1. INTRODUCTION

The development of nanophotonics leads to the emergence of new types of optical signal sources, detectors, and modulators [1,2]. With the increasing complexity of active optical devices, there is a need to adapt existing methods and create new ones for local characterization of the light–matter interaction. Nonlinear [3] and near-field [4,5] microscopy can be considered to map the optical fields in nanostructures, while fluorescence-lifetime imaging microscopy (FLIM) [6], transient reflection/absorption techniques [7,8], and ultrafast perturbation mapping [9,10] provide information about the generation and dynamics of free carriers in nanophotonic devices. However, many of these methods have limitations in quantifying carrier concentration and simultaneously achieving high temporal and spatial resolutions, which restricts their use in such cases where estimation of carrier diffusion on picosecond scales is crucial (e.g., microlasers, micro-modulators, etc.).

Various techniques based on ultrafast electron microscopy with sub-picosecond time resolution and nanoscale space resolution have also been developed to improve the spatial resolution in imaging [11,12]. However, electron microscopy methods are difficult to implement on an optical table and sophisticated to combine with optical measurements. Therefore, it is important to develop a novel all-optical technique for ultrafast characterization of micro- and nanoscale optical devices.

 

Fig. 1. (a) Sketch of the experiment. A pump beam generates the cloud of free carriers in the perovskite microplate, causing lasing. At zero time delay, the free carrier cloud profile coincides with the pump beam intensity profile. The spatiotemporal dynamics of the carrier cloud is visualized using ultrafast OCM. Here, the pump beam is shown in bluish color for clarity. (b) Schematic of the pump–probe OCM setup. OL, objective lens; PBS, polarizing beam splitter; PCF, photonic crystal fiber for supercontinuum generation; RM, reference mirror (not needed in common-path OCM); GM galvanometer-mounted mirror. Pump control: optical chopper, liquid crystal retarder, and steering lens. (c) Data acquisition algorithm.

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The free carriers generated directly affect the material’s refractive index, resulting in the addition of $\Delta n$, the distribution of which could provide information about the concentration and spreading of carriers. A powerful method for studying the spatial distribution of the refractive index is optical coherence tomography (OCT), which has revolutionized the field of biomedical imaging [13,14]. A variation of OCT, optical coherence microscopy (OCM), inherits high interference phase sensitivity and quantitative refractive index measurements from OCT and high transverse resolution from confocal microscopy [13–20]. OCT and OCM have already shown their promise for noninvasive visualization of the internal structure of 3D-printed microstructures [21–23] and integrated nanophotonic passive devices [24,25]. The use of supercontinuum pulses from femtosecond lasers as a radiation source [26–28] allows measurements to be carried out in the ultrafast probe–pump scheme, making OCM an excellent basis for the development of ultrafast $\Delta n$ imaging techniques in nanophotonics. Previously, the pump–probe OCT was only used for molecular contrast imaging at nanosecond timescales [29,30].

Here, we succeeded in implementing the pump–probe scheme with picosecond probe pulses performing OCM. We tested our method on perovskite microlasers, which exhibit high efficiency at room temperature and a wide tuning range, and can be fabricated by low-cost solution-processible methods [31–38]. The efficiency of perovskite microlasers is significantly affected by the density of generated charge carriers, as well as the relaxation processes and the transport of charge carriers related to the crystal structure and its defects [6,36,39]. Carrier dynamics in perovskites is often studied using a transient reflection/absorption method [40–43], which has a femtosecond resolution, usually without giving a spatial picture of the processes and suffering from probe radiation interference inside the sample [44]. Temporally resolved photoluminescence spectroscopy [34,42,45–47] shows the lasing spectral–temporal dynamics without spatial resolution or quantitative carrier information. FLIM [6] has a submicrometer transverse resolution, but is limited by a subnanosecond temporal resolution. However, studying the relaxation of carriers during lasing in perovskites requires picosecond temporal resolution, because the characteristic relaxation times of free carriers through stimulated emission are from 1 ps to 20 ps, while spontaneous ones are from 100 ps to 500 ps upon excitation by femtosecond pulses [42,45,48].

In this work, we develop a novel scheme for pump–probe OCM to register modulation in the refractive index with the resolution of 1 ps in time, 0.5 µm in space, and a sensitivity of down to ${10^{- 3}} \;{\rm RIU}$, improving the previously achieved OCT/OCM temporal resolution by three orders of magnitude. Direct quantitative spatiotemporal visualization of the dynamics of carriers in a perovskite crystal during lasing is carried out, and the relaxation and diffusion constants are estimated simultaneously. The resulting technique is a versatile method for studying ultrafast carrier transport and $\Delta n$ imaging, paving a new avenue for OCT and OCM applications in the research of nanophotonic devices and materials.

2. METHODS

A. Experimental Setup

The experimental setup implements a pump–probe scheme with OCM as the probe (Fig. 1). Figure 1(b) illustrates the schematic of the developed method, where the pump beam is indicated by green lines and the OCM beams are indicated by red lines. Figure 1(c) shows the data acquisition process.

The femtosecond laser (Avesta TEMA) with the center wavelength of 1050 nm, an average power of 3.5 W, a pulse duration of 100 fs (spectrum width of 13 nm), a pulse energy of 50 nJ, and a repetition rate of 70 MHz is used as a source. The source radiation passes through a pulse picker which reduces the pulse repetition rate to 50 kHz to avoid sample overheating. Next, the second harmonic of the source radiation is generated in a nonlinear crystal ($\beta$-barium borate, BBO) and is used as a pump. A part of unconverted source radiation is coupled into a nonlinear photonic crystal fiber (PCF, NKT Photonics SC-5.0-1040), in which a supercontinuum radiation (SC) with a wide spectrum from 925 nm to 1125 nm is generated for the OCM probe based on the Michelson interferometer. The probe pulse duration is 5 ps due to chirping in PCF; a special algorithm was applied to improve the temporal resolution down to 1 ps (see Supplement 1, Section 1 for details). The position of the reference mirror (RM) of the interferometer is chosen to provide pronounced interference between the reference and signal beams, which corresponds to a path difference of approximately 10 µm to 20 µm [14]. The probe beam scans the sample with galvanic mirrors (GM), which makes it possible to construct the spatial distribution of the refractive index at the fixed pump beam position.

The pump beam passes through a delay line based on the linear translation stage (OptoSigma OSMS26-300), providing temporal scanning in the range of 2 ns with a minimum step of 30 fs (note that the time resolution is 1 ps due to the probe pulse width). Pump control includes an optical chopper and a liquid crystal retarder for power tuning and a steering lens for pump beam positioning. The chopper modulates the pump to further separate the signal against the noise background, the retarder together with the polarization beam splitter (PBS) performs automatic adjustment of the pump power, and the steering lens changes the position of the beam spot and its size in the sample area.

The pump and probe beams are aligned with a PBS and focused on the sample using a high-numerical-aperture (NA) objective lens (LOMO, ${\rm NA} = {0.85}$, ${40 \times}$). The waist radii are 2.25 µm and 0.5 µm at $1/{e^2}$ level for pump and prob beams, respectively. Modulation in the refractive index of the sample induced by the pump beam changes the optical path of the probe beam, which is reflected from the sample and interferes with the OCM reference beam. The spectrum resulting from the interference is recorded by the spectrometer (Ocean Optics QE Pro, 400–1150 nm range, 1.5 nm resolution). The interference manifests itself as a cosine wave superimposed on the wide supercontinuum spectrum. The refractive index changes induced in the sample can be calculated from the interference pattern using the fast Fourier transform (FFT). If the sample with 0.5–5 µm thickness is illuminated from below, the beams reflected from the upper and lower surfaces of the sample act as a signal and reference, respectively, and the RM is not needed. In this case, the signal beam passes through the region of the modulated refractive index $\Delta n$, while the reference beam reflects from the lower boundary and therefore does not interact with the $\Delta n$ region. The perovskite crystal samples were studied in this configuration, called common-path OCM [18], since it is more noise tolerant. For the sample with ${\lt}{0.5}\;{\unicode{x00B5}{\rm m}}$ or ${\gt}{5}\;{\unicode{x00B5}{\rm m}}$ thickness, the common-path OCM configuration is not suitable, and the RM is needed (as for a sample of a silicon waveguide in Supplement 1, Section 5 or in Ref. [25]). The lower limit of the suitable sample thickness for a common-path OCM configuration is determined by the width of the OCM spectrum (the wider the spectrum, the thinner the samples that can be examined), while the upper limit is determined by the signal collection depth, which is affected by the lens focus depth and confocal gating [14,18].

B. Data Acquisition Algorithm

The data acquisition algorithm consisting of five stages is shown in Fig. 1(c). (1) The time delay between the pump and probe pulses ($\tau$) is set by positioning the delay line retroreflection mirrors. (2) The positions of the pump and probe beams on the sample are aligned using a steering lens and galvanic mirrors, respectively. (3) The pump pulse affects the refractive index of the sample, which changes the optical path of the signal pulse in the OCM and modifies the interference pattern. The spectrometer takes interferograms with an exposure time of 8 ms. To reduce the noise, the registration is performed in the lock-in mode: the pump beam is modulated (open–close) by an optical chopper with a frequency of 20 Hz (50 ms period, six interferograms per period), and the spectral signal is registered at the chopper frequency. (4) The interferogram is processed by computer software. The spectrum is normalized to an interference-free spectrum, and the Fourier phase is extracted. Then the changes in the amplitude of the Fourier phase ($\Delta \varphi$) at the chopper frequency are determined [20]. The $\Delta \varphi$ carries information about the modulation in the optical path and refractive index $\Delta n$ in the sample under the action of the pump pulse. (5) Measurements in stages 1–4 are repeated for various time delays $\tau$ and different positions ($x$, $y$) of the probe beam on the sample. By varying $\tau$, $x$, $y$, one can obtain a map of modulation in the refractive index $\Delta n(\tau ,x,y)$ with a spatial resolution of 0.5 µm and a temporal resolution of 1 ps, which forms the basis of the experimental data.

The configuration of the experiment makes it possible to record simultaneously the fluorescence spectra of the sample and changes in its refractive index using the same spectrometer.

C. Samples

To demonstrate the capabilities of the developed technique, perovskite microlasers were chosen. The perovskites have a high refractive index, which significantly changes under the action of free carrier generation. On the other hand, the spatiotemporal dynamics of carriers affects the efficiency of laser generation and the quantum yield of fluorescence [6,36,39]. Laser generation in perovskite microcrystals can be caused by a femtosecond pump pulse illuminating either the entire sample or part of it [38]. This enables us to study the spatial dynamics of a locally generated plasma cloud during lasing.

The samples under study were ${{\rm CsPbBr}_3}$ perovskite microcrystals grown by the method of rapid precipitation from dimethyl sulfoxide solution sprayed onto hydrophobic substrates at ambient conditions on a 0.17 mm thick glass substrate (for details on synthesis protocol, see elsewhere [49]), which makes it possible to achieve monocrystalline structures with a low concentration of defect states [49–51]. The sample contains large crystals, microplates (crystals with a length and width of 5–10 µm and a height of 0.5–2 µm), and nanostrips (narrow crystals several micrometers long), as shown in Fig. 1(b). Microplates were chosen as representative samples, since unlike large crystals, lasing can occur in them, and unlike nanostrips, a map of $\Delta n$ can be visualized along both the $x$ and $y$ directions during lasing. A microplate with a size of (${\rm L} \times {\rm W} \times {\rm H}$) $8 \times 8 \times 1.5 \;{{\unicode{x00B5}{\rm m}}^3}$ was chosen as the main sample [Fig. 2(a), upper inset], and the exact size was determined using atomic force microscopy (AFM).

 

Fig. 2. Spectral–temporal characterization of the perovskite sample with pump and probe beams focused at the center of the sample. (a) Dependence of the refractive index modulation $\Delta n$ (and, accordingly, the concentration of electron–hole plasma ${N_{\rm{EHP}}}$) versus the pump fluence. The top inset shows an optical transmission microscope image of the crystal. The yellow dot shows the position and diameter of the pump beam. Scale bar is 10 µm. The lower inset illustrates $\Delta n(\omega)$ from Eq. (1) with ${N_{\rm{EHP}}} = 1.1 \cdot {10^{19}}\; {{\rm cm}^{- 3}}$, the green line denotes pump, and the red band is the spectrum of the probe. (b) Dependence of the fluorescence spectrum of the sample on the pump fluence. (c) Dependence of the relaxation dynamics of the refractive index modulation on the pump fluence (each horizontal line corresponding to a certain fluence value is normalized to the maximum $\Delta n$ value in the line). The green dashed lines mark the lasing threshold, and the red and purple dashed lines indicate fluences, for which the spectra (b) and non-normalized kinetic curves (c) are shown in the bottom subplots.

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3. RESULTS AND DISCUSSION

The main mechanism of laser generation in lead halide perovskite microplates under ultrafast optical pumping is the stimulated emission of the electron–hole plasma (EHP) [42,45]. The contribution of carrier density ${N_{{\rm EHP}}}$ to the complex permittivity $\varepsilon (\omega)$ (dielectric function) is described by the Lorentz model [52–54]

The form of the spectral dependence of the EHP-induced addition $\Delta n$ to the refractive index obtained from Eq. (1) for ${{\rm CsPbBr}_3}$ perovskite at a concentration of ${N_{{\rm EHP}}} = 1.1 \cdot {10^{19}} \;{{\rm cm}^{- 3}}$ is shown in Fig. 2(a) (bottom inset). The pump and probe wavelengths used in the experiment are marked with red and green bands, respectively. For probe wavelengths away from the resonance (as in our case), the refractive index modulation $\Delta n$ depends linearly on the plasma concentration ${N_{{\rm EHP}}}$ (see Supplement 1, Section 2 for details). At long wavelengths $\omega \to 0$, and while ${\omega _p} \ll {\omega _0}$, $\Delta n$ can be estimated from the asymptotic formula

Below the saturation, the number of generated carriers is proportional to the pump intensity [45,56]. Thus a linear dependence of $\Delta n$ and ${N_{{\rm EHP}}}$ on the pump fluence is expected at probe wavelengths.

In the experiment, the microplate sample was excited with focused pump radiation (waist radius is 2.25 µm at $1/{e^2}$ level) located at the center of the sample [Fig. 2(a), top inset; a yellow circle marks the pump position]. Figure 2(a) shows the dependence of $\Delta n$ on the pump fluence measured at zero time delay between the pump and probe. The dependence is linear, which is in excellent agreement with the Lorentz model described above. An additional scale on the $x$ axis converts $\Delta n$ to the concentration of free carriers according to Eq. (1). The linear relationship between $\Delta n$ and the carrier concentration is an advantage for the method allowing quantitative imaging of electron–hole plasma clouds.

To register lasing in the perovskite microcrystal and to determine the lasing threshold, the perovskite fluorescence spectra were recorded for various pump fluences [Fig. 2(b)]. Only a broad band of luminescence is observed in the spectra up to the fluence of $0.57 \;{{\rm mJ/cm}^2}$, which is clearly visible even despite the close pump line. As the fluence exceeds $0.57 \;{{\rm mJ/cm}^2}$, a narrow lasing line appears. Therefore, we will consider this value to be the lasing threshold pump fluence. With a further increase in fluence, the appearance of additional lasing modes and the transfer of energy between them are observed, which is associated with the red shift of the lasing gain profile and band gap renormalization [45]. In addition, a small blue shift for specific modes can be observed in Fig. 2(b), which is caused by a change in the refractive index of the material [34,42,45,47].

Overcoming the threshold fluence is accompanied not only by the appearance of lasing lines in the fluorescence spectrum, but also by the change in the nature of the ultrafast dynamics of $\Delta n$, and, consequently, ultrafast dynamics of the plasma ${N_{{\rm EHP}}}$ [Fig. 2(c)]. In the experiment, we have simultaneously recorded the fluorescence spectra and the sample $\Delta n$ as a function of the time delay between the pump and probe pulses. At fluences below the threshold, the $\Delta n$ dynamics exhibits slow relaxation, while at fluences above, a fast component appears, associated with lasing (stimulated emission).

It should be noted that the rapid relaxation of conductivity and $\Delta n$ in semiconductor crystals can also be associated with the formation of excitons from unbound electrons and holes in plasma even at room temperature [57]. However, in this work, the carrier density ($N \sim {10^{19}} \;{{\rm cm}^{- 3}}$) is higher than the Mott density (${N_{{\rm Mott}}} \sim {10^{17}} \;{{\rm cm}^{- 3}}$) [45], the appearance of the fast relaxation component $\Delta n$ is accompanied by the simultaneous appearance of lasing lines (indicating stimulated carrier relaxation), and a plasma red shift of the lasing gain profile is also observed, which speaks in favor of predominantly plasma carriers and EHP-dependent $\Delta n$ [42,45].

Since $\Delta n$ is directly proportional to ${N_{{\rm EHP}}}$, the dynamics of $\Delta n$ immediately gives the information about the carrier dynamics. Up to the threshold value, the main mechanism of carrier relaxation is spontaneous emission, characterized by relaxation times of the order of 100 ps [34,45]. Above the threshold value, the greatest contribution is made by stimulated emission with characteristic relaxation times from 8 ps to 20 ps [34,45]. Thus, the lasing threshold can be determined not only by the appearance of lasing lines in the fluorescence spectrum of the sample, but also by the occurrence of a fast component in the dynamics of the refractive index, associated with the transition from spontaneous to stimulated emission.

Table 1. Comparison of Measured Perovskite Microlaser Parameters with Literature Data for Various Lead Halide Perovskite Samples^a

View Table

 

Fig. 3. Ultrafast carrier dynamics during lasing. (a) Dependence of the sample fluorescence intensity (blue) and the relaxation time of the refractive index (orange) on the pump fluence. The green dashed line is the lasing threshold, and the red and purple dashed lines mark the fluences at which the relaxation curves were taken in (b). (b) Relaxation curves of the refractive index and, accordingly, of the EHP density at pump fluences below (purple, $0.35 \;{{\rm mJ/cm}^2}$) and above the lasing threshold (red, $0.8\; {{\rm mJ/cm}^2}$). The inset illustrates the same curves in a logarithmic scale fitted with Eq. (3). (c) Zoomed area of relaxation curves and their fit for time delays from 0 ps to 100 ps; the dashed and solid orange curves represent the cavity photon density obtained with Eq. (3) for the fluence of $0.35 \;{{\rm mJ/cm}^2}$ (multiplied by 20 for clarity) and $0.8 \;{{\rm mJ/cm}^2}$, respectively. The inset shows the light–light curve [62], which is the dependence of the sample fluorescence intensity on the pump fluence from (a) (blue circles) along with the fitting by Eq. (3) (gray curve), corresponding to the coupling constant $\beta = 0.03$.

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Comparison of Figs. 2(a)–2(c) allows estimating $\Delta n$ and ${N_{{\rm EHP}}}$ corresponding to the lasing threshold: $\Delta {n^{{\rm th}}} = 1 \cdot {10^{- 2}}$, $N_{{\rm EHP}}^{{\rm th}} = 1.1 \cdot {10^{19}}\; {{\rm cm}^{- 3}}$. The use of asymptotic Eq. (2) makes it possible to estimate the threshold concentration as $N_{{\rm EHP}}^{{\rm th}} = 1.5 \cdot {10^{19}}\; {{\rm cm}^{- 3}}$, which overestimates the value by only 40% compared to the original Eq. (1). Threshold carrier concentrations have been reported ranging from $1.5 \cdot {10^{16}}\; {{\rm cm}^{- 3}}$ for low-threshold nanowires in the exciton lasing mode [34] to $7.9 \cdot {10^{19}} \;{{\rm cm}^{- 3}}$ for 10–20 µm ${{\rm CsPbBr}_3}$ microplate lasers in the electron–hole plasma lasing mode [45] (Table 1). Our data are in good agreement with the latter value, which indicates the applicability of the method for quantitative imaging of carriers.

The lasing threshold value of perovskite microlasers strongly depends on many factors, such as the chemical composition of samples, fabrication process, the quality and material of the substrate, the ratio between the sizes of the pump spot and the structure, as well as the type of excited mode [33,38]. Since the pump spot does not illuminate the structure fully in this experiment, the number of carriers generated in the local region should be sufficient to compensate for losses in the non-excited part of the structure, which leads to a higher lasing threshold.

The transition from spontaneous to stimulated emission is shown in more detail in Fig. 3. Figure 3(a) illustrates two simultaneously existing trends: (1) the threshold character of the dependence of the total fluorescence intensity on the fluence (S-curve), which experiences a characteristic inflection at $0.57 \;{{\rm mJ/cm}^2}$, and (2) a sharp decrease in the relaxation time of $\Delta n$ and carriers at the same fluence from over 100 ps to ${\lt}{20}\;{\rm ps}$.

To describe the ultrafast dynamics of the EHP density ${N_{{\rm EHP}}}$ and photon density ${n_{\rm{ph}}}$ in the lasing mode disregarding diffusion and after pump beam arrival, the system of coupled laser rate equations can be used [62–65]:

Both below the lasing threshold [fluence of $0.35 \;{{\rm mJ/cm}^2}$, purple curve in Fig. 3(b)] and above the lasing threshold [fluence of $0.8 \;{{\rm mJ/cm}^2}$, red curve in Fig. 3(b)], the fitting of the experimental data by the model gives the values of $A = 2 \cdot {10^8}\; {{\rm s}^{- 1}}$, $B = 1.8 \cdot {10^{- 9}} \;{{\rm cm}^3}/{\rm s}$, $C \le 1 \cdot {10^{- 29}} \;{{\rm cm}^6}/{\rm s}$, which are consistent with the literature (Table 1). Auger recombination does not significantly affect the carrier dynamics at the given concentration (${\sim}{10^{19}} \;{{\rm cm}^{- 3}}$), and therefore its contribution can be neglected [34].

Fitting of the experimental data is shown in Fig. 3(b) (inset) in logarithmic scale in the case of both lasing and without lasing. Figure 3(c) also shows the detailed carrier dynamics in the first 100 ps after the arrival of the pump pulse. The model makes it possible to describe quantitatively the dynamics of carriers both at times of the order of several picoseconds [Fig. 3(c)] and at subnanosecond scales [Fig. 3(b), inset]. The photon densities ${n_{\rm{ph}}}$ obtained as the result of the numerical solution of Eq. (3) are also shown in Fig. 3(c) and agree with the experimental data of other works [45].

To determine the characteristic relaxation times and compare the values obtained with literature data, a simplified biexponential decay model is used [45]. As a result of approximation by the biexponential model $\Delta n = a{e^{- t/{\tau _{{\rm lasing}}}}} + b{e^{- t/{\tau _{{\rm fluor}}}}}$, the characteristic relaxation times are: ${\tau _{{\rm lasing}}} = 13.4 \pm 1.3\;{\rm ps} $ for stimulated emission and ${\tau _{{\rm fluor}}} = 170 \pm 20\;{\rm ps} $ for spontaneous emission, which are in good agreement with literature data [34,45].

 

Fig. 4. Ultrafast spatiotemporal dynamics of plasma in a perovskite microcrystal. (a) Schematic of the crystal sample ($8 \times 8 \;{{\unicode{x00B5}{\rm m}}^2}$) and generated plasma cloud. The white dashed square marks the scanning area with the OCM beam ($5 \times 5 \;{{\unicode{x00B5}{\rm m}}^2}$). (b) Spatiotemporal dynamics of $\Delta n$ and corresponding ${N_{\rm{EHP}}}$ in the first 500 ps after the arrival of the pump pulse. (c) Detailed spatiotemporal plasma dynamics up to the first 90 ps. Color bar is common for (b) and (c). (d) Experimental plasma distribution cross section (dots) fitted by a Gaussian function (curves) for clarity. (e) Corresponding simulated plasma distribution cross section according to Eq. (4).

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Finally, we turn to ultrafast spatiotemporal visualization of electron–hole plasma in a perovskite microlaser. The schematic of the sample and the generated cloud are shown in Fig. 4(a). The intensity profile of the beam is Gaussian with radius of 2.25 µm, which leads to a similar distribution of the free carrier cloud ${N_{{\rm EHP}}}$ and refractive index $\Delta n$ at the moment of arrival of the pump pulse $\tau = 0\;{\rm ps} $.

The generated cloud relaxes as a result of diffusion, as well as stimulated, spontaneous, and nonradiative transitions, which leads to a change in both the amplitude and the shape of the EHP cloud profile [6]. The two-dimensional distribution of the plasma cloud in the first 500 ps after the arrival of the pump pulse is shown in Fig. 4(b); the corresponding plasma distribution cross sections are illustrated in Fig. 4(d). The main relaxation occurs in the first 100 ps after the arrival of the pump pulse due to stimulated radiation. Figure 4(c) shows the detailed ultrafast spatiotemporal dynamics of the electron–hole plasma cloud in the interval from ${-}{10}\;{\rm ps}$ to 90 ps in 10 ps steps.

The system of coupled equations Eq. (3) describes the interaction of carriers and photons in the laser mode and yields important characteristics of a microlaser (see Supplement 1, Section 3 for details). However, adding a diffusion term leads to excessive model complexity. To simplify the calculations, it was decided to use the diffusion equation and take into account the interaction of carriers and photons during lasing through a time-dependent coefficient ${k_{{\rm lasing}}}(t)$ (effective stimulated emission coefficient). Thus, to describe the experimental results in Fig. 4, the following equation was used (see Supplement 1, Section 4 for details):

The simulated plasma distribution cross sections are shown in Fig. 4(e). The values of $A$ and ${k_{{\rm fluor}}} = B$ were taken from Eq. (3), and the values of ${k_{{\rm lasing}}}(t)$ and $D$ were used to bring the model into agreement with the experimental data in Fig. 4(d). Since no diffusion is observed at times up to 500 ps, the model helps to give an upper estimate for the carrier diffusion coefficient: $D \le 1\;{{\rm cm}^2}/{\rm s}$. The value $D = 1.1\;{{\rm cm}^2}/{\rm s}$ [61] known for single crystal ${{\rm CsPbBr}_3}$ microplates turns out to be very close to the estimation obtained.

The developed method can also be used to study the carrier diffusion in other nanophotonic devices, for example, in silicon waveguides, for which the diffusion coefficient $D$ is much higher. The determined value of $D = 15\;{{\rm cm}^2}/{\rm s}$ is consistent with the literature [7]; see Supplement 1, Section 5 for details. Thus, the perovskite low carrier diffusion coefficient in comparison with other semiconductors is also preserved during lasing. The generated carrier cloud remains localized for at least a nanosecond, which contributes to the high efficiency of lasing and fluorescence of perovskites. To the best of our knowledge, the diffusion coefficient of a perovskite material in the presence of lasing has been estimated for the first time. A comparison of the perovskite microlaser parameters measured by the ultrafast OCM with the data of other works is shown in Table 1.

Thus, the developed method can be used to simultaneously estimate a whole set of basic parameters of a perovskite microlaser: the carrier relaxation rate as the result of spontaneous and stimulated transitions, the threshold value of the carrier concentration for lasing, and the carrier diffusion coefficient. The high spatial and temporal resolutions of the method makes it possible to visualize the relaxation of a plasma cloud in the presence of laser generation.

The advantages of the method, shown on the perovskite sample, can also be applied to study the ultrafast spatiotemporal dynamics of $\Delta n$ in various nanophotonic devices. The registered $\Delta n$ can be caused by the generation of free carriers (as in the case of the studied perovskite sample), by the optical Kerr effect (without free carrier generation), and by other processes.

The main current limitation is the sensitivity of the method (${10^{- 3}} \;{\rm RIU}$), but this value can be further improved using phase stability compensation algorithms [18] and sources with the wider spectrum. This is especially relevant for thin samples, in which changes in the optical path are small. The temporal resolution can be improved by using wide spectrum ultrashort pulses instead of supercontinuum generation. The limitations on the objects under study include transparency in the sample range (near IR), a flat shape on the waist scale (4.5 µm), as well as the refractive index contrast between the sample and the medium/substrate of at least 0.1 RIU. Limitations can be circumvented by using shorter probe wavelengths in the transparency region of the substrate and sample and high-NA objectives with oil immersion to achieve a smaller waist size, but it should be noted that for ${{\rm CsPbBr}_3}$ perovskites, the $\Delta n$ (${N_{\rm{EHP}}}$) dependence in the visible range will have a complex form due to the resonances of materials (see Supplement 1, Section 2). As a result, many nanophotonic devices that perform field localization and free carrier generation, such as microlasers, nonlinear optical waveguides, microcavities, nanoantennas, detectors, and many other objects, including those of integrated silicon photonics, are suitable objects for study.

Given the possibilities and limitations described above, we believe that the developed method of ultrafast OCM is a promising tool for solving many problems of modern nanophotonics, especially when it is necessary to study the dynamics of carriers with both high spatial and high temporal resolutions.

4. CONCLUSIONS

A novel method for ultrafast visualization of refractive index modulation in nanophotonic devices based on optical coherence microscopy combined with the pump–probe scheme is implemented and tested on perovskite microlasers, whose carrier dynamics during lasing is of great interest. The electron–hole plasma spatiotemporal dynamics in a perovskite microcrystal during lasing is successfully visualized with a resolution of 1 ps in time and 0.5 µm in space. The carrier concentration required to start the laser generation is determined to be $1.1 \cdot {10^{19}} \;{{\rm cm}^{- 3}}$ simultaneously with diffusion coefficient $D \le 1\;{{\rm cm}^2}/{\rm s}$, and the decay times of carriers ${\tau _{{\rm lasing}}} = 13\;{\rm ps} $, ${\tau _{{\rm fluor}}} = 170\;{\rm ps} $. The perovskite carrier diffusion coefficient is low compared to other semiconductors and is found to be maintained during lasing. Therefore, the generated carrier cloud remains localized, which contributes to the high efficiency of lasing and fluorescence. The developed method is promising not only for studying the spatiotemporal dynamics of carriers in microlasers, but also for its further application in the study of the light–matter interaction in various nanophotonic devices and novel photonic materials. This universal approach can solve emerging problems in the rapidly developing field of nanophotonics and expand the scope of OCT-based methods’ applications.