Answer
$L = 3.60~\mu m$
Work Step by Step
We can find an expression for the wavelength in glass:
$\frac{\lambda}{1.60}$
We can find an expression for the wavelength in plastic:
$\frac{\lambda}{1.50}$
We can find $L$ such that the difference in the number of cycles is equal to $\frac{5.65}{2\pi}$:
$\frac{L}{\lambda/1.60}-\frac{L}{\lambda/1.50} = \frac{5.65}{2\pi}$
$1.60~L-1.50~L = \frac{5.65}{2\pi}~\lambda$
$L = (\frac{1}{0.1})(\frac{5.65}{2\pi})~(400~nm)$
$L = 3.60~\mu m$
