How Is the Fresnel Double Prism Equation Derived?

[jh]

Hi,
i need to deduce the Fresnel Double Prism equation, which states:

d = 2a(n-1)alpha,

according to the linked image.
http://www.du.edu/~jcalvert/waves/biprism.gif

The distance between the constructed virtual objects S' and S'' is here called d. The distance from the light source to the prism is called a, the prism has two sides (common base) defined by the common apex angle alpha, and the refractive index n.

Any help appreciated!

 

[clive]

Hi jh,

You probably know that for a single prism the deviation angle is given by
\delta=i+i'-A
The refraction' law in the incidence and emergence points (for very small angles) is
i=nr
nr'=i
and from these two equations you have now
i+i'=n(r+r')=nA.

It is clear now that
\delta=nA-A=A(n-1)

Because you have a double prism,
\delta_{tot}=2\delta.
and \delta_{tot}\cdot a for the distance between the two points (images). ( A=\alpha)