Showing papers on "Fresnel equations published in 1986"

X-ray diffraction of multilayers and superlattices

Abstract: Recursion formulae for calculating the reflected amplitude ratio of multilayers and superlattices have been derived from the Takagi-Taupin differential equations, which describe the dynamical diffraction of X-rays in deformed crystals. Calculated rocking curves of complicated layered structures, such as non-ideal superlattices on perfect crystals, are shown to be in good agreement with observed diffraction profiles. The kinematical theory can save computing time only in the case of an ideal superlattice, for which a geometric series can be used, but the reflectivity must be below 10% so that multiple reflections can be neglected. For a perfect crystal of arbitrary thickness the absorption at the center of the dynamical reflection is found to be proportional to the square root of the reflectivity. Sputter-deposited periodic multilayers of tungsten and carbon can be considered as an artificial crystal, for which dynamical X-ray diffraction calculations give results very similar to those of a macroscopic optical description in terms of the complex index of refraction and Fresnel reflection coefficients.

Angular Dependence of Diffuse Reflectance Infrared Spectra. Part I: FT-IR Spectrogoniophotometer

Abstract: An optical arrangement is described which allows diffuse reflectance infrared spectra to be collected by the use of a variety of incident angles and collection geometries. In the first phase of this study, the reflectance from a nonabsorbing powder and the intensity of the reststrahlen band in quartz were monitored as a function of optical geometry. The contribution of Fresnel reflectance was found to be very dependent on optical geometry, and the effect on the intensity of absorption bands in diffuse reflectance spectra is demonstrated. Based on these preliminary results, some suggestions for minimizing Fresnel reflectance through the optimization of diffuse reflectance optics are made.

Reflection and transmission of beams at a curved interface

Abstract: Gaussian-beam reflection and transmission in the presence of a convex circular boundary separating two different dielectrics is analyzed by the complex ray method whereby rays are traced in a complex coordinate space from the complex source point to the complex extension of the interface and then to the real observation point. The search for the complex parameters describing the ray connecting the source and the observer can be numerically time consuming. Much more efficient are paraxial approximations that express the field as perturbations about its values on the axis of maximum strength. Such paraxial approximations are explored here, using on-axis complex ray and even on-axis real ray solutions with corrections accounting for differences in phase, ray reflection and transmission coefficients, and ray divergence coefficients. Extensive numerical comparisons establish a reasonable range of validity of both paraxial approximations, except in regions characterizing the transition to total reflection, if that can occur.

Characterization of multilayer x-ray analyzers: models and measurements

Abstract: A procedure is described for a detailed characterization of multilayer analyzers that can be effectively applied to their design, optimization, and application for absolute x-ray spectrometry in the 100 to 10,000 eV photon energy region. An accurate analytical model has been developed that is based upon a simple modification of the dynamical Darwin-Prins theory to extend its application to finite multilayer systems and to the low energy x-ray region. Its equivalence to the optical E&M solution of the Fresnel equations at each interface is demonstrated by detailed comparisons for the reflectivity of a multilayer throughout the angular range of incidence of 0° to 90°. A special spectrograph and an experimental method are described for the measurement of the absolute reflectivity characteristics of the multilayer. The experimental measurements at three photon energies in the 100 to 2000 eV region are fit by the analytical modified Darwin-Prins equation (MDP) for 1(0), generating a detailed characterization of two state-of-the-art multilayers: sputtered tungsten-carbon with 2d 70 A and a molecular lead stearate with 2d 100 A. The fitting parameters that are determined by this procedure are applied to help establish the structural characteristics of these multilayers.

Relationship between the p and s Fresnel reflection coefficients of an interface independent of angle of incidence

Abstract: The Fresnel reflection coefficients rp and rs of p- and s-polarized light at the planar interface between two linear isotropic media are found to be interrelated by (rs − rp)/(1 − rsrp) = cos 2β, independent of the angle of incidence ϕ, where tan2β = ∊ and ∊ is the (generally complex) ratio of dielectric constants of the media of refraction and incidence. This complements another relation (found earlier), (rs2 − rp)/(rs − rsrp) = cos 2ϕ, which is valid at a given ϕ independent of ∊ (i.e., for all possible interfaces). Taken together, these two equations specify rp and rs completely and can be used to replace the original Fresnel equations.

Proton-exchanged Fresnel lenses in Ti:LiNbO 3 waveguides

Abstract: Fresnel lenses were fabricated by the patterned proton-exchange technique in titanium-diffused lithium niobate waveguides. The large change in refractive index available with the proton exchange allows fabrication of a high-efficiency thin Fresnel lens. Lens characteristics of nearly diffraction-limited focus spot size and efficiency up to 70% were obtained. The use of the lens to construct an integrated-optic rf spectrum analyzer of 1-GHz bandwidth and 2-MHz resolution is also demonstrated.

Compact Faraday rotator for an optical isolator using magnets arranged with alternating polarities.

Abstract: A method to miniaturize the Faraday rotator for an optical isolator is proposed. The rotator includes a half-wave plate, and magnets arranged with alternating polarities are used. The principle of operation of the rotator is confirmed by measurements. An isolator employing the rotator has a backward loss of 32 dB and a forward loss of 0.65 dB at 0.633-μm wavelength, where the forward loss is mainly due to Fresnel reflection losses.

Switching characteristics of GaAs directional coupler optical switches.

Abstract: A directional coupler optical switch with good switching characteristics (extinction ratio of more than 22 dB at 22.5-V applied voltage) and low-loss guiding properties (total insertion loss of 10.7 dB including Fresnel reflection loss of 4.0 dB) has been developed. The switch element was mounted on a ceramic stem and jointed directly with polarization-maintaining optical fibers. High speed operation of 1.6 GHz was achieved.

Measurement of reflectivities for magnetooptical media

Abstract: Reflectivities for magnetooptical media are measured as functions of angle of incidence in a series of related experiments. Measurements yield sufficient information to determine absolute magnitude and relative phase for the Fresnel amplitude reflection coefficients, r‖(p) and r‖(s), as well as for r⊥, which describes the polar Kerr effect on incident polarized light. Using dielectric tensor values measured at normal incidence, theoretical curves for magnitude and phase of r‖(p),r‖(s), and r⊥ vs angle of incidence are developed. Experimental measurements are compared to theoretical functions of the reflectivities and found to be in good agreement throughout the range of angle of incidence. The theoretical analysis and related experimental procedures explicitly treat the presence of a transparent overcoating layer.

Magnetic depth profiles by neutron reflection

Abstract: Fresnel reflection of polarized neutrons provides direct information on the dependence of the magnetic induction B in materials as a function of the depth z from their surface. The reflectivity of the neutron beam from the surface is sizeable for a glancing angle of a fraction of degree (for neutron wavelengths of the order of 5, A); hence the experiment requires extremely good angular resolution. The requirements have been satisfied in a prototype instrument at the Intense Pulsed Neutron Source at Argonne which is being used to measure the magnetic profile close to the surface of a variety of materials. In superconductors the penetration depth of an applied magnetic field has been probed, as well as the remnant superconducting surface sheath when the applied magnetic field is raised above a critical value (Hc2

Fresnel diffraction in an optical system containing lenses

Abstract: A convenient expression describing the location of Fresnel diffraction patterns when a diffracted spherical wave is focused by a lens is given. Experimental results confirm the predicted positions of 26 axial extrema corresponding to integer numbers of Fresnel half-period zones. Experimental radial intensity profiles are presented for some of these positions.

Planar-Structure Focusing Lens for Operation at 200 MHz and Its Application to the Reflection-Mode Acoustic Microscope

Abstract: In the previous report, t he authors proposed a new planar-structure acoustic-microsc ope lens designed according to the theory of Fresnel zone plate, and demonstrated its good focusing capability. Since the l ens is composed of a number of concentric annular grooves of uniform depth on its radiation plane, it has an advantage over the conventional acoustic lens with a spherical concave in being fabricated easily using a photolithographi c technique. In our first model of the planar lens, which had an aperture angle of 47.3" and operated at 100 MHz, the grooves could be formed successfully using an ordinary chemical etching technique. However, the formation of finer grooves is inevitably limited by the undercutting effect so far as a wet-process is used. Therefore, some advanced photolithographi c technique s hould be employed for fabricating the lens with a wider aperture a ngle and a higher o peration frequency. Recently, the authors have introduced the ECR (electron cyclotron resonance) reactive ion etching technique for forming fine grooves. Using this dry-process, a planar lens designed for an operation frequency of 200 MHz and an aperture a ngle of 90' has been obtained. I n the present report, we describe the fabrication process and some results of imaging experiments by an acoustic microscope equipped with the p lanar lens.

Gradient index lens array with improved image quality due to stray light reduction

Abstract: A gradient index lens array having improved stray light characteristics is enabled by coating the surfaces of the fibers comprising the array with a low transmittance, low viscosity material. The index of this material is matched to that of the fiber surface to prevent Fresnel reflection. The interstitial areas of the lens array comprises a light absorbing material. In another embodiment, stray light caused by light scattering at the end faces of the lens array is reduced by masking the end face with a mask having apertures overlying the fiber faces. The mask apertures have slightly smaller diameters to mask out the rough outer edge of the fibers.

Photoelastic properties of metals measured by off-null ellipsometry

Abstract: The complex coefficients of the reflection of light from an anisotropic conducting medium are derived assuming one of the optical axes to be perpendicular to the reflecting surface. The results represent generalizations of those obtained previously. The theory of reflection is developed with special reference to studies of strain-induced optical anisotropies in metals. How off-null ellipsometry can be used to determine complex photoelastic constants of metals is described. Measurements made on polycrystalline Al and Cu samples at different wavelengths in the visible region are reported.

Variational theory of the reflection of light by interfaces

Abstract: Schwinger’s variational theory of scattering is adapted to the calculation of reflection amplitudes. For an arbitrary transition between two media at a planar interface, we derive variational expressions for the s and p reflection amplitudes that are correct at grazing incidence and correct to second order in the ratio of interface thickness to wavelength. The interface or the substrate (or both) may be absorbing. The Hulthen–Kohn variational theory of is also adapted to reflection. The results are simpler (for the same trial wave function) but not so good as scattering those obtained from the Schwinger method.

Optical joint comprising antireflection films

Abstract: An optical joint for joining a first light transmissive material to a second light transmissive material. The joint comprises a transparent adhesive layer between the first and second materials, and an antireflection film between each of the first and second materials and the adhesive layer. The joint prevents Fresnel reflection which would otherwise occur between the light transmissive materials.

Fresnel Phase Effects For X-Ray Microlithography

Abstract: A study of Fresnel diffraction effects is presented for structures of interest for X-Ray Microlithography. This analysis uses the full optical constants of the mask absorber pattern. Therefore, our calculations take into account the fact that the photons experience a phase shift as they go through the mask's absorbing layer in addition to simple absorption. Results are presented which show the consequences of adding the phase effects to the Fresnel analysis. These results show that phase effects cannot be disregarded in modeling Fresnel intensity profiles on resists.

Plane-wave-spectrum approach for calculations of radiation pattern of junction lasers

Abstract: An analytical expression for mode-conversion coefficients at a laser–air interface is derived using plane-wave decomposition of the modal fields. Both the mode-conversion coefficients and the radiation pattern are explicitly expressed as function of Fresnel reflection coefficients of the interface and the incident mode spectrum.

Computation of Fresnel diffraction

Abstract: A review and commentary is given of practical methods for computing the Fresnel diffraction patterns of circular and linear apertures. The general problem of approximating transcendental functions is also considered.

Spherically pointed optical fiber and its production

Abstract: PURPOSE:To manufacture a spherically pointed optical fiber having high coupling efficiency with good reproducibility and controllability by sticking a high- refractive index liquid having a dopant of a high diffusion rate to the top end of an optical fiber, heating indirectly said part to a spherically pointed shape and changing continuously the refractive index distribution in the spherically pointed part. CONSTITUTION:The optical fiber 1 is inserted into the high fractive index liquid 11 filled into a vessel 10 so that the liquid 11 sticks near to the top end thereof. The optical fiber 1 stuck with the liquid 11 is then inserted into a vessel 8 and the vessel 8 is heated by the flame of a burner 4 while the fiber is held fixed in the space to heat indirectly the top end and to work the top end to a spherical or tapered shape. Alkaline metallic ions diffuse and penetrate quickly into the fiber 1 when an aq. soln. of alkaline metallic silicate is used for the liquid 11. The refractive index is then distributed so as to decrease continuously from the outside surface toward the inside, by which the coupling efficiency to the optical fiber of a semiconductor light emitting element is improved and Fresnel reflection is prevented.

Tunable backscattering for single-mode optical fibers in 1.1-1.6-µm spectral region

Abstract: A very powerful and versatile backscattering apparatus for single-mode fibers has been implemented, which can be tuned to almost every wavelength in the spectral region from 1.06 to 1.6 μm. A Raman source fiber, pumped by a Nd:YAG laser, is filtered by a special conception monochromator (folded path) which, by preserving the image of the output face of the source fiber, allows good launching efficiency into the fiber under test. Directional coupling with suppression of Fresnel reflection is achieved by an acoustooptic modulator, which is insensitive to polarization. Applications to single-mode fibers characterization are discussed, like spectral attenuation, cutoff phenomena as function of length and spot-size propagation.

Steering Mirrors For The Soft X-Ray Range

Abstract: Concave glancing incidence mirrors for deflection and turning of soft X -ray beams bylarge angles are considered. For carbon- coated superpolished concave mirror the reflecti- vity of 60% is measured for 6.76 nm radiation and turning angle T= 29 °, which is two orders higher than for beam deflection by the same angle with one Fresnel reflection from a plane carbon mirror.Concav urfaces of several substances can be used as steering mirrors in the soft X- ray range.'-' This is connected with the following effect. Let a narrow soft X -ray beam fall almost tangentially (with a small sliding angle e ) on a concave cylindrical mirrorwith arc angle ' (Fig. 1). Then after n = /20 reflections the beam will leave themirror at the same sliding angle. The total reflectivity at small G can be expressed as: 1 Ro = lim[Irn(9 )I 21n = exp ( -2 `Y Im 11- , ), (1) 9.0 where r(6) ) is the Fresnel reflection coefficient, is the dielectric constant of mirror' covering. In this paper the results of the reflectivity measurement of the con-cave carbon -coated spherical mirror with D = 40 mm diameter qnd r = 80 mm curvature radi-

Fresnel region imaging with a linear array

Abstract: This paper presents a complete solution to the analysis and synthesis of both angular and axial field patterns in the Fresnel region of linear arrays, and applies the theory to Fresnel region imaging. Both the angular and axial field patterns are shown to be Fourier transforms of the suitably modified aperture distribution, and can therefore also be synthesized by the inverse Fourier transform.