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Resonances of open non-Hermitian systems are associated with the poles of the system scattering matrix. Perturbations of the system cause these poles to shift in the complex frequency plane. In this work, we introduce a novel method for calculating shifts in scattering matrix poles using generalized Wigner-Smith operators. We link our method to traditional cavity perturbation theory and validate its effectiveness through application to complex photonic networks. Our findings underscore the versatility of generalized Wigner-Smith operators for analyzing a broad spectrum of resonant systems and provides new insight into resonant properties of non-Hermitian systems.

Random matrix theory is a useful tool in the study of the physics of multiple scattering systems, often striking a balance between computation speed and physical rigour. Propagation of waves through thick disordered media, as arises in for example optical scattering or electron transport, typically necessitates cascading of multiple random matrices drawn from an underlying ensemble for thin media, greatly increasing computational burden. Here we propose a dual pool based bootstrapping approach to speed up statistical studies of scattering in thick random media. We examine how potential matrix reuse in a pool based approach can impact statistical estimates of population averages. Specifically, we discuss how both bias and additional variance in the sample mean estimator are introduced through bootstrapping. In the diffusive scattering regime, the extra estimator variance is shown to originate from samples in which cascaded transfer matrices are permuted matrix products. Through analysis of the combinatorics and cycle structure of permutations we quantify the resulting correlations. Proofs of several analytic formulae enumerating the frequency with which correlations of different strengths occur are derived. Extension to the ballistic regime is briefly considered., Comment: 1 figure, 3 tables, 13 pages, submitted to journal 11 Sep 23. Supplementary movies available upon request

In this work we present a method for generating random matrices describing electromagnetic scattering from disordered media containing dielectric particles with prescribed single particle scattering characteristics. Resulting scattering matrices automatically satisfy the physical constraints of unitarity, reciprocity and time reversal, whilst also incorporating the polarization properties of electromagnetic waves and scattering anisotropy. Our technique therefore enables statistical study of a variety of polarization phenomena, including depolarization rates and polarization-dependent scattering by chiral particles. In this vein, we perform numerical simulations for media containing isotropic and chiral spherical particles of different sizes for thicknesses ranging from the single to multiple scattering regime and discuss our results, drawing comparisons to established theory.

We report sensing of single nanoparticles using disordered metallic nanoisland substrates supporting surface plasmon polaritons (SPPs). Speckle patterns arising from leakage radiation of elastically scattered SPPs provides a unique fingerprint of the scattering microstructure at the sensor surface. Experimental measurements of the speckle decorrelation are presented and shown to enable detection of sorption of individual gold nanoparticles and polystyrene beads. Our approach is verified through bright-field and fluorescence imaging of particles adhering to the nanoisland substrate., Comment: Supplementary data available from journal website

Speckle patterns produced by disordered scattering systems exhibit a sensitivity to addition of individual particles which can be used for sensing applications. Using a coupled dipole model we investigate how multiple scattering can enhance field perturbations arising in such random scattering based sensors. Three distinct families of multiple scattering paths are shown to contribute and the corresponding complex enhancement factors derived. Probability distributions of individual enhancement factors over the complex plane are characterised numerically within the context of surface plasmon polariton scattering in which absorption is shown to play an important role. We show that enhancements become more strongly dependent on individual scatterer properties when absorption losses are larger, however, amplitude enhancements $\sim 10^2$, comparable to low loss surface plasmons, are achievable through sensor optimisation. Approximate analytic expressions for the complex mean enhancements are also found, which agree well with simulations when loop contributions are negligible., Comment: Supplementary Movies provided

Methods to increase the light scattered from small particles can help improve the sensitivity of many sensing techniques. Here, we investigate the role multiple scattering plays in perturbing the scattered signal when a particle is added to a random scattering environment. Three enhancement factors, parametrising the effect of different classes of multiple scattering trajectories on the field perturbation, are introduced and their mean amplitudes explored numerically in the context of surface plasmon polariton scattering. We demonstrate that there exists an optimum scatterer density at which the sensitivity enhancement is maximised, with factors on the order of $10^2$ achievable. Dependence of the enhancement factors on scatterer properties are also studied.

Small perturbations in the dielectric environment around a high quality whispering gallery mode resonator usually lead to a frequency shift of the resonator modes directly proportional to the polarizability of the perturbation. Here, we report experimental observations of strong frequency shifts that can be opposite and even exceed the contribution of the perturbations' polarizability. The mode frequencies of a lithium niobate whispering gallery mode resonator are shifted using substrates of refractive indices ranging from 1.50 to 4.22. Both blue- and red-shifts are observed, as well as an increase in mode linewidth, when substrates are moved into the evanescent field of the whispering gallery mode. We compare the experimental results to a theoretical model by Foreman et al. and provide an additional intuitive explanation based on the Goos-H\"anchen shift for the optical domain., Comment: 7 pages, 4 figures

In this work we study the scattering and transfer matrices for electric fields defined with respect to an angular spectrum of plane waves. For these matrices, we derive the constraints that are enforced by conservation of energy, reciprocity and time reversal symmetry. Notably, we examine the general case of vector fields in three dimensions and allow for evanescent field components. Moreover, we consider fields described by both continuous and discrete angular spectra, the latter being more relevant to practical applications, such as optical scattering experiments. We compare our results to better-known constraints, such as the unitarity of the scattering matrix for far-field modes, and show that previous results follow from our framework as special cases. Finally, we demonstrate our results numerically with a simple example of wave propagation at a planar glass-air interface, including the effects of total internal reflection. Our formalism makes minimal assumptions about the nature of the scattering medium and is thus applicable to a wide range of scattering problems.

Label free tracking of small bio-particles such as proteins or viruses is of great utility in the study of biological processes, however such experiments are frequently hindered by weak signal strengths and a susceptibility to scattering impurities. To overcome these problems we here propose a novel technique leveraging the enhanced sensitivity of both interferometric detection and the strong field confinement of surface plasmons. Specifically, we show that interference between the field scattered by an analyte particle and a speckle reference field, derived from random scattering of surface plasmons propagating on a rough metal film, enables particle tracking with sub-wavelength accuracy. We present the analytic framework of our technique and verify its robustness to noise through Monte Carlo simulations., Comment: Journal of Lightwave Technology

Polarimetric imaging can provide valuable information about biological samples in a wide range of applications. Detrimental scattering however currently limits the imaging depth of in-vivo imaging to approximately 1 transport mean free path. In this work, single pixel imaging is investigated as a means of reconstructing polarimetric images through scattering media. A theoretical imaging model is presented and the recovery of the spatially resolved Mueller matrix of a hidden test object is demonstrated experimentally for scattering phantoms with thicknesses up to twice the transport mean free path., Comment: 12 pages, 6 figures

In this work we establish universal ensemble independent bounds on the mean and variance of the mutual information and channel capacity for imaging through a complex medium. Both upper and lower bounds are derived and are solely dependent on the mean transmittance of the medium and the number of degrees of freedom $N$. In the asymptotic limit of large $N$, upper bounds on the channel capacity are shown to be well approximated by that of a bimodal channel with independent identically Bernoulli distributed transmission eigenvalues. Reflection based imaging modalities are also considered and permitted regions in the transmission-reflection information plane defined. Numerical examples drawn from the circular and DMPK random matrix ensembles are used to illustrate the validity of the derived bounds. Finally, although the mutual information and channel capacity are shown to be non-linear statistics of the transmission eigenvalues, the existence of central limit theorems is demonstrated and discussed.

Brillouin imaging suffers from intrinsically low signal-to-noise ratios (SNR). Such low SNRs can render common data analysis protocols unreliable, especially for SNRs below $\sim10$. In this work we exploit two denoising algorithms, namely maximum entropy reconstruction (MER) and wavelet analysis (WA), to improve the accuracy and precision in determination of Brillouin shifts and linewidth. Algorithm performance is quantified using Monte-Carlo simulations and benchmarked against the Cram\'er-Rao lower bound. Superior estimation results are demonstrated even at low SNRS ($\geq 1$). Denoising was furthermore applied to experimental Brillouin spectra of distilled water at room temperature, allowing the speed of sound in water to be extracted. Experimental and theoretical values were found to be consistent to within $\pm1\%$ at unity SNR.

Using Fisher information and the Cram\'er-Rao lower bound, we analyse fundamental precision limits in the determination of spectral parameters in inelastic optical scattering. General analytic formulae are derived which account for the instrument response functions of the dispersive element and relay optics found in practical Raman and Brillouin spectrometers. Limiting cases of dispersion and diffraction limited spectrometers, corresponding to measurement of Lorentzian and Voigt lineshapes respectively, are discussed in detail allowing optimal configurations to be identified. Effects of defocus, spherical aberration, detector pixellation and a finite detector size are also considered.

In this work fluctuations in the electric field of surface plasmon polaritons undergoing random scattering on a rough metallic surface are considered. A rigorous closed form analytic expression is derived describing second order correlations in the resulting plasmon speckle pattern assuming statistically stationary and isotropic roughness. Partially coherent planar Schell-model source fields can also be described within the developed framework. Behaviour of the three-dimensional degree of cross polarisation and spectral degree of coherence is also discussed. Expressions derived take full account of dissipation in the metal with non-universal behaviour exhibited within the correlation length of the surface and source fields.

Interactions between whispering gallery modes (WGMs) and small nanoparticles are commonly modelled by treating the particle as a point dipole scatterer. This approach is assumed to be accurate as long as the nanoparticle radius, $a$, is small compared to the WGM wavelength $\lambda$. In this article, however, we show that the large field gradients associated with the evanescent decay of a WGM causes the dipole theory to significantly underestimate the interaction strength, and hence induced WGM resonance shift, even for particles as small as $a\sim \lambda/10$. To mitigate this issue we employ a renormalized Born approximation to more accurately determine nanoparticle induced resonance shifts and hence enable improved particle sizing. The domain of validity of this approximation is investigated and supporting experimental results are presented.

We demonstrate stimulated emission depletion (STED) microscopy with 20 nm gold nanospheres coated by fluorescent silica. Compared with previous demonstrations of STED with a hybrid plasmonic fluorescent label, the current implementation offers a substantially smaller label and a better resolution improvement of up to 2.5-fold beyond the diffraction limit of confocal microscopy. This is achieved at approximately 2 times lower intensity than conventional STED based on dyes alone, and in an aqueous environment, demonstrating the relevance to bio-imaging. Finally, we also show, for the first time in this context, a 3-fold reduction in the rate of photobleaching compared to standard dye-based STED, thus, enabling brighter images.

Whispering gallery mode (WGM) resonators are an important building block for linear, nonlinear and quantum optical experiments. In such experiments, independent control of coupling rates to different modes can lead to improved conversion efficiencies and greater flexibility in generation of non-classical states based on parametric down conversion. In this work, we introduce a scheme which enables selective out-coupling of WGMs belonging to a specific polarization family, while the orthogonally polarized modes remain largely unperturbed. Our technique utilizes material birefringence in both the resonator and coupler such that a negative (positive) birefringence allows selective coupling to TE (TM) polarized WGMs. We formulate a new coupling condition suitable for describing the case where the refractive indices of the resonator and the coupler are almost the same, from which we derive the criterion for polarization-selective coupling. We experimentally demonstrate our proposed method using a lithium niobate disk resonator coupled to a lithium niobate prism, where we show a 22dB suppression of coupling to TM modes relative to TE modes.

Optical whispering gallery mode (WGM) resonators are a powerful and versatile tool used in many branches of science. Fine tuning of the central frequency and line width of individual resonances is however desirable in a number of applications including frequency conversion, optical communications and efficient light-matter coupling. To this end we present a detailed theoretical analysis of dielectric tuning of WGMs supported in axisymmetric resonators. Using the Bethe-Schwinger equation and adopting an angular spectrum field representation we study the resonance shift and mode broadening of high $Q$ WGMs when a planar dielectric substrate is brought close to the resonator. Particular focus is given to use of a uniaxial substrate with an arbitrarily aligned optic axis. Competing red and blue resonance shifts ($\sim 30$ MHz), deriving from generation of a near field material polarisation and back action from the radiation continuum respectively, are found. Anomalous resonance shifts can hence be observed depending on the substrate material, whereas mode broadening on the order of $\sim 50$ MHz can also be simply realised. Furthermore, polarisation selective coupling with extinction ratios of $> 10^4$ can be achieved when the resonator and substrate are of the same composition and their optic axes are chosen correctly. Double refraction and properties of out-coupled beams are also discussed.

Plasmonic metasurfaces enable simultaneous control of the phase, momentum, amplitude and polarisation of light and hence promise great utility in realisation of compact photonic devices. In this paper, we demonstrate a novel chip-scale device suitable for simultaneous polarisation and spectral measurements through use of six integrated plasmonic metasurfaces (IPMs), which diffract light with a given polarisation state and spectral component into well-defined spatial domains. Full calibration and characterisation of our device is presented, whereby good spectral resolution and polarisation accuracy over a wavelength range of 500-700~nm is shown. Functionality of our device in a M\"uller matrix modality is demonstrated through determination of the polarisation properties of a commercially available variable waveplate. Our proposed IPM is robust, compact and can be fabricated with a single photolithography step, promising many applications in polarisation imaging, quantum communication and quantitative sensing.

Complete determination of the polarisation state of light requires at least four distinct projective measurements of the associated Stokes vector. Stability of state reconstruction, however, hinges on the condition number $\kappa$ of the corresponding instrument matrix. Optimisation of redundant measurement frames with an arbitrary number of analysis states, $m$, is considered in this Letter in the sense of minimisation of $\kappa$. The minimum achievable $\kappa$ is analytically found and shown to be independent of $m$, except for $m=5$ where this minimum is unachievable. Distribution of the optimal analysis states over the Poincar\'e sphere is found to be described by spherical 2-designs, including the Platonic solids as special cases. Higher order polarisation properties also play a key role in nonlinear, stochastic and quantum processes. Optimal measurement schemes for nonlinear measurands of degree $t$ are hence also considered and found to correspond to spherical $2t$-designs, thereby constituting a generalisation of the concept of mutually unbiased bases.

We theoretically analyse photonic-plasmonic coupling between a high Q whispering gallery mode (WGM) resonator and a core-shell nanoparticle. Blue and red shifts of WGM resonances are shown to arise from crossing of the photonic and plasmonic modes. Level repulsion in the hybrid system is further seen to enable sensitivity enhancements in WGM sensors: maximal when the two resonators are detuned by half the plasmon linewidth. Approximate bounds are given to quantify possible enhancements. Criteria for reactive vs. resistive coupling are also established.