Peculiarities of nitrogen-vacancy centers’ superluminescence in diamond under optical pumping at 532 nm

In the context of the NV− diamond laser creation (Savvin and Dormidonov in Nat. Commun. 12:7118, 2021), an urgent task is to determine the characteristics of diamonds that can affect laser generation. This work is aimed at investigating the mechanisms of the creation of superluminescence in diamond under the action of optical pumping by the second harmonic of the Nd:YAG laser (λ = 532 nm). It was found that when the HPHT diamond is irradiated by 532 nm radiation with an intensity above ~ 2.0 MW/cm2, a nonlinear intensity increasing in the spectral region 700–750 nm is manifested against the background of the spontaneous photoluminescence spectrum, which, with a further pumping intensity increase turns into a pronounced peak of superluminescence with a maximum of about 718 nm. An increase in the pumping intensity from 2.7 to 46 MW/cm2 widened this peak at half-maximum from 13 to 19 nm. At high levels of pumping intensity, nonlinear pumping radiation absorption and accumulation of NV centers in the excited state were detected. The position of the photoluminescence band was calculated depending on different values of the population inversion density of the color centers, taking into account the diamond's own absorption spectrum. The calculation results are close to the experimental data.


Introduction
The creation of laser media and optical materials with high thermal conductivity, hardness, and chemical and radiation resistance remains an urgent task. Diamond crystals have such characteristics to a significant extent. Monographs [2][3][4][5][6] provide detailed information about the characteristics of diamonds, which can be useful in the analysis of defects and the creation of active media based on diamonds.
The prospects of using various color centers in diamonds to generate laser radiation based on the calculation of the achievable optical gain have been reported in many papers. The analysis of these works [7] showed that NV centers with zero phonon lines (ZPL) at 575 and 638 nm have a high gain, wide absorption and luminescence bands, and a short excited state lifetime. This indicates the prospects of using NV centers as active laser centers in the spectral range 650-750 nm.
There are reports of the cultivation of synthetic diamonds of optical quality with a size of more than 10 mm, and the achievement of the concentration of NV centers up to 10 19 cm −3 , which also indicates good prospects for the use of diamonds as active elements of lasers. Results very close to obtaining laser generation are published in Refs. [8][9][10][11].
It should be noted that to generate laser radiation at the color centers in diamonds, it is necessary to take into account many, not always precisely known, characteristics of the samples. A demonstration of this is the work [1], in which the generation of laser radiation on NV centers was achieved for the first time, after a number of unsuccessful attempts by other authors.
Here, we present the results of a study of superluminescence at NV centers in a band with a maximum of about 718 nm under the action of pulsed Nd:YAG laser at a wavelength of 532 nm, taking into account the characteristics of a particular crystal.

Description of the experiment
The diamond sample was purchased from LLC Velman, Novosibirsk, Russian Federation. A substantially inhomogeneous distribution of C-defects was found in our sample, which implied a substantially inhomogeneous distribution of color centers. Heterogeneities in the distribution of defects in the volume of both natural and synthetic diamonds are well known. This phenomenon is explained from the point of view of growth processes, namely, by the fact that different growth sectors capture and "embed" impurity atoms in their structure in different ways [3]. This is a consequence of the peculiarities in the growth processes of these sectors: as a rule, it is layered, tangential for sectors < 111 > and normal dislocation for sectors < 100 >.
The sample under study was a 4.4 × 4.4 × 0.25 mm plate cut (100) from a diamond synthesized by the HPHT method. After polishing, the plate was irradiated with electrons with an energy of 3 MeV with a fluence of 10 19 cm −2 , then annealed in vacuum at a temperature of 1000 °C for 2 h [12]. After irradiation and annealing, inhomogeneities in the distribution of defects across the plate became clearly visible (Fig. 1).
There is an increased content of defects in dark areas. IR spectroscopy revealed that the sample contains unevenly distributed C-defects in the concentration range 1.8·10 17 -2.55·10 19 cm −3 (1-150 ppm) and has a pronounced zonal-sectorial structure (Fig. 1). The distributions of nitrogen concentrations and NV centers are presented in Table 1.
To register absorption in the IR region of the spectrum, the Bruker Vertex 70 IR Fourier spectrometer was used in conjunction with the Hyperion 2000 microscope. Transmission spectra in the range of 200-900 nm were measured using the Shimadzu spectrometer, from which, taking into account the thickness of the sample, absorption spectra were calculated.
Experiments to obtain superluminescence were carried out on the installation, the scheme of which is shown in Fig. 2. Pulsed radiation with a wavelength of 532 nm (the second harmonic of the Nd:YAG laser) was used to excite the photoluminescence of the sample. The radiation energy of the pump was 80 mJ/imp. with a pulse duration of about 10 ns at half-maximum. The pumping intensity in the range of 0.1-50 MW/cm 2 was varied by neutral light filters. The pumping radiation was focused by a cylindrical lens into a spot of 0.5 × 3.0 mm and directed into one of the zones (Fig. 1). The luminescence of the sample was recorded at the end of the plate at an angle of 10-30 degrees to the plane of the plate (100). The luminescence output in this direction was maximal.
Luminescence radiation from an optic fiber in the range 200-900 nm was fed into the AvaSpec-2048-2 spectrometer. Spectral data were accumulated in a digital file by PC.
Optical absorption spectra calculated from the transmission spectra in individual zones of the sample in the range of 200-750 nm are shown in Fig. 3. In zone 2, the edge of the fundamental absorption of diamond (225 nm), a band with a maximum at 270 nm, as well as a wide band from 440 to 640 nm containing low-intensity absorption bands of NV 0 and NV − -centers were recorded. Thus, the central part of the crystal is an area with a low nitrogen content; therefore, the luminescence intensity in this zone is weak, that is, this area is unsuitable for the creation and study of the required centers and is not further considered. Zones 1 and 2, where the concentration of the centers is increased (see Table 1), as confirmed by the obtained absorption spectra, were selected for research. Zones 1 and 3 have similar absorption spectra, consisting of an intense band in the UV range, and an absorption band from NV centers with an index of ~ 18 cm −1 . In these very zones, superluminescence was observed.

Results
The photoluminescence spectra of the sample depending on the pumping intensity are shown in Fig. 4. The spectra are normalized to the intensity of the maximum of the undisturbed phonon wing of NV centers at 680 nm. With weak pumping, a band of NV centers with a distinguishable ZPL λ = 638 nm is noticeable in photoluminescence. With an increase in the pumping intensity above ~ 2.0 MW/cm 2 , a nonlinear increase in intensity is detected in the 700-760 nm range, and with a pumping intensity above 2.5 MW/cm 2 , this band turns into an intense peak with a maximum of ~ 718 nm. The half-width of this peak increased from 13 to 19 nm with an increase in the pumping intensity from 2.7 to 46 MW/cm 2 . Figure 5 shows the dependence of I phl /I pump (the ratio of the photoluminescence intensity at the maximum to the pumping intensity). This ratio is proportional to the quantum yield of luminescence.
In the range of pumping intensities up to 3 MW/cm 2 , the luminescence intensity increased approximately in proportion to the pumping intensity, at intensities of more than 3-5 MW/cm 2 , the dependence went into saturation, and in

Discussion of the results
Despite the big number of theoretical works on laser generation in diamonds, it has been practically obtained only in laboratory settings. Obviously, it is necessary to take into account not only the characteristics of the centers, but also the characteristics of the crystal matrix in which these centers are contained.
Thus, the analysis of the internal morphology of the crystal enables to select (cut) from the crystal the zone that meets the requirements for uniformity of the distribution and concentration of centers in the best way. Note that the generation obtained in Ref. [1] was detected in a separate growth zone with an increased content of color centers.
An important point should be noted regarding the methods of amplification calculations and terminology. As a criterion for the suitability of using color centers as active laser centers, a high cross section of the radiative junction is used. The cross section of the radiative junction is usually calculated for the maximum of the luminescence band and generation is expected in the same band. In many works (including ours), the cross section of the radiative transition (gain cross section) σ multiplied by the number of centers in the excited state N* is identified with the gain index: In this case, the losses are considered to be spectrally independent. A high cross section of the radiative transition in the color centers is a necessary, but not sufficient condition for obtaining amplification and generation of laser radiation. This calculation does not take into account the fact that the main electronic state of the center can be populated. The population inversion can be obtained relative to it; therefore, amplification and generation in the ZPL region are possible only if more than half of the centers are excited. In addition, the characteristics of the matrix crystal in which the color centers are contained are usually not taken into account. There are losses in the crystal, which are caused both by self-absorption near ZPL, and some other losses unrelated to the color centers. Such losses usually are spectrally dependent and constitute the absorption spectrum of the crystal. Below is a calculation of the possibility of NV − centers laser generation obtained by using the actual measured characteristics of a particular crystal, including the absorption spectrum.
In the absorption spectrum of our synthetic diamond measured at 300 K (Fig. 6a), NV − color centers are manifested by an intense band with a maximum of ~ 560 nm. In addition, the absorption spectrum contains an absorption "step" near the ZPL of centers and a band with a maximum of ~ 785 nm, which is a source of additional losses that negatively affect the generation of laser generation. When registering the spectrum at T ~ 80 K, the ZPL of this band is not registered. Its nature has not yet been established, but the absorption index in the area of the expected generation is high and is about 3 cm −1 . Figure 6b shows the luminescence spectrum of this sample.
The concentration of centers is determined by the ratio: where β is the absorption index and σ is the absorption cross section. In the literature [7][8][9][10][11], sufficiently close values of the absorption and emission cross sections (the cross section of the radiative transition) are given. In our calculations, we (2) N = ∕ , Fig. 6 Absorption spectrum (a) and luminescence spectrum (b) of a crystal with NV centers (300 K) used a certain value of the absorption cross section averaged according to literature data σ = 2.8·10 -17 cm 2 , which gives the value of center concentration N = 6.6·10 17 cm −3 , with an absorption index at the maximum of the band 18.5 cm −1 .
As follows from Fig. 6b, the luminescence maximum is located in the range near ~ 680 nm, which corresponds to a cross section of the radiative transition of 4.3 × 10 -17 cm 2 . In this very region of the spectrum, the appearance of superluminescence should be expected. However, both in Ref. [1] and in our experiments, generation or superluminescence is detected in the region of ~ 720 nm (Fig. 5).
Recall that when propagating in a medium with population inversion, the radiation will be amplified with an amplification factor α(λ) proportional to the concentration of excited centers N* and the radiative transition cross section σ i (λ): where At the same time, the diamond sample has losses in the absorption, scattering, and reflection from the faces, etc. These losses are also spectrally dependent and together constitute the absorption spectrum of the crystal β(λ) (Fig. 6a).
The propagation of radiation in an amplifying medium at a wavelength λ with spectral-dependent losses is written as: In the ratio (2), β(λ) in this case is an indicator of the total optical losses at the wavelength λ. From (4), the threshold density of population inversion is determined, the excess of which will lead to an increase. The threshold density of population inversion is defined as the value of N*, at which I(λ)/I 0 > 1 or or The meaning of this relation is obvious: the intensity of the radiation propagating in the diamond will increase at those wavelengths for which the gain exceeds the loss. In our case, this condition satisfies the range of 700-730 nm, where luminescence is quite intense and absorption losses are minimal (Fig. 7). With a further increase in the inversion density, the radiation intensity increases, while the band, in which the generation frequency can be rearranged, expands. Note that in the region of ZPL the generation of laser radiation is impossible, although the calculation of the cross section of the transition from the upper laser level to the lower gives non-zero values.
Thus, the use of our sample's absorption spectrum takes into account the fact that the basic electronic state of the color center is populated and there is no population inversion relative to this state. In addition, the position of the superluminescence band is determined not by the maximum in the luminescence spectrum, but by the maximum loss ratio. Substituting into formula (4) the real values for zone 1 of our sample at the population inversion density N* = 10 17 cm −3 , we find that the gain will be 1.47 (times). This means that the radiation from the narrow face of the sample will be amplified (in one pass) by about one-and-a-half times, which in the presence of a tuned resonator is sufficient to obtain stable laser generation. Figure 8 shows the calculated radiation spectra calculated according to (4) at population inversion densities N* = 1·10 17 ; 2·10 17 ; 2.5·10 17 cm −3 and the length of the gain medium d = 1 cm. Comparing Figs. 8 and 4, we find a correspondence between the calculated and experimental results.
The dependence of luminescence intensity from the pumping pulse energy (Fig. 6) has a complex character. At low pumping intensities (1-3 MW/cm 2 ), a section of linear intensity growth is observed. The decrease in luminescence yield at the 3-5 MW/cm 2 site can be explained by saturation of the absorption index β(λ). The decrease in the absorption index (saturation) is associated with a decrease in the population of the ground state and the accumulation of centers in the excited state. The absorption saturation is characterized by the saturation parameter I s , numerically equal to the radiation intensity, at which the absorption value is reduced by half: Fig.7 Spectral gain index at NV centers at N* = 1·10 17 ; 2·10 17 ; 2.5·10 17 cm −3 In our case, the calculated saturation parameter was I s = 1.03 MW/cm 2 . In Ref. [9], the experimentally measured saturation parameter of NV − centers was 1.55 MW/ cm 2 , which is a fairly good match to our calculations.
The saturation curve G = 1 -1/(1 + I/I s ) with the parameter I s = 1.03 MW/cm 2 is shown in Fig. 9. The curve of dependence of the luminescence output on the pumping intensity is also shown here. As follows from the figure, the maximum luminescence in the pumping intensity range of 3-5 MW/cm 2 is apparently explained by saturation absorption, but with a further increase in the pumping intensity, the luminescence output drops sharply.
The authors [8] suggest that the decrease in intensity is associated with a decrease in the concentration of color centers due to photolysis of NV − centers and their transformation into NV 0 centers. In this case, the intensity of the phononfree NV 0 centers in the spectrum should increase significantly at 575 nm, as shown in Ref. [1]. Another explanation for the decrease in luminescence intensity may be the effect of burning out a dip in the spectrum of NV centers, found in Ref. [13]. Here, the processes of burning and erasing the dip in the spectrum of NV centers, and its dependence on the temperature of the diamond, were studied. It was found that photochromism occurs when the sample is illuminated by light with a wavelength shorter than 600 nm. It is possible that in our case, a similar phenomenon affects the decrease in luminescence intensity with an increase in pumping.

Conclusion
In connection with the obtained research results, the following conclusions can be drawn. The most probable reason for the decrease in luminescence intensity with an increase in pumping power is the absorption of the color centers in the excited state of the second photon and the transition of the centers to higher metastable levels.
The detected nonlinearities in the absorption of optical pumping energy and the concentration of color centers in the excited state require additional studies of the physics of these processes. In particular, calculations of possible radiation amplification in further studies should be carried out, taking into account the sample's own absorption spectrum. At high levels of pumping intensity, the effect of multiphoton and multistage absorption processes should be studied. Also, it is necessary to study various types of luminescence quenching, for example, concentration quenching, temperature quenching, quenching with nitrogen and other impurity defects, etc. Absorption losses in the field of laser radiation generation should be minimized, which in turn implies an increase in the requirements for the creation of synthetic diamonds and the selection of samples with identical characteristics.
Author contributions VPM: was involved in data processing, participated in the discussion of the results, wrote the main text of the manuscript. ENT: conducted experiments, participated in the discussion of the results. DEG: conducted experiments, processed data, participated in the discussion of the results, translated the manuscript. MAS: participated in the experiments, participated in the discussion of the results, prepared the results for the manuscript, edited the manuscript. EIL: participated in the discussion of the results, prepared the results for the manuscript, edited the manuscript AED: conducted experiments, processed data, participated in the discussion of the results. ADS: Changes in the saturation of the absorption of NV centers (black curve) and the relative luminescence output (ratio of luminescence intensity to pumping intensity) conducted experiments, processed data, participated in the discussion of the results.
Data Availability All data generated or analyzed during this study are included in this published article.