1. Since its discovery [1], infrared radiation was associated predominantly with relatively low-intensity thermal radiation of solids and linear absorption of optical phonon modes in transparent dielectric or semiconductor crystals [2]. Intense laser sources of mid-infrared radiation (3–8 μm, ≈40–100 THz) have been developed in the last decades, such as tunable parametric oscillators [3, 4] and laser generators with a fixed wavelength [5–7], which allow resonant multiphoton excitation of vibrational modes of molecules [8, 9], proteins, and even bacteria [10, 11]. Meanwhile, in wide-gap dielectric and semiconductor crystals, which are promising media for laser writing of optical elements in their bulk through the structural modification of the lattice or impurity defects [12–15], even intense mid-infrared femtosecond laser pulses do not have fundamental advantages over near-infrared pulses (0.8–1.4 μm, ≈200–400 THz). A reason is a giant (about three orders of magnitude) difference between absorption resonances of the electronic subsystem (the band gap width is ~1–10 eV, ~1 PHz) and ion vibrations (the energy of optical phonons of the crystal lattice or local vibrations of impurity atoms is ~0.01–0.1 eV, ~10 THz). In view of this circumstance, existing sources of mid-infrared femtosecond laser pulses do not affect lattice absorption but excite the electronic subsystem of insulators and semiconductors through the trap ionization mechanism or direct tunnel ionization, which is accompanied by impact ionization up to the optical breakdown (formation of a near-critical plasma) [16, 17].

Diamond is a unique optical- and infrared-transparent centrosymmetric wide-gap semiconductor (insulator) with the most rigid crystal lattice, where optical phonons have frequencies up to 40 THz [18]. Correspondingly, according to symmetry selection rules, intrinsic infrared lattice absorption in diamond in the two-phonon region (0.33 eV, 4–5 μm, 60–80 THz) becomes possible under the quasimomentum conservation condition [19]. In addition to linear two-phonon absorption at ~5 cm–1 (≈4 μm, optical biphonon) and 10–100 cm–1 (other pair combinations of optical phonons with short-wavelength acoustic phonons) in weal light fields, the multiphoton absorption of biphonon modes of diamond is possible in strong infrared laser fields; its characteristics are possibly similar to those for multiatomic molecules [8], but they have not yet been studied.

In this work, using the z-scan technique, we study the nonlinear resonant absorption of infrared femtosecond laser pulses in two-phonon absorption bands of synthetic boron-doped diamond (without significant impurity absorption) and estimate the possibility of structural modification of impurity center of synthetic diamond in this regime at higher intensities of laser pulses.

2. Nonlinear absorption in 4 × 4 × 0.3-mm type IIb diamond samples obtained by high-pressure–high-temperature (HPHT) method was measured. The infrared absorption spectrum of diamond is shown in Fig. 1a, where the two-phonon absorption region is given in gray, a laser radiation wavelength of 4673 nm used in the experiment is marked in red, and boron impurity absorption lines are indicated by black dotted lines. The optical image of the diamond sample is shown in the inset of Fig. 1a. A 3.5 × 2.2 × 0.9-mm type-Ib diamond sample, which was also obtained by the HPHT method and was colored in red after irradiation by a 3-МeV electron beam (\(5 \times {{10}^{{18}}}\) cm–2) and 30-min annealing at a temperature of 1200°C (see Fig. 1a), was used as the modified sample [20].

Fig. 1.
figure 1

(Color online) (а) Infrared absorption spectrum of the diamond sample, where two-phonon and impurity absorption regions are indicated; the inset shows the optical image of the sample. (b) Sketch of the experimental setup for the irradiation of diamond: (PM) power meter, (3D-MP) three-coordinate motorized platform, (PA) infrared polarizer, (Mirror) silver mirror, (OPA) optical parametric oscillator, (DFG) difference frequency generator, (Lens) lens, (Sample) diamond sample, and (TETA) laser source of femtosecond pump pulses.

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The sample was exposed to 4673-nm radiation (FWHM ≈ 80 nm), which was obtained by difference frequency generation in a AgGaS2 crystal. To obtain mid-infrared radiation, idler radiation from a PARUS optical parametric oscillator (Avesta, Russia)) was mixed with residual radiation of the first harmonic of a TETA pump laser (Avesta, Russia). The optical parametric oscillator was pumped at a power 5 W and a laser pulse repetition frequency of ν = 10 kHz. The FWHM duration of a 4673-nm mid-infrared laser pulse was τ ≈ 150 fs.

To measure nonlinear optical characteristics, laser radiation was focused by a CaF2 lens with a focal length of 40 mm (the radius of the focal spot at the 1/\({{e}^{2}}\) energy level was \({{w}_{0}} \approx 37.2{\kern 1pt} \)μm, the Rayleigh length is 930 μm). The repetition frequency of laser pulses when estimating multiphoton absorption coefficients was 10 kHz. The sample was displaced with a minimum step of 1 μm by a three-coordinate motorized platform (see Fig. 1b). The laser radiation power was measured by a 3A-P pyroelectric power meter (OPHIR Optronics Ltd., USA), and the energy of radiation incident on the sample was measured by an infrared polarizer.

The modification of diamond was carried out at a wavelength of 4673 nm through a mirror objective with a numerical aperture of NA = 0.28 at focusing in the bulk of the diamond sample at a depth of ~200 μm, point exposure times of 1, 10, 30, 60, 120, and 240 s, a pulse repetition rate of 10 kHz, and a fixed energy density of 2.3 J/cm2. The irradiated sample was characterized by means of visible transmission microspectroscopy with an MSFU-K (LOMO, Russia) spectrometer. Infrared spectra of the modified region were recorded on an FT-805 Fourier-transform infrared spectrometer with a Mikran-3 microscope (Simeks, Russia). Photoluminescence spectra were obtained on a Confotec MR520 3D scanning confocal microscope (SOL Instruments, Belarus).

3. The main method to estimate the multiphoton absorption coefficients is the measurement of the decrease in the energy of laser radiation transmitted through the studied medium in the pre-filamentation regime with the subsequent processing of dependences of nonlinear transmittance. This method is implemented using two main standard technique [21, 22]:

\( \bullet \) the z-scan technique with the open aperture: the measurement of the transmittance of the sample at a fixed radiation energy under the displacement of the sample along the axis of the focusing optical system,

\( \bullet \) the I-scan technique: the measurement of the transmittance of a fixed sample under the variation of the incident radiation energy.

The intensity of laser radiation transmitted through the sample with the thickness L in the case of n-photon absorption can be estimated as [23]

$$I(L) = \frac{{{{{(1 - R)}}^{2}}{{I}_{0}}\exp ( - \alpha L)}}{{{{{(1 + p_{0}^{{n - 1}})}}^{{\frac{1}{{n - 1}}}}}}},$$
(1)

where \(p_{0}^{{n - 1}} = (n - 1){{\beta }_{n}}{{L}_{{{\text{eff}}}}}{{(1 - R)}^{{n - 1}}}I_{0}^{{n - 1}}\), \({{\beta }_{n}}\) is the n-photon absorption coefficient of the medium, α is the linear absorption coefficient, Leff = (1 – exp(–(n – \(1)\alpha L)){\text{/}}(n - 1)\alpha \) is the effective multiphoton absorption length, R is the reflection coefficient at the air–diamond interface, \({{I}_{0}} = \frac{{2{{P}_{0}}}}{{\pi w_{0}^{2}}}\) is the peak intensity of laser radiation in the focal spot, P0 is the peak power of the laser pulse, \({{w}_{0}} = \frac{{\lambda f{\kern 1pt} '}}{{\pi w(0)}}\) is the radius of the focal spot, \(\lambda \) is the wavelength of laser radiation, \(w(0)\) is the radius of the laser beam in front of the focusing system, and \(f{\kern 1pt} '\) is the focal length of the focusing system.

The transmittance of the sample in the z-scan technique with an open aperture [24] can be obtained from Eq. (1) in the form

$$\begin{gathered} {{T}_{{{\text{OA}}}}}(z) \\ = \frac{1}{{{{{\left( {1 + (n - 1){{\beta }_{n}}{{L}_{{{\text{eff}}}}}{{{(1 - R)}}^{{n - 1}}}{{{\left( {\frac{{{{I}_{0}}}}{{1 + {{{\left( {\frac{z}{{{{z}_{R}}}}} \right)}}^{2}}}}} \right)}}^{{n - 1}}}} \right)}}^{{\frac{1}{{n - 1}}}}}}}, \\ \end{gathered} $$
(2)

where \({{z}_{R}} = {{n}_{0}}\frac{{\pi w_{0}^{2}}}{\lambda }\) is the Rayleigh length and n0 is the refractive index of the studied material. The Rayleigh length for the length with a focal length of 40 mm is 0.93 and 2.21 mm in air and in the material, respectively, which are much larger than a sample thickness of 0.3 mm, as assumed in the z-scan technique.

The multiphoton absorption coefficient of the studied sample was estimated from the measured transmittance of the diamond sample for focused ultrashort laser pulses. The approximation of the experimental data by Eq. (2) with various n values showed that the main mechanism of attenuation of 4763-nm ultrashort laser pulses in the considered intensity range in the studied sample is two-photon absorption (n = 2) with the coefficient β2 = (72 ± 7) cm/TW. This coefficient was estimated using the following parameters: the refractive index of diamond n0 = 2.378 at the wavelength \(\lambda = 4673{\kern 1pt} \) nm [25], the linear absorption coefficient \(\alpha = 10{\kern 1pt} \) cm–1 (see Fig. 1a), the reflection coefficient at the air–diamond interface \(R = {{\left( {\frac{{{{n}_{0}} - 1}}{{{{n}_{0}} + 1}}} \right)}^{2}} = 0.166\), and the linear transmittance \({{T}_{0}} = \exp ( - \alpha L) = 0.74\). We note that, according to the theory of multiphoton (non)resonant infrared absorption by the vibrational system of molecules, the degree of absorption nonlinearity is determined by the intensity of incident radiation and is limited by effects of anharmonicity, density of states, and dissociation [8].

Figure 2 presents the dependence of the transmittance on the displacement of the center of the sample from the focal plane of the lens along the optical axis experimentally determined using the z-scan technique and calculated by Eq. (2) for the parameters (а) P = 15.2 mW, P0 = \(\frac{P}{{\tau \nu }}\) = 10 MW, and I0 = 0.47 TW/cm2 and (b) P = 10.9 mW, P0 = \(\frac{P}{{\tau \nu }}\) = 7.2 MW, and I0 = 0.34 TW/cm2.

Fig. 2.
figure 2

(Color online) Transmittance versus the displacement of the center of the sample from the focal plane of the lens at the peak pulse power P0 = (а) 10 and (b) 7.2 MW.

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The intensity-dependent nonlinear transmittance for the case of two-photon absorption can be defined as [23]

$$T(I) = \frac{1}{{\sqrt \pi {{q}_{0}}}}\int\limits_{ - \infty }^\infty \ln (1 + {{q}_{0}}\exp ({{x}^{2}}))dx,$$
(3)

where q0 = β2Leff(1 – R)I.

Figure 3 shows the dependence of the transmittance of the sample on the intensity of laser pulses experimentally determined using the I-scan technique and calculated by Eq. (3) with the above estimate of the two-photon absorption coefficient.

Fig. 3.
figure 3

(Color online) Transmittance of the sample versus the laser radiation intensity.

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Femtosecond laser pulses with a wavelength in the region of the maximum intrinsic lattice absorption of diamonds (in the two-phonon region) can be used to modify diamonds, which was demonstrated for red HPHT diamond rich in nitrogen defects. Using a Fourier-transform infrared microspectrometer, we recorded absorption spectra of the ~200 × 200-μm region modified by the 4673-nm laser in four layers at a peak energy density of 2.3 J/cm2. The spectra were obtained with a 30 × 30-μm diaphragm placed on the microscope (see Fig. 4a) in the modified region and near it with averaging over five measurements. The analysis of infrared transmission spectra revealed a decrease in the 1450 cm–1 absorption center (H1a). The H1a center is characteristic of irradiated and annealed diamonds at temperatures below 1400°C and is strongly manifested in HPHT diamonds. It was shown in [26] that this defect corresponds to interstitial nitrogen [N2I]. Absorption at this peak decreases in the modified region from ~1.05 to ~0.75 cm–1.

Fig. 4.
figure 4

(Color online) (а) Infrared absorption spectrum of diamond in the modified region and near it, where the vertical strip marks the wavelength of laser action. The left inset shows the optical image of the diamond sample. The right inset presents in an increased scale the part of the spectrum indicated by the vertical arrow. (b) (Solid lines, left axis) Ratio of the transmittance spectra in the modified region to the corresponding spectra near it and (orange dashed line, right axis) the absorption spectrum of diamond. The optical microscopy image of the modified region presented between panels (a, b) was obtained in the transmission mode.

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To measure optical transmittance and photoluminescence in the bulk of the sample, we wrote a number of modified microregions at the same laser energy density 2.3 J/cm2 with various exposure times. Figure 4b shows (solid lines, left axis) the ratio of the transmittance spectra in the modified region to the corresponding spectra near it at various exposure times and (orange dashed line, right axis) the absorption spectrum of diamond in the visible band. In the laser action region, “translucence,” i.e., an increase in the transmittance was observed in the spectral range of 480–650 nm with the maximum near 580 nm (corresponding to the neutral, NV0, and negatively charged, NV–, nitrogen vacancy centers [18]). The optical microscopy image shown between Figs. 4a and 4b, which obtained in the transmission mode for the modified region, also demonstrates “bleaching” of diamond.

The resulting regions were also characterized using photoluminescence under excitation by cw laser radiation with wavelengths of 405 and 532 nm (see Fig. 5). Figure 5a presents the luminescence intensity map at a wavelength of 638 nm (NV–), where a decrease in the luminescence intensity in modified microregions is observed. The luminescence spectra (see Fig. 5b) also demonstrate a certain decrease in the luminescence intensity in the region of NV0 and NV– centers at all irradiation parameters compared to the intensity from unmodified diamond.

Fig. 5.
figure 5

(Color online) (a, c) Luminescence intensity maps at a wavelength of (a) 638 and (c) 504 nm under excitation at a wavelength of (a) 532 and (c) 405 nm. (b, d) Photoluminescence spectra in the irradiation region and near it under laser excitation at a wavelength of (b) 532 and (d) 405 nm.

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Figure 5c shows the luminescence intensity map at a wavelength of 504 nm and spectra in the modified region and near it (see Fig. 5d) obtained at an exposure time of 240 s under the excitation by 405-nm radiation. Both a strong inhomogeneity of diamond and an increase in the photoluminescence intensity at modified points near a wavelength of ~500 nm are seen in the intensity distribution map. luminescence spectra demonstrate an increase in the intensity in modified regions for wavelengths 480–520 nm associated with H4 (496-nm zero-phonon line (ZPL)) and H3 color centers (503-nm ZPL) and a simultaneous decrease in the luminescence intensity in the regions of NV0 (>575 nm) and NV– (>637 nm) centers. The above transformations of nitrogen defects in the direction of aggregation correlate with known thermal reactions [26]. In particular, it is known that NV centers in HPHT diamonds can be destroyed at high temperatures ~2300°C up to the complete disappearance, whereas Н3 and H4 centers are formed in synthetic diamonds at temperatures above 1800°C [27] and the H1a center begins to be transformed to other forms at temperatures above 1400°C [27]. Consequently, it can be assumed that NV centers under femtosecond infrared laser irradiation in this work become mobile through the pure mechanism and are joined into larger nitrogen complexes such as H3 or H4 centers. This mechanism fundamentally differs from mechanisms of modification of nitrogen impurity centers through femtosecond laser interband or intracenter electronic excitation in diamonds [28, 29] and can have promising applications in diamond photonics. However, direct experimental evidence of the thermal mechanism of transformation of nitrogen impurity centers and details of its implementation at the direct lattice absorption of intense mid- infrared laser radiation still have to be obtained.

4. To summarize, the nonlinear lattice absorption of ultrashort laser pulses with a wavelength in the region of intrinsic two-phonon absorption in diamond has been studied experimentally for the first time. It has been shown that the nonlinearity of the transmittance of diamond in this case is due to two-photon absorption. It has been demonstrated that the action of more intense femtosecond pulses with such a wavelength results in the local transformation of nitrogen defects.