The Fresnel reflection and transmission amplitude fac- e〓430 e·(4t tors, with the 12 and 23 subscripts for the glass-metal and metal-air boundaries, respectively, are given by /2 cos 81+n cose2 e1/2 12/2 cos e, +n cos e 2 and (9c) Fig. 2. Polar plot of the complex Fresnel reflection factor, r23,at the cos02+E1/2cos03 silver-air interface for angles of incidence around the plasmon angle The amplitude scale is logarithmic The angles B2 and ]3 in the metal and air, respectively, linear reflectivity can be observed only for the case of are defined by nonzero absorption given by E2+0. If E2=0, the reflec cos62=(1-n2sin21/e)/2 (10a) gles of incidence including the surface plasmon resonance however, in nonlinear optics the surface plasmon reso- nance may still be observed through enhanced harmonic generation. The plasmon angle p is still defined by the cosB3=(1-n2sin201)1/ real part of Eq.(13), but r23 now assumes a large nega (10b) tive imaginary value. The amplitude and phase of rain the complex plane as a function of the angle of incidence The complex nature of these angles reflects the real expo- is plotted in Fig. 2 with a logarithmic amplitude scale.At B we now solve for E and find the ratio of the reflected we have r23 =1, but at the plasmon angle ra=-i2E1/E2 nential decay of the fields in the metal and air media optical power to the incident optical power to be given by and beyond this angle r23 rapidly approaches=-1.The amplitude and phase of r23 are a sensitive function of the r12+ regexp 1+y:2723exp( (11) The critical factor in the above formula is 23, which may be rewritten as 23sGn2sin2a1-e)1/2-t(n2sin201-1)/2 (n sin20, -E)/2+E(n ideal free electron plasma, the denominator of r23 vanishes at the plasmon angle 8p given by nsin,=Le/(E+1) which is also the condition for the surface plasmon mode given by Eq. (1). The divergence of r23 at the surface plasmon resonance was recognized by Cardonaand can be described as the ratio of a finite-amplitude reflected field to a zero-amplitude incident field at the metal-air in ect interpretation, without any diver gent factors, is to recognize directly from Eq .( 8a)that yhen 81=p then E2=0 and the only field in the metal has a simple exponential spatial decay from the metal-air boundary to the metal-glass boundary. This description FILM THICKNESS d (A consistent with the surface plasmon mode discussed viously and the surface polariton modes of Agarwal ig. 3. Normalized amplitude of the surface plasmon mode electric field The effect of the surface plasmon resonance on the evaluated at the silver-air interface versus film thickness 632/Am,J.Phys,ol,43,Mo.7,J1975 Timon Mitchell and Watson