Two Fundamental Representations of Localized Pulse Solutions to the Scalar Wave Equation

Besieris, Ioannis M.; Abdelrahman, Mohamed; Shaarawi, Amr M.; Chatzipetros, A.

doi:10.2528/pier97072900

Cited by 101 publications

(110 citation statements)

References 80 publications

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“…The next possible objection to the physical significance of the results obtained might arise from the infinite total energy [11] of the waves given by Eqs. (6), (9), and (10).…”

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confidence: 99%

“…Moreover, physically feasible finite-energy versions of localized waves generally exhibit even better localization properties, although not persistently. A finiteenergy version of the FXW, called the modified focused X wave (MFXW [11]), has the same exponential factor as in Eq. (9), which is multiplied by a fraction that allows to force the axial localization to follow an arbitrarily strong power-law.…”

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confidence: 99%

“…By making use of the historically first representative of localized waves -the so-called focus wave mode (FWM) [22]- [24] (see also Ref. [10] and reviews [11], [15] and references therein) one readily obtains an example of the field that exhibits even much stronger than exponential localization. FWM is given by the scalar function…”

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confidence: 99%

“…The strongly localized waist and the whole amplitude distribution move rigidly and without any spread along the axis z with a superluminal speed c/β. Such wave with intriguing properties, named the focused X wave (FXW) [11], belongs to the so-called propagation-invariant localized solutions to the wave equation -a research field emerged in the 1980-ies (see reviews [11]- [15]) and recently reached its first experimental results [16]- [21]. Hence, in its waist (cross-sectional) plane a one-photon field given by the FXW possesses the same strong localization at any time as the previously considered cylindrical field does in any transversal plane at the instant t = 0.…”

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confidence: 99%

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Photon localization barrier can be overcome

Saari

Menert

Valtna

2005

Optics Communications

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In contradistinction to a widespread belief that the spatial localization of photons is restricted by a power-law falloff of the photon energy density, I.Bialynicki-Birula [Phys. Rev. Lett. 80, 5247 (1998)] has proved that any stronger -up to an almost exponential -falloff is allowed. We are showing that for certain specifically designed cylindrical one-photon states the localization is even better in lateral directions. If the photon state is built from the so-called focus wave mode, the falloff in the waist cross-section plane turns out to be quadratically exponential (Gaussian) and such strong localization persists in the course of propagation.While quantum electrodynamics (QED) underwent an impressive development and reached its maturity in the middle of the last century, one of its basic conceptsthe photon wave function in free space -was deprived of such fortune. Although the photon wave function in coordinate representation was introduced already in 1930 by Landau and Peierls [1] the concept was found to suffer from inherent difficulties that were not overcome during the century (see review [2]). The common explanation presented in textbooks (e.g., [3], [4]) may be summed up as follows: (i) no position operator exists for the photon, (ii) while the position wave function may be localized near a space-time point, the measurable quantities like the electromagnetic field vectors, energy, and the photodetection probability remain spread out due to their non-local relation with the position wave function. However, just before the turn of the century both of these widely-espoused notions were disproved [5], [6] and in the new century a fresh interest in the photon localization problem seems to have been awakened (see, e.g., [7],[8], [9]), meeting the needs of developments in near-field optics, cavity QED, and quantum computing.I. Bialynicki-Birula writes [6] that the statement "even when the position wave function is strongly concentrated near the origin, the energy wave function is spread out over space asymptotically like r −7/2 " (citation from [4], p. 638) is incorrect and that both wave functions may be strongly concentrated near the origin. He demonstrates, on one hand, that photons can be essentially better localized in space -with an exponential falloff of the photon energy density and the photodetection rates. On the other hand, he establishes -and it is even somewhat startling that nobody has done it earlier -that certain localization restrictions arise out of a mathematical property of the positive frequency solutions which therefore are of a universal character and apply not only to photon states but hold for all particles. More specifically, it has been proven in the Letter [6] for the case of spherically imploding-exploding one-photon wavepacket that the Paley-Wiener theorem allows even at instants of maximal localization only such asymptotic decrease of the modulus of the wave function with the radial distance r that is slower than the linear exponential one, i.e., anything slower than ∼...

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“…The next possible objection to the physical significance of the results obtained might arise from the infinite total energy [11] of the waves given by Eqs. (6), (9), and (10).…”

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Photon localization barrier can be overcome

Saari

Menert

Valtna

2005

Optics Communications

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“…These applications may be seen in several researches on electromagnetics fields [33][34][35][36][37][38], considering analysis of their characteristics as self and mutual inductances [39][40][41] or parasitic capacitances [42]. With these feasible applications, an analysis about transformers built with planar coils is presented in this paper, considering the case of square wave excitation, comparing this analysis with the related problems of [19][20][21].…”

Section: Introductionmentioning

confidence: 99%

Planar Transformers Excited by Square Waves

Costa¹

2010

PIER

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Abstract-This paper presents an analysis of found results in experiments developed with transformers built with planar coils, when they are excited by square waves, in comparison with transformers built with planar coils inner ring coils. The transformer was built joining two planar coils one over the other. In this kind of transformer, similar responses as analysis of transformers built with planar coil inner ring coil are found, as well the results of resonance. Because the low self-inductances and parasitic capacitances obtained in these configurations, although the coils resistance is low, which generates low exponential drops on responses. The resonance is found in higher frequencies, but satisfying conditions of sum of responses in resonance.

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Ultrasonic Imaging with Limited‐Diffraction Beams

2007

Localized Waves

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Limited-diffraction beams are a class of waves that may be localized in space and time. Theoretically, these beams are propagation invariant and can propagate to an infinite distance without spreading. In practice, when these beams are produced with wave sources of a finite aperture and energy, they have a very large depth of field, meaning that they can keep a small beam width over a large distance. Because of this property, limited-diffraction beams may have applications in various areas such as medical imaging and tissue characterization. In this paper, fundamentals of limiteddiffraction beams are reviewed and the studies of these beams are put into a unified theoretical framework. Theory of limited-diffraction beams is further developed. New limited-diffraction solutions to Klein-Gordon Equation and Schrodinger Equation, as well as limited-diffraction solutions to these equations in confined spaces are obtained. The relationship between the transformation that converts any solutions to an ( -1)-dimensional wave equation to limited-diffraction solutions of an -dimensional equation and the Lorentz transformation is clarified and extended. The transformation is also applied to the Klein-Gordon Equation. In addition, applications of limited-diffraction beams are summarized. N N

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Two Fundamental Representations of Localized Pulse Solutions to the Scalar Wave Equation

Cited by 101 publications

References 80 publications

Photon localization barrier can be overcome

Photon localization barrier can be overcome

Planar Transformers Excited by Square Waves

Ultrasonic Imaging with Limited‐Diffraction Beams

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