Propagation dynamics of femtosecond laser

RIKEN Review No. 49 (November, 2002): Focused on Ultrafast Optical Sciences

Propagation dynamics of femtosecond laser pulses in hollow fiber filled with argon Muhammad Nurhuda, ∗1,a Akira Suda,∗1 Masatoshi Hatayama,∗2 Keigo Nagasaka,∗2 and Katsumi Midorikawa∗1 ∗1 ∗2

Laser Technology Laboratory, RIKEN

Department of Physics, Tokyo University of Science

We investigate the dynamics of femtosecond laser pulses propagating in a hollow waveguide filled with argon using a simulation model based on the 3D nonlinear Schr¨ odinger equation (NLSE). The results show that if the intensity is low and no ionization takes places, the spatial profile of the beam does not change such that its propagation model may be reduced to simply a 1D model. On the other hand, if the intensity is high and in the presence of ionization, the spatial dynamics as well as temporal dynamics of the propagation become very complicated. By further elaborating the propagation dynamics, it was found that the self-focusing and plasma defocusing are the origin of the complicated structure of the spatio-temporal intensity profiles.

1. Introduction Recent studies have shown that the use of a hollow waveguide filled with a noble gas is an effective way of extending the interaction length of nonlinear optical processes with high-energy laser pulses. As results, the spectral broadening of the pulse through the process of self-phase modulation (SPM) enables one to compress the pulse down to as short as 5 fs and with a power of several tens Gigawatt.1, 2) Recently, this waveguide has also been used for improving the phasematching condition for high-harmonic generation.3) The operation of high-energy laser power in a hollow fiber raises the question whether self-focusing occurs during propagation and complicates the spatial shape of the beam. Self-focusing inside a hollow waveguide has been studied theoretically using a perturbative coupled-mode theory,4) where the electric field inside the hollow waveguide is expanded in terms of leaky modes. It was shown using steady-states analysis that self-focusing takes place if the electric field in fundamental mode is depleted and transfered to higher order modes. This oversimplified model of analysis presents a discrepancy in the critical power Pcr for self-focusing; Pcr is found to be far greater than the value of that of bulk medium ( by a factor greater than 3.5), while in another analysis,5) it was shown that the Pcr in hollow waveguides is only 10% above that of bulk medium. It was also predicted, if the power of the pulses is greater than the critical power Pcr , the third-order nonlinearity alone will lead the pulses to a catastrophic self-focusing.5) However, this catastrophic selffocusing is in fact only an artifact of the solution of the nonlinear Schr¨ odinger equation, and has been proven to be not in accordance with the experimental observations.6) A proper model of the pulse propagation in a hollow waveguide therefore has to include such effects, i.e., the saturation of the nonlinear susceptibility, plasma defocusing, and so forth, in addition to the third-order nonlinearity. Unfortunately, incorporating those effects into the NLSE will coma

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Permanent address: Physics Department, Brawijaya University, Indonesia

plicate the solution, owing to the existing couples between the eigen modes used in the expansion of the electric field. In order to resolve this difficulty, the NLSE is solved according to the following steps; the electric field is expanded in the sum of leaky modes and the diffraction effects and losses due to the leakage are carried out in the frequency domain. After that, we focus on the time domain, and the integration with respect to the nonlinearity is carried out in full 3D space. Using this technique, all nonlinear effects including saturation and plasma defocusing can be apporopriately included. Our paper is organized as follows: In section 2, a short description of the theory underlying the numerical procedures will be reviewed. In section 3, the simulation results obtained by different laser parameters (low- and high-intensity laser fields) will be compared and discussed. We found that if the power of the pulse is greater than the critical power Pcr , the beam undergoes self-focusing. However, the ionizationinduced high-order nonlinearity will compensate the effect of the self-focusing force, and as a result, catastrophic selffocusing inside the hollow waveguide will not be observed. The conclusion of our work will be outlined in section 4.

2. Method Our model for pulse propagation in a hollow waveguide is based on paraxial approximation of the NLSE in the following form (see e.g., Refs. 7 and 8): ∇2⊥ E(r, z, t) −

1 ∂2 2 ∂2 E(r, z, t) = PNL (r, z, t), (1) c ∂z∂t 0 c2 ∂t2

where the first and second terms on the left-hand side stand for diffraction and space time dispersion, respectively, and the right-hand side term stands for the source of nonlinearity PNL (r, z, t). Here c is the speed of light, t is time, and 0 is the permittivity of the vacuum. Note that in Eq. (1), a moving frame is used and no slowly varying envelope (SVE) approximation is made. The source of nonlinearity PNL comes from the nonlinear term and plasma dispersion:

PNL (r, z, t) = ∆χ(r, z, t)E(r, z, t) −

ωp2 E(r, z, t), 0 c2

(2)

where ∆χ(r, z, t)E(r, z, t) is the instantaneous part of the nonlinear susceptibility assumed to be complex, and ωp = p e2 ρ/m 0 accounts for the plasma frequency. Here, ρ is the electron density, e is the electron charge, and m is the electron mass. The imaginary part of ∆χ corresponds to the absorption due to MPI. The evolution of electron density is governed by ∂ρ = N0 (1 − ρ)Γ(I), ∂t

(3)

where N0 is the density of the initial neutral atom and Γ is the rate. The cascade ionization and electron recombinations are neglected since their contributions are negligible compared to that by MPI. Data for the ionization rate of argon was calculated using Perelemov, Popov, and Trent`ev’s (PPT) model,9) which includes the Coulomb tail and has been proven to reproduce the ionization rate of argon accuratelly.10) Equation (1) is solved using the split-step Fourier-transform method; the homogenous part is calculated in the frequency domain, and the nonlinear term is calculated in the time domain. Since we are working with the electric field envelope, in carrying out the integration over the nonlinearity, the SVE method is applied. Within this approximation, only the nonlinearity of the center of the wavelength contributes at most. In order to impose the boundary condition of hollow fiber, the electric field envelope inside the hollow waveguide is written as a sum of eigen modes: A(r, z, ω − ω0 ) =

X



bnm (z, ω − ω0 )Jm

nm

λn r a0



,

(4)

where bnm (z, ω − ω0 ) is the complex amplitude of the mth normalized Bessel function Jm (λn r/a0 ) , where λn is the nth root of the mth Bessel function, and a0 is the hollow bore radius. The summation over m can be truncated to m = 0 because of the linearly polarized electric field, and due to cylindrical symmetry of the space. Thus, to advance a propagation distance of one step, the solution of the homogenous part can be written as: A(r, z + ∆z, ω − ω0 ) =

X

eiβn ∆z−αn ∆z ×

n

bn0 (z, ω − ω0 )J0



λn r a0



,

(5)

where βn = −0.5k(λn /ka0 )2 is the propagation constant, 

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αn = n(ω) λn /ka0 /a0 is the mode loss, and k = ω/c is the wave number. The refractive index of the cladding was calculated using Selmier’s formula.11) Thus, within the framework of eigen mode expansion, we first have to Fourier transform the electric field envelope in the frequency domain, and then project it into the basis J0 (λn r/a0 ). The result is then multiplied by the corresponding phase factor eiβn ∆z−αn ∆z . After that, we have to reconstruct the electric field in the time domain for each position in space and carry out the integral due to nonlinearity. Using the above presented procedure, all effects of nonlinearity, including the saturation and ionizationinduced plasma defocusing, can be apporopriately included.

3. Results and discussion For the simulations presented here, the laser parameters used were divided in two classes, i.e., low-intensity laser field and high-intensity laser field. The motivation behind this choice was that in the low-intensity laser field, the radiation-matter interaction inside the hollow waveguide does not disturb the spatial profile of the beam, whereas in the high-intensity laser field, the self-focusing occurring inside the hollow waveguide and the defocusing due to the saturation of nonlinear susceptibility are expected to greatly change the spatial profile. The laser wavelength was chosen to be 789 nm, and the pulse duration was 60 fs, approximately the same laser parameters used in an experiment currently being conducted at the Laser Technology Laboratory, RIKEN (The Institute of Physical and Chemical Research). For the low-intensity laser field, the laser energy was chosen to be 0.2 mJ, and correspondingly, the peak intensity was 2.36 × 1013 W/cm2 . The power at the center of the time slice was found to be 3.13 GW. For the high-intensity laser field, the laser energy was chosen to be 0.9 mJ, and the calculated peak intensity was 1.06 × 1014 W/cm2 . The peak power at the center of the time slice was found to be 14.1 GW. The gas pressure within the hollow waveguide was chosen to be 1.0 atm and the critical power, calculated using Pcr = 1.99λ2 /4πn2 yielded Pcr = 9.91 GW. The number of eigen modes used in the expansion of the electric field inside the hollow waveguide was 30. The initial profile of the beam was chosen to be the center lobe of the zeroth-order Bessel function, such that the beam is completely coupled to the fundamental mode. On using the simulation parameters described above, one can immediately see that in the case of simulation with low intensity, the peak power at the center of the time slice is much lower than the critical power, such that no self-focusing is expected to occur. However, while using the high-intensity laser field, because the power at the center of the time slice larger than the critical power, one can expect that self-focusing inside the hollow waveguide occurs. In Fig. 1, the spatio-temporal intensity profiles are shown, starting from the propagation distance z = 0 cm, to z = 75 cm with steps of 5 cm. It can be seen that the intensity profiles evaluated at different positions do not change so much. This indicates that there exists an energy exchange between fundamental and higher order modes. Since the strength of the interaction is weak, one may expect that only the lowest excited mode will be coupled, while couplings with higher order modes can principally be neglected. To be clear, in Fig. 2 (a), the probability of energy confined in the fundamental mode is shown, while that in n = 2, due to its very small value, is shown in the inset. It is clear that almost all energy of the electric field remains in the fundamental mode. Furthermore, the energy occupation probability in n = 2 shows an oscillation. The period of this oscillation can be estimated from the perturbative coupled-mode theory,4) yielding πa0 /∆λ12 where ∆λ12 = λ2 − λ1 . In order to derive the relationship of the 3D propagation model with the simplified 1D model, it is necessary to assume that depletion of the fundamental mode is negligible, i.e., the spatial profile of the beam remains constant. Thus, writing E(r, z, t) = R(r)A(z, ω − ω0 )eik0 z the equation governing propagation dynamics then reduces into two secular equations:

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validity of the 1D model. On the other hand, the propagation dynamics of the highintensity laser field in the hollow waveguide is very complicated and possesses highly structured shapes, both temporally and spatially.

Fig. 1. The spatial and temporal dependence of the intensity is displayed in 3D contours for each propagation distance z starting from z = 0 cm to z = 75 cm with steps of 5 cm. The horizontal axis represents the radial distance (−0.125 mm to 0.125 mm) and the vertical axis represents the time (−80 fs to 80 fs). The pulse energy used in simulation was 0.2 mJ, and the gas pressure was 1 atm.

In Fig. 3, the spatial-temporal intensity profiles, evaluated at different z positions, are shown. As can be seen, the spatial profiles change drastically immediately after the pulse interacts with the nonlinear medium inside the hollow waveguide. The reason for such strong turbulence can be traced back to the required condition for self-focusing. Thus, because of laser power higher than the critical power of self-focusing, immediately after the beam enters the nonlinear medium, the beam experiences self-focusing and then defocusing due to saturation of instantaneous susceptibility and ionizationinduced high-order nonlinear susceptibility. As an example, at z = 5 cm the intensity of the beam in the trailing edges is found to be much lower than that in the leading edge. This can be understood since the higher intensity of the leading edges ionizes the medium and the resulting free electron then suppress the beam in the trailing edges. On the other hand, losses due to MPI and leakage result in a decrease of the power in the leading edges, and subsequently also decrease the defocusing force exerted on the beam in the trailing edges. These processes are then followed by power replenishment from the peripheral area to the central area, and the trailing edges of the pulse start to refocus. We have observed that due to refocusing, a peak intensity as high as twice that of the input peak intensity has been reached at a propagation distance about 12 cm. Another interesting feature is the splitting of the pulse. This phenomenon, as can be seen in Fig. 3, emerges at a longitu-

Fig. 2. In (a), the energy confined in the fundamental mode. In the inset, the energy confined in the first excited mode (n = 2). In (b), the comparison of energy transmittance obtained from simulation (black) and the simplified 1D model (red) is shown. 



∇2⊥ R + (ω)k02 − k2 R = 0, 2ik0

∂A(z, ω − ω0 ) + (k2 − k02 )A(z, ω − ω0 ) = 0. ∂z

(6)

The solution for R depends on the initial condition, i.e., the initial beam profile of the pulse, while the solution for A(z, ω − ω0 ) can be solved exactly for the case of the steady-state condition (see e.g., Ref. 12), or using a numerical solution as in the case of the non-steady-state condition. Also, the transmittance in this case can be estimated from this simplified model, providing T (z) = e−2αz , where 

2

α = n(ω0 ) λ1 /k0 a0 /a0 . In Fig. 2 (b), the transmitted energy obtained from simulation and the reduced 1D model are compared; a black line represents that from direct simulation of NLSE, and a red line that from the 1D model. It is seen that both models are in good agreement, indicating the

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Fig. 3. The spatial and temporal dependence of the intensity is displayed in 3D contours for each propagation distance z starting from z = 0 cm to z = 75 cm with steps of 5 cm. All parameters used for simulation were the same as those used in Fig. 1, except the pulse energy in this case is E = 0.9 mJ.

dinal distance of about 15 cm and continues until the end of propagation. This pulse splitting occurring in hollow waveguide is in fact similar to that occurring in free space.13) The mechanism of the pulse splitting can therefore be explained in a manner similar to that in free space. Due to self-focusing or refocusing, the intensity may reach a value where the nonlinear susceptibility becomes saturated, or a value where ionization starts to occur, such that the value of nonlinearity becomes lower than that in the neighboring temporal coordinates. This lesser value of nonlinearity then acts as a hole, and causes the pulse to split. When the beam inside the hollow waveguide undergoes refocusing or defocusing, the energy transfer from fundamental to higher order modes takes place. The occupation of higher order modes strongly depends on the coupling strength. The stronger coupling strength, the higher order of the modes are being occupied, and more complicated the structure of the spatial profile. In this respect, the presence of ripples in the peripheral area can than be understood to be due to the occupation of high-order modes. In Fig. 4, the occupation probabilities of the first four lowest order modes are shown. It can be seen that, the energy occupation probability in higher order modes cannot be neglected. To be more specific, at the end of propagation, about 63% of the total energy remains in the fundamental mode, 26% in the first excited mode (n = 2), 10% in n = 3, and the other 1% is distributed among the rest of the high-order modes. Thus, an approximation used in the perturbative coupled-mode theory4) stating that depletion of the energy in the fundamental mode is negligible cannot be verified here. Finally, it is worth noting that due to strong spatial turbulence occurring inside the hollow waveguide, the outgoing pulse cannot be compressed. The reason for this is because

Fig. 4. In (a), the energy confined in the first four lowest order modes is shown and compared. In (b), the energy transmittance as a function of propagation distance is displayed. In the inset, the accumulated energy losses due to absorption by MPI and to leakage are compared.

the important condition for the compression, i.e., that the spatial-temporal components of the pulse must have equal phases, cannot be satisfied. This constraint clearly hinders us to compress the high-energy laser pulse. To eliminate this unwanted effect, we have proposed a method called “differential pumping technique”, where the pressure of the gas in the hollow waveguide is varied from minimum in the entrance to a given maximum value at the exit. The self-focusing is thus eliminated because of the higher critical power at the entrance, and while the pulse is propagating forward in the hollow waveguide, the absorption due to MPI and losses due to leakage decrease the total power and hence, decrease the possibility of refocusing.

4. Conclusion We have performed simulations of femtosecond laser pulses propagating in a hollow waveguide filled with argon. The method of solving the NLSE in the hollow waveguide utilizes the combination of leaky mode expansion of the electric field and the direct numerical integration in 3D space. The simulation results show that for the low-intensity regime, the spatial profiles of the beam do not change such that the propagation dynamics may be simplified to 1D mode. In the strong field regime, however, third-order nonlinearity, saturation and ionization-induced high-order nonlinear susceptibility play an important role and cause strong turbulence in the propagation dynamics. Self-focusing inside the hollow waveguide can be considered as an energy transfer from fundamental mode to higher order modes. References 1) M. Nisoli, S. De Silvestri, and O. Svelto: Appl. Phys. Lett. 68, 2793 (1966). 2) N. Karasawa, R. Morita, H. Shikegawa, and M. Yamashita: Opt. Lett. 25, 183 (2000). 3) Y. Tamaki, Y. Nagata, M. Obara, and K. Midorikawa: Phys. Rev. A 59, 4041 (1999). 4) G. Tempea and T. Brabec: Opt. Lett. 23, 762 (1998). 5) G. Fibich and A.L. Gaeta: Opt. Lett. 25, 335 (2000). 6) A. L. Gaeta: Phys. Rev. Lett. 84, 3582 (2000). 7) E. Priori, G. Cerullo, M. Nisoli, S. Stagira, S. De Silvestri, P. Villoresi, L. Poletto, P. Ceccherini, and C. Altucci: Phys. Rev. A 61, 1050 (2000). 8) N. G. Shon, A. Suda, Y. Tamaki, and K. Midorikawa: Phys. Rev. A 63, 806 (2000). 9) A. M. Perelemov, V. S. Popov, and M. V. Terent’ev: Sov. Phys. JETP 23, 924 (1966). 10) S. F. J. Larochelle, A. Talebpour, and S. L. Chin: J. Phys. B 31, 1215 (1988). 11) I. H. Malitson: J. Opt. Soc. Am. B 55, 1205 (1965). 12) G. P. Agrawal: in Nonlinear Fiber Optics (Academic Press, San Diego, 1995), p. 35. 13) M. Nurhuda, A. Suda, M. Hatayama, K. Nagasaka, and K. Midorikawa: Phys. Rev. A 66, 023811 (2002).

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