## Ghost Images In the Phasmatron Spectroscope

### Version 1.0 of 6/3/2008-8:10 a.m.

Let xA be a ray through a prism in the minimum deviation position for some known
wavelength (say λ). Then the path of the ray is xABx'. Because of internal
reflections at B, C and A, there is another path that can be traced, namely,
xABCABC...x'. If we increment the incidence angle θ by δθ, so that we
have incidence zA, then the path will primarily be zAB'z', but also there will be
another path, now zAB'C'A'B''...y' which will give a ghost image along with the real
image.

The variation δθ will cause a variation δφ inside the prism,
as follows:

sin(θ)/sin(φ) =n_{λ}=>sin(θ)=n_{λ}*
sin(φ) =>δ(sin(θ))=n_{λ}*δ(sin(φ)) =>
cos(θ)δθ=n_{λ}*cos(φ) δφ
=>δφ = cos(θ)δθ/(n_{λ}*cos(φ))
(1)

Knowing that φ =arcsin(sin(θ)/n_{λ}) (2)

we get: δφ
=cos(θ)δθ/{n_{λ}*sqrt[1-(sin(θ)/n_{λ})^{2}]}
(3)

Now it is easy to see that A2=30°+δφ => B'3=30°-δφ
=> C'1=30°+δφ => A'1=30°-δφ =>
A'2=30°-δφ => B''2=30°+δφ (4)

Therefore, B''1=arcsin{sin(30°+δφ)*n_{λ}}, the
emergence angle of the ghost image,

and B'1=arcsin{sin(30°-δφ )*n_{λ}}, the emergence angle
of the real image.

A different picture of the path of a ghost image can be seen in the following
figure:

The real image path is shown in red. The primary ghost image path is shown in
purple^{[1]}.

Ghost images unfortunately appear along with the real spectroscopic image. The
problem is particularly acute with very bright spectra, as these cause many visible
reflections.

To minimize the problem, the collimator is allowed to turn, so that ghost images
will disappear from a particular section of the spectrum. When we see ghost images in a
certain part of the spectrum, we rotate the collimator slightly, until ghosts shift out
of view. Then we re-adjust the viewing telescope's angle, by bringing the original
image back to view. This way we can have ghost images visible in the violet part of the
spectrum, while we are examining the yellow part, and vice versa. Note also that along
with the main ghost images, there are *secondary* ghost images caused by the fact
that the system contains two prisms. These cannot be remedied.

Notes

- While designing the above figure with EucliDraw, professor Paris Pamfilos discovered that the points ABB'D are
con-cyclic. This implies that the exit ghost angle is equal to the incidence
angle, as shown with EucliDraw on the figure.