Abstract
Optical modulators can have high modulation speed and broad bandwidth, while being compact. However, these optical modulators usually work for lowintensity light beams. Here we present an ultrafast, plasmabased optical modulator, which can directly modulate highpower lasers with intensity up to 10^{16}â€‰Wâ€‰cm^{âˆ’2} to produce an extremely broad spectrum with a fractional bandwidth over 100%, extending to the midinfrared regime in the lowfrequency side. This concept relies on two copropagating laser pulses in a submillimetrescale underdense plasma, where a drive laser pulse first excites an electron plasma wave in its wake while a following carrier laser pulse is modulated by the plasma wave. The laser and plasma parameters suitable for the modulator to work are based on numerical simulations.
Introduction
Optical modulators are key components for manipulating optical signals, which are widely used in scientific and industrial applications. For example, highspeed compact electrooptic modulators (EOMs) are essential for data communications^{1,2,3,4}. EOMs can alter the fundamental characteristics (that is, amplitude, frequency, phase and polarization) of a light beam in a controllable manner, by making use of electrooptic effects to change the refractive index of a material when an external radiofrequency electric field driver is applied. Thanks to the rapid development of the field of radiofrequency photonics^{5,6} together with advanced material and microfabrication technologies^{3,4,7}, the modulation speed of EOMs has dramatically increased from megahertz to 100â€‰gigahertz (refs 8, 9, 10) over the past decade. However, it is still very challenging to successfully achieve terahertz (THz) speed for EOMs using current technologies due to the speed limitation of the highvoltage driver^{1}. On the other hand, the rather low optical damage threshold^{11,12} and low bulk laser damage threshold of EOMs^{13} severely limit their applications in the highintensity regime. For example, a stateoftheart commercially available magnesiumoxidedoped LiNbO_{3} modulator can only handle input light power of 10^{2}â€‰mW level and corresponding light intensity at âˆ¼10^{2}â€‰Wâ€‰cm^{âˆ’2} (ref. 14).
Currently commercial highpower laser systems can deliver peak powers up to petawatts, which can be focused to realize laser intensities from 10^{15} to 10^{21}â€‰Wâ€‰cm^{âˆ’2}. The interaction of such highintensity laser beams with matter is not only of fundamental interest, but also shows prospects of various applications, such as highharmonic generation^{15}, THz radiation generation^{16,17,18}, plasmabased particle accelerators and light sources^{19}, laser fusions^{20}, and laboratory astrophysics^{21}and so on. With such highpower lasers, it has been reported that plasmabased devices have unique advantages in manipulating intense lasers because they have no damage threshold. Typical plasmabased optical devices include plasma channels for the guided propagation of intense laser pulses over many Rayleigh lengths^{22,23}, plasma mirrors to improve the temporal contrast of intense laser pulses^{24,25}, plasma gratings to compress intense laser pulses^{26}, plasma lens to focus intense lasers^{27}, plasma Raman amplifiers to boost the laser power to the multipetawatt regime or higher^{28,29,30} and plasma polarization switching for modulating THz electromagnetic waves^{31}.
In this article, we show a novel ultrafast alloptical plasmabased modulator that can directly modulate the spectrum of intense laser pulses to an extreme broad bandwidth, with a modulation speed of tens of THz and a damage threshold of 10^{16}â€‰Wâ€‰cm^{âˆ’2} level. Because of the ultrafast modulation speed and ultrahigh damage threshold, the plasma optical modulator opens a way to efficiently modulate laser pulses in the highintensity regime. Such highly modulated intense laser pulses may bring a few new physics and applications associated with intense laserâ€“matter interactions. For example, it may be used to produce strong THz radiation via optical rectification as the laser pulses have bandwidth in the THz range^{32}. Another possibility is to produce ultrabright Xray sources via laser interaction with atoms^{33}, since the modulated spectrum by our plasma modulator can be well extended to the midinfrared regime^{34}. The deeply modulated laser pulse exhibits ultrabroad bandwidth, which can suppress the growth rate of the stimulated Raman scattering instability, highly important for laser fusions^{35,36}.
Results
Concept of plasma optical modulators
The concept of the plasma optical modulator is illustrated in Fig. 1: a linearly polarized femtosecond intense drive laser pulse propagates in a submmscale gas, forming an underdense plasma via field ionization. Meanwhile, the ponderomotive force of the laser pulse drives the plasma electrons out of its path. Because the plasma ions are much heavier (by a factor of at least 1,836), they barely move and remain unshielded. The resultant pattern of alternating positive and negative chargeseparation fields behind the laser driver is a plasma wave (also called laser wakefields), which has been welldescribed theoretically^{19} and measured experimentally^{37}. The wave oscillates at the plasma frequency Ï‰_{p}, where , with n_{0} the ambient electron plasma density, the permittivity of free space, and m_{e} and e the electron rest mass and charge, respectively. For the purpose of optical modulation, the plasma wave is driven at a moderate amplitude. A picosecond carrier laser pulse (with an arbitrary polarization direction) copropagates behind the drive laser with a delay of several plasma wavelengths. The amplitude and frequency of the carrier are simultaneously modulated by the plasma wave during its propagation, generating a number of significant frequency sidebands spaced by the plasma frequency in the frequency domain. The modulation speed f_{p} is determined by the plasma frequency Ï‰_{p}, which can be estimated as , for example, f_{p}28â€‰THz for n_{0}=10^{19}â€‰cm^{âˆ’3}, which is several orders of magnitude faster than the speed of an EOM. Particleincell (PIC) simulations show that such plasma modulators can sustain a carrier intensity up to 10^{16}â€‰Wâ€‰cm^{âˆ’2}, which is several orders of magnitude higher than what conventional EOMs can handle.
Parameter dependence of modulation strength
Figure 2 shows an example case to demonstrate the essential features of the modulation obtained from onedimensional (1D) PIC simulations. For simplicity, the vacuum wavelengths of the two laser pulses are both 1â€‰Î¼m in the simulations. In practice, an 800nm Ti:sapphire femtosecond laser pulse can be used to excite the plasma waves, which does not lead to obvious changes of the results presented in the following. The normalized field amplitudes of the driver and the unmodulated carrier are a_{00}=0.8 and a_{10}=0.05, respectively, where a_{i0}=E_{zi0}/E_{n} (i=0,1), and E_{n}=m_{e}cÏ‰_{0}/e, with c and Ï‰_{0} the light speed and angular frequency in vacuum, respectively. For linear polarization, I_{i0}(Wâ€‰cm^{âˆ’2})=1.37 Ã— 10^{18}a_{i0}^{2}/[Î»_{0}(Î¼m)]^{2}, with I_{i0} the peak laser intensity and Î»_{0}=2Ï€c/Ï‰_{0} the laser wavelength in vacuum. Thus, a_{00} and a_{10} correspond to the laser intensities of 8.77 Ã— 10^{17} and 3.43 Ã— 10^{15}â€‰Wâ€‰cm^{âˆ’2}, respectively, for 1â€‰Î¼m laser wavelength. Detailed parameters are given in the Methods. The modulated pulse is well described using the analytical model presented in the Methods. It can be expressed as a_{1}(t)=a_{10}[1+m cos(Ï‰_{p}t)] cos[Ï‰_{0}t+Î² sin(Ï‰_{p}t)], where m and Î² are the amplitude modulation index and the frequency modulation index, respectively. The mixed amplitude and frequency modulation of a sinusoidal carrier by a simple sinusoidal plasma wave yields a mass of sidebands including both Stokes and antiStokes components given by Ï‰_{n}=Ï‰_{0}Â±nÏ‰_{p} (with n a nonzero integer and Ï‰_{p} as a frequency interval). Note that in the quasilinear regime of the plasma wave, that is, where the relativisticelectronmass increase associated with the motion of the plasma electrons can be neglected, the frequency Ï‰_{p} can be calculated as Ï‰_{p}=(n_{0}/n_{c})^{1/2}Ï‰_{0}, with n_{c} (cm^{âˆ’3})=1.1 Ã— 10^{21}/[Î»_{0}(Î¼m)]^{2} the critical plasma density for the corresponding incident laser wavelength Î»_{0}. The spectral bandwidth is defined as B_{Ï‰}=2(Î²+1)Ï‰_{p} (ref. 38), where Î² depends on the amplitude of the drive pulse and the plasma density, in addition to the plasma length.
For high fidelity, we only count the significant sidebands with the amplitudes larger than 1% (âˆ’40â€‰dB) of the amplitude of the unmodulated carrier^{38}. Therefore, the spectral bandwidth of the modulated carrier can be calculated by estimating the number of significant sidebands. As shown in Fig. 2, the higherorder sidebands gradually grow with the laserâ€“plasma interaction time. When the carrier pulse completely passes through the plasma, the maximum significant sidebands for the antiStokes and Stokes components are Ï‰_{+6} and Ï‰_{âˆ’7}, respectively, giving a bandwidth of B_{Ï‰}=13Ï‰_{p}=1.3Ï‰_{0}, accounting for the fact that Ï‰_{p}=0.1Ï‰_{0} for n_{0}/n_{c}=0.01. It is also noted that the sideband spectrum of a mixed modulation is asymmetrical due to the superposition of the sideband components of both amplitude and frequency modulations. The simulation results are in good agreement with the prediction of the analytical model given in the Methods. In this example, the amplitude and frequency modulation indices can be estimated^{38} as m=(a_{10,max}âˆ’a_{10,min})/(a_{10,max}+a_{10,min})=0.42, and Î²=B_{Ï‰}/(2Ï‰_{p})âˆ’1=5.5, respectively. Note that Î²1 corresponds to broadband modulation. The energy transmission rate of the carrier through plasma is âˆ¼94.3% in this example.
We find the modulation is effective for a wide range of laserâ€“plasma parameters. Figure 3 shows the âˆ’40â€‰dB cutoff sidebands, the corresponding fractional bandwidth (Î”Ï‰=B_{Ï‰}/Ï‰_{0}), and the amplitude modulation index m, as a function of the driver intensity, the plasma density and the plasma length. When the driver amplitude is relatively small (for example, a_{00}=0.1), the modulation is quite weak so that the spectrum only consists of the firstorder sidebands. By increasing the driver amplitude, the field strength of the plasma wave is enhanced, and subsequently the modulation indices become larger, leading to higherorder sidebands and a wider bandwidth. The similar scaling law exists when increasing the plasma density. Therefore, by properly increasing the drive laser intensity and the plasma density, one can extend the spectrum of the modulated carrier to the midinfrared regime in the lowfrequency side (or the Stokes waves). We note that the growth of the bandwidth is relatively insensitive to the increase of the plasma length after certain distance, which implies a saturation of modulation. As shown in Fig. 3dâ€“f, the amplitude modulation index m gradually grows with the increase of the driver intensity or the plasma density. When increasing the plasma length, m first grows and then saturates at the 100% level, which indicates that the carrier breaks up into a train of short pulses, and each of these short pulses has a width on the order of the plasma wavelength. According to Fig. 3dâ€“f, Î”Ï‰ and Î² have similar dependence on the driver intensity, the plasma density and the plasma length as m. In the simulation results given above, we have limited the plasma wave excitation to the quasilinear regime (when the driver laser amplitude a_{00}â‰²1). This avoids possible occurrence of curved plasma wave fronts and plasma wave breaking, so that the carrier laser pulse can be modulated efficiently.
For certain applications, it is important to know the parameter range for broad bandwidth generation. Figure 4 illustrates the parameter constraint for generating carrier pulses with ultrabroad bandwidths (for example, Î”Ï‰ â©¾30%), which presents a series of simulations where the threshold for the driver amplitude a_{00,th} is scanned for a given plasma density n_{0}. In general, a broader bandwidth can be achieved at a higher plasma density even if a lower driver intensity is adopted.
In passing, we mention that, even though the frequencies of the drive laser pulse and the carrier laser pulse can be different in a certain range and the time delay between them can also be arbitrary within a picosecond, the plasma optical modulator requires that the two pulses copropagate, that is, there is almost no frequency modulation when they counterpropagate.
Discussion
So far we have discussed the spectrum development of the carrier laser pulse as a function of the drive laser amplitude, the plasma density, and the plasma length. One question still to be answered is the maximum carrier laser intensity allowed in the plasma modulator. A previous study has shown that the resultant pulse train can amplify the field strength of the plasma wake to a wavebreaking level if the initial intensity of the carrier laser is high enough^{39}. We also find the remarkably enhanced plasma waves when the intensity of the pulse train is on the same order of the driver intensity (for example, 10^{17}â€‰Wâ€‰cm^{âˆ’2} level). This can result in severe distortion of the plasma wave and considerable energy loss of the carrier laser to plasma wave excitation as well as electron trapping and acceleration (see Supplementary Fig. 1 and Supplementary Note 1). As a consequence, the frequency modulation of the carrier laser is suppressed. Therefore, the maximum intensity of the carrier laser should be well below 10^{17}â€‰Wâ€‰cm^{âˆ’2} (for example, at 10^{16}â€‰Wâ€‰cm^{âˆ’2} level) to realize an excellent performance of the plasma optical modulator.
The maximum allowed pulse duration of the carrier laser for effective modulation may be interesting for some particular applications. This depends on the life time of the electron plasma waves, which is determined by the collisional damping, Landau damping and phase mixing^{40,41}. Typically the initial electron plasma temperature T_{e} is over 10â€‰eV and the effective T_{e} is over 100â€‰eV when considering the electron quiver motion in the carrier laser with intensity âˆ¼10^{16}â€‰Wâ€‰cm^{âˆ’2}, which leads to a time scale of over 10â€‰ps for the collisional damping under the plasma electron density of âˆ¼10^{19}â€‰cm^{âˆ’3}. The Landau damping time is much longer than the collisional damping time in the present case when the plasma wave is driven at moderate amplitudes. The phase mixing due to the ion motion is the key responsibility for the plasma wave decay since it occurs on a much shorter time scale of (ref. 40) when a highZ gas such as argon is used for the plasma wave excitation, where is the normalized amplitude of the plasma wave, with E_{p}=cm_{e}Ï‰_{p}/e and m_{i} the ion mass. PIC simulations show that the plasma wave starts to decay around 1.65â€‰ps due to the phase mixing, which is in good agreement with the analytical model. The maximum pulse duration for the effective modulation is around 3â€‰ps for the laserâ€“plasma parameters under consideration (see Supplementary Fig. 2, Supplementary Fig. 3 and Supplementary Note 2).
Another issue is the spot sizes of the laser pulses. As we have shown above, the laser pulses need to propagate over a distance of about 1â€‰mm without significant transverse spreading. One needs to take relatively large spot sizes so that the corresponding Rayleigh lengths are long enough, with r_{i} (i=0,1) the spot sizes of the two laser pulses. Meanwhile, selffocusing will occur when the driver power P exceeds a critical power P_{c}, with P_{c} (GW)=17.4(Ï‰_{0}/Ï‰_{p})^{2}. For linear polarization, P/P_{c}=(Ï‰_{p}r_{0}a_{00})^{2}/(32c^{2}) (ref. 19). To avoid strong selffocusing within 1â€‰mm, the spot size of the driver cannot be too large, either. Twodimensional (2D) simulations show that the optimal modulation can be achieved for 1â‰²P/P_{c}â‰²2. An example of 2D simulation is given in Fig. 5. Detailed parameters are given in the Methods. In this example, the driver power is P/P_{c}=1.18 and weak selffocusing occurs during the propagation. As shown in Fig. 5a, the maximum amplitude of the driver increases by 10% (from a_{00}=0.7 to 0.77) at a propagation distance of 392Î»_{0}. The excited plasma wave retains the quasi1D structure, and keeps quite stable during propagation, which is advantageous for the modulation process. It is noted that the driver spot leads to transverse inhomogeneity of the plasma wave, resulting in transverse inhomogeneity of the modulation. By reducing the driver intensity, the corresponding spot size can be increased, and hence, the transverse uniformity of the modulation can be improved.
The technical essentials of realizing the proposed plasma modulators are well within current capabilities. First, the plasma wave excitation by an ultrashort pulse is a wellknown technique, which has been widely adopted for laser wakefield accelerators (LWFAs)^{19}. For the application to plasma optical modulators, the required amplitude of the plasma wave can be much smaller than that for LWFAs, implying that only moderate drive laser power, such as a few terawatts, is required. Second, the frequencies of the carrier pulse and the drive pulse and the time delay between them are relatively flexible, indicating that the experimental configuration is simpler as compared with some other experiments involving two colliding lasers such as Raman amplification^{30}, superradiant amplification^{42} and colliding laser pulses injection in LWFAs^{43}. A test experiment of the plasma optical modulator could be carried out with a carrier laser pulse (for example, Nd:YVO_{4}, 1â€‰Î¼m, âˆ¼15â€‰mJ, âˆ¼1â€‰ps) delayed with respect to a drive laser pulse (for example, Ti:sapphire, 0.8â€‰Î¼m, âˆ¼150â€‰mJ, âˆ¼30â€‰fs), copropagating in a 1mmlong helium gas with a density of âˆ¼10^{19}â€‰cm^{âˆ’3}. The time delay between the two laser pulses can be controlled in a timescale of hundreds of femtoseconds.
In summary, we have illustrated a novel application of the plasma wave as a unique optical modulator for intense lasers. It relies on two copropagating laser pulses in a short underdense plasma: a driver with a typical intensity âˆ¼10^{17}â€‰Wâ€‰cm^{âˆ’2}, which propagates in the plasma and excites a plasma wake, and a carrier, which propagates behind the driver by several plasma wavelengths. Both the amplitude and frequency of the carrier are modulated by the plasma wave, leading to an ultrabroad bandwidth in its spectrum that extends to the midinfrared range. The modulation speed is in the THz regime. Compared with the lowdamage threshold of the conventional EOMs, the plasma modulator allows the carrier intensity as high as up to 10^{16}â€‰Wâ€‰cm^{âˆ’2}. In addition, the plasma modulator offers excellent performance control by changing the driver intensity, the plasma density and the plasma length. The required experimental conditions for such plasma modulators are within current technical capabilities.
Methods
Mathematical model for mixed modulation
Physically, the modulation of the carrier laser pulse by an electron plasma wave is similar to that found for an intense laser propagation in plasma via stimulated Raman forward scattering (coupled with the selfmodulation instability)^{44,45}. The latter leads to a spectrum of Stokes and antiStokes waves when the laser pulse has a duration longer than a plasma wavelength. The evolution of the amount of Stokes/antiStokes modes can be described by photon acceleration and deceleration^{46,47,48}. The dependence of the spectral modulation on the plasma wave amplitude, the plasma density and the interaction time discussed in the above section also qualitatively agrees with the previous theories. The difference between the spectral modulation described in the previous theories and here is that our plasma optical modulator enables the spectral modulation of the carrier laser to be well controlled and to be developed much more efficiently.
The carrier pulse is modulated in the amplitude and frequency by the electron plasma wave, that is, a mixed modulation. Its temporal structure can be written as^{49}
assuming that the excited plasma wave is a simple sinusoidal oscillation, with the normalized axial electric field E_{x}/E_{p}=(E_{max}/E_{p}) cos(Ï‰_{p}t) (ref. 19). Here a_{10} is the normalized amplitude of the unmodulated carrier. For a linearly polarized sinusoidal driver with an optimal pulse length for plasma wave excitation (that is, the pulse length approximate to the plasma wavelength), , yielding the amplitude and frequency modulation indices and , respectively. These two parameters also depend on the interaction time or the plasma length as shown in the simulation results given in Fig. 3. Using simple trigonometrical transformations and a lemma of Bessel function and J_{âˆ’n}(Î²)=(âˆ’1)^{n}J_{n}(Î²), a_{1}(t) can be written as
It is obvious that the spectrum of a_{1}(t) primarily consists of three components: the central frequency Ï‰_{0} that corresponds to the unmodulated carrier, and the two firstorder sidebands Ï‰_{Â±1}=Ï‰_{0}Â±Ï‰_{p} resulting from the modulation process. The amplitudes of the frequency components can be characterized by the expansion in a series of nthorder Bessel function J_{n}. By taking Fourier transformation of a_{1}(t), and considering the kthorder frequency component with k a positive integer, we can get the amplitudes of the upper sideband (Ï‰_{+k}=Ï‰_{0}+kÏ‰_{p}) and the lower sideband (Ï‰_{âˆ’k}=Ï‰_{0}âˆ’kÏ‰_{p}) in the spectrum as follows:
for Ï‰_{Â±k}>0. From equations (3) and (4), it is straightforward to see that the amplitude of the lower sideband is not equal to the amplitude of the corresponding upper sideband, leading to an asymmetrical sideband spectrum. For a weak frequency modulation (0<Î²<<1), the modulation index is so small that the spectrum essentially consists of Ï‰_{0} and only one set of sidebands Ï‰_{Â±1}, with the amplitudes of , and . For a large modulation index (Î²>1), there will be a number of significant sidebands spanning over a broad frequency range.
It is worthwhile to mention that when the excited plasma wave is nonlinear, there are highharmonic components of the plasma oscillations. This leads to an additional frequency modulation at harmonics of the plasma frequency, which are superimposed on the sidebands discussed above, contributing to more energetic sidebands. This nonlinear effect has been included in our selfconsistent PIC simulation results presented above.
PIC simulations
Simulations have been carried out using the code OSIRIS^{50}. In the 1D simulations (for example, in Fig. 2), the temporal profile of the drive pulse is , with 0â‰¤tâ‰¤T_{0} and T_{0}=10T_{L}. The carrier pulse, which is delayed by 40Î»_{0} from the driver, has a duration of T_{1}=303T_{L}. It has a similar profile as the driver at its leading and trailing edges, and a plateau of 283T_{L} in between. The amplitudes of the driver and the carrier are a_{00}=0.8, a_{10}=0.05, respectively. The trapezoidshaped plasma has a length of 400Î»_{0} with a plateau of 380Î»_{0}, located between x=10Î»_{0} and x=410Î»_{0}. The initial plasma electron density in the plateau region is set to be n_{0}/n_{c}=0.01. For laserdriven plasma waves, typically the initial (photoionized) electron plasma temperature is set to be 10â€‰eV. The simulation box size is 800Î»_{0} with 20 macroparticles per cell. The resolution of the computational grid is Î”x=Î»_{0}/40. At t=0, the front of the driver enters the simulation box. In the 2D simulation (Fig. 5), the amplitude of the driver is a_{00}=0.7 and the spot sizes of the driver and the carrier are r_{0}=14Î»_{0} and r_{1}=17Î»_{0}, respectively. The trapezoidshaped plasma has a length of 700Î»_{0}. Other laserâ€“plasma parameters are the same as the 1D simulations. The simulation box size is 1,100Î»_{0} Ã— 100Î»_{0} with four macroparticles per cell. The resolution of the computational grid is Î”x=Î»_{0}/32 and Î”y=Î»_{0}/20.
Data availability
The data that support the findings of this study are available from the corresponding authors upon request.
Additional information
How to cite this article: Yu, L.L. et al. Plasma optical modulators for intense lasers. Nat. Commun. 7:11893 doi: 10.1038/ncomms11893 (2016).
References
Liu, K., Ye, C. R., Khan, S. & Sorger, V. J. Review and perspective on ultrafast wavelengthsize electrooptic modulators. Laser Photon Rev. 9, 172â€“194 (2015).
Reed, G. T., Mashanovich, G., Gardes, F. Y. & Thomson, D. J. Silicon optical modulators. Nat. Photon 4, 518â€“526 (2010).
Liu, M. et al. A graphenebased broadband optical modulator. Nature 474, 64â€“67 (2011).
Phare, C. T., Lee, Y.H. D., Cardenas, J. & Lipson, M. Graphene electrooptic modulator with 30 GHz bandwidth. Nat. Photon 9, 511â€“515 (2015).
Seeds, A. J. & Williams, K. J. Microwave photonics. J. Lightwave Technol. 24, 4628â€“4641 (2006).
Capmany, J. & Novak, D. Microwave photonics combines two world. Nat. Photon 1, 319â€“330 (2007).
Melikyan, A. et al. Highspeed plasmonic phase modulators. Nat. Photon 8, 229â€“233 (2014).
Macario, J. et al. Full spectrum millimeterwave modulation. Opt. Express 20, 23623â€“23629 (2012).
Chen, D. et al. Demonstration of 110 GHz electrooptic polymer modulators. Appl. Phys. Lett. 70, 3335â€“3337 (1997).
Huang, H. et al. Broadband modulation performance of 100GHz EO polymer MZMs. J. Lightwave Technol. 30, 3647â€“3652 (2012).
Bryan, D. A., Gerson, R. & Tomaschke, H. E. Increased optical damage resistance in lithium niobate. Appl. Phys. Lett. 44, 847â€“849 (1984).
Furukawa, Y., Sato, M., Kitamura, K., Yajima, Y. & Minakata, M. Optical damage resistance and crystal quality of LiNbO3 single crystals with various [Li]/[Nb] ratios. J. Appl. Phys. 72, 3250â€“3254 (1992).
Furukawa, Y. et al. Investigation of bulk laser damage threshold of lithium niobate single crystals by Qswitched pulse laser. J. Appl. Phys. 69, 3372â€“3374 (1991).
Newport Corporation. Electrooptic modulator guide. Available at http://www.newport.com/ElectroOpticModulatorSelectionGuide/977460/1033/content.aspx?xcid=bingppc0244.
Paul, A. et al. Quasiphasematched generation of coherent extremeultraviolet light. Nature 421, 51â€“54 (2003).
Thomson, M. D., KreÃŸ, M., LÃ¶ffler, T. & Roskos, H. G. Broadband THz emission from gas plasmas induced by femtosecond optical pulses: From fundamentals to applications. Laser Photon Rev. 1, 349â€“368 (2007).
Chen, Y. et al. Elliptically polarized Terahertz emission in the forward direction of a femtosecond laser filament in air. Appl. Phys. Lett. 93, 231116 (2008).
Cho, M.H. et al. Strong terahertz emission from electromagnetic diffusion near cutoff in plasma. New J. Phys. 17, 043045 (2015).
Esarey, E., Schroeder, C. B. & Leemans, W. P. Physics of laserdriven plasmabased electron accelerators. Rev. Mod. Phys. 81, 1229â€“1285 (2009).
Glenzer, S. H. et al. Symmetric inertial confinement fusion implosions at ultrahigh laser energies. Science 327, 1228â€“1231 (2010).
Zhong, J.Y. et al. Modelling looptop Xray source and reconnection outflows in solar flares with intense lasers. Nat. Phys. 6, 984â€“987 (2010).
Leemans, W. P. et al. GeV electron beams from a centimetrescale accelerator. Nat. Phys. 2, 696â€“699 (2006).
Leemans, W. P. et al. MultiGeV electron beams from capillarydischargeguided subpetawatt laser pulses in the selftrapping regime. Phys. Rev. Lett. 113, 245002 (2014).
Thaury, C. et al. Plasma mirrors for ultrahighintensity optics. Nat. Phys. 3, 424â€“429 (2007).
Andreev, A. A. et al. Optimal ion acceleration from ultrathin foils irradiated by a profiled laser pulse of relativistic intensity. Phys. Plasmas 16, 013103 (2009).
Wu, H.C., Sheng, Z.M., Zhang, Q.J., Cang, Y. & Zhang, J. Manipulating ultrashort intense laser pulses by plasma Bragg gratings. Phys. Plasmas 12, 113103 (2005).
Wang, H.Y. et al. Laser shaping of a relativistic intense, short Gaussian pulse by a plasma lens. Phys. Rev. Lett. 107, 265002 (2011).
Trines, R. M. G. M. et al. Simulations of efficient Raman amplification into the multipetawatt regime. Nat. Phys. 7, 87â€“92 (2011).
Mourou, G. A., Tajima, T. & Bulanov, S. V. Optics in the relativistic regime. Rev. Mod. Phys. 78, 309â€“371 (2006).
Yang, X. et al. Chirped pulse Raman amplification in warm plasma: towards controlling saturation. Sci. Rep. 5, 13333 (2015).
Wen, H., Daranciang, D. & Lindenberg, A. M. Highspeed alloptical terahertz polarization switching by a transient plasma phase modulator. Appl. Phys. Lett. 96, 161103 (2010).
Rice, A. et al. Terahertz optical rectification from <110> zincblende crystals. Appl. Phys. Lett. 64, 1324â€“1326 (1994).
Chang, Z., Rundquist, A., Wang, H., Murnane, M. M. & Kapteyn, H. C. Generation of coherent soft X rays at 2.7 nm using high harmonics. Phys. Rev. Lett. 79, 2967â€“2970 (1997).
Popmintchev, T. et al. Bright coherent ultrahigh harmonics in the keV Xray regime from midinfrared femtosecond lasers. Science 336, 1287â€“1291 (2012).
Thomson, J. & Karush, J. I. Effects of finitebandwidth driver on the parametric instability. Phys. Fluids 17, 1608â€“1613 (1974).
Zhao, Y. et al. Effects of large laser bandwidth on stimulated Raman scattering instability in underdense plasma. Phys. Plasmas 22, 052119(1)â€“052119(7) (2015).
Matlis, N. H. et al. Snapshots of laser wakefields. Nat. Phys 2, 749â€“753 (2006).
Agilent Technologies. Spectrum analysis amplitude and frequency modulation. Available at http://cp.literature.agilent.com/litweb/pdf/59549130.pdf (2001).
Sheng, Z.M., Mima, K., Sentoku, Y., Nishihara, K. & Zhang, J. Generation of highamplitude plasma waves for particle acceleration by crossmodulated laser wake fields. Phys. Plasmas 9, 3147â€“3153 (2002).
Gupta, S. S. & Kaw, P. K. Phase mixing of nonlinear plasma oscillations in an arbitrary mass ratio cold plasma. Phys. Rev. Lett. 82, 1867â€“1870 (1999).
Xu, H., Sheng, Z.M. & Zhang, J. Phase mixing due to ion motion and relativistic effects in nonlinear plasma oscillations. Phys. Scr. 74, 673â€“677 (2006).
Dreher, M., Takahashi, E., MeyerterVehn, J. & Witte, K.J. Observation of superradiant amplification of ultrashort laser pulses in a plasma. Phys. Rev. Lett. 93, 095001 (2004).
Faure, J. et al. Controlled injection and acceleration of electrons in plasma wakefields by colliding laser pulses. Nature 444, 737â€“739 (2006).
Mori, W. B. The physics of the nonlinear optics of plasmas at relativistic intensities for shortpulse lasers. IEEE J. Quantum Electron. 33, 1942â€“1953 (1997).
Tzeng, K.C., Mori, W. B. & Katsouleas, T. Selftrapped electron acceleration from the nonlinear interplay between Raman forward scattering, selffocusing, and hosing. Phys. Plasmas 6, 2105â€“2116 (1999).
Wilks, S. C., Dawson, J. M., Mori, W. B., Katsouleas, T. & Jones, M. E. Photon accelerator. Phys. Rev. Lett. 62, 2600â€“2603 (1989).
Esarey, E., Ting, A. & Sprangle, P. Frequency shifts induced in laser pulses by plasma waves. Phys. Rev. A 42, 3526â€“3531 (1990).
Sheng, Z.M., Ma, J.X., Xu, Z.Z. & Yu, W. Effect of an electron plasma wave on the propagation of an ultrashort laser pulse. J. Opt. Soc. Am. B 10, 122â€“129 (1993).
Ozimek, E. & Sek, A. Perception of amplitude and frequency modulated signals (mixed modulation). J. Acoust. Soc. Am. 82, 1598â€“1603 (1987).
Fonseca, R. A. et al. OSIRIS, a threedimensional fully relativistic particle in cell code for modeling plasma based accelerators. Lect. Notes Comput. Sci. 2331, 342â€“351 (2002).
Acknowledgements
We thank Thomas Sokollik, Yulong Tang, Jun Zheng, Yanping Chen and Guoqiang Xie for useful discussions. This work was supported by the National Basic Research Program of China under grant no. 2013CBA01500, 2014CB339801 and 2015CB859700, the National Natural Science Foundation of China under grant nos 11421064, 11405107 and 11475113. D.A.J. and Z.M.S. acknowledge the support of the U.K. EPSRC (grant no. EP/J018171/1), the EC's LASERLABEUROPE (grant no. 654148), EuCARD2 (grant no. 312453), EuPRAXIA (grant no. 653782) and a Leverhulme Trust Research Project Grant. Simulations have been carried out on the PI supercomputer at Shanghai Jiao Tong University and the Milky Way 2 supercomputer in the National Supercomputer Center in Guangzhou.
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L.L.Y. and Z.M.S. designed the overall concept presented in this paper. L.L.Y. carried out all the simulations and the analytical model, and wrote the main manuscript text. Z.M.S., Y.Z., L.J.Q., M.C., S.M.W., D.A.J., W.B.M. and J.Z. contributed to analyse the results and write the manuscript. All authors discussed the results and commented on the manuscript.
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Supplementary Figures 13, Supplementary Note 12 and Supplementary Reference (PDF 657 kb)
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Yu, LL., Zhao, Y., Qian, LJ. et al. Plasma optical modulators for intense lasers. Nat Commun 7, 11893 (2016). https://doi.org/10.1038/ncomms11893
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DOI: https://doi.org/10.1038/ncomms11893
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