1. Introduction

There are many potential applications for highly localized pulsed beams in free space or linear medium. These include medical applications, fabrication for electronic, optoelectronic or micro-electromechanical systems (MEMS); nano- and microparticle acceleration and manipulation, lidar and laser weapons.

Brittingham [1] proposed the focus wave mode (FWM) solution of Maxwell’s equations, that describes pulses localized in space that propagate at velocity c (luminally) without spreading. This solution created much attention [2–4], but the basic principles behind the concept were rather obscure in Brittingham’s original paper. The solutions include an angular spectrum that contains both forward and backward propagating components. Heyman and coworkers argued that FWMs were therefore “non-causal” and not realizable practically [5–9]. Although it has been argued that these backward propagating components can be made arbitrarily small, nevertheless they are finite in amplitude [10–12]. The same holds in the case of so-called Bessel-Gauss pulse – another luminal localized wave introduced by Overfelt [13]. Besieris, Shaarawi, and Ziolkowski studied different interpretations of the backward-propagating components [14].

The principles behind these localized solutions are now well understood: they can be considered as a coherent superposition of spectral components, each consisting of an angular spectrum that is scaled according to the frequency in a well defined way [15–21]. These have been called iso-diffracting pulses, Type 3 pulses or pulsed beams. Localized solutions then correspond to iso-diffracting wideband superpositions of Bessel beams [19–21]. As to experimental realizability of FWMs, recently it was proven by generation of a partially coherent optical field whose coherence function approximates the FWM within a band in the visible spectral range [22].

The motivation of the present paper is to find analytic expressions for such versions of FWM in free space that are constituted from spectral components propagating into one hemisphere only. The next two sections are devoted to analytical derivations and numerical calculations, respectively. In Section 4 we generalize the results to a prospective model wave pulse possessing a rectangularly cut spectrum. Section 5 considers the results in the wider context of hot topics like “backward light” and indicates possibilities for experimental generation of the wave fields derived.

2. Expressions for FWM with truncated spectrum

The amplitude distribution (wave function) for (axisymmetric) pulsed 0^th-order Bessel beams with arbitrary frequency distribution f(k) can be written in cylindrical coordinates as an integral over wave number k=2π/λ=ω/c:

where θ(k) is a function of wave number. Since kcosθ=k_z=k-2ksin²(θ/2), if the angle θ depends on k in such a way that

where k ₀ is a constant, the group velocity (dk_z/dω)^-1 is equal to c. The minimum value of k, k _min, is then equal to k ₀. This corresponds to a plane wave component propagating at θ=180°, i.e. in the backward direction with k_z=-k ₀ and with the radial wave number $k_{\rho }=\sqrt{k^2-k_z^2}=0$ (see Fig. 1). Forward propagating components correspond to those for θ<90°, i.e. with k_z≥0 and k>2k ₀.

The wave function is

where the retarded time t′=t-z/c has been introduced. Introducing also the variable p given by

and denoting f(k)=g(p), the wave function can be written

We recognize the integral in Eq. (5) as being identical to that which appears when calculating the amplitude in the focal region of a lens with arbitrary pupil function in the paraxial Debye approximation [23].

Equation (5) in general includes backward propagating components. These can be avoided completely by assuming a shifted exponential distribution for the frequency distribution

with parameters z ₀, s and k ₁≥2k ₀ (Fig. 1). This is a Pearson Type 3 distribution and contains no backward propagating spectral components, so that also

The angle θ then varies over the range from 0 to a maximum value (see Fig. 1) given by

 

Fig. 1. Relations between k-space variables k, k _ρ, k _z and parameters k ₀, k ₁. For each plane wave constituent of the axisymmetric luminal localized wave the length k of its wave vector k depends linearly (see the solid line with slope 1 in the upper part of the plot) on the forward-direction axial projection k _z of k, while the radial projection k _ρ depends parabolically (solid curve in the lower part) on k _z. Such relations between k, k _ρ and k _z of the plane wave constituents (or Bessel beam constituents) warrant equality to c of the axial group velocity of the wave packet, i. e. its propagation invariance. The straight line and the parabola become thick where the support of the spectrum starts, i. e. where the spectrum according to Eq. (6) becomes nonzero. The dashed straight lines correspond to strictly backward (θ=180°) and forward (θ=0) propagating plane waves with k _ρ=0. The double-line arrow depicts the wave vector of length k ₁ corresponding to the lowest-frequency plane-wave constituents that propagate under the maximum angle θ_max with respect to the axis z. For general properties of supports of spectra of localized waves see [20, 21].

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Then the wave function is

Introducing the coordinates

instead of the radial and the longitudinal-temporal coordinates, and the new integration variable

we then have

When k ₁=2k ₀, i.e. when θ _max=90°, so that there are just no backward propagating components, Eqs. (11) and (10) reduce to l=p, ν=2k ₀ ρ, and u=2k ₀ (ct′-i(s+1)z ₀).

For the special case when s=0, to which we have restricted ourselves henceforth, Eq. (12) can be written

The integral can be recognized as that in the paraxial theory of diffraction by a circular aperture, and can be expressed analytically in terms of the Lommel functions, so that

where the L(u,ν) is expressed in terms of Lommel functions

Gradshtein and Ryzhik take V ₁ with the negative sign and Born and Wolf with the positive sign. The variable ν is real, but u is in general complex. For points in the moving transversal plane t’=0, which crosses the pulse intensity peak, u is purely imaginary and L(u, ν) is purely real.

For values of ν/|u|>1 it is more convenient to use an expansion in terms of the Lommel functions U ₁ and U ₂:

where

An alternative expansion [24] of the integral in Eq. (13) leads to the expression

where

j_m denoting a spherical Bessel function of order m. Noticing the difference of two terms in the second brackets, it is instructive to rewrite Eq. (18) in the following form

where

and

The first term corresponds to integration from k ₀ to infinity in Eq. (3) for a non-shifted exponential distribution f(k), i. e. it should correspond to well-known focus wave mode (FWM). Indeed, in definition of $U_{k_0,\infty }$ the expression in braces coincides with the well-known expression [2–12] of the normalized FWM if we complex conjugate it and change designations of the parameters k ₀→β and z ₀→a ₁. The second term $U_{k_0,k_1}$ corresponds to integration over exponential spectrum from k ⁰ to k ₁ in Eq. (3) and in order to get the wave function $U_{k_1,\infty }$ sought (whose spectrum spans from k ₁ to infinity) it must be subtracted from the first term as usual in operating with definite integrals.

As a by-product of our derivation, we obtained an analytical expression – $U_{k_0,k_1}$ with definition of S(u, ν) (Eqs. 10 and 18) – for a new FWM-type luminal localized wave, which we name “truncated-spectrum FWM”. Equation (16) can be analogously rewritten as a difference of the common FWM and truncated-spectrum FWM terms.

3. Numerical evaluation of expressions with Lommel functions

For practical calculations it would be of no use to express fields analytically through a special function, if evaluation of the latter was tedious. Therefore we first studied how exact are Eqs. (15), (17), and (19) within the relevant domain of the field arguments ν∈[0,±50], u∈[0-4i,±75-4i] in the case of a finite number of summands.

We established that (i) for majority of the argument values Eqs. (15), (17), and (19) give results coinciding better than 0.1% if the upper limit of the summation index m exceeds a value of 10, (ii) the most preferable expression is Eq. (19), (iii) as checked by tables in [25] the latter gives a relative accuracy better than 10^-5 if m runs up to 30. For all further computations of 3D plots, Eq. (19) with m=0…30 has been used. Computation according to Eq. (18) of one video frame on a grid of 240×80 points with Mathsoft’s package Mathcad 13 took about 9 seconds (with 1.83 GHz CPU).

4. Spatio-temporal behavior of derived wave pulses

For comparison with the common FWM, the most illustrative is the case k ₁=2k ₀, i. e., if the wavefield has all possible forward propagating plane-wave constituents only, the most low-frequency ones propagating perpendicularly to the axis (θ _max=90°, see Fig. 1).

The wave function modulus (Fig. 2) has a sharp central peak characteristic of all axisymmetric wideband localized waves and a double-cone profile, i. e., an X-like profile in any axial (meridian) plane, which is distinctive of all localized waves with no axially propagating plane-wave constituents. The latter is not the case for the FWM (Fig. 3).

The most distinctive difference of the shifted-spectrum FWM (with k ₁ equal to twice the value of k ₀ as in Fig. 2, or greater) from the FWM is complete absence of backward-propagating wave crests or other phase features, while the domination of the backward-propagating plane-wave constituents in the spectrum of the FWM is manifested by ackward-propagation of phase fronts as seen in the animated Fig. 3.

It is interesting to notice in the animated Fig. 2 – and it is checked by plots made under other angles of view as well – that the ripples outside of the axial region move (collapse cylindrically) toward the pulse’s propagation axis z when they are ahead of the pulse’s peak, and expand radially away from the axis when they lag behind the pulse’s peak. This feature reveals the meaningfulness of the mathematical representation of localized waves as generated by a moving sink-emitter pair, which was introduced in [26]. The same feature is clearly seen in ripples filling the space between the cones of the truncated-spectrum FWM (Fig. 4). These ripples, possessing cylindrical coaxial phase surfaces and moving radially in and out, are attributed to the abrupt step in the spectra of both wavefunctions when the angle θ reaches 90°. If the step occurs at a different value of θ, i. e., k ₁≠2k ₀, the phase surfaces of the ripples are no longer cylindrical. In contrast, the ripples are completely absent around the waist of the FWM, as seen in Fig. 3. Moreover, in the waist region the wave function of the FWM decays radially so steeply that it forces us to revise certain undamental notions about the localizability of a photon [27].

 

Fig. 2. The frame t=0 of the animated 3D plot (1.6 MB) of the luminal wavepacket constituted from forward-propagating spectral components only (wavenumbers k≥k ₁=2k ₀), which has been computed according to Eq. (20) and previous Section. Shown are the real part (upper surface plot) and the modulus (pseudocoloured contour plot on the basal plane) of the wave function in dependence on the coordinate z (increasing from the left to the right) and on a transverse coordinate x=±ρ. The value of the spectral decay parameter z ₀=2/k ₀. The distance Δ between the grid lines on the basal plane (x,z) is 2λ ₀, λ ₀=2π/k ₀. Δ=1.25µm if we put k ₀=10⁵ cm^-1, which corresponds to red light. To reveal better the complicated relief of the surface plot a blue diffuse and white specular “lighting”, as well as slight “transparency” and “shininess” have been applied to it. The plane of the contour plot has been slightly risen above the basal plane (x,z) in order to avoid obscuration of the narrow pulse image by the axial grid line, and due to corresponding “parallax” the centrum of the pulse seems to not coincide with the central crossing of the grid lines in the frame shown. [Media 1]

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Fig. 3. The frame t=0 of the animated plot (2.1 MB) of the FWM computed by Eq. (21). For explanatory details see the previous figure caption. [Media 2]

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Fig. 4. The frame t=0 of the animated plot (1.9 MB) of the truncated-spectrum FWM computed by Eq. (21). [Media 3]

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However, the most startling feature of the truncated-spectrum FWM pulse is that it flies with speed c in the direction opposite to the propagation direction of all its plane-wave constituents and of its phase profile. Theoretical prediction and explanation of such phenomenon can be found in [21, 28].

Imaginary parts of the wave functions behave quite similarly, but of course their oscillations are in quadrature with respect to the real ones.

5. Generalization to spectrum with two-sided cutoff

The difference in Eq. (20) corresponding to piecewise integration over the spectral variable hints at a possibility of deriving an analytical expression for a luminal localized wave whose spectrum starts from arbitrary k ₁>k ₀ and ends at arbitrary k ₂<∞. Indeed, if we write instead of Eq. (6) with s=0 the following spectrum expression

and make use of the linearity of the Fourier integral for the third row of Eq. (23), we can write immediately for the wave function corresponding to the band-like spectrum of Eq. (23)

Naturally the same result can be obtained if one generalizes the deduction of Section 2 by defining two sets of intermediate variables instead of Eqs. (10) and (11).

In Figs. 5 and 6, fields with equal bandwiths 0.5k₀ but with propagation directions of all the plane-wave components into opposite hemispheres are presented: in the case of Fig. 5 θ runs from 83° to 74° and of Fig. 5 – from 98° to 126°. The exponential decay parameter was chosen to be rather small z ₀=0.3/k ₀ so that the spectrum is almost rectangular - the spectral amplitude is flat with accuracy <15% between the given boundaries k ₁ and k ₂. Thus, such field is an appropriate model for study of phenomena and applications involving luminal localized waves in the optical region.

 

Fig. 5. The frame t=0 of the animated plot (1.8 MB) of the field computed by Eq. (24) with parameters k ₁=2.25k ₀, k ₂=2.75k ₀, z ₀=0.3/k ₀. [Media 4]

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The spatio-temporal behavior of the fields is similar to that presented in Figs. 2 and 4, except that ripples parallel to the axis z are not seen since now there is no spectral components with θ around 90°. The most remarkable feature seen in Fig. 6 is again the opposite direction of propagation of the pulse with respect to that of its spectral components and the phase pattern.

 

Fig. 6. The same as Fig. 5, but k ₁=1.25k ₀, k ₂=1.75k ₀. [Media 5]

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6. Discussion

The result that with the help of Lommel functions an analytical expression can be written down for an arbitrary luminal localized wave whose spectrum is a piece cut off from the spectrum of the common FWM, considerably promotes further studies of FWM and its applications. The result can be straightforwardly used for obtaining expressions for various kind of derivative fields, including arbitrarily polarized EM vector fields. Of particular importance for this is the study of optical ultrashort localized waves whose spectrum covers the visible region. Such waves can be generated from few-femtosecond-long laser pulses as superluminal localized waves can be launched [29–31]. In realistic cases of finite generating apertures, the expressions obtained give the field inside a cylindrical volume of large but finite length (as is known for all propagation-invariant localized waves).

Fields, depicted in Figs. 4 and 6 – if one reverses the direction of the axis z – are startling examples of “backward light” in vacuum, where negative group velocity (and velocity of the whole pulse intensity profile) originates from angular dispersion in 3D space instead of being caused by anomalies in the dispersion of the 1D propagation medium.

7. Conclusion

In conclusion, we have derived closed-form expressions for FWM-type fields that contain only such spectral components that propagate under arbitrarily prescribed angle interval with respect to the axis. Like the original FWM solution, these expressions describe highly localized pulses that propagate without any lateral or axial spread with luminal velocity c up to infinity (theoretically – for unlimited aperture). Our results facilitate theoretical and possible experimental studies where FWM-type fields are involved and confirm once more that neither backward-propagating (or so-called non-causal) spectral constituents nor any particular spectrum are necessary features of luminal localized waves.