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Perturbing scattering resonances in non-Hermitian systems: a generalized Wigner-Smith operator formulation
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Abstract
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.
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We introduce the refractiveindex.info database, a comprehensive open-source repository containing optical constants for a wide array of materials, and describe in detail the underlying dataset. This collection, derived from a meticulous compilation of data sourced from peer-reviewed publications, manufacturers’ datasheets, and authoritative texts, aims to advance research in optics and photonics. The data is stored using a YAML-based format, ensuring integrity, consistency, and ease of access. Each record is accompanied by detailed metadata, facilitating a comprehensive understanding and efficient utilization of the data. In this descriptor, we outline the data curation protocols and the file format used for data records, and briefly demonstrate how the data can be organized in a user-friendly fashion akin to the books in a traditional library.
We present a graph-based model for multiple scattering of light in integrated lithium niobate on insulator (LNOI) networks, which describes an open network of single-mode integrated waveguides with tunable scattering at the network nodes. We first validate the model at small scale with experimental LNOI resonator devices and show consistent agreement between simulated and measured spectral data. Then, the model is used to demonstrate a novel platform for on-chip multiple scattering in large-scale optical networks up to few hundred nodes, with tunable scattering behaviour and tailored disorder. Combining our simple graph-based model with material properties of LNOI, this platform creates new opportunities to control randomness in large optical networks.
Resonant response of water waves in a narrow gap, the so-called gap resonance, is a hydrodynamic phenomenon with practical applications. When coupled to body motions, the physics becomes rather complicated, involving body motions, fluid resonance and damping that determines the response amplitude. Gap resonances between two identical elongated bodies, with one held fixed and the other floating, are investigated experimentally for unidirectional waves with broadside incidence. Transient wave groups with different maximum surface elevations are generated, to examine the nonlinear physics. Sub- and super-harmonics, which can be substantially larger than the linear component, are observed in the responses. When going from the diffraction problem (both bodies fixed) to the coupled problem (allowing one body to move), the gap responses are significantly amplified, i.e. by a factor of 2. The first resonant mode that dominates the diffraction problem disappears in the coupled problem. A new resonant mode, which is shown to arise from body motions, is excited inside the gap through nonlinear processes. This mode features a double-humped ‘camel-back’ shape, leading to remarkably large responses. The floating body is observed to oscillate in sway with significant amplitudes at two distinct natural frequencies that are far apart. This is of great interest for the design of mooring lines connecting the two bodies, as it appears to cause resonances at the wave frequencies in addition to the low frequencies. A semi-analytical model is developed to investigate the multiple natural frequencies in sway, yielding further insight into gap resonances.
A review is given on the foundations and applications of non-Hermitian classical and quantum physics. First, key theorems and central concepts in non-Hermitian linear algebra, including Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity, and parity-time symmetry, are delineated in a pedagogical and mathematically coherent manner. Building on these, we provide an overview of how diverse classical systems, ranging from photonics, mechanics, electrical circuits, and acoustics to active matter, can be used to simulate non-Hermitian wave physics. In particular, we discuss rich and unique phenomena found therein, such as unidirectional invisibility, enhanced sensitivity, topological energy transfer, coherent perfect absorption, single-mode lasing, and robust biological transport. We then explain in detail how non-Hermitian operators emerge as an effective description of open quantum systems on the basis of the Feshbach projection approach and the quantum trajectory approach. We discuss their applications to physical systems relevant to a variety of fields, including atomic, molecular and optical physics, mesoscopic physics, and nuclear physics with emphasis on prominent phenomena and subjects in quantum regimes, such as quantum resonances, superradiance, the continuous quantum Zeno effect, quantum critical phenomena, Dirac spectra in quantum chromodynamics, and nonunitary conformal field theories. Finally, we introduce the notion of band topology in complex spectra of non-Hermitian systems and present their classifications by providing the proof, first given by this review in a complete manner, as well as a number of instructive examples. Other topics related to non-Hermitian physics, including nonreciprocal transport, speed limits, nonunitary quantum walk, are also reviewed.
Small perturbations in the dielectric environment around resonant dielectric structures 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 oppose and even exceed the contribution of the perturbations’ polarizability. We show in particular how the mode frequencies of a lithium niobate whispering-gallery-mode resonator are shifted by planar substrates—of refractive indices ranging from 1.50 to 4.22—contacting the resonator rim. Both blue- and redshifts 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. [J. Opt. Soc. Am. B 33, 2177 (2016)JOBPDE0740-322410.1364/JOSAB.33.002177] and provide an additional intuitive explanation based on the Goos–Hänchen shift for the optical domain, with applications to dielectric structures ranging from meta-surfaces to photonic crystal cavities.
The use of coherent light for precision measurements has been a key driving force for numerous research directions, ranging from biomedical optics1,2 to semiconductor manufacturing³. Recent work demonstrates that the precision of such measurements can be substantially improved by tailoring the spatial profile of light fields used for estimating an observable system parameter4,5,6,7,8,9,10. These advances naturally raise the intriguing question of which states of light can provide the ultimate measurement precision¹¹. Here we introduce a general approach to determine the optimal coherent states of light for estimating any given parameter, regardless of the complexity of the system. Our analysis reveals that the light fields delivering the ultimate measurement precision are eigenstates of a Hermitian operator that quantifies the Fisher information from the system’s scattering matrix¹². To illustrate this concept, we experimentally show that these maximum information states can probe the phase or the position of an object that is hidden by a disordered medium with a precision improved by an order of magnitude compared with unoptimized states. Our results enable optimally precise measurements in arbitrarily complex systems, thus establishing a new benchmark for metrology and imaging applications3,13.
In the past few years, concepts from non-Hermitian (NH) physics, originally developed within the context of quantum field theories, have been successfully deployed over a wide range of physical settings where wave dynamics are known to play a key role. In optics, a special class of NH Hamiltonians – which respects parity-time symmetry – has been intensely pursued along several fronts. What makes this family of systems so intriguing is the prospect of phase transitions and NH singularities that can in turn lead to a plethora of counterintuitive phenomena. Quite recently, these ideas have permeated several other fields of science and technology in a quest to achieve new behaviors and functionalities in nonconservative environments that would have otherwise been impossible in standard Hermitian arrangements. Here, we provide an overview of recent advancements in these emerging fields, with emphasis on photonic NH platforms, exceptional point dynamics, and the very promising interplay between non-Hermiticity and topological physics.
We present a biorthogonal approach for modeling the response of localized electromagnetic resonators using quasinormal modes, which represent the natural, dissipative eigenmodes of the system with complex frequencies. For many problems of interest in optics and nanophotonics, the quasinormal modes constitute a powerful modeling tool, and the biorthogonal approach provides a coherent, precise, and accessible derivation of the associated theory, enabling an illustrative connection between different modeling approaches that exist in the literature.
The scattering of electromagnetic waves lies at the heart of most experimental techniques over nearly the entire electromagnetic spectrum, ranging from radio waves to optics and x rays. Hence, deep insight into the basics of scattering theory and an understanding of the peculiar features of electromagnetic scattering are necessary for the correct interpretation of experimental data and an understanding of the underlying physics. Recently, a broad spectrum of exceptional scattering phenomena attainable in suitably engineered structures has been predicted and demonstrated. Examples include bound states in the continuum, exceptional points in parity–time ()-symmetrical non-Hermitian systems, coherent perfect absorption, virtual perfect absorption, nontrivial lasing, nonradiating sources, and others. In this paper, we establish a unified description of such exotic scattering phenomena and show that the origin of all these effects can be traced back to the properties of poles and zeros of the underlying scattering matrix. We provide insights on how managing these special points in the complex frequency plane provides a powerful approach to tailor unusual scattering regimes.
The manipulation of small objects with light has become an indispensable tool in many areas of research, ranging from physics to biology and medicine1–7. Here, we demonstrate how to implement micromanipulation at the optimal level of efficiency for arbitrarily shaped targets and inside complex environments such as disordered media. Our approach is to design wavefronts in the far field8–15 with optimal properties in the near field of the target to apply the strongest possible force, pressure or torque as well as to achieve the most efficient focus inside the target. This non-iterative technique only relies on a simple eigenvalue problem established from the system’s scattering matrix and its dependence on small shifts in specific target parameters (access to the near field of the target is not required). To illustrate this concept, we perform a proof-of-principle experiment in the microwave regime, fully confirming our predictions.
Conventional nanophotonic schemes minimise multiple scattering to realise a miniaturised version of beam-splitters, interferometers and optical cavities for light propagation and lasing. Here instead, we introduce a nanophotonic network built from multiple paths and interference, to control and enhance light-matter interaction via light localisation. The network is built from a mesh of subwavelength waveguides, and can sustain localised modes and mirror-less light trapping stemming from interference over hundreds of nodes. With optical gain, these modes can easily lase, reaching ~100 pm linewidths. We introduce a graph solution to the Maxwell’s equation which describes light on the network, and predicts lasing action. In this framework, the network optical modes can be designed via the network connectivity and topology, and lasing can be tailored and enhanced by the network shape. Nanophotonic networks pave the way for new laser device architectures, which can be used for sensitive biosensing and on-chip optical information processing.
We generalize normal mode expansion of Green's tensor to lossy resonators in open systems, resolving a longstanding open challenge. We obtain a simple yet robust formulation, whereby radiation of energy to infinity is captured by a complete, discrete set of modes, rather than a continuum. This enables rapid simulations by providing the spatial variation of over both and in one simulation. Few eigenmodes are often necessary for nanostructures, facilitating both analytic calculations and unified insight into computationally intensive phenomena such as Purcell enhancement, radiative heat transfer, van der Waals forces, and F\"{o}rster resonance energy transfer. We bypass all implementation and completeness issues associated with the alternative quasinormal eigenmode methods, by defining modes with permittivity rather than frequency as the eigenvalue. We obtain true stationary modes that decay rather than diverge at infinity, and are trivially normalized. Completeness is achieved both for sources located within the inclusion and the background through use of the Lippmann-Schwinger equation. Modes are defined by a linear eigenvalue problem, readily implemented using any numerical method. We demonstrate its simple implementation on COMSOL Multiphysics, using the default inbuilt tools. Results were validated against direct scattering simulations, including analytical Mie theory, attaining arbitrarily accurate agreement regardless of source location or detuning from resonance.
The most general expressions for the stored field energies in the frequency domain, which are independent of any constitutive relations and microscopic models, are derived from the time-domain Poynting theorem by using the complex frequency-domain approach. When the complex frequency-domain approach is applied to Maxwell’s equations, two energy balance relations are obtained simultaneously. The first relation is the well-known Poynting theorem, while the second is new in its general form and contains the newly derived expressions for the stored field energies. In contrast to the well-established Poynting theorem in frequency domain, the real part of the second energy balance relation gives an equation for the sum of stored electric and magnetic field energies and provides a natural definition for the stored energy around a radiator; the imaginary part involves an equation related to the difference between the dissipated electric and magnetic field energies. When media are lossless, the new expressions for the stored field energies are shown to agree with all previous studies; when media are lossy, they include new terms that were not shown in previous reports. These additional terms represent the dispersive part of the stored energies and reflect the influence of the losses on the stored energies.
The absorption of electromagnetic energy into a material is a phenomenon that underlies a plethora of applied problems, including sensing and molecular detection, radar detection, photovoltaics and nanophotonics. Commonly, the incident energy is delivered to the system via a single channel, such as by a plane wave incident on one side of an absorber. However, absorption can be made much more efficient by exploiting the interference of more than one incident signal. A ``coherent perfect absorber'' is a system in which complete absorption of electromagnetic radiation is achieved via the interference of several incident waves. Here, we review recent advances in the design and applications of such devices. We present the theoretical principles underlying the phenomenon of coherent perfect absorption, and give an overview of the photonic structures in which it can be realized, including planar and guided-mode structures, graphene-based systems, PT-symmetric structures, 3D structures and quantum-mechanical systems. We also discuss possible applications of the phenomenon in nanophotonics, and give an outlook on the future of this field and its broad implications in different areas of physics and technology.
We introduce a wavefront shaping protocol for focusing inside disordered media based on a generalization of the established Wigner-Smith time-delay operator. The key ingredient for our approach is the scattering (or transmission) matrix of the medium and its derivative with respect to the position of the target one aims to focus on. A specifc experimental realization in the microwave regime is presented showing that the eigenstates of a corresponding operator are sorted by their focusing strength - ranging from strongly focusing on the designated target to completely bypassing it. Our protocol works without optimization or phase-conjugation and we expect it to be particularly attractive for optical imaging in disordered media.
Bound states in the continuum (BICs) are waves that remain localized even though they coexist with a continuous spectrum of radiating waves that can carry energy away. Their very existence defies conventional wisdom. Although BICs were first proposed in quantum mechanics, they are a general wave phenomenon and have since been identified in electromagnetic waves, acoustic waves in air, water waves and elastic waves in solids. These states have been studied in a wide range of material systems, such as piezoelectric materials, dielectric photonic crystals, optical waveguides and fibres, quantum dots, graphene and topological insulators. In this Review, we describe recent developments in this field with an emphasis on the physical mechanisms that lead to BICs across seemingly very different materials and types of waves. We also discuss experimental realizations, existing applications and directions for future work.
The influence of a small perturbation on a cavity mode plays an important
role in fields like optical sensing, cavity quantum electrodynamics and cavity
optomechanics. Typically, the resulting cavity frequency shift directly relates
to the polarizability of the perturbation. Here we demonstrate that particles
perturbing a radiating cavity can induce strong frequency shifts that are
opposite to, and even exceed, the effects based on the particles'
polarizability. A full electrodynamic theory reveals that these anomalous
results rely on a non-trivial phase relation between cavity and nanoparticle
radiation, allowing back-action via the radiation continuum. In addition, an
intuitive model based on coupled mode theory is presented that relates the
phenomenon to retardation. Because of the ubiquity of dissipation, we expect
these findings to benefit the understanding and engineering of a wide class of
systems.
We present a comprehensive overview of sensor technology exploiting optical whispering gallery mode (WGM) resonances. After a short introduction we begin by detailing the fundamental principles and theory of WGMs in optical microcavities and the transduction mechanisms frequently employed for sensing purposes. Key recent theoretical contributions to the modeling and analysis of WGM systems are highlighted. Subsequently we review the state of the art of WGM sensors by outlining efforts made to date to improve current detection limits. Proposals in this vein are numerous and range, for example, from plasmonic enhancements and active cavities to hybrid optomechanical sensors, which are already working in the shot noise limited regime. In parallel to furthering WGM sensitivity, efforts to improve the time resolution are beginning to emerge. We therefore summarize the techniques being pursued in this vein. Ultimately WGM sensors aim for real-world applications, such as measurements of force and temperature, or alternatively gas and biosensing. Each such application is thus reviewed in turn, and important achievements are discussed. Finally, we adopt a more forward-looking perspective and discuss the outlook of WGM sensors within both a physical and biological context and consider how they may yet push the detection envelope further.
An open system is not conservative because energy can escape to the outside. As a result, the time-evolution operator is not Hermitian in the usual sense and the eigenfunctions (factorized solutions in space and time) are no longer normal modes but quasinormal modes (QNMs) whose frequencies omega are complex. Qausinormal-mode analysis has been a powerful tool for investigating open systems. Previous studies have been mostly system specific, and use a few QNMs to provide approximate descriptions. Here the authors review developments that lead to a unifying treatment. The formulation leads to a mathematical structure in close analogy to that in conservative, Hermitian systems. Hence many of the mathematical tools for the latter can be transcribed. Emphasis is placed on those cases in which the QNMs form a complete set and thus give an exact description of the dynamics. In situations where the QNMs are not complete, the ``remainder'' is characterized. Applications to optics in microspheres and to gravitational waves from black holes are given as examples. The second-quantized theory is sketched. Directions for further development are outlined.
We study experimentally and numerically the electromagnetic resonances in split ring resonators SRRs, around 1 GHz. For an individual SRR, we show that both electric and magnetic fields can induce resonances, the magnetic one being the strongest. The utilization of such resonant structures as efficient microwave filter is also demonstrated. The coupling between two or more SRRs can be quite complex and strongly depends on their geometrical arrangement. For small separation distances, very strong coupling, leading to sharp resonances with high quality factors are observed. In that case a magnetic field circulation which connects neighboring elements is established. The practical implications of these results for the fabrication of a left-handed metamaterial are discussed. © 2002 American Institute of Physics.
We propose a hybrid photonic-plasmonic resonant structure which consists of a
metal nanoparticle (MNP) and a whispering gallery mode (WGM) microcavity. It is
found that the hybrid mode enables a strong interaction between the light and
matter, and the single-atom cooperativity is enhanced by more than two orders
of magnitude compared to that in a bare WGM microcavity. This remarkable
improvement originates from two aspects: (1) the MNP offers a highly enhanced
local field in the vicinity of an emitter, and (2), surprisingly, the
high-\textit{Q} property of WGMs can be maintained in the presence of the MNP.
Thus the present system has great advantages over a single microcavity or a
single MNP, and holds great potential in quantum optics, nonlinear optics and
highly sensitive biosening.
The Wigner-Smith (WS) time delay matrix relates a system’s scattering matrix to its frequency derivative and gives rise to so-called WS modes that experience well-defined group delays when interacting with the system. For systems composed of nondispersive and lossless materials, the WS time delay matrix previously was shown to consist of volume integrals of energy-like densities plus correction terms that account for the guiding, scattering, or radiating characteristics of the system. This study extends the use of the WS time delay matrix to systems composed of dispersive and lossy materials. Specifically, it shows that such systems’ WS time delay matrix can be expressed by augmenting the previously derived expressions with terms that account for the dispersive and lossy nature of the system, followed by a transformation that disentangles effects of losses from time delays. Analytical and numerical examples demonstrate the new formulation once again allows for the construction of frequency stable WS modes that experience well-defined group delays upon interacting with a system.
Wigner-Smith (WS) time delay concepts have been used extensively in quantum mechanics to characterize delays experienced by particles interacting with a potential well. This paper formally extends WS time delay theory to Maxwell’s equations and explores its potential applications in electromagnetics. The WS time delay matrix relates a lossless and reciprocal system’s scattering matrix to its frequency derivative and allows for the construction of modes that experience well-defined group delays when interacting with the system. The matrix’ entries for guiding, scattering, and radiating systems are energy-like overlap integrals of the electric and/or magnetic fields that arise upon excitation of the system via its ports. The WS time delay matrix has numerous applications in electromagnetics, including the characterization of group delays in multiport systems, the description of electromagnetic fields in terms of elementary scattering processes, and the characterization of frequency sensitivities of fields and multiport antenna impedance matrices.
Optical biosensors present good performance in detecting biological systems and promote significant advances in clinical diagnostics, drug discovery, food process control, and environmental monitoring. Without complexity in the pretreatment and probable influence on the nature of target molecules, these biosensors have extra strengths including high sensitivity, robustness, reliability and potential to be integrated in one chip. In this review, the state of the art in optical biosensor technologies, including surface plasmon resonance (SPR), optical waveguide, optical resonator, photonic crystal and optical fiber are presented. Principle in terms of each type of biosensor is introduced concisely and particular emphasis has been placed on the achievements which have been gained nowadays. The strength and weakness of each kind of biosensor have been outlined as well. Concluding remarks regarding the perspectives of further developments are discussed.
We discuss the main features of a recently introduced system capable of laser action: the complex active optical network, or lasing network (LANER) [Lepri et al., Phys. Rev. Lett. 118, 123901 (2017)]. The system is experimentally realized with optical fibers linked each other with couplers and with one or more coherently amplifying sections. The LANER displays a standard laser behavior: When the gain provided by the active sections is high enough to overcome the losses, a coherent emission is produced, with a complicated intensity spectrum. A linear theoretical description is discussed in detail, showing how the LANER can be considered as a generalization of the laser with the physical network acting as a complicated cavity. Among its main aspects, the system can be represented by directed graphs disclosing the analogies with the problem of quantum chaos on graphs. Moreover, when the links' lengths are all integer multiples of the same value, the LANER framework corresponds to a lattice problem, with the equivalence of the Brillouin zone with the cavity free spectral range. Experiments in simple configurations are also performed, reporting the evidence of lasing action and its characterization. Examples of spectra of the detected emitted intensity are obtained in different cases, in a phenomenological agreement with the numerical findings of the theory.
Exceptional points in optics
Many complex systems operate with loss. Mathematically, these systems can be described as non-Hermitian. A property of such a system is that there can exist certain conditions—exceptional points—where gain and loss can be perfectly balanced and exotic behavior is predicted to occur. Optical systems generally possess gain and loss and so are ideal systems for exploring exceptional point physics. Miri and Alù review the topic of exceptional points in photonics and explore some of the possible exotic behavior that might be expected from engineering such systems.
Science , this issue p. eaar7709
The newly emerging field of wave front shaping in complex media has recently seen enormous progress. The driving force behind these advances has been the experimental accessibility of the information stored in the scattering matrix of a disordered medium, which can nowadays routinely be exploited to focus light as well as to image or to transmit information even across highly turbid scattering samples. We will provide an overview of these new techniques, of their experimental implementations as well as of the underlying theoretical concepts following from mesoscopic scattering theory. In particular, we will highlight the intimate connections between quantum transport phenomena and the scattering of light fields in disordered media, which can both be described by the same theoretical concepts. We also put particular emphasis on how the above topics relate to application-oriented research fields such as optical imaging, sensing and communication.
The concepts of Wigner time delay and Wigner-Smith matrix allow to
characterize temporal aspects of a quantum scattering process. The article
reviews the statistical properties of the Wigner time delay for disordered
systems~; the case of disorder in 1D with a chiral symmetry is discussed and
the relation with exponential functionals of the Brownian motion underlined.
Another approach for the analysis of time delay statistics is the random matrix
approach, from which we review few results. As a pratical illustration, we
briefly outline a theory of nonlinear transport and AC transport developed by
B\"uttiker and coworkers, where the concept of Wigner-Smith time delay matrix
is a central piece allowing to describe screening properties in
out-of-equilibrium coherent conductors.
We theoretically investigate mode broadening of a high-Q optical whispering-gallery microcavity coupled to a single or multiple dielectric or plasmonic subwavelength particles. The result shows that backscattering contributes dominantly to the mode broadening in both transmission and reflection spectra for dielectric particles binding on the microcavity surface, while absorption also plays an important role for lossy nanoparticles. The mode broadening induced by nanoparticles holds great potential in optical biosensing. For instance, by monitoring the change of mode linewidth, a single 11-nm-radius spherical polystyrene nanoparticle is detectable. This detection breaks through the detection limit of the mode-splitting method using a passive cavity and remains immune to various noises, such as thermal fluctuations and frequency drifts of the probe laser. Finally, the mode broadening is demonstrated to be particularly suitable for detecting lossy nanoparticles, e.g., plasmonic particles.
In this complete introduction to the theory of finding derivatives of scalar-, vector- and matrix-valued functions with respect to complex matrix variables, Hjrungnes describes an essential set of mathematical tools for solving research problems where unknown parameters are contained in complex-valued matrices. The first book examining complex-valued matrix derivatives from an engineering perspective, it uses numerous practical examples from signal processing and communications to demonstrate how these tools can be used to analyze and optimize the performance of engineering systems. Covering un-patterned and certain patterned matrices, this self-contained and easy-to-follow reference deals with applications in a range of areas including wireless communications, control theory, adaptive filtering, resource management and digital signal processing. Over 80 end-of-chapter exercises are provided, with a complete solutions manual available online.
Many-body perturbation theory as formulated by Brueckner and Goldstone is applied to atoms to obtain corrections to Hartree-Fock wave functions and energies. Calculations are made using a complete set of single-particle Hartree-Fock wave functions which includes both the continuum and an infinite number of bound states. It is shown how one may readily perform the sums over an infinite number of bound excited states. In order to demonstrate the usefulness of many-body perturbation theory in atomic problems, calculations are made for a wide variety of properties of the neutral beryllium atom. The calculated 2s-2s correlation energy is -0.0436 atomic unit for l=1 excitations. The calculated dipole and quadrupole polarizabilities are 6.93×10-24 cm3 and 14.1×10-40 cm5, respectively. The calculated dipole and quadrupole shielding factors are 0.972 and 0.75. Results are given for oscillator strengths, photoionization cross sections, and the Thomas-Reiche-Kuhn sum rule, which is 4.14 as compared with 4.00, the theoretical value.
Since the discovery of photosensitivity in optical fibers there has been great interest in the fabrication of Bragg gratings within the core of a fiber. The ability to inscribe intracore Bragg gratings in these photosensitive fibers has revolutionized the field of telecommunications and optical fiber based sensor technology. Over the last few years, the number of researchers investigating fundamental, as well as application aspects of these gratings has increased dramatically. This article reviews the technology of Bragg gratings in optical fibers. It introduces the phenomenon of photosensitivity in optical fibers, examines the properties of Bragg gratings, and presents some of the important developments in devices and applications. The most common fabrication techniques interferometric, phase mask, and point by point are examined in detail with reference to the advantages and the disadvantages in utilizing them for inscribing Bragg gratings. Reflectivity, bandwidth, temperature, and strain sensitivity of the Bragg reflectors are examined and novel and special Bragg grating structures such as chirped gratings, blazed gratings, phase-shifted gratings, and superimposed multiple gratings are discussed. A formalism for calculating the spectral response of Bragg grating structures is described. Finally, devices and applications for telecommunication and fiber-optic sensors are described, and the impact of this technology on the future of the above areas is discussed. © 1997 American Institute of Physics. S0034-67489701312-9
A detailed derivation is given of the perturbation formula for the frequency shift on introducing a sample of ferrite or dielectric material into a resonant cavity. The purpose of this is to make clear what assumptions are involved in the derivation; it is necessary to appreciate what these assumptions are in order to design accurate experiments.
Les problèmes de perturbations et les champs self-consistents
- L Brillouin
L. Brillouin, Les problèmes de perturbations et les champs
self-consistents, J. Phys. Radium 3, 373-389 (1932).
Perturbation theory for cavities
- H A Bethe
- J Schwinger
H. A. Bethe and J. Schwinger, Perturbation theory
for cavities, N.D.R.C. Rpt. D1-117, (Cornell University,
1943).