Ever since the first simple applications of thin film coatings in optics emerged, films with a gradually varying refractive index in the direction of the normal to the film surfaces have appeared particularly intriguing to scientists and engineers. The almost total absence of reflections from a gradual transition between two media with different indices, inspired early interest and hope for other unusual properties that could not be realized by stacks of homogeneous films. This paper will try to explore to what extent such hopes have come true so far, and as a basis for subsequent presentations during this conference, give a general overview of the analytical and computational as well as technical complexities particular to inhomogeneous films.

A set of equivalent solutions to the synthesis of graded index optical coatings is obtained by a combustion of Fourier Transform techniques and special properties of the reflection coefficients in the plane of complex wavenumbers. A mathematical transformation changes the phase shift in reflection without affecting the reflectance, and the resulting refractive index profiles are calculated. Disposing of a set of solutions, one has then the possibility of selecting the most attractive for a practical implementation.

Three main topics are considered in this work. These are the mathematical questions and results related to the concept of Q-function, an exact method for the determination of the refractive index of inhomogeneous coatings, refinement procedures for the synthesis of inhomogeneous coatings.

The optical properties of quasi-inhomogeneous coatings are calculated with standard off-the- shelf optical thin film software running Microsoft Windows. The standard program includes a fast algorithm for repeated groups of layers and allows small index gradations. Index values and layer recipes are generated separately in programs written in Microsoft Excel. These programs approximate inhomogeneous models by the quasi-homogeneous designs required by the standard program. Data is sent to and from the thin film program by dynamic data exchange and the Windows clipboard. The thin film program can be treated as a subroutine. Continuous looping is possible and spectral data is passed back to Excel for analysis. This seamless integration of standard and user-modifiable software was not readily achieved in older operating systems. As an example we discuss an Excel tolerance model for a graded index antireflection design. Thicknesses are subject to random fluctuations simulating manufacturing errors.

A powerful Fourier transform technique used at the NRC for the design of inhomogeneous and quasi-inhomogeneous dielectric optical coatings is reviewed. Its characteristic features and limitations are pointed out and illustrated by numerical examples. A challenging design and fabrication problem is presented with experimental results.

This paper proposes that wavelet construction of the gradient-index refractive index be used to design optical interference coatings. The basic wavelet used is a localized apodized rugate. By incorporating scaled and shifted copies of the basic wavelet, which are other elements of a basis set according to wavelet theory, a variety of interference coatings can be designed.

A series of antireflection coating designs for the 0.4 to 0.8 micrometers spectral region for a substrate of refractive index 1.52 are found by the Quadratic Problem Approximation method. This procedure yields optimal or near-optimal solutions. Parameters varied are the overall optical thickness of the systems and the acceptable ranges of the refractive indices of the coating materials. The calculated merit functions of inhomogeneous layer solutions and of homogeneous multilayer solutions before and after refinement are compared with the quadratic programming merit function.

Prior works by the author and others point to the observation that the ideal antireflection coatings would be a family of inhomogeneous index of refraction versus thickness profiles. Our previous and recent studies have been from the design perspective and of an empirical nature. The pursuit of the understanding of the nature and properties of these profiles has employed optimization tools, various modes of display of the properties, and Fourier techniques. The multiple views of the nature serve to better define and clarify the properties of these profiles. The basic nature and fundamentals of antireflection coatings are described from these several viewpoints. The possibilities and limitations of the inhomogeneous profiles are discussed and compared with their approximation by homogeneous layer systems.

A gradient-index coating design that incorporates broadband antireflection, narrowband reflection, and graded absorbing regions is described. The coating is suitable for deposition on reflective substrates such as diamond-turned aluminum or on glass substrates where a spectrally selective reflector with no transmission is required. Theoretical designs and measured performance are presented.

Quarterwave stacks and quasi-sinusoidal rugate structures are unified by the introduction of the power-sine rugate structures. Changing one parameter modifies the optical admittance profile continuously from quarterwave stack to rugate structure. The fact that power sine rugates have adjustable high order harmonics can be used for the design of non-polarizing coatings.

The paper describes the design and manufacturing techniques used to produce digitized rugate filters for laser rejection in an ophthalmic microscope. The design is synthesized from the refractive index profile necessary to give the required bandwidth, attenuation and angle of incidence constraints. Process control using quartz crystal deposition and broadband diode array radiometer to monitor real time thin film growth is described. The results are presented showing close agreement between theoretical design and measured spectral performance. Further work to extend this technique to multiple laser line rejection is outlined.

This paper describes a transmissive rugate filter which is designed to reflect a portion of the visible spectrum and yet not appear to have a dominant color. The filter design criteria were chosen so that the filter rejects portions of the visible spectrum from 0.38 to 0.78 microns. Observing a scene through this type of optical filter one perceives it to be devoid of coloration, i.e., color neutral, albeit it is somewhat darker. The design constraints assume a solar light source. The eye bounds the wavelength range over which perceived coloration is affected. For this work the spectral characteristics of both the incident light and the standard human eye determine the nature of the spectral tailoring of the reflection bands. Rugate filter design and fabrication technology permits a very wide latitude in the number, location, bandwidth, and peak height of all reflection bands. The result is a color neutral rugate filter having reflection bands tailored to provide the human user with maximum color discrimination capability.

The design and fabrication of inhomogeneous single and multi-line rejection filters with good line rejection and pass band transmission has been investigated. The filters were designed with the help of a filter design program developed in-house and optimized analytically for best conformance of the filter spectrum to the design specifications. The experimental filters obtained in this study achieved optical densities in excess of 4.5 in the design rejection range with good transmission and low ripple in the pass band region. The design rejection peak width of 40 nanometers, which was chosen in order to ensure optimum rejection over the required range of incident angles from 0 to 20 degrees, was readily achieved.

Errors inherent to the approximateness and incompleteness of the inverse Fourier transformation techniques can be partly compensated by using successive approximations. Especially interesting is the situation where it is possible to optimize the primary approximate solution efficiently by means of repeated iterations on unchanged conditions (closed loop optimizations). It is shown that the degree of convergence of the closed loop optimization process and the obtainable result depend on the actual choice of Q-function. The results are similar when the closed loop optimizations are performed prior to and after conversion into a two-indexed quasi-inhomogeneous solution employing double layer equivalents. The subject of the optimization of the number of layers in the two-index solution is discussed and a couple of new Q-functions that are better suited for closed loop optimizations of both converted and unconverted coatings than some of the best known existing Q-functions are presented.

In this work we describe ellipsometric software that has been developed at Moscow State University. This software is based on the assurance that hierarchy of models should be used for the thin film. The hierarchy of layer models and the concept of the model being in the agreement with the accuracy of experimental data are discussed.

Optical properties of silicon-based coatings are investigated in which the refractive index n is varied gradually (SiO_xN_y with continuously changing x and y) or in discrete steps (SiO₂/SiN_1.3 multilayers). The films are prepared at room temperature in a dual- mode microwave/radiofrequency (MW/RF) plasma in which the substrates are placed on an RF powered substrate holder, while simultaneously exposed to a MW discharge. The films' composition is controlled by the working gas mixture using SiH₄, NH₃ and N₂O, and their microstructure, such as packing density and interface roughness, is controlled independently by varying the energy and the flux of bombarding ions. The deposition process is monitored in-situ by optical emission spectroscopy measurements which are related to the compositional depth profiles provided by elastic recoil detection analysis.

The optical and structural properties of TiO₂-MgF₂ mixed films prepared by coevaporation with ion-assisted deposition have been investigated. The composition of coevaporated TiO₂-MgF₂ mixed films can be tuned by controlling the relative deposition rates of the two sources. Most of the films with thicknesses less than 500 nm are found to be homogeneous in optical properties. The optical and structural properties of the mixed films change from MgF₂-dominant to TiO₂-dominant as the TiO₂ content in the mixed film increases. The dependence of the refractive index with composition can be fitted by Bottcher's formula at low TiO₂ content and by Lorentz-Lorentz formula at high TiO₂ content in the mixed films. An application of TiO₂-MgF₂ mixed films for the design and fabrication of antireflection coatings in the near infrared region has also been discussed.

Optical and physical properties of MgF₂, ZnSe, and composite ZnSe-MgF₂ films deposited by co-evaporation of ZnSe and MgF₂ on glass substrates at elevated temperatures are studied. MgF₂ films deposited on glass substrates at temperature of 200 degree(s)C were found durable, resistant to wet environment, high in film packing density, and having a very reproducible refractive index of 1.38 to 0.75 micrometers . ZnSe films formed under same substrate conditions were also found dense and reproducible. The refractive indices of these films were determined to be 2.50 +/- 0.01 at 0.75 micrometers . The adhesion of the films to the substrate declined drastically in hot and wet environment. Composite films deposited on glass substrate at 200 degree(s)C by co-evaporation of ZnSe and MgF₂ with selected individual evaporation rates were observed to be uniform, durable, and enhanced in protection against hazardous environments.

Single phase BaTiO₃ thin films were prepared by metalorganic deposition (MOD) using barium 2-ethylhexanoate and titanium dimethoxy dineodecanoate as the metalorganic precursors. A series of experiments were conducted on the metalorganic spin-coated films and their correspondingly annealed samples by employing experimental techniques ranging from thermogravimetric analysis, fourier transform infrared spectroscopy, X-ray diffraction, scanning electron microscopy, to various optical methods. The conclusions of our experiments are as the following: upon annealing the spin-coated films from the equimolar mixture of barium 2-ethylhexanoate and titanium dimethoxy dineodecanoate formulations, BaTiO₃ is formed around annealing temperature of 600 degree(s)C via solid state reaction between BaCO₃ and TiO₂ (anatase). The structure of MOD prepared BaTiO₃ films show pronounced inhomogeneity, viz. the crystallites are much larger near the surface of the film than near the substrate. Optical characterization shows that the scattering losses contribute dominantly to the total optical losses.

Grading the refractive index period of a rugate is a technique for depositing broad band reflectors using rugate technology. The principle advantage of this technique is the ability to deposit long and short pass reflectors in parallel with other rugate spectral features and thus generate complex performance in a single optical film. Variation of the amplitude of the index profile as the period is changed allows for good edge definition for long or short pass designs. Several of these devices were fabricated and measured performance is presented. These devices demonstrate rugate properties of harmonic suppression and superposition with other rugate structures.

The wide range of optical thin film applications utilizing gradient index coatings has prompted the development of advanced optical control techniques. These include ellipsometric and photometric instruments capable of in-situ measurement of optical performance as the optical structure is being deposited. This paper discusses design sensitivity analysis and instrument configuration for development of a control strategy. The ability to measure optical thickness, refractive index and mechanical thickness is a function of several instrument parameters including wavelength, number of wavelengths, angle of incidence, and complexity of measurement surface. The most critical control data in the fabrication of a particular rugate design, and the instrument parameters and techniques employed and how they affect the control strategy is presented in this discussion.

Fundamental descriptive rugate filter parameters and concepts are shown. A simple rugate filter is derived from the Fourier analysis of a high/low refractive index pair used in a conventional non quarter-wave stack design. Progress in both design and fabrication is demonstrated by representative rugate filters which function in either the visible or infrared. Numerous examples are cited to show the utility of refractive index profile tailoring and the advantages of using this technology.