Scattering at an Interface: Normal Incidence

Transcription

1 Course Insrucor Dr. Raymond C. Rumpf Office: A 337 Phone: (915) Mail: rcrumpf@uep.edu 4347 Applied lecromagneics Topic 3f Scaering a an Inerface: Normal Incidence Scaering These Normal noes Incidence may conain copyrighed maerial obained under fair use rules. Disribuion of hese maerials is sricly prohibied Slide 1 Lecure Ouline Review of lecromagneic Boundary Condiions Problem a an Inerface Reflecion and Transmission Scaering Normal Incidence Slide 2 1

2 Review of lecromagneic Boundary Condiions Scaering Normal Incidence Slide 3 Physical Boundary Condiions and 1 1 and 2 2 Tangenial componens of and are coninuous across an inerface. 1,T 2,T 1,T 2,T and fields normal o he inerface are disconinuous across an inerface. Noe: Normal componens of D and B are coninuous across he inerface. 11,N 11,N D 1,N B 1,N 22,N 22,N D 2,N B 2,N These are more complicaed boundary condiions, physically and analyically. Tangenial componens of he wave vecor are coninuous across an inerface. k1,t k2,t Scaering Normal Incidence Slide 4 2

3 Problem a an Inerface Scaering Normal Incidence Slide 5 Problem a an Inerface (1 of 7) We sar wih a planar inerface beween wo semi infinie half spaces. ach medium has a differen impedance. Scaering Normal Incidence Slide 6 3

4 Problem a an Inerface (2 of 7) A wave is inciden ono he inerface a normal incidence. i 1 i 1 k i Scaering Normal Incidence Slide 7 Problem a an Inerface (3 of 7) Boundary condiions require he angenial componens of and o be coninuous across he inerface. i 1 i 1 k i 1 k 1 Scaering Normal Incidence Slide 8 4

5 Problem a an Inerface (4 of 7) The impedance of Medium 2 requires ha / = 2. i 1 i 1 k i 1 k 1 1 k 0.5 Scaering Normal Incidence Slide 9 Problem a an Inerface (5 of 7) There mus be a refleced wave in Medium 1 in order o reconcile he physics in Medium 2. i 1 i 1 k i 1 k 1 1 k 0.5 Scaering Normal Incidence Slide 10 5

6 Problem a an Inerface (6 of 7) We express he problem and solve simulaneously for he ransmied and refleced wave. i 1 k i i 1 r 13 r k 13 r 43 k 23 Scaering Normal Incidence Slide 11 Problem a an Inerface (7 of 7) I is he oal elecric and magneic fields ha mus be coninuous across he inerface. The individual waves mus saisfy impedance. i 1 k i i 1 r 13 r k 13 r 1 i r k Scaering Normal Incidence 1 i r 23 Slide

7 Noes Abou he Inerface Some of an inciden wave may reflec from he inerface. Some of an inciden wave may ransmi hrough he inerface. While no discussed here, some of an inciden wave may be absorbed a he inerface. The angle of ransmied wave may be differen han he inciden wave. Scaering Normal Incidence Slide 13 Plane Wave a Normal Incidence Scaering Normal Incidence Slide 14 7

8 Problem Seup In general, we have an inciden wave, a refleced wave, and a ransmied wave. i k i k i r r k r Scaering Normal Incidence Slide 15 Deriving Reflecion & Transmission Coefficiens Sep 1 Wrie expression for he applied wave, refleced wave, and ransmied wave. Sep 2 nforce he boundary condiions a he inerface. Sep 3 Solve boundary condiion equaions for r and. Sep 4 Derive a relaionship beween r and. Sep 5 Inspec final equaions and make conclusions. Scaering Normal Incidence Slide 16 8

9 General xpression for he Inciden Wave Wihou loss of generaliy, we can le he inciden wave be linearly polarized along he x direcion. 1z i z 0,ie ax 0,i 1z i z e ay 1 Scaering Normal Incidence Slide 17 General xpression for he Refleced Wave Assuming he inerface is perfecly fla and boh mediums are isoropic, he refleced wave will have he same polarizaion as he inciden wave. Furher, i is ravelling in he same medium as he inciden wave so i has he same propagaion consan, bu opposie sign because i is ravelling backwards. 1z rz 0,re ax 0,r 1z r z e ay 1 This sign enforces he handedness of he problem. in direcion of wave Scaering Normal Incidence Slide 18 9

10 General xpression for he Transmied Wave Assuming he inerface is perfecly fla and boh mediums are isoropic, he ransmied wave will have he same polarizaion as he inciden wave. owever, i is ravelling in a differen medium as he inciden wave so i will have a differen propagaion consan. 2z z 0,e ax 0, 2z z e ay 2 Scaering Normal Incidence Slide 19 Quanifying Reflecion and Transmission We quanify reflecion and ransmission by relaing he various field ampliudes hrough he reflecion coefficien r and ransmission coefficien. 1z 2 z e a x i 0,i 1z rz 0,re a x z e a z 0, x Definiion of Reflecion Coefficien r 0,r 0,i Definiion of Transmission Coefficien 0, 0,i Scaering Normal Incidence Slide 20 10

11 nforce he Boundary Condiions (1 of 2) Boundary condiions require ha he angenial componen of he oal field mus be coninuous across he inerface The oal elecric field in Medium 1 is he sum of he inciden i and refleced r waves i r Subsiue in our general expressions for he elecric field componen of he inciden i, refleced r, and ransmied waves. e a e a e a ,i x 0,r x 0, x 0,i 0,r 0, Scaering Normal Incidence Slide 21 nforce he Boundary Condiions (2 of 2) Boundary condiions require ha he angenial componen of he oal field mus be coninuous across he inerface The oal magneic field in Medium 1 is he sum of he inciden i and refleced r waves i r Subsiue in our general expressions for he magneic field componen of he inciden i, refleced r, and ransmied waves. ea ea ea 0,i 0 0,r 0 0, 0 y y y ,i 0,r 0, Scaering Normal Incidence Slide 22 11

12 Reflecion Coefficien, r 0,i 0,r 0, q. 1a Subsiue q. (1a) ino q. (1b) o eliminae 0,. 0,i 0,r 0,i 0,r q. 2 0,i 0,r 0, q. 1b Solve q. (2) for 0,r / 0,i because his is our definiion of he reflecion coefficien r ,i 0,r 0,i 0,r ,r 2 1 r ,i 2 1 0,r 0,i ,r 2 1 0,i Scaering Normal Incidence Slide 23 Transmission Coefficien, Solve our new equaion for r for 0,r. 0,r 2 1 0,i ,r 0,i 2 1 q. 3 Subsiue q. (3) ino q. (1a) and solve for 0, / 0,i because his is our definiion of he ransmission coefficien. 0, 0,i 0,r , 0,i 0,i 1 0,i , ,i , 2 2 0,i 2 1 Scaering Normal Incidence Slide 24 12

13 Relaion Beween r and Divide q. (1a) by 0,i. 0,i 0,r 0, q. 1a This equals 1 0,i 0,r 0, 0,i 0,i 0,i This is our definiion of This is our definiion of r 1r CAUTION: This equaion looks like conservaion of power, bu i is no! r and are associaed wih field ampliudes, no power quaniies. Scaering Normal Incidence Slide 25 Noes Abou Scaering a Normal Incidence Many imes is used synonymously wih r. We will reserve use of for ransmission lines. The subracion operaion in he expression for reflecion coefficien means ha r can be posiive or negaive. If 1 > 2, he refleced wave will experience a 180 phase shif. 1 + r = (This is no conservaion of power) Boh r and are dimensionless and may be complex. They are complex because boh he phase and ampliude can be affeced a an inerface. 0 r 1 and 0 1 Scaering Normal Incidence Slide 26 13