Electromagnetic waves, in vacuum or dielectrics, can be confined in unbounded
cylinders in such a way that they turn around the main axis. For particular
choices of the cylinder's section, interesting stationary configurations may be
assumed. By refining some results obtained in previous papers, additional more
complex situations are examined here. For such peculiar guided waves an
explicit expression is given in terms of Bessel's functions. Possible
applications are in the development of whispering gallery resonators.Comment: 8 pages, 5 figure

Funaro, Daniele

English

arXiv

Electromagnetic waves in vacuum or dielectrics can be confined in unbounded cylinders in such a way that they turn around the main axis. For particular choices of the cylinder's section, interesting stationary configurations may be assumed. By refining some results obtained in previous papers, additional more complex situations are examined here. For such peculiar guided waves an explicit expression is given in terms of Bessel's functions. Possible applications are in the development of whispering gallery resonators

VGTU Journals (Vilnius Gediminas Technical University - Vilnius Tech)

Trapping Electromagnetic Solitons in Cylinders

FUNARO, Daniele

Archivio istituzionale della ricerca - Università di Modena e Reggio Emilia

This article was downloaded by: [Universita Studi di Pavia]On: 10 March 2014, At: 07:00Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UKMathematical Modelling andAnalysisPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/tmma20Trapping Electromagnetic Solitonsin CylindersDaniele Funaroaa Dipartimento di Fisica, Informatica e Matematica,Università di Modena e Reggio Emilia Via Campi 213/B,41125 Modena, Italy. E-mail:Published online: 06 Mar 2014.To cite this article: Daniele Funaro (2014) Trapping Electromagnetic Solitons in Cylinders,Mathematical Modelling and Analysis, 19:1, 44-51To link to this article:  http://dx.doi.org/10.3846/13926292.2014.892904PLEASE SCROLL DOWN FOR ARTICLETaylor & Francis makes every effort to ensure the accuracy of all the information(the “Content”) contained in the publications on our platform. However, Taylor& Francis, our agents, and our licensors make no representations or warrantieswhatsoever as to the accuracy, completeness, or suitability for any purposeof the Content. Any opinions and views expressed in this publication are theopinions and views of the authors, and are not the views of or endorsed byTaylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. 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Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditionsDownloaded by [Universita Studi di Pavia] at 07:00 10 March 2014 Mathematical Modelling and Analysis Publisher: Taylor&Francis and VGTUVolume 19 Number 1, February 2014, 44–51 http://www.tandfonline.com/TMMAhttp://dx.doi.org/10.3846/13926292.2014.892904 Print ISSN: 1392-6292c©Vilnius Gediminas Technical University, 2014 Online ISSN: 1648-3510Trapping Electromagnetic Solitons in CylindersDaniele FunaroDipartimento di Fisica, Informatica e Matematica, Università di Modena eReggio EmiliaVia Campi 213/B, 41125 Modena, ItalyE-mail: daniele.funaro@unimore.itReceived November 4, 2013; revised January 30, 2014; published online February 20, 2014Abstract. Electromagnetic waves in vacuum or dielectrics can be confined in un-bounded cylinders in such a way that they turn around the main axis. For particularchoices of the cylinder’s section, interesting stationary configurations may be assumed.By refining some results obtained in previous papers, additional more complex situ-ations are examined here. For such peculiar guided waves an explicit expression isgiven in terms of Bessel’s functions. Possible applications are in the development ofwhispering gallery resonators.Keywords: electromagnetism, whispering gallery modes, Bessel functions.AMS Subject Classification: 33F05; 47A75; 78A25.1 IntroductionThe classical equations of electromagnetism allow for solutions confined in ring-shaped domains. In vacuum, this is made possible by the orthogonality of theelectric and magnetic fields (E and B) and by the enforcement of the twodivergence-free conditions (divE = 0 and divB = 0). In fact, if the lines offorce are closed and orthogonal, a toroid is a natural environment to set up theinitial conditions, that successively evolve according to Maxwell’s equations:∂E∂t= c2 curlB,∂B∂t= − curlE, (1.1)where c denotes the speed of light. The above time-dependent equations areeasily put in relation with the vector wave equation and the correspondingeigenmodes.The search for electromagnetic waves trapped in a toroid poses interestingmathematical questions. Numerical computations show a variety of solutions,whose dynamics depends on section’s shape. The behavior is strikingly similarto that of a non-viscous fluid confined in a vortex ring, but with additionalintrinsic constraints.Explicit full solutions in terms of Bessel functions are available in the case ofcylinders, where the magnetic field oscillates parallel to the axis and the electricDownloaded by [Universita Studi di Pavia] at 07:00 10 March 2014 EM Solitons in Cylinders 45field lays on the circular sections. The configuration recalls that of a trainof solitons smoothly circulating inside a 2D circular cavity. As documentedin [1], thin rings with large diameter and circular section can be fairly wellapproximated by the above mentioned solutions.For more compact rings the use of numerical simulations is a necessity. Bythe way, not all the shapes are workable. Indeed, only a restricted range ofsections are compatible with all the electromagnetic constraints. Thus, thesolution process must be implemented together with a sort of shape-detectionalgorithm (see again [1]).Extensions of previous results (briefly recalled in the next section) are hereobtained for electromagnetic waves trapped in unbounded cylinders where, therotation around the axis is combined with a radial oscillation. Here we assumethat the section is an annulus, so that the corresponding domain is a hollowcylinder. Having in mind the vector wave equation, the study is connected tothe search of eigenfunctions for the Laplace operator in such a way that thedimension of the corresponding eigenspace is equal to four. As we will see, thisanalysis leads to the study of specific properties of Bessel functions.2 Electromagnetic Waves Turning Around an AxisWe assume that k ≥ 1 is an integer number. We recall that the k-th Bessel’sfunctions are solutions to the eigenvalue problem (r > 0):u′′(ωr) +u′(ωr)r− k2u(ωr)r2+ ω2u(ωr) = 0 (2.1)for a given parameter ω. Bessel’s functions of the first kind are denoted byu = Jk and those of the second kind by u = Yk. We also recall that Jk tends tozero for r tending to zero, while Yk is unbounded in the neighborhood of r = 0(see [8]).Solutions of the entire set of Maxwell’s equations are obtained in cylindricalcoordinates (r, z, φ) as follows (see [3, 4]):E =(kf(ωr)ωrcos(cωt− kφ+ φ0), 0, f ′(ωr) sin(cωt− kφ+ φ0)),B =1c(0, f(ωr) cos(cωt− kφ+ φ0), 0), (2.2)where f is a linear combination of Jk and Yk and φ0 is a given initial phase.The magnetic field is parallel to the cylinder’s axis and the electric field belongsto the plane (r, φ). Bounded solutions are possible in two cases: the domain issuch that r ≥ Rm > 0 for some given Rm (see for instance the second picturein Fig. 1); the function f is just a multiple of Jk, thus r = 0 may be includedin the domain (see for instance the first picture in Fig. 1).The displacement of the electric field at a given time is shown in Fig. 1for k = 1. During time evolution, the electromagnetic wave rigidly rotatesaround the central point. Two examples are taken into account. In the leftpicture, the domain is the circle 0 ≤ r ≤ RM , where RM is the first zero of J1,Math. Model. Anal., 19(1):44–51, 2014.Downloaded by [Universita Studi di Pavia] at 07:00 10 March 2014 46 D. FunaroFigure 1. Displacement of the electric field at fixed time obtained from the expression in(2.2) subject to different kinds of boundary constraints. The sizes of the domains are notarbitrary, but strictly related to the frequency of rotation as well as boundary conditions.The magnetic field is orthogonal to the page.which takes approximately the value 3.832. In this way, at the boundary, boththe magnetic field and the radial component of the electric field are zero. Inthe right picture, we have the annular region Rm = 1 ≤ r ≤ RM . The outerdiameter is now chosen in such a way that the electric field is orthogonal to theboundary (perfect conductivity).Once the frequency of rotation (depending on cω) is fixed and the typeof boundary conditions decided, the size of the domain is automatically con-strained. This means that there are few domains allowing for the creation ofthese waves. If the proper size is not respected, the wave will not follow a sta-tionary motion, i.e., it does not complete the cycle preserving the phase. Theconditions that permit such a guided behavior depend on the eigenvalues of theLaplace operator on the domain. If the shape is such that some eigenvalues havemultiplicity higher than one, then the construction of rotating solitary wavesis possible (we show later the determination process). An analysis of sphericalvortices has been provided in [1], exactly with the purpose of detecting domainswhere two independent eigenfunctions are related to the same eigenvalue. Inthis paper, we stay for simplicity in the case of the cylinder by analyzing thepeculiar case where four independent eigenfunctions share the same eigenvalue.The evolution in a thin toroid can be approximated, with a rather good levelof accuracy, starting from the cylinder’s version; this extension however is notgoing to be studied here.3 A more involved evolutionWe would like to analyze the situation in which the multiplicity of one of theeigenvalues of the Laplace operator, on a certain annular domain, is equal tofour. This will allow us to built more complex electromagnetic waves turningDownloaded by [Universita Studi di Pavia] at 07:00 10 March 2014 EM Solitons in Cylinders 47–10–8–6–4–20242 3 4 5 6 7 8rFigure 2. Plots of f1 and f4 for λ ≈ 1.0699. In this case both the functions vanish atr = Rm = 1 and r = RM ≈ 7.0927.around an axis. We will need to play with both the functions Jk and Yk,therefore we have to stay away from the point r = 0 (recall that Yk is singularthere). From now on we assume that Rm = 1 ≤ r ≤ RM , where the majordiameter RM has to be properly determined. Of course, if Rm is modified theentire setting scales accordingly, due to the linearity of Maxwell’s equations.We impose Dirichlet boundary conditions on both boundaries (the inner andthe outer circumferences), though other conditions may be considered withoutdifficulty. The use of Dirichlet conditions in the inner part leads us to workwith the function:fn(r) = Jn(λr)Yn(λ)− Yn(λr)Jn(λ), (3.1)where λ is a parameter. Of course, we have fn(1) = 0, ∀n and ∀λ.Now, we would like to have:{fn(RM ) = 0,fj(RM ) = 0,j 6= n, (3.2)which is a nonlinear problem leading to the detection of RM and λ. Thismeans that, once the diameter of the inner circumference has been fixed, thefrequency of the evolving wave and the magnitude of the entire domain aregoing to be uniquely determined by (3.2). Not necessarily such a problem hassolution. However, with the help of numerical tests, we were able to establishsome facts.Math. Model. Anal., 19(1):44–51, 2014.Downloaded by [Universita Studi di Pavia] at 07:00 10 March 2014 48 D. FunaroFigure 3. Plots of two independent eigenfunctions of the Laplace operator withhomogeneous Dirichlet boundary conditions on an annular domain. Despite the fact thatthey look totally different, the domain has been dimensioned in such a way thecorresponding eigenvalues are the same.Here below we report some of the conclusions of our analysis:j = 1, n = 2: no values of RM and λ were found;j = 1, n = 3: no values of RM and λ were found;j = 1, n = 4: RM = 7.0927 for λ = 1.0698;j = 1, n = 5: RM = 3.6761 for λ = 2.3871;j = 1, n = 6: RM = 2.7603 for λ = 3.6054;j = 2, n = 3: no values of RM and λ were found;j = 2, n = 4: no values of RM and λ were found;j = 2, n = 5: RM = 5.1500 for λ = 1.7032.These figures were obtained as the result of a simple bisection algorithm appliedto the difference (fn − fj)(r). When we were not able to find the requestedroot RM it was clear (at least from the numerical viewpoint) that the function|fn − fj |(r), r > Rm, was increasing.Let us better examine a specific situation (j = 1, n = 4). Accordingto Fig. 2, both functions f1 and f4 are zero for r = Rm = 1 and RM ≈7.0927. Thus, by imposing the same boundary constraints, we are able to findout two branches of vector solutions of the type given in (2.2) (k = j andk = n respectively) related to the same parameter ω. The intensity of thecorresponding magnetic fields is given by the level lines of Fig. 3.More explicitly, by setting ω =√λ, we can write:w1 = f1(ωr) sin(cωt− φ+ φ0), w2 = f4(ωr) cos(cωt− 4φ+ φ0),w3 = f1(ωr) cos(cωt− φ+ φ0), w4 = f4(ωr) sin(cωt− 4φ+ φ0). (3.3)The functions wi, i = 1, 2, 3, 4, are schematically reported in Fig. 4, for t = 0and φ0 = 0. In particular, w1 and w3 are shaped as in the first picture ofDownloaded by [Universita Studi di Pavia] at 07:00 10 March 2014 EM Solitons in Cylinders 49Figure 4. Schematic representation of four independent eigenfunctions in (3.3). They allhave the same eigenvalue.Fig. 3 and display a difference of phase of 90 degrees. Using w1 and w3 we canobtain the electromagnetic fields according to the expression (2.2) for k = 1.The other two functions, w2 and w4, are as shown in the second picture ofFig. 3 and differ for an angle of 22.5 degrees. They are also related to (2.2)(this time with k = 4). Finally, we can build general solutions by the linearcombination:γa(w1 + w3) + γb(w2 + w4) (3.4)with γa and γb arbitrary constants. It is just a direct computation verifyingthat the expression provided in (3.4) solves the wave equation in vacuum. Infact, the second derivative in time produces the multiplicative factor −c2λ,while the application of c2 times the Laplace operator provides the same factor(use (2.1)).In Fig. 5, one can see, at different time steps, the evolution of (3.4) forγa = γb = 1 and φ0 = 0. Only half cycle is displayed. From the last picturethe sequence restarts from the beginning with the patterns interchanged. Ananimation is available in [2]. The effect vaguely recalls the juggling of threeclubs.4 ConclusionsFrom the theoretical viewpoint, Maxwell’s equations allow for special solutionstrapped inside an infinite cylinder or, more or less equivalently, in a toroid.The study here developed in vacuum reveals original configurations and mayhelp for instance in the analysis of confined plasma (see, e.g., [5]). Other appli-cations may be found in the field of the so called whispering gallery resonators.Waves trapped in these cavities are smoothly guided to circulate around bycontinuous reflection returning at the origin with the initial phase. Spherical,cylindrical and ring-shaped whispering galleries are commonly produced for abroad range of industrial applications. Typical areas of interest are in fibertelecommunications or biosensing. The literature is very rich. We just mentiona couple of non extremely specialized publications: [6, 7].Math. Model. Anal., 19(1):44–51, 2014.Downloaded by [Universita Studi di Pavia] at 07:00 10 March 2014 50 D. FunaroFigure 5. Evolution at different times of a linear combination of eigenfunctionscorresponding to the same eigenvalue. The two eigenfunctions of Fig. 3 merge and rotatewith an oscillating behavior. The frequency is uniquely established by the global size of thedomain.References[1] C. Chinosi, L. Della Croce and D. Funaro. Rotating electromagnetic waves intoroid-shaped regions. Internat. J. Modern Phys. C, 21(1):11–32, 2010.http://dx.doi.org/10.1142/S0129183110014926.Downloaded by [Universita Studi di Pavia] at 07:00 10 March 2014 EM Solitons in Cylinders 51[2] D. Funaro. http://cdm.unimo.it/home/matematica/funaro.daniele/results.htm[3] D. Funaro. Electromagnetism and the Structure of Matter. World Scientific, Sin-gapore, 2008.[4] D. Funaro. From photons to atoms – the electromagnetic nature of matter, 2012.arXiv:1206.3110[5] R.D. Hazeltine and J.D. Meiss. Plasma Confinement. Dover Pub., Mineola, NY,2003.[6] A.N. Oraevsky. Whispering-gallery waves. Quantum Electronics, 32:377–400,2002. http://dx.doi.org/10.1070/QE2002v032n05ABEH002205.[7] A. Snyder and J. Love. Optical Waveguide Theory. Kluwer Academic Publisher,Norwell, MA, 1983.[8] G.N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge Univ.Press, 1944.Math. Model. Anal., 19(1):44–51, 2014.Downloaded by [Universita Studi di Pavia] at 07:00 10 March 2014 

Electromagnetic waves in vacuum or dielectrics can be confined in unbounded cylinders in such a way that they turn around the main axis. For particular choices of the cylinder's section, interesting stationary configurations may be assumed. By refining some results obtained in previous papers, additional more complex situations are examined here. For such peculiar guided waves an explicit expression is given in terms of Bessel's functions. Possible applications are in the development of whispering gallery resonators.</jats:p

Daniele Funaro

Crossref

Mathematical Modelling and Analysis

TRAPPING ELECTROMAGNETIC SOLITONS IN CYLINDERS

Electromagnetic waves in vacuum or dielectrics can be confined in unbounded cylinders in such a way that they turn around the main axis. For particular choices of the cylinder\u27s section, interesting stationary configurations may be assumed. By refining some results obtained in previous papers, additional more complex situations are examined here. For such peculiar guided waves an explicit expression is given in terms of Bessel\u27s functions. Possible applications are in the development of whispering gallery resonators

Journals Published by Vilnius Tech

Mathematical Modelling and Analysis Publisher: Taylor&Francis and VGTUVolume 19 Number 1, February 2014, 44–51 http://www.tandfonline.com/TMMAhttp://dx.doi.org/10.3846/13926292.2014.892904 Print ISSN: 1392-6292c©Vilnius Gediminas Technical University, 2014 Online ISSN: 1648-3510Trapping Electromagnetic Solitons in CylindersDaniele FunaroDipartimento di Fisica, Informatica e Matematica, Università di Modena eReggio EmiliaVia Campi 213/B, 41125 Modena, ItalyE-mail: daniele.funaro@unimore.itReceived November 4, 2013; revised January 30, 2014; published online February 20, 2014Abstract. Electromagnetic waves in vacuum or dielectrics can be confined in un-bounded cylinders in such a way that they turn around the main axis. For particularchoices of the cylinder’s section, interesting stationary configurations may be assumed.By refining some results obtained in previous papers, additional more complex situ-ations are examined here. For such peculiar guided waves an explicit expression isgiven in terms of Bessel’s functions. Possible applications are in the development ofwhispering gallery resonators.Keywords: electromagnetism, whispering gallery modes, Bessel functions.AMS Subject Classification: 33F05; 47A75; 78A25.1 IntroductionThe classical equations of electromagnetism allow for solutions confined in ring-shaped domains. In vacuum, this is made possible by the orthogonality of theelectric and magnetic fields (E and B) and by the enforcement of the twodivergence-free conditions (divE = 0 and divB = 0). In fact, if the lines offorce are closed and orthogonal, a toroid is a natural environment to set up theinitial conditions, that successively evolve according to Maxwell’s equations:∂E∂t= c2 curlB,∂B∂t= − curlE, (1.1)where c denotes the speed of light. The above time-dependent equations areeasily put in relation with the vector wave equation and the correspondingeigenmodes.The search for electromagnetic waves trapped in a toroid poses interestingmathematical questions. Numerical computations show a variety of solutions,whose dynamics depends on section’s shape. The behavior is strikingly similarto that of a non-viscous fluid confined in a vortex ring, but with additionalintrinsic constraints.Explicit full solutions in terms of Bessel functions are available in the case ofcylinders, where the magnetic field oscillates parallel to the axis and the electricEM Solitons in Cylinders 45field lays on the circular sections. The configuration recalls that of a trainof solitons smoothly circulating inside a 2D circular cavity. As documentedin [1], thin rings with large diameter and circular section can be fairly wellapproximated by the above mentioned solutions.For more compact rings the use of numerical simulations is a necessity. Bythe way, not all the shapes are workable. Indeed, only a restricted range ofsections are compatible with all the electromagnetic constraints. Thus, thesolution process must be implemented together with a sort of shape-detectionalgorithm (see again [1]).Extensions of previous results (briefly recalled in the next section) are hereobtained for electromagnetic waves trapped in unbounded cylinders where, therotation around the axis is combined with a radial oscillation. Here we assumethat the section is an annulus, so that the corresponding domain is a hollowcylinder. Having in mind the vector wave equation, the study is connected tothe search of eigenfunctions for the Laplace operator in such a way that thedimension of the corresponding eigenspace is equal to four. As we will see, thisanalysis leads to the study of specific properties of Bessel functions.2 Electromagnetic Waves Turning Around an AxisWe assume that k ≥ 1 is an integer number. We recall that the k-th Bessel’sfunctions are solutions to the eigenvalue problem (r > 0):u′′(ωr) +u′(ωr)r− k2u(ωr)r2+ ω2u(ωr) = 0 (2.1)for a given parameter ω. Bessel’s functions of the first kind are denoted byu = Jk and those of the second kind by u = Yk. We also recall that Jk tends tozero for r tending to zero, while Yk is unbounded in the neighborhood of r = 0(see [8]).Solutions of the entire set of Maxwell’s equations are obtained in cylindricalcoordinates (r, z, φ) as follows (see [3, 4]):E =(kf(ωr)ωrcos(cωt− kφ+ φ0), 0, f ′(ωr) sin(cωt− kφ+ φ0)),B =1c(0, f(ωr) cos(cωt− kφ+ φ0), 0), (2.2)where f is a linear combination of Jk and Yk and φ0 is a given initial phase.The magnetic field is parallel to the cylinder’s axis and the electric field belongsto the plane (r, φ). Bounded solutions are possible in two cases: the domain issuch that r ≥ Rm > 0 for some given Rm (see for instance the second picturein Fig. 1); the function f is just a multiple of Jk, thus r = 0 may be includedin the domain (see for instance the first picture in Fig. 1).The displacement of the electric field at a given time is shown in Fig. 1for k = 1. During time evolution, the electromagnetic wave rigidly rotatesaround the central point. Two examples are taken into account. In the leftpicture, the domain is the circle 0 ≤ r ≤ RM , where RM is the first zero of J1,Math. Model. Anal., 19(1):44–51, 2014.46 D. FunaroFigure 1. Displacement of the electric field at fixed time obtained from the expression in(2.2) subject to different kinds of boundary constraints. The sizes of the domains are notarbitrary, but strictly related to the frequency of rotation as well as boundary conditions.The magnetic field is orthogonal to the page.which takes approximately the value 3.832. In this way, at the boundary, boththe magnetic field and the radial component of the electric field are zero. Inthe right picture, we have the annular region Rm = 1 ≤ r ≤ RM . The outerdiameter is now chosen in such a way that the electric field is orthogonal to theboundary (perfect conductivity).Once the frequency of rotation (depending on cω) is fixed and the typeof boundary conditions decided, the size of the domain is automatically con-strained. This means that there are few domains allowing for the creation ofthese waves. If the proper size is not respected, the wave will not follow a sta-tionary motion, i.e., it does not complete the cycle preserving the phase. Theconditions that permit such a guided behavior depend on the eigenvalues of theLaplace operator on the domain. If the shape is such that some eigenvalues havemultiplicity higher than one, then the construction of rotating solitary wavesis possible (we show later the determination process). An analysis of sphericalvortices has been provided in [1], exactly with the purpose of detecting domainswhere two independent eigenfunctions are related to the same eigenvalue. Inthis paper, we stay for simplicity in the case of the cylinder by analyzing thepeculiar case where four independent eigenfunctions share the same eigenvalue.The evolution in a thin toroid can be approximated, with a rather good levelof accuracy, starting from the cylinder’s version; this extension however is notgoing to be studied here.3 A more involved evolutionWe would like to analyze the situation in which the multiplicity of one of theeigenvalues of the Laplace operator, on a certain annular domain, is equal tofour. This will allow us to built more complex electromagnetic waves turningEM Solitons in Cylinders 47–10–8–6–4–20242 3 4 5 6 7 8rFigure 2. Plots of f1 and f4 for λ ≈ 1.0699. In this case both the functions vanish atr = Rm = 1 and r = RM ≈ 7.0927.around an axis. We will need to play with both the functions Jk and Yk,therefore we have to stay away from the point r = 0 (recall that Yk is singularthere). From now on we assume that Rm = 1 ≤ r ≤ RM , where the majordiameter RM has to be properly determined. Of course, if Rm is modified theentire setting scales accordingly, due to the linearity of Maxwell’s equations.We impose Dirichlet boundary conditions on both boundaries (the inner andthe outer circumferences), though other conditions may be considered withoutdifficulty. The use of Dirichlet conditions in the inner part leads us to workwith the function:fn(r) = Jn(λr)Yn(λ)− Yn(λr)Jn(λ), (3.1)where λ is a parameter. Of course, we have fn(1) = 0, ∀n and ∀λ.Now, we would like to have:{fn(RM ) = 0,fj(RM ) = 0,j 6= n, (3.2)which is a nonlinear problem leading to the detection of RM and λ. Thismeans that, once the diameter of the inner circumference has been fixed, thefrequency of the evolving wave and the magnitude of the entire domain aregoing to be uniquely determined by (3.2). Not necessarily such a problem hassolution. However, with the help of numerical tests, we were able to establishsome facts.Math. Model. Anal., 19(1):44–51, 2014.48 D. FunaroFigure 3. Plots of two independent eigenfunctions of the Laplace operator withhomogeneous Dirichlet boundary conditions on an annular domain. Despite the fact thatthey look totally different, the domain has been dimensioned in such a way thecorresponding eigenvalues are the same.Here below we report some of the conclusions of our analysis:j = 1, n = 2: no values of RM and λ were found;j = 1, n = 3: no values of RM and λ were found;j = 1, n = 4: RM = 7.0927 for λ = 1.0698;j = 1, n = 5: RM = 3.6761 for λ = 2.3871;j = 1, n = 6: RM = 2.7603 for λ = 3.6054;j = 2, n = 3: no values of RM and λ were found;j = 2, n = 4: no values of RM and λ were found;j = 2, n = 5: RM = 5.1500 for λ = 1.7032.These figures were obtained as the result of a simple bisection algorithm appliedto the difference (fn − fj)(r). When we were not able to find the requestedroot RM it was clear (at least from the numerical viewpoint) that the function|fn − fj |(r), r > Rm, was increasing.Let us better examine a specific situation (j = 1, n = 4). Accordingto Fig. 2, both functions f1 and f4 are zero for r = Rm = 1 and RM ≈7.0927. Thus, by imposing the same boundary constraints, we are able to findout two branches of vector solutions of the type given in (2.2) (k = j andk = n respectively) related to the same parameter ω. The intensity of thecorresponding magnetic fields is given by the level lines of Fig. 3.More explicitly, by setting ω =√λ, we can write:w1 = f1(ωr) sin(cωt− φ+ φ0), w2 = f4(ωr) cos(cωt− 4φ+ φ0),w3 = f1(ωr) cos(cωt− φ+ φ0), w4 = f4(ωr) sin(cωt− 4φ+ φ0). (3.3)The functions wi, i = 1, 2, 3, 4, are schematically reported in Fig. 4, for t = 0and φ0 = 0. In particular, w1 and w3 are shaped as in the first picture ofEM Solitons in Cylinders 49Figure 4. Schematic representation of four independent eigenfunctions in (3.3). They allhave the same eigenvalue.Fig. 3 and display a difference of phase of 90 degrees. Using w1 and w3 we canobtain the electromagnetic fields according to the expression (2.2) for k = 1.The other two functions, w2 and w4, are as shown in the second picture ofFig. 3 and differ for an angle of 22.5 degrees. They are also related to (2.2)(this time with k = 4). Finally, we can build general solutions by the linearcombination:γa(w1 + w3) + γb(w2 + w4) (3.4)with γa and γb arbitrary constants. It is just a direct computation verifyingthat the expression provided in (3.4) solves the wave equation in vacuum. Infact, the second derivative in time produces the multiplicative factor −c2λ,while the application of c2 times the Laplace operator provides the same factor(use (2.1)).In Fig. 5, one can see, at different time steps, the evolution of (3.4) forγa = γb = 1 and φ0 = 0. Only half cycle is displayed. From the last picturethe sequence restarts from the beginning with the patterns interchanged. Ananimation is available in [2]. The effect vaguely recalls the juggling of threeclubs.4 ConclusionsFrom the theoretical viewpoint, Maxwell’s equations allow for special solutionstrapped inside an infinite cylinder or, more or less equivalently, in a toroid.The study here developed in vacuum reveals original configurations and mayhelp for instance in the analysis of confined plasma (see, e.g., [5]). Other appli-cations may be found in the field of the so called whispering gallery resonators.Waves trapped in these cavities are smoothly guided to circulate around bycontinuous reflection returning at the origin with the initial phase. Spherical,cylindrical and ring-shaped whispering galleries are commonly produced for abroad range of industrial applications. Typical areas of interest are in fibertelecommunications or biosensing. The literature is very rich. We just mentiona couple of non extremely specialized publications: [6, 7].Math. Model. Anal., 19(1):44–51, 2014.50 D. FunaroFigure 5. Evolution at different times of a linear combination of eigenfunctionscorresponding to the same eigenvalue. The two eigenfunctions of Fig. 3 merge and rotatewith an oscillating behavior. The frequency is uniquely established by the global size of thedomain.References[1] C. Chinosi, L. Della Croce and D. Funaro. Rotating electromagnetic waves intoroid-shaped regions. Internat. J. Modern Phys. C, 21(1):11–32, 2010.http://dx.doi.org/10.1142/S0129183110014926.EM Solitons in Cylinders 51[2] D. Funaro. http://cdm.unimo.it/home/matematica/funaro.daniele/results.htm[3] D. Funaro. Electromagnetism and the Structure of Matter. World Scientific, Sin-gapore, 2008.[4] D. Funaro. From photons to atoms – the electromagnetic nature of matter, 2012.arXiv:1206.3110[5] R.D. Hazeltine and J.D. Meiss. Plasma Confinement. Dover Pub., Mineola, NY,2003.[6] A.N. Oraevsky. Whispering-gallery waves. Quantum Electronics, 32:377–400,2002. http://dx.doi.org/10.1070/QE2002v032n05ABEH002205.[7] A. Snyder and J. Love. Optical Waveguide Theory. Kluwer Academic Publisher,Norwell, MA, 1983.[8] G.N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge Univ.Press, 1944.Math. Model. Anal., 19(1):44–51, 2014.

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Mathematical Modelling and Analysis Publisher: Taylor&Francis and VGTU
Volume 19 Number 1, February 2014, 44–51 http://www.tandfonline.com/TMMA
http://dx.doi.org/10.3846/13926292.2014.892904 Print ISSN: 1392-6292
c©Vilnius Gediminas Technical University, 2014 Online ISSN: 1648-3510
Trapping Electromagnetic Solitons in Cylinders
Daniele Funaro
Dipartimento di Fisica, Informatica e Matematica, Università di Modena e
Reggio Emilia
Via Campi 213/B, 41125 Modena, Italy
E-mail: daniele.funaro@unimore.it
Received November 4, 2013; revised January 30, 2014; published online February 20, 2014
Abstract. Electromagnetic waves in vacuum or dielectrics can be confined in un-
bounded cylinders in such a way that they turn around the main axis. For particular
choices of the cylinder’s section, interesting stationary configurations may be assumed.
By refining some results obtained in previous papers, additional more complex situ-
ations are examined here. For such peculiar guided waves an explicit expression is
given in terms of Bessel’s functions. Possible applications are in the development of
whispering gallery resonators.
Keywords: electromagnetism, whispering gallery modes, Bessel functions.
AMS Subject Classification: 33F05; 47A75; 78A25.
1 Introduction
The classical equations of electromagnetism allow for solutions confined in ring-
shaped domains. In vacuum, this is made possible by the orthogonality of the
electric and magnetic fields (E and B) and by the enforcement of the two
divergence-free conditions (divE = 0 and divB = 0). In fact, if the lines of
force are closed and orthogonal, a toroid is a natural environment to set up the
initial conditions, that successively evolve according to Maxwell’s equations:
∂E
∂t
= c2 curlB,
∂B
∂t
= − curlE, (1.1)
where c denotes the speed of light. The above time-dependent equations are
easily put in relation with the vector wave equation and the corresponding
eigenmodes.
The search for electromagnetic waves trapped in a toroid poses interesting
mathematical questions. Numerical computations show a variety of solutions,
whose dynamics depends on section’s shape. The behavior is strikingly similar
to that of a non-viscous fluid confined in a vortex ring, but with additional
intrinsic constraints.
Explicit full solutions in terms of Bessel functions are available in the case of
cylinders, where the magnetic field oscillates parallel to the axis and the electric
EM Solitons in Cylinders 45
field lays on the circular sections. The configuration recalls that of a train
of solitons smoothly circulating inside a 2D circular cavity. As documented
in [1], thin rings with large diameter and circular section can be fairly well
approximated by the above mentioned solutions.
For more compact rings the use of numerical simulations is a necessity. By
the way, not all the shapes are workable. Indeed, only a restricted range of
sections are compatible with all the electromagnetic constraints. Thus, the
solution process must be implemented together with a sort of shape-detection
algorithm (see again [1]).
Extensions of previous results (briefly recalled in the next section) are here
obtained for electromagnetic waves trapped in unbounded cylinders where, the
rotation around the axis is combined with a radial oscillation. Here we assume
that the section is an annulus, so that the corresponding domain is a hollow
cylinder. Having in mind the vector wave equation, the study is connected to
the search of eigenfunctions for the Laplace operator in such a way that the
dimension of the corresponding eigenspace is equal to four. As we will see, this
analysis leads to the study of specific properties of Bessel functions.
2 Electromagnetic Waves Turning Around an Axis
We assume that k ≥ 1 is an integer number. We recall that the k-th Bessel’s
functions are solutions to the eigenvalue problem (r > 0):
u′′(ωr) +
u′(ωr)
r
− k
2u(ωr)
r2
+ ω2u(ωr) = 0 (2.1)
for a given parameter ω. Bessel’s functions of the first kind are denoted by
u = Jk and those of the second kind by u = Yk. We also recall that Jk tends to
zero for r tending to zero, while Yk is unbounded in the neighborhood of r = 0
(see [8]).
Solutions of the entire set of Maxwell’s equations are obtained in cylindrical
coordinates (r, z, φ) as follows (see [3, 4]):
E =
(
kf(ωr)
ωr
cos(cωt− kφ+ φ0), 0, f ′(ωr) sin(cωt− kφ+ φ0)
)
,
B =
1
c
(
0, f(ωr) cos(cωt− kφ+ φ0), 0
)
, (2.2)
where f is a linear combination of Jk and Yk and φ0 is a given initial phase.
The magnetic field is parallel to the cylinder’s axis and the electric field belongs
to the plane (r, φ). Bounded solutions are possible in two cases: the domain is
such that r ≥ Rm > 0 for some given Rm (see for instance the second picture
in Fig. 1); the function f is just a multiple of Jk, thus r = 0 may be included
in the domain (see for instance the first picture in Fig. 1).
The displacement of the electric field at a given time is shown in Fig. 1
for k = 1. During time evolution, the electromagnetic wave rigidly rotates
around the central point. Two examples are taken into account. In the left
picture, the domain is the circle 0 ≤ r ≤ RM , where RM is the first zero of J1,
Math. Model. Anal., 19(1):44–51, 2014.
46 D. Funaro
Figure 1. Displacement of the electric field at fixed time obtained from the expression in
(2.2) subject to different kinds of boundary constraints. The sizes of the domains are not
arbitrary, but strictly related to the frequency of rotation as well as boundary conditions.
The magnetic field is orthogonal to the page.
which takes approximately the value 3.832. In this way, at the boundary, both
the magnetic field and the radial component of the electric field are zero. In
the right picture, we have the annular region Rm = 1 ≤ r ≤ RM . The outer
diameter is now chosen in such a way that the electric field is orthogonal to the
boundary (perfect conductivity).
Once the frequency of rotation (depending on cω) is fixed and the type
of boundary conditions decided, the size of the domain is automatically con-
strained. This means that there are few domains allowing for the creation of
these waves. If the proper size is not respected, the wave will not follow a sta-
tionary motion, i.e., it does not complete the cycle preserving the phase. The
conditions that permit such a guided behavior depend on the eigenvalues of the
Laplace operator on the domain. If the shape is such that some eigenvalues have
multiplicity higher than one, then the construction of rotating solitary waves
is possible (we show later the determination process). An analysis of spherical
vortices has been provided in [1], exactly with the purpose of detecting domains
where two independent eigenfunctions are related to the same eigenvalue. In
this paper, we stay for simplicity in the case of the cylinder by analyzing the
peculiar case where four independent eigenfunctions share the same eigenvalue.
The evolution in a thin toroid can be approximated, with a rather good level
of accuracy, starting from the cylinder’s version; this extension however is not
going to be studied here.
3 A more involved evolution
We would like to analyze the situation in which the multiplicity of one of the
eigenvalues of the Laplace operator, on a certain annular domain, is equal to
four. This will allow us to built more complex electromagnetic waves turning
EM Solitons in Cylinders 47
–10
–8
–6
–4
–2
0
2
4
2 3 4 5 6 7 8r
Figure 2. Plots of f1 and f4 for λ ≈ 1.0699. In this case both the functions vanish at
r = Rm = 1 and r = RM ≈ 7.0927.
around an axis. We will need to play with both the functions Jk and Yk,
therefore we have to stay away from the point r = 0 (recall that Yk is singular
there). From now on we assume that Rm = 1 ≤ r ≤ RM , where the major
diameter RM has to be properly determined. Of course, if Rm is modified the
entire setting scales accordingly, due to the linearity of Maxwell’s equations.
We impose Dirichlet boundary conditions on both boundaries (the inner and
the outer circumferences), though other conditions may be considered without
difficulty. The use of Dirichlet conditions in the inner part leads us to work
with the function:
fn(r) = Jn(λr)Yn(λ)− Yn(λr)Jn(λ), (3.1)
where λ is a parameter. Of course, we have fn(1) = 0, ∀n and ∀λ.
Now, we would like to have:
{
fn(RM ) = 0,
fj(RM ) = 0,
j 6= n, (3.2)
which is a nonlinear problem leading to the detection of RM and λ. This
means that, once the diameter of the inner circumference has been fixed, the
frequency of the evolving wave and the magnitude of the entire domain are
going to be uniquely determined by (3.2). Not necessarily such a problem has
solution. However, with the help of numerical tests, we were able to establish
some facts.
Math. Model. Anal., 19(1):44–51, 2014.
48 D. Funaro
Figure 3. Plots of two independent eigenfunctions of the Laplace operator with
homogeneous Dirichlet boundary conditions on an annular domain. Despite the fact that
they look totally different, the domain has been dimensioned in such a way the
corresponding eigenvalues are the same.
Here below we report some of the conclusions of our analysis:
j = 1, n = 2: no values of RM and λ were found;
j = 1, n = 3: no values of RM and λ were found;
j = 1, n = 4: RM = 7.0927 for λ = 1.0698;
j = 1, n = 5: RM = 3.6761 for λ = 2.3871;
j = 1, n = 6: RM = 2.7603 for λ = 3.6054;
j = 2, n = 3: no values of RM and λ were found;
j = 2, n = 4: no values of RM and λ were found;
j = 2, n = 5: RM = 5.1500 for λ = 1.7032.
These figures were obtained as the result of a simple bisection algorithm applied
to the difference (fn − fj)(r). When we were not able to find the requested
root RM it was clear (at least from the numerical viewpoint) that the function
|fn − fj |(r), r > Rm, was increasing.
Let us better examine a specific situation (j = 1, n = 4). According
to Fig. 2, both functions f1 and f4 are zero for r = Rm = 1 and RM ≈
7.0927. Thus, by imposing the same boundary constraints, we are able to find
out two branches of vector solutions of the type given in (2.2) (k = j and
k = n respectively) related to the same parameter ω. The intensity of the
corresponding magnetic fields is given by the level lines of Fig. 3.
More explicitly, by setting ω =
√
λ, we can write:
w1 = f1(ωr) sin(cωt− φ+ φ0), w2 = f4(ωr) cos(cωt− 4φ+ φ0),
w3 = f1(ωr) cos(cωt− φ+ φ0), w4 = f4(ωr) sin(cωt− 4φ+ φ0). (3.3)
The functions wi, i = 1, 2, 3, 4, are schematically reported in Fig. 4, for t = 0
and φ0 = 0. In particular, w1 and w3 are shaped as in the first picture of
EM Solitons in Cylinders 49
Figure 4. Schematic representation of four independent eigenfunctions in (3.3). They all
have the same eigenvalue.
Fig. 3 and display a difference of phase of 90 degrees. Using w1 and w3 we can
obtain the electromagnetic fields according to the expression (2.2) for k = 1.
The other two functions, w2 and w4, are as shown in the second picture of
Fig. 3 and differ for an angle of 22.5 degrees. They are also related to (2.2)
(this time with k = 4). Finally, we can build general solutions by the linear
combination:
γa(w1 + w3) + γb(w2 + w4) (3.4)
with γa and γb arbitrary constants. It is just a direct computation verifying
that the expression provided in (3.4) solves the wave equation in vacuum. In
fact, the second derivative in time produces the multiplicative factor −c2λ,
while the application of c2 times the Laplace operator provides the same factor
(use (2.1)).
In Fig. 5, one can see, at different time steps, the evolution of (3.4) for
γa = γb = 1 and φ0 = 0. Only half cycle is displayed. From the last picture
the sequence restarts from the beginning with the patterns interchanged. An
animation is available in [2]. The effect vaguely recalls the juggling of three
clubs.
4 Conclusions
From the theoretical viewpoint, Maxwell’s equations allow for special solutions
trapped inside an infinite cylinder or, more or less equivalently, in a toroid.
The study here developed in vacuum reveals original configurations and may
help for instance in the analysis of confined plasma (see, e.g., [5]). Other appli-
cations may be found in the field of the so called whispering gallery resonators.
Waves trapped in these cavities are smoothly guided to circulate around by
continuous reflection returning at the origin with the initial phase. Spherical,
cylindrical and ring-shaped whispering galleries are commonly produced for a
broad range of industrial applications. Typical areas of interest are in fiber
telecommunications or biosensing. The literature is very rich. We just mention
a couple of non extremely specialized publications: [6, 7].
Math. Model. Anal., 19(1):44–51, 2014.
50 D. Funaro
Figure 5. Evolution at different times of a linear combination of eigenfunctions
corresponding to the same eigenvalue. The two eigenfunctions of Fig. 3 merge and rotate
with an oscillating behavior. The frequency is uniquely established by the global size of the
domain.
References
[1] C. Chinosi, L. Della Croce and D. Funaro. Rotating electromagnetic waves in
toroid-shaped regions. Internat. J. Modern Phys. C, 21(1):11–32, 2010.
http://dx.doi.org/10.1142/S0129183110014926.
EM Solitons in Cylinders 51
[2] D. Funaro. http://cdm.unimo.it/home/matematica/funaro.daniele/results.htm
[3] D. Funaro. Electromagnetism and the Structure of Matter. World Scientific, Sin-
gapore, 2008.
[4] D. Funaro. From photons to atoms – the electromagnetic nature of matter, 2012.
arXiv:1206.3110
[5] R.D. Hazeltine and J.D. Meiss. Plasma Confinement. Dover Pub., Mineola, NY,
2003.
[6] A.N. Oraevsky. Whispering-gallery waves. Quantum Electronics, 32:377–400,
2002. http://dx.doi.org/10.1070/QE2002v032n05ABEH002205.
[7] A. Snyder and J. Love. Optical Waveguide Theory. Kluwer Academic Publisher,
Norwell, MA, 1983.
[8] G.N. Watson. A Treatise on the Theory of Bessel Functions. Cambridge Univ.
Press, 1944.
Math. Model. Anal., 19(1):44–51, 2014.


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