1. Diamond is a promising material for different optoelectronic devices such as light-emitting diodes, lasers, and photodiodes. This is related to a number of advantageous technical characteristics of diamond [1–3]: (i) the thermal conductivity of diamond is higher than that of copper, which is usually used as a heat-sink material in laser systems (e.g., in disk lasers); (ii) diamond and especially its colorless varieties possess high transparency and optical durability in a broad spectral range from ultraviolet to mid-infrared; (iii) synthetic diamonds obtained by high pressure high temperature synthesis can be controllably doped with various chemical impurities (typically nitrogen) to measurable concentrations of 1–103 ppm (1017–1020 cm–3); (iv) it is possible to synthesize or select diamonds with a single dominant type of impurity color centers, so that centers of other types do not absorb in the emission range of the main center; (v) lasing in the blue, green, and red regions (N3, H3, and NV centers, respectively) of the RGB color gamut can be obtained; (vi) intracenter transitions have direct-gap character and feature high quantum efficiency of photoexcitation and emission (quantum yield ~0.1–1, cross sections on the order of 10–16 cm2), and, since all centers experience homogeneous broadening owing to the high stiffness of the diamond lattice, lasing can be tuned widely across the photoluminescence (PL) spectrum, usually up to 200 nm. Meanwhile, the photophysical parameters of various active media based on natural and synthetic diamonds (e.g., those with N3 and H3 centers, with zero-phonon lines (ZPLs) at about ≈415 and ≈503 nm, respectively [2]), including absorption and emission cross sections, nonradiative and radiative relaxation times, the population inversion of the laser transition and the corresponding gain, are estimated in [3–8] from data of a few indirect measurements of the PL decay rate carried out almost 40 years ago.

Here, using a specially selected sample of natural diamond with predominant H3 and H4 centers and a nearly continuously tuned PL excitation source utilizing frequency-doubled radiation from an optical parametric oscillator producing femtosecond pulses in the spectral range of 600–3000 nm, we measure the absorption cross section of these centers by the saturation of the PL yield as a function of the intensity of excitation at a wavelength of 470 nm and for the first time estimate their concentration.

2. The sample under study (see Fig. 1a) represents a 5 × 3 × 0.9-mm plate of natural IaAB-type diamond containing infrared-active A centers (which are formed by two neighboring nitrogen atoms in substitution positions, 2N) and B1 centers (which are formed by four neighboring nitrogen atoms in substitution positions surrounding a vacancy, 4NV) in concentrations of [A] = 374 ppm (7 × 1019 cm–3) and [B1] = 243 ppm (4.5 × 1019 cm–3), respectively, as well as low concentrations of B2 and H1a centers observed in vibrational infrared absorption spectra (see Fig. 1b) using an FT-805 infrared Fourier-transform spectrometer (Simex, Russia). At the same time, the optical spectra in the region of electronic transitions are dominated by H3 and H4 centers. This can be seen in the absorption spectra in the range of 400–500 nm recorded with an SF-2000 spectrophotometer (OKB Spectr, Russia) (see Fig. 1c) and PL spectra in the range of 500–550/650–750 nm recorded with a Confotec MR520 3D scanning PL/Raman confocal microscope–spectrometer (SOL Instruments, Belarus) (see Fig. 1d). According to the PL data, the concentration of H3 centers is more than an order of magnitude higher than that of H4 centers.

Fig. 1.
figure 1

(Color online) (a) (Top) Experimental setup for the PL spectroscopy of diamond with femtosecond laser excitation, (bottom left) the scheme of energy levels 1–3, and (bottom right) their kinetics. (b) First- and second-order infrared absorption spectra of the diamond plate under study, with the assignment of bands to infrared-active A, B1, B2, and H1a impurity centers (according to [2]). (c) Absorption spectrum of H3 and H4 centers in the sample in the range of 350–550 nm with indicated ZPLs (according to [2]). (d) Photoluminescence/Raman spectra of the sample excited by CW lasers with wavelengths of 405 and 532 nm at a temperature of –70°C; Raman lines and ZPLs for N3, H3, H4, NV0, and NV– centers are indicated (according to [2]).

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The spectral studies of PL from H3 and H4 centers were carried out using an optical setup (see Fig. 1a) based on a PARUS parametric oscillator (Avesta Project, Russia) emitting in the range of 600–2800 nm that is pumped by femtosecond pulses (250 fs, 1030 nm) from a THETA-20 Yb laser (Femtonica, Russia). A 0.5-mm-thick BBO nonlinear crystal was used for the frequency doubling of the PARUS output. Laser radiation at a wavelength of 470 nm (corresponding to the maximum linear absorption of H3 and H4 centers, see Fig. 1c) was loosely focused onto the diamond crystal by a micro-objective with NA = 0.1 (LOMO, Russia) in a spot with a radius of about 0.25 mm (at the 1/e intensity level) and then blocked by an LP500 long-pass filter (Fotooptik, Russia) and an FLP25-VIS-M linear polarizer (LBTEK, China) whose axis was set perpendicular to the direction of polarization of laser radiation. The passed PL/Raman radiation was focused by a fluorite (CaF2) lens with a focal length of 75 mm onto the slit of an ASP-150F spectrometer (Avesta Project, Russia) with the spectral range of 190–1100 nm and a resolution of 0.5 nm. The spectra (see Fig. 2) were collected for 1 s at a repetition rate of ultrashort laser pulses of 10 kHz; after taking each spectrum, the sample was shifted by 50 μm to a new position using a motorized translation stage. The incident pulse energy was varied using a neutral-density filter in the range of E = 0.03–2.5 µJ, so that the peak fluence, intensity, and exposure at the sample are 10–1000 µJ/cm2, 0.1–7 GW/cm2, and (2.5–250) × 1013 photon/cm2, respectively.

Fig. 2.
figure 2

(Color online) Photoluminescence spectra of H3 and H4 centers divided by energies E (listed in the legend) of pump pulses with a wavelength of 470 nm. The FWHM Δλ of the H3 ZPL (503 nm) is shown. The ZPL of the H4 center (496 nm) is believed to be buried in the background signal due to its low intensity (cf. Fig. 1).

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In addition, the spectrally resolved PL kinetics of the diamond sample was studied using a Hamamatsu C5680 streak camera with an S-20 photocathode (sensitivity range 200–900 nm) mounted in the focal plane of an Acton SpectraPro 2500i spectrometer. Photoluminescence was excited by pulses of the second harmonic of a Coherent Mira 900D mode-locked Ti:sapphire laser with a duration of 3 ps and a central wavelength of 400 nm. The laser pulse repetition rate of 180 kHz was set by a pulse picker, and the average power of the beam incident on the sample was 10 µW. Photoluminescence was excited and collected in a confocal scheme with a 5× micro-objective (NA = 0.12). The spectral and time resolution was about 1 nm and 30 ps, respectively.

3. The dynamic PL spectra obtained under pulsed excitation with different pulse energies E are shown in Fig. 2. The PL intensity Φ was divided by E for convenient representation of the entire set of spectral curves and for their comparison with the PL spectrum obtained under CW excitation at 405 nm (see Fig. 1d). The spectra feature strong H3 ZPL, its weak low-frequency anti-Stokes replicas (<500 nm), and strong low- and high-frequency Stokes replicas (>500 nm). At first sight, the ratio Φ/E at the PL peak near 540 nm reflects the relative PL excitation efficiency, disregarding the pump energy: it grows rapidly as E increases up to E = 0.25 µJ, after which the growth slows down considerably (see Fig. 2).

The normalized PL peak intensity as a function of the pump energy E or intensity I saturates above ES ≈ 0.16 µJ (see Fig. 3). This behavior can be interpreted by considering the three-level scheme of PL excitation in H3 and H4 centers shown in Fig. 1a, where levels 1–3 represent, respectively, (i) the zero vibrational level of the nondegenerate electron ground state (with the energy ε1 = 0); (ii) the nondegenerate resonant excited electron state, whose energy equals the ZPL energy plus the energy of phonon replicas associated with H3 and H4 centers, i.e., of optical phonons of different symmetries and vacancy-related local modes [2] (ε2 ≥ εZPL, see the absorption spectrum in Fig. 1c); and (iii) a set of lower-lying nonresonant vibrationally excited states of the same excited term (εZPL < ε3 < ε2), which become populated in the course of nonradiative relaxation from level 2 owing to phonon–phonon anharmonicity and vibrational energy transfer beyond the H3 and H4 centers and from where Stokes-shifted transitions to vibrationally excited states of the ground term finally occur at the spontaneous emission rate γ31 (see Fig. 1a). The corresponding system of kinetic equations can be approximately written as

$$\begin{gathered} \frac{{d{{n}_{1}}}}{{dt}} \approx - {{\kappa }_{{12}}}(I){{n}_{1}} + ({{\gamma }_{{21}}} + {{\kappa }_{{21}}}(I)){{n}_{2}} + {{\gamma }_{{31}}}{{n}_{3}}, \\ \frac{{d{{n}_{2}}}}{{dt}} \approx {{\kappa }_{{12}}}(I){{n}_{1}} - ({{\gamma }_{{21}}} + {{\kappa }_{{21}}}(I) + {{\gamma }_{{23}}}){{n}_{2}}, \\ \frac{{d{{n}_{3}}}}{{dt}} \approx {{\gamma }_{{23}}}{{n}_{2}} - {{\gamma }_{{31}}}{{n}_{3}}, \\ \end{gathered} $$
(1)

where n1 + n2 + n3 = n0. Here, only radiative processes are taken into account for transitions 1 → 2 and 2 → 1, which include relatively rapid stimulated absorption and emission processes (note that the rate constants κ12(I) and κ21(I), related to the Einstein coefficient, are equal if the degeneracies of levels 1 and 2 are the same), as well as much slower spontaneous emission processes (with typical rate constants of γ21, γ31 ~ 109 s–1). Taking into account the large detuning of the pump   wavelength from the corresponding ZPL (470 nm/2.66 eV versus 496 nm/2.52 eV and 503 nm/2.48 eV) and the relatively small width of the pump line (~3–4 nm), it is safe to assert that, at room temperature, the vibrationally excited states of the same excited electron term will be both optical phonons with an energy of about 0.16 eV and phonons of lower energies [2]. Under these conditions, rapid nonradiative relaxation from level 2 to the lower-lying nonresonant vibrationally excited states of the same term (nominal level 3) is in direct analogy with the symmetrical decay of optical phonons into a pair of acoustic phonons with half the energy and opposite wave vectors (Klemens mechanism [9]). Accordingly, we can take the spontaneous decay rate of the optical phonon in diamond at room temperature (decay time on the order of 1–10 ps) to be the lower bound for the nonradiative relaxation rate. We can also estimate this rate by invoking the relaxation rate between vibrational modes (V–V) γ23 ~ 1013 s–1 ≫ γ21 [10] since color centers may be considered as quasi-molecular entities embedded in the diamond lattice. We disregard stimulated radiative transitions from level 2 to higher-energy electron states (e.g., to the conduction band [5]) and spontaneous radiative transitions from level 3 to level 1.

Fig. 3.
figure 3

(Color online) Log–log plots of (right axis) the photoluminescence intensity Φ and (left axis) the ratio Φ/E at a wavelength of 540 nm versus the pump pulse energy E and intensity I (bottom and top axes, respectively). Linear fit is given.

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Under these conditions, in the case of moderate intensities and n1 ≈ n0 ≫ n2, we can retain only the leading terms in system (1) and, considering a model rectangular pulse (I = const), obtain

$$\frac{{d{{n}_{1}}}}{{dt}} \approx - {{\kappa }_{{12}}}(I){{n}_{0}},$$
$$\frac{{d{{n}_{2}}}}{{dt}} \approx {{\kappa }_{{12}}}(I){{n}_{0}},$$
(2)
$$\frac{{d{{n}_{3}}}}{{dt}} \approx {{\gamma }_{{23}}}{{n}_{2}},$$

where, for κ12(I) = κ21(I) ∝ I, n1 decreases linearly during the pulse as n0 – С\(It\), n2 increases linearly (∝It), and, thus, n3 increases quadratically (\( \propto {\kern 1pt} \int {{n}_{2}}(t)dt\, \sim \) It2) and so does the value of Φ, which follows n3. In the case of photoexcitation by rectangular pulses, quantities t and I manifest themselves in similar ways, so it may be expected that model dependences of n1, n2, n3, and Φ on I have a similar shape (see the quasi-linear region and the plateau in the dependence of Φ/E on I in Fig. 3).

Now, it should be expected that the resonant transition is saturated with an increase in I, e.g., when κ12(I)n1S ≈ κ21(I)n2S, κ12(I), κ21(I) ≫ γ23, and n1S ≈ n2S = nS. Under these conditions, system (1) can be written as

$$\frac{{d{{n}_{1}}}}{{dt}} \approx - {{\kappa }_{{12}}}(I){{n}_{{{\text{1S}}}}} + {{\kappa }_{{21}}}(I){{n}_{{{\text{2S}}}}} = 0,$$
$$\frac{{d{{n}_{2}}}}{{dt}} \approx {{\kappa }_{{12}}}(I){{n}_{{{\text{1S}}}}} - {{\kappa }_{{21}}}(I){{n}_{{{\text{2S}}}}},$$
(3)
$$\frac{{d{{n}_{3}}}}{{dt}} \approx {{\gamma }_{{23}}}{{n}_{{\text{S}}}};$$

i.e., n3 increases linearly with t, and so does the value of Φ, and the same is true for the dependence of these quantities on I. Saturation will manifest itself as a plateau in the dependence of Φ/E on I in Fig. 3, where the onset of the plateau corresponds to the saturation intensity IS ≈ 0.3 GW/cm2 (exposure 1.8×1014 photon/cm2). We note that diamond samples can hardly withstand irradiation with nanosecond laser pulses of this intensity level without severe damage; i.e., femtosecond or picosecond laser pump is necessary to attain the saturation of absorption by H3 and H4 centers. Then, using the saturation criterion [11]

$$\frac{{2{{\sigma }_{{12}}}{{I}_{{\text{S}}}}\tau }}{{\hbar \omega }} = 1,$$
(4)

where τ is the laser pulse duration, we can estimate the absorption cross section σ12 for the resonance transition at σ12 ≈ 2.7 × 10–15 cm2 and verify that the rate of radiative transitions 1 → 2 and 2 → 1 in the saturation regime (σ12IS/\(\hbar \)ω) ~ 1013 s–1 becomes comparable to the characteristic rates γ23 of vibrational relaxation, so that the values of n1 and n2 become close. It is noteworthy that the obtained value of σ12 is about an order of magnitude larger than the emission cross section 1.6 × 10–16 cm2 [5, 6] determined from the measurements of PL kinetics almost 40 years ago.

For comparison, close values of the transition cross sections for H3 and H4 centers in this spectral range are obtained from PL kinetics measurements under weak pumping at a wavelength of 400 nm (see Fig. 4), to a certain extent reproducing the conditions of CW pumping at a wavelength of 405 nm (see Fig. 1d). For this reason, it is possible to detect the PL of H4 centers and study its kinetics. In particular, the Φ(t) dependences for H3 and H4 centers are better approximated by two exponentials with decay times τem, 1 = (7.5 ± 0.2) ns and τem, 2 = (22.7 ± 0.5) ns and τem, 1 = (13.1 ± 0.5) ns and τem, 2 = 51.0 ± 0.8 ns, respectively. These times differ noticeably from the values τem = (16.7 ± 0.5) ns and τem = (19.1 ± 1.0) ns measured for H3 and H4 centers, respectively, in much earlier studies [5, 6, 12]. Still, taking into account that the degeneracy of levels 1 and 2 in H3 centers is g1, 2 = 1 [5] and Δλ(H3) ≈ 3 nm (see Fig. 2), the emission cross section corresponding to τem, 1

$${{\sigma }_{{{\text{em}}{\text{,1}}}}}(\lambda ) = \frac{{{{g}_{2}}}}{{{{g}_{1}}}}\frac{{{{\lambda }^{4}}}}{{4\pi }}\frac{1}{{\pi c\Delta \lambda {{\tau }_{{{\text{em}}{\text{,1}}}}}}}$$
(5)

is σem, 1 ≈ 2 × 10–15 cm2, which is in good agreement with σ12 ≈ 2.7 × 10–15 cm2.

Fig. 4.
figure 4

(Color online) Semi-log plots of the time dependences of the photoluminescence intensity for the main H3 center and secondary H4 center in diamond with biexponential fit, obtained by linearization. The inset shows a color wavelength–time map of the photoluminescence kinetics of these centers.

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On the other hand, the knowledge of σ12 makes it possible to estimate the concentration of H3 and H4 centers, taking into account that their optical characteristics are close to each other [5, 6] and the unknown contributions of these centers to the optical absorption coefficient (in particular, at a pump wavelength of 470 nm, Fig. 1c) follow the ratio of the concentrations [A] and [B1] of physically similar optically inactive centers [12, 13], which are measured using infrared spectroscopy (see Fig. 1b):

$$\frac{{{{\alpha }_{{{\text{H}}4}}}}}{{{{\alpha }_{{{\text{H}}3}}}}} = 0.25\frac{{[{\text{B}}1]}}{{[{\text{A}}]}}.$$
(6)

With the known concentrations [A] = 374 ppm and [B1] = 243 ppm, the ratio of the contributions from H3 and H4 centers to α(470 nm) ≈ 24 cm–1 (see Fig. 1c) is 6 : 1; i.e., αH3(470 nm) ≈ 20.5 cm–1 and αH4(470 nm) ≈ 3.5 cm–1. Then, taking into account the dominance of the H3 center in the absorption and, apparently, PL of the diamond sample, we can, with good accuracy, attribute the measured absorption cross section to this center (i.e., σH3(470 nm) ≈ σ12) and estimate its concentration from

$$[{\text{H3}}] = \frac{{{{\alpha }_{{{\text{H3}}}}}(470\,\,{\text{nm}})}}{{{{\sigma }_{{{\text{H3}}}}}(470\,\,{\text{nm}})}}.$$
(7)

In this way, we find [H3] ≈ 7 × 1015 cm–3, i.e., on the order of 0.1 ppm. Taking into account the similarity of the optical properties of H3 and H4 centers [5], we can, using αH4(470 nm) ≈ 3.5 cm–1, in the same way find [H4] ≈ 1 × 1015 cm–3.

Note that the proposed approach to estimate the concentration of H3 centers evidently turns out to be much more versatile and convenient than the one based on measuring the ZPL optical absorption coefficient of these centers (which is unresolved in room-temperature spectra shown in Fig. 1c) according to [12, 13]

$${\text{[H3]}} = (0.67{-} 0.83) \times {{10}^{{15}}}\int {{\alpha }_{{{\text{ZPL}}}}}(\lambda )d\lambda .$$
(8)

The latter method requires that the H3 ZPL be well resolved in the absorption spectrum, which may be possible at liquid-nitrogen temperatures, when the FWHM of this line in the PL spectra of the sample excited at 405 nm decreases to 0.5–0.6 nm (see Fig. 1d).

4. In conclusion, we have studied for the first time the PL spectra of the H3 and H4 centers in diamond excited within their linear absorption band by femtosecond laser pump pulses with a wavelength of 470 nm and varying intensity. We have found that the behavior of the PL intensity as a function of the pump intensity changes from quadratic to linear at a certain pump level. This is associated with the saturation of resonant absorption and have been used for the first time to estimate the process cross section and the individual concentrations of absorbing H3 and H4 centers in diamond.