Design of intense 1.5-cycle pulses generation at 3.6 &#x00B5;m through a pressure gradient hollow-core fiber

Zhiyuan Huang; Ding Wang; Ye Dai; Yanyan Li; Xiaoyang Guo; Wenkai Li; Yun Chen; Jun Lu; Zhengzheng Liu; Ruirui Zhao; Yuxin Leng

doi:10.1364/OE.24.009280

1. Introduction

Ultrashort laser pulses are widely used tools to investigate the atomic and molecular processes in various fields such as physics, chemistry, biology, and material science [1–3]. In particular, isolated attosecond extreme ultraviolet (XUV) pulses make it possible to probe ultrafast dynamics in the attosecond range [4,5]. To obtain the attosecond pulses, a popular method is to focus the intense carrier-envelope phase (CEP) stabilized few-cycle femtosecond laser pulses onto a noble gas target and thereby generated attosecond XUV pulses via high harmonic generation (HHG) [6]. The scaling law of HHG cut-off energy meets this relation $E_{C O} ~ I λ^{2}$ [7]; thus the few-cycle pulses with high peak intensity and long wavelength can generate higher energy harmonics and shorter attosecond pulses. The hollow-core fiber (HCF) compressor filled with noble gas is a universal method to obtain the high-energy few-cycle femtosecond pulses based on self-phase modulation (SPM) [8,9]. Bohman et al. [10] demonstrated the generation of 5-fs, 5-mJ pulses at 800 nm with a repetition rate of 1 kHz using helium gas-filled HCF. However, the central wavelength of driving sources from a CEP stabilized Ti:sapphire laser is usually limited to a narrow range around 800 nm. Optical parametric amplification (OPA) is a typical and reliable approach for generating broadband tunable femtosecond laser pulses in the near-infrared (NIR) or even longer wavelengths in the mid-infrared (MIR) through difference frequency generation (DFG) technique [11–15]. Li et al. [16] obtained the CEP stabilized 0.7-mJ, 1.5-cycle laser pulses at central wavelength 1.75 μm at 1 kHz repetition rate through a three-stage OPA system. Moreover, in [17], the authors used a large diameter HCF filled with argon gas to compress a ~11-mJ, 35-fs, 1.8-µm OPA source [18] to 5-mJ, 2-cycle (12 fs) pulses with clear spatial profile at 100 Hz repetition rate. Further extensions of optical parametric chirped pulse amplification (OPCPA) technology to the MIR regime [19,20], Andriukaitis et al. [21] obtained the 8-mJ, 83-fs, 3.9-µm pulses with a repetition rate of 20 Hz. Zhao et al. [22] achieved the 13.3-mJ, 111-fs MIR pulses with wavelength tuning range from 3.3 to 3.95 µm at 10 Hz repetition rate. In order to gain more insight for future applications, the high-energy few-cycle MIR pulses with a high repetition rate are needed.

Recently Chen et al. [23] reported a three-stage KTA femtosecond OPA system to achieve tunable range of 2.4~4.0 µm; on the next step, in combination with chirped pulse amplification (CPA) [19] technique for generating a 10-mJ, 2.4~4.0 µm OPCPA source with 1 kHz repetition rate, and then experimentally achieving the high-energy few-cycle MIR pulses using HCF compressor. In nonlinear pulses compression based on HCF, the SPM effect gives rise to spectral broadening, but for an intense laser pulses at long central wavelength, the effects such as self-focusing, ionization, and nonlinear mode coupling restrict pulses quality, especially at the input of HCF. Therefore, the large diameter HCF with a pressure gradient is employed to reduce undesirable nonlinear effects.

In this work, we theoretically study the nonlinear compression of 10-mJ, 62-fs, 3.6-µm pulses from an OPCPA source in a pressure gradient HCF filled with argon gas of 1000-µm diameter. After compression with bulk materials, we obtain the 18.3-fs (~1.5 cycle), 3.6-µm sub-cycle pulses with an energy of several millijoule, which is very promising as a driving source to generate the shorter attosecond pulses. Besides, For a given bandwidth of ultrashort pulses, we calculate the optimal fiber length for different coupling efficiency, which can be used in designing hollow-core fiber compressor.

2. Theoretical model

The pulse propagation equation in noble gas-filled HCF can be described as [24–26]

(\partial_{z} - i \hat{L}) E = i \hat{T} [\frac{ω_{0}}{c} n_{2} h {| E |}^{2} E] - \frac{σ}{2} d E - \frac{1}{2} \frac{f}{I} E - \frac{i q_{e}^{2}}{2 c ω_{0} m_{e} ε_{0}} {\hat{T}}^{- 1} [d E],

where the pulse envelope

E

is normalized to the intensity

I

; 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, and instantaneous Kerr nonlinear refractive index, respectively. The linear operator

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

describes the hollow waveguide mode attenuation

α

and the 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}}

;

ω_{0}

is the central angular frequency of the pulse. The retard frame moving at the group velocity of the fundamental mode

τ = t - β^{(1)} z

is introduced. The operator

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

represents the self-steepening effects.

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

, and

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

are the 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;

W

is the ionization rate;

ρ_{n t}

is the neutral density of gas;

I_{p}

is the ionization potential;

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

ρ

evolves as

\frac{\partial ρ}{\partial t} = W (ρ_{n t} - ρ) + \frac{σ}{I_{p}} ρ I,

where the ionization rate

W

is calculated according to Perelomov-Popov-Terentiev (PPT) theory in [27,28]. The pulse envelope

E

at the input of the HCF is expressed as

E (t, z = 0) = \sqrt{η_{m} P_{0}} E_{0} \exp (i φ)

, where

P_{0} = 0.94 E_{i n} / τ

is initial peak power,

E_{i n}

is the input pulse energy,

τ

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

E_{0}

is normalized to the amplitude,

φ

corresponds to pulse phase;

η_{m}

describes the amount of power coupled to the

H E_{1 m}

waveguide modes, and it can be expressed as [29]

η_{m} = \frac{4}{w^{2}} \frac{{[\int_{0}^{a} r J_{0} (u_{m} r / a) e^{- r^{2} / w^{2}} d r]}^{2}}{\int_{0}^{a} r J_{0}^{2} (u_{m} r / a) d r},

where

w

is the 1/e² radius of the beam intensity at the focus just before the input of HCF,

u_{m}

is the mth zero point of

J_{0} (x)

.

3. Results and analysis

In this work, we employ an argon gas-filled HCF with the inner diameter of 1000 µm, and use a 10-mJ, 62-fs, 3.6-µm OPCPA source as input pulses. Figures 1(a) and 1(b) show the temporal and spectral profiles of initial pulses, together with phase, respectively. It should be noted that the initial pulses duration before compensation is 123 fs, after suitable chirp compensation, the compressed pulses duration is 62 fs. Experimentally, the complete compensation is impractical, thus seen from the spectral phase curves in Fig. 1(b), the pulses possess a small positive chirp, and the Fourier transform limit (FTL) is 54 fs.

Fig. 1 (a) and (b) The temporal and spectral intensities of input pulses with an energy of 10 mJ and the duration of 62 fs at 3.6 µm, respectively. Green dotted lines represent the corresponding phase.

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An important step for nonlinear compression in HCF is that the initial pulses are focused into hollow waveguide. As shown in Fig. 2, we calculate the coupling efficiency of the three modes with increasing ratio of spot size to bore size of 1000 µm. The $H E_{1 m}$ modes show different coupling efficiency, and also present the changes for various ratio $w / a$ . In particular, when $w / a = 0.64$ (red dot A), about 98% of the pulse energy is coupled to the fundamental mode $H E_{11}$ , which is the optimal coupling [30]. In addition, the coupling efficiency of the other two modes is close to zero in that case. The superior coupling is a key point in conducting the HCF experiments. On one hand, enhancing the coupling efficiency of $H E_{11}$ achieves high transmission, on the other hand, decreasing other modes coupling improves the compressed pulses quality. Another important point is the energy efficiency. In hollow waveguide, the attenuation coefficients can be described as [29]

α_{m} = {[\frac{u_{m}}{2 π n_{c o r e} (ω)}]}^{2} \frac{λ^{2}}{a^{3}} \frac{n^{2} (ω) + 1}{\sqrt{n^{2} (ω) - 1}},

where

n_{c o r e} (ω)

and

n (ω)

are the refractive index of fiber core and fiber clad, respectively. In Fig. 3(a), we plot the power attenuation of fundamental mode for the three different inner diameter with increasing wavelength from 2 μm to 5 μm. For 1000-μm diameter HCF, the attenuation coefficients (red solid curves) become very small even at long wavelength 3.6 μm. In addition, hollow waveguide has the smallest loss for the lowest order waveguide mode. As shown in Fig. 3(b), the pulses transmittance (blue solid curves) can reach above 90% for the

H E_{11}

mode through the 1000-μm HCF with the length of 1 m.

Fig. 2 The coupling efficiency $η_{m}$ with respect to $w / a$ , $H E_{1 m} (m = 1, 2, 3)$ modes correspond to blue, green, and yellow solid lines, respectively. The red dot represents the optimal coupling of 0.98 at the ratio of 0.64 for fundamental mode.

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Fig. 3 (a) The power attenuation for different wavelength ranging from 2 μm to 5 μm; blue, green, and red solid lines represent various inner diameter of 250 μm, 500 μm, 1000 μm, respectively. Black dotted lines correspond to the wavelength of 3.6 μm. (b) The transmittance of the three modes $H E_{1 m} (m = 1, 2, 3)$ in 1-m HCF with 1000-μm inner diameter.

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Here we first use a 1-m length, 1000-µm diameter HCF with a pressure gradient between 2 mbar at entrance and 800 mbar at the exit side, and keeping $w / a = 0.64$ as the optimal coupling. Figures 4(a) and 4(b) show the temporal and spectral intensities (blue solid curves) of compressed pulses with an energy of 9.4 mJ and the duration of 17.5 fs (~1.5 cycle) at wavelength 3.6 µm, together with −140 fs² group delay dispersion (GDD) and 673 fs³ third-order dispersion (TOD) compensation. The green dotted curves are the corresponding phase after compression in time and frequency domain. The black dotted curves in Fig. 4(a) correspond to FTL profile, and the duration is 15.4 fs (~1.3 cycle).

Fig. 4 (a) and (b) The temporal and spectral profiles of compressed pulses with an energy of 9.4 mJ and the duration of 17.5 fs at 3.6 µm, respectively. Green dotted lines are the corresponding phase; black dotted lines represent FTL pulses, and its duration is 15.4 fs.

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Owing to anomalous dispersion at long wavelength 3.6 μm, the chirp compensation can be done with ordinary bulk materials such as CaF₂ crystal. Figure 5 shows the compressed pulses duration for different thickness with up to 3rd order (blue circle curves) and to 5th order (red square curves) chirp compensation, respectively. We can see these two curves present the similar behavior at the beginning, but there is an obvious difference after 1.5 mm due to the enhanced high order chirp compensation. The optimal duration (marked by black arrow) after chirp compensation are 19.5 fs (~1.6 cycle) and 18.3 fs (~1.5 cycle), corresponding to the thickness of CaF₂ crystal 0.8 mm and 0.7 mm, respectively. However, the superior duration cannot promise the maximum peak intensity. As shown in Figs. 6(a) and 6(b), the largest peak intensities appear at 1.8 mm and 1.6 mm rather than the optimal duration points, respectively. Figures 7(a)-7(b) and 7(c)-7(d) show the temporal (blue solid curves) and spectral (yellow solid curves) intensities for different thickness 0.7 mm and 1.6 mm with up to 5th order chirp compensation, together with the corresponding phase. In general, the quality of compressed pulses mainly include peak power and contrast on the pulses leading edge. In Fig. 7(c), using longer CaF₂ crystal, on one hand, the enhanced high-order dispersion compared with Fig. 7(a) decrease the energy of front wing. On the other hand, the reduced energy is transferred from the leading edge to the central part of pulses, so we can see it obtains the higher peak intensity. Although its duration is 23.0 fs (more than 18.3 fs in Fig. 7(a)), the compressed pulses present a large peak intensity and clean pulses leading edge. Therefore, the compressed pulses in Fig. 7(c) present the better pulses quality.

Fig. 5 The compressed pulses duration for different thickness of CaF₂ crystal. Blue circle and red square lines, with up to 3rd order and to 5th order chirp compensation, respectively. Green dotted lines indicate the two-cycle duration of 24 fs at 3.6 μm.

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Fig. 6 (a) and (b) The temporal intensities of compressed pulses with up to 3rd order and to 5th order chirp compensation, respectively. The intervals marked by white dashed lines correspond to the two-cycle duration at 3.6 μm.

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Fig. 7 (a) and (c) The temporal profiles (blue solid lines) and the phase (green dotted lines) of compressed pulses with up to 5th order chirp compensation. (b) and (d) The corresponding spectral profiles (yellow solid lines) and phase (red dotted lines). (a)-(b) and (c)-(d) represent different thickness of of CaF₂ crystal 0.7 mm and 1.6 mm, respectively.

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However, the optimal coupling efficiency of 98% is very difficult in HCF experiments, especially for large diameter hollow waveguide. In order to be close to experimental conditions, we use a general coupling such as 60%, and increasing propagation length to ensure the broadening spectrum for supporting the bandwidth of sub-cycle (~1.3 cycle) pulses. In addition, 60% coupling efficiency shows the lower peak intensity that decreases the nonlinear effects to some extent. Here we employ the root-mean-square (RMS) method to describe the spectral broadening. The broadening factor after HCF can be expressed 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 output and input pulses, respectively. They can be written as [31,32]

{\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 obtained by

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

.

Figure 8(a) shows the FTL duration (blue circle curves) and spectral broadening factor (green square curves) of output pulses with increasing fiber length from 1 m to 2 m at coupling efficiency 60%, respectively. We can see these two curves present an intersection when the fiber length is 1.6 m, which means the propagation length should be increased to 1.6 m to meet the same FTL duration and spectral broadening for a low coupling efficiency. The corresponding FTL duration and broadening factor are 15.5 fs and 2.1, very approaching the previous case with 1-m HCF at coupling efficiency 98%. Moreover, in Fig. 8(b), we calculate the optimal fiber length (yellow circle curves) when the FTL duration of output pulses approaches ~1.3 cycle and the output energy (red square curves) for various coupling efficiency ranging from 50% to 95%, respectively. As shown in Fig. 8(b), with increasing coupling efficiency, the optimal fiber length shows a dropping trend, while the output energy after HCF presents a linear rising behavior. The other intersection appears at coupling efficiency 62%, and to ensure the enough bandwidth, we need a 1.57-m propagation length and the obtained pulses energy is 5.8 mJ.

Fig. 8 (a) The FTL duration (blue circle lines) and broadening factor (green square lines) of output pulses with different fiber length for coupling efficiency 0.6. (b) The optimal fiber length (yellow circle lines) when the FTL duration of output pulses reaching ~1.3 cycle and the output pulses energy (red square lines) with respect to coupling efficiency.

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4. Conclusion

In conclusion, we have theoretically studied the nonlinear compression of intense MIR pulses based on a pressure gradient HCF filled with argon gas. For a 10-mJ, 62-fs, 3.6-µm OPCPA source, using a pressure gradient from 2 mbar to 800 mbar and 1000-µm diameter HCF, after compression with 0.7-mm CaF₂ crystal, we obtained the 18.3-fs (~1.5 cycle) approaching the FTL duration of 15.4 fs (~1.3 cycle) pulses with high energy at 3.6 µm, which is a promising driving source for strong field research. This method is reliable for generation of high-energy few-cycle or even sub-cycle pulses at long wavelength. Moreover, according to a given bandwidth of sub-cycle (~1.3 cycle) pulses, we showed the optimal fiber length and the corresponding output energy with different coupling efficiency from 50% to 95%, which is meaningful for the design of the HCF compression scheme.

Acknowledgments

This work was partly supported by the National Natural Science Foundation of China (NSFC) (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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Design of intense 1.5-cycle pulses generation at 3.6 µm through a pressure gradient hollow-core fiber

Abstract

1. Introduction

2. Theoretical model

3. Results and analysis

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Equations (5)

Optics Express