Enhanced harmonic generation in relativistic laser plasma interaction
We report the enhancement of individual harmonics generated at a relativistic ultra-steep plasma vacuum interface. Simulations show the harmonic emission to be due to the coupled action of two high velocity oscillations – at the fundamental and at the plasma frequency of the bulk plasma. The synthesis of the enhanced harmonics can be described by the reflection of the incident laser pulse at a relativistic mirror oscillating at and .
Waveform shaping and frequency synthesis based on waveform modulation is ubiquitous in electronics, telecommunication technology, and optics Kundu2013 (); Lee2002 (). For optical waveforms, the carrier frequency is on the order of several hundred THz, while the modulation frequencies used in conventional devices like electro- or acousto-optical modulators are orders of magnitude lower Lee2002 (). As a consequence, any new frequencies are typically very close to the fundamental. The synthesis of new frequencies in the extreme ultraviolet (XUV), e.g. by using relativistic oscillating mirrors Lichters1996 (); Tsakiris2006 (), requires modulation frequencies in the optical regime Dromey2006 (); Thaury2007 () or even in the extreme ultraviolet. The latter has not been proven possible to date.
Phase modulation of light upon reflection from relativistically oscillating plasma surfaces has been proposed as a mechanism for efficient high-harmonic generation (HHG). For reviews see Ref. Quere2010 () & Teubner2009 (). In this process, a dense plasma from a solid density surface located at oscillates such that relativistic velocities are reached. At times when the relativistic Lorentz factor is large and the plasma is moving towards the incident light wave with high velocity , high frequency radiation is generated in the reflected beam by the relativistic Doppler effect Gordienko2004 (); Quere2010 (). Due to the mirror-like reflection with a periodicity of , this process can be interpreted as a phase modulation Bulanov1994 (); Lichters1996 (); Tsakiris2006 (). Consequently, it is often referred to as the Relativistic Oscillating Mirror (ROM). Note that the relativistic plasma oscillation discussed in the literature to date is at the laser frequency or its low order harmonics Lichters1996 (); Tsakiris2006 (), i. e. at optical frequencies. Models that include the relativistic effect of retardation show good agreement with numerical simulations Lichters1996 (); Quere2010 () by accounting for the difference of the time of reflection and the time of observation . In the limit of a strongly relativistic motion, such models predict a harmonic spectral envelope spanning a broad bandwidth in the XUV and soft x-ray range that follows a power law up to some roll-off frequency Baeva2006 () after which the harmonic intensity decreases exponentially. Moreover, the shallow spectral slope not only implies that coherent high frequency radiation can be efficiently generated by this harmonic generation mechanism, but rather is also a signature of a train of extremely short optical pulses in the time domain Baeva2006 (); Quere2010 (); Behmke2011 (). HHG from relativistic surfaces thus is one of the most promising routes for intense attosecond pulse generation Tsakiris2006 (); Sansone2011 ().
In the last decade, relativistic surface high harmonic generation has been demonstrated up to keV photon energies Dromey2006 () in experiments which have also verified the shallow harmonic slope and the scaling of Dromey2007 (). At more moderate intensities of the order of , relativistic surface harmonic radiation has been generated for different laser and plasma parameters Thaury2007 (); Tarasevitch2007 (); Krushelnick2008 (). Recently, efficiencies ranging from – with a strong dependence on the plasma gradient have been reported and generally reveal a decrease of the harmonics’ efficiency for very short scale lengths Rodel2012 (); Kahaly2013 (). For an of to , i.e. for plasma conditions which can be considered to be ideal for efficient generation Rodel2012 (); Dollar2013 (), a constant, nearly diffraction limited beam divergence has been found for ROM harmonics Dromey2009 (). All experiments so far report a decaying spectral envelope towards higher harmonic orders, possibly with small modulations spanning a few harmonic orders Watts2002 (); Teubner2003 (); Behmke2011 () which are explained within the existing theoretical framework of the ROM mechanism where the surface oscillates only at optical frequencies.
Here we report on the strongly enhanced emission of particular harmonic orders under conditions where the plasma density profile is approaching a step function. Depending on the target density, either the 14th (43.4 eV), 16th (49.6 eV), or 18th (55.8 eV) harmonic of a 400-nm driving terawatt laser pulse has been amplified by a factor of up to five as compared to the adjacent harmonics (see Fig. 1A/B). The enhancement is never observed for odd harmonic orders or nonrelativistic intensities. It should be noted that the efficiency of enhanced harmonic generation is remarkably high even though the plasma scale length is very short and thus not optimal for conventional ROM harmonics note (). In fact, the enhanced harmonics contain a pulse energy of and thus qualify for applications like coherent diffraction imaging Sandberg2007 () or seeding of free electron lasers Lambert2008 ().
The experimental setup is identical to the one used in a previous experiment for generating ROM harmonics with 10 Hz Bierbach2012 (): The surface harmonics are generated by focusing 400-nm laser pulses with a pulse duration of 45 fs and a pulse energy of 100 mJ onto rotating plastic or glass targets such that a fresh interaction surface is provided for each laser pulse. The 400-nm radiation is created by a table-top 40-TW laser system operating at 800 nm via second harmonic generation (SHG) in a KDP crystal (0.7 mm). Using an off-axis parabolic mirror, the 400-nm pulses are focused to intensities of (FWHM) which correspond to a normalized vector potential . An angle of incidence of and laser polarization parallel to the plane of incidence is used.
In the following we show that the enhanced harmonics are generated by a relativistic plasma surface oscillating at XUV frequencies. Therefore, we refer to the high-frequency (XUV) ROM as XROM. A first indication of the origin of the enhanced harmonics is found in the observation that fluctuations in the intensity of the 400-nm fundamental do not affect the order of the enhanced harmonic. However, the order of the enhanced harmonic is affected by the target material. For plastic targets (low density), it is the harmonic (28.6 nm), for glass targets (high density) either the or . The enhanced harmonics are in fact conspicuously close to twice the maximum plasma frequency which is the natural frequency for electrons in a plasma with an electron density . Here, is the charge of an electron, the vacuum permittivity and the electron mass. The value of was verified to be near the harmonic for plastic targets or near the harmonic for glass targets, respectively, by measuring the cutoff frequencies for harmonics at lower, non-relativistic intensities. At nonrelativistic conditions it is well established that Coherent Wake Emission (CWE) Quere2006 () becomes dominant and that the high harmonic spectrum ends at Thaury2007 (). It is worth noting that the CWE is strongest for the cutoff harmonic located at , e.g. the harmonic in Fig. 1B. This is a signature of an extremely steep plasma density ramp Dromey2009a (). However, in this frequency range the complex superposition of harmonics generated by different generation mechanisms makes it nearly impossible to distinguish between CWE and ROM harmonics, or – as we will see later – the XROM process.
Before explaining the physical details of the XROM, we briefly review known alternative mechanisms for the generation of radiation at the plasma frequency and its multiples. Plasma radiation at and is a well-known observation in gas discharges, tokamak plasmas Fidone1978 () and astrophysical plasmas such as auroral Forsyth1949 () and solar radio burst Wild1953 () and is, in general, incoherent. In laser produced plasmas, radiation at and Teubner1999 () has been observed and attributed to the coupling of two large-amplitude Langmuir waves together with subsequent two-plasmon decay (TPD) Boyd2000 (); Kunzl2003 (). Common to this type of incoherent, broadband plasma radiation is the requirement of plasma waves generated inside the plasma and, in general, their nonrelativistic origin. Thus such radiation should not be strongly intensity dependent and their spatial and spectral properties would be unlikely to resemble those of ROM harmonic orders. Conversely, the similarity of ROM- and XROM-harmonics in terms of divergence, intensity dependence and bandwidth indicates that their spatial and temporal phases are compatible to the ROM mechanism and hence suggest strongly that the origin of XROM harmonics is not coupled to TPD or CWE but rather to a ROM-like mechanism.
Particle-in-cell (PIC) simulations were performed to further elucidate the origin of the enhanced harmonics. For simulation parameters that closely match our experimental conditions, the temporal evolution of the electron density at the plasma surface is shown in Fig. 2B and clearly reveals an oscillation not only at the optical frequency of the 400-nm laser but also a strong excitation at the plasma frequency in the XUV comment (). Although the amplitude of this oscillation is an order of magnitude smaller than the surface oscillation at , its peak velocity is comparable as seen with the chain rule. Therefore, the surface plasma oscillation at can also be expected to result in relativistic nonlinear effects in the reflected spectrum. The mechanism for the excitation of the strong plasma surface oscillation at is likely due to the jets of electrons which are expelled to the vacuum side by the laser field and which are later reinjected into the dense plasma Brunel1987 (); Geindre2010 () (see Fig. 2B). The computed reflected field analyzed by a time-windowed Fourier transform which is displayed in Fig. 2A. The time integrated spectrum shows the enhanced harmonic at on top of the familiar spectral decay of the harmonics and thus reproduces the experimental result (cf. Fig. 1C). The strongest emission of the enhanced harmonic at can be found in the second half of the laser pulse. A similar effect can be seen for the ROM harmonics at , i.e. the sideband frequencies of the oscillation. This delayed emission of the enhanced harmonics suggests that the relativistic surface plasma mode first needs to grow during a few laser cycles.
For analytically modeling the XROM harmonics, the reflected field is calculated in the spirit of the ROM model Tarasevitch2009 ():
with the retarded time . The trajectory of the mirror
exhibits a low frequency oscillation at and, in addition to the regular ROM model, a high frequency oscillation at . The oscillation amplitudes , and phases , can be estimated using the PIC simulation (see Fig. 2B). The resulting harmonic spectra are shown in Fig. 3. The first signature of the additional modulation to be expected are sidebands at . For parameters which are obtained from the PIC simulation, a strong enhancement spanning several harmonic orders is predicted for the harmonics around . This enhancement, however, can hardly be observed in the experiment because the harmonics up to are dominated by the aforementioned CWE mechanism. More importantly, the harmonic amplitudes at and are strongly increased in agreement with the experiment. Interestingly, the strong feature around which is observed in the experiment and PIC simulations cannot be reproduced when retardation is neglected, see Fig. 3B. Moreover, the harmonics’ efficiency is much lower when retardation is neglected. Accordingly, retardation provides the major contribution to the nonlinearity of the ROM process and is vital for the enhancement of the XROM harmonics.
In conclusion, we found a strong enhancement of harmonic frequencies generated from a plasma surface oscillating at XUV frequencies with relativistic velocity. The enhanced harmonic generation is verified by laser plasma simulations and analytical modeling. The coherent harmonic radiation at may be exploited in the future, for instance, as a novel diagnostic for solid density plasmas. It may further find applications in various scientific fields where intense, coherent XUV radiation of small bandwidth is needed. Examples are the investigation of plasmas in fusion-related research or in laboratory astrophysics since the radiation at is transmitted. Other applications such as XUV spectroscopy, XUV microscopy, coherent diffraction imaging, or the seeding of free-electron lasers are evident.
- (1) S. Kundu, Analog and Digital Communications, Pearson Education India (2013).
- (2) M. Lee, H. E. Katz, C. Erben, D. M. Gill, P. Gopalan, J. D. Heber, and D. J. McGee, Science 298, 1401 (2002).
- (3) R. Lichters, J. Meyer-ter-Vehn, and A. Pukhov, Physics of Plasmas 3, 3425 (1996).
- (4) G. D. Tsakiris, K. Eidmann, J. Meyer-ter-Vehn, and F. Krausz, New Journal of Physics 8, 19 (2006).
- (5) B. Dromey, M. Zepf, A. Gopal, K. Lancaster, M. S. Wei, K. Krushelnick, M. Tatarakis, N. Vakakis, S. Moustaizis, R. Kodama, M. Tampo, C. Stoeckl, R. Clarke, H. Habara, D. Neely, S. Karsch, and P. Norreys, Nature Physics 2, 456 (2006).
- (6) C. Thaury, F. Quéré, J. P. Geindre, A. Levy, T. Ceccotti, P. Monot, M. Bougeard, F. Reau, P. D’Oliveira, P. Audebert, R. Marjoribanks, and P. H. Martin, Nature Physics 3, 424 (2007).
- (7) C. Thaury and F. Quéré, Journal of Physics B-Atomic Molecular and Optical Physics 43, 213001 (2010).
- (8) U. Teubner and P. Gibbon, Reviews of Modern Physics 81, 445 (2009).
- (9) S. Gordienko, A. Pukhov, O. Shorokhov, and T. Baeva, Physical Review Letters 93, 115002 (2004).
- (10) S. V. Bulanov, N. M. Naumova, and F. Pegoraro, Physics of Plasmas 1, 745 (1994).
- (11) T. Baeva, S. Gordienko, and A. Pukhov, Physical Review E 74, 046404 (2006).
- (12) M. Behmke, D. an der Brügge, C. Rödel, M. Cerchez, D. Hemmers, M. Heyer, O. Jäckel, M. Kübel, G. G. Paulus, G. Pretzler, A. Pukhov, M. Toncian, T. Toncian, and O. Willi, Physical Review Letters 106, 185002 (2011).
- (13) G. Sansone, L. Poletto, and M. Nisoli, Nature Photonics 5, 656 (2011).
- (14) B. Dromey, S. Kar, C. Bellei, D. C. Carroll, R. J. Clarke, J. S. Green, S. Kneip, K. Markey, S. R. Nagel, P. T. Simpson, L. Willingale, P. McKenna, D. Neely, Z. Najmudin, K. Krushelnick, P. A. Norreys, and M. Zepf, Physical Review Letters 99, 085001 (2007).
- (15) A. Tarasevitch, K. Lobov, C. Wünsche, and D. von der Linde, Physical Review Letters 98, 103902 (2007).
- (16) K. Krushelnick, W. Rozmus, U. Wagner, F. N. Beg, S. G. Bochkarev, E. L. Clark, A. E. Dangor, R. G. Evans, A. Gopal, H. Habara, S. P. D. Mangles, P. A. Norreys, A. P. L. Robinson, M. Tatarakis, M. S. Wei, and M. Zepf, Physical Review Letters 100, 125005 (2008).
- (17) C. Rödel, D. an der Brügge, J. Bierbach, M. Yeung, T. Hahn, B. Dromey, S. Herzer, S. Fuchs, A. Galestian Pour, E. Eckner, M. Behmke, M. Cerchez, O. Jäckel, D. Hemmers, T. Toncian, M. C. Kaluza, A. Belyanin, G. Pretzler, O. Willi, A. Pukhov, M. Zepf, and G. G. Paulus, Physical Review Letters 109, 125002 (2012).
- (18) S. Kahaly, S. Monchocé, H. Vincenti, T. Dzelzainis, B. Dromey, M. Zepf, Ph. Martin, and F. Quéré, Physical Review Letters 110, 175001 (2013).
- (19) F. Dollar, P. Cummings, V. Chvykov, L. Willingale, M. Vargas, V. Yanovsky, C. Zulick, A. Maksimchuk, A. G. R. Thomas, and K. Krushelnick, Physical Review Letters 110, 175002 (2013).
- (20) B. Dromey, D. Adams, R. Hörlein, Y. Nomura, S. G. Rykovanov, D. C. Carroll, P. S. Foster, S. Kar, K. Markey, P. McKenna, D. Neely, M. Geissler, G. D. Tsakiris, and M. Zepf, Nature Physics 5, 146 (2009).
- (21) J. Bierbach, C. Rödel, M. Yeung, B. Dromey, T. Hahn, A. G. Pour, S. Fuchs, A. E. Paz, S. Herzer, S. Kuschel, O. Jäckel, M. C. Kaluza, G. Pretzler, M. Zepf, and G. G. Paulus, New Journal of Physics 14, 065005 (2012).
- (22) I. Watts, M. Zepf, E. L. Clark, M. Tatarakis, K. Krushelnick, A. E. Dangor, R. M. Allott, R. J. Clarke, D. Neely, and P. A. Norreys, Physical Review Letters 88, 155001 (2002).
- (23) U. Teubner, G. Pretzler, T. Schlegel, K. Eidmann, E. Förster, and K. Witte, Physical Review A 67, 013816 (2003).
- (24) Note that for generating harmonics at a certain photon energy, only half the harmonic order is required when SHG pulses instead of the fundamental are used. This can explain the higher efficiencies of the ROM harmonics in this frequency range when SHG pulses are used.
- (25) R. L. Sandberg, A. Paul, D. A. Raymondson, S. Hädrich, D. M. Gaudiosi, J. Holtsnider, R. I. Tobey, O. Cohen, M. M. Murnane, H. C. Kapteyn, C. G. Song, J. W. Miao, Y. W. Liu, and F. Salmassi, Physical Review Letters 99, 098103 (2007).
- (26) G. Lambert, T. Hara, D. Garzella, T. Tanikawa, M. Labat, B. Carre, H. Kitamura, T. Shintake, M. Bougeard, S. Inoue, Y. Tanaka, P. Salieres, H. Merdji, O. Chubar, O. Gobert, K. Tahara, and M. E. Couprie, Nature Physics 4, 296 (2008).
- (27) F. Quéré, C. Thaury, P. Monot, S. Dobosz, P. Martin, J. P. Geindre, and P. Audebert, Physical Review Letters 96, 125004 (2006).
- (28) B. Dromey, S. G. Rykovanov, D. Adams, R. Hörlein, Y. Nomura, D. C. Carroll, P. S. Foster, S. Kar, K. Markey, P. McKenna, D. Neely, M. Geissler, G. D. Tsakiris, and M. Zepf, Physical Review Letters 102, 225002 (2009).
- (29) I. Fidone, G. Ramponi, and P. Brossier, Physics of Fluids 21, 237 (1978).
- (30) P. A. Forsyth, W. Petrie, and B. W. Currie, Nature 164, 453 (1949).
- (31) J. P. Wild, J. D. Murray, and W. C. Rowe, Nature 172, 533 (1953).
- (32) U. Teubner, P. Gibbon, D. Altenbernd, D. Oberschmidt, E. Förster, A. Mysyrowicz, P. Audebert, J. P. Geindre, and J. C. Gauthier, Laser and Particle Beams 17, 613 (1999).
- (33) T. J. M. Boyd and R. Ondarza-Rovira, Physical Review Letters 85, 1440 (2000).
- (34) T. Kunzl, R. Lichters, and J. Meyer-ter-Vehn. Laser and Particle Beams 21, 583 (2003).
- (35) The plasma resonance has a considerable width due to damping. As it is excited by the laser frequency , the dominant mode will have a frequency equal to an integer multiple of .
- (36) F. Brunel, Physical Review Letters 59, 52 (1987).
- (37) J. P. Geindre, R. S. Marjoribanks, and P. Audebert, Physical Review Letters 104, 135001 (2010).
- (38) A. Tarasevitch, R. Kohn, and D. von der Linde, Journal of Physics B-Atomic Molecular and Optical Physics 42, 134006 (2009).