# Why is the angle of incidence shift in band rejection filter spectra minimized by high refractive index materials?

I would like to know why the angle of incidence shift in band rejection filter spectra and separation of S and P polarized light is minimized by high refractive index materials.

The reason I ask this question is that Thomas D. Rahmlow, Jr., et al. wrote in the 2016 paper Ultra-narrow Bandpass filters for infrared applications with improved angle of incidence performance [https:/cloud.acrobat.com/file/0a135ec5-461d-49a1-8277-1b05301765bd] and the 2006 paper Narrow Band Infrared Filters with Broad Field of View with URL, [https:/cloud.acrobat.com/file/5858a925-ab6e-4b1d-89bc-121c69ea4526]

A thin film’s effective index is defined, for s (perpendicular) polarization, as np = n/cos(incidence angle) and, for p (parallel) polarization, as np = n/cos(incidence angle. Hence, the film’s effective optical thickness decreases with increasing angle of incidence. For a notch filter, this implies that the notch will shift to shorter wavelengths with increasing angles of incidence. The amount of shift is inversely proportional to the effective average index of the composite film.

The reason I ask this question is that even with the advent of optical coatings which have proven to be able to filter green laser pointer rays coming in at a normal or small angle of incidence, large commerical aircraft cockpits remain vulnerable to laser beams arriving at large angles of incidence.

Assuming we can effectively minimize the shift of notch rejection filters due to large angle of incidence, how can we simulataneously achieve very fast tunability(i.e tuning speed of 25 microseconds) of an notch rejection filter to red , green or blue target wavelengths?

Any help is greatly appreciated.

• For a ray from air entering a medium, which direction does it bend? How does the amount of bending change with the index of refraction? How does that angle impact thin film optic effects? – Jon Custer Jul 4 '16 at 10:44
• @Jon Custer, Thank you for your excellent Socrates dialogue. The amount and direction of bending is determined by Snell's law. – Frank Jul 4 '16 at 10:56
• @Jon Custer, 1. Snell's law determines the reflectance and transmittance power for the s-polarized and p-polarized cases. 2." The spectral position of the reflectance minimum red-shifted across the visible with increasing film thickness. The minimum occurs for thicknesses that are much smaller than the quarter-wavelength thickness typical of a conventional AR coating, even when the large refractive index of germanium is taken into account " 3) Because thin films have less phase shift, the shift of notch rejection filters due to large incidence angles is minimized. – Frank Jul 4 '16 at 11:27
• @Jon Custer, The above quote comes from the paper, Optical absorbers based on strong interference in ultra-thin films by Mikhail A. Kats and Federico Capasso. – Frank Jul 4 '16 at 11:29
• @Jon Custer, May I ask if you have time to let us know if I missed any arguments in the above comments? Thank you. – Frank Jul 4 '16 at 11:34

## 1 Answer

Here is why the angle of incidence shift in band rejection filter spectra is minimized by high refractive index materials?

Miroslav wrote in this blog [http://quantum.opticsolomouc.org/archives/464] that "The 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 λ of all types of interference filters decreases as the AOI increases,

λ(θ)=λ * sqrt(1–(sinθ / neff)^2)

where θ stands for the AOI and neff 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 effect on the angular field of the filter and its transmittance.

My friend since college who has a Ph.D in optical physics write yesterday that The reasoning behind the relative insensitivity to angle is given in the paper “Narrow Band Infrared Filters with Broad Field of View” by Thomas D. Rahmlow et al. in section I (“Introduction”), the fifth paragraph:

“For a notch filter, this implies that the notch will shift to shorter wavelengths with increasing angle. The amount of the shift is inversely proportional to the effective average index of the composite film. …..Since the index of the asymmetric filter is dominated by the high index material, the shift on angle is considerably less.” Because the shift is inversely proportional to the effective average refractive index, the higher this is, the better, because the shift with angle will be smaller (but it does not disappear). This is true for a standard quarter-wave filter, as well, but in that case the effective refractive index is NOT dominated by the high index, as it is here. (See this Wikipedia page: https://en.wikipedia.org/wiki/Interference_filter )