Symptom: Image is darker at the Rim of Bandpass Filters

When using dielectrical bandpass filters the images sometimes get too dark at the rim (“vignetting“) or laser lines appear too dark) when the lenses used have a viewing angle that is too large and a bandpass filter that is too “narrow”, say, not enough wavelengths can pass.

Bandpass filters are not only designed for specific frequency ranges, but also for specific incident angles, typically for perpendicular incidence, (=  zero degrees deviation).

When the light does not arrive at zero degrees but under a different angle Alpha, the desired full range of wavelengths won’t pass, but rather lower wavelengths instead.
This is caused by constructive and destructive interference that’s used in dielectrical filters, so the length of the optical path in the various coatings plays an important role. Under 45 degrees the light stays (for example) over 41% longer in the coating than under 0 degrees (factor: squareroot of 2).

The approximation formula reads
new wavelength = Wavelength \sqrt{1-(\frac {\sin(Radians(Alpha))}{(RefractionIndex)})^2}

This effect is called “blue shift” (for a calculator see here)

For wavelength 850nm , 10 degrees and “window glass” you get : new wavelength = 850nm \sqrt{1-(\frac {\sin(Radians(10))}{(1,49)})^2} = 844,21.. When you work with a bandpass filter designed for 830-870nm , 824nm to 864nm will pass under 10 degrees .. Therefore 850nm of Light under 10 degrees can pass.

When you work with a bandpass filter designed for 830-870nm, 782nm to 819nm will pass under 30 degrees  .. Therefore 850nm Light under 30 degrees is blocked
When you work with a bandpass filter designed for 830-870nm, 782nm to 819nm will pass under 30 degrees .. Therefore 800nm Light under 30 degrees can pass. a very unwanted effect.
You can often use dielectrical filters in the camera or even in the lens. You can get them for example here: info@lensation.de
The “optimum” solution would be an image-side telecentric lens, with a CRA (Chief Ray Angle) of zero together with an image-side filter position.

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