Overview of Inhomogeneous Layers
Inhomogeneous layers have refractive indices that vary through their thickness. Many commonly used thin-film materials are slightly inhomogeneous, and for most coatings, this can be treated as negligible. However, significant inhomogeneity in some materials can affect the performance of antireflection coatings.
If such a layer is used in a well-corrected antireflection coating, performance may decrease. Corrections can often be made by adjusting the indices of other layers. For instance, Ogura noted that a slightly decreasing index in the high-index layer of a quarter-half-quarter coating can broaden its characteristic. Zirconium oxide, a widely used material, exhibits varying inhomogeneity depending on deposition temperature, which can be corrected through adjustments, such as using a composite intermediate-index layer.
Creating Highly Inhomogeneous Layers
Highly inhomogeneous layers can be intentionally created by varying the composition during deposition. Jacobsson and Knittl reviewed calculation techniques for such layers, often by dividing the inhomogeneous layer into many thin sublayers, each treated as homogeneous with a mean refractive index. This stepwise approximation can yield highly accurate results, especially with modern computational methods.
Theoretical Basis
The theory of inhomogeneous coatings stems from multilayer antireflection designs for high-index substrates. Adding layers improves performance until the multilayer transitions to a single inhomogeneous layer with a smoothly varying refractive index. Such a layer becomes a near-perfect antireflection coating for wavelengths shorter than twice its optical thickness, with diminishing performance beyond this limit.
The antireflection range is defined by:
\[
\lambda_S = \frac{T}{n + \frac{1}{2}}, \quad \lambda_L = 2T
\]
where \(T\) is the total optical thickness. At wavelengths longer than \(4T\), the coating becomes ineffective.
Practical Applications
While inhomogeneous layers cannot achieve indices as low as air (index 1.0), they can terminate at practical low indices like magnesium fluoride (1.35). Jacobsson and Martensson demonstrated such coatings on germanium plates using evaporated germanium and magnesium fluoride with varying compositions. For example, a coating with a physical thickness of 1.2 μm and a mean index of 2.68 achieved excellent antireflection out to 6.4 μm (Figure 4.65).
Approximation with Homogeneous Layers
Berning proposed using stepwise approximations with alternating homogeneous layers of high- and low-index materials (e.g., germanium and magnesium fluoride) to mimic inhomogeneous layers. This approach simplifies manufacturing while retaining much of the desired performance.
Limitations
The lack of ultra-low-index materials limits the performance of inhomogeneous layers. For high-index substrates, such layers can provide a residual reflectance of about 2.5%. For low-index substrates, their utility lies in enhancing designs that use homogeneous materials.
