Focal Shift of Paraxial Gaussian Beams in a Left-Handed Material Slab Lens

Focal shift is inevitable in conventional lens systems due to the Fresnel number and angular aperture. In this Letter, we demonstrate that there is no focal shift when a paraxial Gaussian beam passes through a left-handed material slab lens without absorption or gain. However, the effect is exhibited in the presence of absorption or gain, and becomes larger as the absorption or gain increases. When the absorption is equal to the gain, the phenomenon of the focal shift caused by the gain is more obvious. In addition, the field distribution is not affected by the absorption or gain and always remains Gaussian both in internal and external focus planes.     

Focusing performance of the closed-boundary cylindrical microlenses analyzed by the boundary element method

In this paper, we investigate the focusing performance of closed-boundary cylindrical microlenses (CBCMs) based on rigorous electromagnetic theory and the boundary element method. The CBCMs with different incident angles, different quantization-level numbers, different microlens diameters, different f-numbers, and different polarizations of incidence are studied. Several focusing performance measures, such as the focal spot size, the diffraction efficiency, the real focal position, and the normalized transmitted power, are presented. It provides very useful information in designing the CBCMs in micro-optical systems.     

Positive and Negative Lateral Shifts from an Anisotropic Metamaterial Slab Backed by a Metal

The lateral shift of a light beam at the surface of an anisotropic metamaterial (AMM) slab backed by a metal is investigated. Analytical expressions of the lateral shifts are derived using the stationary-phase method, in the case that total reflection does and does not occur at the first interface. The sign of the lateral shift in two situations is discussed, and the necessary conditions for the lateral shift to be positive or negative are given. It is shown that the thickness and physical parameters of the AMM slab and the incident angle of the light beam strongly affect the properties of the lateral shift. Numerical results validate these conclusions. The lossy effect of the metamaterial on the lateral shift is also investigated.     

The Imbert-Fedorov shift of paraxial light beams

We present a solution to the problem of reflection and transmission of a polarized paraxial light beam at an interface between two homogeneous media by using a two-form amplitude and an extension matrix to represent the vectorial angular spectrum of a three-dimensional (3D) light beam. We derive general formulas for the Imbert-Fedorov (IF) shift of the reflected and transmitted beams of a polarized paraxial light beam. The IF shift of two different types of polarized beams is calculated, and the influence of the polarization state and the polarization feature of the vectorial angular spectrum on the IF shift is discussed.     

ABCD matrix formalism for propagation of Gaussian beam through left-handed material slab system

We apply a simple ABCD matrix formalism to investigate beam propagation in left-handed material (LHM) slab systems. Firstly, we derive the expressions for the matrix elements for a single LHM slab. Based on the ABCD matrix, we obtain the field distributions both inside and outside the LHM slab, and the conditions for focusing and phase compensation of the Gaussian beam. Secondly, the derived ABCD law is applied to analyze the propagation properties of Gaussian beams through various cascaded LHM slab systems. The corresponding focusing and phase compensation conditions of the Gaussian beam are also obtained. Our theoretic formalism may be applied as a compact and effective tool for the research of light beam propagation in complex LHM slab systems, since it avoids the complicated integral operation.     

Simultaneously Large and Opposite Lateral Beam Shifts for TE and TM Modes on a Double Metal-Cladding Slab

We report simultaneously large and opposite Goos-Hanchen shifts for TE and TM beams on a double metalcladding slab. Theoretical examination shows that both positive and negative lateral shifts are in two orders of the wavelength. It is also found that the magnitude of the lateral beam shift strongly depends on the thickness of the upper metal layer. The optimal thickness of the upper metal layer for zero reflection is found to be the critical thickness above which a negative beam shift occurs. Numerical calculations are in good agreement with the theoretical results.     

Generation of thin and hollow beams by the axicon with a large open angle

We propose a method to produce diffraction-free thin and hollow beams. The method is based on Laguerre-Gaussian (LG) beams incident on a large open-angle axicon. We use the vector diffraction integrals and stationary phase method to deduce a simple and analytical formula of the propagating field of the linearly polarized LG beams through an axicon. The numerical results show that the hollow beams of whose diameter is in the order of the wavelength can be obtained by using the axicon with the refractive index n = 2 and the open angle α = 25°. These diffraction-free thin and hollow beams may be very useful to accurately trap and manipulate atoms. However, when the open angle is over large, the conversion efficiency from the LG beam to the diffraction-free hollow beam will decrease obviously.     

Experimental study of quasi-one-dimensional comb-like photonic crystals containing left-handed material

Quasi-one-dimensional comb-like photonic crystals with backbone waveguide made of left-handed material are physically fabricated by using transmission lines. The transmission properties are experimentally investigated and it is found that the structures possess a special band gap. The gap does not originate from the conventional Bragg scattering mechanism in that it is invariant with the change of scale length. Moreover, the experimental results indicate that the gap-edges are determined by zero-average permittivity of the branch and backbone and zero permeability of backbone, respectively. Because of the characteristics, it will be convenient to realize a special gap using in microwave and optical engineering.     

Double groove silicon grating polarizer in telecommunication wavelength band

In this paper, we propose an ultra broad band polarizer operating in the telecommunication wavelength band, this device consisting a double groove silicon grating is designed with using the inverse mathematical method and rigorous vector diffraction theory. It is shown from our calculations that the device presents extremely high reflection (R > 95%) for TE polarization light and high transmission (T > 95%) for TM polarization over ∼400 nm wavelength range, moreover, the extinction ratio is ∼30 in the central wavelength 1550 nm. Furthermore, it is found with rigorous coupled wave analysis (RCWA) that the extremely wide band property for TE polarization is due to the excitation of strong modulation guided modes in the design wavelength range.     

A Zone Plate as a Tunable Terahertz Filter

A broadband tunable terahertz filter based on a zone plate is demonstrated in our terahertz time domain spec- trometer. The central bandpass frequency covers the whole spectral range of the terahertz wave emitted from a ZnTe emitter, from 0.5 THz to 2.5 THz, and can be tuned continuously by simply moving the zone plate along the terahertz beam path. The peak transmission is about 40% and the bandwidth varies from 0.16 THz to 0.25 THz at different bandpass frequencies when the aperture size is kept constant.     

Broad band beam splitter based on the double-groove fused silica grating

In this paper, we propose a broad band 1 × 3 beam splitter operating in the telecommunication wavelength band under normal incidence, this device consisting of a double-groove fused silica grating layer is designed with using the inverse mathematical method and rigorous vector diffraction theory. It is shown from our calculations that the device presents excellent beam splitter ability for TE polarization light with the average diffraction efficiencies is more than 95% over ∼100 nm wavelength range, moreover, the uniformity of our beam splitter is better than 2% in the whole wavelength band. Furthermore, the physical understanding of the diffraction behaviors taking place inside the beam splitter gratings can be explained by the modal method.     

Wide band polarizer with suspended silicon grating

In this paper, we propose an ultra broadband polarizer operating in the telecommunication wavelength band; this device consisting a single silicon suspended resonant grating layer is designed with using the inverse mathematical method and rigorous vector diffraction theory. It is shown from our calculations that the device presents extremely high reflection (R > 98%) for TE polarization light and high transmission (T > 98%) for TM polarization over ∼330 nm wavelength range; moreover, the extinction ratio is ∼100 in the central wavelength 1550 nm. Furthermore, it is found with Rigorous Coupled Wave Analysis (RCWA) and near field distribution that the extremely wide band property for TE polarization is due to the excitation of strong modulation guided modes in the design wavelength range.     

Electric and magnetic losses and gains in determining the sign of refractive index

Within the framework of classic electromagnetic theories, we have studied the sign of refractive index of optical medias with the emphases on the roles of the electric and magnetic losses and gains. Starting from the Maxwell equations for an isotropic and homogeneous media, we have derived the general form of the complex refractive index and its relation with the complex electric permittivity and magnetic permeability, i.e. , in which the intrinsic electric and magnetic losses and gains are included as the imaginary parts of the complex permittivity and permeability, respectively, as  = _r + i_i and μ = μ_r + iμ_i. The electric and magnetic losses are present in all passive materials, which correspond, respectively, to the positive imaginary permittivity and permeability _i > 0 and μ_i > 0. The electric and magnetic gains are present in materials where external pumping sources enable the light to be amplified instead of attenuated, which correspond, respectively, to the negative imaginary permittivity and permeability _i < 0 and μ_i < 0. We have analyzed and determined uniquely the sign of the refractive index, for all possible combinations of the four parameters _r, μ_r, _i, and μ_i, in light of the relativistic causality. A causal solution requires that the wave impedance be positive Re{Z} > 0. We illustrate the results for all cases in tables of the sign of refractive index. One of the most important messages from the sign tables is that, apart from the well-known case where simultaneously  < 0 and μ < 0, there are other possibilities for the refractive index to be negative n < 0, for example, for _r < 0, μ_r > 0, _i > 0, and μ_i > 0, the refractive index is negative n < 0 provided μ_i/_i > μ_r/_r.     

Analysis of closed-boundary cylindrical microlenses with long focal depth designed by the general focal length function

In this paper, the general focal length function is used to design two-dimensional closed-boundary cylindrical microlenses (CBCMs) with long focal depth. The focusing characteristics of the designed microlenses is investigated by rigorous electromagnetic theory and the boundary element method. A number of focusing performance measures of the designed microlenses, such as the real focal depth, the focal depth range, the focal spot size, and the diffraction efficiency, are presented in detailed. As comparison, the focusing performance of the conventional lenses with the same parameters are investigated simultaneously. Our analysis indicates that the general focal length function is valid in designing CBCMs with larger extended focal depth. Comparing with the open-boundary cylindrical microlenses (OBCMs) designed using the same focal length function, we also find that the designed CBCMs with low f-number exhibit superiority of long focal depth.     

Polarization-sensitive propagation in an anisotropic metamaterial with double-sheeted hyperboloid dispersion relation

The polarization-sensitive propagation in the anisotropic metamaterial (AMM) with double-sheeted hyperboloid dispersion relation is investigated from a purely wave propagation point of view. We show that TE and TM polarized waves present significantly different characteristics which depend on the polarization. The omnidirectional total reflection and oblique total transmission can occur in the interface associated with the AMM. If appropriate conditions are satisfied, one polarized wave exhibits the total refraction, while the other presents the total reflection. We find that the opposite amphoteric refractions can be realized by rotating the principle axis of AMM, such that one polarized wave performs the negative refraction, while the other undergoes positive refraction. The polarization-sensitive characteristics allow us to construct two types of efficient polarizing beam splitters under certain achievable conditions.     

A polarization-dependent wide-angle three-dimensional metamaterial absorber

In this paper, a polarization-dependent wide-angle three-dimensional metamaterial absorber with a near-unity absorbance was presented. The metamaterial absorber structure is composed of coplanar electric and magnetic resonators. By carefully adjusting the structural dimensions, less-than-unity ε and/or μ can be realized. To match the impedance of free space, the structural dimensions were adjusted so that ε=μ, which guarantees minimum reflection. Since the resonance-based structure is made of metallic resonators and lossy substrates, the imaginary part of refractive index is large, which guarantees strong absorption of transmitted waves. Full-wave simulations confirmed the effectiveness of the proposed three-dimensional metamaterial absorber.     

Unique properties of microwave in interlayer exchange-coupled trilayer ferromagnetic films associated with negative imaginary part of permeability

Based upon Landau-Lifshitz equation and Maxwell's equations, we theoretically investigated properties of normally incident microwave propagation in interlayer exchange-coupled trilayer ferromagnetic film. It is found that, near resonance frequency of optic mode, imaginary part of permeability of one ferromagnetic layer is smaller than zero unusually, i.e., the ferromagnetic layer may be taken as an active medium. Thus a number of unique electromagnetic properties are presented, such as, the ferromagnetic layer becomes a left-handed material near low side of the resonance frequency of optic mode, and both phase velocity and time-averaged Poynting flow of the usually defined forward wave are negative simultaneously near high side of the resonance frequency. This work provides a feasible active medium to demonstrate the unique microwave properties.     

Abrupt change of reflectivity from the strongly anisotropic metamaterial

The wave reflection from a non-magnetic anisotropic metamaterial, whose principal elements of the permittivity tensor have different signs, is investigated in this paper. It is found that, if the orientation of the optical axis is properly chosen, an extremely small change of the transverse wave number will lead to a dramatically change of the reflectivity at the glancing incidence. The physical insight for this abrupt change of reflectivity is also given by the analysis of the imaginary part of the k-surface. Since the metamaterial discussed here have been experimental realized from GHz to optical frequencies, the proposed abrupt change property of reflectivity may find some potential applications in various calibration devices, because of its extremely sensitivity to the transverse wave number.     

Multiple omnidirectional resonances in a metamaterial sandwich

Multiple omnidirectional resonances are found in a sandwich structure consisting of a left-handed metamaterial cavity and two single-negative-permittivity reflection walls. These resonances appear because the phase shift of left-handed metamaterial can cancel the reflection phase shift of single-negative-permittivity metamaterials at every incidence angle. By using such particular structure, a kind of structure possessing multiple omnidirectional resonances in both polarizations has been designed. In addition, issues associated with energy loss are discussed.     

Two-Dimensional （2D） Polygonal Electromagnetic Cloaks

Transformation optics offers remarkable control over electromagnetic fields and opens an exciting gateway to design ＇invisible cloak devices＇ recently. We present an important class of two-dimensional （2D） cloaks with polygon geometries. Explicit expressions of transformed medium parameters are derived with their unique properties investigated. It is found that the elements of diagonalized permittivity tensors are always positive within an irregular polygon cloak besides one element diverges to plus infinity and the other two become zero at the inner boundary. At most positions, the principle axes of permittivity tensors do not align with position vectors. An irregular polygon cloak is designed and its invisibility to external electromagnetic waves is numerically verified. Since polygon cloaks can be tailored to resemble any objects, the transformation is finally generalized to the realization of 2D cloaks with arbitrary geometries.