tailieunhanh - Behaviour of Electromagnetic Waves in Different Media and Structures Part 15

Tham khảo tài liệu 'behaviour of electromagnetic waves in different media and structures part 15', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 408 Behaviour of Electromagnetic Waves in Different Media and Structures Substituting Eq. 41 into Eq. 43 we have the following identity 44 which again implies Eq. 33 in a way that is similar to the above case for local response. 4. The equivalence of Lorentz lemma and Green function formulation So far we have shown two different mathematical formulations for discussing the optical reciprocity. Now the question is are these two statements equivalent Now we give a proof. Electrostatics First we demonstrate the equivalence between Lorentz lemma and the symmetry of the scalar Green function in electrostatics by starting with a slightly more general form of Eq. 1 with the surface terms retained . J 1 2 1 2 r 1 2 2 fjda. 45 Note that the above can be applied to the finite boundary region. To demonstrate the equivalence between Eq. 1 and Eq. 6 let US consider two unit point charge distribution as follows 1 f r 2 r r 46 and the potentials at each of their locations are then given by the scalar Green function J G r r 2 G r r . 47 Substituting Eqs. 46 and 47 into Eq. 45 leads to the following result3 G r r G r r 4 G r r G r r G r r G r r 48 3 Note that the proof of the equivalence between the two versions of the reciprocity principle in the previous section remains valid for the case with nonlocal response with Eq. 48 generalized to the following form G r r G f r j daỊ d3f G r r Lq 1G r1 r G f r f f 1G r1 r Ị Reciprocity in Nonlocal Optics and Spectroscopy 409 Here we separate into two different kinds of the boundary conditions to discuss First with the Dirichlet boundary condition given in Eq. 10 substituted into Eq. 48 we obtain Eq. 6 . Thus we have demonstrated the equivalence between the Lorentz lemma in electrostatics and the scalar Green function under the Dirichlet boundary condition. Second the Neumann boundary condition is given by Eq. 11 and thus Eq. 48 becomes the following form If If GN f r GN f r - - GN r r dfl - - GN r r da . 49 By the pervious method we .