Méthodes avancées pour le design de filtres optiques avec des indices de réfraction intermédiaires arbitraires

Optical interference filters can be found everywhere. Their applications range from antireflective coatings present on almost every optical element to narrowband filters used in telecommunication networks and in astronomy.

Most optical filters consist of a stack of homogeneous layers of two or a few materials with discrete refractive indices. They are called multilayer filters. If an appropriate process is available, it is also possible to fabricate graded-index filters, in which the refractive index varies continuously. Another less explored avenue is the conception of multilayer filters, but with layer of arbitrary intermediate refractive indices.

At normal incidence, it has been demonstrated the optimal filter for a given application consists of only two materials with the greatest refractive index contrast. At oblique incidence, the situation is more complex. Electric and magnetic fields continuity conditions at interfaces are different for s and p polarizations, which leads to the definition of different pseudo refractive indices. It is generally accepted that the optimal solution maximizes the pseudo refractive index contrast, and it is therefore probable that the optimal design includes intermediate refractive indices.

The conception of optical filters relies on the use of design, optimization, and synthesis methods. There exist many very effective methods for the conception of multilayer filters with discrete refractive indices. However, the methods are less adapted to the conception of filters with intermediate refractive indices. Graded-index filters can be designed using the approximate Fourier transform relationship between the refractive index profile and the desired spectrum. However, this method has two important drawbacks: (1) it is only approximate and (2) it does not account for the effect of the refractive index dispersion. The former is usually addressed by an iterative approach. However, there was no general solution to the problem of the dispersion.

In this thesis, I present the results of my research on graded-index filters and multilayer filters with arbitrary intermediate refractive indices. My objective was to improve existing conception methods and invent new ones.

I first propose two correction factors to include the effect of refractive index dispersion in the Fourier transform method. Analyzing the effect of dispersion, I was able to identify it is related to a variation of the optical thickness and amplitude of the index profile with wavelength. This leads to, respectively, a scaling of the wavelength axis and a multiplicative correction factor. The efficiency of the correction factors is demonstrated during the design of a filter made of SiO₂/TiO₂ mixtures.

I also propose to design optical filters by optimizing the refractive indices of their layers, while preserving their optical thicknesses constant. When the layers of a filter are of quarterwave thickness, it is possible to control deposition by turning-point monitoring, which is necessary for the fabrication of some types of filters. However, when discrete refractive index materials are used, solutions are discrete and it is impossible to use conventional optimization techniques. If the refractive index is allowed to vary, it is possible to use conventional optimization techniques and the range of solutions is continuous.

I finally propose a whole new synthesis method for multilayer filters with arbitrary refractive index layers: the step method. It consists in adding steps in the index profile at appropriate positions and then optimizing the thickness and refractive index of the layers. This process is repeated until a satisfactory solution is found. The step method provides interesting solutions for many applications. Indeed, in many cases, the solutions include less layers than two-materials solutions and the layers are thicker.

The formalism developed for the step method is also useful to study the conditions leading to an optimal design. It demonstrates that, at oblique incidence, the optimal design consists of layers maximizing the pseudo refractive index contrast, or of graded-index layers.

All the methods I developed have been implemented in a design software I programmed. To help other research groups who wish to develop new optical filter design methods, we have decided to release this software, called OpenFilters, under the GNU General Public License, an open-source license.