VOLUME 85, NUMBER 10 PHYSICAL REVIEW LETTERS 4SEPTEMBER 2000 Creation of Localized Optical Waves that Do Not Obey the Radiation Condition at Infinity Miguel A. Porras and Félix Salazar-Bloise Departamento de Física Aplicada, Escuela Técnica Superior de Ingenieros de Minas, Universidad Politécnica de Madrid, Rios Rosas 21, E-28003 Madrid, Spain Luis Vázquez Departamento de Matemática Aplicada, Facultad de Informática, Universidad Complutense de Madrid, E-28040 Madrid, Spain and Centro de Astrobiología, INTA, 28850 Torrejón de Ardoz, Madrid, Spain (Received 27 December 1999) We show that the diffraction of a shocking optical pulse formed in a nonlinear transparent dielectric creates an optical missile, or localized radiation field whose amplitude and energy decays are slower than 1R and 1R 2 , respectively, far from the aperture. Dispersion does not eliminate, but limits missile behavior to a finite range. Experimental techniques for optical missile generation are suggested. PACS numbers: 42.65.Re, 42.65.Tg The rapid advances of laser technology in the past decade have made the production of intense pulses of light with only a few optical oscillations feasible. Because of the high intensities concentrated in these pulses, of the order of tens of TWcm 2 , the possibility has been recently [1,2] proposed, as first conjectured by Rosen [3], of nonlinearly generating optical shocks; that is, steepening and breaking the optical cycles upon propagation of an intense femtosecond pulse in a nonlinear transparent dielectric. These shocks can form over a propagation distance of a few micrometers, prior to the formation of the more known shocks in the pulse envelope [1]. Optical cycle steepening in realistic nonlinear dielectrics (fused silica), including dispersive and absorption effects of any order, has been recently evidenced from theoretical and numerical studies from Maxwell equations [1,2]. From the early work by Christov [4], on the other hand, it is known that the diffraction properties of few- cycle pulses may differ substantially from those of quasimonochromatic light. Diffraction is a dispersive phenomenon in the sense that different frequencies diffract differently. The diffraction of femtosecond pulses, having a broad content of frequencies, will then induce some dispersionlike transformations in the pulse form—transformations which need to be understood and therefore are being intensively studied [5–7]. The best understood among them is perhaps the differentiation of the pulse temporal form upon propagation from an aper- ture to the far field, or time-derivative effect [5,8]. Closely connected with it [8] is the concept of “electromagnetic missile,” described for the first time by Wu [9] in the context of antenna theory. Electromagnetic missiles are radiation fields from spatially localized sources which do not obey the radiation condition at infinity (1R 2 decay in energy), but decay at slower rates along a specified di- rection away from the source [9]. They can be generated, in principle, by driving the localized source with pulses having a singular derivative stronger than p t , i.e., t m , with m, 12 [10], but their practical realization has run up against the difficulty of physically producing pulses with the required sudden jumps [11]. In this Letter, we show that an optical shock formed in a nonlinear transparent dielectric contains the required jumps to produce the missile effect. The diffraction of a shock pulse by an aperture originates a radiation field whose amplitude and energy present a far field decay sig- nificantly slower than 1z and 1z 2 , respectively, along the perpendicular direction z to the aperture. Numerical simu- lations including dispersion and absorption indicate that the missile effect survives up to a distance proportional to the steepest gradient of the cycles reached in the non- linear dielectric. These optical missiles could be of interest for many practical applications, as distance measurements, alignment, as well as transmission of information and en- ergy over long distances. Let us first derive our basic equation for femtosecond pulse propagation. We consider a polarized pulsed beam of light Ex  , z , t , x  x , y , propagating along the posi- tive z direction according to the wave equation in a ma- terial medium, DE 2 c 22 ≠ tt E  m 0 ≠ tt P, where P is the medium polarization. As shock formation is expected to occur under weak dispersion conditions [1–3], we ne- glect, for the moment, material dispersion by writing the polarization as P  ´ 0 x 1 E 1 P nl E, where P nl is a nonlinear function of E. The linear part of P can be embedded in the second term of the left-hand side of the wave equation by identifying c with the light velocity in the medium. Next, by introducing the local coordinates t 0  t 2 z c, z 0  z , we extract from E its rapid variation with z owing to the pulse transport at c; then the remainder dependence of Ex  , z 0 , t 0  on the new propagation coor- dinate z 0 describes only pulse changes due to diffraction and nonlinearity. If, moreover, these changes are slow enough so that j≠ z 0 Ejøj≠ t 0 Ejc, the following first order propagation equation can be readily derived from the wave equation: 2c≠ z 0 t 0 E  D  E 2m 0 ≠ t 0 t 0 P nl , (1) 2104 0031-9007 00  85(10)  2104(4)$15.00 © 2000 The American Physical Society