_{\text{McIntyre}}=\sqrt{\frac{2\pi}{3}}\frac{{t}_{\mathrm{ref}}}{q}{I}_{\text{noise}}\mathrm{BW}{P}_{\mathrm{RX}}({n}_{\mathrm{th}})\text{\hspace{0.17em}}(\mathrm{Hz}).$$To use Eq. (46), ${t}_{\mathrm{ref}}$ must be explicitly defined. Rice’s equation for FAR based on Gaussiandistributed noise, Eq. (39), is recovered from Eq. (46) if the substitution ${n}_{\mathrm{th}}=\frac{{t}_{\mathrm{ref}}}{q}{I}_{\mathrm{th}}$ is made to express the threshold voltage in terms of an equivalent input current rather than an equivalent input electron count, and a discretized Gaussian distribution of mean $\u27e8{n}_{\mathrm{DC}}\u27e9=\frac{{t}_{\mathrm{ref}}}{q}{I}_{\mathrm{DC}}$ and variance ${n}_{\text{noise}}^{2}=\mathrm{var}({n}_{\text{out}})={(\frac{{t}_{\mathrm{ref}}}{q}{I}_{\text{noise}})}^{2}$ is used in place of the convolution for ${P}_{\mathrm{RX}}({n}_{\mathrm{th}})$. The Burgess variance theorem is used to calculate the APD contribution to ${n}_{\text{noise}}^{2}$, resulting in Eq. (47)$${n}_{\text{noise}}^{2}={n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{TIA}}^{2}+{n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{APD}}^{2}={n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{TIA}}^{2}+\frac{{t}_{\mathrm{ref}}}{q}{I}_{\mathrm{DC}}MF\text{\hspace{0.17em}}({\mathrm{e}}^{2}).$$From Eq. (47), it is found that the relationship ${n}_{\text{noise}}^{2}={(\frac{{t}_{\mathrm{ref}}}{q}{I}_{\text{noise}})}^{2}$ is equivalent to setting ${t}_{\mathrm{ref}}=\frac{1}{2\mathrm{BW}}$. In other words, within the approximations of a white noise spectrum and a flat TIA frequency response, the effective DC current integration time is $\frac{1}{2\mathrm{BW}}$. Alternatively, if ${t}_{\mathrm{DC}}$ and ${n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{TIA}}$ have been obtained from SPICE simulations, the FAR of Eq. (46) can be expressed as Eq. (48)$${\mathrm{FAR}}_{\text{McIntyre}}=\sqrt{\frac{2\pi}{3}}{n}_{\text{noise}}\mathrm{BW}{P}_{\mathrm{RX}}({n}_{\mathrm{th}}).$$It is important that ${n}_{\mathrm{th}}$ be consistently defined if Eq. (48) is used to find a threshold corresponding to a specified FAR and that Eq. (28) then be used to determine the signaldetection probability at that threshold. As noted earlier, if ${V}_{\mathrm{th}}$ is measured in the lab relative to ${V}_{\text{dark}}$, the DC offset $\u27e8{n}_{\mathrm{DC}}\u27e9$ is omitted from the numerator of Eq. (28) for ${P}_{d}$. This treatment is consistent with omitting ${V}_{\text{dark}}$ or $\overline{I}$ from the (${V}_{\mathrm{th}}{V}_{\text{dark}}$) or (${I}_{\mathrm{th}}\overline{I}$) expressions in the Gaussian FAR models of Eqs. (39) and (41). However, when the convolution ${P}_{\mathrm{RX}}({n}_{\text{out}})$ defined in Eq. (4) is used with Eq. (48) for FAR, the value of ${n}_{\text{out}}$ that maps to a given FAR is not referenced to $\u27e8{n}_{\mathrm{DC}}\u27e9$. Consequently, the offset $\u27e8{n}_{\mathrm{DC}}\u27e9$ must be retained in the numerator of Eq. (28) for ${P}_{d}$ calculations based on threshold levels found from Eqs. (4) and (48). In Fig. 5, FARs calculated by the Gaussian approximation of Eq. (39) are compared with those calculated using Eq. (48), based on the same photoreceiver parameters as Fig. 4. FARs in the vicinity of 10 to 100 Hz are of technological interest, and it can be observed from Fig. 5 that, in this case, the Gaussian approximation underestimates the detection threshold required to operate with an FAR below 10 Hz at an APD gain of $M=20$ by about 34%. The size of the discrepancy is strongly dependent on APD gain as well as the relative magnitude of APD shot noise compared with TIA circuit noise. In the case graphed in Fig. 5, amplifier noise dominates, with ${n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{TIA}}=244\text{\hspace{0.17em}}{\mathrm{e}}^{}$ and at $M=10$, ${n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{APD}}=71\text{\hspace{0.17em}}{\mathrm{e}}^{}$; at $M=20$, ${n}_{\text{noise}\text{\hspace{0.17em}}\mathrm{APD}}=160\text{\hspace{0.17em}}{\mathrm{e}}^{}$. When the APD noise is more dominant, such as for photoreceivers assembled from largerdiameter APDs or those characterized by larger values of $k$—or when any APD photoreceiver is operated at higher avalanche gain—the skew of the McIntyre distribution has a larger impact on the FAR versus threshold characteristic. 4.Photoreceiver Performance and Receiver Operating CharacteristicThe signal level required to achieve 99% ${P}_{d}$ is shown in Fig. 6 as a function of FAR, for different APD gains. The photoreceiver parameters are the same as those used in Figs. 4 and 5. Sensitivity improves as the gain is increased from $M=10$ to 20. The plot also shows that the optimal APD gain is closer to $M=15$ than $M=20$ because the threshold required to extinguish false alarms diverges from the Gaussian model above $M=10$, as shown in Fig. 5. This fact is missed by the sensitivity calculation based on the Gaussian approximation, which predict lower overall signal levels and show $M=20$ superior to $M=15$. A plot of ${P}_{d}$ versus FAR at a mean signal level of 250 photons for the same receiver is presented in Fig. 7. An ROC for a specified range gate can be computed from this information using Eq. (2). The NEI and sensitivities for $\mathrm{FAR}=150\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{Hz}$ at 95% and 99% ${P}_{d}$ are plotted in Fig. 8 as functions of avalanche gain, at different operating temperatures. The NEI, which is calculated from the standard deviation of ${V}_{\text{out}}$, is, compared with FAR, less sensitive to the tail of the distribution. This is evident in Fig. 8, where, at 27°C and 50°C, the gain that minimizes NEI is lower than the gain that achieves the best sensitivity at the specified FAR. The optimal gain at $40\xb0\mathrm{C}$ is higher than at the other temperatures owing to lower APD dark current since it is the APD that generates the higheramplitude false alarms that necessitate setting higher detection thresholds than predicted by the Gaussian noise model. The ratios between 95% and 99% sensitivity and NEI are plotted in Fig. 9. This ratio is related to the ratio between threshold and noise given by Eq. (41), being close to $\frac{{n}_{\mathrm{th}}\u27e8{n}_{\mathrm{DC}}\u27e9}{{n}_{\text{noise}}}+1.65$ for ${P}_{d}=95\%$ and $\frac{{n}_{\mathrm{th}}\u27e8{n}_{\mathrm{DC}}\u27e9}{{n}_{\text{noise}}}+2.4$ for ${P}_{d}=99\%$; for ${P}_{d}=50\%$, Eq. (41) is the same as the sensitivitytoNEI ratio. Equation (41) gives $\frac{{n}_{\mathrm{th}}\u27e8{n}_{\mathrm{DC}}\u27e9}{{n}_{\text{noise}}}\approx 4.85$ for $\mathrm{BW}=31\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{MHz}$ and $\mathrm{FAR}=150$, and as $M\to 1$, the 95% and 99% sensitivitytoNEI ratio converge on the values of 6.5 and 7.25 predicted by the Gaussian model. A similar NEItosensitivity curve is presented in Fig. 10, overlaid by the percent error from the Gaussian approximation. The error of the Gaussian approximation increases as the operating gain and temperature increase. 5.ConclusionA correction to the Gaussian FAR model has been presented with estimates of the impact on calculations of sensitivity and ROC. Errors become significant as APD noise starts to dominate the total noise of the photoreceiver, which occurs at higher temperatures and avalanche gains. 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BiographyAndrew S. Huntington manages Voxtel’s detector development group and is responsible for device design, simulation, and manufacture. He specializes in design of lowexcessnoise linearmode InGaAsbased APDs for rangefinding and lidar. His other areas of research activity include development of photodiode arrays for lowlight imaging, sensor system modeling, colloidal quantum dotbased imaging, and new APDs for work beyond $2\text{}\mu \mathrm{m}$ wavelength. George M. Williams, Jr. is the president of Voxtel, Inc. (Beaverton, Oregon, USA). He oversees Voxtel’s research and development and commercial operations and manages Voxtel’s subsidiaries Vadient Optics and NanoVox. Over the last 25 years, he has been involved in the research, design, and manufacture of a variety of infrared, nightvision, silicon, and APD detector technologies, active and active–passive FPA development, and laserranging, lidar, and flashladar system development. Adam O. Lee is the group lead of Voxtel’s readout integrated circuit (ROIC) and CMOS imager design group. In that role, he has developed ROIC architectures for active and active/passive imaging, implementing capabilities, such as realtime laser spot tracking during video imaging, radiation hardness, eventdriven highrate sampling of transient signals, and subnanosecond timeofflight imaging. 