Principal frequency of an ultrashort laser pulse
Enrique G. Neyra, Pablo Vaveliuk, Emilio Pisanty, Andrew S. Maxwell, Maciej Lewenstein, Marcelo F. Ciappina
PPrincipal frequency of an ultrashort laser pulse
Enrique G. Neyra, Pablo Vaveliuk, Emilio Pisanty ,
2, 3
Andrew S.Maxwell ,
2, 4
Maciej Lewenstein ,
2, 5 and Marcelo F. Ciappina
2, 6, 7, ∗ CIOp: Centro de Investigaciones Ópticas, CONICET-CICBA-UNLP,Camino Centenario y 506, M.B. Gonnet (1897), Pcia. Bs. As., Argentina. ICFO – Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona) Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, Max-Born-Straße 2A, Berlin 12489, Germany Department of Physics & Astronomy, University College London,Gower Street, London WC1E 6BT, United Kingdom ICREA, Passeig de Lluís Companys, 23, 08010 Barcelona, Spain Physics Program, Guangdong Technion – Israel Institute of Technology, Shantou, Guangdong 515063, China Technion – Israel Institute of Technology, Haifa, 32000, Israel
We introduce an alternative definition of the main frequency of an ultrashort laser pulse, theprincipal frequency ω P . This parameter is complementary to the most accepted and widely usedcarrier frequency ω . Given the fact that these ultrashort pulses, also known as transients, have atemporal width comprising only few cycles of the carrier wave, corresponding to a spectral bandwidth ∆ ω covering several octaves, ω P describes, in a more precise way, the dynamics driven by thesesources. We present examples where, for instance, ω P is able to correctly predict the high-orderharmonic cutoff independently of the carrier envelope phase. This is confirmed by solving the time-dependent Schrödinger equation in reduced dimensions, supplemented with the time-analysis of thequantum spectra, where it is possible to observe how the sub-cycle electron dynamics is betterdescribed using ω P . The concept of ω P , however, can be applied to a large variety of scenarios, notonly within the strong field physics domain. I. Introduction
During the past two decades we have been witness toa constant development of a varied set of ultrashort laserpulses, with temporal widths well below two optical cy-cles. In general, the main objective of these sources is thetemporal study of diverse physical phenomena on theirnatural scale. The techniques that have been developedto scrutinize dynamics on this territory are based on del-icate control of strong-field laser-atom interactions andconfigure the core of what is known as attosecond spec-troscopy. The spectral range of those pulses is very broad,covering both the THz [1–3], and infrared and visible, [4–7] as well as the XUV [8–12] regions of the electromag-netic spectrum.The driving sources described above also allow thecoherent control of various quantum systems, particu-larly the standard two-level system, widely used as a toymodel for different physical processes. The precise andhigh-speed manipulation over the population of quantumstates has important applications in, e.g., quantum in-formation and spintronics, among others [13]. In addi-tion, the generation of intense ultrashort laser pulses inthe few-cycle, single-cycle and sub-cycle domains has en-abled the study of strongly non-linear light-matter in-teractions and given rise to novel and fascinating phe-nomena [14]. Maybe the most important example arethe so-called isolated attosecond pulses (IAPs). Thesesources are the workhorse to tackle the dynamics of elec-trons under strong fields in its natural temporal, attosec-ond, scale [11, 15]. IAPs are obtained from high-harmonicgeneration (HHG) using a variety of spectral postprocess-ing approaches [16–20].HHG is an extremely non-linear optical process inwhich a strong laser field interacts with atoms, molecules ∗ [email protected] and, recently, bulk materials, and drives the productionof high-frequency ultrashort bursts of coherent electro-magnetic radiation [11, 15]. This emission possesses aset of distinct features, namely, (i) a steadying decreaseof the first harmonics of the driving field yield, (ii) abroad plateau, that can cover up to thousands of har-monic orders of the original driving field, and (iii) a cutoff,where the spectrum suddenly terminates. The underly-ing physics of the HHG in atoms and molecules can betraced out from a sequence involving three steps, whichcan be summarized as follows: (i) the laser ionizes the tar-get via tunnel ionization, (ii) the released electron travelsin the laser continuum gaining kinetic energy and, whenthe laser electric field reverses its direction, (iii) the elec-tron returns back to the parent ion, where it recombines,releasing its kinetic energy as a high energy photon [20].The HHG phenomenon can be modelled using a widerange of approaches, from classical-based schemes [17, 20]to intensive numerical computations involving the numer-ical solution of the time-dependent Schrödinger equation(TDSE), both in one or several spatial dimensions [21].Yet, the quantitative schemes that most closely follow theoverall intuition are the so-called quasi-classical meth-ods, with the Strong-Field Approximation (SFA) beingthe most prominent exponent [17, 18]. Here the emissionamplitude key ingredient is a path-integral sum over dis-crete emission events. Invoking the SFA, the well known3.17-law, which correctly predicts the HHG cutoff law,can be easily obtained [22]. The SFA has been applied toa large variety of strong field processes with undeniablesuccess (for a recent historical review see [18]).The synthesis of ultrashort pulses has advanced quicklyin recent years. Particularly, the generation of high-energy single- and sub-cycle IR laser pulses has been ex-perimentally verified, through the combination and ma-nipulation of the spectral content of laser sources of differ-ent wavelengths. The current generation of these pulsespossesses tremendous technological challenges, due to,amongst other difficulties, the synchronization of their a r X i v : . [ phy s i c s . a t o m - ph ] J a n different sources with sub-fs temporal resolution [23–30].Recently, the generation of a 53-attosecond X-ray pulsewas demonstrated using HHG in noble gases driven by amid-infrared few-cycle laser source [31].When working with ultrashort laser pulses in the few-cycle regime, it is well known that the so-called carrier-envelope phase (CEP), φ , plays an instrumental role inthe resulting laser-matter interaction processes driven bythose sources. This is because the pulse envelope experi-ences appreciable changes within an optical cycle of thecarrier wave. In this way, for instance, the maximum fieldamplitude of a sine-like pulse, typically characterized by φ = 0 , is largely different than that of a cosine-like pulse,where φ = π/ . Furthermore, it has been demonstratedthat not only the CEP, but also the pulse width is relevantin certain strong field processes [32–35].In this work we introduce an alternative definition ofthe main frequency of an ultrashort laser pulse. Thisnew parameter, which we name principal frequency ω P ,appears to be much more appropriate than the standarddefinition, the carrier frequency ω , to correctly charac-terize the interaction of these pulses with matter. Using ω P as the frequency that drives the dynamics, we areable to give a better interpretation of previously pub-lished results, as well as provide more reliable predictionsof strong field processes outcomes. Its definition is basedon a particular way to weight the spectral content of thelaser pulse electric field and it is adequately justified if weresort to the particle nature of light, i.e., if we considerthat light is composed of light-quanta (photons).This article is organized as follows. In Section II, wepresent the mathematical foundations of the principal fre-quency ω P . We present a set of examples based on dif-ferent definitions of the laser electric field. We show how ω P varies as a function of the bandwidth of the pulses forthree archetypal cases. Furthermore we show that thereexists a correlation between the positions of the maximaand minima of the laser pulse’s electric field with the prin-cipal period, defined as T P = 2 π/ω P . In Section III weuse the definition of ω P to characterize the HHG spectraof an atom driven by a series of few-cycle laser pulses.For the computation of the HHG spectra, we use bothquantum mechanical and classical approaches. These twocomplementary schemes allow us to disentangle the un-derlying physics of the HHG process. We end our contri-bution in Section IV presenting our conclusions togetherwith a brief outlook. Atomic units are used throughoutthe article unless otherwise stated. II. Principal frequencyA. Definition
The most accepted definition of carrier frequency, ω ,of an ultrashort laser electric field pulse E ( t ) is the cen-tral frequency of the modulus if its Fourier transform | E ( ω ) | , if the spectrum is symmetric. However, if thespectral content is more complex, ω results from an in-tegral over the density distribution ρ ( ω ) = S ( ω ) , where S ( ω ) = | E ( ω ) | is the spectral power, i.e. ω = (cid:82) ∞−∞ ωS ( ω )d ω (cid:82) ∞−∞ S ( ω )d ω . (1)Considering the integral in the denominator (cid:82) ∞−∞ S ( ω )d ω defines the total number of photons (cid:80) i n i = N , we caninterpret the above definition as an average over the pho-ton energies: (cid:126) ω = (cid:80) i (cid:126) ω i n i (cid:80) i n i . (2)Let us construct a different explanation. The definitionabove establishes that every photon has the same weight in the average of Eq. (2), i.e., there is no difference be-tween n photons with energy ω and n photons withenergy ω , where, e.g., ω < ω . But, on the contrary, itcould be natural to assume that a more energetic photonhas a greater weight in the integral. Therefore, we canchange the density function ρ ( ω ) by an energy distribu-tion, i.e. to use (cid:126) ω i n i instead of the number of photons n i in Eq. (2). In this way, the new density function takesthe following form: ρ ( ω ) = ωS ( ω ) .In this way, our principal frequency ω P results: ω P = (cid:82) ∞−∞ ω S ( ω )d ω (cid:82) ∞−∞ ωS ( ω )d ω . (3)From the above definition we can observe that ω P givesmore weight to photons with greater frequencies (higherenergies). This is so because the actual definition of thenew density function ρ ( ω ) . Let us now see what thismeans in the temporal domain.The electric field of an ultrashort laser pulse can bewritten as E ( t ) = f ( t ) e iω t e iφ , where f ( t ) , ω and φ are the pulse envelope, the carrier frequency and the so-called carrier-enveloped phase (CEP), respectively. If wetake Re[ E ( t )] = 0 , we can find the zeros of E ( t ) , i.e. thetimes where E ( t ) = 0 . As is well known, these zerosare located at nπ/ω and (2 n + 1) π/ ω for sine-like andcosine-like pulses, respectively, and are spaced by π/ω .Here, n = 0 , , . . . . But what happens with the positionof the maxima and minima of the field E ( t ) ? For longpulses, i.e. when the temporal width τ is τ (cid:29) T , where T = 2 π/ω is the carrier period, the envelope in thecentral region varies slowly and the maxima and minimaare spaced by T / . However, if τ is of the order of T ,i.e. τ ∼ T , the situation changes considerably. For thiscase, it is easy to see that the maxima and minima of thefield E ( t ) depends now both on the envelope f ( t ) and theargument of the phase e iω t e iφ . This simple conclusionis important, if we consider that the interaction of ultra-short pulses with matter is dominated by these maximaand minima, and not by the zeros of E ( t ) .In the next section we show how ω P is correlated withthe positions of the maxima and minima of the electricfield E ( t ) , for different cases, each of them with differentspectral content. FIG. 1. (a)-(c) Spectral power S ( ω ) of the fields E R ( ω ) , E R ( ω ) and E G ( ω ) for E = 1 . All the cases are centered at thesame carrier frequency, ω = 2 π . The spectral bandwidth of both E R ( ω ) and E G ( ω ) is ∆ ω = π , meanwhile for E R ( ω ) each‘sub-spectrum’ has a bandwidth ∆ ω = π/ . In (b) δω = π/ . The blue solid (red dashed) represents the ω P ( ω ) frequency(see the text for details). (d)-(f) normalized time-dependent fields E R ( t ) , E R ( t ) and E G ( t ) . Thick blue (thin red) line definesa sine(cosine)-like pulse. In (e) the dashed (dotted) line corresponds to the temporal distance between two consecutive zeros(maximum and minimum) of the field for the sin ( cos )-like pulses, i.e. T / and T P / , respectively (see the text for details). B. Analysis
We start our analysis by considering the following fieldsin the spectral domain: E R ( ω ) = E rect (cid:18) ω − ω ∆ ω (cid:19) (4a) E R ( ω ) = E rect (cid:18) ω − ω + δω ∆ ω (cid:19) + E rect (cid:18) ω − ω − δω ∆ ω (cid:19) (4b) E G ( ω ) = E e − ( ω − ω ω ) rect (cid:18) ω − ω ω (cid:19) , (4c)where E is the peak field strength and the function rect( x/x ) is the so-called rectangle function defined as: rect( x/x ) = (cid:40) | x | ≤ x / | x | > x / . (5)The spectra of Eqs. (4a) and (4c) are centered at thefrequency ω , meanwhile Eq. (4b) is the sum of two carrierwaves with different frequencies, ω + δω and ω − δω . Inall the cases ∆ ω characterizes their respective spectralbandwidths.By taking the Fourier transform we find the electric fields in the temporal domain,i.e. E R ( t ) = E ∆ ω √ π e iω t sinc (cid:18) ∆ ω t (cid:19) (6a) E R ( t ) = E ∆ ω √ π e it ( ω − δω ) sinc (cid:18) ∆ ω t (cid:19) × (1 + e iδωt ) (6b) E G ( t ) = E i ∆ ω e − ( ∆ ωt ) e iω t U ( t )2 √ , (6c)where U ( t ) is given by U ( t ) = erfi (cid:18) ∆ ωt − i ω ω (cid:19) − erfi (cid:18) ∆ ωt i ω ω (cid:19) , (7)and sinc and erfi are the sinc function sinc( x ) = sin( x ) /x and the imaginary error function, respectively.Note that the spectra of the first two fields, Eqs. (4a)and (4b) are symmetric, while the third, Eq. (4c), is asym-metric. Furthermore, the envelope of Eq. (6a), as well asthe one of Eq. (6b), becomes the sinc function. Finally,for Eq. (6c) the envelope results in a Gaussian functionmultiplied by the function U ( t ) , that is composed as asum of two different imaginary error functions erfi( x ) .The fields defined above represent three different situ-ations. The field E R ( t ) is the most common expressionfor an ultrashort pulse, considering its spectral contentis continuous . The spectral function rect( ω − ω ∆ ω ) has theadvantage of possessing a limited bandwidth, given by ∆ ω/ < ω , which prevents the pulse having spectralcontent near zero-frequency. This is particularly relevantin the few-cycle regime, where other envelopes typicallyused, e.g. Gaussian or sech , are unable to fulfill this re-quirement. Near-zero frequencies are correlated with the“zero-area pulse problem” and are incompatible with theparaxial approximation used to focus the laser beams. Inthe E R ( t ) field we observe the so-called frequency beat,considering we are summing up two carriers with frequen-cies separated by δω . These pulses have been alreadyimplemented in the laboratory, and possess interestingproperties [27, 29, 36]. The last one, E G ( t ) , was chosen asa typical example of a field with an asymmetric spectrumand it is a relevant example where to test our hypothesis,since its carrier frequency ω is bandwidth dependent.In Figs. 1(a)-1(c) we plot the spectral power S ( ω ) ofthe fields defined in Eqs. (4a)-(4c), for E = 1 . All thecases are centered at the same carrier frequency, ω =2 π . The spectral bandwidth of both E R ( ω ) and E G ( ω ) is ∆ ω = π , meanwhile for E R ( ω ) each ‘sub-spectrum’has a bandwidth ∆ ω = π/ . For the latter δω = π/ .The temporal counterparts, E R ( t ) , E R ( t ) and E G ( t ) , aredepicted in Figs. 1(d)-1(f), where the thick blue (thin red)line defines a sine(cosine)-like pulse.Through the definition of the principal frequency ω P ,Eq. (3), we determine the principal period T P , T P =2 π/ω P . In this way, we can analyze how the period T ofthe pulses E R ( t ) , E R ( t ) and E G ( t ) , represented as twicethe distance between two adjacent maxima and minimain the temporal central region, is related to T P and thechanges of the bandwidth ∆ ω . In Fig. 1(d) we show how T P / defines more precisely the temporal distance be-tween two adjacent maxima and minima (exemplified forthe cos -like pulse), meanwhile T / is better suitable forthe position of the field zeros (exemplified for the sin -likepulse).Let us now compute how ω P changes as a functionof the bandwidth ∆ ω . For E R ( ω ) is possible to find asimple analytical expression, meanwhile for both E R ( ω ) and E G ( ω ) we deal with its numerical calculation. Forthe case of E R ( ω ) we have: ω P (∆ ω, ω ) = (cid:82) ∞−∞ ω S ( ω )d ω (cid:82) ∞−∞ ωS ( ω )d ω = ω + ∆ ω ω . (8)This last relationship between ω P , ∆ ω and ω has beenshown in [34], for the case where a few-cycle RF pulseinteracts with a two-level system.In Fig. (2) we show how the principal period T P changes as function of the bandwidth ∆ ω . The proce-dure to find the values of T P from the different cases isas follows. Starting at t = 0 we search the position ofa nearest minimum. If we define as t M the time wherethis minimum is located, for a cosine-like pulse we cancompute the period T COS as T COS = 2 t M . Likewise, for asine-like pulse becomes T SIN = 4 t M . Figure 2(a) depictsthe results for E R ( t ) . The dashed green line represents T P computed as T P = 2 π/ω P , meanwhile the thick blue(thin red) solid line corresponds to the T COS ( T SIN ) ex-tracted for the cosine(sine)-like pulses using the proce-dure explained above. We include the value T = 2 π/ω (dotted violet line), which is constant and equal to the FIG. 2. Principal period T P as a function of the pulse band-width ∆ ω for (a) E R ( t ) , (b) E R ( t ) and (c) E G ( t ) . In all thecases, dotted violet line: T = 2 π/ω (note that for the case of E G ( t ) pulses, ω depends on ∆ ω ), dashed green line: T P com-puted from T P = 2 π/ω P and thick blue (thin red) solid line: T COS ( T SIN ) extracted for the time-dependent cosine(sine)-likepulses (see the text for details). unity for this case because the spectrum E R ( ω ) is sym-metrical. We can observe that T P appears to be a muchmore reliable quantity to predict the temporal distancebetween maxima and minima, for both cos- and sine-likepulses, and in a broad range of bandwidths. In Fig. 2(b)we plot the results for E R ( t ) . Here, as well, we findan excellent agreement between the maxima and minimapositions predicted by T P and those computed directlyfrom the time-dependent field. Interestingly, for ∆ ω = 0 ,where the field results a sum of two continuous waveswith frequencies ω − δω and ω + δω , T P allows us toaccurately find the positions of the maxima and minima.Finally, in Fig. 2(c), we illustrate the case of E G ( ω ) . Inthis example, T changes due to the asymmetric characterof the field spectrum. Nevertheless, the positions of themaxima and minima computed using T P are much closerto those extracted from the fields than the ones calcu-lated starting from T . We should note, however, thatthe agreement is not so good as in the previous cases.For asymmetrical pulses, as is this case, it is possible toshow that an improvement in the maxima and minimapositions prediction can be achieved by changing S ( ω ) by S ( ω ) / in the definition of ω P , Eq. (3), although thischoice is relatively difficult to justify from first principles. FIG. 3. Cosine-like (a) and sine-like (b) E R ( t ) pulses for different bandwidths ∆ ω . The red, green and blue lines correspond topulses with a bandwidth of ∆ ω = π , ∆ ω = 2 π and ∆ ω = 3 π in units of ω , which we take corresponding to a laser wavelength of λ =2000 nm, respectively. The labels I-III denote the ionization and recombination regions (see the text for more details). HHGspectra for the Cosine-like (c) and Sine-like (d) pulses plotted in panels (a) and (b), respectively. The dotted lines correspondto the different HHG cutoffs (see the text for more details). III. Results and Discussion
In this section we use some of the pulses previouslydescribed to drive an atomic system and characterize thehigh-order harmonic generation (HHG) spectra in termsof the principal frequency ω P . As examples, we employboth the E R ( t ) and E R ( t ) fields to generate HHG inan hydrogen atom and simulate the dynamics throughthe numerical integration of the one-dimensional time-dependent Schrödinger equation (1D-TDSE). Recently, atheoretical investigation using sinc-shaped pulses for bothHHG and the construction of a single attosecond pulsewas presented [37]. A. E R ( t ) fields To perform the numerical simulations, three valuesof the bandwidth ∆ ω were chosen for the E R ( t ) fields,namely ∆ ω = π , ∆ ω = 2 π and ∆ ω = 3 π in units of ω ,which we take corresponding to a laser wavelength of λ =2000 nm, both for the sine- and cosine-like fields. Tak-ing into account the definition of the full-width at half-maximum (FWHM), we can find its value starting fromthe fields expression, Eq. (6a), as: FWHM = 5 . / ∆ ω .In this way, the corresponding FWHM result 1.77 opt. cy-cles, 0.885 opt. cycles. and 0.59 opt. cycles, for ∆ ω = π , ∆ ω = 2 π and ∆ ω = 3 π , respectively. For all cases, the peak amplitude of the field was heldfixed at E = 0 . a.u., which corresponds to a laserintensity of I = 1 × W/cm . For cosine-like pulses,this value is reached in the central part of the pulse andis independent of the bandwidth. On the contrary, forsine-like pulses, the maximum amplitude of the field de-creases, relative to the maximum of the envelope, as thepulse duration gets shorter; therefore, in order to keep themaximum value of the field at E = 0 . a.u., we mul-tiply the field amplitude by different scaling factors. Inthis way, taking into account that the HHG cutoff scalesas Iλ , any decrease in the maximum harmonic photonenergy it is due to a change in the pulse wavelength orfrequency and not to the peak amplitude of the field,product of the temporary shortening. In all the cases,we use a laser wavelength λ = I P = 0 . a.u.).The different cos- and sine-like pulses are plotted inFigs. 3(a) and 3(b), respectively. The red, green and bluelines correspond to pulses with a bandwidth of ∆ ω = π , ∆ ω = 2 π and ∆ ω = 3 π , respectively. Taking into ac-count Eq. (8), the principal frequency of these pulsestakes the following values: ω P ( π ) = 2 π (1 + 1 / ≈ . ω , ω P (2 π ) = 2 π (1 + 1 / ≈ . ω and ω P (3 π ) =2 π (1+3 / ≈ . ω . For these ω P the associated wave-lengths, thus, result λ P ( π ) = λ / . , λ P (2 π ) = λ / . and λ P (3 π ) = λ / . , respectively.Figures 3(c) and 3(d) show the respective harmonic FIG. 4. Time-frequency analysis extracted from the 1D-TDSE HHG spectra. Superimposed we plot the classically computedrescattering energies of electrons as a function of the ionization time, in white dots, and recombination time, in black dots, forthe laser pulses of Fig. 3(a) (panels (a)-(c)) and Fig. 3(b) (panels (d)-(f)). In panels (a)-(c) the red (green) arrow corresponds tothe electron trajectory that contributes to the more (less) energetic HHG cutoff. Meanwhile, in panels (d)-(f) the green arrowcorresponds to the electron trajectory that contributes to the single HHG cutoff (see the text for more details). The ionizationpotential I p energy is indicated by a black arrow. spectra for the pulses of Figs. 3(a) and 3(b), obtainedsolving the 1D-TDSE (for details about the numericalimplementation see e.g. [38]). The HHG for cosine-likepulses (Fig. 3(c)) show two distinguishable plateaus in allthe cases, with their corresponding cutoffs. The most en-ergetic one, which corresponds to an energy given by I λ ,is marked with a black dotted line. Let us note that for allthe cases this cutoff has the same photon energy (around130 eV). On the contrary, the photon energy values for theless energetic cutoffs depend on the different pulses anddecrease as the pulses become temporarily narrower. Toexplain this behavior, we perform a time-frequency anal-ysis and in this way visualize which temporal regions ofthe pulses are the ones that originate the correspondingcutoffs. The results are shown in Figs. 4(a)-4(c). Fur-thermore, we superimpose the electron kinetic energiesat the recombination time as a function of the ionization(white dots) and recombination times (black dots) calcu-lated classically (for more details see e.g. [39]).From this analysis we observe that the most energeticcutoffs (represented with red arrows) originate in the tem-poral region I-II of the pulse (see Fig. 3(a)). This meansthat the electron is ionized in region I and recombines inregion II. On the contrary, the less energetic ones origi-nate in the region II-III of the pulse (see green arrow inFigs. 4(a)-4(c)), i.e. the electron is ionized in region IIand recombines in region III. Let us notice that the peakamplitude of the field that ionizes the atom in region I issmaller than the one that does so in region II. This makesthe probability of ionization-recombination in region I-II lower than the one in region II-III, [40]. Furthermore,the classical analysis allows us to see what is the excur-sion time of the electron in the continuum for these twodifferent temporal regions, I-II and II-III.In the case of sine-like fields, the features of the har-monic spectra are as follows. An extended plateau, with aclear and single cutoff, is observed in the different spectra(Fig. 3(d)), which decreases as the pulse temporarily be-comes shorter. The time-frequency analysis of these spec-tra is shown in Figs. 4(d)-4(f), respectively. For all thecases, the most probable ionization-recombination event,indicated by a green arrow, occurs at the central part ofthe pulse, i.e. the electron is ionized in region I and re-combines in region II (see Fig. 3(b)). Furthermore, thesepulses are symmetric in the region of ionization and re-combination (I and II), analogously to a continuous field.The introduction of the principal frequency ω P , allowus a better interpretation of the results described above.In Fig. 1, we show that the period of time between a max-imum and a minimum of the field in the central regionof the pulse is accurately described by the period T P , as-sociated with the principal frequency ω P ( T P = 2 π/ω P ).We have also shown that T P decreases as the spectralcontent of the pulse increases (the FWHM value of thepulses gets smaller). This result is clearly visible in thefields represented by Figs. 3(a) and 3(b), where we seethat as the FWHM decreases, the period T P does so aswell.For the case of the spectra generated by the sine-likefields (Fig. 3(d)), the reduction in T P is observed as a FIG. 5. Cosine-like (a) and sine-like (b) E R ( t ) pulses for different values of ∆ ω and δω . The red (blue) line correspondsto pulses with ∆ ω = 0 . and δω = π ( ∆ ω = π/ and δω = π/ ). The labels I-III denote the ionization and recombinationregions (see the text for more details). HHG spectra for the cosine-like (c) and sine-like (d) pulses plotted in panels (a) and (b),respectively. The dotted lines correspond to the different HHG cutoffs (see the text for more details). decrease in the cutoff, as the pulse becomes temporar-ily shorter. Furthermore, the electron time of flight inthe continuum τ , for the most energetic trajectories, de-creases accordingly. This can be seen from the clas-sic analysis shown in Figs. 4(d)-4(f), where the timesof flight, denoted by green arrows, are: τ ( π ) ≈ . opt. cyles, τ (2 π ) ≈ . opt. cycles and τ (3 π ) ≈ . opt. cycles. This analysis is relevant considering the ef-ficiency of the harmonic generation depends on the timeof flight of the electron in the continuum and scales as ∝ λ [41]. To calculate the classical cutoff of the differentspectra, as a function of the frequency ω P , we use the for-mula I P + 3 . U P ( λ P ) . Thus, for λ = 2000 nm, we get: λ P ( π ) = 1960 nm, λ P (2 π ) = 1850 nm and λ P (3 π ) = 1680 nm, which correspond to HHG cutoff at ≈ eV, 115 eVand 97 eV, respectively. These values are indicated by thered, green and blue dotted lines in Fig. 3(d), respectively.In the case of cosine-like pulses the situation is differ-ent. As mentioned before, there are two temporal regionsin the pulses that contribute to the harmonic spectrumcutoff. The region I-II, for all cases, generates a cutoffgiven by I λ and has a value of approximately 132 eV.In contrast, the region II-III of the pulse generates a cut-off that depends on ∆ ω . If we suppose that the kineticenergy the electron acquires in the continuum is given bythe region of the pulse that recombines it, in this caseregion III [42, 43], the intensity I must be adjusted as afunction of the field amplitude in such temporal region, inorder to keep the HHG cutoff ∝ Iλ . Thus, we obtain thefollowing intensities: I ( π ) = 0 . · I , I (2 π ) = 0 . · I and I (3 π ) = 0 . · I (being the relative amplitude ofthe fields in region III equal to 0.9, 0.656, 0.41, respec-tively). If we also assume that the frequency dominating this region is ω P , we obtain the following cutoffs: 105.6eV, 57 eV and 27.6 eV. The position of these cutoffs is de-noted in Fig. 3(c) with a red, green and blue dotted line,respectively. As can be observed, these values for theless energetic cutoff are in very good agreement with the1D-TDSE predictions. Therefore, for the case of cosine-like pulses, the laser-matter interaction seems to be dom-inated simultaneously by a first region (I-II), governedby the carrier frequency ω , and another region (II-III)governed by the principal frequency ω P [44]. B. E R ( t ) fields For the simulations performed with the pulses E R ( t ) ,two representative examples were chosen, based on theparameters ∆ ω and δω (see Eqs. 6b(b) and 4b(b)). Therespective laser electric fields are shown in Figs. 5(a)and 5(b) (for the cosine-like and sine-like pulses, respec-tively), where the red (blue) line denotes the pulse with ∆ ω = 0 . and δω = π ( ∆ ω = π/ and δω = π/ ).As in the case of the E R ( t ) fields, the central wave-length was set to λ = I P = 0 . a.u. The peak electric fieldamplitude is now E = 0 . a.u., which corresponds to alaser peak intensity of I = 5 × W/cm . Analogouslyto the case of the E R ( t ) fields, for the sine-like pulses, thepeak field amplitudes are multiplied by a scale factor, insuch a way that their values are E = 0 . a.u. in all thecases.Figures 5(c) and 5(d) show the harmonic spectra gen-erated by the pulses given by Figs. 5(a) and 5(b), respec-tively. In the case of sine-like pulses, the structure of the FIG. 6. Time-frequency analysis extracted from the 1D-TDSE HHG spectra. Superimposed we plot the classically computedrescattering energies of electrons as a function of the ionization time, in white dots, and recombination time, in black dots, forthe laser pulses of Fig. 5(a) (panels (a) and (b)) and Fig. 5(b) (panels (c) and (d)). In all the panels the green arrow correspondsto the electron trajectory that contributes to the single HHG cutoff (see the text for more details). The ionization potential I p energy is indicated by a black arrow. harmonic spectrum is analogous to that shown for the E R ( t ) pulses. That is, there is a continuous spectrum ofharmonics, which reaches a well-defined maximum value.In the case of cosine-like pulses the situation is differ-ent. The pulse represented by a blue line ( ∆ ω = π/ , δω = π/ ), generates a continuous spectrum similar toa sine-like pulse. On the contrary, for the pulse repre-sented by the red line ( ∆ ω = 0 . and δω = π ), it canbe seen that there are two plateaus with their respectivecutoffs, at energies around 15 eV and 35 eV. To analyzethese structures in the spectra and relate them to theprincipal frequency defined in Eq. (3), we carried out atime-frequency analysis (see Fig. 6)), on which, in addi-tion, we superimpose the classically computed rescatter-ing energies of electrons as a function of the ionizationtime, in white dots, and recombination time, in blackdots. Figs. 6(a) and 6(b), correspond to the cosine-likepulses, red and blue lines, respectively (see Fig. 5(a)) andFig. 6(a). Here we see how the electron ionization prob-ability is greater in region II of the pulse and practicallyzero in region I. This is due to the fact that the amplitudeof the field in region I is not intense enough as to ionizethe electron by tunnelling. In this way, the dominantevent occurs when the electron is ionized in region II and recombines in region III. This event is shown with greenarrows and corresponds to a maximum energy of approx-imately 15 eV. In Fig. 6(b), a similar situation is shown:the ionization probability in region I of the pulse is negli-gible, in relation to the one in region II. In this way thereis only one ionization-recombination event, which takesplace in the temporal region II-III of the pulse. Thisevent is shown with a green arrow and has a maximumenergy of approximately 30 eV.For sine-like pulses, the analysis is shown in Figs. 6(c)and 6(d) (red and blue lines, respectively). For thesecases, we observe there is a predominant ionization-recombination event, which originates in the region I-IIof the pulse, analogously to the case of the pulses E R ( t ) .To explain the spectra obtained as a function of theprincipal frequency ω P , we proceed to compute its valuefor the two pulses presented. Using the definition, Eq. (3),we obtain ω P = 1 . ω for the case ∆ ω = 0 . and δω = π and ω P = 1 . ω for the case ∆ ω = π/ and δω = π/ .Taking into account that for the case of cosine-like pulsesthe peak field amplitude relative to the maximum ampli-tude in region III is: 0.26 and 0.62 (red and blue solidlines, respectively), we proceed to calculate the differentcutoffs in the same way as for the case of the pulses E R ( t ) .These are indicated with a red and blue dotted line inFigs. 5(a) and 5(b), where the cutoff for an intensity of I = 5 × W/cm and a wavelength of λ = 2000 nm isindicated by a black dotted line. For sine-like pulses with ω P (0 . , π ) = 1 . ω , the cutoff results 51.4 eV and with ω P ( π/ , π/
2) = 1 . ω , 66.2 eV. For cosine-like pulses,with ω P (0 . , π ) = 1 . ω we obtain a cutoff at 16 eV andwith ω P ( π/ , π/
2) = 1 . ω at 33.8 eV. These last valuesare in excellent agreement with the quantum mechanicalsimulations. IV. Conclusions and Outlook
In conclusion, a new definition of the principal fre-quency of an ultra-short laser pulse is introduced. Thisnew frequency, called principal frequency, describes in abetter way the interaction between an ultrashort pulsewith matter, particularly when the spectral content of thepulse has more than one octave, which in the temporaldomain corresponds to the single- or sub-cycle regime.In addition, through the principal frequency ω P theCEP effects in the HHG can be interpreted in an alter-native way. For sine-like pulses, a single HHG cutoff isgenerated by a single ionization-recombination event (oneregion of recombination associated to one region of ion-ization) between two symmetric regions at the center ofthe pulse, in which the principal frequency ω P dominatesthe interaction. On the other hand, for cosine-like pulses,there are two ionization-recombination events that con-tribute to the development of several HHG cutoffs. Thefirst ionization-recombination event is governed by thecarrier frequency ω , in the region in which the ampli-tude of the field that ionizes the atom is smaller that themaximum peak field amplitude, responsible to the recom-bination. The second ionization-recombination event isdominated by the principal frequency ω P . Here, the ion-ization of the atom takes place in the region of the pulsewith higher peak amplitude and the recombination in theregion smaller peak amplitude.These two events are clearly visible on the HHG spec-tra as two plateaus, with different cutoff energies and effi-ciency. This is so because the cutoff scales as ∝ λ and thetrajectory excursion time duration as ∝ λ . Because in thecentral part the E G ( t ) pulses have a temporal shape sim-ilar to the E R ( t ) ones, results obtained with this kind ofpulses do not contribute to the final discussion. We havenoted, however, that for pulses with an asymmetric fre-quency spectrum or a more complex frequency contents, a definition of the principal frequency ω P using a density ˜ ρ ( ω ) = ωS ( ω ) / gives better predictions for HHG cutoff.On the other hand, the latter definition would be hard tojustify from first principles.The introduction of the principal frequency ω P sug-gests that the distribution of the photons "inside" of anultra-short pulse is not linear. We show that the prin-cipal frequency is shifted to the higher frequencies whenthe spectral width of the pulse increases. On the otherhand, a similar result was showed in the temporal domainthrough the introduction of an "intrinsic chirp" in THzpulses [45–47], particularly when those pulses are focused.As the non-linear response of matter depends on somepower of the electric field amplitude ( E n ), it is to be ex-pected that for few-cycle pulses, the interaction is domi-nated by the principal frequency ω P (or period T p ), whichis the one that defines the position of the maxima andminima of the field [48]. Acknowledgements
ICFO group acknowledges support from ERC AdGNOQIA, Spanish Ministry of Economy and Com-petitiveness (“Severo Ochoa” Program for Centresof Excellence in R&D (CEX2019-000910-S), PlanNational FISICATEAMO and FIDEUA PID2019-106901GB-I00/10.13039/501100011033, FPI), FundacióPrivada Cellex, Fundació Mir-Puig, and from Gener-alitat de Catalunya (AGAUR Grant No. 2017 SGR1341, CERCA program, QuantumCAT_U16-011424,co-funded by ERDF Operational Program of Cat-alonia 2014-2020), MINECO-EU QUANTERA MAQS(funded by State Research Agency (AEI) PCI2019-111828-2 / 10.13039/501100011033), EU Horizon 2020FET-OPEN OPTOLogic (Grant No 899794), and theNational Science Centre, Poland-Symfonia Grant No.2016/20/W/ST4/00314.
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