Abstract
We derive upper bounds to free-space concentration of electromagnetic waves, mapping out the limits to maximum intensity for any spot size and optical beam-shaping device. For sub-diffraction-limited optical beams, our bounds suggest the possibility for orders-of-magnitude intensity enhancements compared to existing demonstrations, and we use inverse design to discover metasurfaces operating near these new limits. We also demonstrate that our bounds may surprisingly describe maximum concentration defined by a wide variety of metrics. Our bounds require no assumptions about symmetry, scalar waves, or weak scattering, instead relying primarily on the transformation of a quadratic program via orthogonal-projection methods. The bounds and inverse-designed structures presented here can be useful for applications from imaging to 3D printing.
Highlights
We derive upper bounds to free-space concentration of electromagnetic waves, mapping out the limits to the maximal intensity for any spot size and optical-beam-shaping device
For very small spot sizes G, which are most desirable for transformative applications, we show that the focal-point intensity must decrease proportional to G4, a dimension-independent scaling law that cannot be overcome through any form of wave-front engineering
The first nonconstant field dependence in the expansion is quadratic, and since the overlap quantities in the intensity bound are themselves quadratic in the field, the general intensity dependence on spot size always results in an a ζ04 scaling law for small spot sizes. (While this is reminiscent of the quartic dependence between the transmission through a a subwavelength hole in a conducting screen and the hole size [110], we show in Supplemental Material [99] that their physical origins are unrelated.)
Summary
Consider a beam generated by almost any means; for example, an incident wave passing through a scatterer with a complex structural profile [80,81,82], a light beam shaped by precisely controlled spatial light modulators [83,84,85,86], or a light source with a complex spatial emission profile [87,88,89]. Finding the maximal focal intensity at a single point for any desired focal-spot size reduces to determining the optimal effective currents. Equation (6) represents a first key theoretical result of our work It may have an abstract appearance, it is a decisive global bound to the optimization problem, requiring evaluation of only the known free-space dyadic Green’s function at the maximal-intensity point, the zero-field contour, and the effective-current exit surface. The smaller a desired spot size is, the closer these fields are to each other, increasing their overlap and reducing the maximal intensity possible This intuition is furthered by our considering the optimal effective currents that would achieve the bound of Eq (9), which are given by These two fields are almost identical, resulting in a significantly reduced maximal intensity
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