4. Selecting the wavelength [solution]
The first polarizer produces a polarized beam with the polarization
angle at 45° to the optic axis. This means the light incident on the
calcite has equal amplitude components polarized parallel to the optic
axis and perpedicular to it. Both components are undeflected (polarization
along principal axis), but they travel at different speeds and get out
of phase with one another. When they leave the calcite the phase difference
is then constant, so the thickness of the calcite plate will determine
the polarization state of the emerging light. If the accumulated phase
difference is d = p/2,
the wave will be circularly polarized. If d = p, the
polarization is reflected about the optic axis (which in this case will
produce a polarization vector inclined at 135° to the optic axis) but
the light that comes out is linearly polarized. For intermediate values
of d, the polarization will be elliptical. The
value of d produced by a plate of thickness
d is given by
The problem statement tells us that for one of the two Na wavelengths
(l1 = 588.9950
nm and l2 =
589.5924 nm), no light emerges from the second polarizer. This must mean
that for the wavelength that is "eliminated," the light coming out of the
calcite is linearly polarized perpendicular to the second polarizer (which
is oriented the same way as the first). This requres that the total phase
difference d be equal to an odd multiple of
p for this wavelength. But we require not only
that d for one wavelength be an odd multiple
of p (so it is blocked by the second polarizer),
but also that d for the other member of the
doublet be equal (or very close) to an even multiple of p
so that it passes through the second polarizer. Can we achieve both conditions
for a single thickness d? We need to see how d
changes with l, so take the derivative with
respect to l of the equation above:
We have all these quantities, so we just need to know what
value of d gives us
A little number-crunching yields d = 1.563 mm, a reasonably convenient
thickness. What would happen if you rotated the second polarizer by 90°?
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