Majority of interference filters are designed to be used at normal angle of incidence (AOI). The primary effect of an increase in the incident angle on an interference coating is a shift in spectral performance toward shorter wavelengths. In other words, the principal wavelength \(\lambda\) of all types of interference filters decreases as the AOI increases,
\[\lambda(\theta) = \lambda_0 \sqrt{1 – \left(\frac{\sin\theta}{n_{\rm eff}}\right)^2},\] where \(\theta\) stands for the AOI and \(n_{\rm eff}\) is an effective index of refraction with value typically between 1.4 and 2.2. Small tilts are commonly used to tune the peak of a filter to the desired wavelength even though they have an adverse ef fect on the angular field of the filter and its transmittance.
The performance of the filter depends on a polarization of the input light for \(\textrm{AOI}>0\), as S and P polarization modes can be distinguished and interfere differently. The shift of the central or principal wavelength towards shorter wavelengths can be more pronounced for P-polarized light (smaller effective index of refraction). The bandwidth of the filter broadens slightly for P-polarized light while narrowing for S-polarized light. The most striking feature is the decrease in transmission for S-polarized light, while the P-polarized light maintains high transmission.
References
1) M. Bass ed., Handbook of optics, Volume I (2nd edition, McGRAW-HILL, New York 1995), chapter 42: Optical properties of films and coatings, pages 42 .89-91.
2) P.H. Lissberger and W.L. Wilcock, Properties of All-Dielectric Interference Filters. II. Filters in Parallel Beams of Light Incident Obliquely and in Convergent Beams, J. Opt. Soc. Am. 49, 126-128 (1959).
3) Semrock: Filter Spectra at Non-normal Angles of Incidence
4) Omega Optical: Angle of Incidence and Polarization
Interesting idea how to improve a transmittance of IF filter demonstrated experimentally by Paul Kwiat’s group: Novel narrow-band spectral interference filter with very high transmittance, J. Mod. Opt. 58, 306 (2011).
Miroslav, I have two questions. Assuming we can minimize the shift of notch rejection filters due to large angle of incidence by using high refractive index substrate(i.e n > 4) , how can we simultaneously achieve very fast tunability(i.e tuning speed of 100 microseconds) for an notch rejection filter to red , green or blue target wavelengths?
Miroslav, I have another question. During lunch yesterday, I drew on paper an idea of staggering 10 five degree wide bands of increasing angle of incidence shifted by phase delay equal 5 degrees to cover the wide incidence angle range of 0 to 50 degrees. I am hoping to apply Dong Guoyan,et al. idea which the authors wrote in this 2013 paper “Zero phase delay with relax incident condition in photonic crystals(PhC)”. with the URL https://www.osapublishing.org/oe/viewmedia.cfm?uri=oe-21-24-29860&seq=0 that “Since the PhC structure can be engineered easily and have more design flexibility, it is convenient to manipulate the frequency and direction of EMW propagating in PhC with zero phase delay by changing the parameters of PhC configuration or the incident angle. Due to the prosperous nanofabrication technology, this method for engineered control of the phase delay in PhCs may have significant potential applications in chip–scale transmission lines, information processing and the design of new type optical devices” to building 5 degree phase delay devices to create 10 contiguous five degree bands of angle of incidence using an 1 or 2 dimensional array of Fabry Perot interferometers. May I ask if you could critique this idea? Thank you.