Modeling false alarm rate and related characteristics of laser ranging and LIDAR avalanche photodiode photoreceivers

Abstract. An analysis is presented of the error introduced into estimates of avalanche photodiode (APD) laser-rangefinder performance by assuming Gaussian distribution of the APD multiplication gain. The amplitude of current pulses emitted by an APD obeys the McIntyre distribution, the tails of which diverge from the Gaussian distribution having the same mean and variance. Because extinction of false alarms requires setting a discrimination threshold far into the tail of an analog photoreceiver’s output distribution, the threshold level required to achieve a specified false alarm rate (FAR) using an APD-based photoreceiver is often not accurately predicted by the standard FAR model of Rice. Characteristics of APD-based photoreceivers are calculated using the McIntyre distribution and are compared with characteristics calculated using the Gaussian approximation.


Introduction
Many rangefinding photoreceivers designed for the eye-safe spectral region near 1550 nm are assembled from InGaAs avalanche photodiodes (APDs) and resistive-feedback transimpedance amplifiers (RTIAs), followed by threshold pulse-detection and time-stamping circuits. This general photoreceiver configuration-whether deployed as a single-element sensor or as multiple parallel channels in a large-format 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-μm 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 three-dimensional scene data from longer range or using smaller-aperture optics. However, accurate modeling of electro-optic 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 photodiode-based 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., V∕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 leading-edge detector-one of the most common methods of time-offlight rangefinding. The potential at the TIA output (V out ; see Fig. 1) is fed into a threshold comparator that discriminates signal pulses from noise based on amplitude. When V out passes through the detection threshold (V th ) with positive slope, a digital pulse is generated that is time-stamped by a time-to-digital 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 þ dt), the potential at the TIA output transitions through the detection threshold with positive slope.
A receiver operating characteristic (ROC) is a plot of the true-positive rate (TPR) compared with the false-positive rate (FPR) of a photoreceiver. If a rangefinding photoreceiver is operated using a range gate (t gate ) during which reception of a single pulse return is possible, then E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 6 3 ; 6 5 3 TPR ¼ P d t gate ðHzÞ: Poisson statistics are used to calculate the FPR from the FAR, applying the definition that one or more false alarms during t gate constitute a false positive. The probability of false positive (P FP ) is unity minus the probability of zero false alarms occurring during the range gate. The FPR is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 2 ; 6 3 ; 5 6 2 FPR ¼ Due to the term expð−FAR × t gate Þ 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 pulse-height distributions of V 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 out in the absence of a signal return, and the dashed curve is the distribution of V out in the presence of a signal return. When a signal is received, the noise sources that cause V 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 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 out ≥ V 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 out , evaluated at V 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 out is passing through V th . It is also necessary to determine the joint probability that V out has positive slope as it passes through V th . Rice 1 published equations for FAR based on the assumption that both V 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 APD-based photoreceivers, Rice's equation must be modified to account for the amplitude distribution of the APD output, published by McIntyre 2 in 1972 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 3 2 6 ; 5 5 4 where p is the count of primary electrons injected into the APD multiplier, n is the count of output electrons resulting from p, k is the ionization rate ratio of the slower-ionizing carrier type to the faster-ionizing type (typically between 0.2 and 0.4 for InGaAs APDs), Γ is the Euler gamma function, and M is the mean avalanche gain at which the APD is operating. 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 → 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 hni ¼ 600 e − . It is more important to use the McIntyre distribution when calculating FAR than P d   because the number of primary electrons in the zero-signal 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 photon-counting regime (tens of photons or stronger), the Gaussian approximation is sufficiently accurate to calculate P d .

Avalanche Photodiode Photoreceiver Output
Statistics The output of an analog APD photoreceiver is the superposition of the output-voltage 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 electron-count 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 out and V th are represented by equivalent charges at the TIA input (n out ¼ V out ∕G and n th ¼ V th ∕G, respectively). Moreover, although V out is a continuous variable that can take on any value as a result of circuit noise, the fluctuations of V 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 input-referred noise and the APD output symbolically as P TIA and P 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 out , is given by the discrete convolution E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 4 ; 6 3 ; 3 5 5 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 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, charge-integration 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 input-referred noise of the TIA (n TIA , in units of electrons) is restricted to integer values so that the Gaussian distribution function P TIA ðnÞ can be interpreted as the probability of the TIA noise taking on a value within a band of unit width centered on n TIA . For the purpose of convolving P TIA ðn TIA Þ with the output distribution of the APD E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 5 ; 3 2 6 ; 6 4 2 where n TIA is the mean output-voltage level of the TIA in the absence of a signal divided by G and varðn TIA Þ is the square of the output noise referred to the TIA input in units of electrons. It should be noted that although the TIA is characterized by a fixed output-voltage noise, the conversion gain depends on signal pulse shape, so the input-referred noise of the TIA depends on the pulse shape of the signal to which it is referenced. 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 Poisson-weighted sum of McIntyre distributions is used E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 6 ; 3 2 6 ; 4 5 2 In the dark condition, the primary direct-current (DC) dark current and background photocurrent integrate to an average electron count hp DC i. Reception of an optical pulse generates hp signal i primary carriers, and since the DC current is also present, the average primary electron count in the illuminated condition is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 7 ; 3 2 6 ; 3 1 0 In Fig. 2, assuming the TIA contributes no offset, the mean voltage when no signal is present (V dark ) is expressed as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 8 ; 3 2 6 ; 2 4 5 and the mean output voltage when a signal is present (V signal ) is expressed as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 9 ; 3 2 6 ; 1 9 2 To clarify the effective integration times that relate APD currents to the primary electron counts p DC and p 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 capacitive-feedback 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, iðtÞ, to be calculated from the instantaneous count of electrons and holes within its junction, n e ðtÞ and n h ðtÞ, and their respective saturation velocities in units of cm/s, v se and v sh , as 3-5 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 0 ; 6 3 ; 6 3 1 iðtÞ ≈ q w ½v se n e ðtÞ þ v sh n h ðtÞ ðsÞ; where q is the elementary charge in Coulombs and w is the junction width in cm. Equation (10) can be recast in terms of junction transit times for electrons (t e ¼ w∕v se ) and holes (t h ¼ w∕v sh ) as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 1 ; 6 3 ; 5 4 8 iðtÞ ≈ q n e ðtÞ t e þ n h ðtÞ t h : 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 pulse-height distribution can be calculated using the average photon number of the laser pulse (N signal ) and the primary quantum efficiency (QE) of the APD to find the mean primary photoelectron count If, however, the laser pulse duration (t 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 pulse in seconds, the average primary electron count resulting from a signal pulse is approximately E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 4 ; 6 3 ; 2 8 A calculation similar to Eq. (14) applies to the combined dark current (I dark ) and background photocurrent (I background ), regardless of whether the signal pulse is longer or shorter than the junction transit time. I dark and I background are both generated by Poisson processes, and for most SWIR APD designs, the majority of the dark current originates in the InGaAs light-absorption 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 DC . The associated average primary electron count from this combined DC current is then E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 5 ; 3 2 6 ; 7 5 2 where I dark and I background are both in units of amps. The average multiplied electron count from dark current and background photocurrent is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 6 ; 3 2 6 ; 6 8 5 When the TIA does not have a separate output-voltage offset, hn DC i is the input-referred form of V 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 int ) that is longer than t e . In that case, t 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 low-pass 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 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 sparsetypically once per 100 ms, and no faster than once per 1 μs-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 DC ) can be extracted from circuit simulations for use in place of t e in Eq. (15). The same circuit simulation produces a pulseshape-specific 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 dark and I background at the TIA input, it is usually not possible to apply analytic methods to estimate t DC with useful accuracy. Instead, a simulation program with integrated circuit emphasis (SPICE) model of the TIA can be used to arrive at t DC . In such an SPICE model, the APD is represented by a DC current source equal to I DC , a transient-current source waveform, I AC ðtÞ, 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 nanosecond-scale 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 optical-power waveform of the signal pulse, PðtÞ E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 7 ; 6 3 ; 7 3 0 I AC ðtÞ ¼ R × PðtÞ ðAÞ; (17) where PðtÞ is in units of Watts, and the spectral responsivity of the APD is where λ is the laser wavelength in microns. The spectral intensity of the current noise source that models multiplied shot noise on the dark current and background photocurrent is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 9 ; 6 3 ; 5 9 2 S I DC ¼ 2qMFI DC ðA 2 ∕HzÞ; where the excess noise factor (F) is 2 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 0 ; 6 3 ; 5 SPICE models cannot simulate the full amplitude distribution of noise modeled by Eq. (4), but the standard deviation of V out is accessible. The procedure for extracting t DC from an SPICE model of an APD photoreceiver is to simulate the RMS noise on V out , in the absence of an optical signal, both with the noise source representing the APD (V noise ) and without the noise source representing the APD (V noise 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 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 1 ; 6 3 ; 3 8 Conversion gain for the specific pulse shape modeled by I AC ðtÞ is extracted from the SPICE simulation by dividing the swing in V out in response to I AC ðtÞ by the total integrated charge delivered by I AC ðtÞ E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 2 ; 6 3 ; 2 9 9 G ¼ where V signal and V dark are the peak output voltage response from the SPICE simulation and the output voltage in the absence of a signal, respectively. It is important to note that Eq. (17) defines I AC ðtÞ at the output of the APD, and any coupling loss resulting from impedance mismatch between the APD and the TIA must be factored into the SPICE model. The conversion gain is used to express the RMS voltage noise associated with the APD dark current and background photocurrent in units of electrons E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 3 ; 6 3 ; 1 4 2 n noise APD ¼ The same relationship defines the TIA input-referred charge noise (n noise TIA ), the square of which is equal to the variance appearing in Eq. (5) E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 4 ; 3 2 6 ; 7 5 2 varðn TIA Þ ¼ n The noise-equivalent input (NEI) of the photoreceiver is found using G to refer the output-voltage noise in the zerosignal condition (V noise ) to an equivalent amount of input charge (n noise ) and using the product of the mean gain and QE of the APD to refer n noise to an equivalent input level in photons E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 5 ; 3 2 6 ; 6 5 1 The effective DC current integration time for use in Eq. (15) is found by equating the value for n noise APD given by the SPICE model to the value given by the Burgess variance theorem, 6,7 which underlies the noise-current spectral-intensity theorem of Eq. (19) E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 6 ; 3 2 6 ; 5 5 3 This calculation arrives at a value for t DC that is calibrated such that an analytic calculation of the variance of V 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 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 output-voltage levels to input electron count, the input-referred charge noise as well as t 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 input-referred charge noise such as n noise APD and, therefore, larger values of t DC .
These methods were applied to compute P RX ðn out Þ for a rangefinding photoreceiver assembled from a 75-μmdiameter InGaAs APD characterized by 80% QE at 1550 nm, k ¼ 0.2, and I dark ¼ 2.2 nA when operating at M ¼ 10.
The APD was paired with a TIA characterized by a 3-dB bandwidth of 31 MHz, and-when responding to 4-ns full width at half maximum Gaussian-shaped laser pulsest DC ¼ 10.2 ns and n noise TIA ¼ 244 e − . With these parameters, hp DC i ¼ 14 e − at M ¼ 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 ¼ 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 ¼ 10, the divergence of the high-output tails of the photoreceiver distributions from their Gaussian approximations causes an FAR model based on the Gaussian approximation to underpredict the value of V th required to extinguish false alarms below a given rate.

P d and False Alarm Rate for Avalanche
Photodiode Photoreceivers The 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 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 settle . We can estimate t settle ≈ 2t rise ≈ 0.7∕BW and compute where t rise is the 10%-to-90% rise time in seconds and BW is the 3-dB bandwidth in Hertz. Assuming typical design and operation-with FAR < 1 kHz and BW > 10 MHz-the exponential prefactor is essentially unity, and the detection efficiency is given by the second quantity-the CCDF of P RX evaluated at the detection threshold. The exponential prefactor is primarily relevant in the photon-counting 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 photon-counting regime, the Gaussian approximation may be used, resulting in E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 8 ; 3 2 6 ; 5 7 6 where, assuming no offset due to the TIA, hn DC i is found from Eqs. (15) and (16), using t DC and hn signal i ¼ N signal × QE × M. Note that, if the detection threshold (n th ¼ V th ∕G) is referenced to the mean output voltage in the dark condition (V dark ), then the mean DC offset (hn DC i ¼ V dark ∕G) can be omitted from the numerator inside the error function in Eq. (28).
In the case of an RTIA characterized by transimpedance (G Ω in Ohms) and BW in Hz, responding to a laser pulse of peak power P signal in Watts, P d can also be written as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 9 ; 6 3 ; 3 5 6 P d ≈ where it is assumed that a narrow bandpass optical filter restricts the spectrum of the background illumination to the laser wavelength, P background is the transmitted background optical power in units of Watts, and the input-referred noise-current spectral intensity of the TIA (S I TIA ) is in units of A 2 ∕Hz. One merit of Eq. (29) is that G Ω , BW, and S I TIA are specified by most RTIA manufacturers, permitting calculations without SPICE simulation. However, the same transimpedance value (G Ω ) in Eq. (29) is assumed to apply equally to the peak of the signal photocurrent pulse and to the APD current shot noise. The case in which the RTIA bandwidth is too low for V out to track the input photocurrent waveform-which is the configuration that often maximizes receiver sensitivity, given operational constraints on FAR and timing precision-is not modeled by Eq. (29).
In developing an expression for FAR that accounts for the non-Gaussian 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 (η) at every point in time, t. A false alarm occurs when the current transitions through a threshold value (I th ) with a positive slope. Rice shows that the probability of this occurring during the infinitesimal time interval (t, t þ dt) is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 0 ; 3 2 6 ; 2 6 6 PDF FA ¼ dt where PðI ¼ I th ; η; tÞ is the joint probability distribution of the current and its slope at time t, assuming the random variable for the current has the value I th . Rice's classic result for FAR applies to Gaussian-distributed noise, for which PðI ¼ I th ; η; tÞ is the bivariate normal distribution. In the case of two uncorrelated random variables, the bivariate normal distribution is just the product of two single-variable Gaussian distributions E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 1 ; 3 2 6 ; 1 3 5 PðI; η; tÞ Gaussian ¼ 1 Noting that-for IðtÞ not to diverge-the average slope (η) has to be zero, and substitution of Eq. (31) in Eq. (30) gives (32) The FAR is just Eq. (32) without the differential dt.
Rice relates the variances of the current and its slope to its autocorrelation function (ψ) at zero time lag (τ) as (respectively) E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 3 ; 6 3 ; 6 2 2 varðIÞ ¼ ψ 0 ≡ lim t→∞ is the variance of the current in the dark condition, I th ¼ V th G Ω is the detection threshold expressed as an equivalent current at the TIA input, and I ¼ V dark G Ω is the average DC current level in the dark condition. In the absence of a voltage offset associated with the TIA, I ¼ I DC , and if the threshold voltage is referenced to V dark , I ¼ 0.
When the noise spectrum is white (constant S I ) over a finite bandwidth, S I cancels out in the radical and Eq. (38) becomes E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 9 ; 6 3 ; 1 3 8 FAR Gaussian ¼ where the TIA transimpedance relates the input-referred current quantities I th , I, and I noise to the corresponding outputvoltage quantities V th , V dark , and V noise diagramed in Fig. 2.
The variance of the current in the dark condition, varðIÞ, is found from the noise spectral intensity of the dark current and background photocurrent, S I DC , given by Eq. (19), and the input-referred noise-current spectral intensity of the TIA (S I TIA ) E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 4 0 ; 3 2 6 ; 6 6 4 Within the Gaussian approximation, the threshold that must be set to achieve a specified FAR is found from Eq. (39) as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 4 1 ; 3 2 6 ; 5 8 1 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 Gaussian-distributed TIA noise, P RX ðn out Þ, given by Eq. (4), in place of the Gaussian distribution used by Rice. P RX ðn out Þ 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 change-of-variable. Assuming the charge associated with electron count (n out ) is transported in time (t ref ), the current can be rewritten The joint probability distribution of the current and its slope, equivalent to Eq. (31), is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 4 4 ; 3 2 6 ; 2 6 5 PðI; η; Substitution of the modified joint probability distribution into Eq. (30) gives E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 4 5 ; 3 2 6 ; 1 6 It is important that n 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 th is measured in the lab relative to V dark , the DC offset hn DC i is omitted from the numerator of Eq. (28) for P d . This treatment is consistent with omitting V dark or I from the (V th − V dark ) or (I th − I) expressions in the Gaussian FAR models of Eqs. (39) and (41). However, when the convolution P RX ðn out Þ defined in Eq. (4) is used with Eq. (48) for FAR, the value of n out that maps to a given FAR is not referenced to hn DC i. Consequently, the offset hn DC i 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 ¼ 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 noise TIA ¼ 244 e − and at M ¼ 10, n noise APD ¼ 71 e − ; at M ¼ 20, n noise APD ¼ 160 e − . When the APD noise is more dominant, such as for photoreceivers assembled from larger-diameter 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.

Photoreceiver Performance and Receiver
Operating Characteristic The 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 ¼ 10 to 20. The plot also shows that the optimal APD gain is closer to M ¼ 15 than M ¼ 20 because the threshold required to extinguish false alarms diverges from the Gaussian model above M ¼ 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 ¼ 20 superior to M ¼ 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 FAR ¼ 150 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 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   6 Mean signal level required to achieve P d ¼ 99% as calculated at different APD gains using either the convolution model of P RX (solid curves) or its Gaussian approximation (dashed curves).
gain that minimizes NEI is lower than the gain that achieves the best sensitivity at the specified FAR. The optimal gain at −40°C is higher than at the other temperatures owing to lower APD dark current since it is the APD that generates the higher-amplitude 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 n th −hn DC i n noise þ 1.65 for P d ¼ 95% and n th −hn DC i n noise þ 2.4 for P d ¼ 99%; for P d ¼ 50%, Eq. (41) is the same as the sensitivity-to-NEI ratio. Equation (41) gives n th −hn DC i n noise ≈ 4.85 for BW ¼ 31 MHz and FAR ¼ 150, and as M → 1, the 95% and 99% sensitivity-to-NEI ratio converge on the values of 6.5 and 7.25 predicted by the Gaussian model.
A similar NEI-to-sensitivity 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.

Conclusion
A 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. In such cases, numerical convolution of the APD McIntyredistributed dark current with Gaussian-distributed TIA noise can support more accurate modeling, prior to empirical characterization of FAR versus detection threshold. The simulated results presented, which are based on measurements of Voxtel's ROX™ APD photoreceivers, demonstrate where the limitations of the Gaussian model lie for a relevant example. Fig. 8 NEI and sensitivity for FAR ¼ 150 Hz at P d ¼ 95% and 99% versus avalanche gain at −40°C, 27°C, and 50°C, modeled from empirical Voxtel 75-μm ROX photoreceiver dark current, excess noise, and NEI data. Fig. 9 Ratio of sensitivity to NEI, for 150-Hz FAR, modeled from empirical Voxtel 75-μm ROX photoreceiver dark current, excess noise, and NEI data.