Propagation characteristics of Airy-Bessel wave packets in free space

Zhijun Ren; Qiong Wu; Hefa Mao; Yile Shi; Changjiang Fan

doi:10.1364/OE.21.004481

1. Introduction

Laser is widely used in many fields of science research as an information carrier, source of energy and probe. In some real applications, people pay more attention to the laser focus than to the laser system itself because the beam focus of laser systems does all the actual work. For example, as a source of energy in the field of laser-matter interactions, acquiring a focused beam suitable for research demands in different fields has become an important branch of laser technology. Among the laser focus characteristics, the depth of focus is a critical factor. In such research areas as laser filamentation, laser-induced lightning, optical manipulation, and long-range signal transmission, the laser beam is expected to retain stable focused modes over long propagating distances [1–3]. Moreover, as a probe in researching the ultrafast processes of the microcosm, the laser focal beam is important to be spatiotemporally stable because scale stability of the laser focal beam is the primary requirement of accurate measurement. However, diffraction and dispersion are intrinsic properties of light in the propagation process. Hence, general pulsed focal beams are difficult to propagate without spatiotemporal spreading.

By counteracting the diffraction and dispersion of wave packets using the nonlinear effects of some media, three-dimensional (3D) spatiotemporal nonspreading soliton waves were first generated in nonlinear media. These localized wave packets are also called nonlinear light bullets because these types of wave packets can only stably propagate in nonlinear media [4,5], which significantly undermines their application scope [1,5]. Hence, a stable linear localized wave packet without spatiotemporal spread during propagation in free space is needed in some scientific research fields. These 3D wave packets are commonly called linear light bullets (LLBs) [5]. LLBs can usually be used as a spatiotemporally stable focused beam with a great focus depth for physical applications [1,2]. Therefore, LLBs are of unparalleled excellence as a source of energy or for probing in scientific research.

An important breakthrough for generating LLBs in free space was the discovery of spatial diffraction-free mode solutions to the wave equation, which are typically Bessel and Airy function solutions [6,7]. Subsequently, spatial Bessel and Airy beams were experimentally generated [6,8]. The discovery of diffraction-free beams is of great significance and has even fundamentally changed the operating paradigm in some research fields [9–12]. Theoretically, ideal infinite plane waves are also diffraction-free mode solutions of the wave equation. However, Airy and Bessel beams can be regarded as focus wave modes because their central spot radius can be narrowed to several wavelengths [6–8].

Compared with the continuous wave diffraction-free beams, pulsed diffraction-free beams are more useful in some practical applications because of their super intensity power density [13–17]. Although pulsed diffraction-free beams do not diffract during propagation, they cannot easily guarantee stable temporal propagation in a medium and in free space because of the influence of material dispersion and spatially induced dispersion (SID) [18–28]. Given that beams with the Bessel and Airy distribution functions have an invariant propagation mode in free space, these distributions can be potentially used not only in spatial but also in temporal domains. By combining spatial Bessel distribution beams in the transverse plane with temporal Airy distribution pulses in time, Airy-Bessel configuration wave packets with a particle-like nature were first generated on the basis of the grating-telescope combination [1]. As shown in the past research, in free space or in dispersive media, ideal spatial Bessel beams can propagate without diffraction [6,19,20] and ideal temporal Airy pulses can propagate without dispersion [2,7,27]. It is very natural to think that ideal Airy-Bessel wave packets (ABWs) with spatial Bessel distribution and temporal Airy distribution can be regarded as spatiotemporally stable localized optical wave packets, thus ABWs also can be called Airy-Bessel LLBs [1]. However, as spatiotemporal localized Airy-Bessel wave packets, their propagation characteristics becomes complex because of space–time couplings [27,29]. Hence, determining whether or not ideal ABWs are absolutely stable LLBs, with spatiotemporal characteristics similar to those of the ideal spatial Bessel beams and the ideal temporal Airy pulses, is important. In this paper, we build a more comprehensive theoretical model to address the propagation of ABWs in free space. Some significant research results are obtained, which may help us to clearly understand the propagation characteristics of ABWs for the convenience of their effective uses in scientific researches.

2. Theory

The pulsed beam propagation problem in free space is simple since free space is linear and each Fourier component of pulsed beams can be taken independently [18,24,30]. Starting from Maxwell's wave equation for electromagnetic wave propagation, the monochromatic component of pulsed beams is governed by the Helmholtz equation in the frequency domain

[\nabla^{2} + k {(ω)}^{2}] U (x, y, z; ω) = 0,

where

\nabla^{2} = \partial_{x}^{2} + \partial_{y}^{2} + \partial_{z}^{2}

, U is the Fourier component of the pulse at the frequency ω. In free space, the wave number is

k (ω) = ω / c,

c being light speed in free space.

In 1987, Durnin proved that monochromatic axis-symmetric Bessel distribution beams satisfy Eq. (1) [6]. For pulsed Bessel beams, the amplitude of the optical field U(r, z; ω) in the frequency domain can be expressed as [3,20,28,30]

U (r, z; ω) = J_{0} (r / r_{0}) \exp [i β (ω) z] f (0, ω - ω_{0}),

where J₀(·) represents the zero-order Bessel function of first kind, and

r_{0}

is the the radial half-width of the central peak of the spatial Bessel distribution beam. r≡(x + y)^1/2 is the transverse coordinate in a cylindrical system and f(0, ω–ω₀) is the temporal spectrum distribution of pulsed Bessel beams having a central frequency ω₀ at the input plane z = 0.

Pulsed Bessel beams mainly contain two classes: beams with frequency–dependent r₀ and beams with constant waist r₀ [18]. If r₀ is proportional to the frequency ω, the pulsed Bessel beam is called a Bessel-X wave, featuring a profile with a long tail that accompanies the main hump. Bessel-X waves can resist the effects of diffraction and dispersion, and their shape in a plane along the wave axis is X-like [21,28,31–34]. Bessel beams with fixed beam waist r₀ are more useful in some scientific researches because their beam structures are spatially steady [1,19,30]. In this paper, we mainly investigated the latter. Hence, r₀ is constant and independent of the frequency ω. In Eq. (3), the longitudinal wave vector β(ω) is given by [18–20,24,30,35]

β^{2} (ω) = k^{2} (ω) - 1 / r_{0}^{2} = ω^{2} / c^{2} - 1 / r_{0}^{2},

To clarify the propagation characteristics of localized wave packets, β is expanded in a Taylor series around the carrier frequency ω₀, similar to classical theory of pulsed beam propagation in dispersive media [18,19,29,30,36]

β (ω) = β_{0} + \sum_{m = 1}^{\infty} \frac{β_{m}}{m!} {(ω - ω_{0})}^{m},

Where β₀ is the value β at a carrier frequency ω₀. The parameter β_m denotes the mth-order dispersion coefficient; specifically, we obtain

β_{0} = k_{0} \sqrt{1 - \frac{1}{r_{0}^{2} k_{0}^{2}}},

β_{1} = \frac{r_{0} k_{0}}{c \sqrt{r_{0}^{2} k_{0}^{2} - 1}},

β_{2} = - \frac{r_{0}}{c^{2}} {(r_{0}^{2} k_{0}^{2} - 1)}^{- \frac{3}{2}},

β_{3} = \frac{3 r_{0}^{3} k_{0}}{c^{3}} {(r_{0}^{2} k_{0}^{2} - 1)}^{- \frac{5}{2}},

where, k₀ = ω₀/c = 2π/λ₀. For a given ABW, the central wavelength λ₀ is provided. We set the central wavelength λ₀ to 800nm in this paper.

Equation (6) shows that the spatial distribution parameter r₀ of pulsed Bessel beams is the only definitive factor of influencing the dispersion coefficient when the central wavelength of pulsed Bessel beams is given. Hence, such dispersion is referred to as SID [18–20,24,35]. SID inevitably exists, and it is worth pondering why we usually ignore the influence of SID when researching the propagation of other pulsed beams with finite beam width.

From Eq. (6), it is clear that the corresponding dispersion coefficient of SID is insignificant for ordinary pulsed beams with large beam width r₀ and its influence on the pulse shape can be ignored. For focused beams with small beam width r₀, the n-order dispersion value β_nz is still negligible although the coefficient of SID is large. This is primarily because the dispersion is an accumulated process [36], and the effective propagation distance of the focus beams determined by the diffraction length L_diff = γr₀² (see Ref. 18) is very short, generally dozens or hundreds of wavelengths of the carrier. Over the diffraction length, the focus beams quickly broaden. Thus, the effect of SID can still be ignored while studying the propagation of ordinary pulsed beams. SID can evidently influence the temporal characteristics of diffraction-free localized focus beams such as Bessel and Airy beams with a small beam width of the main hump decided by r₀, which generally equals to several wavelengths of the carrier, and a very long diffraction length z, theoretically being infinite [6,20,29,34].

After substitution of Eq. (5) into Eq. (3), we obtain the following equation

U (r, z; ω) = J_{0} (r / r_{0}) f (0, ω - ω_{0}) \exp {i z \cdot [β_{0} + \sum_{n = 1}^{\infty} \frac{β_{n}}{n!} {(ω - ω_{0})}^{n}]},

Compared with ultrashort pulsed beams propagating in a nonlinear medium [5], Eq. (7) shows that beam propagation in free space is comparatively simple because each Fourier component of pulse propagates independently. SID cannot change the initial pulse spectrum distribution f(0, ω–ω₀), i.e., new spectral components are not generated during the propagation of pulsed Bessel beams in free space, which is very consistent with the past research results [18,22,35]. However, SID changes the phase of every spectral component, which depends on the coefficient of SID β_n and the propagation distances z. The change in spectral phase results in a temporal distribution change in pulsed beams. By performing the inverse Fourier transform, the time–space distribution of pulsed Bessel beams is given as

\begin{array}{l} Ψ (r, z; t) = \frac{1}{\sqrt{2 π}} J_{0} (r / r_{0}) \times \\ \int_{- \infty}^{\infty} f (0, ω - ω_{0}) \exp {i z [β_{0} + \sum_{n = 1}^{\infty} \frac{β_{n}}{n!} {(ω - ω_{0})}^{n}]} \exp (- i ω t) d ω . \end{array}

Equation (8) shows that the spatial distribution of a pulsed Bessel beam propagates as a shape-preserving Bessel beam structure during propagation in free space. Similarly to the past researches [1,19,28], pulsed Bessel beams are indeed spatially diffraction-free beams. However, the temporal distribution of pulsed Bessel beams becomes complicated during propagation. We introduce the pulse envelope evolution function

E (r, z; t)

through the following equation

Ψ (t; r, z) = E (r, z; t) \exp [i (β_{0} z - ω_{0} t)] .

Hence, E(r, z; t) can be written in the following form

E (r, z; t) = J_{0} (r / r_{0}) Φ (z, t) .

Apparently, Φ(z, t) represents the temporal profile of pulsed Bessel beams at different propagation distances, as shown by the following equation

Φ (z, t) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} f (0, ω - ω_{0}) \exp [i z \sum_{n = 1}^{\infty} \frac{β_{n}}{n!} {(ω - ω_{0})}^{n} - i (ω - ω_{0}) t] d ω .

In Eq. (11), using methods similar to those being proposed in references [18,36], we introduce the retarded time in the frame traveling at the light group velocity v_g, that is T = t–z/v_g = t–β₁z because β₁ does not influence the temporal profile. Moreover, similar to material dispersion, the high-order dispersion terms is negligible enough and can be ignored [18,36]. Considering only the influence of quadratic and cubic SID, Eq. (11) may be expressed as

\begin{array}{l} Φ (z, T) = \frac{1}{\sqrt{2 π}} \times \\ \int_{- \infty}^{\infty} f (0, ω - ω_{0}) \exp [\frac{i}{2} β_{2} {(ω - ω_{0})}^{2} z + \frac{i}{6} β_{3} {(ω - ω_{0})}^{3} z - i (ω - ω_{0}) T] d ω . \end{array}

Equation (12) is the pivotal expression for studying the temporal propagation characteristics of ABWs in free space. For the purposes of comparison, the temporal propagation characteristics of common Gaussian-Bessel wave packets with temporal Gaussian distributions and spatial Bessel distributions in free space will also be considered. The initial temporal Gaussian distribution pulse is given as [36]

Φ (0, T) = \exp (- \frac{T^{2}}{2 T_{0}^{2}}),

Where T₀ is the full width at half maximum of the Gaussian pulse. By performing Fourier transformation to Eq. (13), then substituting this result into Eq. (12), the temporal distribution of the Gaussian-Bessel beam after propagation at a distance z can be calculated. References [18,21] provide detailed research results on the temporal propagation characteristics of Gaussian-Bessel wave packets under the influence of SID. The temporal broadening factor is given as

η = \frac{T}{T_{0}} = {[1 + {(\frac{z}{L_{2}})}^{2} + \frac{1}{4} {(\frac{z}{L_{3}})}^{2}]}^{1 / 2} .

Both reference [18,21] and Eq. (14) in this paper show that pulsed Bessel beams are not temporally stable although spatially stable when they propagate in free space. Hence, we believe that traditional pulsed Bessel beams with a temporal Gaussian distribution and spatial Bessel distribution are only spatially diffraction-free beams instead of spatiotemporal nonspreading LLBs. Yet, this is what people expect when generating LLBs. An important work is to generate ABWs, which are spatiotemporally stable during propagation, based on the characteristics of Airy and Bessel functions. Imposing a cubic spectral phase on initial Gaussian pulses (which is what pulsed lasers usually produce) and using a grating-telescope compressor are key methods of obtaining temporal Airy distribution profiles. Reference [1] has proven that this can make Gaussian-Bessel beams evolving as ABWs. Using the same method, reference [2] has also proved that they can make Gaussian-Airy beams with temporal Gaussian profile and spatial 2D Airy profile evolving as Airy-Airy wave packets with 3D spatiotemporal Airy profiles. For an ideal ABW, the temporal pulse distribution can be written as the Airy function

Φ (0, T) = A i r y (T / T_{0}),

where T₀ is the temporal scaling parameter that approximately determines the full width at half maximum (FWHM) of the main lobe width of the Airy pulse, since the main lobe of the square of the Airy function has a Gaussian pulse-like intensity distribution [37]. Fourier transform theory can be used to perform a time-frequency transform. If f(0, ω) is the Fourier transform of the incident pulse shape Φ(0, T), we obtain

f (0, ω - ω_{0}) = \int_{- \infty}^{\infty} A i (T / T_{0}) e^{i (ω - ω_{0}) T} d T = \frac{1}{\sqrt{2 π}} T_{0} e^{i T_{0}^{3} {(ω - ω_{0})}^{3} / 3} .

Substituting Eq. (16) into Eq. (12), the following equation is given

Φ (z, T) = \frac{1}{2 π} \int_{- \infty}^{\infty} \exp {i [\frac{{(ω - ω_{0})}^{3}}{3} (T_{0}^{3} + \frac{β_{3} z}{2}) + \frac{β_{2} z}{2} {(ω - ω_{0})}^{2} - T (ω - ω_{0})]} d ω,

where, it is readily known that sgn(β₂) = –1 and sgn(β₃) = 1 from Eqs. (6.2) and (6.3). It is worth noticing that Eq. (17) is also used in examining Airy pulse propagation in linear dispersive media when β₂ and β₃ are respectively quadratic and cubic dispersion coefficient of media [38]. Hence, all the subsequent results are also suitable for analyzing the temporal propagational characteristics of ABWs in dispersive media, with β₂ and β₃ representing respectively the quadratic and cubic dispersion coefficients of media.

To study the temporal propagational characteristics of ABWs in free space based on Eq. (17), we scale the propagational distance of the pulse using the quadratic and cubic SID length from Eq. (18), which is similar to the case of pulse propagation in dispersive media [36].

L_{2} = \frac{T_{0}^{2}}{| β_{2} |} = \frac{c^{2} T_{0}^{2}}{r_{0}} {(r_{0}^{2} k_{0}^{2} - 1)}^{\frac{3}{2}} .

L_{3} = \frac{T_{0}^{3}}{| β_{3} |} = \frac{c^{3} T_{0}^{3}}{3 r_{0}^{3} k_{0}} {(r_{0}^{2} k_{0}^{2} - 1)}^{\frac{5}{2}} .

Let

ω - ω_{0} = {(T_{0}^{3} + \frac{β_{3} z}{2})}^{- 1 / 3} φ = \frac{1}{T_{0}} {(1 + \frac{z}{2 L_{3}})}^{- 1 / 3} φ

Because of

d ω = d (ω - ω_{0})

in Eq. (17), it can be transformed into [38]

\begin{array}{l} Φ (z, T) = \frac{{(1 + z / 2 L_{3})}^{- 1 / 3}}{2 π} \int_{- \infty}^{\infty} \exp {i [\frac{φ^{3}}{3} + \frac{z}{2 L_{2}} {(1 + \frac{z}{2 L_{3}})}^{- \frac{2}{3}} φ^{2} + {(1 + \frac{z}{2 L_{3}})}^{- \frac{1}{3}} \frac{T}{T_{0}} φ]} d φ \\ ^{}^{}^{} = (^{1 + \frac{z}{2 L_{3}}) - 1 / 3} \cdot \exp {i \frac{z}{2 L_{2}} (^{1 + \frac{z}{2 L_{3}}) - \frac{2}{3}} [\frac{z^{2}}{6 L_{2}^{2}} (^{1 + \frac{z}{2 L_{3}}) - \frac{4}{3}} - (^{1 + \frac{z}{2 L_{3}}) - \frac{1}{3}} \frac{T}{T_{0}}]} \\ ^{}^{}^{}^{} \times A i [(^{1 + \frac{z}{2 L_{3}}) - \frac{1}{3}} \frac{T}{T_{0}} - \frac{z^{2}}{4 L_{2}^{2}} (^{1 + \frac{z}{2 L_{3}}) - \frac{4}{3}}] . \end{array}

Equation (20) shows that the temporal distribution of ABWs can still be described using the Airy function after certain propagation distances. However, Eq. (20) is not the propagation-invariant Airy solution because the arguments of the Airy functions in Eqs. (15) and (20) are different. Compared with the incident Airy pulse, quadratic and cubic SID influence the temporal propagation characteristics of ABWs. Further detailed analysis is introduced in the next section.

3. Analysis and discussion

In this section, we first elucidate the temporal propagational characteristics of ABWs under the influence of quadratic and cubic SID separately. The joint influence of both quadratic and cubic SID are discussed subsequently.

3.1 The influence of quadratic SID

To analyze the propagation characteristics of ABWs in free space as influenced by quadratic SID (i.e. spatially induced group velocity dispersion), we take β₃ = 0, that is L₃ = ∞. Equation (20) can then be written as

Φ (z, T) = \exp [i \frac{z}{2 L_{2}} (\frac{z^{2}}{6 L_{2}^{2}} - \frac{T}{T_{0}})] A i (\frac{T}{T_{0}} - \frac{z^{2}}{4 L_{2}^{2}}),

where, the phase term

\frac{z}{2 L_{2}} (\frac{z^{2}}{6 L_{2}^{2}} - \frac{T}{T_{0}})

is a real function and does not influence the intensity distribution of the Airy pulse. Comparing Eqs. (15) and (21), the incident Airy distribution pulse shows an invariable propagation mode after arbitrary propagation distances z in free space. Substituting Eq. (21) into Eq. (10), we know that ideal ABWs can spatiotemporally propagate undistorted at infinite distances from the source. However, a longitudinal displacement of time also exists, which indicates a temporal longitudinal acceleration in the absence of any external potential [1,7,39,40]. The temporal longitudinal acceleration of Airy pulses during propagation is analogous to the spatial transverse displacement characteristics of Airy beams [2,7,8]. The displacement value is z²/4L₂². The main lobe of the Airy pulse moves against the oscillating tail of the Airy pulse. All the sidelobes contribute to the main one through interference. The temporal acceleration of Airy pulses is physically different from the spatial acceleration of Airy beams. The spatial acceleration of an Airy beam means bending its trajectory in space, whereas the temporal acceleration of an Airy pulse means changing its actual group velocity [37]. Hence, it is important to specify the temporal acceleration of ABWs. Equation (21) shown in this paper is identical to Eq. (2) of reference [1], which means that we can theoretically acquire research results identical to those of reference [1] when quadratic SID is considered only. In this situation, ABWs can be regarded as spatiotemporally stable LLBs [1]. Generally, the quadratic dispersion coefficient is much larger than the cubic dispersion coefficient when an optical pulse propagates not only in a medium but also in free space. Obviously, quadratic dispersion is the main factor that influences the pulse propagation. Hence, without particularly stringent conditions, regarding ideal ABWs as absolutely LLBs is reasonable when cubic SID can be ignored.

3.2 The influence of cubic SID

To analyze the influence of cubic SID on ABWs, let β₂ = 0, i.e., L₂ = ∞, Eq. (20) can be written as

Φ (z, T) = {(1 + \frac{z}{2 L_{3}})}^{- 1 / 3} \cdot A i [{(1 + \frac{z}{2 L_{3}})}^{- \frac{1}{3}} \frac{T}{T_{0}}] .

Equation (22) indicates that the main lobe width of the Airy pulse gradually increases as (1 + z/2L₃)^1/3 during propagation, and the amplitude is correspondingly reduced which is in accordance with the law of conservation of energy. The value of L₃ determines the influence extent of the cubic SID [36]. Therefore, despite being ideal ABWs with infinite energy, the temporal Airy distribution pulse is not an absolutely undistorted under the influence of cubic SID that inevitably exists because of space–time coupling effects [27,29,41]. Such research results slightly differ from those of reference [1], the main difference being that the latter did not consider cubic dispersion. It is necessary to note, from Eq. (22), that the broadening velocity of the Airy pulse is comparatively small. For a not extremely long propagation distance, the influence of cubic SID on the Airy distribution pulse is also sufficiently small that it can be ignored because L₂<<L₃. Even when the propagation distance is one cubic SID length L₃, the pulse width only broadens 1.140 times. In some scientific research fields, ABWs can be regarded as spatiotemporal localized LLBs. However, we cannot always regard ABWs as LLBs in all situations because the influence of cubic SID on ABWs cannot be always ignored. What situations where ideal ABWs can be regarded as LLBs, and what situations where ideal ABWs cannot be regarded as LLBs, are worth investigating and will be expounded on in the following.

To determine the relationship between specific physical parameters and the propagating characteristics of ABWs, the broadening factor η of ABWs during propagation is given as

η = \frac{T}{T_{0}} = {(1 + z / 2 L_{3})}^{1 / 3} = {[1 + \frac{3 z r_{0}^{3} k_{0}}{2 c^{3} T_{0}^{3}} {(r_{0}^{2} k_{0}^{2} - 1)}^{\frac{- 5}{2}}]}^{1 / 3} .

Equation (23) shows that, for a given ABW with a certain central wavelength λ₀, the three physical parameters influencing the broadening factor η are respectively z, r₀ and T₀. Usually, r₀ is larger than the wavelength λ₀ of the carrier; consequently, there is

r_{0}^{2} k_{0}^{2} = 4 π^{2} \cdot {(\frac{r_{0}}{λ_{0}})}^{2} > > 1

in Eq. (18.2), and the cubic SID length can be written as

L_{3} \approx c^{3} r_{0}^{2} T_{0}^{3} k_{0}^{4} / 3.

Hence Eq. (23) is transformed into

η \approx {[1 + \frac{3 z}{2 c^{3} r_{0}^{2} T_{0}^{3} k_{0}^{4}}]}^{\frac{1}{3}} .

Apparently, T₀ is the main factor influencing the pulse width because η is related to the third power of T₀. A second important factor is r₀ because η is related to the square of r₀. The least important factor is z because η is only related to the first power of z. For a given propagation distance z, the shorter the main lobe of the temporal Airy pulse, the more quickly the ABW broadens; Additionally, the narrower the hump of the spatial Bessel beams, the more quickly the ABW broadens. The central localized peak hump radius of spatial Bessel beams can be narrowed down to several wavelengths [1,6]. Likewise, the main lobe of the temporal Airy pulse can be narrowed down to few cycles of the carrier wave [1,8]. For these ultra-localized ABWs, L₃ is not too long and the influence of cubic SID on ABWs cannot be simply ignored. For other ABWs with large r₀ and T₀, the influence of cubic SID should be considered if the propagation distance z is long enough.

In different scientific research fields, we require ABWs with different scale stability. Hence, to assure the scale stability of ABWs meeting the special requirements of some research experiments, we have to choose ABWs with proper spatiotemporal parameters for given effective operating distances. Hence, presuming η to be constant, Eq. (25) can then be transformed into

\frac{z}{r_{0}^{2} T_{0}^{3}} \leq \frac{2}{3} \cdot (η^{3} - 1) \cdot c^{3} k_{0}^{4}

Equation (26) gives the mutual influence relations among r₀, T₀ and z for a given η. Generally, the ABW is believed to have nonspreading propagation characteristics and can be regarded as a LLB if its broadening factor is smaller than 1.05 during propagation. When η<1.05, Eq. (26) can be expressed as

\frac{z_{s}}{r_{0}^{2} T_{0}^{3}} \leq 0.1051 \cdot c^{3} k_{0}^{4} .

The propagation distance z_s of an ABW that may be treated as an LLB is inversely related to the square of r₀ or the third power of T₀ from Eq. (27). For example, for an ABW with r₀ = 1μm and T₀ = 10fs, the ABW can be regarded as an LLB in very short propagating distances, i.e., under such condition as z_s≤0.0108m. Once z_s>0.0108m, the ABW cannot continue as a spatiotemporally stable LLB. For example, when we studied the dynamics of microscopic particles by using ABWs as probes, the effective working distance had to be smaller than 0.0108m because this distance is the primary requirement of stability for the size of the probe used. For an ABW with r₀ = 5μm and T₀ = 20fs, the ABW can be regarded as an LLB at very long propagating distances of z_s≤2.16m. Such distances mean that ABWs can be regarded as LLBs for transmission range of about one standard optical platform. For an ABW with r₀ = 40μm and T₀ = 40fs, the ABW can be regarded as an LLB even at long distances of z_s≤1.11 × 10³m, which are adequate for laser-induced lighting.

In short, when an ABW with a small r₀ and T₀ is used in such applied fields as long distance laser-induced lighting, laser filamentation, optical manipulation, and signal transmission in vacuum, the influence of cubic SID on the Airy pulse must be considered.

3.3 The influence of quadratic and cubic SID

After investigating separately the influence of quadratic and cubic SID on ABWs, we analyzed the temporal propagation characteristics of ABWs in free space under the joint influence of quadratic and cubic SID, which is more in line with the actual situation of ABWs propagating in free space. From Eq. (19), we know that cubic SID is the sole reason of influencing the broadening factor. However, both quadratic SID and cubic SID induce the temporal longitudinal acceleration of ABWs. Cubic SID, which is positive dispersion from Eq. (6.4), decreases the longitudinal acceleration of the Airy pulse peak derived from the influence of quadratic SID from a = z²/4L₂² to

a = \frac{z^{2}}{4 L_{2}^{2}} {(1 + \frac{z}{2 L_{3}})}^{- \frac{4}{3}} .

Therefore, the real transmission trajectory of the Airy pulse peak is different from the traditional research results, in which the influence of cubic SID is ignored [1–3,40]. The smaller the L₃ is, the larger the degree of difference will become between the trajectory when cubic SID is considered and the trajectory when cubic SID is ignored. According to Eq. (18), the relationship between L₂ and L₃ can be illustrated as

L_{3} = \frac{c T_{0} \cdot (r_{0}^{2} k_{0}^{2} - 1)}{3 r_{0}^{2} k_{0}} L_{2} .

Given that r₀²k₀²>>1, Eq. (29) is transformed into

L_{3} = \frac{c T_{0} k_{0}}{3} L_{2} .

Equation (30) indicates that the smaller the T₀ of an ABW for a fixed L₂ is, the larger L₃ and the larger influence on the trajectory of the Airy pulse peak will become. For example, the same quadratic SID length L₂ and different cubic SID length L₃ were used among an ABW with r₀ = 5μm and T₀ = 80fs, another ABW with r₀ = 20μm and T₀ = 20fs, and finally one ABW with r₀ = 40μm and T₀ = 10fs. Figure 1 shows the trajectory of the Airy pulse peak for these three cases with cubic SID being ignored. These propagation distances correspond to different quadratic SID lengths (L₂), and time corresponds to pulse width of ABWs (T₀). Figure 1 further illustrates that the influence of cubic SID on the trajectory of the Airy pulse must be considered, especially for ultra-short pulsed ABWs. Otherwise, the actual temporal acceleration of ABWs must deviate from the theoretical prediction given in the past research results because cubic SID is usually ignored [1,7].

Fig. 1 The trajectory of the Airy pulse peak for different cases.

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In reference [1], both theoretical and experimental results show that the peak of the temporal Airy distribution pulse of ABW follows a parabolic trajectory in time. However, the measured Airy pulse peak position distinctly deviated from the theoretical prediction, as shown in Fig. 2(c) of reference [1]. In fact, we readily know the reasons based on the research results in this paper. Apparently, in the process of theoretically calculating the temporal trajectory of ABWs during propagation in reference [1], quadratic dispersion was considered and cubic dispersion was ignored. However, both exist in dispersive media and in free space in practice. Hence, we have to say that, besides influencing the temporal broadening factor of ABWs, cubic SID also influences the temporal trajectory of ABWs, especially for spatiotemporal ultra-localized ABWs.

To show temporal propagation characteristics of ABWs quantitatively, we numerically simulated the propagation of a typical ABW with specific spatiotemporal parameters. For example, the spatial parameter r₀ is 5μm, and the temporal parameter T₀ is 20fs. For these values, one has respectively L₂ = 0.435m, and L₃ = 6.840m from Eq. (18). Based on Eqs. (14) and (23), the temporal broadening factor of Gaussian-Bessel wave packets and ABWs is given in Fig. 2. For the sake of comparison, the FWHM of the initial Gaussian pulse has been chosen to be the same with the FWHM of the main lobe of Airy pulse at z = 0. The Airy pulse broadens by 1.046 times instead of the Gaussian pulse broadening by 4.702 times than its original width for the same propagation distance of 2m. It can be shown that, compared to the Gaussian distribution pulse, the Airy distribution pulse is quite stable during propagation. A deeply comparison of the propagation of initially Gaussian and finite-energy Airy pulse under the influence of both quadratic and cubic dispersion has also been made by Besieris and Shaarawi (see Ref. 38). Thus, ABWs are still superior localized linear wave packets although they cannot be regarded absolutely LLBs. Finally, the numerical simulation revealed the temporal evolution of ideal ABWs propagating in free space. As shown in Fig. 3, the results of the numerical simulation confirm all the above theoretical analysis results.

Fig. 2 Propagating in free space, temporal broadening factor of Gaussian-Bessel wave packet and ideal Airy-Bessel wave packet under the jointly influence both quadratic and cubic SID.

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Fig. 3 The temporal evolution of intensity distribution of an ideal ABW at different propagation distances.

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4. Conclusions

By studying the spatiotemporal propagation characteristics of ideal ABWs in free space, three key conclusions were convincingly obtained. First, quadratic SID does not influence the temporal distribution of ABWs but induces only a temporal longitudinal acceleration of ABWs. Hence, if only quadratic SID is considered, ideal ABWs can be regarded as Airy-Bessel LLBs. Second, cubic SID temporally broadens ABWs in a slight way. This means that, although propagating in free space, ideal ABWs are not absolutely stable LLBs because of the inevitable cubic SID effects originating from space–time couplings. Over short propagation distances, the influence of cubic dispersion on the Airy distribution pulse is small. Hence, an ABW can still basically be regarded as an LLB. However, the influence of cubic SID should be considered when ultra-localized ABWs are used in scientific researches or when other ABWs are used in some research fields such as long-distance filamentation, signal transmission, and laser induced-lightning. A quantitative condition for regarding ABWs as LLBs is thus given. Third, cubic SID decreases the longitudinal acceleration of the temporal Airy pulse peak of ABWs during propagation. The temporal acceleration of an Airy pulse means changing its group velocity. Hence, the influence of cubic SID on the temporal peak of ABWs cannot be simply ignored on some specific occasions. In short, the in-depth study of the propagation characteristics of ABWs has uncovered some new and instructive information, which will lay the foundation for the improved application of ABWs in future scientific research.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No 11274278, the Natural Science Foundation of Zhejiang province of China No Y1100524 and the program for Innovative Research Team, Zhejiang Normal University.

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Propagation characteristics of Airy-Bessel wave packets in free space

Abstract

1. Introduction

2. Theory

3. Analysis and discussion

3.1 The influence of quadratic SID

3.2 The influence of cubic SID

3.3 The influence of quadratic and cubic SID

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (3)

Equations (34)

Optics Express