Electromagnetic scattering. Graduate Course Electrical Engineering (Communications) 1 st Semester, Sharif University of Technology

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1 Electomagnetic scatteing Gaduate Couse Electical Engineeing (Communications) 1 st Semeste, Shaif Univesity of Technology

2 Geneal infomation Infomation about the instucto: Instucto: Behzad Rejaei Affiliation: Shaif Univesity of Technology Room numbe: 60, EE Dept., Shaif Univesity Reseach aeas: Integated passive micowave components Electomagnetic modeling Micowave magnetic devices Integated atificial dielectics Substate integated waveguides Intoduction and fundamentals

3 Geneal infomation Couse stuctue: oal lectues + homewok assignments Couse mateial and efeences Lectue notes (download fom eecouse/emscatteing_ms) Advanced engineeing electomagnetics; Constantine A. Balanis, 1989 Wiley Electomagnetic wave popagation, adiation, and scatteing; Akia Ishimau, 1991 Pentice Hall Scatteing of Electomagnetic Waves: Theoies and Applications; Leung Tsang, Jin Au Kong, Kung-Hau Ding, 000 Wiley Pe-equisites: Electomagnetic theoy, Micowave techniques (at the level of Poza), Diffeential equations, Special functions Intoduction and fundamentals 3

4 Geneal infomation Homewok assignments: Can be downloaded fom download fom eecouse/emscatteing_ms Have to be etuned befoe the homewok class (these ae not sepaate classes but ae pat of the usual classes on Sundays and Tuesdays) Bonus: points added to the final exam gade Times & dates: 1 st semeste, , evey Sunday and Tuesday, 15:00-16:30 Intoduction and fundamentals 4

5 Contents of lectue 1 Contents of lectue 1: Intoduction & motivation Review of Maxell theoy Geen s functions Fa fields and Radiation Intoduction and fundamentals 5

6 Intoduction Electomagnetic scatteing: active intedisciplinay aea with applications in medical imaging, geo-science, emote sensing (weathe, vegetation, etc.), ada Often the aim is to adiate the object with a wave, and gain infomation about the object by analyzing the scatteed wave Intoduction and fundamentals 6

7 Intoduction What does scatteing mean fom a physical point of view?, c c When the incoming (incident, pobing) wave eaches the object, the electic chages inside the object ae set into motion The oscillatoy motion of those chages yield oscillatoy cuents These cuents, in tun, adiate enegy (like in an antenna). These adiated waves constitute the scatteed field. Two types of chage: Fee chages which induce conduction cuents Bound (polaization) chages which induce displacement cuents Electon cloud Positive ions Feely moving electons p, p Intoduction and fundamentals 7

8 Intoduction In this couse we ae inteested in the phenomenon of scatteing of EM waves by dielectic and conductive objects: Imagine a wave, geneated by souces fa away, hits an object These waves can be consideed as plane waves How can we compute the scatteed field fa away fom the object? R D Intoduction and fundamentals 8

9 Intoduction Boad oveview of the couse: Fundamentals Intoduction to scatteing paametes and some concepts Completely solvable cases (layeed media, cylindical objects, wedges, spheical objects*) Geneal fomulation of the scatteing poblem Scatteing fom small objects (Rayleigh scatteing) Shot wavelength appoximation techniques Intoduction and fundamentals 9

10 Review of Maxwell theoy: micoscopic equations Micoscopic Maxwell equations (in vacuum): t 00 0 t Vacuum pemittivity Vacuum pemeability (,t) Electic cuent density (A/m ) (, t) Electic chage density (C/m 3 ) Intoduction and fundamentals 10

11 Review of Maxwell theoy: macoscopic equations These equations ae geneal and fundamental. But solving them inside tue mateials consisting of atomic chages and cuents is almost impossible. Macoscopic appoach: sepaate micoscopic souces (bound to atoms and molecules) fom macoscopic souces (fee conduction electons and thei motion) c, c c p c p Electon cloud Positive ions p, Feely moving electons p Intoduction and fundamentals 11

12 Review of Maxwell theoy: macoscopic equations Micoscopic o polaization chages descibed by volume density of polaization (, t) Related to the fomation of micoscopic dipoles (sepaation of bound positive and negative chages) unde influence of an electic field p i V Micoscopic o polaization chages descibed by p 1 (, t) pi V i Intoduction and fundamentals 1

13 Review of Maxwell theoy: macoscopic equations 3 d Maxwell equation: 0 c p c c 0 Electic flux density (C/m ) Associated polaization cuent p nd Maxwell equation: t p c c 0 t 0 t Intoduction and fundamentals 13

14 Review of Maxwell theoy: macoscopic equations Besides, thee ae micoscopic cuents due to otation of electons and thei spins, unelated to electic polaization. These ae magnetization cuents and induce no local chage. They lead to effective cuent m Equivalent cuent Magnetization Magnetization: volume density of magnetic dipoles Intoduction and fundamentals 14

15 Review of Maxwell theoy: macoscopic equations nd Maxwell equation: 1 0 t t m c c 1 0 Magnetic field (A/m) t c Intoduction and fundamentals 15

16 Review of Maxwell theoy: macoscopic equations Macoscopic Maxwell equations (involving the fee macoscopic souces only): t t c 0 c Intoduction and fundamentals 16

17 Review of Maxwell theoy: macoscopic equations Equations in fequency domain using phaso epesentation E jb H jd J D B 0 We have dopped the subscipt c : all cuents and chages ae consideed fee now. Note also that J j 0 Chage consevation elation Intoduction and fundamentals 17

18 Review of Maxwell theoy: constitutive elations Constitutive elations: elations between flux densities & fields In common mateials the medium is linea and isotopic leading to simple linea elationships P E D E P E E 0 1 e 0 0 e Dielectic constant M H B H M H H m m Pemeability Dielectic const. and pemeability may be fequency dependent and complex. Thei imaginay pats ae elated to loss. Intoduction and fundamentals 18

19 Review of Maxwell theoy: constitutive elations Equivalent polaization and magnetization cuents fo linea, isotopic mateials: p J p jp j0 ee j 0 E t J j E Relative dielectic constant / m J m M m H 1 H 0 J 1 H m p 0 1 Relative pemeability / Intoduction and fundamentals

20 Review of Maxwell theoy: constitutive elations Most conductive media obey Ohm s law: J E Electic conductivity nd Maxwell equation: j H jd J je E j E j E j Complex pemittivity Intoduction and fundamentals 0

21 Review of Maxwell theoy: bounday conditions Bounday conditions at the inteface between two diffeent media nˆ E = nˆ E 1 nˆ B = nˆ B 1 ˆ nˆ D1 s n D ˆ ˆ 1 s n H n H J ˆ : points fom medium 1 to n J s E, D E1, D1, : suface (sheet) cuent and chage density s medium :, H H, B, B 1 1 medium 1: 1, 1 ˆn Intoduction and fundamentals 1

22 Review of Maxwell theoy: bounday conditions If medium 1 is a pefect conducto: nˆ E 0 nˆ B 0 nˆ D nˆ H J s s E, D H, B Pefect conducto ˆn Intoduction and fundamentals

23 Review of Maxwell theoy: plane waves Maxwell equations in a linea, isotopic, homogeneous medium without any conduction cuents and chages: E j H H je Plane wave solutions: E( ) E exp jk H ( ) H exp jk 0 0 Wave vecto Constant vectos Intoduction and fundamentals 3

24 Review of Maxwell theoy: plane waves It follows that k E0 H0 k H0 E0 The electic and magnetic field ae pependicula to the diection of popagation (wave vecto), and to each othe Futhemoe: k k k 1 H ˆ E k E H E k Wave impedance ˆ k k k Intoduction and fundamentals 4

25 Review of Maxwell theoy: enegy and powe Enegy and powe caied by the field: ˆn S Complex Poynting vecto: 1 S E H * V S Total complex powe enteing a volume though its suface: 1 S nˆ E H nˆ ds * ds S S This powe is patially stoed in the volume and patially lost Intoduction and fundamentals 5

26 Review of Maxwell theoy: enegy and powe Complex powe balance: ˆn S S nˆ ds j WM WE Pl S WE E( ) E ( ) dv E( ) dv 4 4 V V V S Aveaged stoed electic enegy WH H ( ) H ( ) dv H ( ) dv 4 4 V V Aveaged stoed magnetic enegy j j Note: hee we have not included conductivity into the dielectic constant Intoduction and fundamentals 6

27 Review of Maxwell theoy: enegy and powe Dissipated powe: P l V V V E E H dv dv dv Using complex pemittivity : Polaization loss Magnetization loss Conduction loss Pl E dv H dv j V Intoduction and fundamentals 7 V

28 Review of Maxwell theoy: vecto wave equation Conside souces in a homogeneous medium Combining Maxwell equations leads to: E j H H j E J, J, V s E k E jj Volume containing souces k Vecto wave equation Intoduction and fundamentals 8

29 Review of Maxwell theoy: vecto potential Conside the same poblem of souces in a unifom medium But now we teat the poblem using the vecto potential E j H H je J, J, V s Intoduce: B H A Vecto potential Volume containing souces Intoduction and fundamentals 9

30 Review of Maxwell theoy: vecto potential Electic field: E j A Scala potential Fom nd and 3 d Maxwell equations: j A A A J j A Intoduction and fundamentals 30

31 Review of Maxwell theoy: vecto potential Gauge feedom: vecto potential is not unique, the tansfomation below yields the same electomagnetic field A A f j f This allows us to impose additional equiements on the vecto potential In Loentz gauge we demand: A j In Coulomb gauge: A 0 Intoduction and fundamentals 31

32 Review of Maxwell theoy: vecto potential Equations in Loentz gauge: k A A J k k Equations in Coulomb gauge: j A A J Intoduction and fundamentals 3

33 Geen s function We estict ouselves to the Loentz gauge: k A A J k The components of the vecto potential and the scala potential basically satisfy the same equation (Helmholtz) Conside the Geen s function satisfying: G(, ) k G(, ) k Intoduction and fundamentals 33

34 Geen s function This function gives the field at geneated by a point souce at (cuent o chage) The solution in infinite space is: exp jk G(, ) 4 Obsevation point R Solution fo potentials: A( ) G (, ) J ( ) dv -function point souce V s 1 ( ) G (, ) ( ) dv V s Intoduction and fundamentals 34

35 Geen s function Expession fo the electic field: 1 E( ) j G(, ) J( ) dv G(, ) ( ) dv V s V s Using chage consevation, patial integation, and including the suface chages, one gets: E ( ) j G (, ) J ( ) dv V s 1 G (, ) G(, ) I G (, ) k Dyadic Geen s function (matix) Intoduction and fundamentals 35

36 Review of Maxwell theoy: vecto potential in D What if the system is unifom in one diection (e.g. z) E H j A 1 A z 0 y Coss section of the cylindical egion containing souces E j A x E j A H y z x y 1 A y x x y Ay x E j A H H z x y 1 A y z z 1 Az x J, Ss x TE z : no electic field along z TM z : no magnetic field along z Intoduction and fundamentals 36

37 Review of Maxwell theoy: vecto potential in D Fo the fist set: x x A k A J y x x x A k A J y y y y k x y Fo the nd set (no potential) x y A z k Az J z k Intoduction and fundamentals 37

38 Review of Maxwell theoy: vecto potential in D These sets ae independent because the souces ae decoupled. Continuity equation: J x J y j x y 0 The souce of the TM z is J z constant along z Thus the D poblem may be decomposed in two sepaate poblems: one fo J z and one fo the othe components of the cuent density Intoduction and fundamentals 38

39 Geen s function in D The Geen s function fo poblems unifom in z-diection satisfies the equation x y G D( ρ, ρ) k G D( ρ, ρ) ( ρ ρ) ρ x, y Obsevation point D Geen s function in homogeneous space y ρ R ρ ρ 1 () GD( ρ, ρ) H0 k ρ ρ 4 j ρ x -function point souce Intoduction and fundamentals 39

40 Geen s function in D The potentials in D A( ρ) G ( ρ, ρ) J ( ρ) dxdy S s D 1 ( ρ) G D ( ρ, ρ) ( ρ) dxdy S s Intoduction and fundamentals 40

41 Fa fields Conside again souces in an infinite medium Quite often we ae inteested in fields geneated by these souces fa away fom themselves Remembe that: E j A A( ) G (, ) J ( ) dv V s 1 ( ) G(, ) ( ) dv V s Obsevation point J V s Intoduction and fundamentals 41

42 Fa fields Lets us inspect the fist tem with vecto potential Obsevation point A( ) G (, ) J ( ) dv V s exp jk G(, ) 4 ˆ 1 J V s ˆ 1 1 ˆ 1 Intoduction and fundamentals 4

43 Fa fields Keeping tems up to the fist ode in / Obsevation point 1 ˆ ˆ jk exp jkˆ exp G(, ) 4 1 ˆ exp jk 1 exp jkˆ O 4 J V s exp jk A( ) exp jkˆ J ( ) dv 4 V s Depends only on diection of Intoduction and fundamentals 43

44 Fa fields The vecto potential tem of the electic field dops as 1/ We can epeat the same pocedue fo the scala potential, but instead we use a diffeent appoach which is faste Since we have the fa field vecto potential, we have the fa magnetic field 1 exp jk H ( ) A( ) exp jkˆ J ( ) dv 4 Vs exp jk exp jkˆ J ( ) dv 4 V s exp jk exp jk ˆ ( ) dv 4 J V s Intoduction and fundamentals 44

45 Fa fields It can be shown that Then: jk jk jk exp exp 1 ˆ O exp jk ˆ O jk exp jk H ( ) ˆ exp jkˆ J ( ) dv O 1 4 V s Fa magnetic field Intoduction and fundamentals 45

46 Fa fields What about the electic field? 1 E H j Then a simila pocedue shows that exp jk ˆ exp jkˆ J ( ) dv 4 V s exp jk ˆ ˆ exp ˆ jk jk ( ) dv O 1 E J 4 V s Fa electic field Intoduction and fundamentals 46

47 Fa fields Let us summaize the esults, but fist look at ˆ jk ˆ F exp J ( ) dv V s Cuent fom facto This vecto only depends on diection of obsevation ˆ Then the fa fields ae f exp jk E ( ) jk ˆ ˆ ˆ 4 F f jk exp jk H ( ) ˆ F 4 ˆ Obsevation point J V s Intoduction and fundamentals 47

48 Fa fields In tems of the component of F nomal to the position vecto: f jk exp jk H ( ) ˆ F 4 f exp jk E ( ) jk F 4 F ˆ ˆ ˆ ˆ ˆ F F ˆ ˆ E f ˆ H f F F The fa fields ae pependicula to the diection vecto ˆ and to each othe (TEM) f 1 ( ) ˆ f H E ˆ J V s Intoduction and fundamentals 48

49 Fa fields The fa field behaves like a TEM wave popagating in the diection of obsevation with the wave vecto k kˆ kˆ ˆ The polaization (electic field) given by E f f exp jk E ( ) jk F 4 ˆ H f Note: fo each diection of obsevation the polaization and popagation diection of TEM wave ae diffeent J V s Intoduction and fundamentals 49

50 Radiation Conside now the fa field Poynting vecto f 1 * 1 E f f f f S ˆ ˆ E H E E E f ˆ ˆ k F S ˆ ˆ 3 H f F ˆ jk ˆ F exp J ( ) dv V s F ˆ ˆ ˆ ˆ ˆ F F ˆ J V s Intoduction and fundamentals 50

51 Radiation Intensity of adiated powe may be diffeent in diffeent diections, best epesentation by using spheical coodinates ˆ sin cos, sin sin, cos F F, z E f ˆ F S k F, 3 ˆ J V s H y f x Intoduction and fundamentals 51

52 Radiation Radiated powe though a small suface fa away ˆ ˆ S n S n sin dp da d d k dp ˆ ˆ F, n sin d d 3 Diffeential solid angle d s z da ˆn S J V s y x Intoduction and fundamentals 5

53 Radiation Total adiated powe: conside a spheical suface k P sin d d 3 F, 0 0 z ˆn S P k 3 F, 0 0 d s y x Intoduction and fundamentals 53

54 Fa fields (D) Let us epeat the above deivation fo a D configuation whee the obsevation point is fa away fom the souce ρ ρ The D Geen s function 1 () GD( ρ, ρ) H0 k ρ ρ 4 j Obsevation point One may poceed as befoe, but to get intuitive esults we estict ouselves to the case whee the distance is so lage that ρ ρ J S s k ρ ρ 1 Intoduction and fundamentals 54

55 Fa fields (D) Unde this condition: G D 1 ( ρ, ρ j ) exp jk 4 j k ρ ρ ρ ρ 4 Next we use ˆ 1 1 j GD( ρ, ρ) exp jk exp jk O 4 j k 4 3/ ˆ Intoduction and fundamentals 55

56 Fa fields (D) Now fist conside the TM z case: A ( ρ) G ( ρ, ρ) J ( ρ) dxdy z D z S s j exp jk exp jk ˆ J z ( ρ) dxdy 4 j k 4 S s F z ˆ The coesponding fa-zone electic field f j E j A exp jk F 4 k 4 z z z ˆ Intoduction and fundamentals 56

57 Fa fields (D) Fa magnetic field best epesented in cylindical coodinates: H f 1 Az O 3 / f 1 Az 1 k j H exp jk F 4 4 z ˆ The fa field behaves like a TEM wave popagating in the diection of obsevation with the wave vecto This is a cylindical wave whose amplitude dops as k k ˆ Intoduction and fundamentals 57

58 Fa fields (D) k k ˆ Cylindical wave: H E z Teatment of the TE z is a bit moe complicated but the final esult fo the fa field is the J z same: The fa field again behaves as a locally TEM, cylindical wave with the magnetic field diected along z and the fa electic field diected along ˆφ TM z Intoduction and fundamentals 58

59 Radiation (D) Conside now the fa field Poynting vecto fo the TM z case Note that f 1 E f f * z k S ˆ E H F 16 z ˆ ˆρ ˆ ˆ F exp jk J ( ρ) dxdy z S s F S s exp jk xcos ysin J z( ρ) dxdy z z J Intoduction and fundamentals 59

60 Radiation (D) Total adiated powe (pe unit length) k P Fz d 16 0 Intoduction and fundamentals 60

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