aa r X i v : . [ phy s i c s . p l a s m - ph ] J un Determination of Carrier-Envelope Phase of RelativisticFew-Cycle Laser Pulses by Thomson Backscattering Spectroscopy
M. Wen,
1, 2
L.L. Jin, H.Y. Wang, Z. Wang, Y.R. Lu, J.E. Chen, and X.Q. Yan
1, 4, ∗ State Key Laboratory of Nuclear Physics and Technology,Peking University, Beijing 100871, China Institute of Photonics & Photon-Technology,Northwest University, Xi’an 710069, China Department of Physics, Northwest University, Xi’an 710069, China Key Lab of High Energy Density Physics Simulation,CAPT, Peking University, Beijing 100871, China (Dated: November 13, 2018)
Abstract
A novel method is proposed to determine the carrier-envelope phase (CEP) of a relativistic few-cycle laser pulse via the central frequency of the isolated light generated from Thomson backscatter-ing (TBS). We theoretically investigate the generation of a uniform flying mirror when a few-cycledrive pulse with relativistic intensity (
I > W (cid:14) cm ) interacts with a target combined with athin and a thick foil. The central frequency of the isolated TBS light generated from the flyingmirror shows a sensitive dependence on the CEP of the drive pulse. The obtained results areverified by one dimensional particle in cell (1D-PIC) simulations. PACS numbers: and duration of 10 fs or shorterare produced [4]. The focusing of the few-cycle pulse can reach ≫ W/cm relativisticintensity on the target [5], which is suitable for laser wakefield acceleration regime to generatemonoenergetic electrons [6] as well as for high harmonic generation on plasma surfaces [7]and gas jets. For relativistic few-cycle laser, carrier-envelop phase (CEP) measurements arestill envisaged for CEP stabilization that will be necessary to generate single attosecondbursts.The electric field of a laser pulse can be written as E ( t ) = E ( t ) cos ( ω L τ + φ ), with E ( t )being the pulse envelope, ω L being the frequency of the carrier wave, and φ being the CEP [8].The CEP φ is defined as the offset between the optical phase and the maximum of the waveenvelope of an optical pulse. The CEP may affect many processes involving instantaneouslaser-matter interaction. On one hand, for few-cycle pulses, it has been proved that theelectric field as a function of time depends on the CEP, although the envelope is the samefor all pulses. The CEP effects of ultrashort laser pulses are widely investigated from thenon-ionizing optics regime [9] to the ionizing intensity regime [10], even to the relativisticregime [11]. On the other hand, with a method for measuring the CEP of a many-cyclepulse [12], CEP effects by intense multi-cycle pulses are experimentally observed [13].So far, a method known as stereo above threshold ionization (ATI) has been demonstratedexperimentally to determine the CEP of few-cycle pulses with intensities up to I = 10 − W/cm [14], at a precision of about π /300 [15]. Other methods of measuring the CEP arepossible through an attosecond photon probe [16] and detection of THz emission generatedin a plasma [17]. However, these methods are not available for laser pulses of intensitiesabove I = 10 W/cm , when relativistic effects become increasingly important. Recently, aquantum method is proposed to determine the CEP of ultra-relativistic intensity by detectingthe angular emission range via multiphoton Compton scattering [18], which is availablewhen the intensity I > W/cm . This Letter reports the CEP of a relativistic intense2 I > W/cm ) few-cycle laser pulse can be determined by detecting the spectroscopy ofthe isolated Thomson Backscattering (TBS) pulse, which is testified by an analytical modeland particle in cell simulations.The corresponding configuration is sketched in Fig. 1(b). In this scheme, an intense few-cycle pulse irradiates a target combined with an ultra-thin (nm) foil and a thick and densefoil (the separation between these two foils is x r ). The electrons of the ultra-thin foil drivenby the intense pulse play the role of a flying mirror. The thick foil behind will reflect the drivepulse and let only the flying mirror pass through. The flying mirror flies with a relativisticfactor γ x = 1 .p − β x , with β x = v x / c being the velocity of the plane flyer in the normaldirection. A counter propagating probe light is then mirrored and frequency upshifted bythe relativistic Doppler factor, which is (1 + β x )/(1 − β x ) ≈ γ x for γ x ≫ γ values andthe heavy ions are left behind unmoved [22, 23] when the charge separation field is muchsmaller than the amplitude of the laser field E L . The charge separation field dependson the area charge density σ = en d , where n and d are the plasma density and thefoil thickness, respectively. In our analytical model, all equations are presented in thenature unit. The normalized quantities are obtained from their counterparts in SI unitsmarked with prime, i.e., time and length are normalized according to t = ω L t ′ and l = k L l ′ ,field E = eE ′ /( mcω L ), vector potential a = eA ′ /( mc ), density n = n ′ / n c and momentum p = p ′ / mc , where e and m are the charge and the mass of the electron, ω L and k L are thelaser frequency and the wave number, c is the speed of light in vacuum and n c = ε mω L / e is the electron critical density. We use a linearly polarized ( E z = 0) pulse with a sine squareenvelop as the drive pulse E y = E y sin ( πτ / T ) cos ( τ + φ ) , (1)with the propagating coordinate τ = t − x and the peak of envelope E y . The dynamics aredescribed by the equations [23, 24] dκdτ = " σ (cid:0) p y (cid:1) , (2)3 IG. 1. (Color online) (a) Schematic showing TBS of a weak probe pulse by a flying mirror surfingon a relativistic few-cycle pulse. The scattered pulse is strongly chirped due to the acceleration ofthe electron layer. (b) Configuration of the CEP measurement with the TBS light. The drive pulseaccelerates the flying mirror and is reflected by the reflect foil, without energy consumed. Afterthe relativistic flying mirror flies to the rear side of the reflect foil, a counter propagating probe ω light is then mirrored and frequency upshifted to ω = f ( φ ) ω L , which highly depends on the CEPof the drive pulse. p y dτ = − E y − h σ p y κ i . (3)Here the terms in the square brackets denote the self-fields of the electron layer, and E y isthe instantaneous laser field when the electron layer surfing in the laser pulse. Analytically, κ = γ − p x and p y can be obtained by integrating Eqs. (2) and (3) over the duration [0 , τ ].With γ = 1 + p x + p y , one can find that the energy of the flying mirror γ is γ = (cid:0) p y (cid:1)(cid:14) κ + κ /2 . (4)Although the self-radiation and the charge separation field are taken into account in thismodel, the dominating force is still from the drive pulse. When the charge surface density σ is considerably small compared with the laser field E L , the analytical model will regressto single electron model κ → γ ) + = p y (cid:14) ≈ (cid:2)R τ E y ( τ, φ ) dτ (cid:3) . a y ( τ, φ ) (cid:14) ∝ cos [2 φ + g ( τ )] , (5)where g ( τ ) is a function of τ . Equation (5) shows the energy gain of the flying mirror (∆ γ ) + varies periodically with the CEP of the drive pulse. In other words, a shift of π in the CEPwould induce the same results. It shows the energy of the flying mirror is mainly dependenton the temporally varied vector potential a y , and carries the detailed information of thedrive pulse. Through a proper process to obtain the energy of the flying mirror (TBS shownlater), we can extract the CEP of the laser pulse.We consider a relativistically intense laser field with a peak amplitude of E y = 3 . I = 2 . × W/cm for λ L = 800nm, with pulse duration T = 2 τ L , where λ L and τ L are laser wavelength and period. The numerical results fromour analytical model are plotted in Fig. 2. Figures 2(a) and (b) show that the temporalvariation of the electric field and the vector potential depend on the CEP. The energy of theflying mirror during the few-cycle laser field with different CE phases are shown in Figs. 2(c)and (d). Figure 2(c) exhibits how the energy of the flying mirror evolves along the laserpropagation x . It is shown that the peak value of γ depends on the CEP of the relativisticallyintense laser. The maximal energy of flying mirror can be almost doubled by choosing theCEP properly, e.g., γ max = 6 . φ = 0 (solid curve), while γ max = 11 . φ = π /25dotted curve). If we detect the energy at a fixed position x , it varies periodically withthe CEP of the drive laser pulse. A similar trend appears in the dependence of the electronlayer energy on time t , as shown in Fig. 2(d).Bright VUV- or X-ray source can be obtained by TBS from the relativistic flying mirror.It has been demonstrated that the TBS light from a flying mirror is chirped and has a broadspectrum [20, 25], sketched in Fig. 1(a), which makes it difficult to find a central frequencyin the spectrum of the TBS light. However, this can be overcome by setting a thick foil asa reflector behind the ultra-thin foil with a distance x r [see Fig. 1(b)], which is a practicalway to generate a uniform flying mirror [25, 26]. After the flying mirror emerges from thereflect foil and divorces from the drive laser, its energy γ becomes a constant and dependson the CEP of the pulse [see Fig. 3(a)]. The solid (blue), dashed (green) and dotted (red)curves correspond to x r = λ , x r = 1 . λ and x r = 2 . λ , respectively. The same with theanalytical prediction from Eq. (5), γ is a periodic function of the CEP φ with a period of π .When a probe light irradiates this flying mirror, the frequency of the probe pulse isupshifted by a fixed factor ω r / ω L = 4 γ x , and an isolated pulse with a narrow spectrumis generated. We verify the results by 1D-PIC simulations [27]. The foil parameters inthe simulations are the same as those in Ref. [25], i.e., density n /n c = 1 and thickness d /λ L = 0 .
001 for the ultra-thin foil, and n /n c = 400 and d /λ L = 0 . x r = λ . We choose φ = 0, φ = π /4, φ = π /2 and φ = 3 π /4 forexamples. As a result of the dependence of the flying mirror energy on the CEP, the centralfrequency of the spectrum is sensitive to the CEP, e.g., varying from ∼ ω L at φ = 0 to ∼ ω L at φ = π/
2. The central frequency of the TBS light as a periodic function of theCEP predicted by the analytic model is shown in Fig. 3(c).It should be noticed that the reflected intense light would interact with the flying mirrorand cause an energy loss (∆ γ ) − ≈ p y = 0, and a very uniform relativistic flying mirror [with the energy γ x = γ − (∆ γ ) − ]is obtained, while the relative energy loss via Coulomb collsion is found to be negligible [25].Taking the CEP of the drive laser into consideration, γ x depends periodically on the CEP φ , illustrated by the right axis in Fig. 3(c). Obeying the relation ω r / ω L = 4 γ x , the central6requency also exhibits periodicity on the CEP. The analytical predictions agree well withthe simulation results.Due to the period of π we will get two possible phases with one measurement, e.g. ω r =200 ω L while φ = 0 . π and φ = 0 . π ( x r = λ ). By introducing a second measurement,this restriction can be removed and the CEP can be determined in the range of π . Thesimulations show when the drive pulse is highly reflected with a limited energy loss [shownin Fig. 3(d)], the drive pulse can be transmitted to another double-foil target again witha different distance x r . For example, if the first measurement gives ω str = 200 ω L , for thesecond measurement with x r = 2 . λ , the CEP is determined to be 0 . π when secondmeasurement gives ω ndr = 297 ω L , or 0 . π when ω ndr = 115 ω L . Moreover, the centralfrequency is very sensitive to the CEP. For example, around the point (cid:16) . π | φ , ω L | ω r (cid:17) ,a difference of 1 ω L in the detectable central frequency introduces a phase shift of 8 × − π in the CEP.In summary, the evolution of a flying mirror driven by a relativistic, few-cycle pulse( I = 2 . × W/cm and T ≈ . λ = 0 . µ m) from an ultra-thin foil is investigatedtheoretically. With the help of a reflect foil, a TBS light pulse with narrow spectrum isobtained when a probe light is reflected from the flying mirror. The central frequency of theTBS light is a periodic function of the CEP, with the period of π . The detection of the centralfrequency of the TBS light makes it possible to determine the CEP of a relativistic few-cyclepulse. We introduce a double-measurement process to determine the CEP in the range of π . In principle, this method is also feasible for a weaker or longer pulses with relativisticintensity ( I > W/cm ), while even thinner foil is needed to generate a uniform flyingmirror.The authors are grateful to Dr. A. Di Piazza for useful discursion. This work was sup-ported by National Nature Science Foundation of China (Grant Nos. 10935002, 11025523,61008016 and 10905003) and National Basic Research Program of China (Grant No.2011CB808104). M.Wen and L.L.Jin acknowledges the support from the Scientific ResearchProgram Funded by Shaanxi Provincial Education Department (Program Nos. 11JK0538and 11JK0529). ∗ [email protected]
1] A. Baltuˇska et al. , Nature (London) , 611 (2003); A. Bonvalet et al. , Appl. Phys. Lett. , 2907 (1995); G. Krauss et al. , Nat. Photon. , 33 (2010); A.L. Cavalieri et al. , New J.Phys. , 242 (2007); E. Goulielmakis et al. , Science , 1614 (2008).[2] F. Krausz and M. Ivanov, Rev. Mod. Phys. , 163 (2009).[3] G. Mourou et al. , Rev. Mod. Phys. , 309 (2006); V. Yanovsky et al. , Opt. Express et al. , Phys. Rep.
41 (2006).[4] A. Stingl et al. , Opt. Lett. , 602 (1995); D. H. Sutter et al. , Appl. Phys. B , S5 (2000);R. Ell et al. , Opt. Lett. , 373 (2001); D. Herrmann et al. , Opt. Lett. , 2459 (2009).[5] F. S. Tsung et al. , Proc. Natl. Acad. Sci. U.S.A. , 29 (2002); L. L. Ji et al. , Phys. Rev. Lett. , 215005 (2009).[6] K. Schmid et al. , Phys. Rev. Lett. , 124801 (2009).[7] P. Heissler et al. , Appl. Phys. B , 511 (2010); C. D. Tsakiris et al. , New J. Phys. , 19(2006).[8] P. Dietrich, F. Krausz, and P. B. Corkum, Opt. Lett. , 16-18 (2000).[9] M. Mehendale et al. , Opt. Lett. , 1672 (2000); U. Morgner et al. , Phys. Rev. Lett. , 5462(2001); Y. Wu and X. X. Yang, Phys. Rev. A , 013832 (2007); X. T. Xie and M. A. Macovei,Phys. Rev. Lett. , 073902 (2010); K. Xia et al. , Phys. Lett. A , 173 (2007).[10] G. G. Paulus et al. , Nature (London) , 182 (2001); F. Krausz et al. , Opt. Photonics News , 46 (1998); A. de Bohan et al. , Phys. Rev. Lett. , 1837 (1998); G. Tempea et al. , J. Opt.Soc. Am. B , 669 (1999); T. Brabec and F. Krausz, Rev. Mod. Phys. , 545 (2000); C. P.J. Martiny and L. B. Madsen, Phys. Rev. Lett. , 093001 (2006); P. Lan et al. , J. Phys B , 403 (2007).[11] E. N. Nerush and I. Yu. Kostyukov, Phys. Rev. Lett. , 035001 (2009); S. Varr´o, LaserPhys. Lett. , 218 (2007).[12] P. Tzallas et al. , Phys. Rev. A 82, 061401 (2010); P. Tzallas et al. , Nat. Phys. 3, 846 (2007).[13] P. K. Jha et al. , Phys. Rev. A 83, 033404 (2011).[14] G.G. Paulus et al. , Phys. Rev. Lett. , 253004 (2003).[15] T. Wittmann et al. , Nature Phys. , 357 (2009).[16] E. Goulielmakis et al. , Science , 1267 (2004).[17] M. Kreß et al. , Nature Phys. , 327 (2006).[18] F. Mackenroth et al. , Phys. Rev. Lett. , 063903 (2010); F. Mackenroth and A. Di Piazza, hys. Rev. A , 032106 (2011).[19] A. Einstein, Ann. Phys. (Leipzig) , 891 (1905).[20] J. Meyer-ter-Vehn and H. C. Wu, Eur. Phys. J. D , 443(2009).[21] Z. Chang et al. , Phys. Rev. Lett. , 2967 (1997).[22] V. V. Kulagin et al. , Phys. Rev. Lett. , 124801 (2007).[23] M. Wen et al. , Eur.Phys. J. D , 451 (2009).[24] H. K. Avetissian, Relativistic Nonlinear Electrodynamics (Springer, New York, 2006).[25] H. -C. Wu et al. , Phys. Rev. Lett. , 234801 (2010).[26] F. Wang et al. , Phys. Plasmas , 083102 (2007).[27] Z. -M. Sheng et al. , Phys. Rev. Lett. , 5340 (2000);X.Q. Yan et al. , Phys. Rev. Lett. ,135003 (2008). (d) t LL (b) e E y / m c L L (a) x/ L (c) a y FIG. 2. (Color online) (a) The electric field, (b) vector potential, (c) spatial and (d) temporal evolu-tion of normalized electron layer energy in the few cycle laser field with φ = 0 [solid (black) curve], π/ π/
200 4000.00.51.0-1.0 -0.5 0.0 0.5 1.00100200300400 0255075100-1.0 -0.5 0.0 0.5 1.004812 0.0 0.5 1.0 1.5 2.0-4-2024 (d)(c) =0 = /4 = /2 = /4 A m p li t ude ( a . u . ) Frequency r L r L (b) x (a) E in -E re - E r e / E E i n / E L FIG. 3. (Color online) (a) The normalized energy of electron layer γ as functions of the initialCEP of the drive ultrashort pulse, when the electron layer reaches the location of the reflect foil x r . The solid (black), dashed (red) and dotted (blue) curves correspond to x r = λ, . λ and2 . λ , respectively. (b) The spectra of Thomson backscattering light with the CE phases of drivepulses φ = 0 , π/ , π/ π/ x r = λ . (c) When x r = λ the central frequency of the TBSlight as a function of the CEP obtained from the model [(black) solid curve], with the matchedsimulation results [(red) open cubes). The (blue) dotted curve represent the dependence of thedetected central frequency on the CEP of the drive pulse when x r = 2 . λ . The electric fields ofthe incident and reflected drive pulse are compared in (d).. The electric fields ofthe incident and reflected drive pulse are compared in (d).