Nonlinear optics using the multipolar Hamiltonian: The Bloch-Maxwell equations and local fields

A systematic method for calculating nonlinear-optical susceptibilities in condensed phases, which incorporates intermolecular forces and spontaneous emission in a consistent way, is developed, using the multipolar ({μ}{·}D) Hamiltonian. Reduced equations of motion that couple the electromagnetic field and material variables are derived for a crystal of point dipoles. The Bloch equations in the local-field approximation, which were derived previously using macroscopic considerations, are obtained as a limiting case of the present microscopic theory. It is shown that correlations among the molecules and the radiation field are not treated rigorously in the local-field approximation, whereas they can be incorporated in a systematic way using the present formalism. An expression for the dielectric function {ε}(k,{ω}) is obtained, which is different from Hopfield's exciton-polariton model. Our result does agree, however, with Hopfield's expression, which is based on the minimal-coupling (p{·}A) Hamiltonian, in the long-wavelength and small-frequency limit (k,{ω}{→}0), provided spontaneous emission is neglected.