This is a recently developed approach to solve Maxwell’s equations [1] on the surface of nano-structures, based on the idea of principal modes. These generalise the analytical Mie modes of spherical particles to describe arbitrary smooth boundaries and surfaces.

This approach is different from more numerical techniques such as Finite-difference time-domain (FDTD), as it decomposes the responses of the nano-system into sets of distinct pairs of optical modes.

The fields at the surface can be decomposed into unique pairs of modes, where one is spatially inside the structure and the other is outside. The two modes in the pair then interact on the surface. The surface integral on the boundary of the nano-structure is used to define the relative orientation in the two sub-spaces (internal and external), $$\int a^\ast \cdot b\;\mathrm{d}s = \langle a | b \rangle = |a| |b| \cos(\xi).$$ The combinations of modes with the smallest angles, i.e. the principal modes, are the sets that are most aligned between the two spaces.

For each pair of modes, their correlation (or equivalently the principal angle \(\xi_n\)) gives information about their sensitivity to excitation by energy which couples to that pair (this is similar to phase matching in non-linear optics). This gives a geometric picture of the light interaction in that channel, and using some simple trigonometry, allows the amplitudes of the modes to be solved analytically.

On the surface of the system the tangential parts of the incident, internal and scattered light obey $$\left| f_\perp^0 + f_\perp^{internal} – f_\perp^{scattered} \right| = 0$$ for each mode pair.

An example of using this information about the modes of a system to control its response to light [2] is shown below, where a resonance is actually due to two types of mode interfering together.

Applications of these modes are to find the optical density of states for quantum optics [3], or to understand non-local and non-linear optics of nano-structures [4, 5].

## References

[Bibtex]

```
@Article{strathprints34708,
author = {Francesco Papoff and Benjamin Hourahine},
journal = {Optics Express},
title = {Geometrical Mie theory for resonances in nanoparticles of any shape},
year = {2011},
month = {October},
number = {22},
pages = {21432--21444},
volume = {19},
abstract = {We give a geometrical theory of resonances in Maxwell?s equations that generalizes the Mie formulae for spheres to all scattering channels of any dielectric or metallic particle without sharp edges. We show that the electromagnetic response of a particle is given by a set of modes of internal and scattered fields that are coupled pairwise on the surface of the particle and reveal that resonances in nanoparticles and excess noise in macroscopic cavities have the same origin. We give examples of two types of optical resonances: those in which a single pair of internal and scattered modes become strongly aligned in the sense defined in this paper, and those resulting from constructive interference of many pairs of weakly aligned modes, an effect relevant for sensing. This approach calculates resonances for every significant mode of particles, demonstrating that modes can be either bright or dark depending on the incident field. Using this extra mode information we then outline how excitation can be optimized. Finally, we apply this theory to gold particles with shapes often used in experiments, demonstrating effects including a Fano-like resonance.},
doi = {10.1364/OE.19.021432},
keywords = {metals, scattering, nanomaterials, scattering theory, Physics, Atomic and Molecular Physics, and Optics},
url = {https://strathprints.strath.ac.uk/34708/},
}
```

[Bibtex]

```
@Article{strathprints44539,
author = {Benjamin Hourahine and Francesco Papoff},
journal = {Optics Express},
title = {Optical control of scattering, absorption and lineshape in nanoparticles},
year = {2013},
month = {August},
note = {9 pages, 5 figures},
number = {17},
pages = {20322--20333},
volume = {21},
abstract = {We find exact conditions for the enhancement or suppression of internal and/or scattered fields in any smooth particle and the determination of their spatial distribution or angular momentum through the combination of simple fields. The incident fields can be generated by a single monochromatic or broad band light source, or by several sources, which may also be impurities embedded in the nanoparticle. We can design the lineshape of a particle introducing very narrow features in its spectral response.},
doi = {10.1364/OE.21.020322},
keywords = {physics, optics, optical control, scattering , absorption, lineshape, nanoparticles, Physics, Atomic and Molecular Physics, and Optics},
url = {https://strathprints.strath.ac.uk/44539/},
}
```

[Bibtex]

```
@Article{strathprints59693,
author = {Duncan McArthur and Benjamin Hourahine and Francesco Papoff},
journal = {Optics Express},
title = {Enhancing ultraviolet spontaneous emission with a designed quantum vacuum},
year = {2017},
month = {February},
number = {4},
pages = {4162--4179},
volume = {25},
abstract = {We determine how to alter the properties of the quantum vacuum at ultraviolet wavelengths to simultaneously enhance the spontaneous transition rates and the far field detection rate of quantum emitters. We find the response of several complex nanostructures in the 200 ? 400 nm range, where many organic molecules have fluorescent responses, using an analytic decomposition of the electromagnetic response in terms of continuous spectra of plane waves and discrete sets of modes. Coupling a nanorod with an aluminum substrate gives decay rates up to 2.7 {$\times$} 103 times larger than the decay rate in vacuum and enhancements of 824 for the far field emission into the entire upper semi-space and of 2.04 {$\times$} 103 for emission within a cone with a 60? semi-angle. This effect is due to both an enhancement of the field at the emitter?s position and a reshaping of the radiation patterns near mode resonances and cannot be obtained by replacing the aluminum substrate with a second nanoparticle or with a fused silica substrate. These large decay rates and far field enhancement factors will be very useful in the detection of fluorescence signals, as these resonances can be shifted by changing the dimensions of th nanorod. Moreover, these nanostructures have potential for nano-lasing because the Q factors of these resonances can reach 107.9, higher than the Q factors observed in nano-lasers.},
doi = {10.1364/OE.25.004162},
keywords = {subwavelength structures , ultraviolet, fluorescence, fluctuations, relaxations, and noise, Optics. Light, Physics and Astronomy(all)},
url = {https://strathprints.strath.ac.uk/59693/},
}
```

[Bibtex]

```
@Article{strathprints57291,
author = {D. McArthur and B. Hourahine and F. Papoff},
journal = {Optics Communications},
title = {Coherent control of plasmons in nanoparticles with nonlocal response},
year = {2017},
month = {January},
pages = {258--265},
volume = {382},
abstract = {We discuss a scheme for the coherent control of light and plasmons in nanoparticles that have nonlocal dielectric permittivity and contain nonlinear impurities or color centers. We consider particles which have a response to light that is strongly influenced by plasmons over a broad range of frequencies. Our coherent control method enables the reduction of absorption and/or suppression of scattering.},
doi = {10.1016/j.optcom.2016.07.032},
keywords = {plasmonics, nanoparticles, nonlocality, optical routing, Optics. Light, Atomic and Molecular Physics, and Optics},
url = {https://strathprints.strath.ac.uk/57291/},
}
```

[Bibtex]

```
@Article{strathprints53589,
author = {Francesco Papoff and Duncan McArthur and Ben Hourahine},
journal = {Scientific Reports},
title = {Coherent control of radiation patterns of nonlinear multiphoton processes in nanoparticles},
year = {2015},
month = {July},
pages = {12040},
volume = {5},
abstract = {We propose a scheme for the coherent control of light waves and currents in metallic nanospheres which applies independently of the nonlinear multiphoton processes at the origin of waves and currents. We derive conditions on the external control field which enable us to change the radiation pattern and suppress radiative losses or to reduce absorption, enabling the particle to behave as a perfect scatterer or as a perfect absorber. The control introduces narrow features in the response of the particles that result in high sensitivity to small variations in the local environment, including subwavelength spatial shifts.},
doi = {10.1038/srep12040},
keywords = {nanoparticles, coherent control, metallic nanospheres, Physics, Physics and Astronomy(all)},
url = {https://strathprints.strath.ac.uk/53589/},
}
```