| Many
of the filters in this catalog (with the exception of dichroic beamsplitters
and the MaxMirror®) are optimized for use with
light at or near normal incidence. However, for some applications it is
desirable to understand how the spectral properties change for a non-zero
angle of incidence (AOI).
There are two main effects
exhibited by the filter spectrum as the angle is increased from normal:
1. the features of the spectrum shift to shorter wavelengths;
2. two distinct spectra emerge – one for s-polarized light
and one for p-polarized light.
As an example, the graph at
the right (Figure A) shows a series of spectra derived from a typical
RazorEdge long-wave-pass (LWP)
filter design. Because the designs are so similar for all of the RazorEdge
filters in the series, the set of curves in the graph can be applied approximately
to any of the filters. Here the wavelength 
is compared to the wavelength 
of a particular spectral feature (in this case the edge location) at normal
incidence. As can be seen from the spectral curves, as the angle is increased
from normal incidence the filter edge shifts toward shorter wavelengths
and the edges associated with s- and p-polarized light shift by different
amounts. For LWP filters, the edge associated with p-polarized light shifts
more than the edge associated with s-polarized light, whereas for short-wave-pass
(SWP) filters the opposite is true. Because of this polarization splitting,
the spectrum for unpolarized light demonstrates a “shelf”
near the 50% transmission point when the splitting significantly exceeds
the edge steepness. However, the edge steepness for polarized light remains
very high.
The shift of almost any spectral
feature can be approximately quantified by a simple model of the wavelength
of the feature vs. angle of incidence  ,
given by the equation:
where neff
is called the effective index of refraction, and
is the wavelength of the spectral feature of interest at normal incidence.
Different shifts that occur for different spectral features and different
filters are described by a different effective index. For the RazorEdge
example above, the shift of the 90% transmission point on the edge is
described by this equation with neff = 2.08 and 1.62
for s- and p-polarized light, respectively. More
details...
Other types of filters
don’t necessarily exhibit such a marked difference in the shift
of features for s- and p-polarized light. For example, the middle graph
(Figure B) shows a series of spectra derived from a typical MaxLine
laser-line filter design curve. As the angle is increased from normal
incidence, the center wavelength shifts toward shorter wavelengths and
the bandwidth broadens slightly for p-polarized light while narrowing
for s-polarized light. The center wavelength shifts are described by the
above equation with neff = 2.19 and 2.13 for s- and
p-polarized light, respectively. The most striking feature is the decrease
in transmission for s-polarized light, whereas the transmission remains
quite high for p-polarized light. More
details...
As another example, the graph (Figure C) shows a series of spectra derived
from a typical U-grade StopLine notch
filter design curve. For the U- and S-grade notch filters, as the
angle is increased from normal incidence, the notch center wavelength
shifts to shorter wavelengths, the notch depth decreases, and the notch
bandwidth decreases (with a greater decrease for p-polarized light than
for s-polarized light). The shift of the notch center wavelength is described
by the above equation with neff = 1.76 for both s-
and p-polarized light.
However, Semrock's
unique E-grade StopLine notch
filters, which exhibit ultrawide passbands (from UV to 1600 nm), behave
somewhat differently from the U-grade filters. Figure D at the bottom
right shows a series of spectra for a typical E-grade notch filter. As
the angle is increased from normal incidence, the notch center wavelength
shifts to shorter wavelengths; however, the shift is greater for p-polarized
light than it is for s-polarized light. The shift is described by the
above equation with neff = 1.71 for p-polarized light
and neff = 1.86 for s-polarized light. Further, whereas
the notch depth and bandwidth both decrease as the angle of incidence
is increased for p-polarized light, as with the U-grade filters, in contrast
the notch depth and bandwidth increase for s-polarized light. More
details...
Note that it is possible
to optimize the design of a notch filter to have a very deep notch even
at a 45° angle of incidence. |
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