Photonic Scattering
Description of an ensemble of spheres on the basis of amplitude scattering matrix theory
Scattering by an aggregate of spheres generally depends on the direction of the incoming light, the sizes, shape and composition of the spheres and the spatial order of the spheres in the aggregate.
Light scattering by a collection of small particles can be described on the basis of Mie theory.
The scattering matrix describes the linear relation between the incident and the scattered far-field components that are parallel and perpendicular to the scattering plane defined by the
direction of propagation of the plane incident wave and the scattering direction,
z is the propagation direction, k is the wavevector of the incident wave. Si are the scattering matrix elements. The scattering matrix contains all information necessary for detection and characterization of scatterers. The results thus can directly be compared to experiments.
The GMM can be extended to the general case of an ensemble of variously shaped particles or to a core-shell system. Furthermore, it allows the individual positioning of spheres which is of importance when comparing ordered with disordered structures.
We investigate the scattering signal of a probe consisting of polymer opal spheres that are arranged in a grid that is very close to the fcc structure (Fig.1).
Generally the optical properties of such a structure are affected by the Bragg resonance condition for the spatial grid. In addition, the spatial order / disorder and the shape / composition of the individual spheres play an important role for the leading to angle and wavelength dependent scattering signals.
Typically the core particles of a core-shell sphere polymer opal sphere ensemble organize in a structure that is very close to an fcc lattice.
The optimization of structure and geometry using Mie theory and corresponding simulations will be of importance for structural colour applications. They allow the tuning of the optical response by varying the geometry and composition. Geometrical boundaries include the size of the sphere which directly affects the lattice constant. The composition of the aggregate can additionally be tuned by adding nano-particles (e.g. carbon) which localize in the interstices of the spatial grid and may narrow the spatial and spectral appearance of the optical signal.
In typical structures the grid constant is e.g.300 nm and the radius r = 100 nm. Technologically polymer opal aggregates may consist of core-shell spheres (Fig.2).
We simulate a system with a size of 100x100x200 lattice constants (200 refers to the propagation direction of incoming radiation). The basis structure is fcc (in 111 direction).
We vary the angle of rotation and calculate the scattering of the polymer opal structure.
Simulation allows to vary and investigate the following aspects:
-Variation of grid constant of polymer opal structure
-Variation of wavelength of incoming radiation.
-Variation of influence of disorder (simulated by variable positioning of the spheres)
Example 1: scattering at a disordered sphere system: randomly positioned points (filling degree 40%)
Example 2: scattering at a grid which is close to fcc (small shift in each sphere position, left) and “ideal” fcc grating (right).
Example3: typical spectrum (width and shape depend on grid constant and nanoparticle concentration)
Main results:
- typical simulation sizes of 100x100x200 (in propagation direction 200 spheres) are sufficient to include the most important properties
- changes in signal difference in simulation of noise: random positioning of spheres leads to stronger disorder than a model using fcc as a start structure and shifting the positions
- scattering signals show typical dependencies on wavelength and angle of incidence
- simulated scattering predicts experimental results and allows optimization of geometry, disorder and material for a design of colour and angular width of scattered light.
