Power Losses in Highly Conducting Lamellar Gratings

Abstract

We compare approaches based on a surface impedance approximation and a perturbation analysis with a rigorous modal technique for the calculation of the diffraction properties of highly conducting lamellar gratings. We show the surface impedance approximation to be inaccurate even for good metals at a wavelength of 10 μm. The perturbation theory results can be quite accurate, but their region of applicability is limited. We demonstrate that the rigorous technique can be used for lamellar gratings made of materials with a complex refractive index of very large modulus.     

A Comparison of Extinction Efficiencies in the Eikonal and the Anomalous Diffraction Approximations for Spheres

Abstract

Extinction curves in the eikonal approximation and the anomalous diffraction approximation have been compared with exact extinction curves for homogeneous spheres of various refractive indices. Our results show the anomalous diffraction approximation to be more suitable for extinction calculations.     

On the Validity of Soft-particle Approximations for the Scattering of Light by Infinitely Long Homogeneous Cylinders

Abstract

The relationship between the Hart-Montroll approximation and various other soft particle approximations has been examined for the scattering of light by infinitely long homogeneous cylinders. Numerical comparisons of this approximation with exact results and other approximations have been presented. It is noted that this approximation can be used to obtain the diameter of the particles to within 1% in the domain 1·05  m  1·15 and 5  x  20·0, m and x being the relative refractive index and the size parameter of the particle respectively.     

Equivalence of interaction hamiltonians in the electric dipole approximation

Abstract

The interaction of an atomic system with an externally applied electromagnetic field can be treated in the electric dipole approximation by means of either the minimal coupling (p · A) or direct coupling (d · E) Hamiltonian. It is shown that both methods lead to identical and unambiguous predictions for observable quantities as long as the atomic wavefunctions are transformed when used in the minimal-coupling formulation. The physical meaning of kinetic momentum is used to show that the atomic states must be described by wavefunctions calculated in the absence of an electromagnetic field when using the d · E (but not the p · A) form of the interaction Hamiltonian. When, however, observables are calculated using the common approximations of resonance atomic physics – the two-level approximation and the rotatingwave approximation – the two formulations can lead to measurably different results. This point is illustrated by calculating the induced polarization (and hence the refractive index) of an atomic system for the two exactly soluble cases of the harmonic oscillator and the hydrogen atom.     

Image Perturbations and Optical Flow in Time-dependent Refraction

Abstract

The ray formulation for light propagation is applied to the problem of image perturbations (‘shimmering’) resulting from perturbations in the refractive index profile of the medium between an object and an observer. Expressions are derived relating the image diagram perturbation and the instantaneous image motion (optical flow) to the change in the refractive index profile, in the paraxial ray approximation. In special circumstances the image perturbation and optical flow can be estimated directly. These results are applied to two simple examples.     

On the Validity of the Anomalous Diffraction Approximation

Abstract

The derivation of the anomalous diffraction approximation from Mie theory has been re-examined. The usual requirement ¦m ? 1¦ ? 1, m being the relative refractive index of the scatterer, is shown to be over-restrictive. A more appropriate requirement is found to be ¦m ? 1¦² ? ¦m + 1¦². The relationship between the anomalous diffraction approximation and another m→1 approximation, namely the eikonal approximation, has been investigated.     

Wave-optical structure design with the local plane-interface approximation

Abstract

The design of an optical element profile with specified transmission function for a given incident wave is of fundamental concern in optical design. A well-known example is the design of an aspherical surface in order to realize a spherical phase-only transmission. Various wave-optical system design methods lead to transmission functions as a first step. Then, often the thin-element approximation is applied in a second step to obtain an element structure with the desired transmission. However, if the refraction at the optical interface cannot be neglected, the thin-element approximation is not valid. In this case, a higher version of the local plane-interface approximation can be used for the element structure design. An algorithm for this model is introduced and its characteristics are discussed for the example of a non-paraxial Gaussian-to-tophat beam shaping element.     

Photonic band structure of guided bloch modes in high index films fully etched through with periodic microstructure

Abstract

By adapting the well-known ‘zigzag’ ray model for use with a periodic waveguide (i.e. replacing the plane wave rays with Bloch wave rays), we show that thin films of high refractive index, supported by a low index substrate and fully etched through with a periodic pattern, can support guided modes. From the dispersion relation of these guided Bloch modes, it is shown that the in-plane modal group velocity can be zero, suggesting applications in enhanced dipole-field interactions and control of spontaneous emission in waveguide lasers.     

Transparent high refractive index nanocomposite thin films

This work reports the preparation of acetic acid-modified TiO₂ nanoparticles by sol–gel synthesis method. The nanoparticles can be incorporated directly into the polymer matrix to form transparent high refractive index nanocomposite thin films. The result shows that increasing the titania content in the hybrid nanocomposite thin films can significantly increase the refractive index. Hybrid nanocomposite thin film with refractive index value of 2.38 had been prepared. All prepared films also exhibit excellent optical transparency in the visible region.     

Solitary Waves of Maxwell's Equations in Nonlinear Anisotropic Media

Abstract

The (1 + 1)-D solitary wave solutions of Maxwell's equations in nonlinearity induced anisotropic media (in liquids such as carbon disulphide, and in crystals, etc.) are investigated. We find that there is no arbitrarily linearly polarized (in the x-y plane perpendicular to the propagation direction z) soliton solution from Maxwell's equations except that with linear polarization either in alignment with or orthogonal to the geometric axis of the light induced refractive index change. This contradicts the prediction of the vector nonlinear Schroedinger equation (an approximation of Maxwell's equations) which yields soliton solutions with an arbitrary linear polarization. However, Maxwell's equations are found to admit stable elliptically polarized solitary wave solutions which reduce to the stable circularly polarized solitary wave solutions of the vector nonlinear Schroedinger equation when the induced refractive index change approaches zero.     

Electromagnetic scattering by refractive index variations over a rough conducting surface

Abstract

This paper presents a numerical method for calculating the scattered field when a vertically polarized Gaussian beam is incident on a flat or slightly rough conducting surface at a grazing angle and the refractive index of the propagation medium has a profile which is not constant. The method is a solution to the parabolic approximation of the full wave equation. The results presented are taken for a linear and log-linear refractive index profile.     

Characterization of vapor-deposited l-leucine nanofilm

X-ray reflective measurements (XRR), atomic force microscopy and single wavelength ellipsometry were used to investigate the optical properties of thin l-leucine films deposited onto silicon substrates. The ellipsometry data (Ψ,Δ) were fitted with a four-layer-model, and the optical refractive index of the l-leucine film measured with ellipsometry was determined to be 1.37. With the conventional effective medium approximation theory and the ellipsometry results, the density of the l-leucine nanofilm was determined to be 70% (0.81 g/cm³) of crystalline l-leucine. This value was in good agreement with the density of 69% (0.80 g/cm³) obtained with XRR measurement. The ellipsometry measurements also enabled us to estimate the surface roughness or absorption layer of the film. This procedure of combined XRR and ellipsometry measurements could be a powerful tool for the determination of the (otherwise hard-to-determine) refractive index in thin organic material films with a rough surface layer.     

Analysis of dark spot formation in absorbing liquid media

Abstract

A theoretical analysis based on coupled field-matter equations is given to describe the recently observed phenomenon of a central dark spot formation of a Gaussian beam transmitted through an absorbing defocusing liquid medium. We find that such a pattern formation, which is accompanied by normal defocusing rings in the far field, originates from interplay between the wave-front curvature of the Gaussian beam and strong spatial self-phase medulation arising from thermally induced refractive index change in the medium. Results of numerical analysis for a thin medium are shown to be in a good quantitative agreement with our experimental findings. Further, the dark spot formation is also predicted by using a focused Gaussian beam and self-focusing medium.     

Optimum corneal asphericity of myopic eyes for refractive surgery

Abstract

In this work, an improvement to the Munnerlyn formula used in refractive surgery is proposed by selecting the final corneal asphericity. A theoretical optical analysis has been performed by adopting the RMS (root mean square) spot size as a merit function. Two different sets of myopes having refractive errors ranging up to —7D have been tested. The results show that a corneal asphericity of around Q=—0.61, with the radius provided by the emmetropization formula based on thin lens theory, may improve image quality after surgery. This value could be used in practical surgery because it is easy to implement in the ablation algorithm and it requires no additional data from other clinical surgery instrumentation.     

Rigorous and approximate analysis of metallic gratings in soft X-ray and EUV regions

Abstract

In deep ultraviolet and soft X-ray regions the complex refractive indices of metals differ substantially from their values in the visible region of the spectrum. The implications of this fact are analysed for metallic gratings illuminated by electromagnetic fields with wavelengths ranging from soft X‐rays to near infrared, concentrating on short wavelengths. In particular, we study metal-stripe gratings (linear polarizers for visible light) by rigorous diffraction theory to determine the short-wavelength region in which a combination of the geometrical thin-element approximation and the theory of single-layer films can be applied. Then we study inductive grid filters for protection of X-ray detectors from infrared radiation in space applications.     

p-Version plate and curved shell element for geometrically non-linear analysis

This paper presents a p-version geometrically non-linear formulation based on the total Lagrangian approach for a nine node three dimensional curved shell element. The element geometry is defined by the coordinates of the nodes located on its middle surface and nodal vectors describing the bottom and top surfaces of the element. The element displacement approximation can be of arbitrary and different polynomial orders in the plane of the element and in the transverse direction. The element approximation functions and the corresponding nodal variables are derived from the Lagrange family of interpolation functions. The resulting approximation functions and the nodal variables are hierarchical and the element displacement approximation ensures C° continuity. The element properties are established using the principle of virtual work and the hierarchical element approximation. In formulating the properties of the element complete three dimensional stresses and strains are considered, hence the element is equally effective for very thin as well as extremely thick shells and plates. Incremental equations of equilibrium are derived and solved using the standard Newton–Raphson method. The total load is divided into increments, and for each increment of load, equilibrium iterations are performed until each component of the residuals is within a preset tolerance. Numerical examples are presented to show the accuracy, efficiency and advantages of the present formulation. The results obtained from the present formulation are compared with those available in the literature.     

Investigations on the optical properties of sol–gel derived lanthanum doped lead titanate thin films

In the present work we studied the optical properties of undoped and La doped lead titanate thin films, and also demonstrated that the optical characterization of thin films can be used as an effective diagnostic tool to assess film quality. The optical properties of Pb_1−xLa_xTi_1−x/4O₃ [where x=0 (undoped), 10, 15, 20, 25 and 30 at.%] thin films were investigated using both transmission and reflection spectra in the 200–900-nm wavelength range. The refractive index (n), extinction coefficient (k) and the thickness of the film (d_f) were determined from the measured transmission spectra. The thickness of the film obtained from the interference fringes in transmission or reflection spectra matched well with those obtained from other methods. The appearance of interference fringes is an indication of the thickness uniformity of the film. The low value of extinction coefficient (in the order of 10⁻²) as observed in our films is a qualitative indication of excellent surface smoothness of the films. The densities of the films were estimated from their refractive indices using effective medium approximation. The average oscillator strength and its associated wavelength were estimated using a Sellmeier-type dispersion equation. Absorption coefficient (α) and the band-gap energy (E_g) were obtained for undoped and La doped films with varying La concentration. It was found that the refractive index and packing fraction values decrease with La doping. La doping was found to decrease the grain size of the films and increase the density of individual grains. Increased La content led to clustering of smaller grains. The observed variation of band-gap energy with La doping has been correlated to the observed microstructure of these films.     

p-version hierarchical three dimensional curved shell element for elastostatics

This paper presents a hierarchical three dimensional curved shell finite element formulation based on the p-approximation concept. The element displacement approximation can be of arbitrary and different polynomial orders in the plane of the shell (ξ, η) and the transverse direction (ξ). The curved shell element approximation functions and the corresponding nodal variables are derived by first constructing the approximation functions of orders p_ξ, p_η and p_ξ and the corresponding nodal variable operators for each of the three directions ξ, η and ξ and then taking their products (sometimes also known as tensor product). This procedure gives the approximation functions and the corresponding nodal variables corresponding to the polynomial orders p_ξ, p_η and p_ξ. Both the element displacement functions and the nodal variables are hierarchical; therefore, the resulting element matrices and the equivalent nodal load vectors are hierarchical also, i.e. the element properties corresponding to the polynomial orders p_ξ, p_η and p_ξ are a subset of those corresponding to the orders (p_ξ + 1), (p_η +1) and (p_ξ +1). The formulation guarantees C° continuity or smoothness of the displacement field across the interelement boundaries. The geometry of the element is described by the co-ordinates of the nodes on its middle surface (ξ = 0) and the nodal vectors describing its bottom (ξ = ?1) and top (ξ = +1) surfaces. The element properties are derived using the principle of virtual work and the hierarchical element approximation. The formulation is equally effective for very thin as well as very thick plates and curved shells. In fact, in many three dimensional applications the element can be used to replace the hierarchical three dimensional solid element without loss of accuracy but significant gain in modelling convenience. Numerical examples are presented to demonstrate the accuracy, efficiency and overall superiority of the present formulation. The results obtained from the present formulation are compared with those available in the literature as well as analytical solutions.     

Equivalent homogeneous finite element for composite materials via reissner principle. Part II: Finite element for shells

The lateral hulls of multi-shelled ships such as Catamarans or Trimarans can be considered as rigid solids, or compared to thin walled beams in multi-layered materials for the global statical or dynamical behaviour of the ship. Unfortunately this approximation no longer hold after a detailed study of the hulls. Working in these problems we proceed to a fine analysis of the stress field at the neighbourhood of the linkage shell-arm or of the holes for access input to the hull itself. Analysis with a beam model is inadequate and the shell theory must be more appropriate. Numerous papers have been devoted to shell elements, see References 1 to 8 to mention just a few. Some of them require a reduced or selective integration scheme.^9-12 We choose to start from the Ahmad element, ^{13, 14} which is suitable for moderate thick shells.^15.16 The final aim of this paper is to explain how to build up a multi-layered equivalent homogeneous shell element which is both economical and accurate. Some examples will be given and compared with those obtained with the Ahmad finite element.