Retarded Electromagnetic Interaction and the Origin of Non-linear Phenomena in Optics

The non-linear relation between electric polarization and electric field strength is achieved through introducing the retarded electromagnetic interactions between classical charge particles. The result agrees with the phenomenological theory in current non-linear optics, means that the non-linear phenomena in optics come from the retarded electromagnetic interaction between charged particles. The current non-linear optics describes interaction between light and materials by the half-classical method, i.e., use the classical theory of electromagnetic field to describe light’s motion and quantum theory to describe interaction between photons and material particles. The theory can deal with non-linear optics well. The Maxwell equation of electromagnetic interaction in medium is ∇× ~ E = ∂ ~ B ∂t ∇× ~ H = ~ Jf + ∂ ~ D ∂t (1) ∇ · ~ D = ρf ∇ · ~ B = 0 (2) The experiments show that for general non-iron electric medium and non-iron magnetic medium, under the condition of weak field, the relations between electric polarization ~ P and electric field strength, as well as magnetic polarization ~ M and magnetic field strength are linear, connected by the so-called constructive equations ~ D = ε ~ E = ε0 ~ E + ~ P ~ B = μ ~ H = μ0( ~ H + ~ M) (3) ~ P = ε0χe ~ E ~ M = χm ~ H (4) But the experiments also show that in the strong fields, the relations between ~ P and ~ E, as well as ~ M and ~ H are non-linear and vary complex. This kind of relations can’t be deduced from theory at present. We now get them by phenomenological hypothesis. For example, in the non-linear optics, the relation between electric polarization ~ P and electric field strength is supposed to be (1) ~ P = ε0(χ 1 e ~ E + ~ χe · ~ E ~ E + ~~ χ 3 e : ~ E ~ E ~ E · ··) (5) Here χe is polarization tensors. By means of the formula and the Maxwell equation, so much non-linear phenomena in optics can be described well. It is proved blow that after the retarded electromagnetic interaction between charged particles are considered, we can obtain Eq.(5) easy and shows that the nonlinear phenomena in optics come from the retarded electromagnetic interaction between charged particles. Let t, ~r, ~v and ~a represent retarded time, coordinate, velocity and acceleration, t, ~r, ~v and ~a represent non-retarded time, coordinate, velocity and acceleration. A particle with charge qi, velocity ~v ′ j and acceleration ~a′j at space point ~r ′ j(t ) and time t would cause retarded potentials as follows at space point ~ri(t) and time t φ′ij = qj 4πε0(1− ~v j ·~nij c )r ′ ij ~ A′ij = qj~v ′ j 4πε0(1− ~ν j ·~nij c )r ′ ij (6) In the formula ~r ij(t, t ) = ~r i(t) − ~r ′ j(t ), r ij =| ~r ′ ij |, ~nij = ~r ′ ij/r ′ ij . When the particle’s speed v << c, the retarded distance r ij(t ) at retarded time t can be replaced approximately by non-retarded distance rij(t), i.e., we can let t′ = t− r′ ij(t ′)/c → t− rij(t)/c (7) r′ ij(t, t ′) =| ~ri(t)− ~r ′ j(t ′) |=| ~ri(t)− ~r ′ j [t− r ′ ij(t ′)/c] |→| ~ri(t)− ~r ′ j [t− rij(t)/c] |= r ′ ij(t) (8) It is noted that r ij(t) 6= rij(t), for r ′ ij(t) is the approximate retarded distance, but rij(t) is not the retarded distance. In this case, we can develop retarded quantities into series in light of small quantity rij/c. By relation ~vj = d~rj/dt, we get (2) ~r′ ij(t, t) ≃ ~ri(t)− ~r ′ j(t− rij/c) = ~ri(t)− ~rj(t) + ~vj(t) c rij(t)− ~aj(t) 2c2 r ij(t) + ~̇aj(t) 6c3 r ij(t) + · · · = ~rij(t) + ~vj(t) c rij(t)− ~aj(t) 2c3 r ij(t) + ~̇aj(t) 6c3 r ij(t) + · · · (9) ~vj(t ′) ≃ ~vj(t)− ~aj(t) c rij(t) + ~̇aj(t) 2c3 r ij(t) + · · · ~a ′ j(t ′) ≃ ~aj(t)− ~̇aj(t) c rij(t) + · · · (10) Mediums are composed of atoms and molecules, and atoms and molecules can be regarded as electrical dipoles composed of two particles with charges±q. Suppose the distance between two charges is rij(t) at time t, the electrical dipole moment is ~ Pj . The direction of ~ Pj = qj~rij is from the point the j -particle located pointing to the mass center point ~ri of electric dipole. By considering the retarded effect of interaction, according to Eq.(9), the retarded electrical dipole moment becomes approximately ~ P ′ j = qj~r ′ ij = qj(~rij + ~νj c rij − ~aj 2c2 r ij + ~̇aj 6c3 r ij) (11) ~vj(t)~aj(t) and ~̇aj(t) are the -particle’s velocity, acceleration and acceleration of acceleration individually. If there is only one electrical dipole in space, the directions of ~rij , ~vj , ~aj and ~̇aj are on the same straight line. In this case, the subscript can be removed and Eq.(11) can be written as P ′ = q(r + v c r − a 2c2 r + ȧ 6c3 r + · · ·) (12) When there exists the action of external electric field E = E0 sin(ωt+ θ ′ 0), the motion equation of electrical dipole is dr dt2 + 2β dr dt + ω 0r = qE0 m sin(ωt− θ′ 0) (13) The second item on the right side is damping force (radiation damping force and damping force caused by collisions between particles). When ω2 > β, the solution of Eq.(13) is (3) r = A sin(ωt− θ0) +Be −βt sin( √ ω2 0 − β 2t− δ) (14)