# Solving electromagnetic scattering problems with scuff-scatter

scuff-scatter is a tool within the scuff-em code suite for solving classical scattering problems involving user-specified incident fields impinging on a material geometry.

To run a scattering calculation using scuff-scatter, you will

• Create a scuff-em geometry file describing the shapes and material properties of the scattering objects in your geometry

• Choose the incident field that will scatter off your objects: a plane wave, a gaussian beam, a point dipole source, or some combination thereof

• Run scuff-scatter with command-line options specifying the geometry, the frequencies, the incident field, and the type of output you wish to get back.

The various output quantities that you can ask scuff-scatter to generate include the following:

• The components of the scattered and total electric and magnetic fields at arbitrary user-specified points away from scattering surfaces. (The points may lie inside or outside the scattering objects).

• The components of the total electric and magnetic fields on the scattering surfaces. (These quantities may alternatively be interpreted as effective surface currents and charges that give rise to the scattered fields.)

• The electric and magnetic dipole moments induced by the incident field on the scattering objects. (These are obtained from the interpretation of the tangential fields as effective sources that radiate the scattered fields.)

• The total power scattered by, and the total power absorbed by, the scattering objects from the incident field.

• The total force and/or torque exerted on the scattering objects by the incident fields (radiation pressure).

• Visualization files plotting the electric and magnetic surface currents, and the associated charge densities, induced by the incident fields on the scattering objects.

• Visualization files plotting field components and Poynting fluxes on arbitrary user-specified surface meshes.

For more sophisticated users, scuff-scatter also offers an advanced mode of operation that exposes---at the command-line level---some of the key efficiencies of the surface-integral-equation formulation implemented by the scuff-em core library. This offers significant speedup for certain types of computations, at the expense of a slightly more effort required to set up your calculation.

# 1. scuff-scatter command-line options

### Common options

scuff-scatter recognizes the following subset of the list of commonly accepted options to scuff-em command-line codes.

--geometry
--EPFile
--Omega
--OmegaFile
--Cache
--WriteCache


## Options defining the scattering problem

--geometry MyGeometry.scuffgeo


Specifies the geometry input file.

--Omega      3.1415
--OmegaFile  MyOmegaFile
--Lambda     0.5
--LambdaFile MyLambdaFile


Specifies the angular frequencies at which to run calculations. (Angular frequencies are interpreted in units of $c/1\,\mu\text{m}=3\cdot 10^{14}$ rad/sec.) The --Omega option may be used more than once to specify multiple frequencies. Alternatively, the --OmegaFile option may be used to specify the name of a file containing a list of frequencies (one per line) at which to run calculations.

The options --Lambda and --LambdaFile may alternatively be used to define the frequencies at which to run calculations in terms of the corresponding free-space wavelength $\lambda=\frac{2\pi}{\omega}$, interpreted in units of microns. Thus --Omega 3.1415 and --Lambda 0.5 are equivalent; both specify an angular frequency $\omega=\pi \cdot 3\cdot 10^{14}$rad/sec, corresponding to a free-space wavelength of $\lambda=0.5\,\mu$m.

Note: Even if you use --Lambda / --LambdaFile instead of --Omega / --OmegaFile to specify the computational frequencies, the output files will still report data in terms of the frequency (the equivalent --Omega values), not the wavelength. To plot your data versus wavelength instead of frequency, just plot versus the quantity $\frac{2\pi}{\omega}$.

## Options defining the incident field

The options for specifying incident fields in scuff-em are described in detail on the page Incident fields in scuff-em; here we just list the available options without commentary.

--pwDirection    nx ny nz
--pwPolarization Ex Ey Ez

--psStrength Px Py Pz
--psLocation xx yy zz

--gbDirection nx ny nz
--gbPolarization Ex Ey Ez
--gbCenter Cx Cy Cz
--gbWaist W


(As in scuff-scatter, these options may occur multiple times to define superpositions of multiple types of incident field.)

## Options requesting scattered and total fields

 --EPFile MyEPFile


Specifies a list of evaluation points at which to compute and report components of the scattered and total fields. This option may be specified more than once to define multiple sets of field evaluation points.

## Options requesting power, force, and torque data

 --PFTFile     MyGeometry.PFT
--EMTPFTFile  MyGeometry.EMTPFT
--OPFTFile    MyGeometry.OPFT
--DSIPFTFile  MyGeometry.DSIPFT


Each of these options requests that power, force, and torque (PFT) data be written to a file of the specified name. The resulting files all have the same file format---reporting absorbed and scattered power, force (radiation pressure), and torque for all objects in the geometry at all frequencies you requested (see the file header for details)---but differ in the algorithm used to compute the force:

• The "energy-momentum-transfer" PFT (EMTPFT) method computes powers, forces and torques by considering the Joule heating of, and Lorentz force on, the surface currents in the presence of the total fields. (This is the default, so the --PFTFile option is synonymous with --EMTPFTFile.)

• The "displaced-surface-integral" PFT (DSIPFT) method computes PFTs by integrating the Poynting vector and Maxwell stress tensor over a bounding surface surrounding the body.

• The "overlap" PFT (OPFT) method computes PFTs directly from the surface currents by exploiting the relationship between the surface currents and the total electric and magnetic fields at body surfaces.

For some types of calculation it is possible to achieve significant computational accelerations by taking advantage of certain efficiencies inherent in the particular mathematical strategy used by the scuff-em core library to solve Maxwell's equations---namely, the discretized surface-integral-equation (SIE) formulation.

You can read about all the gory details of SIE solvers here, but for the purposes of this discussion all you really need to know is this: For a given material geometry irradiated by a given incident field at a given frequency, scuff-em assembles and solves a linear system of the form where

• c represents the unknown surface currents for which we are solving,

• the RHS vector f depends on the geometry, the frequency, and the incident field,

• the matrix M depends on the scattering geometry and the frequency but not on the incident field. More specifically, for a scattering geometry consisting of N objects (or N surfaces in a regions-and-surfaces geometry specification), the matrix M has an N×N block structure in which the (m,n) block describes the interactions of object m with object n.

Armed with just this much knowledge, we can understand the two key efficiencies possible in SIE scattering calculations:

• (1) First, suppose that, in a geometry consisting of 2 or more bodies, we would like to perform calculations for various different relative geometric configurations of the bodies---for example, different separation distances or rotation angles between bodies---at the same frequency. The diagonal blocks of the M matrix, which represent the self-interactions of objects and are the most costly blocks to compute, are independent of the relative configuration of the various objects in the geometry, and thus need only be computed once for a given geometry at a given frequency, after which they may be reused for any number of calculations involving rearrangements of the relative positions and orientations of the bodies.

Thus, if we are interested in running calculations for a sphere-cube geometry at (say) 7 different values of the surface-surface separation, it greatly behooves us to assemble the diagonal (self-interaction) blocks just once per frequency, then reuse these blocks for each of the 7 separation distances. The sphere-cube interaction block of the matrix must be recomputed at each separation distance, but this is relatively cheap compared to the cost of computing the sphere-sphere and cube-cube self-interaction blocks.

• (2) In the equation above, the LHS is independent of the incident field. This means that, once we have assembled and LU-factorized the M matrix for given geometry at a given frequency (a procedure which scales asymptotically like $\sim T^3$ with $T$ the total number of triangles in our surface meshes) we can solve scattering problems for any number of incident fields with cost $\sim T^2$ per incident field---that is, essentially for free compared to the cost of assembling and factorizing the matrix.

Thus, if we are interested in observing the scattering properties of our geometry under irradiation by 7 different types of incident field (say, plane waves originating from 7 different angles) it greatly behooves us to form and LU-factorize the M matrix just once for this frequency, then reuse the factorized matrix to solve the linear system above for the 7 different types of incident field.

To take advantage of efficiency (a), scuff-scatter supports the command-line option

  --transfile MyTransFile


where MyTransFile is a list of geometrical transformations.

To take advantage of efficiency (b), scuff-scatter supports the command-line option

  --IFFile    MyIFFile


where MyIFFile is a list of incident fields.