PHYSICAL REVIEW A 92, 057804 (2015)
Reply to “Comment on ‘Guided electromagnetic waves propagating in a plane dielectric waveguide with nonlinear permittivity’ ” Yury G. Smirnov* and Dmitry V. Valovik† Department of Mathematics and Supercomputing, Penza State University, Krasnaya Street 40, Penza 440026, Russia (Received 18 August 2015; published 23 November 2015) The preceding Comment contains statements that we feel are inaccurate and that lead one to think that the problem we study in Phys. Rev. A 91, 013840 (2015) was solved long ago. However, we argue that our results are new and add to the understanding of the process of transverse electric wave propagation in a Kerr medium. In our Reply we contest the critical statements that are given in the Comment. DOI: 10.1103/PhysRevA.92.057804
PACS number(s): 42.65.Wi, 42.65.Tg, 42.82.Et
In [1], the problem of monochromatic transverse-electric (TE) wave propagation along the boundaries of a plane lossless dielectric layer of thickness h is considered. The layer is located between two half spaces x < 0 and h > h with constant permittivities ε1 and ε3 , respectively; without loss of generality it is possible to assume ε1 ε3 > 0. The permittivity in the layer is described by the Kerr law: ε = ε2 + α|E|2 , where E is the complex amplitude of an electric field, α > 0, ε2 > ε1 . The TE wave has the form E = (0,Ey (x)eiγ z ,0), where Ey (0) > 0 is assumed to be known. We looked for eigenmodes which decay when they move off from the boundaries of the layer. It is necessary to determine real values of the parameter γ (n in [2], and kx in [3]), called propagation constants (PCs), which correspond to guided TE waves. We denote this problem by P . If a PC is known, the corresponding eigenmode is easily calculated numerically with arbitrary accuracy. The problem P was analyzed in many papers; in particular, we are going to discuss [2–4], especially [4]. The problem P is reduced to the dispersion equation (DE), which in [1], see formula (9), has the form k1 +∞ dη dη + n = h, 2 2 2 2 −∞ −k3 k2 + η2 + 2αC k2 +η2 +2αC (1) where n = 0,1,2, . . . is an integer, and C is a known constant, k12 = γ 2 − ε1 , k22 = ε2 − γ 2 , and k32 = γ 2 − ε3 . In [1] (Statement 2) is proved the (spectral) equivalency between the problem P and DE (1), namely each solution to (1) is a PC and each PC is a solution to DE (1). Statement 3 from [1] establishes solvability of the problem P , namely there exists an infinite number of PCs with accumulation point at infinity: a finite number of them reduce to solutions of the corresponding linear problem when α → 0; the rest of PCs (an infinite number) tend to infinity as α → 0. The method used in [1] also allows one to investigate other dependencies in the problem P , in particular, the dependence of γ on A can easily be studied; it also works for any real values of the problem’s parameters.
We begin with the main point: neither in [2] nor in [4] has the aforementioned equivalency been proved; neither Statement 3 nor its part, nor its analog has been proved or formulated in [2,3]. In [2], see formula (40), the DE is found in the form ℘(ωf ± ik0 d; g2f ,g3f ) = λ± + I2f ,
where ℘ is the Weierstrass elliptic function. In fact, this equation is an equation with respect to n(=γ ); however DE (2) was not analyzed in this sense and no results about solvability with respect to n were found in [2]. It is stated in [4] that “According to the periodicity properties of Weierstrass elliptic function ℘, Eq. (1) has infinitely many solutions γi distributed discretely. Thus γi must accumulate at infinity. . . .” It is convenient to rewrite DE (2) in the form ℘(φ(γ ); g2f (γ ),g3f (γ )) = A + Bγ 2 ,
†
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1050-2947/2015/92(5)/057804(2)
(3)
where φ is a known (complicated) function, A,B are constants (see exact expressions for these quantities in [2]). We notice that the left-hand side of Eq. (2) is not periodic with respect to γ . In order to find out the existence of solutions to (2) with respect to γ , it is necessary to give an accurate study of the left-hand side; it is also necessary to take into account the behavior of the right-hand side. Different cases are possible for an equation involving a periodic √and holomorphic function;√for example, the equations sin λ2 + 1 = 2 + λ2 and sin λ2 + 1 = 12 + λ2 have no one and two solutions, respectively. Clearly, it is hasty to claim the existence of infinitely many solutions γi to Eq. (2) [or (3)] without studying its left-hand side. Studying (2), it would be interesting to prove an analog of Statement 3 from [1]. The other issue raised in [4] refers to the corresponding linear problem. The authors of [4] state that DE (2) in the limit α → 0 gives four different DEs; see formulas (2) in [4] (the derivation is not presented). At the same time in [1] only one DE arises in the limit α → 0. This DE is well known in linear theory and has the form tan(k2 h) =
*
(2)
k2 (k1 + k3 ) ; k22 − k1 k3
(4)
the inequality max{ε1 ,ε3 } < γ 2 < ε2 must be fulfilled for any PC γ . 057804-1
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COMMENTS
PHYSICAL REVIEW A 92, 057804 (2015)
Since in [1] the equivalency is proved, then the linear limit presented in [1] is the only one possible. The existence of four different DEs in the linear limit in [4] clearly shows that the discussed equivalency between (2) and the original (nonlinear) problem is absent; obviously, Eq. (1) is not equivalent to Eq. (2). It is easy to verify that under considered restrictions there are no solutions of the linear problem for γ 2 > ε2 . Indeed, solving the original problem P for α = 0 and γ 2 > ε2 , one finds
e2k2 h =
k2 (k1 + k3 ) + k1 k3 k22 − , 2 k2 + k2 (k1 + k3 ) + k1 k3
(5)
where k22 = γ 2 − ε2 . One side of (5) is more than 1 and the other side is less than 1. Obviously, Eq. (5) admits no real solutions γ such that γ 2 > ε2 . This simple analysis completely refutes the necessity of consideration of “additional linear limits with physical significance (not addressed in [1])”; k2 , this is possible due to analyticity see [4]. Replacing k2 with i of tan, Eq. (5) can be derived directly from Eq. (4). It is claimed in [4] that “Statement 3 is a consequence of (1), but (1) has more content. . . . It is inappropriate to claim novelty for the contents of [1], in particular there are no new purely nonlinear TE-guided waves.” Taking into account that the authors of [4] did not prove Statement 3 (or anything similar) in any of their previous works, it is strange that they state the validity of this consequence. We have to agree nevertheless that Eq. (2) has more content but this “additional” content arises due to the absence of the discussed equivalency and has no connections with the original problem. We state that the results proved in [1] are new and had not been published before. The DE, which is similar to (2), has also been found in [3]; however, it has not been studied. Numerically in [3] some
[1] Yu. G. Smirnov and D. V. Valovik, Guided electromagnetic waves propagating in a plane dielectric waveguide with nonlinear permittivity, Phys. Rev. A 91, 013840 (2015). [2] H. W. Sch¨urmann, V. S. Serov, and Yu. V. Shestopalov, Tepolarized waves guided by a lossless nonlinear three-layer structure, Phys. Rev. E 58, 1040 (1998). [3] A. D. Boardman and P. Egan, Optically nonlinear waves in thin films, IEEE J. Quantum Electron. 22, 319 (1986). [4] H. W. Sch¨urmann and V. S. Serov, Comment on “guided electromagnetic waves propagating in a plane dielectric waveguide with nonlinear permittivity”, preceeding paper, Phys. Rev. A 92, 057803 (2015). [5] H. K. Chiang, R. P. Kenan, and C. J. Summers, Spurious roots in nonlinear waveguide calculations and a new format for nonlinear
first new eigenmodes were calculated. It is worth noticing that after [3] at least two papers [5,6] were published in which it was pointed out that the DE from [3] has spurious roots. Existence of spurious roots again points to the absence of equivalency between the DE found in elliptic functions and the original problem. Summarizing, we emphasize the following: (i) Saying that the nonlinear branches “are known for about thirty years. . .” the authors of [4] lean on numerical predictions obtained from the DE expressed through elliptic functions. Existence of infinitely many PCs cannot be found numerically; it is also impossible to derive numerically what happens with the dispersion curves as α → 0. Thus the behavior of the dispersion curves and therefore the influence of the Kerr nonlinearity were not understood up until [1]. (ii) For a wide range of saturable nonlinearities, see [7,8], it is easy to find numerically new guided waves. However, it can be proved that each of these new waves can be considered as a perturbation of the corresponding linearized solution. If a new (nonlinear) solution is just a perturbation of a linear solution, then it cannot be considered as something actually new (existence of perturbed solutions is usually expectable). We also add that the approach used in [1] can be applied for many other types of nonlinearities and can be generalized (the generalization is not obvious) to study more complicated cases; in particular, recently using a generalization of this approach, we have proved results similar to Statement 3 from [1] for the TM case [9]. The TM case cannot be studied using elliptic functions. The authors were supported by the Ministry of Education and Science of the Russian Federation (Goszadanie, Project No. 2.1102.2014K); D.V. was also supported by the Russian Federation President Grant (Project No. MK-90.2014.1).
[6]
[7]
[8]
[9]
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waveguide dispersion equations, IEEE J. Quantum Electron. 28, 1756 (1992). Y.-F. Li and K. Iizuka, Unified nonlinear waveguide dispersion equations without spurious roots, IEEE J. Quantum Electron. 31, 791 (1995). D. V. Valovik, Propagation of electromagnetic te waves in a nonlinear medium with saturation, J. Commun. Technol. Electron. 56, 1311 (2011). V. Yu. Kurseeva and D. V. Valovik, Propagation of TE waves in a plane dielectric waveguide with nonlinear permittivity, in Proceedings of the International Conference DAYS on DIFFRACTION 2014 (IEEE, St. Petersburg, 2014), pp. 177–180. Yu. G. Smirnov and D. V. Valovik, Guided transverse magnetic waves propagating in a plane dielectric waveguide with nonlinear permittivity (Unpublished).
