Effect of Diffraction on Wigner Distributions of Optical Fields and how to Use It in Optical Resonator Theory. I -- Stable Resonators and Gaussian Beams
The first part of the paper is devoted to diffraction phenomena that can be expressed by fractional Fourier transforms whose orders are real numbers. According to a scalar theory, diffraction acts on the amplitude of the electric field as well as on its spherical angular spectrum, and Wigner distributions can be defined on a space-frequency phase space. The phase space is equipped with an Euclidean structure, so that the effects of diffraction are rotations of Wigner distributions associated with optical fields. Such a rotation is shown to split into two specific elliptical rotations. Wigner distributions associated with transverse modes of a resonator are invariant in these rotations, and a complete theory of stable optical resonators and Gaussian beams is developed on the basis of this property, including waist existence and related formulae, and naturally introducing the Gouy phase.
The third part of the paper is devoted to ray tracing in optical resonators. The employed method for dealing with the issue uses the elliptical or hyperbolic rotations that Wigner distributions associated with optical fields undergo during propagation from one spherical mirror of a resonator to the other. It is illustrated by various examples concerning meridional or skew rays, reentrant in stable resonators or propagating in unstable resonators. A classification of optical resonators is eventually deduced.
The second part of the article is devoted to field transfers by diffraction that are represented by fractional Fourier transformations whose orders are complex numbers. The corresponding effects on the Wigner distributions associated with optical fields are still represented by $4\times4$ matrices operating on the scaled phase-space, but unlike matrices involved in the first part, those matrices decompose into two matrices that essentially represent 2--dimensional hyperbolic rotations, not elliptical rotations. The result is applied to the theory of unstable resonators.
Free-space propagation can be described as a shearing of the Wigner distribution function in the spatial coordinate; this shearing is linear in paraxial approximation but assumes a more complex shape for wide-angle propagation. Integration in the frequency domain allows the determination of near-field diffraction, leading to the well known Fresnel diffraction when small angles are considered and allowing exact prediction of wide-angle diffraction. The authors use this technique to demonstrate evanescent wave formation and diffraction elimination for very small apertures.
The eigenmodes of unstable canonical optical resonators possess fractal structure in their transverse intensity cross-sections [Karman et al., Nature 402, 138 (1999)]. In one particular plane, the magnified self-conjugate plane, this structure can be explained in terms of a combination of imaging and diffraction [Courtial and Padgett, PRL 85, 5320 (2000)]. Here we show that this combination of imaging and diffraction simultaneously occurs in the longitudinal direction, resulting in three-dimensional self-similar fractal structure around the centre of the magnified self-conjugate plane.
Existing methods for the localization of light at the nanoscale use either a structure with negative permittivity, by exploiting subwavelength plasmonic resonances, or a dielectric structure with a high refractive index, which reduces the wavelength. In this paper, we provide an alternative to these two methods based on a modification of the modes of dielectric resonators by means of an active medium. We show that an active medium can promote subwavelength light localization in the dielectric structure. We consider a dielectric layer of size substantially smaller than a half-wavelength of ligh…
We introduce the resonant metalens, a cluster of coupled subwavelength resonators. Dispersion allows the conversion of subwavelength wavefields into temporal signatures while the Purcell effect permits an efficient radiation of this information in the far-field. The study of an array of resonant wires using microwaves provides a physical understanding of the underlying mechanism. We experimentally demonstrate imaging and focusing from the far-field with resolutions far below the diffraction limit. This concept is realizable at any frequency where subwavelength resonators can be designed.
We present exact results for the steady-state density matrix of a general class of driven-dissipative systems consisting of a nonlinear Kerr resonator in the presence of both coherent (one-photon) and parametric (two-photon) driving and dissipation. Thanks to the analytical solution, obtained via the complex P-representation formalism, we are able to explore any regime, including photon blockade, multiphoton resonant effects, and a mesoscopic regime with large photon density and quantum correlations. We show how the interplay between one- and two-photon driving provides a way to control the mu…
The problem of pattern formation in resonantly-enhanced near-field lithography by the use of dielectric or plasmonic planar resonators is investigated. Sub-diffraction-limited bright or dark spots can be produced by taking advantage of the rotational invariance of planar resonators. To increase the spatial bandwidth of the resonant enhancement, an array of coupled planar resonators can open up a band of Bloch resonances, analogous to coupled-resonator optical waveguides.
Ptychography employs a set of diffraction patterns that capture redundant information about an illuminated specimen as a localized beam is moved over the specimen. The robustness of this method comes from the redundancy of the dataset that in turn depends on the amount of oversampling and the form of the illumination. Although the role of oversampling in ptychography is fairly well understood, the same cannot be said of the illumination structure. This paper provides a vector space model of ptychography that accounts for the illumination structure in a way that highlights the role of the Wigne…
We consider a nonlinear optical system in general, and a broad aperture laser in particular in a resonator where the diffraction coefficients are of opposite signs along two transverse directions. The system is described by the hyperbolic Maxwell-Bloch equations, where the spatial coupling is provided by the DAlambert operator rather than by the Laplace operator. We show that this system supports hyperbolic transverse patterns residing on hyperbolas in far field domain, and consisting of stretched vortices in near field domain. These structures are described analytically by normal form analysi…
Alternative designs to an electric-LC (ELC) resonator, which is a type of metamaterial inclusion, are presented in this article. Fitting the resonator with an interdigital capacitor (IDC) helps to increase the total capacitance of the structure. In effect, its resonance frequency is shifted downwards. This implies a decreased overall resonator size with respect to its operating wavelength. As a result, the metamaterial, composed of an array of IDC-loaded ELC resonators with their collective electromagnetic response, possesses improved homogeneity and hence is less influenced by diffraction eff…
We study a ring cavity filled with a slab of a right-handed material and a slab of a left-handed material. Both layers are assumed to be nonlinear Kerr media. First, we derive a model for the propagation of light in a left-handed material. By constructing a mean-field model, we show that the sign of diffraction can be made either positive or negative in this resonator, depending on the thicknesses of the layers. Subsequently, we demonstrate that the dynamical behavior of the modulation instability is strongly affected by the sign of the diffraction coefficient. Finally, we study the dissipativ…
The generation of induced radiation in volume resonator formed by metal threads is considered. It is shown that using of such volume diffraction grating allows increasing of lasing efficiency in terahertz range. The requirements on beam and grating parameters are obtained.
We report on the optical characterization of an ultra-high diffraction efficiency grating in 1st order Littrow configuration. The apparatus used was an optical cavity built from the grating under investigation and an additional high reflection mirror. Measurement of the cavity finesse provided precise information about the grating's diffraction efficiency and its optical loss. We measured a finesse of 1580 from which we deduced a diffraction efficiency of (99.635$\pm$0.016)% and an overall optical loss due to scattering and absorption of just 0.185 %. Such high quality gratings, including the …
We investigate the spin-orbit coupling of light in three-dimensional cylindrical and tube-like whispering gallery mode resonators. We show that its origin is the transverse confinement of light in the resonator walls, even in the absence of inhomogeneities or anisotropies. The spin-orbit interaction results in elliptical far-field polarization (spin) states and causes spatial separation of polarization handedness in the far field. The ellipticity and spatial separation are enhanced for whispering gallery modes with higher excitation numbers along the resonator height. We analyze the asymmetry …
Manipulating the polarization of light on the microscale or nanoscale is essential for integrated photonics and quantum optical devices. Nowadays, the metasurface allows one to build on-chip devices that efficiently manipulate the polarization states. However, it remains challenging to generate different types of polarization states simultaneously, which is required to encode information for quantum computing and quantum cryptography applications. By introducing geometrical-scaling-induced (GSI) phase modulations, we demonstrate that an assembly of circularly polarized (CP) and linearly polari…
We encode the sequence of prime numbers into simple superpositions of identical waves, mimicking the archetypal prime number sieve of Eratosthenes. The primes are identified as zeros accompanied by phase singularities in a physically generated wave-field for integer valued momenta. Similarly, primes are encoded in the diffraction pattern from a simple single aperture and in the harmonics of a single vibrating resonator. Further, diffraction physics connections to number theory reveal how to encode all Gaussian primes, twin-primes, and how to construct wave fields with amplitudes equal to the d…
We analyze a simple and feasible practical scheme displaying Zeno, anti-Zeno, and inverse-Zeno effects in the observation of wave-packet spreading caused by free evolution. The scheme is valid both in spatial diffraction of classical optical waves and in time diffraction of a quantum wave packet. In the optical realization, diffraction spreading is observed by placing slits between a light source and a light-power detector. We show that the occurrence of Zeno or anti-Zeno effects depends just on the frequency of observations between the source and detector. These effects are seen to be related…
We describe theoretically and verify experimentally a novel class of diffraction-free pulsed optical beams that are `Omni-resonant': they have the remarkable property of transmission through planar Fabry-Perot resonators without spectral filtering even if their bandwidth far exceeds the cavity resonant linewidth. Ultrashort wave packets endowed with a specific Spatio-temporal structure couple to a \textit{single} resonant mode independently of its linewidth. We confirm that such `space-time' Omni-resonant wave packets retain their bandwidth (1.6~nm), Spatio-temporal profile (1.3-ps pulse width…
A phase space description of the fractional Talbot effect, occurring in a one-dimensional Fresnel diffraction from a periodic grating, is presented. Using the phase space formalism a compact summation formula for the Wigner function at rational multiples of the Talbot distance is derived. The summation formula shows that the fractional Talbot image in the phase space is generated by a finite sum of spatially displaced Wigner functions of the source field.
We propose to use exceptional points (EPs) to construct diffraction-free beam propagation and localized power oscillation in lattices. Specifically, here we propose two systems to utilize EPs for diffraction-free beam propagation, one in synthetic gauge lattices and the other one, in unidirectionally coupled resonators where each resonator individually is capable of creating orbital angular momentum beams (OAM). In the second system, we introduce the concept of robust and tunable OAM beam propagation in discrete lattices. We show that one can create robust OAM beams in an arbitrary number of s…
The integration of nanoscale electronics with conventional optical devices is restricted by the diffraction limit of light. Metals can confine light at the subwavelength scales needed, but they are lossy, while dielectric materials do not confine evanescent waves outside a waveguide or resonator, leading to cross talk between components. We introduce a paradigm shift in light confinement strategy and show that light can be confined below the diffraction limit using completely transparent artificial media. Our approach relies on controlling the optical momentum of evanescent waves, an important…
Dissipative solitons can emerge in a wide variety of dissipative nonlinear systems throughout the fields of optics, medicine or biology. Dissipative solitons can also exist in Kerr-nonlinear optical resonators and rely on the double balance between parametric gain and resonator loss on the one hand and nonlinearity and diffraction or dispersion on the other hand. Mathematically these solitons are solution to the Lugiato-Lefever equation and exist on top of a continuous wave (cw) background. Here we report the observation of temporal dissipative solitons in a high-Q optical microresonator. The …
Electromagnetic resonators, which are based on optical cavities or electronic circuits, are key elements to enhance and control light-matter interaction. In the THz range, current optical cavities exhibit very high-quality factors with $(λ/2)^3$ mode volumes limited by diffraction, whereas resonant electronic circuits show low quality factor but provide strong subwavelength effective volume ($10^{-6} λ^3$). To overcome the limitations of each type of resonator, great efforts are being devoted to improving the performances of current methods or to the emergence of original approaches. Here, we …
We present a novel design of optical micro-cavity where the optical energy resides primarily in free space, therefore is readily accessible to foreign objects such as atoms, molecules, mechanical resonators, etc. We describe the physics of these resonators, and propose a design method based on stochastic optimization. Cavity designs with diffraction-limited mode volumes and quality factors in the range of $10^4$--$10^6$ are presented. With a purely planar geometry, the cavity can be easily integrated on-chip using conventional micro- and nano- fabrication processes.
Growing interest in non-line-of-sight (NLoS) imaging, colloquially referred to as "seeing around corners", has led to the development of phasor-field ($\mathcal{P}$-field) imaging, wherein the field envelope of amplitude-modulated spatially-incoherent light is manipulated like an optical wave to directly probe a space that is otherwise shielded from view by diffuse scattering. Recently, we have established a paraxial theory for $\mathcal{P}$-field imaging in a transmissive geometry that is a proxy for three-bounce NLoS imaging [J. Dove and J. H. Shapiro, Opt. Express {\bf 27}(13) 18016--18037 …
Imaging of the optical properties of individual nanosystems beyond fluorescence can provide a wealth of information. However, the minute signals for absorption and dispersion are challenging to observe, and only specialized techniques requiring sophisticated noise rejection are available. Here we use signal enhancement in a scanning optical microcavity to demonstrate ultra-sensitive imaging. Harnessing multiple interactions of probe light with a sample within an optical resonator, we achieve a 1700-fold signal enhancement compared to diffraction-limited microscopy. We demonstrate quantitative …
An appropriate design of wavefront will enable light fields propagating along arbitrary trajectories thus forming accelerating beams in free space. Previous ways of designing such accelerating beams mainly rely on caustic methods, which start from diffraction integrals and only deal with two-dimensional fields. Here we introduce a new perspective to construct accelerating beams in phase space by designing the corresponding Wigner distribution function (WDF). We find such a WDF-based method is capable of providing both the initial field distribution and the angular spectrum in need by projectin…
A stable optical resonator has been built using a 30 micrometer-wide, metal-coated microcantilever as one mirror. The second mirror was a 12.7 mm-diameter concave dielectric mirror. By positioning the two mirrors 75 mm apart in a near-hemispherical configuration, a Fabry-Perot cavity with a finesse equal to 55 was achieved. The finesse was limited by the optical loss in the cantilever's metal coating; diffraction losses from the small mirror were negligible. The cavity achieved passive laser cooling of the cantilever's Brownian motion.
Nanoscale infrared (IR) resonators with sub-diffraction limited mode volumes and open geometries have emerged as new platforms for implementing cavity quantum electrodynamics (QED) at room temperature. The use of infrared (IR) nano-antennas and tip nanoprobes to study strong light-matter coupling of molecular vibrations with the vacuum field can be exploited for IR quantum control with nanometer and femtosecond resolution. In order to accelerate the development of molecule-based quantum nano-photonic devices in the mid-IR, we develop a generally applicable semi-empirical quantum optics approac…
