Comment on "Direct photodetachment of F − by mid-infrared few-cycle femtosecond laser pulses"
CComment on “Direct photodetachment of F − by mid-infraredfew-cycle femtosecond laser pulses” G. F. Gribakin ∗ and S. M. K. Law † Center for Theoretical Atomic, Molecular and Optical Physics,Queen’s University Belfast, Belfast BT7 1NN, United Kingdom
Abstract
Multiphoton detachment of F − by strong few-cycle laser pulses was studied by Shearer andMonteith using a Keldysh-type approach [Phys. Rev. A , 033415 (2013)]. We believe that thiswork contained errors in the calculation of the detachment amplitude and photoelectron spectra.We describe the necessary corrections to the theory and show that the results, in particular, theinterference features of the photoelectron spectra, appear noticeably different. ∗ [email protected] † [email protected] a r X i v : . [ phy s i c s . a t o m - ph ] A ug n Ref. [1] direct photodetachment of F − by a strong linearly-polarized laser field wasconsidered using the Keldysh-type approach (KTA) [2] generalized to few-cycle pulses [3].Such methods are useful in general for studying strong-field effects in few-cycle pulses, see,e.g., Ref. [4]. The study was performed for an N -cycle pulse with the vector potential of theform A ( t ) = A ˆ z sin (cid:18) ωt N (cid:19) sin( ωt + α ) , (1)where A is the peak amplitude, ω is the carrier frequency and α is the carrier-envelopephase (CEP). Photoelectron momentum, angular and energy distributions were generatedfor a N = 4 cycle laser pulse with a range of peak intensities and mid-infrared wavelengths,while examining effects related to above-threshold channel closures and variation of the CEP.A calculation similar to that of Ref. [1] was also used to identify the effect of electronrescattering in short-pulse multiphoton detachment from F − computed using the R-matrixwith time dependence (RMT) method [5]. Subsequently, an error in the KTA calculationswas uncovered [6]. It concerned the phase factors of the contributions to the detachmentamplitude that arose from successive saddle points in the KTA calculation for a p -waveelectron. Upon correction, the KTA results showed a better agreement with the RMTphotoelectron spectra [5, 6]. We believe that the same error affected the results of Ref. [1].In this comment we show that the interference features of the photoelectron momentum andangular distributions, and the energy spectra are distinctly different from those of Ref. [1]when calculated correctly. We use atomic units throughout, unless stated otherwise.Using the Keldysh-like approach [2] for the N -cycle pulse (1), one finds the detachmentamplitude for an initial state with orbital and magnetic quantum numbers l and m , as (seeEq. (16) of Ref. [1]), A p = − (2 π ) / A N +1) (cid:88) µ =1 ( ± ) l Y lm (ˆ p µ ) exp[ if ( t µ )] (cid:112) − if (cid:48)(cid:48) ( t µ ) , (2)where p is the final photoelectron momentum and A is the asymptotic normalization con-stant of the bound electron wave function (for F − we use A = 0 . N + 1) saddle points t µ in the complex time plane, p µ and f ( t µ ) being theclassical electron momentum and action respectively, evaluated at the saddle points. Theterms in the sum in Eq. (2) contain a phase factor ( ± ) l ≡ ( ± l that alternates (for an odd l ) between the contributions from successive saddle points. When the spherical function2 lm (ˆ p µ ) ≡ Y lm (Θ , ϕ ) in Eq. (2) is evaluated for complex vectors p µ , the polar angle Θ isdetermined by cos Θ = (cid:113) p ⊥ /κ j , sin Θ = ∓ ip ⊥ /κ j , (3)where p ⊥ is the component p perpendicular to the z axis, and κ j = (cid:112) | E j | parameterizesthe energy E j of the bound state for each fine-structure component j = 3 / , / − ( l = 1). The sign in sin Θ alternates in the opposite way to ( ± ) in Eq. (2) and gives rise toan additional m -dependent phase factor. The final expression for the differential detachmentprobability of an electron from the state l, m reads dw ( j ) lm d p = A π (2 l + 1) ( l − | m | )!( l + | m | )! (cid:12)(cid:12)(cid:12) P | m | l (cid:16)(cid:113) p ⊥ /κ j (cid:17)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N +1) (cid:88) µ =1 ( ± ) l + m exp[ if ( t µ )] (cid:112) − if (cid:48)(cid:48) ( t µ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4)where P | m | l is the associated Legendre function. The superscript j is introduced into ex-pression (4) to indicate the detachment from the spin-orbit sublevel j of the ion, whichcontributes with the statistical factor (2 j + 1) / (2 l + 1) to the total detachment probabil-ity. The numerical values of κ j for each fine-structure state of F − are κ / = 0 . κ / = 0 . − from Ref. [8]). Note that Eq. (4) takes asimilar form to Eq. (33) from Ref. [2] in the case of the long periodic pulse.The photoelectron momentum densities are axially symmetric and can be obtained fromEq. (4) by taking p in the Cartesian momentum plane ( p x , p z ), (cid:88) j j + 12 l + 1 l (cid:88) m = − l dw ( j ) lm d p . (5)The photoelectron angular distribution is obtained by integrating Eq. (4) over the electronenergy E e = p / dwdθ = 2 π sin θ (cid:88) j l (cid:88) m = − l j + 12 l + 1 (cid:90) ∞ dw ( j ) lm d p dE e , (6)where θ is the polar angle. The photoelectron energy spectrum is given by dwdE e = 2 π (cid:88) j l (cid:88) m = − l j + 12 l + 1 (cid:90) π dw ( j ) lm d p sin θdθ, (7)and the total detachment probability is w = (cid:90) ∞ dwdE e dE e ≡ (cid:90) π dwdθ dθ. (8)3n Ref. [5] and, we believe, in Ref. [1] too, the presence of the phase factor ( ± ) l + m inthe sum over the saddle points in Eq. (4) was overlooked in the calculations. As a result,the detachment probability for p electrons ( l = 1) was computed correctly for m = ± m = 0, with the interference contributions between the odd and evensaddle points added with the wrong sign. Since m = 0 electron states give a dominantcontribution to the detachment signal, this error affected the interference patterns of thephotoelectron momentum and energy distributions presented in Ref. [1] (see [5] and erratum[6]). In addition, we have found that the magnitudes of the photoelectron angular and energyspectra in Figs. 4–7 of Ref. [1] are incorrect. This is in part due to the extra spin factor 2 inEq. (19) of Ref. [1], which was erroneously retained when accounting for the fine-structuresplitting in Eq. (23) of Ref. [1], and also affected the KTA results in Ref. [5]. The purposeof this Comment is to present correct photoelectron distributions for the same wavelengthsand other laser-pulse parameters as used originally in Ref. [1]. FIG. 1. (Color online) Logarithmic momentum densities for photodetachment of F − by a four-cyclelaser pulse with peak intensity of 1 . × W/cm . The top and bottom rows corresponds to λ = 1300 and 1800 nm, respectively, calculated for α = 0 [(a) and (d)], α = π/ α = 3 π/ − by a four-cycle pulse with peak intensity of 1 . × W/cm and carrier wavelengthof 1300 and 1800 nm, for CEP values α = 0, π/
2, and 3 π/
2. Compared with Fig. 2 ofRef. [1], the correct interference patterns appear more diffuse, lacking any sharp features.Figures 1 (a) and (d) show closer agreement with the momentum densities predicted by theRMT method [5]. At the same time, the overall forward-backward asymmetry along the p z direction (for α = π/ π/
2) is generally unaffected, since this characteristic dependson the symmetry of the laser field only.
Angle ° d w / d θ (a) Angle ° d w / d θ (b) Angle ° d w / d θ (c) Angle ° d w / d θ × -3 (d) Angle ° d w / d θ (e) Angle ° d w / d θ (f) FIG. 2. Photoelectron angular distributions for F − for a four-cycle pulse with λ = 1300 (toprow) and 1800 nm (bottom row), CEP α =0, and peak intensities 7 . × , 1 . × , and1 . × W/cm (left, central, and right columns respectively); (a)–(c) correspond to n min = 5,6, and 6, and (d)–(f) to n min = 9, 10, and 11-photon detachment, respectively. Figure 2 shows photoelectron angular distributions for F − for a four-cycle pulse withCEP α = 0, wavelengths 1300 and 1800 nm and intensities of 7 . × , 1 . × and1 . × W/cm . Because of the errors mentioned earlier, these angular distributions arevery different, both in shape and magnitude, from the (incorrect) results in Fig. 5 of Ref. [1].The oscillatory structure of the distributions is related to the minimum number of photons5hat needs to be absorbed near the peak of the pulse, n min (determined by the integer partof ( U p + | E j | ) /ω + 1 for a given ponderomotive energy U p = A / θ = π/
2) when n min is odd (even). This can be seenin Fig. 2 (a), (d) and (f) corresponding to n min = 5, 9 and 11, respectively, and in contrastto the original (incorrect) results of [1] where a central minimum for odd n min was noted.The effect of channel closure with increasing intensity gives rise to even n min =6, 6 and 10and a minimum at θ = π/
2, as seen in Fig. 2 (b), (c), and (e), respectively. This behaviouris in agreement with the observation that for a long periodic pulse the n -photon detachmentrate is exactly zero at θ = π/ n + l + m [2], and the fact that m = 0 dominatesthe photoelectron spectrum. Figure 2 also indicates that electron emission at angles closeto the direction of the field (i.e., within 0 ≤ θ ≤ ◦ and 135 ◦ ≤ θ ≤ ◦ ) is much strongerhere in comparison to Ref. [1], and in better accord with the momentum maps in Fig. 1.Figure 3 displays the angular distributions computed for CEP values α = π/ π/ α = 0 andthe same wavelengths and intensities as in Fig. 2. It also shows the spectra obtained byincluding only 2, 3 or 4 saddle points closest to the centre of the pulse. Comparing withFig. 7 of Ref. [1], we see that the shapes and magnitudes of the correct spectra are quitedifferent from those reported in Ref. [1]. We note that the spectra shown in Fig. 4 (c)and (f) (corresponding to intensity 1 . × W/cm and wavelength 1300 and 1800 nm,respectively) show better agreement with those calculated using RMT [5] in the low-energyregion (see Fig. 2(c) and (d) of [6]).For completeness, Table I gives the total detachment probability for F − for all wavelengthsand intensities considered, with CEP values α =0 and π/ α = 3 π/ π/ ± ) l + m factor in Eq. (4), the total detachment probability6 ngle ° d w / d θ α = π /2 α =3 π /2 (a) Angle ° d w / d θ α = π /2 α =3 π /2 (b) Angle ° d w / d θ α = π /2 α =3 π /2 (c) Angle ° d w / d θ α = π /2 α =3 π /2 (d) Angle ° d w / d θ α = π /2 α =3 π /2 (e) Angle ° d w / d θ α = π /2 α =3 π /2 (f) FIG. 3. (Color online) Angular distributions as in Fig. 2 but for α = π/ π/ − calculated for a four-cyclepulse at wavelengths 1300 and 1800 nm, and different peak intensities I and CEP phases α = 0and π/
2. The last column shows the detachment probabilities per period for a long pulse. λ I n min w w w no-inta dw/dt (2 π/ω )(nm) (W/cm ) ( α = 0) ( α = π/ × × × × ×
10 0.0468 0.0461 0.0475 0.04971.3 ×
11 0.0752 0.0768 0.0754 0.0789 a Obtained by adding modulus-squared contributions from each saddle point in Eq. (4). nergy (a.u.) d w / d E e Roots 1-10Roots 4-7Roots 5-7Roots 4-6Roots 5-6 (a)
Energy (a.u.) d w / d E e Roots 1-10Roots 4-7Roots 5-7Roots 4-6Roots 5-6 (b)
Energy (a.u.) d w / d E e Roots 1-10Roots 4-7Roots 5-7Roots 4-6Roots 5-6 (c)
Energy (a.u.) d w / d E e Roots 1-10Roots 4-7Roots 5-7Roots 4-6Roots 5-6 (d)
Energy (a.u.) d w / d E e Roots 1-10Roots 4-7Roots 5-7Roots 4-6Roots 5-6 (e)
Energy (a.u.) d w / d E e Roots 1-10Roots 4-7Roots 5-7Roots 4-6Roots 5-6 (f)
FIG. 4. (Color online) Photoelectron energy spectra of F − for the same laser pulse parameters foreach panel as in Fig. 2. Partial contributions from selected saddle points (“roots”) are also shown. is within 1-2% of the values given in Table I. In fact, the total detachment probabilityobtained by neglecting the interference terms [i.e., by adding the modulus-squared valuesof the individual saddle-point contributions in Eq. (4)], w no-int , is within few per cent of thecorrect value. This shows that the interference of the photoelectron wave packets producedat different laser-field maxima does not lead to much suppression of enhancement of electronemission, but only to some spatial redistribution of the photoelectron flux. The values of w no-int shown in the second last column of Table I are also practically independent of theCEP (with differences ∼ . per period deter-mined from the KTA detachment rates in a long periodic pulse, dw/dt [2]. They are closeto the detachment probabilities in the four-cycle pulse, which implies that effectively, thedetachment in the four-cycle pulse is dominated by the central, strongest-field cycle. Asimilar agreement was seen in Ref. [3] (Table II) which compared detachment probabilitiesof H − in a five-cycle pulse with the corresponding one-period long-pulse probabilities fromRef. [2]. 8dditionally, it is interesting to note that the total detachment probabilities in the shortpulse are slightly greater for α = π/ α = 0 if n min is odd, but slightly smaller when n min is even. This effect is entirely due to intereference. It can be explained by the factthat for α = π/ E ( t ) = − d A /dt acquires its maximumpeak magnitude twice within two central half-cycles, whereas for α = 0 the field reaches itspeak value once at the middle of the pulse. By comparing the angular distributions (Figs. 2and 3) for odd and even n min we see that for α = π/
2, constructive interference is moreprominent for odd n min near the θ = π/ n min ,destructive interference is more pronounced for angles near θ = π/
2. This gives slightlyhigher (lower) total detachment probabilities seen in Table I for α = π/ n min is odd(even).In conclusion, the photoelectron spectra presented in Ref. [1] were incorrect, in part dueto the omission of the m -dependent phase factor in the sum over the saddle points that givesthe amplitude. Using the correct phase factor is crucial for obtaining correct interferencefeatures of the photoelectron momentum and angular distributions. ACKNOWLEDGMENTS
We thank H. W. van der Hart and A. C. Brown for useful discussions. S. M. K. Lawacknowledges funding from a DEL-NI studentship. [1] S. F. C. Shearer and M. R. Monteith, Phys. Rev. A , 033415 (2013).[2] G. F. Gribakin and M. Yu. Kuchiev, Phys. Rev. A , 3760 (1997).[3] S. F. C. Shearer, M. C. Smyth and G. F. Gribakin, Phys. Rev. A , 033409 (2011).[4] D. B. Milosevic, G. G. Paulus, D. Bauer and W. Becker, J. Phys. B: At. Mol. Opt. Phys. R203-R262 (2006).[5] O. Hassouneh, S. Law, S. F. C. Shearer, A. C. Brown, and H. W. van der Hart, Phys. Rev. A , 031404(R) (2015).[6] O. Hassouneh, S. M. K. Law, S. F. C. Shearer, A. C. Brown, and H. W. van der Hart, Phys.Rev. A , 069901(E) (2016).
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