Electromagnetic Waves Part 4 docx

Gouy Phase and Matter Waves 25 Hariharan, P., and Robinson, P. A. (1996). The Gouy phase shift as a geometrical quantum effect, J. Mod. Opt. 43, 219. Feng, S., Winful, H. G., and Hellwarth, R. W. (1998). Gouy shift and temporal reshaping of focused single-cycle electromagnetic pulses, Opt. Lett. 23, 385. Feng, S., and Winful, H. G. (2001). Physical origin of the Gouy phase shift, Opt. Lett. 26, 485. Yang, J., and Winful, H. G. (2006). Generakized eikonal treatment of the Gouy phase shift, Opt. Lett. 31, 104. Holme, N. C. R., Daly, B. C., Myaing, M. T., and Norris, T. B. (2003). Gouy phase shift of single-cycle picosecond acoustic pulses, Appl. Phys. Lett. 83, 392. Zhu, W., Agrawal, A., and Nahata, A. (2007). Direct measurement of the Gouy phase shift for surface plasmon-polaritons, Opt. Express 15, 9995. Feurer, T., Stoyanov, N. S., Ward, D. W., and Nelson, K. A. (2002). Direct Visualization of the Gouy Phase by Focusing Phonon Polaritons, Phys. Rev. Lett. 88, 257. Horvath, Z. 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Introduction Present day methodologies for mathematical simulation and computational experiment are generally implemented in electromagnetics through the solution of boundary-value (frequency domain) problems and initial boundary-value (time domain) problems for Maxwell’s equations. Most of the results of this theory concerning open resonators have been obtained by the frequency-domain methods. At the same time, a rich variety of applied problems (analysis of complex electrodynamic structures for the devices of vacuum and solid-state electronics, model synthesis of open dispersive structures for resonant quasi- optics, antenna engineering, and high-power electronics, etc.) can be efficiently solved with the help of more universal time-domain algorithms. The fact that frequency domain approaches are somewhat limited in such problems is the motivation for this study. Moreover, presently known remedies to the various theoretical difficulties in the theory of non-stationary electromagnetic fields are not always satisfactory for practitioners. Such remedies affect the quality of some model problems and limit the capability of time-domain methods for studying transient and stationary processes. One such difficulty is the appropriate and efficient truncation of the computational domain in so-called open problems, i.e. problems where the computational domain is infinite along one or more spatial coordinates. Also, a number of questions occur when solving far-field problems, and problems involving extended sources or sources located in the far-zone. In the present work, we address these difficulties for the case of 0 TE n - and 0 TM n -waves in axially-symmetrical open compact resonators with waveguide feed lines. Sections 2 and 3 are devoted to problem definition. In Sections 4 and 5, we derive exact absorbing conditions for outgoing pulsed waves that enable the replacement of an open problem with an equivalent closed one. In Section 6, we obtain the analytical representation for operators that link the near- and far-field impulsive fields for compact axially-symmetrical structures and consider solutions that allow the use of extended or distant sources. In Section 7, we place some accessory results required for numerical implementation of the approach under Electromagnetic Waves 100 consideration. All analytical results are presented in a form that is suitable for using in the finite-difference method on a finite-sized grid and thus is amenable for software implementation. We develop here the approach initiated in the works by Maikov et al. (1986) and Sirenko et al. (2007) and based on the construction of the exact conditions allowing one to reduce an open problem to an equivalent closed one with a bounded domain of analysis. The derived closed problem can then be solved numerically using the standard finite-difference method (Taflove & Hagness, 2000). In contrast to other well-known approximate methods involving truncation of the computational domain (using, for example, Absorbing Boundary Conditions or Perfectly Matched Layers), our constructed solution is exact, and may be computationally implemented in a way that avoids the problem of unpredictable behavior of computational errors for large observation times. The impact of this approach is most significant in cases of resonant wave scattering, where it results in reliable numerical data. 2. Formulation of the initial boundary-value problem In Fig. 1, the cross-section of a model for an open axially-symmetrical (    0 ) resonant structure is shown, where    ,,z are cylindrical and    , , are spherical coordinates. By     = 0,2 we denote perfectly conducting surfaces obtained by rotating the curve   about the z -axis;        ,, = 0,2 is a similarly defined surface across which the relative Fig. 1. Geometry of the problem in the half-plane 2   . Simulation and Analysis of Transient Processes in Open Axially-symmetrical Structures: Method of Exact Absorbing Boundary Conditions 101 permittivity    g and specific conductivity       1 00 gg change step-wise; these quantities are piecewise constant inside  int and take free space values outside. Here,   , g z ;    12 000 is the impedance of free space;  0 , and  0 are the electric and magnetic constants of vacuum. The two-dimensional initial boundary-value problem describing the pulsed axially- symmetrical 0 TE n - (   0 z EEH ) and 0 TM n - (   0 z HHE ) wave distribution in open structures of this kind is given by                  1 22 22 0 0 1 1 1 0 00 00 0 0 ,, , ,,,, ,,,,, ,, , and , are continuous when crossing ,, , , ,, t t tg pz tg tg i g gg UgtFgttg t tz Ugt g Ugt g g z t Ept t Ept Hpt Uzt z t DUgt U gt                                  2 2 00,,,, g DUgt t                        (1) where      ,, z EEEE and      ,, z HHHH are the electric and magnetic field vectors;      ,,U g tE g t for 0 TE n -waves and      ,,U g tH g t for 0 TM n -waves (Sirenko et al., 2007). The SI system of units is used. The variable t which being the product of the real time by the velocity of light in free space has the dimension of length. The operators 1 D , 2 D will be described in Section 2 and provide an ideal model for fields emitted and absorbed by the waveguides. The domain of analysis  is the part of the half-plane  2 bounded by the contours   together with the artificial boundaries  j (input and output ports) in the virtual waveguides  j ,  1,2j . The regions       int ,:gr rL and  ext (free space), such that    int ext , are separated by the virtual boundary         ,:gr rL . The functions  ,F g t ,    g ,    g ,    g , and     1g which are finite in the closure  of  are supposed to satisfy the hypotheses of the theorem on the unique solvability of problem (1) in the Sobolev space    W 1 2 T ,   0; T T where  T is the observation time (Ladyzhenskaya, 1985). The ‘current’ and ‘instantaneous’ sources given by the functions   ,F g t and    g ,    g as well as all scattering elements given by the functions    g ,    g and by the contours   and     , are located in the region  int . In axially- symmetrical problems, at points g such that 0 , only z H or z E fields components are nonzero. Hence it follows that    0, , 0Uzt ; z ,  0t in (1). 3. Exact absorbing conditions for virtual boundaries in input-output waveguides Equations Electromagnetic Waves 102     2 1 1 12 000,,, ,,. i g g DUgt U gt DUgt t         (2) in (1) give the exact absorbing conditions for the outgoing pulsed waves       11 ,, , si UgtUgtUgt and     2 ,, s UgtUgt traveling into the virtual waveguides  1 and  2 , respectively (Sirenko et al., 2007).   1 , i Ugt is the pulsed wave that excites the axially-symmetrical structure from the circular or coaxial circular waveguide  1 . It is assumed that by the time  0t this wave has not yet reached the boundary  1 . By using conditions (2), we simplify substantially the model simulating an actual electrodynamic structure: the  j -domains are excluded from consideration while the operators j D describe wave transformation on the boundaries  j that separate regular feeding waveguides from the radiating unit. The operators j D are constructed such that a wave incident on  j from the region  int passes into the virtual domain  j as if into a regular waveguide – without deformations or reflections. In other words, it is absorbed completely by the boundary  j . Therefore, we call the boundary conditions (2) as well as the other conditions of this kind ‘exact absorbing conditions’. In the book (Sirenko et al., 2007), one can find six possible versions of the operators j D for virtual boundaries in the cross-sections of circular or coaxial-circular waveguides. We pick out two of them (one for the nonlocal conditions and one for the local conditions) and, taking into consideration the location of the boundaries  j in our problem (in the plane  1 zL for the boundary  1 and in the plane  2 zL for  2 ) as well as the traveling direction for the waves outgoing through these boundaries (towards  z for  1 and towards z for  2 ), write (2) in the form:       1 1 1 1 1 101 1 1 0 11 ,, (, ,) , ,0, a s t s nnn n b zL Uz ULt Jt dd z bat                                (3)     2 2 2 202 2 2 0 22 ,, (, ,) , ,0 a t nnn n b zL Uz ULt J t dd z bat                         (4) (nonlocal absorbing conditions) and         1 2 1 1 111 0 1 2 2 111 2 1 111 0 2 0 1 0 00 ,, ,, , , ,, sin , , , , ,, ,, , , s s zL t Wt ULt dtba t Uzt Wt b at z t Wt Wba t                                             (5) Simulation and Analysis of Transient Processes in Open Axially-symmetrical Structures: Method of Exact Absorbing Boundary Conditions 103       2 2 2 222 0 2 2 222 2 2 222 0 ,, 2 ,, , 0, ,, 1 sin , , , , 0 ,, ,0, 0, zL t Wt ULt d t b a t Uzt Wt b a t z t Wt Wba t                                             (6) (local absorbing conditions). The initial boundary-value problems involved in (5) and (6) with respect to the auxiliary functions    ,, j Wt must be supplemented with the following boundary conditions for all times  0t :         0 0 0, , , , 0 TE -waves ,, 0, , =0 TM -waves j jjj n j jn a Wt Wat Wt Wt                (7) (on the boundaries   0 and   j a of the region  j for a circular waveguide) and           0 0 ,, ,, 0 TE -waves ,, ,, =0 TM -waves jj jj jj n jj n ba Wbt Wat Wt Wt                     (8) (on the boundaries   j b and   j a of the region  j for a coaxial waveguide). In (3) to (8) the following designations are used:   0 Jx is the Bessel function, j a and j b are the radii of the waveguide  j and of its inner conductor respectively (evidently,  0 j b if only  j is a coaxial waveguide),      nj and    nj are the sets of transverse functions and transverse eigenvalues for the waveguide  j . Analytical representations for     nj and  n j are well-known and for 0 TE n -waves take the form:     1 10 1 212 00 ,,, , (circular waveguide) are the roots of the equation , nj nj j nj j j nj j JaJan a Ja                   (9)              12 22 22 100 1 11 212 00 ,,,,,, , (coaxial waveguide) are the roots of the equation , ,. nj nj j nj j j nj j jj nj j qq jq j GaGabGbn ba Ga GJNbNJb                            (10) [...]... McGraw-Hill, ISBN 048 644 614X, New York Borisov, V.V (1996) Electromagnetic fields of transient currents, St Petersburg Univ Press, ISBN 5-288-01256-3, St Petersburg Gradshteyn, I.S & Ryzhik, I.M (2000) Table of Integrals, Series, and Products, ISBN 0-122 947 57-6, Academic Press, San Diego, London Janke, E., Emde, F., & Lösch, F (1960) Tables of Higher Functions, McGraw-Hill, ISBN 007032 245 7, New York Korn,...  0 ,   (48 ) It follows from (47 ), (48 ) that in the domain ext the function U s  g , t   U  g , t   U i  g , t  satisfies the equations 112 Electromagnetic Waves   2 2   1   s    U  g , t   0, t  0, g  ext  2  2         t z    s  s U  g,t   0, g   , z  ext U  g , t  t  0  0, t t 0   s z  L, t  0 U  0, z , t   0,   (49 ) and determines... determines there the pulsed electromagnetic wave crossing the artificial boundary  in one direction only, namely, from int into ext The problems (49 ) and ( 14) are qualitatively the same Therefore, by repeating the transformations of Section 4, we obtain U s  L , , t    t    t   2   1 P  1   t    2    t 1      Pn1  1      n  2 n  1 t  2 L  L 4L2   t   2 2... g , t   g  0,  g  int (42 ) where the operator D is given by (41 ) It is equivalent to the open initial problem (1) This statement can be proved by following the technique developed in (Ladyzhenskaya, 1985) 110 Electromagnetic Waves The initial and the modified problems are equivalent if and only if any solution of the initial problem is a solution to problem (42 ) and at the same time, any solution... 116 Electromagnetic Waves Ladyzhenskaya, O.A (1985) The boundary value problems of mathematical physics, SpringerVerlag, ISBN 3 540 909893, New York Maikov, А.R., Sveshnikov, A.G., & Yakunin, S.A (1986) Difference scheme for the Maxwell transient equations in waveguide systems Journal of Computational Mathematics and Mathematical Physics, Vol 26, pp 851–863, ISSN 0965- 542 5 Mikhailov, V.P (1976) Partial... of (40 ).) The solution of the initial problem is unique and it is evidently the solution to the modified problem according construction In this case, if the solution of (42 ) is unique, it will be a solution to (1) Assume that problem (42 ) has two different solutions U 1  g , t  and U 2  g , t  Then the function u  g , t   U 1  g , t   U 2  g , t  is also the generalized solution to (42 )... F  z , t  , t  0 , t z            z , v  z , t   v  z , t  t  0    z  , t t 0  z  (56) z   1 14 Electromagnetic Waves Here, F  z , t  ,   z  и    z , t  are the amplitude coefficients in the integral presentations    ( 54) for the functions F  g , t  ,   g  , and   g  Now, by extending the functions F  z , t  and v  z , t  with zero on... suitable choice of function,  for 0  t     u  g ,   d   g,t    t  0 for   t  T , (44 ) it is possible to show that every term in (43 ) is nonnegative (Mikhailov, 1976) and therefore u  g , t  is equal to zero for all g  int and 0  t  T , which means that the solution to the problem (42 ) is unique This proves the equivalency of the two problems 6 Far-field zone problem Extended and...1 04 Electromagnetic Waves For TM0n -waves we have:     1     nj     J1 nj  2  a j J1 nj a j  , n  1, 2 , ,    a j (circular waveguide)   are the roots of the equation J 0  a j  0 , nj  0  (11)... fractional derivatives of Green’s function, or by the fractional derivatives of the function at the boundary and the usual Green’s function 1 24 Electromagnetic Waves  If     , i.e   0 , we obtain a representation for the function  ( r ) itself:   (r )  1 4   [  Dx  S 0          G(r , r0 ) 0  Dx0 (r0 )   Dx0 ( r0 ) 0  Dx0 G(r , r0 )]ds0 (13)  This expression means . Quantum jumps of light recording the birth and death of a photon in a cavity, Nature 44 6, 297. 96 Electromagnetic Waves Part 2 Methods of Computational Analysis 5 Simulation and Analysis of Transient.                         (48 ) It follows from (47 ), (48 ) that in the domain ext  the function       ,, , si Ugt Ugt Ugt satisfies the equations Electromagnetic Waves 112      22 ext 22 ext 0 0 1 ,0,. j GaGabGbn ba Ga GJNbNJb                            (10) Electromagnetic Waves 1 04 For 0 TM n -waves we have:     1 11 0 212 00 ,,, , (circular waveguide) are