1. Introduction

The propagation of tightly focused femtosecond (fs) laser pulse in transparent media is very complicated process in laser-matter interactions. Recently, it has attracted extensive attentions due to its potential applications: tissue or single cell surgery in biology and medicine [1], micromachining in transparent material [2], terahertz wave emission from fs laser induced air plasma [3] and laser induced breakdown spectroscopy in analyzing the composition of the material [4].

The mechanism of fs pulse propagation with external focusing has been studied by many groups. In condensed media, these works are focused on laser induced optical breakdown: the breakdown threshold, the energy absorption mechanism and the early-time dynamics of laser-induced breakdown [5, 6]. Liu et al. investigated the coexistence of filamentation and breakdown in water using different focusing condition [7]. The appearance of randomly distributed white light beam which is a characteristic signature of filamentation was observed near the breakdown plasma.

When the high power fs laser pulse propagates in air with external focusing, the self-focusing effect should first lead to filamentation [8–13] before the geometric focus. Theberge et al. investigated the filamentation in air by using different focal length lens and laser energy [14]. They found that the plasma density and plasma column diameter were strongly dependent on the external focusing in single filament condition. Kosareva et al. reviewed many experimental and theoretical works on filamentation of fs laser pulse under external focusing [15]. Their conclusion was that the intensity clamping [16–19] will always limit the laser intensity. But all the experiments reviewed in [15] were performed with small numerical aperture (NA < 0.004). Li et al. investigated the propagation of fs laser in air focused by a lens with NA of 0.08 [20]. They observed an oval-like hollow intensity distribution with the laser energy of a few millijoules. Kiran et al. took the microscopic image of multi-filamentation of fs laser pulse for strong geometric focusing with NA up to 0.1 [21]. The intensity and maximum electron density in the laser focus were numerically calculated to be ${10}^{15}W/c{m}^2$ and above $2.6\times {10}^{19}c{m}^{-3}$ for large numerical aperture.

But the role of intensity clamping during filamentation under strong external focusing is still not quite clear and the plasma density in the breakdown area is not directly measured to the best of our knowledge. In this paper, we present experimental investigation on the propagation of fs laser focused by an Off-Axis Parabolic (OAP) mirror with 127 mm focal length in air. Using time-resolved shadowgraphic measurement and interferometric measurement, intensity clamping is confirmed even in tight focusing condition. The maximum electron density higher than $3\times {10}^{19}c{m}^{-3}$ indicates that each air molecule is ionized in the breakdown spot.

2. Experiment setup

The laser system we used in the experiment is a home-made Ti:sapphire chirped pulse amplification laser system, which can provide laser pulses with up to 640 mJ energy. Its central wavelength is around 800 nm. The pulse duration measured by an autocorrelator is 60 fs. For the experiment reported here, the laser pulse energy was set below 50 mJ to induce breakdown in air.

Figure 1 shows the experimental setup with the initial laser beam profile. The diameter of the laser beam is about 30mm. The distribution of the initial beam intensity was not perfect, and a dark area inside the beam could be seen. The laser pulse was split by a beam splitter. The reflected pump pulse was focused by an OAP mirror with 127 mm focal length to induce violent breakdown in air. A 10 mm aperture was used to select a part of the transmitted beam with good uniformity. This part of transmitted beam was frequency-doubled by a Potassium dihydrogen phosphate (KDP) crystal to a 400 nm probe beam and went through the breakdown area perpendicularly to the pump pulse.

 

Fig. 1 The experimental setup for shadowgraphic and interferometric diagnostics with the initial laser beam profile

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The propagation of tightly focused laser pulses is recorded by time-resolved shadowgraphs. The breakdown area is imaged by a × 10 microscope on the surface of a 16 bit charge-coupled device (CCD) camera. Limited by the size of the CCD chip, the imaging area of each fragment is $1.33mm\times 1.33mm$ with a spatial resolution of $1.3\mu m$ . An optical delay line is used to vary the time delay between the probe pulse and pump pulse. The temporal resolution is approximately the pulse duration of the 400 nm probe pulse, which is estimated to be 170 fs by the group velocity mismatch between the 800 nm pump pulse and 400 nm probe pulse in the KDP crystal. In the mean time, the distribution of electron density in the breakdown area is measured by Nomarski interferometer. In the interferometric measurement, the plasma area is magnified 6 times with an f = 150 mm achromatic lens. The time delay between pump and probe pulse is set at the moment when the filament bunch just ended. The interaction time is only a few picoseconds, so the hydrodynamic effect of plasma on the interferograms can be ignored.

3. Experiment results and discussions

Due to the tight focusing condition, the intensity of the pump laser increases very fast when it approaches to the geometric focus. The laser field can ionize air molecule and induce violent breakdown. In the experiment, bright sparking spots could be seen near the geometric focus, and blast sound could be heard. The induced plasmas can change localized refractive index instantaneously. When the probe laser goes through the ionized region, it can be deflected. So the time-resolved shadowgraphs can reflect the distribution of electron density and the intensity of the pump laser.

Figure 2 shows an example of the time-resolved shadowgraphs of air plasma in breakdown area induced by 16.5 mJ laser pulse, which is incident from the left side. It breaks up into many small filaments with a diameter about $10\mu m$ before the geometric focus. The asymmetry of the initial beam profile and the individual hot spots determine the filamentation pattern presented here and make it reproducible from shot to shot. These small filaments initially point to the geometric focus and then chaotic interactions between these filaments take place. Some of the small filaments deflect from their initial directions and their propagation paths become curved. Some of the small filaments fuse with others or split to new filaments. Notice that a short single filament is visible around the geometrical focus at 5.2 ps. It must be the influence of the small pre-pulse. At the end of the filament bunch, many filaments just cross with others and keep their own directions. The convergence of all the small filaments into the geometric focus is not observed, so the laser intensity in the breakdown area must be much less than that in the linear propagation case.

 

Fig. 2 Time-resolved shadowgraphs of filament bunch in air produced by 16.5 mJ laser pulse (The dash line indicates the position of the focal plane)

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Figure 3 shows the panoramic shadowgraphs of filament bunch induced by pump laser pulse with different laser energy. When the laser energy is higher than 10 mJ, we have to take several fragments of filament bunch at different delay time and combine them to one image due to the limitation of the CCD size. The fluctuation of the laser energy is less than 5%, so the different fragments of filament bunch pumped with the same pulse energy can fit well with each other.

 

Fig. 3 Shadowgraphs of filament bunch produced by laser pulse with different energy of 2.4 mJ, 4.5 mJ, 7.5 mJ, 9 mJ, 16.5 mJ, 22 mJ, 38 mJ and 47 mJ (The dash lines indicate the position of the focal plane).

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As shown in Fig. 3, the length and the initial width of the filament bunch increase with the laser energy. The laser intensity where filament bunch starts to develop can be roughly estimated by the laser energy and the initial transverse size of the filament bunch. Figure 4 shows the width of filament bunch (circle symbols) and the estimated laser intensity (square symbols) at the beginning of the filamentation. Calculated by the transverse size of the filament bunch, the laser intensity which initiates the filamentation is in the range of $2.5\times {10}^{14}~5\times {10}^{14}W/c{m}^2$ , and it has a trend to be a constant value when the laser energy increases to tens of millijoule level. In our opinion, it is the intensity clamping mechanism which leads to the similar laser intensity at the beginning of filament bunch. However, the energy in the filamentation area should be less than the total pulse energy because of two reasons. Firstly, the energy loss due to ionization may be noticeable. Then, the low intensity background around the filamentation bunch may exist. Therefore, the clamping intensity may be overestimated by this method.

 

Fig. 4 The initial width of filament bunch (circle symbols) and the estimated laser intensity from the transverse size of the filament bunch (square symbols) and from the electron density (triangle symbols) as a function of laser energy.

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Besides, the electron density and its distribution in the breakdown area are measured by Nomarski interferometer, which consists of a pair of polarizer and analyzer, a Wollaston prism and an achromatic lens. Using a Wollaston prism, the probe pulse is spatially separated to two parts. One of them goes through the air plasma while the other doesn’t. These two parts propagate with $2°$ angle and form the interference fringes on the CCD. The phase shift of the probe pulse can be expressed as [22]:

Typical interferograms and the corresponding electron density distributions are shown in Fig. 5 with laser energy of 16.5 mJ, 22 mJ, 38 mJ and 47 mJ respectively. The spatial resolution of the interferograms determined by the magnification of the imaging system and the pixel size of the CCD camera is $2.2\mu m$ . But the real spatial resolution of electron density is about $30\mu m$ because the Signal to Noise Ratio of the interferograms is not ideal. The interferograms of laser pulse in lower energy are hardly to be analyzed because the width of plasma column is comparable to the interferometric fringes and the interference fringes are

 

Fig. 5 Interferograms and the corresponding electron density (Ne) of air plasma induced by laser pulse with different energy: (a) 16.5 mJ. (b) 22 mJ. (c) 38 mJ. (d) 47 mJ. (The dash lines indicate the position of the focal plane)

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not continuous. So the electron density distributions for lower laser energies are not presented here.

We can see from Fig. 5 that the electron density in the beginning of filament bunch is around $1\times {10}^{19}c{m}^{-3}$ and it continues to increase when the laser pulse approaches to the geometric focus. The maximum electron density of $3~5\times {10}^{19}c{m}^{-3}$ is obtained in the filament bunch which is in good agreement with the numerical simulation in Ref [21]. This maximum electron density is higher than the molecular density of normal air and it is much higher than the electron density of weakly prefocused or freely propagating fs laser reported in previous works [24–27], but the peak electron density doesn’t increase with the laser energy. Besides, the maximum electron density locates at somewhere inside the breakdown area, not on the top of filament bunch. This points out that the filament bunch cannot converge at the geometric focus.

In order to estimate the laser intensity inside the filament bunch, we have numerically investigated the dependence of electron density on the laser intensity using PPT (Perelemov, Popov, and Trentev’s) optical-field-ionization model [28]. In tight focusing case, two-level ionization should be considered. The time evolution of the density for the charged particles in air can be described by the following set of equations [29]:

Figure 6 shows the calculated electron density as a function of the laser intensity. We can see that the second level ionization becomes considerable when the laser intensity exceeds $3\times {10}^{14}W/c{m}^2$ . The electron density at the beginning stage of filament bunch is about $\text{1}{0}^{\text{19}}{\text{cm}}^{-\text{3}}$ , this electron density corresponds to $1.7\times {10}^{14}W/c{m}^2$ clamping intensity as shown in Fig. 3 (triangle symbols), which is less than the value we estimated by the initial width of filament bunch. In the very focus, the electron density vary from $3\times {10}^{19}~5\times {10}^{19}c{m}^{-3}$ , which corresponds to the laser intensity of $3\times {10}^{14}~6\times {10}^{14}W/c{m}^2$ . The increase of laser intensity near the geometric focus is due to the random overlapping of the filaments in our opinion. At last, we also study the influence of collisional ionization (see the appendix). The calculation result shows that the contribution of collisional ionization to the electron density is very small by comparison with optical field ionization during the laser pulse.

 

Fig. 6 The electron density as a function of the laser intensity.

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4. Conclusion

The propagation of tightly focused high power fs laser pulse by an OAP mirror is investigated experimentally. The dynamic of filament bunch is observed using the time-resolved shadowgraphs for different laser energy from 2.4mJ to 47mJ. The electron density distribution of air plasma is measured by Nomarski interferometer. The initial width of the filament bunch increases with laser energy. The laser intensity which initiates the filamentation is clamped at 10¹⁴ W/cm² level,and this clamping intensity is much higher than the weak prefocusing case [16–19]. The maximum electron density higher than 3 × 10¹⁹ cm⁻³ in the breakdown area indicates that each air molecule is ionized. The overlapping of filaments can only increase laser intensity to 2~3 times of the clamping value. Intensity clamping still works under tight external focusing, but the clamping intensity increases with the NA of external focusing [15,21].

Appendix

The PPT optical field ionization model is used in the numerical simulation to calculate the ionization rate of air. For an atom or ion with ionization potential $U_i$ in external optical field with amplitude F, the ionization rate can be written as [28–30]

(5)$$W={\omega }_{a.u.}\sqrt{3/2\pi }{\left|C_{n*,l*}\right|}^{2}f(l,m)\frac{U_i}{U_H}{(\frac{2F_0}{F\sqrt{1+{\gamma }^2}})}^{2n*-\left|m\right|-3/2}{A}_m({\omega }_0,\gamma )\exp [-\frac{2F_0}{3F}g(\gamma )]$$

Where $n*=z/\sqrt{2U_i}$ is the effective quantum number, z is the charge state of resulting ion, $U_H$ is the ionization potential of hydrogen. The effective orbital quantum number is $l*=n*-1$ and the adiabaticity parameter is $\gamma =\frac{{\omega }_0}{eF}\sqrt{2m_e{U}_i}$ . The constants ${\left|C_{n*,l*}\right|}^2$ and $f(l,m)$ are calculated by

(6)$${\left|C_{n*,l*}\right|}^2=\frac{2^{2n*}}{n*\Gamma (n*+l*+1)\Gamma (n*-l*)}$$

(7)$$f(l,m)=\frac{(2l+1)(l+\left|m\right|)!}{2^{\left|m\right|}\left|m\right|!(l-\left|m\right|)!}$$

The other parameters are $F_0={(2U_i)}^{3/2}$ and ${\omega }_{a.u.}=4.1\times {10}^{16}{\text{s}}^{-1}$ . The functions involved in Eq. (5) is given by

(8)$$A_m({\omega }_0,\gamma )=\frac{4}{\sqrt{3\pi }}\frac{1}{\left|m\right|!}\frac{{\gamma }^2}{1+{\gamma }^2}{\displaystyle \sum _{\kappa \ge v}^{+\infty }\exp [-\alpha (\kappa -v)]}{\varphi }_m[\sqrt{\beta (\kappa -v)}],$$

(9)$$v=\frac{U_i}{\hslash {\omega }_0}(1+\frac{1}{2{\gamma }^2}),$$

(10)$${\varphi }_m[\sqrt{\beta (\kappa -v)}]=\exp (-x^2){\displaystyle {\int }_0^x{(x^2-y^2)}^{{}^{\left|m\right|}}\exp (y^2)dy},$$

(11)$$\alpha (\gamma )=2[\sin h^{-1}\gamma -\frac{\gamma }{\sqrt{1+{\gamma }^2}}],$$

(12)$$\beta (\gamma )=\frac{2\gamma }{\sqrt{1+{\gamma }^2}},$$

(13)$$g(\gamma )=\frac{3}{2\gamma }[(1+\frac{1}{2{\gamma }^2})\sin h^{-1}\gamma -\frac{\sqrt{1+{\gamma }^2}}{2\gamma }].$$

The averaged first level ionization potential of air is 14.6 eV [30] and the averaged second level ionization potentials of air is 25.6 eV, which is calculated by Quantum Chemistry Code GAUSSIAN. The ionization rate of air is then calculated by substituting the ionization potential of air in Eq. (5). Fig. 7 shows the first and second level ionization rate of air.

 

Fig. 7 The first (solid line) and second level (dash line) ionization rate of air.

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Using the calculated ionization rate, we can then investigate the evolution of electron density as a function of laser intensity. But for high laser intensity, the role of collisional ionization should also be considered. The electrons generated by the optical field ionization and collisional ionization can be described as:

(14)$$\frac{\partial \rho }{\partial t}=W_{air}{}^{\prime }(I)({\rho }_{air}-\rho )+\frac{\sigma }{U_i}\rho I.$$

Where $\rho (t)$ is the electron density generated by the femtosecond laser pulse. Here, we only consider the first level ionization, as a result, $W_{air}{}^{\prime }(I)$ is the first level ionization rate of air shown in Fig. 7. $\sigma =5\times {10}^{-20}{\text{cm}}^{\text{2}}$ [21] denotes the inverse Bremsstrahlung coefficient of the air. The initial density of air is ${\rho }_{air}=2.7\times {10}^{19}{\text{cm}}^{\text{-3}}$ . Using a 60 fs Gaussian pulse with temporal envelope $I(t)=I_0\exp (-2t^2/{\tau }^2)$ , the dependence of electron density on the laser intensity with and without collisional ionization is calculated and presented in Fig. 8 . We can see that the contribution of collisional ionization to the induced electron density is very small by comparison with the optical field ionization during the laser pulse. So in our experiment, the electrons are mainly induced by optical field ionization. The effect of collisional ionization can be neglected.

 

Fig. 8 The electron density generated by the first level ionization of air as a function of laser peak intensity with (dash line) and without (solid line) collisional ionization.

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) under grants No. 10734130, No. 10634020, and No. 60978031 and the National Basic Research Programme of China (No. 2007CB815101).

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