Distortion losses of high-speed single-photon avalanche diode optical receivers approaching quantum sensitivity

Abstract

The high internal gain of single-photon avalanche diodes (SPADs) operating in Geiger mode allows the quantum limit of detection to be approached. This offers a significantly improved sensitivity for optical communication over existing photodiodes. A fully integrated CMOS SPAD array receiver (RX) is presented which achieves 500 Mb s⁻¹ four-level pulse amplitude modulation in a visible light communication link within 15.2 dB of the quantum limit. However, SPAD dead time induces around 5.7 dB of transient distortion which restricts error performance and data rate. We propose a model describing a discrete photon counting system which exhibits this nonlinear behaviour and compare it to practical measurements with the RX. A unipolar intensity modulated optical signal is considered, as opposed to bipolar electric fields in conventional radio frequency wireless systems. Intermodulation between the DC and harmonic components of the data-carrying waveform is investigated, and the resulting degradation of signal-to-noise-and-distortion ratio and bit error rate is evaluated. The model is developed as a tool for understanding distortion to ultimately allow rectification through RX architecture, modulation scheme, coding and equalization techniques.

This article is part of the theme issue ‘Optical wireless communication’.

1. Introduction

Integrated CMOS visible light communication (VLC) receivers have recently been developed to enable miniaturized and low-cost links for light fidelity (LiFi) networks, underwater wireless optical communications (UWOC) and plastic optical fibre (POF) communications [1]. Sensitivity of these devices is constrained by electrical noise related to the employment of photodiodes (PDs) or linear avalanche photodiodes (APDs) and their amplification circuits [1,2]. Fast APD receivers (RXs) have improved sensitivities of −38 dBm at 280 Mb s⁻¹ (850 nm) [2], −31.8 dBm at 1 Gb s⁻¹ (675 nm) [3] and −34.6 dBm at 1 Gb s⁻¹ (675 nm) [4] but remain two orders of magnitude above the standard quantum limit determined by photon shot noise [5]. This limit defines the minimum number of photon detections required to ensure a given bit error ratio (BER). The extremely high gain of SPADs in Geiger mode eliminates additive thermal noise in the RX chain since no transimpedance amplifier (TIA) is needed. This allows quantum sensitivity limits to be approached. A SPAD RX for fibre optic applications achieved 200 Mb s⁻¹ at 6.5 × 10^–3 BER within 24 dB of the quantum limit [6]. In this paper, we demonstrate a fully integrated 130 nm CMOS SPAD RX extending these data rate by 2.5× to 500 Mb s⁻¹ with four-level pulse amplitude modulation (4-PAM) at 2 × 10^–3, while improving sensitivity to −46.1 dBm, reducing the margin to the quantum limit to only 15.2 dB [7]. 100 Mb s⁻¹ on–off keying (OOK) is also reported with this RX at −55.2 dBm sensitivity, corresponding to an average of 7.5 detected photons per bit, 13.4 dB from the quantum limit [8]. These results meet the 3.5 × 10^–3 BER threshold required for forward error correction (FEC) to achieve an output BER of 10^–9 using concatenated Bose–Chaudhuri–Hocquenghem inner and outer codes with 6.69% overhead [9].

Effective detection rate is dependent on the dead time caused by quenching and recharging an SPAD after it detects a photon. During this time (3.5 ns reported in [6], 12 ns [7], 10.6 ns [10] and 5 ns [11]), the SPAD is unable to respond to subsequent incident photons. Hence, it is unlikely that more than one photon detection will occur in a symbol period at rates higher than 100 Mb s⁻¹, where PD/APD RXs readily operate. Therefore, a single detector is unable to recover signals above this range with reliable BER—so multiple SPADs are required. Figure 1 shows a representative block diagram of a photon counting RX with this principle compared to a typical PD RX. This presents design challenges to combine, count and sample the outputs of the SPADs with minimal losses and circuit area. Zhang et al. [11] accomplishes this with an analogue silicon photomultiplier (aSiPM) of 60 SPAD currents combined via a node connected to a load resistor. Our RX architecture is digital and permits massively parallel (4096) photon event summation to be achieved at a high fill-factor (43%) and sample rate (800 MHz). The RX operates in a practical, background insensitive VLC link at 1 m in 1 klx ambient conditions using a 450 nm laser diode (LD). The large array provides temporal redundancy to ease the dead time constraint by ensuring active SPADs are always available in a symbol period. This enables reception of symbols with durations (2.5 ns at 400 Mb s⁻¹ OOK, 4 ns at 500 Mb s⁻¹ 4-PAM) shorter than the individual SPAD dead time of 12 ns. Detector redundancy therefore obviates the requirement on current RX implementations that the dead time be matched to the symbol period to achieve the maximum data rate [6,10,11]. A rate of 200 Mb s⁻¹ is attained in [6] with only 4 SPADs and a short 3.5 ns dead time, but at the expense of relying on all SPADs to fire every bit period and compromised sensitivity due to crosstalk between the detectors (3% mean, 1.5–3.5 V excess bias), dark counts (1.46–13.9 kHz) and high afterpulsing probabilities up to 56%. These spurious afterpulse avalanches are caused by the delayed release of trapped carriers around a similar time scale to the dead time [12]. Afterpulsing becomes more significant at shorter dead times and causes inter symbol interference (ISI) at higher speeds. Instead, our SPAD array has an optimal 1.3 V excess bias above 13.9 V breakdown, 6 kHz median dark count rate and an afterpulsing probability of around 1%, albeit with a longer mean dead time of 12 ns. In reality, the dead time is not uniform and may vary depending on voltage and temperature; therefore, it is beneficial to reduce the influence of dead time on data rate and error performance.

Figure 1.

(a) Block diagram of a conventional PD RX and (b) a photon counting RX with summed SPAD array. (Online version in colour.)

Another advantage of our large array is that complex modulation schemes such as PAM or orthogonal frequency division multiplexing (OFDM) can be applied for high spectral efficiency and multipath mitigation. Furthermore, the SPAD RX is direct-to-digital so no analogue-to-digital converter (ADC) is required. We apply adaptive modulation DC biased optical OFDM (DCO-OFDM) with 512 subcarriers and bit loading to attain 350 Mb s⁻¹ and 3.7b/s/Hz mean spectral efficiency at 2 × 10^–3 BER. RX signal-to-noise ratio (SNR) (18 dB peak near DC) is sufficient to convey 32-QAM (quadrature amplitude modulation) and 16-QAM with the lower frequency subcarriers, down to 4-QAM at subcarrier index 416. We attain 5.6 b/s/Hz at 200 Mb s⁻¹.

With 230 pJ/b at best performance, we prove the highest power efficiency of SPAD RXs published to-date [6,10,11]. Although the RX energy per bit is not lower than PD/APD RXs (ranging from 86 pJ/b in [1] to 129 pJ/b [3]), these do not include an ADC function, which is inherently realized by this SPAD RX architecture. Our value of 230 pJ/b could readily be reduced by over an order of magnitude by adopting advanced CMOS nodes. Our sensitivities for 400 Mb s⁻¹ OOK, 350 Mb s⁻¹ DCO-OFDM and 500 Mb s⁻¹ 4-PAM all outperform that of PD, APD and SPAD RXs [1–4,6,10,11] and our maximum bit rate is improved by a factor of 2.5 compared to [6] while maintaining 3.5 × 10^–3 BER suitable for FEC with minimal bit redundancy [9]. The device reaches a sensitivity within 15.2 dB of the quantum limit set by the Poisson statistics of photon arrivals [5]. This gap includes a 1.5 dB power penalty due to the finite extinction ratio (ER) of the LD transmitter (TX) and an 8 dB loss from the SPAD photon detection probability (PDP) (37%) and fill factor (43%). The former is not an RX issue and can be practically eliminated with the use of an external modulator at the source [6]. The latter can be readily reduced with an optimized SPAD technology [13,14] and structure techniques such as three-dimensional stacking [15]. However, the remaining 5.7 dB penalty is unaccounted for and attributed to the dead time inducing transient distortion and ISI. This causes a signal-to-noise-and-distortion ratio (SNDR) limitation which prevents data rate from being further improved by increasing signal power. SPAD intrinsic parasitics evaluated in [16] are not as significant in our RX and too complex with a large array. With OFDM at low frequencies, where there is less signal attenuation, distortion is observed as effective SNR is more impaired from the total power. The dead time pile-up nonlinearity at high signal levels leads us to conclude that practical use of this device is to be in assistance (rather than replacement) of existing APD or PD RXs for LiFi, UWOC and POF applications; with potential to extend range or maintain connectivity in extreme conditions such as dark or diffuse environments.

The aim of this study is to advance understanding of the performance limits of a SPAD RX, to ultimately allow Gb/s data rates to be reached while enhancing sensitivity. This paper introduces an SPAD model to mobile VLC systems which are subject to random blockages and rapidly changing SNR conditions with the aim to ensure connectivity when line-of-sight links are interrupted. A detailed analysis of the main characteristics of an SPAD-based RX is presented. Theoretical equations are developed to gain understanding of the effect of dead time. A single SPAD is considered at first and then expanded to a 64 × 64 array with the objective to accurately predict SNDR. Analytical modelling and simulations are compared to experimental measurements with our integrated device to evaluate the performance of an SPAD RX in a VLC link.

2. Methods

(a). RX implementation

We report a 2.8 mm by 2.6 mm SPAD RX integrated in 130 nm CMOS imaging technology which incorporates 64 × 64 receiver elements at 21 µm pitch [7]. Each element contains a single p-well/deep-n-well SPAD biased at 15.2 V (1.3 V excess bias V_e) with a mean dead time of 12 ns, median dark count rate of 6 kHz at room temperature and PDP of 37% at 450 nm. 1.34 × 1.34 mm active area is chosen for ease of alignment to POF or VLC optics but can be adjusted electronically by disabling areas of detector elements. Figure 2 is a schematic of the chip architecture. A receiver element comprises an SPAD interfaced to an NMOS passive quench, enable SRAM and toggle-flop. The toggling output encodes photon events on both rising and falling edges. Thirty-two elements are combined with an XOR tree into an asynchronous double data rate (DDR) sequence at up to 900 MHz (1.8 Gphotons s⁻¹) limited only by wiring parasitics (not SPAD dead time).

Figure 2.

Simplified chip block diagram.

The array divides into two sets of 64-row XOR trees feeding digital readout chains positioned on the flanks of the active area. Figure 3 shows the row parallel interface circuit, consisting of three 8b ripple counters, sampling and converting the asynchronous DDR row signal into a synchronous binary count without dead time. The three counters operate in a round robin fashion, so that in every clock cycle one counter is reset, one is being read out and one is counting. A local state machine cycles continuously through the counters. The 128 sets of row counters operate in parallel and their outputs are added through a pipelined 7-stage adder tree to give an overall, 16b synchronous sum of SPAD events. The entire digital readout operates from a sample clock generated by an on-chip PLL with programmable frequency up to 800 MHz and distributed through clock trees to the pipelined adder. This digital readout replaces the TIA, analogue signal conditioning and ADC chain of conventional PD/APD-based receivers and will scale favourably to advanced nanometre process nodes. It also lends itself to the integration of the DSP required for complex modulation schemes. After digital gain control and formatting, the summation is conveyed off-chip using two D-PHY transmission blocks, each dual lane, with line rates of 400 Mb s⁻¹, for a total off-chip bandwidth of 1.6 Gb s⁻¹. Figure 4 is an annotated micrograph of the manufactured chip. In addition to the D-PHY blocks, an 8b 400 MHz bandwidth current-steering digital-to-analogue converter generates an analogue output compatible with existing ADC-based RX paths for testability. The total RX power consumption is 115 mW, consisting of 15 mW of SPAD power and 100 mW of DSP. In an improved RX implementation, the XOR tree could be replaced by fast summation placed at the exterior of the array, reducing photon cancellation effects and improving fill factor further. Ultimately, three-dimensional stacking would allow further improvements to SPAD PDP (also in near-infrared) and the placement of combination and summing electronics under the detectors without compromising fill factor.

Figure 3.

Asynchronous interface between (right) SPAD XOR tree and pipelined added tree. Inset: interface timing diagram.

Figure 4.

Annotated chip micrograph and measurement set-up. (Online version in colour.)

We generate sinusoids and OOK and 4-PAM waveforms from a pseudorandom binary sequence (PRBS) via a Mersenne Twister with a pattern length of 2¹⁹⁹³⁷–1 and non-return-to-zero (NRZ) line coding. These are transmitted with an Agilent 81180A Arbitrary Waveform Generator driving an LD (Thorlabs PL450B, 450 nm). A 450 nm optical bandpass filter (10 nm full width half maximum (FWHM)) and two neutral density filters (31% and 10% transmission) are placed above the array to attenuate ambient light and LD power. After frame synchronization, decoding of the received waveform is performed offline. The RX confirms photon counting reception at up to 500 Mb s⁻¹ from 1 cm to 1 m. Figure 4 shows a block diagram of the set-up. Experiments are conducted with the source directly in front of the RX to minimize reflections via the channel. We operate the device below the maximum rate of the XOR tress to avoid readout saturation. The array offers redundancy where detector elements are continually firing and recovering in steady-state conditions. This enables high RX dynamic range and Poisson SNR up to 23 dB per sample in theory. RX sensitivity from 100 Mb s⁻¹ OOK to 500 Mb s⁻¹ 4-PAM is compared to other RXs in figure 5 and a distortion of around 5.7 dB is observed. This penalty cannot be explained with current SPAD models, so a novel method of describing nonlinearity is required.

Figure 5.

Comparison of PD, APD and SPAD visible light RX devices. (Online version in colour.)

(b). Intensity response

We first consider a conventional circuit model and then apply a similar principle to an optical system. Amplifiers experience gain compression where the gain decreases for increasing amplitude since eventually the output signal reaches a limit due to supply voltage (e.g. [17]). This results in a nonlinear input/output relationship. An SPAD has a similar transfer relationship with greater deviation from linearity as the incident photon rate increases (figure 6) due to an undesirable dead time caused by resetting the SPAD after it detects a photon. During this time interval (in the order of 10–14 ns in [7]), the SPAD is unable to respond to impinging photons. A nonlinear system with input x and output y can be represented with a power series [18]

$$y(x,t)=a_{1}x+a_2{x}^2+a_3{x}^3+\cdots ,$$ 2.1

where a₁ is the linear, small signal gain (unity in this case). We intend to describe the large signal curve of an SPAD (figure 6) with dead time τ by this polynomial. The objective is to use this to predict the distortion induced on a modulated input signal. Background noise from dark counts and ambient light is omitted at this point to concentrate on the effect of dead time. Photon shot noise is included in the model since it cannot be eliminated from a practical system. For a passive quench (PQ) SPAD, any photon arriving during the dead time of a previous detection causes τ to be extended. A PQ SPAD output event rate is modelled in [19] as a paralysable detector [20,21]:

$$y_{\text{PQ}}=A{\text{e}}^{-A\tau },$$ 2.2

Figure 6.

Transfer curves of a single PQ SPAD and a 64 × 64 array (a). Corresponding gain (b). Inset: gain as a function of multiples n of 1/τ. −1 dB compression point (highlighted) at n = 0.2303 for a DC arrival rate. (Online version in colour.)

This model assumes a constant (DC) mean incident photon rate A. τ governs effective count rate and the saturation level is scaled by the array size. An expression for the DC gain $G_{\text{PQ}}$ as the input to output ratio is

$$G_{\text{PQ}}=\frac{A{\text{e}}^{-A\tau }}{A}={\text{e}}^{-A\tau }={\text{e}}^{-n}.$$ 2.3

Here G_PQ can be reduced to ${\text{e}}^{-n}$ where $A=n/\tau$ is a normalization based on multiples of the maximum input event rate 1/τ. As the value of factor n increases, the SPAD becomes more nonlinear. Count rate maxima occur at the dead time cut-off point 1/τ marked in figure 6. As a means of quantifying gain compression, the input level where G_PQ has dropped by 1 dB is found: $G_{\text{PQ}}={\text{e}}^{-n}=-1\text{dB}$ when n = 0.2303. Furthermore, the output drops by 4.34 dB when n = 1 and attenuation continues to grow exponentially as n increases, with −43.4 dB a decade higher at n = 10. DC gain is plotted in figure 6 and is identical for any number of SPADs in terms of incident rate per element. Before considering a modulated signal, which is mathematically involved, a single tone sinusoidal input signal with frequency ω₁ is applied

$$x=A\cos ({\omega }_{1}t).$$ 2.4

Here A is the amplitude rate proportional to the incident optical power. The incident photon rate now varies in time for this input. From equations (2.1) and (2.4), up to the third term, the output expansion produces

$$\begin{array}{rl}y& =a_{1}A\cos ({\omega }_{1}t)+a_2{A}^2{\cos }^2({\omega }_{1}t)+a_3{A}^3{\cos }^3({\omega }_{1}t)\\ & =\frac{a_2{A}^2}{2}+\left(a_{1}A+\frac{3a_3{A}^3}{4}\right)\cos ({\omega }_{1}t)+\frac{a_2{A}^2}{2}\cos (2{\omega }_{1}t)+\frac{a_3{A}^3}{4}\cos (3{\omega }_{1}t).\end{array}$$ 2.5

Unwanted higher harmonics (second 2ω₁ and third 3ω₁) are also generated in (2.5) and the second-order nonlinearity causes a DC shift of $a_2{A}^2/2$. Harmonic distortion is due to self-mixing of the signal. It can be suppressed by low-pass filtering the higher-order harmonics. The third order generates both third-order harmonic distortion and a fundamental component which distorts the linear term. The gain of the system is then deduced:

$$G=\frac{y_{{\omega }_1}}{x}=\frac{(a_{1}A+(3a_3{A}^3/4))}{A}=a_1+\frac{3a_3{A}^2}{4}=a_1\left(1+\frac{3a_3{A}^2}{4a_1}\right).$$ 2.6

If $a_3/a_1<0$, the gain compresses with increasing amplitude and since a₁ = 1, a₃ must be negative. The −1 dB compression point can be determined:

$$10{\log }_{10}\left(1+\frac{3a_3{A}^2}{4a_1}\right)=-1\hspace{0.17em}\text{dB}\text{.}$$ 2.7

If two tones are applied to the system,

$$x=B\cos ({\omega }_{1}t)+B\cos ({\omega }_{2}t),$$ 2.8

where B is half of A, such that the peak-to-peak amplitude of the waveform (2B = A) is equal to the rate in (2.2):

$$\begin{array}{rl}y& =a_2{B}^2+\left(a_{1}B+\frac{9a_3{B}^3}{4}\right)\cos ({\omega }_{1,2}t)+a_2{B}^2\cos (({\omega }_1\pm {\omega }_2)t)+\frac{3a_3{B}^3}{4}\cos ((2{\omega }_1\pm {\omega }_2)t)\\ & +\frac{3a_3{B}^3}{4}\cos ((2{\omega }_2\pm {\omega }_1)t)+\frac{a_2{B}^2}{2}\cos (2{\omega }_{1,2}t)+\frac{a_3{B}^3}{4}\cos (3{\omega }_{1,2}t),\end{array}$$ 2.9

where $\cos (k{\omega }_{1,2}t)=\cos (k{\omega }_{1}t)+\cos (k{\omega }_{2}t)$ and k = 1,2 or 3. Intermodulation terms arise at ${\omega }_1\pm {\omega }_2$, $2{\omega }_1\pm {\omega }_2$ and $2{\omega }_2\pm {\omega }_1$. This is caused by the two signals cross-mixing. When B is sufficiently small, the higher-order nonlinear terms are negligible, and the gain remains approximately a₁. As B increases, the fundamentals increase proportionally, whereas the third-order intermodulation (IM3) products increase in proportion to B³. Equation (2.9) is modified by considering an intensity modulated/direct detection (IM/DD) optical signal which is unipolar (as there is no negative light), in contrary to bipolar radio frequency (RF) with electric fields. A unipolar tone with a DC bias is equivalent to setting ω₂ = 0 in (2.8):

$$x=B\cos ({\omega }_{1}t)+B_{\text{DC}}.$$ 2.10

DC bias B_DC is later set to be different to signal amplitude to investigate the effect of a TX with finite ER. Therefore,

$$\begin{array}{rl}y& =a_{1}B+\frac{3a_2{B}^2}{2}+\frac{5a_3{B}^3}{2}+\left(a_{1}B+2a_2{B}^2+\frac{15a_3{B}^3}{4}\right)\cos ({\omega }_{1}t)\\ & +\left(\frac{a_2{B}^2}{2}+\frac{3a_3{B}^3}{2}\right)\cos (2{\omega }_{1}t)+\frac{a_3{B}^3}{4}\cos (3{\omega }_{1}t).\end{array}$$ 2.11

It is seen that intermodulation between the DC and signal components occurs and the higher-order terms in (2.9) fold down to DC and the fundamental, adding distortion. Now, distorted DC and fundamental gains are obtained:

$$\begin{array}{rl}{G}_{\text{DC}}& =\frac{y_{\text{DC}}}{x_{\text{DC}}}=\frac{(a_{1}B+(3a_2{B}^2/2)+(5a_3{B}^3/2))}{B}=1+\frac{3a_{2}B}{2}+\frac{5a_3{B}^2}{2}\\ G_1& =\frac{y_1}{x_1}=\frac{(a_{1}B+2a_2{B}^2+(15a_3{B}^3/4))}{B}=1+2a_{2}B+\frac{15a_3{B}^2}{4}.\end{array}\}$$ 2.12

Gain is no longer dependant on just a₃, as in (2.6), but a₂ as well. This means greater distortion occurs than for a constant rate. Computing the second- and third-order coefficients which delineate both the DC and fundamental distortion

$$\begin{array}{rl}& 10{\log }_{10}\left(1+\frac{3a_{2}B}{2}+\frac{5a_3{B}^2}{2}\right)=-1\hspace{0.17em}\text{dB}10{\log }_{10}\left(1+2a_{2}B+\frac{15a_3{B}^2}{4}\right)=-1\hspace{0.17em}\text{dB}\\ & a_2=\frac{(({10}^{-1/10})-1-15a_3{B}^2/4)}{2B}{a}_3=\frac{(2({10}^{-1/10})-2-3a_{2}B)}{5B^2}.\end{array}$$

Substituting

$$\Rightarrow a_3=\frac{0.1645}{B^2}{a}_2=-\frac{0.4113}{B}.$$

These values hold for −1 dB compression and are generalized by expressing in terms of n,

$$\begin{array}{rl}{a}_3& =\frac{0.1645}{B^2}=\frac{0.7143n}{B^2}\\ \text{and}{a}_2& =-\frac{0.4113}{B}=-\frac{1.7859n}{B}\end{array}\}$$ 2.13

Unlike the conventional polar case (2.6), a₂ is now negative and causing compression and a₃ is positive. The second and third harmonic distortion HD₂ and HD₃ can also be estimated:

$$\begin{array}{rl}\text{H}{\text{D}}_2& =\frac{\text{harmonic amplitude}}{a_{1}B}=\frac{(a_2{B}^2/2+3a_3{B}^3/2)}{a_{1}B}=\frac{a_{2}B}{2}+\frac{3a_3{B}^2}{2}=0.0411=0.1786n\\ \text{H}{\text{D}}_3& =\frac{(a_3{B}^3/4)}{a_{1}B}=\frac{a_3{B}^2}{4}=0.0411=0.1786n.\end{array}\}$$ 2.14

Substituting the coefficients (2.13) into the system polynomial (2.1),

$$y=x-\frac{1.7859n}{B}{x}^2+\frac{0.7143n}{B^2}{x}^3.$$ 2.15

With this general equation, the response can be determined for any input x, such as a sinusoid or data-carrying OOK signal.

(c). Detection statistics

In this section, the model is developed to specify the detection statistics of a SPAD RX. As introduced above, distortion causes a reduction of the received SNDR which cannot be understood with prior constant rate models [18]. Consider a baseband signal w(t) given by

$$w(t)=\sum _{p}a(p)s(t-pT),$$ 2.16

where a(p) is the information symbol sequence, p denotes symbol index, s(t) is pulse shape and T is symbol period. a(p) is a stream of randomly generated bits in Matlab. s(t) sets duty cycle and rectangular NRZ signalling is considered, where the pulse amplitude is held constant throughout the symbol period: $s(t)=1\text{ for }0\le t\le T\text{ and }s(t)=0$ otherwise. T is equal to the bit period since a binary alphabet is used (0 and 1 mapping). With sufficient SNR, higher-order modulation could be realized with M-PAM to increase net bit rate and spectral efficiency, where M is alphabet size. The signal is transmitted with an LD or light emitting diode. Assuming an ideal channel, let peak-to-peak amplitude be A, such that the received signal x is

$$x=Aw(t).$$ 2.17

This can be separated into a DC component B_DC and AC swing B. A TX with finite ER means symbol rate affects AC swing. In contrast to a tone which corresponds to an impulse in the frequency domain, the NRZ spectrum contains many frequencies enveloped with a sinc function. This makes it difficult mathematically to expand polynomial (2.15) with input (2.17), so analysis is shifted to the frequency domain to investigate baseband gain and distortion. We develop a simulator to describe an SPAD RX with this instance and find a solution to the equation from the spectrum rather than the time domain. Considering Parseval's identity, which states that the energy of a signal x(t) is conserved in temporal and spectral space [22]:

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|x(t)|}^2\text{d}t={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{|X(\hspace{0.17em}f)|}^2\hspace{0.17em}\text{d}f,$$ 2.18

where $X(\hspace{0.17em}f)$ is the Fourier transform of x(t), the output response can be determined this way. Photon noise is introduced to the model, but standard Poisson statistics cannot be assumed because the output count distribution is distorted. In the absence of dead time, ideal photon detections follow a Poisson process and the probability of counting k photons during a symbol interval (0,T) is given by [23]

$$p_0(k)=\frac{{(\lambda T)}^k{\text{e}}^{-\lambda T}}{k!},$$ 2.19

where λ is instantaneous mean photon rate, hence λT is the average number of incident photons in T. λ is related to received optical power P_r by [24]

$$\lambda =\frac{P_r\text{PDE}}{h\nu },$$ 2.20

where PDE is SPAD photon detection efficiency; h is Planck's constant and ν is the frequency of the light. In the presence of dead time, however, distortion causes the counts to deviate from a Poisson distribution [25]. It is assumed that an event is counted from each SPAD pulse. Therefore, the total number of events in counting interval (0,T) is obtained by the number of pulse transitions and cannot exceed $k_{\text{max}}=|T/\tau |+1$, where $\lfloor z\rfloor$ denotes the largest integer that is smaller than z. The probability mass function (PMF) of a PQ SPAD during (0,T) is expressed as [26]

$$p_K(k)=\sum _{i=k}^{k_{\text{max}}-1}{(-1)}^{i-k}\left(\begin{array}{c}i\\ k\end{array}\right)\frac{{\lambda }^i{(T-i\tau )}^i{\text{e}}^{-i\lambda \tau }}{i!},$$ 2.21

for $k<k_{\text{max}}$ and $p_K(k)=0$ for $k\ge k_{\text{max}}$. i is integer index. PDE is defined as the product of PDP and fill factor $C_{\text{FF}}$ and treated as a constant 8 dB loss (at 450 nm, V_e = 1.3 V and C_FF = 43%) in this model. It can be seen that the bandwidth of the RX is also affected by τ, so performance is susceptible to both the intensity and frequency of the modulated signal. An array is employed to increase capacity and improve SNR and the output is given by the superposition of the detector elements (2.21) is applied for each SPAD and expanded to a 64 × 64 array of independent variables combined into a single process. The aggregate count distribution of the array is approximated by a Gaussian distribution [27]

$$p_X(x)\sim \mathcal{N}({\mu }_X,{\sigma }_X^2),$$ 2.22

where ${\mu }_X$ and ${\sigma }_X^2$ are the mean and variance of the aggregate distribution, respectively. Numerical methods are developed to determine the achievable BER for OOK modulation. Although τ remains the main limiting factor on RX performance, noise causes BER to increase further. Dominant noise sources are background events from dark counts, afterpulsing, ambient light and ER. ER becomes the principle noise factor at high symbol rates and appears as a DC offset. Furthermore, feedforward ISI arises when $\tau \ge T$, since previous counts from a logical ‘1’ may disperse in time into a subsequent ‘0’ and therefore add to the ‘0’ distribution. These overflow events are almost unnoticeable in the ‘1’ distribution, especially if the mean is large, so ISI is most prevalent for 1 → 0 symbol transitions. Let ${\lambda }_s$ and ${\lambda }_n$ be the mean rates from signal and background noise, respectively. The average rates per element are ${\lambda }_s/N_{\text{SPAD}}$ and ${\lambda }_n/N_{\text{SPAD}}$, where $N_{\text{SPAD}}$ is the number of SPADs in the array. When a ‘0’ bit is transmitted, the average number of counts per symbol is ${\mu }_0={\lambda }_{n}T$, and when a ‘1’ is transmitted, the average is $\mu i_1=({\lambda }_s+{\lambda }_n)T$, where ${\lambda }_{s}T$ is mean signal counts per symbol. We obtain p₀(x) and p₁(x), probabilities that x photons are detected in the counting interval T, when ‘0’ or ‘1’ are sent:

$$p_0(x)\sim \mathcal{N}({\mu }_0,{\sigma }_0^2)\text{and}{p}_1(x)\sim \mathcal{N}({\mu }_1,{\sigma }_1^2),$$ 2.23

with variances ${\sigma }_0^2$ and ${\sigma }_1^2$. Decoding is implemented by comparing the received counts to a threshold x_T [27]

$$x_T=\frac{{\mu }_1{\sigma }_0+{\mu }_0{\sigma }_1}{{\sigma }_1+{\sigma }_0}.$$ 2.24

An error occurs if $x\le x_T$ when a ‘1’ bit is sent and, vice versa, if $x>x_T$ when a ‘0’ is sent. Probability of error is equal to BER for OOK and, assuming equiprobable symbols, is

$$\text{BER}\cong Q\left(\frac{{\mu }_1-{\mu }_0}{{\sigma }_1+{\sigma }_0}\right)=Q\left(\sqrt{\text{SNDR}}\right),$$ 2.25

where $Q(x)=(1/\sqrt{2\pi }){\int }_x^{\mathrm{\infty }}{\text{e}}^{-{\alpha }^2/2}\text{d}\alpha$ is the Q-function. Independent statistics are assumed for each transmitted bit, and it is assumed that the array elements are identical.

(d). SPAD simulator

We develop a simulation model, first presented in [28], to estimate the behaviour of an SPAD RX and compare this to the mathematical framework above. Matlab is used to simulate the stochastic events that occur in the photon counting system. With statistical analysis (2.21), it is shown that the counting process does not follow a Poisson distribution. As in (2.10), a noiseless sinusoidal input signal is generated with an instantaneous optical power at each sample. For a selected time window, the instantaneous mean photon rate is distributed in time with a Poisson process. Our simulator generates a stream of photons, represented by a discrete sequence of ones and zeros corresponding to photons or no photons in a given time step. This is the input to an algorithmic block which describes the statistical detection process of a PQ SPAD. The physical parameters of the SPAD including dead time, PDP and V_e are coded in this block. Each photon event is simulated individually, and the output is expressed as a digital vector where every rising and falling edge represents a detected photon. This is the output of an SPAD paired with a buffer which can be seen as a 1b ADC. The block is initialized in a rest state at zero. Figure 7 displays a block diagram of the simulation method. The output of the algorithmic block is the time domain response and vectors are run for each SPAD in the array and combined with an 800 MHz counter implemented in Matlab to recover the modulated signal. Sampling rate is much higher than τ, so it can be assumed that counting losses arising from finite sampling rates are negligible. The signal is then analysed in the frequency domain by taking a fast Fourier transform (FFT) and distortion metrics are obtained. The RX output is directly compared to its input waveform with respect to count rate and frequency and SNDR is calculated from the signal, noise and harmonic power [29].

Figure 7.

Simulation methodology. Input and output spectra for a unipolar 1 MHz cosine input at 10% and 100% of maximum intensity (top right). (Online version in colour.)

3. Results

(a). Output response

Using the simulator, the response of a PQ SPAD RX is estimated. Figure 7 shows input and output spectra for a unipolar 1 MHz cosine input signal at 10% and 100% of the maximum peak-to-peak rate A = 8.33 × 10⁷ s^–1. Gain is determined from the difference between the input and output tones. 7 dB output SNDR is computed from the output signal power and noise floor at 10% intensity, while arrival SNR = 9 dB. A second harmonic (2 MHz) appears at maximum input intensity and DC (–3 dB) and signal gains (–5 dB) are found from subtracting the corresponding magnitudes. At this intensity, SNDR = 4.7 dB, arrival SNR = 19 dB. Figure 8 shows the output derived from equation (2.15) together with simulated DC and signal gains found for different values of n. These estimates are plotted against the standard paralysable model in figure 8. The corresponding gain is computed from the quotient of the output response calculated in (2.15) and the input. It is concluded that higher distortion occurs with a unipolar waveform than for a constant incident level, which results in SNDR degradation. Near matching between unipolar theory and simulation indicates a reasonably accurate numerical estimate. Simulation accuracy diverges for extremely low event rates due to random noise generation; however, moderate precision is observed around the intensity range (1–100% of capacity) the RX would nominally operate in for VLC. Earlier inflection occurs at approximately n = 0.32 in theory and simulation rather than n = 1 as in the DC model.

Figure 8.

Comparison between the paralysable SPAD model (with DC rate A equal to the peak-to-peak amplitude of the unipolar waveform), analytical prediction and simulation results. (Online version in colour.)

(b). BER performance

The photocount distribution of an SPAD as described in (2.21) is simulated and plotted in figure 9 with mean incident rate A = λ = 8.33 × 10⁶ s^–1, a decade below the maximum of 8.33 × 10⁷ s^–1. Symbol rate R is increased from 1 MBd (T = 1/R = 1 µs) to 10 MBd (T = 100 ns). AT = 8.33 photons/symbol for a logical ‘1’ bit at 1MBd, which ensures integer photon arrivals within a symbol period in the simulation—although SNR is insufficient for 3.5 × 10^–3 BER reception. The average number of events per symbol is inversely proportional to R. Poisson distribution at 1 MBd also plotted (blue) for an ideal RX without dead time. It can be confirmed that the PMF expression (2.21) simplifies to a Poisson form with mean λT when τ is set to zero. At 1 MBd, the ‘1’ level mean is 7 counts/symbol, which is slightly lower than the arrival mean, and this translates to −0.757 dB average loss. This continues with means of 3 counts/symbol, 1 count/symbol and 0.8 counts/symbol at 2 MBd, 5 MBd and 10 MBd, respectively. Thus, loss is approximately unchanged and there is minor nonlinearity at this incident power. In addition, variance diverges to 2 counts/symbol, 0.9 counts/symbol and 0.5 counts/symbol at 2 MBd, 5 MBd and 10 MBd, respectively, whereas arrival variance is 8.33 ($\sqrt{8.33}$ standard deviation). As R increases, the distribution distorts and tends towards a unit Dirac impulse at 0 counts. Hence, the PMF approaches a point where the SPAD is persistently in a recovery state and no following photons are detected. Similar analysis is carried out at maximum intensity A = 8.33 × 10⁷ s^–1. At 1 MBd, ‘1’ mean is 31 counts/symbol, −4.29 dB distortion loss from AT = 83.3 photons/symbol arrival mean. The distribution deviates from Poisson form by −4.29 dB, −4.16 dB, −3.77 dB and −3.19 dB with respect to the corresponding photons/symbol at each R. SNDR = 8 dB at 4 Mb s⁻¹ (2.25), so it is deduced that this is the maximum OOK bit rate of a single SPAD with these parameters to sustain BER for FEC. Figure 10 shows simulated PMF distributions of a 64 × 64 array with τ = 12 ns, A = λ = 8.33 × 10⁶ s^–1, C_FF = 0.43 and R = 100 MBd to 400 MBd (100 Mb/s to 400 Mb/s NRZ). A rate of 400 MBd is chosen because it is the highest baud rate demonstrated by the SPAD RX in practice. Since the number of array elements is large, the output count distribution approaches a Gaussian form (2.23), as according to the central limit theorem [27]. SNDR is calculated from equation (2.25) with 5.78 ER (1.5 dB penalty) added to the model. PMFs shown (figure 10b) with the same parameters, but at maximum intensity A = 8.33 × 10⁷s^–1. A rate of 100 MBd and above results in complete saturation of the array, with a certainty of zero counts per given symbol period, without any chance of an SPAD having time to recover to detect the next symbol. At these rates, the output process is approximated by a Bernoulli distribution, as explained in [30], since T is shorter than the τ of each SPAD.

Figure 9.

Simulated photocount distributions of an SPAD for τ = 12 ns, fixed mean A = 8.33 × 106 s^–1 (a decade below the maximum incident rate) on the left and A = 8.33 × 107 s^–1 on the right. R = 1 MBd to 10 MBd; 1 MBd Poisson distribution without dead time (blue) for reference.

Figure 10.

Simulated and measured PMF count distributions of a 64 × 64 array for τ = 12 ns, A = 8.33 × 106 s^–1 (a) and maximum A = 8.33 × 107 s^–1 (b) at R = 100–400 MBd. (Online version in colour.)

We compare our mathematical framework and simulations to measurements obtained from laboratory experiments with our RX. At 400Mb/s OOK, 3 × 10^–3 BER is reached at −49.9 dBm with no equalization. The RX histogram is overlaid with estimated distributions in figure 10. The lowest peak is the ‘0’ PMF, with mean 4 detected photons/bit due to ISI and ER. The ‘1’ distribution has an average of 23.1 counts/bit and so the RX is operating at around 10% of its peak capacity of 230Gphotons/s. At 100%, the output is saturated as projected with the model. ER = 5.78 from the quotient of ‘1’ and ‘0’ means, which corresponds to 1.5 dB penalty. We sweep the intensity of the optical signal and measure the output count rate of a single detector element (figure 11). For each run, SNDR per symbol is found from the received counts and noise standard deviation. Measurements confirm increased distortion of the signal compared to a constant rate for the standard model with Poisson statistics (blue curve). Maximum SNDR = 6.8 dB at n = 0.3375. The sweep is repeated with all SPADs in the array enabled (figure 12a). Predicted BER from equation (2.25) is plotted along with BER measurements and lines of best fit in figure 12 with incident optical power (measured with a power meter) proportional to n. As shown, the model and measurements have closely matching curves, so we attain an accurate prediction of performance. It can be seen that BER degradation occurs earlier (at −45 dBm) than the paralysable model (upper transfer curve) would indicate because of transient dead time distortion of the signal. This reduces SNDR to approximately 12 dB a decade below the maximum incident rate, despite Poisson SNR = 18 dB at this level (6 dB penalty). SNDR values from numerical estimates, simulations and RX measurements are summarized in table 1. Maximum expected SNR = 23 dB at n = 1, assuming Poisson statistics and no data-carrying signal. Peak (Pk) effective SNDR is 9.6 dB lower than this, at 13.4 dB (when n = 0.21), which defines the BER inflection point. Therefore, 9.6 dB of distortion occurs on the received OOK signal. −49.9 dBm sensitivity at 400 MBd could potentially be reduced to −55.9 dBm if the 6 dB penalty due to nonlinear distortion is mitigated. This would allow a higher symbol rate or modulation depth to improve overall bit rate. A similar penalty occurs with 4-PAM, albeit at a higher average power.

Figure 11.

SNDR response at 1 Mb s⁻¹ OOK with a single SPAD. Input waveform amplitude set by n. Peak SNDR = 6.8 dB at n = 0.3375. (Online version in colour.)

Figure 12.

64 × 64 array SNDR at 400 Mbs−1 OOK against n (a). Theorized and measured BER with TX ER for 400 Mb s⁻¹ OOK and 500 Mb s⁻¹ 4-PAM (R = 400 MBd and 250 MBd, respectively) with received power (b). (Online version in colour.)

Table 1.

Comparison of estimated and measured SNDR for one SPAD and a 64 × 64 array at low and high intensities.

all in dB	10% SNDR	SNDR Pk	array 10% SNDR	array SNDR Pk
theory	4.5	6.5	9	12
simulation	4.7	5.4	9	12.7
measured	4	6.8	9.9	13.4

4. Conclusion

The 130 nm CMOS SPAD RX demonstrates 500 Mb s⁻¹ and a 15.2 dB margin to the quantum limit in a VLC link. Although the device attains the best SPAD RX performance published to date, this is accomplished with a large array of detectors, providing redundancy to ease the dead time constraint. Dead time remains as a 6 dB limitation and performance is highly sensitive to changing signal intensity conditions, so a model is established to aid understanding and predict the resulting distortion. Nonlinearity at high intensities infers that a practical use of this device is to supplement, rather than replace, existing PD/APD RXs. SNDR and BER is estimated and close matching with measurements validates the effectiveness of the simulation algorithm and our analytical model. Techniques such as predistortion at the TX [31] could be implemented to mitigate this distortion, with the metrics determined from our model used for calibration. Alternatively, a digital equivalent of automatic gain control could be employed at the RX to reduce the output level when the input signal intensity is too high. BER could potentially be reduced by optimizing and automatically adjusting the decision threshold for OOK and PAM depending on received count statistics.

Data accessibility

This article has no additional data.

Authors' contributions

J.K. was involved with the acquisition of data from the RX, which was designed by R.H. and those listed in the acknowledgements. R.H. also assisted J.K. with the analysis of data. J.K. drafted the paper and worked on the theory and simulations with support and supervision from K.M. and R.H. H.H. and R.H. contributed to revising the article and final approval of the version to be published.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by the Engineering and Physical Sciences Research Council through the Ultra-Parallel Visible Light Communications Project under grant no. EP/K00042X/1. We are grateful to STMicroelectronics for chip fabrication.