Electrical Engineering 83 2001) 15±20 Ó Springer-Verlag 2001 Optical amplifier performance in digital optical communication systems

Reinhold NoeÂ

15 Contents Optical ampli®ers are important building blocks Other photon numbers need not be considered because for in nowaday's optical communication system. Their noise dt ! 0 the probability of multiple emission/absorption behavior differs considerably from that of electrical events tends towards zero faster than that of single events. ampli®ers. Theory is reviewed and supplemented by This master equation of photon statistics can be written in examples that allow to estimate the performance of optical differential form as transmission systems. In particular, a correct optical noise dP n; t† ®gure de®nition is explained. ˆÀ n a ‡ b†‡c†P n; t†‡ n À 1†a ‡ c† dt 1  P n À 1; t†‡ n ‡ 1†bP n ‡ 1; t† ; 2† Introduction Optical communication via glass ®bers is growing at where a is the stimulated emission rate, b the absorption extraordinary rates. Traditional electrical data regenera- rate and c the spontaneous emission rate. If a particular tors are replaced, where possible, by optical ampli®ers. At photon statistic is to persist it must ful®ll .2) but statistical some place, however, the optical signal has to be detected parameters such as its hni may be time-variable. For and regenerated electrically. Optical ampli®cation has example, if there is only attenuation .b > 0, a ˆ c ˆ 0), a been originally described more than 40 years ago [1]. Poisson distribution Photoelectron statistics are also described in [2] but these n Àl l0 excellent treatments seem to be not widely known today. P n†ˆe 0 3† In particular, a noise ®gure de®nition exists [3] which does n! not permit exact calculation of the resulting receiver sen- is a solution of .2) if its expectation value hniˆl0 t† Àbt sitivity. In a recent publication [4] this was corrected. This decays exponentially with time, l0 t†ˆl0 0†e . This has prompted the author, who has used part of this means a Poisson distribution is conserved under pure manuscript since 1996 in lectures at the Univ. Paderborn, attenuation. Germany, to review this subject. Practical examples are The moment generating function .MGF) of a discrete also given. RV n is X1 2 Às Àsn Àsn Mn †ˆe hie ˆ P n†e ; 4† Photon number distribution nˆÀ1 All random variables .RVs) in this paper are nonnegative, for a continuous random variable x it is except for Gaussian RVs. The interaction of photons with matter is governed by stimulated emission, absorption, Z1 Às Àsx Àsx and spontaneous emission. For in®nitesimal time incre- Mx †ˆe hie ˆ px x†e dx : 5† ments dt the probability P n; t ‡ dt† to ®nd n photons in a À1 medium at time t ‡ dt depends on the probabilities to ®nd The lower summation index/integration boundary is 0 for n À 1, n,orn ‡ 1 photons at time t, and the conditional Às À1 transition probabilities from one of these numbers to n: nonnegative RVs. With e ˆ z , Eq. .4) can be inverted by inverse z, while .5) is inverted by inverse Laplace P n; t ‡ dt†ˆP njn†P n; t†‡P njn À 1†P n À 1; t† transformation. ‡ P njn ‡ 1†P n ‡ 1; t† : 1† Adding statistically independent RVs requires convo- lution of the corresponding PDFs, or multiplication of the corresponding MGFs. As suggested by the name moment generating function, Received: 19 June 2000 the MGF allows to obtain all moments via

 k Às Reinhold Noe k k d M †e Univ. Paderborn, Electrical Engineering and x ˆ À1† : 6† ds†k Information Technology, Optical Communication and sˆ0 High-Frequency Engineering, Warburger Str. 100, D-33098 Paderborn, Germany Distributions, MGFs, mean values and variances of some e-mail: [email protected] discrete and continuous RVs are given in Table 1. Electrical Engineering 83 2001)

When a signal passes an optical ampli®er the proba- The dependence on t can now be dropped. If there is just bilities P n; t† of the discrete number of photons n is time- one optical mode stimulated emission is just n times larger variable. We determine the temporal derivative of the than spontaneous emission, hence c ˆ a. For N noisy Às corresponding time-variable MGF Mn e ; t† [6]. modes o X1 dP n; t† c M †ˆeÀs; t eÀsn : 7† N ˆ 13† ot n dt a nˆ0 holds. The spontaneous emission factor is Using .2) we obtain a nsp ˆ : 14† o Às Às Às s a À b Mn †ˆe ; t c †e À 1 Mn †Àe ; t †a À b e 16 ot The expectation value of the photons in one mode is o  †eÀs À 1 M †eÀs; t : 8† l ˆ n G À 1† : 15† os n sp The signal reaches the ampli®er input at t ˆ 0, with a With above substitutions the MGF is photon distribution P n; 0† and an MGF M eÀs; 0†. The Às Às ÀN n Mn †e ˆ †1 ‡ l †1 À e : 16† solution of .8) is  a À b c=a According to Table 1, the quotient between variance and M †eÀs; t ˆ mean value equals l ‡ 1 and is larger than for the Poisson n a À b ‡ a G À 1† †1 À eÀs distribution where it is 1. This is due to the quantum noise a À b ‡ bG À a† †1 À eÀs of the optical ampli®er. It surpasses the quantum noise of  M ; 0 9† n Às a Poisson distribution which is just shot noise. Inverse z a À b ‡ a G À 1† †1 À e À1 transformation of Mn z † results in a .central) negative with a time-variable .power) gain Binomial distribution  G ˆ G t†ˆe aÀb†t : 10† n ‡ N À 1 ln P n†ˆ n‡N : 17† This means we know the MGF inside or at the output of an n 1 ‡ l† optical ampli®er as a function of the MGF at its input, if Asuf®ciently attenuated signal is always Poisson distrib- Às the term e is replaced by a more complicated one. uted, as will be seen later. For a transmitted one we For a transmitted zero there are no photons at the input, therefore assume a Poisson distribution at the ampli®er  Às l~ eÀsÀ1† input. With Mn e ; 0†ˆe 0 and .9) we ®nd 1 n ˆ 0 Às P n; 0†ˆ ; Mn †ˆe ; 0 1 : 11† 0 n 6ˆ 0 Às Àl0 1Àe † Às Às ÀN Às At time t the signal has passed the ampli®er and Mn †ˆe ; t †1 ‡ l †1 À e e1‡l 1Àe † ; 18† aÀb†t G ˆ e holds. According to .9) the output photon at the ampli®er output with l0 ˆ l~0G. The corresponding MGF is photon distribution is a noncentral negative binomial or Àc=a Às a Às Laguerre distribution Mn †ˆe ; t 1 ‡ G À 1† †1 À e :  a À b ln l Àl 12† P n†ˆ eÀ 0 LNÀ1 0 ; 19† 1 ‡ l†n‡N 1 ‡ l n l 1 ‡ l†

Table 1. Some discrete and continuous distributions P n† n  0† M eÀs†hni 2 Discrete distributions n rn Poisson ln Àl 1 À eÀs† l l eÀl0 0 e 0 0 0 n!  Central negative n ‡ N À 1 ln 1 Nl Nl l ‡ 1† binomial n 1 ‡ l†n‡N 1 ‡ l 1 À eÀs††N l Às Noncentral negative À 0 Àl0 1 À e † l0 ‡ Nl Nl l ‡ 1†‡ 2l ‡ 1†l0 n 1 ‡ l e Às binomial, Laguerre l e NÀ1 Àl0 1 ‡ l 1 À e † n‡N Ln 1 ‡ l† l 1 ‡ l† 1 ‡ l 1 À eÀs††N

p x~† x~  0† M eÀs†hx~i 2 Continuous distributions x~ x~ rx~ Àl~ s Constant d x~ À l~0† e 0 l~0 0 Central v2 , Gamma 1 1 Nl~ ~2 2N l~N x~NÀ1 eÀx~=l~ Nl C N† 1 ‡ l~s†N 2 2 NÀ1 l~ ‡ x~ p!Àl~ s l~ ‡ Nl~ Noncentral v2N À 0 0 0 Nl~ ‡ 2l~l~0 x~ 2 e l~ 2 x~l~ 1 ‡ l~s I 0 e NÀ1 NÀ1 l~ 2 1 ‡ l~s†N l~0 l~ Reinhold NoeÂ:Optical ampli®er performance in digital optical communication systems

with Laguerre polynomials de®ned as l ˆ l2 ‡ l1G2 for the output noise in one mode. In the 1 dn cascade La x†ˆ exxÀa †eÀxxn‡a n n! dxn l G ‡ l F  F ˆ 1 2 2 ˆ F ‡ z2 25† Xn n ‡ a xm z G G z1 G ˆ †À1 m : 20† 1 2 1 m! mˆ0 n À m holds. Recursion allows to obtain Friis' well known cas- There is an important adding property: two statistically cading formula independent RVs n1, n2 with the same noise l and .central F À 1 F À 1 F ˆ F ‡ 2 ‡ 3 ‡ÁÁÁ 26† or noncentral) negative binomial distributions are added 1 G G G to form a new RV n. It obeys a new negative binomial 1 1 2 17 distribution with l0 ˆ l0;1 ‡ l0;2, N ˆ N1 ‡ N2. This for a larger cascade. This is important because optical can be veri®ed by convolving the PDFs or multiplying trunk lines consist alternatively of ampli®ers and attenu- the MGFs. ating ®bers. Apure attenuator with a ˆ 0, b > 0, nsp ˆ 0 has a noise ®gure F ˆ 1 and a ``gain'' G ˆ eÀbt < 1. For 3 example, the cascade of an attenuator .index 1) and a Noise figure subsequent ampli®er .index 2) has the noise ®gure F ˆ 1 ‡ F2À1. If the ampli®er .index 1) is in front of It is useful to divide the output noise photon number l per G1 the attenuator .index 2), l ˆ 0 holds. For increasing mode by the ampli®cation G. Thereby one obtains ®cti- 2 cious input-referred noise which would be ampli®ed by a attenuation, G2 ! 0, we ®nd Às noise-free ampli®er to become the observed output noise. Àl0;1G2 1Àe † This is the additional noise ®gure Às Às ÀN 1‡l G 1ÀeÀs† Mn;2 †ˆe ; t2 ‡ t1 †1 ‡ l1G2 †1 À e e 1 2 ÀÁ l À1 Àl G 1ÀeÀs† F ˆ ˆ 1 À G n : 21†  e 0;1 2 : 27† z G sp The noise ®gure itself is This means a Laguerre becomes a Poisson distribution if ÀÁ attenuation is large. Since the Poisson distribution is also À1 F ˆ 1 ‡ Fz ˆ 1 ‡ 1 À G nsp : 22† conserved under attenuation it has a special importance. This noise ®gure can be called F [4, 5], where ASE An ampli®er in which a, b and c vary as a function of the ASE propagation coordinate may be subdivided into in®nites- stands for ampli®ed spontaneous emission. Due to nsp  1 one ®nds F  1 for ampli®ers. An ideal ampli®er with imal sections and treated by .26) in which summation is replaced by an integral nsp ˆ 1 and ampli®cation G !1possesses F ˆ 2, which can be understood from the following: Zt  d F s†À1† · Without ampli®er just the amplitude but not the phase F ˆ 1 ‡ GÀ1 s†ds 28† dt of an optical signal can be measured. 0 · With ideal ampli®er and a subsequent power splitter both these quantities can be measured simultaneously. with d F À 1† 1 À G t À dt†=G t† Let us cascade two optical ampli®ers. For a Poisson ˆ n t† dt sp dt distribution with mean l~0 at its input the MGF after the ®rst ampli®er .index 1) is a 1 À eÀ aÀb†dt ˆ ˆ a : 29† Às Àl0;1 1Àe † a À b dt Às Às ÀN Às M †ˆe ; t †1 ‡ l †1 À e e 1‡l1 1Àe † 23† n;1 1 1 For constant coef®cients, the integral with l ˆ l~ G . Under the assumption that there is only 0;1 0 1 t one optical ®lter, behind all ampli®ers and directly in front Z of the receiver, the mode number N is identical for both F ˆ 1 ‡ a s†GÀ1 s†ds ampli®ers. The second ampli®er carries the index 2. The 0 MGF at its output is Zt R s ÀN À a #†Àb #††d# M †ˆeÀs; t ‡ t †1 ‡ l †1 À eÀs ˆ 1 ‡ a s†e 0 ds 30† n;2 2 1 2 1 ‡ †l À G †1 À eÀs 0  M 2 2 ; t n;1 Às 1 delivers just the known value F ˆ 1 ‡ 1 À GÀ1†n . The 1 ‡ l2 †1 À e sp longitudinal coordinate L inside the ampli®er is found as 24† R t L ˆ 0 vg ds. If group velocity vg is constant, L ˆ vgt, a Àb †t dL ˆ vg dt, the integration over time can be replaced by an with N ˆ ci=ai, nsp;i ˆ ai= ai À bi†, Gi ˆ e i i i and l ˆ n G À 1†, i ˆ 1; 2. integration over the longitudinal coordinate. i sp;i i As an example, consider Raman ampli®cation with a This can be rewritten as .18) with t ˆ t2 ‡ t1, l ˆ l G ˆ l~ G G and an expectation value pump wavelength roughly 50±100 nm shorter than the 0 0;1 2 0 1 2 signal wavelength. It can be used to render ®bers lossless, Electrical Engineering 83 2001)

Às Àsn a ˆ b. For a constant attenuation of 0.2 dB/km ®ber The MGF of P n† is Mn e †ˆhe i. After inserting .33) ``gain'' without pumping .a ˆ 0) would be the expression ÀbvÀ1L G ˆ e g ˆ 10À 1=10 dB†0:2 dB=km†L.AnL ˆ 50 km long  X1 Às=G À s=G†n Raman ®ber with a ˆ b ideally has the noise ®gure lim Mn e ˆ lim P n†e À1 G!1 G!1 F ˆ 1 ‡ bvg L ˆ 3:3 .if we ignore the dif®culty of nˆ0 achieving a ˆ b over such large lengths). For comparison, becomes an ideal ampli®er .nsp ˆ 1) with G ˆ 10 has F ˆ 1:9. If it is  Às=G Às placed at the input of a 50 km long ®ber without Raman lim Mn e ˆ Mx~ †e ; 34† gain .a ˆ 0) the cascade has a noise ®gure of 1.9. If it is G!1 placed at the ®ber end the cascade noise ®gure is 10. i.e., the MGF of the normalized variable x~. Since MGFs can be calculated from PDFs and vice versa it is indeed 18 Noise ®gure measurement is easy The optical ampli®er is possible to calculate px~ x~† from P n†. For a Laguerre À1 operated without input signal. Gain saturation effects can distribution MGF .18) we set l~ ˆ limG!1 lG ˆ nsp, À1 be taken into account if the input signal is just switched off l~0 ˆ l0G , and ®nd during short measurement intervals. Usually G is nearly Às ÀN Àl~0s polarization-insensitive, hence p ˆ 2 polarization modes Mx~ †ˆe †1 ‡ l~s e1‡l~s : 35† of a monomode ®ber have to be taken into account. The The corresponding PDF is ! output signal is passed through an optical ®lter with NÀ1 p 2 bandwidth Bo. Let the measured noise power be P. The 1 x~ Àl~0‡x~ 2 x~l~0 p x~†ˆ e l~ I ; 36† quotient P=B equals the energy per mode in the frequency x~ NÀ1 o l~ l~0 l~ domain. The photon number is obtained after division by 2 hf . The noise ®gure is therefore a noncentral v2N .chi-square) PDF with 2N degrees- of-freedom .x~  0). For l~ ˆ 0, or if one starts from the P 0 F ˆ 1 ‡ : 31† central negative binomial distribution MGF .16), the MGF pBohfG M †ˆeÀs †1 ‡ l~s ÀN 37† If a polarizer is placed before the power meter only one x~ 2 polarization mode is evaluated and p ˆ 1 holds. of a central v2N or Gamma PDF 1 Caution The hitherto prevailing noise ®gure de®nition [3] p x~†ˆ l~N xNÀ1 eÀx~=l~ : 38† x~ C N† was based on photon number ¯uctuations .pnfs) in the À1 limit of large photon numbers, Fpnf ˆ G ‡ 2 is obtained. It is also found if one inserts À1 1 1 z m 1 À G †nsp. It is measured as Fpnf ˆ G 1 ‡ 2P= pBohf ††. limjzj!0 Im z†ˆC m‡1† 2† into .36). Similar to the addition In an ideal ampli®er it assumes the value 2, in an atten- property of negative binomial distributions, the sum À1 uator it has the value G . It also ful®lls the cascading x ˆ x1 ‡ x2 of two statistically independent RVs x1, x2 2 2 formula .26)! However, it does not describe the noise with v PDFs and the same noise l~ ˆ nsp is a new v PDF of optical ampli®ers exactly. By the way, it also differs with l~0 ˆ l~0;1 ‡ l~0;2, N ˆ N1 ‡ N2, as can be veri®ed by from the classical microwave noise ®gure. Of course we multiplying the corresponding MGFs. can obtain our noise ®gure from the former one, For comparison and in order to apply again .34) we À1 F ˆ 1 ‡ Fpnf À G †=2. assume a Poisson distribution with expectation value l ˆ Gl~ and ®nd 4 0 0 Às Àl~ s Intensity distribution Mx~ †ˆe e 0 39† Assume the photon distribution can be written as the with corresponding PDF so-called Poisson transform [2] Z1 px~ †ˆx~ d †x~ À l~ : 40† xn 0 P n†ˆ p x†eÀx dx~ 32† This is a constant l~ . These ®ndings justify assumption x n! 0 0 .32) and can be interpreted: when the light intensity is subject to a detection process, shot noise is added which is of a PDF px x† that belongs to a nonnegative RV x.Ifwe Poisson distributed for each value the intensity may as- vary the optical power, e.g., by changing the gain G of an sume with a certain probability density. The light intensity optical ampli®er, P n† and px x† will depend on G.Yet G may carry just noise .central v2), signal and noise .non- mainly scales the x range so that the PDF px~ x~† of a nor- central v2), or just signal .constant distribution). The malized variable x~ ˆ x=G depends only weakly on G.From detection, modeled by the Poisson transform, results in equal probabilities px x†dx ˆ px~ x~†dx~ it follows central or noncentral negative binomial, or Poisson dis- À1 px x†ˆG px~ x=G†. This allows to write the photon dis- tributions, respectively. All these can be found in Table 1. tribution as Central and noncentral negative binomial distributions Z1 with 2N ˆ 4 are plotted logarithmically for an ideal ~ n ÀxG~ xG† ampli®er .n ˆ 1) in Fig. 1, but for easier comparison P n†ˆ px~ x~†e dx~ : 33† sp n! normalized probabilities log GP nG† are displayed instead 0 of log P n†. Avalue l~0 ˆ 81:4 was chosen. The higher the Reinhold NoeÂ:Optical ampli®er performance in digital optical communication systems gain G ˆ 1; 2; 4; 1, the higher the noise. In the noise-free 5 case G ˆ 1 Poisson distributions .o symbols) result. For Application 2 À9 in®nite G v4 distributions result with a BER ˆ 10 . We consider now an optical receiver .Fig. 2). Behind the The intensity may be de®ned as the expectation value optical receiver there is an optical ®lter with an impulse with respect to the detection process of the photon num- response which is a cosine oscillation having the fre- ber. This expectation value normally is an RV itself with quency of the received signal and a rectangular envelope respect to ASE, which originates mainly from the ampli®er of duration s1 ˆ 1=Bo. Bo is the optical bandwidth of that input. Since normalization is arbitrary we may likewise ®lter. For constant signal statistically independent ®elds understand the light intensity to be the expectation value E t ‡ is1† are therefore obtained every time interval s1. of the squared electrical ®eld magnitude, hjEj2i, including After photodetection, statistically independent intensity 2 ASE noise. samples hjE t ‡ is1†j i result. In our model the electrical 2 19 AnoncentralP v2N PDF .36) describes an RV part of the optical receiver should possess so little ther- 2 2N 2 v2N  x ˆ iˆ1 xi , where xi are Gaussian RVs with mal noise that it is possible to give it an impulse re- 2 identical variances rx ˆ nsp=2. The square sum of the sponse equal to a comb of M ˆ s2=s1 Dirac pulses with i p P2N expectation values hxiiˆ l0;i of zi is l0 ˆ iˆ1 l0;i. The equal amplitudes, spaced by s1. In that case the baseband electric ®eld is therefore a carrier with constant amplitude, ®lter is not a lowpass ®lter but has in®nite bandwidth! accompanied by Gaussian zero-mean noise xi with The Dirac comb ®lter can in good approximation be r2 ˆ n =2 in two quadratures, exchanged against a ®lter with a continuous impulse re- xi sp p sponse of length s or better s2 À s2, unless M is very pp 2 2 1 small. In order to avoid intersymbol interference s is of E t†ˆ 2 l =M ‡ x1 cos xt†‡x2 sin xt† e1 : 2 0 course chosen smaller than one bit duration T, but not 41† much smaller since signal energy would otherwise be lost. Suitably normalized, the signal at the decision circuit Here e is the unit vector of the signal polarization. For the 1 input is therefore time being, let M ˆ 1. The intensity equals the expectation value of the photon number. One intensity sample is XM 2 x~ ˆ jE t ‡ is1†j 45† p 2 2 2 iˆ1 jE t†j ˆ l0=M ‡ x1 ‡x2 : 42† with PDF .36). In the hypothetical, noise-free case hx~iˆl~0 and contains 2 degrees-of-freedom. If the ampli®er is holds. The decision variable has 2N ˆ 2pM degrees-of- polarization-independent, the orthogonal polarization e2 freedom, where the number of polarizations is usually ‡ p ˆ 2. For l~ ˆ 0 the PDF is given by .38). with e1 e2 ˆ 0 also has to be considered, 0 Thermal noise in the electrical part of the optical pÀÁÀÁ p receiver must also be taken into account. Figure 3 shows E t†ˆ 2 l0=M ‡ x1 cos xt†‡x2 sin xt† e1 log P n† for G ˆ 16; 64 and 256, l~0 ˆ 100 and 2N ˆ 8. ‡ †x3 cos xt†‡x4 sin xt† e2 ; 43† However, Gaussian thermal noise with r ˆ 523 has been added which results in BER = 7 Á 10À2,1Á 10À5, p 2 2:5 Á 10À10, respectively. It is taken into account by 2 2 2 2 jE t†j ˆ l0=M ‡ x1 ‡x2 ‡ x3 ‡ x4 ; 44† convolving the probability distributions are convolved or multiplying the corresponding MGFs. Ahigh gain is needed to make the relative thermal noise contribution and there are 4 degrees-of-freedom. insigni®cant. The chosen thermal noise r ˆ 523 corresponds to a receiver withp a bandwidth of 7 GHz and a thermal noise of 10 pA= Hz, suitable to receive 106 bit/s. BER vs. 10 log l~0=2†, i.e., the mean photon number ex- pressed in dB, has been calculated for l~ ˆ nsp ˆ 1, G !1, 2 i.e., v2N PDFs. Fairly moderate penalties occur for rising N .Fig. 4). The BER curves become steeper as N increases.

Fig. 2. Optical receiver. Apolarizer could, but normally is not, Fig. 1. Transition of normalized central and noncentral placed behind the ampli®er in order to block noise orthogonal to 2 distributions from Poisson via negative binomial to v4 the signal Electrical Engineering 83 2001)

20

Fig. 5. BER vs. mean photon number with degrees-of-freedom as a parameter Fig. 3. Photon distributions with thermal noise added

Fig. 4. BER vs. mean photon number with degrees-of-freedom as Fig. 6. Penalty of average signal power as a function of power a parameter ratio between zeros and ones

With more detail this behavior is investigated in Fig. 5. Large penalties result if there is a bad extinction ratio It shows 10 log l~0=2† vs. N for several constant BER. If between zeros and ones. there is a ®nite power ratio between zeros and ones of the optical signal, a transmitted zero will also have a non- 2 References central binomial or v2N distribution. For the latter case 1. Shimoda K, Takahasi H, Townes CH .1957) Fluctuations in resulting penalties are displayed in Fig. 6. ampli®cations of quanta with application to maser ampli®ers. J. Phys. Soc. Japan 12: S686±S700 6 2. Saleh B .1978) Photoelectron Statistics. Springer-Verlag, Ber- Conclusion lin Heidelberg New York Optical ampli®ers are important building blocks in now- 3. Desurvire E .1994) Erbium Doped Fiber Ampli®ers: Principles aday's optical communication system. The photons at the and Applications. Wiley, New York 4. Haus HA .1998) The noise ®gure of optical ampli®ers. IEEE ampli®er output obey noncentral or central negative bi- Photon. Techn. Lett. 10: S1602±S1604 nomial distributions for transmitted one or zero, respec- 5. Haus HA Private Communication tively. If there is no gain, these become Poisson 6. Saplakoglu G, Celik M .1992) Output Photon Number Distri- distributions. If the gain is in®nite, these become chi- bution of Erbium Doped Fiber Ampli®ers. 3rd Topical Meet- square distributions. The optical noise ®gure, correctly ing of the Optical Society of America on Optical Ampli®ers de®ned, is a direct measure for the photon distribution. and their Applications, Santa Fe, New Mexico, USA, 24±26 Fairly moderate penalties occur if the optical ®lter band- June width is larger than twice the electrical receiver bandwidth.