Approximate fields of an ultra-short, tightly-focused, radially-polarized laser pulse in an under-dense plasma: a Bessel-Bessel light bullet

Yousef I. Salamin

doi:10.1364/OE.25.028990

1. Introduction

The quest for ultra-high power and super-intense laser pulses continues [1,2], as demanded by many applications, including particle laser acceleration [3,4], material processing and the study of the fundamental forces in nature. This work is motivated by the need for analytic expressions to properly model the fields of an ultra-short and tightly-focused laser pulse, the focus of current efforts to reach the desired powers and intensities. It extends earlier vacuum-based results [5] by including a plasma background right from the outset.

A vast amount of work has been published, over the past three decades or so, on Bessel beams, the description of whose fields involves Bessel functions of the first kind [6,7]. Bessel beams have been used in many applications, including optical trapping [8,9], optical tweezing [10,11], and precision drilling [12]. Proposals for the use of Bessel beams in data transfer have recently started to emerge [13,14]. Work on Bessel beams has also led to the introduction of laser bullets [15–20]. This paper aims to present a rigorous derivation for the electromagnetic fields of an ultra-short and tightly-focused pulse of pure radial polarization, a Bessel-Bessel laser bullet, propagating in an under-dense plasma. It has been shown rigorously [21] that a pure radially-polarized mode, the description of whose fields in terms of cylindrical coordinates is independent of the azimuthal coordinate θ, carries no angular momentum. The analytic work to be presented here should be a welcome addition to current efforts dedicated to the understanding of laser-plasma interactions which, otherwise, rely mainly on numerical methods, such as particle-in-cell (PIC) simulations.

When an ultrashort and tightly focused laser pulse impinges upon a target, the ensuing ionization processes generate a plasma of some sort. The charge and current densities, ϱ and J, within the plasma serve as sources in the Maxwell equations which describe the laser-plasma system [22]. For an intense laser pulse the plasma response can be highly nonlinear. However, linear response can still be assumed in some situations, for which Maxwell’s equations are equivalent to the inhomogeneous wave equation [23]

(\nabla^{2} - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} - k_{p}^{2}) A = 0,

for the vector potential A, together with a similar equation for Φ, the scalar potential [24]. In this equation, k_p = ω_p/c is an effective plasma wavenumber and c is the speed of light in vacuum. The plasma wavenumber corresponds to the plasma frequency

ω_{p} = \sqrt{n_{0} e^{2} / m ε_{0}}

, in which n₀ is the number density of the ambient electrons, ε₀ is the permittivity of free-space, and m and −e are the mass and charge, respectively, of the electron. It suffices to solve Eq. (1) for the vector potential and then to use the Lorentz condition to obtain the corresponding scalar potential [5,23–29]. The E and B fields then follow, as usual, from the space-and time-derivatives of the potentials [22].

2. Method

Derivation of the central working expression for the vector potential will follow earlier work [5,23–28]. To make this paper as self-contained as possible, the main steps will be outlined here. First, a change of variables is introduced, from (x, y, z) to ( $r = \sqrt{x^{2} + y^{2}}$ , η =(z+ct)/2, ζ = z−ct). In terms of the new variables, Eq. (1) transforms into

(\frac{1}{r} \frac{\partial}{\partial r} r \frac{\partial}{\partial r} + 2 \frac{\partial^{2}}{\partial η \partial ζ} - k_{p}^{2}) A = 0 .

In Eq. (2) the θ-dependent term in the Laplacian has been dropped. This is equivalent to making the assumption that the vector potential is azimuthally-symmetric. The fields derived below from such a vector potential will be purely radially polarized (no azimuthal electric field component assumed present). Attention in this work will be confined to this case in describing a radially-polarized, ultra-short and tightly-focused laser pulse, which carries no angular momentum (analog of the lowest-order Bessel beam). Description of the angular-momentum-carrying Bessel beams in the standard theory [29,30] results when the θ-dependent term in the Laplacian is retained.

With ẑ a unit vector in the propagation direction and k₀ = 2π/λ₀ a central wavenumber corresponding to a central wavelength λ₀, the ansatz

A (r, η, ζ) = \hat{z} a_{0} a (r, η, ζ) e^{i k_{0} ζ},

is suggested. The vector potential amplitude a(r, η, ζ) is next synthesized from Fourier components, via

a (r, η, ζ) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} a_{k} (r, η, k) e^{i k ζ} d k,

and the corresponding inverse Fourier transform. The steps above yield an equation

(\frac{1}{r} \frac{\partial}{\partial r} r \frac{\partial}{\partial r} + 2 i (k + k_{0}) \frac{\partial}{\partial η} - k_{p}^{2}) a_{k} = 0,

for each Fourier component a_k. One exact analytic solution to Eq. (5) has recently been used to obtain expressions for the fields of an ultrashort and tightly focused laser pulse [5,23,24,27,28]. In this paper, a different solution will be sought, which leads to a bullet structure, using the standard technique of separation of the variables. Inserting a_k(r, η, k) = f_kF(r)G(η) in (5) splits it into

r \frac{d^{2} F}{d r^{2}} + \frac{d F}{d r} + k_{r}^{2} r F = 0, and \frac{d G}{d η} + \frac{i}{2} (\frac{k_{r}^{2} + k_{p}^{2}}{k + k_{0}}) G = 0,

where k_r has been introduced as a separation constant and J₀(x) is an ordinary Bessel function of the first kind and order zero. Equations (6) finally yield

a_{k} (r, η, k) ~ f_{k} J_{0} (k_{r} r) exp [- \frac{i}{2} (\frac{k_{r}^{2} + k_{p}^{2}}{k + k_{0}}) η],

with f_k independent of η and r. We will choose [31]

f_{k} = {\begin{array}{l} \frac{\sqrt{2 π}}{Δ k} & | k | \leq \frac{Δ k}{2}; \\ 0, & elsewhere . \end{array}

This is a uniform distribution of wavenumbers [5,24,27] of width Δk and height

\sqrt{2 π} / Δ k

. Note that this renders a_k(0, 0, k) = f_k, which has the Fourier transform a(0, 0, ζ) = f (ζ) = sinc(ζΔk/2). The quantity |f(ζ)|² represents the initial pulse intensity profile, with an approximate full-width-at-half-maximum ∼ 2π/Δk. Thus, it makes sense to adopt L = 2π/Δk as an initial length (spatial extension) for the pulse in its propagation direction.

The full vector potential amplitude now becomes

a (r, η, ζ) = \frac{J_{0} (k_{r} r)}{Δ k} \int_{- \frac{Δ k}{2}}^{\frac{Δ k}{2}} ϕ_{k} e^{i k ζ} d k; ϕ_{k} (η) = exp [- (\frac{i k_{0} α}{k + k_{0}}) η]; α = \frac{k_{r}^{2} + k_{p}^{2}}{2 k_{0}} .

The integral in (9) cannot be carried out in closed analytic form. However, one can power-series expand ϕ_k, viewed as a function of k′ = k + k₀, around k₀

ϕ_{k} = \sum_{m = 0}^{\infty} \frac{k^{m}}{m!} ϕ_{0}^{(m)}; {ϕ_{0}^{(m)} = \frac{\partial^{m} ϕ_{k}}{\partial k^{m}} |}_{k = 0},

and subsequently carry out the integration, one order at a time [5,24,27,28]. Having done all of this, a truncated series for the amplitude of the full vector potential, to order n

A^{(n)} (r, η, ζ) = a_{0} J_{0} (k_{r} r) e^{i k_{0} ζ} \sum_{m = 0}^{n} \frac{ϕ_{0}^{(m)}}{m! i^{m}} \frac{d^{m} f}{d ζ^{m}},

follows. With L = 2π/Δk adopted for the pulse’s initial axial extension, f(ζ) = sinc(πζ/L). Alternatively, the integration (9) may be carried out in terms of gamma functions, which finally leads to

A^{(n)} = \frac{a_{0} J_{0} (k_{r} r) e^{i k_{0} ζ}}{2 π / L} \sum_{m = 0}^{n} \frac{i^{m + 1} ϕ_{0}^{(m)}}{m! ζ^{m + 1}} [Γ (m + 1, \frac{i π ζ}{L}) - Γ (m + 1, - \frac{i π ζ}{L})] .

It is plausible to think of A⁽ⁿ⁾ as describing an ultra-short and tightly-focused, radially-polarized, pulse. The presence of J₀ suggests that this object is the short-pulse analog of a zero-order Bessel beam. Thus, the running index, m, and the order of the truncation, n, have nothing to do with the indices of a high-order Bessel beam and/or angular momentum. This pulse carries no angular momentum. Moreover, the assumption will be made that the first term in the series contributes the most, while terms beyond the first contribute negligibly [24]. Both equations (11) and (12) yield the following expression for the zeroth-order vector potential

A^{(0)} (r, η, ζ) = a_{0} J_{0} (k_{r} r) j_{0} (\frac{π ζ}{L}) e^{i φ (0)},

in which the sinc function has been replaced by j₀, the zero-order spherical Bessel function of the first kind, and

φ^{(0)} = φ_{0} + k_{0} ζ - α η, φ_{0} = constant .

Surface plots of |A⁽⁰⁾/a₀|² are shown in Fig. 1. This quantity is sharply peaked in the rz–plane. Extensions of its central peak in the r and z directions are determined by the first zeros of J₀ and j₀, respectively. The centroid of the intensity profile advances forward at the group velocity $v_{g}^{(0)}$ (to be introduced shortly) while its overall shape is preserved without distortion. The plasma effects cannot be shown in such a plot, because k_p appears only in the phase factor of the vector potential. Such effects will show up in the field components.

Fig. 1 Surface plots of the pulse’s scaled vector potential |A⁽⁰⁾/a₀|² as a function of the axial coordinate z and the transverse coordinate r, in units of λ₀ = 1μm. The figure displays snapshots at t = 0, 1 fs, 1 ps, and 1 ns. Note that r has been allowed to take on negative values in order to bring out the cylindrical symmetry of the fields, with the understanding that, in reality, r ≥ 0. This is also done in Figs. 3–5 below.

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3. A subluminal group velocity

With current efforts [13] aimed at employing Bessel beams in data transfer in mind, a digression is made next to show that the pulse travels in space at a subluminal group velocity. The frequency and axial wavenumber of the pulse may be obtained, respectively, from [23]

ω_{0} = - \frac{\partial φ^{(0)}}{\partial t} = c k_{0} (1 + ∊); ∊ = \frac{α}{2 k_{0}},

k_{z}^{(0)} = \frac{\partial φ^{(0)}}{\partial k} = k_{0} (1 - ∊) .

From Eqs. (15) and (16) follows the dispersion relation

{[\frac{ω^{(0)}}{c}]}^{2} - {[k_{z}^{(0)}]}^{2} = 4 ∊ k_{0}^{2} .

Next, the phase and group velocities may be found from

v_{p}^{(0)} = \frac{ω^{(0)}}{k_{z}^{(0)}} = c [\frac{1 + ∊}{1 - ∊}] > c .

v_{g}^{(0)} = \frac{d ω^{(0)}}{d k_{z}^{(0)}} = \frac{c^{2}}{v_{p}^{(0)}} \to v_{p}^{(0)} v_{g}^{(0)} = c^{2} .

Equations (18) and (19) together imply that

v_{g}^{(0)} < c

.

4. The zeroth-order fields

Most practical applications require knowledge of the fields E and B which model the laser pulse. The zeroth-order fields are next derived from E = −∇Φ − ∂A/∂t and B = ∇ × A, employing cylindrical coordinates [23,24]. For example, B_θ = −∂A/∂r, according to Eq. (43) in [24]. This leads to two electric field components (axial and radial) and one azimuthal magnetic component, given respectively, by

\begin{array}{l} E_{z}^{(0)} & = & (\frac{E_{0}}{k_{0}}) e^{i φ^{(0)}} J_{0} (k_{r} r) {[Q_{2} + \frac{Q_{3}}{R} - \frac{Q_{1} Q_{4}}{R^{2}}] j_{0} (\frac{π ζ}{L}) \\ + \frac{1}{ζ} [1 - \frac{2 Q_{1}}{R} - \frac{Q_{4}}{R^{2}}] cos (\frac{π ζ}{L})}, \end{array}

E_{r}^{(0)} = E_{0} (\frac{k_{r}}{k_{0}}) e^{i φ^{(0)}} \frac{J_{1} (k_{r} r)}{R} [Q_{1} j_{0} (\frac{π ζ}{L}) + \frac{cos (π ζ / L)}{ζ}],

c B_{θ}^{(0)} = E_{0} (\frac{k_{r}}{k_{0}}) e^{i φ^{(0)}} J_{1} (k_{r} r) j_{0} (\frac{π ζ}{L}) .

In Eqs. (20)–(22) E₀ = ck₀a₀, and

Q_{1} = i k_{0} - \frac{1}{ζ} - \frac{i α}{2}; Q_{2} = i k_{0} - \frac{1}{ζ} + \frac{i α}{2},

R = Q_{2} + \frac{π}{L} cot (\frac{π ζ}{L}),

Q_{3} = \frac{2 Q_{1}}{ζ} + k_{0}^{2} + \frac{π^{2}}{L^{2}} + \frac{α (α - 4 k_{0})}{4}; Q_{4} = - \frac{1}{ζ^{2}} + \frac{π^{2}}{L^{2}} {csc}^{2} (\frac{π ζ}{L}) .

For propagation in vacuum, Eqs. (20)–(25) give the field components in the limit k_p → 0, or

α \to k_{r}^{2} / (2 k_{0})

. Note also that

E_{r}^{(0)}

and

B_{θ}^{(0)}

vanish identically at all points on the z–axis, due to the presence upfront of J₁(k_rr), which vanishes at r = 0, and is independent of the order of truncation, n, as may be inferred from close inspection of Eq. (12) and the fact that ∂J₀(x)/∂x = −J₁(x). These components have the hollow intensity profiles shown in Figs. 2(c)–(f). All profiles exhibit uniformly lit rings, separated by dark ones, whose inner and outer radii may be determined from the zeros of J₀(x) and J₁(x).

Fig. 2 Density plots (upper panel) and surface plots (lower panel) of the initial (t = 0) axial, radial and azimuthal intensity profiles in the focal plane (z = 0) of an ultrashort (L = 0.8λ₀) and tightly focused (w₀ = 0.9λ₀) laser pulse of central wavelength λ₀ = 1μm, propagating in vacuum (n₀ = 0). Other parameters used are φ₀ = 0 and k_r = x₁/w₀, where x₁ = 2.40483 is the first zero of J₀(x).

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Only relatively recently have finite-energy Bessel beams been realized experimentally [15–20]. A pulse of finite extension (or bullet) with which this paper is concerned, does not suffer from the infinite-energy issue. Neither does it get distorted during propagation. In Figs. 3 and 4, vacuum-based intensity profiles of the electric field components are shown at times 0, 1 fs, 1 ps, and 1 ns. It is hard to detect any distortions before the centroid of the pulse has moved a distance of about 30 cm. For the parameters employed in producing Figs. 3 and 4, some distortion becomes evident after 70 cm of propagation (not shown).

Fig. 3 Same as Fig. 1, but for ${| E_{z}^{(0)} / E_{0} |}^{2}$ , and φ₀ = π/2. The pulse is generated at t = 0, with its centroid at the origin of coordinates (r = z = 0). The centroid subsequently advances to positions consistent with z ∼ ct, at (b) t = 1 fs, (c) t = 1 ps, and (d) t = 1 ns.

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Fig. 4 Same as Fig. 3, but for ${| E_{r}^{(0)} / E_{0} |}^{2}$ .

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Figures 1, 3 and 4 demonstrate that the pulse propagates undistorted by diffraction in vacuum. Similar propagation characteristics in an under-dense plasma (n₀ = 10²⁰ cm⁻³) are demonstrated in Figs. 5 and 6. Figure 5, displaying snapshots of |E_r/E₀|² at t = 10, 40, 70, and 100 ps, shows that the radial intensity profile propagates undistorted, neither by diffraction nor by dispersion. Note that the peak intensity in Fig. 5 is almost twice that of Fig. 4, for the parameter sets used. As expected, the plasma electrons and ions act as sources of radiation which add to the incident fields. Figure 6 shows the on-axis (r = 0) axial intensity profile |E_z/E₀|² at different times. Vacuum-and plasma-based profiles are displayed at times t = 0, 1 fs, 1 ps, and 1 ns. The profiles do not seem to suffer from any distortion during propagation. The effect of dispersion due to interaction of the pulse with an under-dense plasma, however, is demonstrated clearly by the red curves (plasma-based) not coinciding with the blue ones (vacuum-based).

Fig. 5 Same as Fig. 4, but for propagation in an under-dense plasma of ambient electron density n₀ = 10²⁰ cm⁻³, and for different times.

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Fig. 6 Snapshots showing the on-axis (r = 0) scaled axial intensity |E_z/E₀|² of a pulse of the same parameters as in Fig. 2, as a function of the propagation distance. Black: in vacuum, and red: in a plasma of ambient electron density n₀ = 10²⁰ cm⁻³.

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5. Concluding remarks

In conclusion, we have derived analytic expressions for the fields of an ultra-short and tightly-focused radially-polarized laser pulse (a Bessel-Bessel laser bullet) from the appropriate vector and scalar potentials. The pulse may be thought of as the finite-extension, finite-energy analog of a zero-order, infinite-energy Bessel beam, which carries no angular momentum. Employing only the leading terms in a power series expansion of the fields, such an object has been shown to propagate, undistorted, in vacuum and in an under-dense plasma. Higher-order corrections to the leading terms in the fields can, in principle, be obtained analytically, but become quite complex with increasing order. Still, the fully analytic expression found in this paper for the vector potential can be used to numerically calculate the fields to any desired order, using a truncated series. The higher-order corrections do not alter the structures of the intensity profiles discussed in this work, nor do they drastically affect the main propagation characteristics of the pulse. All of these findings and conclusions may be traced back to the assumption that the response of the under-dense plasma is linear.

Funding

American University of Sharjah (Faculty Research Grant FRG17-T-14); Alexander von Humboldt Foundation (Re-invitation).

Acknowledgments

Part of this work was done in the group of C. H. Keitel at Max Planck Institute for Nuclear Physics in Heidelberg (Germany). The work has been supported in part by the Alexander von Humboldt Foundation, through a Research Re-invitation, and partially by the American University of Sharjah (United Arab Emirates) through a Faculty Research Grant (FRG17-T-14).

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Approximate fields of an ultra-short, tightly-focused, radially-polarized laser pulse in an under-dense plasma: a Bessel-Bessel light bullet

Abstract

1. Introduction

2. Method

3. A subluminal group velocity

4. The zeroth-order fields

5. Concluding remarks

Funding

Acknowledgments

References and links

Cited By

Figures (6)

Equations (25)

Optics Express