1969). sistive sample which was not studied under pressure. U P. Guetin and G. Schreder, to be published. 14 G. Dolling and R. A. Cowley, Proc. Phys. Soc, 12 Precise determination of the barrier height in simiLondon £8, 463 (1966). lar experimental conditions and quantitative estimation 15 S. S. Mitra, Phys. Rev. L32, 986 (1963). of the differential resistance shape will be the subject 16 M. Hass and B. N. Henvis, J. Phys. Chem. Solids of a future publication. 23, 1099 (1962). 13The P =0 curve was drawn from a slightly more re-"~^L. C. Davis, Phys. Rev. B 2, 4943 (1970).We report exact solutions of the equations of self-induced transparency which generalize the linear theory of refractive index up to theoretical peak intensities of 10 15 W cm" 2 . The refractive indices are real, and there is no dissipation from inhomogeneous broadening. We solve the dielectric-surface problem for these solutions. We note the possibility of regaining previous analytical solutions in a fashion which permits a slight extension and some critical examination of these.The semiclassical equations of motion for a single, undamped two-level atom exposed to a planepolarized electromagnetic field E of arbitrary strength can be expressed in the pseudo Bloch formWe consider a uniform dielectric consisting of n such atoms per unit volume and no host atoms. The two-level atoms couple via the Maxwell wave equation V xv xg + c~2d 2 E/dt 2 = -47rnc-2 d 2 P/dt 2 ,where c is the velocity of light in vacuo. We report exact solutions of this system of equations which contain two free parameters, namely, a frequency or reciprocal temporal length v and the field amplitude E, and which are valid on or off atomic resonance. 1 We also report corresponding rotating solutions. The solutions are distortionless in the exact sense of this, rather than in the sense of Crisp 2 or of Arrechi et al., 3 i.e.,
E(x,/) = E(x-VY), (3) and exhibit self-induced transparency (SIT) first reported by McCall and Hahn. 4 ' 5 They appear to be the natural nonlinear generalizations of linear theory and considerably extend our understanding of the SIT phenomenon. The mathematical theory bears directly on that of all the previous analytical SIT solutions. 2 " 8The notation in (1) is as follows: r= (r l9 r 2 ,r s ); r^p^+p^; r^Op^-p^); and r 3 = p S5 -p 00 . The two atomic states are labeled s (upper) and 0, and p = p(x, t) is the density operator. The atom's energy spacing is hu) s and co 3 = o> 5 . The interaction is contained inx os =x sa is tne matrix element of the dipole operator x. The atoms are inhomogeneously broadenedIn (2), P(x,/) = w^0 5j ['V(^s)r 1 (a; s ,x,/Ka; 5 ,withg(u s ) normalized to unity; u is a unit polarization vector. Since the medium is isotropic, E (x, t) = w£(x, t), and only the magnitudes E(x, t) and P(x, t) appear in the theory.A "sharp-line" solution is one for which ^(ct> s ) = S(w s -a> 0 ). An important result is that if we can find a sharp-line solution of (1) and (2) which is valid on or off resonance, we can always find an inhomo-
330
