The physical aspects of partially coherent radiation interacting with deterministic non-Hermitian periodic materials remain largely unexplored in the statistical optics literature. Here, we consider the scattering of partially coherent radiation by a deterministic periodic medium, symmetric under the simultaneous transformations of parity inversion and time reversal, that is, a parity-time (PT)-symmetric periodic medium. Taking into account light fluctuations, one is able to describe the spectrum changes on propagation and the influence of the coherence-driven angular divergence effect. The far-field spectral density profile is found to depend crucially on the loss/gain properties of the material, giving rise to unexpected and contrasting spectral diffraction profiles when compared to the Hermitian ones.
A theoretical model based on two-point scatterers is suggested to investigate scattering of partially coherent radiation by a non-Hermitian localized structure, invariant under the simultaneous symmetry operations of parity inversion and time reversal. Within the first-order Born approximation, and the formalism of classical coherence theory, the spectral density of the scattered field in the far zone is obtained analytically. We find that the unidirectional character of the scattered radiation, which is one of the principal effects present in non-Hermitian structures, may be suppressed due to…
Band theory for partially coherent light is introduced by using the formalism of second-order classical coherence theory under paraxial approximation. It is demonstrated that the cross-spectral density function, describing correlations between pairs of points in the field, can have bands and gaps and form a correlation band structure. The propagation of a partially coherent beam in non-Hermitian periodic structures is considered to elucidate the interplay between the degree of coherence and the gain/loss present in the lattice. We apply the formalism to study partially coherent Bloch oscillati…
In this study, we propose the application of non-Hermitian photonic crystals (PCs) with anisotropic emissions. Unlike a ring of exceptional points (EPs) in isotropic non-Hermitian PCs, the EPs of anisotropic non-Hermitian PCs appear as lines symmetrical about the $Γ$ point. The non-Hermitian Hamiltonian indicates that the formation of EPs is related to the non-Hermitian strength. The real spectrum appears in the $Γ$Y direction and has been validated as the complex conjugate medium (CCM) by effective medium theory (EMT). But for the $Γ$X direction, EMT indicates that the effective refractive in…
We present a method for the propagation of partially coherent radiation using coherent mode decomposition and wavefront propagation. The radiation field is decomposed into a sum of independent coherent modes. Each mode is then propagated separately using conventional wavefront propagation techniques. The summation of these modes in the plane of observation gives the coherence properties of the propagated radiation. As an example, we analyze propagation of partially coherent radiation transmitted through a single circular aperture.
Lensless imaging methods that account for partial coherence have become very common in the past decade. However, there are no metrics in use for comparing partially coherent light fields, despite the widespread use of such metrics to compare fully coherent objects and wavefields. Here, we show how reformulating the mean squared error and Fourier ring correlation in terms of quantum state fidelity naturally generalizes them to partially coherent wavefields. These results fill an important gap in the lensless imaging literature and will enable quantitative assessments of the reliability and reso…
We present an algorithm for holographic shaping of partially coherent light, bridging the gap between traditional coherent and geometric optical approaches. The description of partially coherent light relies on a mode expansion formalism, with possibly thousands of individual modes, and the inverse-design optimization algorithm is based on gradient descent with the help of algorithmic differentiation. We demonstrate numerically and experimentally that an optical system consisting of two phase patterns can potentially achieve any intensity profile transformation with good accuracy.
A representation of partially spatially coherent and partially polarized stationary electromagnetic fields is given in terms of mutually uncorrelated, transversely shifted, fully coherent and polarized elementary electric-field modes. This representation allows one to propagate non-paraxial partially coherent vector fields using techniques for spatially fully coherent fields, which are numerically far more efficient than methods for propagating correlation functions. A procedure is given to determine the elementary modes from the radiant intensity and the far-zone polarization properties of th…
It has long been assumed that there is an intrinsic conflict between optical vortices and partial coherence, in that deterministic phase vortices do not appear in partially coherent fields. We demonstrate, however, that it is possible to construct a beam that has no deterministic vortices in the source plane yet evolves a deterministic vortex at a specified propagation distance.
Using ultrafast photonic first-order differentiator applied on a partially coherent field, we report the generation of two correlated temporal waveforms and study their correlation properties upon linear and nonlinear propagation along the two orthogonal polarization axis of a dispersive optical fiber. Temporal correlations are maintained in linear propagation whereas Kerr nonlinearity generates anti-correlated temporal intensity patterns for both partially and uncorrelated fields. Experiments are in close agreement with the theoretical analysis.
We consider the propagation of both fully coherent and partially coherent complex scalar fields, through linear shift-invariant imaging systems. The state of such imaging systems is characterized by a countable infinity of aberration coefficients, the values for which can be viewed as coordinates for an infinity of orthogonal axes that span the "aberration space" into which the output propagates. For fully coherent complex scalar disturbances, we study the propagation of the field through the imaging system, while for partially coherent disturbances it is the two-point correlation functions wh…
Here the role and influence of aberrations in optical imaging systems employing partially coherent complex scalar fields is studied. Imaging systems require aberrations to yield contrast in the output image. For linear shift-invariant optical systems, we develop an expression for the output cross-spectral density under the space-frequency formulation of statistically stationary partially coherentfields. We also develop expressions for the output cross{spectral density and associated spectral density for weak-phase, weak-phase-amplitude, and single-material objects in one transverse spatial dim…
We present an investigation of the modulational instability of partially coherent signals in electrical transmission lines. Starting from the modified Ginzburg-Landau equations and the Wigner-Moyal representation, we derive a nonlinear dispersion relation for the modulational instability. It is found that the effect of signal broadbandness reduces the growth rate of the modulational instability.
Programmable spatiotemporal shaping for partially coherent structured wave packets
Shandong Normal University（山东师范大学）；University of Shanghai for Science and Technology（上海理工大学）；Tianjin University of Technology（天津工业大学）；Shanghai Jiao Tong University（上海交通大学）；Westlake University（西湖大学）
Optical control of graphene-based photonic devices and systems has been under extensive explorations, nevertheless, the requirement of high power pump laser due to incoherent modulation makes those schemes low efficient. Here, we demonstrate coherent manipulation of the operating states of partially mode-locked fiber laser based on coherent population oscillation in reduced graphene oxide for the first time. We couple a much weaker continuous wave laser into the resonator operating with parametric instability state, and observe significant depression/enhancement of the sidebands when the coher…
Structured fields that are spatially completely coherent have been extensively studied in the context of long-distance optical communication as the structure in the intensity profile of such fields is used for encoding information. This method of doing optical communication works very well in the absence of turbulence. However, in the presence of turbulence, the intensity structures of such fields start to degrade because of the complete spatial coherence of the field, and this structural degradation increases with the increase in the turbulence strength. On the other hand, several theoretical…
For partially coherent light fields with random fluctuations, the intensity distributions and statistics have been proven to be more propagation robust compared with coherent light. However, its full potential in practical applications has not been realized due to the lack of four-dimensional optical field measurement. Here, a general modal decomposition method of partially coherent light field is proposed and demonstrated. The decomposed random modes can be used to, but not limited to, reconstruct average intensity, cross spectral density and orthogonal decomposition properties of the partial…
On applying a theorem previously derived by Wolf and Collett, we demonstrate that partially coherent Gaussian Schell model uctuating sources (GSMS) produce exactly the same optical forces as a fully coherent laser beam. We also show that this kind of sources helps to control the light-matter interaction in biological samples which are very sensitive to thermal heating induced by higher power intensities; and hence the invasiveness of the manipulation. This is a consequence of the fact that the same photonic force can be obtained with a low intensity GSMS as with a high intensity laser beam.
A new model of physically realizable electromagnetic source is proposed. The source is partially coherent and non-uniformly totally polarized. The coherence and polarization characteristics of this new source are analyzed. The spatial coherence area of the source can be easily modified at will. The state of polarization is linear across the transverse plane of the source with an azimuth that varies from point to point in a different way depending on the selected values of the parameters that define the source.
We consider a realistic model for calculating the cross-spectral density of partially coherent beams from an x-ray undulator in a modern storage ring. This two-point coherence function is seen to have a speckled structure associated with the presence of x-ray coherence vortices and domain walls. Such cross-spectral density speckle is associated with a network of spatial pairs of points for which there is zero correlation. X-ray coherence vortices and domain walls are seen to emerge naturally as the number of coherent modes required increases. An understanding of the existence and nature of suc…
We implement an interferometric method using two angular slits to measure the orbital angular momentum (OAM) mode spectrum of a partially coherent field. As the angular separation of the slits changes, an interference pattern for a particular OAM mode is obtained. The visibility of this interference pattern as a function of angular separation is equivalent to the angular correlation function of the field. By Fourier transforming the angular correlation function obtained from the double angular slit interference, we are able to calculate the OAM spectrum of the partially coherent field. This me…
In many applications of absorbing structures it is important to understand their spatial response to incident fields, for example in thermal solar panels, bolometric imaging and controlling radiative heat transfer. In practice, the illuminating field often originates from thermal sources and is only spatially partially coherent when reaching the absorbing device. In this paper, we present a method to fully characterize the way a structure can absorb such partially coherent fields. The method is presented for any 3D material and accounts for the partial coherence and partial polarization of the…
We investigate the Goos-Hänchen (GH) shifts of partially coherent fields (PCFs) by using the theory of coherence. We derive a formal expression for the GH shifts of PCFs in terms of Mercer's expansion, and then clearly demonstrate the dependence of the GH shift of each mode of PCFs on spatial coherence and beam width. We discuss the effect of spatial coherence on the resultant GH shifts, especially for the cases near the critical angles, such as totally reflection angle.
We present a theory and computation method of radiation pressure from partially coherent light by establishing a coherent mode representation of the radiation forces. This is illustrated with the near field emitted from a Gaussian Schell model source, mechanically acting on a single cylinder with magnetodielectric behavior, or on a photonic molecule constituted by a pair of such cylinders. Thus after studying the force produced by a single particle, we address the effects of the spatial coherence on the bonding and anti-bonding states of two particles. The coherence length manifests the critic…
This is the supplementary information to arXiv:1303.4545v1 [physics.optics] 19 Mar 2013 in which we address the forces exerted by the electromagnetic field emitted by a planar uctuating source on dielectric particles that have arose much interest because of their recently shown magnetodielectric behavior.
Recent investigations of microcavity frequency combs based on cascaded four-wave mixing have revealed a link between the evolution of the optical spectrum and the observed temporal coherence. Here we study a silicon nitride microresonator for which the initial four-wave mixing sidebands are spaced by multiple free spectral ranges (FSRs) from the pump, then fill in to yield a comb with single FSR spacing, resulting in partial coherence. By using a pulse shaper to select and manipulate the phase of various subsets of spectral lines, we are able to probe the structure of the coherence within the …
The Siegert relation connects the first- and second-order coherence properties of light. While strictly valid only in the thermal regime and in the absence of correlations, this relation is routinely extended to the partially coherent regime in the study of high-$β$ nanolasers, where it aids in the identification of the lasing threshold. We re-evaluate the use of a generalized Siegert relation in different device regimes. A full two-time quantum-optical theory is derived as a reference for obtaining first- and second-order correlation functions $g^{(1)}(τ)$ and $g^{(2)}(τ)$ in the steady state…
Comprehensive analysis of non-diffracting optical waves with binominal two-point coherence function (BCF) is presented. This coherence function consist of two terms, each depending on either separation of points or central point. We established the necessary and sufficient conditions for non-negative definiteness of the binominal coherence functions. Spectral analysis, including calculations of the eigenvalues and eigenfunctions for the general BCF case is presented. We considered two simple BCF examples of and analyzed the conditions leading to the maximum irradiance contrast.
The complete characterization of spatial coherence is difficult because the mutual coherence function is a complex-valued function of four independent variables. This difficulty limits the ability of controlling and optimizing spatial coherence in a broad range of key applications. Here we propose a method for measuring the complete mutual coherence function, which does not require any prior knowledge and can be scaled to measure arbitrary coherence properties for any wavelength. Our method can also be used to retrieve objects illuminated by partially coherent beam with unknown coherence prope…
A simple method of phase-and-amplitude extraction is derived that corrects for image blurring induced by partially spatially coherent incident illumination using only a single intensity image as input. The method is based on Fresnel diffraction theory for the case of high Fresnel number, merged with the space-frequency description formalism used to quantify partially coherent fields and assumes the object under study is composed of a single material. A priori knowledge of the object's complex refractive index and information obtained by characterizing the spatial coherence of the source is req…
Radially correlated partially coherent beams with a deterministic vortex structure
Shandong Normal University（山东师范大学）；Shandong Provincial Key Laboratory of Light Field Manipulation Physics and Applications（山东省光场调控物理与应用重点实验室）；Collaborative Innovation Center of Light Manipulations and Applications（光操控与应用协同创新中心）
We put forward a theory on the optical force exerted upon a dipolar particle by a stationary and ergodic partially coherent light field. We show through a rigorous analysis that the ensemble averaged electromagnetic force is given in terms of a partial gradient of the space variable diagonal elements of the coherence tensor. Further, by following this result we characterize the conservative and non-conservative components of this force. In addition, we establish the propagation law for the optical force in terms of the coherence function of light at a diffraction plane. This permits us to eval…
The filamentational instability of spatially broadband femtosecond optical pulses in air is investigated by means of a kinetic wave equation for spatially incoherent photons. An explicit expression for the spatial amplification rate is derived and analyzed. It is found that the spatial spectral broadening of the pulse can lead to stabilization of the filamentation instability. Thus, optical smoothing techniques could optimize current applications of ultra-short laser pulses, such as atmospheric remote sensing.
We introduce Gilmore-Perelomov coherent states for non-unitary representations of non-compact groups, and discuss the main similarities and differences with respect to ordinary unitary Gilmore-Perelomov coherent states. The example of coherent states for the non-unitary finite dimensional representations of $SU(1,1)$ is considered and they are used to describe the propagation of light in coupled PT-symmetric optical devices.
In this paper, we extend the paraxial conical refraction model to the case of the partially coherent light using the unified optical coherence theory. We demonstrate the decomposition of conical refraction correlation functions into well-known conical refraction coherent modes for a Gaussian Schell-model source. Assuming randomness of the electrical field phase of the input beam, we reformulated and significantly simplified the rigorous conical refraction theory. This approach allows us to consider the propagation of light through a conical refraction crystal in exactly the same way as in the …
A matrix representation is introduced for stationary beam-like light fields that allows for simultaneous treatment of the orbital angular momentum (OAM) and the second-order state of spatial coherence. This Coherence-OAM matrix is a counterpart of the 2 x 2 beam Coherence-polarization matrix in a Cartesian polarization basis or Coherence-spin angular momentum (SAM) matrix in a circular polarization basis. The general properties of the Coherence-OAM matrix are discussed and then the Coherence-SAM-OAM matrix is introduced as a unifying entity for treating beams with any coherence, spin and orbit…
Inspired by the concept of coherent frozen waves, this paper introduces one possible theoretical framework of its partially coherent version, a frozen spatial coherence, in which a desired two-point correlation structure of an optical field is created on the propagation axis by superposing partially coherent zero-order Bessel beams. It is shown that the cross-spectral density can be given a description in terms of a two-dimensional Fourier series, analogous to the one-dimensional approach of coherent frozen waves. The formalism is applied to the design of a partially coherent field which is hi…
Physical systems with co-existence and interplay of processes featuring distinct spatio-temporal scales are found in various research areas ranging from studies of brain activity to astrophysics. Complexity of such systems makes their theoretical and experimental analysis technically and conceptually challenging. Here, we discover that radiation of partially mode-locked fibre lasers, while being stochastic and intermittent on short time scale, exhibits periodicity and long scale correlations over slow evolution from one round trip to another. The evolution mapping of intensity auto-correlation…
Recently, a broad spectrum of exceptional scattering effects, including bound states in the continuum, exceptional points in PT-symmetrical non-Hermitian systems, and many others attainable in wisely suitably engineered structures have been predicted and demonstrated. Among these scattering effects, those that rely on coherence properties of light are of a particular interest today. Here, we have discussed coherent scattering effects in photonics. Coherent perfect absorption (CPA) generalizes the concept of perfect absorption to systems with several excitation channels. This effect allows all-…
In this work, we theoretically find that coherent perfect absorption (CPA) and laser modes can be realized in a two-dimensional cylindrical structure composed of conjugate metamaterials (CMs). The required phase factors of CMs for achieving CPA and laser modes are determined by the geometric size of the CM cylinder, which is a unique feature compared with other non-Hermitian optical systems. Based on this property, we also demonstrate that CPA and laser modes can exist simultaneously in a CM cylinder with an extremely large size, where the excitations of CPA and laser modes depend on the angul…
Ultrathin flat meta-optics have shown great promise for holography in recent years. However, most of the reported meta-optical holograms rely on only phase modulation and neglect the amplitude information. Modulation of both amplitude and phase in meta-optics either requires polarization sensitive meta-atoms, or complex scatterers with stringent fabrication requirements. Additionally, almost all the meta-optical holograms were measured under laser illumination. Here we adopt the concept of double-phase holography, to report polarization-independent holography with both amplitude and phase modu…
We simulate the focusing of a partially-coherent X-ray beam emitted by an undulator in a fourth-generation storage ring performing coherent mode decomposition and wave optics propagation. The focus position is shifted, and its size is enlarged when an entrance slit crops the beam. This is the usual case when a slit is used to select the coherent fraction. The pairing of two focusing elements (mirrors, lenses or transfocators) to ensure a fixed focal position is also analyzed. Our results show that the image of a partially coherent source, such an undulator in a low-emittance storage ring, is a…
The competition between staggered imaginary potentials and lattice dimerization results in a unidirectional propagating coalescing state of a non-Hermitian Su-Schrieffer-Heeger (SSH) ring at an exceptional point. A segment of the SSH ring inherits the unidirectional feature and exhibits some intriguing dynamic behaviors when it is embedded in a Hermitian chain as a $\mathcal{PT}$-symmetric scattering center. Based on the Bethe ansatz scattering solution for the interface between Hermitian and non-Hermitian regions, we show that the SSH scattering center supports the following wave-packet dynam…
Non-Hermitian wave engineering has attracted a surge of interest in photonics in recent years. One of the prominent phenomena is coherent perfect absorption (CPA), in which the annihilation of electromagnetic scattering occurs by destructive interference of multiple incident waves. This concept has been implemented in various platforms to demonstrate real-time control of absorption, scattering and radiation by varying the relative phase of the excitation signals. However, so far these studies have been limited to simple photonic systems involving single or few modes at well-defined resonant fr…
We address the forces exerted by the electromagnetic field emitted by a planar uctuating source on dielectric particles that have arose much interest because of their recently shown magnetodielectric behavior. In this context, we analyze as a particular case the modification of the Casimir and Van der Waals forces. We study the effect of the source coherence length as well as the interplay between the force from the radiated field and that from the electric and magnetic dipoles induced on the particle. This allows a control of these interactions as well as of the weight and interference effect…
