760 RECEIVER NOISE STATISTICS 1.2 Amplifier Noise An optical amplifier introduces spontaneous emission noise to the signal in addition to providing gain. Consider a system with an optical preamplifier shown in Figure 4.7. The electric field at the input to the receiver may be written as Eft) ~/-2P cos(2rCfct 4- cb) 4- N(t). Here, P is the signal power, fc is the carrier frequency, and (I) is a random phase uniformly distributed in [0, 2zr]. N(t) represents the amplifier spontaneous emission noise. For our purposes, we will assume that this is a zero-mean Gaussian noise process with autocorrelation RN(r). The received power is given by P(t) - E2(t) - 2P cos2(27Cfct + ~) + 2x/~-fiN(t)cos(27rfct + ~) + N2(t). The mean power is E[P(t)] - P + RN(O ). (I.4) To calculate the autocovariance, note that since N(t) is a Gaussian process, E[N2(t)N2(t + r)] n2(0) + 2R2(r) using the moment formula (H.1). Using this fact, the autocovariance of P(.) can be calculated to be p2 Lp(r) - 2R2(r) + 4PRN(r)cos(2yrfcr) 4- ~ cos(4yrfcr). (I.5) The corresponding spectral density is given by Sp(f) Lp(r)e -i27rfr dr oo = 2SN(f) 9 SN(f) 4- 2P[SN(f - fc) + SN(f + fc)] p2 + -T[~(/- 2fc) + s(U + 2f~)]. (I.6) The 9 denotes the convolution operator, where f (x) 9 g(x) - f-~oo f (u)g(x - u)du. After photodetection, the last term in (I.5) and (I.6) can be omitted because the 2fc components will be filtered out. In order to derive the noise powers, we return to (I.3) and substitute for E[P(.)] and Lp(.) from (I.4) and (I.6), respectively, to obtain Lift) e~[P + RN(0)]8(r) + 7-~2[4PRN(r)cos(2zrfcr)] + ~212R2(r)]. 1.2 Amplifier Noise 761 We also have Si(f) eT-~[P + RN(0)] + 72~,22P[SN(f fc) + SN(f + fc)] + 7"~212SN(f) * SN(f)]. (I.7) The first term on the right-hand side represents the shot noise terms due to the signal and the amplifier noise. The second term represents the signal-spontaneous beat noise, and the last term is the spontaneous-spontaneous beat noise. Note that we have so far assumed that the amplifier noise is Gaussian but with an arbitrary spectral shape SN (f). In practice, it is appropriate to assume that the amplifier noise is centered at f~ and is white over an optical bandwidth Bo < 2f~, with Pn(G-1) SN(f)= O, 2 Bo If + fcl <_ T otherwise. Here, P~ is given by nsphfc, where nsp is the spontaneous emission factor. Corre- spondingly, we have RN(O) f_~c O0 SN(f)df Pn(G- 1)Bo. The spectral density of the photocurrent SI(f) from (I.7) is plotted in Figure 1.1, assuming the preceding value for SN(f). Note that, as before, the shot noise is white, but the signal-spontaneous beat noise spectrum has a rectangular shape, and the spontaneous-spontaneous beat noise a triangular shape. Moreover, the incident optical power P is given by G Pi, where Pi is the input power to the amplifier. Shot / Sponta Signal-spontaneous Power spectral density 2 2 ~ I)]B o 2 29t Pn(G- 9 I B e Bo/2 e~R[ GPi + Pn( G-1)Bo] B o Frequency Figure 1.1 Photocurrent spectral density. 762 RECEIVER NOISE STATISTICS Note that the photocurrent is passed through a low pass filter with bandwidth Be. The noise power at the output of the filter is given by fBe cr 2 SI(f)df 2 2 2 m m O.shot _~_ Crsig_spont q_ O.spont_spont, Be where 2 _ 2eTg[GPi + Pn(G- 1)Bo]Be O'shot 2 47p~2GPi Pn(G - 1)Be, O'sig_spon t and 2 __ ,/pv2 ]2 O'spont_spon t [Pn(G- 1) (2Bo- Be)Be. References [BL90] J.R. Barry and E. A. Lee. Performance of coherent optical receivers. Proceedings of IEEE, 78(8):1369-1394, Aug. 1990. [RH90] R. Ramaswami and P. A. Humblet. Amplifier induced crosstalk in multi-channel optical networks. IEEEIOSA Journal on Lightwave Technology, 8(12):1882-1896, Dec. 1990. Bibliography [Aar95] R. Aaron, editor. 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