Complex source point concept in the modelling of dynamic control for optical beam deflection

Complex source point concept in the modelling of dynamic control for optical beam deflection

A rigorous analytical method for transient dynamics description in a cylindrical lens is developed and used to describe a possibility of beam controlled deflection via material tuning. The complex source point concept is used to simulate beam passing through the lens. Details of both the transien...

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Veröffentlicht: Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 2012
Schriftenreihe:Semiconductor Physics Quantum Electronics & Optoelectronics
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spelling irk-123456789-1183102017-05-30T03:05:28Z Complex source point concept in the modelling of dynamic control for optical beam deflection Sakhnenko, N.K. A rigorous analytical method for transient dynamics description in a cylindrical lens is developed and used to describe a possibility of beam controlled deflection via material tuning. The complex source point concept is used to simulate beam passing through the lens. Details of both the transient response and steady-state regime are described. The excited fields are described using a rigorous mathematical approach based on analytical solution in the Laplace transform domain and accurate evaluation of residues at singular points of the obtained functions. 2012 Article Complex source point concept in the modelling of dynamic control for optical beam deflection / N.K. Sakhnenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 3. — С. 209-213. — Бібліогр.: 14 назв. — англ. 1560-8034 PACS 02.30.Uu, 42.25.Fx, Bs http://dspace.nbuv.gov.ua/handle/123456789/118310 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description A rigorous analytical method for transient dynamics description in a cylindrical lens is developed and used to describe a possibility of beam controlled deflection via material tuning. The complex source point concept is used to simulate beam passing through the lens. Details of both the transient response and steady-state regime are described. The excited fields are described using a rigorous mathematical approach based on analytical solution in the Laplace transform domain and accurate evaluation of residues at singular points of the obtained functions.
format Article
author Sakhnenko, N.K.
spellingShingle Sakhnenko, N.K.
Complex source point concept in the modelling of dynamic control for optical beam deflection
Semiconductor Physics Quantum Electronics & Optoelectronics
author_facet Sakhnenko, N.K.
author_sort Sakhnenko, N.K.
title Complex source point concept in the modelling of dynamic control for optical beam deflection
title_short Complex source point concept in the modelling of dynamic control for optical beam deflection
title_full Complex source point concept in the modelling of dynamic control for optical beam deflection
title_fullStr Complex source point concept in the modelling of dynamic control for optical beam deflection
title_full_unstemmed Complex source point concept in the modelling of dynamic control for optical beam deflection
title_sort complex source point concept in the modelling of dynamic control for optical beam deflection
publisher Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
publishDate 2012
url http://dspace.nbuv.gov.ua/handle/123456789/118310
citation_txt Complex source point concept in the modelling of dynamic control for optical beam deflection / N.K. Sakhnenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2012. — Т. 15, № 3. — С. 209-213. — Бібліогр.: 14 назв. — англ.
series Semiconductor Physics Quantum Electronics & Optoelectronics
work_keys_str_mv AT sakhnenkonk complexsourcepointconceptinthemodellingofdynamiccontrolforopticalbeamdeflection
first_indexed 2025-07-08T13:42:09Z
last_indexed 2025-07-08T13:42:09Z
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fulltext Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 209-213. © 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 209 PACS 02.30.Uu, 42.25.Fx, Bs Complex source point concept in the modelling of dynamic control for optical beam deflection N.K. Sakhnenko Kharkiv National University of Radio Electronics, 14, Lenin Ave., 61166 Kharkiv, Ukraine Phone: 38 (057) 702-13-72; e-mail: n_sakhnenko@yahoo.com Abstract. A rigorous analytical method for transient dynamics description in a cylindrical lens is developed and used to describe a possibility of beam controlled deflection via material tuning. The complex source point concept is used to simulate beam passing through the lens. Details of both the transient response and steady-state regime are described. The excited fields are described using a rigorous mathematical approach based on analytical solution in the Laplace transform domain and accurate evaluation of residues at singular points of the obtained functions. Keywords: time-varying media, optical switching, optical beams, lens. Manuscript received 25.04.12; revised version received 29.05.12; accepted for publication 14.06.12; published online 25.09.12. 1. Introduction Electromagnetic wave propagation in time-varying media yields rise of new physical phenomena and possibilities for novel applications. Tuning the refractive index in time provides a fast frequency shift in the linear material dielectric resonator [1]. Half-restricted time- varying plasma causes focusing of a point source radiation at the plane boundary [2], which resembles action of a lens in the form of a plane layer with double- negative materials. Transient medium is used in the light-modulated photo-induced method for the development of a non-mechanical millimeter wave scanning technique [3]. Plasma based lenses with properties electronically adjusted can offer an alternative to the existing electronic beam steering systems by varying the density of plasma in time [4]. In practice, temporal switching of the material refractive index can be realized by varying the input signal in a nonlinear structure [5], by voltage control [6], by a focused laser beam as a local heat source [7], or by plasma injection of free carriers [8]. The main goal of this paper is to demonstrate a possibility of beam deflection control in a homogeneous lens of simple shape by adjusting its material parameters in time. The investigation based on a rigorous mathematical method that uses the Laplace transformation is aimed at deriving an analytical solution of the problem in a frequency domain. Time domain fields are recovered due to computation of the inverse Laplace transform via evaluation of residues at singular points. This approach provides accurate back transformation of the functions and allows to understand and look inside the fundamental processes. This method has already been successfully applied to solve the various time domain problems with different geometries [1, 9-11]. The accurate solution will reveal peculiarities of nonstationary electromagnetic processes in canonical objects, which will give a possibility to formulate recommendations for applications in new technologies to control electromagnetic radiation. 2. Formulation of the problem and its solution Consider a 2D initial-boundary value problem of exciting a circular cylinder by an incident beam that is modeled by complex source point (CSP). To describe the fields, the cylindrical system of coordinates z,,  centered at the cylinder is introduced. The incident beam is generated by an external source    csH 2 0 with time dependence tie 0 , where v is the phase velocity of background medium. Its position is described by a complex vector cs  with the Cartesian coordinates [12] Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 209-213. © 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 210      ,sin cos 0 0 ibyy ibxx cs cs (1) where ,,, 00 byx are real numbers. In this case, the distance between the source and the point of observation is complex as well 22 )()( cscscs yyxx   . The real point ),( 00 yx corresponds to the center of the beam waist (Fig. 1). The beam width is controlled by the parameter cb [13-15], and the beam direction is defined by the angle  . For the situation depicted in Fig. 1, the value of  is π. Using the addition theorem for Hankel functions, the incident field of CSP can be presented in the following form:                        cscsk cscsk ik cs HJ HJe H cs 2 0 2 0 2 0 (2) where )sincos(2 00 22 0  yxibbcs , 2 0 2 0 2 0 yx  , ))cos((arccos 1 0  cscs ibx , (...) is the Heaviside unit function. The scattered and transmitted fields are expanded, respectively, as follows:   )( 10 ,, csik k k HE k HE evJAU      , if a , (3)   )( 0 )2( 0 ,, csik k HE k HE evHBV      , if a . (4) Here, HEU , or HEV , represents z-coordinate of the electric or magnetic field for E - or H -polarized fields. Unknown expansion coefficients HE kA , and HE kB , are found from the boundary conditions, which requires continuity of tangential components of the total electric and magnetic fields. Assuming the external position of CSP ( a0 ), the unknown coefficients can be obtained in the following form: Fig. 1. Schematic diagram of the problem.         , )( 2 0 )2( 0100 )2( 010 , 0 )2( 0 , vaHvaJvaHvaJ vH a vi A kk HE csk HE k       (5)                 ,)( 0 )2( 0 )2( 0100 )2( 010 , 010 , 010 , vH vaHvaJvaHvaJ vaJvaJvaJvaJ B csk kk HE kk HE kk HE k      (6)        pol., pol,, 1 1, H- E-HE (7) The prime here represents full differentiation with respect to the argument. Fig. 2 represents a near-field portrait of the CSP beam passing through a circular lens. The values of the parameters are: 1 cb , 20 ax , 5.00 ay ,  200 ca , the refractive index of the material is 5.11 n . Suppose that at zero moment of time, the dielectric permittivity value inside the cylinder changes from 1 to 2 in response to some external source. The transformation of the initial CSP field caused by time change of the medium is going to be studied with a particular emphasis on the transient processes and steady-state regime occurring in such a simple dynamic lens. It means that we have an initial-boundary value problem, where transformed fields have to satisfy the wave equations 0),(1),( 22 2  tWvtW tt  , a , (8) 0),(1),( 22  tWvtW tt  , a . (9) Fig. 2. CSP beam passage through a circular dielectric lens (near-field distribution). y x a h 0 0( , )x y Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 209-213. © 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 211 Here, W represents the electric zE or magnetic zH field components after zero moment of time, and 2 2 22 2 11            . The corresponding initial conditions are provided by continuity of the electric flux density D  and the magnetic flux density B  at zero moment of time. In the transient area inside the cylinder, they have the following form [1] )0()0( 21   tEtE , )0()0( 21   tEtE tt ; (10) )0()0(   tHtH , )0()0( 21   tHtH tt . (11) In the steady-state outer region, initial conditions are: )0()0(   tEtE , )0()0(   tEtE tt ; (12) .)0()0( ,)0()0(     tHtH tHtH tt (13) The Laplace transform    0 )()( dtetWpL pt is employed directly to the wave Eqs (8), (9). It follows from the previous works [9-11] that the solution has to be written in the form: , )( )( 2 , , 2 2 2 0 2 1 2 0 2 2, csik k k HE k HEHE e c p nIC U vp ipv L                (14)   . 1 )(, , 0 , csik k k HE k HEHL ecpKD V ip L         (15) The unknown coefficients can be found using the boundary conditions: HE k HE k k HE k R P vpvip vvp ApC , , 2 2 2 0 22 10 2 2 2 1, ))(( )( )(    , (16) HE k HE k k HE k R Q vpvip vvp ApD , , 2 2 2 0 22 10 2 2 2 1, ))(( )( )(    , (17) ,)()( )()( 010 01001 vaKvapJ vaKvaJP kk kk E k   (18) ,)()( )()( 0101 0100 vaKvaJip vaKvaJiP kk kk H k   (19) ,)()( )()( 2102 21001 vpaIvapJ vpaIvaJQ kk kk E k   (20) ,)()( )()( 10202 2101 vaJvpaIi vpaIvaJpiQ kk kk H k   (21) ,)()( )()( 2 2 ,, vpaKvpaI vpaKvpaIR kk kk HEHE k   (22)        .pol, ,pol, 2 2, H- E-HE (23) Behaviour of the back transformed function is defined by its singular points, hence the inverse Laplace transform can be evaluated by calculation of the residues at its poles and the integral along the branch cut. The functions (14) and (15) involve an infinite number of simple poles associated with the eigenfrequencies of the cylinder and given by zeros of the functions HE kR , (22). Besides, apart from these singularities there is another one associated with the carrier frequency 0 of the incident beam. The functions (14) and (15) also have the branch point at p = 0, therefore a branch-cut should be introduced, for instance, along the negative 0)Re( p axis. All residues can be calculated analytically, while the integral along the branch cut should be calculated numerically. After permittivity change, the eigenmodes of the cylinder are excited, which causes transients. It should be mentioned that all eigenfrequencies are complex valued, so this ’ringing’ is observable during limited time interval. Here, the electromagnetic field is defined by the residue at the the singular point 0 ip and coincides with the harmonic source field in the cylinder with dielectric permittivity 2 . So, transformation of the near-field distribution during the transient period is observable. However, in the steady- state regime the field behaviour is determined only by the residue at 0 ip . 3. Numerical results Being based on the above approach, controlling the deflection angle of the beam passing through the isolated dielectric cylinder by adjusting the material parameters in time should be demonstrated. In what follows, with no loss of generality, the E-polarized fields are estimated numerically. Fig. 3 shows the radiation pattern of the CSP beam passing through the cylinder. It is evident that the angle of deflection crucially depends on the position of the incident beam (the value of h in Fig. 1). It is seen Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 209-213. © 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 212 1 2 30 210 60 240 90 270 120 300 150 330 180 0 h=0 1 2 30 210 60 240 90 270 120 300 150 330 180 0 h=0.2 1 2 30 210 60 240 90 270 120 300 150 330 180 0 h=0.4 0.5 1 30 210 60 240 90 270 120 300 150 330 180 0 h=0.6 0.5 1 30 210 60 240 90 270 120 300 150 330 180 0 h=0.8 0.5 1 30 210 60 240 90 270 120 300 150 330 180 0 h=1 Fig. 3. CSP beam passage through a circular dielectric lens (far-field distribution). 170 180 190 200 210 220 230 240 0 0.2 0.4 0.6 0.8 1 polar angle (degrees) fa r fie ld p a tte rn n=1.45 n=1.5 n=1.55 0 5 10 15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 normalized time (T) a bs (E /E 0) Fig. 4. Dependence of the deflection angle on the value Fig. 5. Time representation of the electric field inside the cylinder. of the refractive index inside the cylinder. The refractive index changes from the value 1.4 up to 1.45. that the widest deflection angle is observable for the value of ah close to 0.5. An assumption that the media parameters are tuned in time makes it possible to control the angle of deflection. Thus, Fig. 4 represents the far- field distribution for different values of the refractive index of the lens. It is seen that the rise of the refractive index increases the angle of deflection and vice versa. To estimate the duration of the transient period, the time domain representation of the field was obtained. Fig. 5 shows the dependence of the absolute value of the total electric field normalized by amplitude of the incident beam on the normalized time (T = tc/a). The coordinates of the observation point are: 5.0ax , 0ay . All the beam parameters are the same as in Fig. 2. The initial value of the refractive index is 1.4 that at zero moment of time changes to the value 1.45. It should be noted that although this large jump of permittivity has not been yet attained in practice, it is used here to trace the dynamics of the phenomena in a clearer manner. Before zero moment of time, the incident field is presented. After zero moment of time, the field inside the cylinder is represented by the first term in Eq. (14). The initial wave is split into two waves: the time-transmitted and time- reflected ones with the shifted frequency 120 vv . Within the time interval )1(0 2 anT  , these two waves are observable. The second term in (14) describes the influence of the transient boundary and demonstrates some time delay (for more information see e.g. [ 119 ]). Fig. 5 illustrates the moment of time )1(2 anT  that from the physical viewpoint corresponds to the moment when the wavefront from the transient boundary reaches the point of observation. The transformed field Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 209-213. © 2012, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 213 involves the whole spectrum, which is given by the poles of the function (22). However, it is seen that, after short period of transients, the steady-state regime is achieved, which corresponds to the new position of the deflected angle. 4. Conclusions The temporal analysis of the electromagnetic field transformed by the time change of the permittivity in the dielectric cylinder illuminated by a harmonic CSP beam is carried out. This analysis is based on the exact formulas for the interior and exterior fields that are obtained as the solutions of the initial-boundary value problem. The theory is based on the eigenfunction expansion in the Laplace transform domain and the solution inversion into the time domain through evaluation of residues. The obtained results indicate a possibility of using this simple lens model for very fast beam control. Acknowledgement This work was supported in part by the National Academy of Science of Ukraine in the framework of the State Target Program “Nanotechnologies and Nanomaterials”, and in part by the European Science Foundation via research network project “NEWFOCUS”. References 1. N.K. Sakhnenko, T.M. Benson, P. Sewell, A.G. Nerukh, Transient transformation of whispering gallery resonator modes due to time variations in dielectric permittivity // Opt. and Quant. Electronics, 38, p. 71-81 (2006). 2. A. Nerukh, N.K. Sakhnenko, Formation of point source image by time change of the medium // IEEE J. Selected Topics in Quantum Electronics, 15 (5), p. 1368-1373 (2009). 3. M.R. Chahamir, J. Shaker, M. Cuhaci, A.- R. Sebak, Novel photonically-controlled reflectarray antenna // IEEE Trans. Antennas and Propagation, 54, p. 1134-1141 (2006). 4. P. Linardakis, G. Borg, N. Martin, Plasma-based lens for microwave beam steering // Electronics Lett. 42, p. 444-446 (2006). 5. F. Blom, D. Dijk, H. Hoekstra, M. Driessen, A. Popma, Experimental study of integrated-optics microcavity resonators // Appl. Phys. Lett. 71(6), p. 747-749 (1997). 6. A.A. Savchenkov, V.S. Ilchenko, A.B. Matsko, L. Maleki, High-order tunable filters on a chain of coupled crystalline whispering-gallery-mode resonators // IEEE Photonics Techn. Lett. 17 (1), p. 136-138 (2005). 7. M. Benyoucef, S. Kiravittaya, Y. Mei, A. Rastelli, O. Schmidt, Strongly coupled semiconductor microcavities: A route to couple artificial atoms over micrometric distances // Phys. Rev. B, 77, 035108 (2008). 8. K. Djordjev, S.-J. Choi, R. Dapkus, Microdisk tunable resonant filters and switches // IEEE Photonics Technol. Lett. 14(6), p. 823-830 (2002). 9. J. Brownell, A. Nerukh, N.K. Sakhnenko, S. Zhilkov, A. Alexandrova, Terahertz sensing of non-equilibrium microplasmas // J. Phys. D, 38, p. 1658-1664 (2005). 10. E.V. Bekker, A. Vukovic, P. Sewell, T.M. Benson, N.K. Sakhnenko, A.G. Nerukh, An assessment of coherent coupling through radiation fields in time varying slab waveguides // Opt. and Quant. Electron., 39(7), p. 533-551 (2007). 11. N.K. Sakhnenko, A.G. Nerukh, T. Benson, P. Sewell, Frequency conversion and field pattern rotation in WGM resonator with transient inclusion // Opt. and Quant. Electronics, 39, p. 761-771 (2007). 12. A.V. Boriskin and A.I. Nosich, Whispering-gallery and Luneburg-lens effects in a beam-fed circularly layered dielectric cylinder // IEEE Trans. on Antennas and Propagation, 50(9), p. 1245-1249 (2002). 13. L. Felsen, Complex-point source solutions of the field equations and their relation to the propagation and scattering of the Gaussian beams // Symp. Mathem. 18, p. 39-56 (1975). 14. E. Heyman and L. Felsen, Complex-source pulsed beam fields // J. Opt. Soc. Amer. A, 6(6), p. 806- 817 (1989). Semiconductor Physics, Quantum Electronics & Optoelectronics, 2012. V. 15, N 3. P. 209-213. PACS 02.30.Uu, 42.25.Fx, Bs Complex source point concept in the modelling of dynamic control for optical beam deflection N.K. Sakhnenko Kharkiv National University of Radio Electronics, 14, Lenin Ave., 61166 Kharkiv, Ukraine Phone: 38 (057) 702-13-72; e-mail: n_sakhnenko@yahoo.com Abstract. A rigorous analytical method for transient dynamics description in a cylindrical lens is developed and used to describe a possibility of beam controlled deflection via material tuning. The complex source point concept is used to simulate beam passing through the lens. Details of both the transient response and steady-state regime are described. The excited fields are described using a rigorous mathematical approach based on analytical solution in the Laplace transform domain and accurate evaluation of residues at singular points of the obtained functions. Keywords: time-varying media, optical switching, optical beams, lens. Manuscript received 25.04.12; revised version received 29.05.12; accepted for publication 14.06.12; published online 25.09.12. 1. Introduction Electromagnetic wave propagation in time-varying media yields rise of new physical phenomena and possibilities for novel applications. Tuning the refractive index in time provides a fast frequency shift in the linear material dielectric resonator [1]. Half-restricted time-varying plasma causes focusing of a point source radiation at the plane boundary [2], which resembles action of a lens in the form of a plane layer with double-negative materials. Transient medium is used in the light-modulated photo-induced method for the development of a non-mechanical millimeter wave scanning technique [3]. Plasma based lenses with properties electronically adjusted can offer an alternative to the existing electronic beam steering systems by varying the density of plasma in time [4]. In practice, temporal switching of the material refractive index can be realized by varying the input signal in a nonlinear structure [5], by voltage control [6], by a focused laser beam as a local heat source [7], or by plasma injection of free carriers [8]. The main goal of this paper is to demonstrate a possibility of beam deflection control in a homogeneous lens of simple shape by adjusting its material parameters in time. The investigation based on a rigorous mathematical method that uses the Laplace transformation is aimed at deriving an analytical solution of the problem in a frequency domain. Time domain fields are recovered due to computation of the inverse Laplace transform via evaluation of residues at singular points. This approach provides accurate back transformation of the functions and allows to understand and look inside the fundamental processes. This method has already been successfully applied to solve the various time domain problems with different geometries [1, 9-11]. The accurate solution will reveal peculiarities of nonstationary electromagnetic processes in canonical objects, which will give a possibility to formulate recommendations for applications in new technologies to control electromagnetic radiation. 2. Formulation of the problem and its solution Consider a 2D initial-boundary value problem of exciting a circular cylinder by an incident beam that is modeled by complex source point (CSP). To describe the fields, the cylindrical system of coordinates 00 (,) xy centered at the cylinder is introduced. The incident beam is generated by an external source ( ) ( ) n r - r w cs H r r 2 0 with time dependence t i e 0 w , where v is the phase velocity of background medium. Its position is described by a complex vector cs r r with the Cartesian coordinates [12] î í ì b + = b + = , sin cos 0 0 ib y y ib x x cs cs (1) where b , , , 0 0 b y x are real numbers. In this case, the distance between the source and the point of observation is complex as well 2 2 ) ( ) ( cs cs cs y y x x - + - = r - r r r . The real point ) , ( 0 0 y x corresponds to the center of the beam waist (Fig. 1). The beam width is controlled by the parameter c b w [13-15], and the beam direction is defined by the angle  b . For the situation depicted in Fig. 1, the value of b is π. Using the addition theorem for Hankel functions, the incident field of CSP can be presented in the following form: ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ( ) ( ) ( ) ( ) ] r - r Q n wr n wr + + r - r Q n wr n wr = = n r - r w å j - j cs cs k cs cs k ik cs H J H J e H cs 2 0 2 0 2 0 (2) where ) sin cos ( 2 0 0 2 2 0 b + b + - r = r y x ib b cs , 2 0 2 0 2 0 y x + = r , ) ) cos (( arccos 1 0 - r b + = j cs cs ib x , (...) Q is the Heaviside unit function. The scattered and transmitted fields are expanded, respectively, as follows: ( ) ) ( 1 0 , , cs ik k k H E k H E e v J A U j - j ¥ -¥ = å r w = , if a < r , (3) ( ) ) ( 0 ) 2 ( 0 , , cs ik k H E k H E e v H B V j - j ¥ -¥ = å r w = , if a > r . (4) Here, H E U , or H E V , represents z-coordinate of the electric or magnetic field for E - or H -polarized fields. Unknown expansion coefficients H E k A , and H E k B , are found from the boundary conditions, which requires continuity of tangential components of the total electric and magnetic fields. Assuming the external position of CSP ( a > r 0 ), the unknown coefficients can be obtained in the following form: Fig. 1. Schematic diagram of the problem. ( ) ( ) ( ) ( ) , ) ( 2 0 ) 2 ( 0 1 0 0 ) 2 ( 0 1 0 , 0 ) 2 ( 0 , v a H v a J v a H v a J v H a vi A k k H E cs k H E k w ¢ w - w w ¢ a r w ´ ´ w p = (5) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , ) ( 0 ) 2 ( 0 ) 2 ( 0 1 0 0 ) 2 ( 0 1 0 , 0 1 0 , 0 1 0 , v H v a H v a J v a H v a J v a J v a J v a J v a J B cs k k k H E k k H E k k H E k r w ´ ´ w ¢ w - w w ¢ a w w ¢ a - w ¢ w = (6) ï î ï í ì e e e e = a pol. , pol, , 1 1 , H- E- H E (7) The prime here represents full differentiation with respect to the argument. Fig. 2 represents a near-field portrait of the CSP beam passing through a circular lens. The values of the parameters are: 1 = w c b , 2 0 = a x , 5 . 0 0 = a y , p = w 20 0 c a , the refractive index of the material is 5 . 1 1 = e = n . Suppose that at zero moment of time, the dielectric permittivity value inside the cylinder changes from 1 e to 2 e in response to some external source. The transformation of the initial CSP field caused by time change of the medium is going to be studied with a particular emphasis on the transient processes and steady-state regime occurring in such a simple dynamic lens. It means that we have an initial-boundary value problem, where transformed fields have to satisfy the wave equations 0 ) , ( 1 ) , ( 2 2 2 = r ¶ × - r D t W v t W tt r r , a < r , (8) 0 ) , ( 1 ) , ( 2 2 = r ¶ × - r D t W v t W tt r r , a > r . (9) Fig. 2. CSP beam passage through a circular dielectric lens (near-field distribution). Here, W represents the electric z E or magnetic z H field components after zero moment of time, and 2 2 2 2 2 1 1 j ¶ ¶ r + r ¶ ¶ r + r ¶ ¶ = D . The corresponding initial conditions are provided by continuity of the electric flux density D r and the magnetic flux density B r at zero moment of time. In the transient area inside the cylinder, they have the following form [1] ) 0 ( ) 0 ( 2 1 - + = e e = = t E t E , ) 0 ( ) 0 ( 2 1 - + = ¶ e e = = ¶ t E t E t t ; (10) ) 0 ( ) 0 ( - + = = = t H t H , ) 0 ( ) 0 ( 2 1 - + = ¶ e e = = ¶ t H t H t t . (11) In the steady-state outer region, initial conditions are: ) 0 ( ) 0 ( - + = = = t E t E , ) 0 ( ) 0 ( - + = ¶ = = ¶ t E t E t t ; (12) . ) 0 ( ) 0 ( , ) 0 ( ) 0 ( - + - + = ¶ = = ¶ = = = t H t H t H t H t t (13) The Laplace transform ò ¥ - = 0 ) ( ) ( dt e t W p L pt is employed directly to the wave Eqs (8), (9). It follows from the previous works [9-11] that the solution has to be written in the form: , ) ( ) ( 2 , , 2 2 2 0 2 1 2 0 2 2 , cs ik k k H E k H E H E e c p n I C U v p i p v L j - j ¥ -¥ = å ÷ ø ö ç è æ r + + n w + w + = (14) ( ) . 1 ) ( , , 0 , cs ik k k H E k H E H L e c p K D V i p L j - j ¥ -¥ = å r + + w - = (15) The unknown coefficients can be found using the boundary conditions: H E k H E k k H E k R P v p v i p v v p A p C , , 2 2 2 0 2 2 1 0 2 2 2 1 , ) )( ( ) ( ) ( w + w - - = , (16) H E k H E k k H E k R Q v p v i p v v p A p D , , 2 2 2 0 2 2 1 0 2 2 2 1 , ) )( ( ) ( ) ( w + w - - = , (17) , ) ( ) ( ) ( ) ( 0 1 0 0 1 0 0 1 v a K v a pJ v a K v a J P k k k k E k w ¢ w - - w w ¢ w e e = (18) , ) ( ) ( ) ( ) ( 0 1 0 1 0 1 0 0 v a K v a J ip v a K v a J i P k k k k H k w ¢ w e e - - w w ¢ w - = (19) , ) ( ) ( ) ( ) ( 2 1 0 2 2 1 0 0 1 v pa I v a pJ v pa I v a J Q k k k k E k ¢ w e e - - w ¢ w e e = (20) , ) ( ) ( ) ( ) ( 1 0 2 0 2 2 1 0 1 v a J v pa I i v pa I v a J p i Q k k k k H k w ¢ w e e - - w ¢ e e - = (21) , ) ( ) ( ) ( ) ( 2 2 , , v pa K v pa I v pa K v pa I R k k k k H E H E k ¢ - - ¢ b = (22) ï î ï í ì e e e e = b . pol , , pol , 2 2 , H- E- H E (23) Behaviour of the back transformed function is defined by its singular points, hence the inverse Laplace transform can be evaluated by calculation of the residues at its poles and the integral along the branch cut. The functions (14) and (15) involve an infinite number of simple poles associated with the eigenfrequencies of the cylinder and given by zeros of the functions H E k R , (22). Besides, apart from these singularities there is another one associated with the carrier frequency 0 w of the incident beam. The functions (14) and (15) also have the branch point at p = 0, therefore a branch-cut should be introduced, for instance, along the negative 0 ) Re( = p axis. All residues can be calculated analytically, while the integral along the branch cut should be calculated numerically. After permittivity change, the eigenmodes of the cylinder are excited, which causes transients. It should be mentioned that all eigenfrequencies are complex valued, so this ’ringing’ is observable during limited time interval. Here, the electromagnetic field is defined by the residue at the the singular point 0 w = i p and coincides with the harmonic source field in the cylinder with dielectric permittivity 2 e . So, transformation of the near-field distribution during the transient period is observable. However, in the steady-state regime the field behaviour is determined only by the residue at 0 w = i p . 3. Numerical results Being based on the above approach, controlling the deflection angle of the beam passing through the isolated dielectric cylinder by adjusting the material parameters in time should be demonstrated. In what follows, with no loss of generality, the E-polarized fields are estimated numerically. Fig. 3 shows the radiation pattern of the CSP beam passing through the cylinder. It is evident that the angle of deflection crucially depends on the position of the incident beam (the value of h in Fig. 1). It is seen z , , j r that the widest deflection angle is observable for the value of a h close to 0.5. An assumption that the media parameters are tuned in time makes it possible to control the angle of deflection. Thus, Fig. 4 represents the far-field distribution for different values of the refractive index of the lens. It is seen that the rise of the refractive index increases the angle of deflection and vice versa. To estimate the duration of the transient period, the time domain representation of the field was obtained. Fig. 5 shows the dependence of the absolute value of the total electric field normalized by amplitude of the incident beam on the normalized time (T = tc/a). The coordinates of the observation point are: 5 . 0 - = a x , 0 = a y . All the beam parameters are the same as in Fig. 2. The initial value of the refractive index is 1.4 that at zero moment of time changes to the value 1.45. It should be noted that although this large jump of permittivity has not been yet attained in practice, it is used here to trace the dynamics of the phenomena in a clearer manner. Before zero moment of time, the incident field is presented. After zero moment of time, the field inside the cylinder is represented by the first term in Eq. (14). The initial wave is split into two waves: the time-transmitted and time-reflected ones with the shifted frequency 1 2 0 v v w . Within the time interval ) 1 ( 0 2 a n T r - < < , these two waves are observable. The second term in (14) describes the influence of the transient boundary and demonstrates some time delay (for more information see e.g. [ 11 9 - ]). Fig. 5 illustrates the moment of time ) 1 ( 2 a n T r - > that from the physical viewpoint corresponds to the moment when the wavefront from the transient boundary reaches the point of observation. The transformed field involves the whole spectrum, which is given by the poles of the function (22). However, it is seen that, after short period of transients, the steady-state regime is achieved, which corresponds to the new position of the deflected angle. 4. Conclusions The temporal analysis of the electromagnetic field transformed by the time change of the permittivity in the dielectric cylinder illuminated by a harmonic CSP beam is carried out. This analysis is based on the exact formulas for the interior and exterior fields that are obtained as the solutions of the initial-boundary value problem. The theory is based on the eigenfunction expansion in the Laplace transform domain and the solution inversion into the time domain through evaluation of residues. The obtained results indicate a possibility of using this simple lens model for very fast beam control. Acknowledgement This work was supported in part by the National Academy of Science of Ukraine in the framework of the State Target Program “Nanotechnologies and Nanomaterials”, and in part by the European Science Foundation via research network project “NEWFOCUS”. References 1. N.K. Sakhnenko, T.M. Benson, P. Sewell, A.G. Nerukh, Transient transformation of whispering gallery resonator modes due to time variations in dielectric permittivity // Opt. and Quant. Electronics, 38, p. 71-81 (2006). 2. A. Nerukh, N.K. Sakhnenko, Formation of point source image by time change of the medium // IEEE J. Selected Topics in Quantum Electronics, 15 (5), p. 1368-1373 (2009). 3. M.R. Chahamir, J. Shaker, M. Cuhaci, A.-R. Sebak, Novel photonically-controlled reflectarray antenna // IEEE Trans. Antennas and Propagation, 54, p. 1134-1141 (2006). 4. P. Linardakis, G. Borg, N. Martin, Plasma-based lens for microwave beam steering // Electronics Lett. 42, p. 444-446 (2006). 5. F. Blom, D. Dijk, H. Hoekstra, M. Driessen, A. Popma, Experimental study of integrated-optics microcavity resonators // Appl. Phys. Lett. 71(6), p. 747-749 (1997). 6. A.A. Savchenkov, V.S. Ilchenko, A.B. Matsko, L. Maleki, High-order tunable filters on a chain of coupled crystalline whispering-gallery-mode resonators // IEEE Photonics Techn. Lett. 17 (1), p. 136-138 (2005). 7. M. Benyoucef, S. Kiravittaya, Y. Mei, A. Rastelli, O. Schmidt, Strongly coupled semiconductor microcavities: A route to couple artificial atoms over micrometric distances // Phys. Rev. B, 77, 035108 (2008). 8. K. Djordjev, S.-J. Choi, R. Dapkus, Microdisk tunable resonant filters and switches // IEEE Photonics Technol. Lett. 14(6), p. 823-830 (2002). 9. J. Brownell, A. Nerukh, N.K. Sakhnenko, S. Zhilkov, A. Alexandrova, Terahertz sensing of non-equilibrium microplasmas // J. Phys. D, 38, p. 1658-1664 (2005). 10. E.V. Bekker, A. Vukovic, P. Sewell, T.M. Benson, N.K. Sakhnenko, A.G. Nerukh, An assessment of coherent coupling through radiation fields in time varying slab waveguides // Opt. and Quant. Electron., 39(7), p. 533-551 (2007). 11. N.K. Sakhnenko, A.G. Nerukh, T. Benson, P. Sewell, Frequency conversion and field pattern rotation in WGM resonator with transient inclusion // Opt. and Quant. Electronics, 39, p. 761-771 (2007). 12. A.V. Boriskin and A.I. Nosich, Whispering-gallery and Luneburg-lens effects in a beam-fed circularly layered dielectric cylinder // IEEE Trans. on Antennas and Propagation, 50(9), p. 1245-1249 (2002). 13. L. Felsen, Complex-point source solutions of the field equations and their relation to the propagation and scattering of the Gaussian beams // Symp. Mathem. 18, p. 39-56 (1975). 14. E. Heyman and L. Felsen, Complex-source pulsed beam fields // J. Opt. Soc. Amer. A, 6(6), p. 806-817 (1989). � Fig. 3. CSP beam passage through a circular dielectric lens (far-field distribution). � � Fig. 4. Dependence of the deflection angle on the value Fig. 5. Time representation of the electric field inside the cylinder. of the refractive index inside the cylinder. 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Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine 209 1 2 30 210 60 240 90 270 120 300 150 330 1800 h=0 1 2 30 210 60 240 90 270 120 300 150 330 1800 h=0.2 1 2 30 210 60 240 90 270 120 300 150 330 1800 h=0.4 0.5 1 30 210 60 240 90 270 120 300 150 330 1800 h=0.6 0.5 1 30 210 60 240 90 270 120 300 150 330 1800 h=0.8 0.5 1 30 210 60 240 90 270 120 300 150 330 1800 h=1 170180190200210220230240 0 0.2 0.4 0.6 0.8 1 polar angle (degrees) far field pattern n=1.45 n=1.5 n=1.55 051015 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 normalized time (T) abs(E/E0) 00 (,) xy h a x y y x a h _1404482178.unknown _1404482361.unknown _1404483309.unknown _1404484066.unknown _1406031283.unknown _1406035929.unknown _1413301534.unknown _1406031249.unknown _1404483374.unknown _1404483387.unknown _1404483457.unknown _1404483399.unknown _1404483383.unknown _1404483345.unknown _1404483371.unknown _1404482430.unknown _1404483285.unknown _1404483297.unknown _1404483282.unknown _1404482403.unknown _1404482407.unknown _1404482376.unknown _1404482285.unknown _1404482306.unknown _1404482327.unknown _1404482341.unknown _1404482317.unknown _1404482292.unknown _1404482301.unknown _1404482288.unknown _1404482194.unknown _1404482279.unknown _1404482282.unknown _1404482203.unknown _1404482188.unknown _1404482191.unknown _1404482184.unknown _1403185942.unknown _1404481278.unknown _1404481466.unknown _1404481515.unknown _1404482174.unknown _1404481485.unknown _1404481450.unknown _1404481455.unknown _1404481408.unknown _1403187211.unknown _1403187404.unknown _1403187410.unknown _1403187590.unknown _1403187228.unknown _1403186085.unknown _1403186156.unknown _1403187056.unknown _1403187136.unknown _1403187140.unknown _1403187053.unknown _1403186127.unknown _1403186142.unknown _1403186090.unknown _1403186053.unknown _1403186062.unknown _1403186009.unknown _1403185021.unknown _1403185699.unknown _1403185711.unknown _1403185776.unknown _1403185707.unknown _1403185684.unknown _1403185695.unknown _1403185680.unknown _1403185682.unknown _1403184948.unknown _1403185006.unknown _1403185015.unknown _1403185018.unknown _1403184958.unknown _1399906453.unknown _1403184937.unknown _1403184940.unknown _1403184920.unknown _1399906449.unknown _1399906451.unknown _1399906447.unknown _1399906445.unknown

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