Polarization of harmonics generated from a hydrogen atom in a strong laser field

16 June 1997

PHYSICS

ELSEVIER

LETTERS

A

Physics Letters A 230 ( 1997) 203-208

Polarization of harmonics generated from a hydrogen atom in a strong laser field Vladimir S. Melezhik I Joint Institute for Nuclear Research. Dubna, Moscow Region 141980, Russian Federation Received

14 December

1996; revised manuscript received Communicated

10 March 1997; accepted for publication by B. Fricke

25 March

1997

Abstract

The high-harmonics spectrum of a hydrogen atom subject to an intense (> lOI W/cm’), elliptically polarized laser field is analyzed with a nonperturbative method of global approximation on a subspace grid. A considerable change of the polarization of some harmonics with respect to the laser polarization is found. @ 1997 Published by Elsevier Science B.V. PACT: 42.50.H~;

32.80.Rm

Although the high-harmonics generation by an atom subject to a strong laser field has extensively been investigated both experimentally [l-3] and theoretically [4-lo], most results have been obtained so far only with linearly polarized fields [ 1,4-61. Recently a few experimental groups [2,3] have begun investigating the effects of intense elliptically polarized laser fields in order to clarify the physical nature of atom-laser processes, and to find optimal conditions for high-harmonics generation. In particular, the rotation of polarization of high harmonics generated from argon by an intense ( > 1013 W/cm2), short (200 fs), elliptically polarized 785 nm laser pulse has been measured [ 31. The last achievement in the theoretical treatment of the polarization properties of the emitted radiation is the two-step model [S] generalized recently to the case of elliptically polarized light [ 91, valid in “the tunneling limit” (U,, = @/4wz 2 ZP >

’ E-mail: melezhik@nusun,jinr.dubna.su. 037%9601/97/$17.00 @ 1997 Published PIf SO3759601(97)00250-8

wc, where cl, is the ponderomotive potential and Zr,is the ionization potential). In this paper we present a quantitative analysis, without model simplifications, of the polarization properties of the hydrogen atom’s radiation in an elliptically polarized laser field. A quantitative description of the atom in an elliptically polarized electric field AV(x, Y, Z, t) = ~~~(~)(zsinOof+EYcos~or)

(I)

(where Ea and E are the amplitude and the ellipticity of the pulse, GJOand f(t) denote the laser frequency and the envelope of the pulse) requires a nonperturbative approach for solving the time-dependent fourdimensional Schrijdinger equation. Here an approach to this problem is given using the method of global approximation on a subspace grid originally developed for the stationary Schrijdinger equation [ 111 and applied to atoms in strong nonhomogeneous fields 1121. In principle the method is close to the “discrete vari-

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Letters A 230 (1997) 203-208

and Lagrangeable representation ” , “pseudospectral” mesh methods applied to different stationary problems (the corresponding references can be found in a recent paper [ 13 3 devoted to the Lagrange basis). Following the key idea of Refs. [ 11,141 we seek a solution in spherical coordinates (R, 8, q!~), as an expansion

pp,tJ2) = Plm(8) x cosmqb,

m > 0,

xl/&,

m = 0,

x sin rncp,

m < 0,

(2)

where P[“’(0) are the associated Legendre polynomials, Y = {I, m} and cpvj’ is the N x N matrix inverse to (am} defined on the grid 0, = {ej, 4j}r, coinciding with the nodes of the Gauss quadrature over the variables 0 and 4. Notice that the number of the grid points fij, N = No x N&, is equal to the number of basis functions, v = (I, m} = 1,. . . , N, in expansion (2) [ 121 (No and Nq, are the numbers of the nodes over the variables 8 and 4, respectively). In this way the problem is reduced to a system of Schrodingertype equations with respect to the vector Y( R, t) = {Ai”$j (Rv t)}; = {Ajf2#( R, flj, t)};” of unknown coefficients in expansion (2)) i&8(R,t)

Hkj(R,t)

=B(R,t)W(R,t),

=

-i$

(3)

- f +AV(R,flk,t)

8kj

>

Aj are the weights of the Gauss quadrature 2

If=1

=

o-+-)/2

m=-(&v-I)/2

and

ln4+5+1)/2

l=lml

In such an approach the atom-laser interaction AV( R, f&, t) ( 1) is diagonal, and the nondiagonal,

time-independent part of the matrix Hkj which is proportional to 1/ R2 may be reduced to a diagonal form, since the problem permits splitting off the operator A( R, t). This circumstance has been used for developing an economic algorithm with the computational time proportional to the number of unknowns in Eqs. (3). For approximation of Eq. (3) over the radial variable R a quasi-uniform grid on the interval R E [0, R, = 2501 has been created by mapping (R + x) the initial interval onto x E LO, l] by the formula R = R,(e4” - l)/(e4 - 1). The computations have been performed with a uniform grid of 1000 points for x (Ax = O.OOl), which is more accurate than the analogous grid for the R variable (AR = 0.25). The integration time step At was l/1600 of an optical cycle. To prevent the artificial reflection of the electron from the grid boundary, the wave function was multiplied after each time step by the “mask function” suggested in Ref. [6]. As input to the initial-value problem (3) the ground state of an atom has been used. The spectrum of the emitted radiation [d,~ (w) I2 in the z’/\z’j = (0, sin 19’,cos 0’) direction (the vector z’ is rotated in the plane x = 0 of the polarization ellipse of the laser) has been evaluated by the Fourier transform for the induced dipole of the atom,

(d,,(~)1~=

l&f

dteviw Tl

X

s

dR$* (R, t) =+(R,t) ,z,,

2

(4)

over the final five cycles [ 61. The problem of a hydrogen atom in field (1) is analyzed here for a laser intensity of I = E$z/Sr = 2 x lOi W/cm2 at wavelengths 1064 nm and 532 nm. The parameters Eo and wa of the laser field were chosen in the region U, = Ei/4wi < Z, = l/2, outside the “tunneling limit” (the region of validity of the twostep model [ 91) and the applicability of perturbation theory. However, the limit of the linearly polarized field (E = 0) for these parameters has been investigated by other authors applying different approaches [ 5,6]. The form of the pulse f(t) has also been chosen to coincide with that of Ref. [6], consisting of a linear ramp for the first five optical cycles, followed by 15 additional cycles with f(t) = I, which corre-

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Letters A 230 (1997)

203-208

205

Table I The harmonics intensities Jd, (moo) 1’ calculated at N = 49, Ax = 0.0005 and At = 2n/32OOon for E = 0. The last digits of the values may be subject to computational

A=lO64nm

A = 532 nm

n

I&l”

1

3.328 x 10-j 4.428 x IO-” 1.818 x lo-” 1.96 x 1O-7 5.15 x lo-“’

3 s 7 9

errors

1 3 5 7 9 11 13 IS 17 19

3.098 x IO-’ 3.027 x lO-7 1.S61 x IO-” 2.14 x IO-“’ 1.46 x IO--’ 4.81 x lo-“’ 2.42 x lo-“’ 6.7 x IO-” 3.6 x 10-12 2.4 x IO-”

TTT-TTTT I

5

9

13

17 21 Her manic

25

29

Fig. I. The convergence of the harmonics intensities )d(nwt~))~ (where II = 3,5,. .) with respect to the number, N = Ns, of the terms included in expansion (2) (the grid points over 0). The first harmonic, n = 1. has been omitted and the intensities have been normalized to the third harmonic. The most accurate data (filled circles) are connected by solid lines. The previously published results [ 61 are also presented (open circles connected by solid lines) for comparison (by the courtesy of K.C. Kulander who supplied us with the data from Fig. 3 of Ref. [6] ).

sponds to M 70 fs and z 35 fs duration of the pulse at 1064 nm and 532 nm, respectively. Note also that the parameters of the laser field are rather close to those of the experiment for detecting the polarization rotation of high harmonics [ 31. The result of the calculations for the linearly polarized pulse (E = 0) is given in Fig. 1. In this case the azimuthal variable 4 may be separated, and the twodimensional basis (9” (a) (2) is reduced to the set of Legendre polynomials 9 (cos 19). This figure demonstrates very rapid convergence of the calculated intensities Id: (nwo) /* (n is the harmonic number) in the z direction (8’ = 0) with the number of terms, N = NH = I,,, + 1, in expansion (2) (grid points over 0) at 532 nm. Increasing the laser wavelength to 1064 nm, the convergence becomes slower, but the computation with N = 7 still gives a rather realistic qualitative approximation for the general form of the radiated harmonics spectrum. As shown in Fig. 1, the calculated intensities agree well with the most accurate of the previously published data [ 61. Below n = 11 the open

circles (data of Ref. [ 61) coincide with the filled ones (present calculation, N = No = 36,49) for both A = 532 nm and 1064 nm. There is an insignificant shift of one data set from another only as n > 11 for a few harmonics, which is, however, within the framework defined by the use of different grids over R(x) and t variables in Ref. [ 61 (AR = 0.25, At = 2~/8OOwc) and in the present paper. The ionization rate per optical cycle evaluated here is also close to that of Ref. [ 61. The most accurate values of the intensities Id( nwo) [* for the odd harmonics calculated with AX = 0.0005 and At = 2~/32OOwa are presented in Table 1. After analysing the efficiency of the approach for the particularly well-investigated three-dimensional examples, E = 0, the method has been extended to the general four-dimensional case, E # 0 ( 1). But first, to test the self-consistency of the numerical scheme, the above result (E = 0) obtained with a onedimensional basis of Legendre polynomials ( N = No ) has been reproduced with a two-dimensional basis (N = No x N+) (2) for arbitrary orientation of the field EO in the plane x = 0. For the same grid over 0 the bases give the same results for (d, (0) I2 if &]]z. When rotating the vector Ea in the x = 0 plane the difference A (n, N) between the harmonics intensities calculated with the two-dimensional basis (2) and those presented in Fig. 1 is apparent. This difference is maximal near Ec]]y and A(n, N) -+ 0 with the basis extension, as N = N8 x N+ --f 03. Particularly, thequantityA(n,N) =Id,(nwo)12-Id,(nwo)12cal-

VS. Melezhik/Physics

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Letters A 230 (1997) 203-208

(b)

(a) lo-'

IO-’

t

.:^.‘. &=o.ol

IO4

lo4

2-l lo-*

3 D -

PT 10-O 3 75 ,o-l*

lo-"

?

0

‘0

b

lo+

lo-*

lo-’

0

F

-=I 5

25

15

0

3

1o-s

V u -

i

10-s +y

5

15

25

Harmonic

KoQ

N

72

b

e=O.l

0

lo4

'04

‘0 Q

Harmonic

10-J

D

10-a

b

b

1o-8 5

25

5

Ho’Fmonicz5

HdFmonic

-_L &=0.5 b

i\.

5

5

Ha~monic25

15

25

Harmonic

Fig. 2. (a) Harmonic spectra IdCZ,) (w)12 at 532 nm calculated for a few values E with N = 49, Ax = 0.001 and At = 2?r/16OOoo. The solid lines and the dashed lines correspond to Id; (0) I2 (0’ = 0) (filled circles: odd harmonics w = nwo, n = 1,3,. .) and IdY(w) I2 (8’ = 7r/2) (open circles: odd harmonics o = nwo, n = 1,3,. .), respectively. (b) Same as (a) but for 1064 nm.

207

VS. Melezhik/Physics Letters A 230 (1997)203-208

culated with N = Ne x N4 = 7 x 7 = 49 terms is fan order of magnitude below ( lO-2-1O-3) x Id,(nwo) I2 for n < 11 at 532 nm and for n < 7 at 1064 nm. For some of the harmonics with n b 7 at 1064 nm the deviation A( n, N) becomes visible but remains within the region of the accuracy the one-dimensional basis exhibited on the No = N = 7 grid (see Fig. 1) . In Fig. 2 the intensities (dZ (nwo) I2 and ]d,( nwa) I2 of the harmonics radiated in the z (8’ = 0) and y (0’ = n-/2) directions are presented for elliptically polarized (a #O) fields. The calculations have been performed with N = No x Nb = 7 x 7 = 49, which corresponds to -3 < m < 3, m < 1 6 m + 4. As discussed above, the computation with this basis set is rather accurate at 532 nm and gives a qualitative picture of the total spectrum of harmonics at 1064 nm (the shape and the cut-off position). A more precise analysis of high harmonics n 3 11 (532 nm) and n 2 7 (1064 nm) can be done with the basis extension (No x N4 > 7 x 7) in (2) (see Fig. I). Fig. 2 shows considerable deviation of the ratio p(n) = (d,( nwo) /d, (moo) ) for the generated harmonics (for some n) from the ellipticity of the fundamental, which may indicate a variation of the harmonics polarization. Before performing a quantitative analysis of the harmonics polarization, the convergence of the method with respect to N was tested by analyzing the parameter p( n, N) [ 151, which is in general a function of the ellipticity c’(n) and the polarization ellipse orientation ~+V(rt) and is therefore sensitive to both polarization parameters of the emitted radiation. Here we give a few more convergent results for harmonics polarization. The quantities Id: t (nwa) 1calculated as a function of the angle 8’ are presented in Fig. 3 for E = 0.1 and 0.5 at 532 nm. One can clearly see a significant alteration of harmonics polarization with respect to the polarization of the fundamental laser field for n = 3,5 (E = 0.1) and n = 3,5,9 (E = 0.5). The figure allows one to evaluate the harmonicellipticityc’ = ld(Bi = $‘+90°)I/jd(0~ = $‘)I and the offset angle G’(n) by inspection of the given curves IdZ, (nwo) I. The computations show a considerable change of the ellipticities of the mentioned harmonics and rotation of the polarization ellipse for the 5 (G’(5) = -16O) and 9 ($‘(9) = 6’) harmonics at .s = 0.5. The effect of variation of the ellipticities and rotation of the polarization ellipse has also been shown theoretically for emitted harmonics in the “tun-

0

45

~"""""""""""""""'~

135

90

180

e=.5

1 .o

1.0

0.5

0.5

_ G i! 9

1 n=3

.\ '__.'

/

FLi ~40.0 i

n=9

/

O.O

B’

0

45

90

135

180

&=.l q"""""""""E 1.0

-2

z

0” _

0.5

Fig. 3. The normalized intensities jd,, (nwo) forc=O.l (N=49)and0.5(N=81)at532nm.

/ as a function of 8’

neling limit” region [ 91 with the two-step model [ 81. Both theoretical evaluations (that of Ref. [ 91 and that of the present paper) support the conclusion of Ref. [ 31 that the measured effect of the rotation of the harmonics polarization in argon is due to a single-atom response to a strong laser field. Note also that due to the method stability with respect to rotation of the angle 0’, exhibited in Fig. 3 for the whole range of the E and n considered, a more economical detailed analysis of the dependence on E

208

VS. Melezhik/Physics

of the harmonics polarization is to evaluateg the complex dipole moments dz; (E, moo) and dZ; (E, nwo) for only two different angles 0: and 6;. In this paper an approach for direct numerical integration of the time-dependent four-dimensional Schriidinger equation describing an atom in an elliptically polarized field is suggested. The high efficiency of the method developed here for the problem of highharmonics generation leads to its application to other problems of atomic interaction with elliptic driving fields (particularly for evaluation of the photoelectron spectra). Two circumstances, the rapid convergence over N and the proportionality of the computation time to the number of unknowns of (3), suggest the use of our approach for stronger fields and longer laser pulses. All the computations were performed on a DEC 3000-M600 work-station. I am grateful to L.A. Collins, J.S. Cohen, K.C. Kulander, and A. Scrinzi for useful discussions, and to R.T. Siegel for valuable comments. The warm hospitality and the use of the computer resources of IMEP of the Austrian Academy of Sciences, where part of the work was done, are gratefully acknowledged. The work has been partly supported by INTAS Grant No. 94-2836.

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