1. INTRODUCTION: STRUCTURED NANORING WITH DEFECTS
A recent frontier of optics is the study of the interaction between strong laser pulses and low-dimensional nano-sized systems. Particular interesting structures are the artificial atoms or molecules, quantum confinement of only few electrons, named quantum dots (QDs). In the realization stage, QDs can be tailored almost at will to reproduce characteristics of systems such as atoms, nuclei, metallic clusters, trapped atomic gases, or to implement new desired features. For example, a way to set the electronic properties of a finite man-made low-dimensional system is by electrostatic gating (Kouwenhoven et al., Reference Kouwenhoven, Austing and Tarucha2001; van Kouwen et al., Reference van Kouwen, Reimer, Hidma, van Weert, Algra, Bakkers, Kouwenhoven and Zwiller2010), or by application of an external magnetic field. Thus, these artificial systems can be potentially employed for a large variety of purposes.
The aim of this paper is the investigation of the interaction between a strong laser field and nanorings (Wang, Reference Wang2004; Guo et al., Reference Guo, Kim, Kar, Shafarman, Birkmire, Stach, Agrawal and Hillhouse2008; Hoffmann et al., Reference Hoffmann, Kärnbratt, Chang, Herz, Albinsson and Anderson2008; Hinsche et al., Reference Hinsche, Moskalenko and Berakdar2009; O'Sullivan et al., Reference O'Sullivan, Sprafke, Kondratuk, Rinfray, Claridge, Saywell, Blunt, O'Shea, Beton, Malfois and Anderson2011; Cricchio & Fiordilino, Reference Cricchio and Fiordilino2014, Reference Cricchio and Fiordilino2016): in particular we study a one-dimensional (1D) structured quantum ring (SQR) (Castiglia et al., Reference Castiglia, Corso, De Giovannini, Fiordilino and Frusteri2015), consisting of M point like QDs along an annular structure of radius R. Matter driven by a strong laser field of frequency ωL emits electromagnetic radiation whose spectrum is formed by multiples of ωL with a fast-decreasing emission in the first part of the spectrum, a broad plateau region until a cut-off frequency ω M . This phenomenon is called high-harmonic generation (HHG) (L'Huillier et al., Reference L'Huillier, Schafer and Kulander1991; Gavrila, Reference Gavrila1992; Corkum, Reference Corkum1993; Lewenstein et al., Reference Lewenstein, Balcou, Ivanov, L'Huillier and Corkum1994; Zuo et al., Reference Zuo, Bandrauk, Ivanov and Corkum1995; Alon et al., Reference Alon, Averbukh and Moiseyev1998; Bâldea et al., Reference Bâldea, Gupta, Cederbaum and Moiseyev2004; Xie et al., Reference Xie, Scrinzi, Wickenhauser, Baltuška, Barth and Kitzler2008; Ozaki et al., Reference Ozaki, Elouga Bom, Ganeev, Kieffer, Suzuki and Kuroda2007, Reference Ozaki, Elouga Bom, Abdul-Hadi and Ganeev2010; Ciappina et al., Reference Ciappina, Becker and Jaroń-Becker2008; Ganeev et al., Reference Ganeev, Elouga Bom, Wong, Brichta, Bhardwaj, Redkin and Ozaki2009, Reference Ganeev, Witting, Hutchison, Frank, Redkin, Okell, Lei, Roschuk, Maier, Marangos and Tisch2012; Zhukovsky, Reference Zhukovsky2016). In atoms, HHG is generally explained by the three-step model, thus as an outcome of the ionization and recombination process. Nevertheless the origin of the emission resides in the electron acceleration, whatever its origin is; actually HHG can be obtained also from bound–bound as well as free–free transitions (Daniele & Fiordilino, Reference Daniele and Fiordilino1996; Di Piazza & Fiordilino, Reference Di Piazza and Fiordilino2001; Orlando et al., Reference Orlando, Castiglia, Corso and Fiordilino2008; Fiordilino et al., Reference Fiordilino, Morales, Castiglia, Corso, Daniele and Strelkov2017). Our model does not support ionization, since the active charges are bound along the ring thus the emission is caused by the electron acceleration maybe during violent collisions with the dots. Here, for simplicity sake, we confine ourselves to study the six-dot problem within the single active electron approximation that, in the HHG emission, is largely used (Schafer et al., Reference Schafer, Yang, DiMauro and Kulander1993; Bauer, Reference Bauer1997; Daniele et al., Reference Daniele, Castiglia, Corso, Fiordilino, Morales and Orlando2009).
The HHG is very interesting from many points of view. Various experiments show that to obtain a cutoff of ω M ≈ 100ωL is possible, then HHG allows us to obtain emission in the far-UV or near-X-ray range from an incident laser in the visible region (Baer et al., Reference Baer, Neuhauser, Ždánská and Moiseyev2003; Brandi et al., Reference Brandi, Neshev and Ubachs2003; Chang, Reference Chang2004; Eden, Reference Eden2004; Ganeev et al., Reference Ganeev, Suzuki, Baba, Kuroda and Ozaki2005; Heinrich et al., Reference Heinrich, Kornelis, Anscombe, Hauri, Schlup, Biegert and Keller2006). The coherent properties of the emitted light and the large plateau that, mainly for molecular systems, presents very close lines, allow technological and scientific improvement, such as generation of train of attosecond pulses or of a single isolate attosecond pulse and also the external seeding of free-electron lasers (Orlando et al., Reference Orlando, Corso, Fiordilino and Persico2009a , Reference Orlando, Corso, Fiordilino and Persico b ; Allaria et al., Reference Allaria, Callegari, Cocco, Fawley, Kiskinova, Masciovecchio and Parmigiani2010; Crawford-Uranga et al., Reference Crawford-Uranga, De Giovannini, Räsänen, Oliveira, Mowbray, Nikolopoulos, Karamatskos, Markellos, Lambropoulos, Kurth and Rubio2014; Solanpää et al., Reference Solanpää, Budagosky, Shvetsov-Shilovski, Castro, Rubio and Räsänen2014).
In order to follow in detail the particle dynamics, the necessity of using new methods for producing extremely short light pulses emerges. The most efficient tool is the superposition of a clip of the spectrum emitted by a source, and the shortest possible pulse duration is given by the optical bandwidth Δω of the radiation. Thus, to think that the HHG could be used in the ultrafast science is natural (Farkas & Tóth, Reference Farkas and Tóth1992; Antoine et al., Reference Antoine, L'Huillier and Lewenstein1996; Krausz & Ivanov, Reference Krausz and Ivanov2009). The main benefit of using HHG to make attosecond pulses, is that the presence of a large plateau allows the increase of the bandwidth Δω with a subsequent reduction of the pulse duration τ ≈ 1/(Δω). The principal problem in the use of HHG is the low efficiency of conversion of the energy from the laser to the emitted radiation, thus an eventually extremely short light pulse, made in this way, will be weak. For this reason phase-matching techniques are implemented to increase the efficiency (Rundquist et al., Reference Rundquist, Durfee, Chang, Herne, Backus, Murnane and Kapteyn1998). Actually it is also demonstrated that although the plateau harmonics were not in phase, the single atom spectrum consists of a train of ultrashort pulses (Paul et al., Reference Paul, Toma, Breger, Mullot, Augé, Balcou, Muller and Agostini2001; Lein, Reference Lein2005; Zepf et al., Reference Zepf, Dromey, Landreman, Foster and Hooker2007; Orlando et al., Reference Orlando, Castiglia, Corso and Fiordilino2008; Strelkov, Reference Strelkov2010).
It is well known that the internal symmetry of the matter during HHG strongly affects the emission process and the overall aspect of the spectrum such as plateau extension, yield, and presence of odd and even harmonics. Therefore, if control of the spectrum is looked for, the symmetry of the system provides a versatile knob for the purpose. Both atoms and molecules are very studied, and in particular molecules are sources of very rich HHG spectra, but their applications have been quite limited. For these reasons, the study of the emission by a system that can be easily fabricated such as the SQR can be very promising.
In literature (Castiglia et al., Reference Castiglia, Corso, De Giovannini, Fiordilino and Frusteri2015), the SQR is studied as a 1D system with identical and equidistant scattering centers with the potential energy experienced by the electron taken as:
$${\opf V}_{\rm C} ({\rm \phi} ) = V_0 \cos (M{\rm \phi} ).$$
In this paper, we consider a soft-core potential energy:
$${\opf V}_{\rm S} ({\rm \phi} ) = \sum\limits_{n = 1}^6 - \displaystyle{{K_n} \over {\sqrt {{\rm \eta} _n + 1 - \cos ({\rm \phi} - {\rm \phi} _n )}}} $$
that mimics the presence of M = 6 dots, not necessarily equal, at the angular position ϕ n , where η n are soft-core constants and K n normalization constants. The need for writing the potential energy in this way comes from the requirement of introducing asymmetries to model defects of the system. For comparison sake, we start our calculations by considering the symmetrical case with η n = η0, K n = K 0 and ϕ n = −30° + n60°. The constants in V S must be chosen in such a way to get |V S(30°) − V S(0°)| = 2V 0. For V 0 = 0.33 au, by taking K 0 = 1.9420 au, and η0 = 2/15 the solutions of the time-independent Schrödinger equations:
$${\opf H}_T {\rm \Psi} _{T,n} ({\rm \phi}, 0) = E_{T,n} {\rm \Psi} _{T,n} ({\rm \phi}, 0),$$
$${\opf H}_T = - \displaystyle{{\hbar ^2} \over {2m_{\rm e} R^2}} \displaystyle{{\partial ^2} \over {\partial {\rm \phi} ^2}} + {\opf V}_T ({\rm \phi} ),$$
with T = C, S give E S,n − E S,0 ≅ E C,n − E C,0.
2. THEORY
To evaluate the HHG spectrum, we solve the time-dependent Schrödinger equation in the dipole approximation:
$$i\hbar \displaystyle{\partial \over {\partial t}}{\rm \Psi} ({\rm \phi}, t) = {\opf H}{\rm \Psi} ({\rm \phi}, t)$$
with Hamiltonian:
$${\opf H} = {\opf H}_{\rm S} - e{\bi r} \cdot {\bi {\rm \varepsilon}}, $$
$${\bi r} \cdot {\bi {\rm \varepsilon}} = R\cos ({\rm \phi} ){\rm \varepsilon} _x + R\sin ({\rm \phi} ){\rm \varepsilon} _y, $$
and ε = (ε x , ε y ) the external laser field.
The power emitted by an accelerated electron is given by the Larmor formula:
$$\displaystyle{{dI(t)} \over {dt}} = \displaystyle{{2e^2} \over {3c^3}} \vert {\bi a}(t) \vert ^2, $$
and the energy emitted in the energy range from ω to ω + dω is:
$$\displaystyle{{dI({\rm \omega} )} \over {d{\rm \omega}}} = \displaystyle{{4e^2} \over {3c^3}} \vert {\bi a}({\rm \omega} ) \vert ^2, $$
where a (ω) is the Fourier transform of the average electronic acceleration a (t) = d 2 r (t)/dt 2 and r (t) = 〈Ψ(t)| r |Ψ(t)〉. The electric field emitted by the accelerated electron is:
$${\bi {\rm \varepsilon}} _{\rm e} = - \displaystyle{{e\left\{ {\hat{\bi n}(t) \times [{\hat{\bi n}}(t) \times {\bi a}(t)]} \right\}} \over {r(t)c^2}}, $$
where t is the retarded time,
${\hat{\bi n}}$
is the unit vector in the direction pointing from the position of the electron (at the retarded time t) to the observer; since r(t) ≫ R, to consider
${\hat{\bi n}}$
pointing from the center of the ring is a fair assumption (Jackson, Reference Jackson1999).
A simple way to analyze the amplitudes and the phase of a plane monochromatic wave, defined in the Cartesian system, is to use the four Stokes parameters:
$$\left\{ {\matrix{ {S_0 = {\rm \varepsilon} _{{\rm e}x}^2 ({\rm \omega} ) + {\rm \varepsilon} _{{\rm e}y}^2 ({\rm \omega} ),} \cr {S_1 = {\rm \varepsilon} _{{\rm e}x}^2 ({\rm \omega} ) - {\rm \varepsilon} _{{\rm e}y}^2 ({\rm \omega} ),} \cr {S_2 = 2{\rm \varepsilon} _{{\rm e}x} ({\rm \omega} ){\rm \varepsilon} _{{\rm e}y} ({\rm \omega} )\cos ({\rm \delta} ),} \cr {S_3 = 2{\rm \varepsilon} _{{\rm e}x} ({\rm \omega} ){\rm \varepsilon} _{{\rm e}y} ({\rm \omega} )\sin ({\rm \delta} ),} \cr}} \right.$$
$$S_0^2 = S_1^2 + S_2^2 + S_3^2, $$
where εex (ω) and εey (ω) are the Fourier transform of respectively εex (t) and εey (t); δ is the phase between the two fields (Born & Wolf, Reference Born and Wolf2000). To characterize the polarization of the emission, introducing two independent quantities is expedient: the angle (−π/4) ≤ χ ≤ (π/4), which specifies the orientation of the polarization ellipse (tanχ = ±b/a, where a and b are the major and the minor axes of the polarization ellipse) and the angle 0 ≤ ψ ≤ π between the polarization and the x-direction. We find:
$$S_1 = S_0 \cos 2{\rm \chi} \cos 2{\rm \psi}, $$
$$S_2 = S_0 \cos 2{\rm \chi} \sin 2{\rm \psi}, $$
$$S_3 = S_0 \sin 2{\rm \chi}, $$
and thus for a monochromatic wave:
$$\tan 2{\rm \psi} = \displaystyle{{S_2} \over {S_1}} = \displaystyle{{2{\rm \varepsilon} _{{\rm e}x} ({\rm \omega} ){\rm \varepsilon} _{{\rm e}y} ({\rm \omega} )\cos ({\rm \delta} )} \over {{\rm \varepsilon} _{{\rm e}x}^2 ({\rm \omega} ) - {\rm \varepsilon} _{{\rm e}y}^2 ({\rm \omega} )}},$$
$$\eqalign{& \tan 2{\rm \chi} = \displaystyle{{S_3} \over {\sqrt {S_1^2 + S_2^2 } }} \cr & \qquad\hskip5pt = \displaystyle{{2{\rm \varepsilon} _{{\rm e}x}({\rm \omega} ){\rm \varepsilon} _{{\rm e}y}({\rm \omega} )\sin ({\rm \delta} )} \over {\sqrt {{({\rm \varepsilon} _{{\rm e}x}({\rm \omega} ) + {\rm \varepsilon} _{{\rm e}y}({\rm \omega} ))}^2 - 4{\rm \varepsilon} _{{\rm e}x}^2 ({\rm \omega} ){\rm \varepsilon} _{{\rm e}y}^2 ({\rm \omega} )\mathop {\sin }\nolimits^2 ({\rm \delta} )} }}.} $$
The spectrum obtained by the Fourier transform hides all information about the time dependence of the single harmonic; to get information on the temporal evolution of the emitted spectrum we resort to the Morlet transform of the acceleration, defined as
$$\left\{ {\matrix{ {{\bi a}(t_0 ;{\rm \omega} ) = \int\limits_{ - \infty} ^\infty {{\bi a}(t)M(t_0 ;{\rm \omega}, t)dt,}} \cr {M(t_0 ;{\rm \omega}, t) = \sqrt {\rm \omega} \left\{ {e^{ - i[{\rm \omega} (t - t_0 )]} - e^{ - {\rm \sigma} _0^2 /2}} \right\}e^{ - {\rm \omega} ^2 (t - t_0 )^2 /(2{\rm \sigma} _0^2 )},} \cr}} \right.$$
where σ0 = 15 is the number of oscillations at frequency ω within the Gaussian width and t 0 represents the time for which we require information (De Luca & Fiordilino, Reference De Luca and Fiordilino1996; Chui, Reference Chui2014) coherently with Eq. (9) we define Morlet power spectrum the quantity
$$\displaystyle{{{\rm d}I(t_0 ;{\rm \omega} )} \over {{\rm d}{\rm \omega}}} = \displaystyle{{4e^2} \over {3c^3}} \vert {\bi a}(t_0 ;{\rm \omega} ) \vert ^2 $$
and assume that it gives the energy emitted in the frequency range ω, ω + dω at time t
0. As said before, to look not only for the total intensity of the radiation, but also for other important quantities in order to get important knobs for the improvement of new technology is of paramount importance. In this paper, we also look for the averaged emitted power in a time interval
${\cal T}$
as:
$$P({\cal T}) = \displaystyle{1 \over {\cal T}}\int_{\cal T} \displaystyle{{dI(t)} \over {dt}}dt.$$
3. RESULTS
In all simulations of this work we use a linearly polarized laser with ϕL = 60° (angle between the polarization axis and x-axis), wavelength λ = 780 nm, intensity I 0 = 4 × 1014 W/cm2, and trapezoidal shape of 32 optical cycles (oc), ramping up and down for 3 oc. In the symmetrical case, ℍS and ℍC share the same symmetry, thus presenting the same degeneracy chain in their eigenstates, namely one non-degenerate state, two twofold-degenerate states, and one non-degenerate state. Moreover, always in the same case, with our choice of physical parameters the solutions of the time-independent Schrödinger equation for ℍS are approximately equal to those for ℍC given in (Castiglia et al., Reference Castiglia, Corso, De Giovannini, Fiordilino and Frusteri2015).
In Figure 1, we report, in the first row the Fourier spectrum for the symmetrical ring with the vertical dashed lines corresponding to the energy difference between the excited states and the ground state and, in the second row, V S and the square modulus of the ground state of the Hamiltonian ℍS. Moreover, Figure 1 shows that V S ≅ V 0 cos 6ϕ up to an additive constant when V 0 = 0.33 au. The spectrum contains harmonic as well as non-harmonic lines, which could be used to synthesize short pulses.
Fig. 1. First row: Fourier spectrum, vertical dashed red lines show energy differences between excited states and ground-state; second row: in the left side, V S versus ϕ and in the right side, ground state square modulus versus ϕ. The values of the parameters are: V 0 = 0.33 au, R = 5 au, η1 = η0 = 0.1333, K1 = K0 = 1.9420 au, ϕi = 30°, 90°, 150°, 210°, 270°, 330°, λ = 780 nm, I 0 = 4 · 1014W/cm2, trapezoidal shape(32 oc, 3 up, and 3 down) linearly polarized with ϕL = 60°. System with C6 symmetry.
As a second step, we wish to investigate the possibility of controlling the spectrum by introducing small modifications of V S, thus we evaluate the dynamics of the system by varying η1, K 1, and ϕ1; we start by showing in Figures 2–4 the same quantities of Figure 1, respectively, for η1/η0 = 0.97, 0.99, and 1.03 (the other parameters are not modified). Owing to the reduced symmetry of the potential, the spectra now present both odd and even harmonics, but now the lines are less dense than in the spectrum of Figure 1; the spectrum in Figure 3 is well resolved (especially in the plateau region). However, the cutoff does not appear significantly changed; in agreement with Ref. (Castiglia et al., Reference Castiglia, Corso, De Giovannini, Fiordilino and Frusteri2015) the upper bound for the highest harmonic is:
$${\rm \omega} _{\rm c} \cong \displaystyle{{2{\rm \varepsilon} R} \over \hbar} ;$$
by solving the canonical equations associated with the classical counterpart of ℍ we find that the cutoff is
${\rm \omega} _{\rm c} \le 2.8{\rm \varepsilon} R/ \hbar \cong 30{\rm \omega} _{\rm L}$
(Castiglia et al., Reference Castiglia, Corso, Cricchio, De Giovannini, Frusteri and Fiordilino2016). Attention must be paid to the physical meaning of setting different expressions for V
S. Obviously, by varying V
S we are changing the physical system and the differences are visible observing the ground states reported in Figures 2–4. Very small variations of the potential radically change the physical system, this is favorable to the goal of obtaining a tuning of the radiation properties, but the difference between the ground states can be used for better explaining some results. We note that Figure 4 (η1 = 1.03η0) presents a ground state closer to the symmetrical system of Figure 1 than the other cases; indeed we have significant variations in the ground state for η1 = 0.99η0 and a very different ground state for η1 = 0.97η0. Observing Figures 2 and 3 we note that both spectra present a rich plateau with only harmonic lines, but the first one is less resolved. At the same time, the ground state plots show that, with respect to the symmetrical case (Fig. 1), the population is mainly in the first dot, but in Figure 3 |ΨS(30°, 0)|2/|ΨS(90°, 0)|2 ≅ 3.5 and in Figure 2 |ΨS(30°, 0)|2/|ΨS(90°, 0)|2 ≅ 22; thus, the unbalance of population seems to play an important role.
Fig. 2. Caption as Figure 1 but with η1 = 0.97η0 = 0.1293, which breaks the C6 symmetry.
Fig. 3. Caption as Figure 1 but with η1 = 0.99η0 = 0.1320, which breaks the C6 symmetry.
Fig. 4. Caption as Figure 1 but with η1 = 1.03η0 = 0.1373, which breaks the C6 symmetry.
In Figures 5 and 6, we show the results obtained for K 1/K 0 = 1.01, 1.02 (and the other parameters equal to the ones in Fig. 1); also in this case we could state that, by varying the ground state, the spectrum is, again, more resolved in harmonics, but if the ground state becomes very different with respect to the symmetrical case, the spectrum appears unresolved.
Fig. 5. Caption as Figure 1 but with K 1 = 1.01K 0 = 1.9614 au, which breaks the C6 symmetry.
Fig. 6. Caption as Figure 1 but with K 1 = 1.02K 0 = 1.9808 au, which breaks the C6 symmetry.
The previously analyzed two sets of systems present a specular symmetry for an axes passing along ϕ1 = 30°. In Figures 7 and 8, we break this symmetry by varying ϕ1; with ϕ1 = 32.5° (Fig. 8), we observe that the cutoff is considerably increased. Broad, resonant-like structures are present in all spectra (Figs 1–8); for example in Figure 1, around 7ωL, 13ωL, and 18ωL; they are probably consequence of power broadened, Stark-shifted transitions.
Fig. 7. Caption as Figure 1 but with ϕ i = 30.75°, 90°, 150°, 210°, 270°, 330°, which breaks the C6 symmetry.
Fig. 8. Caption as Figure 1 but with ϕ i = 32.5°, 90°, 150°, 210°, 270°, 330°, which breaks the C6 symmetry.
In Figure 9, we make a systematic study of the cutoff, reporting the logarithm of the spectra by varying: in the first row η1, in the second row K 1 and in the third ϕ1. This analysis is important due to the fact that in the realization of short pulses by superposition of harmonic lines, it has been seen that the harmonics in the cut-off region may play a significant role (Krausz & Ivanov, Reference Krausz and Ivanov2009; Lu et al., Reference Lu, He, Guo and Han2009). Although in Figure 9, there is not a clear behavior, to set parameters, in order to have a cutoff with six or seven harmonics more than in the symmetrical case, is possible. For example, we see an extended cutoff for K 1 = 1.07K 0, for η1 = 0.92η0, and for ϕ1 = 32.5°.
Fig. 9. First row: logarithm of Fourier spectrum versus harmonic order (in the x-axis) versus η1 (in the y-axis) with K 1 = K 0 = 1.9420 au and ϕ i = 30°, 90°, 150°, 210°, 270°, 330°; second row: Fourier spectrum versus harmonic order (in the x-axis) versus K 1 (in the y-axis) with η1 = η0 = 0.1333 and ϕ i = 30°, 90°, 150°, 210°, 270°, 330°; third row: Fourier spectrum versus harmonic order (in the x-axis) versus ϕ1 (in the y-axis) with K 1 = K 0 = 1.9420, η1 = η0 = 0.1333 and ϕ2,…,6 = 90°, 150°, 210°, 270°, 330°. The values of the parameters are: V 0 = 0.33 au, R = 5 au, λ = 780 nm, I 0 = 4 × 1014W/cm2; the laser pulse has a trapezoidal shape (32 oc of full duration with 3 oc of up and down ramping) and is linearly polarized with ϕL = 60°.
The study of the polarization of the emitted fields is reported in Figures 10–12, where we show the change of the ellipse of polarization for different values of, respectively, η1, K 1, and ϕ1. The polarization of the harmonics of the C 6 symmetrical system is the same of the incident laser, instead the other systems present harmonics polarized along different direction and with different ellipticity (1 − b/a = 1 − tanχ); it is interesting to note that, for small variations of the ground state η1 = 1.03η0, we have little variation of the polarization, on the other hand, for major variation we obtain very different ψ and χ.
Fig. 10. First row: ψ versus harmonic order by varying η1; second row: tanχ versus harmonic order by varying η1. The values of the parameters are: V 0 = 0.33 au, R = 5 au, K1 = K0 = 1.9420 au, ϕi = 30°, 90°, 150°, 210°, 270°, 330°, λ = 780 nm, I 0 = 4 × 1014W/cm2, trapezoidal shape(32 oc, 3 up, and 3 down) linearly polarized with ϕ L = 60°.
Fig. 11. First row: ψ versus harmonic order by varying K 1; second row: tanχ versus harmonic order by varying K 1. The values of the parameters are: V 0 = 0.33 au, R = 5 au, η1 = η0 = 1333, ϕ i = 30°, 90°, 150°, 210°, 270°, 330°, λ = 780 nm, I 0 = 4 × 1014W/cm2, trapezoidal shape(32 oc, 3 up, and 3 down) linearly polarized with ϕL = 60°.
Fig. 12. First row: ψ versus harmonic order by varying ϕ1; second row: tanχ versus harmonic order by varying ϕ1. The values of the parameters are: V 0 = 0.33 au, R = 5 au, η1 = η0 = 1333, K 1 = K 0 = 1.9420 au, λ = 780 nm, I 0 = 4 × 1014W/cm2, trapezoidal shape(32 oc, 3 up, and 3 down) linearly polarized with ϕL = 60°.
To study the intensity of a particular harmonic at time t we evaluate the Morlet spectrum and we show it in Figures 13 and 14, respectively, for η1 = 0.97η0 and 0.99η0. In both configurations, we observe that the harmonics from 20th to 29th are emitted mainly in the same time intervals located after the 20th oc. In the bottom part of Figures 13 and 14, we plot the Morlet spectrum (color plot) and the averaged power
$P({\cal T})$
(in black histogram); the last is obtained by dividing the total observation time 32T
L in 64 equal intervals (thus every interval is of the type of
${\cal T}_n = \left[ {(n - 1/2)T_{\rm L}, (n/2)T_{\rm L}} \right]$
with n = 1, …, 64). This analysis allows us to see the correlation between the intensity of the averaged total power and the intensity of the highest-order harmonics. For example, observing the bottom part of Figure 13, we see that in the time interval [20.5, 21] oc the harmonics H21, H22, H23, H25, and H27 are emitted simultaneously and that
$P({\cal T})$
has a maximum. Thus, a strong intensity of the power is associated with a simultaneous emission of high-order harmonics.
Fig. 13. First row: Morlet spectrum versus time in oc; second row: zoom of the Morlet spectrum and (in black histogram)
$P({\cal T})$
versus time in oc. The values of the parameters are: V
0 = 0.33 au, R = 5 au, η1 = 0.97η0 = 0.1293, K
1 = K
0 = 1.9420 au, ϕ
i
= 30°, 90°, 150°, 210°, 270°, 330°, λ = 780 nm, I
0 = 4 · 1014W/cm2, trapezoidal shape(32 oc, 3 up, and 3 down) linearly polarized with ϕL = 60°.
Fig. 14. First row: Morlet spectrum versus time in oc; second row: zoom of the Morlet spectrum and (in black histogram)
$P({\cal T})$
versus time in oc. The values of the parameters are: V
0 = 0.33 au, R = 5 au, η1 = 0.99η0 = 0.1320, K
1 = K
0 = 1.9420 au, ϕ
i
= 30°, 90°, 150°, 210°, 270°, 330°, λ = 780 nm, I
0 = 4 · 1014W/cm2trapezoidal shape (32 oc, 3 up, and 3 down) linearly polarized with ϕL = 60°.
4. CONCLUSION
In this paper, we look for the theoretical basis to design home-made nanosystems; in particular, we analyze the effect on the emission from different systems created by varying the potential felt by an electron. We have numerically studied a SQR driven by a strong laser field by varying its rotational symmetry, and we show that very small modifications of the system structure could give significant variation of the emitted spectrum. The fine tuning of the SQR is a method to set the properties of the home-made emitter. Another motivation to choose SQR comes from the very rich spectrum of the symmetrical case in Figure 1; this system is also a good choice because presents a high-intensity emission in the cut-off region that could be used to realize attosecond pulses, but also, by a fine tuning of the potential, it is possible to obtain other systems having only the harmonic lines. We point out that a system, without symmetry, presents both even and odd harmonics; see, for example, Figure 3. Actually an extension of the cutoff, but with a diminution of resolution, is also possible.
Analyzing the Stokes parameters, we see that within the same spectrum there are harmonics, in the far UV ≈ 39 → 30 nm (27th–20th harmonics), with different polarization. By opportune filtering, we reckon that it possible to isolate from the emitted spectrum a monochromatic radiation with required polarization.
In conclusion, we propose the realization of devices that permit the tuning of the radiation and in particular we analyze the potentiality of the SQRs that have very particular HHG spectrum. We show that with little changes of the symmetry of the system, to get different features in the HHG spectrum is possible and we suggest the manufacturing of similar system to obtain different nanodevices.
