Question

A plate of thickness t behaves as a half wave plate for a light of wavelength λ. ignoring variation in refractive indices with λ, if a light of 2λ is used then plate will behave as? (i know answer is quarter wave plate. But i don't know why... So please explain me)

Answer 1

The thickness of a half-wave plate for wavelength λ is given by:

$$t=\frac{λ}{2(\mu_O-\mu_E)}=\frac{2λ}{4(\mu_O-\mu_E)}=\frac{λ'}{4(\mu_O-\mu_E)}$$

Thus, for wavelength λ' = 2λ, it will behave as a quarter-wave plate.