Lecture 8: Reflection and Transmission of Waves. Normal incidence propagating waves. Normal incidence propagating waves

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1 /8/5 Lecture 8: Reflecton and Transmsson of Waves Instructor: Dr. Gleb V. Tcheslavsk Contact: Offce Hours: Room 3 Class web ste: m/index.htm So far we have consdered plane n an nfnte homogeneous medum. A natural queston would arse: what happens f a plane wave hts some object? Such object can be ether delectrc or conductor. To answer ths queston we need to use boundar condtons. We stud frst normal ncdence on the boundar. We assume that a plane wave s generated n the regon < n a lossless materal wth delectrc constant and that a second lossless materal s n the regon wth a delectrc constant. The permeabltes of both materals are. A porton of the wave s transmtted to the medum another porton s reflected back to medum. 3 4 The drecton of wave s propagaton can be defned usng a rght hand rule. Ths suggests that the polaraton of the reflected feld has to be altered after the ncdent wave strkes the nterface. We assume that the electrc component s unchanged and the magnetc feld wll change ts drecton. j t k Therefore: Incdent wave: E () t Ae (8.3.) Reflected wave: Transmtted wave: j t k Er () t Bre j t k Et () t Ae t (8.3.) (8.3.3) Here k and k are the wave numbers for the regons (meda) and respectvel constants A and B ndcate the terms propagatng n the + and drectons. Snce the materals are assumed as lossless the wll not attenuate (.e. = = k). The magnetc feld ntenstes can be found from the Maxwell s equatons or b usng the characterstc mpedances for two regons. Incdent wave: Reflected wave: Transmtted wave: A Hx () t e Br Hxr () t e At Hxt () t e c jtk jtk jtk (8.4.) (8.4.) (8.4.3) From the boundar condtons the tangental components of the electrc feld must be contnuous and the tangental components of the magnetc feld ntenst must dffer b an surface current that s located at the nterface. Usuall we assume that there are no surface currents whch mples that the tangental components of the magnetc feld ntenst are also contnuous at the nterface. Also snce we chose the nterface between two meda to be at = the exponent wll be j t k jt e e (8.4.4)

2 /8/5 5 6 Therefore at the boundar = we can wrte: E ( t) Er ( t) Et ( t) H ( t) H ( t) H ( t) x xr xt Substtutng the last results to (8.3.x) and (8.4.x) we obtan: A Br At A Br At c c c Therefore f one of three wave s magntudes s known two other can be computed. (8.5.) (8.5.) (8.5.3) (8.5.4) We specf the reflecton and transmsson coeffcents as: Reflecton: Transmsson: B A r c c c c c A t A c c Knowng the characterstc mpedances of the materals allows us to determne the propagaton characterstcs and ampltudes of both : transmtted to the second medum and reflected back. If the characterstc mpedances on both sdes are equal all energ s transmtted nto regon and none s reflected back. Ths s called matchng the meda (lenses glasses etc.). (8.6.) (8.6.) 7 (Example) 8 Snce the characterstc mpedance s: For two delectrc materals: Reflecton: Transmsson: (8.7.) (8.7.) (8.7.3) Example 7.: Descrbe the expected reflectontransmsson characterstc of a tme harmonc EM wave normall ncdent at a laered delectrc. Fnd the total reflecton and transmsson coeffcents of a sngle laer wth known thckness d and delectrc constants are = r ; = 3 =. Plot the frequenc dependence of these coeffcents assumng d =. m and r = 4. The ampltudes of the total transmtted and reflected felds are n E 3 n wave ndex n n - drecton regon number E ( ) ( ) t E ( ) E( ) r (8.8.) (8.8.)

3 /8/5 (Example) (Example) The wave ncdent to the boundar wth the second medum s partall reflected back and partall transmtted to medum. At the nterface between materals and 3 the transmtted to materal feld s partall reflected back and partall transmtted to the materal 3 etc. The phase of ndvdual terms n summatons s dfferent: an addtonal phase dfference of k d appears after each crossng the slab. We assume next that the reflecton coeffcent from the frst nterface s n the + drecton and - n the drecton. Snce the medum 3 s dentcal to medum = - and - =. Smlarl the transmsson coeffcent through the frst nterface wll be + = + n the + drecton and - = - n the - drecton. E E E E E The total reflected electrc feld s: jk d j4kd j6kd Er( ) E e e e... jkd jkd jkd E e e e... The total reflecton coeffcent s: jkd j kd e e Er ( ) E e e Smlarl the total transmtted electrc feld s: j kd jkd jk d j3kd j5kd Et( ) E e e e... jkd j k d j kd E e e e... (8..) (8..) (8..) (Example) The total transmsson coeffcent s: The reflecton coeffcent of the frst boundar s: r (8..) 3 The wave number n the slab: r k k k (8..3) r The frequenc dependence of the total transmsson and reflecton coeffs: e Et ( ) (8..) j kd E e jk d What f the medum where the wave s propagatng s loss.e.? The magnetc feld must be: The Helmholt equaton s: Hr () jer () Er () Er Er () () where the complex propagaton constant s: j j j If the electrc feld s lnearl polared n the x drecton wave equaton reduces to: Ex ( ) E ( ) x x (8..) (8..) (8..3) (8..4) 3

4 /8/5 3 4 The soluton wll be n form: E ( ) E e E e x (8.3.) Travelng exponentall decang The frst term a wave propagatng n + drecton: j E e E e e E e cost (8.3.) The magnetc feld wll be: E e H ( ) E e Alternatvel loss can be ncorporated b: ; ' j" c (8.3.3) (8.3.4) If the materal s a good conductor the characterstc mpedance wll be ver small and t would approach ero as the conductvt approaches nfnt. Therefore t wll be NO transmsson of EM energ nto the conductor t all wll be reflected. In ths case: (8.4.) Polaraton change (8.4.) Consderng the most general soluton (7..3) of the wave equaton and a travelng pulse nstead of a tme-harmonc wave we ma assume that a vrtual pulse wth a negatve ampltude was launched at = + n the drecton smultaneousl wth the real pulse. Both pulses meet at = at the tme t = t + 3t after that moment the keep propagatng but the vrtual pulse becomes real and the real one becomes vrtual. 5 6 At the nterface the ampltudes of both pulses add up to equal ero n order to satsf the requrement that the tangental component of the electrc feld must be ero at a perfect conductor. Snce the Pontng s vector has no real part NO average power s delvered to the regon (perfect conductor)! Felds exst onl n the regon < (medum delectrc): E( ) E ( ) E ( ) E e e u je snk u jkc jkc r x c x E E H H H e e u k ( ) ( ) jkc jkc r( ) cos c Note that at =: E() = and H() = E / c u. The Pontng vector for the frst regon ( < ) wll be: (8.5.) (8.5.) The surface current on the conductor can be found from the boundar condtons on the tangental magnetc feld: E J u H( ) u (8.6.) s x Ths s where no metal plates n a mcrowave oven comes from! E S E H j k k u (8.5.3) * ( ) ( ) ( ) cos sn 4

5 /8/5 (Example) 7 (Example) 8 Example 7.: Pulse radars can be used to determne the veloct of cars. Show how such radars could work. Durng t the car travels a dstance ; therefore the veloct of the car can be estmated. A repettve EM pulse from the radar s ncdent on the car. Because of the hgh conductvt of the car the pulse s reflected back to the radar where the total tme of travel t for the gven pulse can be estmated. The pulse repetton tme (the tme between two consecutve pulses) s t. The dfference n arrval tme for two pulses The veloct s: L ( L ) Vcart t t c c c c v car c t t t The actual dstance between the car and the radar L s not mportant to determne the car s speed; although t can be computed as well. (8.8.) (8.8.) Oblque ncdence propagatng Oblque ncdence propagatng When a plane EM wave ncdent at an oblque angle on a delectrc nterface there are two cases to be consdered: ncdent electrc feld has polaraton parallel to the plane of ncdence and ncdent electrc feld has polaraton that s perpendcular to the plane of ncdence.. Parallel polaraton: The ncdent reflected and transmtted electrc feld vectors le n the plane of ncdence: the x- plane. Note: the angles are measured wth respect to normal. Incdent wave: Reflected wave: Transmtted wave: jk xsn cos E cos sn E ux u e (8..) E jk xsncos H ue (8..) c jk xsn cos cos sn r r Er E rux ru e H E jk xsn cos cos sn t t Et E tux tu e (8..3) jk xsnr cosr r ue (8..4) E Ht u e c jk xsnt cost (8..5) (8..6) 5

6 /8/5 Oblque ncdence propagatng Oblque ncdence propagatng From the boundar condtons:.e. contnut of tangental electrc and magnetc felds at the nterface = we derve: jkx sn jkx snr jkxsnt cose cosre coste jkx sn jkx snr jkxsnt e e e To satsf these condtons the Snell s laws of reflecton and refracton must hold: r; ksn ksnt These smplfcatons lead to: cost cos cost cos cos c ccost cos (8..) (8..) (8..3) (8..4) (8..5) For parallel polaraton a specal angle of ncdence exsts known as the Brewster s angle or polarng angle = B for whch the reflecton coeffcent s ero: =. It happens when Therefore: cos c t c sn B cos If the ncdence angle equals to Brewster s angle the reflected feld wll be polared perpendcularl to the plane of ncdence. (8..) (8..) Oblque ncdence propagatng 3 Oblque ncdence propagatng 4. Perpendcular polaraton: The ncdent reflected and transmtted electrc feld vectors are perpendcular to the plane of ncdence: the x- plane. jk Incdent wave: xsn cos E E u e Reflected wave: Transmtted wave: (8.3.) E H cos u sn u e x jk xsnrcosr Er Eue jk xsn cos jk cos sn x sn r cos r r r x r jk xsnt cost Et Eue (8.3.) (8.3.3) E H u u e (8.3.4) E t t H cos u sn u e t t x t jk xsn cos (8.3.5) (8.3.6) From the contnut of tangental electrc and magnetc felds at the nterface = we agan derve: jkx sn jkx snr jkxsnt e e e (8.4.) jkx sn jkx snr jkxsnt cose cosre coste (8.4.) c c c As before the Snell s laws of reflecton and refracton must hold: ; k sn k sn r t These smplfcatons lead to ver smlar expressons: cos cost cos cost cos cos cos c c t (8.4.3) (8.4.4) (8.4.5) 6

7 /8/5 Oblque ncdence propagatng 5 Oblque ncdence propagatng 6 A magntude of the reflecton coeffcent for the parallel and perpendcular polaraton: Total nternal reflecton and surface Assume that the unform plane wave ncdent on an nterface between two perfect delectrcs wth k > k for nstance water to ar. From the Snell s law: k cost sn k For a partcular angle of ncdence the quantt under the square root becomes ero. Ths angle s called crtcal angle: k c sn k (8.6.) (8.6.) At the ncdence angles exceedng the crtcal angle the phenomenon of total nternal reflecton occurs. Oblque ncdence propagatng 7 Fabr-Perot resonator standng 8 Let us denote: cos j when t c In both cases of parallel and perpendcular polaraton the reflecton coeffcent wll be: j (8.7.) j Here both and are real. As a consequence the magntude of the reflecton coeffcent = and all ncdent power s reflected off the nterface. As a result for nstance for the perpendcular polaraton the transmtted electrc feld s: jkx sn Et TEue e (8.7.) If the ncdence angle exceeds the crtcal angle the feld n regon propagates n the x drecton but rapdl exponentall decas n the drecton awa from the nterface. Ths s a surface wave. (8.7.3) Let us consder agan the electrc component of a lnearl polared EM feld normall ncdent on a perfect conductor wth a reflecton coeffcent = -. Where A = B. j( tk) j( tk) jt jk jk jt j Be k E () t ReB e e ReBe e e Re sn E () t Asntsnk The tangental electrc feld E (t) = at the nterface =. In ths case the sgnal consstng of two oppostel propagatng appears to be statonar n space and oscllatng n tme. Ths s a standng wave. (8.8.) (8.8.) 7

8 /8/5 Fabr-Perot resonator standng Fabr-Perot resonator standng (Example) 3 The standng wave results from the constructve and destructve nterference of the two counter propagatng. Observe that the separaton dstance between two successve null ponts (nodes) equals to the separaton dstance between two successve maxma (antnodes) and equals to one half of the wavelength. Example 7.3: An EM wave propagatng n a vacuum n the regon < s normall ncdent upon a perfect conductor at =. The frequenc of the wave s 3 GH the ampltude of ncdent electrc feld s V/m and t s polared n the u drecton. Fnd the phasor and the nstantaneous expressons for the ncdent and the reflected feld components. f 3 k m (8.3.) 8 c 3 The ncdent feld n the phasor form: jk j E( ) Ae u e uv m u E ( ) ( ) j H e uxa m jt Or: () Re () cos6 E t E e uv m jt H() t Re H() e cos6 uxam (8.3.) (8.3.3) (8.3.4) (8.3.5) Fabr-Perot resonator standng (Example) 3 Fabr-Perot resonator standng 3 The reflected feld n the phasor form: E ( ) E ( ) e u V m j r u E ( ) r j H( ) e uxa m (8.3.) (8.3.) Examnaton of a standng wave suggests that t should be possble to nsert another conductor (a conductve wall) at an of the nodes where the tangental electrc feld s ero wthout changng a structure of electrc feld! The applcable boundar condton therefore s that the tangental electrc feld must be ero at a conductve surface. Or n the nstantaneous form: jt E () t Re E () e cos6 u V m r r jt Hr() t Re Hr() e cos6 uxam (8.3.3) (8.3.4) Let us assume that the plates were nserted nstantaneousl and the EM energ was trapped between the plates. Constructve and destructve nterference wll lead to appearance of a standng wave. 8

9 /8/5 Fabr-Perot resonator standng 33 Fabr-Perot resonator standng 34 For the D Helmholt equaton de( ) ke( ) d and a tme-harmonc sgnal the soluton wll be n a form: E ( ) AsnkBcosk The ntegraton constants A and B can be found from the boundar condton that the tangental electrc feld must be ero at a metal wall. Therefore B = n k L where n s an nteger (resonator mode) and L s the dstance between the metal walls. If the maxmum magntude of electrc feld s E the electrc feld s n E( ) E sn L (8.33.) (8.33.) (8.33.3) (8.33.4) The structure consstng of a parallel plate cavt s called a Fabr-Perot resonator. If the frequenc of a wave matches the dmensons of the resonator (a resonant frequenc) the length of cavt equals an nteger number of half-wavelengths a standng wave wll be formed. All other frequenc components wll be canceled out b a destructve nterference. The Q-factor ( rato of stored energ to the power dsspated per ccle) of ths resonator ma be ver hgh (approaches a mllon). Fabr-Perot resonators are wdel used n EM and optcs: a He-Ne laser s bascall a Fabr-Perot resonator. Fabr-Perot resonator standng 35 Fabr-Perot resonator standng 36 Recall that the wave number s a functon of frequenc and the veloct of lght between plates. r n k (8.35.) c L Therefore we can fnd the resonant frequenc as n r (8.35.) L Consderng two resonators of the same length L but one of the flled wth ar (left) and the other flled wth a delectrc we can fnd that the wll resonate at two dfferent frequences. The frequenc dfference wll be n rr (8.35.3) L r The resonant frequenc for a free space (or ar) can be computed or measured for known resonator s dmensons. The relatve frequenc dfference s r (8.36.) r r r Therefore f we can measure ths frequenc dfference we can estmate the permttvt of unknown materal placed n the resonator and thus dentf the materal. Example 7.4: An empt Fabr-Perot resonator has a resonant frequenc of 35 GH. Determne the thckness of a sheet of paper that s nserted later between the plates f the resonant frequenc changes to 34. GH. The separaton between plates s 5 cm. Assume that the nteger n specfng the mode doesn t change and that there s no reflecton from the paper. 3 paper

10 /8/5 Fabr-Perot resonator standng (Example) 37 Fabr-Perot resonator standng (Example) 38 The relatve delectrc constant separatng the plates wth paper nserted can be approxmated as L L L r paper paper L L L L vacuum paperadded L paper L vacuum r paper L L Therefore: L (3 ) 35.5 Fnall: 4 L.4 m (8.37.) (8.37.) (8.37.3) Example 7.5: A helum-neon laser emts lght at a wavelength of 638 Å n ar. Calculate the frequenc of oscllaton of the laser the perod of oscllaton and the wave number. Å (Åmgstrom) = - m. The frequenc: 8 c.8 4 f H473.8TH The perod: 5 T. s. fs 4 f The wave number: 6 k.3 m ??QUESTIONS??