Answer Happy

Question 2: (Surface plasmons): [85] N Medium 1: vaccum Surface wave Medium 2: (0) Fig. 3. Surface waves (plasmons) at vacuum interface of dielectric material At the interface z = 0 between two dielectric materials we can observe the occurrence of transverse surface waves called surface plasmons. Let us assume that medium 1 for 2 > 0 is vacuum, and that medium 2 for z <0 is a bulk material with dielectric constant (w). Let us also assume that the surface wave is travelling in the -z plane (TM mode) with a phase factor exp(i(k • - wt)).
[14] ai (a) Use the Maxwell equations: üxË = 0 and 7 x B = e(w)€oMo olle to show that the square of the absolute values of the wave vector k in medium I and medium 2 are as follows: k (1) and k?(2) = E(W) (6) Next we want to study transverse magnetic (TM) surface waves with field components (Ex, By, E,). [28] The proper boundary conditions (i.c. continuity of the tangential parts of Ë,K,B) at the interface z = 0 are: Ex(1) = Ex(2); kx(1) = kx(2); B,(1) = B,(2); Show that: (10) 1 (9) (i) k (1) k (2) 02/02 w2(W)/c2 wa (ii) k (1) CE(0))+1 wa (0) (iii) k (2) C (0)+1 w2 EW) (iv) kf(1) = k (2) c? E(0)+1 (5) -
Question 2: (Surface plasmons): 85] Medium 1: vaccum х Surface wave Medium 2: (0) Fig. 3. Surface waves (plasmons) ar vacuum interface of dielectric material At the interface 2 = 0 between two dielectric materials we can observe the occurrence of transverse surface waves called surface plasmons. Let us assume that medium 1 for 2 > 0 is vacuum, and that medium 2 for z < 0 is a bulk material with dielectric constant (w). Let us also assume that the surface wave is travelling in the x – z plane (TM mode) with a phase factor exp(iſk - * - wt)). [14) ExĒ= де (a) Use the Maxwell equations: and 8 x B = e(W)£M* to show that the square of the absolute values of the wave vector k in medium I and medium 2 are as follows: k2(1) and k®(2) (W) (b) Next we want to study transverse magnetic (TM) surface waves with field components (Ex.B.E.). [28] The proper boundary conditions (i.e. continuity of the tangential parts of Ē, k, B) at the interface z = 0 are: E (1) = E(2): kx(1) = k (2): B,(1) = B, (2): Show that: (10) 3. (1) w2/02 (ii) k (1) k (2) ()/ (9) (ii) k (2) == (a) + 1 ()+1 032 (0) (5) (iv) k (1) uk (2) wa ) (0)+1
(c) A propagating surface wave occurs when ky is real and k (1) and k_(2) are purely imaginary. Give [7] a criteria for e(w), which corresponds to this scenario, and state the corresponding wave function(s). [36] (d) Let medium 2 be a model solid containing a plasma of free electrons with e(w) = 1 (8) (1) Show that the dispersion relation of the corresponding surface plasmons is: w”(kx) = 1/2 [0% + 20° k$ + 4c*ki] (ii) Discuss the asymptotic behaviour of wt(kx) for ky → 0 and ky + (Hint: Use V1+xz1+ ***2 + ...). Sketch w+(kx). - 00 ( (24) (4) (iii) Why is w+ (kx) not a surface wave as discussed above?