Open Access
Add to BibSonomyAbstract
We report on experimental results of supercontinuum generation in bulk diamond. The spectrum of supercontinuum generated with 800 nm pump extends up to 600 nm towards short wavelengths. We present the numerical model explaining the phenomenon, in which the role of different nonlinear effects including stimulated Raman scattering is discussed. Unlike in other materials, in diamond the feature of supercontinuum due to stimulated Raman response is apparently visible.
© 2013 Optical Society of America
1. Introduction
The broad band and intense supercontinuum sources are very useful tools in molecular spectroscopy. Fused silica, sapphire and CaF2 are the common materials employed for supercontinuum generation with Ti:Sapphire laser, while YAG, GdVO4 and YVO4 have proven to be useful for Yb:KYW amplifiers. Recently, high energy supercontinuum was generated in KGW crystals using 80 MHz Ti:Sapphire oscillator [1], while supercontinuum spanning 3.3 octave has been generated from mid-IR pulses in YAG plates [2].
Models for supercontinuum generation in solids are also constantly developed. Most of them are based on nonlinear envelope equation (NEE) under slowly-varying-envelope approximation [3], where, besides self-phase modulation, diffraction and dispersion, the self-steepening is addressed. Still those effects themselves are unable to explain the different shapes of the supercontinuum spectra: to this purpose, the effects connected with plasma generation have to be included [4, 5].
It has been argued in many recent works on supercontinuum generation, that in thin solid samples the Raman response of the medium might be neglected [2,4,6–8]. The study on generation of supercontinuum in a solid with high and easily resolved Raman response could bring evidence for verification of this thesis. Diamond is known from its high Raman gain, orders of magnitude higher than that of fused silica [9]. For this reason it is used in Raman lasers [10] and four wave mixing sources [11], in fact nonlinear energy conversion from picosecond laser pulses to Stokes and anti-Stokes components with efficiency as high as 45 % was observed in the past [12]. Therefore it is a perfect material for Raman response study. Thus, our interest in diamond is primarily motivated not by the search for new media for efficient supercontinuum generation, but in a first place we would like to explore the influence of Raman response on the generated supercontinuum.
In this paper we present the experimental indications of the influence of the stimulated Raman scattering (SRS) on the spectrum of the supercontinuum generated in diamond. As expected this influence is weak but unlike in other bulk materials it is clearly visible in diamond. We show that these experimental indications can be reproduced in NEE based model by inclusion of SRS term. We also show the experimental spectra for a range of input pulse energies and discuss the higher order nonlinear effects.
2. Experiment
The 1 kHz Sapphire regenerative amplifier (Coherent Inc. Legend Elite) was used as the source of light. The pulse spectrum was centered at 800 nm and its full width at half maximum was close to 25 nm. The pulse length was about 40 fs. The beam (4.8 mm diameter) was focused on the sample with 125 mm lens. The sample was the 2 mm thick diamond crystal of type IIa. To avoid multiple filamentation we displace the sample from the focus by 3.7 mm towards the lens, so that the beam size at the entrance of the sample is about 150 μm.
The beam was attenuated with neutral density filters so that the sample was illuminated with pulses of energies from 8 to 30 μJ (10 – 40 GW/cm2). In this power density range we do not observe multiple filamentation. The outgoing beam was imaged on a diffuser plate and diffused light was gathered by Ocean Optics USB2000 spectrometer with useful spectral range from 340 to 1020 nm. The spectrometer sensitivity was calibrated with a black body spectrum. The spectra of outgoing light were acquired for different input pulses energy.
In the considered range of input energies the highly temporary and spatially coherent supercontinuum is observed. The characteristic beam pattern is similar as in other experiments with bulk materials [13]. It is yellow in the center and surrounded by a green ring. The sample spectra of supercontinua, generated in the experiment for different input energies, are presented in Fig. 1(a) and all acquired spectra are presented on the map in Fig. 2. For energies below 8 μJ only slight broadening is observed. For energies over 30 μJ the interference fringes appear in the supercontinuum beam pattern, what confirms its coherence.
Fig. 1 Spectra of supercontinua generated in the experiment for different input energies in whole measurement range in logarithmic scale (a). The spectra of supercontinua for input energies over 17 μJ where the peak is observed in linear scale, the inset shows in detail the spectra in vicinity of expected anti-Stokes Raman line (b).
Fig. 2 Spectra of generated supercontinua as a function of input energy measured in experiment (a) and reconstructed from the simulation (b).
Brodeur and Chin [13] showed that energy difference between the blue edge of the supercontinuum and the pump central frequency is the a monotonicaly growing function of the material band gap energy. This observation was confirmed by studies of supercontinuum generation in laser host crystals [1]. In our experiment we found the blue edge of supercontinuum generated in diamond to be close to 600 nm. Although this value is high, it does not stand out from the set of accumulated data for different media [1, 14].
For input energies over 17 μJ we observe a growing peak where the anti-Stokes Raman line is expected to appear (see Fig. 1(b)), corresponding to the frequency separation of 1332 cm−1 from the pulse central frequency. The inset in Fig. 1(b) presents close-up on this peak. We attribute this peak to the stimulated Raman scattering.
Besides the peak at 1332 cm−1, we observe some fringes, visible in Fig. 2 on the short wavelength side of the spectrum. The positions of these fringes are intensity dependent.
3. Model
The chief nonlinear process, without which the supercontinuum generation would not be possible, is the self-phase modulation and it’s spatial manifestation: self-focusing. By reduction of beam size self-focusing causes growth of intensity which initiates the generation of super-continuum [4]. As well important is the photoionization and plasma dynamics, which stops the self-focusing [5]. Those processes are also the major cause of spectrum broadening. Apart from them the important role is played by material refractive index dispersion and it also has to be included in the model. It is our aim to estimate the role of the Raman process.
We base our model of supercontinuum generation on nonlinear envelope equation with slowly-varying-envelope approximation [3], expanded to include free-carrier effects [5]:
where A(z, t) is the slowly-varying-envelope of the electric field, z is the coordinate along the pulse propagation and t is the retarded time [3], is the steepening operator, D̂ is the dispersion operator, ω0 is the pulse central frequency, n0 is the material refractive index at ω0, ε0 and c are the vacuum permittivity and speed of light in vacuum, km = Re (∂mk/∂ωm|0 are the real parts of propagation constant derivatives over the frequency detuning from the central frequency ω0. The imaginary part of propagation constant (linear absorption) is neglected in our study, Pnl and Jfc are the complex envelopes of nonlinear polarization and free-current density.Usually Taylor series expansion of propagation constant with respect to the frequency detun-ing (ω) is used to describe the dispersion in the Fourier space:
where IFT stands for inverse Fourier transform. Since we use the slowly varying envelope and the retarded time, the terms proportional to k0 and k1 are not present in the above expression. We substitute the series in Eq. (2) to get: where k = k(ω0 + ω) is calculated using Sellmair formula [15]. The dispersion curve k − (k0 + k1ω) is presented in Fig. 3. The zero dispersion wavelength of diamond is close to 3.38 μm, thus the supercontinuum generation occurs in positive dispersion regime.The nonlinear polarization is taken in the following form:
The three consecutive nonlinear contributions in Eq. (4) are: i) self-phase modulation with γe = k0ε0cn2/2, where n2 is the nonlinear refractive index; ii) higher order nonlinearity with , where n4 is 4th order nonlinear refractive index; iii) stimulated Raman response with its strength coefficient γR and the medium response function R(t). The value of nonlinear refractive index n2 and Raman response strength coefficient γR at 800 nm were extrapolated from experimental data from reference [16]. There are indications that not only plasma generation but also higher order nonlinearities play a significant role in defocussing of filaments in air [17, 18]. The n4 of diamond [19] is one order of magnitude higher than that of the air [20], thus we decided to check whether inclusion of higher order nonlinearity improves modeling results. As already mentioned, diamond is known for its very strong Raman gain. In agreement with other authors [9, 21] the Raman response is taken in the form: where Θ(t) is the Heaviside step function and τ1 and τ2 are the vibrational period and the decay time of the Raman response, respectively. The Raman spectrum of diamond is dominated by a 1332 cm−1 optical phonon which corresponds to τ1 ≈ 4 fs. The oscillation decay time was measured to be τ2 ≈ 5.7 ps [22].Following the approach of Gulley et al. ([5]) we use the following expression for free-current:
where the first term corresponds to photoionization and is responsible for plasma creation, WPI(|A|) is the photoionization rate, Eg is the diamond band gap energy, I = n0ε0c|A|2/2 is the light intensity. The second term is the plasma fluid contribution according to the Drude model [23], whereas is the cross section of inverse Bremsstrahlung, m is the carrier mass, τc is the carrier collision time, and is the inverse free-charge dispersion operator as derived in [5] with g = (−iω0τc)/(1 − iω0τc). The free-carrier plasma density is represented by ρ. To simplify numerical calculations we use only two first components of Ĝ−1 expansion. This is justified as the collision time in diamond was found to be τc = 360 ns [24], so that g ≈ 1.We use Keldysh theory [25] for calculation of photoionization rate WPI. It was shown that measured values can differ even by four orders of magnitude from the results of Keldysh theory [26]. Although some alternative models have been developed [27], still this theory is used in most of studies on supercontinuum generation [2, 5, 6, 18, 28, 29] and thus we use it as the starting point for this study.
We use the simplest rate equation for free-carrier plasma density:
where we have neglected avalanche ionization and plasma decay, retaining only the photoionization term. Avalanche ionization is considered to have minor effect for pulses shorter than 100 fs [7]. The reduced hole-electron mass and band gap energy Eg can be found in [30].It is worth noting that alternative models, where higher orders of diffraction are taken into account, have also been considered by other authors [8, 31]. These models are particularly interesting in cases of self-focusing of extremely focused beams, where plasma generation is not considered. However, in our model it is not the case, because just plasma generation prevents catastrophic self-focusing.
The input pulse for the simulation is the Gaussian shape chirped pulse:
where A0 is the maximal amplitude, w0 = 7μm is the beam waist in the focus, C is dimensionless chirp parameter and σ is the time width of the pulse; which is back propagated in vacuum conditions by 3.7 mm to generate the pulse of similar waist (150 μm) and convergence as in the experiment.4. Simulation details
The simulation was performed with symmetric split-step Fourier method with local-error step-size control [32]. The error of this method scales with 4th power of step-size, and the goal accuracy can be easily chosen. For local error of 10−3 about 150 steps were made with step sizes going from 40 to 100 μm.
The problem has a cylindrical symmetry so the amplitude of electric field was treated as a function of radial distance from propagation axis and retarded time (A(r, t)) corresponding to the radial component of wavevector kr and to angular frequency detuning ω(Ã(kr, ω)) in Fourier space.
The dispersion and diffraction terms are evaluated in Fourier space where derivatives over time and space change into multiplication by frequencies and wavenumbers, respectively. While transition from time to frequency is done with Fourier transform, transition to Fourier space with cylindrical variables has to be done with Discrete Hankel Transform [33]. The nonlinear polarization and free current terms are evaluated in time and in radial coordinate space. Here time derivatives of the T̂ and Ĝ−1 operators are calculated with five point stencil. The plasma equation (8) is solved at every propagation step for all r’s and t’s with Runge-Kutta method. The field envelope A(r, t) is discretized into grid of 1024 time on 512 radial points.
The calculations are performed on computer graphical card, the method is implemented in MATLAB environment and in the graphical card native CUDA C programming language.
5. Simulation results
Figure 4 presents the simulated spectra for different simulation conditions. The inclusion of higher order nonlinearity was crucial in our calculations; the value of n4 was set as a free parameter. In the present model the strong self-focusing in absence of n4 creates the plateau at the short wavelengths; this feature is created at the very beginning of the sample (first 100μm) and it is smoothened off by further propagation (see blue dashed line in Fig. 4). In such a case, although the plateau is practically cancelled, self-phase modulation broadens substantially the central peak on the blue side, whereas the plasma generation prevents its broadening on the red side. The inclusion of n4 contribution suppresses self-focusing: in our model we had to increase the size of the 4th order nonlinearity by one order of magnitude in order to reproduce the experimental data in the best way. The reason of these discrepancies most likely comes from not accurate treatment of plasma effects, in particular, the possibility of higher (than in Keldysh theory) photoionization rate in high laser field intensities regime has been discussed (see [27] and the references therein). This effect should also suppress self-focusing in our case.
Fig. 4 Experimentally measured spectrum of supercontinuum (solid black line), simulation with all contributions included (red dashed line), without stimulated Raman scattering term (green dashed and dotted line), without n4 nonlinearity (blue dotted line) in linear (a) and logarithmic (b) scale. The inset presents enlarged vicinity of stimulated Raman scattering peak.
We have found that only with inclusion of the stimulated Raman response the model reproduces anti-Stokes peak around 720 nm. This fact confirms our belief that the stimulated Raman scattering can be easily observed in the spectrum of supercontinuum generated in diamond.
By manipulating with the pulse chirp, in the simulation, one can reproduce intensity dependent spectral fringes which are visible on the short wavelength side in the experiment. Figure 5 presents the results of the simulation for different values of chirp parameter. For negative values of chirp the spectral fringes become stronger. However, as the exact shape of experimentally observed fringes (see Fig. 2(a)) could not be reproduced, for clarity, we used unchirped pulses in the above considerations.
Fig. 5 Results of the simulation for different values of chirp parameter C. The fringes on the short wavelength site of the spectrum become more pronounced for negative values of chirp (see the inset). For clarity the spectra are vertically shifted from each other.
The values of parameters not specified in the text are listed in Table 1.
Table 1. The values of parameters used in simulation.
6. Conclusion
In the present paper the results of supercontinuum generation in diamond are presented for a wide range of input energies. We have found that the blue side of the generated spectra is limited to 600 nm, so that diamond can not compete with other broadly used white light sources based on laser amplifier. The experimental results have been described by means of the nonlinear envelope equation Eq. (1). We have analyzed the influence of particular nonlinear effects on the spectrum of the generated supercontinuum, showing that in the case of diamond the effects of stimulated Raman scattering can be observed. In the experiment it manifests itself mainly in the presence of a peak shifted by 1332 cm−1 from the pulse central frequency, what has been successfully reproduced by our model. The fact, that in the case of diamond the stimulated Raman response weakly influences the generated supercontinuum spectrum, confirms reasonability of neglecting it’s influence in other bulks, like fused silica or sapphire, where Raman response is much weaker.
The present study revealed that the supercontinuum generated in diamond is limited to 600 nm. This fact, however, do not exclude the possibility of using diamond supercontinuum in various applications, like spectroscopy, optical coherent tomography or carrier-envelope phase stabilization. In particular the behavior of supercontinuum spectrum in infrared region and generations with different pump sources, like femtosecond laser oscillators, are interesting topics for investigation from the point of view of above applications.
Acknowledgments
We acknowledge support of the Foundation of Polish Science MPD Program co-financed by the EU European Regional Development Fund. This contribution has been supported by Ministry of Science and Higher Education, project no. NN202053140. One of us ( W. G.) was supported by Project NCB:R/ERA-NET MATERA+/01/2011.
References and links
1. M. Bradler, P. Baum, and E. Riedle, “Femtosecond continuum generation in bulk laser host materials with sub-μ J pump pulses,” Appl. Phys. B 97, 561–574 (2009). [CrossRef]
2. F. Silva, D. R. Austin, A. Thai, M. Baudisch, M. Hemmer, D. Faccio, A. Couairon, and J. Biegert, “Multi-octave supercontinuum generation from mid-infrared filamentation in a bulk crystal,” Nature communications 3, 807 (2012). [CrossRef] [PubMed]
3. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997). [CrossRef]
4. A. L. Gaeta, “Catastrophic collapse of ultrashort pulses,” Phys. Rev. Lett. 84, 3582–3585 (2000). [CrossRef] [PubMed]
5. J. R. Gulley and W. M. Dennis, “Ultrashort-pulse propagation through free-carrier plasmas,” Phys. Rev. A 81, 033818 (2010). [CrossRef]
6. N. Y. Vislobokov and A. P. Sukhorukov, “Supercontinuum generation by ultra-high power femtosecond laser pulses in dielectrics,” Phys. Wave Phenom. 17, 11–14 (2009). [CrossRef]
7. A. Wu, I. Chowdhury, and X. Xu, “Femtosecond laser absorption in fused silica: numerical and experimental investigation,” Phys. Rev. B 72, 085128 (2005). [CrossRef]
8. P. Whalen, J. V. Moloney, a. C. Newell, K. Newell, and M. Kolesik, “Optical shock and blow-up of ultrashort pulses in transparent media,” Phys. Rev. A 86, 033806 (2012). [CrossRef]
9. R. H. Stolen, “Raman gain in glass optical waveguides,” Appl. Phys. Lett. 22, 276 (1973). [CrossRef]
10. J.-P. M. Feve, K. E. Shortoff, M. J. Bohn, and J. K. Brasseur, “High average power diamond Raman laser,” Opt. Express 19, 913–22 (2011). [CrossRef] [PubMed]
11. M. Zhi, X. Wang, and A. V. Sokolov, “Broadband coherent light generation in diamond driven by femtosecond pulses,” Opt. Express 16, 12139–47 (2008). [CrossRef] [PubMed]
12. A. A. Kaminskii, R. J. Hemley, J. Lai, C. S. Yan, H. K. Mao, V. G. Ralchenko, H. J. Eichler, and H. Rhee, “High-order stimulated Raman scattering in CVD single crystal diamond,” Laser Phys. Lett. 4, 350–353 (2007). [CrossRef]
13. A. Brodeur and S. L. Chin, “Ultrafast white-light continuum generation and self-focusing in transparent condensed media,” J. Opt. Soc. Am. B 16, 637–650 (1999). [CrossRef]
14. A. Brodeur and S. Chin, “Band-gap dependence of the ultrafast white-light continuum,” Phys. Rev. Lett. 80, 4406–4409 (1998). [CrossRef]
15. G. Ghosh, Handbook of Optical Constants of Solids: Handbook of Thermo-Optic Coefficients of Optical Materials With Applications (Academic Press, 1998).
16. M. D. Levenson and N. Bloembergen, “Dispersion of the nonlinear optical suseptibility tensor in centrosymmetric media,” Phys. Rev. B 10, 4447–4463 (1974). [CrossRef]
17. N. Akozbek, M. Scalora, C. M. Bowden, and S. L. Chin, “White-light continuum generation and filamentation during the propagation of ultra-short laser pulses in air,” Opt. Commun. 191, 353–362 (2001). [CrossRef]
18. C. Brée, A. Demircan, and G. Steinmeyer, “Saturation of the all-optical Kerr effect,” Phys. Rev. Lett. 106, 183902 (2011). [CrossRef] [PubMed]
19. B. Ratajska-Gadomska, “Influence of the interaction between dipoles, optically induced in a crystal lattice, on the nonlinear refractive index of crystals,” Phys. Rev. B 26, 1942–1958 (1982). [CrossRef]
20. V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air component,” Opt. Express 17, 13429–13434 (2009). [CrossRef] [PubMed]
21. A. M. Zheltikov, “Understanding the nonlinear phase and frequency shift of an ultrashort light pulse induced by an inertial third-order optical nonlinearity,” Phys. Rev. A 79, 023823 (2009). [CrossRef]
22. K. Lee, B. J. Sussman, J. Nunn, V. Lorenz, K. Reim, D. Jaksch, I. Walmsley, P. Spizzirri, and S. Prawer, “Comparing phonon dephasing lifetimes in diamond using transient coherent ultrafast phonon spectroscopy,” Diam. Relat. Mater. 19, 1289–1295 (2010). [CrossRef]
23. B. Sapoval, C. Hermann, and C. Hermann, Physics of Semiconductors (Springer, 2003).
24. P. Ščajev, V. Gudelis, E. Ivakin, and K. Jarašiunas, “Nonequilibrium carrier dynamics in bulk HPHT diamond at two-photon carrier generation,” Phys. Status Solidi (a) 208, 2067–2072 (2011). [CrossRef]
25. L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” J. Exptl. Theoret. Phys. 47, 1945–1957 (1964).
26. M. Lenzner, J. Krüger, S. Sartania, Z. Cheng, C. Spielmann, G. Mourou, W. Kautek, and F. Krausz, “Femtosecond optical breakdown in dielectrics,” Phys. Rev. Lett. 80, 4076–4079 (1998). [CrossRef]
27. V. E. Gruzdev, “Analysis of the transparent-crystal ionization model developed by L V Keldysh,” J. Opt. Technol. 71, 504–508 (2004). [CrossRef]
28. M. Mlejnek, E. M. Wright, and J. V. Moloney, “Dynamic spatial replenishment of femtosecond pulses propagating in air,” Opt. Lett. 23, 382–384 (1998). [CrossRef]
29. M. R. Junnarkar, “Short pulse propagation in tight focusing conditions,” Opt. Commun. 195, 273–292 (2001). [CrossRef]
30. H. Löfås, A. Grigoriev, J. Isberg, and R. Ahuja, “Effective masses and electronic structure of diamond including electron correlation effects in first principles calculations using the GW-approximation,” AIP Advances 1, 032139 (2011). [CrossRef]
31. M. Trippenbach, W. Wasilewski, P. Kruk, G. W. Bryant, G. Fibich, and Y. Band, “An improved nonlinear optical pulse propagation equation,” Opt. Commun. 210, 385–391 (2002). [CrossRef]
32. O. V. Sinkin, R. Holzlöhner, J. Zweck, and C. R. Menyuk, “Optimization of the split-step Fourier method in modeling optical-fiber communications systems,” J. of Lightwave Technol. 21, 61–68 (2003). [CrossRef]
33. A. E. Siegman, “Quasi fast Hankel transform.” Opt. Lett. 1, 13–15 (1977). [CrossRef] [PubMed]
