The repulsion between free electrons inside a metal makes its optical response spatially dispersive, so that it is not described by Drude's model but by a hydrodynamic model. We give here fully analytic results for a metallic slab in this framework, thanks to a two-modes cavity formalism leading to a Fabry-Perot formula, and show that a simplification can be made that preserves the accuracy of the results while allowing much simpler analytic expressions. For metallic layers thicker than 2.7 nm modified Fresnel coefficients can actually be used to accurately predict the response of any multilayer with spatially dispersive metals (for reflection, transmission or the guided modes). Finally, this explains why adding a small dielectric layer[Y. Luo et al., Phys. Rev. Lett. 111, 093901 (2013)] allows to reproduce the effects of nonlocality in many cases, and especially for multilayers.
We present a general and useful method to predict the existence, frequency, and spatial properties of gap states in photonic (and other) structures with a gapped spectrum. This method is established using the scattering approach. It offers a viewpoint based on a geometrical Fabry-Perot model. We demonstrate the capabilities of this model by predicting the behaviour of topological edge states in quasi-periodic structures. A proposition to use this model in Casimir physics is presented.
We introduce a theory of optical responses of bianisotropic layers with arbitrary effective medium parameters, which results in generalized Fresnel-Airy equations for reflection and transmission coefficients at all incidence directions and polarizations. The poles of these equations provide explicit expressions for the dispersion of Fabry-Perot resonances and surface electromatic waves in bianisotropic layers and interfaces. The existence conditions of these resonances are topologically related to the zeros of the high-k characteristic function h(k)=0 of bulk bianisotropic materials and Durach…
We propose a novel photonic structure composed of metal nanolayer, Bragg mirror and metal nanolayer. The structure supports resonances that are transitional between Fabry-Perot and Tamm modes. When the dielectric contrast of the DBR is removed these modes are a pair of conventional Fabry-Perot resonances. They spectrally merge into a Tamm mode at high contrast. Such behavior differs from the results for structures supporting Tamm modes reported earlier. The optical properties of the structure in the frequency range of the DBR stop band, including highly beneficial 50% transmittivity through th…
In this study, we theoretically and numerically investigate the resonant modes and reflectance of an optical grating consisting of a wavy dielectric slab by applying the spectral element method. The presence of the wavy shape transforms the waveguide modes into leaky resonant modes. A few resonant modes with specific longitudinal wave number have infinitely large Q factor, while the other resonant modes have finite Q factor. For the leaky resonant mode with zero longitudinal wave number, the Q factor is inversely proportional to the amplitude of the wavy shape. An array of multiple low-Q wavy …
We present a detailed study of the use of Fabry-Perot (FP) cavities for the spectroscopy of single InAs quantum dots (QDs). We derive optimal cavity characteristics and resolution limits, and measure photoluminescence linewidths as low as 0.9 GHz. By embedding the QDs in a planar cavity, we obtain a sufficiently large signal to actively feed back on the length of the FP to lock to the emission of a single QD with a stability below 2% of the QD linewidth. An integration time of approximately two seconds is found to yield an optimum compromise between shot noise and cavity length fluctuations.
In recent years quantum phenomena have been experimentally demonstrated on variety of optomechanical systems ranging from micro-oscillators to photonic crystals. Since single photon couplings are quite small, most experimental approaches rely on the realization of high finesse Fabry-Perot cavities in order to enhance the effective coupling. Here we show that by exploiting a, long path, low finesse fiber Fabry-Perot interferometer ground state cooling can be achieved. We model a 100 m long cavity with a finesse of 10 and analyze the impact of additional noise sources arising from the fiber. As …
We discuss the parametric oscillatory instability in a Fabry-Perot cavity of the Einstein Telescope. Unstable combinations of elastic and optical modes for two possible configurations of gravitational wave third-generation detector are deduced. The results are compared with the results for gravita- tional wave interferometers LIGO and LIGO Voyager.
In this paper we revisited the theory of selective reflection from a dilute vapor cell. A self-consistent theory was developed for reflection spectrum for Fabry-Perot interferometer. Formulas for single and multiple reflections were obtained. We also obtained the effective refractive index for a single selective reflection. The results of the paper are in a good agreement with existing experimental results.
The use of negative index materials is highly efficient for tayloring the spectral dispersion properties of a quarter-wavelength Bragg mirror and for obtaining a resonant behavior of a multilayer Fabry-Perot cavity over a very large spectral range. An optimization method is proposed and validated on some first promising devices.
We address the response of a Fabry-Perot interferometer to a monochromatic point source. We calculate the anticaustics (that is, the virtual wavefronts of null path difference) resulting from the successive internal reflections occurring in the system. They turn to be a family of ellipsoids (or hyperboloids) of revolution, which allows us to reinterpret the operation of the Fabry-Perot from a geometrical point of view that facilitates comparison with other apparently disparate arrangements, such as Young's double slit.
An anisotropic photonic crystal containing two anisotropic defect layers is considered. It is demonstrated that the system under can support a Fabry-Perot bound state in the continuum (FP-BIC). A fully analytic solution of the scattering problem as well as a condition for FP-BIC have been derived in the framework of the temporal coupled-mode theory.
The complex reflected and transmitted amplitudes from a Fabry-Perot interferometer are analyzed using a phase-space approach, in which the real and imaginary parts of those amplitudes are taken as basic variables. As functions of the optical path length of the cavity, the reflected amplitude describes a circle, whereas the transmitted is given by a hippopede. The system performance can be directly related to the geometrical parameters of these curves.
A class of multiwavelength Fabry-Perot lasers is introduced where the spectrum is tailored through a non-periodic patterning of the cavity effective index. The cavity geometry is obtained using an inverse scattering approach and can be designed such that the spacing of discrete Fabry-Perot lasing modes is limited only by the bandwidth of the inverted gain medium. A specific two-color semiconductor laser with a mode spacing in the THz regime is designed, and measurements are presented demonstrating the simultaneous oscillation of the two wavelengths. The extension of the Fabry-Perot laser conce…
We report the first observation of a nonlinear mode in a cylindrical nonlinear Fabry-Perot cavity. The field enhancement from cavity buildup, as well as the large chi3 optical nonlinearity due to resonantly-excited Rb-85 vapor, allows the nonlinear mode to form at low incident optical powers of less than a milliwatt. The mode is observed to occur for both the self-focusing and self-defocusing nonlinearity.
The method of images, when applied to an opto-mechanical Fabry-Perot resonator with a heavy, charged mirror, and a light, charged mirror that can be driven into motion, reveals that the highest order images can move relativistically when the Q of the superconducting resonator is on the order of 10^10. The implications of such relativistic motions of the charge and mass of these images to the emission of electromagnetic and gravitational radiation within the resonator is explored.
We address the response, both in amplitude and intensity, of a Fabry-Perot from a variety of viewpoints. These complementary pictures conspire to achieve a comprehensive and consistent theory of the operation of this system.
In this paper we present a theoretical and experimental study of the dynamical behaviour of birefringent cavities. Our experimental data show that usual hypothesis which provides that a Fabry-Perot cavity is a first-order low pass filter cannot explain the behaviour of a birefringent cavity. We explain this phenomenon, and give the theoretical expression of the equivalent cavity filter which corresponds to a second-order low pass filter.
We demonstrate passive harmonic mode-locking of a quantum well laser diode designed to support a discrete comb of Fabry-Perot modes. Spectral filtering of the mode spectrum was achieved using a non-periodic patterning of the cavity effective index. By selecting six modes spaced at twice the fundamental mode spacing, near-transform limited pulsed output with 2 ps pulse duration was obtained at a repetition rate of 100 GHz.
We experimentally demonstrate that the transmission though rolled-up metal/semiconductor hyperlenses can be enhanced at desired frequencies utilizing Fabry-Pérot resonances. By means of finite difference time domain simulations we prove that hyperlensing occurs at frequencies of high transmission.
The dynamics of light in Fabry-Perot cavities with varying length and input laser frequency are analyzed and the exact condition for resonance is derived. This dynamic resonance depends on the light transit time in the cavity and the Doppler effect due to the mirror motions. The response of the cavity to length variations is very different from its response to laser frequency variations. If the frequency of these variations is equal to multiples of the cavity free spectral range, the response to length is maximized while the response to the laser frequency is zero. Implications of these result…
We analyze an optical 3-port reflection grating by means of a scattering matrix formalism. Amplitude and phase relations between the 3 ports, i.e. the 3 orders of diffraction are derived. Such a grating can be used as an all-reflective, low-loss coupler to Fabry-Perot cavities. We derive the input output relations of a 3-port grating coupled cavity and find distinct properties not present in 2-port coupled cavities. The cavity relations further reveal that the 3-port coupler can be designed such that the additional cavity port interferes destructively. In this case the all-reflective, low-loss…
A spherical Fabry-Perot interferometer with adjustable mirror spacing is used to produce interference fringes with frequency separation (c/2L)/N, N=2-15. The conditions for observation of these fringes are derived from the consideration of the eigenmodes of the cavity with high transverse indices.
One way of optically monitoring molecule concentrations is to utilise the high sensitivity of the transmission and reflection rates of Fabry-Perot cavities to changes of their optical properties. Up to now, intrinsic and extrinsic Fabry-Perot cavity sensors have been considered with analytes either being placed inside the resonator or coupled to evanescent fields on the outside. Here we show that Fabry-Perot cavities can also be used to monitor molecule concentrations non-invasively and remotely, since the reflection of light from the target molecules back into the Fabry-Perot cavity adds upwa…
Various four-mirror optical resonators are studied in the perspective of realizing passive stacking cavities. A comparative study of the mechanical stability is provided. The polarization properties of the cavity eigenmodes are described and it is shown that the effect of mirror misalignments (or motions) induces polarization and stacking power instabilities. These instabilities increase with the finesse of the Fabry-Perot cavity. A tetrahedral configuration of the four mirrors is found to minimize the consequences of the mirrors's motion and misalignment by reducing the instability parameter …
A new set of vector solutions to Maxwell's equations based on solutions to the wave equation in spheroidal coordinates allows laser beams to be described beyond the paraxial approximation. Using these solutions allows us to calculate the complete first-order corrections in the short-wavelength limit to eigenmodes and eigenfrequencies in a Fabry-Perot resonator with perfectly conducting mirrors. Experimentally relevant effects are predicted. Modes which are degenerate according to the paraxial approximation are split according to their total angular momentum. This includes a splitting due to co…
Fabry-Perot interferometers have stimulated numerous scientific and technical applications ranging from high resolution spectroscopy over metrology, optical filters to interfaces of light and matter at the quantum limit and more. End facet machining of optical fibers has enabled the miniaturization of optical Fabry-Perot cavities. Integration with fiber wave guide technology allows for small yet open devices with favorable scaling properties including mechanical stability and compact mode geometry. These Fiber Fabry-Perot Cavities (FFPCs) are stimulating extended applications in many fields in…
We propose to take advantage of the properties of long Josephson junctions to realize a frequency variable Fabry-Perot Filter that operates in the range 100-500 GHz with a bandwidth below 1 GHz. In fact, we show that it is possible to exploit the tunability of the effective impedance of the Josephson component, that is controlled by a dc bias, to tune, up to $10\%$ of the central frequency, the resonance of the system. An analysis of the linearized system indicates the range of operation and the main characteristic parameters Numerical simulations of the full nonlinear Josephson element confir…
In this paper we describe a Fabry-Perot interferometer in the language of quantum optics. We go on to model the Fabry-Perot interferometer as a beam splitter having frequency dependent transmissivity and reflectivity coefficients. The antibunching, a totally non-classical effect, is to be expected if one excites this interferometer with carefully frequency-selected single photons arriving simultaneously from opposite directions. Contrary to a normal beam splitter, even slightly different frequency single photons should be able to show this effect, as long as the photon counters are not frequen…
This document records the results of a comparison of the interferometer simulation Finesse against an analytic (MATLAB based) calculation of the alignment sensing signals of a Fabry Perot cavity. This task was started during the commissioning workshop at the LIGO Livingston site between the 28.1. and 1.02 2013 with the aim of creating a reference example for validating numerical simulation tools. The FFT based simulation OSCAR joined the battle later.
Optical Fabry-Perot cavities always show a non-degeneracy of two orthogonal polarisation states. This is due to the unavoidable birefringence of dielectric mirrors whose effects are extremely important in Fabry-Perot based high-accuracy polarimeters. We have developed and present here a theory of the polarisation state dynamics in a birefringent Fabry-Perot resonator, and we validate it through measurements performed with the polarimeter of the PVLAS experiment. The measurements are performed while a laser is frequency-locked to the cavity, and provide values for the finesse of the cavity and …
A key issue in developing pendular Fabry-Perot interferometers as very accurate displacement measurement devices, is the noise level. The Fabry-Perot pendulums are the most promising device to detect gravitational waves, and therefore the background and the internal noise should be accurately measured and reduced. In fact terminal masses generates additional internal noise mainly due to thermal fluctuations and vibrations. We propose to exploit the reflectivity change, that occurs in some special points, to monitor the pendulums free oscillations and possibly estimate the noise level. We find …
A temporal boundary refers to a specific time at which the properties of an optical medium are abruptly changed. When light interacts with the temporal boundary, its spectral content can be redistributed due to the breaking of continuous time-translational symmetry of the medium where light resides. In this work, we use this principle to demonstrate, at terahertz (THz) frequencies, the resonance-enhanced spectral funneling of light coupled to a Fabry-Perot resonator with a temporal boundary mirror. To produce a temporal boundary effect, we abruptly increase the reflectance of a mirror constitu…
Symmetry-Broken Cavity Solitons and Collective Polarization Conformity in Fabry-Perot Kerr Resonators
Université Bourgogne Europe（勃艮第欧洲大学）；CNRS（法国国家科学研究中心）；Max Planck Institute for the Science of Light（马克斯·普朗克光科学研究所）；Friedrich Alexander University Erlangen-Nuremberg（埃尔兰根-纽伦堡弗里德里希·阿历山大大学）；Alexandria University（亚历山大大学）；Huazhong University of Science and Technology（华中科技大学）；University of Strathclyde（思克莱德大学）；The University of Auckland（奥克兰大学）
The factors limiting the resolution of displacement sensor based on extrinsic Fabry-Perot interferometer were studied. An analytical model giving the dependency of EFPI resolution on the parameters of an optical setup and a sensor interrogator was developed. The proposed model enables one to either estimate the limit of possible resolution achievable with a given setup, or to derive the requirements for optical elements and/or a sensor interrogator necessary for attaining the desired sensor resolution. An experiment supporting the analytical derivations was performed, demonstrating a large dyn…
Factors that contribute to the temperature dependence of a resonant frequency in a low-expansion optical cavity are discussed, including deformation at the cavity ends due to different coefficients of thermal expansion (CTE) of the spacer, optically-contacted mirror substrate and coating. A model of the temperature dependence is presented that incorporates finite-element-analysis of the cavity ends. A measurement of frequency versus temperature of a cavity mode is used along with the model to deduce a spacers CTE versus temperature profile. The measured profile correlates very well with a sepa…
We show that an escape from the potential minimum of Fabry-Perot interferometers can be detected measuring the associated sudden change of reflectivity. We demonstrate that the loss of information that occurs retaining only the sequence of escapes, rather than the full trajectory, can be very mild and can lead to an effective method to reveal the noise intensity or the presence of a coherent signal.
Optical nanofiber cavity research has mainly focused on the fundamental mode. Here, a Fabry-Pérot fiber cavity with an optical nanofiber supporting the higher-order modes, TE01, TM01, HE21o, and HE21e, is demonstrated. Using cavity spectroscopy, with mode imaging and analysis, we observe cavity resonances that exhibit complex, inhomogeneous states of polarization with topological features containing Stokes singularities such as C-points, Poincaré vortices, and L-lines. In situ tuning of the intracavity birefringence enables the desired profile and polarization of the cavity mode to be obtained…
We present a method for remote interrogation of passive fiber Bragg grating Fabry-Perot resonators, employing current modulation of the diode laser source. With the presence of both RF amplitude and frequency modulation, it is a variant of the ultra-sensitive Pound-Drever-Hall laser frequency locking technique. We demonstrate that current modulation and interferometric demodulation removes the need for a phase modulator in the sensing architecture.
We present the first demonstration of real-time closed-loop control and deterministic tuning of an independently suspended Fabry-Perot optical cavity using digitally-enhanced heterodyne interferometry, realising a peak sensitivity of $\sim$10 pm$/\sqrt{\mathrm{Hz}}$ over the 10-1000 Hz frequency band. The methods presented are readily extensible to multiple coupled cavities. As such, we anticipate that refinements of this technique may find application in future interferometric gravitational-wave detectors.
The dynamical response of an optical Fabry-Perot cavity is investigated experimentally. We observe oscillations in the transmitted and reflected light intensity if the frequency of the incoupled light field is rapidly changed. In addition, the decay of a cavity-stored light field is accelerated if the phase and intensity of the incoupled light are switched in an appropriate way. The theoretical model by M. J. Lawrence em et al, JOSA B 16, 523 (1999) agrees with our observations.
In this letter, we demonstrate for the first time that by combining the effects of the Wood-Rayleigh Anomaly (WRA) and the Fabry-Perot (FP) resonance, transmission efficiencies of $1$D metallo-dielectric gratings on substrates can be significantly improved compared to when these two phenomena work separately. Results of combining the WRA and the FP resonance can be utilized to eliminate the necessity of using the index matching technique and the core-shell structure for enhancing the performance of extraordinary optical transmission devices. Further, the outcomes of combining the WRA and the F…
The scattering of weak dispersive waves (DW) on several equally spaced temporal solitons is studied. It is shown by systematic numerical simulations that the reflection of the DWs from the soliton trains strongly depends on the distance between the solitons. The dependence of the reflection and transmission coefficients on the inter-soliton distance and the frequency of the incident waves is studied in detail, revealing fascinating quasi-periodic behavior. The analogy between the observed nonlinear phenomena in temporal domain and usual Fabry-Perot and Bragg resonators is discussed.
We measured the far-field emission patterns in three dimensions of flat organic dye microlasers using a solid angle scanner. Polymer-based microcavities of ribbon shape (i.e., Fabry-Perot type) were investigated. Out of plane emission from the cavities was observed, with significant differences for the two cases of resonators either fully supported by the substrate or sustained by a pedestal. In both cases, the emission diagrams are accounted for by a model combining diffraction at the cavity edges and reflections from the substrate.
We study a dispersion-compensated high-finesse optical Fabry-Perot microcavity under high-intensity cw pumping. The Kerr non-linearity in the optical coatings causes a spontaneous four-wave mixing process, which leads to the emission of time-correlated photon pairs. The photon frequencies are shifted by $\pm 1$ free spectral range relative to the pump frequency. This setup allows for constructing a photon-pair source with precisely adjustable frequency difference between the emitted photons, which may have applications in quantum communication.
Free-running Fabry-Perot lasers normally operate in a single-mode regime until the pumping current is increased beyond the single-mode instability threshold, above which they evolve into a multimode state. As a result of this instability, the single-mode operation of these lasers is typically constrained to few percents of their output power range, this being an undesired limitation in spectroscopy applications. In order to expand the span of single-mode operation, we use an optical injection seed generated by an external-cavity single-mode laser source to force the Fabry-Perot quantum cascade…
We present a method to engineer the frequency splitting of polarization eigenmodes in fiber Fabry-Perot (FFP) cavities. Using specific pattern of multiple CO$_2$ laser pulses, we machine paraboloidal micromirrors with controlled elliptical shape in a large range of radius of curvature. This method is versatile and can be used to produce cavities with maximized or near-zero polarization mode splitting. In addition, we realize dual-wavelength FFP cavities with finesse exceeding $40\,000$ at 780$\,$nm and at 1559$\,$nm in the telecom range. We provide direct evidence that the birefringent frequen…
Anomalous spectral shift of transmission peaks is observed in a Fabry--Pérot cavity filled with a chiral anisotropic medium. The effective refractive index value resides out of the interval between the ordinary and the extraordinary refractive indices. The spectral shift is explained by contribution of a geometric phase. The problem is solved analytically using the approximate Jones matrix method, numerically using the accurate Berreman method and geometrically using the generalized Mauguin--Poincaré rolling cone method. The $o$-mode blue shift is measured for a 4-methoxybenzylidene-4'-$n$-but…
Absorbing metamaterials have attracted a widespread interest but suffer from the metallic ohmic losses that limits their Q-factor to values around 10. In this paper, we report the engineering and experimental characterization of a high Q dark-mode in a metasurface resulting from the coupling of several individual nano-Fabry-Perot structures supporting low-contrast resonances. The resulting resonance has a Q-factor above 20 and a nearly total absorption. Moreover, we also demonstrate that several dark-mode resonances can be excited simultaneously, by exploiting the coupling of various pairs of …
We report on measurements and modeling of the mode structure of tunable Fabry-Pérot optical microcavities with imperfect mirrors. We find that non-spherical mirror shape and finite mirror size lead to loss, mode deformation, and shifted resonance frequencies at particular mirror separations. For small mirror diameters, the useful cavity length is limited to values significantly below the expected stability range. We explain the observations by resonant coupling between different transverse modes of the cavity and mode-dependent diffraction loss. A model based on resonant state expansion that t…
