In this tutorial, Langevin stochastic equations and Markov probability equations are used to model electron- and photon-number fluctuations in three- and four-level lasers. Equations are derived for the moments of the electron and photon numbers (means, variances and correlations). Both approaches produce the same moment equations. In the Langevin approach, the moments of the noise terms must be specified by other means, whereas in the Markov approach, they are determined self-consistently and satisfy the shot-noise rule: For each process that is modeled by the rate equations, the driving terms in the variance equations equal the moduli of the associated terms in the mean equations. The driving terms in the correlation equations have the same magnitudes as the variance terms, but can be positive or negative, depending on whether the changes in the electron and photon numbers are correlated or anti-correlated, respectively. Formulas are derived for the relative intensity noise and its spectrum. The consequences of these results for three- and four-level lasers are discussed.
1. INTRODUCTION
In this tutorial, the noise properties of three- and four-level atom lasers are reviewed in detail. Four-level lasers involve levels 0 (ground), 1 (lower), 2 (upper), and 3 (excited). Type A three-level lasers involve levels 0, 1, and 2, whereas type B lasers involve levels 1, 2, and 3. Let $ S $ be the number of signal (laser) photons in a cavity. Then the laser performance is characterized by the equilibrium number $ {S_0} $, which is proportional to the average photon flux (output power), and the number deviation (fluctuation) $ {S_1} $, which varies randomly in time. Two important metrics are the relative intensity noise (RIN) $ {\left\langle {S_1^2(t)} \right\rangle _t}/S_0^2 $, where $ {\left\langle \right\rangle _t} $ denotes a time average, and the RIN spectrum $ {\left\langle {|{S_1}(\omega {{)|}^2}} \right\rangle _e}/S_0^2T $, where $ {\left\langle \right\rangle _e} $ denotes an ensemble average and $ T $ is the integration time. Both quantities can be measured.
Laser evolution can be modeled by rate equations for the electron and photon numbers [1,2]. These equations, which model absorption, emission, loss, and pumping, are stated in Section 2. In ideal lasers, electrons decay quickly from level 3 to level 2 or from level 1 to level 0, so only the populations of two levels need to be retained: 0 and 2 for a four-level or type A laser, and 1 and 2 for a type B laser. These simplifications are used to derive a set of three general rate equations, which apply to all of the aforementioned lasers.
The general rate equations are deterministic, so they can describe the growth and saturation of the signal-photon number, but cannot describe photon-number fluctuations. One can mimic the effects of quantum fluctuations by adding random source terms to the rate equations, with one source term for each deterministic process. According to the shot-noise rule (which remains to be justified), the source terms are statistically independent, and the variance of each source term is the mean of the deterministic term with which it associated. The stochastic rate equations that result are called Langevin equations [3,4] and are stated in Section 3. They depend nonlinearly on the electron and photon numbers, so in most circumstances one has to solve them numerically. However, one can develop physical insight into laser noise by solving them approximately. By linearizing the Langevin equations about their equilibrium solutions, one can derive stochastic equations for the number deviations, and by treating the noise terms in these equations carefully, one can derive deterministic equations for their means, variances, and correlations. Although the linearized Langevin equations and their associated moment equations are useful, I decided to delay discussions of their consequences until after the source strengths have been determined.
Let $ F(m,n,s) $ be the probability that there are $ m $ lowest-level (0 or 1) electrons, $ n $ upper-level (2) electrons, and $ s $ photons in the cavity, where $ m $, $ n $, and $ s $ are nonnegative integers. The Markov equation [3] for this probability distribution function is stated in Section 4. It is based on the assumption that the electron and photon numbers change randomly, but at the average rates associated with the general rate equations. By calculating the moments of the Markov equation, one can derive equations for the number means, variances, and correlations from first principles. In the Markov approach, the driving terms in the variance and correlation equations appear naturally, and validate the shot-noise formulas upon which the Langevin approach is based. Although the Markov moment equations are exact, they do not form a closed set, because they involve higher-order deviation moments. One can close them by neglecting these deviation moments, in which case they reduce to the linearized Langevin moment equations.
With the validity of the linearized Langevin equations and their source terms established, their consequences are determined in Section 5. By solving the linearized Langevin equations in the frequency domain, one can derive a formula for the RIN spectrum. By inverting this spectrum, one obtains a formula for the autocorrelation function $ {\langle {S_1}(t){S_1}(t + \tau )\rangle _t}/S_0^2 $, where $ \tau $ is the time delay, and by letting $ \tau \to 0 $, one obtains a formula for the RIN. One can also derive a formula for the RIN by solving the Langevin moment equations in steady state. These metrics (and the formulas for them) are standard [5,6]. They allow one to quantify the noise performance of a particular laser and compare the properties of different types of laser. The main conclusions of this article are summarized in Section 6.
In a previous tutorial [7], I reviewed the mathematical methods one requires to analyze a laser-like equation for the photon number, and the physical insights one develops by doing so. These methods and insights are also required for this tutorial. Only those generalizations that are required for multiple-variable systems are described herein. My previous article included a quantum-optical analysis of number fluctuations. Because of length limitations, this article includes only classical analyses (deterministic, stochastic, and probabilistic).
For convenience, this article includes five appendices. In Appendix A, some properties and characterizations of stochastic systems are described. In Appendix B, the rules of stochastic calculus are reviewed for an arbitrary system of differential equations, then used to derive general formulas for the Langevin moment equations. In Appendix C, these formulas (which are not linearized) are applied to the Langevin equations for three- and four-level lasers. Moment equations are stated for the electron and photon numbers that are consistent with the Markov moment equations of Section 4. In Appendix D, some generalizations of the Markov moment equations are derived. These generalized equations are consistent with the formulas of Appendix B. Finally, in Appendix E, contour integration is used to inverse transform the RIN spectrum, in order to determine the autocorrelation function and the RIN.
2. RATE EQUATIONS
In this section, the electron and photon rate equations for three- and four-level lasers are stated and simplified, as described in the introduction. Readers who are familiar with this material [1,2] can proceed directly to Section 2.D.
A. Four-Level Laser
Consider a laser with four electron levels: 0 (ground), 1 (lower), 2 (upper), and 3 (excited), as illustrated in Fig. 1. Examples of such lasers include helium–neon and neodinium–YAG. Let $ {N_j} $ be the number of electrons in level $ j $. Then the rate equations for the electron numbers are
(1)$${d_t}{N_3}\def\LDeqtab{}={a_{30}}P{N_0} - {a_{30}}(P + 1){N_3} - {a_{32}}{N_3},$$
(2)$${d_t}{N_2}\def\LDeqtab{}={a_{32}}{N_3} + {a_{21}}S{N_1} - {a_{21}}(S + 1){N_2},$$
(3)$${d_t}{N_1}\def\LDeqtab{}=- {a_{21}}S{N_1} + {a_{21}}(S + 1){N_2} - {a_{10}}{N_1},$$
(4)$${d_t}{N_0}\def\LDeqtab{}=- {a_{30}}P{N_0} + {a_{30}}(P + 1){N_3} + {a_{10}}{N_1},$$
where $ P $ is the number of pump photons, $ S $ is the number of signal (laser) photons, and $ {a_{jk}} $ is the rate of spontaneous emission between levels $ j $ and $ k $. The transition $ 3 \leftrightarrow 0 $ is stimulated by the pump and the transition $ 2 \leftrightarrow 1 $ is stimulated by the signal, whereas the transitions (decays) $ 3 \to 2 $ and $ 1 \to 0 $ are purely spontaneous. For simplicity, the transition $ 2 \to 0 $ is neglected, as are transitions from the aforementioned levels to other levels. It follows from Eqs. (1)–(4) that (5)$${d_t}({N_0} + {N_1} + {N_2} + {N_3}) = 0.$$
The total electron number is conserved.If $ {a_{32}} \gg {a_{30}}P $, then $ {N_3} $ tends quickly to its quasi-steady-state value
(6)$${N_3} = {a_{30}}P{N_0}/[{a_{32}} + {a_{30}}(P + 1)] \approx {a_{30}}P{N_0}/{a_{32}},$$
which is much less than $ {N_0} $. Likewise, if $ {a_{10}} \gg {a_{21}}S $, then $ {N_1} $ tends quickly to its quasi-steady-state value (7)$${N_1} = {a_{21}}(S + 1){N_2}/({a_{10}} + {a_{21}}S) \approx {a_{21}}(S + 1){N_2}/{a_{10}},$$
which is much less than $ {N_2} $. Electrons in level 3 drop quickly to level 2, and electrons in level 1 drop quickly to level 0. In this case, Eqs. (1)–(4) can be replaced by the reduced set of equations (8)$${d_t}{N_2} \def\LDeqtab{}\approx {a_{30}}P{N_0} - {a_{21}}(S + 1){N_2},$$
(9)$${d_t}{N_0} \def\LDeqtab{}\approx - {a_{30}}P{N_0} + {a_{21}}(S + 1){N_2}.$$
Equations (8) and (9) are what one would obtain by assuming that electrons transition directly between level 0 and level 2. The first terms on the right sides of these equations represent stimulated pump-photon absorption, whereas the second terms represent stimulated and spontaneous signal-photon emission. It follows from Eqs. (8) and (9) that (10)$${d_t}({N_0} + {N_2}) \approx 0.$$
The reduced equations conserve the “total” electron number $ {N_0} + {N_2} $.The rate equation for the signal-photon number is
(11)$$\begin{split}{d_t}S &= {a_{21}}{N_2}(S + 1) - {a_{21}}{N_1}S - bS \\ &\approx {a_{21}}{N_2}(S + 1) - bS,\end{split}$$
where $ b $ is the transmission-loss rate. Stimulated and spontaneous emission increase the signal-photon number, whereas transmission loss decreases the number and stimulated absorption is unimportant. Spontaneous emission seeds the signal growth, but otherwise is unimportant.The emission terms in Eqs. (8), (9), and (11) pertain to one lasing mode. Although the stimulated terms are correct, the spontaneous terms in Eqs. (8) and (9) should be augmented by $ \mp {a_{nl}}{N_2} $, respectively, where $ {a_{nl}} = {a_{21}}I $ is the spontaneous emission rate multiplied by the number of non-lasing (inert) modes into which photons are also emitted. Electrons decay from level 2 to level 1 at the specified rate, then decay immediately to level 0. These additional decays increase the laser threshold (the pump power required for population inversion and signal net gain) but do not affect the well-above-threshold laser evolution significantly (because $ S \gg I $).
B. Type A Three-Level Laser
Now consider a laser with three electron levels: 0 (ground), 1 (lower), and 2 (upper), as illustrated in Fig. 2. In the notation of Section 2.A, the rate equations for the electron numbers are
(12)$$\begin{split}{d_t}{N_2} &= {a_{20}}P{N_0} - {a_{20}}(P + 1){N_2} + {a_{21}}S{N_1}\\&\quad - {a_{21}}(S + 1){N_2} - {a_{nl}}{N_2},\end{split}$$
(13)$${d_t}{N_1}\def\LDeqtab{} = - {a_{21}}S{N_1} + {a_{21}}(S + 1){N_2} + {a_{nl}}{N_2} - {a_{10}}{N_1},$$
(14)$${d_t}{N_0}\def\LDeqtab{} = - {a_{20}}P{N_0} + {a_{20}}(P + 1){N_2} + {a_{10}}{N_1}.$$
The transition $ 2 \leftrightarrow 0 $ is stimulated by the pump and the transition $ 2 \leftrightarrow 1 $ is stimulated by the signal, whereas the transition (decay) $ 1 \to 0 $ is purely spontaneous. It follows from Eqs. (12)–(14) that (15)$${d_t}({N_0} + {N_1} + {N_2}) = 0.$$
The total electron number is conserved. The main difference between this laser and a four-level laser is that pump-photon emission limits the upper-level population.If $ {a_{10}} \gg {a_{21}}S $, electrons in level 1 drop quickly to level 0, so $ {N_1} \ll {N_2},{N_0} $. In this case, Eqs. (12)–(14) can be replaced by the reduced set of equations
(16)$$\begin{split}{d_t}{N_2} &\approx {a_{20}}P{N_0} - {a_{20}}(P + 1){N_2}\\&\quad - {a_{21}}(S + 1){N_2} - {a_{nl}}{N_2},\end{split}$$
(17)$$\begin{split}{d_t}{N_0} &\approx - {a_{20}}P{N_0} + {a_{20}}(P + 1){N_2}\\&\quad + {a_{21}}(S + 1){N_2} + {a_{nl}}{N_2}.\end{split}$$
Equations (16) and (17) are what one would obtain by assuming that electrons transition directly from level 2 to level 0. The first terms on the right sides represent pump-photon absorption, the second terms represent pump-photon emission, the third terms represent signal-photon emission and the fourth terms represent spontaneous emission into non-lasing modes. It follows from Eqs. (16) and (17) that (18)$${d_t}({N_0} + {N_2}) = 0.$$
Once again, the reduced equations conserve the “total” electron number $ {N_0} + {N_2} $.The rate equation for the signal-photon number is
(19)$$\begin{split}{d_t}S& = {a_{21}}{N_2}(S + 1) - {a_{21}}{N_1}S - bS, \\ &\approx {a_{21}}{N_2}(S + 1) - bS.\end{split}$$
Once again, stimulated and spontaneous emission increase the signal-photon number, whereas transmission loss decreases the number and stimulated absorption is unimportant.C. Type B Three-Level Laser
Finally, consider a laser with three different electron levels: 1 (lower), 2 (upper), and 3 (excited), as illustrated in Fig. 3. Examples of such lasers include erbium and ruby. In the notation of Section 2.A, the rate equations for the electron numbers are
(20)$${d_t}{N_3} \def\LDeqtab{}= {a_{31}}P{N_1} - {a_{31}}(P + 1){N_3} - {a_{32}}{N_3},$$
(21)$${d_t}{N_2}\def\LDeqtab{} = {a_{32}}{N_3} + {a_{21}}S{N_1} - {a_{21}}(S + 1){N_2} - {a_{nl}}{N_2},$$
(22)$$\begin{split}{d_t}{N_1} &= - {a_{31}}P{N_1} + {a_{31}}(P + 1){N_3} - {a_{21}}S{N_1}\\&\quad + {a_{21}}(S + 1){N_2} + {a_{nl}}{N_2}.\end{split}$$
The transition $ 3 \leftrightarrow 1 $ is stimulated by the pump and the transition $ 2 \leftrightarrow 1 $ is stimulated by the signal, whereas the transition (decay) $ 3 \to 2 $ is purely spontaneous. It follows from Eqs. (20)–(22) that (23)$${d_t}({N_1} + {N_2} + {N_3}) = 0.$$
The total electron number is conserved.If $ {a_{32}} \gg {a_{31}}P $, then $ {N_3} \ll {N_1} $, $ {N_2} $. In this case, Eqs. (20)–(22) can be replaced by the reduced set of equations
(24)$${d_t}{N_2} \def\LDeqtab{}\approx {a_{31}}P{N_1} + {a_{21}}S{N_1} - {a_{21}}(S + 1){N_2} - {a_{nl}}{N_2},$$
(25)$${d_t}{N_1} \def\LDeqtab{}\approx - {a_{31}}P{N_1} - {a_{21}}S{N_1} + {a_{21}}(S + 1){N_2} + {a_{nl}}{N_2}.$$
Equations (24) and (25) are what one would obtain by assuming that electrons transition directly from level 1 to level 2. The first and second terms on the right sides represent pump- and signal-photon absorption, respectively, whereas the third terms represent signal-photon emission. It follows from Eqs. (24) and (25) that (26)$${d_t}({N_1} + {N_2}) \approx 0.$$
The reduced equations conserve the “total” electron number $ {N_t} = {N_1} + {N_2} $.Table 1. Rate Coefficients for Three Types of Lasera
The rate equation for the signal-photon number is
(27)$${d_t}S = {a_{21}}{N_2}(S + 1) - {a_{21}}{N_1}S - bS.$$
Emission increases the signal-photon number, whereas absorption and loss decrease it. The main difference between this laser and a four-level laser is that $ {N_2} \gt {N_t}/2 $ is required for net gain (so the laser threshold is higher).D. General Rate Equations
For all three types of laser, the rate equations can be written in the forms
(28)$${d_t}S \def\LDeqtab{}= - {a^\prime}MS + aN(S + 1) - bS,$$
(29)$${d_t}N \def\LDeqtab{}= {a^\prime}MS - aN(S + 1) + cM - {c^\prime}N - dN,$$
(30)$${d_t}M \def\LDeqtab{}= - {a^\prime}MS + aN(S + 1) - cM + {c^\prime}N + dN,$$
where $ N = {N_2} $ is the number of upper-level electrons. For ideal four-level and type A three-level lasers, $ M = {N_0} $ is the number of ground-level electrons, whereas for type B three-level lasers, $ M = {N_1} $ is the number of lower-level electrons. The rate coefficients are specified in Table 1. For four-level and type A lasers, $ {a^\prime} = 0 $ (no stimulated signal-photon absorption), and for four-level and type B lasers, $ {c^\prime} = 0 $ (no stimulated pump-photon emission). Only for type A lasers is $ {c^\prime} \ne 0 $ (stimulated pump-photon emission).Equations (28)–(30) can be solved numerically. It is convenient to measure the electron and photon numbers in units of $ {N_t} $, and time in units of $ 1/b $, which is of the order of the transit time (of light across the cavity). The dimensionless rate equations involve three dimensionless parameters: the gain parameter $ a{N_t}/b $, the pumping parameter $ c/b $, and the damping parameter $ d/b $. To simplify my discussions of laser evolution, I will consider only the weak-damping (moderate-to-strong pumping) regime, in which $ d/b $ is much smaller than the other parameters and can be neglected. If $ a{N_t}/b = 10 $, then the inversion $ N $ (or $ N - M $) required to counteract loss is much smaller than $ {N_t} $, which is typical. It only remains to determine how the laser evolution depends on the pump strength.
The evolution of a four-level laser is illustrated in Fig. 4. For short times, the photon number is small and the upper-level-electron number increases linearly with time. If this growth were to continue, all the electrons would be in the upper level after $ b/c $ transit times. When $ aN \gt b $, the photons experience net gain, their number increases rapidly (spikes) and the electron number decreases concomitantly. When $ aN \lt b $, the photons experience net loss and their number decreases rapidly. For moderate pumping (left image), this process repeats itself several times before a steady state is attained, in which the gain rate $ aN $ is clamped at its threshold value $ b $ and the photon flux $ bS $ equals the pumping rate $ cM $. The diminishing oscillations are called relaxation oscillations (ROs) [8,9]. For strong pumping (right image), the photon number spikes, then tends rapidly and almost monotonically to its steady-state value. The evolution of a type A laser is similar. Compared to the previous results, the peak values of $ N/{N_t} $ and $ S/{N_t} $ are slightly lower, because stimulated pump-photon emission limits the number of electrons.
The evolution of a type B laser is illustrated in Fig. 5. Initially, the photon number is small and the upper-level-electron number grows linearly with time. Subsequently, the growth of the upper-electron number slows because the lower-electron number decreases. Like a four-level laser, for moderate pumping (left image), the photon number spikes, then exhibits ROs, whereas for strong pumping (right image), the photon number spikes, then tends almost monotonically to its steady-state value. Compared to the four-level results, the onset of photon generation occurs later, and the peak values of $ N/{N_t} $ and $ S/{N_t} $ are higher, because $ N $ must exceed $ {N_t}/2 $ to produce net gain. In steady state, the net-gain rate $ a(N - M) $ is clamped at its threshold value $ b $. The photon flux $ bS $ still equals the pumping rate $ cM $, but the steady-state value of $ M $ is lower by a factor of 2.
3. LANGEVIN EQUATIONS
The rate equations (28)–(30) are deterministic, so they can describe the growth and saturation of the signal-photon number, but cannot describe photon-number fluctuations. One can mimic the effects of quantum fluctuations by adding random source terms to the rate equations, one source term for each deterministic process. In Section 3.A, the stochastic (Langevin) rate equations are stated and the properties of their source terms are discussed briefly. In Section 3.B, the Langevin equations are perturbed about an above-threshold equilibrium. The result is a set of linear stochastic equations for the electron- and photon-number deviations. The upper- and lowest-level electron deviations are anti-correlated, so the lowest-electron deviation can be eliminated. In Section 3.C, the simplified Langevin equations are used to model ROs, and in Section 3.D, they are used to derive deterministic moment equations for the upper-electron and photon-number variances and correlation.
A. Langevin Equations
The Langevin equations associated with the rate equations are
(31)$${d_t}S \def\LDeqtab{}= - {a^\prime}MS + aN(S + 1) - bS - R_a^\prime + {R_a} - {R_b},$$
(32)$$\begin{split}{d_t}N &= {a^\prime}MS - aN(S + 1) + cM - {c^\prime}N - dN\\&\quad + R_a^\prime - {R_a}+ {R_c} - R_c^\prime - {R_d},\end{split}$$
(33)$$\begin{split}{d_t}M &= - {a^\prime}MS + aN(S + 1) - cM + {c^\prime}N + dN\\&\quad - R_a^\prime + {R_a} - {R_c}+ R_c^\prime + {R_d}.\end{split}$$
On the right side of these equations, each source term $ {R_j} $ is the product of $ S_j^{1/2} $, which is a deterministic function of the electron and photon numbers, and $ {r_j} $, which is a random function of time with the properties (34)$$\langle {r_j}(t)\rangle = 0,\quad \langle {r_j}(t){r_k}({t^\prime})\rangle = {\delta _{jk}}\delta (t - {t^\prime}),$$
where $ \langle \rangle $ denotes an ensemble average. The first part of Eq. (34) states that positive and negative impulses are equally likely, whereas the second part states that different impulses are statistically independent. Notice that Eqs. (32) and (33) conserve the total electron number, even in the presence of noise.Formulas for the source strengths $ {S_j} $ will be derived in Section 4. In the meantime, one can specify the source strengths tentatively by analogy with the known properties of shot noise [10,11]. In this phenomenon, electrons enter a detector at random times, but at a specified mean rate ($ \gamma $). It was shown in [7,12] that the number of electrons in the detector has Poisson statistics, in which the number mean and variance both equal $ \gamma t $. The variance increases (is driven) at the same rate as the mean. These results only apply to a constant driving process (for example, pumping of the upper-level electrons with constant $ M $), but it is reasonable to suppose that similar results apply to the other processes, which are assumed to be independent. With these assumptions, the source strengths
(35)$$\begin{split}S_a^\prime &= {a^\prime}\langle MS\rangle ,\quad {S_a} = a\langle N(S + 1)\rangle , \quad {S_b} = b\langle S\rangle , \\ {S_c} &= c\langle M\rangle ,\quad S_c^\prime = {c^\prime}\langle N\rangle ,\quad {S_d} = d\langle N\rangle \end{split}$$
equal the corresponding rates in the mean equations (provided that $ \langle {R_j}\rangle = 0 $).In Eqs. (32) and (33), the deterministic $ {c^\prime} $ and $ d $ terms appear in the combination $ {c^\prime}N + dN = ({c^\prime} + d)N $, and the random terms appear in the combination $ R_c^\prime + {R_d} $. The random terms both decrease (increase) $ N $ and increase (decrease) $ M $, and neither affects $ S $. Hence, one can replace the deterministic terms by $ {d^\prime}N $, where $ {d^\prime} = {c^\prime} + d $, and the random terms by $ R_d^\prime $, where $ S_d^\prime = {c^\prime}\langle N\rangle + d\langle N\rangle = {d^\prime}\langle N\rangle $.
To be precise, random source terms should be added to the fundamental rate equations (1)–(4) and (11), (12)–(14) and (19), and (20)–(22) and (27). According to Eqs. (35), the source strengths equal the corresponding deterministic rates, so the same approximations that allow one to neglect some of the deterministic terms also allow one to neglect the corresponding source terms. Hence, one need only add to the rate equations source terms that correspond to the retained deterministic terms.
Equations (31)–(33) are stochastic and depend nonlinearly on the electron and photon numbers, so in most circumstances one has to solve them numerically (and repeatedly, to accumulate reliable statistics). However, one can develop physical insight into laser noise by solving them approximately.
B. Linearized Langevin Equations
According to Eq. (28), the growth of the signal-photon number is initiated by spontaneous emission. Below threshold, the output photon flux is proportional to the spontaneous-emission rate. However, above threshold, the photon flux depends only weakly on this rate, so spontaneous emission can be neglected. In this regime, the Langevin equations (31)–(33) can be analyzed perturbatively. Let $ M = {M_0} + {M_1} $, $ N = {N_0} + {N_1} $, and $ S = {S_0} + {S_1} $, where the subscript 0 denotes a (large) zeroth-order quantity and the subscript 1 denotes a (small) first-order quantity (deviation). Then, in zeroth-order equilibrium and without noise, Eqs. (31)–(33) imply that
(36)$$a{N_0} - {a^\prime}{M_0} = b,$$
(37)$$(a{N_0} - {a^\prime}{M_0}){S_0} = b{S_0} = c{M_0} - {d^\prime}{N_0}.$$
Net gain ($ a{N_0} - {a^\prime}{M_0} $) compensates loss ($ b $), and the output photon flux ($ b{S_0} $) equals the difference between the rates of stimulated pump-photon absorption ($ c{M_0} $) and pump- and non-lasing-photon emission ($ {d^\prime}{N_0} $). In the strong-pumping regime, $ c{M_0} \gg {d^\prime}{N_0} $.The first-order equations, with noise, are
(38)$$\begin{split}{d_t}{S_1} &= (a{N_0} - {a^\prime}{M_0} - b){S_1} + (a{S_0}){N_1} - ({a^\prime}{S_0}){M_1}\\&\quad - R_a^\prime + {R_a} - {R_b},\end{split}$$
(39)$$\begin{split}{d_t}{N_1} &= ({a^\prime}{M_0} - a{N_0}){S_1} - (a{S_0} + {d^\prime}){N_1}\\&\quad + ({a^\prime}{S_0} + c){M_1}+ R_a^\prime - {R_a} + {R_c} - R_d^\prime,\end{split}$$
(40)$$\begin{split}{d_t}{M_1} &= (a{N_0} - {a^\prime}{M_0}){S_1} + (a{S_0} + {d^\prime}){N_1}\\&\quad - ({a^\prime}{S_0} + c){M_1} - R_a^\prime + {R_a} - {R_c} + R_d^\prime,\end{split}$$
where the source terms only depend on zeroth-order quantities. One can use Eqs. (38)–(40) to study the evolution of number deviations. However, (41)$${d_t}({M_1} + {N_1}) = 0,$$
so one can conclude that $ {M_1} = - {N_1} $ and eliminate the $ {M_1} $ equation.The remaining rate equations are
(42)$$\begin{split}{d_t}{S_1} &= (a{N_0} - {a^\prime}{M_0} - b){S_1} + (a{S_0} + {a^\prime}{S_0}){N_1}\\&\quad - R_a^\prime + {R_a} - {R_b},\end{split}$$
(43)$$\begin{split}{d_t}{N_1} &= ({a^\prime}{M_0} - a{N_0}){S_1} - (a{S_0} + {a^\prime}{S_0} + c + {d^\prime}){N_1}\\&\quad + R_a^\prime - {R_a} + {R_c} - R_d^\prime.\end{split}$$
These linearized Langevin equations can be rewritten in the compact forms [6] (44)$${d_t}{S_1} \def\LDeqtab{}= {\gamma _{\textit ss}}{S_1} + {\gamma _{\textit sn}}{N_1} + {R_s},$$
(45)$${d_t}{N_1}\def\LDeqtab{} = - {\gamma _{\textit ns}}{S_1} - {\gamma _{\textit nn}}{N_1} + {R_n}.$$
On the right side of these equations, the compound sources have the properties (46)$$\langle {R_j}(t)\rangle = 0,\quad\langle {R_j}(t){R_k}({t^\prime})\rangle = {R_{jk}}\delta (t - {t^\prime}),$$
where the source strengths (variances and correlation) are (47)$$\begin{split}{R_{\textit ss}} &= S_a^\prime + {S_a} + {S_b},\quad {R_{\textit ns}} = - S_a^\prime - {S_a}, \\ {R_{\textit nn}} &= S_a^\prime + {S_a} + {S_c} + S_d^\prime.\end{split}$$
For convenience, the coupling coefficients are specified in Table 2 and the source strengths are specified in Table 3. In the analyses that follow, the $ {\gamma _{\textit ss}} $ terms are retained so that the results remain valid for more complicated models in which $ {\gamma _{\textit ss}} \ne 0 $ [5,6].Table 2. Coupling Coefficients for Three Types of Lasera
Table 3. Source Strengths for Three Types of Lasera
C. Relaxation Oscillations
The characteristic equation associated with Eqs. (44) and (45) can be written in the form
(48)$${\gamma ^2} + 2{\nu _0}\gamma + \omega _0^2 = 0,$$
where the squared frequency and damping parameters are (49)$$\omega _0^2 = {\gamma _{\textit ns}}{\gamma _{\textit sn}} - {\gamma _{\textit nn}}{\gamma _{\textit ss}}, \quad {\nu _0} = ({\gamma _{\textit nn}} - {\gamma _{\textit ss}})/2,$$
respectively [9,13]. If $ {\nu _0} $ is less than, equal to, or greater than $ {\omega _0} $, the laser is termed under-damped, critically damped, or over-damped, respectively. For laser models in which $ {\gamma _{\textit ss}} = 0 $ (which include all the models discussed herein), $ {\omega _0} = ({\gamma _{\textit ns}}{\gamma _{\textit sn}}{)^{1/2}} $ and $ {\nu _0} = {\gamma _{\textit nn}}/2 $. It follows from Eq. (48) that the characteristic exponent $ \gamma = - {\nu _0} \pm i{\omega _r} $, where the resonance frequency $ {\omega _r} = (\omega _0^2 - \nu _0^2{)^{1/2}} $. In terms of the rate coefficients, the exponent (50)$$\gamma = ({\gamma _{\textit ss}} - {\gamma _{\textit nn}})/2 \pm i{[{\gamma _{\textit ns}}{\gamma _{\textit sn}} - {({\gamma _{\textit nn}} + {\gamma _{\textit ss}})^2}/4]^{1/2}}.$$
In the absence of noise ($ {R_s} $, $ {R_n} = 0 $), the solutions of Eqs. (44) and (45) can be written in the forms
(51)$${S_1}(t) \def\LDeqtab{}= {G_{\textit ss}}(t){S_1}(0) + {G_{\textit sn}}(t){N_1}(0),$$
(52)$${N_1}(t) \def\LDeqtab{}= {G_{\textit ns}}(t){S_1}(0) + {G_{\textit nn}}(t){N_1}(0),$$
where the transfer (Green) functions are (53)$${G_{\textit ss}}(t) \def\LDeqtab{}= [\cos ({\omega _r}t) + {\gamma _a}\sin ({\omega _r}t)/{\omega _r}]\exp ( - {\nu _0}t),$$
(54)$${G_{\textit sn}}(t)\def\LDeqtab{} = [{\gamma _{\textit sn}}\sin ({\omega _r}t)/{\omega _r}]\exp ( - {\nu _0}t),$$
(55)$${G_{\textit ns}}(t)\def\LDeqtab{}= - [{\gamma _{\textit ns}}\sin ({\omega _r}t)/{\omega _r}]\exp ( - {\nu _0}t),$$
(56)$${G_{\textit nn}}(t) \def\LDeqtab{}= [\cos ({\omega _r}t) - {\gamma _a}\sin ({\omega _r}t)/{\omega _r}]\exp ( - {\nu _0}t),$$
and the rate coefficient $ {\gamma _a} = ({\gamma _{\textit nn}} + {\gamma _{\textit ss}})/2 $. The Green function $ {G_{jk}}(t) $ describes the effect on deviation $ j $ at time $ t $ of a unit initial value of deviation $ k $. [More generally, the Green function $ {G_{jk}}(t - {t^\prime}) $ describes the effect on deviation $ j $ at time $ t $ of a unit impulse applied to deviation $ k $ at the earlier time $ {t^\prime} $]. Although solutions (51) and (52) cannot describe the first photon-number spike (which is a nonlinear phenomenon), they should describe the subsequent evolutions of the electron and photon numbers. They predict that the electron- and photon-number deviations exhibit ROs if the laser is under-damped and decrease monotonically if the laser is critically damped or over-damped.Consider the four-level laser of Fig. 4. For moderate pumping ($ c/b = 0.01 $), the normalized frequencies predicted by Eqs. (49) are $ {\omega _0}/b = 0.30 $, $ {\nu _0}/b = 0.05 $, and $ {\omega _r}/b = 0.30 $, and the normalized period $ b{\tau _r} = 2\pi b/{\omega _r} = 21 $. For strong pumping ($ c/b = 0.10 $), the frequencies are $ {\omega _0}/b = 0.95 $, $ {\nu _0}/b = 0.50 $, and $ {\omega _r}/b = 0.81 $, and the period $ b{\tau _r} = 7.8 $. The predicted periods are consistent with the numerical results displayed in Fig. 4. The predicted frequencies and periods of a type A laser are similar.
Now consider the type B laser of Fig. 5. For moderate pumping ($ c/b = 0.01 $), the predicted frequencies are $ {\omega _0}/b = 0.30 $, $ {\nu _0}/b = 0.05 $, and $ {\omega _r}/b = 0.30 $, and the predicted period $ b{\tau _r} = 21 $. For strong pumping ($ c/b = 0.10 $), the frequencies are $ {\omega _0}/b = 0.95 $, $ {\nu _0}/b = 0.50 $, and $ {\omega _r}/b = 0.81 $, and the period $ b{\tau _r} = 7.8 $. The predicted periods are consistent with the numerical results displayed in Fig. 5. Thus, perturbation theory describes the approach to equilibrium accurately.
In these examples, the type B frequencies and damping rates are equal to the four-level values, because the examples involve the same pump parameters. The factors of 2 in the coupling coefficients listed in Table 2 (which are due to signal-photon absorption) are compensated by reduced values of $ {S_0} = c{M_0}/b $. In a four-level laser, $ {M_0} = {N_t} - {N_i} $, where $ {N_i} = b/a $ is the inversion, whereas in a type B laser, $ {M_0} = ({N_t} - {N_i})/2 $.
Equations (44) and (45) can also be solved analytically in the presence of noise ($ {R_s} $, $ {R_n} \ne 0 $). However, I will refrain from doing so until the formulas for the source strengths have been validated.
D. Linearized Langevin Moment Equations
One can use Eqs. (44) and (45) to derive equations for the second-order deviation moments $ \langle S_1^2\rangle $, $ \langle {N_1}{S_1}\rangle $, and $ \langle N_1^2\rangle $. Before doing so, a brief discussion of stochastic calculus is required. Let $ \delta t $ be a short time interval and consider the noise increment $ \int _0^{\delta t}{r_j}(t){\rm d}t $. Then it follows from the second of Eqs. (34) that
(57)$$\left\langle {\left[\int_0^{\delta t}{r_j}(t){\rm d}t\right]^2}\right\rangle ={\int _0^{\delta t}} {{\int_0^{\delta t}}}\langle {r_j}(t){r_j}({t^\prime})\rangle {\rm d}t{\rm d}{t^\prime} = \delta t.$$
In a crude sense, the noise increment is of order $ \delta {t^{1/2}} $. This scaling differentiates stochastic calculus from deterministic calculus. Stochastic calculus is reviewed in Appendix B. Ito calculus is based on the approximation that the source functions have zero correlation time, whereas Stratonovich calculus is based on the assumption that they have very short, but nonzero, correlation times. Ito calculus is an idealization, because no noise source has a zero correlation time (infinite frequency bandwidth), and no admitting system has infinite frequency bandwidth (zero response time). Nonetheless, I chose to use it in this article because it is intuitive and its predictions are consistent with the results of Section 4, which are based on the similar assumption of instantaneous changes in photon number.In Ito calculus, the differential equations (44) and (45) are equivalent to the difference equations
(58)$${S_1}(\delta t) \def\LDeqtab{}\approx (1 + {\gamma _{\textit ss}}\delta t){S_1} + ({\gamma _{\textit sn}}\delta t){N_1} + \int _0^{\delta t}{R_s}(t){\rm d}t,$$
(59)$${N_1}(\delta t) \def\LDeqtab{}\approx - ({\gamma _{\textit ns}}\delta t){S_1} + (1 - {\gamma _{\textit nn}}\delta t){N_1} + \int _0^{\delta t}{R_n}(t){\rm d}t.$$
By combining Eqs. (58) and (59), averaging the equations that result and letting $ \delta t \to 0 $, one obtains the variance and correlation equations (60)$${d_t}\langle S_1^2\rangle\def\LDeqtab{} = 2{\gamma _{\textit ss}}\langle S_1^2\rangle + 2{\gamma _{\textit sn}}\langle {N_1}{S_1}\rangle + {R_{\textit ss}},$$
(61)$$\begin{split}{d_t}\langle {N_1}{S_1}\rangle &= - {\gamma _{\textit ns}}\left\langle {S_1^2} \right\rangle + ({\gamma _{\textit ss}} - {\gamma _{\textit nn}})\left\langle {{N_1}{S_1}} \right\rangle\\&\quad + {\gamma _{\textit sn}}\langle N_1^2\rangle + {R_{\textit ns}},\end{split}$$
(62)$${d_t}\langle N_1^2\rangle\def\LDeqtab{} = - 2{\gamma _{\textit ns}}\langle {N_1}{S_1}\rangle - 2{\gamma _{\textit nn}}\langle N_1^2\rangle + {R_{\textit nn}}.$$
Equations (60)–(62) are deterministic, so it is straightforward to solve them analytically. However, I will refrain from doing so until the formulas for the source strengths have been validated.4. MARKOV EQUATIONS
Although the Langevin equations (31)–(33) and (38)–(40) model electron- and photon-number fluctuations, they are based on the premise that these quantities are real numbers that change continuously, whereas they are actually nonnegative integers that change discontinuously (instantaneously). Following Shimoda et al. [14] and McCumber [15], one can replace the Langevin equations by the family of Markov equations
(63)$$\begin{split} {d_t}F(m,n,s) &= - {a^\prime}msF(m,n,s) + {a^\prime}(m + 1)(s + 1) F(m + 1,n - 1,s + 1)\\[-5pt] &\quad - an(s + 1)F(m,n,s) + a(n + 1)sF(m - 1,n + 1,s - 1) \\[-5pt] &\quad- bsF(m,n,s) + b(s + 1)F(m,n,s + 1) \\[-5pt]&\quad - cmF(m,n,s) + c(m + 1)F(m + 1,n - 1,s) \\[-5pt]&\quad- {c^\prime}nF(m,n,s) + {c^\prime}(n + 1)F(m - 1,n + 1,s) \\[-5pt]&\quad- dnF(m,n,s) + d(n + 1)F(m - 1,n + 1,s),\end{split}$$
where $ F(m,n,s) $ is the probability that $ m $ lowest-level electrons, $ n $ upper-level electrons and $ s $ signal photons are in the cavity. (For $ m = 0 $, the second $ a $ term and $ {c^\prime} $ term are omitted, for $ n = 0 $, the second $ {a^\prime} $ and $ c $ terms are omitted, and for $ s = 0 $, the second $ a $ term is omitted.) These equations are based on the assumption that electrons and photons are created and destroyed randomly, but at specified mean rates. The signal-absorption rate is proportional to $ ms $, the signal-emission rate is proportional to $ n(s + 1) $, and the signal-loss rate is proportional to $ s $. (In the absence of spontaneous emission, the signal-emission rate is proportional to $ ns $.) The pump-absorption rate is proportional to $ mp $, the pump-emission rate is proportional to $ n(p + 1) $, and both processes change the number of pump photons. However, if the pump is strong ($ p,p + 1 \approx P \gg 1 $), then both rates are proportional to $ P $, which remains constant and is included in the formulas for $ c $ and $ {c^\prime} $. In this approximation, the effects on the electrons of pump- and non-lasing-photon emission are indistinguishable, so one can replace the $ {c^\prime} $ and $ d $ terms in Eq. (63) by a pair of $ {d^\prime} $ terms ($ {d^\prime} = {c^\prime} + d $).The Markov equation (63) describes the evolution of an ensemble of members. Let $ (m,n,s) $ denote the member with $ m $ lowest-level electrons, $ n $ upper-level electrons, and $ s $ signal photons, and consider the effects of signal absorption. If $ (m,n,s) \to (m - 1,n + 1,s - 1) $, then $ F(m,n,s) $ decreases, whereas if $ (m + 1,n - 1,s + 1) \to (m,n,s) $, then $ F(m,n,s) $ increases. For signal emission, if $ (m,n,s) \to (m + 1,n - 1,s + 1) $, then $ F(m,n,s) $ decreases, whereas if $ (m - 1,n + 1,s - 1) \to (m,n,s) $, then $ F(m,n,s) $ increases. For signal loss, which does not involve the electrons, if $ (m,n,s) \to (m,n,s - 1) $, then $ F(m,n,s) $ decreases, whereas if $ (m,n,s + 1) \to (m,n,s) $, then $ F(m,n,s) $ increases. The descriptions of the electron decay and pumping transitions, which do not involve the signal photons, are similar.
The total probability $ T = {\sum _0^\infty}{\sum _0^\infty}{\sum _0^\infty}F(m,n,s) $, where the summations are over $ m $, $ n $ and $ s $, respectively. It follows from Eq. (63) that
(64)$$\begin{split}{d_t}T &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 msF(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)(s + 1)F(m + 1,n - 1,s + 1) \\[-5pt]&\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 n(s + 1)F(m,n,s)+ a\sum\limits_1 \sum\limits_0 \sum\limits_1 (n + 1)sF(m - 1,n + 1,s - 1) \\[-5pt]&\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_0 sF(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_0 (s + 1)F(m,n,s + 1) \\[-5pt]&\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 mF(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)F(m + 1,n - 1,s) \\[-5pt]&\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 nF(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 (n + 1)F(m - 1,n + 1,s) \\[-5pt]& = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 msF(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 msF(m,n,s) \\[-5pt]&\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 n(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 n(s + 1)F(m,n,s) \\[-5pt] &\quad- b\sum\limits_0 \sum\limits_0 \sum\limits_1 sF(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_1 sF(m,n,s) \\[-5pt]&\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 mF(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 mF(m,n,s) \\[-5pt]&\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 nF(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 nF(m,n,s) \\[-5pt]& = 0.\end{split}$$
Thus, the Markov equation conserves the total probability, which is normalized to 1. Notice that the manipulation of the $ b $ terms in Eq. (64) did not change $ m $ and $ n $, so one can multiply these terms by any polynomial of $ m $ and $ n $ and they would still cancel. Likewise, the manipulation of the $ c $ and $ {d^\prime} $ terms did not change $ s $, so one can multiply these terms by any polynomial of $ s $ and they would still cancel: Signal loss does not affect the electrons directly, and pump-induced electron transitions do not affect the signal photons directly.A. Markov Moment Equations
The number moments $ \langle {m}^{\alpha} {n^\beta }{s^\gamma }\rangle = \sum _0^\infty \sum _0^\infty \sum _0^\infty {m^\alpha }{n^\beta }{s^\gamma }F(m,n,s) $. Three first-order moments and six second-order moments are required to model number fluctuations. While writing this article, I had to decide between doing nine shorter calculations ($ {a^\prime} = 0 $ or $ {c^\prime} = 0 $) three times and doing nine longer calculations ($ {a^\prime} \ne 0 $ and $ {c^\prime} \ne 0 $) once. I chose the latter option, which is the lesser evil. The first signal-photon moment obeys the equation
(65)$$\begin{split}{d_t}\langle s\rangle &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 m{s^2}F(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)s(s + 1)F(m + 1,n - 1,s + 1) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 ns(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 (n + 1){s^2}F(m - 1,n + 1,s - 1) \\[-2pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_0 {s^2}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_0 s(s + 1)F(m,n,s + 1) \\[-2pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m{s^2}F(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 ms(s - 1)F(m,n,s) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 ns(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 n{(s + 1)^2}F(m,n,s) \\[-2pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_1 {s^2}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_1 s(s - 1)F(m,n,s).\end{split}$$
The $ c $ and $ {d^\prime} $ terms were omitted from Eq. (65) because they do not affect $ \langle s\rangle $ directly, as explained above. By combining the terms on the right side, one obtains the first-moment equation (66)$${d_t}\langle s\rangle = - {a^\prime}\langle ms\rangle + a\langle n(s + 1)\rangle - b\langle s\rangle ,$$
where $ a\langle ms\rangle $ is the mean (stimulated) absorption rate, $ a\langle {n(s + 1)} \rangle $ is the mean (stimulated and spontaneous) emission rate, and $ b $ is the mean loss rate.The first upper-level-electron moment obeys the equation
(67)$$\begin{split}{d_t}\langle n\rangle &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 mnsF(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)n(s + 1)F(m + 1,n - 1,s + 1) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 {n^2}(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 n(n + 1)sF(m - 1,n + 1,s - 1) \\[-2pt] &\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 mnF(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)nF(m + 1,n - 1,s) \\[-2pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {n^2}F(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 n(n + 1)F(m - 1,n + 1,s) \\[-2pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 mnsF(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m(n + 1)sF(m,n,s) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 {n^2}(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 n(n - 1)(s + 1)F(m,n,s) \\[-2pt] &\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 mnF(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 m(n + 1)F(m,n,s) \\[-2pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 {n^2}F(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 n(n - 1)F(m,n,s).\end{split}$$
The $ b $ terms were omitted from Eq. (67) because they do not affect $\langle n \rangle $ directly. By combining the terms on the right side, one obtains the first-moment equation (68)$${d_t}\langle n \rangle = {a^\prime}\langle {ms}\rangle - a\langle {n(s + 1)} \rangle + c\langle m \rangle - {d^\prime}\langle n \rangle ,$$
where $ c\langle m \rangle $ is the mean rate of pump-stimulated absorption and $ {d^\prime}\langle n \rangle $ is the mean rate of pump-stimulated emission.The first lowest-level-electron moment obeys the equation
(69)$$\begin{split}{d_t}\langle m \rangle &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}sF(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 m(m + 1)(s + 1)F(m + 1,n - 1,s + 1) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 mn(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 m(n + 1)sF(m - 1,n + 1,s - 1) \\[-2pt] &\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}F(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 m(m + 1)F(m + 1,n - 1,s) \\[-2pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 mnF(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 m(n + 1)F(m - 1,n + 1,s) \\[-2pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 {m^2}sF(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m(m - 1)sF(m,n,s) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 mn(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)n(s + 1)F(m,n,s) \\[-2pt] &\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 {m^2}F(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 m(m - 1)F(m,n,s) \\[-2pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 mnF(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)nF(m,n,s).\end{split}$$
By combining the terms on the right side of Eq. (69), one obtains the first-moment equation (70)$${d_t}\langle m \rangle = - {a^\prime}\langle {ms} \rangle + a\langle {n(s + 1)} \rangle - c\langle m \rangle + {d^\prime}\langle n \rangle .$$
Equations (66), (68) and (70) show that the electron- and photon-number means change at the mean rates associated with the Langevin equations (28)–(30). This commonsense result makes the Markov approach seem reasonable. Furthermore, signal-photon absorption and emission conserve the sums $ \langle m \rangle + \langle n \rangle $ and $ \langle n \rangle + \langle s \rangle $, and the difference $ \langle m \rangle - \langle s \rangle $, and pump-photon absorption and emission conserve the sum $ \langle m \rangle + \langle n \rangle $. These conservation laws were built into Eq. (63), so their appearances serve as algebra checks. Notice that the first moments are coupled to the second moments $ \langle {ms} \rangle $ and $ \langle {ns} \rangle $, which are not yet known. For reference, the $ \pm a\langle n \rangle $ terms in Eqs. (66), (68) and (70) are associated with spontaneous emission.Now consider the second number moments. The variances will be determined first, because the calculations are similar to the first-moment calculations. The second photon moment obeys the equation
(71)$$\begin{split}{d_t}\left\langle {{s^2}} \right\rangle &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 m{s^3}F(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1){s^2}(s + 1)F(m + 1,n - 1,s + 1) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 n{s^2}(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 (n + 1){s^3}F(m - 1,n + 1,s - 1) \\[-2pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_0 {s^3}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_0 {s^2}(s + 1)F(m,n,s + 1) \\[-2pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m{s^3}F(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 ms{(s - 1)^2}F(m,n,s) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 n{s^2}(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 n{(s + 1)^3}F(m,n,s) \\[-2pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_1 {s^3}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_1 s{(s - 1)^2}F(m,n,s).\end{split}$$
The $ c $ and $ {d^\prime} $ terms were omitted from Eq. (71) because they do not affect $ \left\langle {{s^2}} \right\rangle $ directly. By combining the terms on the right side, one obtains the second-moment equation (72)$$ {d_t}\langle {s^2}\rangle = {a^\prime}( - 2\langle m{s^2}\rangle \, + \,\langle ms\rangle ) + a(2\langle n{s^2}\rangle + 3\langle ns\rangle + \langle n\rangle ) + b( - 2\langle {s^2}\rangle + \langle s\rangle ). $$
The second upper-electron moment obeys the equation (73)$$\begin{split}{d_t}\left\langle {{n^2}} \right\rangle &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 m{n^2}sF(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1){n^2}(s + 1)F(m + 1,n - 1,s + 1) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 {n^3}(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 {n^2}(n + 1)sF(m - 1,n + 1,s - 1) \\[-3pt] &\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 m{n^2}F(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1){n^2}F(m + 1,n - 1,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {n^3}F(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 {n^2}(n + 1)F(m - 1,n + 1,s) \\[-3pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m{n^2}sF(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m{(n + 1)^2}sF(m,n,s) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 {n^3}(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 n{(n - 1)^2}(s + 1)F(m,n,s) \\[-3pt] &\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 m{n^2}F(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 m{(n + 1)^2}F(m,n,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 {n^3}F(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 n{(n - 1)^2}F(m,n,s).\end{split}$$
The $ b $ terms were omitted from Eq. (73) because they do not affect $ \left\langle {{n^2}} \right\rangle $ directly. By combining the terms on the right side, one obtains the second-moment equation (74)$$\begin{split}{d_t}\left\langle {{n^2}} \right\rangle& = {a^\prime}(2\langle {mns} \rangle + \langle {ms} \rangle ) + a( - 2\langle {{n^2}s} \rangle - 2\langle {{n^2}} \rangle + \langle {ns} \rangle + \langle n \rangle ) \\[-3pt] &\quad + c(2\langle {mn} \rangle + \langle m \rangle ) + {d^\prime}( - 2\langle {{n^2}} \rangle + \langle n\rangle ).\end{split}$$
The second lowest-electron moment obeys the equation (75)$$\begin{split}{d_t}\left\langle {{m^2}} \right\rangle &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^3}sF(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 {m^2}(m + 1)(s + 1)F(m + 1,n - 1,s + 1) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}n(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 {m^2}(n + 1)sF(m - 1,n + 1,s - 1) \\[-3pt] &\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^3}F(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 {m^2}(m + 1)F(m + 1,n - 1,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}nF(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 {m^2}(n + 1)F(m - 1,n + 1,s) \\[-3pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 {m^3}sF(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m{(m - 1)^2}sF(m,n,s) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 {m^2}n(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 {(m + 1)^2}n(s + 1)F(m,n,s) \\[-3pt] &\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 {m^3}F(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 m{(m - 1)^2}F(m,n,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 {m^2}nF(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 {(m + 1)^2}nF(m,n,s).\end{split}$$
By combining the terms on the right side of Eq. (75), one obtains the second-moment equation (76)$$\begin{split}{d_t}\left\langle {{m^2}} \right\rangle &= {a^\prime}( - 2\left\langle {{m^2}s} \right\rangle + \langle {ms}\rangle ) + a(2\langle {mns} \rangle + 2\langle {mn} \rangle + \langle {ns} \rangle + \langle n \rangle ) \\ &\quad + c( - 2\left\langle {{m^2}} \right\rangle + \langle m \rangle ) + {d^\prime}(2\langle {mn} \rangle + \langle n \rangle ).\end{split}$$
Notice that the second individual moments are coupled to third moments, which are unknown.The number deviations $ \delta s = s - \langle s \rangle $, $ \delta n = n - \langle n \rangle $ and $ \delta m = m - \langle m \rangle $, so the second deviation moments $ \left\langle {\delta {s^2}} \right\rangle = \left\langle {{s^2}} \right\rangle - {\left\langle s \right\rangle ^2} $ and $ \langle {\delta n\delta s} \rangle = \langle {ns} \rangle - \langle n \rangle \langle s \rangle $, and the other deviation moments are defined similarly. By combining Eqs. (66), (68) and (70) with Eqs. (72), (74) and (76), one obtains the variance equations
(77)$$\begin{split}{d_t}\left\langle {\delta {s^2}} \right\rangle &= - 2{a^\prime}\langle {\delta ms\delta s} \rangle + 2a\langle {\delta ns\delta s} \rangle + 2a\langle {\delta n\delta s} \rangle - 2b\left\langle {\delta {s^2}} \right\rangle \\[-2pt] &\quad + {a^\prime}\langle {ms} \rangle + a\langle {n(s + 1)} \rangle + b\langle s \rangle , \end{split}$$
(78)$$\begin{split}{d_t}\left\langle {\delta {n^2}} \right\rangle &= 2{a^\prime}\langle {\delta ms\delta n} \rangle - 2a\langle {\delta ns\delta n} \rangle - 2a\left\langle {\delta {n^2}} \right\rangle + 2c\langle {\delta m\delta n} \rangle - 2{d^\prime}\left\langle {\delta {n^2}} \right\rangle\\&\quad+ {a^\prime}\langle {ms} \rangle + a\langle {n(s + 1)} \rangle + c\langle m \rangle + {d^\prime}\langle n\rangle , \end{split}$$
(79)$$\begin{split}{d_t}\left\langle {\delta {m^2}} \right\rangle &= - 2{a^\prime}\langle {\delta ms\delta m} \rangle + 2a\langle {\delta ns\delta m} \rangle + 2a\langle {\delta m\delta n} \rangle - 2c\left\langle {\delta {m^2}} \right\rangle + 2{d^\prime}\langle {\delta m\delta n}\rangle\\&\quad+ {a^\prime}\langle {ms} \rangle + a\langle {n(s + 1)} \rangle + c\langle m \rangle + {d^\prime}\langle n\rangle .\end{split}$$
Notice that the driving terms in Eqs. (77)–(79) are consistent with the shot-noise rule. For reference, the terms $ 2a\langle {\delta n\delta s} \rangle + a\langle n\rangle $, $ - 2a\left\langle {\delta {n^2}} \right\rangle + a\langle n\rangle $, and $ 2a\langle {\delta m\delta n} \rangle + a\langle n \rangle $ are associated with spontaneous emission.The second photon and upper-electron moment obeys the equation
(80)$$\begin{split}{d_t}\langle {ns} \rangle& = - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 mn{s^2}F(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)ns(s + 1)F(m + 1,n - 1,s + 1) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 {n^2}s(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 n(n + 1){s^2}F(m - 1,n + 1,s - 1) \\[-2pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_0 n{s^2}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_0 ns(s + 1)F(m,n,s + 1) \\[-2pt] &\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 mnsF(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)nsF(m + 1,n - 1,s) \\[-2pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {n^2}sF(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 n(n + 1)sF(m - 1,n + 1,s) \\[-2pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 mn{s^2}F(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m(n + 1)s(s - 1)F(m,n,s) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 {n^2}s(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 n(n - 1)(s + {1)^2}F(m,n,s) \\[-2pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_1 n{s^2}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_1 ns(s - 1)F(m,n,s) \\[-2pt] &\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 mnsF(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 m(n + 1)sF(m,n,s) \\[-2pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 {n^2}sF(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 n(n - 1)sF(m,n,s).\end{split}$$
By combining the terms on the right side of Eq. (80), one obtains the second-moment equation (81)$$\begin{split}{d_t}\langle {ns} \rangle &= {a^\prime}( - \langle {mns}\rangle + \left\langle {m{s^2}} \right\rangle - \langle {ms} \rangle ) + a(\left\langle {{n^2}s} \right\rangle - \left\langle {n{s^2}} \right\rangle + \left\langle {{n^2}} \right\rangle - 2\langle {ns}\rangle -\langle n \rangle ) \\[-2pt] &\quad - b \langle {ns}\rangle + c \langle {ms} \rangle - {d^\prime}\langle {ns} \rangle .\end{split}$$
The second upper- and lowest-level electron moment obeys the equation (82)$$\begin{split}{d_t}\langle {mn} \rangle &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}nsF(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 m(m + 1)n(s + 1)F(m + 1,n - 1,s + 1) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 m{n^2}(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 mn(n + 1)sF(m - 1,n + 1,s - 1) \\[-3pt] &\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}nF(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 m(m + 1)nF(m + 1,n - 1,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 m{n^2}F(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 mn(n + 1)F(m - 1,n + 1,s) \\[-3pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 {m^2}nsF(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m(m - 1)(n + 1)sF(m,n,s) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 m{n^2}(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)n(n - 1)(s + 1)F(m,n,s) \\[-3pt] &\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 {m^2}nF(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 m(m - 1)(n + 1)F(m,n,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 m{n^2}F(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)n(n - 1)F(m,n,s).\end{split}$$
The $ b $ terms were omitted from Eq. (82) because they do not affect $ \langle {mn} \rangle $ directly. By combining the terms on the right side, one obtains the second-moment equation (83)$$\begin{split}{d_t}\langle {mn} \rangle& = {a^\prime}(\left\langle {{m^2}s} \right\rangle - \langle {mns} \rangle - \langle {ms} \rangle ) + a( - \langle {mns} \rangle + \left\langle {{n^2}s} \right\rangle - \langle {mn} \rangle + \left\langle {{n^2}} \right\rangle - \langle {ns}\rangle - \langle n \rangle ) \\[-3pt] &\quad + c(\left\langle {{m^2}} \right\rangle - \langle {mn} \rangle - \langle m \rangle ) + {d^\prime}( - \langle {mn} \rangle + \left\langle {{n^2}} \right\rangle - \langle n \rangle ).\end{split}$$
The second photon and lowest-level electron moment obeys the equation (84)$$\begin{split}{d_t}\langle {ms}\rangle& = - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}{s^2}F(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 m(m + 1)s(s + 1)F(m + 1,n - 1,s + 1) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 mns(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 m(n + 1){s^2}F(m - 1,n + 1,s - 1) \\[-3pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_0 m{s^2}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_0 ms(s + 1)F(m,n,s + 1) \\[-3pt] &\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}sF(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 m(m + 1)sF(m + 1,n - 1,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 mnsF(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 m(n + 1)sF(m - 1,n + 1,s) \\[-3pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 {m^2}{s^2}F(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m(m - 1)s(s - 1)F(m,n,s) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 mns(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)n{(s + 1)^2}F(m,n,s) \\[-3pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_1 m{s^2}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_1 ms(s - 1)F(m,n,s) \\[-3pt] &\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 {m^2}sF(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 m(m - 1)sF(m,n,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 mnsF(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)nsF(m,n,s).\end{split}$$
By combining the terms on the right side of Eq. (84), one obtains the second-moment equation (85)$$\begin{split}{d_t}\langle {ms} \rangle& = {a^\prime}( - \left\langle {{m^2}s} \right\rangle - \left\langle {m{s^2}} \right\rangle + \langle {ms} \rangle )\\ &\quad + a(\langle {mns} \rangle + \left\langle {n{s^2}} \right\rangle + \langle {mn} \rangle + 2\langle {ns} \rangle + \langle n \rangle ) \\ &\quad - b\langle {ms} \rangle - c\langle {ms} \rangle + {d^\prime}\langle {ns} \rangle .\end{split}$$
The second mixed moments are also coupled to third moments, which are unknown.By combining Eqs. (66), (68) and (70) with Eqs. (81), (83) and (85), one obtains the correlation equations
(86)$$\begin{split}{d_t}\langle {\delta n\delta s} \rangle& = {a^\prime}(\langle {\delta ms\delta s} \rangle - \langle {\delta ms\delta n} \rangle )\\ &\quad + a(\langle {\delta ns\delta n}\rangle - \langle {\delta ns\delta s} \rangle ) + a\left\langle {\delta {n^2}} \right\rangle \\ &\quad - (a + b + {d^\prime})\langle {\delta n\delta s} \rangle + c\langle {\delta m\delta s} \rangle\\ &\quad - {a^\prime}\langle {ms} \rangle - a\langle {n(s + 1)} \rangle , \end{split}$$
(87)$$\begin{split}{d_t}\langle {\delta m\delta n} \rangle &= {a^\prime}(\langle {\delta ms\delta m} \rangle - \langle {\delta ms\delta n} \rangle )\\ &\quad + a(\langle {\delta ns\delta n} \rangle - \langle {\delta ns\delta m} \rangle ) \\ &\quad - (a + c + {d^\prime})\langle {\delta m\delta n} \rangle + (a + {d^\prime})\left\langle {\delta {n^2}} \right\rangle \\ &\quad + c\left\langle {\delta {m^2}} \right\rangle - {a^\prime}\langle {ms}\rangle - a\langle {n(s + 1)} \rangle\\ &\quad - c\langle m \rangle - {d^\prime}\langle n \rangle , \end{split}$$
(88)$$ \begin{split}{d_t}\langle {\delta m\delta s} \rangle &= - {a^\prime}(\langle {\delta ms\delta m} \rangle + \langle {\delta ms\delta s} \rangle )\\ &\quad + a(\langle {\delta ns\delta m} \rangle + \langle {\delta ns\delta s} \rangle ) + a\langle {\delta m\delta n} \rangle \\&\quad+ a\langle {\delta n\delta s} \rangle - (b + c)\langle {\delta m\delta s } \rangle \\ &\quad+ {d^\prime}\langle {\delta n\delta s} \rangle + {a^\prime}\langle {ms}\rangle + a\langle {n(s + 1)} \rangle .\end{split}$$
Notice that the driving terms in Eqs. (86)–(88) are consistent with the shot-noise rule. In the first and second of these equations, the driving terms are negative, because the deviations involved are anti-correlated. For reference, the terms $ a\left\langle {\delta {n^2}} \right\rangle - a\langle {\delta n\delta s} \rangle - a\langle n \rangle $, $ - a\langle {\delta m\delta n} \rangle + a\left\langle {\delta {n^2}} \right\rangle - a\langle n \rangle $, and $ a\langle {\delta m\delta n} \rangle + a\langle {\delta n\delta s} \rangle + a\langle n \rangle $ are associated with spontaneous emission.By combining Eqs. (78), (79) and (87), one can show that
(89)$${d_t}\left\langle {{{(\delta m + \delta n)}^2}} \right\rangle = 0.$$
The upper-electron and lowest-electron deviations are completely anti-correlated. If one retains only the absorption and emission terms in the variance and correlation equations, one can show that (90)$${d_t}\left\langle {{{(\delta n + \delta s)}^2}} \right\rangle = 0 = {d_t}\left\langle {{{(\delta m - \delta s)}^2}} \right\rangle .$$
In the absence of loss and pumping, the photon and upper-electron deviations are completely anti-correlated, whereas the photon and lowest-electron deviations are completely correlated. These correlations were built into Eq. (63), so their appearances serve as algebra checks.B. Closed Moment Equations
Within the Markov framework, the mean equations (66), (68) and (70), the variance equations (77)–(79), and the correlation equations (86)–(88) are exact. However, the Markov moment equations do not form a closed set. Fortunately, for a practical laser the first moments (means) are much larger than the deviations and their moments. By making the approximations $\langle {ms} \rangle \approx \langle m \rangle \langle s \rangle $ and $ \langle {ns} \rangle \approx\langle n \rangle \langle s \rangle $ in Eqs. (66), (68) and (70), which amounts to neglecting the correlations $ \delta m\delta s $ and $ \delta n\delta s $, respectively, one obtains the approximate mean equations
(91)$${d_t}\langle s \rangle \def\LDeqtab{}\approx - {a^\prime}\langle m \rangle \langle s \rangle + a\langle n \rangle (\langle s \rangle + 1) - b\langle s\rangle ,$$
(92)$${d_t}\langle n \rangle \def\LDeqtab{}\approx {a^\prime}\langle m \rangle \langle s \rangle - a\langle n \rangle (\langle s \rangle + 1) + c\langle m \rangle - {d^\prime}\langle n \rangle ,$$
(93)$${d_t}\langle m \rangle \def\LDeqtab{}\approx - {a^\prime}\langle m\rangle \langle s \rangle + a\langle n \rangle (\langle s \rangle + 1) - c\langle m \rangle + {d^\prime}\langle n \rangle .$$
Equations (91)–(93) are equivalent to the rate equations (28)–(30), which is a reasonable result. The spontaneous emission terms were retained only to allow the signal number to grow from zero. For all other purposes, they can be neglected, because $ 1 \lt |\delta s| \ll \langle s \rangle $. Notice that the approximate mean equations are closed. Because Eqs. (92) and (93) conserve the total electron number $ \langle m \rangle + \langle n \rangle $, one can omit the latter equation.At this stage, one could use the approximations $ \delta ms \approx \langle m \rangle \delta s + \langle s \rangle \delta m $ and $ \delta ns \approx \langle n \rangle \delta s + \langle s \rangle \delta n $ to simplify all six variance and correlation equations. However, Eq. (89) shows that the electron-number fluctuations are anti-correlated. By making the substitution $ \delta m = - \delta n $ in Eqs. (87) and (88), one obtains the negatives of Eqs. (78) and (86), respectively. Hence, it is sufficient to retain and simplify Eqs. (77), (78) and (86). By making the aforementioned substitution and omitting the spontaneous-emission terms, which are smaller than the other terms by factors of $ \langle m \rangle $, $ \langle n \rangle $ and $\langle s \rangle $, one obtains the intermediate equations
(94)$$\begin{split}{d_t}\left\langle {\delta {s^2}} \right\rangle &= - 2{a^\prime}\langle {\delta ms\delta s} \rangle + 2a\langle {\delta ns\delta s} \rangle - 2b\left\langle {\delta {s^2}} \right\rangle \\ &\quad + {a^\prime}\langle {ms} \rangle + a\langle {ns} \rangle + b\langle s \rangle , \end{split}$$
(95)$$\begin{split} {d_t}\langle {\delta n\delta s} \rangle &= - {a^\prime}\langle {\delta ms\delta n} \rangle + {a^\prime}\langle {\delta ms\delta s} \rangle + a\langle {\delta ns\delta n} \rangle \\&\quad - a\langle {\delta ns\delta s} \rangle - (b + c + {d^\prime})\langle {\delta n\delta s} \rangle\\&\quad - {a^\prime}\langle {ms} \rangle - a\langle {ns} \rangle ,\end{split}$$
(96)$$ \begin{split} {d_t}\left\langle {\delta {n^2}} \right\rangle &= 2{a^\prime}\langle {\delta ms\delta n} \rangle - 2a\langle {\delta ns\delta n} \rangle - 2(c + {d^\prime})\left\langle {\delta {n^2}} \right\rangle \\&\quad+ {a^\prime}\langle {ms} \rangle + a\langle {ns} \rangle + c\langle m \rangle + {d^\prime}\langle n \rangle .\end{split}$$
These equations still satisfy the first of Eqs. (90). By making the aforementioned approximations for $ \langle {ms} \rangle $, $ \langle {ns} \rangle $, $ \delta ms $ and $ \delta ns $, one obtains the approximate variance and correlation equations (97)$$\begin{split}{d_t}\left\langle {\delta {s^2}} \right\rangle &= 2({a^\prime}\langle s \rangle + a\langle s \rangle )\langle {\delta n\delta s} \rangle\\ &\quad + 2(a\langle n \rangle - {a^\prime}\langle m \rangle - b)\left\langle {\delta {s^2}} \right\rangle \\ &\quad + {a^\prime}\langle m \rangle \langle s \rangle + a\langle n \rangle \langle s \rangle + b\langle s \rangle , \end{split}$$
(98)$$\begin{split}{d_t}\langle {\delta n\delta s} \rangle& = ({a^\prime}\langle s \rangle + a\langle s \rangle )\left\langle {\delta {n^2}} \right\rangle\\ &\quad + (a\langle n \rangle - {a^\prime}\langle m \rangle - b)\langle {\delta n\delta s} \rangle \\ &\quad - ({a^\prime}\langle s \rangle + a\langle s \rangle + c + {d^\prime})\langle {\delta n\delta s} \rangle\\ &\quad - (a\langle n \rangle - {a^\prime}\langle m \rangle )\left\langle {\delta {s^2}} \right\rangle\\ &\quad - {a^\prime}\langle m \rangle \langle s \rangle - a\langle n \rangle \langle s \rangle , \end{split}$$
(99)$$ \begin{split}{d_t}\left\langle {\delta {n^2}} \right\rangle &= - 2({a^\prime}\langle s \rangle + a\langle s \rangle + c + {d^\prime})\left\langle {\delta {n^2}} \right\rangle\\ &\quad - 2(a \langle n \rangle - {a^\prime}\langle m \rangle )\langle {\delta n\delta s} \rangle\\&\quad+ {a^\prime}\langle m \rangle \langle s \rangle + a \langle n \rangle \langle s \rangle + c \langle m \rangle + {d^\prime}\langle n \rangle .\end{split}$$
Equations (91), (92) and (97)–(99) form a complete set. The driving terms in Eqs. (97)–(99) justify the shot-noise rule: The driving terms in the variance equations are the sums of the moduli of the associated terms in the mean equations, whereas those in the correlation equation have the same magnitudes but opposite signs, because the changes in the signal-photon and upper-electron numbers are anti-correlated. Furthermore, the coefficients of the variances and correlation on the right sides of Eqs. (97)–(99) are consistent with the coefficients in Eqs. (60)–(62), which were based on the linearized Langevin equations, so the closure approximations are sound.As a bonus, the shot-noise rule derived in this section is consistent with the laws of quantum optics. The Heisenberg-picture calculations [15–19] are based on quantum Langevin equations for the number operators. The operator source terms in these equations have properties that are consistent with the shot-noise rule. The Schrödinger-picture calculations [20–22] are based on an equation for the signal density operator in the number-state representation. The diagonal entries of this equation are consistent with the Markov equation [Eq. (63)].
C. Source Strengths for Different Lasers
It is instructive to consider the source strengths associated with different lasers. In the notation of Table 3, for a four-level laser the compound source strengths $ {R_{\textit ss}} $, $ {R_{\textit sn}} $, and $ {R_{\textit nn}} $ are $ a{N_0}{S_0} + b{S_0} $, $ - a{N_0}{S_0} $, and $ a{N_0}{S_0} + c{M_0} + d{N_0} $, respectively, where the equilibrium conditions are $ a{N_0} = b $ and $ b{S_0} = c{M_0} - d{N_0} $. In terms of the photon flux $ b{S_0} $, the source strengths are $ 2b{S_0} $, $ - b{S_0} $, and $ 2b{S_0} + 2d{N_0} $. In the well-above-threshold regime ($ b{S_0} \gg d{N_0} $), these strengths depend on only the photon flux.
For a type A laser, the compound source strengths are $ a{N_0}{S_0} + b{S_0} $, $ - a{N_0}{S_0} $, and $ a{N_0}{S_0} + c{M_0} + (c + d){N_0} $, where the operating conditions are $ a{N_0} = b $ and $ b{S_0} = c{M_0} - (c + d){N_0} $. In terms of the photon flux, the source strengths are $ 2b{S_0} $, $ - b{S_0} $, and $ 2b{S_0} + 2(c + d){N_0} $. Typically, the electron inversion ($ {N_0} $) is much smaller than the total electron number ($ {N_t} = {M_0} + {N_0} $), so $ c{N_0} \ll c{M_0} \approx b{S_0} $. Type A lasers are only slightly noisier than four-level lasers (and the extra noise is restricted to the electron-number fluctuations).
For a type B laser, the compound source strengths are$ a({M_0} + {N_0}){S_0} + b{S_0} $, $ - a({M_0} + {N_0}){S_0} $, and $ a({M_0} + {N_0}){S_0} $$+\,c{M_0} + d{N_0} $, where the operating conditions are $ a({N_0} - {M_0}) = b $ and $ b{S_0} = c{M_0} - d{N_0} $. In terms of the photon flux, the source strengths are $ ({N_t}/{N_i} + 1)b{S_0} $, $ - ({N_t}/{N_i})b{S_0} $, and $ ({N_t}/{N_i} + 1)b{S_0} + 2d{N_0} $. Typically, the inversion ($ {N_i} = {N_0} - {M_0} $) is much smaller than the total number. Type B lasers are noisier than four-level lasers (and extra noise is present in both the electron- and photon-number fluctuations).
5. RELATIVE INTENSITY NOISE
Now that the shot-noise rule has been established, it is time to discuss the consequences of the stochastic deviation equations (44) and (45), and the deterministic moment equations (60)–(62), which depend upon it. Detailed discussions of RIN, which include typical numbers for specific lasers, are contained in [5,6] and many other textbooks, so in this section I will simply collect the main results and discuss them briefly.
In the absence of noise, the electron- and photon-number deviations exhibit ROs, which attenuate (Section 3.C). In the presence of noise, the number deviations reach a stationary stochastic state: The deviations and their moments fluctuate, but their ensemble or short-time averages are long-time independent. The characteristics of such states are described in Appendix A. Equations (44) and (45) are linear and are driven by additive noise (random impulses with strengths that do not depend on the deviations). Although the initiation of each contribution to $ {S_1} $ and $ {N_1} $ is random, the subsequent evolution is deterministic, so one can use regular calculus to solve the equations. (They are solved using Ito calculus in Appendix B, and equivalent results are obtained.) However, it is common to work in the frequency domain, in which case one defines the Fourier-transform pair
(100)$$\begin{split}F(\omega ) &= \int_{ - T/2}^{T/2} F(t)\exp (i\omega t){\rm d}t,\\ F(t) &= \int_\infty ^\infty F(\omega )\exp ( - i\omega t){\rm d}\omega /2\pi ,\end{split}$$
where $ F $ is an arbitrary function and $ T $ is the integration (measurement) time. For stationary stochastic processes, $ T $ must be finite to prevent infinite integrals [23]. The frequency-domain solutions of Eqs. (44) and (45) are (101)$${S_1}(\omega ) \def\LDeqtab{}= \frac{{({\gamma _{\textit nn}} - i\omega ){R_s}(\omega ) + {\gamma _{\textit sn}}{R_n}(\omega )}}{{\omega _0^2 - 2i{\nu _0}\omega - {\omega ^2}}},$$
(102)$${N_1}(\omega )\def\LDeqtab{} = \frac{{ - {\gamma _{\textit ns}}{R_s}(\omega ) - ({\gamma _{\textit ss}} + i\omega ){R_n}(\omega )}}{{\omega _0^2 - 2i{\nu _0}\omega - {\omega ^2}}},$$
where the transformed source terms have the properties (103)$$\left\langle {{R_j}(\omega )} \right\rangle = 0,\quad \left\langle {{R_j}(\omega ){R_k}(\omega )} \right\rangle = {R_{jk}}T.$$
$ {S_1} $ is dimensionless, and $ {R_j} $ and $ {R_{jk}} $ have units of inverse time, so $ {S_1}(\omega ) $ has units of time and $ {R_j}(\omega ) $ is dimensionless. It follows from Eq. (101) that the RIN spectrum (104)$$Q(\omega ) = \frac{{\left\langle {|{S_1}(\omega {{)|}^2}} \right\rangle }}{{S_0^2T}} = \frac{{A + B{\omega ^2}}}{{S_0^2[(\omega _0^2 - {\omega ^2}{)^2} + 4\nu _0^2{\omega ^2}]}},$$
where the coefficients (105)$$A = \gamma _{\textit nn}^2{R_{\textit ss}} + 2{\gamma _{\textit nn}}{\gamma _{\textit sn}}{R_{\textit ns}} + \gamma _{\textit sn}^2{R_{\textit nn}},\quad B = {R_{\textit ss}}.$$
Equations (104) and (105) are consistent with Eqs. (5.46) and (5.125) of [6]. The RIN spectrum has units of time (inverse frequency), regardless of whether $ S $ represents a photon number or an output power. It is finite at zero frequency, attains its maximum near the resonance (RO) frequency and decreases as $ 1/{\omega ^2} $ for high frequencies [13].According to Eqs. (47) and (105), and the discussion in Section 4.C, the numerator in Eq. (104) has units of frequency cubed and is explicitly proportional to $ {S_0} $, whereas the denominator has units of frequency quadrupled and is explicitly proportional to $ S_0^2 $. Hence, the quantity $ (b{S_0})Q $ is dimensionless and depends on the normalized coupling coefficients $ {\gamma _{jk}}/b $, and the normalized frequencies $ {\omega _0}/b $ and $ {\nu _0}/b $ (which depend implicitly on $ {S_0} $). These quantities are the ones used in Section 2.D to illustrate laser dynamics. Typical values of $ Q $ are very small, because typical values of the photon flux $ b{S_0} $ are very large. The normalized RIN spectrum $ Q(\omega )/Q(0) $ is displayed in Fig. 6. As $ {S_0} $ increases, the resonance frequency increases and the prominence of the RIN maximum decreases.
Formulas for the coupling coefficients and source strengths are listed in Tables 2 and 3, respectively. By rewriting them in terms of the normalized parameters $ {a^\prime} = a{N_t}/b $, $ {c^\prime} = c/b $ and $ S_0^\prime = {S_0}/{N_t} $, and substituting the resulting expressions in Eq. (105), one obtains the four-level RIN-intercept formula
(106)$$(b{S_0})Q(0) = \frac{{{{({c^\prime})}^2} + {{({a^\prime}S_0^\prime)}^2} + {{({a^\prime}S_0^\prime + {c^\prime})}^2}}}{{{{({a^\prime}S_0^\prime)}^2}}},$$
where $ S_0^\prime = {c^\prime}({a^\prime} - 1)/{a^\prime} $. The corresponding type B formula is (107)$$(b{S_0})Q(0) = \frac{{{a^\prime}{{({c^\prime})}^2} + {{(2{a^\prime}S_0^\prime)}^2} + {{(2{a^\prime}S_0^\prime + {c^\prime})}^2}}}{{{{(2{a^\prime}S_0^\prime)}^2}}},$$
where $ S_0^\prime = {c^\prime}({a^\prime} - 1)/2{a^\prime} $. In both formulas, the numerators and denominators are proportional to $ {({c^\prime})^2} $, so when $ {a^\prime} \gg 1 $, $ Q(0) \approx 2/b{S_0} $. Although the three-level noise sources are stronger than their four-level counterparts (Section 4.C), the RIN intercepts of three- and four-level lasers are similar (in the well-above-threshold regime). Some of the large three-level terms cancel, because the electron and photon fluctuations are anti-correlated ($ {R_{\textit ns}}{R_{\textit ss}} \lt 0 $).RIN spectra attain their maxima for frequencies between $ {(\omega _0^2 - 2\nu _0^2)^{1/2}} $ ($ A $-term dominant) and $ {\omega _0} $ ($ B $-term dominant). For four-level lasers,
(108)$$(b{S_0})Q({\omega _0}) = \frac{{{{({c^\prime})}^2} + {{({a^\prime}S_0^\prime)}^2} + {{({a^\prime}S_0^\prime + {c^\prime})}^2} + 2({a^\prime}S_0^\prime)}}{{({a^\prime}S_0^\prime)({a^\prime}S_0^\prime + {c^\prime}{)^2}}},$$
whereas for type B lasers, (109)$$\begin{split}&(b{S_0})Q({\omega _0})\\&\quad = \frac{{{a^\prime}{{({c^\prime})}^2} + {{(2{a^\prime}S_0^\prime)}^2} + {{(2{a^\prime}S_0^\prime + {c^\prime})}^2} + ({a^\prime} + 1)(2{a^\prime}S_0^\prime)}}{{(2{a^\prime}S_0^\prime)(2{a^\prime}S_0^\prime + {c^\prime}{)^2}}}.\end{split}$$
The RIN maxima of type B lasers are more prominent than their four-level counterparts because of the factor $ {a^\prime} + 1 $ in last term in Eq. (109). This term reflects a stronger noise source for type B lasers [$ {R_{\textit ss}} = ({a^\prime} + 1)b{S_0} $ instead of $ 2b{S_0} $].The RIN spectrum (104) is the Fourier transform of the autocorrelation function, so the autocorrelation function is the inverse transform of the RIN spectrum. By using the residue theorem to evaluate the contour integrals (Appendix E), one finds that
(110)$$C(\tau ) = \frac{{\exp ( - {\nu _0}\tau )}}{{4{\nu _0}{\omega _r}S_0^2}}{\rm Re}\left[ {\frac{{A + B{{({\omega _r} - i{\nu _0})}^2}}}{{{\omega _r} - i{\nu _0}}}\exp ( - i{\omega _r}\tau )} \right].$$
Equation (110) is consistent with Eq. (6.5.22) of [5]. The autocorrelation is dimensionless and the quantity $ {S_0}C $ depends on the aforementioned normalized coefficients and frequencies. The normalized autocorrelation function $ C(\tau )/C(0) $ is displayed in Fig. 7. It oscillates, but decreases, as the time delay increases.The RIN is the autocorrelation function evaluated at zero delay. It follows from Eq. (110) that
(111)$$C(0) = \frac{{A + B\omega _0^2}}{{4{\nu _0}\omega _0^2S_0^2}}.$$
The RIN is also the integral of the one-sided RIN spectrum divided by $ \pi $. One can verify this statement approximately by replacing the RIN spectrum with a Lorentz function of height $ (A + B\omega _0^2)/4\nu _0^2\omega _0^2S_0^2 $ and width $ {\nu _0} $. The RIN of a type B laser is stronger than that of a four-level laser, because the spectral peak is higher (as discussed above) and the width is similar.In the preceding analysis, which is based on stochastic differential equations, one only obtains the RIN formula (111) after a sequence of calculations, in which the RIN spectrum and the temporal correlation function are determined first. One can also determine the RIN formula directly, by solving the deterministic moment equations.
Although Eqs. (60)–(62) can be solved analytically, there is no need to do so, because after a short transient period, the second deviation moments attain their steady-state values. This behavior is illustrated in Fig. 8. The normalized moments $ \left\langle {N_1^2} \right\rangle /{N_t} $, $ \left\langle {{N_1}{S_1}} \right\rangle /{N_t} $, and $ \left\langle {S_1^2} \right\rangle /{N_t} $ are directly proportional to $ {S_0}/{N_t} $ and inversely proportional to $ {\gamma _{jk}}/b $.
In steady state, Eqs. (60)–(62) can be rewritten in the matrix form
(112)$$\left[ {\begin{array}{*{20}{c}}{ - 2{\gamma _{\textit ss}}}&{ - 2{\gamma _{\textit sn}}}&0\\{{\gamma _{\textit ns}}}&{{\gamma _{\textit nn}} - {\gamma _{\textit ss}}}&{ - {\gamma _{\textit sn}}}\\0&{2{\gamma _{\textit ns}}}&{2{\gamma _{\textit nn}}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{\left\langle {S_1^2} \right\rangle }\\{\left\langle {{N_1}{S_1}} \right\rangle }\\{\left\langle {N_1^2} \right\rangle }\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{R_{\textit ss}}}\\{{R_{\textit ns}}}\\{{R_{\textit nn}}}\end{array}} \right].$$
The solution of Eq. (112) is (113)$$\left[ {\begin{array}{*{20}{c}}{\left\langle {S_1^2} \right\rangle }\\{\left\langle {{N_1}{S_1}} \right\rangle }\\{\left\langle {N_1^2} \right\rangle }\end{array}} \right] = \frac{1}{\Delta }\left[ {\begin{array}{*{20}{c}}{{\gamma _{\textit ns}}{\gamma _{\textit sn}} + {\gamma _{\textit nn}}({\gamma _{\textit nn}} - {\gamma _{\textit ss}})}&{2{\gamma _{\textit nn}}{\gamma _{\textit sn}}}&{\gamma _{\textit sn}^2}\\{ - {\gamma _{\textit nn}}{\gamma _{\textit ns}}}&{ - 2{\gamma _{\textit nn}}{\gamma _{\textit ss}}}&{ - {\gamma _{\textit sn}}{\gamma _{\textit ss}}}\\{\gamma _{\textit ns}^2}&{2{\gamma _{\textit ns}}{\gamma _{\textit ss}}}&{{\gamma _{\textit ns}}{\gamma _{\textit sn}} - {\gamma _{\textit ss}}({\gamma _{\textit nn}} - {\gamma _{\textit ss}})}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{R_{\textit ss}}}\\{{R_{\textit ns}}}\\{{R_{\textit nn}}}\end{array}} \right],$$
where the determinant (114)$$\Delta = 2({\gamma _{\textit nn}} - {\gamma _{\textit ss}})({\gamma _{\textit ns}}{\gamma _{\textit sn}} - {\gamma _{\textit nn}}{\gamma _{\textit ss}}) = 4{\nu _0}\omega _0^2.$$
Hence, the RIN (115)$$\frac{{\left\langle {S_1^2} \right\rangle }}{{S_0^2}} = \frac{{(\gamma _{\textit nn}^2 + {\gamma _{\textit ns}}{\gamma _{\textit sn}} - {\gamma _{\textit nn}}{\gamma _{\textit ss}}){R_{\textit ss}} + 2{\gamma _{\textit nn}}{\gamma _{\textit sn}}{R_{\textit ns}} + \gamma _{\textit sn}^2{R_{\textit nn}}}}{{2({\gamma _{\textit nn}} - {\gamma _{\textit ss}})({\gamma _{\textit ns}}{\gamma _{\textit sn}} - {\gamma _{\textit nn}}{\gamma _{\textit ss}})S_0^2}}.$$
Equation (115) is equivalent to Eq. (111). However, the derivation based on deterministic moment equations is shorter than the derivation based on stochastic rate equations.The results of this section pertain to photon statistics within the cavity. To determine the output-power statistics, one must model the transmission of photons through the mirrors (facets) [6]. The main effect of the transmission process is the addition of the term $ 1/b{S_0} $ to the RIN spectrum. If the measurement time is $ T $, then the effective bandwidth of the detector is $ 1/T $, so the RIN is increased by the amount $ 1/{S_m} $, where $ {S_m} = b{S_0}T $ is the measured number of photons. This value is the noise-to-signal ratio associated with shot noise (which has Poisson statistics). For frequencies that are significantly higher than the RO frequency, the exterior RIN spectrum tends to the value $ 1/b{S_0} $ (Fig. 6), which is called the standard quantum limit (because it applies to all lasers and is consistent with the quantum theory of transmission through a beam splitter).A photon with a wavelength of 1 µm has an energy of $ 2.0 \times {10^{ - 19}}\;{\rm J} $, from which it follows that the standard quantum limit, measured in dB/Hz, is $ - 187 - 10\mathop {\log }\nolimits_{10} ({P_W}{\lambda _{\mu m}}) $, where $ {P_W} $ is the output power, measured in watts, and $ {\lambda _{\mu m}} $ is the wavelength, measured in micrometers.
The aforementioned results only pertain to the mean and variance of the signal-photon number. Related results for the variance of the signal phase are derived in [24], using the Langevin approach described in Section 3.
6. SUMMARY
In this tutorial, the noise properties of three- and four-level lasers were reviewed in detail. By neglecting the electron populations that decay quickly (those of level 1 or 3), one can derive approximate rate equations for the lowest-level (0 or 1) electron number $ M $, the upper-level (2) electron number $ N $ and the signal-photon number $ S $. These general rate equations [(28)–(30)] apply to all of the aforementioned lasers.
One can mimic the effects of quantum fluctuations (noise) by adding random source terms to the rate equations, with one source term for each process modeled by the equations. The noise impulses are random and instantaneous, and the impulses from different sources are independent. The stochastic rate equations that result [(31)–(33)] are called Langevin equations. These equations depend nonlinearly on the electron and photon numbers, so in most circumstances one has to solve them numerically. However, one can develop considerable insight into laser noise by solving them perturbatively.
In the above-threshold regime, the Langevin equations have equilibrium solutions [(36) and (37)] with nonzero photon flux ($ b{S_0} $). One can model photon-number deviations ($ {S_1} $) by linearizing the Langevin equations about this equilibrium. By using the fact that the upper- and lowest-level electron deviations are anti-correlated ($ {M_1} = - {N_1} $), one can reduce the linearized Langevin equations from a set of three equations for $ {S_1} $, $ {N_1} $, and $ {M_1} $ [(38)–(40)] to a set of two equations for $ {S_1} $ and $ {N_1} $ [(44) and (45)]. By solving these stochastic rate equations in the frequency domain, one can derive a formula for the RIN spectrum (104). This spectrum is finite at zero frequency, attains its maximum near the RO frequency, and decreases as the inverse of the squared frequency for high frequencies. By inverse Fourier transforming the RIN spectrum, one can derive formulas for the (time-delay) autocorrelation function (110) and RIN (111). The autocorrelation function oscillates at the RO frequency and decreases at the RO damping rate. Alternatively, one can use the linearized Langevin equations to derive equations for the variances and correlation of the electron and photon deviations [(60)–(62)]. By solving these deterministic equations in steady state, one can derive the RIN formula (115) directly. Despite the importance of the preceding results, which are standard, they are all based on unproven formulas for the source strengths (variances of the source terms).
Let $ F(m,n,s) $ be the probability that $ m $ lowest-level electrons, $ n $ upper-level electrons and $ s $ signal photons are in the cavity, where $ m $, $ n $ and $ s $ are whole numbers. The Markov equation (63) for this probability distribution function is based on the assumption that changes in the electron and photon numbers occur randomly and instantaneously. By calculating the moments of this probability equation, one obtains deterministic equations for the number means [(66), (68) and (70)], variances [(77)–(79)] and correlations [(86)–(88)]. In the variance and correlation equations, the source strengths appear naturally and justify the shot-noise rule: For each process that is modeled by the rate equations [(28)–(30)], the driving terms in the variance equations equal the moduli of the associated terms in the mean equations. The driving terms in the correlation equations have the same magnitudes as the variance terms, but can be positive or negative depending on whether the changes in the electron and photon numbers are correlated or anti-correlated, respectively. To the best of my knowledge, this result is consistent with the laws of quantum optics [15–22].
Although the aforementioned moment equations are exact, they do not form a closed set because they involve higher-order deviation moments. By using the fact that the upper- and lowest-level electron deviations are anti-correlated ($ \delta m = - \delta n $), one can reduce the set of six equations for the second-order moments of $ \delta s $, $ \delta n $ and $ \delta m $ [(77)–(79) and (86)–(88)] to a set of three equations for the moments of $ \delta s $ and $ \delta n $ [(94)–(96)]. These simplified moment equations are also not closed. One can close them by neglecting deviation moments of orders higher than two, in which case they reduce to the linearized Langevin moment equations [(60)–(62) or (97)–(99)]. This agreement justifies the simplification procedure and closure approximation.
The physical assumptions upon which the Markov equation is based are simple and intuitive. Although the associated number-moment calculations are tedious, they produce self-consistent formulas for the driving terms in the variance and correlation equations, upon which the RIN spectrum and RIN depend. Thus, the Markov approach is useful.
In Appendices B and C, stochastic calculus is used to derive Langevin moment equations for the electron and photon numbers [(C1)–(C9)]. These equations (which are not linearized) are consistent with the Markov moment equations, provided that the source strengths are specified appropriately. Although the Langevin and Markov approaches are based on the complementary assumptions that the electron and photon numbers are real and whole, respectively, their associated moment equations, which involve real averages, are consistent. This consistency reflects the related assumptions that the noise impulses are random and instantaneous, and the number changes are random and instantaneous.
APPENDIX A: STOCHASTIC SYSTEMS
In steady state, lasers emit radiation with a constant average power (photon flux), superimposed on which are random power fluctuations (deviations). It is convenient to define the Fourier-transform pair
(A1)$$\begin{split}F(\omega ) &= \int_{ - T/2}^{T/2} F(t)\exp (i\omega t){\rm d}t,\\F(t) &= \int_\infty ^\infty F(\omega )\exp ( - i\omega t){\rm d}\omega /2\pi ,\end{split}$$
where
$ F(t) $ is an arbitrary function and
$ T $ is the measurement time. The associated Parseval equation is
(A2)$$\int_{ - T/2}^{T/2} |F(t{)|^2}{\rm d}t = \int_\infty ^\infty |F(\omega {)|^2}{\rm d}\omega /2\pi .$$
If the integral quantity is energy, then
$ |F(t{)|^2} $ is the energy per unit time (power) and
$ |F(\omega {)|^2} $ is the energy per unit frequency (spectral density). For an optical pulse, the energy is finite. However, for a laser in steady state, the measured energy (and hence, the spectral density) is proportional to the measurement time (which appears in key formulas).
To be precise, if $ T $ is finite, then $ F(t) $ should be written as a Fourier series, in which the frequencies are separated by $ 2\pi /T $. I assume that this frequency difference is much smaller than any frequency of interest, so the series can be replaced by the integral in the second of Eqs. (A1). In formal discussions of stochastic processes [23], the limit $ T \to \infty $ is taken.
Let $ F $ be a quantity of interest that is associated with a stochastic system (process). Then the process is termed stationary if the ensemble average $ {\left\langle {F(t)} \right\rangle _e} $ is independent of time. It is termed ergodic if
(A3)$${\left\langle {F(t)} \right\rangle _e} = {\left\langle {F(t)} \right\rangle _t} = \int_{-T/2}^{T/2} F(t){\rm d}t/T,$$
where
$ {\left\langle \right\rangle _t} $ denotes a time average. Henceforth, I assume that the processes under consideration are ergodic.
The (temporal) cross correlation of two quantities (such as the electron and photon numbers) is defined as
(A4)$${C_{jk}}(\tau ) = {\left\langle {{F_j}(t){F_k}(t + \tau )} \right\rangle _e} = {\left\langle {{F_j}(t){F_k}(t + \tau )} \right\rangle _t}.$$
By using the second of the preceding definitions, one finds that
(A5)$$\begin{split}{C_{jk}}(\omega ) = {\left\langle {{F_j}(t)\exp ( - i\omega t){F_k}(\omega )} \right\rangle _t} = F_j^*(\omega ){F_k}(\omega )/T.\end{split}$$
The Fourier transform of the cross correlation is the product of the individual Fourier transforms divided by the measurement time. If the quantities are dimensionless (as are the electron and photon numbers), then
$ {C_{jk}}(\tau ) $ is dimensionless and
$ {C_{jk}}(\omega ) $ has the dimension of time (inverse frequency). Notice that
$ {C_{kj}}(\omega ) = C_{jk}^*(\omega ) $ is the Fourier transform of
$ {C_{kj}}(\tau ) = {C_{jk}}( - \tau ) $. If the two quantities are the same (
$ j = k $), then the correlation is called the autocorrelation and its Fourier transform is the spectral density divided by the measurement time. Notice that the autocorrelation is an even function of the time delay
$ \tau $ and its value for zero delay is the integral of the power spectral density
$ |F(\omega {)|^2}/T $ divided by
$ 2\pi $.
In the Langevin approach, one adds to the rate equations random source terms (or combinations thereof), which mimic the effects of quantum fluctuations. Let $ {F_j} $ and $ {F_k} $ be two such terms. Then
(A6)$${\left\langle {{F_j}(t)} \right\rangle _e} = 0,\quad {\left\langle {{F_j}(t){F_k}({t^\prime})} \right\rangle _e} = {S_{jk}}\delta (t - {t^\prime}),$$
where
$ {S_{jk}} $ is a source strength (variance or correlation). In the number rate equations,
$ {F_j} $ and
$ {S_{jk}} $ have the dimension of inverse time. By combining Eqs. (
A1) and (
A6), one finds that
(A7)$$\begin{split}{\left\langle {{F_j}(\omega )F_k^*(\omega )} \right\rangle _e} &= {\left\langle {\iint {F_j}(t){F_k}({t^\prime})\exp [i\omega (t - {t^\prime})]{\rm d}t{\rm d}{t^\prime}} \right\rangle _e} \\ & = \iint {S_{jk}}\delta (t - {t^\prime})\exp [i\omega (t - {t^\prime})]{\rm d}t{\rm d}{t^\prime} \\ & = {S_{jk}}T.\end{split}$$
Notice that this correlation is real and does not depend on frequency. Quantities that are
$ \delta $-correlated in time have infinite frequency bandwidths. For completeness,
(A8)$$\begin{split}{\left\langle {{F_j}(\omega )F_k^*({\omega ^\prime})} \right\rangle _e} &= {S_{jk}}T{\rm sinc}[(\omega - {\omega ^\prime})T/2] \\&\approx 2\pi {S_{jk}}\delta (\omega - {\omega ^\prime}).\end{split}$$
Quantities that are
$ \delta $-correlated in time are also
$ \delta $-correlated in frequency. Equations (
A7) and (
A8) are consistent because the
$ \delta $ function in the latter equation has the required height.
Now let $ F $ be any quantity whose transform is $ \delta $-correlated in frequency (not just a source term). Then some authors use the equation
(A9)$${\left\langle {F(\omega ){F^*}({\omega ^\prime})} \right\rangle _e} = {S_f}(\omega )\delta (\omega - {\omega ^\prime})$$
to define the spectral density
$ {S_f}(\omega ) $. If this density depends on frequency, its inverse transform, which is proportional to the autocorrelation
$ {C_f} $, is not a
$ \delta $ function in time (
$ \tau $).
In Section 5, the linearized Langevin equations were solved by Fourier transformation. Doing so requires one to calculate the Fourier transform of a time derivative. Let $ \dot F = dF/dt $. Then
(A10)$$\begin{split}\int_{-T/2}^{T/2} \dot F(t)\exp (i\omega t){\rm d}t& = F(t)\exp (i\omega t)\Big|_{-T/2}^{T/2}\\&\quad - i\omega \int_{-T/2}^{T/2} F(t)\exp (i\omega t){\rm d}t.\end{split}$$
If
$ F $ were periodic and the frequencies were discrete, the boundary terms would cancel. However, if
$ F $ is random, the sum of the boundary terms is not necessarily zero. I assumed that the boundary terms are much smaller than the integral. This approximation improves as
$ T \to \infty $ [
23].
APPENDIX B: STOCHASTIC DIFFERENTIAL EQUATIONS
Suppose that $ m $ physical quantities ($ {X_i} $) are coupled to each other and driven by $ n $ independent noise sources ($ {r_k} $). Then each quantity satisfies the stochastic differential equation
(B1)$${d_t}{X_i} = {a_i}(X) + {\sum _k}{b_{ik}}(X){r_k}(t),$$
where
$ {a_i} $ and
$ {b_{ik}} $ are deterministic functions of their arguments and
$ X $ is an abbreviation for
$ {X_1},{X_2}, \ldots ,{X_m} $. The sources
$ {r_k} $ are random functions of time with the properties
(B2)$$\langle {{r_k}(t)} \rangle = 0,\quad \left\langle {{r_k}(t){r_l}({t^\prime})} \right\rangle = {\delta _{kl}}\delta (t - {t^\prime}).$$
For each source function, the average impulse is zero (positive and negative impulses are equally likely) and different impulses are independent. The driving terms for
$ {X_i} $ and
$ {X_j} $ are said to be correlated, uncorrelated, or anti-correlated if
$ {b_{ik}}{b_{jk}} $ is positive, zero, or negative, respectively.
Let $ \delta t $ be a short time interval and define the (Wiener) increment $ \delta {w_k} = \int _0^{\delta t}{r_k}(t){\rm d}t $. Then it follows from Eq. (B2) that the increment has the properties
(B3)$$ \langle {\delta {w_k}} \rangle = 0,\quad \left\langle {\delta w_k^2} \right\rangle = \delta t.$$
The second part of Eq. (
B3) implies (in a crude sense) that
$ \delta {w_k} \sim \delta {t^{1/2}} $. Consequently, when one integrates Eq. (
B1) over the time interval
$ \delta t $, one obtains terms of order
$ \delta t $ and
$ \delta {t^{1/2}} $. This fact differentiates stochastic calculus [
3] from regular calculus.
Define the increment $ \delta {X_i} = {X_i}(\delta t) - {X_i}(0) $. Then it follows from Eq. (B1) that
(B4)$$\begin{split}\delta {X_i}& = \int _0^{\delta t}{a_i}[X(t)]{\rm d}t + {\sum _k}\int _0^{\delta t}{b_{ik}}[X(t)]{r_k}(t){\rm d}t \\ &\approx {a_i}(X)\delta t + {\sum _k}\int _0^{\delta t}\Big[{b_{ik}}(X)\\&\quad + {\sum _j}{\partial _j}{b_{ik}}(X){\sum _l}{b_{jl}}(X)\int _0^t{r_l}({t^\prime}){\rm d}{t^\prime}\Big]{r_k}(t){\rm d}t \\ & = {a_i}(X)\delta t + {\sum _k}{b_{ik}}(X)\int _0^{\delta t}{r_k}(t){\rm d}t \\ &\quad + {\sum _k}{\sum _j}{\sum _l}{\partial _j}{b_{ik}}(X){b_{jl}}(X)\int _0^{\delta t}\int _0^t{r_k}(t){r_l}({t^\prime}){\rm d}{t^\prime}{\rm d}t,\end{split}$$
where
$ {X_i} = {X_i}(0) $ and
$ {\partial _j}{b_{ik}} = \partial {b_{ik}}/\partial {X_j} $. When one integrates the random term
$ {b_{ik}}[X(t)]{r_k}(t) $, which depends on
$ {X_j}(t) $, one must retain the largest contribution to
$ {X_j}(t) - {X_j}(0) $, which is
$ {\sum _l}{b_{jl}}[X(0)]\int _0^t{r_l}({t^\prime}){\rm d}{t^\prime} $. The first term on the right side of Eq. (
B4) is deterministic and of order
$ \delta t $, whereas the second term is random and of order
$ \delta {t^{1/2}} $. The third term is of order
$ \delta t $. Consequently, its random component can be neglected compared to the second term, but its deterministic component must be retained. Because the source functions are independent, the formula for this component reduces to
(B5)$${\sum _j}{\sum _k}{b_{jk}}(X){\partial _j}{b_{ik}}(X)\int _0^{\delta t}\int _0^t\left\langle {{r_k}(t){r_k}({t^\prime})} \right\rangle {\rm d}{t^\prime}{\rm d}t.$$
In Ito calculus [
3], one treats the delta function in Eq. (
B2) as a function with zero duration, in which case
$ \left\langle {{r_k}(t){r_k}({t^\prime})} \right\rangle = 0 $ for all
$ {t^\prime} \lt t $. By interpreting the upper limit of the
$ {t^\prime} $ integral as
$ {t_ - } $, one finds that
(B6)$$\left\langle {\delta {X_i}} \right\rangle \approx {a_i}(X)\delta t.$$
In Stratonovich calculus [
3], one treats the delta function as a function with a very short, but nonzero, duration, in which case
$ \left\langle {{r_k}{{(t)}_k}({t^\prime})} \right\rangle \ne 0 $ for
$ {t^\prime} \approx t $. Consequently,
(B7)$$\left\langle {\delta {X_i}} \right\rangle \approx {a_i}(X)\delta t + {\sum _j}{\sum _k}{b_{jk}}(X){\partial _j}{b_{ik}}(X)\delta t/2.$$
In both types of calculus,
(B8)$$\left\langle {\delta X_i^2} \right\rangle \approx {\sum _k}b_{ik}^2(X)\delta t.$$
Equations (
B6) and (
B7) exemplify one of several differences between Ito and Stratonovich calculus.
Another difference involves correlations. It follows from Eq. (B4) that
(B9)$$\begin{split}{X_i}(0){r_l}(0) \approx \left\{ {X_i}( - \delta t) + {\sum _k}{b_{ik}}[X( - \delta t)]\int _{ - \delta t}^0 {r_k}(t){\rm d}t\right\} {r_l}(0).\end{split}$$
The previous values
$ {X_i}( - \delta t) $ and
$ X( - \delta t) $ are independent of the current impulse
$ {r_l}(0) $. In Ito calculus, one interprets the upper limit of the
$ t $ integral as
$ {0_ - } $, in which case
$ \left\langle {{r_k}(t){r_l}(0)} \right\rangle = 0 $ and
(B10)$$\left\langle {{X_i}{r_l}} \right\rangle = 0.$$
The current value
$ {X_i} $ is independent of every current impulse. In Stratonovich calculus,
$ \left\langle {{r_l}(t){r_l}(0)} \right\rangle \ne 0 $ for
$ t \approx 0 $, in which case
(B11)$$\left\langle {{X_i}{r_l}} \right\rangle = {b_{il}}(X)/2.$$
The current value
$ {X_i} $ is (weakly) correlated to every current impulse. Similar results apply to arbitrary functions of
$ {X_i} $. Many authors write the source terms in the Langevin equations as
$ {R_{ik}}(X,t) = {b_{ik}}(X){r_k}(t) $ and assume that they have the properties
(B12)$$\begin{split}\left\langle {{R_{ik}}(X,t)} \right\rangle &= 0,\\ \left\langle {{R_{ik}}(X,t){R_{jl}}(X,{t^\prime})} \right\rangle &= {b_{ik}}(X){b_{jk}}(X){\delta _{kl}}\delta (t - {t^\prime}).\end{split}$$
The first of properties (
B12) is valid only in Ito calculus. Longer tutorial discussions of Ito and Stratonovich calculus are contained in [
7,
25].
I chose to use Ito calculus because its predictions are consistent with the results of Section 4. In Ito calculus, the increment equation (B4) reduces to
(B13)$$\delta {X_i} \approx {a_i}(X)\delta t + {\sum _k}{b_{ik}}(X)\int _0^{\delta t}{r_k}(t){\rm d}t.$$
By using property (
B10) to average Eq. (
B13) and letting
$ \delta t \to 0 $, one obtains the first-moment (mean) equation
(B14)$${d_t}\left\langle {{X_i}} \right\rangle = \left\langle {{a_i}(X)} \right\rangle .$$
Derivations of the second-moment equations are facilitated by the Ito chain rule.
Let $ f $ be an arbitrary deterministic function, and let $ {f_i} = \partial f/\partial {X_i} $ and $ {f_{ij}} = {\partial ^2}f/\partial {X_i}\partial {X_j} $. Then, according to the chain rule for functions of multiple variables,
(B15)$$\delta f(X) \approx {\sum _i}{f_i}(X)\delta {X_i} + {\sum _i}{\sum _j}{f_{ij}}(X)\delta {X_i}\delta {X_j}/2.$$
The first term on the right side of Eq. (
B15) has deterministic contributions of order
$ \delta t $ and random contributions of order
$ \delta {t^{1/2}} $. The contributions to the second terms are of order
$ \delta t $ or smaller. Consequently, one needs to retain only the deterministic contribution
(B16)$$\begin{split}\left\langle {\delta {X_i}\delta {X_j}} \right\rangle& \approx {\sum _k}{\sum _l}{b_{ik}}(X){b_{jl}}(X)\int _0^{\delta t}\int _0^{\delta t}\left\langle {{r_k}(t){r_l}({t^\prime})} \right\rangle {\rm d}t{\rm d}{t^\prime} \\[-5pt] & = {\sum _k}{b_{ik}}(X){b_{jk}}(X)\delta t.\end{split}$$
By combining Eqs. (
B15) and (
B16), and letting
$ \delta t \to 0 $, one obtains the Ito chain rule
(B17)$$\begin{split}{d_t}f(X) &= {\sum _i}{f_i}(X){d_t}{X_i} \\[-5pt]&\quad+ {\sum _i}{\sum _j}{\sum _k}{f_{ij}}(X){b_{ik}}(X){b_{jk}}(X)/2.\end{split}$$
The summations over
$ i $ and
$ j $ stem from the regular chain rule, whereas the summation over
$ k $ stems from the fact that each quantity is driven by multiple sources (
$ {r_k} $ and
$ {r_l} $), but only the correlated products (
$ k = l $) contribute to the derivative. If
$ f $ depends explicitly on time, then the right side of Eq. (
B17) includes the term
$ {f_t}(X,t) $, where
$ {f_t} = \partial f/\partial t $.
By combining Eqs. (B1) and (B17), one obtains the second-moment equations
(B18)$${d_t}\left\langle {X_i^2} \right\rangle \def\LDeqtab{}= 2\left\langle {{X_i}{a_i}(X)} \right\rangle + {\sum _k}\left\langle {b_{ik}^2(X)} \right\rangle ,$$
(B19)$$\begin{split}{d_t}\left\langle {{X_i}{X_j}} \right\rangle &= \left\langle {{X_j}{a_i}(X)} \right\rangle + \left\langle {{X_i}{a_j}(X)} \right\rangle\\&\quad + {\sum _k}\left\langle {{b_{ik}}(X){b_{jk}}(X)} \right\rangle .\end{split}$$
The deviation
$ \delta {X_i} = {X_i} - \left\langle {{X_i}} \right\rangle $, so the variance
$ \left\langle {\delta X_i^2} \right\rangle = \left\langle {X_i^2} \right\rangle - {\left\langle {{X_i}} \right\rangle ^2} $ and the correlation
$ \left\langle {\delta {X_i}\delta {X_j}} \right\rangle = \left\langle {{X_i}{X_j}} \right\rangle - \left\langle {{X_i}} \right\rangle \left\langle {{X_j}} \right\rangle $. By combining Eqs. (
B14), (
B18), and (
B19), one obtains the variance and correlation equations
(B20)$${d_t}\left\langle {\delta X_i^2} \right\rangle \def\LDeqtab{}= 2\left\langle {\delta {X_i}\delta {a_i}(X)} \right\rangle + {\sum _k}\left\langle {b_{ik}^2(X)} \right\rangle ,$$
(B21)$$\begin{split}{d_t}\left\langle {\delta {X_i}\delta {X_j}} \right\rangle &= \left\langle {\delta {X_j}\delta {a_i}(X)} \right\rangle + \left\langle {\delta {X_i}\delta {a_j}(X)} \right\rangle\\&\quad + {\sum _k}\left\langle {{b_{ik}}(X){b_{jk}}(X)} \right\rangle ,\end{split}$$
where
$ \delta {a_i}(X) = {a_i}(X) - \left\langle {{a_i}(X)} \right\rangle $. Equations (
B20) and (
B21) are exact. Notice that the variances are driven by the sums of the squares of the functions
$ {b_{ik}} $ and the correlations are driven by similar combinations.
In Section 3, linearized Langevin equations were derived for the electron- and photon-number deviations associated with an above-threshold laser. These equations are a special case of Eq. (B1), in which $ {a_i}(X) $ depends linearly on $ X $ and $ {b_{ik}} $ is independent of $ X $. For this case and cases like it, the Langevin equations can be written in the matrix form
(B22)$${d_t}X = AX + BR(t),$$
where
$ X = [{X_i}] $ is an
$ m \times 1 $ vector,
$ A = [{a_{ij}}] $ is an
$ m \times m $ coefficient matrix,
$ B = [{b_{ik}}] $ is an
$ m \times n $ coefficient matrix, and
$ R = [{r_k}] $ is an
$ n \times 1 $ vector. The source vector has the properties
(B23)$$\left\langle {R(t)} \right\rangle = 0,\quad \left\langle {R(t){R^T}({t^\prime})} \right\rangle = I\delta (t - {t^\prime}),$$
where
$ T $ denotes a transpose and
$ I $ is the
$ n \times n $ identity matrix.
Let $ Y(X,t) = {e^{ - At}}X(t) $. Then, according to the Ito chain rule (B17), which in this case is identical to the regular chain rule,
(B24)$${d_t}Y = {e^{ - At}}BR(t).$$
The transformed vector
$ Y $ is the sum (integral) of vector impulses multiplied by transfer (Green) matrices. By integrating these contributions to
$ Y $ and rewriting the result in terms of
$X$, one finds that
(B25)$$X(t) = {e^{At}}X(0) + \int _0^t{e^{A(t - {t^\prime})}}BR({t^\prime}){\rm d}{t^\prime}.$$
By using the first of properties (
B23) to average solution (
B25), one obtains the first-moment (mean) vector
(B26)$$\langle {X(t)} \rangle = {e^{At}}\langle {X(0)} \rangle .$$
Equation (
B26) is deterministic, so it follows from the rules of regular calculus that
(B27)$${d_t}\langle X \rangle = A\langle X \rangle .$$
The mean equation (
B27) is consistent with Eq. (
B14).
By combining solution (B25) with its transpose, one finds that
(B28)$$\begin{split}X(t){X^T}(t) &= \left[{e^{At}}X + \int _0^t{e^{A(t - {t^\prime})}}BR({t^\prime}){\rm d}{t^\prime}\right]\\&\quad \times \left[{X^T}{e^{{A^T}t}} + \int _0^t{R^T}({t^{\prime\prime}}){B^T}{e^{{A^T}(t - {t^{\prime\prime}})}}{\rm d}{t^{\prime\prime}}\right],\end{split}$$
where
$ X = X(0) $. By using properties (
B23) to average Eq. (
B28), one obtains the second-moment (correlation) matrix
(B29)$$\begin{split}\left\langle {X(t){X^T}(t)} \right\rangle = {e^{At}}\left\langle {X{X^T}} \right\rangle {e^{{A^T}t}} + \int _0^t{e^{A(t - {t^\prime})}}B{B^T}{e^{{A^T}(t - {t^\prime})}}{\rm d}{t^\prime}.\end{split}$$
It follows from the rules of regular calculus that
(B30)$${d_t}\left\langle {X{X^T}} \right\rangle = A\left\langle {X{X^T}} \right\rangle + \left\langle {X{X^T}} \right\rangle {A^T} + B{B^T}.$$
In component form, the diagonal and off-diagonal entries of Eq. (
B30) are
(B31)$${d_t}\left\langle {X_i^2} \right\rangle \def\LDeqtab{}= 2{\sum _k}{a_{ik}}\left\langle {{X_i}{X_k}} \right\rangle + {\sum _k}b_{ik}^2,$$
(B32)$${d_t}\left\langle {{X_i}{X_j}} \right\rangle \def\LDeqtab{}= {\sum _k}{a_{ik}}\left\langle {{X_j}{X_k}} \right\rangle + {\sum _k}{a_{jk}}\left\langle {{X_i}{X_k}} \right\rangle + {\sum _k}{b_{ik}}{b_{jk}},$$
respectively. The second-moment equations (
B31) and (
B32) are consistent with Eqs. (
B18) and (
B19), respectively, and the associated variance and correlation equations are consistent with Eqs. (
B20) and (
B21).
APPENDIX C: LANGEVIN MOMENT EQUATIONS
Consider the Langevin equations (31)–(33), which depend nonlinearly on the electron and photon numbers (unlike the linearized equations of Section 3.B). Then it follows from Eq. (B14) that the equations for the number means are
(C1)$$\begin{split}{d_t}\langle M \rangle &= - {a^\prime}\langle {MS} \rangle + a\langle {N(S + 1)} \rangle \\&\quad - c\langle M \rangle + {d^\prime}\langle N \rangle ,\end{split}$$
(C2)$$\begin{split}{d_t}\langle N \rangle &= {a^\prime}\langle {MS} \rangle - a\langle {N(S + 1)}\rangle \\&\quad+ c\langle M\rangle - {d^\prime}\langle N \rangle ,\end{split}$$
(C3)$${d_t}\langle S\rangle \def\LDeqtab{}= - {a^\prime}\langle {MS}\rangle + a\langle {N(S + 1)} \rangle - b \langle S \rangle .$$
Equations (
C1)–(
C3) are identical to the Markov mean equations (
66), (
68) and (
70).
It follows from Eq. (B20) that the variance equations are
(C4)$$\begin{split}{d_t}\left\langle {\delta {M^2}} \right\rangle &= - 2{a^\prime} \langle {\delta MS\delta M} \rangle + 2a \langle {\delta NS\delta M} \rangle\\&\quad + 2a \langle {\delta M\delta N} \rangle - 2c\left\langle {\delta {M^2}} \right\rangle + 2{d^\prime}\langle {\delta M\delta N} \rangle \\ &\quad + {a^\prime} \langle {MS} \rangle + a \langle {N(S + 1)} \rangle\\&\quad + c \langle M \rangle + {d^\prime} \langle N \rangle ,\end{split}$$
(C5)$$\begin{split} {d_t}\left\langle {\delta {N^2}} \right\rangle& = 2{a^\prime}\langle {\delta MS\delta N} \rangle - 2a\langle {\delta NS\delta N} \rangle\\&\quad - 2a\left\langle {\delta {N^2}} \right\rangle + 2c\langle {\delta M\delta N} \rangle - 2{d^\prime}\left\langle {\delta {N^2}} \right\rangle \\&\quad+ {a^\prime}\langle {MS} \rangle + a\langle {N(S + 1)} \rangle\\&\quad + c\langle M \rangle + {d^\prime}\langle N \rangle ,\end{split}$$
(C6)$$\begin{split}{d_t}\left\langle {\delta {S^2}} \right\rangle &= - 2{a^\prime}\langle {\delta MS\delta S} \rangle + 2a\langle {\delta NS\delta S} \rangle\\&\quad + 2a\langle {\delta N\delta S} \rangle - 2b\left\langle {\delta {S^2}} \right\rangle\\&\quad+ {a^\prime}\langle {MS} \rangle + a\langle {N(S + 1)} \rangle + b\langle S \rangle ,\end{split}$$
and it follows from Eq. (
B21) that the correlation equations are
(C7)$$\begin{split}{d_t}\langle {\delta M\delta N} \rangle &= - {a^\prime}\langle {\delta MS\delta N} \rangle + a\langle {\delta NS\delta N} \rangle\\&\quad + a\left\langle {\delta {N^2}} \right\rangle - c\langle {\delta M\delta N}\rangle + {d^\prime}\left\langle {\delta {N^2}} \right\rangle \\ &\quad + {a^\prime}\langle {\delta MS\delta M}\rangle - a\langle {\delta NS\delta M} \rangle\\&\quad - a\langle {\delta M\delta N} \rangle + c\left\langle {\delta {M^2}} \right\rangle - {d^\prime}\langle {\delta M\delta N} \rangle \\ &\quad - {a^\prime} \langle {MS} \rangle - a \langle {N(S + 1)} \rangle - c \langle M\rangle - {d^\prime}\langle N \rangle , \end{split}$$
(C8)$$\begin{split} {d_t}\langle {\delta M\delta S} \rangle &= - {a^\prime}\langle {\delta MS\delta S}\rangle + a\langle {\delta NS\delta S} \rangle + a\langle {\delta N\delta S} \rangle\\&\quad - c\langle {\delta M\delta S}\rangle + {d^\prime} \langle {\delta N\delta S} \rangle \\ &\quad - {a^\prime}\langle {\delta MS\delta M} \rangle + a\langle {\delta NS\delta M} \rangle\\&\quad + a\langle {\delta M\delta N} \rangle - b\langle {\delta M\delta S}\rangle\\&\quad+ {a^\prime}\langle {MS} \rangle + a\langle {N(S + 1)} \rangle , \end{split}$$
(C9)$$\begin{split}{d_t} \langle {\delta N\delta S} \rangle &= {a^\prime}\langle {\delta MS\delta S} \rangle - a\langle {\delta NS\delta S}\rangle - a\langle {\delta N\delta S} \rangle\\&\quad + c \langle {\delta M\delta S} \rangle - {d^\prime}\langle {\delta N\delta S} \rangle \\ &\quad - {a^\prime}\langle {\delta MS\delta N} \rangle + a \langle {\delta NS\delta N} \rangle\\&\quad + a\langle {\delta {N^2}} \rangle - b\langle {\delta N\delta S} \rangle \\ &\quad - {a^\prime}\langle {MS} \rangle - a\langle {N(S + 1)} \rangle .\end{split}$$
Equations (
C4)–(
C6) are identical to the Markov variance equations (
77)–(
79) and Eqs. (
C7)–(
C9) are equivalent to the Markov correlation equations (
86)–(
88). (In the former equations, the terms on the right side are grouped differently.) Thus, the Markov approach of Section
4 is consistent with the Langevin approach of Appendix
B, which is based on Ito stochastic calculus and the shot-noise rule. Because the two sets of moment equations are equivalent, the Langevin equations can be closed and simplified in the same way as the Markov equations (Section
4.B).
The compact forms of Eqs. (B20) and (B21) can facilitate analyses of their consequences. For example, let $ {A_m} = {a_m}(M,N,S) $ and $ {A_n} = {a_n}(M,N,S) $. Then
(C10)$$\begin{split}&{d_t}\left\langle {{{(\delta M + \delta N)}^2}} \right\rangle = 2\left\langle {\delta M(\delta {A_m} + \delta {A_n})} \right\rangle \\&\quad + 2\left\langle {\delta N(\delta {A_m} + \delta {A_n})} \right\rangle = 0,\end{split}$$
because
$ \delta {A_n} = - \delta {A_m} $. The noise terms also cancel.
APPENDIX D: MARKOV MOMENT EQUATIONS
In Appendix B, the Langevin approach was used to derive moment (mean, variance and correlation) equations for an arbitrary system. The Markov approach can also be used to derive generalized moment equations. However, in the latter approach, one has to identify each process a priori and specify the correlations between the quantities associated with that process.
As an example, consider the nonlinear rate equations
(D1)$${d_t}N = - a(N,S) + c(N),$$
(D2)$${d_t}S = a(N,S) - b(S),$$
where
$ N $ is the number of upper-level electrons and
$ S $ is the number of signal photons. Equations (
D1) and (
D2) include electron-loss and photon-gain terms that depend on the electron and photon numbers, which are anti-correlated. They also include an electron-gain term that does not depend on the photons and a photon-loss term that does not depend on the electrons.
The probability $ F(n,s) $ obeys the Markov equation
(D3)$$\begin{split}{d_t}F(n,s) &= - a(n,s)F(n,s) + a(n + 1,s - 1)F(n + 1,s - 1)\\&\quad - b(s)F(n,s) + b(s + 1)F(n,s + 1) \\ &\quad - c(n)F(n,s) + c(n - 1)F(n - 1,s),\end{split}$$
which manifests the aforementioned correlations. The total probability was defined in Section
4. It follows from this definition and Eq. (
D3) that
(D4)$$\begin{split}{d_t}T &= - \sum\limits_0 \sum\limits_0 a(n,s)F(n,s) + \sum\limits_0 \sum\limits_1 a(n + 1,s - 1)F(n + 1,s - 1) \\ &\quad - \sum\limits_0 \sum\limits_0 b(s)F(n,s) + \sum\limits_0 \sum\limits_0 b(s + 1)F(n,s + 1) \\ &\quad - \sum\limits_0 \sum\limits_0 c(n)F(n,s) + \sum\limits_1 \sum\limits_0 c(n - 1)F(n - 1,s) \\ & = - \sum\limits_1 \sum\limits_0 a(n,s)F(n,s) + \sum\limits_1 \sum\limits_0 a(n,s)F(n,s) \\ &\quad - \sum\limits_0 \sum\limits_1 b(s)F(n,s) + \sum\limits_0 \sum\limits_1 b(s)F(n,s) \\ &\quad - \sum\limits_0 \sum\limits_0 c(n)F(n,s) + \sum\limits_0 \sum\limits_0 c(n)F(n,s) \\ & = 0,\end{split}$$
provided that
$ a(0,s) = 0 $ and
$ b(0) = 0 $. The first of these conditions states that there cannot be gain (emission) without electrons, whereas the second condition ensures that the signal number is nonnegative (only photons that are present can be lost). In contrast, there is no condition on
$ c(0) $. Nonzero
$ c(0) $ corresponds to direct driving (by an external current, for example).
The number moments were defined in Section 4.A. The first electron moment obeys the equation
(D5)$$\begin{split}{d_t}\langle n \rangle& = - \sum\limits_0 \sum\limits_0 na(n,s)F(n,s) + \sum\limits_0 \sum\limits_1 na(n + 1,s - 1)F(n + 1,s - 1) \\ &\quad - \sum\limits_0 \sum\limits_0 nc(n)F(n,s) + \sum\limits_1 \sum\limits_0 nc(n - 1)F(n - 1,s) \\ & = - \sum\limits_1 \sum\limits_0 na(n,s)F(n,s) + \sum\limits_1 \sum\limits_0 (n - 1)a(n,s)F(n,s) \\ &\quad - \sum\limits_0 \sum\limits_0 nc(n)F(n,s) + \sum\limits_0 \sum\limits_0 (n + 1)c(n)F(n,s).\end{split}$$
As explained after Eq. (
64), the
$ b $ terms were omitted from Eq. (
D5) because they do not affect the electron number directly. By combining the terms on the right side, one obtains the first-moment equation
(D6)$${d_t}\langle n \rangle = - \langle {a(n,s)} \rangle + \langle {c(n)} \rangle ,$$
where
$ \langle {a(n,s)} \rangle $ is the mean emission rate and
$ \langle {c(n)}\rangle $ is the mean pumping rate.
The first photon moment obeys the equation
(D7)$$\begin{split}{d_t}\langle s \rangle &= - \sum\limits_0 \sum\limits_0 sa(n,s)F(n,s) + \sum\limits_0 \sum\limits_1 sa(n + 1,s - 1)F(n + 1,s - 1) \\[-6pt] &\quad - \sum\limits_0 \sum\limits_0 sb(s)F(n,s) + \sum\limits_0 \sum\limits_0 sb(s + 1)F(n,s + 1) \\[-6pt] & = - \sum\limits_1 \sum\limits_0 sa(n,s)F(n,s) + \sum\limits_1 \sum\limits_0 (s + 1)a(n,s)F(n,s) \\[-6pt] &\quad - \sum\limits_0 \sum\limits_1 sb(s)F(n,s) + \sum\limits_0 \sum\limits_1 (s - 1)b(s)F(n,s).\end{split}$$
The
$ c $ terms were omitted from Eq. (
D7) because they do not affect the photon number directly. By combining the terms on the right side, one obtains the first-moment equation
(D8)$${d_t}\langle s \rangle = \langle {a(n,s)} \rangle - \langle {b(s)} \rangle ,$$
where
$ \langle {b(s)} \rangle $ is the mean loss rate. Equations (
D6) and (
D8) are consistent with Eq. (
B14).
By proceeding in a similar way, one obtains the second-moment equations
(D9)$$\begin{split}{d_t}\left\langle {{n^2}} \right\rangle& = - \sum\limits_0 \sum\limits_0 {n^2}a(n,s)F(n,s) + \sum\limits_0 \sum\limits_1 {n^2}a(n + 1,s - 1)F(n + 1,s - 1) \\[-6pt] &\quad - \sum\limits_0 \sum\limits_0 {n^2}c(n)F(n,s) + \sum\limits_1 \sum\limits_0 {n^2}c(n - 1)F(n - 1,s) \\[-6pt] & = - \sum\limits_1 \sum\limits_0 {n^2}a(n,s)F(n,s) + \sum\limits_1 \sum\limits_0 {(n - 1)^2}a(n,s)F(n,s) \\[-6pt] &\quad - \sum\limits_0 \sum\limits_0 {n^2}c(n)F(n,s) + \sum\limits_0 \sum\limits_0 {(n + 1)^2}c(n)F(n,s) \\[-6pt] & = - 2\langle {na(n,s)} \rangle + \langle {a(n,s)} \rangle + 2\langle {nc(n)} \rangle + \langle {c(n)} \rangle , \end{split}$$
(D10)$$\begin{split}{d_t}\left\langle {ns} \right\rangle &= - \sum\limits_0 \sum\limits_0 nsa(n,s)F(n,s) + \sum\limits_0 \sum\limits_1 nsa(n + 1,s - 1)F(n + 1,s - 1) \\[-6pt] &\quad - \sum\limits_0 \sum\limits_0 nsb(s)F(n,s) + \sum\limits_0 \sum\limits_0 nsb(s + 1)F(n,s + 1) \\[-6pt] &\quad - \sum\limits_0 \sum\limits_0 nsc(n)F(n,s) + \sum\limits_1 \sum\limits_0 nsc(n - 1)F(n - 1,s) \\[-6pt] & = - \sum\limits_1 \sum\limits_0 nsa(n,s)F(n,s) + \sum\limits_1 \sum\limits_0 (n - 1)(s + 1)a(n,s)F(n,s) \\[-6pt] &\quad - \sum\limits_0 \sum\limits_1 nsb(s)F(n,s) + \sum\limits_0 \sum\limits_1 n(s - 1)b(s)F(n,s) \\[-6pt] &\quad - \sum\limits_0 \sum\limits_0 nsc(n)F(n,s) + \sum\limits_0 \sum\limits_0 (n + 1)sc(n)F(n,s)\\[-6pt] & = \left\langle {na(n,s)} \right\rangle - \left\langle {sa(n,s)} \right\rangle - \left\langle {a(n,s)} \right\rangle - \left\langle {nb(s)} \right\rangle + \left\langle {sc(n)} \right\rangle , \end{split}$$
(D11)$$\begin{split}{d_t}\left\langle {{s^2}} \right\rangle &= - \sum\limits_0 \sum\limits_0 {s^2}a(n,s)F(n,s) + \sum\limits_0 \sum\limits_1 {s^2}a(n + 1,s - 1)F(n + 1,s - 1) \\[-6pt] &\quad - \sum\limits_0 \sum\limits_0 {s^2}b(s)F(n,s) + \sum\limits_0 \sum\limits_0 {s^2}b(s + 1)F(n,s + 1) \\[-6pt] & = - \sum\limits_1 \sum\limits_0 {s^2}a(n,s)F(n,s) + \sum\limits_1 \sum\limits_0 {(s + 1)^2}a(n,s)F(n,s) \\[-6pt] &\quad - \sum\limits_0 \sum\limits_1 {s^2}b(s)F(n,s) + \sum\limits_0 \sum\limits_1 {(s - 1)^2}b(s)F(n,s) \\&= 2\langle {sa(n,s)} \rangle + \langle {a(n,s)} \rangle - 2\langle {sb(s)} \rangle + \langle {b(s)} \rangle .\end{split}$$
Equations (
D9)–(
D11) are consistent with Eqs. (
B18) and (
B19). By combining them with Eqs. (
D6) and (
D8), one obtains the variance and correlation equations
(D12)$$\begin{split}{d_t}\left\langle {\delta {n^2}} \right\rangle &= - 2\left\langle {\delta n\delta a(n,s)} \right\rangle + 2\left\langle {\delta n\delta c(n)} \right\rangle \\&\quad + \langle {a(n,s)} \rangle + \langle {c(n)} \rangle ,\end{split}$$
(D13)$$\begin{split}{d_t}\left\langle {\delta n\delta s} \right\rangle &= \langle {\delta n\delta a(n,s)} \rangle - \langle {\delta s\delta a(n,s)} \rangle - \langle {\delta n\delta b(s)} \rangle \\&\quad + \langle {\delta s\delta c(n)} \rangle - \langle {a(n,s)} \rangle ,\end{split}$$
(D14)$$\begin{split}{d_t}\left\langle {\delta {s^2}} \right\rangle &= 2\langle {\delta s\delta a(n,s)} \rangle - 2\langle {\delta s\delta b(s)} \rangle \\&\quad+ \langle {a(n,s)}\rangle + \langle {b(s)} \rangle,\end{split}$$
where
$ \delta a(n,s) = a(n,s) - \langle {a(n,s)} \rangle $, and
$ \delta b(s) $ and
$ \delta c(n) $ are defined similarly. The driving terms in Eqs. (
D12)–(
D14) are consistent with the shot-noise rule. This result is a consequence of the quadratic forms that define the moments. It is valid for arbitrary coupling, gain and loss functions. In general, the mean, variance and correlation equations are not closed.
In the preceding example, $ c(n) $ represents electron gain. To model electron loss, one imposes the condition $ c(0) = 0 $ and changes the sign of the $ c $ term in Eq. (D6). One also changes the signs of the $ \delta c $ terms in Eqs. (D12) and (D13). It is straightforward to generalize the preceding results to more complicated systems, which involve more than two quantities.
APPENDIX E: FOURIER INTEGRALS
As explained in Appendix A, the autocorrelation function is the inverse Fourier transform of the RIN spectrum. It follows from Eqs. (100) and (104) that
(E1)$$C(\tau ) = \int_{ - \infty }^\infty \frac{{(A + B{\omega ^2})\exp ( - i\omega \tau )}}{{[(\omega _0^2 - {\omega ^2}{)^2} + 4\nu _0^2{\omega ^2}]}}\frac{{{\rm d}\omega }}{{2\pi }},$$
where the factor of
$ 1/S_0^2 $ was omitted. The Cauchy residue theorem states that
(E2)$$\int_\Gamma F(z){\rm d}z = 2\pi i\sum\limits_j {{\rm res}_j},$$
where
$ \Gamma $ is a closed curve in the complex plane, which is traversed in the anticlockwise direction, and
$ {{\rm res}_j} $ is an abbreviation for the residue of a pole that is enclosed by the curve. The integrand in Eq. (
E1) involves the factor
$ \exp ( - i\omega \tau ) $, so the contour integral must be closed in the lower half-plane. The integrand has simple poles at
$ \pm ({\omega _r} \pm i{\nu _0}) $, where the resonance frequency
$ {\omega _r} $ was defined after Eq. (
49), so the enclosed poles are at
$ {\omega _r} - i{\nu _0} $ and
$ - {\omega _r} - i{\nu _0} $. The denominator in Eq. (
E1) can be rewritten in the form
(E3)$$(\omega - {\omega _r} - i{\nu _0})(\omega - {\omega _r} + i{\nu _0})(\omega + {\omega _r} - i{\nu _0})(\omega + {\omega _r} + i{\nu _0}).$$
Consider the pole at
$ {\omega _r} - i{\nu _0} $. When one calculates the residue at this pole, one evaluates the numerator at this frequency. Then one omits the factor
$ \omega - {\omega _r} + i{\nu _0} $ from the denominator and evaluates the other factors at the aforementioned frequency. It follows from Eq. (
E3) that the residual denominator is
(E4)$$( - 2i{\nu _0})2({\omega _r} - i{\nu _0})(2{\omega _r}).$$
The factor of
$ 1/2\pi $ in Eq. (
E1) cancels the factor of
$ 2\pi $ in Eq. (
E2), and the need to traverse the curve in the clockwise direction introduces a factor of
$ - 1 $. Hence, the residue associated with the pole at
$ {\omega _r} - i{\nu _0} $ is
(E5)$$\frac{{A + B{{({\omega _r} - i{\nu _0})}^2}}}{{8{\nu _0}{\omega _r}({\omega _r} - i{\nu _0})}}\exp ( - i{\omega _r}\tau - {\nu _0}\tau ).$$
By symmetry (
$ {\omega _r} \to - {\omega _r} $), the residue associated with the pole at
$ - {\omega _r} - i{\nu _0} $ is
(E6)$$\frac{{A + B{{({\omega _r} + i{\nu _0})}^2}}}{{8{\nu _0}{\omega _r}({\omega _r} + i{\nu _0})}}\exp (i{\omega _r}\tau - {\nu _0}\tau ).$$
The second residue is the conjugate of the first, so the correlation function is
(E7)$$\begin{split}C(\tau ) = \frac{{\exp ( - {\nu _0}\tau )}}{{4{\nu _0}{\omega _r}}}{\rm Re}\left[ {\frac{{A + B{{({\omega _r} - i{\nu _0})}^2}}}{{{\omega _r} - i{\nu _0}}}\exp ( - i{\omega _r}\tau )} \right].\end{split}$$
In Eqs. (101) and (102), the functions that multiply $ {R_s}(\omega ) $ and $ {R_n}(\omega ) $ are the frequency-domain Green functions $ {G_{\textit ss}}(\omega ) $, $ {G_{\textit sn}}(\omega ) $, $ {G_{\textit ns}}(\omega ) $, and $ {G_{\textit nn}}(\omega ) $. Hence, the time-domain Green functions are combinations of the inverse transforms
(E8)$$\begin{split}{I_1}(t) &= \int_{ - \infty }^\infty \frac{{\exp ( - i\omega t){\rm d}\omega }}{{\omega _0^2 - 2i{\nu _0}\omega - {\omega ^2}}},\\ {I_2}(t)& = \int_{ - \infty }^\infty \frac{{ - i\omega \exp ( - i\omega t){\rm d}\omega }}{{\omega _0^2 - 2i{\nu _0}\omega - {\omega ^2}}}.\end{split}$$
The (common) denominator of these integrals can be written in the form
(E9)$$ - (\omega - {\omega _r} + i{\nu _0})(\omega + {\omega _r} + i{\nu _0}).$$
The pole at
$ {\omega _r} - i{\nu _0} $ has the residual denominator
$ - 2{\omega _r} $, whereas the pole at
$ - {\omega _r} - i{\nu _0} $ has the residual denominator
$ 2{\omega _r} $. By combining the residues and dividing by
$ i $, one obtains the inverse transforms
(E10)$$\begin{split}{I_1}(t) &= - \exp ( - i{\omega _r}t - {\nu _0}t)/2i{\omega _r} + \exp (i{\omega _r}t - {\nu _0}t)/2i{\omega _r} \\ & = \sin ({\omega _r}t)\exp ( - {\nu _0}t)/{\omega _r}, \end{split}$$
(E11)$$\begin{split}{I_2}(t) &= (i{\omega _r} + {\nu _0})\exp ( - i{\omega _r}t - {\nu _0}t)/2i{\omega _r} \\&\quad+ (i{\omega _r} - {\nu _0})\exp (i{\omega _r}t - {\nu _0}t)/2i{\omega _r}\\&= [\cos ({\omega _r}t) - {\nu _0}\sin ({\omega _r}t)/{\omega _r}]\exp ( - {\nu _0}t).\end{split}$$
Notice that
$ {I_2} = {\dot I_1} $, as required by Eqs. (
E8). By combining Eqs. (
101) and (
102) with Eqs. (
E10) and (
E11), one obtains the Green functions (
53)–(
56).
Funding
This work was supported by Futurewei Technologies.
Disclosures
The author declares that there is no conflict of interest.
REFERENCES
1. A. Siegman, Lasers (University Science, 1986).
2. P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1988).
3. C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer, 1985).
4. D. S. Lemons and A. Gythiel, “Paul Langevin’s 1908 paper ‘On the Theory of Brownian Motion’ [‘Sur la théorie du mouvement brownien,’ C. R. Acad. Sci. (Paris) 146, 530–533 (1908)],” Am. J. Phys. 65, 1079–1081 (1997). [CrossRef]
5. G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993).
6. L. C. Coldren, S. W. Corzine, and M. L. Masanovic, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012).
7. C. J. McKinstrie, “Gain, loss and the shot-noise rule,” J. Lightwave Technol., doi:10.1109/JLT.2020.2968720 (to be published).
8. B. van der Pol, “On relaxation oscillations,” Philos. Mag. 2, 978–992 (1926). [CrossRef]
9. R. Dunsmuir, “Theory of relaxation oscillations in optical masers,” J. Electron. Control 10, 453–458 (1961). [CrossRef]
10. W. Schottky, “Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern,” Ann. Phys. 362, 541–567 (1918). [CrossRef]
11. S. O. Rice, “Mathematical analysis of random noise,” Bell Sys. Tech. J. 23, 282–332 (1944). [CrossRef]
12. P. L. Meyer, Introductory Probability and Statistical Applications, 2nd ed. (Addison-Wesley, 1977).
13. T. Ikegami and Y. Suematsu, “Resonance-like characteristics of the direct modulation of a junction laser,” Proc. IEEE 55, 122–123 (1967). [CrossRef]
14. K. Shimoda, H. Takahasi, and C. H. Townes, “Fluctuations in amplification of quanta, with applications to maser amplifiers,” J. Phys. Soc. Jpn. 12, 686–700 (1957). [CrossRef]
15. D. E. McCumber, “Intensity fluctuations in the output of CW laser oscillators,” Phys. Rev. 141, 306–322 (1966). [CrossRef]
16. M. Lax, “Quantum noise IV: Quantum theory of noise sources,” Phys. Rev. 145, 110–129 (1966). [CrossRef]
17. D. E. McCumber, “Intensity fluctuations in the output of laser oscillators,” J. Quantum Electron. 2, 219–221 (1966). [CrossRef]
18. H. Haken, “Theory of intensity and phase fluctuations of a homogeneously broadened laser,” Z. Phys. 190, 327–356 (1966). [CrossRef]
19. M. Lax, “Quantum noise VII: The rate equations and amplitude noise in lasers,” J. Quantum Electron. 3, 37–46 (1967). [CrossRef]
20. M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser,” Phys. Rev. Lett. 16, 853–856 (1966). [CrossRef]
21. M. Lax, “Quantum noise X: Density-matrix treatment of field and population-difference fluctuations,” Phys. Rev. 157, 213–231 (1967). [CrossRef]
22. M. O. Scully and W. E. Lamb, “Quantum theory of an optical maser. 1. General theory,” Phys. Rev. 159, 208–226 (1967). [CrossRef]
23. J. W. Goodman, Statistical Optics (Wiley, 1985).
24. C. H. Henry, “Theory of the phase noise and power spectrum of a single mode injection laser,” J. Quantum Electron. 19, 1391–1397 (1983). [CrossRef]
25. C. J. McKinstrie and T. I. Lakoba, “Probability-density function for energy perturbations of isolated optical pulses,” Opt. Express 11, 3628–3648 (2003). [CrossRef]
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