Modeling the ponderomotive interaction of high-power laser beams with collisional plasma: the FDTD-based approach

Zhili Lin; Xudong Chen; Panfeng Ding; Weibin Qiu; Jixiong Pu

doi:10.1364/OE.25.008440

1. Introduction

Over the past decades, the complex interaction of high-power laser beams with plasma has been a subject of great interest for worldwide physicists since it is relevant for many important applications, such as X-ray laser [1, 2], laser-driven charged-particle or plasma-based accelerators [3, 4], laser-induced nuclear fusion [5, 6], etc. In the presence of intense laser beams, the spatial inhomogeneity of light field exerts a ponderomotive force on electrons. The ponderomotive force pushes the electrons out of the region of high light intensity and changes the electromagnetic field profile and electron density distribution in the plasma [7]. When the ponderomotive force applied to the electrons is comparable to the electron pressure gradient force, a significant nonlinear effect that is highly sensitive to the radiance distribution of laser beam appears [8]. So far the main thrust of theoretical investigations have been directed to the study of nonlinear propagation characteristics of various types of laser beams that leads to the self-focusing phenomenon in collisionless plasma [9, 10] or collisional plasma [11, 12]. However, all the above studies are conducted only by analytically solving the scalar wave equation with paraxial approximation. Therefore, the obtained information is limited and the solutions are not so much accurate for a focused beam whose size is in the order of several wavelengths and with a large diffraction effect.

In recent years, with the rapid progress of computer hardware, the finite-difference time-domain (FDTD) method has become one of the most powerful numerical techniques in directly solving the Maxwell’s curl equations [13] and has been widely applied to model and solve optics and photonics problems [14]. In the present work, the FDTD method is applied to model the ponderomotive interaction of various types of high-power laser beams with collisional plasma for the first time in literature. There are several motivations for this work. Firstly, the FDTD method is powerful in accurately modeling dispersive and dissipative media with the developed approaches including the auxiliary differential equation (ADE) approach [15], the bilinear-transform approach [16], the Z-transform approach [17], the piecewise linear recursive convolution model [18] and the Maclaurin series expansion (MSE) approach [19]. We herewith use the FDTD method to model and implement the deduced nonlinear dielectric constant function of collisional plasma under the ponderomotive regime. Secondly and most importantly, we make a first try to extract the time average of the square of light field by numerically evaluating an integral identity based on the composite trapezoidal rule. Finally, since the total-field/scattered-field (TF/SF) technique can be applied based on the well-known equivalence principle [20], the FDTD method also offers a great degree of convenience and flexibility in the excitation of various types of laser beams [21].

This paper is organized in four sections. In Section 2, the nonlinear dielectric constant function exponentially pertaining to the time average of the square of light field is deduced for the high-power laser beam propagation in the plasma is deduced based on the equilibrium between the ponderomotive force and the plasma pressure gradient force in the nonrelativistic regime. In Section 3, two numerical examples considering two different types of laser beams with different cross-sectional field density profiles are presented and the numerical results are discussed. Finally, a summary is given in Section 4.

2. Physical model

In inhomogeneous oscillating electromagnetic fields, charged particles experience a nonlinear Lorentz force called the ponderomotive force [22]. However, since the ponderomotive force scales with the inverse of the particle mass, the ponderomotive effect on ions is generally negligible with respect to that on free electrons. Thus such a force manifestly redistribute the electron density in plasma until the balance is restored with the plasma pressure gradient force. For the normal incidence of laser beam on plasma, the ponderomotive force per unit volume, $f_{p}$ , is determined proportionally by the gradient of light field intensity,

f_{p} = - \frac{n (r) e^{2}}{2 m_{e} ω_{L}^{2}} \nabla 〈 E^{2} (r, t) 〉,

where

m_{e}

and

e

are the mass and electric charge of electron,

ω_{L}

is the angular frequency of laser,

n (r)

is the space-dependent electron density.

〈 E^{2} (r, t) 〉

is the time average of the square of light field over one laser oscillating period [23]. It is the ponderomotive force due to the spatial variation of

〈 E^{2} (r, t) 〉

makes the slow motion of “oscillation center” of electrons over which a fast oscillation is superposed.

On the other hand, the plasma pressure gradient force per unit volume, $f_{pg}$ , is correlated with the electron temperature of electron plasma and the gradient of electron density,

f_{pg} = - k_{B} T_{e} \nabla n (r),

where

k_{B}

is the Boltzmann's constant and

T_{e}

is the electron temperature given in kelvin and often taken to be constant for a pulsed laser beam incidence. In the steady state with equilibrium, the ponderomotive force balances with the pressure gradient force,

\frac{n (r) e^{2}}{2 m_{e} ω_{L}^{2}} \nabla 〈 E^{2} (r, t) 〉 = k_{B} T_{e} \nabla n (r) .

By integrating Eq. (3) from

n_{0}

to

n

, we obtain the expression for the electron density,

n (r) = n_{0} (r) \exp (- \frac{e^{2} 〈 E^{2} (r, t) 〉}{2 m_{e} k_{B} T_{e} ω_{L}^{2}}),

where

n_{0} (r)

is the initial electron density prior to the incidence of laser beam. Moreover, by solving the equation of motion of the electrons in the plasma, one has the dispersive dielectric constant model of collisional plasma given by

ε (r, ω) = ε_{0} (1 - \frac{ω_{p}^{2} (r)}{ω^{2} - i ν_{c} ω}),

where

ε_{0}

is the permittivity of free space,

ω_{p} (r) = \sqrt{e^{2} n (r) / ε_{0} m_{e}}

is the (electron) plasma resonance frequency and

ν_{c}

is the collision frequency resulting to the absorption of laser energy by the plasma. Consequently, the nonlinear, dispersive and dissipative dielectric constant function of plasma takes the form

ε (r, ω) = ε_{0} [1 - \frac{ω_{p 0}^{2} (r)}{ω^{2} - i ν_{c} ω} \exp (- α 〈 E^{2} (r, t) 〉)]

with

α = e^{2} / (2 m_{e} k_{B} T_{e} ω_{L}^{2})

the exponential coefficient and

ω_{p0} (r) = e \sqrt{n_{0} (r) / ε_{0} m_{e}}

the initial plasma resonance frequency prior to the laser beam incidence. It is noted that the dielectric constant is dependent on the quantity

〈 E^{2} (r, t) 〉

, from which the nonlinear phenomena ensue, such as the self-focusing of laser beams [9, 24-25]. When the light power of laser beam is low, the exponential item in Eq. (6) approaches unit one,

\exp (\cdot) \approx 1

, and Eq. (6) reduces to the well-known Drude model of metals proposed by Paul Drude over one hundred years ago [26].

3. FDTD implementation

In this section, the modified Drude model deduced above is implemented by the FDTD method. Various approaches have been reported on the FDTD implementation of the familiar Drude model, including the MSE approach, the one with highest accuracy. By applying the MSE approach proposed in [19] to realize the dispersive dielectric constant in the FDTD method via the constitutive equation

D (r) = ε (r, ω) E (r),

we have

E^{n + 1} = \frac{b_{0} D^{n + 1} + b_{1} D^{n} + b_{2} D^{n - 1}}{ε_{0} (a_{0} {\bar{ω}}_{p}^{2} + b_{0})} - \frac{a_{1} {\bar{ω}}_{p}^{2} + b_{1}}{a_{0} {\bar{ω}}_{p}^{2} + b_{0}} E^{n} - \frac{a_{2} {\bar{ω}}_{p}^{2} + b_{2}}{a_{0} {\bar{ω}}_{p}^{2} + b_{0}} E^{n - 1},

where

{\bar{ω}}_{p}^{2} = ω_{p 0}^{2} Δ t^{2} \exp (- α 〈 E^{2} (r, t) 〉)

and the normalized coefficients

a_{0} = 6 + 3 {\bar{ν}}_{c} - {\bar{ν}}_{c}^{2}, a_{1} = 60 - 4 {\bar{ν}}_{c}^{2}, a_{2} = 6 - 3 {\bar{ν}}_{c} - {\bar{ν}}_{c}^{2},

b_{0} = 72 + 36 {\bar{ν}}_{c} - 3 {\bar{ν}}_{c}^{3}, b_{1} = - 144, b_{2} = 72 - 36 {\bar{ν}}_{c} + 3 {\bar{ν}}_{c}^{3}

with

{\bar{ν}}_{c} = ν_{c} Δ t

. It should be noted that

{\bar{ω}}_{p}^{2}

is dependent on the time average of the square of light field

〈 E^{2} (r, t) 〉

. So the most important and critical task in this work is how to extract

〈 E^{2} (r, t) 〉

in the discrete framework of FDTD algorithm.

For the FDTD method with spatially interlaced Yee cells and leap-frog temporal updating, both the space and time discretizes with the spatial sizes $Δ x$ , $Δ y$ , $Δ z$ and time step $Δ t$ , respectively. Since the three discrete field components in the $x$ -axis, $y$ -axis, $z$ -axis directions are defined on different sides of each Yee cell, the square of light field $E^{2} (r, t)$ can be approximately and averagely evaluated at the center of the $(i, j, k)$ th cubic Yee cell by

E^{2} (r, t) \approx E^{2} (\tilde{r}, t) = {\bar{E}}_{x}^{2} (\tilde{r}, t) + {\bar{E}}_{y}^{2} (\tilde{r}, t) + {\bar{E}}_{z}^{2} (\tilde{r}, t),

where

\tilde{r} = e_{x} (i + \frac{1}{2}) Δ x + e_{y} (j + \frac{1}{2}) Δ y + e_{z} (k + \frac{1}{2}) Δ z,

{\bar{E}}_{x} (\tilde{r}, t) = \frac{1}{4} {\begin{cases} E_{x} [(i + \frac{1}{2}) Δ x, j Δ y, k Δ z, t] + E_{x} [(i + \frac{1}{2}) Δ x, (j + 1) Δ y, k Δ z, t] + \\ E_{x} [(i + \frac{1}{2}) Δ x, j Δ y, (k + 1) Δ z, t] + E_{x} [(i + \frac{1}{2}) Δ x, (j + 1) Δ y, (k + 1) Δ z, t] \end{cases}},

{\bar{E}}_{y} (\tilde{r}, t) = \frac{1}{4} {\begin{cases} E_{y} [i Δ x, (j + \frac{1}{2}) Δ y, k Δ z, t] + E_{y} [(i + 1) Δ x, (j + \frac{1}{2}) Δ y, k Δ z, t] + \\ E_{y} [i Δ x, (j + \frac{1}{2}) Δ y, (k + 1) Δ z, t] + E_{y} [(i + 1) Δ x, (j + \frac{1}{2}) Δ y, (k + 1) Δ z, t] \end{cases}},

{\bar{E}}_{z} (\tilde{r}, t) = \frac{1}{4} {\begin{cases} E_{z} [i Δ x, j Δ y, (k + \frac{1}{2}) Δ z, t] + E_{z} [(i + 1) Δ x, j Δ y, (k + \frac{1}{2}) Δ z, t] + \\ E_{z} [i Δ x, (j + 1) Δ y, (k + \frac{1}{2}) Δ z, t] + E_{z} [(i + 1) Δ x, (j + 1) Δ y, (k + \frac{1}{2}) Δ z, t] \end{cases}} .

Note that regardless of the polarization state of laser beam with oscillating light field, the time average of the square of light field over one oscillating period is equal to that over one-half period,. Thus $〈 E^{2} (r, t) 〉$ can be calculated and updated in the interval of half period by

〈 E^{2} (r, t) 〉 = \frac{1}{T_{L}} \int_{t_{0}}^{t_{0} + T_{L}} E^{2} (r, t)d t = \frac{2}{T_{L}} \int_{t_{0}}^{t_{0} + T_{L} / 2} E^{2} (r, t)d t,

where

T_{L} = 2 π/ ω_{L}

is the oscillating period and

t_{0}

is the initial time instant of integration. By applying the composite trapezoidal rule to calculate the half-period definite integral in Eq. (15), one obtains

〈 E^{2} (\tilde{r}, t) 〉 \approx \frac{1}{N_{T}} {E^{2} [\tilde{r}, (n - \frac{N_{T}}{2}) Δ t] + E^{2} (\tilde{r}, n Δ t) + 2 \sum_{m = n - N_{T} / 2 + 1}^{n - 1} E^{2} (\tilde{r}, m Δ t)}

in the time interval

[(n - \frac{N_{T}}{2}) Δ t, n Δ t]

, where

N_{T} = T_{L} / Δ t

is the number of time steps per period and is usually an even integer. If and the spatial discretization number

N_{λ} = λ_{L} / Δ x

and the stability number

n_{CFL} = Δ s / c Δ t

are defined, one has

N_{T} = n_{CFL} N_{λ}

, which is preferred an even integer here.

It can be demonstrated [27] that the relative error of Eq. (16) in numerically calculating $〈 E^{2} (\tilde{r}, t) 〉$ as compared with its analytical counterpart $〈 E^{2} (r, t) 〉$ determined by Eq. (15) can be obtained by summing the individual errors for each segment to give

R e l a t i v e E r r o r = - \frac{4}{T_{L}} {\frac{Δ t^{3}}{12} \sum_{m = n - N_{T} / 2 + 1}^{n} {\frac{d^{2} [\cos^{2} (ω_{L} t + φ_{0 α})]}{d t^{2}} |}_{t = m Δ t}} = 0.

Thus the formula given by Eq. (16) is ultimately accurate if the light field of laser beam has a uniform or slowly varying envelope that is much longer than its wavelength.

4. Numerical examples

In this section, two numerical examples are presented to model the complex ponderomotive interactions of two types of laser beams with plasma using the three-dimensional (3D) FDTD method. The first one is configured for simulating the propagation of a Gaussian beam in collisional plasma. The Gaussian beam is considered with an initial boundary light field

E (r, z = 0, t) = e_{x} E_{0} \exp [- {(\frac{r}{w_{0}})}^{2}] \sin (ω_{L} t)

in the source plane

z = 0

, where

E_{0}

and

w_{0}

are the characteristic amplitude and beam size, respectively, and

r

is the radial distance from the center axis of Gaussian beam. In the 3D-FDTD simulation, the Gaussian beam generated by the TF/SF source condition is with the parameters

λ_{L} = 351 nm

,

E_{0} = 2 \times 10^{10} V/m

and

w_{0} = 2 λ_{L}

, and the plasma is with the parameters

n_{0} = 3 \times 10^{27} / m^{3}

,

T_{e} = 1 \times 10^{4} K

and

ν_{c} = 0.01 ω_{p0}

. The total sizes of the computation region is

200

☓

200

☓

440

cells including the surrounding 20-cells-thick convolution PMLs [28] to completely absorb the outgoing light waves. The spatial and temporal discretization parameters are

N_{λ} = 20

and

n_{CFL} = 2

, so that

N_{T} = 40

. The simulation results are shown in Fig. 1 and Fig. 2. The longitudinal-section views of the spatial distributions of transient field component

E_{x}

and the light-field magnitude of laser beam normalized to

E_{0}

and propagating in free space and in plasma over the

x O z

planes are illustrated in Figs. 1(a)-1(b) and Figs. 1(d)-1(e), respectively. The spatial distribution of plasma density normalized to

n_{0}

over the

x O z

plane is also shown in Fig. 1(f), which is determined by Eq. (4) with the extracted values of

〈 E^{2} (r, t) 〉

by Eq. (16). It is noted that the Gaussian laser beam is focused in the plasma and the plasma density distribution is deformed by the ponderomotive force from its initial uniform distribution with a penetration length approximately

12 λ_{L}

. Figure 2 shows the cross-sectional slices of the spatial distributions of light-field magnitudes normalized to

E_{0}

in the collisional plasma on the several cross-sectional planes at

z = 0,

4 λ_{L},

8 λ_{L},

12 λ_{L},

16 λ_{L}

and

20 λ_{L}

, respectively. The light fields are with stochastic distributions on the cross-sectional planes along the propagating direction while the light energy is gradually absorbed by the plasma (See Visualization 3).

Fig. 1 The longitudinal-section views of the simulation results of Example 1 (see Visualization 1 and Visualization 2). (a) The transient distribution of field component $E_{x}$ of Gaussian beam in free space (a) and in plasma (b) over the $x O z$ plane; the normalized distribution of light field magnitude in free space (b) and in plasma (e) over the $x O z$ plane; (c) free space without plasma; (f) the normalized density profile of plasma deformed by the ponderomotive force over the $x O z$ plane.

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Fig. 2 The spatial distribution of light field magnitude of Gaussian laser beam in plasma (see Visualization 3) on several cross-section planes specified by the different $z$ values where $λ_{L} = 20 Δ z$ .

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The second numerical example is designed for investigating the ponderomotive interaction of a vortex Laguerre-Gaussian (vLG) beam [29] with collisional plasma. The boundary light field of the vortex Laguerre-Gaussian beam can be expressed in cylindrical coordinates $(ρ, φ, z)$ with the complex form [30]

E (ρ, z = 0) = e_{x} E_{0} {(\frac{\sqrt{2} ρ}{w_{0}})}^{m} \exp (- \frac{ρ^{2}}{w_{0}^{2}}) e x p (i m φ),

where

E_{0}

and

w_{0}

are the characteristic amplitude and beam size,

m

is the topological charge of the vortex beam that determines its orbital angular momentum (

m ℏ

). In this simulation, the discretization parameters and the parameters of vLG beam and plasma are the same as the first example except that

w_{0} = 1.5 λ_{L}

and

m = 2

. Different initial phases are assigned to the various grids on the source plane according to Eq. (19). The simulation results of the spatial distributions of light fields and plasma density over the

x O z

plane are illustrated in Fig. 3 and Fig. 4. The longitudinal-section views of the spatial distributions of transient field component

E_{x}

and the light-field magnitudes of laser beams normalized to

E_{0}

and propagating in free space and in plasma over the

x O z

planes are illustrated in Figs. 3(a)-3(b) and Figs. 3(d)-3(e), respectively. The spatial distribution of plasma density normalized to

n_{0}

over the

x O z

plane is also shown in Fig. 3(f). Figure 4 shows the cross-sectional views of the spatial distributions of light-field magnitudes of laser beam normalized to

E_{0}

in the collisional plasma on the several cross-section planes at

z = 0,

4 λ_{L},

8 λ_{L},

12 λ_{L},

16 λ_{L}

and

20 λ_{L}

, respectively. It is noted that the rotation of electron plasma and self-focusing phenomenon is manifested that makes the width of laser ring thinner and resists the diffraction effect of laser beam while the light power is gradually attenuated along the propagating direction as the light energy is absorbed. Due to the vortex of laser beam, the penetration depths of laser ring at four azimuthal angles are found longer than that of others, such as the one illustrated in Fig. 3(f) where the upper branch with

ϕ = 0

has more strong penetrating power (see Visualization 6).

Fig. 3 The longitudinal-section views of the simulation results of Example 2 (see Visualization 4 and Visualization 5). (a) The transient distribution of field component $E_{x}$ of vLG laser beam in free space (a) and in plasma (b) over the $x O z$ plane; the normalized distribution of light field magnitude in free space (b) and in plasma (e) over the $x O z$ plane; (c) free space without plasma; (f) the normalized density profile of plasma deformed by the ponderomotive force over the $x O z$ plane.

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Fig. 4 The spatial distribution of light field magnitude of vLG laser beam in plasma (see Visualization 6) on several cross-section planes specified by the different $z$ values where $λ_{L} = 20 Δ z$ .

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5. Summary

In this work, the numerical approach that based on the finite-difference time-domain (FDTD) method is proposed to model and simulate the complex interaction of high-power laser beams with collisional plasma in the ponderomotive regime for the first time in literature. The nonlinear dielectric constant function exponentially pertaining to the time average of the square of light field is deduced by taking the ponderomotive effect into account and using the balance between the ponderomotive force and the plasma pressure gradient force. The composite trapezoidal rule for numerical integration is applied for numerically exacting the time average of the square of light field. The numerical examples of two laser beams with different cross-sectional field density profiles propagating in collisional plasma are presented for the specified laser and plasma parameters. The numerical results show the anticipated self-focusing and attenuation phenomena of laser beams and the deformation of the spatial density distributions of electron plasma along the beam propagation path. Since the FDTD method is powerful and flexible in the light beam excitation, the propagation properties of various laser beams in plasma can be accurately modeled and simulated. The proposed methodology has a wide application prospect in the study of the complex laser-plasma interactions in a scale from several to hundreds of wavelengths that depends on the computer resource.

Funding

National Natural Science Foundation of China (NSFC) (61575070, 61605049, 61101007); Fujian Province Science Fund for Distinguished Young Scholars (2015J06015); Open Research Fund of Key Laboratory of High Power Laser and Physics of CAS (SGKF201305); Program for New Century Excellent Talents of Fujian Provincial Universities (MJK2015-54); Incubation Program for Outstanding Young Researchers of Fujian Provincial Universities (JA14011); Promotion Program for Young and Middle-Aged Teachers in Science and Technology Research of Huaqiao University (ZQN-YX203).

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Name	Description
Visualization 1: MP4 (231 KB)	Gaussian beam propagating in free space
Visualization 2: MP4 (572 KB)	Gaussian beam propagating in plasma
Visualization 3: MP4 (294 KB)	Gaussian beam propagating in plasma-cross section views
Visualization 4: MP4 (182 KB)	vortex Laguerre-Gaussian beam propagating in free space
Visualization 5: MP4 (449 KB)	vortex Laguerre-Gaussian beam propagating in plasma
Visualization 6: MP4 (312 KB)	vortex Laguerre-Gaussian beam propagating in plasma-cross section views

Modeling the ponderomotive interaction of high-power laser beams with collisional plasma: the FDTD-based approach

Abstract

1. Introduction

2. Physical model

3. FDTD implementation

4. Numerical examples

5. Summary

Funding

References and links

Supplementary Material (6)

Cited By

Figures (4)

Equations (20)

Optics Express