Photonic Excess Noise in a p-i-n Photodetector 1

Photonic Excess Noise in a p-i-n Photodetector 1 F. Javier Fraile-Peláez, David J. Santos and José Capmany )+( Dept. de Teoría de la Señal y Comunicaciones, Universidad de Vigo ETSI Telecomunicación, Campus Universitario, E-36200 Vigo, Spain. )+( Dept. de Comunicaciones. Universidad Politécnica de Valencia ETSI Telecomunicación, Camino de Vera s/n, E-46071 Valencia, Spain ABSTRACT e derive an analytical expression for the photonic noise variance of a p-i-n photodetector taking into account the randomness in the shape of the impulse response. The induced excess noise over the quantum limit is computed and the obtained result is contrasted with the standard model of ideal photodetector usually found in the literature. 1 Manuscript of the article published in Optics Communications 135, 37 40 (1997). 1

It is well known that the maximum sensitivity attainable in any photodetection process is determined by the radiation quantum noise. This means that the noise current of any noise-free detecting device should merely be a reproduction of the intrinsic noise of the impinging radiation. In fact, since the instantaneous photon flux is matematically modeled as a train of deltas occurring at random times (representing the photon absorption events), the total radiation noise power is strictly infinite. The actual noise current power is nevertheless finite due to the limited bandwidth of any physical detector, which responds to the each photon absorption by delivering a finite (non-delta) current impulse. The variance of the photocurrent is then given by the well-known shot noise formula [ i(t)] 2 =2ei(t) f, with f the detector bandwidth. The average photocurrent i(t) is proportional to the time-varying average incident photon-flux Q(t) =P (t)/(hν) (with P (t) the optical power and ν the central frequency of the quasi-monochromatic field) if the bandwidth of the photodetector is reasonably greater than that of Q(t), as is expected to be. As opposed to, for example, an APD photodetector, an ideal p-i-n photodetector (i.e., one with unit quantum efficiency, zero dark current and all the absorption taking place within the intrinsic region) is always presented as the prototypical noise-free, quantum-limited detector with a noise current given by the shot noise formula above. It is the purpose of this work to show that this is not true. e shall see that even an ideal p-i-n photodetector can never reach the quantum limit of sensitivity. The considerations that lead to such a result have never been pointed out in the literature, to the authors knowledge. A photodetector is most often modeled as a linear system characterized by an impulse response h(t) which represents the output photocurrent impulse caused by a Dirac delta excitation at the input. For coherent sources (such as lasers) the photon flux is known to follow a Poisson distribution. The problem of obtaining the photonic signal-to-noise ratio at the detector output is stated in terms of computing the statistical mean (γ 1 )andvariance (γ 2 ) of the photocurrent, which, under these circumstances, are known to be [1],[2] γ 1 (t) = γ 2 (t) = Q(τ)h(t τ)dτ (1) Q(τ)h 2 (t τ)dτ. (2) Q(t) will generally be time-varying due to the narrow-band modulation of 2

the optical power. The results (1) (2) have been used to characterize the shot noise current and signal-to-noise ratio of a p-i-n photodiode (Fig. 1). However, such an approach is erroneous since the impulse response of a p-i-n photodiode cannot be defined. The genuine delta input to a photodiode is associated to the absorption of one photon, and the shape of the corresponding output current impulse depends on the coordinate at which the electron-hole pair is created (hereafter, the coordinate at which the photon is absorbed ) [3]. In reference to Fig. 1, and for a photon absorbed at x, the output current is given by i ph (x, t) h(x, t) =e v e + v h u(t) e µ v e u t w x + v h u v e µt xvh, (3) where e is the electron charge, u(t) is the Heaviside function, and v e and v h are the saturation electron and hole velocities, respectively, at which the generated carriers are assumed to travel. The inset of Fig. 1 shows the shape of h(x, t), which is composed of two rectangular current pulses corresponding to the opposite hole and electron displacements. In mathematical terms, we have that h = h(x, t), which means that we are not dealing with a linear, invariant system, but with a linear, variant and stochastic system whose impulse response varies randomly. In studying the photodiode behavior at a text-book level, the approximation that all the flux is absorbed at the edge of the i zone (x =0)is sometimes made. Obviously, such a simplification completely ignores the problem and is useless for our discussion. On the other hand, when volumetric absorption throughout the intrinsic region is considered, another approach is most often used in which the input delta impulse is taken as a macroscopic light impulse conveying many photons which fill the whole volume of the i zone [1], [4]. The response to such an impulse would be the macroscopic h(t). Naturally, this procedure also skips the issue of the x-dependence. In this paper we shall indeed show that the macroscopic h(t) is the statistical average of all the h(x, t). However, the use of an averaged h(t) necessarily leads to underestimation of the variance γ 2, as the randomness in the shape of the photocurrent impulses is ignored. e shall obtain the correct expression of γ 2 and use it in place of (2) to compute the excess noise induced by the dispersive character of h(x, t). 3

The problem addressed here resembles that of obtaining the photocurrent mean and variance of a random gain photodiode (APD), but it is somewhat more complicated in that the x-dependence of h(x, t) does not materialize in the form of a mere multiplying factor, unlike what happens in an APD. For the latter, the instantaneous photocurrent has the form i APD (t) = P k M kh APD (t t k ) P k h(m k,t t k ), with M k the gain corresponding to the k-th photon detection. Thus, M plays a role similar to x in our case, in which the photocurrent can also be written i(t) = X k h(x k,t t k ), (4) where x k is the coordinate of the p-i-n intrinsic region at which the k-th photon is absorbed. Now the procedure employed to calculate γ 1 and γ 2 for an APD relies on the factorization h(m k,t t k )=M k h APD (t t k ) (see for example [1]), which is not valid in (4). The derivation of γ 1 and γ 2 for our case is outlined next. e first compute the characteristic function of i(t),defined as Φ(ω, t) = E [exp{iω i(t)}], with E denoting statistical expectation. Replacing i(t) by its expression (4), the following result can be derived: " Y Z # Φ(ω, t) =E dx p(x)exp{iωh(x k,t t k )}, (5) k 0 where p(x) is the probability density function of x. Following the standard procedure [1], the MacLaurin series expansion for the second characteristic function, Ψ(ω, t) =lnφ(ω, t), is found to be Ψ(ω, t) = X n=1 Z dτ Q(τ) dx p(x)h n (x, t τ) 0 (iω) n. (6) n! The moments γ 1 and γ 2 of i(t) are given by the first and second expansion coefficients of (6), respectively: γ 0 1(t) = γ 0 2(t) = Z dτ Q(τ) dx p(x)h(x, t τ) dτ Q(τ) E x [h(x, t τ)] 0 (7) Z dτ Q(τ) dx p(x)h 2 (x, t τ) dτ Q(τ) E x [h 2 (x, t τ)]. 0 (8) 4

The primes have now been introduced to refer to the true moments, as opposed to those given by (1) (2). Expressions (7) and (8) coincide with (1) and (2) provided the replacements h(t) E x [h(x, t)] and h 2 (t) E x [h 2 (x, t)] are possible. e shall now see that the former is correct, but the latter is not. In order to simplify the algebra, we shall assume uniform illumination throughout the i zone (as would be the case for a front-illuminated p-i-n with << 1/α, α being the absorption coefficient, or a side-illuminated waveguide-photodetector structure). This particularization does not affect the generality of the results in qualitative terms. e thus have ½ 1/, 0 x p(x) = 0, elsewhere. Using (9) and (3), we obtain (9) E x [h(x, t)] = ev e + ev h ³ 1 v t e ³ 1 v t h u(t) u u(t) u µt ve µt vh. (10) E x [h(x, t)] given by (10) coincides with the macroscopic h(t) used heuristically in the literature [1],[4]. Therefore, the expression (1) for γ 1 is correct. After some more algebra involving careful handling of the Heaviside functions, the following result is obtained for E x [h 2 (x, t)]: E x [h 2 (x, t)] = e2 ve 2 ³ 2 1 v t e u(t) u µt ve + e2 vh 2 ³ 2 1 v t h u(t) u µt vh + 2e2 v e v h 2 µ 1 v e + v h µ t u(t) u t v e + v h. (11) E x [h 2 (x, t)] given by (11) differs from h 2 (t) {E x [h(x, t)]} 2, so the result (2) is incorrect. E x [h 2 (x, t)] is naturally larger than h 2 (t) for all t, which reflects the fact that the randomness in the shape of the photodetector response adds noise to the output. The integral (8) can be performed analytically, yielding 1 γ2(t) 0 =Q(t) e2 2 (v e + v h )+ v ev h. (12) v e + v h 5

(Q(τ) is approximated by Q(t) when the photodetector bandwidth is quite larger than that of the optical signal, as expected). The integral (2) can also be calculated, giving 1 γ 2 (t) =Q(t) e2 3 (v e + v h ) 1 3 v h (v h 3v e ). (13) v e riting v h = βv e, with 0 β 1, the following relationship is obtained: γ2 0 (t) γ 2 (t) = 3+12β +3β 2 2+10β +6β 2 2β 3. (14) It is thus seen that the ratio of the correct variance to the underestimated variance is independent of the i-zone width, (because of the uniform illumination assumption), and varies with the carrier velocities ratio. Expression (14) is plotted in Fig. 2. The minimum is found at β =1(v h = v e ), in which case γ2 0 = γ 2 +0.51 db, while the maximum occurs when β =0(the electron contribution completely dominates the photocurrent), for which γ2 0 = γ 2 +1.7 db. This is a reasonable behavior as the shape of the current impulse produced by an electron generated at x ' 0 is very different from that produced by an electron at x '. However, the current impulse produced by the corresponding hole at x ' 0 is similar to that of the electron at x ', and vice versa. So, the simultaneous presence of electrons and holes attenuates the x-dispersion effect, which does not happen when the holes do not contribute. In conclusion, we have shown that some amount of excess noise over the quantum limit is intrinsic to the operation of a p-i-n photodiode. e have found an analytical expression of the noise increase, which turns out to be relatively small, but not negligible. From the theoretical point of view, it follows that some standard formulas describing quantum limits, such as [ i(t)] 2 =2ei(t) f, usually believed to apply to ideal p-i-n photodetectors, should in fact be corrected by a factor greater than one. 6

References [1]. vanetten and J. van der Plaats, Fundamentals of optical fiber communications (Prentice Hall, New York, 1991) p. 386 399. [2] A. Papoulis, Probability, random variables and stochastic processes, 2nd. Ed. (McGraw-Hill, London, 1987) p. 284, 379 385. [3] S. Ramo, J.R. hinnery and Th. van Duzer, Fields and waves in communication electronics, 2nd edn. (iley, New York, 1984) p. 122. [4] B.E.A. Saleh and M.C. Teich, Fundamentals of photonics (iley, New York, 1991) p. 652 654. 7

FIGURES Figure 1: Structure of a front-illuminated p-i-n photodiode. The inset shows the shape of the photocurrent impulse produced by an electron-hole pair generated at a specific x coordinate. Figure 2: Excess noise variance caused by the random impulse response. 8