Open Access
Add to BibSonomy
Abstract
Polycrystalline materials can mediate efficient frequency up-conversion for mid-infrared light. Motivated by the need to understand the properties of the harmonic and supercontinuum radiation from such media, we utilize realistic numerical simulations to reveal its complex temporal and spatial structure. We show that the generated radiation propagates in the form of long-duration pulse trains that can be difficult to compress and that optical filamentation in high-energy pulses gives rise to fine-structured beam profiles. We identify trends concerning pulse energy, sample length, and the microstructure of the material that can inform optimization for different applications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the mid-infrared (MIR) spectral region, ultrafast laser sources are cornerstones for numerous research fields, including strong field physics [1], molecular fingerprint spectroscopy [2] and material processing [3], to name a few. The generation of tunable and coherent MIR radiation relies heavily on nonlinear frequency conversion utilizing $\chi ^{(2)}$ processes [4] which depend on the relative phase velocities of the interacting waves. Ensuring proper phase-matching conditions can be a challenging task, especially for few-cycle pulses with broad spectral bandwidth.
Phase-matching in birefringent crystals and quasi-phase-matching (QPM) [5] in periodically poled materials are two main strategies to optimize the nonlinear conversion yield. As an alternative approach, the random quasi-phase-matching (RQPM) mechanism in polycrystalline materials has attracted considerable interest since M. Baudrier-Raybaut et al. marked its potential of being a building block in a new generation of efficient nonlinear frequency converters [6]. The field has gradually broadened as more solid-state materials [7–9] and waveguides [10] known to exhibit random structures show remarkable capability for supporting efficient frequency conversion over a wide spectral region. Zinc-blende semiconductors in their polycrystalline form have been extensively studied as suitable candidates for implementing RQPM [11–15]. High harmonic generation (HHG) and octave-spanning MIR supercontinua (SC) were observed in polycrystalline zinc selenide (ZnSe) and zinc sulfide (ZnS) [11,13,14]. Ultra-broadband MIR source extending from $2.7$ to $20~\mathrm{\mu}$m was also reported from intra-pulse difference frequency generation (IPDFG) [16]. Optical parametric oscillators (OPO) [17] and amplifiers [18] based on ZnS or ZnSe ceramics present new opportunities for multi-octave frequency comb generation. Second harmonic generation (SHG) in Strontium tetraborate (SBO) offers avenues for reaching the vacuum ultraviolet (VUV) spectral region in a bulk medium [19].
The distinctive features of RQPM include flat response over unparalleled bandwidth extending to the material transparency limit, and linear growth of conversion efficiency with sample thickness. However, the experimental reports in this area are mostly focusing on the observation of spectra, quantification of the conversion efficiency, and observations of beam spatial evolution (including transverse filamentation patterns [12,13] and refocusing-cycles [20]). It was pointed out, in Ref. [13], that RQPM may have a negative impact on the temporal coherence of the generated radiation. To our knowledge, there has been no detailed investigation into the properties of the upconverted light, which can be of crucial importance for practical applications.
The main objective of this paper is to study the spatial and temporal coherence properties of the RQPM-generated harmonics and the accompanying supercontinuum. In our recent work [21], a hierarchy of numerical models were developed for realistic simulations of pulse propagation in polycrystalline media, and we put these capabilities to use in this work. In particular, we use fully resolved in time and space simulations to show that while extremely wide bandwidth and very high efficiency can be readily obtained in samples only a few millimeters thick, the utility of the generated light may vary from case to case. Typically, the temporal profiles of the harmonic and supercontinuum radiation is complex. It is spread to duration significantly beyond that of the pump pulse, and this upconverted radiation exhibits very limited compressibility. In terms of the spatial properties, even modest pulse energies lead to formation of evolving hot spots across the beam. Nevertheless, understanding of these trends makes it possible to optimize experimental setups for a specific goal, and this work aspires to inform such efforts.
2. Numerical simulations
Modeling methods applied in this paper were first introduced in Ref. [21] and we refer the reader to that work for technical details. However, for the sake of completeness, we recap the approach here, and provide typical simulation parameters next.
Our simulation of pulse propagation in polycrystalline media is based on the z-propagated version of unidirectional pulse propagation equation (UPPE) [22,23] in the spectral domain
The nonlinear polarization $\vec {P}$ includes both second and third order nonlinear effects. ZnSe is linearly isotropic due to its cubic structure, but exhibits strong second-order nonlinearity, of which the response is dependent on the grain structure. We use $d = 30$ pm/V as the second-order nonlinear coefficient of ZnSe [25], and the nonlinear refractive index $n_2 = 1.2 \times 10^{-18}~$m$^2$/W is obtained from Z-scan measurements in Ref. [13].
The focus of the numerical model is to provide rigorous description for the RQPM-associated three-wave-mixing process in realistic grain structures. Our approach starts by mapping a 3D polycrystalline structure using Voronoi tessellation [26], and assigning a uniformly sampled random orientation to each grain. Random rotation matrices described in Ref. [27] are used for characterization of the grain orientation with respect to the laboratory frame, and for the transformation of physical quantities between different frames of reference. At each spatial grid point, the local second-order polarization response is calculated from the history of the fields projected to the crystal principal axes, and transformed back to the laboratory frame to be incorporated into the field evolution along the propagation distance.
The main limitation of the fully resolved (3+1)D model comes from the need of fine spatial and temporal resolution, and therefore, massive computation power. A simplified approach, which we also use for certain qualitative simulations, is the single-grain model, in which we assume the beam size is much smaller than the average grain size, and all transverse points in the beam cross section fit into the same grain.
3. Results
3.1 Temporal pulse properties and coherence
The most readily measurable manifestations of the nonlinear light propagation in polycrystalline materials are of course the spectra, and this is where we start our investigations. A number of experiments, modeling [11–14,28], and also our own previous simulations [21] showed that nonlinear propagation in ZnSe gives rise to high harmonic and ultra-broadband supercontinuum generation extending up to the short-wavelength limit of the crystal transparency window. Figure 1 presents simulation results for an incident pulse at the central wavelength of $3.6~\mathrm{\mu}$m, pulse energy of $2.0~\mathrm{\mu}$J, and temporal duration of $60$ fs. These parameters are representative of a typical experimental setup, and the characterization of harmonics in this work are based on these parameters unless otherwise stated. Figure 1(a) depicts the harmonic and supercontinuum spectra for several sample thicknesses, indicating both very high conversion efficiency and extreme bandwidth. These simulated results are very much in line with experimental observations.
Fig. 1. (a) Simulated angle-integrated spectrum in 1 mm, 2 mm, and 3 mm thick ZnSe, with input pulse energy of 2$~\mathrm{\mu}$J and initial duration of 60 fs. The input beam has initial $1/e^2$ intensity radius of 60 $~\mathrm{\mu}$m, and average grain size of the sample is 60$~\mathrm{\mu}$m. Dashed line shows the spectrum of the initial pulse. Each spectrum is normalized to its peak spectral power at the fundamental frequency separately. (b) Energy depletion of the initial pulse and the growth of 2nd to the 5th harmonics with propagation distances.
Figure 1(b) illustrates the RQPM in action for separated frequency bands, showing that the energy in selected harmonics continues to increase with the sample thickness. For this illustration, the energy fractions of the fundamental and of the harmonics are measured after band-pass filters with cutoff frequencies corresponding to the troughs in the spectrum shown in panel (a).
Remarkably, the SHG efficiency can reach up to $21\%$, $32\%$, and $41\%$ for sample thicknesses of $1$, $2$, and $3$ mm, respectively. In the traditional QPM, only nonlinear processes for which the phase lag is precisely engineered to be in tune with the inverse of the periodic domain would be allowed to grow quadratically. Akin to QPM, grain microstructure with randomly orientated crystallites can be seen as an aggregate of randomly alternating domains, which cause the relative phases of interacting waves to experience a random walk [6]. The accumulation of many field increments with random phases results in the amplitude that grows with the square root of the number of crystallites traversed, and thus allows near-to-linear growth of the conversion yield. This is evident in Fig. 1(b), where the second and higher harmonic energies continue to increase nearly linearly with the propagation distance. The deviations from the linear behavior can be ascribed to the depletion of the fundamental energy (in the case of 2nd and 3rd harmonic) and/or increase of the energy in the lower harmonics that “pump” the higher harmonics (4th and 5th).
Incidentally, this high conversion efficiency or the angle-integrated spectra shown above says little about the properties of the newly generated waveforms, in particular their duration or temporal coherence. In contrast, an angularly resolved spectrum can be a powerful tool for interpreting pulse propagation dynamics [29–31]. For homogeneous bulk media in which chromatic dispersion and phase matching play important roles, angular spectra are richly structured and exhibit X-waves [32] or O-waves [33] with well-localized contours along which the radiation energy concentrates. Such contours are witness to the existence of localized sub-pulses propagating with well-defined velocities. Angular resolved spectra can be a useful tool to characterize radiation generated from polycrystalline material, too. Figure 2 compares the simulated angular resolved spectra for samples with average grain sizes of $60~\mathrm{\mu}$m and $20~\mathrm{\mu}$m. First, one can notice immediately that the spectra exhibit overall noisy texture, which is an indication of incoherent superposition of temporally distinct pulses or long-duration pulse trains. As another signature of a complex temporal structure, the spectra exhibit no sharply defined contours, which means that even if well-shaped sub-pulses occur, they do not last for a long propagation distance. Only a vaguely defined contour-structure appears in both spectra, extending from the locus of the second harmonic. It shows a trace of phase-matched growth due to a pulse that propagates with a velocity slightly above the group velocity that would correspond to its central wavelength. It is a sign that for this process the coherence length matches the average grain size [6,15,28]. In Fig. 2(b) the contour-structure becomes more distinct, extending beyond the fourth harmonic, showing that nonlinear processes with relatively large phase-mismatch can benefit from the existence of smaller size grains.
Fig. 2. Logarithmic angularly resolved spectra for input central wavelength of $3.6~\mathrm{\mu}$m after propagation in a 3 mm-thick ZnSe sample with average grain size of (a) $60~\mathrm{\mu}$m and (b) $20~\mathrm{\mu}$m. High harmonic generation up to the 7th order can be observed. The input pulse has initial energy of 2 $\mathrm{\mu}$J, temporal duration of 60 fs, and $1/e^2$ intensity radius of 60 $~\mathrm{\mu}$m.
These observations indicate that simply adding angular resolution to the spectrum already suggests that the coherence of the upconverted radiation will be significantly worse than in homogeneous bulk media. We thus propose that the measurement of the angularly resolved spectrum could become a relatively inexpensive but useful tool for the characterization of pulse features in polycrystalline media. Note that such spectra can be measured easily and have been studied extensively in dispersive bulk media [34,35]. It will be, on the other hand, harder to obtain precise measurements of the temporal profiles of pulses with greatly extended bandwidth. This is where simulations can help, and we will discuss this issue next.
For a broad spectrum, such as that in Fig. 1, temporal pulse walk-off is expected to be significant. As an example, Fig. 3(a-d) compares the real electric field of the whole pulse and the filtered third harmonic (spectral range from $0.99~\mathrm{\mu}$m to 1.4$~\mathrm{\mu}$m) at various propagation distances. As the wavelengths of the fundamental and third harmonics both fall into the material’s normal dispersion range and the third harmonic experiences stronger group velocity dispersion (GVD), every part of previously generated harmonics lags behind, forming a long tail. Meanwhile, the third harmonic can continuously accumulate from either sum frequency generation (SFG) of the fundamental and the second harmonic, or via direct third harmonic generation (THG). New additions of harmonics do not lead to a steady growth of peak intensity, but instead turn into increasing number of sequential sub-pulses with comparable amplitude spreading in time. The temporal dynamics also translate into structured spectral phase, as shown in Fig. 3.
Fig. 3. Top: The temporal profiles of the third harmonic compared to its parent pulse at the propagation distance of a) $z= 0.5$ mm, b) $z=1.0$ mm, c) $z=2.0$ mm, and d) $z=3.0$ mm. The polarization component parallel to the pump is shown. Bottom: Normalized spectral power and corresponding spectral phases of the third harmonic for the sample thickness of e) $z=0.5~$mm, f) $z=1.0~$mm, g) $z=2.0~$mm, and h) $z=3.0~$mm. Simulation parameters are identical to those in Fig. 1.
The growing duration of the 3rd harmonic in Fig. 3 is almost in direct proportion to sample thickness, making clear that material dispersion is the major reason for pulse temporal stretching. Figure 3(e-h) gives the spectral phases after removing envelope phase offset and first-order dispersion around central frequency. At the wavelength of $1.16~\mathrm{\mu}$m ZnSe has a GVD $\sim 556$ fs$^2$/mm and a third-order dispersion (TOD) $\sim$ 474 fs$^3$/mm [36], which dominates the shape of curves in Fig. 3(e,f). In Fig. 3(g) and (h) the newly generated frequency components in the blue-shifted region of the spectrum continue to add more random fluctuations into the spectral phase. The discontinuities in the unwrapped phases exist as an evidence for the loss of coherence, which casts doubt on the compressibility of the pulses.
Studying the evolution of the pulse autocorrelation can help to further quantify the loss of coherence along the propagation distance or sample length. Given the temporal waveform $U(t)$, the autocorrelation of the pulse can be calculated as
where $\tau$ represents the time delay. We have calculated this for the frequency filtered harmonic bands at different propagation distances. Figure 4 shows the change of temporal profiles (black lines) and the absolute value of the autocorrelation (blue lines) for the third harmonic for increasing propagation distances (different columns).As the interaction length increases, the pulse envelope behaves exactly as previously presented in Fig. 3: it splits into multiple sub-peaks and the waveform becomes more and more complex as it propagates through the sample. The autocorrelations also evolve into structures consisting of a progressively narrower central spike indicating the loss of coherence, with a surrounding pedestal revealing the growing waveform duration.Fig. 4. Evolution of the temporal envelope (bottom, black lines), autocorrelation traces (top, blue lines) and the optimally compressed pulse obtained by numerically compensating the second- and third-order dispersion (middle, red lines) of the third harmonic with increasing crystal lengths: a) $z=0.5$ mm, b) $=1$ mm, and c) $z=2$ mm. Simulation parameters are identical to those in Fig. 1.
We found the behaviors of all orders of generated harmonic to be qualitatively very similar to that shown for the third harmonic, which suggests that RQPM is the dominating factor in shaping the pulse’s temporal waveforms. While for higher order harmonics, the number of available paths for frequency up-conversion is larger, the temporal walk-off is faster, which only lead to more structured temporal profiles and quicker deterioration of temporal coherence.
Having seen that the harmonic radiation is hardly organized into simple pulses, one can ask if they are at least compressible. Practically speaking, it is reasonable to assume that in an attempt to optimize the harmonic pulse duration, one can mostly control and compensate the group delay dispersion (GDD) and the TOD. We numerically optimized the compensation of the spectral phase by adding varying amounts of GDD and TOD, while searching for an optimal compressed pulse duration. The cost function to optimize was defined by the full-width at half-maximum (FWHM) pulse duration, which appears to be dominated by the width of the central spike. It turns out that the achievable compression is far from ideal. This is illustrated for the example of the third harmonic for various sample thicknesses in Fig. 4 (red lines). Although the broadened spectrum can, in principle, support shorter transform-limited duration, the loss of temporal coherence has significantly compromised its practical compressibility. As the crystal thickness increases, the overall temporal profile becomes less sensitive to change of additional dispersion. The take away here is that the up-converted light is an unlikely candidate for broadband light-field synthesis unless further refined phase manipulation can be achieved.
However, it is also interesting to note that the coherence properties of the broadened fundamental are, to a great extent, immune to the randomness-induced deterioration. This is in line with the fact that the energy back-flow is hindered by lack of temporal overlap and coherence between different orders of harmonics. Due to the dominating effect of the second-order nonlinearity in shaping the spectral content, the phase acquired from phase-mismatched harmonic generation process mimics an effective negative $n_2$, known as cascading of quadratic nonlinearities (CQN) [37–39]. When the absolute value of $\textrm{n}_{2,CQN}$ approaches the nominal nonlinear refractive index of the material, it counteracts the effect of self-focusing and self phase modulation (SPM). In Fig. 5 we observe the initial five-cycle pulse evolving into a nearly 3-cycle waveform within 2 mm of propagation, and a 2-cycle duration is obtainable with proper dispersion management. The existence of small grains can also be considered an advantage here: grain size control may enable engineering of the effective third-order nonlinearity, without experiencing the manufacturing difficulties of periodically poled structures.
Fig. 5. Evolution of the temporal envelope (bottom, black lines), autocorrelation traces (top, blue lines) and the optimally compressed pulse obtained by numerically compensating the second- and third-order dispersion (middle, red lines) of the fundamental frequency band with increasing crystal lengths: (a) $z=0.5$ mm, (b) $=1$ mm, and (c) $z=2$ mm. Simulation parameters are identical to those in Fig. 1.
3.2 Spatial properties
In the case of relatively loose focusing and strong incident power, beam breaking up into multiple hot spots is almost inevitable, and multi-filamentation will impact the beam uniformity and focusability. Figure 6 depicts the filament traces and hot spot formation in 2-mm-thick samples with different grain sizes. We have previously reported that smaller crystallites result in greater conversion efficiency, especially for higher harmonics [21]. However, the price to pay is the faster accumulation of spatial inhomogeneity, more complicated filamentary structure, and a smaller characteristic hot spot size. As shown in Fig. 6(a) and (f), for both large (a) and small (f) grain sizes the spatial modulations build up and turn into multiple hot spots within a short distance of $\sim 0.2$ mm. The hot spots may circulate around the beam periphery as illustrated in Fig. 6(i), or they may concentrate around the axis as shown in panel (j). Although the beam size remains almost constant and the pointing variations of the centroid remain below 4$\%$ of the rms radius, a considerable fraction of the energy propagates off-axis.
Fig. 6. Multi-filament formation and near-field fluence maps for 2mm-thick ZnSe sample with (a-e) grain size of 60 $\mathrm{\mu}$m, (f-j) grain size of 20 $\mathrm{\mu}$m. Spatial fluence profiles correspond to beam cross sections at sample thicknesses of (b,g) 0.2 mm, (c,h) 0.5 mm, (d,i) 1.0 mm, and (e,j) 2.0 mm. The input pulse has initial energy of 2 $\mathrm{\mu}$J, temporal duration of 60 fs, and $1/e^2$ intensity radius of 60 $~\mathrm{\mu}$m.
Let us next look at the polarization properties of the up-converted radiation. Figure 7 shows the transverse fluence profiles for parallel and perpendicular polarization components of the second and the third harmonics in the sample with 60 $\mathrm{\mu}$m crystallites. Although the number and location of individual filaments show random variation with the propagation distance, within the first two millimeters most of the hot spots arise around the high intensity ring which develops around the edge of the beam and persists over a distance of more than one millimeter (see Fig. 6(a,d,e) ). The approximate size of locally nucleated spots of the third harmonic is clearly smaller than those of the second harmonic, but there is hardly any correlation between the spatial distribution of them. Considerable field depolarization along propagation means that both polarization components of the harmonics attain comparable power, but they are distributed at different random locations and are to evolve independently to some degree.As more and more emerging applications require supercontinua with high spectral power, it is also necessary to study the impact of the grain microstructure on power and energy upscaling. Figure 8 shows a broader beam width using peak power of 115 $\textrm{P}_{\textrm{cr}}$, where $\textrm{P}_{\textrm{cr}}$ is the critical power for self-focusing defined by $\textrm{P}_{\textrm{cr}} = 3.77\lambda ^2/(8\pi n_0n_2)$. $\textrm{P}_{\textrm{cr}}$ is approximately 0.67 MW for ZnSe at the central wavelength of 3.6 $\mathrm{\mu}$m. The beam is fragmented into a larger number of randomly nucleated hot spots exhibiting smaller characteristic sizes in the early stage of propagation. But over longer distances they quickly merge into a few clusters. The final number of structures does not show a sharp increase at this higher input power. Moreover, along the propagation the size of the core area within which the hot spots exist remains almost the same, while the overall beam size increases as it expands into a low intensity background. This suggests that in practice a tight-focusing geometry and/or smaller input beam size can still be effective methods of controlling or even eliminating complicated multi-filamentation patterns.
Fig. 7. Polarization properties. Transverse fluence distribution of the second (a-f) and the third (g-l) harmonics within a 3mm-thick ZnSe sample. The input power $\textrm{P}_{\textrm{in}}\sim 45~\textrm{P}_{\textrm{cr}}$. The corresponding spatial distribution of the fundamental is shown in Fig. 6(a). Each column represents different propagation distance: (a,d,g,j) 1 mm; (b,e,h,k) 2 mm; and (c,f,i,l) 3 mm. And each row represents different polarization components: (a-c) parallel polarization of the second harmonic; (d-f) perpendicular polarization of the second harmonic; (g-i) parallel polarization of the third harmonic; and (j-l) perpendicular polarization of the third harmonic. Simulation parameters are identical to those in Fig. 1.
Fig. 8. Numerically simulated filamentation patterns showing the sum of both parallel and perpendicular polarization components. Top row represents the second harmonic and the second row represents the third harmonic. Each column corresponds to snapshots at different propagation distances within the sample: (a,b) 1 mm; (c,d) 3 mm, (e,f) 5 mm. The initial pulse energy is 5 $\mathrm{\mu}$J, input power $\textrm{P}_{\textrm{in}}\sim 115~\textrm{P}_{\textrm{cr}}$, temporal duration is 60 fs, and the initial beam size is 100 $\mathrm{\mu}$m FWHM.
For a comparison, beam patterns were measured experimentally utilizing laser setups described in details in Ref. [13,40]. Pulses with a central wavelength of 3.6 $\mathrm{\mu}$m were focused to a spot size of 98.5 $\mathrm{\mu}$m FWHM onto ZnSe samples with thicknesses ranging from 5 to 40 mm, and spatial profiles of harmonic bands were imaged. The filtered spatial distributions of the second and the third harmonics in Fig. 9 correspond to spectral ranges of 1.5-1.8 $\mathrm{\mu}$m and 1.0-1.5 $\mathrm{\mu}$m, respectively. These results are in line with the general trends seen in numerical simulations: higher input pulse energies result in a significantly larger number of hot spots, and as the propagation distance increases, they reorganize into fewer and stronger speckles, within which individual filamentary structure becomes indistinguishable. Mismatch of spatial spikes between different orders of harmonics clearly poses difficulty to deterministic control of multi-filamentation pattern and simultaneous manipulation of the broad frequency bands using techniques such as spatial grids [41] or microlens arrays [42]. The number of speckles seen in numerical simulations, however, is considerably fewer as compared to experimental observation. We speculate that this is due to our limited capability of reproducing realistic grain microstructure distribution, or the lack of pump beam fluctuations in the computation model.
Fig. 9. Experimental images of near field intensity profiles of the second (top row) and the third (second row) harmonic at the back surface of ZnSe samples. Intensities indicated in arbitrary units are presented in a linear universal scale for comparison. Each column represents different initial pulse energy or sample thicknesses: (a,b) pulse energy 5 $\mathrm{\mu}$J, sample thickness 5 mm, $\textrm{P}_{\textrm{in}}\sim 30~\textrm{P}_{\textrm{cr}}$; (c,d) pulse energy 22 $\mathrm{\mu}$J, sample thickness 5 mm, $\textrm{P}_{\textrm{in}}\sim 130~\textrm{P}_{\textrm{cr}}$; (e,f) pulse energy 14.2 $\mathrm{\mu}$J, sample thickness 40 mm, $\textrm{P}_{\textrm{in}}\sim 85~{\rm P}_{\textrm{cr}}$. The incident pulses have fixed focused spot sizes of approximately 98.5 $\mathrm{\mu}$m FWHM. The grain size distribution of the samples are $(60 \pm 40)$ $\mathrm{\mu}$m.
3.3 Interplay between spatial evolution and temporal compressibility
It is a well-known fact that when a beam breaks up into a multi-filament structure, its temporal dynamics are strongly correlated to the spatial evolution [12,43,44]. Each hot spot evolves as a distinct source for supercontinuum generation, which leads to not only degraded spatial coherence but also to further loss of the temporal coherence. On the other hand the spatial break-up works together with the RQPM process to enable continuous increase of the upconverted energy, as harmonic radiation from one hot spot is unlikely to be converted back to the fundamental in a spatially distinct hot spot. Clearly, combined influences of multi-filamentation and RQPM can affect pulse compressibility. In this section we show that RQPM is the dominant and harder-to-control process. For this purpose, we turn to single-filament regimes and compare the results with those obtained above from multi-filamentation.
Figure 10 shows the properties of the third harmonic in a single-filament situation, with settings the same as in Fig. 4, except that the pulse energy is lowered to $0.4~\mathrm{\mu}$J to avoid beam break-up. Comparison of Fig. 4(a) and Fig. 10(a) shows that when the spatial beam break-up is relatively weak (see Fig. 6(c) for the spatial profile corresponding to Fig. 4(a) ), the autocorrelations exhibit almost equally broad peaks. Comparing Fig. 4(b,c) to Fig. 10(b,c), one can see that as the spatial complexity increases with propagation, the temporal complexity in Fig. 4(b,c) is due to multiple filaments. However, the pulses generated from single-filaments appear to be completely insensitive to adjustments in GDD and TOD and therefore still incompressible. In Fig. 10(d,e,f) we also illustrate properties of the fundamental which remains well suited for compression as propagation distance increases. However, as it experiences weaker SPM due to lower pulse energy, the compressibility is not as good as that seen in multi-filament situation.
Fig. 10. Evolution of the temporal profiles (black lines), the optimal pulse-compression by compensating the second- and third-order dispersion (red lines) and autocorrelation traces (blue lines) of third harmonics (top row) and the fundamental (second row) generated from a $0.4~\mathrm{\mu}$J input pulse at (a,d) z = 0.5 mm, (b,e) z = 1.0 mm, and (c,f) z = 2.0 mm. Simulation parameters are identical to those in Fig. 1, except that the pulse energy is lowered to $0.4~\mathrm{\mu}$J to avoid spatial break-up.
Clearly, both spatial and temporal dynamics limit the compressibility and focusability of the harmonics. Our observation suggests that a potential avenue to manage the spatial deterioration could be using multiple thinner slabs (of which the optimal thicknesses would be approximately a few hundreds of micrometers) spaced apart to allow for a diffractive healing of the hot spots while the beam propagates between the slabs.
4. Conclusions
We have numerically investigated the temporal coherence and spatial beam quality of harmonics generated from polycrystals possessing strong random $\chi ^{(2)}$ nonlinearity. While the RQPM mechanism underlying the extreme bandwidths and high conversion efficiency in polycrystalline materials is now well understood, less is known about the properties of the generated light, mainly because the RQPM paradigm lacks the spatial and temporal resolution to say more about the quality of the upconverted radiation. Relying on the realistic modeling of intense pulse propagation, we present a case study of the temporal and spatial properties of the harmonic and supercontinuum radiation generated in polycrystalline media pumped by mid-infrared femtosecond pulses. Our findings can be summarized as follows.
The random nature of the frequency conversion process, while enabling the new frequency generation that scales linearly with the sample thickness, is also responsible for what is rather complex temporal structure of the radiation at frequencies higher than the fundamental. Even when band-filtered, harmonic waveforms consist of pulse trains in which the temporal structure becomes finer and finer along the propagation path. Accordingly, the temporal coherence degrades both for higher pulse energies and longer samples. The temporal duration of the output harmonic signal is given more by the propagation length than the pump-pulse duration, which it can exceed significantly. Moreover, we have demonstrated that the radiation appears rather ill-suited for temporal compression. While these trends are common for all high harmonics, the complexity of the temporal structure increases with the harmonic order. We have found that to maintain good temporal coherence properties the input pulse energy should be relatively low, the sample short, and the mean grain size small.
Hand in hand with the complex temporal structure, optical filamentation at higher pulse energies results in a rich spatial structure. The spatial beam break-up is initiated by the randomness of the amplitude across the beam that encompasses multiple crystal grains, each imprinting a different contribution on the profile of the harmonic radiation. This variation subsequently spreads across the spectral bandwidth through both the second- and third-order nonlinearity. While multiple filaments were previously observed in several experiments, our modeling shows that merely few hundred microns long propagation path in a polycrystal medium is sufficient to create multiple hot spots in the beam. We have demonstrated that smaller crystallite size results in finer filament structure. Spatially complex beams are obviously more difficult to handle, focus, image or couple into waveguides. On the other hand, multiple hot spots are positively contributing to the overall frequency conversion, acting as independent parallel sources.
Importantly, we have found that in terms of spatial and temporal properties, one should distinguish between the spectrally broadened fundamental pulse and the other upconverted radiation. The properties of the former are only marginally affected by either RQPM or spatial inhomogeneity. In fact, the same trends that happen to affect the coherence of the harmonics (high pulse energy, long crystal length, small grain size) may be used to enhance the usability of the fundamental, including spectral broadening and pulse compression.
In conclusion, our study lays bare that the mere fact that a polycrystalline medium can act as an extremely efficient up-converter of mid-infrared light does not guarantee highly coherent outputs. In fact, the temporal and spatial properties of the generated radiation are in general less than optimal, in stark contrast to the fundamental frequency band pulse which exhibits clean spectral broadening while preserving good coherence. We have identified trends in terms of the crystallite size, pulse energy and sample length that can be utilized to optimize a setup and the material properties for a specific application.
Funding
Office of Naval Research (N00014-17-1-2705); Air Force Office of Scientific Research (FA9550-16-1-0121); Army Research Laboratory (W911NF1920192).
Disclosures
The authors declare no conflicts of interest.
References
1. B. Wolter, M. G. Pullen, M. Baudisch, M. Sclafani, M. Hemmer, A. Senftleben, C. D. Schröter, J. Ullrich, R. Moshammer, and J. Biegert, “Strong-field physics with mid-IR fields,” Phys. Rev. X 5(2), 021034 (2015). [CrossRef]
2. C. R. Petersen, U. Møller, I. Kubat, B. Zhou, S. Dupont, J. Ramsay, T. Benson, S. Sujecki, N. Abdel-Moneim, Z. Tang, D. Furniss, A. Seddon, and O. Bang, “Mid-infrared supercontinuum covering the 1.4–13.3 µm molecular fingerprint region using ultra-high NA chalcogenide step-index fibre,” Nat. Photonics 8(11), 830–834 (2014). [CrossRef]
3. I. Mingareev, F. Weirauch, A. Olowinsky, L. Shah, P. Kadwani, and M. Richardson, “Welding of polymers using a 2 µm thulium fiber laser,” Opt. Laser Technol. 44(7), 2095–2099 (2012). [CrossRef]
4. B.-H. Chen, E. Wittmann, Y. Morimoto, P. Baum, and E. Riedle, “Octave-spanning single-cycle middle-infrared generation through optical parametric amplification in LiGaS2,” Opt. Express 27(15), 21306–21318 (2019). [CrossRef]
5. M. M. Fejer, G. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28(11), 2631–2654 (1992). [CrossRef]
6. M. Baudrier-Raybaut, R. Haidar, P. Kupecek, P. Lemasson, and E. Rosencher, “Random quasi-phase-matching in bulk polycrystalline isotropic nonlinear materials,” Nature 432(7015), 374–376 (2004). [CrossRef]
7. L. Mateos, P. Molina, J. F. Galisteo-López, C. López, L. E. Bausá, and M. O. Ramírez, “Ultrabroadband generation of multiple concurrent nonlinear coherent interactions in random quadratic media,” Appl. Phys. Lett. 103(10), 101101 (2013). [CrossRef]
8. M. Ayoub, P. Roedig, J. Imbrock, and C. Denz, “Cascaded Čerenkov third-harmonic generation in random quadratic media,” Appl. Phys. Lett. 99(24), 241109 (2011). [CrossRef]
9. R. Fischer, S. Saltiel, D. Neshev, W. Krolikowski, and Y. S. Kivshar, “Broadband femtosecond frequency doubling in random media,” Appl. Phys. Lett. 89(19), 191105 (2006). [CrossRef]
10. A. Aleksandrovsky, A. Vyunishev, A. Zaitsev, and V. Slabko, “Random quasi-phase-matched conversion of broadband radiation in a nonlinear photonic crystal,” Phys. Rev. A 82(5), 055806 (2010). [CrossRef]
11. R. Kupfer, H. J. Quevedo, H. L. Smith, L. A. Lisi, G. Tiwari, C. G. Richmond, B. Bowers, L. Fang, and B. Hegelich, “Cascade random-quasi-phase-matched harmonic generation in polycrystalline ZnSe,” J. Appl. Phys. 124(24), 243102 (2018). [CrossRef]
12. R. Šuminas, G. Tamošauskas, G. Valiulis, V. Jukna, A. Couairon, and A. Dubietis, “Multi-octave spanning nonlinear interactions induced by femtosecond filamentation in polycrystalline ZnSe,” Appl. Phys. Lett. 110(24), 241106 (2017). [CrossRef]
13. K. Werner, M. G. Hastings, A. Schweinsberg, B. L. Wilmer, D. Austin, C. M. Wolfe, M. Kolesik, T. R. Ensley, L. Vanderhoef, A. Valenzuela, and E. Chowdhury, “Ultrafast mid-infrared high harmonic and supercontinuum generation with n2 characterization in zinc selenide,” Opt. Express 27(3), 2867–2885 (2019). [CrossRef]
14. R. Šuminas, A. Marcinkevičiūtė, G. Tamošauskas, and A. Dubietis, “Even and odd harmonics-enhanced supercontinuum generation in zinc-blende semiconductors,” J. Opt. Soc. Am. B 36(2), A22–A27 (2019). [CrossRef]
15. X. Chen and R. Gaume, “Non-stoichiometric grain-growth in ZnSe ceramics for $\chi ^{(2)}$ interaction,” Opt. Mater. Express 9(2), 400–409 (2019). [CrossRef]
16. J. Zhang, K. Fritsch, Q. Wang, F. Krausz, K. F. Mak, and O. Pronin, “Intra-pulse difference-frequency generation of mid-infrared (2.7–20 µm) by random quasi-phase-matching,” Opt. Lett. 44(12), 2986–2989 (2019). [CrossRef]
17. Q. Ru, N. Lee, X. Chen, K. Zhong, G. Tsoy, M. Mirov, S. Vasilyev, S. B. Mirov, and K. L. Vodopyanov, “Optical parametric oscillation in a random polycrystalline medium,” Optica 4(6), 617–618 (2017). [CrossRef]
18. S. Vasilyev, I. Moskalev, M. Mirov, V. Smolski, S. Mirov, and V. Gapontsev, “Ultrafast middle-IR lasers and amplifiers based on polycrystalline Cr: ZnS and Cr: ZnSe,” Opt. Mater. Express 7(7), 2636–2650 (2017). [CrossRef]
19. P. Trabs, F. Noack, A. S. Aleksandrovsky, A. I. Zaitsev, and V. Petrov, “Generation of coherent radiation in the vacuum ultraviolet using randomly quasi-phase-matched strontium tetraborate,” Opt. Lett. 41(3), 618–621 (2016). [CrossRef]
20. R. Šuminas, N. Garejev, A. Šuminienė, V. Jukna, G. Tamošauskas, and A. Dubietis, “Femtosecond filamentation, supercontinuum generation, and determination of n2 in polycrystalline SBN,” J. Opt. Soc. Am. B 37(5), 1530–1534 (2020). [CrossRef]
21. J. Gu, M. G. Hastings, and M. Kolesik, “Simulation of harmonic and supercontinuum generation in polycrystalline media,” J. Opt. Soc. Am. B 37(5), 1510–1517 (2020). [CrossRef]
22. M. Kolesik and J. Moloney, “Nonlinear optical pulse propagation simulation: From Maxwell’s to unidirectional equations,” Phys. Rev. E 70(3), 036604 (2004). [CrossRef]
23. J. Andreasen and M. Kolesik, “Nonlinear propagation of light in structured media: Generalized unidirectional pulse propagation equations,” Phys. Rev. E 86(3), 036706 (2012). [CrossRef]
24. M. R. Querry, Optical constants of minerals and other materials from the millimeter to the ultraviolet (Chemical Research, Development & Engineering Center, US Army Armament, 1987).
25. H. P. Wagner, M. Kühnelt, W. Langbein, and J. M. Hvam, “Dispersion of the second-order nonlinear susceptibility in ZnTe, ZnSe, and ZnS,” Phys. Rev. B 58(16), 10494–10501 (1998). [CrossRef]
26. A. Okabe, B. Boots, K. Sugihara, and S. N. Chiu, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edition (Wiley, 2000), pp. 503–526.
27. J. Arvo, “Fast random rotation matrices,” in Graphics Gems III (IBM Version), (Elsevier, 1992), pp. 117–120.
28. T. Kawamori, Q. Ru, and K. L. Vodopyanov, “Comprehensive model for randomly phase-matched frequency conversion in zinc-blende polycrystals and experimental results for ZnSe,” Phys. Rev. Appl. 11(5), 054015 (2019). [CrossRef]
29. M. Kolesik, E. M. Wright, and J. V. Moloney, “Interpretation of the spectrally resolved far field of femtosecond pulses propagating in bulk nonlinear dispersive media,” Opt. Express 13(26), 10729–10741 (2005). [CrossRef]
30. M. Kolesik, E. M. Wright, and J. V. Moloney, “Supercontinuum and third-harmonic generation accompanying optical filamentation as first-order scattering processes,” Opt. Lett. 32(19), 2816–2818 (2007). [CrossRef]
31. J. M. Brown, A. Couairon, P. Polynkin, and M. B. Gaarde, “Analysis of the angular spectrum for ultrashort laser pulses,” J. Opt. Soc. Am. B 36(2), A105–A111 (2019). [CrossRef]
32. A. Couairon, E. Gaižauskas, D. Faccio, A. Dubietis, and P. Di Trapani, “Nonlinear X-wave formation by femtosecond filamentation in Kerr media,” Phys. Rev. E 73(1), 016608 (2006). [CrossRef]
33. M. A. Porras, A. Parola, and P. Di Trapani, “Nonlinear unbalanced O waves: nonsolitary, conical light bullets in nonlinear dissipative media,” J. Opt. Soc. Am. B 22(7), 1406–1413 (2005). [CrossRef]
34. D. Faccio, M. Clerici, A. Averchi, A. Lotti, O. Jedrkiewicz, A. Dubietis, G. Tamosauskas, A. Couairon, F. Bragheri, D. Papazoglou, S. Tzortzakis, and P. Di Trapani, “Few-cycle laser-pulse collapse in Kerr media: the role of group-velocity dispersion and X-wave formation,” Phys. Rev. A 78(3), 033826 (2008). [CrossRef]
35. D. Faccio, A. Averchi, A. Lotti, M. Kolesik, J. V. Moloney, A. Couairon, and P. Di Trapani, “Generation and control of extreme blueshifted continuum peaks in optical Kerr media,” Phys. Rev. A 78(3), 033825 (2008). [CrossRef]
36. S. Qu, H. Liang, K. Liu, X. Zou, W. Li, Q. J. Wang, and Y. Zhang, “9 µm few-cycle optical parametric chirped-pulse amplifier based on LiGaS2,” Opt. Lett. 44(10), 2422–2425 (2019). [CrossRef]
37. G. Stegeman, D. Hagan, and L. Torner, “$\chi ^{(2)}$ cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons,” Opt. Quantum Electron. 28(12), 1691–1740 (1996). [CrossRef]
38. K. Beckwitt, F. W. Wise, L. Qian, L. A. Walker, and E. Canto-Said, “Compensation for self-focusing by use of cascade quadratic nonlinearity,” Opt. Lett. 26(21), 1696–1698 (2001). [CrossRef]
39. A. S. Mayer, C. R. Phillips, and U. Keller, “Watt-level 10-gigahertz solid-state laser enabled by self-defocusing nonlinearities in an aperiodically poled crystal,” Nat. Commun. 8(1), 1673–1678 (2017). [CrossRef]
40. M. Tripepi, M. Hastings, A. Schweinsberg, L. Vanderhoef, C. Wolfe, T. Ensley, M. Kolesik, J. Moloney, E. Chowdhury, and A. Valenzuela, “Midwave infrared ultrashort pulse laser frequency conversion in single crystal, polycrystalline, and amorphous optical materials,” in Nonlinear Frequency Generation and Conversion: Materials and Devices XIX, vol. 11264 (International Society for Optics and Photonics, 2020), p. 112640Y.
41. A. Dharmadhikari, F. Rajgara, D. Mathur, H. Schröder, and J. Liu, “Efficient broadband emission from condensed media irradiated by low-intensity, unfocused, ultrashort laser light,” Opt. Express 13(21), 8555–8564 (2005). [CrossRef]
42. A. Camino, Z. Hao, X. Liu, and J. Lin, “High spectral power femtosecond supercontinuum source by use of microlens array,” Opt. Lett. 39(4), 747–750 (2014). [CrossRef]
43. A. Voronin and A. Zheltikov, “Power-scalable subcycle pulses from laser filaments,” Sci. Rep. 7(1), 36263 (2017). [CrossRef]
44. A. Dubietis and A. Couairon, “Femtosecond filamentation in solid-state media,” in Ultrafast Supercontinuum Generation in Transparent Solid-State Media, (Springer, 2019), pp. 27–46.
