Abstract

A criticism of the focus wave mode (FWM) solution for localized pulses is that it contains backward propagating components that are difficult to generate in many practical situations. We describe a form of FWM where the strength of the backward propagating components is identically zero and derive special cases where the field can be written in an analytic form. In particular, a free-space version of “backward light” pulse is considered, which moves in the opposite direction with respect to all its spectral constituents.

©2008 Optical Society of America

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 0th-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:

U(ρ,z,t)=kminJ0{kρsin[θ(k)]}exp{ikzcos[θ(k)]}exp(ikct)f(k)dk,

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

ksin2(θ(k)2)=k0

where k 0 is a constant, the group velocity (dkz/)-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 kz=-k 0 and with the radial wave number kρ=k2kz2=0 (see Fig. 1). Forward propagating components correspond to those for θ<90°, i.e. with kz≥0 and k>2k 0.

The wave function is

U(ρ,z,t)=exp[ik0(z+ct)]k0J0(2k0ρcotθ2)exp(ik0ctcot2θ2)f(k)dk,

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

p2=cot2θ2=kk0k0,

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

U(ρ,z,t)=k0exp[ik0(z+ct)]02g(p)J0(2k0ρp)exp(ik0ctp2)pdp.

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

f(k)=z0[(s+1)(kk1)z0]sexp[(s+1)(kk1)z0],kk1,
=0,k<k1,

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

g(p)=z0[(s+1)(k0k1+k0p2)z0]sexp[(s+1)(k1k0)z0]exp[(s+1)k0z0p2],
p[(k1k0)k0]12,
=0,p<[(k1k0)k0]12.

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

θmax=2arccot{[(k1k0)k0]12}=arccos[(k12k0)k1].
 figure: Fig. 1.

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 θmax 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

U(ρ,z,t)=k0z0exp[(s+1)(k1k0)z0]exp[ik0(z+ct)]
×(k1k0)k02[(s+1)(k0p2k1+k0)z0]sJ0(2k0ρp2)
×exp[(s+1)k0z0p2]exp(ik0ctp2)pdp.

Introducing the coordinates

ν=2ρ[k0(k1k0)]12,
u=2(k1k0)[cti(s+1)z0],

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

l=p(k0k1k0)12,

we then have

U=(ρ,z,t)(k1k0)z0[(s+1)(k1k0)z0]sexp[(s+1)(k1k0)z0ik0(z+ct)]
×12(l21)sJ0(νl)exp(12iul2)ldl.

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

U(ρ,z,t)=(k1k0)z0exp[(k1k0)z0]exp[ik0(z+ct)]
×[2iuexp(iv22u)012J0(νl)exp(12iul2)ldl].

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

U(ρ,z,t)=2(k1k0)z0exp[(k1k0)z0]exp[ik0(z+ct)]eiu2iuL(u,ν),

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

L(u,v)=V0(u,v)iV1(u,v)=m=0(1)m(vu)2m[J2m(v)+ivuJ2m+1(v)].

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:

U(ρ,z,t)=2(k1k0)z0exp[(k1k0)z0]exp[ik0(z+ct)]
×[1iuexp(iv22u)eiu2uM(u,v)],

where

M(u,v)=U1(u,v)+iU2(u,v)=m=0(1)m(uv)2m+1[J2m+1(v)+iuvJ2m+2(v)].

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

U(ρ,z,t)=2(k1k0)z0exp[(k1k0)z0ik0(z+ct)]
×[1iuexp(iv22u)exp(iu4)S(u,v)],

where

S(u,v)=m=0im(2m+1)jm(u4)J2m+1(v)v,

jm 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

Uk1,(ρ,z,t)=Uk0,(ρ,z,t)Uk0,k1(ρ,z,t),

where

Uk0,(ρ,z,t)=exp[(k1k0)z0]
×{z0i(ctz)+z0exp[ik0(ct+z)k0ρ2i(ctz)+z0]}

and

Uk0,k1(ρ,z,t)=exp{12[(k1k0)z0+i(k13k0)zi(k1+k0)ct]}
×2(k1k0)z0S(u,v).

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 Uk0, 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 0β and z 0a 1. The second term Uk0,k1 corresponds to integration over exponential spectrum from k 0 to k 1 in Eq. (3) and in order to get the wave function Uk1, 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 – Uk0,k1 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].

 figure: Fig. 2.

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 kk 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 xρ. 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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 figure: Fig. 3.

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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 figure: Fig. 4.

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

f(k)=0,k<k1,
=z0exp[(kk1)z0],k1kk2,
=0=z0exp[(kk1)z0]z0exp[(k1k2)z0]exp[(kk2)z0],k>k2

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)

Uk1,k2(ρ,z,t)=Uk1,(ρ,z,t)exp[(k1k2)z0]Uk2,(ρ,z,t)=
=Uk0,k1(ρ,z,t)+exp[(k1k2)z0]Uk0,k2(ρ,z,t)

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.

 figure: Fig. 5.

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.

 figure: Fig. 6.

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.

Acknowledgments

This research was supported by the Estonian Science Foundation and the Singapore Ministry of Education Tier1 funding.

References and links

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

2. P. A. Bélanger, “Packetlike solutions of the homogeneous-wave equation,” J. Opt. Soc. Am. A 1, 723–724 (1984). [CrossRef]  

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

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

5. E. Heyman and L. B. Felsen, “Propagating pulsed beam solutions by complex source parameter substitution,” IEEE Trans. Antennas Propag. AP-34, 1062–1065 (1986). [CrossRef]  

6. 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]  

7. E. Heyman and B. Z. Steinberg, “Spectral analysis of complex-source pulsed beams,” J. Opt. Soc. Am. A 4, 3–10 (1987). [CrossRef]  

8. E. Heyman and L. B. Felson, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989). [CrossRef]  

9. E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. AP-37, 1604–1608 (1989). [CrossRef]  

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

11. R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993). [CrossRef]  

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

13. P. L. Overfelt, “Bessel-Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991). [CrossRef]   [PubMed]  

14. 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]  

15. E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994). [CrossRef]  

16. C. J. R. Sheppard and X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133, 1–6 (1997). [CrossRef]  

17. Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997). [CrossRef]  

18. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998). [CrossRef]  

19. C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Opt. Soc. Am. A 18, 2594–2600 (2001). [CrossRef]  

20. C. J. R. Sheppard, “Generalized Bessel pulse beams,” J. Opt. Soc. Am. A 19, 2218–2222 (2002). [CrossRef]  

21. P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004). [CrossRef]  

22. K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 056611 (2002). [CrossRef]  

23. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon Press, Oxford, 1975).

24. B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations,” Physica 13, 605–620 (1947). [CrossRef]  

25. J. Boersma “On the computation of Lommel’s functions of two variables” Math. Comput. 16, 232–238 (1962).

26. P. Saari, “Superluminal localized waves of electromagnetic field in vacuo,” in Time’s Arrows, Quantum Measurement and Superluminal Behavior, D. Mugnai, A. Ranfagni, and L. S. Schulman eds. (Scientific Monographs: Physics Sciences Series, Italian CNR, Rome, 2001) http://xxx.lanl.gov/abs/physics/0103054.

27. P. Saari, M. Menert, and H. Valtna, “Photon localization barrier can be overcome,” Opt. Commun. 246, 445–450 (2005). [CrossRef]  

28. M. A. Porras, “Pulsed light beams in vacuum with superluminal and negative group velocities,” Phys. Rev. E 67, 066604 (2003). [CrossRef]  

29. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135 (1997). [CrossRef]  

30. R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized fewcycle optical wave packets,” Phys. Rev. A 67, 063820 (2003). [CrossRef]  

31. H. Valtna, K. Reivelt, and P. Saari, “Methods for generating wideband localized waves of superluminal group velocity,” Opt. Commun. 278, 1–7 (2007). [CrossRef]  

References

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  1. J. N. Brittingham, “Focus wave modes in homogeneous Maxwell’s equations: Transverse electric mode,” J. Appl. Phys. 54, 1179–1189 (1983).
    [Crossref]
  2. P. A. Bélanger, “Packetlike solutions of the homogeneous-wave equation,” J. Opt. Soc. Am. A 1, 723–724 (1984).
    [Crossref]
  3. A. Sezginer, “A general formulation of focus wave modes,” J. Appl. Phys. 57, 678–683 (1985).
    [Crossref]
  4. R. W. Ziolkowski, “Exact solutions of the wave equation with complex source locations,” J. Math. Phys. 26, 861–863 (1985).
    [Crossref]
  5. E. Heyman and L. B. Felsen, “Propagating pulsed beam solutions by complex source parameter substitution,” IEEE Trans. Antennas Propag. AP-34, 1062–1065 (1986).
    [Crossref]
  6. 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]
  7. E. Heyman and B. Z. Steinberg, “Spectral analysis of complex-source pulsed beams,” J. Opt. Soc. Am. A 4, 3–10 (1987).
    [Crossref]
  8. E. Heyman and L. B. Felson, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
    [Crossref]
  9. E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. AP-37, 1604–1608 (1989).
    [Crossref]
  10. I. M. Besieris, A. M. Shaarawi, and R. W. Ziolkowski, “A bidirectional traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
    [Crossref]
  11. R. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
    [Crossref]
  12. R. W. Ziolkowski, “Localized transmission of electromagnetic energy,” Phys. Rev. A 39, 2005–2033 (1989).
    [Crossref] [PubMed]
  13. P. L. Overfelt, “Bessel-Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
    [Crossref] [PubMed]
  14. 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]
  15. E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
    [Crossref]
  16. C. J. R. Sheppard and X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133, 1–6 (1997).
    [Crossref]
  17. Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
    [Crossref]
  18. M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
    [Crossref]
  19. C. J. R. Sheppard, “Bessel pulse beams and focus wave modes,” J. Opt. Soc. Am. A 18, 2594–2600 (2001).
    [Crossref]
  20. C. J. R. Sheppard, “Generalized Bessel pulse beams,” J. Opt. Soc. Am. A 19, 2218–2222 (2002).
    [Crossref]
  21. P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
    [Crossref]
  22. K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 056611 (2002).
    [Crossref]
  23. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon Press, Oxford, 1975).
  24. B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations,” Physica 13, 605–620 (1947).
    [Crossref]
  25. J. Boersma “On the computation of Lommel’s functions of two variables” Math. Comput. 16, 232–238 (1962).
  26. P. Saari, “Superluminal localized waves of electromagnetic field in vacuo,” in Time’s Arrows, Quantum Measurement and Superluminal Behavior, D. Mugnai, A. Ranfagni, and L. S. Schulman eds. (Scientific Monographs: Physics Sciences Series, Italian CNR, Rome, 2001) http://xxx.lanl.gov/abs/physics/0103054.
  27. P. Saari, M. Menert, and H. Valtna, “Photon localization barrier can be overcome,” Opt. Commun. 246, 445–450 (2005).
    [Crossref]
  28. M. A. Porras, “Pulsed light beams in vacuum with superluminal and negative group velocities,” Phys. Rev. E 67, 066604 (2003).
    [Crossref]
  29. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135 (1997).
    [Crossref]
  30. R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized fewcycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
    [Crossref]
  31. H. Valtna, K. Reivelt, and P. Saari, “Methods for generating wideband localized waves of superluminal group velocity,” Opt. Commun. 278, 1–7 (2007).
    [Crossref]

2007 (1)

H. Valtna, K. Reivelt, and P. Saari, “Methods for generating wideband localized waves of superluminal group velocity,” Opt. Commun. 278, 1–7 (2007).
[Crossref]

2005 (1)

P. Saari, M. Menert, and H. Valtna, “Photon localization barrier can be overcome,” Opt. Commun. 246, 445–450 (2005).
[Crossref]

2004 (1)

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
[Crossref]

2003 (2)

M. A. Porras, “Pulsed light beams in vacuum with superluminal and negative group velocities,” Phys. Rev. E 67, 066604 (2003).
[Crossref]

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized fewcycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

2002 (2)

C. J. R. Sheppard, “Generalized Bessel pulse beams,” J. Opt. Soc. Am. A 19, 2218–2222 (2002).
[Crossref]

K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 056611 (2002).
[Crossref]

2001 (1)

1998 (1)

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[Crossref]

1997 (3)

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135 (1997).
[Crossref]

C. J. R. Sheppard and X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133, 1–6 (1997).
[Crossref]

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[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]

1994 (1)

E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
[Crossref]

1993 (1)

1991 (1)

P. L. Overfelt, “Bessel-Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
[Crossref] [PubMed]

1989 (4)

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

E. Heyman and L. B. Felson, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6, 806–817 (1989).
[Crossref]

E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. AP-37, 1604–1608 (1989).
[Crossref]

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

1987 (2)

1986 (1)

E. Heyman and L. B. Felsen, “Propagating pulsed beam solutions by complex source parameter substitution,” IEEE Trans. Antennas Propag. AP-34, 1062–1065 (1986).
[Crossref]

1985 (2)

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

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

1984 (1)

1983 (1)

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

1962 (1)

J. Boersma “On the computation of Lommel’s functions of two variables” Math. Comput. 16, 232–238 (1962).

1947 (1)

B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations,” Physica 13, 605–620 (1947).
[Crossref]

Bélanger, P. A.

Besieris, I. M.

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. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[Crossref]

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

Boersma, J.

J. Boersma “On the computation of Lommel’s functions of two variables” Math. Comput. 16, 232–238 (1962).

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon Press, Oxford, 1975).

Brittingham, J. N.

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

E. Heyman and L. B. Felsen, “Propagating pulsed beam solutions by complex source parameter substitution,” IEEE Trans. Antennas Propag. AP-34, 1062–1065 (1986).
[Crossref]

Felson, L. B.

Fortin, M.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized fewcycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

Gan, X.

C. J. R. Sheppard and X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133, 1–6 (1997).
[Crossref]

Griebner, U.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized fewcycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

Grunwald, R.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized fewcycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

Heyman, E.

E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
[Crossref]

E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. AP-37, 1604–1608 (1989).
[Crossref]

E. Heyman and L. B. Felson, “Complex-source pulsed-beam fields,” J. Opt. Soc. Am. A 6, 806–817 (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]

E. Heyman and B. Z. Steinberg, “Spectral analysis of complex-source pulsed beams,” J. Opt. Soc. Am. A 4, 3–10 (1987).
[Crossref]

E. Heyman and L. B. Felsen, “Propagating pulsed beam solutions by complex source parameter substitution,” IEEE Trans. Antennas Propag. AP-34, 1062–1065 (1986).
[Crossref]

Kebbel, V.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized fewcycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

Kummrow, A.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized fewcycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

Lin, Q.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[Crossref]

Melamed, T.

E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
[Crossref]

Menert, M.

P. Saari, M. Menert, and H. Valtna, “Photon localization barrier can be overcome,” Opt. Commun. 246, 445–450 (2005).
[Crossref]

Neumann, U.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized fewcycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

Nibbering, E. T. J.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized fewcycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

Nijboer, B. R. A.

B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations,” Physica 13, 605–620 (1947).
[Crossref]

Overfelt, P. L.

P. L. Overfelt, “Bessel-Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
[Crossref] [PubMed]

Piché, M.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized fewcycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

Porras, M. A.

M. A. Porras, “Pulsed light beams in vacuum with superluminal and negative group velocities,” Phys. Rev. E 67, 066604 (2003).
[Crossref]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[Crossref]

Reivelt, K.

H. Valtna, K. Reivelt, and P. Saari, “Methods for generating wideband localized waves of superluminal group velocity,” Opt. Commun. 278, 1–7 (2007).
[Crossref]

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
[Crossref]

K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 056611 (2002).
[Crossref]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135 (1997).
[Crossref]

Rini, M.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized fewcycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

Rousseau, G.

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized fewcycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

Saari, P.

H. Valtna, K. Reivelt, and P. Saari, “Methods for generating wideband localized waves of superluminal group velocity,” Opt. Commun. 278, 1–7 (2007).
[Crossref]

P. Saari, M. Menert, and H. Valtna, “Photon localization barrier can be overcome,” Opt. Commun. 246, 445–450 (2005).
[Crossref]

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
[Crossref]

K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 056611 (2002).
[Crossref]

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135 (1997).
[Crossref]

P. Saari, “Superluminal localized waves of electromagnetic field in vacuo,” in Time’s Arrows, Quantum Measurement and Superluminal Behavior, D. Mugnai, A. Ranfagni, and L. S. Schulman eds. (Scientific Monographs: Physics Sciences Series, Italian CNR, Rome, 2001) http://xxx.lanl.gov/abs/physics/0103054.

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, 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. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[Crossref]

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

Sheppard, C. J. R.

Steinberg, B. Z.

Valtna, H.

H. Valtna, K. Reivelt, and P. Saari, “Methods for generating wideband localized waves of superluminal group velocity,” Opt. Commun. 278, 1–7 (2007).
[Crossref]

P. Saari, M. Menert, and H. Valtna, “Photon localization barrier can be overcome,” Opt. Commun. 246, 445–450 (2005).
[Crossref]

Wang, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon Press, Oxford, 1975).

Xu, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[Crossref]

Zhang, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[Crossref]

Ziolkowski, R. W.

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. W. Ziolkowski, I. M. Besieris, and A. M. Shaarawi, “Aperture realizations of exact solutions to homogeneous-wave equations,” J. Opt. Soc. Am. A 10, 75–87 (1993).
[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 traveling plane wave representation of exact solutions of the scalar wave equation,” J. Math. Phys. 30, 1254–1269 (1989).
[Crossref]

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

IEEE J. Quantum Electron. (1)

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Space-time profiles of an ultrashort pulsed Gaussian beam,” IEEE J. Quantum Electron. 33, 566–573 (1997).
[Crossref]

IEEE Trans. Antennas Propag. (3)

E. Heyman and T. Melamed, “Certain considerations in aperture synthesis of ultrawideband/short-pulse radiation,” IEEE Trans. Antennas Propag. 42, 518–525 (1994).
[Crossref]

E. Heyman, “Focus wave modes: a dilemma with causality,” IEEE Trans. Antennas Propag. AP-37, 1604–1608 (1989).
[Crossref]

E. Heyman and L. B. Felsen, “Propagating pulsed beam solutions by complex source parameter substitution,” IEEE Trans. Antennas Propag. AP-34, 1062–1065 (1986).
[Crossref]

J. Appl. Phys. (2)

J. N. Brittingham, “Focus wave 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]

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]

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

J. Opt. Soc. Am. A (7)

Math. Comput. (1)

J. Boersma “On the computation of Lommel’s functions of two variables” Math. Comput. 16, 232–238 (1962).

Opt. Commun. (3)

P. Saari, M. Menert, and H. Valtna, “Photon localization barrier can be overcome,” Opt. Commun. 246, 445–450 (2005).
[Crossref]

H. Valtna, K. Reivelt, and P. Saari, “Methods for generating wideband localized waves of superluminal group velocity,” Opt. Commun. 278, 1–7 (2007).
[Crossref]

C. J. R. Sheppard and X. Gan, “Free-space propagation of femto-second light pulses,” Opt. Commun. 133, 1–6 (1997).
[Crossref]

Phys. Rev. A (3)

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

P. L. Overfelt, “Bessel-Gauss pulses,” Phys. Rev. A 44, 3941–3947 (1991).
[Crossref] [PubMed]

R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized fewcycle optical wave packets,” Phys. Rev. A 67, 063820 (2003).
[Crossref]

Phys. Rev. E (4)

M. A. Porras, “Pulsed light beams in vacuum with superluminal and negative group velocities,” Phys. Rev. E 67, 066604 (2003).
[Crossref]

P. Saari and K. Reivelt, “Generation and classification of localized waves by Lorentz transformations in Fourier space,” Phys. Rev. E 69, 036612 (2004).
[Crossref]

K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E 66, 056611 (2002).
[Crossref]

M. A. Porras, “Ultrashort pulsed Gaussian light beams,” Phys. Rev. E 58, 1086–1093 (1998).
[Crossref]

Phys. Rev. Lett. (1)

P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135 (1997).
[Crossref]

Physica (1)

B. R. A. Nijboer, “The diffraction theory of optical aberrations. Part II: Diffraction pattern in the presence of small aberrations,” Physica 13, 605–620 (1947).
[Crossref]

Other (2)

P. Saari, “Superluminal localized waves of electromagnetic field in vacuo,” in Time’s Arrows, Quantum Measurement and Superluminal Behavior, D. Mugnai, A. Ranfagni, and L. S. Schulman eds. (Scientific Monographs: Physics Sciences Series, Italian CNR, Rome, 2001) http://xxx.lanl.gov/abs/physics/0103054.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon Press, Oxford, 1975).

Supplementary Material (5)

» Media 1: MOV (1675 KB)     
» Media 2: MOV (2108 KB)     
» Media 3: MOV (1976 KB)     
» Media 4: MOV (1700 KB)     
» Media 5: MOV (1839 KB)     

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Figures (6)

Fig. 1.
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 θmax with respect to the axis z. For general properties of supports of spectra of localized waves see [20, 21].
Fig. 2.
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 kk 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 xρ. 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]
Fig. 3.
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]
Fig. 4.
Fig. 4. The frame t=0 of the animated plot (1.9 MB) of the truncated-spectrum FWM computed by Eq. (21). [Media 3]
Fig. 5.
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]
Fig. 6.
Fig. 6. The same as Fig. 5, but k 1=1.25k 0, k 2=1.75k 0. [Media 5]

Equations (39)

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

U ( ρ , z , t ) = k min J 0 { k ρ sin [ θ ( k ) ] } exp { ikz cos [ θ ( k ) ] } exp ( ikct ) f ( k ) d k ,
k sin 2 ( θ ( k ) 2 ) = k 0
U ( ρ , z , t ) = exp [ ik 0 ( z + ct ) ] k 0 J 0 ( 2 k 0 ρ cot θ 2 ) exp ( ik 0 ct cot 2 θ 2 ) f ( k ) d k ,
p 2 = cot 2 θ 2 = k k 0 k 0 ,
U ( ρ , z , t ) = k 0 exp [ i k 0 ( z + ct ) ] 0 2 g ( p ) J 0 ( 2 k 0 ρ p ) exp ( i k 0 ct p 2 ) p d p .
f ( k ) = z 0 [ ( s + 1 ) ( k k 1 ) z 0 ] s exp [ ( s + 1 ) ( k k 1 ) z 0 ] , k k 1 ,
= 0 , k < k 1 ,
g ( p ) = z 0 [ ( s + 1 ) ( k 0 k 1 + k 0 p 2 ) z 0 ] s exp [ ( s + 1 ) ( k 1 k 0 ) z 0 ] exp [ ( s + 1 ) k 0 z 0 p 2 ] ,
p [ ( k 1 k 0 ) k 0 ] 1 2 ,
= 0 , p < [ ( k 1 k 0 ) k 0 ] 1 2 .
θ max = 2arc cot { [ ( k 1 k 0 ) k 0 ] 1 2 } = arccos [ ( k 1 2 k 0 ) k 1 ] .
U ( ρ , z , t ) = k 0 z 0 exp [ ( s + 1 ) ( k 1 k 0 ) z 0 ] exp [ ik 0 ( z + ct ) ]
× ( k 1 k 0 ) k 0 2 [ ( s + 1 ) ( k 0 p 2 k 1 + k 0 ) z 0 ] s J 0 ( 2 k 0 ρ p 2 )
× exp [ ( s + 1 ) k 0 z 0 p 2 ] exp ( ik 0 ct p 2 ) p d p .
ν = 2 ρ [ k 0 ( k 1 k 0 ) ] 1 2 ,
u = 2 ( k 1 k 0 ) [ ct i ( s + 1 ) z 0 ] ,
l = p ( k 0 k 1 k 0 ) 1 2 ,
U = ( ρ , z , t ) ( k 1 k 0 ) z 0 [ ( s + 1 ) ( k 1 k 0 ) z 0 ] s exp [ ( s + 1 ) ( k 1 k 0 ) z 0 ik 0 ( z + ct ) ]
× 1 2 ( l 2 1 ) s J 0 ( ν l ) exp ( 1 2 iul 2 ) l d l .
U ( ρ , z , t ) = ( k 1 k 0 ) z 0 exp [ ( k 1 k 0 ) z 0 ] exp [ ik 0 ( z + ct ) ]
× [ 2 iu exp ( iv 2 2 u ) 0 1 2 J 0 ( ν l ) exp ( 1 2 iul 2 ) l d l ] .
U ( ρ , z , t ) = 2 ( k 1 k 0 ) z 0 exp [ ( k 1 k 0 ) z 0 ] exp [ ik 0 ( z + ct ) ] e iu 2 iu L ( u , ν ) ,
L ( u , v ) = V 0 ( u , v ) iV 1 ( u , v ) = m = 0 ( 1 ) m ( v u ) 2 m [ J 2 m ( v ) + iv u J 2 m + 1 ( v ) ] .
U ( ρ , z , t ) = 2 ( k 1 k 0 ) z 0 exp [ ( k 1 k 0 ) z 0 ] exp [ ik 0 ( z + ct ) ]
× [ 1 iu exp ( iv 2 2 u ) e iu 2 u M ( u , v ) ] ,
M ( u , v ) = U 1 ( u , v ) + iU 2 ( u , v ) = m = 0 ( 1 ) m ( u v ) 2 m + 1 [ J 2 m + 1 ( v ) + iu v J 2 m + 2 ( v ) ] .
U ( ρ , z , t ) = 2 ( k 1 k 0 ) z 0 exp [ ( k 1 k 0 ) z 0 ik 0 ( z + ct ) ]
× [ 1 iu exp ( iv 2 2 u ) exp ( iu 4 ) S ( u , v ) ] ,
S ( u , v ) = m = 0 i m ( 2 m + 1 ) j m ( u 4 ) J 2 m + 1 ( v ) v ,
U k 1 , ( ρ , z , t ) = U k 0 , ( ρ , z , t ) U k 0 , k 1 ( ρ , z , t ) ,
U k 0 , ( ρ , z , t ) = exp [ ( k 1 k 0 ) z 0 ]
× { z 0 i ( ct z ) + z 0 exp [ ik 0 ( ct + z ) k 0 ρ 2 i ( ct z ) + z 0 ] }
U k 0 , k 1 ( ρ , z , t ) = exp { 1 2 [ ( k 1 k 0 ) z 0 + i ( k 1 3 k 0 ) z i ( k 1 + k 0 ) ct ] }
× 2 ( k 1 k 0 ) z 0 S ( u , v ) .
f ( k ) = 0 , k < k 1 ,
= z 0 exp [ ( k k 1 ) z 0 ] , k 1 k k 2 ,
= 0 = z 0 exp [ ( k k 1 ) z 0 ] z 0 exp [ ( k 1 k 2 ) z 0 ] exp [ ( k k 2 ) z 0 ] , k > k 2
U k 1 , k 2 ( ρ , z , t ) = U k 1 , ( ρ , z , t ) exp [ ( k 1 k 2 ) z 0 ] U k 2 , ( ρ , z , t ) =
= U k 0 , k 1 ( ρ , z , t ) + exp [ ( k 1 k 2 ) z 0 ] U k 0 , k 2 ( ρ , z , t )

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