ECE 565 Notes on Shot Noise, Photocurrent Statistics, and Integrate-and-Dump Receivers M. M. Hayat 3/17/5 Let P (t) represent time-varying optical power incident on a semiconductor photodetector with quantum efficiency η. Suppose that each photogenerated electron-hole pair generates an electrical pulse I p (t). Generally speaking, I p (t) is a stochastic process (a random function); it has a random shape and a random duration. In particular, in the absence of any optoelecronic gain the area under I p ( ) is q, the electronic charge. If gain is present, on the other hand, the area would be qg, where G can be random in general. I p is often referred to as the impulseresponse function of the photodetector. Note that photogenerated electron-hole pairs are generated at a rate ηφ(t), where φ(t) P (t)/hν represents the time-varying arrival rate of photons incident on the detector. Let T 1, T 2,..., represent the random generation times of the photogenerated electronhole pairs. With these preliminaries, we can write the the total photocurrent due to all photogenerated pairs up to time t by I(t) I pi (t T i ), (1) i1 where I pi (t T i ) is the photocurrent due to the ith photogenerated pair. Mathematically, the process I(t) is called a shot-noise process with a random filter I p ( ), and it models the photocurrent generated as a result of applying the power P (t) to the detector. We will assume that I p1 ( ), I p2 ( ),..., is a sequence of independent and identically-distributed random processes. This is a meaningful assumption if space-charge effects in the photodetector are ignored, which is typical of low power photodetectors used in optical communication. The randomness in the photocurrent I(t) is attributed to (1) the randomness in the times of the generation of electron-hole pairs and (2) randomness in the functions I pi ( ). Hence, we see that the photocurrent is a random quantity; namely, if we apply the same optical power to the same photodetector at two different times, we will obtain two different photocurrents. However, the statistics of the photocurrents will be the same. These include, for example, the mean photocurrent, i(t) E[I(t)], the variance, σi 2(t) E[I 2 (t) i 2 (t)], and 1
τ τ " τ " # τ τ # $ τ $ % τ %! " # $ τ! Figure 1. Illustration of shot noise. the autocorrelation function, R I (t 1, t 2 ). Note that we can always write I(t) i(t) + (I(t) i(t)), which is the sum of a deterministic quantity and a noise term, which is referred to as shot noise. 1. Photocurrent Statistics We begin by discussing the properties of the sequence T 1, T 2,.... Let N(t, t+ t) represent the number of photogenerated electron-hole pairs that occur in the interval [t, t+ t). It turns out from statistical optics [see for example the Statistical Optics book by J. Goodman] that for coherent light and each t and t, N(t, t + t) is a Poisson random variable with a mean given by n(t, t + t) E[N(t, t + t)] η t+ t t φ(s) ds. This means that P{N(t, t + t) k} n(t, t+ t) k e n(t,t+ t) /k!, which implies that E[N 2 (t, t+ t)] n 2 (t, t+ t)+n(t, t+ t). It is also true that N(t 1, t 2 ) and N(t 3, t 4 ) are independent whenever t 1 < t 2 < t 3 < t 4. The process N(, t) is called an inhomogeneous Poisson process with mean rate ηφ(t) (it is called homogeneous if φ is a constant). 2
1.1. The mean of I(t) First, observe that I(t) lim t j I pj (t j t)n(j t t, j t), (2) where t is the greatest integer less than or equal to t. Let i p (t) E[I pj (t)]. Then, by assuming that N(, ) and I pj ( ) are independent for any j, we obtain E[I(t)] lim t j lim t j lim η t j t E[I pj (t j t)]e[n(j t t, j t)] i pj (t j t)n(j t t, j t) j t i p (t j t)η φ(s) ds j t t In conclusion, the mean of the photocurrent I(t) is i(t) (hν) 1 t i p (t u)φ(u) du. (3) i p (t u)p (u) du (hν) 1 i p (u)p (t u) du. (4) Since i p is a causal function, we can rewrite the above expression as i(t) (hν) 1 i p (t u)p (u) du (hν) 1 i p (u)p (t u) du i p (t) P (t)/hν. (5) 1.2. The variance of I(t) Let us first calculate E[I 2 (t)]. As before, we use the fact that I(t) lim t j I p j (t j t)n(j, j t) and write I 2 (t) lim t j lim t j l I pj (t j t)n(j t t, j t) l I pl (t l t)n(l, l t) I pj (t j t)i pl (t l t)n(j t t, j t)n(l t t, l t).(6) 3
Now, E[I 2 (t)] lim t j l lim t j l,l j + lim t j lim t j l,l j + lim t j E[I pj (t j t)i pl (t l t)]e[n(j t t, j t)n(l t, l t)] E[I pj (t j t)i pl (t l t)]e[n(j t t, j t)n(l t t, l t)] E[N 2 (j t t, j t)]e[i pj 2 (t j t)] E[I pj (t j t)]e[i pl (t l t)]e[n(j t t, j t)]e[n(l t t, l t)] E[N 2 (j t t, j t)]e[i pj 2 (t j t)]. (7) Note that E[N(j t t, j t)] η j t φ(s) ds and E[N 2 (j t t, j t)] η j t φ(s) ds+ j t t j t t (η j t φ(s) j t t ds)2. With this we arrive at ( t 2 t I 2 (t) η E[I 2 p (t u)]φ(u) du + η i p (t u)φ(u) du). (8) Note that the second term on the right is just the mean squared of I(t). Hence, the variance of I(t) is given by σ 2 I(t) E[I 2 (t)] E[I(t)] 2 (hν) 1 t E[I 2 p (t u)]p (u) du (hν) 1 E[I 2 p (u)]p (t u) du. Once again, using the causality of i p, we can express the variance as a convolution: σ 2 I(t) E[I p 2 (t)] P (t)/hν. (1) (9) Example. Avalanche photodiode (APD): The impulse-response function of an APD is often approximated as I p (t) Gqh(t), where G is a random variable representing the APD s gain and h(t) represents the average shape and duration of the response. The area under h is one, so that the area under I p is qg, as expected. We emphasize that in this simplified model h is deterministic. In this case, i(t) ηqgh(t) P (t)/hν (11) 4
and σ 2 I(t) ηq 2 F g 2 h 2 (t) P (t)/hν, (12) where g E[G] is the mean gain (or the multiplication factor) of the APD and F E[G 2 ]/g 2 is the so-called excess noise factor. Example. Slowly-varying power: If the input optical power P (t) is slowly varying with respect to h(t), then we can approximate the convolutions above and obtain i(t) ηqgp (t)/hν (13) and σ 2 I(t) ηq 2 F g 2 P (t)/hν h 2 (t) dt. (14) The quantity represented by the integral has a bandwidth flavor. For example, if h(t) is a rectangular function with width τ and height 1/τ, then h2 (t) dt 1/τ, which is twice the bandwidth associated with the pulse h. Thus, we shall define the detector bandwidth B as B 1 2 or more generally in the case when h is not of unit area as, we define h 2 (t) dt, (15) B h2 (t) dt ( ) 2. (16) 2 h(t) dt as With the above definition of detector bandwidth, we can recast the shot-noise variance σ 2 I(t) 2qF gi(t)b, (17) which states that the noise in the photocurrent is proportional to the mean photocurrent. Namely, shot noise is signal dependent. Moreover, if the detector gain is deterministic and unity, as in a PIN detector, then g F 1, in which case σi 2 (t) 2qgi(t)B, which is the familiar (hopefully) expression for shot noise for detectors without gain. Thus, the multiplicative factor F is the excess noise brought about by the randomness in the photodetector s gain. That is why F is called the excess noise factor. 5
1.3. Signal-to-noise ratio of the photocurrent In the ideal case when dark current and Johnson noise are absent, the signal-to-noise ration (SNR) is generally defined as SNR i2 (t) (18) σi 2 (t). If we now adopt the APD model described in the above example and further adopt the slow-varying-power assumption, we obtain SNR i(t) 2qgF B, (19) that is, the SNR increases with the mean photocurrent. The above expression is called the shot-noise-limited SNR for photocurrent: it is the best SNR we can achieve using classical light. Note that the randomness of the gain (if present) reduces the SNR by a factor F. In actuality, the photocurrent is also corrupted by Johnson noise (which will discuss later) and dark current, which results from the generation of carriers without the involvement of photons (due to thermal excitation or phonon absorption from the lattice, band-to-band tunneling, etc.). With these, the expression for the SNR becomes SNR i 2 s(t) σi 2(t) + σ2 J + σ2 d i 2 s(t), 2qgF (i s (t) + i d )B + σj 2 where i d is the mean dark current and i s is the photocurrent in the absence of dark and Johnson currents. The reason for grouping the dark current with the photocurrent i s is that they are both generated inside the photodetector. For any electron-hole pair generated, the resulting photocurrent will have the same statistics regardless of the source of the pair. Thus, as far as the photocurrent is concerned, dark carriers and photogenerated carriers are indistinguishable. Johnson noise, on the other hand, is different. It is added to the photocurrent and its source is the resistive elements in the electronic circuitry that are connected to the photodetector (primarily the load resistor in the pre-amplifier). 6
1.4. The autocorrelation function of the photocurrent I(t) Let us first derive a general expression for the autocorrelation using (2), R I (t, s) E[I(t)I(s)] E[ I p (t T j )I p (s T l )] (2) E[ j1 l1 I p (t T j )I p (s T l )] + j1 l1,l j j1 I 2 p (t T j ). (21) After some steps (similar to to the ones used in calculating the variance of I), we obtain R I (t, s) η(hν) 1 E[I p (t u)i p (s u)]p (u) du + i(t)i(s). (22) It is difficult to have a simple expression for the autocorrelation function beyond this in the general case. However, when we use the approximation i p (t) qgh(t), one can repeat an analysis similar to that for the variance and obtain R I (t, s) q 2 ηf g 2 (hν) 1 h(t u)h(s u)p (u) du + i(t)i(s), (23) where i(t) is given by (1). 2. Statistics of the Output of Integrate-and-Dump receivers Consider the basic integrate-and-dump receiver diagram shown in Fig. 2. The idea is that instead of sampling the photocurrent at a single instant within a bit and deciding on the presence or absence of a an optical pulse, we would integrate the photocurrent over the bit in question and use the area, S, under the photocurrent as an indicator. Such decision is more robust and would offer better bit error rates. Since the power is digitally modulated in on-off-keying modulation, we will assume in the remaining of this section that over each bit, the power level is constant at P. The thermal noise is assumed to be additive zero-mean Gaussian white noise (AWGN). This means that E[I T (t)] ] and E[I T (t)i T (t + τ)] σ 2 δ(τ), where δ is the Dirac impulse. Intuitively, this means that time samples of I T are uncorrelated, and hence independent since I T is Gaussian. The parameter σ 2 is given by σ 2 4k B tmperature/r, where R is the equivalent resistance of the photodetector circuit, often dominated by the load resistance (where the photocurrent flows), k B is Boltzman s constant, and the temperature is the units of Kelvin. 7
Figure 2. Schematic of an integrate-and-dump receiver. The output S is used to make a decision on the presence or absence of an optical signal in each bit. 2.1. Mean and variance of S For simplicity, let s assume that S corresponds to the zeroth bit, that is, S where Ĩ I + I T is the total photocurrent. Now, E[S] E[ Ĩ(t) dt] E[I(t) + I T (t)] dt Ĩ(t) dt, E[I(t)] dt η(hν) 1 (P (t) i p (t)) dt. By interchanging the convolution and the bit integration, we can recast E[S] as where (24) E[S] η(hν) 1 P (t)î p (t) dt, (25) î p (t) If we use the approximation i p (t) qgh(t), we obtain i p (s t) ds. (26) where E[S] ηqg(hν) 1 ĥ(t) P (t)ĥ(t) dt, (27) h(s t) ds. (28) 8
The variance of S can be calculated as follows. We first calculate E[S 2 ] E[ Ĩ(t) dt Ĩ(s) ds] E[Ĩ(t)Ĩ(s)] dtds E[I(t)I(s) + I(t)I T (s) + I(s)I T (t) + I T (s)i T (t)] dtds [R I (t, s) + σ 2 δ(t s)] dtds R I (t, s) dtds + σ 2 T. (29) However, when we use the approximation i p (t) qgh(t) at this point, we obtain Thus, with this simplification the variance is 2.2. Signa-to-noise ratio of S E[S 2 ] E[S] 2 + η(hν) 1 ĥ 2 (u)p (u) du. (3) E[σS] 2 ηq 2 g 2 F (hν) 1 ĥ 2 (u)p (u) du + σ 2 T. (31) As usual, we define the SNR of S as the mean squared over the variance: 2.2.1. Special case: Instantaneous detector SNR S E[S]2. (32) σs 2 This special case is useful whenever the width of h is small relative to the bit duration T. In such cases we can assume that h(t) δ(t). The integrate-and-dump receiver in this case simply attempts to count the number of photogenerated carriers. It follows from this condition and the definition of ĥ that ĥ(t) 1 if t T and ĥ(t) otherwise. The SNR S simplifies in this case to SNR S (qηgt P/hν) 2 σ 2 T + ηq 2 g 2 F T P/hν. (33) Note that ηt P/hν is the average number of photogenerated electron-hole pairs in each bit, which we now denote by m. With this interpretation, we can recast the SNR as SNR S g 2 m 2, (34) g 2 F m + σq 2 9
where σ 2 q σ 2 T/q 2 is the circuit-noise parameter, which represents the Johnson noise in units of number of electrons. Note that all of the quantities in the above SNR are dimensionless. 1