The development of narrowband filters historically faced challenges due to the difficulty in fabricating low-loss reflecting stacks. The phenomenon of frustrated total internal reflection (FTR) provided a potential solution. Known for some time, FTR occurs when light incident beyond the critical angle penetrates a short distance into the second medium, decaying exponentially. If the second medium is extremely thin, the decay is incomplete, allowing a proportion of the incident light to propagate into the third medium. Baumeister likened this to particle tunneling through a potential barrier, coining the term optical tunneling.
The reflectance in FTR filters can be precisely controlled by varying the thickness of the “frustrating layer” between the first and third media. A filter using this principle is constructed similarly to the polarizing beam splitter. A prism’s hypotenuse is coated with a low-index frustrating layer to ensure incidence beyond the critical angle, followed by a high-index cavity layer, another frustrating layer, and a second prism cemented to form a complete block.
The critical angle \(\psi\) is determined by:
\[
\sin \psi = \frac{n_F}{n_G},
\]
where \(n_F\) and \(n_G\) are the refractive indices of the frustrating layer and the glass prism, respectively. For \(n_F = 1.35\) and \(n_G = 1.52\), \(\psi = 63^\circ\). Using higher-index glass (e.g., \(n_G \approx 1.7\)) reduces this angle.
Challenges and Limitations
While the FTR filter is conceptually simple, it has significant theoretical and practical disadvantages:
1. Polarization Splitting: Large shifts in peak wavelength occur between p- and s-polarized light, often around 100 nm in the visible spectrum. This results from the high angle of incidence and polarization-dependent phase changes at the interfaces.
2. Angular Sensitivity: The filter’s performance is highly angle-dependent, with wavelength shifts of approximately 5 nm/degree of arc.
3. Practical Difficulties: Attempts to fabricate FTR filters have produced disappointing results, falling short of theoretical expectations. Construction challenges are as significant as those for conventional Fabry–Perot filters.
Due to these drawbacks, interest in FTR filters has remained largely theoretical, and they have not seen commercial production.
Theoretical Insights and Alternatives
Baumeister provided a detailed theory for FTR filters, highlighting their similarity to loss-free metal layers in behavior. This opens possibilities for alternative designs, including induced-transmission filters. However, Baumeister concluded that practical applications for single-cavity FTR filters or tunnel-layer filters are limited. He proposed a longwave-pass filter design using multiple tunnel layers separated by dielectric layers, offering limitless shortwave rejection. Yet, these designs still suffer from:
1. Polarization Splitting Near the Edge: This can be mitigated by adding a conventional edge filter at the prism’s front face.
2. Transmission Spikes in the Stop Region: Small transmission spikes appear with minimal tilts, as little as \(1^\circ\) internally or \(2.7^\circ\) externally.
Recent Developments
Li and Dobrowolski revisited the concept for constructing efficient thin-film polarizers. At high angles of incidence, tunnel layers overcome the low admittance contrast challenges of p-polarization near \(45^\circ\). These designs are more complex but benefit from computer-based synthesis and refinement, allowing automatic optimization.
