to be determined For surface waves we are interested in the conditions under which there are no propagating waves in either the metal or air. We assume a and k have values such that in the metal the solution of the wave equation can be written GLAss n(w) k2=-i(k2-u2/c2 (3a) and in air k2=+i(k2-u2/c212 The signs are chosen so that the electric fields in the Fig. I. Attenuated total reflection geometry for a thin metal film be- metal and dielectric describe exponential decay normal to tween a glass prism and air the boundary metals this condition is easily satisfied in the visible region of the spectrum where o< op; in air this condition is satisfied only for evanescent waves in to- EreftectedE ,expli(w/c)n(x sin,+ 2 cose,)].(6b) tal internal reflection The boundary conditions at the plane z =0 are the standard ones: continuity of the tangential component of In the thin metal film we. write the total electric field as a E and H and of the normal component of d and B. We standing wave superposition of two exponentially damped assume the magnetic permeability is equal to unity, and B aves is equal to H. The two transverse modes can be classified by their polarizations, namely, with electric field E para met al=E expli(w/ c )ux sine,exp(z) lel or perpendicular to the plane of incidence defined by k and the z axis. For the case in which Er=Ex=0 Er'expli(w/c)nx sine, exp(-kz).(6c) -polarization, no plasma mode solution is found. The case of interest is E,=0, p-polarization. Now, using con- The transmitted wave in the vacuum is assumed evans tinuity of the tangential component of E and transversality cent of the fields, we have kE|时=k2E,|0- Transmitted=E] expli(w/c)nx sine, exp[(u/lc)n2sin201-1)4/2z](6d) above with continuity of the normal com ponent of D gives Here k is the absorption coefficient at nonnormal inci dence, which by comparison with Eqs.(2)and(3a)is written for the geometry of Fig. 1 as k=-i(w/c e-n'sin0,) Substitution from Eqs.(3a)and(3b)allows the aboy where e is the complex dielectric coet as before, n is the index of refraction glas condition to be rewritten as the dispersion relation given is the angle of incidence. The x dependence of m waves follows from Snell,'s law SURFACE PLASMON EXCITATION known amplitudes E1, Es, E2, and e be related to the incident amplitude e1' through the bound In this section we relate the amplitude of the surface ary conditions. Continuity of the tangential componen plasmon mode to the exciting incident radiation. An elec- of E and h at the i=0 and z=-d boundaries gives metal film at the hypotenuse face of the prism and at an of the fields in the metal at z=010 find the amplitude tromagnetic wave in a glass prism is incident on a thin the required four angle of incidence greater than the critical angle for total internal reflection. The attenuated total reflection geometry is shown in Fig. l. We assume the electric field vectors are p-polarized and described by monochromatic E=E1+71p2x(-2hD万 plane waves. In the glass medium we have an incident and reflected propagating wave of the form d EincidentEr' expli(w/c)n(r sine,-z cos0,))(6a) E,,Et t12Y23 exp(-2kd J. Phys. Vol. 43, No. 7, July 1975 Simon, Mitchell, and Watson/631