# “Amplified Spontaneous Emission” in Micro- and Nanolasers

## Abstract

**:**

## 1. Introduction

## 2. Laser Threshold at the Micro- and Nanoscale

## 3. Summary of Existing Evidence for Photon Bursts

#### 3.1. Experimental Observations

- E.1
- Finite width of ${g}^{\left(2\right)}\left(\tau \right)$ and of ${g}^{\left(2X\right)}\left(\tau \right)$ (the latter measured with a Michelson interferometer) obtained from a photonic crystal nanolaser and ascribed to amplitude fluctuations of a coherent state (cf. Figures 2 and 3 in [71] and Figures 4 and 5 in [74]). Observation compatible with the emission of photon bursts;
- E.2
- Direct observation of photon bursts in a narrow, but well accessible interval of pump values before the onset of (noisy) cw laser oscillation in a microlaser (cf. Figure 3 in [72]);
- E.3
- Narrow, structureless decay of the second-order time-delayed autocorrelation ${g}^{\left(2\right)}\left(\tau \right)$ with typical width $O\left({10}^{-9}s\right)$ both in microlasers and metallo-dielectric nanolasers (cf. Figure 5 in [72], Figure 3 in [75] and Figure 3 in [76]). Similar observation in lasing devices built upon photonic crystal nanostructures (cf. Figure 4 in [77]);
- E.4
- Report of a peak superimposed on the second-order time-delayed autocorrelation, ${g}^{\left(2\right)}\left(\tau \right)$, for same-(pump)-pulse, attributed to ASE in a metallo-dielectric nanolaser (cf. Figure 3 in [75]). This additional peak disappears as lasing is established;
- E.5
- Low-frequency broadband rf spectrum with cutoff compatible with the width of ${g}^{\left(2\right)}\left(\tau \right)$ (cf. Figure 8 in [78]) (micro-VCSEL). The lack of spectral structure (except a gradual decay towards the cutoff) is compatible with irregularly occurring bursts;
- E.6

#### 3.2. Interpretation

## 4. Relationship between Bursts and ${\mathit{g}}^{\left(\mathbf{2}\right)}\left(\mathbf{0}\right)$—Experimental Considerations

## 5. Theoretical Models

#### 5.1. Differential Models

- D.1
- The addition of spontaneous emission, for instance as a constant contribution (cf. Equation (7)), breaks the transcritical bifurcation [97,98,99] which characterizes the standard REs written for macroscopic lasers [50,96]. This is a consequence of the finite cavity volume (expressed by the fraction of spontaneous emission coupled into the lasing mode, $\beta $) and is related to the disappearance of the thermodynamic limit, recovered when $\beta \to 0$ (cf. [67] for details). Its immediate, and partly counterintuitive, consequence is a progressive stabilization of the laser operation as $\beta $ increases (cf. Section 6): nanodevices are more stable than their macroscopic counterparts. For an interesting physical application of imperfect bifurcations, cf. [100].
- D.2
- Differential models have so far considered only the coherent part of the electromagnetic field, without introducing an independent random field for the description of spontaneous emission. The latter is added onto the coherent part (as a coherent contribution from the spontaneous relaxation processes, e.g., in REs) but does not exist as a variable in itself. This is an important conceptual point which prevents a correct description of the below-threshold region. Some stochastic models (Section 5.2) introduce the incoherent field as an independent variable, albeit without the concept of random phase, since they are based on a photon number concept.
- D.3
- As noted in [101] through a numerical integration of discretized REs, the integer nature of photons and emitters makes itself felt at the nanoscale (and even at the mesoscale [102]). This introduces an intrinsic noise, entirely missed by the differential models, and leads to a background granularity which cannot be replaced by other means. In this sense, discrete models (cf. Section 5.2 and Section 5.3) hold superior predictive power for small devices.
- D.4
- The introduction of Langevin terms in differential models to simulate fluctuations has two shortcomings: it may lead to negative photon numbers (thus numerical instabilities) when the latter is very small (thus close to threshold, especially in nanolasers) and to an incorrect approximation of the noise distribution. The former problem could be solved by abandoning the photon number representation, but the latter reposes on the approximation of Poissonian processes (true physical statistics of light-matter interaction) with Gaussian ones; such replacement holds only for large arguments, a condition violated at small photon numbers [103].

#### 5.2. Stochastic Simulators

#### 5.3. Stochastic Predictions

- N.1
- N.2
- N.3
- Prediction of photon bursts in the laser output for semiconductor nanolaser in dynamical regimes between Classes A and B (cf. Figure 7 in [116]);
- N.4
- Prediction of superthermal statistics (free-running, Quantum Well laser, Figure 7 in [119]);
- N.5
- Prediction of superthermal statistics using a Gillespie algorithm in a model with pump blocking, suited to Quantum Dot modelling (cf. Figure 5 in [104]).

## 6. Phase Space Information

#### 6.1. Eigenvalue Analysis

#### 6.2. Eigenvector Analysis

**all**spontaneous emission ends up into the lasing mode. From a more mathematical point of view, this remark matches the fact that the coupling coefficient in Equations (7) and (8) is larger thanks to $\beta =1$. As $\beta $ decreases there is a rapid reduction in coupling between the two variables, represented by the ever smaller component of the eigenvector projection along the photon axis. This holds true even close to the transition—identified by the end of the line traced in each figure—which corresponds to the transformation from a real into a complex eigenvector, or to the appearance of an imaginary part in the eigenvalue. In microlasers (e.g., $\beta ={10}^{-3}$) the amplitude of the photon number component of the eigenvector is only 1% even at threshold (Figure 7).

_{2}lasers considered in [122]).

#### 6.3. Topological Conclusions

## 7. Symmetry Break between Spontaneous and Stimulated Processes

## 8. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Graphical representation of the topics addressed in the manuscript.

**N**and

_{ss}**n**refer to the steady-state operation values of the population inversion and photon number, respectively, around which a fluctuations operates, for a given set of parameters.

_{ss}**Figure 2.**Schematic illustration of the evolution of the time intervals (red islands) in which the photon number exceeds a predetermined threshold for two different values of pump, computed from a Stochastic Laser Simulator [83] for $\beta ={10}^{-4}$. The black lines represent the photon number. The pump in the two panels differs by about 2%. The relative time intervals are ${a}_{c}=0.11$ and ${a}_{c}=0.57$ for the left and right panels, respectively. ${a}_{s}=1-{a}_{c}$. Graphical resolution makes it almost impossible to detect the “holes” in the lower red set (left panel) which match the appearance of the red upper regions—they are, nonetheless, present. Data courtesy of G.P. Puccioni.

**Figure 3.**Zero-delay second-order autocorrelation, Equation (4), in double logarithmic scale, as a function of the duty cycle $\delta $, and for different values of the baseline’s amplitude $\alpha $ (fraction of the peak amplitude A).

**Figure 4.**Real part of the eigenvalues derived from the stability analysis of the rate equations model, Equations (12) and (13). The different curves belong to different values of the fraction of spontaneous emission coupled into the lasing mode (cf. figure legend).

**Figure 5.**Laser characteristic response in double-logarithmic scale as a function of normalized pump, for different values of $\beta $ (cf. legend).

**Figure 6.**Real part of the eigenvalues derived from the stability analysis of the rate equations model, Equations (12) and (13) for a borderline microlaser ($\beta ={10}^{-4}$) and for a macroscopic laser $\beta ={10}^{-6}$)—cf. figure legend.

**Figure 7.**Photon component ${v}_{n}$ of the least stable eigenvector (matching the least stable eigenvalue in Figure 4) for differently-sized lasers. Each curve stops at the point where the eigenvector becomes complex, simultaneously with the appearance of an imaginary component in the corresponding eigenvalue.

**Figure 8.**Comparison of the photon component ${v}_{n}$ of the least stable eigenvector (matching the least stable eigenvalue in Figure 4) for a micro- ($\beta ={10}^{-4}$) and a macroscopic laser ($\beta ={10}^{-6}$). As in Figure 7, each curve stops at the point where the eigenvector becomes complex, simultaneously with the appearance of an imaginary component in the corresponding eigenvalue.

**Figure 9.**Comparison of the population inversion component ${v}_{N}$ (matching the $\mu $ component of the perturbation in the lsa, Equations (14) and (15)) of the least stable eigenvector (matching the least stable eigenvalue in Figure 4) for a micro- ($\beta ={10}^{-4}$) and a macroscopic laser ($\beta ={10}^{-6}$). The closeness of the modulus of this component to 1 (i.e., nearly the entire normalized eigenvector’s amplitude) is clearly visible over the whole pump range. As in Figure 7, each curve stops at the point where the eigenvector becomes complex, simultaneously with the appearance of an imaginary component in the corresponding eigenvalue. The sudden switches between negative and positive unity represent a sudden rotation in the eigendirection.

**Figure 10.**Real part of the eigenvalues derived from the stability analysis of the modified rate equations model, Equations (16) and (17). The different curves belong to different kinds of asymmetry (cf. figure legend). $\beta ={10}^{-1}$. The chosen values of $\xi $, here and in the following figure, are those of [104].

**Figure 11.**Normalized photon component ${v}_{n}$ of the least stable eigenvector (matching the least stable eigenvalue in Figure 10) for different kinds of asymmetry in the emission rates. Each curve stops at the point where the eigenvector becomes complex, simultaneously with the appearance of an imaginary component in the corresponding eigenvalue. $\beta ={10}^{-1}$.

**Table 1.**Summary of the influence of different factors in the (experimental) identification of photon bursts through ${g}^{\left(2\right)}\left(0\right)$.

Feature | Consequence | Comment |
---|---|---|

${g}^{\left(2\right)}\left(0\right)$ | >2 | Allows for a univocal identification of the presence of photon bursts (superthermal bunching) in the temporal laser emission |

${g}^{\left(2\right)}\left(0\right)$ | $1\le {g}^{\left(2\right)}\left(0\right)\le 2$ | Does not exclude the presence of photon bursts, but does not allow the certain identification, since other kinds of signals, such as strongly oscillating photon numbers, may give the same value of ${g}^{\left(2\right)}\left(0\right)$ (cf. e.g., Figure 4c in [76]). |

Frequent bursts | Filling of the measurement window | Frequent bursts increase the fraction of the temporal window in which the photon bursts are measured, reducing the value of ${g}^{\left(2\right)}\left(0\right)$. This is not a parasitic effect and signals convergence towards cw laser emission |

Detection bandwidth | Smaller and broader pulses | Controls the measured value of the autocorrelation signal reducing its value relative to the real one; reduces the range of pump values for the observation of photon bursts |

Detection background | Lower contrast | Reduces the estimated value of ${g}^{\left(2\right)}\left(0\right)$; reduces the range of pump values for the observation of photon bursts |

Signal contrast | Ratio between photon bursts and background | Directly affects the estimate of ${g}^{\left(2\right)}\left(0\right)$. Improvements in height detection or reduction in background lead towards a more realistic evaluation of ${g}^{\left(2\right)}\left(0\right)$ |

Far below threshold | Very infrequent bursts | Likelihood of detecting an actual burst very low compared to the accumulation of background shot noise; results in often-observed drop in ${g}^{\left(2\right)}\left(0\right)\to 1$ at very low pump |

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**MDPI and ACS Style**

Lippi, G.L.
“Amplified Spontaneous Emission” in Micro- and Nanolasers. *Atoms* **2021**, *9*, 6.
https://doi.org/10.3390/atoms9010006

**AMA Style**

Lippi GL.
“Amplified Spontaneous Emission” in Micro- and Nanolasers. *Atoms*. 2021; 9(1):6.
https://doi.org/10.3390/atoms9010006

**Chicago/Turabian Style**

Lippi, Gian Luca.
2021. "“Amplified Spontaneous Emission” in Micro- and Nanolasers" *Atoms* 9, no. 1: 6.
https://doi.org/10.3390/atoms9010006