NONRELATIVISTIC STRONG-FIELD APPROXIMATION (SFA) Note: SFA will automatically be taken to mean Coulomb gauge (relativistic or non-diole) or VG (nonrelativistic, diole-aroximation). If LG is intended (rarely), it will be exlicitly identified. Start with the time-reversed S matrix (temorarily introduce the hat to distinguish oerators from eigenvalues): Aroximate the fully interacting final state by a Volkov solution: Within the matrix element, the interaction Hamiltonian oerator has the Volkov solution as an eigenvector: 1
For the sake of simlicity, one detail of S-matrix theory has been overlooked. Because of the time-reversed nature of the S matrix, the in-state that was used for the Volkov solution should actually be relaced by an out-state. (The final answer is actually unchanged if we ignore this distinction.) Rather than using a Volkov solution that roceeds from t - to the current or laboratory time t, it is necessary to trace the Volkov solution backwards from t + to the laboratory time t. This amounts to the relacement of Volk 1 1, 1/ ex, t f r t i r t d H V I by The transition amlitude is in a comact form: Volk r 1 1, t 1/ ex i r t f d H I, V t M i dth, t r, t, r, t. fi I f i When the initial atomic state is written in stationary-state form, it becomes ossible to searate sace and time arts.
All satial deendence comes from r, t r ex iet i i i This gives the bound-state wave function in momentum sace: ex ir, r Remaining factors constitute the time-deendent art. i i r and from ex(ir ) in the Volkov function. SFA i M 1/ ex, ex, fi i dt i E, i t H I t i d H t I V i This will turn out to be a very imortant feature of the SFA. The momentum-sace wave function of the initial atomic state lays a vital role in the final result. This roerty gives basic information about hotoelectron momentum distributions that is not to be found in the LG. 3
EXAMPLE: CIRCULAR POLARIZATION There is an efficient way to do this that maintains a normalized amlitude: 1 A t A e e it it 0 * * * ; 0; 1. If the laser field roagates in the direction of the z axis, then 1 1/ xˆ iyˆ, where xˆ, yˆ are unit vectors in the directions of the x and y axes. The interaction term is HI, t c cos t z where the hase angle is defined such that is real and c A ex 0 A0 z 4c c U i, 4
The Volkov exonential contains a trigonometric function. This is a sure indicator of Bessel function behavior. The generating function for Bessel functions can be introduced in the form ex i c sin t Jn c ex int n The time-deendent factor in the S matrix now has the form dt... J n c dt ex i / Ei z ze / e / e in t i n 1 t i n 1 t c c A common exonential factor can be achieved if the origin of the sum over the infinite range of the n index is shifted by 1. The result is dt... dt ex i / E i z in t zj n c / Jn 1 Jn 1 5
A Bessel function recursion relation can now be used to give a common Bessel function factor: The time integral is now n J J J n1 c n1 c n c c n c i dt... J ex in n z / E n z. The energy E i is for an initial bound state, so it is negative. It can be relaced by a ositive quantity, sometimes called ionization otential and reresented either by IP or by I. The referred notation here is the binding energy E B. E i E B The delta function from the time integration can be written as the conservation condition n E U B 6
In words, the conservation condition is: kinetic energy of the hotoelectron = energy sulied by n hotons minus energy required to overcome the binding energy minus energy required for interaction energy of the free electron with the field A basically imortant oint, almost never mentioned in laser hysics, is that the need to suly onderomotive energy to maintain a hysical free electron immersed in a strong lane-wave field, cannot be found in any finite order of erturbation theory. See: J. H. Eberly and HRR, Phys. Rev. 145, 1035 (1966). HRR and J. H. Eberly, Phys. Rev. 151, 1058 (1966). The first aer deduced the result from the finite exact sum of an infinite series of divergent Feynman diagrams. The second aer (actually the first chronologically) showed the result by direct analytical means as a finite mass renormalization. This, in turn, was the outcome of the mass-shift effect exlored in two HRR aers of 196. 7
CIRCULAR POLARIZATION S MATRIX AND TRANSITION RATE Substitution of the result of the time-deendent factor into the full S-matrix exression gives SFA i 1 fi 1/ i n c ex B M n z J in E U n V 1 n z n U EB Transition robability is M fi SFA, transition rate is 1 SFA w lim M fi. t t The total transition rate requires an integration over the hase sace of all ossible final states, divided by (πħ) 3, the volume of a unit cell in the hase sace. W 3 wvd 1 wv dd 3 3 where is the solid angle into which the hotoelectron is emitted., 8
DIFFERENTIAL TRANSITION RATE FOR CIRCULAR POLARIZATION 1 cir 0 d EB i Jn EB U n n dw sin, d 1/ 1/ cir z U 1 I 0 0 cir is the classical radius of the circular orbit of an electron in a circularly olarized field, as exressed in several ways. The angle is the angle in which the hotoelectron is emitted with resect to the roagation vector of the laser field. There is a lower limit on the sum over n that follows from the necessity that the final kinetic energy has to be a ositive number. E U 0, B n EB U n n0 where the square bracket means the smallest integer containing the quantity in the bracket. 9
The delta function can be used to erform the integration over. This can be done using the relation 1 E U n n E U n E U B B B The final result for the total ionization (or hotodetachment) rate by a circularly olarized laser field is dw d 1 nn 0 n E U B cir EB i Jn 0 sin. This is an analytically simle result that gives remarkably good agreement with exerimental measurements. 10
VALIDITY CONDITIONS FOR THE SFA FOR CIRCULAR POLARIZATION The analytical nature of the aroximation is elementary:,, M H M H exact exact SFA Volkov fi f I i fi f I i The statement amounts to saying that the effects of the laser field on the detached electron dominate the residual effects of the binding otential. In quantum mechanics, everything is judged in terms of energies and not in terms of field strengths. H = T + V and all that kind of thing. That is, the Schrödinger equation is an energy equation. SFA validity is assured if In ractical alication, it seems to be sufficient to require only in order to obtain accurate sectra. See U. Mohideen et al., Phys. Rev. Lett. 71, 509 (1993). HRR, Phys. Rev. A 54, R1765 (1996). U U E E B B 11
Counts (arb. units) Helium, E B = 0.904 a.u., U =.88 a.u., 815 nm, 1.65e15 w/cm.5 c) 1.65e15 W/cm.0 SFA exeriment 1.5 1.0 0.5 0.0 0 0 40 60 80 100 10 140 Electron energy (ev) MOHCIRC Ar. 7, 010 7:01:11 PM 1
From: HRR, Phys. Rev. A 76, 033404 (007). 13
What the receding slide shows is that the SFA for circular olarization is sufficiently accurate that it could be used to measure the actual eak intensity of the laser to within better accuracy than other means. Part (c) is the best fit. Part (a) assumes 5% less eak intensity and art (b) assumes 5% more eak intensity than the best-fit intensity. The differences are clear. All calculations are done with a Gaussian time rofile of intensity, and with the satial intensity rofile of a Gaussian laser focus. (This is generally called focal averaging.) Deletion effects are fully accounted for. 14
ANOTHER PROPERTY OF CIRCULAR POLARIZATION SPECTRA In the Mohideen helium exeriments, the eak U is 78.5 ev, the eak in the sectrum is at 61.9 ev, or at about 80% of U. In a single-intensity calculation, the high-intensity case would have the eak at U. The reason for the difference is that at the eak intensity for which the actual exeriment was done, less-than-maximum intensities in the focal distribution account for a significant art of the total. The SFA accounts accurately for the focal averaging effect. Mohideen et al. made their own attemt to fit the data using a single intensity, and were unable to find a satisfactory fit. An imortant oint to make here is that the eak of the sectrum occurs at an energy >> E B. This difference is sufficient to justify the SFA without regard to other criteria. Consider the next case from one of the earliest ATI exeriments. 15
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EVALUATION OF THE BUCKSBAUM EXPERIMENT AND CALCULATION Wavelength: = 1064 nm; Peak intensity =.3e13 W/cm ; Xenon: E B = 1.13 ev This amounts to U =.43 ev U << E B That is, the SFA should NOT be alicable. Also, Keldysh = 1.58. According to tunneling terminology, this is in the multihoton domain, so how is it that a Strong-Field Aroximation works so well? The eak in the sectrum occurs at 0 ev, and this is large as comared to the binding energy of 1.13 ev. Hence the SFA validity criterion of U >> E B is not a rigorous limit. An energetic sectrum serves as well. However, this is not so effective with linear olarization. 17
SFA FOR LINEAR POLARIZATION The vector otential for linear olarization can be taken to be A A t 0 cos, 1 An imortant difference between circular and linear olarization comes from A. 1 A t A0 t A0 t cos 1 cos There now exists a double-frequency term cost that did not exist for circular olarization. This is the diole-aroximation residuum of the figure-8 motion of a free electron in a lane-wave field. For each wave eriod, there are two lobes generated in the figure-8, and these come from the non-diole generalization of A. Proceeding as in the circular olarization case, the transition amlitude is i U M fi 1/ i EB ex sin sin dt i E B U t i l t i t V A/ l 0 18
GENERALIZED BESSEL FUNCTION Now the Volkov exonential contains trigonometric functions of t and t. This leads to the generalized Bessel function, with the generating function z 1 ex i l sint sin t, ex J n l z in t n z U / The generalized Bessel function J n (u,v) first arose in the 196 aers of HRR; it was then used also by Nikishov and Ritus in 1964. Functional roerties were systematized in the aer HRR, Phys. Rev. A, 1786 (1980). See Aendices B-D. See also Aendix J in Krainov, Reiss, Smirnov, Radiative Processes in Atomic Physics (Wiley, NY, 1997); ublished online 005. Dattoli and Torre in Frascati have since studied it in great detail, have found further generalizations, and have found recursors in the mathematical literature of the 19 th century. These functions are difficult, but always occur with Volkov solutions for linear olarization. 19
MORE LINEAR POLARIZATION SFA Following the same rocedures as with circular olarization, the total transition rate (that is, after integration over final hase sace) is: 1 lin 1 B i n 0 cos, nn dw E J z d 0 z 1 n E U, 0 U I lin B 1/ 1/ The quantity 0 lin is the classical amlitude of oscillation of a free electron in a linearly olarized lane-wave field. The analogy of this result to the circular olarization case is very close. Will give an examle of alication to sectra; then develo the method for calculating momentum distributions; and then give some details about how deletion effects are taken into consideration. 0
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