1. Introduction

Rapid progress in femtosecond laser physics naturally leads to the problem of propagation of high-power femtosecond laser pulses under atmospheric conditions. High intensity of femtosecond radiation manifests itself through numerous nonlinear phenomena. One of the most interesting effects, which has been actively investigated for about a decade, is the effect of filamentation of a laser beam propagating in atmosphere [1, 2]. In this paper we present the results of experimental investigation of another nonlinear effect discovered by the authors when studying the propagation of intense femtosecond laser radiation: the nonlinear absorption of energy of laser pulses.

A very sensitive photoacoustic method [3] was used to measure the low absorption of laser radiation. The same method was employed in [4] to characterize the length and spatial profile of plasma in the regime of laser beam filamentation. In this paper we made photoacoustic measurements in the range of laser intensities I = 10¹¹-10¹² W/cm² well below the ionization intensity (~10¹⁴W/cm²) and investigated the absorption of non-ionized molecules constituting air.

2. Experimental setup and results

The schematic of the experiment is presented in Fig. 1. Linearly polarized radiation from a femtosecond laser system [5] based on Ti:Sa crystals (wavelength λ₀ ≈ 795 nm, spectral bandwidth Δλ ≈ 18 nm, pulse duration FWHM τ₀ ≈ 80 fs, repetition rate F_n=10 Hz, pulse energy W≤20 mJ) was used in the experiment. A polarization attenuator consisting of a zero order λ/2 wave plate and a thin film polarizer was placed after the regenerative amplifier to change the pulse energy measured by a calibrated photodiode. The radiation from the output of the laser system with Gaussian distribution, beam radius a ₀=4 mm at 1/e² level was slightly focused by a spherical mirror with the focal length f= 86.5 cm.

 

Fig. 1. Experimental setup for photoacoustic absorption measurement in atmosphere.

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The investigation of absorption of laser radiation was performed using a photoacoustic method. A ¼” microphone (MK-301, bandwidth ≈ 100 kHz) was placed 42 cm from the focusing mirror at a distance of r₀ = 1.5 cm from the beam axis. In order to increase the sensitivity, in some experiments we used a photoacoustic cell with a pressure concentrator and a ½” microphone MK-221 [6]. From the geometry considerations the beam radius in the plane of microphone was a ≈ 2 mm. The influence of self-focusing on the beam distribution will be discussed later.

When the femtosecond laser pulse was applied in the focal region, a plasma filament was produced. The maximum length of the filament at maximum energy in a pulse used in our experiment did not exceed 20 cm. The length of the filament was determined both visually and by the photoacoustic method like in [4]. Thus our photoacoustic measurements were performed in the region well beyond the plasma. In addition, the intensity of laser radiation in the plane of the microphone was I≤10¹² W/cm² in all experiments, much lower than ionization intensity. The self-focusing just slightly increases the intensity, as will be shown below.

The absorption of laser radiation in the beam produces an outgoing cylindrical acoustic wave. For a femtosecond laser pulse, the signal from the microphone had the form of a compression wave followed by a rarefaction wave with time delay corresponding to the propagation of sound from the axis of the laser beam to the microphone. At larger delays signals caused by secondary reflections and generated by the plasma filament in the focal region were observed. The dependence of the amplitude of the first peak of the acoustic signal on energy in the femtosecond laser pulse for atmospheric air is shown in Fig. 2. It is a power dependence with power degree β=2.6 ± 0.2. When a nanosecond pulse was used instead of the femtosecond one (pulses with duration τ ≈ 9 ns can be produced when no femtosecond radiation is injected in the amplifying chain, wavelength in this case was λ₀ = 790 nm, spectral bandwidth Δλ ≈ 23 nm), the acoustic signal was also observed (Fig. 2). Its amplitude was much lower and the dependence of the acoustic signal on laser pulse energy was linear. Thus, the experimental results show that the absorption of femtosecond laser pulse energy propagating in atmosphere has a nonlinear behavior and its value essentially exceeds the corresponding linear absorption for nanosecond pulses with the same energy and similar spectral bandwidth.

3. Discussion

It is well known that homonuclear molecules of nitrogen and oxygen, the main components of atmospheric air, have no electron dipole moments. But in the strong electromagnetic field of a femtosecond laser pulse an induced dipole moment is produced, being predominantly oriented along the molecular axis [7]. The interaction of this induced dipole moment with the electric field of a laser pulse leads to the rotational excitation of molecules and, consequently, to the absorption of laser radiation. The rotational energy of the molecules is transformed, through collisions, into heat, which is the source of acoustic wave excitation.

 

Fig. 2. Dependence of the peak of acoustic signal on laser energy. Circles – nanosecond pulses, squares – femtosecond pulses, solid line – model.

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To describe this mechanism quantitatively, one needs to evaluate the probability of rotational excitation of molecule by a short pulse [8]:

Here ħ is Plank constant, V_ij is the matrix element of the energy operator for interaction of the induced dipole moment with the electromagnetic field. For a Gaussian pulse envelope with duration τ and laser frequency ω the matrix element reads [9]:

In these formulas, the frequency of transition ${\omega }_{\mathit{ij}}=\frac{E_i-E_j}{\mathit{ħ}},E_j=\mathit{Bj}\left(j+1\right)$ is the energy of rotation, θ is the angle between the molecular axis and the electric field direction, α is the polarization anisotropy difference between parallel and perpendicular components of the polarizability tensor [9], and E₀ is the field amplitude. To calculate the matrix element V_ij we consider the molecule as a linear rigid rotor with a wavefunction given by spherical functions [8]. According to the properties of spherical functions, only those transitions are not equal to zero for which the projection of the angular momentum on the electric field direction is maintained and the angular momentum selection rules Δj = ±2 are fulfilled. Calculation of the matrix elements and summation over the projection of angular moment yield the following expression for the probability of transition from the rotational level j to the level j + 2 under short-pulse interaction:

where the intensity of electromagnetic radiation $I=\frac{c}{4\bullet \pi }\bullet E_0^2$ (c is the speed of light) is introduced. The number of transitions between levels j and j+2 equals the product of probability of transition (3) and population difference of these levels. Part of the molecules on level j is determined by the Boltsman distribution. By multiplying the number of transitions by the transition energy and summarizing over all rotational levels we find the energy acquired by a single molecule during rotational excitation by a short laser pulse:

where $S={\sum _{j=0}^{\infty }e}^{-2\frac{B^2}{{ħ}^2}{\left(2j+3\right)}^2{\tau }^2}\frac{\left(j+1\right)\left(2j^3+4j^2+26j+15\right)}{{\left(2j+3\right)}^2\left(2j+5\right)}[e^{-\frac{B}{\mathit{kT}}j\left(j+1\right)}-e^{-\frac{B}{\mathit{kT}}\left(j+2\right)\left(j+3\right)}],$ $J=\frac{{ħ}^2}{2B}$ is the momentum of inertia of a molecule, and Q is the rotational partition function, which in case of kT >> B is equal to $Q=\frac{\mathit{kT}}{B}$[7].

For a Gaussian pulse $\left(I=I_0\exp (-\frac{2r^2}{a^2}\right),$ a is the beam radius), by integrating over r and multiplying by the concentration of molecules n , we find an equation for the change of energy of a laser pulse $W=\frac{\pi \bullet a^2\tau \bullet I_0}{2}$ due to rotational excitation of molecules:

Let us now consider a problem of excitation of the photoacoustic signal that arises from the absorption of short-pulse radiation in gases. For linear absorption, this problem has been solved in [3, 10]. Following [10] and introducing nonlinear losses instead of linear ones according to (5), the expression for the amplitude of the acoustic signal can be written as follows:

where c_p =1000 J/kgK⁰ and T =293 K are the heat capacity and temperature of air, v is the velocity of sound, and r₀ = 0.015 m is the distance from the axis of the laser beam to the microphone. Introducing in (6) values for atmospheric air α ≈ 0.8∙10^-24 m³ [11], J=ħ²/2B_e ≈ 2.5∙10^-46 kg∙m² [12], n = 2.67∙10²⁵ m³, and a = 0.002 m, W=1 mJ, we will have for the acoustic signal amplitude p_max ≈ 1.5∙10^-4 Pa, which coincides, within the experimental uncertainties, with the measured value.

Formula (6) gives a dependence of the acoustic signal amplitude proportional to the second power of energy in a pulse, which is lower than that we observed in experiment. The explanation of this discrepancy may be in the fact that the self-focusing of intense laser radiation should be taken into account. Really, power of laser radiation at W=10 mJ exceeds 100 GW, much more than the critical power for self-focusing in atmosphere P_cr = 2-6 GW [13]. The self-focusing reduces the effective beam diameter while laser pulse energy increases. According to (6), this fact should lead to a sharper than quadratic dependence of the acoustic signal amplitude on laser pulse energy.

To take the self-focusing into account, numerical calculation of 2-D nonlinear Shrödinger equation for the slowly varying envelope of the laser electric field was performed, in which diffraction within the transverse plane and instantaneous Kerr nonlinearity were included. The method of numerical calculation has been discussed in detail elsewhere [14]. The initial diameter and convergence of laser beam used in calculation were taken from our experiment. The critical power for self-focusing was chosen P_cr = 5 GW. As a result of the calculations, the intensity distribution of the beam at cross-section corresponding to the position of the microphone was obtained as a function of laser pulse energy. The calculation shows that with growing pulse energy the initially Gaussian beam is transformed to a sharper and narrower form still having a smooth distribution (Fig. 3). At maximum pulse energy the reduction of effective beam diameter (we define it as a FWHM) reaches ~ 30% of initial value. The polynomial approximation of effective beam diameter on laser pulse energy acquired from the numerical calculation can be written as:

It was introduced in (6).

 

Fig. 3. Distribution of normalized intensity of a beam at different energies in a pulse.

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The result of this operation is shown in Fig. 2 as a solid line. Taking the self-focusing into account leads to deviation of this line from a straight line at largest energies. It is worth pointing out that the theoretical curve corrected in such a way corresponds, within a factor of two, to the experimental data in the whole range of measurement, reaching three orders of magnitude without any adjustable parameter.

The proposed model of nonlinear absorption of femtosecond laser pulses in air due to rotational excitation of nitrogen and oxygen molecules with the induced dipole moment works only for linear polarization of electromagnetic radiation. For circular polarization, because of averaging over the angle, the molecule’s rotational energy averaged over the electromagnetic field period is zero and this mechanism is not valid. The waveforms of photoacoustic signals generated by linearly and circularly polarized pulses with equal energy are shown in Fig. 4. It is seen from the figure that the acoustic signal for circular polarization is about an order of magnitude less than for linear polarization. To obtain circular polarization, we used a thin zero order λ/4 wave plate. The non-zero value of the signal for circular polarization can be connected with the ‘non-ideal circularity’ of our radiation.

 

Fig. 4. Acoustic signals for linear (magenta) and circular (blue) polarizations.

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In atomic gases, the mechanism of high intensity radiation absorption considered above is also not valid. Really, in the experiments [15] the injection of atomic Ar in a photoacoustic cell leads to the disappearance of the acoustic signal.

According to existing ideas [16], the absorption of nanosecond pulses, which is linearly dependent on laser pulse energy, is determined by scattering by atmospheric aerosols and absorption by water vapor and oxygen contained in the atmospheric air.

4. Summary

In this paper, we investigated the absorption of intense femtosecond radiation in atmospheric air using a photoacoustic method in the intensity range I = 10¹¹-10¹² W/cm² when the ionization is not important yet. The effect of nonlinear absorption of femtosecond radiation much exceeding the linear absorption of longer (nanosecond) pulses with comparable energy was discovered. A model of the nonlinear absorption based on the rotational excitation of air molecules due to interaction of induced dipole moments with strong ultrashort linearly polarized electromagnetic pulses was developed. It is shown that considering the self-focusing effect is important for better agreement between theory and experimental data.