1. Introduction

New luminescent materials with high gain and quantum yield such as perovskites [1–3] have spurred interest in ASE light sources. The narrow linewidth, high intensity of radiation, and low production cost make these light sources attractive for optoelectronic and optical storage devices [4].

ASE or super-luminescence can be produced by an active medium pumped by incoherent radiation. When a luminescent material is pumped, one of the excited atoms spontaneously emits a photon. Before leaving the active medium, this photon triggers stimulated emission of inverted atoms in its path [5–7]. Moreover, a plane wave traveling over an ASE system is amplified: $E(L)~\exp \left[LG({\gamma }_p-{\gamma }_D)/({\gamma }_p+{\gamma }_D)\right]$, where L is the length of the active medium, $G$ is the gain coefficient which is characteristic of the medium, and ${\gamma }_p$ and ${\gamma }_D$ are the pump and dissipation rates, respectively [5,8]. Above the compensation threshold, which is determined by the relation ${\gamma }_p={\gamma }_D$, a large increase of the output radiation with an increase in pumping is observed. The saturation of the population inversion $D_0=({\gamma }_p-{\gamma }_D)/({\gamma }_p+{\gamma }_D)$ results in an S-shaped dependence of the output intensity on the pump rate. The S-shaped of the input-output curve is also characteristic of lasers, in which it arises due to lasing above the laser threshold ${\gamma }_p\left(r\right)={\gamma }_{las}$, where $r$ is the reflection coefficient from the resonator walls. For the simplest case of the Fabry-Perot resonator, the generation of coherent light begins if $r^2\exp \left[LG({\gamma }_p-{\gamma }_D)/({\gamma }_p+{\gamma }_D)\right]\ge 1$ This inequality determines the threshold of the pump rate [5,8]. In an ASE system, it is assumed that there is no resonator and even if some reflection at boundaries of an active medium exists, then ASE occurs if $r^2\exp \left[LG({\gamma }_p-{\gamma }_D)/({\gamma }_p+{\gamma }_D)\right]\ll 1$.

Thus, similar to a laser, ASE exhibits threshold behavior. Moreover, ASE sources share many features with conventional lasers. Since, above the threshold, the gain of the active medium has a maximum at the transition frequency of inverted atoms ${\omega }_{TLS}$, the amplification of the electromagnetic (EM) field at neighboring frequencies is lower. The spectral linewidths of ASE sources are then narrowed when the volume or the gain coefficient of the active medium increases.

One of the most important characteristics of a light source is its photon statistics. The statistical properties of radiation are characterized by the second-order coherence function $g^{(2)}(\tau )=〈\widehat{I}(t)\widehat{I}(t+\tau )〉/{〈\widehat{I}(t)〉}^2$, where $\widehat{I}(t)$ is the operator of radiation intensity and the time $t$ is such that the system has reached a stationary state and its dynamics does not depend on the initial conditions. Coherent laser light has a Poissonian photon distribution with $g^{\left(2\right)}(0)=1$, while black-body radiation and single-photon sources have super- and sub-Poissonian photon distributions for which $g^{\left(2\right)}(0)$ equals 2 and 0, respectively [9]. Statistical properties of ASE, however, are not well-studied because the determination of the distribution of emitted photons is complex for a wide spectral range of radiation [10]. The question of the statistics expected for ASE is still not settled. There are arguments that ASE is highly coherent with Poissonian photon statistics ($g^{(2)}(0)=1$) [11,12], as well as arguments that ASE has low coherency with $g^{(2)}(0)=2$ [13]. Recent experiments have yielded contradicting results. In [14], it is shown that ASE systems can demonstrate laser-like statistical properties and have $g^{(2)}(0)$ that is close to unity; in [15–17], it is demonstrated that depending on the pump power $g^{(2)}(0)$ changes from 2 to 1, whereas [13,18] reported that $g^{(2)}(0)=2$ as is characteristic of black-body radiation.

In this paper, we investigate the dependence of the coherence properties of ASE on the pump rate. We demonstrate by computer simulations that ASE has the super-Poissonian distribution of photons with $g^{(2)}(0)\approx 2$ for any value of the pump rate. This makes it possible to distinguish ASE sources from lasers.

2. The model for light interaction with active medium

To describe the interaction between an EM field and atoms of a gain medium we use the Jaynes-Cummings Hamiltonian in the rotating wave approximation [19]:

The first and the second terms of Hamiltonian (1) describe the EM field and the active medium, respectively, the third term is responsible for the interaction between the EM field and atoms, the other two terms describe all essential reservoirs and their interactions with the EM field and the active medium. These reservoirs are critical to our considerations since they describe losses in the system, incoherent pumping of the active medium, and the spontaneous emission. ${\widehat{H}}_R$ is a sum of Hamiltonians of reservoirs:

To describe pumping of the active medium atoms, we introduce reservoirs of auxiliary TLSs having negative temperature (see for details [8]).The j-th reservoir interacts with the j-th atom and is described by the Hamiltonian

The Hamiltonian that describes the interaction of the system with these reservoirs is the sum

To obtain the Heisenberg-Langevin equation of motion for the relevant operators of the system, we eliminate the reservoir variables by averaging over the equilibrium states of the reservoirs [19]. In the process of averaging, we assume that, first, the reservoir is much larger than the system, and therefore, one can neglect the influence of the system on the reservoir. Second, since the timescale, characterizing fluctuations in the reservoir (${\tau }_R~\hslash /kT~{10}^{-14}$s [19]), is much shorter than the relaxation time of the system (for a typical gain medium, such as organic semiconductors, the relaxation time is about ${10}^{-13}$s), the fluctuation-dissipation theorem is correct. Consequently, after averaging, both relaxation and noise terms appear in the equations.

The elimination of the reservoir variables, ${\widehat{b}}_{nm}$, ${\widehat{s}}_{jm}$, and ${\widehat{c}}_{jm}$, results in the appearance of the relaxation, ${\gamma }_a$, ${\gamma }_D$, ${\gamma }_{deph}$, ${\gamma }_p$, and fluctuation (noise) terms, ${\widehat{F}}_{an}$, ${\widehat{F}}_{an}^{†}$, ${\widehat{F}}_{\sigma j}$, ${\widehat{F}}_{\sigma j}^{†}$, and ${\widehat{F}}_{Dj}$, in the equation for the operators ${\widehat{a}}_n^{†}$, ${\widehat{a}}_n$, ${\widehat{\sigma }}_j^{†}$, ${\widehat{\sigma }}_j$, and ${\widehat{D}}_j$(see [8,19]). Note that the corresponding terms are connected via the fluctuation-dissipation theorem. The elimination of ${\widehat{b}}_{nm}$ results in the appearance of ${\gamma }_a$ and ${\widehat{F}}_{an}$; the elimination of ${\widehat{c}}_{jm}$ leads to the appearance of ${\gamma }_{deph}$ and ${{\widehat{F}}^{\prime }}_{\sigma j}$; the elimination of ${\widehat{h}}_{jm}$ leads to the appearance of ${\gamma }_{deph}$,${{\widehat{F}}^{\prime }}_{\sigma j}$,${\gamma }_D$, ${{\widehat{F}}^{″}}_{\sigma j}$ and ${{\widehat{F}}^{″}}_{Dj}$. By introducing new noise terms ${\widehat{F}}_{\sigma j}={{\widehat{F}}^{\prime }}_{\sigma j}+{{\widehat{F}}^{″}}_{\sigma j}+{{\widehat{F}}^{‴}}_{\sigma j}$ and ${\widehat{F}}_{Dj}={{\widehat{F}}^{″}}_{Dj}+{{\widehat{F}}^{‴}}_{Dj}$, we can rewrite the final equations for the operators ${\widehat{a}}_n^{†}$, ${\widehat{a}}_n$, ${\widehat{\sigma }}_j^{†}$, ${\widehat{\sigma }}_j$, and ${\widehat{D}}_j$ as follows:

In Eq. (15), the term $-{\gamma }_p\left({\widehat{D}}_j-1\right)$ describes pumping that increases the expectation value of the population inversion operator ${\widehat{D}}_j$ to 1, while the term $-{\gamma }_D\left({\widehat{D}}_j+1\right)$ describes relaxation that decreases $〈{\widehat{D}}_j〉$ to –1. After simple algebra, Eq. (15) may be recast as

Since each atom of the active medium interacts with its own environment described by the corresponding phonon reservoir, the correlation functions of noise operators of different atoms are assumed to be equal to zero [8], i.e.

Since in most ASE experiments, the number $N_c$ of inverted atoms even in a subwavelength volume is large (e.g., in a volume about 100 nm in extent, $N_c$ is of the order of ${10}^3$), we can transition from operators to c-numbers. For this purpose, we use the method of large cells. We divide the whole volume into cells of the size ${\lambda }_{TLS}/10$ (${\lambda }_{TLS}$ is the transition wavelength of gain medium atoms) and switch to the mean operators: ${\widehat{\sigma }}_k^{cell†}={\displaystyle {\sum }_{j\in cel{l}_k}{\widehat{\sigma }}_j^{†}}/N_c$, ${\widehat{\sigma }}_k^{cell}={\displaystyle {\sum }_{j\in cel{l}_k}{\widehat{\sigma }}_j}/N_c$, and ${\widehat{D}}_k^{cell}={\displaystyle {\sum }_{j\in cel{l}_k}{\widehat{D}}_j}/N_c$. Following the method of the system size expansion [19], we consider $1/N_c$ as an expansion parameter and investigate the limiting behavior of the system when $N_c\to \infty$, neglecting higher-order terms with respect to $1/N_c$ terms. In this limit, the expected values of operators grow with $N_c$ faster than the quantum corrections to these expected values. This enables us to transform Eqs. (11)–(15) to a system of equations for c-numbers in the leading order of $1/N_c$, while quantum corrections can be presented as a classical noise in the second order of $1/N_c$.

Since the cell size is smaller than the wavelength of radiation, we assume that the Rabi frequency is the same for all the atoms in the cell. For the n-th mode and the k-th cell, ${\Omega }_{nj}={\Omega }_{nk}$ for any j-th atom of the cell. The corresponding terms in Eqs. (11) and (12) become ${\displaystyle {\sum }_{j\in cel{l}_k}{\Omega }_{nj}{\widehat{\sigma }}_j}={\Omega }_{nk}{N}_c{\widehat{\sigma }}_k^{cell}$, where ${\Omega }_{nk}=-d_{egk}^{}\cdot E_n\left(x_k\right)/\hslash$, $x_k$ is the coordinate of the k-th cell and $d_{egk}^{}$ is the transition dipole moment of atoms averaged over the k-th cell. Finally, we obtain the system of equations for c-numbers:

In the absence of noise terms in Eqs. (24)-(26), Eqs. (22) and (23) and Eqs. (24) and (25) are complex conjugates. If initially $a_n^{cc}(t=0)={\left(a_n(t=0)\right)}^{*}$ and ${\sigma }_k^{cellcc}(t=0)={\left({\sigma }_k^{cell}(t=0)\right)}^{*},$ then these variables remain complex-conjugated all the time. Since operators ${\widehat{a}}_n^{†}$, ${\widehat{a}}_n$ and ${\widehat{\sigma }}_k^{cell†}$, ${\widehat{\sigma }}_k^{cell}$ do not commute, in Eqs. (24)-(25), one needs to consider different realizations for noises. Consequently, c-number noise terms corresponding to conjugated operators though retain correct correlations properties are no longer complex conjugated [8,19]. Then, Eqs. (22) and (23) and Eqs. (24) and (25) are also not complex conjugates. As a result, the variables $a_n^{cc}$, $a_n$ and ${\sigma }_k^{cellcc}$, ${\sigma }_k^{cell}$ corresponding to the operators ${\widehat{a}}_n^{†}$, ${\widehat{a}}_n$ and ${\widehat{\sigma }}_k^{cell†}$, ${\widehat{\sigma }}_k^{cell}$ cease to be complex conjugates.

3. Computer simulation of ASE

As a model for ASE source, we use a 1D waveguide of length $L_0=1800{\lambda }_{TLS}$ which contains a region of length $L=140{\lambda }_{TLS}$ filled with gain medium atoms, ${\lambda }_{TLS}=2\pi c/{\omega }_{TLS}$. Pumping of this region creates a positive population inversion (see Fig. 1).

 

Fig. 1 The sketch of the ASE source based on a single-mode waveguide. The active medium in the central region of the waveguide (shaded by red) with the length L is pumped by an external source; the active medium outside of this region is not pumped.

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In our numerical simulation, the EM field is modeled by 600 modes with equidistant frequencies within the interval $(0.92{\omega }_{TLS},1.08{\omega }_{TLS})$. The gain medium is divided into subwavelength simulation cells of size $\Delta x=\lambda /10$ (the total number of cells is 1400). To verify the validity of the chosen parameters for the computational scheme, namely, the lengths of the waveguide and the active medium cells, we compared the results of simulations for various values of $L_0$ and $\Delta x$. We confirmed that an increase in the waveguide length above $1800{\lambda }_{TLS}$, as well as a decrease of the cell size below $\lambda /10$, did not affect the results of the simulation. We use Eqs. (22)-(26) to find the radiation output, the spectrum, and the second-order coherence function of ASE and laser radiations. Variations in dissipation rates can be compensated for by changes in the pump rate. Therefore, in calculations, the parameters in Eqs. (22)-(26) are chosen to be close to the gain medium based on organic semiconductors [20], while the pump rates are varied over the wide range. The relaxation rates are ${\gamma }_a=2\times {10}^{-3}{\omega }_{TLS}$, ${\gamma }_{deph}=5.2\times {10}^{-2}{\omega }_{TLS}$, ${\gamma }_D={10}^{-6}{\omega }_{TLS}$, and the pump rate varies within the interval ${\gamma }_p=(2\times {10}^{-7}-{10}^{-2}){\omega }_{TLS}$. Such a choice of the modeled medium means that lasing may start if $r^2=\exp \left[-LG\left({\gamma }_p-{\gamma }_D\right)/\left({\gamma }_p+{\gamma }_D\right)\right]\sim 0.05$, and ASE is observable if $|r{|}^2<0.05$.

As mentioned above, the main characteristic that allows one to distinguish coherent laser radiation from incoherent black-body radiation experimentally is the second-order coherence function $g^{(2)}(\tau )$. The value of $g^{(2)}(\tau )$ for any radiation source tends to unity as $\tau \ge 1/\Delta \omega$, where $\Delta \omega$ is the radiation linewidth of the source [9,19]. Consequently, the response time ${\tau }_{res}$ of an experimental setup should be smaller than $1/\Delta \omega$. If ${\tau }_{res}>1/\Delta \omega$, the measured value of $g^{(2)}(0)$ is always about unity. Experimentally, a spectrum of an investigated source is narrowed by filtering. It is preferable to narrow the line as much as possible. For this reason, we study the second-order coherence function at a fixed frequency, $g^{(2)}(\omega ,\tau )=〈I_{\omega }(t)I_{\omega }(t+\tau )〉/{〈I_{\omega }(t)〉}^2$, where $I_{\omega }$ is the output intensity at the frequency $\omega$. Representing the electric field operator as $\widehat{E}\left(x,t\right)={\widehat{E}}^{(+)}\left(x,t\right)+{\widehat{E}}^{(-)}\left(x,t\right)$, where ${\widehat{E}}^{(+)}\left(x,t\right)={\displaystyle {\sum }_n{E}_n(x){\widehat{a}}_n(t)}$, and ${\widehat{E}}^{(-)}\left(x,t\right)={\displaystyle {\sum }_n{E}_n^{*}(x){\widehat{a}}_n^{†}(t)}$, for the average intensity of the electromagnetic field, $〈I\left(x,t\right)〉=〈{\widehat{E}}^{(-)}\left(x,t\right){\widehat{E}}^{(+)}\left(x,t\right)〉$ [9], we arrive at the following expression for $g^{(2)}\left({\omega }_n,\tau \right)$:

Our numerical simulation confirms that the input-output curve of the ASE source has an S-shape (the blue line in Fig. 2) similar to the intensity of conventional laser radiation. The S-shape of the input-output curve of ASE sources arises due to the three regimes of the system behavior corresponding to different pump rates. At a low pump power, the energy loss of the EM wave propagating in the waveguide exceeds its gain in the active medium. In this regime, the EM wave exponentially decays while propagating through the structure; the output power linearly increases with the growth of the pump power. At a higher pump power, the gain exceeds the propagation losses, and the propagating EM wave exponentially intensifies. At this regime, the saturation of the active medium is inessential. This regime corresponds to the shaded area in Fig. 2. A further increase in the pump rate leads to saturation of the active medium (see Fig. 2). The transitional region between the first and third regimes is referred to as the ASE threshold [5]. Note that in all three regimes, $g^{(2)}({\omega }_{TLS},0)$ remains equal to 2 (see Fig. 2.)

 

Fig. 2 The intensity of the EM field at the boundary of the active medium (the blue line) and $g^{(2)}({\omega }_{TLS},0)$ (the red line). The left vertical dashed line shows the compensation threshold, at which pumping compensates for losses in the active medium. The right vertical dashed line represents the pump rate, at which the system transitions to the nonlinear regime due to substantial saturation of the active medium. In this regime, the output power linearly depends on the pump power [5]. The shaded area shows the transitional region in which an increase in the pump rate results in an exponential increase in the radiation intensity.

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In the third region, the decrease of intensity happens because incoherent pumping not only increases $D$ but also causes additional dephasing of the atom dipole moments [see Eq. (24)]. As a result, the stationary value of the dipole moment of atoms of the active medium, ${\sigma }^{cell}$, is inversely proportional to the sum ${\gamma }_{deph}+{\gamma }_p+{\gamma }_D$. In this sum, the relaxation rate of the atomic dipole moment, ${\gamma }_{deph}$, and the relaxation rate of population inversion, ${\gamma }_D$, describe the dephasing that arises due to the interaction of active atoms with phonons, while the pump rate of active atoms, ${\gamma }_p$, describes the dephasing arising due to incoherent pumping. For pumping rates ${\gamma }_p$, which are much greater than both ${\gamma }_{deph}$ and ${\gamma }_D$, the stationary value of the dipole moments of atoms of the active medium is inversely proportional to the pumping rate, ${\sigma }^{cell}~1/{\gamma }_p$. The output power is proportional to ${\left|{\sigma }^{cell}\right|}^2$ and therefore, when the pump rate tends to infinity, the output power monotonically decreases [23], as seen Fig. 2. In this region, $g^{(2)}(\omega ,0)$ also remains equal to 2 for any frequency.

Though, within the transition range, the output intensity increases by more than an order of magnitude (see the blue line in Fig. 2), the value of the second-order coherence function of ASE is independent of the pump rate and the frequency (see the red line in Fig. 2) and is about 2, while for coherent light it should be equal to 1.

In Figs. 3(a),(c) and 3(b),(d), the spectral distribution of the photon number, $n\left(\omega \right)=〈a^{cc}\left(\omega \right)a\left(\omega \right)〉/{〈a^{cc}\left(\omega \right)a\left(\omega \right)〉}_{\max }$, and $g^{(2)}\left(\omega ,0\right)$ are shown at the pump rates below and above the compensation threshold, respectively. For small pump rates, the spectrum of the system is similar to the spectrum of a single atom of the active medium, see Fig. 3(a). For pump rates that are higher than that of the transitional regime, the spectrum of the system markedly narrows, Fig. 3(c). The observed narrowing is a common feature for both lasers and ASE.

 

Fig. 3 Spectra, $n\left(\omega \right)$, normalized by its maximum value, and $g^{(2)}(\omega ,0)$ of an ASE source for the pump rates below (a), (b) and above (c), (d) the compensation threshold. In Figs. (a) and (b), the pump rate ${\gamma }_p$ is equal to ${\gamma }_D$, and in Figs. (c) and (d), it is equal to $20{\gamma }_D$. Dashed red curves show the absorption line of the unpumped medium. The values of $g^{(2)}(\omega ,0)$ are found by using Eq. (27).

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At the same time, we can see that despite the narrowing of the spectrum line, $g^{(2)}(\omega ,0)\approx 2.0$ for any mode, independently of the frequency [see Figs. 3(b) and 3(d)]. As we discuss below, this distinguishes ASE from laser radiation. Visible fluctuations of $g^{(2)}(\omega ,0)$ are due to the finite number of realizations of stochastic process used in the calculations.

As Figs. 2 and 3(b),(d) show, the second-order coherence function $g^{(2)}(\omega ,0)$ depends on neither the pump rate nor the frequency, and it is about 2. This means that the coherent properties of ASE are substantially different from radiation of lasers; they are rather close to the black-body radiation.

4. Laser emission

To test our results, in the system shown in Fig. 1, we add mirrors to the edges of the active medium and make calculations similar to that discussed above. The mirrors introduce positive feedback, and the system is expected to lase. In a laser, the value of $g^{(2)}(0)$ depends on the pump rate [9]: below the generation threshold, $g^{(2)}(0)$ must be close to two, above the threshold, it should tend to one. Our computer simulation confirms this assumption.

In Fig. 4, the dependence of the system output intensity on the pump rate is shown. This dependence has a pronounced S-shape (the blue line). The laser generation threshold (an inflection on the S-shaped curve) arises due to three regimes of a laser. For small pump rates, the amplification of radiation in the gain medium is not sufficient for the loss compensation in the waveguide and mirrors. In this regime, auto-oscillations are not established. For intermediate pump rates, auto-oscillations begin, but the nonlinear contribution is not dominant yet. The beginning of auto-oscillations is defined by the Maxwell-Bloch threshold in a system without noise. At a further increase in the pump rate, the population inversion of the active medium becomes significant causing the transition of the dependence of the radiation intensity from exponential to linear.

 

Fig. 4 Red and black curves represent the coherence function $g^{(2)}(\omega ,0)$ for the dominant modes shown in Fig. 5(c). The blue curve is the generation curve of the laser. The curves are obtained by solving the Maxwell-Bloch equations with noise. The left vertical dashed line shows the lasing threshold given by the Maxwell-Bloch equations without noise. The right vertical dashed line corresponds to the pump rate, for which $g^{(2)}(\omega ,0)-1$ of the modes becomes inversely-proportional to the average number of photons. Grey shading marks the transitional regime of the laser. The parameters of the active structure are the same as for the ASE system discussed above. The amplitude reflectance of the mirrors is 0.8.

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Below the generation threshold, the system spectrum exhibits a large number of lines corresponding to modes of the Fabry-Perot cavity [see Fig. 5(a)]. The value of $g^{(2)}(\omega ,0)$ for all modes is approximately 2 [see Fig. 5(b)]. This is close to $g^{(2)}(\omega ,0)$ for the system without mirrors [see Fig. 3(b)]. Far above the generation threshold, there are only a few lines in the spectrum [see Fig. 5(c)]. These lines correspond to the eigenfrequencies of the Fabry-Perot cavity that are closest to the transition frequency of the active medium, and consequently, have the lowest values of thresholds. For these modes, $g^{(2)}(\omega ,0)$ is close to 1 even though for all other modes it remains close to 2.

 

Fig. 5 Normalized spectra and $g^{(2)}(\omega ,0)$ of laser emission for pump rates ${\gamma }_p={\gamma }_D$ (a), (b) (below the lasing threshold) and ${\gamma }_p=20{\gamma }_D$ (c), (d) (above the laser threshold).

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Near the generation threshold, our model exhibits large fluctuations and a large spread in $g^{(2)}(\omega ,0)$. This happens because, near the threshold, the laser behavior is similar to a second order phase transition [8,19], in which a system undergoes strong fluctuations near the critical point [24]. In our case, the critical region corresponds to pump rates between ${\gamma }_p=1.5{\gamma }_D$ and $6{\gamma }_D$ (the shaded area in Fig. 4).

Thus, in the system with mirrors, a pronounced generation threshold is observed. Our calculations show that below the threshold, for all modes, $g^{(2)}(\omega ,0)$ significantly exceeds 1, while above the threshold, for the modes in which generation occurs, $g^{(2)}(\omega ,0)$ tends to unity. $g^{(2)}(\omega ,0)$ decreases only for the eigenmodes of the system with mirrors and only when there is coherent feedback. This is in agreement with the expected behavior of a laser. We emphasize that in this section we use exactly the same model as in the previous one except that mirrors that create a cavity are added. Then, using the same procedure as in the previous section, we obtain well-known results. This can serve as a confirmation of the correctness of the results obtained for ASE.

5. Discussion and conclusion

In this paper, we compare the coherent properties of a mirrorless ASE system to the properties of a laser built of the same pumped atoms. To make this comparison, it is convenient to use the parameter $\eta =G{D}_{0}L$. At the transition frequency, the gain is $G=4\pi {\omega }_{TLS}{\left|d_j\right|}^2{n}_{TLS}/\left(\hslash {\gamma }_{deph}c\right)$, where $n_{TLS}$ is the concentration of the atoms of active medium (see [25,26]).

Since the refractive index of the active medium $\sqrt{{\epsilon }_{gain}(\omega )}$ differs from the refractive index of vacuum, the reflection from boundaries results in the appearing a Fabry-Perot cavity that, in turn, may cause lasing. The lasing threshold is determined by the following condition [25]:

In our 1D model, each atom radiates in both directions forming incident and refracted waves. Therefore, the difference in refractive indices is automatically taken into account. Thus, the independence of $g^{\left(2\right)}(\omega ,0)~2.0$ on the pump rate and the frequency (see Fig. 2) indicates that lasing is not achieved, and our system may be considered as mirrorless ASE.

An addition of semi-transparent mirrors with $r=0.8$ to the boundaries of the active medium with $\eta =5.6$ converts the system into a laser . As one can see from Fig. 4, for this laser, $g^{(2)}(\omega ,0)$ tends to unity above the laser threshold.

In [15], it has been reported that for a commercial superluminescent light diode (SLD), the value of $g^{\left(2\right)}(0)$ tends to unity with an increase in the pump rate. SLDs are high intensity light sources having a broad spectrum in the output radiation. This type of light source is required for a number of applications [27-29]. Since, even for multimode lasers, the spectra are modulated by lines of Fabry-Perot modes, SLDs are manufactured as ASE systems with high gain. To achieve this, special efforts are taken to reduce reflections inside an SLD. A broad spectrum of the SLD output radiation is often considered as the criterion confirming that an SLD operates as an ASE system [30, 31]. According to [30, 31], to eliminate the spectrum modulation by lines of Fabry-Perot modes, the intrinsic reflection coefficient, ${\left|r\right|}^2$ , should be less than $2.5\cdot {10}^{-6}$ . Thus, if ${\left|r\right|}^2<2.5\cdot {10}^{-6}$ , the SLD may be considered as an ASE system.

In fact, the definition of ASE as the absence of the spectrum modulations is not quite correct. Indeed, for high gain, lasing may start at an even lower value of the reflection coefficient and before the spectrum becomes modulated. For such a small value of the reflection coefficient, the Q-factor of modes is also low, and the linewidth of the modes may be much greater than the distance between them. This means that the modulation in the spectrum is smoothed. For an SLD, a typical value of the gain is $G~1500{\text{cm}}^{-1}$ [20], with the size of active medium of $L~0.01\text{cm}$ . In such a system, the lasing may start at $|r{|}^2\text{=}\text{exp(}-GL\text{)}~3\cdot {10}^{-7}$ . Thus, the system considered in [15] could be a laser with $g^{\left(2\right)}(0)\to 1$ .

To verify this assumption, we perform computer simulation for our model with increased length of the region filled with gain medium atoms. The linear stability analysis of the system shows the existence of the Hopf bifurcation with a lasing solution at the threshold value of the population inversion. The results of computer simulation shown in Fig. 6(a), demonstrate that $g^{(2)}({\omega }_{TLS},0)$ starts decreasing above the lasing threshold while the spectrum width remains constant. The spectrum remains almost without modulation, and for 600 modes taken into account we observe only a weak ripple near ${\omega }_{TLS}$ [see Fig. 6(b)]. Note that an addition of mirrors with high reflection coefficient causes intense modulation of the spectrum shown in Fig. 5(c).

 

Fig. 6 (a) The dependence of ($g^{(2)}\left({\omega }_{TLS},0\right)$the red line) and the spectrum half-width at the half-height $\Delta \omega$ (the blue line) on the pump rate for the mirrorless system with $L=700{\lambda }_{TLS}$ and $G=673c{m}^{-1}$; the dashed black line denotes the lasing threshold derived from linear analysis, ${\gamma }_p=8.66{\gamma }_D$; the threshold value of the population inversion is $D_{th}=0.793$. (b) The normalized spectrum of the system at ${\gamma }_p=800{\gamma }_D$ (the black solid line) and the spectrum of the atom (the red dashed line).

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Thus, according to our calculations, for $\eta ~5$, above the compensation threshold, coherence properties of ASE are still characterized by $g^{\left(2\right)}(\omega ,0)\approx 2$.

It is interesting to study the value of $g^{\left(2\right)}(0)$ for EM waves amplified by stimulated emission in interstellar gases [32–34] and planetary atmospheres [35,36]. Such radiation sources are called astrophysical masers or lasers. In [34, 37], it is claimed that such sources operate as random lasers and should have $g^{\left(2\right)}(0)\approx 1$ . An alternative mechanism could be the phenomenon of ASE [38] with $g^{\left(2\right)}(0)\approx 2$ . An investigation of the second-order coherence function of radiation from an astrophysical “maser” could shed light on the mechanism of formation of such sources. In turn, the understanding of the mechanism of the formation of astrophysical masers and lasers can enable one to estimate the concentration of the molecules, atoms, or ions in interstellar clouds.

In addition to its fundamental significance, the difference between ASE and laser radiation is of practical importance. For ghost imaging applications, light sources with super-Poissonian distributions of photons, a narrow linewidth, and a high intensity are required [39]. Currently, either lasers with rotating ground glass [40–42], producing radiation with $g^{\left(2\right)}(0)$ between 1.25 and 1.9, or incoherent lamps with a frequency filter [43,44] with $g^{\left(2\right)}(0)$ of about 1.05 are used as light sources. In this paper, we demonstrate that the second-order coherence function of ASE sources is about 2 that is closer to the coherence of lasers with rotating ground glass. However, ASE sources are much simpler and easier to manufacture than lasers. This makes ASE light sources promising for ghost imaging.