# Notes on Optical Amplifiers

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1 Notes on Optical Amplifiers Optical amplifiers typically use energy transitions such as those in atomic media or electron/hole recombination in semiconductors. In optical amplifiers that use semiconductor materials, the injected current is the energy source. This injected current produces a nonequilibrium distribution of energy states that provides gain via stimulated emission. In another type of optical amplifier, trace amounts of the gain material are added to the core of a fiber to create a doped-fiber structure. For example, for optical amplification near 1.5 µm, the rare earth metal erbium is used as a dopant with a concentration in the range of 0.1% for the typical silica glass that is used to construct optical amplifiers. In this case, the external energy is provided by an optical source that creates a nonequilibrium distribution of excited energy states that can provide gain. Other types of amplifiers rely on nonlinear processes such as Raman or Brillouin scattering. 1 Physics of Optical Gain The characteristics of phase insensitive amplifiers are specified by the power gain G( f ) = H( f ) 2 and an internal gain per unit length γ int = σ e N 2, where σ e is the stimulated emission cross-section, and the density N 2 is the number of excited energy states per unit volume. The stimulated emission cross section represents the efficiency of a single excited energy state to generate a stimulated emission event. The number of excited states per unit volume, N 2, is a function of the external energy source, and the lightwave signal power in the amplifier. Similarly, the absorption coefficient α, is the product of an absorption cross-section σ a, and the number of absorbing, lower energy states per unit volume. The absorption per unit length is then α = σ a N 1. If there are only two energy levels in the system, then N 2 + N 1 = N where N is the total number of energy states per unit volume. The net gain per unit length, γ is defined as the internal gain minus the loss γ. = γ α = σ e N 2 σ a N 1. (1) A necessary condition to have net gain is σ e N 2 > σ a N 1. We now consider conditions when the gain per unit length γ, is approximately constant. This condition occurs near the input to the amplifier when the number densities, N 1 and N 2, are approximately constant. Referring to (1), if the densities are constant, then the gain per unit length is then a constant γ 0 defined as the small-signal gain per unit length. The differential increase in the lightwave power P(z) over a distance z in the optical amplifier is then 1

2 P(z) = γ 0 P(z) z. The corresponding continuous equation for the evolution of the lightwave power as a function of z is dp(z) dz = γ 0 P(z). (2) Applying the initial condition at the input to the amplifier, P(0) =. P in, the solution to (2) is P(z) = P in e γ0z.. The small signal gain G 0, after a distance z = L, is then defined as G 0 = e γ0l. Gain Saturation The small signal gain G 0 = e γ 0L in (2) is an exponential function of the length L the lightwave signal propagates in the amplifier. As the signal is amplified, the density of the excited upper energy states that provide the gain is eventually depleted for any finite external energy source. Referring to (1), as N 2 decreases, the gain per unit length γ decreases relative to the value γ 0 defined at the input to the amplifier. This nonlinear effect is called gain saturation and occurs in all gain media. The gain at the output of the amplifier G is then reduced relative to the small signal gain G 0. We incorporate gain saturation by modifying (2) to allow the gain per unit length γ to be a function of the lightwave power P(z) dp(z) dz = γp(z) = γ [P(z)/P sat ] P(z), where γ 0 is the constant small signal gain per unit length, and P sat is the saturation power. When P(z) = P sat, γ = γ 0 /2, and the gain per unit length is reduced to half the value relative to the input to the amplifier. Separating the variables we have ( 1 P + 1 ) dp = γ 0 dz. P sat We now integrate the right side from z = 0 to L, and the left side from P(0) = P in to P(L) = GP in where G =. P(L)/P in is the saturated gain. Solving, the saturated gain is [ G = G 0 exp (1 G) P ] in. (3) P sat For typical erbium-doped fiber amplifiers, P sat is a few milliwatts. A plot of the effect of gain saturation is shown in Figure 1. When P in P sat, the gain of the amplifier approaches the small-signal gain G 0. However, as the input power approaches the saturation power, the gain decreases and must be determined by numerical means since (3) is a transcendental equation as a function of G. The characteristic response time of the optical amplification process is governed by the upper state lifetime. This is the average time that the system will remain in an excited energy state before spontaneously relaxing to a lower energy state. The reciprocal of the lifetime is the maximum rate of change of the gain of the amplifier. The amplified lightwave signal will not be distorted by gain saturation as long as the modulation rate is significantly faster than the upper state lifetime. In this case, all the frequency components of the signal experience the same average gain. For example, for erbium-doped fiber amplifiers, the upper state lifetime is approximately 10 ms. This time is significantly slower than typical data rates in lightwave communication systems. The number of excited states per unit volume, and the corresponding 2

3 Gain (db) Output Power (P/Psat) Input Power (P/Psat) Input Power (P/Psat) Figure 1: Gain and output power for an optical amplifier with a small signal gain G 0 = 20 db. gain, is then effectively constant over the bandwidth of the modulated lightwave signal. The long upper state lifetime also means that high frequency noise fluctuations in the external energy source are strongly filtered and are not transferred to the amplified lightwave signal. This approximation is not as accurate for optical amplifiers using semiconductor materials because the lifetime of the upper energy state is of the order of 1 ns. 2 Noise in Optical Amplification The origin of noise in optical amplification is spontaneous emission. The presence of optical gain results in amplified spontaneous emission (ASE). To determine the noise power density spectrum of the amplified spontaneous emission, we use a quantum-optics model. The photon energy for a lightwave signal is much greater than the average thermal energy so that h f kt 0. For a single polarization, the quantum noise per mode is E = h f 2. The corresponding mean number of photons per mode is n = 1 2. This noise is associated with the fundamental vacuum fluctuations of the incident lightwave signal and is always present. Optical amplifiers produce additional fluctuations. The internal energy states of the optical amplifier that produce gain are subject to the same fundamental fluctuations as the lightwave signal. The mean energy per mode for the amplifier fluctuations is then also one half a photon. These fluctuations are independent of the fluctuations in the incident signal. The total noise from the combination of the signal and the amplifier is then one photon per mode referred to the input of the amplifier with a mean energy E = h f. Adding this noise to the input signal, the quantum-optics form of the gain equation listed in (2) for the mean number of signal photons in a mode, n s, becomes d n s dz = γ int ( ns + 1 ), (4) where both the signal and the input noise are amplified by the internal gain. When there is loss, both the signal and the incident noise are attenuated. This loss produces new fluctuations within the amplifier. These fluctuations are fundamental since any system that dissipates energy must also produce fluctuations. These new fluctuations are independent of both the 3

4 signal and the incident noise and thus add to the mean noise energy per mode. The net result is that the the noise is unchanged while the signal is attenuated according to d n s = α n s. dz Adding this loss term to (4), we obtain d n s dz = γ n s + γint, where γ = γ int α is the net gain defined in (1). Let n in be the mean number of photons at the input to the amplifier. The mean number of photons after amplification is ns = G nin +, where = γ int (G 1) γ = n sp (G 1) (5) is the mean number of photons per mode in a single polarization generated by amplified spontaneous emission. This term is then the photon noise density spectrum at the output of the amplifier. The corresponding power density spectrum N sp ( f ), is scaled by the energy of the photon h f The noise power is given by N sp ( f ) = h f n sp (G 1). (6) P ase = ˆ The term n sp is the spontaneous emission factor defined by 0 N sp ( f )d f. (7) n sp. = γ int γ = = γ int γ int α σ e N 2, (8) σ e N 2 σ a N 1 where N 1 and N 2 are constants independent of length so that n sp is a constant. The case when N 1 and N 2 have a spatial dependence will be considered later in this section. If n sp is independent of the input lightwave power to the optical amplifier, then the spontaneous emission is independent of the signal as is the case for thermal noise. When σ e N 2 σ a N 1, then the net gain γ approaches the internal gain γ int. The spontaneous emission factor n sp then approaches unity. When the gain is high and n sp = 1, then the output spontaneous noise power density spectrum becomes N sp ( f ) h f G. This power density spectrum corresponds to the equivalent of one spontaneously emitted photon per mode at the input to the optical amplifier. This is the minimum amount of spontaneous emission that can be generated by the amplifier and is the quantum noise limit. This minimum amount of noise has an equal contribution from the fundamental fluctuations in the input lightwave signal and the internal energy states that provide gain within the amplifier. 4

5 Lightwave Amplifier Statistics The noise statistics for the amplified lightwave signal are determined by assuming that the lightwave power at the output of the amplifier can be represented as a filtered passband signal as shown in Figure 2. The lightwave signal power before the optical sensor is then P(t) = 1 2 s(t) + n(t) 2, where s(t) = 2P s (t) and n(t) = 2P n (t)e jφ(t) are the slowly-varying lightwave signal and noise respectively. 1 Optical Amplifier Optical Filter Square-law Sensor Figure 2: Detection of an amplified lightwave signal using an optical filter with passband bandwidth B o. The characteristics of the optical filter are assumed not to affect the signal s(t) as long as the bandwidth of the filter is large with respect to the bandwidth of the signal. The noise power density spectrum before the filter is approximately the value at the carrier frequency f c so that N sp = N sp ( f c ) and the noise power density spectrum has a constant value. The noise power after the filter can then be written as P ase = B o N sp, where B o is the passband noise bandwidth. To determine the distribution for the amplified lightwave power at a time t, we define z = 2P = s + n 2. The distribution of the random variable z is a noncentral chi-square distribution with two degrees of freedom given by [ f z (z, A) = 1 2σ 2 exp z + ] ( ) A2 A z 2σ 2 I 0 σ 2 z 0 (9) 1 If rms amplitudes are used, the the factor of 2 is not included. 5

6 2 where A 2 = x 2 + y is the mean value. This distribution is a noncentral chi-square distribution with two degrees of freedom. The mean and variance are z = A 2 + 2σ 2 σ 2 z = 4σ 4 + 4σ 2 A 2. (10) The two degrees of freedom are the independent noise amplitudes of the in-phase and quadrature noise components. This distribution can be interpreted as the power in the sum of a constant signal vector s, and a circularly symmetric random noise vector n shown in Figure 3 Many independent contributions Figure 3: The limit of the sum of many independent random vectors superimposed on a constant vector is a Ricean distribution. Using (9), we assign z = 2P, A 2 = 2P s, and σ 2 = P ase. Using these assignments, and f z (z)dz = f P (P)dP, the distribution for the lightwave signal power is f P (P) = 1 [ exp P + P ] ( ) s 2 Ps P I 0 P 0, (11) P ase P ase P ase where P ase = B o N sp is the mean noise power per quadrature component, and P s = 1 2 s 2 = GP in is the amplified power expressed in terms of the mean input signal power, P in, and the amplifier gain G. The mean and the variance for the lightwave power can be determined from (10). Substituting z = 2 P, A 2 = 2P s and σz 2 = P ase = 4σP 2, the mean and variance of the amplified lightwave power are P = P s + P ase (12) σ 2 P = 2P ase P s + P 2 ase. (13) When P s = 0, the distribution reduces to a central chi-square distribution with two degrees of freedom or, equivalently, an exponential distribution with mean P ase and variance P 2 ase. This part of the variance, which does not depend on the signal, is generated from the amplified spontaneous emission mixing or beating with itself and is called spontaneous-spontaneous emission noise. The second part of the variance is a signal dependent noise term that is generated by the amplified signal mixing with the amplified noise. This term is called signal-spontaneous emission noise. When shot noise is included, the variance is modified to σp 2 = P s + P }{{ ase } photon noise + 2P s P }{{ ase } + Pase 2 signal-ase noise ASE-ASE noise (14) where P s + P ase are the additional fluctuations from shot noise. 6

7 3 Output Spectrum The power density spectrum, S y ( f ), of the amplified signal includes both a component from the amplifier and a component from shot noise S y ( f ) = [ er P + R 2 S P ( f ) ] H( f ) 2, (15) where P is the mean lightwave power that produces shot noise, and S P ( f ) is the spectrum of the lightwave power fluctuations generated by spontaneous emission. To determine S y ( f ), we require S P ( f ). This spectrum is the Fourier transform of the power correlation function. Writing the power using rms amplitudes so that P(t) = s(t) + n(t) 2, the power correlation function is R P (τ) = P(t)P(t + τ) = s(t) + n(t) 2 s(t + τ) + n(t + τ) 2, (16) where we allow the signal to be a stationary random process. We now expand the right-hand side of the expression using the fact that the signal s(t) and the noise n(t) are uncorrelated and stationary. Out of the sixteen terms in the expansion, only six are nonzero and may be grouped as follows R P (τ) = 2 P s Pase + R Ps (τ) + 2Re {R s (τ)r n (τ)} + R Pn (τ) (17) where R Ps (τ) and R Pn (τ) are the autocorrelation functions for the signal power and noise power, respectively, and R s (τ) and R n (τ) are the autocorrelation functions for the signal and noise, respectively. The power correlation function, R Pn (τ), for a gaussian noise process can be expressed in terms of the noise autocorrelation function R n (τ). Substituting R Pn (τ) = Pase (1 2 + r n (τ) 2) into (17), we have R P (τ) = 2 P s Pase + R Ps (τ) + 2Re {R s (τ)r n (τ)} + P 2 ase + R n (τ) 2, where R n (τ) = P ase r n (τ). For typical systems, the signal is approximately constant over the width of the noise autocorrelation function R n (τ). The lightwave power correlation function listed in (??) can then be written as R P (τ) = P 2 mean signal +2 2 P s Re {Rn (τ)} signal-noise beat + R n (τ) 2. noise-noise beat where P 2 = ( P s + Pase ) 2, R s (τ)r n (τ) R s (0)R n (τ) = P s Rn (τ), and R Ps (τ) R Ps (0) = Ps 2. The power correlation function consists of three terms. The first term, P 2 = ( P s + Pase ) 2 is the square of the mean lightwave power P. The second term is the beating of the signal with the spontaneous emission, and the last term is the spontaneous emission beating with itself. The density spectrum of the lightwave power fluctuations is the Fourier transform of (18) S P ( f ) = P 2 δ( f ) + P s [Sn ( f ) + S n( f )] + S n ( f ) S n( f ), where 2Re {R n (τ)} = (R n (τ) + R n(τ)) has been used. If the optical filtering is symmetric with respect to the carrier frequency, f c, and the spontaneous emission power density spectrum N sp, is constant over the bandwidth of the filter, then R n (τ) is a real, even function, S n ( f ) = S n( f ), and (18) S P ( f ) = P 2 δ( f ) + 2 P s Sn ( f ) + S n ( f ) S n ( f ). (19) Given this expression, the power density spectrum for the output signal y(t) can be determined using (15). We illustrate by an example. 7

8 Example Let S n ( f ) = N sp over an optical passband bandwidth B o so that S n ( f ) = N sp rect( f /B o ). We assume that the amplified noise is unpolarized, and that the both polarization components are sensed. The spontaneous-spontaneous beat noise term becomes S n ( f ) S n ( f ) = 2N 2 sp (1 f /B o ) for f B o = 0 (otherwise) where the factor of two accounts for twice as much noise being passed through the filter for both polarizations. The density spectrum for the lightwave power fluctuations is then S P ( f ) = P 2 δ( f ) + 2 P s Nsp rect( f /B o ) + 2N 2 sp (1 f /B o ), (20) where we assume that the signal is polarized and thus only beats with one polarization component of the noise. Substituting (20) into (15), we obtain the power density spectrum S y ( f ), of the filtered output signal y(t) S y ( f ) = R 2 P 2 H(0) 2 δ( f ) mean signal +R 2 2P sn sp rect( f /B o ) sig.-spont. noise + er P H( f ) 2 shot noise + 2N 2 sp (1 f /B o ) spont.-spont.-noise H( f ) 2 (21) where H( f ) is the electrical frequency response of the system with a corresponding noise bandwidth B N. Plots of all four terms are shown in Figure 4 for B N < B o. spont. -spont. noise mean signal shot noise sig.-spont. noise f c f Figure 4: The four terms in the current noise density spectrum S y ( f ) of an optically amplified signal where both the optical filter and the electrical filter are ideal low pass filters, and the electrical filter bandwidth B N is less than the optical passband width B o. The electrical power per unit resistance R is determined by integrating (21) P e = ˆ S y ( f )d f = (R [ ]) 2 P 2 H(0) 2 + 2eR P B N + R 2 4P s N sp B N + 4NspB 2 o B N where H( f ) 2 d f = 2B N. 8 (22)

9 Total Noise Noise dbm Hz Sig Sp Noise Sp Sp Noise Thermal Noise Total Shot Noise Amplified Signal Power dbm Figure 5: Noise terms for an optical amplifier at λ = 1.55 µm, for: G 0 = 30 db, n sp = 1, an electrical noise figure F N = 5 db, B o = 0.1 nm, and an output resistance of R = 50Ω. A plot of the noise power density spectrum for each of the three noise terms listed (21) as a function of the amplified signal power P s along with the thermal noise power density spectrum is shown in Figure 5 for values typical of an erbium-doped fiber amplifier. The signal-spontaneous term dominates for large signal levels since it increases linearly with signal level. When the signal dependent noise terms dominate, the signal shot noise term is approximately a factor of 2G less than the signal-spontaneous noise term. For the values used in Figure (5), this corresponds to an offset of a constant offset 33 db. The thermal noise and spontaneous-spontaneous noise are independent of the signal power and are not significant for output signal powers greater than about 1 mw (0 dbm) for the values used in the figure. 4 Noise Figure For historical reasons, the most common definition of the noise figure for optical amplifiers has been the ratio of the input signal-to-noise ratio to the output signal-to-noise ratio where the SNR is defined using the photon number n as the signal. This definition of the SNR is nonstandard with the signal power being equal to the square of the mean number of photons, and the noise power being equal to the variance of the photon number distribution. The SNR using these definitions may be determined by re-scaling the variance listed in by 1/eη so that the distribution for the number of sensed photons m (in units of charge) is then equal to the distribution for the number of incident photons n. The photon noise term is then. If the incident lightwave signal power is constant, then the distribution for the number of photons incident to the amplifier over a time T is Poisson with the mean equaling the variance. The mean value at the input is then n s /G where n s is the mean number of the photons at the output, and G is the amplifier gain. Equating the variance to the mean the definition of the input SNR using photon number becomes SNR in = n2 σ 2 n = (n s/g) 2 n s /G = n s/g. 9

10 In a similar fashion, SNR out = n s 2 /σ 2 n where σ 2 n is the photon number variance after amplification. The noise figure listed is then F NP = SNR in SNR out = n s/g n s 2 /σ 2 n ( ) = n s + 2n s Gn s ( ) n = sp (23) G Gn s where is the number of spontaneously emitted photons per mode at the output of the amplifier. If the signals are measured with a optical spectrum analyzer with a resolution bandwidth B res = 1/2T where T is the observation time, then the relationship between the number of photons and the optical noise power is Similarly, for the signal we have = P aset h f n s = = P ase 2B res h f. (24) GP in 2B res h f. (25) These expressions can be used to determine the noise figure from measured quantities in the lab. The notation F NP is used to indicate that the noise figure is defined in terms of a nonstandard SNR. The definition in (23) is inconsistent with the definition of noise figure defined in for lower frequency systems. That definition used the signal s(t) to determine the SNR, and not a term that is proportional to s(t) 2. As a consequence, using (23) will produce different results than using. Specifically, the last term in (23), which is a combination of the shot noise generated by spontaneous emission and the beating of the spontaneous emission with itself, is signal dependent because of the term n s in the denominator. The cascading property of noise figure is then no longer valid over all operating regimes since the noise is not additive in the photon number n. This inconsistency is a direct consequence of the fact that there is no scaling factor to convert the SNR using the photon number n into a SNR using the lightwave signal s(t). A noise figure that is consistent with lower frequency systems can be defined using power density spectra F N ( f ) = 1 + N [ 1 a G h f nsp (G 1) ] = 1 + = 1 + n sp (1 G 1) (26) N in h f where N a ( f ) is the added amplifier noise power density spectrum given in (6). This spectrum is referred to the input of the amplifier by dividing by G. The term N in ( f ) = h f is the effective input noise power density spectrum given in (4). This is the minimum input noise and corresponds to one photon per mode. For amplifiers with a large gain, the noise figure is F N 1 + n sp where n sp is the spontaneous emission noise factor. When the internal gain of the amplifier is large with respect to the loss, then referring to (8), the spontaneous emission noise factor approaches unity and F N approaches a limiting value of 2. This is the quantum-limited noise figure for a phase insensitive optical amplifier. To date, the inconsistency between (23) and (26) has been mitigated because many lightwave communication systems that use optical amplifiers are operated in a high OSNR regime where n s so that only the the signal-spontaneous beat noise and the signal shot noise are significant. These terms dominate at large gains as can be seen in Figure 5. For these operating 10

11 conditions, the last term in (23) can be neglected. The noise figure based on photon number quantities then becomes F NP n s + 2n s = 1 Gn s G + 2n sp(g 1). (27) G where (5) has been used. This limiting form of the noise figure is independent of the incident lightwave signal n s. For amplifiers with high gain, the limiting form for the noise figure is F NP 2n sp. If the spontaneous emission noise factor approaches unity, then F NP has the same limiting value of 2 as the definition listed in (26) based on the lightwave signal. Therefore, we will use the term F N for either definition recognizing the fundamental inconsistency in the definition listed (23) based on photon number. Using the form for F N listed in (26), the spontaneous emission noise factor may be written as n sp = G(F N 1)/(G 1) and the spontaneous noise power density spectrum N sp listed in (6) can be written as N sp ( f ) = h f (F N 1)G. (28) When F N = 2, which is the quantum limit for a phase insensitive amplifier, N sp ( f ) = h f G and the amplifier adds the equivalent of one photon with energy h f at the input to the amplifier for each noise mode in the system. This fundamental limit will be used to assess the performance of amplified noncoherent lightwave systems with respect to phase synchronous systems limited by photon noise. Spatial Effects The preceding analysis assumed no spatial dependence for the number of energy states per unit volume. In practical optical amplifiers constructed using doped optical fiber, the number densities of the energy states are not constant in space because this would require the external energy source or pump to be constant over the entire length of the amplifier. For most fiber amplifiers, the pump is injected from one side of the amplifier. In order for the densities to be constant, very little of the pump can be absorbed and this leads to inefficient amplification. Accounting for the spatial dependence of the depletion of the pump requires numerical techniques based on a z-dependent form of gain equation listed in (3). The solution depends on both the pump power and the incident signal power and yields z-dependent densities N 1,2 (z). Space-averaged quantities are then defined by averaging over the length of the amplifier. The fact that the space-averaged densities N 1,2 depend on both the pump power and the incident signal power means that the space-averaged spontaneous emission power density spectrum N sp and the noise figure N F also depend on both the signal and the pump power. For a fixed pump power and amplifier length, increasing the signal power increases the probability that an excited energy state will relax via stimulated emission relative to relaxing via spontaneous emission. This reduces the spontaneous emission and results in low-noise operation. Therefore, practical optical amplifiers are operated at a minimum input signal power to produce a minimum amount of spontaneous emission. If the amplifier operates at or above this minimum signal power, the spontaneous emission can be approximated by an AWGN source. For input signal powers below this value, the spontaneous emission power density spectrum, becomes signal dependent as will be shown in lab. 11

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