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) =nλ=>sin(θ)=nλ* sin(φ) =>δ(sin(θ))=nλ*δ(sin(φ)) => cos(θ)δθ=nλ*cos(φ) δφ =>δφ = cos(θ)δθ/(nλ*cos(φ)) (1)
Knowing that φ =arcsin(sin(θ)/nλ) (2)
we get: δφ =cos(θ)δθ/{nλ*sqrt[1-(sin(θ)/nλ)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)*nλ}, the emergence angle of the ghost image,
and B'1=arcsin{sin(30°-δφ )*nλ}, 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.