General theory of sub-quarterwave multilayers with highly absorbing materials

A general theory of multilayers with enhanced reflectance has been developed based on the superposition of sub-quarterwave layers of various highly radiation-absorbing materials. The theory has been developed by second-order expansion of the multilayer reflectance with respect to the optical-constant differences between the materials in the multilayer. The current paper completes and improves the theory that was developed in a previous paper [J. Opt. Soc. Am. A 18, 1406 (2001)] by including the case of nonnormal incidence and general radiation polarization and by providing more-accurate film thickness values of the optimized multilayer than with the previous theory. The theory provides an accurate approach to the design of a new concept of multilayer coatings with more than two materials. The new multilayers are adequate to enhance the reflectance of the materials particularly in the far and the extreme ultraviolet.     

Reflectance optimization of inhomogeneous coatings with continuous variation of the complex refractive index

A model is derived for the reflectance optimization of an inhomogeneous coating made of absorbing materials. The model is applicable mainly for spectral regions where no transparent materials are available, such as in the extreme ultraviolet. The complex refractive index is assumed to take values within a given continuous domain and in a given sequence. The coating design is generated through a series of layer elements with a small refractive-index contrast across interfaces; the thickness of the element is calculated in terms of the refractive-index increment at the interface. The coating is optimized element by element starting from the substrate. When the refractive index varies both continuously and smoothly, the thickness element is of first order in the refractive-index increment. Suggestions are given on how to optimize a more general coating that alternates continuous and smooth refractive-index domains along with discrete indices, which results in a succession of inhomogeneous coatings and finite layers. An example is given to illustrate the model. A new material selection rule is obtained to discriminate whether the addition of a material on top of a partly grown coating will increase or decrease the reflectance of the coating. As a consequence, the model, which is highlighted toward the maximization of reflectance, can be used analogously for reflectance minimization such as for anti-reflection coatings.     

General transfer-matrix method for optical multilayer systems with coherent,partially coherent,and incoherent interference

The optical response of coherent thin-film multilayers is often represented with Fresnel coefficients in a 2 x 2 matrix configuration. Here the usual transfer matrix was modified to a generic form, with the ability to use the absolute squares of the Fresnel coefficients, so as to include incoherent (thick layers) and partially coherent (rough surface or interfaces) reflection and transmission. The method is integrated by use of models for refractive-index depth profiling. The utility of the method is illustrated with various multilayer structures formed by ion implantation into Si, including buried insulating and conducting layers, and multilayers with a thick incoherent layer in an arbitrary position.     

Layer-by-layer design method for multilayers with barrier layers: application to Si/Mo multilayers for extreme-ultraviolet lithography

A previous layer-by-layer multilayer design method [J. Opt. Soc. Am. A 19, 385 (2002)] is completed by adding the possibility of alternating layers with fixed thicknesses along with layers whose thicknesses are optimized for the largest possible reflectance at a desired wavelength. The previous algorithm did not allow for layers with fixed thicknesses. The current formalism is particularly suited for a multilayer design in which barrier layers of given thicknesses are used to prevent diffusion and/or reaction between the multilayer constituents. The design method is also useful both when intermixing zones develop at multilayer interfaces and when capping layers are used. The algorithm allows the design of multilayers with complex barrier layers with any number of layers of any optical constants. The optimization can be performed either for normal incidence or for nonnormal incidence with either s- or p-polarized radiation. The completed method provides a fast and accurate procedure for multilayer optimization regardless of the number of different materials used in the multilayer. The optimum layer thickness is determined by means of functions suitable for implementation in a computer code. The performance of the current algorithm is exemplified through the design of Si/Mo multilayers with intermixing layers or with barrier layers that are optimized for the largest reflectance at 13.4 nm. The use of specific barrier layers on each multilayer interface is also discussed.     

Reflectance enhancement in the extreme ultraviolet and soft x rays by means of multilayers with more than two materials

Sub-quarterwave multilayer coatings with more than two different materials are shown to provide a reflectance enhancement compared with the standard two-material multilayer coatings when reflectance is limited by material absorption. A remarkable reflectance enhancement is obtained when the materials in the multilayer are moderately absorbing. A simple rule based on the material optical constants is provided to select the most suitable materials for the multilayer and to arrange the materials in the correct sequence in order to obtain the highest possible reflectance. It is shown that sub-quarterwave multilayers generalize the concept of multilayers, of which the standard two-material multilayers are a particular case. Various examples illustrate the benefit of sub-quarter-wave multilayer coatings for highest reflectance in the extreme ultraviolet. Applications for sub-quarterwave multilayer coatings are envisaged for astronomy in the extreme ultraviolet (EUV) and soft x rays and also for future EUY lithography.     

Inreflectance: a new function for the optimization of multilayers with absorbing materials

A new magnitude is unveiled from the fact that a multilayer can be optimized for the largest reflectance in a layer-by-layer sequence: This magnitude has been named inreflectance. Inreflectance equals the modulus of the complex reflectance once elevated to the inverse of the complex refractive index of the next outer layer, including the inclination term. The maximization of inreflectance at every internal layer, proceeding sequentially starting with the innermost layer, results in a multilayer with the largest reflectance. At a given layer in the multilayer, inreflectance is different from reflectance when the next outer material absorbs radiation, whereas the two magnitudes are coincident when the next outer layer is transparent. The outermost layer of the multilayer is intrinsically different from the internal layers, and its optimization is performed through reflectance and not through inreflectance. Every maximum of inreflectance is found at a layer thickness that is larger than that for the reflectance maximum. The new magnitude is illustrated with some examples in the extreme ultraviolet, a spectral range in which all materials absorb radiation. Inreflectance can also be used to design multilayers in which some layers have fixed thicknesses and the rest of the layers have to be optimized taking into account the fixed layers.     

Aging and thermal stability of Mg/SiC and Mg/Y2O3 reflection multilayers in the 25-35 nm region

Reflection measurements in the 25-35 nm region were made for Mg/SiC and Mg/Y2O3 multilayers kept in a low-humidity atmosphere for 4 or 5 years. Aged Mg/SiC multilayers keep their reflectances, and the reflectance value at 31.2 nm is 0.44 at 10 degrees of the normal angle of incidence. Aged Mg/Y2O3 multilayers change reflectance as top layer materials, and the best value at 30.1 nm is 0.40 at 10 degrees. Reflection measurements are also made for Mg-based multilayers that are annealed from room temperature to 400 degrees C at 50 degrees C intervals. Both multilayers keep their reflectance at annealing temperatures of 200 degrees C. These results suggest that both Mg-based multilayers can be applied to practical optics.     

Analytical approximate solutions of the time-domain diffusion equation in layered slabs.

Time-domain analytical solutions of the diffusion equation for photon migration through highly scattering two- and three-layered slabs have been obtained. The effect of the refractive-index mismatch with the external medium is taken into account, and approximate boundary conditions at the interface between the diffusive layers have been considered. A Monte Carlo code for photon migration through a layered slab has also been developed. Comparisons with the results of Monte Carlo simulations showed that the analytical solutions correctly describe the mean path length followed by photons inside each diffusive layer and the shape of the temporal profile of received photons, while discrepancies are observed for the continuous-wave reflectance or transmittance.     

Modeling the image quality of enhanced reflectance x-ray multilayers as a surface power spectral density filter function

Residual surface roughness over the entire range of relevant spatial frequencies must be specified and controlled in many high-performance optical systems. This is particularly true for enhanced reflectance multilayers if both high reflectance and high spatial resolution are desired. If we assume that the interfaces making up a multilayer coating are uncorrelated at high spatial frequencies (microroughness) and perfectly correlated at low spatial and midspatial frequencies, then the multilayer can be thought of as a surface power spectral density (PSD) filter function. Multilayer coatings thus behave as a low-pass spatial frequency filter acting on the substrate PSD, with the exact location and shape of this cutoff being material and process dependent. This concept allows us to apply conventional linear systems techniques to the evaluation of image quality and to the derivation of optical fabrication tolerances for applications utilizing enhanced reflectance x-ray multilayers.     

Can multilayers be optimized sequentially when radiation is partially polarized?

For radiation-absorbing materials and nonnormal incidence, multilayers with the largest reflectance at a desired wavelength are here demonstrated to require nonsequential optimization when radiation is partially polarized. The thicknesses of multilayers with the largest possible reflectance at a desired wavelength can be calculated sequentially either for s- or p-polarized radiation or for normal incidence and any polarization. The equations that impose the condition of maximum reflectance in the general case of partial polarization do not allow for the simplification that is performed when radiation is s or p polarized, which implies that multilayer optimization for partially polarized radiation is not sequential. An example is given in which the importance of nonsequential optimization is displayed. In contrast to the case of s- or p-polarized radiation, when radiation is partially polarized extra maxima may be found within an optical path difference of much less than half a wavelength.     

W/SiC x-ray multilayers optimized for use above 100 keV

We have developed a new depth-graded multilayer system comprising W and SiC layers, suitable for use as hard x-ray reflective coatings operating in the energy range 100-200 keV. Grazing-incidence x-ray reflectance at E = 8 keV was used to characterize the interface widths, as well as the temporal and thermal stability in both periodic and depth-graded W/SiC structures, whereas synchrotron radiation was used to measure the hard x-ray reflectance of a depth-graded multilayer designed specifically for use in the range E approximately 150-170 keV. We have modeled the hard x-ray reflectance using newly derived optical constants, which we determined from reflectance versus incidence angle measurements also made using synchrotron radiation, in the range E = 120-180 keV. We describe our experimental investigation in detail compare the new W/SiC multilayers with both W/Si and W/B4C films that have been studied previously, and discuss the significance of these results with regard to the eventual development of a hard x-ray nuclear line telescope.     

Modeling of curvature in multilayered epitaxially grown films

A model was developed to calculate the radius of curvature produced by stresses in lattice-mismatched multilayers grown epitaxially on a planar substrate. The analysis was done for any number (N) of layers using beam bending theory and strain partitioning theory introduced by our group earlier. This model is reduced to a limiting case of a two-layer structure (film-substrate). The variation of the radius of curvature with the relative thicknesses and other material properties of the layers was determined and compared to finite element calculations. The analytical model was further verified by applying it to a symmetric tri-laminate structure.     

Aperiodic multilayers with enhanced reflectivity for extreme ultraviolet lithography

We have developed novel aperiodic multilayers, covered by capping layers resistant to environmental attack, that offer superior performance for extreme ultraviolet lithography. We have designed these coatings using an optimization procedure based on an algorithm able to acquire domain knowledge inside the space of possible solutions. An integrated intensity increase of up to 2.18 times that obtained using standard periodic multilayers has been estimated. The aperiodic structures have minimal absorption in the topmost layers, which makes them especially insensitive to both the choice of capping layer material and any subsequent capping layer degradation due to oxidation or contamination. This property allows for the use of the most resilient capping layer materials available, thereby leading to a significantly improved lifetime. We have produced prototype capped aperiodic coatings and have measured their performance.     

Thermo-elastic moduli of periodic multilayers with wavy architectures

The recently developed parametric finite-volume direct averaging micromechanics theory for periodic materials is employed to investigate the effective moduli and thermal expansion coefficients of lamellar composites with wavy architectures. In the parametric version, a reference square subvolume is mapped onto a quadrilateral subvolume in the actual discretized microstructure to accurately capture the in situ microstructural details. The mapping is used to construct local stiffness matrices of quadrilateral subvolumes which are employed in the local/global stiffness matrix solution strategy for the unit cell problem within a homogenization framework. Complete set of homogenized moduli and thermal expansion coefficients of multilayers comprised of alternating soft and hard laminae with two types of waviness is generated for the first time as a function of the volume content of the hard phase for two amplitude-to-wavelength ratios. The observed changes in the homogenized mechanical and thermal properties relative to the reference flat-layer configuration depend on the wavy microstructure orientation and become greater with increasing amplitude-to-wavelength ratios. Examination of local stress fields explains the differences observed in the homogenized moduli of multilayers with sinusoidal and corrugated waveforms for the two amplitude-to-wavelength ratios.     

Interlayer coupling in ferroelectric bilayer and superlattice heterostructures

Ferroelectric multilayers and superlattices have gained interest for dynamic random access memory (DRAM) applications and as active elements in tunable microwave devices in the telecommunications industry. A number of experimental studies have shown that these materials have many peculiar properties which cannot be described by a simple series connection of the individual layers that make up the heterostructures. A thermodynamic analysis is presented to demonstrate that ferroelectric multilayers interact through internal elastic, electrical, and electromechanical fields and the strength of the coupling can be quantitatively described using Landau theory of phase transformations, theory of elasticity, and principles of electrostatics. The theoretical analysis shows that compositional variations across ferroelectric bilayers result in a broken spatial inversion symmetry that can lead to asymmetric thermodynamic potentials favoring one ferroelectric ground state over the other. Furthermore, the thermodynamic modeling indicates that there is a strong electrostatic coupling between the layers that leads to the suppression of ferroelectricity at a critical paraelectric layer thickness for ferroelectric-paraelectric bilayers. This bilayer is expected to have a gigantic dielectric response similar to the dielectric anomaly near Curie-Weiss temperature in homogeneous ferroelectrics at this critical thickness.     

Model for analyzing optical properties of silicate glasses

A new model for analyzing optical properties of silicate glass materials including borosilicates has been developed. The model is based on computing refractive-index and density values of a given optical glass whose oxide composition is known in terms of weight fractions. The refractive-index variation with wavelength has also been used to predict the chromatic behavior of these glasses. The model has been compared with the existing chromatic model and was found to give more accurate values for borosilicates.     

Determination of optical properties of absorbing materials: a generalized scheme

A generalized reflectance method for determination of optical properties of absorbing materials is developed and compared with other reflectance methods. In the present scheme the specimen is coated with dielectric transparent layer(s) and the reflectance ratios are measured. This novel scheme of specimen preparation and the method of measurement allow the specimen to be free from surface layers and at the same time account for possible effects of surface roughness. It can be applied to a wide variety of materials regardless of their surface conditions and is particularly useful for metals.     

Universal antireflection coatings for substrates for the visible spectral region

It is possible to design normal-incidence antireflection coatings that reduce the reflectance of any substrate with a refractive index that lies in the range of 1.48 to 1.75. The performance of such coatings depends on the width of the spectral region over which the reflectance is to be suppressed, on the coating materials used for their construction, and on the overall optical thickness of the layer system. For example, the calculated average spectral reflectance of a set of six different substrates with refractive indices 1.48, 1.55, 1.60, 1.65, 1.70, and 1.75, when coated with a 0.56-μm-thick, eight-layer antireflection coating designed for the 0.40-0.80-μm spectral region, was 0.34%. This is higher than the average reflectance that is attainable with a conventional antireflection coating of similar optical thicknesses designed for a particular refractive index. However, it is an acceptable value for most applications. With the universal type of antireflection coating described, it is thus possible to coat a number of different refractive-index substrates in one deposition run, and this can result in considerable cost and time savings.     

Topical meeting on optical interference coatings (OIC'2001): manufacturing problem

Measurements are presented of the experimental filters submitted to the first optical thin-film manufacturing problem posed in conjunction with the Topical Meeting on Optical Interference Coatings, in which the object was to produce multilayers with spectral transmittance and reflectance curves that were as close as possible to the target values that were specified in the 400- to 600-nm spectral region. No limit was set on the overall thickness of the solutions or the number of layers used in their construction. The participants were free to use the coating materials of their choice. Six different groups submitted a total of 11 different filters for evaluation. Three different physical vapor deposition processes were used for the manufacture of the coatings: magnetron sputtering, ion-beam sputtering, and plasma-ion-assisted, electron-beam gun evaporation. The solutions ranged in metric thickness from 758 to 4226 nm and consisted of between 8 and 27 layers. For all but two of the samples submitted, the average rms departure of the measured transmittances and reflectances from the target values in the spectral region of interest was between 0.98% and 1.55%.     

Influence of small inhomogeneities on the spectral characteristics of single thin films

It is well known that the spectral transmittance and reflectance of a thin film can be influenced by even small inhomogeneities or variations in its complex refractive-index profile. Formulas are derived that describe the theoretical variation of the spectral characteristics for small changes in the refractive index and the extinction coefficient of a homogeneous thin film. These formulas, accurate to the first order in the change in the complex refractive index, are compared with exact calculations for a number of different types of inhomogeneities. It is shown that specific qualitative features in the refractive-index profile of a nearly homogeneous thin film frequently can be determined from an examination of the change in the spectral transmittance and reflectance at normal incidence.