# Optical amplification

Curator and Contributors

1.00 - Emmanuel Desurvire

## Introduction

There exists several physical processes to achieve optical amplification, namely the action of enhancing or boosting the power of an electromagnetic (EM) light wave. These processes can be grouped into three fundamental categories: laser, scattering (Raman and Brillouin) and parametric. The first is related to the effect of linear polarization due to first-order atomic susceptibility, $$\chi_{at}^{(1)}\ ,$$ in the optical medium. The second two processes are based upon the effect of nonlinear polarization due to second- and third-order susceptibilities $$\chi^{(2)}$$ and $$\chi^{(3)}\ .$$ Here, we shall only be concerned by the first case, i.e. the process of optical amplification in laser media. Parametric amplification should be treated elsewhere and separately under this series, owing to its quite different physics, multiple signalling, materials/tensor characteristics, frequency-agility and noiseless processing, which hold high potential and the promise of future innovations. Consider here that laser amplifiers (not just lasers as light sources) made possible so many applications in physical science, telecoms, and industry, and therefore, that they should be highlighted as a primary field in graduate education.

As well-known to physicists, the two fundamental principles behind the laser are that of light amplification and stimulated and spontaneous emissions. As we shall see, the first principle can be simply described by the so-called semi-classical arguments. But the second principle requires quantum-theory arguments. For any engineering purposes, one might be satisfied by the first level of explanation. The more demanding may wish for the deeper explanation offered at the second level -- but may not wish either being overwhelmed by too many academic details or sophistication. This paper attempts to summarize the basics at both expectation levels, while introducing to, and highlighting, the beautiful simplicity of the higher quantum approach.

## A simple model for light amplification

Classically, light amplification refers to a process according to which the amplitude $$E$$ of the light’s EM field traversing a medium may be enhanced by a certain factor $$\sqrt{G}\ ,$$ with $$G>1$$ being the condition for enhancement. Thus, upon traversing said medium, the incoming light power $$P=|E|^2$$ is boosted by a factor $$G\ ,$$ referred to as the medium gain, yielding $$P'=G\,P$$ as the outcoming light power. So far, this light amplification concept is strictly equivalent to that of light attenuation in an optical medium, whereby the EM field amplitude is decreased by a factor $$\sqrt{T}\ ,$$ with $$T<1\ ,$$ and $$T$$ being referred to as the medium attenuation, also called transmission.

Attenuation is physically intuitive : it can be attributed to the effect of light absorption by the medium constituents (molecules, atoms, electrons..) and back transformation into heat, to the effect of scattering, and other causes. We may thus assume that the fraction of power $$dP$$ which is captured from the EM wave by an elementary slice of medium of thickness $$dz$$ is simply $$dP=-\alpha\,P\,dz\ ,$$ where $$\alpha$$ is the medium’s attenuation/absorption coefficient. Amplification is not so intuitive, but the previous result lead one to infer, similarly, that the fraction of power $$dP$$ which may be transferred to the EM wave from an elementary slice of medium is $$dP=g\,P\,dz\ ,$$ where $$g$$ is the medium’s gain coefficient. We may refine the model by assuming that the medium is “doped” with certain atoms capable to store energy (as supplied by an external source or excitation mechanism), and to release it in presence of the EM wave. Assuming that the density of such atoms be $$N^*$$ [$$m^{-3}$$ ], we thus have $$g=\sigma\,N^*\ ,$$ where $$\sigma$$[ $$m^{-2}$$] is referred to as the gain cross-section.

Summing up the above power differentials and integrating from $$z=0$$ to $$z=L\ ,$$ we obtain

$\tag{1} \frac{P(L)}{P(0)}=exp [\int_0^L (\sigma\,N^*-\alpha ) \,dz ] = e^{g\,L}\,e^{-\alpha\,L}\equiv G\,T$

In the above, the second equality stands in the simplest case where the coefficients are constant along the EM wave path. It was also assumed that the coefficients $$g$$ and $$\alpha$$ are power-independent, i.e. not a function of $$P\ .$$ Concerning the gain coefficient $$g\ ,$$ this is referred to as the small-signal approximation (see below for the general case.). It is seen from the result that the EM wave is amplified under the simple condition $$G\,T>1\ ,$$ meaning that the gain mechanism provides a net energy transfer to the signal over any internal loss.

To complete the model, we may also introduce the effect of the EM beam shape (e.g. a free-space or a guided laser beam), with normalized power surface envelope $$\psi(x,y,z)$$ [$$m^{-2}$$ ] (with transverse profile and longitudinal variation, e.g. a Gaussian beam), as well as of the transverse density-distribution of matter, to obtain the most general expression :

$\tag{2} \frac{P(L)}{P(0)}=exp\left(\int_0^L \int\int_{\infty} \psi (x,y,z) \left(\sigma\,N^*(x,y,z)-\alpha(x,y,z)\right) \,dx\,dy\,dz\right)$

In the most general case, the gain coefficient $$g$$ should be power-dependent, so as express the effect of gain saturation, whereby a substantial fraction of the available medium energy is transferred to the EM wave. Let $$g(P)\equiv\frac{g_0}{1+\frac{P}{P_{sat}}}\ ,$$ where $$P_{sat}$$ is the power threshold for which the gain coefficient is half that of the small-signal regime, i.e. $$g(P_{sat})=\frac{g_0}{2}\ .$$ The differential equation $$dP=P\,(g(P)-\alpha)\,dz$$ must be solved numerically – not really an issue nowadays, especially considering that some numerical treatment is already involved after equation (2) to take into account the transverse dimension. In the low-loss approximation, $$\alpha << g_0\ ,$$ it is yet worth mentioning the “textbook” solution, which is straightforward :

$\tag{3} \log \frac{P(L)}{P(0)} + \frac{P(L)-P(0)}{P_{sat}} \equiv g_0 \,L$

In the high-signal power, or deep-saturation regime, $$\frac{P(L)}{P(0)}>>1\ ,$$ the ‘log’ term may be discarded and we see that, with respect to length, the amplifier power enhancement behaves linearly, according to $$P(L)=P(0)+P_{sat}\,g_0\,L\ ,$$ as opposed to exponentially, according to (1).

Equation (3) is of the type $$f(G)=P_0\ ,$$ where $$f$$ is a transcendental function, i.e. given the input signal power, $$P_0\ ,$$ the gain $$G=f^{-1}(P_0)$$can only be found by numerically solving the reciprocal $$f^{-1}\ .$$ One may alleviate this computational difficulty through the following. Define $$G_{ss} = \exp (g_0\,L)$$ as the small-signal gain, i.e. the signal-invariant gain obtained when $$\frac {P_0}{P_{sat}} << 1\ .$$ Using $$G= \frac{P(L)}{P_0}\ ,$$ we obtain from equation () :

$\tag{4} \frac{1}{G-1}\, \log \frac{G_{ss}}{G} = \frac{P_0}{P_{sat}}$

Equation (4) is of the type $$f(G,G_{ss}) = \frac{P_0}{P_{sat}}\ .$$ This function can be plotted with $$G$$ as the variable, for different values of the small-signal gain $$G_{ss}\ ,$$ as illustrated in Figure 1. It is customary to use the decibel scale, according to $$X(dB) = 10 \log_{10} X\ ,$$ where $$X=G$$ or $$X=\frac{P_0}{P_{sat}}\ .$$ Hence, gains of $$10\ ,$$ $$100$$ and $$1000\ ,$$ correspond to $$G(dB) = 10\,dB\ ,$$ $$G(dB) = 20\,dB\ ,$$ and $$G(dB) = 30\,dB\ ,$$ respectively.

As the figure shows, there exist three amplification regimes :

• the small-signal regime, where the gain is signal-independent, or $$G\simeq G_{ss}\ ;$$
• the saturation regime, where the gain linearly decreases with the input signal power;
• the deep saturation regime, where the amplifier is “useless”, with no gain or $$G\simeq 1\ .$$
Figure 1: Optical amplifier gain $$G$$ as a function of relative input signal power $$\frac{P_0}{P_{sat}}\ ,$$ for different values of the small-signal gain $$G_{ss}\ .$$

One may operate in the small-signal regime in order to maximize the gain; this is the case of optical preamplifiers, where inputs signals are weak. Alternatively, one may operate in the saturation regime, in order to maximize the signal output power; this is the case of optical power amplifiers, also called post-amplifiers or power boosters, where input signals are strong. In optical telecommunications, for instance, post-amplifiers are placed just after the optical transmitter (hence the name), in order to launch high-power signals into the optical fibre ; and pre-amplifiers are places just before the optical receiver, in order to raise the incoming signals from weak to high, prior to photodetection.

## Absorption, stimulated and spontaneous emission

The foregoing provided a flavor about basic optical amplification notions, as well as a first level of engineering worth. Yet, the explanation is clearly incomplete, at least for two unanswered questions : (a) can matter recapture the transferred energy ? and (b) can a fraction of the stored energy be randomly released, independently from EM power ? In a 1917 paper Einstein 1917, Albert Einstein addressed the above issues by elegantly combining the principles of quantization of energy in both atomic matter and EM radiation. The following describes Einstein’s key findings.

Referring to Figure 1, we assume that atoms may dwell into two possible energy “states” or “levels”, namely : a lower state at energy $$E_1\ ,$$ and a higher “excited” state at energy $$E_2>E_1\ ,$$ with the energy-level gap $$\Delta E = E_2-E_1\ .$$ Second, consider that to the EM field at frequency $$\nu\ ,$$ are associated light quanta, or photons, with energy $$h\nu\ ,$$ where $$h$$ is Planck’s constant. Under the resonance condition $$h\nu \simeq \Delta E\ ,$$ an atom in the lower state may capture an incident photon, moving from the lower to the excited state. This is referred to as photon absorption. If the atom is already in the excited state, the incident photon may induce it to return to the lower state, thereby releasing a photon of strictly identical characteristics (i.e. associated with EM having same frequency, phase, polarization and direction). Such a de-excitation process, is referred to as stimulated emission. Since the photons are identical, the EM wave builds up coherently, i.e. through constructive field interference. Alternatively, atoms in the excited state may spontaneously return to the lower state, thereby releasing a photon of random characteristics. This is referred to as spontaneous emission. These three processes thus govern the dynamics of atom-light interaction, causing the EM power to evolve as it propagates through the medium.

Figure 2: (top) two-level atomic energy structure with energy gap $$\Delta E = h\upsilon \ ,$$ and principle of resonant photon absorption (orange); or emission through either stimulated (blue) or spontaneous (purple) processes; (bottom) detail of the photon emission process, showing that stimulated emission produces photons identical to the incident ones, while spontaneous emission produces photons with random characteristics.

Figure 1 : (top) two-level atomic energy structure with energy gap $$\Delta E = h\nu\ ,$$ and principle of resonant photon absorption (orange); or emission through either stimulated (blue) or spontaneous (purple) processes; (bottom) detail of the photon emission process, showing that stimulated emission produces photons identical to the incident ones, while spontaneous emission produces photons with random characteristics.

Let now formalize this interaction process. Assume two atom groups to have densities $$N_1$$ and $$N_2\ ,$$ respectively. In the time interval $$dt\ :$$

• photon absorption causes the changes $$dN_1 = -B_{12}\,N_1\,dt$$ and $$dN_2 = B_{12}\,N_1\,dt$$
• stimulated emission causes the changes $$dN_1 = B_{21}\,N_2\,dt$$ and $$dN_2 = -B_{21}\,N_2\,dt\ .$$
• spontaneous emission causes changes $$dN_1 = A_{21}\,N_2\,dt$$ and $$dN_2 = -A_{21}\,N_2\,dt$$

In the above, $$B_{12},B_{21}$$ and $$A_{21}$$ are called Einstein’ coefficients and have the dimension [$$s^{-1}$$]. Einstein showed that the first two must be equal, i.e. $$B_{12}=B_{21}\ .$$ Combining the differentials, we obtain :

$\tag{5} \frac{dN_1}{dt} = -\frac{dN_2}{dt}=B_{21}\,(N_2-N_1)+A_{21}\,N_2$

Under the change $$-dN_2\ ,$$ the EM wave experiences a net power change (per unit volume) $$dP^* = -h\,\nu\,dN_2\ .$$ Hence, if $$-dN_2>0\ ,$$ the EM power increases. If we overlook the spontaneous contribution, we see from the above result that such a condition requires $$\Delta N = N_2-N_1 > 0\ ,$$ i.e. there must be more atoms in the excited state than in the lower state. This situation is referred to as population inversion.

The coefficient $$B_{21}$$ must be proportional to the photon fluence i.e. the number of incident photons per unit time and surface, namely $$n=\frac{I}{h\nu}\ ,$$ where $$I$$ is the EM intensity. Thus, let $$B_{21}=n\sigma_e \equiv \frac{n\,I}{h\nu}\ ,$$ where $$\sigma_e$$ is called the emission cross-section [$$m^2$$]. Replacing this definition into $$dP^* = h\nu\,B_{21}\,\Delta N$$ and integrating over the plane transverse to the EM wave direction, we obtain the change of power with unit length :

$\tag{6} \frac{dP}{dz} = \int \int_\infty dP^*\,dS = P\sigma_e \int\int_\infty \psi(x,y,z) \,\Delta N (x,y,z)\,ds\equiv P\sigma_e\,\Delta N$

where the last equality stands in the case where the population inversion is uniformly distributed. Integrating over the longitudinal direction, we find that the EM power exponentially increases according to length $$L$$ with gain $$\frac{P(L)}{P(0)}\equiv \exp (\sigma_e\,\Delta \overline{N}\,L)\equiv G\ ,$$ where is the path-averaged population inversion, $$\Delta \overline{N} = \frac{1}{L}\int \Delta N\,dz\ .$$

## Optical pumping

Einstein’s original Quantum theory of radiation has provided a most fundamental explanation for light/matter interaction, leading to the principle of EM amplification through the process of “coherent” photon multiplication. Yet, a problematic issue had to be solved, namely how to achieve the condition of “population inversion”. Indeed, such a condition fully departs from that of thermal equilibrium. This latter is described by Boltzmann’s law, according to which, for two atomic levels of energies $$E_1, E_2>E_1\ ,$$ the corresponding populations must be in the ratio

$\tag{7} \frac{N_2}{N_1}=\exp\left(-\frac{\Delta E}{k_B\,T}\right)$

where $$T$$ is the absolute temperature and $$k_B$$ is Boltzmann’s constant. It is a well-known feature that, at optical frequencies, the photon energy $$h\nu \simeq \Delta E$$ is several orders of magnitude greater than the phonon energy, $$k_B T\ ,$$ thus $$\frac{N_2}{N_1}<<1$$ and $$\Delta N \simeq -N_1\ ,$$ which hopelessly forbids any amplification possibility ! In order to obtain the condition of interest $$N_{2}/N_{1} \ge 1\ ,$$ we need a clever mechanism capable to significantly “populate” the upper level, and possibly so, to significantly “depopulate” the lower one. Such a mechanism is referred to as optical pumping.

The basic concept of optical pumping is to use light absorption to put atoms in the excited state, just as one would fill in a water tower (hence the expression “pumping”). To be efficient, the water (pump photon energy) must be trapped into the tank, without leaking back through the feeding channel (through spontaneous and stimulated emission). This pumping process is illustrated in Figure 2 (top). Ideally, one needs to have a four-level atomic system. The level of lowest energy, referred to as the ground level, is where most atoms may be found, according to Boltzmann’s law. If the atoms are illuminated by a pump EM-wave at frequency $$\nu_{p} = \frac{E_3-E_0}{h}\ ,$$ pump photon absorption occurs between the ground-level “0” and some upper level “3”. According to Einstein, there must be a “pump” stimulated-emission process forcing atomic de-excitation at some rate $$B_{30}$$ (the water leak). But this effect can be alleviated if the atom is able to spontaneously de-excite to level 2 with a much higher rate $$A_{32}\ ,$$ such that $$A_{32}>>B_{30}\ .$$ The cost to pay is some energy loss $$\Delta E' = E_3 - E_2\ ,$$ but the most important is that we now have a mechanism to populate level 2, and hence increase the population $$N_2\ .$$

Atoms thus “pumped” up to the excited level 2 may then spontaneously de-excite to level 1, at a rate $$A_{21} = \frac{1}{\tau_{21}}\ .$$ The parameter is referred to as the atomic excited-state lifetime. Atoms may also de-excite by stimulated emission, at a rate $$B_{21}\ ,$$ when traversed by an EM wave of frequency $$\nu = \frac{E_2-E_1}{h}\ .$$ In order to optimize stimulated emission, one needs atoms having a relatively long excited-state lifetime, such that the condition $$A_{21}<<B_{21}$$ be fulfilled. Finally, atoms having relaxed at lower-level 1 may spontaneously decay to ground-level 0, at some rate $$A_{10}=\frac{1}{\tau_{10}}\ .$$ Should the lower-level lifetime be relatively short, namely $$\tau_{10}<< \tau_{21}\ ,$$ it is clear that the population $$N_1$$ will remain small, if not negligible, in comparison to the population $$N_2\ .$$ Atoms having returned to the ground state are now available for another pumping cycle. If all the above conditions are met, it is clear that the most sought-after condition of population inversion,$$\Delta N = N_2-N_1 > 0\ ,$$ can finally be achieved ! The optimal operation regime is defined by $$\Delta N \simeq N_2 \equiv \rho\ ,$$ meaning that the lower-state population is negligible, and that all atoms (density $$\rho$$) participate in the amplification process.

Figure 3: (top) four-level laser system, showing pump photon absorption (red) from ground-level to level 3, followed by fast energy relaxation to level 2 (grey-dashed), spontaneous/stimulated emission, and fast relaxation from level 1 to ground level (grey-dashed); (bottom) two possible configurations of three-level laser systems.

The above-described principle does not necessarily require four-level atomic systems. Figure 2 (bottom) shows two possible implementations three-level systems. The key difference between the two is whether or not the ground state is the lower-state 1 of the laser transition. If this is not the case, the behaviour is the same as in a 4-level system. In the contrary, the lower-state 1 being “ground” is densely populated, with pump absorption as the only de-populating process. This is the real case of the so-called 3-level systems. Even at relatively high pump fluence, one may not reach the optimal condition $$N_1<< N_2\ .$$ But as we have seen, the regime of population inversion, $$N_2> N_1\ ,$$ whether or not optimal, is the only requirement for optical amplification.

In the case of semiconductor lasers, the model is very similar, except that the system basically reduces to two energy levels : the valence band (level 1) and the conduction band (level 2). Medium inversion is then achieved through electrical pumping. Such a process is the action of an electrical current to inject electrons into the conduction band. Photons are then emitted by stimulated or spontaneous electron-hole recombination. Apart from such conceptual differences, semiconductor lasers strictly respond to the same model as described here.

The introduction of optical pumping represents a first conceptual extension of Einstein’s two-level model of light/matter interaction. The rate-equation system (4) needs then to be completed by the various contributions from/to the different levels involved. The steady-state regime is reached for $$\frac{dN_1}{dt}=0$$ ($$i=1..4$$), leading to a time-invariant inversion parameter, $$\Delta N(x,y,z)\ .$$ Formulas to describe in either 4-level and 3-level laser systems, can be found in textbooks (see Saleh et al. 1991, Desurvire 1994, for instance).

## Cross-section lineshape

The concept of emission cross-section, which was rapidly introduced though the parameter $$\sigma_e\ ,$$ and how it relates to the medium’s physical characteristics (excited-state lifetime, broadening mechanisms, inhomogeneous saturation, temperature dependence, etc.), is a full topic which deserves a separate description. Let recall here only some basics regarding spectral dependence.

The key parameter $$\sigma_e$$ is the one that determines the resonance condition $$h\nu = \frac{hc}{\lambda}\simeq \Delta E\ .$$ This means that the optical amplification process may occur only at wavelengths substantially close to some value $$\lambda_{peak} = \frac{\Delta E}{hc}\ ,$$ and rapidly vanish at wavelengths away from this condition. Thus $$\sigma_e$$ has a spectral dependence, $$\sigma_e(\lambda)\ ,$$ with maximum value at $$\lambda=\lambda_{peak}$$ and a characteristic full-width at half maximum (FWHM), called $$\Delta \lambda_{FWHM}\ .$$ The semi-classical physics of lasers shows that

$\tag{8} \sigma_e(\lambda) = \sigma_{peak} L(\lambda_{peak},\Delta \lambda_{FWHM}) \equiv \sigma_{peak} \frac{1}{1+\left(\frac{\lambda-\lambda_{peak}}{\Delta \lambda_{FWHM}}\right)^2}$

where $$L$$ is a spectral “envelope” of Lorentzian shape. With such a definition, and excluding saturation effects, the gain coefficient of optical amplifiers has a Lorentzian profile, along with a characteristic bandwidth $$\Delta \lambda_{FWHM}\ .$$

In reality, however, laser systems are not this elementary. Most frequently, indeed, that the energy levels 1 and 2 are degenerate, i.e. not single but each forming a plurality of $$g_i$$ ($$i=1,2$$) equally-populated sublevels, referred to as Stark sublevels. The origin of degeneracy is the manifold of arrangements $$J=L+S$$ of the net orbital momentum of the atomic cloud altogether, yielding $$2J+1$$ energy states. The Stark effect, caused by the local, microscopic E-field of the crystalline atomic host, lifts the degeneracy, hence producing energy sublevels spaced by small energy differences $$\delta E_{ij}\ .$$ Because of the ultrafast thermal relaxation between such sublevels, nothing is changed in the above theory : sublevels yet behave collectively as a single groups of “low” and “high” states. If the levels are closely spaced, such that $$\delta E_{ij}<< \Delta E\ ,$$ one may consider that sublevel groups contribute to the absorption/emission processes according to their respective weights $$g1,g2\ ,$$ leading to the effective inversion $$\Delta N' = \frac{g2}{g1} \,N_2-N_1\ ,$$ and gain coefficient $$\sigma_e \,\Delta N'\ ,$$ as still found in most textbooks.

In the general case, however, the above approximation is wholly invalid. Indeed, if $$\delta E_{ij}$$ is of the order of $$k_B T\ ,$$ the population distribution within the Stark sublevels is non-uniform, according to Boltzmann’s law, according to which the sublevel population is exponentially distributed. Second, the split of energy sublevels generates as many different possibilities of absorption/emission events, with photons at energies $$h\nu_{jk} = \Delta E_{jk} = E_{2j}-E_{1k}\ .$$ This is illustrated in Figure 4, in the basic case of a three-level system with sublevel degeneracy. The resulting cross-section lineshapes for emission and absorption,$$\sigma_a(\lambda), \sigma_e(\lambda)\ ,$$ are also shown. We notice a “mirror” effect, which reflects the fact that each of the Stark sub-levels groups are exponentially populated.

Figure 4: (left) three-level laser system with sublevel degeneracies in levels 1 and 2, showing a plurality of photon absorption and emission processes; (right) resulting cross-section lineshapes for absorption, $$\sigma_a(\lambda)\ ,$$ and emission, $$\sigma_e(\lambda)\ ,$$ respectively.

Under the above conditions, light power may be amplified at a rate according to eq.(6), substituting however the new definition for the gain coefficient :

$\tag{9} \sigma_e\,\Delta N(\lambda) \rightarrow \sigma_e(\lambda)\,N_2-\sigma_a(\lambda)\,N_1$

As we have seen, the optimal amplification regime is that whereby the lower-level population $$N_1$$ is reduced to either negligible ($$N_1<<N_2$$), from fast decay to some level 0 (as in 4-level systems), or kept sufficiently small ($$N_1<N_2$$), under high pumping-rate conditions (as in 3-level systems).

Thus, Einstein’s founding concepts of absorption and stimulated emission, along with that of Boltzmann’s thermal equilibrium, as displaced by optical pumping to cause population inversion, are now reconciled in a unified way. The above model suffices to describe the process of optical amplification, whereby an incident EM wave may be powered by photon/atom interaction, a process that elegantly requires a “double quantization” principle.

## Amplified spontaneous emission

The analysis of light amplification by stimulated emission also requires taking into account Einstein’s hypothesis of spontaneous emission. As we have seen, spontaneous emission generates photons at the same frequency as the incident EM wave, but with random characteristics. It is thus expected that such photons contribute in turn to the stimulated emission process, resulting into the buildup of a noisy EM wave background. We shall refer to this latter as amplified spontaneous emission (ASE) noise. Analyzing ASE and related photon statistics require quantum-mechanical principles, as we shall describe in the next section. Because such an analysis is somewhat conceptually demanding, it is possible, for simplicity, to introduce ASE semi-classically, i.e. by modeling atomic spontaneous emission as the noise power associated with a single photon in some EM bandwidth bin $$\Delta \nu\ ,$$ i.e. $$P_0 = h\nu \Delta \nu\ .$$ This noise contributes to an EM power increase $$dP_0 = \sigma_e\,P_0\,N_2\,dz\ ,$$ which can be included in the R.H.S. of eq.(6) to yield

$\tag{10} \frac{dP}{dz} = \sigma_e\, (\Delta N\, P +N_2\,P_0)$

Overlooking transverse effects, basic integration yields the solution

$\tag{11} P(L)=G\,P(0)+\sigma_e\,G\,P(0)\,\int_0^L \frac{N_2(z)}{G(z)}\,dz\equiv G\,P(0) + P_{ASE}$

with $$G=G(L)$$ and $$G(z) = \sigma_e\,\int \Delta N (z) \, dz\ ,$$ as integrated up to coordinate $$z\ .$$ In the simplest case where the populations are coordinate-independent, the ASE power solution nicely reduces to :

$\tag{12} P_{ASE} = \frac{N_2}{N_2-N_1} \, (G-1) \,P_0 \equiv n_{sp} \,(G-1)\,P_0$

In the above, the coefficient $$n_{sp}\ ,$$ called the spontaneous emission factor, represents a measure of medium inversion. It asymptotically converges to unity as the inversion increases, i.e. $$n_{sp}\rightarrow 1$$ when $$N_1\rightarrow 0\ .$$ It is thus concluded that, in this full-inversion regime, the ASE is equivalent to the amplification of one single input photon with gain $$G-1\ .$$ Since the ASE polarization is by definition random, we must double this power to take into account the two orthogonal polarizations, i.e. effect the change $$P_0 \rightarrow 2\,P_0\ .$$

Discussion 6.1. Historically, the above results and the definition of the spontaneous emission factor $$n_{sp}$$ were first derived from the analysis of semiconductor amplifiers. This definition was then widely adopted for describing ASE in other amplifier types, such as fibre amplifiers.

But this was unfortunate, for a simple reason $n_{sp}$ is undefined when $$N_1=N_2\ ,$$ and is negative when $$N_1>N_2\ .$$ We have sought to solve this issue by proposing and advocating an alternative definition of $$n_{sp}$$ Desurvire 1994,Desurvire et al. 2002, which we called the equivalent input noise, and which is simply introduced through :

$\tag{13} P_{ASE} = \left( \frac{\sigma_e}{G} \int \frac{N_2}{G(z)}\,dz\right) GP_{0}\equiv \left[ \frac{N_2}{N_2-N_1}\,\left(1-\frac{1}{G}\right)\right]\,G\,P_0 \equiv n_{eq} \,G\,P_0$

In the above, the second equality stands for populations independent of coordinate $$z\ .$$ It is immediately apparent that $$n_{eq}$$ [the term in brackets] is always positive and finite in any amplification regime, while the two parameters converge ($$n_{eq}\simeq n_{sp}$$) at high gains $$G>>1\ .$$ But unlike $$n_{sp}\ ,$$ the new parameter $$n_{eq}$$ is gain-dependent, which holds further information about the ASE build-up regime. Figure 4 illustrates the previous conclusion with completeness. Before interpretation, consider that the medium inversion $$\Delta N$$ is only a “local”, most generally a coordinate-dependent parameter. Let then introduce :

- the atomic density $$\rho$$ [$$m^{-3}$$], such that $$N_1+N_2=\rho\ ;$$

- the maximum achievable gain, $$G_{max}\ ,$$ at full inversion ($$N_2=\rho$$), i.e. $$G=exp(\rho\,\sigma_e\,L)\ ;$$

- the effective inversion $$\Delta\overline{N} = \overline{N_2}-\overline{N_1}=\log G / \log G_{max}\ ,$$ which ranges from $$\Delta\overline{N}=-1$$ to $$\Delta\overline{N} = +1\ .$$

The figure then compares $$n_{eq} = f( \Delta\overline{N},G(\Delta\overline{N}))$$ and $$n_{sp} = \frac{\overline{N}_2}{\Delta\overline{N}}\ .$$ It is seen that at high inversions $$\Delta\overline{N} \ge 0.5\ ,$$ and at high gains ($$G_{max}>10$$), the two definitions, $$n_{eq},n_{sp}\ ,$$ rapidly converge. In other conditions, the parameter $$n_{sp}$$ is otherwise useless (diverging and/or negative), while $$n_{eq}$$ is valid in any amplification regime. Focusing upon $$n_{eq}\ ,$$ it is seen that ASE noise is minimized to $$n_{eq}=1\ ,$$ the one input photon limit, as inversion is maximized.

Figure 5: Spontaneous emission factor ($${n_{sp}}$$) and equivalent input noise ($${n_{eq}}$$) as functions of the effective medium inversion $$\Delta \bar N = {\bar N_2} - {\bar N_1}\ ;$$ the latter is plotted for different values of the maximum achievable gain $${G_{\max }}\ .$$

To complete, it should be emphasized that $$n_{eq}$$ comes out as a natural parameter in the definition of the amplifier noise Figure 4 Figure 4, namely $$F=2\,n_{eq}+\frac{1}{G}$$ (see next three sections), which has been adopted since as the standard telecom noise measure for optical amplifiers.

## Quantum theory of photon/atom interaction, and STT model for photon amplification

It is clear that the previous semi-classical description of ASE, Albeit accurate, is conceptually incomplete. Furthermore, it yields no information about noise statistics (variance, moments, probability-density function). Here, we shall analyze ASE based upon quantum theory. As we shall see, a most elegant feature of the theory is that it inherently “predicts” the effect of spontaneous emission, without need of any postulate.

Let first define :

• the EM-field (electrical component) operator as $$E=\varepsilon \,\left(a\,e^{-i\omega\,t}-a^{+}\,e^{i\omega\,t}\right)\ ,$$ where $$a,a^{+}$$ are the photon annihilation and creation operators, respectively, and $$\varepsilon$$ the E-field polarization vector ;
• the atomic-dipole operator as $$D = \epsilon'\,\left(|1\rangle\langle {2} |+ |2\rangle \langle 1 |\right)\ ,$$ where $$|1\rangle, |2\rangle$$ are the lower and higher atomic states, respectively ($$\langle i | j \rangle = \delta_{ij}$$), and $$\varepsilon'$$ the atomic-dipole vector orientation;
• the photon/atom interaction Hamiltonian as the operator scalar product $$H = E\cdot D\ .$$

In the above, $$|1\rangle\langle {2} |$$ and $$|2\rangle \langle 1 |$$ and are the lowering and raising operators, respectively. Indeed, applying for instance the first operator to the lower $$|1\rangle$$ and upper $$|2\rangle$$ states, we obtain $$|1\rangle\langle {2} | 1 \rangle = 0$$ (no action) and $$|1\rangle\langle {2} | 2 \rangle = |1\rangle$$(lowering action). Combining the above definitions, we obtain the Hamiltonian

$\tag{14} \begin{array}{lcl} H & = & \varepsilon\cdot \varepsilon'[ae^{-i\omega t}|2\rangle \langle 1|-a^{+}e^{i\omega t}|1\rangle \langle 2|+\{ ae^{-i\omega t}|1\rangle \langle 2|-a^{+}e^{i\omega t}|2\rangle \langle 1|\} ] \\ & \equiv & \varepsilon\cdot \varepsilon'[ae^{-i\omega t}|2\rangle \langle 1|-a^{+}e^{i\omega t}|1\rangle \langle 2|] \end{array}$

The first two terms correspond to photon absorption and emission events, respectively. The last two terms {in braces} corresponding to non-energy-conserving events (photon-absorption from upper-state and photon-creation from lower state) must be discarded, Loudon 1973.

Second, let introduce the photon/atom states, $$|i,n\rangle\ ,$$ where an atom in the atomic-state $$|i\rangle (i = 1,2)$$ interacts with an EM field associated with $$n$$ photons, or in the photon-state $$|n\rangle\ .$$ These states are orthogonal with the property $$\langle j,m|i,n \rangle = \delta_{ij}\delta_{mn}\sqrt{P_{n}}\ ,$$ where $$P_{n}$$ is the probability of finding the EM field in the $$|n\rangle$$ state. By definition, the action of the photon creation and annihilation operators is $$a^{+}|i,n\rangle=\sqrt{n+1}|i,n+1$$ and $$a|i,n\rangle=\sqrt{n}|i,n\rangle\ ,$$ respectively.

The photon/atom system may then evolve from the initial state $$|i,n\rangle$$ to the final state $$|j,m\rangle$$ according to some joint probability $$P_{jm;in}$$ can be calculated from the Hamiltonian matrix element:

$\tag{15} P_{jm;in}=|\langle j,m|H|i,n\rangle|^2$

It is a patient exercise to calculate the above probabilities by substituting the definition in eq.(14) into that of eq.(15). The four non-vanishing ones, whose values are shown in the figure, are found to be the following:

• for absorption events$P_{2,n-1;1,n}$ and $$P_{2,n;1,n+1}$$
• for emission events$P_{1,n;2,n-1}$ and $$P_{1,n+1;2,n}$$
Figure 6: Two possible absorption or emission events in photon/atom interaction, making the system evolve from state $$\left| {i,n} \right\rangle$$ to state $$\left| {j,m} \right\rangle$$ and associated probabilities $${P_{j,m;i,n}}\ .$$ The circle (°) at center stands for the electric-dipole Hamiltonian, $$H\ .$$

The result thus obtained makes possible to construct an equation describing the rate of change $$dP_n$$ between $$z$$ and $$z + dz\ .$$ Indeed, introducing the gain coefficients $$a = \sigma_{e}N_{2}$$ and $$b = \sigma_{a}N_{1}\ ,$$ we sum up the event contributions that increase or decrease the probability $$P_{n}$$ according to the following:

$\tag{16} \begin{array}{lcl} \frac {dP_{n}}{dz} & = & a({P_{1,n;2,n - 1}} - {P_{1,n + 1:2,n}}) - b({P_{2,n;1,n + 1}} - {P_{2,n - 1;1,n}}) \\ & \equiv & a\left\{ {n{P_{n - 1}} - (n + 1){P_n}} \right\} + b\left\{ {(n + 1){P_{n + 1}} - n{P_n}} \right\} \end{array}$

This master equation, which actually constitutes an infinite set of linearly-coupled O.D.E., constitutes the starting point of the seminal 1957 paper by K.Shimoda, H.Takahashi and C.H.Townes (STT) on the analysis of “fluctuations of quanta in maser amplifiersShimoda et al. 1957. The STT paper provided a full analytical solution for the discrete photon statistics $$P_{n}(z)$$ in different amplification regimes (see derivation details and discussion in Desurvire 1994, and Shimoda et al. 1957), a topic which remains beyond the scope this paper. Suffices it to state the key conclusion, according to which, (when input with coherent laser light), optical amplifiers degrade coherence, i.e. the output photon statistics is a combination of amplified-signal of degraded coherence together with incoherent thermal noise (see further below on this issue).

The most immediate feature of interest about the STT model concerns the derivation of the first two statistical moments, i.e. the photon-number mean, i.e. the photon-number mean, <$$n$$>, and mean-square, <$$n^{2}$$>. These can be quite simply extracted from the master equation (16) by effecting the summations ($$p = 1,2$$):

$\tag{17} \frac{ {d < {n^p} > }}{ {dz}} = \sum\limits_n { {n^p}} \frac{{d{P_n} } }{{dz} }$

which yield:

$\tag{18} \begin{cases} \frac {d<n>}{dz} = (a-b)<n> + a \\ \frac {d<n^2>}{dz} = 2(a-b)<n^{2}> + (3a + b)<n> + a \end{cases}$

In the above system, the first equation is the photon-mean equivalent to previous eq.(10) about EM power in section 6. It is clear that the first term in the R.H.S., $$(a-b)<n>\ ,$$ corresponds to the effect of stimulated emission (as proportional to the net gain coefficient and to the photon mean), and that the second term, $$a = a x 1\ ,$$ corresponds to the effect of spontaneous emission (as proportional to the emission coefficient and to the “mean power” of a single photon). Integration of this first equation yields

$\tag{19} < n > = G < {n_0} > + N$

where $$n_{0} = <n(z=0)>$$ is the input photon number, and $$N$$ is the mean ASE defined by

$\tag{20} N = G\int_L {\frac{ {a(z)}}{{G(z)} }dz}$

We have reached the same result as previously found in section 6: the amplifier output is the sum of the amplified input signal $$(G<n_{0}>)$$ and the ASE ($$N$$), here both with the meaning of mean photon numbers. To recall, with coordinate-independent coefficients $$a,b$$ we find $$N = n_{sp}(G-1)$$ with $$n_{sp} = a/(a-b)\ .$$ At full inversion, or in the ideal lossless/absorption-less case $$(b \rightarrow 0)\ ,$$ we have $$n_{sp} \rightarrow 1$$ and $$N = N_{ideal} = G-1\ ,$$ which represents the case of the “ideal” low-noise optical amplifier. Such a notion or “ideal amplifier” will be addressed again in Section 8.

A key difference introduced the STT model is the possibility to predict, not only the mean $$<n>\ ,$$ but the mean-square, $$<n^{2}>\ ,$$ hence, the statistical variance $$\sigma^{2} = <n^{2}> - <n>^{2}$$ of the photon probability distribution function (PDF). Overlooking fine integration details for $$n^{2}$$ in eq.$$(17)^1\ ,$$ we shall directly provide the result, which takes the canonical form:

$\tag{21} {\sigma ^2} = {G^2}\left( {\sigma _0^2 - {n_0}} \right) + G{n_0} + N + 2G{n_0}N + {N^2}$

The photon-statistics variance (commonly referred to as “amplifier noise”), is seen to be made of three contributions:

• an excess noise, $$\sigma _{excess}^2 = {G^2}\left( {\sigma _0^2 - {n_0}} \right)$$
• a shot noise, $$\sigma _{shot}^2 = G{n_0} + N$$
• a beat noise, $$\sigma _{beat}^2 = 2G{n_0}N + {N^2}$$

It is a well-known feature that coherent light, as produced by lasers, is distributed according to Poisson statistics. In this case, the variance is simply $$\sigma_{0}^{2} = n_{0}\ ,$$ showing that if the optical amplifier is input with (Poisson-distributed) coherent light, the above excess noise contribution $$\sigma_{excess}^{2}$$ vanishes. The shot noise contribution is seen to be equal to the mean amplifier output, i.e. $$Gn_{0} + N\ .$$ The amplified-signal contribution $$Gn_{0}$$ may be seen as pure “Poisson noise”, as if the amplifier would output purely coherent light! But because of spontaneous emission, the amplifier outputs $$N$$ as the unavoidable ASE, regardless of input. Finally, we observe beat noise. It is made of two independent contributions, i.e. $$insert formula$$ and $$insert formula\ .$$ They appear as the products of amplified signal with ASE, and ASE with itself, respectively, hence the name.

It should be emphasized that the canonical form of amplifier noise, $$\sigma^{2}$$ (eq.(21)), has made possible to define a key and standard parameter for optical amplifiers : the noise figure, which will be formally derived in Section 9.

Discussion 7.1. Originally, the concept of beat noise was derived from classical photodetection analysis : under square-law detection, the linear superimposition of amplified signal and ASE fields generates said s-ASE and ASE-ASE beat-noises. But here, no considerations of photodetection are involved in the STT model; these noise components should be regarded as intrinsic to the output EM photon statistics. We may also regroup the pure ASE under the form $$\sigma_{thermal}^{2} = N^{2} + N = N(N+1)$$ where we recognize $$\sigma_{thermal}^{2}$$ the variance of thermal light statistics (also referred to as Bose-Einstein). Then the contribution of spontaneous emission to output amplifier noise makes better physical sense. The remaining terms are $$Gn_{0}$$ and $$2Gn_{0}N\ ,$$ which we may regroup as “signal-stimulated noise” according to

$\tag{22} \sigma _{s - stim}^2 = G{n_0}(1 + 2N) = {G^2}{n_0}\frac{ {1 + 2N}}{G} \equiv {G^2}{n_0}\left( {2{n_{eq}} + \frac{1}{G} } \right) \equiv G{n_0}GF$

where $$F = 2n_{eq}+1/G$$ is the amplifier noise figure, and $$GF$$ appears as a noise enhancement factor from a fictitious, pure coherent-noise, $$\sigma^{2} = Gn_{0}\ .$$ With these new definitions, we may conveniently rewrite eq.(21) into the form :

$\tag{23} {\sigma ^2} = \sigma _{excess}^2 + \sigma _{thermal}^2 + \sigma _{s - stim}^2$

Concerning the s-ASE “beat noise” term, $$\sigma_{s-ASE}^{2} = 2Gn_{0}N\ ,$$ now integrated into $$\sigma_{s-stim}^{2}\ ,$$ we have shown [3] that it is in fact not attributable to the mixing with spontaneous emission (SE), despite the factor $$N\ ,$$ which bears the same thermal origin as SE. Indeed, if SE is artificially withdrawn from the STT master equation (15), accounting only for stimulated-emission events, one obtains the output signal $$<n> = Gn_{0}$$ (i.e. without ASE, $$N$$), and for the amplifier noise, one obtains for the canonical form $$\sigma^{2}(noASE) = \sigma_{excess}^{2} + \sigma_{s-stim}^{2}$$ (i.e., without thermal noise, $$\sigma_{thermal}^{2}$$). At the time (1994), we thus concluded that the so-called s-ASE contribution was a noise intrinsic to the stimulated emission process. Later on (1999), based upon a more detailed analysis using probability-generating functions (PGF) by T.Li and M.Teich (see description in Desurvire et al. 2002), we could confirm such a conclusion. The analysis showed that the PGF is the product of two independent processes : ASE and stimulated emission, pointing to the fact that the output PDF is a convolution of the two processes. As we have shown, removing the effect of SE from the model does not get rid of the noise $$\sigma_{s-ASE}^{2} = 2Gn_{0}N\ ;$$ hence, the latter must be attributed to a non-deterministic attribute of stimulated emission, heretofore considered being simply deterministic. To date, this discussion has not been pursued. This can be attributed to the fact that the whole issue of re-interpreting optical amplifier noise may be considered as of minor interest for any engineering purposes (considering that the model for square-law detection of optically-amplified signals is adequate, or simply too academic for the subject.

## Three-dimensional quantum beamsplitter model for E-field amplification

In this section, we describe E-field amplification through the “three-dimensional quantum beamsplitter” model. Such a model represents an extension to the general case, which we presented in 1999 Desurvire et al. 2002,Desurvire 1999, of well-known 2D models for the passive attenuator (PA) and the ideal amplifier (IA). As we have established it, non-ideal optical amplifiers (NIA), namely those with internal loss, represent a most general case situated between PA and IA limits. While PA and IA are modeled with 2D beamsplitters, the NIA requires a 3D beamsplitter.

### Passive attenuator

To clarify the above notions, consider first the PA case, as illustrated in Figure 6 (top). We assume a “signal” input EM wave, which is represented by the photon-field operator $$a\ .$$ By definition, the commutation property between the boson-operator $$a$$ and its Hermitian-conjugate $$a^{+}\ ,$$ is $$[a,a^{+}]=1\ .$$ In the Heisenberg picture, the EM quantum state of the system $$|n\rangle$$ remains constant, while the initial operator $$a(t_{0})$$ must evolve to some operator $$a(t_{0} + L/c) = b\ ,$$ after the EM wave has traveled through some distance $$L$$ through the device. Here, the device is a PA with field transmission $$\sqrt{T}(T\le 1)\ .$$ The case of a 50/50 photon beamsplitter corresponds to $$T = 1/2\ .$$ As we know, splitting photons (or say, selecting a mean fraction $$T$$ to pass through the lossy channel), is not deterministic. There must be an extra boson-field operator, $$\gamma\ ,$$ to randomly assist the “loss decision” : it is the so-called vacuum-noise field, which enters into a “third” device port, as illustrated in Figure 6 (top). The evolution equation may be written $$b = \sqrt{T}a + \lambda \gamma\ ,$$ where $$\lambda$$ is a complex number and $$\gamma$$ the vacuum field. The commuting condition $$[b,b^{+}] = 1$$ imposes $$\lambda = \sqrt{1-T}\ ,$$ within an arbitrary phase factor. We thus have

$\tag{24} b = \sqrt T a + \sqrt {1 - T} \gamma$

Figure 7: Quantum beamsplitter model for passive attenuator with transmission $$T < 1$$ (top) and ideal amplifier with gain $$G > 1$$(bottom). The signal field, $$a\ ,$$ is input from the left, and resulting in field, $$b\ ,$$ as output at right; vacuum noise fields,$$\gamma ,{\gamma ^ + }$$ (respectively), are input through a third port (blue arrow).

The vacuum-field commutes with the signal field ($$a,a^{+}$$) and carries no energy, hence satisfying:

$\tag{25} [\gamma ,{\gamma ^ + }] = {1,_{}}[{\gamma ^{( + )}},{a^{( + )}}] = {0,_{}} < {\gamma ^ + }\gamma > = {0,_{}} < \gamma {\gamma ^ + } > = 1$

From the above, one easily obtains the PA mean output photon number:

$\tag{26} < {b^ + }b > = T < {a^ + }a > \equiv T{n_0}$

and mean-square

$\tag{27} < {({b^ + }b)^2} > = {T^2} < {({a^ + }a)^2} > + T(1 - T){n_0}$

yielding the PA output variance

$\tag{28} \Delta {({b^ + }b)^2} \equiv < {({b^ + }b)^2} > - < {b^ + }b{ > ^2} = {T^2}\left[ {\Delta { {({a^ + }a)}^2} - {n_0} } \right] > + T{n_0}$

It is also readily checked that given input signals with coherent/Poisson ($$\Delta(a^{+}a)^{2} = n_{0}$$) or thermal/Bose-Einstein ($$\Delta(a^{+}a)^{2} = n_{0}(n_{0}+1)$$) statistics, the PA output signals with same statistics, leading to the conclusion that photon statistics are invariant under passive attenuation.

The above demonstration clearly introduces the role of vacuum noise in the photon partition (beamsplitting) or attenuation (PA) processes, and hence, a 2D coupling device.

### Ideal amplifier

Consider next the IA case, as illustrated in Figure 6 (bottom). The IA has a field transmission $$\sqrt{G}(G\ge 1)\ .$$ The conditions $$[b,b^{+}] = 1\ ,$$ along with $$G\ge 1\ ,$$ imposes an evolution equation of the form $$b = \sqrt{G}a + \lambda \gamma^{+}$$ (note conjugate vacuum-field, $$\gamma^{+}$$), where $$\lambda = \sqrt{G - 1}\ ,$$ within an arbitrary phase factor. We thus have

$\tag{29} b = \sqrt G a + \sqrt {G - 1} {\gamma ^ + }$

From the above, one easily obtains the IA mean output photon number :

$\tag{30} < n > \equiv < {b^ + }b > = G < {a^ + }a > + G - 1 \equiv G{n_0} + G - 1$

and mean-square

$\tag{31} < {({b^ + }b)^2} > = {G^2} < {({a^ + }a)^2} > + 3G(G - 1){n_0} + (G - 1)(2G - 1)$

yielding the IA output variance

$\tag{32} {\sigma ^2} = \Delta {({b^ + }b)^2} = {G^2}\left[ {\Delta { {({a^ + }a)}^2} - {n_0} } \right] > + G{n_0} + 2G(G - 1){n_0} + G(G - 1)$

The first term in brackets in the R.H.S. is the amplifier excess noise, which vanishes for Poisson/coherent photon statistics. The remaining terms makes sense with respect to previous canonical form (20), should we substitute the minimal noise of ideal amplifiers,$${N_{ideal}} = G - 1$$

$\tag{33} {\sigma ^2} \equiv {G^2}(\sigma _0^2 - {n_0}) + G{n_0} + {N_{ideal}}2G{n_0}{N_{ideal}} + N_{ideal}^2$

Quite remarkable indeed, is that the above IA model, only through a few lines of elementary computation, lead to the same exact result for output noise, as in the more elaborate STT model. Furthermore, the demonstration clearly introduces, again, the role of vacuum noise in the photon multiplication process (stimulated and spontaneous), and hence a 2D coupling device different from the PA.

The formula (31) has an interesting limiting case: high coherent signals inputs ($${n_0} > > 1$$) and high gains $$(G > > 1)\ .$$ In such a limit, we obtain for output noise${\sigma ^2} = 2{G^2}{n_0}\ ,$ meaning ultimately that ideal amplifier noise is quadratic in gain ($$G$$) and linear in input power ($${n_0}$$).

### Non-ideal amplifier

The NIA case combines the two previous PA and IA cases, in a unified and continuous way, a discovery of Desurvire 1999. The ground idea is that the two processes of photon loss/absorption and amplification could be combined in said sequence, as illustrated in Figure 7. Hence, the whole may be described in a single evolution equation, covering systems characterized from passive loss to amplification, and the latter from non-ideal (incomplete inversion) to ideal. Here, we shall directly provide the result for the evolution equation:

$\tag{34} b = \sqrt G a + \sqrt {g(1 - T)} {\gamma _1} + \sqrt {T(g - 1)} \gamma _2^ +$

In the above, - $$T$$is the PA transmission - $$g$$is the IA (fully-inverted, lossless) gain - $$gT = G$$ is the NIA “net gain” - $${\gamma _1},\gamma _2^ +$$ are non-commuting vacuum-fields, corresponding to PA and IA, respectively, together with same properties as $$\gamma \ .$$

Figure 8: Quantum beamsplitter model for non-ideal optical amplifiers, as made from a passive attenuator with transmission $$T < 1$$ (blue, left), followed by an ideal amplifier with gain $$g > 1$$(yellow, right), altogether yielding net gain or transmission $$G = gT\ .$$

We note in eq.(34) that the PA case is obtained with $$g = 1$$ (a truly “noiseless” amplifier !) and the IA case with $$T = 1\ .$$

Owing to earlier analysis, it may not be a surprise that the photon mean and variance of the PA-IA system output take the form:

$\tag{35} \begin{cases} <n> = Gn_{0} + N \\ \sigma^{2} = G^{2}(\sigma^{2}_{0} - n_{0}) + Gn_{0} + N + 2Gn_{0}N + N^{2} \end{cases}$

where $$N$$has the meaning of (19), along with $$N = {n_{sp}}(G - 1) = {n_{eq}}G\ .$$

Remarkably enough, it is seen that the NIA model yields the same exact result for output noise (eq.(35)), as in the more elaborate STT model (eq.(19),(21)). We should emphasize here that, to describe the general NIA case, passive attenuation must precede ideal amplification. The reverse arrangement (PA following IA) is inadequate : indeed, passive attenuation is photon-statistics-preserving, and that IA-PA system would have the statistics of an IA – and not that of the general case, regardless how much attenuation be introduced.

The NIA model shows that optical (laser) amplification is necessarily mediated by two vacuum-noise fields (exception being PA and IA). Optical amplifiers may thus be seen as 4-port devices (signal input/output and two vacuum-noise ports), as represented by the 3D beamsplitter in Figure 8. Such a conclusion applies to a single mode of the quantized EM field. Because there exist two degenerate polarization modes, we may view optical amplifiers as two-channel PA-IA systems, with four independent vacuum-noise couplings.

Figure 9: 3D beamsplitter representation of non-ideal optical (laser) amplifiers, showing two independent vacuum-noise fields ($${\gamma _1},\gamma _2^ +$$) controlling passive attenuation ($${\gamma _1}$$), with transmission $$T\ ,$$ and ideal amplification ($$\gamma _2^ +$$), with gain $$g\ ,$$ yielding the nest system gain $$G = gT\ .$$

## Amplified quantum noise

In this section, we describe how the E-field quantum noise is enhanced by optical amplification. This requires first to recall some basics concerning noise in coherent states. All derivations details about noise amplification may be found in Desurvire et al. 2002, for instance.

Most generally, the scalar E-field at frequency $$\omega$$ and phase $$\varphi$$ is represented by the operator

$\tag{36} E = a{e^{ - i(\omega t - \varphi )}} - {a^ + }{e^{i(\omega t - \varphi )}}$

Its mean observable amplitude is $$< E > = \sqrt { {n_0} } \sin (\omega t - \varphi )\ ,$$ where $${n_0}$$ is the mean photon number associated with the EM wave. Furthermore, the E-field is the superposition of two quadratures : one real for the “cosine” component,$${E_2}\ ,$$ and one imaginary for the “sine” component, $${E_1}\ .$$ The quadrature operators are Hermitian, with the following definition :

$\tag{37} \begin{cases} E_{1} = \frac{1}{2i} \lbrack a^{+}e^{i(\omega t - \phi)} - ae^{-i(\omega t - \phi)} \rbrack \\ E_{2} = \frac{1}{2} \lbrack a^{+}e^{i(\omega t - \phi)} + ae^{-i(\omega t - \phi)} \rbrack \end{cases}$

It is straightforward to check that the quadratures have the commuting relation $$[{E_1},{E_2}] = i/2\ .$$ It is easily established that if the above operators are applied to a coherent state, the mean quadrature amplitudes are $$< {E_1} > = \sqrt { {n_0}} \sin (\omega t - \varphi )$$ and $$< {E_2} > = \sqrt {{n_0} } \sin (\omega t - \varphi )\ .$$ The quadratures are also characterized each by quantum noise, with variances $${(\Delta {E_1})^2},{(\Delta {E_2})^2}\ ,$$ as defined by $${(\Delta {E_i})^2} = < E_i^ + {E_i} > - < {E_i}{ > ^2}\ ,$$ with $$i = 1,2\ .$$ The phasor diagram of the E-field, with quadrature amplitudes and quantum noise, is shown in Figure 9. The green cloud represents the uncertainty associated with the E-field phasor, with $${(\Delta E)^2} = {(\Delta {E_1})^2} + {(\Delta {E_2})^2}$$ for the amplitude noise and $${(\Delta \varphi )^2}$$ for the phase noise. Here, we only focus on amplitude (see Desurvire et al. 2002 for detained analysis of the phase).

Figure 10: Phasor diagram of the electric field $$E\ ,$$ having mean amplitude $$< E >$$ and phase $$\varphi \ ,$$ showing the two quadratures $${E_1},{E_2}$$ of mean amplitudes $$< {E_1} > , < {E_2} >$$ and deviations $$\Delta {E_1},\Delta {E_2}\ .$$

It is also easily established, by application of the operators in eq. (37) to a coherent state, that

$\tag{38} \begin{cases} <E_{1}^{+}E_{1}> = \frac{1}{4} + n_{0} \sin^{2} (\omega t - \phi) \\ <E_{2}^{+}E_{2}> = \frac{1}{4} + n_{0} \cos^{2} (\omega t - \phi) \end{cases}$

From the above, we obtain $${(\Delta {E_1})^2} = {(\Delta {E_2})^2} = 1/4\ ,$$ and hence

$\tag{39} \Delta {E_1}\Delta {E_2} = \frac{1}{4}$

The above equality relation represents the minimum of Heisenberg’s uncertainty principle, as applied to the two observables $${E_1},{E_2}$$ (namely, if $$A,B$$are Hermitian operators, then $$\Delta A\Delta B \geqslant \left| {[A,B]} \right|/2$$). This is the reason why coherent states are called “minimum-uncertainty states”.

### Amplified E-field

Next, consider the E-field quadratures after amplification in a PA-IA system of net gain $$G$$ and amplified spontaneous emission $$N = {n_{sp}}(G - 1) = {n_{eq}}G\ .$$ The E-field output being $$b\ ,$$ we may define the quadrature operators :

$\tag{40} \begin{cases} Q_{1} = \frac{1}{2i} \lbrack b^{+}e^{ i( {\omega t}- \phi ) } - be^{ -i( {\omega t}- \phi ) } \rbrack \\ Q_{2} = \frac{1}{2} \lbrack b^{+}e^{ i( {\omega t}- \phi ) } + be^{ -i( {\omega t}- \phi ) } \rbrack \end{cases}$

It is straightforward to establish that the output mean amplitudes are given by $$< {Q_i} > = \sqrt G < {E_i} > \ .$$ Albeit elementary, however, the computation of $$< Q_i^ + {Q_i} >$$ is not straightforward (see details in Desurvire et al. 2002). Suffices it to provide directly the result, which nicely reduces to :

$\tag{41} \begin{cases} <Q_{1}^{+}Q_{1}> = \frac{1+2N}{4} + Gn_{0} \sin^{2} (\omega t - \phi) \\ <Q_{2}^{+}Q_{2}> = \frac{1+2N}{4} + Gn_{0} \cos^{2} (\omega t - \phi) \end{cases}$

Combining the above, we obtain $${(\Delta {Q_1})^2} = < Q_i^ + {Q_i} > - < {Q_i}{ > ^2} \equiv (1 + 2N)/4\ ,$$ and hence the minimal Heisenberg uncertainty relation :

$\tag{42} \Delta {Q_1}\Delta {Q_2} = \frac{ {1 + 2N} }{4} \geqslant \frac{1}{4}$

In the case of IA, where $$N = G - 1\ ,$$ the above reduces to $$\Delta {Q_1}\Delta {Q_2} = (2G - 1)/4\ .$$ This last result was shown by Yamamoto and Mukai Yamamoto et al. 1989, while eq. (42) represents the result we showed in 1999 for the general case. Furthermore, this last result nicely relates to the concept of optical amplifier noise figure, $$F\ ,$$ which has appeared a few times in the forgoing, and is described in the last section. Indeed, given the definition

$\tag{43} F = \frac{ {1 + 2N} }{G}$

the minimal uncertainty of optical amplifiers becomes

$\tag{44} \Delta {Q_1}\Delta {Q_2} = \frac{ {GF} }{4} = (GF)(\Delta {E_1}\Delta {E_2}) \geqslant \frac{1}{4}$

meaning that amplification magnifies the coherent-state uncertainty by the factor $$GF$$ = gain times noise figure. We also note the input/output noise correspondence $$\Delta {Q_i} = \sqrt {GF} \Delta {E_i} \geqslant 1/2\ .$$ The phasor diagram of input/output E-field is shown in Figure 10.

Figure 11: Phasor diagram of the input and output electric fields in the general case of non-ideal laser amplifiers, showing input and output amplitudes et quantum uncertainties for the two quadratures.

It is worth mentioning here, without demonstration, that the amplifier phase noise, may be approximated by the relation $$\Delta \varphi = \sqrt F \Delta {\varphi _0}\ ,$$ where $$\Delta {\varphi _0} \equiv 1/(2\sqrt { {n_0} } )$$ is the phase uncertainty associated with a coherent state with $${n_0}$$ photons.

## Noise figure

The previous sections have shown that optical amplifier noise, considering its conceptual complexity in the quantum-operator formalism and physical interpretation (multiple vacuum-noise couplings), may yet be described by very simple formulas ! Furthermore, we have seen that the noise figure $$F$$ appears as a fundamental parameter to characterize amplifier quantum noise, both in amplitude and phase.

In this concluding section, we provide a physical interpretation of noise figure (NF). The NF concept is as old as 1944, with the Friis definition, named after its inventor. Simply defined, it is the ratio of input-to-output signal-to-noise-ratio (SNR), or

$\tag{45} F = \frac{ {SN{R_{in}}}}{ {SN{R_{out}} } }$

where, for a given input/output power $$P$$ with noise variance $${(\Delta P)^2}\ ,$$ the SNR is defined as $$SNR = < P > /\sqrt { { {(\Delta P)}^2}} \ .$$ Because of amplifier noise, $$SN{R_{out} } < SN{R_{in} }\ ,$$ and hence, as an overriding rule. The NF may thus be regarded as an unavoidable “penalty” factor, say a “tax to pay”, for boosting EM power in any microwave/RF/optical circuits. In the optical domain, several theoretical analyses, semi-classical and quantum, pointed out the property according to which $$F \geqslant 2\ ,$$ i.e. the minimal NF, in ideal conditions is “two”, representing in engineering terms, a 3dB tax penalty.

With the advent, through years 1980-90, of optical amplifiers, first as semiconductor optical amplifiers (SOA), then as erbium-doped fibre amplifiers (EDFA), did resurface the key issue of NF definition and characterization. The definition crystallized in 1989 Desurvire 1990, from the Friis formula :

$\tag{46} F* = \frac{ {SN{R_{in}}}}{ {SN{R_{out}}}} \equiv \left( {\frac{ { {n_0}}}{ {\sigma _0^2}} } \right)\left( {\frac{{{\sigma ^2} } }{{ < n > } } } \right)$

where the STT results could be substituted to yield

$\tag{47} F* = \frac{ { {n_0}}}{ {\sigma _0^2}}\left( {\frac{ {\sigma _{excess}^2 + \sigma _{shot}^2 + \sigma _{beat}^2}}{ { < {n_0} > }}} \right) \equiv \frac{ { {n_0} } }{{\sigma _0^2} }\left( {\frac{{\sigma _{excess}^2 + G{n_0} + N + 2G{n_0}N + {N^2} } }{{G{n_0} + N} } } \right)$

Letting $$\sigma _0^2 = {n_0}\ ,$$ and $$\sigma _{excess}^2 = 0\ ,$$ for a coherent/Poisson input EM signals, and factorizing $$G\ ,$$ we obtain

$\tag{48} F* = \frac{ {1 + 2N}}{G} + \frac{ {N(N + 1)}}{{{G^2}{n_0} } } \equiv F + \delta F$

with

$\tag{49} \delta F = \frac{ {N(N + 1)}}{ { {G^2}{n_0}}} = \frac{ { {n_{eq}}}}{ {{n_0} } }\left( {\frac{{1 + {n_{eq} } } }{G} } \right)$

The Friis definition for NF, as applied to optical amplifiers, $$F*\ ,$$ thus reduces to the earlier fundamental (quantum) measure, $$F = (1 + 2N)/G\ ,$$ augmented by some penalty factor $$\delta F\ .$$ This factor, which is referred to as signal-dependent noise figure, is of the order of $$n_{eq}^2/(G{n_0})\ .$$ As we have seen, the equivalent input noise $${n_{eq}}$$ has a magnitude of unity, while in most amplifier applications, the input signal photon number, $${n_0}\ ,$$ may be several powers of ten (e.g. a 1GHz EM wave at a 1µm wavelength, with 1µW mean power holds 5,000 photons). Thus, for most cases of interest, we have $${n_{eq}} < < {n_0}\ ,$$ possibly combined with the usual high-gain condition, $$G > > 1\ ,$$ we may safely neglect $$\delta F\ ,$$ and hence $$F* \approx F\ .$$ We conclude that the noise figure $$F$$ is a truly fundamental parameter in optical amplifiers, as opposed to a mere “performance indicator” for telecom standards.

Since there exists a conceptual continuity between PA and NIA/IA, we may see any arrangement of passive attenuators and optical amplifiers, as a global, single system characterized by a net transmission $$\tilde G$$ and an overall noise figure $$\tilde F\ .$$ For instance, multi-channel lightwave transmission systems are made of the concatenation of $$k$$ lossy segments of optical fibre (transmission $$\tilde T$$), whose loss is compensated “in-line” with $$k$$ “lumped” optical amplifiers (non-ideal, net gain $$\tilde g$$), all elements being assumed to have identical loss/gain characteristics. The system transparency condition (for each wavelength channel, the power at the receiver end must equals the power input at the transmitter end), is $$\tilde G = {(\tilde g\tilde T)^k} = 1\ .$$ Optical amplification may also be continuous along the line, which is referred to as distributed amplification, as opposed to lumped amplification. Analytical formulas for system NF in several possible configurations can be found in Desurvire 1994, Desurvire et al. 2002.

Another application of interest if the amplification of light pulses. Here, the goal is not the transmission of data over distance, but the boosting of laser light in order to achieve, though free-space, multiple-stage optical amplification, what is referred to as “extreme” peak powers, -- namely in the class of terawatt ($${10^{12}}$$W) to petawatt ($${10^{15}}$$W). Thus, the system condition is $$\tilde G = {(\tilde g\tilde T)^k} > > 1\ ,$$ where $$\tilde T$$ represents the loss experienced by the signal pulse due to passive optics (lenses, gratings, beamsplitters, apertures, mirrors, isolators..) and $$\tilde g$$ the net gain of in-line optical boosters (in reality, each optics/booster stage having its optimized loss/gain characteristics).

Whether in telecom or in extreme-light systems, the above-described general rules for NF remain the same. In both cases, the systems are made of the concatenation of passive/active components. It is worth concluding here on his elegant property of amplifiers, which comes from the microwave domain and extends to lightwave.

Assume a concatenation or chain of amplifiers, labelled “1” and “2”, with gain and noise-figure characteristics ($${G_1},{F_1}$$), and ($${G_2},{F_2}$$), respectively. Which chain arrangement minimizes the overall NF, between “1” followed by “2” ($$1 \to 2$$), or the reverse ($$2 \to 1$$)? A classic textbook demonstration leads to the concatenated noise figure formula:

$\tag{50} {F_{1 \to 2}} = {F_1} + \frac{ { {F_2} - 1}}{{{G_1} } }$

The instant conclusion of the above is that the NF is minimized when the amplifier in the chain’s front-end is the one with highest gain ($${G_1} > {G_2}$$) and lowest noise ($${F_1} < {F_2}$$). For a chain of $$k$$ amplifiers, we have

$\tag{51} {F_{1 \to k}} = {F_1} + \frac{ { {F_2} - 1}}{ { {G_1}}} + \frac{ { {F_3} - 1}}{{{G_1}{G_2} } } + ... + \frac{{{F_k} - 1} }{{{G_1}{G_2}...{G_{k - 1} } } }$

which shows that the system NF, no matter how many elements in the chain, is mostly dominated by (gain/NF) characteristics of the first amplifier. In the optical amplification case, the formula in eq.(51) applies, should one substitute the definition $$F*$$ in eq.(48) for each element $$k$$ ($$G \to {G_k}\ ,$$ $$N \to {N_k}$$ for gain and ASE, and $${n_0} \to {n_{0,k - 1}}$$ for input) Desurvire 1994. However, the formula is only of academic interest, considering that any amplifier chains, including lossy elements, may be viewed as a single device with overall gain/NF characteristics : ($$\tilde G,\tilde F$$).

## References

[1] A.Einstein, « Zur Quantumtheorie der Strahlung », 1917

[2] B.E.A. Saleh and M.C. Teich, “Fundamentals of photonics”, Wiley, 1991

[3] E. Desurvire, “Erbium-doped fiber amplifiers, Principle and Applications”, Wiley, 1994

[4] E. Desurvire, D.Bayart, B.Desthieux and S.Bigo, “Erbium-doped fiber amplifiers, Device and System Developments”, Wiley, 2002

[5] See discussion in R.Loudon, “The quantum theory of light”, Oxford, 1973, pp. 174 and 190

[6] K.Shimoda, H.Takahashi and C.H.Townes, “Fluctuations in amplification of quanta with applications to maser amplifiers,” J.Phys. Soc. Japan, Vol.17, N°6, 686 (1957)

[7] E.Desurvire, “A three-dimensional quantum vacuum-noise beamsplitter model for non-ideal linear optical amplifiers,” Optical Fiber Technology, Vol.5, N.1, 81, Academic Press /Elsevier, 1999

[8] Y.Yamamoto and T.Mukai, “Fundamentals of optical amplifiers”, Optical and Quantum Electron., Vol.21, 1, 1989

[9] E. Desurvire, “Analysis of noise figure spectral distribution in erbium-doped fiber amplifiers pumped near 980 and 1480 nm”, Appl. Opt, Vol. 29, N.21, 3118 (1990)