Definition: mirrors consisting of multiple thin layers of different transparent optical materials
More general term: mirrors
More specific terms: Bragg mirrors, quarter-wave mirrors, dichroic mirrors, dispersive mirrors, chirped mirrors, cold mirrors, hot mirrors, supermirrors
A dielectric mirror is a mirror based on multiple thin layers of (usually two) different transparent dielectric optical materials (→ dielectric coatings, thin-film coatings, interference coatings). Even if the Fresnel reflection coefficient from a single interface between two materials is small (due to a small difference in refractive indices), the reflections from many interfaces can (in a certain wavelength range) constructively interfere to result in a very high overall reflectance (reflectivity) of the device. The simplest and most common design is that of a Bragg mirror, where all optical layer thickness values are just one-quarter of the design wavelength. That design leads to the highest possible reflectance for a given number of layer pairs and given materials (assuming normal incidence). It is also possible to design dichroic mirrors with controlled properties for different wavelengths.
The higher the refractive index contrast of the used layer materials, the smaller is the number of layer pairs required for a high reflectance, and the higher will be the reflection bandwidth.
More complicated multilayer structure designs can be used to obtain certain functions, such as for example
a broader reflection bandwidth
a combination of desirable reflectance values in different wavelength ranges
special polarization properties (for non-normal incidence): thin-film polarizers, polarizing and non-polarizing beam splitters
edge filters, e.g. long-pass filters, high-pass filters, band-pass filters
tailored chromatic dispersion properties (see the article on dispersive mirrors)
The number of thin-film layers required depends very much on the required function and on the refractive index difference between the coating materials. Few layer pairs are sufficient in some cases, whereas more than 100 layers are required in other cases.
The resonator mirrors of a laser are almost always dielectric mirrors, because such devices routinely achieve a very high reflectance of > 99.9%, and their limited reflection bandwidth can be convenient because it allows the transmission of pump light (at a shorter wavelength) through a folding mirror of the resonator (→ dichroic mirrors). Dielectric mirrors used for lasers and for reflecting laser beams are often called laser mirrors.
A characteristic property of dielectric mirrors is that they are optical properties depend substantially on the angle of incidence. As an example, Figure 1 shows reflectance spectra of a simple Bragg mirror for different incidence angles. The larger that angle, the more the reflection spectrum is shifted towards shorter wavelengths. This is essentially because the component of the wave vector perpendicular to the layer surfaces becomes smaller for a given wavelength, which can be compensated by reducing the wavelength [9].
Figure 1: The reflectance spectrum of a Bragg mirror for different incidence angles from normal incidence (red) up to 60° (blue) in steps of 10°.
Optimized Bragg mirrors, also called supermirrors, can even have much higher reflectivities – in extreme cases, even larger than 99.9999%, allowing e.g. the construction of optical resonators (cavities) with extremely high Q factor. Such performance requires high quality mirror coatings with very low absorption and scattering, also with high uniformity of the layer thickness.
As dielectric mirrors usually provide a high reflectance at most in a smaller part of the visible spectrum, they often do not appear like other mirrors such as silver mirrors (or other metal-coated mirrors), as used in households: dielectric mirrors are usually transparent to visible light and shine in colors which depend on the angle of view. It can even be difficult to see which side of the substrate has the mirror coating.
Dielectric multilayer mirrors can be made on both plane and curved surfaces. In the latter case, the mirrors are focusing or defocusing. For example, a concave surface with radius of curvature R focuses with a focal length R / 2 for normal incidence. For small radii of curvature (e.g. below 10 mm), there may be problems with the coating quality in terms of homogeneity and stability.
Calculating Mirror Properties
Food for Thought
For angled incidence of light on a dielectric mirror, the optical path lengths appear to be longer. One may thus expect that all reflection features are pushed toward longer wavelengths when a dielectric mirror is tilted against the incident beam. What is wrong with this argument?
The reflection properties (including the dispersion) of a dielectric multilayer mirror can be calculated with modeling software e.g. based on a matrix method, where each layer is associated with a 2-by-2 complex matrix, and all matrices are multiplied together to result in a matrix of the whole layer structure. From that matrix, the complex amplitudes of reflected and transmitted waves can be calculated, and also the field distribution within the structure. The chromatic dispersion properties result from the frequency dependence of the complex reflection or transmission coefficients, which can be calculated based on the Fresnel equations.
Some non-trivial mathematical aspects come into play when materials are absorbing. A problem can also be to obtain sufficiently precise material data, particularly for materials where the obtained refractive index has a significant dependence on the fabrication method (see below).
In most cases, the optical intensities within dielectric mirrors are too low to cause substantial changes of the refractive index. In some special cases, however, the Kerr effect becomes significant [5, 6]. Here, one may apply an iterative numerical procedure, where after each step the refractive index profile of the coding structure is updated according to the calculated optical intensities.
Designing Dielectric Mirrors
It can be a difficult task to find a dielectric mirror design which satisfies certain criteria, such as
a combination of reflectivities at different wavelengths
very broadband reflection ranges
anti-reflection properties
certain polarization properties (for non-normal incidence; → thin-film polarizers)
a certain chromatic dispersion profile
minimum sensitivity to growth errors
Such dielectric mirror designs can often only be found by using numerical optimization algorithms, although analytical design strategies are known for some design targets (e.g. chirped mirror designs for dispersive mirrors). Technical challenges arise from the high dimensionality of the searched parameter space, and from the myriads of local optima which make it difficult to find the global optimum. An efficient optimization requires advanced mirror design software with features like efficient multi-dimensional optimization with Monte Carlo methods, definition of sophisticated figure-of-merit functions (also taking into account the sensitivity to growth errors), etc.
Beyond the technical optimization problems, there are of course also fundamental limitations. In many cases, the design involves a compromise between the obtained optical properties, the required number of layers, and the required growth precision.
For details on the fabrication of dielectric mirrors, see the article on dielectric coatings.
Long Wavelengths
Dielectric mirrors are mostly used for the visible, ultraviolet and near infrared spectral region. For a rather long wavelength radiation, e.g. light at 10.6 μm from CO2 lasers, it is hard to fabricate well performing dielectric mirrors due to a lack of suitable dielectric materials. Note that these do not only need to exhibit sufficiently small absorption, but must also be suitable for controlled deposition on a mirror. Therefore, metal-coated mirrors instead of dielectric mirrors are widely used in infrared optics.
Bibliography
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