Abstract

Propagation characteristics of truncated Localized Waves propagating in dispersive silica and free space are numerically analyzed. It is shown that those characteristics are affected by the changes in the relation between the transverse spatial spectral components and the wave vector. Numerical experiments demonstrate that as the non-linearity of this relation gets stronger, the pulses propagating in silica become more immune to decay and distortion whereas the pulses propagating in free-space suffer from early decay and distortion.

© 2010 Optical Society of America

1. Introduction

Undistorted pulse propagation is a subject of great interest due to its various applications in high-speed and long-distance communication [1], energy transfer [2], medical radiography [3], and optical lithography [4]. Localized Waves (LWs) [5], which represent a family of solutions to the homogeneous scalar wave equation, can propagate long distances without distortion. Peaks of an ideal and infinite-energy LW propagate with an arbitrary but constant speed along a straight line; and the ideal LW maintains its shape rigidly or with only local deformations for all time. This is a result of the inherent coupling between the wave’s temporal and spatial spectra. Depending on the type of this coupling, LWs can be categorized into two main groups: (i) X-waves [6] and (ii) Focus Wave Modes (FWM) [5, 7]. For X-waves, the coupling relation is linear; and the waves in this group are solely composed of forward propagating components. The coupling relation for FWMs is parabolic [8]; and the waves in this group have backward propagating components [9, 10]. Accordingly, X-waves were widely favored in studies due to their causal structure, and the linear nature of their spectra coupling relation (see, e.g. [11] and references therein). Initially FWMs were thought to be physically non-realizable because of the presence of backward propagating wave components, but it was later shown that a physically realizable approximation an be obtained by letting the forward propagating components dominate over the backward propagating ones [12]. Moreover, several LW solutions from the FWM family were constructed to be naturally free from backward propagating components [13, 14].

An important critique to the physical realization of ideal LW solution is the infinite energy requirement [15]. Wu and Lehmann [16] proved that the energy density of any finite-energy solution to the source-free Maxwell’s equations approaches zero as time tends to infinity, thus no finite-energy LW solution that can propagate to infinity without distortion could physically exist. To achieve an approximately-undistorted propagation while meeting the finite-energy requirement, several finite-energy LW solutions were constructed [13, 17]. Additionally, the ideal infinite-energy LW solution could be truncated in space and time (e.g., using dynamic Gaussian apertures [18]) to achieve the same goal.

A special area of interest for LW applications is in the field of optics and photonics [14]. A considerable amount of work was published on the characterization of optical (high-frequency) LW propagation (see chapters 7–12 in [11]); however only a few studies focused on their propagation in dispersive media [1922]. In [1921] closed form expressions for LWs propagating in dispersive media are presented; however, these expressions are assumed to hold only within a limited spectral band. In [22], generic spectral expressions are given only for superluminal infinite-energy X-waves propagating in a dispersive medium.

In this paper, the propagation characteristics of truncated (finite-energy) LWs propagating in dispersive silica are analyzed and compared to those of truncated LWs propagating in free-space. The truncation, which is necessary for physical realization of the pulses, is achieved by multiplying the ideal infinite-energy LW solution by spatial and temporal Gaussian windows. This procedure is equivalent to truncating the spatial and temporal spectra of the LW as discussed later on. The spectral truncation is carried out in such a way that (possible) backward propagating components are filtered out. Because of the truncation, the inseparable coupling of the spatial and temporal spectra is not rigorously enforced; and hence the resulting pulses suffer from distortion and decay. Several numerical experiments were performed to quantify those effects and relate them to the spectral structure of the pulses.

The remainder of the paper is organized as follows: Section 2 starts with the general expression of LWs; describes the necessary condition for undistorted propagation, and discusses the effects of the spatial and the temporal truncation on the spectral structure of the LW. Section 3 provides several numerical experiments where the propagation characteristics of LWs propagating in free-space and silica are compared. Furthermore, it discusses the relation between those characteristics and the LW spectral structure. Section 4 summarizes the discussions on truncated LWs propagation in dispersive silica and provides future research avenues.

2. Formulation and discussion

A cylindrically symmetric scalar field ψ(ρ, z, t), which represents the LW-type solution of the homogeneous wave equation and propagates along the z-direction, can be constructed by a polychromatic superposition of Bessel beams [18], using

ψ(ρ,z,t)=0kρdkρdkzdωψ˜(kρ,kz,ω)J0(kρρ)ei(kzzωt).
Here, {kρ, kz, ω} and {ρ, z, t} are the spectral, and spatial and temporal variables, respectively, ψ̃(kρ, kz, ω) is the spectral amplitude of the LW, and J0(·) is the zeroth order Bessel function of the first kind. The spectral variables satisfy the dispersion relation
kρ2+kz2=(n(ω)ωc)2,
where n(ω) is the refractive index of the medium and c is speed of light in free-space. Fully-undistorted pulse propagation can only be achieved if the pulse is capable of reconstructing itself periodically. In other words ω and kz must satisfy [23]
ω=α+v0kz,
where v0 is the pulse peak velocity and α is an arbitrary constant that quantifies the periodicity of the field shape along the z-direction. For undistorted LW propagation, conditions (2) and (3) must both hold true for all values of the spectral variables, kρ, kz and ω. In mathematical terms, this can be achieved using Dirac-delta functions (in the generalized sense); i.e., choosing
ψ˜(kρ,kz,ω)=ψ˜(ω)δ(kρ(n(ω)ωc)2kz2)δ(kzωαv0),
where ψ˜′(ω) is an arbitrary spectral amplitude function. Inserting Eq. (4) into Eq. (1) yields
ψ(ρ,z,t)=ωminωmaxdωψ˜(ω)ξ1J0(ξ1ρ)ei(ξ2zωt),ξ1=(n(ω)ωc)2ξ22,ξ2=ωαv0.
The limits of integration ωmin and ωmax are chosen as
ωmin=αcmax(cn(ω)v0,ωmax=αcmin(cn(ω)v0),
to ensure that ξ1 ≤ 0, i.e., to avoid non-physical solutions.

Several observations about Eqs. (4) and (5) are in order:

  1. The ‘ordinary’ X-wave [6] and FWM waves [5, 7] can be recovered by setting the spectral amplitude function to ψ̃′(ω) = (ω/v0)m exp(−), where a is an arbitrary positive real constant and m = 0, 1,2,... is the order of the pulse. X-wave solutions are obtained by setting α = 0, while for FWM solutions α > 0 [18].
  2. The presence of a non-zero α in the pulse spectrum yields backward propagating (acuasal) field components, thus especially for non-truncated LWs, it is important to choose the pulse parameters such that the causal components dominate over the acausal ones. This is typically achieved by setting a α ≪ 1 [18].
  3. Any ψ (ρ, z, t) satisfying Eq. (5) (including X-wave and FWM solutions) extend from –∞ to +∞ in both space and time [18]. To achieve practically realizable pulses, ψ ( ρ, z, t) has to be truncated in space and time. In this work, the LWs are truncated in space and time via multiplication with
    ft(t)={e(t/T)|t|2T0elsewhereandfρ(ρ)={e(ρ/R)ρ2R0elsewhere.
  4. Multiplying ψ(ρ, z, t) with functions ft(t) and fρ(ρ) given in Eq. (7) is equivalent to convolving ψ̃(kρ, kz, ω) with their spectra, t(ω) and ρ (kρ), respectively. Writing the spectrum [Eq. (4)] in terms of kz as ψ̃(kρ, kz, ω) = ψ̃″(kz) δ (kρ – χ1) δ (ω – χ2) with χ1={(n(χ2)/c)2α(α+2v0kz)+[(n(χ2)v0/c)21kz2}1/2 and χ2 = α + v0kz, then performing the convolution with the truncating functions spectra yields
    ψ¯˜(kρ,kz,ω=ψ˜(kz)f˜ρ(kρχ1)f˜t(ωχ2).
    Considering that t(·) and ρ(·) are Gaussian type functions, it becomes clear that the convolution operations truncate the spectra of ψ (ρ, z, t). As a result, one can choose T and R properly to eliminate/suppress the spectral content corresponding to backward propagating components (This procedure is analogous to the finite-time dynamic aperture excitation analyzed in [8]). In the remainder of the paper, the truncated pulse and its spectrum are represented by ψ̄(ρ, z, t) and ψ¯˜(kρ,kz,ω), respectively.
  5. It should be noted here that replacing the spectrum [Eq. (4)] with Eq. (8) yields a field that is not an exact solution to the homogeneous wave equation. Yet this field is expanded in terms of Bessel beams, which are exact solutions to the homogeneous wave equation, and propagated as such. Thus the truncated spectrum should be understood as an imposed initial condition, whereas the propagated field rigidly satisfies the wave equation.
  6. As a result of truncation, the undistorted propagation condition (3) is not rigorously enforced for all values of the spectral variables in the spectrum [Eq. (8)]. This eventually results in distortion and decay of the pulses during propagation [16].

3. Numerical experiments

Several numerical experiments are performed to understand the relation between the spectral structure and depletion, and the propagation characteristics of truncated LWs propagating in a dispersive medium; specifically to quantify the effect of the parameter α on the propagation characteristics of truncated X-wave (α = 0) and FWM waves ( = {0.02, 2.0, 5.0, 10.0, 20.0}) in bulk fused silica. In all experiments, the pulses are also propagated in free-space, as a non-dispersive medium, for comparison purpose. The silica refractive index is modeled using the empirical Sellmeier equation [24]

n2(ω)=1+j=1NBjωj2ωj2ω2,
where ωj is the j-th resonance frequency, Bj is the strength of the j-th resonance, and N is the total number of material resonances in the frequency range of interest. For a typical fused silica medium [24], N = 3, and the values of Bj and ωj are the ones listed in Table 1. It should be noted that with the parameters given in Table 1, Eq. (9) is only accurate when used in the frequency band ω ∈ [8.196, 51.643] × 1014rad [24].

Tables Icon

Table 1. Values of Bj and ωj for typical bulk fused silica [see Eq. (9)]

For all pulses investigated in the experiments, m = 28 and a = 10−14s−1 (to confine the pulse’s temporal spectrum within the validity range of (9)); for pulses propagating in silica, v0 = c/1.44, R = 5 × 10−5m, and T = 2.5 × 10−13s; and for the pulses propagating in free-space, v0 = c, R = 7.5 × 10−5m, and T = 1.75 × 10−13s. The truncation windows for pulses propagating in free-space are modified to compensate for the longer wavelengths in free-space. All pulses are assumed to be launched from an aperture at z = 0.

3.1. Spectral structure and depletion

As discussed above, the undistorted propagation relation (3) is not rigorously enforced when the LW is truncated. However, because the truncation windows ft(t) and fρ(ρ) are sufficiently wide enough, the spectrum is still confined around the kρω line, described by Eqs. (2) and (3), at a constant kz. The amount of deviation in the spectrum from this line determines the level of distortion and decay that occurs during the propagation of the truncated LWs. Figure 1 depicts the spectrum of the truncated LWs, |ψ¯˜(kρ,kz,ω)| at kz = 0. The figure shows that the spectra of the FWM pulses with higher α values follow more closely the kρω line within the range with the significant spectral amplitude concentration. On the other hand, with the increase in α, the deviation from this line within the range, where the spectral amplitudes are smaller, becomes larger. As will be shown in Section 3.2, this deviation from the ideal kρω line deteriorates the field depth of the pulse if it occurs within the range with higher spectral amplitudes. It should be noted that as the parameter α increases, the value of ωmin also increases according to Eq. (6). Such increase in the lower limit of the integral (5) may result in truncation in the spectra as observed in Figs. 2(f) and 3(f) for the FWM pulse with ( = 20.0).

 figure: Fig. 1

Fig. 1 The spectrum of the pulses |ψ¯˜(kρ,0,ω)| showing the coupling between the spatial kρ and temporal ω spectral components of the (a) X-wave (α = 0), (b) FWM with ( = 0.02), (c) FWM with ( = 2.0), (d) FWM with ( = 5.0), (e) FWM with ( = 10.0) and (f) FWM with ( = 20.0). Exact coupling relation, that is obtained by substituting Eq. (3) into Eq. (2) is shown in red circles for reference.

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 figure: Fig. 2

Fig. 2 Comparison of the spatial spectral depletion of the pulses propagating in free-space and silica at z′ = 0, z′ = zD and z′ = 2zD for (a) X-wave (α = 0), (b) FWM with ( = 0.02), (c) FWM with ( = 2.0), (d) FWM with ( = 5.0), (e) FWM with ( = 10.0) and (f) FWM with ( = 20.0).

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In [25], the spectral structure of LWs launched from a dynamic aperture in free-space was investigated and a relation between the spectral depletion and the field depth was established.

Thus, in order to extend this investigation to LW pulses propagating in silica as well as to quantify the effect of the parameter α on the relation between the pulse’s spectral depletion and its propagation characteristics, the following experiments are performed.

First, the behavior of the transverse spatial spectra of the pulses is investigated. The transverse spatial spectrum, Φρ(kρ;z′), is obtained by computing the Bessel-Fourier transform of the truncated pulse at the pulse peak, (z = z′,t = z′/v0), viz.

Φρ(kρ;z)=0ρdρψ¯(ρ,z,zc0)J0(kρρ).
The absolute value of the spectrum [Eq. (10)] is then normalized with respect to its maximum, viz., Φρn(kρ;z′) = |Φρ(kρ; z′)|/ max(|Φρ(kρ;0)|). Figure 2 presents Φρn(kρ;z′) computed for all six types of pulses propagating in free-space and silica, at z′ = 0, z′ = zD and at z′ = 2zD, where zD is the distance at which the pulse intensity at the centroid drops to 1/e of its initial value [see Section 3.2]. The figure shows that the spatial spectra are expanded and shifted towards higher frequencies as α increases, yet the expansion and shift are more significant in the free-space case. Comparison of the plots in Figs. 2(a) and 2(b) show that the spectra of the X-wave and the FWM with = 0.02 are very similar for the pulses propagating in silica, yet there is a difference of several orders of magnitude in terms of bandwidth and frequency range for those propagating in free-space. The nature of the spatial spectral depletion is revealed by comparing the spectra of the pulses at z′ = 0, z′ = zD and z′ = 2zD as shown in the figure. The plots show that the spatial spectra deplete differently for pulses propagating in free-space and silica. In free-space, all spatial frequency components deplete equally or with a slight tendency to deplete more at the lower frequency components of the spectrum; however, in silica, the spatial spectra consistently deplete significantly more at the higher frequency components of the spectrum.

The relation between the depletion of the transverse spatial spectrum and that of the temporal spectrum determines the deviation from the ideal kρω line as the pulses propagate. Thus in what follows, the behavior of the temporal spectra of the pulses propagating in free-space and silica are investigated. The temporal spectrum Φt(ω; z′) is computed from the Fourier transform of the truncated pulse expressions at ρ= 0 and centered around the pulse peak, (z = z′, t = tz′/v0), viz.

Φt(ω;z)=dtψ¯(0,z,tzv0)eiωt.
The absolute value of the spectrum [Eq. (11)] is normalized with respect to its maximum, viz., Φtn(ω;z′) = |Φt(ω;z′)|/max(|Φt (ω;0)|). Figure 3 compares Φtn(ω;z′) for all six types of pulses propagating in free-space and silica, respectively. The figure shows that all temporal spectra are almost identical, with the exception of that of the FWM with = 20.0, where the lower temporal frequency components of the spectrum are truncated at ω ≈ 2 × 1015rad, which is equal to ωmin given by Eq. (6). The figure shows that all temporal spectra deplete uniformally as the pulses propagate to z = 2zD.

 figure: Fig. 3

Fig. 3 Comparison of the temporal spectral depletion of the truncated pulses in free-space and silica at z′ = 0, z′ = zD and z′ = 2zD for (a) X-wave, (b) FWM with ( = 0.02),(c) FWM with ( = 2.0), (d) FWM with ( = 5.0), (e) FWM with ( = 10.0) and (f) FWM with ( = 20.0).

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So far, the effects of α on the spectral structure and its depletion of pulses propagating in dispersive silica and free-space are demonstrated numerically. In the remainder of this section these attributes will be related to the effect of α on the propagation characteristics (i.e., field depth and distortion) of the pulses.

3.2. Field Depth

The normalized intensity at ρ = 0, viz., In(z′) = | ψ̄(0, z′,z′/v0)|2/| ψ̄(0, 0, 0)|2, for all pulses propagating in free-space and silica, is plotted in Fig. 4. The field depth, defined by the distance at which the intensity drops to half of its initial value, for all six types of pulses in free-space and silica, is presented in Table 2. It was previously established in [25] that for the LW pulses launched from a dynamic aperture in free-space, the low spatial spectral components deplete first with distance. Results obtained for pulses propagating in free-space agree with [25] as presented in Figs. 2 and 4(a). On the other hand, for the LW pulses propagating in silica, results show that higher spatial frequency components deplete first as shown in Fig. 2. Thus for LW pulses propagating in silica, the presence of α enhances the field depth. This could be explained by the change in the nature of the coupling relation between ω and kρ. For X-wave and FWM pulses propagating in free-space ωkρ and ωkρ2, respectively [8]. This linear relation enhances the field localization of X-wave pulses because as both spatial and temporal spectra deplete, the undistorted propagation condition (3) is still preserved. The same relations do not hold true for the pulses propagating in a dispersive medium due to the frequency-dependent refractive index. It should be noted that for the FWM with = 20.0 the field depth is less than that of the FWM with = 10.0; this is due to the truncation in the spectrum imposed by the physical requirement of the choice of ωmin.

 figure: Fig. 4

Fig. 4 Normalized intensities at the centroid of the truncated pulses vesrus the propagation distance in (a) free-space and (b) silica.

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Tables Icon

Table 2. Field depth of X-wave and FWM pulses propagating in free-space and silica

3.3. Pulse spreading

Truncated LW pulses spread in both time and space because of their finite energy content [16]. To characterize the pulse spreading, the temporal and spatial pulse intensity profiles are computed and compared for all six types of pulses. Figure 5 presents the percentage change in the full-width half-maximum (FWHM) of pulses intensity at ρ= 0, defined as Δ% = 100 × |(I(z) – I(0))|/I(0), where I(z) is the intensity FWHM at distance z, computed along the propagation interval z ∈ [0, zD]. The figure shows that in free-space the truncated LW pulses only suffer from dispersion when > 1, where the pulse’s FWHM increases exponentially as the pulse propagates. For these large values of α and with the chosen truncation windows, the FWM pulses behave like truncated Gaussian pulses. In silica, all pulses suffer from temporal spreading as they propagate, but contrary to those propagating in free-space, the FWM pulses with > 1 exhibit the least spreading. The pulse spreading in silica could be readily explained by the nature of their spectral depletion shown in Figs. 2 and 3, where it is shown that the higher spatial frequency components are depleted before the lower components, while all temporal frequency components deplete almost uniformally. This uneven depletion in the transverse spatial spectra and temporal spectra results in a non-linear relation between ω and kz; hence the different spectral components of the pulse propagate with different velocities resulting in temporal spreading.

 figure: Fig. 5

Fig. 5 Percentage change of the FWHM of the pulses along the propagation distance from z = 0 to z = zD in (a) free-space and (b) silica.

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The pulse spot-size, S(z), is defined by the FWHM of the intensity in the transverse plane at distance z. Figure 6 depicts the percentage change in the spot-size, defined by Δ% = 100 × |S(z) – S(0)|/S(0), computed along the same propagation interval z ∈ [0, zD], for all six types of pulses. The figure shows that in free-space and silica, the FWM pulses with increasing α show less spreading in their spot-sizes. The presence of α in the LW expression forces a parabolic relation between ω and kρ in the free-space case, this non-linearity makes the kρω more immune to perturbations, resulting in less change in the spot-size. For pulses propagating in silica, the relation between kρ and ω is not parabolic because of the frequency-dependent refractive index, yet the presence of α in this relation still increases the non-linearity and results in less change in the spot-size.

 figure: Fig. 6

Fig. 6 Percentage change of the spot-size of the pulses along the propagation distance from z = 0 to z = zD in (a) free-space and (b) silica.

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4. Conclusion

In this paper, the spectral structures of truncated LWs propagating in free-space and in dispersive silica are analyzed. The analysis shows that in contrast to the pulses propagating in free-space, the presence of a parabolic relation between the wave vector and its transverse component enhances the field depth and suppresses pulse distortion.

Several numerical experiments are performed to determine the effect of the parameter α, which defines the periodicity of the field shape along the z-direction, on the spectral structure of the truncated LWs propagating in free-space and silica. Results show that the field depth is enhanced when > 1 for pulses propagating in silica, in contrast to pulses propagating in free-space, where the field depth deteriorates for > 1. Results also show that the spectra with higher values of α show stronger non-linearity in the relation between their spectral components, which renders them more immune to dispersion and distortion. On the other hand, the spectra are shifted towards higher values as α increases, which could lead to clipping in the spectra, and accordingly deterioration in the propagation characteristics.

Results presented in this paper are valuable for LW applications in various fields of science and engineering, including but not limited to, optical energy transfer, high-speed and long-distance communication systems, medical radiography and hyperthermia.

Analysis of the propagation characteristics of LWs propagating through slabs and refracting at interfaces of meta-materials, as well as back-scattering from penetrable objects, are underway.

References and links

1. H. A. Willebrand and B. S. Ghuman, “Fiber optics without fiber,” IEEE Spectr. 38, 40–45 (2001). [CrossRef]  

2. L. B. Felsen, “Phase space issues in ultrawideband/short pulse wave modeling,” in Ultra-Wideband, Short-Pulse Electromagnetics, H. Bertoni, L. Carin, and L. B. Felsen, eds. (Plenum Press, New York, 1993).

3. J.-Y. Lu, J. Cheng, and B. Cameron, “Low sidelobe limited diffraction optical coherence tomography,” in “Coherence Domain Optical Methods in Biomedical Science and Clinical Applications VI, Proc. of SPIE,”, vol. 4619, V. V. Tuchin, J. A. Izatt, and J. G. Fujimoto, eds. (SPIE, 2006), vol. 4619, pp. 300–311.

4. T. Ito and S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000). [CrossRef]   [PubMed]  

5. J. N. Brittingham, “Focus waves modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983). [CrossRef]  

6. J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves: exact solutions to free-space scalar wave equation and their infinite realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992). [CrossRef]   [PubMed]  

7. R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26, 861–863 (1985). [CrossRef]  

8. A. M. Shaarawi, “Comparison of two localized wave fields generated from dynamic apertures,” J. Opt. Soc. Am. A 14, 1804–1816 (1997). [CrossRef]  

9. E. Heyman, B. Z. Steinberg, and L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A 4, 2081–2091 (1987). [CrossRef]  

10. E. Heyman, “The focus wave mode: a dilemma with causality,” IEEE Trans. Antenn. Propag. 37, 1604–1608 (1989). [CrossRef]  

11. H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized waves (J. Wiley & Sons, New York, NY, 2008). [CrossRef]  

12. A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, “On the evanescent fields and the causality of the focus wave modes,” J. Math. Phys. 36, 5565–5587 (1995). [CrossRef]  

13. M. Zamboni-Rached, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008). [CrossRef]  

14. M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A. 79, 013816 (2009). [CrossRef]  

15. A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985). [CrossRef]  

16. T. T. Wu and H. Lehmann, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985). [CrossRef]  

17. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A. 39, 2005–2033 (1989). [CrossRef]   [PubMed]  

18. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989). [CrossRef]  

19. H. Sõnajalg and P. Saari, “Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators,” Opt. Lett. 21, 1162–1164 (1996). [CrossRef]   [PubMed]  

20. M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. 26, 1364–1366 (2001). [CrossRef]  

21. S. Orlov, A. Piskarskas, and A. Stabinis, “Localized optical subcycle pulses in dispersive media,” Opt. Lett. 27, 2167–2169 (2002). [CrossRef]  

22. M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003). [CrossRef]  

23. R. Donnelly and R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. Lond. A 440, 541–565 (1993). [CrossRef]  

24. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, New York, 1995), 2nd ed.

25. A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, “The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses,” J. Phys. Math. Gen. 29, 5157–5179 (1996). [CrossRef]  

References

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  1. H. A. Willebrand and B. S. Ghuman, “Fiber optics without fiber,” IEEE Spectr. 38, 40–45 (2001).
    [Crossref]
  2. L. B. Felsen, “Phase space issues in ultrawideband/short pulse wave modeling,” in Ultra-Wideband, Short-Pulse Electromagnetics, H. Bertoni, L. Carin, and L. B. Felsen, eds. (Plenum Press, New York, 1993).
  3. J.-Y. Lu, J. Cheng, and B. Cameron, “Low sidelobe limited diffraction optical coherence tomography,” in “Coherence Domain Optical Methods in Biomedical Science and Clinical Applications VI, Proc. of SPIE,”, vol. 4619, V. V. Tuchin, J. A. Izatt, and J. G. Fujimoto, eds. (SPIE, 2006), vol. 4619, pp. 300–311.
  4. T. Ito and S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000).
    [Crossref] [PubMed]
  5. J. N. Brittingham, “Focus waves modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
    [Crossref]
  6. J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves: exact solutions to free-space scalar wave equation and their infinite realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
    [Crossref] [PubMed]
  7. R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26, 861–863 (1985).
    [Crossref]
  8. A. M. Shaarawi, “Comparison of two localized wave fields generated from dynamic apertures,” J. Opt. Soc. Am. A 14, 1804–1816 (1997).
    [Crossref]
  9. E. Heyman, B. Z. Steinberg, and L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A 4, 2081–2091 (1987).
    [Crossref]
  10. E. Heyman, “The focus wave mode: a dilemma with causality,” IEEE Trans. Antenn. Propag. 37, 1604–1608 (1989).
    [Crossref]
  11. H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized waves (J. Wiley & Sons, New York, NY, 2008).
    [Crossref]
  12. A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, “On the evanescent fields and the causality of the focus wave modes,” J. Math. Phys. 36, 5565–5587 (1995).
    [Crossref]
  13. M. Zamboni-Rached, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
    [Crossref]
  14. M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A. 79, 013816 (2009).
    [Crossref]
  15. A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
    [Crossref]
  16. T. T. Wu and H. Lehmann, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985).
    [Crossref]
  17. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A. 39, 2005–2033 (1989).
    [Crossref] [PubMed]
  18. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
    [Crossref]
  19. H. Sõnajalg and P. Saari, “Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators,” Opt. Lett. 21, 1162–1164 (1996).
    [Crossref] [PubMed]
  20. M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. 26, 1364–1366 (2001).
    [Crossref]
  21. S. Orlov, A. Piskarskas, and A. Stabinis, “Localized optical subcycle pulses in dispersive media,” Opt. Lett. 27, 2167–2169 (2002).
    [Crossref]
  22. M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
    [Crossref]
  23. R. Donnelly and R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. Lond. A 440, 541–565 (1993).
    [Crossref]
  24. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, New York, 1995), 2nd ed.
  25. A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, “The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses,” J. Phys. Math. Gen. 29, 5157–5179 (1996).
    [Crossref]

2009 (1)

M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A. 79, 013816 (2009).
[Crossref]

2008 (1)

M. Zamboni-Rached, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[Crossref]

2003 (1)

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[Crossref]

2002 (1)

2001 (2)

2000 (1)

T. Ito and S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000).
[Crossref] [PubMed]

1997 (1)

1996 (2)

H. Sõnajalg and P. Saari, “Suppression of temporal spread of ultrashort pulses in dispersive media by Bessel beam generators,” Opt. Lett. 21, 1162–1164 (1996).
[Crossref] [PubMed]

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, “The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses,” J. Phys. Math. Gen. 29, 5157–5179 (1996).
[Crossref]

1995 (1)

A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, “On the evanescent fields and the causality of the focus wave modes,” J. Math. Phys. 36, 5565–5587 (1995).
[Crossref]

1993 (1)

R. Donnelly and R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. Lond. A 440, 541–565 (1993).
[Crossref]

1992 (1)

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves: exact solutions to free-space scalar wave equation and their infinite realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[Crossref] [PubMed]

1989 (3)

E. Heyman, “The focus wave mode: a dilemma with causality,” IEEE Trans. Antenn. Propag. 37, 1604–1608 (1989).
[Crossref]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A. 39, 2005–2033 (1989).
[Crossref] [PubMed]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[Crossref]

1987 (1)

1985 (3)

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26, 861–863 (1985).
[Crossref]

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[Crossref]

T. T. Wu and H. Lehmann, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985).
[Crossref]

1983 (1)

J. N. Brittingham, “Focus waves modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, New York, 1995), 2nd ed.

Besieris, I. M.

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, “The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses,” J. Phys. Math. Gen. 29, 5157–5179 (1996).
[Crossref]

A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, “On the evanescent fields and the causality of the focus wave modes,” J. Math. Phys. 36, 5565–5587 (1995).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[Crossref]

Brittingham, J. N.

J. N. Brittingham, “Focus waves modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[Crossref]

Cameron, B.

J.-Y. Lu, J. Cheng, and B. Cameron, “Low sidelobe limited diffraction optical coherence tomography,” in “Coherence Domain Optical Methods in Biomedical Science and Clinical Applications VI, Proc. of SPIE,”, vol. 4619, V. V. Tuchin, J. A. Izatt, and J. G. Fujimoto, eds. (SPIE, 2006), vol. 4619, pp. 300–311.

Cheng, J.

J.-Y. Lu, J. Cheng, and B. Cameron, “Low sidelobe limited diffraction optical coherence tomography,” in “Coherence Domain Optical Methods in Biomedical Science and Clinical Applications VI, Proc. of SPIE,”, vol. 4619, V. V. Tuchin, J. A. Izatt, and J. G. Fujimoto, eds. (SPIE, 2006), vol. 4619, pp. 300–311.

Donnelly, R.

R. Donnelly and R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. Lond. A 440, 541–565 (1993).
[Crossref]

Felsen, L. B.

E. Heyman, B. Z. Steinberg, and L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A 4, 2081–2091 (1987).
[Crossref]

L. B. Felsen, “Phase space issues in ultrawideband/short pulse wave modeling,” in Ultra-Wideband, Short-Pulse Electromagnetics, H. Bertoni, L. Carin, and L. B. Felsen, eds. (Plenum Press, New York, 1993).

Ghuman, B. S.

H. A. Willebrand and B. S. Ghuman, “Fiber optics without fiber,” IEEE Spectr. 38, 40–45 (2001).
[Crossref]

Greenleaf, J. F.

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves: exact solutions to free-space scalar wave equation and their infinite realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[Crossref] [PubMed]

Hernández-Figueroa, H. E.

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[Crossref]

Heyman, E.

E. Heyman, “The focus wave mode: a dilemma with causality,” IEEE Trans. Antenn. Propag. 37, 1604–1608 (1989).
[Crossref]

E. Heyman, B. Z. Steinberg, and L. B. Felsen, “Spectral analysis of focus wave modes,” J. Opt. Soc. Am. A 4, 2081–2091 (1987).
[Crossref]

Ito, T.

T. Ito and S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000).
[Crossref] [PubMed]

Lehmann, H.

T. T. Wu and H. Lehmann, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985).
[Crossref]

Lu, J.-Y.

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves: exact solutions to free-space scalar wave equation and their infinite realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[Crossref] [PubMed]

J.-Y. Lu, J. Cheng, and B. Cameron, “Low sidelobe limited diffraction optical coherence tomography,” in “Coherence Domain Optical Methods in Biomedical Science and Clinical Applications VI, Proc. of SPIE,”, vol. 4619, V. V. Tuchin, J. A. Izatt, and J. G. Fujimoto, eds. (SPIE, 2006), vol. 4619, pp. 300–311.

Nóbrega, K. Z.

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[Crossref]

Okazaki, S.

T. Ito and S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000).
[Crossref] [PubMed]

Orlov, S.

Piskarskas, A.

Porras, M. A.

Recami, E.

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[Crossref]

Saari, P.

Sedky, S. M.

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, “The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses,” J. Phys. Math. Gen. 29, 5157–5179 (1996).
[Crossref]

Sezginer, A.

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[Crossref]

Shaarawi, A. M.

A. M. Shaarawi, “Comparison of two localized wave fields generated from dynamic apertures,” J. Opt. Soc. Am. A 14, 1804–1816 (1997).
[Crossref]

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, “The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses,” J. Phys. Math. Gen. 29, 5157–5179 (1996).
[Crossref]

A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, “On the evanescent fields and the causality of the focus wave modes,” J. Math. Phys. 36, 5565–5587 (1995).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[Crossref]

Sõnajalg, H.

Stabinis, A.

Steinberg, B. Z.

Willebrand, H. A.

H. A. Willebrand and B. S. Ghuman, “Fiber optics without fiber,” IEEE Spectr. 38, 40–45 (2001).
[Crossref]

Wu, T. T.

T. T. Wu and H. Lehmann, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985).
[Crossref]

Zamboni-Rached, M.

M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A. 79, 013816 (2009).
[Crossref]

M. Zamboni-Rached, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[Crossref]

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[Crossref]

Ziolkowski, R. W.

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, “The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses,” J. Phys. Math. Gen. 29, 5157–5179 (1996).
[Crossref]

A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, “On the evanescent fields and the causality of the focus wave modes,” J. Math. Phys. 36, 5565–5587 (1995).
[Crossref]

R. Donnelly and R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. Lond. A 440, 541–565 (1993).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[Crossref]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A. 39, 2005–2033 (1989).
[Crossref] [PubMed]

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26, 861–863 (1985).
[Crossref]

IEEE Spectr. (1)

H. A. Willebrand and B. S. Ghuman, “Fiber optics without fiber,” IEEE Spectr. 38, 40–45 (2001).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

E. Heyman, “The focus wave mode: a dilemma with causality,” IEEE Trans. Antenn. Propag. 37, 1604–1608 (1989).
[Crossref]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

J.-Y. Lu and J. F. Greenleaf, “Nondiffracting X waves: exact solutions to free-space scalar wave equation and their infinite realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[Crossref] [PubMed]

J. Appl. Phys. (3)

J. N. Brittingham, “Focus waves modes in homogeneous Maxwell’s equations: transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
[Crossref]

A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
[Crossref]

T. T. Wu and H. Lehmann, “Spreading of electromagnetic pulses,” J. Appl. Phys. 58, 2064–2065 (1985).
[Crossref]

J. Math. Phys. (3)

R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26, 861–863 (1985).
[Crossref]

A. M. Shaarawi, R. W. Ziolkowski, and I. M. Besieris, “On the evanescent fields and the causality of the focus wave modes,” J. Math. Phys. 36, 5565–5587 (1995).
[Crossref]

I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional travelling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Phys. Math. Gen. (1)

A. M. Shaarawi, S. M. Sedky, R. W. Ziolkowski, and I. M. Besieris, “The spatial distribution of the illumination of dynamic apertures and its effect on the decay rate of the radiated localized pulses,” J. Phys. Math. Gen. 29, 5157–5179 (1996).
[Crossref]

Nature (1)

T. Ito and S. Okazaki, “Pushing the limits of lithography,” Nature 406, 1027–1031 (2000).
[Crossref] [PubMed]

Opt. Commun. (1)

M. Zamboni-Rached, K. Z. Nóbrega, H. E. Hernández-Figueroa, and E. Recami, “Localized superluminal solutions to the wave equation in (vacuum or) dispersive media, for arbitrary frequencies and with adjustable bandwidth,” Opt. Commun. 226, 15–23 (2003).
[Crossref]

Opt. Lett. (3)

Phys. Rev. A (1)

M. Zamboni-Rached, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[Crossref]

Phys. Rev. A. (2)

M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A. 79, 013816 (2009).
[Crossref]

R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A. 39, 2005–2033 (1989).
[Crossref] [PubMed]

Proc. R. Soc. Lond. A (1)

R. Donnelly and R. W. Ziolkowski, “Designing localized waves,” Proc. R. Soc. Lond. A 440, 541–565 (1993).
[Crossref]

Other (4)

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, New York, 1995), 2nd ed.

H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized waves (J. Wiley & Sons, New York, NY, 2008).
[Crossref]

L. B. Felsen, “Phase space issues in ultrawideband/short pulse wave modeling,” in Ultra-Wideband, Short-Pulse Electromagnetics, H. Bertoni, L. Carin, and L. B. Felsen, eds. (Plenum Press, New York, 1993).

J.-Y. Lu, J. Cheng, and B. Cameron, “Low sidelobe limited diffraction optical coherence tomography,” in “Coherence Domain Optical Methods in Biomedical Science and Clinical Applications VI, Proc. of SPIE,”, vol. 4619, V. V. Tuchin, J. A. Izatt, and J. G. Fujimoto, eds. (SPIE, 2006), vol. 4619, pp. 300–311.

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Figures (6)

Fig. 1
Fig. 1 The spectrum of the pulses | ψ ¯ ˜ ( k ρ , 0 , ω ) | showing the coupling between the spatial kρ and temporal ω spectral components of the (a) X-wave (α = 0), (b) FWM with ( = 0.02), (c) FWM with ( = 2.0), (d) FWM with ( = 5.0), (e) FWM with ( = 10.0) and (f) FWM with ( = 20.0). Exact coupling relation, that is obtained by substituting Eq. (3) into Eq. (2) is shown in red circles for reference.
Fig. 2
Fig. 2 Comparison of the spatial spectral depletion of the pulses propagating in free-space and silica at z′ = 0, z′ = zD and z′ = 2zD for (a) X-wave (α = 0), (b) FWM with ( = 0.02), (c) FWM with ( = 2.0), (d) FWM with ( = 5.0), (e) FWM with ( = 10.0) and (f) FWM with ( = 20.0).
Fig. 3
Fig. 3 Comparison of the temporal spectral depletion of the truncated pulses in free-space and silica at z′ = 0, z′ = zD and z′ = 2zD for (a) X-wave, (b) FWM with ( = 0.02),(c) FWM with ( = 2.0), (d) FWM with ( = 5.0), (e) FWM with ( = 10.0) and (f) FWM with ( = 20.0).
Fig. 4
Fig. 4 Normalized intensities at the centroid of the truncated pulses vesrus the propagation distance in (a) free-space and (b) silica.
Fig. 5
Fig. 5 Percentage change of the FWHM of the pulses along the propagation distance from z = 0 to z = zD in (a) free-space and (b) silica.
Fig. 6
Fig. 6 Percentage change of the spot-size of the pulses along the propagation distance from z = 0 to z = zD in (a) free-space and (b) silica.

Tables (2)

Tables Icon

Table 1 Values of Bj and ωj for typical bulk fused silica [see Eq. (9)]

Tables Icon

Table 2 Field depth of X-wave and FWM pulses propagating in free-space and silica

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

ψ ( ρ , z , t ) = 0 k ρ d k ρ d k z d ω ψ ˜ ( k ρ , k z , ω ) J 0 ( k ρ ρ ) e i ( k z z ω t ) .
k ρ 2 + k z 2 = ( n ( ω ) ω c ) 2 ,
ω = α + v 0 k z ,
ψ ˜ ( k ρ , k z , ω ) = ψ ˜ ( ω ) δ ( k ρ ( n ( ω ) ω c ) 2 k z 2 ) δ ( k z ω α v 0 ) ,
ψ ( ρ , z , t ) = ω min ω max d ω ψ ˜ ( ω ) ξ 1 J 0 ( ξ 1 ρ ) e i ( ξ 2 z ω t ) , ξ 1 = ( n ( ω ) ω c ) 2 ξ 2 2 , ξ 2 = ω α v 0 .
ω min = α c max ( c n ( ω ) v 0 , ω max = α c min ( c n ( ω ) v 0 ) ,
f t ( t ) = { e ( t / T ) | t | 2 T 0 elsewhere and f ρ ( ρ ) = { e ( ρ / R ) ρ 2 R 0 elsewhere .
ψ ¯ ˜ ( k ρ , k z , ω = ψ ˜ ( k z ) f ˜ ρ ( k ρ χ 1 ) f ˜ t ( ω χ 2 ) .
n 2 ( ω ) = 1 + j = 1 N B j ω j 2 ω j 2 ω 2 ,
Φ ρ ( k ρ ; z ) = 0 ρ d ρ ψ ¯ ( ρ , z , z c 0 ) J 0 ( k ρ ρ ) .
Φ t ( ω ; z ) = dt ψ ¯ ( 0 , z , t z v 0 ) e i ω t .

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