We present an optimization framework based on Lagrange duality and the scattering $mathbb{T}$ operator of electromagnetism to construct limits on the possible features that may be imparted to a collection of output fields from a collection of input fields, i.e., constraints on achievable optical transformations and the characteristics of structured materials as communication channels. Implications of these bounds on the performance of representative optical devices having multi-wavelength or multiport functionalities are examined in the context of electromagnetic shielding, focusing, near-field resolution, and linear computing.
The inverse problem in optics, which is closely related to the classical question of the resolving power, is reconsidered as a communication channel problem. The main result is the evaluation of the maximum number $M_epsilon$ of $epsilon$-distinguishable messages ($epsilon$ being a bound on the noise of the image) which can be conveyed back from the image to reconstruct the object. We study the case of coherent illumination. By using the concept of Kolmogorovs $epsilon$-capacity, we obtain: $M_epsilon ~ 2^{S log(1/epsilon)} to infty$ as $epsilon to 0$, where S is the Shannon number. Moreover, we show that the $epsilon$-capacity in inverse optical imaging is nearly equal to the amount of information on the object which is contained in the image. We thus compare the results obtained through the classical information theory, which is based on the probability theory, with those derived from a form of topological information theory, based on Kolmogorovs $epsilon$-entropy and $epsilon$-capacity, which are concepts related to the evaluation of the massiveness of compact sets.
We present a method based on the scattering $mathbb{T}$ operator, and conservation of net real and reactive power, to provide physical bounds on any electromagnetic design objective that can be framed as a net radiative emission, scattering or absorption process. Application of this approach to planewave scattering from an arbitrarily shaped, compact body of homogeneous electric susceptibility $chi$ is found to predictively quantify and differentiate the relative performance of dielectric and metallic materials across all optical length scales. When the size of a device is restricted to be much smaller than the wavelength (a subwavelength cavity, antenna, nanoparticle, etc.), the maximum cross section enhancement that may be achieved via material structuring is found to be much weaker than prior predictions: the response of strong metals ($mathrm{Re}[chi] < 0$) exhibits a diluted (homogenized) effective medium scaling $propto |chi| / mathrm{Im}[chi]$; below a threshold size inversely proportional to the index of refraction (consistent with the half-wavelength resonance condition), the maximum cross section enhancement possible with dielectrics ($mathrm{Re}[chi] > 0$) shows the same material dependence as Rayleigh scattering. In the limit of a bounding volume much larger than the wavelength in all dimensions, achievable scattering interactions asymptote to the geometric area, as predicted by ray optics. For representative metal and dielectric materials, geometries capable of scattering power from an incident plane wave within an order of magnitude (typically a factor of two) of the bound are discovered by inverse design. The basis of the method rests entirely on scattering theory, and can thus likely be applied to acoustics, quantum mechanics, and other wave physics.
We show how the central equality of scattering theory, the definition of the $mathbb{T}$ operator, can be used to generate hierarchies of mean-field constraints that act as natural complements to the standard electromagnetic design problem of optimizing some objective with respect to structural degrees of freedom. Proof-of-concept application to the problem of maximizing radiative Purcell enhancement for a dipolar current source in the vicinity of a structured medium, an effect central to many sensing and quantum technologies, yields performance bounds that are frequently more than an order of magnitude tighter than all current frameworks, highlighting the irreality of these models in the presence of differing domain and field-localization length scales. Closely related to domain decomposition and multi-grid methods, similar constructions are possible in any branch of wave physics, paving the way for systematic evaluations of fundamental limits beyond electromagnetism.
Optical metasurfaces have been heralded as the platform to integrate multiple functionalities in a compact form-factor, potentially replacing bulky components. A central stepping stone towards realizing this promise is the demonstration of multifunctionality under several constraints (e.g. at multiple incident wavelengths and/or angles) in a single device -- an achievement being hampered by design limitations inherent to single-layer planar geometries. Here, we propose a general framework for the inverse design of volumetric 3D metaoptics via topology optimization, showing that even few-wavelength thick devices can achieve high-efficiency multifunctionality. We embody our framework in multiple closely-spaced patterned layers of a low-index polymer. We experimentally demonstrate our approach with an inverse-designed 3d-printed light concentrator working at five different non-paraxial angles of incidence. Our framework paves the way towards realizing multifunctional ultra-compact 3D nanophotonic devices.
The polarization of light is utilized in many technologies throughout science and engineering. The ability to transform one state of polarization to another is a key enabling technology. Common polarization transformers are simple polarizers and polarization rotators. Simple polarizers change the intensity depending on the input state and can only output a fixed polarized state, while polarization rotators rotates the input Stokes vector in the 3D Stokes space. We demonstrate an all-optical input-agnostic polarization transformer (AI-APT), which transforms all input states of polarization to a particular state that can be polarized or partially polarized. The output state of polarization and intensity depends solely on setup parameters, and not on the input state, thereby the AI-APT functions differently from simple polarizers and polarization rotators. The AI-APT is completely passive, and thus can be used as a polarization controller or stabilizer for single photons and ultrafast pulses. The AI-APT may open a new frontier of partially polarized ultrafast optics.