Abstract

Ultrafast visible-range excitation in natural diamond produces UV-VIS A-band photoluminescence with marginal zero-phonon line and intense regular multi-peak optical-phonon progression. The A-band photoexcitation can occur via band-center, center-center and band-band transitions, being related to two-photon (center-terminated), or interband three-photon and impact-ionization processes. Minor thermal pre-heating of the diamond (< 55 °C) demonstrates strong damping of A-band progression intensities with two different activation energies.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Luminescence is well-recognized ultrasensitive method of tracking numerous optically-active impurities in diamonds, being realized in cathodo-, photo-, thermostimulated and other luminescence spectrometers (for large bibliography, see [1]). Usually, it is acquired as photoluminescence (PL), providing broadly spectrally-tunable excitation and its typically 200–300-nm wide red-shifted Stokes emission [2,3]. In the PL spectra, zero-phonon line (ZPL) commonly appears as the most prominent (intense) one, while the following Stokes phonon progressions exhibit much lower intensities [4].

Recently, broadband structured PL spectra were detected in natural diamonds excited by near-IR (NIR, 1030 nm) and visible-range (515 nm) ultrashort (femtosecond, fs) laser pulses [5,6]. Comparing to previous non- and resonant ultrashort-laser photoluminescence studies in diamond [79], intense UV-visible [5] or UV-NIR [6] PL bands of diverse nitrogen- and vacancy-based optical centers almost without ZPL and very prominent phonon progressions, involving different bulk or local atomic vibrational modes [1,4]. These PL photoexcitation regimes hold a promise on micro-scale mapping of diverse nitrogen centers and, potentially, other impurities in natural and synthetic diamonds, which may be differently distributed along different crystallographic axes and in different growth zones [10]. However, despite the obvious perspectives of the revealed PL photoexcitation regimes by fs-lasers [5,6], photo-physical mechanisms of ultrafast excitation in host diamond and its buried optical centers, as well as the origin of the structured PL spectra were not studied in details yet.

In this study, we explore ultrafast 515-nm intensity-dependent PL photoexcitation in a natural diamond in order to reveal its atomistic origin and underlying photo-processes, coupling laser radiation and bulk transient electron-hole plasma to its electronic states. Pre-heating of the diamond is utilized to enlighten through the thermal effect the fine structure of the related vibronic progressions in the PL spectra, as atomic-scale probe of photoexcitation conditions.

2. Experimental details

Ultrashort-pulse laser PL photoexcitation of the natural colorless diamond (direct bandgap - 6.5 eV [11], mass – 14 mg, dimensions - 2 × 2×1 mm3) brick occurred in the absorption band near ≈ 30 000 cm−1 (Fig. 1(a)), related to optical color centers in the bandgap [1]. According to IR-transmission spectroscopy (Fig. 1(b)) and the common classification [12], this nitrogen-doped (>700 ppm) IaAB-type diamond contained about 650 ppm of non-luminous low-aggregation A-centers (substitutional doublets 2N) with their IR-peak at 1282 cm−1, minor concentration (≈50 ppm) of B1-centers (4NV) with their peak at 1175 cm−1 and marginal B2-centers (platelets) (peak at 1365 cm−1 [1]). 10-kHz train of 300-fs, 515-nm (≈19000 cm−1) pulses from Yb-doped fiber laser Satsuma at different pulse energies E = 20–400 nJ were focused by a 0.3-NA Nikon micro-objective into a 3-micron (1/e-intensity level) focal spot inside the diamond through its polished side windows (Fig. 1(c)). The resulting room-temperature photoluminescence (PL) was collected by a 0.2-NA UV-transparent quartz/fluorite objective and then focused, as a luminous track, onto an input slit of a broadband spectrometer ASP-150F (spectral resolution – 0.5 nm, non-gated CCD-array, Fig. 1(a)), or onto a color camera (Fig. 1(c), inset). In temperature-dependent PL studies the laterally thermally-isolated diamond sample was resistively heated in 5°C steps from 24 till 55°C, with the stationary sample temperature measured at the maximal fs-laser intensity I0 ≈ 16 TW/cm2 both by a built-in thermocouple and a remote directional-sampling digital infrared thermometer CA380 (CASON, operation range: –32°C - 380°C) with accuracy ≈0.3 °C.

 figure: Fig. 1.

Fig. 1. (a) UV-near-IR transmittance and absorption coefficient of the natural diamond with the positions of one- (ω) and two-photon (2ω) excitation. (b) Optical density of IaAB diamond (curve 3, left axis) and its photoluminescence spectra (right axis) excited at peak laser intensities of 0.5 (curve 1) and 3.1 (curve 2) TW/cm2, with the expected position of two-photon excitation (2ω). (c) IR spectrum with the spectral assignment of A and B1,2 bands. (d) Excitation/PL collection layout. Inset (top): color image of the luminous track in the focal region inside the diamond (image size – 300 × 60 microns).

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3. Experimental results and discussion

3.1 Assignment of PL origin

First, we have analyzed photo-physical mechanisms of the observed room-temperature UV-vis structured (multi-peak, visible peak numbers N’) PL (Figs. 1(b),2) by varying incident fs-laser intensity. These structured PL spectra were not related to interference of the PL radiation in the brick along the axis of its collection because of temporal PL incoherence. In principle, one can decompose the acquired PL spectra, similarly to [5], into a series of nine prominent Lorentzian lines with the intensity Φ1−9, full-width at half maximum (FWHM) Γ1−9, and inter-peak spacing Ω1−9 (Fig. 2), where the inter-peak spacing (phonon energy) weakly varies vs. N, but in this study, for the sake of accuracy, we analyzed the spectrally-integrated (300–550 nm, 20 000 −30 000 cm−1) A-band intensity ΦΣ.

 figure: Fig. 2.

Fig. 2. Multi-peak PL spectrum (dark pints) decomposed into colored and numbered eleven fitting Lorentzian lines and their cumulative violet spectrum, the possible A-band central position [24,25,29] and mirror-like anti-Stokes absorption spectrum (dashed curve). The green tringles show the single-photon and double-photon (TPA) laser energy. The pump peak was almost totally attenuated by a notch filter.

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The PL origin can be identified as A-band with its peak at 415–445 nm [1,13], which is especially very intense in low-nitrogen diamonds because of its strong nitrogen A-center damping [14,15]. The corresponding PL excitation band is known to be symmetric to A-band, occupying the range of 3–4 eV [16]. Meanwhile, in contrast to our room-temperature observation of the multi-peak PL spectra (Figs. 1(b),2), commonly this band was observed as structureless even at low temperature in very perfect diamonds, but was thought, e.g., to be a vibronic side-band of a center with ZPL at about 3.0 eV, where the ZPL is not observed due to the very large Huang-Rhys factor of the center and the strong nonhomogeneous stress around dislocations [16]. In our case, the structured PL band spectrum can be approximated by a standard intensity distribution for phonon progression with the peak number N (single-mode coupling) [4]

$$\Phi (N) \propto {{{e^{ - S}}{S^N}} \over {N!}},$$
where the Huang-Rhys factor S ≈ 6, evaluated for the three most intense peaks N’ = 4’−6’, using Eq. (1), since high- and low-N peaks are too low in their intensity to be accurately acquired. For this reason, the all actual numbers N could be shifted by some integer number regarding N’ assigned for the visible peaks.

Importantly, A-bands may have different nature [1]. One of the A-band model is radiative recombination at dislocations, observed as a relatively narrow A-band peaked at 440 nm in low-nitrogen type II diamonds. There are different opinions about the dislocation-related centers: donor-acceptor (D-A) pairs decorating dislocations [13], vacancies bound to dislocations [17,18], dislocations decorated with D-A pairs [19], on pure non-decorated dislocations [2023]. The second model of the broad A-band with a maximum at 480 nm observed in natural type I diamonds is intra-center transitions at the B2-centers (platelets). Finally, the A-band is also assigned to electron-hole recombination at deep centers, the energy levels of which lie in the middle of the bandgap [24,25]: EA = (Eg/2) ± 0.75 eV, occurring through the formation of free excitons [26] as a two-stage process [27,28], but could be related to the 4 eV band (possibly a transition from the conduction band to the dislocation band localized at about 1.8 eV above the valence band [29]. In our study, the observed A-band extends in the range of 2–4 eV with its peak at ≈3 eV at the presence of high (650 ppm) A-center density and marginal B2-center density in the high-nitrogen natural diamond, and exhibits the highly structured character of the vibronic progression. Hence, in some aspects it resembles the previously observed A-bands, but in other aspect strongly differs and requires its further studies.

3.2 Ultrafast photoexcitation mechanisms

Both intra-center and interband ultrashort-pulse laser photoexcitation paths were recently demonstrated [6,30,31] via 1) direct external (valence band-excited state of the center, Fig. 3(a)) or 2) internal (ground state- excited state of the center, Fig. 3(b)) excitation of intra-gap optical centers, or 3) coupling bulk electron-hole plasma to intra-gap optical centers via relaxation of photoexcited electrons from the conduction band into excited center states (holes - from the valence band into the ground center state, respectively), as shown in Fig. 3(c).

 figure: Fig. 3.

Fig. 3. Possible two-photon external (a) and internal (b) intra-gap and three-photon interband (c) excitation schemes of the A-center photoluminescence. The green arrows show the laser photons, blue and red arrows - actual and possible photoluminescence, curves arrows – non-radiative relaxation, dark and light circles – electron and hole, respectively.

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Considering the first and second PL excitation paths (Figs. 3(a),(b)), the corresponding basic PL excitation band is known to be symmetric to A-band, occupying the range of 3–4 eV [16], with the sample-specific additional UV excitation ranges > 4.6 eV (C-centers in type Ib diamonds) and > 5.2 eV (B2-centers) [1]. Then, direct two-photon absorption (Fig. 1(b), TPA, total energy ≈ 2 × 19 × 103 cm−1) provides excitation in the former range. In this case, the PL yield Φ as a function of the incident peak fs-laser intensity I0 and the transient occupation of the excited state ρexc regarding its maximum occupation ρ0 reads

$$\frac{{d{\rho _{exc}}}}{{dt}} = {\sigma _2}{I^2}({{\rho_0} - {\rho_{exc}}} )- \frac{{{\rho _{exc}}}}{\tau },\;\;\Phi \propto \int {{\rho _{exc}}} dt,$$
where the consequent terms describe the rates of the two-photon photoexcitation at the instantaneous laser-field intensity I with the cross-section σ2, and the non-radiative relaxation of the excited state with the time constant τ. At lower laser intensities, when the saturation of the excited state is negligible, one can expect the PL yield Φ scaling as I02 in the weak/medium excitation regime (ρexc « ρ0), while at higher laser intensities the saturation effects will diminish this dependence toward linear or sub-linear relationships. Also, for the strong observed PL bands, one can expect minor non-radiative relaxation channels.

In our experimental PL measurements, the spectrally-integrated (300–550 nm) PL intensity ΦΣ presented versus the peak intensity I0 (Fig. 4), demonstrated in lgΦΣ-lgI0 coordinates a series of slopes, gradually changing from ≈3.1 through 2.0 till ≈0.8. The medium- and high-intensity parts of the dependence are in the perfect agreement with the trend ∝I02 and its (sub)linear continuation predicted above, potentially indicating that both the two-photon external and internal intra-center excitation (the internal photoexcitation paths #1,2 in Figs. 3(a),(b)) appear to be quite possible in our experiments. However, the initial, lower-intensity higher slope of 3.1 doesn’t fit into this picture, implying the potential interband photoexcitation path (path #3 in Fig. 3(c)). Actually, at different, lower focusing NA=0.25, the same laser wavelength and pulsewidth our recent experiments demonstrated the continuous variation of such nonlinearity slopes from 12 to 1 versus the pulse energy increasing in the same range (not presented here).

 figure: Fig. 4.

Fig. 4. Spectrally-integrated (the entire A-band, spectral range ≈300–550 nm) PL intensity versus incident pulse energy E (bottom axis) and peak intensity I0 (top axis), with their corresponding sectional linear fits in lgΦi-lgE coordinates in the different ranges.

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As a result, alternatively one can consider coupling of transient dynamics of the fs-laser excited bulk electron-hole plasma to the (non)radiative dynamics in the center underlying the A-band photoluminescence (path #3 in Fig. 3(c)), with some relationship ρexcCρe,h (where C is some numerical factor « 1) between the transient EHP density and the saturable population of the excited state of the center. One can consider three-photon inter-band photoexcitation of electron-hole pairs via direct transitions (3 × 2.4 eV > 6.5 eV in the Γ-point and around [11]) and their radiative/non-radiative “band-to-state” evolution into the enter excited state (electron) and ground state (hole); radiative or non-radiative recombination transition of electron from the ground state to the valence band (Fig. 3(c)). This related EHP dynamics in diamond, based on three-photon e-h pair generation can be described by the kinetic rate equation for EHP density ρe,h [32], and the related expression for the PL yield Φ

$$\frac{{d{\rho _{e,h}}}}{{dt}} = {\sigma _3}{I^3} + \alpha I{\rho _{e,h}} - \gamma \rho _{e,h}^3 - \beta \rho _{e,h}^2 - \frac{{{\rho _{e,h}}}}{{{\tau _{ST}}}},\;\;\Phi \propto \int {\beta \rho _{e,h}^2} dt$$
where the consequent terms describe the rates of the three-photon internal ionization at the instantaneous laser intensity I with the cross-section σ3, impact ionization rate by hot carriers with the coefficient α, three-body Auger recombination with the coefficient γ, much slower radiative two-body recombination with the coefficient β, measured in this study as the time-integrated PL yield, and the non-radiative carrier trapping process with the time constant τST.

According to our generalized analysis, in low-intensity fs-laser photoexcitation regime, the photogeneration rate is low and the resulting low-density EHP (ρeh « 1020 cm−3) relaxes via the fast trapping of near-edge free carriers, accompanied by slow radiative recombination. In terms of EHP dynamics, the PL yield exhibits the steep raise in this regime

$$\frac{{d{\rho _{e,h}}}}{{dt}} = {\sigma _3}{I^3} - \beta \rho _{e,h}^2 - \frac{{{\rho _{e,h}}}}{{{\tau _{ST}}}},\mathop {}\limits_{} \;\;{\sigma _3}{I^3} \approx \frac{{{\rho _{e,h}}}}{{{\tau _{ST}}}},\mathop {}\limits_{} \;\;{\rho _{e,h}} \propto I_0^3,\;\;\;\Phi \propto \int {\beta \rho _{e,h}^2} dt \propto I_0^6.$$

At higher fs-laser intensities, the photogeneration rate could be balanced by radiative recombination in EHP

$$\frac{{d{\rho _{e,h}}}}{{dt}} = {\sigma _3}{I^3} - \beta \rho _{e,h}^2,\mathop {}\limits_{} \;\;{\sigma _3}{I^3} \approx \beta \rho _{e,h}^2,\mathop {}\limits_{} \;\;{\rho _{e,h}} \propto I_0^{3/2},\;\;\;\Phi \propto \int {\beta \rho _{e,h}^2} dt \propto I_0^3,$$
indicating the third-power PL yield. Nevertheless, in the next, medium-intensity photoexcitation regime ultrafast three-body Auger recombination limits the resulting medium-density EHP (ηeh∼1021 cm−3) and related PL yield as follows
$$\frac{{d{\rho _{e,h}}}}{{dt}} = {\sigma _3}{I^3} - \gamma \rho _{e,h}^3 - \beta \rho _{e,h}^2,\mathop {}\limits_{} \;\;{\sigma _3}{I^3} \approx \gamma \;\;\rho _{e,h}^3,\mathop {}\limits_{} {\rho _{e,h}} \propto I_0^1,\;\;\;\Phi \propto \int {\beta \rho _{e,h}^2} dt \propto I_0^2.$$

Finally, in the high-intensity bound dense EHP (ρeh∼1021 - 1022 cm−3) supports both predominating free-carrier absorption and related impact ionization, hindered by the counteracting conjugated three-body Auger recombination, yielding in even slower raise of EHP density versus I0 (E) and similar trend for the PL yield

$$\frac{{d{\rho _{e,h}}}}{{dt}} = \alpha I{\rho _{e,h}} - \gamma \rho _{e,h}^3 - \beta \rho _{e,h}^2,\alpha I{\rho _{e,h}} \approx \gamma \;\;\rho _{e,h}^3,{\rho _{e,h}} \propto I_0^{1/2},\;\;\;\Phi \propto \int {\beta \rho _{e,h}^2} dt \propto I_0^1.$$

Hence, these predicted external photoexcitation regimes via coupling of the electron-hole plasma to the A-band related intra-gap optical center also demonstrate reasonably good qualitative agreement with our experimental observations presented in Fig. 4 and previous studies of EHP dynamics in transparent dielectrics [3335]. Meanwhile, the EHP-center coupling mechanisms, bringing electrons and holes to the excited and ground center states, respectively, was not specified above and should definitely include the saturation effect for the latter states at ρexc∼re,h, in the context of the low measured intra-gap optical absorption coefficients (∼1–10 cm−1, Fig. 1(b)). Regarding more firm distinguishing the considered interband and intragap [7] photoexcitation paths for the A-band photoluminescence in the medium- and high-intensity ranges, additional focused studies are still required.

3.3 Thermal impact on PL spectral parameters

The related stationary resistive heating of the diamond from 24°C till 55°C significantly modified the basic PL spectral characteristics. The spectrally-integrated PL intensity ΦΣ was damped very strongly versus the increasing temperature (Figs. 5(a),(b)), in god agreement with the previous studies for A-band PL both in natural and synthetic diamonds. Specifically, it is well known that in most natural diamonds the A-band intensity also falls almost to zero by a temperature above 400 K [28,3639]. Such strong hysteresis-like temperature dependence [13] indicates thermally-activated process [40] such as either 1) thermal dissociation of the laser excited center state, or, for the crossing potential curves, 2) thermally-induced structural transformation from the excited state, or 3) non-radiative transition to the high vibrational levels of the ground state. The corresponding activation energy was previously determined as 0.3 eV [16], while in our study for the dependence lnΦΣ−1/kBT we derived two – low- and high-temperature – activation energies of ≈0.2 and ≈1.4 eV (Fig. 5(c)), respectively, which are yet to be understood.

 figure: Fig. 5.

Fig. 5. (a) Multi-peak A-band PL spectra at different sample temperatures, where the pump peak was almost totally attenuated by a notch filter. (b) Temperature dependences of integrated PL intensity ΦΣ. (c) lnΦΣ - 1/kBT curves and damping energies of 1.4 and 0.2 eV as their linear slopes.

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Hence, the observed and identified strong pre-heating effect opens the way to strong manipulation by the A-band photoluminescence and its control in optical tracing of PL-marked natural and synthetic diamonds [41].

4. Conclusions

Ultrafast visible-range laser excitation of nitrogen centers in natural diamond produces UV-vis A-band photoluminescence with marginal zero-phonon line and intense regular multi-peak optical-phonon progression. The A-band photoexcitation could occur via three possible mechanisms, involving band-center, center-center and band-band transitions, and related to two-photon (center-terminated), or interband three-photon and impact-ionization processes. Similarly to the previous stationary PL studies, in our case of the ultrashort-pulse laser PL excitation the minor thermal pre-heating of the diamond (< 55 °C) demonstrates the expected strong damping of progression intensities of the A-band, but with two different activation energies.

Funding

Russian Science Foundation (21-79-30063).

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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References

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  1. A. M. Zaitsev, Optical Properties of Diamond: A Data Handbook (Springer Science & Business Media, 2013).
  2. H.-C. Lu, M.-Y. Lin, S.-L. Chou, Y.-C. Peng, J.-I. Lo, and B.-M. Cheng, “Identification of nitrogen defects in diamond with photoluminescence excited in the 160–240 nm region,” Anal. Chem. 84(21), 9596–9600 (2012).
    [Crossref]
  3. Y. Luo and C. M. Breeding, “Fluorescence produced by optical defects in diamond: measurement, characterization and challenges,” Gems Gemol. 49(2), 82–97 (2013).
    [Crossref]
  4. G. Davies, “The Jahn-Teller effect and vibronic coupling at deep levels in diamond,” Rep. Prog. Phys. 44(7), 787–830 (1981).
    [Crossref]
  5. S. I. Kudryashov, A. O. Levchenko, P. A. Danilov, N. A. Smirnov, A. E. Rupasov, R. A. Khmel’nitskii, O. E. Koval’chuk, and A. A. Ionin, “Fine structure of the photoluminescence spectrum of diamond under the multiple emission of an optical phonon during the autolocalization of photoexcited electrons,” JETP Lett. 112(9), 533–536 (2020).
    [Crossref]
  6. S. I. Kudryashov, R. A. Khmelnitskii, P. A. Danilov, N. A. Smirnov, A. O. Levchenko, O. E. Kovalchuk, M. V. Uspenskaya, E. A. Oleynichuk, and M. S. Kovalev, “Broadband and fine-structured luminescence in diamond facilitated by femtosecond laser driven electron impact and injection of “vacancy-interstitial” pairs,” Opt. Lett. 46(6), 1438–1441 (2021).
    [Crossref]
  7. I. V. Fedotov and A. M. Zheltikov, “Background-free two-photon fluorescence readout via a three-photon charge-state modulation of nitrogen-vacancy centers in diamond,” Opt. Lett. 44(15), 3737–3740 (2019).
    [Crossref]
  8. S. M. Pimenov, I. I. Vlasov, A. A. Khomich, B. Neuenschwander, M. Muralt, and V. Romano, “Picosecond-laser-induced structural modifications in the bulk of single-crystal diamond,” Appl. Phys. A 105(3), 673–677 (2011).
    [Crossref]
  9. S. Castelletto, J. Maksimovic, T. Katkus, T. Ohshima, B. C. Johnson, and S. Juodkazis, “Color centers enabled by direct femto-second laser writing in wide bandgap semiconductors,” Nanomaterials 11(1), 72 (2020).
    [Crossref]
  10. Y. N. Palyanov, Y. M. Borzdov, A. F. Khokhryakov, I. N. Kupriyanov, and A. G. Sokol, “Effect of nitrogen impurity on diamond crystal growth processes,” Cryst. Growth Des. 10(7), 3169–3175 (2010).
    [Crossref]
  11. H. Landoldt and R. Boernstein, Semiconductors: Physics of Group IV Elements and III-V Compounds, O. Madelung, ed. (Springer, 1991).
  12. T. Hainschwang, F. Notari, and G. Pamies, “A defect study and classification of brown diamonds with deformation-related color,” Minerals 10(10), 903 (2020).
    [Crossref]
  13. N. Yamamoto, J. C. H. Spence, and D. Fathy, “Cathodoluminescence and polarization studies from individual dislocations in diamond,” Philos. Mag. B 49(6), 609–629 (1984).
    [Crossref]
  14. Y. Yokota, H. Kotsuka, T. Sogi, J. S. Ma, A. Hiraki, H. Kawarada, K. Matsuda, and M. Hatada, “Formation of optical centers in CVD diamond by electron and neutron irradiation,” Diamond and Relat. Mater. 1(5-6), 470–477 (1992).
    [Crossref]
  15. S. C. Lawson, H. Kanda, K. Era, and Y. Sato, “A study of broad band cathodoluminescence from boron-doped high-pressure synthetic and chemical vapor deposited diamond,” (personal communication, 1994).
  16. K. Iakoubovskii and G. J. Adriaenssens, “Luminescence excitation spectra in diamond,” Phys. Rev. B 61(15), 10174–10182 (2000).
    [Crossref]
  17. J. F. Prins, “Ion-implanted n-type diamond: optical evidence, ” in Int. Conf. Diamond Films (1995).
  18. J. F. Prins, “Increased band A cathodoluminescence after carbon ion implantation and annealing of diamond,” Diamond and Relat. Mater. 5(9), 907–913 (1996).
    [Crossref]
  19. J. Bruley and P. E. Batson, “Electronic structure near individual dislocations in diamond by energy-loss spectroscopy,” Mater. Res. Soc. Symp. Proc. 162, 255 (1989).
    [Crossref]
  20. J. Ruan, K. Kobashi, and W. J. Choyke, “On the “band-A” emission and boron related luminescence in diamond,” Appl. Phys. Lett. 60(25), 3138–3140 (1992).
    [Crossref]
  21. P. J. Dean and J. C. Male, “Luminescence and birefringence in a semiconducting diamond,” Br. J. Appl. Phys. 15(1), 101–102 (1964).
    [Crossref]
  22. E. V. Sobolev and Y. I. Dubov, “On the nature of X-ray luminescence,” Fiz. Tverd. Tela 17, 1142–1148 (1975).
  23. E. V. Sobolev and Y. I. Dubov, “On some features of the X-ray luminescence of natural and synthetic diamonds,” Synthetic Diamonds 2, 3–11 (1979).
  24. R. Heiderhoff, “Analyse der elektronischen Eigenschaften von Diamant mittels rastermikroskopischer Methoden,” PhD thesis, University of Wuppertal (1997) [in German].
  25. C. Manfredotti, F. Wang, P. Polesello, E. Vittone, and F. Fizzotti, “Blue-violet electroluminescence and photocurrent spectra from polycrystalline chemical vapor deposited diamond film,” Appl. Phys. Lett. 67(23), 3376–3378 (1995).
    [Crossref]
  26. H. Kawarada and A. Yamaguchi, “Excitonic recombination radiation as characterization of diamonds using cathodoluminescence,” Diamond Relat. Mater. 2(2-4), 100–105 (1993).
    [Crossref]
  27. E. F. Martynovich, L. V. Morozhnikova, Yu. A. Klyuev, and S. P. Plotnikova, “X-ray luminescence in various kinds of natural diamonds,” in: Problems of Theory and Practice of Diamond Treatment (NIIMASH, Moscow, 1977) [in Russian].
  28. A.V. Kurdumov, V.G. Malogolovets, N.V. Novikov, A.H. Piljankevich, and L.A. Shulman, Polymorphous Modification of Carbon and Boron Nitride, (Metallurgija, Moscow, 1994), p.318.
  29. R. Jones and T. King, “Calculation of local densities of states at defects in diamond and silicon,” Physica B + C 116(1-3), 72–75 (1983).
    [Crossref]
  30. S. I. Kudryashov, N. G. Stsepuro, P. A. Danilov, N. A. Smirnov, A. O. Levchenko, and M. S. Kovalev, “Cumulative defocusing of sub-MHz-rate femtosecond-laser pulses in bulk diamond envisioned by transient A-center photoluminescence,” Opt. Mater. Express 11(7), 2234–2241 (2021).
    [Crossref]
  31. G. K. Krasin, S. I. Kudryashov, P. A. Danilov, N. A. Smirnov, A. O. Levchenko, and M. S. Kovalev, “Ultrashort-laser electron-hole plasma and intragap states in diamond,” Eur. Phys. J. D, submitted for publication.
  32. M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B 71(11), 115109 (2005).
    [Crossref]
  33. V. V. Temnov, K. Sokolowski-Tinten, P. Zhou, A. El-Khamhawy, and D. von der Linde, “Multiphoton ionization in dielectrics: comparison of circular and linear polarization,” Phys. Rev. Lett. 97(23), 237403 (2006).
    [Crossref]
  34. S. Kudryashov, A. Samokhvalov, S. Shelygina, A. Karabutov, G. D. Tsibidis, D. Pankin, and V. Veiko, “Electronic and vibrational processes in absorbing liquids in femtosecond laser sub- and filamentation regimes: ultrasonic and optical characterization,” Laser Phys. Lett. 17(10), 105302 (2020).
    [Crossref]
  35. S. I. Kudryashov, A. O. Levchenko, P. A. Danilov, N. A. Smirnov, and A. A. Ionin, “IR femtosecond laser micro-filaments in diamond visualized by inter-band UV photoluminescence,” Opt. Lett. 45(7), 2026–2029 (2020).
    [Crossref]
  36. P. J. Dean and J. C. Male, “Some properties of the visible luminescence excited in diamond by irradiation in the fundamental absorption edge,” J. Phys. Chem. Solids 25(12), 1369–1383 (1964).
    [Crossref]
  37. A. T. Collins, “The electronic and optical properties of diamond; do they favour device applications?” Mat. Res. Soc. Symp. Proc. 162, 3 (1989).
    [Crossref]
  38. P. J. Dean, P. J. Kennedy, and J. E. Ralph, “Particle excited luminescence in diamond,” Proc. Phys. Soc. 76(5), 670–687 (1960).
    [Crossref]
  39. E. S. Vilutis and V. G. Krongauz, Optics and Spectroscopy 15, 39 (1963).
  40. L.V. Levshin and A.M. Saletskiy, Luminescence and its Measurements: Molecular Luminescence (Publishing house of Moscow State University, 2013) [in Russian].
  41. V. V. Vasil’ev, V. L. Velichansky, S. A. Zibrov, A. A. Ionin, S. I. Kudryashov, A. O. Levchenko, L. V. Seleznev, and D.V. Sinitsyn, “Method for generation of optically transparent image inside diamond and apparatus for its visualization,” Russian Patent RU#02465377 (30 June 2011) [in Russian].

2021 (2)

2020 (5)

S. Kudryashov, A. Samokhvalov, S. Shelygina, A. Karabutov, G. D. Tsibidis, D. Pankin, and V. Veiko, “Electronic and vibrational processes in absorbing liquids in femtosecond laser sub- and filamentation regimes: ultrasonic and optical characterization,” Laser Phys. Lett. 17(10), 105302 (2020).
[Crossref]

S. I. Kudryashov, A. O. Levchenko, P. A. Danilov, N. A. Smirnov, and A. A. Ionin, “IR femtosecond laser micro-filaments in diamond visualized by inter-band UV photoluminescence,” Opt. Lett. 45(7), 2026–2029 (2020).
[Crossref]

S. Castelletto, J. Maksimovic, T. Katkus, T. Ohshima, B. C. Johnson, and S. Juodkazis, “Color centers enabled by direct femto-second laser writing in wide bandgap semiconductors,” Nanomaterials 11(1), 72 (2020).
[Crossref]

S. I. Kudryashov, A. O. Levchenko, P. A. Danilov, N. A. Smirnov, A. E. Rupasov, R. A. Khmel’nitskii, O. E. Koval’chuk, and A. A. Ionin, “Fine structure of the photoluminescence spectrum of diamond under the multiple emission of an optical phonon during the autolocalization of photoexcited electrons,” JETP Lett. 112(9), 533–536 (2020).
[Crossref]

T. Hainschwang, F. Notari, and G. Pamies, “A defect study and classification of brown diamonds with deformation-related color,” Minerals 10(10), 903 (2020).
[Crossref]

2019 (1)

2013 (1)

Y. Luo and C. M. Breeding, “Fluorescence produced by optical defects in diamond: measurement, characterization and challenges,” Gems Gemol. 49(2), 82–97 (2013).
[Crossref]

2012 (1)

H.-C. Lu, M.-Y. Lin, S.-L. Chou, Y.-C. Peng, J.-I. Lo, and B.-M. Cheng, “Identification of nitrogen defects in diamond with photoluminescence excited in the 160–240 nm region,” Anal. Chem. 84(21), 9596–9600 (2012).
[Crossref]

2011 (1)

S. M. Pimenov, I. I. Vlasov, A. A. Khomich, B. Neuenschwander, M. Muralt, and V. Romano, “Picosecond-laser-induced structural modifications in the bulk of single-crystal diamond,” Appl. Phys. A 105(3), 673–677 (2011).
[Crossref]

2010 (1)

Y. N. Palyanov, Y. M. Borzdov, A. F. Khokhryakov, I. N. Kupriyanov, and A. G. Sokol, “Effect of nitrogen impurity on diamond crystal growth processes,” Cryst. Growth Des. 10(7), 3169–3175 (2010).
[Crossref]

2006 (1)

V. V. Temnov, K. Sokolowski-Tinten, P. Zhou, A. El-Khamhawy, and D. von der Linde, “Multiphoton ionization in dielectrics: comparison of circular and linear polarization,” Phys. Rev. Lett. 97(23), 237403 (2006).
[Crossref]

2005 (1)

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B 71(11), 115109 (2005).
[Crossref]

2000 (1)

K. Iakoubovskii and G. J. Adriaenssens, “Luminescence excitation spectra in diamond,” Phys. Rev. B 61(15), 10174–10182 (2000).
[Crossref]

1996 (1)

J. F. Prins, “Increased band A cathodoluminescence after carbon ion implantation and annealing of diamond,” Diamond and Relat. Mater. 5(9), 907–913 (1996).
[Crossref]

1995 (1)

C. Manfredotti, F. Wang, P. Polesello, E. Vittone, and F. Fizzotti, “Blue-violet electroluminescence and photocurrent spectra from polycrystalline chemical vapor deposited diamond film,” Appl. Phys. Lett. 67(23), 3376–3378 (1995).
[Crossref]

1993 (1)

H. Kawarada and A. Yamaguchi, “Excitonic recombination radiation as characterization of diamonds using cathodoluminescence,” Diamond Relat. Mater. 2(2-4), 100–105 (1993).
[Crossref]

1992 (2)

J. Ruan, K. Kobashi, and W. J. Choyke, “On the “band-A” emission and boron related luminescence in diamond,” Appl. Phys. Lett. 60(25), 3138–3140 (1992).
[Crossref]

Y. Yokota, H. Kotsuka, T. Sogi, J. S. Ma, A. Hiraki, H. Kawarada, K. Matsuda, and M. Hatada, “Formation of optical centers in CVD diamond by electron and neutron irradiation,” Diamond and Relat. Mater. 1(5-6), 470–477 (1992).
[Crossref]

1989 (2)

J. Bruley and P. E. Batson, “Electronic structure near individual dislocations in diamond by energy-loss spectroscopy,” Mater. Res. Soc. Symp. Proc. 162, 255 (1989).
[Crossref]

A. T. Collins, “The electronic and optical properties of diamond; do they favour device applications?” Mat. Res. Soc. Symp. Proc. 162, 3 (1989).
[Crossref]

1984 (1)

N. Yamamoto, J. C. H. Spence, and D. Fathy, “Cathodoluminescence and polarization studies from individual dislocations in diamond,” Philos. Mag. B 49(6), 609–629 (1984).
[Crossref]

1983 (1)

R. Jones and T. King, “Calculation of local densities of states at defects in diamond and silicon,” Physica B + C 116(1-3), 72–75 (1983).
[Crossref]

1981 (1)

G. Davies, “The Jahn-Teller effect and vibronic coupling at deep levels in diamond,” Rep. Prog. Phys. 44(7), 787–830 (1981).
[Crossref]

1979 (1)

E. V. Sobolev and Y. I. Dubov, “On some features of the X-ray luminescence of natural and synthetic diamonds,” Synthetic Diamonds 2, 3–11 (1979).

1975 (1)

E. V. Sobolev and Y. I. Dubov, “On the nature of X-ray luminescence,” Fiz. Tverd. Tela 17, 1142–1148 (1975).

1964 (2)

P. J. Dean and J. C. Male, “Luminescence and birefringence in a semiconducting diamond,” Br. J. Appl. Phys. 15(1), 101–102 (1964).
[Crossref]

P. J. Dean and J. C. Male, “Some properties of the visible luminescence excited in diamond by irradiation in the fundamental absorption edge,” J. Phys. Chem. Solids 25(12), 1369–1383 (1964).
[Crossref]

1960 (1)

P. J. Dean, P. J. Kennedy, and J. E. Ralph, “Particle excited luminescence in diamond,” Proc. Phys. Soc. 76(5), 670–687 (1960).
[Crossref]

Adriaenssens, G. J.

K. Iakoubovskii and G. J. Adriaenssens, “Luminescence excitation spectra in diamond,” Phys. Rev. B 61(15), 10174–10182 (2000).
[Crossref]

Batson, P. E.

J. Bruley and P. E. Batson, “Electronic structure near individual dislocations in diamond by energy-loss spectroscopy,” Mater. Res. Soc. Symp. Proc. 162, 255 (1989).
[Crossref]

Boernstein, R.

H. Landoldt and R. Boernstein, Semiconductors: Physics of Group IV Elements and III-V Compounds, O. Madelung, ed. (Springer, 1991).

Borzdov, Y. M.

Y. N. Palyanov, Y. M. Borzdov, A. F. Khokhryakov, I. N. Kupriyanov, and A. G. Sokol, “Effect of nitrogen impurity on diamond crystal growth processes,” Cryst. Growth Des. 10(7), 3169–3175 (2010).
[Crossref]

Breeding, C. M.

Y. Luo and C. M. Breeding, “Fluorescence produced by optical defects in diamond: measurement, characterization and challenges,” Gems Gemol. 49(2), 82–97 (2013).
[Crossref]

Bruley, J.

J. Bruley and P. E. Batson, “Electronic structure near individual dislocations in diamond by energy-loss spectroscopy,” Mater. Res. Soc. Symp. Proc. 162, 255 (1989).
[Crossref]

Castelletto, S.

S. Castelletto, J. Maksimovic, T. Katkus, T. Ohshima, B. C. Johnson, and S. Juodkazis, “Color centers enabled by direct femto-second laser writing in wide bandgap semiconductors,” Nanomaterials 11(1), 72 (2020).
[Crossref]

Cheng, B.-M.

H.-C. Lu, M.-Y. Lin, S.-L. Chou, Y.-C. Peng, J.-I. Lo, and B.-M. Cheng, “Identification of nitrogen defects in diamond with photoluminescence excited in the 160–240 nm region,” Anal. Chem. 84(21), 9596–9600 (2012).
[Crossref]

Chou, S.-L.

H.-C. Lu, M.-Y. Lin, S.-L. Chou, Y.-C. Peng, J.-I. Lo, and B.-M. Cheng, “Identification of nitrogen defects in diamond with photoluminescence excited in the 160–240 nm region,” Anal. Chem. 84(21), 9596–9600 (2012).
[Crossref]

Choyke, W. J.

J. Ruan, K. Kobashi, and W. J. Choyke, “On the “band-A” emission and boron related luminescence in diamond,” Appl. Phys. Lett. 60(25), 3138–3140 (1992).
[Crossref]

Collins, A. T.

A. T. Collins, “The electronic and optical properties of diamond; do they favour device applications?” Mat. Res. Soc. Symp. Proc. 162, 3 (1989).
[Crossref]

Danilov, P. A.

Davies, G.

G. Davies, “The Jahn-Teller effect and vibronic coupling at deep levels in diamond,” Rep. Prog. Phys. 44(7), 787–830 (1981).
[Crossref]

Dean, P. J.

P. J. Dean and J. C. Male, “Some properties of the visible luminescence excited in diamond by irradiation in the fundamental absorption edge,” J. Phys. Chem. Solids 25(12), 1369–1383 (1964).
[Crossref]

P. J. Dean and J. C. Male, “Luminescence and birefringence in a semiconducting diamond,” Br. J. Appl. Phys. 15(1), 101–102 (1964).
[Crossref]

P. J. Dean, P. J. Kennedy, and J. E. Ralph, “Particle excited luminescence in diamond,” Proc. Phys. Soc. 76(5), 670–687 (1960).
[Crossref]

Dubov, Y. I.

E. V. Sobolev and Y. I. Dubov, “On some features of the X-ray luminescence of natural and synthetic diamonds,” Synthetic Diamonds 2, 3–11 (1979).

E. V. Sobolev and Y. I. Dubov, “On the nature of X-ray luminescence,” Fiz. Tverd. Tela 17, 1142–1148 (1975).

El-Khamhawy, A.

V. V. Temnov, K. Sokolowski-Tinten, P. Zhou, A. El-Khamhawy, and D. von der Linde, “Multiphoton ionization in dielectrics: comparison of circular and linear polarization,” Phys. Rev. Lett. 97(23), 237403 (2006).
[Crossref]

Era, K.

S. C. Lawson, H. Kanda, K. Era, and Y. Sato, “A study of broad band cathodoluminescence from boron-doped high-pressure synthetic and chemical vapor deposited diamond,” (personal communication, 1994).

Fathy, D.

N. Yamamoto, J. C. H. Spence, and D. Fathy, “Cathodoluminescence and polarization studies from individual dislocations in diamond,” Philos. Mag. B 49(6), 609–629 (1984).
[Crossref]

Fedotov, I. V.

Fizzotti, F.

C. Manfredotti, F. Wang, P. Polesello, E. Vittone, and F. Fizzotti, “Blue-violet electroluminescence and photocurrent spectra from polycrystalline chemical vapor deposited diamond film,” Appl. Phys. Lett. 67(23), 3376–3378 (1995).
[Crossref]

Hainschwang, T.

T. Hainschwang, F. Notari, and G. Pamies, “A defect study and classification of brown diamonds with deformation-related color,” Minerals 10(10), 903 (2020).
[Crossref]

Hatada, M.

Y. Yokota, H. Kotsuka, T. Sogi, J. S. Ma, A. Hiraki, H. Kawarada, K. Matsuda, and M. Hatada, “Formation of optical centers in CVD diamond by electron and neutron irradiation,” Diamond and Relat. Mater. 1(5-6), 470–477 (1992).
[Crossref]

Heiderhoff, R.

R. Heiderhoff, “Analyse der elektronischen Eigenschaften von Diamant mittels rastermikroskopischer Methoden,” PhD thesis, University of Wuppertal (1997) [in German].

Hiraki, A.

Y. Yokota, H. Kotsuka, T. Sogi, J. S. Ma, A. Hiraki, H. Kawarada, K. Matsuda, and M. Hatada, “Formation of optical centers in CVD diamond by electron and neutron irradiation,” Diamond and Relat. Mater. 1(5-6), 470–477 (1992).
[Crossref]

Iakoubovskii, K.

K. Iakoubovskii and G. J. Adriaenssens, “Luminescence excitation spectra in diamond,” Phys. Rev. B 61(15), 10174–10182 (2000).
[Crossref]

Ionin, A. A.

S. I. Kudryashov, A. O. Levchenko, P. A. Danilov, N. A. Smirnov, A. E. Rupasov, R. A. Khmel’nitskii, O. E. Koval’chuk, and A. A. Ionin, “Fine structure of the photoluminescence spectrum of diamond under the multiple emission of an optical phonon during the autolocalization of photoexcited electrons,” JETP Lett. 112(9), 533–536 (2020).
[Crossref]

S. I. Kudryashov, A. O. Levchenko, P. A. Danilov, N. A. Smirnov, and A. A. Ionin, “IR femtosecond laser micro-filaments in diamond visualized by inter-band UV photoluminescence,” Opt. Lett. 45(7), 2026–2029 (2020).
[Crossref]

V. V. Vasil’ev, V. L. Velichansky, S. A. Zibrov, A. A. Ionin, S. I. Kudryashov, A. O. Levchenko, L. V. Seleznev, and D.V. Sinitsyn, “Method for generation of optically transparent image inside diamond and apparatus for its visualization,” Russian Patent RU#02465377 (30 June 2011) [in Russian].

Johnson, B. C.

S. Castelletto, J. Maksimovic, T. Katkus, T. Ohshima, B. C. Johnson, and S. Juodkazis, “Color centers enabled by direct femto-second laser writing in wide bandgap semiconductors,” Nanomaterials 11(1), 72 (2020).
[Crossref]

Jones, R.

R. Jones and T. King, “Calculation of local densities of states at defects in diamond and silicon,” Physica B + C 116(1-3), 72–75 (1983).
[Crossref]

Juodkazis, S.

S. Castelletto, J. Maksimovic, T. Katkus, T. Ohshima, B. C. Johnson, and S. Juodkazis, “Color centers enabled by direct femto-second laser writing in wide bandgap semiconductors,” Nanomaterials 11(1), 72 (2020).
[Crossref]

Kanda, H.

S. C. Lawson, H. Kanda, K. Era, and Y. Sato, “A study of broad band cathodoluminescence from boron-doped high-pressure synthetic and chemical vapor deposited diamond,” (personal communication, 1994).

Karabutov, A.

S. Kudryashov, A. Samokhvalov, S. Shelygina, A. Karabutov, G. D. Tsibidis, D. Pankin, and V. Veiko, “Electronic and vibrational processes in absorbing liquids in femtosecond laser sub- and filamentation regimes: ultrasonic and optical characterization,” Laser Phys. Lett. 17(10), 105302 (2020).
[Crossref]

Katkus, T.

S. Castelletto, J. Maksimovic, T. Katkus, T. Ohshima, B. C. Johnson, and S. Juodkazis, “Color centers enabled by direct femto-second laser writing in wide bandgap semiconductors,” Nanomaterials 11(1), 72 (2020).
[Crossref]

Kawarada, H.

H. Kawarada and A. Yamaguchi, “Excitonic recombination radiation as characterization of diamonds using cathodoluminescence,” Diamond Relat. Mater. 2(2-4), 100–105 (1993).
[Crossref]

Y. Yokota, H. Kotsuka, T. Sogi, J. S. Ma, A. Hiraki, H. Kawarada, K. Matsuda, and M. Hatada, “Formation of optical centers in CVD diamond by electron and neutron irradiation,” Diamond and Relat. Mater. 1(5-6), 470–477 (1992).
[Crossref]

Kennedy, P. J.

P. J. Dean, P. J. Kennedy, and J. E. Ralph, “Particle excited luminescence in diamond,” Proc. Phys. Soc. 76(5), 670–687 (1960).
[Crossref]

Khmel’nitskii, R. A.

S. I. Kudryashov, A. O. Levchenko, P. A. Danilov, N. A. Smirnov, A. E. Rupasov, R. A. Khmel’nitskii, O. E. Koval’chuk, and A. A. Ionin, “Fine structure of the photoluminescence spectrum of diamond under the multiple emission of an optical phonon during the autolocalization of photoexcited electrons,” JETP Lett. 112(9), 533–536 (2020).
[Crossref]

Khmelnitskii, R. A.

Khokhryakov, A. F.

Y. N. Palyanov, Y. M. Borzdov, A. F. Khokhryakov, I. N. Kupriyanov, and A. G. Sokol, “Effect of nitrogen impurity on diamond crystal growth processes,” Cryst. Growth Des. 10(7), 3169–3175 (2010).
[Crossref]

Khomich, A. A.

S. M. Pimenov, I. I. Vlasov, A. A. Khomich, B. Neuenschwander, M. Muralt, and V. Romano, “Picosecond-laser-induced structural modifications in the bulk of single-crystal diamond,” Appl. Phys. A 105(3), 673–677 (2011).
[Crossref]

King, T.

R. Jones and T. King, “Calculation of local densities of states at defects in diamond and silicon,” Physica B + C 116(1-3), 72–75 (1983).
[Crossref]

Klyuev, Yu. A.

E. F. Martynovich, L. V. Morozhnikova, Yu. A. Klyuev, and S. P. Plotnikova, “X-ray luminescence in various kinds of natural diamonds,” in: Problems of Theory and Practice of Diamond Treatment (NIIMASH, Moscow, 1977) [in Russian].

Kobashi, K.

J. Ruan, K. Kobashi, and W. J. Choyke, “On the “band-A” emission and boron related luminescence in diamond,” Appl. Phys. Lett. 60(25), 3138–3140 (1992).
[Crossref]

Kotsuka, H.

Y. Yokota, H. Kotsuka, T. Sogi, J. S. Ma, A. Hiraki, H. Kawarada, K. Matsuda, and M. Hatada, “Formation of optical centers in CVD diamond by electron and neutron irradiation,” Diamond and Relat. Mater. 1(5-6), 470–477 (1992).
[Crossref]

Koval’chuk, O. E.

S. I. Kudryashov, A. O. Levchenko, P. A. Danilov, N. A. Smirnov, A. E. Rupasov, R. A. Khmel’nitskii, O. E. Koval’chuk, and A. A. Ionin, “Fine structure of the photoluminescence spectrum of diamond under the multiple emission of an optical phonon during the autolocalization of photoexcited electrons,” JETP Lett. 112(9), 533–536 (2020).
[Crossref]

Kovalchuk, O. E.

Kovalev, M. S.

Krasin, G. K.

G. K. Krasin, S. I. Kudryashov, P. A. Danilov, N. A. Smirnov, A. O. Levchenko, and M. S. Kovalev, “Ultrashort-laser electron-hole plasma and intragap states in diamond,” Eur. Phys. J. D, submitted for publication.

Krongauz, V. G.

E. S. Vilutis and V. G. Krongauz, Optics and Spectroscopy 15, 39 (1963).

Kudryashov, S.

S. Kudryashov, A. Samokhvalov, S. Shelygina, A. Karabutov, G. D. Tsibidis, D. Pankin, and V. Veiko, “Electronic and vibrational processes in absorbing liquids in femtosecond laser sub- and filamentation regimes: ultrasonic and optical characterization,” Laser Phys. Lett. 17(10), 105302 (2020).
[Crossref]

Kudryashov, S. I.

S. I. Kudryashov, N. G. Stsepuro, P. A. Danilov, N. A. Smirnov, A. O. Levchenko, and M. S. Kovalev, “Cumulative defocusing of sub-MHz-rate femtosecond-laser pulses in bulk diamond envisioned by transient A-center photoluminescence,” Opt. Mater. Express 11(7), 2234–2241 (2021).
[Crossref]

S. I. Kudryashov, R. A. Khmelnitskii, P. A. Danilov, N. A. Smirnov, A. O. Levchenko, O. E. Kovalchuk, M. V. Uspenskaya, E. A. Oleynichuk, and M. S. Kovalev, “Broadband and fine-structured luminescence in diamond facilitated by femtosecond laser driven electron impact and injection of “vacancy-interstitial” pairs,” Opt. Lett. 46(6), 1438–1441 (2021).
[Crossref]

S. I. Kudryashov, A. O. Levchenko, P. A. Danilov, N. A. Smirnov, A. E. Rupasov, R. A. Khmel’nitskii, O. E. Koval’chuk, and A. A. Ionin, “Fine structure of the photoluminescence spectrum of diamond under the multiple emission of an optical phonon during the autolocalization of photoexcited electrons,” JETP Lett. 112(9), 533–536 (2020).
[Crossref]

S. I. Kudryashov, A. O. Levchenko, P. A. Danilov, N. A. Smirnov, and A. A. Ionin, “IR femtosecond laser micro-filaments in diamond visualized by inter-band UV photoluminescence,” Opt. Lett. 45(7), 2026–2029 (2020).
[Crossref]

G. K. Krasin, S. I. Kudryashov, P. A. Danilov, N. A. Smirnov, A. O. Levchenko, and M. S. Kovalev, “Ultrashort-laser electron-hole plasma and intragap states in diamond,” Eur. Phys. J. D, submitted for publication.

V. V. Vasil’ev, V. L. Velichansky, S. A. Zibrov, A. A. Ionin, S. I. Kudryashov, A. O. Levchenko, L. V. Seleznev, and D.V. Sinitsyn, “Method for generation of optically transparent image inside diamond and apparatus for its visualization,” Russian Patent RU#02465377 (30 June 2011) [in Russian].

Kupriyanov, I. N.

Y. N. Palyanov, Y. M. Borzdov, A. F. Khokhryakov, I. N. Kupriyanov, and A. G. Sokol, “Effect of nitrogen impurity on diamond crystal growth processes,” Cryst. Growth Des. 10(7), 3169–3175 (2010).
[Crossref]

Kurdumov, A.V.

A.V. Kurdumov, V.G. Malogolovets, N.V. Novikov, A.H. Piljankevich, and L.A. Shulman, Polymorphous Modification of Carbon and Boron Nitride, (Metallurgija, Moscow, 1994), p.318.

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S. C. Lawson, H. Kanda, K. Era, and Y. Sato, “A study of broad band cathodoluminescence from boron-doped high-pressure synthetic and chemical vapor deposited diamond,” (personal communication, 1994).

Levchenko, A. O.

S. I. Kudryashov, R. A. Khmelnitskii, P. A. Danilov, N. A. Smirnov, A. O. Levchenko, O. E. Kovalchuk, M. V. Uspenskaya, E. A. Oleynichuk, and M. S. Kovalev, “Broadband and fine-structured luminescence in diamond facilitated by femtosecond laser driven electron impact and injection of “vacancy-interstitial” pairs,” Opt. Lett. 46(6), 1438–1441 (2021).
[Crossref]

S. I. Kudryashov, N. G. Stsepuro, P. A. Danilov, N. A. Smirnov, A. O. Levchenko, and M. S. Kovalev, “Cumulative defocusing of sub-MHz-rate femtosecond-laser pulses in bulk diamond envisioned by transient A-center photoluminescence,” Opt. Mater. Express 11(7), 2234–2241 (2021).
[Crossref]

S. I. Kudryashov, A. O. Levchenko, P. A. Danilov, N. A. Smirnov, and A. A. Ionin, “IR femtosecond laser micro-filaments in diamond visualized by inter-band UV photoluminescence,” Opt. Lett. 45(7), 2026–2029 (2020).
[Crossref]

S. I. Kudryashov, A. O. Levchenko, P. A. Danilov, N. A. Smirnov, A. E. Rupasov, R. A. Khmel’nitskii, O. E. Koval’chuk, and A. A. Ionin, “Fine structure of the photoluminescence spectrum of diamond under the multiple emission of an optical phonon during the autolocalization of photoexcited electrons,” JETP Lett. 112(9), 533–536 (2020).
[Crossref]

G. K. Krasin, S. I. Kudryashov, P. A. Danilov, N. A. Smirnov, A. O. Levchenko, and M. S. Kovalev, “Ultrashort-laser electron-hole plasma and intragap states in diamond,” Eur. Phys. J. D, submitted for publication.

V. V. Vasil’ev, V. L. Velichansky, S. A. Zibrov, A. A. Ionin, S. I. Kudryashov, A. O. Levchenko, L. V. Seleznev, and D.V. Sinitsyn, “Method for generation of optically transparent image inside diamond and apparatus for its visualization,” Russian Patent RU#02465377 (30 June 2011) [in Russian].

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L.V. Levshin and A.M. Saletskiy, Luminescence and its Measurements: Molecular Luminescence (Publishing house of Moscow State University, 2013) [in Russian].

Lin, M.-Y.

H.-C. Lu, M.-Y. Lin, S.-L. Chou, Y.-C. Peng, J.-I. Lo, and B.-M. Cheng, “Identification of nitrogen defects in diamond with photoluminescence excited in the 160–240 nm region,” Anal. Chem. 84(21), 9596–9600 (2012).
[Crossref]

Liu, J.

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B 71(11), 115109 (2005).
[Crossref]

Lo, J.-I.

H.-C. Lu, M.-Y. Lin, S.-L. Chou, Y.-C. Peng, J.-I. Lo, and B.-M. Cheng, “Identification of nitrogen defects in diamond with photoluminescence excited in the 160–240 nm region,” Anal. Chem. 84(21), 9596–9600 (2012).
[Crossref]

Lu, H.-C.

H.-C. Lu, M.-Y. Lin, S.-L. Chou, Y.-C. Peng, J.-I. Lo, and B.-M. Cheng, “Identification of nitrogen defects in diamond with photoluminescence excited in the 160–240 nm region,” Anal. Chem. 84(21), 9596–9600 (2012).
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Y. Luo and C. M. Breeding, “Fluorescence produced by optical defects in diamond: measurement, characterization and challenges,” Gems Gemol. 49(2), 82–97 (2013).
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Y. Yokota, H. Kotsuka, T. Sogi, J. S. Ma, A. Hiraki, H. Kawarada, K. Matsuda, and M. Hatada, “Formation of optical centers in CVD diamond by electron and neutron irradiation,” Diamond and Relat. Mater. 1(5-6), 470–477 (1992).
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S. Castelletto, J. Maksimovic, T. Katkus, T. Ohshima, B. C. Johnson, and S. Juodkazis, “Color centers enabled by direct femto-second laser writing in wide bandgap semiconductors,” Nanomaterials 11(1), 72 (2020).
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A.V. Kurdumov, V.G. Malogolovets, N.V. Novikov, A.H. Piljankevich, and L.A. Shulman, Polymorphous Modification of Carbon and Boron Nitride, (Metallurgija, Moscow, 1994), p.318.

Manfredotti, C.

C. Manfredotti, F. Wang, P. Polesello, E. Vittone, and F. Fizzotti, “Blue-violet electroluminescence and photocurrent spectra from polycrystalline chemical vapor deposited diamond film,” Appl. Phys. Lett. 67(23), 3376–3378 (1995).
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E. F. Martynovich, L. V. Morozhnikova, Yu. A. Klyuev, and S. P. Plotnikova, “X-ray luminescence in various kinds of natural diamonds,” in: Problems of Theory and Practice of Diamond Treatment (NIIMASH, Moscow, 1977) [in Russian].

Matsuda, K.

Y. Yokota, H. Kotsuka, T. Sogi, J. S. Ma, A. Hiraki, H. Kawarada, K. Matsuda, and M. Hatada, “Formation of optical centers in CVD diamond by electron and neutron irradiation,” Diamond and Relat. Mater. 1(5-6), 470–477 (1992).
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M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B 71(11), 115109 (2005).
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E. F. Martynovich, L. V. Morozhnikova, Yu. A. Klyuev, and S. P. Plotnikova, “X-ray luminescence in various kinds of natural diamonds,” in: Problems of Theory and Practice of Diamond Treatment (NIIMASH, Moscow, 1977) [in Russian].

Muralt, M.

S. M. Pimenov, I. I. Vlasov, A. A. Khomich, B. Neuenschwander, M. Muralt, and V. Romano, “Picosecond-laser-induced structural modifications in the bulk of single-crystal diamond,” Appl. Phys. A 105(3), 673–677 (2011).
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Neuenschwander, B.

S. M. Pimenov, I. I. Vlasov, A. A. Khomich, B. Neuenschwander, M. Muralt, and V. Romano, “Picosecond-laser-induced structural modifications in the bulk of single-crystal diamond,” Appl. Phys. A 105(3), 673–677 (2011).
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T. Hainschwang, F. Notari, and G. Pamies, “A defect study and classification of brown diamonds with deformation-related color,” Minerals 10(10), 903 (2020).
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A.V. Kurdumov, V.G. Malogolovets, N.V. Novikov, A.H. Piljankevich, and L.A. Shulman, Polymorphous Modification of Carbon and Boron Nitride, (Metallurgija, Moscow, 1994), p.318.

Ohshima, T.

S. Castelletto, J. Maksimovic, T. Katkus, T. Ohshima, B. C. Johnson, and S. Juodkazis, “Color centers enabled by direct femto-second laser writing in wide bandgap semiconductors,” Nanomaterials 11(1), 72 (2020).
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Palyanov, Y. N.

Y. N. Palyanov, Y. M. Borzdov, A. F. Khokhryakov, I. N. Kupriyanov, and A. G. Sokol, “Effect of nitrogen impurity on diamond crystal growth processes,” Cryst. Growth Des. 10(7), 3169–3175 (2010).
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T. Hainschwang, F. Notari, and G. Pamies, “A defect study and classification of brown diamonds with deformation-related color,” Minerals 10(10), 903 (2020).
[Crossref]

Pankin, D.

S. Kudryashov, A. Samokhvalov, S. Shelygina, A. Karabutov, G. D. Tsibidis, D. Pankin, and V. Veiko, “Electronic and vibrational processes in absorbing liquids in femtosecond laser sub- and filamentation regimes: ultrasonic and optical characterization,” Laser Phys. Lett. 17(10), 105302 (2020).
[Crossref]

Peng, Y.-C.

H.-C. Lu, M.-Y. Lin, S.-L. Chou, Y.-C. Peng, J.-I. Lo, and B.-M. Cheng, “Identification of nitrogen defects in diamond with photoluminescence excited in the 160–240 nm region,” Anal. Chem. 84(21), 9596–9600 (2012).
[Crossref]

Piljankevich, A.H.

A.V. Kurdumov, V.G. Malogolovets, N.V. Novikov, A.H. Piljankevich, and L.A. Shulman, Polymorphous Modification of Carbon and Boron Nitride, (Metallurgija, Moscow, 1994), p.318.

Pimenov, S. M.

S. M. Pimenov, I. I. Vlasov, A. A. Khomich, B. Neuenschwander, M. Muralt, and V. Romano, “Picosecond-laser-induced structural modifications in the bulk of single-crystal diamond,” Appl. Phys. A 105(3), 673–677 (2011).
[Crossref]

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E. F. Martynovich, L. V. Morozhnikova, Yu. A. Klyuev, and S. P. Plotnikova, “X-ray luminescence in various kinds of natural diamonds,” in: Problems of Theory and Practice of Diamond Treatment (NIIMASH, Moscow, 1977) [in Russian].

Polesello, P.

C. Manfredotti, F. Wang, P. Polesello, E. Vittone, and F. Fizzotti, “Blue-violet electroluminescence and photocurrent spectra from polycrystalline chemical vapor deposited diamond film,” Appl. Phys. Lett. 67(23), 3376–3378 (1995).
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M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B 71(11), 115109 (2005).
[Crossref]

Romano, V.

S. M. Pimenov, I. I. Vlasov, A. A. Khomich, B. Neuenschwander, M. Muralt, and V. Romano, “Picosecond-laser-induced structural modifications in the bulk of single-crystal diamond,” Appl. Phys. A 105(3), 673–677 (2011).
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Ruan, J.

J. Ruan, K. Kobashi, and W. J. Choyke, “On the “band-A” emission and boron related luminescence in diamond,” Appl. Phys. Lett. 60(25), 3138–3140 (1992).
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Rudolph, W.

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B 71(11), 115109 (2005).
[Crossref]

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S. I. Kudryashov, A. O. Levchenko, P. A. Danilov, N. A. Smirnov, A. E. Rupasov, R. A. Khmel’nitskii, O. E. Koval’chuk, and A. A. Ionin, “Fine structure of the photoluminescence spectrum of diamond under the multiple emission of an optical phonon during the autolocalization of photoexcited electrons,” JETP Lett. 112(9), 533–536 (2020).
[Crossref]

Saletskiy, A.M.

L.V. Levshin and A.M. Saletskiy, Luminescence and its Measurements: Molecular Luminescence (Publishing house of Moscow State University, 2013) [in Russian].

Samokhvalov, A.

S. Kudryashov, A. Samokhvalov, S. Shelygina, A. Karabutov, G. D. Tsibidis, D. Pankin, and V. Veiko, “Electronic and vibrational processes in absorbing liquids in femtosecond laser sub- and filamentation regimes: ultrasonic and optical characterization,” Laser Phys. Lett. 17(10), 105302 (2020).
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S. C. Lawson, H. Kanda, K. Era, and Y. Sato, “A study of broad band cathodoluminescence from boron-doped high-pressure synthetic and chemical vapor deposited diamond,” (personal communication, 1994).

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V. V. Vasil’ev, V. L. Velichansky, S. A. Zibrov, A. A. Ionin, S. I. Kudryashov, A. O. Levchenko, L. V. Seleznev, and D.V. Sinitsyn, “Method for generation of optically transparent image inside diamond and apparatus for its visualization,” Russian Patent RU#02465377 (30 June 2011) [in Russian].

Shelygina, S.

S. Kudryashov, A. Samokhvalov, S. Shelygina, A. Karabutov, G. D. Tsibidis, D. Pankin, and V. Veiko, “Electronic and vibrational processes in absorbing liquids in femtosecond laser sub- and filamentation regimes: ultrasonic and optical characterization,” Laser Phys. Lett. 17(10), 105302 (2020).
[Crossref]

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A.V. Kurdumov, V.G. Malogolovets, N.V. Novikov, A.H. Piljankevich, and L.A. Shulman, Polymorphous Modification of Carbon and Boron Nitride, (Metallurgija, Moscow, 1994), p.318.

Sinitsyn, D.V.

V. V. Vasil’ev, V. L. Velichansky, S. A. Zibrov, A. A. Ionin, S. I. Kudryashov, A. O. Levchenko, L. V. Seleznev, and D.V. Sinitsyn, “Method for generation of optically transparent image inside diamond and apparatus for its visualization,” Russian Patent RU#02465377 (30 June 2011) [in Russian].

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[Crossref]

Sokol, A. G.

Y. N. Palyanov, Y. M. Borzdov, A. F. Khokhryakov, I. N. Kupriyanov, and A. G. Sokol, “Effect of nitrogen impurity on diamond crystal growth processes,” Cryst. Growth Des. 10(7), 3169–3175 (2010).
[Crossref]

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V. V. Temnov, K. Sokolowski-Tinten, P. Zhou, A. El-Khamhawy, and D. von der Linde, “Multiphoton ionization in dielectrics: comparison of circular and linear polarization,” Phys. Rev. Lett. 97(23), 237403 (2006).
[Crossref]

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N. Yamamoto, J. C. H. Spence, and D. Fathy, “Cathodoluminescence and polarization studies from individual dislocations in diamond,” Philos. Mag. B 49(6), 609–629 (1984).
[Crossref]

Starke, K.

M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B 71(11), 115109 (2005).
[Crossref]

Stsepuro, N. G.

Temnov, V. V.

V. V. Temnov, K. Sokolowski-Tinten, P. Zhou, A. El-Khamhawy, and D. von der Linde, “Multiphoton ionization in dielectrics: comparison of circular and linear polarization,” Phys. Rev. Lett. 97(23), 237403 (2006).
[Crossref]

Tsibidis, G. D.

S. Kudryashov, A. Samokhvalov, S. Shelygina, A. Karabutov, G. D. Tsibidis, D. Pankin, and V. Veiko, “Electronic and vibrational processes in absorbing liquids in femtosecond laser sub- and filamentation regimes: ultrasonic and optical characterization,” Laser Phys. Lett. 17(10), 105302 (2020).
[Crossref]

Uspenskaya, M. V.

Vasil’ev, V. V.

V. V. Vasil’ev, V. L. Velichansky, S. A. Zibrov, A. A. Ionin, S. I. Kudryashov, A. O. Levchenko, L. V. Seleznev, and D.V. Sinitsyn, “Method for generation of optically transparent image inside diamond and apparatus for its visualization,” Russian Patent RU#02465377 (30 June 2011) [in Russian].

Veiko, V.

S. Kudryashov, A. Samokhvalov, S. Shelygina, A. Karabutov, G. D. Tsibidis, D. Pankin, and V. Veiko, “Electronic and vibrational processes in absorbing liquids in femtosecond laser sub- and filamentation regimes: ultrasonic and optical characterization,” Laser Phys. Lett. 17(10), 105302 (2020).
[Crossref]

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V. V. Vasil’ev, V. L. Velichansky, S. A. Zibrov, A. A. Ionin, S. I. Kudryashov, A. O. Levchenko, L. V. Seleznev, and D.V. Sinitsyn, “Method for generation of optically transparent image inside diamond and apparatus for its visualization,” Russian Patent RU#02465377 (30 June 2011) [in Russian].

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E. S. Vilutis and V. G. Krongauz, Optics and Spectroscopy 15, 39 (1963).

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C. Manfredotti, F. Wang, P. Polesello, E. Vittone, and F. Fizzotti, “Blue-violet electroluminescence and photocurrent spectra from polycrystalline chemical vapor deposited diamond film,” Appl. Phys. Lett. 67(23), 3376–3378 (1995).
[Crossref]

Vlasov, I. I.

S. M. Pimenov, I. I. Vlasov, A. A. Khomich, B. Neuenschwander, M. Muralt, and V. Romano, “Picosecond-laser-induced structural modifications in the bulk of single-crystal diamond,” Appl. Phys. A 105(3), 673–677 (2011).
[Crossref]

von der Linde, D.

V. V. Temnov, K. Sokolowski-Tinten, P. Zhou, A. El-Khamhawy, and D. von der Linde, “Multiphoton ionization in dielectrics: comparison of circular and linear polarization,” Phys. Rev. Lett. 97(23), 237403 (2006).
[Crossref]

Wang, F.

C. Manfredotti, F. Wang, P. Polesello, E. Vittone, and F. Fizzotti, “Blue-violet electroluminescence and photocurrent spectra from polycrystalline chemical vapor deposited diamond film,” Appl. Phys. Lett. 67(23), 3376–3378 (1995).
[Crossref]

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Yamamoto, N.

N. Yamamoto, J. C. H. Spence, and D. Fathy, “Cathodoluminescence and polarization studies from individual dislocations in diamond,” Philos. Mag. B 49(6), 609–629 (1984).
[Crossref]

Yokota, Y.

Y. Yokota, H. Kotsuka, T. Sogi, J. S. Ma, A. Hiraki, H. Kawarada, K. Matsuda, and M. Hatada, “Formation of optical centers in CVD diamond by electron and neutron irradiation,” Diamond and Relat. Mater. 1(5-6), 470–477 (1992).
[Crossref]

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A. M. Zaitsev, Optical Properties of Diamond: A Data Handbook (Springer Science & Business Media, 2013).

Zheltikov, A. M.

Zhou, P.

V. V. Temnov, K. Sokolowski-Tinten, P. Zhou, A. El-Khamhawy, and D. von der Linde, “Multiphoton ionization in dielectrics: comparison of circular and linear polarization,” Phys. Rev. Lett. 97(23), 237403 (2006).
[Crossref]

Zibrov, S. A.

V. V. Vasil’ev, V. L. Velichansky, S. A. Zibrov, A. A. Ionin, S. I. Kudryashov, A. O. Levchenko, L. V. Seleznev, and D.V. Sinitsyn, “Method for generation of optically transparent image inside diamond and apparatus for its visualization,” Russian Patent RU#02465377 (30 June 2011) [in Russian].

Anal. Chem. (1)

H.-C. Lu, M.-Y. Lin, S.-L. Chou, Y.-C. Peng, J.-I. Lo, and B.-M. Cheng, “Identification of nitrogen defects in diamond with photoluminescence excited in the 160–240 nm region,” Anal. Chem. 84(21), 9596–9600 (2012).
[Crossref]

Appl. Phys. A (1)

S. M. Pimenov, I. I. Vlasov, A. A. Khomich, B. Neuenschwander, M. Muralt, and V. Romano, “Picosecond-laser-induced structural modifications in the bulk of single-crystal diamond,” Appl. Phys. A 105(3), 673–677 (2011).
[Crossref]

Appl. Phys. Lett. (2)

J. Ruan, K. Kobashi, and W. J. Choyke, “On the “band-A” emission and boron related luminescence in diamond,” Appl. Phys. Lett. 60(25), 3138–3140 (1992).
[Crossref]

C. Manfredotti, F. Wang, P. Polesello, E. Vittone, and F. Fizzotti, “Blue-violet electroluminescence and photocurrent spectra from polycrystalline chemical vapor deposited diamond film,” Appl. Phys. Lett. 67(23), 3376–3378 (1995).
[Crossref]

Br. J. Appl. Phys. (1)

P. J. Dean and J. C. Male, “Luminescence and birefringence in a semiconducting diamond,” Br. J. Appl. Phys. 15(1), 101–102 (1964).
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Data availability

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Figures (5)

Fig. 1.
Fig. 1. (a) UV-near-IR transmittance and absorption coefficient of the natural diamond with the positions of one- (ω) and two-photon (2ω) excitation. (b) Optical density of IaAB diamond (curve 3, left axis) and its photoluminescence spectra (right axis) excited at peak laser intensities of 0.5 (curve 1) and 3.1 (curve 2) TW/cm2, with the expected position of two-photon excitation (2ω). (c) IR spectrum with the spectral assignment of A and B1,2 bands. (d) Excitation/PL collection layout. Inset (top): color image of the luminous track in the focal region inside the diamond (image size – 300 × 60 microns).
Fig. 2.
Fig. 2. Multi-peak PL spectrum (dark pints) decomposed into colored and numbered eleven fitting Lorentzian lines and their cumulative violet spectrum, the possible A-band central position [24,25,29] and mirror-like anti-Stokes absorption spectrum (dashed curve). The green tringles show the single-photon and double-photon (TPA) laser energy. The pump peak was almost totally attenuated by a notch filter.
Fig. 3.
Fig. 3. Possible two-photon external (a) and internal (b) intra-gap and three-photon interband (c) excitation schemes of the A-center photoluminescence. The green arrows show the laser photons, blue and red arrows - actual and possible photoluminescence, curves arrows – non-radiative relaxation, dark and light circles – electron and hole, respectively.
Fig. 4.
Fig. 4. Spectrally-integrated (the entire A-band, spectral range ≈300–550 nm) PL intensity versus incident pulse energy E (bottom axis) and peak intensity I0 (top axis), with their corresponding sectional linear fits in lgΦi-lgE coordinates in the different ranges.
Fig. 5.
Fig. 5. (a) Multi-peak A-band PL spectra at different sample temperatures, where the pump peak was almost totally attenuated by a notch filter. (b) Temperature dependences of integrated PL intensity ΦΣ. (c) lnΦΣ - 1/kBT curves and damping energies of 1.4 and 0.2 eV as their linear slopes.

Equations (7)

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Φ ( N ) e S S N N ! ,
d ρ e x c d t = σ 2 I 2 ( ρ 0 ρ e x c ) ρ e x c τ , Φ ρ e x c d t ,
d ρ e , h d t = σ 3 I 3 + α I ρ e , h γ ρ e , h 3 β ρ e , h 2 ρ e , h τ S T , Φ β ρ e , h 2 d t
d ρ e , h d t = σ 3 I 3 β ρ e , h 2 ρ e , h τ S T , σ 3 I 3 ρ e , h τ S T , ρ e , h I 0 3 , Φ β ρ e , h 2 d t I 0 6 .
d ρ e , h d t = σ 3 I 3 β ρ e , h 2 , σ 3 I 3 β ρ e , h 2 , ρ e , h I 0 3 / 2 , Φ β ρ e , h 2 d t I 0 3 ,
d ρ e , h d t = σ 3 I 3 γ ρ e , h 3 β ρ e , h 2 , σ 3 I 3 γ ρ e , h 3 , ρ e , h I 0 1 , Φ β ρ e , h 2 d t I 0 2 .
d ρ e , h d t = α I ρ e , h γ ρ e , h 3 β ρ e , h 2 , α I ρ e , h γ ρ e , h 3 , ρ e , h I 0 1 / 2 , Φ β ρ e , h 2 d t I 0 1 .