J. Phys. B: At. Mol. Opt. Phys. 30 (1997) 2505–2515. Printed in the UK
PII: S0953-4075(97)81809-3
R-matrix Floquet theory of multiphoton processes: XII.
Harmonic generation in magnesium
R Gȩbarowski, K T Taylor and P G Burke
Department of Applied Mathematics and Theoretical Physics, The Queen’s University, Belfast
BT7 1NN, UK
Received 18 February 1997
Abstract. The newly developed R-matrix Floquet theory of harmonic generation is applied
to magnesium in its ground state for the case where three photons are needed to reach the
first ionization threshold in the perturbative limit. Results are presented both for multiphoton
ionization and for harmonic generation where the influence of the low-lying autoionizing states
is studied. The results are compared at a qualitative level with those from recent experiments
and from other calculations.
1. Introduction
The R-matrix Floquet theory of harmonic generation has been described in the first of two
preceding papers (Gȩbarowski et al (1997), hereafter referred to as RMF X). In the second
paper (Bensaid et al (1997), hereafter referred to as RMF XI) the application to one-electron
systems interacting with an external field has been considered. In this paper we report the
first application of this theory to a many-electron atom where we consider third harmonic
generation (THG) in the vicinity of a three-photon resonance with an autoionizing (AI) state
in magnesium. This work is motivated by recent experimental results for systems with the
outer electron shell configuration ns2 , such as magnesium (Shao et al 1993, Karapanagioti
et al 1995, 1996) and calcium (Faucher et al 1994) where both THG and multiphoton
ionization (MPI) rates were investigated. Both simultaneously measured spectra of THG
and MPI have revealed Fano-type line profiles (Faucher et al 1994). In these experiments
two-colour excitation processes are used in order to take advantage of a ladder excitation
scheme. Since we are interested in the qualitative behaviour of the THG spectra rather than
comparing results at a quantitative level, we will confine ourselves to a simpler situation.
Namely we will consider monochromatic, linearly polarized, three-photon coupling of the
ground state with an AI state. Since the pioneering work of Fano (1961), there have been
many valuable contributions to the theory of photoionization in the neighbourhood of an
autoionizing resonance and the resulting ejected electron spectra (e.g. Lambropoulos and
Zoller 1981, Rza̧żewski and Eberly 1983). In some cases, taking a particular analytical
expression for the laser pulse envelope enables analytic expressions to be obtained for
electron and photon spectra (Rza̧żewski et al 1985, Zakrzewski 1986) as a solution of the
time-dependent problem.
There have been many contributions to the theory of THG (Georges et al 1977, Alber
and Zoller 1983, van Enk et al 1994, Zhang and van Enk 1995) in the presence of
autoionizing states. Interest in THG is also driven by the need to understand the role
c 1997 IOP Publishing Ltd
0953-4075/97/102505+11$19.50 2505
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R Gȩbarowski et al
of electron–electron interactions in above-threshold ionization (ATI) spectra. The system
consisting of a two-valence electron atom in the presence of an intense laser field has always
been a challenge for theory and experiment in so far as electron–electron interaction effects
impact on the double-ionization process (see e.g. Bauer et al 1995, Charalambidis et al
1994, Parker et al 1996). Finally, we mention that an enhancement in the THG rate in the
vicinity of the two-photon resonance with an AI state has been found by Zhang and van
Enk (1995). The influence of a double resonance on third-order susceptibility and THG
near an AI resonance has also been investigated in detail in helium, where apart from the
three-photon resonance there is an additional single-photon resonance (van Enk et al 1995,
1996a, b).
The present paper is organized as follows. In section 2 we describe our basis
approximation. In section 3 we present results for the magnesium atom in the situation
where the frequency of the intense laser field is close to resonance with the 3p3d 1 Po AI
state. Finally, in section 4 we draw our conclusions.
2. The approximation
We now apply the theory discussed in RMF X to the magnesium atom. The notation
introduced in that paper will be adhered to here. Let us first discuss the basis approximation
involved in the present calculations. In order to study the qualitative behaviour of the THG
rates near the the 3p3d 1 Po AI state, we have retained only the 3s 2 S and 3p 2 Po residual
ion states of Mg+ . The 1s, 2s, 3s, 2p orbitals are those given by Clementi and Roetti
(1974) whereas the 3p orbital was taken from a CI calculation performed recently (Vaeck
1995). As a result of coupling the residual ion states with the (N +1)-electron we obtain Mg
configurations corresponding to 3snl and 3pn0 l 0 . In the present three-photon calculations we
take the maximum total angular momentum up to L = 3 and consequently 1 S, 1 Po , 1 D, 1 Fo
symmetries are accessible to the system.
To have a manageable size of calculation, we take typically nco = 30 continuum orbitals
in the basis expansion (see equation (22) in RMF X) for an internal region size of a = 50 au.
In our calculations we take at least nine Floquet blocks (corresponding to five absorption
and three emission blocks). For these parameters, the resulting Floquet Hamiltonian in the
internal region is a (sparse) matrix of order 1517. A discussion of the accuracy of the results
and the validity of the approximations used will be presented in the next section.
3. Results and discussion
A schematic energy level diagram for the field-free magnesium atom is presented in figure 1
and is based on previous theoretical calculations (Moccia and Spizzo 1988a). In particular,
we investigate the interaction between the 3s2 1 S ground state plus three photons and the
3p3d 1 Po AI state, and how this affects THG. In this system the excitation to the continuum
may proceed by two ionization paths—either through a direct transition from an excited
Rydberg level (the transition induced by the field) or via the 3p3d 1 Po AI state (the transition
due to Coulombic electron–electron interactions). The excitation to the AI state of interest
from the ground state is obtained via a three-photon absorption process. The quantum
interference between these two paths results in an asymmetric shape for the resonance—
characterized by a Fano q parameter. In recent experiments (Karapanagioti et al 1995,
1996) a slightly more complicated model was considered where the 3p3d 1 Po AI state was
coupled to the ground state via another, much broader AI state, namely 3s2 1 S, with the help
Harmonic generation in magnesium
2507
Figure 1. An energy level diagram for the field-free Mg atom with the excitation scheme
possible using linearly polarized laser light of frequency ω so that a near three-photon resonance
is obtained with the 3p3d 1 Po AI state. The broken thick arrow indicates possible excitation
directly to the one-electron continuum from a Rydberg level 1 S or 1 D. Energy level values
shown are those from theoretical calculations by Moccia and Spizzo (1988a).
of a two-colour ladder excitation. The two-colour R-matrix approach (van der Hart 1996)
for that particular experimental situation is beyond the scope of this paper. Nevertheless,
some preliminary results on multiphoton ionization rates within the framework of R-matrix
theory in the two-colour scheme have been obtained quite recently (van der Hart et al
1996). Furthermore, let us note that the system depicted in figure 1 offers an interesting
possibility to study the influence of an intermediate two-photon resonance on HG rates.
The intermediate two-photon resonance may be due to a highly excited Rydberg state. A
study of a similar configuration in helium but with double one-photon and three-photon
resonances has been investigated recently (van Enk et al 1996b, Glass et al 1997).
As pointed out in section 2, we take the maximum total angular momentum in the
calculations as L = 3, since for the three-photon process from the 1 S ground state we
have to allow, in principle, coupling of the initial Li = 0 ground state with a Lf = 3 state.
However, we only have contributions to the direct THG rate (the lowest-order process of reemission from the 3p3d 1 Po AI state to the ground state in the electric dipole approximation)
from the part of the initial population which has been transferred to 1 Po symmetry. Hence
after absorption of three photons, the part of the atomic wavefunction which has total angular
momentum L = 3 (see figure 2) becomes ‘trapped’ against re-emission of one photon with
frequency 3ω. On the other hand, we expect levels belonging to 1 Fo symmetry to play
a more important role in the ionization process, as we have an additional channel open,
through which it is possible to reach the one-electron continuum. In addition, we note that
this channel may be greatly enhanced by an intermediate resonance with one of the 1 D
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R Gȩbarowski et al
Figure 2. The excitation scheme from the magnesium ground state 3s2 1 S made possible by three
photons of frequency ω. Note that the direct THG process in the electric dipole approximation
may occur only from levels with 1 Po symmetry down to the ground state. Therefore a part of
the initial population of the ground state excited to 1 Fo symmetry through the 1 D symmetry
will not contribute to THG rates originating from the direct harmonic generation process, that
is, from de-excitation by a photon of energy 3ω.
Rydberg levels. The importance of nd orbitals and Rydberg levels of total 1 D symmetry in
a quantitative description is supported by the generalized cross sections obtained by Chang
and Tang (1992) for the case of three-photon ionization. Finally, we note that admixture
of Rydberg levels in the ‘dressed’ ground state expansion and hence in the dipole matrix
elements makes it necessary to keep a large internal region radius a in our approximation
since we neglect the contribution to the dipole matrix elements from the external region as
discussed in RMF X.
3.1. The photoionization cross section and AI widths
Within our approximation we obtain a value of the ground state energy Eg = −0.274 93 au
(measured relative to the first ionization limit) which should be compared to the value of
−0.276 84 au obtained by Moccia and Spizzo (1988a). Nevertheless, there is a very good
agreement between the photoionization one-photon spectra presented in Moccia and Spizzo
(1988b) and in Chang and Tang (1992) with our data, shown in figure 3. In this figure the
photoionization cross section σ is obtained from the total ionization rate 0 induced by a
laser field of intensity I = 1011 W cm−2 . In the case of one-photon processes, the maximum
value of angular momentum for the continuum orbitals was taken to be lmax = 2, and for
each angular momentum nco = 30 continuum orbitals were included. The maximum total
angular symmetry was L = 2 and a total of four Floquet blocks (two absorption and one
emission) were included in the Floquet–Fourier expansion. We remark that the position of
the 3p3d 1 Po AI state and its width as well as the overall behaviour of the cross section are
all well reproduced with this model.
The AI width 0 of the Fano resonance in the R-matrix Floquet approach can be obtained
from the corresponding AI pole position in the complex energy plane and is taken as
Harmonic generation in magnesium
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Figure 3. Photoionization cross section σ versus frequency ω. Total symmetries 1 S, 1 Po , 1 D
are taken into account. The AI resonance 3p3d 1 Po is labelled in the figure.
0 = −2 Im(EAI ). We note here that if we tune the field frequency so that we are close
to the AI resonance, both poles corresponding to the ground state dressed by the field and
the AI state have approximately the same real part but the imaginary parts differ by several
orders of magnitude. In the one-photon coupling case the resulting width of the 3p3d 1 Po
AI state has a value 0 = 2.88 × 10−4 au. A comparison of our results with those of other
workers for the widths of some AI states of figure 1 is given in table 1. There is clearly
a wide diversity of theoretical values for the width of the 3p3d 1 Po resonance, with our
value being approximately two times larger than the value obtained by Bates and Altick
(1973). However, in view of the relatively simple model that we have used we regard our
result as satisfactory and certainly accurate enough to make our THG results physically
meaningful.
3.2. Convergence criteria for harmonic generation rates
In the following we will discuss the sensitivity of our THG results to changes in various
parameters of the model. Apart from demonstrating the accuracy of the present approach, we
will point out some interesting physical properties of the THG process. Results throughout
this paper will be shown, where possible, as a function of scaled laser frequency detuning
21/ 0 = 2(ω − ωr )/ 0 from the three-photon resonance with the AI state, where ωr is the
resonant laser frequency.
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R Gȩbarowski et al
Table 1. Widths of AI 3p2 and 3p3d states expressed in atomic units. Numbers in parentheses
represent powers of ten.
Level
a
b
c
d
3p2 1 S
1.51(−3)
0.92(−4)
3.18(−3)
1.50(−4)
0.6(−4)
1.8(−3)
0.57(−4)
3p3d 1 Po
3p3d 1 Fo
a
b
c
d
e
e
2.88(−4)
3.16(−3)
Moccia and Spizzo (1988a).
Bates and Altick (1973).
Chang (1986).
Mengali and Moccia (1996).
Present results, pole position in the complex plane.
3.2.1. The maximum total symmetry. First, let us discuss the influence of the total symmetry
1 o
F on the THG process. Figure 4 shows the influence of the maximum angular momentum
retained in the calculations on the THG spectrum near the AI state. In this figure the scaled
THG spectrum (the S(3) matrix element is scaled by the fundamental power spectrum matrix
element S(1)) near the 3p3d 1 Po AI resonance for a laser field of intensity I = 1011 W cm−2
is shown on a logarithmic scale. Two curves in this figure correspond to cases with different
maximum angular momentum in the R-matrix expansion basis. For the 1 Po case the (N +1)electron has maximum angular momentum lmax = 2 and for the 1 Fo case it has maximum
angular momentum lmax = 3. The data for maximum Lf = 1 are shown by a broken curve
whereas the case with maximum Lf = 3 is depicted by a full curve. It is evident from
figure 4 that apart from a slight shift in position of the maximum of the THG spectrum and an
additional maximum arising away from the AI resonance, both results are in good qualitative
agreement. Both curves exhibit a distinctive maximum near the resonance 1 = 0. Clearly in
the vicinity of the AI resonance THG rates become enhanced by several orders of magnitude.
The results presented in figure 4 are shown for a = 50 au and 30 continuum orbitals. Thus
we may conclude that at this intensity, higher-order processes (like continuum–continuum
transitions depicted in figure 2 with straight broken arrows) are not important for THG rates
near the AI state.
3.2.2. The number of Floquet blocks. We have compared results for the THG rates when
five Floquet absorption blocks are included with the case when seven absorption blocks
are retained (in both cases three emission blocks are retained). We have not observed any
significant difference for field intensities up to I = 1011 W cm−2 when 1 Po is the maximum
total symmetry included in the calculation. However, we have found that the inclusion of
only three absorption blocks, which is the minimum number which must be included to
describe the physical process, is not sufficient to obtain converged ionization and harmonic
generation rates.
3.2.3. The internal region radius. We now discuss the importance of the corrections due to
the finite internal region radius. In figure 5 results are shown for the case where the internal
region radius is a = 50 (broken curve) and a = 60 (full curve). These test calculations have
been carried out for a field intensity I = 1011 W cm−2 with maximum total symmetry 1 Po .
The position and height of the maximum in the THG spectrum is modified when the larger
internal region radius is used. However, the qualitative agreement between the two results
indicates that a = 50 au should be large enough for the purpose of our qualitative discussion.
Harmonic generation in magnesium
2511
Figure 4. The THG spectrum near the 3p3d 1 Po AI resonance for laser intensity I =
1011 W cm−2 versus the scaled detuning 21/ 0 = 2(ω − ωr )/ 0 where ωr is the frequency
for which there is a three-photon resonance with the AI state and ω is the laser frequency. The
full curve shows data for maximum total angular momentum L = 3 whereas the broken curve
pertains to L = 1.
Figure 5. Comparison of the THG spectrum for different values of the internal region radius
a: results for a = 50 au are denoted by the broken curve and for a = 60 au by the full curve.
In each case 25 continuum orbitals have been retained. In these test calculations, the intensity
is set to be I = 1011 W cm−2 and the total maximum symmetry L = 1 is taken (only 1 S and
1 Po symmetries are included).
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R Gȩbarowski et al
3.3. Total three-photon ionization rates
Multiphoton processes near the AI resonance could be very sensitive to the intensity of
the applied field. In particular, it is undesirable to have excessively strong laser fields for
the purpose of studying various effects near the resonance with an AI state. In figure 6
total ionization rates 0 are shown as a function of intensity I in a double-logarithmic plot.
The field frequency is ω = 0.128 152 au and the maximum symmetry is taken to be 1 Fo .
The internal region size is a = 50 au. In the perturbative regime we expect that the total
ionization rate behaves according to the power law 0 ∼ I α , where α is the number of
photons needed to ionize the atom. The linear fit to the data presented in figure 6 yields
the value α = 2.76 ± 0.02. This indicates that the system follows a perturbative prediction
(the dotted curve in the figure corresponds to α = 3). Note that the ground state width in
the most intense field considered here is four orders of magnitude smaller than the width
of the AI state of interest. This ensures that the depletion of the ground state is negligible
on the time scale of the AI state decay time.
Figure 6. The three-photon total ionization rate 0 versus laser intensity I expressed in W cm−2
for a fixed laser frequency ω = 0.128 152 au. Note the double-log scale. Full circles show
numerical data which are fitted with a line of slope 2.76 ± 0.02. The reference dotted line has
a slope equal to 3.
3.4. Intensity dependence of the THG spectrum near the AI state
Figure 7 again shows the behaviour of the THG spectrum in the vicinity of the AI resonance.
The same parameters as before are used and, in particular, symmetries 1 S, 1 Po , 1 D and 1 Fo
are included. From this figure it is evident that there is a distinct enhancement by several
orders of magnitude in the THG rate for a laser frequency close to the resonance frequency
1 = 0. The non-symmetrical shape of the THG curves may be due to quantum interference
Harmonic generation in magnesium
2513
Figure 7. The THG spectrum near the 3p3d 1 Po AI resonance at various laser field intensities
I = 1011 W cm−2 (full curve), I = 5 × 1010 W cm−2 (dotted curve) and I = 1010 W cm−2
(broken curve) versus the scaled detuning 21/ 0 = 2(ω − ωr )/ 0, where ωr is the frequency for
which there is a three-photon resonance with the AI state and ω is the laser frequency.
Figure 8. Same as in figure 7 but intensities I = 1010 W cm−2 (broken curve) and
I = 8 × 109 W cm−2 (dotted curve) are shown.
2514
R Gȩbarowski et al
of the type resulting in a Fano resonance shape. The quantum interference may occur
between two excitation paths leading to a continuum level which is then coupled with the
ground state through the emission of a photon corresponding to the third harmonic of the
laser frequency ω. Note also that the peak position of the THG spectrum moves away from
1 = 0 with decreasing field intensity, as has been observed in a situation investigated earlier
(Zhang and van Enk 1995), where two-photon coupling of the ground state with a AI state
was considered. Another interesting feature visible in figure 7 is the sensitivity of the overall
shape of the THG spectrum to the field intensity. For a field intensity I = 1010 W cm−2
the spectrum takes a dispersive shape, characterized not only by a maximum but also by a
distinctive minimum. Moreover, data for even lower intensity, I = 8 × 109 W cm−2 (dotted
curve in figure 8), also exhibit a similar dispersive shape for the THG spectrum. This
indicates that the minimum is a robust feature existing over a range of field intensities. As
has been pointed out earlier (Faucher et al 1996) this minimum may result from competition
between the coherent (harmonic generation) and the incoherent (ionization) decay channels
of the ground state.
4. Conclusions
We have carried out an application of the R-matrix Floquet theory of harmonic generation for
an atomic system with an outer sub-shell configuration ns2 . Although our approach allows
only a single ionizing electron and explicitly takes into account only the internal region
contribution to harmonic generation rates, we are able to observe qualitative enhancement
of the THG rates close to a three-photon resonance with an AI state.
Our approximation of only retaining the contribution to the THG rate from the internal
region is consistent with the well established understanding that the most important region
for harmonic generation lies in the vicinity of the nucleus where it is possible for the electron
to radiate the energy gained from the field when far from the nucleus.
Using this approach we will be able to obtain higher than third-order harmonics, provided
we include more Floquet blocks and total symmetries in the basis expansion and consider
a more complete outer sub-shell description of the residual ion.
Acknowledgments
We are indebted to Nathalie Vaeck for generating the parameters of the 3p orbital used in
the internal region basis expansion. We would also like to thank Alain Cyr, Martin Dörr,
David Glass, Hugo van der Hart and Kuba Zakrzewski for many enlightening discussions.
Financial support through EPSRC grant no GR/K 24741, the Stefan Batory Foundation
(RG) and the EC HCM Network ERB CHRX CT 920013 is gratefully acknowledged. The
calculations were carried out on the Cray Y-MP/EL at Queen’s University.
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