Solution:

Given, thickness of phase retardation plate (d) = 0.1436 mm
μo = 1.5443
μe = 1.5533
Wavelength range (λ) = 400 nm to 750 nm

To determine the thickness of the plate required for a quarter-wave plate and half-wave plate, we use the following formulas:

(i) For a quarter-wave plate, the thickness (d) is given by:
d = λ / 4 * (μe - μo)

(ii) For a half-wave plate, the thickness (d) is given by:
d = λ / 2 * (μe - μo)

(i) For a quarter-wave plate:
d = λ / 4 * (μe - μo)
=> λ = 4d / (μe - μo)
=> λ = 4 * 0.1436 / (1.5533 - 1.5443)
=> λ = 587.6 nm

Therefore, the thickness of the quartz plate is such that it acts as a quarter-wave plate for a wavelength of 587.6 nm.

(ii) For a half-wave plate:
d = λ / 2 * (μe - μo)
=> λ = 2d / (μe - μo)
=> λ = 2 * 0.1436 / (1.5533 - 1.5443)
=> λ = 410.5 nm

Therefore, the thickness of the quartz plate is such that it acts as a half-wave plate for a wavelength of 410.5 nm.

Explanation:

- Phase retardation plate: A phase retardation plate is a transparent optical element that alters the polarization state of light passing through it.
- Quartz: Quartz is a common material used in phase retardation plates due to its birefringent properties, which means it has two different indices of refraction for light polarized in different directions.
- Quarter-wave plate: A quarter-wave plate is a type of phase retardation plate that introduces a phase shift of 90 degrees between the two orthogonal polarization components of light passing through it.
- Half-wave plate: A half-wave plate is a type of phase retardation plate that introduces a phase shift of 180 degrees between the two orthogonal polarization components of light passing through it.
- Formula: The thickness of a phase retardation plate required to act as a quarter-wave plate or half-wave plate depends on the wavelength of light passing through it and the birefringence of the material. The formulas used to calculate the thickness are based on the phase shift introduced by the plate and the difference in refractive indices for the two polarization components of light.
- Calculation: In this problem, we are given the thickness of the phase retardation plate made of quartz and the refractive indices of the material. Using the formulas for a quarter-wave plate and half-wave plate, we can calculate the wavelength of light for which the plate acts as each type of plate. The wavelength range given is in the visible range, which is from 400 nm to 750 nm.