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The generation and focusing properties of higher-order polarized beams have attracted lots of interests due to its significant applications. In this paper,we derived the formula of transforming linear polarization into higher-order polarization, which is applicable to generating arbitrary order polarization. Based on the derived formula, the focusing properties of higher-order polarization by dielectric metasurface lens are studied , which exhibit an Abbe-limit-breaking feature for small numerical aperture, i.e., NA<0.6. When a binary phase (0 & {\pi}) is further imposed on the aperture of metasurface lens, the focusing spot of fourth-order polarization breaks Abbe limit even by 14.3% at NA= 0.6. In addition, the effect of fabrication tolerance, say, substrate thickness and central deviation, on the focusing feature of higher-order polarization is also investigated. Our study may find significant applications in achieving higher-resolution lithography and imaging, say, by just replacing conventional linear or circular polarization with higher-order polarization., Comment: 8 pages, 4 figures

Fractional-order vector vortex beams are recently demonstrated to be new carriers of fractional-strength optical vortices. However, why can those new vortex beams formed by the combination of both unstable states propagate stably in free space? Here, we solve this scientific problem by revealing the physical essence of propagable fractional-strength optical vortices in free space.Three new understandings regarding those peculiar vortex beams are therefore proposed, namely Abbe diffraction limit, phase evolution of vortex beam, and phase binary time vector property.For the first one, owing to Abbe diffraction limit, the inherent polarization modes are intertwined together, thereby maintaining the entire peculiar vortex beams in free space. For the second one, we demonstrate the phase evolution of vortex beam, which is the physical reason of polarization rotation of fractional-order VVBs. For the third one, the phase is not merely a scalar attribute of light beam, but manifests a binary time vector property. This work provides entirely different physical viewpoints on the phase of vortex beam and Abbe diffraction limit, which may deepen our knowledge on the behavior of light beam in classical optics.

A channel should be built to transmit information from one place to another. Imaging is 2 or higher dimensional information communication. Conventionally, an imaging channel comprises a lens and free spaces of its both sides. The transfer function of each part is known; thus, the response of a conventional imaging channel is known as well. Replacing the lens with a scattering layer, the image can still be extracted from the detection plane. That is to say, the scattering medium reconstructs the channel for imaging. Aided by deep learning, we find that different from the lens there are different channels in a scattering medium, i.e., the same scattering medium can construct different channels to match different manners of source encoding. Moreover, we found that without a valid channel the convolution law for a shift-invariant system, i.e., the output is the convolution of its point spread function (PSF) and the input object, is broken, and information cannot be transmitted onto the detection plane. In other words, valid channels are essential to transmit image information through even a shift-invariant system., Comment: 10 pages, 4 figures

Scattering of light in complex media scrambles optical wavefronts and breaks the principles of conventional imaging methods. For decades, researchers have endeavored to conquer the problem by inventing approaches such as adaptive optics, iterative wavefront shaping, and transmission matrix measurement. That said, imaging through/into thick scattering media remains challenging to date. With the rapid development of computing power, deep learning has been introduced and shown potentials to reconstruct target information through complex media or from rough surfaces. But it also fails once coming to optically thick media where ballistic photons become negligible. Here, instead of treating deep learning only as an image extraction method, whose best-selling advantage is to avoid complicate physical models, we exploit it as a tool to explore the underlying physical principles. By adjusting the weights of ballistic and scattered photons through a random phasemask, it is found that although deep learning can extract images from both scattered and ballistic light, the mechanisms are different: scattering may function as an encryption key and decryption from scattered light is key sensitive, while extraction from ballistic light is stable. Based on this finding, it is hypothesized and experimentally confirmed that the foundation of the generalization capability of trained neural networks for different diffusers can trace back to the contribution of ballistic photons, even though their weights of photon counting in detection are not that significant. Moreover, the study may pave an avenue for using deep learning as a probe in exploring the unknown physical principles in various fields., Comment: 12 pages, 6 figures

Quantum interference occurs frequently in the interaction of laser radiation with materials, leading to a series of fascinating effects such as lasing without inversion, electromagnetically induced transparency, Fano resonance, etc. Such quantum interference effects are mostly enabled by single-photon resonance with transitions in the matter, regardless of how many optical frequencies are involved. Here, we demonstrate quantum interference driven by multiple photons in the emission spectroscopy of nitrogen ions that are resonantly pumped by ultrafast infrared laser pulses. In the spectral domain, Fano resonance is observed in the emission spectrum, where a laser-assisted dynamic Stark effect creates the continuum. In the time domain, the fast-evolving emission is measured, revealing the nature of free-induction decay (FID) arising from quantum radiation and molecular cooperativity. These findings clarify the mechanism of coherent emission of nitrogen ions pumped with MIR pump laser and are likely to be universal. The present work opens a route to explore the important role of quantum interference during the interaction of intense laser pulses with materials near multiple photon resonance., Comment: 20 pages, 8 figures

Photonic lattices have been widely used for simulating quantum physics, owing to the similar evolutions of paraxial waves and quantum particles. However, nonparaxial wave propagations in photonic lattices break the paradigm of the quantum-optical analogy. Here, we reveal that nonparaxiality exerts stretched and compressed forces on the energy spectrum in the celebrated Aubry-Andre-Harper model. By exploring the mini-gaps induced by the finite size of the different effects of nonparaxiality, we experimentally present that the expansion of one band gap supports the adiabatic transfer of boundary states while Landau-Zener transition occurs at the narrowing of the other gap, whereas identical transport behaviors are expected for the two gaps under paraxial approximation. Our results not only serve as a foundation of future studies of dynamic state transfer but also inspire applications leveraging nonparaxial transitions as a new degree of freedom., Comment: 17 pages, 4 figures

Is it possible to modulate the inherent properties of a single light beam, namely amplitude, phase and polarization, simultaneously, by merely its phase? Here, we solve this scientific problem by unifying all these three properties of a single light beam using phase vectorization and phase version of Malus's law. Full-property spatial light modulator is therefore developed based on the unification of these fundament links, which enables pixel-level polarization, amplitude and phase manipulation of light beams in a real-time dynamic way. This work not only implies that the amplitude, phase and polarization of a single light beam are interconnected, but also offers a solid answer on how to modulate these three natures of a single light beam simultaneously, which will deepen our understanding about the behavior of light beam, and facilitating extensive developments in optics and relate fields.

Light beam with optical vortices can propagate in free space only with integer orbital angular momentum. Here, we invert this scientific consensus theoretically and experimentally by proposing light beams carrying natural non-integer orbital angular momentum. These peculiar light beams are actually special solutions of wave function, which possess optical vortices with the topological charge l+0.5, where l is an integer. Owing to the interaction of phase and polarization singularity, these vortex beams with fractional topological charge can maintain their amplitude and vortex phase even when they propagate to an infinite distance. This work demonstrates another state of optical vortices in free space, which will fundamentally inject new vigor into optics, and other relate scientific fields.

Modern scattering-type scanning near-field optical microscopy (s-SNOM) has become an indispensable tool in material research. However, as the s-SNOM technique marches into the far-infrared (IR) and terahertz (THz) regimes, emerging experiments sometimes produce puzzling results. For example, anomalies in the near-field optical contrast have been widely reported. In this Letter, we systematically investigate a series of extreme subwavelength metallic nanostructures via s-SNOM near-field imaging in the GHz to THz frequency range. We find that the near-field material contrast is greatly impacted by the lateral size of the nanostructure, while the spatial resolution is practically independent of it. The contrast is also strongly affected by the connectivity of the metallic structures to a larger metallic ground plane. The observed effect can be largely explained by a quasi-electrostatic analysis. We also compare the THz s-SNOM results to those of the mid-IR regime, where the size-dependence becomes significant only for smaller structures. Our results reveal that the quantitative analysis of the near-field optical material contrasts in the long-wavelength regime requires a careful assessment of the size and configuration of metallic (optically conductive) structures.

Superposition of two independent orthogonally polarized beams is a conventional principle of creating a new light beam. Here, we intend to achieve the inverse process, namely, extracting inherent polarization modes from a single light beam. However, inherent polarization modes within a light beam are always entangled so that a stable polarization is maintained during propagation in free space. To overcome this limitation, we report an approach that breaks the modulation symmetry of a light beam, thereby disentangling the inherent polarization modes. Using polarization mode competition along with an optical pen, polarization modes are extracted at will in the focal region of an objective lens. This work demonstrates polarization mode extraction from a light beam, which will not only provide an entirely new principle of polarization modulation but also pave the way for multidimensional manipulation of light fields, thereby facilitating extensive developments in optics.

Wavefront aberrations can reflect the imaging quality of high-performance optical systems better than geometric aberrations. Although laser interferometers have emerged as the main tool for measurement of transmitted wavefronts, their application is greatly limited, as they are typically designed for operation at specific wavelengths. In a previous study, we proposed a method for determining the wavefront transmitted by an optical system at any wavelength in a certain band. Although this method works well for most monochromatic systems, where the image plane is at the focal point for the transmission wavelength, for general multi-color systems, it is more practical to measure the wavefront at the defocused image plane. Hence, in this paper, we have developed a complete method for determining transmitted wavefronts in a broad bandwidth at any defocused position, enabling wavefront measurements for multi-color systems. Here, we assume that in small ranges, the Zernike coefficients have a linear relationship with position, such that Zernike coefficients at defocused positions can be derived from measurements performed at the focal point. We conducted experiments to verify these assumptions, validating the new method. The experimental setup has been improved so that it can handle multi-color systems, and a detailed experimental process is summarized. With this technique, application of broadband transmission wavefront measurement can be extended to most general optical systems, which is of great significance for characterization of achromatic and apochromatic optical lenses., Comment: 18 pages, 14 figures

Vector polarization induced by a change in scalar phase has been far beyond our understanding of the relationship between polarization and phase in classical optics due to the entanglement of inherent polarized modes of light beams. To overcome this limitation, we establish this miraculous relationship by the principle of phase vectorization, which can transform three particular phases into linear, circular and elliptical polarizations. A polarized-spatial light modulator based on the principle of phase vectorization can be achieved using a phase-only spatial light modulator, which not only enables pixel-level polarization manipulation of light beams in a real-time dynamic way but also simultaneously retains complete phase control. This work demonstrates the phase-to-polarization link and reports the creation of polarized-spatial light modulator, which will transform our view of the natural properties of light beam and pave the way for the era of vector optics.

Singly ionized nitrogen molecules in ambient air pumped by near-infrared femtosecond laser give rise to superradiant emission. Here we demonstrate coherent control of this superradiance by injecting a pair of resonant seeding pulses inside the nitrogen gas plasma. Strong modulation of the 391.4 nm superradiance with a period of 1.3 fs is observed when the delay between the two seeding pulses are finely tuned, pinpointing the essential role of macroscopic coherence in this lasing process. Based on this time-resolved method, the complex temporal evolution of the macroscopic coherence between two involved energy levels has been experimentally revealed, which is found to last for around 10 picoseconds in the low gas pressure range. These observations provide a new level of control on the "air lasing" based on nitrogen ions, which can find potential applications in optical remote sensing.

A cavity-free laser in the sky could lead to revolutionary improvements in optical remote sensing for atmospheric science. Abundant in air, nitrogen molecules are prime candidates as an active medium for such a laser. Nitrogen molecules, either neutral or singly ionized, exhibit a significant optical gain in the UV regime when pumped by an intense femtosecond laser. Despite numerous works, the physical origin of this lasing is still not understood. Here we offer a consistent interpretation of all experimental facts by introducing a concept drawn from the field of quantum optics: transient lasing without inversion. Our experiments and simulations identify the crucial role of macroscopic coherence, instead of population inversion, in the nitrogen ions lasing system. Manipulating the macroscopic coherence provides a new level of control over nitrogen air lasing and offers new perspectives for the generation of very short wavelength laser where population inversion is difficult to achieve.

A light beam propagating with an infinite anti-diffracting distance requires infinite power to preserve its shape. However, the fundamental barrier of finite power in free space has made the problem of diffraction insurmountable over the past few decades. Here, to overcome this limitation, we report an approach that employs the multiple energy oscillation mechanism, thereby permitting the creation of a light beam with an ultralong anti-diffracting distance in free space. The anti-diffracting distance is no longer restricted by finite power in free space but instead depends on the number of energy oscillations. This unprecedented propagation behavior is attributed to a new understanding of non-diffractive light beam: when an anti-diffracting light beam completely discharges its energy, it cannot recharge again. A versatile optical pen is therefore developed to manipulate the number, amplitude, position and phase of energy oscillations for an arbitrary numerical aperture of a focusing lens so that energy recharge can occur in free space and multiple energy oscillations can be realized. A light beam with a tunable number of energy oscillations is eventually generated in free space and propagates along a wavy trajectory. This work will enable extending non-diffractive light beams to an expanded realm and facilitate extensive developments in optics and other research fields, such as electronics and acoustics.

We present an analytical nonlocal effective medium approximation to describe the optical nonlocal effects in metallic nanorod metamaterials based on Mie scattering theory. It is shown that the developed nonlocal effective medium theory can predict a coexistence state of two modes around the epsilon-near-zero region, where strong optical nonlocal effects lead to the behavior of both positive refraction and negative refraction in the nanorod metamaterials. Outside of the coexistence region, only one mode can be excited and its behavior can be well described using the local effective medium theory., Comment: 13 pages, 7 figures

The energy flux patterns of inverse Goos-Hanchen (GH) shift around the interface between air and negatively refractive photonic crystal (NRPhC) with different surface terminations is investigated. Results show that NRPhC exhibits inverse GH shift in TM and TE polarization, and the localization and pattern of energy flux differ in TM and TE polarizations and are strongly affected by surface termination. This is different to the condition of negative permittivity materials (i.e., metal), which only presents inverse GH shift in TM polarization. In the case of TE polarization, the energy flux pattern exhibits the flux of backward wave whose localization changes from the surface to inside of NRPhC with the variation of surface termination. In the case of TM polarization, the energy flux pattern is always confined within the surface of NRPhC, whereas its pattern changes from the flux of backward wave to vortices at the surface of NRPhC, which is different to the energy flux of TM polarization of metal. By properly truncating the surface of NRPhC we can control the magnitude of inverse GH shift for TM and TE polarized light., Comment: 5 pages,6 figures

The effect of surface termination on the inverse Goos Hanchen (GH) shift of two dimensional (2D) negatively refractive photonic crystal (NRPhC) containing air holes arranged in hexagonal lattice in a dielectric background is investigated for TM polarization. Results show that the magnitude of the inverse GH shift of 2DNRPhC strongly depends on surface termination even for an incident beam with a fixed frequency and incidence angle. Further study by calculating the dispersion of surface mode of 2D-NRPhC as a function of surface termination reveals that 2DNRPhC presents large inverse Goos Hanchen shift at those terminations where surface mode is excited, that is, large inverse Goos-Hanchen shift originates from backward surface mode of 2DNRPhC. In addition, the coupling coefficient of incident field into the field of surface mode as a function of surface termination is studied and demonstrates above results. This paper provides technical information regarding the combination of various functional photonic elements in the design of integrated optical circuits., Comment: 5 pages,6 figures