REFLECTION FROM A METALLIC SURFACE For a metallc medum the delectrc functon and the ndex of refracton are complex valued functons. Ths s also the case for semconductors and nsulators n certan frequency ranges near and at absorpton bands. Fresnel's equatons are stll val but the angles n the equatons are now complex valued and do no longer have the obvous geometrcal nterpretaton. For normal ncence we have Ê n - n ˆ n n R Á Æ - Ë n + n n + n where ñ n + k n( + k ) ( ) + ( - ) ( ) + + n - n k k n + n ( k k) where k s the so called extncton coeffcent. For metallc systems n e e + 4ps w For normal ncence the refracted, or rather transmtted, wave wll vary as ( ) - ( - ) nkz E 0 -wt kkz 0 nkz 0 wt ~ e e e
Bo E. Sernelus 3: REFRACTION INTO A CONDUCTING MEDIUM Navely one would beleve that the plane wave nse the metal would vary lke n the precedng secton but n a new drecton. The dampng would be along the drecton of propagaton. Ths s however not so! The dampng s stll n the z-drecton only. Ths means that the surfaces of constant ampltude are no longer the same as the surfaces of constant phase. The surfaces of constant ampltude are parallel to the nterface and the surfaces of constant phase are perpendcular to the drecton of propagaton. The whole problem s a bt awkward. Approxmately the angle of refracton and the wavelength are determned by the real part of the ndex of refracton and the dampng n the z-drecton by the magnary part of the ndex of refracton. Note however that ths s just approxmate! The correct dependence s ( ) ( ) k E 0snq0 -x k0 n sn q 0 z wt ~ e ( )+ - - -k0 Im n -sn q 0 z k 0 snq 0 x k 0 Re n sn e e (- )+ - q 0 z-wt and the real transmsson angle s q real È sn q arctaní 0 Í ÎRe n - sn q 0 Note that Fresnel's coeffcents are stll val for complex valued refractve ndces but the angles are complex valued.
Bo E. Sernelus 3:3 TOTAL INTERNAL REFLECTION I wll just brefly touch upon ths subject, because of lack of tme. It s very mportant though. I treat ths n more detal n my other course, TFYY70 fundamentals of surface modes. If an electromagnetc wave s mpngng on an nterface from a materal of hgher refractve ndex to a one wth lower, the refracton angle s greater than the angle of ncence. If we let the angle of ncence ncrease we reach a crtcal angle, - q c sn ( n n ), when all energy s reflected. For all angles greater than ths crtcal angle we have total reflecton. The nterestng thng s that there s also a wave parallel to the nterface, whch decays exponentally away from the nterface. If we have a steady state condton and an nfnte nterface all energy s reflected. If we start the rradaton, at frst some energy s used to create ths evanescent wave but as soon as steady state condtons are reached no more energy s needed for ths. If however the nterface s fnte ths evanescent wave can radate out at the edge of the nterface and some energy s needed to compensate for ths. There can also be mperfectons at the nterface makng the wave radate. There can also be dampng or losses n e.g. a metallc system whch means that energy has to be fed nto the mode. There are other nterestng effects arsng from that the mode has felds extendng outse the nterface. If the nterface s a glass-prsm ar nterface the felds extend n the ar outse the prsm. Puttng another prsm close to the frst allows the mode to "jump" across the ar gap and be emtted through the second prsm. Smlar effects are utlzed n the so called ATR-experment, used to study surface modes.
Bo E. Sernelus 3:4 MULTILAYERS Now we wll study the reflecton and transmsson between two parallel nterfaces between three meda. It could for nstance be an ant reflecton coatng on an ar-glass nterface. We have such a problem to solve n the problem solvng sesson. I want to demonstrate that the problem can be solved n dfferent ways and also show how the soluton can be extended to the case of many layers. The frst approach s a straght forward extenson of what we have done for the sngle nterface. In the frst medum we have an ncomng wave and a reflected. In the sandwched layer we also have two waves, one gong n the postve z- drecton and one n the negatve z-drecton. In the thrd medum we only have one wave travellng n the postve z-drecton. We use the same type of boundary condtons as we used n the sngle nterface geometry at both boundares and fnd the ampltudes and angles for the waves. The procedure s straght forward and can be extended to a geometry wth more than two nterfaces. However, the soluton becomes ncreasngly cumbersome the more nterfaces we have. We wll use ths method n the problem solvng sesson and wll not dwell on t here. Instead we use another method. All the waves n our ansatz, except the ncomng wave on the frst nterface, can be vewed as a superposton of waves havng been multple reflected a varyng number of tmes between the nterfaces. The ampltudes of these waves, takng the phases nto account are added together and ths leads to nterference effects We wll here follow the outlne n Surface Modes n Physcs. Bo E. Sernelus, Surface Modes n Physc, Wley-VCH, Berln 00.
Bo E. Sernelus 3:5 Fresnel's coeffcents gve the relatve ampltudes of the reflected and transmtted waves at an nterface between two meda. We have derved these. They are: n tj s sn qj cosq cosq sn q + qj ncosq nj cosqj ( ) + ( ) n n rj s sn q - qj j j - ( + j ) cosq - cosq sn q q ncosq + nj cosqj n tj p sn q j cosq cosq sn q + qj cos q qj nj cosq + ncosqj ( ) ( - ) ( q - q ) n cosq ( q + qj ) - n cosq + rj p tan j tan j ncosqj j ncosqj where the superscrpts s and p represent s-polarzed and p-polarzed waves, respectvely and the angles q and qj are the angle of ncence and angle of transmsson, respectvely. The optcal propertes of each materal enter n the form of the refractve ndex, n. These coeffcents are val also for complex valued refractve ndces. In that case the angles are not to be nterpreted as the geometrcal angles. They are complex valued, and the sne and cosne of the angles are also complex valued. The sne functons are obtaned from Snell's law njsn qj nsn q and the cosne functons are obtaned from relatons of the type: cosq - sn q For s-polarzed waves the electrc feld vector s perpendcular to the plane of ncence and for p-polarzed waves t s n the plane of ncence. The plane of ncence s the plane defned by the ncomng wave and the normal to the surface.
Bo E. Sernelus 3:6 q j q j The nterface between two meda and j, dscussed n the text.
Bo E. Sernelus 3:7 TWO INTERFACES BETWEEN THREE MEDIA We treat two nterfaces as shown n the fgure below. We need to take multple reflectons n the mdle layer nto account. 3 Waves contrbutng to the total reflected and transmtted waves n a three layer structure as dscussed n the text. The total reflected ampltude, r, s obtaned from an nfnte summaton of waves due to the multple reflectons n the mdle layer. Each tme a wave mpnges on an nterface the Fresnel equatons are used and the phases of the waves are taken nto account. All coeffcents are complexvalued. The phase d s the phase dfference for two waves: one wave that s transmtted through the frst nterface, passng through layer, s reflected at the second nterface, passng through layer agan and s fnally transmtted through the frst nterface; the second wave s one that s reflected at the frst nterface. Ths second wave has furthermore traveled a longer dstance n layer before t mpnges on the nterface.
Bo E. Sernelus 3:8 The results are obtaned as follows: e r r r + t xt x e d ; r e 3 [ + rx] ; x 3 - e r3r e d t t yt y [ e d 3 + yr3e d ; re ]; y - e r3r pd d n cos( q ) l e r t t r - r e r r + e 3 3 r3tt r r + - e r3r - e r3r r + e r3( tt -rr) r + e r 3 - e d r3r - e r 3 r r + e r 3 + e rr3 e t t t 3 + d e rr3 We have made use of the followng useful relatons: r -r tt - rr These results are val for both polarzaton drectons. The reflecton, transmsson and absorpton are [( ) ] Ï * Re n Ô 3 cosq 3 t, p - polarzaton Ô * R r ; T Re[ ( n ) Ì cosq ] ; Ô ( n Ô Re 3 cosq 3) t, s - polarzaton ÓÔ Re( n cosq) eff n sn q sn q N 0 ; real real sn q sn q eff real eff real sn q N sn q ; Re ˆ w 0 k n c N cosq [ ] A -R -T
Bo E. Sernelus 3:9 GENERAL NUMBER OF LAYERS The procedure descrbed n the precedng secton can be extended to more layers, but becomes too cumbersome f we have many layers. Here we wll ntroduce a more sutable approach. Assume that we have N layers sandwched between medum 0 and N+. So we have N+ meda. Then n general layer n wll have an ncomng and a reflected wave on the left se. We denote these wth x n and y n, respectvely. On the rght hand se there wll be an ncomng and an outgong wave as well. These are reflected and ncomng waves on layer n+. x n y n+ y n x n+ n The waves on the two ses of the layer are related to each other. Ths relaton can be expressed as Ê xn ˆ Á Ë yn Ê xn+ ˆ M n Á Ë yn+ where
Bo E. Sernelus 3:0 Ê rn-, nˆ Ê - e n 0 ˆ M n Á tn, n r Ë n, n Á - - d Ë 0 e n Ê - e n rn, ne d n ˆ - Á t - n, n rn, ne n d e n - Ë - Each nterface s shared between two neghborng layers. We have let the left most nterface belong to the layer. Ths means that the felds to the rght n the fgure are the felds nse layer n just to the left of the rght nterface. To get the felds just nse layer n+ we have to multply from the rght wth the matrx Ê Á tnn, + Ërnn, + rnn, + ˆ We have N number of layers and N+ number of nterfaces n our problem. Ths means that we n the end have to multply wth a matrx of ths knd to take care of the rght most nterface. Ê x ˆ rnn xn Á n Ê, + ˆ N y Á Ë t NN rnn Ê + ˆ M M KM KM Á, + Ë, + Ë yn + Ê xn + ˆ M Á Ë yn + Thus f we know the felds on the rght hand se of our layers we get the felds on the left hand se. Ths s not exactly what we want. We want y and x N+ as functons of x when y N+ 0. Ths s not mpossble to fnd. We have: x MxN+ + MyN+ MxN+ y MxN+ + MyN+ MxN+ and
Bo E. Sernelus 3: x t N + x M y M r x M Let us see f we reproduce our prevous result wth one layer. Then we have M Ê - e r ˆ Ê e Á - Á t Ër e e t3 Ër3 r3 ˆ Ê - + - e r + ˆ r3e r3e r e t Á - - t 3 Ër e + r 3e rr3e + e and M M and - e + rr3e tt3 - re + r3e tt3 t r tt3 - e + rr3e - re + r3e - e + rr3e e t t 3 + rr3e d r r e + 3 d + rr3e d