Few-cycle laser pulses generation with frequency tuning in a molecular gas-filled hollow-core fiber

Zhiyuan Huang; Ding Wang; Yuxin Leng; Ye Dai

doi:10.1364/OE.23.017711

1. Introduction

The few-cycle laser pulses are powerful tools for the investigation of the ultrafast dynamics process in various fields including high-field laser science, particularly the high harmonic generation (HHG) [1] and attosecond pulse generation [2,3 ], chemical reaction dynamics [4,5 ], and time-resolved measurements in atomic and molecular physics [6]. Since the advent of chirped pulse amplification (CPA) technique [7], the Ti:sapphire lasers have become the widely used sources of stable, energetic pulses, while the pulse duration is usually limited to more than 20 fs due to the finite bandwidth of the gain medium. Most commonly, the energetic few-cycle pulses can be obtained by using noncollinear optical parametric amplifiers [8] or hollow-core fiber (HCF) filled with noble gas [9,10 ] or filamentation in noble gas [11–13 ] or the divided-pulse nonlinear compression based on HCF [14,15 ]. However, the central wavelength tuning of generated few-cycle pulses is limited to a narrow range around the fundamental wavelength of 800 nm. An alternative approach to generate few-cycle pulses through molecular phase modulation (MPM) has been demonstrated [16–19 ]. This mechanism is based on rotational effect in Raman-active gases. For the molecular rotation, a pump-probe configuration is used where an intense pump pulse is first coupled into the HCF to excite molecules and then a relatively weak time-delayed probe pulse obtains a molecular phase modulation due to refractive index variation induced by the rotational effect [20–22 ]. With this mechanism, the central wavelength of few-cycle pulses can be effectively tuned, which is very promising as a tunable source.

In this paper, we numerically study the propagation dynamics of an ultrashort laser pulse coupling into a HCF filled with prealigned nitrogen gas. Based on the MPM effect, we obtain the few-cycle pulses with frequency tuning by controlling the time delay between the pump and probe pulses. This paper is organized as follows. Section 2 presents the detailed model of pulse propagation with molecular gas-filled HCF. Section 3 specially investigates the influence of MPM effect created by molecular alignment revivals on probe pulse. Section 4 discusses the potential and limitations of this method. The paper ends with a conclusion in section 5.

2. Theoretical model

The pulse propagation equation in HCF filled with molecular gases can be written as [14,23 ]

\begin{array}{l} (\partial_{z} - i \hat{D}) E = i \hat{T} [\frac{ω_{0}}{c} n_{2} h {| E |}^{2} E] - \frac{σ}{2} d E - \frac{1}{2} \frac{f}{{| E |}^{2}} E \\ - \frac{i q_{e}^{2}}{2 c ω_{0} m_{e} ε_{0}} {\hat{T}}^{- 1} [d E] + i \frac{ω_{0}}{c} \hat{T} [δ n_{r o t} E], \end{array}

where the pulse field envelope

E

is normalized to the intensity

I

. Linear operator

\hat{D} = i α^{(0)} - α^{(1)} \partial_{τ} + \sum_{m = 2}^{\infty} (i \partial_{τ})^{m} [β^{(m)} + i α^{(m)}] / m!

shows the waveguide mode attenuation

α

and dispersion

β

where

α^{(m)} = d^{m} α / d ω^{m} |_{ω = ω_{0}}

and

β^{(m)} = d^{m} β / d ω^{m} |_{ω = ω_{0}}

, respectively. In the frequency domain, the expression of

\hat{D}

is much simpler, and it consists of all orders of dispersion. The retard frame moving at the group velocity of the fundamental mode

τ = t - β^{(1)} z

is introduced. The constants

c

,

σ

,

q_{e}

,

m_{e}

,

ε_{0}

, and

n_{2}

are light speed in vacuum, impact ionization cross section, electron charge and mass, vacuum permittivity, instantaneous Kerr nonlinear refractive index, respectively.

δ n_{r o t} = \int_{0}^{a} Δ n V^{2} \cdot r d r / \int_{0}^{a} V^{2} \cdot r d r

describes the effect of change of refractive index

Δ n (t, r, z)

created by molecular rotational effect, which can either be obtained by simple “weak field” model or “strong field” model as

Δ n (t, r, z) = 2 π (ρ_{0} Δ α / n_{0}) [〈 〈 \cos^{2} θ 〉 〉 (t, r, z) - 1 / 3]

where

ρ_{0}

is the neutral density of gas [19,24 ];

Δ α

is the polarizability difference between parallel to and perpendicular to the molecular axis;

n_{0}

is the linear refractive index; the molecular alignment

〈 〈 \cos^{2} θ 〉 〉

is averaged over the Boltzmann distribution. Here, for pump pulse, the three-step simplified model is used; for the probe pulse,

Δ n

is calculated by strong field model. The operator

\hat{T} = 1 + (i / ω_{0}) \partial_{τ}

represents self-steepening effects. The coefficients

h = \int_{0}^{a} V^{4} \cdot r d r / \int_{0}^{a} V^{2} \cdot r d r

,

d = \int_{0}^{a} ρ V^{2} \cdot r d r / \int_{0}^{a} V^{2} \cdot r d r

,

f = \int_{0}^{a} η W (ρ_{0} - ρ) U_{p} \cdot r d r / \int_{0}^{a} V^{2} \cdot r d r

describe mode coupling involving instantaneous Kerr, plasma and ionization effects encoded in the pulse transverse mode

V = J_{0} (u r / a)

, where

r

is transverse coordinate;

a

is fiber inner radius;

ρ

is electron density;

W

is the ionization;

J_{0} (x)

is the 0th order Bessel function of the first kind;

u

is the first zero point of

J_{0} (x)

. Assuming electrons born at rest, the electron density can be expressed as

\frac{\partial ρ}{\partial t} = η W (I) (ρ_{0} - ρ) + \frac{σ}{U_{p}} ρ I,

where the ionization rate

W (I)

is calculated according to Perelomov, Popov and Terent’ev (PPT)’s theory presented in [25];

η

is a modulation factor;

U_{p}

is the ionization potential. The coefficient

η = 1 + (1.5 a_{2} - 3.75 a_{4}) (〈 〈 \cos^{2} θ 〉 〉 - 1 / 3) + 4.375 a_{4} (〈 〈 \cos^{4} θ 〉 〉 - 1 / 5)

describes the dependence of ionization rate on molecular alignment, where

a_{2} = 0.39

and

a_{4} = - 0.21

[26]. The pulse envelope

E (t, z = 0)

at the input of the HCF is expressed as

E (t, z = 0) = f_{c o u p} \sqrt{\frac{P_{p e a k}}{S_{a r e a}}} U (t, z = 0),

where

P_{p e a k} = 0.94 E_{i n} / τ

is input peak power,

E_{i n}

is the initial pulse energy,

τ

is the pulse full width at half maximum (FWHM);

S_{a r e a} = 0.5 π w_{0}^{2}

is the fundamental mode area,

w_{0}

is the 1/e² radius of the beam intensity Gaussian profile at the focus just before the input of HCF;

U (t, z = 0) = \exp [- (2 \ln 2) t^{2} / τ^{2}]

is the normalized envelope;

f_{c o u p}

is the fraction of the pulse that is coupled to the fundamental mode. The optimal coupling, i.e., most of the energy is coupled to fundamental mode, is met when

w_{0} = 0.65 a

, which corresponds to

f_{c o u p} ≅ 0.88

[27].

It should be noted that the propagation dynamics of both the pump and probe pulses are obtained by integration of Eq. (1) using the 4th order Runge-Kutta method with self-adaptive steps. The detailed steps to solve the pump-probe model are given in [28].

3. Results and analysis

In this work, we use the HCF with the length of 1 m and the inner diameter of 250 µm, and consider molecular nitrogen at gas pressure of 1 bar, together with 1800-nm pump pulse and 800-nm, 40-fs probe pulse.

Figures 1(a) and 1(b) show the first revival of molecular alignment $〈〈 \cos^{2} θ 〉〉 (t)$ created by 1800-nm, 150-fs pump pulse of intensity 3 × 10¹³ W/cm², the corresponding change of refractive index $Δ n (t)$ (yellow solid curves) and its derivative with respect to time $\partial Δ n (t) / \partial t$ (green solid curves) are presented in Figs. 1(c) and 1(d). The experimental and theoretical results have demonstrated that an ultrashort laser pulse shows a blueshift or redshift as its temporal peak is tuned to the falling (between time A and C) or rising (between time D and F) edge of the molecular alignment [20,28 ]. In fact, the frequency shift of laser pulse induced by MPM effect meets such a relationship $δ ω (t) ~ - \partial Δ n (t) / \partial t$ , thus the pulse obtains the maximum blueshift and redshift at time B and E, corresponding to the negative (time B₂) and positive (time E₂) maximum value of $\partial Δ n (t) / \partial t$ .

Fig. 1 (a) and (b) The time-dependent molecular alignment (blue solid lines). (c) and (d) The change of refractive index (yellow solid lines) and its derivative with respect to time (green solid lines). The red dots A-F correspond to A_k-F_k (k = 1, 2).

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The abilities of spectral broadening and wavelength tuning depend on the value of $\partial Δ n (t) / \partial t$ which relates to the pump pulse duration and intensity [29]. Figures 2(a) and 2(b) plot the positive (red square curves) and negative (blue circle curves) maximum value of $\partial Δ n (t) / \partial t$ with various pulse duration. For pump intensity 3 × 10¹³ W/cm², as shown in Fig. 2(a), the maximum value of ${(\partial Δ n / \partial t)}_{\max}$ appears at $τ = 120$ fs. While for intensity 1 × 10¹⁴ W/cm² in Fig. 2(b), the optimal pulse duration is $τ = 90$ fs, and its maximum value is more than double compared to Fig. 2(a).

Fig. 2 (a) and (b) The positive (red square lines) and negative (blue circle lines) maximum value of $\partial Δ n (t) / \partial t$ versus pulse duration with intensities of 3 × 10¹³ W/cm² and 1 × 10¹⁴ W/cm² at 1800 nm, respectively.

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Here we use the root-mean-square (RMS) to describe accurately the spectral broadening and the pulse duration. The spectral broadening factor after HCF can be written as $δ = Δ ω_{r m s} / Δ ω_{0}$ , where $Δ ω_{r m s}$ and $Δ ω_{0}$ are the RMS spectral widths of the output and input pulses, and they are defied as [30,31 ]

{\begin{cases} (^{Δ ω) 2} = 〈 (^{ω - ω_{0}) 2} 〉 - 〈^{(ω - ω_{0}) 〉 2} \\ 〈 (^{ω - ω_{0}) n} 〉 = \frac{\int_{- \infty}^{\infty} {(ω - ω_{0})}^{n} I (ω) d ω}{\int_{- \infty}^{\infty} I (ω) d ω} \end{cases},

where

I (ω)

is the spectral intensity,

ω_{0}

is calculated by

ω_{0} = \int_{- \infty}^{\infty} ω I (ω) d ω / \int_{- \infty}^{\infty} I (ω) d ω

. Similarly, the RMS pulse duration is expressed as [31,32 ]

{\begin{cases} (^{Δ τ) 2} = 〈 t^{2} 〉 - 〈^{t 〉 2} \\ 〈 t^{n} 〉 = \frac{\int_{- \infty}^{\infty} t^{n} I (z, t) d t}{\int_{- \infty}^{\infty} I (z, t) d t} \end{cases},

where

I (z, t)

is the temporal intensity.

In Figs. 3(a) and 3(c) , we calculate the wavelength shift (blue circle curves) and spectral broadening factor (green square curves) of probe pulse with initial energy of 50 µJ depending on the time from delay A to C and delay D to F induced by the pump pulse with input energy of 1.3 mJ (intensity 1 × 10¹⁴ W/cm²) and duration of 90 fs, respectively. As shown in Fig. 3(a), the wavelength shift of probe pulse is first enhanced and then suppressed with increasing the time delay between pump and probe pulses, the maximum wavelength tuning is ~200 nm. Besides, the spectral broadening displays the opposite variation compared to the wavelength shift. In Fig. 3(c), the wavelength shift appears the similar behavior and the maximum shift is ~250 nm. Figures 3(b) and 3(d) show the probe pulse RMS duration (yellow circle curves) after compression and the group delay dispersion (GDD) compensation (red square curves) used to compress probe pulse. In Fig. 3(b), seen from the curves of GDD compensation, the prob pulse obtains a positive or negative chirp. Although the spectrum of probe pulse is broadened both from delay A to B and from delay B to C, it obtains the opposite frequency chirp in this two different periods. As shown in Fig. 3(d), we can also observe the probe pulse experiences the opposite chirp, but the evolution of GDD compensation reflects the opposite result with respect to time delay compared to the former case.

Fig. 3 (a) and (c) The wavelength shift (blue circle lines) and broadening factor (green square lines) of probe pulses. (b) and (d) The pulse duration (yellow circle lines) and the GDD compensation (red square lines) of compressed probe pulses.

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Figures 4(a) and 4(d) show the temporal profiles of probe pulse after GDD compensation for different time delays 4.14 ps (black solid curves), 4.27 ps (green dotted curves), 8.34 ps (blue solid curves), 8.47 ps (red dotted curves) and the compressed pulse duration 6.5 fs, 8.0 fs, 8.5 fs, 7.0 fs, together with the corresponding wavelength tuning 84 nm, 102 nm, 50 nm, 56 nm, respectively. The spectral intensities (green solid curves), phase before (blue solid curves) and after (red dotted curves) compensation representing the four delays are given in Figs. 4(b)-4(f). In Figs. 4(c) and 4(e), we can observe the probe pulse after propagating through HCF exhibits a smooth spectrum without disruptive modulation. After optimal positive or negative GDD compensation, the probe pulse is compressed < 9 fs.

Fig. 4 (a) and (d) The temporal profiles of compressed probe pulses for different time delays 4.14 ps (black solid lines), 4.27 ps (greed dotted lines), 8.34 ps (blue solid lines), and 8.47 ps (red dotted lines). (b)-(c) and (e)-(f) The corresponding spectral intensities (green solid lines), phase before (blue solid lines) and after (red dotted lines) compensation with various time delays, respectively.

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In order to compress the probe pulse, we use different glasses to compensate the negative chirp obtained by adjusting the time delay to proper values. Table 1 presents the ratio of third-order dispersion (TOD) to GDD of different glass types at the central wavelength 698 nm and 850 nm, corresponding to the delays 4.27 ps and 8.34, respectively. Considering the probe pulse gains almost linear chirp in HCF, we choose the minimum ratio. Thus FK51A is used for delay 4.27 ps, and LASF9 is picked for delay 8.34 ps. As shown in Figs. 5(a) and 5(c) , the compressed probe pulses are 7.8 fs (blue solid curves) and 9.2 fs (red solid curves), and the thickness of the glasses are 1.83 mm (FK51A, 74.8 fs² GDD, 37.3 fs³ TOD) and 0.88 mm (LASF9, 132.4 fs² GDD, 88.2 fs³ TOD), respectively. Although the TOD exists in the compressed pulses, the probe pulses (blue or red solid curves) after compensation in Figs. 5(a) and 5(c) exhibit clean profiles.

Table 1. The ratio of TOD to GDD for different glass types at central wavelength 698 nm and 850 nm.

View Table

Fig. 5 (a) and (c) The temporal intensities of compressed probe pulses. (b) and (d) The spectral intensities (green solid lines) and the phase (blue and red solid lines) of probe pulses after compensation. (a)-(b) and (c)-(d) correspond to various time delays 4.27 ps and 8.34 ps, respectively.

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4. Discussion

Using the pump-probe technique, we can obtain the pulse with wavelength tuning and spectrum broadening by controlling the time delay between pump and probe pulses. It should be noted that the spectral broadening survives in the process of wavelength tuning, and it exhibits the opposite behavior relative to wavelength shift, which means the pulse experiences the maximum spectral broadening at time A (or C, D, F) and hardly broadened at time B (or E) in Fig. 1. Based on the MPM effect, it can impart a positive or negative chirp on the probe pulse. In particular, when the time delay is tuned to the positive interval, the chirp is created by SPM and MPM effects, while for negative interval, the chirp induced by MPM effect is partly balanced by the positive chirp from SPM effect. As shown in Fig. 6(b) , the probe pulse possessing the positive chirp (4.14 ps, blue solid curves) obtains a broader spectrum compared to the case with the negative chirp (4.27 ps, green solid curves). After switching off the SPM effect in the simulations, we observe the spectral profiles (black and red solid curves) exhibit the similar width. Moreover, in Fig. 6(a), we can see the compressed probe pulse (blue solid curves) possesses a pedestal located in the pulse leading edge, which may be due to the SPM effect. Since SPM is a third-order nonlinear process, especially when the initial energy of probe pulse increased large enough, the phenomenons are more obvious. Note that in Fig. 7 , for input energy of 0.2 mJ, the simulation results show that the spectrum undergoes a larger modulation due to the SPM effect, while the plasma-induced phase change or molecular ionization loss effects have little influences with such pulse energy.

Fig. 6 (a) and (b) The temporal and spectral profiles of compressed probe pulses for different time delays with (blue and green solid lines) or without (black and red solid lines) considering SPM effect, respectively.

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Fig. 7 (a) and (c) The temporal intensities of compressed probe pulses with input energy 0.2 mJ for different time delays 4.14 ps and 4.27 ps, respectively, together with no plasma-induced phase change effect (black solid lines), no SPM effect (green solid lines), no molecular ionization loss (red solid lines). (b) and (d) The corresponding spectral profiles.

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5. Conclusion

In conclusion, we have numerically studied the pulse compression method for generation of few-cycle laser pulses by using ultrafast molecular phase modulation in HCF filled with preligned nitrogen gas. We find that the MPM effect can impart a positive or negative chirp on probe pulse and allow for tens or even hundreds of nanometers of the compressed pulse by controlling the time delay between pump and probe pulses. With this approach, we obtain the frequency-tunable few-cycle pulses with smooth spectra, which is very promising as tunable source.

Acknowledgments

This work was partly supported by the National Natural Science Foundation of China (Grant Nos. 11204328, 61221064, 61078037, 11127901, 11134010, 61205208), the National Basic Research Program of China (Grant No. 2011CB808101), the Natural Science Foundation of Shanghai, China (Grant No. 13ZR1414800).

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Glass types	TOD/GDD (698 nm)	TOD/GDD (850 nm)
BK7	0.5421	0.8457
BAF10	0.5309	0.7035
BAK1	0.5103	0.7163
FK51A	0.4979	0.7327
LASF9	0.5620	0.6658
SF5	0.5900	0.7137
SF10	0.5955	0.6977
SF11	0.6091	0.6998
Fused silica	0.5539	0.9124

Few-cycle laser pulses generation with frequency tuning in a molecular gas-filled hollow-core fiber

Abstract

1. Introduction

2. Theoretical model

3. Results and analysis

4. Discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (5)

Optics Express