Behavioral study of high-order harmonics and attosecond pulse generation via bichromatic driving laser fields

Journal Pre-proof Behavioral study of high-order harmonics and attosecond pulse generation via bichromatic driving laser fields M. Ebrahimzadeh, S. Batebi

PII:

S0030-4026(19)31611-0

DOI:

https://doi.org/10.1016/j.ijleo.2019.163713

Reference:

IJLEO 163713

To appear in:

Optik

Received Date:

4 August 2019

Revised Date:

27 October 2019

Accepted Date:

5 November 2019

Please cite this article as: Ebrahimzadeh M, Batebi S, Behavioral study of high-order harmonics and attosecond pulse generation via bichromatic driving laser fields, Optik (2019), doi: https://doi.org/10.1016/j.ijleo.2019.163713

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Behavioral study of high-order harmonics and attosecond pulse generation via bichromatic driving laser fields M. Ebrahimzadeh and S. Batebi* Department of Physics, Faculty of Science, University of Guilan, P.O. Box 41335-19141, Rasht, Iran [email protected]

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Abstract: High-order harmonics intensity and subsequently attosecond pulses which are generated after the interaction of some two color femtosecond pulses with helium ions are studied in this paper. Cutoff frequency in the high-order harmonics' intensity spectrum, maximum ionization probability, attosecond pulse intensity and duration in this research are investigated versus the frequency spacing of the two color driving pulse elements (δω). It is seen that the cutoff frequency decays exponentially by δω, whereas generation of the most powerful and shortest attosecond pulses as well as the maximum ionization probability occurs around the value of δω = 0 (i.e., for the single color pulses). However, the longest attosecond pulses are generated for the case of δω = 0.8 ω0 (in which ω0 is the frequency of the fundamental pulse). Finally, the most isolated attosecond pulses are generated for the case in which δω = 1.25 ω0.

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Keywords: High order harmonic generation; Attosecond; helium ion; two color

1. Introduction

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Many nonlinear phenomena such as high-order harmonic generation (HHG) [1,2] multi-photon ionization [3,4], and above threshold ionization [5,6], can be achieved by exposing molecular or atomic targets to an intense (>1013 W/cm2) and ultra-short field. Over the past decade, Due to the high efficiency of HHG, this method has been extensively studied in the production of trains or isolated attosecond pulses (IAP) in the ultraviolet (UV) to extreme ultraviolet (XUV) spectral range [7-11], which has been a subject of much interest in ultrafast science and technology. Since the motion of the electron around the atom is at attosecond time-scales, the produced attosecond pulses can be used to investigate atomic physics at sub-angstrom scales. The HHG process for a single atomic system is well described by the semi classical three-step model [5,12]. In the above model, the maximum cutoff energy of the electron is 𝐸𝑐 = 𝐼𝑝 + 3.17𝑈𝑝 , where 𝐼𝑝 represents the atomic ionization potential and 𝑈𝑝 = 𝐼/4𝜔2 shows the free electron ponderomotive energy in the driving laser field with the frequency of 𝜔 and intensity of 𝐼. Initial experimental studies on HHG and attosecond pulse generation which performed in 2001 [8,13] were based on theoretical researches previously investigated [6,12]. Stronger and shorter pulses can be extracted from harmonic spectra which have the broader plateaus. Hence, the harmonics spectra that have broader plateaus (larger cutoff order) are more attractive. Therefore, different approaches have been used to broaden the plateau of the harmonics intensity spectrum, such as applying some chirp effects to the field of the excitation laser pulse [14,15], creating an inhomogeneity in the spatial distribution of the incoming laser field, which are performed using metal nano-structures to employ the surface plasmons [16-18], or by using multi-color driving pulses [16,19-21], and Etc. Various systems from very simple gas tsrgets such as hydrogen atom, H [20], hydrogen molecular ion, 𝐻2+ [22,23], or helium ion, 𝐻𝑒 + [11,24-26], to very complex molecules such as bulk crystals [27], graphene flakes

[28], and dielectric or metallic surfaces [29,30], theoretically or experimentally have been considered for investigation of the HHG and attosecond metrology. In this paper, we theoretically study the HHG and the produced attosecond pulse behavior, where an 𝐻𝑒 + ion is subjected to some two-color laser radiations, such that these pulses are generated by superposition of two concentric laser pulses at different wavelengths. The wavelength of one of these pulses is fixed at 800 nm, but the wavelength of the second pulse changes. Therefore, the spectrum of harmonics intensities and cutoff harmonics, as well as pulse train output characteristics, were investigated in terms of the spectral difference of these two pulses. Over-the-barrier ionization (OBI) threshold, 𝐼𝑏 , for 𝐻𝑒 + , which is about 8.76 𝑃𝑊/𝑐𝑚2 (corresponding to 𝐸𝑏 = 0.5 𝑎. 𝑢.), is also considered in this research and the intensity of the driving pulse is set to be lower than 𝐼𝑏 . OBI intensity threshold for atoms can be calculated from 𝐼𝑏 = 4 × 109

4 𝐼𝑃

𝒵2

[31] where 𝒵 is the charge state for the atomic target and 𝐼𝑃 , in the units of eV, represents the ionization

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potential.

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As you know, the Keldysh parameter 𝛾 = 𝜔√2𝐼𝑃 /𝐸0 determines the ionization process [32]. Accordingly, when the value of this parameter is less than the unity (𝛾 ≪ 1), the tunneling ionization, and if this parameter is greater than the unity (𝛾 ≫ 1), the multi-photon ionization is the dominant process. In the present study, peak intensity of the driving femtosecond laser pulse is chosen so that the tunneling ionization is the dominant ionization process.

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2. Theoretical method

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Time dependent Schrödinger equation (TDSE) and its numerical solution in investigating the HHG is one of the most efficient methods that can be used. On the other hand, because of the spherical symmetry of the helium ion, the Schrödinger equation can be solved in one dimension, which additional to simplicity, the calculated results is very close to the results obtained from the three-dimensional solution of this equation. In this paper, we are not looking for a method to isolate the output attosecond pulses, and only examine the effect of the wavelength difference of the combining pulses.

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In this study, two-color driving laser fields are created theoretically via superposition of a fundamental field with a constant oscillation frequency of 𝜔0 = 0.057 𝑎. 𝑢. (corresponding to a wavelength of 800 nm) with a variable frequency controlling field, which its frequency varies from 𝛿𝜔 = 1.5𝜔0 to −0.5𝜔0 (respectively from 320 nm in UV-A to 1600 nm in SWIR ranges). Both fields set to be linearly polarized in the same direction. The final electric field of the pulse can be written as 𝐸(𝑡) = 𝐸0 𝑓(𝑡) cos(𝜔0 𝑡) +

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2

1 2

𝐸0 𝑓(𝑡) cos[(𝜔0 + 𝛿𝜔)𝑡] .

(1)

𝐸0 = 0.2924 𝑎. 𝑢. is the peak amplitude of the laser field which is corresponding to the intensity of 300 TW/cm2. Also 𝑓(𝑡) is the Gaussian shaped envelope which covers 4.5 cycle of the fundamental field and takes ~4.5 𝑓𝑠 at the full width at half maximum (FWHM). The Keldysh parameter for these bichromatic fields (𝛾 < 0.4 𝑎. 𝑢.) shows that the ionization processes are of the tunneling type. The intensity of the electric field is also selected so that not to be within the OBI range. Figure 1 shows a typical bichrimatic pulsed laser field composed of a fundamental laser field and a controlling field with a frequency difference of 𝛿𝜔 = 1.25 𝜔0 relative to the main field.

II

Interaction of the introduced laser field with noble gas atoms can be studied using the 1D-TDSE under the dipole approximation. Thus, the temporal behavior of the electronic waveform can be investigated: 𝜕𝜓(𝓏,𝑡) 𝜕𝑡

= [−

1 𝜕2 2 𝜕𝓏2

+ 𝑉𝐶 (𝓏) + 𝑉𝐿 (𝓏, 𝑡)] 𝜓(𝓏, 𝑡),

(2)

where, 𝑉𝐶 (𝓏) = −2/√(𝓏 − 𝑑)2 + 𝛼 2 denotes the soft-core Coulomb potential. 𝛼 is a softening parameter that must be lower than unity in order to deepen the Coulomb potential beyond the ionization potential of the helium ion, 𝐼𝑃 = −2 𝑎. 𝑢., to contain the electron state. 𝑉𝐿 (𝓏, 𝑡) = 𝓏𝐸(𝓏, 𝑡) shows the time dependent interaction between the external field and the atom. The second-order split-operator method: 𝜓(𝓏, 𝑡 + Δ𝑡) ≈

̂ 𝛥𝑡

̂ 𝛥𝑡

𝑒 −𝑖𝑇𝓏 2 𝑒 −𝑖𝑈(𝓏,𝑡+Δ𝑡/2)Δ𝑡 𝑒 −𝑖𝑇𝓏 2 𝜓(𝓏, 𝑡),

(3)

𝑁0 (𝑡) = |⟨𝜓0 (𝓏)|𝜓(𝓏, 𝑡)⟩|2 ,

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is used to solve the 1D-TDSE in Eq. (2). In the above equation, 𝑈(𝑧, 𝑡) = 𝑉𝐶 (𝓏) + 𝑉𝐿 (𝓏, 𝑡). In order to avoid spurious unwanted reflections from the spatial calculation walls, the electronic wave function at any temporal step is multiplied in a mask function of the form cos1/8 [33]. By using the electron wave function, other physical parameters can be accessed. The ground state, and the total populations, respectively can be calculated as:

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(4)

and 𝑁(𝑡) = |⟨𝜓(𝓏, 𝑡)|𝜓(𝓏, 𝑡)⟩|2 .

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(5)

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Also we can introduce the ionization probability, P(t), via 𝑃(𝑡) = 1 − 𝑁(𝑡). Ehrenfest's theorem introduces the dipole momentum d(t) and dipole acceleration a(t) which are induced by the incomming laser field [16], as follows: 𝑑(𝑡) = ⟨𝜓(𝓏, 𝑡)|𝓏|𝜓(𝓏, 𝑡)⟩,

(6)

and

𝜕

𝜕𝓏

𝑈(𝓏, 𝑡)|𝜓(𝓏, 𝑡)⟩.

(7)

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= − ⟨𝜓(𝓏, 𝑡)|

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𝑑 2 〈𝓏〉 = −⟨𝜓(𝓏, 𝑡)|[𝐻(𝑡), [𝐻(𝑡), 𝓏 ]]|𝜓(𝓏, 𝑡)⟩ 𝑑𝑡 2

𝑎(𝑡) =

Calculating the Fourier transform of the dipole acceleration expectation value leads to the intensity spectrum of high order harmonics: 𝑇/2

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1

2

𝒮(𝜔) = | ∫−𝑇/2 𝑎(𝑡)𝐻(𝑡)𝑒 −𝑖𝜔𝑡 𝑑𝑡| , 𝑇

(8a)

where,

1

𝑡

2

𝑇

𝐻(𝑡) = (1 − cos 2𝜋 ),

(8b)

Is the Hanning filter [34]. Also, by using the dipole acceleration, attosecond pulse trains can be calculated from superposing several harmonics: 2

ℐ(𝑡) = |∑𝑞 𝑎𝑞 𝑒 −𝑖𝑞𝜔0 𝑡 | ,

(9) III

in which, 𝑡

𝑎𝑞 (𝑡) = ∫0 𝑎(𝑡′)𝑒 −𝑖𝑞𝜔0 𝑡′ 𝑑𝑡′.

(10)

High-order harmonics’ time-frequency distribution, can be studied by the use of the Morlet-wavelet transform W[a]=w(ω,t) of the dipole acceleration: 𝒲(𝜔, 𝑡) =

𝑡 − √𝜎√𝜋 ∫0 𝑎(𝑡′)𝑒 𝜔

2 𝜔2 (𝑡−𝑡′ ) 2 2𝜎

′

𝑒 𝑖𝜔(𝑡−𝑡 ) 𝑑𝑡′

(11)

where, σ is a constant which is set to achieve the best balance between the resolutions in frequency and time domains.

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Here, in the numerical calculations, time and space steps set to be 0.05 a.u. As it is seen in Fig. 1, total duration of the interaction is 496 a.u. (or 12.0 fs) time interval, which is corresponding to 4.5 optical cycle oscillation for the incomming laser field. Also we set the spatial width of the calculation window to be 300 a.u. (or 16 nm). It is further assumed that the helium ion is at its ground state, |1𝑠⟩, before the driving laser turns on (𝑡 = −∞).

3. Numerical Results

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Using the aforementioned calculation tools we can study the behavior of the HHG and attosecond pulses with respect to the difference between the frequencies of the main and probe pulses. In this study the calculated data can be compared with the situation of the single color pulse excitation (when 𝛿𝜔 = 0). Figure 2 shows the results of solving the TDSE, which indicates the effect of the linearly polarized fields on the absolute waveform of the electron. In this figure time behavior of the electron wavefunction affected by a two-color driving pulse (𝛿𝜔 = 1.25 𝜔0 ) is compared with the case of the single color excitation pulse. Furthermore, electronic population depletion rate of the ground state, i.e. N0, and the total population, N, are depicted in figure 3 for some color differences (𝛿𝜔) for the combining pulses. HHG spectra corresponding to some 𝛿𝜔 are depicted in figure 4.

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HHG spectrum for the absence of the probe pulse (𝛿𝜔 = 0) is also plotted in this figure for the comparison. In this figure, the main cutoff frequencies are specified for different colors, but no particular order can be found. Therefore, we study the high-order harmonics with greater precision. Figure 5 shows the distribution of high-order harmonics' intensity spectrum with respect to 𝛿𝜔. From this figure, easily we can extract the behavior of the main and the second cutoff frequencies in terms of 𝛿𝜔 which is illustrated in figure 6. At the first glance, it can be expected that the attosecond pulse intensities obeys the behavior of the final cutoff frequency, but we will see that it is not correct in this respect. From this figure we can expect that the generated attosecond pulses regarding to two color differences around 𝛿𝜔 = 0 are the shorter and most powerful pulses. Figure 7 shows the maximum ionization probability which is calculated at the end of the excitation process. As shown in Fig. 2, parts of the electron wave function that are removed from the nucleus and which do not return to it in the next fluctuations of the electric field, will eventually be removed from the computational window. Therefore, to calculate the ground state and total population, and also to calculate the ionization probability, it is enough to study the narrow range near the nucleus. This point is taken in Fig. IV

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3, and also here, to calculate the total ionization probability. Although the total variation of the ionization probability (Δ𝑃) is about 15%, but as shown in Fig. 7, the most ionization occurs when a single-color monochrome pulse with a fundamental frequency (𝜔0 ) excites the helium ion. Time-frequency distribution which can be obtained via Eq. (11) is depicted in figure 8 for some different values of 𝛿𝜔. As it can be seen, long trajectories for the main peak of the distribution, have been significantly weakened, and it gives us hope to get isolated pulses. Figure 9 shows the attosecond pulse generation related to two different values of 𝛿𝜔. Although we did not seek to reach the isolated pulses, with more than 85% of intensity difference between the main pulse and the other pulses of the attosecond pulse train, what are seen in this figure are very similar to isolated pulses. Now we can find the behavior of the attosecond pulse intensity and duration versus the frequency difference (𝛿𝜔). Figure 10 represents the attosecond pulse intensity and duration versus 𝛿𝜔. These pulses are generated via the superposing some harmonics from 50th to the first or second cutoff points in the HHG spectra. In this figure, we can see that shortest and most powerful extracted attosecond pulses are related to values of frequency difference around 𝛿𝜔 = 0. On the other hand, the widest and weakest pulses are related to 𝛿𝜔~0.8.

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4. Conclusion

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Generated high order harmonics and the extracted attosecond pulses during the exposure an He+ to some two color laser fields with different frequency difference between the elements (𝛿𝜔) have been studied in this paper. As the intensity of the driving pulses is a crucial parameter on the behavior of the cutoff and the other parameters of the extracted harmonics and attosecond pulses, intensity of the incoming femtosecond pulses in this study are kept constant. It is seen that the final cutoff frequency decays exponentially with 𝛿𝜔. Long trajectories of the electron in the time-frequency distributions of high order harmonics were almost eliminated. Thus, although we did not attend to achieve the single isolated attosecond pulses, but, the attosecond pulse trains which are generated here, almost show the single isolated pulses specially for 𝛿𝜔 = 1.25 𝜔0 . Despite the cutoff frequency is exponentially reduced by 𝛿𝜔, the pattern of intensity of the attosecond pulses behaves like the first cutoff frequency, which indicates that the harmonics before the first cutoff frequency have a greater effect on the intensity of the output attosecond pulse intensities. Thus, in this study we concluded that the most powerful and shortest attosecond pulses are generated in the situation for which, targets of He+ are exposed to two color laser fields with 𝛿𝜔 about zero. Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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[1] [2] [3] [4] [5] [6] [7] [8]

VI

E0 (  0 )

Electric Field, E (a.u.)

0.3

E1 ( 0+ )

0.2

E=E0+E1

0.1 0 -0.1 -0.2 -0.3

 = 1.25  0

-6

-4

-2

0

Time (fs)

2

4

6

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(a)

-100

 = 0 0

100

(b)

-100

 = 1.25 0

-p

P = ||2 (arb.u.)

0

100 0.5

1

1.5

2

2.5

Time, t (OC)

3

3.5

4

4.5

0

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0

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Electronic Coordinate, z (a.u.)

Figure 1. A typical two-color electric field. The oscillation frequency of the fundamental element is 𝜔0 and another is 2.25𝜔0 , i.e., the frequency difference in this example is 𝛿𝜔 = 1.25𝜔0 .

(a)

.8

 = 0  = 0.5  0  =  0

.6

 = 1.5  0

ur

.4 .2 1

Total Population, N (arb. u.)

na

1

.8 .6 .4 .2

(b)

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Ground State Population N0 (arb. u.)

Figure 2. Probability density of the electron presence around the nucleus in an He+ ion under the excitation of (a) a monochromatic (the case of 𝛿𝜔 = 0), and (b) a typical two-color (the case of 𝛿𝜔 = 1.25𝜔0 ) driving laser field.

0 -6

-4

-2

0 Time, t (fs)

 = 0  = 0.5 

0

 =  0  =1.5  0

2

4

6

Figure 3. (a) Population of the ground state, N0 and (b) the total population, N, calculated versus time for some values of 𝛿𝜔.

VII

0  = -0.5 

Intensity (arb.u.)

 = 

-10

=0 =0.5 

0

-5

 = 1.5 

0

0

0

=1.00 

0

 = 0.5 0

=1.50 0

 = 0

-15

=-0.50 0

-20 -25 -30 -35 -40 0

100

200

300

Frequencies, / 0

400

500

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Figure 4. High order harmonic intensity spectra obtained via the exposure of an He+ to some two color driving femtosecond intense pulses related to different values of 𝛿𝜔. First cutoff frequency for each spectrum is indicated.

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Figure 5. High order harmonics intensity distribution with respect to 𝛿𝜔.

Figure 6. First and second cutoff frequencies of the HHG spectra vs. 𝛿𝜔.

VIII

(a)

400  = -0.5  0

(b)

(f)

 = 0

 = 0.5 0

200

(c)

(e)

(d)

400  =  0

 = 1.25 0

300

100 -4

-2

0

2

lP

200

0 -6

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100

re

Frequency, / 0

300

4

Time-Frequency Distribution Function (arb. u.)

500

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Figure 7. Maximum ionization probability of an He+ after the excitation by two color driving pulses with different values of the color differences (𝛿𝜔).

-4

-2

0

2

 = 1.5 0

4

-4

-2

0

2

4

500 400 300 200 100 0

6

Time, t (fs)

Jo

Intensity, I (arb. u.)

200 150

250

(b)

200 150

60

6.8 as

80

Intensity, I (arb. u.)

(a)

250  = 0

80

300

Intensity, I (arb. u.)

ur

300

na

Figure 8. Time-frequency distribution of the harmonics intensity for some different values of 𝛿𝜔 from (a) 𝛿𝜔 = −0.5 𝜔0 to (e) 𝛿𝜔 = 1.5 𝜔0 .

 = 1.25  0

100 50 0 -1.6

-1.4

40

-1.2

60 40

20 as

20 0

-0.8

-0.6

Time, t (fs)

Time, t (fs)

-0.4

100

20 50 0 -6

-4

-2

0

Time, t (fs)

2

4

6

IX

0 -6

-4

-2

0

Time, t (fs)

2

4

6

lP

re

-p

ro of

Figure 9. Attosecond pulse trains generated via exciting a helium ion by (a) a single color (𝛿𝜔 = 0), and (b) a two color (𝛿𝜔 = 1.25𝜔0 ) femtosecond driving pulse. The main pulse in these trains are dominantly greater than the others so that we can consider them as isolated attosecond pulses.

Jo

ur

na

Figure 10. Attosecond pulse intensity (a) and duration (b) which are generated via exposing helium ions to some bichromatic driving pulses. These values are illustrated versus the frequency difference between the color elements of the pulse (𝛿𝜔). Attosecond pulses are generated by superposing the harmonics from 50𝜔0 to the first (blue squares) and second (brawn circles) cutoff frequencies.

X