Lecue Elecomagneic Waves Pogam: 1. Enegy caied by he wave (Poyning veco).. Maxwell s equaions and Bounday condiions a inefaces. 3. Maeials boundaies: eflecion and efacion. Snell s Law. Quesions you should be able o answe by he end of oday s lecue: 1. Wha is he diecion of enegy flux of he EM wave?. Wha is he basic pinciple behind he bounday condiions fo EM waves? 3. Coninuiy of wha wave paamee is esponsible fo eflecion and Snell s laws? 1
Reflecion and Refacion a Maeials Inefaces Bounday Condiions: Coninuiy condiions fo he fields obeying Maxwell s Equaions. These condiions can be deived fom applicaion of Maxwell s equaions, Gauss and Sokes Theoems and have o be saisfied a any maeials bounday. 1. nˆ B B 1 0 B B 1 The componen of he magneic inducion pependicula (nomal) o he ineface is coninuous acoss he ineface.. nˆ D D 1 D D ( suface chage densiy) 1 In he pesence of suface chage a he maeial ineface, he nomal componen of he elecic displacemen changes abuply by an amoun equal o suface chage densiy. 3. nˆ E E 1 0 E E 1 The componen of he elecic field paallel (angenial) o he ineface is coninuous acoss he ineface. 4. nˆ H H K H H K (K suface cuen densiy) 1 1 In he pesence of a suface cuen a he ineface, he componen of he magneic inducion paallel (angenial) o he ineface changes abuply by he amoun equal o suface cuen K. In many cases in opics, he suface chage densiy and suface cuen densiy ae zeo, and consequenly he nomal componens of D and B and he angenial componens of E and H ae coninuous. Conside a chage-fee cuen-fee ineface beween maeials wih efacive indices, n. Conside a siuaion whee a wave is inciden fom he op ono he ineface ik i i E i e inciden wave E efleced wave ik i e ik e E ansmied wave The dispesion elaions fo a homogeneous n ki k 1 c medium ell us: 0 n k
Due o phase coninuiy, he phases of all hee waves (inciden, efleced and ansmied) have o be equal a he ineface plane x 0. Consequenly he phases of he efleced and ansmied waves ae compleely deemined by he phase of he inciden wave. k i k k k iy y k iz z x0 x0 x0 k y k z z k y y kiy k y ky y k z z k k k iz z z We wee able o make hese conclusions because z and y ae abiay coodinaes. The equaions above have wo impoan consequences: 1. The vecos k i, k, k all lie in a plane called he plane of incidence. We have oiened ou coodinae sysem such ha he plane of incidence coincides wih he x-z plane. Then elecic ikxk x z z field can wien in he following fom: E E 0 e. The angenial componens of he waveveco (componens lying wihin he plane of incidence) ae idenical egadless of he medium ha hey ae in: k iz k k. z z Then: n k si i k, k iz i and k z sin n n k, k sin z k k k iz z z Fom hese equaions we find: 1. Angle of eflecion equals angle of incidence: k iz k z sin i sin i. Snell s law: k iz sin c i sin i n sin k n sin z c 3
Toal inenal eflecion. Waveguides. As you emembe fom he Snell s law: sin i n sin sin sin i n This implies ha if n hen i and if n hen i. Which means ha in he case of n fo a ceain angle of incidence i c he efacion angle becomes equal o 90, which in pacice means ha he ligh canno escape hough he ineface and will say inside he maeial wih highe efacive index. This effec is called oal inenal eflecion, and he ciical angle is simply: sin c n sin n n 1 Opical fibes and waveguides used fo ansmission of infomaion ove he long disances use his pinciple o keep he EM waves inside. Waveguides consis of highe efacive index ( ) coe and lowe efacive index ( n ) cladding: Then we can find he maximum angle a which we can sill couple EM waves ino he waveguide so we can ake advanage of he oal inenal eflecion. The efacive index fo ai is n 1, hen we can find: n sin c n sinmax cos c 1 sin c 1 1sinmax sin c cosc sin max n Anohe impoan chaaceisic conneced o he maximum angle is numeical apeue (NA), and in case of coupling beween ai ( ) and fibe: NA 1sin max n 4
s-p Polaizaion: ansmission and eflecion coefficiens In geneal we expess he elecic fields on boh sides of he ineface as: ik i ik E ei i e E e x 0 E ik i E e e x 0 I is useful o sepaae he elecic field ino componens. These componens ae ohogonal o each ohe and ae called Polaizaions. Evey wave can be epesened as a supeposiion of polaizaions. (p) In he plane of incidence ( x z plane) (s) Pependicula o he plane of incidence ( y -diecion). Le s conside boh polaizaions sepaaely. s-polaized elecic field: E x, y, z s 0, E s x, y, z,0 The elecic field above he ineface in he maeial will be equal o: ik ix x ik x x iiz E y ˆ E e E e e s is s inciden efleced This componen is angenial (paallel) o he maeial ineface and hence, o he bounday condiion E 1 E, will be coninuous acoss he ineface: E iy E y E y. Recall ha: B E 0 E H 0 H x, y, z ik Subsiuing he plane wavefom: H i 0 e ino he equaion above, we find: E ih 0 5
E ˆx ŷ ẑ x y z 0 E y 0 Since E H, and he elecic field is pependicula o he plane of incidence, he magneic field will lie in he plane of incidence and pependicula o he waveveco. Recall he bounday condiion fo he angenial componen (paallel o he ineface, z in ou case) of he magneic field: The angenial componen of he magneic field H H J. 1 In he absence of cuens ( H 1 H ) he condiion above implies coninuiy of he z-componen of he magneic field acoss he ineface (in x 0 plane): H iz H z H z 1 1 ik ix xk iz z 1 ik ix xk iz z H iz E y E x E i e ikix E i e i x y i x i 0 0 0 Recall: k ix k i cos i n n 1 cos i and k x k cos cos ik ix xk iz z ik ix xz H iz cos i E i e cos i E i e c0 0 0 ik ix xk iz z H cos i E e cos i E e z c c 0 0 0 0 ik ix x z n z n i k x xk z xz ik x H cos E e cos E e z 0 0 Consequenly he bounday condiions will lead us o wo equaions fo elecic field ampliudes fo he efleced and ansmied waves: E iy E y E y E iy E y E y H iz H z H z cos i E iy cos i E y n cos E y Then we can find eflecion and ansmission coefficiens fo he s-polaized wave: Ey cos i n cos s E cos i n cos iy E y cos i s E n1 cos i n cos iy Then he efleciviy of he ineface is simply: R s 6
p-polaizaion: E x, y, z E x, y, z,0, E x, y, z p px pz In he absence of ineface chage he bounday condiion elecical fields pependicula o he maeial ineface is: D 1 D 1 E 1 E This implies ha he z-componen of elecic field has o be conseved acoss he ineface: E E E 1 iz 1 z z We also know ha he componens of he elecical field paallel o he ineface have o be conseved acoss he ineface: E E 1 Which in his case means ha x-componen of he elecic field have o be conseved acoss he ineface: E ix E x E x Combining he equaions we go fom he bounday condiions we ge: 1 E iz 1 E z E z 1 sin i E i 1 sin i E sin E sin i n sin and 1, n n 1 E i E n E E ix E x E x cos i E i E cos E Leading o he ansmission and eflecion coefficiens: p E E i n cos n i 1 cos n cos i cos p E E i cos i n cos i cos Noe ha fo p-polaizaion hee exiss an angle known as Bewse angle a which he eflecion coefficien is zeo and all he ligh is ansmied: 7
n cos i cos 0 n cos i cos n cos i cos p n cos i cos n 1 sin i 1 sin n 1 sin i 1 sin i n 4 4 n sin i n n n sin i n 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. R s 0.1 R p 0.0 0 o 10 o 0 o 30 o 40 o 50 o 60 o 70 o 80 o 90 o Angle of incidence θ Image by MIT OpenCouseWae. I I Ani-eflecion coaings: maximizing he coupling of ligh ino he maeial. Many applicaions such as sola panels, opical ineconnecs equie maximum coupling of he incoming ligh ino he maeial. As you have seen above a inesecion of any wo maeials wih diffeen efacive indices hee is significan eflecion, which is highly undesiable fo he above-menioned applicaions. I is possible, howeve o ceae a coaing a he ineface of he wo maeials ha would minimize he eflecion n 0 n S beween hem. By using he eflecion and ansmission coefficiens a T.I boh sufaces one can find ha he eflecion coefficien will be minimal when: R.I n I n 0 n S R 01.I R 1S.T 01.I T 1S.T 01.I Image by MIT OpenCouseWae. 8
Alenaively one can use quae-wave coaings. These coaings ae pecisely d hick and 4n I hey wok by making he waves efleced fom he fis and second inefaces be exacly ou of phase and hence annihilae each ohe. Howeve, noe ha quae-wave coaings wok bes fo a paicula wavelengh, which hey have been designed fo bu maching indices of efacion is geneal soluion ha woks fo mos maeials wih low dispesion. 0 9
MIT OpenCouseWae hp://ocw.mi.edu 3.04 Eleconic, Opical and Magneic Popeies of Maeials Sping 013 Fo infomaion abou ciing hese maeials o ou Tems of Use, visi: hp://ocw.mi.edu/ems.