Form and structure factors for impedance and reflection from periodic layers

In an exact treatment of the Maxwell equations, we derive form and structure factors for reflection from periodic layers, and we show that these factors are significantly different from their analogs in kinematic x-ray diffraction. Quite generally, we show that reflection and impedance can be written precisely as the sum of an additive form factor and the product of a structure factor and a second form factor. This additive form factor does not have an analog in kinematic x-ray diffraction. It is demonstrated that the form factors are found by analytic continuation to an arbitrary wavelength of expressions for the impedance both at long wavelengths and at quarter wavelengths. A correction to the Bragg law relating fringe spacing to the total structure thickness is derived. We go beyond previous numerical work by deriving simple analytic exact expressions for reflection and impedance of periodic layers for all frequencies within the reflection passband, and for an arbitrary number of periods in the structure, an arbitrary index profile within each period, arbitrary layer thicknesses (not just quarter-wave layers), and for arbitrary sizes of the refractive index differences.     

Distributed bragg reflector resonators with cylindrical symmetry and extremely high Q-factors

A simple non-Maxwellian method is presented that allows the approximate solution of all the dimensions of a multilayered dielectric TE0qp mode cylindrical resonant cavity that constitutes a distributed Bragg reflection (DBR) resonator. The analysis considers an arbitrary number of alternating dielectric and free-space layers of cylindrical geometry enclosed by a metal cylinder. The layers may be arranged along the axial direction, the radial direction, or both. Given only the aspect ratio of the cavity, the desired frequency and the dielectric constants of the material layers, the relevant dimensions are determined from only a set of simultaneous equations, and iterative techniques are not required. The formulas were verified using rigorous method of lines (MoL) calculations and previously published experimental work. We show that the simple approximation gives dimensions close to the values of the optimum Bragg reflection condition determined by the rigorous analysis. The resulting solution is more compact with a higher Q-factor when compared to other reported cylindrical DBR structures. This is because it properly takes into account the effect of the aspect ratio on the Bragg antiresonance condition along the z-axis of the resonator. Previous analyses assumed the propagation in the z-direction was independent of the aspect ratio, and the layers of the Bragg reflector were a quarter of a wavelength thick along the z-direction. When the aspect ratio is properly taken into account, we show that the thickness of the Bragg reflectors are equivalent to the thickness of plane wave Bragg reflectors (or quarter wavelength plates). Thus it turns out that the sizes of the reflectors are related to the free-space propagation constant rather than the propagation constant in the z-direction.     

Omni-directional bands in multilayer structure containing exponentially graded materials

The omni-directional reflection (ODR) in a one-dimensional multilayered structure containing exponentially graded material is studied theoretically using the transfer matrix method. We propose a periodic multiplayer structure containing alternate layers of the exponentially graded refractive index and layers of constant refractive index. The reflectance for TE- and TM-modes of the structure are studied for different angles of incidence. We obtained two ODR bands, one in the visible and the other in the infrared region for both polarizations. The behavior of the ODR band in the infrared region is different from the usual Bragg ODR band in the visible region. The width of the ODR band for the TM-mode is larger that for the TE-mode in the wavelength range 850 nm–1050 nm. Such a structure with a large ODR band may be useful in the design of a broad infrared reflector.     

Transmittance analysis of three-dimensional photonic crystals by the effective medium theory

A simple method for calculating the transmittance of three-dimensional photonic crystals is proposed. The crystals are divided into multilayer thin films, and each film is divided into rectangles with a minute width to calculate the effective permittivity of the film by the effective medium theory. Transmittance of the multilayer thin films is calculated with the matrix method. As the number of atomic layers increases, remarkable stop bands appear. When the refractive index of photonic atoms increases, the stop band shifts to a lower frequency, the band widens, and the number of bands increases. Polarization and incident angle dependences are also analyzed. The limit of application for this calculation method is also discussed.     

Plane waves in regular arrays of dipole scatterers and effective-medium modeling

A simple analytical theory for finding eigensolutions for plane electromagnetic waves propagating along an axis in infinite regular arrays of small dipole particles is presented. The spacing between dipoles in every plane is assumed to be smaller than the wavelength; separation between the planes is arbitrary. The influence of evanescent modes is taken into account. This theory gives a model for an effective propagation constant that can be applied in a wide frequency range from the quasi-static regime to the Bragg reflection (photonic bandgap) region.     

Discrete thin-film layer thickness modulation

A novel analytical thin-film design method is presented that is based on electrical engineering communication theory. The proposed thickness modulation describes the thickness modulation of discrete, homogeneous thin-film layers of a multilayer coating. One modulation scheme, amplitude modulation, is presented in which analytical equations determine individual layer thicknesses for a given modulation amplitude, carrier frequency (f(c)), direct-current bias, as well as several layers and refractive indices. The spectral performance (especially stop bands) of multilayer coatings with alternating layers of two refractive indices is presented for different carrier frequencies and modulation amplitudes. For f(c) 相似文献   

基于镜像法提取多层介质中3D电容参数的研究

提出了一种适用于任意多层电介质分层结构中的3D互连电容参数提取的新方法--反射镜像法.该方法无需对格林函数进行推导,而是根据自由电荷反射原理计算反射镜像的位置和镜像电荷系数从而获得各级镜像分布,继而计算任意多层电介质环境下的3D电容,包括底层或顶层介质接地情况下的3D电容.该方法可克服级数形式的格林函数因受限于电介质分层数目而难以适应大规模集成电路中复杂多变电介质环境下的3D电容参数提取问题.通过对导体面元建立层次式数据结构,这一方法可以利用层次式算法(hierarchical method)实现计算加速.实验证明,该方法能在保证可靠精度的情况下达到迅速收敛,与层次式加速算法结合后,计算效率可达到FastCap法的数倍.     

Simple expressions for Bragg reflection from axially excited chiral sculptured thin films

Reported here is the derivation of analytic expressions for the reflection coeffcients of a halfspace filled by a chiral sculptured thin film (STF) that is axially excited by a normally incident plane wave at the centrewavelength of the Bragg regime. Although these expressions are correct to the second order with respect to a small parameter that delineates the local anisotropy of the chiral STF, their first-order reductions are particularly simple and very adequate for practical purposes. A small difference between the two cross-polarized reflection coeffcients has also been found. The results are explained on the basis of a simple model which utilizes the standard Fresnel equations for a scalar dielectric halfspace, combined with coupled wave theory.     

Understanding profile-induced group-delay ripple in Bragg gratings

The relationship between group-delay ripple and the apodization profile of chirped Bragg gratings is analyzed. Simple physical explanations are given for departures from ideal linear group delay by use of only the concepts of reflection at discontinuities and band gaps and the optical path lengths of cavities. Quantitative expressions are obtained for the amplitudes, phases, and periods of both the fast and slow components of the ripple.     

High Q-factor distributed bragg reflector resonators with reflectors of arbitrary thickness

The Bragg reflection technique improves the Q-factor of a resonator by reducing conductor and dielectric losses. This is achieved by designing a low-loss inner resonant region (usually free space) surrounded by an outer anti-resonant region made of distributed Bragg reflector layers. In this paper we develop a simple non-Maxwellian model and apply it to design three distinct cylindrical Bragg resonators based on the same set of single-crystal sapphire plates and rings by changing only the dimension of the cavity that supports the structure. To accomplish this, the simple model allows an arbitrary thickness for either the horizontal or the cylindrical dielectric reflectors by relaxing the condition that they must be lambda/4 thick. The model also allows for higher-order field variations in both the resonant and the anti-resonant regions. The resonators were constructed and experimental results were compared with the simple model and the rigorous method of lines analysis. For the fundamental mode, an unloaded Q-factor of 234,000 at 9.7 GHz was obtained. This is larger than that for a whispering gallery mode resonator. The resonator also exhibited a greatly reduced spurious mode density when compared to an overmoded whispering gallery mode resonator.     

Analysis and fabrication of hybrid metal-dielectric omnidirectional Bragg reflectors

We describe chalcogenide glass and polymer based Bragg reflectors with a metallic underlayer and use a transfer matrix model to analyze their performance. The angle-averaged reflectance of a hybrid mirror approaches unity for only a few periods and is much higher than that for a nonmetallized Bragg reflector or for the metallic layer alone. For an angle-averaged reflectance greater than 0.99, the addition of a metallic underlayer enables nearly a tripling of the omnidirectional bandwidth (from approximately 110 to approximately 305 nm) concurrent with a significant reduction in the number of required periods (from 10.5 to 4.5). Hybrid mirrors of 4.5 periods, with a 50 nm Au underlayer and overall thickness of approximately 2 microm, were fabricated atop silicon substrates and characterized. They exhibit an omnidirectional stop band in the 1450-1750 nm wavelength range, in good agreement with theoretical predictions.     

A treatment of the general volume holographic grating as an array of parallel stacked mirrors

An alternative model to Kogelnik's coupled wave theory of the volume holographic grating is developed in terms of an infinite array of parallel stacked mirrors. The model is based on summing the individual Fresnel reflections from an infinite number of infinitesimal discontinuities in the permittivity profile. The resulting first-order coupled partial differential equations are solved in a rotated frame of reference in order to derive analytical expressions for the diffraction efficiency of the general slanted grating at an arbitrary angle of incidence. The model has been tested using computational solutions of the Helmholtz equation for the unslanted reflection grating. For index modulations characteristic of modern silver halide and photopolymer materials used in display and optical element holography the new model shows excellent agreement with the numerical results. Kogelnik's model also provides good agreement as long as the dephasing parameter is not too large. The model has been tested against Kogelnik's theory for a variety of cases with finite fringe slant with good agreement for typical index modulations. A further advantage of the new model is that colour holographic gratings may be treated at and away from Bragg resonance. Numerical and analytical results are presented concerning the diffractive efficiency of two- and three-colour holographic gratings.     

Highly dispersive mirror in Ta2O5/SiO2 for femtosecond lasers designed by inverse spectral theory

A highly dispersive mirror for dispersion compensation in femtosecond lasers is designed by inverse spectral theory. The design of a simple quarter-wave Bragg reflector can be modified by moving the poles in the optical impedance found in the photonic stop band. These spectral quantities are used as independent variables in the numerical optimization because they have no effect on the location of the photonic stop band, and so the design requirements to obtain a high reflectivity and a specific delay spectrum are decoupled. The design was fabricated by ion-beam sputtering. A group delay dispersion of -300 fs(2) was measured over a bandwidth of 28 nm, with a remaining reflectivity of greater than 99% in this range. The mirrors were used to make two Ti:sapphire lasers with 10- and 4-mm-long crystals, both of which generated near-transform-limited pulses of 35-fs duration. Because of the high dispersion of the mirrors, the laser cavities needed only five and three bounces from the mirrors, thus keeping reflection losses to a minimum.     

采用DBR LD的正弦相位调制激光干涉仪

分析影响正弦相位调制半导体激光干涉仪测量精度和系统分辨力的因素,提出了用分布布拉格反射半导体激光器DBR LD实现高分辨力亚纳米精度测量的方案。理论计算表明,DBR LD的波长连续调制深度比F-P腔LD高一个量级。指出邮于DBR LD的特殊结构可通过简单的反馈回路稳定输出光功率,有效地避免了光强波动对测量精度提高的限制。     

Distributed Bragg reflector based on porous anodic alumina fabricated by pulse anodization

In this paper, we demonstrate a distributed Bragg reflector (DBR) based on nanoporous anodic aluminum oxide (AAO) formed by pulse anodization. The AAO structure with alternating mild anodized (MA) and hard anodized (HA) layers having different porosities and thereby different refractive indices was fabricated in 0.3 M H?SO? using potential pulses of 25 and 35 V. The effective refractive index of the HA layers can be tailored by changing the porosity of the HA layers. The porosity of the HA layers can be significantly increased by selective chemical etching of HA segments in 0.52 M H?PO?. Before etching, the porous AAO structure was supported by a polymer nanorod frame. On the selected surface area pores were infiltrated with polymers (polystyrene and PMMA). The designed AAO structure consists of alternating high and low refractive index layers and behaves as a distributed Bragg mirror reflecting light in two different ranges of wavelength. This behavior is extremely important in optical communication lines where two separate spectral bands of high reflectivity in the infrared region are desired.     

Distributed Bragg reflector made of anodic alumina membrane

A new kind of distributed Bragg reflector is made of layer-by-layer anodic alumina membrane using electrochemical anodization, which is consisted of periodically stacked main stem channel layers and branched channel layers with the period comparable to the optical wavelength. The first Bragg condition peak, which is characterized for main inhibition of incident light perpendicular to the surface of anodic alumina membrane, could be modulated from 727 to 1200 nm by modifying the anodizing voltage waveform. It possibly provides a plan to fabricate light filters in large area, which have very low transmissivity within stop band but high transmissivity in other ranges.     

Multielement hypersonic piezotransducers with slowly varying parameters for acoustooptic devices

The frequency bands of acoustooptic devices can be extended appreciably by using multielement transducers in the form of multielement chains with variable parameters such as the grating step, phase shift per cell, piezolayer thickness, cell impedance, and electrode length of a single element. When the optimum law of variation of the parameters along the transducer length has been found, the law of variation of the angle of inclination of the acoustic wavefront can be accurately matched with the law of variation of the Bragg condition with frequency, which can increase the working frequency band of an acoustooptic device (in particular, a deflector) while maintaining a high diffraction efficiency. In the present paper a relationship obtained earlier to determine the frequency dependence of the structure step, which ensures accurate self-tuning to the Bragg angle, and also a solution of the dispersion equation obtained for one variant of a multielement structure are used to analyze the frequency dependence of the length of isolated electrodes, which determines the inductance of the element, and also the impedance of isolated cells contained in the multielement structure. Pis’ma Zh. Tekh. Fiz. 25, 70–75 (March 12, 1999)     

Analysis of the vibrational mode spectrum of a linear chain with spatially exponential properties

We deduce the dynamic frequency-domain-lattice Green’s function of a linear chain with properties (masses and next-neighbor spring constants) of exponential spatial dependence. We analyze the system as discrete chain as well as the continuous limiting case which represents an elastic 1D exponentially graded material. The discrete model yields closed form expressions for the N×N Green’s function for an arbitrary number N=2,…,∞ of particles of the chain. Utilizing this Green’s function yields an explicit expression for the vibrational mode density. Despite of its simplicity the model reflects some characteristics of the dynamics of a 1D exponentially graded elastic material. As a special case the well-known expressions for the Green’s function and oscillator density of the homogeneous linear chain are contained in the model. The width of the frequency band is determined by the grading parameter which characterizes the exponential spatial dependence of the properties. In the limiting case of large grading parameter, the frequency band is localized around a single finite frequency where the band width tends to zero inversely with the grading parameter. In the continuum limit the discrete Green’s function recovers the Green’s function of the continuous equation of motion which takes in the time domain the form of a Klein-Gordon equation.     

Optical properties of graphene based annular photonic crystals

The optical properties of a graphene based annular photonic crystal (APC) are theoretically investigated. The proposed structure is a hollow core cylindrical shell consists of the alternate dielectric layer and graphene monolayer immersed in free space. In order to study the photonic band structures of the APC, we obtained the optical spectra of the graphene based APC by employing the transfer matrix method in the cylindrical waves for both TE and TM polarizations. In this work we study the effect of different geometrical and optical parameters of the structure on the low loss high reflectance graphene induced band gap. It is found that the graphene induced band gap which appeared in the frequency below 10 THz is polarization independent and remains almost invariant with the change in the period number, the radius of the inner core region and the refractive indices of the inner core region and the surrounding medium. However, its width increases by increasing the azimuthal mode number and the chemical potential of the graphene monolayers and decreases by increasing the refractive index and the thickness of the dielectric layers.     

Derivation of COM equations using the surface impedance method

The surface impedance method is used for the consistent derivation of coupling of modes equations which describe the interaction of SAW with a periodical system of electrodes of finite thickness. The exact analytic solution of the electrostatic problem in the presence of an arbitrary external electric field for a plane system of electrodes is applied to the calculation of the charge and electric field distributions. Mechanical perturbations are taken into account to first order of the thickness of the electrodes. As a result the scalar self-consistent equation for the electric potential of acoustic waves in the gratings is obtained. For the periodic structure this equation is reduced to the form of COM equations for slowly varying amplitudes. Analytical expressions for all coefficients of the COM equations connecting them with geometrical and material parameters are found. The NSPUDT effect can be considered. Dissipation and energy storage terms can be introduced empirically. The solution of the COM equations is represented in the form of a P matrix with elements written in a convenient form. A simple formula for calculating the location of maximum transducer frequency response is proposed, The balance of energy is considered. Some new relations among the elements of P matrix are found