118 J. Opt. Soc. Am. B/Vol. 15, No. 1 /January 1998 Fiordilino et al.

Fluctuating laser field that induces a blueshift inharmonic generation

Emilio Fiordilino

Istituto Nazionale Fisica della Materia and Istituto di Fisica, via Archirafi 36, 90123 Palermo, Italy

Francesca Morales

Istituto Nazionale Fisica della Materia and Dipartimento di Energetica ed Applicazioni della Fisica,Universita di Palermo, Viale delle Scienze, 90128 Palermo Italy

Rosalba Daniele

Istituto Nazionale Fisica della Materia and Istituto di Fisica, via Archirafi 36, 90123 Palermo, Italy

Received March 31, 1997; revised manuscript received September 23, 1997

The spectrum of a two-level atom in the presence of a multimode laser pulse is calculated. The field is allowedto fluctuate in amplitude or in phase; the emitted spectrum has richer emission lines than in the case of thenonfluctuating field and shows peaks shifted toward the blue with respect to the traditional harmonic peaks.The position of the lines is predicted by the formula v2n11 5 (2n 1 1)(1 1 D)vL with D being a parameterthat can be found numerically. In this way the fluctuations seem to result in an effective increase of the laserfrequency. © 1998 Optical Society of America [S0740-3224(98)04401-4]

OCIS codes: 140.3460, 140.1340, 190.4160.

1. INTRODUCTIONIt is already a well-established experimental fact that anatom in the presence of a strong laser field of frequencyvL becomes the source of new waves. The experimentalspectrum of the scattered radiation contains only odd har-monics of the pumping field: vn 5 nvL , with n being anodd integer.1–8 Three parts can be recognized in theemitted spectrum: a zone of strong decrease in the inten-sity of the first few lines, a broad plateau of lines at al-most the same intensity, and a zone of rapid quenching ofthe emission. Also, blueshift or redshift of the spectrallines with respect to the pure harmonic vn have been re-ported. The amount and direction of the shift is relatedto the actual experiment.6

As soon as the findings were well established the pos-sibility of designing a new high-frequency tunable laserdevice, soon baptized as tabletop synchrotron,9 was fore-seen; already, waves obtained during higher-order gen-eration had been used in actual experiments.10,11 As-suming a one-electron atom, the physical mechanismproducing the diffusion of the harmonics can be found inthe strong modifications and oscillations induced in theatomic wave function by the external laser field. The in-tensity of the lines can be calculated by means of the clas-sical formula12–14:

d

dvS~v! 5

4e2

3c3 uaF~v!u2, (1)

where (d/dv)S(v) is the total energy irradiated in the an-gular frequency range v to v 1 dv so that the number ofphotons emitted during the whole laser shot in the samerange is given by

0740-3224/98/010118-04$10.00 ©

n~v! 51

\v

ddv

S~v!. (2)

In Eq. (1), a is the quantum average of the electron accel-eration; its Fourier transform is denoted by the subscriptF.

To have a high-frequency, tunable laser, we need tocontrol the emitted spectrum and the extension of the pla-teau. To represent the extension of the plateau, the so-called simple-man model is most popular. This model re-lates the maximum detectable harmonic \vM to the laserintensity by the expression

\vM 5 UI 1 bUp , (3)

where UI is the atomic ionization energy, Up5 e2E0

2/(4mvL2) is the ponderomotive potential, and E 0

is the maximum laser electric field.15,16 The physical ex-planation of such a model can be found in the maximumpossible harmonic that is emitted when the active elec-tron is extracted from the atom while at rest and is thenpushed back to the parent ion by the field. In fact, if theelectron is ionized with a residual positive amount of en-ergy, it will essentially run away from the atomic core andwill not emit. Newtonian laws applied to this classicalpicture yield a value for b of roughly 3. Equation (3)gives the theoretical ground to the universal race to useincreasingly powerful lasers to get a broad plateau.

Other ways have been introduced to control the emittedspectrum. It can be shown that a two-level atom gives allthe features of the higher-order harmonic generation, i.e.,emission of odd harmonic frequencies, existence of a pla-teau, and rapid quenching of the emission.17–21 Thephysical reason of this success is that the transition is

1998 Optical Society of America

Fiordilino et al. Vol. 15, No. 1 /January 1998 /J. Opt. Soc. Am. B 119

broadened by the laser field by a quantity proportional tothe laser–atom interaction energy so that the discretetwo-level atom becomes in reality a system with a con-tinual energy band not different from the classical picturedescribed above. The latter system leads to Eq. (3),where the ionized immobile electron is allowed to exploreonly a limited portion of the space and of the energy. Byusing the simplified two-level model it has been seen that,with well-chosen laser pulse shape and duration, signifi-cant harmonic emission even at an intermediate laserintensity21 is obtained and that rapid fluctuations of theexternal field modify the spectrum and enhance higher-order emission.22 In this respect often the laser temporalvariation seems more important than the maximum in-tensity.

A wavelet analysis of the temporal evolution of thespectrum emitted by a two-level atom in presence of ir-regular laser pulses has shown harmonic pulse durationsof the order of few optical cycles of the pumping field.22,23

So far the experimental results24,25 have confirmed theemission enhancement with respect to what can be pre-dicted by means of Eq. (3), when the laser field is allowedto undergo fast changes of the intensity.

The simple-man model has been employed to determinethe extension of the plateau for laser intensities thatchange slowly with time. Its cautious use can also sug-gest an explanation of the redshift or blueshift of the har-monics from the theoretical odd frequency vn , when arapid time-varying pulse profile is used. What essen-tially happens in this case is that the field intensity at theionization time is different from the field value at the re-capture time. So the ponderomotive potential enteringthe calculations acquires two different values in theenergy-conservation law leading to Eq. (3). The emittedfrequencies are therefore shifted.

In the present paper we study the spectrum emitted byan atom in the presence of a multimode laser pulse to seehow the inclusion of rapid and random fluctuating fieldsmodifies the spectrum. We will show that the emittedfrequencies are shifted toward the blue according to theexpression

v2n11 5 ~2n 1 1 !~1 1 D!vL , (4)

exactly as if the laser frequency were larger than thenominal one of the quantity DvL ; here 2n 1 1 gives thenumber of emitted lines, and D is a shift parameter thatcan be obtained from the numerical results.

2. THEORY AND RESULTSHere we show the calculated spectra in the case of anatom in the presence of a laser electric field of the form

E~t ! 5 z1K

F~t !(k51

K

hk~t !cos@vLt 1 fk~t !#, (5)

where F(t) is the pulse profile describing the temporalevolution of the laser envelope; for it we have chosen thefollowing form:

F~t ! 5 S tt D 2

expF1 2 S tt D 2G , t > 0, (6)

for the sake of comparison with other calculations carriedout in different physical situations and because it gives agood harmonic yield.21 t is the time required to reach themaximum value of 1 and gives the approximate laser-pulse duration; in the present calculations, t is always as-signed the value of 100 optical cycles. The 2K randomtime-dependent quantities hk(t) and fk(t) describe thefluctuating amplitude and phase of the modes; K is thenumber of the modes of the field, and we will display plotsfor K 5 10. The two fluctuating functions hk(t) andfk(t), generically denoted with Rk(t), were generated inthe following way:

R1~t ! 5 R0 ,

Rk11~t ! 5 Rk~t !@1 1 gk~t !#, 1 < k , K, (7)

with the constant R0 giving the mean value of the quan-tity under consideration. gk(t) is a number randomlygenerated with zero mean value and obeying Gaussianstatistics. After having generated the functions to be en-tered in the calculations we used the expression26

h~t ! 5 h~t !cos@vt 1 f~t !#, (8)

with

h2~t ! 5 (k51

K

hk2~t ! 1 (

k.j51

K

2hk~t !h j~t !cos~fk 2 f j!,

tan f 5

(k51

K

hk~t !sin fk

(k51

K

hk~t !cos fk

. (9)

The Hamiltonian of the problem is H(t) 5 Ha

1 V(r, t), where Ha is the bare atom wave function andV(r, t) 5 E (t)–r is the laser–atom interaction energy.Coherently to the two-level approximation, the wave func-tion ut& of the atoms in the presence of the laser field is alinear combination of the two active atomic states u0& andu1&:

ut& 5 c0 exp~2iv0t !u0& 1 c1 exp~2iv1t !u1&. (10)

Then the time-dependent Schrodinger equation gives theusual system of two coupled differential equations:

i\ c0 5 E~t ! • r10 exp~2iv10t !c1~t !,

i\ c1 5 E~t ! • r10 exp~iv10t !c0~t !, (11)

with v10 [ v1 2 v0 and r10 [ ^1uru0&.We solved this set of coupled differential equations by

following an iterative method, described in (Refs. 27 and28), that can be considered exact and used in the contextof higher order harmonic generation in Refs. 21, 22, and29.

120 J. Opt. Soc. Am. B/Vol. 15, No. 1 /January 1998 Fiordilino et al.

By denoting W0 [ h0z • r10 , the adimensional param-eters used are v10 /vL 5 10 and W0 /\vL > 8.3 3 1022;with the reasonable hypothesis that u^0uru1&u > a0 andthe choices of \v10 5 0.12 a.u.53.27 eV we obtain a fieldstrength h0 5 1023 a.u. or I 5 3.5 3 1010 W/cm2, whichcan be considered of intermediate intensity.

In Fig. 1 we show the spectrum emitted by the modelatom when no fluctuations are introduced; the spectrum

Fig. 1. Fourier acceleration spectrum for a two-level atom in thepresence of a laser field without fluctuations. The parametersfor the calculations are v10 5 0.12 a.u., v10 /vL 5 10, W0 /\vL5 8.3 3 1022, and t 5 100 optical cycles.

Fig. 2. Fourier acceleration spectrum for a two-level atom in thepresence of a laser electric field fluctuating with Gaussian statis-tics; the width of the Gaussian is s 5 1023. The parameters forthe calculations are K 5 10, v10 /vL 5 10, W0 /\vL 5 8.33 1022, and t 5 100 optical cycles.

is not rich of harmonics and shows bunches of odd linesaround the frequency 2nv10 : the reason for such a fea-ture is that the field is kept at low intensity30; however,the 43rd harmonic is definitely seen. In Fig. 2 we showthe spectrum emitted by our model atom when the fieldintensity is allowed to fluctuate in the way described inexpression (7) while the phase is taken constant: fk(t)5 0. In Fig. 3 we show the spectrum when the phasehas been allowed to fluctuate in the same way and thefield has been kept to the constant value of 1023 a.u. Inboth plots the width of the Gaussian was taken as 1023.A comparison of Figs. 2 and 3 with Fig. 1 tells immedi-ately that the spectrum in the fluctuating cases presentsa larger number of lines that are uniformly distributed.In the case of Fig. 2 we observe a shift of the lines towardthe blue with D 5 0.12, while for Fig. 3 we observe D5 0.14. We note that the shift is not constant frompulse shot to pulse shot but depends on the particular re-alization of the shot simulation; however, for a definite re-alization the value of D is clearly deducible. The fact thata suitable choice of the pulse profile F(t) can improve theharmonic yield and the number of lines has been alreadypredicted by previous calculations21,22; in actual experi-ments this fact has been seen, together with a shift of thelines from the harmonic position.6,24 In the present cal-culations we see that the shift is proportional to the har-monic position through expression (4), so we can say thatfluctuations behave as if the effective laser frequencywere larger than the nominal one. The amount of blue-shift found in the present calculations is much largerthan in the experimental value as reported in Ref. 6. Itis not easy to give a numerical comparison with the re-sults of Ref. 24 because a value of the shift is not given;from the reported plots it cannot be determined if the

Fig. 3. Fourier acceleration spectrum for a two-level atom in thepresence of a laser field with fluctuating phase. The phase fluc-tuates with Gaussian statistics; the width of the Gaussian is s5 1023. The parameters for the calculations are K 5 10,v10 /vL 5 10, W0 /\vL 5 8.3 3 1022, and t 5 100 optical cycles.

Fiordilino et al. Vol. 15, No. 1 /January 1998 /J. Opt. Soc. Am. B 121

shift is constant with the harmonic order or if it changes,as in our case; apparently our shift is larger than this one.However, from the physical point of view it should bestressed that the resonant condition used in the presentcalculations induces a fast evolution on the populations ofthe atomic levels and produces relatively large electronaccelerations; as already stated this characteristic resultsin a larger harmonic yield with respect to the nonreso-nant case and could cause also a larger blueshift; the im-portant point here is that the shift is due to variations inlaser intensity and becomes larger with fluctuating fields.

The particular simplicity of the model and the rapidfluctuations of the field do not allow explanations of theresult in Eq. (4) on the ground of generalizations of thesimple-man model. However, in the nonfluctuating caseit can be shown that the acceleration obeys the followingequation:

r 9 1 $D~z ! 1 E~z !cos@2~z 1 f!#%r

5 22\v10

\vL

V01~z !

\vLr01

1 4V01~z !

\vLE

0

z

r~z1!V018 ~z1!

\vLdz1 , (12)

where the apices denote the derivative with respect to z5 vLt and

E~z ! 5 2W0

2

\2vL2 F2~z !, D~z ! 5

\2v102

\2vL2 1 E~z !. (13)

From Eq. (12) we see that a rapidly changing pulse profilemakes richer the harmonic spectrum of the acceleration.The left-hand side of Eq. (12) is the equation of a para-metric oscillator with time-dependent frequency; if thepulse profile is taken as constant, then the left-hand sideof the equation becomes the Mathieu equation, which isknown to give quasi-chaotic behavior. This fact shouldexplain why the value of D is dependent on the particularrealization of the laser shot. We might say that it is al-ready an interesting feature that a value of D can be ac-tually found even with a fluctuating field.

REFERENCES1. M. Ferray, A. L’Huillier, X. F. Li, L. A. Lompre, and C.

Manus, J. Phys. B 21, L31 (1988).2. A. L’Huillier, K. J. Schafer, and K. C. Kulander, J. Phys. B

24, 3315 (1991).3. K. Miyazaki and H. Sakai, J. Phys. B 25, L83 (1992).4. C.-G. Wahlstrom, J. Larsson, A. Persson, T. Starczewski, S.

Svanberg, P. Salieres, Ph. Balcou, and A. L’Huillier, Phys.Rev. A 48, 4709 (1993).

5. P. Balcou and A. L’Huillier, Phys. Rev. Lett. 70, 774 (1993).6. K. Miyazaki and H. Takada, Phys. Rev. A 52, 3007 (1995).7. C. Lynga, A. L’Huillier, and C.-G. Wahlstrom, J. Phys. B

29, 3293 (1996).8. P. Salieres, T. Ditmire, M. D. Perry, A. L’Huillier, and M.

Lewenstein, J. Phys. B 29, 4771 (1996).9. C. Kan, C. E. Capjack, R. Rankin, and N. H. Burnett, Phys.

Rev. A 52, 4336 (1995).10. R. Haight and D. R. Peale, Phys. Rev. Lett. 70, 3979 (1993).11. N. E. Karapanagioti, D. Xenakis, D. Charalambidis, and C.

Fotakis, J. Phys. B 29, 3599 (1996).12. J. D. Jackson, Classical Electrodynamics (Wiley, New York,

1975).13. K. Burnett, V. C. Reed, J. Cooper, and P. L. Knight, Phys.

Rev. A 45, 3347 (1992).14. T. F. Jiang and S. I. Chu, Phys. Rev. A 46, 7322 (1992).15. P. B. Corkum, Phys. Rev. Lett. 71, 1994 (1993).16. M. Lewenstein, P. Balcou, M. Yu. Ivanov, A. L’Huillier, and

P. B. Corkum, Phys. Rev. A 48, 2117 (1994).17. B. Sundaram and P. W. Milonni, Phys. Rev. A 41, 6571

(1990).18. L. Plaja and L. Roso-Franco, J. Opt. Soc. Am. B 9, 2210

(1992).19. V. C. Reed, K. Burnett, and P. L. Knight, Phys. Rev. A 47,

34 (1993).20. L. Plaja and L. Roso, J. Mod. Opt. 40, 793 (1993).21. E. Fiordilino and V. Miceli, J. Mod. Opt. 41, 1415 (1994).22. S. De Luca and E. Fiordilino, J. Phys. B 29, 3277 (1996).23. P. Antoine, B. Piraux, and A. Maquet, Phys. Rev. A 51,

1750 (1995).24. J. Zhou, J. Peatross, M. M. Murnane, H. C. Kapteyn, and I.

P. Christov, Phys. Rev. Lett. 77, 1743 (1996).25. I. P. Christov, J. Zhou, J. Peatross, A. Rundquist, M. M.

Murnane, and H. C. Kapteyn, Phys. Rev. Lett. 76, 752(1996).

26. M. H. Mittleman, Theory of Laser–Atom Interactions (Ple-num, New York, 1982).

27. E. Fiordilino, Nuovo Cimento D 9, 599 (1987).28. E. Fiordilino and F. Trombetta, J. Mod. Opt. 35, 1635

(1988).29. E. Fiordilino and V. Miceli, J. Mod. Opt. 43, 735 (1996).30. E. Fiordilino and V. Miceli, Laser Phys. 6, 1204 (1996).