Electromagnetic Beam Propagation in Nonlinear Media
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High Power Laser Science and Engineering, (2015), Vol. 3, e11, 8 pages. © The Author(s) 2015. The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence <http://creativecommons.org/licenses/by/3.0/>. doi:10.1017/hpl.2015.3 Electromagnetic beam propagation in nonlinear media V.V. Semak1 and M.N. Shneider2 1Signature Science, LLC, NJ 08234, USA 2Department of Mechanical and Aerospace Engineering, Princeton University, NJ 08544, USA (Received 30 October 2015; accepted 12 January 2015) Abstract We deduce a complete wave propagation equation that includes inhomogeneity of the dielectric constant and present this propagation equation in compact vector form. Although similar equations are known in narrow fields such as radio wave propagation in the ionosphere and electromagnetic and acoustic wave propagation in stratified media, we develop here a novel approach of using such equations in the modeling of laser beam propagation in nonlinear media. Our approach satisfies the correspondence principle since in the limit of zero-length wavelength it reduces from physical to geometrical optics. Keywords: light propagation; self-focusing; wavefront correction 1. Introduction and other extraordinary particularities of nonlinear beam propagation[6, 9, 10]. Therefore, understandably, we did not The science of nonlinear optics is one of the rapidly growing expect to discover that the results obtained with our recent fields driven by multiple important technological applica- straightforward theoretical model and numerical simulation tions. Since the invention of lasers this field has experienced of ultrahigh intensity laser pulse propagation in gases[11, 12] revolutionary progress fueled by splendid experimental and contradict the established models of self-focusing, beam theoretical results that provide deep understanding of the self-trapping, filamentation and continuum generation. nonlinear response of matter to high intensity electromag- Being unable to find either errors or invalidating as- netic waves. However, as we will demonstrate here, this achievement was accompanied by the fundamental failure sumptions in our forthright approach we journeyed back of one particular subdiscipline – the propagation of a laser to the source (Maxwell’s equations) in order to review the beam in nonlinear media. foundational scientific principles. This examination exposed Many theoretical works describing laser propagation in the assumptions in the original physical concept that, in our nonlinear media have been published since the concept of opinion, are inconsistent and self-contradictory. Also, this laser beam self-focusing and self-trapping was proposed[1]. examination provided theoretical justification and supports After more than 50 years of intense research the theoretical the validity of our previously published approach[11, 12]. concepts and models of self-focusing, beam self-trapping, Below is the account of the results of this journey. filamentation and filament plasma defocusing and the corre- sponding mathematical models were formulated. The books and extensive reviews (for example, Refs. [2–8]) written 2. Formulation of general equation for beam of electro- within the past two decades devote chapters to the detailed magnetic wave propagation in nonlinear media description of the peculiarities of this physical phenomenon. Current frontier research of laser beam propagation in As is well known, the electric and magnetic fields in di- nonlinear media deals with ingenious formulations and electric media are described by the macroscopic Maxwell’s creative solutions of the nonlinear Schrodinger equation equations as follows: that describes laser beam collapse, self-trapping, disper- sion, filamentation, modulation instability, pulse splitting ∇ · D = 0, (1) ∇ · B = 0, (2) Correspondence to: M.N. Shneider, Department of Mechanical and Aerospace Engineering, Princeton University, NJ 08544, USA. Email: ∂ B ∇×E =− , (3) [email protected] ∂t 1 Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 18:23:49, subject to the Cambridge Core terms of use. 2 V.V. Semak and M.N. Shneider ∂ D an inadequate description of wave propagation in nonlinear ∇×H = , (4) ∂t media. For the conditions of inhomogeneity of the properties where E and B are the electric and the magnetic fields, cor- of electrically neutral media, such as in the case of prop- respondingly, while D and H are, respectively, the displace- agation of short wavelength waves in the ionosphere[13], ment and magnetization fields. The latter fields, sometimes Equation (9) can be rewritten in the following form: called ‘macroscopic’ fields, reflect the effect from matter and are defined using phenomenological constituent equations 2 2 n ∂ E ∇ε · E that relate them to the ‘microscopic’ electric field, E, and E − +∇ = 0, (10) c2 ∂t2 ε the magnetic field, B: where we use Equations (1) and (5) from which it follows D = ε εE, (5) 0 that ∇ε·E+ε∇·E = 0 and, therefore, ∇·E =−(∇ε · E/ε). = μ μ , B 0 H (6) Finally, recalling that ε = n2 and using the convenient expression for the displacement field D = ε0εE = ε0(1 + ε μ where 0 and 0 are the permittivity and the permeability χ)E = ε0 E + P, where χ is the electric susceptibility and of free space and ε and μ are the permittivity and the P is the polarization density, we rewrite Equation (7)inthe permeability of material. following form: Following the known procedure we take the curl vector operator of both sides of Equation (3) and the time derivative ∂2 ∇ · ∂2 ∂2 1 E n E PL PNL of both sides of Equation (4) and assume that the magnetic E − + 2∇ = μ0 + μ0 , c2 ∂t2 n ∂t2 ∂t2 effect of the media is negligible, i.e., μ = 1. Then, using (11) Equations (5) and (6), we can eliminate the magnetic and where the polarization density vector is represented by the magnetization fields, obtaining the equation for the electric sum of linear and nonlinear components denoted using the field subscripts ‘L’ and ‘NL’, respectively. 2 ∂ (εE) The solution of the propagation equation expressed in the ∇×(∇×E) + ε0μ0 = 0. (7) ∂t2 form of either Equation (10)or(11) can be expressed in Assuming that the material effect on the electric field is terms of the slowly varying amplitude function slow compared to the period of optical oscillation or the (()·−ω ) laser pulse duration, i.e., assuming time independence of the E(r, t) = A(r)ei k r r t + c.c. (12) permittivity of material, assuming that the material is weakly absorbing, and recalling that√ the speed of the electromagnetic In this solution the vector amplitudes of the electric field = / ε μ wave in a vacuum is c √1 0 0 and the index of refraction and the wavevector are coordinate dependent, i.e., expres- of the material is n = ε, we rewrite Equation (7)as sion (12) represents the electric field of a nonplanar wave. Below we will demonstrate that, within paraxial approx- n2 ∂2 E imation and considering the propagation range in which ∇×(∇×E) + = 0. (8) c2 ∂t2 variation of the spatial profile of laser beam irradiance due to the effect of nonlinear induced refraction is small and, Now, using the identity ∇×(∇×A) =∇(∇ · A) − A we thus, can be considered as perturbation, the propagation rewrite Equation (8) in the following form: Equations (10)or(11) can be straightforwardly modified into an equation that, as proposed in Refs. [11, 12], has a solution 2 2 represented by the blending of the solution of the Helmholtz n ∂ E E − −∇(∇ · E) = 0. (9) equation for propagation of a laser beam in a medium with a c2 ∂t2 uniform and irradiance independent refractive index similar From this point the derivations will significantly deviate to the one obtained by Kogelnik and Li[14] and a correction from the procedure commonly performed in all of the books term that represents nonlinear field perturbation expressed in and journal publications on nonlinear optics since we will terms of paraxial ray optics (the eikonal equation)[15]. be considering the permittivity of the material and, conse- quently, the index of refraction, as a coordinate dependent function. In contrast to our approach, customary considera- 3. Examination of current theory for laser beam propa- tions treat permittivity as a constant, zeroing the third term in gation in nonlinear media the left-hand side of Equation (9) and reducing this equation to the commonly known form that contains only two first Now we will demonstrate that the current formulations of terms and is called the wave equation. As we show below, the propagation equation in nonlinear (self-induced inho- neglecting the third term is a significant mistake that leads to mogeneity) media are based on two inconsistent and self- Downloaded from https://www.cambridge.org/core. 02 Oct 2021 at 18:23:49, subject to the Cambridge Core terms of use. Electromagnetic beam propagation in nonlinear media 3 contradictory assumptions. First, in all theoretical works work[16]. All of the subsequent works, except for our recent known to us it is assumed that the laser beam has a plane publications[11, 12], followed the path laid[16] and labored on wavefront. Second, as mentioned above, the term responsi- various mathematical treatments of equations that can be ble for refraction due to media inhomogeneity is disregarded, traced to the propagation equation deduced for homogeneous i.e., the media are assumed as homogeneous. media. Thus, the ‘foundational’ propagation equation in Under the first assumption, i.e., assumption of a plane nonlinear optics was deduced while disregarding media wavefront, the solution of the propagation equation is inhomogeneity (inherent, induced or self-induced), and has expressed in the form of the slowly varying amplitude the form of Equation (15) missing the ‘refraction’ term (see function for example, Equation 7.2.9 in Ref. [3]). At this point one should wonder how an equation with a i(kz z−ωt) E(x, y, z, t) = Ax (x, y, z)e xˆ term missing the refraction due to media inhomogeneity can i(kz z−ωt) + Ay(x, y, z)e yˆ + c.c.