Friedrich-Alexander-Universität Erlangen-Nürnberg

Molecular Mie Model for Second Harmonic Generation at The Surface of Spherical Nanoparticles

Molecular Mie Model for Second Harmonic Generation at The Surface of Spherical Nanoparticles

Optical Second Harmonic generation is a widely used tool to study the surface of colloidal spherical nanoparticles because Second Harmonic generation is intrinsically surface sensitive, and can be applied in situ and gives real time information about surface properties and processes.

Fig. 1: Measurement geometry and typical scattering profile IFH,SH with polarization of fundamental and second harmonic either parallel (p) or perpendicular (s) to the scattering plane. The position of the first maximum in pp and sp polarization and the ratio of maximum intensities Ipp/Isp are useful experimental quantities.

Experiments are performed within the SAOT:


Our research is aimed at the simulation of Second Harmonic Generation and the evaluation of respective experimental data. A numerical method based on linear Mie scattering was developed and compared to experimental results to determine surface properties.


The steps of the developed nonlinear Mie model are:

1.    The electric field of a plane wave scattered at the particle is calculated using Mie theory. Mie theory relies on spherical symmetry and all fields are expanded into vector spherical harmonics, thus reducing the boundary conditions at the surface of the particle to a set of simple linear equations

2.    The nonlinear polarization P(2) at the surface of the particle is determined from the second harmonic surface susceptibility tensor χ(2)xyz and the electric field vectors of the FH.

3.    The surface polarization, which is identical to a dipole density, is modeled as an ensemble of elementary electric dipoles placed at the positions where P(2) is nonzero. An electric dipole placed at the origin is given by the first order vector spherical harmonic. Using addition theorems for translation and rotation this dipole can be placed at any position near the surface of the sphere.

4.    The dipoles radiate at second harmonic frequency and their fields are also scattered by the sphere. The total second harmonic emission is the coherent sum of all dipole fields.

This theory offers flexibility to model systems, where χ(2)xyz is not limited to a thin homogeneous layer at the surface of the sphere, for example brush-particles. It is also possible to study plasmonic core shell particles with a dielectric core with a metallic shell.


Fig. 2: Simulation steps of the nonlinear Mie model (|E|²). a) Scattered fundamental harmonic field close to the surface of the particle. b) Dipole placed at the surface of the particle. c) Dipole scattered by the sphere. d) Dipole rotated. e) Coherent sum of all dipole fields.
Fig. 3: Position of the first maximum in sp and pp polarization and ratio of maximum intensities Ipp/Isp for varying ratio of the three nonzero second harmonic susceptibilities χ(2)zzz, χ(2)zxx and χ(2)xxz.


  1. Wunderlich, S.; Schürer, B.; Sauerbeck, C.; Peukert, W.; Peschel, U. (2011): Molecular Mie model for second harmonic generation and sum frequency generation. In: Phys. Rev. B 84 (23), S. 235403.
  2. Schürer, B.; Wunderlich, S.; Sauerbeck, C.; Peschel, U.; Peukert, W. (2010): Probing colloidal interfaces by angle-resolved second harmonic light scattering. In: Phys. Rev. B 82 (24), S. 241404.
  3. Schürer, B.; Hoffmann, M.; Wunderlich, S.; Harnau, L.; Peschel, U.; Ballauff, M.; Peukert, W. (2011): Second harmonic light scattering from spherical polyelectrolyte brushes. In: J. Phys. Chem. C 115 (37), S. 18302–18309.


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Optical Metrology
Optical Material Processing
Optics in Medicine
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and Computational Optics.