From: ellocomateo@free.fr   
      
   Hello,   
      
   I have a question regarding nonlinear optics, and in particular the   
   contributions from magnetic dipole (MD) and electric quadrupole (EQ) in   
   second harmonic generation (SHG).   
   I am aware, one usually places himself in the electric dipole   
   approximation, yet I have to check for higher order terms.   
      
   My trouble arises since I am no good at theory: I get lost in the   
   indexes when I handle the nablas.   
   Could someone please explain me how to work with them in the following   
   (what is their effect on the indices)?   
      
      
   I consider the nonlinear wave equation source term:   
      
   S = \mu_0 (\partial^2 P / \partial t^2)   
        + \mu_0 (\nabla x \partial M / \partial t)   
        - \mu_0 \nabla \partial^2 Q / \partial t^2)   
      
   Taking terms of the order 0 and 1 in the expansion of the   
   electromagnetic field into account, the following contributions to the   
   source term, associated to the ED, MD, and EQ, are obtained (w stands   
   for \omega):   
      
   	0th order:   
      
   ED contribution   
   P_i(2w) = \chi_{ijk}^{2,ED} E_j(w) E_k(w)   
      
   	1st order:   
      
   MD and EQ contribution at the fundamental frequency   
   P_i(2w) =   
   	\chi_{ijk}^{MD} E_j(w) B_k(w)   
   	+\chi_{ijkl}^{EQ} E_j(w) \nabla_k E_l(w)   
      
   MD contribution at the SHG frequency   
   M_i(2w) = \chi_{ijk}^{MD} E_j(w) E_k(w)   
      
   EQ contribution at the SHG frequency   
   Q_{ij}(2w) = \chi_{ijkl}^{EQ} E_k(w) E_l(w)   
      
   So far so good, I know how to get the allowed tensor components for each   
   terms, given the point group symmetry of my sample (say mm2):   
   I have a list of allowed \chi_{ijk}^{ED}, \chi_{ijk}^{MD}, \chi_{ijkl}^{EQ}.   
      
   Yet I have to consider these in my actual setup: I am using a single   
   laser beam in a simple polarizer & analyzer setup with linear polarized   
   light. As such, I am considering the /electric/ field polarization.   
   My list of allowed tensor components is reduced by considering I cannot   
   project the electric field along the wave vector direction.   
   Equivalently I cannot analyze SHG light polarized along that propagation   
   direction.   
   Eg. if k//z, all ED tensor component presenting an index z are inaccessible.   
   Because I am unfamiliar in handling nabla, I am blocked in carrying out   
   such an analysis with the MD and EQ contributions.   
      
   Please someone help me, I realize this is basic, but I am blocked, yet   
   willing to learn.   
      
   Thank you for reading this through.   
      
   Mat   
      
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