One crucial information for a pedestrian crowd simulation is the number of agents moving from an origin to a certain target. While this setup has a large impact on the simulation, it is in most setups challenging to find the number of agents that should be spawned at a source in the simulation. Often, number are chosen based on surveys and experience of modelers and event organizers. These approaches are important and useful but reach their limits when we want to perform real-time predictions. In this case, a static information about the inflow is not sufficient. Instead, we need a dynamic information that can be retrieved each time the prediction is started. Nowadays,  sensor data such as video footage or GPS tracks of a crowd are often available. If we can estimate the number of pedestrians who stem from a certain origin from this sensor data, we can dynamically initialize the simulation.
In this study, we use density heatmaps that can be derived from sensor data as input for a random forest regressor to predict the origin distributions. We study three different datasets: A simulated dataset, experimental data, and a hybrid approach with both experimental and simulated data. In the hybrid setup, the model is trained with simulated data and then tested on experimental data. The results demonstrate that the random forest model is able to predict the origin distribution based on a single density heatmap for all three configurations. This is especially promising for applying the approach on real data since there is often only a limited amount of data available.

§ INTRODUCTION

One crucial initial condition for a pedestrian crowd simulation is the number of agents moving from an origin to a certain target. This setting has a high impact on the simulation result.
At the same time, it is in most setups challenging to find the number of agents that should be generated at a source in the simulation. Often, number are chosen based on surveys and experience of modelers and event organizers.
These approaches are important and useful but reach their limits when we want to perform real-time predictions. In this case, a static information about the inflow is not sufficient. Instead, we need an updated information each time the prediction is calculated.
We want to use available sensor data of the pedestrians currently present in an observation area to estimate the number of pedestrians who stem from a certain origin. This can serve as an approximation of the number of pedestrians entering the scenario at this origin for the next prediction.
As sensors often video cameras are present on site of urban events. Outside of controlled experiments, it is highly challenging to estimate the trajectories of single pedestrians from the video footage. Nevertheless, there are several approaches to estimate the current density from video footage. Consequently, we use density heatmaps as input for this study.

§.§ Research question

Can we use machine learning methods to estimate the distributions of pedestrians on the origins based on a single density heatmap?

§.§ Background

In previous work, we were able to demonstrate that it is possible to prediction direction distributions at crossing based on simulated data with an accuracy  about $80$% as a proof-of-concept [2].
Now, we want to test the methodology on data of pedestrians as e.g., surveillance data, experimental data, or similar. We need trajectories to create the heatmaps and for the response, the actual distributions of the origins.
Unfortunately, there are only a few public data sets of pedestrian trajectories available. In particular, to our knowledge there is no public dataset of a crossing scenario with unidirectional flow available.
There are, on the other hand, several datasets on merging behavior of pedestrians. Therefore, we decided to test our routines with simulated and experimental data on a merging scenario. In this setup, the distribution on the destination is trivial since everyone heads to the same target. Nevertheless, it is interesting to study if we can find out from which origins pedestrians come.
We hope that the problems are similar enough that if the retracing of the pedestrians to the origin distributions is successful, the prediction of the target distributions is also successful on real data.
In both cases, the available input data are density heatmaps and we are trying to infer the percentage of pedestrians coming from or heading to a known list of origins or destinations, respectively.

§ METHODS AND MATERIALS

We choose the T-Junction experiments from [1, 8] as scenario. The trajectories as well as video footage of the experiment are available online [<https://doi.org/10.34735/ped.2009.7>].

§.§ Simulation setup

All simulations are carried out with the Optimal Steps Model [5, 7] of the open-source simulation framework Vadere [3]. We performed a manual calibration of the parameters to the experimental data. In particular, the navigation field was adapted to the data. It is calculated with obstacles method  with factor . That means that the obstacles are regarded in particular way in the calculation of the navigation field. As a result, agents keep more distance to the walls. For the comparison of simulated and experimental data, we used the measurement areas defined in

Snapshot of the simulation with Vadere. Agents are moving from two origins (green) to one destination (orange) through a T-junction setup. Measurement areas are shown in red.

§.§.§ Calibration of the simulation: navigation field

The Voronoi diagrams are calculated with the package  of the Python library  [6].

Experimental data set.

Simulated data set: static floor field.

Simulated data set: dynamic floor field.

Simulated data set: static floor field with obstacle density $0.3$.

Simulated data set: static floor field with obstacle density $0.9$.

Average Voronoi density for the experimental data set (a) compared to the simulation with $4$ different navigation fields (b-e).

We found the best agreement between simulated data and the experimental data for the static floor field with obstacle density with factor $0.3$. For the following simulations, this configuration is used.

§.§.§ Calculation density heatmaps

Analogue to [2], the density heatmaps are calculated from the pedestrian trajectories with a Gaussian density function [5].

Experimental dataset: $40\%$ of participants come from the left origin, $60\%$ come from the right origin.

Experimental dataset: $50\%$ of participants come from the left origin, $50\%$ come from the right origin.

Simulated dataset: $0\%$ of agents come from the left origin, $100\%$ come from the right origin.

Simulated dataset: $17\%$ of agents come from the left origin, $83\%$ come from the right origin.

Exemplary density heatmaps for both experimental and simulated datasets.

§.§.§ Comparison of simulation and experiment

We compare the obtained simulated data with the experimental dataset. In Figure <ref>, the trajectories for two configurations (entrance widths $0.5\,m$ and $2.4\,m$) are compared. We observe that the agent flows / trajectories from left and right origin are more mixed in the simulation compared to the pedestrian trajectories from the experiment. For the experimental data, we observe an area free of pedestrians below the corridor. This effect is stronger for smaller entrance widths. We can mainly observe the effect in the experimental data set, only slightly in the simulated data.

Trajectories of $134$ participants in the experiment ($b_{\text{entrance}}=0.5\,\text{m}$).

Trajectories of a simulation with  $134$ agents ($b_{\text{entrance}}=0.5\,\text{m}$).

Trajectories of $303$ participants in the experiment ($b_{\text{entrance}}=2.40\,\text{m}$).

Trajectories of a simulation with  $303$ agents ($b_{\text{entrance}}=2.40\,\text{m}$).

Comparison of trajectories between experiment and simulation.

In addition, we show the fundamental diagrams for the simulated data. It can be compared to the results from the experiment in [8]. In Figure <ref>, the density-speed relationship is evaluated for the three measurement areas defined in [1] that are depicted in red in Figure <ref>.

Fundamental diagrams for simulated data in the T-junction setup. The simulation was set up according to the experiment in [8] with $303$ agents and an entrance width of $2.4\,m$.

§.§.§ Simulated dataset

Our simulated data set consists of three simulation runs for each of the seven scenarios (240-50-240, 240-60-240, 240-80-240, 240-100-240, 240-120-240, 240-150-240, 240-240-240), analogue to [1]. The tuple of numbers refers to the size of the entrance of the waiting area $b_{\text{entrance}}$, the width of the corridor $b_{\text{cor1}}$ and the width of the corridor towards the exit $b_{\text{exit}}$. In total, we obtain $14.714$ samples from the simulations.

Due to the functional whiteness of the simulator, we may have many identical density heatmaps. Since this increases the noise in the relationship between input and output, we decided to throw out identical consecutive heatmaps.

After removing duplicates, we end up with $7.895$ heatmaps. When looking closer at Table <ref>, we notice the heatmaps are not equally distributed between the existing origin distributions and the majority of heatmaps correspond to a $50\%\mid50\%$ distribution. To prevent the random forest model to only predicts $50\%\mid50\%$, we reduce the number of heatmaps for the equal case, see Table <ref>. In the end, $7.023$ heatmaps remain.

Origin distributions in full dataset.

Left         Right      # Heatmaps

0.000000      1.000000         557

0.125000      0.875000           2

0.142857      0.857143           7

0.166667      0.833333          70

0.200000      0.800000         245

0.222222      0.777778           1

0.250000      0.750000         444

0.285714      0.714286          64

0.333333      0.666667         853

0.375000      0.625000          23

0.400000      0.600000         592

0.428571      0.571429         154

0.444444      0.555556           3

0.500000      0.500000        1759

0.555556      0.444444           4

0.571429      0.428571         111

0.600000      0.400000         652

0.625000      0.375000          22

0.666667      0.333333         931

0.714286      0.285714          54

0.750000      0.250000         469

0.800000      0.200000         220

0.833333      0.166667          57

0.857143      0.142857          11

0.875000      0.125000           4

1.000000      0.000000         586

Origin distributions in reduced dataset.

Left         Right      # Heatmaps

0.000000      1.000000         557

0.125000      0.875000           2

0.142857      0.857143           7

0.166667      0.833333          70

0.200000      0.800000         245

0.222222      0.777778           1

0.250000      0.750000         444

0.285714      0.714286          64

0.333333      0.666667         853

0.375000      0.625000          23

0.400000      0.600000         592

0.428571      0.571429         154

0.444444      0.555556           3

0.500000      0.500000         887

0.555556      0.444444           4

0.571429      0.428571         111

0.600000      0.400000         652

0.625000      0.375000          22

0.666667      0.333333         931

0.714286      0.285714          54

0.750000      0.250000         469

0.800000      0.200000         220

0.833333      0.166667          57

0.857143      0.142857          11

0.875000      0.125000           4

1.000000      0.000000         586

§.§ Random forest

We decided to use random forest as machine learning model as in [2]. It is an easy and robust algorithm that needs only a few parameters and often provides good results. Consequently, the configuration is easy, and the model can be used right away. In addition, the model is robust on the parameters which is not necessarily the case for machine learning models. Furthermore, the output is interpretable.

The input to our random forest routine is a single density heatmap, generated as described before, from pedestrian positions using a Gaussian function. We want to predict the percentage of pedestrians stemming from each of the two origins from the density heatmap.

For the random forest we use the  from the Python library scikit-learn [4]. Before training, we shuffle the heatmaps and divide them into training and test sets with $80\%$ and $20\%$ respectively. For each origin, we train one forest with $50$ trees. Consequently, when evaluating the model, each origin is also predicted separately,  and the predictions are normalized over both directions to assure that they sum up to $100\%$.

§ RESULTS AND DISCUSSION

Now, we use our Random forest setup to predict the origin distributions from density heatmaps. We use three different data set: First, the dataset that contains simulations performed with Vadere. Second, the experimental dataset calculated from the trajectory data. Third, a hybrid dataset where we train the random forest with simulated data but evaluate its performance on the experimental data. For each setup, we perform five runs to study the variation in the results.

§.§ Performance measure

The random forest routines can provide an out of bag error estimate as well as the coefficient of determination. However, both quantities are applied before the model predictions are normalized. Therefore, we split our dataset into a training and a test set and compare the prediction on the test set to the corresponding response to evaluate the model. In a first step, we evaluate the Euclidean norm of the difference vector. In the second step, we derive a relative error. Since we normalize the prediction, the maximum error

\begin{equation}
e_{max} = \sqrt{2 \cdot 100^2} \approx 141.42
\end{equation}

occurs if we predict that all pedestrians within the cutout head left while, in fact, they head right (or straight). The relative error is then

\begin{equation}
e = 100 \cdot \frac{y - \hat{y}}{e_{max}} ,
\end{equation}

where $y$ is the response on the test set and $\hat{y}$ is the prediction on the test set. Thereby we obtain a relative error of the prediction that is easy to interpret. Thus, the relative error is used as a measure.

§.§ Retracing using simulated data

For the first setup, we train and test the random forest solely on simulated data. We generate simulation data for all distributions of pedestrians on the origins. We expect an accuracy of  roughly $80$%, similar to our observations in [2] where we predicted the target distributions. The results in Table <ref> show an accuracy of about $80\%$ of the setup with a standard deviation of about $15\%$.

Relative error of  random forest prediction using simulated data ($7.023$ density heatmaps).

Run 1      Run 2      Run 3      Run 4      Run 5

Mean Euclidean error     19.23%     20.15%     19.85%     19.78%     19.57%

Stdev  Euclidean error     14.98%     15.08%     15.93%     15.47%     15.29%

§.§ Retracing using experimental data

In this section, we analyze the results of the random forest prediction based on experimental data. The dataset from the experiments shows only a few  different distributions of the pedestrians on the origins.
In experimental data set, mainly an equal distribution of the pedestrians on the origins is observed. We mainly observe the following distributions: $100\mid0$, $50\mid50$, $66\mid34$, $75\mid25$. This is a consequence of the experimental setup and the size of the observation area.
Table <ref> summarizes the results for the experimental dataset for both training and evaluation. The prediction accuracy is lower than when we used solely simulated data. We observe that the equal distribution is estimated well, due to the large data base. Other distributions are estimated with lower accuracy. Most of the predicted distributions are targets from the training set even though we use a regressor, not a classifier.

Relative error of  random forest prediction using the experiment data ($6.678$ density heatmaps).

Run 1      Run 2      Run 3      Run 4      Run 5

Mean Euclidean error     11.12%     9.68%     9.66%     10.33%     9.82%

Stdev  Euclidean error     15.89%     13.46%     13.19%     14.48%     13.89%

§.§ Retracing using hybrid approach

We observed that the available experimental data has only a few different distributions. Unfortunately, performing experiments is expensive, time-consuming, and complicated. Ideally, we would have a heterogeneous set of participants to reflect the average population well. In addition, results may strongly depend on the setup, how the subjects are primed, if and how subjects know each other, their form of the day and many more. In addition, the culture may have an impact on the behavior, so we need comparable studies.

Simulated data on the other hand can be generated easily and cheap. For applications, none or only limited experimental data is available.

As a result, we decided to complement the experimental data with simulated data: The algorithm is trained solely on simulated data. The testing is performed in two steps:
First, testing of the trained random forest on the simulated data. This aims to find out how well the forest is trained. The results should be identical to Section <ref>. Second, testing of the trained random forest on the experimental data. The goal of this part is to find out if the gained knowledge from the simulations can be extrapolated to real data.

Table <ref> lists the results for $5$ trainings of the random forest algorithm. As the input data is shuffled, the separation in training and test data varies among the runs. We observe a mean error about $30\%$ with a standard deviation of approximately $25\%$. As expected, the results are inferior to training and testing with the same data set. Both the size of the error and the size of the standard deviation have almost doubled. Nevertheless, we still obtain an accuracy of about $70\%$. That means, it is possible to train with simulated data and use the trained forest for predictions on experimental data.

Relative error of  random forest prediction using the hybrid dataset.

Run 1      Run 2      Run 3      Run 4      Run 5

Mean Euclidean error     29.36%     29.39%     29.24%     29.43%     29.40%

Stdev  Euclidean error     24.48%     24.73%     24.51%     24.70%     24.80%

§ EXTENDING THE OBSERVATION AREA

In the previous chapters, we saw limitations of the results due to the low variation of the origin distributions present in the data sets. Therefore, we decided to use a larger observation area so that more than a maximum of two pedestrians can be present at one time. That allows a larger variation in the distributions on the origins. The new observation area has the size of $2.4\,\text{m} \times 2\,\text{m} (B \times H)$. It is again placed over the full corridor width just before the target.

In Figure <ref>, we depicted two exemplary heatmaps for the larger observation area. Due to the larger observation area (2), more variation in the data is seen. Nevertheless, the equal distribution on both origins ($50\%\mid 50\%$) and distributions close to it are still most common.

Experimental dataset: $60\%$ of participants come from the left origin, $40\%$ come from the right origin.

Simulated dataset: $25\%$ of agents come from the left origin, $75\%$ come from the right origin.

Exemplary density heatmaps for the larger observation area (#2).

In Tables <ref>, <ref>, <ref> the results for the random forest prediction of the origin distributions are shown for solely simulated data, solely experimental data, and the hybrid dataset with simulated data for training and experimental data for testing, respectively. The results for all datasets have improved compared to the smaller observation area (#1). The improvement can be seen from a decrease in the mean error in combination with a decrease of the standard deviation. That means, that there is less variation of the quality of the prediction within the test set.

Relative error of  random forest prediction using simulated data generated with Vadere (observation area #2, $9.666$ samples).

Run 1      Run 2      Run 3      Run 4      Run 5

Mean Euclidean error     12.02%     12.03%     12.09%     11.61%     11.82%

Stdev  Euclidean error     9.64%     10.07%     9.82%     9.56%     9.39%

Relative error of random forest prediction using experiment data (observation area #2, $7.221$ samples).

Run 1      Run 2      Run 3      Run 4      Run 5

Mean Euclidean error     7.65%     7.63%     7.70%     7.85%     8.02%

Stdev  Euclidean error     9.33%     9.27%     8.37%     8.88%     9.55%

Relative error of  random forest prediction using the hybrid dataset (observation area #2).

Run 1      Run 2      Run 3      Run 4      Run 5

Mean Euclidean error     19.78%     19.65%     19.67%     19.80%     19.56%

Stdev  Euclidean error     18.63%     18.42%     18.31%     18.77%     18.15%

§ CONCLUSION AND OUTLOOK

We were able to retrace the origin distributions based on single density heatmap. Basis of this study was a merging experiment (T-junction) where two pedestrian streams head for the same destination. We trained as random forest regressor with three different data sets: The first data set stems from simulations with the simulation framework Vadere. The setup of the simulation resembles the T-junction experiment. The second dataset is the publicly available data of the experiment. The last dataset is a hybrid dataset of both sources that means we used simulated data for training the random forest and experimental data for testing.

We observed the best results ($\approx 92\%$) on experimental data which is most likely  due to the limited set of origin distributions in the data set. The accuracy of the random forest model on simulated data was at about $88\%$. Finally, the hybrid approach yielded an accuracy of approximately $80\%$. This is a promising result that motivates the application of the approach on real data. Especially for applications, we believe the results of the hybrid approach are encouraging because often  there will  not be enough training data available.

In the future, we would like to look into other scenarios and topographies more complicated than the one studied here. Unfortunately, there is only limited amount of public data available. In addition, we want to try other learning models to see how they perform on the problem, in particular convolutional neural networks that are designed to work on image data. Furthermore, we believe that for more complicated scenarios, it will be beneficial or maybe even necessary to consider a time-dependent input, that is a time series of consecutive heatmaps.

This research was funded by the German Federal Ministry of Education and Research (BMBF) grant number 13N14464 (S2UCRE).

§ ORIGIN DISTRIBUTIONS OBSERVED IN DIFFERENT DATASETS (SIMULATED DATA, EXPERIMENTAL DATA) FOR BOTH OBSERVATION AREAS (1,2)

Origin distributions in experiment dataset using observation area #1 ($2.4\,m \times 1\,m$).

Left      Right      #Frames

0       0.000000      1.000000        1372

1       0.166667      0.833333          11

2       0.200000      0.800000          45

3       0.250000      0.750000         294

4       0.285714      0.714286           3

5       0.333333      0.666667         881

6       0.400000      0.600000         517

7       0.428571      0.571429          20

8       0.500000      0.500000        2241

9       0.571429      0.428571          23

10      0.600000      0.400000         471

11      0.625000      0.375000           2

12      0.666667      0.333333         838

13      0.714286      0.285714           2

14      0.750000      0.250000         304

15      0.800000      0.200000          35

16      1.000000      0.000000         738

Origin distributions in experiment dataset using observation area #2 ($2.4\,\text{m} \times 2\,\text{m}$).

Left      Right      #Frames

0       0.000000      1.000000         873

1       0.125000      0.875000           1

2       0.142857      0.857143           6

3       0.166667      0.833333          16

4       0.200000      0.800000          45

5       0.222222      0.777778          15

6       0.250000      0.750000         166

7       0.285714      0.714286         164

8       0.300000      0.700000          25

9       0.333333      0.666667         471

10      0.363636      0.636364          62

11      0.375000      0.625000         347

12      0.400000      0.600000         458

13      0.416667      0.583333          22

14      0.428571      0.571429         473

15      0.444444      0.555556         478

16      0.454545      0.545455          81

17      0.461538      0.538462           1

18      0.500000      0.500000         829

19      0.538462      0.461538           1

20      0.545455      0.454545          76

21      0.555556      0.444444         381

22      0.571429      0.428571         386

23      0.583333      0.416667          22

24      0.600000      0.400000         461

25      0.615385      0.384615           8

26      0.625000      0.375000         200

27      0.636364      0.363636          38

28      0.666667      0.333333         504

29      0.700000      0.300000          20

30      0.714286      0.285714          55

31      0.750000      0.250000         221

32      0.777778      0.222222           1

33      0.800000      0.200000          72

34      0.833333      0.166667          27

35      1.000000      0.000000         215

Origin distributions observed in simulated dataset using observation area #1 ($2.4\,\text{m} \times 1\,\text{m}$).

Left      Right      #Frames

0       0.000000      1.000000         557

1       0.125000      0.875000           2

2       0.142857      0.857143           7

3       0.166667      0.833333          70

4       0.200000      0.800000         245

5       0.222222      0.777778           1

6       0.250000      0.750000         444

7       0.285714      0.714286          64

8       0.333333      0.666667         853

9       0.375000      0.625000          23

10      0.400000      0.600000         592

11      0.428571      0.571429         154

12      0.444444      0.555556           3

13      0.500000      0.500000         887

14      0.555556      0.444444           4

15      0.571429      0.428571         111

16      0.600000      0.400000         652

17      0.625000      0.375000          22

18      0.666667      0.333333         931

19      0.714286      0.285714          54

20      0.750000      0.250000         469

21      0.800000      0.200000         220

22      0.833333      0.166667          57

23      0.857143      0.142857          11

24      0.875000      0.125000           4

25      1.000000      0.000000         586

Origin distributions in simulated dataset using observation area #2 ($2.4\,\text{m} \times 2\,\text{m}$).

Left      Right      #Frames

0       0.000000      1.000000          70

1       0.100000      0.900000           5

2       0.111111      0.888889           6

3       0.125000      0.875000          14

4       0.142857      0.857143          14

5       0.166667      0.833333          67

6       0.181818      0.818182          18

7       0.200000      0.800000          89

8       0.222222      0.777778          45

9       0.230769      0.769231           6

10      0.250000      0.750000         179

11      0.272727      0.727273          79

12      0.285714      0.714286         122

13      0.300000      0.700000         126

14      0.307692      0.692308          31

15      0.333333      0.666667         565

16      0.357143      0.642857          37

17      0.363636      0.636364         237

18      0.375000      0.625000         246

19      0.384615      0.615385         138

20      0.400000      0.600000         612

21      0.416667      0.583333         295

22      0.428571      0.571429         431

23      0.437500      0.562500           3

24      0.444444      0.555556         326

25      0.454545      0.545455         405

26      0.461538      0.538462         228

27      0.466667      0.533333          23

28      0.470588      0.529412           2

Left      Right      #Frames

28      0.470588      0.529412           2

29      0.500000      0.500000         920

30      0.533333      0.466667          17

31      0.538462      0.461538         187

32      0.545455      0.454545         383

33      0.555556      0.444444         285

34      0.562500      0.437500           3

35      0.571429      0.428571         482

36      0.583333      0.416667         270

37      0.600000      0.400000         548

38      0.615385      0.384615         133

39      0.625000      0.375000         264

40      0.636364      0.363636         204

41      0.642857      0.357143          36

42      0.666667      0.333333         594

43      0.692308      0.307692          31

44      0.700000      0.300000          98

45      0.714286      0.285714         178

46      0.727273      0.272727          64

47      0.733333      0.266667           1

48      0.750000      0.250000         185

49      0.769231      0.230769           4

50      0.777778      0.222222          35

51      0.800000      0.200000         102

52      0.833333      0.166667          93

53      0.857143      0.142857          39

54      0.875000      0.125000           8

55      0.888889      0.111111           2

56      1.000000      0.000000          81

[1]
Maik Boltes, Jun Zhang, Armin Seyfried, and Bernhard Steffen.
T-junction: Experiments, trajectory collection, and analysis.
In 2011 IEEE International Conference on Computer Vision
Workshops (ICCV Workshops), pages 158–165, 11 2011.

[2]
Marion Gödel, Gerta Köster, Daniel Lehmberg, Manfred Gruber, Angelika
Kneidl, and Florian Sesser.
Can we learn where people go?
Collective Dynamics, 2020.

[3]
Benedikt Kleinmeier, Benedikt Zönnchen, Marion Gödel, and Gerta Köster.
Vadere: An open-source simulation framework to promote
interdisciplinary understanding.
Collective Dynamics, 4, 2019.

[4]
Fabian Pedregosa, Gaël Varoquaux, Alexandre Gramfort, Vincent Michel,
Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron
Weiss, Vincent Dubourg, Jake Vanderplas, Alexandre Passos, David Cournapeau,
Matthieu Brucher, Matthieu Perrot, and Édouard Duchesnay.
Scikit-learn: Machine learning in Python.
Journal of Machine Learning Research, 12:2825–2830, 2011.

[5]
Michael J. Seitz and Gerta Köster.
Natural discretization of pedestrian movement in continuous space.
Physical Review E, 86(4):046108, 2012.

[6]
Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy,
David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan
Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod
Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric
Larson, C J Carey, İlhan Polat, Yu Feng, Eric W. Moore, Jake
VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen,
E. A. Quintero, Charles R. Harris, Anne M. Archibald, Antônio H. Ribeiro,
Fabian Pedregosa, Paul van Mulbregt, and SciPy 1.0 Contributors.
SciPy 1.0: Fundamental Algorithms for Scientific Computing in
Nature Methods, 17:261–272, 2020.

[7]
Isabella von Sivers and Gerta Köster.
Dynamic stride length adaptation according to utility and personal
Transportation Research Part B: Methodological, 74:104–117,

[8]
Jun Zhang, Wolfram Klingsch, Andreas Schadschneider, and Armin Seyfried.
Transitions in pedestrian fundamental diagrams of straight corridors
and t-junctions.
Journal of Statistical Mechanics: Theory and Experiment,
2011(06):P06004, 2011.


††thanks: These authors contributed equally††thanks: These authors contributed
equally

# Low-threshold nanolasers based on miniaturized bound states in the continuum

Yuhao Ren  State Key Laboratory of Optoelectronic Materials and Technologies,
School of Physics, Sun Yat-sen University, Guangzhou 510275, China  Peishen Li
State Key Laboratory of Advanced Optical Communication Systems and Networks,
School of Electronics & Frontiers Science Center for Nano-optoelectronics,
Peking University, 100871, Beijing, China  Zhuojun Liu  State Key Laboratory
for Mesoscopic Physics and Frontiers Science Center for Nano-optoelectronics,
School of Physics, Peking University, 100871, Beijing, China  Zihao Chen
State Key Laboratory of Advanced Optical Communication Systems and Networks,
School of Electronics & Frontiers Science Center for Nano-optoelectronics,
Peking University, 100871, Beijing, China  You-Ling Chen  State Key Laboratory
on Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of
Sciences, Beijing 100083, China  Chao Peng  State Key Laboratory of Advanced
Optical Communication Systems and Networks, School of Electronics & Frontiers
Science Center for Nano-optoelectronics, Peking University, 100871, Beijing,
China  Peng Cheng Laboratory, Shenzhen 518055, China  Jin Liu  State Key
Laboratory of Optoelectronic Materials and Technologies, School of Physics,
Sun Yat-sen University, Guangzhou 510275, China

###### Abstract

The pursuit of compact lasers with low-thresholds has imposed strict
requirements on tight light confinements with minimized radiation losses.
Bound states in the continuum (BICs) have been recently demonstrated as an
effective mechanism to trap light along the out-of-plane direction, paving the
way to low-threshold lasers. To date, most reported BIC lasers are still bulky
due to the absence of in-plane light confinement. In this work, we combine
BICs and photonic band gaps to realize three-dimensional (3D) light
confinements, as referred to miniaturized (mini-) BICs. Together with 3D
carrier confinements provided by quantum dots (QDs) as optical gain materials,
we have realized highly-compact active BIC resonators with a record-high
quality ($Q$) factor up to 32500, which enables single-mode continuous wave
(CW) lasing with the lowest threshold of 80 W/cm2 among the reported BIC
lasers. In addidtion, our photon statistics measurements under both CW and
pulsed excitations confirm the occurence of the phase transition from
spontaneous emission to stimulated emission, further suggesting that
conventional criteria of input-output and linewidth are not sufficient for
claiming nanoscale lasing. Our work reveal a via path towards compact BIC
lasers with ultra-low power consumption and potentially boost the applications
in cavity quantum electrodynamics (QEDs), nonlinear optics and integrated
photonics.

keywords: Nanolaser, Bound states in the continuum, Photon statistics.

##  I Introduction

Nanoscale coherent light generations via stimulated emissions have been the
scientific frontier of nanophononics, topological photonicsLu et al. (2014);
Ozawa et al. (2019); Ota et al. (2020), non-Hermitian physicsHodaei et al.
(2014); Peng et al. (2014); Özdemir et al. (2019) and optics in random
mediaWiersma (2008); Sapienza (2019). From the viewpoint of technology, the
scalable creations of miniaturized lasers with low-power consumption enable a
variety of important applications across optical interconnectsTucker (2010);
Tatum et al. (2015), bio-sensingWang et al. (2017) and far-field beam
synthesis Bahari et al. (2017); Zhang et al. (2020) etc. To achieve lasing at
the extreme sub-wavelength scale, plasmonic cavities are usually employed
however they unavoidably suffer from high Ohmic losses associated with
metalsWang et al. (2020); Khurgin and Sun (2014). While at the wavelength
scale, dielectric nanolasers have been realized with the assistance of high-
quality ($Q$) cavities utilizing total internal reflection or photonic
bandgaps (PBGs)Kim et al. (2012); Painter et al. (1999); Imada et al. (1999),
such as micro-disks or photonic crystal (PhC) defect cavities. However, due to
the limited lasing volume, their emission power is still not quite sufficient
in driving the applications, for instance, on-chip optical communications.
Recently, several new designs such as random laserLee et al. (2019); Trivedi
et al. (2022), topological laserSmirnova et al. (2020); Shao et al. (2020) and
moire lattice laserMao et al. (2021) had been proposed to achieve lasing
behavior at ten-wavelength scale, to best compromise the footprint and power.

Trapping light is no doubt the first step towards nanoscale lasers. As an
emerging mechanism, BICs have been demonstrated as a very powerful tool to
suppress out-of-plane radiations and consequently boost the Q-factors of
planar optical resonatorsHsu et al. (2013, 2016). In addition, the vectorial
nature of BICs enables emissions of structured light from chip-scale devices,
leading to ultra-fast switchable nanolasers Huang et al. (2020) and
multiplexed nanolasers carrying orbital angular momentaBahari et al. (2021).
In principle, ideal BICs with infinite $Q$s only exist in periodic and
symmetric structures. Therefore, early demonstrations of BICs employed
relatively large sample sizes ranging from a-few-tens to hundreds of periodic
unit-cell in order to maintain the high-$Q$ featureKodigala et al. (2017); Ge
et al. (2019). One of the successful efforts in promoting the $Q$s is to
topologically merge a set of BICs into the so-called super-BIC regime Jin et
al. (2019), which dramatically minimizes the radiation loss and therefore
significantly reduces the thresholds of BIC lasers Hwang et al. (2021).

To make the nanolaser suitable for the practical applications, we look for the
BICs in miniaturized sizes, namely ten-wavelength scale. For active devices,
the benefits of shrinking the mode sizes lies on two folds: first, in the
spontaneous emission regime, the small mode volumes can greatly enhance the
strength of light-matter interactions at single-photon level, enabling the
explorations of cavity QED effects in both weak- and strong-coupling
regimesGerard (2003); second, in the stimulated emission regime, miniaturized
mode volumes produce strong light trapping which significantly reduces the
lasing thresholdsMa and Oulton (2019). However, the symmetry breaking or
truncating the infinite size, transiting ideal BICs to quasi-onesKoshelev et
al. (2018); Liu et al. (2019), unavoidably lowers down the $Q$-factors
accordingly and makes them less favorable for low-threshold lasing. Thus it is
highly-desirable yet an on-going challenge to achieve mini-BICs with high-$Q$
factors.

Figure 1: Design of mini-BIC lasers. a, Schematic of mini-BIC laser device
consisting of a suspended GaAs thin membrane with periodically etched air
holes. The PhCs in regime A is surrounded by another heterogeneous PCs in
region B. Three layers of QDs are embedded in the center of the membrane as
optical gain materials. Inset: cross-section of the etched membrane with three
layers of QDs. b, The band diagrams of PhCs in regime A (with infinite size).
The continuous band (TE A, represented by the red line) associated with ideal
PhCs in regime A was quantized into discrete modes above the light line and
located in the bandgap of PhCs in region B (represented by the yellow area).
c, The momentum distribution of each mode, modes are labeled as Mpq, according
to their momentum peak positions in the first quadrant. d, The near-field
($6~{}\rm{\mu m}\times 6~{}\rm{\mu m}$) and far-field patterns
($30^{\circ}\times 30^{\circ}$) of four modes M11 through M22.

In this work, we simultaneously employ the BICs to suppress the out-of-plane
radiation and utilize the PBGs to minimize the in-plane optical disspations
for achieving high-$Q$ optical resonators with small footprints. We fabricated
active GaAs membranes supporting mini-BICs with $Q$s as high as $\sim$32500
and exploited high-density InAs QDs as optical gain materials. Laser
oscillations at telecom O-band under both CW and pulsed optical pumping were
observed with a threshold down to 80 W/cm2, which is nearly two order of
magnitude lower than the previous reported BIC lasersHwang et al. (2021). We
systematically compare the lasing and non-lasing behaviors by using time-
resolved and photon statistics measurements, further revealing the phase
transition from spontaneous to stimulated emission in our devices. This work
serves as a crucial step towards understanding and realizing high-performance
BIC lasers with miniaturized footprints and may further boost the applications
of BICs in cavity QED and integrated nonlinear photonics which demand for both
high-$Q$ and low mode volume cavities.

Figure 2: Optical characterizations of the active mini-BIC cavity. a, $\mu$-PL
spectra of QD ensemble (black) and the cavity modes (blue) of the mini-BIC.
Inset shows top-view SEM image of the fabricated mini-BIC device. b, Zoomed-in
$\mu$-PL spectrum of the cavity modes associated with mini-BICs. The inset
shows experimental far-field patterns (x/y-polarized) of modes M11 through
M22. c, Simulated (upper) and measured (lower) $Q$-factors of the M11 mode as
a function of the lattice constant $a$. Inset: high-Q rings corresponding to
the constellation of multiple BICs in momentum space for three different
lattice constants.

##  II Design

Our laser cavity is based on PhCs consisting of a suspended GaAs membrane with
periodically etched air holes in a square lattice array, as schematically
shown in Fig. 1(a). Three layers of high-density ($10^{10}/\rm{cm}^{2}$) InAs
QDs are embedded in the center of GaAs membranes as optical gain materials at
telecom O-band. The details of the QD epitaxial wafer are presented in Fig. S1
of the supplementary information (SI). To tightly localize the light in three
dimensions, we explored a recent proposal of mini-BICs in which the out-of-
plane radiation is suppressed by BICs while the in-plane light confinement is
achieved by PBGs Chen et al. (2022). The BICs associated with the PhCs in
region A (lattice constant $a$) is surrounded by heterogeneous PhCs in region
B (lattice constant $b$) with a gap size of $g$ in between. The band diagram
of the designed structure is shown in Fig. 1(b), in which the TE-A mode of
PhCs in region A sits in the band gap of PhCs in region B. Therefore, region B
serves as highly reflective mirrors to suppress the in-plane light leakage
from the mini-BICs residing in region A, and thus significantly improves the
$Q$-factor and reduces mode volume $V$. In addition, the finite size of the
cavity region A quantizes the continuous TE-A band into discrete modes with a
mode spacing of $\delta k$ = $\pi/L$ , where $L$ is the cavity length of
region A. Each mode can thus be labeled by a pair of integers $(p,q)$,
indicating that its momentum is mostly localized near $p\pi/L$; $q\pi/L$ in
the first quadrant of the momentum space. In Fig. 1(c), we plot the highly
momentum-localized modes of M11, M12/M21 and M22 in the first quadrant of
momentum space, in which M12 and M21 are degenerated in frequency due to the
90-degree rotation symmetry of the structure ($C_{4}$). The near-field mode
distributions reveal that the mini-BICs are highly spatially localized in the
cavity region enclosed by the PBG mirrors, while such momentum space
localizations result in highly-directional emissions towards specific angles,
as presented in Fig. 1(d).

##  III Device fabrication and characterizations

Experimentally, the mini-BIC patterns were defined on the electron beam
(E-beam) resist by using a 100 kV E-beam writer and then transferred to the
GaAs layer via a chloride-based dry etch process. The GaAs membranes were
released by selectively wet etching 1500-nm-thick AlGaAs sacrificial layer
underneath. The full fabrication flow of the devices is presented in Fig. S2
of SI. Sharp resonances corresponding to the cavity modes were identified by
measuring micro-photoluminescence ($\mu$-PL) from the cavity area at room
temperature. The details of the home-made $\mu$-PL setup is presented in Fig.
S3 of SI. For comparison, the $\mu$-PL of QD ensemble, indicating the optical
gain spectrum, was obtained from the area without any cavity structures. As
presented in Fig. 2(a), by choosing a proper lattice constant $a$ = 445 nm
with 17 perodicities, $b$ = 463 nm with 15 perdicities and $g$ = 455 nm,
cavity modes are spectrally aligned with the gain spectrum to facilitate the
lasing oscillations. The sharp resonances measured from the cavity area were
further zoomed in Fig. 2(b), where the optical characteristics such as
resonant wavelength and mode spacing agreed very well with the simulations in
Fig. 1. In additions, we performed the far-field characterizations of each
mode, further revealing the vectorial nature and momentum localization of the
mini-BIC modes, as predicted in the Fig. 1(d). The dependence of the
$Q$-factor (measured at the transparent excitation power) on the constant a of
the M11 mode is presented in Fig. 2(c). As shown in the simulations, the $Q$s
of mini-BIC can be engineered by tailoring the topological charges of the BICs
via tuning the latticed constant of etched air holes.

We measured the $\mu$-PL spectra of one cavity under CW excitation with
different excitation powers, as prseented in Fig. 3(a). Sharp cavity modes on
a broad emission background can be identified under low excitation powers. By
increasing the excitation power, M11 mode dominated the emission spectrum and
its line-width reduced significantly. The intensity and line-width of the
cavity modes as a function of the excitation power were plotted in Fig. 3(b).
A sharp increase in the input-output (IO) curve together with reduction of the
cavity line-width, commonly believed as signatures of lasing, was observed,
suggesting the occurrence of lasing oscillation.

To reliably identify the lasing oscillation, we characterized photon
statistics of the emitted photons by measuring the second-order
coherenceKreinberg et al. (2017); Jagsch et al. (2018), which can rigorously
quatify the quantum nature of the phase transition from spontaneous emission
to stimulated emission. In Fig. 3(c), at very low excitation powers, the
emission was in a thermal state but only exhibited a slight bunching behavior
in g2(0) due to the relatively short coherence time of the emitted photons.
With further increase of the excitation powers, the coherence time got
significantly prolonged and therefore appreciable photon bunching effects were
observed. As long as the excitation power crossed the lasing threshold of 41
$\rm{\mu}$W with a beam diameter of 8.11 $\mu\rm{m}$ (see Fig. S4),
corresponding to a power density of 80 W/cm2, the thermal state evolved
towards a coherent state and the g2(0) gradually lowered down to $1$ when far-
above threshold. The photon statistics evidence together with the I-O curve
and line-width reduction unequivocally demonstrated the realization of lasing
oscillation in our mini-BIC devices.

On the other hand, there have been increasing debates on using the kink
feature upon I-O curves as the single criterion of nanoscale lasing,
especially when the device experiences cavity QED effectsSamuel et al. (2009);
Reeves et al. (2018); Ning (2013). To that end, we characterized a less
optimal mini-BIC device similarly and presented the experimental data in Fig.
3(e-h). This non-lasing device also exhibited a “threshold” feature upon the
I-O curve as well as a line-width narrowing behavior, which are very similar
to the lasing device shown in Fig. 3(a-b). However, no signatures of phase
transition from thermal emission to a coherent state were observed from the
photon statistics measurements. As shown in Fig. 3(f), the g2(0) raised
monotonously with the increase of the excitation power, which indicates that
the device was operating as a nano-light-emitting-diode (nano-LED) instead a
nanolaser.

Figure 3: Characterizations of lasing and non-lasing behavior under CW
excitations. a,e, Evolution of the normalized emission spectra of the lasing
device (a) and non-lasing device (e) with the increased excitation power. b,f,
Integrated output intensity and line-width of the cavity mode M11 as a
function of the excitation power, showing ”threshold-like” behaviors at 80
W/cm2 (b) and 184 W/cm2. The red dashlines shows different slopes the output
power when inceasing the excitation power. The shaded areas represnet the
excitatiton powers at which the g2 trace are presented. (f), respectively.
c,g, Second-order correlation functions at zero delay time g2(0) as a function
of the excitation power. The lasing device exhibited a clear phase transition
from spontaneous emission (a thermal state with g2(0)$>$1) to stimulated
emission (a coherent state with g2(0)=1) while the non-lasing emissions
remained in a thermal state for all the excitation powers. d,h, Auto-
correlation traces taken at the pump power densities marked in (b,f)

Indeed, it is not surprising that the non-lasing device can also exhibit
nonlinear output intensities upon the excitation power. Such a fact has been
observed in a coupled single-QD cavity system without involving lasing
oscillations, as a consequence of the non-resonant couplings between different
excitonic states of QDs and the cavity modeWinger et al. (2009). At low
excitation powers, the intensity of the cavity mode followed the linearly
increased emission of exciton state formed in the singe QD. Under high
excitation powers, the cavity mode showed a super-linear power-dependence
owing to the non-resonant couplings to the bi-exciton states whose intensities
grow nearly as twice as the exciton states. Such a mechanism could also be
responsible for the nonlinear output intensities of the many QDs-cavity
coupled system, as observed in Fig. 3(d). The line-width narrowing shown in
Fig. 3(e) is easier to be understood since the active cavity experienced more
absorption from the gain material under low excitation powers. With the
increase of excitation power, the excited states of QDs get populated,
resulting in the reduction of the cavity line-width.

Figure 4: Characterizations of lasing and non-lasing devices under pulsed
excitation. a, Second-order correlation function at zero delay time g2(0) as a
function of the excitation power. The lasing device exhibited a clear phase
transition from spontaneous emission (a thermal state with g2(0)$>$1) to
stimulated emission (a coherent state with g2(0)=1). Auto-correlation traces
taken at the pump power densities marked in (a). c, Second-order correlation
function at zero delay time g2(0) as a function of the excitation power. The
non-lasing device remained in a thermal state (g2(0)$>$1) through all the
excitation powers. The shaded areas W X Y Z represnet the excitatiton powers
at which the g2 trace are presented. (d), Auto-correlation traces taken at the
pump power densities marked in (c). e,f The photon lifetime traces of lasing
and non-lasing devices under different excitation powers. g, The lifetimes of
lasing and non-lasing devices as a function of the excitation power. The
orange dash line indicates the time resolution of our measurement system.

To investigate the possibility of turning the non-lasing behavior to a lasing
oscillation with higher-peak powers and less thermal effects, we further
performed pulsed excitation on the same devices. The CW lasing device also
exhibited clear a nonlinear IO curve and line-width reductions, as presented
in Fig. S5(a,b) of SI. The pulsed g2(0) clear exhibited the phase transition
from a thermal state to a coherent states when increasing the excitation
power, as shown in Fig. 4(a,b). On the contrary, the nano-LED device showed
“laser-like” behaviors in terms of nolinear incerease in the I-O curve and
line-width reductions, but its photon statistics remained in a thermal state
across all the excitation power, as shown in Fig. 4(c,d). One of the
advantages of pulsed excitation scheme is that we are able to directly measure
the photon lifetimeAtlasov et al. (2009) which can serve as an additional
evidence of stimulated emissions. The decay curves under different excitation
powers for the laser and LED devices are presented in Fig. 4(e,f),
respectively. Their emissions lifetimes were shortened during the increasing
of pulse power, as quantitatively presented in Fig. 4(g). For the lasing
device, the lifetime reduced more rapidly than those of the nano-LED till the
measurement was beyond the time resolution of our photon-detectors. Such fast
decay rates above threshold are strong indications of stimulated emissions.
The lifetime of photons emitted by the nano-LED device also decreased, which
is expected due to the different carrier relaxation dynamics under high
excitation powersSchwab et al. (2006). However, the photon lifetime of nano-
LED reduced rather slowly as compared to the lasing device and became
saturated (not reaching the time resolution limit of our measurement) at high
excitation powers, which is more preferably to ascribe to spontaneous
emission.

##  IV Conclusion

We have demonstrated III-V active mini-BIC resonators with a quality factor as
high as $\sim$32500, which enables the realizations of lasing oscillations
with a record-low threshold of 80 W/cm2 that is nearly two orders of magnitude
lower than the state-of-the-art BIC lasers (see table I in SI for comparison).
Both CW and pulsed lasing were unequivocally realized by systematically
measuring the device characteristics including $\mu$-PL spectra, I-O curves,
emission line-width, photon statistics and photon lifetimes. Our
investigations suggest that any claim of nanolasing with cavity QED effects
should be very careful and further comparisons between more advanced theory
and experiments are indispensable. Moving forwards, the vectorial nature of
mini-BICs could be exploited to build chip-scale lasers capable of emitting
coherent structured light for high-capacity optical communicationWillner et
al. (2021) and high-dimension quantum information processingErhard et al.
(2018). It is also highly desirable to implement the electrical injections in
active BIC devcies, as successfully demonstrated for PhC defect and surface
emitting lasers in similar structuresEllis et al. (2011); Morita et al.
(2021); Itoh et al. (2022), From the perspective of applications, the
realization of mini-BIC laser may immediately boost the development of on-chip
cavity QEDVuckovic (2014) as well as the integrated nonlinear
photonicsHendrickson et al. (2014) in which high-Q factors and small mode
volumes are highly beneficial.

Acknowledgements This research was supported by National Key Research and
Development Program of China (2018YFA0306103); National Natural Science
Foundation of China (11874437, 62035017, 61922004, 62135001), Major Key
Project of PCL (PCL2021A14) and the national super-computer center in
Guangzhou.

Device fabrication: The samples are fabricated using a 500 nm-thick GaAs/1.5
$\mu$m-thick Al0.8Ga0.2As/GaAs substrate wafer. Three layers InAs/GaAs QDs are
embedded in the middle of the first GaAs layer, whose central emission
wavelength is 1.3 $\mu$m. The AlGaAs layer act as sacrificial and etch stop
layer. First, a layer of Si3N4 with a thickness of 200 nm was deposited on the
wafer by Inductively coupled plasma chemical vapor deposition (ICP-CVD) as a
hard mask for PhC dry etching. A 400 nm ARP6200 electron beam resist was spin
coated on the surface of the hard mask. Subsequently, the electron beam
lithography was used to define the PhC pattern in ARP6200. The PhC pattern was
transferred from resist into the hard mask layer using reactive ion etching
(RIE). Afterwards, the electron beam resist was removed by Inductively coupled
plasma etching (ICP) with O2 plasma. Then ICP etching was performed to obtain
the air holes through the active layer. The residual Si3N4 hard mask was
removed by RIE dry etching. Finally, the sacrificial layer was wet etching by
immersing the sample in 10$\%$ hydrofluoric acid solution for 1min30s to form
an air region slab underneath the PhC membrane.

Optical measurement: The fabricated mini-BIC devices were characterized with a
customized micro-photoluminescence ($\mu$-PL) measurement system in a surface-
normal pump configuration. 780 nm CW laser diode and pulse laser diode (10 ps,
86 MHz period) were used to optically pumped the sample via a 50x objective
lens with a numerical aperture of 0.65. The emission spectra were collected by
the same objective and analyzed by a spectrometer (iHR550) with a resolution
of 0.025 nm.

## References

  * Lu et al. (2014) L. Lu, J. D. Joannopoulos, M. Soljačić,  Topological photonics,  Nature photonics 8 (2014) 821–829.
  * Ozawa et al. (2019) T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, et al.,  Topological photonics,  Reviews of Modern Physics 91 (2019) 015006.
  * Ota et al. (2020) Y. Ota, K. Takata, T. Ozawa, A. Amo, Z. Jia, B. Kante, M. Notomi, Y. Arakawa, S. Iwamoto,  Active topological photonics,  Nanophotonics 9 (2020) 547–567.
  * Hodaei et al. (2014) H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, M. Khajavikhan,  Parity-time–symmetric microring lasers,  Science 346 (2014) 975–978.
  * Peng et al. (2014) B. Peng, Ş. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. Bender, F. Nori, L. Yang,  Loss-induced suppression and revival of lasing,  Science 346 (2014) 328–332.
  * Özdemir et al. (2019) Ş. K. Özdemir, S. Rotter, F. Nori, L. Yang,  Parity–time symmetry and exceptional points in photonics,  Nature materials 18 (2019) 783–798.
  * Wiersma (2008) D. S. Wiersma,  The physics and applications of random lasers,  Nature physics 4 (2008) 359–367.
  * Sapienza (2019) R. Sapienza,  Determining random lasing action,  Nature Reviews Physics 1 (2019) 690–695.
  * Tucker (2010) R. S. Tucker,  Green optical communications—part ii: Energy limitations in networks,  IEEE Journal of Selected Topics in Quantum Electronics 17 (2010) 261–274.
  * Tatum et al. (2015) J. A. Tatum, D. Gazula, L. A. Graham, J. K. Guenter, R. H. Johnson, J. King, C. Kocot, G. D. Landry, I. Lyubomirsky, A. N. MacInnes, et al.,  Vcsel-based interconnects for current and future data centers,  Journal of Lightwave Technology 33 (2015) 727–732.
  * Wang et al. (2017) X.-Y. Wang, Y.-L. Wang, S. Wang, B. Li, X.-W. Zhang, L. Dai, R.-M. Ma,  Lasing enhanced surface plasmon resonance sensing,  Nanophotonics 6 (2017) 472–478.
  * Bahari et al. (2017) B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, B. Kanté,  Nonreciprocal lasing in topological cavities of arbitrary geometries,  Science 358 (2017) 636–640.
  * Zhang et al. (2020) Z. Zhang, H. Zhao, D. G. Pires, X. Qiao, Z. Gao, J. M. Jornet, S. Longhi, N. M. Litchinitser, L. Feng,  Ultrafast control of fractional orbital angular momentum of microlaser emissions,  Light: Science & Applications 9 (2020) 1–9.
  * Wang et al. (2020) S.-L. Wang, S. Wang, X.-K. Man, R.-M. Ma,  Loss and gain in a plasmonic nanolaser,  Nanophotonics 9 (2020) 3403–3408.
  * Khurgin and Sun (2014) J. B. Khurgin, G. Sun,  Comparative analysis of spasers, vertical-cavity surface-emitting lasers and surface-plasmon-emitting diodes,  Nature Photonics 8 (2014) 468–473.
  * Kim et al. (2012) Y.-H. Kim, S.-H. Kwon, J. M. Lee, M.-S. Hwang, J.-H. Kang, W. I. Park, H.-G. Park,  Graphene-contact electrically driven microdisk lasers,  Nature communications 3 (2012) 1–6.
  * Painter et al. (1999) O. Painter, R. Lee, A. Scherer, A. Yariv, J. O’brien, P. Dapkus, I. Kim,  Two-dimensional photonic band-gap defect mode laser,  Science 284 (1999) 1819–1821.
  * Imada et al. (1999) M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, G. Sasaki,  Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,  Applied physics letters 75 (1999) 316–318.
  * Lee et al. (2019) M. Lee, S. Callard, C. Seassal, H. Jeon,  Taming of random lasers,  Nature Photonics 13 (2019) 445–448.
  * Trivedi et al. (2022) M. Trivedi, D. Saxena, W. K. Ng, R. Sapienza, G. Volpe,  Self-organized lasers from reconfigurable colloidal assemblies,  Nature Physics (2022) 1–6.
  * Smirnova et al. (2020) D. Smirnova, A. Tripathi, S. Kruk, M.-S. Hwang, H.-R. Kim, H.-G. Park, Y. Kivshar,  Room-temperature lasing from nanophotonic topological cavities,  Light: Science & Applications 9 (2020) 1–8.
  * Shao et al. (2020) Z.-K. Shao, H.-Z. Chen, S. Wang, X.-R. Mao, Z.-Q. Yang, S.-L. Wang, X.-X. Wang, X. Hu, R.-M. Ma,  A high-performance topological bulk laser based on band-inversion-induced reflection,  Nature nanotechnology 15 (2020) 67–72.
  * Mao et al. (2021) X.-R. Mao, Z.-K. Shao, H.-Y. Luan, S.-L. Wang, R.-M. Ma,  Magic-angle lasers in nanostructured moiré superlattice,  Nature Nanotechnology 16 (2021) 1099–1105.
  * Hsu et al. (2013) C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, M. Soljačić,  Observation of trapped light within the radiation continuum,  Nature 499 (2013) 188–191.
  * Hsu et al. (2016) C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, M. Soljačić,  Bound states in the continuum,  Nature Reviews Materials 1 (2016) 1–13.
  * Huang et al. (2020) C. Huang, C. Zhang, S. Xiao, Y. Wang, Y. Fan, Y. Liu, N. Zhang, G. Qu, H. Ji, J. Han, et al.,  Ultrafast control of vortex microlasers,  Science 367 (2020) 1018–1021.
  * Bahari et al. (2021) B. Bahari, L. Hsu, S. H. Pan, D. Preece, A. Ndao, A. El Amili, Y. Fainman, B. Kanté,  Photonic quantum hall effect and multiplexed light sources of large orbital angular momenta,  Nature Physics 17 (2021) 700–703.
  * Kodigala et al. (2017) A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, B. Kanté,  Lasing action from photonic bound states in continuum,  Nature 541 (2017) 196–199.
  * Ge et al. (2019) X. Ge, M. Minkov, S. Fan, X. Li, W. Zhou,  Laterally confined photonic crystal surface emitting laser incorporating monolayer tungsten disulfide,  npj 2D Materials and Applications 3 (2019) 1–5.
  * Jin et al. (2019) J. Jin, X. Yin, L. Ni, M. Soljačić, B. Zhen, C. Peng,  Topologically enabled ultrahigh-q guided resonances robust to out-of-plane scattering,  Nature 574 (2019) 501–504.
  * Hwang et al. (2021) M.-S. Hwang, H.-C. Lee, K.-H. Kim, K.-Y. Jeong, S.-H. Kwon, K. Koshelev, Y. Kivshar, H.-G. Park,  Ultralow-threshold laser using super-bound states in the continuum,  Nature communications 12 (2021) 1–9.
  * Gerard (2003) J.-M. Gerard,  Solid-state cavity-quantum electrodynamics with self-assembled quantum dots,  in: Single Quantum Dots, Springer, 2003, pp. 269–314.
  * Ma and Oulton (2019) R.-M. Ma, R. F. Oulton,  Applications of nanolasers,  Nature Nanotechnology 14 (2019) 12–22.
  * Koshelev et al. (2018) K. Koshelev, S. Lepeshov, M. Liu, A. Bogdanov, Y. Kivshar,  Asymmetric metasurfaces with high-q resonances governed by bound states in the continuum,  Physical review letters 121 (2018) 193903.
  * Liu et al. (2019) Z. Liu, Y. Xu, Y. Lin, J. Xiang, T. Feng, Q. Cao, J. Li, S. Lan, J. Liu,  High-q quasibound states in the continuum for nonlinear metasurfaces,  Physical review letters 123 (2019) 253901.
  * Chen et al. (2022) Z. Chen, X. Yin, J. Jin, Z. Zheng, Z. Zhang, F. Wang, L. He, B. Zhen, C. Peng,  Observation of miniaturized bound states in the continuum with ultra-high quality factors,  Science Bulletin 67 (2022) 359–366.
  * Kreinberg et al. (2017) S. Kreinberg, W. W. Chow, J. Wolters, C. Schneider, C. Gies, F. Jahnke, S. Höfling, M. Kamp, S. Reitzenstein,  Emission from quantum-dot high-$\beta$ microcavities: transition from spontaneous emission to lasing and the effects of superradiant emitter coupling,  Light: Science & Applications 6 (2017) e17030–e17030.
  * Jagsch et al. (2018) S. T. Jagsch, N. V. Triviño, F. Lohof, G. Callsen, S. Kalinowski, I. M. Rousseau, R. Barzel, J.-F. Carlin, F. Jahnke, R. Butté, et al.,  A quantum optical study of thresholdless lasing features in high-$\beta$ nitride nanobeam cavities,  Nature communications 9 (2018) 1–7.
  * Samuel et al. (2009) I. D. Samuel, E. B. Namdas, G. A. Turnbull,  How to recognize lasing,  Nature Photonics 3 (2009) 546–549.
  * Reeves et al. (2018) L. Reeves, Y. Wang, T. F. Krauss,  2d material microcavity light emitters: to lase or not to lase?,  Advanced Optical Materials 6 (2018) 1800272.
  * Ning (2013) C.-Z. Ning,  What is laser threshold?,  IEEE Journal of Selected Topics in Quantum Electronics 19 (2013) 1503604–1503604.
  * Winger et al. (2009) M. Winger, T. Volz, G. Tarel, S. Portolan, A. Badolato, K. J. Hennessy, E. L. Hu, A. Beveratos, J. Finley, V. Savona, et al.,  Explanation of photon correlations in the far-off-resonance optical emission from a quantum-dot–cavity system,  Physical review letters 103 (2009) 207403.
  * Atlasov et al. (2009) K. A. Atlasov, M. Calic, K. F. Karlsson, P. Gallo, A. Rudra, B. Dwir, E. Kapon,  Photonic-crystal microcavity laser with site-controlled quantum-wire active medium,  Optics Express 17 (2009) 18178–18183.
  * Schwab et al. (2006) M. Schwab, H. Kurtze, T. Auer, T. Berstermann, M. Bayer, J. Wiersig, N. Baer, C. Gies, F. Jahnke, J. Reithmaier, et al.,  Radiative emission dynamics of quantum dots in a single cavity micropillar,  Physical Review B 74 (2006) 045323.
  * Willner et al. (2021) A. E. Willner, K. Pang, H. Song, K. Zou, H. Zhou,  Orbital angular momentum of light for communications,  Applied Physics Reviews 8 (2021) 041312.
  * Erhard et al. (2018) M. Erhard, R. Fickler, M. Krenn, A. Zeilinger,  Twisted photons: new quantum perspectives in high dimensions,  Light: Science & Applications 7 (2018) 17146–17146.
  * Ellis et al. (2011) B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, J. Vučković,  Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,  Nature photonics 5 (2011) 297–300.
  * Morita et al. (2021) R. Morita, T. Inoue, M. De Zoysa, K. Ishizaki, S. Noda,  Photonic-crystal lasers with two-dimensionally arranged gain and loss sections for high-peak-power short-pulse operation,  Nature Photonics 15 (2021) 311–318.
  * Itoh et al. (2022) Y. Itoh, N. Kono, D. Inoue, N. Fujiwara, M. Ogasawara, K. Fujii, H. Yoshinaga, H. Yagi, M. Yanagisawa, M. Yoshida, et al.,  High-power cw oscillation of 1.3-$\mu$m wavelength inp-based photonic-crystal surface-emitting lasers,  Optics Express 30 (2022) 29539–29545.
  * Vuckovic (2014) J. Vuckovic,  Quantum optics and cavity qed with quantum dots in photonic crystals,  Quantum Optics and Nanophotonics, C. Fabre, V. Sandoghdar, N. Treps, and LF Cugliandolo, eds.(Oxford University Press, 2017) (2014).
  * Hendrickson et al. (2014) S. M. Hendrickson, A. C. Foster, R. M. Camacho, B. D. Clader,  Integrated nonlinear photonics: emerging applications and ongoing challenges,  JOSA B 31 (2014) 3193–3203.

SUPPLEMENTARY INFORMATION

##  I Quantum dot wafer

The QD wafer was grown via a molecular beam epitaxy. As schematically shown in
Fig. S1(a), 300 nm GaAs buffer layer was firstly grown on a 350-$\mu$m-thick
GaAs substrate followed by a 1500 nm AlGaAs sacrificial layer. 500 nm GaAs
with three layers of high-density InAs QDs was finally grown on top. An atomic
force microscope (AFM) image of uncapped InAs/GaAs QDs indicates a QD density
of $\sim$1010/cm2 is presented in Fig. S1(b).

Figure S1: QD wafer. a, Schematic epitaxial structure of wafer for mini-BIC
laser. b, AFM image of the uncapped QDs.

##  II Device fabrication

The full fabrication flow of the mini-BIC laser is shown in Fig.S2. First, a
layer of Si3N4 with a thickness of 200 nm was deposited on the wafer
byiInductively coupled plasma chemical vapor deposition (ICP-CVD) as a hard
mask for PhC dry etching. A 400nm ARP6200 electron beam resist was spin coated
on the surface of the hard mask. Subsequently, the electron beam lithography
(EBL) was used to define the PhC pattern in ARP6200. The PhC pattern was
transferred from resist into the hard mask layer using reactive ion etching
(RIE). Afterwards, the electron beam resist was removed by Inductively coupled
plasma etching (ICP) with O2 plasma. Then a chloride-based etching was
performed to obtain the air holes through the active layer. The residual Si3N4
hard mask was removed by RIE dry etching. Finally, the sacrificial layer was
undercut by immersing the sample in 10% hydrofluoric acid solution for 1min30s
to form a suspended PhC membrane.

Figure S2: Fabrication flow of the mimi-BIC laser. a, As-grown epitaxial
wafer. b, Deposition of Si3N4 hard mask. c, Spin-coating of E-beam resist as a
soft mask. d, Patter definition with EBL. e, Transfering PhC pattern to hard
mask layer by RIE. f, Removal of the E-beam resist. g, TransferING pattern to
active layer by ICP. h, Removal of the hard mask by RIE. i, Creation of the
suspended memebrane with a selective wet etching.

##  III Optical setup

The cuostomied confocal $\mu$PL setup for optical characterizations is shown
in Fig. S3. The device is excited optically by a 780 nm CW laser diode or a
pulse laser (10 ps, 86MHz period) through a microscope objective with a
numerical aperture (NA) of 0.65. The spot size of the pump laser is about 8.11
$\mu$m, as shown in Fig. S4. The signal emitted from the sample is collected
by the same objective, and then analyzed by a spectrometer (iHR550) with a
resolution of 0.025 nm or guided to time correlated single-photon-counting
(TCSPC) measurements for the second-order correlation or lifetime measurements
via a flip mirror. For TCSPC measurements, the signal first passes through a
home made grating filter with a spectral resolution of 1 nm to isolate the
laser mode. Then the filtered signal went into either one superconducting
nanowire single-photon detectors (SNSPD) for lifetime measurements or into a
50:50 fiber beam splitter (BS) and two SNSPDs for intensity autocorrelation
measurements.

Figure S3: Experimental setup for Optical characterization.  Figure S4: The
estimate of the pump spot sizes.

##  IV Pulsed excitation

Fig.S5 presents the comparison of the lasing and non-lasing devices under
pulsed excitation. The lasing device exhibited single-mode operation with
typical lasing signatures of a nonlinear increased IO cureve and a line-width
reduction behavior. Similarly, the nano-LED also showed a conventional ”laser-
like” behaivor in both the IO curve and line-width.

Figure S5: Characterizations of lasing and non-lasing devices under pulsed
excitation. a,b, Evolution of the normalized emission spectra of the lasing
device (a) and non-lasing device (b) with the increased excitation power. c,d,
Integrated output intensity and line-width of the cavity mode M11 as a
function of the excitation power, both showing “threshold-like” behaviors.

##  V Comparion of mini-BIC laser with other BIC lasers

Table 1: Comparison of the mini-BIC laser with other BIC lasers. | Laser type |  | Pump
---
method
| Threshold
---
peak power (mW)
| Threshold power
---
density (kW/cm2)
Q factor
Nature 541, 196 (2017) | BIC | Pulse | 15.6 |  $\sim$4 |  $\sim$4701
arXiv:1707.00181 | BIC | Pulse | 73 | - | -
Nature Nanotech. 13 (2018) | BIC | Pulse | $8.80\times 10^{5}$ | $7.0\times 10^{4}$ | 2750
npj 2D Materials and Applications.3 (2019) | BIC | CW | - | 0.144 | 2500
Science 367 (2020) | BIC | Pulse | $5.28\times 10^{5}$ | $4.2\times 10^{4}$ | -
Nano Lett. 20 (2020) | BIC | Pulse | $5.09\times 10^{8}$ | $1.8\times 10^{5}$ | 2590
arXiv:2012.15642 | BIC | Pulse | $\sim 2.16\times 10^{6}$ | $\sim 2.75\times 10^{8}$ |  $\sim$2883
Nature Commun. 12 (2021) | Super-BIC | Pulse | 0.34 | 1.47 |  $\sim$7300
This work | Mini-BIC | CW | 0.041 | 0.08 |  $\sim$32500